LMFDB - Embedded newform 7225.2.a.bq.1.10

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:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4 x^{11} - 10 x^{10} + 52 x^{9} + 21 x^{8} - 232 x^{7} + 44 x^{6} + 424 x^{5} - 137 x^{4} + \cdots + 17 $ x^12 - 4x^11 - 10x^10 + 52x^9 + 21x^8 - 232x^7 + 44x^6 + 424x^5 - 137x^4 - 316x^3 + 46x^2 + 92*x + 17
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 85)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$1.80583$ 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)

$f(q)$	$=$	\(q+1.80583 q^{2} +0.687917 q^{3} +1.26102 q^{4} +1.24226 q^{6} -4.34193 q^{7} -1.33447 q^{8} -2.52677 q^{9} +0.0525004 q^{11} +0.867478 q^{12} -3.02508 q^{13} -7.84078 q^{14} -4.93187 q^{16} -4.56292 q^{18} +7.82043 q^{19} -2.98689 q^{21} +0.0948069 q^{22} +1.04197 q^{23} -0.918004 q^{24} -5.46278 q^{26} -3.80196 q^{27} -5.47526 q^{28} +0.420754 q^{29} -1.38429 q^{31} -6.23717 q^{32} +0.0361159 q^{33} -3.18631 q^{36} +0.336949 q^{37} +14.1224 q^{38} -2.08100 q^{39} +6.59268 q^{41} -5.39381 q^{42} +9.99466 q^{43} +0.0662042 q^{44} +1.88162 q^{46} +6.13168 q^{47} -3.39272 q^{48} +11.8523 q^{49} -3.81469 q^{52} +12.0629 q^{53} -6.86569 q^{54} +5.79417 q^{56} +5.37981 q^{57} +0.759811 q^{58} -5.09779 q^{59} +5.97063 q^{61} -2.49979 q^{62} +10.9711 q^{63} -1.39954 q^{64} +0.0652192 q^{66} +0.916040 q^{67} +0.716788 q^{69} -4.17986 q^{71} +3.37190 q^{72} -5.39059 q^{73} +0.608473 q^{74} +9.86173 q^{76} -0.227953 q^{77} -3.75794 q^{78} +9.98296 q^{79} +4.96488 q^{81} +11.9053 q^{82} -6.53008 q^{83} -3.76653 q^{84} +18.0487 q^{86} +0.289444 q^{87} -0.0700602 q^{88} +10.2159 q^{89} +13.1347 q^{91} +1.31395 q^{92} -0.952278 q^{93} +11.0728 q^{94} -4.29066 q^{96} -19.2238 q^{97} +21.4033 q^{98} -0.132657 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 4 q^{2} - 8 q^{3} + 12 q^{4} + 8 q^{6} - 16 q^{7} + 12 q^{8} + 12 q^{9} + 16 q^{11} - 16 q^{12} + 8 q^{13} - 16 q^{14} + 12 q^{16} - 4 q^{18} + 16 q^{21} - 16 q^{22} - 16 q^{23} + 16 q^{26} - 32 q^{27}+ \cdots + 56 q^{99}+O(q^{100}) $ 12 * q + 4 * q^2 - 8 * q^3 + 12 * q^4 + 8 * q^6 - 16 * q^7 + 12 * q^8 + 12 * q^9 + 16 * q^11 - 16 * q^12 + 8 * q^13 - 16 * q^14 + 12 * q^16 - 4 * q^18 + 16 * q^21 - 16 * q^22 - 16 * q^23 + 16 * q^26 - 32 * q^27 - 40 * q^28 + 16 * q^29 + 24 * q^31 + 28 * q^32 + 12 * q^36 - 24 * q^37 + 24 * q^38 + 8 * q^39 + 8 * q^41 + 16 * q^43 + 8 * q^44 + 40 * q^46 + 32 * q^47 + 24 * q^48 + 20 * q^49 + 24 * q^52 + 8 * q^54 - 24 * q^56 - 32 * q^57 + 16 * q^58 + 8 * q^59 + 24 * q^61 - 8 * q^62 - 48 * q^63 + 36 * q^64 + 40 * q^66 + 8 * q^67 + 48 * q^69 - 16 * q^71 - 12 * q^72 - 16 * q^73 + 16 * q^76 - 24 * q^77 + 24 * q^78 + 40 * q^79 + 36 * q^81 + 16 * q^82 + 40 * q^83 + 32 * q^84 - 8 * q^86 + 32 * q^87 - 48 * q^88 + 8 * q^89 + 72 * q^91 + 8 * q^92 - 24 * q^93 - 16 * q^94 + 8 * q^96 - 32 * q^97 + 60 * q^98 + 56 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.80583	1.27691	0.638457	−	0.769657i	$-0.279573\pi$
$2$	1.80583	1.27691	0.638457	+	0.769657i	$0.279573\pi$
$3$	0.687917	0.397169	0.198585	−	0.980084i	$-0.436366\pi$
$3$	0.687917	0.397169	0.198585	+	0.980084i	$0.436366\pi$
$4$	1.26102	0.630511
$5$	0	0
$5$	0	0
$6$	1.24226	0.507151
$7$	−4.34193	−1.64109	−0.820547	−	0.571579i	$-0.806331\pi$
$7$	−4.34193	−1.64109	−0.820547	+	0.571579i	$0.806331\pi$
$8$	−1.33447	−0.471806
$9$	−2.52677	−0.842257
$10$	0	0
$11$	0.0525004	0.0158295	0.00791474	−	0.999969i	$-0.497481\pi$
$11$	0.0525004	0.0158295	0.00791474	+	0.999969i	$0.497481\pi$
$12$	0.867478	0.250419
$13$	−3.02508	−0.839006	−0.419503	−	0.907754i	$-0.637796\pi$
$13$	−3.02508	−0.839006	−0.419503	+	0.907754i	$0.637796\pi$
$14$	−7.84078	−2.09554
$15$	0	0
$16$	−4.93187	−1.23297
$17$	0	0
$17$	0	0
$18$	−4.56292	−1.07549
$19$	7.82043	1.79413	0.897065	−	0.441898i	$-0.145695\pi$
$19$	7.82043	1.79413	0.897065	+	0.441898i	$0.145695\pi$
$20$	0	0
$21$	−2.98689	−0.651792
$22$	0.0948069	0.0202129
$23$	1.04197	0.217266	0.108633	−	0.994082i	$-0.465353\pi$
$23$	1.04197	0.217266	0.108633	+	0.994082i	$0.465353\pi$
$24$	−0.918004	−0.187387
$25$	0	0
$26$	−5.46278	−1.07134
$27$	−3.80196	−0.731687
$28$	−5.47526	−1.03473
$29$	0.420754	0.0781321	0.0390661	−	0.999237i	$-0.487562\pi$
$29$	0.420754	0.0781321	0.0390661	+	0.999237i	$0.487562\pi$
$30$	0	0
$31$	−1.38429	−0.248626	−0.124313	−	0.992243i	$-0.539673\pi$
$31$	−1.38429	−0.248626	−0.124313	+	0.992243i	$0.539673\pi$
$32$	−6.23717	−1.10259
$33$	0.0361159	0.00628698
$34$	0	0
$35$	0	0
$36$	−3.18631	−0.531052
$37$	0.336949	0.0553941	0.0276971	−	0.999616i	$-0.491183\pi$
$37$	0.336949	0.0553941	0.0276971	+	0.999616i	$0.491183\pi$
$38$	14.1224	2.29095
$39$	−2.08100	−0.333227
$40$	0	0
$41$	6.59268	1.02960	0.514802	−	0.857309i	$-0.327865\pi$
$41$	6.59268	1.02960	0.514802	+	0.857309i	$0.327865\pi$
$42$	−5.39381	−0.832283
$43$	9.99466	1.52417	0.762086	−	0.647476i	$-0.224175\pi$
$43$	9.99466	1.52417	0.762086	+	0.647476i	$0.224175\pi$
$44$	0.0662042	0.00998065
$45$	0	0
$46$	1.88162	0.277430
$47$	6.13168	0.894398	0.447199	−	0.894435i	$-0.352422\pi$
$47$	6.13168	0.894398	0.447199	+	0.894435i	$0.352422\pi$
$48$	−3.39272	−0.489696
$49$	11.8523	1.69319
$50$	0	0
$51$	0	0
$52$	−3.81469	−0.529002
$53$	12.0629	1.65696	0.828482	−	0.560016i	$-0.189205\pi$
$53$	12.0629	1.65696	0.828482	+	0.560016i	$0.189205\pi$
$54$	−6.86569	−0.934302
$55$	0	0
$56$	5.79417	0.774279
$57$	5.37981	0.712573
$58$	0.759811	0.0997681
$59$	−5.09779	−0.663676	−0.331838	−	0.943336i	$-0.607669\pi$
$59$	−5.09779	−0.663676	−0.331838	+	0.943336i	$0.607669\pi$
$60$	0	0
$61$	5.97063	0.764461	0.382230	−	0.924067i	$-0.375156\pi$
$61$	5.97063	0.764461	0.382230	+	0.924067i	$0.375156\pi$
$62$	−2.49979	−0.317474
$63$	10.9711	1.38222
$64$	−1.39954	−0.174943
$65$	0	0
$66$	0.0652192	0.00802793
$67$	0.916040	0.111912	0.0559561	−	0.998433i	$-0.482179\pi$
$67$	0.916040	0.111912	0.0559561	+	0.998433i	$0.482179\pi$
$68$	0	0
$69$	0.716788	0.0862912
$70$	0	0
$71$	−4.17986	−0.496058	−0.248029	−	0.968753i	$-0.579783\pi$
$71$	−4.17986	−0.496058	−0.248029	+	0.968753i	$0.579783\pi$
$72$	3.37190	0.397382
$73$	−5.39059	−0.630920	−0.315460	−	0.948939i	$-0.602159\pi$
$73$	−5.39059	−0.630920	−0.315460	+	0.948939i	$0.602159\pi$
$74$	0.608473	0.0707336
$75$	0	0
$76$	9.86173	1.13122
$77$	−0.227953	−0.0259777
$78$	−3.75794	−0.425503
$79$	9.98296	1.12317	0.561585	−	0.827419i	$-0.310192\pi$
$79$	9.98296	1.12317	0.561585	+	0.827419i	$0.310192\pi$
$80$	0	0
$81$	4.96488	0.551653
$82$	11.9053	1.31472
$83$	−6.53008	−0.716770	−0.358385	−	0.933574i	$-0.616672\pi$
$83$	−6.53008	−0.716770	−0.358385	+	0.933574i	$0.616672\pi$
$84$	−3.76653	−0.410962
$85$	0	0
$86$	18.0487	1.94624
$87$	0.289444	0.0310317
$88$	−0.0700602	−0.00746845
$89$	10.2159	1.08289	0.541443	−	0.840738i	$-0.317878\pi$
$89$	10.2159	1.08289	0.541443	+	0.840738i	$0.317878\pi$
$90$	0	0
$91$	13.1347	1.37689
$92$	1.31395	0.136988
$93$	−0.952278	−0.0987466
$94$	11.0728	1.14207
$95$	0	0
$96$	−4.29066	−0.437914
$97$	−19.2238	−1.95188	−0.975940	−	0.218037i	$-0.930035\pi$
$97$	−19.2238	−1.95188	−0.975940	+	0.218037i	$0.930035\pi$
$98$	21.4033	2.16206
$99$	−0.132657	−0.0133325
$100$	0	0
$101$	13.2926	1.32266	0.661331	−	0.750094i	$-0.269992\pi$
$101$	13.2926	1.32266	0.661331	+	0.750094i	$0.269992\pi$
$102$	0	0
$103$	6.91299	0.681157	0.340579	−	0.940216i	$-0.389377\pi$
$103$	6.91299	0.681157	0.340579	+	0.940216i	$0.389377\pi$
$104$	4.03687	0.395848
$105$	0	0
$106$	21.7835	2.11580
$107$	−14.7407	−1.42504	−0.712520	−	0.701651i	$-0.752446\pi$
$107$	−14.7407	−1.42504	−0.712520	+	0.701651i	$0.752446\pi$
$108$	−4.79435	−0.461337
$109$	−4.74828	−0.454803	−0.227402	−	0.973801i	$-0.573023\pi$
$109$	−4.74828	−0.454803	−0.227402	+	0.973801i	$0.573023\pi$
$110$	0	0
$111$	0.231793	0.0220008
$112$	21.4138	2.02342
$113$	−3.01331	−0.283468	−0.141734	−	0.989905i	$-0.545268\pi$
$113$	−3.01331	−0.283468	−0.141734	+	0.989905i	$0.545268\pi$
$114$	9.71502	0.909895
$115$	0	0
$116$	0.530580	0.0492631
$117$	7.64368	0.706658
$118$	−9.20574	−0.847457
$119$	0	0
$120$	0	0
$121$	−10.9972	−0.999749
$122$	10.7819	0.976151
$123$	4.53522	0.408927
$124$	−1.74562	−0.156761
$125$	0	0
$126$	19.8119	1.76498
$127$	0.00828706	0.000735358 0	0.000367679	−	1.00000i	$-0.499883\pi$
$127$	0.00828706	0.000735358 0	0.000367679		1.00000i	$0.499883\pi$
$128$	9.94702	0.879200
$129$	6.87550	0.605354
$130$	0	0
$131$	−8.18458	−0.715090	−0.357545	−	0.933896i	$-0.616386\pi$
$131$	−8.18458	−0.715090	−0.357545	+	0.933896i	$0.616386\pi$
$132$	0.0455430	0.00396401
$133$	−33.9558	−2.94434
$134$	1.65421	0.142902
$135$	0	0
$136$	0	0
$137$	2.23387	0.190852	0.0954260	−	0.995437i	$-0.469579\pi$
$137$	2.23387	0.190852	0.0954260	+	0.995437i	$0.469579\pi$
$138$	1.29440	0.110186
$139$	19.7358	1.67397	0.836985	−	0.547225i	$-0.184316\pi$
$139$	19.7358	1.67397	0.836985	+	0.547225i	$0.184316\pi$
$140$	0	0
$141$	4.21809	0.355227
$142$	−7.54812	−0.633424
$143$	−0.158818	−0.0132810
$144$	12.4617	1.03847
$145$	0	0
$146$	−9.73448	−0.805632
$147$	8.15343	0.672483
$148$	0.424900	0.0349266
$149$	10.1835	0.834263	0.417132	−	0.908846i	$-0.363035\pi$
$149$	10.1835	0.834263	0.417132	+	0.908846i	$0.363035\pi$
$150$	0	0
$151$	−15.3484	−1.24903	−0.624516	−	0.781012i	$-0.714704\pi$
$151$	−15.3484	−1.24903	−0.624516	+	0.781012i	$0.714704\pi$
$152$	−10.4361	−0.846482
$153$	0	0
$154$	−0.411645	−0.0331713
$155$	0	0
$156$	−2.62419	−0.210103
$157$	−9.41222	−0.751177	−0.375588	−	0.926787i	$-0.622559\pi$
$157$	−9.41222	−0.751177	−0.375588	+	0.926787i	$0.622559\pi$
$158$	18.0275	1.43419
$159$	8.29826	0.658095
$160$	0	0
$161$	−4.52415	−0.356553
$162$	8.96572	0.704414
$163$	−12.4453	−0.974793	−0.487397	−	0.873181i	$-0.662053\pi$
$163$	−12.4453	−0.974793	−0.487397	+	0.873181i	$0.662053\pi$
$164$	8.31351	0.649176
$165$	0	0
$166$	−11.7922	−0.915253
$167$	−12.6025	−0.975211	−0.487605	−	0.873064i	$-0.662130\pi$
$167$	−12.6025	−0.975211	−0.487605	+	0.873064i	$0.662130\pi$
$168$	3.98591	0.307520
$169$	−3.84890	−0.296069
$170$	0	0
$171$	−19.7604	−1.51112
$172$	12.6035	0.961007
$173$	12.1965	0.927284	0.463642	−	0.886023i	$-0.346542\pi$
$173$	12.1965	0.927284	0.463642	+	0.886023i	$0.346542\pi$
$174$	0.522687	0.0396248
$175$	0	0
$176$	−0.258925	−0.0195172
$177$	−3.50686	−0.263592
$178$	18.4482	1.38275
$179$	13.8608	1.03600	0.518002	−	0.855379i	$-0.326676\pi$
$179$	13.8608	1.03600	0.518002	+	0.855379i	$0.326676\pi$
$180$	0	0
$181$	2.04424	0.151947	0.0759736	−	0.997110i	$-0.475794\pi$
$181$	2.04424	0.151947	0.0759736	+	0.997110i	$0.475794\pi$
$182$	23.7190	1.75817
$183$	4.10730	0.303620
$184$	−1.39048	−0.102507
$185$	0	0
$186$	−1.71965	−0.126091
$187$	0	0
$188$	7.73218	0.563927
$189$	16.5078	1.20077
$190$	0	0
$191$	−3.08056	−0.222902	−0.111451	−	0.993770i	$-0.535550\pi$
$191$	−3.08056	−0.222902	−0.111451	+	0.993770i	$0.535550\pi$
$192$	−0.962768	−0.0694818
$193$	20.6419	1.48584	0.742918	−	0.669382i	$-0.233441\pi$
$193$	20.6419	1.48584	0.742918	+	0.669382i	$0.233441\pi$
$194$	−34.7149	−2.49239
$195$	0	0
$196$	14.9461	1.06758
$197$	−18.2365	−1.29930	−0.649649	−	0.760235i	$-0.725084\pi$
$197$	−18.2365	−1.29930	−0.649649	+	0.760235i	$0.725084\pi$
$198$	−0.239555	−0.0170244
$199$	26.6677	1.89042	0.945210	−	0.326463i	$-0.105857\pi$
$199$	26.6677	1.89042	0.945210	+	0.326463i	$0.105857\pi$
$200$	0	0
$201$	0.630160	0.0444480
$202$	24.0041	1.68893
$203$	−1.82689	−0.128222
$204$	0	0
$205$	0	0
$206$	12.4837	0.869779
$207$	−2.63282	−0.182993
$208$	14.9193	1.03447
$209$	0.410576	0.0284001
$210$	0	0
$211$	7.80566	0.537364	0.268682	−	0.963229i	$-0.413412\pi$
$211$	7.80566	0.537364	0.268682	+	0.963229i	$0.413412\pi$
$212$	15.2115	1.04473
$213$	−2.87540	−0.197019
$214$	−26.6193	−1.81966
$215$	0	0
$216$	5.07360	0.345215
$217$	6.01049	0.408019
$218$	−8.57459	−0.580745
$219$	−3.70828	−0.250582
$220$	0	0
$221$	0	0
$222$	0.418579	0.0280932
$223$	−4.76891	−0.319350	−0.159675	−	0.987170i	$-0.551045\pi$
$223$	−4.76891	−0.319350	−0.159675	+	0.987170i	$0.551045\pi$
$224$	27.0814	1.80945
$225$	0	0
$226$	−5.44152	−0.361964
$227$	−11.2111	−0.744105	−0.372053	−	0.928212i	$-0.621346\pi$
$227$	−11.2111	−0.744105	−0.372053	+	0.928212i	$0.621346\pi$
$228$	6.78405	0.449285
$229$	20.6484	1.36449	0.682244	−	0.731125i	$-0.261004\pi$
$229$	20.6484	1.36449	0.682244	+	0.731125i	$0.261004\pi$
$230$	0	0
$231$	−0.156813	−0.0103175
$232$	−0.561484	−0.0368632
$233$	−3.05851	−0.200370	−0.100185	−	0.994969i	$-0.531943\pi$
$233$	−3.05851	−0.200370	−0.100185	+	0.994969i	$0.531943\pi$
$234$	13.8032	0.902342
$235$	0	0
$236$	−6.42842	−0.418455
$237$	6.86745	0.446089
$238$	0	0
$239$	4.94072	0.319588	0.159794	−	0.987150i	$-0.448917\pi$
$239$	4.94072	0.319588	0.159794	+	0.987150i	$0.448917\pi$
$240$	0	0
$241$	−1.66176	−0.107043	−0.0535217	−	0.998567i	$-0.517045\pi$
$241$	−1.66176	−0.107043	−0.0535217	+	0.998567i	$0.517045\pi$
$242$	−19.8592	−1.27659
$243$	14.8213	0.950787
$244$	7.52909	0.482001
$245$	0	0
$246$	8.18983	0.522165
$247$	−23.6574	−1.50529
$248$	1.84729	0.117303
$249$	−4.49215	−0.284679
$250$	0	0
$251$	9.14240	0.577063	0.288531	−	0.957470i	$-0.406833\pi$
$251$	9.14240	0.577063	0.288531	+	0.957470i	$0.406833\pi$
$252$	13.8347	0.871506
$253$	0.0547038	0.00343920
$254$	0.0149650	0.000938989 0
$255$	0	0
$256$	20.7617	1.29761
$257$	16.6522	1.03874	0.519369	−	0.854550i	$-0.326167\pi$
$257$	16.6522	1.03874	0.519369	+	0.854550i	$0.326167\pi$
$258$	12.4160	0.772985
$259$	−1.46301	−0.0909070
$260$	0	0
$261$	−1.06315	−0.0658073
$262$	−14.7800	−0.913109
$263$	21.7280	1.33981	0.669903	−	0.742448i	$-0.266335\pi$
$263$	21.7280	1.33981	0.669903	+	0.742448i	$0.266335\pi$
$264$	−0.0481956	−0.00296624
$265$	0	0
$266$	−61.3183	−3.75967
$267$	7.02771	0.430089
$268$	1.15515	0.0705618
$269$	−23.1721	−1.41283	−0.706414	−	0.707799i	$-0.749688\pi$
$269$	−23.1721	−1.41283	−0.706414	+	0.707799i	$0.749688\pi$
$270$	0	0
$271$	3.95595	0.240307	0.120153	−	0.992755i	$-0.461661\pi$
$271$	3.95595	0.240307	0.120153	+	0.992755i	$0.461661\pi$
$272$	0	0
$273$	9.03556	0.546857
$274$	4.03398	0.243702
$275$	0	0
$276$	0.903885	0.0544075
$277$	16.9099	1.01602	0.508009	−	0.861352i	$-0.330382\pi$
$277$	16.9099	1.01602	0.508009	+	0.861352i	$0.330382\pi$
$278$	35.6395	2.13752
$279$	3.49779	0.209407
$280$	0	0
$281$	4.66987	0.278581	0.139291	−	0.990252i	$-0.455518\pi$
$281$	4.66987	0.278581	0.139291	+	0.990252i	$0.455518\pi$
$282$	7.61715	0.453595
$283$	8.70036	0.517183	0.258591	−	0.965987i	$-0.416742\pi$
$283$	8.70036	0.517183	0.258591	+	0.965987i	$0.416742\pi$
$284$	−5.27089	−0.312770
$285$	0	0
$286$	−0.286798	−0.0169587
$287$	−28.6250	−1.68968
$288$	15.7599	0.928661
$289$	0	0
$290$	0	0
$291$	−13.2244	−0.775227
$292$	−6.79765	−0.397802
$293$	−0.739100	−0.0431787	−0.0215893	−	0.999767i	$-0.506873\pi$
$293$	−0.739100	−0.0431787	−0.0215893	+	0.999767i	$0.506873\pi$
$294$	14.7237	0.858704
$295$	0	0
$296$	−0.449649	−0.0261353
$297$	−0.199605	−0.0115822
$298$	18.3896	1.06528
$299$	−3.15204	−0.182287
$300$	0	0
$301$	−43.3961	−2.50131
$302$	−27.7165	−1.59491
$303$	9.14419	0.525320
$304$	−38.5693	−2.21210
$305$	0	0
$306$	0	0
$307$	2.86108	0.163290	0.0816451	−	0.996661i	$-0.473983\pi$
$307$	2.86108	0.163290	0.0816451	+	0.996661i	$0.473983\pi$
$308$	−0.287454	−0.0163792
$309$	4.75556	0.270535
$310$	0	0
$311$	20.4980	1.16233	0.581167	−	0.813784i	$-0.302596\pi$
$311$	20.4980	1.16233	0.581167	+	0.813784i	$0.302596\pi$
$312$	2.77703	0.157219
$313$	−8.46543	−0.478494	−0.239247	−	0.970959i	$-0.576901\pi$
$313$	−8.46543	−0.478494	−0.239247	+	0.970959i	$0.576901\pi$
$314$	−16.9969	−0.959189
$315$	0	0
$316$	12.5887	0.708171
$317$	21.0918	1.18463	0.592317	−	0.805705i	$-0.298213\pi$
$317$	21.0918	1.18463	0.592317	+	0.805705i	$0.298213\pi$
$318$	14.9852	0.840331
$319$	0.0220898	0.00123679
$320$	0	0
$321$	−10.1404	−0.565982
$322$	−8.16985	−0.455288
$323$	0	0
$324$	6.26082	0.347823
$325$	0	0
$326$	−22.4741	−1.24473
$327$	−3.26643	−0.180634
$328$	−8.79773	−0.485774
$329$	−26.6233	−1.46779
$330$	0	0
$331$	−15.2261	−0.836903	−0.418451	−	0.908239i	$-0.637427\pi$
$331$	−15.2261	−0.836903	−0.418451	+	0.908239i	$0.637427\pi$
$332$	−8.23457	−0.451931
$333$	−0.851393	−0.0466561
$334$	−22.7580	−1.24526
$335$	0	0
$336$	14.7309	0.803638
$337$	−11.4737	−0.625011	−0.312506	−	0.949916i	$-0.601168\pi$
$337$	−11.4737	−0.625011	−0.312506	+	0.949916i	$0.601168\pi$
$338$	−6.95046	−0.378055
$339$	−2.07290	−0.112585
$340$	0	0
$341$	−0.0726759	−0.00393562
$342$	−35.6840	−1.92957
$343$	−21.0685	−1.13759
$344$	−13.3376	−0.719114
$345$	0	0
$346$	22.0248	1.18406
$347$	14.9843	0.804399	0.402199	−	0.915552i	$-0.368246\pi$
$347$	14.9843	0.804399	0.402199	+	0.915552i	$0.368246\pi$
$348$	0.364995	0.0195658
$349$	−4.53667	−0.242842	−0.121421	−	0.992601i	$-0.538745\pi$
$349$	−4.53667	−0.242842	−0.121421	+	0.992601i	$0.538745\pi$
$350$	0	0
$351$	11.5012	0.613890
$352$	−0.327454	−0.0174534
$353$	7.71469	0.410612	0.205306	−	0.978698i	$-0.434181\pi$
$353$	7.71469	0.410612	0.205306	+	0.978698i	$0.434181\pi$
$354$	−6.33279	−0.336584
$355$	0	0
$356$	12.8825	0.682771
$357$	0	0
$358$	25.0302	1.32289
$359$	−11.0284	−0.582057	−0.291029	−	0.956714i	$-0.593997\pi$
$359$	−11.0284	−0.582057	−0.291029	+	0.956714i	$0.593997\pi$
$360$	0	0
$361$	42.1592	2.21890
$362$	3.69155	0.194023
$363$	−7.56519	−0.397070
$364$	16.5631	0.868142
$365$	0	0
$366$	7.41708	0.387697
$367$	−16.0285	−0.836683	−0.418342	−	0.908290i	$-0.637389\pi$
$367$	−16.0285	−0.836683	−0.418342	+	0.908290i	$0.637389\pi$
$368$	−5.13885	−0.267881
$369$	−16.6582	−0.867191
$370$	0	0
$371$	−52.3761	−2.71923
$372$	−1.20084	−0.0622608
$373$	−10.2501	−0.530732	−0.265366	−	0.964148i	$-0.585493\pi$
$373$	−10.2501	−0.530732	−0.265366	+	0.964148i	$0.585493\pi$
$374$	0	0
$375$	0	0
$376$	−8.18254	−0.421982
$377$	−1.27281	−0.0655533
$378$	29.8103	1.53328
$379$	−13.9520	−0.716664	−0.358332	−	0.933594i	$-0.616654\pi$
$379$	−13.9520	−0.716664	−0.358332	+	0.933594i	$0.616654\pi$
$380$	0	0
$381$	0.00570081	0.000292061 0
$382$	−5.56297	−0.284626
$383$	5.62186	0.287264	0.143632	−	0.989631i	$-0.454122\pi$
$383$	5.62186	0.287264	0.143632	+	0.989631i	$0.454122\pi$
$384$	6.84272	0.349191
$385$	0	0
$386$	37.2757	1.89729
$387$	−25.2542	−1.28374
$388$	−24.2416	−1.23068
$389$	35.8157	1.81593	0.907965	−	0.419047i	$-0.137636\pi$
$389$	35.8157	1.81593	0.907965	+	0.419047i	$0.137636\pi$
$390$	0	0
$391$	0	0
$392$	−15.8166	−0.798858
$393$	−5.63032	−0.284012
$394$	−32.9320	−1.65909
$395$	0	0
$396$	−0.167283	−0.00840627
$397$	3.89017	0.195242	0.0976211	−	0.995224i	$-0.468877\pi$
$397$	3.89017	0.195242	0.0976211	+	0.995224i	$0.468877\pi$
$398$	48.1573	2.41390
$399$	−23.3587	−1.16940
$400$	0	0
$401$	−36.7386	−1.83464	−0.917318	−	0.398156i	$-0.869650\pi$
$401$	−36.7386	−1.83464	−0.917318	+	0.398156i	$0.869650\pi$
$402$	1.13796	0.0567564
$403$	4.18759	0.208599
$404$	16.7622	0.833952
$405$	0	0
$406$	−3.29904	−0.163729
$407$	0.0176900	0.000876860 0
$408$	0	0
$409$	−22.3529	−1.10528	−0.552641	−	0.833419i	$-0.686380\pi$
$409$	−22.3529	−1.10528	−0.552641	+	0.833419i	$0.686380\pi$
$410$	0	0
$411$	1.53671	0.0758005
$412$	8.71743	0.429477
$413$	22.1342	1.08915
$414$	−4.75442	−0.233667
$415$	0	0
$416$	18.8679	0.925077
$417$	13.5766	0.664849
$418$	0.741431	0.0362646
$419$	21.8658	1.06821	0.534107	−	0.845417i	$-0.320648\pi$
$419$	21.8658	1.06821	0.534107	+	0.845417i	$0.320648\pi$
$420$	0	0
$421$	20.6111	1.00453	0.502263	−	0.864715i	$-0.332501\pi$
$421$	20.6111	1.00453	0.502263	+	0.864715i	$0.332501\pi$
$422$	14.0957	0.686168
$423$	−15.4934	−0.753313
$424$	−16.0975	−0.781766
$425$	0	0
$426$	−5.19248	−0.251576
$427$	−25.9241	−1.25455
$428$	−18.5884	−0.898503
$429$	−0.109254	−0.00527481
$430$	0	0
$431$	−22.3052	−1.07441	−0.537203	−	0.843453i	$-0.680519\pi$
$431$	−22.3052	−1.07441	−0.537203	+	0.843453i	$0.680519\pi$
$432$	18.7508	0.902146
$433$	−8.93127	−0.429210	−0.214605	−	0.976701i	$-0.568846\pi$
$433$	−8.93127	−0.429210	−0.214605	+	0.976701i	$0.568846\pi$
$434$	10.8539	0.521005
$435$	0	0
$436$	−5.98769	−0.286758
$437$	8.14865	0.389803
$438$	−6.69652	−0.319972
$439$	22.2643	1.06262	0.531308	−	0.847179i	$-0.321701\pi$
$439$	22.2643	1.06262	0.531308	+	0.847179i	$0.321701\pi$
$440$	0	0
$441$	−29.9481	−1.42610
$442$	0	0
$443$	6.17421	0.293346	0.146673	−	0.989185i	$-0.453144\pi$
$443$	6.17421	0.293346	0.146673	+	0.989185i	$0.453144\pi$
$444$	0.292296	0.0138718
$445$	0	0
$446$	−8.61184	−0.407783
$447$	7.00539	0.331344
$448$	6.07671	0.287097
$449$	30.2985	1.42988	0.714938	−	0.699188i	$-0.246455\pi$
$449$	30.2985	1.42988	0.714938	+	0.699188i	$0.246455\pi$
$450$	0	0
$451$	0.346119	0.0162981
$452$	−3.79984	−0.178730
$453$	−10.5584	−0.496077
$454$	−20.2453	−0.950159
$455$	0	0
$456$	−7.17919	−0.336196
$457$	25.5756	1.19638	0.598188	−	0.801356i	$-0.295888\pi$
$457$	25.5756	1.19638	0.598188	+	0.801356i	$0.295888\pi$
$458$	37.2876	1.74233
$459$	0	0
$460$	0	0
$461$	−14.0565	−0.654677	−0.327338	−	0.944907i	$-0.606152\pi$
$461$	−14.0565	−0.654677	−0.327338	+	0.944907i	$0.606152\pi$
$462$	−0.283177	−0.0131746
$463$	25.1233	1.16758	0.583789	−	0.811905i	$-0.301569\pi$
$463$	25.1233	1.16758	0.583789	+	0.811905i	$0.301569\pi$
$464$	−2.07511	−0.0963343
$465$	0	0
$466$	−5.52315	−0.255855
$467$	−22.2900	−1.03146	−0.515730	−	0.856751i	$-0.672479\pi$
$467$	−22.2900	−1.03146	−0.515730	+	0.856751i	$0.672479\pi$
$468$	9.63884	0.445556
$469$	−3.97738	−0.183658
$470$	0	0
$471$	−6.47483	−0.298344
$472$	6.80285	0.313126
$473$	0.524724	0.0241268
$474$	12.4014	0.569617
$475$	0	0
$476$	0	0
$477$	−30.4801	−1.39559
$478$	8.92209	0.408087
$479$	−26.7496	−1.22222	−0.611111	−	0.791545i	$-0.709277\pi$
$479$	−26.7496	−1.22222	−0.611111	+	0.791545i	$0.709277\pi$
$480$	0	0
$481$	−1.01930	−0.0464760
$482$	−3.00086	−0.136685
$483$	−3.11224	−0.141612
$484$	−13.8678	−0.630353
$485$	0	0
$486$	26.7647	1.21407
$487$	23.1460	1.04885	0.524423	−	0.851458i	$-0.324281\pi$
$487$	23.1460	1.04885	0.524423	+	0.851458i	$0.324281\pi$
$488$	−7.96763	−0.360677
$489$	−8.56135	−0.387158
$490$	0	0
$491$	7.78981	0.351549	0.175775	−	0.984430i	$-0.443757\pi$
$491$	7.78981	0.351549	0.175775	+	0.984430i	$0.443757\pi$
$492$	5.71901	0.257833
$493$	0	0
$494$	−42.7213	−1.92212
$495$	0	0
$496$	6.82714	0.306548
$497$	18.1487	0.814079
$498$	−8.11207	−0.363510
$499$	8.79764	0.393837	0.196918	−	0.980420i	$-0.436907\pi$
$499$	8.79764	0.393837	0.196918	+	0.980420i	$0.436907\pi$
$500$	0	0
$501$	−8.66948	−0.387324
$502$	16.5096	0.736860
$503$	−7.83946	−0.349544	−0.174772	−	0.984609i	$-0.555919\pi$
$503$	−7.83946	−0.349544	−0.174772	+	0.984609i	$0.555919\pi$
$504$	−14.6405	−0.652141
$505$	0	0
$506$	0.0987858	0.00439156
$507$	−2.64773	−0.117590
$508$	0.0104502	0.000463651 0
$509$	−2.52868	−0.112082	−0.0560409	−	0.998428i	$-0.517848\pi$
$509$	−2.52868	−0.112082	−0.0560409	+	0.998428i	$0.517848\pi$
$510$	0	0
$511$	23.4055	1.03540
$512$	17.5981	0.777732
$513$	−29.7330	−1.31274
$514$	30.0711	1.32638
$515$	0	0
$516$	8.67015	0.381682
$517$	0.321916	0.0141578
$518$	−2.64195	−0.116080
$519$	8.39019	0.368289
$520$	0	0
$521$	10.1083	0.442853	0.221426	−	0.975177i	$-0.428929\pi$
$521$	10.1083	0.442853	0.221426	+	0.975177i	$0.428929\pi$
$522$	−1.91987	−0.0840303
$523$	5.80494	0.253833	0.126916	−	0.991913i	$-0.459492\pi$
$523$	5.80494	0.253833	0.126916	+	0.991913i	$0.459492\pi$
$524$	−10.3209	−0.450872
$525$	0	0
$526$	39.2371	1.71082
$527$	0	0
$528$	−0.178119	−0.00775164
$529$	−21.9143	−0.952796
$530$	0	0
$531$	12.8809	0.558985
$532$	−42.8189	−1.85644
$533$	−19.9434	−0.863844
$534$	12.6908	0.549187
$535$	0	0
$536$	−1.22243	−0.0528009
$537$	9.53508	0.411469
$538$	−41.8449	−1.80406
$539$	0.622253	0.0268023
$540$	0	0
$541$	42.3404	1.82035	0.910177	−	0.414219i	$-0.135945\pi$
$541$	42.3404	1.82035	0.910177	+	0.414219i	$0.135945\pi$
$542$	7.14377	0.306851
$543$	1.40627	0.0603487
$544$	0	0
$545$	0	0
$546$	16.3167	0.698290
$547$	20.0061	0.855398	0.427699	−	0.903921i	$-0.359324\pi$
$547$	20.0061	0.855398	0.427699	+	0.903921i	$0.359324\pi$
$548$	2.81695	0.120334
$549$	−15.0864	−0.643872
$550$	0	0
$551$	3.29048	0.140179
$552$	−0.956532	−0.0407127
$553$	−43.3453	−1.84323
$554$	30.5364	1.29737
$555$	0	0
$556$	24.8873	1.05546
$557$	25.8965	1.09727	0.548636	−	0.836062i	$-0.315147\pi$
$557$	25.8965	1.09727	0.548636	+	0.836062i	$0.315147\pi$
$558$	6.31641	0.267395
$559$	−30.2346	−1.27879
$560$	0	0
$561$	0	0
$562$	8.43299	0.355724
$563$	−28.0309	−1.18136	−0.590681	−	0.806905i	$-0.701141\pi$
$563$	−28.0309	−1.18136	−0.590681	+	0.806905i	$0.701141\pi$
$564$	5.31910	0.223975
$565$	0	0
$566$	15.7114	0.660398
$567$	−21.5571	−0.905315
$568$	5.57790	0.234043
$569$	39.1921	1.64302	0.821509	−	0.570195i	$-0.193132\pi$
$569$	39.1921	1.64302	0.821509	+	0.570195i	$0.193132\pi$
$570$	0	0
$571$	−3.86762	−0.161855	−0.0809275	−	0.996720i	$-0.525788\pi$
$571$	−3.86762	−0.161855	−0.0809275	+	0.996720i	$0.525788\pi$
$572$	−0.200273	−0.00837383
$573$	−2.11917	−0.0885297
$574$	−51.6918	−2.15757
$575$	0	0
$576$	3.53632	0.147347
$577$	14.4808	0.602845	0.301423	−	0.953491i	$-0.402539\pi$
$577$	14.4808	0.602845	0.301423	+	0.953491i	$0.402539\pi$
$578$	0	0
$579$	14.1999	0.590128
$580$	0	0
$581$	28.3531	1.17629
$582$	−23.8810	−0.989898
$583$	0.633306	0.0262289
$584$	7.19357	0.297672
$585$	0	0
$586$	−1.33469	−0.0551355
$587$	−31.2452	−1.28963	−0.644814	−	0.764339i	$-0.723065\pi$
$587$	−31.2452	−1.28963	−0.644814	+	0.764339i	$0.723065\pi$
$588$	10.2816	0.424008
$589$	−10.8258	−0.446068
$590$	0	0
$591$	−12.5452	−0.516041
$592$	−1.66179	−0.0682991
$593$	2.00543	0.0823530	0.0411765	−	0.999152i	$-0.486889\pi$
$593$	2.00543	0.0823530	0.0411765	+	0.999152i	$0.486889\pi$
$594$	−0.360452	−0.0147895
$595$	0	0
$596$	12.8416	0.526012
$597$	18.3451	0.750816
$598$	−5.69204	−0.232765
$599$	36.0451	1.47276	0.736381	−	0.676567i	$-0.236533\pi$
$599$	36.0451	1.47276	0.736381	+	0.676567i	$0.236533\pi$
$600$	0	0
$601$	38.7948	1.58247	0.791237	−	0.611510i	$-0.209438\pi$
$601$	38.7948	1.58247	0.791237	+	0.611510i	$0.209438\pi$
$602$	−78.3660	−3.19396
$603$	−2.31462	−0.0942588
$604$	−19.3546	−0.787529
$605$	0	0
$606$	16.5129	0.670789
$607$	−28.9535	−1.17519	−0.587594	−	0.809156i	$-0.699925\pi$
$607$	−28.9535	−1.17519	−0.587594	+	0.809156i	$0.699925\pi$
$608$	−48.7774	−1.97818
$609$	−1.25675	−0.0509259
$610$	0	0
$611$	−18.5488	−0.750405
$612$	0	0
$613$	−9.10707	−0.367831	−0.183915	−	0.982942i	$-0.558877\pi$
$613$	−9.10707	−0.367831	−0.183915	+	0.982942i	$0.558877\pi$
$614$	5.16662	0.208508
$615$	0	0
$616$	0.304197	0.0122564
$617$	5.05551	0.203527	0.101764	−	0.994809i	$-0.467551\pi$
$617$	5.05551	0.203527	0.101764	+	0.994809i	$0.467551\pi$
$618$	8.58774	0.345450
$619$	16.8593	0.677632	0.338816	−	0.940853i	$-0.389974\pi$
$619$	16.8593	0.677632	0.338816	+	0.940853i	$0.389974\pi$
$620$	0	0
$621$	−3.96152	−0.158970
$622$	37.0159	1.48420
$623$	−44.3568	−1.77712
$624$	10.2632	0.410858
$625$	0	0
$626$	−15.2871	−0.610996
$627$	0.282442	0.0112797
$628$	−11.8690	−0.473625
$629$	0	0
$630$	0	0
$631$	2.71309	0.108007	0.0540033	−	0.998541i	$-0.482802\pi$
$631$	2.71309	0.108007	0.0540033	+	0.998541i	$0.482802\pi$
$632$	−13.3220	−0.529919
$633$	5.36965	0.213424
$634$	38.0882	1.51268
$635$	0	0
$636$	10.4643	0.414936
$637$	−35.8543	−1.42060
$638$	0.0398904	0.00157928
$639$	10.5615	0.417808
$640$	0	0
$641$	22.8123	0.901030	0.450515	−	0.892769i	$-0.351240\pi$
$641$	22.8123	0.901030	0.450515	+	0.892769i	$0.351240\pi$
$642$	−18.3118	−0.722711
$643$	19.2771	0.760214	0.380107	−	0.924942i	$-0.375887\pi$
$643$	19.2771	0.760214	0.380107	+	0.924942i	$0.375887\pi$
$644$	−5.70506	−0.224811
$645$	0	0
$646$	0	0
$647$	−12.8098	−0.503606	−0.251803	−	0.967778i	$-0.581024\pi$
$647$	−12.8098	−0.503606	−0.251803	+	0.967778i	$0.581024\pi$
$648$	−6.62548	−0.260273
$649$	−0.267636	−0.0105056
$650$	0	0
$651$	4.13472	0.162052
$652$	−15.6938	−0.614618
$653$	41.7200	1.63263	0.816315	−	0.577607i	$-0.196013\pi$
$653$	41.7200	1.63263	0.816315	+	0.577607i	$0.196013\pi$
$654$	−5.89861	−0.230654
$655$	0	0
$656$	−32.5142	−1.26947
$657$	13.6208	0.531397
$658$	−48.0772	−1.87424
$659$	−34.3290	−1.33727	−0.668633	−	0.743592i	$-0.733120\pi$
$659$	−34.3290	−1.33727	−0.668633	+	0.743592i	$0.733120\pi$
$660$	0	0
$661$	27.7009	1.07744	0.538720	−	0.842485i	$-0.318908\pi$
$661$	27.7009	1.07744	0.538720	+	0.842485i	$0.318908\pi$
$662$	−27.4958	−1.06865
$663$	0	0
$664$	8.71419	0.338176
$665$	0	0
$666$	−1.53747	−0.0595758
$667$	0.438413	0.0169754
$668$	−15.8920	−0.614881
$669$	−3.28062	−0.126836
$670$	0	0
$671$	0.313461	0.0121010
$672$	18.6297	0.718658
$673$	36.2078	1.39571	0.697854	−	0.716240i	$-0.254138\pi$
$673$	36.2078	1.39571	0.697854	+	0.716240i	$0.254138\pi$
$674$	−20.7195	−0.798086
$675$	0	0
$676$	−4.85355	−0.186675
$677$	−17.1158	−0.657812	−0.328906	−	0.944363i	$-0.606680\pi$
$677$	−17.1158	−0.657812	−0.328906	+	0.944363i	$0.606680\pi$
$678$	−3.74331	−0.143761
$679$	83.4684	3.20322
$680$	0	0
$681$	−7.71229	−0.295536
$682$	−0.131240	−0.00502545
$683$	8.85022	0.338644	0.169322	−	0.985561i	$-0.445842\pi$
$683$	8.85022	0.338644	0.169322	+	0.985561i	$0.445842\pi$
$684$	−24.9183	−0.952776
$685$	0	0
$686$	−38.0462	−1.45261
$687$	14.2044	0.541932
$688$	−49.2924	−1.87925
$689$	−36.4911	−1.39020
$690$	0	0
$691$	1.39834	0.0531953	0.0265977	−	0.999646i	$-0.491533\pi$
$691$	1.39834	0.0531953	0.0265977	+	0.999646i	$0.491533\pi$
$692$	15.3801	0.584663
$693$	0.575985	0.0218799
$694$	27.0591	1.02715
$695$	0	0
$696$	−0.386254	−0.0146409
$697$	0	0
$698$	−8.19245	−0.310089
$699$	−2.10400	−0.0795806
$700$	0	0
$701$	2.36331	0.0892608	0.0446304	−	0.999004i	$-0.485789\pi$
$701$	2.36331	0.0892608	0.0446304	+	0.999004i	$0.485789\pi$
$702$	20.7693	0.783885
$703$	2.63509	0.0993843
$704$	−0.0734765	−0.00276925
$705$	0	0
$706$	13.9314	0.524316
$707$	−57.7154	−2.17061
$708$	−4.42222	−0.166197
$709$	−32.5266	−1.22156	−0.610782	−	0.791799i	$-0.709145\pi$
$709$	−32.5266	−1.22156	−0.610782	+	0.791799i	$0.709145\pi$
$710$	0	0
$711$	−25.2247	−0.945998
$712$	−13.6328	−0.510912
$713$	−1.44239	−0.0540179
$714$	0	0
$715$	0	0
$716$	17.4788	0.653212
$717$	3.39880	0.126931
$718$	−19.9154	−0.743237
$719$	−16.4302	−0.612744	−0.306372	−	0.951912i	$-0.599115\pi$
$719$	−16.4302	−0.612744	−0.306372	+	0.951912i	$0.599115\pi$
$720$	0	0
$721$	−30.0157	−1.11784
$722$	76.1323	2.83335
$723$	−1.14315	−0.0425143
$724$	2.57783	0.0958043
$725$	0	0
$726$	−13.6614	−0.507024
$727$	32.4709	1.20428	0.602139	−	0.798391i	$-0.294315\pi$
$727$	32.4709	1.20428	0.602139	+	0.798391i	$0.294315\pi$
$728$	−17.5278	−0.649624
$729$	−4.69881	−0.174030
$730$	0	0
$731$	0	0
$732$	5.17939	0.191436
$733$	8.63424	0.318913	0.159456	−	0.987205i	$-0.449026\pi$
$733$	8.63424	0.318913	0.159456	+	0.987205i	$0.449026\pi$
$734$	−28.9448	−1.06837
$735$	0	0
$736$	−6.49894	−0.239554
$737$	0.0480925	0.00177151
$738$	−30.0819	−1.10733
$739$	10.6026	0.390022	0.195011	−	0.980801i	$-0.437526\pi$
$739$	10.6026	0.390022	0.195011	+	0.980801i	$0.437526\pi$
$740$	0	0
$741$	−16.2743	−0.597853
$742$	−94.5824	−3.47223
$743$	−22.3424	−0.819665	−0.409832	−	0.912161i	$-0.634413\pi$
$743$	−22.3424	−0.819665	−0.409832	+	0.912161i	$0.634413\pi$
$744$	1.27079	0.0465893
$745$	0	0
$746$	−18.5100	−0.677700
$747$	16.5000	0.603704
$748$	0	0
$749$	64.0032	2.33863
$750$	0	0
$751$	−7.13989	−0.260538	−0.130269	−	0.991479i	$-0.541584\pi$
$751$	−7.13989	−0.260538	−0.130269	+	0.991479i	$0.541584\pi$
$752$	−30.2406	−1.10276
$753$	6.28921	0.229192
$754$	−2.29849	−0.0837060
$755$	0	0
$756$	20.8167	0.757097
$757$	−18.0266	−0.655186	−0.327593	−	0.944819i	$-0.606238\pi$
$757$	−18.0266	−0.655186	−0.327593	+	0.944819i	$0.606238\pi$
$758$	−25.1949	−0.915119
$759$	0.0376317	0.00136594
$760$	0	0
$761$	−49.9437	−1.81046	−0.905230	−	0.424923i	$-0.860301\pi$
$761$	−49.9437	−1.81046	−0.905230	+	0.424923i	$0.860301\pi$
$762$	0.0102947	0.000372937 0
$763$	20.6167	0.746375
$764$	−3.88466	−0.140542
$765$	0	0
$766$	10.1521	0.366811
$767$	15.4212	0.556828
$768$	14.2823	0.515369
$769$	−43.9115	−1.58349	−0.791745	−	0.610852i	$-0.790827\pi$
$769$	−43.9115	−1.58349	−0.791745	+	0.610852i	$0.790827\pi$
$770$	0	0
$771$	11.4554	0.412554
$772$	26.0299	0.936836
$773$	9.67759	0.348079	0.174039	−	0.984739i	$-0.444318\pi$
$773$	9.67759	0.348079	0.174039	+	0.984739i	$0.444318\pi$
$774$	−45.6048	−1.63923
$775$	0	0
$776$	25.6536	0.920910
$777$	−1.00643	−0.0361055
$778$	64.6771	2.31879
$779$	51.5576	1.84724
$780$	0	0
$781$	−0.219445	−0.00785234
$782$	0	0
$783$	−1.59969	−0.0571683
$784$	−58.4542	−2.08765
$785$	0	0
$786$	−10.1674	−0.362659
$787$	−19.8384	−0.707161	−0.353581	−	0.935404i	$-0.615036\pi$
$787$	−19.8384	−0.707161	−0.353581	+	0.935404i	$0.615036\pi$
$788$	−22.9966	−0.819221
$789$	14.9471	0.532130
$790$	0	0
$791$	13.0836	0.465198
$792$	0.177026	0.00629035
$793$	−18.0616	−0.641387
$794$	7.02499	0.249308
$795$	0	0
$796$	33.6285	1.19193
$797$	−5.56111	−0.196984	−0.0984922	−	0.995138i	$-0.531402\pi$
$797$	−5.56111	−0.196984	−0.0984922	+	0.995138i	$0.531402\pi$
$798$	−42.1819	−1.49322
$799$	0	0
$800$	0	0
$801$	−25.8133	−0.912068
$802$	−66.3436	−2.34267
$803$	−0.283008	−0.00998714
$804$	0.794645	0.0280250
$805$	0	0
$806$	7.56207	0.266363
$807$	−15.9405	−0.561132
$808$	−17.7385	−0.624040
$809$	−31.0442	−1.09146	−0.545729	−	0.837962i	$-0.683747\pi$
$809$	−31.0442	−1.09146	−0.545729	+	0.837962i	$0.683747\pi$
$810$	0	0
$811$	46.9708	1.64937	0.824683	−	0.565595i	$-0.191353\pi$
$811$	46.9708	1.64937	0.824683	+	0.565595i	$0.191353\pi$
$812$	−2.30374	−0.0808455
$813$	2.72136	0.0954424
$814$	0.0319451	0.00111968
$815$	0	0
$816$	0	0
$817$	78.1626	2.73456
$818$	−40.3656	−1.41135
$819$	−33.1883	−1.15969
$820$	0	0
$821$	16.7433	0.584346	0.292173	−	0.956366i	$-0.405622\pi$
$821$	16.7433	0.584346	0.292173	+	0.956366i	$0.405622\pi$
$822$	2.77505	0.0967908
$823$	26.5996	0.927205	0.463602	−	0.886043i	$-0.346557\pi$
$823$	26.5996	0.927205	0.463602	+	0.886043i	$0.346557\pi$
$824$	−9.22517	−0.321374
$825$	0	0
$826$	39.9707	1.39076
$827$	−34.4855	−1.19918	−0.599590	−	0.800307i	$-0.704670\pi$
$827$	−34.4855	−1.19918	−0.599590	+	0.800307i	$0.704670\pi$
$828$	−3.32004	−0.115379
$829$	−7.29346	−0.253312	−0.126656	−	0.991947i	$-0.540424\pi$
$829$	−7.29346	−0.253312	−0.126656	+	0.991947i	$0.540424\pi$
$830$	0	0
$831$	11.6326	0.403531
$832$	4.23372	0.146778
$833$	0	0
$834$	24.5170	0.848956
$835$	0	0
$836$	0.517745	0.0179066
$837$	5.26302	0.181917
$838$	39.4859	1.36402
$839$	35.5082	1.22588	0.612940	−	0.790129i	$-0.289987\pi$
$839$	35.5082	1.22588	0.612940	+	0.790129i	$0.289987\pi$
$840$	0	0
$841$	−28.8230	−0.993895
$842$	37.2202	1.28269
$843$	3.21248	0.110644
$844$	9.84311	0.338814
$845$	0	0
$846$	−27.9784	−0.961916
$847$	47.7492	1.64068
$848$	−59.4925	−2.04298
$849$	5.98513	0.205409
$850$	0	0
$851$	0.351091	0.0120352
$852$	−3.62594	−0.124223
$853$	0.238822	0.00817712	0.00408856	−	0.999992i	$-0.498699\pi$
$853$	0.238822	0.00817712	0.00408856	+	0.999992i	$0.498699\pi$
$854$	−46.8144	−1.60196
$855$	0	0
$856$	19.6711	0.672343
$857$	18.1950	0.621528	0.310764	−	0.950487i	$-0.399415\pi$
$857$	18.1950	0.621528	0.310764	+	0.950487i	$0.399415\pi$
$858$	−0.197293	−0.00673548
$859$	−20.1560	−0.687715	−0.343857	−	0.939022i	$-0.611734\pi$
$859$	−20.1560	−0.687715	−0.343857	+	0.939022i	$0.611734\pi$
$860$	0	0
$861$	−19.6916	−0.671088
$862$	−40.2795	−1.37192
$863$	−14.7597	−0.502424	−0.251212	−	0.967932i	$-0.580829\pi$
$863$	−14.7597	−0.502424	−0.251212	+	0.967932i	$0.580829\pi$
$864$	23.7135	0.806749
$865$	0	0
$866$	−16.1284	−0.548064
$867$	0	0
$868$	7.57936	0.257260
$869$	0.524110	0.0177792
$870$	0	0
$871$	−2.77109	−0.0938949
$872$	6.33644	0.214579
$873$	48.5741	1.64398
$874$	14.7151	0.497745
$875$	0	0
$876$	−4.67622	−0.157995
$877$	42.7577	1.44382	0.721912	−	0.691984i	$-0.243263\pi$
$877$	42.7577	1.44382	0.721912	+	0.691984i	$0.243263\pi$
$878$	40.2055	1.35687
$879$	−0.508439	−0.0171492
$880$	0	0
$881$	−43.6832	−1.47172	−0.735862	−	0.677131i	$-0.763223\pi$
$881$	−43.6832	−1.47172	−0.735862	+	0.677131i	$0.763223\pi$
$882$	−54.0813	−1.82101
$883$	41.4824	1.39599	0.697997	−	0.716101i	$-0.254075\pi$
$883$	41.4824	1.39599	0.697997	+	0.716101i	$0.254075\pi$
$884$	0	0
$885$	0	0
$886$	11.1496	0.374577
$887$	32.8798	1.10400	0.551998	−	0.833845i	$-0.313866\pi$
$887$	32.8798	1.10400	0.551998	+	0.833845i	$0.313866\pi$
$888$	−0.309321	−0.0103801
$889$	−0.0359818	−0.00120679
$890$	0	0
$891$	0.260658	0.00873238
$892$	−6.01370	−0.201354
$893$	47.9524	1.60467
$894$	12.6505	0.423097
$895$	0	0
$896$	−43.1892	−1.44285
$897$	−2.16834	−0.0723988
$898$	54.7140	1.82583
$899$	−0.582447	−0.0194257
$900$	0	0
$901$	0	0
$902$	0.625031	0.0208113
$903$	−29.8529	−0.993443
$904$	4.02116	0.133742
$905$	0	0
$906$	−19.0667	−0.633448
$907$	36.4475	1.21022	0.605110	−	0.796142i	$-0.293129\pi$
$907$	36.4475	1.21022	0.605110	+	0.796142i	$0.293129\pi$
$908$	−14.1374	−0.469166
$909$	−33.5873	−1.11402
$910$	0	0
$911$	29.8739	0.989767	0.494883	−	0.868959i	$-0.335211\pi$
$911$	29.8739	0.989767	0.494883	+	0.868959i	$0.335211\pi$
$912$	−26.5325	−0.878579
$913$	−0.342832	−0.0113461
$914$	46.1852	1.52767
$915$	0	0
$916$	26.0381	0.860324
$917$	35.5369	1.17353
$918$	0	0
$919$	11.6218	0.383368	0.191684	−	0.981457i	$-0.438605\pi$
$919$	11.6218	0.383368	0.191684	+	0.981457i	$0.438605\pi$
$920$	0	0
$921$	1.96818	0.0648539
$922$	−25.3837	−0.835966
$923$	12.6444	0.416196
$924$	−0.197744	−0.00650531
$925$	0	0
$926$	45.3684	1.49090
$927$	−17.4675	−0.573709
$928$	−2.62432	−0.0861475
$929$	−5.04232	−0.165433	−0.0827165	−	0.996573i	$-0.526360\pi$
$929$	−5.04232	−0.165433	−0.0827165	+	0.996573i	$0.526360\pi$
$930$	0	0
$931$	92.6904	3.03781
$932$	−3.85685	−0.126335
$933$	14.1009	0.461643
$934$	−40.2520	−1.31709
$935$	0	0
$936$	−10.2003	−0.333406
$937$	−34.9116	−1.14051	−0.570257	−	0.821466i	$-0.693156\pi$
$937$	−34.9116	−1.14051	−0.570257	+	0.821466i	$0.693156\pi$
$938$	−7.18248	−0.234516
$939$	−5.82351	−0.190043
$940$	0	0
$941$	−11.0586	−0.360502	−0.180251	−	0.983621i	$-0.557691\pi$
$941$	−11.0586	−0.360502	−0.180251	+	0.983621i	$0.557691\pi$
$942$	−11.6924	−0.380960
$943$	6.86937	0.223697
$944$	25.1416	0.818290
$945$	0	0
$946$	0.947562	0.0308079
$947$	11.4918	0.373435	0.186717	−	0.982414i	$-0.440215\pi$
$947$	11.4918	0.373435	0.186717	+	0.982414i	$0.440215\pi$
$948$	8.66000	0.281264
$949$	16.3069	0.529346
$950$	0	0
$951$	14.5094	0.470500
$952$	0	0
$953$	0.704896	0.0228338	0.0114169	−	0.999935i	$-0.496366\pi$
$953$	0.704896	0.0228338	0.0114169	+	0.999935i	$0.496366\pi$
$954$	−55.0419	−1.78205
$955$	0	0
$956$	6.23035	0.201504
$957$	0.0151959	0.000491215 0
$958$	−48.3053	−1.56067
$959$	−9.69929	−0.313206
$960$	0	0
$961$	−29.0837	−0.938185
$962$	−1.84068	−0.0593459
$963$	37.2465	1.20025
$964$	−2.09552	−0.0674920
$965$	0	0
$966$	−5.62018	−0.180826
$967$	39.9545	1.28485	0.642425	−	0.766348i	$-0.277928\pi$
$967$	39.9545	1.28485	0.642425	+	0.766348i	$0.277928\pi$
$968$	14.6755	0.471688
$969$	0	0
$970$	0	0
$971$	43.2538	1.38808	0.694040	−	0.719936i	$-0.255829\pi$
$971$	43.2538	1.38808	0.694040	+	0.719936i	$0.255829\pi$
$972$	18.6900	0.599481
$973$	−85.6915	−2.74714
$974$	41.7978	1.33929
$975$	0	0
$976$	−29.4464	−0.942555
$977$	−48.8671	−1.56340	−0.781699	−	0.623656i	$-0.785647\pi$
$977$	−48.8671	−1.56340	−0.781699	+	0.623656i	$0.785647\pi$
$978$	−15.4603	−0.494367
$979$	0.536340	0.0171415
$980$	0	0
$981$	11.9978	0.383061
$982$	14.0671	0.448898
$983$	48.8832	1.55913	0.779565	−	0.626321i	$-0.215440\pi$
$983$	48.8832	1.55913	0.779565	+	0.626321i	$0.215440\pi$
$984$	−6.05211	−0.192934
$985$	0	0
$986$	0	0
$987$	−18.3146	−0.582961
$988$	−29.8325	−0.949099
$989$	10.4141	0.331150
$990$	0	0
$991$	14.5340	0.461687	0.230843	−	0.972991i	$-0.425851\pi$
$991$	14.5340	0.461687	0.230843	+	0.972991i	$0.425851\pi$
$992$	8.63407	0.274132
$993$	−10.4743	−0.332392
$994$	32.7734	1.03951
$995$	0	0
$996$	−5.66470	−0.179493
$997$	44.2040	1.39995	0.699977	−	0.714165i	$-0.253193\pi$
$997$	44.2040	1.39995	0.699977	+	0.714165i	$0.253193\pi$
$998$	15.8870	0.502896
$999$	−1.28107	−0.0405312

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7225.2.a.bq.1.10		12
5.4	even	2		1445.2.a.q.1.3		12
17.11	odd	16		425.2.m.b.376.5		24
17.14	odd	16		425.2.m.b.26.5		24
17.16	even	2		7225.2.a.bs.1.10		12
85.4	even	4		1445.2.d.j.866.19		24
85.14	odd	16		85.2.l.a.26.2	✓	24
85.28	even	16		425.2.n.c.274.5		24
85.48	even	16		425.2.n.f.349.2		24
85.62	even	16		425.2.n.f.274.2		24
85.64	even	4		1445.2.d.j.866.20		24
85.79	odd	16		85.2.l.a.36.2	yes	24
85.82	even	16		425.2.n.c.349.5		24
85.84	even	2		1445.2.a.p.1.3		12
255.14	even	16		765.2.be.b.451.5		24
255.164	even	16		765.2.be.b.631.5		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
85.2.l.a.26.2	✓	24	85.14	odd	16
85.2.l.a.36.2	yes	24	85.79	odd	16
425.2.m.b.26.5		24	17.14	odd	16
425.2.m.b.376.5		24	17.11	odd	16
425.2.n.c.274.5		24	85.28	even	16
425.2.n.c.349.5		24	85.82	even	16
425.2.n.f.274.2		24	85.62	even	16
425.2.n.f.349.2		24	85.48	even	16
765.2.be.b.451.5		24	255.14	even	16
765.2.be.b.631.5		24	255.164	even	16
1445.2.a.p.1.3		12	85.84	even	2
1445.2.a.q.1.3		12	5.4	even	2
1445.2.d.j.866.19		24	85.4	even	4
1445.2.d.j.866.20		24	85.64	even	4
7225.2.a.bq.1.10		12	1.1	even	1	trivial
7225.2.a.bs.1.10		12	17.16	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)

\(f(q)\)	\(=\)	\(q+1.80583 q^{2} +0.687917 q^{3} +1.26102 q^{4} +1.24226 q^{6} -4.34193 q^{7} -1.33447 q^{8} -2.52677 q^{9} +0.0525004 q^{11} +0.867478 q^{12} -3.02508 q^{13} -7.84078 q^{14} -4.93187 q^{16} -4.56292 q^{18} +7.82043 q^{19} -2.98689 q^{21} +0.0948069 q^{22} +1.04197 q^{23} -0.918004 q^{24} -5.46278 q^{26} -3.80196 q^{27} -5.47526 q^{28} +0.420754 q^{29} -1.38429 q^{31} -6.23717 q^{32} +0.0361159 q^{33} -3.18631 q^{36} +0.336949 q^{37} +14.1224 q^{38} -2.08100 q^{39} +6.59268 q^{41} -5.39381 q^{42} +9.99466 q^{43} +0.0662042 q^{44} +1.88162 q^{46} +6.13168 q^{47} -3.39272 q^{48} +11.8523 q^{49} -3.81469 q^{52} +12.0629 q^{53} -6.86569 q^{54} +5.79417 q^{56} +5.37981 q^{57} +0.759811 q^{58} -5.09779 q^{59} +5.97063 q^{61} -2.49979 q^{62} +10.9711 q^{63} -1.39954 q^{64} +0.0652192 q^{66} +0.916040 q^{67} +0.716788 q^{69} -4.17986 q^{71} +3.37190 q^{72} -5.39059 q^{73} +0.608473 q^{74} +9.86173 q^{76} -0.227953 q^{77} -3.75794 q^{78} +9.98296 q^{79} +4.96488 q^{81} +11.9053 q^{82} -6.53008 q^{83} -3.76653 q^{84} +18.0487 q^{86} +0.289444 q^{87} -0.0700602 q^{88} +10.2159 q^{89} +13.1347 q^{91} +1.31395 q^{92} -0.952278 q^{93} +11.0728 q^{94} -4.29066 q^{96} -19.2238 q^{97} +21.4033 q^{98} -0.132657 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 4 q^{2} - 8 q^{3} + 12 q^{4} + 8 q^{6} - 16 q^{7} + 12 q^{8} + 12 q^{9} + 16 q^{11} - 16 q^{12} + 8 q^{13} - 16 q^{14} + 12 q^{16} - 4 q^{18} + 16 q^{21} - 16 q^{22} - 16 q^{23} + 16 q^{26} - 32 q^{27}+ \cdots + 56 q^{99}+O(q^{100}) \) 12 * q + 4 * q^2 - 8 * q^3 + 12 * q^4 + 8 * q^6 - 16 * q^7 + 12 * q^8 + 12 * q^9 + 16 * q^11 - 16 * q^12 + 8 * q^13 - 16 * q^14 + 12 * q^16 - 4 * q^18 + 16 * q^21 - 16 * q^22 - 16 * q^23 + 16 * q^26 - 32 * q^27 - 40 * q^28 + 16 * q^29 + 24 * q^31 + 28 * q^32 + 12 * q^36 - 24 * q^37 + 24 * q^38 + 8 * q^39 + 8 * q^41 + 16 * q^43 + 8 * q^44 + 40 * q^46 + 32 * q^47 + 24 * q^48 + 20 * q^49 + 24 * q^52 + 8 * q^54 - 24 * q^56 - 32 * q^57 + 16 * q^58 + 8 * q^59 + 24 * q^61 - 8 * q^62 - 48 * q^63 + 36 * q^64 + 40 * q^66 + 8 * q^67 + 48 * q^69 - 16 * q^71 - 12 * q^72 - 16 * q^73 + 16 * q^76 - 24 * q^77 + 24 * q^78 + 40 * q^79 + 36 * q^81 + 16 * q^82 + 40 * q^83 + 32 * q^84 - 8 * q^86 + 32 * q^87 - 48 * q^88 + 8 * q^89 + 72 * q^91 + 8 * q^92 - 24 * q^93 - 16 * q^94 + 8 * q^96 - 32 * q^97 + 60 * q^98 + 56 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more