LMFDB - Embedded newform 5225.2.a.s.1.7

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			1.7
Root			$1.10124$ 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)

$f(q)$	$=$	\(q-1.10124 q^{2} +1.12617 q^{3} -0.787279 q^{4} -1.24018 q^{6} +1.81708 q^{7} +3.06945 q^{8} -1.73174 q^{9} +1.00000 q^{11} -0.886611 q^{12} +3.00590 q^{13} -2.00104 q^{14} -1.80563 q^{16} +0.396094 q^{17} +1.90705 q^{18} -1.00000 q^{19} +2.04635 q^{21} -1.10124 q^{22} +2.47213 q^{23} +3.45673 q^{24} -3.31020 q^{26} -5.32875 q^{27} -1.43055 q^{28} -4.66419 q^{29} -10.5803 q^{31} -4.15048 q^{32} +1.12617 q^{33} -0.436192 q^{34} +1.36336 q^{36} -3.18261 q^{37} +1.10124 q^{38} +3.38516 q^{39} -9.74534 q^{41} -2.25351 q^{42} -3.52635 q^{43} -0.787279 q^{44} -2.72240 q^{46} +3.07500 q^{47} -2.03345 q^{48} -3.69820 q^{49} +0.446069 q^{51} -2.36648 q^{52} -1.32628 q^{53} +5.86821 q^{54} +5.57745 q^{56} -1.12617 q^{57} +5.13637 q^{58} -6.08186 q^{59} +13.2731 q^{61} +11.6514 q^{62} -3.14672 q^{63} +8.18192 q^{64} -1.24018 q^{66} -9.86746 q^{67} -0.311836 q^{68} +2.78404 q^{69} +3.14591 q^{71} -5.31549 q^{72} +10.7607 q^{73} +3.50480 q^{74} +0.787279 q^{76} +1.81708 q^{77} -3.72786 q^{78} +2.48793 q^{79} -0.805865 q^{81} +10.7319 q^{82} +8.40591 q^{83} -1.61105 q^{84} +3.88335 q^{86} -5.25268 q^{87} +3.06945 q^{88} -4.18002 q^{89} +5.46197 q^{91} -1.94626 q^{92} -11.9152 q^{93} -3.38630 q^{94} -4.67415 q^{96} -5.66478 q^{97} +4.07260 q^{98} -1.73174 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.10124	−0.778691	−0.389346	−	0.921092i	$-0.627299\pi$
$2$	−1.10124	−0.778691	−0.389346	+	0.921092i	$0.627299\pi$
$3$	1.12617	0.650195	0.325098	−	0.945680i	$-0.394603\pi$
$3$	1.12617	0.650195	0.325098	+	0.945680i	$0.394603\pi$
$4$	−0.787279	−0.393640
$5$	0	0
$5$	0	0
$6$	−1.24018	−0.506301
$7$	1.81708	0.686793	0.343397	−	0.939190i	$-0.388422\pi$
$7$	1.81708	0.686793	0.343397	+	0.939190i	$0.388422\pi$
$8$	3.06945	1.08522
$9$	−1.73174	−0.577246
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	−0.886611	−0.255943
$13$	3.00590	0.833687	0.416843	−	0.908978i	$-0.363136\pi$
$13$	3.00590	0.833687	0.416843	+	0.908978i	$0.363136\pi$
$14$	−2.00104	−0.534800
$15$	0	0
$16$	−1.80563	−0.451408
$17$	0.396094	0.0960668	0.0480334	−	0.998846i	$-0.484705\pi$
$17$	0.396094	0.0960668	0.0480334	+	0.998846i	$0.484705\pi$
$18$	1.90705	0.449497
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	2.04635	0.446550
$22$	−1.10124	−0.234784
$23$	2.47213	0.515475	0.257737	−	0.966215i	$-0.417023\pi$
$23$	2.47213	0.515475	0.257737	+	0.966215i	$0.417023\pi$
$24$	3.45673	0.705602
$25$	0	0
$26$	−3.31020	−0.649185
$27$	−5.32875	−1.02552
$28$	−1.43055	−0.270349
$29$	−4.66419	−0.866118	−0.433059	−	0.901366i	$-0.642566\pi$
$29$	−4.66419	−0.866118	−0.433059	+	0.901366i	$0.642566\pi$
$30$	0	0
$31$	−10.5803	−1.90027	−0.950135	−	0.311838i	$-0.899055\pi$
$31$	−10.5803	−1.90027	−0.950135	+	0.311838i	$0.899055\pi$
$32$	−4.15048	−0.733708
$33$	1.12617	0.196041
$34$	−0.436192	−0.0748064
$35$	0	0
$36$	1.36336	0.227227
$37$	−3.18261	−0.523217	−0.261609	−	0.965174i	$-0.584253\pi$
$37$	−3.18261	−0.523217	−0.261609	+	0.965174i	$0.584253\pi$
$38$	1.10124	0.178644
$39$	3.38516	0.542059
$40$	0	0
$41$	−9.74534	−1.52197	−0.760983	−	0.648772i	$-0.775283\pi$
$41$	−9.74534	−1.52197	−0.760983	+	0.648772i	$0.775283\pi$
$42$	−2.25351	−0.347724
$43$	−3.52635	−0.537764	−0.268882	−	0.963173i	$-0.586654\pi$
$43$	−3.52635	−0.537764	−0.268882	+	0.963173i	$0.586654\pi$
$44$	−0.787279	−0.118687
$45$	0	0
$46$	−2.72240	−0.401396
$47$	3.07500	0.448535	0.224267	−	0.974528i	$-0.428001\pi$
$47$	3.07500	0.448535	0.224267	+	0.974528i	$0.428001\pi$
$48$	−2.03345	−0.293503
$49$	−3.69820	−0.528315
$50$	0	0
$51$	0.446069	0.0624622
$52$	−2.36648	−0.328172
$53$	−1.32628	−0.182179	−0.0910893	−	0.995843i	$-0.529035\pi$
$53$	−1.32628	−0.182179	−0.0910893	+	0.995843i	$0.529035\pi$
$54$	5.86821	0.798562
$55$	0	0
$56$	5.57745	0.745319
$57$	−1.12617	−0.149165
$58$	5.13637	0.674439
$59$	−6.08186	−0.791791	−0.395895	−	0.918296i	$-0.629566\pi$
$59$	−6.08186	−0.791791	−0.395895	+	0.918296i	$0.629566\pi$
$60$	0	0
$61$	13.2731	1.69944	0.849721	−	0.527233i	$-0.176770\pi$
$61$	13.2731	1.69944	0.849721	+	0.527233i	$0.176770\pi$
$62$	11.6514	1.47972
$63$	−3.14672	−0.396449
$64$	8.18192	1.02274
$65$	0	0
$66$	−1.24018	−0.152656
$67$	−9.86746	−1.20550	−0.602751	−	0.797930i	$-0.705929\pi$
$67$	−9.86746	−1.20550	−0.602751	+	0.797930i	$0.705929\pi$
$68$	−0.311836	−0.0378157
$69$	2.78404	0.335159
$70$	0	0
$71$	3.14591	0.373351	0.186675	−	0.982422i	$-0.440229\pi$
$71$	3.14591	0.373351	0.186675	+	0.982422i	$0.440229\pi$
$72$	−5.31549	−0.626436
$73$	10.7607	1.25944	0.629720	−	0.776822i	$-0.283170\pi$
$73$	10.7607	1.25944	0.629720	+	0.776822i	$0.283170\pi$
$74$	3.50480	0.407425
$75$	0	0
$76$	0.787279	0.0903071
$77$	1.81708	0.207076
$78$	−3.72786	−0.422097
$79$	2.48793	0.279914	0.139957	−	0.990158i	$-0.455304\pi$
$79$	2.48793	0.279914	0.139957	+	0.990158i	$0.455304\pi$
$80$	0	0
$81$	−0.805865	−0.0895405
$82$	10.7319	1.18514
$83$	8.40591	0.922668	0.461334	−	0.887227i	$-0.347371\pi$
$83$	8.40591	0.922668	0.461334	+	0.887227i	$0.347371\pi$
$84$	−1.61105	−0.175780
$85$	0	0
$86$	3.88335	0.418752
$87$	−5.25268	−0.563146
$88$	3.06945	0.327205
$89$	−4.18002	−0.443081	−0.221540	−	0.975151i	$-0.571108\pi$
$89$	−4.18002	−0.443081	−0.221540	+	0.975151i	$0.571108\pi$
$90$	0	0
$91$	5.46197	0.572570
$92$	−1.94626	−0.202911
$93$	−11.9152	−1.23555
$94$	−3.38630	−0.349270
$95$	0	0
$96$	−4.67415	−0.477053
$97$	−5.66478	−0.575172	−0.287586	−	0.957755i	$-0.592853\pi$
$97$	−5.66478	−0.575172	−0.287586	+	0.957755i	$0.592853\pi$
$98$	4.07260	0.411394
$99$	−1.73174	−0.174046
$100$	0	0
$101$	8.17189	0.813134	0.406567	−	0.913621i	$-0.366726\pi$
$101$	8.17189	0.813134	0.406567	+	0.913621i	$0.366726\pi$
$102$	−0.491227	−0.0486388
$103$	5.11727	0.504219	0.252110	−	0.967699i	$-0.418876\pi$
$103$	5.11727	0.504219	0.252110	+	0.967699i	$0.418876\pi$
$104$	9.22647	0.904729
$105$	0	0
$106$	1.46055	0.141861
$107$	−11.0610	−1.06931	−0.534653	−	0.845072i	$-0.679558\pi$
$107$	−11.0610	−1.06931	−0.534653	+	0.845072i	$0.679558\pi$
$108$	4.19521	0.403684
$109$	1.77465	0.169981	0.0849904	−	0.996382i	$-0.472914\pi$
$109$	1.77465	0.169981	0.0849904	+	0.996382i	$0.472914\pi$
$110$	0	0
$111$	−3.58416	−0.340193
$112$	−3.28099	−0.310024
$113$	−17.0343	−1.60245	−0.801225	−	0.598363i	$-0.795818\pi$
$113$	−17.0343	−1.60245	−0.801225	+	0.598363i	$0.795818\pi$
$114$	1.24018	0.116154
$115$	0	0
$116$	3.67202	0.340938
$117$	−5.20543	−0.481242
$118$	6.69756	0.616561
$119$	0.719735	0.0659780
$120$	0	0
$121$	1.00000	0.0909091
$122$	−14.6168	−1.32334
$123$	−10.9749	−0.989575
$124$	8.32962	0.748022
$125$	0	0
$126$	3.46528	0.308711
$127$	−8.14604	−0.722844	−0.361422	−	0.932402i	$-0.617709\pi$
$127$	−8.14604	−0.722844	−0.361422	+	0.932402i	$0.617709\pi$
$128$	−0.709272	−0.0626913
$129$	−3.97128	−0.349651
$130$	0	0
$131$	−10.3533	−0.904569	−0.452284	−	0.891874i	$-0.649391\pi$
$131$	−10.3533	−0.904569	−0.452284	+	0.891874i	$0.649391\pi$
$132$	−0.886611	−0.0771696
$133$	−1.81708	−0.157561
$134$	10.8664	0.938714
$135$	0	0
$136$	1.21579	0.104253
$137$	0.222099	0.0189752	0.00948761	−	0.999955i	$-0.496980\pi$
$137$	0.222099	0.0189752	0.00948761	+	0.999955i	$0.496980\pi$
$138$	−3.06589	−0.260986
$139$	1.52039	0.128958	0.0644790	−	0.997919i	$-0.479461\pi$
$139$	1.52039	0.128958	0.0644790	+	0.997919i	$0.479461\pi$
$140$	0	0
$141$	3.46298	0.291635
$142$	−3.46439	−0.290725
$143$	3.00590	0.251366
$144$	3.12688	0.260574
$145$	0	0
$146$	−11.8500	−0.980715
$147$	−4.16481	−0.343508
$148$	2.50560	0.205959
$149$	13.4208	1.09947	0.549737	−	0.835338i	$-0.314728\pi$
$149$	13.4208	1.09947	0.549737	+	0.835338i	$0.314728\pi$
$150$	0	0
$151$	−1.77240	−0.144236	−0.0721180	−	0.997396i	$-0.522976\pi$
$151$	−1.77240	−0.144236	−0.0721180	+	0.997396i	$0.522976\pi$
$152$	−3.06945	−0.248965
$153$	−0.685931	−0.0554542
$154$	−2.00104	−0.161248
$155$	0	0
$156$	−2.66506	−0.213376
$157$	−17.6002	−1.40465	−0.702326	−	0.711856i	$-0.747855\pi$
$157$	−17.6002	−1.40465	−0.702326	+	0.711856i	$0.747855\pi$
$158$	−2.73980	−0.217967
$159$	−1.49362	−0.118452
$160$	0	0
$161$	4.49207	0.354024
$162$	0.887447	0.0697244
$163$	−13.7583	−1.07763	−0.538816	−	0.842423i	$-0.681128\pi$
$163$	−13.7583	−1.07763	−0.538816	+	0.842423i	$0.681128\pi$
$164$	7.67230	0.599106
$165$	0	0
$166$	−9.25689	−0.718474
$167$	4.42753	0.342612	0.171306	−	0.985218i	$-0.445201\pi$
$167$	4.42753	0.342612	0.171306	+	0.985218i	$0.445201\pi$
$168$	6.28117	0.484603
$169$	−3.96457	−0.304967
$170$	0	0
$171$	1.73174	0.132429
$172$	2.77622	0.211685
$173$	−8.62772	−0.655953	−0.327977	−	0.944686i	$-0.606367\pi$
$173$	−8.62772	−0.655953	−0.327977	+	0.944686i	$0.606367\pi$
$174$	5.78444	0.438517
$175$	0	0
$176$	−1.80563	−0.136105
$177$	−6.84922	−0.514819
$178$	4.60319	0.345023
$179$	4.62285	0.345528	0.172764	−	0.984963i	$-0.444730\pi$
$179$	4.62285	0.345528	0.172764	+	0.984963i	$0.444730\pi$
$180$	0	0
$181$	−3.43691	−0.255464	−0.127732	−	0.991809i	$-0.540770\pi$
$181$	−3.43691	−0.255464	−0.127732	+	0.991809i	$0.540770\pi$
$182$	−6.01492	−0.445856
$183$	14.9477	1.10497
$184$	7.58808	0.559401
$185$	0	0
$186$	13.1214	0.962110
$187$	0.396094	0.0289652
$188$	−2.42088	−0.176561
$189$	−9.68278	−0.704319
$190$	0	0
$191$	5.94862	0.430427	0.215213	−	0.976567i	$-0.430955\pi$
$191$	5.94862	0.430427	0.215213	+	0.976567i	$0.430955\pi$
$192$	9.21424	0.664981
$193$	6.33082	0.455702	0.227851	−	0.973696i	$-0.426830\pi$
$193$	6.33082	0.455702	0.227851	+	0.973696i	$0.426830\pi$
$194$	6.23826	0.447881
$195$	0	0
$196$	2.91152	0.207966
$197$	10.9220	0.778163	0.389082	−	0.921203i	$-0.372792\pi$
$197$	10.9220	0.778163	0.389082	+	0.921203i	$0.372792\pi$
$198$	1.90705	0.135528
$199$	−17.0111	−1.20588	−0.602941	−	0.797786i	$-0.706005\pi$
$199$	−17.0111	−1.20588	−0.602941	+	0.797786i	$0.706005\pi$
$200$	0	0
$201$	−11.1124	−0.783811
$202$	−8.99918	−0.633180
$203$	−8.47523	−0.594844
$204$	−0.351181	−0.0245876
$205$	0	0
$206$	−5.63532	−0.392631
$207$	−4.28108	−0.297556
$208$	−5.42755	−0.376333
$209$	−1.00000	−0.0691714
$210$	0	0
$211$	4.62775	0.318588	0.159294	−	0.987231i	$-0.449078\pi$
$211$	4.62775	0.318588	0.159294	+	0.987231i	$0.449078\pi$
$212$	1.04415	0.0717127
$213$	3.54283	0.242751
$214$	12.1808	0.832659
$215$	0	0
$216$	−16.3563	−1.11291
$217$	−19.2252	−1.30509
$218$	−1.95431	−0.132363
$219$	12.1183	0.818882
$220$	0	0
$221$	1.19062	0.0800896
$222$	3.94700	0.264906
$223$	6.92101	0.463465	0.231733	−	0.972780i	$-0.425561\pi$
$223$	6.92101	0.463465	0.231733	+	0.972780i	$0.425561\pi$
$224$	−7.54177	−0.503905
$225$	0	0
$226$	18.7588	1.24781
$227$	−20.9784	−1.39238	−0.696192	−	0.717855i	$-0.745124\pi$
$227$	−20.9784	−1.39238	−0.696192	+	0.717855i	$0.745124\pi$
$228$	0.886611	0.0587173
$229$	−4.01914	−0.265592	−0.132796	−	0.991143i	$-0.542396\pi$
$229$	−4.01914	−0.265592	−0.132796	+	0.991143i	$0.542396\pi$
$230$	0	0
$231$	2.04635	0.134640
$232$	−14.3165	−0.939925
$233$	−6.92311	−0.453548	−0.226774	−	0.973947i	$-0.572818\pi$
$233$	−6.92311	−0.453548	−0.226774	+	0.973947i	$0.572818\pi$
$234$	5.73241	0.374739
$235$	0	0
$236$	4.78812	0.311680
$237$	2.80184	0.181999
$238$	−0.792598	−0.0513765
$239$	19.6937	1.27388	0.636941	−	0.770913i	$-0.280200\pi$
$239$	19.6937	1.27388	0.636941	+	0.770913i	$0.280200\pi$
$240$	0	0
$241$	0.140555	0.00905396	0.00452698	−	0.999990i	$-0.498559\pi$
$241$	0.140555	0.00905396	0.00452698	+	0.999990i	$0.498559\pi$
$242$	−1.10124	−0.0707901
$243$	15.0787	0.967299
$244$	−10.4496	−0.668968
$245$	0	0
$246$	12.0860	0.770574
$247$	−3.00590	−0.191261
$248$	−32.4756	−2.06220
$249$	9.46649	0.599914
$250$	0	0
$251$	2.70575	0.170786	0.0853928	−	0.996347i	$-0.472785\pi$
$251$	2.70575	0.170786	0.0853928	+	0.996347i	$0.472785\pi$
$252$	2.47734	0.156058
$253$	2.47213	0.155421
$254$	8.97071	0.562872
$255$	0	0
$256$	−15.5828	−0.973923
$257$	−10.3982	−0.648622	−0.324311	−	0.945950i	$-0.605132\pi$
$257$	−10.3982	−0.648622	−0.324311	+	0.945950i	$0.605132\pi$
$258$	4.37331	0.272271
$259$	−5.78306	−0.359342
$260$	0	0
$261$	8.07716	0.499964
$262$	11.4014	0.704380
$263$	23.0218	1.41959	0.709793	−	0.704410i	$-0.248789\pi$
$263$	23.0218	1.41959	0.709793	+	0.704410i	$0.248789\pi$
$264$	3.45673	0.212747
$265$	0	0
$266$	2.00104	0.122692
$267$	−4.70741	−0.288089
$268$	7.76844	0.474533
$269$	−5.97611	−0.364370	−0.182185	−	0.983264i	$-0.558317\pi$
$269$	−5.97611	−0.364370	−0.182185	+	0.983264i	$0.558317\pi$
$270$	0	0
$271$	−30.7969	−1.87078	−0.935388	−	0.353623i	$-0.884950\pi$
$271$	−30.7969	−1.87078	−0.935388	+	0.353623i	$0.884950\pi$
$272$	−0.715200	−0.0433653
$273$	6.15112	0.372282
$274$	−0.244584	−0.0147758
$275$	0	0
$276$	−2.19182	−0.131932
$277$	14.0355	0.843311	0.421655	−	0.906756i	$-0.361449\pi$
$277$	14.0355	0.843311	0.421655	+	0.906756i	$0.361449\pi$
$278$	−1.67431	−0.100419
$279$	18.3222	1.09692
$280$	0	0
$281$	−12.9615	−0.773221	−0.386610	−	0.922243i	$-0.626354\pi$
$281$	−12.9615	−0.773221	−0.386610	+	0.922243i	$0.626354\pi$
$282$	−3.81355	−0.227094
$283$	−25.3050	−1.50423	−0.752114	−	0.659033i	$-0.770966\pi$
$283$	−25.3050	−1.50423	−0.752114	+	0.659033i	$0.770966\pi$
$284$	−2.47671	−0.146966
$285$	0	0
$286$	−3.31020	−0.195737
$287$	−17.7081	−1.04528
$288$	7.18754	0.423530
$289$	−16.8431	−0.990771
$290$	0	0
$291$	−6.37952	−0.373974
$292$	−8.47164	−0.495765
$293$	−22.9413	−1.34025	−0.670123	−	0.742250i	$-0.733759\pi$
$293$	−22.9413	−1.34025	−0.670123	+	0.742250i	$0.733759\pi$
$294$	4.58644	0.267487
$295$	0	0
$296$	−9.76886	−0.567803
$297$	−5.32875	−0.309205
$298$	−14.7795	−0.856151
$299$	7.43097	0.429744
$300$	0	0
$301$	−6.40768	−0.369333
$302$	1.95183	0.112315
$303$	9.20295	0.528696
$304$	1.80563	0.103560
$305$	0	0
$306$	0.755371	0.0431817
$307$	−11.0205	−0.628971	−0.314485	−	0.949262i	$-0.601832\pi$
$307$	−11.0205	−0.628971	−0.314485	+	0.949262i	$0.601832\pi$
$308$	−1.43055	−0.0815133
$309$	5.76292	0.327841
$310$	0	0
$311$	14.0537	0.796913	0.398456	−	0.917187i	$-0.369546\pi$
$311$	14.0537	0.796913	0.398456	+	0.917187i	$0.369546\pi$
$312$	10.3906	0.588251
$313$	15.0295	0.849519	0.424759	−	0.905306i	$-0.360359\pi$
$313$	15.0295	0.849519	0.424759	+	0.905306i	$0.360359\pi$
$314$	19.3820	1.09379
$315$	0	0
$316$	−1.95870	−0.110185
$317$	26.7125	1.50033	0.750163	−	0.661253i	$-0.229975\pi$
$317$	26.7125	1.50033	0.750163	+	0.661253i	$0.229975\pi$
$318$	1.64483	0.0922373
$319$	−4.66419	−0.261144
$320$	0	0
$321$	−12.4566	−0.695257
$322$	−4.94683	−0.275676
$323$	−0.396094	−0.0220392
$324$	0.634441	0.0352467
$325$	0	0
$326$	15.1511	0.839143
$327$	1.99856	0.110521
$328$	−29.9129	−1.65166
$329$	5.58753	0.308051
$330$	0	0
$331$	18.3065	1.00622	0.503109	−	0.864223i	$-0.332190\pi$
$331$	18.3065	1.00622	0.503109	+	0.864223i	$0.332190\pi$
$332$	−6.61780	−0.363199
$333$	5.51144	0.302025
$334$	−4.87575	−0.266789
$335$	0	0
$336$	−3.69495	−0.201576
$337$	−3.55269	−0.193527	−0.0967636	−	0.995307i	$-0.530849\pi$
$337$	−3.55269	−0.193527	−0.0967636	+	0.995307i	$0.530849\pi$
$338$	4.36592	0.237475
$339$	−19.1835	−1.04191
$340$	0	0
$341$	−10.5803	−0.572953
$342$	−1.90705	−0.103122
$343$	−19.4395	−1.04964
$344$	−10.8240	−0.583589
$345$	0	0
$346$	9.50116	0.510785
$347$	13.9284	0.747713	0.373857	−	0.927486i	$-0.378035\pi$
$347$	13.9284	0.747713	0.373857	+	0.927486i	$0.378035\pi$
$348$	4.13532	0.221677
$349$	−2.94165	−0.157463	−0.0787313	−	0.996896i	$-0.525087\pi$
$349$	−2.94165	−0.157463	−0.0787313	+	0.996896i	$0.525087\pi$
$350$	0	0
$351$	−16.0177	−0.854960
$352$	−4.15048	−0.221221
$353$	3.34914	0.178257	0.0891283	−	0.996020i	$-0.471592\pi$
$353$	3.34914	0.178257	0.0891283	+	0.996020i	$0.471592\pi$
$354$	7.54260	0.400885
$355$	0	0
$356$	3.29084	0.174414
$357$	0.810545	0.0428986
$358$	−5.09085	−0.269060
$359$	−17.3050	−0.913321	−0.456660	−	0.889641i	$-0.650955\pi$
$359$	−17.3050	−0.913321	−0.456660	+	0.889641i	$0.650955\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	3.78485	0.198927
$363$	1.12617	0.0591087
$364$	−4.30010	−0.225386
$365$	0	0
$366$	−16.4610	−0.860430
$367$	−2.84237	−0.148370	−0.0741851	−	0.997244i	$-0.523636\pi$
$367$	−2.84237	−0.148370	−0.0741851	+	0.997244i	$0.523636\pi$
$368$	−4.46376	−0.232689
$369$	16.8764	0.878549
$370$	0	0
$371$	−2.40996	−0.125119
$372$	9.38057	0.486360
$373$	−31.3251	−1.62195	−0.810976	−	0.585079i	$-0.801064\pi$
$373$	−31.3251	−1.62195	−0.810976	+	0.585079i	$0.801064\pi$
$374$	−0.436192	−0.0225550
$375$	0	0
$376$	9.43856	0.486757
$377$	−14.0201	−0.722071
$378$	10.6630	0.548447
$379$	15.3932	0.790697	0.395349	−	0.918531i	$-0.370624\pi$
$379$	15.3932	0.790697	0.395349	+	0.918531i	$0.370624\pi$
$380$	0	0
$381$	−9.17383	−0.469990
$382$	−6.55083	−0.335170
$383$	−18.8395	−0.962654	−0.481327	−	0.876541i	$-0.659845\pi$
$383$	−18.8395	−0.962654	−0.481327	+	0.876541i	$0.659845\pi$
$384$	−0.798761	−0.0407616
$385$	0	0
$386$	−6.97173	−0.354852
$387$	6.10672	0.310422
$388$	4.45977	0.226410
$389$	−30.9839	−1.57095	−0.785473	−	0.618896i	$-0.787580\pi$
$389$	−30.9839	−1.57095	−0.785473	+	0.618896i	$0.787580\pi$
$390$	0	0
$391$	0.979194	0.0495200
$392$	−11.3515	−0.573335
$393$	−11.6595	−0.588146
$394$	−12.0277	−0.605949
$395$	0	0
$396$	1.36336	0.0685115
$397$	−9.73881	−0.488777	−0.244388	−	0.969677i	$-0.578587\pi$
$397$	−9.73881	−0.488777	−0.244388	+	0.969677i	$0.578587\pi$
$398$	18.7332	0.939010
$399$	−2.04635	−0.102446
$400$	0	0
$401$	−16.8539	−0.841642	−0.420821	−	0.907144i	$-0.638258\pi$
$401$	−16.8539	−0.841642	−0.420821	+	0.907144i	$0.638258\pi$
$402$	12.2374	0.610347
$403$	−31.8032	−1.58423
$404$	−6.43356	−0.320082
$405$	0	0
$406$	9.33322	0.463200
$407$	−3.18261	−0.157756
$408$	1.36919	0.0677849
$409$	14.4968	0.716822	0.358411	−	0.933564i	$-0.383319\pi$
$409$	14.4968	0.716822	0.358411	+	0.933564i	$0.383319\pi$
$410$	0	0
$411$	0.250122	0.0123376
$412$	−4.02872	−0.198481
$413$	−11.0513	−0.543797
$414$	4.71448	0.231704
$415$	0	0
$416$	−12.4759	−0.611682
$417$	1.71222	0.0838479
$418$	1.10124	0.0538632
$419$	23.5582	1.15089	0.575447	−	0.817839i	$-0.304828\pi$
$419$	23.5582	1.15089	0.575447	+	0.817839i	$0.304828\pi$
$420$	0	0
$421$	3.16377	0.154193	0.0770964	−	0.997024i	$-0.475435\pi$
$421$	3.16377	0.154193	0.0770964	+	0.997024i	$0.475435\pi$
$422$	−5.09625	−0.248081
$423$	−5.32510	−0.258915
$424$	−4.07095	−0.197703
$425$	0	0
$426$	−3.90149	−0.189028
$427$	24.1183	1.16717
$428$	8.70808	0.420921
$429$	3.38516	0.163437
$430$	0	0
$431$	−22.3848	−1.07824	−0.539120	−	0.842229i	$-0.681243\pi$
$431$	−22.3848	−1.07824	−0.539120	+	0.842229i	$0.681243\pi$
$432$	9.62176	0.462927
$433$	33.0415	1.58787	0.793937	−	0.608000i	$-0.208028\pi$
$433$	33.0415	1.58787	0.793937	+	0.608000i	$0.208028\pi$
$434$	21.1715	1.01626
$435$	0	0
$436$	−1.39715	−0.0669111
$437$	−2.47213	−0.118258
$438$	−13.3452	−0.637656
$439$	−13.5183	−0.645193	−0.322597	−	0.946537i	$-0.604556\pi$
$439$	−13.5183	−0.645193	−0.322597	+	0.946537i	$0.604556\pi$
$440$	0	0
$441$	6.40432	0.304968
$442$	−1.31115	−0.0623651
$443$	−26.2368	−1.24655	−0.623273	−	0.782004i	$-0.714197\pi$
$443$	−26.2368	−1.24655	−0.623273	+	0.782004i	$0.714197\pi$
$444$	2.82173	0.133914
$445$	0	0
$446$	−7.62167	−0.360896
$447$	15.1141	0.714873
$448$	14.8672	0.702411
$449$	−37.8759	−1.78747	−0.893737	−	0.448591i	$-0.851926\pi$
$449$	−37.8759	−1.78747	−0.893737	+	0.448591i	$0.851926\pi$
$450$	0	0
$451$	−9.74534	−0.458890
$452$	13.4107	0.630788
$453$	−1.99603	−0.0937815
$454$	23.1022	1.08424
$455$	0	0
$456$	−3.45673	−0.161876
$457$	8.11897	0.379789	0.189895	−	0.981804i	$-0.439185\pi$
$457$	8.11897	0.379789	0.189895	+	0.981804i	$0.439185\pi$
$458$	4.42602	0.206814
$459$	−2.11068	−0.0985182
$460$	0	0
$461$	2.99911	0.139683	0.0698413	−	0.997558i	$-0.477751\pi$
$461$	2.99911	0.139683	0.0698413	+	0.997558i	$0.477751\pi$
$462$	−2.25351	−0.104843
$463$	−20.0068	−0.929795	−0.464897	−	0.885365i	$-0.653909\pi$
$463$	−20.0068	−0.929795	−0.464897	+	0.885365i	$0.653909\pi$
$464$	8.42181	0.390973
$465$	0	0
$466$	7.62397	0.353174
$467$	−2.07746	−0.0961333	−0.0480667	−	0.998844i	$-0.515306\pi$
$467$	−2.07746	−0.0961333	−0.0480667	+	0.998844i	$0.515306\pi$
$468$	4.09813	0.189436
$469$	−17.9300	−0.827930
$470$	0	0
$471$	−19.8209	−0.913298
$472$	−18.6680	−0.859264
$473$	−3.52635	−0.162142
$474$	−3.08548	−0.141721
$475$	0	0
$476$	−0.566633	−0.0259716
$477$	2.29677	0.105162
$478$	−21.6874	−0.991960
$479$	39.1671	1.78959	0.894795	−	0.446478i	$-0.147322\pi$
$479$	39.1671	1.78959	0.894795	+	0.446478i	$0.147322\pi$
$480$	0	0
$481$	−9.56659	−0.436199
$482$	−0.154784	−0.00705024
$483$	5.05884	0.230185
$484$	−0.787279	−0.0357854
$485$	0	0
$486$	−16.6052	−0.753228
$487$	40.8687	1.85194	0.925968	−	0.377603i	$-0.123251\pi$
$487$	40.8687	1.85194	0.925968	+	0.377603i	$0.123251\pi$
$488$	40.7410	1.84426
$489$	−15.4942	−0.700671
$490$	0	0
$491$	−6.86419	−0.309777	−0.154888	−	0.987932i	$-0.549502\pi$
$491$	−6.86419	−0.309777	−0.154888	+	0.987932i	$0.549502\pi$
$492$	8.64033	0.389536
$493$	−1.84746	−0.0832052
$494$	3.31020	0.148933
$495$	0	0
$496$	19.1041	0.857798
$497$	5.71638	0.256415
$498$	−10.4248	−0.467148
$499$	−28.2774	−1.26587	−0.632936	−	0.774204i	$-0.718150\pi$
$499$	−28.2774	−1.26587	−0.632936	+	0.774204i	$0.718150\pi$
$500$	0	0
$501$	4.98615	0.222765
$502$	−2.97967	−0.132989
$503$	0.0217862	0.000971397 0	0.000485698	−	1.00000i	$-0.499845\pi$
$503$	0.0217862	0.000971397 0	0.000485698		1.00000i	$0.499845\pi$
$504$	−9.65869	−0.430232
$505$	0	0
$506$	−2.72240	−0.121025
$507$	−4.46478	−0.198288
$508$	6.41321	0.284540
$509$	12.8569	0.569873	0.284936	−	0.958546i	$-0.408028\pi$
$509$	12.8569	0.569873	0.284936	+	0.958546i	$0.408028\pi$
$510$	0	0
$511$	19.5530	0.864975
$512$	18.5788	0.821077
$513$	5.32875	0.235270
$514$	11.4509	0.505077
$515$	0	0
$516$	3.12650	0.137637
$517$	3.07500	0.135238
$518$	6.36852	0.279817
$519$	−9.71629	−0.426498
$520$	0	0
$521$	35.4800	1.55441	0.777203	−	0.629250i	$-0.216638\pi$
$521$	35.4800	1.55441	0.777203	+	0.629250i	$0.216638\pi$
$522$	−8.89486	−0.389317
$523$	31.9759	1.39821	0.699104	−	0.715020i	$-0.253583\pi$
$523$	31.9759	1.39821	0.699104	+	0.715020i	$0.253583\pi$
$524$	8.15091	0.356074
$525$	0	0
$526$	−25.3524	−1.10542
$527$	−4.19077	−0.182553
$528$	−2.03345	−0.0884946
$529$	−16.8886	−0.734286
$530$	0	0
$531$	10.5322	0.457058
$532$	1.43055	0.0620223
$533$	−29.2935	−1.26884
$534$	5.18397	0.224333
$535$	0	0
$536$	−30.2877	−1.30823
$537$	5.20612	0.224661
$538$	6.58111	0.283732
$539$	−3.69820	−0.159293
$540$	0	0
$541$	19.1990	0.825432	0.412716	−	0.910860i	$-0.364580\pi$
$541$	19.1990	0.825432	0.412716	+	0.910860i	$0.364580\pi$
$542$	33.9146	1.45676
$543$	−3.87055	−0.166101
$544$	−1.64398	−0.0704849
$545$	0	0
$546$	−6.77383	−0.289893
$547$	−19.1493	−0.818764	−0.409382	−	0.912363i	$-0.634256\pi$
$547$	−19.1493	−0.818764	−0.409382	+	0.912363i	$0.634256\pi$
$548$	−0.174854	−0.00746940
$549$	−22.9855	−0.980996
$550$	0	0
$551$	4.66419	0.198701
$552$	8.54548	0.363720
$553$	4.52078	0.192243
$554$	−15.4564	−0.656679
$555$	0	0
$556$	−1.19697	−0.0507630
$557$	20.7708	0.880085	0.440043	−	0.897977i	$-0.354963\pi$
$557$	20.7708	0.880085	0.440043	+	0.897977i	$0.354963\pi$
$558$	−20.1771	−0.854165
$559$	−10.5999	−0.448326
$560$	0	0
$561$	0.446069	0.0188331
$562$	14.2737	0.602100
$563$	−2.56588	−0.108139	−0.0540694	−	0.998537i	$-0.517219\pi$
$563$	−2.56588	−0.108139	−0.0540694	+	0.998537i	$0.517219\pi$
$564$	−2.72633	−0.114799
$565$	0	0
$566$	27.8668	1.17133
$567$	−1.46432	−0.0614958
$568$	9.65621	0.405166
$569$	32.6425	1.36844	0.684221	−	0.729274i	$-0.260142\pi$
$569$	32.6425	1.36844	0.684221	+	0.729274i	$0.260142\pi$
$570$	0	0
$571$	26.8048	1.12175	0.560873	−	0.827902i	$-0.310465\pi$
$571$	26.8048	1.12175	0.560873	+	0.827902i	$0.310465\pi$
$572$	−2.36648	−0.0989476
$573$	6.69916	0.279861
$574$	19.5008	0.813948
$575$	0	0
$576$	−14.1689	−0.590373
$577$	35.1268	1.46235	0.731174	−	0.682191i	$-0.238973\pi$
$577$	35.1268	1.46235	0.731174	+	0.682191i	$0.238973\pi$
$578$	18.5482	0.771505
$579$	7.12959	0.296295
$580$	0	0
$581$	15.2742	0.633682
$582$	7.02535	0.291210
$583$	−1.32628	−0.0549289
$584$	33.0293	1.36676
$585$	0	0
$586$	25.2638	1.04364
$587$	23.9743	0.989523	0.494762	−	0.869029i	$-0.335255\pi$
$587$	23.9743	0.989523	0.494762	+	0.869029i	$0.335255\pi$
$588$	3.27887	0.135218
$589$	10.5803	0.435952
$590$	0	0
$591$	12.3001	0.505958
$592$	5.74662	0.236185
$593$	−2.13888	−0.0878333	−0.0439166	−	0.999035i	$-0.513984\pi$
$593$	−2.13888	−0.0878333	−0.0439166	+	0.999035i	$0.513984\pi$
$594$	5.86821	0.240776
$595$	0	0
$596$	−10.5659	−0.432796
$597$	−19.1574	−0.784059
$598$	−8.18325	−0.334638
$599$	34.6600	1.41617	0.708084	−	0.706128i	$-0.249560\pi$
$599$	34.6600	1.41617	0.708084	+	0.706128i	$0.249560\pi$
$600$	0	0
$601$	−40.8041	−1.66443	−0.832216	−	0.554451i	$-0.812928\pi$
$601$	−40.8041	−1.66443	−0.832216	+	0.554451i	$0.812928\pi$
$602$	7.05637	0.287596
$603$	17.0879	0.695871
$604$	1.39537	0.0567770
$605$	0	0
$606$	−10.1346	−0.411691
$607$	−35.9560	−1.45941	−0.729704	−	0.683763i	$-0.760342\pi$
$607$	−35.9560	−1.45941	−0.729704	+	0.683763i	$0.760342\pi$
$608$	4.15048	0.168324
$609$	−9.54455	−0.386765
$610$	0	0
$611$	9.24314	0.373937
$612$	0.540019	0.0218290
$613$	−37.5253	−1.51563	−0.757816	−	0.652468i	$-0.773734\pi$
$613$	−37.5253	−1.51563	−0.757816	+	0.652468i	$0.773734\pi$
$614$	12.1361	0.489774
$615$	0	0
$616$	5.57745	0.224722
$617$	−10.0204	−0.403404	−0.201702	−	0.979447i	$-0.564647\pi$
$617$	−10.0204	−0.403404	−0.201702	+	0.979447i	$0.564647\pi$
$618$	−6.34633	−0.255287
$619$	−29.6702	−1.19255	−0.596274	−	0.802781i	$-0.703353\pi$
$619$	−29.6702	−1.19255	−0.596274	+	0.802781i	$0.703353\pi$
$620$	0	0
$621$	−13.1734	−0.528628
$622$	−15.4764	−0.620549
$623$	−7.59544	−0.304305
$624$	−6.11235	−0.244690
$625$	0	0
$626$	−16.5511	−0.661513
$627$	−1.12617	−0.0449749
$628$	13.8563	0.552927
$629$	−1.26061	−0.0502638
$630$	0	0
$631$	18.7254	0.745447	0.372723	−	0.927943i	$-0.378424\pi$
$631$	18.7254	0.745447	0.372723	+	0.927943i	$0.378424\pi$
$632$	7.63658	0.303767
$633$	5.21164	0.207144
$634$	−29.4168	−1.16829
$635$	0	0
$636$	1.17590	0.0466273
$637$	−11.1164	−0.440449
$638$	5.13637	0.203351
$639$	−5.44789	−0.215515
$640$	0	0
$641$	−25.4242	−1.00420	−0.502098	−	0.864811i	$-0.667438\pi$
$641$	−25.4242	−1.00420	−0.502098	+	0.864811i	$0.667438\pi$
$642$	13.7176	0.541391
$643$	−17.8870	−0.705395	−0.352697	−	0.935737i	$-0.614735\pi$
$643$	−17.8870	−0.705395	−0.352697	+	0.935737i	$0.614735\pi$
$644$	−3.53651	−0.139358
$645$	0	0
$646$	0.436192	0.0171618
$647$	7.30694	0.287265	0.143633	−	0.989631i	$-0.454122\pi$
$647$	7.30694	0.287265	0.143633	+	0.989631i	$0.454122\pi$
$648$	−2.47356	−0.0971708
$649$	−6.08186	−0.238734
$650$	0	0
$651$	−21.6509	−0.848565
$652$	10.8316	0.424199
$653$	−4.36571	−0.170843	−0.0854217	−	0.996345i	$-0.527224\pi$
$653$	−4.36571	−0.170843	−0.0854217	+	0.996345i	$0.527224\pi$
$654$	−2.20089	−0.0860615
$655$	0	0
$656$	17.5965	0.687028
$657$	−18.6346	−0.727007
$658$	−6.15319	−0.239876
$659$	−41.6526	−1.62255	−0.811277	−	0.584662i	$-0.801227\pi$
$659$	−41.6526	−1.62255	−0.811277	+	0.584662i	$0.801227\pi$
$660$	0	0
$661$	5.30624	0.206389	0.103194	−	0.994661i	$-0.467094\pi$
$661$	5.30624	0.206389	0.103194	+	0.994661i	$0.467094\pi$
$662$	−20.1598	−0.783533
$663$	1.34084	0.0520739
$664$	25.8015	1.00129
$665$	0	0
$666$	−6.06940	−0.235184
$667$	−11.5305	−0.446462
$668$	−3.48570	−0.134866
$669$	7.79424	0.301343
$670$	0	0
$671$	13.2731	0.512401
$672$	−8.49332	−0.327637
$673$	−30.5025	−1.17578	−0.587892	−	0.808940i	$-0.700042\pi$
$673$	−30.5025	−1.17578	−0.587892	+	0.808940i	$0.700042\pi$
$674$	3.91235	0.150698
$675$	0	0
$676$	3.12122	0.120047
$677$	−30.1893	−1.16027	−0.580134	−	0.814521i	$-0.697000\pi$
$677$	−30.1893	−1.16027	−0.580134	+	0.814521i	$0.697000\pi$
$678$	21.1256	0.811323
$679$	−10.2934	−0.395024
$680$	0	0
$681$	−23.6253	−0.905322
$682$	11.6514	0.446154
$683$	−20.2522	−0.774929	−0.387464	−	0.921885i	$-0.626649\pi$
$683$	−20.2522	−0.774929	−0.387464	+	0.921885i	$0.626649\pi$
$684$	−1.36336	−0.0521294
$685$	0	0
$686$	21.4075	0.817343
$687$	−4.52624	−0.172687
$688$	6.36730	0.242751
$689$	−3.98667	−0.151880
$690$	0	0
$691$	−5.43936	−0.206923	−0.103462	−	0.994633i	$-0.532992\pi$
$691$	−5.43936	−0.206923	−0.103462	+	0.994633i	$0.532992\pi$
$692$	6.79243	0.258209
$693$	−3.14672	−0.119534
$694$	−15.3384	−0.582238
$695$	0	0
$696$	−16.1228	−0.611135
$697$	−3.86007	−0.146210
$698$	3.23945	0.122615
$699$	−7.79660	−0.294895
$700$	0	0
$701$	−19.7193	−0.744789	−0.372394	−	0.928075i	$-0.621463\pi$
$701$	−19.7193	−0.744789	−0.372394	+	0.928075i	$0.621463\pi$
$702$	17.6392	0.665750
$703$	3.18261	0.120034
$704$	8.18192	0.308368
$705$	0	0
$706$	−3.68819	−0.138807
$707$	14.8490	0.558455
$708$	5.39225	0.202653
$709$	18.2446	0.685191	0.342595	−	0.939483i	$-0.388694\pi$
$709$	18.2446	0.685191	0.342595	+	0.939483i	$0.388694\pi$
$710$	0	0
$711$	−4.30845	−0.161579
$712$	−12.8304	−0.480838
$713$	−26.1558	−0.979541
$714$	−0.892601	−0.0334048
$715$	0	0
$716$	−3.63947	−0.136013
$717$	22.1785	0.828271
$718$	19.0568	0.711195
$719$	36.8064	1.37265	0.686323	−	0.727296i	$-0.259223\pi$
$719$	36.8064	1.37265	0.686323	+	0.727296i	$0.259223\pi$
$720$	0	0
$721$	9.29851	0.346295
$722$	−1.10124	−0.0409838
$723$	0.158289	0.00588684
$724$	2.70581	0.100561
$725$	0	0
$726$	−1.24018	−0.0460274
$727$	−41.1500	−1.52617	−0.763085	−	0.646299i	$-0.776316\pi$
$727$	−41.1500	−1.52617	−0.763085	+	0.646299i	$0.776316\pi$
$728$	16.7653	0.621362
$729$	19.3988	0.718474
$730$	0	0
$731$	−1.39677	−0.0516612
$732$	−11.7680	−0.434960
$733$	41.1579	1.52020	0.760100	−	0.649806i	$-0.225150\pi$
$733$	41.1579	1.52020	0.760100	+	0.649806i	$0.225150\pi$
$734$	3.13012	0.115535
$735$	0	0
$736$	−10.2605	−0.378208
$737$	−9.86746	−0.363472
$738$	−18.5849	−0.684119
$739$	−29.5872	−1.08838	−0.544192	−	0.838961i	$-0.683164\pi$
$739$	−29.5872	−1.08838	−0.544192	+	0.838961i	$0.683164\pi$
$740$	0	0
$741$	−3.38516	−0.124357
$742$	2.65394	0.0974291
$743$	−15.0223	−0.551113	−0.275557	−	0.961285i	$-0.588862\pi$
$743$	−15.0223	−0.551113	−0.275557	+	0.961285i	$0.588862\pi$
$744$	−36.5731	−1.34083
$745$	0	0
$746$	34.4963	1.26300
$747$	−14.5568	−0.532607
$748$	−0.311836	−0.0114019
$749$	−20.0987	−0.734392
$750$	0	0
$751$	38.6495	1.41034	0.705171	−	0.709038i	$-0.250870\pi$
$751$	38.6495	1.41034	0.705171	+	0.709038i	$0.250870\pi$
$752$	−5.55232	−0.202472
$753$	3.04714	0.111044
$754$	15.4394	0.562271
$755$	0	0
$756$	7.62305	0.277248
$757$	−17.8737	−0.649631	−0.324815	−	0.945777i	$-0.605302\pi$
$757$	−17.8737	−0.649631	−0.324815	+	0.945777i	$0.605302\pi$
$758$	−16.9516	−0.615709
$759$	2.78404	0.101054
$760$	0	0
$761$	−39.7350	−1.44039	−0.720195	−	0.693771i	$-0.755948\pi$
$761$	−39.7350	−1.44039	−0.720195	+	0.693771i	$0.755948\pi$
$762$	10.1026	0.365977
$763$	3.22469	0.116742
$764$	−4.68322	−0.169433
$765$	0	0
$766$	20.7467	0.749610
$767$	−18.2815	−0.660105
$768$	−17.5489	−0.633240
$769$	−27.3872	−0.987609	−0.493804	−	0.869573i	$-0.664394\pi$
$769$	−27.3872	−0.987609	−0.493804	+	0.869573i	$0.664394\pi$
$770$	0	0
$771$	−11.7102	−0.421731
$772$	−4.98412	−0.179382
$773$	35.8608	1.28982	0.644911	−	0.764258i	$-0.276894\pi$
$773$	35.8608	1.28982	0.644911	+	0.764258i	$0.276894\pi$
$774$	−6.72494	−0.241723
$775$	0	0
$776$	−17.3878	−0.624185
$777$	−6.51272	−0.233642
$778$	34.1206	1.22328
$779$	9.74534	0.349163
$780$	0	0
$781$	3.14591	0.112569
$782$	−1.07832	−0.0385608
$783$	24.8543	0.888220
$784$	6.67760	0.238486
$785$	0	0
$786$	12.8399	0.457984
$787$	10.1175	0.360649	0.180324	−	0.983607i	$-0.442285\pi$
$787$	10.1175	0.360649	0.180324	+	0.983607i	$0.442285\pi$
$788$	−8.59870	−0.306316
$789$	25.9265	0.923008
$790$	0	0
$791$	−30.9527	−1.10055
$792$	−5.31549	−0.188878
$793$	39.8975	1.41680
$794$	10.7247	0.380606
$795$	0	0
$796$	13.3925	0.474683
$797$	41.7207	1.47782	0.738911	−	0.673803i	$-0.235340\pi$
$797$	41.7207	1.47782	0.738911	+	0.673803i	$0.235340\pi$
$798$	2.25351	0.0797735
$799$	1.21799	0.0430893
$800$	0	0
$801$	7.23870	0.255767
$802$	18.5601	0.655379
$803$	10.7607	0.379735
$804$	8.74860	0.308539
$805$	0	0
$806$	35.0228	1.23363
$807$	−6.73013	−0.236912
$808$	25.0832	0.882425
$809$	48.0804	1.69042	0.845209	−	0.534436i	$-0.179476\pi$
$809$	48.0804	1.69042	0.845209	+	0.534436i	$0.179476\pi$
$810$	0	0
$811$	−33.1054	−1.16249	−0.581243	−	0.813730i	$-0.697434\pi$
$811$	−33.1054	−1.16249	−0.581243	+	0.813730i	$0.697434\pi$
$812$	6.67237	0.234154
$813$	−34.6825	−1.21637
$814$	3.50480	0.122843
$815$	0	0
$816$	−0.805437	−0.0281959
$817$	3.52635	0.123371
$818$	−15.9644	−0.558183
$819$	−9.45871	−0.330514
$820$	0	0
$821$	36.5920	1.27707	0.638535	−	0.769593i	$-0.279541\pi$
$821$	36.5920	1.27707	0.638535	+	0.769593i	$0.279541\pi$
$822$	−0.275443	−0.00960718
$823$	12.9128	0.450110	0.225055	−	0.974346i	$-0.427744\pi$
$823$	12.9128	0.450110	0.225055	+	0.974346i	$0.427744\pi$
$824$	15.7072	0.547187
$825$	0	0
$826$	12.1700	0.423450
$827$	14.2210	0.494512	0.247256	−	0.968950i	$-0.420471\pi$
$827$	14.2210	0.494512	0.247256	+	0.968950i	$0.420471\pi$
$828$	3.37041	0.117130
$829$	−49.9790	−1.73584	−0.867922	−	0.496701i	$-0.834545\pi$
$829$	−49.9790	−1.73584	−0.867922	+	0.496701i	$0.834545\pi$
$830$	0	0
$831$	15.8064	0.548317
$832$	24.5940	0.852645
$833$	−1.46484	−0.0507535
$834$	−1.88556	−0.0652916
$835$	0	0
$836$	0.787279	0.0272286
$837$	56.3795	1.94876
$838$	−25.9431	−0.896191
$839$	−35.3809	−1.22149	−0.610743	−	0.791829i	$-0.709129\pi$
$839$	−35.3809	−1.22149	−0.610743	+	0.791829i	$0.709129\pi$
$840$	0	0
$841$	−7.24533	−0.249839
$842$	−3.48406	−0.120069
$843$	−14.5969	−0.502744
$844$	−3.64333	−0.125409
$845$	0	0
$846$	5.86419	0.201615
$847$	1.81708	0.0624358
$848$	2.39478	0.0822369
$849$	−28.4978	−0.978042
$850$	0	0
$851$	−7.86781	−0.269705
$852$	−2.78920	−0.0955563
$853$	−20.9094	−0.715924	−0.357962	−	0.933736i	$-0.616528\pi$
$853$	−20.9094	−0.715924	−0.357962	+	0.933736i	$0.616528\pi$
$854$	−26.5599	−0.908862
$855$	0	0
$856$	−33.9512	−1.16043
$857$	21.5067	0.734654	0.367327	−	0.930092i	$-0.380273\pi$
$857$	21.5067	0.734654	0.367327	+	0.930092i	$0.380273\pi$
$858$	−3.72786	−0.127267
$859$	47.9041	1.63447	0.817233	−	0.576307i	$-0.195507\pi$
$859$	47.9041	1.63447	0.817233	+	0.576307i	$0.195507\pi$
$860$	0	0
$861$	−19.9424	−0.679633
$862$	24.6510	0.839616
$863$	46.5232	1.58367	0.791834	−	0.610737i	$-0.209127\pi$
$863$	46.5232	1.58367	0.791834	+	0.610737i	$0.209127\pi$
$864$	22.1168	0.752430
$865$	0	0
$866$	−36.3865	−1.23646
$867$	−18.9682	−0.644195
$868$	15.1356	0.513736
$869$	2.48793	0.0843973
$870$	0	0
$871$	−29.6606	−1.00501
$872$	5.44721	0.184466
$873$	9.80993	0.332016
$874$	2.72240	0.0920865
$875$	0	0
$876$	−9.54052	−0.322344
$877$	−9.27067	−0.313048	−0.156524	−	0.987674i	$-0.550029\pi$
$877$	−9.27067	−0.313048	−0.156524	+	0.987674i	$0.550029\pi$
$878$	14.8868	0.502406
$879$	−25.8358	−0.871421
$880$	0	0
$881$	−16.1306	−0.543454	−0.271727	−	0.962374i	$-0.587595\pi$
$881$	−16.1306	−0.543454	−0.271727	+	0.962374i	$0.587595\pi$
$882$	−7.05267	−0.237476
$883$	0.0216069	0.000727130 0	0.000363565	−	1.00000i	$-0.499884\pi$
$883$	0.0216069	0.000727130 0	0.000363565		1.00000i	$0.499884\pi$
$884$	−0.937348	−0.0315264
$885$	0	0
$886$	28.8929	0.970674
$887$	42.0739	1.41270	0.706352	−	0.707860i	$-0.250339\pi$
$887$	42.0739	1.41270	0.706352	+	0.707860i	$0.250339\pi$
$888$	−11.0014	−0.369183
$889$	−14.8020	−0.496444
$890$	0	0
$891$	−0.805865	−0.0269975
$892$	−5.44877	−0.182438
$893$	−3.07500	−0.102901
$894$	−16.6442	−0.556665
$895$	0	0
$896$	−1.28881	−0.0430560
$897$	8.36855	0.279418
$898$	41.7103	1.39189
$899$	49.3483	1.64586
$900$	0	0
$901$	−0.525331	−0.0175013
$902$	10.7319	0.357334
$903$	−7.21614	−0.240138
$904$	−52.2859	−1.73900
$905$	0	0
$906$	2.19810	0.0730269
$907$	4.10956	0.136456	0.0682279	−	0.997670i	$-0.478265\pi$
$907$	4.10956	0.136456	0.0682279	+	0.997670i	$0.478265\pi$
$908$	16.5159	0.548098
$909$	−14.1516	−0.469378
$910$	0	0
$911$	10.7860	0.357356	0.178678	−	0.983908i	$-0.442818\pi$
$911$	10.7860	0.357356	0.178678	+	0.983908i	$0.442818\pi$
$912$	2.03345	0.0673343
$913$	8.40591	0.278195
$914$	−8.94090	−0.295739
$915$	0	0
$916$	3.16418	0.104548
$917$	−18.8127	−0.621252
$918$	2.32436	0.0767153
$919$	−32.3453	−1.06697	−0.533486	−	0.845809i	$-0.679118\pi$
$919$	−32.3453	−1.06697	−0.533486	+	0.845809i	$0.679118\pi$
$920$	0	0
$921$	−12.4109	−0.408954
$922$	−3.30273	−0.108770
$923$	9.45628	0.311257
$924$	−1.61105	−0.0529996
$925$	0	0
$926$	22.0322	0.724023
$927$	−8.86177	−0.291059
$928$	19.3586	0.635478
$929$	−41.8124	−1.37182	−0.685910	−	0.727686i	$-0.740596\pi$
$929$	−41.8124	−1.37182	−0.685910	+	0.727686i	$0.740596\pi$
$930$	0	0
$931$	3.69820	0.121204
$932$	5.45042	0.178534
$933$	15.8269	0.518149
$934$	2.28777	0.0748582
$935$	0	0
$936$	−15.9778	−0.522252
$937$	18.0548	0.589825	0.294913	−	0.955524i	$-0.404709\pi$
$937$	18.0548	0.589825	0.294913	+	0.955524i	$0.404709\pi$
$938$	19.7452	0.644702
$939$	16.9258	0.552353
$940$	0	0
$941$	19.9391	0.649997	0.324999	−	0.945714i	$-0.394636\pi$
$941$	19.9391	0.649997	0.324999	+	0.945714i	$0.394636\pi$
$942$	21.8275	0.711177
$943$	−24.0917	−0.784535
$944$	10.9816	0.357421
$945$	0	0
$946$	3.88335	0.126259
$947$	−7.78827	−0.253085	−0.126542	−	0.991961i	$-0.540388\pi$
$947$	−7.78827	−0.253085	−0.126542	+	0.991961i	$0.540388\pi$
$948$	−2.20583	−0.0716419
$949$	32.3455	1.04998
$950$	0	0
$951$	30.0829	0.975504
$952$	2.20919	0.0716004
$953$	−47.5841	−1.54140	−0.770700	−	0.637199i	$-0.780093\pi$
$953$	−47.5841	−1.54140	−0.770700	+	0.637199i	$0.780093\pi$
$954$	−2.52929	−0.0818887
$955$	0	0
$956$	−15.5045	−0.501450
$957$	−5.25268	−0.169795
$958$	−43.1322	−1.39354
$959$	0.403573	0.0130321
$960$	0	0
$961$	80.9419	2.61103
$962$	10.5351	0.339664
$963$	19.1547	0.617253
$964$	−0.110656	−0.00356400
$965$	0	0
$966$	−5.57097	−0.179243
$967$	−52.8234	−1.69869	−0.849343	−	0.527841i	$-0.823002\pi$
$967$	−52.8234	−1.69869	−0.849343	+	0.527841i	$0.823002\pi$
$968$	3.06945	0.0986559
$969$	−0.446069	−0.0143298
$970$	0	0
$971$	13.9120	0.446456	0.223228	−	0.974766i	$-0.428341\pi$
$971$	13.9120	0.446456	0.223228	+	0.974766i	$0.428341\pi$
$972$	−11.8711	−0.380767
$973$	2.76268	0.0885675
$974$	−45.0060	−1.44209
$975$	0	0
$976$	−23.9663	−0.767142
$977$	6.97874	0.223270	0.111635	−	0.993749i	$-0.464391\pi$
$977$	6.97874	0.223270	0.111635	+	0.993749i	$0.464391\pi$
$978$	17.0628	0.545607
$979$	−4.18002	−0.133594
$980$	0	0
$981$	−3.07323	−0.0981207
$982$	7.55910	0.241220
$983$	28.5372	0.910197	0.455098	−	0.890441i	$-0.349604\pi$
$983$	28.5372	0.910197	0.455098	+	0.890441i	$0.349604\pi$
$984$	−33.6870	−1.07390
$985$	0	0
$986$	2.03448	0.0647912
$987$	6.29252	0.200293
$988$	2.36648	0.0752878
$989$	−8.71760	−0.277204
$990$	0	0
$991$	20.5503	0.652802	0.326401	−	0.945231i	$-0.394164\pi$
$991$	20.5503	0.652802	0.326401	+	0.945231i	$0.394164\pi$
$992$	43.9131	1.39424
$993$	20.6163	0.654238
$994$	−6.29508	−0.199668
$995$	0	0
$996$	−7.45277	−0.236150
$997$	17.1671	0.543688	0.271844	−	0.962341i	$-0.412366\pi$
$997$	17.1671	0.543688	0.271844	+	0.962341i	$0.412366\pi$
$998$	31.1401	0.985724
$999$	16.9593	0.536569

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5225.2.a.s.1.7	✓	15
5.4	even	2		5225.2.a.x.1.9	yes	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5225.2.a.s.1.7	✓	15	1.1	even	1	trivial
5225.2.a.x.1.9	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)

\(f(q)\)	\(=\)	\(q-1.10124 q^{2} +1.12617 q^{3} -0.787279 q^{4} -1.24018 q^{6} +1.81708 q^{7} +3.06945 q^{8} -1.73174 q^{9} +1.00000 q^{11} -0.886611 q^{12} +3.00590 q^{13} -2.00104 q^{14} -1.80563 q^{16} +0.396094 q^{17} +1.90705 q^{18} -1.00000 q^{19} +2.04635 q^{21} -1.10124 q^{22} +2.47213 q^{23} +3.45673 q^{24} -3.31020 q^{26} -5.32875 q^{27} -1.43055 q^{28} -4.66419 q^{29} -10.5803 q^{31} -4.15048 q^{32} +1.12617 q^{33} -0.436192 q^{34} +1.36336 q^{36} -3.18261 q^{37} +1.10124 q^{38} +3.38516 q^{39} -9.74534 q^{41} -2.25351 q^{42} -3.52635 q^{43} -0.787279 q^{44} -2.72240 q^{46} +3.07500 q^{47} -2.03345 q^{48} -3.69820 q^{49} +0.446069 q^{51} -2.36648 q^{52} -1.32628 q^{53} +5.86821 q^{54} +5.57745 q^{56} -1.12617 q^{57} +5.13637 q^{58} -6.08186 q^{59} +13.2731 q^{61} +11.6514 q^{62} -3.14672 q^{63} +8.18192 q^{64} -1.24018 q^{66} -9.86746 q^{67} -0.311836 q^{68} +2.78404 q^{69} +3.14591 q^{71} -5.31549 q^{72} +10.7607 q^{73} +3.50480 q^{74} +0.787279 q^{76} +1.81708 q^{77} -3.72786 q^{78} +2.48793 q^{79} -0.805865 q^{81} +10.7319 q^{82} +8.40591 q^{83} -1.61105 q^{84} +3.88335 q^{86} -5.25268 q^{87} +3.06945 q^{88} -4.18002 q^{89} +5.46197 q^{91} -1.94626 q^{92} -11.9152 q^{93} -3.38630 q^{94} -4.67415 q^{96} -5.66478 q^{97} +4.07260 q^{98} -1.73174 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q - 5 q^{2} - 4 q^{3} + 17 q^{4} - q^{6} - 15 q^{7} - 15 q^{8} + 19 q^{9} + 15 q^{11} - 9 q^{12} - 13 q^{13} + 9 q^{14} + 21 q^{16} - 11 q^{17} - 16 q^{18} - 15 q^{19} - 5 q^{22} - 10 q^{23} + 17 q^{24}+ \cdots + 19 q^{99}+O(q^{100}) \) 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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