LMFDB - Embedded newform 775.2.a.l.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [775,2,Mod(1,775)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(775, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("775.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 775 = 5^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	775.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$6.18840615665$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 16x^{8} + 88x^{6} - 183x^{4} + 92x^{2} - 4 $ x^10 - 16x^8 + 88x^6 - 183x^4 + 92x^2 - 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 155)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$2.02998$ of defining polynomial
Character	$\chi$	$=$	775.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.02998 q^{2} +3.01138 q^{3} +2.12080 q^{4} +6.11304 q^{6} -3.60704 q^{7} +0.245225 q^{8} +6.06843 q^{9} +O(q^{10})$	\(q+2.02998 q^{2} +3.01138 q^{3} +2.12080 q^{4} +6.11304 q^{6} -3.60704 q^{7} +0.245225 q^{8} +6.06843 q^{9} +5.37457 q^{11} +6.38655 q^{12} +0.621453 q^{13} -7.32220 q^{14} -3.74380 q^{16} -0.427115 q^{17} +12.3188 q^{18} -2.18390 q^{19} -10.8622 q^{21} +10.9102 q^{22} -6.22884 q^{23} +0.738466 q^{24} +1.26153 q^{26} +9.24023 q^{27} -7.64981 q^{28} -2.11304 q^{29} -1.00000 q^{31} -8.09028 q^{32} +16.1849 q^{33} -0.867033 q^{34} +12.8699 q^{36} -3.14239 q^{37} -4.43326 q^{38} +1.87143 q^{39} +7.31686 q^{41} -22.0500 q^{42} +6.81366 q^{43} +11.3984 q^{44} -12.6444 q^{46} -1.60669 q^{47} -11.2740 q^{48} +6.01073 q^{49} -1.28621 q^{51} +1.31798 q^{52} -6.94467 q^{53} +18.7574 q^{54} -0.884535 q^{56} -6.57655 q^{57} -4.28941 q^{58} -8.71526 q^{59} +1.35074 q^{61} -2.02998 q^{62} -21.8891 q^{63} -8.93547 q^{64} +32.8549 q^{66} +9.79835 q^{67} -0.905827 q^{68} -18.7574 q^{69} -4.44983 q^{71} +1.48813 q^{72} -1.82773 q^{73} -6.37897 q^{74} -4.63161 q^{76} -19.3863 q^{77} +3.79896 q^{78} +0.867033 q^{79} +9.62057 q^{81} +14.8531 q^{82} -10.9736 q^{83} -23.0365 q^{84} +13.8316 q^{86} -6.36316 q^{87} +1.31798 q^{88} +10.1130 q^{89} -2.24160 q^{91} -13.2101 q^{92} -3.01138 q^{93} -3.26153 q^{94} -24.3629 q^{96} -1.50557 q^{97} +12.2016 q^{98} +32.6152 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 12 q^{4} + 8 q^{6} + 22 q^{9}+O(q^{10}) $ 10 * q + 12 * q^4 + 8 * q^6 + 22 * q^9	$ 10 q + 12 q^{4} + 8 q^{6} + 22 q^{9} + 16 q^{11} - 6 q^{14} + 8 q^{16} - 4 q^{19} + 20 q^{21} - 8 q^{24} + 28 q^{26} + 32 q^{29} - 10 q^{31} - 28 q^{34} + 44 q^{36} - 16 q^{39} + 36 q^{41} + 52 q^{44} + 8 q^{46} + 22 q^{49} + 20 q^{51} + 12 q^{56} + 12 q^{59} - 70 q^{64} + 12 q^{71} + 28 q^{74} - 30 q^{76} + 28 q^{79} - 14 q^{81} - 40 q^{84} + 36 q^{86} + 48 q^{89} - 4 q^{91} - 48 q^{94} - 60 q^{96} + 64 q^{99}+O(q^{100}) $ 10 * q + 12 * q^4 + 8 * q^6 + 22 * q^9 + 16 * q^11 - 6 * q^14 + 8 * q^16 - 4 * q^19 + 20 * q^21 - 8 * q^24 + 28 * q^26 + 32 * q^29 - 10 * q^31 - 28 * q^34 + 44 * q^36 - 16 * q^39 + 36 * q^41 + 52 * q^44 + 8 * q^46 + 22 * q^49 + 20 * q^51 + 12 * q^56 + 12 * q^59 - 70 * q^64 + 12 * q^71 + 28 * q^74 - 30 * q^76 + 28 * q^79 - 14 * q^81 - 40 * q^84 + 36 * q^86 + 48 * q^89 - 4 * q^91 - 48 * q^94 - 60 * q^96 + 64 * q^99

Display 100 coefficients 10 coefficients

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 $

$2$	2.02998	1.43541	0.717705	−	0.696347i	$-0.245193\pi$
$2$	2.02998	1.43541	0.717705	+	0.696347i	$0.245193\pi$
$3$	3.01138	1.73862	0.869312	−	0.494264i	$-0.164563\pi$
$3$	3.01138	1.73862	0.869312	+	0.494264i	$0.164563\pi$
$4$	2.12080	1.06040
$5$	0	0
$5$	0	0
$6$	6.11304	2.49564
$7$	−3.60704	−1.36333	−0.681666	−	0.731663i	$-0.738744\pi$
$7$	−3.60704	−1.36333	−0.681666	+	0.731663i	$0.738744\pi$
$8$	0.245225	0.0867001
$9$	6.06843	2.02281
$10$	0	0
$11$	5.37457	1.62049	0.810247	−	0.586089i	$-0.199333\pi$
$11$	5.37457	1.62049	0.810247	+	0.586089i	$0.199333\pi$
$12$	6.38655	1.84364
$13$	0.621453	0.172360	0.0861800	−	0.996280i	$-0.472534\pi$
$13$	0.621453	0.172360	0.0861800	+	0.996280i	$0.472534\pi$
$14$	−7.32220	−1.95694
$15$	0	0
$16$	−3.74380	−0.935951
$17$	−0.427115	−0.103591	−0.0517953	−	0.998658i	$-0.516494\pi$
$17$	−0.427115	−0.103591	−0.0517953	+	0.998658i	$0.516494\pi$
$18$	12.3188	2.90356
$19$	−2.18390	−0.501020	−0.250510	−	0.968114i	$-0.580598\pi$
$19$	−2.18390	−0.501020	−0.250510	+	0.968114i	$0.580598\pi$
$20$	0	0
$21$	−10.8622	−2.37032
$22$	10.9102	2.32607
$23$	−6.22884	−1.29880	−0.649402	−	0.760445i	$-0.724981\pi$
$23$	−6.22884	−1.29880	−0.649402	+	0.760445i	$0.724981\pi$
$24$	0.738466	0.150739
$25$	0	0
$26$	1.26153	0.247407
$27$	9.24023	1.77828
$28$	−7.64981	−1.44568
$29$	−2.11304	−0.392381	−0.196190	−	0.980566i	$-0.562857\pi$
$29$	−2.11304	−0.392381	−0.196190	+	0.980566i	$0.562857\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	−8.09028	−1.43017
$33$	16.1849	2.81743
$34$	−0.867033	−0.148695
$35$	0	0
$36$	12.8699	2.14499
$37$	−3.14239	−0.516605	−0.258303	−	0.966064i	$-0.583163\pi$
$37$	−3.14239	−0.516605	−0.258303	+	0.966064i	$0.583163\pi$
$38$	−4.43326	−0.719169
$39$	1.87143	0.299669
$40$	0	0
$41$	7.31686	1.14270	0.571351	−	0.820706i	$-0.306419\pi$
$41$	7.31686	1.14270	0.571351	+	0.820706i	$0.306419\pi$
$42$	−22.0500	−3.40238
$43$	6.81366	1.03907	0.519537	−	0.854448i	$-0.326104\pi$
$43$	6.81366	1.03907	0.519537	+	0.854448i	$0.326104\pi$
$44$	11.3984	1.71837
$45$	0	0
$46$	−12.6444	−1.86432
$47$	−1.60669	−0.234359	−0.117180	−	0.993111i	$-0.537385\pi$
$47$	−1.60669	−0.234359	−0.117180	+	0.993111i	$0.537385\pi$
$48$	−11.2740	−1.62727
$49$	6.01073	0.858675
$50$	0	0
$51$	−1.28621	−0.180105
$52$	1.31798	0.182771
$53$	−6.94467	−0.953923	−0.476962	−	0.878924i	$-0.658262\pi$
$53$	−6.94467	−0.953923	−0.476962	+	0.878924i	$0.658262\pi$
$54$	18.7574	2.55256
$55$	0	0
$56$	−0.884535	−0.118201
$57$	−6.57655	−0.871086
$58$	−4.28941	−0.563227
$59$	−8.71526	−1.13463	−0.567315	−	0.823501i	$-0.692018\pi$
$59$	−8.71526	−1.13463	−0.567315	+	0.823501i	$0.692018\pi$
$60$	0	0
$61$	1.35074	0.172945	0.0864724	−	0.996254i	$-0.472441\pi$
$61$	1.35074	0.172945	0.0864724	+	0.996254i	$0.472441\pi$
$62$	−2.02998	−0.257807
$63$	−21.8891	−2.75776
$64$	−8.93547	−1.11693
$65$	0	0
$66$	32.8549	4.04416
$67$	9.79835	1.19706	0.598529	−	0.801101i	$-0.295752\pi$
$67$	9.79835	1.19706	0.598529	+	0.801101i	$0.295752\pi$
$68$	−0.905827	−0.109848
$69$	−18.7574	−2.25813
$70$	0	0
$71$	−4.44983	−0.528098	−0.264049	−	0.964509i	$-0.585058\pi$
$71$	−4.44983	−0.528098	−0.264049	+	0.964509i	$0.585058\pi$
$72$	1.48813	0.175378
$73$	−1.82773	−0.213919	−0.106960	−	0.994263i	$-0.534112\pi$
$73$	−1.82773	−0.213919	−0.106960	+	0.994263i	$0.534112\pi$
$74$	−6.37897	−0.741540
$75$	0	0
$76$	−4.63161	−0.531282
$77$	−19.3863	−2.20927
$78$	3.79896	0.430148
$79$	0.867033	0.0975489	0.0487744	−	0.998810i	$-0.484468\pi$
$79$	0.867033	0.0975489	0.0487744	+	0.998810i	$0.484468\pi$
$80$	0	0
$81$	9.62057	1.06895
$82$	14.8531	1.64025
$83$	−10.9736	−1.20451	−0.602254	−	0.798305i	$-0.705730\pi$
$83$	−10.9736	−1.20451	−0.602254	+	0.798305i	$0.705730\pi$
$84$	−23.0365	−2.51349
$85$	0	0
$86$	13.8316	1.49150
$87$	−6.36316	−0.682203
$88$	1.31798	0.140497
$89$	10.1130	1.07198	0.535990	−	0.844224i	$-0.319939\pi$
$89$	10.1130	1.07198	0.535990	+	0.844224i	$0.319939\pi$
$90$	0	0
$91$	−2.24160	−0.234984
$92$	−13.2101	−1.37725
$93$	−3.01138	−0.312266
$94$	−3.26153	−0.336402
$95$	0	0
$96$	−24.3629	−2.48653
$97$	−1.50557	−0.152867	−0.0764337	−	0.997075i	$-0.524353\pi$
$97$	−1.50557	−0.152867	−0.0764337	+	0.997075i	$0.524353\pi$
$98$	12.2016	1.23255
$99$	32.6152	3.27795
$100$	0	0
$101$	10.2838	1.02327	0.511637	−	0.859202i	$-0.329039\pi$
$101$	10.2838	1.02327	0.511637	+	0.859202i	$0.329039\pi$
$102$	−2.61097	−0.258525
$103$	7.79799	0.768359	0.384180	−	0.923258i	$-0.374484\pi$
$103$	7.79799	0.768359	0.384180	+	0.923258i	$0.374484\pi$
$104$	0.152396	0.0149436
$105$	0	0
$106$	−14.0975	−1.36927
$107$	11.7987	1.14062	0.570312	−	0.821428i	$-0.306822\pi$
$107$	11.7987	1.14062	0.570312	+	0.821428i	$0.306822\pi$
$108$	19.5967	1.88569
$109$	13.8482	1.32642	0.663210	−	0.748433i	$-0.269194\pi$
$109$	13.8482	1.32642	0.663210	+	0.748433i	$0.269194\pi$
$110$	0	0
$111$	−9.46293	−0.898182
$112$	13.5040	1.27601
$113$	5.64493	0.531030	0.265515	−	0.964107i	$-0.414458\pi$
$113$	5.64493	0.531030	0.265515	+	0.964107i	$0.414458\pi$
$114$	−13.3502	−1.25036
$115$	0	0
$116$	−4.48133	−0.416081
$117$	3.77124	0.348652
$118$	−17.6918	−1.62866
$119$	1.54062	0.141228
$120$	0	0
$121$	17.8860	1.62600
$122$	2.74197	0.248247
$123$	22.0339	1.98673
$124$	−2.12080	−0.190454
$125$	0	0
$126$	−44.4343	−3.95852
$127$	−8.94411	−0.793662	−0.396831	−	0.917892i	$-0.629890\pi$
$127$	−8.94411	−0.793662	−0.396831	+	0.917892i	$0.629890\pi$
$128$	−1.95822	−0.173084
$129$	20.5186	1.80656
$130$	0	0
$131$	−17.2945	−1.51103	−0.755514	−	0.655133i	$-0.772613\pi$
$131$	−17.2945	−1.51103	−0.755514	+	0.655133i	$0.772613\pi$
$132$	34.3250	2.98760
$133$	7.87740	0.683057
$134$	19.8904	1.71827
$135$	0	0
$136$	−0.104739	−0.00898132
$137$	−15.7578	−1.34628	−0.673139	−	0.739516i	$-0.735054\pi$
$137$	−15.7578	−1.34628	−0.673139	+	0.739516i	$0.735054\pi$
$138$	−38.0771	−3.24134
$139$	−11.0282	−0.935402	−0.467701	−	0.883887i	$-0.654918\pi$
$139$	−11.0282	−0.935402	−0.467701	+	0.883887i	$0.654918\pi$
$140$	0	0
$141$	−4.83835	−0.407462
$142$	−9.03305	−0.758036
$143$	3.34004	0.279308
$144$	−22.7190	−1.89325
$145$	0	0
$146$	−3.71024	−0.307062
$147$	18.1006	1.49291
$148$	−6.66438	−0.547809
$149$	19.5123	1.59851	0.799254	−	0.600993i	$-0.205228\pi$
$149$	19.5123	1.59851	0.799254	+	0.600993i	$0.205228\pi$
$150$	0	0
$151$	11.2300	0.913882	0.456941	−	0.889497i	$-0.348945\pi$
$151$	11.2300	0.913882	0.456941	+	0.889497i	$0.348945\pi$
$152$	−0.535546	−0.0434385
$153$	−2.59192	−0.209544
$154$	−39.3537	−3.17121
$155$	0	0
$156$	3.96894	0.317769
$157$	6.81276	0.543717	0.271859	−	0.962337i	$-0.412362\pi$
$157$	6.81276	0.543717	0.271859	+	0.962337i	$0.412362\pi$
$158$	1.76006	0.140023
$159$	−20.9131	−1.65851
$160$	0	0
$161$	22.4677	1.77070
$162$	19.5295	1.53438
$163$	−15.6559	−1.22626	−0.613132	−	0.789980i	$-0.710091\pi$
$163$	−15.6559	−1.22626	−0.613132	+	0.789980i	$0.710091\pi$
$164$	15.5176	1.21172
$165$	0	0
$166$	−22.2761	−1.72896
$167$	16.7946	1.29961	0.649803	−	0.760103i	$-0.274851\pi$
$167$	16.7946	1.29961	0.649803	+	0.760103i	$0.274851\pi$
$168$	−2.66368	−0.205507
$169$	−12.6138	−0.970292
$170$	0	0
$171$	−13.2528	−1.01347
$172$	14.4504	1.10183
$173$	−0.0717558	−0.00545549	−0.00272775	−	0.999996i	$-0.500868\pi$
$173$	−0.0717558	−0.00545549	−0.00272775	+	0.999996i	$0.500868\pi$
$174$	−12.9171	−0.979240
$175$	0	0
$176$	−20.1213	−1.51670
$177$	−26.2450	−1.97269
$178$	20.5292	1.53873
$179$	14.3790	1.07473	0.537367	−	0.843348i	$-0.319419\pi$
$179$	14.3790	1.07473	0.537367	+	0.843348i	$0.319419\pi$
$180$	0	0
$181$	−9.22167	−0.685442	−0.342721	−	0.939437i	$-0.611348\pi$
$181$	−9.22167	−0.685442	−0.342721	+	0.939437i	$0.611348\pi$
$182$	−4.55040	−0.337298
$183$	4.06760	0.300686
$184$	−1.52747	−0.112606
$185$	0	0
$186$	−6.11304	−0.448230
$187$	−2.29556	−0.167868
$188$	−3.40746	−0.248515
$189$	−33.3299	−2.42439
$190$	0	0
$191$	1.61233	0.116664	0.0583319	−	0.998297i	$-0.481422\pi$
$191$	1.61233	0.116664	0.0583319	+	0.998297i	$0.481422\pi$
$192$	−26.9081	−1.94193
$193$	2.35980	0.169862	0.0849310	−	0.996387i	$-0.472933\pi$
$193$	2.35980	0.169862	0.0849310	+	0.996387i	$0.472933\pi$
$194$	−3.05627	−0.219427
$195$	0	0
$196$	12.7476	0.910540
$197$	18.6942	1.33191	0.665953	−	0.745994i	$-0.268025\pi$
$197$	18.6942	1.33191	0.665953	+	0.745994i	$0.268025\pi$
$198$	66.2081	4.70520
$199$	19.7375	1.39915	0.699577	−	0.714557i	$-0.253372\pi$
$199$	19.7375	1.39915	0.699577	+	0.714557i	$0.253372\pi$
$200$	0	0
$201$	29.5066	2.08123
$202$	20.8758	1.46882
$203$	7.62180	0.534946
$204$	−2.72779	−0.190984
$205$	0	0
$206$	15.8297	1.10291
$207$	−37.7993	−2.62723
$208$	−2.32660	−0.161320
$209$	−11.7375	−0.811900
$210$	0	0
$211$	23.8199	1.63983	0.819914	−	0.572486i	$-0.194021\pi$
$211$	23.8199	1.63983	0.819914	+	0.572486i	$0.194021\pi$
$212$	−14.7283	−1.01154
$213$	−13.4001	−0.918163
$214$	23.9511	1.63726
$215$	0	0
$216$	2.26593	0.154177
$217$	3.60704	0.244862
$218$	28.1116	1.90396
$219$	−5.50398	−0.371925
$220$	0	0
$221$	−0.265432	−0.0178549
$222$	−19.2095	−1.28926
$223$	12.6787	0.849030	0.424515	−	0.905421i	$-0.360445\pi$
$223$	12.6787	0.849030	0.424515	+	0.905421i	$0.360445\pi$
$224$	29.1819	1.94980
$225$	0	0
$226$	11.4591	0.762246
$227$	10.6526	0.707036	0.353518	−	0.935428i	$-0.384985\pi$
$227$	10.6526	0.707036	0.353518	+	0.935428i	$0.384985\pi$
$228$	−13.9476	−0.923700
$229$	2.70098	0.178486	0.0892430	−	0.996010i	$-0.471555\pi$
$229$	2.70098	0.178486	0.0892430	+	0.996010i	$0.471555\pi$
$230$	0	0
$231$	−58.3795	−3.84109
$232$	−0.518169	−0.0340195
$233$	−16.8362	−1.10298	−0.551489	−	0.834182i	$-0.685940\pi$
$233$	−16.8362	−1.10298	−0.551489	+	0.834182i	$0.685940\pi$
$234$	7.65553	0.500458
$235$	0	0
$236$	−18.4833	−1.20316
$237$	2.61097	0.169601
$238$	3.12742	0.202721
$239$	8.52019	0.551125	0.275563	−	0.961283i	$-0.411136\pi$
$239$	8.52019	0.551125	0.275563	+	0.961283i	$0.411136\pi$
$240$	0	0
$241$	−1.01457	−0.0653543	−0.0326772	−	0.999466i	$-0.510403\pi$
$241$	−1.01457	−0.0653543	−0.0326772	+	0.999466i	$0.510403\pi$
$242$	36.3082	2.33398
$243$	1.25056	0.0802232
$244$	2.86466	0.183391
$245$	0	0
$246$	44.7283	2.85177
$247$	−1.35719	−0.0863558
$248$	−0.245225	−0.0155718
$249$	−33.0457	−2.09418
$250$	0	0
$251$	−1.69708	−0.107119	−0.0535595	−	0.998565i	$-0.517057\pi$
$251$	−1.69708	−0.107119	−0.0535595	+	0.998565i	$0.517057\pi$
$252$	−46.4224	−2.92433
$253$	−33.4774	−2.10470
$254$	−18.1563	−1.13923
$255$	0	0
$256$	13.8958	0.868487
$257$	−28.2211	−1.76038	−0.880192	−	0.474618i	$-0.842586\pi$
$257$	−28.2211	−1.76038	−0.880192	+	0.474618i	$0.842586\pi$
$258$	41.6522	2.59315
$259$	11.3347	0.704305
$260$	0	0
$261$	−12.8228	−0.793712
$262$	−35.1074	−2.16894
$263$	−17.5427	−1.08173	−0.540866	−	0.841109i	$-0.681903\pi$
$263$	−17.5427	−1.08173	−0.540866	+	0.841109i	$0.681903\pi$
$264$	3.96894	0.244271
$265$	0	0
$266$	15.9909	0.980467
$267$	30.4542	1.86377
$268$	20.7803	1.26936
$269$	10.6342	0.648380	0.324190	−	0.945992i	$-0.394908\pi$
$269$	10.6342	0.648380	0.324190	+	0.945992i	$0.394908\pi$
$270$	0	0
$271$	−28.6806	−1.74222	−0.871111	−	0.491086i	$-0.836600\pi$
$271$	−28.6806	−1.74222	−0.871111	+	0.491086i	$0.836600\pi$
$272$	1.59904	0.0969557
$273$	−6.75033	−0.408548
$274$	−31.9879	−1.93246
$275$	0	0
$276$	−39.7808	−2.39452
$277$	−15.5474	−0.934151	−0.467075	−	0.884217i	$-0.654692\pi$
$277$	−15.5474	−0.934151	−0.467075	+	0.884217i	$0.654692\pi$
$278$	−22.3870	−1.34269
$279$	−6.06843	−0.363308
$280$	0	0
$281$	−3.39292	−0.202404	−0.101202	−	0.994866i	$-0.532269\pi$
$281$	−3.39292	−0.202404	−0.101202	+	0.994866i	$0.532269\pi$
$282$	−9.82173	−0.584876
$283$	−14.6418	−0.870364	−0.435182	−	0.900343i	$-0.643316\pi$
$283$	−14.6418	−0.870364	−0.435182	+	0.900343i	$0.643316\pi$
$284$	−9.43721	−0.559995
$285$	0	0
$286$	6.78020	0.400922
$287$	−26.3922	−1.55788
$288$	−49.0953	−2.89297
$289$	−16.8176	−0.989269
$290$	0	0
$291$	−4.53384	−0.265779
$292$	−3.87624	−0.226840
$293$	17.5545	1.02554	0.512771	−	0.858525i	$-0.328619\pi$
$293$	17.5545	1.02554	0.512771	+	0.858525i	$0.328619\pi$
$294$	36.7438	2.14294
$295$	0	0
$296$	−0.770591	−0.0447897
$297$	49.6622	2.88170
$298$	39.6095	2.29451
$299$	−3.87093	−0.223862
$300$	0	0
$301$	−24.5771	−1.41660
$302$	22.7966	1.31180
$303$	30.9684	1.77909
$304$	8.17608	0.468930
$305$	0	0
$306$	−5.26153	−0.300782
$307$	12.1391	0.692815	0.346407	−	0.938084i	$-0.387402\pi$
$307$	12.1391	0.692815	0.346407	+	0.938084i	$0.387402\pi$
$308$	−41.1145	−2.34271
$309$	23.4827	1.33589
$310$	0	0
$311$	24.3028	1.37808	0.689041	−	0.724722i	$-0.258032\pi$
$311$	24.3028	1.37808	0.689041	+	0.724722i	$0.258032\pi$
$312$	0.458922	0.0259813
$313$	−4.28099	−0.241976	−0.120988	−	0.992654i	$-0.538606\pi$
$313$	−4.28099	−0.241976	−0.120988	+	0.992654i	$0.538606\pi$
$314$	13.8297	0.780457
$315$	0	0
$316$	1.83881	0.103441
$317$	16.7645	0.941587	0.470793	−	0.882243i	$-0.343968\pi$
$317$	16.7645	0.941587	0.470793	+	0.882243i	$0.343968\pi$
$318$	−42.4530	−2.38065
$319$	−11.3567	−0.635851
$320$	0	0
$321$	35.5304	1.98311
$322$	45.6088	2.54168
$323$	0.932776	0.0519010
$324$	20.4033	1.13352
$325$	0	0
$326$	−31.7811	−1.76019
$327$	41.7023	2.30614
$328$	1.79428	0.0990723
$329$	5.79538	0.319510
$330$	0	0
$331$	31.8749	1.75200	0.876001	−	0.482310i	$-0.160202\pi$
$331$	31.8749	1.75200	0.876001	+	0.482310i	$0.160202\pi$
$332$	−23.2728	−1.27726
$333$	−19.0694	−1.04499
$334$	34.0926	1.86547
$335$	0	0
$336$	40.6659	2.21850
$337$	−14.6535	−0.798228	−0.399114	−	0.916901i	$-0.630682\pi$
$337$	−14.6535	−0.798228	−0.399114	+	0.916901i	$0.630682\pi$
$338$	−25.6057	−1.39277
$339$	16.9990	0.923261
$340$	0	0
$341$	−5.37457	−0.291049
$342$	−26.9029	−1.45474
$343$	3.56835	0.192673
$344$	1.67088	0.0900878
$345$	0	0
$346$	−0.145663	−0.00783087
$347$	−0.332745	−0.0178627	−0.00893133	−	0.999960i	$-0.502843\pi$
$347$	−0.332745	−0.0178627	−0.00893133	+	0.999960i	$0.502843\pi$
$348$	−13.4950	−0.723408
$349$	−17.1970	−0.920534	−0.460267	−	0.887780i	$-0.652246\pi$
$349$	−17.1970	−0.920534	−0.460267	+	0.887780i	$0.652246\pi$
$350$	0	0
$351$	5.74236	0.306505
$352$	−43.4818	−2.31759
$353$	−2.50778	−0.133476	−0.0667378	−	0.997771i	$-0.521259\pi$
$353$	−2.50778	−0.133476	−0.0667378	+	0.997771i	$0.521259\pi$
$354$	−53.2767	−2.83163
$355$	0	0
$356$	21.4477	1.13673
$357$	4.63940	0.245543
$358$	29.1890	1.54268
$359$	−13.0691	−0.689758	−0.344879	−	0.938647i	$-0.612080\pi$
$359$	−13.0691	−0.689758	−0.344879	+	0.938647i	$0.612080\pi$
$360$	0	0
$361$	−14.2306	−0.748979
$362$	−18.7198	−0.983889
$363$	53.8616	2.82700
$364$	−4.75400	−0.249177
$365$	0	0
$366$	8.25713	0.431607
$367$	−9.26512	−0.483635	−0.241818	−	0.970322i	$-0.577744\pi$
$367$	−9.26512	−0.483635	−0.241818	+	0.970322i	$0.577744\pi$
$368$	23.3196	1.21562
$369$	44.4019	2.31147
$370$	0	0
$371$	25.0497	1.30051
$372$	−6.38655	−0.331127
$373$	2.74696	0.142232	0.0711162	−	0.997468i	$-0.477344\pi$
$373$	2.74696	0.142232	0.0711162	+	0.997468i	$0.477344\pi$
$374$	−4.65993	−0.240959
$375$	0	0
$376$	−0.393999	−0.0203190
$377$	−1.31315	−0.0676308
$378$	−67.6588	−3.47999
$379$	−24.2614	−1.24623	−0.623113	−	0.782132i	$-0.714132\pi$
$379$	−24.2614	−1.24623	−0.623113	+	0.782132i	$0.714132\pi$
$380$	0	0
$381$	−26.9342	−1.37988
$382$	3.27298	0.167460
$383$	−33.0311	−1.68781	−0.843905	−	0.536492i	$-0.819749\pi$
$383$	−33.0311	−1.68781	−0.843905	+	0.536492i	$0.819749\pi$
$384$	−5.89695	−0.300928
$385$	0	0
$386$	4.79033	0.243822
$387$	41.3483	2.10185
$388$	−3.19301	−0.162101
$389$	27.1257	1.37533	0.687665	−	0.726028i	$-0.258636\pi$
$389$	27.1257	1.37533	0.687665	+	0.726028i	$0.258636\pi$
$390$	0	0
$391$	2.66043	0.134544
$392$	1.47398	0.0744472
$393$	−52.0804	−2.62711
$394$	37.9487	1.91183
$395$	0	0
$396$	69.1704	3.47594
$397$	−6.56274	−0.329374	−0.164687	−	0.986346i	$-0.552661\pi$
$397$	−6.56274	−0.329374	−0.164687	+	0.986346i	$0.552661\pi$
$398$	40.0667	2.00836
$399$	23.7219	1.18758
$400$	0	0
$401$	−0.733108	−0.0366097	−0.0183048	−	0.999832i	$-0.505827\pi$
$401$	−0.733108	−0.0366097	−0.0183048	+	0.999832i	$0.505827\pi$
$402$	59.8976	2.98742
$403$	−0.621453	−0.0309568
$404$	21.8099	1.08508
$405$	0	0
$406$	15.4721	0.767866
$407$	−16.8890	−0.837156
$408$	−0.315410	−0.0156151
$409$	20.5700	1.01712	0.508561	−	0.861026i	$-0.330177\pi$
$409$	20.5700	1.01712	0.508561	+	0.861026i	$0.330177\pi$
$410$	0	0
$411$	−47.4527	−2.34067
$412$	16.5380	0.814769
$413$	31.4363	1.54688
$414$	−76.7317	−3.77116
$415$	0	0
$416$	−5.02773	−0.246505
$417$	−33.2102	−1.62631
$418$	−23.8269	−1.16541
$419$	−34.9015	−1.70505	−0.852524	−	0.522688i	$-0.824929\pi$
$419$	−34.9015	−1.70505	−0.852524	+	0.522688i	$0.824929\pi$
$420$	0	0
$421$	4.55407	0.221952	0.110976	−	0.993823i	$-0.464602\pi$
$421$	4.55407	0.221952	0.110976	+	0.993823i	$0.464602\pi$
$422$	48.3538	2.35383
$423$	−9.75007	−0.474064
$424$	−1.70300	−0.0827052
$425$	0	0
$426$	−27.2020	−1.31794
$427$	−4.87218	−0.235781
$428$	25.0227	1.20952
$429$	10.0581	0.485612
$430$	0	0
$431$	−10.1639	−0.489579	−0.244789	−	0.969576i	$-0.578719\pi$
$431$	−10.1639	−0.489579	−0.244789	+	0.969576i	$0.578719\pi$
$432$	−34.5936	−1.66438
$433$	27.2734	1.31068	0.655338	−	0.755336i	$-0.272526\pi$
$433$	27.2734	1.31068	0.655338	+	0.755336i	$0.272526\pi$
$434$	7.32220	0.351477
$435$	0	0
$436$	29.3693	1.40654
$437$	13.6032	0.650727
$438$	−11.1730	−0.533864
$439$	−11.4167	−0.544892	−0.272446	−	0.962171i	$-0.587833\pi$
$439$	−11.4167	−0.544892	−0.272446	+	0.962171i	$0.587833\pi$
$440$	0	0
$441$	36.4757	1.73694
$442$	−0.538820	−0.0256291
$443$	−24.3749	−1.15809	−0.579043	−	0.815297i	$-0.696574\pi$
$443$	−24.3749	−1.15809	−0.579043	+	0.815297i	$0.696574\pi$
$444$	−20.0690	−0.952433
$445$	0	0
$446$	25.7375	1.21871
$447$	58.7590	2.77920
$448$	32.2306	1.52275
$449$	11.7817	0.556014	0.278007	−	0.960579i	$-0.410326\pi$
$449$	11.7817	0.556014	0.278007	+	0.960579i	$0.410326\pi$
$450$	0	0
$451$	39.3250	1.85174
$452$	11.9718	0.563105
$453$	33.8178	1.58890
$454$	21.6245	1.01489
$455$	0	0
$456$	−1.61273	−0.0755232
$457$	20.2372	0.946656	0.473328	−	0.880886i	$-0.343053\pi$
$457$	20.2372	0.946656	0.473328	+	0.880886i	$0.343053\pi$
$458$	5.48293	0.256200
$459$	−3.94664	−0.184213
$460$	0	0
$461$	−21.6221	−1.00704	−0.503521	−	0.863983i	$-0.667962\pi$
$461$	−21.6221	−1.00704	−0.503521	+	0.863983i	$0.667962\pi$
$462$	−118.509	−5.51354
$463$	−25.8793	−1.20271	−0.601357	−	0.798980i	$-0.705373\pi$
$463$	−25.8793	−1.20271	−0.601357	+	0.798980i	$0.705373\pi$
$464$	7.91079	0.367249
$465$	0	0
$466$	−34.1771	−1.58323
$467$	−5.85967	−0.271153	−0.135577	−	0.990767i	$-0.543289\pi$
$467$	−5.85967	−0.271153	−0.135577	+	0.990767i	$0.543289\pi$
$468$	7.99806	0.369710
$469$	−35.3430	−1.63199
$470$	0	0
$471$	20.5158	0.945320
$472$	−2.13720	−0.0983725
$473$	36.6205	1.68381
$474$	5.30021	0.243447
$475$	0	0
$476$	3.26735	0.149759
$477$	−42.1432	−1.92961
$478$	17.2958	0.791090
$479$	2.64005	0.120627	0.0603136	−	0.998179i	$-0.480790\pi$
$479$	2.64005	0.120627	0.0603136	+	0.998179i	$0.480790\pi$
$480$	0	0
$481$	−1.95284	−0.0890421
$482$	−2.05956	−0.0938103
$483$	67.6588	3.07858
$484$	37.9327	1.72421
$485$	0	0
$486$	2.53860	0.115153
$487$	19.1480	0.867679	0.433839	−	0.900990i	$-0.357159\pi$
$487$	19.1480	0.867679	0.433839	+	0.900990i	$0.357159\pi$
$488$	0.331235	0.0149943
$489$	−47.1459	−2.13201
$490$	0	0
$491$	−35.0395	−1.58131	−0.790654	−	0.612263i	$-0.790260\pi$
$491$	−35.0395	−1.58131	−0.790654	+	0.612263i	$0.790260\pi$
$492$	46.7295	2.10673
$493$	0.902510	0.0406470
$494$	−2.75506	−0.123956
$495$	0	0
$496$	3.74380	0.168102
$497$	16.0507	0.719973
$498$	−67.0819	−3.00601
$499$	−15.2066	−0.680743	−0.340371	−	0.940291i	$-0.610553\pi$
$499$	−15.2066	−0.680743	−0.340371	+	0.940291i	$0.610553\pi$
$500$	0	0
$501$	50.5750	2.25952
$502$	−3.44504	−0.153760
$503$	14.2812	0.636767	0.318383	−	0.947962i	$-0.396860\pi$
$503$	14.2812	0.636767	0.318383	+	0.947962i	$0.396860\pi$
$504$	−5.36774	−0.239098
$505$	0	0
$506$	−67.9582	−3.02111
$507$	−37.9850	−1.68697
$508$	−18.9687	−0.841600
$509$	14.4812	0.641869	0.320935	−	0.947101i	$-0.396003\pi$
$509$	14.4812	0.641869	0.320935	+	0.947101i	$0.396003\pi$
$510$	0	0
$511$	6.59268	0.291643
$512$	32.1246	1.41972
$513$	−20.1797	−0.890956
$514$	−57.2882	−2.52687
$515$	0	0
$516$	43.5158	1.91568
$517$	−8.63525	−0.379778
$518$	23.0092	1.01097
$519$	−0.216084	−0.00948505
$520$	0	0
$521$	−10.3008	−0.451285	−0.225643	−	0.974210i	$-0.572448\pi$
$521$	−10.3008	−0.451285	−0.225643	+	0.974210i	$0.572448\pi$
$522$	−26.0300	−1.13930
$523$	7.00060	0.306115	0.153057	−	0.988217i	$-0.451088\pi$
$523$	7.00060	0.306115	0.153057	+	0.988217i	$0.451088\pi$
$524$	−36.6782	−1.60230
$525$	0	0
$526$	−35.6113	−1.55273
$527$	0.427115	0.0186054
$528$	−60.5931	−2.63697
$529$	15.7985	0.686891
$530$	0	0
$531$	−52.8880	−2.29514
$532$	16.7064	0.724314
$533$	4.54708	0.196956
$534$	61.8214	2.67527
$535$	0	0
$536$	2.40280	0.103785
$537$	43.3006	1.86856
$538$	21.5872	0.930691
$539$	32.3051	1.39148
$540$	0	0
$541$	22.3411	0.960518	0.480259	−	0.877127i	$-0.340543\pi$
$541$	22.3411	0.960518	0.480259	+	0.877127i	$0.340543\pi$
$542$	−58.2209	−2.50080
$543$	−27.7700	−1.19172
$544$	3.45548	0.148153
$545$	0	0
$546$	−13.7030	−0.586434
$547$	−16.8457	−0.720270	−0.360135	−	0.932900i	$-0.617269\pi$
$547$	−16.8457	−0.720270	−0.360135	+	0.932900i	$0.617269\pi$
$548$	−33.4191	−1.42759
$549$	8.19689	0.349835
$550$	0	0
$551$	4.61465	0.196591
$552$	−4.59979	−0.195780
$553$	−3.12742	−0.132992
$554$	−31.5608	−1.34089
$555$	0	0
$556$	−23.3887	−0.991901
$557$	6.14793	0.260496	0.130248	−	0.991481i	$-0.458423\pi$
$557$	6.14793	0.260496	0.130248	+	0.991481i	$0.458423\pi$
$558$	−12.3188	−0.521495
$559$	4.23437	0.179095
$560$	0	0
$561$	−6.91281	−0.291859
$562$	−6.88754	−0.290533
$563$	36.0414	1.51896	0.759482	−	0.650528i	$-0.225452\pi$
$563$	36.0414	1.51896	0.759482	+	0.650528i	$0.225452\pi$
$564$	−10.2612	−0.432074
$565$	0	0
$566$	−29.7225	−1.24933
$567$	−34.7018	−1.45734
$568$	−1.09121	−0.0457861
$569$	39.1798	1.64250	0.821252	−	0.570566i	$-0.193276\pi$
$569$	39.1798	1.64250	0.821252	+	0.570566i	$0.193276\pi$
$570$	0	0
$571$	10.0732	0.421549	0.210775	−	0.977535i	$-0.432401\pi$
$571$	10.0732	0.421549	0.210775	+	0.977535i	$0.432401\pi$
$572$	7.08357	0.296179
$573$	4.85534	0.202835
$574$	−53.5755	−2.23620
$575$	0	0
$576$	−54.2243	−2.25934
$577$	−44.3050	−1.84444	−0.922220	−	0.386666i	$-0.873627\pi$
$577$	−44.3050	−1.84444	−0.922220	+	0.386666i	$0.873627\pi$
$578$	−34.1393	−1.42001
$579$	7.10626	0.295326
$580$	0	0
$581$	39.5821	1.64214
$582$	−9.20359	−0.381501
$583$	−37.3246	−1.54583
$584$	−0.448204	−0.0185468
$585$	0	0
$586$	35.6351	1.47207
$587$	15.7608	0.650520	0.325260	−	0.945625i	$-0.394548\pi$
$587$	15.7608	0.650520	0.325260	+	0.945625i	$0.394548\pi$
$588$	38.3878	1.58309
$589$	2.18390	0.0899859
$590$	0	0
$591$	56.2954	2.31568
$592$	11.7645	0.483517
$593$	5.11628	0.210101	0.105050	−	0.994467i	$-0.466500\pi$
$593$	5.11628	0.210101	0.105050	+	0.994467i	$0.466500\pi$
$594$	100.813	4.13641
$595$	0	0
$596$	41.3817	1.69506
$597$	59.4372	2.43260
$598$	−7.85790	−0.321333
$599$	−47.0659	−1.92306	−0.961530	−	0.274700i	$-0.911422\pi$
$599$	−47.0659	−1.92306	−0.961530	+	0.274700i	$0.911422\pi$
$600$	0	0
$601$	−43.1495	−1.76010	−0.880052	−	0.474878i	$-0.842492\pi$
$601$	−43.1495	−1.76010	−0.880052	+	0.474878i	$0.842492\pi$
$602$	−49.8910	−2.03341
$603$	59.4606	2.42142
$604$	23.8165	0.969081
$605$	0	0
$606$	62.8651	2.55372
$607$	−30.5560	−1.24023	−0.620115	−	0.784511i	$-0.712914\pi$
$607$	−30.5560	−1.24023	−0.620115	+	0.784511i	$0.712914\pi$
$608$	17.6683	0.716546
$609$	22.9522	0.930069
$610$	0	0
$611$	−0.998479	−0.0403942
$612$	−5.49695	−0.222201
$613$	2.60357	0.105157	0.0525785	−	0.998617i	$-0.483256\pi$
$613$	2.60357	0.105157	0.0525785	+	0.998617i	$0.483256\pi$
$614$	24.6421	0.994473
$615$	0	0
$616$	−4.75400	−0.191544
$617$	11.5938	0.466748	0.233374	−	0.972387i	$-0.425023\pi$
$617$	11.5938	0.466748	0.233374	+	0.972387i	$0.425023\pi$
$618$	47.6694	1.91755
$619$	−36.9684	−1.48588	−0.742942	−	0.669355i	$-0.766570\pi$
$619$	−36.9684	−1.48588	−0.742942	+	0.669355i	$0.766570\pi$
$620$	0	0
$621$	−57.5559	−2.30964
$622$	49.3340	1.97811
$623$	−36.4781	−1.46146
$624$	−7.00628	−0.280475
$625$	0	0
$626$	−8.69031	−0.347335
$627$	−35.3461	−1.41159
$628$	14.4485	0.576558
$629$	1.34216	0.0535155
$630$	0	0
$631$	−36.0350	−1.43453	−0.717265	−	0.696800i	$-0.754606\pi$
$631$	−36.0350	−1.43453	−0.717265	+	0.696800i	$0.754606\pi$
$632$	0.212618	0.00845750
$633$	71.7308	2.85104
$634$	34.0315	1.35156
$635$	0	0
$636$	−44.3524	−1.75869
$637$	3.73538	0.148001
$638$	−23.0537	−0.912707
$639$	−27.0035	−1.06824
$640$	0	0
$641$	30.3678	1.19946	0.599728	−	0.800204i	$-0.295275\pi$
$641$	30.3678	1.19946	0.599728	+	0.800204i	$0.295275\pi$
$642$	72.1259	2.84658
$643$	1.32529	0.0522644	0.0261322	−	0.999658i	$-0.491681\pi$
$643$	1.32529	0.0522644	0.0261322	+	0.999658i	$0.491681\pi$
$644$	47.6495	1.87765
$645$	0	0
$646$	1.89351	0.0744992
$647$	−44.8501	−1.76324	−0.881619	−	0.471961i	$-0.843546\pi$
$647$	−44.8501	−1.76324	−0.881619	+	0.471961i	$0.843546\pi$
$648$	2.35920	0.0926783
$649$	−46.8408	−1.83866
$650$	0	0
$651$	10.8622	0.425722
$652$	−33.2030	−1.30033
$653$	9.14757	0.357972	0.178986	−	0.983852i	$-0.442718\pi$
$653$	9.14757	0.357972	0.178986	+	0.983852i	$0.442718\pi$
$654$	84.6547	3.31026
$655$	0	0
$656$	−27.3929	−1.06951
$657$	−11.0914	−0.432718
$658$	11.7645	0.458627
$659$	32.9973	1.28539	0.642696	−	0.766121i	$-0.277816\pi$
$659$	32.9973	1.28539	0.642696	+	0.766121i	$0.277816\pi$
$660$	0	0
$661$	−8.09926	−0.315025	−0.157512	−	0.987517i	$-0.550347\pi$
$661$	−8.09926	−0.315025	−0.157512	+	0.987517i	$0.550347\pi$
$662$	64.7052	2.51484
$663$	−0.799317	−0.0310429
$664$	−2.69099	−0.104431
$665$	0	0
$666$	−38.7103	−1.50000
$667$	13.1618	0.509626
$668$	35.6180	1.37810
$669$	38.1805	1.47614
$670$	0	0
$671$	7.25966	0.280256
$672$	87.8780	3.38997
$673$	−7.62589	−0.293956	−0.146978	−	0.989140i	$-0.546955\pi$
$673$	−7.62589	−0.293956	−0.146978	+	0.989140i	$0.546955\pi$
$674$	−29.7463	−1.14578
$675$	0	0
$676$	−26.7514	−1.02890
$677$	−2.60357	−0.100063	−0.0500316	−	0.998748i	$-0.515932\pi$
$677$	−2.60357	−0.100063	−0.0500316	+	0.998748i	$0.515932\pi$
$678$	34.5076	1.32526
$679$	5.43064	0.208409
$680$	0	0
$681$	32.0790	1.22927
$682$	−10.9102	−0.417775
$683$	42.9054	1.64173	0.820865	−	0.571122i	$-0.193492\pi$
$683$	42.9054	1.64173	0.820865	+	0.571122i	$0.193492\pi$
$684$	−28.1066	−1.07468
$685$	0	0
$686$	7.24367	0.276564
$687$	8.13370	0.310320
$688$	−25.5090	−0.972522
$689$	−4.31578	−0.164418
$690$	0	0
$691$	−2.85088	−0.108453	−0.0542263	−	0.998529i	$-0.517269\pi$
$691$	−2.85088	−0.108453	−0.0542263	+	0.998529i	$0.517269\pi$
$692$	−0.152180	−0.00578501
$693$	−117.644	−4.46894
$694$	−0.675464	−0.0256402
$695$	0	0
$696$	−1.56041	−0.0591470
$697$	−3.12514	−0.118373
$698$	−34.9095	−1.32134
$699$	−50.7004	−1.91766
$700$	0	0
$701$	42.8583	1.61874	0.809368	−	0.587302i	$-0.199810\pi$
$701$	42.8583	1.61874	0.809368	+	0.587302i	$0.199810\pi$
$702$	11.6569	0.439960
$703$	6.86265	0.258830
$704$	−48.0243	−1.80998
$705$	0	0
$706$	−5.09073	−0.191592
$707$	−37.0940	−1.39506
$708$	−55.6604	−2.09185
$709$	−21.1169	−0.793063	−0.396532	−	0.918021i	$-0.629786\pi$
$709$	−21.1169	−0.793063	−0.396532	+	0.918021i	$0.629786\pi$
$710$	0	0
$711$	5.26153	0.197323
$712$	2.47997	0.0929407
$713$	6.22884	0.233272
$714$	9.41787	0.352455
$715$	0	0
$716$	30.4949	1.13965
$717$	25.6576	0.958199
$718$	−26.5299	−0.990086
$719$	−49.2198	−1.83559	−0.917794	−	0.397058i	$-0.870031\pi$
$719$	−49.2198	−1.83559	−0.917794	+	0.397058i	$0.870031\pi$
$720$	0	0
$721$	−28.1277	−1.04753
$722$	−28.8878	−1.07509
$723$	−3.05527	−0.113627
$724$	−19.5573	−0.726843
$725$	0	0
$726$	109.338	4.05791
$727$	−10.4123	−0.386170	−0.193085	−	0.981182i	$-0.561849\pi$
$727$	−10.4123	−0.386170	−0.193085	+	0.981182i	$0.561849\pi$
$728$	−0.549697	−0.0203731
$729$	−25.0958	−0.929475
$730$	0	0
$731$	−2.91022	−0.107638
$732$	8.62658	0.318848
$733$	−6.22566	−0.229950	−0.114975	−	0.993368i	$-0.536679\pi$
$733$	−6.22566	−0.229950	−0.114975	+	0.993368i	$0.536679\pi$
$734$	−18.8080	−0.694215
$735$	0	0
$736$	50.3931	1.85751
$737$	52.6619	1.93983
$738$	90.1348	3.31791
$739$	−4.95574	−0.182300	−0.0911499	−	0.995837i	$-0.529054\pi$
$739$	−4.95574	−0.182300	−0.0911499	+	0.995837i	$0.529054\pi$
$740$	0	0
$741$	−4.08702	−0.150140
$742$	50.8502	1.86677
$743$	22.3581	0.820237	0.410119	−	0.912032i	$-0.365487\pi$
$743$	22.3581	0.820237	0.410119	+	0.912032i	$0.365487\pi$
$744$	−0.738466	−0.0270735
$745$	0	0
$746$	5.57627	0.204162
$747$	−66.5924	−2.43649
$748$	−4.86843	−0.178007
$749$	−42.5584	−1.55505
$750$	0	0
$751$	−43.5737	−1.59003	−0.795014	−	0.606591i	$-0.792536\pi$
$751$	−43.5737	−1.59003	−0.795014	+	0.606591i	$0.792536\pi$
$752$	6.01512	0.219349
$753$	−5.11057	−0.186240
$754$	−2.66567	−0.0970779
$755$	0	0
$756$	−70.6860	−2.57083
$757$	16.0563	0.583575	0.291787	−	0.956483i	$-0.405750\pi$
$757$	16.0563	0.583575	0.291787	+	0.956483i	$0.405750\pi$
$758$	−49.2501	−1.78884
$759$	−100.813	−3.65929
$760$	0	0
$761$	−30.2154	−1.09531	−0.547653	−	0.836706i	$-0.684479\pi$
$761$	−30.2154	−1.09531	−0.547653	+	0.836706i	$0.684479\pi$
$762$	−54.6757	−1.98069
$763$	−49.9511	−1.80835
$764$	3.41943	0.123710
$765$	0	0
$766$	−67.0523	−2.42270
$767$	−5.41612	−0.195565
$768$	41.8456	1.50997
$769$	22.2921	0.803873	0.401936	−	0.915668i	$-0.368337\pi$
$769$	22.2921	0.803873	0.401936	+	0.915668i	$0.368337\pi$
$770$	0	0
$771$	−84.9846	−3.06064
$772$	5.00467	0.180122
$773$	30.6691	1.10309	0.551545	−	0.834145i	$-0.314039\pi$
$773$	30.6691	1.10309	0.551545	+	0.834145i	$0.314039\pi$
$774$	83.9360	3.01702
$775$	0	0
$776$	−0.369203	−0.0132536
$777$	34.1332	1.22452
$778$	55.0646	1.97416
$779$	−15.9793	−0.572517
$780$	0	0
$781$	−23.9159	−0.855779
$782$	5.40062	0.193126
$783$	−19.5249	−0.697764
$784$	−22.5030	−0.803678
$785$	0	0
$786$	−105.722	−3.77098
$787$	32.5003	1.15851	0.579255	−	0.815146i	$-0.303343\pi$
$787$	32.5003	1.15851	0.579255	+	0.815146i	$0.303343\pi$
$788$	39.6467	1.41235
$789$	−52.8279	−1.88072
$790$	0	0
$791$	−20.3615	−0.723970
$792$	7.99806	0.284199
$793$	0.839422	0.0298088
$794$	−13.3222	−0.472787
$795$	0	0
$796$	41.8593	1.48367
$797$	−39.3420	−1.39357	−0.696783	−	0.717282i	$-0.745386\pi$
$797$	−39.3420	−1.39357	−0.696783	+	0.717282i	$0.745386\pi$
$798$	48.1548	1.70466
$799$	0.686240	0.0242774
$800$	0	0
$801$	61.3703	2.16841
$802$	−1.48819	−0.0525499
$803$	−9.82324	−0.346655
$804$	62.5776	2.20694
$805$	0	0
$806$	−1.26153	−0.0444356
$807$	32.0237	1.12729
$808$	2.52184	0.0887179
$809$	−8.30992	−0.292161	−0.146081	−	0.989273i	$-0.546666\pi$
$809$	−8.30992	−0.292161	−0.146081	+	0.989273i	$0.546666\pi$
$810$	0	0
$811$	−36.5351	−1.28292	−0.641461	−	0.767155i	$-0.721672\pi$
$811$	−36.5351	−1.28292	−0.641461	+	0.767155i	$0.721672\pi$
$812$	16.1643	0.567257
$813$	−86.3683	−3.02907
$814$	−34.2842	−1.20166
$815$	0	0
$816$	4.81531	0.168570
$817$	−14.8803	−0.520597
$818$	41.7567	1.45999
$819$	−13.6030	−0.475328
$820$	0	0
$821$	−0.121834	−0.00425205	−0.00212603	−	0.999998i	$-0.500677\pi$
$821$	−0.121834	−0.00425205	−0.00212603	+	0.999998i	$0.500677\pi$
$822$	−96.3279	−3.35982
$823$	−9.87971	−0.344385	−0.172193	−	0.985063i	$-0.555085\pi$
$823$	−9.87971	−0.344385	−0.172193	+	0.985063i	$0.555085\pi$
$824$	1.91226	0.0666168
$825$	0	0
$826$	63.8149	2.22040
$827$	−13.4253	−0.466845	−0.233422	−	0.972375i	$-0.574992\pi$
$827$	−13.4253	−0.466845	−0.233422	+	0.972375i	$0.574992\pi$
$828$	−80.1649	−2.78592
$829$	39.0793	1.35728	0.678641	−	0.734471i	$-0.262569\pi$
$829$	39.0793	1.35728	0.678641	+	0.734471i	$0.262569\pi$
$830$	0	0
$831$	−46.8191	−1.62414
$832$	−5.55297	−0.192515
$833$	−2.56727	−0.0889507
$834$	−67.4160	−2.33442
$835$	0	0
$836$	−24.8929	−0.860940
$837$	−9.24023	−0.319389
$838$	−70.8491	−2.44744
$839$	50.9588	1.75929	0.879646	−	0.475628i	$-0.157779\pi$
$839$	50.9588	1.75929	0.879646	+	0.475628i	$0.157779\pi$
$840$	0	0
$841$	−24.5351	−0.846037
$842$	9.24465	0.318592
$843$	−10.2174	−0.351905
$844$	50.5173	1.73888
$845$	0	0
$846$	−19.7924	−0.680477
$847$	−64.5155	−2.21678
$848$	25.9995	0.892825
$849$	−44.0920	−1.51324
$850$	0	0
$851$	19.5734	0.670969
$852$	−28.4191	−0.973621
$853$	−43.3724	−1.48504	−0.742520	−	0.669823i	$-0.766370\pi$
$853$	−43.3724	−1.48504	−0.742520	+	0.669823i	$0.766370\pi$
$854$	−9.89040	−0.338443
$855$	0	0
$856$	2.89333	0.0988921
$857$	30.9366	1.05678	0.528388	−	0.849003i	$-0.322797\pi$
$857$	30.9366	1.05678	0.528388	+	0.849003i	$0.322797\pi$
$858$	20.4178	0.697052
$859$	37.0464	1.26401	0.632004	−	0.774965i	$-0.282233\pi$
$859$	37.0464	1.26401	0.632004	+	0.774965i	$0.282233\pi$
$860$	0	0
$861$	−79.4771	−2.70857
$862$	−20.6325	−0.702746
$863$	8.17506	0.278282	0.139141	−	0.990273i	$-0.455566\pi$
$863$	8.17506	0.278282	0.139141	+	0.990273i	$0.455566\pi$
$864$	−74.7560	−2.54325
$865$	0	0
$866$	55.3644	1.88136
$867$	−50.6442	−1.71997
$868$	7.64981	0.259652
$869$	4.65993	0.158077
$870$	0	0
$871$	6.08921	0.206325
$872$	3.39593	0.115001
$873$	−9.13644	−0.309222
$874$	27.6141	0.934060
$875$	0	0
$876$	−11.6729	−0.394389
$877$	10.6199	0.358608	0.179304	−	0.983794i	$-0.442615\pi$
$877$	10.6199	0.358608	0.179304	+	0.983794i	$0.442615\pi$
$878$	−23.1757	−0.782143
$879$	52.8632	1.78303
$880$	0	0
$881$	0.374741	0.0126253	0.00631267	−	0.999980i	$-0.497991\pi$
$881$	0.374741	0.0126253	0.00631267	+	0.999980i	$0.497991\pi$
$882$	74.0447	2.49322
$883$	−12.0046	−0.403986	−0.201993	−	0.979387i	$-0.564742\pi$
$883$	−12.0046	−0.403986	−0.201993	+	0.979387i	$0.564742\pi$
$884$	−0.562928	−0.0189333
$885$	0	0
$886$	−49.4804	−1.66233
$887$	−16.8820	−0.566842	−0.283421	−	0.958996i	$-0.591469\pi$
$887$	−16.8820	−0.566842	−0.283421	+	0.958996i	$0.591469\pi$
$888$	−2.32055	−0.0778724
$889$	32.2618	1.08202
$890$	0	0
$891$	51.7064	1.73223
$892$	26.8891	0.900312
$893$	3.50884	0.117419
$894$	119.279	3.98930
$895$	0	0
$896$	7.06337	0.235971
$897$	−11.6569	−0.389211
$898$	23.9166	0.798108
$899$	2.11304	0.0704737
$900$	0	0
$901$	2.96617	0.0988175
$902$	79.8288	2.65801
$903$	−74.0112	−2.46294
$904$	1.38428	0.0460403
$905$	0	0
$906$	68.6492	2.28072
$907$	−17.9763	−0.596892	−0.298446	−	0.954427i	$-0.596468\pi$
$907$	−17.9763	−0.596892	−0.298446	+	0.954427i	$0.596468\pi$
$908$	22.5920	0.749742
$909$	62.4064	2.06989
$910$	0	0
$911$	43.8315	1.45220	0.726102	−	0.687587i	$-0.241330\pi$
$911$	43.8315	1.45220	0.726102	+	0.687587i	$0.241330\pi$
$912$	24.6213	0.815293
$913$	−58.9783	−1.95190
$914$	41.0810	1.35884
$915$	0	0
$916$	5.72825	0.189267
$917$	62.3819	2.06003
$918$	−8.01159	−0.264422
$919$	23.5175	0.775772	0.387886	−	0.921707i	$-0.373205\pi$
$919$	23.5175	0.775772	0.387886	+	0.921707i	$0.373205\pi$
$920$	0	0
$921$	36.5555	1.20454
$922$	−43.8923	−1.44552
$923$	−2.76536	−0.0910229
$924$	−123.811	−4.07310
$925$	0	0
$926$	−52.5344	−1.72639
$927$	47.3216	1.55425
$928$	17.0951	0.561173
$929$	18.6473	0.611797	0.305899	−	0.952064i	$-0.401043\pi$
$929$	18.6473	0.611797	0.305899	+	0.952064i	$0.401043\pi$
$930$	0	0
$931$	−13.1268	−0.430214
$932$	−35.7063	−1.16960
$933$	73.1849	2.39597
$934$	−11.8950	−0.389216
$935$	0	0
$936$	0.924803	0.0302281
$937$	−30.9366	−1.01066	−0.505328	−	0.862927i	$-0.668628\pi$
$937$	−30.9366	−1.01066	−0.505328	+	0.862927i	$0.668628\pi$
$938$	−71.7454	−2.34257
$939$	−12.8917	−0.420705
$940$	0	0
$941$	41.8389	1.36391	0.681954	−	0.731395i	$-0.261130\pi$
$941$	41.8389	1.36391	0.681954	+	0.731395i	$0.261130\pi$
$942$	41.6466	1.35692
$943$	−45.5756	−1.48415
$944$	32.6282	1.06196
$945$	0	0
$946$	74.3388	2.41696
$947$	1.57934	0.0513215	0.0256608	−	0.999671i	$-0.491831\pi$
$947$	1.57934	0.0513215	0.0256608	+	0.999671i	$0.491831\pi$
$948$	5.53735	0.179845
$949$	−1.13585	−0.0368711
$950$	0	0
$951$	50.4843	1.63706
$952$	0.377798	0.0122445
$953$	53.0407	1.71816	0.859079	−	0.511843i	$-0.171037\pi$
$953$	53.0407	1.71816	0.859079	+	0.511843i	$0.171037\pi$
$954$	−85.5498	−2.76978
$955$	0	0
$956$	18.0696	0.584414
$957$	−34.1993	−1.10551
$958$	5.35924	0.173149
$959$	56.8389	1.83542
$960$	0	0
$961$	1.00000	0.0322581
$962$	−3.96423	−0.127812
$963$	71.5996	2.30726
$964$	−2.15171	−0.0693018
$965$	0	0
$966$	137.346	4.41903
$967$	−42.7504	−1.37476	−0.687380	−	0.726298i	$-0.741239\pi$
$967$	−42.7504	−1.37476	−0.687380	+	0.726298i	$0.741239\pi$
$968$	4.38609	0.140974
$969$	2.80895	0.0902363
$970$	0	0
$971$	−33.5552	−1.07684	−0.538419	−	0.842678i	$-0.680978\pi$
$971$	−33.5552	−1.07684	−0.538419	+	0.842678i	$0.680978\pi$
$972$	2.65218	0.0850688
$973$	39.7792	1.27526
$974$	38.8700	1.24547
$975$	0	0
$976$	−5.05691	−0.161868
$977$	17.9826	0.575314	0.287657	−	0.957734i	$-0.407124\pi$
$977$	17.9826	0.575314	0.287657	+	0.957734i	$0.407124\pi$
$978$	−95.7050	−3.06031
$979$	54.3532	1.73714
$980$	0	0
$981$	84.0370	2.68310
$982$	−71.1293	−2.26983
$983$	−60.4165	−1.92699	−0.963494	−	0.267731i	$-0.913726\pi$
$983$	−60.4165	−1.92699	−0.963494	+	0.267731i	$0.913726\pi$
$984$	5.40326	0.172249
$985$	0	0
$986$	1.83207	0.0583451
$987$	17.4521	0.555507
$988$	−2.87833	−0.0915718
$989$	−42.4412	−1.34955
$990$	0	0
$991$	−26.7497	−0.849732	−0.424866	−	0.905256i	$-0.639679\pi$
$991$	−26.7497	−0.849732	−0.424866	+	0.905256i	$0.639679\pi$
$992$	8.09028	0.256867
$993$	95.9875	3.04607
$994$	32.5825	1.03346
$995$	0	0
$996$	−70.0833	−2.22067
$997$	−20.7787	−0.658067	−0.329034	−	0.944318i	$-0.606723\pi$
$997$	−20.7787	−0.658067	−0.329034	+	0.944318i	$0.606723\pi$
$998$	−30.8691	−0.977145
$999$	−29.0364	−0.918670

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	775.2.a.l.1.8		10
3.2	odd	2		6975.2.a.ch.1.3		10
5.2	odd	4		155.2.b.b.94.8	yes	10
5.3	odd	4		155.2.b.b.94.3	✓	10
5.4	even	2	inner	775.2.a.l.1.3		10
15.2	even	4		1395.2.c.e.559.3		10
15.8	even	4		1395.2.c.e.559.8		10
15.14	odd	2		6975.2.a.ch.1.8		10
20.3	even	4		2480.2.d.g.1489.1		10
20.7	even	4		2480.2.d.g.1489.10		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
155.2.b.b.94.3	✓	10	5.3	odd	4
155.2.b.b.94.8	yes	10	5.2	odd	4
775.2.a.l.1.3		10	5.4	even	2	inner
775.2.a.l.1.8		10	1.1	even	1	trivial
1395.2.c.e.559.3		10	15.2	even	4
1395.2.c.e.559.8		10	15.8	even	4
2480.2.d.g.1489.1		10	20.3	even	4
2480.2.d.g.1489.10		10	20.7	even	4
6975.2.a.ch.1.3		10	3.2	odd	2
6975.2.a.ch.1.8		10	15.14	odd	2

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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 775 = 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	775.a (trivial)

\(f(q)\)	\(=\)	\(q+2.02998 q^{2} +3.01138 q^{3} +2.12080 q^{4} +6.11304 q^{6} -3.60704 q^{7} +0.245225 q^{8} +6.06843 q^{9} +O(q^{10})\)	\(q+2.02998 q^{2} +3.01138 q^{3} +2.12080 q^{4} +6.11304 q^{6} -3.60704 q^{7} +0.245225 q^{8} +6.06843 q^{9} +5.37457 q^{11} +6.38655 q^{12} +0.621453 q^{13} -7.32220 q^{14} -3.74380 q^{16} -0.427115 q^{17} +12.3188 q^{18} -2.18390 q^{19} -10.8622 q^{21} +10.9102 q^{22} -6.22884 q^{23} +0.738466 q^{24} +1.26153 q^{26} +9.24023 q^{27} -7.64981 q^{28} -2.11304 q^{29} -1.00000 q^{31} -8.09028 q^{32} +16.1849 q^{33} -0.867033 q^{34} +12.8699 q^{36} -3.14239 q^{37} -4.43326 q^{38} +1.87143 q^{39} +7.31686 q^{41} -22.0500 q^{42} +6.81366 q^{43} +11.3984 q^{44} -12.6444 q^{46} -1.60669 q^{47} -11.2740 q^{48} +6.01073 q^{49} -1.28621 q^{51} +1.31798 q^{52} -6.94467 q^{53} +18.7574 q^{54} -0.884535 q^{56} -6.57655 q^{57} -4.28941 q^{58} -8.71526 q^{59} +1.35074 q^{61} -2.02998 q^{62} -21.8891 q^{63} -8.93547 q^{64} +32.8549 q^{66} +9.79835 q^{67} -0.905827 q^{68} -18.7574 q^{69} -4.44983 q^{71} +1.48813 q^{72} -1.82773 q^{73} -6.37897 q^{74} -4.63161 q^{76} -19.3863 q^{77} +3.79896 q^{78} +0.867033 q^{79} +9.62057 q^{81} +14.8531 q^{82} -10.9736 q^{83} -23.0365 q^{84} +13.8316 q^{86} -6.36316 q^{87} +1.31798 q^{88} +10.1130 q^{89} -2.24160 q^{91} -13.2101 q^{92} -3.01138 q^{93} -3.26153 q^{94} -24.3629 q^{96} -1.50557 q^{97} +12.2016 q^{98} +32.6152 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 12 q^{4} + 8 q^{6} + 22 q^{9}+O(q^{10}) \) 10 * q + 12 * q^4 + 8 * q^6 + 22 * q^9	\( 10 q + 12 q^{4} + 8 q^{6} + 22 q^{9} + 16 q^{11} - 6 q^{14} + 8 q^{16} - 4 q^{19} + 20 q^{21} - 8 q^{24} + 28 q^{26} + 32 q^{29} - 10 q^{31} - 28 q^{34} + 44 q^{36} - 16 q^{39} + 36 q^{41} + 52 q^{44} + 8 q^{46} + 22 q^{49} + 20 q^{51} + 12 q^{56} + 12 q^{59} - 70 q^{64} + 12 q^{71} + 28 q^{74} - 30 q^{76} + 28 q^{79} - 14 q^{81} - 40 q^{84} + 36 q^{86} + 48 q^{89} - 4 q^{91} - 48 q^{94} - 60 q^{96} + 64 q^{99}+O(q^{100}) \) 10 * q + 12 * q^4 + 8 * q^6 + 22 * q^9 + 16 * q^11 - 6 * q^14 + 8 * q^16 - 4 * q^19 + 20 * q^21 - 8 * q^24 + 28 * q^26 + 32 * q^29 - 10 * q^31 - 28 * q^34 + 44 * q^36 - 16 * q^39 + 36 * q^41 + 52 * q^44 + 8 * q^46 + 22 * q^49 + 20 * q^51 + 12 * q^56 + 12 * q^59 - 70 * q^64 + 12 * q^71 + 28 * q^74 - 30 * q^76 + 28 * q^79 - 14 * q^81 - 40 * q^84 + 36 * q^86 + 48 * q^89 - 4 * q^91 - 48 * q^94 - 60 * q^96 + 64 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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