LMFDB - Embedded newform 7225.2.a.bv.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7225,2,Mod(1,7225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7225, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("7225.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7225 = 5^{2} \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7225.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:	$57.6919154604$
Analytic rank:	$0$
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} - 21 x^{13} - 2 x^{12} + 171 x^{11} + 30 x^{10} - 678 x^{9} - 153 x^{8} + 1350 x^{7} + 301 x^{6} + \cdots + 3 $ x^15 - 21x^13 - 2x^12 + 171x^11 + 30x^10 - 678x^9 - 153x^8 + 1350x^7 + 301x^6 - 1284x^5 - 183x^4 + 555x^3 + 18x^2 - 81*x + 3
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			1.9
Root			$-0.525320$ of defining polynomial
Character	$\chi$	$=$	7225.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.525320 q^{2} -1.36683 q^{3} -1.72404 q^{4} -0.718021 q^{6} +0.426746 q^{7} -1.95631 q^{8} -1.13179 q^{9} +O(q^{10})$	\(q+0.525320 q^{2} -1.36683 q^{3} -1.72404 q^{4} -0.718021 q^{6} +0.426746 q^{7} -1.95631 q^{8} -1.13179 q^{9} -1.05071 q^{11} +2.35646 q^{12} -2.19489 q^{13} +0.224179 q^{14} +2.42039 q^{16} -0.594552 q^{18} +2.52103 q^{19} -0.583288 q^{21} -0.551961 q^{22} -3.45732 q^{23} +2.67394 q^{24} -1.15302 q^{26} +5.64743 q^{27} -0.735727 q^{28} -6.88478 q^{29} +1.94611 q^{31} +5.18411 q^{32} +1.43614 q^{33} +1.95125 q^{36} +1.86393 q^{37} +1.32435 q^{38} +3.00003 q^{39} -9.98081 q^{41} -0.306413 q^{42} -8.66192 q^{43} +1.81147 q^{44} -1.81620 q^{46} -2.79588 q^{47} -3.30824 q^{48} -6.81789 q^{49} +3.78407 q^{52} +7.96642 q^{53} +2.96671 q^{54} -0.834850 q^{56} -3.44581 q^{57} -3.61672 q^{58} -8.61359 q^{59} -0.659622 q^{61} +1.02233 q^{62} -0.482987 q^{63} -2.11745 q^{64} +0.754434 q^{66} -13.6882 q^{67} +4.72555 q^{69} -3.86656 q^{71} +2.21413 q^{72} -10.9241 q^{73} +0.979159 q^{74} -4.34636 q^{76} -0.448388 q^{77} +1.57598 q^{78} +16.9973 q^{79} -4.32369 q^{81} -5.24313 q^{82} -2.51998 q^{83} +1.00561 q^{84} -4.55028 q^{86} +9.41029 q^{87} +2.05552 q^{88} -5.82597 q^{89} -0.936661 q^{91} +5.96055 q^{92} -2.65999 q^{93} -1.46873 q^{94} -7.08577 q^{96} +13.6876 q^{97} -3.58158 q^{98} +1.18918 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q + 9 q^{3} + 12 q^{4} - 9 q^{6} + 12 q^{7} - 6 q^{8} + 12 q^{9}+O(q^{10}) $ 15 * q + 9 * q^3 + 12 * q^4 - 9 * q^6 + 12 * q^7 - 6 * q^8 + 12 * q^9	$ 15 q + 9 q^{3} + 12 q^{4} - 9 q^{6} + 12 q^{7} - 6 q^{8} + 12 q^{9} - 6 q^{11} + 24 q^{12} + 6 q^{16} - 12 q^{18} + 6 q^{19} + 30 q^{21} + 12 q^{22} + 36 q^{23} - 18 q^{24} + 36 q^{26} + 36 q^{27} + 24 q^{28} + 18 q^{29} - 12 q^{32} - 12 q^{33} - 9 q^{36} + 12 q^{37} + 6 q^{38} - 9 q^{39} + 18 q^{41} - 36 q^{42} + 3 q^{43} + 12 q^{44} - 21 q^{46} + 3 q^{47} - 12 q^{48} + 15 q^{49} + 27 q^{52} - 21 q^{54} + 6 q^{56} + 39 q^{57} + 18 q^{58} - 12 q^{59} + 15 q^{61} + 54 q^{62} + 60 q^{63} - 36 q^{64} + 18 q^{66} + 24 q^{67} + 42 q^{69} - 6 q^{71} - 66 q^{72} - 9 q^{73} + 36 q^{74} - 18 q^{76} - 30 q^{77} + 30 q^{78} + 9 q^{79} + 51 q^{81} - 36 q^{82} - 15 q^{83} + 9 q^{84} - 36 q^{86} + 51 q^{87} + 30 q^{88} - 24 q^{89} - 27 q^{91} + 15 q^{92} + 42 q^{93} - 57 q^{94} - 42 q^{96} + 48 q^{97} - 12 q^{99}+O(q^{100}) $ 15 * q + 9 * q^3 + 12 * q^4 - 9 * q^6 + 12 * q^7 - 6 * q^8 + 12 * q^9 - 6 * q^11 + 24 * q^12 + 6 * q^16 - 12 * q^18 + 6 * q^19 + 30 * q^21 + 12 * q^22 + 36 * q^23 - 18 * q^24 + 36 * q^26 + 36 * q^27 + 24 * q^28 + 18 * q^29 - 12 * q^32 - 12 * q^33 - 9 * q^36 + 12 * q^37 + 6 * q^38 - 9 * q^39 + 18 * q^41 - 36 * q^42 + 3 * q^43 + 12 * q^44 - 21 * q^46 + 3 * q^47 - 12 * q^48 + 15 * q^49 + 27 * q^52 - 21 * q^54 + 6 * q^56 + 39 * q^57 + 18 * q^58 - 12 * q^59 + 15 * q^61 + 54 * q^62 + 60 * q^63 - 36 * q^64 + 18 * q^66 + 24 * q^67 + 42 * q^69 - 6 * q^71 - 66 * q^72 - 9 * q^73 + 36 * q^74 - 18 * q^76 - 30 * q^77 + 30 * q^78 + 9 * q^79 + 51 * q^81 - 36 * q^82 - 15 * q^83 + 9 * q^84 - 36 * q^86 + 51 * q^87 + 30 * q^88 - 24 * q^89 - 27 * q^91 + 15 * q^92 + 42 * q^93 - 57 * q^94 - 42 * q^96 + 48 * q^97 - 12 * q^99

Display 100 coefficients 10 coefficients

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 $

$2$	0.525320	0.371458	0.185729	−	0.982601i	$-0.440535\pi$
$2$	0.525320	0.371458	0.185729	+	0.982601i	$0.440535\pi$
$3$	−1.36683	−0.789137	−0.394568	−	0.918867i	$-0.629106\pi$
$3$	−1.36683	−0.789137	−0.394568	+	0.918867i	$0.629106\pi$
$4$	−1.72404	−0.862019
$5$	0	0
$5$	0	0
$6$	−0.718021	−0.293131
$7$	0.426746	0.161295	0.0806475	−	0.996743i	$-0.474301\pi$
$7$	0.426746	0.161295	0.0806475	+	0.996743i	$0.474301\pi$
$8$	−1.95631	−0.691661
$9$	−1.13179	−0.377263
$10$	0	0
$11$	−1.05071	−0.316802	−0.158401	−	0.987375i	$-0.550634\pi$
$11$	−1.05071	−0.316802	−0.158401	+	0.987375i	$0.550634\pi$
$12$	2.35646	0.680251
$13$	−2.19489	−0.608753	−0.304376	−	0.952552i	$-0.598448\pi$
$13$	−2.19489	−0.608753	−0.304376	+	0.952552i	$0.598448\pi$
$14$	0.224179	0.0599143
$15$	0	0
$16$	2.42039	0.605096
$17$	0	0
$17$	0	0
$18$	−0.594552	−0.140137
$19$	2.52103	0.578365	0.289182	−	0.957274i	$-0.406617\pi$
$19$	2.52103	0.578365	0.289182	+	0.957274i	$0.406617\pi$
$20$	0	0
$21$	−0.583288	−0.127284
$22$	−0.551961	−0.117678
$23$	−3.45732	−0.720901	−0.360450	−	0.932778i	$-0.617377\pi$
$23$	−3.45732	−0.720901	−0.360450	+	0.932778i	$0.617377\pi$
$24$	2.67394	0.545816
$25$	0	0
$26$	−1.15302	−0.226126
$27$	5.64743	1.08685
$28$	−0.735727	−0.139039
$29$	−6.88478	−1.27847	−0.639236	−	0.769011i	$-0.720749\pi$
$29$	−6.88478	−1.27847	−0.639236	+	0.769011i	$0.720749\pi$
$30$	0	0
$31$	1.94611	0.349531	0.174765	−	0.984610i	$-0.444083\pi$
$31$	1.94611	0.349531	0.174765	+	0.984610i	$0.444083\pi$
$32$	5.18411	0.916429
$33$	1.43614	0.250000
$34$	0	0
$35$	0	0
$36$	1.95125	0.325208
$37$	1.86393	0.306428	0.153214	−	0.988193i	$-0.451038\pi$
$37$	1.86393	0.306428	0.153214	+	0.988193i	$0.451038\pi$
$38$	1.32435	0.214838
$39$	3.00003	0.480389
$40$	0	0
$41$	−9.98081	−1.55874	−0.779371	−	0.626563i	$-0.784461\pi$
$41$	−9.98081	−1.55874	−0.779371	+	0.626563i	$0.784461\pi$
$42$	−0.306413	−0.0472806
$43$	−8.66192	−1.32093	−0.660465	−	0.750857i	$-0.729641\pi$
$43$	−8.66192	−1.32093	−0.660465	+	0.750857i	$0.729641\pi$
$44$	1.81147	0.273089
$45$	0	0
$46$	−1.81620	−0.267784
$47$	−2.79588	−0.407821	−0.203910	−	0.978990i	$-0.565365\pi$
$47$	−2.79588	−0.407821	−0.203910	+	0.978990i	$0.565365\pi$
$48$	−3.30824	−0.477504
$49$	−6.81789	−0.973984
$50$	0	0
$51$	0	0
$52$	3.78407	0.524756
$53$	7.96642	1.09427	0.547136	−	0.837044i	$-0.315718\pi$
$53$	7.96642	1.09427	0.547136	+	0.837044i	$0.315718\pi$
$54$	2.96671	0.403718
$55$	0	0
$56$	−0.834850	−0.111562
$57$	−3.44581	−0.456409
$58$	−3.61672	−0.474898
$59$	−8.61359	−1.12139	−0.560697	−	0.828021i	$-0.689467\pi$
$59$	−8.61359	−1.12139	−0.560697	+	0.828021i	$0.689467\pi$
$60$	0	0
$61$	−0.659622	−0.0844560	−0.0422280	−	0.999108i	$-0.513446\pi$
$61$	−0.659622	−0.0844560	−0.0422280	+	0.999108i	$0.513446\pi$
$62$	1.02233	0.129836
$63$	−0.482987	−0.0608506
$64$	−2.11745	−0.264682
$65$	0	0
$66$	0.754434	0.0928644
$67$	−13.6882	−1.67228	−0.836139	−	0.548517i	$-0.815192\pi$
$67$	−13.6882	−1.67228	−0.836139	+	0.548517i	$0.815192\pi$
$68$	0	0
$69$	4.72555	0.568890
$70$	0	0
$71$	−3.86656	−0.458876	−0.229438	−	0.973323i	$-0.573689\pi$
$71$	−3.86656	−0.458876	−0.229438	+	0.973323i	$0.573689\pi$
$72$	2.21413	0.260938
$73$	−10.9241	−1.27856	−0.639282	−	0.768973i	$-0.720768\pi$
$73$	−10.9241	−1.27856	−0.639282	+	0.768973i	$0.720768\pi$
$74$	0.979159	0.113825
$75$	0	0
$76$	−4.34636	−0.498562
$77$	−0.448388	−0.0510985
$78$	1.57598	0.178444
$79$	16.9973	1.91235	0.956175	−	0.292795i	$-0.0945852\pi$
$79$	16.9973	1.91235	0.956175	+	0.292795i	$0.0945852\pi$
$80$	0	0
$81$	−4.32369	−0.480410
$82$	−5.24313	−0.579006
$83$	−2.51998	−0.276604	−0.138302	−	0.990390i	$-0.544164\pi$
$83$	−2.51998	−0.276604	−0.138302	+	0.990390i	$0.544164\pi$
$84$	1.00561	0.109721
$85$	0	0
$86$	−4.55028	−0.490669
$87$	9.41029	1.00889
$88$	2.05552	0.219119
$89$	−5.82597	−0.617552	−0.308776	−	0.951135i	$-0.599919\pi$
$89$	−5.82597	−0.617552	−0.308776	+	0.951135i	$0.599919\pi$
$90$	0	0
$91$	−0.936661	−0.0981887
$92$	5.96055	0.621430
$93$	−2.65999	−0.275828
$94$	−1.46873	−0.151488
$95$	0	0
$96$	−7.08577	−0.723188
$97$	13.6876	1.38977	0.694885	−	0.719121i	$-0.255455\pi$
$97$	13.6876	1.38977	0.694885	+	0.719121i	$0.255455\pi$
$98$	−3.58158	−0.361794
$99$	1.18918	0.119518
$100$	0	0
$101$	14.2955	1.42245	0.711226	−	0.702963i	$-0.248140\pi$
$101$	14.2955	1.42245	0.711226	+	0.702963i	$0.248140\pi$
$102$	0	0
$103$	−6.12045	−0.603065	−0.301533	−	0.953456i	$-0.597498\pi$
$103$	−6.12045	−0.603065	−0.301533	+	0.953456i	$0.597498\pi$
$104$	4.29389	0.421051
$105$	0	0
$106$	4.18492	0.406476
$107$	4.63694	0.448270	0.224135	−	0.974558i	$-0.428044\pi$
$107$	4.63694	0.448270	0.224135	+	0.974558i	$0.428044\pi$
$108$	−9.73639	−0.936885
$109$	5.15369	0.493634	0.246817	−	0.969062i	$-0.420615\pi$
$109$	5.15369	0.493634	0.246817	+	0.969062i	$0.420615\pi$
$110$	0	0
$111$	−2.54766	−0.241813
$112$	1.03289	0.0975990
$113$	7.93719	0.746668	0.373334	−	0.927697i	$-0.378215\pi$
$113$	7.93719	0.746668	0.373334	+	0.927697i	$0.378215\pi$
$114$	−1.81016	−0.169537
$115$	0	0
$116$	11.8696	1.10207
$117$	2.48415	0.229660
$118$	−4.52489	−0.416550
$119$	0	0
$120$	0	0
$121$	−9.89600	−0.899637
$122$	−0.346513	−0.0313718
$123$	13.6420	1.23006
$124$	−3.35516	−0.301302
$125$	0	0
$126$	−0.253723	−0.0226034
$127$	−11.0274	−0.978521	−0.489260	−	0.872138i	$-0.662733\pi$
$127$	−11.0274	−0.978521	−0.489260	+	0.872138i	$0.662733\pi$
$128$	−11.4806	−1.01475
$129$	11.8393	1.04239
$130$	0	0
$131$	11.3633	0.992812	0.496406	−	0.868090i	$-0.334653\pi$
$131$	11.3633	0.992812	0.496406	+	0.868090i	$0.334653\pi$
$132$	−2.47596	−0.215505
$133$	1.07584	0.0932874
$134$	−7.19068	−0.621181
$135$	0	0
$136$	0	0
$137$	21.3512	1.82415	0.912077	−	0.410020i	$-0.134478\pi$
$137$	21.3512	1.82415	0.912077	+	0.410020i	$0.134478\pi$
$138$	2.48243	0.211318
$139$	7.86879	0.667422	0.333711	−	0.942675i	$-0.391699\pi$
$139$	7.86879	0.667422	0.333711	+	0.942675i	$0.391699\pi$
$140$	0	0
$141$	3.82148	0.321827
$142$	−2.03118	−0.170453
$143$	2.30620	0.192854
$144$	−2.73936	−0.228280
$145$	0	0
$146$	−5.73863	−0.474932
$147$	9.31886	0.768607
$148$	−3.21348	−0.264147
$149$	3.51249	0.287754	0.143877	−	0.989596i	$-0.454043\pi$
$149$	3.51249	0.287754	0.143877	+	0.989596i	$0.454043\pi$
$150$	0	0
$151$	−18.4357	−1.50028	−0.750138	−	0.661281i	$-0.770013\pi$
$151$	−18.4357	−1.50028	−0.750138	+	0.661281i	$0.770013\pi$
$152$	−4.93193	−0.400033
$153$	0	0
$154$	−0.235547	−0.0189809
$155$	0	0
$156$	−5.17217	−0.414105
$157$	10.2544	0.818389	0.409194	−	0.912447i	$-0.365810\pi$
$157$	10.2544	0.818389	0.409194	+	0.912447i	$0.365810\pi$
$158$	8.92905	0.710357
$159$	−10.8887	−0.863531
$160$	0	0
$161$	−1.47540	−0.116278
$162$	−2.27132	−0.178452
$163$	11.6376	0.911529	0.455764	−	0.890100i	$-0.349366\pi$
$163$	11.6376	0.911529	0.455764	+	0.890100i	$0.349366\pi$
$164$	17.2073	1.34367
$165$	0	0
$166$	−1.32380	−0.102747
$167$	−18.4280	−1.42600	−0.713002	−	0.701162i	$-0.752665\pi$
$167$	−18.4280	−1.42600	−0.713002	+	0.701162i	$0.752665\pi$
$168$	1.14109	0.0880373
$169$	−8.18246	−0.629420
$170$	0	0
$171$	−2.85328	−0.218196
$172$	14.9335	1.13867
$173$	10.5648	0.803226	0.401613	−	0.915810i	$-0.368450\pi$
$173$	10.5648	0.803226	0.401613	+	0.915810i	$0.368450\pi$
$174$	4.94342	0.374760
$175$	0	0
$176$	−2.54313	−0.191696
$177$	11.7733	0.884933
$178$	−3.06050	−0.229394
$179$	−7.84681	−0.586498	−0.293249	−	0.956036i	$-0.594736\pi$
$179$	−7.84681	−0.586498	−0.293249	+	0.956036i	$0.594736\pi$
$180$	0	0
$181$	−9.61242	−0.714485	−0.357243	−	0.934012i	$-0.616283\pi$
$181$	−9.61242	−0.714485	−0.357243	+	0.934012i	$0.616283\pi$
$182$	−0.492047	−0.0364730
$183$	0.901589	0.0666473
$184$	6.76360	0.498619
$185$	0	0
$186$	−1.39735	−0.102458
$187$	0	0
$188$	4.82020	0.351549
$189$	2.41002	0.175303
$190$	0	0
$191$	−13.1215	−0.949441	−0.474721	−	0.880136i	$-0.657451\pi$
$191$	−13.1215	−0.949441	−0.474721	+	0.880136i	$0.657451\pi$
$192$	2.89419	0.208870
$193$	1.06152	0.0764097	0.0382048	−	0.999270i	$-0.487836\pi$
$193$	1.06152	0.0764097	0.0382048	+	0.999270i	$0.487836\pi$
$194$	7.19040	0.516241
$195$	0	0
$196$	11.7543	0.839593
$197$	23.1485	1.64926	0.824630	−	0.565672i	$-0.191383\pi$
$197$	23.1485	1.64926	0.824630	+	0.565672i	$0.191383\pi$
$198$	0.624703	0.0443957
$199$	9.09163	0.644489	0.322244	−	0.946657i	$-0.395563\pi$
$199$	9.09163	0.644489	0.322244	+	0.946657i	$0.395563\pi$
$200$	0	0
$201$	18.7094	1.31966
$202$	7.50971	0.528381
$203$	−2.93805	−0.206211
$204$	0	0
$205$	0	0
$206$	−3.21520	−0.224013
$207$	3.91295	0.271969
$208$	−5.31248	−0.368354
$209$	−2.64888	−0.183227
$210$	0	0
$211$	−22.7874	−1.56875	−0.784375	−	0.620287i	$-0.787016\pi$
$211$	−22.7874	−1.56875	−0.784375	+	0.620287i	$0.787016\pi$
$212$	−13.7344	−0.943284
$213$	5.28491	0.362116
$214$	2.43588	0.166513
$215$	0	0
$216$	−11.0482	−0.751731
$217$	0.830494	0.0563776
$218$	2.70734	0.183364
$219$	14.9313	1.00896
$220$	0	0
$221$	0	0
$222$	−1.33834	−0.0898234
$223$	4.99269	0.334336	0.167168	−	0.985928i	$-0.446538\pi$
$223$	4.99269	0.334336	0.167168	+	0.985928i	$0.446538\pi$
$224$	2.21230	0.147815
$225$	0	0
$226$	4.16957	0.277355
$227$	19.3088	1.28157	0.640785	−	0.767720i	$-0.278609\pi$
$227$	19.3088	1.28157	0.640785	+	0.767720i	$0.278609\pi$
$228$	5.94071	0.393433
$229$	−25.4274	−1.68029	−0.840146	−	0.542360i	$-0.817531\pi$
$229$	−25.4274	−1.68029	−0.840146	+	0.542360i	$0.817531\pi$
$230$	0	0
$231$	0.612868	0.0403237
$232$	13.4688	0.884269
$233$	3.89150	0.254940	0.127470	−	0.991842i	$-0.459314\pi$
$233$	3.89150	0.254940	0.127470	+	0.991842i	$0.459314\pi$
$234$	1.30497	0.0853089
$235$	0	0
$236$	14.8502	0.966663
$237$	−23.2324	−1.50911
$238$	0	0
$239$	0.772632	0.0499774	0.0249887	−	0.999688i	$-0.492045\pi$
$239$	0.772632	0.0499774	0.0249887	+	0.999688i	$0.492045\pi$
$240$	0	0
$241$	−0.889832	−0.0573191	−0.0286596	−	0.999589i	$-0.509124\pi$
$241$	−0.889832	−0.0573191	−0.0286596	+	0.999589i	$0.509124\pi$
$242$	−5.19857	−0.334177
$243$	−11.0326	−0.707740
$244$	1.13721	0.0728027
$245$	0	0
$246$	7.16644	0.456915
$247$	−5.53339	−0.352081
$248$	−3.80719	−0.241757
$249$	3.44438	0.218279
$250$	0	0
$251$	8.77153	0.553654	0.276827	−	0.960920i	$-0.410717\pi$
$251$	8.77153	0.553654	0.276827	+	0.960920i	$0.410717\pi$
$252$	0.832688	0.0524544
$253$	3.63265	0.228383
$254$	−5.79290	−0.363479
$255$	0	0
$256$	−1.79606	−0.112254
$257$	28.6140	1.78489	0.892446	−	0.451154i	$-0.148987\pi$
$257$	28.6140	1.78489	0.892446	+	0.451154i	$0.148987\pi$
$258$	6.21944	0.387205
$259$	0.795424	0.0494252
$260$	0	0
$261$	7.79211	0.482320
$262$	5.96935	0.368788
$263$	−0.388165	−0.0239353	−0.0119676	−	0.999928i	$-0.503810\pi$
$263$	−0.388165	−0.0239353	−0.0119676	+	0.999928i	$0.503810\pi$
$264$	−2.80954	−0.172915
$265$	0	0
$266$	0.565162	0.0346523
$267$	7.96309	0.487333
$268$	23.5990	1.44154
$269$	−27.8846	−1.70015	−0.850077	−	0.526658i	$-0.823445\pi$
$269$	−27.8846	−1.70015	−0.850077	+	0.526658i	$0.823445\pi$
$270$	0	0
$271$	8.03421	0.488044	0.244022	−	0.969770i	$-0.421533\pi$
$271$	8.03421	0.488044	0.244022	+	0.969770i	$0.421533\pi$
$272$	0	0
$273$	1.28025	0.0774844
$274$	11.2162	0.677596
$275$	0	0
$276$	−8.14703	−0.490394
$277$	0.395467	0.0237613	0.0118807	−	0.999929i	$-0.496218\pi$
$277$	0.395467	0.0237613	0.0118807	+	0.999929i	$0.496218\pi$
$278$	4.13364	0.247919
$279$	−2.20258	−0.131865
$280$	0	0
$281$	6.81801	0.406728	0.203364	−	0.979103i	$-0.434812\pi$
$281$	6.81801	0.406728	0.203364	+	0.979103i	$0.434812\pi$
$282$	2.00750	0.119545
$283$	22.3535	1.32878	0.664389	−	0.747387i	$-0.268692\pi$
$283$	22.3535	1.32878	0.664389	+	0.747387i	$0.268692\pi$
$284$	6.66610	0.395560
$285$	0	0
$286$	1.21149	0.0716370
$287$	−4.25928	−0.251417
$288$	−5.86731	−0.345735
$289$	0	0
$290$	0	0
$291$	−18.7086	−1.09672
$292$	18.8335	1.10215
$293$	−5.27232	−0.308012	−0.154006	−	0.988070i	$-0.549218\pi$
$293$	−5.27232	−0.308012	−0.154006	+	0.988070i	$0.549218\pi$
$294$	4.89539	0.285505
$295$	0	0
$296$	−3.64643	−0.211944
$297$	−5.93383	−0.344316
$298$	1.84518	0.106889
$299$	7.58843	0.438850
$300$	0	0
$301$	−3.69644	−0.213059
$302$	−9.68465	−0.557289
$303$	−19.5394	−1.12251
$304$	6.10187	0.349966
$305$	0	0
$306$	0	0
$307$	10.0566	0.573960	0.286980	−	0.957937i	$-0.407349\pi$
$307$	10.0566	0.573960	0.286980	+	0.957937i	$0.407349\pi$
$308$	0.773038	0.0440479
$309$	8.36558	0.475901
$310$	0	0
$311$	2.87791	0.163191	0.0815956	−	0.996666i	$-0.473998\pi$
$311$	2.87791	0.163191	0.0815956	+	0.996666i	$0.473998\pi$
$312$	−5.86900	−0.332267
$313$	31.3122	1.76987	0.884934	−	0.465716i	$-0.154203\pi$
$313$	31.3122	1.76987	0.884934	+	0.465716i	$0.154203\pi$
$314$	5.38684	0.303997
$315$	0	0
$316$	−29.3041	−1.64848
$317$	31.2348	1.75432	0.877159	−	0.480199i	$-0.159436\pi$
$317$	31.2348	1.75432	0.877159	+	0.480199i	$0.159436\pi$
$318$	−5.72006	−0.320765
$319$	7.23392	0.405022
$320$	0	0
$321$	−6.33789	−0.353747
$322$	−0.775057	−0.0431922
$323$	0	0
$324$	7.45421	0.414123
$325$	0	0
$326$	6.11348	0.338594
$327$	−7.04420	−0.389545
$328$	19.5256	1.07812
$329$	−1.19313	−0.0657795
$330$	0	0
$331$	26.8730	1.47707	0.738536	−	0.674214i	$-0.235517\pi$
$331$	26.8730	1.47707	0.738536	+	0.674214i	$0.235517\pi$
$332$	4.34455	0.238438
$333$	−2.10957	−0.115604
$334$	−9.68062	−0.529700
$335$	0	0
$336$	−1.41178	−0.0770190
$337$	−18.5655	−1.01133	−0.505664	−	0.862731i	$-0.668752\pi$
$337$	−18.5655	−1.01133	−0.505664	+	0.862731i	$0.668752\pi$
$338$	−4.29842	−0.233803
$339$	−10.8487	−0.589223
$340$	0	0
$341$	−2.04480	−0.110732
$342$	−1.49888	−0.0810504
$343$	−5.89673	−0.318394
$344$	16.9454	0.913636
$345$	0	0
$346$	5.54990	0.298364
$347$	19.6030	1.05235	0.526173	−	0.850378i	$-0.323626\pi$
$347$	19.6030	1.05235	0.526173	+	0.850378i	$0.323626\pi$
$348$	−16.2237	−0.869682
$349$	0.810028	0.0433598	0.0216799	−	0.999765i	$-0.493099\pi$
$349$	0.810028	0.0433598	0.0216799	+	0.999765i	$0.493099\pi$
$350$	0	0
$351$	−12.3955	−0.661622
$352$	−5.44700	−0.290326
$353$	−19.8152	−1.05466	−0.527328	−	0.849662i	$-0.676806\pi$
$353$	−19.8152	−1.05466	−0.527328	+	0.849662i	$0.676806\pi$
$354$	6.18474	0.328715
$355$	0	0
$356$	10.0442	0.532342
$357$	0	0
$358$	−4.12209	−0.217859
$359$	33.7814	1.78292	0.891458	−	0.453104i	$-0.149683\pi$
$359$	33.7814	1.78292	0.891458	+	0.453104i	$0.149683\pi$
$360$	0	0
$361$	−12.6444	−0.665494
$362$	−5.04960	−0.265401
$363$	13.5261	0.709937
$364$	1.61484	0.0846406
$365$	0	0
$366$	0.473623	0.0247567
$367$	31.5083	1.64472	0.822359	−	0.568969i	$-0.192658\pi$
$367$	31.5083	1.64472	0.822359	+	0.568969i	$0.192658\pi$
$368$	−8.36804	−0.436214
$369$	11.2962	0.588055
$370$	0	0
$371$	3.39964	0.176501
$372$	4.58592	0.237769
$373$	−12.4545	−0.644872	−0.322436	−	0.946591i	$-0.604502\pi$
$373$	−12.4545	−0.644872	−0.322436	+	0.946591i	$0.604502\pi$
$374$	0	0
$375$	0	0
$376$	5.46962	0.282074
$377$	15.1113	0.778273
$378$	1.26603	0.0651178
$379$	6.83988	0.351341	0.175670	−	0.984449i	$-0.443791\pi$
$379$	6.83988	0.351341	0.175670	+	0.984449i	$0.443791\pi$
$380$	0	0
$381$	15.0725	0.772187
$382$	−6.89301	−0.352677
$383$	27.2277	1.39127	0.695635	−	0.718396i	$-0.255123\pi$
$383$	27.2277	1.39127	0.695635	+	0.718396i	$0.255123\pi$
$384$	15.6919	0.800774
$385$	0	0
$386$	0.557637	0.0283830
$387$	9.80346	0.498338
$388$	−23.5980	−1.19801
$389$	13.3615	0.677453	0.338727	−	0.940885i	$-0.390004\pi$
$389$	13.3615	0.677453	0.338727	+	0.940885i	$0.390004\pi$
$390$	0	0
$391$	0	0
$392$	13.3379	0.673667
$393$	−15.5316	−0.783465
$394$	12.1604	0.612630
$395$	0	0
$396$	−2.05020	−0.103026
$397$	−24.2040	−1.21476	−0.607382	−	0.794410i	$-0.707780\pi$
$397$	−24.2040	−1.21476	−0.607382	+	0.794410i	$0.707780\pi$
$398$	4.77602	0.239400
$399$	−1.47049	−0.0736165
$400$	0	0
$401$	12.4648	0.622461	0.311230	−	0.950334i	$-0.399259\pi$
$401$	12.4648	0.622461	0.311230	+	0.950334i	$0.399259\pi$
$402$	9.82841	0.490197
$403$	−4.27148	−0.212778
$404$	−24.6459	−1.22618
$405$	0	0
$406$	−1.54342	−0.0765987
$407$	−1.95845	−0.0970768
$408$	0	0
$409$	27.7331	1.37131	0.685657	−	0.727924i	$-0.259515\pi$
$409$	27.7331	1.37131	0.685657	+	0.727924i	$0.259515\pi$
$410$	0	0
$411$	−29.1833	−1.43951
$412$	10.5519	0.519854
$413$	−3.67582	−0.180875
$414$	2.05555	0.101025
$415$	0	0
$416$	−11.3785	−0.557878
$417$	−10.7553	−0.526687
$418$	−1.39151	−0.0680611
$419$	−12.9059	−0.630496	−0.315248	−	0.949009i	$-0.602088\pi$
$419$	−12.9059	−0.630496	−0.315248	+	0.949009i	$0.602088\pi$
$420$	0	0
$421$	12.7003	0.618977	0.309488	−	0.950903i	$-0.399842\pi$
$421$	12.7003	0.618977	0.309488	+	0.950903i	$0.399842\pi$
$422$	−11.9707	−0.582724
$423$	3.16434	0.153856
$424$	−15.5848	−0.756866
$425$	0	0
$426$	2.77627	0.134511
$427$	−0.281492	−0.0136223
$428$	−7.99427	−0.386418
$429$	−3.15217	−0.152188
$430$	0	0
$431$	20.1531	0.970743	0.485371	−	0.874308i	$-0.338684\pi$
$431$	20.1531	0.970743	0.485371	+	0.874308i	$0.338684\pi$
$432$	13.6690	0.657648
$433$	24.2191	1.16390	0.581948	−	0.813226i	$-0.302291\pi$
$433$	24.2191	1.16390	0.581948	+	0.813226i	$0.302291\pi$
$434$	0.436275	0.0209419
$435$	0	0
$436$	−8.88516	−0.425522
$437$	−8.71602	−0.416944
$438$	7.84370	0.374787
$439$	−27.3778	−1.30667	−0.653336	−	0.757068i	$-0.726631\pi$
$439$	−27.3778	−1.30667	−0.653336	+	0.757068i	$0.726631\pi$
$440$	0	0
$441$	7.71641	0.367448
$442$	0	0
$443$	36.2102	1.72040	0.860199	−	0.509959i	$-0.170339\pi$
$443$	36.2102	1.72040	0.860199	+	0.509959i	$0.170339\pi$
$444$	4.39227	0.208448
$445$	0	0
$446$	2.62276	0.124192
$447$	−4.80096	−0.227078
$448$	−0.903616	−0.0426918
$449$	10.5372	0.497281	0.248640	−	0.968596i	$-0.420016\pi$
$449$	10.5372	0.497281	0.248640	+	0.968596i	$0.420016\pi$
$450$	0	0
$451$	10.4870	0.493812
$452$	−13.6840	−0.643642
$453$	25.1984	1.18392
$454$	10.1433	0.476049
$455$	0	0
$456$	6.74109	0.315681
$457$	−13.9010	−0.650261	−0.325131	−	0.945669i	$-0.605408\pi$
$457$	−13.9010	−0.650261	−0.325131	+	0.945669i	$0.605408\pi$
$458$	−13.3576	−0.624158
$459$	0	0
$460$	0	0
$461$	−33.4413	−1.55752	−0.778758	−	0.627324i	$-0.784150\pi$
$461$	−33.4413	−1.55752	−0.778758	+	0.627324i	$0.784150\pi$
$462$	0.321952	0.0149786
$463$	−9.25282	−0.430015	−0.215007	−	0.976612i	$-0.568978\pi$
$463$	−9.25282	−0.430015	−0.215007	+	0.976612i	$0.568978\pi$
$464$	−16.6638	−0.773598
$465$	0	0
$466$	2.04428	0.0946996
$467$	−26.3851	−1.22096	−0.610478	−	0.792033i	$-0.709023\pi$
$467$	−26.3851	−1.22096	−0.610478	+	0.792033i	$0.709023\pi$
$468$	−4.28277	−0.197971
$469$	−5.84139	−0.269730
$470$	0	0
$471$	−14.0159	−0.645821
$472$	16.8509	0.775624
$473$	9.10118	0.418473
$474$	−12.2045	−0.560569
$475$	0	0
$476$	0	0
$477$	−9.01630	−0.412828
$478$	0.405879	0.0185645
$479$	−34.1460	−1.56017	−0.780085	−	0.625673i	$-0.784824\pi$
$479$	−34.1460	−1.56017	−0.780085	+	0.625673i	$0.784824\pi$
$480$	0	0
$481$	−4.09111	−0.186539
$482$	−0.467447	−0.0212916
$483$	2.01661	0.0917590
$484$	17.0611	0.775504
$485$	0	0
$486$	−5.79564	−0.262895
$487$	31.0117	1.40527	0.702636	−	0.711549i	$-0.252006\pi$
$487$	31.0117	1.40527	0.702636	+	0.711549i	$0.252006\pi$
$488$	1.29043	0.0584149
$489$	−15.9066	−0.719321
$490$	0	0
$491$	3.56118	0.160714	0.0803569	−	0.996766i	$-0.474394\pi$
$491$	3.56118	0.160714	0.0803569	+	0.996766i	$0.474394\pi$
$492$	−23.5194	−1.06034
$493$	0	0
$494$	−2.90680	−0.130783
$495$	0	0
$496$	4.71033	0.211500
$497$	−1.65004	−0.0740145
$498$	1.80940	0.0810813
$499$	−0.645096	−0.0288785	−0.0144392	−	0.999896i	$-0.504596\pi$
$499$	−0.645096	−0.0288785	−0.0144392	+	0.999896i	$0.504596\pi$
$500$	0	0
$501$	25.1879	1.12531
$502$	4.60787	0.205659
$503$	24.2030	1.07916	0.539580	−	0.841934i	$-0.318583\pi$
$503$	24.2030	1.07916	0.539580	+	0.841934i	$0.318583\pi$
$504$	0.944873	0.0420880
$505$	0	0
$506$	1.90830	0.0848345
$507$	11.1840	0.496699
$508$	19.0116	0.843504
$509$	−24.7497	−1.09701	−0.548505	−	0.836147i	$-0.684803\pi$
$509$	−24.7497	−1.09701	−0.548505	+	0.836147i	$0.684803\pi$
$510$	0	0
$511$	−4.66180	−0.206226
$512$	22.0176	0.973050
$513$	14.2374	0.628595
$514$	15.0315	0.663012
$515$	0	0
$516$	−20.4115	−0.898564
$517$	2.93767	0.129198
$518$	0.417853	0.0183594
$519$	−14.4402	−0.633855
$520$	0	0
$521$	−1.23458	−0.0540878	−0.0270439	−	0.999634i	$-0.508609\pi$
$521$	−1.23458	−0.0540878	−0.0270439	+	0.999634i	$0.508609\pi$
$522$	4.09336	0.179161
$523$	−14.8930	−0.651225	−0.325612	−	0.945503i	$-0.605570\pi$
$523$	−14.8930	−0.651225	−0.325612	+	0.945503i	$0.605570\pi$
$524$	−19.5907	−0.855823
$525$	0	0
$526$	−0.203911	−0.00889094
$527$	0	0
$528$	3.47601	0.151274
$529$	−11.0469	−0.480302
$530$	0	0
$531$	9.74876	0.423060
$532$	−1.85479	−0.0804155
$533$	21.9068	0.948888
$534$	4.18317	0.181024
$535$	0	0
$536$	26.7784	1.15665
$537$	10.7252	0.462827
$538$	−14.6484	−0.631535
$539$	7.16364	0.308560
$540$	0	0
$541$	−31.7233	−1.36389	−0.681946	−	0.731402i	$-0.738866\pi$
$541$	−31.7233	−1.36389	−0.681946	+	0.731402i	$0.738866\pi$
$542$	4.22054	0.181288
$543$	13.1385	0.563827
$544$	0	0
$545$	0	0
$546$	0.672542	0.0287822
$547$	23.5068	1.00508	0.502539	−	0.864554i	$-0.332399\pi$
$547$	23.5068	1.00508	0.502539	+	0.864554i	$0.332399\pi$
$548$	−36.8102	−1.57246
$549$	0.746553	0.0318621
$550$	0	0
$551$	−17.3568	−0.739423
$552$	−9.24466	−0.393479
$553$	7.25356	0.308453
$554$	0.207747	0.00882633
$555$	0	0
$556$	−13.5661	−0.575331
$557$	−16.4566	−0.697286	−0.348643	−	0.937256i	$-0.613357\pi$
$557$	−16.4566	−0.697286	−0.348643	+	0.937256i	$0.613357\pi$
$558$	−1.15706	−0.0489823
$559$	19.0119	0.804119
$560$	0	0
$561$	0	0
$562$	3.58164	0.151082
$563$	33.0781	1.39408	0.697038	−	0.717034i	$-0.254501\pi$
$563$	33.0781	1.39408	0.697038	+	0.717034i	$0.254501\pi$
$564$	−6.58838	−0.277421
$565$	0	0
$566$	11.7428	0.493585
$567$	−1.84512	−0.0774877
$568$	7.56421	0.317387
$569$	13.2798	0.556720	0.278360	−	0.960477i	$-0.410209\pi$
$569$	13.2798	0.556720	0.278360	+	0.960477i	$0.410209\pi$
$570$	0	0
$571$	−7.73126	−0.323543	−0.161772	−	0.986828i	$-0.551721\pi$
$571$	−7.73126	−0.323543	−0.161772	+	0.986828i	$0.551721\pi$
$572$	−3.97597	−0.166244
$573$	17.9349	0.749239
$574$	−2.23749	−0.0933908
$575$	0	0
$576$	2.39651	0.0998546
$577$	−8.60574	−0.358262	−0.179131	−	0.983825i	$-0.557329\pi$
$577$	−8.60574	−0.358262	−0.179131	+	0.983825i	$0.557329\pi$
$578$	0	0
$579$	−1.45091	−0.0602977
$580$	0	0
$581$	−1.07539	−0.0446149
$582$	−9.82802	−0.407385
$583$	−8.37042	−0.346667
$584$	21.3709	0.884333
$585$	0	0
$586$	−2.76966	−0.114413
$587$	−33.1043	−1.36636	−0.683180	−	0.730250i	$-0.739404\pi$
$587$	−33.1043	−1.36636	−0.683180	+	0.730250i	$0.739404\pi$
$588$	−16.0661	−0.662554
$589$	4.90620	0.202156
$590$	0	0
$591$	−31.6399	−1.30149
$592$	4.51142	0.185418
$593$	−3.21197	−0.131900	−0.0659499	−	0.997823i	$-0.521008\pi$
$593$	−3.21197	−0.131900	−0.0659499	+	0.997823i	$0.521008\pi$
$594$	−3.11716	−0.127899
$595$	0	0
$596$	−6.05567	−0.248050
$597$	−12.4267	−0.508590
$598$	3.98636	0.163014
$599$	16.4121	0.670582	0.335291	−	0.942115i	$-0.391165\pi$
$599$	16.4121	0.670582	0.335291	+	0.942115i	$0.391165\pi$
$600$	0	0
$601$	28.7176	1.17142	0.585708	−	0.810522i	$-0.300816\pi$
$601$	28.7176	1.17142	0.585708	+	0.810522i	$0.300816\pi$
$602$	−1.94182	−0.0791425
$603$	15.4921	0.630888
$604$	31.7838	1.29327
$605$	0	0
$606$	−10.2645	−0.416965
$607$	−5.81026	−0.235831	−0.117916	−	0.993024i	$-0.537621\pi$
$607$	−5.81026	−0.235831	−0.117916	+	0.993024i	$0.537621\pi$
$608$	13.0693	0.530030
$609$	4.01581	0.162729
$610$	0	0
$611$	6.13664	0.248262
$612$	0	0
$613$	9.22803	0.372716	0.186358	−	0.982482i	$-0.440331\pi$
$613$	9.22803	0.372716	0.186358	+	0.982482i	$0.440331\pi$
$614$	5.28293	0.213202
$615$	0	0
$616$	0.877187	0.0353429
$617$	18.4256	0.741787	0.370894	−	0.928675i	$-0.379051\pi$
$617$	18.4256	0.741787	0.370894	+	0.928675i	$0.379051\pi$
$618$	4.39461	0.176777
$619$	17.6479	0.709330	0.354665	−	0.934993i	$-0.384595\pi$
$619$	17.6479	0.709330	0.354665	+	0.934993i	$0.384595\pi$
$620$	0	0
$621$	−19.5250	−0.783510
$622$	1.51182	0.0606186
$623$	−2.48621	−0.0996080
$624$	7.26123	0.290682
$625$	0	0
$626$	16.4489	0.657431
$627$	3.62056	0.144591
$628$	−17.6789	−0.705467
$629$	0	0
$630$	0	0
$631$	−10.5071	−0.418281	−0.209141	−	0.977886i	$-0.567067\pi$
$631$	−10.5071	−0.418281	−0.209141	+	0.977886i	$0.567067\pi$
$632$	−33.2521	−1.32270
$633$	31.1464	1.23796
$634$	16.4083	0.651655
$635$	0	0
$636$	18.7725	0.744380
$637$	14.9645	0.592915
$638$	3.80013	0.150448
$639$	4.37613	0.173117
$640$	0	0
$641$	30.6891	1.21215	0.606073	−	0.795409i	$-0.292744\pi$
$641$	30.6891	1.21215	0.606073	+	0.795409i	$0.292744\pi$
$642$	−3.32943	−0.131402
$643$	16.6193	0.655403	0.327701	−	0.944781i	$-0.393726\pi$
$643$	16.6193	0.655403	0.327701	+	0.944781i	$0.393726\pi$
$644$	2.54364	0.100234
$645$	0	0
$646$	0	0
$647$	−5.35376	−0.210478	−0.105239	−	0.994447i	$-0.533561\pi$
$647$	−5.35376	−0.210478	−0.105239	+	0.994447i	$0.533561\pi$
$648$	8.45849	0.332281
$649$	9.05040	0.355259
$650$	0	0
$651$	−1.13514	−0.0444896
$652$	−20.0637	−0.785755
$653$	−31.0039	−1.21328	−0.606639	−	0.794978i	$-0.707483\pi$
$653$	−31.0039	−1.21328	−0.606639	+	0.794978i	$0.707483\pi$
$654$	−3.70046	−0.144699
$655$	0	0
$656$	−24.1574	−0.943189
$657$	12.3637	0.482355
$658$	−0.626776	−0.0244343
$659$	17.4175	0.678490	0.339245	−	0.940698i	$-0.389828\pi$
$659$	17.4175	0.678490	0.339245	+	0.940698i	$0.389828\pi$
$660$	0	0
$661$	−4.19942	−0.163339	−0.0816693	−	0.996659i	$-0.526025\pi$
$661$	−4.19942	−0.163339	−0.0816693	+	0.996659i	$0.526025\pi$
$662$	14.1169	0.548670
$663$	0	0
$664$	4.92988	0.191316
$665$	0	0
$666$	−1.10820	−0.0429419
$667$	23.8029	0.921651
$668$	31.7706	1.22924
$669$	−6.82414	−0.263837
$670$	0	0
$671$	0.693073	0.0267558
$672$	−3.02383	−0.116647
$673$	14.4500	0.557008	0.278504	−	0.960435i	$-0.410161\pi$
$673$	14.4500	0.557008	0.278504	+	0.960435i	$0.410161\pi$
$674$	−9.75284	−0.375665
$675$	0	0
$676$	14.1069	0.542572
$677$	−7.90720	−0.303898	−0.151949	−	0.988388i	$-0.548555\pi$
$677$	−7.90720	−0.303898	−0.151949	+	0.988388i	$0.548555\pi$
$678$	−5.69907	−0.218871
$679$	5.84115	0.224163
$680$	0	0
$681$	−26.3918	−1.01133
$682$	−1.07417	−0.0411322
$683$	−25.5957	−0.979393	−0.489697	−	0.871893i	$-0.662892\pi$
$683$	−25.5957	−0.979393	−0.489697	+	0.871893i	$0.662892\pi$
$684$	4.91916	0.188089
$685$	0	0
$686$	−3.09768	−0.118270
$687$	34.7549	1.32598
$688$	−20.9652	−0.799290
$689$	−17.4854	−0.666141
$690$	0	0
$691$	8.26213	0.314306	0.157153	−	0.987574i	$-0.449768\pi$
$691$	8.26213	0.314306	0.157153	+	0.987574i	$0.449768\pi$
$692$	−18.2141	−0.692396
$693$	0.507480	0.0192776
$694$	10.2979	0.390902
$695$	0	0
$696$	−18.4095	−0.697809
$697$	0	0
$698$	0.425524	0.0161063
$699$	−5.31900	−0.201183
$700$	0	0
$701$	27.2367	1.02872	0.514358	−	0.857576i	$-0.328030\pi$
$701$	27.2367	1.02872	0.514358	+	0.857576i	$0.328030\pi$
$702$	−6.51160	−0.245765
$703$	4.69902	0.177227
$704$	2.22484	0.0838516
$705$	0	0
$706$	−10.4093	−0.391760
$707$	6.10054	0.229435
$708$	−20.2976	−0.762829
$709$	−5.24111	−0.196834	−0.0984169	−	0.995145i	$-0.531378\pi$
$709$	−5.24111	−0.196834	−0.0984169	+	0.995145i	$0.531378\pi$
$710$	0	0
$711$	−19.2374	−0.721459
$712$	11.3974	0.427137
$713$	−6.72831	−0.251977
$714$	0	0
$715$	0	0
$716$	13.5282	0.505572
$717$	−1.05605	−0.0394390
$718$	17.7461	0.662278
$719$	10.7001	0.399048	0.199524	−	0.979893i	$-0.436060\pi$
$719$	10.7001	0.399048	0.199524	+	0.979893i	$0.436060\pi$
$720$	0	0
$721$	−2.61188	−0.0972714
$722$	−6.64236	−0.247203
$723$	1.21624	0.0452326
$724$	16.5722	0.615900
$725$	0	0
$726$	7.10554	0.263711
$727$	−3.03596	−0.112597	−0.0562987	−	0.998414i	$-0.517930\pi$
$727$	−3.03596	−0.112597	−0.0562987	+	0.998414i	$0.517930\pi$
$728$	1.83240	0.0679133
$729$	28.0507	1.03891
$730$	0	0
$731$	0	0
$732$	−1.55437	−0.0574513
$733$	−34.4475	−1.27235	−0.636173	−	0.771546i	$-0.719484\pi$
$733$	−34.4475	−1.27235	−0.636173	+	0.771546i	$0.719484\pi$
$734$	16.5519	0.610943
$735$	0	0
$736$	−17.9231	−0.660654
$737$	14.3823	0.529781
$738$	5.93411	0.218438
$739$	−0.320020	−0.0117721	−0.00588606	−	0.999983i	$-0.501874\pi$
$739$	−0.320020	−0.0117721	−0.00588606	+	0.999983i	$0.501874\pi$
$740$	0	0
$741$	7.56318	0.277840
$742$	1.78590	0.0655625
$743$	35.0265	1.28500	0.642498	−	0.766287i	$-0.277898\pi$
$743$	35.0265	1.28500	0.642498	+	0.766287i	$0.277898\pi$
$744$	5.20377	0.190779
$745$	0	0
$746$	−6.54262	−0.239543
$747$	2.85209	0.104352
$748$	0	0
$749$	1.97880	0.0723038
$750$	0	0
$751$	20.8216	0.759790	0.379895	−	0.925030i	$-0.375960\pi$
$751$	20.8216	0.759790	0.379895	+	0.925030i	$0.375960\pi$
$752$	−6.76711	−0.246771
$753$	−11.9892	−0.436909
$754$	7.93829	0.289095
$755$	0	0
$756$	−4.15497	−0.151115
$757$	−8.96497	−0.325837	−0.162919	−	0.986640i	$-0.552091\pi$
$757$	−8.96497	−0.325837	−0.162919	+	0.986640i	$0.552091\pi$
$758$	3.59313	0.130508
$759$	−4.96520	−0.180225
$760$	0	0
$761$	26.9925	0.978479	0.489240	−	0.872149i	$-0.337274\pi$
$761$	26.9925	0.978479	0.489240	+	0.872149i	$0.337274\pi$
$762$	7.91789	0.286835
$763$	2.19932	0.0796207
$764$	22.6220	0.818437
$765$	0	0
$766$	14.3033	0.516798
$767$	18.9059	0.682651
$768$	2.45490	0.0885836
$769$	−41.4938	−1.49630	−0.748152	−	0.663528i	$-0.769059\pi$
$769$	−41.4938	−1.49630	−0.748152	+	0.663528i	$0.769059\pi$
$770$	0	0
$771$	−39.1103	−1.40852
$772$	−1.83010	−0.0658666
$773$	−22.4915	−0.808965	−0.404482	−	0.914546i	$-0.632548\pi$
$773$	−22.4915	−0.808965	−0.404482	+	0.914546i	$0.632548\pi$
$774$	5.14996	0.185111
$775$	0	0
$776$	−26.7773	−0.961250
$777$	−1.08721	−0.0390033
$778$	7.01905	0.251645
$779$	−25.1620	−0.901521
$780$	0	0
$781$	4.06264	0.145373
$782$	0	0
$783$	−38.8813	−1.38951
$784$	−16.5019	−0.589354
$785$	0	0
$786$	−8.15906	−0.291024
$787$	−21.3705	−0.761776	−0.380888	−	0.924621i	$-0.624382\pi$
$787$	−21.3705	−0.761776	−0.380888	+	0.924621i	$0.624382\pi$
$788$	−39.9089	−1.42169
$789$	0.530554	0.0188882
$790$	0	0
$791$	3.38717	0.120434
$792$	−2.32642	−0.0826656
$793$	1.44780	0.0514128
$794$	−12.7148	−0.451233
$795$	0	0
$796$	−15.6743	−0.555562
$797$	−41.4653	−1.46878	−0.734388	−	0.678730i	$-0.762531\pi$
$797$	−41.4653	−1.46878	−0.734388	+	0.678730i	$0.762531\pi$
$798$	−0.772478	−0.0273454
$799$	0	0
$800$	0	0
$801$	6.59377	0.232979
$802$	6.54800	0.231218
$803$	11.4780	0.405051
$804$	−32.2557	−1.13757
$805$	0	0
$806$	−2.24390	−0.0790379
$807$	38.1134	1.34165
$808$	−27.9664	−0.983856
$809$	−39.6607	−1.39439	−0.697197	−	0.716879i	$-0.745570\pi$
$809$	−39.6607	−1.39439	−0.697197	+	0.716879i	$0.745570\pi$
$810$	0	0
$811$	−21.9630	−0.771224	−0.385612	−	0.922661i	$-0.626010\pi$
$811$	−21.9630	−0.771224	−0.385612	+	0.922661i	$0.626010\pi$
$812$	5.06532	0.177758
$813$	−10.9814	−0.385133
$814$	−1.02881	−0.0360599
$815$	0	0
$816$	0	0
$817$	−21.8370	−0.763979
$818$	14.5688	0.509385
$819$	1.06010	0.0370430
$820$	0	0
$821$	−26.0854	−0.910388	−0.455194	−	0.890392i	$-0.650430\pi$
$821$	−26.0854	−0.910388	−0.455194	+	0.890392i	$0.650430\pi$
$822$	−15.3306	−0.534716
$823$	10.7108	0.373355	0.186678	−	0.982421i	$-0.440228\pi$
$823$	10.7108	0.373355	0.186678	+	0.982421i	$0.440228\pi$
$824$	11.9735	0.417117
$825$	0	0
$826$	−1.93098	−0.0671875
$827$	−43.0498	−1.49699	−0.748495	−	0.663141i	$-0.769223\pi$
$827$	−43.0498	−1.49699	−0.748495	+	0.663141i	$0.769223\pi$
$828$	−6.74608	−0.234443
$829$	38.1054	1.32345	0.661727	−	0.749745i	$-0.269824\pi$
$829$	38.1054	1.32345	0.661727	+	0.749745i	$0.269824\pi$
$830$	0	0
$831$	−0.540535	−0.0187509
$832$	4.64758	0.161126
$833$	0	0
$834$	−5.64996	−0.195642
$835$	0	0
$836$	4.56677	0.157945
$837$	10.9905	0.379887
$838$	−6.77975	−0.234203
$839$	−27.6128	−0.953299	−0.476649	−	0.879093i	$-0.658149\pi$
$839$	−27.6128	−0.953299	−0.476649	+	0.879093i	$0.658149\pi$
$840$	0	0
$841$	18.4002	0.634489
$842$	6.67175	0.229924
$843$	−9.31903	−0.320964
$844$	39.2864	1.35229
$845$	0	0
$846$	1.66229	0.0571509
$847$	−4.22308	−0.145107
$848$	19.2818	0.662140
$849$	−30.5533	−1.04859
$850$	0	0
$851$	−6.44419	−0.220904
$852$	−9.11139	−0.312151
$853$	15.2286	0.521418	0.260709	−	0.965417i	$-0.416044\pi$
$853$	15.2286	0.521418	0.260709	+	0.965417i	$0.416044\pi$
$854$	−0.147873	−0.00506012
$855$	0	0
$856$	−9.07132	−0.310051
$857$	−47.0658	−1.60774	−0.803868	−	0.594807i	$-0.797228\pi$
$857$	−47.0658	−1.60774	−0.803868	+	0.594807i	$0.797228\pi$
$858$	−1.65590	−0.0565314
$859$	33.7888	1.15286	0.576430	−	0.817147i	$-0.304446\pi$
$859$	33.7888	1.15286	0.576430	+	0.817147i	$0.304446\pi$
$860$	0	0
$861$	5.82169	0.198403
$862$	10.5869	0.360590
$863$	−51.0120	−1.73647	−0.868235	−	0.496153i	$-0.834745\pi$
$863$	−51.0120	−1.73647	−0.868235	+	0.496153i	$0.834745\pi$
$864$	29.2769	0.996020
$865$	0	0
$866$	12.7228	0.432338
$867$	0	0
$868$	−1.43180	−0.0485986
$869$	−17.8593	−0.605836
$870$	0	0
$871$	30.0440	1.01800
$872$	−10.0822	−0.341428
$873$	−15.4915	−0.524309
$874$	−4.57870	−0.154877
$875$	0	0
$876$	−25.7421	−0.869745
$877$	−28.6658	−0.967975	−0.483988	−	0.875075i	$-0.660812\pi$
$877$	−28.6658	−0.967975	−0.483988	+	0.875075i	$0.660812\pi$
$878$	−14.3821	−0.485373
$879$	7.20634	0.243064
$880$	0	0
$881$	19.5557	0.658847	0.329423	−	0.944182i	$-0.393146\pi$
$881$	19.5557	0.658847	0.329423	+	0.944182i	$0.393146\pi$
$882$	4.05359	0.136491
$883$	16.3225	0.549295	0.274647	−	0.961545i	$-0.411439\pi$
$883$	16.3225	0.549295	0.274647	+	0.961545i	$0.411439\pi$
$884$	0	0
$885$	0	0
$886$	19.0220	0.639055
$887$	−16.6193	−0.558023	−0.279011	−	0.960288i	$-0.590007\pi$
$887$	−16.6193	−0.558023	−0.279011	+	0.960288i	$0.590007\pi$
$888$	4.98403	0.167253
$889$	−4.70589	−0.157831
$890$	0	0
$891$	4.54295	0.152195
$892$	−8.60760	−0.288204
$893$	−7.04851	−0.235869
$894$	−2.52204	−0.0843497
$895$	0	0
$896$	−4.89928	−0.163674
$897$	−10.3721	−0.346313
$898$	5.53540	0.184719
$899$	−13.3985	−0.446865
$900$	0	0
$901$	0	0
$902$	5.50902	0.183430
$903$	5.05239	0.168133
$904$	−15.5276	−0.516441
$905$	0	0
$906$	13.2372	0.439777
$907$	40.8198	1.35540	0.677700	−	0.735339i	$-0.262977\pi$
$907$	40.8198	1.35540	0.677700	+	0.735339i	$0.262977\pi$
$908$	−33.2891	−1.10474
$909$	−16.1795	−0.536639
$910$	0	0
$911$	39.7910	1.31834	0.659168	−	0.751996i	$-0.270909\pi$
$911$	39.7910	1.31834	0.659168	+	0.751996i	$0.270909\pi$
$912$	−8.34020	−0.276171
$913$	2.64778	0.0876287
$914$	−7.30248	−0.241545
$915$	0	0
$916$	43.8379	1.44844
$917$	4.84923	0.160136
$918$	0	0
$919$	−27.5142	−0.907609	−0.453804	−	0.891101i	$-0.649933\pi$
$919$	−27.5142	−0.907609	−0.453804	+	0.891101i	$0.649933\pi$
$920$	0	0
$921$	−13.7456	−0.452933
$922$	−17.5674	−0.578552
$923$	8.48667	0.279342
$924$	−1.05661	−0.0347598
$925$	0	0
$926$	−4.86069	−0.159732
$927$	6.92705	0.227514
$928$	−35.6914	−1.17163
$929$	−19.2743	−0.632370	−0.316185	−	0.948698i	$-0.602402\pi$
$929$	−19.2743	−0.632370	−0.316185	+	0.948698i	$0.602402\pi$
$930$	0	0
$931$	−17.1881	−0.563318
$932$	−6.70909	−0.219764
$933$	−3.93360	−0.128780
$934$	−13.8606	−0.453534
$935$	0	0
$936$	−4.85978	−0.158847
$937$	−2.94916	−0.0963447	−0.0481724	−	0.998839i	$-0.515340\pi$
$937$	−2.94916	−0.0963447	−0.0481724	+	0.998839i	$0.515340\pi$
$938$	−3.06860	−0.100193
$939$	−42.7983	−1.39667
$940$	0	0
$941$	2.73460	0.0891453	0.0445727	−	0.999006i	$-0.485807\pi$
$941$	2.73460	0.0891453	0.0445727	+	0.999006i	$0.485807\pi$
$942$	−7.36286	−0.239895
$943$	34.5069	1.12370
$944$	−20.8482	−0.678551
$945$	0	0
$946$	4.78104	0.155445
$947$	−5.00319	−0.162582	−0.0812909	−	0.996690i	$-0.525904\pi$
$947$	−5.00319	−0.162582	−0.0812909	+	0.996690i	$0.525904\pi$
$948$	40.0536	1.30088
$949$	23.9771	0.778329
$950$	0	0
$951$	−42.6925	−1.38440
$952$	0	0
$953$	32.8175	1.06306	0.531531	−	0.847039i	$-0.321617\pi$
$953$	32.8175	1.06306	0.531531	+	0.847039i	$0.321617\pi$
$954$	−4.73645	−0.153348
$955$	0	0
$956$	−1.33205	−0.0430815
$957$	−9.88751	−0.319618
$958$	−17.9376	−0.579537
$959$	9.11154	0.294227
$960$	0	0
$961$	−27.2127	−0.877828
$962$	−2.14914	−0.0692912
$963$	−5.24804	−0.169116
$964$	1.53410	0.0494102
$965$	0	0
$966$	1.05937	0.0340846
$967$	32.7981	1.05472	0.527358	−	0.849643i	$-0.323183\pi$
$967$	32.7981	1.05472	0.527358	+	0.849643i	$0.323183\pi$
$968$	19.3597	0.622244
$969$	0	0
$970$	0	0
$971$	21.3103	0.683882	0.341941	−	0.939721i	$-0.388916\pi$
$971$	21.3103	0.683882	0.341941	+	0.939721i	$0.388916\pi$
$972$	19.0206	0.610085
$973$	3.35798	0.107652
$974$	16.2911	0.521999
$975$	0	0
$976$	−1.59654	−0.0511040
$977$	34.3683	1.09954	0.549769	−	0.835317i	$-0.314716\pi$
$977$	34.3683	1.09954	0.549769	+	0.835317i	$0.314716\pi$
$978$	−8.35606	−0.267197
$979$	6.12142	0.195641
$980$	0	0
$981$	−5.83289	−0.186230
$982$	1.87076	0.0596984
$983$	53.6887	1.71240	0.856202	−	0.516640i	$-0.172818\pi$
$983$	53.6887	1.71240	0.856202	+	0.516640i	$0.172818\pi$
$984$	−26.6881	−0.850785
$985$	0	0
$986$	0	0
$987$	1.63080	0.0519090
$988$	9.53977	0.303501
$989$	29.9470	0.952259
$990$	0	0
$991$	−28.7325	−0.912718	−0.456359	−	0.889796i	$-0.650847\pi$
$991$	−28.7325	−0.912718	−0.456359	+	0.889796i	$0.650847\pi$
$992$	10.0888	0.320320
$993$	−36.7307	−1.16561
$994$	−0.866800	−0.0274932
$995$	0	0
$996$	−5.93824	−0.188160
$997$	−33.9344	−1.07471	−0.537357	−	0.843355i	$-0.680577\pi$
$997$	−33.9344	−1.07471	−0.537357	+	0.843355i	$0.680577\pi$
$998$	−0.338882	−0.0107271
$999$	10.5264	0.333041

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7225.2.a.bv.1.9	yes	15
5.4	even	2		7225.2.a.bt.1.7	✓	15
17.16	even	2		7225.2.a.bu.1.9	yes	15
85.84	even	2		7225.2.a.bw.1.7	yes	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7225.2.a.bt.1.7	✓	15	5.4	even	2
7225.2.a.bu.1.9	yes	15	17.16	even	2
7225.2.a.bv.1.9	yes	15	1.1	even	1	trivial
7225.2.a.bw.1.7	yes	15	85.84	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 7225 = 5^{2} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7225.a (trivial)

\(f(q)\)	\(=\)	\(q+0.525320 q^{2} -1.36683 q^{3} -1.72404 q^{4} -0.718021 q^{6} +0.426746 q^{7} -1.95631 q^{8} -1.13179 q^{9} +O(q^{10})\)	\(q+0.525320 q^{2} -1.36683 q^{3} -1.72404 q^{4} -0.718021 q^{6} +0.426746 q^{7} -1.95631 q^{8} -1.13179 q^{9} -1.05071 q^{11} +2.35646 q^{12} -2.19489 q^{13} +0.224179 q^{14} +2.42039 q^{16} -0.594552 q^{18} +2.52103 q^{19} -0.583288 q^{21} -0.551961 q^{22} -3.45732 q^{23} +2.67394 q^{24} -1.15302 q^{26} +5.64743 q^{27} -0.735727 q^{28} -6.88478 q^{29} +1.94611 q^{31} +5.18411 q^{32} +1.43614 q^{33} +1.95125 q^{36} +1.86393 q^{37} +1.32435 q^{38} +3.00003 q^{39} -9.98081 q^{41} -0.306413 q^{42} -8.66192 q^{43} +1.81147 q^{44} -1.81620 q^{46} -2.79588 q^{47} -3.30824 q^{48} -6.81789 q^{49} +3.78407 q^{52} +7.96642 q^{53} +2.96671 q^{54} -0.834850 q^{56} -3.44581 q^{57} -3.61672 q^{58} -8.61359 q^{59} -0.659622 q^{61} +1.02233 q^{62} -0.482987 q^{63} -2.11745 q^{64} +0.754434 q^{66} -13.6882 q^{67} +4.72555 q^{69} -3.86656 q^{71} +2.21413 q^{72} -10.9241 q^{73} +0.979159 q^{74} -4.34636 q^{76} -0.448388 q^{77} +1.57598 q^{78} +16.9973 q^{79} -4.32369 q^{81} -5.24313 q^{82} -2.51998 q^{83} +1.00561 q^{84} -4.55028 q^{86} +9.41029 q^{87} +2.05552 q^{88} -5.82597 q^{89} -0.936661 q^{91} +5.96055 q^{92} -2.65999 q^{93} -1.46873 q^{94} -7.08577 q^{96} +13.6876 q^{97} -3.58158 q^{98} +1.18918 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + 9 q^{3} + 12 q^{4} - 9 q^{6} + 12 q^{7} - 6 q^{8} + 12 q^{9}+O(q^{10}) \) 15 * q + 9 * q^3 + 12 * q^4 - 9 * q^6 + 12 * q^7 - 6 * q^8 + 12 * q^9	\( 15 q + 9 q^{3} + 12 q^{4} - 9 q^{6} + 12 q^{7} - 6 q^{8} + 12 q^{9} - 6 q^{11} + 24 q^{12} + 6 q^{16} - 12 q^{18} + 6 q^{19} + 30 q^{21} + 12 q^{22} + 36 q^{23} - 18 q^{24} + 36 q^{26} + 36 q^{27} + 24 q^{28} + 18 q^{29} - 12 q^{32} - 12 q^{33} - 9 q^{36} + 12 q^{37} + 6 q^{38} - 9 q^{39} + 18 q^{41} - 36 q^{42} + 3 q^{43} + 12 q^{44} - 21 q^{46} + 3 q^{47} - 12 q^{48} + 15 q^{49} + 27 q^{52} - 21 q^{54} + 6 q^{56} + 39 q^{57} + 18 q^{58} - 12 q^{59} + 15 q^{61} + 54 q^{62} + 60 q^{63} - 36 q^{64} + 18 q^{66} + 24 q^{67} + 42 q^{69} - 6 q^{71} - 66 q^{72} - 9 q^{73} + 36 q^{74} - 18 q^{76} - 30 q^{77} + 30 q^{78} + 9 q^{79} + 51 q^{81} - 36 q^{82} - 15 q^{83} + 9 q^{84} - 36 q^{86} + 51 q^{87} + 30 q^{88} - 24 q^{89} - 27 q^{91} + 15 q^{92} + 42 q^{93} - 57 q^{94} - 42 q^{96} + 48 q^{97} - 12 q^{99}+O(q^{100}) \) 15 * q + 9 * q^3 + 12 * q^4 - 9 * q^6 + 12 * q^7 - 6 * q^8 + 12 * q^9 - 6 * q^11 + 24 * q^12 + 6 * q^16 - 12 * q^18 + 6 * q^19 + 30 * q^21 + 12 * q^22 + 36 * q^23 - 18 * q^24 + 36 * q^26 + 36 * q^27 + 24 * q^28 + 18 * q^29 - 12 * q^32 - 12 * q^33 - 9 * q^36 + 12 * q^37 + 6 * q^38 - 9 * q^39 + 18 * q^41 - 36 * q^42 + 3 * q^43 + 12 * q^44 - 21 * q^46 + 3 * q^47 - 12 * q^48 + 15 * q^49 + 27 * q^52 - 21 * q^54 + 6 * q^56 + 39 * q^57 + 18 * q^58 - 12 * q^59 + 15 * q^61 + 54 * q^62 + 60 * q^63 - 36 * q^64 + 18 * q^66 + 24 * q^67 + 42 * q^69 - 6 * q^71 - 66 * q^72 - 9 * q^73 + 36 * q^74 - 18 * q^76 - 30 * q^77 + 30 * q^78 + 9 * q^79 + 51 * q^81 - 36 * q^82 - 15 * q^83 + 9 * q^84 - 36 * q^86 + 51 * q^87 + 30 * q^88 - 24 * q^89 - 27 * q^91 + 15 * q^92 + 42 * q^93 - 57 * q^94 - 42 * q^96 + 48 * q^97 - 12 * q^99

Introduction

L-functions

Modular forms

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Database

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