LMFDB - Embedded newform 5625.2.a.o.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5625,2,Mod(1,5625)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5625, 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("5625.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5625 = 3^{2} \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5625.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:	$44.9158511370$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.46840000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 11x^{4} + 8x^{3} + 31x^{2} - 15x - 9 $ x^6 - x^5 - 11x^4 + 8x^3 + 31x^2 - 15x - 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1875)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-2.38719$ of defining polynomial
Character	$\chi$	$=$	5625.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.38719 q^{2} +3.69868 q^{4} -3.31671 q^{7} +4.05506 q^{8} +O(q^{10})$	$q+2.38719 q^{2} +3.69868 q^{4} -3.31671 q^{7} +4.05506 q^{8} +4.36655 q^{11} -5.85502 q^{13} -7.91762 q^{14} +2.28286 q^{16} +0.407830 q^{17} -6.64447 q^{19} +10.4238 q^{22} +4.61860 q^{23} -13.9771 q^{26} -12.2674 q^{28} -3.30132 q^{29} -8.77495 q^{31} -2.66052 q^{32} +0.973567 q^{34} -3.09911 q^{37} -15.8616 q^{38} -2.89349 q^{41} -1.33270 q^{43} +16.1505 q^{44} +11.0255 q^{46} -11.0609 q^{47} +4.00057 q^{49} -21.6558 q^{52} +2.70021 q^{53} -13.4495 q^{56} -7.88089 q^{58} +5.80518 q^{59} +11.2366 q^{61} -20.9475 q^{62} -10.9169 q^{64} +0.0418484 q^{67} +1.50843 q^{68} +16.2062 q^{71} -5.35799 q^{73} -7.39817 q^{74} -24.5757 q^{76} -14.4826 q^{77} -3.55370 q^{79} -6.90732 q^{82} +4.98952 q^{83} -3.18140 q^{86} +17.7066 q^{88} +1.94025 q^{89} +19.4194 q^{91} +17.0827 q^{92} -26.4044 q^{94} -5.99258 q^{97} +9.55012 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - q^{2} + 11 q^{4} - 2 q^{7} - 6 q^{8}+O(q^{10}) $ 6 * q - q^2 + 11 * q^4 - 2 * q^7 - 6 * q^8	$ 6 q - q^{2} + 11 q^{4} - 2 q^{7} - 6 q^{8} + 4 q^{14} + 17 q^{16} - 2 q^{17} - 2 q^{19} + 9 q^{22} + q^{23} - 37 q^{26} - 44 q^{28} - 31 q^{29} - 2 q^{31} - 33 q^{32} + 37 q^{34} - 22 q^{37} - 27 q^{38} - 33 q^{41} - 3 q^{43} + 11 q^{44} - 12 q^{46} + 6 q^{47} + 4 q^{49} - 33 q^{52} - 14 q^{53} + 30 q^{56} - q^{58} + 8 q^{59} + 34 q^{61} - 31 q^{62} + 12 q^{64} + 2 q^{67} - 27 q^{68} + 3 q^{71} - 36 q^{73} - 36 q^{74} + 27 q^{76} - 16 q^{77} + 25 q^{79} + 36 q^{82} + 12 q^{83} + 30 q^{86} + 56 q^{88} - 18 q^{89} + 28 q^{91} - 3 q^{92} - 50 q^{94} + 7 q^{97} - 15 q^{98}+O(q^{100}) $ 6 * q - q^2 + 11 * q^4 - 2 * q^7 - 6 * q^8 + 4 * q^14 + 17 * q^16 - 2 * q^17 - 2 * q^19 + 9 * q^22 + q^23 - 37 * q^26 - 44 * q^28 - 31 * q^29 - 2 * q^31 - 33 * q^32 + 37 * q^34 - 22 * q^37 - 27 * q^38 - 33 * q^41 - 3 * q^43 + 11 * q^44 - 12 * q^46 + 6 * q^47 + 4 * q^49 - 33 * q^52 - 14 * q^53 + 30 * q^56 - q^58 + 8 * q^59 + 34 * q^61 - 31 * q^62 + 12 * q^64 + 2 * q^67 - 27 * q^68 + 3 * q^71 - 36 * q^73 - 36 * q^74 + 27 * q^76 - 16 * q^77 + 25 * q^79 + 36 * q^82 + 12 * q^83 + 30 * q^86 + 56 * q^88 - 18 * q^89 + 28 * q^91 - 3 * q^92 - 50 * q^94 + 7 * q^97 - 15 * q^98

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 $

$2$	2.38719	1.68800	0.843999	−	0.536345i	$-0.180195\pi$
$2$	2.38719	1.68800	0.843999	+	0.536345i	$0.180195\pi$
$3$	0	0
$3$	0	0
$4$	3.69868	1.84934
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−3.31671	−1.25360	−0.626799	−	0.779181i	$-0.715635\pi$
$7$	−3.31671	−1.25360	−0.626799	+	0.779181i	$0.715635\pi$
$8$	4.05506	1.43368
$9$	0	0
$10$	0	0
$11$	4.36655	1.31656	0.658282	−	0.752771i	$-0.271283\pi$
$11$	4.36655	1.31656	0.658282	+	0.752771i	$0.271283\pi$
$12$	0	0
$13$	−5.85502	−1.62389	−0.811946	−	0.583733i	$-0.801591\pi$
$13$	−5.85502	−1.62389	−0.811946	+	0.583733i	$0.801591\pi$
$14$	−7.91762	−2.11607
$15$	0	0
$16$	2.28286	0.570714
$17$	0.407830	0.0989132	0.0494566	−	0.998776i	$-0.484251\pi$
$17$	0.407830	0.0989132	0.0494566	+	0.998776i	$0.484251\pi$
$18$	0	0
$19$	−6.64447	−1.52435	−0.762173	−	0.647374i	$-0.775867\pi$
$19$	−6.64447	−1.52435	−0.762173	+	0.647374i	$0.775867\pi$
$20$	0	0
$21$	0	0
$22$	10.4238	2.22236
$23$	4.61860	0.963045	0.481523	−	0.876434i	$-0.340084\pi$
$23$	4.61860	0.963045	0.481523	+	0.876434i	$0.340084\pi$
$24$	0	0
$25$	0	0
$26$	−13.9771	−2.74113
$27$	0	0
$28$	−12.2674	−2.31833
$29$	−3.30132	−0.613040	−0.306520	−	0.951864i	$-0.599165\pi$
$29$	−3.30132	−0.613040	−0.306520	+	0.951864i	$0.599165\pi$
$30$	0	0
$31$	−8.77495	−1.57603	−0.788014	−	0.615658i	$-0.788890\pi$
$31$	−8.77495	−1.57603	−0.788014	+	0.615658i	$0.788890\pi$
$32$	−2.66052	−0.470318
$33$	0	0
$34$	0.973567	0.166965
$35$	0	0
$36$	0	0
$37$	−3.09911	−0.509491	−0.254745	−	0.967008i	$-0.581992\pi$
$37$	−3.09911	−0.509491	−0.254745	+	0.967008i	$0.581992\pi$
$38$	−15.8616	−2.57309
$39$	0	0
$40$	0	0
$41$	−2.89349	−0.451888	−0.225944	−	0.974140i	$-0.572547\pi$
$41$	−2.89349	−0.451888	−0.225944	+	0.974140i	$0.572547\pi$
$42$	0	0
$43$	−1.33270	−0.203234	−0.101617	−	0.994824i	$-0.532402\pi$
$43$	−1.33270	−0.203234	−0.101617	+	0.994824i	$0.532402\pi$
$44$	16.1505	2.43477
$45$	0	0
$46$	11.0255	1.62562
$47$	−11.0609	−1.61339	−0.806696	−	0.590967i	$-0.798746\pi$
$47$	−11.0609	−1.61339	−0.806696	+	0.590967i	$0.798746\pi$
$48$	0	0
$49$	4.00057	0.571510
$50$	0	0
$51$	0	0
$52$	−21.6558	−3.00312
$53$	2.70021	0.370903	0.185451	−	0.982653i	$-0.440625\pi$
$53$	2.70021	0.370903	0.185451	+	0.982653i	$0.440625\pi$
$54$	0	0
$55$	0	0
$56$	−13.4495	−1.79726
$57$	0	0
$58$	−7.88089	−1.03481
$59$	5.80518	0.755770	0.377885	−	0.925852i	$-0.376651\pi$
$59$	5.80518	0.755770	0.377885	+	0.925852i	$0.376651\pi$
$60$	0	0
$61$	11.2366	1.43870	0.719352	−	0.694646i	$-0.244439\pi$
$61$	11.2366	1.43870	0.719352	+	0.694646i	$0.244439\pi$
$62$	−20.9475	−2.66033
$63$	0	0
$64$	−10.9169	−1.36461
$65$	0	0
$66$	0	0
$67$	0.0418484	0.00511260	0.00255630	−	0.999997i	$-0.499186\pi$
$67$	0.0418484	0.00511260	0.00255630	+	0.999997i	$0.499186\pi$
$68$	1.50843	0.182924
$69$	0	0
$70$	0	0
$71$	16.2062	1.92332	0.961659	−	0.274250i	$-0.0884295\pi$
$71$	16.2062	1.92332	0.961659	+	0.274250i	$0.0884295\pi$
$72$	0	0
$73$	−5.35799	−0.627105	−0.313553	−	0.949571i	$-0.601519\pi$
$73$	−5.35799	−0.627105	−0.313553	+	0.949571i	$0.601519\pi$
$74$	−7.39817	−0.860019
$75$	0	0
$76$	−24.5757	−2.81903
$77$	−14.4826	−1.65044
$78$	0	0
$79$	−3.55370	−0.399822	−0.199911	−	0.979814i	$-0.564065\pi$
$79$	−3.55370	−0.399822	−0.199911	+	0.979814i	$0.564065\pi$
$80$	0	0
$81$	0	0
$82$	−6.90732	−0.762786
$83$	4.98952	0.547671	0.273836	−	0.961776i	$-0.411708\pi$
$83$	4.98952	0.547671	0.273836	+	0.961776i	$0.411708\pi$
$84$	0	0
$85$	0	0
$86$	−3.18140	−0.343059
$87$	0	0
$88$	17.7066	1.88753
$89$	1.94025	0.205666	0.102833	−	0.994699i	$-0.467209\pi$
$89$	1.94025	0.205666	0.102833	+	0.994699i	$0.467209\pi$
$90$	0	0
$91$	19.4194	2.03571
$92$	17.0827	1.78100
$93$	0	0
$94$	−26.4044	−2.72340
$95$	0	0
$96$	0	0
$97$	−5.99258	−0.608454	−0.304227	−	0.952600i	$-0.598398\pi$
$97$	−5.99258	−0.608454	−0.304227	+	0.952600i	$0.598398\pi$
$98$	9.55012	0.964708
$99$	0	0
$100$	0	0
$101$	−5.06181	−0.503669	−0.251834	−	0.967770i	$-0.581034\pi$
$101$	−5.06181	−0.503669	−0.251834	+	0.967770i	$0.581034\pi$
$102$	0	0
$103$	−6.90359	−0.680231	−0.340115	−	0.940384i	$-0.610466\pi$
$103$	−6.90359	−0.680231	−0.340115	+	0.940384i	$0.610466\pi$
$104$	−23.7425	−2.32814
$105$	0	0
$106$	6.44592	0.626083
$107$	−17.2723	−1.66978	−0.834890	−	0.550417i	$-0.814469\pi$
$107$	−17.2723	−1.66978	−0.834890	+	0.550417i	$0.814469\pi$
$108$	0	0
$109$	−6.54439	−0.626839	−0.313419	−	0.949615i	$-0.601475\pi$
$109$	−6.54439	−0.626839	−0.313419	+	0.949615i	$0.601475\pi$
$110$	0	0
$111$	0	0
$112$	−7.57157	−0.715446
$113$	8.66993	0.815599	0.407799	−	0.913072i	$-0.366296\pi$
$113$	8.66993	0.815599	0.407799	+	0.913072i	$0.366296\pi$
$114$	0	0
$115$	0	0
$116$	−12.2105	−1.13372
$117$	0	0
$118$	13.8581	1.27574
$119$	−1.35265	−0.123998
$120$	0	0
$121$	8.06676	0.733342
$122$	26.8240	2.42853
$123$	0	0
$124$	−32.4557	−2.91461
$125$	0	0
$126$	0	0
$127$	3.24595	0.288031	0.144016	−	0.989575i	$-0.453998\pi$
$127$	3.24595	0.288031	0.144016	+	0.989575i	$0.453998\pi$
$128$	−20.7396	−1.83314
$129$	0	0
$130$	0	0
$131$	4.40192	0.384598	0.192299	−	0.981336i	$-0.438406\pi$
$131$	4.40192	0.384598	0.192299	+	0.981336i	$0.438406\pi$
$132$	0	0
$133$	22.0378	1.91092
$134$	0.0999001	0.00863005
$135$	0	0
$136$	1.65378	0.141810
$137$	1.06523	0.0910085	0.0455042	−	0.998964i	$-0.485511\pi$
$137$	1.06523	0.0910085	0.0455042	+	0.998964i	$0.485511\pi$
$138$	0	0
$139$	−3.77286	−0.320010	−0.160005	−	0.987116i	$-0.551151\pi$
$139$	−3.77286	−0.320010	−0.160005	+	0.987116i	$0.551151\pi$
$140$	0	0
$141$	0	0
$142$	38.6872	3.24656
$143$	−25.5663	−2.13796
$144$	0	0
$145$	0	0
$146$	−12.7905	−1.05855
$147$	0	0
$148$	−11.4626	−0.942221
$149$	7.88089	0.645627	0.322814	−	0.946463i	$-0.395371\pi$
$149$	7.88089	0.645627	0.322814	+	0.946463i	$0.395371\pi$
$150$	0	0
$151$	7.46432	0.607437	0.303719	−	0.952762i	$-0.401772\pi$
$151$	7.46432	0.607437	0.303719	+	0.952762i	$0.401772\pi$
$152$	−26.9437	−2.18543
$153$	0	0
$154$	−34.5727	−2.78595
$155$	0	0
$156$	0	0
$157$	−0.268958	−0.0214652	−0.0107326	−	0.999942i	$-0.503416\pi$
$157$	−0.268958	−0.0214652	−0.0107326	+	0.999942i	$0.503416\pi$
$158$	−8.48336	−0.674900
$159$	0	0
$160$	0	0
$161$	−15.3186	−1.20727
$162$	0	0
$163$	−10.3496	−0.810645	−0.405322	−	0.914174i	$-0.632841\pi$
$163$	−10.3496	−0.810645	−0.405322	+	0.914174i	$0.632841\pi$
$164$	−10.7021	−0.835693
$165$	0	0
$166$	11.9109	0.924468
$167$	−4.26896	−0.330342	−0.165171	−	0.986265i	$-0.552818\pi$
$167$	−4.26896	−0.330342	−0.165171	+	0.986265i	$0.552818\pi$
$168$	0	0
$169$	21.2813	1.63702
$170$	0	0
$171$	0	0
$172$	−4.92921	−0.375849
$173$	−23.5031	−1.78691	−0.893454	−	0.449154i	$-0.851725\pi$
$173$	−23.5031	−1.78691	−0.893454	+	0.449154i	$0.851725\pi$
$174$	0	0
$175$	0	0
$176$	9.96820	0.751382
$177$	0	0
$178$	4.63175	0.347164
$179$	−19.6897	−1.47168	−0.735840	−	0.677155i	$-0.763212\pi$
$179$	−19.6897	−1.47168	−0.735840	+	0.677155i	$0.763212\pi$
$180$	0	0
$181$	−12.8221	−0.953060	−0.476530	−	0.879158i	$-0.658106\pi$
$181$	−12.8221	−0.953060	−0.476530	+	0.879158i	$0.658106\pi$
$182$	46.3578	3.43627
$183$	0	0
$184$	18.7287	1.38070
$185$	0	0
$186$	0	0
$187$	1.78081	0.130226
$188$	−40.9105	−2.98371
$189$	0	0
$190$	0	0
$191$	19.0583	1.37901	0.689507	−	0.724279i	$-0.257827\pi$
$191$	19.0583	1.37901	0.689507	+	0.724279i	$0.257827\pi$
$192$	0	0
$193$	−16.9842	−1.22255	−0.611275	−	0.791418i	$-0.709343\pi$
$193$	−16.9842	−1.22255	−0.611275	+	0.791418i	$0.709343\pi$
$194$	−14.3054	−1.02707
$195$	0	0
$196$	14.7968	1.05692
$197$	−0.469462	−0.0334478	−0.0167239	−	0.999860i	$-0.505324\pi$
$197$	−0.469462	−0.0334478	−0.0167239	+	0.999860i	$0.505324\pi$
$198$	0	0
$199$	7.02491	0.497983	0.248991	−	0.968506i	$-0.419901\pi$
$199$	7.02491	0.497983	0.248991	+	0.968506i	$0.419901\pi$
$200$	0	0
$201$	0	0
$202$	−12.0835	−0.850192
$203$	10.9495	0.768507
$204$	0	0
$205$	0	0
$206$	−16.4802	−1.14823
$207$	0	0
$208$	−13.3662	−0.926777
$209$	−29.0134	−2.00690
$210$	0	0
$211$	15.7531	1.08449	0.542243	−	0.840222i	$-0.317575\pi$
$211$	15.7531	1.08449	0.542243	+	0.840222i	$0.317575\pi$
$212$	9.98721	0.685925
$213$	0	0
$214$	−41.2323	−2.81859
$215$	0	0
$216$	0	0
$217$	29.1040	1.97571
$218$	−15.6227	−1.05810
$219$	0	0
$220$	0	0
$221$	−2.38785	−0.160624
$222$	0	0
$223$	2.27697	0.152477	0.0762385	−	0.997090i	$-0.475709\pi$
$223$	2.27697	0.152477	0.0762385	+	0.997090i	$0.475709\pi$
$224$	8.82417	0.589590
$225$	0	0
$226$	20.6968	1.37673
$227$	−20.2674	−1.34520	−0.672599	−	0.740007i	$-0.734822\pi$
$227$	−20.2674	−1.34520	−0.672599	+	0.740007i	$0.734822\pi$
$228$	0	0
$229$	19.6358	1.29757	0.648786	−	0.760971i	$-0.275277\pi$
$229$	19.6358	1.29757	0.648786	+	0.760971i	$0.275277\pi$
$230$	0	0
$231$	0	0
$232$	−13.3871	−0.878905
$233$	−17.1468	−1.12333	−0.561663	−	0.827366i	$-0.689838\pi$
$233$	−17.1468	−1.12333	−0.561663	+	0.827366i	$0.689838\pi$
$234$	0	0
$235$	0	0
$236$	21.4715	1.39768
$237$	0	0
$238$	−3.22904	−0.209308
$239$	−21.7654	−1.40789	−0.703945	−	0.710255i	$-0.748580\pi$
$239$	−21.7654	−1.40789	−0.703945	+	0.710255i	$0.748580\pi$
$240$	0	0
$241$	3.90067	0.251264	0.125632	−	0.992077i	$-0.459904\pi$
$241$	3.90067	0.251264	0.125632	+	0.992077i	$0.459904\pi$
$242$	19.2569	1.23788
$243$	0	0
$244$	41.5607	2.66065
$245$	0	0
$246$	0	0
$247$	38.9035	2.47537
$248$	−35.5830	−2.25952
$249$	0	0
$250$	0	0
$251$	21.8389	1.37846	0.689230	−	0.724543i	$-0.257949\pi$
$251$	21.8389	1.37846	0.689230	+	0.724543i	$0.257949\pi$
$252$	0	0
$253$	20.1674	1.26791
$254$	7.74869	0.486196
$255$	0	0
$256$	−27.6757	−1.72973
$257$	17.1453	1.06949	0.534747	−	0.845012i	$-0.320407\pi$
$257$	17.1453	1.06949	0.534747	+	0.845012i	$0.320407\pi$
$258$	0	0
$259$	10.2789	0.638697
$260$	0	0
$261$	0	0
$262$	10.5082	0.649201
$263$	17.6266	1.08690	0.543451	−	0.839441i	$-0.317117\pi$
$263$	17.6266	1.08690	0.543451	+	0.839441i	$0.317117\pi$
$264$	0	0
$265$	0	0
$266$	52.6084	3.22563
$267$	0	0
$268$	0.154784	0.00945492
$269$	−1.94772	−0.118755	−0.0593773	−	0.998236i	$-0.518912\pi$
$269$	−1.94772	−0.118755	−0.0593773	+	0.998236i	$0.518912\pi$
$270$	0	0
$271$	−13.8707	−0.842583	−0.421292	−	0.906925i	$-0.638423\pi$
$271$	−13.8707	−0.842583	−0.421292	+	0.906925i	$0.638423\pi$
$272$	0.931016	0.0564512
$273$	0	0
$274$	2.54290	0.153622
$275$	0	0
$276$	0	0
$277$	26.4814	1.59111	0.795556	−	0.605880i	$-0.207179\pi$
$277$	26.4814	1.59111	0.795556	+	0.605880i	$0.207179\pi$
$278$	−9.00654	−0.540177
$279$	0	0
$280$	0	0
$281$	21.6050	1.28885	0.644425	−	0.764668i	$-0.277097\pi$
$281$	21.6050	1.28885	0.644425	+	0.764668i	$0.277097\pi$
$282$	0	0
$283$	2.05287	0.122030	0.0610151	−	0.998137i	$-0.480566\pi$
$283$	2.05287	0.122030	0.0610151	+	0.998137i	$0.480566\pi$
$284$	59.9413	3.55686
$285$	0	0
$286$	−61.0315	−3.60887
$287$	9.59688	0.566486
$288$	0	0
$289$	−16.8337	−0.990216
$290$	0	0
$291$	0	0
$292$	−19.8175	−1.15973
$293$	11.9787	0.699805	0.349902	−	0.936786i	$-0.386215\pi$
$293$	11.9787	0.699805	0.349902	+	0.936786i	$0.386215\pi$
$294$	0	0
$295$	0	0
$296$	−12.5671	−0.730447
$297$	0	0
$298$	18.8132	1.08982
$299$	−27.0420	−1.56388
$300$	0	0
$301$	4.42016	0.254774
$302$	17.8187	1.02535
$303$	0	0
$304$	−15.1684	−0.869965
$305$	0	0
$306$	0	0
$307$	−7.07969	−0.404059	−0.202030	−	0.979379i	$-0.564754\pi$
$307$	−7.07969	−0.404059	−0.202030	+	0.979379i	$0.564754\pi$
$308$	−53.5664	−3.05223
$309$	0	0
$310$	0	0
$311$	11.4130	0.647171	0.323586	−	0.946199i	$-0.395112\pi$
$311$	11.4130	0.647171	0.323586	+	0.946199i	$0.395112\pi$
$312$	0	0
$313$	11.9077	0.673064	0.336532	−	0.941672i	$-0.390746\pi$
$313$	11.9077	0.673064	0.336532	+	0.941672i	$0.390746\pi$
$314$	−0.642054	−0.0362332
$315$	0	0
$316$	−13.1440	−0.739407
$317$	−12.7719	−0.717340	−0.358670	−	0.933465i	$-0.616770\pi$
$317$	−12.7719	−0.717340	−0.358670	+	0.933465i	$0.616770\pi$
$318$	0	0
$319$	−14.4154	−0.807107
$320$	0	0
$321$	0	0
$322$	−36.5683	−2.03787
$323$	−2.70981	−0.150778
$324$	0	0
$325$	0	0
$326$	−24.7065	−1.36837
$327$	0	0
$328$	−11.7333	−0.647863
$329$	36.6857	2.02255
$330$	0	0
$331$	8.66041	0.476019	0.238010	−	0.971263i	$-0.423505\pi$
$331$	8.66041	0.476019	0.238010	+	0.971263i	$0.423505\pi$
$332$	18.4546	1.01283
$333$	0	0
$334$	−10.1908	−0.557617
$335$	0	0
$336$	0	0
$337$	−15.2564	−0.831069	−0.415534	−	0.909577i	$-0.636405\pi$
$337$	−15.2564	−0.831069	−0.415534	+	0.909577i	$0.636405\pi$
$338$	50.8025	2.76329
$339$	0	0
$340$	0	0
$341$	−38.3163	−2.07494
$342$	0	0
$343$	9.94824	0.537155
$344$	−5.40416	−0.291373
$345$	0	0
$346$	−56.1064	−3.01630
$347$	21.8540	1.17319	0.586593	−	0.809882i	$-0.300469\pi$
$347$	21.8540	1.17319	0.586593	+	0.809882i	$0.300469\pi$
$348$	0	0
$349$	7.29614	0.390553	0.195277	−	0.980748i	$-0.437440\pi$
$349$	7.29614	0.390553	0.195277	+	0.980748i	$0.437440\pi$
$350$	0	0
$351$	0	0
$352$	−11.6173	−0.619203
$353$	30.5320	1.62506	0.812528	−	0.582923i	$-0.198091\pi$
$353$	30.5320	1.62506	0.812528	+	0.582923i	$0.198091\pi$
$354$	0	0
$355$	0	0
$356$	7.17637	0.380347
$357$	0	0
$358$	−47.0031	−2.48419
$359$	−3.06747	−0.161895	−0.0809473	−	0.996718i	$-0.525795\pi$
$359$	−3.06747	−0.161895	−0.0809473	+	0.996718i	$0.525795\pi$
$360$	0	0
$361$	25.1489	1.32363
$362$	−30.6088	−1.60876
$363$	0	0
$364$	71.8261	3.76471
$365$	0	0
$366$	0	0
$367$	−31.6035	−1.64969	−0.824845	−	0.565358i	$-0.808738\pi$
$367$	−31.6035	−1.64969	−0.824845	+	0.565358i	$0.808738\pi$
$368$	10.5436	0.549623
$369$	0	0
$370$	0	0
$371$	−8.95582	−0.464963
$372$	0	0
$373$	−22.4904	−1.16451	−0.582254	−	0.813007i	$-0.697829\pi$
$373$	−22.4904	−1.16451	−0.582254	+	0.813007i	$0.697829\pi$
$374$	4.25113	0.219821
$375$	0	0
$376$	−44.8525	−2.31309
$377$	19.3293	0.995511
$378$	0	0
$379$	30.1612	1.54928	0.774638	−	0.632404i	$-0.217932\pi$
$379$	30.1612	1.54928	0.774638	+	0.632404i	$0.217932\pi$
$380$	0	0
$381$	0	0
$382$	45.4959	2.32777
$383$	−16.7725	−0.857034	−0.428517	−	0.903534i	$-0.640964\pi$
$383$	−16.7725	−0.857034	−0.428517	+	0.903534i	$0.640964\pi$
$384$	0	0
$385$	0	0
$386$	−40.5445	−2.06366
$387$	0	0
$388$	−22.1646	−1.12524
$389$	−22.4835	−1.13996	−0.569980	−	0.821659i	$-0.693049\pi$
$389$	−22.4835	−1.13996	−0.569980	+	0.821659i	$0.693049\pi$
$390$	0	0
$391$	1.88360	0.0952579
$392$	16.2226	0.819363
$393$	0	0
$394$	−1.12069	−0.0564598
$395$	0	0
$396$	0	0
$397$	3.65800	0.183590	0.0917948	−	0.995778i	$-0.470740\pi$
$397$	3.65800	0.183590	0.0917948	+	0.995778i	$0.470740\pi$
$398$	16.7698	0.840594
$399$	0	0
$400$	0	0
$401$	25.3774	1.26729	0.633643	−	0.773626i	$-0.281559\pi$
$401$	25.3774	1.26729	0.633643	+	0.773626i	$0.281559\pi$
$402$	0	0
$403$	51.3775	2.55930
$404$	−18.7220	−0.931454
$405$	0	0
$406$	26.1386	1.29724
$407$	−13.5324	−0.670777
$408$	0	0
$409$	−27.8011	−1.37468	−0.687339	−	0.726337i	$-0.741221\pi$
$409$	−27.8011	−1.37468	−0.687339	+	0.726337i	$0.741221\pi$
$410$	0	0
$411$	0	0
$412$	−25.5341	−1.25798
$413$	−19.2541	−0.947433
$414$	0	0
$415$	0	0
$416$	15.5774	0.763745
$417$	0	0
$418$	−69.2605	−3.38764
$419$	0.119817	0.00585344	0.00292672	−	0.999996i	$-0.499068\pi$
$419$	0.119817	0.00585344	0.00292672	+	0.999996i	$0.499068\pi$
$420$	0	0
$421$	−0.0393633	−0.00191845	−0.000959225	−	1.00000i	$-0.500305\pi$
$421$	−0.0393633	−0.00191845	−0.000959225		1.00000i	$0.500305\pi$
$422$	37.6056	1.83061
$423$	0	0
$424$	10.9495	0.531756
$425$	0	0
$426$	0	0
$427$	−37.2687	−1.80356
$428$	−63.8848	−3.08799
$429$	0	0
$430$	0	0
$431$	−16.3869	−0.789328	−0.394664	−	0.918825i	$-0.629139\pi$
$431$	−16.3869	−0.789328	−0.394664	+	0.918825i	$0.629139\pi$
$432$	0	0
$433$	−17.5610	−0.843930	−0.421965	−	0.906612i	$-0.638660\pi$
$433$	−17.5610	−0.843930	−0.421965	+	0.906612i	$0.638660\pi$
$434$	69.4767	3.33499
$435$	0	0
$436$	−24.2056	−1.15924
$437$	−30.6882	−1.46801
$438$	0	0
$439$	−31.1577	−1.48708	−0.743538	−	0.668694i	$-0.766854\pi$
$439$	−31.1577	−1.48708	−0.743538	+	0.668694i	$0.766854\pi$
$440$	0	0
$441$	0	0
$442$	−5.70026	−0.271134
$443$	13.1787	0.626141	0.313071	−	0.949730i	$-0.398642\pi$
$443$	13.1787	0.626141	0.313071	+	0.949730i	$0.398642\pi$
$444$	0	0
$445$	0	0
$446$	5.43555	0.257381
$447$	0	0
$448$	36.2081	1.71067
$449$	29.2119	1.37859	0.689297	−	0.724479i	$-0.257920\pi$
$449$	29.2119	1.37859	0.689297	+	0.724479i	$0.257920\pi$
$450$	0	0
$451$	−12.6346	−0.594939
$452$	32.0673	1.50832
$453$	0	0
$454$	−48.3822	−2.27069
$455$	0	0
$456$	0	0
$457$	35.8001	1.67466	0.837330	−	0.546698i	$-0.184115\pi$
$457$	35.8001	1.67466	0.837330	+	0.546698i	$0.184115\pi$
$458$	46.8745	2.19030
$459$	0	0
$460$	0	0
$461$	−6.01315	−0.280060	−0.140030	−	0.990147i	$-0.544720\pi$
$461$	−6.01315	−0.280060	−0.140030	+	0.990147i	$0.544720\pi$
$462$	0	0
$463$	−41.2139	−1.91537	−0.957687	−	0.287813i	$-0.907072\pi$
$463$	−41.2139	−1.91537	−0.957687	+	0.287813i	$0.907072\pi$
$464$	−7.53644	−0.349871
$465$	0	0
$466$	−40.9328	−1.89617
$467$	29.9663	1.38667	0.693337	−	0.720614i	$-0.256140\pi$
$467$	29.9663	1.38667	0.693337	+	0.720614i	$0.256140\pi$
$468$	0	0
$469$	−0.138799	−0.00640914
$470$	0	0
$471$	0	0
$472$	23.5404	1.08353
$473$	−5.81928	−0.267571
$474$	0	0
$475$	0	0
$476$	−5.00303	−0.229313
$477$	0	0
$478$	−51.9582	−2.37651
$479$	−8.33937	−0.381036	−0.190518	−	0.981684i	$-0.561017\pi$
$479$	−8.33937	−0.381036	−0.190518	+	0.981684i	$0.561017\pi$
$480$	0	0
$481$	18.1454	0.827357
$482$	9.31165	0.424134
$483$	0	0
$484$	29.8363	1.35620
$485$	0	0
$486$	0	0
$487$	−4.91110	−0.222543	−0.111272	−	0.993790i	$-0.535492\pi$
$487$	−4.91110	−0.222543	−0.111272	+	0.993790i	$0.535492\pi$
$488$	45.5653	2.06264
$489$	0	0
$490$	0	0
$491$	−4.33484	−0.195629	−0.0978144	−	0.995205i	$-0.531185\pi$
$491$	−4.33484	−0.195629	−0.0978144	+	0.995205i	$0.531185\pi$
$492$	0	0
$493$	−1.34638	−0.0606378
$494$	92.8701	4.17842
$495$	0	0
$496$	−20.0319	−0.899461
$497$	−53.7511	−2.41107
$498$	0	0
$499$	36.1803	1.61965	0.809826	−	0.586671i	$-0.199562\pi$
$499$	36.1803	1.61965	0.809826	+	0.586671i	$0.199562\pi$
$500$	0	0
$501$	0	0
$502$	52.1336	2.32684
$503$	37.3990	1.66754	0.833769	−	0.552113i	$-0.186178\pi$
$503$	37.3990	1.66754	0.833769	+	0.552113i	$0.186178\pi$
$504$	0	0
$505$	0	0
$506$	48.1433	2.14023
$507$	0	0
$508$	12.0057	0.532667
$509$	26.2959	1.16555	0.582773	−	0.812635i	$-0.301968\pi$
$509$	26.2959	1.16555	0.582773	+	0.812635i	$0.301968\pi$
$510$	0	0
$511$	17.7709	0.786138
$512$	−24.5878	−1.08664
$513$	0	0
$514$	40.9291	1.80530
$515$	0	0
$516$	0	0
$517$	−48.2978	−2.12413
$518$	24.5376	1.07812
$519$	0	0
$520$	0	0
$521$	−7.02231	−0.307653	−0.153826	−	0.988098i	$-0.549160\pi$
$521$	−7.02231	−0.307653	−0.153826	+	0.988098i	$0.549160\pi$
$522$	0	0
$523$	13.0796	0.571933	0.285967	−	0.958240i	$-0.407685\pi$
$523$	13.0796	0.571933	0.285967	+	0.958240i	$0.407685\pi$
$524$	16.2813	0.711252
$525$	0	0
$526$	42.0780	1.83469
$527$	−3.57869	−0.155890
$528$	0	0
$529$	−1.66851	−0.0725437
$530$	0	0
$531$	0	0
$532$	81.5106	3.53393
$533$	16.9415	0.733817
$534$	0	0
$535$	0	0
$536$	0.169698	0.00732984
$537$	0	0
$538$	−4.64958	−0.200458
$539$	17.4687	0.752430
$540$	0	0
$541$	−2.08066	−0.0894546	−0.0447273	−	0.998999i	$-0.514242\pi$
$541$	−2.08066	−0.0894546	−0.0447273	+	0.998999i	$0.514242\pi$
$542$	−33.1119	−1.42228
$543$	0	0
$544$	−1.08504	−0.0465206
$545$	0	0
$546$	0	0
$547$	−36.9553	−1.58009	−0.790046	−	0.613047i	$-0.789944\pi$
$547$	−36.9553	−1.58009	−0.790046	+	0.613047i	$0.789944\pi$
$548$	3.93993	0.168305
$549$	0	0
$550$	0	0
$551$	21.9355	0.934485
$552$	0	0
$553$	11.7866	0.501217
$554$	63.2161	2.68579
$555$	0	0
$556$	−13.9546	−0.591807
$557$	2.67455	0.113324	0.0566621	−	0.998393i	$-0.481954\pi$
$557$	2.67455	0.113324	0.0566621	+	0.998393i	$0.481954\pi$
$558$	0	0
$559$	7.80296	0.330030
$560$	0	0
$561$	0	0
$562$	51.5754	2.17557
$563$	−13.8746	−0.584745	−0.292372	−	0.956305i	$-0.594445\pi$
$563$	−13.8746	−0.584745	−0.292372	+	0.956305i	$0.594445\pi$
$564$	0	0
$565$	0	0
$566$	4.90058	0.205987
$567$	0	0
$568$	65.7170	2.75742
$569$	−33.3404	−1.39770	−0.698852	−	0.715266i	$-0.746305\pi$
$569$	−33.3404	−1.39770	−0.698852	+	0.715266i	$0.746305\pi$
$570$	0	0
$571$	35.3794	1.48058	0.740291	−	0.672287i	$-0.234688\pi$
$571$	35.3794	1.48058	0.740291	+	0.672287i	$0.234688\pi$
$572$	−94.5613	−3.95381
$573$	0	0
$574$	22.9096	0.956227
$575$	0	0
$576$	0	0
$577$	16.6041	0.691239	0.345620	−	0.938375i	$-0.387669\pi$
$577$	16.6041	0.691239	0.345620	+	0.938375i	$0.387669\pi$
$578$	−40.1852	−1.67148
$579$	0	0
$580$	0	0
$581$	−16.5488	−0.686560
$582$	0	0
$583$	11.7906	0.488317
$584$	−21.7270	−0.899069
$585$	0	0
$586$	28.5955	1.18127
$587$	26.9696	1.11315	0.556576	−	0.830796i	$-0.312115\pi$
$587$	26.9696	1.11315	0.556576	+	0.830796i	$0.312115\pi$
$588$	0	0
$589$	58.3049	2.40241
$590$	0	0
$591$	0	0
$592$	−7.07482	−0.290773
$593$	−12.7895	−0.525201	−0.262600	−	0.964905i	$-0.584580\pi$
$593$	−12.7895	−0.525201	−0.262600	+	0.964905i	$0.584580\pi$
$594$	0	0
$595$	0	0
$596$	29.1489	1.19398
$597$	0	0
$598$	−64.5545	−2.63983
$599$	−31.1978	−1.27471	−0.637354	−	0.770571i	$-0.719971\pi$
$599$	−31.1978	−1.27471	−0.637354	+	0.770571i	$0.719971\pi$
$600$	0	0
$601$	19.2310	0.784450	0.392225	−	0.919869i	$-0.371705\pi$
$601$	19.2310	0.784450	0.392225	+	0.919869i	$0.371705\pi$
$602$	10.5518	0.430058
$603$	0	0
$604$	27.6081	1.12336
$605$	0	0
$606$	0	0
$607$	−8.01104	−0.325158	−0.162579	−	0.986696i	$-0.551981\pi$
$607$	−8.01104	−0.325158	−0.162579	+	0.986696i	$0.551981\pi$
$608$	17.6777	0.716926
$609$	0	0
$610$	0	0
$611$	64.7616	2.61997
$612$	0	0
$613$	11.3203	0.457224	0.228612	−	0.973518i	$-0.426581\pi$
$613$	11.3203	0.457224	0.228612	+	0.973518i	$0.426581\pi$
$614$	−16.9006	−0.682052
$615$	0	0
$616$	−58.7278	−2.36621
$617$	9.71325	0.391041	0.195520	−	0.980700i	$-0.437360\pi$
$617$	9.71325	0.391041	0.195520	+	0.980700i	$0.437360\pi$
$618$	0	0
$619$	20.4652	0.822567	0.411284	−	0.911507i	$-0.365081\pi$
$619$	20.4652	0.822567	0.411284	+	0.911507i	$0.365081\pi$
$620$	0	0
$621$	0	0
$622$	27.2450	1.09242
$623$	−6.43526	−0.257823
$624$	0	0
$625$	0	0
$626$	28.4260	1.13613
$627$	0	0
$628$	−0.994789	−0.0396964
$629$	−1.26391	−0.0503954
$630$	0	0
$631$	2.46405	0.0980924	0.0490462	−	0.998797i	$-0.484382\pi$
$631$	2.46405	0.0980924	0.0490462	+	0.998797i	$0.484382\pi$
$632$	−14.4105	−0.573218
$633$	0	0
$634$	−30.4889	−1.21087
$635$	0	0
$636$	0	0
$637$	−23.4234	−0.928070
$638$	−34.4123	−1.36240
$639$	0	0
$640$	0	0
$641$	−31.7217	−1.25293	−0.626465	−	0.779449i	$-0.715499\pi$
$641$	−31.7217	−1.25293	−0.626465	+	0.779449i	$0.715499\pi$
$642$	0	0
$643$	−11.5574	−0.455779	−0.227889	−	0.973687i	$-0.573182\pi$
$643$	−11.5574	−0.455779	−0.227889	+	0.973687i	$0.573182\pi$
$644$	−56.6584	−2.23266
$645$	0	0
$646$	−6.46883	−0.254513
$647$	−6.06222	−0.238330	−0.119165	−	0.992874i	$-0.538022\pi$
$647$	−6.06222	−0.238330	−0.119165	+	0.992874i	$0.538022\pi$
$648$	0	0
$649$	25.3486	0.995021
$650$	0	0
$651$	0	0
$652$	−38.2799	−1.49916
$653$	−26.3569	−1.03143	−0.515713	−	0.856762i	$-0.672473\pi$
$653$	−26.3569	−1.03143	−0.515713	+	0.856762i	$0.672473\pi$
$654$	0	0
$655$	0	0
$656$	−6.60543	−0.257899
$657$	0	0
$658$	87.5756	3.41405
$659$	−16.2879	−0.634487	−0.317243	−	0.948344i	$-0.602757\pi$
$659$	−16.2879	−0.634487	−0.317243	+	0.948344i	$0.602757\pi$
$660$	0	0
$661$	−20.4308	−0.794665	−0.397332	−	0.917675i	$-0.630064\pi$
$661$	−20.4308	−0.794665	−0.397332	+	0.917675i	$0.630064\pi$
$662$	20.6740	0.803519
$663$	0	0
$664$	20.2328	0.785186
$665$	0	0
$666$	0	0
$667$	−15.2475	−0.590386
$668$	−15.7895	−0.610914
$669$	0	0
$670$	0	0
$671$	49.0653	1.89415
$672$	0	0
$673$	−20.7479	−0.799772	−0.399886	−	0.916565i	$-0.630950\pi$
$673$	−20.7479	−0.799772	−0.399886	+	0.916565i	$0.630950\pi$
$674$	−36.4199	−1.40284
$675$	0	0
$676$	78.7126	3.02741
$677$	−20.1892	−0.775935	−0.387967	−	0.921673i	$-0.626823\pi$
$677$	−20.1892	−0.775935	−0.387967	+	0.921673i	$0.626823\pi$
$678$	0	0
$679$	19.8756	0.762757
$680$	0	0
$681$	0	0
$682$	−91.4682	−3.50250
$683$	6.69431	0.256151	0.128075	−	0.991764i	$-0.459120\pi$
$683$	6.69431	0.256151	0.128075	+	0.991764i	$0.459120\pi$
$684$	0	0
$685$	0	0
$686$	23.7484	0.906716
$687$	0	0
$688$	−3.04235	−0.115989
$689$	−15.8098	−0.602306
$690$	0	0
$691$	−7.86690	−0.299271	−0.149635	−	0.988741i	$-0.547810\pi$
$691$	−7.86690	−0.299271	−0.149635	+	0.988741i	$0.547810\pi$
$692$	−86.9304	−3.30460
$693$	0	0
$694$	52.1697	1.98034
$695$	0	0
$696$	0	0
$697$	−1.18005	−0.0446977
$698$	17.4173	0.659254
$699$	0	0
$700$	0	0
$701$	−11.5944	−0.437916	−0.218958	−	0.975734i	$-0.570266\pi$
$701$	−11.5944	−0.437916	−0.218958	+	0.975734i	$0.570266\pi$
$702$	0	0
$703$	20.5919	0.776640
$704$	−47.6691	−1.79660
$705$	0	0
$706$	72.8857	2.74309
$707$	16.7885	0.631398
$708$	0	0
$709$	37.6176	1.41276	0.706379	−	0.707834i	$-0.250327\pi$
$709$	37.6176	1.41276	0.706379	+	0.707834i	$0.250327\pi$
$710$	0	0
$711$	0	0
$712$	7.86785	0.294860
$713$	−40.5280	−1.51779
$714$	0	0
$715$	0	0
$716$	−72.8260	−2.72163
$717$	0	0
$718$	−7.32263	−0.273278
$719$	21.0855	0.786355	0.393177	−	0.919463i	$-0.371376\pi$
$719$	21.0855	0.786355	0.393177	+	0.919463i	$0.371376\pi$
$720$	0	0
$721$	22.8972	0.852737
$722$	60.0353	2.23428
$723$	0	0
$724$	−47.4248	−1.76253
$725$	0	0
$726$	0	0
$727$	26.1113	0.968416	0.484208	−	0.874953i	$-0.339108\pi$
$727$	26.1113	0.968416	0.484208	+	0.874953i	$0.339108\pi$
$728$	78.7470	2.91856
$729$	0	0
$730$	0	0
$731$	−0.543513	−0.0201025
$732$	0	0
$733$	13.4461	0.496642	0.248321	−	0.968678i	$-0.420121\pi$
$733$	13.4461	0.496642	0.248321	+	0.968678i	$0.420121\pi$
$734$	−75.4436	−2.78468
$735$	0	0
$736$	−12.2879	−0.452937
$737$	0.182733	0.00673106
$738$	0	0
$739$	−53.9014	−1.98279	−0.991397	−	0.130887i	$-0.958218\pi$
$739$	−53.9014	−1.98279	−0.991397	+	0.130887i	$0.958218\pi$
$740$	0	0
$741$	0	0
$742$	−21.3793	−0.784857
$743$	−4.33521	−0.159043	−0.0795216	−	0.996833i	$-0.525339\pi$
$743$	−4.33521	−0.159043	−0.0795216	+	0.996833i	$0.525339\pi$
$744$	0	0
$745$	0	0
$746$	−53.6888	−1.96569
$747$	0	0
$748$	6.58664	0.240831
$749$	57.2873	2.09323
$750$	0	0
$751$	−2.87653	−0.104966	−0.0524831	−	0.998622i	$-0.516714\pi$
$751$	−2.87653	−0.104966	−0.0524831	+	0.998622i	$0.516714\pi$
$752$	−25.2503	−0.920785
$753$	0	0
$754$	46.1428	1.68042
$755$	0	0
$756$	0	0
$757$	−16.0480	−0.583275	−0.291637	−	0.956529i	$-0.594200\pi$
$757$	−16.0480	−0.583275	−0.291637	+	0.956529i	$0.594200\pi$
$758$	72.0005	2.61518
$759$	0	0
$760$	0	0
$761$	43.5182	1.57753	0.788767	−	0.614692i	$-0.210720\pi$
$761$	43.5182	1.57753	0.788767	+	0.614692i	$0.210720\pi$
$762$	0	0
$763$	21.7058	0.785804
$764$	70.4907	2.55026
$765$	0	0
$766$	−40.0391	−1.44667
$767$	−33.9895	−1.22729
$768$	0	0
$769$	−25.9575	−0.936052	−0.468026	−	0.883715i	$-0.655035\pi$
$769$	−25.9575	−0.936052	−0.468026	+	0.883715i	$0.655035\pi$
$770$	0	0
$771$	0	0
$772$	−62.8191	−2.26091
$773$	52.9965	1.90615	0.953076	−	0.302731i	$-0.0978984\pi$
$773$	52.9965	1.90615	0.953076	+	0.302731i	$0.0978984\pi$
$774$	0	0
$775$	0	0
$776$	−24.3003	−0.872329
$777$	0	0
$778$	−53.6724	−1.92425
$779$	19.2257	0.688833
$780$	0	0
$781$	70.7650	2.53217
$782$	4.49652	0.160795
$783$	0	0
$784$	9.13272	0.326169
$785$	0	0
$786$	0	0
$787$	−13.1832	−0.469929	−0.234964	−	0.972004i	$-0.575497\pi$
$787$	−13.1832	−0.469929	−0.234964	+	0.972004i	$0.575497\pi$
$788$	−1.73639	−0.0618562
$789$	0	0
$790$	0	0
$791$	−28.7557	−1.02243
$792$	0	0
$793$	−65.7908	−2.33630
$794$	8.73233	0.309899
$795$	0	0
$796$	25.9829	0.920939
$797$	−15.6197	−0.553280	−0.276640	−	0.960974i	$-0.589221\pi$
$797$	−15.6197	−0.553280	−0.276640	+	0.960974i	$0.589221\pi$
$798$	0	0
$799$	−4.51095	−0.159586
$800$	0	0
$801$	0	0
$802$	60.5806	2.13918
$803$	−23.3959	−0.825625
$804$	0	0
$805$	0	0
$806$	122.648	4.32009
$807$	0	0
$808$	−20.5259	−0.722100
$809$	−11.7683	−0.413750	−0.206875	−	0.978367i	$-0.566329\pi$
$809$	−11.7683	−0.413750	−0.206875	+	0.978367i	$0.566329\pi$
$810$	0	0
$811$	−38.3050	−1.34507	−0.672536	−	0.740065i	$-0.734795\pi$
$811$	−38.3050	−1.34507	−0.672536	+	0.740065i	$0.734795\pi$
$812$	40.4988	1.42123
$813$	0	0
$814$	−32.3045	−1.13227
$815$	0	0
$816$	0	0
$817$	8.85505	0.309799
$818$	−66.3666	−2.32045
$819$	0	0
$820$	0	0
$821$	−22.4713	−0.784254	−0.392127	−	0.919911i	$-0.628261\pi$
$821$	−22.4713	−0.784254	−0.392127	+	0.919911i	$0.628261\pi$
$822$	0	0
$823$	19.8645	0.692434	0.346217	−	0.938155i	$-0.387466\pi$
$823$	19.8645	0.692434	0.346217	+	0.938155i	$0.387466\pi$
$824$	−27.9945	−0.975235
$825$	0	0
$826$	−45.9632	−1.59927
$827$	−21.4039	−0.744288	−0.372144	−	0.928175i	$-0.621377\pi$
$827$	−21.4039	−0.744288	−0.372144	+	0.928175i	$0.621377\pi$
$828$	0	0
$829$	−24.6676	−0.856743	−0.428371	−	0.903603i	$-0.640912\pi$
$829$	−24.6676	−0.856743	−0.428371	+	0.903603i	$0.640912\pi$
$830$	0	0
$831$	0	0
$832$	63.9185	2.21598
$833$	1.63155	0.0565299
$834$	0	0
$835$	0	0
$836$	−107.311	−3.71143
$837$	0	0
$838$	0.286026	0.00988060
$839$	4.60054	0.158828	0.0794142	−	0.996842i	$-0.474695\pi$
$839$	4.60054	0.158828	0.0794142	+	0.996842i	$0.474695\pi$
$840$	0	0
$841$	−18.1013	−0.624182
$842$	−0.0939677	−0.00323834
$843$	0	0
$844$	58.2655	2.00558
$845$	0	0
$846$	0	0
$847$	−26.7551	−0.919317
$848$	6.16419	0.211679
$849$	0	0
$850$	0	0
$851$	−14.3136	−0.490663
$852$	0	0
$853$	−27.1689	−0.930245	−0.465122	−	0.885246i	$-0.653990\pi$
$853$	−27.1689	−0.930245	−0.465122	+	0.885246i	$0.653990\pi$
$854$	−88.9674	−3.04440
$855$	0	0
$856$	−70.0404	−2.39393
$857$	−7.66692	−0.261897	−0.130949	−	0.991389i	$-0.541802\pi$
$857$	−7.66692	−0.261897	−0.130949	+	0.991389i	$0.541802\pi$
$858$	0	0
$859$	9.09944	0.310469	0.155234	−	0.987878i	$-0.450387\pi$
$859$	9.09944	0.310469	0.155234	+	0.987878i	$0.450387\pi$
$860$	0	0
$861$	0	0
$862$	−39.1186	−1.33238
$863$	−21.7471	−0.740279	−0.370140	−	0.928976i	$-0.620690\pi$
$863$	−21.7471	−0.740279	−0.370140	+	0.928976i	$0.620690\pi$
$864$	0	0
$865$	0	0
$866$	−41.9216	−1.42455
$867$	0	0
$868$	107.646	3.65375
$869$	−15.5174	−0.526392
$870$	0	0
$871$	−0.245023	−0.00830230
$872$	−26.5379	−0.898687
$873$	0	0
$874$	−73.2585	−2.47800
$875$	0	0
$876$	0	0
$877$	−20.9781	−0.708379	−0.354190	−	0.935174i	$-0.615243\pi$
$877$	−20.9781	−0.708379	−0.354190	+	0.935174i	$0.615243\pi$
$878$	−74.3794	−2.51018
$879$	0	0
$880$	0	0
$881$	−40.6001	−1.36785	−0.683927	−	0.729551i	$-0.739729\pi$
$881$	−40.6001	−1.36785	−0.683927	+	0.729551i	$0.739729\pi$
$882$	0	0
$883$	−44.2689	−1.48977	−0.744883	−	0.667195i	$-0.767495\pi$
$883$	−44.2689	−1.48977	−0.744883	+	0.667195i	$0.767495\pi$
$884$	−8.83189	−0.297049
$885$	0	0
$886$	31.4602	1.05693
$887$	−21.2137	−0.712285	−0.356142	−	0.934432i	$-0.615908\pi$
$887$	−21.2137	−0.712285	−0.356142	+	0.934432i	$0.615908\pi$
$888$	0	0
$889$	−10.7659	−0.361076
$890$	0	0
$891$	0	0
$892$	8.42177	0.281982
$893$	73.4935	2.45937
$894$	0	0
$895$	0	0
$896$	68.7873	2.29802
$897$	0	0
$898$	69.7343	2.32706
$899$	28.9689	0.966168
$900$	0	0
$901$	1.10123	0.0366872
$902$	−30.1612	−1.00426
$903$	0	0
$904$	35.1571	1.16931
$905$	0	0
$906$	0	0
$907$	−42.8598	−1.42314	−0.711568	−	0.702617i	$-0.752015\pi$
$907$	−42.8598	−1.42314	−0.711568	+	0.702617i	$0.752015\pi$
$908$	−74.9627	−2.48772
$909$	0	0
$910$	0	0
$911$	−52.6870	−1.74560	−0.872800	−	0.488078i	$-0.837698\pi$
$911$	−52.6870	−1.74560	−0.872800	+	0.488078i	$0.837698\pi$
$912$	0	0
$913$	21.7870	0.721045
$914$	85.4617	2.82682
$915$	0	0
$916$	72.6266	2.39965
$917$	−14.5999	−0.482131
$918$	0	0
$919$	−17.5655	−0.579434	−0.289717	−	0.957112i	$-0.593561\pi$
$919$	−17.5655	−0.579434	−0.289717	+	0.957112i	$0.593561\pi$
$920$	0	0
$921$	0	0
$922$	−14.3545	−0.472741
$923$	−94.8874	−3.12326
$924$	0	0
$925$	0	0
$926$	−98.3855	−3.23315
$927$	0	0
$928$	8.78323	0.288324
$929$	58.2546	1.91127	0.955636	−	0.294550i	$-0.0951699\pi$
$929$	58.2546	1.91127	0.955636	+	0.294550i	$0.0951699\pi$
$930$	0	0
$931$	−26.5816	−0.871178
$932$	−63.4206	−2.07741
$933$	0	0
$934$	71.5352	2.34070
$935$	0	0
$936$	0	0
$937$	21.6766	0.708142	0.354071	−	0.935218i	$-0.384797\pi$
$937$	21.6766	0.708142	0.354071	+	0.935218i	$0.384797\pi$
$938$	−0.331340	−0.0108186
$939$	0	0
$940$	0	0
$941$	41.9457	1.36739	0.683695	−	0.729768i	$-0.260372\pi$
$941$	41.9457	1.36739	0.683695	+	0.729768i	$0.260372\pi$
$942$	0	0
$943$	−13.3639	−0.435188
$944$	13.2524	0.431329
$945$	0	0
$946$	−13.8917	−0.451659
$947$	40.0301	1.30080	0.650401	−	0.759591i	$-0.274601\pi$
$947$	40.0301	1.30080	0.650401	+	0.759591i	$0.274601\pi$
$948$	0	0
$949$	31.3712	1.01835
$950$	0	0
$951$	0	0
$952$	−5.48510	−0.177773
$953$	33.1969	1.07535	0.537677	−	0.843151i	$-0.319302\pi$
$953$	33.1969	1.07535	0.537677	+	0.843151i	$0.319302\pi$
$954$	0	0
$955$	0	0
$956$	−80.5033	−2.60366
$957$	0	0
$958$	−19.9077	−0.643187
$959$	−3.53305	−0.114088
$960$	0	0
$961$	45.9997	1.48386
$962$	43.3164	1.39658
$963$	0	0
$964$	14.4273	0.464673
$965$	0	0
$966$	0	0
$967$	20.5477	0.660770	0.330385	−	0.943846i	$-0.392821\pi$
$967$	20.5477	0.660770	0.330385	+	0.943846i	$0.392821\pi$
$968$	32.7112	1.05138
$969$	0	0
$970$	0	0
$971$	−20.4201	−0.655312	−0.327656	−	0.944797i	$-0.606259\pi$
$971$	−20.4201	−0.655312	−0.327656	+	0.944797i	$0.606259\pi$
$972$	0	0
$973$	12.5135	0.401164
$974$	−11.7237	−0.375653
$975$	0	0
$976$	25.6516	0.821088
$977$	27.7767	0.888655	0.444327	−	0.895864i	$-0.353443\pi$
$977$	27.7767	0.888655	0.444327	+	0.895864i	$0.353443\pi$
$978$	0	0
$979$	8.47221	0.270773
$980$	0	0
$981$	0	0
$982$	−10.3481	−0.330221
$983$	15.7171	0.501298	0.250649	−	0.968078i	$-0.419356\pi$
$983$	15.7171	0.501298	0.250649	+	0.968078i	$0.419356\pi$
$984$	0	0
$985$	0	0
$986$	−3.21406	−0.102357
$987$	0	0
$988$	143.891	4.57780
$989$	−6.15519	−0.195724
$990$	0	0
$991$	−6.18035	−0.196325	−0.0981627	−	0.995170i	$-0.531297\pi$
$991$	−6.18035	−0.196325	−0.0981627	+	0.995170i	$0.531297\pi$
$992$	23.3459	0.741233
$993$	0	0
$994$	−128.314	−4.06988
$995$	0	0
$996$	0	0
$997$	−60.9412	−1.93003	−0.965014	−	0.262200i	$-0.915552\pi$
$997$	−60.9412	−1.93003	−0.965014	+	0.262200i	$0.915552\pi$
$998$	86.3692	2.73397
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5625.2.a.o.1.6		6
3.2	odd	2		1875.2.a.l.1.1	yes	6
5.4	even	2		5625.2.a.r.1.1		6
15.2	even	4		1875.2.b.e.1249.2		12
15.8	even	4		1875.2.b.e.1249.11		12
15.14	odd	2		1875.2.a.i.1.6	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1875.2.a.i.1.6	✓	6	15.14	odd	2
1875.2.a.l.1.1	yes	6	3.2	odd	2
1875.2.b.e.1249.2		12	15.2	even	4
1875.2.b.e.1249.11		12	15.8	even	4
5625.2.a.o.1.6		6	1.1	even	1	trivial
5625.2.a.r.1.1		6	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 5625 = 3^{2} \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5625.a (trivial)

\(f(q)\)	\(=\)	\(q+2.38719 q^{2} +3.69868 q^{4} -3.31671 q^{7} +4.05506 q^{8} +O(q^{10})\)	\(q+2.38719 q^{2} +3.69868 q^{4} -3.31671 q^{7} +4.05506 q^{8} +4.36655 q^{11} -5.85502 q^{13} -7.91762 q^{14} +2.28286 q^{16} +0.407830 q^{17} -6.64447 q^{19} +10.4238 q^{22} +4.61860 q^{23} -13.9771 q^{26} -12.2674 q^{28} -3.30132 q^{29} -8.77495 q^{31} -2.66052 q^{32} +0.973567 q^{34} -3.09911 q^{37} -15.8616 q^{38} -2.89349 q^{41} -1.33270 q^{43} +16.1505 q^{44} +11.0255 q^{46} -11.0609 q^{47} +4.00057 q^{49} -21.6558 q^{52} +2.70021 q^{53} -13.4495 q^{56} -7.88089 q^{58} +5.80518 q^{59} +11.2366 q^{61} -20.9475 q^{62} -10.9169 q^{64} +0.0418484 q^{67} +1.50843 q^{68} +16.2062 q^{71} -5.35799 q^{73} -7.39817 q^{74} -24.5757 q^{76} -14.4826 q^{77} -3.55370 q^{79} -6.90732 q^{82} +4.98952 q^{83} -3.18140 q^{86} +17.7066 q^{88} +1.94025 q^{89} +19.4194 q^{91} +17.0827 q^{92} -26.4044 q^{94} -5.99258 q^{97} +9.55012 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - q^{2} + 11 q^{4} - 2 q^{7} - 6 q^{8}+O(q^{10}) \) 6 * q - q^2 + 11 * q^4 - 2 * q^7 - 6 * q^8	\( 6 q - q^{2} + 11 q^{4} - 2 q^{7} - 6 q^{8} + 4 q^{14} + 17 q^{16} - 2 q^{17} - 2 q^{19} + 9 q^{22} + q^{23} - 37 q^{26} - 44 q^{28} - 31 q^{29} - 2 q^{31} - 33 q^{32} + 37 q^{34} - 22 q^{37} - 27 q^{38} - 33 q^{41} - 3 q^{43} + 11 q^{44} - 12 q^{46} + 6 q^{47} + 4 q^{49} - 33 q^{52} - 14 q^{53} + 30 q^{56} - q^{58} + 8 q^{59} + 34 q^{61} - 31 q^{62} + 12 q^{64} + 2 q^{67} - 27 q^{68} + 3 q^{71} - 36 q^{73} - 36 q^{74} + 27 q^{76} - 16 q^{77} + 25 q^{79} + 36 q^{82} + 12 q^{83} + 30 q^{86} + 56 q^{88} - 18 q^{89} + 28 q^{91} - 3 q^{92} - 50 q^{94} + 7 q^{97} - 15 q^{98}+O(q^{100}) \) 6 * q - q^2 + 11 * q^4 - 2 * q^7 - 6 * q^8 + 4 * q^14 + 17 * q^16 - 2 * q^17 - 2 * q^19 + 9 * q^22 + q^23 - 37 * q^26 - 44 * q^28 - 31 * q^29 - 2 * q^31 - 33 * q^32 + 37 * q^34 - 22 * q^37 - 27 * q^38 - 33 * q^41 - 3 * q^43 + 11 * q^44 - 12 * q^46 + 6 * q^47 + 4 * q^49 - 33 * q^52 - 14 * q^53 + 30 * q^56 - q^58 + 8 * q^59 + 34 * q^61 - 31 * q^62 + 12 * q^64 + 2 * q^67 - 27 * q^68 + 3 * q^71 - 36 * q^73 - 36 * q^74 + 27 * q^76 - 16 * q^77 + 25 * q^79 + 36 * q^82 + 12 * q^83 + 30 * q^86 + 56 * q^88 - 18 * q^89 + 28 * q^91 - 3 * q^92 - 50 * q^94 + 7 * q^97 - 15 * q^98

Introduction

L-functions

Modular forms

Varieties

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