LMFDB - Embedded newform 756.4.x.a.125.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [756,4,Mod(125,756)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(756, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 5, 3]))

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

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

chi := DirichletCharacter("756.125");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 756 = 2^{2} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	756.x (of order $6$, degree $2$, not 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:	$44.6054439643$
Analytic rank:	$0$
Dimension:	$48$
Relative dimension:	$24$ over $\Q(\zeta_{6})$
Twist minimal:	no (minimal twist has level 252)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			125.12
Character	$\chi$	$=$	756.125
Dual form			756.4.x.a.629.12

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.330097 + 0.571745i) q^{5} +(-15.6440 - 9.91296i) q^{7} +O(q^{10})$	$q+(-0.330097 + 0.571745i) q^{5} +(-15.6440 - 9.91296i) q^{7} +(-21.4150 + 12.3640i) q^{11} +(-43.5283 - 25.1311i) q^{13} +67.5530 q^{17} +62.9647i q^{19} +(-135.811 - 78.4107i) q^{23} +(62.2821 + 107.876i) q^{25} +(129.859 - 74.9743i) q^{29} +(139.518 + 80.5507i) q^{31} +(10.8317 - 5.67211i) q^{35} -16.2664 q^{37} +(134.901 - 233.655i) q^{41} +(188.088 + 325.779i) q^{43} +(31.3735 + 54.3405i) q^{47} +(146.467 + 310.156i) q^{49} +136.820i q^{53} -16.3253i q^{55} +(358.829 - 621.509i) q^{59} +(23.9682 - 13.8380i) q^{61} +(28.7371 - 16.5914i) q^{65} +(-163.773 + 283.664i) q^{67} +246.304i q^{71} +261.253i q^{73} +(457.579 + 18.8649i) q^{77} +(-391.980 - 678.930i) q^{79} +(599.424 + 1038.23i) q^{83} +(-22.2990 + 38.6231i) q^{85} +968.771 q^{89} +(431.831 + 824.643i) q^{91} +(-35.9997 - 20.7845i) q^{95} +(1106.79 - 639.005i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q + 6 q^{7}+O(q^{10}) $ 48 * q + 6 * q^7	$ 48 q + 6 q^{7} + 12 q^{11} + 408 q^{23} - 600 q^{25} + 84 q^{29} + 336 q^{37} + 84 q^{43} + 318 q^{49} - 2964 q^{65} - 588 q^{67} - 2400 q^{77} + 204 q^{79} - 360 q^{85} - 1080 q^{91} - 300 q^{95}+O(q^{100}) $ 48 * q + 6 * q^7 + 12 * q^11 + 408 * q^23 - 600 * q^25 + 84 * q^29 + 336 * q^37 + 84 * q^43 + 318 * q^49 - 2964 * q^65 - 588 * q^67 - 2400 * q^77 + 204 * q^79 - 360 * q^85 - 1080 * q^91 - 300 * q^95

Display 100 coefficients 10 coefficients

Character values

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

$n$	$29$	$325$	$379$
$\chi(n)$	$e\left(\frac{5}{6}\right)$	$-1$	$1$

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 $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−0.330097	+	0.571745i	−0.0295248	+	0.0511384i	−0.880410	−	0.474213i	$-0.842733\pi$
$5$	−0.330097	+	0.571745i	−0.0295248	+	0.0511384i	0.850885	+	0.525351i	$0.176066\pi$
$6$		0			0
$7$	−15.6440	−	9.91296i	−0.844694	−	0.535249i
$7$	−15.6440	−	9.91296i	−0.844694	−	0.535249i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−21.4150	+	12.3640i	−0.586989	+	0.338898i	−0.763906	−	0.645328i	$-0.776721\pi$
$11$	−21.4150	+	12.3640i	−0.586989	+	0.338898i	0.176917	+	0.984226i	$0.443388\pi$
$12$		0			0
$13$	−43.5283	−	25.1311i	−0.928659	−	0.536162i	−0.0422718	−	0.999106i	$-0.513460\pi$
$13$	−43.5283	−	25.1311i	−0.928659	−	0.536162i	−0.886387	+	0.462945i	$0.846793\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	67.5530			0.963765			0.481882	−	0.876236i	$-0.339953\pi$
$17$	67.5530			0.963765			0.481882	+	0.876236i	$0.339953\pi$
$18$		0			0
$19$			62.9647i			0.760268i	0.924932	+	0.380134i	$0.124122\pi$
$19$			62.9647i			0.760268i	−0.924932	+	0.380134i	$0.875878\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−135.811	−	78.4107i	−1.23124	−	0.710859i	−0.263954	−	0.964535i	$-0.585027\pi$
$23$	−135.811	−	78.4107i	−1.23124	−	0.710859i	−0.967289	+	0.253676i	$0.918360\pi$
$24$		0			0
$25$	62.2821	+	107.876i	0.498257	+	0.863006i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	129.859	−	74.9743i	0.831527	−	0.480082i	−0.0228484	−	0.999739i	$-0.507273\pi$
$29$	129.859	−	74.9743i	0.831527	−	0.480082i	0.854375	+	0.519657i	$0.173940\pi$
$30$		0			0
$31$	139.518	+	80.5507i	0.808327	+	0.466688i	0.846375	−	0.532588i	$-0.178780\pi$
$31$	139.518	+	80.5507i	0.808327	+	0.466688i	−0.0380475	+	0.999276i	$0.512114\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	10.8317	−	5.67211i	0.0523112	−	0.0273932i
$36$		0			0
$37$	−16.2664			−0.0722753			−0.0361377	−	0.999347i	$-0.511505\pi$
$37$	−16.2664			−0.0722753			−0.0361377	+	0.999347i	$0.511505\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	134.901	−	233.655i	0.513853	−	0.890019i	−0.486018	−	0.873949i	$-0.661551\pi$
$41$	134.901	−	233.655i	0.513853	−	0.890019i	0.999871	−	0.0160703i	$-0.00511554\pi$
$42$		0			0
$43$	188.088	+	325.779i	0.667051	+	1.15537i	0.978725	+	0.205178i	$0.0657772\pi$
$43$	188.088	+	325.779i	0.667051	+	1.15537i	−0.311673	+	0.950189i	$0.600889\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	31.3735	+	54.3405i	0.0973679	+	0.168646i	0.910594	−	0.413301i	$-0.135624\pi$
$47$	31.3735	+	54.3405i	0.0973679	+	0.168646i	−0.813226	+	0.581947i	$0.802291\pi$
$48$		0			0
$49$	146.467	+	310.156i	0.427016	+	0.904244i
$50$		0			0
$51$		0			0
$52$		0			0
$53$			136.820i			0.354597i	0.984157	+	0.177299i	$0.0567358\pi$
$53$			136.820i			0.354597i	−0.984157	+	0.177299i	$0.943264\pi$
$54$		0			0
$55$		−	16.3253i		−	0.0400236i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	358.829	−	621.509i	0.791788	−	1.37142i	−0.133070	−	0.991107i	$-0.542484\pi$
$59$	358.829	−	621.509i	0.791788	−	1.37142i	0.924859	−	0.380311i	$-0.124183\pi$
$60$		0			0
$61$	23.9682	−	13.8380i	0.0503083	−	0.0290455i	−0.474635	−	0.880183i	$-0.657420\pi$
$61$	23.9682	−	13.8380i	0.0503083	−	0.0290455i	0.524943	+	0.851137i	$0.324087\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	28.7371	−	16.5914i	0.0548369	−	0.0316601i
$66$		0			0
$67$	−163.773	+	283.664i	−0.298628	+	0.517239i	−0.975822	−	0.218565i	$-0.929862\pi$
$67$	−163.773	+	283.664i	−0.298628	+	0.517239i	0.677194	+	0.735804i	$0.263196\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			246.304i			0.411703i	0.978583	+	0.205852i	$0.0659965\pi$
$71$			246.304i			0.411703i	−0.978583	+	0.205852i	$0.934004\pi$
$72$		0			0
$73$			261.253i			0.418868i	0.977823	+	0.209434i	$0.0671622\pi$
$73$			261.253i			0.418868i	−0.977823	+	0.209434i	$0.932838\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	457.579	+	18.8649i	0.677221	+	0.0279201i
$78$		0			0
$79$	−391.980	−	678.930i	−0.558244	−	0.966906i	−0.997643	−	0.0686145i	$-0.978142\pi$
$79$	−391.980	−	678.930i	−0.558244	−	0.966906i	0.439400	−	0.898292i	$-0.355191\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	599.424	+	1038.23i	0.792715	+	1.37302i	0.924280	+	0.381715i	$0.124666\pi$
$83$	599.424	+	1038.23i	0.792715	+	1.37302i	−0.131565	+	0.991308i	$0.542000\pi$
$84$		0			0
$85$	−22.2990	+	38.6231i	−0.0284550	+	0.0492854i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	968.771			1.15381			0.576907	−	0.816810i	$-0.304259\pi$
$89$	968.771			1.15381			0.576907	+	0.816810i	$0.304259\pi$
$90$		0			0
$91$	431.831	+	824.643i	0.497453	+	0.949957i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−35.9997	−	20.7845i	−0.0388789	−	0.0224467i
$96$		0			0
$97$	1106.79	−	639.005i	1.15853	−	0.668878i	0.207579	−	0.978218i	$-0.433442\pi$
$97$	1106.79	−	639.005i	1.15853	−	0.668878i	0.950951	+	0.309340i	$0.100108\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−285.125	−	493.852i	−0.280901	−	0.486535i	0.690706	−	0.723136i	$-0.257300\pi$
$101$	−285.125	−	493.852i	−0.280901	−	0.486535i	−0.971607	+	0.236601i	$0.923967\pi$
$102$		0			0
$103$	1559.24	+	900.228i	1.49162	+	0.861186i	0.999954	−	0.00959990i	$-0.00305579\pi$
$103$	1559.24	+	900.228i	1.49162	+	0.861186i	0.491663	+	0.870785i	$0.336389\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			2038.95i			1.84218i	0.389350	+	0.921090i	$0.372700\pi$
$107$			2038.95i			1.84218i	−0.389350	+	0.921090i	$0.627300\pi$
$108$		0			0
$109$	−1189.58			−1.04533			−0.522666	−	0.852538i	$-0.675062\pi$
$109$	−1189.58			−1.04533			−0.522666	+	0.852538i	$0.675062\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	646.454	+	373.230i	0.538171	+	0.310713i	0.744337	−	0.667804i	$-0.232766\pi$
$113$	646.454	+	373.230i	0.538171	+	0.310713i	−0.206167	+	0.978517i	$0.566099\pi$
$114$		0			0
$115$	89.6618	−	51.7663i	0.0727044	−	0.0419759i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−1056.80	−	669.650i	−0.814086	−	0.515854i
$120$		0			0
$121$	−359.764	+	623.130i	−0.270296	+	0.468167i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−164.761			−0.117893
$126$		0			0
$127$	−1832.75			−1.28055			−0.640277	−	0.768144i	$-0.721180\pi$
$127$	−1832.75			−1.28055			−0.640277	+	0.768144i	$0.721180\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	235.639	−	408.138i	0.157159	−	0.272208i	−0.776684	−	0.629890i	$-0.783100\pi$
$131$	235.639	−	408.138i	0.157159	−	0.272208i	0.933843	+	0.357683i	$0.116433\pi$
$132$		0			0
$133$	624.166	−	985.016i	0.406933	−	0.642194i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	2271.58	−	1311.50i	1.41660	−	0.817874i	0.420601	−	0.907246i	$-0.361819\pi$
$137$	2271.58	−	1311.50i	1.41660	−	0.817874i	0.995998	+	0.0893720i	$0.0284860\pi$
$138$		0			0
$139$	164.415	+	94.9252i	0.100328	+	0.0579241i	0.549324	−	0.835609i	$-0.314885\pi$
$139$	164.415	+	94.9252i	0.100328	+	0.0579241i	−0.448997	+	0.893533i	$0.648219\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	1242.88			0.726817
$144$		0			0
$145$			98.9953i			0.0566973i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	383.974	+	221.687i	0.211117	+	0.121888i	0.601830	−	0.798624i	$-0.294438\pi$
$149$	383.974	+	221.687i	0.211117	+	0.121888i	−0.390714	+	0.920512i	$0.627772\pi$
$150$		0			0
$151$	943.459	+	1634.12i	0.508461	+	0.880680i	0.999952	+	0.00979739i	$0.00311865\pi$
$151$	943.459	+	1634.12i	0.508461	+	0.880680i	−0.491491	+	0.870883i	$0.663548\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−92.1089	+	53.1791i	−0.0477314	+	0.0275577i
$156$		0			0
$157$	2133.22	+	1231.62i	1.08439	+	0.626075i	0.932078	−	0.362257i	$-0.117994\pi$
$157$	2133.22	+	1231.62i	1.08439	+	0.626075i	0.152316	+	0.988332i	$0.451327\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	1347.34	+	2572.94i	0.659537	+	1.25948i
$162$		0			0
$163$	2514.36			1.20822			0.604111	−	0.796900i	$-0.293528\pi$
$163$	2514.36			1.20822			0.604111	+	0.796900i	$0.293528\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	1415.81	−	2452.25i	0.656038	−	1.13629i	−0.325594	−	0.945510i	$-0.605564\pi$
$167$	1415.81	−	2452.25i	0.656038	−	1.13629i	0.981632	−	0.190782i	$-0.0611024\pi$
$168$		0			0
$169$	164.640	+	285.164i	0.0749385	+	0.129797i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	667.439	+	1156.04i	0.293320	+	0.508046i	0.974593	−	0.223984i	$-0.0719064\pi$
$173$	667.439	+	1156.04i	0.293320	+	0.508046i	−0.681272	+	0.732030i	$0.738573\pi$
$174$		0			0
$175$	95.0295	−	2305.00i	0.0410489	−	0.995667i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	1431.23i		−	0.597626i	−0.954312	−	0.298813i	$-0.903409\pi$
$179$		−	1431.23i		−	0.597626i	0.954312	−	0.298813i	$-0.0965907\pi$
$180$		0			0
$181$			262.769i			0.107909i	0.998543	+	0.0539543i	$0.0171825\pi$
$181$			262.769i			0.107909i	−0.998543	+	0.0539543i	$0.982817\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	5.36951	−	9.30026i	0.00213391	−	0.00369605i
$186$		0			0
$187$	−1446.65	+	835.223i	−0.565719	+	0.326618i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	97.8990	−	56.5220i	0.0370876	−	0.0214125i	−0.481342	−	0.876533i	$-0.659850\pi$
$191$	97.8990	−	56.5220i	0.0370876	−	0.0214125i	0.518429	+	0.855121i	$0.326517\pi$
$192$		0			0
$193$	−2091.54	+	3622.65i	−0.780063	+	1.35111i	0.151842	+	0.988405i	$0.451480\pi$
$193$	−2091.54	+	3622.65i	−0.780063	+	1.35111i	−0.931905	+	0.362704i	$0.881854\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	314.320i		−	0.113677i	−0.998383	−	0.0568385i	$-0.981898\pi$
$197$		−	314.320i		−	0.113677i	0.998383	−	0.0568385i	$-0.0181020\pi$
$198$		0			0
$199$			1407.97i			0.501550i	0.968045	+	0.250775i	$0.0806855\pi$
$199$			1407.97i			0.501550i	−0.968045	+	0.250775i	$0.919314\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−2774.73	−	114.395i	−0.959349	−	0.0395516i
$204$		0			0
$205$	89.0607	+	154.258i	0.0303428	+	0.0525552i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−778.494	−	1348.39i	−0.257653	−	0.446269i
$210$		0			0
$211$	−73.9358	+	128.061i	−0.0241230	+	0.0417822i	−0.877835	−	0.478963i	$-0.841013\pi$
$211$	−73.9358	+	128.061i	−0.0241230	+	0.0417822i	0.853712	+	0.520746i	$0.174346\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−248.350			−0.0787782
$216$		0			0
$217$	−1384.12	−	2643.16i	−0.432995	−	0.826865i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−2940.46	−	1697.68i	−0.895009	−	0.516734i
$222$		0			0
$223$	−2005.48	+	1157.86i	−0.602228	+	0.347696i	−0.769917	−	0.638143i	$-0.779703\pi$
$223$	−2005.48	+	1157.86i	−0.602228	+	0.347696i	0.167690	+	0.985840i	$0.446369\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	2233.35	+	3868.27i	0.653006	+	1.13104i	0.982390	+	0.186843i	$0.0598257\pi$
$227$	2233.35	+	3868.27i	0.653006	+	1.13104i	−0.329384	+	0.944196i	$0.606841\pi$
$228$		0			0
$229$	−1016.15	−	586.677i	−0.293229	−	0.169296i	0.346168	−	0.938172i	$-0.387483\pi$
$229$	−1016.15	−	586.677i	−0.293229	−	0.169296i	−0.639397	+	0.768877i	$0.720816\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			1751.08i			0.492347i	0.969226	+	0.246173i	$0.0791733\pi$
$233$			1751.08i			0.492347i	−0.969226	+	0.246173i	$0.920827\pi$
$234$		0			0
$235$	−41.4252			−0.0114991
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−5007.40	−	2891.03i	−1.35524	−	0.782447i	−0.366261	−	0.930512i	$-0.619362\pi$
$239$	−5007.40	−	2891.03i	−1.35524	−	0.782447i	−0.988978	+	0.148065i	$0.952695\pi$
$240$		0			0
$241$	230.468	−	133.061i	0.0616005	−	0.0355651i	−0.468883	−	0.883260i	$-0.655344\pi$
$241$	230.468	−	133.061i	0.0616005	−	0.0355651i	0.530484	+	0.847695i	$0.322010\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−225.678	−	18.6400i	−0.0588492	−	0.00486067i
$246$		0			0
$247$	1582.37	−	2740.74i	0.407626	−	0.706029i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−7212.14			−1.81365			−0.906826	−	0.421506i	$-0.861502\pi$
$251$	−7212.14			−1.81365			−0.906826	+	0.421506i	$0.861502\pi$
$252$		0			0
$253$	3877.87			0.963635
$254$		0			0
$255$		0			0
$256$		0			0
$257$	2752.31	−	4767.14i	0.668033	−	1.15707i	−0.310421	−	0.950599i	$-0.600470\pi$
$257$	2752.31	−	4767.14i	0.668033	−	1.15707i	0.978453	−	0.206468i	$-0.0661968\pi$
$258$		0			0
$259$	254.471	+	161.249i	0.0610505	+	0.0386853i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	4088.19	−	2360.32i	0.958511	−	0.553396i	0.0627964	−	0.998026i	$-0.479998\pi$
$263$	4088.19	−	2360.32i	0.958511	−	0.553396i	0.895714	+	0.444630i	$0.146665\pi$
$264$		0			0
$265$	−78.2261	−	45.1639i	−0.0181336	−	0.0104694i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−3595.18			−0.814877			−0.407439	−	0.913233i	$-0.633578\pi$
$269$	−3595.18			−0.814877			−0.407439	+	0.913233i	$0.633578\pi$
$270$		0			0
$271$			3673.66i			0.823466i	0.911305	+	0.411733i	$0.135076\pi$
$271$			3673.66i			0.823466i	−0.911305	+	0.411733i	$0.864924\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−2667.55	−	1540.11i	−0.584942	−	0.337716i
$276$		0			0
$277$	880.297	+	1524.72i	0.190946	+	0.330727i	0.945564	−	0.325437i	$-0.105511\pi$
$277$	880.297	+	1524.72i	0.190946	+	0.330727i	−0.754618	+	0.656164i	$0.772178\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	6960.95	−	4018.91i	1.47778	−	0.853195i	0.478093	−	0.878309i	$-0.341328\pi$
$281$	6960.95	−	4018.91i	1.47778	−	0.853195i	0.999685	+	0.0251141i	$0.00799491\pi$
$282$		0			0
$283$	−6519.11	−	3763.81i	−1.36933	−	0.790584i	−0.378488	−	0.925606i	$-0.623556\pi$
$283$	−6519.11	−	3763.81i	−1.36933	−	0.790584i	−0.990843	+	0.135022i	$0.956889\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−4426.59	+	2318.02i	−0.910430	+	0.476754i
$288$		0			0
$289$	−349.597			−0.0711575
$290$		0			0
$291$		0			0
$292$		0			0
$293$	105.807	−	183.262i	0.0210965	−	0.0365403i	−0.855284	−	0.518159i	$-0.826618\pi$
$293$	105.807	−	183.262i	0.0210965	−	0.0365403i	0.876381	+	0.481619i	$0.159951\pi$
$294$		0			0
$295$	236.897	+	410.317i	0.0467548	+	0.0809816i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	3941.08	+	6826.16i	0.762270	+	1.32029i
$300$		0			0
$301$	286.984	−	6960.98i	0.0549551	−	1.33297i
$302$		0			0
$303$		0			0
$304$		0			0
$305$			18.2716i			0.00343025i
$306$		0			0
$307$		−	2701.00i		−	0.502131i	−0.967970	−	0.251065i	$-0.919219\pi$
$307$		−	2701.00i		−	0.502131i	0.967970	−	0.251065i	$-0.0807809\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−3234.86	+	5602.95i	−0.589815	+	1.02159i	0.404442	+	0.914564i	$0.367466\pi$
$311$	−3234.86	+	5602.95i	−0.589815	+	1.02159i	−0.994256	+	0.107025i	$0.965867\pi$
$312$		0			0
$313$	−1476.26	+	852.319i	−0.266592	+	0.153917i	−0.627338	−	0.778747i	$-0.715855\pi$
$313$	−1476.26	+	852.319i	−0.266592	+	0.153917i	0.360746	+	0.932664i	$0.382522\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−1247.17	+	720.056i	−0.220972	+	0.127578i	−0.606400	−	0.795159i	$-0.707387\pi$
$317$	−1247.17	+	720.056i	−0.220972	+	0.127578i	0.385428	+	0.922738i	$0.374054\pi$
$318$		0			0
$319$	−1853.96	+	3211.16i	−0.325398	+	0.563606i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			4253.45i			0.732719i
$324$		0			0
$325$		−	6260.86i		−	1.06858i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	47.8694	−	1161.10i	0.00802166	−	0.194571i
$330$		0			0
$331$	−5661.53	−	9806.05i	−0.940138	−	1.62837i	−0.765205	−	0.643786i	$-0.777363\pi$
$331$	−5661.53	−	9806.05i	−0.940138	−	1.62837i	−0.174932	−	0.984580i	$-0.555971\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−108.122	−	187.273i	−0.0176339	−	0.0305428i
$336$		0			0
$337$	−1069.77	+	1852.89i	−0.172920	+	0.299505i	−0.939439	−	0.342715i	$-0.888653\pi$
$337$	−1069.77	+	1852.89i	−0.172920	+	0.299505i	0.766520	+	0.642221i	$0.221987\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−3983.71			−0.632639
$342$		0			0
$343$	783.244	−	6303.98i	0.123298	−	0.992370i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−3237.97	−	1869.44i	−0.500931	−	0.289213i	0.228167	−	0.973622i	$-0.426727\pi$
$347$	−3237.97	−	1869.44i	−0.500931	−	0.289213i	−0.729098	+	0.684409i	$0.760060\pi$
$348$		0			0
$349$	5644.28	−	3258.72i	0.865705	−	0.499815i	−0.000213394	−	1.00000i	$-0.500068\pi$
$349$	5644.28	−	3258.72i	0.865705	−	0.499815i	0.865919	+	0.500185i	$0.166735\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−1772.88	−	3070.72i	−0.267312	−	0.462998i	0.700855	−	0.713304i	$-0.252802\pi$
$353$	−1772.88	−	3070.72i	−0.267312	−	0.462998i	−0.968167	+	0.250306i	$0.919469\pi$
$354$		0			0
$355$	−140.823	−	81.3044i	−0.0210539	−	0.0121555i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			6414.84i			0.943071i	0.881847	+	0.471535i	$0.156300\pi$
$359$			6414.84i			0.943071i	−0.881847	+	0.471535i	$0.843700\pi$
$360$		0			0
$361$	2894.45			0.421993
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−149.370	−	86.2390i	−0.0214203	−	0.0123670i
$366$		0			0
$367$	−6771.55	+	3909.55i	−0.963139	+	0.556068i	−0.897138	−	0.441751i	$-0.854357\pi$
$367$	−6771.55	+	3909.55i	−0.963139	+	0.556068i	−0.0660011	+	0.997820i	$0.521024\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	1356.29	−	2140.40i	0.189798	−	0.299526i
$372$		0			0
$373$	5548.13	−	9609.65i	0.770165	−	1.33396i	−0.167307	−	0.985905i	$-0.553507\pi$
$373$	5548.13	−	9609.65i	0.770165	−	1.33396i	0.937472	−	0.348060i	$-0.113159\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−7536.74			−1.02961
$378$		0			0
$379$	−5192.03			−0.703686			−0.351843	−	0.936059i	$-0.614445\pi$
$379$	−5192.03			−0.703686			−0.351843	+	0.936059i	$0.614445\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−6875.13	+	11908.1i	−0.917240	+	1.58871i	−0.113652	+	0.993521i	$0.536255\pi$
$383$	−6875.13	+	11908.1i	−0.917240	+	1.58871i	−0.803588	+	0.595186i	$0.797078\pi$
$384$		0			0
$385$	−161.832	+	255.392i	−0.0214226	+	0.0338077i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	8425.29	−	4864.34i	1.09815	−	0.634016i	0.162413	−	0.986723i	$-0.448072\pi$
$389$	8425.29	−	4864.34i	1.09815	−	0.634016i	0.935734	+	0.352707i	$0.114739\pi$
$390$		0			0
$391$	−9174.45	−	5296.87i	−1.18663	−	0.685101i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	517.567			0.0659281
$396$		0			0
$397$			9192.44i			1.16210i	0.813866	+	0.581052i	$0.197359\pi$
$397$			9192.44i			1.16210i	−0.813866	+	0.581052i	$0.802641\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−7796.50	−	4501.31i	−0.970919	−	0.560560i	−0.0714027	−	0.997448i	$-0.522748\pi$
$401$	−7796.50	−	4501.31i	−0.970919	−	0.560560i	−0.899516	+	0.436887i	$0.856081\pi$
$402$		0			0
$403$	−4048.65	−	7012.46i	−0.500440	−	0.866788i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	348.346	−	201.118i	0.0424248	−	0.0244940i
$408$		0			0
$409$	4580.39	+	2644.49i	0.553755	+	0.319711i	0.750635	−	0.660717i	$-0.229748\pi$
$409$	4580.39	+	2644.49i	0.553755	+	0.319711i	−0.196880	+	0.980428i	$0.563081\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−11774.5	+	6165.81i	−1.40287	+	0.734624i
$414$		0			0
$415$	−791.473			−0.0936190
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−2805.44	+	4859.17i	−0.327100	+	0.566554i	−0.981935	−	0.189218i	$-0.939405\pi$
$419$	−2805.44	+	4859.17i	−0.327100	+	0.566554i	0.654835	+	0.755772i	$0.272738\pi$
$420$		0			0
$421$	−641.365	−	1110.88i	−0.0742475	−	0.128600i	0.826511	−	0.562920i	$-0.190322\pi$
$421$	−641.365	−	1110.88i	−0.0742475	−	0.128600i	−0.900759	+	0.434320i	$0.856989\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	4207.34	+	7287.32i	0.480202	+	0.831734i
$426$		0			0
$427$	−512.133	−	21.1140i	−0.0580418	−	0.00239292i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	7183.17i		−	0.802787i	−0.915906	−	0.401393i	$-0.868526\pi$
$431$		−	7183.17i		−	0.802787i	0.915906	−	0.401393i	$-0.131474\pi$
$432$		0			0
$433$		−	15271.9i		−	1.69497i	−0.530820	−	0.847485i	$-0.678116\pi$
$433$		−	15271.9i		−	1.69497i	0.530820	−	0.847485i	$-0.321884\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	4937.10	−	8551.31i	0.540443	−	0.936075i
$438$		0			0
$439$	−5050.88	+	2916.13i	−0.549123	+	0.317037i	−0.748768	−	0.662832i	$-0.769354\pi$
$439$	−5050.88	+	2916.13i	−0.549123	+	0.317037i	0.199645	+	0.979868i	$0.436021\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−2636.52	+	1522.20i	−0.282765	+	0.163255i	−0.634675	−	0.772780i	$-0.718866\pi$
$443$	−2636.52	+	1522.20i	−0.282765	+	0.163255i	0.351909	+	0.936034i	$0.385533\pi$
$444$		0			0
$445$	−319.788	+	553.890i	−0.0340661	+	0.0590043i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	1445.89i		−	0.151973i	−0.997109	−	0.0759863i	$-0.975789\pi$
$449$		−	1445.89i		−	0.151973i	0.997109	−	0.0759863i	$-0.0242105\pi$
$450$		0			0
$451$			6671.64i			0.696575i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−614.032	−	25.3150i	−0.0632665	−	0.00260832i
$456$		0			0
$457$	8001.23	+	13858.5i	0.818998	+	1.41855i	0.906422	+	0.422374i	$0.138803\pi$
$457$	8001.23	+	13858.5i	0.818998	+	1.41855i	−0.0874239	+	0.996171i	$0.527863\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−6375.19	−	11042.2i	−0.644083	−	1.11559i	−0.984512	−	0.175315i	$-0.943906\pi$
$461$	−6375.19	−	11042.2i	−0.644083	−	1.11559i	0.340429	−	0.940270i	$-0.389428\pi$
$462$		0			0
$463$	3779.86	−	6546.91i	0.379406	−	0.657151i	−0.611570	−	0.791191i	$-0.709462\pi$
$463$	3779.86	−	6546.91i	0.379406	−	0.657151i	0.990976	+	0.134040i	$0.0427949\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	15808.4			1.56643			0.783217	−	0.621749i	$-0.213577\pi$
$467$	15808.4			1.56643			0.783217	+	0.621749i	$0.213577\pi$
$468$		0			0
$469$	5374.01	−	2814.14i	0.529102	−	0.277068i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−8055.84	−	4651.04i	−0.783103	−	0.452125i
$474$		0			0
$475$	−6792.36	+	3921.57i	−0.656115	+	0.378808i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	4236.78	+	7338.32i	0.404141	+	0.699992i	0.994221	−	0.107352i	$-0.0342372\pi$
$479$	4236.78	+	7338.32i	0.404141	+	0.699992i	−0.590080	+	0.807345i	$0.700904\pi$
$480$		0			0
$481$	708.050	+	408.793i	0.0671191	+	0.0387512i
$482$		0			0
$483$		0			0
$484$		0			0
$485$			843.735i			0.0789939i
$486$		0			0
$487$	−5933.46			−0.552096			−0.276048	−	0.961144i	$-0.589025\pi$
$487$	−5933.46			−0.552096			−0.276048	+	0.961144i	$0.589025\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−232.094	−	133.999i	−0.0213325	−	0.0123163i	0.489296	−	0.872118i	$-0.337254\pi$
$491$	−232.094	−	133.999i	−0.0213325	−	0.0123163i	−0.510628	+	0.859802i	$0.670587\pi$
$492$		0			0
$493$	8772.38	−	5064.74i	0.801396	−	0.462686i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	2441.60	−	3853.17i	0.220364	−	0.347763i
$498$		0			0
$499$	−5365.31	+	9292.98i	−0.481331	+	0.833689i	−0.999770	−	0.0214248i	$-0.993180\pi$
$499$	−5365.31	+	9292.98i	−0.481331	+	0.833689i	0.518440	+	0.855114i	$0.326513\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	1490.98			0.132166			0.0660830	−	0.997814i	$-0.478950\pi$
$503$	1490.98			0.132166			0.0660830	+	0.997814i	$0.478950\pi$
$504$		0			0
$505$	376.476			0.0331742
$506$		0			0
$507$		0			0
$508$		0			0
$509$	5918.88	−	10251.8i	0.515422	−	0.892736i	−0.484418	−	0.874837i	$-0.660969\pi$
$509$	5918.88	−	10251.8i	0.515422	−	0.892736i	0.999840	−	0.0178998i	$-0.00569800\pi$
$510$		0			0
$511$	2589.79	−	4087.04i	0.224199	−	0.353816i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−1029.40	+	594.326i	−0.0880794	+	0.0508526i
$516$		0			0
$517$	−1343.73	−	775.802i	−0.114308	−	0.0659956i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	5868.12			0.493449			0.246724	−	0.969086i	$-0.420646\pi$
$521$	5868.12			0.493449			0.246724	+	0.969086i	$0.420646\pi$
$522$		0			0
$523$			20680.0i			1.72901i	0.502622	+	0.864507i	$0.332369\pi$
$523$			20680.0i			1.72901i	−0.502622	+	0.864507i	$0.667631\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	9424.84	+	5441.44i	0.779037	+	0.449777i
$528$		0			0
$529$	6212.96	+	10761.2i	0.510640	+	0.884455i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−11744.0	+	6780.40i	−0.954388	+	0.551016i
$534$		0			0
$535$	−1165.76	−	673.053i	−0.0942062	−	0.0543900i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−6971.35	−	4831.09i	−0.557100	−	0.386066i
$540$		0			0
$541$	7401.95			0.588234			0.294117	−	0.955769i	$-0.404974\pi$
$541$	7401.95			0.588234			0.294117	+	0.955769i	$0.404974\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	392.677	−	680.137i	0.0308632	−	0.0534566i
$546$		0			0
$547$	5185.72	+	8981.93i	0.405348	+	0.702083i	0.994362	−	0.106040i	$-0.0338171\pi$
$547$	5185.72	+	8981.93i	0.405348	+	0.702083i	−0.589014	+	0.808123i	$0.700484\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	4720.73	+	8176.55i	0.364991	+	0.632183i
$552$		0			0
$553$	−598.081	+	14506.8i	−0.0459909	+	1.11554i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			23317.1i			1.77375i	0.462014	+	0.886873i	$0.347127\pi$
$557$			23317.1i			1.77375i	−0.462014	+	0.886873i	$0.652873\pi$
$558$		0			0
$559$		−	18907.4i		−	1.43059i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	4168.16	−	7219.47i	0.312020	−	0.540434i	−0.666780	−	0.745255i	$-0.732328\pi$
$563$	4168.16	−	7219.47i	0.312020	−	0.540434i	0.978799	+	0.204821i	$0.0656611\pi$
$564$		0			0
$565$	−426.785	+	246.405i	−0.0317788	+	0.0183475i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	19760.4	−	11408.7i	1.45589	−	0.840556i	0.457080	−	0.889425i	$-0.348895\pi$
$569$	19760.4	−	11408.7i	1.45589	−	0.840556i	0.998805	+	0.0488697i	$0.0155619\pi$
$570$		0			0
$571$	4072.82	−	7054.34i	0.298498	−	0.517014i	−0.677295	−	0.735712i	$-0.736848\pi$
$571$	4072.82	−	7054.34i	0.298498	−	0.517014i	0.975793	+	0.218698i	$0.0701810\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	19534.3i		−	1.41676i
$576$		0			0
$577$			19823.7i			1.43028i	0.698982	+	0.715140i	$0.253637\pi$
$577$			19823.7i			1.43028i	−0.698982	+	0.715140i	$0.746363\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	914.597	−	22184.1i	0.0653079	−	1.58408i
$582$		0			0
$583$	−1691.64	−	2930.00i	−0.120172	−	0.208145i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	349.569	+	605.472i	0.0245797	+	0.0425732i	0.878054	−	0.478562i	$-0.158842\pi$
$587$	349.569	+	605.472i	0.0245797	+	0.0425732i	−0.853474	+	0.521136i	$0.825509\pi$
$588$		0			0
$589$	−5071.84	+	8784.69i	−0.354808	+	0.614545i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	22770.8			1.57687			0.788434	−	0.615119i	$-0.210892\pi$
$593$	22770.8			1.57687			0.788434	+	0.615119i	$0.210892\pi$
$594$		0			0
$595$	731.714	−	383.168i	0.0504157	−	0.0264006i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	1737.23	+	1002.99i	0.118500	+	0.0684159i	0.558078	−	0.829788i	$-0.311539\pi$
$599$	1737.23	+	1002.99i	0.118500	+	0.0684159i	−0.439579	+	0.898204i	$0.644872\pi$
$600$		0			0
$601$	12437.9	−	7181.05i	0.844184	−	0.487390i	−0.0145004	−	0.999895i	$-0.504616\pi$
$601$	12437.9	−	7181.05i	0.844184	−	0.487390i	0.858684	+	0.512505i	$0.171282\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−237.514	−	411.387i	−0.0159609	−	0.0276450i
$606$		0			0
$607$	13864.4	+	8004.64i	0.927085	+	0.535253i	0.885888	−	0.463898i	$-0.153550\pi$
$607$	13864.4	+	8004.64i	0.927085	+	0.535253i	0.0411965	+	0.999151i	$0.486883\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	3153.79i		−	0.208820i
$612$		0			0
$613$	721.073			0.0475104			0.0237552	−	0.999718i	$-0.492438\pi$
$613$	721.073			0.0475104			0.0237552	+	0.999718i	$0.492438\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	7798.16	+	4502.27i	0.508820	+	0.293767i	0.732349	−	0.680930i	$-0.238424\pi$
$617$	7798.16	+	4502.27i	0.508820	+	0.293767i	−0.223528	+	0.974697i	$0.571758\pi$
$618$		0			0
$619$	−9685.79	+	5592.09i	−0.628925	+	0.363110i	−0.780336	−	0.625361i	$-0.784952\pi$
$619$	−9685.79	+	5592.09i	−0.628925	+	0.363110i	0.151411	+	0.988471i	$0.451618\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−15155.4	−	9603.38i	−0.974620	−	0.617578i
$624$		0			0
$625$	−7730.87	+	13390.3i	−0.494776	+	0.856977i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−1098.85			−0.0696564
$630$		0			0
$631$	8969.76			0.565897			0.282948	−	0.959135i	$-0.408688\pi$
$631$	8969.76			0.565897			0.282948	+	0.959135i	$0.408688\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	604.986	−	1047.87i	0.0378081	−	0.0654855i
$636$		0			0
$637$	1419.11	−	17181.4i	0.0882684	−	1.06868i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	1495.77	−	863.586i	0.0921678	−	0.0532131i	−0.453208	−	0.891405i	$-0.649720\pi$
$641$	1495.77	−	863.586i	0.0921678	−	0.0532131i	0.545375	+	0.838192i	$0.316387\pi$
$642$		0			0
$643$	15392.5	+	8886.86i	0.944045	+	0.545045i	0.891226	−	0.453560i	$-0.149846\pi$
$643$	15392.5	+	8886.86i	0.944045	+	0.545045i	0.0528189	+	0.998604i	$0.483179\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−7518.12			−0.456828			−0.228414	−	0.973564i	$-0.573354\pi$
$647$	−7518.12			−0.456828			−0.228414	+	0.973564i	$0.573354\pi$
$648$		0			0
$649$			17746.2i			1.07334i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−5483.68	−	3166.00i	−0.328626	−	0.189732i	0.326605	−	0.945161i	$-0.394095\pi$
$653$	−5483.68	−	3166.00i	−0.328626	−	0.189732i	−0.655231	+	0.755429i	$0.727429\pi$
$654$		0			0
$655$	155.567	+	269.451i	0.00928018	+	0.0160737i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−5478.28	+	3162.89i	−0.323829	+	0.186963i	−0.653098	−	0.757273i	$-0.726531\pi$
$659$	−5478.28	+	3162.89i	−0.323829	+	0.186963i	0.329269	+	0.944236i	$0.393198\pi$
$660$		0			0
$661$	18882.2	+	10901.7i	1.11110	+	0.641491i	0.939113	−	0.343608i	$-0.111649\pi$
$661$	18882.2	+	10901.7i	1.11110	+	0.641491i	0.171983	+	0.985100i	$0.444983\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	357.143	+	682.015i	0.0208262	+	0.0397705i
$666$		0			0
$667$	−23515.1			−1.36508
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−342.186	+	592.684i	−0.0196870	+	0.0340988i
$672$		0			0
$673$	−12150.3	−	21044.9i	−0.695926	−	1.20538i	−0.969868	−	0.243632i	$-0.921661\pi$
$673$	−12150.3	−	21044.9i	−0.695926	−	1.20538i	0.273942	−	0.961746i	$-0.411672\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	15606.5	+	27031.3i	0.885979	+	1.53456i	0.844587	+	0.535418i	$0.179846\pi$
$677$	15606.5	+	27031.3i	0.885979	+	1.53456i	0.0413922	+	0.999143i	$0.486821\pi$
$678$		0			0
$679$	−23649.0	−	974.990i	−1.33662	−	0.0551056i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			32330.6i			1.81127i	0.424063	+	0.905633i	$0.360603\pi$
$683$			32330.6i			1.81127i	−0.424063	+	0.905633i	$0.639397\pi$
$684$		0			0
$685$			1731.69i			0.0965902i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	3438.43	−	5955.53i	0.190121	−	0.329300i
$690$		0			0
$691$	−10098.5	+	5830.40i	−0.555958	+	0.320982i	−0.751521	−	0.659709i	$-0.770680\pi$
$691$	−10098.5	+	5830.40i	−0.555958	+	0.320982i	0.195564	+	0.980691i	$0.437346\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−108.546	+	62.6691i	−0.00592430	+	0.00342040i
$696$		0			0
$697$	9112.95	−	15784.1i	0.495233	−	0.857769i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			35502.0i			1.91283i	0.292015	+	0.956414i	$0.405674\pi$
$701$			35502.0i			1.91283i	−0.292015	+	0.956414i	$0.594326\pi$
$702$		0			0
$703$		−	1024.21i		−	0.0549486i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−435.042	+	10552.2i	−0.0231421	+	0.561326i
$708$		0			0
$709$	6912.86	+	11973.4i	0.366175	+	0.634233i	0.988964	−	0.148156i	$-0.0473339\pi$
$709$	6912.86	+	11973.4i	0.366175	+	0.634233i	−0.622789	+	0.782390i	$0.714001\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−12632.1	−	21879.4i	−0.663498	−	1.14921i
$714$		0			0
$715$	−410.271	+	710.610i	−0.0214591	+	0.0371683i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	35196.0			1.82558			0.912788	−	0.408434i	$-0.133925\pi$
$719$	35196.0			1.82558			0.912788	+	0.408434i	$0.133925\pi$
$720$		0			0
$721$	−15468.8	−	29539.8i	−0.799011	−	1.52583i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	16175.8	+	9339.11i	0.828627	+	0.478408i
$726$		0			0
$727$	27427.8	−	15835.4i	1.39923	−	0.807846i	0.404918	−	0.914353i	$-0.367300\pi$
$727$	27427.8	−	15835.4i	1.39923	−	0.807846i	0.994312	+	0.106507i	$0.0339667\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	12705.9	+	22007.3i	0.642881	+	1.11350i
$732$		0			0
$733$	33629.6	+	19416.1i	1.69460	+	0.978375i	0.950717	+	0.310060i	$0.100349\pi$
$733$	33629.6	+	19416.1i	1.69460	+	0.978375i	0.743878	+	0.668315i	$0.232984\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	8099.56i		−	0.404818i
$738$		0			0
$739$	−63.2905			−0.00315044			−0.00157522	−	0.999999i	$-0.500501\pi$
$739$	−63.2905			−0.00315044			−0.00157522	+	0.999999i	$0.500501\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	17506.0	+	10107.1i	0.864376	+	0.499048i	0.865475	−	0.500952i	$-0.167017\pi$
$743$	17506.0	+	10107.1i	0.864376	+	0.499048i	−0.00109946	+	0.999999i	$0.500350\pi$
$744$		0			0
$745$	−253.497	+	146.357i	−0.0124663	+	0.00719745i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	20212.1	−	31897.3i	0.986025	−	1.55608i
$750$		0			0
$751$	2887.55	−	5001.39i	0.140304	−	0.243014i	−0.787307	−	0.616561i	$-0.788525\pi$
$751$	2887.55	−	5001.39i	0.140304	−	0.243014i	0.927611	+	0.373547i	$0.121859\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−1245.73			−0.0600488
$756$		0			0
$757$	−10392.0			−0.498949			−0.249474	−	0.968381i	$-0.580258\pi$
$757$	−10392.0			−0.498949			−0.249474	+	0.968381i	$0.580258\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	4343.28	−	7522.79i	0.206891	−	0.358345i	−0.743843	−	0.668355i	$-0.766999\pi$
$761$	4343.28	−	7522.79i	0.206891	−	0.358345i	0.950733	+	0.310009i	$0.100332\pi$
$762$		0			0
$763$	18609.7	+	11792.3i	0.882985	+	0.559513i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−31238.4	+	18035.5i	−1.47060	+	0.849053i
$768$		0			0
$769$	−24932.6	−	14394.8i	−1.16917	−	0.675021i	−0.215687	−	0.976463i	$-0.569199\pi$
$769$	−24932.6	−	14394.8i	−1.16917	−	0.675021i	−0.953485	+	0.301441i	$0.902532\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	12521.5			0.582622			0.291311	−	0.956628i	$-0.405909\pi$
$773$	12521.5			0.582622			0.291311	+	0.956628i	$0.405909\pi$
$774$		0			0
$775$			20067.4i			0.930121i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	14712.0	+	8493.98i	0.676653	+	0.390666i
$780$		0			0
$781$	−3045.30	−	5274.62i	−0.139526	−	0.241665i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−1408.34	+	813.107i	−0.0640330	+	0.0369695i
$786$		0			0
$787$	15973.1	+	9222.08i	0.723482	+	0.417702i	0.816033	−	0.578006i	$-0.196169\pi$
$787$	15973.1	+	9222.08i	0.723482	+	0.417702i	−0.0925512	+	0.995708i	$0.529502\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−6413.28	−	12247.1i	−0.288281	−	0.550513i
$792$		0			0
$793$	−1391.06			−0.0622924
$794$		0			0
$795$		0			0
$796$		0			0
$797$	4810.14	−	8331.41i	0.213782	−	0.370281i	−0.739113	−	0.673581i	$-0.764755\pi$
$797$	4810.14	−	8331.41i	0.213782	−	0.370281i	0.952895	+	0.303300i	$0.0980886\pi$
$798$		0			0
$799$	2119.37	+	3670.86i	0.0938398	+	0.162535i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−3230.13	−	5594.75i	−0.141954	−	0.245871i
$804$		0			0
$805$	−1915.82	−	78.9846i	−0.0838806	−	0.00345819i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	18955.4i		−	0.823776i	−0.911234	−	0.411888i	$-0.864869\pi$
$809$		−	18955.4i		−	0.823776i	0.911234	−	0.411888i	$-0.135131\pi$
$810$		0			0
$811$			16233.0i			0.702857i	0.936215	+	0.351428i	$0.114304\pi$
$811$			16233.0i			0.702857i	−0.936215	+	0.351428i	$0.885696\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−829.984	+	1437.58i	−0.0356725	+	0.0617866i
$816$		0			0
$817$	−20512.5	+	11842.9i	−0.878388	+	0.507138i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−15840.8	+	9145.67i	−0.673382	+	0.388777i	−0.797357	−	0.603508i	$-0.793769\pi$
$821$	−15840.8	+	9145.67i	−0.673382	+	0.388777i	0.123975	+	0.992285i	$0.460436\pi$
$822$		0			0
$823$	9144.29	−	15838.4i	0.387302	−	0.670827i	−0.604783	−	0.796390i	$-0.706740\pi$
$823$	9144.29	−	15838.4i	0.387302	−	0.670827i	0.992086	+	0.125563i	$0.0400737\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	26216.6i		−	1.10234i	−0.834392	−	0.551172i	$-0.814181\pi$
$827$		−	26216.6i		−	1.10234i	0.834392	−	0.551172i	$-0.185819\pi$
$828$		0			0
$829$		−	7768.92i		−	0.325483i	−0.986669	−	0.162742i	$-0.947966\pi$
$829$		−	7768.92i		−	0.325483i	0.986669	−	0.162742i	$-0.0520337\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	9894.25	+	20951.9i	0.411543	+	0.871478i
$834$		0			0
$835$	934.708	+	1618.96i	0.0387388	+	0.0670976i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−22208.7	−	38466.7i	−0.913863	−	1.58286i	−0.808558	−	0.588416i	$-0.799752\pi$
$839$	−22208.7	−	38466.7i	−0.913863	−	1.58286i	−0.105304	−	0.994440i	$-0.533582\pi$
$840$		0			0
$841$	−952.198	+	1649.26i	−0.0390421	+	0.0676229i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−217.389			−0.00885017
$846$		0			0
$847$	11805.2	−	6181.89i	0.478903	−	0.250782i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	2209.17	+	1275.46i	0.0889885	+	0.0513775i
$852$		0			0
$853$	1982.58	−	1144.64i	0.0795806	−	0.0459459i	−0.459682	−	0.888084i	$-0.652036\pi$
$853$	1982.58	−	1144.64i	0.0795806	−	0.0459459i	0.539262	+	0.842138i	$0.318703\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	10021.5	+	17357.7i	0.399448	+	0.691864i	0.993658	−	0.112446i	$-0.0358685\pi$
$857$	10021.5	+	17357.7i	0.399448	+	0.691864i	−0.594210	+	0.804310i	$0.702535\pi$
$858$		0			0
$859$	−2786.72	−	1608.91i	−0.110689	−	0.0639062i	0.443634	−	0.896208i	$-0.353689\pi$
$859$	−2786.72	−	1608.91i	−0.110689	−	0.0639062i	−0.554322	+	0.832302i	$0.687023\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	30086.4i		−	1.18674i	−0.804931	−	0.593368i	$-0.797798\pi$
$863$		−	30086.4i		−	1.18674i	0.804931	−	0.593368i	$-0.202202\pi$
$864$		0			0
$865$	−881.279			−0.0346409
$866$		0			0
$867$		0			0
$868$		0			0
$869$	16788.6	+	9692.87i	0.655365	+	0.378375i
$870$		0			0
$871$	14257.5	−	8231.59i	0.554648	−	0.320226i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	2577.51	+	1633.27i	0.0995837	+	0.0631023i
$876$		0			0
$877$	−13897.7	+	24071.6i	−0.535112	+	0.926841i	0.464046	+	0.885811i	$0.346397\pi$
$877$	−13897.7	+	24071.6i	−0.535112	+	0.926841i	−0.999158	+	0.0410299i	$0.986936\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−43478.8			−1.66270			−0.831349	−	0.555751i	$-0.812431\pi$
$881$	−43478.8			−1.66270			−0.831349	+	0.555751i	$0.812431\pi$
$882$		0			0
$883$	18034.0			0.687308			0.343654	−	0.939096i	$-0.388335\pi$
$883$	18034.0			0.687308			0.343654	+	0.939096i	$0.388335\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−4520.21	+	7829.24i	−0.171109	+	0.296370i	−0.938808	−	0.344441i	$-0.888068\pi$
$887$	−4520.21	+	7829.24i	−0.171109	+	0.296370i	0.767699	+	0.640811i	$0.221402\pi$
$888$		0			0
$889$	28671.5	+	18168.0i	1.08168	+	0.685416i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−3421.53	+	1975.42i	−0.128216	+	0.0740257i
$894$		0			0
$895$	818.298	+	472.445i	0.0305617	+	0.0176448i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	24156.9			0.896194
$900$		0			0
$901$			9242.59i			0.341748i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−150.237	−	86.7392i	−0.00551827	−	0.00318598i
$906$		0			0
$907$	−20675.3	−	35810.7i	−0.756905	−	1.31100i	−0.944422	−	0.328737i	$-0.893377\pi$
$907$	−20675.3	−	35810.7i	−0.756905	−	1.31100i	0.187516	−	0.982261i	$-0.439956\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−36143.5	+	20867.5i	−1.31448	+	0.758913i	−0.982834	−	0.184493i	$-0.940936\pi$
$911$	−36143.5	+	20867.5i	−1.31448	+	0.758913i	−0.331642	+	0.943405i	$0.607602\pi$
$912$		0			0
$913$	−25673.4	−	14822.5i	−0.930630	−	0.537299i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−7732.18	+	4049.02i	−0.278450	+	0.145813i
$918$		0			0
$919$	−4424.74			−0.158824			−0.0794118	−	0.996842i	$-0.525304\pi$
$919$	−4424.74			−0.158824			−0.0794118	+	0.996842i	$0.525304\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	6189.89	−	10721.2i	0.220740	−	0.382332i
$924$		0			0
$925$	−1013.11	−	1754.75i	−0.0360117	−	0.0623740i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	16928.2	+	29320.5i	0.597843	+	1.03550i	0.993139	+	0.116941i	$0.0373089\pi$
$929$	16928.2	+	29320.5i	0.597843	+	1.03550i	−0.395295	+	0.918554i	$0.629358\pi$
$930$		0			0
$931$	−19528.8	+	9222.22i	−0.687467	+	0.324647i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	1102.82i		−	0.0385733i
$936$		0			0
$937$		−	26589.0i		−	0.927026i	−0.886090	−	0.463513i	$-0.846589\pi$
$937$		−	26589.0i		−	0.927026i	0.886090	−	0.463513i	$-0.153411\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−19164.6	+	33194.1i	−0.663919	+	1.14994i	0.315658	+	0.948873i	$0.397775\pi$
$941$	−19164.6	+	33194.1i	−0.663919	+	1.14994i	−0.979577	+	0.201069i	$0.935558\pi$
$942$		0			0
$943$	−36642.1	+	21155.3i	−1.26536	+	0.730553i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−27807.6	+	16054.7i	−0.954196	+	0.550906i	−0.894382	−	0.447304i	$-0.852384\pi$
$947$	−27807.6	+	16054.7i	−0.954196	+	0.550906i	−0.0598145	+	0.998210i	$0.519051\pi$
$948$		0			0
$949$	6565.57	−	11371.9i	0.224581	−	0.388986i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	40510.3i		−	1.37697i	−0.725249	−	0.688487i	$-0.758275\pi$
$953$		−	40510.3i		−	1.37697i	0.725249	−	0.688487i	$-0.241725\pi$
$954$		0			0
$955$			74.6311i			0.00252880i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−48537.3	−	2001.07i	−1.63436	−	0.0673806i
$960$		0			0
$961$	−1918.68	−	3323.26i	−0.0644048	−	0.111552i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−1380.82	−	2391.65i	−0.0460624	−	0.0797824i
$966$		0			0
$967$	−2453.20	+	4249.08i	−0.0815820	+	0.141304i	−0.903930	−	0.427681i	$-0.859331\pi$
$967$	−2453.20	+	4249.08i	−0.0815820	+	0.141304i	0.822348	+	0.568985i	$0.192664\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	41704.4			1.37833			0.689165	−	0.724604i	$-0.257977\pi$
$971$	41704.4			1.37833			0.689165	+	0.724604i	$0.257977\pi$
$972$		0			0
$973$	−1631.12	−	3114.85i	−0.0537422	−	0.102628i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−39726.6	−	22936.2i	−1.30089	−	0.751067i	−0.320330	−	0.947306i	$-0.603794\pi$
$977$	−39726.6	−	22936.2i	−1.30089	−	0.751067i	−0.980556	+	0.196239i	$0.937127\pi$
$978$		0			0
$979$	−20746.3	+	11977.9i	−0.677276	+	0.391026i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−29949.7	−	51874.3i	−0.971766	−	1.68315i	−0.690219	−	0.723600i	$-0.742486\pi$
$983$	−29949.7	−	51874.3i	−0.971766	−	1.68315i	−0.281546	−	0.959548i	$-0.590847\pi$
$984$		0			0
$985$	179.711	+	103.756i	0.00581326	+	0.00335629i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	58992.5i		−	1.89672i
$990$		0			0
$991$	37752.2			1.21013			0.605065	−	0.796176i	$-0.293147\pi$
$991$	37752.2			1.21013			0.605065	+	0.796176i	$0.293147\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−805.002	−	464.768i	−0.0256485	−	0.0148082i
$996$		0			0
$997$	−42235.3	+	24384.6i	−1.34163	+	0.774591i	−0.987047	−	0.160433i	$-0.948711\pi$
$997$	−42235.3	+	24384.6i	−1.34163	+	0.774591i	−0.354585	+	0.935024i	$0.615378\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	756.4.x.a.125.12		48
3.2	odd	2		252.4.x.a.41.7	✓	48
7.6	odd	2	inner	756.4.x.a.125.13		48
9.2	odd	6	inner	756.4.x.a.629.13		48
9.4	even	3		2268.4.f.a.1133.26		48
9.5	odd	6		2268.4.f.a.1133.23		48
9.7	even	3		252.4.x.a.209.18	yes	48
21.20	even	2		252.4.x.a.41.18	yes	48
63.13	odd	6		2268.4.f.a.1133.24		48
63.20	even	6	inner	756.4.x.a.629.12		48
63.34	odd	6		252.4.x.a.209.7	yes	48
63.41	even	6		2268.4.f.a.1133.25		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.4.x.a.41.7	✓	48	3.2	odd	2
252.4.x.a.41.18	yes	48	21.20	even	2
252.4.x.a.209.7	yes	48	63.34	odd	6
252.4.x.a.209.18	yes	48	9.7	even	3
756.4.x.a.125.12		48	1.1	even	1	trivial
756.4.x.a.125.13		48	7.6	odd	2	inner
756.4.x.a.629.12		48	63.20	even	6	inner
756.4.x.a.629.13		48	9.2	odd	6	inner
2268.4.f.a.1133.23		48	9.5	odd	6
2268.4.f.a.1133.24		48	63.13	odd	6
2268.4.f.a.1133.25		48	63.41	even	6
2268.4.f.a.1133.26		48	9.4	even	3

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:	\( N \)	\(=\)	\( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	756.x (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.330097 + 0.571745i) q^{5} +(-15.6440 - 9.91296i) q^{7} +O(q^{10})\)	\(q+(-0.330097 + 0.571745i) q^{5} +(-15.6440 - 9.91296i) q^{7} +(-21.4150 + 12.3640i) q^{11} +(-43.5283 - 25.1311i) q^{13} +67.5530 q^{17} +62.9647i q^{19} +(-135.811 - 78.4107i) q^{23} +(62.2821 + 107.876i) q^{25} +(129.859 - 74.9743i) q^{29} +(139.518 + 80.5507i) q^{31} +(10.8317 - 5.67211i) q^{35} -16.2664 q^{37} +(134.901 - 233.655i) q^{41} +(188.088 + 325.779i) q^{43} +(31.3735 + 54.3405i) q^{47} +(146.467 + 310.156i) q^{49} +136.820i q^{53} -16.3253i q^{55} +(358.829 - 621.509i) q^{59} +(23.9682 - 13.8380i) q^{61} +(28.7371 - 16.5914i) q^{65} +(-163.773 + 283.664i) q^{67} +246.304i q^{71} +261.253i q^{73} +(457.579 + 18.8649i) q^{77} +(-391.980 - 678.930i) q^{79} +(599.424 + 1038.23i) q^{83} +(-22.2990 + 38.6231i) q^{85} +968.771 q^{89} +(431.831 + 824.643i) q^{91} +(-35.9997 - 20.7845i) q^{95} +(1106.79 - 639.005i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 6 q^{7}+O(q^{10}) \) 48 * q + 6 * q^7	\( 48 q + 6 q^{7} + 12 q^{11} + 408 q^{23} - 600 q^{25} + 84 q^{29} + 336 q^{37} + 84 q^{43} + 318 q^{49} - 2964 q^{65} - 588 q^{67} - 2400 q^{77} + 204 q^{79} - 360 q^{85} - 1080 q^{91} - 300 q^{95}+O(q^{100}) \) 48 * q + 6 * q^7 + 12 * q^11 + 408 * q^23 - 600 * q^25 + 84 * q^29 + 336 * q^37 + 84 * q^43 + 318 * q^49 - 2964 * q^65 - 588 * q^67 - 2400 * q^77 + 204 * q^79 - 360 * q^85 - 1080 * q^91 - 300 * q^95

Introduction

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