LMFDB - Embedded newform 7225.2.a.bw.1.4

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			1.4
Root			$-1.76581$ 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)

$f(q)$	$=$	$q-1.76581 q^{2} +2.30878 q^{3} +1.11810 q^{4} -4.07688 q^{6} -3.96414 q^{7} +1.55728 q^{8} +2.33047 q^{9} +O(q^{10})$	\(q-1.76581 q^{2} +2.30878 q^{3} +1.11810 q^{4} -4.07688 q^{6} -3.96414 q^{7} +1.55728 q^{8} +2.33047 q^{9} +1.16390 q^{11} +2.58144 q^{12} +0.159139 q^{13} +6.99993 q^{14} -4.98605 q^{16} -4.11518 q^{18} +5.95425 q^{19} -9.15233 q^{21} -2.05524 q^{22} +5.43883 q^{23} +3.59541 q^{24} -0.281010 q^{26} -1.54580 q^{27} -4.43229 q^{28} -8.43313 q^{29} -10.9310 q^{31} +5.68989 q^{32} +2.68720 q^{33} +2.60569 q^{36} +11.8504 q^{37} -10.5141 q^{38} +0.367417 q^{39} +2.52928 q^{41} +16.1613 q^{42} -3.99444 q^{43} +1.30136 q^{44} -9.60395 q^{46} -7.52141 q^{47} -11.5117 q^{48} +8.71439 q^{49} +0.177933 q^{52} +7.51807 q^{53} +2.72959 q^{54} -6.17326 q^{56} +13.7471 q^{57} +14.8913 q^{58} -0.790493 q^{59} +6.30517 q^{61} +19.3020 q^{62} -9.23831 q^{63} -0.0751761 q^{64} -4.74509 q^{66} -6.53340 q^{67} +12.5571 q^{69} +12.5000 q^{71} +3.62919 q^{72} +5.16427 q^{73} -20.9255 q^{74} +6.65743 q^{76} -4.61387 q^{77} -0.648790 q^{78} +8.96704 q^{79} -10.5603 q^{81} -4.46623 q^{82} -8.65347 q^{83} -10.2332 q^{84} +7.05343 q^{86} -19.4703 q^{87} +1.81252 q^{88} -2.22994 q^{89} -0.630849 q^{91} +6.08114 q^{92} -25.2372 q^{93} +13.2814 q^{94} +13.1367 q^{96} +2.92234 q^{97} -15.3880 q^{98} +2.71244 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.76581	−1.24862	−0.624309	−	0.781177i	$-0.714620\pi$
$2$	−1.76581	−1.24862	−0.624309	+	0.781177i	$0.714620\pi$
$3$	2.30878	1.33298	0.666488	−	0.745516i	$-0.267797\pi$
$3$	2.30878	1.33298	0.666488	+	0.745516i	$0.267797\pi$
$4$	1.11810	0.559049
$5$	0	0
$5$	0	0
$6$	−4.07688	−1.66438
$7$	−3.96414	−1.49830	−0.749152	−	0.662399i	$-0.769539\pi$
$7$	−3.96414	−1.49830	−0.749152	+	0.662399i	$0.769539\pi$
$8$	1.55728	0.550580
$9$	2.33047	0.776824
$10$	0	0
$11$	1.16390	0.350930	0.175465	−	0.984486i	$-0.443857\pi$
$11$	1.16390	0.350930	0.175465	+	0.984486i	$0.443857\pi$
$12$	2.58144	0.745198
$13$	0.159139	0.0441372	0.0220686	−	0.999756i	$-0.492975\pi$
$13$	0.159139	0.0441372	0.0220686	+	0.999756i	$0.492975\pi$
$14$	6.99993	1.87081
$15$	0	0
$16$	−4.98605	−1.24651
$17$	0	0
$17$	0	0
$18$	−4.11518	−0.969957
$19$	5.95425	1.36600	0.682999	−	0.730419i	$-0.260675\pi$
$19$	5.95425	1.36600	0.682999	+	0.730419i	$0.260675\pi$
$20$	0	0
$21$	−9.15233	−1.99720
$22$	−2.05524	−0.438178
$23$	5.43883	1.13407	0.567037	−	0.823692i	$-0.308090\pi$
$23$	5.43883	1.13407	0.567037	+	0.823692i	$0.308090\pi$
$24$	3.59541	0.733910
$25$	0	0
$26$	−0.281010	−0.0551105
$27$	−1.54580	−0.297489
$28$	−4.43229	−0.837624
$29$	−8.43313	−1.56599	−0.782997	−	0.622026i	$-0.786310\pi$
$29$	−8.43313	−1.56599	−0.782997	+	0.622026i	$0.786310\pi$
$30$	0	0
$31$	−10.9310	−1.96326	−0.981630	−	0.190796i	$-0.938893\pi$
$31$	−10.9310	−1.96326	−0.981630	+	0.190796i	$0.938893\pi$
$32$	5.68989	1.00584
$33$	2.68720	0.467781
$34$	0	0
$35$	0	0
$36$	2.60569	0.434282
$37$	11.8504	1.94819	0.974094	−	0.226142i	$-0.0726114\pi$
$37$	11.8504	1.94819	0.974094	+	0.226142i	$0.0726114\pi$
$38$	−10.5141	−1.70561
$39$	0.367417	0.0588338
$40$	0	0
$41$	2.52928	0.395007	0.197504	−	0.980302i	$-0.436717\pi$
$41$	2.52928	0.395007	0.197504	+	0.980302i	$0.436717\pi$
$42$	16.1613	2.49374
$43$	−3.99444	−0.609146	−0.304573	−	0.952489i	$-0.598514\pi$
$43$	−3.99444	−0.609146	−0.304573	+	0.952489i	$0.598514\pi$
$44$	1.30136	0.196187
$45$	0	0
$46$	−9.60395	−1.41603
$47$	−7.52141	−1.09711	−0.548555	−	0.836114i	$-0.684822\pi$
$47$	−7.52141	−1.09711	−0.548555	+	0.836114i	$0.684822\pi$
$48$	−11.5117	−1.66157
$49$	8.71439	1.24491
$50$	0	0
$51$	0	0
$52$	0.177933	0.0246748
$53$	7.51807	1.03269	0.516343	−	0.856382i	$-0.327293\pi$
$53$	7.51807	1.03269	0.516343	+	0.856382i	$0.327293\pi$
$54$	2.72959	0.371450
$55$	0	0
$56$	−6.17326	−0.824936
$57$	13.7471	1.82084
$58$	14.8913	1.95533
$59$	−0.790493	−0.102913	−0.0514567	−	0.998675i	$-0.516386\pi$
$59$	−0.790493	−0.102913	−0.0514567	+	0.998675i	$0.516386\pi$
$60$	0	0
$61$	6.30517	0.807294	0.403647	−	0.914915i	$-0.367742\pi$
$61$	6.30517	0.807294	0.403647	+	0.914915i	$0.367742\pi$
$62$	19.3020	2.45136
$63$	−9.23831	−1.16392
$64$	−0.0751761	−0.00939701
$65$	0	0
$66$	−4.74509	−0.584080
$67$	−6.53340	−0.798182	−0.399091	−	0.916911i	$-0.630674\pi$
$67$	−6.53340	−0.798182	−0.399091	+	0.916911i	$0.630674\pi$
$68$	0	0
$69$	12.5571	1.51169
$70$	0	0
$71$	12.5000	1.48347	0.741737	−	0.670691i	$-0.234002\pi$
$71$	12.5000	1.48347	0.741737	+	0.670691i	$0.234002\pi$
$72$	3.62919	0.427704
$73$	5.16427	0.604433	0.302216	−	0.953239i	$-0.402274\pi$
$73$	5.16427	0.604433	0.302216	+	0.953239i	$0.402274\pi$
$74$	−20.9255	−2.43254
$75$	0	0
$76$	6.65743	0.763659
$77$	−4.61387	−0.525800
$78$	−0.648790	−0.0734610
$79$	8.96704	1.00887	0.504435	−	0.863449i	$-0.331701\pi$
$79$	8.96704	1.00887	0.504435	+	0.863449i	$0.331701\pi$
$80$	0	0
$81$	−10.5603	−1.17337
$82$	−4.46623	−0.493213
$83$	−8.65347	−0.949841	−0.474921	−	0.880029i	$-0.657523\pi$
$83$	−8.65347	−0.949841	−0.474921	+	0.880029i	$0.657523\pi$
$84$	−10.2332	−1.11653
$85$	0	0
$86$	7.05343	0.760591
$87$	−19.4703	−2.08743
$88$	1.81252	0.193215
$89$	−2.22994	−0.236373	−0.118186	−	0.992991i	$-0.537708\pi$
$89$	−2.22994	−0.236373	−0.118186	+	0.992991i	$0.537708\pi$
$90$	0	0
$91$	−0.630849	−0.0661309
$92$	6.08114	0.634002
$93$	−25.2372	−2.61698
$94$	13.2814	1.36987
$95$	0	0
$96$	13.1367	1.34076
$97$	2.92234	0.296719	0.148359	−	0.988933i	$-0.452601\pi$
$97$	2.92234	0.296719	0.148359	+	0.988933i	$0.452601\pi$
$98$	−15.3880	−1.55442
$99$	2.71244	0.272611
$100$	0	0
$101$	−2.20379	−0.219286	−0.109643	−	0.993971i	$-0.534971\pi$
$101$	−2.20379	−0.219286	−0.109643	+	0.993971i	$0.534971\pi$
$102$	0	0
$103$	13.1183	1.29259	0.646293	−	0.763089i	$-0.276318\pi$
$103$	13.1183	1.29259	0.646293	+	0.763089i	$0.276318\pi$
$104$	0.247823	0.0243011
$105$	0	0
$106$	−13.2755	−1.28943
$107$	13.7101	1.32540	0.662702	−	0.748883i	$-0.269410\pi$
$107$	13.7101	1.32540	0.662702	+	0.748883i	$0.269410\pi$
$108$	−1.72835	−0.166311
$109$	−5.52256	−0.528965	−0.264483	−	0.964390i	$-0.585201\pi$
$109$	−5.52256	−0.528965	−0.264483	+	0.964390i	$0.585201\pi$
$110$	0	0
$111$	27.3599	2.59689
$112$	19.7654	1.86765
$113$	11.6026	1.09148	0.545742	−	0.837954i	$-0.316248\pi$
$113$	11.6026	1.09148	0.545742	+	0.837954i	$0.316248\pi$
$114$	−24.2747	−2.27354
$115$	0	0
$116$	−9.42907	−0.875467
$117$	0.370869	0.0342868
$118$	1.39586	0.128500
$119$	0	0
$120$	0	0
$121$	−9.64533	−0.876848
$122$	−11.1338	−1.00800
$123$	5.83955	0.526535
$124$	−12.2219	−1.09756
$125$	0	0
$126$	16.3131	1.45329
$127$	19.6911	1.74730	0.873650	−	0.486555i	$-0.161747\pi$
$127$	19.6911	1.74730	0.873650	+	0.486555i	$0.161747\pi$
$128$	−11.2470	−0.994106
$129$	−9.22228	−0.811976
$130$	0	0
$131$	−11.8933	−1.03912	−0.519562	−	0.854433i	$-0.673905\pi$
$131$	−11.8933	−1.03912	−0.519562	+	0.854433i	$0.673905\pi$
$132$	3.00455	0.261512
$133$	−23.6035	−2.04668
$134$	11.5368	0.996625
$135$	0	0
$136$	0	0
$137$	−11.6571	−0.995932	−0.497966	−	0.867197i	$-0.665919\pi$
$137$	−11.6571	−0.995932	−0.497966	+	0.867197i	$0.665919\pi$
$138$	−22.1734	−1.88753
$139$	0.813980	0.0690409	0.0345204	−	0.999404i	$-0.489010\pi$
$139$	0.813980	0.0690409	0.0345204	+	0.999404i	$0.489010\pi$
$140$	0	0
$141$	−17.3653	−1.46242
$142$	−22.0726	−1.85229
$143$	0.185222	0.0154891
$144$	−11.6199	−0.968321
$145$	0	0
$146$	−9.11915	−0.754706
$147$	20.1196	1.65944
$148$	13.2499	1.08913
$149$	−5.26470	−0.431301	−0.215650	−	0.976471i	$-0.569187\pi$
$149$	−5.26470	−0.431301	−0.215650	+	0.976471i	$0.569187\pi$
$150$	0	0
$151$	6.59029	0.536310	0.268155	−	0.963376i	$-0.413586\pi$
$151$	6.59029	0.536310	0.268155	+	0.963376i	$0.413586\pi$
$152$	9.27240	0.752091
$153$	0	0
$154$	8.14724	0.656523
$155$	0	0
$156$	0.410808	0.0328910
$157$	−1.41508	−0.112935	−0.0564677	−	0.998404i	$-0.517984\pi$
$157$	−1.41508	−0.112935	−0.0564677	+	0.998404i	$0.517984\pi$
$158$	−15.8341	−1.25970
$159$	17.3576	1.37655
$160$	0	0
$161$	−21.5603	−1.69919
$162$	18.6476	1.46509
$163$	1.00357	0.0786060	0.0393030	−	0.999227i	$-0.487486\pi$
$163$	1.00357	0.0786060	0.0393030	+	0.999227i	$0.487486\pi$
$164$	2.82798	0.220828
$165$	0	0
$166$	15.2804	1.18599
$167$	17.6935	1.36916	0.684581	−	0.728937i	$-0.259985\pi$
$167$	17.6935	1.36916	0.684581	+	0.728937i	$0.259985\pi$
$168$	−14.2527	−1.09962
$169$	−12.9747	−0.998052
$170$	0	0
$171$	13.8762	1.06114
$172$	−4.46617	−0.340542
$173$	−8.56090	−0.650873	−0.325437	−	0.945564i	$-0.605511\pi$
$173$	−8.56090	−0.650873	−0.325437	+	0.945564i	$0.605511\pi$
$174$	34.3809	2.60641
$175$	0	0
$176$	−5.80328	−0.437439
$177$	−1.82508	−0.137181
$178$	3.93765	0.295139
$179$	−11.7413	−0.877585	−0.438793	−	0.898588i	$-0.644594\pi$
$179$	−11.7413	−0.877585	−0.438793	+	0.898588i	$0.644594\pi$
$180$	0	0
$181$	−9.11961	−0.677855	−0.338928	−	0.940812i	$-0.610064\pi$
$181$	−9.11961	−0.677855	−0.338928	+	0.940812i	$0.610064\pi$
$182$	1.11396	0.0825723
$183$	14.5573	1.07610
$184$	8.46975	0.624398
$185$	0	0
$186$	44.5642	3.26761
$187$	0	0
$188$	−8.40967	−0.613338
$189$	6.12775	0.445728
$190$	0	0
$191$	3.32018	0.240240	0.120120	−	0.992759i	$-0.461672\pi$
$191$	3.32018	0.240240	0.120120	+	0.992759i	$0.461672\pi$
$192$	−0.173565	−0.0125260
$193$	−2.03365	−0.146385	−0.0731927	−	0.997318i	$-0.523319\pi$
$193$	−2.03365	−0.146385	−0.0731927	+	0.997318i	$0.523319\pi$
$194$	−5.16031	−0.370489
$195$	0	0
$196$	9.74353	0.695967
$197$	14.6969	1.04711	0.523555	−	0.851992i	$-0.324605\pi$
$197$	14.6969	1.04711	0.523555	+	0.851992i	$0.324605\pi$
$198$	−4.78967	−0.340387
$199$	−2.41972	−0.171529	−0.0857646	−	0.996315i	$-0.527333\pi$
$199$	−2.41972	−0.171529	−0.0857646	+	0.996315i	$0.527333\pi$
$200$	0	0
$201$	−15.0842	−1.06396
$202$	3.89149	0.273804
$203$	33.4301	2.34633
$204$	0	0
$205$	0	0
$206$	−23.1645	−1.61395
$207$	12.6750	0.880975
$208$	−0.793475	−0.0550176
$209$	6.93017	0.479370
$210$	0	0
$211$	11.6071	0.799065	0.399533	−	0.916719i	$-0.369172\pi$
$211$	11.6071	0.799065	0.399533	+	0.916719i	$0.369172\pi$
$212$	8.40594	0.577322
$213$	28.8597	1.97743
$214$	−24.2094	−1.65492
$215$	0	0
$216$	−2.40723	−0.163791
$217$	43.3319	2.94156
$218$	9.75181	0.660476
$219$	11.9232	0.805694
$220$	0	0
$221$	0	0
$222$	−48.3125	−3.24252
$223$	22.6170	1.51455	0.757274	−	0.653098i	$-0.226531\pi$
$223$	22.6170	1.51455	0.757274	+	0.653098i	$0.226531\pi$
$224$	−22.5555	−1.50705
$225$	0	0
$226$	−20.4881	−1.36285
$227$	−18.3149	−1.21560	−0.607801	−	0.794089i	$-0.707948\pi$
$227$	−18.3149	−1.21560	−0.607801	+	0.794089i	$0.707948\pi$
$228$	15.3705	1.01794
$229$	14.4978	0.958045	0.479022	−	0.877803i	$-0.340991\pi$
$229$	14.4978	0.958045	0.479022	+	0.877803i	$0.340991\pi$
$230$	0	0
$231$	−10.6524	−0.700878
$232$	−13.1327	−0.862205
$233$	13.6952	0.897203	0.448601	−	0.893732i	$-0.351922\pi$
$233$	13.6952	0.897203	0.448601	+	0.893732i	$0.351922\pi$
$234$	−0.654885	−0.0428112
$235$	0	0
$236$	−0.883848	−0.0575336
$237$	20.7029	1.34480
$238$	0	0
$239$	25.7020	1.66252	0.831261	−	0.555883i	$-0.187620\pi$
$239$	25.7020	1.66252	0.831261	+	0.555883i	$0.187620\pi$
$240$	0	0
$241$	0.764589	0.0492515	0.0246258	−	0.999697i	$-0.492161\pi$
$241$	0.764589	0.0492515	0.0246258	+	0.999697i	$0.492161\pi$
$242$	17.0319	1.09485
$243$	−19.7441	−1.26658
$244$	7.04979	0.451317
$245$	0	0
$246$	−10.3116	−0.657441
$247$	0.947552	0.0602913
$248$	−17.0225	−1.08093
$249$	−19.9790	−1.26612
$250$	0	0
$251$	−27.3585	−1.72685	−0.863427	−	0.504473i	$-0.831687\pi$
$251$	−27.3585	−1.72685	−0.863427	+	0.504473i	$0.831687\pi$
$252$	−10.3293	−0.650687
$253$	6.33027	0.397980
$254$	−34.7708	−2.18171
$255$	0	0
$256$	20.0105	1.25066
$257$	−0.192378	−0.0120002	−0.00600010	−	0.999982i	$-0.501910\pi$
$257$	−0.192378	−0.0120002	−0.00600010	+	0.999982i	$0.501910\pi$
$258$	16.2848	1.01385
$259$	−46.9765	−2.91898
$260$	0	0
$261$	−19.6532	−1.21650
$262$	21.0014	1.29747
$263$	21.7389	1.34048	0.670240	−	0.742144i	$-0.266191\pi$
$263$	21.7389	1.34048	0.670240	+	0.742144i	$0.266191\pi$
$264$	4.18471	0.257551
$265$	0	0
$266$	41.6793	2.55552
$267$	−5.14844	−0.315079
$268$	−7.30498	−0.446223
$269$	−21.0646	−1.28433	−0.642166	−	0.766566i	$-0.721964\pi$
$269$	−21.0646	−1.28433	−0.642166	+	0.766566i	$0.721964\pi$
$270$	0	0
$271$	13.3522	0.811086	0.405543	−	0.914076i	$-0.367082\pi$
$271$	13.3522	0.811086	0.405543	+	0.914076i	$0.367082\pi$
$272$	0	0
$273$	−1.45649	−0.0881509
$274$	20.5842	1.24354
$275$	0	0
$276$	14.0400	0.845110
$277$	2.01604	0.121132	0.0605660	−	0.998164i	$-0.480709\pi$
$277$	2.01604	0.121132	0.0605660	+	0.998164i	$0.480709\pi$
$278$	−1.43734	−0.0862057
$279$	−25.4743	−1.52511
$280$	0	0
$281$	5.54922	0.331039	0.165519	−	0.986207i	$-0.447070\pi$
$281$	5.54922	0.331039	0.165519	+	0.986207i	$0.447070\pi$
$282$	30.6639	1.82601
$283$	20.1586	1.19830	0.599151	−	0.800636i	$-0.295505\pi$
$283$	20.1586	1.19830	0.599151	+	0.800636i	$0.295505\pi$
$284$	13.9762	0.829334
$285$	0	0
$286$	−0.327068	−0.0193399
$287$	−10.0264	−0.591840
$288$	13.2601	0.781360
$289$	0	0
$290$	0	0
$291$	6.74705	0.395519
$292$	5.77416	0.337907
$293$	0.840362	0.0490944	0.0245472	−	0.999699i	$-0.492186\pi$
$293$	0.840362	0.0490944	0.0245472	+	0.999699i	$0.492186\pi$
$294$	−35.5275	−2.07201
$295$	0	0
$296$	18.4543	1.07263
$297$	−1.79916	−0.104398
$298$	9.29647	0.538530
$299$	0.865529	0.0500548
$300$	0	0
$301$	15.8345	0.912685
$302$	−11.6372	−0.669647
$303$	−5.08808	−0.292303
$304$	−29.6882	−1.70273
$305$	0	0
$306$	0	0
$307$	−4.39971	−0.251105	−0.125552	−	0.992087i	$-0.540070\pi$
$307$	−4.39971	−0.251105	−0.125552	+	0.992087i	$0.540070\pi$
$308$	−5.15876	−0.293948
$309$	30.2873	1.72299
$310$	0	0
$311$	28.8540	1.63616	0.818079	−	0.575106i	$-0.195039\pi$
$311$	28.8540	1.63616	0.818079	+	0.575106i	$0.195039\pi$
$312$	0.572170	0.0323927
$313$	9.64425	0.545126	0.272563	−	0.962138i	$-0.412129\pi$
$313$	9.64425	0.545126	0.272563	+	0.962138i	$0.412129\pi$
$314$	2.49876	0.141013
$315$	0	0
$316$	10.0260	0.564008
$317$	18.8207	1.05708	0.528538	−	0.848910i	$-0.322740\pi$
$317$	18.8207	1.05708	0.528538	+	0.848910i	$0.322740\pi$
$318$	−30.6503	−1.71878
$319$	−9.81535	−0.549554
$320$	0	0
$321$	31.6536	1.76673
$322$	38.0714	2.12164
$323$	0	0
$324$	−11.8075	−0.655970
$325$	0	0
$326$	−1.77213	−0.0981490
$327$	−12.7504	−0.705098
$328$	3.93878	0.217483
$329$	29.8159	1.64380
$330$	0	0
$331$	13.8371	0.760555	0.380277	−	0.924872i	$-0.375829\pi$
$331$	13.8371	0.760555	0.380277	+	0.924872i	$0.375829\pi$
$332$	−9.67542	−0.531008
$333$	27.6169	1.51340
$334$	−31.2434	−1.70956
$335$	0	0
$336$	45.6340	2.48954
$337$	−1.73190	−0.0943424	−0.0471712	−	0.998887i	$-0.515021\pi$
$337$	−1.73190	−0.0943424	−0.0471712	+	0.998887i	$0.515021\pi$
$338$	22.9109	1.24619
$339$	26.7879	1.45492
$340$	0	0
$341$	−12.7226	−0.688967
$342$	−24.5028	−1.32496
$343$	−6.79607	−0.366953
$344$	−6.22044	−0.335383
$345$	0	0
$346$	15.1170	0.812693
$347$	33.5022	1.79849	0.899246	−	0.437443i	$-0.144116\pi$
$347$	33.5022	1.79849	0.899246	+	0.437443i	$0.144116\pi$
$348$	−21.7697	−1.16698
$349$	28.7452	1.53870	0.769348	−	0.638830i	$-0.220581\pi$
$349$	28.7452	1.53870	0.769348	+	0.638830i	$0.220581\pi$
$350$	0	0
$351$	−0.245996	−0.0131303
$352$	6.62248	0.352979
$353$	27.8319	1.48134	0.740672	−	0.671866i	$-0.234507\pi$
$353$	27.8319	1.48134	0.740672	+	0.671866i	$0.234507\pi$
$354$	3.22274	0.171287
$355$	0	0
$356$	−2.49329	−0.132144
$357$	0	0
$358$	20.7329	1.09577
$359$	−20.0128	−1.05624	−0.528118	−	0.849171i	$-0.677102\pi$
$359$	−20.0128	−1.05624	−0.528118	+	0.849171i	$0.677102\pi$
$360$	0	0
$361$	16.4530	0.865950
$362$	16.1035	0.846383
$363$	−22.2690	−1.16882
$364$	−0.705350	−0.0369704
$365$	0	0
$366$	−25.7054	−1.34364
$367$	7.09090	0.370142	0.185071	−	0.982725i	$-0.440749\pi$
$367$	7.09090	0.370142	0.185071	+	0.982725i	$0.440749\pi$
$368$	−27.1183	−1.41364
$369$	5.89441	0.306851
$370$	0	0
$371$	−29.8027	−1.54728
$372$	−28.2177	−1.46302
$373$	−13.3552	−0.691505	−0.345752	−	0.938326i	$-0.612376\pi$
$373$	−13.3552	−0.691505	−0.345752	+	0.938326i	$0.612376\pi$
$374$	0	0
$375$	0	0
$376$	−11.7129	−0.604047
$377$	−1.34204	−0.0691186
$378$	−10.8205	−0.556544
$379$	18.4128	0.945803	0.472901	−	0.881115i	$-0.343207\pi$
$379$	18.4128	0.945803	0.472901	+	0.881115i	$0.343207\pi$
$380$	0	0
$381$	45.4624	2.32911
$382$	−5.86281	−0.299968
$383$	20.2300	1.03370	0.516852	−	0.856075i	$-0.327104\pi$
$383$	20.2300	1.03370	0.516852	+	0.856075i	$0.327104\pi$
$384$	−25.9669	−1.32512
$385$	0	0
$386$	3.59105	0.182780
$387$	−9.30892	−0.473199
$388$	3.26746	0.165880
$389$	13.9935	0.709501	0.354751	−	0.934961i	$-0.384566\pi$
$389$	13.9935	0.709501	0.354751	+	0.934961i	$0.384566\pi$
$390$	0	0
$391$	0	0
$392$	13.5707	0.685424
$393$	−27.4591	−1.38513
$394$	−25.9520	−1.30744
$395$	0	0
$396$	3.03278	0.152403
$397$	9.90960	0.497349	0.248674	−	0.968587i	$-0.420005\pi$
$397$	9.90960	0.497349	0.248674	+	0.968587i	$0.420005\pi$
$398$	4.27277	0.214175
$399$	−54.4952	−2.72817
$400$	0	0
$401$	19.4158	0.969577	0.484789	−	0.874631i	$-0.338897\pi$
$401$	19.4158	0.969577	0.484789	+	0.874631i	$0.338897\pi$
$402$	26.6359	1.32848
$403$	−1.73954	−0.0866528
$404$	−2.46406	−0.122591
$405$	0	0
$406$	−59.0313	−2.92968
$407$	13.7927	0.683678
$408$	0	0
$409$	−8.58738	−0.424619	−0.212309	−	0.977203i	$-0.568098\pi$
$409$	−8.58738	−0.424619	−0.212309	+	0.977203i	$0.568098\pi$
$410$	0	0
$411$	−26.9136	−1.32755
$412$	14.6676	0.722619
$413$	3.13362	0.154196
$414$	−22.3817	−1.10000
$415$	0	0
$416$	0.905483	0.0443950
$417$	1.87930	0.0920298
$418$	−12.2374	−0.598550
$419$	−2.01237	−0.0983108	−0.0491554	−	0.998791i	$-0.515653\pi$
$419$	−2.01237	−0.0983108	−0.0491554	+	0.998791i	$0.515653\pi$
$420$	0	0
$421$	−28.6304	−1.39536	−0.697680	−	0.716409i	$-0.745784\pi$
$421$	−28.6304	−1.39536	−0.697680	+	0.716409i	$0.745784\pi$
$422$	−20.4960	−0.997728
$423$	−17.5284	−0.852261
$424$	11.7077	0.568577
$425$	0	0
$426$	−50.9608	−2.46906
$427$	−24.9946	−1.20957
$428$	15.3292	0.740965
$429$	0.427638	0.0206466
$430$	0	0
$431$	9.51635	0.458387	0.229193	−	0.973381i	$-0.426391\pi$
$431$	9.51635	0.458387	0.229193	+	0.973381i	$0.426391\pi$
$432$	7.70742	0.370823
$433$	−1.90151	−0.0913809	−0.0456905	−	0.998956i	$-0.514549\pi$
$433$	−1.90151	−0.0913809	−0.0456905	+	0.998956i	$0.514549\pi$
$434$	−76.5160	−3.67288
$435$	0	0
$436$	−6.17476	−0.295717
$437$	32.3841	1.54914
$438$	−21.0541	−1.00600
$439$	12.7612	0.609057	0.304529	−	0.952503i	$-0.401501\pi$
$439$	12.7612	0.609057	0.304529	+	0.952503i	$0.401501\pi$
$440$	0	0
$441$	20.3086	0.967078
$442$	0	0
$443$	3.03153	0.144032	0.0720162	−	0.997403i	$-0.477057\pi$
$443$	3.03153	0.144032	0.0720162	+	0.997403i	$0.477057\pi$
$444$	30.5910	1.45179
$445$	0	0
$446$	−39.9374	−1.89109
$447$	−12.1550	−0.574913
$448$	0.298008	0.0140796
$449$	4.12856	0.194839	0.0974195	−	0.995243i	$-0.468941\pi$
$449$	4.12856	0.194839	0.0974195	+	0.995243i	$0.468941\pi$
$450$	0	0
$451$	2.94384	0.138620
$452$	12.9729	0.610192
$453$	15.2155	0.714889
$454$	32.3407	1.51782
$455$	0	0
$456$	21.4079	1.00252
$457$	3.70691	0.173402	0.0867010	−	0.996234i	$-0.472368\pi$
$457$	3.70691	0.173402	0.0867010	+	0.996234i	$0.472368\pi$
$458$	−25.6005	−1.19623
$459$	0	0
$460$	0	0
$461$	29.1985	1.35991	0.679956	−	0.733253i	$-0.261999\pi$
$461$	29.1985	1.35991	0.679956	+	0.733253i	$0.261999\pi$
$462$	18.8102	0.875129
$463$	−21.6458	−1.00596	−0.502982	−	0.864297i	$-0.667764\pi$
$463$	−21.6458	−1.00596	−0.502982	+	0.864297i	$0.667764\pi$
$464$	42.0481	1.95203
$465$	0	0
$466$	−24.1832	−1.12026
$467$	−0.642849	−0.0297475	−0.0148737	−	0.999889i	$-0.504735\pi$
$467$	−0.642849	−0.0297475	−0.0148737	+	0.999889i	$0.504735\pi$
$468$	0.414667	0.0191680
$469$	25.8993	1.19592
$470$	0	0
$471$	−3.26710	−0.150540
$472$	−1.23102	−0.0566621
$473$	−4.64914	−0.213768
$474$	−36.5575	−1.67914
$475$	0	0
$476$	0	0
$477$	17.5207	0.802216
$478$	−45.3849	−2.07585
$479$	−8.01664	−0.366290	−0.183145	−	0.983086i	$-0.558628\pi$
$479$	−8.01664	−0.366290	−0.183145	+	0.983086i	$0.558628\pi$
$480$	0	0
$481$	1.88586	0.0859876
$482$	−1.35012	−0.0614964
$483$	−49.7779	−2.26497
$484$	−10.7844	−0.490201
$485$	0	0
$486$	34.8644	1.58148
$487$	−32.6884	−1.48125	−0.740626	−	0.671918i	$-0.765471\pi$
$487$	−32.6884	−1.48125	−0.740626	+	0.671918i	$0.765471\pi$
$488$	9.81889	0.444480
$489$	2.31703	0.104780
$490$	0	0
$491$	−27.8240	−1.25568	−0.627839	−	0.778343i	$-0.716060\pi$
$491$	−27.8240	−1.25568	−0.627839	+	0.778343i	$0.716060\pi$
$492$	6.52919	0.294359
$493$	0	0
$494$	−1.67320	−0.0752809
$495$	0	0
$496$	54.5024	2.44723
$497$	−49.5516	−2.22269
$498$	35.2791	1.58090
$499$	−31.3904	−1.40523	−0.702614	−	0.711572i	$-0.747984\pi$
$499$	−31.3904	−1.40523	−0.702614	+	0.711572i	$0.747984\pi$
$500$	0	0
$501$	40.8503	1.82506
$502$	48.3101	2.15618
$503$	38.6876	1.72500	0.862498	−	0.506061i	$-0.168899\pi$
$503$	38.6876	1.72500	0.862498	+	0.506061i	$0.168899\pi$
$504$	−14.3866	−0.640830
$505$	0	0
$506$	−11.1781	−0.496926
$507$	−29.9557	−1.33038
$508$	22.0165	0.976826
$509$	4.99261	0.221294	0.110647	−	0.993860i	$-0.464708\pi$
$509$	4.99261	0.221294	0.110647	+	0.993860i	$0.464708\pi$
$510$	0	0
$511$	−20.4719	−0.905623
$512$	−12.8408	−0.567487
$513$	−9.20405	−0.406369
$514$	0.339703	0.0149837
$515$	0	0
$516$	−10.3114	−0.453934
$517$	−8.75419	−0.385009
$518$	82.9517	3.64469
$519$	−19.7653	−0.867598
$520$	0	0
$521$	−20.3481	−0.891466	−0.445733	−	0.895166i	$-0.647057\pi$
$521$	−20.3481	−0.891466	−0.445733	+	0.895166i	$0.647057\pi$
$522$	34.7038	1.51895
$523$	−4.12826	−0.180516	−0.0902581	−	0.995918i	$-0.528769\pi$
$523$	−4.12826	−0.180516	−0.0902581	+	0.995918i	$0.528769\pi$
$524$	−13.2979	−0.580921
$525$	0	0
$526$	−38.3869	−1.67375
$527$	0	0
$528$	−13.3985	−0.583095
$529$	6.58082	0.286123
$530$	0	0
$531$	−1.84222	−0.0799456
$532$	−26.3910	−1.14419
$533$	0.402507	0.0174345
$534$	9.09118	0.393414
$535$	0	0
$536$	−10.1743	−0.439463
$537$	−27.1081	−1.16980
$538$	37.1961	1.60364
$539$	10.1427	0.436877
$540$	0	0
$541$	2.98336	0.128265	0.0641323	−	0.997941i	$-0.479572\pi$
$541$	2.98336	0.128265	0.0641323	+	0.997941i	$0.479572\pi$
$542$	−23.5774	−1.01274
$543$	−21.0552	−0.903564
$544$	0	0
$545$	0	0
$546$	2.57189	0.110067
$547$	−28.1432	−1.20331	−0.601657	−	0.798754i	$-0.705493\pi$
$547$	−28.1432	−1.20331	−0.601657	+	0.798754i	$0.705493\pi$
$548$	−13.0338	−0.556774
$549$	14.6940	0.627125
$550$	0	0
$551$	−50.2130	−2.13914
$552$	19.5548	0.832308
$553$	−35.5466	−1.51159
$554$	−3.55995	−0.151248
$555$	0	0
$556$	0.910109	0.0385972
$557$	6.99605	0.296432	0.148216	−	0.988955i	$-0.452647\pi$
$557$	6.99605	0.296432	0.148216	+	0.988955i	$0.452647\pi$
$558$	44.9829	1.90428
$559$	−0.635670	−0.0268860
$560$	0	0
$561$	0	0
$562$	−9.79889	−0.413341
$563$	24.0210	1.01236	0.506182	−	0.862427i	$-0.331056\pi$
$563$	24.0210	1.01236	0.506182	+	0.862427i	$0.331056\pi$
$564$	−19.4161	−0.817564
$565$	0	0
$566$	−35.5963	−1.49622
$567$	41.8626	1.75806
$568$	19.4659	0.816771
$569$	−26.0829	−1.09345	−0.546727	−	0.837311i	$-0.684126\pi$
$569$	−26.0829	−1.09345	−0.546727	+	0.837311i	$0.684126\pi$
$570$	0	0
$571$	14.9358	0.625042	0.312521	−	0.949911i	$-0.398827\pi$
$571$	14.9358	0.625042	0.312521	+	0.949911i	$0.398827\pi$
$572$	0.207097	0.00865914
$573$	7.66556	0.320233
$574$	17.7048	0.738983
$575$	0	0
$576$	−0.175196	−0.00729982
$577$	18.1157	0.754168	0.377084	−	0.926179i	$-0.376927\pi$
$577$	18.1157	0.754168	0.377084	+	0.926179i	$0.376927\pi$
$578$	0	0
$579$	−4.69525	−0.195128
$580$	0	0
$581$	34.3035	1.42315
$582$	−11.9140	−0.493853
$583$	8.75031	0.362401
$584$	8.04220	0.332789
$585$	0	0
$586$	−1.48392	−0.0613002
$587$	−43.9157	−1.81259	−0.906297	−	0.422641i	$-0.861103\pi$
$587$	−43.9157	−1.81259	−0.906297	+	0.422641i	$0.861103\pi$
$588$	22.4957	0.927707
$589$	−65.0857	−2.68181
$590$	0	0
$591$	33.9319	1.39577
$592$	−59.0866	−2.42844
$593$	29.1441	1.19680	0.598402	−	0.801196i	$-0.295802\pi$
$593$	29.1441	1.19680	0.598402	+	0.801196i	$0.295802\pi$
$594$	3.17697	0.130353
$595$	0	0
$596$	−5.88644	−0.241118
$597$	−5.58660	−0.228644
$598$	−1.52836	−0.0624994
$599$	31.3423	1.28061	0.640305	−	0.768121i	$-0.278808\pi$
$599$	31.3423	1.28061	0.640305	+	0.768121i	$0.278808\pi$
$600$	0	0
$601$	27.1894	1.10908	0.554539	−	0.832157i	$-0.312894\pi$
$601$	27.1894	1.10908	0.554539	+	0.832157i	$0.312894\pi$
$602$	−27.9608	−1.13960
$603$	−15.2259	−0.620047
$604$	7.36859	0.299824
$605$	0	0
$606$	8.98460	0.364974
$607$	23.1597	0.940022	0.470011	−	0.882660i	$-0.344250\pi$
$607$	23.1597	0.940022	0.470011	+	0.882660i	$0.344250\pi$
$608$	33.8790	1.37397
$609$	77.1828	3.12761
$610$	0	0
$611$	−1.19695	−0.0484234
$612$	0	0
$613$	−35.3663	−1.42843	−0.714216	−	0.699926i	$-0.753216\pi$
$613$	−35.3663	−1.42843	−0.714216	+	0.699926i	$0.753216\pi$
$614$	7.76907	0.313534
$615$	0	0
$616$	−7.18507	−0.289495
$617$	0.688801	0.0277301	0.0138650	−	0.999904i	$-0.495586\pi$
$617$	0.688801	0.0277301	0.0138650	+	0.999904i	$0.495586\pi$
$618$	−53.4818	−2.15135
$619$	−23.4208	−0.941360	−0.470680	−	0.882304i	$-0.655991\pi$
$619$	−23.4208	−0.941360	−0.470680	+	0.882304i	$0.655991\pi$
$620$	0	0
$621$	−8.40731	−0.337374
$622$	−50.9507	−2.04294
$623$	8.83977	0.354158
$624$	−1.83196	−0.0733371
$625$	0	0
$626$	−17.0300	−0.680654
$627$	16.0002	0.638988
$628$	−1.58219	−0.0631364
$629$	0	0
$630$	0	0
$631$	−37.1206	−1.47775	−0.738874	−	0.673844i	$-0.764642\pi$
$631$	−37.1206	−1.47775	−0.738874	+	0.673844i	$0.764642\pi$
$632$	13.9642	0.555464
$633$	26.7982	1.06513
$634$	−33.2338	−1.31988
$635$	0	0
$636$	19.4075	0.769556
$637$	1.38680	0.0549470
$638$	17.3321	0.686184
$639$	29.1308	1.15240
$640$	0	0
$641$	14.1361	0.558344	0.279172	−	0.960241i	$-0.409940\pi$
$641$	14.1361	0.558344	0.279172	+	0.960241i	$0.409940\pi$
$642$	−55.8943	−2.20597
$643$	−21.0943	−0.831876	−0.415938	−	0.909393i	$-0.636547\pi$
$643$	−21.0943	−0.831876	−0.415938	+	0.909393i	$0.636547\pi$
$644$	−24.1065	−0.949928
$645$	0	0
$646$	0	0
$647$	9.72333	0.382264	0.191132	−	0.981564i	$-0.438784\pi$
$647$	9.72333	0.382264	0.191132	+	0.981564i	$0.438784\pi$
$648$	−16.4453	−0.646033
$649$	−0.920057	−0.0361154
$650$	0	0
$651$	100.044	3.92102
$652$	1.12209	0.0439446
$653$	−33.9833	−1.32987	−0.664934	−	0.746902i	$-0.731540\pi$
$653$	−33.9833	−1.32987	−0.664934	+	0.746902i	$0.731540\pi$
$654$	22.5148	0.880398
$655$	0	0
$656$	−12.6111	−0.492381
$657$	12.0352	0.469538
$658$	−52.6493	−2.05248
$659$	31.0552	1.20974	0.604869	−	0.796325i	$-0.293226\pi$
$659$	31.0552	1.20974	0.604869	+	0.796325i	$0.293226\pi$
$660$	0	0
$661$	20.2090	0.786040	0.393020	−	0.919530i	$-0.371430\pi$
$661$	20.2090	0.786040	0.393020	+	0.919530i	$0.371430\pi$
$662$	−24.4337	−0.949643
$663$	0	0
$664$	−13.4758	−0.522964
$665$	0	0
$666$	−48.7664	−1.88966
$667$	−45.8663	−1.77595
$668$	19.7830	0.765428
$669$	52.2177	2.01885
$670$	0	0
$671$	7.33861	0.283304
$672$	−52.0757	−2.00886
$673$	2.75626	0.106246	0.0531229	−	0.998588i	$-0.483082\pi$
$673$	2.75626	0.106246	0.0531229	+	0.998588i	$0.483082\pi$
$674$	3.05821	0.117798
$675$	0	0
$676$	−14.5069	−0.557960
$677$	12.3826	0.475902	0.237951	−	0.971277i	$-0.423524\pi$
$677$	12.3826	0.475902	0.237951	+	0.971277i	$0.423524\pi$
$678$	−47.3025	−1.81664
$679$	−11.5846	−0.444575
$680$	0	0
$681$	−42.2851	−1.62037
$682$	22.4657	0.860257
$683$	16.7571	0.641194	0.320597	−	0.947216i	$-0.396116\pi$
$683$	16.7571	0.641194	0.320597	+	0.947216i	$0.396116\pi$
$684$	15.5149	0.593229
$685$	0	0
$686$	12.0006	0.458185
$687$	33.4724	1.27705
$688$	19.9165	0.759308
$689$	1.19642	0.0455799
$690$	0	0
$691$	10.1110	0.384641	0.192321	−	0.981332i	$-0.438399\pi$
$691$	10.1110	0.384641	0.192321	+	0.981332i	$0.438399\pi$
$692$	−9.57192	−0.363870
$693$	−10.7525	−0.408454
$694$	−59.1586	−2.24563
$695$	0	0
$696$	−30.3206	−1.14930
$697$	0	0
$698$	−50.7587	−1.92124
$699$	31.6192	1.19595
$700$	0	0
$701$	−6.68932	−0.252652	−0.126326	−	0.991989i	$-0.540319\pi$
$701$	−6.68932	−0.252652	−0.126326	+	0.991989i	$0.540319\pi$
$702$	0.434384	0.0163948
$703$	70.5600	2.66122
$704$	−0.0874977	−0.00329769
$705$	0	0
$706$	−49.1460	−1.84963
$707$	8.73615	0.328557
$708$	−2.04061	−0.0766909
$709$	20.8547	0.783216	0.391608	−	0.920132i	$-0.371919\pi$
$709$	20.8547	0.783216	0.391608	+	0.920132i	$0.371919\pi$
$710$	0	0
$711$	20.8974	0.783715
$712$	−3.47263	−0.130142
$713$	−59.4516	−2.22648
$714$	0	0
$715$	0	0
$716$	−13.1279	−0.490613
$717$	59.3402	2.21610
$718$	35.3389	1.31884
$719$	44.9662	1.67696	0.838478	−	0.544935i	$-0.183446\pi$
$719$	44.9662	1.67696	0.838478	+	0.544935i	$0.183446\pi$
$720$	0	0
$721$	−52.0028	−1.93669
$722$	−29.0530	−1.08124
$723$	1.76527	0.0656511
$724$	−10.1966	−0.378954
$725$	0	0
$726$	39.3228	1.45941
$727$	−32.9441	−1.22183	−0.610915	−	0.791696i	$-0.709198\pi$
$727$	−32.9441	−1.22183	−0.610915	+	0.791696i	$0.709198\pi$
$728$	−0.982405	−0.0364104
$729$	−13.9038	−0.514956
$730$	0	0
$731$	0	0
$732$	16.2764	0.601594
$733$	40.6329	1.50081	0.750404	−	0.660979i	$-0.229859\pi$
$733$	40.6329	1.50081	0.750404	+	0.660979i	$0.229859\pi$
$734$	−12.5212	−0.462166
$735$	0	0
$736$	30.9463	1.14070
$737$	−7.60425	−0.280106
$738$	−10.4084	−0.383140
$739$	−1.75111	−0.0644156	−0.0322078	−	0.999481i	$-0.510254\pi$
$739$	−1.75111	−0.0644156	−0.0322078	+	0.999481i	$0.510254\pi$
$740$	0	0
$741$	2.18769	0.0803669
$742$	52.6260	1.93196
$743$	49.5195	1.81669	0.908346	−	0.418219i	$-0.137345\pi$
$743$	49.5195	1.81669	0.908346	+	0.418219i	$0.137345\pi$
$744$	−39.3013	−1.44086
$745$	0	0
$746$	23.5827	0.863426
$747$	−20.1667	−0.737859
$748$	0	0
$749$	−54.3487	−1.98586
$750$	0	0
$751$	−10.9498	−0.399566	−0.199783	−	0.979840i	$-0.564024\pi$
$751$	−10.9498	−0.399566	−0.199783	+	0.979840i	$0.564024\pi$
$752$	37.5021	1.36756
$753$	−63.1649	−2.30186
$754$	2.36979	0.0863028
$755$	0	0
$756$	6.85142	0.249184
$757$	11.2480	0.408815	0.204408	−	0.978886i	$-0.434473\pi$
$757$	11.2480	0.408815	0.204408	+	0.978886i	$0.434473\pi$
$758$	−32.5136	−1.18095
$759$	14.6152	0.530498
$760$	0	0
$761$	−39.8436	−1.44433	−0.722165	−	0.691720i	$-0.756853\pi$
$761$	−39.8436	−1.44433	−0.722165	+	0.691720i	$0.756853\pi$
$762$	−80.2781	−2.90817
$763$	21.8922	0.792551
$764$	3.71228	0.134306
$765$	0	0
$766$	−35.7224	−1.29070
$767$	−0.125798	−0.00454231
$768$	46.1999	1.66709
$769$	−31.4576	−1.13439	−0.567195	−	0.823584i	$-0.691971\pi$
$769$	−31.4576	−1.13439	−0.567195	+	0.823584i	$0.691971\pi$
$770$	0	0
$771$	−0.444159	−0.0159960
$772$	−2.27382	−0.0818366
$773$	17.7459	0.638276	0.319138	−	0.947708i	$-0.396607\pi$
$773$	17.7459	0.638276	0.319138	+	0.947708i	$0.396607\pi$
$774$	16.4378	0.590845
$775$	0	0
$776$	4.55089	0.163368
$777$	−108.458	−3.89093
$778$	−24.7100	−0.885896
$779$	15.0599	0.539579
$780$	0	0
$781$	14.5488	0.520595
$782$	0	0
$783$	13.0359	0.465865
$784$	−43.4504	−1.55180
$785$	0	0
$786$	48.4876	1.72950
$787$	−28.4355	−1.01362	−0.506808	−	0.862059i	$-0.669175\pi$
$787$	−28.4355	−1.01362	−0.506808	+	0.862059i	$0.669175\pi$
$788$	16.4325	0.585385
$789$	50.1905	1.78683
$790$	0	0
$791$	−45.9944	−1.63537
$792$	4.22402	0.150094
$793$	1.00340	0.0356317
$794$	−17.4985	−0.620999
$795$	0	0
$796$	−2.70548	−0.0958932
$797$	12.9298	0.457996	0.228998	−	0.973427i	$-0.426455\pi$
$797$	12.9298	0.457996	0.228998	+	0.973427i	$0.426455\pi$
$798$	96.2284	3.40645
$799$	0	0
$800$	0	0
$801$	−5.19680	−0.183620
$802$	−34.2846	−1.21063
$803$	6.01072	0.212114
$804$	−16.8656	−0.594804
$805$	0	0
$806$	3.07171	0.108196
$807$	−48.6335	−1.71198
$808$	−3.43192	−0.120734
$809$	−18.7621	−0.659640	−0.329820	−	0.944044i	$-0.606988\pi$
$809$	−18.7621	−0.659640	−0.329820	+	0.944044i	$0.606988\pi$
$810$	0	0
$811$	34.4296	1.20899	0.604493	−	0.796610i	$-0.293376\pi$
$811$	34.4296	1.20899	0.604493	+	0.796610i	$0.293376\pi$
$812$	37.3781	1.31171
$813$	30.8272	1.08116
$814$	−24.3553	−0.853653
$815$	0	0
$816$	0	0
$817$	−23.7839	−0.832092
$818$	15.1637	0.530187
$819$	−1.47017	−0.0513721
$820$	0	0
$821$	−0.0467805	−0.00163265	−0.000816326	−	1.00000i	$-0.500260\pi$
$821$	−0.0467805	−0.00163265	−0.000816326		1.00000i	$0.500260\pi$
$822$	47.5245	1.65761
$823$	32.5408	1.13430	0.567151	−	0.823614i	$-0.308046\pi$
$823$	32.5408	1.13430	0.567151	+	0.823614i	$0.308046\pi$
$824$	20.4288	0.711673
$825$	0	0
$826$	−5.53339	−0.192531
$827$	−12.3947	−0.431004	−0.215502	−	0.976503i	$-0.569139\pi$
$827$	−12.3947	−0.431004	−0.215502	+	0.976503i	$0.569139\pi$
$828$	14.1719	0.492508
$829$	45.3870	1.57636	0.788178	−	0.615447i	$-0.211025\pi$
$829$	45.3870	1.57636	0.788178	+	0.615447i	$0.211025\pi$
$830$	0	0
$831$	4.65459	0.161466
$832$	−0.0119634	−0.000414758 0
$833$	0	0
$834$	−3.31850	−0.114910
$835$	0	0
$836$	7.74860	0.267991
$837$	16.8970	0.584047
$838$	3.55347	0.122753
$839$	−16.9408	−0.584860	−0.292430	−	0.956287i	$-0.594464\pi$
$839$	−16.9408	−0.584860	−0.292430	+	0.956287i	$0.594464\pi$
$840$	0	0
$841$	42.1178	1.45234
$842$	50.5560	1.74227
$843$	12.8119	0.441267
$844$	12.9779	0.446716
$845$	0	0
$846$	30.9519	1.06415
$847$	38.2354	1.31378
$848$	−37.4855	−1.28726
$849$	46.5417	1.59731
$850$	0	0
$851$	64.4521	2.20939
$852$	32.2679	1.10548
$853$	−35.8467	−1.22737	−0.613684	−	0.789552i	$-0.710313\pi$
$853$	−35.8467	−1.22737	−0.613684	+	0.789552i	$0.710313\pi$
$854$	44.1357	1.51029
$855$	0	0
$856$	21.3504	0.729741
$857$	28.1102	0.960226	0.480113	−	0.877207i	$-0.340596\pi$
$857$	28.1102	0.960226	0.480113	+	0.877207i	$0.340596\pi$
$858$	−0.755129	−0.0257797
$859$	2.58598	0.0882327	0.0441163	−	0.999026i	$-0.485953\pi$
$859$	2.58598	0.0882327	0.0441163	+	0.999026i	$0.485953\pi$
$860$	0	0
$861$	−23.1488	−0.788909
$862$	−16.8041	−0.572350
$863$	−6.65163	−0.226424	−0.113212	−	0.993571i	$-0.536114\pi$
$863$	−6.65163	−0.226424	−0.113212	+	0.993571i	$0.536114\pi$
$864$	−8.79540	−0.299226
$865$	0	0
$866$	3.35772	0.114100
$867$	0	0
$868$	48.4492	1.64447
$869$	10.4368	0.354043
$870$	0	0
$871$	−1.03972	−0.0352295
$872$	−8.60015	−0.291238
$873$	6.81044	0.230498
$874$	−57.1843	−1.93429
$875$	0	0
$876$	13.3313	0.450422
$877$	−14.3578	−0.484828	−0.242414	−	0.970173i	$-0.577939\pi$
$877$	−14.3578	−0.484828	−0.242414	+	0.970173i	$0.577939\pi$
$878$	−22.5338	−0.760480
$879$	1.94021	0.0654417
$880$	0	0
$881$	58.0232	1.95485	0.977425	−	0.211282i	$-0.0677639\pi$
$881$	58.0232	1.95485	0.977425	+	0.211282i	$0.0677639\pi$
$882$	−35.8613	−1.20751
$883$	−20.2776	−0.682394	−0.341197	−	0.939992i	$-0.610832\pi$
$883$	−20.2776	−0.682394	−0.341197	+	0.939992i	$0.610832\pi$
$884$	0	0
$885$	0	0
$886$	−5.35312	−0.179841
$887$	9.19529	0.308748	0.154374	−	0.988013i	$-0.450664\pi$
$887$	9.19529	0.308748	0.154374	+	0.988013i	$0.450664\pi$
$888$	42.6069	1.42979
$889$	−78.0581	−2.61799
$890$	0	0
$891$	−12.2912	−0.411770
$892$	25.2880	0.846706
$893$	−44.7843	−1.49865
$894$	21.4635	0.717847
$895$	0	0
$896$	44.5848	1.48947
$897$	1.99832	0.0667219
$898$	−7.29027	−0.243280
$899$	92.1823	3.07445
$900$	0	0
$901$	0	0
$902$	−5.19827	−0.173083
$903$	36.5584	1.21659
$904$	18.0685	0.600949
$905$	0	0
$906$	−26.8678	−0.892623
$907$	−21.8148	−0.724348	−0.362174	−	0.932110i	$-0.617965\pi$
$907$	−21.8148	−0.724348	−0.362174	+	0.932110i	$0.617965\pi$
$908$	−20.4778	−0.679581
$909$	−5.13588	−0.170346
$910$	0	0
$911$	24.6136	0.815484	0.407742	−	0.913097i	$-0.366316\pi$
$911$	24.6136	0.815484	0.407742	+	0.913097i	$0.366316\pi$
$912$	−68.5435	−2.26970
$913$	−10.0718	−0.333328
$914$	−6.54571	−0.216513
$915$	0	0
$916$	16.2100	0.535594
$917$	47.1467	1.55692
$918$	0	0
$919$	50.1165	1.65319	0.826596	−	0.562796i	$-0.190274\pi$
$919$	50.1165	1.65319	0.826596	+	0.562796i	$0.190274\pi$
$920$	0	0
$921$	−10.1580	−0.334717
$922$	−51.5592	−1.69801
$923$	1.98923	0.0654764
$924$	−11.9104	−0.391825
$925$	0	0
$926$	38.2224	1.25606
$927$	30.5719	1.00411
$928$	−47.9836	−1.57514
$929$	49.9234	1.63793	0.818966	−	0.573842i	$-0.194548\pi$
$929$	49.9234	1.63793	0.818966	+	0.573842i	$0.194548\pi$
$930$	0	0
$931$	51.8876	1.70055
$932$	15.3126	0.501580
$933$	66.6175	2.18096
$934$	1.13515	0.0371433
$935$	0	0
$936$	0.577545	0.0188776
$937$	7.15643	0.233790	0.116895	−	0.993144i	$-0.462706\pi$
$937$	7.15643	0.233790	0.116895	+	0.993144i	$0.462706\pi$
$938$	−45.7333	−1.49325
$939$	22.2665	0.726639
$940$	0	0
$941$	−28.5619	−0.931092	−0.465546	−	0.885024i	$-0.654142\pi$
$941$	−28.5619	−0.931092	−0.465546	+	0.885024i	$0.654142\pi$
$942$	5.76909	0.187967
$943$	13.7563	0.447967
$944$	3.94144	0.128283
$945$	0	0
$946$	8.20951	0.266914
$947$	25.9542	0.843397	0.421699	−	0.906736i	$-0.361434\pi$
$947$	25.9542	0.843397	0.421699	+	0.906736i	$0.361434\pi$
$948$	23.1479	0.751809
$949$	0.821837	0.0266780
$950$	0	0
$951$	43.4529	1.40906
$952$	0	0
$953$	−20.8716	−0.676096	−0.338048	−	0.941129i	$-0.609767\pi$
$953$	−20.8716	−0.676096	−0.338048	+	0.941129i	$0.609767\pi$
$954$	−30.9382	−1.00166
$955$	0	0
$956$	28.7373	0.929430
$957$	−22.6615	−0.732542
$958$	14.1559	0.457356
$959$	46.2103	1.49221
$960$	0	0
$961$	88.4860	2.85439
$962$	−3.33007	−0.107366
$963$	31.9510	1.02960
$964$	0.854885	0.0275340
$965$	0	0
$966$	87.8985	2.82809
$967$	−2.22275	−0.0714790	−0.0357395	−	0.999361i	$-0.511379\pi$
$967$	−2.22275	−0.0714790	−0.0357395	+	0.999361i	$0.511379\pi$
$968$	−15.0204	−0.482775
$969$	0	0
$970$	0	0
$971$	−29.2447	−0.938507	−0.469254	−	0.883063i	$-0.655477\pi$
$971$	−29.2447	−0.938507	−0.469254	+	0.883063i	$0.655477\pi$
$972$	−22.0758	−0.708082
$973$	−3.22673	−0.103444
$974$	57.7216	1.84952
$975$	0	0
$976$	−31.4379	−1.00630
$977$	−4.50160	−0.144019	−0.0720095	−	0.997404i	$-0.522941\pi$
$977$	−4.50160	−0.144019	−0.0720095	+	0.997404i	$0.522941\pi$
$978$	−4.09145	−0.130830
$979$	−2.59543	−0.0829503
$980$	0	0
$981$	−12.8702	−0.410913
$982$	49.1319	1.56786
$983$	49.6767	1.58444	0.792220	−	0.610236i	$-0.208925\pi$
$983$	49.6767	1.58444	0.792220	+	0.610236i	$0.208925\pi$
$984$	9.09379	0.289899
$985$	0	0
$986$	0	0
$987$	68.8384	2.19115
$988$	1.05946	0.0337058
$989$	−21.7250	−0.690816
$990$	0	0
$991$	−9.39571	−0.298464	−0.149232	−	0.988802i	$-0.547680\pi$
$991$	−9.39571	−0.298464	−0.149232	+	0.988802i	$0.547680\pi$
$992$	−62.1960	−1.97472
$993$	31.9468	1.01380
$994$	87.4989	2.77530
$995$	0	0
$996$	−22.3384	−0.707820
$997$	−38.5343	−1.22039	−0.610196	−	0.792250i	$-0.708909\pi$
$997$	−38.5343	−1.22039	−0.610196	+	0.792250i	$0.708909\pi$
$998$	55.4296	1.75459
$999$	−18.3182	−0.579564

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7225.2.a.bw.1.4	yes	15
5.4	even	2		7225.2.a.bu.1.12	yes	15
17.16	even	2		7225.2.a.bt.1.4	✓	15
85.84	even	2		7225.2.a.bv.1.12	yes	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7225.2.a.bt.1.4	✓	15	17.16	even	2
7225.2.a.bu.1.12	yes	15	5.4	even	2
7225.2.a.bv.1.12	yes	15	85.84	even	2
7225.2.a.bw.1.4	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)

\(f(q)\)	\(=\)	\(q-1.76581 q^{2} +2.30878 q^{3} +1.11810 q^{4} -4.07688 q^{6} -3.96414 q^{7} +1.55728 q^{8} +2.33047 q^{9} +O(q^{10})\)	\(q-1.76581 q^{2} +2.30878 q^{3} +1.11810 q^{4} -4.07688 q^{6} -3.96414 q^{7} +1.55728 q^{8} +2.33047 q^{9} +1.16390 q^{11} +2.58144 q^{12} +0.159139 q^{13} +6.99993 q^{14} -4.98605 q^{16} -4.11518 q^{18} +5.95425 q^{19} -9.15233 q^{21} -2.05524 q^{22} +5.43883 q^{23} +3.59541 q^{24} -0.281010 q^{26} -1.54580 q^{27} -4.43229 q^{28} -8.43313 q^{29} -10.9310 q^{31} +5.68989 q^{32} +2.68720 q^{33} +2.60569 q^{36} +11.8504 q^{37} -10.5141 q^{38} +0.367417 q^{39} +2.52928 q^{41} +16.1613 q^{42} -3.99444 q^{43} +1.30136 q^{44} -9.60395 q^{46} -7.52141 q^{47} -11.5117 q^{48} +8.71439 q^{49} +0.177933 q^{52} +7.51807 q^{53} +2.72959 q^{54} -6.17326 q^{56} +13.7471 q^{57} +14.8913 q^{58} -0.790493 q^{59} +6.30517 q^{61} +19.3020 q^{62} -9.23831 q^{63} -0.0751761 q^{64} -4.74509 q^{66} -6.53340 q^{67} +12.5571 q^{69} +12.5000 q^{71} +3.62919 q^{72} +5.16427 q^{73} -20.9255 q^{74} +6.65743 q^{76} -4.61387 q^{77} -0.648790 q^{78} +8.96704 q^{79} -10.5603 q^{81} -4.46623 q^{82} -8.65347 q^{83} -10.2332 q^{84} +7.05343 q^{86} -19.4703 q^{87} +1.81252 q^{88} -2.22994 q^{89} -0.630849 q^{91} +6.08114 q^{92} -25.2372 q^{93} +13.2814 q^{94} +13.1367 q^{96} +2.92234 q^{97} -15.3880 q^{98} +2.71244 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + 9 q^{3} + 12 q^{4} + 9 q^{6} + 12 q^{7} + 6 q^{8} + 12 q^{9}+O(q^{10}) \) 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

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