LMFDB - Embedded newform 7225.2.a.bw.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			$0.567962$ 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)

$f(q)$	$=$	$q+0.567962 q^{2} +2.88559 q^{3} -1.67742 q^{4} +1.63891 q^{6} +5.00907 q^{7} -2.08863 q^{8} +5.32662 q^{9} +O(q^{10})$	\(q+0.567962 q^{2} +2.88559 q^{3} -1.67742 q^{4} +1.63891 q^{6} +5.00907 q^{7} -2.08863 q^{8} +5.32662 q^{9} +4.17554 q^{11} -4.84034 q^{12} -0.657310 q^{13} +2.84496 q^{14} +2.16857 q^{16} +3.02532 q^{18} -5.62036 q^{19} +14.4541 q^{21} +2.37155 q^{22} +4.27833 q^{23} -6.02694 q^{24} -0.373327 q^{26} +6.71368 q^{27} -8.40231 q^{28} -5.59957 q^{29} +0.143757 q^{31} +5.40894 q^{32} +12.0489 q^{33} -8.93498 q^{36} +4.88950 q^{37} -3.19215 q^{38} -1.89673 q^{39} -1.55997 q^{41} +8.20939 q^{42} +2.97180 q^{43} -7.00414 q^{44} +2.42993 q^{46} +11.4416 q^{47} +6.25761 q^{48} +18.0908 q^{49} +1.10259 q^{52} -13.3194 q^{53} +3.81312 q^{54} -10.4621 q^{56} -16.2181 q^{57} -3.18034 q^{58} +10.1079 q^{59} -0.961538 q^{61} +0.0816486 q^{62} +26.6814 q^{63} -1.26508 q^{64} +6.84332 q^{66} -0.747944 q^{67} +12.3455 q^{69} +0.0687573 q^{71} -11.1254 q^{72} +1.46435 q^{73} +2.77705 q^{74} +9.42771 q^{76} +20.9156 q^{77} -1.07727 q^{78} +6.72529 q^{79} +3.39306 q^{81} -0.886004 q^{82} -6.20936 q^{83} -24.2456 q^{84} +1.68787 q^{86} -16.1581 q^{87} -8.72118 q^{88} +10.9442 q^{89} -3.29252 q^{91} -7.17655 q^{92} +0.414824 q^{93} +6.49840 q^{94} +15.6080 q^{96} -7.76437 q^{97} +10.2749 q^{98} +22.2416 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	0.567962	0.401610	0.200805	−	0.979631i	$-0.435644\pi$
$2$	0.567962	0.401610	0.200805	+	0.979631i	$0.435644\pi$
$3$	2.88559	1.66600	0.832998	−	0.553276i	$-0.186623\pi$
$3$	2.88559	1.66600	0.832998	+	0.553276i	$0.186623\pi$
$4$	−1.67742	−0.838710
$5$	0	0
$5$	0	0
$6$	1.63891	0.669080
$7$	5.00907	1.89325	0.946626	−	0.322335i	$-0.104468\pi$
$7$	5.00907	1.89325	0.946626	+	0.322335i	$0.104468\pi$
$8$	−2.08863	−0.738444
$9$	5.32662	1.77554
$10$	0	0
$11$	4.17554	1.25897	0.629487	−	0.777011i	$-0.283265\pi$
$11$	4.17554	1.25897	0.629487	+	0.777011i	$0.283265\pi$
$12$	−4.84034	−1.39729
$13$	−0.657310	−0.182305	−0.0911526	−	0.995837i	$-0.529055\pi$
$13$	−0.657310	−0.182305	−0.0911526	+	0.995837i	$0.529055\pi$
$14$	2.84496	0.760348
$15$	0	0
$16$	2.16857	0.542143
$17$	0	0
$17$	0	0
$18$	3.02532	0.713075
$19$	−5.62036	−1.28940	−0.644700	−	0.764436i	$-0.723018\pi$
$19$	−5.62036	−1.28940	−0.644700	+	0.764436i	$0.723018\pi$
$20$	0	0
$21$	14.4541	3.15415
$22$	2.37155	0.505616
$23$	4.27833	0.892093	0.446046	−	0.895010i	$-0.352832\pi$
$23$	4.27833	0.892093	0.446046	+	0.895010i	$0.352832\pi$
$24$	−6.02694	−1.23024
$25$	0	0
$26$	−0.373327	−0.0732155
$27$	6.71368	1.29205
$28$	−8.40231	−1.58789
$29$	−5.59957	−1.03981	−0.519907	−	0.854223i	$-0.674033\pi$
$29$	−5.59957	−1.03981	−0.519907	+	0.854223i	$0.674033\pi$
$30$	0	0
$31$	0.143757	0.0258195	0.0129098	−	0.999917i	$-0.495891\pi$
$31$	0.143757	0.0258195	0.0129098	+	0.999917i	$0.495891\pi$
$32$	5.40894	0.956174
$33$	12.0489	2.09744
$34$	0	0
$35$	0	0
$36$	−8.93498	−1.48916
$37$	4.88950	0.803828	0.401914	−	0.915677i	$-0.368345\pi$
$37$	4.88950	0.803828	0.401914	+	0.915677i	$0.368345\pi$
$38$	−3.19215	−0.517836
$39$	−1.89673	−0.303720
$40$	0	0
$41$	−1.55997	−0.243627	−0.121813	−	0.992553i	$-0.538871\pi$
$41$	−1.55997	−0.243627	−0.121813	+	0.992553i	$0.538871\pi$
$42$	8.20939	1.26674
$43$	2.97180	0.453196	0.226598	−	0.973988i	$-0.427240\pi$
$43$	2.97180	0.453196	0.226598	+	0.973988i	$0.427240\pi$
$44$	−7.00414	−1.05591
$45$	0	0
$46$	2.42993	0.358273
$47$	11.4416	1.66893	0.834466	−	0.551060i	$-0.185777\pi$
$47$	11.4416	1.66893	0.834466	+	0.551060i	$0.185777\pi$
$48$	6.25761	0.903208
$49$	18.0908	2.58440
$50$	0	0
$51$	0	0
$52$	1.10259	0.152901
$53$	−13.3194	−1.82956	−0.914778	−	0.403957i	$-0.867635\pi$
$53$	−13.3194	−1.82956	−0.914778	+	0.403957i	$0.867635\pi$
$54$	3.81312	0.518900
$55$	0	0
$56$	−10.4621	−1.39806
$57$	−16.2181	−2.14813
$58$	−3.18034	−0.417599
$59$	10.1079	1.31594	0.657968	−	0.753045i	$-0.271416\pi$
$59$	10.1079	1.31594	0.657968	+	0.753045i	$0.271416\pi$
$60$	0	0
$61$	−0.961538	−0.123112	−0.0615562	−	0.998104i	$-0.519606\pi$
$61$	−0.961538	−0.123112	−0.0615562	+	0.998104i	$0.519606\pi$
$62$	0.0816486	0.0103694
$63$	26.6814	3.36155
$64$	−1.26508	−0.158135
$65$	0	0
$66$	6.84332	0.842354
$67$	−0.747944	−0.0913759	−0.0456880	−	0.998956i	$-0.514548\pi$
$67$	−0.747944	−0.0913759	−0.0456880	+	0.998956i	$0.514548\pi$
$68$	0	0
$69$	12.3455	1.48622
$70$	0	0
$71$	0.0687573	0.00815999	0.00408000	−	0.999992i	$-0.498701\pi$
$71$	0.0687573	0.00815999	0.00408000	+	0.999992i	$0.498701\pi$
$72$	−11.1254	−1.31114
$73$	1.46435	0.171389	0.0856944	−	0.996321i	$-0.472689\pi$
$73$	1.46435	0.171389	0.0856944	+	0.996321i	$0.472689\pi$
$74$	2.77705	0.322825
$75$	0	0
$76$	9.42771	1.08143
$77$	20.9156	2.38355
$78$	−1.07727	−0.121977
$79$	6.72529	0.756654	0.378327	−	0.925672i	$-0.376500\pi$
$79$	6.72529	0.756654	0.378327	+	0.925672i	$0.376500\pi$
$80$	0	0
$81$	3.39306	0.377007
$82$	−0.886004	−0.0978428
$83$	−6.20936	−0.681566	−0.340783	−	0.940142i	$-0.610692\pi$
$83$	−6.20936	−0.681566	−0.340783	+	0.940142i	$0.610692\pi$
$84$	−24.2456	−2.64541
$85$	0	0
$86$	1.68787	0.182008
$87$	−16.1581	−1.73233
$88$	−8.72118	−0.929681
$89$	10.9442	1.16008	0.580041	−	0.814587i	$-0.303037\pi$
$89$	10.9442	1.16008	0.580041	+	0.814587i	$0.303037\pi$
$90$	0	0
$91$	−3.29252	−0.345149
$92$	−7.17655	−0.748207
$93$	0.414824	0.0430153
$94$	6.49840	0.670259
$95$	0	0
$96$	15.6080	1.59298
$97$	−7.76437	−0.788352	−0.394176	−	0.919035i	$-0.628970\pi$
$97$	−7.76437	−0.788352	−0.394176	+	0.919035i	$0.628970\pi$
$98$	10.2749	1.03792
$99$	22.2416	2.23536
$100$	0	0
$101$	6.07228	0.604215	0.302107	−	0.953274i	$-0.402310\pi$
$101$	6.07228	0.604215	0.302107	+	0.953274i	$0.402310\pi$
$102$	0	0
$103$	−12.0314	−1.18549	−0.592744	−	0.805391i	$-0.701956\pi$
$103$	−12.0314	−1.18549	−0.592744	+	0.805391i	$0.701956\pi$
$104$	1.37288	0.134622
$105$	0	0
$106$	−7.56490	−0.734768
$107$	−10.9225	−1.05591	−0.527957	−	0.849271i	$-0.677042\pi$
$107$	−10.9225	−1.05591	−0.527957	+	0.849271i	$0.677042\pi$
$108$	−11.2617	−1.08365
$109$	−14.5989	−1.39832	−0.699162	−	0.714964i	$-0.746443\pi$
$109$	−14.5989	−1.39832	−0.699162	+	0.714964i	$0.746443\pi$
$110$	0	0
$111$	14.1091	1.33917
$112$	10.8625	1.02641
$113$	−1.40986	−0.132628	−0.0663141	−	0.997799i	$-0.521124\pi$
$113$	−1.40986	−0.132628	−0.0663141	+	0.997799i	$0.521124\pi$
$114$	−9.21124	−0.862712
$115$	0	0
$116$	9.39282	0.872102
$117$	−3.50125	−0.323690
$118$	5.74091	0.528493
$119$	0	0
$120$	0	0
$121$	6.43516	0.585015
$122$	−0.546117	−0.0494431
$123$	−4.50143	−0.405881
$124$	−0.241141	−0.0216551
$125$	0	0
$126$	15.1541	1.35003
$127$	−1.54162	−0.136797	−0.0683984	−	0.997658i	$-0.521789\pi$
$127$	−1.54162	−0.136797	−0.0683984	+	0.997658i	$0.521789\pi$
$128$	−11.5364	−1.01968
$129$	8.57540	0.755022
$130$	0	0
$131$	−2.08132	−0.181845	−0.0909227	−	0.995858i	$-0.528982\pi$
$131$	−2.08132	−0.181845	−0.0909227	+	0.995858i	$0.528982\pi$
$132$	−20.2111	−1.75915
$133$	−28.1528	−2.44116
$134$	−0.424804	−0.0366975
$135$	0	0
$136$	0	0
$137$	−11.1320	−0.951072	−0.475536	−	0.879696i	$-0.657746\pi$
$137$	−11.1320	−0.951072	−0.475536	+	0.879696i	$0.657746\pi$
$138$	7.01177	0.596882
$139$	−15.0404	−1.27571	−0.637853	−	0.770158i	$-0.720177\pi$
$139$	−15.0404	−1.27571	−0.637853	+	0.770158i	$0.720177\pi$
$140$	0	0
$141$	33.0158	2.78043
$142$	0.0390515	0.00327713
$143$	−2.74463	−0.229517
$144$	11.5512	0.962598
$145$	0	0
$146$	0.831694	0.0688314
$147$	52.2026	4.30560
$148$	−8.20174	−0.674178
$149$	9.68788	0.793662	0.396831	−	0.917892i	$-0.370110\pi$
$149$	9.68788	0.793662	0.396831	+	0.917892i	$0.370110\pi$
$150$	0	0
$151$	12.9184	1.05129	0.525644	−	0.850705i	$-0.323824\pi$
$151$	12.9184	1.05129	0.525644	+	0.850705i	$0.323824\pi$
$152$	11.7389	0.952149
$153$	0	0
$154$	11.8793	0.957259
$155$	0	0
$156$	3.18161	0.254732
$157$	14.3030	1.14150	0.570752	−	0.821122i	$-0.306652\pi$
$157$	14.3030	1.14150	0.570752	+	0.821122i	$0.306652\pi$
$158$	3.81971	0.303880
$159$	−38.4342	−3.04803
$160$	0	0
$161$	21.4304	1.68896
$162$	1.92713	0.151410
$163$	14.2490	1.11607	0.558035	−	0.829817i	$-0.311555\pi$
$163$	14.2490	1.11607	0.558035	+	0.829817i	$0.311555\pi$
$164$	2.61672	0.204332
$165$	0	0
$166$	−3.52668	−0.273724
$167$	−0.117789	−0.00911480	−0.00455740	−	0.999990i	$-0.501451\pi$
$167$	−0.117789	−0.00911480	−0.00455740	+	0.999990i	$0.501451\pi$
$168$	−30.1894	−2.32916
$169$	−12.5679	−0.966765
$170$	0	0
$171$	−29.9376	−2.28938
$172$	−4.98496	−0.380099
$173$	−12.6922	−0.964971	−0.482486	−	0.875904i	$-0.660266\pi$
$173$	−12.6922	−0.964971	−0.482486	+	0.875904i	$0.660266\pi$
$174$	−9.17716	−0.695719
$175$	0	0
$176$	9.05497	0.682544
$177$	29.1673	2.19235
$178$	6.21588	0.465900
$179$	10.8855	0.813618	0.406809	−	0.913513i	$-0.366641\pi$
$179$	10.8855	0.813618	0.406809	+	0.913513i	$0.366641\pi$
$180$	0	0
$181$	13.1464	0.977163	0.488581	−	0.872518i	$-0.337515\pi$
$181$	13.1464	0.977163	0.488581	+	0.872518i	$0.337515\pi$
$182$	−1.87002	−0.138615
$183$	−2.77460	−0.205105
$184$	−8.93586	−0.658760
$185$	0	0
$186$	0.235604	0.0172753
$187$	0	0
$188$	−19.1924	−1.39975
$189$	33.6293	2.44617
$190$	0	0
$191$	−15.1085	−1.09321	−0.546605	−	0.837391i	$-0.684080\pi$
$191$	−15.1085	−1.09321	−0.546605	+	0.837391i	$0.684080\pi$
$192$	−3.65049	−0.263451
$193$	12.5625	0.904266	0.452133	−	0.891950i	$-0.350663\pi$
$193$	12.5625	0.904266	0.452133	+	0.891950i	$0.350663\pi$
$194$	−4.40987	−0.316610
$195$	0	0
$196$	−30.3459	−2.16756
$197$	−11.4550	−0.816134	−0.408067	−	0.912952i	$-0.633797\pi$
$197$	−11.4550	−0.816134	−0.408067	+	0.912952i	$0.633797\pi$
$198$	12.6324	0.897743
$199$	−5.54827	−0.393306	−0.196653	−	0.980473i	$-0.563007\pi$
$199$	−5.54827	−0.393306	−0.196653	+	0.980473i	$0.563007\pi$
$200$	0	0
$201$	−2.15826	−0.152232
$202$	3.44883	0.242658
$203$	−28.0486	−1.96863
$204$	0	0
$205$	0	0
$206$	−6.83338	−0.476104
$207$	22.7890	1.58395
$208$	−1.42543	−0.0988355
$209$	−23.4681	−1.62332
$210$	0	0
$211$	3.30121	0.227265	0.113632	−	0.993523i	$-0.463751\pi$
$211$	3.30121	0.227265	0.113632	+	0.993523i	$0.463751\pi$
$212$	22.3422	1.53447
$213$	0.198405	0.0135945
$214$	−6.20354	−0.424065
$215$	0	0
$216$	−14.0224	−0.954106
$217$	0.720090	0.0488829
$218$	−8.29163	−0.561580
$219$	4.22550	0.285533
$220$	0	0
$221$	0	0
$222$	8.01342	0.537826
$223$	18.2613	1.22287	0.611434	−	0.791295i	$-0.290593\pi$
$223$	18.2613	1.22287	0.611434	+	0.791295i	$0.290593\pi$
$224$	27.0938	1.81028
$225$	0	0
$226$	−0.800745	−0.0532648
$227$	12.4335	0.825237	0.412619	−	0.910904i	$-0.364614\pi$
$227$	12.4335	0.825237	0.412619	+	0.910904i	$0.364614\pi$
$228$	27.2045	1.80166
$229$	−18.6762	−1.23416	−0.617080	−	0.786900i	$-0.711685\pi$
$229$	−18.6762	−1.23416	−0.617080	+	0.786900i	$0.711685\pi$
$230$	0	0
$231$	60.3538	3.97099
$232$	11.6955	0.767844
$233$	−1.53341	−0.100457	−0.0502284	−	0.998738i	$-0.515995\pi$
$233$	−1.53341	−0.100457	−0.0502284	+	0.998738i	$0.515995\pi$
$234$	−1.98857	−0.129997
$235$	0	0
$236$	−16.9552	−1.10369
$237$	19.4064	1.26058
$238$	0	0
$239$	−9.27893	−0.600204	−0.300102	−	0.953907i	$-0.597021\pi$
$239$	−9.27893	−0.600204	−0.300102	+	0.953907i	$0.597021\pi$
$240$	0	0
$241$	28.0757	1.80851	0.904257	−	0.426988i	$-0.140425\pi$
$241$	28.0757	1.80851	0.904257	+	0.426988i	$0.140425\pi$
$242$	3.65493	0.234948
$243$	−10.3501	−0.663958
$244$	1.61290	0.103256
$245$	0	0
$246$	−2.55664	−0.163006
$247$	3.69432	0.235064
$248$	−0.300256	−0.0190663
$249$	−17.9177	−1.13549
$250$	0	0
$251$	−10.3009	−0.650189	−0.325094	−	0.945682i	$-0.605396\pi$
$251$	−10.3009	−0.650189	−0.325094	+	0.945682i	$0.605396\pi$
$252$	−44.7560	−2.81936
$253$	17.8643	1.12312
$254$	−0.875582	−0.0549389
$255$	0	0
$256$	−4.02208	−0.251380
$257$	−24.5419	−1.53088	−0.765441	−	0.643506i	$-0.777479\pi$
$257$	−24.5419	−1.53088	−0.765441	+	0.643506i	$0.777479\pi$
$258$	4.87050	0.303224
$259$	24.4918	1.52185
$260$	0	0
$261$	−29.8268	−1.84623
$262$	−1.18211	−0.0730309
$263$	−2.84217	−0.175256	−0.0876279	−	0.996153i	$-0.527929\pi$
$263$	−2.84217	−0.175256	−0.0876279	+	0.996153i	$0.527929\pi$
$264$	−25.1658	−1.54884
$265$	0	0
$266$	−15.9897	−0.980393
$267$	31.5804	1.93269
$268$	1.25462	0.0766378
$269$	25.6796	1.56571	0.782857	−	0.622201i	$-0.213761\pi$
$269$	25.6796	1.56571	0.782857	+	0.622201i	$0.213761\pi$
$270$	0	0
$271$	−11.7663	−0.714752	−0.357376	−	0.933961i	$-0.616329\pi$
$271$	−11.7663	−0.714752	−0.357376	+	0.933961i	$0.616329\pi$
$272$	0	0
$273$	−9.50085	−0.575017
$274$	−6.32256	−0.381960
$275$	0	0
$276$	−20.7086	−1.24651
$277$	−11.5928	−0.696545	−0.348273	−	0.937393i	$-0.613232\pi$
$277$	−11.5928	−0.696545	−0.348273	+	0.937393i	$0.613232\pi$
$278$	−8.54235	−0.512336
$279$	0.765741	0.0458437
$280$	0	0
$281$	29.5107	1.76046	0.880229	−	0.474549i	$-0.157389\pi$
$281$	29.5107	1.76046	0.880229	+	0.474549i	$0.157389\pi$
$282$	18.7517	1.11665
$283$	0.0152294	0.000905295 0	0.000452648	−	1.00000i	$-0.499856\pi$
$283$	0.0152294	0.000905295 0	0.000452648		1.00000i	$0.499856\pi$
$284$	−0.115335	−0.00684386
$285$	0	0
$286$	−1.55884	−0.0921764
$287$	−7.81401	−0.461246
$288$	28.8114	1.69773
$289$	0	0
$290$	0	0
$291$	−22.4048	−1.31339
$292$	−2.45632	−0.143745
$293$	3.52542	0.205957	0.102979	−	0.994684i	$-0.467163\pi$
$293$	3.52542	0.205957	0.102979	+	0.994684i	$0.467163\pi$
$294$	29.6491	1.72917
$295$	0	0
$296$	−10.2124	−0.593582
$297$	28.0333	1.62666
$298$	5.50235	0.318743
$299$	−2.81219	−0.162633
$300$	0	0
$301$	14.8860	0.858013
$302$	7.33719	0.422208
$303$	17.5221	1.00662
$304$	−12.1882	−0.699040
$305$	0	0
$306$	0	0
$307$	21.7368	1.24058	0.620292	−	0.784371i	$-0.287014\pi$
$307$	21.7368	1.24058	0.620292	+	0.784371i	$0.287014\pi$
$308$	−35.0842	−1.99911
$309$	−34.7177	−1.97502
$310$	0	0
$311$	−1.74863	−0.0991556	−0.0495778	−	0.998770i	$-0.515788\pi$
$311$	−1.74863	−0.0991556	−0.0495778	+	0.998770i	$0.515788\pi$
$312$	3.96157	0.224280
$313$	−16.6552	−0.941406	−0.470703	−	0.882292i	$-0.656000\pi$
$313$	−16.6552	−0.941406	−0.470703	+	0.882292i	$0.656000\pi$
$314$	8.12357	0.458440
$315$	0	0
$316$	−11.2811	−0.634613
$317$	−21.9922	−1.23520	−0.617602	−	0.786491i	$-0.711896\pi$
$317$	−21.9922	−1.23520	−0.617602	+	0.786491i	$0.711896\pi$
$318$	−21.8292	−1.22412
$319$	−23.3812	−1.30910
$320$	0	0
$321$	−31.5177	−1.75915
$322$	12.1717	0.678301
$323$	0	0
$324$	−5.69158	−0.316199
$325$	0	0
$326$	8.09291	0.448225
$327$	−42.1265	−2.32960
$328$	3.25821	0.179904
$329$	57.3119	3.15971
$330$	0	0
$331$	5.73526	0.315238	0.157619	−	0.987500i	$-0.449618\pi$
$331$	5.73526	0.315238	0.157619	+	0.987500i	$0.449618\pi$
$332$	10.4157	0.571636
$333$	26.0445	1.42723
$334$	−0.0668998	−0.00366059
$335$	0	0
$336$	31.3448	1.71000
$337$	−8.30683	−0.452502	−0.226251	−	0.974069i	$-0.572647\pi$
$337$	−8.30683	−0.452502	−0.226251	+	0.974069i	$0.572647\pi$
$338$	−7.13811	−0.388262
$339$	−4.06827	−0.220958
$340$	0	0
$341$	0.600264	0.0325061
$342$	−17.0034	−0.919439
$343$	55.5547	2.99967
$344$	−6.20701	−0.334659
$345$	0	0
$346$	−7.20870	−0.387542
$347$	16.6251	0.892485	0.446242	−	0.894912i	$-0.352762\pi$
$347$	16.6251	0.892485	0.446242	+	0.894912i	$0.352762\pi$
$348$	27.1038	1.45292
$349$	−18.6990	−1.00093	−0.500467	−	0.865755i	$-0.666839\pi$
$349$	−18.6990	−1.00093	−0.500467	+	0.865755i	$0.666839\pi$
$350$	0	0
$351$	−4.41297	−0.235547
$352$	22.5852	1.20380
$353$	−5.78418	−0.307861	−0.153930	−	0.988082i	$-0.549193\pi$
$353$	−5.78418	−0.307861	−0.153930	+	0.988082i	$0.549193\pi$
$354$	16.5659	0.880467
$355$	0	0
$356$	−18.3580	−0.972971
$357$	0	0
$358$	6.18253	0.326757
$359$	34.2408	1.80716	0.903581	−	0.428417i	$-0.140928\pi$
$359$	34.2408	1.80716	0.903581	+	0.428417i	$0.140928\pi$
$360$	0	0
$361$	12.5885	0.662552
$362$	7.46665	0.392438
$363$	18.5692	0.974632
$364$	5.52293	0.289480
$365$	0	0
$366$	−1.57587	−0.0823720
$367$	17.4496	0.910860	0.455430	−	0.890272i	$-0.349486\pi$
$367$	17.4496	0.910860	0.455430	+	0.890272i	$0.349486\pi$
$368$	9.27787	0.483642
$369$	−8.30938	−0.432569
$370$	0	0
$371$	−66.7177	−3.46381
$372$	−0.695834	−0.0360773
$373$	3.90239	0.202058	0.101029	−	0.994883i	$-0.467787\pi$
$373$	3.90239	0.202058	0.101029	+	0.994883i	$0.467787\pi$
$374$	0	0
$375$	0	0
$376$	−23.8974	−1.23241
$377$	3.68065	0.189563
$378$	19.1002	0.982407
$379$	32.4372	1.66619	0.833094	−	0.553132i	$-0.186567\pi$
$379$	32.4372	1.66619	0.833094	+	0.553132i	$0.186567\pi$
$380$	0	0
$381$	−4.44848	−0.227903
$382$	−8.58103	−0.439044
$383$	−12.4887	−0.638140	−0.319070	−	0.947731i	$-0.603371\pi$
$383$	−12.4887	−0.638140	−0.319070	+	0.947731i	$0.603371\pi$
$384$	−33.2893	−1.69879
$385$	0	0
$386$	7.13501	0.363162
$387$	15.8297	0.804668
$388$	13.0241	0.661199
$389$	−12.0159	−0.609232	−0.304616	−	0.952475i	$-0.598528\pi$
$389$	−12.0159	−0.609232	−0.304616	+	0.952475i	$0.598528\pi$
$390$	0	0
$391$	0	0
$392$	−37.7851	−1.90843
$393$	−6.00582	−0.302954
$394$	−6.50600	−0.327768
$395$	0	0
$396$	−37.3084	−1.87482
$397$	−10.6585	−0.534934	−0.267467	−	0.963567i	$-0.586187\pi$
$397$	−10.6585	−0.534934	−0.267467	+	0.963567i	$0.586187\pi$
$398$	−3.15121	−0.157956
$399$	−81.2374	−4.06696
$400$	0	0
$401$	−10.0381	−0.501278	−0.250639	−	0.968081i	$-0.580641\pi$
$401$	−10.0381	−0.501278	−0.250639	+	0.968081i	$0.580641\pi$
$402$	−1.22581	−0.0611378
$403$	−0.0944931	−0.00470704
$404$	−10.1858	−0.506761
$405$	0	0
$406$	−15.9306	−0.790621
$407$	20.4163	1.01200
$408$	0	0
$409$	−13.1053	−0.648017	−0.324008	−	0.946054i	$-0.605031\pi$
$409$	−13.1053	−0.648017	−0.324008	+	0.946054i	$0.605031\pi$
$410$	0	0
$411$	−32.1224	−1.58448
$412$	20.1817	0.994281
$413$	50.6312	2.49140
$414$	12.9433	0.636129
$415$	0	0
$416$	−3.55535	−0.174315
$417$	−43.4003	−2.12532
$418$	−13.3290	−0.651941
$419$	−37.0499	−1.81001	−0.905003	−	0.425406i	$-0.860131\pi$
$419$	−37.0499	−1.81001	−0.905003	+	0.425406i	$0.860131\pi$
$420$	0	0
$421$	−4.49427	−0.219037	−0.109519	−	0.993985i	$-0.534931\pi$
$421$	−4.49427	−0.219037	−0.109519	+	0.993985i	$0.534931\pi$
$422$	1.87496	0.0912717
$423$	60.9452	2.96326
$424$	27.8193	1.35102
$425$	0	0
$426$	0.112687	0.00545969
$427$	−4.81642	−0.233083
$428$	18.3215	0.885605
$429$	−7.91987	−0.382375
$430$	0	0
$431$	39.1192	1.88430	0.942152	−	0.335185i	$-0.108799\pi$
$431$	39.1192	1.88430	0.942152	+	0.335185i	$0.108799\pi$
$432$	14.5591	0.700476
$433$	−8.46352	−0.406731	−0.203366	−	0.979103i	$-0.565188\pi$
$433$	−8.46352	−0.406731	−0.203366	+	0.979103i	$0.565188\pi$
$434$	0.408984	0.0196319
$435$	0	0
$436$	24.4885	1.17279
$437$	−24.0458	−1.15026
$438$	2.39993	0.114673
$439$	−32.1434	−1.53412	−0.767061	−	0.641574i	$-0.778282\pi$
$439$	−32.1434	−1.53412	−0.767061	+	0.641574i	$0.778282\pi$
$440$	0	0
$441$	96.3629	4.58871
$442$	0	0
$443$	24.3440	1.15662	0.578308	−	0.815818i	$-0.303713\pi$
$443$	24.3440	1.15662	0.578308	+	0.815818i	$0.303713\pi$
$444$	−23.6668	−1.12318
$445$	0	0
$446$	10.3717	0.491116
$447$	27.9553	1.32224
$448$	−6.33686	−0.299388
$449$	−13.6669	−0.644980	−0.322490	−	0.946573i	$-0.604520\pi$
$449$	−13.6669	−0.644980	−0.322490	+	0.946573i	$0.604520\pi$
$450$	0	0
$451$	−6.51373	−0.306719
$452$	2.36492	0.111237
$453$	37.2773	1.75144
$454$	7.06173	0.331423
$455$	0	0
$456$	33.8736	1.58628
$457$	−9.61288	−0.449672	−0.224836	−	0.974397i	$-0.572185\pi$
$457$	−9.61288	−0.449672	−0.224836	+	0.974397i	$0.572185\pi$
$458$	−10.6074	−0.495651
$459$	0	0
$460$	0	0
$461$	19.3179	0.899725	0.449862	−	0.893098i	$-0.351473\pi$
$461$	19.3179	0.899725	0.449862	+	0.893098i	$0.351473\pi$
$462$	34.2787	1.59479
$463$	−20.8479	−0.968883	−0.484442	−	0.874824i	$-0.660977\pi$
$463$	−20.8479	−0.968883	−0.484442	+	0.874824i	$0.660977\pi$
$464$	−12.1431	−0.563728
$465$	0	0
$466$	−0.870916	−0.0403444
$467$	−22.0407	−1.01992	−0.509962	−	0.860197i	$-0.670341\pi$
$467$	−22.0407	−1.01992	−0.509962	+	0.860197i	$0.670341\pi$
$468$	5.87306	0.271482
$469$	−3.74651	−0.172998
$470$	0	0
$471$	41.2726	1.90174
$472$	−21.1117	−0.971745
$473$	12.4089	0.570561
$474$	11.0221	0.506262
$475$	0	0
$476$	0	0
$477$	−70.9473	−3.24845
$478$	−5.27008	−0.241048
$479$	8.16793	0.373202	0.186601	−	0.982436i	$-0.440253\pi$
$479$	8.16793	0.373202	0.186601	+	0.982436i	$0.440253\pi$
$480$	0	0
$481$	−3.21392	−0.146542
$482$	15.9459	0.726317
$483$	61.8395	2.81379
$484$	−10.7945	−0.490657
$485$	0	0
$486$	−5.87845	−0.266652
$487$	13.5824	0.615476	0.307738	−	0.951471i	$-0.400428\pi$
$487$	13.5824	0.615476	0.307738	+	0.951471i	$0.400428\pi$
$488$	2.00830	0.0909116
$489$	41.1169	1.85937
$490$	0	0
$491$	41.4309	1.86975	0.934875	−	0.354978i	$-0.115512\pi$
$491$	41.4309	1.86975	0.934875	+	0.354978i	$0.115512\pi$
$492$	7.55079	0.340416
$493$	0	0
$494$	2.09824	0.0944041
$495$	0	0
$496$	0.311748	0.0139979
$497$	0.344410	0.0154489
$498$	−10.1766	−0.456022
$499$	2.98794	0.133759	0.0668793	−	0.997761i	$-0.478696\pi$
$499$	2.98794	0.133759	0.0668793	+	0.997761i	$0.478696\pi$
$500$	0	0
$501$	−0.339891	−0.0151852
$502$	−5.85053	−0.261122
$503$	22.7093	1.01256	0.506279	−	0.862370i	$-0.331021\pi$
$503$	22.7093	1.01256	0.506279	+	0.862370i	$0.331021\pi$
$504$	−55.7278	−2.48231
$505$	0	0
$506$	10.1463	0.451057
$507$	−36.2659	−1.61063
$508$	2.58594	0.114733
$509$	12.4475	0.551727	0.275863	−	0.961197i	$-0.411036\pi$
$509$	12.4475	0.551727	0.275863	+	0.961197i	$0.411036\pi$
$510$	0	0
$511$	7.33502	0.324482
$512$	20.7884	0.918726
$513$	−37.7333	−1.66597
$514$	−13.9389	−0.614817
$515$	0	0
$516$	−14.3845	−0.633244
$517$	47.7750	2.10114
$518$	13.9104	0.611189
$519$	−36.6245	−1.60764
$520$	0	0
$521$	2.89567	0.126862	0.0634308	−	0.997986i	$-0.479796\pi$
$521$	2.89567	0.126862	0.0634308	+	0.997986i	$0.479796\pi$
$522$	−16.9405	−0.741465
$523$	−43.9292	−1.92089	−0.960444	−	0.278472i	$-0.910172\pi$
$523$	−43.9292	−1.92089	−0.960444	+	0.278472i	$0.910172\pi$
$524$	3.49124	0.152515
$525$	0	0
$526$	−1.61424	−0.0703844
$527$	0	0
$528$	26.1289	1.13712
$529$	−4.69592	−0.204170
$530$	0	0
$531$	53.8410	2.33650
$532$	47.2241	2.04742
$533$	1.02538	0.0444144
$534$	17.9365	0.776188
$535$	0	0
$536$	1.56218	0.0674760
$537$	31.4110	1.35548
$538$	14.5851	0.628806
$539$	75.5389	3.25369
$540$	0	0
$541$	−27.3972	−1.17790	−0.588949	−	0.808170i	$-0.700458\pi$
$541$	−27.3972	−1.17790	−0.588949	+	0.808170i	$0.700458\pi$
$542$	−6.68281	−0.287051
$543$	37.9351	1.62795
$544$	0	0
$545$	0	0
$546$	−5.39612	−0.230933
$547$	−26.0311	−1.11301	−0.556506	−	0.830844i	$-0.687858\pi$
$547$	−26.0311	−1.11301	−0.556506	+	0.830844i	$0.687858\pi$
$548$	18.6731	0.797674
$549$	−5.12175	−0.218591
$550$	0	0
$551$	31.4716	1.34074
$552$	−25.7852	−1.09749
$553$	33.6874	1.43254
$554$	−6.58428	−0.279739
$555$	0	0
$556$	25.2290	1.06995
$557$	28.7744	1.21921	0.609605	−	0.792705i	$-0.291328\pi$
$557$	28.7744	1.21921	0.609605	+	0.792705i	$0.291328\pi$
$558$	0.434912	0.0184113
$559$	−1.95340	−0.0826199
$560$	0	0
$561$	0	0
$562$	16.7609	0.707017
$563$	15.0383	0.633790	0.316895	−	0.948461i	$-0.397360\pi$
$563$	15.0383	0.633790	0.316895	+	0.948461i	$0.397360\pi$
$564$	−55.3813	−2.33197
$565$	0	0
$566$	0.00864974	0.000363576 0
$567$	16.9961	0.713768
$568$	−0.143609	−0.00602570
$569$	37.3329	1.56508	0.782539	−	0.622602i	$-0.213924\pi$
$569$	37.3329	1.56508	0.782539	+	0.622602i	$0.213924\pi$
$570$	0	0
$571$	−38.0534	−1.59248	−0.796242	−	0.604978i	$-0.793182\pi$
$571$	−38.0534	−1.59248	−0.796242	+	0.604978i	$0.793182\pi$
$572$	4.60389	0.192498
$573$	−43.5968	−1.82128
$574$	−4.43806	−0.185241
$575$	0	0
$576$	−6.73859	−0.280774
$577$	37.8644	1.57632	0.788159	−	0.615472i	$-0.211035\pi$
$577$	37.8644	1.57632	0.788159	+	0.615472i	$0.211035\pi$
$578$	0	0
$579$	36.2501	1.50650
$580$	0	0
$581$	−31.1031	−1.29038
$582$	−12.7251	−0.527471
$583$	−55.6156	−2.30336
$584$	−3.05849	−0.126561
$585$	0	0
$586$	2.00231	0.0827145
$587$	−32.8884	−1.35745	−0.678725	−	0.734393i	$-0.737467\pi$
$587$	−32.8884	−1.35745	−0.678725	+	0.734393i	$0.737467\pi$
$588$	−87.5657	−3.61115
$589$	−0.807968	−0.0332917
$590$	0	0
$591$	−33.0544	−1.35968
$592$	10.6032	0.435790
$593$	29.6499	1.21757	0.608787	−	0.793334i	$-0.291656\pi$
$593$	29.6499	1.21757	0.608787	+	0.793334i	$0.291656\pi$
$594$	15.9218	0.653281
$595$	0	0
$596$	−16.2506	−0.665652
$597$	−16.0100	−0.655247
$598$	−1.59722	−0.0653150
$599$	−7.52613	−0.307509	−0.153755	−	0.988109i	$-0.549137\pi$
$599$	−7.52613	−0.307509	−0.153755	+	0.988109i	$0.549137\pi$
$600$	0	0
$601$	18.0062	0.734488	0.367244	−	0.930125i	$-0.380301\pi$
$601$	18.0062	0.734488	0.367244	+	0.930125i	$0.380301\pi$
$602$	8.45467	0.344586
$603$	−3.98402	−0.162242
$604$	−21.6696	−0.881725
$605$	0	0
$606$	9.95189	0.404268
$607$	1.75527	0.0712444	0.0356222	−	0.999365i	$-0.488659\pi$
$607$	1.75527	0.0712444	0.0356222	+	0.999365i	$0.488659\pi$
$608$	−30.4002	−1.23289
$609$	−80.9369	−3.27973
$610$	0	0
$611$	−7.52069	−0.304255
$612$	0	0
$613$	16.6412	0.672131	0.336066	−	0.941839i	$-0.390904\pi$
$613$	16.6412	0.672131	0.336066	+	0.941839i	$0.390904\pi$
$614$	12.3457	0.498231
$615$	0	0
$616$	−43.6850	−1.76012
$617$	−41.7577	−1.68110	−0.840551	−	0.541733i	$-0.817768\pi$
$617$	−41.7577	−1.68110	−0.840551	+	0.541733i	$0.817768\pi$
$618$	−19.7183	−0.793187
$619$	9.18747	0.369276	0.184638	−	0.982807i	$-0.440889\pi$
$619$	9.18747	0.369276	0.184638	+	0.982807i	$0.440889\pi$
$620$	0	0
$621$	28.7233	1.15263
$622$	−0.993154	−0.0398219
$623$	54.8202	2.19633
$624$	−4.11319	−0.164660
$625$	0	0
$626$	−9.45950	−0.378078
$627$	−67.7192	−2.70444
$628$	−23.9922	−0.957391
$629$	0	0
$630$	0	0
$631$	−29.9015	−1.19036	−0.595181	−	0.803592i	$-0.702920\pi$
$631$	−29.9015	−1.19036	−0.595181	+	0.803592i	$0.702920\pi$
$632$	−14.0467	−0.558746
$633$	9.52593	0.378622
$634$	−12.4907	−0.496070
$635$	0	0
$636$	64.4703	2.55641
$637$	−11.8913	−0.471149
$638$	−13.2797	−0.525747
$639$	0.366244	0.0144884
$640$	0	0
$641$	−13.2872	−0.524811	−0.262406	−	0.964958i	$-0.584516\pi$
$641$	−13.2872	−0.524811	−0.262406	+	0.964958i	$0.584516\pi$
$642$	−17.9009	−0.706491
$643$	−13.1341	−0.517959	−0.258979	−	0.965883i	$-0.583386\pi$
$643$	−13.1341	−0.517959	−0.258979	+	0.965883i	$0.583386\pi$
$644$	−35.9478	−1.41654
$645$	0	0
$646$	0	0
$647$	−20.8117	−0.818191	−0.409095	−	0.912492i	$-0.634156\pi$
$647$	−20.8117	−0.818191	−0.409095	+	0.912492i	$0.634156\pi$
$648$	−7.08686	−0.278398
$649$	42.2060	1.65673
$650$	0	0
$651$	2.07788	0.0814387
$652$	−23.9016	−0.936059
$653$	−35.4359	−1.38671	−0.693356	−	0.720595i	$-0.743869\pi$
$653$	−35.4359	−1.38671	−0.693356	+	0.720595i	$0.743869\pi$
$654$	−23.9262	−0.935590
$655$	0	0
$656$	−3.38291	−0.132080
$657$	7.80003	0.304308
$658$	32.5510	1.26897
$659$	−22.4546	−0.874706	−0.437353	−	0.899290i	$-0.644084\pi$
$659$	−22.4546	−0.874706	−0.437353	+	0.899290i	$0.644084\pi$
$660$	0	0
$661$	−5.69679	−0.221579	−0.110790	−	0.993844i	$-0.535338\pi$
$661$	−5.69679	−0.221579	−0.110790	+	0.993844i	$0.535338\pi$
$662$	3.25741	0.126603
$663$	0	0
$664$	12.9691	0.503298
$665$	0	0
$666$	14.7923	0.573190
$667$	−23.9568	−0.927610
$668$	0.197582	0.00764467
$669$	52.6947	2.03729
$670$	0	0
$671$	−4.01495	−0.154995
$672$	78.1814	3.01591
$673$	15.1352	0.583420	0.291710	−	0.956507i	$-0.405776\pi$
$673$	15.1352	0.583420	0.291710	+	0.956507i	$0.405776\pi$
$674$	−4.71796	−0.181729
$675$	0	0
$676$	21.0817	0.810835
$677$	19.9248	0.765774	0.382887	−	0.923795i	$-0.374930\pi$
$677$	19.9248	0.765774	0.382887	+	0.923795i	$0.374930\pi$
$678$	−2.31062	−0.0887389
$679$	−38.8923	−1.49255
$680$	0	0
$681$	35.8778	1.37484
$682$	0.340927	0.0130548
$683$	−14.2091	−0.543694	−0.271847	−	0.962340i	$-0.587635\pi$
$683$	−14.2091	−0.543694	−0.271847	+	0.962340i	$0.587635\pi$
$684$	50.2179	1.92013
$685$	0	0
$686$	31.5529	1.20470
$687$	−53.8920	−2.05611
$688$	6.44457	0.245697
$689$	8.75496	0.333537
$690$	0	0
$691$	−35.5099	−1.35086	−0.675431	−	0.737424i	$-0.736042\pi$
$691$	−35.5099	−1.35086	−0.675431	+	0.737424i	$0.736042\pi$
$692$	21.2902	0.809331
$693$	111.410	4.23210
$694$	9.44245	0.358431
$695$	0	0
$696$	33.7483	1.27922
$697$	0	0
$698$	−10.6203	−0.401985
$699$	−4.42478	−0.167360
$700$	0	0
$701$	−51.3506	−1.93949	−0.969743	−	0.244130i	$-0.921498\pi$
$701$	−51.3506	−1.93949	−0.969743	+	0.244130i	$0.921498\pi$
$702$	−2.50640	−0.0945980
$703$	−27.4807	−1.03646
$704$	−5.28238	−0.199087
$705$	0	0
$706$	−3.28519	−0.123640
$707$	30.4165	1.14393
$708$	−48.9257	−1.83874
$709$	−2.35324	−0.0883780	−0.0441890	−	0.999023i	$-0.514070\pi$
$709$	−2.35324	−0.0883780	−0.0441890	+	0.999023i	$0.514070\pi$
$710$	0	0
$711$	35.8231	1.34347
$712$	−22.8584	−0.856655
$713$	0.615040	0.0230334
$714$	0	0
$715$	0	0
$716$	−18.2595	−0.682389
$717$	−26.7752	−0.999937
$718$	19.4475	0.725774
$719$	−19.2163	−0.716648	−0.358324	−	0.933597i	$-0.616652\pi$
$719$	−19.2163	−0.716648	−0.358324	+	0.933597i	$0.616652\pi$
$720$	0	0
$721$	−60.2661	−2.24443
$722$	7.14978	0.266087
$723$	81.0149	3.01298
$724$	−22.0520	−0.819556
$725$	0	0
$726$	10.5466	0.391422
$727$	−16.4927	−0.611680	−0.305840	−	0.952083i	$-0.598937\pi$
$727$	−16.4927	−0.611680	−0.305840	+	0.952083i	$0.598937\pi$
$728$	6.87686	0.254873
$729$	−40.0452	−1.48316
$730$	0	0
$731$	0	0
$732$	4.65418	0.172023
$733$	−6.80546	−0.251365	−0.125683	−	0.992070i	$-0.540112\pi$
$733$	−6.80546	−0.251365	−0.125683	+	0.992070i	$0.540112\pi$
$734$	9.91069	0.365810
$735$	0	0
$736$	23.1412	0.852996
$737$	−3.12307	−0.115040
$738$	−4.71941	−0.173724
$739$	−24.7430	−0.910188	−0.455094	−	0.890443i	$-0.650394\pi$
$739$	−24.7430	−0.910188	−0.455094	+	0.890443i	$0.650394\pi$
$740$	0	0
$741$	10.6603	0.391616
$742$	−37.8931	−1.39110
$743$	27.0871	0.993729	0.496865	−	0.867828i	$-0.334485\pi$
$743$	27.0871	0.993729	0.496865	+	0.867828i	$0.334485\pi$
$744$	−0.866416	−0.0317643
$745$	0	0
$746$	2.21641	0.0811485
$747$	−33.0749	−1.21015
$748$	0	0
$749$	−54.7114	−1.99911
$750$	0	0
$751$	−14.6034	−0.532885	−0.266443	−	0.963851i	$-0.585848\pi$
$751$	−14.6034	−0.532885	−0.266443	+	0.963851i	$0.585848\pi$
$752$	24.8120	0.904800
$753$	−29.7242	−1.08321
$754$	2.09047	0.0761305
$755$	0	0
$756$	−56.4105	−2.05163
$757$	5.50378	0.200038	0.100019	−	0.994986i	$-0.468110\pi$
$757$	5.50378	0.200038	0.100019	+	0.994986i	$0.468110\pi$
$758$	18.4231	0.669157
$759$	51.5491	1.87112
$760$	0	0
$761$	−12.4420	−0.451023	−0.225512	−	0.974240i	$-0.572405\pi$
$761$	−12.4420	−0.451023	−0.225512	+	0.974240i	$0.572405\pi$
$762$	−2.52657	−0.0915280
$763$	−73.1271	−2.64738
$764$	25.3432	0.916885
$765$	0	0
$766$	−7.09308	−0.256283
$767$	−6.64403	−0.239902
$768$	−11.6061	−0.418798
$769$	−25.9053	−0.934170	−0.467085	−	0.884212i	$-0.654696\pi$
$769$	−25.9053	−0.934170	−0.467085	+	0.884212i	$0.654696\pi$
$770$	0	0
$771$	−70.8179	−2.55044
$772$	−21.0725	−0.758417
$773$	10.7423	0.386372	0.193186	−	0.981162i	$-0.438118\pi$
$773$	10.7423	0.386372	0.193186	+	0.981162i	$0.438118\pi$
$774$	8.99065	0.323162
$775$	0	0
$776$	16.2169	0.582154
$777$	70.6734	2.53539
$778$	−6.82459	−0.244673
$779$	8.76760	0.314132
$780$	0	0
$781$	0.287099	0.0102732
$782$	0	0
$783$	−37.5937	−1.34349
$784$	39.2312	1.40112
$785$	0	0
$786$	−3.41108	−0.121669
$787$	−13.9051	−0.495662	−0.247831	−	0.968803i	$-0.579718\pi$
$787$	−13.9051	−0.495662	−0.247831	+	0.968803i	$0.579718\pi$
$788$	19.2148	0.684500
$789$	−8.20134	−0.291975
$790$	0	0
$791$	−7.06208	−0.251099
$792$	−46.4545	−1.65069
$793$	0.632029	0.0224440
$794$	−6.05361	−0.214835
$795$	0	0
$796$	9.30677	0.329870
$797$	36.5024	1.29298	0.646490	−	0.762922i	$-0.276236\pi$
$797$	36.5024	1.29298	0.646490	+	0.762922i	$0.276236\pi$
$798$	−46.1398	−1.63333
$799$	0	0
$800$	0	0
$801$	58.2956	2.05977
$802$	−5.70126	−0.201318
$803$	6.11444	0.215774
$804$	3.62031	0.127678
$805$	0	0
$806$	−0.0536685	−0.00189039
$807$	74.1009	2.60847
$808$	−12.6828	−0.446179
$809$	10.9881	0.386321	0.193161	−	0.981167i	$-0.438126\pi$
$809$	10.9881	0.386321	0.193161	+	0.981167i	$0.438126\pi$
$810$	0	0
$811$	−37.3689	−1.31220	−0.656100	−	0.754674i	$-0.727795\pi$
$811$	−37.3689	−1.31220	−0.656100	+	0.754674i	$0.727795\pi$
$812$	47.0493	1.65111
$813$	−33.9527	−1.19077
$814$	11.5957	0.406429
$815$	0	0
$816$	0	0
$817$	−16.7026	−0.584350
$818$	−7.44333	−0.260250
$819$	−17.5380	−0.612827
$820$	0	0
$821$	−2.68224	−0.0936107	−0.0468054	−	0.998904i	$-0.514904\pi$
$821$	−2.68224	−0.0936107	−0.0468054	+	0.998904i	$0.514904\pi$
$822$	−18.2443	−0.636344
$823$	−33.0593	−1.15237	−0.576187	−	0.817318i	$-0.695460\pi$
$823$	−33.0593	−1.15237	−0.576187	+	0.817318i	$0.695460\pi$
$824$	25.1292	0.875417
$825$	0	0
$826$	28.7566	1.00057
$827$	2.05218	0.0713614	0.0356807	−	0.999363i	$-0.488640\pi$
$827$	2.05218	0.0713614	0.0356807	+	0.999363i	$0.488640\pi$
$828$	−38.2268	−1.32847
$829$	−11.7740	−0.408928	−0.204464	−	0.978874i	$-0.565545\pi$
$829$	−11.7740	−0.408928	−0.204464	+	0.978874i	$0.565545\pi$
$830$	0	0
$831$	−33.4521	−1.16044
$832$	0.831548	0.0288287
$833$	0	0
$834$	−24.6497	−0.853550
$835$	0	0
$836$	39.3658	1.36149
$837$	0.965140	0.0333601
$838$	−21.0429	−0.726916
$839$	−51.0024	−1.76080	−0.880400	−	0.474232i	$-0.842726\pi$
$839$	−51.0024	−1.76080	−0.880400	+	0.474232i	$0.842726\pi$
$840$	0	0
$841$	2.35516	0.0812126
$842$	−2.55258	−0.0879676
$843$	85.1556	2.93292
$844$	−5.53751	−0.190609
$845$	0	0
$846$	34.6146	1.19007
$847$	32.2342	1.10758
$848$	−28.8840	−0.991882
$849$	0.0439459	0.00150822
$850$	0	0
$851$	20.9189	0.717089
$852$	−0.332809	−0.0114018
$853$	6.87916	0.235538	0.117769	−	0.993041i	$-0.462426\pi$
$853$	6.87916	0.235538	0.117769	+	0.993041i	$0.462426\pi$
$854$	−2.73554	−0.0936083
$855$	0	0
$856$	22.8130	0.779733
$857$	−22.0791	−0.754207	−0.377103	−	0.926171i	$-0.623080\pi$
$857$	−22.0791	−0.754207	−0.377103	+	0.926171i	$0.623080\pi$
$858$	−4.49818	−0.153565
$859$	−27.4713	−0.937308	−0.468654	−	0.883382i	$-0.655261\pi$
$859$	−27.4713	−0.937308	−0.468654	+	0.883382i	$0.655261\pi$
$860$	0	0
$861$	−22.5480	−0.768434
$862$	22.2182	0.756755
$863$	−4.04381	−0.137653	−0.0688265	−	0.997629i	$-0.521925\pi$
$863$	−4.04381	−0.137653	−0.0688265	+	0.997629i	$0.521925\pi$
$864$	36.3139	1.23542
$865$	0	0
$866$	−4.80696	−0.163347
$867$	0	0
$868$	−1.20789	−0.0409986
$869$	28.0817	0.952607
$870$	0	0
$871$	0.491631	0.0166583
$872$	30.4918	1.03258
$873$	−41.3579	−1.39975
$874$	−13.6571	−0.461957
$875$	0	0
$876$	−7.08794	−0.239479
$877$	−16.7190	−0.564560	−0.282280	−	0.959332i	$-0.591091\pi$
$877$	−16.7190	−0.564560	−0.282280	+	0.959332i	$0.591091\pi$
$878$	−18.2563	−0.616119
$879$	10.1729	0.343124
$880$	0	0
$881$	−44.2673	−1.49140	−0.745702	−	0.666279i	$-0.767886\pi$
$881$	−44.2673	−1.49140	−0.745702	+	0.666279i	$0.767886\pi$
$882$	54.7305	1.84287
$883$	2.86400	0.0963812	0.0481906	−	0.998838i	$-0.484655\pi$
$883$	2.86400	0.0963812	0.0481906	+	0.998838i	$0.484655\pi$
$884$	0	0
$885$	0	0
$886$	13.8264	0.464509
$887$	−4.69527	−0.157652	−0.0788259	−	0.996888i	$-0.525117\pi$
$887$	−4.69527	−0.157652	−0.0788259	+	0.996888i	$0.525117\pi$
$888$	−29.4687	−0.988905
$889$	−7.72209	−0.258991
$890$	0	0
$891$	14.1679	0.474641
$892$	−30.6319	−1.02563
$893$	−64.3060	−2.15192
$894$	15.8775	0.531024
$895$	0	0
$896$	−57.7866	−1.93051
$897$	−8.11482	−0.270946
$898$	−7.76227	−0.259030
$899$	−0.804978	−0.0268475
$900$	0	0
$901$	0	0
$902$	−3.69955	−0.123181
$903$	42.9548	1.42945
$904$	2.94468	0.0979385
$905$	0	0
$906$	21.1721	0.703396
$907$	−17.4972	−0.580985	−0.290493	−	0.956877i	$-0.593819\pi$
$907$	−17.4972	−0.580985	−0.290493	+	0.956877i	$0.593819\pi$
$908$	−20.8561	−0.692135
$909$	32.3448	1.07281
$910$	0	0
$911$	−14.2233	−0.471238	−0.235619	−	0.971846i	$-0.575712\pi$
$911$	−14.2233	−0.471238	−0.235619	+	0.971846i	$0.575712\pi$
$912$	−35.1701	−1.16460
$913$	−25.9275	−0.858073
$914$	−5.45975	−0.180593
$915$	0	0
$916$	31.3279	1.03510
$917$	−10.4255	−0.344279
$918$	0	0
$919$	30.1242	0.993707	0.496854	−	0.867834i	$-0.334489\pi$
$919$	30.1242	0.993707	0.496854	+	0.867834i	$0.334489\pi$
$920$	0	0
$921$	62.7235	2.06681
$922$	10.9718	0.361338
$923$	−0.0451949	−0.00148761
$924$	−101.239	−3.33051
$925$	0	0
$926$	−11.8408	−0.389113
$927$	−64.0867	−2.10488
$928$	−30.2877	−0.994243
$929$	−1.83388	−0.0601675	−0.0300838	−	0.999547i	$-0.509577\pi$
$929$	−1.83388	−0.0601675	−0.0300838	+	0.999547i	$0.509577\pi$
$930$	0	0
$931$	−101.677	−3.33233
$932$	2.57216	0.0842540
$933$	−5.04582	−0.165193
$934$	−12.5183	−0.409612
$935$	0	0
$936$	7.31282	0.239027
$937$	53.6062	1.75124	0.875618	−	0.483004i	$-0.160454\pi$
$937$	53.6062	1.75124	0.875618	+	0.483004i	$0.160454\pi$
$938$	−2.12787	−0.0694775
$939$	−48.0600	−1.56838
$940$	0	0
$941$	−6.39257	−0.208392	−0.104196	−	0.994557i	$-0.533227\pi$
$941$	−6.39257	−0.208392	−0.104196	+	0.994557i	$0.533227\pi$
$942$	23.4413	0.763758
$943$	−6.67406	−0.217337
$944$	21.9197	0.713426
$945$	0	0
$946$	7.04778	0.229143
$947$	−49.7691	−1.61728	−0.808640	−	0.588304i	$-0.799796\pi$
$947$	−49.7691	−1.61728	−0.808640	+	0.588304i	$0.799796\pi$
$948$	−32.5527	−1.05726
$949$	−0.962531	−0.0312451
$950$	0	0
$951$	−63.4604	−2.05784
$952$	0	0
$953$	−6.67955	−0.216372	−0.108186	−	0.994131i	$-0.534504\pi$
$953$	−6.67955	−0.216372	−0.108186	+	0.994131i	$0.534504\pi$
$954$	−40.2954	−1.30461
$955$	0	0
$956$	15.5646	0.503397
$957$	−67.4687	−2.18095
$958$	4.63907	0.149882
$959$	−55.7611	−1.80062
$960$	0	0
$961$	−30.9793	−0.999333
$962$	−1.82538	−0.0588527
$963$	−58.1799	−1.87482
$964$	−47.0947	−1.51682
$965$	0	0
$966$	35.1225	1.13005
$967$	3.07928	0.0990230	0.0495115	−	0.998774i	$-0.484234\pi$
$967$	3.07928	0.0990230	0.0495115	+	0.998774i	$0.484234\pi$
$968$	−13.4407	−0.432000
$969$	0	0
$970$	0	0
$971$	−18.8610	−0.605279	−0.302640	−	0.953105i	$-0.597868\pi$
$971$	−18.8610	−0.605279	−0.302640	+	0.953105i	$0.597868\pi$
$972$	17.3614	0.556868
$973$	−75.3382	−2.41523
$974$	7.71427	0.247181
$975$	0	0
$976$	−2.08517	−0.0667446
$977$	−12.5553	−0.401679	−0.200839	−	0.979624i	$-0.564367\pi$
$977$	−12.5553	−0.401679	−0.200839	+	0.979624i	$0.564367\pi$
$978$	23.3528	0.746741
$979$	45.6979	1.46051
$980$	0	0
$981$	−77.7630	−2.48278
$982$	23.5312	0.750910
$983$	9.84262	0.313931	0.156965	−	0.987604i	$-0.449829\pi$
$983$	9.84262	0.313931	0.156965	+	0.987604i	$0.449829\pi$
$984$	9.40185	0.299720
$985$	0	0
$986$	0	0
$987$	165.379	5.26406
$988$	−6.19693	−0.197151
$989$	12.7143	0.404293
$990$	0	0
$991$	−7.29887	−0.231856	−0.115928	−	0.993258i	$-0.536984\pi$
$991$	−7.29887	−0.231856	−0.115928	+	0.993258i	$0.536984\pi$
$992$	0.777573	0.0246880
$993$	16.5496	0.525186
$994$	0.195612	0.00620444
$995$	0	0
$996$	30.0554	0.952343
$997$	62.4018	1.97629	0.988143	−	0.153537i	$-0.0490664\pi$
$997$	62.4018	1.97629	0.988143	+	0.153537i	$0.0490664\pi$
$998$	1.69704	0.0537187
$999$	32.8265	1.03859

Twists

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

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

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)

\(f(q)\)	\(=\)	\(q+0.567962 q^{2} +2.88559 q^{3} -1.67742 q^{4} +1.63891 q^{6} +5.00907 q^{7} -2.08863 q^{8} +5.32662 q^{9} +O(q^{10})\)	\(q+0.567962 q^{2} +2.88559 q^{3} -1.67742 q^{4} +1.63891 q^{6} +5.00907 q^{7} -2.08863 q^{8} +5.32662 q^{9} +4.17554 q^{11} -4.84034 q^{12} -0.657310 q^{13} +2.84496 q^{14} +2.16857 q^{16} +3.02532 q^{18} -5.62036 q^{19} +14.4541 q^{21} +2.37155 q^{22} +4.27833 q^{23} -6.02694 q^{24} -0.373327 q^{26} +6.71368 q^{27} -8.40231 q^{28} -5.59957 q^{29} +0.143757 q^{31} +5.40894 q^{32} +12.0489 q^{33} -8.93498 q^{36} +4.88950 q^{37} -3.19215 q^{38} -1.89673 q^{39} -1.55997 q^{41} +8.20939 q^{42} +2.97180 q^{43} -7.00414 q^{44} +2.42993 q^{46} +11.4416 q^{47} +6.25761 q^{48} +18.0908 q^{49} +1.10259 q^{52} -13.3194 q^{53} +3.81312 q^{54} -10.4621 q^{56} -16.2181 q^{57} -3.18034 q^{58} +10.1079 q^{59} -0.961538 q^{61} +0.0816486 q^{62} +26.6814 q^{63} -1.26508 q^{64} +6.84332 q^{66} -0.747944 q^{67} +12.3455 q^{69} +0.0687573 q^{71} -11.1254 q^{72} +1.46435 q^{73} +2.77705 q^{74} +9.42771 q^{76} +20.9156 q^{77} -1.07727 q^{78} +6.72529 q^{79} +3.39306 q^{81} -0.886004 q^{82} -6.20936 q^{83} -24.2456 q^{84} +1.68787 q^{86} -16.1581 q^{87} -8.72118 q^{88} +10.9442 q^{89} -3.29252 q^{91} -7.17655 q^{92} +0.414824 q^{93} +6.49840 q^{94} +15.6080 q^{96} -7.76437 q^{97} +10.2749 q^{98} +22.2416 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + 9 q^{3} + 12 q^{4} + 9 q^{6} + 12 q^{7} + 6 q^{8} + 12 q^{9}+O(q^{10}) \) 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

Varieties

Fields

Representations

Groups

Database

Properties

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