LMFDB - Embedded newform 312.4.m.a.181.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [312,4,Mod(181,312)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(312, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0, 1]))

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

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

chi := DirichletCharacter("312.181");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 312 = 2^{3} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	312.m (of order $2$, degree $1$, 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:	$18.4085959218$
Analytic rank:	$0$
Dimension:	$84$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			181.11
Character	$\chi$	$=$	312.181
Dual form			312.4.m.a.181.12

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-2.48687 - 1.34740i) q^{2} +3.00000i q^{3} +(4.36905 + 6.70160i) q^{4} +8.37088 q^{5} +(4.04219 - 7.46061i) q^{6} -16.5971i q^{7} +(-1.83557 - 22.5528i) q^{8} -9.00000 q^{9} +(-20.8173 - 11.2789i) q^{10} -23.2256 q^{11} +(-20.1048 + 13.1072i) q^{12} +(-46.2169 + 7.80987i) q^{13} +(-22.3629 + 41.2748i) q^{14} +25.1126i q^{15} +(-25.8228 + 58.5592i) q^{16} -14.7565 q^{17} +(22.3818 + 12.1266i) q^{18} +124.395 q^{19} +(36.5728 + 56.0983i) q^{20} +49.7913 q^{21} +(57.7591 + 31.2941i) q^{22} -173.645 q^{23} +(67.6585 - 5.50670i) q^{24} -54.9283 q^{25} +(125.459 + 42.8503i) q^{26} -27.0000i q^{27} +(111.227 - 72.5136i) q^{28} -126.409i q^{29} +(33.8367 - 62.4519i) q^{30} +27.9669i q^{31} +(143.120 - 110.836i) q^{32} -69.6768i q^{33} +(36.6976 + 19.8829i) q^{34} -138.932i q^{35} +(-39.3215 - 60.3144i) q^{36} -362.661 q^{37} +(-309.355 - 167.610i) q^{38} +(-23.4296 - 138.651i) q^{39} +(-15.3653 - 188.787i) q^{40} -309.872i q^{41} +(-123.825 - 67.0886i) q^{42} +556.370i q^{43} +(-101.474 - 155.649i) q^{44} -75.3379 q^{45} +(431.834 + 233.969i) q^{46} +163.307i q^{47} +(-175.678 - 77.4683i) q^{48} +67.5362 q^{49} +(136.600 + 74.0102i) q^{50} -44.2696i q^{51} +(-254.263 - 275.606i) q^{52} -334.459i q^{53} +(-36.3797 + 67.1455i) q^{54} -194.419 q^{55} +(-374.312 + 30.4651i) q^{56} +373.186i q^{57} +(-170.323 + 314.363i) q^{58} -809.123 q^{59} +(-168.295 + 109.718i) q^{60} -6.20734i q^{61} +(37.6825 - 69.5500i) q^{62} +149.374i q^{63} +(-505.261 + 82.7945i) q^{64} +(-386.877 + 65.3755i) q^{65} +(-93.8822 + 173.277i) q^{66} -252.494 q^{67} +(-64.4721 - 98.8923i) q^{68} -520.936i q^{69} +(-187.197 + 345.507i) q^{70} -822.887i q^{71} +(16.5201 + 202.976i) q^{72} -449.440i q^{73} +(901.892 + 488.648i) q^{74} -164.785i q^{75} +(543.489 + 833.647i) q^{76} +385.478i q^{77} +(-128.551 + 376.376i) q^{78} -547.122 q^{79} +(-216.159 + 490.192i) q^{80} +81.0000 q^{81} +(-417.521 + 770.613i) q^{82} -959.070 q^{83} +(217.541 + 333.681i) q^{84} -123.525 q^{85} +(749.651 - 1383.62i) q^{86} +379.227 q^{87} +(42.6322 + 523.803i) q^{88} -656.191i q^{89} +(187.356 + 101.510i) q^{90} +(129.621 + 767.067i) q^{91} +(-758.666 - 1163.70i) q^{92} -83.9007 q^{93} +(220.039 - 406.123i) q^{94} +1041.30 q^{95} +(332.507 + 429.361i) q^{96} -417.791i q^{97} +(-167.954 - 90.9980i) q^{98} +209.030 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 84 q - 756 q^{9} - 36 q^{10} + 12 q^{12} - 208 q^{14} - 148 q^{16} - 104 q^{17} + 620 q^{22} + 2188 q^{25} + 444 q^{26} - 204 q^{30} - 40 q^{38} - 1924 q^{40} + 192 q^{42} + 624 q^{48} - 3396 q^{49} - 1292 q^{52}+ \cdots + 2480 q^{95}+O(q^{100}) $ 84 * q - 756 * q^9 - 36 * q^10 + 12 * q^12 - 208 * q^14 - 148 * q^16 - 104 * q^17 + 620 * q^22 + 2188 * q^25 + 444 * q^26 - 204 * q^30 - 40 * q^38 - 1924 * q^40 + 192 * q^42 + 624 * q^48 - 3396 * q^49 - 1292 * q^52 - 1616 * q^55 + 608 * q^56 - 2120 * q^62 - 2856 * q^64 + 696 * q^65 - 396 * q^66 - 2536 * q^68 - 3936 * q^74 - 156 * q^78 + 3160 * q^79 + 6804 * q^81 + 4276 * q^82 - 2088 * q^87 + 1780 * q^88 + 324 * q^90 + 4792 * q^92 - 860 * q^94 + 2480 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/312\mathbb{Z}\right)^\times$.

$n$	$79$	$145$	$157$	$209$
$\chi(n)$	$1$	$-1$	$-1$	$1$

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 $

$2$	−2.48687	−	1.34740i	−0.879242	−	0.476376i
$2$	−2.48687	−	1.34740i	−0.879242	−	0.476376i
$3$			3.00000i			0.577350i
$3$			3.00000i			0.577350i
$4$	4.36905	+	6.70160i	0.546131	+	0.837699i
$5$	8.37088			0.748714			0.374357	−	0.927285i	$-0.377863\pi$
$5$	8.37088			0.748714			0.374357	+	0.927285i	$0.377863\pi$
$6$	4.04219	−	7.46061i	0.275036	−	0.507630i
$7$		−	16.5971i		−	0.896159i	−0.893994	−	0.448080i	$-0.852108\pi$
$7$		−	16.5971i		−	0.896159i	0.893994	−	0.448080i	$-0.147892\pi$
$8$	−1.83557	−	22.5528i	−0.0811214	−	0.996704i
$9$	−9.00000			−0.333333
$10$	−20.8173	−	11.2789i	−0.658301	−	0.356670i
$11$	−23.2256			−0.636616			−0.318308	−	0.947987i	$-0.603115\pi$
$11$	−23.2256			−0.636616			−0.318308	+	0.947987i	$0.603115\pi$
$12$	−20.1048	+	13.1072i	−0.483646	+	0.315309i
$13$	−46.2169	+	7.80987i	−0.986021	+	0.166621i
$13$	−46.2169	+	7.80987i	−0.986021	+	0.166621i
$14$	−22.3629	+	41.2748i	−0.426909	+	0.787940i
$15$			25.1126i			0.432270i
$16$	−25.8228	+	58.5592i	−0.403481	+	0.914988i
$17$	−14.7565			−0.210529			−0.105264	−	0.994444i	$-0.533569\pi$
$17$	−14.7565			−0.210529			−0.105264	+	0.994444i	$0.533569\pi$
$18$	22.3818	+	12.1266i	0.293081	+	0.158792i
$19$	124.395			1.50201			0.751006	−	0.660295i	$-0.229569\pi$
$19$	124.395			1.50201			0.751006	+	0.660295i	$0.229569\pi$
$20$	36.5728	+	56.0983i	0.408896	+	0.627198i
$21$	49.7913			0.517398
$22$	57.7591	+	31.2941i	0.559740	+	0.303269i
$23$	−173.645			−1.57424			−0.787121	−	0.616798i	$-0.788429\pi$
$23$	−173.645			−1.57424			−0.787121	+	0.616798i	$0.788429\pi$
$24$	67.6585	−	5.50670i	0.575447	−	0.0468355i
$25$	−54.9283			−0.439427
$26$	125.459	+	42.8503i	0.946325	+	0.323217i
$27$		−	27.0000i		−	0.192450i
$28$	111.227	−	72.5136i	0.750712	−	0.489421i
$29$		−	126.409i		−	0.809433i	−0.914442	−	0.404716i	$-0.867370\pi$
$29$		−	126.409i		−	0.809433i	0.914442	−	0.404716i	$-0.132630\pi$
$30$	33.8367	−	62.4519i	0.205923	−	0.380070i
$31$			27.9669i			0.162032i	0.996713	+	0.0810162i	$0.0258165\pi$
$31$			27.9669i			0.162032i	−0.996713	+	0.0810162i	$0.974183\pi$
$32$	143.120	−	110.836i	0.790636	−	0.612287i
$33$		−	69.6768i		−	0.367551i
$34$	36.6976	+	19.8829i	0.185105	+	0.100291i
$35$		−	138.932i		−	0.670967i
$36$	−39.3215	−	60.3144i	−0.182044	−	0.279233i
$37$	−362.661			−1.61138			−0.805692	−	0.592335i	$-0.798206\pi$
$37$	−362.661			−1.61138			−0.805692	+	0.592335i	$0.798206\pi$
$38$	−309.355	−	167.610i	−1.32063	−	0.715523i
$39$	−23.4296	−	138.651i	−0.0961985	−	0.569280i
$40$	−15.3653	−	188.787i	−0.0607368	−	0.746247i
$41$		−	309.872i		−	1.18034i	−0.807279	−	0.590170i	$-0.799061\pi$
$41$		−	309.872i		−	1.18034i	0.807279	−	0.590170i	$-0.200939\pi$
$42$	−123.825	−	67.0886i	−0.454918	−	0.246476i
$43$			556.370i			1.97316i	0.163293	+	0.986578i	$0.447788\pi$
$43$			556.370i			1.97316i	−0.163293	+	0.986578i	$0.552212\pi$
$44$	−101.474	−	155.649i	−0.347676	−	0.533293i
$45$	−75.3379			−0.249571
$46$	431.834	+	233.969i	1.38414	+	0.749932i
$47$			163.307i			0.506825i	0.967358	+	0.253412i	$0.0815530\pi$
$47$			163.307i			0.506825i	−0.967358	+	0.253412i	$0.918447\pi$
$48$	−175.678	−	77.4683i	−0.528269	−	0.232950i
$49$	67.5362			0.196899
$50$	136.600	+	74.0102i	0.386362	+	0.209332i
$51$		−	44.2696i		−	0.121549i
$52$	−254.263	−	275.606i	−0.678075	−	0.734993i
$53$		−	334.459i		−	0.866821i	−0.901197	−	0.433411i	$-0.857310\pi$
$53$		−	334.459i		−	0.866821i	0.901197	−	0.433411i	$-0.142690\pi$
$54$	−36.3797	+	67.1455i	−0.0916786	+	0.169210i
$55$	−194.419			−0.476644
$56$	−374.312	+	30.4651i	−0.893206	+	0.0726977i
$57$			373.186i			0.867187i
$58$	−170.323	+	314.363i	−0.385595	+	0.711687i
$59$	−809.123			−1.78540			−0.892702	−	0.450647i	$-0.851193\pi$
$59$	−809.123			−1.78540			−0.892702	+	0.450647i	$0.851193\pi$
$60$	−168.295	+	109.718i	−0.362113	+	0.236077i
$61$		−	6.20734i		−	0.0130290i	−0.999979	−	0.00651450i	$-0.997926\pi$
$61$		−	6.20734i		−	0.0130290i	0.999979	−	0.00651450i	$-0.00207364\pi$
$62$	37.6825	−	69.5500i	0.0771883	−	0.142466i
$63$			149.374i			0.298720i
$64$	−505.261	+	82.7945i	−0.986839	+	0.161708i
$65$	−386.877	+	65.3755i	−0.738248	+	0.124751i
$66$	−93.8822	+	173.277i	−0.175092	+	0.323166i
$67$	−252.494			−0.460403			−0.230202	−	0.973143i	$-0.573939\pi$
$67$	−252.494			−0.460403			−0.230202	+	0.973143i	$0.573939\pi$
$68$	−64.4721	−	98.8923i	−0.114976	−	0.176360i
$69$		−	520.936i		−	0.908889i
$70$	−187.197	+	345.507i	−0.319633	+	0.589942i
$71$		−	822.887i		−	1.37548i	−0.725959	−	0.687738i	$-0.758604\pi$
$71$		−	822.887i		−	1.37548i	0.725959	−	0.687738i	$-0.241396\pi$
$72$	16.5201	+	202.976i	0.0270405	+	0.332235i
$73$		−	449.440i		−	0.720589i	−0.932839	−	0.360294i	$-0.882676\pi$
$73$		−	449.440i		−	0.720589i	0.932839	−	0.360294i	$-0.117324\pi$
$74$	901.892	+	488.648i	1.41679	+	0.767624i
$75$		−	164.785i		−	0.253703i
$76$	543.489	+	833.647i	0.820296	+	1.25823i
$77$			385.478i			0.570510i
$78$	−128.551	+	376.376i	−0.186609	+	0.546361i
$79$	−547.122			−0.779191			−0.389595	−	0.920986i	$-0.627385\pi$
$79$	−547.122			−0.779191			−0.389595	+	0.920986i	$0.627385\pi$
$80$	−216.159	+	490.192i	−0.302092	+	0.685065i
$81$	81.0000			0.111111
$82$	−417.521	+	770.613i	−0.562286	+	1.03780i
$83$	−959.070			−1.26833			−0.634166	−	0.773197i	$-0.718656\pi$
$83$	−959.070			−1.26833			−0.634166	+	0.773197i	$0.718656\pi$
$84$	217.541	+	333.681i	0.282567	+	0.433424i
$85$	−123.525			−0.157626
$86$	749.651	−	1383.62i	0.939964	−	1.73488i
$87$	379.227			0.467326
$88$	42.6322	+	523.803i	0.0516432	+	0.634518i
$89$		−	656.191i		−	0.781529i	−0.920491	−	0.390765i	$-0.872211\pi$
$89$		−	656.191i		−	0.781529i	0.920491	−	0.390765i	$-0.127789\pi$
$90$	187.356	+	101.510i	0.219434	+	0.118890i
$91$	129.621	+	767.067i	0.149319	+	0.883632i
$92$	−758.666	−	1163.70i	−0.859743	−	1.31874i
$93$	−83.9007			−0.0935494
$94$	220.039	−	406.123i	0.241439	−	0.445621i
$95$	1041.30			1.12458
$96$	332.507	+	429.361i	0.353504	+	0.456474i
$97$		−	417.791i		−	0.437322i	−0.975801	−	0.218661i	$-0.929831\pi$
$97$		−	417.791i		−	0.437322i	0.975801	−	0.218661i	$-0.0701689\pi$
$98$	−167.954	−	90.9980i	−0.173121	−	0.0937978i
$99$	209.030			0.212205
$100$	−239.985	−	368.108i	−0.239985	−	0.368108i
$101$		−	1487.66i		−	1.46562i	−0.680432	−	0.732811i	$-0.738208\pi$
$101$		−	1487.66i		−	1.46562i	0.680432	−	0.732811i	$-0.261792\pi$
$102$	−59.6487	+	110.093i	−0.0579029	+	0.106871i
$103$	−220.278			−0.210724			−0.105362	−	0.994434i	$-0.533600\pi$
$103$	−220.278			−0.210724			−0.105362	+	0.994434i	$0.533600\pi$
$104$	260.969	+	1027.99i	0.246059	+	0.969255i
$105$	416.797			0.387383
$106$	−450.649	+	831.757i	−0.412933	+	0.762145i
$107$			871.468i			0.787365i	0.919247	+	0.393682i	$0.128799\pi$
$107$			871.468i			0.787365i	−0.919247	+	0.393682i	$0.871201\pi$
$108$	180.943	−	117.964i	0.161215	−	0.105103i
$109$	152.585			0.134082			0.0670411	−	0.997750i	$-0.478644\pi$
$109$	152.585			0.134082			0.0670411	+	0.997750i	$0.478644\pi$
$110$	483.494	+	261.959i	0.419085	+	0.227062i
$111$		−	1087.98i		−	0.930332i
$112$	971.914	+	428.583i	0.819975	+	0.361583i
$113$	1053.23			0.876810			0.438405	−	0.898778i	$-0.355544\pi$
$113$	1053.23			0.876810			0.438405	+	0.898778i	$0.355544\pi$
$114$	502.829	−	928.065i	0.413107	−	0.762467i
$115$	−1453.57			−1.17866
$116$	847.142	−	552.287i	0.678061	−	0.442057i
$117$	415.952	−	70.2889i	0.328674	−	0.0555402i
$118$	2012.18	+	1090.21i	1.56980	+	0.850524i
$119$			244.916i			0.188667i
$120$	566.362	−	46.0960i	0.430846	−	0.0350664i
$121$	−791.572			−0.594720
$122$	−8.36374	+	15.4369i	−0.00620670	+	0.0114556i
$123$	929.617			0.681470
$124$	−187.423	+	122.189i	−0.135734	+	0.0884910i
$125$	−1506.16			−1.07772
$126$	201.266	−	371.474i	0.142303	−	0.262647i
$127$	2021.53			1.41245			0.706226	−	0.707987i	$-0.250396\pi$
$127$	2021.53			1.41245			0.706226	+	0.707987i	$0.250396\pi$
$128$	1368.08	+	474.887i	0.944703	+	0.327926i
$129$	−1669.11			−1.13920
$130$	1050.20	+	358.695i	0.708527	+	0.241997i
$131$		−	1126.13i		−	0.751073i	−0.926808	−	0.375537i	$-0.877458\pi$
$131$		−	1126.13i		−	0.751073i	0.926808	−	0.375537i	$-0.122542\pi$
$132$	466.946	−	304.422i	0.307897	−	0.200731i
$133$		−	2064.60i		−	1.34604i
$134$	627.919	+	340.209i	0.404806	+	0.219325i
$135$		−	226.014i		−	0.144090i
$136$	27.0866	+	332.802i	0.0170784	+	0.209835i
$137$			882.672i			0.550451i	0.961380	+	0.275225i	$0.0887525\pi$
$137$			882.672i			0.550451i	−0.961380	+	0.275225i	$0.911248\pi$
$138$	−701.907	+	1295.50i	−0.432973	+	0.799133i
$139$			2278.54i			1.39039i	0.718823	+	0.695193i	$0.244681\pi$
$139$			2278.54i			1.39039i	−0.718823	+	0.695193i	$0.755319\pi$
$140$	931.069	−	607.003i	0.562069	−	0.366436i
$141$	−489.921			−0.292615
$142$	−1108.75	+	2046.41i	−0.655244	+	1.20938i
$143$	1073.42	−	181.389i	0.627717	−	0.106073i
$144$	232.405	−	527.033i	0.134494	−	0.304996i
$145$		−	1058.15i		−	0.606034i
$146$	−605.573	+	1117.70i	−0.343271	+	0.633571i
$147$			202.609i			0.113679i
$148$	−1584.49	−	2430.41i	−0.880027	−	1.34985i
$149$	1732.23			0.952413			0.476207	−	0.879333i	$-0.342011\pi$
$149$	1732.23			0.952413			0.476207	+	0.879333i	$0.342011\pi$
$150$	−222.031	+	409.799i	−0.120858	+	0.223066i
$151$		−	2401.99i		−	1.29451i	−0.762274	−	0.647255i	$-0.775917\pi$
$151$		−	2401.99i		−	1.29451i	0.762274	−	0.647255i	$-0.224083\pi$
$152$	−228.336	−	2805.47i	−0.121845	−	1.49706i
$153$	132.809			0.0701762
$154$	519.391	−	958.633i	0.271777	−	0.501616i
$155$			234.108i			0.121316i
$156$	826.817	−	762.789i	0.424348	−	0.391487i
$157$			3021.09i			1.53573i	0.640612	+	0.767865i	$0.278681\pi$
$157$			3021.09i			1.53573i	−0.640612	+	0.767865i	$0.721319\pi$
$158$	1360.62	+	737.190i	0.685097	+	0.371188i
$159$	1003.38			0.500460
$160$	1198.04	−	927.793i	0.591960	−	0.458428i
$161$			2882.01i			1.41077i
$162$	−201.437	−	109.139i	−0.0976935	−	0.0529307i
$163$	1603.60			0.770577			0.385288	−	0.922796i	$-0.374102\pi$
$163$	1603.60			0.770577			0.385288	+	0.922796i	$0.374102\pi$
$164$	2076.64	−	1353.85i	0.988770	−	0.644621i
$165$		−	583.256i		−	0.275190i
$166$	2385.08	+	1292.25i	1.11517	+	0.604203i
$167$			3215.29i			1.48986i	0.667143	+	0.744930i	$0.267517\pi$
$167$			3215.29i			1.48986i	−0.667143	+	0.744930i	$0.732483\pi$
$168$	−91.3953	−	1122.94i	−0.0419720	−	0.515693i
$169$	2075.01	−	721.897i	0.944475	−	0.328583i
$170$	307.191	+	166.437i	0.138591	+	0.0750892i
$171$	−1119.56			−0.500671
$172$	−3728.57	+	2430.81i	−1.65291	+	1.07760i
$173$		−	2437.42i		−	1.07118i	−0.844479	−	0.535589i	$-0.820090\pi$
$173$		−	2437.42i		−	1.07118i	0.844479	−	0.535589i	$-0.179910\pi$
$174$	−943.088	−	510.968i	−0.410893	−	0.222623i
$175$			911.651i			0.393796i
$176$	599.749	−	1360.07i	0.256863	−	0.582496i
$177$		−	2427.37i		−	1.03080i
$178$	−884.148	+	1631.86i	−0.372302	+	0.687153i
$179$			3033.74i			1.26677i	0.773835	+	0.633387i	$0.218336\pi$
$179$			3033.74i			1.26677i	−0.773835	+	0.633387i	$0.781664\pi$
$180$	−329.155	−	504.884i	−0.136299	−	0.209066i
$181$		−	1824.97i		−	0.749442i	−0.927138	−	0.374721i	$-0.877738\pi$
$181$		−	1824.97i		−	0.749442i	0.927138	−	0.374721i	$-0.122262\pi$
$182$	711.192	−	2082.25i	0.289654	−	0.848058i
$183$	18.6220			0.00752229
$184$	318.738	+	3916.20i	0.127705	+	1.56905i
$185$	−3035.80			−1.20647
$186$	208.650	+	113.047i	0.0822525	+	0.0445647i
$187$	342.729			0.134026
$188$	−1094.42	+	713.496i	−0.424567	+	0.276793i
$189$	−448.122			−0.172466
$190$	−2589.57	−	1403.04i	−0.988776	−	0.535722i
$191$	2060.69			0.780662			0.390331	−	0.920675i	$-0.372361\pi$
$191$	2060.69			0.780662			0.390331	+	0.920675i	$0.372361\pi$
$192$	−248.384	−	1515.78i	−0.0933622	−	0.569752i
$193$			2776.88i			1.03567i	0.855480	+	0.517835i	$0.173262\pi$
$193$			2776.88i			1.03567i	−0.855480	+	0.517835i	$0.826738\pi$
$194$	−562.930	+	1038.99i	−0.208330	+	0.384512i
$195$	−196.127	−	1160.63i	−0.0720252	−	0.426228i
$196$	295.069	+	452.600i	0.107532	+	0.164942i
$197$	−2780.38			−1.00555			−0.502776	−	0.864417i	$-0.667688\pi$
$197$	−2780.38			−1.00555			−0.502776	+	0.864417i	$0.667688\pi$
$198$	−519.832	−	281.647i	−0.186580	−	0.101090i
$199$	3560.57			1.26835			0.634175	−	0.773189i	$-0.281340\pi$
$199$	3560.57			1.26835			0.634175	+	0.773189i	$0.281340\pi$
$200$	100.825	+	1238.79i	0.0356469	+	0.437979i
$201$		−	757.481i		−	0.265814i
$202$	−2004.47	+	3699.62i	−0.698187	+	1.28864i
$203$	−2098.02			−0.725381
$204$	296.677	−	193.416i	0.101821	−	0.0663816i
$205$		−	2593.91i		−	0.883738i
$206$	547.802	+	296.801i	0.185278	+	0.100384i
$207$	1562.81			0.524748
$208$	736.109	−	2908.10i	0.245385	−	0.969426i
$209$	−2889.15			−0.956205
$210$	−1036.52	−	561.590i	−0.340603	−	0.184540i
$211$			3569.23i			1.16453i	0.812999	+	0.582265i	$0.197833\pi$
$211$			3569.23i			1.16453i	−0.812999	+	0.582265i	$0.802167\pi$
$212$	2241.41	−	1461.27i	0.726136	−	0.473398i
$213$	2468.66			0.794131
$214$	1174.21	−	2167.23i	0.375082	−	0.692284i
$215$			4657.31i			1.47733i
$216$	−608.927	+	49.5603i	−0.191816	+	0.0156118i
$217$	464.169			0.145207
$218$	−379.458	−	205.592i	−0.117891	−	0.0638735i
$219$	1348.32			0.416032
$220$	−849.425	−	1302.92i	−0.260310	−	0.399284i
$221$	682.002	−	115.247i	0.207586	−	0.0350784i
$222$	−1465.94	+	2705.68i	−0.443188	+	0.817987i
$223$		−	1406.96i		−	0.422498i	−0.977432	−	0.211249i	$-0.932247\pi$
$223$		−	1406.96i		−	0.422498i	0.977432	−	0.211249i	$-0.0677531\pi$
$224$	−1839.55	−	2375.38i	−0.548707	−	0.708535i
$225$	494.355			0.146476
$226$	−2619.24	−	1419.12i	−0.770927	−	0.417691i
$227$	−70.1646			−0.0205154			−0.0102577	−	0.999947i	$-0.503265\pi$
$227$	−70.1646			−0.0205154			−0.0102577	+	0.999947i	$0.503265\pi$
$228$	−2500.94	+	1630.47i	−0.726442	+	0.473598i
$229$	−837.989			−0.241816			−0.120908	−	0.992664i	$-0.538581\pi$
$229$	−837.989			−0.241816			−0.120908	+	0.992664i	$0.538581\pi$
$230$	3614.83	+	1958.53i	1.03633	+	0.561485i
$231$	−1156.43			−0.329384
$232$	−2850.88	+	232.032i	−0.806765	+	0.0656623i
$233$	−6308.04			−1.77362			−0.886809	−	0.462136i	$-0.847083\pi$
$233$	−6308.04			−1.77362			−0.886809	+	0.462136i	$0.847083\pi$
$234$	−1129.13	−	385.653i	−0.315442	−	0.107739i
$235$			1367.02i			0.379467i
$236$	−3535.10	−	5422.42i	−0.975065	−	1.49563i
$237$		−	1641.37i		−	0.449866i
$238$	329.998	−	609.074i	0.0898765	−	0.165884i
$239$			5112.79i			1.38376i	0.722012	+	0.691880i	$0.243217\pi$
$239$			5112.79i			1.38376i	−0.722012	+	0.691880i	$0.756783\pi$
$240$	−1470.58	−	648.478i	−0.395522	−	0.174413i
$241$			276.608i			0.0739330i	0.999317	+	0.0369665i	$0.0117695\pi$
$241$			276.608i			0.0739330i	−0.999317	+	0.0369665i	$0.988231\pi$
$242$	1968.54	+	1066.56i	0.522902	+	0.283310i
$243$			243.000i			0.0641500i
$244$	41.5991	−	27.1202i	0.0109144	−	0.00711554i
$245$	565.338			0.147421
$246$	−2311.84	−	1252.56i	−0.599176	−	0.324636i
$247$	−5749.17	+	971.511i	−1.48102	+	0.250266i
$248$	630.733	−	51.3351i	0.161498	−	0.0131443i
$249$		−	2877.21i		−	0.732272i
$250$	3745.62	+	2029.39i	0.947576	+	0.513400i
$251$		−	3768.11i		−	0.947574i	−0.880640	−	0.473787i	$-0.842887\pi$
$251$		−	3768.11i		−	0.947574i	0.880640	−	0.473787i	$-0.157113\pi$
$252$	−1001.04	+	652.622i	−0.250237	+	0.163140i
$253$	4033.02			1.00219
$254$	−5027.27	−	2723.79i	−1.24189	−	0.672858i
$255$		−	370.576i		−	0.0910053i
$256$	−2762.37	−	3024.32i	−0.674406	−	0.738360i
$257$	1835.81			0.445583			0.222792	−	0.974866i	$-0.428483\pi$
$257$	1835.81			0.445583			0.222792	+	0.974866i	$0.428483\pi$
$258$	4150.86	+	2248.95i	1.00163	+	0.542689i
$259$			6019.13i			1.44406i
$260$	−2128.40	−	2307.06i	−0.507685	−	0.550299i
$261$			1137.68i			0.269811i
$262$	−1517.34	+	2800.54i	−0.357793	+	0.660375i
$263$	−2858.96			−0.670308			−0.335154	−	0.942163i	$-0.608788\pi$
$263$	−2858.96			−0.670308			−0.335154	+	0.942163i	$0.608788\pi$
$264$	−1571.41	+	127.896i	−0.366339	+	0.0298162i
$265$		−	2799.72i		−	0.649002i
$266$	−2781.83	+	5134.39i	−0.641222	+	1.18350i
$267$	1968.57			0.451216
$268$	−1103.16	−	1692.11i	−0.251441	−	0.385679i
$269$		−	1513.76i		−	0.343106i	−0.985175	−	0.171553i	$-0.945122\pi$
$269$		−	1513.76i		−	0.343106i	0.985175	−	0.171553i	$-0.0548784\pi$
$270$	−304.530	+	562.067i	−0.0686411	+	0.126690i
$271$		−	3396.37i		−	0.761310i	−0.924717	−	0.380655i	$-0.875699\pi$
$271$		−	3396.37i		−	0.761310i	0.924717	−	0.380655i	$-0.124301\pi$
$272$	381.055	−	864.132i	0.0849443	−	0.192631i
$273$	−2301.20	+	388.864i	−0.510165	+	0.0862092i
$274$	1189.31	−	2195.09i	0.262222	−	0.483979i
$275$	1275.74			0.279746
$276$	3491.10	−	2276.00i	0.761376	−	0.496373i
$277$		−	1744.88i		−	0.378482i	−0.981931	−	0.189241i	$-0.939397\pi$
$277$		−	1744.88i		−	0.378482i	0.981931	−	0.189241i	$-0.0606027\pi$
$278$	3070.10	−	5666.44i	0.662346	−	1.22248i
$279$		−	251.702i		−	0.0540108i
$280$	−3133.32	+	255.020i	−0.668756	+	0.0544298i
$281$			3474.67i			0.737657i	0.929498	+	0.368828i	$0.120241\pi$
$281$			3474.67i			0.737657i	−0.929498	+	0.368828i	$0.879759\pi$
$282$	1218.37	+	660.117i	0.257280	+	0.139395i
$283$		−	1706.09i		−	0.358363i	−0.983816	−	0.179181i	$-0.942655\pi$
$283$		−	1706.09i		−	0.358363i	0.983816	−	0.179181i	$-0.0573449\pi$
$284$	5514.66	−	3595.24i	1.15223	−	0.751190i
$285$			3123.89i			0.649275i
$286$	−2913.85	−	995.225i	−0.602446	−	0.205765i
$287$	−5142.98			−1.05777
$288$	−1288.08	+	997.522i	−0.263545	+	0.204096i
$289$	−4695.24			−0.955678
$290$	−1425.75	+	2631.49i	−0.288700	+	0.532850i
$291$	1253.37			0.252488
$292$	3011.96	−	1963.63i	0.603637	−	0.393536i
$293$	3584.54			0.714714			0.357357	−	0.933968i	$-0.383678\pi$
$293$	3584.54			0.714714			0.357357	+	0.933968i	$0.383678\pi$
$294$	272.994	−	503.861i	0.0541542	−	0.0999517i
$295$	−6773.07			−1.33676
$296$	665.690	+	8179.05i	0.130718	+	1.60607i
$297$			627.091i			0.122517i
$298$	−4307.83	−	2333.99i	−0.837401	−	0.453707i
$299$	8025.36	−	1356.15i	1.55224	−	0.262301i
$300$	1104.32	−	719.954i	0.212527	−	0.138555i
$301$	9234.13			1.76826
$302$	−3236.43	+	5973.43i	−0.616674	+	1.13819i
$303$	4462.98			0.846177
$304$	−3212.23	+	7284.49i	−0.606033	+	1.37432i
$305$		−	51.9609i		−	0.00975499i
$306$	−330.278	−	178.946i	−0.0617018	−	0.0334303i
$307$	−4601.91			−0.855520			−0.427760	−	0.903892i	$-0.640697\pi$
$307$	−4601.91			−0.855520			−0.427760	+	0.903892i	$0.640697\pi$
$308$	−2583.32	+	1684.17i	−0.477916	+	0.311573i
$309$		−	660.833i		−	0.121662i
$310$	315.435	−	582.195i	0.0577920	−	0.106666i
$311$	9575.88			1.74597			0.872987	−	0.487743i	$-0.162180\pi$
$311$	9575.88			1.74597			0.872987	+	0.487743i	$0.162180\pi$
$312$	−3083.96	+	782.908i	−0.559600	+	0.142062i
$313$	−640.816			−0.115722			−0.0578611	−	0.998325i	$-0.518428\pi$
$313$	−640.816			−0.115722			−0.0578611	+	0.998325i	$0.518428\pi$
$314$	4070.61	−	7513.07i	0.731585	−	1.35028i
$315$			1250.39i			0.223656i
$316$	−2390.41	−	3666.59i	−0.425541	−	0.652728i
$317$	−3844.07			−0.681086			−0.340543	−	0.940229i	$-0.610611\pi$
$317$	−3844.07			−0.681086			−0.340543	+	0.940229i	$0.610611\pi$
$318$	−2495.27	−	1351.95i	−0.440025	−	0.238407i
$319$			2935.92i			0.515298i
$320$	−4229.48	+	693.063i	−0.738860	+	0.121073i
$321$	−2614.41			−0.454585
$322$	3883.21	−	7167.19i	0.672058	−	1.24041i
$323$	−1835.64			−0.316216
$324$	353.893	+	542.829i	0.0606813	+	0.0930777i
$325$	2538.62	−	428.983i	0.433284	−	0.0732176i
$326$	−3987.96	−	2160.69i	−0.677523	−	0.367084i
$327$			457.754i			0.0774124i
$328$	−6988.50	+	568.792i	−1.17645	+	0.0957508i
$329$	2710.42			0.454196
$330$	−785.877	+	1450.48i	−0.131094	+	0.241959i
$331$	9122.40			1.51484			0.757421	−	0.652927i	$-0.226459\pi$
$331$	9122.40			1.51484			0.757421	+	0.652927i	$0.226459\pi$
$332$	−4190.22	−	6427.30i	−0.692676	−	1.06248i
$333$	3263.95			0.537128
$334$	4332.26	−	7996.01i	0.709733	−	1.30995i
$335$	−2113.59			−0.344710
$336$	−1285.75	+	2915.74i	−0.208760	+	0.473413i
$337$	5151.68			0.832730			0.416365	−	0.909198i	$-0.363304\pi$
$337$	5151.68			0.832730			0.416365	+	0.909198i	$0.363304\pi$
$338$	−6132.97	−	1000.60i	−0.986951	−	0.161022i
$339$			3159.69i			0.506226i
$340$	−539.688	−	827.816i	−0.0860844	−	0.132043i
$341$		−	649.548i		−	0.103152i
$342$	2784.19	+	1508.49i	0.440210	+	0.238508i
$343$		−	6813.71i		−	1.07261i
$344$	12547.7	−	1021.26i	1.96665	−	0.160065i
$345$		−	4360.70i		−	0.680499i
$346$	−3284.17	+	6061.55i	−0.510284	+	0.941824i
$347$			11604.4i			1.79527i	0.440742	+	0.897634i	$0.354716\pi$
$347$			11604.4i			1.79527i	−0.440742	+	0.897634i	$0.645284\pi$
$348$	1656.86	+	2541.42i	0.255222	+	0.391479i
$349$	2017.61			0.309456			0.154728	−	0.987957i	$-0.450550\pi$
$349$	2017.61			0.309456			0.154728	+	0.987957i	$0.450550\pi$
$350$	1228.35	−	2267.16i	0.187595	−	0.346242i
$351$	210.867	+	1247.86i	0.0320662	+	0.189760i
$352$	−3324.06	+	2574.23i	−0.503332	+	0.389792i
$353$			6809.89i			1.02678i	0.858155	+	0.513391i	$0.171611\pi$
$353$			6809.89i			1.02678i	−0.858155	+	0.513391i	$0.828389\pi$
$354$	−3270.63	+	6036.55i	−0.491050	+	0.906325i
$355$		−	6888.29i		−	1.02984i
$356$	4397.53	−	2866.93i	0.654686	−	0.426818i
$357$	−734.747			−0.108927
$358$	4087.65	−	7544.53i	0.603461	−	1.11380i
$359$		−	4459.33i		−	0.655584i	−0.944750	−	0.327792i	$-0.893696\pi$
$359$		−	4459.33i		−	0.655584i	0.944750	−	0.327792i	$-0.106304\pi$
$360$	138.288	+	1699.08i	0.0202456	+	0.248749i
$361$	8615.18			1.25604
$362$	−2458.96	+	4538.47i	−0.357016	+	0.658941i
$363$		−	2374.71i		−	0.343361i
$364$	−4574.25	+	4220.03i	−0.658670	+	0.607663i
$365$		−	3762.21i		−	0.539515i
$366$	−46.3106	−	25.0912i	−0.00661391	−	0.00358344i
$367$	−4612.98			−0.656118			−0.328059	−	0.944657i	$-0.606395\pi$
$367$	−4612.98			−0.656118			−0.328059	+	0.944657i	$0.606395\pi$
$368$	4484.01	−	10168.5i	0.635177	−	1.44041i
$369$			2788.85i			0.393447i
$370$	7549.63	+	4090.42i	1.06077	+	0.574732i
$371$	−5551.06			−0.776810
$372$	−366.566	−	562.268i	−0.0510903	−	0.0783663i
$373$		−	11536.7i		−	1.60147i	−0.599018	−	0.800735i	$-0.704442\pi$
$373$		−	11536.7i		−	1.60147i	0.599018	−	0.800735i	$-0.295558\pi$
$374$	−852.324	−	461.792i	−0.117841	−	0.0638468i
$375$		−	4518.48i		−	0.622222i
$376$	3683.04	−	299.761i	0.505154	−	0.0411143i
$377$	987.238	+	5842.24i	0.134868	+	0.798118i
$378$	1114.42	+	603.797i	0.151639	+	0.0821587i
$379$	3947.90			0.535066			0.267533	−	0.963549i	$-0.413792\pi$
$379$	3947.90			0.535066			0.267533	+	0.963549i	$0.413792\pi$
$380$	4549.48	+	6978.36i	0.614167	+	0.942058i
$381$			6064.58i			0.815479i
$382$	−5124.67	−	2776.57i	−0.686390	−	0.371889i
$383$		−	1328.83i		−	0.177285i	−0.996064	−	0.0886425i	$-0.971747\pi$
$383$		−	1328.83i		−	0.177285i	0.996064	−	0.0886425i	$-0.0282529\pi$
$384$	−1424.66	+	4104.23i	−0.189328	+	0.545425i
$385$			3226.79i			0.427149i
$386$	3741.56	−	6905.75i	0.493369	−	0.910605i
$387$		−	5007.33i		−	0.657718i
$388$	2799.87	−	1825.35i	0.366344	−	0.238835i
$389$			8622.02i			1.12379i	0.827209	+	0.561894i	$0.189927\pi$
$389$			8622.02i			1.12379i	−0.827209	+	0.561894i	$0.810073\pi$
$390$	−1076.09	+	3150.60i	−0.139717	+	0.409068i
$391$	2562.41			0.331423
$392$	−123.967	−	1523.13i	−0.0159727	−	0.196250i
$393$	3378.39			0.433632
$394$	6914.43	+	3746.27i	0.884122	+	0.479021i
$395$	−4579.90			−0.583391
$396$	913.265	+	1400.84i	0.115892	+	0.177764i
$397$	−12450.9			−1.57403			−0.787017	−	0.616931i	$-0.788376\pi$
$397$	−12450.9			−1.57403			−0.787017	+	0.616931i	$0.788376\pi$
$398$	−8854.67	−	4797.49i	−1.11519	−	0.604212i
$399$	6193.80			0.777138
$400$	1418.40	−	3216.56i	0.177300	−	0.402070i
$401$			2689.39i			0.334917i	0.985879	+	0.167458i	$0.0535560\pi$
$401$			2689.39i			0.334917i	−0.985879	+	0.167458i	$0.946444\pi$
$402$	−1020.63	+	1883.76i	−0.126627	+	0.233715i
$403$	−218.418	−	1292.54i	−0.0269979	−	0.159767i
$404$	9969.70	−	6499.67i	1.22775	−	0.800422i
$405$	678.041			0.0831905
$406$	5217.51	+	2826.87i	0.637785	+	0.345554i
$407$	8423.03			1.02583
$408$	−998.406	+	81.2599i	−0.121148	+	0.00986020i
$409$			9558.37i			1.15558i	0.816186	+	0.577789i	$0.196084\pi$
$409$			9558.37i			1.15558i	−0.816186	+	0.577789i	$0.803916\pi$
$410$	−3495.02	+	6450.71i	−0.420992	+	0.777019i
$411$	−2648.02			−0.317803
$412$	−962.405	−	1476.21i	−0.115083	−	0.176524i
$413$			13429.1i			1.60001i
$414$	−3886.50	−	2105.72i	−0.461380	−	0.249977i
$415$	−8028.26			−0.949619
$416$	−5748.97	+	6240.24i	−0.677564	+	0.735464i
$417$	−6835.63			−0.802739
$418$	7184.95	+	3892.83i	0.840736	+	0.455514i
$419$		−	2835.98i		−	0.330660i	−0.986238	−	0.165330i	$-0.947131\pi$
$419$		−	2835.98i		−	0.330660i	0.986238	−	0.165330i	$-0.0528689\pi$
$420$	1821.01	+	2793.21i	0.211562	+	0.324511i
$421$	6691.51			0.774642			0.387321	−	0.921945i	$-0.373400\pi$
$421$	6691.51			0.774642			0.387321	+	0.921945i	$0.373400\pi$
$422$	4809.16	−	8876.21i	0.554754	−	1.02390i
$423$		−	1469.76i		−	0.168942i
$424$	−7543.01	+	613.923i	−0.863965	+	0.0703178i
$425$	810.552			0.0925119
$426$	−6139.24	−	3326.26i	−0.698233	−	0.378305i
$427$	−103.024			−0.0116761
$428$	−5840.23	+	3807.49i	−0.659575	+	0.430005i
$429$	544.167	+	3220.25i	0.0612416	+	0.362413i
$430$	6275.24	−	11582.1i	0.703765	−	1.29893i
$431$			4507.64i			0.503771i	0.967757	+	0.251886i	$0.0810507\pi$
$431$			4507.64i			0.503771i	−0.967757	+	0.251886i	$0.918949\pi$
$432$	1581.10	+	697.215i	0.176090	+	0.0776499i
$433$	15177.6			1.68450			0.842248	−	0.539090i	$-0.181232\pi$
$433$	15177.6			1.68450			0.842248	+	0.539090i	$0.181232\pi$
$434$	−1154.33	−	625.420i	−0.127672	−	0.0691730i
$435$	3174.46			0.349894
$436$	666.650	+	1022.56i	0.0732265	+	0.112321i
$437$	−21600.7			−2.36453
$438$	−3353.10	−	1816.72i	−0.365793	−	0.198188i
$439$	−6916.43			−0.751943			−0.375972	−	0.926631i	$-0.622691\pi$
$439$	−6916.43			−0.751943			−0.375972	+	0.926631i	$0.622691\pi$
$440$	356.869	+	4384.69i	0.0386660	+	0.475073i
$441$	−607.826			−0.0656328
$442$	−1851.33	−	632.323i	−0.199228	−	0.0680464i
$443$		−	935.734i		−	0.100357i	−0.998740	−	0.0501784i	$-0.984021\pi$
$443$		−	935.734i		−	0.100357i	0.998740	−	0.0501784i	$-0.0159790\pi$
$444$	7291.23	−	4753.46i	0.779339	−	0.508084i
$445$		−	5492.90i		−	0.585142i
$446$	−1895.73	+	3498.93i	−0.201268	+	0.371478i
$447$			5196.68i			0.549876i
$448$	1374.15	+	8385.87i	0.144916	+	0.884365i
$449$		−	3019.21i		−	0.317339i	−0.987332	−	0.158669i	$-0.949280\pi$
$449$		−	3019.21i		−	0.317339i	0.987332	−	0.158669i	$-0.0507204\pi$
$450$	−1229.40	−	666.092i	−0.128787	−	0.0697775i
$451$			7196.97i			0.751424i
$452$	4601.61	+	7058.32i	0.478853	+	0.734503i
$453$	7205.96			0.747385
$454$	174.490	+	94.5395i	0.0180380	+	0.00977304i
$455$	1085.04	+	6421.03i	0.111797	+	0.661588i
$456$	8416.40	−	685.008i	0.864329	−	0.0703474i
$457$		−	13732.7i		−	1.40567i	−0.711354	−	0.702834i	$-0.751918\pi$
$457$		−	13732.7i		−	1.40567i	0.711354	−	0.702834i	$-0.248082\pi$
$458$	2083.97	+	1129.10i	0.212615	+	0.115195i
$459$			398.427i			0.0405162i
$460$	−6350.70	−	9741.21i	−0.643702	−	0.987361i
$461$	−9639.21			−0.973846			−0.486923	−	0.873445i	$-0.661881\pi$
$461$	−9639.21			−0.973846			−0.486923	+	0.873445i	$0.661881\pi$
$462$	2875.90	+	1558.17i	0.289608	+	0.156911i
$463$			6341.49i			0.636532i	0.948001	+	0.318266i	$0.103100\pi$
$463$			6341.49i			0.636532i	−0.948001	+	0.318266i	$0.896900\pi$
$464$	7402.41	+	3264.23i	0.740621	+	0.326591i
$465$	−702.323			−0.0700418
$466$	15687.3	+	8499.42i	1.55944	+	0.844910i
$467$		−	5746.60i		−	0.569424i	−0.958613	−	0.284712i	$-0.908102\pi$
$467$		−	5746.60i		−	0.569424i	0.958613	−	0.284712i	$-0.0918980\pi$
$468$	2288.37	+	2480.45i	0.226025	+	0.244998i
$469$			4190.66i			0.412595i
$470$	1841.92	−	3399.61i	0.180769	−	0.333643i
$471$	−9063.28			−0.886654
$472$	1485.20	+	18248.0i	0.144834	+	1.77952i
$473$		−	12922.0i		−	1.25614i
$474$	−2211.57	+	4081.87i	−0.214305	+	0.395541i
$475$	−6832.82			−0.660024
$476$	−1641.33	+	1070.05i	−0.158046	+	0.103037i
$477$			3010.13i			0.288940i
$478$	6888.95	−	12714.8i	0.659191	−	1.21666i
$479$		−	7754.98i		−	0.739737i	−0.929084	−	0.369869i	$-0.879403\pi$
$479$		−	7754.98i		−	0.739737i	0.929084	−	0.369869i	$-0.120597\pi$
$480$	2783.38	+	3594.13i	0.264674	+	0.341768i
$481$	16761.1	−	2832.34i	1.58886	−	0.268490i
$482$	372.700	−	687.887i	0.0352199	−	0.0650050i
$483$	−8646.03			−0.814510
$484$	−3458.42	−	5304.79i	−0.324795	−	0.498196i
$485$		−	3497.28i		−	0.327429i
$486$	327.417	−	604.310i	0.0305595	−	0.0564034i
$487$			10370.4i			0.964947i	0.875911	+	0.482474i	$0.160262\pi$
$487$			10370.4i			0.964947i	−0.875911	+	0.482474i	$0.839738\pi$
$488$	−139.993	+	11.3940i	−0.0129861	+	0.00105693i
$489$			4810.81i			0.444893i
$490$	−1405.92	−	761.733i	−0.129618	−	0.0702277i
$491$		−	18737.7i		−	1.72225i	−0.508397	−	0.861123i	$-0.669762\pi$
$491$		−	18737.7i		−	1.72225i	0.508397	−	0.861123i	$-0.330238\pi$
$492$	4061.55	+	6229.92i	0.372172	+	0.570867i
$493$			1865.36i			0.170409i
$494$	15606.4	+	5330.38i	1.42139	+	0.485476i
$495$	1749.77			0.158881
$496$	−1637.72	−	722.183i	−0.148258	−	0.0653769i
$497$	−13657.5			−1.23264
$498$	−3876.74	+	7155.25i	−0.348837	+	0.643844i
$499$	−9332.52			−0.837236			−0.418618	−	0.908162i	$-0.637485\pi$
$499$	−9332.52			−0.837236			−0.418618	+	0.908162i	$0.637485\pi$
$500$	−6580.49	−	10093.7i	−0.588577	−	0.902805i
$501$	−9645.86			−0.860171
$502$	−5077.13	+	9370.80i	−0.451402	+	0.833146i
$503$	−85.6243			−0.00759006			−0.00379503	−	0.999993i	$-0.501208\pi$
$503$	−85.6243			−0.00759006			−0.00379503	+	0.999993i	$0.501208\pi$
$504$	3368.81	−	274.186i	0.297735	−	0.0242326i
$505$		−	12453.0i		−	1.09733i
$506$	−10029.6	−	5434.07i	−0.881166	−	0.477419i
$507$	2165.69	+	6225.04i	0.189708	+	0.545293i
$508$	8832.15	+	13547.4i	0.771384	+	1.18321i
$509$	1278.96			0.111373			0.0556867	−	0.998448i	$-0.482265\pi$
$509$	1278.96			0.111373			0.0556867	+	0.998448i	$0.482265\pi$
$510$	−499.312	+	921.574i	−0.0433528	+	0.0800156i
$511$	−7459.40			−0.645762
$512$	2794.69	+	11243.1i	0.241229	+	0.970468i
$513$		−	3358.67i		−	0.289062i
$514$	−4565.43	−	2473.57i	−0.391775	−	0.212265i
$515$	−1843.92			−0.157772
$516$	−7292.43	−	11185.7i	−0.622154	−	0.954309i
$517$		−	3792.90i		−	0.322653i
$518$	8110.15	−	14968.8i	0.687914	−	1.26967i
$519$	7312.27			0.618445
$520$	2184.54	+	8605.16i	0.184228	+	0.725695i
$521$	−1766.24			−0.148523			−0.0742616	−	0.997239i	$-0.523660\pi$
$521$	−1766.24			−0.148523			−0.0742616	+	0.997239i	$0.523660\pi$
$522$	1532.91	−	2829.26i	0.128532	−	0.237229i
$523$		−	15522.3i		−	1.29779i	−0.760878	−	0.648894i	$-0.775232\pi$
$523$		−	15522.3i		−	1.29779i	0.760878	−	0.648894i	$-0.224768\pi$
$524$	7546.88	−	4920.13i	0.629174	−	0.410185i
$525$	−2734.95			−0.227358
$526$	7109.86	+	3852.15i	0.589363	+	0.319319i
$527$		−	412.694i		−	0.0341124i
$528$	4080.22	+	1799.25i	0.336304	+	0.148300i
$529$	17985.7			1.47824
$530$	−3772.33	+	6962.54i	−0.309169	+	0.570629i
$531$	7282.11			0.595135
$532$	13836.1	−	9020.35i	1.12758	−	0.735116i
$533$	2420.06	+	14321.4i	0.196669	+	1.16384i
$534$	−4895.59	−	2652.45i	−0.396728	−	0.214949i
$535$			7294.96i			0.589511i
$536$	463.469	+	5694.45i	0.0373486	+	0.458886i
$537$	−9101.23			−0.731373
$538$	−2039.63	+	3764.52i	−0.163447	+	0.301673i
$539$	−1568.57			−0.125349
$540$	1514.65	−	987.466i	0.120704	−	0.0786922i
$541$	−22081.4			−1.75481			−0.877407	−	0.479748i	$-0.840728\pi$
$541$	−22081.4			−1.75481			−0.877407	+	0.479748i	$0.840728\pi$
$542$	−4576.26	+	8446.34i	−0.362670	+	0.669375i
$543$	5474.91			0.432691
$544$	−2111.96	+	1635.55i	−0.166451	+	0.128904i
$545$	1277.27			0.100389
$546$	6246.75	+	2133.57i	0.489626	+	0.167232i
$547$		−	21980.9i		−	1.71817i	−0.511837	−	0.859083i	$-0.671035\pi$
$547$		−	21980.9i		−	1.71817i	0.511837	−	0.859083i	$-0.328965\pi$
$548$	−5915.31	+	3856.44i	−0.461112	+	0.300618i
$549$			55.8661i			0.00434300i
$550$	−3172.61	−	1718.93i	−0.245965	−	0.133264i
$551$		−	15724.7i		−	1.21578i
$552$	−11748.6	+	956.214i	−0.905894	+	0.0737304i
$553$			9080.64i			0.698279i
$554$	−2351.04	+	4339.28i	−0.180300	+	0.332777i
$555$		−	9107.39i		−	0.696553i
$556$	−15269.9	+	9955.08i	−1.16473	+	0.759333i
$557$	23410.1			1.78082			0.890412	−	0.455156i	$-0.150416\pi$
$557$	23410.1			1.78082			0.890412	+	0.455156i	$0.150416\pi$
$558$	−339.142	+	625.950i	−0.0257294	+	0.0474885i
$559$	−4345.18	−	25713.7i	−0.328769	−	1.94557i
$560$	8135.77	+	3587.62i	0.613927	+	0.270722i
$561$			1028.19i			0.0773799i
$562$	4681.76	−	8641.06i	0.351402	−	0.648578i
$563$		−	11642.7i		−	0.871551i	−0.900056	−	0.435775i	$-0.856474\pi$
$563$		−	11642.7i		−	0.871551i	0.900056	−	0.435775i	$-0.143526\pi$
$564$	−2140.49	−	3283.25i	−0.159806	−	0.245124i
$565$	8816.46			0.656480
$566$	−2298.78	+	4242.83i	−0.170715	+	0.315087i
$567$		−	1344.37i		−	0.0995733i
$568$	−18558.4	+	1510.47i	−1.37094	+	0.111580i
$569$	13777.7			1.01510			0.507548	−	0.861624i	$-0.330552\pi$
$569$	13777.7			1.01510			0.507548	+	0.861624i	$0.330552\pi$
$570$	4209.12	−	7768.72i	0.309299	−	0.570870i
$571$			9112.18i			0.667834i	0.942603	+	0.333917i	$0.108370\pi$
$571$			9112.18i			0.667834i	−0.942603	+	0.333917i	$0.891630\pi$
$572$	5905.41	+	6401.10i	0.431674	+	0.467908i
$573$			6182.08i			0.450715i
$574$	12789.9	+	6929.63i	0.930038	+	0.503898i
$575$	9538.06			0.691764
$576$	4547.35	−	745.151i	0.328946	−	0.0539027i
$577$			3992.15i			0.288033i	0.989575	+	0.144017i	$0.0460019\pi$
$577$			3992.15i			0.288033i	−0.989575	+	0.144017i	$0.953998\pi$
$578$	11676.5	+	6326.35i	0.840272	+	0.455262i
$579$	−8330.65			−0.597945
$580$	7091.32	−	4623.13i	0.507674	−	0.330974i
$581$			15917.8i			1.13663i
$582$	−3116.98	−	1688.79i	−0.221998	−	0.120279i
$583$			7768.02i			0.551833i
$584$	−10136.1	+	824.978i	−0.718214	+	0.0584552i
$585$	3481.89	−	588.380i	0.246083	−	0.0415838i
$586$	−8914.29	−	4829.79i	−0.628406	−	0.340473i
$587$	−4964.98			−0.349109			−0.174554	−	0.984648i	$-0.555848\pi$
$587$	−4964.98			−0.349109			−0.174554	+	0.984648i	$0.555848\pi$
$588$	−1357.80	+	885.207i	−0.0952292	+	0.0620839i
$589$			3478.95i			0.243374i
$590$	16843.8	+	9126.01i	1.17533	+	0.636800i
$591$		−	8341.13i		−	0.580555i
$592$	9364.92	−	21237.2i	0.650162	−	1.47440i
$593$		−	6012.41i		−	0.416357i	−0.978091	−	0.208179i	$-0.933246\pi$
$593$		−	6012.41i		−	0.416357i	0.978091	−	0.208179i	$-0.0667536\pi$
$594$	844.940	−	1559.49i	0.0583641	−	0.107722i
$595$			2050.16i			0.141258i
$596$	7568.19	+	11608.7i	0.520143	+	0.797836i
$597$			10681.7i			0.732283i
$598$	−21785.3	−	7440.77i	−1.48974	−	0.508822i
$599$	−5330.42			−0.363598			−0.181799	−	0.983336i	$-0.558192\pi$
$599$	−5330.42			−0.363598			−0.181799	+	0.983336i	$0.558192\pi$
$600$	−3716.37	+	302.474i	−0.252867	+	0.0205808i
$601$	−4976.57			−0.337768			−0.168884	−	0.985636i	$-0.554016\pi$
$601$	−4976.57			−0.337768			−0.168884	+	0.985636i	$0.554016\pi$
$602$	−22964.1	−	12442.0i	−1.55473	−	0.842358i
$603$	2272.44			0.153468
$604$	16097.1	−	10494.4i	1.08441	−	0.706972i
$605$	−6626.15			−0.445275
$606$	−11098.9	−	6013.40i	−0.743994	−	0.403099i
$607$	−3168.09			−0.211843			−0.105922	−	0.994374i	$-0.533779\pi$
$607$	−3168.09			−0.211843			−0.105922	+	0.994374i	$0.533779\pi$
$608$	17803.5	−	13787.4i	1.18754	−	0.919662i
$609$		−	6294.07i		−	0.418799i
$610$	−70.0119	+	129.220i	−0.00464705	+	0.00857700i
$611$	−1275.41	−	7547.55i	−0.0844475	−	0.499740i
$612$	580.249	+	890.031i	0.0383254	+	0.0587866i
$613$	−3807.27			−0.250855			−0.125428	−	0.992103i	$-0.540030\pi$
$613$	−3807.27			−0.250855			−0.125428	+	0.992103i	$0.540030\pi$
$614$	11444.3	+	6200.59i	0.752209	+	0.407549i
$615$	7781.72			0.510226
$616$	8693.62	−	707.570i	0.568629	−	0.0462806i
$617$		−	8976.23i		−	0.585688i	−0.956160	−	0.292844i	$-0.905398\pi$
$617$		−	8976.23i		−	0.585688i	0.956160	−	0.292844i	$-0.0946017\pi$
$618$	−890.404	+	1643.41i	−0.0579568	+	0.106970i
$619$	1253.41			0.0813872			0.0406936	−	0.999172i	$-0.487043\pi$
$619$	1253.41			0.0813872			0.0406936	+	0.999172i	$0.487043\pi$
$620$	−1568.89	+	1022.83i	−0.101626	+	0.0662544i
$621$			4688.43i			0.302963i
$622$	−23814.0	−	12902.5i	−1.53513	−	0.831741i
$623$	−10890.9			−0.700375
$624$	8724.30	+	2208.33i	0.559698	+	0.141673i
$625$	−5741.83			−0.367477
$626$	1593.63	+	863.432i	0.101748	+	0.0551273i
$627$		−	8667.46i		−	0.552065i
$628$	−20246.2	+	13199.3i	−1.28648	+	0.838710i
$629$	5351.63			0.339242
$630$	1684.77	−	3109.56i	0.106544	−	0.196647i
$631$		−	22115.1i		−	1.39523i	−0.716474	−	0.697614i	$-0.754245\pi$
$631$		−	22115.1i		−	1.39523i	0.716474	−	0.697614i	$-0.245755\pi$
$632$	1004.28	+	12339.2i	0.0632090	+	0.776623i
$633$	−10707.7			−0.672342
$634$	9559.69	+	5179.48i	0.598839	+	0.324453i
$635$	16921.9			1.05752
$636$	4383.81	+	6724.24i	0.273317	+	0.419235i
$637$	−3121.32	+	527.449i	−0.194146	+	0.0328074i
$638$	3955.85	−	7301.26i	0.245476	−	0.453072i
$639$			7405.98i			0.458492i
$640$	11452.0	+	3975.23i	0.707313	+	0.245523i
$641$	−26606.7			−1.63947			−0.819736	−	0.572741i	$-0.805880\pi$
$641$	−26606.7			−1.63947			−0.819736	+	0.572741i	$0.805880\pi$
$642$	6501.69	+	3522.64i	0.399690	+	0.216554i
$643$	15622.5			0.958152			0.479076	−	0.877773i	$-0.340972\pi$
$643$	15622.5			0.958152			0.479076	+	0.877773i	$0.340972\pi$
$644$	−19314.1	+	12591.7i	−1.18180	+	0.770467i
$645$	−13971.9			−0.852937
$646$	4565.01	+	2473.34i	0.278031	+	0.150638i
$647$	−8706.60			−0.529045			−0.264522	−	0.964380i	$-0.585214\pi$
$647$	−8706.60			−0.529045			−0.264522	+	0.964380i	$0.585214\pi$
$648$	−148.681	−	1826.78i	−0.00901349	−	0.110745i
$649$	18792.4			1.13662
$650$	−6891.23	−	2353.70i	−0.415840	−	0.142030i
$651$			1392.51i			0.0838352i
$652$	7006.23	+	10746.7i	0.420836	+	0.645512i
$653$		−	3649.42i		−	0.218703i	−0.994003	−	0.109351i	$-0.965123\pi$
$653$		−	3649.42i		−	0.218703i	0.994003	−	0.109351i	$-0.0348773\pi$
$654$	616.775	−	1138.37i	0.0368774	−	0.0680642i
$655$		−	9426.71i		−	0.562339i
$656$	18145.9	+	8001.76i	1.08000	+	0.476245i
$657$			4044.96i			0.240196i
$658$	−6740.47	−	3652.01i	−0.399348	−	0.216368i
$659$			17881.8i			1.05702i	0.848927	+	0.528510i	$0.177249\pi$
$659$			17881.8i			1.05702i	−0.848927	+	0.528510i	$0.822751\pi$
$660$	3908.75	−	2548.28i	0.230527	−	0.150290i
$661$	−4411.35			−0.259579			−0.129789	−	0.991542i	$-0.541430\pi$
$661$	−4411.35			−0.259579			−0.129789	+	0.991542i	$0.541430\pi$
$662$	−22686.2	−	12291.5i	−1.33191	−	0.721634i
$663$	345.740	+	2046.01i	0.0202525	+	0.119850i
$664$	1760.44	+	21629.7i	0.102889	+	1.26415i
$665$		−	17282.5i		−	1.00780i
$666$	−8117.03	−	4397.83i	−0.472265	−	0.255875i
$667$			21950.3i			1.27424i
$668$	−21547.6	+	14047.8i	−1.24805	+	0.813659i
$669$	4220.88			0.243929
$670$	5256.24	+	2847.85i	0.303084	+	0.164212i
$671$			144.169i			0.00829447i
$672$	7126.15	−	5518.66i	0.409073	−	0.316796i
$673$	−18882.8			−1.08154			−0.540771	−	0.841170i	$-0.681868\pi$
$673$	−18882.8			−1.08154			−0.540771	+	0.841170i	$0.681868\pi$
$674$	−12811.6	−	6941.35i	−0.732171	−	0.396693i
$675$			1483.07i			0.0845677i
$676$	13903.7	+	10751.9i	0.791061	+	0.611737i
$677$			8607.17i			0.488627i	0.969696	+	0.244313i	$0.0785625\pi$
$677$			8607.17i			0.488627i	−0.969696	+	0.244313i	$0.921437\pi$
$678$	4257.35	−	7857.73i	0.241154	−	0.445095i
$679$	−6934.12			−0.391910
$680$	226.739	+	2785.84i	0.0127868	+	0.157106i
$681$		−	210.494i		−	0.0118446i
$682$	−875.198	+	1615.34i	−0.0491394	+	0.0906959i
$683$	8615.26			0.482656			0.241328	−	0.970444i	$-0.422417\pi$
$683$	8615.26			0.482656			0.241328	+	0.970444i	$0.422417\pi$
$684$	−4891.40	−	7502.82i	−0.273432	−	0.419412i
$685$			7388.74i			0.412130i
$686$	−9180.76	+	16944.8i	−0.510967	+	0.943085i
$687$		−	2513.97i		−	0.139613i
$688$	−32580.6	−	14367.0i	−1.80541	−	0.796130i
$689$	2612.09	+	15457.7i	0.144430	+	0.854704i
$690$	−5875.58	+	10844.5i	−0.324173	+	0.598323i
$691$	−3884.91			−0.213877			−0.106938	−	0.994266i	$-0.534105\pi$
$691$	−3884.91			−0.213877			−0.106938	+	0.994266i	$0.534105\pi$
$692$	16334.6	−	10649.2i	0.897325	−	0.585004i
$693$		−	3469.30i		−	0.190170i
$694$	15635.7	−	28858.7i	0.855223	−	1.57847i
$695$			19073.4i			1.04100i
$696$	−696.097	−	8552.64i	−0.0379102	−	0.465786i
$697$			4572.64i			0.248495i
$698$	−5017.53	−	2718.52i	−0.272087	−	0.147418i
$699$		−	18924.1i		−	1.02400i
$700$	−6109.52	+	3983.05i	−0.329883	+	0.215065i
$701$			5168.26i			0.278463i	0.990260	+	0.139231i	$0.0444632\pi$
$701$			5168.26i			0.278463i	−0.990260	+	0.139231i	$0.955537\pi$
$702$	1156.96	−	3387.38i	0.0622032	−	0.182120i
$703$	−45113.4			−2.42032
$704$	11735.0	−	1922.95i	0.628238	−	0.102946i
$705$	−4101.07			−0.219085
$706$	9175.62	−	16935.3i	0.489135	−	0.902789i
$707$	−24690.9			−1.31343
$708$	16267.2	−	10605.3i	0.863504	−	0.562954i
$709$	−12559.3			−0.665269			−0.332635	−	0.943056i	$-0.607938\pi$
$709$	−12559.3			−0.665269			−0.332635	+	0.943056i	$0.607938\pi$
$710$	−9281.25	+	17130.3i	−0.490590	+	0.905476i
$711$	4924.10			0.259730
$712$	−14799.0	+	1204.48i	−0.778953	+	0.0633987i
$713$		−	4856.32i		−	0.255078i
$714$	1827.22	+	989.995i	0.0957732	+	0.0518902i
$715$	8985.44	−	1518.39i	0.469981	−	0.0794187i
$716$	−20330.9	+	13254.6i	−1.06118	+	0.691826i
$717$	−15338.4			−0.798915
$718$	−6008.48	+	11089.8i	−0.312304	+	0.576416i
$719$	−595.772			−0.0309020			−0.0154510	−	0.999881i	$-0.504918\pi$
$719$	−595.772			−0.0309020			−0.0154510	+	0.999881i	$0.504918\pi$
$720$	1945.43	−	4411.73i	0.100697	−	0.228355i
$721$			3655.97i			0.188843i
$722$	−21424.8	−	11608.0i	−1.10436	−	0.598347i
$723$	−829.823			−0.0426852
$724$	12230.2	−	7973.39i	0.627807	−	0.409294i
$725$			6943.43i			0.355686i
$726$	−3199.68	+	5905.61i	−0.163569	+	0.301898i
$727$	21554.1			1.09958			0.549792	−	0.835302i	$-0.314707\pi$
$727$	21554.1			1.09958			0.549792	+	0.835302i	$0.314707\pi$
$728$	17061.6	−	4331.33i	0.868607	−	0.220508i
$729$	−729.000			−0.0370370
$730$	−5069.18	+	9356.13i	−0.257012	+	0.474364i
$731$		−	8210.10i		−	0.415406i
$732$	81.3606	+	124.797i	0.00410816	+	0.00630142i
$733$	−264.927			−0.0133496			−0.00667482	−	0.999978i	$-0.502125\pi$
$733$	−264.927			−0.0133496			−0.00667482	+	0.999978i	$0.502125\pi$
$734$	11471.9	+	6215.50i	0.576887	+	0.312559i
$735$			1696.01i			0.0851134i
$736$	−24852.2	+	19246.1i	−1.24465	+	0.963888i
$737$	5864.32			0.293100
$738$	3757.69	−	6935.51i	0.187429	−	0.345935i
$739$	405.069			0.0201633			0.0100817	−	0.999949i	$-0.496791\pi$
$739$	405.069			0.0201633			0.0100817	+	0.999949i	$0.496791\pi$
$740$	−13263.5	−	20344.7i	−0.658889	−	1.01066i
$741$	−2914.53	−	17247.5i	−0.144491	−	0.855065i
$742$	13804.8	+	7479.47i	0.683004	+	0.370054i
$743$			16691.2i			0.824149i	0.911150	+	0.412074i	$0.135196\pi$
$743$			16691.2i			0.824149i	−0.911150	+	0.412074i	$0.864804\pi$
$744$	154.005	+	1892.20i	0.00758886	+	0.0932411i
$745$	14500.3			0.713086
$746$	−15544.5	+	28690.3i	−0.762902	+	1.40808i
$747$	8631.63			0.422777
$748$	1497.40	+	2296.83i	0.0731958	+	0.112273i
$749$	14463.8			0.705604
$750$	−6088.17	+	11236.9i	−0.296412	+	0.547083i
$751$	−30944.9			−1.50359			−0.751796	−	0.659396i	$-0.770812\pi$
$751$	−30944.9			−1.50359			−0.751796	+	0.659396i	$0.770812\pi$
$752$	−9563.13	−	4217.04i	−0.463739	−	0.204494i
$753$	11304.3			0.547082
$754$	5416.67	−	15859.1i	0.261623	−	0.765986i
$755$		−	20106.7i		−	0.969218i
$756$	−1957.87	−	3003.13i	−0.0941891	−	0.144475i
$757$		−	22212.4i		−	1.06648i	−0.845965	−	0.533238i	$-0.820975\pi$
$757$		−	22212.4i		−	1.06648i	0.845965	−	0.533238i	$-0.179025\pi$
$758$	−9817.92	−	5319.38i	−0.470452	−	0.254893i
$759$			12099.1i			0.578614i
$760$	−1911.37	−	23484.2i	−0.0912273	−	1.12087i
$761$		−	20301.1i		−	0.967034i	−0.875335	−	0.483517i	$-0.839359\pi$
$761$		−	20301.1i		−	0.967034i	0.875335	−	0.483517i	$-0.160641\pi$
$762$	8171.38	−	15081.8i	0.388475	−	0.717003i
$763$		−	2532.46i		−	0.120159i
$764$	9003.27	+	13809.9i	0.426344	+	0.653960i
$765$	1111.73			0.0525419
$766$	−1790.46	+	3304.63i	−0.0844544	+	0.155876i
$767$	37395.2	−	6319.15i	1.76045	−	0.297485i
$768$	9072.97	−	8287.11i	0.426293	−	0.389369i
$769$			25514.7i			1.19647i	0.801322	+	0.598233i	$0.204130\pi$
$769$			25514.7i			1.19647i	−0.801322	+	0.598233i	$0.795870\pi$
$770$	4347.76	−	8024.60i	0.203484	−	0.375567i
$771$			5507.44i			0.257258i
$772$	−18609.6	+	12132.4i	−0.867581	+	0.565612i
$773$	1757.92			0.0817955			0.0408977	−	0.999163i	$-0.486978\pi$
$773$	1757.92			0.0817955			0.0408977	+	0.999163i	$0.486978\pi$
$774$	−6746.86	+	12452.6i	−0.313321	+

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:	\( N \)	\(=\)	\( 312 = 2^{3} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	312.m (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-2.48687 - 1.34740i) q^{2} +3.00000i q^{3} +(4.36905 + 6.70160i) q^{4} +8.37088 q^{5} +(4.04219 - 7.46061i) q^{6} -16.5971i q^{7} +(-1.83557 - 22.5528i) q^{8} -9.00000 q^{9} +(-20.8173 - 11.2789i) q^{10} -23.2256 q^{11} +(-20.1048 + 13.1072i) q^{12} +(-46.2169 + 7.80987i) q^{13} +(-22.3629 + 41.2748i) q^{14} +25.1126i q^{15} +(-25.8228 + 58.5592i) q^{16} -14.7565 q^{17} +(22.3818 + 12.1266i) q^{18} +124.395 q^{19} +(36.5728 + 56.0983i) q^{20} +49.7913 q^{21} +(57.7591 + 31.2941i) q^{22} -173.645 q^{23} +(67.6585 - 5.50670i) q^{24} -54.9283 q^{25} +(125.459 + 42.8503i) q^{26} -27.0000i q^{27} +(111.227 - 72.5136i) q^{28} -126.409i q^{29} +(33.8367 - 62.4519i) q^{30} +27.9669i q^{31} +(143.120 - 110.836i) q^{32} -69.6768i q^{33} +(36.6976 + 19.8829i) q^{34} -138.932i q^{35} +(-39.3215 - 60.3144i) q^{36} -362.661 q^{37} +(-309.355 - 167.610i) q^{38} +(-23.4296 - 138.651i) q^{39} +(-15.3653 - 188.787i) q^{40} -309.872i q^{41} +(-123.825 - 67.0886i) q^{42} +556.370i q^{43} +(-101.474 - 155.649i) q^{44} -75.3379 q^{45} +(431.834 + 233.969i) q^{46} +163.307i q^{47} +(-175.678 - 77.4683i) q^{48} +67.5362 q^{49} +(136.600 + 74.0102i) q^{50} -44.2696i q^{51} +(-254.263 - 275.606i) q^{52} -334.459i q^{53} +(-36.3797 + 67.1455i) q^{54} -194.419 q^{55} +(-374.312 + 30.4651i) q^{56} +373.186i q^{57} +(-170.323 + 314.363i) q^{58} -809.123 q^{59} +(-168.295 + 109.718i) q^{60} -6.20734i q^{61} +(37.6825 - 69.5500i) q^{62} +149.374i q^{63} +(-505.261 + 82.7945i) q^{64} +(-386.877 + 65.3755i) q^{65} +(-93.8822 + 173.277i) q^{66} -252.494 q^{67} +(-64.4721 - 98.8923i) q^{68} -520.936i q^{69} +(-187.197 + 345.507i) q^{70} -822.887i q^{71} +(16.5201 + 202.976i) q^{72} -449.440i q^{73} +(901.892 + 488.648i) q^{74} -164.785i q^{75} +(543.489 + 833.647i) q^{76} +385.478i q^{77} +(-128.551 + 376.376i) q^{78} -547.122 q^{79} +(-216.159 + 490.192i) q^{80} +81.0000 q^{81} +(-417.521 + 770.613i) q^{82} -959.070 q^{83} +(217.541 + 333.681i) q^{84} -123.525 q^{85} +(749.651 - 1383.62i) q^{86} +379.227 q^{87} +(42.6322 + 523.803i) q^{88} -656.191i q^{89} +(187.356 + 101.510i) q^{90} +(129.621 + 767.067i) q^{91} +(-758.666 - 1163.70i) q^{92} -83.9007 q^{93} +(220.039 - 406.123i) q^{94} +1041.30 q^{95} +(332.507 + 429.361i) q^{96} -417.791i q^{97} +(-167.954 - 90.9980i) q^{98} +209.030 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 84 q - 756 q^{9} - 36 q^{10} + 12 q^{12} - 208 q^{14} - 148 q^{16} - 104 q^{17} + 620 q^{22} + 2188 q^{25} + 444 q^{26} - 204 q^{30} - 40 q^{38} - 1924 q^{40} + 192 q^{42} + 624 q^{48} - 3396 q^{49} - 1292 q^{52}+ \cdots + 2480 q^{95}+O(q^{100}) \) 84 * q - 756 * q^9 - 36 * q^10 + 12 * q^12 - 208 * q^14 - 148 * q^16 - 104 * q^17 + 620 * q^22 + 2188 * q^25 + 444 * q^26 - 204 * q^30 - 40 * q^38 - 1924 * q^40 + 192 * q^42 + 624 * q^48 - 3396 * q^49 - 1292 * q^52 - 1616 * q^55 + 608 * q^56 - 2120 * q^62 - 2856 * q^64 + 696 * q^65 - 396 * q^66 - 2536 * q^68 - 3936 * q^74 - 156 * q^78 + 3160 * q^79 + 6804 * q^81 + 4276 * q^82 - 2088 * q^87 + 1780 * q^88 + 324 * q^90 + 4792 * q^92 - 860 * q^94 + 2480 * q^95

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