LMFDB - Embedded newform 9025.2.a.co.1.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9025,2,Mod(1,9025)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9025, 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("9025.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9025 = 5^{2} \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9025.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:	$72.0649878242$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 32 x^{18} + 426 x^{16} - 3061 x^{14} + 12909 x^{12} - 32678 x^{10} + 49159 x^{8} - 42549 x^{6} + \cdots + 405 $ x^20 - 32x^18 + 426x^16 - 3061x^14 + 12909x^12 - 32678x^10 + 49159x^8 - 42549x^6 + 20056x^4 - 4635*x^2 + 405
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			1.13
Root			$0.810675$ of defining polynomial
Character	$\chi$	$=$	9025.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.810675 q^{2} -1.66033 q^{3} -1.34281 q^{4} -1.34598 q^{6} +5.21276 q^{7} -2.70993 q^{8} -0.243318 q^{9} +O(q^{10})$	\(q+0.810675 q^{2} -1.66033 q^{3} -1.34281 q^{4} -1.34598 q^{6} +5.21276 q^{7} -2.70993 q^{8} -0.243318 q^{9} +3.14148 q^{11} +2.22950 q^{12} -5.93417 q^{13} +4.22586 q^{14} +0.488739 q^{16} -3.59387 q^{17} -0.197252 q^{18} -8.65489 q^{21} +2.54672 q^{22} +8.45583 q^{23} +4.49937 q^{24} -4.81068 q^{26} +5.38496 q^{27} -6.99973 q^{28} -6.84132 q^{29} -3.11860 q^{31} +5.81607 q^{32} -5.21588 q^{33} -2.91346 q^{34} +0.326728 q^{36} -0.373041 q^{37} +9.85266 q^{39} +3.60203 q^{41} -7.01630 q^{42} -5.51851 q^{43} -4.21839 q^{44} +6.85493 q^{46} +1.04603 q^{47} -0.811466 q^{48} +20.1729 q^{49} +5.96700 q^{51} +7.96844 q^{52} -2.81271 q^{53} +4.36546 q^{54} -14.1262 q^{56} -5.54609 q^{58} -7.22336 q^{59} +6.96653 q^{61} -2.52817 q^{62} -1.26836 q^{63} +3.73746 q^{64} -4.22838 q^{66} -0.749780 q^{67} +4.82587 q^{68} -14.0394 q^{69} -2.54581 q^{71} +0.659374 q^{72} -7.36554 q^{73} -0.302415 q^{74} +16.3758 q^{77} +7.98730 q^{78} +9.10321 q^{79} -8.21084 q^{81} +2.92007 q^{82} +9.47217 q^{83} +11.6218 q^{84} -4.47372 q^{86} +11.3588 q^{87} -8.51318 q^{88} +9.87979 q^{89} -30.9334 q^{91} -11.3545 q^{92} +5.17789 q^{93} +0.847991 q^{94} -9.65657 q^{96} +5.88772 q^{97} +16.3537 q^{98} -0.764377 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 24 q^{4} - 20 q^{6} + 8 q^{7} + 16 q^{9}+O(q^{10}) $ 20 * q + 24 * q^4 - 20 * q^6 + 8 * q^7 + 16 * q^9	$ 20 q + 24 q^{4} - 20 q^{6} + 8 q^{7} + 16 q^{9} + 10 q^{11} + 40 q^{16} + 38 q^{17} + 56 q^{23} - 54 q^{24} - 22 q^{26} - 16 q^{28} - 32 q^{36} + 80 q^{39} + 44 q^{43} - 16 q^{44} + 98 q^{47} - 4 q^{49} - 54 q^{54} + 50 q^{58} + 40 q^{61} + 90 q^{62} + 6 q^{63} + 46 q^{64} + 8 q^{66} + 4 q^{68} + 34 q^{73} + 20 q^{74} + 80 q^{77} + 60 q^{81} - 10 q^{82} + 84 q^{83} + 90 q^{87} + 78 q^{92} + 20 q^{93} - 120 q^{96} - 76 q^{99}+O(q^{100}) $ 20 * q + 24 * q^4 - 20 * q^6 + 8 * q^7 + 16 * q^9 + 10 * q^11 + 40 * q^16 + 38 * q^17 + 56 * q^23 - 54 * q^24 - 22 * q^26 - 16 * q^28 - 32 * q^36 + 80 * q^39 + 44 * q^43 - 16 * q^44 + 98 * q^47 - 4 * q^49 - 54 * q^54 + 50 * q^58 + 40 * q^61 + 90 * q^62 + 6 * q^63 + 46 * q^64 + 8 * q^66 + 4 * q^68 + 34 * q^73 + 20 * q^74 + 80 * q^77 + 60 * q^81 - 10 * q^82 + 84 * q^83 + 90 * q^87 + 78 * q^92 + 20 * q^93 - 120 * q^96 - 76 * 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 $

$2$	0.810675	0.573234	0.286617	−	0.958045i	$-0.407469\pi$
$2$	0.810675	0.573234	0.286617	+	0.958045i	$0.407469\pi$
$3$	−1.66033	−0.958590	−0.479295	−	0.877654i	$-0.659108\pi$
$3$	−1.66033	−0.958590	−0.479295	+	0.877654i	$0.659108\pi$
$4$	−1.34281	−0.671403
$5$	0	0
$5$	0	0
$6$	−1.34598	−0.549496
$7$	5.21276	1.97024	0.985120	−	0.171870i	$-0.0549811\pi$
$7$	5.21276	1.97024	0.985120	+	0.171870i	$0.0549811\pi$
$8$	−2.70993	−0.958105
$9$	−0.243318	−0.0811059
$10$	0	0
$11$	3.14148	0.947191	0.473595	−	0.880743i	$-0.342956\pi$
$11$	3.14148	0.947191	0.473595	+	0.880743i	$0.342956\pi$
$12$	2.22950	0.643600
$13$	−5.93417	−1.64584	−0.822921	−	0.568155i	$-0.807657\pi$
$13$	−5.93417	−1.64584	−0.822921	+	0.568155i	$0.807657\pi$
$14$	4.22586	1.12941
$15$	0	0
$16$	0.488739	0.122185
$17$	−3.59387	−0.871642	−0.435821	−	0.900033i	$-0.643542\pi$
$17$	−3.59387	−0.871642	−0.435821	+	0.900033i	$0.643542\pi$
$18$	−0.197252	−0.0464926
$19$	0	0
$19$	0	0
$20$	0	0
$21$	−8.65489	−1.88865
$22$	2.54672	0.542962
$23$	8.45583	1.76316	0.881581	−	0.472032i	$-0.156479\pi$
$23$	8.45583	1.76316	0.881581	+	0.472032i	$0.156479\pi$
$24$	4.49937	0.918429
$25$	0	0
$26$	−4.81068	−0.943453
$27$	5.38496	1.03634
$28$	−6.99973	−1.32282
$29$	−6.84132	−1.27040	−0.635200	−	0.772347i	$-0.719082\pi$
$29$	−6.84132	−1.27040	−0.635200	+	0.772347i	$0.719082\pi$
$30$	0	0
$31$	−3.11860	−0.560117	−0.280059	−	0.959983i	$-0.590354\pi$
$31$	−3.11860	−0.560117	−0.280059	+	0.959983i	$0.590354\pi$
$32$	5.81607	1.02815
$33$	−5.21588	−0.907967
$34$	−2.91346	−0.499655
$35$	0	0
$36$	0.326728	0.0544547
$37$	−0.373041	−0.0613275	−0.0306638	−	0.999530i	$-0.509762\pi$
$37$	−0.373041	−0.0613275	−0.0306638	+	0.999530i	$0.509762\pi$
$38$	0	0
$39$	9.85266	1.57769
$40$	0	0
$41$	3.60203	0.562542	0.281271	−	0.959628i	$-0.409244\pi$
$41$	3.60203	0.562542	0.281271	+	0.959628i	$0.409244\pi$
$42$	−7.01630	−1.08264
$43$	−5.51851	−0.841565	−0.420783	−	0.907162i	$-0.638244\pi$
$43$	−5.51851	−0.841565	−0.420783	+	0.907162i	$0.638244\pi$
$44$	−4.21839	−0.635947
$45$	0	0
$46$	6.85493	1.01070
$47$	1.04603	0.152579	0.0762896	−	0.997086i	$-0.475693\pi$
$47$	1.04603	0.152579	0.0762896	+	0.997086i	$0.475693\pi$
$48$	−0.811466	−0.117125
$49$	20.1729	2.88184
$50$	0	0
$51$	5.96700	0.835547
$52$	7.96844	1.10502
$53$	−2.81271	−0.386356	−0.193178	−	0.981164i	$-0.561879\pi$
$53$	−2.81271	−0.386356	−0.193178	+	0.981164i	$0.561879\pi$
$54$	4.36546	0.594063
$55$	0	0
$56$	−14.1262	−1.88770
$57$	0	0
$58$	−5.54609	−0.728237
$59$	−7.22336	−0.940402	−0.470201	−	0.882559i	$-0.655819\pi$
$59$	−7.22336	−0.940402	−0.470201	+	0.882559i	$0.655819\pi$
$60$	0	0
$61$	6.96653	0.891973	0.445986	−	0.895040i	$-0.352853\pi$
$61$	6.96653	0.891973	0.445986	+	0.895040i	$0.352853\pi$
$62$	−2.52817	−0.321078
$63$	−1.26836	−0.159798
$64$	3.73746	0.467183
$65$	0	0
$66$	−4.22838	−0.520478
$67$	−0.749780	−0.0916002	−0.0458001	−	0.998951i	$-0.514584\pi$
$67$	−0.749780	−0.0916002	−0.0458001	+	0.998951i	$0.514584\pi$
$68$	4.82587	0.585223
$69$	−14.0394	−1.69015
$70$	0	0
$71$	−2.54581	−0.302132	−0.151066	−	0.988524i	$-0.548271\pi$
$71$	−2.54581	−0.302132	−0.151066	+	0.988524i	$0.548271\pi$
$72$	0.659374	0.0777079
$73$	−7.36554	−0.862071	−0.431035	−	0.902335i	$-0.641852\pi$
$73$	−7.36554	−0.862071	−0.431035	+	0.902335i	$0.641852\pi$
$74$	−0.302415	−0.0351550
$75$	0	0
$76$	0	0
$77$	16.3758	1.86619
$78$	7.98730	0.904384
$79$	9.10321	1.02419	0.512096	−	0.858928i	$-0.328869\pi$
$79$	9.10321	1.02419	0.512096	+	0.858928i	$0.328869\pi$
$80$	0	0
$81$	−8.21084	−0.912316
$82$	2.92007	0.322468
$83$	9.47217	1.03971	0.519853	−	0.854256i	$-0.325987\pi$
$83$	9.47217	1.03971	0.519853	+	0.854256i	$0.325987\pi$
$84$	11.6218	1.26805
$85$	0	0
$86$	−4.47372	−0.482414
$87$	11.3588	1.21779
$88$	−8.51318	−0.907508
$89$	9.87979	1.04726	0.523628	−	0.851947i	$-0.324578\pi$
$89$	9.87979	1.04726	0.523628	+	0.851947i	$0.324578\pi$
$90$	0	0
$91$	−30.9334	−3.24270
$92$	−11.3545	−1.18379
$93$	5.17789	0.536922
$94$	0.847991	0.0874636
$95$	0	0
$96$	−9.65657	−0.985569
$97$	5.88772	0.597808	0.298904	−	0.954283i	$-0.403379\pi$
$97$	5.88772	0.597808	0.298904	+	0.954283i	$0.403379\pi$
$98$	16.3537	1.65197
$99$	−0.764377	−0.0768228
$100$	0	0
$101$	−3.42323	−0.340624	−0.170312	−	0.985390i	$-0.554478\pi$
$101$	−3.42323	−0.340624	−0.170312	+	0.985390i	$0.554478\pi$
$102$	4.83730	0.478964
$103$	−2.67723	−0.263796	−0.131898	−	0.991263i	$-0.542107\pi$
$103$	−2.67723	−0.263796	−0.131898	+	0.991263i	$0.542107\pi$
$104$	16.0812	1.57689
$105$	0	0
$106$	−2.28020	−0.221472
$107$	−5.31595	−0.513912	−0.256956	−	0.966423i	$-0.582720\pi$
$107$	−5.31595	−0.513912	−0.256956	+	0.966423i	$0.582720\pi$
$108$	−7.23096	−0.695800
$109$	−0.350173	−0.0335405	−0.0167702	−	0.999859i	$-0.505338\pi$
$109$	−0.350173	−0.0335405	−0.0167702	+	0.999859i	$0.505338\pi$
$110$	0	0
$111$	0.619369	0.0587879
$112$	2.54768	0.240733
$113$	16.1663	1.52079	0.760397	−	0.649459i	$-0.225005\pi$
$113$	16.1663	1.52079	0.760397	+	0.649459i	$0.225005\pi$
$114$	0	0
$115$	0	0
$116$	9.18656	0.852951
$117$	1.44389	0.133488
$118$	−5.85580	−0.539070
$119$	−18.7340	−1.71734
$120$	0	0
$121$	−1.13112	−0.102829
$122$	5.64759	0.511309
$123$	−5.98054	−0.539247
$124$	4.18767	0.376064
$125$	0	0
$126$	−1.02823	−0.0916016
$127$	3.88495	0.344733	0.172367	−	0.985033i	$-0.444859\pi$
$127$	3.88495	0.344733	0.172367	+	0.985033i	$0.444859\pi$
$128$	−8.60227	−0.760340
$129$	9.16253	0.806716
$130$	0	0
$131$	0.485978	0.0424601	0.0212301	−	0.999775i	$-0.493242\pi$
$131$	0.485978	0.0424601	0.0212301	+	0.999775i	$0.493242\pi$
$132$	7.00391	0.609612
$133$	0	0
$134$	−0.607828	−0.0525084
$135$	0	0
$136$	9.73914	0.835124
$137$	−9.86523	−0.842844	−0.421422	−	0.906865i	$-0.638469\pi$
$137$	−9.86523	−0.842844	−0.421422	+	0.906865i	$0.638469\pi$
$138$	−11.3814	−0.968851
$139$	−3.59641	−0.305044	−0.152522	−	0.988300i	$-0.548739\pi$
$139$	−3.59641	−0.305044	−0.152522	+	0.988300i	$0.548739\pi$
$140$	0	0
$141$	−1.73675	−0.146261
$142$	−2.06383	−0.173193
$143$	−18.6421	−1.55893
$144$	−0.118919	−0.00990991
$145$	0	0
$146$	−5.97106	−0.494168
$147$	−33.4936	−2.76250
$148$	0.500921	0.0411755
$149$	3.40487	0.278938	0.139469	−	0.990226i	$-0.455460\pi$
$149$	3.40487	0.278938	0.139469	+	0.990226i	$0.455460\pi$
$150$	0	0
$151$	7.26888	0.591533	0.295766	−	0.955260i	$-0.404425\pi$
$151$	7.26888	0.591533	0.295766	+	0.955260i	$0.404425\pi$
$152$	0	0
$153$	0.874453	0.0706953
$154$	13.2754	1.06976
$155$	0	0
$156$	−13.2302	−1.05926
$157$	−1.35096	−0.107818	−0.0539092	−	0.998546i	$-0.517168\pi$
$157$	−1.35096	−0.107818	−0.0539092	+	0.998546i	$0.517168\pi$
$158$	7.37975	0.587101
$159$	4.67002	0.370357
$160$	0	0
$161$	44.0782	3.47385
$162$	−6.65633	−0.522970
$163$	−9.78493	−0.766415	−0.383207	−	0.923662i	$-0.625181\pi$
$163$	−9.78493	−0.766415	−0.383207	+	0.923662i	$0.625181\pi$
$164$	−4.83682	−0.377692
$165$	0	0
$166$	7.67885	0.595994
$167$	13.6349	1.05510	0.527551	−	0.849523i	$-0.323110\pi$
$167$	13.6349	1.05510	0.527551	+	0.849523i	$0.323110\pi$
$168$	23.4541	1.80953
$169$	22.2144	1.70880
$170$	0	0
$171$	0	0
$172$	7.41029	0.565029
$173$	−12.7965	−0.972898	−0.486449	−	0.873709i	$-0.661708\pi$
$173$	−12.7965	−0.972898	−0.486449	+	0.873709i	$0.661708\pi$
$174$	9.20831	0.698080
$175$	0	0
$176$	1.53536	0.115732
$177$	11.9931	0.901459
$178$	8.00930	0.600322
$179$	−21.2798	−1.59052	−0.795262	−	0.606266i	$-0.792667\pi$
$179$	−21.2798	−1.59052	−0.795262	+	0.606266i	$0.792667\pi$
$180$	0	0
$181$	6.22350	0.462589	0.231295	−	0.972884i	$-0.425704\pi$
$181$	6.22350	0.462589	0.231295	+	0.972884i	$0.425704\pi$
$182$	−25.0770	−1.85883
$183$	−11.5667	−0.855036
$184$	−22.9147	−1.68929
$185$	0	0
$186$	4.19759	0.307782
$187$	−11.2901	−0.825611
$188$	−1.40462	−0.102442
$189$	28.0705	2.04183
$190$	0	0
$191$	−23.3581	−1.69014	−0.845068	−	0.534660i	$-0.820440\pi$
$191$	−23.3581	−1.69014	−0.845068	+	0.534660i	$0.820440\pi$
$192$	−6.20541	−0.447837
$193$	21.2862	1.53221	0.766107	−	0.642713i	$-0.222191\pi$
$193$	21.2862	1.53221	0.766107	+	0.642713i	$0.222191\pi$
$194$	4.77303	0.342684
$195$	0	0
$196$	−27.0883	−1.93488
$197$	−7.25543	−0.516928	−0.258464	−	0.966021i	$-0.583216\pi$
$197$	−7.25543	−0.516928	−0.258464	+	0.966021i	$0.583216\pi$
$198$	−0.619661	−0.0440374
$199$	−9.25931	−0.656375	−0.328187	−	0.944613i	$-0.606438\pi$
$199$	−9.25931	−0.656375	−0.328187	+	0.944613i	$0.606438\pi$
$200$	0	0
$201$	1.24488	0.0878070
$202$	−2.77513	−0.195257
$203$	−35.6622	−2.50299
$204$	−8.01252	−0.560989
$205$	0	0
$206$	−2.17037	−0.151217
$207$	−2.05745	−0.143003
$208$	−2.90026	−0.201097
$209$	0	0
$210$	0	0
$211$	−2.27544	−0.156648	−0.0783238	−	0.996928i	$-0.524957\pi$
$211$	−2.27544	−0.156648	−0.0783238	+	0.996928i	$0.524957\pi$
$212$	3.77693	0.259400
$213$	4.22688	0.289621
$214$	−4.30951	−0.294592
$215$	0	0
$216$	−14.5929	−0.992919
$217$	−16.2565	−1.10356
$218$	−0.283876	−0.0192265
$219$	12.2292	0.826372
$220$	0	0
$221$	21.3267	1.43459
$222$	0.502107	0.0336992
$223$	22.7572	1.52394	0.761969	−	0.647613i	$-0.224233\pi$
$223$	22.7572	1.52394	0.761969	+	0.647613i	$0.224233\pi$
$224$	30.3178	2.02569
$225$	0	0
$226$	13.1056	0.871770
$227$	−5.81897	−0.386219	−0.193109	−	0.981177i	$-0.561857\pi$
$227$	−5.81897	−0.386219	−0.193109	+	0.981177i	$0.561857\pi$
$228$	0	0
$229$	−20.0623	−1.32575	−0.662877	−	0.748728i	$-0.730665\pi$
$229$	−20.0623	−1.32575	−0.662877	+	0.748728i	$0.730665\pi$
$230$	0	0
$231$	−27.1891	−1.78891
$232$	18.5395	1.21718
$233$	10.8033	0.707751	0.353875	−	0.935293i	$-0.384864\pi$
$233$	10.8033	0.707751	0.353875	+	0.935293i	$0.384864\pi$
$234$	1.17052	0.0765196
$235$	0	0
$236$	9.69957	0.631388
$237$	−15.1143	−0.981779
$238$	−15.1872	−0.984439
$239$	−17.5462	−1.13497	−0.567484	−	0.823384i	$-0.692083\pi$
$239$	−17.5462	−1.13497	−0.567484	+	0.823384i	$0.692083\pi$
$240$	0	0
$241$	13.7035	0.882723	0.441361	−	0.897329i	$-0.354496\pi$
$241$	13.7035	0.882723	0.441361	+	0.897329i	$0.354496\pi$
$242$	−0.916974	−0.0589453
$243$	−2.52222	−0.161800
$244$	−9.35470	−0.598873
$245$	0	0
$246$	−4.84827	−0.309115
$247$	0	0
$248$	8.45119	0.536651
$249$	−15.7269	−0.996651
$250$	0	0
$251$	26.3313	1.66202	0.831009	−	0.556259i	$-0.187764\pi$
$251$	26.3313	1.66202	0.831009	+	0.556259i	$0.187764\pi$
$252$	1.70316	0.107289
$253$	26.5638	1.67005
$254$	3.14943	0.197613
$255$	0	0
$256$	−14.4486	−0.903036
$257$	21.0221	1.31132	0.655660	−	0.755056i	$-0.272391\pi$
$257$	21.0221	1.31132	0.655660	+	0.755056i	$0.272391\pi$
$258$	7.42783	0.462437
$259$	−1.94457	−0.120830
$260$	0	0
$261$	1.66461	0.103037
$262$	0.393971	0.0243396
$263$	25.1989	1.55383	0.776916	−	0.629605i	$-0.216783\pi$
$263$	25.1989	1.55383	0.776916	+	0.629605i	$0.216783\pi$
$264$	14.1347	0.869928
$265$	0	0
$266$	0	0
$267$	−16.4037	−1.00389
$268$	1.00681	0.0615007
$269$	5.92302	0.361133	0.180566	−	0.983563i	$-0.442207\pi$
$269$	5.92302	0.361133	0.180566	+	0.983563i	$0.442207\pi$
$270$	0	0
$271$	5.25200	0.319036	0.159518	−	0.987195i	$-0.449006\pi$
$271$	5.25200	0.319036	0.159518	+	0.987195i	$0.449006\pi$
$272$	−1.75647	−0.106501
$273$	51.3596	3.10842
$274$	−7.99750	−0.483147
$275$	0	0
$276$	18.8522	1.13477
$277$	30.0369	1.80474	0.902371	−	0.430960i	$-0.141825\pi$
$277$	30.0369	1.80474	0.902371	+	0.430960i	$0.141825\pi$
$278$	−2.91552	−0.174861
$279$	0.758810	0.0454288
$280$	0	0
$281$	11.4942	0.685684	0.342842	−	0.939393i	$-0.388610\pi$
$281$	11.4942	0.685684	0.342842	+	0.939393i	$0.388610\pi$
$282$	−1.40794	−0.0838417
$283$	26.7888	1.59243	0.796215	−	0.605013i	$-0.206832\pi$
$283$	26.7888	1.59243	0.796215	+	0.605013i	$0.206832\pi$
$284$	3.41853	0.202853
$285$	0	0
$286$	−15.1127	−0.893630
$287$	18.7765	1.10834
$288$	−1.41515	−0.0833886
$289$	−4.08408	−0.240240
$290$	0	0
$291$	−9.77554	−0.573052
$292$	9.89048	0.578797
$293$	25.1893	1.47157	0.735786	−	0.677214i	$-0.236813\pi$
$293$	25.1893	1.47157	0.735786	+	0.677214i	$0.236813\pi$
$294$	−27.1524	−1.58356
$295$	0	0
$296$	1.01091	0.0587582
$297$	16.9167	0.981609
$298$	2.76025	0.159897
$299$	−50.1783	−2.90189
$300$	0	0
$301$	−28.7667	−1.65808
$302$	5.89270	0.339087
$303$	5.68368	0.326519
$304$	0	0
$305$	0	0
$306$	0.708897	0.0405249
$307$	23.4360	1.33756	0.668781	−	0.743459i	$-0.266816\pi$
$307$	23.4360	1.33756	0.668781	+	0.743459i	$0.266816\pi$
$308$	−21.9895	−1.25297
$309$	4.44508	0.252872
$310$	0	0
$311$	−3.88324	−0.220198	−0.110099	−	0.993921i	$-0.535117\pi$
$311$	−3.88324	−0.220198	−0.110099	+	0.993921i	$0.535117\pi$
$312$	−26.7000	−1.51159
$313$	−14.3819	−0.812913	−0.406456	−	0.913670i	$-0.633236\pi$
$313$	−14.3819	−0.812913	−0.406456	+	0.913670i	$0.633236\pi$
$314$	−1.09519	−0.0618051
$315$	0	0
$316$	−12.2238	−0.687645
$317$	−6.03891	−0.339179	−0.169590	−	0.985515i	$-0.554244\pi$
$317$	−6.03891	−0.339179	−0.169590	+	0.985515i	$0.554244\pi$
$318$	3.78587	0.212301
$319$	−21.4918	−1.20331
$320$	0	0
$321$	8.82621	0.492631
$322$	35.7331	1.99133
$323$	0	0
$324$	11.0256	0.612532
$325$	0	0
$326$	−7.93240	−0.439335
$327$	0.581401	0.0321515
$328$	−9.76123	−0.538974
$329$	5.45271	0.300618
$330$	0	0
$331$	15.6048	0.857716	0.428858	−	0.903372i	$-0.358916\pi$
$331$	15.6048	0.857716	0.428858	+	0.903372i	$0.358916\pi$
$332$	−12.7193	−0.698061
$333$	0.0907674	0.00497402
$334$	11.0535	0.604820
$335$	0	0
$336$	−4.22998	−0.230764
$337$	6.57360	0.358087	0.179044	−	0.983841i	$-0.442700\pi$
$337$	6.57360	0.358087	0.179044	+	0.983841i	$0.442700\pi$
$338$	18.0086	0.979541
$339$	−26.8413	−1.45782
$340$	0	0
$341$	−9.79701	−0.530538
$342$	0	0
$343$	68.6672	3.70768
$344$	14.9548	0.806308
$345$	0	0
$346$	−10.3738	−0.557698
$347$	−26.4761	−1.42131	−0.710655	−	0.703540i	$-0.751601\pi$
$347$	−26.4761	−1.42131	−0.710655	+	0.703540i	$0.751601\pi$
$348$	−15.2527	−0.817630
$349$	3.67426	0.196679	0.0983394	−	0.995153i	$-0.468647\pi$
$349$	3.67426	0.196679	0.0983394	+	0.995153i	$0.468647\pi$
$350$	0	0
$351$	−31.9553	−1.70565
$352$	18.2710	0.973850
$353$	23.9603	1.27528	0.637639	−	0.770335i	$-0.279911\pi$
$353$	23.9603	1.27528	0.637639	+	0.770335i	$0.279911\pi$
$354$	9.72254	0.516747
$355$	0	0
$356$	−13.2666	−0.703130
$357$	31.1046	1.64623
$358$	−17.2510	−0.911743
$359$	15.2947	0.807221	0.403610	−	0.914931i	$-0.367755\pi$
$359$	15.2947	0.807221	0.403610	+	0.914931i	$0.367755\pi$
$360$	0	0
$361$	0	0
$362$	5.04524	0.265172
$363$	1.87803	0.0985712
$364$	41.5376	2.17716
$365$	0	0
$366$	−9.37685	−0.490136
$367$	−1.60652	−0.0838596	−0.0419298	−	0.999121i	$-0.513351\pi$
$367$	−1.60652	−0.0838596	−0.0419298	+	0.999121i	$0.513351\pi$
$368$	4.13270	0.215432
$369$	−0.876436	−0.0456255
$370$	0	0
$371$	−14.6620	−0.761213
$372$	−6.95290	−0.360491
$373$	−10.7159	−0.554850	−0.277425	−	0.960747i	$-0.589481\pi$
$373$	−10.7159	−0.554850	−0.277425	+	0.960747i	$0.589481\pi$
$374$	−9.15258	−0.473268
$375$	0	0
$376$	−2.83467	−0.146187
$377$	40.5975	2.09088
$378$	22.7561	1.17045
$379$	24.9469	1.28144	0.640719	−	0.767776i	$-0.278637\pi$
$379$	24.9469	1.28144	0.640719	+	0.767776i	$0.278637\pi$
$380$	0	0
$381$	−6.45028	−0.330458
$382$	−18.9359	−0.968843
$383$	31.8243	1.62614	0.813072	−	0.582163i	$-0.197793\pi$
$383$	31.8243	1.62614	0.813072	+	0.582163i	$0.197793\pi$
$384$	14.2826	0.728854
$385$	0	0
$386$	17.2562	0.878317
$387$	1.34275	0.0682559
$388$	−7.90607	−0.401370
$389$	9.40808	0.477008	0.238504	−	0.971141i	$-0.423343\pi$
$389$	9.40808	0.477008	0.238504	+	0.971141i	$0.423343\pi$
$390$	0	0
$391$	−30.3892	−1.53685
$392$	−54.6671	−2.76111
$393$	−0.806883	−0.0407018
$394$	−5.88180	−0.296321
$395$	0	0
$396$	1.02641	0.0515790
$397$	19.7057	0.988999	0.494500	−	0.869178i	$-0.335351\pi$
$397$	19.7057	0.988999	0.494500	+	0.869178i	$0.335351\pi$
$398$	−7.50629	−0.376256
$399$	0	0
$400$	0	0
$401$	−24.8633	−1.24161	−0.620806	−	0.783964i	$-0.713195\pi$
$401$	−24.8633	−1.24161	−0.620806	+	0.783964i	$0.713195\pi$
$402$	1.00919	0.0503340
$403$	18.5063	0.921865
$404$	4.59673	0.228696
$405$	0	0
$406$	−28.9104	−1.43480
$407$	−1.17190	−0.0580889
$408$	−16.1701	−0.800542
$409$	13.4013	0.662650	0.331325	−	0.943517i	$-0.392504\pi$
$409$	13.4013	0.662650	0.331325	+	0.943517i	$0.392504\pi$
$410$	0	0
$411$	16.3795	0.807941
$412$	3.59501	0.177113
$413$	−37.6537	−1.85282
$414$	−1.66793	−0.0819741
$415$	0	0
$416$	−34.5135	−1.69217
$417$	5.97122	0.292412
$418$	0	0
$419$	11.4696	0.560326	0.280163	−	0.959952i	$-0.409611\pi$
$419$	11.4696	0.560326	0.280163	+	0.959952i	$0.409611\pi$
$420$	0	0
$421$	−30.3063	−1.47704	−0.738518	−	0.674234i	$-0.764474\pi$
$421$	−30.3063	−1.47704	−0.738518	+	0.674234i	$0.764474\pi$
$422$	−1.84464	−0.0897957
$423$	−0.254518	−0.0123751
$424$	7.62225	0.370169
$425$	0	0
$426$	3.42663	0.166021
$427$	36.3149	1.75740
$428$	7.13829	0.345042
$429$	30.9519	1.49437
$430$	0	0
$431$	7.14175	0.344006	0.172003	−	0.985096i	$-0.444976\pi$
$431$	7.14175	0.344006	0.172003	+	0.985096i	$0.444976\pi$
$432$	2.63184	0.126625
$433$	19.8013	0.951589	0.475795	−	0.879556i	$-0.342161\pi$
$433$	19.8013	0.951589	0.475795	+	0.879556i	$0.342161\pi$
$434$	−13.1788	−0.632601
$435$	0	0
$436$	0.470214	0.0225192
$437$	0	0
$438$	9.91390	0.473704
$439$	18.4741	0.881722	0.440861	−	0.897575i	$-0.354673\pi$
$439$	18.4741	0.881722	0.440861	+	0.897575i	$0.354673\pi$
$440$	0	0
$441$	−4.90842	−0.233734
$442$	17.2890	0.822353
$443$	14.4823	0.688077	0.344039	−	0.938956i	$-0.388205\pi$
$443$	14.4823	0.688077	0.344039	+	0.938956i	$0.388205\pi$
$444$	−0.831693	−0.0394704
$445$	0	0
$446$	18.4487	0.873573
$447$	−5.65320	−0.267387
$448$	19.4825	0.920462
$449$	−3.74142	−0.176568	−0.0882842	−	0.996095i	$-0.528138\pi$
$449$	−3.74142	−0.176568	−0.0882842	+	0.996095i	$0.528138\pi$
$450$	0	0
$451$	11.3157	0.532834
$452$	−21.7081	−1.02107
$453$	−12.0687	−0.567037
$454$	−4.71730	−0.221394
$455$	0	0
$456$	0	0
$457$	−28.4372	−1.33023	−0.665117	−	0.746739i	$-0.731618\pi$
$457$	−28.4372	−1.33023	−0.665117	+	0.746739i	$0.731618\pi$
$458$	−16.2640	−0.759967
$459$	−19.3529	−0.903315
$460$	0	0
$461$	33.2711	1.54959	0.774796	−	0.632212i	$-0.217853\pi$
$461$	33.2711	1.54959	0.774796	+	0.632212i	$0.217853\pi$
$462$	−22.0415	−1.02547
$463$	14.8019	0.687904	0.343952	−	0.938987i	$-0.388234\pi$
$463$	14.8019	0.687904	0.343952	+	0.938987i	$0.388234\pi$
$464$	−3.34362	−0.155224
$465$	0	0
$466$	8.75801	0.405707
$467$	18.7126	0.865916	0.432958	−	0.901414i	$-0.357470\pi$
$467$	18.7126	0.865916	0.432958	+	0.901414i	$0.357470\pi$
$468$	−1.93886	−0.0896239
$469$	−3.90843	−0.180474
$470$	0	0
$471$	2.24303	0.103354
$472$	19.5748	0.901003
$473$	−17.3363	−0.797123
$474$	−12.2528	−0.562789
$475$	0	0
$476$	25.1561	1.15303
$477$	0.684383	0.0313357
$478$	−14.2243	−0.650602
$479$	−28.9800	−1.32413	−0.662065	−	0.749446i	$-0.730320\pi$
$479$	−28.9800	−1.32413	−0.662065	+	0.749446i	$0.730320\pi$
$480$	0	0
$481$	2.21369	0.100935
$482$	11.1091	0.506007
$483$	−73.1842	−3.33000
$484$	1.51888	0.0690400
$485$	0	0
$486$	−2.04470	−0.0927494
$487$	35.1415	1.59242	0.796208	−	0.605024i	$-0.206836\pi$
$487$	35.1415	1.59242	0.796208	+	0.605024i	$0.206836\pi$
$488$	−18.8788	−0.854603
$489$	16.2462	0.734677
$490$	0	0
$491$	17.1383	0.773441	0.386721	−	0.922197i	$-0.373608\pi$
$491$	17.1383	0.773441	0.386721	+	0.922197i	$0.373608\pi$
$492$	8.03070	0.362052
$493$	24.5868	1.10733
$494$	0	0
$495$	0	0
$496$	−1.52418	−0.0684378
$497$	−13.2707	−0.595273
$498$	−12.7494	−0.571314
$499$	−32.8782	−1.47183	−0.735914	−	0.677075i	$-0.763247\pi$
$499$	−32.8782	−1.47183	−0.735914	+	0.677075i	$0.763247\pi$
$500$	0	0
$501$	−22.6384	−1.01141
$502$	21.3461	0.952725
$503$	9.82245	0.437961	0.218981	−	0.975729i	$-0.429727\pi$
$503$	9.82245	0.437961	0.218981	+	0.975729i	$0.429727\pi$
$504$	3.43716	0.153103
$505$	0	0
$506$	21.5346	0.957330
$507$	−36.8831	−1.63804
$508$	−5.21673	−0.231455
$509$	24.8280	1.10048	0.550241	−	0.835006i	$-0.314536\pi$
$509$	24.8280	1.10048	0.550241	+	0.835006i	$0.314536\pi$
$510$	0	0
$511$	−38.3948	−1.69849
$512$	5.49144	0.242690
$513$	0	0
$514$	17.0421	0.751693
$515$	0	0
$516$	−12.3035	−0.541631
$517$	3.28608	0.144522
$518$	−1.57642	−0.0692638
$519$	21.2463	0.932610
$520$	0	0
$521$	−22.6940	−0.994242	−0.497121	−	0.867681i	$-0.665610\pi$
$521$	−22.6940	−0.994242	−0.497121	+	0.867681i	$0.665610\pi$
$522$	1.34946	0.0590643
$523$	9.96175	0.435597	0.217798	−	0.975994i	$-0.430112\pi$
$523$	9.96175	0.435597	0.217798	+	0.975994i	$0.430112\pi$
$524$	−0.652575	−0.0285079
$525$	0	0
$526$	20.4281	0.890709
$527$	11.2079	0.488222
$528$	−2.54920	−0.110940
$529$	48.5011	2.10874
$530$	0	0
$531$	1.75757	0.0762721
$532$	0	0
$533$	−21.3750	−0.925855
$534$	−13.2980	−0.575463
$535$	0	0
$536$	2.03185	0.0877626
$537$	35.3314	1.52466
$538$	4.80164	0.207013
$539$	63.3727	2.72965
$540$	0	0
$541$	40.8858	1.75782	0.878909	−	0.476989i	$-0.158272\pi$
$541$	40.8858	1.75782	0.878909	+	0.476989i	$0.158272\pi$
$542$	4.25767	0.182883
$543$	−10.3330	−0.443433
$544$	−20.9022	−0.896175
$545$	0	0
$546$	41.6359	1.78185
$547$	−9.66465	−0.413231	−0.206615	−	0.978422i	$-0.566245\pi$
$547$	−9.66465	−0.413231	−0.206615	+	0.978422i	$0.566245\pi$
$548$	13.2471	0.565888
$549$	−1.69508	−0.0723443
$550$	0	0
$551$	0	0
$552$	38.0459	1.61934
$553$	47.4529	2.01790
$554$	24.3502	1.03454
$555$	0	0
$556$	4.82928	0.204807
$557$	7.10942	0.301236	0.150618	−	0.988592i	$-0.451874\pi$
$557$	7.10942	0.301236	0.150618	+	0.988592i	$0.451874\pi$
$558$	0.615149	0.0260413
$559$	32.7478	1.38508
$560$	0	0
$561$	18.7452	0.791423
$562$	9.31803	0.393057
$563$	−4.97569	−0.209700	−0.104850	−	0.994488i	$-0.533436\pi$
$563$	−4.97569	−0.209700	−0.104850	+	0.994488i	$0.533436\pi$
$564$	2.33212	0.0982000
$565$	0	0
$566$	21.7170	0.912835
$567$	−42.8012	−1.79748
$568$	6.89897	0.289475
$569$	1.74276	0.0730602	0.0365301	−	0.999333i	$-0.488370\pi$
$569$	1.74276	0.0730602	0.0365301	+	0.999333i	$0.488370\pi$
$570$	0	0
$571$	22.7468	0.951923	0.475961	−	0.879466i	$-0.342100\pi$
$571$	22.7468	0.951923	0.475961	+	0.879466i	$0.342100\pi$
$572$	25.0327	1.04667
$573$	38.7821	1.62015
$574$	15.2216	0.635339
$575$	0	0
$576$	−0.909391	−0.0378913
$577$	19.1215	0.796040	0.398020	−	0.917377i	$-0.369697\pi$
$577$	19.1215	0.796040	0.398020	+	0.917377i	$0.369697\pi$
$578$	−3.31086	−0.137714
$579$	−35.3420	−1.46877
$580$	0	0
$581$	49.3762	2.04847
$582$	−7.92479	−0.328493
$583$	−8.83607	−0.365953
$584$	19.9601	0.825954
$585$	0	0
$586$	20.4203	0.843555
$587$	21.1399	0.872539	0.436269	−	0.899816i	$-0.356299\pi$
$587$	21.1399	0.872539	0.436269	+	0.899816i	$0.356299\pi$
$588$	44.9754	1.85475
$589$	0	0
$590$	0	0
$591$	12.0464	0.495522
$592$	−0.182320	−0.00749329
$593$	22.3994	0.919833	0.459917	−	0.887962i	$-0.347879\pi$
$593$	22.3994	0.919833	0.459917	+	0.887962i	$0.347879\pi$
$594$	13.7140	0.562691
$595$	0	0
$596$	−4.57208	−0.187280
$597$	15.3735	0.629194
$598$	−40.6783	−1.66346
$599$	−40.6177	−1.65959	−0.829797	−	0.558066i	$-0.811544\pi$
$599$	−40.6177	−1.65959	−0.829797	+	0.558066i	$0.811544\pi$
$600$	0	0
$601$	−25.3233	−1.03296	−0.516479	−	0.856300i	$-0.672758\pi$
$601$	−25.3233	−1.03296	−0.516479	+	0.856300i	$0.672758\pi$
$602$	−23.3204	−0.950470
$603$	0.182435	0.00742932
$604$	−9.76069	−0.397157
$605$	0	0
$606$	4.60762	0.187172
$607$	−22.0057	−0.893185	−0.446593	−	0.894737i	$-0.647363\pi$
$607$	−22.0057	−0.893185	−0.446593	+	0.894737i	$0.647363\pi$
$608$	0	0
$609$	59.2108	2.39934
$610$	0	0
$611$	−6.20732	−0.251121
$612$	−1.17422	−0.0474650
$613$	18.1382	0.732596	0.366298	−	0.930498i	$-0.380625\pi$
$613$	18.1382	0.732596	0.366298	+	0.930498i	$0.380625\pi$
$614$	18.9990	0.766736
$615$	0	0
$616$	−44.3772	−1.78801
$617$	10.4828	0.422020	0.211010	−	0.977484i	$-0.432325\pi$
$617$	10.4828	0.422020	0.211010	+	0.977484i	$0.432325\pi$
$618$	3.60352	0.144955
$619$	−27.3590	−1.09965	−0.549826	−	0.835279i	$-0.685306\pi$
$619$	−27.3590	−1.09965	−0.549826	+	0.835279i	$0.685306\pi$
$620$	0	0
$621$	45.5343	1.82723
$622$	−3.14805	−0.126225
$623$	51.5010	2.06334
$624$	4.81538	0.192769
$625$	0	0
$626$	−11.6590	−0.465989
$627$	0	0
$628$	1.81408	0.0723895
$629$	1.34066	0.0534557
$630$	0	0
$631$	−21.2842	−0.847311	−0.423655	−	0.905823i	$-0.639253\pi$
$631$	−21.2842	−0.847311	−0.423655	+	0.905823i	$0.639253\pi$
$632$	−24.6691	−0.981283
$633$	3.77797	0.150161
$634$	−4.89560	−0.194429
$635$	0	0
$636$	−6.27093	−0.248659
$637$	−119.709	−4.74306
$638$	−17.4229	−0.689779
$639$	0.619441	0.0245047
$640$	0	0
$641$	−31.4849	−1.24358	−0.621790	−	0.783184i	$-0.713594\pi$
$641$	−31.4849	−1.24358	−0.621790	+	0.783184i	$0.713594\pi$
$642$	7.15519	0.282393
$643$	28.7933	1.13550	0.567749	−	0.823202i	$-0.307814\pi$
$643$	28.7933	1.13550	0.567749	+	0.823202i	$0.307814\pi$
$644$	−59.1885	−2.33235
$645$	0	0
$646$	0	0
$647$	33.0422	1.29902	0.649512	−	0.760352i	$-0.274973\pi$
$647$	33.0422	1.29902	0.649512	+	0.760352i	$0.274973\pi$
$648$	22.2508	0.874094
$649$	−22.6920	−0.890740
$650$	0	0
$651$	26.9911	1.05787
$652$	13.1393	0.514573
$653$	−5.57173	−0.218038	−0.109019	−	0.994040i	$-0.534771\pi$
$653$	−5.57173	−0.218038	−0.109019	+	0.994040i	$0.534771\pi$
$654$	0.471327	0.0184304
$655$	0	0
$656$	1.76045	0.0687341
$657$	1.79216	0.0699190
$658$	4.42038	0.172324
$659$	13.6005	0.529801	0.264901	−	0.964276i	$-0.414661\pi$
$659$	13.6005	0.529801	0.264901	+	0.964276i	$0.414661\pi$
$660$	0	0
$661$	17.3451	0.674645	0.337322	−	0.941389i	$-0.390479\pi$
$661$	17.3451	0.674645	0.337322	+	0.941389i	$0.390479\pi$
$662$	12.6504	0.491672
$663$	−35.4092	−1.37518
$664$	−25.6689	−0.996147
$665$	0	0
$666$	0.0735829	0.00285128
$667$	−57.8490	−2.23992
$668$	−18.3091	−0.708398
$669$	−37.7845	−1.46083
$670$	0	0
$671$	21.8852	0.844869
$672$	−50.3374	−1.94181
$673$	−21.2208	−0.818001	−0.409000	−	0.912534i	$-0.634123\pi$
$673$	−21.2208	−0.818001	−0.409000	+	0.912534i	$0.634123\pi$
$674$	5.32906	0.205268
$675$	0	0
$676$	−29.8296	−1.14729
$677$	−19.2359	−0.739295	−0.369648	−	0.929172i	$-0.620522\pi$
$677$	−19.2359	−0.739295	−0.369648	+	0.929172i	$0.620522\pi$
$678$	−21.7595	−0.835670
$679$	30.6913	1.17782
$680$	0	0
$681$	9.66139	0.370225
$682$	−7.94219	−0.304122
$683$	26.7765	1.02458	0.512288	−	0.858814i	$-0.328798\pi$
$683$	26.7765	1.02458	0.512288	+	0.858814i	$0.328798\pi$
$684$	0	0
$685$	0	0
$686$	55.6668	2.12537
$687$	33.3100	1.27085
$688$	−2.69711	−0.102826
$689$	16.6911	0.635881
$690$	0	0
$691$	9.12738	0.347222	0.173611	−	0.984814i	$-0.444457\pi$
$691$	9.12738	0.347222	0.173611	+	0.984814i	$0.444457\pi$
$692$	17.1832	0.653207
$693$	−3.98452	−0.151359
$694$	−21.4635	−0.814743
$695$	0	0
$696$	−30.7816	−1.16677
$697$	−12.9452	−0.490335
$698$	2.97864	0.112743
$699$	−17.9371	−0.678443
$700$	0	0
$701$	−26.3690	−0.995945	−0.497972	−	0.867193i	$-0.665922\pi$
$701$	−26.3690	−0.995945	−0.497972	+	0.867193i	$0.665922\pi$
$702$	−25.9054	−0.977735
$703$	0	0
$704$	11.7412	0.442511
$705$	0	0
$706$	19.4240	0.731033
$707$	−17.8445	−0.671111
$708$	−16.1045	−0.605242
$709$	−10.7590	−0.404062	−0.202031	−	0.979379i	$-0.564754\pi$
$709$	−10.7590	−0.404062	−0.202031	+	0.979379i	$0.564754\pi$
$710$	0	0
$711$	−2.21497	−0.0830680
$712$	−26.7735	−1.00338
$713$	−26.3704	−0.987577
$714$	25.2157	0.943673
$715$	0	0
$716$	28.5746	1.06788
$717$	29.1324	1.08797
$718$	12.3990	0.462726
$719$	23.3758	0.871771	0.435886	−	0.900002i	$-0.356435\pi$
$719$	23.3758	0.871771	0.435886	+	0.900002i	$0.356435\pi$
$720$	0	0
$721$	−13.9558	−0.519741
$722$	0	0
$723$	−22.7523	−0.846169
$724$	−8.35696	−0.310584
$725$	0	0
$726$	1.52248	0.0565044
$727$	−21.9311	−0.813380	−0.406690	−	0.913566i	$-0.633317\pi$
$727$	−21.9311	−0.813380	−0.406690	+	0.913566i	$0.633317\pi$
$728$	83.8274	3.10685
$729$	28.8202	1.06742
$730$	0	0
$731$	19.8328	0.733544
$732$	15.5318	0.574074
$733$	−4.10167	−0.151499	−0.0757493	−	0.997127i	$-0.524135\pi$
$733$	−4.10167	−0.151499	−0.0757493	+	0.997127i	$0.524135\pi$
$734$	−1.30237	−0.0480712
$735$	0	0
$736$	49.1797	1.81279
$737$	−2.35542	−0.0867629
$738$	−0.710505	−0.0261541
$739$	−27.0374	−0.994587	−0.497293	−	0.867582i	$-0.665673\pi$
$739$	−27.0374	−0.994587	−0.497293	+	0.867582i	$0.665673\pi$
$740$	0	0
$741$	0	0
$742$	−11.8861	−0.436353
$743$	−22.7737	−0.835485	−0.417742	−	0.908566i	$-0.637179\pi$
$743$	−22.7737	−0.835485	−0.417742	+	0.908566i	$0.637179\pi$
$744$	−14.0317	−0.514428
$745$	0	0
$746$	−8.68714	−0.318059
$747$	−2.30475	−0.0843262
$748$	15.1604	0.554318
$749$	−27.7108	−1.01253
$750$	0	0
$751$	−9.20689	−0.335964	−0.167982	−	0.985790i	$-0.553725\pi$
$751$	−9.20689	−0.335964	−0.167982	+	0.985790i	$0.553725\pi$
$752$	0.511236	0.0186429
$753$	−43.7186	−1.59319
$754$	32.9114	1.19856
$755$	0	0
$756$	−37.6933	−1.37089
$757$	49.8137	1.81051	0.905255	−	0.424869i	$-0.139680\pi$
$757$	49.8137	1.81051	0.905255	+	0.424869i	$0.139680\pi$
$758$	20.2238	0.734563
$759$	−44.1046	−1.60089
$760$	0	0
$761$	5.86513	0.212611	0.106305	−	0.994334i	$-0.466098\pi$
$761$	5.86513	0.212611	0.106305	+	0.994334i	$0.466098\pi$
$762$	−5.22908	−0.189430
$763$	−1.82537	−0.0660827
$764$	31.3654	1.13476
$765$	0	0
$766$	25.7992	0.932161
$767$	42.8647	1.54775
$768$	23.9893	0.865641
$769$	7.92140	0.285653	0.142826	−	0.989748i	$-0.454381\pi$
$769$	7.92140	0.285653	0.142826	+	0.989748i	$0.454381\pi$
$770$	0	0
$771$	−34.9035	−1.25702
$772$	−28.5832	−1.02873
$773$	40.8315	1.46861	0.734303	−	0.678822i	$-0.237509\pi$
$773$	40.8315	1.46861	0.734303	+	0.678822i	$0.237509\pi$
$774$	1.08854	0.0391266
$775$	0	0
$776$	−15.9553	−0.572762
$777$	3.22863	0.115826
$778$	7.62689	0.273437
$779$	0	0
$780$	0	0
$781$	−7.99761	−0.286177
$782$	−24.6358	−0.880973
$783$	−36.8403	−1.31656
$784$	9.85928	0.352117
$785$	0	0
$786$	−0.654120	−0.0233317
$787$	−25.0801	−0.894009	−0.447005	−	0.894532i	$-0.647509\pi$
$787$	−25.0801	−0.894009	−0.447005	+	0.894532i	$0.647509\pi$
$788$	9.74264	0.347067
$789$	−41.8384	−1.48949
$790$	0	0
$791$	84.2709	2.99633
$792$	2.07141	0.0736043
$793$	−41.3406	−1.46805
$794$	15.9749	0.566928
$795$	0	0
$796$	12.4335	0.440692
$797$	33.6403	1.19160	0.595800	−	0.803133i	$-0.296835\pi$
$797$	33.6403	1.19160	0.595800	+	0.803133i	$0.296835\pi$
$798$	0	0
$799$	−3.75930	−0.132994
$800$	0	0
$801$	−2.40393	−0.0849386
$802$	−20.1560	−0.711734
$803$	−23.1387	−0.816546
$804$	−1.67163	−0.0589539
$805$	0	0
$806$	15.0026	0.528444
$807$	−9.83414	−0.346178
$808$	9.27671	0.326354
$809$	2.48812	0.0874778	0.0437389	−	0.999043i	$-0.486073\pi$
$809$	2.48812	0.0874778	0.0437389	+	0.999043i	$0.486073\pi$
$810$	0	0
$811$	−8.75981	−0.307598	−0.153799	−	0.988102i	$-0.549151\pi$
$811$	−8.75981	−0.307598	−0.153799	+	0.988102i	$0.549151\pi$
$812$	47.8874	1.68052
$813$	−8.72004	−0.305825
$814$	−0.950029	−0.0332985
$815$	0	0
$816$	2.91631	0.102091
$817$	0	0
$818$	10.8641	0.379854
$819$	7.52665	0.263002
$820$	0	0
$821$	−35.7896	−1.24907	−0.624533	−	0.780998i	$-0.714711\pi$
$821$	−35.7896	−1.24907	−0.624533	+	0.780998i	$0.714711\pi$
$822$	13.2785	0.463139
$823$	−51.8547	−1.80754	−0.903771	−	0.428015i	$-0.859213\pi$
$823$	−51.8547	−1.80754	−0.903771	+	0.428015i	$0.859213\pi$
$824$	7.25512	0.252744
$825$	0	0
$826$	−30.5249	−1.06210
$827$	18.3836	0.639262	0.319631	−	0.947542i	$-0.396441\pi$
$827$	18.3836	0.639262	0.319631	+	0.947542i	$0.396441\pi$
$828$	2.76276	0.0960125
$829$	−48.3995	−1.68098	−0.840491	−	0.541825i	$-0.817734\pi$
$829$	−48.3995	−1.68098	−0.840491	+	0.541825i	$0.817734\pi$
$830$	0	0
$831$	−49.8710	−1.73001
$832$	−22.1787	−0.768910
$833$	−72.4988	−2.51194
$834$	4.84072	0.167620
$835$	0	0
$836$	0	0
$837$	−16.7935	−0.580470
$838$	9.29811	0.321198
$839$	51.9032	1.79190	0.895948	−	0.444158i	$-0.146497\pi$
$839$	51.9032	1.79190	0.895948	+	0.444158i	$0.146497\pi$
$840$	0	0
$841$	17.8036	0.613918
$842$	−24.5685	−0.846687
$843$	−19.0840	−0.657290
$844$	3.05547	0.105174
$845$	0	0
$846$	−0.206331	−0.00709381
$847$	−5.89628	−0.202599
$848$	−1.37468	−0.0472068
$849$	−44.4782	−1.52649
$850$	0	0
$851$	−3.15437	−0.108130
$852$	−5.67588	−0.194452
$853$	−22.3160	−0.764087	−0.382043	−	0.924144i	$-0.624779\pi$
$853$	−22.3160	−0.764087	−0.382043	+	0.924144i	$0.624779\pi$
$854$	29.4396	1.00740
$855$	0	0
$856$	14.4058	0.492382
$857$	−9.28809	−0.317275	−0.158638	−	0.987337i	$-0.550710\pi$
$857$	−9.28809	−0.317275	−0.158638	+	0.987337i	$0.550710\pi$
$858$	25.0919	0.856624
$859$	−29.6520	−1.01171	−0.505856	−	0.862618i	$-0.668823\pi$
$859$	−29.6520	−1.01171	−0.505856	+	0.862618i	$0.668823\pi$
$860$	0	0
$861$	−31.1751	−1.06245
$862$	5.78964	0.197196
$863$	49.5092	1.68531	0.842656	−	0.538452i	$-0.180991\pi$
$863$	49.5092	1.68531	0.842656	+	0.538452i	$0.180991\pi$
$864$	31.3193	1.06550
$865$	0	0
$866$	16.0524	0.545483
$867$	6.78090	0.230292
$868$	21.8294	0.740936
$869$	28.5975	0.970105
$870$	0	0
$871$	4.44932	0.150760
$872$	0.948943	0.0321353
$873$	−1.43259	−0.0484857
$874$	0	0
$875$	0	0
$876$	−16.4214	−0.554829
$877$	−4.48885	−0.151578	−0.0757888	−	0.997124i	$-0.524147\pi$
$877$	−4.48885	−0.151578	−0.0757888	+	0.997124i	$0.524147\pi$
$878$	14.9765	0.505433
$879$	−41.8224	−1.41063
$880$	0	0
$881$	48.7018	1.64081	0.820403	−	0.571786i	$-0.193749\pi$
$881$	48.7018	1.64081	0.820403	+	0.571786i	$0.193749\pi$
$882$	−3.97914	−0.133984
$883$	9.86848	0.332101	0.166050	−	0.986117i	$-0.446899\pi$
$883$	9.86848	0.332101	0.166050	+	0.986117i	$0.446899\pi$
$884$	−28.6376	−0.963185
$885$	0	0
$886$	11.7405	0.394429
$887$	32.8117	1.10171	0.550854	−	0.834601i	$-0.314302\pi$
$887$	32.8117	1.10171	0.550854	+	0.834601i	$0.314302\pi$
$888$	−1.67845	−0.0563250
$889$	20.2513	0.679207
$890$	0	0
$891$	−25.7942	−0.864137
$892$	−30.5586	−1.02318
$893$	0	0
$894$	−4.58291	−0.153275
$895$	0	0
$896$	−44.8416	−1.49805
$897$	83.3124	2.78172
$898$	−3.03307	−0.101215
$899$	21.3353	0.711573
$900$	0	0
$901$	10.1085	0.336764
$902$	9.17334	0.305439
$903$	47.7621	1.58942
$904$	−43.8094	−1.45708
$905$	0	0
$906$	−9.78380	−0.325045
$907$	−40.1137	−1.33195	−0.665977	−	0.745972i	$-0.731985\pi$
$907$	−40.1137	−1.33195	−0.665977	+	0.745972i	$0.731985\pi$
$908$	7.81375	0.259309
$909$	0.832932	0.0276266
$910$	0	0
$911$	−35.7221	−1.18353	−0.591763	−	0.806112i	$-0.701568\pi$
$911$	−35.7221	−1.18353	−0.591763	+	0.806112i	$0.701568\pi$
$912$	0	0
$913$	29.7566	0.984799
$914$	−23.0533	−0.762535
$915$	0	0
$916$	26.9398	0.890115
$917$	2.53329	0.0836566
$918$	−15.6889	−0.517811
$919$	−16.3383	−0.538952	−0.269476	−	0.963007i	$-0.586850\pi$
$919$	−16.3383	−0.538952	−0.269476	+	0.963007i	$0.586850\pi$
$920$	0	0
$921$	−38.9114	−1.28217
$922$	26.9721	0.888278
$923$	15.1073	0.497262
$924$	36.5097	1.20108
$925$	0	0
$926$	11.9996	0.394330
$927$	0.651419	0.0213954
$928$	−39.7896	−1.30616
$929$	27.3872	0.898546	0.449273	−	0.893394i	$-0.351683\pi$
$929$	27.3872	0.898546	0.449273	+	0.893394i	$0.351683\pi$
$930$	0	0
$931$	0	0
$932$	−14.5068	−0.475186
$933$	6.44744	0.211080
$934$	15.1698	0.496372
$935$	0	0
$936$	−3.91284	−0.127895
$937$	−23.1863	−0.757465	−0.378732	−	0.925506i	$-0.623640\pi$
$937$	−23.1863	−0.757465	−0.378732	+	0.925506i	$0.623640\pi$
$938$	−3.16846	−0.103454
$939$	23.8786	0.779250
$940$	0	0
$941$	30.5107	0.994621	0.497310	−	0.867573i	$-0.334321\pi$
$941$	30.5107	0.994621	0.497310	+	0.867573i	$0.334321\pi$
$942$	1.81837	0.0592457
$943$	30.4581	0.991853
$944$	−3.53034	−0.114903
$945$	0	0
$946$	−14.0541	−0.456938
$947$	9.95293	0.323427	0.161713	−	0.986838i	$-0.448298\pi$
$947$	9.95293	0.323427	0.161713	+	0.986838i	$0.448298\pi$
$948$	20.2956	0.659170
$949$	43.7083	1.41883
$950$	0	0
$951$	10.0266	0.325134
$952$	50.7678	1.64539
$953$	−32.8613	−1.06448	−0.532241	−	0.846593i	$-0.678650\pi$
$953$	−32.8613	−1.06448	−0.532241	+	0.846593i	$0.678650\pi$
$954$	0.554812	0.0179627
$955$	0	0
$956$	23.5611	0.762021
$957$	35.6835	1.15348
$958$	−23.4934	−0.759036
$959$	−51.4251	−1.66060
$960$	0	0
$961$	−21.2743	−0.686269
$962$	1.79458	0.0578596
$963$	1.29346	0.0416813
$964$	−18.4012	−0.592663
$965$	0	0
$966$	−59.3286	−1.90887
$967$	−44.3051	−1.42476	−0.712378	−	0.701796i	$-0.752382\pi$
$967$	−44.3051	−1.42476	−0.712378	+	0.701796i	$0.752382\pi$
$968$	3.06527	0.0985214
$969$	0	0
$970$	0	0
$971$	−18.9040	−0.606657	−0.303328	−	0.952886i	$-0.598098\pi$
$971$	−18.9040	−0.606657	−0.303328	+	0.952886i	$0.598098\pi$
$972$	3.38685	0.108633
$973$	−18.7472	−0.601009
$974$	28.4884	0.912826
$975$	0	0
$976$	3.40482	0.108986
$977$	23.7328	0.759280	0.379640	−	0.925134i	$-0.376048\pi$
$977$	23.7328	0.759280	0.379640	+	0.925134i	$0.376048\pi$
$978$	13.1704	0.421142
$979$	31.0371	0.991951
$980$	0	0
$981$	0.0852032	0.00272033
$982$	13.8936	0.443363
$983$	−27.7056	−0.883671	−0.441835	−	0.897096i	$-0.645672\pi$
$983$	−27.7056	−0.883671	−0.441835	+	0.897096i	$0.645672\pi$
$984$	16.2068	0.516655
$985$	0	0
$986$	19.9319	0.634762
$987$	−9.05327	−0.288169
$988$	0	0
$989$	−46.6636	−1.48382
$990$	0	0
$991$	15.1329	0.480712	0.240356	−	0.970685i	$-0.422736\pi$
$991$	15.1329	0.480712	0.240356	+	0.970685i	$0.422736\pi$
$992$	−18.1380	−0.575882
$993$	−25.9090	−0.822198
$994$	−10.7582	−0.341231
$995$	0	0
$996$	21.1181	0.669154
$997$	24.4715	0.775020	0.387510	−	0.921866i	$-0.373335\pi$
$997$	24.4715	0.775020	0.387510	+	0.921866i	$0.373335\pi$
$998$	−26.6535	−0.843702
$999$	−2.00881	−0.0635560

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9025.2.a.co.1.13	yes	20
5.4	even	2		9025.2.a.cn.1.8	✓	20
19.18	odd	2	inner	9025.2.a.co.1.8	yes	20
95.94	odd	2		9025.2.a.cn.1.13	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9025.2.a.cn.1.8	✓	20	5.4	even	2
9025.2.a.cn.1.13	yes	20	95.94	odd	2
9025.2.a.co.1.8	yes	20	19.18	odd	2	inner
9025.2.a.co.1.13	yes	20	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:	\( N \)	\(=\)	\( 9025 = 5^{2} \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9025.a (trivial)

\(f(q)\)	\(=\)	\(q+0.810675 q^{2} -1.66033 q^{3} -1.34281 q^{4} -1.34598 q^{6} +5.21276 q^{7} -2.70993 q^{8} -0.243318 q^{9} +O(q^{10})\)	\(q+0.810675 q^{2} -1.66033 q^{3} -1.34281 q^{4} -1.34598 q^{6} +5.21276 q^{7} -2.70993 q^{8} -0.243318 q^{9} +3.14148 q^{11} +2.22950 q^{12} -5.93417 q^{13} +4.22586 q^{14} +0.488739 q^{16} -3.59387 q^{17} -0.197252 q^{18} -8.65489 q^{21} +2.54672 q^{22} +8.45583 q^{23} +4.49937 q^{24} -4.81068 q^{26} +5.38496 q^{27} -6.99973 q^{28} -6.84132 q^{29} -3.11860 q^{31} +5.81607 q^{32} -5.21588 q^{33} -2.91346 q^{34} +0.326728 q^{36} -0.373041 q^{37} +9.85266 q^{39} +3.60203 q^{41} -7.01630 q^{42} -5.51851 q^{43} -4.21839 q^{44} +6.85493 q^{46} +1.04603 q^{47} -0.811466 q^{48} +20.1729 q^{49} +5.96700 q^{51} +7.96844 q^{52} -2.81271 q^{53} +4.36546 q^{54} -14.1262 q^{56} -5.54609 q^{58} -7.22336 q^{59} +6.96653 q^{61} -2.52817 q^{62} -1.26836 q^{63} +3.73746 q^{64} -4.22838 q^{66} -0.749780 q^{67} +4.82587 q^{68} -14.0394 q^{69} -2.54581 q^{71} +0.659374 q^{72} -7.36554 q^{73} -0.302415 q^{74} +16.3758 q^{77} +7.98730 q^{78} +9.10321 q^{79} -8.21084 q^{81} +2.92007 q^{82} +9.47217 q^{83} +11.6218 q^{84} -4.47372 q^{86} +11.3588 q^{87} -8.51318 q^{88} +9.87979 q^{89} -30.9334 q^{91} -11.3545 q^{92} +5.17789 q^{93} +0.847991 q^{94} -9.65657 q^{96} +5.88772 q^{97} +16.3537 q^{98} -0.764377 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 24 q^{4} - 20 q^{6} + 8 q^{7} + 16 q^{9}+O(q^{10}) \) 20 * q + 24 * q^4 - 20 * q^6 + 8 * q^7 + 16 * q^9	\( 20 q + 24 q^{4} - 20 q^{6} + 8 q^{7} + 16 q^{9} + 10 q^{11} + 40 q^{16} + 38 q^{17} + 56 q^{23} - 54 q^{24} - 22 q^{26} - 16 q^{28} - 32 q^{36} + 80 q^{39} + 44 q^{43} - 16 q^{44} + 98 q^{47} - 4 q^{49} - 54 q^{54} + 50 q^{58} + 40 q^{61} + 90 q^{62} + 6 q^{63} + 46 q^{64} + 8 q^{66} + 4 q^{68} + 34 q^{73} + 20 q^{74} + 80 q^{77} + 60 q^{81} - 10 q^{82} + 84 q^{83} + 90 q^{87} + 78 q^{92} + 20 q^{93} - 120 q^{96} - 76 q^{99}+O(q^{100}) \) 20 * q + 24 * q^4 - 20 * q^6 + 8 * q^7 + 16 * q^9 + 10 * q^11 + 40 * q^16 + 38 * q^17 + 56 * q^23 - 54 * q^24 - 22 * q^26 - 16 * q^28 - 32 * q^36 + 80 * q^39 + 44 * q^43 - 16 * q^44 + 98 * q^47 - 4 * q^49 - 54 * q^54 + 50 * q^58 + 40 * q^61 + 90 * q^62 + 6 * q^63 + 46 * q^64 + 8 * q^66 + 4 * q^68 + 34 * q^73 + 20 * q^74 + 80 * q^77 + 60 * q^81 - 10 * q^82 + 84 * q^83 + 90 * q^87 + 78 * q^92 + 20 * q^93 - 120 * q^96 - 76 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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