LMFDB - Embedded newform 9025.2.a.bo.1.4

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:	$4$
Coefficient field:	4.4.2225.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 5x^{2} + 2x + 4 $ x^4 - x^3 - 5x^2 + 2x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1805)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.43828$ 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)

$f(q)$	$=$	\(q+2.43828 q^{2} -2.94523 q^{3} +3.94523 q^{4} -7.18129 q^{6} +3.82025 q^{7} +4.74301 q^{8} +5.67435 q^{9} -2.12498 q^{11} -11.6196 q^{12} +3.65438 q^{13} +9.31485 q^{14} +3.67435 q^{16} +3.04243 q^{17} +13.8357 q^{18} -11.2515 q^{21} -5.18129 q^{22} +4.81167 q^{23} -13.9692 q^{24} +8.91042 q^{26} -7.87657 q^{27} +15.0717 q^{28} +6.03385 q^{29} +3.57184 q^{31} -0.526911 q^{32} +6.25853 q^{33} +7.41831 q^{34} +22.3866 q^{36} -3.93134 q^{37} -10.7630 q^{39} -7.60945 q^{41} -27.4343 q^{42} +5.60415 q^{43} -8.38351 q^{44} +11.7322 q^{46} +8.41831 q^{47} -10.8218 q^{48} +7.59430 q^{49} -8.96065 q^{51} +14.4174 q^{52} -4.80028 q^{53} -19.2053 q^{54} +18.1195 q^{56} +14.7122 q^{58} -5.13510 q^{59} -13.5100 q^{61} +8.70916 q^{62} +21.6774 q^{63} -8.63346 q^{64} +15.2601 q^{66} +5.38101 q^{67} +12.0031 q^{68} -14.1714 q^{69} -0.123434 q^{71} +26.9135 q^{72} +12.4860 q^{73} -9.58572 q^{74} -8.11794 q^{77} -26.2432 q^{78} -4.25699 q^{79} +6.17521 q^{81} -18.5540 q^{82} -4.39739 q^{83} -44.3897 q^{84} +13.6645 q^{86} -17.7711 q^{87} -10.0788 q^{88} -0.0772394 q^{89} +13.9607 q^{91} +18.9831 q^{92} -10.5199 q^{93} +20.5262 q^{94} +1.55187 q^{96} +18.0231 q^{97} +18.5171 q^{98} -12.0579 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + q^{2} + q^{3} + 3 q^{4} - 7 q^{6} + 11 q^{7} + 6 q^{8} + 5 q^{9} - 16 q^{12} - 2 q^{13} + 11 q^{14} - 3 q^{16} + 7 q^{17} + 17 q^{18} - 2 q^{21} + q^{22} + 11 q^{23} - 13 q^{24} + 9 q^{26} - 14 q^{27}+ \cdots - 9 q^{99}+O(q^{100}) $ 4 * q + q^2 + q^3 + 3 * q^4 - 7 * q^6 + 11 * q^7 + 6 * q^8 + 5 * q^9 - 16 * q^12 - 2 * q^13 + 11 * q^14 - 3 * q^16 + 7 * q^17 + 17 * q^18 - 2 * q^21 + q^22 + 11 * q^23 - 13 * q^24 + 9 * q^26 - 14 * q^27 + 13 * q^28 + 15 * q^29 + q^31 + 3 * q^32 + 12 * q^33 + 22 * q^34 + 16 * q^36 - 11 * q^37 - 29 * q^39 - 22 * q^41 - 19 * q^42 + 26 * q^43 - 12 * q^44 - 10 * q^46 + 26 * q^47 - 13 * q^48 + 13 * q^49 + 11 * q^51 + 27 * q^52 - 16 * q^53 - 25 * q^54 + 8 * q^56 + 3 * q^58 + 10 * q^59 + 2 * q^61 + 31 * q^62 + 17 * q^63 + 4 * q^64 + 22 * q^66 + 3 * q^67 - 4 * q^68 - 14 * q^69 - 18 * q^71 + 29 * q^72 + 24 * q^73 - 17 * q^74 + 6 * q^77 - 15 * q^78 - 30 * q^79 - 4 * q^81 + 13 * q^82 + 12 * q^83 - 52 * q^84 + 16 * q^86 + q^87 - 23 * q^88 - 9 * q^89 + 9 * q^91 + 25 * q^92 - 7 * q^93 + 11 * q^94 - 6 * q^96 + 19 * q^97 + 48 * q^98 - 9 * 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 $

$2$	2.43828	1.72413	0.862063	−	0.506801i	$-0.169172\pi$
$2$	2.43828	1.72413	0.862063	+	0.506801i	$0.169172\pi$
$3$	−2.94523	−1.70043	−0.850213	−	0.526438i	$-0.823527\pi$
$3$	−2.94523	−1.70043	−0.850213	+	0.526438i	$0.823527\pi$
$4$	3.94523	1.97261
$5$	0	0
$5$	0	0
$6$	−7.18129	−2.93175
$7$	3.82025	1.44392	0.721959	−	0.691936i	$-0.243242\pi$
$7$	3.82025	1.44392	0.721959	+	0.691936i	$0.243242\pi$
$8$	4.74301	1.67691
$9$	5.67435	1.89145
$10$	0	0
$11$	−2.12498	−0.640704	−0.320352	−	0.947299i	$-0.603801\pi$
$11$	−2.12498	−0.640704	−0.320352	+	0.947299i	$0.603801\pi$
$12$	−11.6196	−3.35428
$13$	3.65438	1.01354	0.506772	−	0.862080i	$-0.330839\pi$
$13$	3.65438	1.01354	0.506772	+	0.862080i	$0.330839\pi$
$14$	9.31485	2.48950
$15$	0	0
$16$	3.67435	0.918588
$17$	3.04243	0.737898	0.368949	−	0.929450i	$-0.379718\pi$
$17$	3.04243	0.737898	0.368949	+	0.929450i	$0.379718\pi$
$18$	13.8357	3.26110
$19$	0	0
$19$	0	0
$20$	0	0
$21$	−11.2515	−2.45528
$22$	−5.18129	−1.10466
$23$	4.81167	1.00330	0.501651	−	0.865070i	$-0.332726\pi$
$23$	4.81167	1.00330	0.501651	+	0.865070i	$0.332726\pi$
$24$	−13.9692	−2.85146
$25$	0	0
$26$	8.91042	1.74748
$27$	−7.87657	−1.51585
$28$	15.0717	2.84829
$29$	6.03385	1.12046	0.560229	−	0.828338i	$-0.310713\pi$
$29$	6.03385	1.12046	0.560229	+	0.828338i	$0.310713\pi$
$30$	0	0
$31$	3.57184	0.641521	0.320761	−	0.947160i	$-0.396061\pi$
$31$	3.57184	0.641521	0.320761	+	0.947160i	$0.396061\pi$
$32$	−0.526911	−0.0931455
$33$	6.25853	1.08947
$34$	7.41831	1.27223
$35$	0	0
$36$	22.3866	3.73110
$37$	−3.93134	−0.646309	−0.323154	−	0.946346i	$-0.604743\pi$
$37$	−3.93134	−0.646309	−0.323154	+	0.946346i	$0.604743\pi$
$38$	0	0
$39$	−10.7630	−1.72346
$40$	0	0
$41$	−7.60945	−1.18840	−0.594198	−	0.804318i	$-0.702531\pi$
$41$	−7.60945	−1.18840	−0.594198	+	0.804318i	$0.702531\pi$
$42$	−27.4343	−4.23321
$43$	5.60415	0.854625	0.427312	−	0.904104i	$-0.359460\pi$
$43$	5.60415	0.854625	0.427312	+	0.904104i	$0.359460\pi$
$44$	−8.38351	−1.26386
$45$	0	0
$46$	11.7322	1.72982
$47$	8.41831	1.22794	0.613969	−	0.789330i	$-0.289572\pi$
$47$	8.41831	1.22794	0.613969	+	0.789330i	$0.289572\pi$
$48$	−10.8218	−1.56199
$49$	7.59430	1.08490
$50$	0	0
$51$	−8.96065	−1.25474
$52$	14.4174	1.99933
$53$	−4.80028	−0.659369	−0.329685	−	0.944091i	$-0.606942\pi$
$53$	−4.80028	−0.659369	−0.329685	+	0.944091i	$0.606942\pi$
$54$	−19.2053	−2.61351
$55$	0	0
$56$	18.1195	2.42132
$57$	0	0
$58$	14.7122	1.93181
$59$	−5.13510	−0.668533	−0.334266	−	0.942479i	$-0.608489\pi$
$59$	−5.13510	−0.668533	−0.334266	+	0.942479i	$0.608489\pi$
$60$	0	0
$61$	−13.5100	−1.72978	−0.864891	−	0.501960i	$-0.832612\pi$
$61$	−13.5100	−1.72978	−0.864891	+	0.501960i	$0.832612\pi$
$62$	8.70916	1.10606
$63$	21.6774	2.73110
$64$	−8.63346	−1.07918
$65$	0	0
$66$	15.2601	1.87839
$67$	5.38101	0.657395	0.328698	−	0.944435i	$-0.393390\pi$
$67$	5.38101	0.657395	0.328698	+	0.944435i	$0.393390\pi$
$68$	12.0031	1.45559
$69$	−14.1714	−1.70604
$70$	0	0
$71$	−0.123434	−0.0146489	−0.00732443	−	0.999973i	$-0.502331\pi$
$71$	−0.123434	−0.0146489	−0.00732443	+	0.999973i	$0.502331\pi$
$72$	26.9135	3.17179
$73$	12.4860	1.46138	0.730689	−	0.682710i	$-0.239199\pi$
$73$	12.4860	1.46138	0.730689	+	0.682710i	$0.239199\pi$
$74$	−9.58572	−1.11432
$75$	0	0
$76$	0	0
$77$	−8.11794	−0.925125
$78$	−26.2432	−2.97146
$79$	−4.25699	−0.478949	−0.239474	−	0.970903i	$-0.576975\pi$
$79$	−4.25699	−0.478949	−0.239474	+	0.970903i	$0.576975\pi$
$80$	0	0
$81$	6.17521	0.686134
$82$	−18.5540	−2.04895
$83$	−4.39739	−0.482677	−0.241338	−	0.970441i	$-0.577586\pi$
$83$	−4.39739	−0.482677	−0.241338	+	0.970441i	$0.577586\pi$
$84$	−44.3897	−4.84331
$85$	0	0
$86$	13.6645	1.47348
$87$	−17.7711	−1.90526
$88$	−10.0788	−1.07440
$89$	−0.0772394	−0.00818736	−0.00409368	−	0.999992i	$-0.501303\pi$
$89$	−0.0772394	−0.00818736	−0.00409368	+	0.999992i	$0.501303\pi$
$90$	0	0
$91$	13.9607	1.46347
$92$	18.9831	1.97913
$93$	−10.5199	−1.09086
$94$	20.5262	2.11712
$95$	0	0
$96$	1.55187	0.158387
$97$	18.0231	1.82996	0.914982	−	0.403495i	$-0.132205\pi$
$97$	18.0231	1.82996	0.914982	+	0.403495i	$0.132205\pi$
$98$	18.5171	1.87051
$99$	−12.0579	−1.21186
$100$	0	0
$101$	18.6498	1.85573	0.927864	−	0.372918i	$-0.121643\pi$
$101$	18.6498	1.85573	0.927864	+	0.372918i	$0.121643\pi$
$102$	−21.8486	−2.16333
$103$	−0.377423	−0.0371886	−0.0185943	−	0.999827i	$-0.505919\pi$
$103$	−0.377423	−0.0371886	−0.0185943	+	0.999827i	$0.505919\pi$
$104$	17.3328	1.69962
$105$	0	0
$106$	−11.7044	−1.13684
$107$	−2.76144	−0.266958	−0.133479	−	0.991052i	$-0.542615\pi$
$107$	−2.76144	−0.266958	−0.133479	+	0.991052i	$0.542615\pi$
$108$	−31.0748	−2.99018
$109$	−3.00308	−0.287643	−0.143822	−	0.989604i	$-0.545939\pi$
$109$	−3.00308	−0.287643	−0.143822	+	0.989604i	$0.545939\pi$
$110$	0	0
$111$	11.5787	1.09900
$112$	14.0369	1.32637
$113$	−3.61708	−0.340266	−0.170133	−	0.985421i	$-0.554420\pi$
$113$	−3.61708	−0.340266	−0.170133	+	0.985421i	$0.554420\pi$
$114$	0	0
$115$	0	0
$116$	23.8049	2.21023
$117$	20.7362	1.91707
$118$	−12.5208	−1.15264
$119$	11.6229	1.06547
$120$	0	0
$121$	−6.48448	−0.589498
$122$	−32.9413	−2.98236
$123$	22.4116	2.02078
$124$	14.0917	1.26547
$125$	0	0
$126$	52.8557	4.70876
$127$	−13.6188	−1.20847	−0.604236	−	0.796805i	$-0.706522\pi$
$127$	−13.6188	−1.20847	−0.604236	+	0.796805i	$0.706522\pi$
$128$	−19.9970	−1.76750
$129$	−16.5055	−1.45323
$130$	0	0
$131$	5.67059	0.495442	0.247721	−	0.968831i	$-0.420318\pi$
$131$	5.67059	0.495442	0.247721	+	0.968831i	$0.420318\pi$
$132$	24.6913	2.14910
$133$	0	0
$134$	13.1204	1.13343
$135$	0	0
$136$	14.4303	1.23739
$137$	8.10501	0.692457	0.346229	−	0.938150i	$-0.387462\pi$
$137$	8.10501	0.692457	0.346229	+	0.938150i	$0.387462\pi$
$138$	−34.5540	−2.94143
$139$	−17.5325	−1.48709	−0.743543	−	0.668688i	$-0.766856\pi$
$139$	−17.5325	−1.48709	−0.743543	+	0.668688i	$0.766856\pi$
$140$	0	0
$141$	−24.7938	−2.08802
$142$	−0.300966	−0.0252565
$143$	−7.76547	−0.649382
$144$	20.8496	1.73746
$145$	0	0
$146$	30.4445	2.51960
$147$	−22.3669	−1.84479
$148$	−15.5100	−1.27492
$149$	19.1952	1.57253	0.786265	−	0.617889i	$-0.212012\pi$
$149$	19.1952	1.57253	0.786265	+	0.617889i	$0.212012\pi$
$150$	0	0
$151$	−11.1473	−0.907152	−0.453576	−	0.891218i	$-0.649852\pi$
$151$	−11.1473	−0.907152	−0.453576	+	0.891218i	$0.649852\pi$
$152$	0	0
$153$	17.2638	1.39570
$154$	−19.7938	−1.59503
$155$	0	0
$156$	−42.4624	−3.39971
$157$	−11.1452	−0.889486	−0.444743	−	0.895658i	$-0.646705\pi$
$157$	−11.1452	−0.889486	−0.444743	+	0.895658i	$0.646705\pi$
$158$	−10.3797	−0.825768
$159$	14.1379	1.12121
$160$	0	0
$161$	18.3818	1.44869
$162$	15.0569	1.18298
$163$	6.68669	0.523742	0.261871	−	0.965103i	$-0.415660\pi$
$163$	6.68669	0.523742	0.261871	+	0.965103i	$0.415660\pi$
$164$	−30.0210	−2.34425
$165$	0	0
$166$	−10.7221	−0.832195
$167$	18.6569	1.44371	0.721856	−	0.692043i	$-0.243289\pi$
$167$	18.6569	1.44371	0.721856	+	0.692043i	$0.243289\pi$
$168$	−53.3659	−4.11727
$169$	0.354510	0.0272700
$170$	0	0
$171$	0	0
$172$	22.1096	1.68584
$173$	−14.7385	−1.12054	−0.560272	−	0.828308i	$-0.689304\pi$
$173$	−14.7385	−1.12054	−0.560272	+	0.828308i	$0.689304\pi$
$174$	−43.3309	−3.28490
$175$	0	0
$176$	−7.80791	−0.588543
$177$	15.1240	1.13679
$178$	−0.188331	−0.0141160
$179$	12.1961	0.911582	0.455791	−	0.890087i	$-0.349357\pi$
$179$	12.1961	0.911582	0.455791	+	0.890087i	$0.349357\pi$
$180$	0	0
$181$	13.1841	0.979966	0.489983	−	0.871732i	$-0.337003\pi$
$181$	13.1841	0.979966	0.489983	+	0.871732i	$0.337003\pi$
$182$	34.0400	2.52321
$183$	39.7901	2.94137
$184$	22.8218	1.68244
$185$	0	0
$186$	−25.6504	−1.88078
$187$	−6.46510	−0.472775
$188$	33.2121	2.42224
$189$	−30.0904	−2.18876
$190$	0	0
$191$	4.61708	0.334080	0.167040	−	0.985950i	$-0.446579\pi$
$191$	4.61708	0.334080	0.167040	+	0.985950i	$0.446579\pi$
$192$	25.4275	1.83507
$193$	−5.22468	−0.376081	−0.188040	−	0.982161i	$-0.560214\pi$
$193$	−5.22468	−0.376081	−0.188040	+	0.982161i	$0.560214\pi$
$194$	43.9453	3.15509
$195$	0	0
$196$	29.9612	2.14009
$197$	−21.1783	−1.50889	−0.754445	−	0.656363i	$-0.772094\pi$
$197$	−21.1783	−1.50889	−0.754445	+	0.656363i	$0.772094\pi$
$198$	−29.4005	−2.08940
$199$	15.3752	1.08992	0.544960	−	0.838462i	$-0.316545\pi$
$199$	15.3752	1.08992	0.544960	+	0.838462i	$0.316545\pi$
$200$	0	0
$201$	−15.8483	−1.11785
$202$	45.4736	3.19951
$203$	23.0508	1.61785
$204$	−35.3518	−2.47512
$205$	0	0
$206$	−0.920265	−0.0641179
$207$	27.3031	1.89770
$208$	13.4275	0.931029
$209$	0	0
$210$	0	0
$211$	−20.8659	−1.43647	−0.718235	−	0.695800i	$-0.755050\pi$
$211$	−20.8659	−1.43647	−0.718235	+	0.695800i	$0.755050\pi$
$212$	−18.9382	−1.30068
$213$	0.363540	0.0249093
$214$	−6.73316	−0.460270
$215$	0	0
$216$	−37.3586	−2.54193
$217$	13.6453	0.926305
$218$	−7.32237	−0.495934
$219$	−36.7741	−2.48497
$220$	0	0
$221$	11.1182	0.747892
$222$	28.2321	1.89482
$223$	13.6759	0.915806	0.457903	−	0.889002i	$-0.348601\pi$
$223$	13.6759	0.915806	0.457903	+	0.889002i	$0.348601\pi$
$224$	−2.01293	−0.134495
$225$	0	0
$226$	−8.81947	−0.586662
$227$	3.71196	0.246372	0.123186	−	0.992384i	$-0.460689\pi$
$227$	3.71196	0.246372	0.123186	+	0.992384i	$0.460689\pi$
$228$	0	0
$229$	11.3086	0.747293	0.373646	−	0.927571i	$-0.378107\pi$
$229$	11.3086	0.747293	0.373646	+	0.927571i	$0.378107\pi$
$230$	0	0
$231$	23.9092	1.57311
$232$	28.6186	1.87890
$233$	10.6753	0.699362	0.349681	−	0.936869i	$-0.386290\pi$
$233$	10.6753	0.699362	0.349681	+	0.936869i	$0.386290\pi$
$234$	50.5608	3.30527
$235$	0	0
$236$	−20.2591	−1.31876
$237$	12.5378	0.814417
$238$	28.3398	1.83700
$239$	−17.2813	−1.11783	−0.558916	−	0.829224i	$-0.688783\pi$
$239$	−17.2813	−1.11783	−0.558916	+	0.829224i	$0.688783\pi$
$240$	0	0
$241$	−17.4986	−1.12719	−0.563593	−	0.826052i	$-0.690581\pi$
$241$	−17.4986	−1.12719	−0.563593	+	0.826052i	$0.690581\pi$
$242$	−15.8110	−1.01637
$243$	5.44232	0.349125
$244$	−53.3001	−3.41219
$245$	0	0
$246$	54.6457	3.48408
$247$	0	0
$248$	16.9413	1.07577
$249$	12.9513	0.820756
$250$	0	0
$251$	17.9012	1.12991	0.564956	−	0.825121i	$-0.308893\pi$
$251$	17.9012	1.12991	0.564956	+	0.825121i	$0.308893\pi$
$252$	85.5224	5.38740
$253$	−10.2247	−0.642820
$254$	−33.2065	−2.08356
$255$	0	0
$256$	−31.4914	−1.96821
$257$	−26.7208	−1.66680	−0.833400	−	0.552671i	$-0.813609\pi$
$257$	−26.7208	−1.66680	−0.833400	+	0.552671i	$0.813609\pi$
$258$	−40.2450	−2.50555
$259$	−15.0187	−0.933217
$260$	0	0
$261$	34.2382	2.11929
$262$	13.8265	0.854204
$263$	19.7817	1.21979	0.609895	−	0.792482i	$-0.291211\pi$
$263$	19.7817	1.21979	0.609895	+	0.792482i	$0.291211\pi$
$264$	29.6843	1.82694
$265$	0	0
$266$	0	0
$267$	0.227487	0.0139220
$268$	21.2293	1.29679
$269$	−7.04833	−0.429744	−0.214872	−	0.976642i	$-0.568933\pi$
$269$	−7.04833	−0.429744	−0.214872	+	0.976642i	$0.568933\pi$
$270$	0	0
$271$	26.1672	1.58955	0.794773	−	0.606907i	$-0.207590\pi$
$271$	26.1672	1.58955	0.794773	+	0.606907i	$0.207590\pi$
$272$	11.1790	0.677825
$273$	−41.1173	−2.48853
$274$	19.7623	1.19388
$275$	0	0
$276$	−55.9096	−3.36536
$277$	−2.86953	−0.172413	−0.0862066	−	0.996277i	$-0.527475\pi$
$277$	−2.86953	−0.172413	−0.0862066	+	0.996277i	$0.527475\pi$
$278$	−42.7492	−2.56393
$279$	20.2679	1.21341
$280$	0	0
$281$	22.4401	1.33866	0.669332	−	0.742963i	$-0.266580\pi$
$281$	22.4401	1.33866	0.669332	+	0.742963i	$0.266580\pi$
$282$	−60.4544	−3.60001
$283$	4.68139	0.278280	0.139140	−	0.990273i	$-0.455566\pi$
$283$	4.68139	0.278280	0.139140	+	0.990273i	$0.455566\pi$
$284$	−0.486973	−0.0288965
$285$	0	0
$286$	−18.9344	−1.11962
$287$	−29.0700	−1.71595
$288$	−2.98988	−0.176180
$289$	−7.74360	−0.455506
$290$	0	0
$291$	−53.0820	−3.11172
$292$	49.2602	2.88273
$293$	3.26229	0.190585	0.0952926	−	0.995449i	$-0.469621\pi$
$293$	3.26229	0.190585	0.0952926	+	0.995449i	$0.469621\pi$
$294$	−54.5369	−3.18066
$295$	0	0
$296$	−18.6464	−1.08380
$297$	16.7375	0.971209
$298$	46.8033	2.71124
$299$	17.5837	1.01689
$300$	0	0
$301$	21.4093	1.23401
$302$	−27.1802	−1.56404
$303$	−54.9280	−3.15553
$304$	0	0
$305$	0	0
$306$	42.0941	2.40636
$307$	−1.71746	−0.0980207	−0.0490103	−	0.998798i	$-0.515607\pi$
$307$	−1.71746	−0.0980207	−0.0490103	+	0.998798i	$0.515607\pi$
$308$	−32.0271	−1.82491
$309$	1.11160	0.0632365
$310$	0	0
$311$	−0.892946	−0.0506343	−0.0253172	−	0.999679i	$-0.508060\pi$
$311$	−0.892946	−0.0506343	−0.0253172	+	0.999679i	$0.508060\pi$
$312$	−51.0489	−2.89008
$313$	−11.5358	−0.652040	−0.326020	−	0.945363i	$-0.605708\pi$
$313$	−11.5358	−0.652040	−0.326020	+	0.945363i	$0.605708\pi$
$314$	−27.1752	−1.53359
$315$	0	0
$316$	−16.7948	−0.944780
$317$	18.8246	1.05729	0.528647	−	0.848842i	$-0.322699\pi$
$317$	18.8246	1.05729	0.528647	+	0.848842i	$0.322699\pi$
$318$	34.4722	1.93311
$319$	−12.8218	−0.717883
$320$	0	0
$321$	8.13305	0.453943
$322$	44.8200	2.49772
$323$	0	0
$324$	24.3626	1.35348
$325$	0	0
$326$	16.3041	0.902998
$327$	8.84476	0.489116
$328$	−36.0917	−1.99283
$329$	32.1601	1.77304
$330$	0	0
$331$	15.5779	0.856240	0.428120	−	0.903722i	$-0.359176\pi$
$331$	15.5779	0.856240	0.428120	+	0.903722i	$0.359176\pi$
$332$	−17.3487	−0.952134
$333$	−22.3078	−1.22246
$334$	45.4908	2.48914
$335$	0	0
$336$	−41.3419	−2.25539
$337$	−6.43425	−0.350496	−0.175248	−	0.984524i	$-0.556073\pi$
$337$	−6.43425	−0.350496	−0.175248	+	0.984524i	$0.556073\pi$
$338$	0.864396	0.0470170
$339$	10.6531	0.578598
$340$	0	0
$341$	−7.59007	−0.411026
$342$	0	0
$343$	2.27039	0.122590
$344$	26.5805	1.43313
$345$	0	0
$346$	−35.9366	−1.93196
$347$	19.3367	1.03805	0.519023	−	0.854760i	$-0.326296\pi$
$347$	19.3367	1.03805	0.519023	+	0.854760i	$0.326296\pi$
$348$	−70.1108	−3.75833
$349$	7.78870	0.416920	0.208460	−	0.978031i	$-0.433155\pi$
$349$	7.78870	0.416920	0.208460	+	0.978031i	$0.433155\pi$
$350$	0	0
$351$	−28.7840	−1.53638
$352$	1.11967	0.0596788
$353$	24.7007	1.31468	0.657342	−	0.753593i	$-0.271681\pi$
$353$	24.7007	1.31468	0.657342	+	0.753593i	$0.271681\pi$
$354$	36.8767	1.95997
$355$	0	0
$356$	−0.304727	−0.0161505
$357$	−34.2319	−1.81175
$358$	29.7376	1.57168
$359$	24.8224	1.31008	0.655038	−	0.755596i	$-0.272653\pi$
$359$	24.8224	1.31008	0.655038	+	0.755596i	$0.272653\pi$
$360$	0	0
$361$	0	0
$362$	32.1466	1.68959
$363$	19.0982	1.00240
$364$	55.0779	2.88687
$365$	0	0
$366$	97.0195	5.07129
$367$	7.01716	0.366293	0.183146	−	0.983086i	$-0.441372\pi$
$367$	7.01716	0.366293	0.183146	+	0.983086i	$0.441372\pi$
$368$	17.6798	0.921621
$369$	−43.1787	−2.24779
$370$	0	0
$371$	−18.3383	−0.952075
$372$	−41.5033	−2.15184
$373$	−36.2105	−1.87491	−0.937455	−	0.348107i	$-0.886825\pi$
$373$	−36.2105	−1.87491	−0.937455	+	0.348107i	$0.886825\pi$
$374$	−15.7637	−0.815124
$375$	0	0
$376$	39.9281	2.05914
$377$	22.0500	1.13563
$378$	−73.3690	−3.77370
$379$	28.2455	1.45087	0.725437	−	0.688288i	$-0.241638\pi$
$379$	28.2455	1.45087	0.725437	+	0.688288i	$0.241638\pi$
$380$	0	0
$381$	40.1104	2.05492
$382$	11.2578	0.575997
$383$	26.4312	1.35057	0.675287	−	0.737555i	$-0.264020\pi$
$383$	26.4312	1.35057	0.675287	+	0.737555i	$0.264020\pi$
$384$	58.8957	3.00551
$385$	0	0
$386$	−12.7392	−0.648411
$387$	31.7999	1.61648
$388$	71.1050	3.60981
$389$	−5.61736	−0.284811	−0.142406	−	0.989808i	$-0.545484\pi$
$389$	−5.61736	−0.284811	−0.142406	+	0.989808i	$0.545484\pi$
$390$	0	0
$391$	14.6392	0.740335
$392$	36.0199	1.81928
$393$	−16.7012	−0.842462
$394$	−51.6387	−2.60152
$395$	0	0
$396$	−47.5710	−2.39053
$397$	29.0391	1.45743	0.728715	−	0.684818i	$-0.240118\pi$
$397$	29.0391	1.45743	0.728715	+	0.684818i	$0.240118\pi$
$398$	37.4891	1.87916
$399$	0	0
$400$	0	0
$401$	−25.4613	−1.27148	−0.635739	−	0.771904i	$-0.719305\pi$
$401$	−25.4613	−1.27148	−0.635739	+	0.771904i	$0.719305\pi$
$402$	−38.6426	−1.92732
$403$	13.0529	0.650210
$404$	73.5778	3.66063
$405$	0	0
$406$	56.2044	2.78938
$407$	8.35401	0.414093
$408$	−42.5005	−2.10409
$409$	19.8024	0.979166	0.489583	−	0.871957i	$-0.337149\pi$
$409$	19.8024	0.979166	0.489583	+	0.871957i	$0.337149\pi$
$410$	0	0
$411$	−23.8711	−1.17747
$412$	−1.48902	−0.0733588
$413$	−19.6174	−0.965307
$414$	66.5727	3.27187
$415$	0	0
$416$	−1.92553	−0.0944070
$417$	51.6371	2.52868
$418$	0	0
$419$	0.172123	0.00840877	0.00420439	−	0.999991i	$-0.498662\pi$
$419$	0.172123	0.00840877	0.00420439	+	0.999991i	$0.498662\pi$
$420$	0	0
$421$	−17.4056	−0.848297	−0.424149	−	0.905593i	$-0.639427\pi$
$421$	−17.4056	−0.848297	−0.424149	+	0.905593i	$0.639427\pi$
$422$	−50.8771	−2.47666
$423$	47.7685	2.32258
$424$	−22.7678	−1.10570
$425$	0	0
$426$	0.886412	0.0429468
$427$	−51.6117	−2.49766
$428$	−10.8945	−0.526605
$429$	22.8711	1.10423
$430$	0	0
$431$	5.06518	0.243981	0.121990	−	0.992531i	$-0.461072\pi$
$431$	5.06518	0.243981	0.121990	+	0.992531i	$0.461072\pi$
$432$	−28.9413	−1.39244
$433$	12.5169	0.601524	0.300762	−	0.953699i	$-0.402759\pi$
$433$	12.5169	0.601524	0.300762	+	0.953699i	$0.402759\pi$
$434$	33.2712	1.59707
$435$	0	0
$436$	−11.8478	−0.567409
$437$	0	0
$438$	−89.6658	−4.28440
$439$	14.7581	0.704367	0.352183	−	0.935931i	$-0.385439\pi$
$439$	14.7581	0.704367	0.352183	+	0.935931i	$0.385439\pi$
$440$	0	0
$441$	43.0928	2.05204
$442$	27.1094	1.28946
$443$	−5.35950	−0.254638	−0.127319	−	0.991862i	$-0.540637\pi$
$443$	−5.35950	−0.254638	−0.127319	+	0.991862i	$0.540637\pi$
$444$	45.6805	2.16790
$445$	0	0
$446$	33.3457	1.57896
$447$	−56.5341	−2.67397
$448$	−32.9820	−1.55825
$449$	7.16065	0.337932	0.168966	−	0.985622i	$-0.445957\pi$
$449$	7.16065	0.337932	0.168966	+	0.985622i	$0.445957\pi$
$450$	0	0
$451$	16.1699	0.761411
$452$	−14.2702	−0.671214
$453$	32.8312	1.54255
$454$	9.05082	0.424776
$455$	0	0
$456$	0	0
$457$	−32.3884	−1.51507	−0.757534	−	0.652796i	$-0.773596\pi$
$457$	−32.3884	−1.51507	−0.757534	+	0.652796i	$0.773596\pi$
$458$	27.5736	1.28843
$459$	−23.9639	−1.11854
$460$	0	0
$461$	−25.2124	−1.17426	−0.587130	−	0.809493i	$-0.699742\pi$
$461$	−25.2124	−1.17426	−0.587130	+	0.809493i	$0.699742\pi$
$462$	58.2973	2.71224
$463$	40.8494	1.89843	0.949216	−	0.314627i	$-0.101879\pi$
$463$	40.8494	1.89843	0.949216	+	0.314627i	$0.101879\pi$
$464$	22.1705	1.02924
$465$	0	0
$466$	26.0294	1.20579
$467$	−14.6209	−0.676577	−0.338288	−	0.941042i	$-0.609848\pi$
$467$	−14.6209	−0.676577	−0.338288	+	0.941042i	$0.609848\pi$
$468$	81.8092	3.78163
$469$	20.5568	0.949225
$470$	0	0
$471$	32.8252	1.51250
$472$	−24.3558	−1.12107
$473$	−11.9087	−0.547562
$474$	30.5707	1.40416
$475$	0	0
$476$	45.8548	2.10175
$477$	−27.2385	−1.24716
$478$	−42.1366	−1.92729
$479$	1.06944	0.0488640	0.0244320	−	0.999701i	$-0.492222\pi$
$479$	1.06944	0.0488640	0.0244320	+	0.999701i	$0.492222\pi$
$480$	0	0
$481$	−14.3666	−0.655062
$482$	−42.6666	−1.94341
$483$	−54.1385	−2.46339
$484$	−25.5827	−1.16285
$485$	0	0
$486$	13.2699	0.601936
$487$	−6.25304	−0.283352	−0.141676	−	0.989913i	$-0.545249\pi$
$487$	−6.25304	−0.283352	−0.141676	+	0.989913i	$0.545249\pi$
$488$	−64.0782	−2.90068
$489$	−19.6938	−0.890585
$490$	0	0
$491$	−0.813681	−0.0367209	−0.0183605	−	0.999831i	$-0.505845\pi$
$491$	−0.813681	−0.0367209	−0.0183605	+	0.999831i	$0.505845\pi$
$492$	88.4186	3.98622
$493$	18.3576	0.826784
$494$	0	0
$495$	0	0
$496$	13.1242	0.589294
$497$	−0.471547	−0.0211518
$498$	31.5790	1.41509
$499$	1.57636	0.0705676	0.0352838	−	0.999377i	$-0.488766\pi$
$499$	1.57636	0.0705676	0.0352838	+	0.999377i	$0.488766\pi$
$500$	0	0
$501$	−54.9487	−2.45493
$502$	43.6481	1.94811
$503$	−3.00182	−0.133845	−0.0669223	−	0.997758i	$-0.521318\pi$
$503$	−3.00182	−0.133845	−0.0669223	+	0.997758i	$0.521318\pi$
$504$	102.816	4.57980
$505$	0	0
$506$	−24.9307	−1.10830
$507$	−1.04411	−0.0463707
$508$	−53.7292	−2.38385
$509$	32.7217	1.45036	0.725182	−	0.688557i	$-0.241755\pi$
$509$	32.7217	1.45036	0.725182	+	0.688557i	$0.241755\pi$
$510$	0	0
$511$	47.6997	2.11011
$512$	−36.7910	−1.62595
$513$	0	0
$514$	−65.1529	−2.87377
$515$	0	0
$516$	−65.1179	−2.86665
$517$	−17.8887	−0.786745
$518$	−36.6199	−1.60898
$519$	43.4081	1.90540
$520$	0	0
$521$	16.4073	0.718819	0.359409	−	0.933180i	$-0.382978\pi$
$521$	16.4073	0.718819	0.359409	+	0.933180i	$0.382978\pi$
$522$	83.4824	3.65393
$523$	−20.4437	−0.893940	−0.446970	−	0.894549i	$-0.647497\pi$
$523$	−20.4437	−0.893940	−0.446970	+	0.894549i	$0.647497\pi$
$524$	22.3718	0.977315
$525$	0	0
$526$	48.2333	2.10307
$527$	10.8671	0.473378
$528$	22.9960	1.00077
$529$	0.152154	0.00661541
$530$	0	0
$531$	−29.1384	−1.26450
$532$	0	0
$533$	−27.8079	−1.20449
$534$	0.554678	0.0240033
$535$	0	0
$536$	25.5222	1.10239
$537$	−35.9203	−1.55008
$538$	−17.1858	−0.740933
$539$	−16.1377	−0.695101
$540$	0	0
$541$	−28.8016	−1.23828	−0.619139	−	0.785281i	$-0.712518\pi$
$541$	−28.8016	−1.23828	−0.619139	+	0.785281i	$0.712518\pi$
$542$	63.8031	2.74058
$543$	−38.8301	−1.66636
$544$	−1.60309	−0.0687320
$545$	0	0
$546$	−100.256	−4.29054
$547$	26.9553	1.15253	0.576264	−	0.817264i	$-0.304510\pi$
$547$	26.9553	1.15253	0.576264	+	0.817264i	$0.304510\pi$
$548$	31.9761	1.36595
$549$	−76.6606	−3.27180
$550$	0	0
$551$	0	0
$552$	−67.2153	−2.86087
$553$	−16.2628	−0.691563
$554$	−6.99672	−0.297262
$555$	0	0
$556$	−69.1696	−2.93345
$557$	−7.95439	−0.337039	−0.168519	−	0.985698i	$-0.553899\pi$
$557$	−7.95439	−0.337039	−0.168519	+	0.985698i	$0.553899\pi$
$558$	49.4188	2.09207
$559$	20.4797	0.866199
$560$	0	0
$561$	19.0412	0.803919
$562$	54.7153	2.30803
$563$	−34.0931	−1.43685	−0.718426	−	0.695603i	$-0.755137\pi$
$563$	−34.0931	−1.43685	−0.718426	+	0.695603i	$0.755137\pi$
$564$	−97.8172	−4.11885
$565$	0	0
$566$	11.4146	0.479789
$567$	23.5908	0.990722
$568$	−0.585446	−0.0245648
$569$	−25.7236	−1.07839	−0.539195	−	0.842181i	$-0.681271\pi$
$569$	−25.7236	−1.07839	−0.539195	+	0.842181i	$0.681271\pi$
$570$	0	0
$571$	−31.6325	−1.32378	−0.661890	−	0.749601i	$-0.730245\pi$
$571$	−31.6325	−1.32378	−0.661890	+	0.749601i	$0.730245\pi$
$572$	−30.6365	−1.28098
$573$	−13.5983	−0.568079
$574$	−70.8809	−2.95851
$575$	0	0
$576$	−48.9893	−2.04122
$577$	−26.8689	−1.11856	−0.559282	−	0.828977i	$-0.688923\pi$
$577$	−26.8689	−1.11856	−0.559282	+	0.828977i	$0.688923\pi$
$578$	−18.8811	−0.785350
$579$	15.3879	0.639498
$580$	0	0
$581$	−16.7991	−0.696946
$582$	−129.429	−5.36500
$583$	10.2005	0.422461
$584$	59.2213	2.45060
$585$	0	0
$586$	7.95439	0.328593
$587$	−42.5145	−1.75476	−0.877380	−	0.479796i	$-0.840711\pi$
$587$	−42.5145	−1.75476	−0.877380	+	0.479796i	$0.840711\pi$
$588$	−88.2426	−3.63906
$589$	0	0
$590$	0	0
$591$	62.3748	2.56576
$592$	−14.4451	−0.593691
$593$	23.2139	0.953280	0.476640	−	0.879099i	$-0.341855\pi$
$593$	23.2139	0.953280	0.476640	+	0.879099i	$0.341855\pi$
$594$	40.8108	1.67449
$595$	0	0
$596$	75.7293	3.10199
$597$	−45.2834	−1.85333
$598$	42.8740	1.75325
$599$	0.313115	0.0127935	0.00639676	−	0.999980i	$-0.497964\pi$
$599$	0.313115	0.0127935	0.00639676	+	0.999980i	$0.497964\pi$
$600$	0	0
$601$	20.9580	0.854893	0.427447	−	0.904041i	$-0.359413\pi$
$601$	20.9580	0.854893	0.427447	+	0.904041i	$0.359413\pi$
$602$	52.2018	2.12759
$603$	30.5338	1.24343
$604$	−43.9785	−1.78946
$605$	0	0
$606$	−133.930	−5.44053
$607$	−10.1085	−0.410290	−0.205145	−	0.978732i	$-0.565767\pi$
$607$	−10.1085	−0.410290	−0.205145	+	0.978732i	$0.565767\pi$
$608$	0	0
$609$	−67.8899	−2.75104
$610$	0	0
$611$	30.7637	1.24457
$612$	68.1097	2.75317
$613$	36.3560	1.46840	0.734202	−	0.678931i	$-0.237556\pi$
$613$	36.3560	1.46840	0.734202	+	0.678931i	$0.237556\pi$
$614$	−4.18766	−0.169000
$615$	0	0
$616$	−38.5035	−1.55135
$617$	−35.3399	−1.42273	−0.711365	−	0.702822i	$-0.751923\pi$
$617$	−35.3399	−1.42273	−0.711365	+	0.702822i	$0.751923\pi$
$618$	2.71039	0.109028
$619$	−32.5878	−1.30982	−0.654908	−	0.755709i	$-0.727292\pi$
$619$	−32.5878	−1.30982	−0.654908	+	0.755709i	$0.727292\pi$
$620$	0	0
$621$	−37.8994	−1.52085
$622$	−2.17726	−0.0873000
$623$	−0.295074	−0.0118219
$624$	−39.5470	−1.58315
$625$	0	0
$626$	−28.1275	−1.12420
$627$	0	0
$628$	−43.9704	−1.75461
$629$	−11.9608	−0.476910
$630$	0	0
$631$	1.66950	0.0664616	0.0332308	−	0.999448i	$-0.489420\pi$
$631$	1.66950	0.0664616	0.0332308	+	0.999448i	$0.489420\pi$
$632$	−20.1909	−0.803153
$633$	61.4549	2.44261
$634$	45.8997	1.82291
$635$	0	0
$636$	55.7772	2.21171
$637$	27.7525	1.09959
$638$	−31.2632	−1.23772
$639$	−0.700405	−0.0277076
$640$	0	0
$641$	−21.3997	−0.845238	−0.422619	−	0.906307i	$-0.638889\pi$
$641$	−21.3997	−0.845238	−0.422619	+	0.906307i	$0.638889\pi$
$642$	19.8307	0.782655
$643$	8.43473	0.332633	0.166317	−	0.986072i	$-0.446813\pi$
$643$	8.43473	0.332633	0.166317	+	0.986072i	$0.446813\pi$
$644$	72.5202	2.85770
$645$	0	0
$646$	0	0
$647$	29.2025	1.14807	0.574034	−	0.818831i	$-0.305378\pi$
$647$	29.2025	1.14807	0.574034	+	0.818831i	$0.305378\pi$
$648$	29.2891	1.15058
$649$	10.9120	0.428332
$650$	0	0
$651$	−40.1885	−1.57511
$652$	26.3805	1.03314
$653$	7.61427	0.297970	0.148985	−	0.988839i	$-0.452399\pi$
$653$	7.61427	0.297970	0.148985	+	0.988839i	$0.452399\pi$
$654$	21.5660	0.843299
$655$	0	0
$656$	−27.9598	−1.09165
$657$	70.8501	2.76412
$658$	78.4153	3.05695
$659$	18.2941	0.712637	0.356318	−	0.934365i	$-0.384032\pi$
$659$	18.2941	0.712637	0.356318	+	0.934365i	$0.384032\pi$
$660$	0	0
$661$	20.3586	0.791859	0.395930	−	0.918281i	$-0.370422\pi$
$661$	20.3586	0.791859	0.395930	+	0.918281i	$0.370422\pi$
$662$	37.9834	1.47627
$663$	−32.7456	−1.27174
$664$	−20.8569	−0.809404
$665$	0	0
$666$	−54.3928	−2.10768
$667$	29.0329	1.12416
$668$	73.6056	2.84789
$669$	−40.2786	−1.55726
$670$	0	0
$671$	28.7085	1.10828
$672$	5.92853	0.228698
$673$	29.6829	1.14419	0.572096	−	0.820186i	$-0.306130\pi$
$673$	29.6829	1.14419	0.572096	+	0.820186i	$0.306130\pi$
$674$	−15.6885	−0.604299
$675$	0	0
$676$	1.39862	0.0537932
$677$	9.17128	0.352481	0.176240	−	0.984347i	$-0.443606\pi$
$677$	9.17128	0.352481	0.176240	+	0.984347i	$0.443606\pi$
$678$	25.9753	0.997576
$679$	68.8526	2.64232
$680$	0	0
$681$	−10.9326	−0.418937
$682$	−18.5067	−0.708660
$683$	−15.6556	−0.599043	−0.299522	−	0.954089i	$-0.596827\pi$
$683$	−15.6556	−0.599043	−0.299522	+	0.954089i	$0.596827\pi$
$684$	0	0
$685$	0	0
$686$	5.53586	0.211360
$687$	−33.3064	−1.27072
$688$	20.5916	0.785048
$689$	−17.5421	−0.668299
$690$	0	0
$691$	−13.7912	−0.524643	−0.262321	−	0.964981i	$-0.584488\pi$
$691$	−13.7912	−0.524643	−0.262321	+	0.964981i	$0.584488\pi$
$692$	−58.1466	−2.21040
$693$	−46.0640	−1.74983
$694$	47.1483	1.78972
$695$	0	0
$696$	−84.2883	−3.19494
$697$	−23.1513	−0.876916
$698$	18.9911	0.718822
$699$	−31.4412	−1.18921
$700$	0	0
$701$	−13.7313	−0.518622	−0.259311	−	0.965794i	$-0.583496\pi$
$701$	−13.7313	−0.518622	−0.259311	+	0.965794i	$0.583496\pi$
$702$	−70.1835	−2.64891
$703$	0	0
$704$	18.3459	0.691437
$705$	0	0
$706$	60.2272	2.26668
$707$	71.2470	2.67952
$708$	59.6677	2.24245
$709$	−12.3146	−0.462483	−0.231242	−	0.972896i	$-0.574279\pi$
$709$	−12.3146	−0.462483	−0.231242	+	0.972896i	$0.574279\pi$
$710$	0	0
$711$	−24.1557	−0.905908
$712$	−0.366347	−0.0137294
$713$	17.1865	0.643640
$714$	−83.4671	−3.12368
$715$	0	0
$716$	48.1165	1.79820
$717$	50.8972	1.90079
$718$	60.5240	2.25874
$719$	−6.92307	−0.258187	−0.129094	−	0.991632i	$-0.541207\pi$
$719$	−6.92307	−0.258187	−0.129094	+	0.991632i	$0.541207\pi$
$720$	0	0
$721$	−1.44185	−0.0536973
$722$	0	0
$723$	51.5374	1.91670
$724$	52.0142	1.93309
$725$	0	0
$726$	46.5669	1.72826
$727$	−34.2263	−1.26938	−0.634692	−	0.772765i	$-0.718873\pi$
$727$	−34.2263	−1.26938	−0.634692	+	0.772765i	$0.718873\pi$
$728$	66.2155	2.45411
$729$	−34.5545	−1.27980
$730$	0	0
$731$	17.0503	0.630626
$732$	156.981	5.80218
$733$	−37.5214	−1.38589	−0.692943	−	0.720993i	$-0.743686\pi$
$733$	−37.5214	−1.38589	−0.692943	+	0.720993i	$0.743686\pi$
$734$	17.1098	0.631535
$735$	0	0
$736$	−2.53532	−0.0934531
$737$	−11.4345	−0.421196
$738$	−105.282	−3.87548
$739$	−38.6683	−1.42244	−0.711218	−	0.702972i	$-0.751856\pi$
$739$	−38.6683	−1.42244	−0.711218	+	0.702972i	$0.751856\pi$
$740$	0	0
$741$	0	0
$742$	−44.7139	−1.64150
$743$	34.5415	1.26720	0.633602	−	0.773659i	$-0.281576\pi$
$743$	34.5415	1.26720	0.633602	+	0.773659i	$0.281576\pi$
$744$	−49.8959	−1.82927
$745$	0	0
$746$	−88.2915	−3.23258
$747$	−24.9523	−0.912959
$748$	−25.5063	−0.932601
$749$	−10.5494	−0.385466
$750$	0	0
$751$	−33.3178	−1.21578	−0.607892	−	0.794020i	$-0.707985\pi$
$751$	−33.3178	−1.21578	−0.607892	+	0.794020i	$0.707985\pi$
$752$	30.9318	1.12797
$753$	−52.7230	−1.92133
$754$	53.7642	1.95798
$755$	0	0
$756$	−118.714	−4.31757
$757$	−27.2310	−0.989728	−0.494864	−	0.868970i	$-0.664782\pi$
$757$	−27.2310	−0.989728	−0.494864	+	0.868970i	$0.664782\pi$
$758$	68.8706	2.50149
$759$	30.1140	1.09307
$760$	0	0
$761$	20.1113	0.729033	0.364516	−	0.931197i	$-0.381234\pi$
$761$	20.1113	0.729033	0.364516	+	0.931197i	$0.381234\pi$
$762$	97.8006	3.54294
$763$	−11.4725	−0.415334
$764$	18.2154	0.659011
$765$	0	0
$766$	64.4469	2.32856
$767$	−18.7656	−0.677587
$768$	92.7493	3.34680
$769$	−3.25931	−0.117534	−0.0587669	−	0.998272i	$-0.518717\pi$
$769$	−3.25931	−0.117534	−0.0587669	+	0.998272i	$0.518717\pi$
$770$	0	0
$771$	78.6988	2.83427
$772$	−20.6125	−0.741861
$773$	−49.6874	−1.78713	−0.893565	−	0.448933i	$-0.851804\pi$
$773$	−49.6874	−1.78713	−0.893565	+	0.448933i	$0.851804\pi$
$774$	77.5372	2.78702
$775$	0	0
$776$	85.4835	3.06868
$777$	44.2335	1.58687
$778$	−13.6967	−0.491051
$779$	0	0
$780$	0	0
$781$	0.262293	0.00938559
$782$	35.6945	1.27643
$783$	−47.5260	−1.69844
$784$	27.9041	0.996576
$785$	0	0
$786$	−40.7222	−1.45251
$787$	−27.1159	−0.966578	−0.483289	−	0.875461i	$-0.660558\pi$
$787$	−27.1159	−0.966578	−0.483289	+	0.875461i	$0.660558\pi$
$788$	−83.5531	−2.97646
$789$	−58.2615	−2.07416
$790$	0	0
$791$	−13.8181	−0.491317
$792$	−57.1905	−2.03218
$793$	−49.3708	−1.75321
$794$	70.8055	2.51279
$795$	0	0
$796$	60.6586	2.14999
$797$	−27.3573	−0.969044	−0.484522	−	0.874779i	$-0.661006\pi$
$797$	−27.3573	−0.969044	−0.484522	+	0.874779i	$0.661006\pi$
$798$	0	0
$799$	25.6122	0.906093
$800$	0	0
$801$	−0.438283	−0.0154860
$802$	−62.0820	−2.19219
$803$	−26.5325	−0.936311
$804$	−62.5251	−2.20509
$805$	0	0
$806$	31.8266	1.12104
$807$	20.7589	0.730748
$808$	88.4564	3.11188
$809$	−12.3922	−0.435686	−0.217843	−	0.975984i	$-0.569902\pi$
$809$	−12.3922	−0.435686	−0.217843	+	0.975984i	$0.569902\pi$
$810$	0	0
$811$	−14.5389	−0.510528	−0.255264	−	0.966871i	$-0.582162\pi$
$811$	−14.5389	−0.510528	−0.255264	+	0.966871i	$0.582162\pi$
$812$	90.9407	3.19139
$813$	−77.0683	−2.70290
$814$	20.3694	0.713948
$815$	0	0
$816$	−32.9246	−1.15259
$817$	0	0
$818$	48.2839	1.68821
$819$	79.2176	2.76809
$820$	0	0
$821$	6.47357	0.225929	0.112965	−	0.993599i	$-0.463965\pi$
$821$	6.47357	0.225929	0.112965	+	0.993599i	$0.463965\pi$
$822$	−58.2044	−2.03011
$823$	−0.210155	−0.00732553	−0.00366276	−	0.999993i	$-0.501166\pi$
$823$	−0.210155	−0.00732553	−0.00366276	+	0.999993i	$0.501166\pi$
$824$	−1.79012	−0.0623619
$825$	0	0
$826$	−47.8327	−1.66431
$827$	23.1396	0.804643	0.402322	−	0.915498i	$-0.368203\pi$
$827$	23.1396	0.804643	0.402322	+	0.915498i	$0.368203\pi$
$828$	107.717	3.74342
$829$	−24.6843	−0.857321	−0.428660	−	0.903466i	$-0.641014\pi$
$829$	−24.6843	−0.857321	−0.428660	+	0.903466i	$0.641014\pi$
$830$	0	0
$831$	8.45141	0.293176
$832$	−31.5500	−1.09380
$833$	23.1052	0.800547
$834$	125.906	4.35977
$835$	0	0
$836$	0	0
$837$	−28.1338	−0.972448
$838$	0.419685	0.0144978
$839$	−31.2238	−1.07797	−0.538983	−	0.842317i	$-0.681191\pi$
$839$	−31.2238	−1.07797	−0.538983	+	0.842317i	$0.681191\pi$
$840$	0	0
$841$	7.40738	0.255427
$842$	−42.4398	−1.46257
$843$	−66.0912	−2.27630
$844$	−82.3208	−2.83360
$845$	0	0
$846$	116.473	4.00443
$847$	−24.7723	−0.851187
$848$	−17.6379	−0.605689
$849$	−13.7877	−0.473194
$850$	0	0
$851$	−18.9163	−0.648443
$852$	1.43425	0.0491364
$853$	−27.1366	−0.929140	−0.464570	−	0.885536i	$-0.653791\pi$
$853$	−27.1366	−0.929140	−0.464570	+	0.885536i	$0.653791\pi$
$854$	−125.844	−4.30629
$855$	0	0
$856$	−13.0975	−0.447664
$857$	−47.6457	−1.62755	−0.813773	−	0.581183i	$-0.802590\pi$
$857$	−47.6457	−1.62755	−0.813773	+	0.581183i	$0.802590\pi$
$858$	55.7661	1.90382
$859$	−19.6361	−0.669974	−0.334987	−	0.942223i	$-0.608732\pi$
$859$	−19.6361	−0.669974	−0.334987	+	0.942223i	$0.608732\pi$
$860$	0	0
$861$	85.6177	2.91784
$862$	12.3503	0.420654
$863$	49.3684	1.68052	0.840260	−	0.542183i	$-0.182402\pi$
$863$	49.3684	1.68052	0.840260	+	0.542183i	$0.182402\pi$
$864$	4.15025	0.141194
$865$	0	0
$866$	30.5197	1.03710
$867$	22.8066	0.774554
$868$	53.8339	1.82724
$869$	9.04600	0.306865
$870$	0	0
$871$	19.6643	0.666299
$872$	−14.2437	−0.482351
$873$	102.269	3.46129
$874$	0	0
$875$	0	0
$876$	−145.082	−4.90188
$877$	−41.1968	−1.39112	−0.695559	−	0.718469i	$-0.744843\pi$
$877$	−41.1968	−1.39112	−0.695559	+	0.718469i	$0.744843\pi$
$878$	35.9845	1.21442
$879$	−9.60819	−0.324076
$880$	0	0
$881$	−10.4510	−0.352103	−0.176052	−	0.984381i	$-0.556333\pi$
$881$	−10.4510	−0.352103	−0.176052	+	0.984381i	$0.556333\pi$
$882$	105.072	3.53797
$883$	50.2957	1.69259	0.846293	−	0.532717i	$-0.178829\pi$
$883$	50.2957	1.69259	0.846293	+	0.532717i	$0.178829\pi$
$884$	43.8639	1.47530
$885$	0	0
$886$	−13.0680	−0.439027
$887$	0.201711	0.00677278	0.00338639	−	0.999994i	$-0.498922\pi$
$887$	0.201711	0.00677278	0.00338639	+	0.999994i	$0.498922\pi$
$888$	54.9178	1.84292
$889$	−52.0272	−1.74494
$890$	0	0
$891$	−13.1222	−0.439609
$892$	53.9545	1.80653
$893$	0	0
$894$	−137.846	−4.61027
$895$	0	0
$896$	−76.3935	−2.55213
$897$	−51.7879	−1.72915
$898$	17.4597	0.582637
$899$	21.5520	0.718798
$900$	0	0
$901$	−14.6045	−0.486548
$902$	39.4268	1.31277
$903$	−63.0551	−2.09834
$904$	−17.1558	−0.570595
$905$	0	0
$906$	80.0518	2.65954
$907$	−6.11841	−0.203158	−0.101579	−	0.994827i	$-0.532390\pi$
$907$	−6.11841	−0.203158	−0.101579	+	0.994827i	$0.532390\pi$
$908$	14.6445	0.485996
$909$	105.826	3.51002
$910$	0	0
$911$	−0.960234	−0.0318140	−0.0159070	−	0.999873i	$-0.505064\pi$
$911$	−0.960234	−0.0318140	−0.0159070	+	0.999873i	$0.505064\pi$
$912$	0	0
$913$	9.34435	0.309253
$914$	−78.9722	−2.61217
$915$	0	0
$916$	44.6149	1.47412
$917$	21.6631	0.715378
$918$	−58.4308	−1.92851
$919$	−51.3264	−1.69310	−0.846551	−	0.532307i	$-0.821325\pi$
$919$	−51.3264	−1.69310	−0.846551	+	0.532307i	$0.821325\pi$
$920$	0	0
$921$	5.05831	0.166677
$922$	−61.4750	−2.02457
$923$	−0.451073	−0.0148473
$924$	94.3270	3.10313
$925$	0	0
$926$	99.6023	3.27314
$927$	−2.14163	−0.0703404
$928$	−3.17930	−0.104366
$929$	−46.0177	−1.50979	−0.754896	−	0.655844i	$-0.772313\pi$
$929$	−46.0177	−1.50979	−0.754896	+	0.655844i	$0.772313\pi$
$930$	0	0
$931$	0	0
$932$	42.1165	1.37957
$933$	2.62993	0.0860999
$934$	−35.6500	−1.16650
$935$	0	0
$936$	98.3522	3.21474
$937$	3.79467	0.123966	0.0619832	−	0.998077i	$-0.480257\pi$
$937$	3.79467	0.123966	0.0619832	+	0.998077i	$0.480257\pi$
$938$	50.1233	1.63658
$939$	33.9754	1.10875
$940$	0	0
$941$	23.1763	0.755526	0.377763	−	0.925902i	$-0.376693\pi$
$941$	23.1763	0.755526	0.377763	+	0.925902i	$0.376693\pi$
$942$	80.0371	2.60775
$943$	−36.6142	−1.19232
$944$	−18.8682	−0.614106
$945$	0	0
$946$	−29.0367	−0.944066
$947$	−18.3891	−0.597565	−0.298783	−	0.954321i	$-0.596581\pi$
$947$	−18.3891	−0.597565	−0.298783	+	0.954321i	$0.596581\pi$
$948$	49.4644	1.60653
$949$	45.6287	1.48117
$950$	0	0
$951$	−55.4427	−1.79785
$952$	55.1273	1.78669
$953$	−14.6072	−0.473174	−0.236587	−	0.971610i	$-0.576029\pi$
$953$	−14.6072	−0.473174	−0.236587	+	0.971610i	$0.576029\pi$
$954$	−66.4151	−2.15027
$955$	0	0
$956$	−68.1785	−2.20505
$957$	37.7631	1.22071
$958$	2.60760	0.0842477
$959$	30.9631	0.999852
$960$	0	0
$961$	−18.2420	−0.588450
$962$	−35.0299	−1.12941
$963$	−15.6694	−0.504938
$964$	−69.0361	−2.22350
$965$	0	0
$966$	−132.005	−4.24719
$967$	−59.4737	−1.91254	−0.956272	−	0.292478i	$-0.905520\pi$
$967$	−59.4737	−1.91254	−0.956272	+	0.292478i	$0.905520\pi$
$968$	−30.7559	−0.988533
$969$	0	0
$970$	0	0
$971$	22.3008	0.715667	0.357833	−	0.933785i	$-0.383516\pi$
$971$	22.3008	0.715667	0.357833	+	0.933785i	$0.383516\pi$
$972$	21.4712	0.688689
$973$	−66.9785	−2.14723
$974$	−15.2467	−0.488535
$975$	0	0
$976$	−49.6406	−1.58896
$977$	15.2734	0.488638	0.244319	−	0.969695i	$-0.421436\pi$
$977$	15.2734	0.488638	0.244319	+	0.969695i	$0.421436\pi$
$978$	−48.0191	−1.53548
$979$	0.164132	0.00524567
$980$	0	0
$981$	−17.0406	−0.544063
$982$	−1.98398	−0.0633115
$983$	41.4127	1.32086	0.660431	−	0.750887i	$-0.270374\pi$
$983$	41.4127	1.32086	0.660431	+	0.750887i	$0.270374\pi$
$984$	106.298	3.38866
$985$	0	0
$986$	44.7610	1.42548
$987$	−94.7186	−3.01493
$988$	0	0
$989$	26.9653	0.857447
$990$	0	0
$991$	−18.9609	−0.602314	−0.301157	−	0.953575i	$-0.597373\pi$
$991$	−18.9609	−0.602314	−0.301157	+	0.953575i	$0.597373\pi$
$992$	−1.88204	−0.0597549
$993$	−45.8805	−1.45597
$994$	−1.14976	−0.0364683
$995$	0	0
$996$	51.0958	1.61903
$997$	22.5133	0.713002	0.356501	−	0.934295i	$-0.383970\pi$
$997$	22.5133	0.713002	0.356501	+	0.934295i	$0.383970\pi$
$998$	3.84361	0.121668
$999$	30.9655	0.979704

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9025.2.a.bo.1.4		4
5.4	even	2		1805.2.a.j.1.1	✓	4
19.18	odd	2		9025.2.a.bh.1.1		4
95.94	odd	2		1805.2.a.n.1.4	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1805.2.a.j.1.1	✓	4	5.4	even	2
1805.2.a.n.1.4	yes	4	95.94	odd	2
9025.2.a.bh.1.1		4	19.18	odd	2
9025.2.a.bo.1.4		4	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)

\(f(q)\)	\(=\)	\(q+2.43828 q^{2} -2.94523 q^{3} +3.94523 q^{4} -7.18129 q^{6} +3.82025 q^{7} +4.74301 q^{8} +5.67435 q^{9} -2.12498 q^{11} -11.6196 q^{12} +3.65438 q^{13} +9.31485 q^{14} +3.67435 q^{16} +3.04243 q^{17} +13.8357 q^{18} -11.2515 q^{21} -5.18129 q^{22} +4.81167 q^{23} -13.9692 q^{24} +8.91042 q^{26} -7.87657 q^{27} +15.0717 q^{28} +6.03385 q^{29} +3.57184 q^{31} -0.526911 q^{32} +6.25853 q^{33} +7.41831 q^{34} +22.3866 q^{36} -3.93134 q^{37} -10.7630 q^{39} -7.60945 q^{41} -27.4343 q^{42} +5.60415 q^{43} -8.38351 q^{44} +11.7322 q^{46} +8.41831 q^{47} -10.8218 q^{48} +7.59430 q^{49} -8.96065 q^{51} +14.4174 q^{52} -4.80028 q^{53} -19.2053 q^{54} +18.1195 q^{56} +14.7122 q^{58} -5.13510 q^{59} -13.5100 q^{61} +8.70916 q^{62} +21.6774 q^{63} -8.63346 q^{64} +15.2601 q^{66} +5.38101 q^{67} +12.0031 q^{68} -14.1714 q^{69} -0.123434 q^{71} +26.9135 q^{72} +12.4860 q^{73} -9.58572 q^{74} -8.11794 q^{77} -26.2432 q^{78} -4.25699 q^{79} +6.17521 q^{81} -18.5540 q^{82} -4.39739 q^{83} -44.3897 q^{84} +13.6645 q^{86} -17.7711 q^{87} -10.0788 q^{88} -0.0772394 q^{89} +13.9607 q^{91} +18.9831 q^{92} -10.5199 q^{93} +20.5262 q^{94} +1.55187 q^{96} +18.0231 q^{97} +18.5171 q^{98} -12.0579 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + q^{2} + q^{3} + 3 q^{4} - 7 q^{6} + 11 q^{7} + 6 q^{8} + 5 q^{9} - 16 q^{12} - 2 q^{13} + 11 q^{14} - 3 q^{16} + 7 q^{17} + 17 q^{18} - 2 q^{21} + q^{22} + 11 q^{23} - 13 q^{24} + 9 q^{26} - 14 q^{27}+ \cdots - 9 q^{99}+O(q^{100}) \) 4 * q + q^2 + q^3 + 3 * q^4 - 7 * q^6 + 11 * q^7 + 6 * q^8 + 5 * q^9 - 16 * q^12 - 2 * q^13 + 11 * q^14 - 3 * q^16 + 7 * q^17 + 17 * q^18 - 2 * q^21 + q^22 + 11 * q^23 - 13 * q^24 + 9 * q^26 - 14 * q^27 + 13 * q^28 + 15 * q^29 + q^31 + 3 * q^32 + 12 * q^33 + 22 * q^34 + 16 * q^36 - 11 * q^37 - 29 * q^39 - 22 * q^41 - 19 * q^42 + 26 * q^43 - 12 * q^44 - 10 * q^46 + 26 * q^47 - 13 * q^48 + 13 * q^49 + 11 * q^51 + 27 * q^52 - 16 * q^53 - 25 * q^54 + 8 * q^56 + 3 * q^58 + 10 * q^59 + 2 * q^61 + 31 * q^62 + 17 * q^63 + 4 * q^64 + 22 * q^66 + 3 * q^67 - 4 * q^68 - 14 * q^69 - 18 * q^71 + 29 * q^72 + 24 * q^73 - 17 * q^74 + 6 * q^77 - 15 * q^78 - 30 * q^79 - 4 * q^81 + 13 * q^82 + 12 * q^83 - 52 * q^84 + 16 * q^86 + q^87 - 23 * q^88 - 9 * q^89 + 9 * q^91 + 25 * q^92 - 7 * q^93 + 11 * q^94 - 6 * q^96 + 19 * q^97 + 48 * q^98 - 9 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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