LMFDB - Embedded newform 378.3.s.d.53.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [378,3,Mod(53,378)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(378, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 4]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("378.53");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 378 = 2 \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	378.s (of order $6$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$10.2997539928$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	8.0.621801639936.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} - 34x^{6} + 116x^{5} + 413x^{4} - 1024x^{3} - 1664x^{2} + 2196x + 4467 $ x^8 - 4x^7 - 34x^6 + 116x^5 + 413x^4 - 1024x^3 - 1664x^2 + 2196*x + 4467
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			53.4
Root			$2.31664 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	378.53
Dual form			378.3.s.d.107.4

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.22474 - 0.707107i) q^{2} +(1.00000 - 1.73205i) q^{4} +(4.33729 - 2.50413i) q^{5} +(0.0413813 - 6.99988i) q^{7} -2.82843i q^{8} +O(q^{10})$	\(q+(1.22474 - 0.707107i) q^{2} +(1.00000 - 1.73205i) q^{4} +(4.33729 - 2.50413i) q^{5} +(0.0413813 - 6.99988i) q^{7} -2.82843i q^{8} +(3.54138 - 6.13385i) q^{10} +(-1.32611 - 0.765629i) q^{11} +0.917237 q^{13} +(-4.89898 - 8.60233i) q^{14} +(-2.00000 - 3.46410i) q^{16} +(-14.3380 - 8.27803i) q^{17} +(6.50000 + 11.2583i) q^{19} -10.0165i q^{20} -2.16553 q^{22} +(10.3596 - 5.98115i) q^{23} +(0.0413813 - 0.0716745i) q^{25} +(1.12338 - 0.648585i) q^{26} +(-12.0828 - 7.07155i) q^{28} -33.5266i q^{29} +(8.66553 - 15.0091i) q^{31} +(-4.89898 - 2.82843i) q^{32} -23.4138 q^{34} +(-17.3492 - 30.4641i) q^{35} +(27.8724 + 48.2765i) q^{37} +(15.9217 + 9.19239i) q^{38} +(-7.08276 - 12.2677i) q^{40} -6.53953i q^{41} +32.7449 q^{43} +(-2.65222 + 1.53126i) q^{44} +(8.45862 - 14.6508i) q^{46} +(46.3841 - 26.7799i) q^{47} +(-48.9966 - 0.579328i) q^{49} -0.117044i q^{50} +(0.917237 - 1.58870i) q^{52} +(-72.4078 - 41.8047i) q^{53} -7.66895 q^{55} +(-19.7986 - 0.117044i) q^{56} +(-23.7069 - 41.0616i) q^{58} +(41.4386 + 23.9246i) q^{59} +(40.2897 + 69.7838i) q^{61} -24.5098i q^{62} -8.00000 q^{64} +(3.97832 - 2.29689i) q^{65} +(1.21033 - 2.09636i) q^{67} +(-28.6759 + 16.5561i) q^{68} +(-42.7897 - 25.0431i) q^{70} -83.3216i q^{71} +(-36.0414 + 62.4255i) q^{73} +(68.2732 + 39.4176i) q^{74} +26.0000 q^{76} +(-5.41418 + 9.25091i) q^{77} +(61.1655 + 105.942i) q^{79} +(-17.3492 - 10.0165i) q^{80} +(-4.62414 - 8.00925i) q^{82} +158.860i q^{83} -82.9172 q^{85} +(40.1041 - 23.1541i) q^{86} +(-2.16553 + 3.75080i) q^{88} +(-46.9922 + 27.1310i) q^{89} +(0.0379564 - 6.42055i) q^{91} -23.9246i q^{92} +(37.8724 - 65.5970i) q^{94} +(56.3848 + 32.5538i) q^{95} +12.0828 q^{97} +(-60.4180 + 33.9363i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{4} - 24 q^{7}+O(q^{10}) $ 8 * q + 8 * q^4 - 24 * q^7	$ 8 q + 8 q^{4} - 24 q^{7} + 4 q^{10} + 56 q^{13} - 16 q^{16} + 52 q^{19} + 80 q^{22} - 24 q^{25} - 48 q^{28} - 28 q^{31} + 56 q^{34} + 4 q^{37} - 8 q^{40} - 176 q^{43} + 92 q^{46} - 100 q^{49} + 56 q^{52} - 256 q^{55} - 68 q^{58} + 152 q^{61} - 64 q^{64} + 180 q^{67} - 172 q^{70} - 264 q^{73} + 208 q^{76} + 392 q^{79} + 36 q^{82} - 712 q^{85} + 80 q^{88} - 316 q^{91} + 84 q^{94} + 48 q^{97}+O(q^{100}) $ 8 * q + 8 * q^4 - 24 * q^7 + 4 * q^10 + 56 * q^13 - 16 * q^16 + 52 * q^19 + 80 * q^22 - 24 * q^25 - 48 * q^28 - 28 * q^31 + 56 * q^34 + 4 * q^37 - 8 * q^40 - 176 * q^43 + 92 * q^46 - 100 * q^49 + 56 * q^52 - 256 * q^55 - 68 * q^58 + 152 * q^61 - 64 * q^64 + 180 * q^67 - 172 * q^70 - 264 * q^73 + 208 * q^76 + 392 * q^79 + 36 * q^82 - 712 * q^85 + 80 * q^88 - 316 * q^91 + 84 * q^94 + 48 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/378\mathbb{Z}\right)^\times$.

$n$	$29$	$325$
$\chi(n)$	$-1$	$e\left(\frac{2}{3}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	1.22474	−	0.707107i	0.612372	−	0.353553i
$2$	1.22474	−	0.707107i	0.612372	−	0.353553i
$3$		0			0
$3$		0			0
$4$	1.00000	−	1.73205i	0.250000	−	0.433013i
$5$	4.33729	−	2.50413i	0.867458	−	0.500827i	0.000955133	−	1.00000i	$-0.499696\pi$
$5$	4.33729	−	2.50413i	0.867458	−	0.500827i	0.866503	+	0.499173i	$0.166363\pi$
$6$		0			0
$7$	0.0413813	−	6.99988i	0.00591161	−	0.999983i
$7$	0.0413813	−	6.99988i	0.00591161	−	0.999983i
$8$		−	2.82843i		−	0.353553i
$9$		0			0
$10$	3.54138	−	6.13385i	0.354138	−	0.613385i
$11$	−1.32611	−	0.765629i	−0.120555	−	0.0696026i	0.438510	−	0.898726i	$-0.355506\pi$
$11$	−1.32611	−	0.765629i	−0.120555	−	0.0696026i	−0.559065	+	0.829124i	$0.688840\pi$
$12$		0			0
$13$	0.917237			0.0705567			0.0352784	−	0.999378i	$-0.488768\pi$
$13$	0.917237			0.0705567			0.0352784	+	0.999378i	$0.488768\pi$
$14$	−4.89898	−	8.60233i	−0.349927	−	0.614452i
$15$		0			0
$16$	−2.00000	−	3.46410i	−0.125000	−	0.216506i
$17$	−14.3380	−	8.27803i	−0.843410	−	0.486943i	0.0150117	−	0.999887i	$-0.495221\pi$
$17$	−14.3380	−	8.27803i	−0.843410	−	0.486943i	−0.858422	+	0.512944i	$0.828555\pi$
$18$		0			0
$19$	6.50000	+	11.2583i	0.342105	+	0.592544i	0.984823	−	0.173559i	$-0.0555267\pi$
$19$	6.50000	+	11.2583i	0.342105	+	0.592544i	−0.642718	+	0.766103i	$0.722193\pi$
$20$		−	10.0165i		−	0.500827i
$21$		0			0
$22$	−2.16553			−0.0984330
$23$	10.3596	−	5.98115i	0.450420	−	0.260050i	−0.257588	−	0.966255i	$-0.582928\pi$
$23$	10.3596	−	5.98115i	0.450420	−	0.260050i	0.708007	+	0.706205i	$0.249594\pi$
$24$		0			0
$25$	0.0413813	−	0.0716745i	0.00165525	−	0.00286698i
$26$	1.12338	−	0.648585i	0.0432070	−	0.0249456i
$27$		0			0
$28$	−12.0828	−	7.07155i	−0.431527	−	0.252555i
$29$		−	33.5266i		−	1.15609i	−0.816005	−	0.578045i	$-0.803816\pi$
$29$		−	33.5266i		−	1.15609i	0.816005	−	0.578045i	$-0.196184\pi$
$30$		0			0
$31$	8.66553	−	15.0091i	0.279533	−	0.484165i	−0.691736	−	0.722151i	$-0.743154\pi$
$31$	8.66553	−	15.0091i	0.279533	−	0.484165i	0.971269	+	0.237985i	$0.0764870\pi$
$32$	−4.89898	−	2.82843i	−0.153093	−	0.0883883i
$33$		0			0
$34$	−23.4138			−0.688642
$35$	−17.3492	−	30.4641i	−0.495690	−	0.870403i
$36$		0			0
$37$	27.8724	+	48.2765i	0.753309	+	1.30477i	0.946211	+	0.323551i	$0.104877\pi$
$37$	27.8724	+	48.2765i	0.753309	+	1.30477i	−0.192902	+	0.981218i	$0.561790\pi$
$38$	15.9217	+	9.19239i	0.418992	+	0.241905i
$39$		0			0
$40$	−7.08276	−	12.2677i	−0.177069	−	0.306693i
$41$		−	6.53953i		−	0.159501i	−0.996815	−	0.0797503i	$-0.974588\pi$
$41$		−	6.53953i		−	0.159501i	0.996815	−	0.0797503i	$-0.0254123\pi$
$42$		0			0
$43$	32.7449			0.761508			0.380754	−	0.924676i	$-0.375664\pi$
$43$	32.7449			0.761508			0.380754	+	0.924676i	$0.375664\pi$
$44$	−2.65222	+	1.53126i	−0.0602776	+	0.0348013i
$45$		0			0
$46$	8.45862	−	14.6508i	0.183883	−	0.318495i
$47$	46.3841	−	26.7799i	0.986895	−	0.569784i	0.0825503	−	0.996587i	$-0.473693\pi$
$47$	46.3841	−	26.7799i	0.986895	−	0.569784i	0.904345	+	0.426803i	$0.140360\pi$
$48$		0			0
$49$	−48.9966	−	0.579328i	−0.999930	−	0.0118230i
$50$		−	0.117044i		−	0.00234088i
$51$		0			0
$52$	0.917237	−	1.58870i	0.0176392	−	0.0305520i
$53$	−72.4078	−	41.8047i	−1.36618	−	0.788767i	−0.375746	−	0.926723i	$-0.622614\pi$
$53$	−72.4078	−	41.8047i	−1.36618	−	0.788767i	−0.990438	+	0.137955i	$0.955947\pi$
$54$		0			0
$55$	−7.66895			−0.139435
$56$	−19.7986	−	0.117044i	−0.353547	−	0.00209007i
$57$		0			0
$58$	−23.7069	−	41.0616i	−0.408740	−	0.707958i
$59$	41.4386	+	23.9246i	0.702349	+	0.405501i	0.808222	−	0.588878i	$-0.200430\pi$
$59$	41.4386	+	23.9246i	0.702349	+	0.405501i	−0.105873	+	0.994380i	$0.533764\pi$
$60$		0			0
$61$	40.2897	+	69.7838i	0.660486	+	1.14400i	0.980488	+	0.196579i	$0.0629832\pi$
$61$	40.2897	+	69.7838i	0.660486	+	1.14400i	−0.320002	+	0.947417i	$0.603683\pi$
$62$		−	24.5098i		−	0.395319i
$63$		0			0
$64$	−8.00000			−0.125000
$65$	3.97832	−	2.29689i	0.0612050	−	0.0353367i
$66$		0			0
$67$	1.21033	−	2.09636i	0.0180646	−	0.0312889i	−0.856852	−	0.515563i	$-0.827583\pi$
$67$	1.21033	−	2.09636i	0.0180646	−	0.0312889i	0.874916	+	0.484274i	$0.160916\pi$
$68$	−28.6759	+	16.5561i	−0.421705	+	0.243472i
$69$		0			0
$70$	−42.7897	−	25.0431i	−0.611281	−	0.357758i
$71$		−	83.3216i		−	1.17354i	−0.809753	−	0.586772i	$-0.800399\pi$
$71$		−	83.3216i		−	1.17354i	0.809753	−	0.586772i	$-0.199601\pi$
$72$		0			0
$73$	−36.0414	+	62.4255i	−0.493718	+	0.855144i	−0.999974	−	0.00723922i	$-0.997696\pi$
$73$	−36.0414	+	62.4255i	−0.493718	+	0.855144i	0.506256	+	0.862383i	$0.331029\pi$
$74$	68.2732	+	39.4176i	0.922611	+	0.532670i
$75$		0			0
$76$	26.0000			0.342105
$77$	−5.41418	+	9.25091i	−0.0703141	+	0.120142i
$78$		0			0
$79$	61.1655	+	105.942i	0.774247	+	1.34104i	0.935217	+	0.354076i	$0.115205\pi$
$79$	61.1655	+	105.942i	0.774247	+	1.34104i	−0.160969	+	0.986959i	$0.551462\pi$
$80$	−17.3492	−	10.0165i	−0.216864	−	0.125207i
$81$		0			0
$82$	−4.62414	−	8.00925i	−0.0563920	−	0.0976738i
$83$			158.860i			1.91398i	0.290126	+	0.956989i	$0.406303\pi$
$83$			158.860i			1.91398i	−0.290126	+	0.956989i	$0.593697\pi$
$84$		0			0
$85$	−82.9172			−0.975497
$86$	40.1041	−	23.1541i	0.466327	−	0.269234i
$87$		0			0
$88$	−2.16553	+	3.75080i	−0.0246082	+	0.0426227i
$89$	−46.9922	+	27.1310i	−0.528003	+	0.304843i	−0.740203	−	0.672384i	$-0.765271\pi$
$89$	−46.9922	+	27.1310i	−0.528003	+	0.304843i	0.212200	+	0.977226i	$0.431937\pi$
$90$		0			0
$91$	0.0379564	−	6.42055i	0.000417104	−	0.0705555i
$92$		−	23.9246i		−	0.260050i
$93$		0			0
$94$	37.8724	−	65.5970i	0.402898	−	0.697840i
$95$	56.3848	+	32.5538i	0.593524	+	0.342671i
$96$		0			0
$97$	12.0828			0.124565			0.0622823	−	0.998059i	$-0.480162\pi$
$97$	12.0828			0.124565			0.0622823	+	0.998059i	$0.480162\pi$
$98$	−60.4180	+	33.9363i	−0.616510	+	0.346289i
$99$		0			0
$100$	−0.0827625	−	0.143349i	−0.000827625	−	0.00143349i
$101$	157.468	+	90.9145i	1.55909	+	0.900143i	0.997344	+	0.0728364i	$0.0232051\pi$
$101$	157.468	+	90.9145i	1.55909	+	0.900143i	0.561750	+	0.827307i	$0.310128\pi$
$102$		0			0
$103$	57.8724	+	100.238i	0.561868	+	0.973184i	0.997333	+	0.0729788i	$0.0232506\pi$
$103$	57.8724	+	100.238i	0.561868	+	0.973184i	−0.435465	+	0.900206i	$0.643416\pi$
$104$		−	2.59434i		−	0.0249456i
$105$		0			0
$106$	−118.241			−1.11549
$107$	−148.186	+	85.5551i	−1.38491	+	0.799580i	−0.992736	−	0.120310i	$-0.961611\pi$
$107$	−148.186	+	85.5551i	−1.38491	+	0.799580i	−0.392177	+	0.919890i	$0.628278\pi$
$108$		0			0
$109$	65.9586	−	114.244i	0.605125	−	1.04811i	−0.386907	−	0.922119i	$-0.626456\pi$
$109$	65.9586	−	114.244i	0.605125	−	1.04811i	0.992032	−	0.125988i	$-0.0402102\pi$
$110$	−9.39251	+	5.42277i	−0.0853864	+	0.0492979i
$111$		0			0
$112$	−24.3311	+	13.8564i	−0.217242	+	0.123718i
$113$			7.65629i			0.0677548i	0.999426	+	0.0338774i	$0.0107856\pi$
$113$			7.65629i			0.0677548i	−0.999426	+	0.0338774i	$0.989214\pi$
$114$		0			0
$115$	29.9552	−	51.8839i	0.260480	−	0.451165i
$116$	−58.0698	−	33.5266i	−0.500602	−	0.289023i
$117$		0			0
$118$	67.6689			0.573466
$119$	−58.5385	+	100.022i	−0.491921	+	0.840517i
$120$		0			0
$121$	−59.3276	−	102.758i	−0.490311	−	0.849243i
$122$	98.6891	+	56.9782i	0.808927	+	0.467034i
$123$		0			0
$124$	−17.3311	−	30.0183i	−0.139767	−	0.242083i
$125$			124.792i			0.998338i
$126$		0			0
$127$	−8.57934			−0.0675538			−0.0337769	−	0.999429i	$-0.510754\pi$
$127$	−8.57934			−0.0675538			−0.0337769	+	0.999429i	$0.510754\pi$
$128$	−9.79796	+	5.65685i	−0.0765466	+	0.0441942i
$129$		0			0
$130$	3.24829	−	5.62620i	0.0249868	−	0.0432785i
$131$	60.9713	−	35.2018i	0.465429	−	0.268716i	−0.248895	−	0.968530i	$-0.580067\pi$
$131$	60.9713	−	35.2018i	0.465429	−	0.268716i	0.714324	+	0.699815i	$0.246734\pi$
$132$		0			0
$133$	79.0759	−	45.0333i	0.594556	−	0.338596i
$134$		−	3.42333i		−	0.0255473i
$135$		0			0
$136$	−23.4138	+	40.5539i	−0.172160	+	0.298191i
$137$	58.4288	+	33.7339i	0.426488	+	0.246233i	0.697849	−	0.716245i	$-0.254141\pi$
$137$	58.4288	+	33.7339i	0.426488	+	0.246233i	−0.271362	+	0.962477i	$0.587474\pi$
$138$		0			0
$139$	166.572			1.19836			0.599182	−	0.800613i	$-0.295493\pi$
$139$	166.572			1.19836			0.599182	+	0.800613i	$0.295493\pi$
$140$	−70.1145	−	0.414497i	−0.500818	−	0.00296069i
$141$		0			0
$142$	−58.9172	−	102.048i	−0.414910	−	0.718645i
$143$	−1.21636	−	0.702263i	−0.00850598	−	0.00491093i
$144$		0			0
$145$	−83.9552	−	145.415i	−0.579001	−	1.00286i
$146$			101.940i			0.698222i
$147$		0			0
$148$	111.490			0.753309
$149$	−196.724	+	113.578i	−1.32029	+	0.762271i	−0.983775	−	0.179406i	$-0.942582\pi$
$149$	−196.724	+	113.578i	−1.32029	+	0.762271i	−0.336517	+	0.941677i	$0.609249\pi$
$150$		0			0
$151$	137.824	−	238.719i	0.912743	−	1.58092i	0.102571	−	0.994726i	$-0.467293\pi$
$151$	137.824	−	238.719i	0.912743	−	1.58092i	0.810172	−	0.586192i	$-0.199373\pi$
$152$	31.8434	−	18.3848i	0.209496	−	0.120952i
$153$		0			0
$154$	−0.0896122	+	15.1584i	−0.000581897	+	0.0984312i
$155$		−	86.7986i		−	0.559991i
$156$		0			0
$157$	−99.6173	+	172.542i	−0.634505	+	1.09899i	0.352115	+	0.935957i	$0.385463\pi$
$157$	−99.6173	+	172.542i	−0.634505	+	1.09899i	−0.986620	+	0.163038i	$0.947871\pi$
$158$	149.824	+	86.5011i	0.948255	+	0.547475i
$159$		0			0
$160$	−28.3311			−0.177069
$161$	−41.4386	−	72.7638i	−0.257383	−	0.451949i
$162$		0			0
$163$	48.5000	+	84.0045i	0.297546	+	0.515365i	0.975574	−	0.219671i	$-0.0704985\pi$
$163$	48.5000	+	84.0045i	0.297546	+	0.515365i	−0.678028	+	0.735036i	$0.737165\pi$
$164$	−11.3268	−	6.53953i	−0.0690658	−	0.0398752i
$165$		0			0
$166$	112.331	+	194.563i	0.676693	+	1.17207i
$167$		−	21.6911i		−	0.129887i	−0.997889	−	0.0649433i	$-0.979313\pi$
$167$		−	21.6911i		−	0.129887i	0.997889	−	0.0649433i	$-0.0206867\pi$
$168$		0			0
$169$	−168.159			−0.995022
$170$	−101.552	+	58.6313i	−0.597367	+	0.344890i
$171$		0			0
$172$	32.7449	−	56.7158i	0.190377	−	0.329743i
$173$	180.701	−	104.328i	1.04451	−	0.603049i	0.123404	−	0.992357i	$-0.460619\pi$
$173$	180.701	−	104.328i	1.04451	−	0.603049i	0.921108	+	0.389307i	$0.127286\pi$
$174$		0			0
$175$	−0.500000	−	0.292630i	−0.00285714	−	0.00167217i
$176$			6.12503i			0.0348013i
$177$		0			0
$178$	−38.3690	+	66.4571i	−0.215556	+	0.373354i
$179$	54.9489	+	31.7248i	0.306977	+	0.177233i	0.645573	−	0.763699i	$-0.276619\pi$
$179$	54.9489	+	31.7248i	0.306977	+	0.177233i	−0.338596	+	0.940932i	$0.609952\pi$
$180$		0			0
$181$	−212.311			−1.17299			−0.586493	−	0.809954i	$-0.699492\pi$
$181$	−212.311			−1.17299			−0.586493	+	0.809954i	$0.699492\pi$
$182$	−4.49353	−	7.89038i	−0.0246897	−	0.0433537i
$183$		0			0
$184$	−16.9172	−	29.3015i	−0.0919415	−	0.159247i
$185$	241.782	+	139.593i	1.30693	+	0.754555i
$186$		0			0
$187$	12.6758	+	21.9551i	0.0677850	+	0.117407i
$188$		−	107.119i		−	0.569784i
$189$		0			0
$190$	92.0759			0.484610
$191$	50.9706	−	29.4279i	0.266862	−	0.154073i	−0.360599	−	0.932721i	$-0.617428\pi$
$191$	50.9706	−	29.4279i	0.266862	−	0.154073i	0.627461	+	0.778648i	$0.284094\pi$
$192$		0			0
$193$	−69.6173	+	120.581i	−0.360711	+	0.624770i	−0.988078	−	0.153954i	$-0.950799\pi$
$193$	−69.6173	+	120.581i	−0.360711	+	0.624770i	0.627367	+	0.778724i	$0.284133\pi$
$194$	14.7983	−	8.54380i	0.0762799	−	0.0440402i
$195$		0			0
$196$	−50.0000	+	84.2852i	−0.255102	+	0.430027i
$197$		−	317.720i		−	1.61279i	−0.591375	−	0.806396i	$-0.701415\pi$
$197$		−	317.720i		−	1.61279i	0.591375	−	0.806396i	$-0.298585\pi$
$198$		0			0
$199$	−124.869	+	216.279i	−0.627482	+	1.08683i	0.360573	+	0.932731i	$0.382581\pi$
$199$	−124.869	+	216.279i	−0.627482	+	1.08683i	−0.988055	+	0.154100i	$0.950752\pi$
$200$	−0.202726	−	0.117044i	−0.00101363	−	0.000585219i
$201$		0			0
$202$	257.145			1.27299
$203$	−234.682	−	1.38737i	−1.15607	−	0.00683436i
$204$		0			0
$205$	−16.3759	−	28.3638i	−0.0798822	−	0.138360i
$206$	141.758	+	81.8440i	0.688145	+	0.397301i
$207$		0			0
$208$	−1.83447	−	3.17740i	−0.00881959	−	0.0152760i
$209$		−	19.9063i		−	0.0952457i
$210$		0			0
$211$	−112.248			−0.531982			−0.265991	−	0.963975i	$-0.585699\pi$
$211$	−112.248			−0.531982			−0.265991	+	0.963975i	$0.585699\pi$
$212$	−144.816	+	83.6093i	−0.683092	+	0.394384i
$213$		0			0
$214$	−120.993	+	209.566i	−0.565389	+	0.979282i
$215$	142.024	−	81.9975i	0.660576	−	0.381384i
$216$		0			0
$217$	−104.703	−	61.2787i	−0.482505	−	0.282390i
$218$		−	186.559i		−	0.855776i
$219$		0			0
$220$	−7.66895	+	13.2830i	−0.0348589	+	0.0603773i
$221$	−13.1513	−	7.59292i	−0.0595083	−	0.0343571i
$222$		0			0
$223$	79.2346			0.355312			0.177656	−	0.984093i	$-0.443149\pi$
$223$	79.2346			0.355312			0.177656	+	0.984093i	$0.443149\pi$
$224$	−20.0014	+	34.1752i	−0.0892918	+	0.152568i
$225$		0			0
$226$	5.41381	+	9.37700i	0.0239549	+	0.0414911i
$227$	−310.600	−	179.325i	−1.36828	−	0.789977i	−0.377572	−	0.925980i	$-0.623241\pi$
$227$	−310.600	−	179.325i	−1.36828	−	0.789977i	−0.990708	+	0.136003i	$0.956574\pi$
$228$		0			0
$229$	−61.1689	−	105.948i	−0.267113	−	0.462654i	0.701002	−	0.713159i	$-0.252736\pi$
$229$	−61.1689	−	105.948i	−0.267113	−	0.462654i	−0.968115	+	0.250506i	$0.919403\pi$
$230$		−	84.7261i		−	0.368374i
$231$		0			0
$232$	−94.8276			−0.408740
$233$	−184.071	+	106.273i	−0.790003	+	0.456108i	−0.839964	−	0.542643i	$-0.817424\pi$
$233$	−184.071	+	106.273i	−0.790003	+	0.456108i	0.0499606	+	0.998751i	$0.484090\pi$
$234$		0			0
$235$	134.121	−	232.304i	0.570726	−	0.988527i
$236$	82.8772	−	47.8492i	0.351175	−	0.202751i
$237$		0			0
$238$	−0.968893	+	163.894i	−0.00407098	+	0.688630i
$239$		−	342.762i		−	1.43415i	−0.696997	−	0.717074i	$-0.745481\pi$
$239$		−	342.762i		−	1.43415i	0.696997	−	0.717074i	$-0.254519\pi$
$240$		0			0
$241$	−16.3724	+	28.3579i	−0.0679354	+	0.117668i	−0.897992	−	0.440011i	$-0.854975\pi$
$241$	−16.3724	+	28.3579i	−0.0679354	+	0.117668i	0.830057	+	0.557679i	$0.188308\pi$
$242$	−145.322	−	83.9019i	−0.600506	−	0.346702i
$243$		0			0
$244$	161.159			0.660486
$245$	−213.963	+	120.181i	−0.873318	+	0.490536i
$246$		0			0
$247$	5.96204	+	10.3266i	0.0241378	+	0.0418079i
$248$	−42.4522	−	24.5098i	−0.171178	−	0.0988299i
$249$		0			0
$250$	88.2414	+	152.839i	0.352966	+	0.611355i
$251$			8.32425i			0.0331643i	0.999863	+	0.0165822i	$0.00527851\pi$
$251$			8.32425i			0.0331643i	−0.999863	+	0.0165822i	$0.994721\pi$
$252$		0			0
$253$	−18.3174			−0.0724006
$254$	−10.5075	+	6.06651i	−0.0413681	+	0.0238839i
$255$		0			0
$256$	−8.00000	+	13.8564i	−0.0312500	+	0.0541266i
$257$	13.4806	−	7.78302i	0.0524536	−	0.0302841i	−0.473544	−	0.880770i	$-0.657025\pi$
$257$	13.4806	−	7.78302i	0.0524536	−	0.0302841i	0.525997	+	0.850486i	$0.323692\pi$
$258$		0			0
$259$	339.083	−	193.106i	1.30920	−	0.745583i
$260$		−	9.18754i		−	0.0353367i
$261$		0			0
$262$	49.7828	−	86.2264i	0.190011	−	0.329108i
$263$	5.52394	+	3.18925i	0.0210036	+	0.0121264i	0.510465	−	0.859899i	$-0.329473\pi$
$263$	5.52394	+	3.18925i	0.0210036	+	0.0121264i	−0.489461	+	0.872025i	$0.662807\pi$
$264$		0			0
$265$	−418.738			−1.58014
$266$	65.0045	−	111.069i	0.244378	−	0.417554i
$267$		0			0
$268$	−2.42066	−	4.19271i	−0.00903232	−	0.0156444i
$269$	299.243	+	172.768i	1.11243	+	0.642261i	0.939457	−	0.342666i	$-0.111330\pi$
$269$	299.243	+	172.768i	1.11243	+	0.642261i	0.172971	+	0.984927i	$0.444663\pi$
$270$		0			0
$271$	−109.214	−	189.164i	−0.403003	−	0.698021i	0.591084	−	0.806610i	$-0.298700\pi$
$271$	−109.214	−	189.164i	−0.403003	−	0.698021i	−0.994087	+	0.108589i	$0.965367\pi$
$272$			66.2243i			0.243472i
$273$		0			0
$274$	95.4138			0.348226
$275$	−0.109752	+	0.0633654i	−0.000399098	+	0.000230420i
$276$		0			0
$277$	−65.3656	+	113.217i	−0.235977	+	0.408724i	−0.959556	−	0.281517i	$-0.909162\pi$
$277$	−65.3656	+	113.217i	−0.235977	+	0.408724i	0.723579	+	0.690241i	$0.242496\pi$
$278$	204.009	−	117.785i	0.733845	−	0.423685i
$279$		0			0
$280$	−86.1655	+	49.0708i	−0.307734	+	0.175253i
$281$		−	430.516i		−	1.53208i	−0.642790	−	0.766042i	$-0.722223\pi$
$281$		−	430.516i		−	1.53208i	0.642790	−	0.766042i	$-0.277777\pi$
$282$		0			0
$283$	193.945	−	335.922i	0.685318	−	1.18701i	−0.288019	−	0.957625i	$-0.592997\pi$
$283$	193.945	−	335.922i	0.685318	−	1.18701i	0.973337	−	0.229380i	$-0.0736700\pi$
$284$	−144.317	−	83.3216i	−0.508159	−	0.293386i
$285$		0			0
$286$	−1.98630			−0.00694511
$287$	−45.7759	−	0.270614i	−0.159498	−	0.000942906i
$288$		0			0
$289$	−7.44834	−	12.9009i	−0.0257728	−	0.0446398i
$290$	−205.647	−	118.731i	−0.709129	−	0.409416i
$291$		0			0
$292$	72.0828	+	124.851i	0.246859	+	0.427572i
$293$		−	397.726i		−	1.35743i	−0.734404	−	0.678713i	$-0.762538\pi$
$293$		−	397.726i		−	1.35743i	0.734404	−	0.678713i	$-0.237462\pi$
$294$		0			0
$295$	239.642			0.812344
$296$	136.546	−	78.8351i	0.461306	−	0.266335i
$297$		0			0
$298$	−160.624	+	278.209i	−0.539007	+	0.933588i
$299$	9.50226	−	5.48613i	0.0317801	−	0.0183483i
$300$		0			0
$301$	1.35502	−	229.210i	0.00450174	−	0.761495i
$302$		−	389.826i		−	1.29081i
$303$		0			0
$304$	26.0000	−	45.0333i	0.0855263	−	0.148136i
$305$	349.496	+	201.782i	1.14589	+	0.661579i
$306$		0			0
$307$	−116.669			−0.380029			−0.190015	−	0.981781i	$-0.560854\pi$
$307$	−116.669			−0.380029			−0.190015	+	0.981781i	$0.560854\pi$
$308$	10.6089	+	18.6286i	0.0344444	+	0.0604823i
$309$		0			0
$310$	−61.3759	−	106.306i	−0.197987	−	0.342923i
$311$	420.797	+	242.947i	1.35305	+	0.781181i	0.988675	−	0.150073i	$-0.0479509\pi$
$311$	420.797	+	242.947i	1.35305	+	0.781181i	0.364370	+	0.931254i	$0.381284\pi$
$312$		0			0
$313$	76.8311	+	133.075i	0.245467	+	0.425161i	0.962263	−	0.272122i	$-0.0877255\pi$
$313$	76.8311	+	133.075i	0.245467	+	0.425161i	−0.716796	+	0.697283i	$0.754392\pi$
$314$			281.760i			0.897326i
$315$		0			0
$316$	244.662			0.774247
$317$	−83.8741	+	48.4247i	−0.264587	+	0.152759i	−0.626425	−	0.779482i	$-0.715483\pi$
$317$	−83.8741	+	48.4247i	−0.264587	+	0.152759i	0.361838	+	0.932241i	$0.382149\pi$
$318$		0			0
$319$	−25.6689	+	44.4599i	−0.0804669	+	0.139373i
$320$	−34.6983	+	20.0331i	−0.108432	+	0.0626034i
$321$		0			0
$322$	−102.203	−	59.8156i	−0.317402	−	0.185763i
$323$		−	215.229i		−	0.666343i
$324$		0			0
$325$	0.0379564	−	0.0657425i	0.000116789	−	0.000202285i
$326$	118.800	+	68.5894i	0.364418	+	0.210397i
$327$		0			0
$328$	−18.4966			−0.0563920
$329$	−185.536	−	325.791i	−0.563940	−	0.990246i
$330$		0			0
$331$	36.6344	+	63.4527i	0.110678	+	0.191700i	0.916044	−	0.401078i	$-0.131364\pi$
$331$	36.6344	+	63.4527i	0.110678	+	0.191700i	−0.805366	+	0.592778i	$0.798031\pi$
$332$	275.154	+	158.860i	0.828776	+	0.478494i
$333$		0			0
$334$	−15.3379	−	26.5660i	−0.0459219	−	0.0795390i
$335$		−	12.1233i		−	0.0361890i
$336$		0			0
$337$	−165.304			−0.490515			−0.245258	−	0.969458i	$-0.578873\pi$
$337$	−165.304			−0.490515			−0.245258	+	0.969458i	$0.578873\pi$
$338$	−205.951	+	118.906i	−0.609324	+	0.351793i
$339$		0			0
$340$	−82.9172	+	143.617i	−0.243874	+	0.422403i
$341$	−22.9828	+	13.2691i	−0.0673984	+	0.0389125i
$342$		0			0
$343$	−6.08276	+	342.946i	−0.0177340	+	0.999843i
$344$		−	92.6165i		−	0.269234i
$345$		0			0
$346$	147.541	−	255.549i	0.426420	−	0.738581i
$347$	−181.558	−	104.823i	−0.523222	−	0.302082i	0.215030	−	0.976607i	$-0.431015\pi$
$347$	−181.558	−	104.823i	−0.523222	−	0.302082i	−0.738252	+	0.674525i	$0.764348\pi$
$348$		0			0
$349$	−442.655			−1.26835			−0.634177	−	0.773188i	$-0.718661\pi$
$349$	−442.655			−1.26835			−0.634177	+	0.773188i	$0.718661\pi$
$350$	−0.819293	−	0.00484342i	−0.00234084	−	1.38384e-5i
$351$		0			0
$352$	4.33105	+	7.50160i	0.0123041	+	0.0213114i
$353$	−144.537	−	83.4483i	−0.409452	−	0.236397i	0.281102	−	0.959678i	$-0.409300\pi$
$353$	−144.537	−	83.4483i	−0.409452	−	0.236397i	−0.690554	+	0.723280i	$0.742633\pi$
$354$		0			0
$355$	−208.648	−	361.390i	−0.587742	−	1.01800i
$356$			108.524i			0.304843i
$357$		0			0
$358$	89.7312			0.250646
$359$	−16.3523	+	9.44101i	−0.0455496	+	0.0262981i	−0.522602	−	0.852577i	$-0.675039\pi$
$359$	−16.3523	+	9.44101i	−0.0455496	+	0.0262981i	0.477052	+	0.878875i	$0.341705\pi$
$360$		0			0
$361$	96.0000	−	166.277i	0.265928	−	0.460601i
$362$	−260.026	+	150.126i	−0.718304	+	0.414713i
$363$		0			0
$364$	−11.0828	−	6.48629i	−0.0304471	−	0.0178195i
$365$			361.010i			0.989068i
$366$		0			0
$367$	196.110	−	339.673i	0.534361	−	0.925540i	−0.464833	−	0.885398i	$-0.653886\pi$
$367$	196.110	−	339.673i	0.534361	−	0.925540i	0.999194	−	0.0401419i	$-0.0127810\pi$
$368$	−41.4386	−	23.9246i	−0.112605	−	0.0650125i
$369$		0			0
$370$	394.828			1.06710
$371$	−295.624	+	505.116i	−0.796830	+	1.36150i
$372$		0			0
$373$	277.811	+	481.182i	0.744800	+	1.29003i	0.950288	+	0.311372i	$0.100789\pi$
$373$	277.811	+	481.182i	0.744800	+	1.29003i	−0.205488	+	0.978660i	$0.565878\pi$
$374$	31.0492	+	17.9263i	0.0830194	+	0.0479313i
$375$		0			0
$376$	−75.7449	−	131.194i	−0.201449	−	0.348920i
$377$		−	30.7519i		−	0.0815700i
$378$		0			0
$379$	433.552			1.14394			0.571968	−	0.820276i	$-0.306180\pi$
$379$	433.552			1.14394			0.571968	+	0.820276i	$0.306180\pi$
$380$	112.770	−	65.1075i	0.296762	−	0.171336i
$381$		0			0
$382$	41.6173	−	72.0833i	0.108946	−	0.188700i
$383$	−118.682	+	68.5211i	−0.309875	+	0.178906i	−0.646871	−	0.762600i	$-0.723923\pi$
$383$	−118.682	+	68.5211i	−0.309875	+	0.178906i	0.336996	+	0.941506i	$0.390589\pi$
$384$		0			0
$385$	−0.317351	+	53.6817i	−0.000824288	+	0.139433i
$386$			196.907i			0.510123i
$387$		0			0
$388$	12.0828	−	20.9280i	0.0311411	−	0.0539380i
$389$	−301.317	−	173.965i	−0.774594	−	0.447212i	0.0599172	−	0.998203i	$-0.480916\pi$
$389$	−301.317	−	173.965i	−0.774594	−	0.447212i	−0.834511	+	0.550991i	$0.814250\pi$
$390$		0			0
$391$	−198.049			−0.506518
$392$	−1.63859	+	138.583i	−0.00418007	+	0.353529i
$393$		0			0
$394$	−224.662	−	389.126i	−0.570208	−	0.987630i
$395$	530.585	+	306.333i	1.34325	+	0.775528i
$396$		0			0
$397$	−231.200	−	400.450i	−0.582368	−	1.00869i	−0.995198	−	0.0978827i	$-0.968793\pi$
$397$	−231.200	−	400.450i	−0.582368	−	1.00869i	0.412830	−	0.910808i	$-0.364540\pi$
$398$			353.183i			0.887394i
$399$		0			0
$400$	−0.331050			−0.000827625
$401$	−383.337	+	221.320i	−0.955952	+	0.551919i	−0.894925	−	0.446217i	$-0.852771\pi$
$401$	−383.337	+	221.320i	−0.955952	+	0.551919i	−0.0610272	+	0.998136i	$0.519438\pi$
$402$		0			0
$403$	7.94834	−	13.7669i	0.0197229	−	0.0341611i
$404$	314.937	−	181.829i	0.779547	−	0.450072i
$405$		0			0
$406$	−288.407	+	164.246i	−0.710362	+	0.404547i
$407$		−	85.3597i		−	0.209729i
$408$		0			0
$409$	−28.1139	+	48.6947i	−0.0687381	+	0.119058i	−0.898346	−	0.439288i	$-0.855231\pi$
$409$	−28.1139	+	48.6947i	−0.0687381	+	0.119058i	0.829608	+	0.558346i	$0.188564\pi$
$410$	−40.1125	−	23.1590i	−0.0978353	−	0.0564853i
$411$		0			0
$412$	231.490			0.561868
$413$	169.184	−	289.075i	0.409646	−	0.699940i
$414$		0			0
$415$	397.807	+	689.022i	0.958571	+	1.66029i
$416$	−4.49353	−	2.59434i	−0.0108017	−	0.00623639i
$417$		0			0
$418$	−14.0759	−	24.3802i	−0.0336744	−	0.0583258i
$419$			330.707i			0.789276i	0.918837	+	0.394638i	$0.129130\pi$
$419$			330.707i			0.789276i	−0.918837	+	0.394638i	$0.870870\pi$
$420$		0			0
$421$	−30.5930			−0.0726675			−0.0363338	−	0.999340i	$-0.511568\pi$
$421$	−30.5930			−0.0726675			−0.0363338	+	0.999340i	$0.511568\pi$
$422$	−137.476	+	79.3715i	−0.325771	+	0.188084i
$423$		0			0
$424$	−118.241	+	204.800i	−0.278871	+	0.483019i
$425$	−1.18665	+	0.685111i	−0.00279211	+	0.00161203i
$426$		0			0
$427$	490.145	−	279.135i	1.14788	−	0.653712i
$428$			342.220i			0.799580i
$429$		0			0
$430$	115.962	−	200.852i	0.269679	−	0.467098i
$431$	580.479	+	335.140i	1.34682	+	0.777586i	0.987798	−	0.155744i	$-0.0497773\pi$
$431$	580.479	+	335.140i	1.34682	+	0.777586i	0.359021	+	0.933330i	$0.383111\pi$
$432$		0			0
$433$	−192.276			−0.444055			−0.222027	−	0.975040i	$-0.571267\pi$
$433$	−192.276			−0.444055			−0.222027	+	0.975040i	$0.571267\pi$
$434$	−171.566	−	1.01425i	−0.395313	−	0.00233697i
$435$		0			0
$436$	−131.917	−	228.487i	−0.302562	−	0.524054i
$437$	134.675	+	77.7549i	0.308182	+	0.177929i
$438$		0			0
$439$	80.5208	+	139.466i	0.183419	+	0.317691i	0.943043	−	0.332672i	$-0.107950\pi$
$439$	80.5208	+	139.466i	0.183419	+	0.317691i	−0.759624	+	0.650363i	$0.774617\pi$
$440$			21.6911i			0.0492979i
$441$		0			0
$442$	−21.4760			−0.0485883
$443$	590.281	−	340.799i	1.33246	−	0.769297i	0.346785	−	0.937945i	$-0.387273\pi$
$443$	590.281	−	340.799i	1.33246	−	0.769297i	0.985676	+	0.168647i	$0.0539399\pi$
$444$		0			0
$445$	−135.879	+	235.350i	−0.305347	+	0.528876i
$446$	97.0422	−	56.0273i	0.217583	−	0.125622i
$447$		0			0
$448$	−0.331050	+	55.9990i	−0.000738951	+	0.124998i
$449$			804.351i			1.79143i	0.444630	+	0.895714i	$0.353335\pi$
$449$			804.351i			1.79143i	−0.444630	+	0.895714i	$0.646665\pi$
$450$		0			0
$451$	−5.00685	+	8.67212i	−0.0111017	+	0.0192286i
$452$	13.2611	+	7.65629i	0.0293387	+	0.0169387i
$453$		0			0
$454$	−507.207			−1.11720
$455$	−15.9133	−	27.9428i	−0.0349743	−	0.0614128i
$456$		0			0
$457$	44.4415	+	76.9749i	0.0972462	+	0.168435i	0.910544	−	0.413413i	$-0.135663\pi$
$457$	44.4415	+	76.9749i	0.0972462	+	0.168435i	−0.813298	+	0.581848i	$0.802330\pi$
$458$	−149.833	−	86.5060i	−0.327146	−	0.188878i
$459$		0			0
$460$	−59.9104	−	103.768i	−0.130240	−	0.225582i
$461$			862.183i			1.87024i	0.354325	+	0.935122i	$0.384711\pi$
$461$			862.183i			1.87024i	−0.354325	+	0.935122i	$0.615289\pi$
$462$		0			0
$463$	−140.938			−0.304401			−0.152201	−	0.988350i	$-0.548636\pi$
$463$	−140.938			−0.304401			−0.152201	+	0.988350i	$0.548636\pi$
$464$	−116.140	+	67.0533i	−0.250301	+	0.144511i
$465$		0			0
$466$	−150.293	+	260.315i	−0.322517	+	0.558616i
$467$	−233.715	+	134.936i	−0.500461	+	0.288941i	−0.728904	−	0.684616i	$-0.759970\pi$
$467$	−233.715	+	134.936i	−0.500461	+	0.288941i	0.228443	+	0.973557i	$0.426637\pi$
$468$		0			0
$469$	−14.6241	−	8.55892i	−0.0311815	−	0.0182493i
$470$		−	379.351i		−	0.807129i
$471$		0			0
$472$	67.6689	−	117.206i	0.143366	−	0.248318i
$473$	−43.4232	−	25.0704i	−0.0918038	−	0.0530030i
$474$		0			0
$475$	1.07591			0.00226508
$476$	114.704	+	201.413i	0.240974	+	0.423137i
$477$		0			0
$478$	−242.369	−	419.795i	−0.507048	−	0.878233i
$479$	−652.109	−	376.495i	−1.36140	−	0.786003i	−0.371587	−	0.928398i	$-0.621186\pi$
$479$	−652.109	−	376.495i	−1.36140	−	0.786003i	−0.989810	+	0.142395i	$0.954520\pi$
$480$		0			0
$481$	25.5656	+	44.2810i	0.0531510	+	0.0920603i
$482$			46.3082i			0.0960752i
$483$		0			0
$484$	−237.311			−0.490311
$485$	52.4064	−	30.2569i	0.108054	−	0.0623853i
$486$		0			0
$487$	−375.969	+	651.198i	−0.772011	+	1.33716i	0.164449	+	0.986386i	$0.447415\pi$
$487$	−375.969	+	651.198i	−0.772011	+	1.33716i	−0.936459	+	0.350776i	$0.885918\pi$
$488$	197.378	−	113.956i	0.404464	−	0.233517i
$489$		0			0
$490$	−177.069	+	298.486i	−0.361365	+	0.609155i
$491$			165.492i			0.337051i	0.985697	+	0.168526i	$0.0539006\pi$
$491$			165.492i			0.337051i	−0.985697	+	0.168526i	$0.946099\pi$
$492$		0			0
$493$	−277.535	+	480.704i	−0.562950	+	0.975059i
$494$	14.6040	+	8.43160i	0.0295627	+	0.0170680i
$495$		0			0
$496$	−69.3242			−0.139767
$497$	−583.241	−	3.44795i	−1.17352	−	0.00693753i
$498$		0			0
$499$	213.341	+	369.518i	0.427538	+	0.740517i	0.996654	−	0.0817402i	$-0.0260478\pi$
$499$	213.341	+	369.518i	0.427538	+	0.740517i	−0.569116	+	0.822257i	$0.692714\pi$
$500$	216.146	+	124.792i	0.432293	+	0.249584i
$501$		0			0
$502$	5.88613	+	10.1951i	0.0117254	+	0.0203089i
$503$			302.442i			0.601276i	0.953738	+	0.300638i	$0.0971996\pi$
$503$			302.442i			0.601276i	−0.953738	+	0.300638i	$0.902800\pi$
$504$		0			0
$505$	910.648			1.80326
$506$	−22.4341	+	12.9523i	−0.0443361	+	0.0255975i
$507$		0			0
$508$	−8.57934	+	14.8598i	−0.0168885	+	0.0292517i
$509$	−122.770	+	70.8814i	−0.241199	+	0.139256i	−0.615728	−	0.787959i	$-0.711138\pi$
$509$	−122.770	+	70.8814i	−0.241199	+	0.139256i	0.374529	+	0.927215i	$0.377804\pi$
$510$		0			0
$511$	435.479	+	254.869i	0.852210	+	0.498764i
$512$			22.6274i			0.0441942i
$513$		0			0
$514$	11.0068	−	19.0644i	0.0214141	−	0.0370903i
$515$	502.019	+	289.841i	0.974794	+	0.562798i
$516$		0			0
$517$	−82.0137			−0.158634
$518$	278.743	−	476.273i	0.538115	−	0.919446i
$519$		0			0
$520$	−6.49658	−	11.2524i	−0.0124934	−	0.0216392i
$521$	390.077	+	225.211i	0.748708	+	0.432267i	0.825227	−	0.564801i	$-0.191047\pi$
$521$	390.077	+	225.211i	0.748708	+	0.432267i	−0.0765187	+	0.997068i	$0.524380\pi$
$522$		0			0
$523$	156.221	+	270.583i	0.298702	+	0.517366i	0.975839	−	0.218490i	$-0.0701132\pi$
$523$	156.221	+	270.583i	0.298702	+	0.517366i	−0.677138	+	0.735856i	$0.736780\pi$
$524$		−	140.807i		−	0.268716i
$525$		0			0
$526$	9.02055			0.0171493
$527$	−248.492	+	143.467i	−0.471522	+	0.272233i
$528$		0			0
$529$	−192.952	+	334.202i	−0.364748	+	0.631762i
$530$	−512.847	+	296.092i	−0.967636	+	0.558665i
$531$		0			0
$532$	1.07591	−	181.997i	0.00202239	−	0.342099i
$533$		−	5.99830i		−	0.0112538i
$534$		0			0
$535$	−428.483	+	742.154i	−0.800903	+	1.38720i
$536$	−5.92939	−	3.42333i	−0.0110623	−	0.00638682i
$537$		0			0
$538$	488.662			0.908294
$539$	64.5312	+	38.2814i	0.119724	+	0.0710231i
$540$		0			0
$541$	−307.173	−	532.039i	−0.567787	−	0.983436i	−0.996784	−	0.0801300i	$-0.974466\pi$
$541$	−307.173	−	532.039i	−0.567787	−	0.983436i	0.428998	−	0.903306i	$-0.358867\pi$
$542$	−267.518	−	154.452i	−0.493576	−	0.284966i
$543$		0			0
$544$	46.8276	+	81.1078i	0.0860802	+	0.149095i
$545$		−	660.677i		−	1.21225i
$546$		0			0
$547$	−675.263			−1.23448			−0.617242	−	0.786773i	$-0.711750\pi$
$547$	−675.263			−1.23448			−0.617242	+	0.786773i	$0.711750\pi$
$548$	116.858	−	67.4678i	0.213244	−	0.123116i
$549$		0			0
$550$	−0.0896122	+	0.155213i	−0.000162931	+	0.000282205i
$551$	377.454	−	217.923i	0.685034	−	0.395505i
$552$		0			0
$553$	744.111	−	423.767i	1.34559	−	0.766306i
$554$			184.882i			0.333722i
$555$		0			0
$556$	166.572	−	288.512i	0.299591	−	0.518906i
$557$	−579.013	−	334.293i	−1.03952	−	0.600168i	−0.119823	−	0.992795i	$-0.538233\pi$
$557$	−579.013	−	334.293i	−1.03952	−	0.600168i	−0.919698	+	0.392628i	$0.871566\pi$
$558$		0			0
$559$	30.0348			0.0537295
$560$	−70.8325	+	121.027i	−0.126487	+	0.216120i
$561$		0			0
$562$	−304.421	−	527.272i	−0.541674	−	0.938207i
$563$	−10.9678	−	6.33228i	−0.0194810	−	0.0112474i	0.490228	−	0.871594i	$-0.336914\pi$
$563$	−10.9678	−	6.33228i	−0.0194810	−	0.0112474i	−0.509709	+	0.860347i	$0.670247\pi$
$564$		0			0
$565$	19.1724	+	33.2075i	0.0339334	+	0.0587744i
$566$		−	548.559i		−	0.969186i
$567$		0			0
$568$	−235.669			−0.414910
$569$	509.198	−	293.986i	0.894900	−	0.516671i	0.0193580	−	0.999813i	$-0.493838\pi$
$569$	509.198	−	293.986i	0.894900	−	0.516671i	0.875542	+	0.483142i	$0.160504\pi$
$570$		0			0
$571$	386.759	−	669.886i	0.677336	−	1.17318i	−0.298444	−	0.954427i	$-0.596468\pi$
$571$	386.759	−	669.886i	0.677336	−	1.17318i	0.975780	−	0.218753i	$-0.0701990\pi$
$572$	−2.43271	+	1.40453i	−0.00425299	+	0.00245547i
$573$		0			0
$574$	−56.2551	+	32.0370i	−0.0980055	+	0.0558136i
$575$		−	0.990030i		−	0.00172179i
$576$		0			0
$577$	−45.7965	+	79.3219i	−0.0793700	+	0.137473i	−0.902978	−	0.429686i	$-0.858624\pi$
$577$	−45.7965	+	79.3219i	−0.0793700	+	0.137473i	0.823608	+	0.567159i	$0.191958\pi$
$578$	−18.2446	−	10.5335i	−0.0315651	−	0.0182241i
$579$		0			0
$580$	−335.821			−0.579001
$581$	1112.00	+	6.57383i	1.91394	+	0.0113147i
$582$		0			0
$583$	64.0137	+	110.875i	0.109801	+	0.190180i
$584$	176.566	+	101.940i	0.302339	+	0.174556i
$585$		0			0
$586$	−281.235	−	487.113i	−0.479923	−	0.831250i
$587$			237.495i			0.404592i	0.979324	+	0.202296i	$0.0648403\pi$
$587$			237.495i			0.404592i	−0.979324	+	0.202296i	$0.935160\pi$
$588$		0			0
$589$	225.304			0.382519
$590$	293.500	−	169.452i	0.497457	−	0.287207i
$591$		0			0
$592$	111.490	−	193.106i	0.188327	−	0.326192i
$593$	−23.8699	+	13.7813i	−0.0402529	+	0.0232400i	−0.519991	−	0.854172i	$-0.674065\pi$
$593$	−23.8699	+	13.7813i	−0.0402529	+	0.0232400i	0.479739	+	0.877412i	$0.340732\pi$
$594$		0			0
$595$	−3.43122	+	580.411i	−0.00576676	+	0.975480i
$596$			454.314i			0.762271i
$597$		0			0
$598$	7.75856	−	13.4382i	0.0129742	−	0.0224719i
$599$	−815.769	−	470.985i	−1.36189	−	0.786285i	−0.372010	−	0.928229i	$-0.621331\pi$
$599$	−815.769	−	470.985i	−1.36189	−	0.786285i	−0.989875	+	0.141944i	$0.954665\pi$
$600$		0			0
$601$	−374.262			−0.622732			−0.311366	−	0.950290i	$-0.600787\pi$
$601$	−374.262			−0.622732			−0.311366	+	0.950290i	$0.600787\pi$
$602$	−160.416	−	281.682i	−0.266472	−	0.467910i
$603$		0			0
$604$	−275.648	−	477.437i	−0.456372	−	0.790459i
$605$	−514.642	−	297.129i	−0.850648	−	0.491122i
$606$		0			0
$607$	−190.328	−	329.657i	−0.313555	−	0.543092i	0.665575	−	0.746331i	$-0.268187\pi$
$607$	−190.328	−	329.657i	−0.313555	−	0.543092i	−0.979129	+	0.203239i	$0.934853\pi$
$608$		−	73.5391i		−	0.120952i
$609$		0			0
$610$	570.724			0.935614
$611$	42.5452	−	24.5635i	0.0696321	−	0.0402021i
$612$		0			0
$613$	455.283	−	788.573i	0.742713	−	1.28642i	−0.208543	−	0.978013i	$-0.566872\pi$
$613$	455.283	−	788.573i	0.742713	−	1.28642i	0.951256	−	0.308403i	$-0.0997945\pi$
$614$	−142.890	+	82.4974i	−0.232719	+	0.134361i
$615$		0			0
$616$	26.1655	+	15.3136i	0.0424765	+	0.0248598i
$617$		−	16.8438i		−	0.0272996i	−0.999907	−	0.0136498i	$-0.995655\pi$
$617$		−	16.8438i		−	0.0272996i	0.999907	−	0.0136498i	$-0.00434500\pi$
$618$		0			0
$619$	391.707	−	678.456i	0.632806	−	1.09605i	−0.354169	−	0.935181i	$-0.615236\pi$
$619$	391.707	−	678.456i	0.632806	−	1.09605i	0.986975	−	0.160871i	$-0.0514303\pi$
$620$	−150.340	−	86.7986i	−0.242483	−	0.139998i
$621$		0			0
$622$	687.159			1.10476
$623$	187.969	+	330.063i	0.301716	+	0.529796i
$624$		0			0
$625$	313.531	+	543.052i	0.501650	+	0.868883i
$626$	188.197	+	108.656i	0.300634	+	0.173571i
$627$		0			0
$628$	199.235	+	345.084i	0.317253	+	0.549497i
$629$		−	922.916i		−	1.46727i
$630$		0			0
$631$	−679.228			−1.07643			−0.538215	−	0.842807i	$-0.680901\pi$
$631$	−679.228			−1.07643			−0.538215	+	0.842807i	$0.680901\pi$
$632$	299.649	−	173.002i	0.474128	−	0.273738i
$633$		0			0
$634$	−68.4829	+	118.616i	−0.108017	+	0.187091i
$635$	−37.2111	+	21.4838i	−0.0586001	+	0.0338328i
$636$		0			0
$637$	−44.9415	−	0.531381i	−0.0705518	−	0.000834193i
$638$			72.6028i			0.113797i
$639$		0			0
$640$	−28.3311	+	49.0708i	−0.0442673	+	0.0766732i
$641$	589.124	+	340.131i	0.919070	+	0.530625i	0.883338	−	0.468736i	$-0.155291\pi$
$641$	589.124	+	340.131i	0.919070	+	0.530625i	0.0357315	+	0.999361i	$0.488624\pi$
$642$		0			0
$643$	843.966			1.31254			0.656272	−	0.754524i	$-0.272132\pi$
$643$	843.966			1.31254			0.656272	+	0.754524i	$0.272132\pi$
$644$	−167.469	−	0.990030i	−0.260045	−	0.00153731i
$645$		0			0
$646$	−152.190	−	263.600i	−0.235588	−	0.408050i
$647$	−1111.67	−	641.824i	−1.71819	−	0.992000i	−0.922216	−	0.386676i	$-0.873623\pi$
$647$	−1111.67	−	641.824i	−1.71819	−	0.992000i	−0.795979	−	0.605324i	$-0.793044\pi$
$648$		0			0
$649$	−36.6347	−	63.4532i	−0.0564479	−	0.0977707i
$650$		−	0.107357i		−	0.000165165i
$651$		0			0
$652$	194.000			0.297546
$653$	−276.898	+	159.867i	−0.424040	+	0.244820i	−0.696804	−	0.717261i	$-0.745395\pi$
$653$	−276.898	+	159.867i	−0.424040	+	0.244820i	0.272764	+	0.962081i	$0.412062\pi$
$654$		0			0
$655$	176.300	−	305.360i	0.269160	−	0.466199i
$656$	−22.6536	+	13.0791i	−0.0345329	+	0.0199376i
$657$		0			0
$658$	−457.604	−	267.817i	−0.695446	−	0.407017i
$659$		−	915.684i		−	1.38951i	−0.719249	−	0.694753i	$-0.755514\pi$
$659$		−	915.684i		−	1.38951i	0.719249	−	0.694753i	$-0.244486\pi$
$660$		0			0
$661$	−251.183	+	435.061i	−0.380004	+	0.658186i	−0.991062	−	0.133399i	$-0.957411\pi$
$661$	−251.183	+	435.061i	−0.380004	+	0.658186i	0.611058	+	0.791586i	$0.290744\pi$
$662$	89.7356	+	51.8089i	0.135552	+	0.0782612i
$663$		0			0
$664$	449.324			0.676693
$665$	230.206	−	393.339i	0.346174	−	0.591488i
$666$		0			0
$667$	−200.528	−	347.324i	−0.300641	−	0.520726i
$668$	−37.5700	−	21.6911i	−0.0562426	−	0.0324717i
$669$		0			0
$670$	−8.57249	−	14.8480i	−0.0127948	−	0.0221612i
$671$		−	123.388i		−	0.183886i
$672$		0			0
$673$	299.724			0.445356			0.222678	−	0.974892i	$-0.428520\pi$
$673$	299.724			0.445356			0.222678	+	0.974892i	$0.428520\pi$
$674$	−202.455	+	116.887i	−0.300378	+	0.173423i
$675$		0			0
$676$	−168.159	+	291.259i	−0.248755	+	0.430857i
$677$	−286.869	+	165.624i	−0.423736	+	0.244644i	−0.696674	−	0.717387i	$-0.745338\pi$
$677$	−286.869	+	165.624i	−0.423736	+	0.244644i	0.272939	+	0.962031i	$0.412004\pi$
$678$		0			0
$679$	0.500000	−	84.5779i	0.000736377	−	0.124562i
$680$			234.525i			0.344890i
$681$		0			0
$682$	−18.7654	+	32.5026i	−0.0275153	+	0.0476578i
$683$	−858.175	−	495.468i	−1.25648	−	0.725428i	−0.284090	−	0.958797i	$-0.591692\pi$
$683$	−858.175	−	495.468i	−1.25648	−	0.725428i	−0.972388	+	0.233369i	$0.925025\pi$
$684$		0			0
$685$	337.897			0.493280
$686$	235.050	+	424.323i	0.342638	+	0.618546i
$687$		0			0
$688$	−65.4897	−	113.432i	−0.0951886	−	0.164871i
$689$	−66.4151	−	38.3448i	−0.0963935	−	0.0556528i
$690$		0			0
$691$	−326.842	−	566.106i	−0.472998	−	0.819257i	0.526524	−	0.850160i	$-0.323495\pi$
$691$	−326.842	−	566.106i	−0.472998	−	0.819257i	−0.999522	+	0.0309035i	$0.990162\pi$
$692$		−	417.310i		−	0.603049i
$693$		0			0
$694$	−296.483			−0.427209
$695$	722.473	−	417.120i	1.03953	−	0.600173i
$696$		0			0
$697$	−54.1344	+	93.7636i	−0.0776677	+	0.134524i
$698$	−542.140	+	313.005i	−0.776705	+	0.448431i
$699$		0			0
$700$	−1.00685	+	0.573396i	−0.00143836	+	0.000819137i
$701$		−	281.648i		−	0.401781i	−0.979614	−	0.200890i	$-0.935616\pi$
$701$		−	281.648i		−	0.401781i	0.979614	−	0.200890i	$-0.0643835\pi$
$702$		0			0
$703$	−362.342	+	627.594i	−0.515422	+	0.892737i
$704$	10.6089	+	6.12503i	0.0150694	+	0.00870033i
$705$		0			0
$706$	−236.027			−0.334316
$707$	642.906	−	1098.50i	0.909344	−	1.55375i
$708$		0			0
$709$	−250.372	−	433.658i	−0.353135	−	0.611647i	0.633662	−	0.773610i	$-0.281551\pi$
$709$	−250.372	−	433.658i	−0.353135	−	0.611647i	−0.986797	+	0.161963i	$0.948218\pi$
$710$	−511.082	−	295.073i	−0.719834	−	0.415596i
$711$		0			0
$712$	76.7380	+	132.914i	0.107778	+	0.186677i
$713$		−	207.319i		−	0.290770i
$714$		0			0
$715$	−7.03425			−0.00983811
$716$	109.898	−	63.4495i	0.153489	−	0.0886166i
$717$		0			0
$718$	−13.3516	+	23.1256i	−0.0185955	+	0.0322084i
$719$	−371.073	+	214.239i	−0.516095	+	0.297968i	−0.735336	−	0.677703i	$-0.762975\pi$
$719$	−371.073	+	214.239i	−0.516095	+	0.297968i	0.219240	+	0.975671i	$0.429642\pi$
$720$		0			0
$721$	704.049	−	400.952i	0.976489	−	0.556105i
$722$		−	271.529i		−	0.376079i
$723$		0			0
$724$	−212.311	+	367.733i	−0.293247	+	0.507918i
$725$	−2.40300	−	1.38737i	−0.00331449	−	0.00191362i
$726$		0			0
$727$	−1134.87			−1.56103			−0.780515	−	0.625137i	$-0.785043\pi$
$727$	−1134.87			−1.56103			−0.780515	+	0.625137i	$0.785043\pi$
$728$	−18.1601	−	0.107357i	−0.0249451	−	0.000147468i
$729$		0			0
$730$	255.273	+	442.145i	0.349688	+	0.605678i
$731$	−469.495	−	271.063i	−0.642264	−	0.370811i
$732$		0			0
$733$	322.786	+	559.082i	0.440363	+	0.762731i	0.997716	−	0.0675441i	$-0.0215163\pi$
$733$	322.786	+	559.082i	0.440363	+	0.762731i	−0.557353	+	0.830276i	$0.688183\pi$
$734$		−	554.684i		−	0.755700i
$735$		0			0
$736$	−67.6689			−0.0919415
$737$	−3.21006	+	1.85333i	−0.00435558	+	0.00251469i
$738$		0			0
$739$	396.638	−	686.997i	0.536723	−	0.929631i	−0.462355	−	0.886695i	$-0.652995\pi$
$739$	396.638	−	686.997i	0.536723	−	0.929631i	0.999078	−	0.0429362i	$-0.0136712\pi$
$740$	483.563	−	279.185i	0.653464	−	0.377277i
$741$		0			0
$742$	−4.89298	+	827.676i	−0.00659431	+	1.11547i
$743$			1254.90i			1.68896i	0.535584	+	0.844482i	$0.320092\pi$
$743$			1254.90i			1.68896i	−0.535584	+	0.844482i	$0.679908\pi$
$744$		0			0
$745$	−568.831	+	985.245i	−0.763532	+	1.32248i
$746$	680.494	+	392.883i	0.912190	+	0.526653i
$747$		0			0
$748$	50.7032			0.0677850
$749$	592.743	+	1040.82i	0.791379	+	1.38962i
$750$		0			0
$751$	427.766	+	740.912i	0.569595	+	0.986567i	0.996606	+	0.0823207i	$0.0262332\pi$
$751$	427.766	+	740.912i	0.569595	+	0.986567i	−0.427011	+	0.904246i	$0.640433\pi$
$752$	−185.536	−	107.119i	−0.246724	−	0.142446i
$753$		0			0
$754$	−21.7449	−	37.6632i	−0.0288393	−	0.0499512i
$755$		−	1380.52i		−	1.82851i
$756$		0			0
$757$	254.428			0.336100			0.168050	−	0.985778i	$-0.446253\pi$
$757$	254.428			0.336100			0.168050	+	0.985778i	$0.446253\pi$
$758$	530.991	−	306.568i	0.700515	−	0.404443i
$759$		0			0
$760$	92.0759	−	159.480i	0.121153	−	0.209842i
$761$	831.324	−	479.965i	1.09241	−	0.630703i	0.158192	−	0.987408i	$-0.449433\pi$
$761$	831.324	−	479.965i	1.09241	−	0.630703i	0.934217	+	0.356705i	$0.116100\pi$
$762$		0			0
$763$	−796.962	−	466.430i	−1.04451	−	0.611310i
$764$		−	117.711i		−	0.154073i
$765$		0			0
$766$	−96.9035	+	167.842i	−0.126506	+	0.219115i
$767$	38.0090	+	21.9445i	0.0495555	+	0.0286109i
$768$		0			0
$769$	563.090			0.732236			0.366118	−	0.930568i	$-0.380687\pi$
$769$	563.090			0.732236			0.366118	+	0.930568i	$0.380687\pi$
$770$	37.5700	+	65.9708i	0.0487922	+	0.0856764i
$771$		0			0
$772$	139.235	+	241.161i	0.180356	+	0.312385i
$773$	−164.508	−	94.9789i	−0.212818	−	0.122871i	0.389802	−	0.920899i	$-0.372543\pi$
$773$	−164.508	−	94.9789i	−0.212818	−	0.122871i	−0.602620	+	0.798028i	$0.705877\pi$
$774$		0			0
$775$	−0.717181	−	1.24219i	−0.000925395	−	0.00160283i
$776$		−	34.1752i		−	0.0440402i
$777$		0			0
$778$	−492.049			−0.632453
$779$	73.6242	−	42.5069i	0.0945111	−	0.0545660i
$780$		0			0
$781$	−63.7934	+	110.493i	−0.0816817	+	0.141477i
$782$	−242.559	+	140.041i	−0.310178	+	0.179081i
$783$		0			0
$784$	95.9863	+	170.888i	0.122432	+	0.217969i
$785$			997.820i			1.27111i
$786$		0			0
$787$	484.155	−	838.581i	0.615191	−	1.06554i	−0.375160	−	0.926960i	$-0.622412\pi$
$787$	484.155	−	838.581i	0.615191	−	1.06554i	0.990351	−	0.138582i	$-0.0442544\pi$
$788$	−550.308	−	317.720i	−0.698360	−	0.403198i
$789$		0			0
$790$	866.442			1.09676
$791$	53.5931	+	0.316827i	0.0677536	+	0.000400540i
$792$		0			0
$793$	36.9552	+	64.0083i	0.0466018	+	0.0807166i
$794$	−566.322	−	326.966i	−0.713252	−	0.411796i
$795$		0			0
$796$	249.738	+	432.559i	0.313741	+	0.543416i
$797$			155.544i			0.195162i	0.995228	+	0.0975810i	$0.0311105\pi$
$797$			155.544i			0.195162i	−0.995228	+	0.0975810i	$0.968889\pi$
$798$		0			0
$799$	−886.738			−1.10981
$800$	−0.405452	+	0.234088i	−0.000506815	+	0.000292610i
$801$		0			0
$802$	−312.993	+	542.120i	−0.390266	+	0.675960i
$803$	95.5895	−	55.1886i	0.119040	−	0.0687281i
$804$		0			0
$805$	−361.941	−	211.830i	−0.449617	−	0.263143i
$806$		−	22.4813i		−	0.0278924i
$807$		0			0
$808$	257.145	−	445.388i	0.318249	−	0.551223i
$809$	−996.749	−	575.473i	−1.23208	−	0.711339i	−0.264613	−	0.964355i	$-0.585244\pi$
$809$	−996.749	−	575.473i	−1.23208	−	0.711339i	−0.967462	+	0.253016i	$0.918578\pi$
$810$		0			0
$811$	−1084.32			−1.33702			−0.668511	−	0.743702i	$-0.733068\pi$
$811$	−1084.32			−1.33702			−0.668511	+	0.743702i	$0.733068\pi$
$812$	−237.085	+	405.094i	−0.291977	+	0.498885i
$813$		0			0
$814$	−60.3584	−	104.544i	−0.0741504	−	0.128432i
$815$	420.717	+	242.901i	0.516217	+	0.298038i
$816$		0			0
$817$	212.842	+	368.652i	0.260516	+	0.451227i
$818$			79.5180i			0.0972103i
$819$		0			0
$820$	−65.5034			−0.0798822
$821$	1146.70	−	662.047i	1.39671	−	0.806391i	0.402664	−	0.915348i	$-0.368084\pi$
$821$	1146.70	−	662.047i	1.39671	−	0.806391i	0.994047	+	0.108956i	$0.0347509\pi$
$822$		0			0
$823$	−342.000	+	592.362i	−0.415553	+	0.719759i	−0.995486	−	0.0949048i	$-0.969745\pi$
$823$	−342.000	+	592.362i	−0.415553	+	0.719759i	0.579933	+	0.814664i	$0.303079\pi$
$824$	283.516	−	163.688i	0.344073	−	0.198650i
$825$		0			0
$826$	2.80023	−	473.674i	0.00339011	−	0.573456i
$827$			1121.73i			1.35638i	0.734887	+	0.678190i	$0.237235\pi$
$827$			1121.73i			1.35638i	−0.734887	+	0.678190i	$0.762765\pi$
$828$		0			0
$829$	577.273	−	999.866i	0.696348	−	1.20611i	−0.273376	−	0.961907i	$-0.588140\pi$
$829$	577.273	−	999.866i	0.696348	−	1.20611i	0.969724	−	0.244203i	$-0.0785265\pi$
$830$	974.424	+	562.584i	1.17401	+	0.677812i
$831$		0			0
$832$	−7.33790			−0.00881959
$833$	697.716	+	413.902i	0.837594	+	0.496881i
$834$		0			0
$835$	−54.3174	−	94.0804i	−0.0650507	−	0.112671i
$836$	−34.4788	−	19.9063i	−0.0412426	−	0.0238114i
$837$		0			0
$838$	233.845	+	405.031i	0.279051	+	0.483331i
$839$			761.637i			0.907792i	0.891055	+	0.453896i	$0.149966\pi$
$839$			761.637i			0.907792i	−0.891055	+	0.453896i	$0.850034\pi$
$840$		0			0
$841$	−283.035			−0.336546
$842$	−37.4687	+	21.6325i	−0.0444996	+	0.0256919i
$843$		0			0
$844$	−112.248	+	194.420i	−0.132996	+	0.230355i
$845$	−729.353	+	421.092i	−0.863139	+	0.498334i
$846$		0			0
$847$	−721.752	+	411.034i	−0.852127	+	0.485282i
$848$			334.437i			0.394384i
$849$		0			0
$850$	−0.968893	+	1.67817i	−0.00113987	+	0.00197432i
$851$	577.497	+	333.418i	0.678610	+	0.391796i
$852$		0			0
$853$	−186.235			−0.218329			−0.109164	−	0.994024i	$-0.534818\pi$
$853$	−186.235			−0.218329			−0.109164	+	0.994024i	$0.534818\pi$
$854$	402.924	−	688.454i	0.471808	−	0.806152i
$855$		0			0
$856$	241.986	+	419.133i	0.282694	+	0.489641i
$857$	−1393.45	−	804.507i	−1.62596	−	0.938748i	−0.985282	−	0.170938i	$-0.945320\pi$
$857$	−1393.45	−	804.507i	−1.62596	−	0.938748i	−0.640678	−	0.767810i	$-0.721346\pi$
$858$		0			0
$859$	−257.997	−	446.863i	−0.300345	−	0.520213i	0.675869	−	0.737022i	$-0.263769\pi$
$859$	−257.997	−	446.863i	−0.300345	−	0.520213i	−0.976214	+	0.216809i	$0.930435\pi$
$860$		−	327.990i		−	0.381384i
$861$		0			0
$862$	947.918			1.09967
$863$	−794.112	+	458.481i	−0.920177	+	0.531264i	−0.883691	−	0.468070i	$-0.844950\pi$
$863$	−794.112	+	458.481i	−0.920177	+	0.531264i	−0.0364853	+	0.999334i	$0.511616\pi$
$864$		0			0
$865$	522.500	−	904.997i	0.604047	−	1.04624i
$866$	−235.489	+	135.959i	−0.271927	+	0.156997i
$867$		0			0
$868$	−210.841	+	120.073i	−0.242905	+	0.138333i
$869$		−	187.320i		−	0.215558i
$870$		0			0
$871$	1.11016	−	1.92286i	0.00127458	−	0.00220764i
$872$	−323.130	−	186.559i	−0.370562	−	0.213944i
$873$		0			0
$874$	219.924			0.251629
$875$	873.530	+	5.16406i	0.998320	+	0.00590178i
$876$		0			0
$877$	−183.652	−	318.094i	−0.209409	−	0.362707i	0.742119	−	0.670268i	$-0.233821\pi$
$877$	−183.652	−	318.094i	−0.209409	−	0.362707i	−0.951529	+	0.307560i	$0.900487\pi$
$878$	197.235	+	113.874i	0.224641	+	0.129697i
$879$		0			0
$880$	15.3379	+	26.5660i	0.0174294	+	0.0301887i
$881$			987.585i			1.12098i	0.828161	+	0.560491i	$0.189387\pi$
$881$			987.585i			1.12098i	−0.828161	+	0.560491i	$0.810613\pi$
$882$		0			0
$883$	−1337.97			−1.51525			−0.757625	−	0.652690i	$-0.773640\pi$
$883$	−1337.97			−1.51525			−0.757625	+	0.652690i	$0.773640\pi$
$884$	−26.3027	+	15.1858i	−0.0297541	+	0.0171786i
$885$		0			0
$886$	481.962	−	834.783i	0.543975	−	0.942193i
$887$	−697.496	+	402.700i	−0.786354	+	0.454002i	−0.838678	−	0.544628i	$-0.816671\pi$
$887$	−697.496	+	402.700i	−0.786354	+	0.454002i	0.0523231	+	0.998630i	$0.483337\pi$
$888$		0			0
$889$	−0.355024	+	60.0543i	−0.000399352	+	0.0675527i
$890$			384.325i			0.431825i
$891$		0			0
$892$	79.2346	−	137.238i	0.0888280	−	0.153855i
$893$	602.993	+	348.138i	0.675244	+	0.389852i
$894$		0			0
$895$	317.772			0.355053
$896$	39.1918	+	68.8186i	0.0437409	+	0.0768065i
$897$		0			0
$898$	568.762	+	985.125i	0.633366	+	1.09702i
$899$	−503.206	−	290.526i	−0.559739	−	0.323166i
$900$		0			0
$901$	692.121	+	1198.79i	0.768169	+	1.33051i
$902$			14.1615i			0.0157001i
$903$		0			0
$904$	21.6553			0.0239549
$905$	−920.852	+	531.654i	−1.01752	+	0.587463i
$906$		0			0
$907$	422.924	−	732.526i	0.466289	−	0.807636i	−0.532970	−	0.846134i	$-0.678924\pi$
$907$	422.924	−	732.526i	0.466289	−	0.807636i	0.999259	+	0.0384981i	$0.0122574\pi$
$908$	−621.199	+	358.650i	−0.684140	+	0.394989i
$909$		0			0
$910$	−39.2483	−	22.9704i	−0.0431300	−	0.0252422i
$911$			1512.59i			1.66036i	0.557494	+	0.830181i	$0.311763\pi$
$911$			1512.59i			1.66036i	−0.557494	+	0.830181i	$0.688237\pi$
$912$		0			0
$913$	121.628	−	210.666i	0.133218	−	0.230740i
$914$	108.859	+	62.8498i	0.119102	+	0.0687634i
$915$		0			0
$916$	−244.676			−0.267113
$917$	−243.885	−	428.248i	−0.265960	−	0.467010i
$918$		0			0
$919$	437.287	+	757.402i	0.475829	+	0.824159i	0.999617	−	0.0276893i	$-0.00881491\pi$
$919$	437.287	+	757.402i	0.475829	+	0.824159i	−0.523788	+	0.851849i	$0.675482\pi$
$920$	−146.750	−	84.7261i	−0.159511	−	0.0920936i
$921$		0			0
$922$	609.655	+	1055.95i	0.661231	+	1.14529i
$923$		−	76.4257i		−	0.0828014i
$924$		0			0
$925$	4.61359			0.00498766
$926$	−172.613	+	99.6581i	−0.186407	+	0.107622i
$927$		0			0
$928$	−94.8276	+	164.246i	−0.102185	+	0.176990i
$929$	62.6266	−	36.1575i	0.0674129	−	0.0389209i	−0.465915	−	0.884830i	$-0.654275\pi$
$929$	62.6266	−	36.1575i	0.0674129	−	0.0389209i	0.533328	+	0.845909i	$0.320941\pi$
$930$		0			0
$931$	−311.955	−	555.385i	−0.335076	−	0.596547i
$932$			425.093i			0.456108i
$933$		0			0
$934$	−190.828	+	330.523i	−0.204312	+	0.353879i
$935$	109.957	+	63.4838i	0.117601	+	0.0678971i
$936$		0			0
$937$	−372.711			−0.397770			−0.198885	−	0.980023i	$-0.563732\pi$
$937$	−372.711			−0.397770			−0.198885	+	0.980023i	$0.563732\pi$
$938$	−23.9629	−	0.141662i	−0.0255468	−	0.000151025i
$939$		0			0
$940$	−268.241	−	464.608i	−0.285363	−	0.494264i
$941$	875.744	+	505.611i	0.930652	+	0.537312i	0.887018	−	0.461735i	$-0.152773\pi$
$941$	875.744	+	505.611i	0.930652	+	0.537312i	0.0436343	+	0.999048i	$0.486106\pi$
$942$		0			0
$943$	−39.1139	−	67.7472i	−0.0414781	−	0.0718422i
$944$		−	191.397i		−	0.202751i
$945$		0			0
$946$	−70.9098			−0.0749575
$947$	−199.216	+	115.017i	−0.210365	+	0.121454i	−0.601481	−	0.798887i	$-0.705422\pi$
$947$	−199.216	+	115.017i	−0.210365	+	0.121454i	0.391116	+	0.920341i	$0.372089\pi$
$948$		0			0
$949$	−33.0585	+	57.2590i	−0.0348351	+	0.0603362i
$950$	1.31772	−	0.760785i	0.00138707	−	0.000800827i
$951$		0			0
$952$	282.904	+	165.572i	0.297168	+	0.173920i
$953$			1273.02i			1.33580i	0.744249	+	0.667902i	$0.232808\pi$
$953$			1273.02i			1.33580i	−0.744249	+	0.667902i	$0.767192\pi$
$954$		0			0
$955$	147.383	−	255.274i	0.154327	−	0.267303i
$956$	−593.680	−	342.762i	−0.621005	−	0.358537i
$957$		0			0
$958$	−1064.89			−1.11158
$959$	238.551	−	407.598i	0.248750	−	0.425024i
$960$		0			0
$961$	330.317	+	572.126i	0.343723	+	0.595345i
$962$	62.6228	+	36.1553i	0.0650964	+	0.0375834i
$963$		0			0
$964$	32.7449	+	56.7158i	0.0339677	+	0.0588338i
$965$			697.324i			0.722616i
$966$		0			0
$967$	−1273.70			−1.31717			−0.658585	−	0.752506i	$-0.728845\pi$
$967$	−1273.70			−1.31717			−0.658585	+	0.752506i	$0.728845\pi$
$968$	−290.645	+	167.804i	−0.300253	+	0.173351i
$969$		0			0
$970$	42.7897	−	74.1139i	0.0441131	−	0.0764061i
$971$	863.509	−	498.547i	0.889299	−	0.513437i	0.0155857	−	0.999879i	$-0.495039\pi$
$971$	863.509	−	498.547i	0.889299	−	0.513437i	0.873713	+	0.486442i	$0.161705\pi$
$972$		0			0
$973$	6.89298	−	1165.99i	0.00708426	−	1.19834i
$974$			1063.40i			1.09179i
$975$		0			0
$976$	161.159	−	279.135i	0.165122	−	0.285999i
$977$	129.181	+	74.5828i	0.132222	+	0.0763386i	0.564652	−	0.825329i	$-0.309010\pi$
$977$	129.181	+	74.5828i	0.132222	+	0.0763386i	−0.432430	+	0.901668i	$0.642344\pi$
$978$		0			0
$979$	83.0890			0.0848713
$980$	−5.80286	+	490.776i	−0.00592128	+	0.500792i
$981$		0			0
$982$	117.021	+	202.686i	0.119166	+	0.206401i
$983$	−726.232	−	419.290i	−0.738791	−	0.426541i	0.0828384	−	0.996563i	$-0.473601\pi$
$983$	−726.232	−	419.290i	−0.738791	−	0.426541i	−0.821630	+	0.570022i	$0.806935\pi$
$984$		0			0
$985$	−795.614	−	1378.04i	−0.807730	−	1.39903i
$986$			784.986i			0.796132i
$987$		0			0
$988$	23.8482			0.0241378
$989$	339.225	−	195.852i	0.342998	−	0.198030i
$990$		0			0
$991$	−438.059	+	758.741i	−0.442037	+	0.765631i	−0.997841	−	0.0656827i	$-0.979077\pi$
$991$	−438.059	+	758.741i	−0.442037	+	0.765631i	0.555803	+	0.831314i	$0.312411\pi$
$992$	−84.9045	+	49.0196i	−0.0855892	+	0.0494149i
$993$		0			0
$994$	−716.759	+	408.191i	−0.721086	+	0.410655i
$995$			1250.76i			1.25704i
$996$		0			0
$997$	−347.903	+	602.586i	−0.348950	+	0.604399i	−0.986063	−	0.166371i	$-0.946795\pi$
$997$	−347.903	+	602.586i	−0.348950	+	0.604399i	0.637113	+	0.770770i	$0.280128\pi$
$998$	522.577	+	301.710i	0.523625	+	0.302315i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	378.3.s.d.53.4	yes	8
3.2	odd	2	inner	378.3.s.d.53.1	✓	8
7.2	even	3	inner	378.3.s.d.107.1	yes	8
21.2	odd	6	inner	378.3.s.d.107.4	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
378.3.s.d.53.1	✓	8	3.2	odd	2	inner
378.3.s.d.53.4	yes	8	1.1	even	1	trivial
378.3.s.d.107.1	yes	8	7.2	even	3	inner
378.3.s.d.107.4	yes	8	21.2	odd	6	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	378.s (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.22474 - 0.707107i) q^{2} +(1.00000 - 1.73205i) q^{4} +(4.33729 - 2.50413i) q^{5} +(0.0413813 - 6.99988i) q^{7} -2.82843i q^{8} +O(q^{10})\)	\(q+(1.22474 - 0.707107i) q^{2} +(1.00000 - 1.73205i) q^{4} +(4.33729 - 2.50413i) q^{5} +(0.0413813 - 6.99988i) q^{7} -2.82843i q^{8} +(3.54138 - 6.13385i) q^{10} +(-1.32611 - 0.765629i) q^{11} +0.917237 q^{13} +(-4.89898 - 8.60233i) q^{14} +(-2.00000 - 3.46410i) q^{16} +(-14.3380 - 8.27803i) q^{17} +(6.50000 + 11.2583i) q^{19} -10.0165i q^{20} -2.16553 q^{22} +(10.3596 - 5.98115i) q^{23} +(0.0413813 - 0.0716745i) q^{25} +(1.12338 - 0.648585i) q^{26} +(-12.0828 - 7.07155i) q^{28} -33.5266i q^{29} +(8.66553 - 15.0091i) q^{31} +(-4.89898 - 2.82843i) q^{32} -23.4138 q^{34} +(-17.3492 - 30.4641i) q^{35} +(27.8724 + 48.2765i) q^{37} +(15.9217 + 9.19239i) q^{38} +(-7.08276 - 12.2677i) q^{40} -6.53953i q^{41} +32.7449 q^{43} +(-2.65222 + 1.53126i) q^{44} +(8.45862 - 14.6508i) q^{46} +(46.3841 - 26.7799i) q^{47} +(-48.9966 - 0.579328i) q^{49} -0.117044i q^{50} +(0.917237 - 1.58870i) q^{52} +(-72.4078 - 41.8047i) q^{53} -7.66895 q^{55} +(-19.7986 - 0.117044i) q^{56} +(-23.7069 - 41.0616i) q^{58} +(41.4386 + 23.9246i) q^{59} +(40.2897 + 69.7838i) q^{61} -24.5098i q^{62} -8.00000 q^{64} +(3.97832 - 2.29689i) q^{65} +(1.21033 - 2.09636i) q^{67} +(-28.6759 + 16.5561i) q^{68} +(-42.7897 - 25.0431i) q^{70} -83.3216i q^{71} +(-36.0414 + 62.4255i) q^{73} +(68.2732 + 39.4176i) q^{74} +26.0000 q^{76} +(-5.41418 + 9.25091i) q^{77} +(61.1655 + 105.942i) q^{79} +(-17.3492 - 10.0165i) q^{80} +(-4.62414 - 8.00925i) q^{82} +158.860i q^{83} -82.9172 q^{85} +(40.1041 - 23.1541i) q^{86} +(-2.16553 + 3.75080i) q^{88} +(-46.9922 + 27.1310i) q^{89} +(0.0379564 - 6.42055i) q^{91} -23.9246i q^{92} +(37.8724 - 65.5970i) q^{94} +(56.3848 + 32.5538i) q^{95} +12.0828 q^{97} +(-60.4180 + 33.9363i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{4} - 24 q^{7}+O(q^{10}) \) 8 * q + 8 * q^4 - 24 * q^7	\( 8 q + 8 q^{4} - 24 q^{7} + 4 q^{10} + 56 q^{13} - 16 q^{16} + 52 q^{19} + 80 q^{22} - 24 q^{25} - 48 q^{28} - 28 q^{31} + 56 q^{34} + 4 q^{37} - 8 q^{40} - 176 q^{43} + 92 q^{46} - 100 q^{49} + 56 q^{52} - 256 q^{55} - 68 q^{58} + 152 q^{61} - 64 q^{64} + 180 q^{67} - 172 q^{70} - 264 q^{73} + 208 q^{76} + 392 q^{79} + 36 q^{82} - 712 q^{85} + 80 q^{88} - 316 q^{91} + 84 q^{94} + 48 q^{97}+O(q^{100}) \) 8 * q + 8 * q^4 - 24 * q^7 + 4 * q^10 + 56 * q^13 - 16 * q^16 + 52 * q^19 + 80 * q^22 - 24 * q^25 - 48 * q^28 - 28 * q^31 + 56 * q^34 + 4 * q^37 - 8 * q^40 - 176 * q^43 + 92 * q^46 - 100 * q^49 + 56 * q^52 - 256 * q^55 - 68 * q^58 + 152 * q^61 - 64 * q^64 + 180 * q^67 - 172 * q^70 - 264 * q^73 + 208 * q^76 + 392 * q^79 + 36 * q^82 - 712 * q^85 + 80 * q^88 - 316 * q^91 + 84 * q^94 + 48 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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