LMFDB - Embedded newform 1925.2.b.q.1849.8

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1925,2,Mod(1849,1925)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1925.1849"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1925, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1925 = 5^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1925.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [14,0,0,-26,0,-2,0,0,-26,0,-14] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$15.3712023891$
Analytic rank:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} + 27x^{12} + 283x^{10} + 1442x^{8} + 3659x^{6} + 4315x^{4} + 2225x^{2} + 400 $ x^14 + 27x^12 + 283x^10 + 1442x^8 + 3659x^6 + 4315x^4 + 2225x^2 + 400
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1849.8
Root			$0.633891i$ of defining polynomial
Character	$\chi$	$=$	1925.1849
Dual form			1925.2.b.q.1849.7

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+0.633891i q^{2} +0.425088i q^{3} +1.59818 q^{4} -0.269459 q^{6} +1.00000i q^{7} +2.28085i q^{8} +2.81930 q^{9} -1.00000 q^{11} +0.679368i q^{12} -4.61413i q^{13} -0.633891 q^{14} +1.75055 q^{16} -0.0874635i q^{17} +1.78713i q^{18} +2.76975 q^{19} -0.425088 q^{21} -0.633891i q^{22} +4.36257i q^{23} -0.969564 q^{24} +2.92485 q^{26} +2.47371i q^{27} +1.59818i q^{28} +0.498559 q^{29} +7.10325 q^{31} +5.67137i q^{32} -0.425088i q^{33} +0.0554423 q^{34} +4.50576 q^{36} -1.50274i q^{37} +1.75572i q^{38} +1.96141 q^{39} +6.97670 q^{41} -0.269459i q^{42} -5.66659i q^{43} -1.59818 q^{44} -2.76539 q^{46} +0.269459i q^{47} +0.744139i q^{48} -1.00000 q^{49} +0.0371797 q^{51} -7.37421i q^{52} +7.48460i q^{53} -1.56806 q^{54} -2.28085 q^{56} +1.17739i q^{57} +0.316032i q^{58} -11.7082 q^{59} +4.43160 q^{61} +4.50269i q^{62} +2.81930i q^{63} -0.0939234 q^{64} +0.269459 q^{66} +13.4088i q^{67} -0.139783i q^{68} -1.85448 q^{69} -4.40847 q^{71} +6.43041i q^{72} -5.93802i q^{73} +0.952571 q^{74} +4.42657 q^{76} -1.00000i q^{77} +1.24332i q^{78} -11.4349 q^{79} +7.40635 q^{81} +4.42246i q^{82} -2.39958i q^{83} -0.679368 q^{84} +3.59200 q^{86} +0.211931i q^{87} -2.28085i q^{88} +2.80623 q^{89} +4.61413 q^{91} +6.97219i q^{92} +3.01951i q^{93} -0.170808 q^{94} -2.41083 q^{96} +0.504324i q^{97} -0.633891i q^{98} -2.81930 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 26 q^{4} - 2 q^{6} - 26 q^{9} - 14 q^{11} - 2 q^{14} + 58 q^{16} - 36 q^{19} + 44 q^{24} + 26 q^{26} + 4 q^{29} + 48 q^{31} - 66 q^{34} + 88 q^{36} - 20 q^{39} - 20 q^{41} + 26 q^{44} + 6 q^{46}+ \cdots + 26 q^{99}+O(q^{100}) $ 14 * q - 26 * q^4 - 2 * q^6 - 26 * q^9 - 14 * q^11 - 2 * q^14 + 58 * q^16 - 36 * q^19 + 44 * q^24 + 26 * q^26 + 4 * q^29 + 48 * q^31 - 66 * q^34 + 88 * q^36 - 20 * q^39 - 20 * q^41 + 26 * q^44 + 6 * q^46 - 14 * q^49 + 58 * q^51 + 94 * q^54 + 12 * q^56 - 2 * q^59 + 56 * q^61 - 96 * q^64 + 2 * q^66 - 66 * q^69 - 20 * q^71 + 12 * q^74 + 40 * q^76 - 38 * q^79 + 54 * q^81 + 42 * q^84 + 110 * q^86 + 4 * q^89 - 6 * q^91 - 42 * q^94 - 46 * q^96 + 26 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1925\mathbb{Z}\right)^\times$.

$n$	$276$	$1002$	$1751$
$\chi(n)$	$1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0.633891i	0.448228i	0.974563	+	0.224114i	$0.0719489\pi$
$2$	0.633891i	0.448228i	−0.974563	+	0.224114i	$0.928051\pi$
$3$	0.425088i	0.245425i	0.992442	+	0.122712i	$0.0391593\pi$
$3$	0.425088i	0.245425i	−0.992442	+	0.122712i	$0.960841\pi$
$4$	1.59818	0.799091
$5$	0	0
$5$	0	0
$6$	−0.269459	−0.110006
$7$	1.00000i	0.377964i
$7$	1.00000i	0.377964i
$8$	2.28085i	0.806404i
$9$	2.81930	0.939767
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0.679368i	0.196117i
$13$	− 4.61413i	− 1.27973i	−0.768488	−	0.639864i	$-0.778991\pi$
$13$	− 4.61413i	− 1.27973i	0.768488	−	0.639864i	$-0.221009\pi$
$14$	−0.633891	−0.169414
$15$	0	0
$16$	1.75055	0.437638
$17$	− 0.0874635i	− 0.0212130i	−0.999944	−	0.0106065i	$-0.996624\pi$
$17$	− 0.0874635i	− 0.0212130i	0.999944	−	0.0106065i	$-0.00337622\pi$
$18$	1.78713i	0.421230i
$19$	2.76975	0.635425	0.317713	−	0.948187i	$-0.397085\pi$
$19$	2.76975	0.635425	0.317713	+	0.948187i	$0.397085\pi$
$20$	0	0
$21$	−0.425088	−0.0927618
$22$	− 0.633891i	− 0.135146i
$23$	4.36257i	0.909659i	0.890578	+	0.454830i	$0.150300\pi$
$23$	4.36257i	0.909659i	−0.890578	+	0.454830i	$0.849700\pi$
$24$	−0.969564	−0.197911
$25$	0	0
$26$	2.92485	0.573611
$27$	2.47371i	0.476067i
$28$	1.59818i	0.302028i
$29$	0.498559	0.0925800	0.0462900	−	0.998928i	$-0.485260\pi$
$29$	0.498559	0.0925800	0.0462900	+	0.998928i	$0.485260\pi$
$30$	0	0
$31$	7.10325	1.27578	0.637891	−	0.770127i	$-0.279807\pi$
$31$	7.10325	1.27578	0.637891	+	0.770127i	$0.279807\pi$
$32$	5.67137i	1.00257i
$33$	− 0.425088i	− 0.0739983i
$34$	0.0554423	0.00950828
$35$	0	0
$36$	4.50576	0.750959
$37$	− 1.50274i	− 0.247048i	−0.992342	−	0.123524i	$-0.960580\pi$
$37$	− 1.50274i	− 0.247048i	0.992342	−	0.123524i	$-0.0394197\pi$
$38$	1.75572i	0.284816i
$39$	1.96141	0.314077
$40$	0	0
$41$	6.97670	1.08958	0.544789	−	0.838573i	$-0.316610\pi$
$41$	6.97670	1.08958	0.544789	+	0.838573i	$0.316610\pi$
$42$	− 0.269459i	− 0.0415785i
$43$	− 5.66659i	− 0.864147i	−0.901838	−	0.432074i	$-0.857782\pi$
$43$	− 5.66659i	− 0.864147i	0.901838	−	0.432074i	$-0.142218\pi$
$44$	−1.59818	−0.240935
$45$	0	0
$46$	−2.76539	−0.407735
$47$	0.269459i	0.0393047i	0.999807	+	0.0196523i	$0.00625594\pi$
$47$	0.269459i	0.0393047i	−0.999807	+	0.0196523i	$0.993744\pi$
$48$	0.744139i	0.107407i
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0.0371797	0.00520620
$52$	− 7.37421i	− 1.02262i
$53$	7.48460i	1.02809i	0.857764	+	0.514044i	$0.171853\pi$
$53$	7.48460i	1.02809i	−0.857764	+	0.514044i	$0.828147\pi$
$54$	−1.56806	−0.213387
$55$	0	0
$56$	−2.28085	−0.304792
$57$	1.17739i	0.155949i
$58$	0.316032i	0.0414970i
$59$	−11.7082	−1.52428	−0.762142	−	0.647410i	$-0.775852\pi$
$59$	−11.7082	−1.52428	−0.762142	+	0.647410i	$0.775852\pi$
$60$	0	0
$61$	4.43160	0.567408	0.283704	−	0.958912i	$-0.408437\pi$
$61$	4.43160	0.567408	0.283704	+	0.958912i	$0.408437\pi$
$62$	4.50269i	0.571842i
$63$	2.81930i	0.355198i
$64$	−0.0939234	−0.0117404
$65$	0	0
$66$	0.269459	0.0331682
$67$	13.4088i	1.63815i	0.573687	+	0.819075i	$0.305513\pi$
$67$	13.4088i	1.63815i	−0.573687	+	0.819075i	$0.694487\pi$
$68$	− 0.139783i	− 0.0169511i
$69$	−1.85448	−0.223253
$70$	0	0
$71$	−4.40847	−0.523189	−0.261595	−	0.965178i	$-0.584248\pi$
$71$	−4.40847	−0.523189	−0.261595	+	0.965178i	$0.584248\pi$
$72$	6.43041i	0.757832i
$73$	− 5.93802i	− 0.694992i	−0.937681	−	0.347496i	$-0.887032\pi$
$73$	− 5.93802i	− 0.694992i	0.937681	−	0.347496i	$-0.112968\pi$
$74$	0.952571	0.110734
$75$	0	0
$76$	4.42657	0.507763
$77$	− 1.00000i	− 0.113961i
$78$	1.24332i	0.140778i
$79$	−11.4349	−1.28652	−0.643262	−	0.765646i	$-0.722419\pi$
$79$	−11.4349	−1.28652	−0.643262	+	0.765646i	$0.722419\pi$
$80$	0	0
$81$	7.40635	0.822928
$82$	4.42246i	0.488380i
$83$	− 2.39958i	− 0.263388i	−0.991290	−	0.131694i	$-0.957958\pi$
$83$	− 2.39958i	− 0.263388i	0.991290	−	0.131694i	$-0.0420416\pi$
$84$	−0.679368	−0.0741251
$85$	0	0
$86$	3.59200	0.387335
$87$	0.211931i	0.0227214i
$88$	− 2.28085i	− 0.243140i
$89$	2.80623	0.297459	0.148730	−	0.988878i	$-0.452482\pi$
$89$	2.80623	0.297459	0.148730	+	0.988878i	$0.452482\pi$
$90$	0	0
$91$	4.61413	0.483692
$92$	6.97219i	0.726901i
$93$	3.01951i	0.313108i
$94$	−0.170808	−0.0176175
$95$	0	0
$96$	−2.41083	−0.246054
$97$	0.504324i	0.0512063i	0.999672	+	0.0256032i	$0.00815063\pi$
$97$	0.504324i	0.0512063i	−0.999672	+	0.0256032i	$0.991849\pi$
$98$	− 0.633891i	− 0.0640326i
$99$	−2.81930	−0.283350
$100$	0	0
$101$	15.9406	1.58615	0.793075	−	0.609124i	$-0.208479\pi$
$101$	15.9406	1.58615	0.793075	+	0.609124i	$0.208479\pi$
$102$	0.0235679i	0.00233357i
$103$	− 1.37151i	− 0.135139i	−0.997715	−	0.0675693i	$-0.978476\pi$
$103$	− 1.37151i	− 0.135139i	0.997715	−	0.0675693i	$-0.0215244\pi$
$104$	10.5241	1.03198
$105$	0	0
$106$	−4.74442	−0.460819
$107$	− 13.9079i	− 1.34452i	−0.740314	−	0.672262i	$-0.765323\pi$
$107$	− 13.9079i	− 1.34452i	0.740314	−	0.672262i	$-0.234677\pi$
$108$	3.95345i	0.380421i
$109$	−11.6899	−1.11969	−0.559845	−	0.828597i	$-0.689139\pi$
$109$	−11.6899	−1.11969	−0.559845	+	0.828597i	$0.689139\pi$
$110$	0	0
$111$	0.638796	0.0606318
$112$	1.75055i	0.165412i
$113$	7.87792i	0.741093i	0.928814	+	0.370546i	$0.120830\pi$
$113$	7.87792i	0.741093i	−0.928814	+	0.370546i	$0.879170\pi$
$114$	−0.746336	−0.0699008
$115$	0	0
$116$	0.796788	0.0739799
$117$	− 13.0086i	− 1.20265i
$118$	− 7.42175i	− 0.683227i
$119$	0.0874635	0.00801777
$120$	0	0
$121$	1.00000	0.0909091
$122$	2.80915i	0.254328i
$123$	2.96571i	0.267409i
$124$	11.3523	1.01947
$125$	0	0
$126$	−1.78713	−0.159210
$127$	3.84487i	0.341177i	0.985342	+	0.170589i	$0.0545669\pi$
$127$	3.84487i	0.341177i	−0.985342	+	0.170589i	$0.945433\pi$
$128$	11.2832i	0.997303i
$129$	2.40880	0.212083
$130$	0	0
$131$	−1.35840	−0.118684	−0.0593420	−	0.998238i	$-0.518900\pi$
$131$	−1.35840	−0.118684	−0.0593420	+	0.998238i	$0.518900\pi$
$132$	− 0.679368i	− 0.0591314i
$133$	2.76975i	0.240168i
$134$	−8.49973	−0.734265
$135$	0	0
$136$	0.199492	0.0171063
$137$	2.25610i	0.192752i	0.995345	+	0.0963759i	$0.0307251\pi$
$137$	2.25610i	0.192752i	−0.995345	+	0.0963759i	$0.969275\pi$
$138$	− 1.17554i	− 0.100068i
$139$	−0.275141	−0.0233371	−0.0116686	−	0.999932i	$-0.503714\pi$
$139$	−0.275141	−0.0233371	−0.0116686	+	0.999932i	$0.503714\pi$
$140$	0	0
$141$	−0.114544	−0.00964634
$142$	− 2.79449i	− 0.234508i
$143$	4.61413i	0.385853i
$144$	4.93533	0.411278
$145$	0	0
$146$	3.76405	0.311515
$147$	− 0.425088i	− 0.0350607i
$148$	− 2.40165i	− 0.197414i
$149$	3.21776	0.263609	0.131804	−	0.991276i	$-0.457923\pi$
$149$	3.21776	0.263609	0.131804	+	0.991276i	$0.457923\pi$
$150$	0	0
$151$	12.9470	1.05361	0.526805	−	0.849986i	$-0.323390\pi$
$151$	12.9470	1.05361	0.526805	+	0.849986i	$0.323390\pi$
$152$	6.31741i	0.512409i
$153$	− 0.246586i	− 0.0199353i
$154$	0.633891	0.0510804
$155$	0	0
$156$	3.13469	0.250976
$157$	− 19.8615i	− 1.58512i	−0.609794	−	0.792560i	$-0.708748\pi$
$157$	− 19.8615i	− 1.58512i	0.609794	−	0.792560i	$-0.291252\pi$
$158$	− 7.24846i	− 0.576656i
$159$	−3.18161	−0.252318
$160$	0	0
$161$	−4.36257	−0.343819
$162$	4.69482i	0.368860i
$163$	14.3543i	1.12431i	0.827031	+	0.562157i	$0.190028\pi$
$163$	14.3543i	1.12431i	−0.827031	+	0.562157i	$0.809972\pi$
$164$	11.1500	0.870672
$165$	0	0
$166$	1.52107	0.118058
$167$	23.2790i	1.80138i	0.434457	+	0.900692i	$0.356940\pi$
$167$	23.2790i	1.80138i	−0.434457	+	0.900692i	$0.643060\pi$
$168$	− 0.969564i	− 0.0748035i
$169$	−8.29015	−0.637704
$170$	0	0
$171$	7.80877	0.597151
$172$	− 9.05625i	− 0.690533i
$173$	10.2501i	0.779303i	0.920962	+	0.389651i	$0.127405\pi$
$173$	10.2501i	0.779303i	−0.920962	+	0.389651i	$0.872595\pi$
$174$	−0.134341	−0.0101844
$175$	0	0
$176$	−1.75055	−0.131953
$177$	− 4.97703i	− 0.374097i
$178$	1.77884i	0.133330i
$179$	−22.5775	−1.68752	−0.843761	−	0.536718i	$-0.819664\pi$
$179$	−22.5775	−1.68752	−0.843761	+	0.536718i	$0.819664\pi$
$180$	0	0
$181$	−13.7545	−1.02236	−0.511181	−	0.859473i	$-0.670792\pi$
$181$	−13.7545	−1.02236	−0.511181	+	0.859473i	$0.670792\pi$
$182$	2.92485i	0.216804i
$183$	1.88382i	0.139256i
$184$	−9.95039	−0.733553
$185$	0	0
$186$	−1.91404	−0.140344
$187$	0.0874635i	0.00639597i
$188$	0.430645i	0.0314080i
$189$	−2.47371	−0.179936
$190$	0	0
$191$	4.74700	0.343481	0.171741	−	0.985142i	$-0.445061\pi$
$191$	4.74700	0.343481	0.171741	+	0.985142i	$0.445061\pi$
$192$	− 0.0399257i	− 0.00288139i
$193$	− 15.6899i	− 1.12938i	−0.825301	−	0.564692i	$-0.808995\pi$
$193$	− 15.6899i	− 1.12938i	0.825301	−	0.564692i	$-0.191005\pi$
$194$	−0.319686	−0.0229521
$195$	0	0
$196$	−1.59818	−0.114156
$197$	− 6.40911i	− 0.456630i	−0.973587	−	0.228315i	$-0.926678\pi$
$197$	− 6.40911i	− 0.456630i	0.973587	−	0.228315i	$-0.0733217\pi$
$198$	− 1.78713i	− 0.127006i
$199$	10.4872	0.743419	0.371710	−	0.928349i	$-0.378772\pi$
$199$	10.4872	0.743419	0.371710	+	0.928349i	$0.378772\pi$
$200$	0	0
$201$	−5.69993	−0.402042
$202$	10.1046i	0.710957i
$203$	0.498559i	0.0349920i
$204$	0.0594199	0.00416023
$205$	0	0
$206$	0.869385	0.0605729
$207$	12.2994i	0.854868i
$208$	− 8.07727i	− 0.560058i
$209$	−2.76975	−0.191588
$210$	0	0
$211$	−9.57881	−0.659432	−0.329716	−	0.944080i	$-0.606953\pi$
$211$	−9.57881	−0.659432	−0.329716	+	0.944080i	$0.606953\pi$
$212$	11.9618i	0.821537i
$213$	− 1.87399i	− 0.128404i
$214$	8.81606	0.602654
$215$	0	0
$216$	−5.64218	−0.383902
$217$	7.10325i	0.482200i
$218$	− 7.41013i	− 0.501877i
$219$	2.52418	0.170568
$220$	0	0
$221$	−0.403568	−0.0271469
$222$	0.404927i	0.0271769i
$223$	10.0219i	0.671115i	0.942020	+	0.335558i	$0.108925\pi$
$223$	10.0219i	0.671115i	−0.942020	+	0.335558i	$0.891075\pi$
$224$	−5.67137	−0.378934
$225$	0	0
$226$	−4.99374	−0.332179
$227$	− 6.51139i	− 0.432176i	−0.976374	−	0.216088i	$-0.930670\pi$
$227$	− 6.51139i	− 0.432176i	0.976374	−	0.216088i	$-0.0693299\pi$
$228$	1.88168i	0.124617i
$229$	−28.5066	−1.88377	−0.941886	−	0.335934i	$-0.890948\pi$
$229$	−28.5066	−1.88377	−0.941886	+	0.335934i	$0.890948\pi$
$230$	0	0
$231$	0.425088	0.0279687
$232$	1.13714i	0.0746569i
$233$	− 13.7790i	− 0.902693i	−0.892349	−	0.451346i	$-0.850944\pi$
$233$	− 13.7790i	− 0.902693i	0.892349	−	0.451346i	$-0.149056\pi$
$234$	8.24603	0.539060
$235$	0	0
$236$	−18.7119	−1.21804
$237$	− 4.86083i	− 0.315745i
$238$	0.0554423i	0.00359379i
$239$	−11.9099	−0.770389	−0.385195	−	0.922835i	$-0.625866\pi$
$239$	−11.9099	−0.770389	−0.385195	+	0.922835i	$0.625866\pi$
$240$	0	0
$241$	7.72250	0.497450	0.248725	−	0.968574i	$-0.419989\pi$
$241$	7.72250	0.497450	0.248725	+	0.968574i	$0.419989\pi$
$242$	0.633891i	0.0407480i
$243$	10.5695i	0.678034i
$244$	7.08250	0.453411
$245$	0	0
$246$	−1.87994	−0.119860
$247$	− 12.7800i	− 0.813171i
$248$	16.2015i	1.02880i
$249$	1.02003	0.0646419
$250$	0	0
$251$	−7.54774	−0.476409	−0.238205	−	0.971215i	$-0.576559\pi$
$251$	−7.54774	−0.476409	−0.238205	+	0.971215i	$0.576559\pi$
$252$	4.50576i	0.283836i
$253$	− 4.36257i	− 0.274273i
$254$	−2.43723	−0.152925
$255$	0	0
$256$	−7.34016	−0.458760
$257$	− 16.9146i	− 1.05511i	−0.849522	−	0.527553i	$-0.823110\pi$
$257$	− 16.9146i	− 1.05511i	0.849522	−	0.527553i	$-0.176890\pi$
$258$	1.52692i	0.0950617i
$259$	1.50274	0.0933755
$260$	0	0
$261$	1.40559	0.0870036
$262$	− 0.861077i	− 0.0531975i
$263$	− 11.0615i	− 0.682082i	−0.940048	−	0.341041i	$-0.889220\pi$
$263$	− 11.0615i	− 0.682082i	0.940048	−	0.341041i	$-0.110780\pi$
$264$	0.969564	0.0596725
$265$	0	0
$266$	−1.75572	−0.107650
$267$	1.19289i	0.0730039i
$268$	21.4298i	1.30903i
$269$	27.2259	1.65999	0.829997	−	0.557768i	$-0.188342\pi$
$269$	27.2259	1.65999	0.829997	+	0.557768i	$0.188342\pi$
$270$	0	0
$271$	−14.4274	−0.876405	−0.438203	−	0.898876i	$-0.644385\pi$
$271$	−14.4274	−0.876405	−0.438203	+	0.898876i	$0.644385\pi$
$272$	− 0.153109i	− 0.00928363i
$273$	1.96141i	0.118710i
$274$	−1.43012	−0.0863968
$275$	0	0
$276$	−2.96379	−0.178399
$277$	− 3.80344i	− 0.228527i	−0.993450	−	0.114263i	$-0.963549\pi$
$277$	− 3.80344i	− 0.228527i	0.993450	−	0.114263i	$-0.0364508\pi$
$278$	− 0.174409i	− 0.0104604i
$279$	20.0262	1.19894
$280$	0	0
$281$	−20.3505	−1.21401	−0.607005	−	0.794698i	$-0.707629\pi$
$281$	−20.3505	−1.21401	−0.607005	+	0.794698i	$0.707629\pi$
$282$	− 0.0726083i	− 0.00432376i
$283$	4.21465i	0.250535i	0.992123	+	0.125267i	$0.0399789\pi$
$283$	4.21465i	0.250535i	−0.992123	+	0.125267i	$0.960021\pi$
$284$	−7.04554	−0.418076
$285$	0	0
$286$	−2.92485	−0.172950
$287$	6.97670i	0.411821i
$288$	15.9893i	0.942178i
$289$	16.9924	0.999550
$290$	0	0
$291$	−0.214382	−0.0125673
$292$	− 9.49003i	− 0.555362i
$293$	− 33.1882i	− 1.93887i	−0.245340	−	0.969437i	$-0.578900\pi$
$293$	− 33.1882i	− 1.93887i	0.245340	−	0.969437i	$-0.421100\pi$
$294$	0.269459	0.0157152
$295$	0	0
$296$	3.42753	0.199221
$297$	− 2.47371i	− 0.143539i
$298$	2.03971i	0.118157i
$299$	20.1295	1.16412
$300$	0	0
$301$	5.66659	0.326617
$302$	8.20697i	0.472258i
$303$	6.77616i	0.389280i
$304$	4.84860	0.278086
$305$	0	0
$306$	0.156309	0.00893556
$307$	− 24.5918i	− 1.40353i	−0.712409	−	0.701765i	$-0.752396\pi$
$307$	− 24.5918i	− 1.40353i	0.712409	−	0.701765i	$-0.247604\pi$
$308$	− 1.59818i	− 0.0910649i
$309$	0.583011	0.0331663
$310$	0	0
$311$	6.42367	0.364253	0.182126	−	0.983275i	$-0.441702\pi$
$311$	6.42367	0.364253	0.182126	+	0.983275i	$0.441702\pi$
$312$	4.47369i	0.253273i
$313$	− 5.05001i	− 0.285443i	−0.989763	−	0.142722i	$-0.954415\pi$
$313$	− 5.05001i	− 0.285443i	0.989763	−	0.142722i	$-0.0455854\pi$
$314$	12.5900	0.710496
$315$	0	0
$316$	−18.2750	−1.02805
$317$	12.6498i	0.710481i	0.934775	+	0.355240i	$0.115601\pi$
$317$	12.6498i	0.710481i	−0.934775	+	0.355240i	$0.884399\pi$
$318$	− 2.01679i	− 0.113096i
$319$	−0.498559	−0.0279139
$320$	0	0
$321$	5.91206	0.329979
$322$	− 2.76539i	− 0.154109i
$323$	− 0.242252i	− 0.0134793i
$324$	11.8367	0.657595
$325$	0	0
$326$	−9.09905	−0.503950
$327$	− 4.96924i	− 0.274800i
$328$	15.9128i	0.878639i
$329$	−0.269459	−0.0148558
$330$	0	0
$331$	31.6097	1.73743	0.868714	−	0.495314i	$-0.164947\pi$
$331$	31.6097	1.73743	0.868714	+	0.495314i	$0.164947\pi$
$332$	− 3.83496i	− 0.210471i
$333$	− 4.23667i	− 0.232168i
$334$	−14.7564	−0.807432
$335$	0	0
$336$	−0.744139	−0.0405961
$337$	− 15.8472i	− 0.863251i	−0.902053	−	0.431625i	$-0.857940\pi$
$337$	− 15.8472i	− 0.863251i	0.902053	−	0.431625i	$-0.142060\pi$
$338$	− 5.25505i	− 0.285837i
$339$	−3.34881	−0.181882
$340$	0	0
$341$	−7.10325	−0.384663
$342$	4.94991i	0.267660i
$343$	− 1.00000i	− 0.0539949i
$344$	12.9247	0.696852
$345$	0	0
$346$	−6.49746	−0.349306
$347$	8.60284i	0.461825i	0.972975	+	0.230912i	$0.0741711\pi$
$347$	8.60284i	0.461825i	−0.972975	+	0.230912i	$0.925829\pi$
$348$	0.338705i	0.0181565i
$349$	−31.7291	−1.69842	−0.849209	−	0.528057i	$-0.822921\pi$
$349$	−31.7291	−1.69842	−0.849209	+	0.528057i	$0.822921\pi$
$350$	0	0
$351$	11.4140	0.609236
$352$	− 5.67137i	− 0.302285i
$353$	17.4824i	0.930494i	0.885181	+	0.465247i	$0.154035\pi$
$353$	17.4824i	0.930494i	−0.885181	+	0.465247i	$0.845965\pi$
$354$	3.15490	0.167681
$355$	0	0
$356$	4.48486	0.237697
$357$	0.0371797i	0.00196776i
$358$	− 14.3117i	− 0.756396i
$359$	29.4116	1.55229	0.776143	−	0.630557i	$-0.217174\pi$
$359$	29.4116	1.55229	0.776143	+	0.630557i	$0.217174\pi$
$360$	0	0
$361$	−11.3285	−0.596235
$362$	− 8.71884i	− 0.458252i
$363$	0.425088i	0.0223113i
$364$	7.37421	0.386514
$365$	0	0
$366$	−1.19414	−0.0624185
$367$	− 29.0423i	− 1.51600i	−0.652256	−	0.757999i	$-0.726177\pi$
$367$	− 29.0423i	− 1.51600i	0.652256	−	0.757999i	$-0.273823\pi$
$368$	7.63691i	0.398102i
$369$	19.6694	1.02395
$370$	0	0
$371$	−7.48460	−0.388581
$372$	4.82572i	0.250202i
$373$	− 22.5467i	− 1.16743i	−0.811960	−	0.583713i	$-0.801599\pi$
$373$	− 22.5467i	− 1.16743i	0.811960	−	0.583713i	$-0.198401\pi$
$374$	−0.0554423	−0.00286685
$375$	0	0
$376$	−0.614598	−0.0316955
$377$	− 2.30041i	− 0.118477i
$378$	− 1.56806i	− 0.0806526i
$379$	−23.7421	−1.21955	−0.609775	−	0.792574i	$-0.708740\pi$
$379$	−23.7421	−1.21955	−0.609775	+	0.792574i	$0.708740\pi$
$380$	0	0
$381$	−1.63441	−0.0837333
$382$	3.00908i	0.153958i
$383$	− 4.69277i	− 0.239790i	−0.992787	−	0.119895i	$-0.961744\pi$
$383$	− 4.69277i	− 0.239790i	0.992787	−	0.119895i	$-0.0382557\pi$
$384$	−4.79635	−0.244763
$385$	0	0
$386$	9.94569	0.506222
$387$	− 15.9758i	− 0.812097i
$388$	0.806001i	0.0409185i
$389$	−13.3329	−0.676007	−0.338003	−	0.941145i	$-0.609752\pi$
$389$	−13.3329	−0.676007	−0.338003	+	0.941145i	$0.609752\pi$
$390$	0	0
$391$	0.381566	0.0192966
$392$	− 2.28085i	− 0.115201i
$393$	− 0.577439i	− 0.0291280i
$394$	4.06268	0.204675
$395$	0	0
$396$	−4.50576	−0.226423
$397$	− 18.4351i	− 0.925229i	−0.886560	−	0.462614i	$-0.846911\pi$
$397$	− 18.4351i	− 0.925229i	0.886560	−	0.462614i	$-0.153089\pi$
$398$	6.64775i	0.333222i
$399$	−1.17739	−0.0589432
$400$	0	0
$401$	−24.8247	−1.23968	−0.619842	−	0.784726i	$-0.712803\pi$
$401$	−24.8247	−1.23968	−0.619842	+	0.784726i	$0.712803\pi$
$402$	− 3.61313i	− 0.180207i
$403$	− 32.7753i	− 1.63265i
$404$	25.4760	1.26748
$405$	0	0
$406$	−0.316032	−0.0156844
$407$	1.50274i	0.0744879i
$408$	0.0848015i	0.00419830i
$409$	22.1055	1.09305	0.546524	−	0.837443i	$-0.315951\pi$
$409$	22.1055	1.09305	0.546524	+	0.837443i	$0.315951\pi$
$410$	0	0
$411$	−0.959042	−0.0473060
$412$	− 2.19192i	− 0.107988i
$413$	− 11.7082i	− 0.576125i
$414$	−7.79648	−0.383176
$415$	0	0
$416$	26.1684	1.28301
$417$	− 0.116959i	− 0.00572751i
$418$	− 1.75572i	− 0.0858751i
$419$	−29.4106	−1.43680	−0.718401	−	0.695629i	$-0.755126\pi$
$419$	−29.4106	−1.43680	−0.718401	+	0.695629i	$0.755126\pi$
$420$	0	0
$421$	−22.9317	−1.11762	−0.558812	−	0.829295i	$-0.688743\pi$
$421$	−22.9317	−1.11762	−0.558812	+	0.829295i	$0.688743\pi$
$422$	− 6.07192i	− 0.295576i
$423$	0.759687i	0.0369372i
$424$	−17.0713	−0.829055
$425$	0	0
$426$	1.18790	0.0575541
$427$	4.43160i	0.214460i
$428$	− 22.2273i	− 1.07440i
$429$	−1.96141	−0.0946977
$430$	0	0
$431$	31.0474	1.49550	0.747750	−	0.663981i	$-0.231134\pi$
$431$	31.0474	1.49550	0.747750	+	0.663981i	$0.231134\pi$
$432$	4.33037i	0.208345i
$433$	− 27.3772i	− 1.31566i	−0.753165	−	0.657832i	$-0.771474\pi$
$433$	− 27.3772i	− 1.31566i	0.753165	−	0.657832i	$-0.228526\pi$
$434$	−4.50269	−0.216136
$435$	0	0
$436$	−18.6826	−0.894735
$437$	12.0833i	0.578020i
$438$	1.60005i	0.0764535i
$439$	−5.43727	−0.259507	−0.129753	−	0.991546i	$-0.541419\pi$
$439$	−5.43727	−0.259507	−0.129753	+	0.991546i	$0.541419\pi$
$440$	0	0
$441$	−2.81930	−0.134252
$442$	− 0.255818i	− 0.0121680i
$443$	− 10.3375i	− 0.491148i	−0.969378	−	0.245574i	$-0.921024\pi$
$443$	− 10.3375i	− 0.491148i	0.969378	−	0.245574i	$-0.0789765\pi$
$444$	1.02091	0.0484503
$445$	0	0
$446$	−6.35278	−0.300813
$447$	1.36783i	0.0646961i
$448$	− 0.0939234i	− 0.00443747i
$449$	−17.5398	−0.827754	−0.413877	−	0.910333i	$-0.635826\pi$
$449$	−17.5398	−0.827754	−0.413877	+	0.910333i	$0.635826\pi$
$450$	0	0
$451$	−6.97670	−0.328520
$452$	12.5904i	0.592201i
$453$	5.50360i	0.258582i
$454$	4.12751	0.193714
$455$	0	0
$456$	−2.68545	−0.125758
$457$	− 1.34919i	− 0.0631124i	−0.999502	−	0.0315562i	$-0.989954\pi$
$457$	− 1.34919i	− 0.0631124i	0.999502	−	0.0315562i	$-0.0100463\pi$
$458$	− 18.0701i	− 0.844360i
$459$	0.216360	0.0100988
$460$	0	0
$461$	−30.5881	−1.42463	−0.712316	−	0.701859i	$-0.752354\pi$
$461$	−30.5881	−1.42463	−0.712316	+	0.701859i	$0.752354\pi$
$462$	0.269459i	0.0125364i
$463$	− 40.4250i	− 1.87871i	−0.342949	−	0.939354i	$-0.611426\pi$
$463$	− 40.4250i	− 1.87871i	0.342949	−	0.939354i	$-0.388574\pi$
$464$	0.872753	0.0405165
$465$	0	0
$466$	8.73438	0.404613
$467$	− 26.3158i	− 1.21775i	−0.793266	−	0.608875i	$-0.791621\pi$
$467$	− 26.3158i	− 1.21775i	0.793266	−	0.608875i	$-0.208379\pi$
$468$	− 20.7901i	− 0.961024i
$469$	−13.4088	−0.619162
$470$	0	0
$471$	8.44288	0.389027
$472$	− 26.7048i	− 1.22919i
$473$	5.66659i	0.260550i
$474$	3.08123	0.141526
$475$	0	0
$476$	0.139783	0.00640693
$477$	21.1013i	0.966163i
$478$	− 7.54960i	− 0.345310i
$479$	−3.35349	−0.153225	−0.0766123	−	0.997061i	$-0.524410\pi$
$479$	−3.35349	−0.153225	−0.0766123	+	0.997061i	$0.524410\pi$
$480$	0	0
$481$	−6.93382	−0.316155
$482$	4.89522i	0.222971i
$483$	− 1.85448i	− 0.0843816i
$484$	1.59818	0.0726447
$485$	0	0
$486$	−6.69991	−0.303914
$487$	13.7963i	0.625169i	0.949890	+	0.312585i	$0.101195\pi$
$487$	13.7963i	0.625169i	−0.949890	+	0.312585i	$0.898805\pi$
$488$	10.1078i	0.457560i
$489$	−6.10183	−0.275934
$490$	0	0
$491$	−37.0100	−1.67024	−0.835119	−	0.550070i	$-0.814601\pi$
$491$	−37.0100	−1.67024	−0.835119	+	0.550070i	$0.814601\pi$
$492$	4.73975i	0.213684i
$493$	− 0.0436057i	− 0.00196390i
$494$	8.10112	0.364487
$495$	0	0
$496$	12.4346	0.558331
$497$	− 4.40847i	− 0.197747i
$498$	0.646589i	0.0289743i
$499$	17.0397	0.762803	0.381401	−	0.924410i	$-0.375442\pi$
$499$	17.0397	0.762803	0.381401	+	0.924410i	$0.375442\pi$
$500$	0	0
$501$	−9.89563	−0.442104
$502$	− 4.78444i	− 0.213540i
$503$	− 7.14111i	− 0.318406i	−0.987246	−	0.159203i	$-0.949108\pi$
$503$	− 7.14111i	− 0.318406i	0.987246	−	0.159203i	$-0.0508925\pi$
$504$	−6.43041	−0.286433
$505$	0	0
$506$	2.76539	0.122937
$507$	− 3.52404i	− 0.156508i
$508$	6.14480i	0.272632i
$509$	19.5536	0.866697	0.433348	−	0.901226i	$-0.357332\pi$
$509$	19.5536	0.866697	0.433348	+	0.901226i	$0.357332\pi$
$510$	0	0
$511$	5.93802	0.262682
$512$	17.9135i	0.791674i
$513$	6.85158i	0.302505i
$514$	10.7220	0.472928
$515$	0	0
$516$	3.84970	0.169474
$517$	− 0.269459i	− 0.0118508i
$518$	0.952571i	0.0418536i
$519$	−4.35721	−0.191260
$520$	0	0
$521$	19.9867	0.875635	0.437818	−	0.899064i	$-0.355752\pi$
$521$	19.9867	0.875635	0.437818	+	0.899064i	$0.355752\pi$
$522$	0.890989i	0.0389975i
$523$	20.3762i	0.890990i	0.895284	+	0.445495i	$0.146972\pi$
$523$	20.3762i	0.890990i	−0.895284	+	0.445495i	$0.853028\pi$
$524$	−2.17097	−0.0948393
$525$	0	0
$526$	7.01179	0.305729
$527$	− 0.621275i	− 0.0270632i
$528$	− 0.744139i	− 0.0323845i
$529$	3.96796	0.172520
$530$	0	0
$531$	−33.0091	−1.43247
$532$	4.42657i	0.191916i
$533$	− 32.1914i	− 1.39436i
$534$	−0.756164	−0.0327224
$535$	0	0
$536$	−30.5836	−1.32101
$537$	− 9.59743i	− 0.414160i
$538$	17.2583i	0.744057i
$539$	1.00000	0.0430730
$540$	0	0
$541$	37.7722	1.62395	0.811977	−	0.583689i	$-0.198391\pi$
$541$	37.7722	1.62395	0.811977	+	0.583689i	$0.198391\pi$
$542$	− 9.14543i	− 0.392830i
$543$	− 5.84687i	− 0.250913i
$544$	0.496038	0.0212674
$545$	0	0
$546$	−1.24332	−0.0532092
$547$	− 17.2831i	− 0.738972i	−0.929236	−	0.369486i	$-0.879534\pi$
$547$	− 17.2831i	− 0.738972i	0.929236	−	0.369486i	$-0.120466\pi$
$548$	3.60566i	0.154026i
$549$	12.4940	0.533231
$550$	0	0
$551$	1.38089	0.0588277
$552$	− 4.22979i	− 0.180032i
$553$	− 11.4349i	− 0.486260i
$554$	2.41097	0.102432
$555$	0	0
$556$	−0.439725	−0.0186485
$557$	6.79002i	0.287703i	0.989599	+	0.143851i	$0.0459487\pi$
$557$	6.79002i	0.287703i	−0.989599	+	0.143851i	$0.954051\pi$
$558$	12.6944i	0.537398i
$559$	−26.1464	−1.10587
$560$	0	0
$561$	−0.0371797	−0.00156973
$562$	− 12.9000i	− 0.544154i
$563$	− 4.49862i	− 0.189594i	−0.995497	−	0.0947972i	$-0.969780\pi$
$563$	− 4.49862i	− 0.189594i	0.995497	−	0.0947972i	$-0.0302203\pi$
$564$	−0.183062	−0.00770831
$565$	0	0
$566$	−2.67163	−0.112297
$567$	7.40635i	0.311038i
$568$	− 10.0551i	− 0.421902i
$569$	−27.7691	−1.16414	−0.582070	−	0.813139i	$-0.697757\pi$
$569$	−27.7691	−1.16414	−0.582070	+	0.813139i	$0.697757\pi$
$570$	0	0
$571$	−17.7674	−0.743542	−0.371771	−	0.928325i	$-0.621249\pi$
$571$	−17.7674	−0.743542	−0.371771	+	0.928325i	$0.621249\pi$
$572$	7.37421i	0.308331i
$573$	2.01789i	0.0842987i
$574$	−4.42246	−0.184590
$575$	0	0
$576$	−0.264798	−0.0110333
$577$	− 44.9562i	− 1.87155i	−0.352595	−	0.935776i	$-0.614701\pi$
$577$	− 44.9562i	− 1.87155i	0.352595	−	0.935776i	$-0.385299\pi$
$578$	10.7713i	0.448027i
$579$	6.66959	0.277179
$580$	0	0
$581$	2.39958	0.0995512
$582$	− 0.135895i	− 0.00563302i
$583$	− 7.48460i	− 0.309980i
$584$	13.5438	0.560444
$585$	0	0
$586$	21.0377	0.869059
$587$	17.9592i	0.741255i	0.928782	+	0.370627i	$0.120857\pi$
$587$	17.9592i	0.741255i	−0.928782	+	0.370627i	$0.879143\pi$
$588$	− 0.679368i	− 0.0280167i
$589$	19.6743	0.810664
$590$	0	0
$591$	2.72444	0.112068
$592$	− 2.63062i	− 0.108118i
$593$	− 33.4098i	− 1.37198i	−0.727613	−	0.685988i	$-0.759370\pi$
$593$	− 33.4098i	− 1.37198i	0.727613	−	0.685988i	$-0.240630\pi$
$594$	1.56806	0.0643385
$595$	0	0
$596$	5.14256	0.210648
$597$	4.45799i	0.182453i
$598$	12.7599i	0.521790i
$599$	20.7221	0.846681	0.423341	−	0.905971i	$-0.360857\pi$
$599$	20.7221	0.846681	0.423341	+	0.905971i	$0.360857\pi$
$600$	0	0
$601$	−28.3592	−1.15680	−0.578398	−	0.815755i	$-0.696322\pi$
$601$	−28.3592	−1.15680	−0.578398	+	0.815755i	$0.696322\pi$
$602$	3.59200i	0.146399i
$603$	37.8035i	1.53948i
$604$	20.6916	0.841930
$605$	0	0
$606$	−4.29535	−0.174487
$607$	46.1211i	1.87200i	0.352004	+	0.935998i	$0.385500\pi$
$607$	46.1211i	1.87200i	−0.352004	+	0.935998i	$0.614500\pi$
$608$	15.7083i	0.637055i
$609$	−0.211931	−0.00858789
$610$	0	0
$611$	1.24332	0.0502993
$612$	− 0.394089i	− 0.0159301i
$613$	− 0.542682i	− 0.0219187i	−0.999940	−	0.0109594i	$-0.996511\pi$
$613$	− 0.542682i	− 0.0219187i	0.999940	−	0.0109594i	$-0.00348854\pi$
$614$	15.5885	0.629102
$615$	0	0
$616$	2.28085	0.0918983
$617$	19.0924i	0.768630i	0.923202	+	0.384315i	$0.125562\pi$
$617$	19.0924i	0.768630i	−0.923202	+	0.384315i	$0.874438\pi$
$618$	0.369565i	0.0148661i
$619$	−32.9718	−1.32525	−0.662624	−	0.748952i	$-0.730557\pi$
$619$	−32.9718	−1.32525	−0.662624	+	0.748952i	$0.730557\pi$
$620$	0	0
$621$	−10.7918	−0.433058
$622$	4.07190i	0.163268i
$623$	2.80623i	0.112429i
$624$	3.43355	0.137452
$625$	0	0
$626$	3.20115	0.127944
$627$	− 1.17739i	− 0.0470204i
$628$	− 31.7423i	− 1.26665i
$629$	−0.131435	−0.00524064
$630$	0	0
$631$	29.5879	1.17787	0.588937	−	0.808179i	$-0.299546\pi$
$631$	29.5879	1.17787	0.588937	+	0.808179i	$0.299546\pi$
$632$	− 26.0813i	− 1.03746i
$633$	− 4.07184i	− 0.161841i
$634$	−8.01856	−0.318458
$635$	0	0
$636$	−5.08480	−0.201625
$637$	4.61413i	0.182818i
$638$	− 0.316032i	− 0.0125118i
$639$	−12.4288	−0.491676
$640$	0	0
$641$	−13.8383	−0.546579	−0.273290	−	0.961932i	$-0.588112\pi$
$641$	−13.8383	−0.546579	−0.273290	+	0.961932i	$0.588112\pi$
$642$	3.74760i	0.147906i
$643$	− 9.95758i	− 0.392689i	−0.980535	−	0.196344i	$-0.937093\pi$
$643$	− 9.95758i	− 0.392689i	0.980535	−	0.196344i	$-0.0629070\pi$
$644$	−6.97219	−0.274743
$645$	0	0
$646$	0.153562	0.00604180
$647$	− 28.3400i	− 1.11416i	−0.830459	−	0.557080i	$-0.811921\pi$
$647$	− 28.3400i	− 1.11416i	0.830459	−	0.557080i	$-0.188079\pi$
$648$	16.8928i	0.663613i
$649$	11.7082	0.459589
$650$	0	0
$651$	−3.01951	−0.118344
$652$	22.9408i	0.898430i
$653$	44.6604i	1.74770i	0.486199	+	0.873848i	$0.338383\pi$
$653$	44.6604i	1.74770i	−0.486199	+	0.873848i	$0.661617\pi$
$654$	3.14996	0.123173
$655$	0	0
$656$	12.2131	0.476840
$657$	− 16.7410i	− 0.653131i
$658$	− 0.170808i	− 0.00665878i
$659$	1.54774	0.0602913	0.0301456	−	0.999546i	$-0.490403\pi$
$659$	1.54774	0.0602913	0.0301456	+	0.999546i	$0.490403\pi$
$660$	0	0
$661$	31.8659	1.23944	0.619721	−	0.784822i	$-0.287246\pi$
$661$	31.8659	1.23944	0.619721	+	0.784822i	$0.287246\pi$
$662$	20.0371i	0.778765i
$663$	− 0.171552i	− 0.00666252i
$664$	5.47309	0.212397
$665$	0	0
$666$	2.68558	0.104064
$667$	2.17500i	0.0842163i
$668$	37.2041i	1.43947i
$669$	−4.26019	−0.164708
$670$	0	0
$671$	−4.43160	−0.171080
$672$	− 2.41083i	− 0.0929998i
$673$	26.5399i	1.02304i	0.859272	+	0.511519i	$0.170917\pi$
$673$	26.5399i	1.02304i	−0.859272	+	0.511519i	$0.829083\pi$
$674$	10.0454	0.386934
$675$	0	0
$676$	−13.2492	−0.509584
$677$	16.5717i	0.636904i	0.947939	+	0.318452i	$0.103163\pi$
$677$	16.5717i	0.636904i	−0.947939	+	0.318452i	$0.896837\pi$
$678$	− 2.12278i	− 0.0815249i
$679$	−0.504324	−0.0193542
$680$	0	0
$681$	2.76791	0.106067
$682$	− 4.50269i	− 0.172417i
$683$	3.06152i	0.117146i	0.998283	+	0.0585730i	$0.0186550\pi$
$683$	3.06152i	0.117146i	−0.998283	+	0.0585730i	$0.981345\pi$
$684$	12.4798	0.477178
$685$	0	0
$686$	0.633891	0.0242021
$687$	− 12.1178i	− 0.462324i
$688$	− 9.91967i	− 0.378184i
$689$	34.5349	1.31567
$690$	0	0
$691$	29.7730	1.13262	0.566309	−	0.824193i	$-0.308371\pi$
$691$	29.7730	1.13262	0.566309	+	0.824193i	$0.308371\pi$
$692$	16.3816i	0.622734i
$693$	− 2.81930i	− 0.107096i
$694$	−5.45326	−0.207003
$695$	0	0
$696$	−0.483385	−0.0183226
$697$	− 0.610207i	− 0.0231132i
$698$	− 20.1128i	− 0.761279i
$699$	5.85729	0.221543
$700$	0	0
$701$	2.55517	0.0965076	0.0482538	−	0.998835i	$-0.484634\pi$
$701$	2.55517	0.0965076	0.0482538	+	0.998835i	$0.484634\pi$
$702$	7.23525i	0.273077i
$703$	− 4.16221i	− 0.156981i
$704$	0.0939234	0.00353987
$705$	0	0
$706$	−11.0819	−0.417074
$707$	15.9406i	0.599508i
$708$	− 7.95421i	− 0.298937i
$709$	6.53595	0.245463	0.122731	−	0.992440i	$-0.460835\pi$
$709$	6.53595	0.245463	0.122731	+	0.992440i	$0.460835\pi$
$710$	0	0
$711$	−32.2383	−1.20903
$712$	6.40060i	0.239872i
$713$	30.9885i	1.16053i
$714$	−0.0235679	−0.000882005 0
$715$	0	0
$716$	−36.0830	−1.34848
$717$	− 5.06277i	− 0.189073i
$718$	18.6437i	0.695778i
$719$	−27.4125	−1.02231	−0.511157	−	0.859487i	$-0.670783\pi$
$719$	−27.4125	−1.02231	−0.511157	+	0.859487i	$0.670783\pi$
$720$	0	0
$721$	1.37151	0.0510776
$722$	− 7.18101i	− 0.267249i
$723$	3.28274i	0.122086i
$724$	−21.9822	−0.816961
$725$	0	0
$726$	−0.269459	−0.0100006
$727$	30.1724i	1.11903i	0.828819	+	0.559516i	$0.189013\pi$
$727$	30.1724i	1.11903i	−0.828819	+	0.559516i	$0.810987\pi$
$728$	10.5241i	0.390051i
$729$	17.7261	0.656522
$730$	0	0
$731$	−0.495620	−0.0183312
$732$	3.01069i	0.111278i
$733$	− 14.4616i	− 0.534152i	−0.963675	−	0.267076i	$-0.913942\pi$
$733$	− 14.4616i	− 0.534152i	0.963675	−	0.267076i	$-0.0860575\pi$
$734$	18.4097	0.679513
$735$	0	0
$736$	−24.7418	−0.911993
$737$	− 13.4088i	− 0.493921i
$738$	12.4683i	0.458963i
$739$	−32.9543	−1.21224	−0.606121	−	0.795372i	$-0.707275\pi$
$739$	−32.9543	−1.21224	−0.606121	+	0.795372i	$0.707275\pi$
$740$	0	0
$741$	5.43262	0.199572
$742$	− 4.74442i	− 0.174173i
$743$	− 3.76396i	− 0.138086i	−0.997614	−	0.0690432i	$-0.978005\pi$
$743$	− 3.76396i	− 0.138086i	0.997614	−	0.0690432i	$-0.0219946\pi$
$744$	−6.88706	−0.252492
$745$	0	0
$746$	14.2922	0.523274
$747$	− 6.76513i	− 0.247523i
$748$	0.139783i	0.00511096i
$749$	13.9079	0.508182
$750$	0	0
$751$	45.9444	1.67653	0.838267	−	0.545260i	$-0.183569\pi$
$751$	45.9444	1.67653	0.838267	+	0.545260i	$0.183569\pi$
$752$	0.471703i	0.0172012i
$753$	− 3.20845i	− 0.116923i
$754$	1.45821	0.0531049
$755$	0	0
$756$	−3.95345	−0.143785
$757$	40.3467i	1.46643i	0.679999	+	0.733213i	$0.261980\pi$
$757$	40.3467i	1.46643i	−0.679999	+	0.733213i	$0.738020\pi$
$758$	− 15.0499i	− 0.546637i
$759$	1.85448	0.0673133
$760$	0	0
$761$	−13.6871	−0.496157	−0.248079	−	0.968740i	$-0.579799\pi$
$761$	−13.6871	−0.496157	−0.248079	+	0.968740i	$0.579799\pi$
$762$	− 1.03604i	− 0.0375316i
$763$	− 11.6899i	− 0.423203i
$764$	7.58658	0.274473
$765$	0	0
$766$	2.97471	0.107480
$767$	54.0233i	1.95067i
$768$	− 3.12022i	− 0.112591i
$769$	−38.5474	−1.39006	−0.695028	−	0.718983i	$-0.744608\pi$
$769$	−38.5474	−1.39006	−0.695028	+	0.718983i	$0.744608\pi$
$770$	0	0
$771$	7.19021	0.258949
$772$	− 25.0753i	− 0.902482i
$773$	− 5.51659i	− 0.198418i	−0.995067	−	0.0992090i	$-0.968369\pi$
$773$	− 5.51659i	− 0.198418i	0.995067	−	0.0992090i	$-0.0316312\pi$
$774$	10.1269	0.364005
$775$	0	0
$776$	−1.15029	−0.0412930
$777$	0.638796i	0.0229167i
$778$	− 8.45163i	− 0.303006i
$779$	19.3237	0.692345
$780$	0	0
$781$	4.40847	0.157748
$782$	0.241871i	0.00864930i
$783$	1.23329i	0.0440743i
$784$	−1.75055	−0.0625197
$785$	0	0
$786$	0.366034	0.0130560
$787$	− 29.7792i	− 1.06151i	−0.847524	−	0.530756i	$-0.821908\pi$
$787$	− 29.7792i	− 1.06151i	0.847524	−	0.530756i	$-0.178092\pi$
$788$	− 10.2429i	− 0.364889i
$789$	4.70212	0.167400
$790$	0	0
$791$	−7.87792	−0.280107
$792$	− 6.43041i	− 0.228495i
$793$	− 20.4479i	− 0.726128i
$794$	11.6858	0.414714
$795$	0	0
$796$	16.7605	0.594060
$797$	− 17.9922i	− 0.637315i	−0.947870	−	0.318658i	$-0.896768\pi$
$797$	− 17.9922i	− 0.637315i	0.947870	−	0.318658i	$-0.103232\pi$
$798$	− 0.746336i	− 0.0264200i
$799$	0.0235679	0.000833771 0
$800$	0	0
$801$	7.91160	0.279543
$802$	− 15.7361i	− 0.555662i
$803$	5.93802i	0.209548i
$804$	−9.10953	−0.321268
$805$	0	0
$806$	20.7760	0.731802
$807$	11.5734i	0.407404i
$808$	36.3582i	1.27908i
$809$	−10.4613	−0.367801	−0.183901	−	0.982945i	$-0.558872\pi$
$809$	−10.4613	−0.367801	−0.183901	+	0.982945i	$0.558872\pi$
$810$	0	0
$811$	43.7701	1.53698	0.768489	−	0.639863i	$-0.221009\pi$
$811$	43.7701	1.53698	0.768489	+	0.639863i	$0.221009\pi$
$812$	0.796788i	0.0279618i
$813$	− 6.13293i	− 0.215091i
$814$	−0.952571	−0.0333876
$815$	0	0
$816$	0.0650850	0.00227843
$817$	− 15.6951i	− 0.549101i
$818$	14.0125i	0.489936i
$819$	13.0086	0.454557
$820$	0	0
$821$	−11.8371	−0.413119	−0.206559	−	0.978434i	$-0.566227\pi$
$821$	−11.8371	−0.413119	−0.206559	+	0.978434i	$0.566227\pi$
$822$	− 0.607928i	− 0.0212039i
$823$	46.7082i	1.62815i	0.580762	+	0.814073i	$0.302755\pi$
$823$	46.7082i	1.62815i	−0.580762	+	0.814073i	$0.697245\pi$
$824$	3.12821	0.108976
$825$	0	0
$826$	7.42175	0.258236
$827$	40.2990i	1.40133i	0.713489	+	0.700666i	$0.247114\pi$
$827$	40.2990i	1.40133i	−0.713489	+	0.700666i	$0.752886\pi$
$828$	19.6567i	0.683117i
$829$	−7.42077	−0.257734	−0.128867	−	0.991662i	$-0.541134\pi$
$829$	−7.42077	−0.257734	−0.128867	+	0.991662i	$0.541134\pi$
$830$	0	0
$831$	1.61680	0.0560861
$832$	0.433374i	0.0150246i
$833$	0.0874635i	0.00303043i
$834$	0.0741392	0.00256723
$835$	0	0
$836$	−4.42657	−0.153096
$837$	17.5714i	0.607357i
$838$	− 18.6431i	− 0.644016i
$839$	−23.6372	−0.816046	−0.408023	−	0.912972i	$-0.633782\pi$
$839$	−23.6372	−0.816046	−0.408023	+	0.912972i	$0.633782\pi$
$840$	0	0
$841$	−28.7514	−0.991429
$842$	− 14.5362i	− 0.500951i
$843$	− 8.65076i	− 0.297948i
$844$	−15.3087	−0.526947
$845$	0	0
$846$	−0.481558	−0.0165563
$847$	1.00000i	0.0343604i
$848$	13.1022i	0.449931i
$849$	−1.79160	−0.0614874
$850$	0	0
$851$	6.55580	0.224730
$852$	− 2.99498i	− 0.102606i
$853$	31.5024i	1.07862i	0.842106	+	0.539311i	$0.181315\pi$
$853$	31.5024i	1.07862i	−0.842106	+	0.539311i	$0.818685\pi$
$854$	−2.80915	−0.0961271
$855$	0	0
$856$	31.7218	1.08423
$857$	30.8943i	1.05533i	0.849452	+	0.527665i	$0.176932\pi$
$857$	30.8943i	1.05533i	−0.849452	+	0.527665i	$0.823068\pi$
$858$	− 1.24332i	− 0.0424462i
$859$	−35.8938	−1.22468	−0.612340	−	0.790594i	$-0.709772\pi$
$859$	−35.8938	−1.22468	−0.612340	+	0.790594i	$0.709772\pi$
$860$	0	0
$861$	−2.96571	−0.101071
$862$	19.6806i	0.670325i
$863$	16.3679i	0.557170i	0.960412	+	0.278585i	$0.0898653\pi$
$863$	16.3679i	0.557170i	−0.960412	+	0.278585i	$0.910135\pi$
$864$	−14.0293	−0.477288
$865$	0	0
$866$	17.3542	0.589718
$867$	7.22324i	0.245314i
$868$	11.3523i	0.385322i
$869$	11.4349	0.387901
$870$	0	0
$871$	61.8700	2.09639
$872$	− 26.6630i	− 0.902923i
$873$	1.42184i	0.0481220i
$874$	−7.65946	−0.259085
$875$	0	0
$876$	4.03410	0.136300
$877$	− 19.6026i	− 0.661932i	−0.943643	−	0.330966i	$-0.892625\pi$
$877$	− 19.6026i	− 0.661932i	0.943643	−	0.330966i	$-0.107375\pi$
$878$	− 3.44664i	− 0.116318i
$879$	14.1079	0.475848
$880$	0	0
$881$	19.6500	0.662025	0.331012	−	0.943626i	$-0.392610\pi$
$881$	19.6500	0.662025	0.331012	+	0.943626i	$0.392610\pi$
$882$	− 1.78713i	− 0.0601757i
$883$	− 35.2229i	− 1.18534i	−0.805444	−	0.592672i	$-0.798073\pi$
$883$	− 35.2229i	− 1.18534i	0.805444	−	0.592672i	$-0.201927\pi$
$884$	−0.644975	−0.0216928
$885$	0	0
$886$	6.55283	0.220147
$887$	− 38.4382i	− 1.29063i	−0.763918	−	0.645314i	$-0.776727\pi$
$887$	− 38.4382i	− 1.29063i	0.763918	−	0.645314i	$-0.223273\pi$
$888$	1.45700i	0.0488937i
$889$	−3.84487	−0.128953
$890$	0	0
$891$	−7.40635	−0.248122
$892$	16.0168i	0.536282i
$893$	0.746336i	0.0249752i
$894$	−0.867055	−0.0289986
$895$	0	0
$896$	−11.2832	−0.376945
$897$	8.55679i	0.285703i
$898$	− 11.1183i	− 0.371023i
$899$	3.54139	0.118112
$900$	0	0
$901$	0.654629	0.0218089
$902$	− 4.42246i	− 0.147252i
$903$	2.40880i	0.0801599i
$904$	−17.9684	−0.597620
$905$	0	0
$906$	−3.48868	−0.115904
$907$	− 40.3084i	− 1.33842i	−0.743075	−	0.669209i	$-0.766633\pi$
$907$	− 40.3084i	− 1.33842i	0.743075	−	0.669209i	$-0.233367\pi$
$908$	− 10.4064i	− 0.345348i
$909$	44.9414	1.49061
$910$	0	0
$911$	18.7318	0.620612	0.310306	−	0.950637i	$-0.399568\pi$
$911$	18.7318	0.620612	0.310306	+	0.950637i	$0.399568\pi$
$912$	2.06108i	0.0682492i
$913$	2.39958i	0.0794144i
$914$	0.855238	0.0282888
$915$	0	0
$916$	−45.5588	−1.50531
$917$	− 1.35840i	− 0.0448583i
$918$	0.137148i	0.00452657i
$919$	−21.0980	−0.695958	−0.347979	−	0.937502i	$-0.613132\pi$
$919$	−21.0980	−0.695958	−0.347979	+	0.937502i	$0.613132\pi$
$920$	0	0
$921$	10.4537	0.344461
$922$	− 19.3895i	− 0.638561i
$923$	20.3412i	0.669540i
$924$	0.679368	0.0223496
$925$	0	0
$926$	25.6250	0.842090
$927$	− 3.86669i	− 0.126999i
$928$	2.82751i	0.0928176i
$929$	−42.6410	−1.39901	−0.699503	−	0.714629i	$-0.746595\pi$
$929$	−42.6410	−1.39901	−0.699503	+	0.714629i	$0.746595\pi$
$930$	0	0
$931$	−2.76975	−0.0907750
$932$	− 22.0214i	− 0.721334i
$933$	2.73062i	0.0893966i
$934$	16.6814	0.545831
$935$	0	0
$936$	29.6707	0.969818
$937$	5.15757i	0.168491i	0.996445	+	0.0842453i	$0.0268479\pi$
$937$	5.15757i	0.168491i	−0.996445	+	0.0842453i	$0.973152\pi$
$938$	− 8.49973i	− 0.277526i
$939$	2.14670	0.0700548
$940$	0	0
$941$	−53.5224	−1.74478	−0.872390	−	0.488810i	$-0.837431\pi$
$941$	−53.5224	−1.74478	−0.872390	+	0.488810i	$0.837431\pi$
$942$	5.35186i	0.174373i
$943$	30.4364i	0.991144i
$944$	−20.4959	−0.667084
$945$	0	0
$946$	−3.59200	−0.116786
$947$	16.2031i	0.526530i	0.964724	+	0.263265i	$0.0847994\pi$
$947$	16.2031i	0.526530i	−0.964724	+	0.263265i	$0.915201\pi$
$948$	− 7.76849i	− 0.252309i
$949$	−27.3987	−0.889401
$950$	0	0
$951$	−5.37726	−0.174370
$952$	0.199492i	0.00646556i
$953$	35.2173i	1.14080i	0.821366	+	0.570401i	$0.193212\pi$
$953$	35.2173i	1.14080i	−0.821366	+	0.570401i	$0.806788\pi$
$954$	−13.3759	−0.433062
$955$	0	0
$956$	−19.0342	−0.615611
$957$	− 0.211931i	− 0.00685077i
$958$	− 2.12574i	− 0.0686797i
$959$	−2.25610	−0.0728533
$960$	0	0
$961$	19.4562	0.627619
$962$	− 4.39528i	− 0.141710i
$963$	− 39.2104i	− 1.26354i
$964$	12.3420	0.397508
$965$	0	0
$966$	1.17554	0.0378223
$967$	50.4619i	1.62275i	0.584529	+	0.811373i	$0.301279\pi$
$967$	50.4619i	1.62275i	−0.584529	+	0.811373i	$0.698721\pi$
$968$	2.28085i	0.0733094i
$969$	0.102979	0.00330815
$970$	0	0
$971$	37.8328	1.21411	0.607057	−	0.794659i	$-0.292350\pi$
$971$	37.8328	1.21411	0.607057	+	0.794659i	$0.292350\pi$
$972$	16.8920i	0.541811i
$973$	− 0.275141i	− 0.00882061i
$974$	−8.74534	−0.280219
$975$	0	0
$976$	7.75774	0.248319
$977$	19.6920i	0.630002i	0.949091	+	0.315001i	$0.102005\pi$
$977$	19.6920i	0.630002i	−0.949091	+	0.315001i	$0.897995\pi$
$978$	− 3.86790i	− 0.123682i
$979$	−2.80623	−0.0896874
$980$	0	0
$981$	−32.9574	−1.05225
$982$	− 23.4603i	− 0.748648i
$983$	61.7486i	1.96947i	0.174052	+	0.984736i	$0.444314\pi$
$983$	61.7486i	1.96947i	−0.174052	+	0.984736i	$0.555686\pi$
$984$	−6.76435	−0.215640
$985$	0	0
$986$	0.0276413	0.000880277 0
$987$	− 0.114544i	− 0.00364597i
$988$	− 20.4248i	− 0.649798i
$989$	24.7209	0.786080
$990$	0	0
$991$	27.0205	0.858336	0.429168	−	0.903225i	$-0.358807\pi$
$991$	27.0205	0.858336	0.429168	+	0.903225i	$0.358807\pi$
$992$	40.2852i	1.27905i
$993$	13.4369i	0.426408i
$994$	2.79449	0.0886358
$995$	0	0
$996$	1.63020	0.0516547
$997$	− 49.4068i	− 1.56473i	−0.622821	−	0.782364i	$-0.714014\pi$
$997$	− 49.4068i	− 1.56473i	0.622821	−	0.782364i	$-0.285986\pi$
$998$	10.8013i	0.341910i
$999$	3.71734	0.117612

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1925.2.b.q.1849.8		14
5.2	odd	4		1925.2.a.ba.1.4	✓	7
5.3	odd	4		1925.2.a.bc.1.4	yes	7
5.4	even	2	inner	1925.2.b.q.1849.7		14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1925.2.a.ba.1.4	✓	7	5.2	odd	4
1925.2.a.bc.1.4	yes	7	5.3	odd	4
1925.2.b.q.1849.7		14	5.4	even	2	inner
1925.2.b.q.1849.8		14	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1925 = 5^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1925.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.633891i q^{2} +0.425088i q^{3} +1.59818 q^{4} -0.269459 q^{6} +1.00000i q^{7} +2.28085i q^{8} +2.81930 q^{9} -1.00000 q^{11} +0.679368i q^{12} -4.61413i q^{13} -0.633891 q^{14} +1.75055 q^{16} -0.0874635i q^{17} +1.78713i q^{18} +2.76975 q^{19} -0.425088 q^{21} -0.633891i q^{22} +4.36257i q^{23} -0.969564 q^{24} +2.92485 q^{26} +2.47371i q^{27} +1.59818i q^{28} +0.498559 q^{29} +7.10325 q^{31} +5.67137i q^{32} -0.425088i q^{33} +0.0554423 q^{34} +4.50576 q^{36} -1.50274i q^{37} +1.75572i q^{38} +1.96141 q^{39} +6.97670 q^{41} -0.269459i q^{42} -5.66659i q^{43} -1.59818 q^{44} -2.76539 q^{46} +0.269459i q^{47} +0.744139i q^{48} -1.00000 q^{49} +0.0371797 q^{51} -7.37421i q^{52} +7.48460i q^{53} -1.56806 q^{54} -2.28085 q^{56} +1.17739i q^{57} +0.316032i q^{58} -11.7082 q^{59} +4.43160 q^{61} +4.50269i q^{62} +2.81930i q^{63} -0.0939234 q^{64} +0.269459 q^{66} +13.4088i q^{67} -0.139783i q^{68} -1.85448 q^{69} -4.40847 q^{71} +6.43041i q^{72} -5.93802i q^{73} +0.952571 q^{74} +4.42657 q^{76} -1.00000i q^{77} +1.24332i q^{78} -11.4349 q^{79} +7.40635 q^{81} +4.42246i q^{82} -2.39958i q^{83} -0.679368 q^{84} +3.59200 q^{86} +0.211931i q^{87} -2.28085i q^{88} +2.80623 q^{89} +4.61413 q^{91} +6.97219i q^{92} +3.01951i q^{93} -0.170808 q^{94} -2.41083 q^{96} +0.504324i q^{97} -0.633891i q^{98} -2.81930 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 26 q^{4} - 2 q^{6} - 26 q^{9} - 14 q^{11} - 2 q^{14} + 58 q^{16} - 36 q^{19} + 44 q^{24} + 26 q^{26} + 4 q^{29} + 48 q^{31} - 66 q^{34} + 88 q^{36} - 20 q^{39} - 20 q^{41} + 26 q^{44} + 6 q^{46}+ \cdots + 26 q^{99}+O(q^{100}) \) 14 * q - 26 * q^4 - 2 * q^6 - 26 * q^9 - 14 * q^11 - 2 * q^14 + 58 * q^16 - 36 * q^19 + 44 * q^24 + 26 * q^26 + 4 * q^29 + 48 * q^31 - 66 * q^34 + 88 * q^36 - 20 * q^39 - 20 * q^41 + 26 * q^44 + 6 * q^46 - 14 * q^49 + 58 * q^51 + 94 * q^54 + 12 * q^56 - 2 * q^59 + 56 * q^61 - 96 * q^64 + 2 * q^66 - 66 * q^69 - 20 * q^71 + 12 * q^74 + 40 * q^76 - 38 * q^79 + 54 * q^81 + 42 * q^84 + 110 * q^86 + 4 * q^89 - 6 * q^91 - 42 * q^94 - 46 * q^96 + 26 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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