LMFDB - Embedded newform 925.2.a.m.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [925,2,Mod(1,925)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("925.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(925, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 925 = 5^{2} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	925.a (trivial)

Newform invariants

comment:select newform

sage:traces = [9,5,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$7.38616218697$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 4x^{8} - 6x^{7} + 30x^{6} + 15x^{5} - 70x^{4} - 22x^{3} + 44x^{2} + 4x - 4 $ x^9 - 4x^8 - 6x^7 + 30x^6 + 15x^5 - 70x^4 - 22x^3 + 44x^2 + 4x - 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 185)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.753109$ of defining polynomial
Character	$\chi$	$=$	925.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+0.246891 q^{2} +3.34515 q^{3} -1.93904 q^{4} +0.825888 q^{6} +2.66723 q^{7} -0.972515 q^{8} +8.19006 q^{9} -1.12244 q^{11} -6.48640 q^{12} -4.38171 q^{13} +0.658515 q^{14} +3.63798 q^{16} +1.15960 q^{17} +2.02205 q^{18} +6.43164 q^{19} +8.92229 q^{21} -0.277121 q^{22} +3.45756 q^{23} -3.25321 q^{24} -1.08181 q^{26} +17.3615 q^{27} -5.17188 q^{28} -0.729661 q^{29} -7.22523 q^{31} +2.84321 q^{32} -3.75474 q^{33} +0.286295 q^{34} -15.8809 q^{36} -1.00000 q^{37} +1.58791 q^{38} -14.6575 q^{39} +4.61897 q^{41} +2.20283 q^{42} -1.32712 q^{43} +2.17646 q^{44} +0.853641 q^{46} -10.3318 q^{47} +12.1696 q^{48} +0.114110 q^{49} +3.87905 q^{51} +8.49634 q^{52} +2.77765 q^{53} +4.28641 q^{54} -2.59392 q^{56} +21.5148 q^{57} -0.180147 q^{58} -4.32778 q^{59} -4.73004 q^{61} -1.78384 q^{62} +21.8448 q^{63} -6.57401 q^{64} -0.927011 q^{66} +2.64090 q^{67} -2.24852 q^{68} +11.5661 q^{69} +3.33860 q^{71} -7.96495 q^{72} -5.87153 q^{73} -0.246891 q^{74} -12.4712 q^{76} -2.99381 q^{77} -3.61881 q^{78} -6.77753 q^{79} +33.5068 q^{81} +1.14038 q^{82} +10.3882 q^{83} -17.3007 q^{84} -0.327654 q^{86} -2.44083 q^{87} +1.09159 q^{88} -3.34606 q^{89} -11.6870 q^{91} -6.70437 q^{92} -24.1695 q^{93} -2.55084 q^{94} +9.51099 q^{96} +3.02933 q^{97} +0.0281728 q^{98} -9.19285 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 5 q^{2} + 8 q^{3} + 11 q^{4} + 2 q^{6} + 8 q^{7} + 15 q^{8} + 13 q^{9} + 16 q^{12} + 6 q^{13} - 4 q^{14} + 11 q^{16} + 18 q^{17} - 3 q^{18} - 4 q^{19} + 4 q^{21} + 6 q^{22} + 16 q^{23} + 6 q^{24}+ \cdots + 8 q^{99}+O(q^{100}) $ 9 * q + 5 * q^2 + 8 * q^3 + 11 * q^4 + 2 * q^6 + 8 * q^7 + 15 * q^8 + 13 * q^9 + 16 * q^12 + 6 * q^13 - 4 * q^14 + 11 * q^16 + 18 * q^17 - 3 * q^18 - 4 * q^19 + 4 * q^21 + 6 * q^22 + 16 * q^23 + 6 * q^24 - 6 * q^26 + 32 * q^27 - 20 * q^28 - 2 * q^29 - 6 * q^31 + 35 * q^32 + 6 * q^33 + 6 * q^34 + 11 * q^36 - 9 * q^37 + 4 * q^38 - 16 * q^39 + 2 * q^41 + 4 * q^42 + 14 * q^43 + 4 * q^44 + 16 * q^46 + 34 * q^47 + 6 * q^48 - 7 * q^49 + 8 * q^51 + 16 * q^52 - 4 * q^53 - 18 * q^54 - 24 * q^56 - 4 * q^57 - 22 * q^58 + 10 * q^59 + 8 * q^61 + 26 * q^62 + 18 * q^63 + 15 * q^64 - 34 * q^66 + 16 * q^67 + 36 * q^68 - 10 * q^69 + 8 * q^71 + 9 * q^72 + 8 * q^73 - 5 * q^74 - 48 * q^76 + 14 * q^77 - 76 * q^78 - 26 * q^79 + 13 * q^81 - 4 * q^82 + 70 * q^83 - 24 * q^84 - 16 * q^86 - 12 * q^87 - 36 * q^88 - 8 * q^89 - 18 * q^91 + 16 * q^92 + 2 * q^93 - 26 * q^94 + 52 * q^96 - 2 * q^97 + 21 * q^98 + 8 * 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 $

$2$	0.246891	0.174578	0.0872891	−	0.996183i	$-0.472180\pi$
$2$	0.246891	0.174578	0.0872891	+	0.996183i	$0.472180\pi$
$3$	3.34515	1.93133	0.965663	−	0.259799i	$-0.0836562\pi$
$3$	3.34515	1.93133	0.965663	+	0.259799i	$0.0836562\pi$
$4$	−1.93904	−0.969522
$5$	0	0
$5$	0	0
$6$	0.825888	0.337167
$7$	2.66723	1.00812	0.504059	−	0.863669i	$-0.331839\pi$
$7$	2.66723	1.00812	0.504059	+	0.863669i	$0.331839\pi$
$8$	−0.972515	−0.343836
$9$	8.19006	2.73002
$10$	0	0
$11$	−1.12244	−0.338429	−0.169214	−	0.985579i	$-0.554123\pi$
$11$	−1.12244	−0.338429	−0.169214	+	0.985579i	$0.554123\pi$
$12$	−6.48640	−1.87246
$13$	−4.38171	−1.21527	−0.607635	−	0.794217i	$-0.707881\pi$
$13$	−4.38171	−1.21527	−0.607635	+	0.794217i	$0.707881\pi$
$14$	0.658515	0.175995
$15$	0	0
$16$	3.63798	0.909496
$17$	1.15960	0.281245	0.140622	−	0.990063i	$-0.455090\pi$
$17$	1.15960	0.281245	0.140622	+	0.990063i	$0.455090\pi$
$18$	2.02205	0.476602
$19$	6.43164	1.47552	0.737760	−	0.675063i	$-0.235884\pi$
$19$	6.43164	1.47552	0.737760	+	0.675063i	$0.235884\pi$
$20$	0	0
$21$	8.92229	1.94700
$22$	−0.277121	−0.0590823
$23$	3.45756	0.720952	0.360476	−	0.932769i	$-0.382614\pi$
$23$	3.45756	0.720952	0.360476	+	0.932769i	$0.382614\pi$
$24$	−3.25321	−0.664059
$25$	0	0
$26$	−1.08181	−0.212160
$27$	17.3615	3.34123
$28$	−5.17188	−0.977393
$29$	−0.729661	−0.135495	−0.0677473	−	0.997703i	$-0.521581\pi$
$29$	−0.729661	−0.135495	−0.0677473	+	0.997703i	$0.521581\pi$
$30$	0	0
$31$	−7.22523	−1.29769	−0.648845	−	0.760920i	$-0.724748\pi$
$31$	−7.22523	−1.29769	−0.648845	+	0.760920i	$0.724748\pi$
$32$	2.84321	0.502614
$33$	−3.75474	−0.653616
$34$	0.286295	0.0490992
$35$	0	0
$36$	−15.8809	−2.64681
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	1.58791	0.257594
$39$	−14.6575	−2.34708
$40$	0	0
$41$	4.61897	0.721361	0.360681	−	0.932689i	$-0.382544\pi$
$41$	4.61897	0.721361	0.360681	+	0.932689i	$0.382544\pi$
$42$	2.20283	0.339905
$43$	−1.32712	−0.202384	−0.101192	−	0.994867i	$-0.532266\pi$
$43$	−1.32712	−0.202384	−0.101192	+	0.994867i	$0.532266\pi$
$44$	2.17646	0.328114
$45$	0	0
$46$	0.853641	0.125863
$47$	−10.3318	−1.50705	−0.753527	−	0.657417i	$-0.771649\pi$
$47$	−10.3318	−1.50705	−0.753527	+	0.657417i	$0.771649\pi$
$48$	12.1696	1.75653
$49$	0.114110	0.0163014
$50$	0	0
$51$	3.87905	0.543175
$52$	8.49634	1.17823
$53$	2.77765	0.381539	0.190770	−	0.981635i	$-0.438902\pi$
$53$	2.77765	0.381539	0.190770	+	0.981635i	$0.438902\pi$
$54$	4.28641	0.583306
$55$	0	0
$56$	−2.59392	−0.346627
$57$	21.5148	2.84971
$58$	−0.180147	−0.0236544
$59$	−4.32778	−0.563429	−0.281714	−	0.959498i	$-0.590903\pi$
$59$	−4.32778	−0.563429	−0.281714	+	0.959498i	$0.590903\pi$
$60$	0	0
$61$	−4.73004	−0.605619	−0.302810	−	0.953051i	$-0.597925\pi$
$61$	−4.73004	−0.605619	−0.302810	+	0.953051i	$0.597925\pi$
$62$	−1.78384	−0.226549
$63$	21.8448	2.75218
$64$	−6.57401	−0.821751
$65$	0	0
$66$	−0.927011	−0.114107
$67$	2.64090	0.322637	0.161319	−	0.986902i	$-0.448425\pi$
$67$	2.64090	0.322637	0.161319	+	0.986902i	$0.448425\pi$
$68$	−2.24852	−0.272673
$69$	11.5661	1.39239
$70$	0	0
$71$	3.33860	0.396219	0.198109	−	0.980180i	$-0.436520\pi$
$71$	3.33860	0.396219	0.198109	+	0.980180i	$0.436520\pi$
$72$	−7.96495	−0.938678
$73$	−5.87153	−0.687210	−0.343605	−	0.939114i	$-0.611648\pi$
$73$	−5.87153	−0.687210	−0.343605	+	0.939114i	$0.611648\pi$
$74$	−0.246891	−0.0287005
$75$	0	0
$76$	−12.4712	−1.43055
$77$	−2.99381	−0.341176
$78$	−3.61881	−0.409749
$79$	−6.77753	−0.762532	−0.381266	−	0.924465i	$-0.624512\pi$
$79$	−6.77753	−0.762532	−0.381266	+	0.924465i	$0.624512\pi$
$80$	0	0
$81$	33.5068	3.72298
$82$	1.14038	0.125934
$83$	10.3882	1.14026	0.570129	−	0.821555i	$-0.306893\pi$
$83$	10.3882	1.14026	0.570129	+	0.821555i	$0.306893\pi$
$84$	−17.3007	−1.88766
$85$	0	0
$86$	−0.327654	−0.0353318
$87$	−2.44083	−0.261684
$88$	1.09159	0.116364
$89$	−3.34606	−0.354682	−0.177341	−	0.984149i	$-0.556750\pi$
$89$	−3.34606	−0.354682	−0.177341	+	0.984149i	$0.556750\pi$
$90$	0	0
$91$	−11.6870	−1.22513
$92$	−6.70437	−0.698979
$93$	−24.1695	−2.50626
$94$	−2.55084	−0.263099
$95$	0	0
$96$	9.51099	0.970711
$97$	3.02933	0.307582	0.153791	−	0.988103i	$-0.450852\pi$
$97$	3.02933	0.307582	0.153791	+	0.988103i	$0.450852\pi$
$98$	0.0281728	0.00284588
$99$	−9.19285	−0.923916
$100$	0	0
$101$	−10.3972	−1.03456	−0.517279	−	0.855817i	$-0.673055\pi$
$101$	−10.3972	−1.03456	−0.517279	+	0.855817i	$0.673055\pi$
$102$	0.957701	0.0948266
$103$	3.69429	0.364009	0.182004	−	0.983298i	$-0.441741\pi$
$103$	3.69429	0.364009	0.182004	+	0.983298i	$0.441741\pi$
$104$	4.26128	0.417853
$105$	0	0
$106$	0.685776	0.0666085
$107$	−6.75105	−0.652649	−0.326324	−	0.945258i	$-0.605810\pi$
$107$	−6.75105	−0.652649	−0.326324	+	0.945258i	$0.605810\pi$
$108$	−33.6648	−3.23940
$109$	−6.13569	−0.587693	−0.293846	−	0.955853i	$-0.594935\pi$
$109$	−6.13569	−0.587693	−0.293846	+	0.955853i	$0.594935\pi$
$110$	0	0
$111$	−3.34515	−0.317508
$112$	9.70334	0.916879
$113$	2.09180	0.196780	0.0983898	−	0.995148i	$-0.468631\pi$
$113$	2.09180	0.196780	0.0983898	+	0.995148i	$0.468631\pi$
$114$	5.31182	0.497497
$115$	0	0
$116$	1.41484	0.131365
$117$	−35.8865	−3.31771
$118$	−1.06849	−0.0983624
$119$	3.09292	0.283528
$120$	0	0
$121$	−9.74013	−0.885466
$122$	−1.16780	−0.105728
$123$	15.4512	1.39318
$124$	14.0101	1.25814
$125$	0	0
$126$	5.39327	0.480471
$127$	−16.7945	−1.49027	−0.745135	−	0.666914i	$-0.767615\pi$
$127$	−16.7945	−1.49027	−0.745135	+	0.666914i	$0.767615\pi$
$128$	−7.30949	−0.646074
$129$	−4.43942	−0.390869
$130$	0	0
$131$	−16.4522	−1.43743	−0.718716	−	0.695304i	$-0.755270\pi$
$131$	−16.4522	−1.43743	−0.718716	+	0.695304i	$0.755270\pi$
$132$	7.28060	0.633695
$133$	17.1547	1.48750
$134$	0.652015	0.0563255
$135$	0	0
$136$	−1.12773	−0.0967020
$137$	7.51317	0.641894	0.320947	−	0.947097i	$-0.395999\pi$
$137$	7.51317	0.641894	0.320947	+	0.947097i	$0.395999\pi$
$138$	2.85556	0.243082
$139$	10.8184	0.917608	0.458804	−	0.888538i	$-0.348278\pi$
$139$	10.8184	0.917608	0.458804	+	0.888538i	$0.348278\pi$
$140$	0	0
$141$	−34.5616	−2.91061
$142$	0.824270	0.0691712
$143$	4.91822	0.411282
$144$	29.7953	2.48294
$145$	0	0
$146$	−1.44963	−0.119972
$147$	0.381716	0.0314834
$148$	1.93904	0.159389
$149$	14.7133	1.20536	0.602682	−	0.797981i	$-0.294099\pi$
$149$	14.7133	1.20536	0.602682	+	0.797981i	$0.294099\pi$
$150$	0	0
$151$	16.9858	1.38229	0.691144	−	0.722717i	$-0.257107\pi$
$151$	16.9858	1.38229	0.691144	+	0.722717i	$0.257107\pi$
$152$	−6.25487	−0.507337
$153$	9.49720	0.767803
$154$	−0.739144	−0.0595619
$155$	0	0
$156$	28.4216	2.27555
$157$	11.4204	0.911450	0.455725	−	0.890121i	$-0.349380\pi$
$157$	11.4204	0.911450	0.455725	+	0.890121i	$0.349380\pi$
$158$	−1.67331	−0.133122
$159$	9.29166	0.736877
$160$	0	0
$161$	9.22211	0.726804
$162$	8.27254	0.649952
$163$	−17.1484	−1.34316	−0.671581	−	0.740931i	$-0.734385\pi$
$163$	−17.1484	−1.34316	−0.671581	+	0.740931i	$0.734385\pi$
$164$	−8.95638	−0.699376
$165$	0	0
$166$	2.56476	0.199064
$167$	8.51875	0.659201	0.329600	−	0.944121i	$-0.393086\pi$
$167$	8.51875	0.659201	0.329600	+	0.944121i	$0.393086\pi$
$168$	−8.67706	−0.669450
$169$	6.19942	0.476879
$170$	0	0
$171$	52.6755	4.02820
$172$	2.57334	0.196216
$173$	−4.79414	−0.364492	−0.182246	−	0.983253i	$-0.558337\pi$
$173$	−4.79414	−0.364492	−0.182246	+	0.983253i	$0.558337\pi$
$174$	−0.602618	−0.0456844
$175$	0	0
$176$	−4.08342	−0.307800
$177$	−14.4771	−1.08816
$178$	−0.826113	−0.0619198
$179$	−23.8605	−1.78342	−0.891708	−	0.452611i	$-0.850493\pi$
$179$	−23.8605	−1.78342	−0.891708	+	0.452611i	$0.850493\pi$
$180$	0	0
$181$	2.23666	0.166250	0.0831248	−	0.996539i	$-0.473510\pi$
$181$	2.23666	0.166250	0.0831248	+	0.996539i	$0.473510\pi$
$182$	−2.88542	−0.213882
$183$	−15.8227	−1.16965
$184$	−3.36253	−0.247889
$185$	0	0
$186$	−5.96724	−0.437539
$187$	−1.30158	−0.0951813
$188$	20.0339	1.46112
$189$	46.3072	3.36835
$190$	0	0
$191$	−16.3404	−1.18235	−0.591176	−	0.806543i	$-0.701336\pi$
$191$	−16.3404	−1.18235	−0.591176	+	0.806543i	$0.701336\pi$
$192$	−21.9911	−1.58707
$193$	−18.1778	−1.30847	−0.654234	−	0.756292i	$-0.727009\pi$
$193$	−18.1778	−1.30847	−0.654234	+	0.756292i	$0.727009\pi$
$194$	0.747914	0.0536971
$195$	0	0
$196$	−0.221265	−0.0158046
$197$	1.71045	0.121864	0.0609322	−	0.998142i	$-0.480593\pi$
$197$	1.71045	0.121864	0.0609322	+	0.998142i	$0.480593\pi$
$198$	−2.26963	−0.161296
$199$	−5.86596	−0.415827	−0.207913	−	0.978147i	$-0.566667\pi$
$199$	−5.86596	−0.415827	−0.207913	+	0.978147i	$0.566667\pi$
$200$	0	0
$201$	8.83422	0.623118
$202$	−2.56697	−0.180611
$203$	−1.94617	−0.136595
$204$	−7.52164	−0.526620
$205$	0	0
$206$	0.912086	0.0635481
$207$	28.3176	1.96821
$208$	−15.9406	−1.10528
$209$	−7.21914	−0.499358
$210$	0	0
$211$	−17.0431	−1.17330	−0.586648	−	0.809842i	$-0.699553\pi$
$211$	−17.0431	−1.17330	−0.586648	+	0.809842i	$0.699553\pi$
$212$	−5.38598	−0.369911
$213$	11.1681	0.765228
$214$	−1.66677	−0.113938
$215$	0	0
$216$	−16.8843	−1.14883
$217$	−19.2714	−1.30822
$218$	−1.51485	−0.102598
$219$	−19.6412	−1.32723
$220$	0	0
$221$	−5.08104	−0.341788
$222$	−0.825888	−0.0554300
$223$	3.73873	0.250364	0.125182	−	0.992134i	$-0.460049\pi$
$223$	3.73873	0.250364	0.125182	+	0.992134i	$0.460049\pi$
$224$	7.58350	0.506694
$225$	0	0
$226$	0.516446	0.0343535
$227$	18.3517	1.21805	0.609023	−	0.793153i	$-0.291562\pi$
$227$	18.3517	1.21805	0.609023	+	0.793153i	$0.291562\pi$
$228$	−41.7182	−2.76286
$229$	−15.8336	−1.04631	−0.523156	−	0.852237i	$-0.675245\pi$
$229$	−15.8336	−1.04631	−0.523156	+	0.852237i	$0.675245\pi$
$230$	0	0
$231$	−10.0147	−0.658922
$232$	0.709606	0.0465879
$233$	23.3893	1.53229	0.766143	−	0.642670i	$-0.222174\pi$
$233$	23.3893	1.53229	0.766143	+	0.642670i	$0.222174\pi$
$234$	−8.86005	−0.579200
$235$	0	0
$236$	8.39175	0.546257
$237$	−22.6719	−1.47270
$238$	0.763615	0.0494978
$239$	−8.61855	−0.557488	−0.278744	−	0.960365i	$-0.589918\pi$
$239$	−8.61855	−0.557488	−0.278744	+	0.960365i	$0.589918\pi$
$240$	0	0
$241$	−0.491805	−0.0316799	−0.0158400	−	0.999875i	$-0.505042\pi$
$241$	−0.491805	−0.0316799	−0.0158400	+	0.999875i	$0.505042\pi$
$242$	−2.40475	−0.154583
$243$	60.0009	3.84906
$244$	9.17175	0.587161
$245$	0	0
$246$	3.81475	0.243220
$247$	−28.1816	−1.79315
$248$	7.02665	0.446192
$249$	34.7503	2.20221
$250$	0	0
$251$	−14.9337	−0.942609	−0.471305	−	0.881971i	$-0.656217\pi$
$251$	−14.9337	−0.942609	−0.471305	+	0.881971i	$0.656217\pi$
$252$	−42.3580	−2.66830
$253$	−3.88091	−0.243991
$254$	−4.14641	−0.260169
$255$	0	0
$256$	11.3434	0.708960
$257$	21.3577	1.33225	0.666127	−	0.745838i	$-0.267951\pi$
$257$	21.3577	1.33225	0.666127	+	0.745838i	$0.267951\pi$
$258$	−1.09605	−0.0682372
$259$	−2.66723	−0.165734
$260$	0	0
$261$	−5.97596	−0.369903
$262$	−4.06189	−0.250944
$263$	−9.37607	−0.578153	−0.289077	−	0.957306i	$-0.593348\pi$
$263$	−9.37607	−0.578153	−0.289077	+	0.957306i	$0.593348\pi$
$264$	3.65154	0.224737
$265$	0	0
$266$	4.23533	0.259685
$267$	−11.1931	−0.685006
$268$	−5.12083	−0.312804
$269$	2.91988	0.178028	0.0890141	−	0.996030i	$-0.471628\pi$
$269$	2.91988	0.178028	0.0890141	+	0.996030i	$0.471628\pi$
$270$	0	0
$271$	2.62192	0.159270	0.0796350	−	0.996824i	$-0.474625\pi$
$271$	2.62192	0.159270	0.0796350	+	0.996824i	$0.474625\pi$
$272$	4.21861	0.255791
$273$	−39.0949	−2.36613
$274$	1.85493	0.112061
$275$	0	0
$276$	−22.4272	−1.34996
$277$	8.75381	0.525965	0.262983	−	0.964801i	$-0.415294\pi$
$277$	8.75381	0.525965	0.262983	+	0.964801i	$0.415294\pi$
$278$	2.67097	0.160194
$279$	−59.1751	−3.54272
$280$	0	0
$281$	22.9508	1.36913	0.684565	−	0.728952i	$-0.259992\pi$
$281$	22.9508	1.36913	0.684565	+	0.728952i	$0.259992\pi$
$282$	−8.53295	−0.508129
$283$	8.60777	0.511679	0.255839	−	0.966719i	$-0.417648\pi$
$283$	8.60777	0.511679	0.255839	+	0.966719i	$0.417648\pi$
$284$	−6.47369	−0.384143
$285$	0	0
$286$	1.21426	0.0718009
$287$	12.3198	0.727217
$288$	23.2861	1.37215
$289$	−15.6553	−0.920901
$290$	0	0
$291$	10.1336	0.594041
$292$	11.3852	0.666266
$293$	31.2104	1.82333	0.911667	−	0.410931i	$-0.134796\pi$
$293$	31.2104	1.82333	0.911667	+	0.410931i	$0.134796\pi$
$294$	0.0942422	0.00549632
$295$	0	0
$296$	0.972515	0.0565263
$297$	−19.4873	−1.13077
$298$	3.63259	0.210430
$299$	−15.1501	−0.876150
$300$	0	0
$301$	−3.53973	−0.204027
$302$	4.19365	0.241318
$303$	−34.7802	−1.99807
$304$	23.3982	1.34198
$305$	0	0
$306$	2.34477	0.134042
$307$	22.1209	1.26251	0.631253	−	0.775577i	$-0.282541\pi$
$307$	22.1209	1.26251	0.631253	+	0.775577i	$0.282541\pi$
$308$	5.80513	0.330778
$309$	12.3580	0.703020
$310$	0	0
$311$	−13.5854	−0.770359	−0.385179	−	0.922842i	$-0.625860\pi$
$311$	−13.5854	−0.770359	−0.385179	+	0.922842i	$0.625860\pi$
$312$	14.2546	0.807010
$313$	23.9064	1.35127	0.675635	−	0.737236i	$-0.263870\pi$
$313$	23.9064	1.35127	0.675635	+	0.737236i	$0.263870\pi$
$314$	2.81960	0.159119
$315$	0	0
$316$	13.1419	0.739292
$317$	32.4299	1.82145	0.910724	−	0.413016i	$-0.135525\pi$
$317$	32.4299	1.82145	0.910724	+	0.413016i	$0.135525\pi$
$318$	2.29403	0.128643
$319$	0.819001	0.0458553
$320$	0	0
$321$	−22.5833	−1.26048
$322$	2.27686	0.126884
$323$	7.45814	0.414982
$324$	−64.9713	−3.60951
$325$	0	0
$326$	−4.23377	−0.234487
$327$	−20.5248	−1.13503
$328$	−4.49201	−0.248030
$329$	−27.5574	−1.51929
$330$	0	0
$331$	12.3756	0.680224	0.340112	−	0.940385i	$-0.389535\pi$
$331$	12.3756	0.680224	0.340112	+	0.940385i	$0.389535\pi$
$332$	−20.1433	−1.10551
$333$	−8.19006	−0.448812
$334$	2.10320	0.115082
$335$	0	0
$336$	32.4592	1.77079
$337$	−23.4947	−1.27984	−0.639918	−	0.768444i	$-0.721032\pi$
$337$	−23.4947	−1.27984	−0.639918	+	0.768444i	$0.721032\pi$
$338$	1.53058	0.0832527
$339$	6.99738	0.380046
$340$	0	0
$341$	8.10990	0.439176
$342$	13.0051	0.703236
$343$	−18.3662	−0.991684
$344$	1.29064	0.0695868
$345$	0	0
$346$	−1.18363	−0.0636324
$347$	−2.80759	−0.150720	−0.0753598	−	0.997156i	$-0.524011\pi$
$347$	−2.80759	−0.150720	−0.0753598	+	0.997156i	$0.524011\pi$
$348$	4.73287	0.253709
$349$	−2.42088	−0.129587	−0.0647933	−	0.997899i	$-0.520639\pi$
$349$	−2.42088	−0.129587	−0.0647933	+	0.997899i	$0.520639\pi$
$350$	0	0
$351$	−76.0733	−4.06049
$352$	−3.19134	−0.170099
$353$	1.35444	0.0720894	0.0360447	−	0.999350i	$-0.488524\pi$
$353$	1.35444	0.0720894	0.0360447	+	0.999350i	$0.488524\pi$
$354$	−3.57426	−0.189970
$355$	0	0
$356$	6.48817	0.343872
$357$	10.3463	0.547584
$358$	−5.89094	−0.311346
$359$	13.9347	0.735448	0.367724	−	0.929935i	$-0.380137\pi$
$359$	13.9347	0.735448	0.367724	+	0.929935i	$0.380137\pi$
$360$	0	0
$361$	22.3660	1.17716
$362$	0.552211	0.0290236
$363$	−32.5822	−1.71012
$364$	22.6617	1.18780
$365$	0	0
$366$	−3.90648	−0.204195
$367$	−7.76283	−0.405216	−0.202608	−	0.979260i	$-0.564942\pi$
$367$	−7.76283	−0.405216	−0.202608	+	0.979260i	$0.564942\pi$
$368$	12.5786	0.655703
$369$	37.8296	1.96933
$370$	0	0
$371$	7.40862	0.384637
$372$	46.8658	2.42988
$373$	24.3774	1.26222	0.631108	−	0.775695i	$-0.282600\pi$
$373$	24.3774	1.26222	0.631108	+	0.775695i	$0.282600\pi$
$374$	−0.321349	−0.0166166
$375$	0	0
$376$	10.0479	0.518179
$377$	3.19717	0.164662
$378$	11.4328	0.588041
$379$	−13.7125	−0.704363	−0.352181	−	0.935932i	$-0.614560\pi$
$379$	−13.7125	−0.704363	−0.352181	+	0.935932i	$0.614560\pi$
$380$	0	0
$381$	−56.1801	−2.87819
$382$	−4.03430	−0.206413
$383$	−30.6230	−1.56476	−0.782381	−	0.622800i	$-0.785995\pi$
$383$	−30.6230	−1.56476	−0.782381	+	0.622800i	$0.785995\pi$
$384$	−24.4514	−1.24778
$385$	0	0
$386$	−4.48794	−0.228430
$387$	−10.8692	−0.552511
$388$	−5.87401	−0.298207
$389$	16.6593	0.844659	0.422330	−	0.906442i	$-0.361213\pi$
$389$	16.6593	0.844659	0.422330	+	0.906442i	$0.361213\pi$
$390$	0	0
$391$	4.00940	0.202764
$392$	−0.110974	−0.00560502
$393$	−55.0350	−2.77615
$394$	0.422294	0.0212749
$395$	0	0
$396$	17.8254	0.895758
$397$	−26.1701	−1.31344	−0.656720	−	0.754135i	$-0.728056\pi$
$397$	−26.1701	−1.31344	−0.656720	+	0.754135i	$0.728056\pi$
$398$	−1.44825	−0.0725943
$399$	57.3850	2.87284
$400$	0	0
$401$	−15.7569	−0.786860	−0.393430	−	0.919354i	$-0.628712\pi$
$401$	−15.7569	−0.786860	−0.393430	+	0.919354i	$0.628712\pi$
$402$	2.18109	0.108783
$403$	31.6589	1.57704
$404$	20.1606	1.00303
$405$	0	0
$406$	−0.480492	−0.0238464
$407$	1.12244	0.0556373
$408$	−3.77243	−0.186763
$409$	−27.8854	−1.37884	−0.689421	−	0.724361i	$-0.742135\pi$
$409$	−27.8854	−1.37884	−0.689421	+	0.724361i	$0.742135\pi$
$410$	0	0
$411$	25.1327	1.23971
$412$	−7.16339	−0.352915
$413$	−11.5432	−0.568002
$414$	6.99137	0.343607
$415$	0	0
$416$	−12.4582	−0.610811
$417$	36.1893	1.77220
$418$	−1.78234	−0.0871771
$419$	17.4407	0.852035	0.426017	−	0.904715i	$-0.359916\pi$
$419$	17.4407	0.852035	0.426017	+	0.904715i	$0.359916\pi$
$420$	0	0
$421$	17.2720	0.841788	0.420894	−	0.907110i	$-0.361716\pi$
$421$	17.2720	0.841788	0.420894	+	0.907110i	$0.361716\pi$
$422$	−4.20779	−0.204832
$423$	−84.6183	−4.11428
$424$	−2.70130	−0.131187
$425$	0	0
$426$	2.75731	0.133592
$427$	−12.6161	−0.610536
$428$	13.0906	0.632758
$429$	16.4522	0.794319
$430$	0	0
$431$	15.2304	0.733624	0.366812	−	0.930295i	$-0.380449\pi$
$431$	15.2304	0.733624	0.366812	+	0.930295i	$0.380449\pi$
$432$	63.1610	3.03883
$433$	4.59721	0.220928	0.110464	−	0.993880i	$-0.464766\pi$
$433$	4.59721	0.220928	0.110464	+	0.993880i	$0.464766\pi$
$434$	−4.75792	−0.228388
$435$	0	0
$436$	11.8974	0.569781
$437$	22.2378	1.06378
$438$	−4.84923	−0.231705
$439$	13.6389	0.650950	0.325475	−	0.945551i	$-0.394476\pi$
$439$	13.6389	0.650950	0.325475	+	0.945551i	$0.394476\pi$
$440$	0	0
$441$	0.934568	0.0445032
$442$	−1.25446	−0.0596688
$443$	18.9480	0.900249	0.450124	−	0.892966i	$-0.351380\pi$
$443$	18.9480	0.900249	0.450124	+	0.892966i	$0.351380\pi$
$444$	6.48640	0.307831
$445$	0	0
$446$	0.923059	0.0437081
$447$	49.2184	2.32795
$448$	−17.5344	−0.828421
$449$	0.199955	0.00943648	0.00471824	−	0.999989i	$-0.498498\pi$
$449$	0.199955	0.00943648	0.00471824	+	0.999989i	$0.498498\pi$
$450$	0	0
$451$	−5.18452	−0.244129
$452$	−4.05609	−0.190782
$453$	56.8203	2.66965
$454$	4.53087	0.212644
$455$	0	0
$456$	−20.9235	−0.979832
$457$	−29.0697	−1.35982	−0.679910	−	0.733295i	$-0.737981\pi$
$457$	−29.0697	−1.35982	−0.679910	+	0.733295i	$0.737981\pi$
$458$	−3.90916	−0.182663
$459$	20.1325	0.939703
$460$	0	0
$461$	22.2419	1.03591	0.517954	−	0.855408i	$-0.326694\pi$
$461$	22.2419	1.03591	0.517954	+	0.855408i	$0.326694\pi$
$462$	−2.47255	−0.115033
$463$	−11.0948	−0.515617	−0.257809	−	0.966196i	$-0.583000\pi$
$463$	−11.0948	−0.515617	−0.257809	+	0.966196i	$0.583000\pi$
$464$	−2.65449	−0.123232
$465$	0	0
$466$	5.77461	0.267504
$467$	32.4417	1.50122	0.750612	−	0.660744i	$-0.229759\pi$
$467$	32.4417	1.50122	0.750612	+	0.660744i	$0.229759\pi$
$468$	69.5855	3.21659
$469$	7.04389	0.325257
$470$	0	0
$471$	38.2031	1.76031
$472$	4.20883	0.193727
$473$	1.48961	0.0684925
$474$	−5.59748	−0.257101
$475$	0	0
$476$	−5.99732	−0.274887
$477$	22.7491	1.04161
$478$	−2.12784	−0.0973252
$479$	38.0979	1.74074	0.870370	−	0.492399i	$-0.163880\pi$
$479$	38.0979	1.74074	0.870370	+	0.492399i	$0.163880\pi$
$480$	0	0
$481$	4.38171	0.199789
$482$	−0.121422	−0.00553062
$483$	30.8494	1.40370
$484$	18.8865	0.858479
$485$	0	0
$486$	14.8137	0.671963
$487$	14.8929	0.674861	0.337430	−	0.941350i	$-0.390442\pi$
$487$	14.8929	0.674861	0.337430	+	0.941350i	$0.390442\pi$
$488$	4.60003	0.208234
$489$	−57.3639	−2.59408
$490$	0	0
$491$	34.0392	1.53617	0.768085	−	0.640348i	$-0.221210\pi$
$491$	34.0392	1.53617	0.768085	+	0.640348i	$0.221210\pi$
$492$	−29.9605	−1.35072
$493$	−0.846116	−0.0381071
$494$	−6.95779	−0.313046
$495$	0	0
$496$	−26.2853	−1.18024
$497$	8.90481	0.399435
$498$	8.57953	0.384458
$499$	12.6605	0.566762	0.283381	−	0.959007i	$-0.408544\pi$
$499$	12.6605	0.566762	0.283381	+	0.959007i	$0.408544\pi$
$500$	0	0
$501$	28.4965	1.27313
$502$	−3.68700	−0.164559
$503$	26.1961	1.16803	0.584014	−	0.811744i	$-0.301481\pi$
$503$	26.1961	1.16803	0.584014	+	0.811744i	$0.301481\pi$
$504$	−21.2443	−0.946298
$505$	0	0
$506$	−0.958162	−0.0425955
$507$	20.7380	0.921008
$508$	32.5653	1.44485
$509$	−7.79659	−0.345578	−0.172789	−	0.984959i	$-0.555278\pi$
$509$	−7.79659	−0.345578	−0.172789	+	0.984959i	$0.555278\pi$
$510$	0	0
$511$	−15.6607	−0.692789
$512$	17.4196	0.769843
$513$	111.663	4.93005
$514$	5.27301	0.232583
$515$	0	0
$516$	8.60823	0.378956
$517$	11.5969	0.510030
$518$	−0.658515	−0.0289335
$519$	−16.0371	−0.703953
$520$	0	0
$521$	−31.3029	−1.37141	−0.685703	−	0.727881i	$-0.740505\pi$
$521$	−31.3029	−1.37141	−0.685703	+	0.727881i	$0.740505\pi$
$522$	−1.47541	−0.0645770
$523$	−10.1864	−0.445419	−0.222709	−	0.974885i	$-0.571490\pi$
$523$	−10.1864	−0.445419	−0.222709	+	0.974885i	$0.571490\pi$
$524$	31.9015	1.39362
$525$	0	0
$526$	−2.31487	−0.100933
$527$	−8.37839	−0.364969
$528$	−13.6597	−0.594461
$529$	−11.0453	−0.480228
$530$	0	0
$531$	−35.4447	−1.53817
$532$	−33.2637	−1.44216
$533$	−20.2390	−0.876648
$534$	−2.76347	−0.119587
$535$	0	0
$536$	−2.56831	−0.110934
$537$	−79.8170	−3.44436
$538$	0.720891	0.0310798
$539$	−0.128082	−0.00551688
$540$	0	0
$541$	−27.5572	−1.18478	−0.592389	−	0.805652i	$-0.701815\pi$
$541$	−27.5572	−1.18478	−0.592389	+	0.805652i	$0.701815\pi$
$542$	0.647327	0.0278051
$543$	7.48197	0.321082
$544$	3.29700	0.141358
$545$	0	0
$546$	−9.65219	−0.413075
$547$	37.9483	1.62255	0.811277	−	0.584662i	$-0.198773\pi$
$547$	37.9483	1.62255	0.811277	+	0.584662i	$0.198773\pi$
$548$	−14.5684	−0.622330
$549$	−38.7393	−1.65335
$550$	0	0
$551$	−4.69292	−0.199925
$552$	−11.2482	−0.478754
$553$	−18.0772	−0.768722
$554$	2.16124	0.0918221
$555$	0	0
$556$	−20.9774	−0.889641
$557$	−40.1642	−1.70181	−0.850906	−	0.525319i	$-0.823946\pi$
$557$	−40.1642	−1.70181	−0.850906	+	0.525319i	$0.823946\pi$
$558$	−14.6098	−0.618482
$559$	5.81506	0.245951
$560$	0	0
$561$	−4.35400	−0.183826
$562$	5.66635	0.239020
$563$	24.9958	1.05345	0.526723	−	0.850037i	$-0.323421\pi$
$563$	24.9958	1.05345	0.526723	+	0.850037i	$0.323421\pi$
$564$	67.0165	2.82190
$565$	0	0
$566$	2.12518	0.0893280
$567$	89.3704	3.75320
$568$	−3.24684	−0.136234
$569$	28.7495	1.20524	0.602620	−	0.798028i	$-0.294123\pi$
$569$	28.7495	1.20524	0.602620	+	0.798028i	$0.294123\pi$
$570$	0	0
$571$	−11.4297	−0.478318	−0.239159	−	0.970980i	$-0.576872\pi$
$571$	−11.4297	−0.478318	−0.239159	+	0.970980i	$0.576872\pi$
$572$	−9.53664	−0.398747
$573$	−54.6612	−2.28351
$574$	3.04166	0.126956
$575$	0	0
$576$	−53.8415	−2.24339
$577$	−16.0777	−0.669322	−0.334661	−	0.942339i	$-0.608622\pi$
$577$	−16.0777	−0.669322	−0.334661	+	0.942339i	$0.608622\pi$
$578$	−3.86516	−0.160769
$579$	−60.8076	−2.52708
$580$	0	0
$581$	27.7078	1.14951
$582$	2.50189	0.103707
$583$	−3.11775	−0.129124
$584$	5.71015	0.236287
$585$	0	0
$586$	7.70557	0.318314
$587$	30.9265	1.27647	0.638237	−	0.769840i	$-0.279664\pi$
$587$	30.9265	1.27647	0.638237	+	0.769840i	$0.279664\pi$
$588$	−0.740164	−0.0305239
$589$	−46.4701	−1.91477
$590$	0	0
$591$	5.72171	0.235360
$592$	−3.63798	−0.149520
$593$	26.0479	1.06966	0.534829	−	0.844960i	$-0.320376\pi$
$593$	26.0479	1.06966	0.534829	+	0.844960i	$0.320376\pi$
$594$	−4.81124	−0.197407
$595$	0	0
$596$	−28.5298	−1.16863
$597$	−19.6225	−0.803097
$598$	−3.74041	−0.152957
$599$	−13.0774	−0.534328	−0.267164	−	0.963651i	$-0.586087\pi$
$599$	−13.0774	−0.534328	−0.267164	+	0.963651i	$0.586087\pi$
$600$	0	0
$601$	33.9125	1.38332	0.691659	−	0.722224i	$-0.256880\pi$
$601$	33.9125	1.38332	0.691659	+	0.722224i	$0.256880\pi$
$602$	−0.873927	−0.0356186
$603$	21.6291	0.880806
$604$	−32.9363	−1.34016
$605$	0	0
$606$	−8.58691	−0.348819
$607$	−9.17202	−0.372281	−0.186140	−	0.982523i	$-0.559598\pi$
$607$	−9.17202	−0.372281	−0.186140	+	0.982523i	$0.559598\pi$
$608$	18.2865	0.741617
$609$	−6.51025	−0.263808
$610$	0	0
$611$	45.2712	1.83148
$612$	−18.4155	−0.744402
$613$	−6.08037	−0.245584	−0.122792	−	0.992432i	$-0.539185\pi$
$613$	−6.08037	−0.245584	−0.122792	+	0.992432i	$0.539185\pi$
$614$	5.46145	0.220406
$615$	0	0
$616$	2.91152	0.117309
$617$	45.4032	1.82786	0.913931	−	0.405869i	$-0.133031\pi$
$617$	45.4032	1.82786	0.913931	+	0.405869i	$0.133031\pi$
$618$	3.05107	0.122732
$619$	16.8229	0.676170	0.338085	−	0.941116i	$-0.390221\pi$
$619$	16.8229	0.676170	0.338085	+	0.941116i	$0.390221\pi$
$620$	0	0
$621$	60.0286	2.40887
$622$	−3.35412	−0.134488
$623$	−8.92472	−0.357561
$624$	−53.3238	−2.13466
$625$	0	0
$626$	5.90228	0.235902
$627$	−24.1491	−0.964423
$628$	−22.1447	−0.883671
$629$	−1.15960	−0.0462363
$630$	0	0
$631$	43.4329	1.72904	0.864518	−	0.502602i	$-0.167624\pi$
$631$	43.4329	1.72904	0.864518	+	0.502602i	$0.167624\pi$
$632$	6.59125	0.262186
$633$	−57.0119	−2.26602
$634$	8.00666	0.317985
$635$	0	0
$636$	−18.0169	−0.714418
$637$	−0.499998	−0.0198106
$638$	0.202204	0.00800533
$639$	27.3433	1.08168
$640$	0	0
$641$	38.3171	1.51344	0.756718	−	0.653742i	$-0.226802\pi$
$641$	38.3171	1.51344	0.756718	+	0.653742i	$0.226802\pi$
$642$	−5.57561	−0.220052
$643$	−18.2679	−0.720415	−0.360208	−	0.932872i	$-0.617294\pi$
$643$	−18.2679	−0.720415	−0.360208	+	0.932872i	$0.617294\pi$
$644$	−17.8821	−0.704653
$645$	0	0
$646$	1.84135	0.0724469
$647$	−50.7089	−1.99357	−0.996786	−	0.0801053i	$-0.974474\pi$
$647$	−50.7089	−1.99357	−0.996786	+	0.0801053i	$0.974474\pi$
$648$	−32.5859	−1.28009
$649$	4.85767	0.190680
$650$	0	0
$651$	−64.4656	−2.52661
$652$	33.2514	1.30223
$653$	−4.99994	−0.195663	−0.0978315	−	0.995203i	$-0.531191\pi$
$653$	−4.99994	−0.195663	−0.0978315	+	0.995203i	$0.531191\pi$
$654$	−5.06740	−0.198151
$655$	0	0
$656$	16.8037	0.656075
$657$	−48.0881	−1.87610
$658$	−6.80367	−0.265235
$659$	5.50306	0.214369	0.107184	−	0.994239i	$-0.465817\pi$
$659$	5.50306	0.214369	0.107184	+	0.994239i	$0.465817\pi$
$660$	0	0
$661$	−34.7100	−1.35006	−0.675032	−	0.737789i	$-0.735870\pi$
$661$	−34.7100	−1.35006	−0.675032	+	0.737789i	$0.735870\pi$
$662$	3.05542	0.118752
$663$	−16.9969	−0.660104
$664$	−10.1027	−0.392062
$665$	0	0
$666$	−2.02205	−0.0783529
$667$	−2.52285	−0.0976851
$668$	−16.5182	−0.639110
$669$	12.5066	0.483534
$670$	0	0
$671$	5.30919	0.204959
$672$	25.3680	0.978591
$673$	−17.1380	−0.660622	−0.330311	−	0.943872i	$-0.607154\pi$
$673$	−17.1380	−0.660622	−0.330311	+	0.943872i	$0.607154\pi$
$674$	−5.80062	−0.223431
$675$	0	0
$676$	−12.0210	−0.462345
$677$	38.1417	1.46590	0.732951	−	0.680281i	$-0.238142\pi$
$677$	38.1417	1.46590	0.732951	+	0.680281i	$0.238142\pi$
$678$	1.72759	0.0663477
$679$	8.07992	0.310079
$680$	0	0
$681$	61.3893	2.35244
$682$	2.00226	0.0766705
$683$	39.1257	1.49710	0.748552	−	0.663076i	$-0.230750\pi$
$683$	39.1257	1.49710	0.748552	+	0.663076i	$0.230750\pi$
$684$	−102.140	−3.90543
$685$	0	0
$686$	−4.53446	−0.173126
$687$	−52.9657	−2.02077
$688$	−4.82804	−0.184067
$689$	−12.1709	−0.463673
$690$	0	0
$691$	−18.3155	−0.696753	−0.348376	−	0.937355i	$-0.613267\pi$
$691$	−18.3155	−0.696753	−0.348376	+	0.937355i	$0.613267\pi$
$692$	9.29606	0.353383
$693$	−24.5194	−0.931417
$694$	−0.693169	−0.0263124
$695$	0	0
$696$	2.37374	0.0899764
$697$	5.35616	0.202879
$698$	−0.597693	−0.0226230
$699$	78.2409	2.95934
$700$	0	0
$701$	−16.7803	−0.633782	−0.316891	−	0.948462i	$-0.602639\pi$
$701$	−16.7803	−0.633782	−0.316891	+	0.948462i	$0.602639\pi$
$702$	−18.7818	−0.708874
$703$	−6.43164	−0.242574
$704$	7.37893	0.278104
$705$	0	0
$706$	0.334398	0.0125852
$707$	−27.7317	−1.04296
$708$	28.0717	1.05500
$709$	2.26861	0.0851995	0.0425997	−	0.999092i	$-0.486436\pi$
$709$	2.26861	0.0851995	0.0425997	+	0.999092i	$0.486436\pi$
$710$	0	0
$711$	−55.5084	−2.08173
$712$	3.25409	0.121952
$713$	−24.9817	−0.935572
$714$	2.55441	0.0955963
$715$	0	0
$716$	46.2666	1.72906
$717$	−28.8304	−1.07669
$718$	3.44036	0.128393
$719$	−53.0051	−1.97676	−0.988379	−	0.152010i	$-0.951425\pi$
$719$	−53.0051	−1.97676	−0.988379	+	0.152010i	$0.951425\pi$
$720$	0	0
$721$	9.85351	0.366964
$722$	5.52197	0.205506
$723$	−1.64516	−0.0611842
$724$	−4.33698	−0.161183
$725$	0	0
$726$	−8.04426	−0.298550
$727$	−14.0953	−0.522766	−0.261383	−	0.965235i	$-0.584179\pi$
$727$	−14.0953	−0.522766	−0.261383	+	0.965235i	$0.584179\pi$
$728$	11.3658	0.421245
$729$	100.192	3.71081
$730$	0	0
$731$	−1.53893	−0.0569194
$732$	30.6809	1.13400
$733$	−14.1555	−0.522846	−0.261423	−	0.965224i	$-0.584192\pi$
$733$	−14.1555	−0.522846	−0.261423	+	0.965224i	$0.584192\pi$
$734$	−1.91657	−0.0707420
$735$	0	0
$736$	9.83059	0.362361
$737$	−2.96426	−0.109190
$738$	9.33978	0.343802
$739$	−14.4933	−0.533146	−0.266573	−	0.963815i	$-0.585891\pi$
$739$	−14.4933	−0.533146	−0.266573	+	0.963815i	$0.585891\pi$
$740$	0	0
$741$	−94.2719	−3.46316
$742$	1.82912	0.0671492
$743$	44.5090	1.63288	0.816439	−	0.577431i	$-0.195945\pi$
$743$	44.5090	1.63288	0.816439	+	0.577431i	$0.195945\pi$
$744$	23.5052	0.861743
$745$	0	0
$746$	6.01857	0.220356
$747$	85.0803	3.11293
$748$	2.52383	0.0922804
$749$	−18.0066	−0.657947
$750$	0	0
$751$	32.6002	1.18960	0.594800	−	0.803874i	$-0.297231\pi$
$751$	32.6002	1.18960	0.594800	+	0.803874i	$0.297231\pi$
$752$	−37.5871	−1.37066
$753$	−49.9557	−1.82049
$754$	0.789351	0.0287465
$755$	0	0
$756$	−89.7917	−3.26569
$757$	29.1322	1.05883	0.529413	−	0.848364i	$-0.322412\pi$
$757$	29.1322	1.05883	0.529413	+	0.848364i	$0.322412\pi$
$758$	−3.38549	−0.122966
$759$	−12.9822	−0.471226
$760$	0	0
$761$	−20.1054	−0.728820	−0.364410	−	0.931239i	$-0.618729\pi$
$761$	−20.1054	−0.728820	−0.364410	+	0.931239i	$0.618729\pi$
$762$	−13.8704	−0.502470
$763$	−16.3653	−0.592464
$764$	31.6848	1.14632
$765$	0	0
$766$	−7.56054	−0.273173
$767$	18.9631	0.684717
$768$	37.9453	1.36923
$769$	−1.06557	−0.0384254	−0.0192127	−	0.999815i	$-0.506116\pi$
$769$	−1.06557	−0.0384254	−0.0192127	+	0.999815i	$0.506116\pi$
$770$	0	0
$771$	71.4447	2.57302
$772$	35.2476	1.26859
$773$	15.1360	0.544404	0.272202	−	0.962240i	$-0.412248\pi$
$773$	15.1360	0.544404	0.272202	+	0.962240i	$0.412248\pi$
$774$	−2.68350	−0.0964565
$775$	0	0
$776$	−2.94607	−0.105758
$777$	−8.92229	−0.320085
$778$	4.11303	0.147459
$779$	29.7075	1.06438
$780$	0	0
$781$	−3.74738	−0.134092
$782$	0.989884	0.0353982
$783$	−12.6680	−0.452718
$784$	0.415131	0.0148261
$785$	0	0
$786$	−13.5877	−0.484655
$787$	18.4833	0.658858	0.329429	−	0.944180i	$-0.393144\pi$
$787$	18.4833	0.658858	0.329429	+	0.944180i	$0.393144\pi$
$788$	−3.31663	−0.118150
$789$	−31.3644	−1.11660
$790$	0	0
$791$	5.57930	0.198377
$792$	8.94018	0.317676
$793$	20.7257	0.735990
$794$	−6.46116	−0.229298
$795$	0	0
$796$	11.3744	0.403153
$797$	−42.9927	−1.52288	−0.761441	−	0.648235i	$-0.775508\pi$
$797$	−42.9927	−1.52288	−0.761441	+	0.648235i	$0.775508\pi$
$798$	14.1678	0.501536
$799$	−11.9808	−0.423851
$800$	0	0
$801$	−27.4044	−0.968288
$802$	−3.89023	−0.137369
$803$	6.59044	0.232572
$804$	−17.1300	−0.604127
$805$	0	0
$806$	7.81630	0.275317
$807$	9.76744	0.343830
$808$	10.1114	0.355718
$809$	7.24815	0.254831	0.127416	−	0.991849i	$-0.459332\pi$
$809$	7.24815	0.254831	0.127416	+	0.991849i	$0.459332\pi$
$810$	0	0
$811$	19.2340	0.675399	0.337699	−	0.941254i	$-0.390351\pi$
$811$	19.2340	0.675399	0.337699	+	0.941254i	$0.390351\pi$
$812$	3.77372	0.132431
$813$	8.77071	0.307602
$814$	0.277121	0.00971307
$815$	0	0
$816$	14.1119	0.494016
$817$	−8.53555	−0.298621
$818$	−6.88464	−0.240716
$819$	−95.7175	−3.34464
$820$	0	0
$821$	−14.1258	−0.492994	−0.246497	−	0.969144i	$-0.579280\pi$
$821$	−14.1258	−0.492994	−0.246497	+	0.969144i	$0.579280\pi$
$822$	6.20504	0.216426
$823$	−47.4962	−1.65561	−0.827807	−	0.561014i	$-0.810412\pi$
$823$	−47.4962	−1.65561	−0.827807	+	0.561014i	$0.810412\pi$
$824$	−3.59275	−0.125159
$825$	0	0
$826$	−2.84990	−0.0991609
$827$	−30.5117	−1.06100	−0.530498	−	0.847686i	$-0.677995\pi$
$827$	−30.5117	−1.06100	−0.530498	+	0.847686i	$0.677995\pi$
$828$	−54.9092	−1.90823
$829$	1.66809	0.0579353	0.0289676	−	0.999580i	$-0.490778\pi$
$829$	1.66809	0.0579353	0.0289676	+	0.999580i	$0.490778\pi$
$830$	0	0
$831$	29.2828	1.01581
$832$	28.8054	0.998648
$833$	0.132322	0.00458469
$834$	8.93482	0.309388
$835$	0	0
$836$	13.9982	0.484139
$837$	−125.441	−4.33588
$838$	4.30596	0.148747
$839$	−24.7993	−0.856166	−0.428083	−	0.903739i	$-0.640811\pi$
$839$	−24.7993	−0.856166	−0.428083	+	0.903739i	$0.640811\pi$
$840$	0	0
$841$	−28.4676	−0.981641
$842$	4.26431	0.146958
$843$	76.7740	2.64424
$844$	33.0474	1.13754
$845$	0	0
$846$	−20.8915	−0.718265
$847$	−25.9791	−0.892654
$848$	10.1050	0.347009
$849$	28.7943	0.988219
$850$	0	0
$851$	−3.45756	−0.118524
$852$	−21.6555	−0.741905
$853$	18.7133	0.640732	0.320366	−	0.947294i	$-0.396194\pi$
$853$	18.7133	0.640732	0.320366	+	0.947294i	$0.396194\pi$
$854$	−3.11480	−0.106586
$855$	0	0
$856$	6.56549	0.224404
$857$	−3.17695	−0.108522	−0.0542612	−	0.998527i	$-0.517280\pi$
$857$	−3.17695	−0.108522	−0.0542612	+	0.998527i	$0.517280\pi$
$858$	4.06190	0.138671
$859$	−12.7152	−0.433838	−0.216919	−	0.976190i	$-0.569601\pi$
$859$	−12.7152	−0.433838	−0.216919	+	0.976190i	$0.569601\pi$
$860$	0	0
$861$	41.2118	1.40449
$862$	3.76026	0.128075
$863$	6.35507	0.216329	0.108165	−	0.994133i	$-0.465503\pi$
$863$	6.35507	0.216329	0.108165	+	0.994133i	$0.465503\pi$
$864$	49.3626	1.67935
$865$	0	0
$866$	1.13501	0.0385692
$867$	−52.3695	−1.77856
$868$	37.3680	1.26835
$869$	7.60738	0.258063
$870$	0	0
$871$	−11.5717	−0.392091
$872$	5.96705	0.202070
$873$	24.8104	0.839704
$874$	5.49031	0.185713
$875$	0	0
$876$	38.0851	1.28678
$877$	−27.3415	−0.923255	−0.461628	−	0.887074i	$-0.652734\pi$
$877$	−27.3415	−0.923255	−0.461628	+	0.887074i	$0.652734\pi$
$878$	3.36732	0.113642
$879$	104.404	3.52145
$880$	0	0
$881$	35.3107	1.18965	0.594823	−	0.803856i	$-0.297222\pi$
$881$	35.3107	1.18965	0.594823	+	0.803856i	$0.297222\pi$
$882$	0.230736	0.00776930
$883$	−44.8243	−1.50846	−0.754229	−	0.656612i	$-0.771989\pi$
$883$	−44.8243	−1.50846	−0.754229	+	0.656612i	$0.771989\pi$
$884$	9.85237	0.331371
$885$	0	0
$886$	4.67810	0.157164
$887$	24.8600	0.834716	0.417358	−	0.908742i	$-0.362956\pi$
$887$	24.8600	0.834716	0.417358	+	0.908742i	$0.362956\pi$
$888$	3.25321	0.109171
$889$	−44.7947	−1.50237
$890$	0	0
$891$	−37.6094	−1.25996
$892$	−7.24957	−0.242733
$893$	−66.4507	−2.22369
$894$	12.1516	0.406410
$895$	0	0
$896$	−19.4961	−0.651319
$897$	−50.6793	−1.69213
$898$	0.0493672	0.00164740
$899$	5.27197	0.175830
$900$	0	0
$901$	3.22096	0.107306
$902$	−1.28001	−0.0426197
$903$	−11.8409	−0.394042
$904$	−2.03430	−0.0676599
$905$	0	0
$906$	14.0284	0.466063
$907$	9.40054	0.312140	0.156070	−	0.987746i	$-0.450117\pi$
$907$	9.40054	0.312140	0.156070	+	0.987746i	$0.450117\pi$
$908$	−35.5848	−1.18092
$909$	−85.1535	−2.82436
$910$	0	0
$911$	−49.3826	−1.63612	−0.818059	−	0.575134i	$-0.804950\pi$
$911$	−49.3826	−1.63612	−0.818059	+	0.575134i	$0.804950\pi$
$912$	78.2706	2.59180
$913$	−11.6602	−0.385896
$914$	−7.17703	−0.237395
$915$	0	0
$916$	30.7020	1.01442
$917$	−43.8817	−1.44910
$918$	4.97052	0.164052
$919$	51.3494	1.69386	0.846931	−	0.531703i	$-0.178448\pi$
$919$	51.3494	1.69386	0.846931	+	0.531703i	$0.178448\pi$
$920$	0	0
$921$	73.9978	2.43831
$922$	5.49132	0.180847
$923$	−14.6288	−0.481513
$924$	19.4190	0.638839
$925$	0	0
$926$	−2.73919	−0.0900155
$927$	30.2564	0.993751
$928$	−2.07458	−0.0681015
$929$	19.8561	0.651458	0.325729	−	0.945463i	$-0.394390\pi$
$929$	19.8561	0.651458	0.325729	+	0.945463i	$0.394390\pi$
$930$	0	0
$931$	0.733915	0.0240531
$932$	−45.3529	−1.48559
$933$	−45.4453	−1.48781
$934$	8.00956	0.262081
$935$	0	0
$936$	34.9001	1.14075
$937$	−34.6189	−1.13095	−0.565475	−	0.824765i	$-0.691307\pi$
$937$	−34.6189	−1.13095	−0.565475	+	0.824765i	$0.691307\pi$
$938$	1.73907	0.0567827
$939$	79.9706	2.60974
$940$	0	0
$941$	−13.2367	−0.431506	−0.215753	−	0.976448i	$-0.569221\pi$
$941$	−13.2367	−0.431506	−0.215753	+	0.976448i	$0.569221\pi$
$942$	9.43201	0.307311
$943$	15.9704	0.520067
$944$	−15.7444	−0.512436
$945$	0	0
$946$	0.367772	0.0119573
$947$	23.4210	0.761080	0.380540	−	0.924765i	$-0.375738\pi$
$947$	23.4210	0.761080	0.380540	+	0.924765i	$0.375738\pi$
$948$	43.9618	1.42781
$949$	25.7274	0.835145
$950$	0	0
$951$	108.483	3.51781
$952$	−3.00791	−0.0974870
$953$	−35.3381	−1.14471	−0.572357	−	0.820005i	$-0.693971\pi$
$953$	−35.3381	−1.14471	−0.572357	+	0.820005i	$0.693971\pi$
$954$	5.61654	0.181842
$955$	0	0
$956$	16.7118	0.540497
$957$	2.73968	0.0885614
$958$	9.40604	0.303895
$959$	20.0394	0.647105
$960$	0	0
$961$	21.2040	0.684000
$962$	1.08181	0.0348788
$963$	−55.2915	−1.78174
$964$	0.953631	0.0307144
$965$	0	0
$966$	7.61644	0.245055
$967$	41.7549	1.34275	0.671374	−	0.741119i	$-0.265705\pi$
$967$	41.7549	1.34275	0.671374	+	0.741119i	$0.265705\pi$
$968$	9.47241	0.304455
$969$	24.9486	0.801466
$970$	0	0
$971$	−40.4393	−1.29776	−0.648879	−	0.760892i	$-0.724762\pi$
$971$	−40.4393	−1.29776	−0.648879	+	0.760892i	$0.724762\pi$
$972$	−116.345	−3.73175
$973$	28.8552	0.925057
$974$	3.67692	0.117816
$975$	0	0
$976$	−17.2078	−0.550808
$977$	−14.1984	−0.454249	−0.227124	−	0.973866i	$-0.572932\pi$
$977$	−14.1984	−0.454249	−0.227124	+	0.973866i	$0.572932\pi$
$978$	−14.1626	−0.452871
$979$	3.75576	0.120035
$980$	0	0
$981$	−50.2517	−1.60441
$982$	8.40398	0.268182
$983$	23.4641	0.748390	0.374195	−	0.927350i	$-0.377919\pi$
$983$	23.4641	0.748390	0.374195	+	0.927350i	$0.377919\pi$
$984$	−15.0265	−0.479026
$985$	0	0
$986$	−0.208898	−0.00665268
$987$	−92.1837	−2.93424
$988$	54.6454	1.73850
$989$	−4.58860	−0.145909
$990$	0	0
$991$	31.8649	1.01222	0.506111	−	0.862469i	$-0.331083\pi$
$991$	31.8649	1.01222	0.506111	+	0.862469i	$0.331083\pi$
$992$	−20.5429	−0.652237
$993$	41.3983	1.31373
$994$	2.19852	0.0697327
$995$	0	0
$996$	−67.3823	−2.13509
$997$	30.6540	0.970821	0.485411	−	0.874286i	$-0.338670\pi$
$997$	30.6540	0.970821	0.485411	+	0.874286i	$0.338670\pi$
$998$	3.12576	0.0989442
$999$	−17.3615	−0.549295

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	925.2.a.m.1.4		9
3.2	odd	2		8325.2.a.cq.1.6		9
5.2	odd	4		185.2.b.a.149.10	yes	18
5.3	odd	4		185.2.b.a.149.9	✓	18
5.4	even	2		925.2.a.l.1.6		9
15.2	even	4		1665.2.c.e.334.9		18
15.8	even	4		1665.2.c.e.334.10		18
15.14	odd	2		8325.2.a.cr.1.4		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
185.2.b.a.149.9	✓	18	5.3	odd	4
185.2.b.a.149.10	yes	18	5.2	odd	4
925.2.a.l.1.6		9	5.4	even	2
925.2.a.m.1.4		9	1.1	even	1	trivial
1665.2.c.e.334.9		18	15.2	even	4
1665.2.c.e.334.10		18	15.8	even	4
8325.2.a.cq.1.6		9	3.2	odd	2
8325.2.a.cr.1.4		9	15.14	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 925 = 5^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	925.a (trivial)

\(f(q)\)	\(=\)	\(q+0.246891 q^{2} +3.34515 q^{3} -1.93904 q^{4} +0.825888 q^{6} +2.66723 q^{7} -0.972515 q^{8} +8.19006 q^{9} -1.12244 q^{11} -6.48640 q^{12} -4.38171 q^{13} +0.658515 q^{14} +3.63798 q^{16} +1.15960 q^{17} +2.02205 q^{18} +6.43164 q^{19} +8.92229 q^{21} -0.277121 q^{22} +3.45756 q^{23} -3.25321 q^{24} -1.08181 q^{26} +17.3615 q^{27} -5.17188 q^{28} -0.729661 q^{29} -7.22523 q^{31} +2.84321 q^{32} -3.75474 q^{33} +0.286295 q^{34} -15.8809 q^{36} -1.00000 q^{37} +1.58791 q^{38} -14.6575 q^{39} +4.61897 q^{41} +2.20283 q^{42} -1.32712 q^{43} +2.17646 q^{44} +0.853641 q^{46} -10.3318 q^{47} +12.1696 q^{48} +0.114110 q^{49} +3.87905 q^{51} +8.49634 q^{52} +2.77765 q^{53} +4.28641 q^{54} -2.59392 q^{56} +21.5148 q^{57} -0.180147 q^{58} -4.32778 q^{59} -4.73004 q^{61} -1.78384 q^{62} +21.8448 q^{63} -6.57401 q^{64} -0.927011 q^{66} +2.64090 q^{67} -2.24852 q^{68} +11.5661 q^{69} +3.33860 q^{71} -7.96495 q^{72} -5.87153 q^{73} -0.246891 q^{74} -12.4712 q^{76} -2.99381 q^{77} -3.61881 q^{78} -6.77753 q^{79} +33.5068 q^{81} +1.14038 q^{82} +10.3882 q^{83} -17.3007 q^{84} -0.327654 q^{86} -2.44083 q^{87} +1.09159 q^{88} -3.34606 q^{89} -11.6870 q^{91} -6.70437 q^{92} -24.1695 q^{93} -2.55084 q^{94} +9.51099 q^{96} +3.02933 q^{97} +0.0281728 q^{98} -9.19285 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 5 q^{2} + 8 q^{3} + 11 q^{4} + 2 q^{6} + 8 q^{7} + 15 q^{8} + 13 q^{9} + 16 q^{12} + 6 q^{13} - 4 q^{14} + 11 q^{16} + 18 q^{17} - 3 q^{18} - 4 q^{19} + 4 q^{21} + 6 q^{22} + 16 q^{23} + 6 q^{24}+ \cdots + 8 q^{99}+O(q^{100}) \) 9 * q + 5 * q^2 + 8 * q^3 + 11 * q^4 + 2 * q^6 + 8 * q^7 + 15 * q^8 + 13 * q^9 + 16 * q^12 + 6 * q^13 - 4 * q^14 + 11 * q^16 + 18 * q^17 - 3 * q^18 - 4 * q^19 + 4 * q^21 + 6 * q^22 + 16 * q^23 + 6 * q^24 - 6 * q^26 + 32 * q^27 - 20 * q^28 - 2 * q^29 - 6 * q^31 + 35 * q^32 + 6 * q^33 + 6 * q^34 + 11 * q^36 - 9 * q^37 + 4 * q^38 - 16 * q^39 + 2 * q^41 + 4 * q^42 + 14 * q^43 + 4 * q^44 + 16 * q^46 + 34 * q^47 + 6 * q^48 - 7 * q^49 + 8 * q^51 + 16 * q^52 - 4 * q^53 - 18 * q^54 - 24 * q^56 - 4 * q^57 - 22 * q^58 + 10 * q^59 + 8 * q^61 + 26 * q^62 + 18 * q^63 + 15 * q^64 - 34 * q^66 + 16 * q^67 + 36 * q^68 - 10 * q^69 + 8 * q^71 + 9 * q^72 + 8 * q^73 - 5 * q^74 - 48 * q^76 + 14 * q^77 - 76 * q^78 - 26 * q^79 + 13 * q^81 - 4 * q^82 + 70 * q^83 - 24 * q^84 - 16 * q^86 - 12 * q^87 - 36 * q^88 - 8 * q^89 - 18 * q^91 + 16 * q^92 + 2 * q^93 - 26 * q^94 + 52 * q^96 - 2 * q^97 + 21 * q^98 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

Properties

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