LMFDB - Embedded newform 5225.2.a.j.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5225,2,Mod(1,5225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5225, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("5225.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5225 = 5^{2} \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5225.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:	$41.7218350561$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	$\Q(\zeta_{22})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 4x^{3} + 3x^{2} + 3x - 1 $ x^5 - x^4 - 4x^3 + 3x^2 + 3*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1045)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.830830$ of defining polynomial
Character	$\chi$	$=$	5225.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.37279 q^{2} +1.23648 q^{3} -0.115460 q^{4} +1.69742 q^{6} -2.43232 q^{7} -2.90407 q^{8} -1.47112 q^{9} -1.00000 q^{11} -0.142763 q^{12} +4.84815 q^{13} -3.33906 q^{14} -3.75575 q^{16} +1.04001 q^{17} -2.01953 q^{18} +1.00000 q^{19} -3.00752 q^{21} -1.37279 q^{22} -0.377030 q^{23} -3.59082 q^{24} +6.65547 q^{26} -5.52845 q^{27} +0.280835 q^{28} +4.50688 q^{29} +4.76341 q^{31} +0.652306 q^{32} -1.23648 q^{33} +1.42770 q^{34} +0.169855 q^{36} -3.01428 q^{37} +1.37279 q^{38} +5.99464 q^{39} +0.790451 q^{41} -4.12867 q^{42} +7.46777 q^{43} +0.115460 q^{44} -0.517582 q^{46} +9.67305 q^{47} -4.64391 q^{48} -1.08380 q^{49} +1.28594 q^{51} -0.559766 q^{52} +12.7465 q^{53} -7.58937 q^{54} +7.06364 q^{56} +1.23648 q^{57} +6.18698 q^{58} +1.32111 q^{59} -5.58779 q^{61} +6.53915 q^{62} +3.57824 q^{63} +8.40698 q^{64} -1.69742 q^{66} +1.20728 q^{67} -0.120079 q^{68} -0.466190 q^{69} -0.259538 q^{71} +4.27224 q^{72} +6.27686 q^{73} -4.13797 q^{74} -0.115460 q^{76} +2.43232 q^{77} +8.22935 q^{78} -9.16926 q^{79} -2.42245 q^{81} +1.08512 q^{82} +14.2430 q^{83} +0.347247 q^{84} +10.2517 q^{86} +5.57266 q^{87} +2.90407 q^{88} -1.23983 q^{89} -11.7923 q^{91} +0.0435318 q^{92} +5.88986 q^{93} +13.2790 q^{94} +0.806563 q^{96} -2.95633 q^{97} -1.48783 q^{98} +1.47112 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 3 q^{2} + 7 q^{3} + 5 q^{4} + 2 q^{6} + 11 q^{7} - 3 q^{8} + 8 q^{9} - 5 q^{11} + 7 q^{12} - q^{13} - 3 q^{16} + 3 q^{17} + 7 q^{18} + 5 q^{19} + 11 q^{21} - 3 q^{22} + 8 q^{23} + 9 q^{24} - 16 q^{26}+ \cdots - 8 q^{99}+O(q^{100}) $ 5 * q + 3 * q^2 + 7 * q^3 + 5 * q^4 + 2 * q^6 + 11 * q^7 - 3 * q^8 + 8 * q^9 - 5 * q^11 + 7 * q^12 - q^13 - 3 * q^16 + 3 * q^17 + 7 * q^18 + 5 * q^19 + 11 * q^21 - 3 * q^22 + 8 * q^23 + 9 * q^24 - 16 * q^26 + 10 * q^27 + 22 * q^28 + 11 * q^29 - 5 * q^31 + 2 * q^32 - 7 * q^33 + 4 * q^34 - 3 * q^36 + 9 * q^37 + 3 * q^38 - 8 * q^39 + 15 * q^41 - 11 * q^42 + 13 * q^43 - 5 * q^44 + 18 * q^46 + 20 * q^47 + 20 * q^48 + 20 * q^49 + 24 * q^51 - q^52 + 5 * q^53 + 17 * q^54 + 7 * q^57 + 33 * q^58 - 17 * q^59 + 3 * q^61 - 14 * q^62 + 22 * q^63 - 17 * q^64 - 2 * q^66 + 28 * q^67 + 25 * q^68 - 2 * q^69 - 6 * q^71 + 26 * q^72 + 16 * q^73 - 21 * q^74 + 5 * q^76 - 11 * q^77 - 29 * q^78 + 3 * q^79 + q^81 - 2 * q^82 + 33 * q^83 + 33 * q^84 + 10 * q^86 + 3 * q^88 - 16 * q^89 - 22 * q^91 + 19 * q^92 + 26 * q^93 - 10 * q^94 + 5 * q^96 + 14 * q^97 - 10 * q^98 - 8 * 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 $

$2$	1.37279	0.970706	0.485353	−	0.874318i	$-0.338691\pi$
$2$	1.37279	0.970706	0.485353	+	0.874318i	$0.338691\pi$
$3$	1.23648	0.713881	0.356941	−	0.934127i	$-0.383820\pi$
$3$	1.23648	0.713881	0.356941	+	0.934127i	$0.383820\pi$
$4$	−0.115460	−0.0577299
$5$	0	0
$5$	0	0
$6$	1.69742	0.692969
$7$	−2.43232	−0.919332	−0.459666	−	0.888092i	$-0.652031\pi$
$7$	−2.43232	−0.919332	−0.459666	+	0.888092i	$0.652031\pi$
$8$	−2.90407	−1.02674
$9$	−1.47112	−0.490373
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	−0.142763	−0.0412123
$13$	4.84815	1.34463	0.672317	−	0.740263i	$-0.265299\pi$
$13$	4.84815	1.34463	0.672317	+	0.740263i	$0.265299\pi$
$14$	−3.33906	−0.892401
$15$	0	0
$16$	−3.75575	−0.938937
$17$	1.04001	0.252238	0.126119	−	0.992015i	$-0.459748\pi$
$17$	1.04001	0.252238	0.126119	+	0.992015i	$0.459748\pi$
$18$	−2.01953	−0.476008
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−3.00752	−0.656294
$22$	−1.37279	−0.292679
$23$	−0.377030	−0.0786163	−0.0393081	−	0.999227i	$-0.512515\pi$
$23$	−0.377030	−0.0786163	−0.0393081	+	0.999227i	$0.512515\pi$
$24$	−3.59082	−0.732974
$25$	0	0
$26$	6.65547	1.30525
$27$	−5.52845	−1.06395
$28$	0.280835	0.0530729
$29$	4.50688	0.836907	0.418453	−	0.908238i	$-0.362572\pi$
$29$	4.50688	0.836907	0.418453	+	0.908238i	$0.362572\pi$
$30$	0	0
$31$	4.76341	0.855535	0.427767	−	0.903889i	$-0.359300\pi$
$31$	4.76341	0.855535	0.427767	+	0.903889i	$0.359300\pi$
$32$	0.652306	0.115313
$33$	−1.23648	−0.215243
$34$	1.42770	0.244849
$35$	0	0
$36$	0.169855	0.0283092
$37$	−3.01428	−0.495545	−0.247773	−	0.968818i	$-0.579699\pi$
$37$	−3.01428	−0.495545	−0.247773	+	0.968818i	$0.579699\pi$
$38$	1.37279	0.222695
$39$	5.99464	0.959910
$40$	0	0
$41$	0.790451	0.123448	0.0617238	−	0.998093i	$-0.480340\pi$
$41$	0.790451	0.123448	0.0617238	+	0.998093i	$0.480340\pi$
$42$	−4.12867	−0.637068
$43$	7.46777	1.13882	0.569412	−	0.822052i	$-0.307171\pi$
$43$	7.46777	1.13882	0.569412	+	0.822052i	$0.307171\pi$
$44$	0.115460	0.0174062
$45$	0	0
$46$	−0.517582	−0.0763133
$47$	9.67305	1.41096	0.705479	−	0.708730i	$-0.250732\pi$
$47$	9.67305	1.41096	0.705479	+	0.708730i	$0.250732\pi$
$48$	−4.64391	−0.670290
$49$	−1.08380	−0.154829
$50$	0	0
$51$	1.28594	0.180068
$52$	−0.559766	−0.0776256
$53$	12.7465	1.75087	0.875435	−	0.483336i	$-0.160575\pi$
$53$	12.7465	1.75087	0.875435	+	0.483336i	$0.160575\pi$
$54$	−7.58937	−1.03278
$55$	0	0
$56$	7.06364	0.943919
$57$	1.23648	0.163776
$58$	6.18698	0.812390
$59$	1.32111	0.171994	0.0859968	−	0.996295i	$-0.472593\pi$
$59$	1.32111	0.171994	0.0859968	+	0.996295i	$0.472593\pi$
$60$	0	0
$61$	−5.58779	−0.715443	−0.357721	−	0.933828i	$-0.616446\pi$
$61$	−5.58779	−0.715443	−0.357721	+	0.933828i	$0.616446\pi$
$62$	6.53915	0.830472
$63$	3.57824	0.450816
$64$	8.40698	1.05087
$65$	0	0
$66$	−1.69742	−0.208938
$67$	1.20728	0.147493	0.0737465	−	0.997277i	$-0.476504\pi$
$67$	1.20728	0.147493	0.0737465	+	0.997277i	$0.476504\pi$
$68$	−0.120079	−0.0145617
$69$	−0.466190	−0.0561227
$70$	0	0
$71$	−0.259538	−0.0308015	−0.0154007	−	0.999881i	$-0.504902\pi$
$71$	−0.259538	−0.0308015	−0.0154007	+	0.999881i	$0.504902\pi$
$72$	4.27224	0.503488
$73$	6.27686	0.734651	0.367325	−	0.930093i	$-0.380274\pi$
$73$	6.27686	0.734651	0.367325	+	0.930093i	$0.380274\pi$
$74$	−4.13797	−0.481029
$75$	0	0
$76$	−0.115460	−0.0132441
$77$	2.43232	0.277189
$78$	8.22935	0.931790
$79$	−9.16926	−1.03162	−0.515811	−	0.856702i	$-0.672509\pi$
$79$	−9.16926	−1.03162	−0.515811	+	0.856702i	$0.672509\pi$
$80$	0	0
$81$	−2.42245	−0.269161
$82$	1.08512	0.119831
$83$	14.2430	1.56337	0.781686	−	0.623673i	$-0.214360\pi$
$83$	14.2430	1.56337	0.781686	+	0.623673i	$0.214360\pi$
$84$	0.347247	0.0378877
$85$	0	0
$86$	10.2517	1.10546
$87$	5.57266	0.597452
$88$	2.90407	0.309575
$89$	−1.23983	−0.131421	−0.0657107	−	0.997839i	$-0.520931\pi$
$89$	−1.23983	−0.131421	−0.0657107	+	0.997839i	$0.520931\pi$
$90$	0	0
$91$	−11.7923	−1.23617
$92$	0.0435318	0.00453851
$93$	5.88986	0.610750
$94$	13.2790	1.36963
$95$	0	0
$96$	0.806563	0.0823195
$97$	−2.95633	−0.300170	−0.150085	−	0.988673i	$-0.547955\pi$
$97$	−2.95633	−0.300170	−0.150085	+	0.988673i	$0.547955\pi$
$98$	−1.48783	−0.150294
$99$	1.47112	0.147853
$100$	0	0
$101$	−1.55010	−0.154241	−0.0771204	−	0.997022i	$-0.524573\pi$
$101$	−1.55010	−0.154241	−0.0771204	+	0.997022i	$0.524573\pi$
$102$	1.76533	0.174793
$103$	0.339313	0.0334335	0.0167168	−	0.999860i	$-0.494679\pi$
$103$	0.339313	0.0334335	0.0167168	+	0.999860i	$0.494679\pi$
$104$	−14.0794	−1.38060
$105$	0	0
$106$	17.4982	1.69958
$107$	9.05124	0.875017	0.437508	−	0.899214i	$-0.355861\pi$
$107$	9.05124	0.875017	0.437508	+	0.899214i	$0.355861\pi$
$108$	0.638313	0.0614217
$109$	14.7515	1.41294	0.706471	−	0.707742i	$-0.250286\pi$
$109$	14.7515	1.41294	0.706471	+	0.707742i	$0.250286\pi$
$110$	0	0
$111$	−3.72710	−0.353761
$112$	9.13520	0.863195
$113$	4.71030	0.443108	0.221554	−	0.975148i	$-0.428887\pi$
$113$	4.71030	0.443108	0.221554	+	0.975148i	$0.428887\pi$
$114$	1.69742	0.158978
$115$	0	0
$116$	−0.520363	−0.0483145
$117$	−7.13221	−0.659373
$118$	1.81360	0.166955
$119$	−2.52963	−0.231891
$120$	0	0
$121$	1.00000	0.0909091
$122$	−7.67083	−0.694485
$123$	0.977376	0.0881270
$124$	−0.549982	−0.0493899
$125$	0	0
$126$	4.91215	0.437610
$127$	−14.7860	−1.31205	−0.656024	−	0.754740i	$-0.727763\pi$
$127$	−14.7860	−1.31205	−0.656024	+	0.754740i	$0.727763\pi$
$128$	10.2364	0.904775
$129$	9.23374	0.812986
$130$	0	0
$131$	−7.82065	−0.683294	−0.341647	−	0.939828i	$-0.610985\pi$
$131$	−7.82065	−0.683294	−0.341647	+	0.939828i	$0.610985\pi$
$132$	0.142763	0.0124260
$133$	−2.43232	−0.210909
$134$	1.65734	0.143172
$135$	0	0
$136$	−3.02025	−0.258984
$137$	7.70625	0.658389	0.329195	−	0.944262i	$-0.393223\pi$
$137$	7.70625	0.658389	0.329195	+	0.944262i	$0.393223\pi$
$138$	−0.639979	−0.0544786
$139$	14.3953	1.22099	0.610495	−	0.792020i	$-0.290970\pi$
$139$	14.3953	1.22099	0.610495	+	0.792020i	$0.290970\pi$
$140$	0	0
$141$	11.9605	1.00726
$142$	−0.356290	−0.0298992
$143$	−4.84815	−0.405423
$144$	5.52516	0.460430
$145$	0	0
$146$	8.61678	0.713130
$147$	−1.34010	−0.110530
$148$	0.348028	0.0286078
$149$	−13.8399	−1.13381	−0.566906	−	0.823783i	$-0.691860\pi$
$149$	−13.8399	−1.13381	−0.566906	+	0.823783i	$0.691860\pi$
$150$	0	0
$151$	−8.10281	−0.659398	−0.329699	−	0.944086i	$-0.606947\pi$
$151$	−8.10281	−0.659398	−0.329699	+	0.944086i	$0.606947\pi$
$152$	−2.90407	−0.235551
$153$	−1.52997	−0.123691
$154$	3.33906	0.269069
$155$	0	0
$156$	−0.692139	−0.0554155
$157$	7.13378	0.569338	0.284669	−	0.958626i	$-0.408116\pi$
$157$	7.13378	0.569338	0.284669	+	0.958626i	$0.408116\pi$
$158$	−12.5874	−1.00140
$159$	15.7608	1.24991
$160$	0	0
$161$	0.917060	0.0722744
$162$	−3.32550	−0.261276
$163$	23.8705	1.86968	0.934840	−	0.355068i	$-0.115542\pi$
$163$	23.8705	1.86968	0.934840	+	0.355068i	$0.115542\pi$
$164$	−0.0912652	−0.00712661
$165$	0	0
$166$	19.5526	1.51757
$167$	8.49981	0.657735	0.328868	−	0.944376i	$-0.393333\pi$
$167$	8.49981	0.657735	0.328868	+	0.944376i	$0.393333\pi$
$168$	8.73405	0.673846
$169$	10.5046	0.808043
$170$	0	0
$171$	−1.47112	−0.112499
$172$	−0.862227	−0.0657442
$173$	3.76732	0.286424	0.143212	−	0.989692i	$-0.454257\pi$
$173$	3.76732	0.286424	0.143212	+	0.989692i	$0.454257\pi$
$174$	7.65007	0.579950
$175$	0	0
$176$	3.75575	0.283100
$177$	1.63352	0.122783
$178$	−1.70202	−0.127571
$179$	−12.0321	−0.899323	−0.449661	−	0.893199i	$-0.648455\pi$
$179$	−12.0321	−0.899323	−0.449661	+	0.893199i	$0.648455\pi$
$180$	0	0
$181$	−9.01903	−0.670380	−0.335190	−	0.942151i	$-0.608800\pi$
$181$	−9.01903	−0.670380	−0.335190	+	0.942151i	$0.608800\pi$
$182$	−16.1883	−1.19995
$183$	−6.90918	−0.510741
$184$	1.09492	0.0807188
$185$	0	0
$186$	8.08552	0.592859
$187$	−1.04001	−0.0760527
$188$	−1.11685	−0.0814544
$189$	13.4470	0.978123
$190$	0	0
$191$	−15.2598	−1.10416	−0.552079	−	0.833792i	$-0.686165\pi$
$191$	−15.2598	−1.10416	−0.552079	+	0.833792i	$0.686165\pi$
$192$	10.3950	0.750198
$193$	−3.17246	−0.228359	−0.114179	−	0.993460i	$-0.536424\pi$
$193$	−3.17246	−0.228359	−0.114179	+	0.993460i	$0.536424\pi$
$194$	−4.05841	−0.291377
$195$	0	0
$196$	0.125136	0.00893827
$197$	6.09397	0.434178	0.217089	−	0.976152i	$-0.430344\pi$
$197$	6.09397	0.434178	0.217089	+	0.976152i	$0.430344\pi$
$198$	2.01953	0.143522
$199$	−15.8148	−1.12108	−0.560541	−	0.828127i	$-0.689407\pi$
$199$	−15.8148	−1.12108	−0.560541	+	0.828127i	$0.689407\pi$
$200$	0	0
$201$	1.49278	0.105292
$202$	−2.12796	−0.149723
$203$	−10.9622	−0.769395
$204$	−0.148475	−0.0103953
$205$	0	0
$206$	0.465805	0.0324541
$207$	0.554657	0.0385513
$208$	−18.2084	−1.26253
$209$	−1.00000	−0.0691714
$210$	0	0
$211$	10.4371	0.718522	0.359261	−	0.933237i	$-0.383029\pi$
$211$	10.4371	0.718522	0.359261	+	0.933237i	$0.383029\pi$
$212$	−1.47171	−0.101077
$213$	−0.320913	−0.0219886
$214$	12.4254	0.849384
$215$	0	0
$216$	16.0550	1.09240
$217$	−11.5862	−0.786520
$218$	20.2507	1.37155
$219$	7.76120	0.524453
$220$	0	0
$221$	5.04210	0.339168
$222$	−5.11651	−0.343397
$223$	15.2161	1.01895	0.509474	−	0.860486i	$-0.329840\pi$
$223$	15.2161	1.01895	0.509474	+	0.860486i	$0.329840\pi$
$224$	−1.58662	−0.106010
$225$	0	0
$226$	6.46623	0.430127
$227$	15.4170	1.02326	0.511630	−	0.859206i	$-0.329042\pi$
$227$	15.4170	1.02326	0.511630	+	0.859206i	$0.329042\pi$
$228$	−0.142763	−0.00945474
$229$	18.2529	1.20618	0.603091	−	0.797672i	$-0.293935\pi$
$229$	18.2529	1.20618	0.603091	+	0.797672i	$0.293935\pi$
$230$	0	0
$231$	3.00752	0.197880
$232$	−13.0883	−0.859289
$233$	−19.1041	−1.25155	−0.625776	−	0.780003i	$-0.715218\pi$
$233$	−19.1041	−1.25155	−0.625776	+	0.780003i	$0.715218\pi$
$234$	−9.79100	−0.640057
$235$	0	0
$236$	−0.152535	−0.00992917
$237$	−11.3376	−0.736456
$238$	−3.47264	−0.225098
$239$	20.6551	1.33607	0.668034	−	0.744131i	$-0.267136\pi$
$239$	20.6551	1.33607	0.668034	+	0.744131i	$0.267136\pi$
$240$	0	0
$241$	15.9963	1.03042	0.515208	−	0.857065i	$-0.327715\pi$
$241$	15.9963	1.03042	0.515208	+	0.857065i	$0.327715\pi$
$242$	1.37279	0.0882460
$243$	13.5900	0.871801
$244$	0.645164	0.0413024
$245$	0	0
$246$	1.34173	0.0855454
$247$	4.84815	0.308480
$248$	−13.8333	−0.878416
$249$	17.6112	1.11606
$250$	0	0
$251$	−11.3379	−0.715639	−0.357820	−	0.933791i	$-0.616480\pi$
$251$	−11.3379	−0.715639	−0.357820	+	0.933791i	$0.616480\pi$
$252$	−0.413142	−0.0260255
$253$	0.377030	0.0237037
$254$	−20.2981	−1.27361
$255$	0	0
$256$	−2.76162	−0.172601
$257$	−9.80023	−0.611322	−0.305661	−	0.952140i	$-0.598877\pi$
$257$	−9.80023	−0.611322	−0.305661	+	0.952140i	$0.598877\pi$
$258$	12.6759	0.789170
$259$	7.33171	0.455570
$260$	0	0
$261$	−6.63016	−0.410397
$262$	−10.7361	−0.663277
$263$	21.2899	1.31279	0.656395	−	0.754418i	$-0.272081\pi$
$263$	21.2899	1.31279	0.656395	+	0.754418i	$0.272081\pi$
$264$	3.59082	0.221000
$265$	0	0
$266$	−3.33906	−0.204731
$267$	−1.53302	−0.0938192
$268$	−0.139392	−0.00851475
$269$	−18.2788	−1.11448	−0.557240	−	0.830352i	$-0.688140\pi$
$269$	−18.2788	−1.11448	−0.557240	+	0.830352i	$0.688140\pi$
$270$	0	0
$271$	19.7872	1.20199	0.600994	−	0.799253i	$-0.294772\pi$
$271$	19.7872	1.20199	0.600994	+	0.799253i	$0.294772\pi$
$272$	−3.90600	−0.236836
$273$	−14.5809	−0.882476
$274$	10.5790	0.639103
$275$	0	0
$276$	0.0538262	0.00323995
$277$	−24.5658	−1.47602	−0.738008	−	0.674792i	$-0.764233\pi$
$277$	−24.5658	−1.47602	−0.738008	+	0.674792i	$0.764233\pi$
$278$	19.7616	1.18522
$279$	−7.00755	−0.419531
$280$	0	0
$281$	14.0898	0.840524	0.420262	−	0.907403i	$-0.361938\pi$
$281$	14.0898	0.840524	0.420262	+	0.907403i	$0.361938\pi$
$282$	16.4192	0.977751
$283$	18.8367	1.11973	0.559863	−	0.828585i	$-0.310854\pi$
$283$	18.8367	1.11973	0.559863	+	0.828585i	$0.310854\pi$
$284$	0.0299662	0.00177816
$285$	0	0
$286$	−6.65547	−0.393546
$287$	−1.92263	−0.113489
$288$	−0.959621	−0.0565462
$289$	−15.9184	−0.936376
$290$	0	0
$291$	−3.65544	−0.214286
$292$	−0.724724	−0.0424113
$293$	−2.08270	−0.121673	−0.0608364	−	0.998148i	$-0.519377\pi$
$293$	−2.08270	−0.121673	−0.0608364	+	0.998148i	$0.519377\pi$
$294$	−1.83967	−0.107292
$295$	0	0
$296$	8.75370	0.508798
$297$	5.52845	0.320793
$298$	−18.9993	−1.10060
$299$	−1.82790	−0.105710
$300$	0	0
$301$	−18.1640	−1.04696
$302$	−11.1234	−0.640081
$303$	−1.91667	−0.110110
$304$	−3.75575	−0.215407
$305$	0	0
$306$	−2.10032	−0.120068
$307$	−19.8442	−1.13257	−0.566284	−	0.824210i	$-0.691619\pi$
$307$	−19.8442	−1.13257	−0.566284	+	0.824210i	$0.691619\pi$
$308$	−0.280835	−0.0160021
$309$	0.419554	0.0238676
$310$	0	0
$311$	3.71814	0.210836	0.105418	−	0.994428i	$-0.466382\pi$
$311$	3.71814	0.210836	0.105418	+	0.994428i	$0.466382\pi$
$312$	−17.4089	−0.985582
$313$	−27.0866	−1.53102	−0.765511	−	0.643423i	$-0.777514\pi$
$313$	−27.0866	−1.53102	−0.765511	+	0.643423i	$0.777514\pi$
$314$	9.79316	0.552660
$315$	0	0
$316$	1.05868	0.0595554
$317$	−9.50445	−0.533823	−0.266911	−	0.963721i	$-0.586003\pi$
$317$	−9.50445	−0.533823	−0.266911	+	0.963721i	$0.586003\pi$
$318$	21.6362	1.21330
$319$	−4.50688	−0.252337
$320$	0	0
$321$	11.1917	0.624658
$322$	1.25893	0.0701572
$323$	1.04001	0.0578674
$324$	0.279695	0.0155386
$325$	0	0
$326$	32.7691	1.81491
$327$	18.2400	1.00867
$328$	−2.29553	−0.126749
$329$	−23.5280	−1.29714
$330$	0	0
$331$	7.27869	0.400073	0.200036	−	0.979788i	$-0.435894\pi$
$331$	7.27869	0.400073	0.200036	+	0.979788i	$0.435894\pi$
$332$	−1.64449	−0.0902532
$333$	4.43437	0.243002
$334$	11.6684	0.638467
$335$	0	0
$336$	11.2955	0.616219
$337$	−29.3790	−1.60037	−0.800187	−	0.599750i	$-0.795267\pi$
$337$	−29.3790	−1.60037	−0.800187	+	0.599750i	$0.795267\pi$
$338$	14.4205	0.784372
$339$	5.82419	0.316326
$340$	0	0
$341$	−4.76341	−0.257953
$342$	−2.01953	−0.109204
$343$	19.6624	1.06167
$344$	−21.6870	−1.16928
$345$	0	0
$346$	5.17173	0.278034
$347$	−28.1028	−1.50864	−0.754318	−	0.656509i	$-0.772032\pi$
$347$	−28.1028	−1.50864	−0.754318	+	0.656509i	$0.772032\pi$
$348$	−0.643418	−0.0344908
$349$	20.9948	1.12383	0.561913	−	0.827196i	$-0.310065\pi$
$349$	20.9948	1.12383	0.561913	+	0.827196i	$0.310065\pi$
$350$	0	0
$351$	−26.8027	−1.43062
$352$	−0.652306	−0.0347680
$353$	−15.1874	−0.808345	−0.404172	−	0.914683i	$-0.632440\pi$
$353$	−15.1874	−0.808345	−0.404172	+	0.914683i	$0.632440\pi$
$354$	2.24248	0.119186
$355$	0	0
$356$	0.143150	0.00758693
$357$	−3.12783	−0.165542
$358$	−16.5175	−0.872978
$359$	11.5028	0.607093	0.303546	−	0.952817i	$-0.401829\pi$
$359$	11.5028	0.607093	0.303546	+	0.952817i	$0.401829\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−12.3812	−0.650741
$363$	1.23648	0.0648983
$364$	1.36153	0.0713637
$365$	0	0
$366$	−9.48483	−0.495780
$367$	1.52355	0.0795284	0.0397642	−	0.999209i	$-0.487339\pi$
$367$	1.52355	0.0795284	0.0397642	+	0.999209i	$0.487339\pi$
$368$	1.41603	0.0738158
$369$	−1.16285	−0.0605354
$370$	0	0
$371$	−31.0037	−1.60963
$372$	−0.680042	−0.0352585
$373$	−16.5646	−0.857682	−0.428841	−	0.903380i	$-0.641078\pi$
$373$	−16.5646	−0.857682	−0.428841	+	0.903380i	$0.641078\pi$
$374$	−1.42770	−0.0738248
$375$	0	0
$376$	−28.0912	−1.44869
$377$	21.8500	1.12533
$378$	18.4598	0.949470
$379$	−0.0190602	−0.000979058 0	−0.000489529	−	1.00000i	$-0.500156\pi$
$379$	−0.0190602	−0.000979058 0	−0.000489529		1.00000i	$0.500156\pi$
$380$	0	0
$381$	−18.2826	−0.936647
$382$	−20.9484	−1.07181
$383$	−18.2085	−0.930412	−0.465206	−	0.885202i	$-0.654020\pi$
$383$	−18.2085	−0.930412	−0.465206	+	0.885202i	$0.654020\pi$
$384$	12.6570	0.645902
$385$	0	0
$386$	−4.35511	−0.221669
$387$	−10.9860	−0.558449
$388$	0.341337	0.0173288
$389$	−31.4031	−1.59220	−0.796099	−	0.605166i	$-0.793107\pi$
$389$	−31.4031	−1.59220	−0.796099	+	0.605166i	$0.793107\pi$
$390$	0	0
$391$	−0.392114	−0.0198300
$392$	3.14745	0.158970
$393$	−9.67007	−0.487791
$394$	8.36572	0.421459
$395$	0	0
$396$	−0.169855	−0.00853554
$397$	25.8619	1.29797	0.648985	−	0.760801i	$-0.275194\pi$
$397$	25.8619	1.29797	0.648985	+	0.760801i	$0.275194\pi$
$398$	−21.7103	−1.08824
$399$	−3.00752	−0.150564
$400$	0	0
$401$	−15.5141	−0.774737	−0.387369	−	0.921925i	$-0.626616\pi$
$401$	−15.5141	−0.774737	−0.387369	+	0.921925i	$0.626616\pi$
$402$	2.04926	0.102208
$403$	23.0938	1.15038
$404$	0.178974	0.00890430
$405$	0	0
$406$	−15.0487	−0.746856
$407$	3.01428	0.149412
$408$	−3.73448	−0.184884
$409$	20.7451	1.02578	0.512889	−	0.858455i	$-0.328575\pi$
$409$	20.7451	1.02578	0.512889	+	0.858455i	$0.328575\pi$
$410$	0	0
$411$	9.52862	0.470012
$412$	−0.0391770	−0.00193011
$413$	−3.21336	−0.158119
$414$	0.761425	0.0374220
$415$	0	0
$416$	3.16248	0.155053
$417$	17.7994	0.871643
$418$	−1.37279	−0.0671451
$419$	6.94955	0.339508	0.169754	−	0.985487i	$-0.445703\pi$
$419$	6.94955	0.339508	0.169754	+	0.985487i	$0.445703\pi$
$420$	0	0
$421$	17.0486	0.830897	0.415448	−	0.909617i	$-0.363625\pi$
$421$	17.0486	0.830897	0.415448	+	0.909617i	$0.363625\pi$
$422$	14.3280	0.697474
$423$	−14.2302	−0.691897
$424$	−37.0168	−1.79770
$425$	0	0
$426$	−0.440545	−0.0213445
$427$	13.5913	0.657729
$428$	−1.04505	−0.0505146
$429$	−5.99464	−0.289424
$430$	0	0
$431$	10.6174	0.511421	0.255711	−	0.966753i	$-0.417691\pi$
$431$	10.6174	0.511421	0.255711	+	0.966753i	$0.417691\pi$
$432$	20.7635	0.998982
$433$	5.21952	0.250834	0.125417	−	0.992104i	$-0.459973\pi$
$433$	5.21952	0.250834	0.125417	+	0.992104i	$0.459973\pi$
$434$	−15.9053	−0.763480
$435$	0	0
$436$	−1.70321	−0.0815689
$437$	−0.377030	−0.0180358
$438$	10.6545	0.509090
$439$	−25.8096	−1.23182	−0.615912	−	0.787815i	$-0.711212\pi$
$439$	−25.8096	−1.23182	−0.615912	+	0.787815i	$0.711212\pi$
$440$	0	0
$441$	1.59441	0.0759241
$442$	6.92172	0.329233
$443$	30.3917	1.44395	0.721977	−	0.691917i	$-0.243234\pi$
$443$	30.3917	1.44395	0.721977	+	0.691917i	$0.243234\pi$
$444$	0.430330	0.0204225
$445$	0	0
$446$	20.8885	0.989099
$447$	−17.1128	−0.809407
$448$	−20.4485	−0.966100
$449$	10.8574	0.512394	0.256197	−	0.966625i	$-0.417530\pi$
$449$	10.8574	0.512394	0.256197	+	0.966625i	$0.417530\pi$
$450$	0	0
$451$	−0.790451	−0.0372209
$452$	−0.543850	−0.0255805
$453$	−10.0190	−0.470732
$454$	21.1642	0.993285
$455$	0	0
$456$	−3.59082	−0.168156
$457$	−18.3849	−0.860008	−0.430004	−	0.902827i	$-0.641488\pi$
$457$	−18.3849	−0.860008	−0.430004	+	0.902827i	$0.641488\pi$
$458$	25.0573	1.17085
$459$	−5.74961	−0.268369
$460$	0	0
$461$	−28.2662	−1.31649	−0.658245	−	0.752804i	$-0.728701\pi$
$461$	−28.2662	−1.31649	−0.658245	+	0.752804i	$0.728701\pi$
$462$	4.12867	0.192083
$463$	−33.4266	−1.55346	−0.776732	−	0.629831i	$-0.783124\pi$
$463$	−33.4266	−1.55346	−0.776732	+	0.629831i	$0.783124\pi$
$464$	−16.9267	−0.785803
$465$	0	0
$466$	−26.2258	−1.21489
$467$	1.21297	0.0561297	0.0280648	−	0.999606i	$-0.491066\pi$
$467$	1.21297	0.0561297	0.0280648	+	0.999606i	$0.491066\pi$
$468$	0.823483	0.0380655
$469$	−2.93650	−0.135595
$470$	0	0
$471$	8.82077	0.406440
$472$	−3.83659	−0.176594
$473$	−7.46777	−0.343369
$474$	−15.5641	−0.714882
$475$	0	0
$476$	0.292070	0.0133870
$477$	−18.7517	−0.858580
$478$	28.3550	1.29693
$479$	−25.9098	−1.18385	−0.591923	−	0.805994i	$-0.701631\pi$
$479$	−25.9098	−1.18385	−0.591923	+	0.805994i	$0.701631\pi$
$480$	0	0
$481$	−14.6137	−0.666327
$482$	21.9596	1.00023
$483$	1.13392	0.0515954
$484$	−0.115460	−0.00524817
$485$	0	0
$486$	18.6562	0.846262
$487$	3.09538	0.140265	0.0701326	−	0.997538i	$-0.477658\pi$
$487$	3.09538	0.140265	0.0701326	+	0.997538i	$0.477658\pi$
$488$	16.2273	0.734577
$489$	29.5154	1.33473
$490$	0	0
$491$	17.4687	0.788352	0.394176	−	0.919035i	$-0.371030\pi$
$491$	17.4687	0.788352	0.394176	+	0.919035i	$0.371030\pi$
$492$	−0.112847	−0.00508756
$493$	4.68718	0.211100
$494$	6.65547	0.299444
$495$	0	0
$496$	−17.8902	−0.803293
$497$	0.631280	0.0283168
$498$	24.1763	1.08337
$499$	−6.47276	−0.289760	−0.144880	−	0.989449i	$-0.546280\pi$
$499$	−6.47276	−0.289760	−0.144880	+	0.989449i	$0.546280\pi$
$500$	0	0
$501$	10.5098	0.469545
$502$	−15.5644	−0.694675
$503$	15.5931	0.695262	0.347631	−	0.937631i	$-0.386986\pi$
$503$	15.5931	0.695262	0.347631	+	0.937631i	$0.386986\pi$
$504$	−10.3915	−0.462873
$505$	0	0
$506$	0.517582	0.0230093
$507$	12.9887	0.576847
$508$	1.70719	0.0757444
$509$	6.14404	0.272330	0.136165	−	0.990686i	$-0.456522\pi$
$509$	6.14404	0.272330	0.136165	+	0.990686i	$0.456522\pi$
$510$	0	0
$511$	−15.2673	−0.675388
$512$	−24.2638	−1.07232
$513$	−5.52845	−0.244087
$514$	−13.4536	−0.593414
$515$	0	0
$516$	−1.06613	−0.0469335
$517$	−9.67305	−0.425420
$518$	10.0649	0.442225
$519$	4.65822	0.204473
$520$	0	0
$521$	41.4684	1.81676	0.908382	−	0.418141i	$-0.137318\pi$
$521$	41.4684	1.81676	0.908382	+	0.418141i	$0.137318\pi$
$522$	−9.10179	−0.398375
$523$	−2.76287	−0.120812	−0.0604059	−	0.998174i	$-0.519240\pi$
$523$	−2.76287	−0.120812	−0.0604059	+	0.998174i	$0.519240\pi$
$524$	0.902970	0.0394464
$525$	0	0
$526$	29.2264	1.27433
$527$	4.95398	0.215799
$528$	4.64391	0.202100
$529$	−22.8578	−0.993819
$530$	0	0
$531$	−1.94351	−0.0843411
$532$	0.280835	0.0121758
$533$	3.83222	0.165992
$534$	−2.10451	−0.0910709
$535$	0	0
$536$	−3.50603	−0.151438
$537$	−14.8775	−0.642010
$538$	−25.0929	−1.08183
$539$	1.08380	0.0466828
$540$	0	0
$541$	29.6566	1.27504	0.637519	−	0.770435i	$-0.279961\pi$
$541$	29.6566	1.27504	0.637519	+	0.770435i	$0.279961\pi$
$542$	27.1636	1.16678
$543$	−11.1518	−0.478572
$544$	0.678402	0.0290862
$545$	0	0
$546$	−20.0164	−0.856624
$547$	21.7318	0.929184	0.464592	−	0.885525i	$-0.346201\pi$
$547$	21.7318	0.929184	0.464592	+	0.885525i	$0.346201\pi$
$548$	−0.889761	−0.0380087
$549$	8.22031	0.350834
$550$	0	0
$551$	4.50688	0.192000
$552$	1.35385	0.0576237
$553$	22.3026	0.948403
$554$	−33.7236	−1.43278
$555$	0	0
$556$	−1.66207	−0.0704876
$557$	−8.08244	−0.342464	−0.171232	−	0.985231i	$-0.554775\pi$
$557$	−8.08244	−0.342464	−0.171232	+	0.985231i	$0.554775\pi$
$558$	−9.61987	−0.407242
$559$	36.2049	1.53130
$560$	0	0
$561$	−1.28594	−0.0542926
$562$	19.3422	0.815902
$563$	25.7685	1.08601	0.543006	−	0.839729i	$-0.317286\pi$
$563$	25.7685	1.08601	0.543006	+	0.839729i	$0.317286\pi$
$564$	−1.38096	−0.0581488
$565$	0	0
$566$	25.8588	1.08693
$567$	5.89217	0.247448
$568$	0.753717	0.0316253
$569$	−19.2047	−0.805102	−0.402551	−	0.915398i	$-0.631876\pi$
$569$	−19.2047	−0.805102	−0.402551	+	0.915398i	$0.631876\pi$
$570$	0	0
$571$	−28.0284	−1.17295	−0.586477	−	0.809966i	$-0.699486\pi$
$571$	−28.0284	−1.17295	−0.586477	+	0.809966i	$0.699486\pi$
$572$	0.559766	0.0234050
$573$	−18.8684	−0.788237
$574$	−2.63936	−0.110165
$575$	0	0
$576$	−12.3677	−0.515320
$577$	34.8547	1.45102	0.725509	−	0.688212i	$-0.241604\pi$
$577$	34.8547	1.45102	0.725509	+	0.688212i	$0.241604\pi$
$578$	−21.8525	−0.908946
$579$	−3.92268	−0.163021
$580$	0	0
$581$	−34.6436	−1.43726
$582$	−5.01813	−0.208008
$583$	−12.7465	−0.527907
$584$	−18.2285	−0.754299
$585$	0	0
$586$	−2.85910	−0.118109
$587$	−7.98152	−0.329433	−0.164716	−	0.986341i	$-0.552671\pi$
$587$	−7.98152	−0.329433	−0.164716	+	0.986341i	$0.552671\pi$
$588$	0.154728	0.00638086
$589$	4.76341	0.196273
$590$	0	0
$591$	7.53507	0.309951
$592$	11.3209	0.465286
$593$	−11.6876	−0.479951	−0.239976	−	0.970779i	$-0.577139\pi$
$593$	−11.6876	−0.479951	−0.239976	+	0.970779i	$0.577139\pi$
$594$	7.58937	0.311396
$595$	0	0
$596$	1.59796	0.0654548
$597$	−19.5547	−0.800320
$598$	−2.50931	−0.102614
$599$	−20.3132	−0.829973	−0.414986	−	0.909828i	$-0.636214\pi$
$599$	−20.3132	−0.829973	−0.414986	+	0.909828i	$0.636214\pi$
$600$	0	0
$601$	32.4805	1.32491	0.662453	−	0.749104i	$-0.269516\pi$
$601$	32.4805	1.32491	0.662453	+	0.749104i	$0.269516\pi$
$602$	−24.9353	−1.01629
$603$	−1.77606	−0.0723266
$604$	0.935548	0.0380669
$605$	0	0
$606$	−2.63117	−0.106884
$607$	45.4701	1.84557	0.922787	−	0.385311i	$-0.125906\pi$
$607$	45.4701	1.84557	0.922787	+	0.385311i	$0.125906\pi$
$608$	0.652306	0.0264545
$609$	−13.5545	−0.549257
$610$	0	0
$611$	46.8964	1.89722
$612$	0.176650	0.00714066
$613$	−4.06460	−0.164168	−0.0820839	−	0.996625i	$-0.526158\pi$
$613$	−4.06460	−0.164168	−0.0820839	+	0.996625i	$0.526158\pi$
$614$	−27.2418	−1.09939
$615$	0	0
$616$	−7.06364	−0.284602
$617$	27.7980	1.11911	0.559554	−	0.828794i	$-0.310973\pi$
$617$	27.7980	1.11911	0.559554	+	0.828794i	$0.310973\pi$
$618$	0.575958	0.0231684
$619$	31.8288	1.27931	0.639654	−	0.768663i	$-0.279077\pi$
$619$	31.8288	1.27931	0.639654	+	0.768663i	$0.279077\pi$
$620$	0	0
$621$	2.08439	0.0836438
$622$	5.10420	0.204660
$623$	3.01566	0.120820
$624$	−22.5144	−0.901295
$625$	0	0
$626$	−37.1840	−1.48617
$627$	−1.23648	−0.0493802
$628$	−0.823665	−0.0328678
$629$	−3.13487	−0.124995
$630$	0	0
$631$	32.8526	1.30784	0.653920	−	0.756564i	$-0.273123\pi$
$631$	32.8526	1.30784	0.653920	+	0.756564i	$0.273123\pi$
$632$	26.6282	1.05921
$633$	12.9053	0.512940
$634$	−13.0476	−0.518185
$635$	0	0
$636$	−1.81974	−0.0721573
$637$	−5.25445	−0.208189
$638$	−6.18698	−0.244945
$639$	0.381811	0.0151042
$640$	0	0
$641$	31.2727	1.23520	0.617599	−	0.786493i	$-0.288106\pi$
$641$	31.2727	1.23520	0.617599	+	0.786493i	$0.288106\pi$
$642$	15.3638	0.606359
$643$	13.7579	0.542557	0.271279	−	0.962501i	$-0.412554\pi$
$643$	13.7579	0.542557	0.271279	+	0.962501i	$0.412554\pi$
$644$	−0.105883	−0.00417239
$645$	0	0
$646$	1.42770	0.0561723
$647$	−20.3525	−0.800140	−0.400070	−	0.916485i	$-0.631014\pi$
$647$	−20.3525	−0.800140	−0.400070	+	0.916485i	$0.631014\pi$
$648$	7.03496	0.276359
$649$	−1.32111	−0.0518580
$650$	0	0
$651$	−14.3260	−0.561482
$652$	−2.75608	−0.107936
$653$	4.04869	0.158437	0.0792187	−	0.996857i	$-0.474757\pi$
$653$	4.04869	0.158437	0.0792187	+	0.996857i	$0.474757\pi$
$654$	25.0396	0.979124
$655$	0	0
$656$	−2.96873	−0.115910
$657$	−9.23401	−0.360253
$658$	−32.2989	−1.25914
$659$	−13.0892	−0.509882	−0.254941	−	0.966957i	$-0.582056\pi$
$659$	−13.0892	−0.509882	−0.254941	+	0.966957i	$0.582056\pi$
$660$	0	0
$661$	9.55227	0.371540	0.185770	−	0.982593i	$-0.440522\pi$
$661$	9.55227	0.371540	0.185770	+	0.982593i	$0.440522\pi$
$662$	9.99208	0.388353
$663$	6.23445	0.242126
$664$	−41.3627	−1.60518
$665$	0	0
$666$	6.08744	0.235884
$667$	−1.69923	−0.0657945
$668$	−0.981386	−0.0379710
$669$	18.8144	0.727408
$670$	0	0
$671$	5.58779	0.215714
$672$	−1.96182	−0.0756789
$673$	−31.4730	−1.21320	−0.606598	−	0.795009i	$-0.707466\pi$
$673$	−31.4730	−1.21320	−0.606598	+	0.795009i	$0.707466\pi$
$674$	−40.3310	−1.55349
$675$	0	0
$676$	−1.21285	−0.0466482
$677$	35.4776	1.36352	0.681758	−	0.731578i	$-0.261216\pi$
$677$	35.4776	1.36352	0.681758	+	0.731578i	$0.261216\pi$
$678$	7.99536	0.307060
$679$	7.19075	0.275956
$680$	0	0
$681$	19.0628	0.730487
$682$	−6.53915	−0.250397
$683$	40.6216	1.55434	0.777171	−	0.629289i	$-0.216654\pi$
$683$	40.6216	1.55434	0.777171	+	0.629289i	$0.216654\pi$
$684$	0.169855	0.00649457
$685$	0	0
$686$	26.9923	1.03057
$687$	22.5693	0.861072
$688$	−28.0471	−1.06929
$689$	61.7971	2.35428
$690$	0	0
$691$	−12.5025	−0.475616	−0.237808	−	0.971312i	$-0.576429\pi$
$691$	−12.5025	−0.475616	−0.237808	+	0.971312i	$0.576429\pi$
$692$	−0.434974	−0.0165352
$693$	−3.57824	−0.135926
$694$	−38.5791	−1.46444
$695$	0	0
$696$	−16.1834	−0.613431
$697$	0.822073	0.0311382
$698$	28.8214	1.09091
$699$	−23.6218	−0.893460
$700$	0	0
$701$	−22.1578	−0.836887	−0.418443	−	0.908243i	$-0.637424\pi$
$701$	−22.1578	−0.836887	−0.418443	+	0.908243i	$0.637424\pi$
$702$	−36.7944	−1.38872
$703$	−3.01428	−0.113686
$704$	−8.40698	−0.316850
$705$	0	0
$706$	−20.8491	−0.784665
$707$	3.77035	0.141798
$708$	−0.188606	−0.00708825
$709$	−32.1717	−1.20824	−0.604118	−	0.796895i	$-0.706474\pi$
$709$	−32.1717	−1.20824	−0.604118	+	0.796895i	$0.706474\pi$
$710$	0	0
$711$	13.4891	0.505880
$712$	3.60055	0.134936
$713$	−1.79595	−0.0672589
$714$	−4.29384	−0.160693
$715$	0	0
$716$	1.38922	0.0519178
$717$	25.5396	0.953794
$718$	15.7908	0.589308
$719$	4.63511	0.172860	0.0864302	−	0.996258i	$-0.472454\pi$
$719$	4.63511	0.172860	0.0864302	+	0.996258i	$0.472454\pi$
$720$	0	0
$721$	−0.825320	−0.0307365
$722$	1.37279	0.0510898
$723$	19.7791	0.735594
$724$	1.04134	0.0387009
$725$	0	0
$726$	1.69742	0.0629972
$727$	29.6755	1.10060	0.550302	−	0.834966i	$-0.314513\pi$
$727$	29.6755	1.10060	0.550302	+	0.834966i	$0.314513\pi$
$728$	34.2456	1.26923
$729$	24.0711	0.891523
$730$	0	0
$731$	7.76652	0.287255
$732$	0.797732	0.0294850
$733$	−36.0642	−1.33206	−0.666031	−	0.745924i	$-0.732008\pi$
$733$	−36.0642	−1.33206	−0.666031	+	0.745924i	$0.732008\pi$
$734$	2.09150	0.0771987
$735$	0	0
$736$	−0.245939	−0.00906544
$737$	−1.20728	−0.0444708
$738$	−1.59634	−0.0587621
$739$	13.1048	0.482066	0.241033	−	0.970517i	$-0.422514\pi$
$739$	13.1048	0.482066	0.241033	+	0.970517i	$0.422514\pi$
$740$	0	0
$741$	5.99464	0.220218
$742$	−42.5614	−1.56248
$743$	−49.4215	−1.81310	−0.906550	−	0.422098i	$-0.861294\pi$
$743$	−49.4215	−1.81310	−0.906550	+	0.422098i	$0.861294\pi$
$744$	−17.1046	−0.627085
$745$	0	0
$746$	−22.7396	−0.832557
$747$	−20.9531	−0.766636
$748$	0.120079	0.00439051
$749$	−22.0155	−0.804431
$750$	0	0
$751$	5.10832	0.186405	0.0932026	−	0.995647i	$-0.470290\pi$
$751$	5.10832	0.186405	0.0932026	+	0.995647i	$0.470290\pi$
$752$	−36.3295	−1.32480
$753$	−14.0190	−0.510882
$754$	29.9954	1.09237
$755$	0	0
$756$	−1.55258	−0.0564669
$757$	38.3696	1.39457	0.697283	−	0.716796i	$-0.254392\pi$
$757$	38.3696	1.39457	0.697283	+	0.716796i	$0.254392\pi$
$758$	−0.0261656	−0.000950378 0
$759$	0.466190	0.0169216
$760$	0	0
$761$	−34.9845	−1.26819	−0.634093	−	0.773257i	$-0.718626\pi$
$761$	−34.9845	−1.26819	−0.634093	+	0.773257i	$0.718626\pi$
$762$	−25.0981	−0.909209
$763$	−35.8805	−1.29896
$764$	1.76189	0.0637428
$765$	0	0
$766$	−24.9964	−0.903157
$767$	6.40493	0.231269
$768$	−3.41468	−0.123217
$769$	4.72143	0.170259	0.0851296	−	0.996370i	$-0.472870\pi$
$769$	4.72143	0.170259	0.0851296	+	0.996370i	$0.472870\pi$
$770$	0	0
$771$	−12.1178	−0.436411
$772$	0.366292	0.0131831
$773$	−47.5946	−1.71186	−0.855930	−	0.517092i	$-0.827014\pi$
$773$	−47.5946	−1.71186	−0.855930	+	0.517092i	$0.827014\pi$
$774$	−15.0814	−0.542090
$775$	0	0
$776$	8.58539	0.308198
$777$	9.06551	0.325223
$778$	−43.1097	−1.54556
$779$	0.790451	0.0283208
$780$	0	0
$781$	0.259538	0.00928700
$782$	−0.538288	−0.0192491
$783$	−24.9160	−0.890427
$784$	4.07050	0.145375
$785$	0	0
$786$	−13.2749	−0.473501
$787$	−1.02158	−0.0364154	−0.0182077	−	0.999834i	$-0.505796\pi$
$787$	−1.02158	−0.0364154	−0.0182077	+	0.999834i	$0.505796\pi$
$788$	−0.703608	−0.0250650
$789$	26.3245	0.937176
$790$	0	0
$791$	−11.4570	−0.407363
$792$	−4.27224	−0.151807
$793$	−27.0904	−0.962010
$794$	35.5028	1.25995
$795$	0	0
$796$	1.82597	0.0647199
$797$	37.8469	1.34061	0.670304	−	0.742087i	$-0.266164\pi$
$797$	37.8469	1.34061	0.670304	+	0.742087i	$0.266164\pi$
$798$	−4.12867	−0.146154
$799$	10.0600	0.355898
$800$	0	0
$801$	1.82393	0.0644455
$802$	−21.2975	−0.752042
$803$	−6.27686	−0.221506
$804$	−0.172356	−0.00607852
$805$	0	0
$806$	31.7028	1.11668
$807$	−22.6014	−0.795606
$808$	4.50161	0.158366
$809$	25.6686	0.902461	0.451231	−	0.892407i	$-0.350985\pi$
$809$	25.6686	0.902461	0.451231	+	0.892407i	$0.350985\pi$
$810$	0	0
$811$	−6.16852	−0.216606	−0.108303	−	0.994118i	$-0.534542\pi$
$811$	−6.16852	−0.216606	−0.108303	+	0.994118i	$0.534542\pi$
$812$	1.26569	0.0444170
$813$	24.4665	0.858077
$814$	4.13797	0.145036
$815$	0	0
$816$	−4.82969	−0.169073
$817$	7.46777	0.261264
$818$	28.4785	0.995729
$819$	17.3478	0.606183
$820$	0	0
$821$	−30.9910	−1.08159	−0.540796	−	0.841154i	$-0.681877\pi$
$821$	−30.9910	−1.08159	−0.540796	+	0.841154i	$0.681877\pi$
$822$	13.0807	0.456243
$823$	40.7706	1.42117	0.710587	−	0.703609i	$-0.248429\pi$
$823$	40.7706	1.42117	0.710587	+	0.703609i	$0.248429\pi$
$824$	−0.985391	−0.0343277
$825$	0	0
$826$	−4.41126	−0.153487
$827$	−46.2940	−1.60980	−0.804900	−	0.593410i	$-0.797781\pi$
$827$	−46.2940	−1.60980	−0.804900	+	0.593410i	$0.797781\pi$
$828$	−0.0640405	−0.00222556
$829$	−14.8471	−0.515660	−0.257830	−	0.966190i	$-0.583007\pi$
$829$	−14.8471	−0.515660	−0.257830	+	0.966190i	$0.583007\pi$
$830$	0	0
$831$	−30.3751	−1.05370
$832$	40.7583	1.41304
$833$	−1.12716	−0.0390539
$834$	24.4348	0.846109
$835$	0	0
$836$	0.115460	0.00399326
$837$	−26.3343	−0.910246
$838$	9.54024	0.329562
$839$	36.8854	1.27343	0.636713	−	0.771101i	$-0.280294\pi$
$839$	36.8854	1.27343	0.636713	+	0.771101i	$0.280294\pi$
$840$	0	0
$841$	−8.68803	−0.299587
$842$	23.4040	0.806556
$843$	17.4217	0.600035
$844$	−1.20507	−0.0414802
$845$	0	0
$846$	−19.5350	−0.671628
$847$	−2.43232	−0.0835756
$848$	−47.8728	−1.64396
$849$	23.2912	0.799352
$850$	0	0
$851$	1.13648	0.0389579
$852$	0.0370525	0.00126940
$853$	−9.84716	−0.337160	−0.168580	−	0.985688i	$-0.553918\pi$
$853$	−9.84716	−0.337160	−0.168580	+	0.985688i	$0.553918\pi$
$854$	18.6579	0.638462
$855$	0	0
$856$	−26.2855	−0.898419
$857$	25.6874	0.877464	0.438732	−	0.898618i	$-0.355428\pi$
$857$	25.6874	0.877464	0.438732	+	0.898618i	$0.355428\pi$
$858$	−8.22935	−0.280945
$859$	48.4118	1.65179	0.825894	−	0.563825i	$-0.190671\pi$
$859$	48.4118	1.65179	0.825894	+	0.563825i	$0.190671\pi$
$860$	0	0
$861$	−2.37729	−0.0810179
$862$	14.5754	0.496439
$863$	−15.3028	−0.520913	−0.260457	−	0.965486i	$-0.583873\pi$
$863$	−15.3028	−0.520913	−0.260457	+	0.965486i	$0.583873\pi$
$864$	−3.60624	−0.122687
$865$	0	0
$866$	7.16528	0.243486
$867$	−19.6828	−0.668461
$868$	1.33773	0.0454057
$869$	9.16926	0.311046
$870$	0	0
$871$	5.85308	0.198324
$872$	−42.8395	−1.45073
$873$	4.34911	0.147195
$874$	−0.517582	−0.0175075
$875$	0	0
$876$	−0.896106	−0.0302766
$877$	34.7213	1.17245	0.586227	−	0.810147i	$-0.300613\pi$
$877$	34.7213	1.17245	0.586227	+	0.810147i	$0.300613\pi$
$878$	−35.4310	−1.19574
$879$	−2.57522	−0.0868600
$880$	0	0
$881$	33.7901	1.13842	0.569208	−	0.822194i	$-0.307250\pi$
$881$	33.7901	1.13842	0.569208	+	0.822194i	$0.307250\pi$
$882$	2.18878	0.0737000
$883$	−54.4286	−1.83167	−0.915835	−	0.401556i	$-0.868470\pi$
$883$	−54.4286	−1.83167	−0.915835	+	0.401556i	$0.868470\pi$
$884$	−0.582160	−0.0195801
$885$	0	0
$886$	41.7213	1.40166
$887$	−14.5748	−0.489373	−0.244686	−	0.969602i	$-0.578685\pi$
$887$	−14.5748	−0.489373	−0.244686	+	0.969602i	$0.578685\pi$
$888$	10.8238	0.363222
$889$	35.9644	1.20621
$890$	0	0
$891$	2.42245	0.0811550
$892$	−1.75685	−0.0588237
$893$	9.67305	0.323696
$894$	−23.4922	−0.785696
$895$	0	0
$896$	−24.8981	−0.831789
$897$	−2.26016	−0.0754645
$898$	14.9049	0.497384
$899$	21.4681	0.716003
$900$	0	0
$901$	13.2565	0.441636
$902$	−1.08512	−0.0361305
$903$	−22.4594	−0.747404
$904$	−13.6791	−0.454959
$905$	0	0
$906$	−13.7539	−0.456942
$907$	36.7115	1.21898	0.609492	−	0.792792i	$-0.291373\pi$
$907$	36.7115	1.21898	0.609492	+	0.792792i	$0.291373\pi$
$908$	−1.78004	−0.0590727
$909$	2.28038	0.0756356
$910$	0	0
$911$	18.5321	0.613997	0.306999	−	0.951710i	$-0.400675\pi$
$911$	18.5321	0.613997	0.306999	+	0.951710i	$0.400675\pi$
$912$	−4.64391	−0.153775
$913$	−14.2430	−0.471374
$914$	−25.2385	−0.834815
$915$	0	0
$916$	−2.10747	−0.0696328
$917$	19.0224	0.628173
$918$	−7.89298	−0.260507
$919$	−30.1650	−0.995052	−0.497526	−	0.867449i	$-0.665758\pi$
$919$	−30.1650	−0.995052	−0.497526	+	0.867449i	$0.665758\pi$
$920$	0	0
$921$	−24.5369	−0.808519
$922$	−38.8035	−1.27792
$923$	−1.25828	−0.0414167
$924$	−0.347247	−0.0114236
$925$	0	0
$926$	−45.8875	−1.50796
$927$	−0.499171	−0.0163949
$928$	2.93987	0.0965058
$929$	−53.1253	−1.74298	−0.871492	−	0.490409i	$-0.836847\pi$
$929$	−53.1253	−1.74298	−0.871492	+	0.490409i	$0.836847\pi$
$930$	0	0
$931$	−1.08380	−0.0355203
$932$	2.20575	0.0722519
$933$	4.59740	0.150512
$934$	1.66515	0.0544854
$935$	0	0
$936$	20.7125	0.677008
$937$	−46.9097	−1.53247	−0.766237	−	0.642558i	$-0.777873\pi$
$937$	−46.9097	−1.53247	−0.766237	+	0.642558i	$0.777873\pi$
$938$	−4.03118	−0.131623
$939$	−33.4919	−1.09297
$940$	0	0
$941$	36.0038	1.17369	0.586845	−	0.809699i	$-0.300370\pi$
$941$	36.0038	1.17369	0.586845	+	0.809699i	$0.300370\pi$
$942$	12.1090	0.394534
$943$	−0.298024	−0.00970499
$944$	−4.96175	−0.161491
$945$	0	0
$946$	−10.2517	−0.333310
$947$	21.3336	0.693249	0.346625	−	0.938004i	$-0.387328\pi$
$947$	21.3336	0.693249	0.346625	+	0.938004i	$0.387328\pi$
$948$	1.30904	0.0425155
$949$	30.4312	0.987837
$950$	0	0
$951$	−11.7520	−0.381086
$952$	7.34622	0.238093
$953$	−0.777926	−0.0251995	−0.0125997	−	0.999921i	$-0.504011\pi$
$953$	−0.777926	−0.0251995	−0.0125997	+	0.999921i	$0.504011\pi$
$954$	−25.7420	−0.833429
$955$	0	0
$956$	−2.38483	−0.0771310
$957$	−5.57266	−0.180139
$958$	−35.5685	−1.14917
$959$	−18.7441	−0.605278
$960$	0	0
$961$	−8.30988	−0.268061
$962$	−20.0615	−0.646808
$963$	−13.3155	−0.429085
$964$	−1.84693	−0.0594857
$965$	0	0
$966$	1.55664	0.0500839
$967$	−41.5502	−1.33616	−0.668082	−	0.744088i	$-0.732884\pi$
$967$	−41.5502	−1.33616	−0.668082	+	0.744088i	$0.732884\pi$
$968$	−2.90407	−0.0933404
$969$	1.28594	0.0413105
$970$	0	0
$971$	41.5722	1.33412	0.667058	−	0.745006i	$-0.267553\pi$
$971$	41.5722	1.33412	0.667058	+	0.745006i	$0.267553\pi$
$972$	−1.56910	−0.0503289
$973$	−35.0139	−1.12250
$974$	4.24930	0.136156
$975$	0	0
$976$	20.9863	0.671756
$977$	4.43754	0.141969	0.0709847	−	0.997477i	$-0.477386\pi$
$977$	4.43754	0.141969	0.0709847	+	0.997477i	$0.477386\pi$
$978$	40.5183	1.29563
$979$	1.23983	0.0396250
$980$	0	0
$981$	−21.7013	−0.692869
$982$	23.9808	0.765258
$983$	−4.98576	−0.159021	−0.0795106	−	0.996834i	$-0.525336\pi$
$983$	−4.98576	−0.159021	−0.0795106	+	0.996834i	$0.525336\pi$
$984$	−2.83837	−0.0904839
$985$	0	0
$986$	6.43449	0.204916
$987$	−29.0918	−0.926004
$988$	−0.559766	−0.0178085
$989$	−2.81558	−0.0895301
$990$	0	0
$991$	−39.9837	−1.27012	−0.635062	−	0.772461i	$-0.719025\pi$
$991$	−39.9837	−1.27012	−0.635062	+	0.772461i	$0.719025\pi$
$992$	3.10720	0.0986539
$993$	8.99994	0.285605
$994$	0.866612	0.0274873
$995$	0	0
$996$	−2.03338	−0.0644301
$997$	31.3940	0.994258	0.497129	−	0.867677i	$-0.334388\pi$
$997$	31.3940	0.994258	0.497129	+	0.867677i	$0.334388\pi$
$998$	−8.88571	−0.281272
$999$	16.6643	0.527235

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5225.2.a.j.1.3		5
5.4	even	2		1045.2.a.d.1.3	✓	5
15.14	odd	2		9405.2.a.v.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.2.a.d.1.3	✓	5	5.4	even	2
5225.2.a.j.1.3		5	1.1	even	1	trivial
9405.2.a.v.1.3		5	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:	\( N \)	\(=\)	\( 5225 = 5^{2} \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5225.a (trivial)

\(f(q)\)	\(=\)	\(q+1.37279 q^{2} +1.23648 q^{3} -0.115460 q^{4} +1.69742 q^{6} -2.43232 q^{7} -2.90407 q^{8} -1.47112 q^{9} -1.00000 q^{11} -0.142763 q^{12} +4.84815 q^{13} -3.33906 q^{14} -3.75575 q^{16} +1.04001 q^{17} -2.01953 q^{18} +1.00000 q^{19} -3.00752 q^{21} -1.37279 q^{22} -0.377030 q^{23} -3.59082 q^{24} +6.65547 q^{26} -5.52845 q^{27} +0.280835 q^{28} +4.50688 q^{29} +4.76341 q^{31} +0.652306 q^{32} -1.23648 q^{33} +1.42770 q^{34} +0.169855 q^{36} -3.01428 q^{37} +1.37279 q^{38} +5.99464 q^{39} +0.790451 q^{41} -4.12867 q^{42} +7.46777 q^{43} +0.115460 q^{44} -0.517582 q^{46} +9.67305 q^{47} -4.64391 q^{48} -1.08380 q^{49} +1.28594 q^{51} -0.559766 q^{52} +12.7465 q^{53} -7.58937 q^{54} +7.06364 q^{56} +1.23648 q^{57} +6.18698 q^{58} +1.32111 q^{59} -5.58779 q^{61} +6.53915 q^{62} +3.57824 q^{63} +8.40698 q^{64} -1.69742 q^{66} +1.20728 q^{67} -0.120079 q^{68} -0.466190 q^{69} -0.259538 q^{71} +4.27224 q^{72} +6.27686 q^{73} -4.13797 q^{74} -0.115460 q^{76} +2.43232 q^{77} +8.22935 q^{78} -9.16926 q^{79} -2.42245 q^{81} +1.08512 q^{82} +14.2430 q^{83} +0.347247 q^{84} +10.2517 q^{86} +5.57266 q^{87} +2.90407 q^{88} -1.23983 q^{89} -11.7923 q^{91} +0.0435318 q^{92} +5.88986 q^{93} +13.2790 q^{94} +0.806563 q^{96} -2.95633 q^{97} -1.48783 q^{98} +1.47112 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 3 q^{2} + 7 q^{3} + 5 q^{4} + 2 q^{6} + 11 q^{7} - 3 q^{8} + 8 q^{9} - 5 q^{11} + 7 q^{12} - q^{13} - 3 q^{16} + 3 q^{17} + 7 q^{18} + 5 q^{19} + 11 q^{21} - 3 q^{22} + 8 q^{23} + 9 q^{24} - 16 q^{26}+ \cdots - 8 q^{99}+O(q^{100}) \) 5 * q + 3 * q^2 + 7 * q^3 + 5 * q^4 + 2 * q^6 + 11 * q^7 - 3 * q^8 + 8 * q^9 - 5 * q^11 + 7 * q^12 - q^13 - 3 * q^16 + 3 * q^17 + 7 * q^18 + 5 * q^19 + 11 * q^21 - 3 * q^22 + 8 * q^23 + 9 * q^24 - 16 * q^26 + 10 * q^27 + 22 * q^28 + 11 * q^29 - 5 * q^31 + 2 * q^32 - 7 * q^33 + 4 * q^34 - 3 * q^36 + 9 * q^37 + 3 * q^38 - 8 * q^39 + 15 * q^41 - 11 * q^42 + 13 * q^43 - 5 * q^44 + 18 * q^46 + 20 * q^47 + 20 * q^48 + 20 * q^49 + 24 * q^51 - q^52 + 5 * q^53 + 17 * q^54 + 7 * q^57 + 33 * q^58 - 17 * q^59 + 3 * q^61 - 14 * q^62 + 22 * q^63 - 17 * q^64 - 2 * q^66 + 28 * q^67 + 25 * q^68 - 2 * q^69 - 6 * q^71 + 26 * q^72 + 16 * q^73 - 21 * q^74 + 5 * q^76 - 11 * q^77 - 29 * q^78 + 3 * q^79 + q^81 - 2 * q^82 + 33 * q^83 + 33 * q^84 + 10 * q^86 + 3 * q^88 - 16 * q^89 - 22 * q^91 + 19 * q^92 + 26 * q^93 - 10 * q^94 + 5 * q^96 + 14 * q^97 - 10 * q^98 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more