LMFDB - Embedded newform 756.4.k.h.109.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [756,4,Mod(109,756)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(756, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 4])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("756.109"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 756 = 2^{2} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	756.k (of order $3$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,0,7,0,-11] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$44.6054439643$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 198 x^{14} + 15603 x^{12} + 623882 x^{10} + 13296387 x^{8} + 143321550 x^{6} + 644261986 x^{4} + \cdots + 9753129 $ x^16 + 198x^14 + 15603x^12 + 623882x^10 + 13296387x^8 + 143321550x^6 + 644261986x^4 + 674526288*x^2 + 9753129
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{16} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			109.3
Root			$-7.59982i$ of defining polynomial
Character	$\chi$	$=$	756.109
Dual form			756.4.k.h.541.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-4.37589 - 7.57927i) q^{5} +(18.4405 + 1.71677i) q^{7} +(-15.5735 + 26.9741i) q^{11} -4.09748 q^{13} +(-36.5246 + 63.2624i) q^{17} +(-65.1108 - 112.775i) q^{19} +(26.5021 + 45.9030i) q^{23} +(24.2031 - 41.9210i) q^{25} -165.352 q^{29} +(-33.5406 + 58.0939i) q^{31} +(-67.6819 - 147.278i) q^{35} +(-20.2607 - 35.0926i) q^{37} +247.624 q^{41} -164.059 q^{43} +(-34.9356 - 60.5102i) q^{47} +(337.105 + 63.3161i) q^{49} +(-291.593 + 505.053i) q^{53} +272.592 q^{55} +(-114.993 + 199.173i) q^{59} +(10.4236 + 18.0541i) q^{61} +(17.9301 + 31.0559i) q^{65} +(-452.903 + 784.451i) q^{67} -412.489 q^{71} +(-78.0706 + 135.222i) q^{73} +(-333.492 + 470.681i) q^{77} +(-92.3545 - 159.963i) q^{79} -1355.25 q^{83} +639.311 q^{85} +(82.2747 + 142.504i) q^{89} +(-75.5597 - 7.03441i) q^{91} +(-569.836 + 986.985i) q^{95} +474.685 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 7 q^{5} - 11 q^{7} - 43 q^{11} - 52 q^{13} + 38 q^{17} + 32 q^{19} - 118 q^{23} - 143 q^{25} + 160 q^{29} + 5 q^{31} + 161 q^{35} - 136 q^{37} - 216 q^{41} + 800 q^{43} - 78 q^{47} + 253 q^{49} + 105 q^{53}+ \cdots - 838 q^{97}+O(q^{100}) $ 16 * q + 7 * q^5 - 11 * q^7 - 43 * q^11 - 52 * q^13 + 38 * q^17 + 32 * q^19 - 118 * q^23 - 143 * q^25 + 160 * q^29 + 5 * q^31 + 161 * q^35 - 136 * q^37 - 216 * q^41 + 800 * q^43 - 78 * q^47 + 253 * q^49 + 105 * q^53 - 948 * q^55 + 574 * q^59 - 424 * q^61 - 408 * q^65 - 88 * q^67 - 856 * q^71 + 644 * q^73 - 656 * q^77 + 266 * q^79 + 2054 * q^83 + 384 * q^85 + 732 * q^89 + 1022 * q^91 - 364 * q^95 - 838 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/756\mathbb{Z}\right)^\times$.

$n$	$29$	$325$	$379$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$	$1$

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 $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−4.37589	−	7.57927i	−0.391392	−	0.677911i	0.601242	−	0.799067i	$-0.294673\pi$
$5$	−4.37589	−	7.57927i	−0.391392	−	0.677911i	−0.992633	+	0.121157i	$0.961340\pi$
$6$		0			0
$7$	18.4405	+	1.71677i	0.995694	+	0.0926966i
$7$	18.4405	+	1.71677i	0.995694	+	0.0926966i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−15.5735	+	26.9741i	−0.426872	+	0.739364i	−0.996593	−	0.0824740i	$-0.973718\pi$
$11$	−15.5735	+	26.9741i	−0.426872	+	0.739364i	0.569721	+	0.821838i	$0.307051\pi$
$12$		0			0
$13$	−4.09748			−0.0874182			−0.0437091	−	0.999044i	$-0.513917\pi$
$13$	−4.09748			−0.0874182			−0.0437091	+	0.999044i	$0.513917\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−36.5246	+	63.2624i	−0.521089	+	0.902553i	0.478610	+	0.878027i	$0.341141\pi$
$17$	−36.5246	+	63.2624i	−0.521089	+	0.902553i	−0.999699	+	0.0245251i	$0.992193\pi$
$18$		0			0
$19$	−65.1108	−	112.775i	−0.786181	−	1.36171i	−0.928291	−	0.371855i	$-0.878722\pi$
$19$	−65.1108	−	112.775i	−0.786181	−	1.36171i	0.142109	−	0.989851i	$-0.454612\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	26.5021	+	45.9030i	0.240264	+	0.416150i	0.960789	−	0.277279i	$-0.0894326\pi$
$23$	26.5021	+	45.9030i	0.240264	+	0.416150i	−0.720525	+	0.693429i	$0.756099\pi$
$24$		0			0
$25$	24.2031	−	41.9210i	0.193625	−	0.335368i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−165.352			−1.05879			−0.529397	−	0.848374i	$-0.677582\pi$
$29$	−165.352			−1.05879			−0.529397	+	0.848374i	$0.677582\pi$
$30$		0			0
$31$	−33.5406	+	58.0939i	−0.194325	+	0.336580i	−0.946679	−	0.322179i	$-0.895585\pi$
$31$	−33.5406	+	58.0939i	−0.194325	+	0.336580i	0.752354	+	0.658759i	$0.228918\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−67.6819	−	147.278i	−0.326867	−	0.711272i
$36$		0			0
$37$	−20.2607	−	35.0926i	−0.0900228	−	0.155924i	0.817498	−	0.575932i	$-0.195361\pi$
$37$	−20.2607	−	35.0926i	−0.0900228	−	0.155924i	−0.907520	+	0.420008i	$0.862027\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	247.624			0.943229			0.471615	−	0.881805i	$-0.343671\pi$
$41$	247.624			0.943229			0.471615	+	0.881805i	$0.343671\pi$
$42$		0			0
$43$	−164.059			−0.581830			−0.290915	−	0.956749i	$-0.593960\pi$
$43$	−164.059			−0.581830			−0.290915	+	0.956749i	$0.593960\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−34.9356	−	60.5102i	−0.108423	−	0.187794i	0.806709	−	0.590949i	$-0.201247\pi$
$47$	−34.9356	−	60.5102i	−0.108423	−	0.187794i	−0.915132	+	0.403155i	$0.867913\pi$
$48$		0			0
$49$	337.105	+	63.3161i	0.982815	+	0.184595i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−291.593	+	505.053i	−0.755723	+	1.30895i	0.189291	+	0.981921i	$0.439381\pi$
$53$	−291.593	+	505.053i	−0.755723	+	1.30895i	−0.945014	+	0.327030i	$0.893952\pi$
$54$		0			0
$55$	272.592			0.668297
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−114.993	+	199.173i	−0.253742	+	0.439494i	−0.964553	−	0.263889i	$-0.914995\pi$
$59$	−114.993	+	199.173i	−0.253742	+	0.439494i	0.710811	+	0.703383i	$0.248328\pi$
$60$		0			0
$61$	10.4236	+	18.0541i	0.0218787	+	0.0378950i	0.876757	−	0.480933i	$-0.159702\pi$
$61$	10.4236	+	18.0541i	0.0218787	+	0.0378950i	−0.854879	+	0.518828i	$0.826369\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	17.9301	+	31.0559i	0.0342148	+	0.0592617i
$66$		0			0
$67$	−452.903	+	784.451i	−0.825835	+	1.43039i	0.0754446	+	0.997150i	$0.475962\pi$
$67$	−452.903	+	784.451i	−0.825835	+	1.43039i	−0.901280	+	0.433238i	$0.857371\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−412.489			−0.689485			−0.344742	−	0.938697i	$-0.612034\pi$
$71$	−412.489			−0.689485			−0.344742	+	0.938697i	$0.612034\pi$
$72$		0			0
$73$	−78.0706	+	135.222i	−0.125171	+	0.216802i	−0.921800	−	0.387666i	$-0.873281\pi$
$73$	−78.0706	+	135.222i	−0.125171	+	0.216802i	0.796629	+	0.604469i	$0.206615\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−333.492	+	470.681i	−0.493571	+	0.696611i
$78$		0			0
$79$	−92.3545	−	159.963i	−0.131528	−	0.227813i	0.792738	−	0.609563i	$-0.208655\pi$
$79$	−92.3545	−	159.963i	−0.131528	−	0.227813i	−0.924266	+	0.381750i	$0.875322\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−1355.25			−1.79227			−0.896134	−	0.443784i	$-0.853636\pi$
$83$	−1355.25			−1.79227			−0.896134	+	0.443784i	$0.853636\pi$
$84$		0			0
$85$	639.311			0.815800
$86$		0			0
$87$		0			0
$88$		0			0
$89$	82.2747	+	142.504i	0.0979899	+	0.169724i	0.910853	−	0.412732i	$-0.135425\pi$
$89$	82.2747	+	142.504i	0.0979899	+	0.169724i	−0.812863	+	0.582455i	$0.802092\pi$
$90$		0			0
$91$	−75.5597	−	7.03441i	−0.0870418	−	0.00810337i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−569.836	+	986.985i	−0.615410	+	1.06592i
$96$		0			0
$97$	474.685			0.496875			0.248438	−	0.968648i	$-0.420083\pi$
$97$	474.685			0.496875			0.248438	+	0.968648i	$0.420083\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−736.223	+	1275.18i	−0.725316	+	1.25628i	0.233528	+	0.972350i	$0.424973\pi$
$101$	−736.223	+	1275.18i	−0.725316	+	1.25628i	−0.958844	+	0.283934i	$0.908360\pi$
$102$		0			0
$103$	−858.620	−	1487.17i	−0.821382	−	1.42267i	−0.904654	−	0.426148i	$-0.859870\pi$
$103$	−858.620	−	1487.17i	−0.821382	−	1.42267i	0.0832720	−	0.996527i	$-0.473463\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	134.180	+	232.406i	0.121230	+	0.209977i	0.920253	−	0.391324i	$-0.127983\pi$
$107$	134.180	+	232.406i	0.121230	+	0.209977i	−0.799023	+	0.601301i	$0.794649\pi$
$108$		0			0
$109$	−613.690	+	1062.94i	−0.539273	+	0.934049i	0.459670	+	0.888090i	$0.347968\pi$
$109$	−613.690	+	1062.94i	−0.539273	+	0.934049i	−0.998943	+	0.0459592i	$0.985366\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−997.330			−0.830274			−0.415137	−	0.909759i	$-0.636266\pi$
$113$	−997.330			−0.830274			−0.415137	+	0.909759i	$0.636266\pi$
$114$		0			0
$115$	231.941	−	401.734i	0.188075	−	0.325755i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−782.139	+	1103.89i	−0.602509	+	0.850363i
$120$		0			0
$121$	180.431	+	312.516i	0.135561	+	0.234798i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−1517.61			−1.08592
$126$		0			0
$127$	−2.43376			−0.00170048			−0.000850241	−	1.00000i	$-0.500271\pi$
$127$	−2.43376			−0.00170048			−0.000850241		1.00000i	$0.500271\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	991.340	+	1717.05i	0.661174	+	1.14519i	0.980307	+	0.197477i	$0.0632749\pi$
$131$	991.340	+	1717.05i	0.661174	+	1.14519i	−0.319133	+	0.947710i	$0.603392\pi$
$132$		0			0
$133$	−1007.07	−	2191.41i	−0.656571	−	1.42872i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	374.349	−	648.392i	0.233451	−	0.404349i	−0.725370	−	0.688359i	$-0.758331\pi$
$137$	374.349	−	648.392i	0.233451	−	0.404349i	0.958821	+	0.284010i	$0.0916648\pi$
$138$		0			0
$139$	−139.476			−0.0851094			−0.0425547	−	0.999094i	$-0.513550\pi$
$139$	−139.476			−0.0851094			−0.0425547	+	0.999094i	$0.513550\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	63.8122	−	110.526i	0.0373164	−	0.0646339i
$144$		0			0
$145$	723.561	+	1253.24i	0.414403	+	0.717768i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−1480.29	−	2563.94i	−0.813893	−	1.40970i	−0.910120	−	0.414345i	$-0.864011\pi$
$149$	−1480.29	−	2563.94i	−0.813893	−	1.40970i	0.0962263	−	0.995359i	$-0.469323\pi$
$150$		0			0
$151$	1004.33	−	1739.55i	0.541267	−	0.937502i	−0.457565	−	0.889176i	$-0.651278\pi$
$151$	1004.33	−	1739.55i	0.541267	−	0.937502i	0.998832	−	0.0483253i	$-0.0153884\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	587.080			0.304228
$156$		0			0
$157$	555.556	−	962.251i	0.282409	−	0.489147i	−0.689569	−	0.724220i	$-0.742200\pi$
$157$	555.556	−	962.251i	0.282409	−	0.489147i	0.971978	+	0.235074i	$0.0755332\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	409.908	+	891.974i	0.200654	+	0.436630i
$162$		0			0
$163$	1894.78	+	3281.86i	0.910495	+	1.57702i	0.813367	+	0.581751i	$0.197632\pi$
$163$	1894.78	+	3281.86i	0.910495	+	1.57702i	0.0971277	+	0.995272i	$0.469034\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	163.938			0.0759634			0.0379817	−	0.999278i	$-0.487907\pi$
$167$	163.938			0.0759634			0.0379817	+	0.999278i	$0.487907\pi$
$168$		0			0
$169$	−2180.21			−0.992358
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−858.615	−	1487.16i	−0.377337	−	0.653567i	0.613337	−	0.789821i	$-0.289827\pi$
$173$	−858.615	−	1487.16i	−0.377337	−	0.653567i	−0.990674	+	0.136255i	$0.956493\pi$
$174$		0			0
$175$	518.286	−	731.494i	0.223879	−	0.315976i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−1530.26	+	2650.49i	−0.638978	+	1.10674i	0.346679	+	0.937984i	$0.387309\pi$
$179$	−1530.26	+	2650.49i	−0.638978	+	1.10674i	−0.985657	+	0.168759i	$0.946024\pi$
$180$		0			0
$181$	1661.54			0.682327			0.341163	−	0.940004i	$-0.389179\pi$
$181$	1661.54			0.682327			0.341163	+	0.940004i	$0.389179\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−177.318	+	307.123i	−0.0704683	+	0.122055i
$186$		0			0
$187$	−1137.63	−	1970.44i	−0.444877	−	0.770549i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	1425.87	+	2469.68i	0.540170	+	0.935602i	0.998894	+	0.0470228i	$0.0149734\pi$
$191$	1425.87	+	2469.68i	0.540170	+	0.935602i	−0.458724	+	0.888579i	$0.651693\pi$
$192$		0			0
$193$	−1415.95	+	2452.50i	−0.528096	+	0.914690i	0.471367	+	0.881937i	$0.343761\pi$
$193$	−1415.95	+	2452.50i	−0.528096	+	0.914690i	−0.999463	+	0.0327528i	$0.989573\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−1312.38			−0.474637			−0.237318	−	0.971432i	$-0.576268\pi$
$197$	−1312.38			−0.474637			−0.237318	+	0.971432i	$0.576268\pi$
$198$		0			0
$199$	−1750.79	+	3032.46i	−0.623669	+	1.08023i	0.365127	+	0.930958i	$0.381025\pi$
$199$	−1750.79	+	3032.46i	−0.623669	+	1.08023i	−0.988797	+	0.149269i	$0.952308\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−3049.17	−	283.870i	−1.05424	−	0.0981466i
$204$		0			0
$205$	−1083.58	−	1876.81i	−0.369172	−	0.639425i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	4056.02			1.34240
$210$		0			0
$211$	−898.695			−0.293217			−0.146608	−	0.989195i	$-0.546836\pi$
$211$	−898.695			−0.293217			−0.146608	+	0.989195i	$0.546836\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	717.903	+	1243.44i	0.227724	+	0.394429i
$216$		0			0
$217$	−718.239	+	1013.70i	−0.224688	+	0.317118i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	149.659	−	259.217i	0.0455527	−	0.0788995i
$222$		0			0
$223$	2228.51			0.669203			0.334602	−	0.942360i	$-0.391398\pi$
$223$	2228.51			0.669203			0.334602	+	0.942360i	$0.391398\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−451.677	+	782.327i	−0.132065	+	0.228744i	−0.924473	−	0.381249i	$-0.875494\pi$
$227$	−451.677	+	782.327i	−0.132065	+	0.228744i	0.792407	+	0.609992i	$0.208828\pi$
$228$		0			0
$229$	1316.46	+	2280.18i	0.379888	+	0.657985i	0.991046	−	0.133524i	$-0.0426293\pi$
$229$	1316.46	+	2280.18i	0.379888	+	0.657985i	−0.611158	+	0.791509i	$0.709296\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−1794.70	−	3108.52i	−0.504614	−	0.874017i	−0.999986	−	0.00533580i	$-0.998302\pi$
$233$	−1794.70	−	3108.52i	−0.504614	−	0.874017i	0.495372	−	0.868681i	$-0.335032\pi$
$234$		0			0
$235$	−305.749	+	529.572i	−0.0848717	+	0.147002i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−51.2598			−0.0138733			−0.00693665	−	0.999976i	$-0.502208\pi$
$239$	−51.2598			−0.0138733			−0.00693665	+	0.999976i	$0.502208\pi$
$240$		0			0
$241$	1353.08	−	2343.60i	0.361658	−	0.626409i	−0.626576	−	0.779360i	$-0.715544\pi$
$241$	1353.08	−	2343.60i	0.361658	−	0.626409i	0.988234	+	0.152951i	$0.0488776\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−995.248	−	2832.08i	−0.259527	−	0.738509i
$246$		0			0
$247$	266.790	+	462.094i	0.0687266	+	0.119038i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	3198.35			0.804295			0.402147	−	0.915575i	$-0.368264\pi$
$251$	3198.35			0.804295			0.402147	+	0.915575i	$0.368264\pi$
$252$		0			0
$253$	−1650.93			−0.410248
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−976.932	−	1692.10i	−0.237118	−	0.410701i	0.722768	−	0.691091i	$-0.242870\pi$
$257$	−976.932	−	1692.10i	−0.237118	−	0.410701i	−0.959886	+	0.280390i	$0.909536\pi$
$258$		0			0
$259$	−313.372	−	681.909i	−0.0751815	−	0.163597i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−1254.82	+	2173.42i	−0.294204	+	0.509576i	−0.974799	−	0.223083i	$-0.928388\pi$
$263$	−1254.82	+	2173.42i	−0.294204	+	0.509576i	0.680595	+	0.732659i	$0.261721\pi$
$264$		0			0
$265$	5103.91			1.18314
$266$		0			0
$267$		0			0
$268$		0			0
$269$	3598.65	−	6233.04i	0.815663	−	1.41277i	−0.0931875	−	0.995649i	$-0.529706\pi$
$269$	3598.65	−	6233.04i	0.815663	−	1.41277i	0.908851	−	0.417122i	$-0.136961\pi$
$270$		0			0
$271$	1467.99	+	2542.64i	0.329056	+	0.569942i	0.982325	−	0.187184i	$-0.0599362\pi$
$271$	1467.99	+	2542.64i	0.329056	+	0.569942i	−0.653269	+	0.757126i	$0.726603\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	753.855	+	1305.72i	0.165306	+	0.286318i
$276$		0			0
$277$	2390.58	−	4140.60i	0.518541	−	0.898140i	−0.481227	−	0.876596i	$-0.659809\pi$
$277$	2390.58	−	4140.60i	0.518541	−	0.898140i	0.999768	−	0.0215434i	$-0.00685802\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	3904.37			0.828879			0.414439	−	0.910077i	$-0.363978\pi$
$281$	3904.37			0.828879			0.414439	+	0.910077i	$0.363978\pi$
$282$		0			0
$283$	−3473.96	+	6017.07i	−0.729701	+	1.26388i	0.227309	+	0.973823i	$0.427007\pi$
$283$	−3473.96	+	6017.07i	−0.729701	+	1.26388i	−0.957010	+	0.290056i	$0.906326\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	4566.32	+	425.113i	0.939168	+	0.0874342i
$288$		0			0
$289$	−211.590	−	366.485i	−0.0430675	−	0.0745950i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	6370.38			1.27018			0.635088	−	0.772440i	$-0.280964\pi$
$293$	6370.38			1.27018			0.635088	+	0.772440i	$0.280964\pi$
$294$		0			0
$295$	2012.78			0.397250
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−108.592	−	188.087i	−0.0210035	−	0.0363791i
$300$		0			0
$301$	−3025.33	−	281.650i	−0.579325	−	0.0539337i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	91.2248	−	158.006i	0.0171263	−	0.0296636i
$306$		0			0
$307$	−136.784			−0.0254288			−0.0127144	−	0.999919i	$-0.504047\pi$
$307$	−136.784			−0.0254288			−0.0127144	+	0.999919i	$0.504047\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	2448.71	−	4241.29i	0.446475	−	0.773318i	−0.551679	−	0.834057i	$-0.686012\pi$
$311$	2448.71	−	4241.29i	0.446475	−	0.773318i	0.998154	+	0.0607393i	$0.0193458\pi$
$312$		0			0
$313$	−4636.12	−	8030.00i	−0.837217	−	1.45010i	−0.892212	−	0.451616i	$-0.850848\pi$
$313$	−4636.12	−	8030.00i	−0.837217	−	1.45010i	0.0549948	−	0.998487i	$-0.482486\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−4401.03	−	7622.81i	−0.779768	−	1.35060i	−0.932075	−	0.362265i	$-0.882003\pi$
$317$	−4401.03	−	7622.81i	−0.779768	−	1.35060i	0.152307	−	0.988333i	$-0.451330\pi$
$318$		0			0
$319$	2575.11	−	4460.21i	0.451969	−	0.782834i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	9512.58			1.63868
$324$		0			0
$325$	−99.1717	+	171.771i	−0.0169263	+	0.0293173i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−540.348	−	1175.82i	−0.0905482	−	0.197036i
$330$		0			0
$331$	−5764.23	−	9983.94i	−0.957193	−	1.65791i	−0.729268	−	0.684228i	$-0.760139\pi$
$331$	−5764.23	−	9983.94i	−0.957193	−	1.65791i	−0.227925	−	0.973679i	$-0.573194\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	7927.43			1.29290
$336$		0			0
$337$	−1301.84			−0.210433			−0.105216	−	0.994449i	$-0.533554\pi$
$337$	−1301.84			−0.210433			−0.105216	+	0.994449i	$0.533554\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−1044.69	−	1809.45i	−0.165903	−	0.287353i
$342$		0			0
$343$	6107.70	+	1746.31i	0.961472	+	0.274904i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	4096.12	−	7094.69i	0.633692	−	1.09759i	−0.353098	−	0.935586i	$-0.614872\pi$
$347$	4096.12	−	7094.69i	0.633692	−	1.09759i	0.986791	−	0.162001i	$-0.0517947\pi$
$348$		0			0
$349$	−11739.7			−1.80061			−0.900305	−	0.435259i	$-0.856657\pi$
$349$	−11739.7			−1.80061			−0.900305	+	0.435259i	$0.856657\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	5167.92	−	8951.09i	0.779208	−	1.34963i	−0.153191	−	0.988197i	$-0.548955\pi$
$353$	5167.92	−	8951.09i	0.779208	−	1.34963i	0.932399	−	0.361431i	$-0.117712\pi$
$354$		0			0
$355$	1805.01	+	3126.36i	0.269859	+	0.467409i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−5354.35	−	9274.01i	−0.787164	−	1.36341i	−0.927698	−	0.373332i	$-0.878215\pi$
$359$	−5354.35	−	9274.01i	−0.787164	−	1.36341i	0.140534	−	0.990076i	$-0.455118\pi$
$360$		0			0
$361$	−5049.34	+	8745.71i	−0.736162	+	1.27507i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	1366.51			0.195963
$366$		0			0
$367$	4373.75	−	7575.56i	0.622093	−	1.07750i	−0.367003	−	0.930220i	$-0.619616\pi$
$367$	4373.75	−	7575.56i	0.622093	−	1.07750i	0.989095	−	0.147276i	$-0.0470506\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−6244.18	+	8812.85i	−0.873805	+	1.23326i
$372$		0			0
$373$	2854.40	+	4943.97i	0.396234	+	0.686298i	0.993258	−	0.115926i	$-0.0369835\pi$
$373$	2854.40	+	4943.97i	0.396234	+	0.686298i	−0.597024	+	0.802224i	$0.703650\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	677.525			0.0925579
$378$		0			0
$379$	4068.56			0.551419			0.275709	−	0.961241i	$-0.411087\pi$
$379$	4068.56			0.551419			0.275709	+	0.961241i	$0.411087\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	7356.61	+	12742.0i	0.981475	+	1.69997i	0.656658	+	0.754188i	$0.271969\pi$
$383$	7356.61	+	12742.0i	0.981475	+	1.69997i	0.324817	+	0.945777i	$0.394697\pi$
$384$		0			0
$385$	5026.74	+	467.977i	0.665419	+	0.0619489i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	3452.14	−	5979.28i	0.449950	−	0.779336i	−0.548433	−	0.836195i	$-0.684775\pi$
$389$	3452.14	−	5979.28i	0.449950	−	0.779336i	0.998382	+	0.0568591i	$0.0181086\pi$
$390$		0			0
$391$	−3871.92			−0.500796
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−808.267	+	1399.96i	−0.102958	+	0.178328i
$396$		0			0
$397$	4344.11	+	7524.22i	0.549180	+	0.951208i	0.998331	+	0.0577522i	$0.0183933\pi$
$397$	4344.11	+	7524.22i	0.549180	+	0.951208i	−0.449151	+	0.893456i	$0.648273\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−946.137	−	1638.76i	−0.117825	−	0.204079i	0.801081	−	0.598557i	$-0.204259\pi$
$401$	−946.137	−	1638.76i	−0.117825	−	0.204079i	−0.918905	+	0.394478i	$0.870926\pi$
$402$		0			0
$403$	137.432	−	238.039i	0.0169875	−	0.0294232i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	1262.12			0.153713
$408$		0			0
$409$	2507.84	−	4343.71i	0.303190	−	0.525141i	−0.673666	−	0.739035i	$-0.735282\pi$
$409$	2507.84	−	4343.71i	0.303190	−	0.525141i	0.976857	+	0.213895i	$0.0686149\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−2462.46	+	3475.44i	−0.293389	+	0.414081i
$414$		0			0
$415$	5930.44	+	10271.8i	0.701479	+	1.21500i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	4510.00			0.525842			0.262921	−	0.964817i	$-0.415314\pi$
$419$	4510.00			0.525842			0.262921	+	0.964817i	$0.415314\pi$
$420$		0			0
$421$	10160.9			1.17628			0.588140	−	0.808759i	$-0.299860\pi$
$421$	10160.9			1.17628			0.588140	+	0.808759i	$0.299860\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	1768.02	+	3062.29i	0.201792	+	0.349513i
$426$		0			0
$427$	161.221	+	350.822i	0.0182717	+	0.0397599i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−6987.82	+	12103.3i	−0.780955	+	1.35265i	0.150431	+	0.988620i	$0.451934\pi$
$431$	−6987.82	+	12103.3i	−0.780955	+	1.35265i	−0.931386	+	0.364033i	$0.881400\pi$
$432$		0			0
$433$	−6363.55			−0.706265			−0.353132	−	0.935573i	$-0.614883\pi$
$433$	−6363.55			−0.706265			−0.353132	+	0.935573i	$0.614883\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	3451.15	−	5977.57i	0.377782	−	0.654338i
$438$		0			0
$439$	−3422.89	−	5928.62i	−0.372131	−	0.644551i	0.617762	−	0.786365i	$-0.288040\pi$
$439$	−3422.89	−	5928.62i	−0.372131	−	0.644551i	−0.989893	+	0.141815i	$0.954706\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−5590.98	−	9683.86i	−0.599628	−	1.03859i	−0.992876	−	0.119154i	$-0.961982\pi$
$443$	−5590.98	−	9683.86i	−0.599628	−	1.03859i	0.393247	−	0.919433i	$-0.371352\pi$
$444$		0			0
$445$	720.051	−	1247.16i	0.0767049	−	0.132857i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	3351.98			0.352316			0.176158	−	0.984362i	$-0.443633\pi$
$449$	3351.98			0.352316			0.176158	+	0.984362i	$0.443633\pi$
$450$		0			0
$451$	−3856.38	+	6679.44i	−0.402638	+	0.697390i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	277.325	+	603.469i	0.0285741	+	0.0621782i
$456$		0			0
$457$	−744.580	−	1289.65i	−0.0762144	−	0.132007i	0.825399	−	0.564549i	$-0.190950\pi$
$457$	−744.580	−	1289.65i	−0.0762144	−	0.132007i	−0.901614	+	0.432542i	$0.857617\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−957.491			−0.0967349			−0.0483675	−	0.998830i	$-0.515402\pi$
$461$	−957.491			−0.0967349			−0.0483675	+	0.998830i	$0.515402\pi$
$462$		0			0
$463$	−5753.05			−0.577466			−0.288733	−	0.957410i	$-0.593234\pi$
$463$	−5753.05			−0.577466			−0.288733	+	0.957410i	$0.593234\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−2128.84	−	3687.27i	−0.210945	−	0.365367i	0.741066	−	0.671432i	$-0.234321\pi$
$467$	−2128.84	−	3687.27i	−0.210945	−	0.365367i	−0.952010	+	0.306066i	$0.900987\pi$
$468$		0			0
$469$	−9698.49	+	13688.2i	−0.954871	+	1.34768i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	2554.97	−	4425.34i	0.248367	−	0.430184i
$474$		0			0
$475$	−6303.54			−0.608897
$476$		0			0
$477$		0			0
$478$		0			0
$479$	149.647	−	259.196i	0.0142746	−	0.0247243i	−0.858800	−	0.512311i	$-0.828789\pi$
$479$	149.647	−	259.196i	0.0142746	−	0.0247243i	0.873074	+	0.487587i	$0.162123\pi$
$480$		0			0
$481$	83.0179	+	143.791i	0.00786963	+	0.0136306i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−2077.17	−	3597.76i	−0.194473	−	0.336837i
$486$		0			0
$487$	343.055	−	594.188i	0.0319205	−	0.0552880i	−0.849624	−	0.527389i	$-0.823171\pi$
$487$	343.055	−	594.188i	0.0319205	−	0.0552880i	0.881544	+	0.472101i	$0.156504\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	118.556			0.0108969			0.00544843	−	0.999985i	$-0.498266\pi$
$491$	118.556			0.0108969			0.00544843	+	0.999985i	$0.498266\pi$
$492$		0			0
$493$	6039.40	−	10460.5i	0.551726	−	0.955617i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−7606.51	−	708.147i	−0.686516	−	0.0639129i
$498$		0			0
$499$	−2667.81	−	4620.78i	−0.239334	−	0.414538i	0.721190	−	0.692738i	$-0.243596\pi$
$499$	−2667.81	−	4620.78i	−0.239334	−	0.414538i	−0.960523	+	0.278200i	$0.910262\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	13651.8			1.21014			0.605072	−	0.796171i	$-0.293144\pi$
$503$	13651.8			1.21014			0.605072	+	0.796171i	$0.293144\pi$
$504$		0			0
$505$	12886.5			1.13553
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−1147.20	−	1987.01i	−0.0998994	−	0.173031i	0.811743	−	0.584014i	$-0.198519\pi$
$509$	−1147.20	−	1987.01i	−0.0998994	−	0.173031i	−0.911643	+	0.410983i	$0.865185\pi$
$510$		0			0
$511$	−1671.81	+	2359.54i	−0.144729	+	0.204266i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−7514.46	+	13015.4i	−0.642964	+	1.11365i
$516$		0			0
$517$	2176.28			0.185131
$518$		0			0
$519$		0			0
$520$		0			0
$521$	9663.83	−	16738.2i	0.812630	−	1.40752i	−0.0983873	−	0.995148i	$-0.531368\pi$
$521$	9663.83	−	16738.2i	0.812630	−	1.40752i	0.911017	−	0.412368i	$-0.135298\pi$
$522$		0			0
$523$	2782.31	+	4819.10i	0.232623	+	0.402915i	0.958579	−	0.284826i	$-0.0919358\pi$
$523$	2782.31	+	4819.10i	0.232623	+	0.402915i	−0.725956	+	0.687741i	$0.758602\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−2450.11	−	4243.71i	−0.202521	−	0.350776i
$528$		0			0
$529$	4678.77	−	8103.88i	0.384546	−	0.666054i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−1014.64			−0.0824554
$534$		0			0
$535$	1174.31	−	2033.97i	0.0948972	−	0.164367i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−6957.81	+	8107.07i	−0.556019	+	0.647859i
$540$		0			0
$541$	−3609.69	−	6252.17i	−0.286863	−	0.496861i	0.686196	−	0.727416i	$-0.259279\pi$
$541$	−3609.69	−	6252.17i	−0.286863	−	0.496861i	−0.973059	+	0.230555i	$0.925946\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	10741.8			0.844269
$546$		0			0
$547$	−13852.1			−1.08277			−0.541383	−	0.840776i	$-0.682099\pi$
$547$	−13852.1			−1.08277			−0.541383	+	0.840776i	$0.682099\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	10766.2	+	18647.6i	0.832404	+	1.44177i
$552$		0			0
$553$	−1428.45	−	3108.35i	−0.109844	−	0.239024i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−9539.00	+	16522.0i	−0.725638	+	1.25684i	0.233072	+	0.972459i	$0.425122\pi$
$557$	−9539.00	+	16522.0i	−0.725638	+	1.25684i	−0.958711	+	0.284383i	$0.908211\pi$
$558$		0			0
$559$	672.227			0.0508625
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−12253.5	+	21223.6i	−0.917268	+	1.58876i	−0.113722	+	0.993513i	$0.536277\pi$
$563$	−12253.5	+	21223.6i	−0.917268	+	1.58876i	−0.803546	+	0.595243i	$0.797056\pi$
$564$		0			0
$565$	4364.21	+	7559.04i	0.324962	+	0.562851i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−8769.77	−	15189.7i	−0.646130	−	1.11913i	−0.984039	−	0.177951i	$-0.943053\pi$
$569$	−8769.77	−	15189.7i	−0.646130	−	1.11913i	0.337910	−	0.941179i	$-0.390280\pi$
$570$		0			0
$571$	−2553.59	+	4422.95i	−0.187153	+	0.324159i	−0.944300	−	0.329086i	$-0.893259\pi$
$571$	−2553.59	+	4422.95i	−0.187153	+	0.324159i	0.757147	+	0.653245i	$0.226593\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	2565.74			0.186084
$576$		0			0
$577$	1915.48	−	3317.71i	0.138202	−	0.239373i	−0.788614	−	0.614888i	$-0.789201\pi$
$577$	1915.48	−	3317.71i	0.138202	−	0.239373i	0.926816	+	0.375516i	$0.122534\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−24991.5	−	2326.65i	−1.78455	−	0.166137i
$582$		0			0
$583$	−9082.24	−	15730.9i	−0.645194	−	1.11751i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	16845.1			1.18445			0.592225	−	0.805773i	$-0.298250\pi$
$587$	16845.1			1.18445			0.592225	+	0.805773i	$0.298250\pi$
$588$		0			0
$589$	8735.41			0.611097
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−9821.01	−	17010.5i	−0.680102	−	1.17797i	−0.974949	−	0.222427i	$-0.928602\pi$
$593$	−9821.01	−	17010.5i	−0.680102	−	1.17797i	0.294847	−	0.955544i	$-0.404731\pi$
$594$		0			0
$595$	11789.2	+	1097.55i	0.812287	+	0.0756219i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−12029.8	+	20836.2i	−0.820573	+	1.42127i	0.0846826	+	0.996408i	$0.473012\pi$
$599$	−12029.8	+	20836.2i	−0.820573	+	1.42127i	−0.905256	+	0.424867i	$0.860321\pi$
$600$		0			0
$601$	−1349.87			−0.0916179			−0.0458089	−	0.998950i	$-0.514587\pi$
$601$	−1349.87			−0.0916179			−0.0458089	+	0.998950i	$0.514587\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	1579.10	−	2735.07i	0.106115	−	0.183796i
$606$		0			0
$607$	−2969.97	−	5144.15i	−0.198596	−	0.343978i	0.749478	−	0.662030i	$-0.230305\pi$
$607$	−2969.97	−	5144.15i	−0.198596	−	0.343978i	−0.948073	+	0.318052i	$0.896971\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	143.148	+	247.939i	0.00947814	+	0.0164166i
$612$		0			0
$613$	6026.40	−	10438.0i	0.397070	−	0.687746i	−0.596293	−	0.802767i	$-0.703360\pi$
$613$	6026.40	−	10438.0i	0.397070	−	0.687746i	0.993363	+	0.115021i	$0.0366936\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−17556.9			−1.14557			−0.572784	−	0.819706i	$-0.694137\pi$
$617$	−17556.9			−1.14557			−0.572784	+	0.819706i	$0.694137\pi$
$618$		0			0
$619$	−4933.36	+	8544.82i	−0.320337	+	0.554839i	−0.980557	−	0.196232i	$-0.937129\pi$
$619$	−4933.36	+	8544.82i	−0.320337	+	0.554839i	0.660221	+	0.751071i	$0.270463\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	1272.54	+	2769.09i	0.0818352	+	0.178076i
$624$		0			0
$625$	3615.53	+	6262.28i	0.231394	+	0.400786i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	2960.06			0.187639
$630$		0			0
$631$	−12457.3			−0.785924			−0.392962	−	0.919555i	$-0.628550\pi$
$631$	−12457.3			−0.785924			−0.392962	+	0.919555i	$0.628550\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	10.6499	+	18.4461i	0.000665555	+	0.00115277i
$636$		0			0
$637$	−1381.28	−	259.436i	−0.0859159	−	0.0161370i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−14152.4	+	24512.7i	−0.872055	+	1.51044i	−0.0121888	+	0.999926i	$0.503880\pi$
$641$	−14152.4	+	24512.7i	−0.872055	+	1.51044i	−0.859867	+	0.510519i	$0.829453\pi$
$642$		0			0
$643$	6375.26			0.391004			0.195502	−	0.980703i	$-0.437366\pi$
$643$	6375.26			0.391004			0.195502	+	0.980703i	$0.437366\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−9488.62	+	16434.8i	−0.576563	+	0.998637i	0.419307	+	0.907845i	$0.362273\pi$
$647$	−9488.62	+	16434.8i	−0.576563	+	0.998637i	−0.995870	+	0.0907920i	$0.971060\pi$
$648$		0			0
$649$	−3581.68	−	6203.65i	−0.216631	−	0.375215i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−11559.9	−	20022.3i	−0.692762	−	1.19990i	−0.970929	−	0.239367i	$-0.923060\pi$
$653$	−11559.9	−	20022.3i	−0.692762	−	1.19990i	0.278167	−	0.960533i	$-0.410273\pi$
$654$		0			0
$655$	8676.00	−	15027.3i	0.517556	−	0.896434i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−18814.3			−1.11214			−0.556072	−	0.831134i	$-0.687692\pi$
$659$	−18814.3			−1.11214			−0.556072	+	0.831134i	$0.687692\pi$
$660$		0			0
$661$	−8646.73	+	14976.6i	−0.508803	+	0.881273i	0.491145	+	0.871078i	$0.336579\pi$
$661$	−8646.73	+	14976.6i	−0.508803	+	0.881273i	−0.999948	+	0.0101948i	$0.996755\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−12202.5	+	17222.2i	−0.711568	+	1.00429i
$666$		0			0
$667$	−4382.17	−	7590.14i	−0.254390	−	0.440617i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−649.326			−0.0373576
$672$		0			0
$673$	471.164			0.0269867			0.0134933	−	0.999909i	$-0.495705\pi$
$673$	471.164			0.0269867			0.0134933	+	0.999909i	$0.495705\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	6540.30	+	11328.1i	0.371291	+	0.643095i	0.989764	−	0.142711i	$-0.0455818\pi$
$677$	6540.30	+	11328.1i	0.371291	+	0.643095i	−0.618473	+	0.785806i	$0.712248\pi$
$678$		0			0
$679$	8753.43	+	814.922i	0.494736	+	0.0460587i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−12021.6	+	20822.1i	−0.673492	+	1.16652i	0.303416	+	0.952858i	$0.401873\pi$
$683$	−12021.6	+	20822.1i	−0.673492	+	1.16652i	−0.976907	+	0.213663i	$0.931460\pi$
$684$		0			0
$685$	−6552.45			−0.365483
$686$		0			0
$687$		0			0
$688$		0			0
$689$	1194.80	−	2069.45i	0.0660640	−	0.114426i
$690$		0			0
$691$	14489.0	+	25095.7i	0.797667	+	1.38160i	0.921132	+	0.389250i	$0.127266\pi$
$691$	14489.0	+	25095.7i	0.797667	+	1.38160i	−0.123465	+	0.992349i	$0.539401\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	610.333	+	1057.13i	0.0333111	+	0.0576966i
$696$		0			0
$697$	−9044.37	+	15665.3i	−0.491506	+	0.851314i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	11710.4			0.630948			0.315474	−	0.948934i	$-0.397836\pi$
$701$	11710.4			0.630948			0.315474	+	0.948934i	$0.397836\pi$
$702$		0			0
$703$	−2638.38	+	4569.82i	−0.141548	+	0.245169i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−15765.5	+	22251.0i	−0.838646	+	1.18364i
$708$		0			0
$709$	15118.2	+	26185.4i	0.800810	+	1.38704i	0.919084	+	0.394063i	$0.128931\pi$
$709$	15118.2	+	26185.4i	0.800810	+	1.38704i	−0.118274	+	0.992981i	$0.537736\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−3555.58			−0.186757
$714$		0			0
$715$	−1116.94			−0.0584213
$716$		0			0
$717$		0			0
$718$		0			0
$719$	10202.7	+	17671.6i	0.529202	+	0.916604i	0.999420	+	0.0340540i	$0.0108418\pi$
$719$	10202.7	+	17671.6i	0.529202	+	0.916604i	−0.470218	+	0.882550i	$0.655825\pi$
$720$		0			0
$721$	−13280.3	−	28898.3i	−0.685968	−	1.49269i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−4002.02	+	6931.71i	−0.205009	+	0.355086i
$726$		0			0
$727$	9255.41			0.472165			0.236083	−	0.971733i	$-0.424136\pi$
$727$	9255.41			0.472165			0.236083	+	0.971733i	$0.424136\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	5992.17	−	10378.7i	0.303185	−	0.525132i
$732$		0			0
$733$	−6332.55	−	10968.3i	−0.319097	−	0.552692i	0.661203	−	0.750207i	$-0.270046\pi$
$733$	−6332.55	−	10968.3i	−0.319097	−	0.552692i	−0.980300	+	0.197515i	$0.936713\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−14106.6	−	24433.3i	−0.705052	−	1.22119i
$738$		0			0
$739$	10189.9	−	17649.5i	0.507229	−	0.878547i	−0.492736	−	0.870179i	$-0.664003\pi$
$739$	10189.9	−	17649.5i	0.507229	−	0.878547i	0.999965	−	0.00836784i	$-0.00266360\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	6237.53			0.307985			0.153993	−	0.988072i	$-0.450787\pi$
$743$	6237.53			0.307985			0.153993	+	0.988072i	$0.450787\pi$
$744$		0			0
$745$	−12955.2	+	22439.0i	−0.637103	+	1.10349i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	2075.36	+	4516.05i	0.101244	+	0.220311i
$750$		0			0
$751$	915.324	+	1585.39i	0.0444749	+	0.0770327i	0.887406	−	0.460989i	$-0.152505\pi$
$751$	915.324	+	1585.39i	0.0444749	+	0.0770327i	−0.842931	+	0.538022i	$0.819172\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−17579.4			−0.847390
$756$		0			0
$757$	−25055.9			−1.20300			−0.601500	−	0.798873i	$-0.705430\pi$
$757$	−25055.9			−1.20300			−0.601500	+	0.798873i	$0.705430\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	7998.04	+	13853.0i	0.380984	+	0.659883i	0.991203	−	0.132349i	$-0.0422520\pi$
$761$	7998.04	+	13853.0i	0.380984	+	0.659883i	−0.610219	+	0.792232i	$0.708919\pi$
$762$		0			0
$763$	−13141.6	+	18547.6i	−0.623535	+	0.880039i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	471.180	−	816.108i	0.0221817	−	0.0384198i
$768$		0			0
$769$	13978.5			0.655499			0.327749	−	0.944765i	$-0.393710\pi$
$769$	13978.5			0.655499			0.327749	+	0.944765i	$0.393710\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	2081.41	−	3605.10i	0.0968473	−	0.167744i	−0.813531	−	0.581522i	$-0.802457\pi$
$773$	2081.41	−	3605.10i	0.0968473	−	0.167744i	0.910378	+	0.413777i	$0.135791\pi$
$774$		0			0
$775$	1623.57	+	2812.11i	0.0752521	+	0.130341i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−16123.0	−	27925.9i	−0.741549	−	1.28440i
$780$		0			0
$781$	6423.90	−	11126.5i	0.294322	−	0.509780i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−9724.22			−0.442130
$786$		0			0
$787$	5108.19	−	8847.65i	0.231369	−	0.400743i	−0.726842	−	0.686804i	$-0.759013\pi$
$787$	5108.19	−	8847.65i	0.231369	−	0.400743i	0.958211	+	0.286062i	$0.0923462\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−18391.3	−	1712.18i	−0.826699	−	0.0769636i
$792$		0			0
$793$	−42.7103	−	73.9764i	−0.00191260	−	0.00331271i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	36066.8			1.60295			0.801474	−	0.598029i	$-0.204049\pi$
$797$	36066.8			1.60295			0.801474	+	0.598029i	$0.204049\pi$
$798$		0			0
$799$	5104.03			0.225992
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−2431.67	−	4211.77i	−0.106864	−	0.185094i
$804$		0			0
$805$	4966.79	−	7009.99i	0.217462	−	0.306919i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	9438.62	−	16348.2i	0.410191	−	0.710471i	−0.584720	−	0.811235i	$-0.698796\pi$
$809$	9438.62	−	16348.2i	0.410191	−	0.710471i	0.994910	+	0.100764i	$0.0321288\pi$
$810$		0			0
$811$	22508.5			0.974575			0.487288	−	0.873242i	$-0.337986\pi$
$811$	22508.5			0.974575			0.487288	+	0.873242i	$0.337986\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	16582.7	−	28722.1i	0.712720	−	1.23447i
$816$		0			0
$817$	10682.0	+	18501.7i	0.457424	+	0.792282i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	91.8340	+	159.061i	0.00390381	+	0.00676160i	0.867971	−	0.496615i	$-0.165424\pi$
$821$	91.8340	+	159.061i	0.00390381	+	0.00676160i	−0.864067	+	0.503377i	$0.832091\pi$
$822$		0			0
$823$	−19548.1	+	33858.3i	−0.827951	+	1.43405i	0.0716918	+	0.997427i	$0.477160\pi$
$823$	−19548.1	+	33858.3i	−0.827951	+	1.43405i	−0.899643	+	0.436626i	$0.856173\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	19050.5			0.801031			0.400515	−	0.916290i	$-0.368831\pi$
$827$	19050.5			0.801031			0.400515	+	0.916290i	$0.368831\pi$
$828$		0			0
$829$	9501.36	−	16456.8i	0.398065	−	0.689469i	−0.595422	−	0.803413i	$-0.703015\pi$
$829$	9501.36	−	16456.8i	0.398065	−	0.689469i	0.993487	+	0.113944i	$0.0363485\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−16318.2	+	19013.5i	−0.678741	+	0.790851i
$834$		0			0
$835$	−717.374	−	1242.53i	−0.0297315	−	0.0514964i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	10157.2			0.417958			0.208979	−	0.977920i	$-0.432986\pi$
$839$	10157.2			0.417958			0.208979	+	0.977920i	$0.432986\pi$
$840$		0			0
$841$	2952.15			0.121044
$842$		0			0
$843$		0			0
$844$		0			0
$845$	9540.37	+	16524.4i	0.388401	+	0.672730i
$846$		0			0
$847$	2790.73	+	6072.71i	0.113212	+	0.246353i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	1073.90	−	1860.06i	0.0432585	−	0.0749259i
$852$		0			0
$853$	−17282.0			−0.693700			−0.346850	−	0.937921i	$-0.612749\pi$
$853$	−17282.0			−0.693700			−0.346850	+	0.937921i	$0.612749\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−9478.85	+	16417.9i	−0.377820	+	0.654403i	−0.990745	−	0.135738i	$-0.956659\pi$
$857$	−9478.85	+	16417.9i	−0.377820	+	0.654403i	0.612925	+	0.790141i	$0.289993\pi$
$858$		0			0
$859$	−24501.4	−	42437.6i	−0.973197	−	1.68563i	−0.685760	−	0.727827i	$-0.740530\pi$
$859$	−24501.4	−	42437.6i	−0.973197	−	1.68563i	−0.287437	−	0.957800i	$-0.592803\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−11544.5	−	19995.7i	−0.455364	−	0.788714i	0.543345	−	0.839510i	$-0.317158\pi$
$863$	−11544.5	−	19995.7i	−0.455364	−	0.788714i	−0.998709	+	0.0507956i	$0.983824\pi$
$864$		0			0
$865$	−7514.42	+	13015.4i	−0.295373	+	0.511601i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	5753.14			0.224582
$870$		0			0
$871$	1855.76	−	3214.27i	0.0721930	−	0.125042i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−27985.6	−	2605.39i	−1.08124	−	0.100661i
$876$		0			0
$877$	3822.56	+	6620.88i	0.147182	+	0.254927i	0.930185	−	0.367091i	$-0.119646\pi$
$877$	3822.56	+	6620.88i	0.147182	+	0.254927i	−0.783003	+	0.622018i	$0.786313\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	9616.44			0.367748			0.183874	−	0.982950i	$-0.441136\pi$
$881$	9616.44			0.367748			0.183874	+	0.982950i	$0.441136\pi$
$882$		0			0
$883$	43669.8			1.66433			0.832166	−	0.554527i	$-0.187101\pi$
$883$	43669.8			1.66433			0.832166	+	0.554527i	$0.187101\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	18807.3	+	32575.3i	0.711938	+	1.23311i	0.964129	+	0.265435i	$0.0855156\pi$
$887$	18807.3	+	32575.3i	0.711938	+	1.23311i	−0.252191	+	0.967678i	$0.581151\pi$
$888$		0			0
$889$	−44.8798	−	4.17819i	−0.00169316	−	0.000157629i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−4549.37	+	7879.74i	−0.170480	+	0.295280i
$894$		0			0
$895$	26785.1			1.00036
$896$		0			0
$897$		0			0
$898$		0			0
$899$	5545.98	−	9605.93i	0.205750	−	0.356369i
$900$		0			0
$901$	−21300.6	−	36893.7i	−0.787598	−	1.36416i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−7270.71	−	12593.2i	−0.267057	−	0.462556i
$906$		0			0
$907$	−18868.9	+	32681.8i	−0.690772	+	1.19645i	0.280813	+	0.959763i	$0.409396\pi$
$907$	−18868.9	+	32681.8i	−0.690772	+	1.19645i	−0.971585	+	0.236690i	$0.923937\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	30477.4			1.10841			0.554205	−	0.832380i	$-0.313022\pi$
$911$	30477.4			1.10841			0.554205	+	0.832380i	$0.313022\pi$
$912$		0			0
$913$	21106.0	−	36556.7i	0.765069	−	1.32514i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	15333.1	+	33365.2i	0.552172	+	1.20154i
$918$		0			0
$919$	−2079.26	−	3601.38i	−0.0746336	−	0.129269i	0.826293	−	0.563240i	$-0.190445\pi$
$919$	−2079.26	−	3601.38i	−0.0746336	−	0.129269i	−0.900927	+	0.433971i	$0.857112\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	1690.17			0.0602735
$924$		0			0
$925$	−1961.49			−0.0697226
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−5967.51	−	10336.0i	−0.210751	−	0.365032i	0.741199	−	0.671286i	$-0.234258\pi$
$929$	−5967.51	−	10336.0i	−0.210751	−	0.365032i	−0.951950	+	0.306254i	$0.900924\pi$
$930$		0			0
$931$	−14808.7	−	42139.7i	−0.521306	−	1.48343i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−9956.32	+	17244.8i	−0.348242	+	0.603173i
$936$		0			0
$937$	−30162.5			−1.05162			−0.525809	−	0.850603i	$-0.676237\pi$
$937$	−30162.5			−1.05162			−0.525809	+	0.850603i	$0.676237\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	18275.7	−	31654.4i	0.633124	−	1.09660i	−0.353786	−	0.935327i	$-0.615106\pi$
$941$	18275.7	−	31654.4i	0.633124	−	1.09660i	0.986909	−	0.161276i	$-0.0515609\pi$
$942$		0			0
$943$	6562.57	+	11366.7i	0.226624	+	0.392525i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−24322.1	−	42127.1i	−0.834595	−	1.44556i	−0.894359	−	0.447349i	$-0.852368\pi$
$947$	−24322.1	−	42127.1i	−0.834595	−	1.44556i	0.0597639	−	0.998213i	$-0.480965\pi$
$948$		0			0
$949$	319.893	−	554.070i	0.0109422	−	0.0189525i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−20167.7			−0.685514			−0.342757	−	0.939424i	$-0.611361\pi$
$953$	−20167.7			−0.685514			−0.342757	+	0.939424i	$0.611361\pi$
$954$		0			0
$955$	12478.9	−	21614.1i	0.422836	−	0.732374i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	8016.33	−	11314.0i	0.269928	−	0.380968i
$960$		0			0
$961$	12645.6	+	21902.8i	0.424476	+	0.735214i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	24784.3			0.826771
$966$		0			0
$967$	−51088.0			−1.69895			−0.849473	−	0.527633i	$-0.823080\pi$
$967$	−51088.0			−1.69895			−0.849473	+	0.527633i	$0.823080\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−16321.7	−	28270.0i	−0.539431	−	0.934321i	−0.998935	−	0.0461457i	$-0.985306\pi$
$971$	−16321.7	−	28270.0i	−0.539431	−	0.934321i	0.459504	−	0.888176i	$-0.348027\pi$
$972$		0			0
$973$	−2572.01	−	239.448i	−0.0847430	−	0.00788936i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−26792.3	+	46405.6i	−0.877339	+	1.51960i	−0.0230899	+	0.999733i	$0.507350\pi$
$977$	−26792.3	+	46405.6i	−0.877339	+	1.51960i	−0.854250	+	0.519863i	$0.825983\pi$
$978$		0			0
$979$	−5125.23			−0.167317
$980$		0			0
$981$		0			0
$982$		0			0
$983$	19673.4	−	34075.2i	0.638334	−	1.10563i	−0.347464	−	0.937693i	$-0.612957\pi$
$983$	19673.4	−	34075.2i	0.638334	−	1.10563i	0.985798	−	0.167934i	$-0.0537095\pi$
$984$		0			0
$985$	5742.85	+	9946.91i	0.185769	+	0.321761i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−4347.90	−	7530.79i	−0.139793	−	0.242128i
$990$		0			0
$991$	1336.27	−	2314.48i	0.0428334	−	0.0741896i	−0.843814	−	0.536636i	$-0.819695\pi$
$991$	1336.27	−	2314.48i	0.0428334	−	0.0741896i	0.886647	+	0.462446i	$0.153028\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	30645.1			0.976396
$996$		0			0
$997$	585.644	−	1014.37i	0.0186034	−	0.0322220i	−0.856574	−	0.516024i	$-0.827411\pi$
$997$	585.644	−	1014.37i	0.0186034	−	0.0322220i	0.875177	+	0.483802i	$0.160745\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	756.4.k.h.109.3	yes	16
3.2	odd	2		756.4.k.f.109.6	✓	16
7.2	even	3	inner	756.4.k.h.541.3	yes	16
21.2	odd	6		756.4.k.f.541.6	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
756.4.k.f.109.6	✓	16	3.2	odd	2
756.4.k.f.541.6	yes	16	21.2	odd	6
756.4.k.h.109.3	yes	16	1.1	even	1	trivial
756.4.k.h.541.3	yes	16	7.2	even	3	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	756.k (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-4.37589 - 7.57927i) q^{5} +(18.4405 + 1.71677i) q^{7} +(-15.5735 + 26.9741i) q^{11} -4.09748 q^{13} +(-36.5246 + 63.2624i) q^{17} +(-65.1108 - 112.775i) q^{19} +(26.5021 + 45.9030i) q^{23} +(24.2031 - 41.9210i) q^{25} -165.352 q^{29} +(-33.5406 + 58.0939i) q^{31} +(-67.6819 - 147.278i) q^{35} +(-20.2607 - 35.0926i) q^{37} +247.624 q^{41} -164.059 q^{43} +(-34.9356 - 60.5102i) q^{47} +(337.105 + 63.3161i) q^{49} +(-291.593 + 505.053i) q^{53} +272.592 q^{55} +(-114.993 + 199.173i) q^{59} +(10.4236 + 18.0541i) q^{61} +(17.9301 + 31.0559i) q^{65} +(-452.903 + 784.451i) q^{67} -412.489 q^{71} +(-78.0706 + 135.222i) q^{73} +(-333.492 + 470.681i) q^{77} +(-92.3545 - 159.963i) q^{79} -1355.25 q^{83} +639.311 q^{85} +(82.2747 + 142.504i) q^{89} +(-75.5597 - 7.03441i) q^{91} +(-569.836 + 986.985i) q^{95} +474.685 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 7 q^{5} - 11 q^{7} - 43 q^{11} - 52 q^{13} + 38 q^{17} + 32 q^{19} - 118 q^{23} - 143 q^{25} + 160 q^{29} + 5 q^{31} + 161 q^{35} - 136 q^{37} - 216 q^{41} + 800 q^{43} - 78 q^{47} + 253 q^{49} + 105 q^{53}+ \cdots - 838 q^{97}+O(q^{100}) \) 16 * q + 7 * q^5 - 11 * q^7 - 43 * q^11 - 52 * q^13 + 38 * q^17 + 32 * q^19 - 118 * q^23 - 143 * q^25 + 160 * q^29 + 5 * q^31 + 161 * q^35 - 136 * q^37 - 216 * q^41 + 800 * q^43 - 78 * q^47 + 253 * q^49 + 105 * q^53 - 948 * q^55 + 574 * q^59 - 424 * q^61 - 408 * q^65 - 88 * q^67 - 856 * q^71 + 644 * q^73 - 656 * q^77 + 266 * q^79 + 2054 * q^83 + 384 * q^85 + 732 * q^89 + 1022 * q^91 - 364 * q^95 - 838 * q^97

Introduction

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