LMFDB - Embedded newform 378.3.s.d.53.1

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			53.1
Root			$4.76613 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	378.53
Dual form			378.3.s.d.107.1

$q$-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	−0.866503	−	0.499173i	$-0.833637\pi$
$5$	−4.33729	+	2.50413i	−0.867458	+	0.500827i	−0.000955133		1.00000i	$0.500304\pi$
$6$		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	0.559065	−	0.829124i	$-0.311160\pi$
$11$	1.32611	+	0.765629i	0.120555	+	0.0696026i	−0.438510	+	0.898726i	$0.644494\pi$
$12$		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	0.858422	−	0.512944i	$-0.171445\pi$
$17$	14.3380	+	8.27803i	0.843410	+	0.486943i	−0.0150117	+	0.999887i	$0.504779\pi$
$18$		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	−0.708007	−	0.706205i	$-0.750406\pi$
$23$	−10.3596	+	5.98115i	−0.450420	+	0.260050i	0.257588	+	0.966255i	$0.417072\pi$
$24$		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	$0.196184\pi$
$29$			33.5266i			1.15609i	−0.816005	+	0.578045i	$0.803816\pi$
$30$		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	$0.0254123\pi$
$41$			6.53953i			0.159501i	−0.996815	+	0.0797503i	$0.974588\pi$
$42$		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	−0.904345	−	0.426803i	$-0.859640\pi$
$47$	−46.3841	+	26.7799i	−0.986895	+	0.569784i	−0.0825503	+	0.996587i	$0.526307\pi$
$48$		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	0.990438	−	0.137955i	$-0.0440530\pi$
$53$	72.4078	+	41.8047i	1.36618	+	0.788767i	0.375746	+	0.926723i	$0.377386\pi$
$54$		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	0.105873	−	0.994380i	$-0.466236\pi$
$59$	−41.4386	−	23.9246i	−0.702349	−	0.405501i	−0.808222	+	0.588878i	$0.799570\pi$
$60$		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	$0.199601\pi$
$71$			83.3216i			1.17354i	−0.809753	+	0.586772i	$0.800399\pi$
$72$		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	$-0.593697\pi$
$83$		−	158.860i		−	1.91398i	0.290126	−	0.956989i	$-0.406303\pi$
$84$		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	−0.212200	−	0.977226i	$-0.568063\pi$
$89$	46.9922	−	27.1310i	0.528003	−	0.304843i	0.740203	+	0.672384i	$0.234729\pi$
$90$		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	$-0.976795\pi$
$101$	−157.468	−	90.9145i	−1.55909	−	0.900143i	−0.561750	−	0.827307i	$-0.689872\pi$
$102$		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	0.392177	−	0.919890i	$-0.371722\pi$
$107$	148.186	−	85.5551i	1.38491	−	0.799580i	0.992736	+	0.120310i	$0.0383887\pi$
$108$		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	$-0.989214\pi$
$113$		−	7.65629i		−	0.0677548i	0.999426	−	0.0338774i	$-0.0107856\pi$
$114$		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	−0.714324	−	0.699815i	$-0.753266\pi$
$131$	−60.9713	+	35.2018i	−0.465429	+	0.268716i	0.248895	+	0.968530i	$0.419933\pi$
$132$		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	0.271362	−	0.962477i	$-0.412526\pi$
$137$	−58.4288	−	33.7339i	−0.426488	−	0.246233i	−0.697849	+	0.716245i	$0.745859\pi$
$138$		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	0.336517	−	0.941677i	$-0.390751\pi$
$149$	196.724	−	113.578i	1.32029	−	0.762271i	0.983775	+	0.179406i	$0.0574176\pi$
$150$		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	$0.0206867\pi$
$167$			21.6911i			0.129887i	−0.997889	+	0.0649433i	$0.979313\pi$
$168$		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	−0.921108	−	0.389307i	$-0.872714\pi$
$173$	−180.701	+	104.328i	−1.04451	+	0.603049i	−0.123404	+	0.992357i	$0.539381\pi$
$174$		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	0.338596	−	0.940932i	$-0.390048\pi$
$179$	−54.9489	−	31.7248i	−0.306977	−	0.177233i	−0.645573	+	0.763699i	$0.723381\pi$
$180$		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	−0.627461	−	0.778648i	$-0.715906\pi$
$191$	−50.9706	+	29.4279i	−0.266862	+	0.154073i	0.360599	+	0.932721i	$0.382572\pi$
$192$		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	$0.298585\pi$
$197$			317.720i			1.61279i	−0.591375	+	0.806396i	$0.701415\pi$
$198$		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	0.990708	−	0.136003i	$-0.0434257\pi$
$227$	310.600	+	179.325i	1.36828	+	0.789977i	0.377572	+	0.925980i	$0.376759\pi$
$228$		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	−0.0499606	−	0.998751i	$-0.515910\pi$
$233$	184.071	−	106.273i	0.790003	−	0.456108i	0.839964	+	0.542643i	$0.182576\pi$
$234$		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	$0.254519\pi$
$239$			342.762i			1.43415i	−0.696997	+	0.717074i	$0.745481\pi$
$240$		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	$-0.994721\pi$
$251$		−	8.32425i		−	0.0331643i	0.999863	−	0.0165822i	$-0.00527851\pi$
$252$		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	−0.525997	−	0.850486i	$-0.676308\pi$
$257$	−13.4806	+	7.78302i	−0.0524536	+	0.0302841i	0.473544	+	0.880770i	$0.342975\pi$
$258$		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	0.489461	−	0.872025i	$-0.337193\pi$
$263$	−5.52394	−	3.18925i	−0.0210036	−	0.0121264i	−0.510465	+	0.859899i	$0.670527\pi$
$264$		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	−0.172971	−	0.984927i	$-0.555337\pi$
$269$	−299.243	−	172.768i	−1.11243	−	0.642261i	−0.939457	+	0.342666i	$0.888670\pi$
$270$		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	$0.277777\pi$
$281$			430.516i			1.53208i	−0.642790	+	0.766042i	$0.722223\pi$
$282$		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	$0.237462\pi$
$293$			397.726i			1.35743i	−0.734404	+	0.678713i	$0.762538\pi$
$294$		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	−0.364370	−	0.931254i	$-0.618716\pi$
$311$	−420.797	−	242.947i	−1.35305	−	0.781181i	−0.988675	+	0.150073i	$0.952049\pi$
$312$		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	−0.361838	−	0.932241i	$-0.617851\pi$
$317$	83.8741	−	48.4247i	0.264587	−	0.152759i	0.626425	+	0.779482i	$0.284517\pi$
$318$		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	0.738252	−	0.674525i	$-0.235652\pi$
$347$	181.558	+	104.823i	0.523222	+	0.302082i	−0.215030	+	0.976607i	$0.568985\pi$
$348$		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	0.690554	−	0.723280i	$-0.257367\pi$
$353$	144.537	+	83.4483i	0.409452	+	0.236397i	−0.281102	+	0.959678i	$0.590700\pi$
$354$		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	−0.477052	−	0.878875i	$-0.658295\pi$
$359$	16.3523	−	9.44101i	0.0455496	−	0.0262981i	0.522602	+	0.852577i	$0.324961\pi$
$360$		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	−0.336996	−	0.941506i	$-0.609411\pi$
$383$	118.682	−	68.5211i	0.309875	−	0.178906i	0.646871	+	0.762600i	$0.276077\pi$
$384$		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	0.834511	−	0.550991i	$-0.185750\pi$
$389$	301.317	+	173.965i	0.774594	+	0.447212i	−0.0599172	+	0.998203i	$0.519084\pi$
$390$		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	0.0610272	−	0.998136i	$-0.480562\pi$
$401$	383.337	−	221.320i	0.955952	−	0.551919i	0.894925	+	0.446217i	$0.147229\pi$
$402$		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	$-0.870870\pi$
$419$		−	330.707i		−	0.789276i	0.918837	−	0.394638i	$-0.129130\pi$
$420$		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	−0.359021	−	0.933330i	$-0.616889\pi$
$431$	−580.479	−	335.140i	−1.34682	−	0.777586i	−0.987798	+	0.155744i	$0.950223\pi$
$432$		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	−0.985676	−	0.168647i	$-0.946060\pi$
$443$	−590.281	+	340.799i	−1.33246	+	0.769297i	−0.346785	+	0.937945i	$0.612727\pi$
$444$		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	$-0.646665\pi$
$449$		−	804.351i		−	1.79143i	0.444630	−	0.895714i	$-0.353335\pi$
$450$		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	$-0.615289\pi$
$461$		−	862.183i		−	1.87024i	0.354325	−	0.935122i	$-0.384711\pi$
$462$		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	−0.228443	−	0.973557i	$-0.573363\pi$
$467$	233.715	−	134.936i	0.500461	−	0.288941i	0.728904	+	0.684616i	$0.240030\pi$
$468$		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	0.989810	−	0.142395i	$-0.0454804\pi$
$479$	652.109	+	376.495i	1.36140	+	0.786003i	0.371587	+	0.928398i	$0.378814\pi$
$480$		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	$-0.946099\pi$
$491$		−	165.492i		−	0.337051i	0.985697	−	0.168526i	$-0.0539006\pi$
$492$		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	$-0.902800\pi$
$503$		−	302.442i		−	0.601276i	0.953738	−	0.300638i	$-0.0971996\pi$
$504$		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	−0.374529	−	0.927215i	$-0.622196\pi$
$509$	122.770	−	70.8814i	0.241199	−	0.139256i	0.615728	+	0.787959i	$0.288862\pi$
$510$		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	0.0765187	−	0.997068i	$-0.475620\pi$
$521$	−390.077	−	225.211i	−0.748708	−	0.432267i	−0.825227	+	0.564801i	$0.808953\pi$
$522$		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	0.919698	−	0.392628i	$-0.128434\pi$
$557$	579.013	+	334.293i	1.03952	+	0.600168i	0.119823	+	0.992795i	$0.461767\pi$
$558$		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	0.509709	−	0.860347i	$-0.329753\pi$
$563$	10.9678	+	6.33228i	0.0194810	+	0.0112474i	−0.490228	+	0.871594i	$0.663086\pi$
$564$		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	−0.875542	−	0.483142i	$-0.839496\pi$
$569$	−509.198	+	293.986i	−0.894900	+	0.516671i	−0.0193580	+	0.999813i	$0.506162\pi$
$570$		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	$-0.935160\pi$
$587$		−	237.495i		−	0.404592i	0.979324	−	0.202296i	$-0.0648403\pi$
$588$		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	−0.479739	−	0.877412i	$-0.659268\pi$
$593$	23.8699	−	13.7813i	0.0402529	−	0.0232400i	0.519991	+	0.854172i	$0.325935\pi$
$594$		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	0.989875	−	0.141944i	$-0.0453352\pi$
$599$	815.769	+	470.985i	1.36189	+	0.786285i	0.372010	+	0.928229i	$0.378669\pi$
$600$		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	$0.00434500\pi$
$617$			16.8438i			0.0272996i	−0.999907	+	0.0136498i	$0.995655\pi$
$618$		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	−0.0357315	−	0.999361i	$-0.511376\pi$
$641$	−589.124	−	340.131i	−0.919070	−	0.530625i	−0.883338	+	0.468736i	$0.844709\pi$
$642$		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	$0.126377\pi$
$647$	1111.67	+	641.824i	1.71819	+	0.992000i	0.795979	+	0.605324i	$0.206956\pi$
$648$		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	−0.272764	−	0.962081i	$-0.587938\pi$
$653$	276.898	−	159.867i	0.424040	−	0.244820i	0.696804	+	0.717261i	$0.254605\pi$
$654$		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	$0.244486\pi$
$659$			915.684i			1.38951i	−0.719249	+	0.694753i	$0.755514\pi$
$660$		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	−0.272939	−	0.962031i	$-0.587996\pi$
$677$	286.869	−	165.624i	0.423736	−	0.244644i	0.696674	+	0.717387i	$0.254662\pi$
$678$		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	0.972388	−	0.233369i	$-0.0749751\pi$
$683$	858.175	+	495.468i	1.25648	+	0.725428i	0.284090	+	0.958797i	$0.408308\pi$
$684$		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	$0.0643835\pi$
$701$			281.648i			0.401781i	−0.979614	+	0.200890i	$0.935616\pi$
$702$		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	−0.219240	−	0.975671i	$-0.570358\pi$
$719$	371.073	−	214.239i	0.516095	−	0.297968i	0.735336	+	0.677703i	$0.237025\pi$
$720$		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	$-0.679908\pi$
$743$		−	1254.90i		−	1.68896i	0.535584	−	0.844482i	$-0.320092\pi$
$744$		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	−0.934217	−	0.356705i	$-0.883900\pi$
$761$	−831.324	+	479.965i	−1.09241	+	0.630703i	−0.158192	+	0.987408i	$0.550567\pi$
$762$		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	0.602620	−	0.798028i	$-0.294123\pi$
$773$	164.508	+	94.9789i	0.212818	+	0.122871i	−0.389802	+	0.920899i	$0.627457\pi$
$774$		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
$782$

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

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