LMFDB - Embedded newform 755.2.f.d.603.1

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			603.1
Character	$\chi$	$=$	755.603
Dual form			755.2.f.d.452.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-1.93553 + 1.93553i) q^{2} -5.49259i q^{4} +(0.577204 - 2.16029i) q^{5} +(6.76003 + 6.76003i) q^{8} +3.00000i q^{9} +(3.06411 + 5.29851i) q^{10} -4.14607 q^{11} -15.1834 q^{16} +(4.92652 - 4.92652i) q^{17} +(-5.80660 - 5.80660i) q^{18} +7.85139i q^{19} +(-11.8656 - 3.17034i) q^{20} +(8.02486 - 8.02486i) q^{22} +(-4.33367 - 2.49385i) q^{25} -1.08956i q^{29} +10.1567 q^{31} +(15.8679 - 15.8679i) q^{32} +19.0709i q^{34} +16.4778 q^{36} +(6.20472 - 6.20472i) q^{37} +(-15.1966 - 15.1966i) q^{38} +(18.5055 - 10.7017i) q^{40} +(8.27063 + 8.27063i) q^{43} +22.7726i q^{44} +(6.48086 + 1.73161i) q^{45} +(7.83491 - 7.83491i) q^{47} -7.00000i q^{49} +(13.2149 - 3.56104i) q^{50} +(-2.39313 + 8.95669i) q^{55} +(2.10889 + 2.10889i) q^{58} -13.2065i q^{59} +(-19.6586 + 19.6586i) q^{62} +31.0589i q^{64} +(-27.0594 - 27.0594i) q^{68} +(-20.2801 + 20.2801i) q^{72} +24.0189i q^{74} +43.1245 q^{76} +(-8.76390 + 32.8004i) q^{80} -9.00000 q^{81} +(-7.79909 - 13.4863i) q^{85} -32.0162 q^{86} +(-28.0275 - 28.0275i) q^{88} +(-15.8955 + 9.19233i) q^{90} +30.3295i q^{94} +(16.9612 + 4.53185i) q^{95} +(2.79799 - 2.79799i) q^{97} +(13.5487 + 13.5487i) q^{98} -12.4382i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.93553	+	1.93553i	−1.36863	+	1.36863i	−0.506233	+	0.862397i	$0.668962\pi$
$2$	−1.93553	+	1.93553i	−1.36863	+	1.36863i	−0.862397	+	0.506233i	$0.831038\pi$
$3$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$3$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$4$		−	5.49259i		−	2.74629i
$5$	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	$0.750000\pi$
$8$	6.76003	+	6.76003i	2.39003	+	2.39003i
$9$			3.00000i			1.00000i
$10$	3.06411	+	5.29851i	0.968957	+	1.67554i
$11$	−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
$16$	−15.1834			−3.79584
$17$	4.92652	−	4.92652i	1.19486	−	1.19486i	0.219171	−	0.975686i	$-0.429665\pi$
$17$	4.92652	−	4.92652i	1.19486	−	1.19486i	0.975686	−	0.219171i	$-0.0703352\pi$
$18$	−5.80660	−	5.80660i	−1.36863	−	1.36863i
$19$			7.85139i			1.80123i	0.434615	+	0.900616i	$0.356884\pi$
$19$			7.85139i			1.80123i	−0.434615	+	0.900616i	$0.643116\pi$
$20$	−11.8656	−	3.17034i	−2.65322	−	0.708911i
$21$		0			0
$22$	8.02486	−	8.02486i	1.71091	−	1.71091i
$23$		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	$-0.134464\pi$
$31$	10.1567			1.82419			0.912094	+	0.409980i	$0.134464\pi$
$32$	15.8679	−	15.8679i	2.80507	−	2.80507i
$33$		0			0
$34$			19.0709i			3.27063i
$35$		0			0
$36$	16.4778			2.74629
$37$	6.20472	−	6.20472i	1.02005	−	1.02005i	0.0202554	−	0.999795i	$-0.493552\pi$
$37$	6.20472	−	6.20472i	1.02005	−	1.02005i	0.999795	−	0.0202554i	$-0.00644794\pi$
$38$	−15.1966	−	15.1966i	−2.46522	−	2.46522i
$39$		0			0
$40$	18.5055	−	10.7017i	2.92598	−	1.69208i
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$	8.27063	+	8.27063i	1.26126	+	1.26126i	0.950483	+	0.310776i	$0.100589\pi$
$43$	8.27063	+	8.27063i	1.26126	+	1.26126i	0.310776	+	0.950483i	$0.399411\pi$
$44$			22.7726i			3.43311i
$45$	6.48086	+	1.73161i	0.966109	+	0.258133i
$46$		0			0
$47$	7.83491	−	7.83491i	1.14284	−	1.14284i	0.154910	−	0.987929i	$-0.450491\pi$
$47$	7.83491	−	7.83491i	1.14284	−	1.14284i	0.987929	−	0.154910i	$-0.0495087\pi$
$48$		0			0
$49$		−	7.00000i		−	1.00000i
$50$	13.2149	−	3.56104i	1.86887	−	0.503606i
$51$		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
$58$	2.10889	+	2.10889i	0.276910	+	0.276910i
$59$		−	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			$\pi$
$62$	−19.6586	+	19.6586i	−2.49664	+	2.49664i
$63$		0			0
$64$			31.0589i			3.88236i
$65$		0			0
$66$		0			0
$67$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$67$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$68$	−27.0594	−	27.0594i	−3.28143	−	3.28143i
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$	−20.2801	+	20.2801i	−2.39003	+	2.39003i
$73$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$73$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$74$			24.0189i			2.79214i
$75$		0			0
$76$	43.1245			4.94671
$77$		0			0
$78$		0			0
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$	−8.76390	+	32.8004i	−0.979834	+	3.66720i
$81$	−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
$85$	−7.79909	−	13.4863i	−0.845930	−	1.46280i
$86$	−32.0162			−3.45239
$87$		0			0
$88$	−28.0275	−	28.0275i	−2.98775	−	2.98775i
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$	−15.8955	+	9.19233i	−1.67554	+	0.968957i
$91$		0			0
$92$		0			0
$93$		0			0
$94$			30.3295i			3.12825i
$95$	16.9612	+	4.53185i	1.74019	+	0.464958i
$96$		0			0
$97$	2.79799	−	2.79799i	0.284093	−	0.284093i	−0.550646	−	0.834739i	$-0.685619\pi$
$97$	2.79799	−	2.79799i	0.284093	−	0.284093i	0.834739	+	0.550646i	$0.185619\pi$
$98$	13.5487	+	13.5487i	1.36863	+	1.36863i
$99$		−	12.4382i		−	1.25009i
$100$	−13.6977	+	23.8031i	−1.36977	+	2.38031i
$101$		0			0		1.00000			$0$
$101$		0			0		−1.00000			$\pi$
$102$		0			0
$103$	10.9003	+	10.9003i	1.07404	+	1.07404i	0.997030	+	0.0770128i	$0.0245382\pi$
$103$	10.9003	+	10.9003i	1.07404	+	1.07404i	0.0770128	+	0.997030i	$0.475462\pi$
$104$		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	$0.5\pi$
$110$	−12.7040	−	21.9680i	−1.21128	−	2.09456i
$111$		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
$116$	−5.98452			−0.555649
$117$		0			0
$118$	25.5616	+	25.5616i	2.35314	+	2.35314i
$119$		0			0
$120$		0			0
$121$	6.18987			0.562716
$122$		0			0
$123$		0			0
$124$		−	55.7863i		−	5.00976i
$125$	−7.88884	+	7.92251i	−0.705600	+	0.708611i
$126$		0			0
$127$	13.6933	−	13.6933i	1.21509	−	1.21509i	0.245754	−	0.969332i	$-0.420964\pi$
$127$	13.6933	−	13.6933i	1.21509	−	1.21509i	0.969332	−	0.245754i	$-0.0790356\pi$
$128$	−28.3798	−	28.3798i	−2.50845	−	2.50845i
$129$		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
$136$	66.6069			5.71149
$137$	2.85381	−	2.85381i	0.243818	−	0.243818i	−0.574610	−	0.818427i	$-0.694846\pi$
$137$	2.85381	−	2.85381i	0.243818	−	0.243818i	0.818427	+	0.574610i	$0.194846\pi$
$138$		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
$144$		−	45.5501i		−	3.79584i
$145$	−2.35377	−	0.628900i	−0.195470	−	0.0522273i
$146$		0			0
$147$		0			0
$148$	−34.0800	−	34.0800i	−2.80136	−	2.80136i
$149$		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
$152$	−53.0756	+	53.0756i	−4.30500	+	4.30500i
$153$	14.7796	+	14.7796i	1.19486	+	1.19486i
$154$		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
$160$	−25.1201	−	43.4381i	−1.98592	−	3.43409i
$161$		0			0
$162$	17.4198	−	17.4198i	1.36863	−	1.36863i
$163$		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
$167$	−8.28821	+	8.28821i	−0.641361	+	0.641361i	−0.950890	−	0.309529i	$-0.899829\pi$
$167$	−8.28821	+	8.28821i	−0.641361	+	0.641361i	0.309529	+	0.950890i	$0.399829\pi$
$168$		0			0
$169$			13.0000i			1.00000i
$170$	41.1986	+	11.0078i	3.15979	+	0.844260i
$171$	−23.5542			−1.80123
$172$	45.4272	−	45.4272i	3.46379	−	3.46379i
$173$	18.4678	+	18.4678i	1.40408	+	1.40408i	0.786534	+	0.617547i	$0.211873\pi$
$173$	18.4678	+	18.4678i	1.40408	+	1.40408i	0.617547	+	0.786534i	$0.288127\pi$
$174$		0			0
$175$		0			0
$176$	62.9512			4.74513
$177$		0			0
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$	9.51103	−	35.5967i	0.708911	−	2.65322i
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−9.82259	−	16.9854i	−0.722171	−	1.24879i
$186$		0			0
$187$	−20.4257	+	20.4257i	−1.49367	+	1.49367i
$188$	−43.0339	−	43.0339i	−3.13857	−	3.13857i
$189$		0			0
$190$	−41.6006	+	24.0575i	−3.01803	+	1.74532i
$191$	−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
$193$	−19.4712	−	19.4712i	−1.40157	−	1.40157i	−0.795123	−	0.606448i	$-0.792594\pi$
$193$	−19.4712	−	19.4712i	−1.40157	−	1.40157i	−0.606448	−	0.795123i	$-0.707406\pi$
$194$			10.8312i			0.777636i
$195$		0			0
$196$	−38.4481			−2.74629
$197$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$197$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$198$	24.0746	+	24.0746i	1.71091	+	1.71091i
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$	−12.4372	−	46.1542i	−0.879445	−	3.26360i
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$	−42.1960			−2.93993
$207$		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
$215$	22.6408	−	13.0931i	1.54409	−	0.892941i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$	49.1954	+	13.1445i	3.31676	+	0.886199i
$221$		0			0
$222$		0			0
$223$	−12.4325	−	12.4325i	−0.832542	−	0.832542i	0.155322	−	0.987864i	$-0.450359\pi$
$223$	−12.4325	−	12.4325i	−0.832542	−	0.832542i	−0.987864	+	0.155322i	$0.950359\pi$
$224$		0			0
$225$	7.48155	−	13.0010i	0.498770	−	0.866734i
$226$		0			0
$227$	−1.50910	+	1.50910i	−0.100163	+	0.100163i	−0.755412	−	0.655250i	$-0.772563\pi$
$227$	−1.50910	+	1.50910i	−0.100163	+	0.100163i	0.655250	+	0.755412i	$0.272563\pi$
$228$		0			0
$229$			22.1698i			1.46502i	0.680755	+	0.732512i	$0.261652\pi$
$229$			22.1698i			1.46502i	−0.680755	+	0.732512i	$0.738348\pi$
$230$		0			0
$231$		0			0
$232$	7.36548	−	7.36548i	0.483567	−	0.483567i
$233$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$233$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$234$		0			0
$235$	−12.4033	−	21.4480i	−0.809102	−	1.39911i
$236$	−72.5379			−4.72181
$237$		0			0
$238$		0			0
$239$		−	11.9189i		−	0.770969i	−0.922714	−	0.385484i	$-0.874034\pi$
$239$		−	11.9189i		−	0.770969i	0.922714	−	0.385484i	$-0.125966\pi$
$240$		0			0
$241$	18.9041			1.21772			0.608859	−	0.793278i	$-0.291627\pi$
$241$	18.9041			1.21772			0.608859	+	0.793278i	$0.291627\pi$
$242$	−11.9807	+	11.9807i	−0.770150	+	0.770150i
$243$		0			0
$244$		0			0
$245$	−15.1220	−	4.04043i	−0.966109	−	0.258133i
$246$		0			0
$247$		0			0
$248$	68.6593	+	68.6593i	4.35987	+	4.35987i
$249$		0			0
$250$	−0.0651568	−	30.6034i	−0.00412088	−	1.93553i
$251$	24.5764			1.55125			0.775625	−	0.631194i	$-0.217435\pi$
$251$	24.5764			1.55125			0.775625	+	0.631194i	$0.217435\pi$
$252$		0			0
$253$		0			0
$254$			53.0078i			3.32601i
$255$		0			0
$256$	47.7425			2.98391
$257$		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	$0.0480583\pi$
$269$			4.93371i			0.300814i	−0.988624	+	0.150407i	$0.951942\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$	−74.8012	+	74.8012i	−4.53549	+	4.53549i
$273$		0			0
$274$			11.0473i			0.667392i
$275$	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	$0.750000\pi$
$278$	−45.1032	−	45.1032i	−2.70511	−	2.70511i
$279$			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
$288$	47.6036	+	47.6036i	2.80507	+	2.80507i
$289$		−	31.5413i		−	1.85537i
$290$	5.77306	−	3.33854i	0.339006	−	0.196046i
$291$		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
$296$	83.8882			4.87590
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$	−23.7842	+	23.7842i	−1.36863	+	1.36863i
$303$		0			0
$304$		−	119.210i		−	6.83719i
$305$		0			0
$306$	−57.2127			−3.27063
$307$	−22.2855	+	22.2855i	−1.27190	+	1.27190i	−0.326815	+	0.945088i	$0.605975\pi$
$307$	−22.2855	+	22.2855i	−1.27190	+	1.27190i	−0.945088	+	0.326815i	$0.894025\pi$
$308$		0			0
$309$		0			0
$310$	31.1211	+	53.8151i	1.76756	+	3.05649i
$311$	−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
$313$	−3.64043	−	3.64043i	−0.205769	−	0.205769i	0.596697	−	0.802466i	$-0.296479\pi$
$313$	−3.64043	−	3.64043i	−0.205769	−	0.205769i	−0.802466	+	0.596697i	$0.796479\pi$
$314$		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
$320$	67.0961	+	17.9273i	3.75079	+	1.00217i
$321$		0			0
$322$		0			0
$323$	38.6801	+	38.6801i	2.15222	+	2.15222i
$324$			49.4333i			2.74629i
$325$		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	$-0.482044\pi$
$331$	2.05147			0.112759			0.0563795	+	0.998409i	$0.482044\pi$
$332$		0			0
$333$	18.6142	+	18.6142i	1.02005	+	1.02005i
$334$		−	32.0842i		−	1.75557i
$335$		0			0
$336$		0			0
$337$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$337$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$338$	−25.1620	−	25.1620i	−1.36863	−	1.36863i
$339$		0			0
$340$	−74.0748	+	42.8372i	−4.01727	+	2.32317i
$341$	−42.1102			−2.28039
$342$	45.5899	−	45.5899i	2.46522	−	2.46522i
$343$		0			0
$344$			111.819i			6.02890i
$345$		0			0
$346$	−71.4901			−3.84333
$347$	1.71179	−	1.71179i	0.0918939	−	0.0918939i	−0.659665	−	0.751559i	$-0.729302\pi$
$347$	1.71179	−	1.71179i	0.0918939	−	0.0918939i	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
$352$	−65.7893	+	65.7893i	−3.50658	+	3.50658i
$353$		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	$0.5\pi$
$360$	32.1050	+	55.5165i	1.69208	+	2.92598i
$361$	−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
$370$	51.8877	+	13.8638i	2.69751	+	0.720745i
$371$		0			0
$372$		0			0
$373$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$373$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$374$		−	79.0693i		−	4.08858i
$375$		0			0
$376$	105.928			5.46284
$377$		0			0
$378$		0			0
$379$		0			0			−	1.00000i	$-0.5\pi$
$379$		0			0				1.00000i	$0.5\pi$
$380$	24.8916	−	93.1612i	1.27691	−	4.77907i
$381$		0			0
$382$	40.8854	−	40.8854i	2.09188	−	2.09188i
$383$	−19.1043	−	19.1043i	−0.976183	−	0.976183i	0.0235404	−	0.999723i	$-0.492506\pi$
$383$	−19.1043	−	19.1043i	−0.976183	−	0.976183i	−0.999723	+	0.0235404i	$0.992506\pi$
$384$		0			0
$385$		0			0
$386$	75.3746			3.83646
$387$	−24.8119	+	24.8119i	−1.26126	+	1.26126i
$388$	−15.3682	−	15.3682i	−0.780203	−	0.780203i
$389$		0			0			−	1.00000i	$-0.5\pi$
$389$		0			0				1.00000i	$0.5\pi$
$390$		0			0
$391$		0			0
$392$	47.3202	−	47.3202i	2.39003	−	2.39003i
$393$		0			0
$394$		0			0
$395$		0			0
$396$	−68.3179			−3.43311
$397$	12.8718	−	12.8718i	0.646018	−	0.646018i	−0.306010	−	0.952028i	$-0.598994\pi$
$397$	12.8718	−	12.8718i	0.646018	−	0.646018i	0.952028	+	0.306010i	$0.0989941\pi$
$398$		0			0
$399$		0			0
$400$	65.7997	+	37.8651i	3.28999	+	1.89325i
$401$	−37.8726			−1.89127			−0.945634	−	0.325234i	$-0.894557\pi$
$401$	−37.8726			−1.89127			−0.945634	+	0.325234i	$0.894557\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	−5.19484	+	19.4426i	−0.258133	+	0.966109i
$406$		0			0
$407$	−25.7252	+	25.7252i	−1.27515	+	1.27515i
$408$		0			0
$409$		0			0			−	1.00000i	$-0.5\pi$
$409$		0			0				1.00000i	$0.5\pi$
$410$		0			0
$411$		0			0
$412$	59.8711	−	59.8711i	2.94964	−	2.94964i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$	63.0063	+	63.0063i	3.08174	+	3.08174i
$419$		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
$423$	23.5047	+	23.5047i	1.14284	+	1.14284i
$424$		0			0
$425$	−33.6359	+	9.06391i	−1.63158	+	0.439664i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$	−18.4799	+	69.1641i	−0.891179	+	3.33539i
$431$		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	$0.400145\pi$
$439$			39.8596i			1.90240i	−0.308582	+	0.951198i	$0.599855\pi$
$440$	−76.7251	+	44.3699i	−3.65773	+	2.11525i
$441$	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
$446$	48.1271			2.27888
$447$		0			0
$448$		0			0
$449$		0			0			−	1.00000i	$-0.5\pi$
$449$		0			0				1.00000i	$0.5\pi$
$450$	10.6831	+	39.6447i	0.503606	+	1.86887i
$451$		0			0
$452$		0			0
$453$		0			0
$454$		−	5.84184i		−	0.274171i
$455$		0			0
$456$		0			0
$457$	30.2287	−	30.2287i	1.41404	−	1.41404i	0.695927	−	0.718112i	$-0.254994\pi$
$457$	30.2287	−	30.2287i	1.41404	−	1.41404i	0.718112	−	0.695927i	$-0.245006\pi$
$458$	−42.9105	−	42.9105i	−2.00507	−	2.00507i
$459$		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
$463$	−7.27359	−	7.27359i	−0.338032	−	0.338032i	0.517594	−	0.855626i	$-0.326828\pi$
$463$	−7.27359	−	7.27359i	−0.338032	−	0.338032i	−0.855626	+	0.517594i	$0.826828\pi$
$464$			16.5432i			0.768000i
$465$		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
$470$	65.5203	+	17.5063i	3.02223	+	0.807505i
$471$		0			0
$472$	89.2763	−	89.2763i	4.10927	−	4.10927i
$473$	−34.2906	−	34.2906i	−1.57668	−	1.57668i
$474$		0			0
$475$	19.5802	−	34.0253i	0.898401	−	1.56119i
$476$		0			0
$477$		0			0
$478$	23.0694	+	23.0694i	1.05517	+	1.05517i
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$		0			0
$482$	−36.5895	+	36.5895i	−1.66661	+	1.66661i
$483$		0			0
$484$		−	33.9984i		−	1.54538i
$485$	−4.42945	−	7.65947i	−0.201131	−	0.347799i
$486$		0			0
$487$	−2.41411	+	2.41411i	−0.109394	+	0.109394i	−0.759685	−	0.650291i	$-0.774647\pi$
$487$	−2.41411	+	2.41411i	−0.109394	+	0.109394i	0.650291	+	0.759685i	$0.274647\pi$
$488$		0			0
$489$		0			0
$490$	37.0895	−	21.4488i	1.67554	−	0.968957i
$491$	21.7007			0.979340			0.489670	−	0.871908i	$-0.337117\pi$
$491$	21.7007			0.979340			0.489670	+	0.871908i	$0.337117\pi$
$492$		0			0
$493$	−5.36776	−	5.36776i	−0.241752	−	0.241752i
$494$		0			0
$495$	−26.8701	−	7.17938i	−1.20772	−	0.322689i
$496$	−154.212			−6.92433
$497$		0			0
$498$		0			0
$499$		0			0			−	1.00000i	$-0.5\pi$
$499$		0			0				1.00000i	$0.5\pi$
$500$	43.5151	+	43.3302i	1.94605	+	1.93778i
$501$		0			0
$502$	−47.5685	+	47.5685i	−2.12309	+	2.12309i
$503$	9.41835	+	9.41835i	0.419943	+	0.419943i	0.885184	−	0.465241i	$-0.154032\pi$
$503$	9.41835	+	9.41835i	0.419943	+	0.419943i	−0.465241	+	0.885184i	$0.654032\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$	−75.2118	−	75.2118i	−3.33699	−	3.33699i
$509$		0			0			−	1.00000i	$-0.5\pi$
$509$		0			0				1.00000i	$0.5\pi$
$510$		0			0
$511$		0			0
$512$	−35.6477	+	35.6477i	−1.57542	+	1.57542i
$513$		0			0
$514$		0			0
$515$	29.8396	−	17.2561i	1.31489	−	0.760396i
$516$		0			0
$517$	−32.4840	+	32.4840i	−1.42865	+	1.42865i
$518$		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	$0.975513\pi$
$522$	−6.32666	+	6.32666i	−0.276910	+	0.276910i
$523$		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
$527$	50.0370	−	50.0370i	2.17965	−	2.17965i
$528$		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
$538$	−9.54937	−	9.54937i	−0.411702	−	0.411702i
$539$			29.0225i			1.25009i
$540$		0			0
$541$	−46.3986			−1.99483			−0.997417	−	0.0718303i	$-0.977116\pi$
$541$	−46.3986			−1.99483			−0.997417	+	0.0718303i	$0.977116\pi$
$542$		0			0
$543$		0			0
$544$		−	156.347i		−	6.70332i
$545$		0			0
$546$		0			0
$547$	−2.23892	+	2.23892i	−0.0957295	+	0.0957295i	−0.753350	−	0.657620i	$-0.771563\pi$
$547$	−2.23892	+	2.23892i	−0.0957295	+	0.0957295i	0.657620	+	0.753350i	$0.271563\pi$
$548$	−15.6748	−	15.6748i	−0.669595	−	0.669595i
$549$		0			0
$550$	−54.7899	+	14.7643i	−2.33625	+	0.629551i
$551$	8.55458			0.364437
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	127.992			5.42808
$557$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$557$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$558$	−58.9757	−	58.9757i	−2.49664	−	2.49664i
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−5.32803	−	5.32803i	−0.224550	−	0.224550i	0.585862	−	0.810411i	$-0.300756\pi$
$563$	−5.32803	−	5.32803i	−0.224550	−	0.224550i	−0.810411	+	0.585862i	$0.800756\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	46.8448i		−	1.96384i	−0.189308	−	0.981918i	$-0.560624\pi$
$569$		−	46.8448i		−	1.96384i	0.189308	−	0.981918i	$-0.439376\pi$
$570$		0			0
$571$	−1.38743			−0.0580620			−0.0290310	−	0.999579i	$-0.509242\pi$
$571$	−1.38743			−0.0580620			−0.0290310	+	0.999579i	$0.509242\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−93.1767			−3.88236
$577$	−21.7695	+	21.7695i	−0.906274	+	0.906274i	−0.995969	−	0.0896949i	$-0.971411\pi$
$577$	−21.7695	+	21.7695i	−0.906274	+	0.906274i	0.0896949	+	0.995969i	$0.471411\pi$
$578$	61.0492	+	61.0492i	2.53931	+	2.53931i
$579$		0			0
$580$	−3.45429	+	12.9283i	−0.143432	+	0.536818i
$581$		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
$590$	69.9747	−	40.4662i	2.88081	−	1.66597i
$591$		0			0
$592$	−94.2086	+	94.2086i	−3.87195	+	3.87195i
$593$		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
$604$		−	67.4941i		−	2.74629i
$605$	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	$0.750000\pi$
$608$	124.585	+	124.585i	5.05258	+	5.05258i
$609$		0			0
$610$		0			0
$611$		0			0
$612$	81.1781	−	81.1781i	3.28143	−	3.28143i
$613$	21.5764	+	21.5764i	0.871463	+	0.871463i	0.992632	−	0.121169i	$-0.0386643\pi$
$613$	21.5764	+	21.5764i	0.871463	+	0.871463i	−0.121169	+	0.992632i	$0.538664\pi$
$614$		−	86.2689i		−	3.48153i
$615$		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	$0.5\pi$
$620$	−120.514	−	32.2001i	−4.83998	−	1.29319i
$621$		0			0
$622$	6.94129	−	6.94129i	0.278320	−	0.278320i
$623$		0			0
$624$		0			0
$625$	12.5614	+	21.6151i	0.502456	+	0.864603i
$626$	14.0923			0.563244
$627$		0			0
$628$		0			0
$629$		−	61.1354i		−	2.43763i
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−21.6777	−	37.4854i	−0.860252	−	1.48756i
$636$		0			0
$637$		0			0
$638$	−8.74359	−	8.74359i	−0.346162	−	0.346162i
$639$		0			0
$640$	−77.6895	+	44.9276i	−3.07095	+	1.77592i
$641$	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
$643$	−25.6405	−	25.6405i	−1.01116	−	1.01116i	−0.999937	−	0.0112267i	$-0.996426\pi$
$643$	−25.6405	−	25.6405i	−1.01116	−	1.01116i	−0.0112267	−	0.999937i	$-0.503574\pi$
$644$		0			0
$645$		0			0
$646$	−149.733			−5.89117
$647$	13.9185	−	13.9185i	0.547195	−	0.547195i	−0.378434	−	0.925628i	$-0.623537\pi$
$647$	13.9185	−	13.9185i	0.547195	−	0.547195i	0.925628	+	0.378434i	$0.123537\pi$
$648$	−60.8403	−	60.8403i	−2.39003	−	2.39003i
$649$			54.7550i			2.14932i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	31.5764	+	31.5764i	1.23568	+	1.23568i	0.961749	+	0.273931i	$0.0883240\pi$
$653$	31.5764	+	31.5764i	1.23568	+	1.23568i	0.273931	+	0.961749i	$0.411676\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$			42.0140i			1.63663i	0.574767	+	0.818317i	$0.305093\pi$
$659$			42.0140i			1.63663i	−0.574767	+	0.818317i	$0.694907\pi$
$660$		0			0
$661$		0			0		1.00000			$0$
$661$		0			0		−1.00000			$\pi$
$662$	−3.97069	+	3.97069i	−0.154325	+	0.154325i
$663$		0			0
$664$		0			0
$665$		0			0
$666$	−72.0567			−2.79214
$667$		0			0
$668$	45.5237	+	45.5237i	1.76137	+	1.76137i
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	3.18652	+	3.18652i	0.122831	+	0.122831i	0.765850	−	0.643019i	$-0.222318\pi$
$673$	3.18652	+	3.18652i	0.122831	+	0.122831i	−0.643019	+	0.765850i	$0.722318\pi$
$674$		0			0
$675$		0			0
$676$	71.4037			2.74629
$677$		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
$680$	38.4458	−	143.890i	1.47433	−	5.51793i
$681$		0			0
$682$	81.5057	−	81.5057i	3.12101	−	3.12101i
$683$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$683$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$684$			129.373i			4.94671i
$685$	−4.51782	−	7.81228i	−0.172617	−	0.298492i
$686$		0			0
$687$		0			0
$688$	−125.576	−	125.576i	−4.78754	−	4.78754i
$689$		0			0
$690$		0			0
$691$		0			0		1.00000			$0$
$691$		0			0		−1.00000			$\pi$
$692$	101.436	−	101.436i	3.85602	−	3.85602i
$693$		0			0
$694$			6.62647i			0.251538i
$695$	50.3405	+	13.4504i	1.90952	+	0.510203i
$696$		0			0
$697$		0			0
$698$	61.8141	+	61.8141i	2.33970	+	2.33970i
$699$		0			0
$700$		0			0
$701$	−34.8546			−1.31644			−0.658219	−	0.752826i	$-0.728690\pi$
$701$	−34.8546			−1.31644			−0.658219	+	0.752826i	$0.728690\pi$
$702$		0			0
$703$	48.7157	+	48.7157i	1.83735	+	1.83735i
$704$		−	128.772i		−	4.85329i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$			23.0070i			0.864046i	0.901863	+	0.432023i	$0.142200\pi$
$709$			23.0070i			0.864046i	−0.901863	+	0.432023i	$0.857800\pi$
$710$		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	$0.5\pi$
$720$	−98.4012	−	26.2917i	−3.66720	−	0.979834i
$721$		0			0
$722$	82.5396	−	82.5396i	3.07180	−	3.07180i
$723$		0			0
$724$		0			0
$725$	−2.71721	+	4.72181i	−0.100915	+	0.175364i
$726$		0			0
$727$	11.7118	−	11.7118i	0.434366	−	0.434366i	−0.455744	−	0.890111i	$-0.650627\pi$
$727$	11.7118	−	11.7118i	0.434366	−	0.434366i	0.890111	+	0.455744i	$0.150627\pi$
$728$		0			0
$729$		−	27.0000i		−	1.00000i
$730$		0			0
$731$	81.4909			3.01405
$732$		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	$0.5\pi$
$740$	−93.2937	+	53.9514i	−3.42954	+	1.98329i
$741$		0			0
$742$		0			0
$743$	−35.7192	−	35.7192i	−1.31041	−	1.31041i	−0.921110	−	0.389301i	$-0.872716\pi$
$743$	−35.7192	−	35.7192i	−1.31041	−	1.31041i	−0.389301	−	0.921110i	$-0.627284\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$	112.190	+	112.190i	4.10207	+	4.10207i
$749$		0			0
$750$		0			0
$751$		0			0		1.00000			$0$
$751$		0			0		−1.00000			$\pi$
$752$	−118.960	+	118.960i	−4.33803	+	4.33803i
$753$		0			0
$754$		0			0
$755$	7.09280	−	26.5460i	0.258133	−	0.966109i
$756$		0			0
$757$	−15.1963	+	15.1963i	−0.552318	+	0.552318i	−0.927109	−	0.374791i	$-0.877714\pi$
$757$	−15.1963	+	15.1963i	−0.552318	+	0.552318i	0.374791	+	0.927109i	$0.377714\pi$
$758$		0			0
$759$		0			0
$760$	84.0231	+	145.294i	3.04784	+	5.27037i
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$		0			0
$763$		0			0
$764$			116.023i			4.19757i
$765$	40.4589	−	23.3973i	1.46280	−	0.845930i
$766$	73.9540			2.67206
$767$		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
$772$	−106.948	+	106.948i	−3.84913	+	3.84913i
$773$	11.5764	+	11.5764i	0.416375	+	0.416375i	0.883952	−	0.467578i	$-0.154873\pi$
$773$	11.5764	+	11.5764i	0.416375	+	0.416375i	−0.467578	+	0.883952i	$0.654873\pi$
$774$		−	96.0486i		−	3.45239i
$775$	−44.0156	−	25.3292i	−1.58109	−	0.909851i
$776$	37.8290			1.35798
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$			106.284i			3.79584i
$785$		0			0
$786$		0			0
$787$	−17.1266	+	17.1266i	−0.610499	+	0.610499i	−0.943076	−	0.332577i	$-0.892082\pi$
$787$	−17.1266	+	17.1266i	−0.610499	+	0.610499i	0.332577	+	0.943076i	$0.392082\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$	84.0826	−	84.0826i	2.98775	−	2.98775i
$793$		0			0
$794$			49.8277i			1.76832i
$795$		0			0
$796$		0			0
$797$	16.9412	−	16.9412i	0.600089	−	0.600089i	−0.340247	−	0.940336i	$-0.610511\pi$
$797$	16.9412	−	16.9412i	0.600089	−	0.600089i	0.940336	+	0.340247i	$0.110511\pi$
$798$		0			0
$799$		−	77.1977i		−	2.73106i
$800$	−108.338	+	29.1940i	−3.83033	+	1.03216i
$801$		0			0
$802$	73.3037	−	73.3037i	2.58844	−	2.58844i
$803$		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	$0.5\pi$
$810$	−27.5770	−	47.6866i	−0.968957	−	1.67554i
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		0			0
$813$		0			0
$814$		−	99.5840i		−	3.49042i
$815$		0			0
$816$		0			0
$817$	−64.9359	+	64.9359i	−2.27182	+	2.27182i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		1.00000			$0$
$821$		0			0		−1.00000			$\pi$
$822$		0			0
$823$	33.3827	+	33.3827i	1.16365	+	1.16365i	0.983671	+	0.179976i	$0.0576019\pi$
$823$	33.3827	+	33.3827i	1.16365	+	1.16365i	0.179976	+	0.983671i	$0.442398\pi$
$824$			147.373i			5.13399i
$825$		0			0
$826$		0			0
$827$	−38.2882	+	38.2882i	−1.33141	+	1.33141i	−0.427303	+	0.904109i	$0.640536\pi$
$827$	−38.2882	+	38.2882i	−1.33141	+	1.33141i	−0.904109	+	0.427303i	$0.859464\pi$
$828$		0			0
$829$		−	49.1528i		−	1.70715i	−0.520972	−	0.853574i	$-0.674430\pi$
$829$		−	49.1528i		−	1.70715i	0.520972	−	0.853574i	$-0.325570\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−34.4857	−	34.4857i	−1.19486	−	1.19486i
$834$		0			0
$835$	13.1209	+	22.6889i	0.454068	+	0.785181i
$836$	−178.797			−6.18382
$837$		0			0
$838$		0			0
$839$			36.4883i			1.25971i	0.776711	+	0.629857i	$0.216887\pi$
$839$			36.4883i			1.25971i	−0.776711	+	0.629857i	$0.783113\pi$
$840$		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
$846$	−90.9884			−3.12825
$847$		0			0
$848$		0			0
$849$		0			0
$850$	47.5600	−	82.6471i	1.63130	−	2.83477i
$851$		0			0
$852$		0			0
$853$	7.97982	+	7.97982i	0.273224	+	0.273224i	0.830397	−	0.557173i	$-0.188114\pi$
$853$	7.97982	+	7.97982i	0.273224	+	0.273224i	−0.557173	+	0.830397i	$0.688114\pi$
$854$		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	$0.5\pi$
$860$	−71.9150	−	124.356i	−2.45228	−	4.24052i
$861$		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
$865$	50.5554	−	29.2360i	1.71894	−	0.994055i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	8.39397	+	8.39397i	0.284093	+	0.284093i
$874$		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	$0.750000\pi$
$878$	−77.1497	−	77.1497i	−2.60367	−	2.60367i
$879$		0			0
$880$	36.3357	−	135.993i	1.22488	−	4.58431i
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$	−40.6462	+	40.6462i	−1.36863	+	1.36863i
$883$	41.9916	+	41.9916i	1.41313	+	1.41313i	0.734224	+	0.678907i	$0.237546\pi$
$883$	41.9916	+	41.9916i	1.41313	+	1.41313i	0.678907	+	0.734224i	$0.262454\pi$
$884$		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
$892$	−68.2866	+	68.2866i	−2.28641	+	2.28641i
$893$	61.5149	+	61.5149i	2.05852	+	2.05852i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	11.0663i		−	0.369082i
$900$	−71.4092	−	41.0931i	−2.38031	−	1.36977i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−20.8123	+	20.8123i	−0.691061	+	0.691061i	−0.962465	−	0.271404i	$-0.912512\pi$
$907$	−20.8123	+	20.8123i	−0.691061	+	0.691061i	0.271404	+	0.962465i	$0.412512\pi$
$908$	8.28888	+	8.28888i	0.275076	+	0.275076i
$909$		0			0
$910$		0			0
$911$	5.08270			0.168397			0.0841986	−	0.996449i	$-0.473167\pi$
$911$	5.08270			0.168397			0.0841986	+	0.996449i	$0.473167\pi$
$912$		0			0
$913$		0			0
$914$			117.017i			3.87059i
$915$		0			0
$916$	121.770			4.02339
$917$		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
$922$	−29.7147	+	29.7147i	−0.978601	+	0.978601i
$923$		0			0
$924$		0			0
$925$	−42.3629	+	11.4156i	−1.39288	+	0.375342i
$926$	28.1566			0.925282
$927$	−32.7010	+	32.7010i	−1.07404	+	1.07404i
$928$	−17.2890	−	17.2890i	−0.567541	−	0.567541i
$929$		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
$935$	32.3356	+	55.9151i	1.05749	+	1.82862i
$936$		0			0
$937$	37.2537	−	37.2537i	1.21703	−	1.21703i	0.248358	−	0.968668i	$-0.420109\pi$
$937$	37.2537	−	37.2537i	1.21703	−	1.21703i	0.968668	−	0.248358i	$-0.0798910\pi$
$938$		0			0
$939$		0			0
$940$	−117.805	+	68.1262i	−3.84237	+	2.22203i
$941$		0			0		1.00000			$0$
$941$		0			0		−1.00000			$\pi$
$942$		0			0
$943$		0			0
$944$			200.519i			6.52634i
$945$		0			0
$946$	132.741			4.31579
$947$		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
$950$	27.9591	+	103.755i	0.907112	+	3.36627i
$951$		0			0
$952$		0			0
$953$	−39.7336	−	39.7336i	−1.28710	−	1.28710i	−0.936542	−	0.350555i	$-0.885993\pi$
$953$	−39.7336	−	39.7336i	−1.28710	−	1.28710i	−0.350555	−	0.936542i	$-0.614007\pi$
$954$		0			0
$955$	−12.1926	+	45.6330i	−0.394544	+	1.47665i
$956$	−65.4656			−2.11731
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	72.1576			2.32766
$962$		0			0
$963$		0			0
$964$		−	103.832i		−	3.34421i
$965$	−53.3023	+	30.8246i	−1.71586	+	0.992278i
$966$		0			0
$967$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$967$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$968$	41.8437	+	41.8437i	1.34491	+	1.34491i
$969$		0			0
$970$	23.3985	+	6.25182i	0.751281	+	0.200734i
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$		0			0
$973$		0			0
$974$		−	9.34519i		−	0.299439i
$975$		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
$980$	−22.1924	+	83.0590i	−0.708911	+	2.65322i
$981$		0			0
$982$	−42.0025	+	42.0025i	−1.34035	+	1.34035i
$983$		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
$986$	20.7790			0.661737
$987$		0			0
$988$		0			0
$989$		0			0
$990$	65.9039	−	38.1120i	2.09456	−	1.21128i
$991$	16.7653			0.532568			0.266284	−	0.963895i	$-0.414204\pi$
$991$	16.7653			0.532568			0.266284	+	0.963895i	$0.414204\pi$
$992$	161.164	−	161.164i	5.11698	−	5.11698i
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	10.0854	−	10.0854i	0.319409	−	0.319409i	−0.529131	−	0.848540i	$-0.677482\pi$
$997$	10.0854	−	10.0854i	0.319409	−	0.319409i	0.848540	+	0.529131i	$0.177482\pi$
$998$		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.603.1	yes	28
5.2	odd	4	inner	755.2.f.d.452.1	✓	28
151.150	odd	2	CM	755.2.f.d.603.1	yes	28
755.452	even	4	inner	755.2.f.d.452.1	✓	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
755.2.f.d.452.1	✓	28	5.2	odd	4	inner
755.2.f.d.452.1	✓	28	755.452	even	4	inner
755.2.f.d.603.1	yes	28	1.1	even	1	trivial
755.2.f.d.603.1	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)

\(f(q)\)	\(=\)	\(q+(-1.93553 + 1.93553i) q^{2} -5.49259i q^{4} +(0.577204 - 2.16029i) q^{5} +(6.76003 + 6.76003i) q^{8} +3.00000i q^{9} +(3.06411 + 5.29851i) q^{10} -4.14607 q^{11} -15.1834 q^{16} +(4.92652 - 4.92652i) q^{17} +(-5.80660 - 5.80660i) q^{18} +7.85139i q^{19} +(-11.8656 - 3.17034i) q^{20} +(8.02486 - 8.02486i) q^{22} +(-4.33367 - 2.49385i) q^{25} -1.08956i q^{29} +10.1567 q^{31} +(15.8679 - 15.8679i) q^{32} +19.0709i q^{34} +16.4778 q^{36} +(6.20472 - 6.20472i) q^{37} +(-15.1966 - 15.1966i) q^{38} +(18.5055 - 10.7017i) q^{40} +(8.27063 + 8.27063i) q^{43} +22.7726i q^{44} +(6.48086 + 1.73161i) q^{45} +(7.83491 - 7.83491i) q^{47} -7.00000i q^{49} +(13.2149 - 3.56104i) q^{50} +(-2.39313 + 8.95669i) q^{55} +(2.10889 + 2.10889i) q^{58} -13.2065i q^{59} +(-19.6586 + 19.6586i) q^{62} +31.0589i q^{64} +(-27.0594 - 27.0594i) q^{68} +(-20.2801 + 20.2801i) q^{72} +24.0189i q^{74} +43.1245 q^{76} +(-8.76390 + 32.8004i) q^{80} -9.00000 q^{81} +(-7.79909 - 13.4863i) q^{85} -32.0162 q^{86} +(-28.0275 - 28.0275i) q^{88} +(-15.8955 + 9.19233i) q^{90} +30.3295i q^{94} +(16.9612 + 4.53185i) q^{95} +(2.79799 - 2.79799i) q^{97} +(13.5487 + 13.5487i) q^{98} -12.4382i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 112 q^{16} + 168 q^{36} - 14 q^{38} + 126 q^{58} - 154 q^{68} - 252 q^{81}+O(q^{100}) \) 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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