LMFDB - Embedded newform 9025.2.a.ct.1.12

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:	$1$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 95)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
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)

$f(q)$	$=$	$q-0.244477 q^{2} -2.73837 q^{3} -1.94023 q^{4} +0.669469 q^{6} +1.94027 q^{7} +0.963297 q^{8} +4.49866 q^{9} +O(q^{10})$	\(q-0.244477 q^{2} -2.73837 q^{3} -1.94023 q^{4} +0.669469 q^{6} +1.94027 q^{7} +0.963297 q^{8} +4.49866 q^{9} +4.23333 q^{11} +5.31307 q^{12} -1.26982 q^{13} -0.474352 q^{14} +3.64496 q^{16} +2.46021 q^{17} -1.09982 q^{18} -5.31318 q^{21} -1.03495 q^{22} +5.13716 q^{23} -2.63786 q^{24} +0.310443 q^{26} -4.10387 q^{27} -3.76457 q^{28} -8.64123 q^{29} -5.10084 q^{31} -2.81770 q^{32} -11.5924 q^{33} -0.601466 q^{34} -8.72843 q^{36} +11.0305 q^{37} +3.47724 q^{39} -2.49838 q^{41} +1.29895 q^{42} -4.46879 q^{43} -8.21364 q^{44} -1.25592 q^{46} -6.76901 q^{47} -9.98123 q^{48} -3.23535 q^{49} -6.73697 q^{51} +2.46375 q^{52} +0.689802 q^{53} +1.00330 q^{54} +1.86906 q^{56} +2.11258 q^{58} -5.19336 q^{59} +2.80281 q^{61} +1.24704 q^{62} +8.72862 q^{63} -6.60105 q^{64} +2.83408 q^{66} -5.21227 q^{67} -4.77338 q^{68} -14.0674 q^{69} -0.791916 q^{71} +4.33354 q^{72} -1.41576 q^{73} -2.69671 q^{74} +8.21381 q^{77} -0.850107 q^{78} +6.06294 q^{79} -2.25806 q^{81} +0.610798 q^{82} -11.8522 q^{83} +10.3088 q^{84} +1.09252 q^{86} +23.6629 q^{87} +4.07795 q^{88} +2.23291 q^{89} -2.46380 q^{91} -9.96727 q^{92} +13.9680 q^{93} +1.65487 q^{94} +7.71591 q^{96} -8.60981 q^{97} +0.790969 q^{98} +19.0443 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 18 q^{4} - 12 q^{6} + 12 q^{9}+O(q^{10}) $ 24 * q + 18 * q^4 - 12 * q^6 + 12 * q^9	$ 24 q + 18 q^{4} - 12 q^{6} + 12 q^{9} + 12 q^{11} - 24 q^{14} + 6 q^{16} - 6 q^{21} - 42 q^{24} - 12 q^{26} - 36 q^{29} - 42 q^{31} - 6 q^{34} - 6 q^{36} + 24 q^{39} - 60 q^{41} - 30 q^{44} - 6 q^{46} + 12 q^{49} - 30 q^{51} - 24 q^{54} - 18 q^{56} - 60 q^{59} + 30 q^{61} + 36 q^{66} - 66 q^{69} - 96 q^{71} + 24 q^{74} - 72 q^{79} - 96 q^{81} + 54 q^{84} - 108 q^{86} - 84 q^{89} - 96 q^{91} - 36 q^{94} - 120 q^{96}+O(q^{100}) $ 24 * q + 18 * q^4 - 12 * q^6 + 12 * q^9 + 12 * q^11 - 24 * q^14 + 6 * q^16 - 6 * q^21 - 42 * q^24 - 12 * q^26 - 36 * q^29 - 42 * q^31 - 6 * q^34 - 6 * q^36 + 24 * q^39 - 60 * q^41 - 30 * q^44 - 6 * q^46 + 12 * q^49 - 30 * q^51 - 24 * q^54 - 18 * q^56 - 60 * q^59 + 30 * q^61 + 36 * q^66 - 66 * q^69 - 96 * q^71 + 24 * q^74 - 72 * q^79 - 96 * q^81 + 54 * q^84 - 108 * q^86 - 84 * q^89 - 96 * q^91 - 36 * q^94 - 120 * q^96

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 $

$2$	−0.244477	−0.172872	−0.0864358	−	0.996257i	$-0.527548\pi$
$2$	−0.244477	−0.172872	−0.0864358	+	0.996257i	$0.527548\pi$
$3$	−2.73837	−1.58100	−0.790499	−	0.612464i	$-0.790179\pi$
$3$	−2.73837	−1.58100	−0.790499	+	0.612464i	$0.790179\pi$
$4$	−1.94023	−0.970115
$5$	0	0
$5$	0	0
$6$	0.669469	0.273309
$7$	1.94027	0.733354	0.366677	−	0.930348i	$-0.380495\pi$
$7$	1.94027	0.733354	0.366677	+	0.930348i	$0.380495\pi$
$8$	0.963297	0.340577
$9$	4.49866	1.49955
$10$	0	0
$11$	4.23333	1.27640	0.638198	−	0.769872i	$-0.279680\pi$
$11$	4.23333	1.27640	0.638198	+	0.769872i	$0.279680\pi$
$12$	5.31307	1.53375
$13$	−1.26982	−0.352186	−0.176093	−	0.984374i	$-0.556346\pi$
$13$	−1.26982	−0.352186	−0.176093	+	0.984374i	$0.556346\pi$
$14$	−0.474352	−0.126776
$15$	0	0
$16$	3.64496	0.911239
$17$	2.46021	0.596689	0.298345	−	0.954458i	$-0.403566\pi$
$17$	2.46021	0.596689	0.298345	+	0.954458i	$0.403566\pi$
$18$	−1.09982	−0.259230
$19$	0	0
$19$	0	0
$20$	0	0
$21$	−5.31318	−1.15943
$22$	−1.03495	−0.220653
$23$	5.13716	1.07117	0.535586	−	0.844481i	$-0.320091\pi$
$23$	5.13716	1.07117	0.535586	+	0.844481i	$0.320091\pi$
$24$	−2.63786	−0.538451
$25$	0	0
$26$	0.310443	0.0608829
$27$	−4.10387	−0.789791
$28$	−3.76457	−0.711438
$29$	−8.64123	−1.60464	−0.802318	−	0.596897i	$-0.796400\pi$
$29$	−8.64123	−1.60464	−0.802318	+	0.596897i	$0.796400\pi$
$30$	0	0
$31$	−5.10084	−0.916138	−0.458069	−	0.888917i	$-0.651459\pi$
$31$	−5.10084	−0.916138	−0.458069	+	0.888917i	$0.651459\pi$
$32$	−2.81770	−0.498104
$33$	−11.5924	−2.01798
$34$	−0.601466	−0.103151
$35$	0	0
$36$	−8.72843	−1.45474
$37$	11.0305	1.81341	0.906703	−	0.421770i	$-0.138591\pi$
$37$	11.0305	1.81341	0.906703	+	0.421770i	$0.138591\pi$
$38$	0	0
$39$	3.47724	0.556805
$40$	0	0
$41$	−2.49838	−0.390182	−0.195091	−	0.980785i	$-0.562500\pi$
$41$	−2.49838	−0.390182	−0.195091	+	0.980785i	$0.562500\pi$
$42$	1.29895	0.200432
$43$	−4.46879	−0.681484	−0.340742	−	0.940157i	$-0.610678\pi$
$43$	−4.46879	−0.681484	−0.340742	+	0.940157i	$0.610678\pi$
$44$	−8.21364	−1.23825
$45$	0	0
$46$	−1.25592	−0.185175
$47$	−6.76901	−0.987362	−0.493681	−	0.869643i	$-0.664349\pi$
$47$	−6.76901	−0.987362	−0.493681	+	0.869643i	$0.664349\pi$
$48$	−9.98123	−1.44067
$49$	−3.23535	−0.462192
$50$	0	0
$51$	−6.73697	−0.943364
$52$	2.46375	0.341661
$53$	0.689802	0.0947515	0.0473758	−	0.998877i	$-0.484914\pi$
$53$	0.689802	0.0947515	0.0473758	+	0.998877i	$0.484914\pi$
$54$	1.00330	0.136532
$55$	0	0
$56$	1.86906	0.249763
$57$	0	0
$58$	2.11258	0.277396
$59$	−5.19336	−0.676118	−0.338059	−	0.941125i	$-0.609770\pi$
$59$	−5.19336	−0.676118	−0.338059	+	0.941125i	$0.609770\pi$
$60$	0	0
$61$	2.80281	0.358864	0.179432	−	0.983770i	$-0.442574\pi$
$61$	2.80281	0.358864	0.179432	+	0.983770i	$0.442574\pi$
$62$	1.24704	0.158374
$63$	8.72862	1.09970
$64$	−6.60105	−0.825131
$65$	0	0
$66$	2.83408	0.348851
$67$	−5.21227	−0.636780	−0.318390	−	0.947960i	$-0.603142\pi$
$67$	−5.21227	−0.636780	−0.318390	+	0.947960i	$0.603142\pi$
$68$	−4.77338	−0.578858
$69$	−14.0674	−1.69352
$70$	0	0
$71$	−0.791916	−0.0939832	−0.0469916	−	0.998895i	$-0.514963\pi$
$71$	−0.791916	−0.0939832	−0.0469916	+	0.998895i	$0.514963\pi$
$72$	4.33354	0.510713
$73$	−1.41576	−0.165702	−0.0828512	−	0.996562i	$-0.526403\pi$
$73$	−1.41576	−0.165702	−0.0828512	+	0.996562i	$0.526403\pi$
$74$	−2.69671	−0.313486
$75$	0	0
$76$	0	0
$77$	8.21381	0.936050
$78$	−0.850107	−0.0962557
$79$	6.06294	0.682134	0.341067	−	0.940039i	$-0.389212\pi$
$79$	6.06294	0.682134	0.341067	+	0.940039i	$0.389212\pi$
$80$	0	0
$81$	−2.25806	−0.250895
$82$	0.610798	0.0674514
$83$	−11.8522	−1.30095	−0.650476	−	0.759527i	$-0.725430\pi$
$83$	−11.8522	−1.30095	−0.650476	+	0.759527i	$0.725430\pi$
$84$	10.3088	1.12478
$85$	0	0
$86$	1.09252	0.117809
$87$	23.6629	2.53692
$88$	4.07795	0.434711
$89$	2.23291	0.236688	0.118344	−	0.992973i	$-0.462241\pi$
$89$	2.23291	0.236688	0.118344	+	0.992973i	$0.462241\pi$
$90$	0	0
$91$	−2.46380	−0.258277
$92$	−9.96727	−1.03916
$93$	13.9680	1.44841
$94$	1.65487	0.170687
$95$	0	0
$96$	7.71591	0.787501
$97$	−8.60981	−0.874194	−0.437097	−	0.899414i	$-0.643993\pi$
$97$	−8.60981	−0.874194	−0.437097	+	0.899414i	$0.643993\pi$
$98$	0.790969	0.0798999
$99$	19.0443	1.91402
$100$	0	0
$101$	−13.4510	−1.33842	−0.669212	−	0.743071i	$-0.733368\pi$
$101$	−13.4510	−1.33842	−0.669212	+	0.743071i	$0.733368\pi$
$102$	1.64704	0.163081
$103$	4.03579	0.397658	0.198829	−	0.980034i	$-0.436286\pi$
$103$	4.03579	0.397658	0.198829	+	0.980034i	$0.436286\pi$
$104$	−1.22322	−0.119946
$105$	0	0
$106$	−0.168641	−0.0163798
$107$	16.9589	1.63948	0.819741	−	0.572734i	$-0.194117\pi$
$107$	16.9589	1.63948	0.819741	+	0.572734i	$0.194117\pi$
$108$	7.96246	0.766188
$109$	−6.83567	−0.654739	−0.327369	−	0.944896i	$-0.606162\pi$
$109$	−6.83567	−0.654739	−0.327369	+	0.944896i	$0.606162\pi$
$110$	0	0
$111$	−30.2056	−2.86699
$112$	7.07221	0.668261
$113$	5.92416	0.557298	0.278649	−	0.960393i	$-0.410113\pi$
$113$	5.92416	0.557298	0.278649	+	0.960393i	$0.410113\pi$
$114$	0	0
$115$	0	0
$116$	16.7660	1.55668
$117$	−5.71250	−0.528121
$118$	1.26966	0.116882
$119$	4.77348	0.437584
$120$	0	0
$121$	6.92107	0.629188
$122$	−0.685224	−0.0620373
$123$	6.84150	0.616877
$124$	9.89681	0.888759
$125$	0	0
$126$	−2.13395	−0.190107
$127$	−19.1802	−1.70197	−0.850983	−	0.525193i	$-0.823993\pi$
$127$	−19.1802	−1.70197	−0.850983	+	0.525193i	$0.823993\pi$
$128$	7.24921	0.640746
$129$	12.2372	1.07742
$130$	0	0
$131$	−0.307990	−0.0269092	−0.0134546	−	0.999909i	$-0.504283\pi$
$131$	−0.307990	−0.0269092	−0.0134546	+	0.999909i	$0.504283\pi$
$132$	22.4920	1.95767
$133$	0	0
$134$	1.27428	0.110081
$135$	0	0
$136$	2.36992	0.203219
$137$	14.4994	1.23876	0.619382	−	0.785090i	$-0.287383\pi$
$137$	14.4994	1.23876	0.619382	+	0.785090i	$0.287383\pi$
$138$	3.43917	0.292761
$139$	15.7973	1.33991	0.669955	−	0.742401i	$-0.266313\pi$
$139$	15.7973	1.33991	0.669955	+	0.742401i	$0.266313\pi$
$140$	0	0
$141$	18.5360	1.56102
$142$	0.193606	0.0162470
$143$	−5.37558	−0.449529
$144$	16.3974	1.36645
$145$	0	0
$146$	0.346122	0.0286452
$147$	8.85957	0.730725
$148$	−21.4017	−1.75921
$149$	−12.4650	−1.02117	−0.510585	−	0.859827i	$-0.670571\pi$
$149$	−12.4650	−1.02117	−0.510585	+	0.859827i	$0.670571\pi$
$150$	0	0
$151$	−13.8400	−1.12629	−0.563143	−	0.826360i	$-0.690408\pi$
$151$	−13.8400	−1.12629	−0.563143	+	0.826360i	$0.690408\pi$
$152$	0	0
$153$	11.0677	0.894767
$154$	−2.00809	−0.161816
$155$	0	0
$156$	−6.74666	−0.540165
$157$	18.1970	1.45228	0.726138	−	0.687549i	$-0.241313\pi$
$157$	18.1970	1.45228	0.726138	+	0.687549i	$0.241313\pi$
$158$	−1.48225	−0.117922
$159$	−1.88893	−0.149802
$160$	0	0
$161$	9.96748	0.785548
$162$	0.552043	0.0433726
$163$	15.3337	1.20103	0.600516	−	0.799613i	$-0.294962\pi$
$163$	15.3337	1.20103	0.600516	+	0.799613i	$0.294962\pi$
$164$	4.84744	0.378522
$165$	0	0
$166$	2.89760	0.224897
$167$	−6.21690	−0.481079	−0.240539	−	0.970639i	$-0.577324\pi$
$167$	−6.21690	−0.481079	−0.240539	+	0.970639i	$0.577324\pi$
$168$	−5.11817	−0.394875
$169$	−11.3875	−0.875965
$170$	0	0
$171$	0	0
$172$	8.67048	0.661118
$173$	2.76604	0.210298	0.105149	−	0.994456i	$-0.466468\pi$
$173$	2.76604	0.210298	0.105149	+	0.994456i	$0.466468\pi$
$174$	−5.78503	−0.438562
$175$	0	0
$176$	15.4303	1.16310
$177$	14.2213	1.06894
$178$	−0.545896	−0.0409167
$179$	11.3783	0.850456	0.425228	−	0.905086i	$-0.360194\pi$
$179$	11.3783	0.850456	0.425228	+	0.905086i	$0.360194\pi$
$180$	0	0
$181$	−3.56654	−0.265099	−0.132549	−	0.991176i	$-0.542316\pi$
$181$	−3.56654	−0.265099	−0.132549	+	0.991176i	$0.542316\pi$
$182$	0.602344	0.0446487
$183$	−7.67513	−0.567362
$184$	4.94861	0.364816
$185$	0	0
$186$	−3.41485	−0.250389
$187$	10.4149	0.761612
$188$	13.1334	0.957855
$189$	−7.96263	−0.579196
$190$	0	0
$191$	−8.67323	−0.627573	−0.313786	−	0.949494i	$-0.601598\pi$
$191$	−8.67323	−0.627573	−0.313786	+	0.949494i	$0.601598\pi$
$192$	18.0761	1.30453
$193$	−0.422110	−0.0303841	−0.0151921	−	0.999885i	$-0.504836\pi$
$193$	−0.422110	−0.0303841	−0.0151921	+	0.999885i	$0.504836\pi$
$194$	2.10490	0.151123
$195$	0	0
$196$	6.27732	0.448380
$197$	−15.2415	−1.08591	−0.542956	−	0.839761i	$-0.682695\pi$
$197$	−15.2415	−1.08591	−0.542956	+	0.839761i	$0.682695\pi$
$198$	−4.65590	−0.330880
$199$	12.9264	0.916331	0.458166	−	0.888867i	$-0.348507\pi$
$199$	12.9264	0.916331	0.458166	+	0.888867i	$0.348507\pi$
$200$	0	0
$201$	14.2731	1.00675
$202$	3.28846	0.231376
$203$	−16.7663	−1.17677
$204$	13.0713	0.915172
$205$	0	0
$206$	−0.986659	−0.0687438
$207$	23.1103	1.60628
$208$	−4.62845	−0.320926
$209$	0	0
$210$	0	0
$211$	4.44619	0.306089	0.153044	−	0.988219i	$-0.451092\pi$
$211$	4.44619	0.306089	0.153044	+	0.988219i	$0.451092\pi$
$212$	−1.33837	−0.0919199
$213$	2.16856	0.148587
$214$	−4.14608	−0.283420
$215$	0	0
$216$	−3.95325	−0.268985
$217$	−9.89701	−0.671853
$218$	1.67117	0.113186
$219$	3.87688	0.261975
$220$	0	0
$221$	−3.12404	−0.210146
$222$	7.38459	0.495621
$223$	9.19893	0.616006	0.308003	−	0.951385i	$-0.400339\pi$
$223$	9.19893	0.616006	0.308003	+	0.951385i	$0.400339\pi$
$224$	−5.46711	−0.365287
$225$	0	0
$226$	−1.44832	−0.0963411
$227$	0.600308	0.0398438	0.0199219	−	0.999802i	$-0.493658\pi$
$227$	0.600308	0.0398438	0.0199219	+	0.999802i	$0.493658\pi$
$228$	0	0
$229$	−7.11499	−0.470172	−0.235086	−	0.971975i	$-0.575537\pi$
$229$	−7.11499	−0.470172	−0.235086	+	0.971975i	$0.575537\pi$
$230$	0	0
$231$	−22.4924	−1.47989
$232$	−8.32407	−0.546502
$233$	20.1662	1.32113	0.660567	−	0.750767i	$-0.270316\pi$
$233$	20.1662	1.32113	0.660567	+	0.750767i	$0.270316\pi$
$234$	1.39658	0.0912971
$235$	0	0
$236$	10.0763	0.655912
$237$	−16.6026	−1.07845
$238$	−1.16701	−0.0756459
$239$	−11.6342	−0.752554	−0.376277	−	0.926507i	$-0.622796\pi$
$239$	−11.6342	−0.752554	−0.376277	+	0.926507i	$0.622796\pi$
$240$	0	0
$241$	−25.5301	−1.64454	−0.822269	−	0.569100i	$-0.807292\pi$
$241$	−25.5301	−1.64454	−0.822269	+	0.569100i	$0.807292\pi$
$242$	−1.69205	−0.108769
$243$	18.4950	1.18646
$244$	−5.43811	−0.348139
$245$	0	0
$246$	−1.67259	−0.106640
$247$	0	0
$248$	−4.91362	−0.312015
$249$	32.4558	2.05680
$250$	0	0
$251$	15.3783	0.970669	0.485335	−	0.874328i	$-0.338698\pi$
$251$	15.3783	0.970669	0.485335	+	0.874328i	$0.338698\pi$
$252$	−16.9355	−1.06684
$253$	21.7473	1.36724
$254$	4.68912	0.294221
$255$	0	0
$256$	11.4298	0.714365
$257$	−16.0534	−1.00139	−0.500693	−	0.865625i	$-0.666921\pi$
$257$	−16.0534	−1.00139	−0.500693	+	0.865625i	$0.666921\pi$
$258$	−2.99172	−0.186256
$259$	21.4022	1.32987
$260$	0	0
$261$	−38.8739	−2.40624
$262$	0.0752965	0.00465183
$263$	14.2397	0.878055	0.439027	−	0.898474i	$-0.355323\pi$
$263$	14.2397	0.878055	0.439027	+	0.898474i	$0.355323\pi$
$264$	−11.1669	−0.687277
$265$	0	0
$266$	0	0
$267$	−6.11453	−0.374203
$268$	10.1130	0.617750
$269$	−16.2414	−0.990254	−0.495127	−	0.868821i	$-0.664879\pi$
$269$	−16.2414	−0.990254	−0.495127	+	0.868821i	$0.664879\pi$
$270$	0	0
$271$	−5.92530	−0.359936	−0.179968	−	0.983672i	$-0.557599\pi$
$271$	−5.92530	−0.359936	−0.179968	+	0.983672i	$0.557599\pi$
$272$	8.96737	0.543727
$273$	6.74680	0.408335
$274$	−3.54476	−0.214147
$275$	0	0
$276$	27.2941	1.64291
$277$	−6.30936	−0.379093	−0.189546	−	0.981872i	$-0.560702\pi$
$277$	−6.30936	−0.379093	−0.189546	+	0.981872i	$0.560702\pi$
$278$	−3.86208	−0.231632
$279$	−22.9469	−1.37380
$280$	0	0
$281$	2.67961	0.159852	0.0799261	−	0.996801i	$-0.474532\pi$
$281$	2.67961	0.159852	0.0799261	+	0.996801i	$0.474532\pi$
$282$	−4.53164	−0.269855
$283$	−31.0643	−1.84658	−0.923290	−	0.384105i	$-0.874510\pi$
$283$	−31.0643	−1.84658	−0.923290	+	0.384105i	$0.874510\pi$
$284$	1.53650	0.0911745
$285$	0	0
$286$	1.31421	0.0777107
$287$	−4.84754	−0.286142
$288$	−12.6759	−0.746933
$289$	−10.9474	−0.643962
$290$	0	0
$291$	23.5768	1.38210
$292$	2.74690	0.160750
$293$	0.865650	0.0505718	0.0252859	−	0.999680i	$-0.491950\pi$
$293$	0.865650	0.0505718	0.0252859	+	0.999680i	$0.491950\pi$
$294$	−2.16596	−0.126322
$295$	0	0
$296$	10.6257	0.617604
$297$	−17.3730	−1.00809
$298$	3.04740	0.176531
$299$	−6.52329	−0.377251
$300$	0	0
$301$	−8.67067	−0.499769
$302$	3.38357	0.194703
$303$	36.8338	2.11605
$304$	0	0
$305$	0	0
$306$	−2.70579	−0.154680
$307$	−8.99201	−0.513201	−0.256601	−	0.966518i	$-0.582603\pi$
$307$	−8.99201	−0.513201	−0.256601	+	0.966518i	$0.582603\pi$
$308$	−15.9367	−0.908077
$309$	−11.0515	−0.628696
$310$	0	0
$311$	−11.4407	−0.648742	−0.324371	−	0.945930i	$-0.605153\pi$
$311$	−11.4407	−0.648742	−0.324371	+	0.945930i	$0.605153\pi$
$312$	3.34962	0.189635
$313$	−7.86521	−0.444568	−0.222284	−	0.974982i	$-0.571351\pi$
$313$	−7.86521	−0.444568	−0.222284	+	0.974982i	$0.571351\pi$
$314$	−4.44875	−0.251057
$315$	0	0
$316$	−11.7635	−0.661749
$317$	16.5265	0.928218	0.464109	−	0.885778i	$-0.346375\pi$
$317$	16.5265	0.928218	0.464109	+	0.885778i	$0.346375\pi$
$318$	0.461801	0.0258965
$319$	−36.5812	−2.04815
$320$	0	0
$321$	−46.4398	−2.59202
$322$	−2.43682	−0.135799
$323$	0	0
$324$	4.38115	0.243397
$325$	0	0
$326$	−3.74875	−0.207624
$327$	18.7186	1.03514
$328$	−2.40669	−0.132887
$329$	−13.1337	−0.724085
$330$	0	0
$331$	−7.91446	−0.435018	−0.217509	−	0.976058i	$-0.569793\pi$
$331$	−7.91446	−0.435018	−0.217509	+	0.976058i	$0.569793\pi$
$332$	22.9961	1.26207
$333$	49.6225	2.71930
$334$	1.51989	0.0831648
$335$	0	0
$336$	−19.3663	−1.05652
$337$	−19.0361	−1.03696	−0.518480	−	0.855090i	$-0.673502\pi$
$337$	−19.0361	−1.03696	−0.518480	+	0.855090i	$0.673502\pi$
$338$	2.78400	0.151429
$339$	−16.2225	−0.881087
$340$	0	0
$341$	−21.5935	−1.16936
$342$	0	0
$343$	−19.8594	−1.07230
$344$	−4.30477	−0.232098
$345$	0	0
$346$	−0.676233	−0.0363545
$347$	5.46420	0.293334	0.146667	−	0.989186i	$-0.453145\pi$
$347$	5.46420	0.293334	0.146667	+	0.989186i	$0.453145\pi$
$348$	−45.9114	−2.46111
$349$	−6.26435	−0.335323	−0.167662	−	0.985845i	$-0.553622\pi$
$349$	−6.26435	−0.335323	−0.167662	+	0.985845i	$0.553622\pi$
$350$	0	0
$351$	5.21120	0.278153
$352$	−11.9283	−0.635779
$353$	8.29398	0.441444	0.220722	−	0.975337i	$-0.429159\pi$
$353$	8.29398	0.441444	0.220722	+	0.975337i	$0.429159\pi$
$354$	−3.47679	−0.184789
$355$	0	0
$356$	−4.33237	−0.229615
$357$	−13.0715	−0.691820
$358$	−2.78174	−0.147020
$359$	13.0857	0.690637	0.345318	−	0.938486i	$-0.387771\pi$
$359$	13.0857	0.690637	0.345318	+	0.938486i	$0.387771\pi$
$360$	0	0
$361$	0	0
$362$	0.871938	0.0458281
$363$	−18.9524	−0.994745
$364$	4.78035	0.250558
$365$	0	0
$366$	1.87640	0.0980808
$367$	−26.0942	−1.36211	−0.681053	−	0.732234i	$-0.738478\pi$
$367$	−26.0942	−1.36211	−0.681053	+	0.732234i	$0.738478\pi$
$368$	18.7247	0.976094
$369$	−11.2394	−0.585099
$370$	0	0
$371$	1.33840	0.0694864
$372$	−27.1011	−1.40513
$373$	27.1969	1.40820	0.704100	−	0.710101i	$-0.251351\pi$
$373$	27.1969	1.40820	0.704100	+	0.710101i	$0.251351\pi$
$374$	−2.54620	−0.131661
$375$	0	0
$376$	−6.52057	−0.336273
$377$	10.9728	0.565130
$378$	1.94668	0.100127
$379$	−13.6999	−0.703714	−0.351857	−	0.936054i	$-0.614450\pi$
$379$	−13.6999	−0.703714	−0.351857	+	0.936054i	$0.614450\pi$
$380$	0	0
$381$	52.5224	2.69080
$382$	2.12041	0.108490
$383$	10.2515	0.523827	0.261913	−	0.965091i	$-0.415647\pi$
$383$	10.2515	0.523827	0.261913	+	0.965091i	$0.415647\pi$
$384$	−19.8510	−1.01302
$385$	0	0
$386$	0.103196	0.00525255
$387$	−20.1036	−1.02192
$388$	16.7050	0.848069
$389$	−5.05671	−0.256385	−0.128193	−	0.991749i	$-0.540918\pi$
$389$	−5.05671	−0.256385	−0.128193	+	0.991749i	$0.540918\pi$
$390$	0	0
$391$	12.6385	0.639157
$392$	−3.11660	−0.157412
$393$	0.843389	0.0425434
$394$	3.72620	0.187723
$395$	0	0
$396$	−36.9503	−1.85682
$397$	−23.8758	−1.19829	−0.599146	−	0.800640i	$-0.704493\pi$
$397$	−23.8758	−1.19829	−0.599146	+	0.800640i	$0.704493\pi$
$398$	−3.16022	−0.158408
$399$	0	0
$400$	0	0
$401$	24.7655	1.23673	0.618366	−	0.785891i	$-0.287795\pi$
$401$	24.7655	1.23673	0.618366	+	0.785891i	$0.287795\pi$
$402$	−3.48945	−0.174038
$403$	6.47717	0.322651
$404$	26.0981	1.29843
$405$	0	0
$406$	4.09899	0.203429
$407$	46.6958	2.31463
$408$	−6.48970	−0.321288
$409$	9.89998	0.489523	0.244761	−	0.969583i	$-0.421290\pi$
$409$	9.89998	0.489523	0.244761	+	0.969583i	$0.421290\pi$
$410$	0	0
$411$	−39.7046	−1.95848
$412$	−7.83036	−0.385774
$413$	−10.0765	−0.495833
$414$	−5.64995	−0.277680
$415$	0	0
$416$	3.57799	0.175425
$417$	−43.2588	−2.11839
$418$	0	0
$419$	4.00810	0.195808	0.0979042	−	0.995196i	$-0.468786\pi$
$419$	4.00810	0.195808	0.0979042	+	0.995196i	$0.468786\pi$
$420$	0	0
$421$	−20.2781	−0.988294	−0.494147	−	0.869378i	$-0.664520\pi$
$421$	−20.2781	−0.988294	−0.494147	+	0.869378i	$0.664520\pi$
$422$	−1.08699	−0.0529140
$423$	−30.4515	−1.48060
$424$	0.664484	0.0322702
$425$	0	0
$426$	−0.530163	−0.0256865
$427$	5.43822	0.263174
$428$	−32.9043	−1.59049
$429$	14.7203	0.710704
$430$	0	0
$431$	−13.4783	−0.649229	−0.324615	−	0.945846i	$-0.605235\pi$
$431$	−13.4783	−0.649229	−0.324615	+	0.945846i	$0.605235\pi$
$432$	−14.9584	−0.719689
$433$	34.3776	1.65208	0.826040	−	0.563611i	$-0.190588\pi$
$433$	34.3776	1.65208	0.826040	+	0.563611i	$0.190588\pi$
$434$	2.41959	0.116144
$435$	0	0
$436$	13.2628	0.635172
$437$	0	0
$438$	−0.947808	−0.0452880
$439$	−23.4179	−1.11768	−0.558838	−	0.829277i	$-0.688753\pi$
$439$	−23.4179	−1.11768	−0.558838	+	0.829277i	$0.688753\pi$
$440$	0	0
$441$	−14.5547	−0.693082
$442$	0.763756	0.0363282
$443$	2.39888	0.113974	0.0569871	−	0.998375i	$-0.481851\pi$
$443$	2.39888	0.113974	0.0569871	+	0.998375i	$0.481851\pi$
$444$	58.6059	2.78131
$445$	0	0
$446$	−2.24893	−0.106490
$447$	34.1337	1.61447
$448$	−12.8078	−0.605113
$449$	−38.2019	−1.80286	−0.901430	−	0.432924i	$-0.857482\pi$
$449$	−38.2019	−1.80286	−0.901430	+	0.432924i	$0.857482\pi$
$450$	0	0
$451$	−10.5765	−0.498027
$452$	−11.4942	−0.540644
$453$	37.8991	1.78065
$454$	−0.146762	−0.00688786
$455$	0	0
$456$	0	0
$457$	−1.87605	−0.0877581	−0.0438790	−	0.999037i	$-0.513972\pi$
$457$	−1.87605	−0.0877581	−0.0438790	+	0.999037i	$0.513972\pi$
$458$	1.73945	0.0812793
$459$	−10.0964	−0.471260
$460$	0	0
$461$	16.8204	0.783402	0.391701	−	0.920092i	$-0.371887\pi$
$461$	16.8204	0.783402	0.391701	+	0.920092i	$0.371887\pi$
$462$	5.49889	0.255831
$463$	24.3641	1.13230	0.566148	−	0.824303i	$-0.308433\pi$
$463$	24.3641	1.13230	0.566148	+	0.824303i	$0.308433\pi$
$464$	−31.4969	−1.46221
$465$	0	0
$466$	−4.93019	−0.228386
$467$	−13.6511	−0.631696	−0.315848	−	0.948810i	$-0.602289\pi$
$467$	−13.6511	−0.631696	−0.315848	+	0.948810i	$0.602289\pi$
$468$	11.0836	0.512338
$469$	−10.1132	−0.466985
$470$	0	0
$471$	−49.8300	−2.29604
$472$	−5.00275	−0.230270
$473$	−18.9179	−0.869844
$474$	4.05895	0.186434
$475$	0	0
$476$	−9.26166	−0.424507
$477$	3.10318	0.142085
$478$	2.84430	0.130095
$479$	−21.7405	−0.993351	−0.496675	−	0.867936i	$-0.665446\pi$
$479$	−21.7405	−0.993351	−0.496675	+	0.867936i	$0.665446\pi$
$480$	0	0
$481$	−14.0068	−0.638656
$482$	6.24153	0.284294
$483$	−27.2946	−1.24195
$484$	−13.4285	−0.610385
$485$	0	0
$486$	−4.52161	−0.205104
$487$	−29.2454	−1.32524	−0.662618	−	0.748957i	$-0.730555\pi$
$487$	−29.2454	−1.32524	−0.662618	+	0.748957i	$0.730555\pi$
$488$	2.69994	0.122221
$489$	−41.9894	−1.89883
$490$	0	0
$491$	−18.3556	−0.828375	−0.414188	−	0.910191i	$-0.635934\pi$
$491$	−18.3556	−0.828375	−0.414188	+	0.910191i	$0.635934\pi$
$492$	−13.2741	−0.598442
$493$	−21.2593	−0.957469
$494$	0	0
$495$	0	0
$496$	−18.5923	−0.834821
$497$	−1.53653	−0.0689229
$498$	−7.93470	−0.355562
$499$	10.4159	0.466279	0.233140	−	0.972443i	$-0.425100\pi$
$499$	10.4159	0.466279	0.233140	+	0.972443i	$0.425100\pi$
$500$	0	0
$501$	17.0242	0.760584
$502$	−3.75964	−0.167801
$503$	−4.35731	−0.194283	−0.0971413	−	0.995271i	$-0.530970\pi$
$503$	−4.35731	−0.194283	−0.0971413	+	0.995271i	$0.530970\pi$
$504$	8.40825	0.374533
$505$	0	0
$506$	−5.31672	−0.236357
$507$	31.1833	1.38490
$508$	37.2140	1.65110
$509$	38.2608	1.69588	0.847941	−	0.530091i	$-0.177842\pi$
$509$	38.2608	1.69588	0.847941	+	0.530091i	$0.177842\pi$
$510$	0	0
$511$	−2.74696	−0.121518
$512$	−17.2928	−0.764239
$513$	0	0
$514$	3.92470	0.173111
$515$	0	0
$516$	−23.7430	−1.04523
$517$	−28.6555	−1.26027
$518$	−5.23235	−0.229896
$519$	−7.57442	−0.332480
$520$	0	0
$521$	2.48812	0.109007	0.0545033	−	0.998514i	$-0.482642\pi$
$521$	2.48812	0.109007	0.0545033	+	0.998514i	$0.482642\pi$
$522$	9.50379	0.415970
$523$	19.4533	0.850635	0.425317	−	0.905044i	$-0.360163\pi$
$523$	19.4533	0.850635	0.425317	+	0.905044i	$0.360163\pi$
$524$	0.597571	0.0261050
$525$	0	0
$526$	−3.48127	−0.151791
$527$	−12.5492	−0.546650
$528$	−42.2538	−1.83886
$529$	3.39040	0.147409
$530$	0	0
$531$	−23.3631	−1.01387
$532$	0	0
$533$	3.17251	0.137417
$534$	1.49486	0.0646891
$535$	0	0
$536$	−5.02096	−0.216873
$537$	−31.1580	−1.34457
$538$	3.97065	0.171187
$539$	−13.6963	−0.589941
$540$	0	0
$541$	14.4565	0.621535	0.310768	−	0.950486i	$-0.399414\pi$
$541$	14.4565	0.621535	0.310768	+	0.950486i	$0.399414\pi$
$542$	1.44860	0.0622228
$543$	9.76650	0.419121
$544$	−6.93215	−0.297214
$545$	0	0
$546$	−1.64944	−0.0705895
$547$	31.3961	1.34240	0.671200	−	0.741276i	$-0.265779\pi$
$547$	31.3961	1.34240	0.671200	+	0.741276i	$0.265779\pi$
$548$	−28.1321	−1.20174
$549$	12.6089	0.538135
$550$	0	0
$551$	0	0
$552$	−13.5511	−0.576774
$553$	11.7637	0.500245
$554$	1.54250	0.0655343
$555$	0	0
$556$	−30.6504	−1.29987
$557$	30.4484	1.29014	0.645070	−	0.764124i	$-0.276828\pi$
$557$	30.4484	1.29014	0.645070	+	0.764124i	$0.276828\pi$
$558$	5.61000	0.237490
$559$	5.67458	0.240009
$560$	0	0
$561$	−28.5198	−1.20411
$562$	−0.655104	−0.0276339
$563$	−26.7012	−1.12532	−0.562661	−	0.826687i	$-0.690223\pi$
$563$	−26.7012	−1.12532	−0.562661	+	0.826687i	$0.690223\pi$
$564$	−35.9642	−1.51437
$565$	0	0
$566$	7.59451	0.319221
$567$	−4.38124	−0.183995
$568$	−0.762850	−0.0320085
$569$	25.6513	1.07536	0.537679	−	0.843150i	$-0.319301\pi$
$569$	25.6513	1.07536	0.537679	+	0.843150i	$0.319301\pi$
$570$	0	0
$571$	2.93559	0.122851	0.0614253	−	0.998112i	$-0.480435\pi$
$571$	2.93559	0.122851	0.0614253	+	0.998112i	$0.480435\pi$
$572$	10.4299	0.436095
$573$	23.7505	0.992191
$574$	1.18511	0.0494657
$575$	0	0
$576$	−29.6959	−1.23733
$577$	17.7732	0.739908	0.369954	−	0.929050i	$-0.379373\pi$
$577$	17.7732	0.739908	0.369954	+	0.929050i	$0.379373\pi$
$578$	2.67638	0.111323
$579$	1.15589	0.0480373
$580$	0	0
$581$	−22.9965	−0.954057
$582$	−5.76400	−0.238925
$583$	2.92016	0.120941
$584$	−1.36380	−0.0564344
$585$	0	0
$586$	−0.211632	−0.00874243
$587$	−23.1236	−0.954415	−0.477207	−	0.878791i	$-0.658351\pi$
$587$	−23.1236	−0.954415	−0.477207	+	0.878791i	$0.658351\pi$
$588$	−17.1896	−0.708887
$589$	0	0
$590$	0	0
$591$	41.7368	1.71682
$592$	40.2058	1.65245
$593$	−17.1574	−0.704569	−0.352284	−	0.935893i	$-0.614595\pi$
$593$	−17.1574	−0.704569	−0.352284	+	0.935893i	$0.614595\pi$
$594$	4.24732	0.174269
$595$	0	0
$596$	24.1849	0.990653
$597$	−35.3973	−1.44872
$598$	1.59480	0.0652160
$599$	13.3967	0.547376	0.273688	−	0.961819i	$-0.411756\pi$
$599$	13.3967	0.547376	0.273688	+	0.961819i	$0.411756\pi$
$600$	0	0
$601$	−8.26251	−0.337035	−0.168517	−	0.985699i	$-0.553898\pi$
$601$	−8.26251	−0.337035	−0.168517	+	0.985699i	$0.553898\pi$
$602$	2.11978	0.0863958
$603$	−23.4482	−0.954885
$604$	26.8528	1.09263
$605$	0	0
$606$	−9.00503	−0.365804
$607$	−23.5191	−0.954612	−0.477306	−	0.878737i	$-0.658387\pi$
$607$	−23.5191	−0.954612	−0.477306	+	0.878737i	$0.658387\pi$
$608$	0	0
$609$	45.9124	1.86046
$610$	0	0
$611$	8.59545	0.347735
$612$	−21.4738	−0.868027
$613$	29.8313	1.20488	0.602438	−	0.798166i	$-0.294196\pi$
$613$	29.8313	1.20488	0.602438	+	0.798166i	$0.294196\pi$
$614$	2.19834	0.0887179
$615$	0	0
$616$	7.91234	0.318797
$617$	−11.5902	−0.466605	−0.233303	−	0.972404i	$-0.574953\pi$
$617$	−11.5902	−0.466605	−0.233303	+	0.972404i	$0.574953\pi$
$618$	2.70183	0.108684
$619$	29.6697	1.19253	0.596264	−	0.802789i	$-0.296651\pi$
$619$	29.6697	1.19253	0.596264	+	0.802789i	$0.296651\pi$
$620$	0	0
$621$	−21.0823	−0.846002
$622$	2.79699	0.112149
$623$	4.33246	0.173576
$624$	12.6744	0.507382
$625$	0	0
$626$	1.92287	0.0768532
$627$	0	0
$628$	−35.3063	−1.40888
$629$	27.1374	1.08204
$630$	0	0
$631$	−24.4964	−0.975187	−0.487593	−	0.873071i	$-0.662125\pi$
$631$	−24.4964	−0.975187	−0.487593	+	0.873071i	$0.662125\pi$
$632$	5.84041	0.232319
$633$	−12.1753	−0.483925
$634$	−4.04034	−0.160463
$635$	0	0
$636$	3.66496	0.145325
$637$	4.10832	0.162778
$638$	8.94326	0.354067
$639$	−3.56256	−0.140933
$640$	0	0
$641$	−12.3802	−0.488989	−0.244495	−	0.969651i	$-0.578622\pi$
$641$	−12.3802	−0.488989	−0.244495	+	0.969651i	$0.578622\pi$
$642$	11.3535	0.448086
$643$	−45.9092	−1.81048	−0.905241	−	0.424898i	$-0.860310\pi$
$643$	−45.9092	−1.81048	−0.905241	+	0.424898i	$0.860310\pi$
$644$	−19.3392	−0.762072
$645$	0	0
$646$	0	0
$647$	−33.6773	−1.32399	−0.661996	−	0.749508i	$-0.730290\pi$
$647$	−33.6773	−1.32399	−0.661996	+	0.749508i	$0.730290\pi$
$648$	−2.17518	−0.0854491
$649$	−21.9852	−0.862994
$650$	0	0
$651$	27.1017	1.06220
$652$	−29.7510	−1.16514
$653$	−19.6365	−0.768437	−0.384219	−	0.923242i	$-0.625529\pi$
$653$	−19.6365	−0.768437	−0.384219	+	0.923242i	$0.625529\pi$
$654$	−4.57627	−0.178946
$655$	0	0
$656$	−9.10651	−0.355549
$657$	−6.36903	−0.248479
$658$	3.21090	0.125174
$659$	−20.7693	−0.809059	−0.404529	−	0.914525i	$-0.632565\pi$
$659$	−20.7693	−0.809059	−0.404529	+	0.914525i	$0.632565\pi$
$660$	0	0
$661$	−13.2561	−0.515603	−0.257802	−	0.966198i	$-0.582998\pi$
$661$	−13.2561	−0.515603	−0.257802	+	0.966198i	$0.582998\pi$
$662$	1.93490	0.0752022
$663$	8.55476	0.332239
$664$	−11.4172	−0.443074
$665$	0	0
$666$	−12.1316	−0.470089
$667$	−44.3914	−1.71884
$668$	12.0622	0.466702
$669$	−25.1901	−0.973904
$670$	0	0
$671$	11.8652	0.458052
$672$	14.9710	0.577517
$673$	12.8576	0.495624	0.247812	−	0.968808i	$-0.420289\pi$
$673$	12.8576	0.495624	0.247812	+	0.968808i	$0.420289\pi$
$674$	4.65388	0.179261
$675$	0	0
$676$	22.0945	0.849787
$677$	−25.5573	−0.982245	−0.491122	−	0.871091i	$-0.663413\pi$
$677$	−25.5573	−0.982245	−0.491122	+	0.871091i	$0.663413\pi$
$678$	3.96604	0.152315
$679$	−16.7054	−0.641093
$680$	0	0
$681$	−1.64386	−0.0629930
$682$	5.27913	0.202148
$683$	−22.8692	−0.875064	−0.437532	−	0.899203i	$-0.644147\pi$
$683$	−22.8692	−0.875064	−0.437532	+	0.899203i	$0.644147\pi$
$684$	0	0
$685$	0	0
$686$	4.85516	0.185371
$687$	19.4835	0.743340
$688$	−16.2886	−0.620995
$689$	−0.875927	−0.0333701
$690$	0	0
$691$	−32.3148	−1.22931	−0.614657	−	0.788795i	$-0.710705\pi$
$691$	−32.3148	−1.22931	−0.614657	+	0.788795i	$0.710705\pi$
$692$	−5.36675	−0.204013
$693$	36.9511	1.40366
$694$	−1.33587	−0.0507091
$695$	0	0
$696$	22.7944	0.864018
$697$	−6.14656	−0.232818
$698$	1.53149	0.0579678
$699$	−55.2226	−2.08871
$700$	0	0
$701$	34.7457	1.31233	0.656164	−	0.754618i	$-0.272178\pi$
$701$	34.7457	1.31233	0.656164	+	0.754618i	$0.272178\pi$
$702$	−1.27402	−0.0480848
$703$	0	0
$704$	−27.9444	−1.05319
$705$	0	0
$706$	−2.02769	−0.0763131
$707$	−26.0986	−0.981539
$708$	−27.5927	−1.03700
$709$	8.00778	0.300738	0.150369	−	0.988630i	$-0.451954\pi$
$709$	8.00778	0.300738	0.150369	+	0.988630i	$0.451954\pi$
$710$	0	0
$711$	27.2751	1.02290
$712$	2.15096	0.0806105
$713$	−26.2038	−0.981341
$714$	3.19570	0.119596
$715$	0	0
$716$	−22.0766	−0.825040
$717$	31.8587	1.18979
$718$	−3.19916	−0.119391
$719$	14.2690	0.532143	0.266071	−	0.963953i	$-0.414274\pi$
$719$	14.2690	0.532143	0.266071	+	0.963953i	$0.414274\pi$
$720$	0	0
$721$	7.83053	0.291624
$722$	0	0
$723$	69.9107	2.60001
$724$	6.91991	0.257177
$725$	0	0
$726$	4.63344	0.171963
$727$	−3.13639	−0.116322	−0.0581612	−	0.998307i	$-0.518524\pi$
$727$	−3.13639	−0.116322	−0.0581612	+	0.998307i	$0.518524\pi$
$728$	−2.37337	−0.0879631
$729$	−43.8720	−1.62489
$730$	0	0
$731$	−10.9942	−0.406634
$732$	14.8915	0.550407
$733$	−12.4004	−0.458020	−0.229010	−	0.973424i	$-0.573549\pi$
$733$	−12.4004	−0.458020	−0.229010	+	0.973424i	$0.573549\pi$
$734$	6.37944	0.235469
$735$	0	0
$736$	−14.4750	−0.533555
$737$	−22.0653	−0.812784
$738$	2.74777	0.101147
$739$	−9.34314	−0.343693	−0.171846	−	0.985124i	$-0.554973\pi$
$739$	−9.34314	−0.343693	−0.171846	+	0.985124i	$0.554973\pi$
$740$	0	0
$741$	0	0
$742$	−0.327209	−0.0120122
$743$	−38.3186	−1.40577	−0.702887	−	0.711302i	$-0.748106\pi$
$743$	−38.3186	−1.40577	−0.702887	+	0.711302i	$0.748106\pi$
$744$	13.4553	0.493295
$745$	0	0
$746$	−6.64901	−0.243438
$747$	−53.3191	−1.95084
$748$	−20.2073	−0.738852
$749$	32.9050	1.20232
$750$	0	0
$751$	49.0370	1.78939	0.894693	−	0.446681i	$-0.147394\pi$
$751$	49.0370	1.78939	0.894693	+	0.446681i	$0.147394\pi$
$752$	−24.6728	−0.899723
$753$	−42.1114	−1.53463
$754$	−2.68261	−0.0976949
$755$	0	0
$756$	15.4493	0.561887
$757$	36.7936	1.33728	0.668642	−	0.743584i	$-0.266876\pi$
$757$	36.7936	1.33728	0.668642	+	0.743584i	$0.266876\pi$
$758$	3.34930	0.121652
$759$	−59.5521	−2.16160
$760$	0	0
$761$	−0.255560	−0.00926406	−0.00463203	−	0.999989i	$-0.501474\pi$
$761$	−0.255560	−0.00926406	−0.00463203	+	0.999989i	$0.501474\pi$
$762$	−12.8405	−0.465163
$763$	−13.2631	−0.480155
$764$	16.8281	0.608818
$765$	0	0
$766$	−2.50626	−0.0905548
$767$	6.59465	0.238119
$768$	−31.2991	−1.12941
$769$	−9.45643	−0.341008	−0.170504	−	0.985357i	$-0.554540\pi$
$769$	−9.45643	−0.341008	−0.170504	+	0.985357i	$0.554540\pi$
$770$	0	0
$771$	43.9602	1.58319
$772$	0.818991	0.0294761
$773$	−30.2259	−1.08715	−0.543575	−	0.839360i	$-0.682930\pi$
$773$	−30.2259	−1.08715	−0.543575	+	0.839360i	$0.682930\pi$
$774$	4.91486	0.176661
$775$	0	0
$776$	−8.29381	−0.297730
$777$	−58.6071	−2.10252
$778$	1.23625	0.0443217
$779$	0	0
$780$	0	0
$781$	−3.35244	−0.119960
$782$	−3.08983	−0.110492
$783$	35.4625	1.26733
$784$	−11.7927	−0.421168
$785$	0	0
$786$	−0.206190	−0.00735454
$787$	36.3976	1.29743	0.648717	−	0.761030i	$-0.275306\pi$
$787$	36.3976	1.29743	0.648717	+	0.761030i	$0.275306\pi$
$788$	29.5720	1.05346
$789$	−38.9934	−1.38820
$790$	0	0
$791$	11.4945	0.408697
$792$	18.3453	0.651872
$793$	−3.55908	−0.126387
$794$	5.83709	0.207151
$795$	0	0
$796$	−25.0803	−0.888947
$797$	−19.2118	−0.680517	−0.340258	−	0.940332i	$-0.610515\pi$
$797$	−19.2118	−0.680517	−0.340258	+	0.940332i	$0.610515\pi$
$798$	0	0
$799$	−16.6532	−0.589148
$800$	0	0
$801$	10.0451	0.354926
$802$	−6.05461	−0.213796
$803$	−5.99338	−0.211502
$804$	−27.6931	−0.976661
$805$	0	0
$806$	−1.58352	−0.0557771
$807$	44.4749	1.56559
$808$	−12.9573	−0.455837
$809$	−43.0501	−1.51356	−0.756781	−	0.653669i	$-0.773229\pi$
$809$	−43.0501	−1.51356	−0.756781	+	0.653669i	$0.773229\pi$
$810$	0	0
$811$	31.3282	1.10008	0.550040	−	0.835138i	$-0.314612\pi$
$811$	31.3282	1.10008	0.550040	+	0.835138i	$0.314612\pi$
$812$	32.5305	1.14160
$813$	16.2257	0.569058
$814$	−11.4161	−0.400133
$815$	0	0
$816$	−24.5560	−0.859631
$817$	0	0
$818$	−2.42032	−0.0846245
$819$	−11.0838	−0.387299
$820$	0	0
$821$	17.0708	0.595776	0.297888	−	0.954601i	$-0.403718\pi$
$821$	17.0708	0.595776	0.297888	+	0.954601i	$0.403718\pi$
$822$	9.70686	0.338566
$823$	36.3136	1.26581	0.632907	−	0.774228i	$-0.281862\pi$
$823$	36.3136	1.26581	0.632907	+	0.774228i	$0.281862\pi$
$824$	3.88766	0.135433
$825$	0	0
$826$	2.46348	0.0857155
$827$	19.3430	0.672623	0.336311	−	0.941751i	$-0.390821\pi$
$827$	19.3430	0.672623	0.336311	+	0.941751i	$0.390821\pi$
$828$	−44.8393	−1.55828
$829$	20.1843	0.701029	0.350514	−	0.936557i	$-0.386007\pi$
$829$	20.1843	0.701029	0.350514	+	0.936557i	$0.386007\pi$
$830$	0	0
$831$	17.2773	0.599345
$832$	8.38217	0.290600
$833$	−7.95964	−0.275785
$834$	10.5758	0.366210
$835$	0	0
$836$	0	0
$837$	20.9332	0.723557
$838$	−0.979889	−0.0338497
$839$	−25.8209	−0.891437	−0.445719	−	0.895173i	$-0.647052\pi$
$839$	−25.8209	−0.891437	−0.445719	+	0.895173i	$0.647052\pi$
$840$	0	0
$841$	45.6708	1.57486
$842$	4.95753	0.170848
$843$	−7.33776	−0.252726
$844$	−8.62664	−0.296941
$845$	0	0
$846$	7.44469	0.255954
$847$	13.4288	0.461418
$848$	2.51430	0.0863413
$849$	85.0654	2.91944
$850$	0	0
$851$	56.6655	1.94247
$852$	−4.20750	−0.144147
$853$	−14.3234	−0.490425	−0.245212	−	0.969469i	$-0.578858\pi$
$853$	−14.3234	−0.490425	−0.245212	+	0.969469i	$0.578858\pi$
$854$	−1.32952	−0.0454953
$855$	0	0
$856$	16.3365	0.558370
$857$	24.2649	0.828874	0.414437	−	0.910078i	$-0.363979\pi$
$857$	24.2649	0.828874	0.414437	+	0.910078i	$0.363979\pi$
$858$	−3.59878	−0.122860
$859$	40.5232	1.38263	0.691317	−	0.722551i	$-0.257031\pi$
$859$	40.5232	1.38263	0.691317	+	0.722551i	$0.257031\pi$
$860$	0	0
$861$	13.2744	0.452389
$862$	3.29515	0.112233
$863$	22.9160	0.780069	0.390035	−	0.920800i	$-0.372463\pi$
$863$	22.9160	0.780069	0.390035	+	0.920800i	$0.372463\pi$
$864$	11.5635	0.393398
$865$	0	0
$866$	−8.40453	−0.285598
$867$	29.9779	1.01810
$868$	19.2025	0.651775
$869$	25.6664	0.870673
$870$	0	0
$871$	6.61866	0.224265
$872$	−6.58478	−0.222989
$873$	−38.7326	−1.31090
$874$	0	0
$875$	0	0
$876$	−7.52203	−0.254146
$877$	49.6951	1.67808	0.839042	−	0.544067i	$-0.183116\pi$
$877$	49.6951	1.67808	0.839042	+	0.544067i	$0.183116\pi$
$878$	5.72515	0.193214
$879$	−2.37047	−0.0799539
$880$	0	0
$881$	−11.0358	−0.371804	−0.185902	−	0.982568i	$-0.559521\pi$
$881$	−11.0358	−0.371804	−0.185902	+	0.982568i	$0.559521\pi$
$882$	3.55830	0.119814
$883$	−22.1244	−0.744547	−0.372273	−	0.928123i	$-0.621422\pi$
$883$	−22.1244	−0.744547	−0.372273	+	0.928123i	$0.621422\pi$
$884$	6.06135	0.203865
$885$	0	0
$886$	−0.586472	−0.0197029
$887$	−20.0329	−0.672640	−0.336320	−	0.941748i	$-0.609182\pi$
$887$	−20.0329	−0.672640	−0.336320	+	0.941748i	$0.609182\pi$
$888$	−29.0970	−0.976430
$889$	−37.2148	−1.24814
$890$	0	0
$891$	−9.55909	−0.320242
$892$	−17.8481	−0.597597
$893$	0	0
$894$	−8.34491	−0.279095
$895$	0	0
$896$	14.0654	0.469893
$897$	17.8632	0.596433
$898$	9.33951	0.311663
$899$	44.0775	1.47007
$900$	0	0
$901$	1.69706	0.0565372
$902$	2.58571	0.0860947
$903$	23.7435	0.790133
$904$	5.70673	0.189803
$905$	0	0
$906$	−9.26546	−0.307824
$907$	0.681366	0.0226244	0.0113122	−	0.999936i	$-0.496399\pi$
$907$	0.681366	0.0226244	0.0113122	+	0.999936i	$0.496399\pi$
$908$	−1.16474	−0.0386531
$909$	−60.5114	−2.00704
$910$	0	0
$911$	−28.7169	−0.951433	−0.475716	−	0.879599i	$-0.657811\pi$
$911$	−28.7169	−0.951433	−0.475716	+	0.879599i	$0.657811\pi$
$912$	0	0
$913$	−50.1744	−1.66053
$914$	0.458652	0.0151709
$915$	0	0
$916$	13.8047	0.456121
$917$	−0.597584	−0.0197340
$918$	2.46834	0.0814674
$919$	6.91259	0.228025	0.114013	−	0.993479i	$-0.463630\pi$
$919$	6.91259	0.228025	0.114013	+	0.993479i	$0.463630\pi$
$920$	0	0
$921$	24.6234	0.811370
$922$	−4.11220	−0.135428
$923$	1.00559	0.0330995
$924$	43.6405	1.43567
$925$	0	0
$926$	−5.95647	−0.195742
$927$	18.1556	0.596309
$928$	24.3484	0.799276
$929$	40.8302	1.33960	0.669798	−	0.742544i	$-0.266381\pi$
$929$	40.8302	1.33960	0.669798	+	0.742544i	$0.266381\pi$
$930$	0	0
$931$	0	0
$932$	−39.1272	−1.28165
$933$	31.3288	1.02566
$934$	3.33737	0.109202
$935$	0	0
$936$	−5.50284	−0.179866
$937$	−14.7439	−0.481664	−0.240832	−	0.970567i	$-0.577420\pi$
$937$	−14.7439	−0.481664	−0.240832	+	0.970567i	$0.577420\pi$
$938$	2.47245	0.0807284
$939$	21.5378	0.702861
$940$	0	0
$941$	52.2137	1.70212	0.851059	−	0.525071i	$-0.175961\pi$
$941$	52.2137	1.70212	0.851059	+	0.525071i	$0.175961\pi$
$942$	12.1823	0.396921
$943$	−12.8346	−0.417952
$944$	−18.9296	−0.616105
$945$	0	0
$946$	4.62499	0.150371
$947$	−57.3403	−1.86331	−0.931655	−	0.363343i	$-0.881635\pi$
$947$	−57.3403	−1.86331	−0.931655	+	0.363343i	$0.881635\pi$
$948$	32.2128	1.04622
$949$	1.79777	0.0583580
$950$	0	0
$951$	−45.2555	−1.46751
$952$	4.59828	0.149031
$953$	9.31765	0.301828	0.150914	−	0.988547i	$-0.451778\pi$
$953$	9.31765	0.301828	0.150914	+	0.988547i	$0.451778\pi$
$954$	−0.758657	−0.0245624
$955$	0	0
$956$	22.5731	0.730065
$957$	100.173	3.23812
$958$	5.31507	0.171722
$959$	28.1327	0.908452
$960$	0	0
$961$	−4.98144	−0.160692
$962$	3.42435	0.110405
$963$	76.2925	2.45849
$964$	49.5343	1.59539
$965$	0	0
$966$	6.67292	0.214698
$967$	8.82697	0.283856	0.141928	−	0.989877i	$-0.454670\pi$
$967$	8.82697	0.283856	0.141928	+	0.989877i	$0.454670\pi$
$968$	6.66705	0.214287
$969$	0	0
$970$	0	0
$971$	−55.9918	−1.79686	−0.898431	−	0.439114i	$-0.855292\pi$
$971$	−55.9918	−1.79686	−0.898431	+	0.439114i	$0.855292\pi$
$972$	−35.8846	−1.15100
$973$	30.6511	0.982628
$974$	7.14984	0.229096
$975$	0	0
$976$	10.2161	0.327011
$977$	−41.1034	−1.31501	−0.657506	−	0.753449i	$-0.728389\pi$
$977$	−41.1034	−1.31501	−0.657506	+	0.753449i	$0.728389\pi$
$978$	10.2655	0.328253
$979$	9.45265	0.302108
$980$	0	0
$981$	−30.7513	−0.981815
$982$	4.48752	0.143203
$983$	2.49864	0.0796941	0.0398471	−	0.999206i	$-0.487313\pi$
$983$	2.49864	0.0796941	0.0398471	+	0.999206i	$0.487313\pi$
$984$	6.59039	0.210094
$985$	0	0
$986$	5.19741	0.165519
$987$	35.9650	1.14478
$988$	0	0
$989$	−22.9569	−0.729986
$990$	0	0
$991$	−17.4234	−0.553471	−0.276736	−	0.960946i	$-0.589253\pi$
$991$	−17.4234	−0.553471	−0.276736	+	0.960946i	$0.589253\pi$
$992$	14.3727	0.456332
$993$	21.6727	0.687762
$994$	0.375647	0.0119148
$995$	0	0
$996$	−62.9717	−1.99533
$997$	−12.8010	−0.405412	−0.202706	−	0.979240i	$-0.564974\pi$
$997$	−12.8010	−0.405412	−0.202706	+	0.979240i	$0.564974\pi$
$998$	−2.54645	−0.0806065
$999$	−45.2679	−1.43221

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9025.2.a.ct.1.12		24
5.2	odd	4		1805.2.b.l.1084.12		24
5.3	odd	4		1805.2.b.l.1084.13		24
5.4	even	2	inner	9025.2.a.ct.1.13		24
19.3	odd	18		475.2.l.f.351.5		48
19.13	odd	18		475.2.l.f.226.5		48
19.18	odd	2		9025.2.a.cu.1.13		24
95.3	even	36		95.2.p.a.9.5	yes	48
95.13	even	36		95.2.p.a.74.4	yes	48
95.18	even	4		1805.2.b.k.1084.12		24
95.22	even	36		95.2.p.a.9.4	✓	48
95.32	even	36		95.2.p.a.74.5	yes	48
95.37	even	4		1805.2.b.k.1084.13		24
95.79	odd	18		475.2.l.f.351.4		48
95.89	odd	18		475.2.l.f.226.4		48
95.94	odd	2		9025.2.a.cu.1.12		24
285.32	odd	36		855.2.da.b.739.4		48
285.98	odd	36		855.2.da.b.199.4		48
285.203	odd	36		855.2.da.b.739.5		48
285.212	odd	36		855.2.da.b.199.5		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
95.2.p.a.9.4	✓	48	95.22	even	36
95.2.p.a.9.5	yes	48	95.3	even	36
95.2.p.a.74.4	yes	48	95.13	even	36
95.2.p.a.74.5	yes	48	95.32	even	36
475.2.l.f.226.4		48	95.89	odd	18
475.2.l.f.226.5		48	19.13	odd	18
475.2.l.f.351.4		48	95.79	odd	18
475.2.l.f.351.5		48	19.3	odd	18
855.2.da.b.199.4		48	285.98	odd	36
855.2.da.b.199.5		48	285.212	odd	36
855.2.da.b.739.4		48	285.32	odd	36
855.2.da.b.739.5		48	285.203	odd	36
1805.2.b.k.1084.12		24	95.18	even	4
1805.2.b.k.1084.13		24	95.37	even	4
1805.2.b.l.1084.12		24	5.2	odd	4
1805.2.b.l.1084.13		24	5.3	odd	4
9025.2.a.ct.1.12		24	1.1	even	1	trivial
9025.2.a.ct.1.13		24	5.4	even	2	inner
9025.2.a.cu.1.12		24	95.94	odd	2
9025.2.a.cu.1.13		24	19.18	odd	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:	\( N \)	\(=\)	\( 9025 = 5^{2} \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9025.a (trivial)

\(f(q)\)	\(=\)	\(q-0.244477 q^{2} -2.73837 q^{3} -1.94023 q^{4} +0.669469 q^{6} +1.94027 q^{7} +0.963297 q^{8} +4.49866 q^{9} +O(q^{10})\)	\(q-0.244477 q^{2} -2.73837 q^{3} -1.94023 q^{4} +0.669469 q^{6} +1.94027 q^{7} +0.963297 q^{8} +4.49866 q^{9} +4.23333 q^{11} +5.31307 q^{12} -1.26982 q^{13} -0.474352 q^{14} +3.64496 q^{16} +2.46021 q^{17} -1.09982 q^{18} -5.31318 q^{21} -1.03495 q^{22} +5.13716 q^{23} -2.63786 q^{24} +0.310443 q^{26} -4.10387 q^{27} -3.76457 q^{28} -8.64123 q^{29} -5.10084 q^{31} -2.81770 q^{32} -11.5924 q^{33} -0.601466 q^{34} -8.72843 q^{36} +11.0305 q^{37} +3.47724 q^{39} -2.49838 q^{41} +1.29895 q^{42} -4.46879 q^{43} -8.21364 q^{44} -1.25592 q^{46} -6.76901 q^{47} -9.98123 q^{48} -3.23535 q^{49} -6.73697 q^{51} +2.46375 q^{52} +0.689802 q^{53} +1.00330 q^{54} +1.86906 q^{56} +2.11258 q^{58} -5.19336 q^{59} +2.80281 q^{61} +1.24704 q^{62} +8.72862 q^{63} -6.60105 q^{64} +2.83408 q^{66} -5.21227 q^{67} -4.77338 q^{68} -14.0674 q^{69} -0.791916 q^{71} +4.33354 q^{72} -1.41576 q^{73} -2.69671 q^{74} +8.21381 q^{77} -0.850107 q^{78} +6.06294 q^{79} -2.25806 q^{81} +0.610798 q^{82} -11.8522 q^{83} +10.3088 q^{84} +1.09252 q^{86} +23.6629 q^{87} +4.07795 q^{88} +2.23291 q^{89} -2.46380 q^{91} -9.96727 q^{92} +13.9680 q^{93} +1.65487 q^{94} +7.71591 q^{96} -8.60981 q^{97} +0.790969 q^{98} +19.0443 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 18 q^{4} - 12 q^{6} + 12 q^{9}+O(q^{10}) \) 24 * q + 18 * q^4 - 12 * q^6 + 12 * q^9	\( 24 q + 18 q^{4} - 12 q^{6} + 12 q^{9} + 12 q^{11} - 24 q^{14} + 6 q^{16} - 6 q^{21} - 42 q^{24} - 12 q^{26} - 36 q^{29} - 42 q^{31} - 6 q^{34} - 6 q^{36} + 24 q^{39} - 60 q^{41} - 30 q^{44} - 6 q^{46} + 12 q^{49} - 30 q^{51} - 24 q^{54} - 18 q^{56} - 60 q^{59} + 30 q^{61} + 36 q^{66} - 66 q^{69} - 96 q^{71} + 24 q^{74} - 72 q^{79} - 96 q^{81} + 54 q^{84} - 108 q^{86} - 84 q^{89} - 96 q^{91} - 36 q^{94} - 120 q^{96}+O(q^{100}) \) 24 * q + 18 * q^4 - 12 * q^6 + 12 * q^9 + 12 * q^11 - 24 * q^14 + 6 * q^16 - 6 * q^21 - 42 * q^24 - 12 * q^26 - 36 * q^29 - 42 * q^31 - 6 * q^34 - 6 * q^36 + 24 * q^39 - 60 * q^41 - 30 * q^44 - 6 * q^46 + 12 * q^49 - 30 * q^51 - 24 * q^54 - 18 * q^56 - 60 * q^59 + 30 * q^61 + 36 * q^66 - 66 * q^69 - 96 * q^71 + 24 * q^74 - 72 * q^79 - 96 * q^81 + 54 * q^84 - 108 * q^86 - 84 * q^89 - 96 * q^91 - 36 * q^94 - 120 * q^96

Introduction

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