LMFDB - Embedded newform 755.2.f.d.603.10

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			603.10
Character	$\chi$	$=$	755.603
Dual form			755.2.f.d.452.10

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(1.19922 - 1.19922i) q^{2} -0.876236i q^{4} +(-0.417270 + 2.19679i) q^{5} +(1.34764 + 1.34764i) q^{8} +3.00000i q^{9} +(2.13403 + 3.13482i) q^{10} -4.12544 q^{11} +4.98468 q^{16} +(-5.49709 + 5.49709i) q^{17} +(3.59765 + 3.59765i) q^{18} -1.93299i q^{19} +(1.92491 + 0.365627i) q^{20} +(-4.94730 + 4.94730i) q^{22} +(-4.65177 - 1.83331i) q^{25} +7.69805i q^{29} +11.1317 q^{31} +(3.28244 - 3.28244i) q^{32} +13.1844i q^{34} +2.62871 q^{36} +(7.37501 - 7.37501i) q^{37} +(-2.31807 - 2.31807i) q^{38} +(-3.52280 + 2.39814i) q^{40} +(0.190925 + 0.190925i) q^{43} +3.61486i q^{44} +(-6.59037 - 1.25181i) q^{45} +(9.68626 - 9.68626i) q^{47} -7.00000i q^{49} +(-7.77701 + 3.37994i) q^{50} +(1.72142 - 9.06273i) q^{55} +(9.23162 + 9.23162i) q^{58} +8.49359i q^{59} +(13.3493 - 13.3493i) q^{62} +2.09667i q^{64} +(4.81675 + 4.81675i) q^{68} +(-4.04291 + 4.04291i) q^{72} -17.6884i q^{74} -1.69375 q^{76} +(-2.07996 + 10.9503i) q^{80} -9.00000 q^{81} +(-9.78218 - 14.3697i) q^{85} +0.457920 q^{86} +(-5.55959 - 5.55959i) q^{88} +(-9.40446 + 6.40208i) q^{90} -23.2318i q^{94} +(4.24637 + 0.806578i) q^{95} +(-6.31962 + 6.31962i) q^{97} +(-8.39451 - 8.39451i) q^{98} -12.3763i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.19922	−	1.19922i	0.847973	−	0.847973i	−0.141907	−	0.989880i	$-0.545323\pi$
$2$	1.19922	−	1.19922i	0.847973	−	0.847973i	0.989880	+	0.141907i	$0.0453233\pi$
$3$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$3$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$4$		−	0.876236i		−	0.438118i
$5$	−0.417270	+	2.19679i	−0.186609	+	0.982434i
$5$	−0.417270	+	2.19679i	−0.186609	+	0.982434i
$6$		0			0
$7$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$7$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$8$	1.34764	+	1.34764i	0.476461	+	0.476461i
$9$			3.00000i			1.00000i
$10$	2.13403	+	3.13482i	0.674839	+	0.991318i
$11$	−4.12544			−1.24387			−0.621934	−	0.783070i	$-0.713653\pi$
$11$	−4.12544			−1.24387			−0.621934	+	0.783070i	$0.713653\pi$
$12$		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
$16$	4.98468			1.24617
$17$	−5.49709	+	5.49709i	−1.33324	+	1.33324i	−0.430787	+	0.902454i	$0.641764\pi$
$17$	−5.49709	+	5.49709i	−1.33324	+	1.33324i	−0.902454	+	0.430787i	$0.858236\pi$
$18$	3.59765	+	3.59765i	0.847973	+	0.847973i
$19$		−	1.93299i		−	0.443458i	−0.975108	−	0.221729i	$-0.928830\pi$
$19$		−	1.93299i		−	0.443458i	0.975108	−	0.221729i	$-0.0711700\pi$
$20$	1.92491	+	0.365627i	0.430422	+	0.0817567i
$21$		0			0
$22$	−4.94730	+	4.94730i	−1.05477	+	1.05477i
$23$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$23$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$24$		0			0
$25$	−4.65177	−	1.83331i	−0.930354	−	0.366662i
$26$		0			0
$27$		0			0
$28$		0			0
$29$			7.69805i			1.42949i	0.699384	+	0.714746i	$0.253458\pi$
$29$			7.69805i			1.42949i	−0.699384	+	0.714746i	$0.746542\pi$
$30$		0			0
$31$	11.1317			1.99930			0.999652	−	0.0263632i	$-0.00839265\pi$
$31$	11.1317			1.99930			0.999652	+	0.0263632i	$0.00839265\pi$
$32$	3.28244	−	3.28244i	0.580258	−	0.580258i
$33$		0			0
$34$			13.1844i			2.26110i
$35$		0			0
$36$	2.62871			0.438118
$37$	7.37501	−	7.37501i	1.21244	−	1.21244i	0.242223	−	0.970221i	$-0.422123\pi$
$37$	7.37501	−	7.37501i	1.21244	−	1.21244i	0.970221	−	0.242223i	$-0.0778765\pi$
$38$	−2.31807	−	2.31807i	−0.376040	−	0.376040i
$39$		0			0
$40$	−3.52280	+	2.39814i	−0.557004	+	0.379180i
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$	0.190925	+	0.190925i	0.0291158	+	0.0291158i	0.721515	−	0.692399i	$-0.243446\pi$
$43$	0.190925	+	0.190925i	0.0291158	+	0.0291158i	−0.692399	+	0.721515i	$0.743446\pi$
$44$			3.61486i			0.544961i
$45$	−6.59037	−	1.25181i	−0.982434	−	0.186609i
$46$		0			0
$47$	9.68626	−	9.68626i	1.41289	−	1.41289i	0.675809	−	0.737077i	$-0.263794\pi$
$47$	9.68626	−	9.68626i	1.41289	−	1.41289i	0.737077	−	0.675809i	$-0.236206\pi$
$48$		0			0
$49$		−	7.00000i		−	1.00000i
$50$	−7.77701	+	3.37994i	−1.09984	+	0.477996i
$51$		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
$55$	1.72142	−	9.06273i	0.232117	−	1.22202i
$56$		0			0
$57$		0			0
$58$	9.23162	+	9.23162i	1.21217	+	1.21217i
$59$			8.49359i			1.10577i	0.833257	+	0.552886i	$0.186473\pi$
$59$			8.49359i			1.10577i	−0.833257	+	0.552886i	$0.813527\pi$
$60$		0			0
$61$		0			0		1.00000			$0$
$61$		0			0		−1.00000			$\pi$
$62$	13.3493	−	13.3493i	1.69536	−	1.69536i
$63$		0			0
$64$			2.09667i			0.262083i
$65$		0			0
$66$		0			0
$67$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$67$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$68$	4.81675	+	4.81675i	0.584116	+	0.584116i
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$	−4.04291	+	4.04291i	−0.476461	+	0.476461i
$73$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$73$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$74$		−	17.6884i		−	2.05624i
$75$		0			0
$76$	−1.69375			−0.194287
$77$		0			0
$78$		0			0
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$	−2.07996	+	10.9503i	−0.232546	+	1.22428i
$81$	−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
$85$	−9.78218	−	14.3697i	−1.06103	−	1.55862i
$86$	0.457920			0.0493788
$87$		0			0
$88$	−5.55959	−	5.55959i	−0.592655	−	0.592655i
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$	−9.40446	+	6.40208i	−0.991318	+	0.674839i
$91$		0			0
$92$		0			0
$93$		0			0
$94$		−	23.2318i		−	2.39618i
$95$	4.24637	+	0.806578i	0.435668	+	0.0827532i
$96$		0			0
$97$	−6.31962	+	6.31962i	−0.641661	+	0.641661i	−0.950964	−	0.309303i	$-0.899904\pi$
$97$	−6.31962	+	6.31962i	−0.641661	+	0.641661i	0.309303	+	0.950964i	$0.399904\pi$
$98$	−8.39451	−	8.39451i	−0.847973	−	0.847973i
$99$		−	12.3763i		−	1.24387i
$100$	−1.60641	+	4.07605i	−0.160641	+	0.407605i
$101$		0			0		1.00000			$0$
$101$		0			0		−1.00000			$\pi$
$102$		0			0
$103$	8.54934	+	8.54934i	0.842391	+	0.842391i	0.989169	−	0.146778i	$-0.0468903\pi$
$103$	8.54934	+	8.54934i	0.842391	+	0.842391i	−0.146778	+	0.989169i	$0.546890\pi$
$104$		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	$0.5\pi$
$110$	−8.80381	−	12.9325i	−0.839410	−	1.23307i
$111$		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
$116$	6.74531			0.626286
$117$		0			0
$118$	10.1856	+	10.1856i	0.937665	+	0.937665i
$119$		0			0
$120$		0			0
$121$	6.01928			0.547208
$122$		0			0
$123$		0			0
$124$		−	9.75396i		−	0.875931i
$125$	5.96844	−	9.45398i	0.533834	−	0.845590i
$126$		0			0
$127$	−1.40547	+	1.40547i	−0.124715	+	0.124715i	−0.766709	−	0.641994i	$-0.778107\pi$
$127$	−1.40547	+	1.40547i	−0.124715	+	0.124715i	0.641994	+	0.766709i	$0.278107\pi$
$128$	9.07923	+	9.07923i	0.802498	+	0.802498i
$129$		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
$136$	−14.8162			−1.27047
$137$	13.4521	−	13.4521i	1.14929	−	1.14929i	0.162603	−	0.986692i	$-0.448011\pi$
$137$	13.4521	−	13.4521i	1.14929	−	1.14929i	0.986692	−	0.162603i	$-0.0519890\pi$
$138$		0			0
$139$			11.7118i			0.993380i	0.867928	+	0.496690i	$0.165451\pi$
$139$			11.7118i			0.993380i	−0.867928	+	0.496690i	$0.834549\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$			14.9540i			1.24617i
$145$	−16.9110	−	3.21217i	−1.40438	−	0.266756i
$146$		0			0
$147$		0			0
$148$	−6.46224	−	6.46224i	−0.531193	−	0.531193i
$149$		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
$152$	2.60496	−	2.60496i	0.211290	−	0.211290i
$153$	−16.4913	−	16.4913i	−1.33324	−	1.33324i
$154$		0			0
$155$	−4.64491	+	24.4539i	−0.373088	+	1.96419i
$156$		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
$160$	5.84116	+	8.58049i	0.461784	+	0.678347i
$161$		0			0
$162$	−10.7929	+	10.7929i	−0.847973	+	0.847973i
$163$		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
$170$	−28.9633	−	5.50145i	−2.22139	−	0.421942i
$171$	5.79896			0.443458
$172$	0.167295	−	0.167295i	0.0127561	−	0.0127561i
$173$	13.0529	+	13.0529i	0.992393	+	0.992393i	0.999971	−	0.00757841i	$-0.00241231\pi$
$173$	13.0529	+	13.0529i	0.992393	+	0.992393i	−0.00757841	+	0.999971i	$0.502412\pi$
$174$		0			0
$175$		0			0
$176$	−20.5640			−1.55007
$177$		0			0
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$	−1.09688	+	5.77472i	−0.0817567	+	0.430422i
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	13.1240	+	19.2787i	0.964893	+	1.41740i
$186$		0			0
$187$	22.6779	−	22.6779i	1.65838	−	1.65838i
$188$	−8.48744	−	8.48744i	−0.619011	−	0.619011i
$189$		0			0
$190$	6.05957	−	4.12505i	0.439608	−	0.299263i
$191$	−27.1079			−1.96146			−0.980729	−	0.195374i	$-0.937408\pi$
$191$	−27.1079			−1.96146			−0.980729	+	0.195374i	$0.937408\pi$
$192$		0			0
$193$	−6.08667	−	6.08667i	−0.438128	−	0.438128i	0.453254	−	0.891382i	$-0.350263\pi$
$193$	−6.08667	−	6.08667i	−0.438128	−	0.438128i	−0.891382	+	0.453254i	$0.850263\pi$
$194$			15.1572i			1.08822i
$195$		0			0
$196$	−6.13365			−0.438118
$197$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$197$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$198$	−14.8419	−	14.8419i	−1.05477	−	1.05477i
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$	−3.79826	−	8.73953i	−0.268577	−	0.617978i
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$	20.5050			1.42865
$207$		0			0
$208$		0			0
$209$			7.97443i			0.551603i
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−0.499089	+	0.339755i	−0.0340376	+	0.0231711i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$	−7.94109	−	1.50837i	−0.535388	−	0.101695i
$221$		0			0
$222$		0			0
$223$	8.32205	+	8.32205i	0.557286	+	0.557286i	0.928534	−	0.371248i	$-0.121070\pi$
$223$	8.32205	+	8.32205i	0.557286	+	0.557286i	−0.371248	+	0.928534i	$0.621070\pi$
$224$		0			0
$225$	5.49993	−	13.9553i	0.366662	−	0.930354i
$226$		0			0
$227$	−6.20067	+	6.20067i	−0.411553	+	0.411553i	−0.882279	−	0.470726i	$-0.843992\pi$
$227$	−6.20067	+	6.20067i	−0.411553	+	0.411553i	0.470726	+	0.882279i	$0.343992\pi$
$228$		0			0
$229$		−	29.9310i		−	1.97790i	−0.148257	−	0.988949i	$-0.547366\pi$
$229$		−	29.9310i		−	1.97790i	0.148257	−	0.988949i	$-0.452634\pi$
$230$		0			0
$231$		0			0
$232$	−10.3742	+	10.3742i	−0.681098	+	0.681098i
$233$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$233$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$234$		0			0
$235$	17.2369	+	25.3205i	1.12441	+	1.65172i
$236$	7.44239			0.484458
$237$		0			0
$238$		0			0
$239$		−	30.4665i		−	1.97072i	−0.170496	−	0.985358i	$-0.554537\pi$
$239$		−	30.4665i		−	1.97072i	0.170496	−	0.985358i	$-0.445463\pi$
$240$		0			0
$241$	28.2190			1.81775			0.908873	−	0.417073i	$-0.136944\pi$
$241$	28.2190			1.81775			0.908873	+	0.417073i	$0.136944\pi$
$242$	7.21842	−	7.21842i	0.464017	−	0.464017i
$243$		0			0
$244$		0			0
$245$	15.3775	+	2.92089i	0.982434	+	0.186609i
$246$		0			0
$247$		0			0
$248$	15.0014	+	15.0014i	0.952591	+	0.952591i
$249$		0			0
$250$	−4.17991	−	18.4948i	−0.264361	−	1.16971i
$251$	−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
$254$			3.37091i			0.211510i
$255$		0			0
$256$	17.5826			1.09891
$257$		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
$261$	−23.0942			−1.42949
$262$		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
$269$			32.7141i			1.99461i	0.0733538	+	0.997306i	$0.476630\pi$
$269$			32.7141i			1.99461i	−0.0733538	+	0.997306i	$0.523370\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$	−27.4013	+	27.4013i	−1.66144	+	1.66144i
$273$		0			0
$274$		−	32.2641i		−	1.94914i
$275$	19.1906	+	7.56321i	1.15724	+	0.456079i
$276$		0			0
$277$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$277$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$278$	14.0449	+	14.0449i	0.842360	+	0.842360i
$279$			33.3950i			1.99930i
$280$		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
$288$	9.84731	+	9.84731i	0.580258	+	0.580258i
$289$		−	43.4360i		−	2.55506i
$290$	−24.1320	+	16.4279i	−1.41708	+	0.964677i
$291$		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
$295$	−18.6586	−	3.54412i	−1.08635	−	0.206347i
$296$	19.8776			1.15536
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$	−14.7362	+	14.7362i	−0.847973	+	0.847973i
$303$		0			0
$304$		−	9.63533i		−	0.552624i
$305$		0			0
$306$	−39.5532			−2.26110
$307$	19.3162	−	19.3162i	1.10244	−	1.10244i	0.108319	−	0.994116i	$-0.465453\pi$
$307$	19.3162	−	19.3162i	1.10244	−	1.10244i	0.994116	−	0.108319i	$-0.0345469\pi$
$308$		0			0
$309$		0			0
$310$	23.7553	+	34.8958i	1.34921	+	1.98195i
$311$	18.4550			1.04649			0.523244	−	0.852183i	$-0.324722\pi$
$311$	18.4550			1.04649			0.523244	+	0.852183i	$0.324722\pi$
$312$		0			0
$313$	−23.8819	−	23.8819i	−1.34988	−	1.34988i	−0.885783	−	0.464100i	$-0.846378\pi$
$313$	−23.8819	−	23.8819i	−1.34988	−	1.34988i	−0.464100	−	0.885783i	$-0.653622\pi$
$314$		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
$319$		−	31.7579i		−	1.77810i
$320$	−4.60593	−	0.874876i	−0.257479	−	0.0489070i
$321$		0			0
$322$		0			0
$323$	10.6258	+	10.6258i	0.591236	+	0.591236i
$324$			7.88612i			0.438118i
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−34.9616			−1.92166			−0.960832	−	0.277133i	$-0.910616\pi$
$331$	−34.9616			−1.92166			−0.960832	+	0.277133i	$0.910616\pi$
$332$		0			0
$333$	22.1250	+	22.1250i	1.21244	+	1.21244i
$334$		−	39.0661i		−	2.13760i
$335$		0			0
$336$		0			0
$337$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$337$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$338$	15.5898	+	15.5898i	0.847973	+	0.847973i
$339$		0			0
$340$	−12.5913	+	8.57150i	−0.682857	+	0.464855i
$341$	−45.9230			−2.48687
$342$	6.95421	−	6.95421i	0.376040	−	0.376040i
$343$		0			0
$344$			0.514594i			0.0277451i
$345$		0			0
$346$	31.3065			1.68305
$347$	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
$349$			26.0127i			1.39243i	0.717835	+	0.696213i	$0.245133\pi$
$349$			26.0127i			1.39243i	−0.717835	+	0.696213i	$0.754867\pi$
$350$		0			0
$351$		0			0
$352$	−13.5415	+	13.5415i	−0.721765	+	0.721765i
$353$		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	$0.5\pi$
$360$	−7.19443	−	10.5684i	−0.379180	−	0.557004i
$361$	15.2636			0.803345
$362$		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
$370$	38.8578	+	7.38086i	2.02012	+	0.383713i
$371$		0			0
$372$		0			0
$373$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$373$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$374$		−	54.3915i		−	2.81252i
$375$		0			0
$376$	26.1071			1.34637
$377$		0			0
$378$		0			0
$379$		0			0			−	1.00000i	$-0.5\pi$
$379$		0			0				1.00000i	$0.5\pi$
$380$	0.706753	−	3.72082i	0.0362556	−	0.190874i
$381$		0			0
$382$	−32.5082	+	32.5082i	−1.66326	+	1.66326i
$383$	26.3315	+	26.3315i	1.34548	+	1.34548i	0.890505	+	0.454973i	$0.150351\pi$
$383$	26.3315	+	26.3315i	1.34548	+	1.34548i	0.454973	+	0.890505i	$0.349649\pi$
$384$		0			0
$385$		0			0
$386$	−14.5985			−0.743042
$387$	−0.572775	+	0.572775i	−0.0291158	+	0.0291158i
$388$	5.53748	+	5.53748i	0.281123	+	0.281123i
$389$		0			0			−	1.00000i	$-0.5\pi$
$389$		0			0				1.00000i	$0.5\pi$
$390$		0			0
$391$		0			0
$392$	9.43345	−	9.43345i	0.476461	−	0.476461i
$393$		0			0
$394$		0			0
$395$		0			0
$396$	−10.8446			−0.544961
$397$	25.6921	−	25.6921i	1.28945	−	1.28945i	0.354332	−	0.935120i	$-0.384708\pi$
$397$	25.6921	−	25.6921i	1.28945	−	1.28945i	0.935120	−	0.354332i	$-0.115292\pi$
$398$		0			0
$399$		0			0
$400$	−23.1876	−	9.13847i	−1.15938	−	0.456923i
$401$	33.7970			1.68774			0.843871	−	0.536546i	$-0.180271\pi$
$401$	33.7970			1.68774			0.843871	+	0.536546i	$0.180271\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	3.75543	−	19.7711i	0.186609	−	0.982434i
$406$		0			0
$407$	−30.4252	+	30.4252i	−1.50812	+	1.50812i
$408$		0			0
$409$		0			0			−	1.00000i	$-0.5\pi$
$409$		0			0				1.00000i	$0.5\pi$
$410$		0			0
$411$		0			0
$412$	7.49124	−	7.49124i	0.369067	−	0.369067i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$	9.56306	+	9.56306i	0.467745	+	0.467745i
$419$		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
$423$	29.0588	+	29.0588i	1.41289	+	1.41289i
$424$		0			0
$425$	35.6491	−	15.4933i	1.72923	−	0.751537i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$	−0.191076	+	1.00595i	−0.00921452	+	0.0485114i
$431$		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
$439$			21.4764i			1.02501i	0.858683	+	0.512507i	$0.171283\pi$
$439$			21.4764i			1.02501i	−0.858683	+	0.512507i	$0.828717\pi$
$440$	14.5331	−	9.89341i	0.692839	−	0.471650i
$441$	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
$446$	19.9599			0.945127
$447$		0			0
$448$		0			0
$449$		0			0			−	1.00000i	$-0.5\pi$
$449$		0			0				1.00000i	$0.5\pi$
$450$	−10.1398	−	23.3310i	−0.477996	−	1.09984i
$451$		0			0
$452$		0			0
$453$		0			0
$454$			14.8719i			0.697972i
$455$		0			0
$456$		0			0
$457$	−12.6885	+	12.6885i	−0.593541	+	0.593541i	−0.938586	−	0.345045i	$-0.887864\pi$
$457$	−12.6885	+	12.6885i	−0.593541	+	0.593541i	0.345045	+	0.938586i	$0.387864\pi$
$458$	−35.8937	−	35.8937i	−1.67720	−	1.67720i
$459$		0			0
$460$		0			0
$461$	−42.5144			−1.98010			−0.990048	−	0.140733i	$-0.955054\pi$
$461$	−42.5144			−1.98010			−0.990048	+	0.140733i	$0.955054\pi$
$462$		0			0
$463$	13.6663	+	13.6663i	0.635128	+	0.635128i	0.949350	−	0.314222i	$-0.101744\pi$
$463$	13.6663	+	13.6663i	0.635128	+	0.635128i	−0.314222	+	0.949350i	$0.601744\pi$
$464$			38.3723i			1.78139i
$465$		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
$470$	51.0354	+	9.69394i	2.35409	+	0.447148i
$471$		0			0
$472$	−11.4463	+	11.4463i	−0.526857	+	0.526857i
$473$	−0.787650	−	0.787650i	−0.0362162	−	0.0362162i
$474$		0			0
$475$	−3.54377	+	8.99182i	−0.162599	+	0.412573i
$476$		0			0
$477$		0			0
$478$	−36.5359	−	36.5359i	−1.67112	−	1.67112i
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$		0			0
$482$	33.8407	−	33.8407i	1.54140	−	1.54140i
$483$		0			0
$484$		−	5.27431i		−	0.239741i
$485$	−11.2459	−	16.5199i	−0.510650	−	0.750129i
$486$		0			0
$487$	21.2876	−	21.2876i	0.964633	−	0.964633i	−0.0347623	−	0.999396i	$-0.511067\pi$
$487$	21.2876	−	21.2876i	0.964633	−	0.964633i	0.999396	+	0.0347623i	$0.0110674\pi$
$488$		0			0
$489$		0			0
$490$	21.9437	−	14.9382i	0.991318	−	0.674839i
$491$	2.78626			0.125742			0.0628710	−	0.998022i	$-0.479974\pi$
$491$	2.78626			0.125742			0.0628710	+	0.998022i	$0.479974\pi$
$492$		0			0
$493$	−42.3169	−	42.3169i	−1.90586	−	1.90586i
$494$		0			0
$495$	27.1882	+	5.16427i	1.22202	+	0.232117i
$496$	55.4878			2.49147
$497$		0			0
$498$		0			0
$499$		0			0			−	1.00000i	$-0.5\pi$
$499$		0			0				1.00000i	$0.5\pi$
$500$	−8.28391	−	5.22976i	−0.370468	−	0.233882i
$501$		0			0
$502$	−29.4724	+	29.4724i	−1.31542	+	1.31542i
$503$	31.3740	+	31.3740i	1.39890	+	1.39890i	0.803235	+	0.595663i	$0.203111\pi$
$503$	31.3740	+	31.3740i	1.39890	+	1.39890i	0.595663	+	0.803235i	$0.296889\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$	1.23152	+	1.23152i	0.0546398	+	0.0546398i
$509$		0			0			−	1.00000i	$-0.5\pi$
$509$		0			0				1.00000i	$0.5\pi$
$510$		0			0
$511$		0			0
$512$	2.92684	−	2.92684i	0.129349	−	0.129349i
$513$		0			0
$514$		0			0
$515$	−22.3485	+	15.2137i	−0.984792	+	0.670396i
$516$		0			0
$517$	−39.9601	+	39.9601i	−1.75744	+	1.75744i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−31.1216			−1.36346			−0.681731	−	0.731603i	$-0.738773\pi$
$521$	−31.1216			−1.36346			−0.681731	+	0.731603i	$0.738773\pi$
$522$	−27.6949	+	27.6949i	−1.21217	+	1.21217i
$523$		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
$527$	−61.1917	+	61.1917i	−2.66555	+	2.66555i
$528$		0			0
$529$			23.0000i			1.00000i
$530$		0			0
$531$	−25.4808			−1.10577
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$	39.2312	+	39.2312i	1.69138	+	1.69138i
$539$			28.8781i			1.24387i
$540$		0			0
$541$	−40.3539			−1.73495			−0.867476	−	0.497480i	$-0.834259\pi$
$541$	−40.3539			−1.73495			−0.867476	+	0.497480i	$0.834259\pi$
$542$		0			0
$543$		0			0
$544$			36.0877i			1.54725i
$545$		0			0
$546$		0			0
$547$	−22.3255	+	22.3255i	−0.954570	+	0.954570i	−0.999012	−	0.0444423i	$-0.985849\pi$
$547$	−22.3255	+	22.3255i	−0.954570	+	0.954570i	0.0444423	+	0.999012i	$0.485849\pi$
$548$	−11.7873	−	11.7873i	−0.503527	−	0.503527i
$549$		0			0
$550$	32.0836	−	13.9438i	1.36805	−	0.594564i
$551$	14.8802			0.633920
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	10.2623			0.435218
$557$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$557$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$558$	40.0478	+	40.0478i	1.69536	+	1.69536i
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−16.4907	−	16.4907i	−0.695002	−	0.695002i	0.268326	−	0.963328i	$-0.413529\pi$
$563$	−16.4907	−	16.4907i	−0.695002	−	0.695002i	−0.963328	+	0.268326i	$0.913529\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	19.2289i		−	0.806117i	−0.915174	−	0.403059i	$-0.867947\pi$
$569$		−	19.2289i		−	0.806117i	0.915174	−	0.403059i	$-0.132053\pi$
$570$		0			0
$571$	19.4771			0.815090			0.407545	−	0.913185i	$-0.366385\pi$
$571$	19.4771			0.815090			0.407545	+	0.913185i	$0.366385\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−6.29000			−0.262083
$577$	33.2798	−	33.2798i	1.38545	−	1.38545i	0.550850	−	0.834604i	$-0.314303\pi$
$577$	33.2798	−	33.2798i	1.38545	−	1.38545i	0.834604	−	0.550850i	$-0.185697\pi$
$578$	−52.0891	−	52.0891i	−2.16662	−	2.16662i
$579$		0			0
$580$	−2.81462	+	14.8180i	−0.116871	+	0.615285i
$581$		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
$589$		−	21.5174i		−	0.886607i
$590$	−26.6259	+	18.1256i	−1.09617	+	0.746217i
$591$		0			0
$592$	36.7621	−	36.7621i	1.51091	−	1.51091i
$593$		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
$601$	38.9551			1.58901			0.794505	−	0.607257i	$-0.207730\pi$
$601$	38.9551			1.58901			0.794505	+	0.607257i	$0.207730\pi$
$602$		0			0
$603$		0			0
$604$			10.7674i			0.438118i
$605$	−2.51167	+	13.2231i	−0.102114	+	0.537595i
$606$		0			0
$607$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$607$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$608$	−6.34491	−	6.34491i	−0.257320	−	0.257320i
$609$		0			0
$610$		0			0
$611$		0			0
$612$	−14.4502	+	14.4502i	−0.584116	+	0.584116i
$613$	−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	$-0.538664\pi$
$614$		−	46.3286i		−	1.86967i
$615$		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	$0.5\pi$
$620$	21.4274	+	4.07003i	0.860545	+	0.163457i
$621$		0			0
$622$	22.1316	−	22.1316i	0.887394	−	0.887394i
$623$		0			0
$624$		0			0
$625$	18.2780	+	17.0563i	0.731118	+	0.682251i
$626$	−57.2790			−2.28933
$627$		0			0
$628$		0			0
$629$			81.0822i			3.23296i
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−2.50105	−	3.67397i	−0.0992513	−	0.145797i
$636$		0			0
$637$		0			0
$638$	−38.0845	−	38.0845i	−1.50778	−	1.50778i
$639$		0			0
$640$	−23.7336	+	16.1567i	−0.938155	+	0.638648i
$641$	−48.9165			−1.93209			−0.966043	−	0.258381i	$-0.916811\pi$
$641$	−48.9165			−1.93209			−0.966043	+	0.258381i	$0.916811\pi$
$642$		0			0
$643$	35.6782	+	35.6782i	1.40701	+	1.40701i	0.774782	+	0.632228i	$0.217859\pi$
$643$	35.6782	+	35.6782i	1.40701	+	1.40701i	0.632228	+	0.774782i	$0.282141\pi$
$644$		0			0
$645$		0			0
$646$	25.4853			1.00270
$647$	20.9507	−	20.9507i	0.823656	−	0.823656i	−0.162974	−	0.986630i	$-0.552109\pi$
$647$	20.9507	−	20.9507i	0.823656	−	0.823656i	0.986630	+	0.162974i	$0.0521087\pi$
$648$	−12.1287	−	12.1287i	−0.476461	−	0.476461i
$649$		−	35.0398i		−	1.37543i
$650$		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
$659$			25.0496i			0.975793i	0.872902	+	0.487896i	$0.162236\pi$
$659$			25.0496i			0.975793i	−0.872902	+	0.487896i	$0.837764\pi$
$660$		0			0
$661$		0			0		1.00000			$0$
$661$		0			0		−1.00000			$\pi$
$662$	−41.9265	+	41.9265i	−1.62952	+	1.62952i
$663$		0			0
$664$		0			0
$665$		0			0
$666$	53.0653			2.05624
$667$		0			0
$668$	−14.2723	−	14.2723i	−0.552212	−	0.552212i
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−5.02634	−	5.02634i	−0.193751	−	0.193751i	0.603564	−	0.797315i	$-0.293747\pi$
$673$	−5.02634	−	5.02634i	−0.193751	−	0.193751i	−0.797315	+	0.603564i	$0.793747\pi$
$674$		0			0
$675$		0			0
$676$	11.3911			0.438118
$677$		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
$680$	6.18234	−	32.5480i	0.237082	−	1.24816i
$681$		0			0
$682$	−55.0716	+	55.0716i	−2.10880	+	2.10880i
$683$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$683$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$684$		−	5.08126i		−	0.194287i
$685$	23.9384	+	35.1647i	0.914638	+	1.34358i
$686$		0			0
$687$		0			0
$688$	0.951700	+	0.951700i	0.0362832	+	0.0362832i
$689$		0			0
$690$		0			0
$691$		0			0		1.00000			$0$
$691$		0			0		−1.00000			$\pi$
$692$	11.4374	−	11.4374i	0.434785	−	0.434785i
$693$		0			0
$694$		−	63.0504i		−	2.39336i
$695$	−25.7283	−	4.88698i	−0.975931	−	0.185374i
$696$		0			0
$697$		0			0
$698$	31.1948	+	31.1948i	1.18074	+	1.18074i
$699$		0			0
$700$		0			0
$701$	−48.6993			−1.83935			−0.919674	−	0.392682i	$-0.871547\pi$
$701$	−48.6993			−1.83935			−0.919674	+	0.392682i	$0.871547\pi$
$702$		0			0
$703$	−14.2558	−	14.2558i	−0.537668	−	0.537668i
$704$		−	8.64967i		−	0.325997i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$		−	41.7042i		−	1.56623i	−0.621875	−	0.783117i	$-0.713629\pi$
$709$		−	41.7042i		−	1.56623i	0.621875	−	0.783117i	$-0.286371\pi$
$710$		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	$0.5\pi$
$720$	−32.8509	−	6.23988i	−1.22428	−	0.232546i
$721$		0			0
$722$	18.3043	−	18.3043i	0.681215	−	0.681215i
$723$		0			0
$724$		0			0
$725$	14.1129	−	35.8096i	0.524140	−	1.32993i
$726$		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
$731$	−2.09906			−0.0776367
$732$		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	$0.5\pi$
$740$	16.8927	−	11.4997i	0.620988	−	0.422737i
$741$		0			0
$742$		0			0
$743$	−38.0494	−	38.0494i	−1.39590	−	1.39590i	−0.811389	−	0.584507i	$-0.801288\pi$
$743$	−38.0494	−	38.0494i	−1.39590	−	1.39590i	−0.584507	−	0.811389i	$-0.698712\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$	−19.8712	−	19.8712i	−0.726564	−	0.726564i
$749$		0			0
$750$		0			0
$751$		0			0		1.00000			$0$
$751$		0			0		−1.00000			$\pi$
$752$	48.2829	−	48.2829i	1.76070	−	1.76070i
$753$		0			0
$754$		0			0
$755$	5.12750	−	26.9946i	0.186609	−	0.982434i
$756$		0			0
$757$	−38.8661	+	38.8661i	−1.41261	+	1.41261i	−0.672681	+	0.739933i	$0.734857\pi$
$757$	−38.8661	+	38.8661i	−1.41261	+	1.41261i	−0.739933	+	0.672681i	$0.765143\pi$
$758$		0			0
$759$		0			0
$760$	4.63558	+	6.80953i	0.168150	+	0.247008i
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$		0			0
$763$		0			0
$764$			23.7529i			0.859350i
$765$	43.1092	−	29.3465i	1.55862	−	1.06103i
$766$	63.1543			2.28186
$767$		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
$772$	−5.33336	+	5.33336i	−0.191952	+	0.191952i
$773$	−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	$-0.654873\pi$
$774$			1.37376i			0.0493788i
$775$	−51.7819	−	20.4078i	−1.86006	−	0.733069i
$776$	−17.0331			−0.611453
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		−	34.8928i		−	1.24617i
$785$		0			0
$786$		0			0
$787$	24.6605	−	24.6605i	0.879052	−	0.879052i	−0.114385	−	0.993437i	$-0.536490\pi$
$787$	24.6605	−	24.6605i	0.879052	−	0.879052i	0.993437	+	0.114385i	$0.0364897\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$	16.6788	−	16.6788i	0.592655	−	0.592655i
$793$		0			0
$794$		−	61.6208i		−	2.18684i
$795$		0			0
$796$		0			0
$797$	9.29545	−	9.29545i	0.329262	−	0.329262i	−0.523044	−	0.852306i	$-0.675204\pi$
$797$	9.29545	−	9.29545i	0.329262	−	0.329262i	0.852306	+	0.523044i	$0.175204\pi$
$798$		0			0
$799$			106.492i			3.76743i
$800$	−21.2869	+	9.25142i	−0.752605	+	0.327087i
$801$		0			0
$802$	40.5299	−	40.5299i	1.43116	−	1.43116i
$803$		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	$0.5\pi$
$810$	−19.2062	−	28.2134i	−0.674839	−	0.991318i
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		0			0
$813$		0			0
$814$			72.9727i			2.55769i
$815$		0			0
$816$		0			0
$817$	0.369056	−	0.369056i	0.0129116	−	0.0129116i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		1.00000			$0$
$821$		0			0		−1.00000			$\pi$
$822$		0			0
$823$	−40.4750	−	40.4750i	−1.41087	−	1.41087i	−0.754019	−	0.656852i	$-0.771888\pi$
$823$	−40.4750	−	40.4750i	−1.41087	−	1.41087i	−0.656852	−	0.754019i	$-0.728112\pi$
$824$			23.0428i			0.802733i
$825$		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
$833$	38.4796	+	38.4796i	1.33324	+	1.33324i
$834$		0			0
$835$	28.9852	+	42.5783i	1.00307	+	1.47348i
$836$	6.98748			0.241667
$837$		0			0
$838$		0			0
$839$		−	57.9291i		−	1.99993i	−0.00817087	−	0.999967i	$-0.502601\pi$
$839$		−	57.9291i		−	1.99993i	0.00817087	−	0.999967i	$-0.497399\pi$
$840$		0			0
$841$	−30.2600			−1.04345
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−28.5583	−	5.42451i	−0.982434	−	0.186609i
$846$	69.6955			2.39618
$847$		0			0
$848$		0			0
$849$		0			0
$850$	24.1711	−	61.3308i	0.829061	−	2.10363i
$851$		0			0
$852$		0			0
$853$	−39.9746	−	39.9746i	−1.36870	−	1.36870i	−0.862291	−	0.506413i	$-0.830971\pi$
$853$	−39.9746	−	39.9746i	−1.36870	−	1.36870i	−0.506413	−	0.862291i	$-0.669029\pi$
$854$		0			0
$855$	−2.41973	+	12.7391i	−0.0827532	+	0.435668i
$856$		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	$0.5\pi$
$860$	0.297705	+	0.437320i	0.0101517	+	0.0149125i
$861$		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
$865$	−34.1210	+	23.2279i	−1.16015	+	0.789771i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	−18.9589	−	18.9589i	−0.641661	−	0.641661i
$874$		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	$0.750000\pi$
$878$	25.7549	+	25.7549i	0.869185	+	0.869185i
$879$		0			0
$880$	8.58075	−	45.1748i	0.289257	−	1.52284i
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$	25.1835	−	25.1835i	0.847973	−	0.847973i
$883$	−41.3046	−	41.3046i	−1.39001	−	1.39001i	−0.825266	−	0.564744i	$-0.808975\pi$
$883$	−41.3046	−	41.3046i	−1.39001	−	1.39001i	−0.564744	−	0.825266i	$-0.691025\pi$
$884$		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
$891$	37.1290			1.24387
$892$	7.29208	−	7.29208i	0.244157	−	0.244157i
$893$	−18.7234	−	18.7234i	−0.626555	−	0.626555i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$			85.6921i			2.85799i
$900$	−12.2281	−	4.81923i	−0.407605	−	0.160641i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−6.89702	+	6.89702i	−0.229012	+	0.229012i	−0.812280	−	0.583268i	$-0.801774\pi$
$907$	−6.89702	+	6.89702i	−0.229012	+	0.229012i	0.583268	+	0.812280i	$0.301774\pi$
$908$	5.43325	+	5.43325i	0.180309	+	0.180309i
$909$		0			0
$910$		0			0
$911$	−50.1971			−1.66310			−0.831552	−	0.555447i	$-0.812547\pi$
$911$	−50.1971			−1.66310			−0.831552	+	0.555447i	$0.812547\pi$
$912$		0			0
$913$		0			0
$914$			30.4324i			1.00661i
$915$		0			0
$916$	−26.2266			−0.866552
$917$		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
$922$	−50.9840	+	50.9840i	−1.67907	+	1.67907i
$923$		0			0
$924$		0			0
$925$	−47.8275	+	20.7862i	−1.57256	+	0.683445i
$926$	32.7777			1.07714
$927$	−25.6480	+	25.6480i	−0.842391	+	0.842391i
$928$	25.2684	+	25.2684i	0.829475	+	0.829475i
$929$		0			0			−	1.00000i	$-0.5\pi$
$929$		0			0				1.00000i	$0.5\pi$
$930$		0			0
$931$	−13.5309			−0.443458
$932$		0			0
$933$		0			0
$934$		0			0
$935$	40.3558	+	59.2815i	1.31978	+	1.93871i
$936$		0			0
$937$	−42.8735	+	42.8735i	−1.40062	+	1.40062i	−0.602486	+	0.798129i	$0.705823\pi$
$937$	−42.8735	+	42.8735i	−1.40062	+	1.40062i	−0.798129	+	0.602486i	$0.794177\pi$
$938$		0			0
$939$		0			0
$940$	22.1867	−	15.1036i	0.723650	−	0.492624i
$941$		0			0		1.00000			$0$
$941$		0			0		−1.00000			$\pi$
$942$		0			0
$943$		0			0
$944$			42.3379i			1.37798i
$945$		0			0
$946$	−1.88912			−0.0614207
$947$		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
$950$	6.53339	+	15.0329i	0.211971	+	0.487731i
$951$		0			0
$952$		0			0
$953$	−19.7862	−	19.7862i	−0.640936	−	0.640936i	0.309849	−	0.950786i	$-0.399721\pi$
$953$	−19.7862	−	19.7862i	−0.640936	−	0.640936i	−0.950786	+	0.309849i	$0.899721\pi$
$954$		0			0
$955$	11.3113	−	59.5503i	0.366025	−	1.92700i
$956$	−26.6959			−0.863406
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	92.9138			2.99722
$962$		0			0
$963$		0			0
$964$		−	24.7265i		−	0.796387i
$965$	15.9109	−	10.8313i	0.512190	−	0.348673i
$966$		0			0
$967$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$967$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$968$	8.11180	+	8.11180i	0.260723	+	0.260723i
$969$		0			0
$970$	−33.2971	−	6.32464i	−1.06911	−	0.203072i
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$		0			0
$973$		0			0
$974$		−	51.0568i		−	1.63597i
$975$		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
$980$	2.55939	−	13.4743i	0.0817567	−	0.430422i
$981$		0			0
$982$	3.34132	−	3.34132i	0.106626	−	0.106626i
$983$		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
$986$	−101.494			−3.23223
$987$		0			0
$988$		0			0
$989$		0			0
$990$	38.7976	−	26.4114i	1.23307	−	0.839410i
$991$	57.9001			1.83926			0.919629	−	0.392789i	$-0.128490\pi$
$991$	57.9001			1.83926			0.919629	+	0.392789i	$0.128490\pi$
$992$	36.5390	−	36.5390i	1.16011	−	1.16011i
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−0.152825	+	0.152825i	−0.00484001	+	0.00484001i	−0.709523	−	0.704683i	$-0.751089\pi$
$997$	−0.152825	+	0.152825i	−0.00484001	+	0.00484001i	0.704683	+	0.709523i	$0.251089\pi$
$998$		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.603.10	yes	28
5.2	odd	4	inner	755.2.f.d.452.10	✓	28
151.150	odd	2	CM	755.2.f.d.603.10	yes	28
755.452	even	4	inner	755.2.f.d.452.10	✓	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
755.2.f.d.452.10	✓	28	5.2	odd	4	inner
755.2.f.d.452.10	✓	28	755.452	even	4	inner
755.2.f.d.603.10	yes	28	1.1	even	1	trivial
755.2.f.d.603.10	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)

\(f(q)\)	\(=\)	\(q+(1.19922 - 1.19922i) q^{2} -0.876236i q^{4} +(-0.417270 + 2.19679i) q^{5} +(1.34764 + 1.34764i) q^{8} +3.00000i q^{9} +(2.13403 + 3.13482i) q^{10} -4.12544 q^{11} +4.98468 q^{16} +(-5.49709 + 5.49709i) q^{17} +(3.59765 + 3.59765i) q^{18} -1.93299i q^{19} +(1.92491 + 0.365627i) q^{20} +(-4.94730 + 4.94730i) q^{22} +(-4.65177 - 1.83331i) q^{25} +7.69805i q^{29} +11.1317 q^{31} +(3.28244 - 3.28244i) q^{32} +13.1844i q^{34} +2.62871 q^{36} +(7.37501 - 7.37501i) q^{37} +(-2.31807 - 2.31807i) q^{38} +(-3.52280 + 2.39814i) q^{40} +(0.190925 + 0.190925i) q^{43} +3.61486i q^{44} +(-6.59037 - 1.25181i) q^{45} +(9.68626 - 9.68626i) q^{47} -7.00000i q^{49} +(-7.77701 + 3.37994i) q^{50} +(1.72142 - 9.06273i) q^{55} +(9.23162 + 9.23162i) q^{58} +8.49359i q^{59} +(13.3493 - 13.3493i) q^{62} +2.09667i q^{64} +(4.81675 + 4.81675i) q^{68} +(-4.04291 + 4.04291i) q^{72} -17.6884i q^{74} -1.69375 q^{76} +(-2.07996 + 10.9503i) q^{80} -9.00000 q^{81} +(-9.78218 - 14.3697i) q^{85} +0.457920 q^{86} +(-5.55959 - 5.55959i) q^{88} +(-9.40446 + 6.40208i) q^{90} -23.2318i q^{94} +(4.24637 + 0.806578i) q^{95} +(-6.31962 + 6.31962i) q^{97} +(-8.39451 - 8.39451i) q^{98} -12.3763i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 112 q^{16} + 168 q^{36} - 14 q^{38} + 126 q^{58} - 154 q^{68} - 252 q^{81}+O(q^{100}) \) 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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