LMFDB - Embedded newform 5225.2.a.s.1.10

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5225,2,Mod(1,5225)] 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("5225.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage: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")

Level:	$ N $	$=$	$ 5225 = 5^{2} \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5225.a (trivial)

Newform invariants

comment:select newform

sage:traces = [15,-5,-4,17,0,-1,-15] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$41.7218350561$
Analytic rank:	$1$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 5 x^{14} - 11 x^{13} + 85 x^{12} + 6 x^{11} - 537 x^{10} + 327 x^{9} + 1556 x^{8} - 1451 x^{7} + \cdots - 13 $ x^15 - 5x^14 - 11x^13 + 85x^12 + 6x^11 - 537x^10 + 327x^9 + 1556x^8 - 1451x^7 - 2033x^6 + 2316x^5 + 927x^4 - 1295x^3 + 17x^2 + 103x - 13
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			1.10
Root			$-0.324814$ 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)

$f(q)$	$=$	\(q+0.324814 q^{2} -3.24319 q^{3} -1.89450 q^{4} -1.05343 q^{6} -0.166007 q^{7} -1.26499 q^{8} +7.51826 q^{9} +1.00000 q^{11} +6.14420 q^{12} -6.05331 q^{13} -0.0539213 q^{14} +3.37811 q^{16} -0.875666 q^{17} +2.44204 q^{18} -1.00000 q^{19} +0.538390 q^{21} +0.324814 q^{22} -4.61502 q^{23} +4.10259 q^{24} -1.96620 q^{26} -14.6536 q^{27} +0.314499 q^{28} -1.00709 q^{29} +8.35612 q^{31} +3.62723 q^{32} -3.24319 q^{33} -0.284429 q^{34} -14.2433 q^{36} +7.03003 q^{37} -0.324814 q^{38} +19.6320 q^{39} -8.32057 q^{41} +0.174877 q^{42} -6.62275 q^{43} -1.89450 q^{44} -1.49902 q^{46} +8.43004 q^{47} -10.9558 q^{48} -6.97244 q^{49} +2.83995 q^{51} +11.4680 q^{52} +10.0757 q^{53} -4.75968 q^{54} +0.209996 q^{56} +3.24319 q^{57} -0.327117 q^{58} +4.98050 q^{59} +7.32714 q^{61} +2.71418 q^{62} -1.24808 q^{63} -5.57804 q^{64} -1.05343 q^{66} +0.360331 q^{67} +1.65895 q^{68} +14.9674 q^{69} -7.74042 q^{71} -9.51050 q^{72} +6.39400 q^{73} +2.28345 q^{74} +1.89450 q^{76} -0.166007 q^{77} +6.37676 q^{78} -0.309265 q^{79} +24.9695 q^{81} -2.70264 q^{82} +3.15788 q^{83} -1.01998 q^{84} -2.15116 q^{86} +3.26618 q^{87} -1.26499 q^{88} +13.0206 q^{89} +1.00489 q^{91} +8.74314 q^{92} -27.1005 q^{93} +2.73819 q^{94} -11.7638 q^{96} -10.6444 q^{97} -2.26475 q^{98} +7.51826 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q - 5 q^{2} - 4 q^{3} + 17 q^{4} - q^{6} - 15 q^{7} - 15 q^{8} + 19 q^{9} + 15 q^{11} - 9 q^{12} - 13 q^{13} + 9 q^{14} + 21 q^{16} - 11 q^{17} - 16 q^{18} - 15 q^{19} - 5 q^{22} - 10 q^{23} + 17 q^{24}+ \cdots + 19 q^{99}+O(q^{100}) $ 15 * q - 5 * q^2 - 4 * q^3 + 17 * q^4 - q^6 - 15 * q^7 - 15 * q^8 + 19 * q^9 + 15 * q^11 - 9 * q^12 - 13 * q^13 + 9 * q^14 + 21 * q^16 - 11 * q^17 - 16 * q^18 - 15 * q^19 - 5 * q^22 - 10 * q^23 + 17 * q^24 - 17 * q^26 - 13 * q^27 - 30 * q^28 + 5 * q^29 + 6 * q^31 - 40 * q^32 - 4 * q^33 + 17 * q^34 + 28 * q^36 - 13 * q^37 + 5 * q^38 - 22 * q^39 - 30 * q^42 - 36 * q^43 + 17 * q^44 + 13 * q^46 + 6 * q^47 - 14 * q^48 + 16 * q^49 + 4 * q^51 - 50 * q^52 - 9 * q^53 + 9 * q^54 - 18 * q^56 + 4 * q^57 - 2 * q^58 - 7 * q^59 - 2 * q^61 - 11 * q^62 - 39 * q^63 + 17 * q^64 - q^66 - 35 * q^67 + 18 * q^68 - 9 * q^69 + 13 * q^71 - 68 * q^72 - 2 * q^73 + 13 * q^74 - 17 * q^76 - 15 * q^77 + 10 * q^78 + 6 * q^79 + 11 * q^81 - 14 * q^82 - 30 * q^83 - 6 * q^84 - 25 * q^86 - 19 * q^87 - 15 * q^88 + 55 * q^89 + 26 * q^91 + 18 * q^92 - 14 * q^93 - 22 * q^94 - 17 * q^96 - 28 * q^97 + 22 * q^98 + 19 * 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 $

$2$	0.324814	0.229678	0.114839	−	0.993384i	$-0.463365\pi$
$2$	0.324814	0.229678	0.114839	+	0.993384i	$0.463365\pi$
$3$	−3.24319	−1.87245	−0.936227	−	0.351395i	$-0.885708\pi$
$3$	−3.24319	−1.87245	−0.936227	+	0.351395i	$0.885708\pi$
$4$	−1.89450	−0.947248
$5$	0	0
$5$	0	0
$6$	−1.05343	−0.430062
$7$	−0.166007	−0.0627446	−0.0313723	−	0.999508i	$-0.509988\pi$
$7$	−0.166007	−0.0627446	−0.0313723	+	0.999508i	$0.509988\pi$
$8$	−1.26499	−0.447240
$9$	7.51826	2.50609
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	6.14420	1.77368
$13$	−6.05331	−1.67889	−0.839443	−	0.543447i	$-0.817119\pi$
$13$	−6.05331	−1.67889	−0.839443	+	0.543447i	$0.817119\pi$
$14$	−0.0539213	−0.0144111
$15$	0	0
$16$	3.37811	0.844527
$17$	−0.875666	−0.212380	−0.106190	−	0.994346i	$-0.533865\pi$
$17$	−0.875666	−0.212380	−0.106190	+	0.994346i	$0.533865\pi$
$18$	2.44204	0.575593
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0.538390	0.117486
$22$	0.324814	0.0692506
$23$	−4.61502	−0.962299	−0.481149	−	0.876639i	$-0.659781\pi$
$23$	−4.61502	−0.962299	−0.481149	+	0.876639i	$0.659781\pi$
$24$	4.10259	0.837437
$25$	0	0
$26$	−1.96620	−0.385604
$27$	−14.6536	−2.82008
$28$	0.314499	0.0594347
$29$	−1.00709	−0.187012	−0.0935059	−	0.995619i	$-0.529807\pi$
$29$	−1.00709	−0.187012	−0.0935059	+	0.995619i	$0.529807\pi$
$30$	0	0
$31$	8.35612	1.50080	0.750402	−	0.660982i	$-0.229860\pi$
$31$	8.35612	1.50080	0.750402	+	0.660982i	$0.229860\pi$
$32$	3.62723	0.641210
$33$	−3.24319	−0.564566
$34$	−0.284429	−0.0487791
$35$	0	0
$36$	−14.2433	−2.37389
$37$	7.03003	1.15573	0.577865	−	0.816133i	$-0.303886\pi$
$37$	7.03003	1.15573	0.577865	+	0.816133i	$0.303886\pi$
$38$	−0.324814	−0.0526918
$39$	19.6320	3.14364
$40$	0	0
$41$	−8.32057	−1.29945	−0.649727	−	0.760167i	$-0.725117\pi$
$41$	−8.32057	−1.29945	−0.649727	+	0.760167i	$0.725117\pi$
$42$	0.174877	0.0269841
$43$	−6.62275	−1.00996	−0.504980	−	0.863131i	$-0.668500\pi$
$43$	−6.62275	−1.00996	−0.504980	+	0.863131i	$0.668500\pi$
$44$	−1.89450	−0.285606
$45$	0	0
$46$	−1.49902	−0.221019
$47$	8.43004	1.22965	0.614824	−	0.788664i	$-0.289227\pi$
$47$	8.43004	1.22965	0.614824	+	0.788664i	$0.289227\pi$
$48$	−10.9558	−1.58134
$49$	−6.97244	−0.996063
$50$	0	0
$51$	2.83995	0.397673
$52$	11.4680	1.59032
$53$	10.0757	1.38401	0.692005	−	0.721893i	$-0.256728\pi$
$53$	10.0757	1.38401	0.692005	+	0.721893i	$0.256728\pi$
$54$	−4.75968	−0.647711
$55$	0	0
$56$	0.209996	0.0280619
$57$	3.24319	0.429571
$58$	−0.327117	−0.0429525
$59$	4.98050	0.648406	0.324203	−	0.945987i	$-0.394904\pi$
$59$	4.98050	0.648406	0.324203	+	0.945987i	$0.394904\pi$
$60$	0	0
$61$	7.32714	0.938144	0.469072	−	0.883160i	$-0.344588\pi$
$61$	7.32714	0.938144	0.469072	+	0.883160i	$0.344588\pi$
$62$	2.71418	0.344702
$63$	−1.24808	−0.157243
$64$	−5.57804	−0.697255
$65$	0	0
$66$	−1.05343	−0.129669
$67$	0.360331	0.0440214	0.0220107	−	0.999758i	$-0.492993\pi$
$67$	0.360331	0.0440214	0.0220107	+	0.999758i	$0.492993\pi$
$68$	1.65895	0.201177
$69$	14.9674	1.80186
$70$	0	0
$71$	−7.74042	−0.918619	−0.459310	−	0.888276i	$-0.651903\pi$
$71$	−7.74042	−0.918619	−0.459310	+	0.888276i	$0.651903\pi$
$72$	−9.51050	−1.12082
$73$	6.39400	0.748361	0.374180	−	0.927356i	$-0.377924\pi$
$73$	6.39400	0.748361	0.374180	+	0.927356i	$0.377924\pi$
$74$	2.28345	0.265446
$75$	0	0
$76$	1.89450	0.217314
$77$	−0.166007	−0.0189182
$78$	6.37676	0.722025
$79$	−0.309265	−0.0347950	−0.0173975	−	0.999849i	$-0.505538\pi$
$79$	−0.309265	−0.0347950	−0.0173975	+	0.999849i	$0.505538\pi$
$80$	0	0
$81$	24.9695	2.77438
$82$	−2.70264	−0.298456
$83$	3.15788	0.346623	0.173311	−	0.984867i	$-0.444553\pi$
$83$	3.15788	0.346623	0.173311	+	0.984867i	$0.444553\pi$
$84$	−1.01998	−0.111289
$85$	0	0
$86$	−2.15116	−0.231966
$87$	3.26618	0.350171
$88$	−1.26499	−0.134848
$89$	13.0206	1.38018	0.690088	−	0.723726i	$-0.257572\pi$
$89$	13.0206	1.38018	0.690088	+	0.723726i	$0.257572\pi$
$90$	0	0
$91$	1.00489	0.105341
$92$	8.74314	0.911535
$93$	−27.1005	−2.81019
$94$	2.73819	0.282423
$95$	0	0
$96$	−11.7638	−1.20064
$97$	−10.6444	−1.08077	−0.540385	−	0.841418i	$-0.681721\pi$
$97$	−10.6444	−1.08077	−0.540385	+	0.841418i	$0.681721\pi$
$98$	−2.26475	−0.228774
$99$	7.51826	0.755614
$100$	0	0
$101$	10.8769	1.08229	0.541145	−	0.840929i	$-0.317991\pi$
$101$	10.8769	1.08229	0.541145	+	0.840929i	$0.317991\pi$
$102$	0.922455	0.0913367
$103$	5.78563	0.570075	0.285037	−	0.958516i	$-0.407994\pi$
$103$	5.78563	0.570075	0.285037	+	0.958516i	$0.407994\pi$
$104$	7.65736	0.750866
$105$	0	0
$106$	3.27274	0.317877
$107$	4.75657	0.459835	0.229918	−	0.973210i	$-0.426154\pi$
$107$	4.75657	0.459835	0.229918	+	0.973210i	$0.426154\pi$
$108$	27.7611	2.67131
$109$	14.0827	1.34888	0.674438	−	0.738332i	$-0.264386\pi$
$109$	14.0827	1.34888	0.674438	+	0.738332i	$0.264386\pi$
$110$	0	0
$111$	−22.7997	−2.16405
$112$	−0.560788	−0.0529895
$113$	16.5897	1.56063	0.780314	−	0.625387i	$-0.215059\pi$
$113$	16.5897	1.56063	0.780314	+	0.625387i	$0.215059\pi$
$114$	1.05343	0.0986630
$115$	0	0
$116$	1.90793	0.177147
$117$	−45.5104	−4.20744
$118$	1.61774	0.148925
$119$	0.145366	0.0133257
$120$	0	0
$121$	1.00000	0.0909091
$122$	2.37996	0.215471
$123$	26.9852	2.43317
$124$	−15.8306	−1.42163
$125$	0	0
$126$	−0.405394	−0.0361154
$127$	−16.0352	−1.42290	−0.711448	−	0.702738i	$-0.751961\pi$
$127$	−16.0352	−1.42290	−0.711448	+	0.702738i	$0.751961\pi$
$128$	−9.06628	−0.801354
$129$	21.4788	1.89110
$130$	0	0
$131$	9.50050	0.830063	0.415031	−	0.909807i	$-0.363771\pi$
$131$	9.50050	0.830063	0.415031	+	0.909807i	$0.363771\pi$
$132$	6.14420	0.534784
$133$	0.166007	0.0143946
$134$	0.117040	0.0101108
$135$	0	0
$136$	1.10771	0.0949850
$137$	−14.1245	−1.20674	−0.603368	−	0.797463i	$-0.706175\pi$
$137$	−14.1245	−1.20674	−0.603368	+	0.797463i	$0.706175\pi$
$138$	4.86161	0.413848
$139$	8.78232	0.744907	0.372453	−	0.928051i	$-0.378517\pi$
$139$	8.78232	0.744907	0.372453	+	0.928051i	$0.378517\pi$
$140$	0	0
$141$	−27.3402	−2.30246
$142$	−2.51420	−0.210987
$143$	−6.05331	−0.506203
$144$	25.3975	2.11646
$145$	0	0
$146$	2.07686	0.171882
$147$	22.6129	1.86508
$148$	−13.3184	−1.09476
$149$	−6.56136	−0.537528	−0.268764	−	0.963206i	$-0.586615\pi$
$149$	−6.56136	−0.537528	−0.268764	+	0.963206i	$0.586615\pi$
$150$	0	0
$151$	−15.9594	−1.29876	−0.649380	−	0.760464i	$-0.724971\pi$
$151$	−15.9594	−1.29876	−0.649380	+	0.760464i	$0.724971\pi$
$152$	1.26499	0.102604
$153$	−6.58349	−0.532244
$154$	−0.0539213	−0.00434510
$155$	0	0
$156$	−37.1928	−2.97781
$157$	13.8417	1.10469	0.552343	−	0.833617i	$-0.313734\pi$
$157$	13.8417	1.10469	0.552343	+	0.833617i	$0.313734\pi$
$158$	−0.100454	−0.00799166
$159$	−32.6775	−2.59150
$160$	0	0
$161$	0.766124	0.0603790
$162$	8.11043	0.637216
$163$	0.673322	0.0527386	0.0263693	−	0.999652i	$-0.491605\pi$
$163$	0.673322	0.0527386	0.0263693	+	0.999652i	$0.491605\pi$
$164$	15.7633	1.23091
$165$	0	0
$166$	1.02572	0.0796117
$167$	9.05874	0.700986	0.350493	−	0.936565i	$-0.386014\pi$
$167$	9.05874	0.700986	0.350493	+	0.936565i	$0.386014\pi$
$168$	−0.681057	−0.0525447
$169$	23.6426	1.81866
$170$	0	0
$171$	−7.51826	−0.574936
$172$	12.5468	0.956682
$173$	−21.7359	−1.65255	−0.826275	−	0.563267i	$-0.809544\pi$
$173$	−21.7359	−1.65255	−0.826275	+	0.563267i	$0.809544\pi$
$174$	1.06090	0.0804267
$175$	0	0
$176$	3.37811	0.254634
$177$	−16.1527	−1.21411
$178$	4.22926	0.316996
$179$	−1.39240	−0.104073	−0.0520363	−	0.998645i	$-0.516571\pi$
$179$	−1.39240	−0.104073	−0.0520363	+	0.998645i	$0.516571\pi$
$180$	0	0
$181$	12.5467	0.932587	0.466294	−	0.884630i	$-0.345589\pi$
$181$	12.5467	0.932587	0.466294	+	0.884630i	$0.345589\pi$
$182$	0.326402	0.0241945
$183$	−23.7633	−1.75663
$184$	5.83794	0.430379
$185$	0	0
$186$	−8.80261	−0.645438
$187$	−0.875666	−0.0640351
$188$	−15.9707	−1.16478
$189$	2.43259	0.176945
$190$	0	0
$191$	−14.7271	−1.06561	−0.532807	−	0.846237i	$-0.678863\pi$
$191$	−14.7271	−1.06561	−0.532807	+	0.846237i	$0.678863\pi$
$192$	18.0906	1.30558
$193$	−11.7487	−0.845689	−0.422845	−	0.906202i	$-0.638968\pi$
$193$	−11.7487	−0.845689	−0.422845	+	0.906202i	$0.638968\pi$
$194$	−3.45744	−0.248229
$195$	0	0
$196$	13.2093	0.943519
$197$	−13.0704	−0.931224	−0.465612	−	0.884989i	$-0.654166\pi$
$197$	−13.0704	−0.931224	−0.465612	+	0.884989i	$0.654166\pi$
$198$	2.44204	0.173548
$199$	−16.7629	−1.18829	−0.594145	−	0.804358i	$-0.702509\pi$
$199$	−16.7629	−1.18829	−0.594145	+	0.804358i	$0.702509\pi$
$200$	0	0
$201$	−1.16862	−0.0824281
$202$	3.53297	0.248579
$203$	0.167183	0.0117340
$204$	−5.38027	−0.376694
$205$	0	0
$206$	1.87925	0.130934
$207$	−34.6969	−2.41160
$208$	−20.4487	−1.41786
$209$	−1.00000	−0.0691714
$210$	0	0
$211$	−19.4607	−1.33973	−0.669866	−	0.742482i	$-0.733648\pi$
$211$	−19.4607	−1.33973	−0.669866	+	0.742482i	$0.733648\pi$
$212$	−19.0885	−1.31100
$213$	25.1036	1.72007
$214$	1.54500	0.105614
$215$	0	0
$216$	18.5366	1.26125
$217$	−1.38717	−0.0941673
$218$	4.57425	0.309807
$219$	−20.7369	−1.40127
$220$	0	0
$221$	5.30068	0.356563
$222$	−7.40566	−0.497035
$223$	−11.4123	−0.764224	−0.382112	−	0.924116i	$-0.624803\pi$
$223$	−11.4123	−0.764224	−0.382112	+	0.924116i	$0.624803\pi$
$224$	−0.602144	−0.0402324
$225$	0	0
$226$	5.38857	0.358442
$227$	−4.01475	−0.266468	−0.133234	−	0.991085i	$-0.542536\pi$
$227$	−4.01475	−0.266468	−0.133234	+	0.991085i	$0.542536\pi$
$228$	−6.14420	−0.406910
$229$	26.3140	1.73888	0.869440	−	0.494038i	$-0.164480\pi$
$229$	26.3140	1.73888	0.869440	+	0.494038i	$0.164480\pi$
$230$	0	0
$231$	0.538390	0.0354235
$232$	1.27395	0.0836392
$233$	14.6801	0.961724	0.480862	−	0.876796i	$-0.340324\pi$
$233$	14.6801	0.961724	0.480862	+	0.876796i	$0.340324\pi$
$234$	−14.7824	−0.966356
$235$	0	0
$236$	−9.43554	−0.614202
$237$	1.00300	0.0651521
$238$	0.0472170	0.00306063
$239$	−28.5560	−1.84713	−0.923567	−	0.383437i	$-0.874740\pi$
$239$	−28.5560	−1.84713	−0.923567	+	0.383437i	$0.874740\pi$
$240$	0	0
$241$	−22.3426	−1.43921	−0.719607	−	0.694381i	$-0.755678\pi$
$241$	−22.3426	−1.43921	−0.719607	+	0.694381i	$0.755678\pi$
$242$	0.324814	0.0208798
$243$	−37.0199	−2.37483
$244$	−13.8812	−0.888655
$245$	0	0
$246$	8.76516	0.558846
$247$	6.05331	0.385163
$248$	−10.5704	−0.671220
$249$	−10.2416	−0.649035
$250$	0	0
$251$	11.5146	0.726795	0.363398	−	0.931634i	$-0.381617\pi$
$251$	11.5146	0.726795	0.363398	+	0.931634i	$0.381617\pi$
$252$	2.36448	0.148948
$253$	−4.61502	−0.290144
$254$	−5.20847	−0.326808
$255$	0	0
$256$	8.21122	0.513201
$257$	−10.5478	−0.657955	−0.328978	−	0.944338i	$-0.606704\pi$
$257$	−10.5478	−0.657955	−0.328978	+	0.944338i	$0.606704\pi$
$258$	6.97662	0.434345
$259$	−1.16703	−0.0725158
$260$	0	0
$261$	−7.57156	−0.468668
$262$	3.08590	0.190647
$263$	−20.3165	−1.25277	−0.626386	−	0.779513i	$-0.715466\pi$
$263$	−20.3165	−1.25277	−0.626386	+	0.779513i	$0.715466\pi$
$264$	4.10259	0.252497
$265$	0	0
$266$	0.0539213	0.00330612
$267$	−42.2281	−2.58432
$268$	−0.682645	−0.0416992
$269$	−16.4623	−1.00373	−0.501863	−	0.864947i	$-0.667352\pi$
$269$	−16.4623	−1.00373	−0.501863	+	0.864947i	$0.667352\pi$
$270$	0	0
$271$	23.2292	1.41107	0.705537	−	0.708673i	$-0.250706\pi$
$271$	23.2292	1.41107	0.705537	+	0.708673i	$0.250706\pi$
$272$	−2.95809	−0.179361
$273$	−3.25904	−0.197246
$274$	−4.58783	−0.277161
$275$	0	0
$276$	−28.3556	−1.70681
$277$	17.7363	1.06567	0.532837	−	0.846218i	$-0.321126\pi$
$277$	17.7363	1.06567	0.532837	+	0.846218i	$0.321126\pi$
$278$	2.85262	0.171089
$279$	62.8235	3.76114
$280$	0	0
$281$	−25.6914	−1.53262	−0.766310	−	0.642471i	$-0.777909\pi$
$281$	−25.6914	−1.53262	−0.766310	+	0.642471i	$0.777909\pi$
$282$	−8.88048	−0.528825
$283$	−5.17198	−0.307442	−0.153721	−	0.988114i	$-0.549126\pi$
$283$	−5.17198	−0.307442	−0.153721	+	0.988114i	$0.549126\pi$
$284$	14.6642	0.870160
$285$	0	0
$286$	−1.96620	−0.116264
$287$	1.38127	0.0815337
$288$	27.2705	1.60693
$289$	−16.2332	−0.954895
$290$	0	0
$291$	34.5216	2.02369
$292$	−12.1134	−0.708883
$293$	−24.8835	−1.45371	−0.726854	−	0.686792i	$-0.759018\pi$
$293$	−24.8835	−1.45371	−0.726854	+	0.686792i	$0.759018\pi$
$294$	7.34500	0.428369
$295$	0	0
$296$	−8.89289	−0.516889
$297$	−14.6536	−0.850286
$298$	−2.13122	−0.123458
$299$	27.9362	1.61559
$300$	0	0
$301$	1.09942	0.0633695
$302$	−5.18384	−0.298297
$303$	−35.2758	−2.02654
$304$	−3.37811	−0.193748
$305$	0	0
$306$	−2.13841	−0.122245
$307$	1.37200	0.0783042	0.0391521	−	0.999233i	$-0.487534\pi$
$307$	1.37200	0.0783042	0.0391521	+	0.999233i	$0.487534\pi$
$308$	0.314499	0.0179202
$309$	−18.7639	−1.06744
$310$	0	0
$311$	1.75534	0.0995360	0.0497680	−	0.998761i	$-0.484152\pi$
$311$	1.75534	0.0995360	0.0497680	+	0.998761i	$0.484152\pi$
$312$	−24.8343	−1.40596
$313$	18.5787	1.05013	0.525065	−	0.851062i	$-0.324041\pi$
$313$	18.5787	1.05013	0.525065	+	0.851062i	$0.324041\pi$
$314$	4.49597	0.253722
$315$	0	0
$316$	0.585901	0.0329595
$317$	−4.00886	−0.225160	−0.112580	−	0.993643i	$-0.535911\pi$
$317$	−4.00886	−0.225160	−0.112580	+	0.993643i	$0.535911\pi$
$318$	−10.6141	−0.595210
$319$	−1.00709	−0.0563862
$320$	0	0
$321$	−15.4265	−0.861021
$322$	0.248848	0.0138677
$323$	0.875666	0.0487234
$324$	−47.3045	−2.62803
$325$	0	0
$326$	0.218704	0.0121129
$327$	−45.6728	−2.52571
$328$	10.5254	0.581169
$329$	−1.39944	−0.0771537
$330$	0	0
$331$	6.40426	0.352010	0.176005	−	0.984389i	$-0.443682\pi$
$331$	6.40426	0.352010	0.176005	+	0.984389i	$0.443682\pi$
$332$	−5.98260	−0.328338
$333$	52.8536	2.89636
$334$	2.94240	0.161001
$335$	0	0
$336$	1.81874	0.0992204
$337$	−26.4195	−1.43916	−0.719582	−	0.694407i	$-0.755667\pi$
$337$	−26.4195	−1.43916	−0.719582	+	0.694407i	$0.755667\pi$
$338$	7.67945	0.417707
$339$	−53.8035	−2.92221
$340$	0	0
$341$	8.35612	0.452509
$342$	−2.44204	−0.132050
$343$	2.31952	0.125242
$344$	8.37769	0.451695
$345$	0	0
$346$	−7.06013	−0.379555
$347$	6.14023	0.329625	0.164812	−	0.986325i	$-0.447298\pi$
$347$	6.14023	0.329625	0.164812	+	0.986325i	$0.447298\pi$
$348$	−6.18776	−0.331699
$349$	17.0701	0.913742	0.456871	−	0.889533i	$-0.348970\pi$
$349$	17.0701	0.913742	0.456871	+	0.889533i	$0.348970\pi$
$350$	0	0
$351$	88.7026	4.73459
$352$	3.62723	0.193332
$353$	24.1803	1.28699	0.643495	−	0.765451i	$-0.277484\pi$
$353$	24.1803	1.28699	0.643495	+	0.765451i	$0.277484\pi$
$354$	−5.24662	−0.278855
$355$	0	0
$356$	−24.6674	−1.30737
$357$	−0.471450	−0.0249518
$358$	−0.452270	−0.0239032
$359$	−27.5984	−1.45659	−0.728294	−	0.685265i	$-0.759687\pi$
$359$	−27.5984	−1.45659	−0.728294	+	0.685265i	$0.759687\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	4.07534	0.214195
$363$	−3.24319	−0.170223
$364$	−1.90376	−0.0997841
$365$	0	0
$366$	−7.71865	−0.403460
$367$	−0.102872	−0.00536987	−0.00268494	−	0.999996i	$-0.500855\pi$
$367$	−0.102872	−0.00536987	−0.00268494	+	0.999996i	$0.500855\pi$
$368$	−15.5900	−0.812687
$369$	−62.5562	−3.25655
$370$	0	0
$371$	−1.67264	−0.0868392
$372$	51.3417	2.66194
$373$	−11.1255	−0.576055	−0.288027	−	0.957622i	$-0.592999\pi$
$373$	−11.1255	−0.576055	−0.288027	+	0.957622i	$0.592999\pi$
$374$	−0.284429	−0.0147075
$375$	0	0
$376$	−10.6639	−0.549948
$377$	6.09623	0.313972
$378$	0.790138	0.0406403
$379$	−19.2468	−0.988643	−0.494321	−	0.869279i	$-0.664583\pi$
$379$	−19.2468	−0.988643	−0.494321	+	0.869279i	$0.664583\pi$
$380$	0	0
$381$	52.0053	2.66431
$382$	−4.78356	−0.244748
$383$	−8.55556	−0.437169	−0.218584	−	0.975818i	$-0.570144\pi$
$383$	−8.55556	−0.437169	−0.218584	+	0.975818i	$0.570144\pi$
$384$	29.4037	1.50050
$385$	0	0
$386$	−3.81614	−0.194236
$387$	−49.7916	−2.53105
$388$	20.1657	1.02376
$389$	16.7276	0.848125	0.424063	−	0.905633i	$-0.360604\pi$
$389$	16.7276	0.848125	0.424063	+	0.905633i	$0.360604\pi$
$390$	0	0
$391$	4.04122	0.204373
$392$	8.82005	0.445480
$393$	−30.8119	−1.55425
$394$	−4.24543	−0.213882
$395$	0	0
$396$	−14.2433	−0.715753
$397$	37.0868	1.86133	0.930667	−	0.365867i	$-0.119228\pi$
$397$	37.0868	1.86133	0.930667	+	0.365867i	$0.119228\pi$
$398$	−5.44482	−0.272924
$399$	−0.538390	−0.0269532
$400$	0	0
$401$	−4.15586	−0.207534	−0.103767	−	0.994602i	$-0.533090\pi$
$401$	−4.15586	−0.207534	−0.103767	+	0.994602i	$0.533090\pi$
$402$	−0.379584	−0.0189319
$403$	−50.5822	−2.51968
$404$	−20.6062	−1.02520
$405$	0	0
$406$	0.0543035	0.00269504
$407$	7.03003	0.348466
$408$	−3.59250	−0.177855
$409$	−19.6949	−0.973851	−0.486926	−	0.873443i	$-0.661882\pi$
$409$	−19.6949	−0.973851	−0.486926	+	0.873443i	$0.661882\pi$
$410$	0	0
$411$	45.8083	2.25956
$412$	−10.9608	−0.540002
$413$	−0.826796	−0.0406840
$414$	−11.2701	−0.553893
$415$	0	0
$416$	−21.9568	−1.07652
$417$	−28.4827	−1.39480
$418$	−0.324814	−0.0158872
$419$	−4.19043	−0.204716	−0.102358	−	0.994748i	$-0.532639\pi$
$419$	−4.19043	−0.204716	−0.102358	+	0.994748i	$0.532639\pi$
$420$	0	0
$421$	22.5663	1.09981	0.549907	−	0.835226i	$-0.314663\pi$
$421$	22.5663	1.09981	0.549907	+	0.835226i	$0.314663\pi$
$422$	−6.32111	−0.307707
$423$	63.3792	3.08160
$424$	−12.7457	−0.618985
$425$	0	0
$426$	8.15401	0.395063
$427$	−1.21635	−0.0588635
$428$	−9.01131	−0.435578
$429$	19.6320	0.947843
$430$	0	0
$431$	−18.3524	−0.884002	−0.442001	−	0.897014i	$-0.645731\pi$
$431$	−18.3524	−0.884002	−0.442001	+	0.897014i	$0.645731\pi$
$432$	−49.5013	−2.38163
$433$	1.78082	0.0855810	0.0427905	−	0.999084i	$-0.486375\pi$
$433$	1.78082	0.0855810	0.0427905	+	0.999084i	$0.486375\pi$
$434$	−0.450572	−0.0216282
$435$	0	0
$436$	−26.6796	−1.27772
$437$	4.61502	0.220766
$438$	−6.73564	−0.321841
$439$	−26.8306	−1.28055	−0.640277	−	0.768144i	$-0.721180\pi$
$439$	−26.8306	−1.28055	−0.640277	+	0.768144i	$0.721180\pi$
$440$	0	0
$441$	−52.4206	−2.49622
$442$	1.72174	0.0818946
$443$	15.4460	0.733862	0.366931	−	0.930248i	$-0.380409\pi$
$443$	15.4460	0.733862	0.366931	+	0.930248i	$0.380409\pi$
$444$	43.1939	2.04989
$445$	0	0
$446$	−3.70687	−0.175525
$447$	21.2797	1.00650
$448$	0.925991	0.0437490
$449$	−14.7640	−0.696758	−0.348379	−	0.937354i	$-0.613268\pi$
$449$	−14.7640	−0.696758	−0.348379	+	0.937354i	$0.613268\pi$
$450$	0	0
$451$	−8.32057	−0.391800
$452$	−31.4291	−1.47830
$453$	51.7594	2.43187
$454$	−1.30405	−0.0612020
$455$	0	0
$456$	−4.10259	−0.192121
$457$	16.2215	0.758811	0.379406	−	0.925230i	$-0.376128\pi$
$457$	16.2215	0.758811	0.379406	+	0.925230i	$0.376128\pi$
$458$	8.54717	0.399383
$459$	12.8316	0.598929
$460$	0	0
$461$	13.4764	0.627659	0.313830	−	0.949479i	$-0.398388\pi$
$461$	13.4764	0.627659	0.313830	+	0.949479i	$0.398388\pi$
$462$	0.174877	0.00813600
$463$	28.6461	1.33130	0.665648	−	0.746266i	$-0.268155\pi$
$463$	28.6461	1.33130	0.665648	+	0.746266i	$0.268155\pi$
$464$	−3.40205	−0.157936
$465$	0	0
$466$	4.76829	0.220887
$467$	−17.0725	−0.790023	−0.395011	−	0.918676i	$-0.629259\pi$
$467$	−17.0725	−0.790023	−0.395011	+	0.918676i	$0.629259\pi$
$468$	86.2192	3.98549
$469$	−0.0598173	−0.00276211
$470$	0	0
$471$	−44.8911	−2.06847
$472$	−6.30027	−0.289993
$473$	−6.62275	−0.304514
$474$	0.325790	0.0149640
$475$	0	0
$476$	−0.275396	−0.0126228
$477$	75.7521	3.46845
$478$	−9.27539	−0.424246
$479$	28.3507	1.29538	0.647689	−	0.761905i	$-0.275736\pi$
$479$	28.3507	1.29538	0.647689	+	0.761905i	$0.275736\pi$
$480$	0	0
$481$	−42.5550	−1.94034
$482$	−7.25720	−0.330556
$483$	−2.48468	−0.113057
$484$	−1.89450	−0.0861134
$485$	0	0
$486$	−12.0246	−0.545447
$487$	11.4047	0.516798	0.258399	−	0.966038i	$-0.416805\pi$
$487$	11.4047	0.516798	0.258399	+	0.966038i	$0.416805\pi$
$488$	−9.26874	−0.419576
$489$	−2.18371	−0.0987507
$490$	0	0
$491$	−29.9154	−1.35006	−0.675032	−	0.737788i	$-0.735870\pi$
$491$	−29.9154	−1.35006	−0.675032	+	0.737788i	$0.735870\pi$
$492$	−51.1233	−2.30482
$493$	0.881874	0.0397176
$494$	1.96620	0.0884636
$495$	0	0
$496$	28.2279	1.26747
$497$	1.28496	0.0576384
$498$	−3.32662	−0.149069
$499$	−37.9248	−1.69775	−0.848875	−	0.528594i	$-0.822719\pi$
$499$	−37.9248	−1.69775	−0.848875	+	0.528594i	$0.822719\pi$
$500$	0	0
$501$	−29.3792	−1.31256
$502$	3.74010	0.166929
$503$	−2.74270	−0.122291	−0.0611456	−	0.998129i	$-0.519475\pi$
$503$	−2.74270	−0.122291	−0.0611456	+	0.998129i	$0.519475\pi$
$504$	1.57881	0.0703256
$505$	0	0
$506$	−1.49902	−0.0666397
$507$	−76.6774	−3.40536
$508$	30.3787	1.34784
$509$	0.635877	0.0281847	0.0140924	−	0.999901i	$-0.495514\pi$
$509$	0.635877	0.0281847	0.0140924	+	0.999901i	$0.495514\pi$
$510$	0	0
$511$	−1.06145	−0.0469556
$512$	20.7997	0.919225
$513$	14.6536	0.646971
$514$	−3.42608	−0.151118
$515$	0	0
$516$	−40.6915	−1.79134
$517$	8.43004	0.370753
$518$	−0.379068	−0.0166553
$519$	70.4936	3.09433
$520$	0	0
$521$	28.8718	1.26490	0.632449	−	0.774602i	$-0.282050\pi$
$521$	28.8718	1.26490	0.632449	+	0.774602i	$0.282050\pi$
$522$	−2.45935	−0.107643
$523$	20.2611	0.885956	0.442978	−	0.896532i	$-0.353922\pi$
$523$	20.2611	0.885956	0.442978	+	0.896532i	$0.353922\pi$
$524$	−17.9987	−0.786275
$525$	0	0
$526$	−6.59910	−0.287734
$527$	−7.31717	−0.318741
$528$	−10.9558	−0.476791
$529$	−1.70157	−0.0739814
$530$	0	0
$531$	37.4447	1.62496
$532$	−0.314499	−0.0136352
$533$	50.3670	2.18164
$534$	−13.7163	−0.593561
$535$	0	0
$536$	−0.455814	−0.0196882
$537$	4.51580	0.194871
$538$	−5.34720	−0.230534
$539$	−6.97244	−0.300324
$540$	0	0
$541$	25.4105	1.09248	0.546242	−	0.837628i	$-0.316058\pi$
$541$	25.4105	1.09248	0.546242	+	0.837628i	$0.316058\pi$
$542$	7.54518	0.324093
$543$	−40.6912	−1.74623
$544$	−3.17624	−0.136180
$545$	0	0
$546$	−1.05858	−0.0453032
$547$	0.458489	0.0196036	0.00980178	−	0.999952i	$-0.496880\pi$
$547$	0.458489	0.0196036	0.00980178	+	0.999952i	$0.496880\pi$
$548$	26.7588	1.14308
$549$	55.0874	2.35107
$550$	0	0
$551$	1.00709	0.0429034
$552$	−18.9335	−0.805865
$553$	0.0513400	0.00218320
$554$	5.76101	0.244762
$555$	0	0
$556$	−16.6381	−0.705611
$557$	43.5515	1.84534	0.922669	−	0.385593i	$-0.126003\pi$
$557$	43.5515	1.84534	0.922669	+	0.385593i	$0.126003\pi$
$558$	20.4059	0.863852
$559$	40.0896	1.69561
$560$	0	0
$561$	2.83995	0.119903
$562$	−8.34492	−0.352009
$563$	−46.9523	−1.97880	−0.989401	−	0.145208i	$-0.953615\pi$
$563$	−46.9523	−1.97880	−0.989401	+	0.145208i	$0.953615\pi$
$564$	51.7959	2.18100
$565$	0	0
$566$	−1.67993	−0.0706128
$567$	−4.14509	−0.174078
$568$	9.79153	0.410844
$569$	30.2122	1.26656	0.633282	−	0.773922i	$-0.281708\pi$
$569$	30.2122	1.26656	0.633282	+	0.773922i	$0.281708\pi$
$570$	0	0
$571$	23.2284	0.972077	0.486038	−	0.873938i	$-0.338442\pi$
$571$	23.2284	0.972077	0.486038	+	0.873938i	$0.338442\pi$
$572$	11.4680	0.479500
$573$	47.7627	1.99531
$574$	0.448656	0.0187265
$575$	0	0
$576$	−41.9371	−1.74738
$577$	−9.07629	−0.377851	−0.188926	−	0.981991i	$-0.560500\pi$
$577$	−9.07629	−0.377851	−0.188926	+	0.981991i	$0.560500\pi$
$578$	−5.27277	−0.219318
$579$	38.1032	1.58351
$580$	0	0
$581$	−0.524229	−0.0217487
$582$	11.2131	0.464798
$583$	10.0757	0.417295
$584$	−8.08832	−0.334697
$585$	0	0
$586$	−8.08250	−0.333885
$587$	36.7331	1.51614	0.758069	−	0.652174i	$-0.226143\pi$
$587$	36.7331	1.51614	0.758069	+	0.652174i	$0.226143\pi$
$588$	−42.8401	−1.76670
$589$	−8.35612	−0.344308
$590$	0	0
$591$	42.3896	1.74368
$592$	23.7482	0.976044
$593$	28.5431	1.17213	0.586063	−	0.810266i	$-0.300677\pi$
$593$	28.5431	1.17213	0.586063	+	0.810266i	$0.300677\pi$
$594$	−4.75968	−0.195292
$595$	0	0
$596$	12.4305	0.509172
$597$	54.3652	2.22502
$598$	9.07406	0.371066
$599$	−45.9055	−1.87565	−0.937825	−	0.347109i	$-0.887164\pi$
$599$	−45.9055	−1.87565	−0.937825	+	0.347109i	$0.887164\pi$
$600$	0	0
$601$	34.0331	1.38824	0.694120	−	0.719860i	$-0.255794\pi$
$601$	34.0331	1.38824	0.694120	+	0.719860i	$0.255794\pi$
$602$	0.357107	0.0145546
$603$	2.70906	0.110321
$604$	30.2351	1.23025
$605$	0	0
$606$	−11.4581	−0.465452
$607$	−3.84641	−0.156121	−0.0780606	−	0.996949i	$-0.524873\pi$
$607$	−3.84641	−0.156121	−0.0780606	+	0.996949i	$0.524873\pi$
$608$	−3.62723	−0.147104
$609$	−0.542207	−0.0219713
$610$	0	0
$611$	−51.0297	−2.06444
$612$	12.4724	0.504167
$613$	−17.5232	−0.707754	−0.353877	−	0.935292i	$-0.615137\pi$
$613$	−17.5232	−0.707754	−0.353877	+	0.935292i	$0.615137\pi$
$614$	0.445645	0.0179848
$615$	0	0
$616$	0.209996	0.00846098
$617$	14.0815	0.566900	0.283450	−	0.958987i	$-0.408521\pi$
$617$	14.0815	0.566900	0.283450	+	0.958987i	$0.408521\pi$
$618$	−6.09477	−0.245167
$619$	−23.1027	−0.928577	−0.464288	−	0.885684i	$-0.653690\pi$
$619$	−23.1027	−0.928577	−0.464288	+	0.885684i	$0.653690\pi$
$620$	0	0
$621$	67.6265	2.71376
$622$	0.570158	0.0228613
$623$	−2.16150	−0.0865985
$624$	66.3191	2.65489
$625$	0	0
$626$	6.03462	0.241192
$627$	3.24319	0.129520
$628$	−26.2230	−1.04641
$629$	−6.15596	−0.245454
$630$	0	0
$631$	3.60839	0.143648	0.0718239	−	0.997417i	$-0.477118\pi$
$631$	3.60839	0.143648	0.0718239	+	0.997417i	$0.477118\pi$
$632$	0.391216	0.0155617
$633$	63.1148	2.50859
$634$	−1.30213	−0.0517144
$635$	0	0
$636$	61.9074	2.45479
$637$	42.2064	1.67228
$638$	−0.327117	−0.0129507
$639$	−58.1945	−2.30214
$640$	0	0
$641$	−17.5350	−0.692589	−0.346294	−	0.938126i	$-0.612560\pi$
$641$	−17.5350	−0.692589	−0.346294	+	0.938126i	$0.612560\pi$
$642$	−5.01073	−0.197758
$643$	−19.6051	−0.773151	−0.386575	−	0.922258i	$-0.626342\pi$
$643$	−19.6051	−0.773151	−0.386575	+	0.922258i	$0.626342\pi$
$644$	−1.45142	−0.0571939
$645$	0	0
$646$	0.284429	0.0111907
$647$	−41.2895	−1.62326	−0.811629	−	0.584173i	$-0.801419\pi$
$647$	−41.2895	−1.62326	−0.811629	+	0.584173i	$0.801419\pi$
$648$	−31.5860	−1.24082
$649$	4.98050	0.195502
$650$	0	0
$651$	4.49885	0.176324
$652$	−1.27561	−0.0499566
$653$	8.64148	0.338167	0.169084	−	0.985602i	$-0.445919\pi$
$653$	8.64148	0.338167	0.169084	+	0.985602i	$0.445919\pi$
$654$	−14.8351	−0.580100
$655$	0	0
$656$	−28.1078	−1.09742
$657$	48.0717	1.87546
$658$	−0.454558	−0.0177205
$659$	−2.34694	−0.0914237	−0.0457118	−	0.998955i	$-0.514556\pi$
$659$	−2.34694	−0.0914237	−0.0457118	+	0.998955i	$0.514556\pi$
$660$	0	0
$661$	−37.8564	−1.47244	−0.736222	−	0.676740i	$-0.763392\pi$
$661$	−37.8564	−1.47244	−0.736222	+	0.676740i	$0.763392\pi$
$662$	2.08019	0.0808490
$663$	−17.1911	−0.667647
$664$	−3.99468	−0.155024
$665$	0	0
$666$	17.1676	0.665230
$667$	4.64774	0.179961
$668$	−17.1617	−0.664008
$669$	37.0122	1.43097
$670$	0	0
$671$	7.32714	0.282861
$672$	1.95287	0.0753334
$673$	−32.1381	−1.23883	−0.619417	−	0.785062i	$-0.712631\pi$
$673$	−32.1381	−1.23883	−0.619417	+	0.785062i	$0.712631\pi$
$674$	−8.58144	−0.330545
$675$	0	0
$676$	−44.7908	−1.72272
$677$	−46.2653	−1.77812	−0.889060	−	0.457791i	$-0.848641\pi$
$677$	−46.2653	−1.77812	−0.889060	+	0.457791i	$0.848641\pi$
$678$	−17.4761	−0.671167
$679$	1.76703	0.0678125
$680$	0	0
$681$	13.0206	0.498950
$682$	2.71418	0.103931
$683$	3.51505	0.134500	0.0672498	−	0.997736i	$-0.478578\pi$
$683$	3.51505	0.134500	0.0672498	+	0.997736i	$0.478578\pi$
$684$	14.2433	0.544607
$685$	0	0
$686$	0.753412	0.0287654
$687$	−85.3414	−3.25598
$688$	−22.3724	−0.852938
$689$	−60.9917	−2.32360
$690$	0	0
$691$	6.47549	0.246339	0.123170	−	0.992386i	$-0.460694\pi$
$691$	6.47549	0.246339	0.123170	+	0.992386i	$0.460694\pi$
$692$	41.1786	1.56537
$693$	−1.24808	−0.0474107
$694$	1.99443	0.0757077
$695$	0	0
$696$	−4.13167	−0.156611
$697$	7.28604	0.275979
$698$	5.54461	0.209867
$699$	−47.6102	−1.80078
$700$	0	0
$701$	−14.8438	−0.560643	−0.280321	−	0.959906i	$-0.590441\pi$
$701$	−14.8438	−0.560643	−0.280321	+	0.959906i	$0.590441\pi$
$702$	28.8118	1.08743
$703$	−7.03003	−0.265142
$704$	−5.57804	−0.210230
$705$	0	0
$706$	7.85411	0.295593
$707$	−1.80564	−0.0679079
$708$	30.6012	1.15006
$709$	−25.8169	−0.969572	−0.484786	−	0.874633i	$-0.661103\pi$
$709$	−25.8169	−0.969572	−0.484786	+	0.874633i	$0.661103\pi$
$710$	0	0
$711$	−2.32514	−0.0871994
$712$	−16.4708	−0.617270
$713$	−38.5637	−1.44422
$714$	−0.153134	−0.00573088
$715$	0	0
$716$	2.63789	0.0985826
$717$	92.6124	3.45868
$718$	−8.96435	−0.334547
$719$	25.3406	0.945044	0.472522	−	0.881319i	$-0.343344\pi$
$719$	25.3406	0.945044	0.472522	+	0.881319i	$0.343344\pi$
$720$	0	0
$721$	−0.960452	−0.0357691
$722$	0.324814	0.0120883
$723$	72.4613	2.69486
$724$	−23.7696	−0.883391
$725$	0	0
$726$	−1.05343	−0.0390965
$727$	−3.83993	−0.142415	−0.0712075	−	0.997462i	$-0.522685\pi$
$727$	−3.83993	−0.142415	−0.0712075	+	0.997462i	$0.522685\pi$
$728$	−1.27117	−0.0471128
$729$	45.1542	1.67238
$730$	0	0
$731$	5.79932	0.214496
$732$	45.0195	1.66397
$733$	−19.8037	−0.731466	−0.365733	−	0.930720i	$-0.619182\pi$
$733$	−19.8037	−0.731466	−0.365733	+	0.930720i	$0.619182\pi$
$734$	−0.0334142	−0.00123334
$735$	0	0
$736$	−16.7397	−0.617035
$737$	0.360331	0.0132730
$738$	−20.3191	−0.747958
$739$	−43.0878	−1.58501	−0.792506	−	0.609864i	$-0.791224\pi$
$739$	−43.0878	−1.58501	−0.792506	+	0.609864i	$0.791224\pi$
$740$	0	0
$741$	−19.6320	−0.721200
$742$	−0.543297	−0.0199451
$743$	16.0472	0.588714	0.294357	−	0.955696i	$-0.404895\pi$
$743$	16.0472	0.588714	0.294357	+	0.955696i	$0.404895\pi$
$744$	34.2817	1.25683
$745$	0	0
$746$	−3.61371	−0.132307
$747$	23.7418	0.868667
$748$	1.65895	0.0606571
$749$	−0.789622	−0.0288522
$750$	0	0
$751$	−34.8124	−1.27032	−0.635161	−	0.772380i	$-0.719066\pi$
$751$	−34.8124	−1.27032	−0.635161	+	0.772380i	$0.719066\pi$
$752$	28.4776	1.03847
$753$	−37.3440	−1.36089
$754$	1.98014	0.0721124
$755$	0	0
$756$	−4.60853	−0.167611
$757$	31.5794	1.14777	0.573886	−	0.818935i	$-0.305435\pi$
$757$	31.5794	1.14777	0.573886	+	0.818935i	$0.305435\pi$
$758$	−6.25164	−0.227070
$759$	14.9674	0.543281
$760$	0	0
$761$	−26.0445	−0.944113	−0.472056	−	0.881568i	$-0.656488\pi$
$761$	−26.0445	−0.944113	−0.472056	+	0.881568i	$0.656488\pi$
$762$	16.8920	0.611934
$763$	−2.33782	−0.0846346
$764$	27.9004	1.00940
$765$	0	0
$766$	−2.77897	−0.100408
$767$	−30.1485	−1.08860
$768$	−26.6305	−0.960946
$769$	−1.27892	−0.0461189	−0.0230595	−	0.999734i	$-0.507341\pi$
$769$	−1.27892	−0.0461189	−0.0230595	+	0.999734i	$0.507341\pi$
$770$	0	0
$771$	34.2086	1.23199
$772$	22.2578	0.801077
$773$	43.3371	1.55873	0.779364	−	0.626571i	$-0.215542\pi$
$773$	43.3371	1.55873	0.779364	+	0.626571i	$0.215542\pi$
$774$	−16.1730	−0.581326
$775$	0	0
$776$	13.4650	0.483364
$777$	3.78490	0.135782
$778$	5.43337	0.194796
$779$	8.32057	0.298115
$780$	0	0
$781$	−7.74042	−0.276974
$782$	1.31264	0.0469401
$783$	14.7574	0.527388
$784$	−23.5537	−0.841202
$785$	0	0
$786$	−10.0081	−0.356978
$787$	−15.9911	−0.570022	−0.285011	−	0.958524i	$-0.591997\pi$
$787$	−15.9911	−0.570022	−0.285011	+	0.958524i	$0.591997\pi$
$788$	24.7617	0.882100
$789$	65.8903	2.34576
$790$	0	0
$791$	−2.75400	−0.0979210
$792$	−9.51050	−0.337941
$793$	−44.3535	−1.57504
$794$	12.0463	0.427508
$795$	0	0
$796$	31.7572	1.12561
$797$	−28.8110	−1.02054	−0.510269	−	0.860015i	$-0.670454\pi$
$797$	−28.8110	−1.02054	−0.510269	+	0.860015i	$0.670454\pi$
$798$	−0.174877	−0.00619057
$799$	−7.38190	−0.261153
$800$	0	0
$801$	97.8919	3.45884
$802$	−1.34988	−0.0476659
$803$	6.39400	0.225639
$804$	2.21395	0.0780799
$805$	0	0
$806$	−16.4298	−0.578715
$807$	53.3904	1.87943
$808$	−13.7591	−0.484044
$809$	47.1962	1.65933	0.829666	−	0.558261i	$-0.188531\pi$
$809$	47.1962	1.65933	0.829666	+	0.558261i	$0.188531\pi$
$810$	0	0
$811$	21.4242	0.752304	0.376152	−	0.926558i	$-0.377247\pi$
$811$	21.4242	0.752304	0.376152	+	0.926558i	$0.377247\pi$
$812$	−0.316728	−0.0111150
$813$	−75.3367	−2.64217
$814$	2.28345	0.0800349
$815$	0	0
$816$	9.59365	0.335845
$817$	6.62275	0.231701
$818$	−6.39718	−0.223672
$819$	7.55502	0.263994
$820$	0	0
$821$	9.70806	0.338814	0.169407	−	0.985546i	$-0.445815\pi$
$821$	9.70806	0.338814	0.169407	+	0.985546i	$0.445815\pi$
$822$	14.8792	0.518971
$823$	−41.7049	−1.45374	−0.726871	−	0.686774i	$-0.759026\pi$
$823$	−41.7049	−1.45374	−0.726871	+	0.686774i	$0.759026\pi$
$824$	−7.31874	−0.254960
$825$	0	0
$826$	−0.268555	−0.00934422
$827$	19.1359	0.665420	0.332710	−	0.943029i	$-0.392037\pi$
$827$	19.1359	0.665420	0.332710	+	0.943029i	$0.392037\pi$
$828$	65.7332	2.28439
$829$	−53.7809	−1.86789	−0.933943	−	0.357421i	$-0.883656\pi$
$829$	−53.7809	−1.86789	−0.933943	+	0.357421i	$0.883656\pi$
$830$	0	0
$831$	−57.5222	−1.99542
$832$	33.7656	1.17061
$833$	6.10553	0.211544
$834$	−9.25158	−0.320356
$835$	0	0
$836$	1.89450	0.0655225
$837$	−122.447	−4.23238
$838$	−1.36111	−0.0470188
$839$	−11.9762	−0.413464	−0.206732	−	0.978398i	$-0.566283\pi$
$839$	−11.9762	−0.413464	−0.206732	+	0.978398i	$0.566283\pi$
$840$	0	0
$841$	−27.9858	−0.965027
$842$	7.32985	0.252603
$843$	83.3220	2.86976
$844$	36.8683	1.26906
$845$	0	0
$846$	20.5865	0.707777
$847$	−0.166007	−0.00570405
$848$	34.0369	1.16883
$849$	16.7737	0.575672
$850$	0	0
$851$	−32.4437	−1.11216
$852$	−47.5587	−1.62934
$853$	−36.1717	−1.23850	−0.619248	−	0.785196i	$-0.712562\pi$
$853$	−36.1717	−1.23850	−0.619248	+	0.785196i	$0.712562\pi$
$854$	−0.395089	−0.0135197
$855$	0	0
$856$	−6.01700	−0.205657
$857$	13.2431	0.452375	0.226187	−	0.974084i	$-0.427374\pi$
$857$	13.2431	0.452375	0.226187	+	0.974084i	$0.427374\pi$
$858$	6.37676	0.217699
$859$	−18.5582	−0.633198	−0.316599	−	0.948560i	$-0.602541\pi$
$859$	−18.5582	−0.633198	−0.316599	+	0.948560i	$0.602541\pi$
$860$	0	0
$861$	−4.47971	−0.152668
$862$	−5.96111	−0.203036
$863$	5.72788	0.194979	0.0974896	−	0.995237i	$-0.468919\pi$
$863$	5.72788	0.194979	0.0974896	+	0.995237i	$0.468919\pi$
$864$	−53.1518	−1.80826
$865$	0	0
$866$	0.578437	0.0196561
$867$	52.6473	1.78800
$868$	2.62799	0.0891997
$869$	−0.309265	−0.0104911
$870$	0	0
$871$	−2.18120	−0.0739070
$872$	−17.8144	−0.603272
$873$	−80.0270	−2.70850
$874$	1.49902	0.0507052
$875$	0	0
$876$	39.2860	1.32735
$877$	−1.40932	−0.0475893	−0.0237947	−	0.999717i	$-0.507575\pi$
$877$	−1.40932	−0.0475893	−0.0237947	+	0.999717i	$0.507575\pi$
$878$	−8.71495	−0.294115
$879$	80.7017	2.72200
$880$	0	0
$881$	21.6242	0.728538	0.364269	−	0.931294i	$-0.381319\pi$
$881$	21.6242	0.728538	0.364269	+	0.931294i	$0.381319\pi$
$882$	−17.0270	−0.573327
$883$	−45.4323	−1.52892	−0.764460	−	0.644671i	$-0.776994\pi$
$883$	−45.4323	−1.52892	−0.764460	+	0.644671i	$0.776994\pi$
$884$	−10.0421	−0.337753
$885$	0	0
$886$	5.01708	0.168552
$887$	−35.7575	−1.20062	−0.600310	−	0.799767i	$-0.704956\pi$
$887$	−35.7575	−1.20062	−0.600310	+	0.799767i	$0.704956\pi$
$888$	28.8413	0.967851
$889$	2.66195	0.0892791
$890$	0	0
$891$	24.9695	0.836508
$892$	21.6205	0.723909
$893$	−8.43004	−0.282101
$894$	6.91195	0.231170
$895$	0	0
$896$	1.50506	0.0502806
$897$	−90.6022	−3.02512
$898$	−4.79557	−0.160030
$899$	−8.41536	−0.280668
$900$	0	0
$901$	−8.82299	−0.293937
$902$	−2.70264	−0.0899880
$903$	−3.56562	−0.118657
$904$	−20.9858	−0.697976
$905$	0	0
$906$	16.8122	0.558547
$907$	−11.5725	−0.384257	−0.192129	−	0.981370i	$-0.561539\pi$
$907$	−11.5725	−0.384257	−0.192129	+	0.981370i	$0.561539\pi$
$908$	7.60593	0.252412
$909$	81.7753	2.71232
$910$	0	0
$911$	19.3672	0.641665	0.320832	−	0.947136i	$-0.396037\pi$
$911$	19.3672	0.641665	0.320832	+	0.947136i	$0.396037\pi$
$912$	10.9558	0.362784
$913$	3.15788	0.104511
$914$	5.26898	0.174282
$915$	0	0
$916$	−49.8518	−1.64715
$917$	−1.57715	−0.0520819
$918$	4.16789	0.137561
$919$	35.9029	1.18433	0.592163	−	0.805818i	$-0.298274\pi$
$919$	35.9029	1.18433	0.592163	+	0.805818i	$0.298274\pi$
$920$	0	0
$921$	−4.44966	−0.146621
$922$	4.37733	0.144160
$923$	46.8552	1.54226
$924$	−1.01998	−0.0335548
$925$	0	0
$926$	9.30464	0.305769
$927$	43.4979	1.42866
$928$	−3.65294	−0.119914
$929$	−26.0456	−0.854529	−0.427265	−	0.904127i	$-0.640523\pi$
$929$	−26.0456	−0.854529	−0.427265	+	0.904127i	$0.640523\pi$
$930$	0	0
$931$	6.97244	0.228513
$932$	−27.8113	−0.910991
$933$	−5.69289	−0.186377
$934$	−5.54540	−0.181451
$935$	0	0
$936$	57.5700	1.88174
$937$	−59.8785	−1.95615	−0.978073	−	0.208262i	$-0.933219\pi$
$937$	−59.8785	−1.95615	−0.978073	+	0.208262i	$0.933219\pi$
$938$	−0.0194295	−0.000634395 0
$939$	−60.2542	−1.96632
$940$	0	0
$941$	17.0743	0.556606	0.278303	−	0.960493i	$-0.410228\pi$
$941$	17.0743	0.556606	0.278303	+	0.960493i	$0.410228\pi$
$942$	−14.5813	−0.475083
$943$	38.3996	1.25046
$944$	16.8247	0.547596
$945$	0	0
$946$	−2.15116	−0.0699403
$947$	−9.94311	−0.323108	−0.161554	−	0.986864i	$-0.551651\pi$
$947$	−9.94311	−0.323108	−0.161554	+	0.986864i	$0.551651\pi$
$948$	−1.90019	−0.0617152
$949$	−38.7049	−1.25641
$950$	0	0
$951$	13.0015	0.421602
$952$	−0.183887	−0.00595980
$953$	−29.7554	−0.963871	−0.481935	−	0.876207i	$-0.660066\pi$
$953$	−29.7554	−0.963871	−0.481935	+	0.876207i	$0.660066\pi$
$954$	24.6053	0.796627
$955$	0	0
$956$	54.0992	1.74969
$957$	3.26618	0.105581
$958$	9.20872	0.297520
$959$	2.34476	0.0757161
$960$	0	0
$961$	38.8247	1.25241
$962$	−13.8224	−0.445654
$963$	35.7612	1.15239
$964$	42.3280	1.36329
$965$	0	0
$966$	−0.807060	−0.0259667
$967$	−21.9469	−0.705765	−0.352883	−	0.935668i	$-0.614798\pi$
$967$	−21.9469	−0.705765	−0.352883	+	0.935668i	$0.614798\pi$
$968$	−1.26499	−0.0406582
$969$	−2.83995	−0.0912323
$970$	0	0
$971$	14.5115	0.465695	0.232847	−	0.972513i	$-0.425196\pi$
$971$	14.5115	0.465695	0.232847	+	0.972513i	$0.425196\pi$
$972$	70.1341	2.24955
$973$	−1.45792	−0.0467389
$974$	3.70442	0.118697
$975$	0	0
$976$	24.7519	0.792288
$977$	50.3307	1.61022	0.805111	−	0.593124i	$-0.202106\pi$
$977$	50.3307	1.61022	0.805111	+	0.593124i	$0.202106\pi$
$978$	−0.709299	−0.0226809
$979$	13.0206	0.416139
$980$	0	0
$981$	105.877	3.38040
$982$	−9.71695	−0.310080
$983$	−9.71499	−0.309860	−0.154930	−	0.987925i	$-0.549515\pi$
$983$	−9.71499	−0.309860	−0.154930	+	0.987925i	$0.549515\pi$
$984$	−34.1359	−1.08821
$985$	0	0
$986$	0.286445	0.00912227
$987$	4.53865	0.144467
$988$	−11.4680	−0.364845
$989$	30.5641	0.971883
$990$	0	0
$991$	24.1420	0.766897	0.383449	−	0.923562i	$-0.374736\pi$
$991$	24.1420	0.766897	0.383449	+	0.923562i	$0.374736\pi$
$992$	30.3096	0.962330
$993$	−20.7702	−0.659123
$994$	0.417373	0.0132383
$995$	0	0
$996$	19.4027	0.614797
$997$	5.23428	0.165771	0.0828856	−	0.996559i	$-0.473586\pi$
$997$	5.23428	0.165771	0.0828856	+	0.996559i	$0.473586\pi$
$998$	−12.3185	−0.389936
$999$	−103.015	−3.25925

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5225.2.a.s.1.10	✓	15
5.4	even	2		5225.2.a.x.1.6	yes	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5225.2.a.s.1.10	✓	15	1.1	even	1	trivial
5225.2.a.x.1.6	yes	15	5.4	even	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 5225 = 5^{2} \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5225.a (trivial)

\(f(q)\)	\(=\)	\(q+0.324814 q^{2} -3.24319 q^{3} -1.89450 q^{4} -1.05343 q^{6} -0.166007 q^{7} -1.26499 q^{8} +7.51826 q^{9} +1.00000 q^{11} +6.14420 q^{12} -6.05331 q^{13} -0.0539213 q^{14} +3.37811 q^{16} -0.875666 q^{17} +2.44204 q^{18} -1.00000 q^{19} +0.538390 q^{21} +0.324814 q^{22} -4.61502 q^{23} +4.10259 q^{24} -1.96620 q^{26} -14.6536 q^{27} +0.314499 q^{28} -1.00709 q^{29} +8.35612 q^{31} +3.62723 q^{32} -3.24319 q^{33} -0.284429 q^{34} -14.2433 q^{36} +7.03003 q^{37} -0.324814 q^{38} +19.6320 q^{39} -8.32057 q^{41} +0.174877 q^{42} -6.62275 q^{43} -1.89450 q^{44} -1.49902 q^{46} +8.43004 q^{47} -10.9558 q^{48} -6.97244 q^{49} +2.83995 q^{51} +11.4680 q^{52} +10.0757 q^{53} -4.75968 q^{54} +0.209996 q^{56} +3.24319 q^{57} -0.327117 q^{58} +4.98050 q^{59} +7.32714 q^{61} +2.71418 q^{62} -1.24808 q^{63} -5.57804 q^{64} -1.05343 q^{66} +0.360331 q^{67} +1.65895 q^{68} +14.9674 q^{69} -7.74042 q^{71} -9.51050 q^{72} +6.39400 q^{73} +2.28345 q^{74} +1.89450 q^{76} -0.166007 q^{77} +6.37676 q^{78} -0.309265 q^{79} +24.9695 q^{81} -2.70264 q^{82} +3.15788 q^{83} -1.01998 q^{84} -2.15116 q^{86} +3.26618 q^{87} -1.26499 q^{88} +13.0206 q^{89} +1.00489 q^{91} +8.74314 q^{92} -27.1005 q^{93} +2.73819 q^{94} -11.7638 q^{96} -10.6444 q^{97} -2.26475 q^{98} +7.51826 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q - 5 q^{2} - 4 q^{3} + 17 q^{4} - q^{6} - 15 q^{7} - 15 q^{8} + 19 q^{9} + 15 q^{11} - 9 q^{12} - 13 q^{13} + 9 q^{14} + 21 q^{16} - 11 q^{17} - 16 q^{18} - 15 q^{19} - 5 q^{22} - 10 q^{23} + 17 q^{24}+ \cdots + 19 q^{99}+O(q^{100}) \) 15 * q - 5 * q^2 - 4 * q^3 + 17 * q^4 - q^6 - 15 * q^7 - 15 * q^8 + 19 * q^9 + 15 * q^11 - 9 * q^12 - 13 * q^13 + 9 * q^14 + 21 * q^16 - 11 * q^17 - 16 * q^18 - 15 * q^19 - 5 * q^22 - 10 * q^23 + 17 * q^24 - 17 * q^26 - 13 * q^27 - 30 * q^28 + 5 * q^29 + 6 * q^31 - 40 * q^32 - 4 * q^33 + 17 * q^34 + 28 * q^36 - 13 * q^37 + 5 * q^38 - 22 * q^39 - 30 * q^42 - 36 * q^43 + 17 * q^44 + 13 * q^46 + 6 * q^47 - 14 * q^48 + 16 * q^49 + 4 * q^51 - 50 * q^52 - 9 * q^53 + 9 * q^54 - 18 * q^56 + 4 * q^57 - 2 * q^58 - 7 * q^59 - 2 * q^61 - 11 * q^62 - 39 * q^63 + 17 * q^64 - q^66 - 35 * q^67 + 18 * q^68 - 9 * q^69 + 13 * q^71 - 68 * q^72 - 2 * q^73 + 13 * q^74 - 17 * q^76 - 15 * q^77 + 10 * q^78 + 6 * q^79 + 11 * q^81 - 14 * q^82 - 30 * q^83 - 6 * q^84 - 25 * q^86 - 19 * q^87 - 15 * q^88 + 55 * q^89 + 26 * q^91 + 18 * q^92 - 14 * q^93 - 22 * q^94 - 17 * q^96 - 28 * q^97 + 22 * q^98 + 19 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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