LMFDB - Embedded newform 755.2.f.d.603.3

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			603.3
Character	$\chi$	$=$	755.603
Dual form			755.2.f.d.452.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-1.77492 + 1.77492i) q^{2} -4.30072i q^{4} +(-2.04886 - 0.895640i) q^{5} +(4.08360 + 4.08360i) q^{8} +3.00000i q^{9} +(5.22627 - 2.04688i) q^{10} +1.48889 q^{11} -5.89474 q^{16} +(-4.94783 + 4.94783i) q^{17} +(-5.32477 - 5.32477i) q^{18} -1.94680i q^{19} +(-3.85189 + 8.81157i) q^{20} +(-2.64267 + 2.64267i) q^{22} +(3.39566 + 3.67008i) q^{25} -10.2040i q^{29} -2.76324 q^{31} +(2.29551 - 2.29551i) q^{32} -17.5641i q^{34} +12.9022 q^{36} +(8.06038 - 8.06038i) q^{37} +(3.45542 + 3.45542i) q^{38} +(-4.70929 - 12.0242i) q^{40} +(-9.08168 - 9.08168i) q^{43} -6.40331i q^{44} +(2.68692 - 6.14658i) q^{45} +(-8.90926 + 8.90926i) q^{47} -7.00000i q^{49} +(-12.5412 - 0.487084i) q^{50} +(-3.05053 - 1.33351i) q^{55} +(18.1113 + 18.1113i) q^{58} -14.3698i q^{59} +(4.90454 - 4.90454i) q^{62} -3.64075i q^{64} +(21.2792 + 21.2792i) q^{68} +(-12.2508 + 12.2508i) q^{72} +28.6131i q^{74} -8.37262 q^{76} +(12.0775 + 5.27956i) q^{80} -9.00000 q^{81} +(14.5689 - 5.70594i) q^{85} +32.2386 q^{86} +(6.08004 + 6.08004i) q^{88} +(6.14064 + 15.6788i) q^{90} -31.6265i q^{94} +(-1.74363 + 3.98871i) q^{95} +(0.308339 - 0.308339i) q^{97} +(12.4245 + 12.4245i) q^{98} +4.46668i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.77492	+	1.77492i	−1.25506	+	1.25506i	−0.301639	+	0.953422i	$0.597534\pi$
$2$	−1.77492	+	1.77492i	−1.25506	+	1.25506i	−0.953422	+	0.301639i	$0.902466\pi$
$3$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$3$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$4$		−	4.30072i		−	2.15036i
$5$	−2.04886	−	0.895640i	−0.916278	−	0.400542i
$5$	−2.04886	−	0.895640i	−0.916278	−	0.400542i
$6$		0			0
$7$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$7$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$8$	4.08360	+	4.08360i	1.44377	+	1.44377i
$9$			3.00000i			1.00000i
$10$	5.22627	−	2.04688i	1.65269	−	0.647280i
$11$	1.48889			0.448918			0.224459	−	0.974484i	$-0.427939\pi$
$11$	1.48889			0.448918			0.224459	+	0.974484i	$0.427939\pi$
$12$		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
$16$	−5.89474			−1.47368
$17$	−4.94783	+	4.94783i	−1.20003	+	1.20003i	−0.225868	+	0.974158i	$0.572522\pi$
$17$	−4.94783	+	4.94783i	−1.20003	+	1.20003i	−0.974158	+	0.225868i	$0.927478\pi$
$18$	−5.32477	−	5.32477i	−1.25506	−	1.25506i
$19$		−	1.94680i		−	0.446626i	−0.974747	−	0.223313i	$-0.928313\pi$
$19$		−	1.94680i		−	0.446626i	0.974747	−	0.223313i	$-0.0716871\pi$
$20$	−3.85189	+	8.81157i	−0.861310	+	1.97033i
$21$		0			0
$22$	−2.64267	+	2.64267i	−0.563420	+	0.563420i
$23$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$23$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$24$		0			0
$25$	3.39566	+	3.67008i	0.679132	+	0.734017i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	10.2040i		−	1.89483i	−0.320006	−	0.947415i	$-0.603685\pi$
$29$		−	10.2040i		−	1.89483i	0.320006	−	0.947415i	$-0.396315\pi$
$30$		0			0
$31$	−2.76324			−0.496292			−0.248146	−	0.968723i	$-0.579821\pi$
$31$	−2.76324			−0.496292			−0.248146	+	0.968723i	$0.579821\pi$
$32$	2.29551	−	2.29551i	0.405793	−	0.405793i
$33$		0			0
$34$		−	17.5641i		−	3.01221i
$35$		0			0
$36$	12.9022			2.15036
$37$	8.06038	−	8.06038i	1.32512	−	1.32512i	0.415545	−	0.909573i	$-0.363591\pi$
$37$	8.06038	−	8.06038i	1.32512	−	1.32512i	0.909573	−	0.415545i	$-0.136409\pi$
$38$	3.45542	+	3.45542i	0.560543	+	0.560543i
$39$		0			0
$40$	−4.70929	−	12.0242i	−0.744604	−	1.90119i
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$	−9.08168	−	9.08168i	−1.38494	−	1.38494i	−0.835591	−	0.549352i	$-0.814875\pi$
$43$	−9.08168	−	9.08168i	−1.38494	−	1.38494i	−0.549352	−	0.835591i	$-0.685125\pi$
$44$		−	6.40331i		−	0.965335i
$45$	2.68692	−	6.14658i	0.400542	−	0.916278i
$46$		0			0
$47$	−8.90926	+	8.90926i	−1.29955	+	1.29955i	−0.370861	+	0.928689i	$0.620937\pi$
$47$	−8.90926	+	8.90926i	−1.29955	+	1.29955i	−0.928689	+	0.370861i	$0.879063\pi$
$48$		0			0
$49$		−	7.00000i		−	1.00000i
$50$	−12.5412	−	0.487084i	−1.77359	−	0.0688840i
$51$		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
$55$	−3.05053	−	1.33351i	−0.411334	−	0.179811i
$56$		0			0
$57$		0			0
$58$	18.1113	+	18.1113i	2.37813	+	2.37813i
$59$		−	14.3698i		−	1.87079i	−0.353607	−	0.935394i	$-0.615045\pi$
$59$		−	14.3698i		−	1.87079i	0.353607	−	0.935394i	$-0.384955\pi$
$60$		0			0
$61$		0			0		1.00000			$0$
$61$		0			0		−1.00000			$\pi$
$62$	4.90454	−	4.90454i	0.622877	−	0.622877i
$63$		0			0
$64$		−	3.64075i		−	0.455094i
$65$		0			0
$66$		0			0
$67$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$67$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$68$	21.2792	+	21.2792i	2.58049	+	2.58049i
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$	−12.2508	+	12.2508i	−1.44377	+	1.44377i
$73$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$73$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$74$			28.6131i			3.32621i
$75$		0			0
$76$	−8.37262			−0.960405
$77$		0			0
$78$		0			0
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$	12.0775	+	5.27956i	1.35030	+	0.590273i
$81$	−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
$85$	14.5689	−	5.70594i	1.58022	−	0.618896i
$86$	32.2386			3.47638
$87$		0			0
$88$	6.08004	+	6.08004i	0.648135	+	0.648135i
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$	6.14064	+	15.6788i	0.647280	+	1.65269i
$91$		0			0
$92$		0			0
$93$		0			0
$94$		−	31.6265i		−	3.26203i
$95$	−1.74363	+	3.98871i	−0.178892	+	0.409233i
$96$		0			0
$97$	0.308339	−	0.308339i	0.0313070	−	0.0313070i	−0.691280	−	0.722587i	$-0.742953\pi$
$97$	0.308339	−	0.308339i	0.0313070	−	0.0313070i	0.722587	+	0.691280i	$0.242953\pi$
$98$	12.4245	+	12.4245i	1.25506	+	1.25506i
$99$			4.46668i			0.448918i
$100$	15.7840	−	14.6038i	1.57840	−	1.46038i
$101$		0			0		1.00000			$0$
$101$		0			0		−1.00000			$\pi$
$102$		0			0
$103$	−3.68300	−	3.68300i	−0.362897	−	0.362897i	0.501981	−	0.864878i	$-0.332605\pi$
$103$	−3.68300	−	3.68300i	−0.362897	−	0.362897i	−0.864878	+	0.501981i	$0.832605\pi$
$104$		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	$0.5\pi$
$110$	7.78135	−	3.04758i	0.741923	−	0.290576i
$111$		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
$116$	−43.8844			−4.07457
$117$		0			0
$118$	25.5053	+	25.5053i	2.34795	+	2.34795i
$119$		0			0
$120$		0			0
$121$	−8.78320			−0.798473
$122$		0			0
$123$		0			0
$124$			11.8839i			1.06721i
$125$	−3.67016	−	10.5608i	−0.328269	−	0.944584i
$126$		0			0
$127$	15.7900	−	15.7900i	1.40114	−	1.40114i	0.604629	−	0.796507i	$-0.293321\pi$
$127$	15.7900	−	15.7900i	1.40114	−	1.40114i	0.796507	−	0.604629i	$-0.206679\pi$
$128$	11.0531	+	11.0531i	0.976964	+	0.976964i
$129$		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
$136$	−40.4100			−3.46513
$137$	−12.3973	+	12.3973i	−1.05917	+	1.05917i	−0.0610332	+	0.998136i	$0.519440\pi$
$137$	−12.3973	+	12.3973i	−1.05917	+	1.05917i	−0.998136	+	0.0610332i	$0.980560\pi$
$138$		0			0
$139$		−	1.67233i		−	0.141845i	−0.997482	−	0.0709224i	$-0.977406\pi$
$139$		−	1.67233i		−	0.141845i	0.997482	−	0.0709224i	$-0.0225943\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		−	17.6842i		−	1.47368i
$145$	−9.13909	+	20.9065i	−0.758960	+	1.73619i
$146$		0			0
$147$		0			0
$148$	−34.6654	−	34.6654i	−2.84948	−	2.84948i
$149$		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
$152$	7.94994	−	7.94994i	0.644825	−	0.644825i
$153$	−14.8435	−	14.8435i	−1.20003	−	1.20003i
$154$		0			0
$155$	5.66148	+	2.47486i	0.454741	+	0.198786i
$156$		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
$160$	−6.75913	+	2.64723i	−0.534356	+	0.209282i
$161$		0			0
$162$	15.9743	−	15.9743i	1.25506	−	1.25506i
$163$		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
$170$	−15.7311	+	35.9863i	−1.20652	+	2.76003i
$171$	5.84039			0.446626
$172$	−39.0577	+	39.0577i	−2.97812	+	2.97812i
$173$	−17.5102	−	17.5102i	−1.33127	−	1.33127i	−0.904231	−	0.427043i	$-0.859555\pi$
$173$	−17.5102	−	17.5102i	−1.33127	−	1.33127i	−0.427043	−	0.904231i	$-0.640445\pi$
$174$		0			0
$175$		0			0
$176$	−8.77663			−0.661563
$177$		0			0
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$	−26.4347	−	11.5557i	−1.97033	−	0.861310i
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−23.7338	+	9.29539i	−1.74494	+	0.683411i
$186$		0			0
$187$	−7.36679	+	7.36679i	−0.538713	+	0.538713i
$188$	38.3162	+	38.3162i	2.79450	+	2.79450i
$189$		0			0
$190$	−3.98486	−	10.1745i	−0.289092	−	0.738134i
$191$	22.0803			1.59767			0.798836	−	0.601548i	$-0.205449\pi$
$191$	22.0803			1.59767			0.798836	+	0.601548i	$0.205449\pi$
$192$		0			0
$193$	−16.8575	−	16.8575i	−1.21343	−	1.21343i	−0.969891	−	0.243539i	$-0.921692\pi$
$193$	−16.8575	−	16.8575i	−1.21343	−	1.21343i	−0.243539	−	0.969891i	$-0.578308\pi$
$194$			1.09456i			0.0785845i
$195$		0			0
$196$	−30.1050			−2.15036
$197$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$197$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$198$	−7.92802	−	7.92802i	−0.563420	−	0.563420i
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$	−1.12064	+	28.8537i	−0.0792413	+	2.04026i
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$	13.0741			0.910916
$207$		0			0
$208$		0			0
$209$		−	2.89857i		−	0.200498i
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	10.4732	+	26.7410i	0.714265	+	1.82372i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$	−5.73506	+	13.1195i	−0.386658	+	0.884515i
$221$		0			0
$222$		0			0
$223$	−9.98654	−	9.98654i	−0.668748	−	0.668748i	0.288678	−	0.957426i	$-0.406784\pi$
$223$	−9.98654	−	9.98654i	−0.668748	−	0.668748i	−0.957426	+	0.288678i	$0.906784\pi$
$224$		0			0
$225$	−11.0102	+	10.1870i	−0.734017	+	0.679132i
$226$		0			0
$227$	−19.8038	+	19.8038i	−1.31442	+	1.31442i	−0.396301	+	0.918121i	$0.629706\pi$
$227$	−19.8038	+	19.8038i	−1.31442	+	1.31442i	−0.918121	+	0.396301i	$0.870294\pi$
$228$		0			0
$229$			25.0201i			1.65337i	0.562664	+	0.826686i	$0.309777\pi$
$229$			25.0201i			1.65337i	−0.562664	+	0.826686i	$0.690223\pi$
$230$		0			0
$231$		0			0
$232$	41.6690	−	41.6690i	2.73570	−	2.73570i
$233$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$233$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$234$		0			0
$235$	26.2333	−	10.2743i	1.71127	−	0.670224i
$236$	−61.8004			−4.02287
$237$		0			0
$238$		0			0
$239$		−	23.1171i		−	1.49532i	−0.664081	−	0.747660i	$-0.731177\pi$
$239$		−	23.1171i		−	1.49532i	0.664081	−	0.747660i	$-0.268823\pi$
$240$		0			0
$241$	27.7185			1.78551			0.892754	−	0.450545i	$-0.148770\pi$
$241$	27.7185			1.78551			0.892754	+	0.450545i	$0.148770\pi$
$242$	15.5895	−	15.5895i	1.00213	−	1.00213i
$243$		0			0
$244$		0			0
$245$	−6.26948	+	14.3420i	−0.400542	+	0.916278i
$246$		0			0
$247$		0			0
$248$	−11.2839	−	11.2839i	−0.716532	−	0.716532i
$249$		0			0
$250$	25.2588	+	12.2303i	1.59751	+	0.773514i
$251$	−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
$254$			56.0521i			3.51702i
$255$		0			0
$256$	−31.9553			−1.99720
$257$		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
$261$	30.6119			1.89483
$262$		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
$269$			18.5157i			1.12892i	0.825461	+	0.564460i	$0.190915\pi$
$269$			18.5157i			1.12892i	−0.825461	+	0.564460i	$0.809085\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$	29.1662	−	29.1662i	1.76846	−	1.76846i
$273$		0			0
$274$		−	44.0084i		−	2.65864i
$275$	5.05577	+	5.46436i	0.304874	+	0.329513i
$276$		0			0
$277$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$277$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$278$	2.96825	+	2.96825i	0.178024	+	0.178024i
$279$		−	8.28971i		−	0.496292i
$280$		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
$288$	6.88653	+	6.88653i	0.405793	+	0.405793i
$289$		−	31.9621i		−	1.88013i
$290$	−20.8863	−	53.3287i	−1.22649	−	3.13157i
$291$		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
$295$	−12.8702	+	29.4417i	−0.749330	+	1.71416i
$296$	65.8307			3.82633
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$	21.8106	−	21.8106i	1.25506	−	1.25506i
$303$		0			0
$304$			11.4758i			0.658185i
$305$		0			0
$306$	52.6922			3.01221
$307$	−0.0908747	+	0.0908747i	−0.00518650	+	0.00518650i	−0.709695	−	0.704509i	$-0.751167\pi$
$307$	−0.0908747	+	0.0908747i	−0.00518650	+	0.00518650i	0.704509	+	0.709695i	$0.251167\pi$
$308$		0			0
$309$		0			0
$310$	−14.4414	+	5.65601i	−0.820217	+	0.321240i
$311$	25.1966			1.42877			0.714384	−	0.699754i	$-0.246707\pi$
$311$	25.1966			1.42877			0.714384	+	0.699754i	$0.246707\pi$
$312$		0			0
$313$	12.5875	+	12.5875i	0.711488	+	0.711488i	0.966846	−	0.255359i	$-0.0821936\pi$
$313$	12.5875	+	12.5875i	0.711488	+	0.711488i	−0.255359	+	0.966846i	$0.582194\pi$
$314$		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
$319$		−	15.1926i		−	0.850624i
$320$	−3.26080	+	7.45939i	−0.182284	+	0.416992i
$321$		0			0
$322$		0			0
$323$	9.63242	+	9.63242i	0.535962	+	0.535962i
$324$			38.7065i			2.15036i
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−13.9142			−0.764795			−0.382397	−	0.923998i	$-0.624901\pi$
$331$	−13.9142			−0.764795			−0.382397	+	0.923998i	$0.624901\pi$
$332$		0			0
$333$	24.1811	+	24.1811i	1.32512	+	1.32512i
$334$			57.8207i			3.16381i
$335$		0			0
$336$		0			0
$337$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$337$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$338$	−23.0740	−	23.0740i	−1.25506	−	1.25506i
$339$		0			0
$340$	−24.5397	−	62.6567i	−1.33085	−	3.39804i
$341$	−4.11416			−0.222794
$342$	−10.3662	+	10.3662i	−0.560543	+	0.560543i
$343$		0			0
$344$		−	74.1719i		−	3.99908i
$345$		0			0
$346$	62.1585			3.34166
$347$	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
$349$			37.1878i			1.99062i	0.0967585	+	0.995308i	$0.469153\pi$
$349$			37.1878i			1.99062i	−0.0967585	+	0.995308i	$0.530847\pi$
$350$		0			0
$351$		0			0
$352$	3.41777	−	3.41777i	0.182168	−	0.182168i
$353$		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	$0.5\pi$
$360$	36.0725	−	14.1279i	1.90119	−	0.744604i
$361$	15.2100			0.800526
$362$		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
$370$	25.6271	−	58.6243i	1.33229	−	3.04773i
$371$		0			0
$372$		0			0
$373$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$373$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$374$		−	26.1510i		−	1.35224i
$375$		0			0
$376$	−72.7637			−3.75250
$377$		0			0
$378$		0			0
$379$		0			0			−	1.00000i	$-0.5\pi$
$379$		0			0				1.00000i	$0.5\pi$
$380$	17.1543	+	7.49885i	0.879998	+	0.384683i
$381$		0			0
$382$	−39.1908	+	39.1908i	−2.00518	+	2.00518i
$383$	2.45053	+	2.45053i	0.125216	+	0.125216i	0.766938	−	0.641722i	$-0.221780\pi$
$383$	2.45053	+	2.45053i	0.125216	+	0.125216i	−0.641722	+	0.766938i	$0.721780\pi$
$384$		0			0
$385$		0			0
$386$	59.8416			3.04586
$387$	27.2450	−	27.2450i	1.38494	−	1.38494i
$388$	−1.32608	−	1.32608i	−0.0673214	−	0.0673214i
$389$		0			0			−	1.00000i	$-0.5\pi$
$389$		0			0				1.00000i	$0.5\pi$
$390$		0			0
$391$		0			0
$392$	28.5852	−	28.5852i	1.44377	−	1.44377i
$393$		0			0
$394$		0			0
$395$		0			0
$396$	19.2099			0.965335
$397$	−18.1269	+	18.1269i	−0.909761	+	0.909761i	−0.996253	−	0.0864916i	$-0.972434\pi$
$397$	−18.1269	+	18.1269i	−0.909761	+	0.909761i	0.0864916	+	0.996253i	$0.472434\pi$
$398$		0			0
$399$		0			0
$400$	−20.0165	−	21.6342i	−1.00083	−	1.08171i
$401$	−21.1265			−1.05501			−0.527503	−	0.849553i	$-0.676871\pi$
$401$	−21.1265			−1.05501			−0.527503	+	0.849553i	$0.676871\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	18.4397	+	8.06076i	0.916278	+	0.400542i
$406$		0			0
$407$	12.0010	−	12.0010i	0.594869	−	0.594869i
$408$		0			0
$409$		0			0			−	1.00000i	$-0.5\pi$
$409$		0			0				1.00000i	$0.5\pi$
$410$		0			0
$411$		0			0
$412$	−15.8396	+	15.8396i	−0.780359	+	0.780359i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$	5.14474	+	5.14474i	0.251638	+	0.251638i
$419$		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
$423$	−26.7278	−	26.7278i	−1.29955	−	1.29955i
$424$		0			0
$425$	−34.9601	−	1.35781i	−1.69581	−	0.0658634i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$	−66.0524	−	28.8742i	−3.18533	−	1.39244i
$431$		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
$439$			41.5228i			1.98178i	0.134686	+	0.990888i	$0.456997\pi$
$439$			41.5228i			1.98178i	−0.134686	+	0.990888i	$0.543003\pi$
$440$	−7.01163	−	17.9027i	−0.334266	−	0.853477i
$441$	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
$446$	35.4507			1.67864
$447$		0			0
$448$		0			0
$449$		0			0			−	1.00000i	$-0.5\pi$
$449$		0			0				1.00000i	$0.5\pi$
$450$	1.46125	−	37.6235i	0.0688840	−	1.77359i
$451$		0			0
$452$		0			0
$453$		0			0
$454$		−	70.3004i		−	3.29936i
$455$		0			0
$456$		0			0
$457$	−23.9295	+	23.9295i	−1.11937	+	1.11937i	−0.127539	+	0.991834i	$0.540708\pi$
$457$	−23.9295	+	23.9295i	−1.11937	+	1.11937i	−0.991834	+	0.127539i	$0.959292\pi$
$458$	−44.4087	−	44.4087i	−2.07508	−	2.07508i
$459$		0			0
$460$		0			0
$461$	−31.2322			−1.45463			−0.727314	−	0.686305i	$-0.759232\pi$
$461$	−31.2322			−1.45463			−0.727314	+	0.686305i	$0.759232\pi$
$462$		0			0
$463$	29.7779	+	29.7779i	1.38390	+	1.38390i	0.837578	+	0.546317i	$0.183971\pi$
$463$	29.7779	+	29.7779i	1.38390	+	1.38390i	0.546317	+	0.837578i	$0.316029\pi$
$464$			60.1497i			2.79238i
$465$		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
$470$	−28.3260	+	64.7984i	−1.30658	+	2.98893i
$471$		0			0
$472$	58.6805	−	58.6805i	2.70099	−	2.70099i
$473$	−13.5216	−	13.5216i	−0.621726	−	0.621726i
$474$		0			0
$475$	7.14490	−	6.61065i	0.327831	−	0.303318i
$476$		0			0
$477$		0			0
$478$	41.0311	+	41.0311i	1.87672	+	1.87672i
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$		0			0
$482$	−49.1983	+	49.1983i	−2.24092	+	2.24092i
$483$		0			0
$484$			37.7741i			1.71700i
$485$	−0.907903	+	0.355582i	−0.0412257	+	0.0161462i
$486$		0			0
$487$	−9.27743	+	9.27743i	−0.420400	+	0.420400i	−0.885342	−	0.464941i	$-0.846075\pi$
$487$	−9.27743	+	9.27743i	−0.420400	+	0.420400i	0.464941	+	0.885342i	$0.346075\pi$
$488$		0			0
$489$		0			0
$490$	−14.3282	−	36.5839i	−0.647280	−	1.65269i
$491$	−43.7404			−1.97398			−0.986989	−	0.160786i	$-0.948597\pi$
$491$	−43.7404			−1.97398			−0.986989	+	0.160786i	$0.948597\pi$
$492$		0			0
$493$	50.4876	+	50.4876i	2.27385	+	2.27385i
$494$		0			0
$495$	4.00054	−	9.15160i	0.179811	−	0.411334i
$496$	16.2885			0.731377
$497$		0			0
$498$		0			0
$499$		0			0			−	1.00000i	$-0.5\pi$
$499$		0			0				1.00000i	$0.5\pi$
$500$	−45.4189	+	15.7843i	−2.03120	+	0.705896i
$501$		0			0
$502$	43.6213	−	43.6213i	1.94691	−	1.94691i
$503$	−11.5200	−	11.5200i	−0.513651	−	0.513651i	0.401992	−	0.915643i	$-0.368318\pi$
$503$	−11.5200	−	11.5200i	−0.513651	−	0.513651i	−0.915643	+	0.401992i	$0.868318\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$	−67.9083	−	67.9083i	−3.01295	−	3.01295i
$509$		0			0			−	1.00000i	$-0.5\pi$
$509$		0			0				1.00000i	$0.5\pi$
$510$		0			0
$511$		0			0
$512$	34.6121	−	34.6121i	1.52965	−	1.52965i
$513$		0			0
$514$		0			0
$515$	4.24732	+	10.8446i	0.187159	+	0.477870i
$516$		0			0
$517$	−13.2649	+	13.2649i	−0.583391	+	0.583391i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	13.5486			0.593576			0.296788	−	0.954943i	$-0.404085\pi$
$521$	13.5486			0.593576			0.296788	+	0.954943i	$0.404085\pi$
$522$	−54.3339	+	54.3339i	−2.37813	+	2.37813i
$523$		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
$527$	13.6720	−	13.6720i	0.595563	−	0.595563i
$528$		0			0
$529$			23.0000i			1.00000i
$530$		0			0
$531$	43.1094			1.87079
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$	−32.8639	−	32.8639i	−1.41686	−	1.41686i
$539$		−	10.4222i		−	0.448918i
$540$		0			0
$541$	31.5415			1.35608			0.678038	−	0.735026i	$-0.262830\pi$
$541$	31.5415			1.35608			0.678038	+	0.735026i	$0.262830\pi$
$542$		0			0
$543$		0			0
$544$			22.7156i			0.973924i
$545$		0			0
$546$		0			0
$547$	9.52594	−	9.52594i	0.407300	−	0.407300i	−0.473496	−	0.880796i	$-0.657008\pi$
$547$	9.52594	−	9.52594i	0.407300	−	0.407300i	0.880796	+	0.473496i	$0.157008\pi$
$548$	53.3171	+	53.3171i	2.27759	+	2.27759i
$549$		0			0
$550$	−18.6724	−	0.725215i	−0.796196	−	0.0309233i
$551$	−19.8651			−0.846280
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	−7.19220			−0.305017
$557$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$557$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$558$	14.7136	+	14.7136i	0.622877	+	0.622877i
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$	2.17773	+	2.17773i	0.0917802	+	0.0917802i	0.751506	−	0.659726i	$-0.229328\pi$
$563$	2.17773	+	2.17773i	0.0917802	+	0.0917802i	−0.659726	+	0.751506i	$0.729328\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	46.1243i		−	1.93363i	−0.255478	−	0.966815i	$-0.582233\pi$
$569$		−	46.1243i		−	1.93363i	0.255478	−	0.966815i	$-0.417767\pi$
$570$		0			0
$571$	38.2140			1.59920			0.799602	−	0.600530i	$-0.205044\pi$
$571$	38.2140			1.59920			0.799602	+	0.600530i	$0.205044\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	10.9222			0.455094
$577$	−27.0267	+	27.0267i	−1.12514	+	1.12514i	−0.134178	+	0.990957i	$0.542839\pi$
$577$	−27.0267	+	27.0267i	−1.12514	+	1.12514i	−0.990957	+	0.134178i	$0.957161\pi$
$578$	56.7304	+	56.7304i	2.35967	+	2.35967i
$579$		0			0
$580$	89.9130	+	39.3046i	3.73344	+	1.63204i
$581$		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
$589$			5.37945i			0.221657i
$590$	−29.4132	−	75.1004i	−1.21092	−	3.09184i
$591$		0			0
$592$	−47.5138	+	47.5138i	−1.95280	+	1.95280i
$593$		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
$601$	−37.6960			−1.53765			−0.768826	−	0.639458i	$-0.779159\pi$
$601$	−37.6960			−1.53765			−0.768826	+	0.639458i	$0.779159\pi$
$602$		0			0
$603$		0			0
$604$			52.8481i			2.15036i
$605$	17.9955	+	7.86658i	0.731623	+	0.319822i
$606$		0			0
$607$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$607$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$608$	−4.46889	−	4.46889i	−0.181237	−	0.181237i
$609$		0			0
$610$		0			0
$611$		0			0
$612$	−63.8377	+	63.8377i	−2.58049	+	2.58049i
$613$	−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	$-0.538664\pi$
$614$		−	0.322592i		−	0.0130187i
$615$		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	$0.5\pi$
$620$	10.6437	−	24.3484i	0.427461	−	0.977857i
$621$		0			0
$622$	−44.7221	+	44.7221i	−1.79319	+	1.79319i
$623$		0			0
$624$		0			0
$625$	−1.93902	+	24.9247i	−0.0775606	+	0.996988i
$626$	−44.6837			−1.78592
$627$		0			0
$628$		0			0
$629$			79.7628i			3.18035i
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−46.4937	+	18.2094i	−1.84505	+	0.722616i
$636$		0			0
$637$		0			0
$638$	26.9658	+	26.9658i	1.06759	+	1.06759i
$639$		0			0
$640$	−12.7466	−	32.5458i	−0.503855	−	1.28649i
$641$	−1.87040			−0.0738763			−0.0369381	−	0.999318i	$-0.511760\pi$
$641$	−1.87040			−0.0738763			−0.0369381	+	0.999318i	$0.511760\pi$
$642$		0			0
$643$	−30.5765	−	30.5765i	−1.20582	−	1.20582i	−0.972368	−	0.233452i	$-0.924998\pi$
$643$	−30.5765	−	30.5765i	−1.20582	−	1.20582i	−0.233452	−	0.972368i	$-0.575002\pi$
$644$		0			0
$645$		0			0
$646$	−34.1937			−1.34533
$647$	35.9245	−	35.9245i	1.41234	−	1.41234i	0.669766	−	0.742572i	$-0.266394\pi$
$647$	35.9245	−	35.9245i	1.41234	−	1.41234i	0.742572	−	0.669766i	$-0.233606\pi$
$648$	−36.7524	−	36.7524i	−1.44377	−	1.44377i
$649$		−	21.3951i		−	0.839831i
$650$		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
$659$		−	49.2669i		−	1.91917i	−0.281425	−	0.959583i	$-0.590807\pi$
$659$		−	49.2669i		−	1.91917i	0.281425	−	0.959583i	$-0.409193\pi$
$660$		0			0
$661$		0			0		1.00000			$0$
$661$		0			0		−1.00000			$\pi$
$662$	24.6967	−	24.6967i	0.959865	−	0.959865i
$663$		0			0
$664$		0			0
$665$		0			0
$666$	−85.8394			−3.32621
$667$		0			0
$668$	−70.0510	−	70.0510i	−2.71035	−	2.71035i
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−31.5471	−	31.5471i	−1.21605	−	1.21605i	−0.969003	−	0.247050i	$-0.920539\pi$
$673$	−31.5471	−	31.5471i	−1.21605	−	1.21605i	−0.247050	−	0.969003i	$-0.579461\pi$
$674$		0			0
$675$		0			0
$676$	55.9093			2.15036
$677$		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
$680$	82.7944	+	36.1928i	3.17502	+	1.38793i
$681$		0			0
$682$	7.30233	−	7.30233i	0.279621	−	0.279621i
$683$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$683$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$684$		−	25.1179i		−	0.960405i
$685$	36.5037	−	14.2968i	1.39474	−	0.546251i
$686$		0			0
$687$		0			0
$688$	53.5341	+	53.5341i	2.04097	+	2.04097i
$689$		0			0
$690$		0			0
$691$		0			0		1.00000			$0$
$691$		0			0		−1.00000			$\pi$
$692$	−75.3063	+	75.3063i	−2.86272	+	2.86272i
$693$		0			0
$694$			93.3192i			3.54235i
$695$	−1.49780	+	3.42636i	−0.0568148	+	0.129969i
$696$		0			0
$697$		0			0
$698$	−66.0055	−	66.0055i	−2.49835	−	2.49835i
$699$		0			0
$700$		0			0
$701$	−9.43566			−0.356380			−0.178190	−	0.983996i	$-0.557024\pi$
$701$	−9.43566			−0.356380			−0.178190	+	0.983996i	$0.557024\pi$
$702$		0			0
$703$	−15.6919	−	15.6919i	−0.591831	−	0.591831i
$704$		−	5.42069i		−	0.204300i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$		−	0.109919i		−	0.00412809i	−0.999998	−	0.00206405i	$-0.999343\pi$
$709$		−	0.109919i		−	0.00412809i	0.999998	−	0.00206405i	$-0.000657007\pi$
$710$		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	$0.5\pi$
$720$	−15.8387	+	36.2325i	−0.590273	+	1.35030i
$721$		0			0
$722$	−26.9966	+	26.9966i	−1.00471	+	1.00471i
$723$		0			0
$724$		0			0
$725$	37.4494	−	34.6492i	1.39084	−	1.28684i
$726$		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
$731$	89.8693			3.32394
$732$		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	$0.5\pi$
$740$	39.9768	+	102.072i	1.46958	+	3.75225i
$741$		0			0
$742$		0			0
$743$	−28.5585	−	28.5585i	−1.04771	−	1.04771i	−0.998803	−	0.0489065i	$-0.984426\pi$
$743$	−28.5585	−	28.5585i	−1.04771	−	1.04771i	−0.0489065	−	0.998803i	$-0.515574\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$	31.6825	+	31.6825i	1.15843	+	1.15843i
$749$		0			0
$750$		0			0
$751$		0			0		1.00000			$0$
$751$		0			0		−1.00000			$\pi$
$752$	52.5177	−	52.5177i	1.91512	−	1.91512i
$753$		0			0
$754$		0			0
$755$	25.1768	+	11.0058i	0.916278	+	0.400542i
$756$		0			0
$757$	10.4525	−	10.4525i	0.379902	−	0.379902i	−0.491164	−	0.871067i	$-0.663429\pi$
$757$	10.4525	−	10.4525i	0.379902	−	0.379902i	0.871067	+	0.491164i	$0.163429\pi$
$758$		0			0
$759$		0			0
$760$	−23.4086	+	9.16803i	−0.849119	+	0.332559i
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$		0			0
$763$		0			0
$764$		−	94.9610i		−	3.43557i
$765$	17.1178	+	43.7067i	0.618896	+	1.58022i
$766$	−8.69900			−0.314308
$767$		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
$772$	−72.4994	+	72.4994i	−2.60931	+	2.60931i
$773$	−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	$-0.654873\pi$
$774$			96.7158i			3.47638i
$775$	−9.38300	−	10.1413i	−0.337047	−	0.364286i
$776$	2.51826			0.0904004
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$			41.2631i			1.47368i
$785$		0			0
$786$		0			0
$787$	39.6736	−	39.6736i	1.41421	−	1.41421i	0.705382	−	0.708827i	$-0.250775\pi$
$787$	39.6736	−	39.6736i	1.41421	−	1.41421i	0.708827	−	0.705382i	$-0.249225\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$	−18.2401	+	18.2401i	−0.648135	+	0.648135i
$793$		0			0
$794$		−	64.3476i		−	2.28361i
$795$		0			0
$796$		0			0
$797$	8.47182	−	8.47182i	0.300087	−	0.300087i	−0.540961	−	0.841048i	$-0.681939\pi$
$797$	8.47182	−	8.47182i	0.300087	−	0.300087i	0.841048	+	0.540961i	$0.181939\pi$
$798$		0			0
$799$		−	88.1631i		−	3.11899i
$800$	16.2195	+	0.629945i	0.573445	+	0.0222719i
$801$		0			0
$802$	37.4979	−	37.4979i	1.32410	−	1.32410i
$803$		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	$0.5\pi$
$810$	−47.0364	+	18.4219i	−1.65269	+	0.647280i
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		0			0
$813$		0			0
$814$			42.6019i			1.49320i
$815$		0			0
$816$		0			0
$817$	−17.6802	+	17.6802i	−0.618551	+	0.618551i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		1.00000			$0$
$821$		0			0		−1.00000			$\pi$
$822$		0			0
$823$	37.6762	+	37.6762i	1.31331	+	1.31331i	0.918960	+	0.394351i	$0.129030\pi$
$823$	37.6762	+	37.6762i	1.31331	+	1.31331i	0.394351	+	0.918960i	$0.370970\pi$
$824$		−	30.0798i		−	1.04788i
$825$		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
$833$	34.6348	+	34.6348i	1.20003	+	1.20003i
$834$		0			0
$835$	−47.9606	+	18.7839i	−1.65975	+	0.650043i
$836$	−12.4659			−0.431143
$837$		0			0
$838$		0			0
$839$			51.9869i			1.79479i	0.441231	+	0.897394i	$0.354542\pi$
$839$			51.9869i			1.79479i	−0.441231	+	0.897394i	$0.645458\pi$
$840$		0			0
$841$	−75.1211			−2.59038
$842$		0			0
$843$		0			0
$844$		0			0
$845$	11.6433	−	26.6352i	0.400542	−	0.916278i
$846$	94.8796			3.26203
$847$		0			0
$848$		0			0
$849$		0			0
$850$	64.4616	−	59.6416i	2.21101	−	2.04569i
$851$		0			0
$852$		0			0
$853$	19.0284	+	19.0284i	0.651520	+	0.651520i	0.953359	−	0.301839i	$-0.0976004\pi$
$853$	19.0284	+	19.0284i	0.651520	+	0.651520i	−0.301839	+	0.953359i	$0.597600\pi$
$854$		0			0
$855$	−11.9661	−	5.23088i	−0.409233	−	0.178892i
$856$		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	$0.5\pi$
$860$	115.006	−	45.0422i	3.92166	−	1.53593i
$861$		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
$865$	20.1931	+	51.5587i	0.686586	+	1.75305i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	0.925016	+	0.925016i	0.0313070	+	0.0313070i
$874$		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	$0.750000\pi$
$878$	−73.6999	−	73.6999i	−2.48725	−	2.48725i
$879$		0			0
$880$	17.9821	+	7.86070i	0.606176	+	0.264984i
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$	−37.2734	+	37.2734i	−1.25506	+	1.25506i
$883$	−19.7005	−	19.7005i	−0.662973	−	0.662973i	0.293106	−	0.956080i	$-0.405311\pi$
$883$	−19.7005	−	19.7005i	−0.662973	−	0.662973i	−0.956080	+	0.293106i	$0.905311\pi$
$884$		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
$891$	−13.4000			−0.448918
$892$	−42.9493	+	42.9493i	−1.43805	+	1.43805i
$893$	17.3445	+	17.3445i	0.580412	+	0.580412i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$			28.1960i			0.940389i
$900$	43.8113	+	47.3520i	1.46038	+	1.57840i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−12.0217	+	12.0217i	−0.399173	+	0.399173i	−0.877941	−	0.478768i	$-0.841083\pi$
$907$	−12.0217	+	12.0217i	−0.399173	+	0.399173i	0.478768	+	0.877941i	$0.341083\pi$
$908$	85.1704	+	85.1704i	2.82648	+	2.82648i
$909$		0			0
$910$		0			0
$911$	59.7741			1.98040			0.990202	−	0.139643i	$-0.0445955\pi$
$911$	59.7741			1.98040			0.990202	+	0.139643i	$0.0445955\pi$
$912$		0			0
$913$		0			0
$914$		−	84.9460i		−	2.80976i
$915$		0			0
$916$	107.604			3.55534
$917$		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
$922$	55.4348	−	55.4348i	1.82565	−	1.82565i
$923$		0			0
$924$		0			0
$925$	56.9525	+	2.21197i	1.87259	+	0.0727291i
$926$	−105.707			−3.47375
$927$	11.0490	−	11.0490i	0.362897	−	0.362897i
$928$	−23.4233	−	23.4233i	−0.768909	−	0.768909i
$929$		0			0			−	1.00000i	$-0.5\pi$
$929$		0			0				1.00000i	$0.5\pi$
$930$		0			0
$931$	−13.6276			−0.446626
$932$		0			0
$933$		0			0
$934$		0			0
$935$	21.6915	−	8.49554i	0.709389	−	0.277834i
$936$		0			0
$937$	41.2261	−	41.2261i	1.34680	−	1.34680i	0.457680	−	0.889117i	$-0.348680\pi$
$937$	41.2261	−	41.2261i	1.34680	−	1.34680i	0.889117	−	0.457680i	$-0.151320\pi$
$938$		0			0
$939$		0			0
$940$	−44.1870	−	112.822i	−1.44122	−	3.67985i
$941$		0			0		1.00000			$0$
$941$		0			0		−1.00000			$\pi$
$942$		0			0
$943$		0			0
$944$			84.7062i			2.75695i
$945$		0			0
$946$	47.9998			1.56061
$947$		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
$950$	−0.948252	+	24.4151i	−0.0307654	+	0.792130i
$951$		0			0
$952$		0			0
$953$	34.7120	+	34.7120i	1.12443	+	1.12443i	0.991067	+	0.133366i	$0.0425786\pi$
$953$	34.7120	+	34.7120i	1.12443	+	1.12443i	0.133366	+	0.991067i	$0.457421\pi$
$954$		0			0
$955$	−45.2394	−	19.7760i	−1.46391	−	0.639936i
$956$	−99.4201			−3.21548
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−23.3645			−0.753695
$962$		0			0
$963$		0			0
$964$		−	119.210i		−	3.83948i
$965$	19.4404	+	49.6369i	0.625809	+	1.59787i
$966$		0			0
$967$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$967$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$968$	−35.8671	−	35.8671i	−1.15281	−	1.15281i
$969$		0			0
$970$	0.980328	−	2.24259i	0.0314764	−	0.0720053i
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$		0			0
$973$		0			0
$974$		−	32.9335i		−	1.05526i
$975$		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
$980$	61.6810	+	26.9633i	1.97033	+	0.861310i
$981$		0			0
$982$	77.6360	−	77.6360i	2.47746	−	2.47746i
$983$		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
$986$	−179.223			−5.70763
$987$		0			0
$988$		0			0
$989$		0			0
$990$	9.14275	+	23.3441i	0.290576	+	0.741923i
$991$	−62.8962			−1.99796			−0.998982	−	0.0451210i	$-0.985633\pi$
$991$	−62.8962			−1.99796			−0.998982	+	0.0451210i	$0.985633\pi$
$992$	−6.34304	+	6.34304i	−0.201392	+	0.201392i
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	34.8166	−	34.8166i	1.10265	−	1.10265i	0.108563	−	0.994090i	$-0.465375\pi$
$997$	34.8166	−	34.8166i	1.10265	−	1.10265i	0.994090	−	0.108563i	$-0.0346249\pi$
$998$		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.603.3	yes	28
5.2	odd	4	inner	755.2.f.d.452.3	✓	28
151.150	odd	2	CM	755.2.f.d.603.3	yes	28
755.452	even	4	inner	755.2.f.d.452.3	✓	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
755.2.f.d.452.3	✓	28	5.2	odd	4	inner
755.2.f.d.452.3	✓	28	755.452	even	4	inner
755.2.f.d.603.3	yes	28	1.1	even	1	trivial
755.2.f.d.603.3	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)

\(f(q)\)	\(=\)	\(q+(-1.77492 + 1.77492i) q^{2} -4.30072i q^{4} +(-2.04886 - 0.895640i) q^{5} +(4.08360 + 4.08360i) q^{8} +3.00000i q^{9} +(5.22627 - 2.04688i) q^{10} +1.48889 q^{11} -5.89474 q^{16} +(-4.94783 + 4.94783i) q^{17} +(-5.32477 - 5.32477i) q^{18} -1.94680i q^{19} +(-3.85189 + 8.81157i) q^{20} +(-2.64267 + 2.64267i) q^{22} +(3.39566 + 3.67008i) q^{25} -10.2040i q^{29} -2.76324 q^{31} +(2.29551 - 2.29551i) q^{32} -17.5641i q^{34} +12.9022 q^{36} +(8.06038 - 8.06038i) q^{37} +(3.45542 + 3.45542i) q^{38} +(-4.70929 - 12.0242i) q^{40} +(-9.08168 - 9.08168i) q^{43} -6.40331i q^{44} +(2.68692 - 6.14658i) q^{45} +(-8.90926 + 8.90926i) q^{47} -7.00000i q^{49} +(-12.5412 - 0.487084i) q^{50} +(-3.05053 - 1.33351i) q^{55} +(18.1113 + 18.1113i) q^{58} -14.3698i q^{59} +(4.90454 - 4.90454i) q^{62} -3.64075i q^{64} +(21.2792 + 21.2792i) q^{68} +(-12.2508 + 12.2508i) q^{72} +28.6131i q^{74} -8.37262 q^{76} +(12.0775 + 5.27956i) q^{80} -9.00000 q^{81} +(14.5689 - 5.70594i) q^{85} +32.2386 q^{86} +(6.08004 + 6.08004i) q^{88} +(6.14064 + 15.6788i) q^{90} -31.6265i q^{94} +(-1.74363 + 3.98871i) q^{95} +(0.308339 - 0.308339i) q^{97} +(12.4245 + 12.4245i) q^{98} +4.46668i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 112 q^{16} + 168 q^{36} - 14 q^{38} + 126 q^{58} - 154 q^{68} - 252 q^{81}+O(q^{100}) \) 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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