LMFDB - Embedded newform 312.4.m.a.181.74

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			181.74
Character	$\chi$	$=$	312.181
Dual form			312.4.m.a.181.73

$q$-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} +(104.413 + 81.6946i) 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} +(149.586 + 343.849i) 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} +(-245.084 + 313.238i) 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	$-0.622137\pi$
$5$	−8.37088			−0.748714			−0.374357	+	0.927285i	$0.622137\pi$
$6$	−4.04219	+	7.46061i	−0.275036	+	0.507630i
$7$			16.5971i			0.896159i	0.893994	+	0.448080i	$0.147892\pi$
$7$			16.5971i			0.896159i	−0.893994	+	0.448080i	$0.852108\pi$
$8$	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	$-0.396885\pi$
$11$	23.2256			0.636616			0.318308	+	0.947987i	$0.396885\pi$
$12$	−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	$-0.770431\pi$
$19$	−124.395			−1.50201			−0.751006	+	0.660295i	$0.770431\pi$
$20$	−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
$26$	104.413	+	81.6946i	0.787577	+	0.616217i
$27$		−	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	$-0.974183\pi$
$31$		−	27.9669i		−	0.162032i	0.996713	−	0.0810162i	$-0.0258165\pi$
$32$	−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	$-0.201794\pi$
$37$	362.661			1.61138			0.805692	+	0.592335i	$0.201794\pi$
$38$	−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	$0.200939\pi$
$41$			309.872i			1.18034i	−0.807279	+	0.590170i	$0.799061\pi$
$42$	−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	$-0.918447\pi$
$47$		−	163.307i		−	0.506825i	0.967358	−	0.253412i	$-0.0815530\pi$
$48$	−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
$52$	149.586	+	343.849i	0.398919	+	0.916986i
$53$		−	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	$-0.148807\pi$
$59$	809.123			1.78540			0.892702	+	0.450647i	$0.148807\pi$
$60$	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	$-0.426061\pi$
$67$	252.494			0.460403			0.230202	+	0.973143i	$0.426061\pi$
$68$	−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	$0.241396\pi$
$71$			822.887i			1.37548i	−0.725959	+	0.687738i	$0.758604\pi$
$72$	−16.5201	−	202.976i	−0.0270405	−	0.332235i
$73$			449.440i			0.720589i	0.932839	+	0.360294i	$0.117324\pi$
$73$			449.440i			0.720589i	−0.932839	+	0.360294i	$0.882676\pi$
$74$	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
$78$	−245.084	+	313.238i	−0.355773	+	0.454708i
$79$	−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	$-0.281344\pi$
$83$	959.070			1.26833			0.634166	+	0.773197i	$0.281344\pi$
$84$	−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	$0.127789\pi$
$89$			656.191i			0.781529i	−0.920491	+	0.390765i	$0.872211\pi$
$90$	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	$0.0701689\pi$
$97$			417.791i			0.437322i	−0.975801	+	0.218661i	$0.929831\pi$
$98$	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	$0.533600\pi$
$104$	−91.3005	+	1056.66i	−0.0860842	+	0.996288i
$105$	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	$-0.521356\pi$
$109$	−152.585			−0.134082			−0.0670411	+	0.997750i	$0.521356\pi$
$110$	−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
$130$	−874.025	−	683.856i	−0.589670	−	0.461370i
$131$		−	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	$-0.911248\pi$
$137$		−	882.672i		−	0.550451i	0.961380	−	0.275225i	$-0.0887525\pi$
$138$	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	$-0.657989\pi$
$149$	−1732.23			−0.952413			−0.476207	+	0.879333i	$0.657989\pi$
$150$	222.031	−	409.799i	0.120858	−	0.223066i
$151$			2401.99i			1.29451i	0.762274	+	0.647255i	$0.224083\pi$
$151$			2401.99i			1.29451i	−0.762274	+	0.647255i	$0.775917\pi$
$152$	−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
$156$	−1031.55	+	448.757i	−0.529422	+	0.230316i
$157$			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	$-0.625898\pi$
$163$	−1603.60			−0.770577			−0.385288	+	0.922796i	$0.625898\pi$
$164$	−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	$-0.732483\pi$
$167$		−	3215.29i		−	1.48986i	0.667143	−	0.744930i	$-0.267517\pi$
$168$	−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	$-0.122262\pi$
$182$	−1355.89	+	1732.95i	−0.552228	+	0.705794i
$183$	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	$-0.826738\pi$
$193$		−	2776.88i		−	1.03567i	0.855480	−	0.517835i	$-0.173262\pi$
$194$	−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	$-0.332312\pi$
$197$	2780.38			1.00555			0.502776	+	0.864417i	$0.332312\pi$
$198$	−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
$208$	−1650.79	+	2504.76i	−0.550297	+	0.834969i
$209$	−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	$0.0677531\pi$
$223$			1406.96i			0.422498i	−0.977432	+	0.211249i	$0.932247\pi$
$224$	−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	$-0.496735\pi$
$227$	70.1646			0.0205154			0.0102577	+	0.999947i	$0.496735\pi$
$228$	2500.94	−	1630.47i	0.726442	−	0.473598i
$229$	837.989			0.241816			0.120908	−	0.992664i	$-0.461419\pi$
$229$	837.989			0.241816			0.120908	+	0.992664i	$0.461419\pi$
$230$	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	$0.847083\pi$
$234$	−939.713	−	735.252i	−0.262526	−	0.205406i
$235$			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	$-0.756783\pi$
$239$		−	5112.79i		−	1.38376i	0.722012	−	0.691880i	$-0.243217\pi$
$240$	1470.58	+	648.478i	0.395522	+	0.174413i
$241$		−	276.608i		−	0.0739330i	−0.999317	−	0.0369665i	$-0.988231\pi$
$241$		−	276.608i		−	0.0739330i	0.999317	−	0.0369665i	$-0.0117695\pi$
$242$	−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
$260$	−1252.16	−	2878.32i	−0.298676	−	0.686561i
$261$			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	$0.124301\pi$
$271$			3396.37i			0.761310i	−0.924717	+	0.380655i	$0.875699\pi$
$272$	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	$-0.879759\pi$
$281$		−	3474.67i		−	0.737657i	0.929498	−	0.368828i	$-0.120241\pi$
$282$	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
$286$	2425.04	+	1897.41i	0.501384	+	0.392294i
$287$	−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	$-0.616322\pi$
$293$	−3584.54			−0.714714			−0.357357	+	0.933968i	$0.616322\pi$
$294$	−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	$-0.359303\pi$
$307$	4601.91			0.855520			0.427760	+	0.903892i	$0.359303\pi$
$308$	−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	$0.162180\pi$
$312$	−3169.98	−	273.902i	−0.575207	−	0.0497007i
$313$	−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	$-0.389389\pi$
$317$	3844.07			0.681086			0.340543	+	0.940229i	$0.389389\pi$
$318$	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	$-0.773541\pi$
$331$	−9122.40			−1.51484			−0.757421	+	0.652927i	$0.773541\pi$
$332$	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	$0.363304\pi$
$338$	4187.61	+	4591.13i	0.673893	+	0.738829i
$339$			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	$-0.549450\pi$
$349$	−2017.61			−0.309456			−0.154728	+	0.987957i	$0.549450\pi$
$350$	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	$-0.828389\pi$
$353$		−	6809.89i		−	1.02678i	0.858155	−	0.513391i	$-0.171611\pi$
$354$	−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	$0.106304\pi$
$359$			4459.33i			0.655584i	−0.944750	+	0.327792i	$0.893696\pi$
$360$	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
$364$	−5706.90	+	2482.69i	−0.821766	+	0.357495i
$365$		−	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	$-0.586208\pi$
$379$	−3947.90			−0.535066			−0.267533	+	0.963549i	$0.586208\pi$
$380$	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	$0.0282529\pi$
$383$			1328.83i			0.177285i	−0.996064	+	0.0886425i	$0.971747\pi$
$384$	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	$0.810073\pi$
$390$	2051.57	−	2622.08i	0.266372	−	0.340446i
$391$	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	$-0.211624\pi$
$397$	12450.9			1.57403			0.787017	+	0.616931i	$0.211624\pi$
$398$	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	$-0.946444\pi$
$401$		−	2689.39i		−	0.334917i	0.985879	−	0.167458i	$-0.0535560\pi$
$402$	−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	$-0.803916\pi$
$409$		−	9558.37i		−	1.15558i	0.816186	−	0.577789i	$-0.196084\pi$
$410$	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
$416$	−7480.20	+	4004.74i	−0.881603	+	0.471992i
$417$	−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	$-0.626600\pi$
$421$	−6691.51			−0.774642			−0.387321	+	0.921945i	$0.626600\pi$
$422$	−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	$-0.918949\pi$
$431$		−	4507.64i		−	0.503771i	0.967757	−	0.251886i	$-0.0810507\pi$
$432$	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
$442$	−1540.77	−	1205.53i	−0.165807	−	0.129731i
$443$		−	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	$0.0507204\pi$
$449$			3019.21i			0.317339i	−0.987332	+	0.158669i	$0.949280\pi$
$450$	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	$0.248082\pi$
$457$			13732.7i			1.40567i	−0.711354	+	0.702834i	$0.751918\pi$
$458$	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	$-0.338119\pi$
$461$	9639.21			0.973846			0.486923	+	0.873445i	$0.338119\pi$
$462$	−2875.90	−	1558.17i	−0.289608	−	0.156911i
$463$		−	6341.49i		−	0.636532i	−0.948001	−	0.318266i	$-0.896900\pi$
$463$		−	6341.49i		−	0.636532i	0.948001	−	0.318266i	$-0.103100\pi$
$464$	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	$-0.0918980\pi$
$468$	−1346.27	−	3094.64i	−0.132973	−	0.305662i
$469$			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	$0.120597\pi$
$479$			7754.98i			0.739737i	−0.929084	+	0.369869i	$0.879403\pi$
$480$	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	$-0.839738\pi$
$487$		−	10370.4i		−	0.964947i	0.875911	−	0.482474i	$-0.160262\pi$
$488$	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
$494$	−12988.4	−	10162.4i	−1.18295	−	0.925565i
$495$	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	$-0.362515\pi$
$499$	9332.52			0.837236			0.418618	+	0.908162i	$0.362515\pi$
$500$	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	$-0.517735\pi$
$509$	−1278.96			−0.111373			−0.0556867	+	0.998448i	$0.517735\pi$
$510$	−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
$520$	764.266	−	8845.17i	0.0644524	−	0.745935i
$521$	−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	$-0.159272\pi$
$541$	22081.4			1.75481			0.877407	+	0.479748i	$0.159272\pi$
$542$	−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
$546$	−5198.84	−	4067.68i	−0.407490	−	0.318829i
$547$		−	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	$-0.849584\pi$
$557$	−23410.1			−1.78082			−0.890412	+	0.455156i	$0.849584\pi$
$558$	−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	$0.891630\pi$
$572$	3474.21	+	7986.10i	0.253958	+	0.583768i
$573$			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	$-0.953998\pi$
$577$		−	3992.15i		−	0.288033i	0.989575	−	0.144017i	$-0.0460019\pi$
$578$	−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	$-0.444152\pi$
$587$	4964.98			0.349109			0.174554	+	0.984648i	$0.444152\pi$
$588$	−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	$0.0667536\pi$
$593$			6012.41i			0.416357i	−0.978091	+	0.208179i	$0.933246\pi$
$594$	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
$598$	−18130.8	−	14185.9i	−1.23984	−	0.970075i
$599$	−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	$-0.459970\pi$
$613$	3807.27			0.250855			0.125428	+	0.992103i	$0.459970\pi$
$614$	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	$0.0946017\pi$
$617$			8976.23i			0.585688i	−0.956160	+	0.292844i	$0.905398\pi$
$618$	890.404	−	1643.41i	0.0579568	−	0.106970i
$619$	−1253.41			−0.0813872			−0.0406936	−	0.999172i	$-0.512957\pi$
$619$	−1253.41			−0.0813872			−0.0406936	+	0.999172i	$0.512957\pi$
$620$	−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
$624$	−7514.27	−	4952.37i	−0.482070	−	0.317714i
$625$	−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	$0.245755\pi$
$631$			22115.1i			1.39523i	−0.716474	+	0.697614i	$0.754245\pi$
$632$	−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	$-0.659028\pi$
$643$	−15622.5			−0.958152			−0.479076	+	0.877773i	$0.659028\pi$
$644$	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
$650$	−5735.21	−	4487.35i	−0.346082	−	0.270782i
$651$			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	$-0.458570\pi$
$661$	4411.35			0.259579			0.129789	+	0.991542i	$0.458570\pi$
$662$	−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
$676$	4227.97	+	17059.9i	0.240554	+	0.970636i
$677$			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	$-0.577583\pi$
$683$	−8615.26			−0.482656			−0.241328	+	0.970444i	$0.577583\pi$
$684$	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	$-0.465895\pi$
$691$	3884.91			0.213877			0.106938	+	0.994266i	$0.465895\pi$
$692$	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	$0.955537\pi$
$702$	2205.76	−	2819.14i	0.118591	−	0.151569i
$703$	−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	$-0.392062\pi$
$709$	12559.3			0.665269			0.332635	+	0.943056i	$0.392062\pi$
$710$	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	$0.314707\pi$
$728$	−17537.5	−	1515.32i	−0.892833	−	0.0771451i
$729$	−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	$-0.497875\pi$
$733$	264.927			0.0133496			0.00667482	+	0.999978i	$0.497875\pi$
$734$	−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	$-0.503209\pi$
$739$	−405.069			−0.0201633			−0.0100817	+	0.999949i	$0.503209\pi$
$740$	−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	$-0.864804\pi$
$743$		−	16691.2i		−	0.824149i	0.911150	−	0.412074i	$-0.135196\pi$
$744$	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
$754$	10326.9	−	13198.7i	0.498786	−	0.637490i
$755$		−	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	$0.160641\pi$
$761$			20301.1i			0.967034i	−0.875335	+	0.483517i	$0.839359\pi$
$762$	−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	$-0.795870\pi$
$769$		−	25514.7i		−	1.19647i	0.801322	−	0.598233i	$-0.204130\pi$
$770$	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	$-0.513022\pi$
$773$	−1757.92			−0.0817955			−0.0408977	+	0.999163i	$0.513022\pi$
$774$	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} +(104.413 + 81.6946i) 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} +(149.586 + 343.849i) 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} +(-245.084 + 313.238i) 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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