LMFDB - Embedded newform 756.4.t.e.269.14

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [756,4,Mod(269,756)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("756.269"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(756, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 3, 1])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 756 = 2^{2} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	756.t (of order $6$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [32,0,0,0,0,0,10] 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:	$32$
Relative dimension:	$16$ over $\Q(\zeta_{6})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			269.14
Character	$\chi$	$=$	756.269
Dual form			756.4.t.e.593.14

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(7.20808 + 12.4848i) q^{5} +(-12.6183 - 13.5565i) q^{7} +(22.8968 + 13.2194i) q^{11} -67.4385i q^{13} +(-25.9770 + 44.9935i) q^{17} +(73.1587 - 42.2382i) q^{19} +(-56.3688 + 32.5446i) q^{23} +(-41.4129 + 71.7292i) q^{25} -195.057i q^{29} +(-203.415 - 117.442i) q^{31} +(78.2955 - 255.253i) q^{35} +(-92.8867 - 160.885i) q^{37} +482.993 q^{41} +367.461 q^{43} +(173.462 + 300.444i) q^{47} +(-24.5561 + 342.120i) q^{49} +(-506.821 - 292.613i) q^{53} +381.147i q^{55} +(231.361 - 400.728i) q^{59} +(-81.8846 + 47.2761i) q^{61} +(841.953 - 486.102i) q^{65} +(257.763 - 446.458i) q^{67} +91.6789i q^{71} +(-6.35841 - 3.67103i) q^{73} +(-109.709 - 477.207i) q^{77} +(690.901 + 1196.68i) q^{79} +919.589 q^{83} -748.978 q^{85} +(-400.130 - 693.046i) q^{89} +(-914.228 + 850.960i) q^{91} +(1054.67 + 608.912i) q^{95} -682.165i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q + 10 q^{7} + 72 q^{19} - 514 q^{25} - 714 q^{31} + 8 q^{37} + 368 q^{43} + 890 q^{49} + 1272 q^{61} + 1840 q^{67} + 324 q^{73} + 532 q^{79} - 1344 q^{85} - 4596 q^{91}+O(q^{100}) $ 32 * q + 10 * q^7 + 72 * q^19 - 514 * q^25 - 714 * q^31 + 8 * q^37 + 368 * q^43 + 890 * q^49 + 1272 * q^61 + 1840 * q^67 + 324 * q^73 + 532 * q^79 - 1344 * q^85 - 4596 * q^91

Character values

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

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

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
$5$	7.20808	+	12.4848i	0.644710	+	1.11667i	0.984368	+	0.176122i	$0.0563554\pi$
$5$	7.20808	+	12.4848i	0.644710	+	1.11667i	−0.339658	+	0.940549i	$0.610311\pi$
$6$		0			0
$7$	−12.6183	−	13.5565i	−0.681325	−	0.731981i
$7$	−12.6183	−	13.5565i	−0.681325	−	0.731981i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	22.8968	+	13.2194i	0.627603	+	0.362347i	0.779823	−	0.626000i	$-0.215309\pi$
$11$	22.8968	+	13.2194i	0.627603	+	0.362347i	−0.152220	+	0.988347i	$0.548642\pi$
$12$		0			0
$13$		−	67.4385i		−	1.43877i	−0.694609	−	0.719387i	$-0.744423\pi$
$13$		−	67.4385i		−	1.43877i	0.694609	−	0.719387i	$-0.255577\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−25.9770	+	44.9935i	−0.370609	+	0.641914i	−0.989659	−	0.143438i	$-0.954184\pi$
$17$	−25.9770	+	44.9935i	−0.370609	+	0.641914i	0.619050	+	0.785351i	$0.287518\pi$
$18$		0			0
$19$	73.1587	−	42.2382i	0.883355	−	0.510005i	0.0115920	−	0.999933i	$-0.496310\pi$
$19$	73.1587	−	42.2382i	0.883355	−	0.510005i	0.871763	+	0.489927i	$0.162977\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−56.3688	+	32.5446i	−0.511031	+	0.295044i	−0.733257	−	0.679951i	$-0.762001\pi$
$23$	−56.3688	+	32.5446i	−0.511031	+	0.295044i	0.222226	+	0.974995i	$0.428668\pi$
$24$		0			0
$25$	−41.4129	+	71.7292i	−0.331303	+	0.573834i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	195.057i		−	1.24901i	−0.781021	−	0.624504i	$-0.785301\pi$
$29$		−	195.057i		−	1.24901i	0.781021	−	0.624504i	$-0.214699\pi$
$30$		0			0
$31$	−203.415	−	117.442i	−1.17853	−	0.680426i	−0.222857	−	0.974851i	$-0.571538\pi$
$31$	−203.415	−	117.442i	−1.17853	−	0.680426i	−0.955674	+	0.294425i	$0.904872\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	78.2955	−	255.253i	0.378125	−	1.23273i
$36$		0			0
$37$	−92.8867	−	160.885i	−0.412716	−	0.714845i	0.582470	−	0.812852i	$-0.302086\pi$
$37$	−92.8867	−	160.885i	−0.412716	−	0.714845i	−0.995186	+	0.0980076i	$0.968753\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	482.993			1.83978			0.919888	−	0.392181i	$-0.128279\pi$
$41$	482.993			1.83978			0.919888	+	0.392181i	$0.128279\pi$
$42$		0			0
$43$	367.461			1.30319			0.651597	−	0.758566i	$-0.274100\pi$
$43$	367.461			1.30319			0.651597	+	0.758566i	$0.274100\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	173.462	+	300.444i	0.538340	+	0.932432i	0.998994	+	0.0448522i	$0.0142817\pi$
$47$	173.462	+	300.444i	0.538340	+	0.932432i	−0.460654	+	0.887580i	$0.652385\pi$
$48$		0			0
$49$	−24.5561	+	342.120i	−0.0715921	+	0.997434i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−506.821	−	292.613i	−1.31353	−	0.758369i	−0.330854	−	0.943682i	$-0.607337\pi$
$53$	−506.821	−	292.613i	−1.31353	−	0.758369i	−0.982679	+	0.185313i	$0.940670\pi$
$54$		0			0
$55$			381.147i			0.934435i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	231.361	−	400.728i	0.510518	−	0.884244i	−0.489407	−	0.872055i	$-0.662787\pi$
$59$	231.361	−	400.728i	0.510518	−	0.884244i	0.999926	−	0.0121884i	$-0.00387980\pi$
$60$		0			0
$61$	−81.8846	+	47.2761i	−0.171873	+	0.0992309i	−0.583469	−	0.812136i	$-0.698305\pi$
$61$	−81.8846	+	47.2761i	−0.171873	+	0.0992309i	0.411596	+	0.911367i	$0.364972\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	841.953	−	486.102i	1.60664	−	0.927593i
$66$		0			0
$67$	257.763	−	446.458i	0.470011	−	0.814082i	−0.529401	−	0.848371i	$-0.677583\pi$
$67$	257.763	−	446.458i	0.470011	−	0.814082i	0.999412	+	0.0342893i	$0.0109168\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			91.6789i			0.153243i	0.997060	+	0.0766217i	$0.0244134\pi$
$71$			91.6789i			0.153243i	−0.997060	+	0.0766217i	$0.975587\pi$
$72$		0			0
$73$	−6.35841	−	3.67103i	−0.0101945	−	0.00588578i	0.494894	−	0.868953i	$-0.335207\pi$
$73$	−6.35841	−	3.67103i	−0.0101945	−	0.00588578i	−0.505089	+	0.863068i	$0.668540\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−109.709	−	477.207i	−0.162371	−	0.706269i
$78$		0			0
$79$	690.901	+	1196.68i	0.983956	+	1.70426i	0.646489	+	0.762924i	$0.276237\pi$
$79$	690.901	+	1196.68i	0.983956	+	1.70426i	0.337467	+	0.941337i	$0.390430\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	919.589			1.21612			0.608060	−	0.793891i	$-0.291948\pi$
$83$	919.589			1.21612			0.608060	+	0.793891i	$0.291948\pi$
$84$		0			0
$85$	−748.978			−0.955742
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−400.130	−	693.046i	−0.476559	−	0.825424i	0.523081	−	0.852283i	$-0.324783\pi$
$89$	−400.130	−	693.046i	−0.476559	−	0.825424i	−0.999639	+	0.0268595i	$0.991449\pi$
$90$		0			0
$91$	−914.228	+	850.960i	−1.05316	+	0.980273i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	1054.67	+	608.912i	1.13902	+	0.657612i
$96$		0			0
$97$		−	682.165i		−	0.714056i	−0.934094	−	0.357028i	$-0.883790\pi$
$97$		−	682.165i		−	0.714056i	0.934094	−	0.357028i	$-0.116210\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−176.897	+	306.395i	−0.174276	+	0.301856i	−0.939911	−	0.341421i	$-0.889092\pi$
$101$	−176.897	+	306.395i	−0.174276	+	0.301856i	0.765634	+	0.643276i	$0.222425\pi$
$102$		0			0
$103$	1359.55	−	784.937i	1.30059	−	0.750895i	0.320083	−	0.947389i	$-0.396289\pi$
$103$	1359.55	−	784.937i	1.30059	−	0.750895i	0.980505	+	0.196494i	$0.0629558\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	1071.06	−	618.375i	0.967691	−	0.558697i	0.0691597	−	0.997606i	$-0.477968\pi$
$107$	1071.06	−	618.375i	0.967691	−	0.558697i	0.898532	+	0.438909i	$0.144635\pi$
$108$		0			0
$109$	−12.1052	+	20.9668i	−0.0106373	+	0.0184244i	−0.871295	−	0.490760i	$-0.836719\pi$
$109$	−12.1052	+	20.9668i	−0.0106373	+	0.0184244i	0.860658	+	0.509184i	$0.170053\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	748.064i		−	0.622761i	−0.950285	−	0.311380i	$-0.899209\pi$
$113$		−	748.064i		−	0.622761i	0.950285	−	0.311380i	$-0.100791\pi$
$114$		0			0
$115$	−812.622	−	469.168i	−0.658934	−	0.380436i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	937.740	−	215.586i	0.722374	−	0.166073i
$120$		0			0
$121$	−315.992	−	547.315i	−0.237410	−	0.411206i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	607.990			0.435043
$126$		0			0
$127$	609.849			0.426105			0.213052	−	0.977041i	$-0.431659\pi$
$127$	609.849			0.426105			0.213052	+	0.977041i	$0.431659\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−50.4605	−	87.4001i	−0.0336546	−	0.0582915i	0.848708	−	0.528862i	$-0.177381\pi$
$131$	−50.4605	−	87.4001i	−0.0336546	−	0.0582915i	−0.882362	+	0.470571i	$0.844048\pi$
$132$		0			0
$133$	−1495.74	−	458.799i	−0.975166	−	0.299120i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	667.573	+	385.424i	0.416311	+	0.240357i	0.693498	−	0.720459i	$-0.256069\pi$
$137$	667.573	+	385.424i	0.416311	+	0.240357i	−0.277187	+	0.960816i	$0.589402\pi$
$138$		0			0
$139$		−	1317.89i		−	0.804188i	−0.915598	−	0.402094i	$-0.868282\pi$
$139$		−	1317.89i		−	0.804188i	0.915598	−	0.402094i	$-0.131718\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	891.500	−	1544.12i	0.521335	−	0.902979i
$144$		0			0
$145$	2435.25	−	1405.99i	1.39473	−	0.805249i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	2406.09	−	1389.16i	1.32292	−	0.763787i	0.338725	−	0.940885i	$-0.390004\pi$
$149$	2406.09	−	1389.16i	1.32292	−	0.763787i	0.984193	+	0.177098i	$0.0566710\pi$
$150$		0			0
$151$	528.173	−	914.822i	0.284650	−	0.493027i	−0.687875	−	0.725830i	$-0.741456\pi$
$151$	528.173	−	914.822i	0.284650	−	0.493027i	0.972524	+	0.232802i	$0.0747894\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	3386.12i		−	1.75471i
$156$		0			0
$157$	−2093.78	−	1208.84i	−1.06434	−	0.614497i	−0.137711	−	0.990472i	$-0.543974\pi$
$157$	−2093.78	−	1208.84i	−1.06434	−	0.614497i	−0.926630	+	0.375975i	$0.877308\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	1152.47	+	353.505i	0.564145	+	0.173044i
$162$		0			0
$163$	1516.60	+	2626.82i	0.728768	+	1.26226i	0.957404	+	0.288751i	$0.0932399\pi$
$163$	1516.60	+	2626.82i	0.728768	+	1.26226i	−0.228637	+	0.973512i	$0.573427\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−2494.60			−1.15592			−0.577958	−	0.816067i	$-0.696150\pi$
$167$	−2494.60			−1.15592			−0.577958	+	0.816067i	$0.696150\pi$
$168$		0			0
$169$	−2350.95			−1.07007
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−1621.57	−	2808.65i	−0.712635	−	1.23432i	−0.963865	−	0.266392i	$-0.914168\pi$
$173$	−1621.57	−	2808.65i	−0.712635	−	1.23432i	0.251230	−	0.967927i	$-0.419165\pi$
$174$		0			0
$175$	1494.96	−	343.689i	0.645761	−	0.148460i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	1850.07	+	1068.14i	0.772516	+	0.446012i	0.833771	−	0.552110i	$-0.186177\pi$
$179$	1850.07	+	1068.14i	0.772516	+	0.446012i	−0.0612553	+	0.998122i	$0.519510\pi$
$180$		0			0
$181$			1751.31i			0.719194i	0.933108	+	0.359597i	$0.117086\pi$
$181$			1751.31i			0.719194i	−0.933108	+	0.359597i	$0.882914\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	1339.07	−	2319.34i	0.532164	−	0.921736i
$186$		0			0
$187$	−1189.58	+	686.804i	−0.465191	+	0.268578i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−3375.53	+	1948.86i	−1.27877	+	0.738296i	−0.976622	−	0.214965i	$-0.931036\pi$
$191$	−3375.53	+	1948.86i	−1.27877	+	0.738296i	−0.302145	+	0.953262i	$0.597703\pi$
$192$		0			0
$193$	−175.245	+	303.532i	−0.0653595	+	0.113206i	−0.896853	−	0.442328i	$-0.854153\pi$
$193$	−175.245	+	303.532i	−0.0653595	+	0.113206i	0.831494	+	0.555534i	$0.187486\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	139.013i		−	0.0502755i	−0.999684	−	0.0251377i	$-0.991998\pi$
$197$		−	139.013i		−	0.0502755i	0.999684	−	0.0251377i	$-0.00800244\pi$
$198$		0			0
$199$	−1933.25	−	1116.17i	−0.688667	−	0.397602i	0.114445	−	0.993430i	$-0.463491\pi$
$199$	−1933.25	−	1116.17i	−0.688667	−	0.397602i	−0.803113	+	0.595827i	$0.796824\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−2644.29	+	2461.30i	−0.914250	+	0.850981i
$204$		0			0
$205$	3481.45	+	6030.05i	1.18612	+	2.05443i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	2233.46			0.739195
$210$		0			0
$211$	−1852.94			−0.604557			−0.302279	−	0.953220i	$-0.597747\pi$
$211$	−1852.94			−0.604557			−0.302279	+	0.953220i	$0.597747\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	2648.69	+	4587.67i	0.840182	+	1.45524i
$216$		0			0
$217$	974.662	+	4239.52i	0.304905	+	1.32625i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	3034.30	+	1751.85i	0.923569	+	0.533223i
$222$		0			0
$223$		−	1583.76i		−	0.475588i	−0.971316	−	0.237794i	$-0.923576\pi$
$223$		−	1583.76i		−	0.475588i	0.971316	−	0.237794i	$-0.0764244\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−1384.34	+	2397.75i	−0.404767	+	0.701076i	−0.994294	−	0.106672i	$-0.965981\pi$
$227$	−1384.34	+	2397.75i	−0.404767	+	0.701076i	0.589528	+	0.807748i	$0.299314\pi$
$228$		0			0
$229$	4669.12	−	2695.72i	1.34736	−	0.777896i	0.359481	−	0.933152i	$-0.382954\pi$
$229$	4669.12	−	2695.72i	1.34736	−	0.777896i	0.987874	+	0.155257i	$0.0496204\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	461.026	−	266.174i	0.129626	−	0.0748395i	−0.433785	−	0.901016i	$-0.642822\pi$
$233$	461.026	−	266.174i	0.129626	−	0.0748395i	0.563411	+	0.826177i	$0.309489\pi$
$234$		0			0
$235$	−2500.65	+	4331.25i	−0.694147	+	1.20230i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			2324.65i			0.629159i	0.949231	+	0.314579i	$0.101863\pi$
$239$			2324.65i			0.629159i	−0.949231	+	0.314579i	$0.898137\pi$
$240$		0			0
$241$	5182.58	+	2992.17i	1.38523	+	0.799761i	0.992773	−	0.120011i	$-0.0382930\pi$
$241$	5182.58	+	2992.17i	1.38523	+	0.799761i	0.392454	+	0.919772i	$0.371626\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−4448.29	+	2159.45i	−1.15996	+	0.563111i
$246$		0			0
$247$	−2848.48	−	4933.71i	−0.733783	−	1.27095i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	1729.90			0.435021			0.217510	−	0.976058i	$-0.430206\pi$
$251$	1729.90			0.435021			0.217510	+	0.976058i	$0.430206\pi$
$252$		0			0
$253$	−1720.88			−0.427633
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−2563.06	−	4439.34i	−0.622097	−	1.07750i	−0.989095	−	0.147282i	$-0.952948\pi$
$257$	−2563.06	−	4439.34i	−0.622097	−	1.07750i	0.366997	−	0.930222i	$-0.380386\pi$
$258$		0			0
$259$	−1008.95	+	3289.31i	−0.242059	+	0.789142i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−540.225	−	311.899i	−0.126661	−	0.0731275i	0.435331	−	0.900270i	$-0.356631\pi$
$263$	−540.225	−	311.899i	−0.126661	−	0.0731275i	−0.561992	+	0.827143i	$0.689965\pi$
$264$		0			0
$265$		−	8436.73i		−	1.95571i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−2620.06	+	4538.07i	−0.593858	+	1.02859i	0.399849	+	0.916581i	$0.369062\pi$
$269$	−2620.06	+	4538.07i	−0.593858	+	1.02859i	−0.993707	+	0.112011i	$0.964271\pi$
$270$		0			0
$271$	−2754.67	+	1590.41i	−0.617471	+	0.356497i	−0.775884	−	0.630876i	$-0.782696\pi$
$271$	−2754.67	+	1590.41i	−0.617471	+	0.356497i	0.158413	+	0.987373i	$0.449362\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−1896.44	+	1094.91i	−0.415854	+	0.240093i
$276$		0			0
$277$	2142.77	−	3711.39i	0.464789	−	0.805039i	−0.534403	−	0.845230i	$-0.679463\pi$
$277$	2142.77	−	3711.39i	0.464789	−	0.805039i	0.999192	+	0.0401913i	$0.0127967\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	3158.57i		−	0.670549i	−0.942120	−	0.335275i	$-0.891171\pi$
$281$		−	3158.57i		−	0.670549i	0.942120	−	0.335275i	$-0.108829\pi$
$282$		0			0
$283$	−4756.39	−	2746.11i	−0.999075	−	0.576816i	−0.0911008	−	0.995842i	$-0.529039\pi$
$283$	−4756.39	−	2746.11i	−0.999075	−	0.576816i	−0.907975	+	0.419025i	$0.862372\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−6094.56	−	6547.68i	−1.25349	−	1.34668i
$288$		0			0
$289$	1106.89	+	1917.19i	0.225298	+	0.390227i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−7977.45			−1.59061			−0.795303	−	0.606211i	$-0.792689\pi$
$293$	−7977.45			−1.59061			−0.795303	+	0.606211i	$0.792689\pi$
$294$		0			0
$295$	6670.66			1.31655
$296$		0			0
$297$		0			0
$298$		0			0
$299$	2194.76	+	3801.43i	0.424502	+	0.735258i
$300$		0			0
$301$	−4636.74	−	4981.48i	−0.887898	−	0.953913i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−1180.46	−	681.540i	−0.221617	−	0.127950i
$306$		0			0
$307$			109.666i			0.0203875i	0.999948	+	0.0101937i	$0.00324482\pi$
$307$			109.666i			0.0203875i	−0.999948	+	0.0101937i	$0.996755\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−5068.16	+	8778.31i	−0.924081	+	1.60055i	−0.131048	+	0.991376i	$0.541834\pi$
$311$	−5068.16	+	8778.31i	−0.924081	+	1.60055i	−0.793033	+	0.609179i	$0.791499\pi$
$312$		0			0
$313$	−4977.70	+	2873.88i	−0.898902	+	0.518981i	−0.876844	−	0.480775i	$-0.840355\pi$
$313$	−4977.70	+	2873.88i	−0.898902	+	0.518981i	−0.0220583	+	0.999757i	$0.507022\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−1809.45	+	1044.69i	−0.320596	+	0.185096i	−0.651658	−	0.758513i	$-0.725926\pi$
$317$	−1809.45	+	1044.69i	−0.320596	+	0.185096i	0.331062	+	0.943609i	$0.392593\pi$
$318$		0			0
$319$	2578.55	−	4466.18i	0.452574	−	0.783881i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			4388.89i			0.756051i
$324$		0			0
$325$	4837.31	+	2792.82i	0.825617	+	0.476670i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	1884.17	−	6142.63i	0.315738	−	1.02934i
$330$		0			0
$331$	−2451.81	−	4246.66i	−0.407141	−	0.705188i	0.587427	−	0.809277i	$-0.300141\pi$
$331$	−2451.81	−	4246.66i	−0.407141	−	0.705188i	−0.994568	+	0.104089i	$0.966807\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	7431.89			1.21208
$336$		0			0
$337$	−106.849			−0.0172713			−0.00863564	−	0.999963i	$-0.502749\pi$
$337$	−106.849			−0.0172713			−0.00863564	+	0.999963i	$0.502749\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−3105.04	−	5378.08i	−0.493100	−	0.854074i
$342$		0			0
$343$	4947.80	−	3984.08i	0.778880	−	0.627173i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	3852.00	+	2223.95i	0.595926	+	0.344058i	0.767437	−	0.641124i	$-0.221532\pi$
$347$	3852.00	+	2223.95i	0.595926	+	0.344058i	−0.171511	+	0.985182i	$0.554865\pi$
$348$		0			0
$349$			11885.5i			1.82297i	0.411337	+	0.911483i	$0.365062\pi$
$349$			11885.5i			1.82297i	−0.411337	+	0.911483i	$0.634938\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−4881.90	+	8455.69i	−0.736082	+	1.27493i	0.218164	+	0.975912i	$0.429993\pi$
$353$	−4881.90	+	8455.69i	−0.736082	+	1.27493i	−0.954247	+	0.299020i	$0.903340\pi$
$354$		0			0
$355$	−1144.59	+	660.829i	−0.171123	+	0.0987977i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−1124.21	+	649.062i	−0.165274	+	0.0954212i	−0.580356	−	0.814363i	$-0.697086\pi$
$359$	−1124.21	+	649.062i	−0.165274	+	0.0954212i	0.415081	+	0.909784i	$0.363753\pi$
$360$		0			0
$361$	138.627	−	240.110i	0.0202110	−	0.0350065i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		−	105.844i		−	0.0151785i
$366$		0			0
$367$	−4909.59	−	2834.55i	−0.698306	−	0.403167i	0.108410	−	0.994106i	$-0.465424\pi$
$367$	−4909.59	−	2834.55i	−0.698306	−	0.403167i	−0.806716	+	0.590939i	$0.798757\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	2428.43	+	10563.0i	0.339832	+	1.47818i
$372$		0			0
$373$	−5015.85	−	8687.71i	−0.696276	−	1.20599i	−0.969749	−	0.244105i	$-0.921506\pi$
$373$	−5015.85	−	8687.71i	−0.696276	−	1.20599i	0.273473	−	0.961880i	$-0.411828\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−13154.4			−1.79704
$378$		0			0
$379$	−247.479			−0.0335413			−0.0167707	−	0.999859i	$-0.505339\pi$
$379$	−247.479			−0.0335413			−0.0167707	+	0.999859i	$0.505339\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	5620.07	+	9734.25i	0.749797	+	1.29869i	0.947920	+	0.318509i	$0.103182\pi$
$383$	5620.07	+	9734.25i	0.749797	+	1.29869i	−0.198123	+	0.980177i	$0.563485\pi$
$384$		0			0
$385$	5167.02	−	4809.44i	0.683988	−	0.636654i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−10416.4	−	6013.93i	−1.35767	−	0.783852i	−0.368362	−	0.929682i	$-0.620081\pi$
$389$	−10416.4	−	6013.93i	−1.35767	−	0.783852i	−0.989310	+	0.145830i	$0.953415\pi$
$390$		0			0
$391$		−	3381.64i		−	0.437384i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−9960.15	+	17251.5i	−1.26873	+	2.19751i
$396$		0			0
$397$	−1590.74	+	918.415i	−0.201101	+	0.116106i	−0.597169	−	0.802116i	$-0.703708\pi$
$397$	−1590.74	+	918.415i	−0.201101	+	0.116106i	0.396068	+	0.918221i	$0.370374\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	5462.80	−	3153.95i	0.680298	−	0.392770i	−0.119669	−	0.992814i	$-0.538183\pi$
$401$	5462.80	−	3153.95i	0.680298	−	0.392770i	0.799967	+	0.600044i	$0.204850\pi$
$402$		0			0
$403$	−7920.11	+	13718.0i	−0.978979	+	1.69564i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	4911.65i		−	0.598185i
$408$		0			0
$409$	909.488	+	525.093i	0.109954	+	0.0634821i	0.553969	−	0.832538i	$-0.313113\pi$
$409$	909.488	+	525.093i	0.109954	+	0.0634821i	−0.444014	+	0.896020i	$0.646446\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−8351.85	+	1920.08i	−0.995079	+	0.228768i
$414$		0			0
$415$	6628.47	+	11480.8i	0.784045	+	1.35801i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	167.910			0.0195774			0.00978871	−	0.999952i	$-0.496884\pi$
$419$	167.910			0.0195774			0.00978871	+	0.999952i	$0.496884\pi$
$420$		0			0
$421$	1746.98			0.202239			0.101120	−	0.994874i	$-0.467757\pi$
$421$	1746.98			0.202239			0.101120	+	0.994874i	$0.467757\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−2151.57	−	3726.62i	−0.245568	−	0.425336i
$426$		0			0
$427$	1674.14	+	513.522i	0.189736	+	0.0581992i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	9045.88	+	5222.64i	1.01096	+	0.583679i	0.911473	−	0.411360i	$-0.134946\pi$
$431$	9045.88	+	5222.64i	1.01096	+	0.583679i	0.0994886	+	0.995039i	$0.468279\pi$
$432$		0			0
$433$			16316.4i			1.81090i	0.424456	+	0.905448i	$0.360465\pi$
$433$			16316.4i			1.81090i	−0.424456	+	0.905448i	$0.639535\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−2749.25	+	4761.83i	−0.300948	+	0.521257i
$438$		0			0
$439$	2692.73	−	1554.65i	0.292750	−	0.169019i	−0.346432	−	0.938075i	$-0.612607\pi$
$439$	2692.73	−	1554.65i	0.292750	−	0.169019i	0.639181	+	0.769056i	$0.279273\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−2966.03	+	1712.44i	−0.318104	+	0.183658i	−0.650547	−	0.759466i	$-0.725460\pi$
$443$	−2966.03	+	1712.44i	−0.318104	+	0.183658i	0.332443	+	0.943123i	$0.392127\pi$
$444$		0			0
$445$	5768.34	−	9991.06i	0.614485	−	1.06432i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	12905.8i		−	1.35648i	−0.734840	−	0.678241i	$-0.762743\pi$
$449$		−	12905.8i		−	1.35648i	0.734840	−	0.678241i	$-0.237257\pi$
$450$		0			0
$451$	11059.0	+	6384.90i	1.15465	+	0.666637i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−17213.9	−	5280.13i	−1.77362	−	0.544036i
$456$		0			0
$457$	−8161.27	−	14135.7i	−0.835379	−	1.44692i	−0.893721	−	0.448623i	$-0.851915\pi$
$457$	−8161.27	−	14135.7i	−0.835379	−	1.44692i	0.0583421	−	0.998297i	$-0.481419\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	18427.3			1.86170			0.930850	−	0.365402i	$-0.119069\pi$
$461$	18427.3			1.86170			0.930850	+	0.365402i	$0.119069\pi$
$462$		0			0
$463$	12510.6			1.25576			0.627881	−	0.778310i	$-0.283923\pi$
$463$	12510.6			1.25576			0.627881	+	0.778310i	$0.283923\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−5742.61	−	9946.49i	−0.569029	−	0.985586i	−0.996662	−	0.0816344i	$-0.973986\pi$
$467$	−5742.61	−	9946.49i	−0.569029	−	0.985586i	0.427634	−	0.903952i	$-0.359347\pi$
$468$		0			0
$469$	−9304.93	+	2139.20i	−0.916123	+	0.210616i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	8413.67	+	4857.63i	0.817888	+	0.472208i
$474$		0			0
$475$			6996.82i			0.675866i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−7009.77	+	12141.3i	−0.668653	+	1.15814i	0.309629	+	0.950858i	$0.399795\pi$
$479$	−7009.77	+	12141.3i	−0.668653	+	1.15814i	−0.978281	+	0.207283i	$0.933538\pi$
$480$		0			0
$481$	−10849.8	+	6264.14i	−1.02850	+	0.593805i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	8516.67	−	4917.10i	0.797365	−	0.460359i
$486$		0			0
$487$	−2400.43	+	4157.66i	−0.223355	+	0.386862i	−0.955825	−	0.293938i	$-0.905034\pi$
$487$	−2400.43	+	4157.66i	−0.223355	+	0.386862i	0.732470	+	0.680799i	$0.238367\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			14112.9i			1.29717i	0.761144	+	0.648583i	$0.224638\pi$
$491$			14112.9i			1.29717i	−0.761144	+	0.648583i	$0.775362\pi$
$492$		0			0
$493$	8776.32	+	5067.01i	0.801756	+	0.462894i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	1242.84	−	1156.83i	0.112171	−	0.104409i
$498$		0			0
$499$	−7997.90	−	13852.8i	−0.717506	−	1.24276i	−0.961985	−	0.273102i	$-0.911950\pi$
$499$	−7997.90	−	13852.8i	−0.717506	−	1.24276i	0.244480	−	0.969654i	$-0.421383\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	1347.36			0.119435			0.0597173	−	0.998215i	$-0.480980\pi$
$503$	1347.36			0.119435			0.0597173	+	0.998215i	$0.480980\pi$
$504$		0			0
$505$	−5100.35			−0.449431
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−3587.33	−	6213.44i	−0.312388	−	0.541073i	0.666490	−	0.745514i	$-0.267796\pi$
$509$	−3587.33	−	6213.44i	−0.312388	−	0.541073i	−0.978879	+	0.204441i	$0.934462\pi$
$510$		0			0
$511$	30.4662	+	132.520i	0.00263747	+	0.0114723i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	19599.5	+	11315.8i	1.67701	+	0.968220i
$516$		0			0
$517$			9172.27i			0.780263i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	5766.64	−	9988.12i	0.484916	−	0.839899i	−0.514934	−	0.857230i	$-0.672183\pi$
$521$	5766.64	−	9988.12i	0.484916	−	0.839899i	0.999850	+	0.0173307i	$0.00551682\pi$
$522$		0			0
$523$	−8238.61	+	4756.56i	−0.688813	+	0.397686i	−0.803167	−	0.595754i	$-0.796853\pi$
$523$	−8238.61	+	4756.56i	−0.688813	+	0.397686i	0.114354	+	0.993440i	$0.463520\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	10568.3	−	6101.59i	0.873549	−	0.504344i
$528$		0			0
$529$	−3965.20	+	6867.94i	−0.325898	+	0.564472i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	32572.3i		−	2.64702i
$534$		0			0
$535$	15440.5	+	8914.59i	1.24776	+	0.720395i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−5084.89	+	7508.82i	−0.406348	+	0.600051i
$540$		0			0
$541$	−4538.52	−	7860.95i	−0.360677	−	0.624711i	0.627395	−	0.778701i	$-0.284121\pi$
$541$	−4538.52	−	7860.95i	−0.360677	−	0.624711i	−0.988072	+	0.153990i	$0.950788\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−349.021			−0.0274320
$546$		0			0
$547$	−2085.88			−0.163045			−0.0815227	−	0.996671i	$-0.525978\pi$
$547$	−2085.88			−0.163045			−0.0815227	+	0.996671i	$0.525978\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−8238.87	−	14270.1i	−0.637001	−	1.10332i
$552$		0			0
$553$	7504.70	−	24466.2i	0.577093	−	1.88139i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	16794.3	+	9696.17i	1.27755	+	0.737594i	0.976397	−	0.215982i	$-0.0692952\pi$
$557$	16794.3	+	9696.17i	1.27755	+	0.737594i	0.301153	+	0.953576i	$0.402629\pi$
$558$		0			0
$559$		−	24781.0i		−	1.87500i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	10006.8	−	17332.3i	0.749087	−	1.29746i	−0.199173	−	0.979964i	$-0.563826\pi$
$563$	10006.8	−	17332.3i	0.749087	−	1.29746i	0.948261	−	0.317493i	$-0.102841\pi$
$564$		0			0
$565$	9339.40	−	5392.11i	0.695419	−	0.401500i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−15536.4	+	8969.96i	−1.14468	+	0.660879i	−0.947584	−	0.319506i	$-0.896483\pi$
$569$	−15536.4	+	8969.96i	−1.14468	+	0.660879i	−0.197092	+	0.980385i	$0.563150\pi$
$570$		0			0
$571$	6927.55	−	11998.9i	0.507721	−	0.879399i	−0.492239	−	0.870460i	$-0.663821\pi$
$571$	6927.55	−	11998.9i	0.507721	−	0.879399i	0.999960	−	0.00893868i	$-0.00284531\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	5391.06i		−	0.390996i
$576$		0			0
$577$	−9548.15	−	5512.63i	−0.688899	−	0.397736i	0.114300	−	0.993446i	$-0.463537\pi$
$577$	−9548.15	−	5512.63i	−0.688899	−	0.397736i	−0.803200	+	0.595710i	$0.796871\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−11603.7	−	12466.4i	−0.828573	−	0.890177i
$582$		0			0
$583$	−7736.38	−	13399.8i	−0.549585	−	0.951909i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−6502.77			−0.457237			−0.228618	−	0.973516i	$-0.573421\pi$
$587$	−6502.77			−0.457237			−0.228618	+	0.973516i	$0.573421\pi$
$588$		0			0
$589$	−19842.1			−1.38808
$590$		0			0
$591$		0			0
$592$		0			0
$593$	6411.44	+	11104.9i	0.443990	+	0.769013i	0.997981	−	0.0635092i	$-0.0202292\pi$
$593$	6411.44	+	11104.9i	0.443990	+	0.769013i	−0.553991	+	0.832523i	$0.686896\pi$
$594$		0			0
$595$	9450.84	+	10153.5i	0.651171	+	0.699585i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	5354.89	+	3091.65i	0.365267	+	0.210887i	0.671389	−	0.741105i	$-0.265698\pi$
$599$	5354.89	+	3091.65i	0.365267	+	0.210887i	−0.306122	+	0.951992i	$0.599031\pi$
$600$		0			0
$601$		−	3115.47i		−	0.211452i	−0.994395	−	0.105726i	$-0.966283\pi$
$601$		−	3115.47i		−	0.211452i	0.994395	−	0.105726i	$-0.0337167\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	4555.40	−	7890.18i	0.306121	−	0.530217i
$606$		0			0
$607$	−18993.0	+	10965.6i	−1.27002	+	0.733246i	−0.974991	−	0.222243i	$-0.928662\pi$
$607$	−18993.0	+	10965.6i	−1.27002	+	0.733246i	−0.295028	+	0.955489i	$0.595329\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	20261.5	−	11698.0i	1.34156	−	0.774550i
$612$		0			0
$613$	2882.20	−	4992.11i	0.189903	−	0.328922i	−0.755314	−	0.655363i	$-0.772516\pi$
$613$	2882.20	−	4992.11i	0.189903	−	0.328922i	0.945218	+	0.326440i	$0.105849\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	22744.4i		−	1.48405i	−0.670374	−	0.742023i	$-0.733866\pi$
$617$		−	22744.4i		−	1.48405i	0.670374	−	0.742023i	$-0.266134\pi$
$618$		0			0
$619$	−2684.28	−	1549.77i	−0.174298	−	0.100631i	0.410313	−	0.911945i	$-0.365419\pi$
$619$	−2684.28	−	1549.77i	−0.174298	−	0.100631i	−0.584611	+	0.811314i	$0.698753\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−4346.29	+	14169.4i	−0.279503	+	0.911214i
$624$		0			0
$625$	9559.06	+	16556.8i	0.611780	+	1.05963i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	9651.68			0.611825
$630$		0			0
$631$	24192.9			1.52631			0.763157	−	0.646213i	$-0.223648\pi$
$631$	24192.9			1.52631			0.763157	+	0.646213i	$0.223648\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	4395.84	+	7613.82i	0.274714	+	0.475819i
$636$		0			0
$637$	23072.0	+	1656.03i	1.43508	+	0.103005i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	25451.7	+	14694.5i	1.56830	+	0.905459i	0.996367	+	0.0851596i	$0.0271400\pi$
$641$	25451.7	+	14694.5i	1.56830	+	0.905459i	0.571934	+	0.820300i	$0.306193\pi$
$642$		0			0
$643$		−	13562.2i		−	0.831791i	−0.909412	−	0.415896i	$-0.863468\pi$
$643$		−	13562.2i		−	0.831791i	0.909412	−	0.415896i	$-0.136532\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−1405.74	+	2434.81i	−0.0854176	+	0.147948i	−0.905569	−	0.424199i	$-0.860556\pi$
$647$	−1405.74	+	2434.81i	−0.0854176	+	0.147948i	0.820152	+	0.572146i	$0.193889\pi$
$648$		0			0
$649$	10594.8	−	6116.92i	0.640806	−	0.369969i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	12738.9	−	7354.83i	0.763420	−	0.440761i	−0.0671024	−	0.997746i	$-0.521375\pi$
$653$	12738.9	−	7354.83i	0.763420	−	0.440761i	0.830522	+	0.556985i	$0.188042\pi$
$654$		0			0
$655$	727.446	−	1259.97i	0.0433949	−	0.0751622i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	22037.4i		−	1.30266i	−0.758794	−	0.651331i	$-0.774211\pi$
$659$		−	22037.4i		−	1.30266i	0.758794	−	0.651331i	$-0.225789\pi$
$660$		0			0
$661$	−20052.8	−	11577.5i	−1.17998	−	0.681260i	−0.223967	−	0.974597i	$-0.571901\pi$
$661$	−20052.8	−	11577.5i	−1.17998	−	0.681260i	−0.956009	+	0.293337i	$0.905234\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−5053.42	−	21981.0i	−0.294682	−	1.28179i
$666$		0			0
$667$	6348.06	+	10995.2i	0.368512	+	0.638282i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−2499.86			−0.143824
$672$		0			0
$673$	−10001.3			−0.572843			−0.286421	−	0.958104i	$-0.592466\pi$
$673$	−10001.3			−0.572843			−0.286421	+	0.958104i	$0.592466\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−3229.72	−	5594.03i	−0.183350	−	0.317572i	0.759669	−	0.650310i	$-0.225361\pi$
$677$	−3229.72	−	5594.03i	−0.183350	−	0.317572i	−0.943019	+	0.332738i	$0.892028\pi$
$678$		0			0
$679$	−9247.76	+	8607.78i	−0.522675	+	0.486504i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	29593.9	+	17086.0i	1.65795	+	0.957217i	0.973660	+	0.228006i	$0.0732206\pi$
$683$	29593.9	+	17086.0i	1.65795	+	0.957217i	0.684289	+	0.729211i	$0.260113\pi$
$684$		0			0
$685$			11112.7i			0.619844i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−19733.4	+	34179.3i	−1.09112	+	1.88988i
$690$		0			0
$691$	−24700.4	+	14260.8i	−1.35984	+	0.785104i	−0.989602	−	0.143833i	$-0.954057\pi$
$691$	−24700.4	+	14260.8i	−1.35984	+	0.785104i	−0.370238	+	0.928937i	$0.620724\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	16453.6	−	9499.47i	0.898014	−	0.518468i
$696$		0			0
$697$	−12546.7	+	21731.6i	−0.681838	+	1.18098i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			5740.86i			0.309314i	0.987968	+	0.154657i	$0.0494273\pi$
$701$			5740.86i			0.309314i	−0.987968	+	0.154657i	$0.950573\pi$
$702$		0			0
$703$	−13590.9	−	7846.73i	−0.729149	−	0.420975i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	6385.77	−	1468.08i	0.339691	−	0.0780948i
$708$		0			0
$709$	10839.9	+	18775.3i	0.574193	+	0.994531i	0.996129	+	0.0879058i	$0.0280174\pi$
$709$	10839.9	+	18775.3i	0.574193	+	0.994531i	−0.421936	+	0.906626i	$0.638649\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	15288.4			0.803022
$714$		0			0
$715$	25704.0			1.34444
$716$		0			0
$717$		0			0
$718$		0			0
$719$	63.8988	+	110.676i	0.00331436	+	0.00574063i	0.867678	−	0.497127i	$-0.165612\pi$
$719$	63.8988	+	110.676i	0.00331436	+	0.00574063i	−0.864363	+	0.502868i	$0.832278\pi$
$720$		0			0
$721$	−27796.2	−	8526.14i	−1.43576	−	0.440402i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	13991.3	+	8077.89i	0.716723	+	0.413800i
$726$		0			0
$727$			165.101i			0.00842264i	0.999991	+	0.00421132i	$0.00134051\pi$
$727$			165.101i			0.00842264i	−0.999991	+	0.00421132i	$0.998659\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−9545.55	+	16533.4i	−0.482975	+	0.836538i
$732$		0			0
$733$	10591.8	−	6115.17i	0.533720	−	0.308143i	−0.208810	−	0.977956i	$-0.566959\pi$
$733$	10591.8	−	6115.17i	0.533720	−	0.308143i	0.742530	+	0.669813i	$0.233626\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	11803.9	−	6814.96i	0.589960	−	0.340614i
$738$		0			0
$739$	−2182.80	+	3780.72i	−0.108654	+	0.188195i	−0.915225	−	0.402942i	$-0.867988\pi$
$739$	−2182.80	+	3780.72i	−0.108654	+	0.188195i	0.806571	+	0.591137i	$0.201321\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	26621.6i		−	1.31447i	−0.753686	−	0.657234i	$-0.771726\pi$
$743$		−	26621.6i		−	1.31447i	0.753686	−	0.657234i	$-0.228274\pi$
$744$		0			0
$745$	34686.6	+	20026.3i	1.70580	+	0.984843i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−21897.9	−	6716.90i	−1.06827	−	0.327677i
$750$		0			0
$751$	12129.2	+	21008.4i	0.589349	+	1.02078i	0.994318	+	0.106452i	$0.0339491\pi$
$751$	12129.2	+	21008.4i	0.589349	+	1.02078i	−0.404969	+	0.914331i	$0.632718\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	15228.4			0.734066
$756$		0			0
$757$	−8702.59			−0.417835			−0.208917	−	0.977933i	$-0.566994\pi$
$757$	−8702.59			−0.417835			−0.208917	+	0.977933i	$0.566994\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−16901.0	−	29273.5i	−0.805075	−	1.39443i	−0.916240	−	0.400629i	$-0.868792\pi$
$761$	−16901.0	−	29273.5i	−0.805075	−	1.39443i	0.111165	−	0.993802i	$-0.464542\pi$
$762$		0			0
$763$	436.984	−	100.462i	0.0207338	−	0.00476668i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−27024.5	−	15602.6i	−1.27223	−	0.734521i
$768$		0			0
$769$			3781.67i			0.177335i	0.996061	+	0.0886675i	$0.0282608\pi$
$769$			3781.67i			0.177335i	−0.996061	+	0.0886675i	$0.971739\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	7306.10	−	12654.5i	0.339951	−	0.588813i	−0.644472	−	0.764628i	$-0.722923\pi$
$773$	7306.10	−	12654.5i	0.339951	−	0.588813i	0.984423	+	0.175815i	$0.0562561\pi$
$774$		0			0
$775$	16848.0	−	9727.22i	0.780903	−	0.450854i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	35335.1	−	20400.7i	1.62518	−	0.938296i
$780$		0			0
$781$	−1211.94	+	2099.15i	−0.0555273	+	0.0961760i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	34853.7i		−	1.58469i
$786$		0			0
$787$	9515.26	+	5493.64i	0.430982	+	0.248827i	0.699765	−	0.714373i	$-0.253288\pi$
$787$	9515.26	+	5493.64i	0.430982	+	0.248827i	−0.268783	+	0.963201i	$0.586621\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−10141.1	+	9439.31i	−0.455849	+	0.424302i
$792$		0			0
$793$	3188.23	+	5522.17i	0.142771	+	0.247286i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	26825.7			1.19224			0.596119	−	0.802896i	$-0.296709\pi$
$797$	26825.7			1.19224			0.596119	+	0.802896i	$0.296709\pi$
$798$		0			0
$799$	−18024.1			−0.798055
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−97.0580	−	168.109i	−0.00426538	−	0.00738786i
$804$		0			0
$805$	3893.67	+	16936.4i	0.170477	+	0.741527i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	1110.89	+	641.370i	0.0482777	+	0.0278731i	0.523945	−	0.851752i	$-0.324460\pi$
$809$	1110.89	+	641.370i	0.0482777	+	0.0278731i	−0.475667	+	0.879625i	$0.657793\pi$
$810$		0			0
$811$		−	16526.3i		−	0.715557i	−0.933806	−	0.357779i	$-0.883534\pi$
$811$		−	16526.3i		−	0.715557i	0.933806	−	0.357779i	$-0.116466\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−21863.5	+	37868.7i	−0.939688	+	1.62759i
$816$		0			0
$817$	26883.0	−	15520.9i	1.15118	−	0.664636i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	19174.8	−	11070.6i	0.815111	−	0.470605i	−0.0336167	−	0.999435i	$-0.510703\pi$
$821$	19174.8	−	11070.6i	0.815111	−	0.470605i	0.848728	+	0.528830i	$0.177369\pi$
$822$		0			0
$823$	−14924.7	+	25850.3i	−0.632129	+	1.09488i	0.354986	+	0.934872i	$0.384486\pi$
$823$	−14924.7	+	25850.3i	−0.632129	+	1.09488i	−0.987116	+	0.160009i	$0.948848\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	25515.2i		−	1.07285i	−0.843946	−	0.536427i	$-0.819774\pi$
$827$		−	25515.2i		−	1.07285i	0.843946	−	0.536427i	$-0.180226\pi$
$828$		0			0
$829$	−18525.0	−	10695.4i	−0.776116	−	0.448091i	0.0589359	−	0.998262i	$-0.481229\pi$
$829$	−18525.0	−	10695.4i	−0.776116	−	0.448091i	−0.835052	+	0.550171i	$0.814563\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−14755.3	−	9992.12i	−0.613734	−	0.415614i
$834$		0			0
$835$	−17981.3	−	31144.5i	−0.745231	−	1.29078i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	25411.3			1.04564			0.522821	−	0.852442i	$-0.324879\pi$
$839$	25411.3			1.04564			0.522821	+	0.852442i	$0.324879\pi$
$840$		0			0
$841$	−13658.4			−0.560023
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−16945.8	−	29351.0i	−0.689887	−	1.19492i
$846$		0			0
$847$	−3432.37	+	11189.9i	−0.139242	+	0.453944i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	10471.8	+	6045.91i	0.421821	+	0.243538i
$852$		0			0
$853$		−	26049.3i		−	1.04562i	−0.852450	−	0.522809i	$-0.824884\pi$
$853$		−	26049.3i		−	1.04562i	0.852450	−	0.522809i	$-0.175116\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−14248.5	+	24679.2i	−0.567935	+	0.983692i	0.428835	+	0.903383i	$0.358924\pi$
$857$	−14248.5	+	24679.2i	−0.567935	+	0.983692i	−0.996770	+	0.0803090i	$0.974409\pi$
$858$		0			0
$859$	17656.8	−	10194.1i	0.701329	−	0.404913i	−0.106513	−	0.994311i	$-0.533969\pi$
$859$	17656.8	−	10194.1i	0.701329	−	0.404913i	0.807842	+	0.589399i	$0.200635\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−25128.5	+	14507.9i	−0.991174	+	0.572255i	−0.905625	−	0.424079i	$-0.860598\pi$
$863$	−25128.5	+	14507.9i	−0.991174	+	0.572255i	−0.0855492	+	0.996334i	$0.527264\pi$
$864$		0			0
$865$	23376.8	−	40489.9i	0.918886	−	1.59156i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			36533.3i			1.42613i
$870$		0			0
$871$	−30108.4	−	17383.1i	−1.17128	−	0.676239i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−7671.82	−	8242.21i	−0.296405	−	0.318443i
$876$		0			0
$877$	5471.32	+	9476.60i	0.210665	+	0.364883i	0.951923	−	0.306338i	$-0.0991037\pi$
$877$	5471.32	+	9476.60i	0.210665	+	0.364883i	−0.741258	+	0.671220i	$0.765770\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−13436.8			−0.513845			−0.256922	−	0.966432i	$-0.582708\pi$
$881$	−13436.8			−0.513845			−0.256922	+	0.966432i	$0.582708\pi$
$882$		0			0
$883$	−42178.7			−1.60751			−0.803753	−	0.594963i	$-0.797167\pi$
$883$	−42178.7			−1.60751			−0.803753	+	0.594963i	$0.797167\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−4490.76	−	7778.23i	−0.169994	−	0.294439i	0.768423	−	0.639942i	$-0.221042\pi$
$887$	−4490.76	−	7778.23i	−0.169994	−	0.294439i	−0.938418	+	0.345503i	$0.887708\pi$
$888$		0			0
$889$	−7695.26	−	8267.40i	−0.290316	−	0.311901i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	25380.4	+	14653.4i	0.951091	+	0.549113i
$894$		0			0
$895$			30796.8i			1.15020i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−22907.9	+	39677.7i	−0.849858	+	1.47200i
$900$		0			0
$901$	26331.4	−	15202.5i	0.973615	−	0.562117i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−21864.7	+	12623.6i	−0.803104	+	0.463672i
$906$		0			0
$907$	22692.2	−	39304.0i	0.830740	−	1.43888i	−0.0667129	−	0.997772i	$-0.521251\pi$
$907$	22692.2	−	39304.0i	0.830740	−	1.43888i	0.897453	−	0.441111i	$-0.145416\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			33841.9i			1.23077i	0.788226	+	0.615386i	$0.211000\pi$
$911$			33841.9i			1.23077i	−0.788226	+	0.615386i	$0.789000\pi$
$912$		0			0
$913$	21055.6	+	12156.5i	0.763240	+	0.440657i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−548.111	+	1786.91i	−0.0197385	+	0.0643499i
$918$		0			0
$919$	605.250	+	1048.32i	0.0217251	+	0.0376290i	0.876684	−	0.481068i	$-0.159751\pi$
$919$	605.250	+	1048.32i	0.0217251	+	0.0376290i	−0.854958	+	0.518696i	$0.826417\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	6182.69			0.220483
$924$		0			0
$925$	15386.8			0.546936
$926$		0			0
$927$		0			0
$928$		0			0
$929$	17197.4	+	29786.7i	0.607350	+	1.05196i	0.991675	+	0.128763i	$0.0411006\pi$
$929$	17197.4	+	29786.7i	0.607350	+	1.05196i	−0.384326	+	0.923198i	$0.625566\pi$
$930$		0			0
$931$	12654.0	+	26066.2i	0.445455	+	0.917601i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−17149.2	−	9901.08i	−0.599827	−	0.346310i
$936$		0			0
$937$		−	7755.34i		−	0.270391i	−0.990819	−	0.135195i	$-0.956834\pi$
$937$		−	7755.34i		−	0.270391i	0.990819	−	0.135195i	$-0.0431662\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−9177.67	+	15896.2i	−0.317942	+	0.550692i	−0.980058	−	0.198709i	$-0.936325\pi$
$941$	−9177.67	+	15896.2i	−0.317942	+	0.550692i	0.662116	+	0.749401i	$0.269658\pi$
$942$		0			0
$943$	−27225.7	+	15718.8i	−0.940182	+	0.542815i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	19440.4	−	11223.9i	0.667082	−	0.385140i	−0.127888	−	0.991789i	$-0.540820\pi$
$947$	19440.4	−	11223.9i	0.667082	−	0.385140i	0.794970	+	0.606649i	$0.207486\pi$
$948$		0			0
$949$	−247.569	+	428.802i	−0.00846830	+	0.0146675i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	725.850i		−	0.0246722i	−0.999924	−	0.0123361i	$-0.996073\pi$
$953$		−	725.850i		−	0.0246722i	0.999924	−	0.0123361i	$-0.00392680\pi$
$954$		0			0
$955$	−48662.1	−	28095.1i	−1.64887	−	0.951975i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−3198.67	−	13913.3i	−0.107706	−	0.468493i
$960$		0			0
$961$	12689.7	+	21979.2i	0.425958	+	0.737781i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−5052.71			−0.168552
$966$		0			0
$967$	−12685.9			−0.421872			−0.210936	−	0.977500i	$-0.567651\pi$
$967$	−12685.9			−0.421872			−0.210936	+	0.977500i	$0.567651\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	10598.8	+	18357.7i	0.350291	+	0.606722i	0.986300	−	0.164959i	$-0.0527492\pi$
$971$	10598.8	+	18357.7i	0.350291	+	0.606722i	−0.636009	+	0.771682i	$0.719416\pi$
$972$		0			0
$973$	−17866.0	+	16629.6i	−0.588650	+	0.547914i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	12691.6	+	7327.51i	0.415599	+	0.239946i	0.693193	−	0.720752i	$-0.256203\pi$
$977$	12691.6	+	7327.51i	0.415599	+	0.239946i	−0.277593	+	0.960699i	$0.589537\pi$
$978$		0			0
$979$		−	21158.0i		−	0.690718i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−18204.1	+	31530.4i	−0.590661	+	1.02305i	0.403483	+	0.914987i	$0.367799\pi$
$983$	−18204.1	+	31530.4i	−0.590661	+	1.02305i	−0.994144	+	0.108067i	$0.965534\pi$
$984$		0			0
$985$	1735.55	−	1002.02i	0.0561412	−	0.0324131i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−20713.4	+	11958.9i	−0.665972	+	0.384499i
$990$		0			0
$991$	12174.7	−	21087.2i	0.390254	−	0.675940i	−0.602229	−	0.798324i	$-0.705720\pi$
$991$	12174.7	−	21087.2i	0.390254	−	0.675940i	0.992483	+	0.122384i	$0.0390538\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	32181.6i		−	1.02535i
$996$		0			0
$997$	−36125.2	−	20856.9i	−1.14754	−	0.662532i	−0.199253	−	0.979948i	$-0.563851\pi$
$997$	−36125.2	−	20856.9i	−1.14754	−	0.662532i	−0.948286	+	0.317416i	$0.897185\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	756.4.t.e.269.14	yes	32
3.2	odd	2	inner	756.4.t.e.269.3	✓	32
7.5	odd	6	inner	756.4.t.e.593.3	yes	32
21.5	even	6	inner	756.4.t.e.593.14	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
756.4.t.e.269.3	✓	32	3.2	odd	2	inner
756.4.t.e.269.14	yes	32	1.1	even	1	trivial
756.4.t.e.593.3	yes	32	7.5	odd	6	inner
756.4.t.e.593.14	yes	32	21.5	even	6	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.t (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(7.20808 + 12.4848i) q^{5} +(-12.6183 - 13.5565i) q^{7} +(22.8968 + 13.2194i) q^{11} -67.4385i q^{13} +(-25.9770 + 44.9935i) q^{17} +(73.1587 - 42.2382i) q^{19} +(-56.3688 + 32.5446i) q^{23} +(-41.4129 + 71.7292i) q^{25} -195.057i q^{29} +(-203.415 - 117.442i) q^{31} +(78.2955 - 255.253i) q^{35} +(-92.8867 - 160.885i) q^{37} +482.993 q^{41} +367.461 q^{43} +(173.462 + 300.444i) q^{47} +(-24.5561 + 342.120i) q^{49} +(-506.821 - 292.613i) q^{53} +381.147i q^{55} +(231.361 - 400.728i) q^{59} +(-81.8846 + 47.2761i) q^{61} +(841.953 - 486.102i) q^{65} +(257.763 - 446.458i) q^{67} +91.6789i q^{71} +(-6.35841 - 3.67103i) q^{73} +(-109.709 - 477.207i) q^{77} +(690.901 + 1196.68i) q^{79} +919.589 q^{83} -748.978 q^{85} +(-400.130 - 693.046i) q^{89} +(-914.228 + 850.960i) q^{91} +(1054.67 + 608.912i) q^{95} -682.165i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 10 q^{7} + 72 q^{19} - 514 q^{25} - 714 q^{31} + 8 q^{37} + 368 q^{43} + 890 q^{49} + 1272 q^{61} + 1840 q^{67} + 324 q^{73} + 532 q^{79} - 1344 q^{85} - 4596 q^{91}+O(q^{100}) \) 32 * q + 10 * q^7 + 72 * q^19 - 514 * q^25 - 714 * q^31 + 8 * q^37 + 368 * q^43 + 890 * q^49 + 1272 * q^61 + 1840 * q^67 + 324 * q^73 + 532 * q^79 - 1344 * q^85 - 4596 * q^91

Introduction

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