LMFDB - Embedded newform 1225.2.a.bc.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1225,2,Mod(1,1225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1225, 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("1225.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1225 = 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1225.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:	$9.78167424761$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 6x^{2} + 4 $ x^4 - 6*x^2 + 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.874032$ of defining polynomial
Character	$\chi$	$=$	1225.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+0.381966 q^{2} +0.874032 q^{3} -1.85410 q^{4} +0.333851 q^{6} -1.47214 q^{8} -2.23607 q^{9} +2.23607 q^{11} -1.62054 q^{12} -4.03631 q^{13} +3.14590 q^{16} +7.40492 q^{17} -0.854102 q^{18} +4.24264 q^{19} +0.854102 q^{22} +3.76393 q^{23} -1.28669 q^{24} -1.54173 q^{26} -4.57649 q^{27} +2.23607 q^{29} +6.86474 q^{31} +4.14590 q^{32} +1.95440 q^{33} +2.82843 q^{34} +4.14590 q^{36} +10.7082 q^{37} +1.62054 q^{38} -3.52786 q^{39} -4.78282 q^{41} +5.00000 q^{43} -4.14590 q^{44} +1.43769 q^{46} -9.48683 q^{47} +2.74962 q^{48} +6.47214 q^{51} +7.48373 q^{52} +9.70820 q^{53} -1.74806 q^{54} +3.70820 q^{57} +0.854102 q^{58} -13.1893 q^{59} -3.03476 q^{61} +2.62210 q^{62} -4.70820 q^{64} +0.746512 q^{66} +8.70820 q^{67} -13.7295 q^{68} +3.28980 q^{69} -7.47214 q^{71} +3.29180 q^{72} +2.62210 q^{73} +4.09017 q^{74} -7.86629 q^{76} -1.34752 q^{78} -4.70820 q^{79} +2.70820 q^{81} -1.82688 q^{82} +6.86474 q^{83} +1.90983 q^{86} +1.95440 q^{87} -3.29180 q^{88} +7.94510 q^{89} -6.97871 q^{92} +6.00000 q^{93} -3.62365 q^{94} +3.62365 q^{96} -10.1058 q^{97} -5.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 6 q^{2} + 6 q^{4} + 12 q^{8} + 26 q^{16} + 10 q^{18} - 10 q^{22} + 24 q^{23} + 30 q^{32} + 30 q^{36} + 16 q^{37} - 32 q^{39} + 20 q^{43} - 30 q^{44} + 46 q^{46} + 8 q^{51} + 12 q^{53} - 12 q^{57}+ \cdots - 20 q^{99}+O(q^{100}) $ 4 * q + 6 * q^2 + 6 * q^4 + 12 * q^8 + 26 * q^16 + 10 * q^18 - 10 * q^22 + 24 * q^23 + 30 * q^32 + 30 * q^36 + 16 * q^37 - 32 * q^39 + 20 * q^43 - 30 * q^44 + 46 * q^46 + 8 * q^51 + 12 * q^53 - 12 * q^57 - 10 * q^58 + 8 * q^64 + 8 * q^67 - 12 * q^71 + 40 * q^72 - 6 * q^74 - 68 * q^78 + 8 * q^79 - 16 * q^81 + 30 * q^86 - 40 * q^88 + 66 * q^92 + 24 * q^93 - 20 * q^99

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 $

$2$	0.381966	0.270091	0.135045	−	0.990839i	$-0.456882\pi$
$2$	0.381966	0.270091	0.135045	+	0.990839i	$0.456882\pi$
$3$	0.874032	0.504623	0.252311	−	0.967646i	$-0.418809\pi$
$3$	0.874032	0.504623	0.252311	+	0.967646i	$0.418809\pi$
$4$	−1.85410	−0.927051
$5$	0	0
$5$	0	0
$6$	0.333851	0.136294
$7$	0	0
$7$	0	0
$8$	−1.47214	−0.520479
$9$	−2.23607	−0.745356
$10$	0	0
$11$	2.23607	0.674200	0.337100	−	0.941469i	$-0.390554\pi$
$11$	2.23607	0.674200	0.337100	+	0.941469i	$0.390554\pi$
$12$	−1.62054	−0.467811
$13$	−4.03631	−1.11947	−0.559735	−	0.828671i	$-0.689097\pi$
$13$	−4.03631	−1.11947	−0.559735	+	0.828671i	$0.689097\pi$
$14$	0	0
$15$	0	0
$16$	3.14590	0.786475
$17$	7.40492	1.79596	0.897978	−	0.440040i	$-0.145036\pi$
$17$	7.40492	1.79596	0.897978	+	0.440040i	$0.145036\pi$
$18$	−0.854102	−0.201314
$19$	4.24264	0.973329	0.486664	−	0.873589i	$-0.338214\pi$
$19$	4.24264	0.973329	0.486664	+	0.873589i	$0.338214\pi$
$20$	0	0
$21$	0	0
$22$	0.854102	0.182095
$23$	3.76393	0.784834	0.392417	−	0.919787i	$-0.371639\pi$
$23$	3.76393	0.784834	0.392417	+	0.919787i	$0.371639\pi$
$24$	−1.28669	−0.262645
$25$	0	0
$26$	−1.54173	−0.302359
$27$	−4.57649	−0.880746
$28$	0	0
$29$	2.23607	0.415227	0.207614	−	0.978211i	$-0.433430\pi$
$29$	2.23607	0.415227	0.207614	+	0.978211i	$0.433430\pi$
$30$	0	0
$31$	6.86474	1.23294	0.616472	−	0.787377i	$-0.288562\pi$
$31$	6.86474	1.23294	0.616472	+	0.787377i	$0.288562\pi$
$32$	4.14590	0.732898
$33$	1.95440	0.340217
$34$	2.82843	0.485071
$35$	0	0
$36$	4.14590	0.690983
$37$	10.7082	1.76042	0.880209	−	0.474586i	$-0.157402\pi$
$37$	10.7082	1.76042	0.880209	+	0.474586i	$0.157402\pi$
$38$	1.62054	0.262887
$39$	−3.52786	−0.564910
$40$	0	0
$41$	−4.78282	−0.746951	−0.373476	−	0.927640i	$-0.621834\pi$
$41$	−4.78282	−0.746951	−0.373476	+	0.927640i	$0.621834\pi$
$42$	0	0
$43$	5.00000	0.762493	0.381246	−	0.924473i	$-0.375495\pi$
$43$	5.00000	0.762493	0.381246	+	0.924473i	$0.375495\pi$
$44$	−4.14590	−0.625018
$45$	0	0
$46$	1.43769	0.211976
$47$	−9.48683	−1.38380	−0.691898	−	0.721995i	$-0.743225\pi$
$47$	−9.48683	−1.38380	−0.691898	+	0.721995i	$0.743225\pi$
$48$	2.74962	0.396873
$49$	0	0
$50$	0	0
$51$	6.47214	0.906280
$52$	7.48373	1.03781
$53$	9.70820	1.33352	0.666762	−	0.745271i	$-0.267680\pi$
$53$	9.70820	1.33352	0.666762	+	0.745271i	$0.267680\pi$
$54$	−1.74806	−0.237881
$55$	0	0
$56$	0	0
$57$	3.70820	0.491164
$58$	0.854102	0.112149
$59$	−13.1893	−1.71710	−0.858550	−	0.512730i	$-0.828634\pi$
$59$	−13.1893	−1.71710	−0.858550	+	0.512730i	$0.828634\pi$
$60$	0	0
$61$	−3.03476	−0.388561	−0.194280	−	0.980946i	$-0.562237\pi$
$61$	−3.03476	−0.388561	−0.194280	+	0.980946i	$0.562237\pi$
$62$	2.62210	0.333007
$63$	0	0
$64$	−4.70820	−0.588525
$65$	0	0
$66$	0.746512	0.0918893
$67$	8.70820	1.06388	0.531938	−	0.846783i	$-0.321464\pi$
$67$	8.70820	1.06388	0.531938	+	0.846783i	$0.321464\pi$
$68$	−13.7295	−1.66494
$69$	3.28980	0.396045
$70$	0	0
$71$	−7.47214	−0.886779	−0.443390	−	0.896329i	$-0.646224\pi$
$71$	−7.47214	−0.886779	−0.443390	+	0.896329i	$0.646224\pi$
$72$	3.29180	0.387942
$73$	2.62210	0.306893	0.153447	−	0.988157i	$-0.450963\pi$
$73$	2.62210	0.306893	0.153447	+	0.988157i	$0.450963\pi$
$74$	4.09017	0.475473
$75$	0	0
$76$	−7.86629	−0.902325
$77$	0	0
$78$	−1.34752	−0.152577
$79$	−4.70820	−0.529714	−0.264857	−	0.964288i	$-0.585325\pi$
$79$	−4.70820	−0.529714	−0.264857	+	0.964288i	$0.585325\pi$
$80$	0	0
$81$	2.70820	0.300912
$82$	−1.82688	−0.201745
$83$	6.86474	0.753503	0.376751	−	0.926314i	$-0.377041\pi$
$83$	6.86474	0.753503	0.376751	+	0.926314i	$0.377041\pi$
$84$	0	0
$85$	0	0
$86$	1.90983	0.205942
$87$	1.95440	0.209533
$88$	−3.29180	−0.350907
$89$	7.94510	0.842179	0.421089	−	0.907019i	$-0.361648\pi$
$89$	7.94510	0.842179	0.421089	+	0.907019i	$0.361648\pi$
$90$	0	0
$91$	0	0
$92$	−6.97871	−0.727581
$93$	6.00000	0.622171
$94$	−3.62365	−0.373751
$95$	0	0
$96$	3.62365	0.369837
$97$	−10.1058	−1.02609	−0.513046	−	0.858361i	$-0.671483\pi$
$97$	−10.1058	−1.02609	−0.513046	+	0.858361i	$0.671483\pi$
$98$	0	0
$99$	−5.00000	−0.502519
$100$	0	0
$101$	11.1074	1.10523	0.552613	−	0.833438i	$-0.313631\pi$
$101$	11.1074	1.10523	0.552613	+	0.833438i	$0.313631\pi$
$102$	2.47214	0.244778
$103$	9.48683	0.934765	0.467383	−	0.884055i	$-0.345197\pi$
$103$	9.48683	0.934765	0.467383	+	0.884055i	$0.345197\pi$
$104$	5.94200	0.582661
$105$	0	0
$106$	3.70820	0.360173
$107$	6.00000	0.580042	0.290021	−	0.957020i	$-0.406338\pi$
$107$	6.00000	0.580042	0.290021	+	0.957020i	$0.406338\pi$
$108$	8.48528	0.816497
$109$	−1.00000	−0.0957826	−0.0478913	−	0.998853i	$-0.515250\pi$
$109$	−1.00000	−0.0957826	−0.0478913	+	0.998853i	$0.515250\pi$
$110$	0	0
$111$	9.35931	0.888347
$112$	0	0
$113$	9.76393	0.918513	0.459257	−	0.888304i	$-0.348116\pi$
$113$	9.76393	0.918513	0.459257	+	0.888304i	$0.348116\pi$
$114$	1.41641	0.132659
$115$	0	0
$116$	−4.14590	−0.384937
$117$	9.02546	0.834404
$118$	−5.03786	−0.463773
$119$	0	0
$120$	0	0
$121$	−6.00000	−0.545455
$122$	−1.15917	−0.104947
$123$	−4.18034	−0.376929
$124$	−12.7279	−1.14300
$125$	0	0
$126$	0	0
$127$	7.00000	0.621150	0.310575	−	0.950549i	$-0.399478\pi$
$127$	7.00000	0.621150	0.310575	+	0.950549i	$0.399478\pi$
$128$	−10.0902	−0.891853
$129$	4.37016	0.384771
$130$	0	0
$131$	−14.8098	−1.29394	−0.646971	−	0.762515i	$-0.723964\pi$
$131$	−14.8098	−1.29394	−0.646971	+	0.762515i	$0.723964\pi$
$132$	−3.62365	−0.315398
$133$	0	0
$134$	3.32624	0.287343
$135$	0	0
$136$	−10.9010	−0.934757
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	1.25659	0.106968
$139$	−19.5927	−1.66183	−0.830914	−	0.556401i	$-0.812182\pi$
$139$	−19.5927	−1.66183	−0.830914	+	0.556401i	$0.812182\pi$
$140$	0	0
$141$	−8.29180	−0.698295
$142$	−2.85410	−0.239511
$143$	−9.02546	−0.754747
$144$	−7.03444	−0.586203
$145$	0	0
$146$	1.00155	0.0828890
$147$	0	0
$148$	−19.8541	−1.63200
$149$	9.65248	0.790762	0.395381	−	0.918517i	$-0.370613\pi$
$149$	9.65248	0.790762	0.395381	+	0.918517i	$0.370613\pi$
$150$	0	0
$151$	8.41641	0.684918	0.342459	−	0.939533i	$-0.388740\pi$
$151$	8.41641	0.684918	0.342459	+	0.939533i	$0.388740\pi$
$152$	−6.24574	−0.506597
$153$	−16.5579	−1.33863
$154$	0	0
$155$	0	0
$156$	6.54102	0.523701
$157$	−5.45052	−0.434999	−0.217500	−	0.976060i	$-0.569790\pi$
$157$	−5.45052	−0.434999	−0.217500	+	0.976060i	$0.569790\pi$
$158$	−1.79837	−0.143071
$159$	8.48528	0.672927
$160$	0	0
$161$	0	0
$162$	1.03444	0.0812734
$163$	−2.29180	−0.179507	−0.0897537	−	0.995964i	$-0.528608\pi$
$163$	−2.29180	−0.179507	−0.0897537	+	0.995964i	$0.528608\pi$
$164$	8.86784	0.692462
$165$	0	0
$166$	2.62210	0.203514
$167$	14.2697	1.10422	0.552110	−	0.833772i	$-0.313823\pi$
$167$	14.2697	1.10422	0.552110	+	0.833772i	$0.313823\pi$
$168$	0	0
$169$	3.29180	0.253215
$170$	0	0
$171$	−9.48683	−0.725476
$172$	−9.27051	−0.706870
$173$	4.24264	0.322562	0.161281	−	0.986909i	$-0.448437\pi$
$173$	4.24264	0.322562	0.161281	+	0.986909i	$0.448437\pi$
$174$	0.746512	0.0565930
$175$	0	0
$176$	7.03444	0.530241
$177$	−11.5279	−0.866487
$178$	3.03476	0.227465
$179$	3.70820	0.277164	0.138582	−	0.990351i	$-0.455746\pi$
$179$	3.70820	0.277164	0.138582	+	0.990351i	$0.455746\pi$
$180$	0	0
$181$	11.1074	0.825605	0.412802	−	0.910821i	$-0.364550\pi$
$181$	11.1074	0.825605	0.412802	+	0.910821i	$0.364550\pi$
$182$	0	0
$183$	−2.65248	−0.196077
$184$	−5.54102	−0.408489
$185$	0	0
$186$	2.29180	0.168043
$187$	16.5579	1.21083
$188$	17.5896	1.28285
$189$	0	0
$190$	0	0
$191$	−23.1246	−1.67324	−0.836619	−	0.547785i	$-0.815471\pi$
$191$	−23.1246	−1.67324	−0.836619	+	0.547785i	$0.815471\pi$
$192$	−4.11512	−0.296983
$193$	−10.7082	−0.770793	−0.385397	−	0.922751i	$-0.625935\pi$
$193$	−10.7082	−0.770793	−0.385397	+	0.922751i	$0.625935\pi$
$194$	−3.86008	−0.277138
$195$	0	0
$196$	0	0
$197$	−5.94427	−0.423512	−0.211756	−	0.977323i	$-0.567918\pi$
$197$	−5.94427	−0.423512	−0.211756	+	0.977323i	$0.567918\pi$
$198$	−1.90983	−0.135726
$199$	25.2495	1.78989	0.894945	−	0.446176i	$-0.147214\pi$
$199$	25.2495	1.78989	0.894945	+	0.446176i	$0.147214\pi$
$200$	0	0
$201$	7.61125	0.536856
$202$	4.24264	0.298511
$203$	0	0
$204$	−12.0000	−0.840168
$205$	0	0
$206$	3.62365	0.252472
$207$	−8.41641	−0.584981
$208$	−12.6978	−0.880435
$209$	9.48683	0.656218
$210$	0	0
$211$	7.41641	0.510567	0.255283	−	0.966866i	$-0.417831\pi$
$211$	7.41641	0.510567	0.255283	+	0.966866i	$0.417831\pi$
$212$	−18.0000	−1.23625
$213$	−6.53089	−0.447489
$214$	2.29180	0.156664
$215$	0	0
$216$	6.73722	0.458410
$217$	0	0
$218$	−0.381966	−0.0258700
$219$	2.29180	0.154865
$220$	0	0
$221$	−29.8885	−2.01052
$222$	3.57494	0.239934
$223$	−5.86319	−0.392628	−0.196314	−	0.980541i	$-0.562897\pi$
$223$	−5.86319	−0.392628	−0.196314	+	0.980541i	$0.562897\pi$
$224$	0	0
$225$	0	0
$226$	3.72949	0.248082
$227$	−26.4574	−1.75604	−0.878020	−	0.478625i	$-0.841135\pi$
$227$	−26.4574	−1.75604	−0.878020	+	0.478625i	$0.841135\pi$
$228$	−6.87539	−0.455334
$229$	−7.07107	−0.467269	−0.233635	−	0.972324i	$-0.575062\pi$
$229$	−7.07107	−0.467269	−0.233635	+	0.972324i	$0.575062\pi$
$230$	0	0
$231$	0	0
$232$	−3.29180	−0.216117
$233$	4.52786	0.296630	0.148315	−	0.988940i	$-0.452615\pi$
$233$	4.52786	0.296630	0.148315	+	0.988940i	$0.452615\pi$
$234$	3.44742	0.225365
$235$	0	0
$236$	24.4543	1.59184
$237$	−4.11512	−0.267306
$238$	0	0
$239$	29.1246	1.88391	0.941957	−	0.335733i	$-0.108984\pi$
$239$	29.1246	1.88391	0.941957	+	0.335733i	$0.108984\pi$
$240$	0	0
$241$	−14.7310	−0.948909	−0.474454	−	0.880280i	$-0.657355\pi$
$241$	−14.7310	−0.948909	−0.474454	+	0.880280i	$0.657355\pi$
$242$	−2.29180	−0.147322
$243$	16.0965	1.03259
$244$	5.62675	0.360216
$245$	0	0
$246$	−1.59675	−0.101805
$247$	−17.1246	−1.08961
$248$	−10.1058	−0.641721
$249$	6.00000	0.380235
$250$	0	0
$251$	−11.5687	−0.730213	−0.365106	−	0.930966i	$-0.618967\pi$
$251$	−11.5687	−0.730213	−0.365106	+	0.930966i	$0.618967\pi$
$252$	0	0
$253$	8.41641	0.529135
$254$	2.67376	0.167767
$255$	0	0
$256$	5.56231	0.347644
$257$	12.1089	0.755334	0.377667	−	0.925941i	$-0.376726\pi$
$257$	12.1089	0.755334	0.377667	+	0.925941i	$0.376726\pi$
$258$	1.66925	0.103923
$259$	0	0
$260$	0	0
$261$	−5.00000	−0.309492
$262$	−5.65685	−0.349482
$263$	29.9443	1.84644	0.923221	−	0.384268i	$-0.125546\pi$
$263$	29.9443	1.84644	0.923221	+	0.384268i	$0.125546\pi$
$264$	−2.87714	−0.177075
$265$	0	0
$266$	0	0
$267$	6.94427	0.424983
$268$	−16.1459	−0.986268
$269$	−15.3500	−0.935907	−0.467954	−	0.883753i	$-0.655009\pi$
$269$	−15.3500	−0.935907	−0.467954	+	0.883753i	$0.655009\pi$
$270$	0	0
$271$	−4.44897	−0.270256	−0.135128	−	0.990828i	$-0.543145\pi$
$271$	−4.44897	−0.270256	−0.135128	+	0.990828i	$0.543145\pi$
$272$	23.2951	1.41247
$273$	0	0
$274$	−2.29180	−0.138452
$275$	0	0
$276$	−6.09962	−0.367154
$277$	−24.0000	−1.44202	−0.721010	−	0.692925i	$-0.756322\pi$
$277$	−24.0000	−1.44202	−0.721010	+	0.692925i	$0.756322\pi$
$278$	−7.48373	−0.448844
$279$	−15.3500	−0.918982
$280$	0	0
$281$	−24.5967	−1.46732	−0.733659	−	0.679517i	$-0.762189\pi$
$281$	−24.5967	−1.46732	−0.733659	+	0.679517i	$0.762189\pi$
$282$	−3.16718	−0.188603
$283$	−20.8005	−1.23646	−0.618232	−	0.785996i	$-0.712151\pi$
$283$	−20.8005	−1.23646	−0.618232	+	0.785996i	$0.712151\pi$
$284$	13.8541	0.822090
$285$	0	0
$286$	−3.44742	−0.203850
$287$	0	0
$288$	−9.27051	−0.546270
$289$	37.8328	2.22546
$290$	0	0
$291$	−8.83282	−0.517789
$292$	−4.86163	−0.284506
$293$	18.4335	1.07690	0.538448	−	0.842659i	$-0.319011\pi$
$293$	18.4335	1.07690	0.538448	+	0.842659i	$0.319011\pi$
$294$	0	0
$295$	0	0
$296$	−15.7639	−0.916260
$297$	−10.2333	−0.593799
$298$	3.68692	0.213577
$299$	−15.1924	−0.878599
$300$	0	0
$301$	0	0
$302$	3.21478	0.184990
$303$	9.70820	0.557722
$304$	13.3469	0.765498
$305$	0	0
$306$	−6.32456	−0.361551
$307$	14.5548	0.830686	0.415343	−	0.909665i	$-0.363662\pi$
$307$	14.5548	0.830686	0.415343	+	0.909665i	$0.363662\pi$
$308$	0	0
$309$	8.29180	0.471704
$310$	0	0
$311$	−9.56564	−0.542418	−0.271209	−	0.962520i	$-0.587423\pi$
$311$	−9.56564	−0.542418	−0.271209	+	0.962520i	$0.587423\pi$
$312$	5.19350	0.294024
$313$	21.2132	1.19904	0.599521	−	0.800359i	$-0.295358\pi$
$313$	21.2132	1.19904	0.599521	+	0.800359i	$0.295358\pi$
$314$	−2.08191	−0.117489
$315$	0	0
$316$	8.72949	0.491072
$317$	−11.9443	−0.670857	−0.335429	−	0.942066i	$-0.608881\pi$
$317$	−11.9443	−0.670857	−0.335429	+	0.942066i	$0.608881\pi$
$318$	3.24109	0.181751
$319$	5.00000	0.279946
$320$	0	0
$321$	5.24419	0.292702
$322$	0	0
$323$	31.4164	1.74806
$324$	−5.02129	−0.278960
$325$	0	0
$326$	−0.875388	−0.0484833
$327$	−0.874032	−0.0483341
$328$	7.04096	0.388772
$329$	0	0
$330$	0	0
$331$	−1.29180	−0.0710035	−0.0355018	−	0.999370i	$-0.511303\pi$
$331$	−1.29180	−0.0710035	−0.0355018	+	0.999370i	$0.511303\pi$
$332$	−12.7279	−0.698535
$333$	−23.9443	−1.31214
$334$	5.45052	0.298239
$335$	0	0
$336$	0	0
$337$	−23.1246	−1.25968	−0.629839	−	0.776726i	$-0.716879\pi$
$337$	−23.1246	−1.25968	−0.629839	+	0.776726i	$0.716879\pi$
$338$	1.25735	0.0683911
$339$	8.53399	0.463503
$340$	0	0
$341$	15.3500	0.831250
$342$	−3.62365	−0.195944
$343$	0	0
$344$	−7.36068	−0.396861
$345$	0	0
$346$	1.62054	0.0871210
$347$	27.6525	1.48446	0.742231	−	0.670144i	$-0.233768\pi$
$347$	27.6525	1.48446	0.742231	+	0.670144i	$0.233768\pi$
$348$	−3.62365	−0.194248
$349$	−1.00155	−0.0536118	−0.0268059	−	0.999641i	$-0.508534\pi$
$349$	−1.00155	−0.0536118	−0.0268059	+	0.999641i	$0.508534\pi$
$350$	0	0
$351$	18.4721	0.985970
$352$	9.27051	0.494120
$353$	−12.1089	−0.644493	−0.322247	−	0.946656i	$-0.604438\pi$
$353$	−12.1089	−0.644493	−0.322247	+	0.946656i	$0.604438\pi$
$354$	−4.40325	−0.234030
$355$	0	0
$356$	−14.7310	−0.780743
$357$	0	0
$358$	1.41641	0.0748595
$359$	0.0557281	0.00294122	0.00147061	−	0.999999i	$-0.499532\pi$
$359$	0.0557281	0.00294122	0.00147061	+	0.999999i	$0.499532\pi$
$360$	0	0
$361$	−1.00000	−0.0526316
$362$	4.24264	0.222988
$363$	−5.24419	−0.275249
$364$	0	0
$365$	0	0
$366$	−1.01316	−0.0529585
$367$	−8.69161	−0.453698	−0.226849	−	0.973930i	$-0.572842\pi$
$367$	−8.69161	−0.453698	−0.226849	+	0.973930i	$0.572842\pi$
$368$	11.8409	0.617252
$369$	10.6947	0.556745
$370$	0	0
$371$	0	0
$372$	−11.1246	−0.576784
$373$	−16.1246	−0.834901	−0.417450	−	0.908700i	$-0.637076\pi$
$373$	−16.1246	−0.834901	−0.417450	+	0.908700i	$0.637076\pi$
$374$	6.32456	0.327035
$375$	0	0
$376$	13.9659	0.720237
$377$	−9.02546	−0.464835
$378$	0	0
$379$	−19.0000	−0.975964	−0.487982	−	0.872854i	$-0.662267\pi$
$379$	−19.0000	−0.975964	−0.487982	+	0.872854i	$0.662267\pi$
$380$	0	0
$381$	6.11822	0.313446
$382$	−8.83282	−0.451926
$383$	−13.1893	−0.673941	−0.336971	−	0.941515i	$-0.609402\pi$
$383$	−13.1893	−0.673941	−0.336971	+	0.941515i	$0.609402\pi$
$384$	−8.81913	−0.450049
$385$	0	0
$386$	−4.09017	−0.208184
$387$	−11.1803	−0.568329
$388$	18.7372	0.951239
$389$	27.7639	1.40769	0.703844	−	0.710355i	$-0.251466\pi$
$389$	27.7639	1.40769	0.703844	+	0.710355i	$0.251466\pi$
$390$	0	0
$391$	27.8716	1.40953
$392$	0	0
$393$	−12.9443	−0.652952
$394$	−2.27051	−0.114387
$395$	0	0
$396$	9.27051	0.465861
$397$	9.89949	0.496841	0.248421	−	0.968652i	$-0.420088\pi$
$397$	9.89949	0.496841	0.248421	+	0.968652i	$0.420088\pi$
$398$	9.64446	0.483433
$399$	0	0
$400$	0	0
$401$	23.0689	1.15201	0.576003	−	0.817448i	$-0.304612\pi$
$401$	23.0689	1.15201	0.576003	+	0.817448i	$0.304612\pi$
$402$	2.90724	0.145000
$403$	−27.7082	−1.38024
$404$	−20.5942	−1.02460
$405$	0	0
$406$	0	0
$407$	23.9443	1.18687
$408$	−9.52786	−0.471700
$409$	−14.7310	−0.728402	−0.364201	−	0.931320i	$-0.618658\pi$
$409$	−14.7310	−0.728402	−0.364201	+	0.931320i	$0.618658\pi$
$410$	0	0
$411$	−5.24419	−0.258677
$412$	−17.5896	−0.866575
$413$	0	0
$414$	−3.21478	−0.157998
$415$	0	0
$416$	−16.7341	−0.820458
$417$	−17.1246	−0.838596
$418$	3.62365	0.177238
$419$	4.86163	0.237506	0.118753	−	0.992924i	$-0.462110\pi$
$419$	4.86163	0.237506	0.118753	+	0.992924i	$0.462110\pi$
$420$	0	0
$421$	−9.29180	−0.452854	−0.226427	−	0.974028i	$-0.572705\pi$
$421$	−9.29180	−0.452854	−0.226427	+	0.974028i	$0.572705\pi$
$422$	2.83282	0.137899
$423$	21.2132	1.03142
$424$	−14.2918	−0.694071
$425$	0	0
$426$	−2.49458	−0.120863
$427$	0	0
$428$	−11.1246	−0.537728
$429$	−7.88854	−0.380862
$430$	0	0
$431$	6.76393	0.325807	0.162904	−	0.986642i	$-0.447914\pi$
$431$	6.76393	0.325807	0.162904	+	0.986642i	$0.447914\pi$
$432$	−14.3972	−0.692684
$433$	−32.7031	−1.57161	−0.785806	−	0.618473i	$-0.787752\pi$
$433$	−32.7031	−1.57161	−0.785806	+	0.618473i	$0.787752\pi$
$434$	0	0
$435$	0	0
$436$	1.85410	0.0887954
$437$	15.9690	0.763901
$438$	0.875388	0.0418277
$439$	17.7658	0.847915	0.423957	−	0.905682i	$-0.360641\pi$
$439$	17.7658	0.847915	0.423957	+	0.905682i	$0.360641\pi$
$440$	0	0
$441$	0	0
$442$	−11.4164	−0.543023
$443$	20.1803	0.958797	0.479398	−	0.877597i	$-0.340855\pi$
$443$	20.1803	0.958797	0.479398	+	0.877597i	$0.340855\pi$
$444$	−17.3531	−0.823543
$445$	0	0
$446$	−2.23954	−0.106045
$447$	8.43657	0.399036
$448$	0	0
$449$	−12.5967	−0.594477	−0.297239	−	0.954803i	$-0.596066\pi$
$449$	−12.5967	−0.594477	−0.297239	+	0.954803i	$0.596066\pi$
$450$	0	0
$451$	−10.6947	−0.503594
$452$	−18.1033	−0.851509
$453$	7.35621	0.345625
$454$	−10.1058	−0.474290
$455$	0	0
$456$	−5.45898	−0.255640
$457$	7.29180	0.341096	0.170548	−	0.985349i	$-0.445446\pi$
$457$	7.29180	0.341096	0.170548	+	0.985349i	$0.445446\pi$
$458$	−2.70091	−0.126205
$459$	−33.8885	−1.58178
$460$	0	0
$461$	−33.4009	−1.55564	−0.777819	−	0.628489i	$-0.783674\pi$
$461$	−33.4009	−1.55564	−0.777819	+	0.628489i	$0.783674\pi$
$462$	0	0
$463$	−14.0000	−0.650635	−0.325318	−	0.945605i	$-0.605471\pi$
$463$	−14.0000	−0.650635	−0.325318	+	0.945605i	$0.605471\pi$
$464$	7.03444	0.326566
$465$	0	0
$466$	1.72949	0.0801171
$467$	−9.64446	−0.446292	−0.223146	−	0.974785i	$-0.571633\pi$
$467$	−9.64446	−0.446292	−0.223146	+	0.974785i	$0.571633\pi$
$468$	−16.7341	−0.773535
$469$	0	0
$470$	0	0
$471$	−4.76393	−0.219510
$472$	19.4164	0.893714
$473$	11.1803	0.514073
$474$	−1.57184	−0.0721968
$475$	0	0
$476$	0	0
$477$	−21.7082	−0.993950
$478$	11.1246	0.508828
$479$	−9.02546	−0.412384	−0.206192	−	0.978512i	$-0.566107\pi$
$479$	−9.02546	−0.412384	−0.206192	+	0.978512i	$0.566107\pi$
$480$	0	0
$481$	−43.2216	−1.97074
$482$	−5.62675	−0.256291
$483$	0	0
$484$	11.1246	0.505664
$485$	0	0
$486$	6.14833	0.278894
$487$	28.7082	1.30089	0.650446	−	0.759552i	$-0.274582\pi$
$487$	28.7082	1.30089	0.650446	+	0.759552i	$0.274582\pi$
$488$	4.46758	0.202238
$489$	−2.00310	−0.0905835
$490$	0	0
$491$	7.47214	0.337213	0.168606	−	0.985683i	$-0.446073\pi$
$491$	7.47214	0.337213	0.168606	+	0.985683i	$0.446073\pi$
$492$	7.75078	0.349432
$493$	16.5579	0.745730
$494$	−6.54102	−0.294294
$495$	0	0
$496$	21.5958	0.969678
$497$	0	0
$498$	2.29180	0.102698
$499$	−25.4164	−1.13779	−0.568897	−	0.822409i	$-0.692630\pi$
$499$	−25.4164	−1.13779	−0.568897	+	0.822409i	$0.692630\pi$
$500$	0	0
$501$	12.4721	0.557214
$502$	−4.41887	−0.197224
$503$	16.8918	0.753166	0.376583	−	0.926383i	$-0.377099\pi$
$503$	16.8918	0.753166	0.376583	+	0.926383i	$0.377099\pi$
$504$	0	0
$505$	0	0
$506$	3.21478	0.142914
$507$	2.87714	0.127778
$508$	−12.9787	−0.575837
$509$	5.78437	0.256388	0.128194	−	0.991749i	$-0.459082\pi$
$509$	5.78437	0.256388	0.128194	+	0.991749i	$0.459082\pi$
$510$	0	0
$511$	0	0
$512$	22.3050	0.985749
$513$	−19.4164	−0.857255
$514$	4.62520	0.204009
$515$	0	0
$516$	−8.10272	−0.356702
$517$	−21.2132	−0.932956
$518$	0	0
$519$	3.70820	0.162772
$520$	0	0
$521$	36.0230	1.57820	0.789099	−	0.614266i	$-0.210548\pi$
$521$	36.0230	1.57820	0.789099	+	0.614266i	$0.210548\pi$
$522$	−1.90983	−0.0835910
$523$	−3.24109	−0.141723	−0.0708615	−	0.997486i	$-0.522575\pi$
$523$	−3.24109	−0.141723	−0.0708615	+	0.997486i	$0.522575\pi$
$524$	27.4589	1.19955
$525$	0	0
$526$	11.4377	0.498707
$527$	50.8328	2.21431
$528$	6.14833	0.267572
$529$	−8.83282	−0.384035
$530$	0	0
$531$	29.4922	1.27985
$532$	0	0
$533$	19.3050	0.836190
$534$	2.65248	0.114784
$535$	0	0
$536$	−12.8197	−0.553725
$537$	3.24109	0.139863
$538$	−5.86319	−0.252780
$539$	0	0
$540$	0	0
$541$	−8.70820	−0.374395	−0.187197	−	0.982322i	$-0.559940\pi$
$541$	−8.70820	−0.374395	−0.187197	+	0.982322i	$0.559940\pi$
$542$	−1.69936	−0.0729936
$543$	9.70820	0.416619
$544$	30.7000	1.31625
$545$	0	0
$546$	0	0
$547$	29.0000	1.23995	0.619975	−	0.784621i	$-0.287143\pi$
$547$	29.0000	1.23995	0.619975	+	0.784621i	$0.287143\pi$
$548$	11.1246	0.475220
$549$	6.78593	0.289616
$550$	0	0
$551$	9.48683	0.404153
$552$	−4.84303	−0.206133
$553$	0	0
$554$	−9.16718	−0.389476
$555$	0	0
$556$	36.3268	1.54060
$557$	24.5967	1.04220	0.521099	−	0.853496i	$-0.325522\pi$
$557$	24.5967	1.04220	0.521099	+	0.853496i	$0.325522\pi$
$558$	−5.86319	−0.248208
$559$	−20.1815	−0.853589
$560$	0	0
$561$	14.4721	0.611014
$562$	−9.39512	−0.396309
$563$	−13.8083	−0.581950	−0.290975	−	0.956731i	$-0.593980\pi$
$563$	−13.8083	−0.581950	−0.290975	+	0.956731i	$0.593980\pi$
$564$	15.3738	0.647355
$565$	0	0
$566$	−7.94510	−0.333957
$567$	0	0
$568$	11.0000	0.461550
$569$	−29.9443	−1.25533	−0.627665	−	0.778484i	$-0.715989\pi$
$569$	−29.9443	−1.25533	−0.627665	+	0.778484i	$0.715989\pi$
$570$	0	0
$571$	26.4164	1.10549	0.552746	−	0.833350i	$-0.313580\pi$
$571$	26.4164	1.10549	0.552746	+	0.833350i	$0.313580\pi$
$572$	16.7341	0.699689
$573$	−20.2117	−0.844354
$574$	0	0
$575$	0	0
$576$	10.5279	0.438661
$577$	42.1900	1.75639	0.878196	−	0.478301i	$-0.158747\pi$
$577$	42.1900	1.75639	0.878196	+	0.478301i	$0.158747\pi$
$578$	14.4508	0.601076
$579$	−9.35931	−0.388960
$580$	0	0
$581$	0	0
$582$	−3.37384	−0.139850
$583$	21.7082	0.899062
$584$	−3.86008	−0.159731
$585$	0	0
$586$	7.04096	0.290860
$587$	5.86319	0.242000	0.121000	−	0.992653i	$-0.461390\pi$
$587$	5.86319	0.242000	0.121000	+	0.992653i	$0.461390\pi$
$588$	0	0
$589$	29.1246	1.20006
$590$	0	0
$591$	−5.19548	−0.213714
$592$	33.6869	1.38452
$593$	−30.7000	−1.26070	−0.630350	−	0.776311i	$-0.717088\pi$
$593$	−30.7000	−1.26070	−0.630350	+	0.776311i	$0.717088\pi$
$594$	−3.90879	−0.160380
$595$	0	0
$596$	−17.8967	−0.733076
$597$	22.0689	0.903219
$598$	−5.80298	−0.237301
$599$	−6.81966	−0.278644	−0.139322	−	0.990247i	$-0.544492\pi$
$599$	−6.81966	−0.278644	−0.139322	+	0.990247i	$0.544492\pi$
$600$	0	0
$601$	−29.4621	−1.20178	−0.600891	−	0.799331i	$-0.705187\pi$
$601$	−29.4621	−1.20178	−0.600891	+	0.799331i	$0.705187\pi$
$602$	0	0
$603$	−19.4721	−0.792967
$604$	−15.6049	−0.634953
$605$	0	0
$606$	3.70820	0.150635
$607$	−45.2247	−1.83562	−0.917808	−	0.397025i	$-0.870042\pi$
$607$	−45.2247	−1.83562	−0.917808	+	0.397025i	$0.870042\pi$
$608$	17.5896	0.713351
$609$	0	0
$610$	0	0
$611$	38.2918	1.54912
$612$	30.7000	1.24098
$613$	14.4164	0.582273	0.291137	−	0.956681i	$-0.405967\pi$
$613$	14.4164	0.582273	0.291137	+	0.956681i	$0.405967\pi$
$614$	5.55944	0.224361
$615$	0	0
$616$	0	0
$617$	19.3607	0.779432	0.389716	−	0.920935i	$-0.372573\pi$
$617$	19.3607	0.779432	0.389716	+	0.920935i	$0.372573\pi$
$618$	3.16718	0.127403
$619$	−9.28050	−0.373015	−0.186507	−	0.982454i	$-0.559717\pi$
$619$	−9.28050	−0.373015	−0.186507	+	0.982454i	$0.559717\pi$
$620$	0	0
$621$	−17.2256	−0.691240
$622$	−3.65375	−0.146502
$623$	0	0
$624$	−11.0983	−0.444288
$625$	0	0
$626$	8.10272	0.323850
$627$	8.29180	0.331142
$628$	10.1058	0.403266
$629$	79.2934	3.16163
$630$	0	0
$631$	−28.1246	−1.11962	−0.559812	−	0.828620i	$-0.689126\pi$
$631$	−28.1246	−1.11962	−0.559812	+	0.828620i	$0.689126\pi$
$632$	6.93112	0.275705
$633$	6.48218	0.257643
$634$	−4.56231	−0.181192
$635$	0	0
$636$	−15.7326	−0.623837
$637$	0	0
$638$	1.90983	0.0756109
$639$	16.7082	0.660966
$640$	0	0
$641$	−16.5279	−0.652811	−0.326406	−	0.945230i	$-0.605838\pi$
$641$	−16.5279	−0.652811	−0.326406	+	0.945230i	$0.605838\pi$
$642$	2.00310	0.0790562
$643$	22.2148	0.876064	0.438032	−	0.898959i	$-0.355676\pi$
$643$	22.2148	0.876064	0.438032	+	0.898959i	$0.355676\pi$
$644$	0	0
$645$	0	0
$646$	12.0000	0.472134
$647$	19.0525	0.749030	0.374515	−	0.927221i	$-0.377809\pi$
$647$	19.0525	0.749030	0.374515	+	0.927221i	$0.377809\pi$
$648$	−3.98684	−0.156618
$649$	−29.4922	−1.15767
$650$	0	0
$651$	0	0
$652$	4.24922	0.166412
$653$	1.41641	0.0554283	0.0277142	−	0.999616i	$-0.491177\pi$
$653$	1.41641	0.0554283	0.0277142	+	0.999616i	$0.491177\pi$
$654$	−0.333851	−0.0130546
$655$	0	0
$656$	−15.0463	−0.587458
$657$	−5.86319	−0.228745
$658$	0	0
$659$	−8.83282	−0.344078	−0.172039	−	0.985090i	$-0.555035\pi$
$659$	−8.83282	−0.344078	−0.172039	+	0.985090i	$0.555035\pi$
$660$	0	0
$661$	19.9752	0.776946	0.388473	−	0.921460i	$-0.373003\pi$
$661$	19.9752	0.776946	0.388473	+	0.921460i	$0.373003\pi$
$662$	−0.493422	−0.0191774
$663$	−26.1235	−1.01455
$664$	−10.1058	−0.392182
$665$	0	0
$666$	−9.14590	−0.354396
$667$	8.41641	0.325885
$668$	−26.4574	−1.02367
$669$	−5.12461	−0.198129
$670$	0	0
$671$	−6.78593	−0.261968
$672$	0	0
$673$	−41.1246	−1.58524	−0.792619	−	0.609718i	$-0.791283\pi$
$673$	−41.1246	−1.58524	−0.792619	+	0.609718i	$0.791283\pi$
$674$	−8.83282	−0.340227
$675$	0	0
$676$	−6.10333	−0.234743
$677$	−13.8083	−0.530696	−0.265348	−	0.964153i	$-0.585487\pi$
$677$	−13.8083	−0.530696	−0.265348	+	0.964153i	$0.585487\pi$
$678$	3.25969	0.125188
$679$	0	0
$680$	0	0
$681$	−23.1246	−0.886137
$682$	5.86319	0.224513
$683$	41.0689	1.57146	0.785729	−	0.618571i	$-0.212288\pi$
$683$	41.0689	1.57146	0.785729	+	0.618571i	$0.212288\pi$
$684$	17.5896	0.672553
$685$	0	0
$686$	0	0
$687$	−6.18034	−0.235795
$688$	15.7295	0.599681
$689$	−39.1853	−1.49284
$690$	0	0
$691$	1.79677	0.0683524	0.0341762	−	0.999416i	$-0.489119\pi$
$691$	1.79677	0.0683524	0.0341762	+	0.999416i	$0.489119\pi$
$692$	−7.86629	−0.299031
$693$	0	0
$694$	10.5623	0.400940
$695$	0	0
$696$	−2.87714	−0.109058
$697$	−35.4164	−1.34149
$698$	−0.382559	−0.0144801
$699$	3.95750	0.149686
$700$	0	0
$701$	31.4164	1.18658	0.593291	−	0.804988i	$-0.297828\pi$
$701$	31.4164	1.18658	0.593291	+	0.804988i	$0.297828\pi$
$702$	7.05573	0.266301
$703$	45.4311	1.71346
$704$	−10.5279	−0.396784
$705$	0	0
$706$	−4.62520	−0.174072
$707$	0	0
$708$	21.3738	0.803278
$709$	−10.0000	−0.375558	−0.187779	−	0.982211i	$-0.560129\pi$
$709$	−10.0000	−0.375558	−0.187779	+	0.982211i	$0.560129\pi$
$710$	0	0
$711$	10.5279	0.394826
$712$	−11.6963	−0.438336
$713$	25.8384	0.967656
$714$	0	0
$715$	0	0
$716$	−6.87539	−0.256945
$717$	25.4558	0.950666
$718$	0.0212862	0.000794395 0
$719$	1.54173	0.0574969	0.0287485	−	0.999587i	$-0.490848\pi$
$719$	1.54173	0.0574969	0.0287485	+	0.999587i	$0.490848\pi$
$720$	0	0
$721$	0	0
$722$	−0.381966	−0.0142153
$723$	−12.8754	−0.478841
$724$	−20.5942	−0.765378
$725$	0	0
$726$	−2.00310	−0.0743421
$727$	5.65685	0.209801	0.104901	−	0.994483i	$-0.466548\pi$
$727$	5.65685	0.209801	0.104901	+	0.994483i	$0.466548\pi$
$728$	0	0
$729$	5.94427	0.220158
$730$	0	0
$731$	37.0246	1.36940
$732$	4.91796	0.181773
$733$	−6.48218	−0.239425	−0.119712	−	0.992809i	$-0.538197\pi$
$733$	−6.48218	−0.239425	−0.119712	+	0.992809i	$0.538197\pi$
$734$	−3.31990	−0.122540
$735$	0	0
$736$	15.6049	0.575203
$737$	19.4721	0.717265
$738$	4.08502	0.150372
$739$	−7.29180	−0.268233	−0.134117	−	0.990966i	$-0.542820\pi$
$739$	−7.29180	−0.268233	−0.134117	+	0.990966i	$0.542820\pi$
$740$	0	0
$741$	−14.9675	−0.549843
$742$	0	0
$743$	33.7082	1.23663	0.618317	−	0.785929i	$-0.287815\pi$
$743$	33.7082	1.23663	0.618317	+	0.785929i	$0.287815\pi$
$744$	−8.83282	−0.323827
$745$	0	0
$746$	−6.15905	−0.225499
$747$	−15.3500	−0.561628
$748$	−30.7000	−1.12250
$749$	0	0
$750$	0	0
$751$	26.8328	0.979143	0.489572	−	0.871963i	$-0.337153\pi$
$751$	26.8328	0.979143	0.489572	+	0.871963i	$0.337153\pi$
$752$	−29.8446	−1.08832
$753$	−10.1115	−0.368482
$754$	−3.44742	−0.125548
$755$	0	0
$756$	0	0
$757$	−12.4164	−0.451282	−0.225641	−	0.974211i	$-0.572448\pi$
$757$	−12.4164	−0.451282	−0.225641	+	0.974211i	$0.572448\pi$
$758$	−7.25735	−0.263599
$759$	7.35621	0.267014
$760$	0	0
$761$	52.9148	1.91816	0.959080	−	0.283136i	$-0.0913747\pi$
$761$	52.9148	1.91816	0.959080	+	0.283136i	$0.0913747\pi$
$762$	2.33695	0.0846589
$763$	0	0
$764$	42.8754	1.55118
$765$	0	0
$766$	−5.03786	−0.182025
$767$	53.2361	1.92224
$768$	4.86163	0.175429
$769$	8.69161	0.313428	0.156714	−	0.987644i	$-0.449910\pi$
$769$	8.69161	0.313428	0.156714	+	0.987644i	$0.449910\pi$
$770$	0	0
$771$	10.5836	0.381159
$772$	19.8541	0.714565
$773$	15.3500	0.552102	0.276051	−	0.961143i	$-0.410974\pi$
$773$	15.3500	0.552102	0.276051	+	0.961143i	$0.410974\pi$
$774$	−4.27051	−0.153500
$775$	0	0
$776$	14.8771	0.534059
$777$	0	0
$778$	10.6049	0.380203
$779$	−20.2918	−0.727029
$780$	0	0
$781$	−16.7082	−0.597867
$782$	10.6460	0.380700
$783$	−10.2333	−0.365710
$784$	0	0
$785$	0	0
$786$	−4.94427	−0.176356
$787$	−16.3516	−0.582871	−0.291435	−	0.956591i	$-0.594133\pi$
$787$	−16.3516	−0.582871	−0.291435	+	0.956591i	$0.594133\pi$
$788$	11.0213	0.392617
$789$	26.1723	0.931757
$790$	0	0
$791$	0	0
$792$	7.36068	0.261550
$793$	12.2492	0.434983
$794$	3.78127	0.134192
$795$	0	0
$796$	−46.8152	−1.65932
$797$	−39.0277	−1.38243	−0.691216	−	0.722648i	$-0.742925\pi$
$797$	−39.0277	−1.38243	−0.691216	+	0.722648i	$0.742925\pi$
$798$	0	0
$799$	−70.2492	−2.48524
$800$	0	0
$801$	−17.7658	−0.627723
$802$	8.81153	0.311146
$803$	5.86319	0.206907
$804$	−14.1120	−0.497693
$805$	0	0
$806$	−10.5836	−0.372791
$807$	−13.4164	−0.472280
$808$	−16.3516	−0.575246
$809$	29.0689	1.02201	0.511004	−	0.859578i	$-0.329274\pi$
$809$	29.0689	1.02201	0.511004	+	0.859578i	$0.329274\pi$
$810$	0	0
$811$	7.07107	0.248299	0.124149	−	0.992264i	$-0.460380\pi$
$811$	7.07107	0.248299	0.124149	+	0.992264i	$0.460380\pi$
$812$	0	0
$813$	−3.88854	−0.136377
$814$	9.14590	0.320564
$815$	0	0
$816$	20.3607	0.712766
$817$	21.2132	0.742156
$818$	−5.62675	−0.196735
$819$	0	0
$820$	0	0
$821$	10.5836	0.369370	0.184685	−	0.982798i	$-0.440874\pi$
$821$	10.5836	0.369370	0.184685	+	0.982798i	$0.440874\pi$
$822$	−2.00310	−0.0698662
$823$	−29.5410	−1.02974	−0.514868	−	0.857270i	$-0.672159\pi$
$823$	−29.5410	−1.02974	−0.514868	+	0.857270i	$0.672159\pi$
$824$	−13.9659	−0.486525
$825$	0	0
$826$	0	0
$827$	−1.47214	−0.0511912	−0.0255956	−	0.999672i	$-0.508148\pi$
$827$	−1.47214	−0.0511912	−0.0255956	+	0.999672i	$0.508148\pi$
$828$	15.6049	0.542307
$829$	10.0757	0.349944	0.174972	−	0.984573i	$-0.444016\pi$
$829$	10.0757	0.349944	0.174972	+	0.984573i	$0.444016\pi$
$830$	0	0
$831$	−20.9768	−0.727676
$832$	19.0038	0.658837
$833$	0	0
$834$	−6.54102	−0.226497
$835$	0	0
$836$	−17.5896	−0.608348
$837$	−31.4164	−1.08591
$838$	1.85698	0.0641483
$839$	−31.3190	−1.08125	−0.540626	−	0.841263i	$-0.681813\pi$
$839$	−31.3190	−1.08125	−0.540626	+	0.841263i	$0.681813\pi$
$840$	0	0
$841$	−24.0000	−0.827586
$842$	−3.54915	−0.122312
$843$	−21.4983	−0.740442
$844$	−13.7508	−0.473321
$845$	0	0
$846$	8.10272	0.278577
$847$	0	0
$848$	30.5410	1.04878
$849$	−18.1803	−0.623948
$850$	0	0
$851$	40.3050	1.38164
$852$	12.1089	0.414845
$853$	−22.2148	−0.760619	−0.380309	−	0.924859i	$-0.624182\pi$
$853$	−22.2148	−0.760619	−0.380309	+	0.924859i	$0.624182\pi$
$854$	0	0
$855$	0	0
$856$	−8.83282	−0.301899
$857$	20.6730	0.706177	0.353088	−	0.935590i	$-0.385131\pi$
$857$	20.6730	0.706177	0.353088	+	0.935590i	$0.385131\pi$
$858$	−3.01316	−0.102867
$859$	39.7742	1.35708	0.678539	−	0.734564i	$-0.262613\pi$
$859$	39.7742	1.35708	0.678539	+	0.734564i	$0.262613\pi$
$860$	0	0
$861$	0	0
$862$	2.58359	0.0879975
$863$	−1.36068	−0.0463181	−0.0231590	−	0.999732i	$-0.507372\pi$
$863$	−1.36068	−0.0463181	−0.0231590	+	0.999732i	$0.507372\pi$
$864$	−18.9737	−0.645497
$865$	0	0
$866$	−12.4915	−0.424478
$867$	33.0671	1.12302
$868$	0	0
$869$	−10.5279	−0.357133
$870$	0	0
$871$	−35.1490	−1.19098
$872$	1.47214	0.0498528
$873$	22.5973	0.764803
$874$	6.09962	0.206323
$875$	0	0
$876$	−4.24922	−0.143568
$877$	−38.2918	−1.29302	−0.646511	−	0.762905i	$-0.723773\pi$
$877$	−38.2918	−1.29302	−0.646511	+	0.762905i	$0.723773\pi$
$878$	6.78593	0.229014
$879$	16.1115	0.543426
$880$	0	0
$881$	−1.62054	−0.0545975	−0.0272988	−	0.999627i	$-0.508691\pi$
$881$	−1.62054	−0.0545975	−0.0272988	+	0.999627i	$0.508691\pi$
$882$	0	0
$883$	−32.7082	−1.10072	−0.550359	−	0.834928i	$-0.685509\pi$
$883$	−32.7082	−1.10072	−0.550359	+	0.834928i	$0.685509\pi$
$884$	55.4164	1.86386
$885$	0	0
$886$	7.70820	0.258962
$887$	−24.9945	−0.839232	−0.419616	−	0.907702i	$-0.637835\pi$
$887$	−24.9945	−0.839232	−0.419616	+	0.907702i	$0.637835\pi$
$888$	−13.7782	−0.462366
$889$	0	0
$890$	0	0
$891$	6.05573	0.202875
$892$	10.8709	0.363986
$893$	−40.2492	−1.34689
$894$	3.22248	0.107776
$895$	0	0
$896$	0	0
$897$	−13.2786	−0.443361
$898$	−4.81153	−0.160563
$899$	15.3500	0.511952
$900$	0	0
$901$	71.8885	2.39495
$902$	−4.08502	−0.136016
$903$	0	0
$904$	−14.3738	−0.478067
$905$	0	0
$906$	2.80982	0.0933501
$907$	−14.8328	−0.492516	−0.246258	−	0.969204i	$-0.579201\pi$
$907$	−14.8328	−0.492516	−0.246258	+	0.969204i	$0.579201\pi$
$908$	49.0547	1.62794
$909$	−24.8369	−0.823786
$910$	0	0
$911$	21.7639	0.721071	0.360536	−	0.932745i	$-0.382594\pi$
$911$	21.7639	0.721071	0.360536	+	0.932745i	$0.382594\pi$
$912$	11.6656	0.386288
$913$	15.3500	0.508011
$914$	2.78522	0.0921268
$915$	0	0
$916$	13.1105	0.433182
$917$	0	0
$918$	−12.9443	−0.427225
$919$	31.8328	1.05007	0.525034	−	0.851081i	$-0.324053\pi$
$919$	31.8328	1.05007	0.525034	+	0.851081i	$0.324053\pi$
$920$	0	0
$921$	12.7214	0.419183
$922$	−12.7580	−0.420163
$923$	30.1599	0.992724
$924$	0	0
$925$	0	0
$926$	−5.34752	−0.175731
$927$	−21.2132	−0.696733
$928$	9.27051	0.304319
$929$	−47.0516	−1.54371	−0.771857	−	0.635797i	$-0.780672\pi$
$929$	−47.0516	−1.54371	−0.771857	+	0.635797i	$0.780672\pi$
$930$	0	0
$931$	0	0
$932$	−8.39512	−0.274991
$933$	−8.36068	−0.273716
$934$	−3.68385	−0.120539
$935$	0	0
$936$	−13.2867	−0.434290
$937$	−28.4906	−0.930747	−0.465374	−	0.885114i	$-0.654080\pi$
$937$	−28.4906	−0.930747	−0.465374	+	0.885114i	$0.654080\pi$
$938$	0	0
$939$	18.5410	0.605063
$940$	0	0
$941$	3.62365	0.118128	0.0590638	−	0.998254i	$-0.481188\pi$
$941$	3.62365	0.118128	0.0590638	+	0.998254i	$0.481188\pi$
$942$	−1.81966	−0.0592877
$943$	−18.0022	−0.586233
$944$	−41.4922	−1.35046
$945$	0	0
$946$	4.27051	0.138846
$947$	−12.0000	−0.389948	−0.194974	−	0.980808i	$-0.562462\pi$
$947$	−12.0000	−0.389948	−0.194974	+	0.980808i	$0.562462\pi$
$948$	7.62985	0.247806
$949$	−10.5836	−0.343558
$950$	0	0
$951$	−10.4397	−0.338530
$952$	0	0
$953$	−36.5967	−1.18548	−0.592742	−	0.805392i	$-0.701955\pi$
$953$	−36.5967	−1.18548	−0.592742	+	0.805392i	$0.701955\pi$
$954$	−8.29180	−0.268457
$955$	0	0
$956$	−54.0000	−1.74648
$957$	4.37016	0.141267
$958$	−3.44742	−0.111381
$959$	0	0
$960$	0	0
$961$	16.1246	0.520149
$962$	−16.5092	−0.532278
$963$	−13.4164	−0.432338
$964$	27.3128	0.879687
$965$	0	0
$966$	0	0
$967$	−46.2492	−1.48727	−0.743637	−	0.668583i	$-0.766901\pi$
$967$	−46.2492	−1.48727	−0.743637	+	0.668583i	$0.766901\pi$
$968$	8.83282	0.283897
$969$	27.4589	0.882108
$970$	0	0
$971$	−12.7279	−0.408458	−0.204229	−	0.978923i	$-0.565469\pi$
$971$	−12.7279	−0.408458	−0.204229	+	0.978923i	$0.565469\pi$
$972$	−29.8446	−0.957266
$973$	0	0
$974$	10.9656	0.351359
$975$	0	0
$976$	−9.54704	−0.305593
$977$	−27.7639	−0.888247	−0.444123	−	0.895966i	$-0.646485\pi$
$977$	−27.7639	−0.888247	−0.444123	+	0.895966i	$0.646485\pi$
$978$	−0.765117	−0.0244658
$979$	17.7658	0.567797
$980$	0	0
$981$	2.23607	0.0713922
$982$	2.85410	0.0910781
$983$	8.10272	0.258437	0.129218	−	0.991616i	$-0.458753\pi$
$983$	8.10272	0.258437	0.129218	+	0.991616i	$0.458753\pi$
$984$	6.15403	0.196183
$985$	0	0
$986$	6.32456	0.201415
$987$	0	0
$988$	31.7508	1.01013
$989$	18.8197	0.598430
$990$	0	0
$991$	−5.00000	−0.158830	−0.0794151	−	0.996842i	$-0.525305\pi$
$991$	−5.00000	−0.158830	−0.0794151	+	0.996842i	$0.525305\pi$
$992$	28.4605	0.903622
$993$	−1.12907	−0.0358300
$994$	0	0
$995$	0	0
$996$	−11.1246	−0.352497
$997$	−48.6722	−1.54146	−0.770731	−	0.637160i	$-0.780109\pi$
$997$	−48.6722	−1.54146	−0.770731	+	0.637160i	$0.780109\pi$
$998$	−9.70820	−0.307308
$999$	−49.0060	−1.55048

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1225.2.a.bc.1.2	yes	4
5.2	odd	4		1225.2.b.n.99.5		8
5.3	odd	4		1225.2.b.n.99.4		8
5.4	even	2		1225.2.a.ba.1.3	✓	4
7.6	odd	2	inner	1225.2.a.bc.1.1	yes	4
35.13	even	4		1225.2.b.n.99.3		8
35.27	even	4		1225.2.b.n.99.6		8
35.34	odd	2		1225.2.a.ba.1.4	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1225.2.a.ba.1.3	✓	4	5.4	even	2
1225.2.a.ba.1.4	yes	4	35.34	odd	2
1225.2.a.bc.1.1	yes	4	7.6	odd	2	inner
1225.2.a.bc.1.2	yes	4	1.1	even	1	trivial
1225.2.b.n.99.3		8	35.13	even	4
1225.2.b.n.99.4		8	5.3	odd	4
1225.2.b.n.99.5		8	5.2	odd	4
1225.2.b.n.99.6		8	35.27	even	4

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:	\( N \)	\(=\)	\( 1225 = 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1225.a (trivial)

\(f(q)\)	\(=\)	\(q+0.381966 q^{2} +0.874032 q^{3} -1.85410 q^{4} +0.333851 q^{6} -1.47214 q^{8} -2.23607 q^{9} +2.23607 q^{11} -1.62054 q^{12} -4.03631 q^{13} +3.14590 q^{16} +7.40492 q^{17} -0.854102 q^{18} +4.24264 q^{19} +0.854102 q^{22} +3.76393 q^{23} -1.28669 q^{24} -1.54173 q^{26} -4.57649 q^{27} +2.23607 q^{29} +6.86474 q^{31} +4.14590 q^{32} +1.95440 q^{33} +2.82843 q^{34} +4.14590 q^{36} +10.7082 q^{37} +1.62054 q^{38} -3.52786 q^{39} -4.78282 q^{41} +5.00000 q^{43} -4.14590 q^{44} +1.43769 q^{46} -9.48683 q^{47} +2.74962 q^{48} +6.47214 q^{51} +7.48373 q^{52} +9.70820 q^{53} -1.74806 q^{54} +3.70820 q^{57} +0.854102 q^{58} -13.1893 q^{59} -3.03476 q^{61} +2.62210 q^{62} -4.70820 q^{64} +0.746512 q^{66} +8.70820 q^{67} -13.7295 q^{68} +3.28980 q^{69} -7.47214 q^{71} +3.29180 q^{72} +2.62210 q^{73} +4.09017 q^{74} -7.86629 q^{76} -1.34752 q^{78} -4.70820 q^{79} +2.70820 q^{81} -1.82688 q^{82} +6.86474 q^{83} +1.90983 q^{86} +1.95440 q^{87} -3.29180 q^{88} +7.94510 q^{89} -6.97871 q^{92} +6.00000 q^{93} -3.62365 q^{94} +3.62365 q^{96} -10.1058 q^{97} -5.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 6 q^{2} + 6 q^{4} + 12 q^{8} + 26 q^{16} + 10 q^{18} - 10 q^{22} + 24 q^{23} + 30 q^{32} + 30 q^{36} + 16 q^{37} - 32 q^{39} + 20 q^{43} - 30 q^{44} + 46 q^{46} + 8 q^{51} + 12 q^{53} - 12 q^{57}+ \cdots - 20 q^{99}+O(q^{100}) \) 4 * q + 6 * q^2 + 6 * q^4 + 12 * q^8 + 26 * q^16 + 10 * q^18 - 10 * q^22 + 24 * q^23 + 30 * q^32 + 30 * q^36 + 16 * q^37 - 32 * q^39 + 20 * q^43 - 30 * q^44 + 46 * q^46 + 8 * q^51 + 12 * q^53 - 12 * q^57 - 10 * q^58 + 8 * q^64 + 8 * q^67 - 12 * q^71 + 40 * q^72 - 6 * q^74 - 68 * q^78 + 8 * q^79 - 16 * q^81 + 30 * q^86 - 40 * q^88 + 66 * q^92 + 24 * q^93 - 20 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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