LMFDB - Embedded newform 378.4.k.d.269.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [378,4,Mod(215,378)]

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, 5]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("378.215");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 378 = 2 \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	378.k (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:	$22.3027219822$
Analytic rank:	$0$
Dimension:	$20$
Relative dimension:	$10$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 2 x^{19} + 2 x^{18} - 32 x^{17} - 669 x^{16} + 1752 x^{15} - 1654 x^{14} + 13878 x^{13} + \cdots + 2458624 $ x^20 - 2x^19 + 2x^18 - 32x^17 - 669x^16 + 1752x^15 - 1654x^14 + 13878x^13 + 494085x^12 - 780096x^11 + 556152x^10 - 2898760x^9 - 1242580x^8 + 9104368x^7 - 4109728x^6 + 12152000x^5 + 28375312x^4 - 23774016x^3 + 7683200x^2 - 6146560*x + 2458624
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 2^{16}\cdot 3^{18}\cdot 7^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			269.7
Root			$-1.38263 - 5.16003i$ of defining polynomial
Character	$\chi$	$=$	378.269
Dual form			378.4.k.d.215.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.73205 - 1.00000i) q^{2} +(2.00000 - 3.46410i) q^{4} +(-3.80525 - 6.59089i) q^{5} +(18.4140 + 1.98070i) q^{7} -8.00000i q^{8} +O(q^{10})$	\(q+(1.73205 - 1.00000i) q^{2} +(2.00000 - 3.46410i) q^{4} +(-3.80525 - 6.59089i) q^{5} +(18.4140 + 1.98070i) q^{7} -8.00000i q^{8} +(-13.1818 - 7.61050i) q^{10} +(35.9276 + 20.7428i) q^{11} +69.6608i q^{13} +(33.8747 - 14.9834i) q^{14} +(-8.00000 - 13.8564i) q^{16} +(54.3382 - 94.1166i) q^{17} +(82.5626 - 47.6675i) q^{19} -30.4420 q^{20} +82.9712 q^{22} +(-109.711 + 63.3419i) q^{23} +(33.5401 - 58.0932i) q^{25} +(69.6608 + 120.656i) q^{26} +(43.6894 - 59.8267i) q^{28} -208.824i q^{29} +(-243.112 - 140.361i) q^{31} +(-27.7128 - 16.0000i) q^{32} -217.353i q^{34} +(-57.0155 - 128.902i) q^{35} +(113.109 + 195.911i) q^{37} +(95.3350 - 165.125i) q^{38} +(-52.7271 + 30.4420i) q^{40} +446.019 q^{41} +228.824 q^{43} +(143.710 - 82.9712i) q^{44} +(-126.684 + 219.423i) q^{46} +(-139.590 - 241.777i) q^{47} +(335.154 + 72.9452i) q^{49} -134.160i q^{50} +(241.312 + 139.322i) q^{52} +(-118.028 - 68.1437i) q^{53} -315.726i q^{55} +(15.8456 - 147.312i) q^{56} +(-208.824 - 361.694i) q^{58} +(405.464 - 702.284i) q^{59} +(-489.580 + 282.659i) q^{61} -561.442 q^{62} -64.0000 q^{64} +(459.127 - 265.077i) q^{65} +(-475.109 + 822.914i) q^{67} +(-217.353 - 376.466i) q^{68} +(-227.656 - 166.249i) q^{70} +448.145i q^{71} +(267.005 + 154.155i) q^{73} +(391.822 + 226.218i) q^{74} -381.340i q^{76} +(620.487 + 453.120i) q^{77} +(139.734 + 242.026i) q^{79} +(-60.8840 + 105.454i) q^{80} +(772.528 - 446.019i) q^{82} +495.955 q^{83} -827.083 q^{85} +(396.335 - 228.824i) q^{86} +(165.942 - 287.421i) q^{88} +(-248.020 - 429.584i) q^{89} +(-137.977 + 1282.74i) q^{91} +506.735i q^{92} +(-483.554 - 279.180i) q^{94} +(-628.343 - 362.774i) q^{95} +389.224i q^{97} +(653.448 - 208.809i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 40 q^{4} + 4 q^{7}+O(q^{10}) $ 20 * q + 40 * q^4 + 4 * q^7	$ 20 q + 40 q^{4} + 4 q^{7} - 24 q^{10} - 160 q^{16} + 408 q^{19} + 240 q^{22} - 646 q^{25} + 56 q^{28} - 102 q^{31} + 194 q^{37} - 96 q^{40} - 2332 q^{43} - 624 q^{46} + 2840 q^{49} - 648 q^{52} + 96 q^{58} + 1878 q^{61} - 1280 q^{64} - 386 q^{67} + 3672 q^{70} + 1788 q^{73} + 814 q^{79} - 672 q^{82} - 4560 q^{85} + 480 q^{88} + 2724 q^{91} - 1536 q^{94}+O(q^{100}) $ 20 * q + 40 * q^4 + 4 * q^7 - 24 * q^10 - 160 * q^16 + 408 * q^19 + 240 * q^22 - 646 * q^25 + 56 * q^28 - 102 * q^31 + 194 * q^37 - 96 * q^40 - 2332 * q^43 - 624 * q^46 + 2840 * q^49 - 648 * q^52 + 96 * q^58 + 1878 * q^61 - 1280 * q^64 - 386 * q^67 + 3672 * q^70 + 1788 * q^73 + 814 * q^79 - 672 * q^82 - 4560 * q^85 + 480 * q^88 + 2724 * q^91 - 1536 * q^94

Display 100 coefficients 10 coefficients

Character values

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

$n$	$29$	$325$
$\chi(n)$	$-1$	$e\left(\frac{1}{6}\right)$

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 $

$2$	1.73205	−	1.00000i	0.612372	−	0.353553i
$2$	1.73205	−	1.00000i	0.612372	−	0.353553i
$3$		0			0
$3$		0			0
$4$	2.00000	−	3.46410i	0.250000	−	0.433013i
$5$	−3.80525	−	6.59089i	−0.340352	−	0.589507i	0.644146	−	0.764903i	$-0.277213\pi$
$5$	−3.80525	−	6.59089i	−0.340352	−	0.589507i	−0.984498	+	0.175395i	$0.943880\pi$
$6$		0			0
$7$	18.4140	+	1.98070i	0.994265	+	0.106947i
$7$	18.4140	+	1.98070i	0.994265	+	0.106947i
$8$		−	8.00000i		−	0.353553i
$9$		0			0
$10$	−13.1818	−	7.61050i	−0.416845	−	0.240665i
$11$	35.9276	+	20.7428i	0.984780	+	0.568563i	0.903710	−	0.428146i	$-0.140833\pi$
$11$	35.9276	+	20.7428i	0.984780	+	0.568563i	0.0810699	+	0.996708i	$0.474166\pi$
$12$		0			0
$13$			69.6608i			1.48619i	0.669188	+	0.743093i	$0.266642\pi$
$13$			69.6608i			1.48619i	−0.669188	+	0.743093i	$0.733358\pi$
$14$	33.8747	−	14.9834i	0.646672	−	0.286034i
$15$		0			0
$16$	−8.00000	−	13.8564i	−0.125000	−	0.216506i
$17$	54.3382	−	94.1166i	0.775233	−	1.34274i	−0.159430	−	0.987209i	$-0.550966\pi$
$17$	54.3382	−	94.1166i	0.775233	−	1.34274i	0.934663	−	0.355534i	$-0.115701\pi$
$18$		0			0
$19$	82.5626	−	47.6675i	0.996903	−	0.575562i	0.0895723	−	0.995980i	$-0.471450\pi$
$19$	82.5626	−	47.6675i	0.996903	−	0.575562i	0.907330	+	0.420418i	$0.138117\pi$
$20$	−30.4420			−0.340352
$21$		0			0
$22$	82.9712			0.804069
$23$	−109.711	+	63.3419i	−0.994627	+	0.574248i	−0.906654	−	0.421875i	$-0.861372\pi$
$23$	−109.711	+	63.3419i	−0.994627	+	0.574248i	−0.0879727	+	0.996123i	$0.528039\pi$
$24$		0			0
$25$	33.5401	−	58.0932i	0.268321	−	0.464745i
$26$	69.6608	+	120.656i	0.525446	+	0.910100i
$27$		0			0
$28$	43.6894	−	59.8267i	0.294876	−	0.403792i
$29$		−	208.824i		−	1.33716i	−0.743639	−	0.668581i	$-0.766902\pi$
$29$		−	208.824i		−	1.33716i	0.743639	−	0.668581i	$-0.233098\pi$
$30$		0			0
$31$	−243.112	−	140.361i	−1.40852	−	0.813209i	−0.413274	−	0.910607i	$-0.635615\pi$
$31$	−243.112	−	140.361i	−1.40852	−	0.813209i	−0.995246	+	0.0973972i	$0.968948\pi$
$32$	−27.7128	−	16.0000i	−0.153093	−	0.0883883i
$33$		0			0
$34$		−	217.353i		−	1.09635i
$35$	−57.0155	−	128.902i	−0.275354	−	0.622526i
$36$		0			0
$37$	113.109	+	195.911i	0.502569	+	0.870474i	0.999996	+	0.00296850i	$0.000944905\pi$
$37$	113.109	+	195.911i	0.502569	+	0.870474i	−0.497427	+	0.867506i	$0.665722\pi$
$38$	95.3350	−	165.125i	0.406984	−	0.704917i
$39$		0			0
$40$	−52.7271	+	30.4420i	−0.208422	+	0.120333i
$41$	446.019			1.69894			0.849469	−	0.527638i	$-0.176922\pi$
$41$	446.019			1.69894			0.849469	+	0.527638i	$0.176922\pi$
$42$		0			0
$43$	228.824			0.811519			0.405760	−	0.913980i	$-0.367007\pi$
$43$	228.824			0.811519			0.405760	+	0.913980i	$0.367007\pi$
$44$	143.710	−	82.9712i	0.492390	−	0.284281i
$45$		0			0
$46$	−126.684	+	219.423i	−0.406055	+	0.703307i
$47$	−139.590	−	241.777i	−0.433219	−	0.750358i	0.563929	−	0.825823i	$-0.309289\pi$
$47$	−139.590	−	241.777i	−0.433219	−	0.750358i	−0.997148	+	0.0754652i	$0.975956\pi$
$48$		0			0
$49$	335.154	+	72.9452i	0.977124	+	0.212668i
$50$		−	134.160i		−	0.379463i
$51$		0			0
$52$	241.312	+	139.322i	0.643538	+	0.371547i
$53$	−118.028	−	68.1437i	−0.305895	−	0.176609i	0.339193	−	0.940717i	$-0.389846\pi$
$53$	−118.028	−	68.1437i	−0.305895	−	0.176609i	−0.645088	+	0.764108i	$0.723179\pi$
$54$		0			0
$55$		−	315.726i		−	0.774046i
$56$	15.8456	−	147.312i	0.0378116	−	0.351526i
$57$		0			0
$58$	−208.824	−	361.694i	−0.472758	−	0.818842i
$59$	405.464	−	702.284i	0.894693	−	1.54965i	0.0605095	−	0.998168i	$-0.480727\pi$
$59$	405.464	−	702.284i	0.894693	−	1.54965i	0.834184	−	0.551487i	$-0.185939\pi$
$60$		0			0
$61$	−489.580	+	282.659i	−1.02761	+	0.593292i	−0.916300	−	0.400494i	$-0.868839\pi$
$61$	−489.580	+	282.659i	−1.02761	+	0.593292i	−0.111312	+	0.993785i	$0.535505\pi$
$62$	−561.442			−1.15005
$63$		0			0
$64$	−64.0000			−0.125000
$65$	459.127	−	265.077i	0.876117	−	0.505827i
$66$		0			0
$67$	−475.109	+	822.914i	−0.866326	+	1.50052i	−0.000601731		1.00000i	$0.500192\pi$
$67$	−475.109	+	822.914i	−0.866326	+	1.50052i	−0.865724	+	0.500521i	$0.833142\pi$
$68$	−217.353	−	376.466i	−0.387617	−	0.671371i
$69$		0			0
$70$	−227.656	−	166.249i	−0.388715	−	0.283865i
$71$			448.145i			0.749084i	0.927210	+	0.374542i	$0.122200\pi$
$71$			448.145i			0.749084i	−0.927210	+	0.374542i	$0.877800\pi$
$72$		0			0
$73$	267.005	+	154.155i	0.428090	+	0.247158i	0.698532	−	0.715578i	$-0.253837\pi$
$73$	267.005	+	154.155i	0.428090	+	0.247158i	−0.270443	+	0.962736i	$0.587170\pi$
$74$	391.822	+	226.218i	0.615518	+	0.355370i
$75$		0			0
$76$		−	381.340i		−	0.575562i
$77$	620.487	+	453.120i	0.918325	+	0.670622i
$78$		0			0
$79$	139.734	+	242.026i	0.199003	+	0.344684i	0.948205	−	0.317658i	$-0.102896\pi$
$79$	139.734	+	242.026i	0.199003	+	0.344684i	−0.749202	+	0.662341i	$0.769563\pi$
$80$	−60.8840	+	105.454i	−0.0850880	+	0.147377i
$81$		0			0
$82$	772.528	−	446.019i	1.04038	−	0.600665i
$83$	495.955			0.655882			0.327941	−	0.944698i	$-0.393645\pi$
$83$	495.955			0.655882			0.327941	+	0.944698i	$0.393645\pi$
$84$		0			0
$85$	−827.083			−1.05541
$86$	396.335	−	228.824i	0.496952	−	0.286915i
$87$		0			0
$88$	165.942	−	287.421i	0.201017	−	0.348172i
$89$	−248.020	−	429.584i	−0.295395	−	0.511638i	0.679682	−	0.733507i	$-0.262118\pi$
$89$	−248.020	−	429.584i	−0.295395	−	0.511638i	−0.975077	+	0.221868i	$0.928784\pi$
$90$		0			0
$91$	−137.977	+	1282.74i	−0.158944	+	1.47766i
$92$			506.735i			0.574248i
$93$		0			0
$94$	−483.554	−	279.180i	−0.530583	−	0.306332i
$95$	−628.343	−	362.774i	−0.678596	−	0.391788i
$96$		0			0
$97$			389.224i			0.407419i	0.979031	+	0.203710i	$0.0652998\pi$
$97$			389.224i			0.407419i	−0.979031	+	0.203710i	$0.934700\pi$
$98$	653.448	−	208.809i	0.673554	−	0.215234i
$99$		0			0
$100$	−134.160	−	232.373i	−0.134160	−	0.232373i
$101$	40.0868	−	69.4324i	0.0394929	−	0.0684038i	−0.845603	−	0.533812i	$-0.820759\pi$
$101$	40.0868	−	69.4324i	0.0394929	−	0.0684038i	0.885096	+	0.465408i	$0.154092\pi$
$102$		0			0
$103$	−951.673	+	549.449i	−0.910400	+	0.525619i	−0.880560	−	0.473935i	$-0.842833\pi$
$103$	−951.673	+	549.449i	−0.910400	+	0.525619i	−0.0298399	+	0.999555i	$0.509500\pi$
$104$	557.286			0.525446
$105$		0			0
$106$	−272.575			−0.249762
$107$	−1274.15	+	735.633i	−1.15119	+	0.664639i	−0.949177	−	0.314744i	$-0.898081\pi$
$107$	−1274.15	+	735.633i	−1.15119	+	0.664639i	−0.202012	+	0.979383i	$0.564748\pi$
$108$		0			0
$109$	260.408	−	451.039i	0.228831	−	0.396346i	−0.728631	−	0.684906i	$-0.759843\pi$
$109$	260.408	−	451.039i	0.228831	−	0.396346i	0.957462	+	0.288560i	$0.0931765\pi$
$110$	−315.726	−	546.854i	−0.273667	−	0.474005i
$111$		0			0
$112$	−119.867	−	270.998i	−0.101128	−	0.228633i
$113$		−	1073.12i		−	0.893372i	−0.894691	−	0.446686i	$-0.852604\pi$
$113$		−	1073.12i		−	0.893372i	0.894691	−	0.446686i	$-0.147396\pi$
$114$		0			0
$115$	834.959	+	482.064i	0.677047	+	0.390893i
$116$	−723.389	−	417.649i	−0.579008	−	0.334291i
$117$		0			0
$118$		−	1621.86i		−	1.26529i
$119$	1187.00	−	1625.44i	0.914390	−	1.25213i
$120$		0			0
$121$	195.028	+	337.798i	0.146527	+	0.253793i
$122$	−565.319	+	979.160i	−0.419521	+	0.726631i
$123$		0			0
$124$	−972.446	+	561.442i	−0.704260	+	0.406605i
$125$	−1461.83			−1.04600
$126$		0			0
$127$	−178.570			−0.124768			−0.0623838	−	0.998052i	$-0.519870\pi$
$127$	−178.570			−0.124768			−0.0623838	+	0.998052i	$0.519870\pi$
$128$	−110.851	+	64.0000i	−0.0765466	+	0.0441942i
$129$		0			0
$130$	530.154	−	918.253i	0.357673	−	0.619509i
$131$	845.844	+	1465.05i	0.564136	+	0.977111i	0.997130	+	0.0757146i	$0.0241238\pi$
$131$	845.844	+	1465.05i	0.564136	+	0.977111i	−0.432994	+	0.901397i	$0.642543\pi$
$132$		0			0
$133$	1614.73	−	714.220i	1.05274	−	0.465645i
$134$			1900.44i			1.22517i
$135$		0			0
$136$	−752.933	−	434.706i	−0.474731	−	0.274086i
$137$	1210.65	+	698.967i	0.754982	+	0.435889i	0.827491	−	0.561479i	$-0.189767\pi$
$137$	1210.65	+	698.967i	0.754982	+	0.435889i	−0.0725095	+	0.997368i	$0.523101\pi$
$138$		0			0
$139$		−	909.881i		−	0.555217i	−0.960694	−	0.277608i	$-0.910458\pi$
$139$		−	909.881i		−	0.555217i	0.960694	−	0.277608i	$-0.0895418\pi$
$140$	−560.561	−	60.2964i	−0.338400	−	0.0363998i
$141$		0			0
$142$	448.145	+	776.209i	0.264841	+	0.458718i
$143$	−1444.96	+	2502.74i	−0.844990	+	1.46357i
$144$		0			0
$145$	−1376.34	+	794.630i	−0.788267	+	0.455106i
$146$	616.621			0.349534
$147$		0			0
$148$	904.874			0.502569
$149$	−2823.70	+	1630.27i	−1.55253	+	0.896353i	−0.554594	+	0.832121i	$0.687126\pi$
$149$	−2823.70	+	1630.27i	−1.55253	+	0.896353i	−0.997935	+	0.0642322i	$0.979540\pi$
$150$		0			0
$151$	150.359	−	260.430i	0.0810335	−	0.140354i	−0.822661	−	0.568533i	$-0.807511\pi$
$151$	150.359	−	260.430i	0.0810335	−	0.140354i	0.903694	+	0.428179i	$0.140845\pi$
$152$	−381.340	−	660.501i	−0.203492	−	0.352458i
$153$		0			0
$154$	1527.84	+	164.341i	0.799458	+	0.0859932i
$155$			2136.43i			1.10711i
$156$		0			0
$157$	1040.01	+	600.448i	0.528672	+	0.305229i	0.740476	−	0.672083i	$-0.234600\pi$
$157$	1040.01	+	600.448i	0.528672	+	0.305229i	−0.211803	+	0.977312i	$0.567934\pi$
$158$	484.051	+	279.467i	0.243728	+	0.140716i
$159$		0			0
$160$			243.536i			0.120333i
$161$	−2145.69	+	949.076i	−1.05034	+	0.464582i
$162$		0			0
$163$	1802.92	+	3122.75i	0.866353	+	1.50057i	0.865697	+	0.500568i	$0.166875\pi$
$163$	1802.92	+	3122.75i	0.866353	+	1.50057i	0.000655724		1.00000i	$0.499791\pi$
$164$	892.038	−	1545.06i	0.424735	−	0.735662i
$165$		0			0
$166$	859.020	−	495.955i	0.401644	−	0.231889i
$167$	−365.366			−0.169299			−0.0846494	−	0.996411i	$-0.526977\pi$
$167$	−365.366			−0.169299			−0.0846494	+	0.996411i	$0.526977\pi$
$168$		0			0
$169$	−2655.62			−1.20875
$170$	−1432.55	+	827.083i	−0.646303	+	0.373143i
$171$		0			0
$172$	457.648	−	792.670i	0.202880	−	0.351398i
$173$	−109.334	−	189.372i	−0.0480491	−	0.0832234i	0.841001	−	0.541034i	$-0.181967\pi$
$173$	−109.334	−	189.372i	−0.0480491	−	0.0832234i	−0.889050	+	0.457811i	$0.848634\pi$
$174$		0			0
$175$	732.674	−	1003.30i	0.316485	−	0.433384i
$176$		−	663.770i		−	0.284281i
$177$		0			0
$178$	−859.168	−	496.041i	−0.361783	−	0.208876i
$179$	481.269	+	277.861i	0.200959	+	0.116024i	0.597103	−	0.802165i	$-0.296318\pi$
$179$	481.269	+	277.861i	0.200959	+	0.116024i	−0.396144	+	0.918189i	$0.629652\pi$
$180$		0			0
$181$		−	1268.55i		−	0.520942i	−0.965482	−	0.260471i	$-0.916122\pi$
$181$		−	1268.55i		−	0.520942i	0.965482	−	0.260471i	$-0.0838778\pi$
$182$	1043.75	+	2359.74i	0.425100	+	0.961075i
$183$		0			0
$184$	506.735	+	877.691i	0.203027	+	0.351654i
$185$	860.818	−	1490.98i	0.342101	−	0.592536i
$186$		0			0
$187$	3904.48	−	2254.26i	1.52687	−	0.881537i
$188$	−1116.72			−0.433219
$189$		0			0
$190$	−1451.10			−0.554071
$191$	−2009.26	+	1160.05i	−0.761178	+	0.439466i	−0.829718	−	0.558182i	$-0.811499\pi$
$191$	−2009.26	+	1160.05i	−0.761178	+	0.439466i	0.0685408	+	0.997648i	$0.478166\pi$
$192$		0			0
$193$	−1971.20	+	3414.22i	−0.735183	+	1.27337i	0.219461	+	0.975621i	$0.429570\pi$
$193$	−1971.20	+	3414.22i	−0.735183	+	1.27337i	−0.954643	+	0.297752i	$0.903763\pi$
$194$	389.224	+	674.155i	0.144044	+	0.249492i
$195$		0			0
$196$	922.997	−	1015.12i	0.336369	−	0.369940i
$197$			2423.88i			0.876620i	0.898824	+	0.438310i	$0.144423\pi$
$197$			2423.88i			0.876620i	−0.898824	+	0.438310i	$0.855577\pi$
$198$		0			0
$199$	2813.50	+	1624.37i	1.00223	+	0.578637i	0.908907	−	0.416999i	$-0.136918\pi$
$199$	2813.50	+	1624.37i	1.00223	+	0.578637i	0.0933217	+	0.995636i	$0.470251\pi$
$200$	−464.745	−	268.321i	−0.164312	−	0.0948658i
$201$		0			0
$202$		−	160.347i		−	0.0558515i
$203$	413.617	−	3845.30i	0.143006	−	1.32949i
$204$		0			0
$205$	−1697.22	−	2939.66i	−0.578237	−	1.00154i
$206$	−1098.90	+	1903.35i	−0.371669	+	0.643750i
$207$		0			0
$208$	965.248	−	557.286i	0.321769	−	0.185773i
$209$	3955.03			1.30897
$210$		0			0
$211$	1432.71			0.467449			0.233724	−	0.972303i	$-0.424909\pi$
$211$	1432.71			0.467449			0.233724	+	0.972303i	$0.424909\pi$
$212$	−472.113	+	272.575i	−0.152948	+	0.0883043i
$213$		0			0
$214$	−1471.27	+	2548.31i	−0.469971	+	0.814013i
$215$	−870.733	−	1508.15i	−0.276202	−	0.478396i
$216$		0			0
$217$	−4198.65	−	3066.13i	−1.31347	−	0.959183i
$218$		−	1041.63i		−	0.323615i
$219$		0			0
$220$	−1093.71	−	631.453i	−0.335172	−	0.193512i
$221$	6556.24	+	3785.24i	1.99557	+	1.15214i
$222$		0			0
$223$		−	2663.28i		−	0.799761i	−0.916567	−	0.399880i	$-0.869052\pi$
$223$		−	2663.28i		−	0.799761i	0.916567	−	0.399880i	$-0.130948\pi$
$224$	−478.614	−	349.515i	−0.142762	−	0.104254i
$225$		0			0
$226$	−1073.12	−	1858.71i	−0.315855	−	0.547076i
$227$	387.483	−	671.140i	0.113296	−	0.196234i	−0.803801	−	0.594898i	$-0.797193\pi$
$227$	387.483	−	671.140i	0.113296	−	0.196234i	0.917097	+	0.398664i	$0.130526\pi$
$228$		0			0
$229$	−233.017	+	134.532i	−0.0672409	+	0.0388216i	−0.533244	−	0.845962i	$-0.679027\pi$
$229$	−233.017	+	134.532i	−0.0672409	+	0.0388216i	0.466003	+	0.884783i	$0.345694\pi$
$230$	1928.26			0.552806
$231$		0			0
$232$	−1670.60			−0.472758
$233$	−5053.40	+	2917.58i	−1.42086	+	0.820332i	−0.996372	−	0.0851039i	$-0.972878\pi$
$233$	−5053.40	+	2917.58i	−1.42086	+	0.820332i	−0.424484	+	0.905435i	$0.639544\pi$
$234$		0			0
$235$	−1062.35	+	1840.05i	−0.294894	+	0.510772i
$236$	−1621.86	−	2809.14i	−0.447347	−	0.774827i
$237$		0			0
$238$	430.510	−	4002.35i	0.117251	−	1.09006i
$239$			4306.97i			1.16567i	0.812591	+	0.582835i	$0.198056\pi$
$239$			4306.97i			1.16567i	−0.812591	+	0.582835i	$0.801944\pi$
$240$		0			0
$241$	−458.177	−	264.529i	−0.122464	−	0.0707045i	0.437517	−	0.899210i	$-0.355858\pi$
$241$	−458.177	−	264.529i	−0.122464	−	0.0707045i	−0.559981	+	0.828506i	$0.689191\pi$
$242$	675.596	+	390.056i	0.179459	+	0.103610i
$243$		0			0
$244$			2261.27i			0.593292i
$245$	−794.571	−	2486.54i	−0.207197	−	0.648404i
$246$		0			0
$247$	3320.56	+	5751.37i	0.855392	+	1.48158i
$248$	−1122.88	+	1944.89i	−0.287513	+	0.497987i
$249$		0			0
$250$	−2531.96	+	1461.83i	−0.640541	+	0.369816i
$251$	−1949.18			−0.490164			−0.245082	−	0.969502i	$-0.578815\pi$
$251$	−1949.18			−0.490164			−0.245082	+	0.969502i	$0.578815\pi$
$252$		0			0
$253$	−5255.56			−1.30598
$254$	−309.292	+	178.570i	−0.0764043	+	0.0441120i
$255$		0			0
$256$	−128.000	+	221.703i	−0.0312500	+	0.0541266i
$257$	−3433.49	−	5946.98i	−0.833367	−	1.44343i	−0.895353	−	0.445357i	$-0.853077\pi$
$257$	−3433.49	−	5946.98i	−0.833367	−	1.44343i	0.0619856	−	0.998077i	$-0.480257\pi$
$258$		0			0
$259$	1694.76	+	3831.55i	0.406591	+	0.919230i
$260$		−	2120.61i		−	0.505827i
$261$		0			0
$262$	2930.09	+	1691.69i	0.690922	+	0.398904i
$263$	−298.788	−	172.505i	−0.0700533	−	0.0404453i	0.464564	−	0.885539i	$-0.346211\pi$
$263$	−298.788	−	172.505i	−0.0700533	−	0.0404453i	−0.534618	+	0.845094i	$0.679544\pi$
$264$		0			0
$265$			1037.22i			0.240436i
$266$	2082.57	−	2851.79i	0.480039	−	0.657348i
$267$		0			0
$268$	1900.44	+	3291.65i	0.433163	+	0.750260i
$269$	−1610.60	+	2789.64i	−0.365056	+	0.632295i	−0.988785	−	0.149345i	$-0.952283\pi$
$269$	−1610.60	+	2789.64i	−0.365056	+	0.632295i	0.623729	+	0.781640i	$0.285617\pi$
$270$		0			0
$271$	−1254.43	+	724.245i	−0.281185	+	0.162342i	−0.633960	−	0.773366i	$-0.718572\pi$
$271$	−1254.43	+	724.245i	−0.281185	+	0.162342i	0.352775	+	0.935708i	$0.385238\pi$
$272$	−1738.82			−0.387617
$273$		0			0
$274$	2795.87			0.616440
$275$	2410.03	−	1391.43i	0.528474	−	0.305115i
$276$		0			0
$277$	2907.24	−	5035.48i	0.630610	−	1.09225i	−0.356817	−	0.934174i	$-0.616138\pi$
$277$	2907.24	−	5035.48i	0.630610	−	1.09225i	0.987427	−	0.158074i	$-0.0505285\pi$
$278$	−909.881	−	1575.96i	−0.196299	−	0.339999i
$279$		0			0
$280$	−1031.22	+	456.124i	−0.220096	+	0.0973523i
$281$			601.050i			0.127600i	0.997963	+	0.0638001i	$0.0203220\pi$
$281$			601.050i			0.127600i	−0.997963	+	0.0638001i	$0.979678\pi$
$282$		0			0
$283$	−4004.55	−	2312.03i	−0.841152	−	0.485639i	0.0165040	−	0.999864i	$-0.494746\pi$
$283$	−4004.55	−	2312.03i	−0.841152	−	0.485639i	−0.857655	+	0.514225i	$0.828080\pi$
$284$	1552.42	+	896.289i	0.324363	+	0.187271i
$285$		0			0
$286$			5779.84i			1.19500i
$287$	8213.01	+	883.428i	1.68919	+	0.181697i
$288$		0			0
$289$	−3448.79	−	5973.48i	−0.701972	−	1.21585i
$290$	−1589.26	+	2752.68i	−0.321809	+	0.557389i
$291$		0			0
$292$	1068.02	−	616.621i	0.214045	−	0.123579i
$293$	−2487.86			−0.496049			−0.248024	−	0.968754i	$-0.579781\pi$
$293$	−2487.86			−0.496049			−0.248024	+	0.968754i	$0.579781\pi$
$294$		0			0
$295$	−6171.57			−1.21804
$296$	1567.29	−	904.874i	0.307759	−	0.177685i
$297$		0			0
$298$	−3260.53	+	5647.41i	−0.633817	+	1.09780i
$299$	−4412.45	−	7642.58i	−0.853440	−	1.47820i
$300$		0			0
$301$	4213.57	+	453.231i	0.806865	+	0.0867899i
$302$		−	601.437i		−	0.114599i
$303$		0			0
$304$	−1321.00	−	762.680i	−0.249226	−	0.143891i
$305$	3725.95	+	2151.18i	0.699500	+	0.403856i
$306$		0			0
$307$		−	3921.70i		−	0.729065i	−0.931191	−	0.364533i	$-0.881229\pi$
$307$		−	3921.70i		−	0.729065i	0.931191	−	0.364533i	$-0.118771\pi$
$308$	2810.63	−	1243.19i	0.519969	−	0.229991i
$309$		0			0
$310$	2136.43	+	3700.40i	0.391423	+	0.677964i
$311$	1503.52	−	2604.17i	0.274137	−	0.474820i	−0.695780	−	0.718255i	$-0.744941\pi$
$311$	1503.52	−	2604.17i	0.274137	−	0.474820i	0.969917	+	0.243435i	$0.0782744\pi$
$312$		0			0
$313$	−221.058	+	127.628i	−0.0399199	+	0.0230478i	−0.519827	−	0.854271i	$-0.674004\pi$
$313$	−221.058	+	127.628i	−0.0399199	+	0.0230478i	0.479907	+	0.877319i	$0.340670\pi$
$314$	2401.79			0.431659
$315$		0			0
$316$	1117.87			0.199003
$317$	6253.04	−	3610.19i	1.10790	−	0.639649i	0.169618	−	0.985510i	$-0.445747\pi$
$317$	6253.04	−	3610.19i	1.10790	−	0.639649i	0.938286	+	0.345861i	$0.112413\pi$
$318$		0			0
$319$	4331.60	−	7502.56i	0.760261	−	1.31681i
$320$	243.536	+	421.817i	0.0425440	+	0.0736884i
$321$		0			0
$322$	−2767.37	+	3789.54i	−0.478943	+	0.655847i
$323$		−	10360.7i		−	1.78478i
$324$		0			0
$325$	4046.82	+	2336.43i	0.690698	+	0.398775i
$326$	6245.50	+	3605.84i	1.06106	+	0.612604i
$327$		0			0
$328$		−	3568.15i		−	0.600665i
$329$	−2091.53	−	4728.58i	−0.350486	−	0.792386i
$330$		0			0
$331$	967.390	+	1675.57i	0.160642	+	0.278241i	0.935099	−	0.354386i	$-0.115310\pi$
$331$	967.390	+	1675.57i	0.160642	+	0.278241i	−0.774457	+	0.632627i	$0.781977\pi$
$332$	991.911	−	1718.04i	0.163970	−	0.284005i
$333$		0			0
$334$	−632.833	+	365.366i	−0.103674	+	0.0598561i
$335$	7231.64			1.17942
$336$		0			0
$337$	−7421.63			−1.19965			−0.599825	−	0.800131i	$-0.704763\pi$
$337$	−7421.63			−1.19965			−0.599825	+	0.800131i	$0.704763\pi$
$338$	−4599.67	+	2655.62i	−0.740205	+	0.427358i
$339$		0			0
$340$	−1654.17	+	2865.10i	−0.263852	+	0.457005i
$341$	−5822.94	−	10085.6i	−0.924721	−	1.60166i
$342$		0			0
$343$	6027.05	+	2007.05i	0.948776	+	0.315949i
$344$		−	1830.59i		−	0.286915i
$345$		0			0
$346$	−378.743	−	218.667i	−0.0588478	−	0.0339758i
$347$	2663.98	+	1538.05i	0.412132	+	0.237944i	0.691705	−	0.722180i	$-0.256860\pi$
$347$	2663.98	+	1538.05i	0.412132	+	0.237944i	−0.279574	+	0.960124i	$0.590193\pi$
$348$		0			0
$349$		−	5872.31i		−	0.900681i	−0.892857	−	0.450340i	$-0.851303\pi$
$349$		−	5872.31i		−	0.900681i	0.892857	−	0.450340i	$-0.148697\pi$
$350$	265.731	−	2470.44i	0.0405826	−	0.377287i
$351$		0			0
$352$	−663.770	−	1149.68i	−0.100509	−	0.174086i
$353$	−2032.65	+	3520.65i	−0.306479	+	0.530837i	−0.977589	−	0.210520i	$-0.932484\pi$
$353$	−2032.65	+	3520.65i	−0.306479	+	0.530837i	0.671111	+	0.741357i	$0.265818\pi$
$354$		0			0
$355$	2953.67	−	1705.30i	0.441590	−	0.254952i
$356$	−1984.16			−0.295395
$357$		0			0
$358$	1111.44			0.164083
$359$	−3283.42	+	1895.69i	−0.482709	+	0.278692i	−0.721545	−	0.692368i	$-0.756568\pi$
$359$	−3283.42	+	1895.69i	−0.482709	+	0.278692i	0.238836	+	0.971060i	$0.423234\pi$
$360$		0			0
$361$	1114.89	−	1931.04i	0.162543	−	0.281533i
$362$	−1268.55	−	2197.19i	−0.184181	−	0.319010i
$363$		0			0
$364$	4167.57	+	3043.44i	0.600111	+	0.438240i
$365$		−	2346.40i		−	0.336482i
$366$		0			0
$367$	2090.27	+	1206.82i	0.297306	+	0.171650i	0.641232	−	0.767347i	$-0.278424\pi$
$367$	2090.27	+	1206.82i	0.297306	+	0.171650i	−0.343926	+	0.938997i	$0.611757\pi$
$368$	1755.38	+	1013.47i	0.248657	+	0.143562i
$369$		0			0
$370$		−	3443.27i		−	0.483803i
$371$	−2038.41	−	1488.58i	−0.285253	−	0.208310i
$372$		0			0
$373$	−5755.03	−	9968.00i	−0.798885	−	1.38371i	−0.920343	−	0.391112i	$-0.872090\pi$
$373$	−5755.03	−	9968.00i	−0.798885	−	1.38371i	0.121458	−	0.992597i	$-0.461243\pi$
$374$	4508.51	−	7808.97i	0.623341	−	1.07966i
$375$		0			0
$376$	−1934.22	+	1116.72i	−0.265292	+	0.153166i
$377$	14546.9			1.98727
$378$		0			0
$379$	228.682			0.0309937			0.0154969	−	0.999880i	$-0.495067\pi$
$379$	228.682			0.0309937			0.0154969	+	0.999880i	$0.495067\pi$
$380$	−2513.37	+	1451.10i	−0.339298	+	0.195894i
$381$		0			0
$382$	−2320.09	+	4018.52i	−0.310749	+	0.538234i
$383$	2943.87	+	5098.94i	0.392754	+	0.680270i	0.992812	−	0.119687i	$-0.0381890\pi$
$383$	2943.87	+	5098.94i	0.392754	+	0.680270i	−0.600058	+	0.799957i	$0.704856\pi$
$384$		0			0
$385$	625.358	−	5813.80i	0.0827823	−	0.769607i
$386$			7884.81i			1.03971i
$387$		0			0
$388$	1348.31	+	778.447i	0.176418	+	0.101855i
$389$	1522.49	+	879.012i	0.198441	+	0.114570i	0.595928	−	0.803038i	$-0.296784\pi$
$389$	1522.49	+	879.012i	0.198441	+	0.114570i	−0.397487	+	0.917608i	$0.630118\pi$
$390$		0			0
$391$			13767.6i			1.78070i
$392$	583.562	−	2681.23i	0.0751896	−	0.345466i
$393$		0			0
$394$	2423.88	+	4198.28i	0.309932	+	0.536818i
$395$	1063.44	−	1841.94i	0.135462	−	0.234628i
$396$		0			0
$397$	−3314.81	+	1913.81i	−0.419057	+	0.241943i	−0.694674	−	0.719325i	$-0.744451\pi$
$397$	−3314.81	+	1913.81i	−0.419057	+	0.241943i	0.275617	+	0.961268i	$0.411118\pi$
$398$	6497.49			0.818316
$399$		0			0
$400$	−1073.28			−0.134160
$401$	−11018.7	+	6361.67i	−1.37219	+	0.792237i	−0.991204	−	0.132343i	$-0.957750\pi$
$401$	−11018.7	+	6361.67i	−1.37219	+	0.792237i	−0.380990	+	0.924579i	$0.624417\pi$
$402$		0			0
$403$	9777.62	−	16935.3i	1.20858	−	2.09332i
$404$	−160.347	−	277.730i	−0.0197465	−	0.0342019i
$405$		0			0
$406$	−3128.89	−	7073.87i	−0.382474	−	0.864706i
$407$			9384.81i			1.14297i
$408$		0			0
$409$	−1172.04	−	676.676i	−0.141696	−	0.0818080i	0.427476	−	0.904027i	$-0.359403\pi$
$409$	−1172.04	−	676.676i	−0.141696	−	0.0818080i	−0.569172	+	0.822219i	$0.692736\pi$
$410$	−5879.33	−	3394.43i	−0.708193	−	0.408876i
$411$		0			0
$412$			4395.59i			0.525619i
$413$	8857.24	−	12128.8i	1.05529	−	1.44508i
$414$		0			0
$415$	−1887.24	−	3268.79i	−0.223231	−	0.386647i
$416$	1114.57	−	1930.50i	0.131362	−	0.227525i
$417$		0			0
$418$	6850.32	−	3955.03i	0.801579	−	0.462792i
$419$	3643.82			0.424850			0.212425	−	0.977177i	$-0.431864\pi$
$419$	3643.82			0.424850			0.212425	+	0.977177i	$0.431864\pi$
$420$		0			0
$421$	15760.7			1.82454			0.912268	−	0.409594i	$-0.134330\pi$
$421$	15760.7			1.82454			0.912268	+	0.409594i	$0.134330\pi$
$422$	2481.52	−	1432.71i	0.286253	−	0.165268i
$423$		0			0
$424$	−545.149	+	944.227i	−0.0624406	+	0.108150i
$425$	−3645.02	−	6313.36i	−0.416022	−	0.720572i
$426$		0			0
$427$	−9575.01	+	4235.19i	−1.08517	+	0.479989i
$428$			5885.07i			0.664639i
$429$		0			0
$430$	−3016.31	−	1741.47i	−0.338277	−	0.195305i
$431$	4005.60	+	2312.63i	0.447663	+	0.258459i	0.706843	−	0.707371i	$-0.250119\pi$
$431$	4005.60	+	2312.63i	0.447663	+	0.258459i	−0.259179	+	0.965829i	$0.583452\pi$
$432$		0			0
$433$		−	9233.87i		−	1.02483i	−0.858738	−	0.512415i	$-0.828751\pi$
$433$		−	9233.87i		−	1.02483i	0.858738	−	0.512415i	$-0.171249\pi$
$434$	−10338.4	−	1112.05i	−1.14346	−	0.122995i
$435$		0			0
$436$	−1041.63	−	1804.16i	−0.114415	−	0.198173i
$437$	−6038.71	+	10459.3i	−0.661031	+	1.14494i
$438$		0			0
$439$	−2349.15	+	1356.28i	−0.255396	+	0.147453i	−0.622233	−	0.782832i	$-0.713774\pi$
$439$	−2349.15	+	1356.28i	−0.255396	+	0.147453i	0.366836	+	0.930285i	$0.380441\pi$
$440$	−2525.81			−0.273667
$441$		0			0
$442$	15141.0			1.62937
$443$	−15697.4	+	9062.88i	−1.68353	+	0.971987i	−0.724248	+	0.689539i	$0.757813\pi$
$443$	−15697.4	+	9062.88i	−1.68353	+	0.971987i	−0.959283	+	0.282448i	$0.908854\pi$
$444$		0			0
$445$	−1887.56	+	3269.35i	−0.201076	+	0.348274i
$446$	−2663.28	−	4612.94i	−0.282758	−	0.489751i
$447$		0			0
$448$	−1178.50	−	126.764i	−0.124283	−	0.0133684i
$449$			5238.46i			0.550598i	0.961359	+	0.275299i	$0.0887768\pi$
$449$			5238.46i			0.550598i	−0.961359	+	0.275299i	$0.911223\pi$
$450$		0			0
$451$	16024.4	+	9251.69i	1.67308	+	0.965953i
$452$	−3717.41	−	2146.25i	−0.386841	−	0.223343i
$453$		0			0
$454$		−	1549.93i		−	0.160224i
$455$	8979.41	−	3971.75i	0.925190	−	0.409227i
$456$		0			0
$457$	2006.30	+	3475.01i	0.205363	+	0.355699i	0.950248	−	0.311494i	$-0.100829\pi$
$457$	2006.30	+	3475.01i	0.205363	+	0.355699i	−0.744886	+	0.667192i	$0.767496\pi$
$458$	−269.064	+	466.033i	−0.0274510	+	0.0475465i
$459$		0			0
$460$	3339.84	−	1928.26i	0.338523	−	0.195447i
$461$	−9396.48			−0.949322			−0.474661	−	0.880169i	$-0.657429\pi$
$461$	−9396.48			−0.949322			−0.474661	+	0.880169i	$0.657429\pi$
$462$		0			0
$463$	10531.9			1.05715			0.528573	−	0.848888i	$-0.322727\pi$
$463$	10531.9			1.05715			0.528573	+	0.848888i	$0.322727\pi$
$464$	−2893.56	+	1670.60i	−0.289504	+	0.167145i
$465$		0			0
$466$	−5835.17	+	10106.8i	−0.580062	+	1.00470i
$467$	−1087.99	−	1884.46i	−0.107808	−	0.186729i	0.807074	−	0.590450i	$-0.201050\pi$
$467$	−1087.99	−	1884.46i	−0.107808	−	0.186729i	−0.914882	+	0.403721i	$0.867717\pi$
$468$		0			0
$469$	−10378.6	+	14212.1i	−1.02183	+	1.39926i
$470$			4249.41i			0.417044i
$471$		0			0
$472$	−5618.27	−	3243.71i	−0.547885	−	0.316322i
$473$	8221.09	+	4746.45i	0.799168	+	0.461400i
$474$		0			0
$475$		−	6395.10i		−	0.617741i
$476$	−3256.68	−	7362.78i	−0.313592	−	0.708976i
$477$		0			0
$478$	4306.97	+	7459.90i	0.412126	+	0.713824i
$479$	393.499	−	681.560i	0.0375353	−	0.0650131i	−0.846647	−	0.532154i	$-0.821383\pi$
$479$	393.499	−	681.560i	0.0375353	−	0.0650131i	0.884183	+	0.467141i	$0.154716\pi$
$480$		0			0
$481$	−13647.3	+	7879.27i	−1.29369	+	0.746911i
$482$	−1058.11			−0.0999913
$483$		0			0
$484$	1560.22			0.146527
$485$	2565.33	−	1481.09i	0.240177	−	0.138666i
$486$		0			0
$487$	−7016.77	+	12153.4i	−0.652895	+	1.13085i	0.329522	+	0.944148i	$0.393112\pi$
$487$	−7016.77	+	12153.4i	−0.652895	+	1.13085i	−0.982417	+	0.186700i	$0.940221\pi$
$488$	2261.27	+	3916.64i	0.209760	+	0.363316i
$489$		0			0
$490$	−3862.77	−	3512.24i	−0.356127	−	0.323810i
$491$		−	13511.7i		−	1.24191i	−0.783848	−	0.620953i	$-0.786746\pi$
$491$		−	13511.7i		−	1.24191i	0.783848	−	0.620953i	$-0.213254\pi$
$492$		0			0
$493$	−19653.8	−	11347.2i	−1.79547	−	1.03661i
$494$	11502.7	+	6641.11i	1.04764	+	0.604854i
$495$		0			0
$496$			4491.54i			0.406605i
$497$	−887.638	+	8252.15i	−0.0801127	+	0.744788i
$498$		0			0
$499$	−1573.47	−	2725.33i	−0.141159	−	0.244494i	0.786775	−	0.617241i	$-0.211749\pi$
$499$	−1573.47	−	2725.33i	−0.141159	−	0.244494i	−0.927933	+	0.372746i	$0.878416\pi$
$500$	−2923.65	+	5063.92i	−0.261500	+	0.452931i
$501$		0			0
$502$	−3376.08	+	1949.18i	−0.300163	+	0.173299i
$503$	−11458.6			−1.01574			−0.507868	−	0.861435i	$-0.669566\pi$
$503$	−11458.6			−1.01574			−0.507868	+	0.861435i	$0.669566\pi$
$504$		0			0
$505$	−610.162			−0.0537660
$506$	−9102.89	+	5255.56i	−0.799749	+	0.461735i
$507$		0			0
$508$	−357.139	+	618.583i	−0.0311919	+	0.0540260i
$509$	−4545.13	−	7872.40i	−0.395795	−	0.685537i	0.597407	−	0.801938i	$-0.296198\pi$
$509$	−4545.13	−	7872.40i	−0.395795	−	0.685537i	−0.993202	+	0.116401i	$0.962864\pi$
$510$		0			0
$511$	4611.30	+	3367.48i	0.399201	+	0.291523i
$512$			512.000i			0.0441942i
$513$		0			0
$514$	−11894.0	−	6866.99i	−1.02066	−	0.589280i
$515$	7242.71	+	4181.58i	0.619713	+	0.357791i
$516$		0			0
$517$		−	11582.0i		−	0.985250i
$518$	6766.95	+	4941.67i	0.573982	+	0.419160i
$519$		0			0
$520$	−2120.61	−	3673.01i	−0.178837	−	0.309754i
$521$	3297.78	−	5711.92i	0.277310	−	0.480314i	−0.693406	−	0.720547i	$-0.743891\pi$
$521$	3297.78	−	5711.92i	0.277310	−	0.480314i	0.970715	+	0.240233i	$0.0772239\pi$
$522$		0			0
$523$	15343.8	−	8858.74i	1.28286	−	0.740661i	0.305491	−	0.952195i	$-0.401179\pi$
$523$	15343.8	−	8858.74i	1.28286	−	0.740661i	0.977371	+	0.211534i	$0.0678460\pi$
$524$	6766.75			0.564136
$525$		0			0
$526$	−690.020			−0.0571983
$527$	−26420.5	+	15253.9i	−2.18386	+	1.26085i
$528$		0			0
$529$	1940.90	−	3361.74i	0.159522	−	0.276299i
$530$	1037.22	+	1796.51i	0.0850071	+	0.147237i
$531$		0			0
$532$	755.319	−	7022.01i	0.0615549	−	0.572261i
$533$			31070.0i			2.52494i
$534$		0			0
$535$	9696.96	+	5598.54i	0.783619	+	0.452423i
$536$	6583.31	+	3800.88i	0.530514	+	0.306293i
$537$		0			0
$538$			6442.40i			0.516267i
$539$	10528.2	+	9572.77i	0.841337	+	0.764988i
$540$		0			0
$541$	−5217.11	−	9036.30i	−0.414605	−	0.718117i	0.580782	−	0.814059i	$-0.302747\pi$
$541$	−5217.11	−	9036.30i	−0.414605	−	0.718117i	−0.995387	+	0.0959425i	$0.969413\pi$
$542$	−1448.49	+	2508.86i	−0.114793	+	0.198828i
$543$		0			0
$544$	−3011.73	+	1738.82i	−0.237366	+	0.137043i
$545$	−3963.67			−0.311532
$546$		0			0
$547$	5534.29			0.432595			0.216297	−	0.976328i	$-0.430602\pi$
$547$	5534.29			0.432595			0.216297	+	0.976328i	$0.430602\pi$
$548$	4842.58	−	2795.87i	0.377491	−	0.217944i
$549$		0			0
$550$	2782.86	−	4820.06i	0.215749	−	0.373687i
$551$	−9954.14	−	17241.1i	−0.769620	−	1.33302i
$552$		0			0
$553$	2093.68	+	4733.44i	0.160999	+	0.363990i
$554$		−	11628.9i		−	0.891817i
$555$		0			0
$556$	−3151.92	−	1819.76i	−0.240416	−	0.138804i
$557$	−2791.15	−	1611.47i	−0.212324	−	0.122586i	0.390067	−	0.920787i	$-0.372452\pi$
$557$	−2791.15	−	1611.47i	−0.212324	−	0.122586i	−0.602391	+	0.798201i	$0.705785\pi$
$558$		0			0
$559$			15940.1i			1.20607i
$560$	−1329.99	+	1821.25i	−0.100362	+	0.137432i
$561$		0			0
$562$	601.050	+	1041.05i	0.0451135	+	0.0781388i
$563$	−12025.9	+	20829.4i	−0.900229	+	1.55924i	−0.0730337	+	0.997329i	$0.523268\pi$
$563$	−12025.9	+	20829.4i	−0.900229	+	1.55924i	−0.827196	+	0.561914i	$0.810065\pi$
$564$		0			0
$565$	−7072.84	+	4083.51i	−0.526649	+	0.304061i
$566$	−9248.12			−0.686797
$567$		0			0
$568$	3585.16			0.264841
$569$	−908.051	+	524.264i	−0.0669025	+	0.0386262i	−0.533078	−	0.846066i	$-0.678965\pi$
$569$	−908.051	+	524.264i	−0.0669025	+	0.0386262i	0.466176	+	0.884692i	$0.345631\pi$
$570$		0			0
$571$	−860.310	+	1490.10i	−0.0630523	+	0.109210i	−0.895828	−	0.444400i	$-0.853417\pi$
$571$	−860.310	+	1490.10i	−0.0630523	+	0.109210i	0.832776	+	0.553610i	$0.186750\pi$
$572$	5779.84	+	10011.0i	0.422495	+	0.731783i
$573$		0			0
$574$	15108.8	−	6682.87i	1.09866	−	0.485954i
$575$			8497.98i			0.616331i
$576$		0			0
$577$	−10601.5	−	6120.76i	−0.764895	−	0.441612i	0.0661554	−	0.997809i	$-0.478927\pi$
$577$	−10601.5	−	6120.76i	−0.764895	−	0.441612i	−0.831051	+	0.556197i	$0.812260\pi$
$578$	−11947.0	−	6897.58i	−0.859737	−	0.496369i
$579$		0			0
$580$			6357.04i			0.455106i
$581$	9132.54	+	982.336i	0.652120	+	0.0701449i
$582$		0			0
$583$	−2826.98	−	4896.48i	−0.200826	−	0.347841i
$584$	1233.24	−	2136.04i	0.0873834	−	0.151353i
$585$		0			0
$586$	−4309.10	+	2487.86i	−0.303766	+	0.175380i
$587$	−10849.1			−0.762845			−0.381422	−	0.924401i	$-0.624566\pi$
$587$	−10849.1			−0.762845			−0.381422	+	0.924401i	$0.624566\pi$
$588$		0			0
$589$	−26762.6			−1.87221
$590$	−10689.5	+	6171.57i	−0.745896	+	0.430643i
$591$		0			0
$592$	1809.75	−	3134.57i	0.125642	−	0.217619i
$593$	6900.76	+	11952.5i	0.477876	+	0.827705i	0.999678	−	0.0253614i	$-0.00807365\pi$
$593$	6900.76	+	11952.5i	0.477876	+	0.827705i	−0.521803	+	0.853066i	$0.674740\pi$
$594$		0			0
$595$	−15229.9	−	1638.20i	−1.04936	−	0.112873i
$596$			13042.1i			0.896353i
$597$		0			0
$598$	−15285.2	−	8824.89i	−1.04525	−	0.603473i
$599$	−2377.72	−	1372.78i	−0.162189	−	0.0936397i	0.416709	−	0.909040i	$-0.363184\pi$
$599$	−2377.72	−	1372.78i	−0.162189	−	0.0936397i	−0.578897	+	0.815400i	$0.696517\pi$
$600$		0			0
$601$			27491.8i			1.86591i	0.359992	+	0.932955i	$0.382779\pi$
$601$			27491.8i			1.86591i	−0.359992	+	0.932955i	$0.617221\pi$
$602$	7751.35	−	3428.56i	0.524787	−	0.232122i
$603$		0			0
$604$	−601.437	−	1041.72i	−0.0405168	−	0.0701771i
$605$	1484.26	−	2570.81i	0.0997417	−	0.172758i
$606$		0			0
$607$	−15811.8	+	9128.95i	−1.05730	+	0.610433i	−0.924684	−	0.380735i	$-0.875671\pi$
$607$	−15811.8	+	9128.95i	−1.05730	+	0.610433i	−0.132616	+	0.991167i	$0.542338\pi$
$608$	−3050.72			−0.203492
$609$		0			0
$610$	8604.72			0.571139
$611$	16842.4	−	9723.96i	1.11517	−	0.643845i
$612$		0			0
$613$	163.820	−	283.744i	0.0107938	−	0.0186955i	−0.860578	−	0.509319i	$-0.829897\pi$
$613$	163.820	−	283.744i	0.0107938	−	0.0186955i	0.871372	+	0.490623i	$0.163231\pi$
$614$	−3921.70	−	6792.58i	−0.257764	−	0.446460i
$615$		0			0
$616$	3624.96	−	4963.90i	0.237101	−	0.324677i
$617$			8466.10i			0.552403i	0.961100	+	0.276201i	$0.0890757\pi$
$617$			8466.10i			0.552403i	−0.961100	+	0.276201i	$0.910924\pi$
$618$		0			0
$619$	22315.8	+	12884.1i	1.44903	+	0.836598i	0.998424	−	0.0561255i	$-0.0178747\pi$
$619$	22315.8	+	12884.1i	1.44903	+	0.836598i	0.450606	+	0.892723i	$0.351208\pi$
$620$	7400.81	+	4272.86i	0.479393	+	0.276778i
$621$		0			0
$622$		−	6014.07i		−	0.387689i
$623$	−3716.18	−	8401.63i	−0.238982	−	0.540296i
$624$		0			0
$625$	1370.11	+	2373.10i	0.0876869	+	0.151878i
$626$	−255.256	+	442.116i	−0.0162972	+	0.0282276i
$627$		0			0
$628$	4160.03	−	2401.79i	0.264336	−	0.152615i
$629$	24584.6			1.55843
$630$		0			0
$631$	4711.88			0.297269			0.148635	−	0.988892i	$-0.452512\pi$
$631$	4711.88			0.297269			0.148635	+	0.988892i	$0.452512\pi$
$632$	1936.20	−	1117.87i	0.121864	−	0.0703582i
$633$		0			0
$634$	7220.38	−	12506.1i	0.452300	−	0.783406i
$635$	679.503	+	1176.93i	0.0424649	+	0.0735514i
$636$		0			0
$637$	−5081.42	+	23347.1i	−0.316065	+	1.45219i
$638$		−	17326.4i		−	1.07517i
$639$		0			0
$640$	843.634	+	487.072i	0.0521056	+	0.0300832i
$641$	−26282.9	−	15174.5i	−1.61952	−	0.935031i	−0.987043	−	0.160453i	$-0.948704\pi$
$641$	−26282.9	−	15174.5i	−1.61952	−	0.935031i	−0.632478	−	0.774578i	$-0.717962\pi$
$642$		0			0
$643$			16400.8i			1.00588i	0.864320	+	0.502941i	$0.167749\pi$
$643$			16400.8i			1.00588i	−0.864320	+	0.502941i	$0.832251\pi$
$644$	−1003.69	+	9331.05i	−0.0614144	+	0.570955i
$645$		0			0
$646$	−10360.7	−	17945.2i	−0.631015	−	1.09295i
$647$	773.759	−	1340.19i	0.0470164	−	0.0814347i	−0.841559	−	0.540164i	$-0.818362\pi$
$647$	773.759	−	1340.19i	0.0470164	−	0.0814347i	0.888576	+	0.458730i	$0.151695\pi$
$648$		0			0
$649$	29134.7	−	16820.9i	1.76215	−	1.01738i
$650$	9345.72			0.563953
$651$		0			0
$652$	14423.4			0.866353
$653$	21349.9	−	12326.4i	1.27946	−	0.738697i	0.302712	−	0.953082i	$-0.402108\pi$
$653$	21349.9	−	12326.4i	1.27946	−	0.738697i	0.976749	+	0.214385i	$0.0687748\pi$
$654$		0			0
$655$	6437.30	−	11149.7i	0.384009	−	0.665124i
$656$	−3568.15	−	6180.22i	−0.212367	−	0.367831i
$657$		0			0
$658$	−8351.22	−	6098.61i	−0.494779	−	0.361320i
$659$			23870.0i			1.41099i	0.708713	+	0.705497i	$0.249276\pi$
$659$			23870.0i			1.41099i	−0.708713	+	0.705497i	$0.750724\pi$
$660$		0			0
$661$	13488.0	+	7787.28i	0.793678	+	0.458230i	0.841256	−	0.540637i	$-0.181817\pi$
$661$	13488.0	+	7787.28i	0.793678	+	0.458230i	−0.0475777	+	0.998868i	$0.515150\pi$
$662$	3351.14	+	1934.78i	0.196746	+	0.113591i
$663$		0			0
$664$		−	3967.64i		−	0.231889i
$665$	−10851.8	−	7924.69i	−0.632803	−	0.462115i
$666$		0			0
$667$	13227.3	+	22910.4i	0.767863	+	1.32998i
$668$	−730.732	+	1265.67i	−0.0423247	+	0.0733085i
$669$		0			0
$670$	12525.6	−	7231.64i	0.722247	−	0.416989i
$671$	−23452.6			−1.34929
$672$		0			0
$673$	−20734.4			−1.18760			−0.593799	−	0.804613i	$-0.702373\pi$
$673$	−20734.4			−1.18760			−0.593799	+	0.804613i	$0.702373\pi$
$674$	−12854.6	+	7421.63i	−0.734633	+	0.424141i
$675$		0			0
$676$	−5311.25	+	9199.35i	−0.302187	+	0.523404i
$677$	−8440.21	−	14618.9i	−0.479149	−	0.829910i	0.520565	−	0.853822i	$-0.325721\pi$
$677$	−8440.21	−	14618.9i	−0.479149	−	0.829910i	−0.999714	+	0.0239117i	$0.992388\pi$
$678$		0			0
$679$	−770.933	+	7167.18i	−0.0435725	+	0.405083i
$680$			6616.66i			0.373143i
$681$		0			0
$682$	−20171.3	−	11645.9i	−1.13255	−	0.653877i
$683$	15466.7	+	8929.70i	0.866496	+	0.500272i	0.866182	−	0.499728i	$-0.166567\pi$
$683$	15466.7	+	8929.70i	0.866496	+	0.500272i	0.000313819		1.00000i	$0.499900\pi$
$684$		0			0
$685$		−	10639.0i		−	0.593423i
$686$	12446.2	−	2550.73i	0.692709	−	0.141964i
$687$		0			0
$688$	−1830.59	−	3170.68i	−0.101440	−	0.175699i
$689$	4746.94	−	8221.94i	0.262473	−	0.454617i
$690$		0			0
$691$	−7207.82	+	4161.44i	−0.396814	+	0.229101i	−0.685108	−	0.728441i	$-0.740245\pi$
$691$	−7207.82	+	4161.44i	−0.396814	+	0.229101i	0.288294	+	0.957542i	$0.406912\pi$
$692$	−874.670			−0.0480491
$693$		0			0
$694$	6152.19			0.336504
$695$	−5996.93	+	3462.33i	−0.327304	+	0.188969i
$696$		0			0
$697$	24235.9	−	41977.8i	1.31707	−	2.28124i
$698$	−5872.31	−	10171.1i	−0.318439	−	0.551552i
$699$		0			0
$700$	−2010.18	−	4544.65i	−0.108539	−	0.245388i
$701$		−	18425.3i		−	0.992743i	−0.868110	−	0.496372i	$-0.834665\pi$
$701$		−	18425.3i		−	0.992743i	0.868110	−	0.496372i	$-0.165335\pi$
$702$		0			0
$703$	18677.2	+	10783.3i	1.00202	+	0.578519i
$704$	−2299.37	−	1327.54i	−0.123097	−	0.0710703i
$705$		0			0
$706$			8130.60i			0.433426i
$707$	875.685	−	1199.13i	0.0465821	−	0.0637878i
$708$		0			0
$709$	−7379.90	−	12782.4i	−0.390914	−	0.677082i	0.601657	−	0.798755i	$-0.294508\pi$
$709$	−7379.90	−	12782.4i	−0.390914	−	0.677082i	−0.992570	+	0.121672i	$0.961174\pi$
$710$	3410.61	−	5907.34i	0.180279	−	0.312252i
$711$		0			0
$712$	−3436.67	+	1984.16i	−0.180891	+	0.104438i
$713$	35562.8			1.86794
$714$		0			0
$715$	21993.7			1.15038
$716$	1925.07	−	1111.44i	0.100480	−	0.0580119i
$717$		0			0
$718$	−3791.37	+	6566.85i	−0.197065	+	0.341327i
$719$	−5617.07	−	9729.05i	−0.291351	−	0.504635i	0.682778	−	0.730626i	$-0.260771\pi$
$719$	−5617.07	−	9729.05i	−0.291351	−	0.504635i	−0.974129	+	0.225991i	$0.927438\pi$
$720$		0			0
$721$	−18612.4	+	8232.60i	−0.961392	+	0.425240i
$722$		−	4459.54i		−	0.229871i
$723$		0			0
$724$	−4394.38	−	2537.10i	−0.225574	−	0.130235i
$725$	−12131.3	−	7003.99i	−0.621440	−	0.358789i
$726$		0			0
$727$			15222.7i			0.776589i	0.921535	+	0.388294i	$0.126936\pi$
$727$			15222.7i			0.776589i	−0.921535	+	0.388294i	$0.873064\pi$
$728$	10261.9	+	1103.81i	0.522433	+	0.0561951i
$729$		0			0
$730$	−2346.40	−	4064.08i	−0.118965	−	0.206053i
$731$	12433.9	−	21536.1i	0.629117	−	1.08966i
$732$		0			0
$733$	−1373.40	+	792.935i	−0.0692058	+	0.0399560i	−0.534204	−	0.845356i	$-0.679388\pi$
$733$	−1373.40	+	792.935i	−0.0692058	+	0.0399560i	0.464998	+	0.885312i	$0.346055\pi$
$734$	4827.28			0.242749
$735$		0			0
$736$	4053.88			0.203027
$737$	−34139.1	+	19710.2i	−1.70628	+	0.985122i
$738$		0			0
$739$	−6391.83	+	11071.0i	−0.318169	+	0.551086i	−0.980106	−	0.198474i	$-0.936401\pi$
$739$	−6391.83	+	11071.0i	−0.318169	+	0.551086i	0.661937	+	0.749560i	$0.269735\pi$
$740$	−3443.27	−	5963.92i	−0.171050	−	0.296268i
$741$		0			0
$742$	−5019.20	−	539.887i	−0.248330	−	0.0267114i
$743$		−	17184.7i		−	0.848513i	−0.905542	−	0.424257i	$-0.860535\pi$
$743$		−	17184.7i		−	0.848513i	0.905542	−	0.424257i	$-0.139465\pi$
$744$		0			0
$745$	21489.8	+	12407.1i	1.05681	+	0.610151i
$746$	−19936.0	−	11510.1i	−0.978430	−	0.564897i
$747$		0			0
$748$		−	18034.0i		−	0.881537i
$749$	−24919.4	+	11022.3i	−1.21567	+	0.537710i
$750$		0			0
$751$	−15084.1	−	26126.5i	−0.732926	−	1.26947i	−0.955627	−	0.294579i	$-0.904821\pi$
$751$	−15084.1	−	26126.5i	−0.732926	−	1.26947i	0.222701	−	0.974887i	$-0.428513\pi$
$752$	−2233.44	+	3868.44i	−0.108305	+	0.187590i
$753$		0			0
$754$	25195.9	−	14546.9i	1.21695	−	0.702607i
$755$	−2288.62			−0.110320
$756$		0			0
$757$	1617.03			0.0776378			0.0388189	−	0.999246i	$-0.487640\pi$
$757$	1617.03			0.0776378			0.0388189	+	0.999246i	$0.487640\pi$
$758$	396.089	−	228.682i	0.0189797	−	0.0109579i
$759$		0			0
$760$	−2902.19	+	5026.74i	−0.138518	+	0.239920i
$761$	12383.7	+	21449.2i	0.589895	+	1.02173i	0.994246	+	0.107123i	$0.0341640\pi$
$761$	12383.7	+	21449.2i	0.589895	+	1.02173i	−0.404351	+	0.914604i	$0.632503\pi$
$762$		0			0
$763$	5688.53	−	7789.67i	0.269906	−	0.369600i
$764$			9280.37i			0.439466i
$765$		0			0
$766$	10197.9	+	5887.74i	0.481024	+	0.277719i
$767$	48921.6	+	28244.9i	2.30307	+	1.32968i
$768$		0			0
$769$			7601.01i			0.356436i	0.983991	+	0.178218i	$0.0570333\pi$
$769$			7601.01i			0.356436i	−0.983991	+	0.178218i	$0.942967\pi$
$770$	−4730.65	−	10695.2i	−0.221403	−	0.500554i
$771$		0			0
$772$	7884.81	+	13656.9i	0.367591	+	0.636687i
$773$	−105.272	+	182.337i	−0.00489828	+	0.00848408i	−0.868464	−	0.495752i	$-0.834893\pi$
$773$	−105.272	+	182.337i	−0.00489828	+	0.00848408i	0.863566	+	0.504236i	$0.168226\pi$
$774$		0			0
$775$	−16308.0	+	9415.41i	−0.755871	+	0.436402i
$776$	3113.79			0.144044
$777$		0			0
$778$	3516.05			0.162026
$779$	36824.5	−	21260.6i	1.69368	−	0.977845i
$780$		0			0
$781$	−9295.77	+	16100.8i	−0.425901	+	0.737683i
$782$	13767.6	+	23846.1i	0.629574	+	1.09045i
$783$		0			0
$784$	−1670.47	−	5227.59i	−0.0760965	−	0.238137i
$785$		−	9139.42i		−	0.415541i
$786$		0			0
$787$	26049.2	+	15039.5i	1.17986	+	0.681195i	0.955983	−	0.293422i	$-0.0947941\pi$
$787$	26049.2	+	15039.5i	1.17986	+	0.681195i	0.223881	+	0.974617i	$0.428127\pi$
$788$	8396.56	+	4847.75i	0.379588	+	0.219155i
$789$		0			0
$790$		−	4253.77i		−	0.191573i
$791$	2125.53	−	19760.6i	0.0955439	−	0.888248i
$792$		0			0
$793$	−19690.3	−	34104.5i	−0.881742	−	1.52722i
$794$	−3827.61	+	6629.62i	−0.171079	+	0.296318i
$795$		0			0
$796$	11254.0	−	6497.49i	0.501114	−	0.289318i
$797$	−9787.93			−0.435014			−0.217507	−	0.976059i	$-0.569792\pi$
$797$	−9787.93			−0.435014			−0.217507	+	0.976059i	$0.569792\pi$
$798$		0			0
$799$	−30340.3			−1.34338
$800$	−1858.98	+	1073.28i	−0.0821562	+	0.0474329i
$801$		0			0
$802$	−12723.3	+	22037.5i	−0.560196	+	0.970288i
$803$	6395.22	+	11076.9i	0.281049	+	0.486792i
$804$		0			0
$805$	14420.2	+	10530.5i	0.631359	+	0.461060i
$806$		−	39110.5i		−	1.70919i
$807$		0			0
$808$	−555.459	−	320.695i	−0.0241844	−	0.0139629i
$809$	−1888.55	−	1090.35i	−0.0820738	−	0.0473853i	0.458401	−	0.888745i	$-0.348422\pi$
$809$	−1888.55	−	1090.35i	−0.0820738	−	0.0473853i	−0.540475	+	0.841360i	$0.681756\pi$
$810$		0			0
$811$		−	25003.4i		−	1.08260i	−0.840830	−	0.541299i	$-0.817933\pi$
$811$		−	25003.4i		−	1.08260i	0.840830	−	0.541299i	$-0.182067\pi$
$812$	−12493.3	−	9123.41i	−0.539936	−	0.394297i
$813$		0			0
$814$	9384.81	+	16255.0i	0.404100	+	0.699922i
$815$	13721.1	−	23765.7i	0.589730	−	1.02144i
$816$		0			0
$817$	18892.3	−	10907.5i	0.809006	−	0.467080i
$818$	−2706.70			−0.115694
$819$		0			0
$820$	−13577.7			−0.578237
$821$	12815.9	−	7399.28i	0.544798	−	0.314539i	−0.202223	−	0.979339i	$-0.564817\pi$
$821$	12815.9	−	7399.28i	0.544798	−	0.314539i	0.747021	+	0.664800i	$0.231483\pi$
$822$		0			0
$823$	14438.2	−	25007.7i	0.611524	−	1.05919i	−0.379459	−	0.925208i	$-0.623890\pi$
$823$	14438.2	−	25007.7i	0.611524	−	1.05919i	0.990984	−	0.133983i	$-0.0427767\pi$
$824$	4395.59	+	7613.39i	0.185835	+	0.321875i
$825$		0			0
$826$	3212.40	−	29864.9i	0.135319	−	1.25803i
$827$		−	37153.8i		−	1.56223i	−0.624388	−	0.781114i	$-0.714652\pi$
$827$		−	37153.8i		−	1.56223i	0.624388	−	0.781114i	$-0.285348\pi$
$828$		0			0
$829$	−16977.8	−	9802.12i	−0.711293	−	0.410665i	0.100247	−	0.994963i	$-0.468037\pi$
$829$	−16977.8	−	9802.12i	−0.711293	−	0.410665i	−0.811540	+	0.584297i	$0.801370\pi$
$830$	−6537.58	−	3774.47i	−0.273401	−	0.157848i
$831$		0			0
$832$		−	4458.29i		−	0.185773i
$833$	25077.0	−	27579.8i	1.04306	−	1.14716i
$834$		0			0
$835$	1390.31	+	2408.09i	0.0576212	+	0.0998028i
$836$	7910.06	−	13700.6i	0.327243	−	0.566802i
$837$		0			0
$838$	6311.28	−	3643.82i	0.260167	−	0.150207i
$839$	42978.7			1.76852			0.884260	−	0.466995i	$-0.154664\pi$
$839$	42978.7			1.76852			0.884260	+	0.466995i	$0.154664\pi$
$840$		0			0
$841$	−19218.6			−0.788004
$842$	27298.3	−	15760.7i	1.11730	−	0.645071i
$843$		0			0
$844$	2865.42	−	4963.05i	0.116862	−	0.202411i
$845$	10105.3	+	17502.9i	0.411400	+	0.712567i
$846$		0			0
$847$	2922.17	+	6606.52i	0.118544	+	0.268008i
$848$			2180.60i			0.0883043i
$849$		0			0
$850$	−12626.7	−	7290.04i	−0.509521	−	0.294172i
$851$	−24818.7	−	14329.1i	−0.999736	−	0.577198i
$852$		0			0
$853$			25902.2i			1.03971i	0.854254	+	0.519855i	$0.174014\pi$
$853$			25902.2i			1.03971i	−0.854254	+	0.519855i	$0.825986\pi$
$854$	−12349.2	+	16910.6i	−0.494826	+	0.677597i
$855$		0			0
$856$	5885.07	+	10193.2i	0.234985	+	0.407007i
$857$	18373.0	−	31823.0i	0.732335	−	1.26844i	−0.223548	−	0.974693i	$-0.571764\pi$
$857$	18373.0	−	31823.0i	0.732335	−	1.26844i	0.955883	−	0.293748i	$-0.0949026\pi$
$858$		0			0
$859$	−105.465	+	60.8903i	−0.00418908	+	0.00241857i	−0.502093	−	0.864814i	$-0.667437\pi$
$859$	−105.465	+	60.8903i	−0.00418908	+	0.00241857i	0.497904	+	0.867232i	$0.334103\pi$
$860$	−6965.86			−0.276202
$861$		0			0
$862$	9250.54			0.365516
$863$	8597.16	−	4963.57i	0.339109	−	0.195784i	−0.320769	−	0.947157i	$-0.603941\pi$
$863$	8597.16	−	4963.57i	0.339109	−	0.195784i	0.659878	+	0.751373i	$0.270608\pi$
$864$		0			0
$865$	−832.085	+	1441.21i	−0.0327072	+	0.0566505i
$866$	−9233.87	−	15993.5i	−0.362332	−	0.627578i
$867$		0			0
$868$	−19018.7	+	8412.30i	−0.743706	+	0.328954i
$869$			11593.9i			0.452583i
$870$		0			0
$871$	−57324.8	−	33096.5i	−2.23005	−	1.28752i
$872$	−3608.32	−	2083.26i	−0.140130	−	0.0809038i
$873$		0			0
$874$			24154.8i			0.934839i
$875$	−26918.1	−	2895.43i	−1.04000	−	0.111867i
$876$		0			0
$877$	21368.9	+	37011.9i	0.822777	+	1.42509i	0.903607	+	0.428363i	$0.140910\pi$
$877$	21368.9	+	37011.9i	0.822777	+	1.42509i	−0.0808302	+	0.996728i	$0.525757\pi$
$878$	−2712.57	+	4698.30i	−0.104265	+	0.180592i
$879$		0			0
$880$	−4374.83	+	2525.81i	−0.167586	+	0.0967558i
$881$	−48398.2			−1.85082			−0.925412	−	0.378964i	$-0.876281\pi$
$881$	−48398.2			−1.85082			−0.925412	+	0.378964i	$0.876281\pi$
$882$		0			0
$883$	−2006.61			−0.0764755			−0.0382378	−	0.999269i	$-0.512174\pi$
$883$	−2006.61			−0.0764755			−0.0382378	+	0.999269i	$0.512174\pi$
$884$	26224.9	−	15141.0i	0.997783	−	0.576070i
$885$		0			0
$886$	−18125.8	+	31394.7i	−0.687299	+	1.19044i
$887$	16220.8	+	28095.3i	0.614027	+	1.06353i	0.990554	+	0.137121i	$0.0437848\pi$
$887$	16220.8	+	28095.3i	0.614027	+	1.06353i	−0.376527	+	0.926406i	$0.622882\pi$
$888$		0			0
$889$	−3288.19	−	353.692i	−0.124052	−	0.0133436i
$890$			7550.24i			0.284365i
$891$		0			0
$892$	−9225.88	−	5326.57i	−0.346307	−	0.199940i
$893$	−23049.8	−	13307.8i	−0.863755	−	0.498689i
$894$		0			0
$895$		−	4229.32i		−	0.157956i
$896$	−2167.98	+	958.936i	−0.0808340	+	0.0357542i
$897$		0			0
$898$	5238.46	+	9073.29i	0.194666	+	0.337171i
$899$	−29310.7	+	50767.6i	−1.08739	+	1.88342i
$900$		0			0
$901$	−12826.9	+	7405.62i	−0.474280	+	0.273826i
$902$	37006.7			1.36606
$903$		0			0
$904$	−8584.99			−0.315855
$905$	−8360.86	+	4827.14i	−0.307099	+	0.177304i
$906$		0			0
$907$	14199.4	−	24594.1i	0.519827	−	0.900367i	−0.479907	−	0.877319i	$-0.659330\pi$
$907$	14199.4	−	24594.1i	0.519827	−	0.900367i	0.999734	−	0.0230477i	$-0.00733696\pi$
$908$	−1549.93	−	2684.56i	−0.0566479	−	0.0981170i
$909$		0			0
$910$	11581.0	−	15858.7i	0.421877	−	0.577703i
$911$		−	33935.2i		−	1.23416i	−0.786899	−	0.617082i	$-0.788315\pi$
$911$		−	33935.2i		−	1.23416i	0.786899	−	0.617082i	$-0.211685\pi$
$912$		0			0
$913$	17818.5	+	10287.5i	0.645899	+	0.372910i
$914$	6950.03	+	4012.60i	0.251517	+	0.145213i
$915$		0			0
$916$			1076.26i			0.0388216i
$917$	12673.6	+	28652.8i	0.456400	+	1.03184i
$918$		0			0
$919$	15515.7	+	26874.0i	0.556926	+	0.964625i	0.997751	+	0.0670315i	$0.0213528\pi$
$919$	15515.7	+	26874.0i	0.556926	+	0.964625i	−0.440824	+	0.897593i	$0.645314\pi$
$920$	3856.51	−	6679.67i	0.138202	−	0.239372i
$921$		0			0
$922$	−16275.2	+	9396.48i	−0.581339	+	0.335636i
$923$	−31218.1			−1.11328
$924$		0			0
$925$	15174.8			0.539399
$926$	18241.8	−	10531.9i	0.647367	−	0.373758i
$927$		0			0
$928$	−3341.19	+	5787.11i	−0.118190	+	0.204710i
$929$	−8981.07	−	15555.7i	−0.317179	−	0.549370i	0.662719	−	0.748868i	$-0.269402\pi$
$929$	−8981.07	−	15555.7i	−0.317179	−	0.549370i	−0.979898	+	0.199498i	$0.936069\pi$
$930$		0			0
$931$	31148.3	−	9953.40i	1.09650	−	0.350386i
$932$			23340.7i			0.820332i
$933$		0			0
$934$	−3768.92	−	2175.99i	−0.132037	−	0.0762318i
$935$	−29715.1	−	17156.0i	−1.03935	−	0.600066i
$936$		0			0
$937$			16916.5i			0.589794i	0.955529	+	0.294897i	$0.0952853\pi$
$937$			16916.5i			0.589794i	−0.955529	+	0.294897i	$0.904715\pi$
$938$	−3764.19	+	34994.7i	−0.131029	+	1.21814i
$939$		0			0
$940$	4249.41	+	7360.19i	0.147447	+	0.255386i
$941$	−12922.0	+	22381.6i	−0.447657	+	0.775365i	−0.998233	−	0.0594204i	$-0.981075\pi$
$941$	−12922.0	+	22381.6i	−0.447657	+	0.775365i	0.550576	+	0.834785i	$0.314408\pi$
$942$		0			0
$943$	−48933.4	+	28251.7i	−1.68981	+	0.975612i
$944$	−12974.8			−0.447347
$945$		0			0
$946$	18985.8			0.652518
$947$	29636.0	−	17110.4i	1.01694	−	0.587130i	0.103724	−	0.994606i	$-0.466924\pi$
$947$	29636.0	−	17110.4i	1.01694	−	0.587130i	0.913216	+	0.407476i	$0.133591\pi$
$948$		0			0
$949$	−10738.6	+	18599.8i	−0.367322	+	0.636221i
$950$	−6395.10	−	11076.6i	−0.218405	−	0.378288i
$951$		0			0
$952$	−13003.5	−	9496.02i	−0.442696	−	0.323286i
$953$		−	27581.0i		−	0.937499i	−0.883331	−	0.468749i	$-0.844705\pi$
$953$		−	27581.0i		−	0.937499i	0.883331	−	0.468749i	$-0.155295\pi$
$954$		0			0
$955$	15291.5	+	8828.54i	0.518137	+	0.299146i
$956$	14919.8	+	8613.95i	0.504750	+	0.291417i
$957$		0			0
$958$		−	1574.00i		−	0.0530830i
$959$	20908.4	+	15268.7i	0.704034	+	0.514132i
$960$		0			0
$961$	24506.6	+	42446.8i	0.822619	+	1.42482i
$962$	−15758.5	+	27294.6i	−0.528146	+	0.914775i
$963$		0			0
$964$	−1832.71	+	1058.11i	−0.0612319	+	0.0353522i
$965$	30003.7			1.00088
$966$		0			0
$967$	−28587.4			−0.950682			−0.475341	−	0.879802i	$-0.657675\pi$
$967$	−28587.4			−0.950682			−0.475341	+	0.879802i	$0.657675\pi$
$968$	2702.38	−	1560.22i	0.0897293	−	0.0518052i
$969$		0			0
$970$	2962.19	−	5130.66i	0.0980517	−	0.169830i
$971$	15680.4	+	27159.2i	0.518237	+	0.897612i	0.999776	+	0.0211873i	$0.00674463\pi$
$971$	15680.4	+	27159.2i	0.518237	+	0.897612i	−0.481539	+	0.876425i	$0.659922\pi$
$972$		0			0
$973$	1802.20	−	16754.6i	0.0593790	−	0.552032i
$974$			28067.1i			0.923333i
$975$		0			0
$976$	7833.28	+	4522.55i	0.256903	+	0.148323i
$977$	47672.5	+	27523.7i	1.56108	+	0.901292i	0.997148	+	0.0754729i	$0.0240466\pi$
$977$	47672.5	+	27523.7i	1.56108	+	0.901292i	0.563935	+	0.825819i	$0.309287\pi$
$978$		0			0
$979$		−	20578.6i		−	0.671801i
$980$	−10202.8	−	2220.60i	−0.332566	−	0.0723821i
$981$		0			0
$982$	−13511.7	−	23403.0i	−0.439080	−	0.760509i
$983$	−8513.59	+	14746.0i	−0.276237	+	0.478457i	−0.970447	−	0.241316i	$-0.922421\pi$
$983$	−8513.59	+	14746.0i	−0.276237	+	0.478457i	0.694209	+	0.719773i	$0.255754\pi$
$984$		0			0
$985$	15975.5	−	9223.47i	0.516774	−	0.298359i
$986$	−45388.6			−1.46599
$987$		0			0
$988$	26564.5			0.855392
$989$	−25104.6	+	14494.2i	−0.807159	+	0.466013i
$990$		0			0
$991$	15372.8	−	26626.5i	0.492769	−	0.853502i	−0.507196	−	0.861831i	$-0.669318\pi$
$991$	15372.8	−	26626.5i	0.492769	−	0.853502i	0.999965	+	0.00832913i	$0.00265128\pi$
$992$	4491.54	+	7779.57i	0.143756	+	0.248994i
$993$		0			0
$994$	6714.72	+	15180.8i	0.214263	+	0.484412i
$995$		−	24724.6i		−	0.787761i
$996$		0			0
$997$	7523.54	+	4343.72i	0.238990	+	0.137981i	0.614712	−	0.788751i	$-0.289272\pi$
$997$	7523.54	+	4343.72i	0.238990	+	0.137981i	−0.375723	+	0.926732i	$0.622605\pi$
$998$	−5450.66	−	3146.94i	−0.172883	−	0.0998143i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	378.4.k.d.269.7	yes	20
3.2	odd	2	inner	378.4.k.d.269.4	yes	20
7.5	odd	6	inner	378.4.k.d.215.4	✓	20
21.5	even	6	inner	378.4.k.d.215.7	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
378.4.k.d.215.4	✓	20	7.5	odd	6	inner
378.4.k.d.215.7	yes	20	21.5	even	6	inner
378.4.k.d.269.4	yes	20	3.2	odd	2	inner
378.4.k.d.269.7	yes	20	1.1	even	1	trivial

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:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	378.k (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.73205 - 1.00000i) q^{2} +(2.00000 - 3.46410i) q^{4} +(-3.80525 - 6.59089i) q^{5} +(18.4140 + 1.98070i) q^{7} -8.00000i q^{8} +O(q^{10})\)	\(q+(1.73205 - 1.00000i) q^{2} +(2.00000 - 3.46410i) q^{4} +(-3.80525 - 6.59089i) q^{5} +(18.4140 + 1.98070i) q^{7} -8.00000i q^{8} +(-13.1818 - 7.61050i) q^{10} +(35.9276 + 20.7428i) q^{11} +69.6608i q^{13} +(33.8747 - 14.9834i) q^{14} +(-8.00000 - 13.8564i) q^{16} +(54.3382 - 94.1166i) q^{17} +(82.5626 - 47.6675i) q^{19} -30.4420 q^{20} +82.9712 q^{22} +(-109.711 + 63.3419i) q^{23} +(33.5401 - 58.0932i) q^{25} +(69.6608 + 120.656i) q^{26} +(43.6894 - 59.8267i) q^{28} -208.824i q^{29} +(-243.112 - 140.361i) q^{31} +(-27.7128 - 16.0000i) q^{32} -217.353i q^{34} +(-57.0155 - 128.902i) q^{35} +(113.109 + 195.911i) q^{37} +(95.3350 - 165.125i) q^{38} +(-52.7271 + 30.4420i) q^{40} +446.019 q^{41} +228.824 q^{43} +(143.710 - 82.9712i) q^{44} +(-126.684 + 219.423i) q^{46} +(-139.590 - 241.777i) q^{47} +(335.154 + 72.9452i) q^{49} -134.160i q^{50} +(241.312 + 139.322i) q^{52} +(-118.028 - 68.1437i) q^{53} -315.726i q^{55} +(15.8456 - 147.312i) q^{56} +(-208.824 - 361.694i) q^{58} +(405.464 - 702.284i) q^{59} +(-489.580 + 282.659i) q^{61} -561.442 q^{62} -64.0000 q^{64} +(459.127 - 265.077i) q^{65} +(-475.109 + 822.914i) q^{67} +(-217.353 - 376.466i) q^{68} +(-227.656 - 166.249i) q^{70} +448.145i q^{71} +(267.005 + 154.155i) q^{73} +(391.822 + 226.218i) q^{74} -381.340i q^{76} +(620.487 + 453.120i) q^{77} +(139.734 + 242.026i) q^{79} +(-60.8840 + 105.454i) q^{80} +(772.528 - 446.019i) q^{82} +495.955 q^{83} -827.083 q^{85} +(396.335 - 228.824i) q^{86} +(165.942 - 287.421i) q^{88} +(-248.020 - 429.584i) q^{89} +(-137.977 + 1282.74i) q^{91} +506.735i q^{92} +(-483.554 - 279.180i) q^{94} +(-628.343 - 362.774i) q^{95} +389.224i q^{97} +(653.448 - 208.809i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 40 q^{4} + 4 q^{7}+O(q^{10}) \) 20 * q + 40 * q^4 + 4 * q^7	\( 20 q + 40 q^{4} + 4 q^{7} - 24 q^{10} - 160 q^{16} + 408 q^{19} + 240 q^{22} - 646 q^{25} + 56 q^{28} - 102 q^{31} + 194 q^{37} - 96 q^{40} - 2332 q^{43} - 624 q^{46} + 2840 q^{49} - 648 q^{52} + 96 q^{58} + 1878 q^{61} - 1280 q^{64} - 386 q^{67} + 3672 q^{70} + 1788 q^{73} + 814 q^{79} - 672 q^{82} - 4560 q^{85} + 480 q^{88} + 2724 q^{91} - 1536 q^{94}+O(q^{100}) \) 20 * q + 40 * q^4 + 4 * q^7 - 24 * q^10 - 160 * q^16 + 408 * q^19 + 240 * q^22 - 646 * q^25 + 56 * q^28 - 102 * q^31 + 194 * q^37 - 96 * q^40 - 2332 * q^43 - 624 * q^46 + 2840 * q^49 - 648 * q^52 + 96 * q^58 + 1878 * q^61 - 1280 * q^64 - 386 * q^67 + 3672 * q^70 + 1788 * q^73 + 814 * q^79 - 672 * q^82 - 4560 * q^85 + 480 * q^88 + 2724 * q^91 - 1536 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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