LMFDB - Embedded newform 9025.2.a.ca.1.2

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:	$6$
Coefficient field:	6.6.5822000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 8x^{4} + 5x^{3} + 14x^{2} + x - 1 $ x^6 - x^5 - 8x^4 + 5x^3 + 14*x^2 + x - 1
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			1.2
Root			$2.47848$ of defining polynomial
Character	$\chi$	$=$	9025.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.860448 q^{2} +0.867394 q^{3} -1.25963 q^{4} -0.746348 q^{6} +0.263921 q^{7} +2.80474 q^{8} -2.24763 q^{9} +O(q^{10})$	\(q-0.860448 q^{2} +0.867394 q^{3} -1.25963 q^{4} -0.746348 q^{6} +0.263921 q^{7} +2.80474 q^{8} -2.24763 q^{9} -5.32135 q^{11} -1.09259 q^{12} +1.19606 q^{13} -0.227090 q^{14} +0.105921 q^{16} -4.46010 q^{17} +1.93397 q^{18} +0.228924 q^{21} +4.57874 q^{22} +0.920897 q^{23} +2.43282 q^{24} -1.02914 q^{26} -4.55176 q^{27} -0.332443 q^{28} +9.87581 q^{29} +0.0400121 q^{31} -5.70062 q^{32} -4.61570 q^{33} +3.83769 q^{34} +2.83118 q^{36} +7.76091 q^{37} +1.03745 q^{39} -6.27419 q^{41} -0.196977 q^{42} -10.0853 q^{43} +6.70292 q^{44} -0.792384 q^{46} -8.44271 q^{47} +0.0918755 q^{48} -6.93035 q^{49} -3.86866 q^{51} -1.50659 q^{52} +4.20047 q^{53} +3.91655 q^{54} +0.740231 q^{56} -8.49762 q^{58} +8.10775 q^{59} +0.218505 q^{61} -0.0344283 q^{62} -0.593196 q^{63} +4.69325 q^{64} +3.97157 q^{66} +3.76600 q^{67} +5.61807 q^{68} +0.798780 q^{69} -10.1077 q^{71} -6.30402 q^{72} -9.91093 q^{73} -6.67786 q^{74} -1.40442 q^{77} -0.892673 q^{78} -0.00558375 q^{79} +2.79472 q^{81} +5.39862 q^{82} -8.82190 q^{83} -0.288359 q^{84} +8.67786 q^{86} +8.56621 q^{87} -14.9250 q^{88} +0.431648 q^{89} +0.315664 q^{91} -1.15999 q^{92} +0.0347063 q^{93} +7.26452 q^{94} -4.94469 q^{96} -17.3729 q^{97} +5.96320 q^{98} +11.9604 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{2} + 4 q^{3} + 8 q^{4} + 8 q^{6} - 7 q^{7} + 10 q^{9}+O(q^{10}) $ 6 * q + 2 * q^2 + 4 * q^3 + 8 * q^4 + 8 * q^6 - 7 * q^7 + 10 * q^9	$ 6 q + 2 q^{2} + 4 q^{3} + 8 q^{4} + 8 q^{6} - 7 q^{7} + 10 q^{9} + 15 q^{11} - 20 q^{12} + 6 q^{13} - 20 q^{14} + 20 q^{16} - 5 q^{17} - 8 q^{18} - 7 q^{21} + 14 q^{22} + 18 q^{24} - 22 q^{26} + 28 q^{27} - 10 q^{28} + 20 q^{29} - 12 q^{31} - 8 q^{32} + q^{33} - 14 q^{36} + 20 q^{37} + 16 q^{39} - 8 q^{41} + 30 q^{42} - 27 q^{43} + 46 q^{44} - 16 q^{46} + 8 q^{47} - 24 q^{48} + 11 q^{49} + 11 q^{51} + 4 q^{52} + 19 q^{53} + 30 q^{54} - 62 q^{56} - 20 q^{58} + 41 q^{59} + 6 q^{61} + 30 q^{62} + 18 q^{63} + 48 q^{64} + 46 q^{66} - 15 q^{67} + 14 q^{68} + 10 q^{69} - 2 q^{71} - 68 q^{72} - 8 q^{73} + 2 q^{74} - 21 q^{77} + 46 q^{78} - 18 q^{79} + 50 q^{81} + 18 q^{82} + 10 q^{83} - 4 q^{84} + 10 q^{86} + 14 q^{87} - 52 q^{88} - 17 q^{89} + 23 q^{91} - 28 q^{92} - 10 q^{93} + 28 q^{94} + 26 q^{96} - 4 q^{97} + 50 q^{98} + 58 q^{99}+O(q^{100}) $ 6 * q + 2 * q^2 + 4 * q^3 + 8 * q^4 + 8 * q^6 - 7 * q^7 + 10 * q^9 + 15 * q^11 - 20 * q^12 + 6 * q^13 - 20 * q^14 + 20 * q^16 - 5 * q^17 - 8 * q^18 - 7 * q^21 + 14 * q^22 + 18 * q^24 - 22 * q^26 + 28 * q^27 - 10 * q^28 + 20 * q^29 - 12 * q^31 - 8 * q^32 + q^33 - 14 * q^36 + 20 * q^37 + 16 * q^39 - 8 * q^41 + 30 * q^42 - 27 * q^43 + 46 * q^44 - 16 * q^46 + 8 * q^47 - 24 * q^48 + 11 * q^49 + 11 * q^51 + 4 * q^52 + 19 * q^53 + 30 * q^54 - 62 * q^56 - 20 * q^58 + 41 * q^59 + 6 * q^61 + 30 * q^62 + 18 * q^63 + 48 * q^64 + 46 * q^66 - 15 * q^67 + 14 * q^68 + 10 * q^69 - 2 * q^71 - 68 * q^72 - 8 * q^73 + 2 * q^74 - 21 * q^77 + 46 * q^78 - 18 * q^79 + 50 * q^81 + 18 * q^82 + 10 * q^83 - 4 * q^84 + 10 * q^86 + 14 * q^87 - 52 * q^88 - 17 * q^89 + 23 * q^91 - 28 * q^92 - 10 * q^93 + 28 * q^94 + 26 * q^96 - 4 * q^97 + 50 * q^98 + 58 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−0.860448	−0.608429	−0.304214	−	0.952604i	$-0.598394\pi$
$2$	−0.860448	−0.608429	−0.304214	+	0.952604i	$0.598394\pi$
$3$	0.867394	0.500790	0.250395	−	0.968144i	$-0.419440\pi$
$3$	0.867394	0.500790	0.250395	+	0.968144i	$0.419440\pi$
$4$	−1.25963	−0.629814
$5$	0	0
$5$	0	0
$6$	−0.746348	−0.304695
$7$	0.263921	0.0997528	0.0498764	−	0.998755i	$-0.484117\pi$
$7$	0.263921	0.0997528	0.0498764	+	0.998755i	$0.484117\pi$
$8$	2.80474	0.991626
$9$	−2.24763	−0.749209
$10$	0	0
$11$	−5.32135	−1.60445	−0.802223	−	0.597024i	$-0.796350\pi$
$11$	−5.32135	−1.60445	−0.802223	+	0.597024i	$0.796350\pi$
$12$	−1.09259	−0.315405
$13$	1.19606	0.331726	0.165863	−	0.986149i	$-0.446959\pi$
$13$	1.19606	0.331726	0.165863	+	0.986149i	$0.446959\pi$
$14$	−0.227090	−0.0606925
$15$	0	0
$16$	0.105921	0.0264803
$17$	−4.46010	−1.08173	−0.540867	−	0.841108i	$-0.681904\pi$
$17$	−4.46010	−1.08173	−0.540867	+	0.841108i	$0.681904\pi$
$18$	1.93397	0.455841
$19$	0	0
$19$	0	0
$20$	0	0
$21$	0.228924	0.0499552
$22$	4.57874	0.976191
$23$	0.920897	0.192020	0.0960101	−	0.995380i	$-0.469392\pi$
$23$	0.920897	0.192020	0.0960101	+	0.995380i	$0.469392\pi$
$24$	2.43282	0.496596
$25$	0	0
$26$	−1.02914	−0.201832
$27$	−4.55176	−0.875987
$28$	−0.332443	−0.0628257
$29$	9.87581	1.83389	0.916946	−	0.399012i	$-0.130647\pi$
$29$	9.87581	1.83389	0.916946	+	0.399012i	$0.130647\pi$
$30$	0	0
$31$	0.0400121	0.00718639	0.00359319	−	0.999994i	$-0.498856\pi$
$31$	0.0400121	0.00718639	0.00359319	+	0.999994i	$0.498856\pi$
$32$	−5.70062	−1.00774
$33$	−4.61570	−0.803491
$34$	3.83769	0.658158
$35$	0	0
$36$	2.83118	0.471863
$37$	7.76091	1.27589	0.637943	−	0.770084i	$-0.279786\pi$
$37$	7.76091	1.27589	0.637943	+	0.770084i	$0.279786\pi$
$38$	0	0
$39$	1.03745	0.166125
$40$	0	0
$41$	−6.27419	−0.979864	−0.489932	−	0.871761i	$-0.662978\pi$
$41$	−6.27419	−0.979864	−0.489932	+	0.871761i	$0.662978\pi$
$42$	−0.196977	−0.0303942
$43$	−10.0853	−1.53799	−0.768995	−	0.639254i	$-0.779243\pi$
$43$	−10.0853	−1.53799	−0.768995	+	0.639254i	$0.779243\pi$
$44$	6.70292	1.01050
$45$	0	0
$46$	−0.792384	−0.116831
$47$	−8.44271	−1.23150	−0.615748	−	0.787943i	$-0.711146\pi$
$47$	−8.44271	−1.23150	−0.615748	+	0.787943i	$0.711146\pi$
$48$	0.0918755	0.0132611
$49$	−6.93035	−0.990049
$50$	0	0
$51$	−3.86866	−0.541721
$52$	−1.50659	−0.208926
$53$	4.20047	0.576979	0.288490	−	0.957483i	$-0.406847\pi$
$53$	4.20047	0.576979	0.288490	+	0.957483i	$0.406847\pi$
$54$	3.91655	0.532976
$55$	0	0
$56$	0.740231	0.0989175
$57$	0	0
$58$	−8.49762	−1.11579
$59$	8.10775	1.05554	0.527770	−	0.849387i	$-0.323028\pi$
$59$	8.10775	1.05554	0.527770	+	0.849387i	$0.323028\pi$
$60$	0	0
$61$	0.218505	0.0279768	0.0139884	−	0.999902i	$-0.495547\pi$
$61$	0.218505	0.0279768	0.0139884	+	0.999902i	$0.495547\pi$
$62$	−0.0344283	−0.00437240
$63$	−0.593196	−0.0747357
$64$	4.69325	0.586656
$65$	0	0
$66$	3.97157	0.488867
$67$	3.76600	0.460090	0.230045	−	0.973180i	$-0.426113\pi$
$67$	3.76600	0.460090	0.230045	+	0.973180i	$0.426113\pi$
$68$	5.61807	0.681291
$69$	0.798780	0.0961618
$70$	0	0
$71$	−10.1077	−1.19956	−0.599780	−	0.800165i	$-0.704745\pi$
$71$	−10.1077	−1.19956	−0.599780	+	0.800165i	$0.704745\pi$
$72$	−6.30402	−0.742936
$73$	−9.91093	−1.15999	−0.579993	−	0.814621i	$-0.696945\pi$
$73$	−9.91093	−1.15999	−0.579993	+	0.814621i	$0.696945\pi$
$74$	−6.67786	−0.776286
$75$	0	0
$76$	0	0
$77$	−1.40442	−0.160048
$78$	−0.892673	−0.101075
$79$	−0.00558375	−0.000628221 0	−0.000314111	−	1.00000i	$-0.500100\pi$
$79$	−0.00558375	−0.000628221 0	−0.000314111		1.00000i	$0.500100\pi$
$80$	0	0
$81$	2.79472	0.310524
$82$	5.39862	0.596177
$83$	−8.82190	−0.968329	−0.484165	−	0.874977i	$-0.660876\pi$
$83$	−8.82190	−0.968329	−0.484165	+	0.874977i	$0.660876\pi$
$84$	−0.288359	−0.0314625
$85$	0	0
$86$	8.67786	0.935758
$87$	8.56621	0.918395
$88$	−14.9250	−1.59101
$89$	0.431648	0.0457546	0.0228773	−	0.999738i	$-0.492717\pi$
$89$	0.431648	0.0457546	0.0228773	+	0.999738i	$0.492717\pi$
$90$	0	0
$91$	0.315664	0.0330906
$92$	−1.15999	−0.120937
$93$	0.0347063	0.00359887
$94$	7.26452	0.749278
$95$	0	0
$96$	−4.94469	−0.504665
$97$	−17.3729	−1.76395	−0.881973	−	0.471299i	$-0.843785\pi$
$97$	−17.3729	−1.76395	−0.881973	+	0.471299i	$0.843785\pi$
$98$	5.96320	0.602375
$99$	11.9604	1.20207
$100$	0	0
$101$	−8.95142	−0.890699	−0.445350	−	0.895357i	$-0.646921\pi$
$101$	−8.95142	−0.890699	−0.445350	+	0.895357i	$0.646921\pi$
$102$	3.32879	0.329599
$103$	5.66535	0.558224	0.279112	−	0.960259i	$-0.409960\pi$
$103$	5.66535	0.558224	0.279112	+	0.960259i	$0.409960\pi$
$104$	3.35463	0.328948
$105$	0	0
$106$	−3.61429	−0.351051
$107$	8.52693	0.824329	0.412165	−	0.911109i	$-0.364773\pi$
$107$	8.52693	0.824329	0.412165	+	0.911109i	$0.364773\pi$
$108$	5.73353	0.551709
$109$	−7.01751	−0.672156	−0.336078	−	0.941834i	$-0.609101\pi$
$109$	−7.01751	−0.672156	−0.336078	+	0.941834i	$0.609101\pi$
$110$	0	0
$111$	6.73176	0.638951
$112$	0.0279549	0.00264149
$113$	8.74027	0.822215	0.411108	−	0.911587i	$-0.365142\pi$
$113$	8.74027	0.822215	0.411108	+	0.911587i	$0.365142\pi$
$114$	0	0
$115$	0	0
$116$	−12.4398	−1.15501
$117$	−2.68829	−0.248532
$118$	−6.97630	−0.642221
$119$	−1.17711	−0.107906
$120$	0	0
$121$	17.3167	1.57425
$122$	−0.188013	−0.0170219
$123$	−5.44219	−0.490706
$124$	−0.0504004	−0.00452609
$125$	0	0
$126$	0.510415	0.0454714
$127$	6.80930	0.604228	0.302114	−	0.953272i	$-0.402308\pi$
$127$	6.80930	0.604228	0.302114	+	0.953272i	$0.402308\pi$
$128$	7.36295	0.650799
$129$	−8.74791	−0.770210
$130$	0	0
$131$	11.4004	0.996055	0.498028	−	0.867161i	$-0.334058\pi$
$131$	11.4004	0.996055	0.498028	+	0.867161i	$0.334058\pi$
$132$	5.81407	0.506050
$133$	0	0
$134$	−3.24045	−0.279932
$135$	0	0
$136$	−12.5094	−1.07267
$137$	17.1356	1.46400	0.731998	−	0.681307i	$-0.238588\pi$
$137$	17.1356	1.46400	0.731998	+	0.681307i	$0.238588\pi$
$138$	−0.687309	−0.0585076
$139$	13.4272	1.13888	0.569439	−	0.822034i	$-0.307160\pi$
$139$	13.4272	1.13888	0.569439	+	0.822034i	$0.307160\pi$
$140$	0	0
$141$	−7.32316	−0.616721
$142$	8.69713	0.729847
$143$	−6.36463	−0.532237
$144$	−0.238072	−0.0198393
$145$	0	0
$146$	8.52784	0.705769
$147$	−6.01134	−0.495807
$148$	−9.77586	−0.803571
$149$	19.3771	1.58743	0.793716	−	0.608289i	$-0.208144\pi$
$149$	19.3771	1.58743	0.793716	+	0.608289i	$0.208144\pi$
$150$	0	0
$151$	15.2903	1.24431	0.622153	−	0.782896i	$-0.286258\pi$
$151$	15.2903	1.24431	0.622153	+	0.782896i	$0.286258\pi$
$152$	0	0
$153$	10.0246	0.810445
$154$	1.20843	0.0973778
$155$	0	0
$156$	−1.30680	−0.104628
$157$	−16.7845	−1.33955	−0.669773	−	0.742566i	$-0.733608\pi$
$157$	−16.7845	−1.33955	−0.669773	+	0.742566i	$0.733608\pi$
$158$	0.00480453	0.000382228 0
$159$	3.64346	0.288945
$160$	0	0
$161$	0.243044	0.0191546
$162$	−2.40471	−0.188932
$163$	−20.1558	−1.57872	−0.789361	−	0.613929i	$-0.789588\pi$
$163$	−20.1558	−1.57872	−0.789361	+	0.613929i	$0.789588\pi$
$164$	7.90315	0.617132
$165$	0	0
$166$	7.59079	0.589159
$167$	−23.6089	−1.82691	−0.913454	−	0.406942i	$-0.866595\pi$
$167$	−23.6089	−1.82691	−0.913454	+	0.406942i	$0.866595\pi$
$168$	0.642071	0.0495369
$169$	−11.5695	−0.889958
$170$	0	0
$171$	0	0
$172$	12.7037	0.968648
$173$	7.10025	0.539822	0.269911	−	0.962885i	$-0.413006\pi$
$173$	7.10025	0.539822	0.269911	+	0.962885i	$0.413006\pi$
$174$	−7.37079	−0.558778
$175$	0	0
$176$	−0.563644	−0.0424863
$177$	7.03262	0.528604
$178$	−0.371411	−0.0278384
$179$	3.65176	0.272946	0.136473	−	0.990644i	$-0.456423\pi$
$179$	3.65176	0.272946	0.136473	+	0.990644i	$0.456423\pi$
$180$	0	0
$181$	−8.74210	−0.649796	−0.324898	−	0.945749i	$-0.605330\pi$
$181$	−8.74210	−0.649796	−0.324898	+	0.945749i	$0.605330\pi$
$182$	−0.271613	−0.0201333
$183$	0.189530	0.0140105
$184$	2.58288	0.190412
$185$	0	0
$186$	−0.0298629	−0.00218966
$187$	23.7337	1.73558
$188$	10.6347	0.775614
$189$	−1.20131	−0.0873821
$190$	0	0
$191$	3.41340	0.246985	0.123492	−	0.992346i	$-0.460591\pi$
$191$	3.41340	0.246985	0.123492	+	0.992346i	$0.460591\pi$
$192$	4.07090	0.293792
$193$	−13.7389	−0.988946	−0.494473	−	0.869193i	$-0.664639\pi$
$193$	−13.7389	−0.988946	−0.494473	+	0.869193i	$0.664639\pi$
$194$	14.9485	1.07324
$195$	0	0
$196$	8.72966	0.623547
$197$	12.4598	0.887722	0.443861	−	0.896096i	$-0.353608\pi$
$197$	12.4598	0.887722	0.443861	+	0.896096i	$0.353608\pi$
$198$	−10.2913	−0.731372
$199$	27.1939	1.92772	0.963861	−	0.266407i	$-0.0858364\pi$
$199$	27.1939	1.92772	0.963861	+	0.266407i	$0.0858364\pi$
$200$	0	0
$201$	3.26660	0.230409
$202$	7.70223	0.541927
$203$	2.60643	0.182936
$204$	4.87308	0.341184
$205$	0	0
$206$	−4.87474	−0.339640
$207$	−2.06983	−0.143863
$208$	0.126688	0.00878422
$209$	0	0
$210$	0	0
$211$	15.6319	1.07614	0.538071	−	0.842899i	$-0.319153\pi$
$211$	15.6319	1.07614	0.538071	+	0.842899i	$0.319153\pi$
$212$	−5.29103	−0.363390
$213$	−8.76733	−0.600728
$214$	−7.33698	−0.501546
$215$	0	0
$216$	−12.7665	−0.868651
$217$	0.0105600	0.000716862 0
$218$	6.03821	0.408959
$219$	−8.59668	−0.580910
$220$	0	0
$221$	−5.33453	−0.358839
$222$	−5.79234	−0.388756
$223$	19.1486	1.28229	0.641143	−	0.767421i	$-0.278460\pi$
$223$	19.1486	1.28229	0.641143	+	0.767421i	$0.278460\pi$
$224$	−1.50451	−0.100525
$225$	0	0
$226$	−7.52055	−0.500259
$227$	27.9989	1.85835	0.929175	−	0.369641i	$-0.120519\pi$
$227$	27.9989	1.85835	0.929175	+	0.369641i	$0.120519\pi$
$228$	0	0
$229$	7.43209	0.491126	0.245563	−	0.969381i	$-0.421027\pi$
$229$	7.43209	0.491126	0.245563	+	0.969381i	$0.421027\pi$
$230$	0	0
$231$	−1.21818	−0.0801505
$232$	27.6991	1.81853
$233$	14.2940	0.936431	0.468215	−	0.883614i	$-0.344897\pi$
$233$	14.2940	0.936431	0.468215	+	0.883614i	$0.344897\pi$
$234$	2.31313	0.151214
$235$	0	0
$236$	−10.2128	−0.664794
$237$	−0.00484331	−0.000314607 0
$238$	1.01285	0.0656531
$239$	1.04233	0.0674225	0.0337113	−	0.999432i	$-0.489267\pi$
$239$	1.04233	0.0674225	0.0337113	+	0.999432i	$0.489267\pi$
$240$	0	0
$241$	−18.2225	−1.17381	−0.586907	−	0.809655i	$-0.699654\pi$
$241$	−18.2225	−1.17381	−0.586907	+	0.809655i	$0.699654\pi$
$242$	−14.9001	−0.957818
$243$	16.0794	1.03149
$244$	−0.275236	−0.0176202
$245$	0	0
$246$	4.68273	0.298560
$247$	0	0
$248$	0.112224	0.00712621
$249$	−7.65206	−0.484930
$250$	0	0
$251$	2.92399	0.184561	0.0922804	−	0.995733i	$-0.470584\pi$
$251$	2.92399	0.184561	0.0922804	+	0.995733i	$0.470584\pi$
$252$	0.747207	0.0470696
$253$	−4.90041	−0.308086
$254$	−5.85905	−0.367630
$255$	0	0
$256$	−15.7219	−0.982621
$257$	13.8615	0.864659	0.432330	−	0.901716i	$-0.357692\pi$
$257$	13.8615	0.864659	0.432330	+	0.901716i	$0.357692\pi$
$258$	7.52712	0.468618
$259$	2.04827	0.127273
$260$	0	0
$261$	−22.1971	−1.37397
$262$	−9.80944	−0.606029
$263$	12.4674	0.768776	0.384388	−	0.923172i	$-0.374413\pi$
$263$	12.4674	0.768776	0.384388	+	0.923172i	$0.374413\pi$
$264$	−12.9459	−0.796762
$265$	0	0
$266$	0	0
$267$	0.374409	0.0229134
$268$	−4.74376	−0.289771
$269$	27.2482	1.66135	0.830676	−	0.556755i	$-0.187954\pi$
$269$	27.2482	1.66135	0.830676	+	0.556755i	$0.187954\pi$
$270$	0	0
$271$	−25.6315	−1.55700	−0.778501	−	0.627644i	$-0.784020\pi$
$271$	−25.6315	−1.55700	−0.778501	+	0.627644i	$0.784020\pi$
$272$	−0.472420	−0.0286446
$273$	0.273805	0.0165715
$274$	−14.7443	−0.890737
$275$	0	0
$276$	−1.00617	−0.0605641
$277$	−21.1317	−1.26968	−0.634841	−	0.772643i	$-0.718934\pi$
$277$	−21.1317	−1.26968	−0.634841	+	0.772643i	$0.718934\pi$
$278$	−11.5534	−0.692926
$279$	−0.0899323	−0.00538411
$280$	0	0
$281$	1.59925	0.0954033	0.0477017	−	0.998862i	$-0.484810\pi$
$281$	1.59925	0.0954033	0.0477017	+	0.998862i	$0.484810\pi$
$282$	6.30120	0.375231
$283$	−2.19141	−0.130266	−0.0651328	−	0.997877i	$-0.520747\pi$
$283$	−2.19141	−0.130266	−0.0651328	+	0.997877i	$0.520747\pi$
$284$	12.7319	0.755500
$285$	0	0
$286$	5.47643	0.323828
$287$	−1.65589	−0.0977442
$288$	12.8129	0.755006
$289$	2.89249	0.170147
$290$	0	0
$291$	−15.0691	−0.883367
$292$	12.4841	0.730576
$293$	−14.0044	−0.818145	−0.409073	−	0.912502i	$-0.634148\pi$
$293$	−14.0044	−0.818145	−0.409073	+	0.912502i	$0.634148\pi$
$294$	5.17245	0.301663
$295$	0	0
$296$	21.7673	1.26520
$297$	24.2215	1.40547
$298$	−16.6730	−0.965839
$299$	1.10144	0.0636982
$300$	0	0
$301$	−2.66172	−0.153419
$302$	−13.1565	−0.757072
$303$	−7.76440	−0.446053
$304$	0	0
$305$	0	0
$306$	−8.62569	−0.493098
$307$	4.81646	0.274890	0.137445	−	0.990509i	$-0.456111\pi$
$307$	4.81646	0.274890	0.137445	+	0.990509i	$0.456111\pi$
$308$	1.76904	0.100801
$309$	4.91409	0.279553
$310$	0	0
$311$	19.4101	1.10064	0.550322	−	0.834952i	$-0.314505\pi$
$311$	19.4101	1.10064	0.550322	+	0.834952i	$0.314505\pi$
$312$	2.90978	0.164734
$313$	5.71724	0.323157	0.161579	−	0.986860i	$-0.448341\pi$
$313$	5.71724	0.323157	0.161579	+	0.986860i	$0.448341\pi$
$314$	14.4422	0.815018
$315$	0	0
$316$	0.00703346	0.000395663 0
$317$	−14.8729	−0.835348	−0.417674	−	0.908597i	$-0.637155\pi$
$317$	−14.8729	−0.835348	−0.417674	+	0.908597i	$0.637155\pi$
$318$	−3.13501	−0.175803
$319$	−52.5526	−2.94238
$320$	0	0
$321$	7.39621	0.412816
$322$	−0.209127	−0.0116542
$323$	0	0
$324$	−3.52030	−0.195572
$325$	0	0
$326$	17.3430	0.960540
$327$	−6.08695	−0.336609
$328$	−17.5975	−0.971659
$329$	−2.22821	−0.122845
$330$	0	0
$331$	−0.0342375	−0.00188187	−0.000940933	−	1.00000i	$-0.500300\pi$
$331$	−0.0342375	−0.00188187	−0.000940933		1.00000i	$0.500300\pi$
$332$	11.1123	0.609868
$333$	−17.4436	−0.955905
$334$	20.3142	1.11154
$335$	0	0
$336$	0.0242479	0.00132283
$337$	−23.8249	−1.29782	−0.648911	−	0.760864i	$-0.724775\pi$
$337$	−23.8249	−1.29782	−0.648911	+	0.760864i	$0.724775\pi$
$338$	9.95491	0.541476
$339$	7.58125	0.411757
$340$	0	0
$341$	−0.212918	−0.0115302
$342$	0	0
$343$	−3.67651	−0.198513
$344$	−28.2866	−1.52511
$345$	0	0
$346$	−6.10940	−0.328443
$347$	32.4568	1.74237	0.871186	−	0.490952i	$-0.163351\pi$
$347$	32.4568	1.74237	0.871186	+	0.490952i	$0.163351\pi$
$348$	−10.7902	−0.578418
$349$	2.59566	0.138943	0.0694713	−	0.997584i	$-0.477869\pi$
$349$	2.59566	0.138943	0.0694713	+	0.997584i	$0.477869\pi$
$350$	0	0
$351$	−5.44416	−0.290588
$352$	30.3350	1.61686
$353$	−2.39072	−0.127245	−0.0636226	−	0.997974i	$-0.520265\pi$
$353$	−2.39072	−0.127245	−0.0636226	+	0.997974i	$0.520265\pi$
$354$	−6.05120	−0.321618
$355$	0	0
$356$	−0.543716	−0.0288169
$357$	−1.02102	−0.0540382
$358$	−3.14215	−0.166068
$359$	−15.6300	−0.824918	−0.412459	−	0.910976i	$-0.635330\pi$
$359$	−15.6300	−0.824918	−0.412459	+	0.910976i	$0.635330\pi$
$360$	0	0
$361$	0	0
$362$	7.52213	0.395354
$363$	15.0204	0.788368
$364$	−0.397620	−0.0208409
$365$	0	0
$366$	−0.163081	−0.00852438
$367$	6.19364	0.323305	0.161653	−	0.986848i	$-0.448318\pi$
$367$	6.19364	0.323305	0.161653	+	0.986848i	$0.448318\pi$
$368$	0.0975426	0.00508476
$369$	14.1020	0.734123
$370$	0	0
$371$	1.10859	0.0575553
$372$	−0.0437170	−0.00226662
$373$	26.9063	1.39316	0.696578	−	0.717481i	$-0.254705\pi$
$373$	26.9063	1.39316	0.696578	+	0.717481i	$0.254705\pi$
$374$	−20.4217	−1.05598
$375$	0	0
$376$	−23.6796	−1.22118
$377$	11.8120	0.608350
$378$	1.03366	0.0531658
$379$	27.0655	1.39026	0.695129	−	0.718885i	$-0.255347\pi$
$379$	27.0655	1.39026	0.695129	+	0.718885i	$0.255347\pi$
$380$	0	0
$381$	5.90634	0.302591
$382$	−2.93705	−0.150273
$383$	−30.5061	−1.55879	−0.779394	−	0.626533i	$-0.784473\pi$
$383$	−30.5061	−1.55879	−0.779394	+	0.626533i	$0.784473\pi$
$384$	6.38658	0.325914
$385$	0	0
$386$	11.8216	0.601704
$387$	22.6680	1.15228
$388$	21.8834	1.11096
$389$	−13.3023	−0.674453	−0.337226	−	0.941424i	$-0.609489\pi$
$389$	−13.3023	−0.674453	−0.337226	+	0.941424i	$0.609489\pi$
$390$	0	0
$391$	−4.10729	−0.207715
$392$	−19.4378	−0.981759
$393$	9.88862	0.498815
$394$	−10.7210	−0.540116
$395$	0	0
$396$	−15.0657	−0.757078
$397$	−23.0108	−1.15488	−0.577440	−	0.816433i	$-0.695948\pi$
$397$	−23.0108	−1.15488	−0.577440	+	0.816433i	$0.695948\pi$
$398$	−23.3989	−1.17288
$399$	0	0
$400$	0	0
$401$	−35.9215	−1.79383	−0.896917	−	0.442199i	$-0.854198\pi$
$401$	−35.9215	−1.79383	−0.896917	+	0.442199i	$0.854198\pi$
$402$	−2.81074	−0.140187
$403$	0.0478567	0.00238391
$404$	11.2755	0.560975
$405$	0	0
$406$	−2.24270	−0.111303
$407$	−41.2985	−2.04709
$408$	−10.8506	−0.537185
$409$	32.3379	1.59900	0.799502	−	0.600663i	$-0.205097\pi$
$409$	32.3379	1.59900	0.799502	+	0.600663i	$0.205097\pi$
$410$	0	0
$411$	14.8633	0.733154
$412$	−7.13624	−0.351577
$413$	2.13981	0.105293
$414$	1.78098	0.0875306
$415$	0	0
$416$	−6.81826	−0.334293
$417$	11.6466	0.570339
$418$	0	0
$419$	12.6510	0.618041	0.309021	−	0.951055i	$-0.399999\pi$
$419$	12.6510	0.618041	0.309021	+	0.951055i	$0.399999\pi$
$420$	0	0
$421$	14.6787	0.715396	0.357698	−	0.933837i	$-0.383562\pi$
$421$	14.6787	0.715396	0.357698	+	0.933837i	$0.383562\pi$
$422$	−13.4504	−0.654756
$423$	18.9761	0.922649
$424$	11.7812	0.572148
$425$	0	0
$426$	7.54384	0.365500
$427$	0.0576682	0.00279076
$428$	−10.7408	−0.519174
$429$	−5.52064	−0.266539
$430$	0	0
$431$	12.9971	0.626049	0.313025	−	0.949745i	$-0.398658\pi$
$431$	12.9971	0.626049	0.313025	+	0.949745i	$0.398658\pi$
$432$	−0.482128	−0.0231964
$433$	25.3993	1.22061	0.610305	−	0.792166i	$-0.291047\pi$
$433$	25.3993	1.22061	0.610305	+	0.792166i	$0.291047\pi$
$434$	−0.00908637	−0.000436160 0
$435$	0	0
$436$	8.83946	0.423333
$437$	0	0
$438$	7.39700	0.353442
$439$	18.6099	0.888203	0.444101	−	0.895977i	$-0.353523\pi$
$439$	18.6099	0.888203	0.444101	+	0.895977i	$0.353523\pi$
$440$	0	0
$441$	15.5768	0.741754
$442$	4.59009	0.218328
$443$	10.3665	0.492527	0.246263	−	0.969203i	$-0.420797\pi$
$443$	10.3665	0.492527	0.246263	+	0.969203i	$0.420797\pi$
$444$	−8.47952	−0.402420
$445$	0	0
$446$	−16.4764	−0.780180
$447$	16.8075	0.794970
$448$	1.23865	0.0585206
$449$	12.9229	0.609869	0.304934	−	0.952373i	$-0.401365\pi$
$449$	12.9229	0.609869	0.304934	+	0.952373i	$0.401365\pi$
$450$	0	0
$451$	33.3871	1.57214
$452$	−11.0095	−0.517843
$453$	13.2627	0.623136
$454$	−24.0916	−1.13067
$455$	0	0
$456$	0	0
$457$	−21.6446	−1.01249	−0.506246	−	0.862389i	$-0.668967\pi$
$457$	−21.6446	−1.01249	−0.506246	+	0.862389i	$0.668967\pi$
$458$	−6.39493	−0.298815
$459$	20.3013	0.947584
$460$	0	0
$461$	25.6902	1.19651	0.598255	−	0.801306i	$-0.295861\pi$
$461$	25.6902	1.19651	0.598255	+	0.801306i	$0.295861\pi$
$462$	1.04818	0.0487659
$463$	5.30550	0.246567	0.123284	−	0.992371i	$-0.460657\pi$
$463$	5.30550	0.246567	0.123284	+	0.992371i	$0.460657\pi$
$464$	1.04606	0.0485620
$465$	0	0
$466$	−12.2992	−0.569751
$467$	−0.970711	−0.0449191	−0.0224596	−	0.999748i	$-0.507150\pi$
$467$	−0.970711	−0.0449191	−0.0224596	+	0.999748i	$0.507150\pi$
$468$	3.38625	0.156529
$469$	0.993927	0.0458953
$470$	0	0
$471$	−14.5587	−0.670831
$472$	22.7402	1.04670
$473$	53.6673	2.46762
$474$	0.00416742	0.000191416 0
$475$	0	0
$476$	1.48273	0.0679607
$477$	−9.44110	−0.432278
$478$	−0.896868	−0.0410218
$479$	0.227190	0.0103806	0.00519030	−	0.999987i	$-0.498348\pi$
$479$	0.227190	0.0103806	0.00519030	+	0.999987i	$0.498348\pi$
$480$	0	0
$481$	9.28248	0.423245
$482$	15.6795	0.714182
$483$	0.210815	0.00959241
$484$	−21.8126	−0.991484
$485$	0	0
$486$	−13.8355	−0.627591
$487$	−16.5677	−0.750753	−0.375376	−	0.926872i	$-0.622487\pi$
$487$	−16.5677	−0.750753	−0.375376	+	0.926872i	$0.622487\pi$
$488$	0.612851	0.0277425
$489$	−17.4830	−0.790608
$490$	0	0
$491$	22.0413	0.994709	0.497354	−	0.867547i	$-0.334305\pi$
$491$	22.0413	0.994709	0.497354	+	0.867547i	$0.334305\pi$
$492$	6.85514	0.309054
$493$	−44.0471	−1.98378
$494$	0	0
$495$	0	0
$496$	0.00423813	0.000190298 0
$497$	−2.66763	−0.119659
$498$	6.58420	0.295045
$499$	25.3168	1.13334	0.566669	−	0.823946i	$-0.308232\pi$
$499$	25.3168	1.13334	0.566669	+	0.823946i	$0.308232\pi$
$500$	0	0
$501$	−20.4782	−0.914897
$502$	−2.51595	−0.112292
$503$	−17.2941	−0.771104	−0.385552	−	0.922686i	$-0.625989\pi$
$503$	−17.2941	−0.771104	−0.385552	+	0.922686i	$0.625989\pi$
$504$	−1.66376	−0.0741099
$505$	0	0
$506$	4.21655	0.187449
$507$	−10.0353	−0.445682
$508$	−8.57719	−0.380551
$509$	−11.2867	−0.500272	−0.250136	−	0.968211i	$-0.580475\pi$
$509$	−11.2867	−0.500272	−0.250136	+	0.968211i	$0.580475\pi$
$510$	0	0
$511$	−2.61570	−0.115712
$512$	−1.19798	−0.0529438
$513$	0	0
$514$	−11.9271	−0.526084
$515$	0	0
$516$	11.0191	0.485090
$517$	44.9266	1.97587
$518$	−1.76243	−0.0774367
$519$	6.15872	0.270338
$520$	0	0
$521$	19.6658	0.861573	0.430787	−	0.902454i	$-0.358236\pi$
$521$	19.6658	0.861573	0.430787	+	0.902454i	$0.358236\pi$
$522$	19.0995	0.835962
$523$	35.4632	1.55070	0.775349	−	0.631534i	$-0.217574\pi$
$523$	35.4632	1.55070	0.775349	+	0.631534i	$0.217574\pi$
$524$	−14.3602	−0.627330
$525$	0	0
$526$	−10.7276	−0.467745
$527$	−0.178458	−0.00777375
$528$	−0.488901	−0.0212767
$529$	−22.1519	−0.963128
$530$	0	0
$531$	−18.2232	−0.790820
$532$	0	0
$533$	−7.50428	−0.325047
$534$	−0.322159	−0.0139412
$535$	0	0
$536$	10.5627	0.456237
$537$	3.16752	0.136688
$538$	−23.4457	−1.01082
$539$	36.8788	1.58848
$540$	0	0
$541$	7.58701	0.326191	0.163095	−	0.986610i	$-0.447852\pi$
$541$	7.58701	0.326191	0.163095	+	0.986610i	$0.447852\pi$
$542$	22.0546	0.947325
$543$	−7.58285	−0.325411
$544$	25.4254	1.09010
$545$	0	0
$546$	−0.235595	−0.0100826
$547$	−16.6909	−0.713650	−0.356825	−	0.934171i	$-0.616141\pi$
$547$	−16.6909	−0.713650	−0.356825	+	0.934171i	$0.616141\pi$
$548$	−21.5845	−0.922045
$549$	−0.491119	−0.0209604
$550$	0	0
$551$	0	0
$552$	2.24037	0.0953566
$553$	−0.00147367	−6.26668e−5 0
$554$	18.1828	0.772511
$555$	0	0
$556$	−16.9133	−0.717282
$557$	18.6607	0.790680	0.395340	−	0.918535i	$-0.370627\pi$
$557$	18.6607	0.790680	0.395340	+	0.918535i	$0.370627\pi$
$558$	0.0773821	0.00327585
$559$	−12.0626	−0.510192
$560$	0	0
$561$	20.5865	0.869163
$562$	−1.37607	−0.0580462
$563$	−6.21513	−0.261936	−0.130968	−	0.991387i	$-0.541809\pi$
$563$	−6.21513	−0.261936	−0.130968	+	0.991387i	$0.541809\pi$
$564$	9.22446	0.388420
$565$	0	0
$566$	1.88559	0.0792573
$567$	0.737584	0.0309756
$568$	−28.3494	−1.18951
$569$	16.6738	0.699003	0.349501	−	0.936936i	$-0.386351\pi$
$569$	16.6738	0.699003	0.349501	+	0.936936i	$0.386351\pi$
$570$	0	0
$571$	−14.4171	−0.603337	−0.301669	−	0.953413i	$-0.597544\pi$
$571$	−14.4171	−0.603337	−0.301669	+	0.953413i	$0.597544\pi$
$572$	8.01707	0.335210
$573$	2.96076	0.123688
$574$	1.42481	0.0594704
$575$	0	0
$576$	−10.5487	−0.439528
$577$	−22.2199	−0.925028	−0.462514	−	0.886612i	$-0.653053\pi$
$577$	−22.2199	−0.925028	−0.462514	+	0.886612i	$0.653053\pi$
$578$	−2.48884	−0.103522
$579$	−11.9170	−0.495255
$580$	0	0
$581$	−2.32829	−0.0965936
$582$	12.9662	0.537466
$583$	−22.3522	−0.925732
$584$	−27.7976	−1.15027
$585$	0	0
$586$	12.0501	0.497783
$587$	−41.5186	−1.71366	−0.856828	−	0.515602i	$-0.827568\pi$
$587$	−41.5186	−1.71366	−0.856828	+	0.515602i	$0.827568\pi$
$588$	7.57205	0.312266
$589$	0	0
$590$	0	0
$591$	10.8075	0.444562
$592$	0.822045	0.0337859
$593$	8.48062	0.348257	0.174129	−	0.984723i	$-0.444289\pi$
$593$	8.48062	0.348257	0.174129	+	0.984723i	$0.444289\pi$
$594$	−20.8413	−0.855131
$595$	0	0
$596$	−24.4079	−0.999787
$597$	23.5878	0.965384
$598$	−0.947736	−0.0387558
$599$	15.1403	0.618614	0.309307	−	0.950962i	$-0.399903\pi$
$599$	15.1403	0.618614	0.309307	+	0.950962i	$0.399903\pi$
$600$	0	0
$601$	−24.4130	−0.995826	−0.497913	−	0.867227i	$-0.665900\pi$
$601$	−24.4130	−0.995826	−0.497913	+	0.867227i	$0.665900\pi$
$602$	2.29027	0.0933445
$603$	−8.46457	−0.344704
$604$	−19.2601	−0.783682
$605$	0	0
$606$	6.68087	0.271392
$607$	22.1867	0.900532	0.450266	−	0.892894i	$-0.351329\pi$
$607$	22.1867	0.900532	0.450266	+	0.892894i	$0.351329\pi$
$608$	0	0
$609$	2.26080	0.0916124
$610$	0	0
$611$	−10.0980	−0.408520
$612$	−12.6273	−0.510430
$613$	−11.2934	−0.456135	−0.228068	−	0.973645i	$-0.573241\pi$
$613$	−11.2934	−0.456135	−0.228068	+	0.973645i	$0.573241\pi$
$614$	−4.14432	−0.167251
$615$	0	0
$616$	−3.93902	−0.158708
$617$	8.72864	0.351402	0.175701	−	0.984444i	$-0.443781\pi$
$617$	8.72864	0.351402	0.175701	+	0.984444i	$0.443781\pi$
$618$	−4.22832	−0.170088
$619$	46.3421	1.86265	0.931323	−	0.364194i	$-0.118655\pi$
$619$	46.3421	1.86265	0.931323	+	0.364194i	$0.118655\pi$
$620$	0	0
$621$	−4.19170	−0.168207
$622$	−16.7014	−0.669664
$623$	0.113921	0.00456415
$624$	0.109888	0.00439905
$625$	0	0
$626$	−4.91939	−0.196618
$627$	0	0
$628$	21.1422	0.843665
$629$	−34.6144	−1.38017
$630$	0	0
$631$	14.1200	0.562108	0.281054	−	0.959692i	$-0.409316\pi$
$631$	14.1200	0.562108	0.281054	+	0.959692i	$0.409316\pi$
$632$	−0.0156610	−0.000622961 0
$633$	13.5590	0.538922
$634$	12.7974	0.508250
$635$	0	0
$636$	−4.58941	−0.181982
$637$	−8.28908	−0.328425
$638$	45.2188	1.79023
$639$	22.7183	0.898721
$640$	0	0
$641$	14.0331	0.554273	0.277136	−	0.960831i	$-0.410615\pi$
$641$	14.0331	0.554273	0.277136	+	0.960831i	$0.410615\pi$
$642$	−6.36405	−0.251169
$643$	−20.4083	−0.804823	−0.402411	−	0.915459i	$-0.631828\pi$
$643$	−20.4083	−0.804823	−0.402411	+	0.915459i	$0.631828\pi$
$644$	−0.306145	−0.0120638
$645$	0	0
$646$	0	0
$647$	14.5688	0.572757	0.286379	−	0.958117i	$-0.407548\pi$
$647$	14.5688	0.572757	0.286379	+	0.958117i	$0.407548\pi$
$648$	7.83846	0.307924
$649$	−43.1442	−1.69356
$650$	0	0
$651$	0.00915971	0.000358997 0
$652$	25.3888	0.994302
$653$	26.9592	1.05500	0.527498	−	0.849556i	$-0.323130\pi$
$653$	26.9592	1.05500	0.527498	+	0.849556i	$0.323130\pi$
$654$	5.23751	0.204803
$655$	0	0
$656$	−0.664570	−0.0259471
$657$	22.2761	0.869073
$658$	1.91726	0.0747426
$659$	14.3259	0.558059	0.279029	−	0.960283i	$-0.409987\pi$
$659$	14.3259	0.558059	0.279029	+	0.960283i	$0.409987\pi$
$660$	0	0
$661$	45.9785	1.78836	0.894179	−	0.447710i	$-0.147760\pi$
$661$	45.9785	1.78836	0.894179	+	0.447710i	$0.147760\pi$
$662$	0.0294596	0.00114498
$663$	−4.62714	−0.179703
$664$	−24.7432	−0.960221
$665$	0	0
$666$	15.0093	0.581600
$667$	9.09460	0.352144
$668$	29.7384	1.15061
$669$	16.6094	0.642156
$670$	0	0
$671$	−1.16274	−0.0448872
$672$	−1.30501	−0.0503417
$673$	3.81843	0.147190	0.0735948	−	0.997288i	$-0.476553\pi$
$673$	3.81843	0.147190	0.0735948	+	0.997288i	$0.476553\pi$
$674$	20.5001	0.789633
$675$	0	0
$676$	14.5732	0.560508
$677$	48.8890	1.87896	0.939479	−	0.342607i	$-0.111310\pi$
$677$	48.8890	1.87896	0.939479	+	0.342607i	$0.111310\pi$
$678$	−6.52328	−0.250525
$679$	−4.58506	−0.175959
$680$	0	0
$681$	24.2860	0.930643
$682$	0.183205	0.00701529
$683$	−15.9118	−0.608849	−0.304425	−	0.952536i	$-0.598464\pi$
$683$	−15.9118	−0.608849	−0.304425	+	0.952536i	$0.598464\pi$
$684$	0	0
$685$	0	0
$686$	3.16345	0.120781
$687$	6.44655	0.245951
$688$	−1.06825	−0.0407265
$689$	5.02400	0.191399
$690$	0	0
$691$	36.6387	1.39380	0.696902	−	0.717167i	$-0.254561\pi$
$691$	36.6387	1.39380	0.696902	+	0.717167i	$0.254561\pi$
$692$	−8.94368	−0.339988
$693$	3.15660	0.119909
$694$	−27.9274	−1.06011
$695$	0	0
$696$	24.0260	0.910704
$697$	27.9835	1.05995
$698$	−2.23343	−0.0845367
$699$	12.3985	0.468955
$700$	0	0
$701$	−17.6715	−0.667442	−0.333721	−	0.942672i	$-0.608304\pi$
$701$	−17.6715	−0.667442	−0.333721	+	0.942672i	$0.608304\pi$
$702$	4.68442	0.176802
$703$	0	0
$704$	−24.9744	−0.941258
$705$	0	0
$706$	2.05709	0.0774196
$707$	−2.36247	−0.0888497
$708$	−8.85848	−0.332922
$709$	15.3223	0.575442	0.287721	−	0.957714i	$-0.407102\pi$
$709$	15.3223	0.575442	0.287721	+	0.957714i	$0.407102\pi$
$710$	0	0
$711$	0.0125502	0.000470669 0
$712$	1.21066	0.0453714
$713$	0.0368470	0.00137993
$714$	0.878537	0.0328784
$715$	0	0
$716$	−4.59986	−0.171905
$717$	0.904108	0.0337645
$718$	13.4488	0.501904
$719$	1.62082	0.0604462	0.0302231	−	0.999543i	$-0.490378\pi$
$719$	1.62082	0.0604462	0.0302231	+	0.999543i	$0.490378\pi$
$720$	0	0
$721$	1.49521	0.0556844
$722$	0	0
$723$	−15.8061	−0.587834
$724$	11.0118	0.409250
$725$	0	0
$726$	−12.9243	−0.479666
$727$	−11.1635	−0.414031	−0.207016	−	0.978338i	$-0.566375\pi$
$727$	−11.1635	−0.414031	−0.207016	+	0.978338i	$0.566375\pi$
$728$	0.885357	0.0328135
$729$	5.56303	0.206038
$730$	0	0
$731$	44.9813	1.66370
$732$	−0.238738	−0.00882400
$733$	−12.1110	−0.447331	−0.223666	−	0.974666i	$-0.571802\pi$
$733$	−12.1110	−0.447331	−0.223666	+	0.974666i	$0.571802\pi$
$734$	−5.32931	−0.196708
$735$	0	0
$736$	−5.24969	−0.193506
$737$	−20.0402	−0.738190
$738$	−12.1341	−0.446662
$739$	40.0688	1.47395	0.736977	−	0.675918i	$-0.236253\pi$
$739$	40.0688	1.47395	0.736977	+	0.675918i	$0.236253\pi$
$740$	0	0
$741$	0	0
$742$	−0.953887	−0.0350183
$743$	−23.3715	−0.857416	−0.428708	−	0.903443i	$-0.641031\pi$
$743$	−23.3715	−0.857416	−0.428708	+	0.903443i	$0.641031\pi$
$744$	0.0973421	0.00356873
$745$	0	0
$746$	−23.1515	−0.847636
$747$	19.8283	0.725481
$748$	−29.8957	−1.09309
$749$	2.25044	0.0822292
$750$	0	0
$751$	−16.6615	−0.607986	−0.303993	−	0.952674i	$-0.598320\pi$
$751$	−16.6615	−0.607986	−0.303993	+	0.952674i	$0.598320\pi$
$752$	−0.894263	−0.0326104
$753$	2.53625	0.0924262
$754$	−10.1636	−0.370138
$755$	0	0
$756$	1.51320	0.0550345
$757$	−23.5257	−0.855057	−0.427528	−	0.904002i	$-0.640616\pi$
$757$	−23.5257	−0.855057	−0.427528	+	0.904002i	$0.640616\pi$
$758$	−23.2884	−0.845874
$759$	−4.25059	−0.154287
$760$	0	0
$761$	−8.20495	−0.297429	−0.148715	−	0.988880i	$-0.547514\pi$
$761$	−8.20495	−0.297429	−0.148715	+	0.988880i	$0.547514\pi$
$762$	−5.08210	−0.184105
$763$	−1.85207	−0.0670494
$764$	−4.29961	−0.155555
$765$	0	0
$766$	26.2489	0.948412
$767$	9.69733	0.350150
$768$	−13.6371	−0.492087
$769$	18.8871	0.681087	0.340544	−	0.940229i	$-0.389389\pi$
$769$	18.8871	0.681087	0.340544	+	0.940229i	$0.389389\pi$
$770$	0	0
$771$	12.0234	0.433013
$772$	17.3059	0.622853
$773$	42.4983	1.52856	0.764279	−	0.644886i	$-0.223095\pi$
$773$	42.4983	1.52856	0.764279	+	0.644886i	$0.223095\pi$
$774$	−19.5046	−0.701079
$775$	0	0
$776$	−48.7264	−1.74918
$777$	1.77665	0.0637371
$778$	11.4459	0.410356
$779$	0	0
$780$	0	0
$781$	53.7864	1.92463
$782$	3.53411	0.126380
$783$	−44.9523	−1.60646
$784$	−0.734071	−0.0262168
$785$	0	0
$786$	−8.50864	−0.303493
$787$	44.4450	1.58429	0.792146	−	0.610331i	$-0.208964\pi$
$787$	44.4450	1.58429	0.792146	+	0.610331i	$0.208964\pi$
$788$	−15.6947	−0.559100
$789$	10.8142	0.384995
$790$	0	0
$791$	2.30674	0.0820183
$792$	33.5459	1.19200
$793$	0.261345	0.00928062
$794$	19.7996	0.702662
$795$	0	0
$796$	−34.2542	−1.21411
$797$	−0.711060	−0.0251870	−0.0125935	−	0.999921i	$-0.504009\pi$
$797$	−0.711060	−0.0251870	−0.0125935	+	0.999921i	$0.504009\pi$
$798$	0	0
$799$	37.6554	1.33215
$800$	0	0
$801$	−0.970184	−0.0342798
$802$	30.9086	1.09142
$803$	52.7395	1.86114
$804$	−4.11471	−0.145115
$805$	0	0
$806$	−0.0411782	−0.00145044
$807$	23.6349	0.831989
$808$	−25.1064	−0.883241
$809$	−37.2669	−1.31023	−0.655117	−	0.755528i	$-0.727381\pi$
$809$	−37.2669	−1.31023	−0.655117	+	0.755528i	$0.727381\pi$
$810$	0	0
$811$	26.0182	0.913622	0.456811	−	0.889564i	$-0.348992\pi$
$811$	26.0182	0.913622	0.456811	+	0.889564i	$0.348992\pi$
$812$	−3.28314	−0.115216
$813$	−22.2326	−0.779731
$814$	35.5352	1.24551
$815$	0	0
$816$	−0.409774	−0.0143450
$817$	0	0
$818$	−27.8251	−0.972881
$819$	−0.709496	−0.0247918
$820$	0	0
$821$	1.15939	0.0404631	0.0202316	−	0.999795i	$-0.493560\pi$
$821$	1.15939	0.0404631	0.0202316	+	0.999795i	$0.493560\pi$
$822$	−12.7891	−0.446072
$823$	−15.4136	−0.537284	−0.268642	−	0.963240i	$-0.586575\pi$
$823$	−15.4136	−0.537284	−0.268642	+	0.963240i	$0.586575\pi$
$824$	15.8899	0.553549
$825$	0	0
$826$	−1.84119	−0.0640633
$827$	−37.9042	−1.31806	−0.659029	−	0.752118i	$-0.729032\pi$
$827$	−37.9042	−1.31806	−0.659029	+	0.752118i	$0.729032\pi$
$828$	2.60722	0.0906072
$829$	12.1577	0.422254	0.211127	−	0.977459i	$-0.432287\pi$
$829$	12.1577	0.422254	0.211127	+	0.977459i	$0.432287\pi$
$830$	0	0
$831$	−18.3295	−0.635844
$832$	5.61339	0.194609
$833$	30.9100	1.07097
$834$	−10.0213	−0.347011
$835$	0	0
$836$	0	0
$837$	−0.182125	−0.00629518
$838$	−10.8855	−0.376034
$839$	−47.7649	−1.64903	−0.824513	−	0.565843i	$-0.808551\pi$
$839$	−47.7649	−1.64903	−0.824513	+	0.565843i	$0.808551\pi$
$840$	0	0
$841$	68.5316	2.36316
$842$	−12.6303	−0.435267
$843$	1.38718	0.0477770
$844$	−19.6904	−0.677770
$845$	0	0
$846$	−16.3279	−0.561366
$847$	4.57025	0.157036
$848$	0.444919	0.0152786
$849$	−1.90081	−0.0652357
$850$	0	0
$851$	7.14700	0.244996
$852$	11.0436	0.378347
$853$	27.5081	0.941859	0.470930	−	0.882171i	$-0.343919\pi$
$853$	27.5081	0.941859	0.470930	+	0.882171i	$0.343919\pi$
$854$	−0.0496205	−0.00169798
$855$	0	0
$856$	23.9158	0.817427
$857$	−7.97761	−0.272510	−0.136255	−	0.990674i	$-0.543507\pi$
$857$	−7.97761	−0.272510	−0.136255	+	0.990674i	$0.543507\pi$
$858$	4.75022	0.162170
$859$	2.43789	0.0831798	0.0415899	−	0.999135i	$-0.486758\pi$
$859$	2.43789	0.0831798	0.0415899	+	0.999135i	$0.486758\pi$
$860$	0	0
$861$	−1.43631	−0.0489493
$862$	−11.1833	−0.380906
$863$	19.8250	0.674852	0.337426	−	0.941352i	$-0.390444\pi$
$863$	19.8250	0.674852	0.337426	+	0.941352i	$0.390444\pi$
$864$	25.9479	0.882765
$865$	0	0
$866$	−21.8547	−0.742655
$867$	2.50893	0.0852077
$868$	−0.0133017	−0.000451490 0
$869$	0.0297131	0.00100795
$870$	0	0
$871$	4.50435	0.152624
$872$	−19.6823	−0.666527
$873$	39.0477	1.32157
$874$	0	0
$875$	0	0
$876$	10.8286	0.365865
$877$	8.63651	0.291634	0.145817	−	0.989312i	$-0.453419\pi$
$877$	8.63651	0.291634	0.145817	+	0.989312i	$0.453419\pi$
$878$	−16.0129	−0.540408
$879$	−12.1473	−0.409719
$880$	0	0
$881$	−6.78634	−0.228638	−0.114319	−	0.993444i	$-0.536469\pi$
$881$	−6.78634	−0.228638	−0.114319	+	0.993444i	$0.536469\pi$
$882$	−13.4031	−0.451305
$883$	−22.2213	−0.747807	−0.373904	−	0.927468i	$-0.621981\pi$
$883$	−22.2213	−0.747807	−0.373904	+	0.927468i	$0.621981\pi$
$884$	6.71952	0.226002
$885$	0	0
$886$	−8.91983	−0.299668
$887$	−3.22267	−0.108207	−0.0541033	−	0.998535i	$-0.517230\pi$
$887$	−3.22267	−0.108207	−0.0541033	+	0.998535i	$0.517230\pi$
$888$	18.8809	0.633600
$889$	1.79712	0.0602734
$890$	0	0
$891$	−14.8716	−0.498219
$892$	−24.1202	−0.807602
$893$	0	0
$894$	−14.4620	−0.483683
$895$	0	0
$896$	1.94324	0.0649190
$897$	0.955386	0.0318994
$898$	−11.1195	−0.371062
$899$	0.395152	0.0131791
$900$	0	0
$901$	−18.7345	−0.624137
$902$	−28.7279	−0.956535
$903$	−2.30876	−0.0768306
$904$	24.5142	0.815330
$905$	0	0
$906$	−11.4119	−0.379134
$907$	0.911956	0.0302810	0.0151405	−	0.999885i	$-0.495180\pi$
$907$	0.911956	0.0302810	0.0151405	+	0.999885i	$0.495180\pi$
$908$	−35.2682	−1.17042
$909$	20.1195	0.667320
$910$	0	0
$911$	−51.6358	−1.71077	−0.855385	−	0.517992i	$-0.826680\pi$
$911$	−51.6358	−1.71077	−0.855385	+	0.517992i	$0.826680\pi$
$912$	0	0
$913$	46.9444	1.55363
$914$	18.6241	0.616029
$915$	0	0
$916$	−9.36167	−0.309318
$917$	3.00880	0.0993593
$918$	−17.4682	−0.576537
$919$	−16.5629	−0.546361	−0.273180	−	0.961963i	$-0.588076\pi$
$919$	−16.5629	−0.546361	−0.273180	+	0.961963i	$0.588076\pi$
$920$	0	0
$921$	4.17777	0.137662
$922$	−22.1051	−0.727991
$923$	−12.0893	−0.397925
$924$	1.53446	0.0504799
$925$	0	0
$926$	−4.56511	−0.150019
$927$	−12.7336	−0.418227
$928$	−56.2983	−1.84808
$929$	−25.5753	−0.839098	−0.419549	−	0.907733i	$-0.637812\pi$
$929$	−25.5753	−0.839098	−0.419549	+	0.907733i	$0.637812\pi$
$930$	0	0
$931$	0	0
$932$	−18.0051	−0.589777
$933$	16.8362	0.551192
$934$	0.835247	0.0273301
$935$	0	0
$936$	−7.53996	−0.246451
$937$	1.00369	0.0327890	0.0163945	−	0.999866i	$-0.494781\pi$
$937$	1.00369	0.0327890	0.0163945	+	0.999866i	$0.494781\pi$
$938$	−0.855223	−0.0279240
$939$	4.95910	0.161834
$940$	0	0
$941$	55.6685	1.81474	0.907371	−	0.420331i	$-0.138086\pi$
$941$	55.6685	1.81474	0.907371	+	0.420331i	$0.138086\pi$
$942$	12.5270	0.408153
$943$	−5.77788	−0.188154
$944$	0.858784	0.0279510
$945$	0	0
$946$	−46.1779	−1.50137
$947$	−28.8174	−0.936438	−0.468219	−	0.883612i	$-0.655104\pi$
$947$	−28.8174	−0.936438	−0.468219	+	0.883612i	$0.655104\pi$
$948$	0.00610078	0.000198144 0
$949$	−11.8540	−0.384798
$950$	0	0
$951$	−12.9007	−0.418334
$952$	−3.30150	−0.107002
$953$	−8.08279	−0.261827	−0.130914	−	0.991394i	$-0.541791\pi$
$953$	−8.08279	−0.261827	−0.130914	+	0.991394i	$0.541791\pi$
$954$	8.12358	0.263011
$955$	0	0
$956$	−1.31294	−0.0424637
$957$	−45.5838	−1.47351
$958$	−0.195486	−0.00631585
$959$	4.52245	0.146038
$960$	0	0
$961$	−30.9984	−0.999948
$962$	−7.98710	−0.257514
$963$	−19.1654	−0.617595
$964$	22.9536	0.739285
$965$	0	0
$966$	−0.181395	−0.00583630
$967$	−9.84792	−0.316688	−0.158344	−	0.987384i	$-0.550615\pi$
$967$	−9.84792	−0.316688	−0.158344	+	0.987384i	$0.550615\pi$
$968$	48.5690	1.56107
$969$	0	0
$970$	0	0
$971$	1.84777	0.0592979	0.0296489	−	0.999560i	$-0.490561\pi$
$971$	1.84777	0.0592979	0.0296489	+	0.999560i	$0.490561\pi$
$972$	−20.2541	−0.649650
$973$	3.54371	0.113606
$974$	14.2556	0.456780
$975$	0	0
$976$	0.0231444	0.000740833 0
$977$	26.4611	0.846567	0.423283	−	0.905997i	$-0.360877\pi$
$977$	26.4611	0.846567	0.423283	+	0.905997i	$0.360877\pi$
$978$	15.0432	0.481029
$979$	−2.29695	−0.0734108
$980$	0	0
$981$	15.7728	0.503586
$982$	−18.9654	−0.605210
$983$	35.5543	1.13401	0.567003	−	0.823716i	$-0.308103\pi$
$983$	35.5543	1.13401	0.567003	+	0.823716i	$0.308103\pi$
$984$	−15.2639	−0.486597
$985$	0	0
$986$	37.9002	1.20699
$987$	−1.93274	−0.0615197
$988$	0	0
$989$	−9.28750	−0.295325
$990$	0	0
$991$	−39.1942	−1.24504	−0.622522	−	0.782603i	$-0.713892\pi$
$991$	−39.1942	−1.24504	−0.622522	+	0.782603i	$0.713892\pi$
$992$	−0.228094	−0.00724199
$993$	−0.0296974	−0.000942420 0
$994$	2.29536	0.0728043
$995$	0	0
$996$	9.63875	0.305416
$997$	3.55869	0.112705	0.0563524	−	0.998411i	$-0.482053\pi$
$997$	3.55869	0.112705	0.0563524	+	0.998411i	$0.482053\pi$
$998$	−21.7838	−0.689555
$999$	−35.3258	−1.11766

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9025.2.a.ca.1.2		6
5.4	even	2		1805.2.a.q.1.5	✓	6
19.18	odd	2		9025.2.a.bq.1.5		6
95.94	odd	2		1805.2.a.r.1.2	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1805.2.a.q.1.5	✓	6	5.4	even	2
1805.2.a.r.1.2	yes	6	95.94	odd	2
9025.2.a.bq.1.5		6	19.18	odd	2
9025.2.a.ca.1.2		6	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 9025 = 5^{2} \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9025.a (trivial)

\(f(q)\)	\(=\)	\(q-0.860448 q^{2} +0.867394 q^{3} -1.25963 q^{4} -0.746348 q^{6} +0.263921 q^{7} +2.80474 q^{8} -2.24763 q^{9} +O(q^{10})\)	\(q-0.860448 q^{2} +0.867394 q^{3} -1.25963 q^{4} -0.746348 q^{6} +0.263921 q^{7} +2.80474 q^{8} -2.24763 q^{9} -5.32135 q^{11} -1.09259 q^{12} +1.19606 q^{13} -0.227090 q^{14} +0.105921 q^{16} -4.46010 q^{17} +1.93397 q^{18} +0.228924 q^{21} +4.57874 q^{22} +0.920897 q^{23} +2.43282 q^{24} -1.02914 q^{26} -4.55176 q^{27} -0.332443 q^{28} +9.87581 q^{29} +0.0400121 q^{31} -5.70062 q^{32} -4.61570 q^{33} +3.83769 q^{34} +2.83118 q^{36} +7.76091 q^{37} +1.03745 q^{39} -6.27419 q^{41} -0.196977 q^{42} -10.0853 q^{43} +6.70292 q^{44} -0.792384 q^{46} -8.44271 q^{47} +0.0918755 q^{48} -6.93035 q^{49} -3.86866 q^{51} -1.50659 q^{52} +4.20047 q^{53} +3.91655 q^{54} +0.740231 q^{56} -8.49762 q^{58} +8.10775 q^{59} +0.218505 q^{61} -0.0344283 q^{62} -0.593196 q^{63} +4.69325 q^{64} +3.97157 q^{66} +3.76600 q^{67} +5.61807 q^{68} +0.798780 q^{69} -10.1077 q^{71} -6.30402 q^{72} -9.91093 q^{73} -6.67786 q^{74} -1.40442 q^{77} -0.892673 q^{78} -0.00558375 q^{79} +2.79472 q^{81} +5.39862 q^{82} -8.82190 q^{83} -0.288359 q^{84} +8.67786 q^{86} +8.56621 q^{87} -14.9250 q^{88} +0.431648 q^{89} +0.315664 q^{91} -1.15999 q^{92} +0.0347063 q^{93} +7.26452 q^{94} -4.94469 q^{96} -17.3729 q^{97} +5.96320 q^{98} +11.9604 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{2} + 4 q^{3} + 8 q^{4} + 8 q^{6} - 7 q^{7} + 10 q^{9}+O(q^{10}) \) 6 * q + 2 * q^2 + 4 * q^3 + 8 * q^4 + 8 * q^6 - 7 * q^7 + 10 * q^9	\( 6 q + 2 q^{2} + 4 q^{3} + 8 q^{4} + 8 q^{6} - 7 q^{7} + 10 q^{9} + 15 q^{11} - 20 q^{12} + 6 q^{13} - 20 q^{14} + 20 q^{16} - 5 q^{17} - 8 q^{18} - 7 q^{21} + 14 q^{22} + 18 q^{24} - 22 q^{26} + 28 q^{27} - 10 q^{28} + 20 q^{29} - 12 q^{31} - 8 q^{32} + q^{33} - 14 q^{36} + 20 q^{37} + 16 q^{39} - 8 q^{41} + 30 q^{42} - 27 q^{43} + 46 q^{44} - 16 q^{46} + 8 q^{47} - 24 q^{48} + 11 q^{49} + 11 q^{51} + 4 q^{52} + 19 q^{53} + 30 q^{54} - 62 q^{56} - 20 q^{58} + 41 q^{59} + 6 q^{61} + 30 q^{62} + 18 q^{63} + 48 q^{64} + 46 q^{66} - 15 q^{67} + 14 q^{68} + 10 q^{69} - 2 q^{71} - 68 q^{72} - 8 q^{73} + 2 q^{74} - 21 q^{77} + 46 q^{78} - 18 q^{79} + 50 q^{81} + 18 q^{82} + 10 q^{83} - 4 q^{84} + 10 q^{86} + 14 q^{87} - 52 q^{88} - 17 q^{89} + 23 q^{91} - 28 q^{92} - 10 q^{93} + 28 q^{94} + 26 q^{96} - 4 q^{97} + 50 q^{98} + 58 q^{99}+O(q^{100}) \) 6 * q + 2 * q^2 + 4 * q^3 + 8 * q^4 + 8 * q^6 - 7 * q^7 + 10 * q^9 + 15 * q^11 - 20 * q^12 + 6 * q^13 - 20 * q^14 + 20 * q^16 - 5 * q^17 - 8 * q^18 - 7 * q^21 + 14 * q^22 + 18 * q^24 - 22 * q^26 + 28 * q^27 - 10 * q^28 + 20 * q^29 - 12 * q^31 - 8 * q^32 + q^33 - 14 * q^36 + 20 * q^37 + 16 * q^39 - 8 * q^41 + 30 * q^42 - 27 * q^43 + 46 * q^44 - 16 * q^46 + 8 * q^47 - 24 * q^48 + 11 * q^49 + 11 * q^51 + 4 * q^52 + 19 * q^53 + 30 * q^54 - 62 * q^56 - 20 * q^58 + 41 * q^59 + 6 * q^61 + 30 * q^62 + 18 * q^63 + 48 * q^64 + 46 * q^66 - 15 * q^67 + 14 * q^68 + 10 * q^69 - 2 * q^71 - 68 * q^72 - 8 * q^73 + 2 * q^74 - 21 * q^77 + 46 * q^78 - 18 * q^79 + 50 * q^81 + 18 * q^82 + 10 * q^83 - 4 * q^84 + 10 * q^86 + 14 * q^87 - 52 * q^88 - 17 * q^89 + 23 * q^91 - 28 * q^92 - 10 * q^93 + 28 * q^94 + 26 * q^96 - 4 * q^97 + 50 * q^98 + 58 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more