LMFDB - Embedded newform 755.2.f.d.603.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [755,2,Mod(452,755)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(755, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([1, 2])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("755.452"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 755 = 5 \cdot 151 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	755.f (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [28,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$6.02870535261$
Analytic rank:	$0$
Dimension:	$28$
Relative dimension:	$14$ over $\Q(i)$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label			603.6
Character	$\chi$	$=$	755.603
Dual form			755.2.f.d.452.6

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-0.503691 + 0.503691i) q^{2} +1.49259i q^{4} +(-0.577204 - 2.16029i) q^{5} +(-1.75919 - 1.75919i) q^{8} +3.00000i q^{9} +(1.37885 + 0.797385i) q^{10} -4.14607 q^{11} -1.21300 q^{16} +(3.11919 - 3.11919i) q^{17} +(-1.51107 - 1.51107i) q^{18} -7.85139i q^{19} +(3.22442 - 0.861529i) q^{20} +(2.08834 - 2.08834i) q^{22} +(-4.33367 + 2.49385i) q^{25} -1.08956i q^{29} -10.1567 q^{31} +(4.12936 - 4.12936i) q^{32} +3.14222i q^{34} -4.47777 q^{36} +(-5.95831 + 5.95831i) q^{37} +(3.95468 + 3.95468i) q^{38} +(-2.78494 + 4.81576i) q^{40} +(-4.19484 - 4.19484i) q^{43} -6.18838i q^{44} +(6.48086 - 1.73161i) q^{45} +(5.71089 - 5.71089i) q^{47} -7.00000i q^{49} +(0.926702 - 3.43896i) q^{50} +(2.39313 + 8.95669i) q^{55} +(0.548804 + 0.548804i) q^{58} -13.2065i q^{59} +(5.11582 - 5.11582i) q^{62} +1.73383i q^{64} +(4.65568 + 4.65568i) q^{68} +(5.27756 - 5.27756i) q^{72} -6.00230i q^{74} +11.7189 q^{76} +(0.700151 + 2.62044i) q^{80} -9.00000 q^{81} +(-8.53876 - 4.93794i) q^{85} +4.22581 q^{86} +(7.29371 + 7.29371i) q^{88} +(-2.39215 + 4.13655i) q^{90} +5.75305i q^{94} +(-16.9612 + 4.53185i) q^{95} +(-13.6445 + 13.6445i) q^{97} +(3.52584 + 3.52584i) q^{98} -12.4382i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q - 112 q^{16} + 168 q^{36} - 14 q^{38} + 126 q^{58} - 154 q^{68} - 252 q^{81}+O(q^{100}) $ 28 * q - 112 * q^16 + 168 * q^36 - 14 * q^38 + 126 * q^58 - 154 * q^68 - 252 * q^81

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/755\mathbb{Z}\right)^\times$.

$n$	$6$	$152$
$\chi(n)$	$-1$	$e\left(\frac{3}{4}\right)$

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 $

$2$	−0.503691	+	0.503691i	−0.356164	+	0.356164i	−0.862397	−	0.506233i	$-0.831038\pi$
$2$	−0.503691	+	0.503691i	−0.356164	+	0.356164i	0.506233	+	0.862397i	$0.331038\pi$
$3$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$3$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$4$			1.49259i			0.746295i
$5$	−0.577204	−	2.16029i	−0.258133	−	0.966109i
$5$	−0.577204	−	2.16029i	−0.258133	−	0.966109i
$6$		0			0
$7$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$7$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$8$	−1.75919	−	1.75919i	−0.621967	−	0.621967i
$9$			3.00000i			1.00000i
$10$	1.37885	+	0.797385i	0.436031	+	0.252155i
$11$	−4.14607			−1.25009			−0.625043	−	0.780590i	$-0.714919\pi$
$11$	−4.14607			−1.25009			−0.625043	+	0.780590i	$0.714919\pi$
$12$		0			0
$13$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$13$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$14$		0			0
$15$		0			0
$16$	−1.21300			−0.303251
$17$	3.11919	−	3.11919i	0.756516	−	0.756516i	−0.219171	−	0.975686i	$-0.570335\pi$
$17$	3.11919	−	3.11919i	0.756516	−	0.756516i	0.975686	+	0.219171i	$0.0703352\pi$
$18$	−1.51107	−	1.51107i	−0.356164	−	0.356164i
$19$		−	7.85139i		−	1.80123i	−0.434615	−	0.900616i	$-0.643116\pi$
$19$		−	7.85139i		−	1.80123i	0.434615	−	0.900616i	$-0.356884\pi$
$20$	3.22442	−	0.861529i	0.721002	−	0.192644i
$21$		0			0
$22$	2.08834	−	2.08834i	0.445235	−	0.445235i
$23$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$23$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$24$		0			0
$25$	−4.33367	+	2.49385i	−0.866734	+	0.498770i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	1.08956i		−	0.202327i	−0.994870	−	0.101163i	$-0.967744\pi$
$29$		−	1.08956i		−	0.202327i	0.994870	−	0.101163i	$-0.0322565\pi$
$30$		0			0
$31$	−10.1567			−1.82419			−0.912094	−	0.409980i	$-0.865536\pi$
$31$	−10.1567			−1.82419			−0.912094	+	0.409980i	$0.865536\pi$
$32$	4.12936	−	4.12936i	0.729974	−	0.729974i
$33$		0			0
$34$			3.14222i			0.538887i
$35$		0			0
$36$	−4.47777			−0.746295
$37$	−5.95831	+	5.95831i	−0.979539	+	0.979539i	−0.999795	−	0.0202554i	$-0.993552\pi$
$37$	−5.95831	+	5.95831i	−0.979539	+	0.979539i	0.0202554	+	0.999795i	$0.493552\pi$
$38$	3.95468	+	3.95468i	0.641533	+	0.641533i
$39$		0			0
$40$	−2.78494	+	4.81576i	−0.440337	+	0.761438i
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$	−4.19484	−	4.19484i	−0.639707	−	0.639707i	0.310776	−	0.950483i	$-0.399411\pi$
$43$	−4.19484	−	4.19484i	−0.639707	−	0.639707i	−0.950483	+	0.310776i	$0.899411\pi$
$44$		−	6.18838i		−	0.932933i
$45$	6.48086	−	1.73161i	0.966109	−	0.258133i
$46$		0			0
$47$	5.71089	−	5.71089i	0.833019	−	0.833019i	−0.154910	−	0.987929i	$-0.549509\pi$
$47$	5.71089	−	5.71089i	0.833019	−	0.833019i	0.987929	+	0.154910i	$0.0495087\pi$
$48$		0			0
$49$		−	7.00000i		−	1.00000i
$50$	0.926702	−	3.43896i	0.131055	−	0.486343i
$51$		0			0
$52$		0			0
$53$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$53$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$54$		0			0
$55$	2.39313	+	8.95669i	0.322689	+	1.20772i
$56$		0			0
$57$		0			0
$58$	0.548804	+	0.548804i	0.0720614	+	0.0720614i
$59$		−	13.2065i		−	1.71934i	−0.510851	−	0.859669i	$-0.670669\pi$
$59$		−	13.2065i		−	1.71934i	0.510851	−	0.859669i	$-0.329331\pi$
$60$		0			0
$61$		0			0		1.00000			$0$
$61$		0			0		−1.00000			$\pi$
$62$	5.11582	−	5.11582i	0.649710	−	0.649710i
$63$		0			0
$64$			1.73383i			0.216729i
$65$		0			0
$66$		0			0
$67$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$67$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$68$	4.65568	+	4.65568i	0.564584	+	0.564584i
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$	5.27756	−	5.27756i	0.621967	−	0.621967i
$73$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$73$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$74$		−	6.00230i		−	0.697753i
$75$		0			0
$76$	11.7189			1.34425
$77$		0			0
$78$		0			0
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$	0.700151	+	2.62044i	0.0782792	+	0.292974i
$81$	−9.00000			−1.00000
$82$		0			0
$83$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$83$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$84$		0			0
$85$	−8.53876	−	4.93794i	−0.926159	−	0.535595i
$86$	4.22581			0.455681
$87$		0			0
$88$	7.29371	+	7.29371i	0.777512	+	0.777512i
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$	−2.39215	+	4.13655i	−0.252155	+	0.436031i
$91$		0			0
$92$		0			0
$93$		0			0
$94$			5.75305i			0.593382i
$95$	−16.9612	+	4.53185i	−1.74019	+	0.464958i
$96$		0			0
$97$	−13.6445	+	13.6445i	−1.38538	+	1.38538i	−0.550646	+	0.834739i	$0.685619\pi$
$97$	−13.6445	+	13.6445i	−1.38538	+	1.38538i	−0.834739	+	0.550646i	$0.814381\pi$
$98$	3.52584	+	3.52584i	0.356164	+	0.356164i
$99$		−	12.4382i		−	1.25009i
$100$	−3.72230	−	6.46839i	−0.372230	−	0.646839i
$101$		0			0		1.00000			$0$
$101$		0			0		−1.00000			$\pi$
$102$		0			0
$103$	9.33716	+	9.33716i	0.920017	+	0.920017i	0.997030	−	0.0770128i	$-0.0245382\pi$
$103$	9.33716	+	9.33716i	0.920017	+	0.920017i	−0.0770128	+	0.997030i	$0.524538\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$107$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$108$		0			0
$109$		0			0			−	1.00000i	$-0.5\pi$
$109$		0			0				1.00000i	$0.5\pi$
$110$	−5.71681	−	3.30601i	−0.545076	−	0.315216i
$111$		0			0
$112$		0			0
$113$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$113$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$114$		0			0
$115$		0			0
$116$	1.62627			0.150995
$117$		0			0
$118$	6.65200	+	6.65200i	0.612366	+	0.612366i
$119$		0			0
$120$		0			0
$121$	6.18987			0.562716
$122$		0			0
$123$		0			0
$124$		−	15.1597i		−	1.36138i
$125$	7.88884	+	7.92251i	0.705600	+	0.708611i
$126$		0			0
$127$	8.15431	−	8.15431i	0.723578	−	0.723578i	−0.245754	−	0.969332i	$-0.579036\pi$
$127$	8.15431	−	8.15431i	0.723578	−	0.723578i	0.969332	+	0.245754i	$0.0790356\pi$
$128$	7.38539	+	7.38539i	0.652783	+	0.652783i
$129$		0			0
$130$		0			0
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	−10.9745			−0.941055
$137$	−16.3051	+	16.3051i	−1.39304	+	1.39304i	−0.574610	+	0.818427i	$0.694846\pi$
$137$	−16.3051	+	16.3051i	−1.39304	+	1.39304i	−0.818427	+	0.574610i	$0.805154\pi$
$138$		0			0
$139$			23.3027i			1.97651i	0.152816	+	0.988255i	$0.451166\pi$
$139$			23.3027i			1.97651i	−0.152816	+	0.988255i	$0.548834\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		−	3.63901i		−	0.303251i
$145$	−2.35377	+	0.628900i	−0.195470	+	0.0522273i
$146$		0			0
$147$		0			0
$148$	−8.89331	−	8.89331i	−0.731025	−	0.731025i
$149$		0			0			−	1.00000i	$-0.5\pi$
$149$		0			0				1.00000i	$0.5\pi$
$150$		0			0
$151$	−12.2882			−1.00000
$151$	−12.2882			−1.00000
$152$	−13.8121	+	13.8121i	−1.12031	+	1.12031i
$153$	9.35758	+	9.35758i	0.756516	+	0.756516i
$154$		0			0
$155$	5.86246	+	21.9413i	0.470884	+	1.76237i
$156$		0			0
$157$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$157$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$158$		0			0
$159$		0			0
$160$	−11.3041	−	6.53711i	−0.893665	−	0.516804i
$161$		0			0
$162$	4.53322	−	4.53322i	0.356164	−	0.356164i
$163$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$163$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	16.2882	−	16.2882i	1.26042	−	1.26042i	0.309529	−	0.950890i	$-0.399829\pi$
$167$	16.2882	−	16.2882i	1.26042	−	1.26042i	0.950890	−	0.309529i	$-0.100171\pi$
$168$		0			0
$169$			13.0000i			1.00000i
$170$	6.78810	−	1.81370i	0.520623	−	0.139105i
$171$	23.5542			1.80123
$172$	6.26117	−	6.26117i	0.477410	−	0.477410i
$173$	−2.22268	−	2.22268i	−0.168987	−	0.168987i	0.617547	−	0.786534i	$-0.288127\pi$
$173$	−2.22268	−	2.22268i	−0.168987	−	0.168987i	−0.786534	+	0.617547i	$0.788127\pi$
$174$		0			0
$175$		0			0
$176$	5.02920			0.379090
$177$		0			0
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$	2.58459	+	9.67326i	0.192644	+	0.721002i
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	16.3108	+	9.43249i	1.19919	+	0.693490i
$186$		0			0
$187$	−12.9324	+	12.9324i	−0.945710	+	0.945710i
$188$	8.52401	+	8.52401i	0.621678	+	0.621678i
$189$		0			0
$190$	6.26058	−	10.8259i	0.454190	−	0.785393i
$191$	−21.1236			−1.52845			−0.764224	−	0.644951i	$-0.776878\pi$
$191$	−21.1236			−1.52845			−0.764224	+	0.644951i	$0.776878\pi$
$192$		0			0
$193$	−2.62117	−	2.62117i	−0.188676	−	0.188676i	0.606448	−	0.795123i	$-0.292594\pi$
$193$	−2.62117	−	2.62117i	−0.188676	−	0.188676i	−0.795123	+	0.606448i	$0.792594\pi$
$194$		−	13.7452i		−	0.986848i
$195$		0			0
$196$	10.4481			0.746295
$197$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$197$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$198$	6.26502	+	6.26502i	0.445235	+	0.445235i
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$	12.0109	+	3.23659i	0.849298	+	0.228861i
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$	−9.40609			−0.655353
$207$		0			0
$208$		0			0
$209$			32.5524i			2.25170i
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−6.64077	+	11.4833i	−0.452897	+	0.783156i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$	−13.3687	+	3.57196i	−0.901315	+	0.240821i
$221$		0			0
$222$		0			0
$223$	17.0714	+	17.0714i	1.14319	+	1.14319i	0.987864	+	0.155322i	$0.0496415\pi$
$223$	17.0714	+	17.0714i	1.14319	+	1.14319i	0.155322	+	0.987864i	$0.450359\pi$
$224$		0			0
$225$	−7.48155	−	13.0010i	−0.498770	−	0.866734i
$226$		0			0
$227$	21.2538	−	21.2538i	1.41066	−	1.41066i	0.655250	−	0.755412i	$-0.272563\pi$
$227$	21.2538	−	21.2538i	1.41066	−	1.41066i	0.755412	−	0.655250i	$-0.227437\pi$
$228$		0			0
$229$		−	22.1698i		−	1.46502i	−0.680755	−	0.732512i	$-0.738348\pi$
$229$		−	22.1698i		−	1.46502i	0.680755	−	0.732512i	$-0.261652\pi$
$230$		0			0
$231$		0			0
$232$	−1.91675	+	1.91675i	−0.125841	+	0.125841i
$233$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$233$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$234$		0			0
$235$	−15.6335	−	9.04081i	−1.01982	−	0.589757i
$236$	19.7119			1.28313
$237$		0			0
$238$		0			0
$239$			11.9189i			0.770969i	0.922714	+	0.385484i	$0.125966\pi$
$239$			11.9189i			0.770969i	−0.922714	+	0.385484i	$0.874034\pi$
$240$		0			0
$241$	−18.9041			−1.21772			−0.608859	−	0.793278i	$-0.708373\pi$
$241$	−18.9041			−1.21772			−0.608859	+	0.793278i	$0.708373\pi$
$242$	−3.11779	+	3.11779i	−0.200419	+	0.200419i
$243$		0			0
$244$		0			0
$245$	−15.1220	+	4.04043i	−0.966109	+	0.258133i
$246$		0			0
$247$		0			0
$248$	17.8675	+	17.8675i	1.13458	+	1.13458i
$249$		0			0
$250$	−7.96404	−	0.0169560i	−0.503690	−	0.00107239i
$251$	−24.5764			−1.55125			−0.775625	−	0.631194i	$-0.782565\pi$
$251$	−24.5764			−1.55125			−0.775625	+	0.631194i	$0.782565\pi$
$252$		0			0
$253$		0			0
$254$			8.21451i			0.515424i
$255$		0			0
$256$	−10.9076			−0.681724
$257$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$257$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	3.26869			0.202327
$262$		0			0
$263$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$263$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	4.93371i		−	0.300814i	−0.988624	−	0.150407i	$-0.951942\pi$
$269$		−	4.93371i		−	0.300814i	0.988624	−	0.150407i	$-0.0480583\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$	−3.78359	+	3.78359i	−0.229414	+	0.229414i
$273$		0			0
$274$		−	16.4255i		−	0.992299i
$275$	17.9677	−	10.3397i	1.08349	−	0.623506i
$276$		0			0
$277$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$277$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$278$	−11.7374	−	11.7374i	−0.703961	−	0.703961i
$279$		−	30.4700i		−	1.82419i
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$283$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	12.3881	+	12.3881i	0.729974	+	0.729974i
$289$		−	2.45874i		−	0.144631i
$290$	0.868801	−	1.50234i	0.0510178	−	0.0882207i
$291$		0			0
$292$		0			0
$293$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$293$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$294$		0			0
$295$	−28.5298	+	7.62284i	−1.66107	+	0.443819i
$296$	20.9636			1.21848
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$	6.18946	−	6.18946i	0.356164	−	0.356164i
$303$		0			0
$304$			9.52376i			0.546225i
$305$		0			0
$306$	−9.42667			−0.538887
$307$	10.8330	−	10.8330i	0.618273	−	0.618273i	−0.326815	−	0.945088i	$-0.605975\pi$
$307$	10.8330	−	10.8330i	0.618273	−	0.618273i	0.945088	+	0.326815i	$0.105975\pi$
$308$		0			0
$309$		0			0
$310$	−14.0045	−	8.09876i	−0.795402	−	0.459979i
$311$	−3.58624			−0.203357			−0.101679	−	0.994817i	$-0.532421\pi$
$311$	−3.58624			−0.203357			−0.101679	+	0.994817i	$0.532421\pi$
$312$		0			0
$313$	24.7537	+	24.7537i	1.39916	+	1.39916i	0.802466	+	0.596697i	$0.203521\pi$
$313$	24.7537	+	24.7537i	1.39916	+	1.39916i	0.596697	+	0.802466i	$0.296479\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$317$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$318$		0			0
$319$			4.51740i			0.252926i
$320$	3.74558	−	1.00078i	0.209384	−	0.0559451i
$321$		0			0
$322$		0			0
$323$	−24.4900	−	24.4900i	−1.36266	−	1.36266i
$324$		−	13.4333i		−	0.746295i
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−2.05147			−0.112759			−0.0563795	−	0.998409i	$-0.517956\pi$
$331$	−2.05147			−0.112759			−0.0563795	+	0.998409i	$0.517956\pi$
$332$		0			0
$333$	−17.8749	−	17.8749i	−0.979539	−	0.979539i
$334$			16.4085i			0.897831i
$335$		0			0
$336$		0			0
$337$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$337$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$338$	−6.54799	−	6.54799i	−0.356164	−	0.356164i
$339$		0			0
$340$	7.37032	−	12.7449i	0.399712	−	0.691187i
$341$	42.1102			2.28039
$342$	−11.8640	+	11.8640i	−0.641533	+	0.641533i
$343$		0			0
$344$			14.7590i			0.795753i
$345$		0			0
$346$	2.23909			0.120374
$347$	26.2882	−	26.2882i	1.41122	−	1.41122i	0.659665	−	0.751559i	$-0.270698\pi$
$347$	26.2882	−	26.2882i	1.41122	−	1.41122i	0.751559	−	0.659665i	$-0.229302\pi$
$348$		0			0
$349$		−	31.9364i		−	1.70952i	−0.519024	−	0.854759i	$-0.673705\pi$
$349$		−	31.9364i		−	1.70952i	0.519024	−	0.854759i	$-0.326295\pi$
$350$		0			0
$351$		0			0
$352$	−17.1206	+	17.1206i	−0.912530	+	0.912530i
$353$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$353$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$	−14.4473	−	8.35482i	−0.761438	−	0.440337i
$361$	−42.6443			−2.24444
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$367$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$368$		0			0
$369$		0			0
$370$	−12.9667	+	3.46455i	−0.674105	+	0.180113i
$371$		0			0
$372$		0			0
$373$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$373$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$374$		−	13.0279i		−	0.673655i
$375$		0			0
$376$	−20.0931			−1.03622
$377$		0			0
$378$		0			0
$379$		0			0			−	1.00000i	$-0.5\pi$
$379$		0			0				1.00000i	$0.5\pi$
$380$	−6.76420	−	25.3162i	−0.346996	−	1.29869i
$381$		0			0
$382$	10.6398	−	10.6398i	0.544378	−	0.544378i
$383$	−20.0257	−	20.0257i	−1.02326	−	1.02326i	−0.999723	−	0.0235404i	$-0.992506\pi$
$383$	−20.0257	−	20.0257i	−1.02326	−	1.02326i	−0.0235404	−	0.999723i	$-0.507494\pi$
$384$		0			0
$385$		0			0
$386$	2.64052			0.134399
$387$	12.5845	−	12.5845i	0.639707	−	0.639707i
$388$	−20.3656	−	20.3656i	−1.03391	−	1.03391i
$389$		0			0			−	1.00000i	$-0.5\pi$
$389$		0			0				1.00000i	$0.5\pi$
$390$		0			0
$391$		0			0
$392$	−12.3143	+	12.3143i	−0.621967	+	0.621967i
$393$		0			0
$394$		0			0
$395$		0			0
$396$	18.5651			0.932933
$397$	25.0662	−	25.0662i	1.25804	−	1.25804i	0.306010	−	0.952028i	$-0.401006\pi$
$397$	25.0662	−	25.0662i	1.25804	−	1.25804i	0.952028	−	0.306010i	$-0.0989941\pi$
$398$		0			0
$399$		0			0
$400$	5.25676	−	3.02505i	0.262838	−	0.151253i
$401$	37.8726			1.89127			0.945634	−	0.325234i	$-0.105443\pi$
$401$	37.8726			1.89127			0.945634	+	0.325234i	$0.105443\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	5.19484	+	19.4426i	0.258133	+	0.966109i
$406$		0			0
$407$	24.7035	−	24.7035i	1.22451	−	1.22451i
$408$		0			0
$409$		0			0			−	1.00000i	$-0.5\pi$
$409$		0			0				1.00000i	$0.5\pi$
$410$		0			0
$411$		0			0
$412$	−13.9365	+	13.9365i	−0.686604	+	0.686604i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$	−16.3964	−	16.3964i	−0.801972	−	0.801972i
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$		0			0
$423$	17.1327	+	17.1327i	0.833019	+	0.833019i
$424$		0			0
$425$	−5.73875	+	21.2964i	−0.278370	+	1.03303i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$	−2.43915	−	9.12895i	−0.117626	−	0.440237i
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$		0			0
$433$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$433$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		−	39.8596i		−	1.90240i	−0.308582	−	0.951198i	$-0.599855\pi$
$439$		−	39.8596i		−	1.90240i	0.308582	−	0.951198i	$-0.400145\pi$
$440$	11.5465	−	19.9665i	0.550460	−	0.951864i
$441$	21.0000			1.00000
$442$		0			0
$443$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$443$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$444$		0			0
$445$		0			0
$446$	−17.1974			−0.814323
$447$		0			0
$448$		0			0
$449$		0			0			−	1.00000i	$-0.5\pi$
$449$		0			0				1.00000i	$0.5\pi$
$450$	10.3169	+	2.78010i	0.486343	+	0.131055i
$451$		0			0
$452$		0			0
$453$		0			0
$454$			21.4107i			1.00485i
$455$		0			0
$456$		0			0
$457$	−0.474264	+	0.474264i	−0.0221851	+	0.0221851i	−0.718112	−	0.695927i	$-0.754994\pi$
$457$	−0.474264	+	0.474264i	−0.0221851	+	0.0221851i	0.695927	+	0.718112i	$0.254994\pi$
$458$	11.1668	+	11.1668i	0.521788	+	0.521788i
$459$		0			0
$460$		0			0
$461$	15.3522			0.715022			0.357511	−	0.933909i	$-0.383625\pi$
$461$	15.3522			0.715022			0.357511	+	0.933909i	$0.383625\pi$
$462$		0			0
$463$	−29.5482	−	29.5482i	−1.37322	−	1.37322i	−0.855626	−	0.517594i	$-0.826828\pi$
$463$	−29.5482	−	29.5482i	−1.37322	−	1.37322i	−0.517594	−	0.855626i	$-0.673172\pi$
$464$			1.32164i			0.0613558i
$465$		0			0
$466$		0			0
$467$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$467$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$468$		0			0
$469$		0			0
$470$	12.4282	−	3.32068i	0.573272	−	0.153172i
$471$		0			0
$472$	−23.2327	+	23.2327i	−1.06937	+	1.06937i
$473$	17.3921	+	17.3921i	0.799688	+	0.799688i
$474$		0			0
$475$	19.5802	+	34.0253i	0.898401	+	1.56119i
$476$		0			0
$477$		0			0
$478$	−6.00344	−	6.00344i	−0.274591	−	0.274591i
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$		0			0
$482$	9.52182	−	9.52182i	0.433707	−	0.433707i
$483$		0			0
$484$			9.23894i			0.419952i
$485$	37.3516	+	21.6003i	1.69605	+	0.980819i
$486$		0			0
$487$	31.1155	−	31.1155i	1.40998	−	1.40998i	0.650291	−	0.759685i	$-0.274647\pi$
$487$	31.1155	−	31.1155i	1.40998	−	1.40998i	0.759685	−	0.650291i	$-0.225353\pi$
$488$		0			0
$489$		0			0
$490$	5.58169	−	9.65195i	0.252155	−	0.436031i
$491$	−21.7007			−0.979340			−0.489670	−	0.871908i	$-0.662883\pi$
$491$	−21.7007			−0.979340			−0.489670	+	0.871908i	$0.662883\pi$
$492$		0			0
$493$	−3.39856	−	3.39856i	−0.153063	−	0.153063i
$494$		0			0
$495$	−26.8701	+	7.17938i	−1.20772	+	0.322689i
$496$	12.3201			0.553187
$497$		0			0
$498$		0			0
$499$		0			0			−	1.00000i	$-0.5\pi$
$499$		0			0				1.00000i	$0.5\pi$
$500$	−11.8251	+	11.7748i	−0.528833	+	0.526585i
$501$		0			0
$502$	12.3789	−	12.3789i	0.552499	−	0.552499i
$503$	−30.2869	−	30.2869i	−1.35042	−	1.35042i	−0.885184	−	0.465241i	$-0.845968\pi$
$503$	−30.2869	−	30.2869i	−1.35042	−	1.35042i	−0.465241	−	0.885184i	$-0.654032\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$	12.1710	+	12.1710i	0.540002	+	0.540002i
$509$		0			0			−	1.00000i	$-0.5\pi$
$509$		0			0				1.00000i	$0.5\pi$
$510$		0			0
$511$		0			0
$512$	−9.27673	+	9.27673i	−0.409977	+	0.409977i
$513$		0			0
$514$		0			0
$515$	14.7815	−	25.5604i	0.651350	−	1.12632i
$516$		0			0
$517$	−23.6777	+	23.6777i	−1.04135	+	1.04135i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−45.5158			−1.99409			−0.997043	−	0.0768521i	$-0.975513\pi$
$521$	−45.5158			−1.99409			−0.997043	+	0.0768521i	$0.975513\pi$
$522$	−1.64641	+	1.64641i	−0.0720614	+	0.0720614i
$523$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$523$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−31.6806	+	31.6806i	−1.38003	+	1.38003i
$528$		0			0
$529$			23.0000i			1.00000i
$530$		0			0
$531$	39.6195			1.71934
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$	2.48507	+	2.48507i	0.107139	+	0.107139i
$539$			29.0225i			1.25009i
$540$		0			0
$541$	46.3986			1.99483			0.997417	−	0.0718303i	$-0.0228840\pi$
$541$	46.3986			1.99483			0.997417	+	0.0718303i	$0.0228840\pi$
$542$		0			0
$543$		0			0
$544$		−	25.7605i		−	1.10447i
$545$		0			0
$546$		0			0
$547$	−32.9998	+	32.9998i	−1.41097	+	1.41097i	−0.657620	+	0.753350i	$0.728437\pi$
$547$	−32.9998	+	32.9998i	−1.41097	+	1.41097i	−0.753350	+	0.657620i	$0.771563\pi$
$548$	−24.3368	−	24.3368i	−1.03962	−	1.03962i
$549$		0			0
$550$	−3.84217	+	14.2582i	−0.163831	+	0.607971i
$551$	−8.55458			−0.364437
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	−34.7814			−1.47506
$557$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$557$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$558$	15.3475	+	15.3475i	0.649710	+	0.649710i
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−33.1302	−	33.1302i	−1.39627	−	1.39627i	−0.810411	−	0.585862i	$-0.800756\pi$
$563$	−33.1302	−	33.1302i	−1.39627	−	1.39627i	−0.585862	−	0.810411i	$-0.699244\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$			46.8448i			1.96384i	0.189308	+	0.981918i	$0.439376\pi$
$569$			46.8448i			1.96384i	−0.189308	+	0.981918i	$0.560624\pi$
$570$		0			0
$571$	1.38743			0.0580620			0.0290310	−	0.999579i	$-0.490758\pi$
$571$	1.38743			0.0580620			0.0290310	+	0.999579i	$0.490758\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−5.20150			−0.216729
$577$	26.0785	−	26.0785i	1.08566	−	1.08566i	0.0896949	−	0.995969i	$-0.471411\pi$
$577$	26.0785	−	26.0785i	1.08566	−	1.08566i	0.995969	−	0.0896949i	$-0.0285892\pi$
$578$	1.23844	+	1.23844i	0.0515125	+	0.0515125i
$579$		0			0
$580$	−0.938690	−	3.51321i	−0.0389770	−	0.145878i
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$587$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$588$		0			0
$589$			79.7438i			3.28579i
$590$	10.5307	−	18.2098i	0.433540	−	0.749685i
$591$		0			0
$592$	7.22745	−	7.22745i	0.297046	−	0.297046i
$593$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$593$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$	−48.0158			−1.95861			−0.979304	−	0.202397i	$-0.935127\pi$
$601$	−48.0158			−1.95861			−0.979304	+	0.202397i	$0.935127\pi$
$602$		0			0
$603$		0			0
$604$		−	18.3413i		−	0.746295i
$605$	−3.57282	−	13.3719i	−0.145256	−	0.543645i
$606$		0			0
$607$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$607$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$608$	−32.4212	−	32.4212i	−1.31485	−	1.31485i
$609$		0			0
$610$		0			0
$611$		0			0
$612$	−13.9670	+	13.9670i	−0.564584	+	0.564584i
$613$	−27.5764	−	27.5764i	−1.11380	−	1.11380i	−0.992632	−	0.121169i	$-0.961336\pi$
$613$	−27.5764	−	27.5764i	−1.11380	−	1.11380i	−0.121169	−	0.992632i	$-0.538664\pi$
$614$			10.9130i			0.440413i
$615$		0			0
$616$		0			0
$617$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$617$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$618$		0			0
$619$		0			0			−	1.00000i	$-0.5\pi$
$619$		0			0				1.00000i	$0.5\pi$
$620$	−32.7493	+	8.75025i	−1.31524	+	0.351418i
$621$		0			0
$622$	1.80636	−	1.80636i	0.0724284	−	0.0724284i
$623$		0			0
$624$		0			0
$625$	12.5614	−	21.6151i	0.502456	−	0.864603i
$626$	−24.9365			−0.996663
$627$		0			0
$628$		0			0
$629$			37.1702i			1.48207i
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−22.3223	−	12.9089i	−0.885835	−	0.512276i
$636$		0			0
$637$		0			0
$638$	−2.27538	−	2.27538i	−0.0900830	−	0.0900830i
$639$		0			0
$640$	11.6917	−	20.2174i	0.462154	−	0.799164i
$641$	38.3956			1.51653			0.758267	−	0.651944i	$-0.226046\pi$
$641$	38.3956			1.51653			0.758267	+	0.651944i	$0.226046\pi$
$642$		0			0
$643$	25.0712	+	25.0712i	0.988710	+	0.988710i	0.999937	−	0.0112267i	$-0.00357365\pi$
$643$	25.0712	+	25.0712i	0.988710	+	0.988710i	−0.0112267	+	0.999937i	$0.503574\pi$
$644$		0			0
$645$		0			0
$646$	24.6708			0.970660
$647$	−33.1704	+	33.1704i	−1.30406	+	1.30406i	−0.378434	+	0.925628i	$0.623537\pi$
$647$	−33.1704	+	33.1704i	−1.30406	+	1.30406i	−0.925628	+	0.378434i	$0.876463\pi$
$648$	15.8327	+	15.8327i	0.621967	+	0.621967i
$649$			54.7550i			2.14932i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−17.5764	−	17.5764i	−0.687818	−	0.687818i	0.273931	−	0.961749i	$-0.411676\pi$
$653$	−17.5764	−	17.5764i	−0.687818	−	0.687818i	−0.961749	+	0.273931i	$0.911676\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	42.0140i		−	1.63663i	−0.574767	−	0.818317i	$-0.694907\pi$
$659$		−	42.0140i		−	1.63663i	0.574767	−	0.818317i	$-0.305093\pi$
$660$		0			0
$661$		0			0		1.00000			$0$
$661$		0			0		−1.00000			$\pi$
$662$	1.03331	−	1.03331i	0.0401607	−	0.0401607i
$663$		0			0
$664$		0			0
$665$		0			0
$666$	18.0069			0.697753
$667$		0			0
$668$	24.3116	+	24.3116i	0.940644	+	0.940644i
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	36.5492	+	36.5492i	1.40887	+	1.40887i	0.765850	+	0.643019i	$0.222318\pi$
$673$	36.5492	+	36.5492i	1.40887	+	1.40887i	0.643019	+	0.765850i	$0.277682\pi$
$674$		0			0
$675$		0			0
$676$	−19.4037			−0.746295
$677$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$677$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$678$		0			0
$679$		0			0
$680$	6.33452	+	23.7080i	0.242918	+	0.909162i
$681$		0			0
$682$	−21.2105	+	21.2105i	−0.812193	+	0.812193i
$683$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$683$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$684$			35.1567i			1.34425i
$685$	44.6350	+	25.8123i	1.70542	+	0.986237i
$686$		0			0
$687$		0			0
$688$	5.08835	+	5.08835i	0.193992	+	0.193992i
$689$		0			0
$690$		0			0
$691$		0			0		1.00000			$0$
$691$		0			0		−1.00000			$\pi$
$692$	3.31755	−	3.31755i	0.126114	−	0.126114i
$693$		0			0
$694$			26.4823i			1.00525i
$695$	50.3405	−	13.4504i	1.90952	−	0.510203i
$696$		0			0
$697$		0			0
$698$	16.0861	+	16.0861i	0.608869	+	0.608869i
$699$		0			0
$700$		0			0
$701$	34.8546			1.31644			0.658219	−	0.752826i	$-0.271310\pi$
$701$	34.8546			1.31644			0.658219	+	0.752826i	$0.271310\pi$
$702$		0			0
$703$	46.7810	+	46.7810i	1.76438	+	1.76438i
$704$		−	7.18859i		−	0.270930i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$		−	23.0070i		−	0.864046i	−0.901863	−	0.432023i	$-0.857800\pi$
$709$		−	23.0070i		−	0.864046i	0.901863	−	0.432023i	$-0.142200\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$	−7.86131	+	2.10045i	−0.292974	+	0.0782792i
$721$		0			0
$722$	21.4796	−	21.4796i	0.799387	−	0.799387i
$723$		0			0
$724$		0			0
$725$	2.71721	+	4.72181i	0.100915	+	0.175364i
$726$		0			0
$727$	36.2882	−	36.2882i	1.34586	−	1.34586i	0.455744	−	0.890111i	$-0.349373\pi$
$727$	36.2882	−	36.2882i	1.34586	−	1.34586i	0.890111	−	0.455744i	$-0.150627\pi$
$728$		0			0
$729$		−	27.0000i		−	1.00000i
$730$		0			0
$731$	−26.1690			−0.967896
$732$		0			0
$733$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$733$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0			−	1.00000i	$-0.5\pi$
$739$		0			0				1.00000i	$0.5\pi$
$740$	−14.0788	+	24.3453i	−0.517548	+	0.894952i
$741$		0			0
$742$		0			0
$743$	14.4961	+	14.4961i	0.531809	+	0.531809i	0.921110	−	0.389301i	$-0.127284\pi$
$743$	14.4961	+	14.4961i	0.531809	+	0.531809i	−0.389301	+	0.921110i	$0.627284\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$	−19.3027	−	19.3027i	−0.705778	−	0.705778i
$749$		0			0
$750$		0			0
$751$		0			0		1.00000			$0$
$751$		0			0		−1.00000			$\pi$
$752$	−6.92733	+	6.92733i	−0.252614	+	0.252614i
$753$		0			0
$754$		0			0
$755$	7.09280	+	26.5460i	0.258133	+	0.966109i
$756$		0			0
$757$	35.8200	−	35.8200i	1.30190	−	1.30190i	0.374791	−	0.927109i	$-0.377714\pi$
$757$	35.8200	−	35.8200i	1.30190	−	1.30190i	0.927109	−	0.374791i	$-0.122286\pi$
$758$		0			0
$759$		0			0
$760$	37.8104	+	21.8656i	1.37153	+	0.793150i
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$		0			0
$763$		0			0
$764$		−	31.5288i		−	1.14067i
$765$	14.8138	−	25.6163i	0.535595	−	0.926159i
$766$	20.1735			0.728898
$767$		0			0
$768$		0			0
$769$		0			0			−	1.00000i	$-0.5\pi$
$769$		0			0				1.00000i	$0.5\pi$
$770$		0			0
$771$		0			0
$772$	3.91233	−	3.91233i	0.140808	−	0.140808i
$773$	−37.5764	−	37.5764i	−1.35153	−	1.35153i	−0.883952	−	0.467578i	$-0.845127\pi$
$773$	−37.5764	−	37.5764i	−1.35153	−	1.35153i	−0.467578	−	0.883952i	$-0.654873\pi$
$774$			12.6774i			0.455681i
$775$	44.0156	−	25.3292i	1.58109	−	0.909851i
$776$	48.0063			1.72333
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$			8.49103i			0.303251i
$785$		0			0
$786$		0			0
$787$	−35.7866	+	35.7866i	−1.27565	+	1.27565i	−0.332577	+	0.943076i	$0.607918\pi$
$787$	−35.7866	+	35.7866i	−1.27565	+	1.27565i	−0.943076	+	0.332577i	$0.892082\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$	−21.8811	+	21.8811i	−0.777512	+	0.777512i
$793$		0			0
$794$			25.2513i			0.896135i
$795$		0			0
$796$		0			0
$797$	36.1524	−	36.1524i	1.28058	−	1.28058i	0.340247	−	0.940336i	$-0.389489\pi$
$797$	36.1524	−	36.1524i	1.28058	−	1.28058i	0.940336	−	0.340247i	$-0.110511\pi$
$798$		0			0
$799$		−	35.6267i		−	1.26038i
$800$	−7.59727	+	28.1933i	−0.268604	+	0.996782i
$801$		0			0
$802$	−19.0761	+	19.0761i	−0.673601	+	0.673601i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0			−	1.00000i	$-0.5\pi$
$809$		0			0				1.00000i	$0.5\pi$
$810$	−12.4097	−	7.17646i	−0.436031	−	0.252155i
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		0			0
$813$		0			0
$814$			24.8859i			0.872251i
$815$		0			0
$816$		0			0
$817$	−32.9353	+	32.9353i	−1.15226	+	1.15226i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		1.00000			$0$
$821$		0			0		−1.00000			$\pi$
$822$		0			0
$823$	−23.0564	−	23.0564i	−0.803696	−	0.803696i	0.179976	−	0.983671i	$-0.442398\pi$
$823$	−23.0564	−	23.0564i	−0.803696	−	0.803696i	−0.983671	+	0.179976i	$0.942398\pi$
$824$		−	32.8516i		−	1.14444i
$825$		0			0
$826$		0			0
$827$	−13.7118	+	13.7118i	−0.476806	+	0.476806i	−0.904109	−	0.427303i	$-0.859464\pi$
$827$	−13.7118	+	13.7118i	−0.476806	+	0.476806i	0.427303	+	0.904109i	$0.359464\pi$
$828$		0			0
$829$			49.1528i			1.70715i	0.520972	+	0.853574i	$0.325570\pi$
$829$			49.1528i			1.70715i	−0.520972	+	0.853574i	$0.674430\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−21.8344	−	21.8344i	−0.756516	−	0.756516i
$834$		0			0
$835$	−44.5888	−	25.7856i	−1.54306	−	0.892346i
$836$	−48.5874			−1.68043
$837$		0			0
$838$		0			0
$839$		−	36.4883i		−	1.25971i	−0.776711	−	0.629857i	$-0.783113\pi$
$839$		−	36.4883i		−	1.25971i	0.776711	−	0.629857i	$-0.216887\pi$
$840$		0			0
$841$	27.8129			0.959064
$842$		0			0
$843$		0			0
$844$		0			0
$845$	28.0837	−	7.50365i	0.966109	−	0.258133i
$846$	−17.2592			−0.593382
$847$		0			0
$848$		0			0
$849$		0			0
$850$	−7.83624	−	13.6174i	−0.268781	−	0.467072i
$851$		0			0
$852$		0			0
$853$	40.5256	+	40.5256i	1.38757	+	1.38757i	0.830397	+	0.557173i	$0.188114\pi$
$853$	40.5256	+	40.5256i	1.38757	+	1.38757i	0.557173	+	0.830397i	$0.311886\pi$
$854$		0			0
$855$	−13.5956	−	50.8837i	−0.464958	−	1.74019i
$856$		0			0
$857$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$857$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$858$		0			0
$859$		0			0			−	1.00000i	$-0.5\pi$
$859$		0			0				1.00000i	$0.5\pi$
$860$	−17.1399	−	9.91195i	−0.584465	−	0.337995i
$861$		0			0
$862$		0			0
$863$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$863$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$864$		0			0
$865$	−3.51868	+	6.08456i	−0.119639	+	0.206881i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	−40.9334	−	40.9334i	−1.38538	−	1.38538i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$877$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$878$	20.0769	+	20.0769i	0.677564	+	0.677564i
$879$		0			0
$880$	−2.90287	−	10.8645i	−0.0978558	−	0.366242i
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$	−10.5775	+	10.5775i	−0.356164	+	0.356164i
$883$	1.64376	+	1.64376i	0.0553168	+	0.0553168i	0.734224	−	0.678907i	$-0.237546\pi$
$883$	1.64376	+	1.64376i	0.0553168	+	0.0553168i	−0.678907	+	0.734224i	$0.737546\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$887$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	37.3146			1.25009
$892$	−25.4806	+	25.4806i	−0.853154	+	0.853154i
$893$	−44.8384	−	44.8384i	−1.50046	−	1.50046i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$			11.0663i			0.369082i
$900$	19.4052	−	11.1669i	0.646839	−	0.372230i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−37.1598	+	37.1598i	−1.23387	+	1.23387i	−0.271404	+	0.962465i	$0.587488\pi$
$907$	−37.1598	+	37.1598i	−1.23387	+	1.23387i	−0.962465	+	0.271404i	$0.912512\pi$
$908$	31.7232	+	31.7232i	1.05277	+	1.05277i
$909$		0			0
$910$		0			0
$911$	−5.08270			−0.168397			−0.0841986	−	0.996449i	$-0.526833\pi$
$911$	−5.08270			−0.168397			−0.0841986	+	0.996449i	$0.526833\pi$
$912$		0			0
$913$		0			0
$914$		−	0.477765i		−	0.0158031i
$915$		0			0
$916$	33.0904			1.09334
$917$		0			0
$918$		0			0
$919$		0			0			−	1.00000i	$-0.5\pi$
$919$		0			0				1.00000i	$0.5\pi$
$920$		0			0
$921$		0			0
$922$	−7.73276	+	7.73276i	−0.254665	+	0.254665i
$923$		0			0
$924$		0			0
$925$	10.9622	−	40.6805i	0.360435	−	1.33757i
$926$	29.7663			0.978182
$927$	−28.0115	+	28.0115i	−0.920017	+	0.920017i
$928$	−4.49919	−	4.49919i	−0.147693	−	0.147693i
$929$		0			0			−	1.00000i	$-0.5\pi$
$929$		0			0				1.00000i	$0.5\pi$
$930$		0			0
$931$	−54.9597			−1.80123
$932$		0			0
$933$		0			0
$934$		0			0
$935$	35.4023	+	20.4730i	1.15778	+	0.669540i
$936$		0			0
$937$	−22.0490	+	22.0490i	−0.720310	+	0.720310i	−0.968668	−	0.248358i	$-0.920109\pi$
$937$	−22.0490	+	22.0490i	−0.720310	+	0.720310i	0.248358	+	0.968668i	$0.420109\pi$
$938$		0			0
$939$		0			0
$940$	13.4942	−	23.3344i	0.440133	−	0.761084i
$941$		0			0		1.00000			$0$
$941$		0			0		−1.00000			$\pi$
$942$		0			0
$943$		0			0
$944$			16.0195i			0.521391i
$945$		0			0
$946$	−17.5205			−0.569640
$947$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$947$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$948$		0			0
$949$		0			0
$950$	−27.0007	−	7.27589i	−0.876017	−	0.236061i
$951$		0			0
$952$		0			0
$953$	18.0898	+	18.0898i	0.585987	+	0.585987i	0.936542	−	0.350555i	$-0.114007\pi$
$953$	18.0898	+	18.0898i	0.585987	+	0.585987i	−0.350555	+	0.936542i	$0.614007\pi$
$954$		0			0
$955$	12.1926	+	45.6330i	0.394544	+	1.47665i
$956$	−17.7900			−0.575370
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	72.1576			2.32766
$962$		0			0
$963$		0			0
$964$		−	28.2160i		−	0.908777i
$965$	−4.14952	+	7.17542i	−0.133578	+	0.230985i
$966$		0			0
$967$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$967$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$968$	−10.8891	−	10.8891i	−0.349991	−	0.349991i
$969$		0			0
$970$	−29.6936	+	7.93378i	−0.953403	+	0.254738i
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$		0			0
$973$		0			0
$974$			31.3452i			1.00436i
$975$		0			0
$976$		0			0
$977$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$977$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$978$		0			0
$979$		0			0
$980$	−6.03070	−	22.5709i	−0.192644	−	0.721002i
$981$		0			0
$982$	10.9305	−	10.9305i	0.348805	−	0.348805i
$983$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$983$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$984$		0			0
$985$		0			0
$986$	3.42365			0.109031
$987$		0			0
$988$		0			0
$989$		0			0
$990$	9.91803	−	17.1504i	0.315216	−	0.545076i
$991$	16.7653			0.532568			0.266284	−	0.963895i	$-0.414204\pi$
$991$	16.7653			0.532568			0.266284	+	0.963895i	$0.414204\pi$
$992$	−41.9404	+	41.9404i	−1.33161	+	1.33161i
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−43.5004	+	43.5004i	−1.37767	+	1.37767i	−0.529131	+	0.848540i	$0.677482\pi$
$997$	−43.5004	+	43.5004i	−1.37767	+	1.37767i	−0.848540	+	0.529131i	$0.822518\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	755.2.f.d.603.6	yes	28
5.2	odd	4	inner	755.2.f.d.452.6	✓	28
151.150	odd	2	CM	755.2.f.d.603.6	yes	28
755.452	even	4	inner	755.2.f.d.452.6	✓	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
755.2.f.d.452.6	✓	28	5.2	odd	4	inner
755.2.f.d.452.6	✓	28	755.452	even	4	inner
755.2.f.d.603.6	yes	28	1.1	even	1	trivial
755.2.f.d.603.6	yes	28	151.150	odd	2	CM

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:	\( N \)	\(=\)	\( 755 = 5 \cdot 151 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	755.f (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.503691 + 0.503691i) q^{2} +1.49259i q^{4} +(-0.577204 - 2.16029i) q^{5} +(-1.75919 - 1.75919i) q^{8} +3.00000i q^{9} +(1.37885 + 0.797385i) q^{10} -4.14607 q^{11} -1.21300 q^{16} +(3.11919 - 3.11919i) q^{17} +(-1.51107 - 1.51107i) q^{18} -7.85139i q^{19} +(3.22442 - 0.861529i) q^{20} +(2.08834 - 2.08834i) q^{22} +(-4.33367 + 2.49385i) q^{25} -1.08956i q^{29} -10.1567 q^{31} +(4.12936 - 4.12936i) q^{32} +3.14222i q^{34} -4.47777 q^{36} +(-5.95831 + 5.95831i) q^{37} +(3.95468 + 3.95468i) q^{38} +(-2.78494 + 4.81576i) q^{40} +(-4.19484 - 4.19484i) q^{43} -6.18838i q^{44} +(6.48086 - 1.73161i) q^{45} +(5.71089 - 5.71089i) q^{47} -7.00000i q^{49} +(0.926702 - 3.43896i) q^{50} +(2.39313 + 8.95669i) q^{55} +(0.548804 + 0.548804i) q^{58} -13.2065i q^{59} +(5.11582 - 5.11582i) q^{62} +1.73383i q^{64} +(4.65568 + 4.65568i) q^{68} +(5.27756 - 5.27756i) q^{72} -6.00230i q^{74} +11.7189 q^{76} +(0.700151 + 2.62044i) q^{80} -9.00000 q^{81} +(-8.53876 - 4.93794i) q^{85} +4.22581 q^{86} +(7.29371 + 7.29371i) q^{88} +(-2.39215 + 4.13655i) q^{90} +5.75305i q^{94} +(-16.9612 + 4.53185i) q^{95} +(-13.6445 + 13.6445i) q^{97} +(3.52584 + 3.52584i) q^{98} -12.4382i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 112 q^{16} + 168 q^{36} - 14 q^{38} + 126 q^{58} - 154 q^{68} - 252 q^{81}+O(q^{100}) \) 28 * q - 112 * q^16 + 168 * q^36 - 14 * q^38 + 126 * q^58 - 154 * q^68 - 252 * q^81

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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