LMFDB - Embedded newform 5625.2.a.y.1.5

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:	$8$
Coefficient field:	8.8.46980000000.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 15x^{6} + 80x^{4} - 180x^{2} + 145 $ x^8 - 15x^6 + 80x^4 - 180*x^2 + 145
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$1.47408$ 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)

$f(q)$	$=$	$q+1.47408 q^{2} +0.172909 q^{4} -4.57433 q^{7} -2.69328 q^{8} +O(q^{10})$	$q+1.47408 q^{2} +0.172909 q^{4} -4.57433 q^{7} -2.69328 q^{8} +5.57701 q^{11} -0.102193 q^{13} -6.74292 q^{14} -4.31592 q^{16} +5.83189 q^{17} +2.31592 q^{19} +8.22095 q^{22} -6.14006 q^{23} -0.150641 q^{26} -0.790943 q^{28} -3.74148 q^{29} +8.49606 q^{31} -0.975455 q^{32} +8.59667 q^{34} -1.46649 q^{37} +3.41385 q^{38} -2.75770 q^{41} -2.49639 q^{43} +0.964316 q^{44} -9.05093 q^{46} -8.61138 q^{47} +13.9245 q^{49} -0.0176701 q^{52} -6.37033 q^{53} +12.3199 q^{56} -5.51524 q^{58} -6.92506 q^{59} +4.37283 q^{61} +12.5239 q^{62} +7.19394 q^{64} -8.01381 q^{67} +1.00839 q^{68} -14.7514 q^{71} -10.9217 q^{73} -2.16172 q^{74} +0.400444 q^{76} -25.5111 q^{77} -8.92640 q^{79} -4.06507 q^{82} +12.3927 q^{83} -3.67987 q^{86} -15.0204 q^{88} +5.18921 q^{89} +0.467465 q^{91} -1.06167 q^{92} -12.6939 q^{94} +6.81762 q^{97} +20.5258 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 14 q^{4} - 10 q^{7}+O(q^{10}) $ 8 * q + 14 * q^4 - 10 * q^7	$ 8 q + 14 q^{4} - 10 q^{7} - 10 q^{13} - 18 q^{16} + 2 q^{19} - 20 q^{22} - 10 q^{28} + 4 q^{31} + 50 q^{34} - 50 q^{43} - 30 q^{46} - 6 q^{49} - 30 q^{52} - 60 q^{58} + 46 q^{61} - 14 q^{64} - 40 q^{67} - 50 q^{73} - 34 q^{76} - 12 q^{79} - 60 q^{82} - 70 q^{88} - 10 q^{91} - 20 q^{94} - 50 q^{97}+O(q^{100}) $ 8 * q + 14 * q^4 - 10 * q^7 - 10 * q^13 - 18 * q^16 + 2 * q^19 - 20 * q^22 - 10 * q^28 + 4 * q^31 + 50 * q^34 - 50 * q^43 - 30 * q^46 - 6 * q^49 - 30 * q^52 - 60 * q^58 + 46 * q^61 - 14 * q^64 - 40 * q^67 - 50 * q^73 - 34 * q^76 - 12 * q^79 - 60 * q^82 - 70 * q^88 - 10 * q^91 - 20 * q^94 - 50 * q^97

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 $

$2$	1.47408	1.04233	0.521166	−	0.853456i	$-0.325497\pi$
$2$	1.47408	1.04233	0.521166	+	0.853456i	$0.325497\pi$
$3$	0	0
$3$	0	0
$4$	0.172909	0.0864545
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−4.57433	−1.72893	−0.864467	−	0.502690i	$-0.832344\pi$
$7$	−4.57433	−1.72893	−0.864467	+	0.502690i	$0.832344\pi$
$8$	−2.69328	−0.952217
$9$	0	0
$10$	0	0
$11$	5.57701	1.68153	0.840766	−	0.541399i	$-0.182105\pi$
$11$	5.57701	1.68153	0.840766	+	0.541399i	$0.182105\pi$
$12$	0	0
$13$	−0.102193	−0.0283433	−0.0141717	−	0.999900i	$-0.504511\pi$
$13$	−0.102193	−0.0283433	−0.0141717	+	0.999900i	$0.504511\pi$
$14$	−6.74292	−1.80212
$15$	0	0
$16$	−4.31592	−1.07898
$17$	5.83189	1.41444	0.707221	−	0.706993i	$-0.249949\pi$
$17$	5.83189	1.41444	0.707221	+	0.706993i	$0.249949\pi$
$18$	0	0
$19$	2.31592	0.531309	0.265654	−	0.964068i	$-0.414412\pi$
$19$	2.31592	0.531309	0.265654	+	0.964068i	$0.414412\pi$
$20$	0	0
$21$	0	0
$22$	8.22095	1.75271
$23$	−6.14006	−1.28029	−0.640145	−	0.768254i	$-0.721126\pi$
$23$	−6.14006	−1.28029	−0.640145	+	0.768254i	$0.721126\pi$
$24$	0	0
$25$	0	0
$26$	−0.150641	−0.0295431
$27$	0	0
$28$	−0.790943	−0.149474
$29$	−3.74148	−0.694775	−0.347388	−	0.937722i	$-0.612931\pi$
$29$	−3.74148	−0.694775	−0.347388	+	0.937722i	$0.612931\pi$
$30$	0	0
$31$	8.49606	1.52594	0.762968	−	0.646436i	$-0.223741\pi$
$31$	8.49606	1.52594	0.762968	+	0.646436i	$0.223741\pi$
$32$	−0.975455	−0.172438
$33$	0	0
$34$	8.59667	1.47432
$35$	0	0
$36$	0	0
$37$	−1.46649	−0.241089	−0.120545	−	0.992708i	$-0.538464\pi$
$37$	−1.46649	−0.241089	−0.120545	+	0.992708i	$0.538464\pi$
$38$	3.41385	0.553800
$39$	0	0
$40$	0	0
$41$	−2.75770	−0.430681	−0.215340	−	0.976539i	$-0.569086\pi$
$41$	−2.75770	−0.430681	−0.215340	+	0.976539i	$0.569086\pi$
$42$	0	0
$43$	−2.49639	−0.380695	−0.190348	−	0.981717i	$-0.560962\pi$
$43$	−2.49639	−0.380695	−0.190348	+	0.981717i	$0.560962\pi$
$44$	0.964316	0.145376
$45$	0	0
$46$	−9.05093	−1.33449
$47$	−8.61138	−1.25610	−0.628050	−	0.778173i	$-0.716147\pi$
$47$	−8.61138	−1.25610	−0.628050	+	0.778173i	$0.716147\pi$
$48$	0	0
$49$	13.9245	1.98921
$50$	0	0
$51$	0	0
$52$	−0.0176701	−0.00245041
$53$	−6.37033	−0.875032	−0.437516	−	0.899211i	$-0.644142\pi$
$53$	−6.37033	−0.875032	−0.437516	+	0.899211i	$0.644142\pi$
$54$	0	0
$55$	0	0
$56$	12.3199	1.64632
$57$	0	0
$58$	−5.51524	−0.724186
$59$	−6.92506	−0.901566	−0.450783	−	0.892634i	$-0.648855\pi$
$59$	−6.92506	−0.901566	−0.450783	+	0.892634i	$0.648855\pi$
$60$	0	0
$61$	4.37283	0.559883	0.279942	−	0.960017i	$-0.409685\pi$
$61$	4.37283	0.559883	0.279942	+	0.960017i	$0.409685\pi$
$62$	12.5239	1.59053
$63$	0	0
$64$	7.19394	0.899243
$65$	0	0
$66$	0	0
$67$	−8.01381	−0.979042	−0.489521	−	0.871991i	$-0.662828\pi$
$67$	−8.01381	−0.979042	−0.489521	+	0.871991i	$0.662828\pi$
$68$	1.00839	0.122285
$69$	0	0
$70$	0	0
$71$	−14.7514	−1.75067	−0.875337	−	0.483513i	$-0.839361\pi$
$71$	−14.7514	−1.75067	−0.875337	+	0.483513i	$0.839361\pi$
$72$	0	0
$73$	−10.9217	−1.27829	−0.639146	−	0.769086i	$-0.720712\pi$
$73$	−10.9217	−1.27829	−0.639146	+	0.769086i	$0.720712\pi$
$74$	−2.16172	−0.251295
$75$	0	0
$76$	0.400444	0.0459340
$77$	−25.5111	−2.90726
$78$	0	0
$79$	−8.92640	−1.00430	−0.502149	−	0.864781i	$-0.667457\pi$
$79$	−8.92640	−1.00430	−0.502149	+	0.864781i	$0.667457\pi$
$80$	0	0
$81$	0	0
$82$	−4.06507	−0.448912
$83$	12.3927	1.36027	0.680137	−	0.733085i	$-0.261920\pi$
$83$	12.3927	1.36027	0.680137	+	0.733085i	$0.261920\pi$
$84$	0	0
$85$	0	0
$86$	−3.67987	−0.396811
$87$	0	0
$88$	−15.0204	−1.60118
$89$	5.18921	0.550055	0.275028	−	0.961436i	$-0.411313\pi$
$89$	5.18921	0.550055	0.275028	+	0.961436i	$0.411313\pi$
$90$	0	0
$91$	0.467465	0.0490037
$92$	−1.06167	−0.110687
$93$	0	0
$94$	−12.6939	−1.30927
$95$	0	0
$96$	0	0
$97$	6.81762	0.692225	0.346112	−	0.938193i	$-0.387502\pi$
$97$	6.81762	0.692225	0.346112	+	0.938193i	$0.387502\pi$
$98$	20.5258	2.07342
$99$	0	0
$100$	0	0
$101$	0.0509560	0.00507031	0.00253515	−	0.999997i	$-0.499193\pi$
$101$	0.0509560	0.00507031	0.00253515	+	0.999997i	$0.499193\pi$
$102$	0	0
$103$	−13.5363	−1.33377	−0.666884	−	0.745161i	$-0.732372\pi$
$103$	−13.5363	−1.33377	−0.666884	+	0.745161i	$0.732372\pi$
$104$	0.275235	0.0269890
$105$	0	0
$106$	−9.39037	−0.912074
$107$	−2.67981	−0.259067	−0.129533	−	0.991575i	$-0.541348\pi$
$107$	−2.67981	−0.259067	−0.129533	+	0.991575i	$0.541348\pi$
$108$	0	0
$109$	0.408977	0.0391729	0.0195864	−	0.999808i	$-0.493765\pi$
$109$	0.408977	0.0391729	0.0195864	+	0.999808i	$0.493765\pi$
$110$	0	0
$111$	0	0
$112$	19.7424	1.86549
$113$	−7.29250	−0.686021	−0.343010	−	0.939332i	$-0.611447\pi$
$113$	−7.29250	−0.686021	−0.343010	+	0.939332i	$0.611447\pi$
$114$	0	0
$115$	0	0
$116$	−0.646936	−0.0600665
$117$	0	0
$118$	−10.2081	−0.939730
$119$	−26.6770	−2.44548
$120$	0	0
$121$	20.1030	1.82755
$122$	6.44590	0.583584
$123$	0	0
$124$	1.46905	0.131924
$125$	0	0
$126$	0	0
$127$	−10.7350	−0.952578	−0.476289	−	0.879289i	$-0.658018\pi$
$127$	−10.7350	−0.952578	−0.476289	+	0.879289i	$0.658018\pi$
$128$	12.5554	1.10975
$129$	0	0
$130$	0	0
$131$	12.7077	1.11028	0.555140	−	0.831757i	$-0.312665\pi$
$131$	12.7077	1.11028	0.555140	+	0.831757i	$0.312665\pi$
$132$	0	0
$133$	−10.5938	−0.918598
$134$	−11.8130	−1.02049
$135$	0	0
$136$	−15.7069	−1.34686
$137$	11.1225	0.950262	0.475131	−	0.879915i	$-0.342401\pi$
$137$	11.1225	0.950262	0.475131	+	0.879915i	$0.342401\pi$
$138$	0	0
$139$	−23.3910	−1.98400	−0.991998	−	0.126251i	$-0.959706\pi$
$139$	−23.3910	−1.98400	−0.991998	+	0.126251i	$0.959706\pi$
$140$	0	0
$141$	0	0
$142$	−21.7448	−1.82478
$143$	−0.569933	−0.0476602
$144$	0	0
$145$	0	0
$146$	−16.0995	−1.33240
$147$	0	0
$148$	−0.253569	−0.0208433
$149$	−23.2617	−1.90567	−0.952837	−	0.303484i	$-0.901850\pi$
$149$	−23.2617	−1.90567	−0.952837	+	0.303484i	$0.901850\pi$
$150$	0	0
$151$	−4.15431	−0.338073	−0.169036	−	0.985610i	$-0.554066\pi$
$151$	−4.15431	−0.338073	−0.169036	+	0.985610i	$0.554066\pi$
$152$	−6.23741	−0.505921
$153$	0	0
$154$	−37.6054	−3.03033
$155$	0	0
$156$	0	0
$157$	1.51275	0.120731	0.0603653	−	0.998176i	$-0.480773\pi$
$157$	1.51275	0.120731	0.0603653	+	0.998176i	$0.480773\pi$
$158$	−13.1582	−1.04681
$159$	0	0
$160$	0	0
$161$	28.0866	2.21354
$162$	0	0
$163$	13.5303	1.05977	0.529887	−	0.848068i	$-0.322234\pi$
$163$	13.5303	1.05977	0.529887	+	0.848068i	$0.322234\pi$
$164$	−0.476832	−0.0372343
$165$	0	0
$166$	18.2678	1.41786
$167$	7.58121	0.586651	0.293326	−	0.956013i	$-0.405238\pi$
$167$	7.58121	0.586651	0.293326	+	0.956013i	$0.405238\pi$
$168$	0	0
$169$	−12.9896	−0.999197
$170$	0	0
$171$	0	0
$172$	−0.431648	−0.0329128
$173$	−10.8593	−0.825619	−0.412809	−	0.910817i	$-0.635452\pi$
$173$	−10.8593	−0.825619	−0.412809	+	0.910817i	$0.635452\pi$
$174$	0	0
$175$	0	0
$176$	−24.0699	−1.81434
$177$	0	0
$178$	7.64931	0.573340
$179$	−15.7601	−1.17797	−0.588984	−	0.808145i	$-0.700472\pi$
$179$	−15.7601	−1.17797	−0.588984	+	0.808145i	$0.700472\pi$
$180$	0	0
$181$	−18.4683	−1.37274	−0.686370	−	0.727253i	$-0.740797\pi$
$181$	−18.4683	−1.37274	−0.686370	+	0.727253i	$0.740797\pi$
$182$	0.689081	0.0510781
$183$	0	0
$184$	16.5369	1.21911
$185$	0	0
$186$	0	0
$187$	32.5245	2.37843
$188$	−1.48899	−0.108596
$189$	0	0
$190$	0	0
$191$	−15.5035	−1.12179	−0.560897	−	0.827885i	$-0.689544\pi$
$191$	−15.5035	−1.12179	−0.560897	+	0.827885i	$0.689544\pi$
$192$	0	0
$193$	−7.55803	−0.544039	−0.272020	−	0.962292i	$-0.587692\pi$
$193$	−7.55803	−0.544039	−0.272020	+	0.962292i	$0.587692\pi$
$194$	10.0497	0.721528
$195$	0	0
$196$	2.40767	0.171976
$197$	−0.879538	−0.0626645	−0.0313323	−	0.999509i	$-0.509975\pi$
$197$	−0.879538	−0.0626645	−0.0313323	+	0.999509i	$0.509975\pi$
$198$	0	0
$199$	13.6734	0.969284	0.484642	−	0.874713i	$-0.338950\pi$
$199$	13.6734	0.969284	0.484642	+	0.874713i	$0.338950\pi$
$200$	0	0
$201$	0	0
$202$	0.0751131	0.00528494
$203$	17.1148	1.20122
$204$	0	0
$205$	0	0
$206$	−19.9535	−1.39023
$207$	0	0
$208$	0.441058	0.0305819
$209$	12.9159	0.893412
$210$	0	0
$211$	16.2272	1.11713	0.558563	−	0.829462i	$-0.311353\pi$
$211$	16.2272	1.11713	0.558563	+	0.829462i	$0.311353\pi$
$212$	−1.10149	−0.0756505
$213$	0	0
$214$	−3.95025	−0.270033
$215$	0	0
$216$	0	0
$217$	−38.8638	−2.63824
$218$	0.602865	0.0408311
$219$	0	0
$220$	0	0
$221$	−0.595980	−0.0400899
$222$	0	0
$223$	−6.96852	−0.466647	−0.233323	−	0.972399i	$-0.574960\pi$
$223$	−6.96852	−0.466647	−0.233323	+	0.972399i	$0.574960\pi$
$224$	4.46205	0.298133
$225$	0	0
$226$	−10.7497	−0.715061
$227$	16.8793	1.12032	0.560161	−	0.828384i	$-0.310739\pi$
$227$	16.8793	1.12032	0.560161	+	0.828384i	$0.310739\pi$
$228$	0	0
$229$	−4.65613	−0.307686	−0.153843	−	0.988095i	$-0.549165\pi$
$229$	−4.65613	−0.307686	−0.153843	+	0.988095i	$0.549165\pi$
$230$	0	0
$231$	0	0
$232$	10.0768	0.661577
$233$	−6.44822	−0.422437	−0.211219	−	0.977439i	$-0.567743\pi$
$233$	−6.44822	−0.422437	−0.211219	+	0.977439i	$0.567743\pi$
$234$	0	0
$235$	0	0
$236$	−1.19741	−0.0779444
$237$	0	0
$238$	−39.3240	−2.54900
$239$	−21.5875	−1.39638	−0.698189	−	0.715914i	$-0.746010\pi$
$239$	−21.5875	−1.39638	−0.698189	+	0.715914i	$0.746010\pi$
$240$	0	0
$241$	−0.597069	−0.0384606	−0.0192303	−	0.999815i	$-0.506122\pi$
$241$	−0.597069	−0.0384606	−0.0192303	+	0.999815i	$0.506122\pi$
$242$	29.6335	1.90491
$243$	0	0
$244$	0.756102	0.0484045
$245$	0	0
$246$	0	0
$247$	−0.236671	−0.0150590
$248$	−22.8822	−1.45302
$249$	0	0
$250$	0	0
$251$	18.6685	1.17834	0.589172	−	0.808008i	$-0.299454\pi$
$251$	18.6685	1.17834	0.589172	+	0.808008i	$0.299454\pi$
$252$	0	0
$253$	−34.2432	−2.15285
$254$	−15.8243	−0.992902
$255$	0	0
$256$	4.11969	0.257481
$257$	−0.137173	−0.00855659	−0.00427829	−	0.999991i	$-0.501362\pi$
$257$	−0.137173	−0.00855659	−0.00427829	+	0.999991i	$0.501362\pi$
$258$	0	0
$259$	6.70820	0.416828
$260$	0	0
$261$	0	0
$262$	18.7322	1.15728
$263$	9.48260	0.584722	0.292361	−	0.956308i	$-0.405559\pi$
$263$	9.48260	0.584722	0.292361	+	0.956308i	$0.405559\pi$
$264$	0	0
$265$	0	0
$266$	−15.6161	−0.957483
$267$	0	0
$268$	−1.38566	−0.0846427
$269$	14.5901	0.889577	0.444789	−	0.895636i	$-0.353279\pi$
$269$	14.5901	0.889577	0.444789	+	0.895636i	$0.353279\pi$
$270$	0	0
$271$	−10.2751	−0.624168	−0.312084	−	0.950055i	$-0.601027\pi$
$271$	−10.2751	−0.624168	−0.312084	+	0.950055i	$0.601027\pi$
$272$	−25.1700	−1.52615
$273$	0	0
$274$	16.3955	0.990488
$275$	0	0
$276$	0	0
$277$	1.26715	0.0761354	0.0380677	−	0.999275i	$-0.487880\pi$
$277$	1.26715	0.0761354	0.0380677	+	0.999275i	$0.487880\pi$
$278$	−34.4801	−2.06798
$279$	0	0
$280$	0	0
$281$	20.6741	1.23331	0.616657	−	0.787232i	$-0.288487\pi$
$281$	20.6741	1.23331	0.616657	+	0.787232i	$0.288487\pi$
$282$	0	0
$283$	−14.8827	−0.884687	−0.442343	−	0.896846i	$-0.645853\pi$
$283$	−14.8827	−0.884687	−0.442343	+	0.896846i	$0.645853\pi$
$284$	−2.55066	−0.151354
$285$	0	0
$286$	−0.840126	−0.0496777
$287$	12.6146	0.744618
$288$	0	0
$289$	17.0110	1.00064
$290$	0	0
$291$	0	0
$292$	−1.88847	−0.110514
$293$	−18.8011	−1.09837	−0.549186	−	0.835700i	$-0.685062\pi$
$293$	−18.8011	−1.09837	−0.549186	+	0.835700i	$0.685062\pi$
$294$	0	0
$295$	0	0
$296$	3.94966	0.229569
$297$	0	0
$298$	−34.2896	−1.98634
$299$	0.627472	0.0362877
$300$	0	0
$301$	11.4193	0.658197
$302$	−6.12377	−0.352384
$303$	0	0
$304$	−9.99533	−0.573271
$305$	0	0
$306$	0	0
$307$	−20.6614	−1.17921	−0.589604	−	0.807692i	$-0.700716\pi$
$307$	−20.6614	−1.17921	−0.589604	+	0.807692i	$0.700716\pi$
$308$	−4.41110	−0.251346
$309$	0	0
$310$	0	0
$311$	−14.4381	−0.818711	−0.409356	−	0.912375i	$-0.634246\pi$
$311$	−14.4381	−0.818711	−0.409356	+	0.912375i	$0.634246\pi$
$312$	0	0
$313$	8.51064	0.481050	0.240525	−	0.970643i	$-0.422680\pi$
$313$	8.51064	0.481050	0.240525	+	0.970643i	$0.422680\pi$
$314$	2.22991	0.125841
$315$	0	0
$316$	−1.54346	−0.0868261
$317$	−8.96625	−0.503595	−0.251797	−	0.967780i	$-0.581022\pi$
$317$	−8.96625	−0.503595	−0.251797	+	0.967780i	$0.581022\pi$
$318$	0	0
$319$	−20.8663	−1.16829
$320$	0	0
$321$	0	0
$322$	41.4019	2.30724
$323$	13.5062	0.751505
$324$	0	0
$325$	0	0
$326$	19.9447	1.10464
$327$	0	0
$328$	7.42725	0.410101
$329$	39.3913	2.17171
$330$	0	0
$331$	14.9211	0.820139	0.410070	−	0.912054i	$-0.365504\pi$
$331$	14.9211	0.820139	0.410070	+	0.912054i	$0.365504\pi$
$332$	2.14281	0.117602
$333$	0	0
$334$	11.1753	0.611485
$335$	0	0
$336$	0	0
$337$	−21.5645	−1.17469	−0.587347	−	0.809335i	$-0.699828\pi$
$337$	−21.5645	−1.17469	−0.587347	+	0.809335i	$0.699828\pi$
$338$	−19.1476	−1.04149
$339$	0	0
$340$	0	0
$341$	47.3826	2.56591
$342$	0	0
$343$	−31.6749	−1.71028
$344$	6.72346	0.362505
$345$	0	0
$346$	−16.0075	−0.860569
$347$	−9.28149	−0.498256	−0.249128	−	0.968471i	$-0.580144\pi$
$347$	−9.28149	−0.498256	−0.249128	+	0.968471i	$0.580144\pi$
$348$	0	0
$349$	31.6754	1.69555	0.847773	−	0.530359i	$-0.177943\pi$
$349$	31.6754	1.69555	0.847773	+	0.530359i	$0.177943\pi$
$350$	0	0
$351$	0	0
$352$	−5.44012	−0.289960
$353$	−10.5976	−0.564051	−0.282026	−	0.959407i	$-0.591006\pi$
$353$	−10.5976	−0.564051	−0.282026	+	0.959407i	$0.591006\pi$
$354$	0	0
$355$	0	0
$356$	0.897262	0.0475548
$357$	0	0
$358$	−23.2317	−1.22783
$359$	28.0738	1.48168	0.740838	−	0.671683i	$-0.234428\pi$
$359$	28.0738	1.48168	0.740838	+	0.671683i	$0.234428\pi$
$360$	0	0
$361$	−13.6365	−0.717711
$362$	−27.2238	−1.43085
$363$	0	0
$364$	0.0808290	0.00423659
$365$	0	0
$366$	0	0
$367$	20.7679	1.08407	0.542037	−	0.840355i	$-0.317653\pi$
$367$	20.7679	1.08407	0.542037	+	0.840355i	$0.317653\pi$
$368$	26.5000	1.38141
$369$	0	0
$370$	0	0
$371$	29.1400	1.51287
$372$	0	0
$373$	35.1281	1.81887	0.909433	−	0.415850i	$-0.136516\pi$
$373$	35.1281	1.81887	0.909433	+	0.415850i	$0.136516\pi$
$374$	47.9437	2.47911
$375$	0	0
$376$	23.1928	1.19608
$377$	0.382354	0.0196922
$378$	0	0
$379$	−29.6962	−1.52539	−0.762696	−	0.646757i	$-0.776125\pi$
$379$	−29.6962	−1.52539	−0.762696	+	0.646757i	$0.776125\pi$
$380$	0	0
$381$	0	0
$382$	−22.8534	−1.16928
$383$	33.8083	1.72752	0.863762	−	0.503900i	$-0.168102\pi$
$383$	33.8083	1.72752	0.863762	+	0.503900i	$0.168102\pi$
$384$	0	0
$385$	0	0
$386$	−11.1411	−0.567069
$387$	0	0
$388$	1.17883	0.0598460
$389$	−6.12203	−0.310399	−0.155200	−	0.987883i	$-0.549602\pi$
$389$	−6.12203	−0.310399	−0.155200	+	0.987883i	$0.549602\pi$
$390$	0	0
$391$	−35.8082	−1.81090
$392$	−37.5025	−1.89416
$393$	0	0
$394$	−1.29651	−0.0653172
$395$	0	0
$396$	0	0
$397$	33.6557	1.68913	0.844565	−	0.535453i	$-0.179859\pi$
$397$	33.6557	1.68913	0.844565	+	0.535453i	$0.179859\pi$
$398$	20.1557	1.01031
$399$	0	0
$400$	0	0
$401$	−10.9037	−0.544505	−0.272252	−	0.962226i	$-0.587769\pi$
$401$	−10.9037	−0.544505	−0.272252	+	0.962226i	$0.587769\pi$
$402$	0	0
$403$	−0.868239	−0.0432501
$404$	0.00881075	0.000438351 0
$405$	0	0
$406$	25.2285	1.25207
$407$	−8.17862	−0.405399
$408$	0	0
$409$	27.7499	1.37214	0.686071	−	0.727534i	$-0.259334\pi$
$409$	27.7499	1.37214	0.686071	+	0.727534i	$0.259334\pi$
$410$	0	0
$411$	0	0
$412$	−2.34054	−0.115310
$413$	31.6775	1.55875
$414$	0	0
$415$	0	0
$416$	0.0996849	0.00488746
$417$	0	0
$418$	19.0391	0.931232
$419$	−14.6939	−0.717844	−0.358922	−	0.933368i	$-0.616856\pi$
$419$	−14.6939	−0.717844	−0.358922	+	0.933368i	$0.616856\pi$
$420$	0	0
$421$	17.3285	0.844538	0.422269	−	0.906471i	$-0.361234\pi$
$421$	17.3285	0.844538	0.422269	+	0.906471i	$0.361234\pi$
$422$	23.9202	1.16442
$423$	0	0
$424$	17.1571	0.833221
$425$	0	0
$426$	0	0
$427$	−20.0028	−0.968001
$428$	−0.463363	−0.0223975
$429$	0	0
$430$	0	0
$431$	8.27310	0.398501	0.199251	−	0.979949i	$-0.436149\pi$
$431$	8.27310	0.398501	0.199251	+	0.979949i	$0.436149\pi$
$432$	0	0
$433$	25.4312	1.22215	0.611073	−	0.791575i	$-0.290738\pi$
$433$	25.4312	1.22215	0.611073	+	0.791575i	$0.290738\pi$
$434$	−57.2883	−2.74992
$435$	0	0
$436$	0.0707158	0.00338667
$437$	−14.2199	−0.680229
$438$	0	0
$439$	−1.41035	−0.0673125	−0.0336562	−	0.999433i	$-0.510715\pi$
$439$	−1.41035	−0.0673125	−0.0336562	+	0.999433i	$0.510715\pi$
$440$	0	0
$441$	0	0
$442$	−0.878521	−0.0417870
$443$	−1.68914	−0.0802537	−0.0401268	−	0.999195i	$-0.512776\pi$
$443$	−1.68914	−0.0802537	−0.0401268	+	0.999195i	$0.512776\pi$
$444$	0	0
$445$	0	0
$446$	−10.2722	−0.486401
$447$	0	0
$448$	−32.9075	−1.55473
$449$	−21.2232	−1.00158	−0.500792	−	0.865567i	$-0.666958\pi$
$449$	−21.2232	−1.00158	−0.500792	+	0.865567i	$0.666958\pi$
$450$	0	0
$451$	−15.3797	−0.724203
$452$	−1.26094	−0.0593096
$453$	0	0
$454$	24.8815	1.16775
$455$	0	0
$456$	0	0
$457$	−27.1534	−1.27018	−0.635091	−	0.772438i	$-0.719037\pi$
$457$	−27.1534	−1.27018	−0.635091	+	0.772438i	$0.719037\pi$
$458$	−6.86351	−0.320711
$459$	0	0
$460$	0	0
$461$	−25.3906	−1.18256	−0.591278	−	0.806468i	$-0.701376\pi$
$461$	−25.3906	−1.18256	−0.591278	+	0.806468i	$0.701376\pi$
$462$	0	0
$463$	−22.2273	−1.03299	−0.516496	−	0.856290i	$-0.672764\pi$
$463$	−22.2273	−1.03299	−0.516496	+	0.856290i	$0.672764\pi$
$464$	16.1479	0.749649
$465$	0	0
$466$	−9.50519	−0.440320
$467$	−23.2288	−1.07490	−0.537450	−	0.843296i	$-0.680612\pi$
$467$	−23.2288	−1.07490	−0.537450	+	0.843296i	$0.680612\pi$
$468$	0	0
$469$	36.6578	1.69270
$470$	0	0
$471$	0	0
$472$	18.6511	0.858486
$473$	−13.9224	−0.640151
$474$	0	0
$475$	0	0
$476$	−4.61269	−0.211423
$477$	0	0
$478$	−31.8216	−1.45549
$479$	10.3023	0.470723	0.235361	−	0.971908i	$-0.424373\pi$
$479$	10.3023	0.470723	0.235361	+	0.971908i	$0.424373\pi$
$480$	0	0
$481$	0.149865	0.00683327
$482$	−0.880127	−0.0400887
$483$	0	0
$484$	3.47600	0.158000
$485$	0	0
$486$	0	0
$487$	−1.26808	−0.0574622	−0.0287311	−	0.999587i	$-0.509147\pi$
$487$	−1.26808	−0.0574622	−0.0287311	+	0.999587i	$0.509147\pi$
$488$	−11.7772	−0.533131
$489$	0	0
$490$	0	0
$491$	19.4660	0.878491	0.439245	−	0.898367i	$-0.355246\pi$
$491$	19.4660	0.878491	0.439245	+	0.898367i	$0.355246\pi$
$492$	0	0
$493$	−21.8199	−0.982719
$494$	−0.348872	−0.0156965
$495$	0	0
$496$	−36.6683	−1.64646
$497$	67.4780	3.02680
$498$	0	0
$499$	−20.4104	−0.913693	−0.456847	−	0.889545i	$-0.651021\pi$
$499$	−20.4104	−0.913693	−0.456847	+	0.889545i	$0.651021\pi$
$500$	0	0
$501$	0	0
$502$	27.5188	1.22822
$503$	15.1327	0.674732	0.337366	−	0.941374i	$-0.390464\pi$
$503$	15.1327	0.674732	0.337366	+	0.941374i	$0.390464\pi$
$504$	0	0
$505$	0	0
$506$	−50.4771	−2.24398
$507$	0	0
$508$	−1.85618	−0.0823547
$509$	10.4033	0.461120	0.230560	−	0.973058i	$-0.425944\pi$
$509$	10.4033	0.461120	0.230560	+	0.973058i	$0.425944\pi$
$510$	0	0
$511$	49.9596	2.21008
$512$	−19.0379	−0.841366
$513$	0	0
$514$	−0.202203	−0.00891880
$515$	0	0
$516$	0	0
$517$	−48.0258	−2.11217
$518$	9.88842	0.434472
$519$	0	0
$520$	0	0
$521$	42.3626	1.85594	0.927970	−	0.372654i	$-0.121552\pi$
$521$	42.3626	1.85594	0.927970	+	0.372654i	$0.121552\pi$
$522$	0	0
$523$	15.9612	0.697936	0.348968	−	0.937135i	$-0.386532\pi$
$523$	15.9612	0.697936	0.348968	+	0.937135i	$0.386532\pi$
$524$	2.19728	0.0959887
$525$	0	0
$526$	13.9781	0.609474
$527$	49.5481	2.15835
$528$	0	0
$529$	14.7003	0.639144
$530$	0	0
$531$	0	0
$532$	−1.83176	−0.0794169
$533$	0.281818	0.0122069
$534$	0	0
$535$	0	0
$536$	21.5834	0.932261
$537$	0	0
$538$	21.5070	0.927234
$539$	77.6570	3.34492
$540$	0	0
$541$	−8.87271	−0.381467	−0.190734	−	0.981642i	$-0.561087\pi$
$541$	−8.87271	−0.381467	−0.190734	+	0.981642i	$0.561087\pi$
$542$	−15.1463	−0.650590
$543$	0	0
$544$	−5.68875	−0.243903
$545$	0	0
$546$	0	0
$547$	−33.9532	−1.45173	−0.725867	−	0.687835i	$-0.758561\pi$
$547$	−33.9532	−1.45173	−0.725867	+	0.687835i	$0.758561\pi$
$548$	1.92319	0.0821544
$549$	0	0
$550$	0	0
$551$	−8.66497	−0.369140
$552$	0	0
$553$	40.8323	1.73637
$554$	1.86787	0.0793583
$555$	0	0
$556$	−4.04451	−0.171526
$557$	19.1218	0.810215	0.405108	−	0.914269i	$-0.367234\pi$
$557$	19.1218	0.810215	0.405108	+	0.914269i	$0.367234\pi$
$558$	0	0
$559$	0.255114	0.0107902
$560$	0	0
$561$	0	0
$562$	30.4753	1.28552
$563$	35.6665	1.50316	0.751581	−	0.659640i	$-0.229291\pi$
$563$	35.6665	1.50316	0.751581	+	0.659640i	$0.229291\pi$
$564$	0	0
$565$	0	0
$566$	−21.9383	−0.922137
$567$	0	0
$568$	39.7297	1.66702
$569$	19.7413	0.827598	0.413799	−	0.910368i	$-0.364202\pi$
$569$	19.7413	0.827598	0.413799	+	0.910368i	$0.364202\pi$
$570$	0	0
$571$	−5.42108	−0.226865	−0.113433	−	0.993546i	$-0.536185\pi$
$571$	−5.42108	−0.226865	−0.113433	+	0.993546i	$0.536185\pi$
$572$	−0.0985465	−0.00412044
$573$	0	0
$574$	18.5950	0.776139
$575$	0	0
$576$	0	0
$577$	9.55239	0.397671	0.198836	−	0.980033i	$-0.436284\pi$
$577$	9.55239	0.397671	0.198836	+	0.980033i	$0.436284\pi$
$578$	25.0755	1.04300
$579$	0	0
$580$	0	0
$581$	−56.6882	−2.35182
$582$	0	0
$583$	−35.5274	−1.47139
$584$	29.4152	1.21721
$585$	0	0
$586$	−27.7143	−1.14487
$587$	−13.2510	−0.546928	−0.273464	−	0.961882i	$-0.588169\pi$
$587$	−13.2510	−0.546928	−0.273464	+	0.961882i	$0.588169\pi$
$588$	0	0
$589$	19.6762	0.810743
$590$	0	0
$591$	0	0
$592$	6.32925	0.260131
$593$	−11.9367	−0.490181	−0.245091	−	0.969500i	$-0.578818\pi$
$593$	−11.9367	−0.490181	−0.245091	+	0.969500i	$0.578818\pi$
$594$	0	0
$595$	0	0
$596$	−4.02216	−0.164754
$597$	0	0
$598$	0.924944	0.0378238
$599$	23.8683	0.975234	0.487617	−	0.873058i	$-0.337866\pi$
$599$	23.8683	0.975234	0.487617	+	0.873058i	$0.337866\pi$
$600$	0	0
$601$	−24.9569	−1.01801	−0.509007	−	0.860762i	$-0.669987\pi$
$601$	−24.9569	−1.01801	−0.509007	+	0.860762i	$0.669987\pi$
$602$	16.8329	0.686060
$603$	0	0
$604$	−0.718317	−0.0292279
$605$	0	0
$606$	0	0
$607$	14.9667	0.607479	0.303739	−	0.952755i	$-0.401765\pi$
$607$	14.9667	0.607479	0.303739	+	0.952755i	$0.401765\pi$
$608$	−2.25908	−0.0916177
$609$	0	0
$610$	0	0
$611$	0.880025	0.0356020
$612$	0	0
$613$	−3.24907	−0.131229	−0.0656143	−	0.997845i	$-0.520901\pi$
$613$	−3.24907	−0.131229	−0.0656143	+	0.997845i	$0.520901\pi$
$614$	−30.4565	−1.22913
$615$	0	0
$616$	68.7084	2.76834
$617$	12.2787	0.494321	0.247160	−	0.968975i	$-0.420503\pi$
$617$	12.2787	0.494321	0.247160	+	0.968975i	$0.420503\pi$
$618$	0	0
$619$	−3.00525	−0.120791	−0.0603955	−	0.998175i	$-0.519236\pi$
$619$	−3.00525	−0.120791	−0.0603955	+	0.998175i	$0.519236\pi$
$620$	0	0
$621$	0	0
$622$	−21.2829	−0.853368
$623$	−23.7372	−0.951009
$624$	0	0
$625$	0	0
$626$	12.5454	0.501413
$627$	0	0
$628$	0.261568	0.0104377
$629$	−8.55241	−0.341007
$630$	0	0
$631$	2.22160	0.0884405	0.0442203	−	0.999022i	$-0.485920\pi$
$631$	2.22160	0.0884405	0.0442203	+	0.999022i	$0.485920\pi$
$632$	24.0413	0.956310
$633$	0	0
$634$	−13.2170	−0.524913
$635$	0	0
$636$	0	0
$637$	−1.42299	−0.0563809
$638$	−30.7585	−1.21774
$639$	0	0
$640$	0	0
$641$	−23.9614	−0.946420	−0.473210	−	0.880950i	$-0.656905\pi$
$641$	−23.9614	−0.946420	−0.473210	+	0.880950i	$0.656905\pi$
$642$	0	0
$643$	−36.6154	−1.44397	−0.721986	−	0.691908i	$-0.756770\pi$
$643$	−36.6154	−1.44397	−0.721986	+	0.691908i	$0.756770\pi$
$644$	4.85644	0.191370
$645$	0	0
$646$	19.9092	0.783317
$647$	−11.9069	−0.468110	−0.234055	−	0.972223i	$-0.575200\pi$
$647$	−11.9069	−0.468110	−0.234055	+	0.972223i	$0.575200\pi$
$648$	0	0
$649$	−38.6211	−1.51601
$650$	0	0
$651$	0	0
$652$	2.33951	0.0916223
$653$	28.7997	1.12702	0.563511	−	0.826109i	$-0.309450\pi$
$653$	28.7997	1.12702	0.563511	+	0.826109i	$0.309450\pi$
$654$	0	0
$655$	0	0
$656$	11.9020	0.464696
$657$	0	0
$658$	58.0659	2.26364
$659$	40.4340	1.57509	0.787543	−	0.616260i	$-0.211353\pi$
$659$	40.4340	1.57509	0.787543	+	0.616260i	$0.211353\pi$
$660$	0	0
$661$	−30.1350	−1.17212	−0.586059	−	0.810269i	$-0.699321\pi$
$661$	−30.1350	−1.17212	−0.586059	+	0.810269i	$0.699321\pi$
$662$	21.9949	0.854857
$663$	0	0
$664$	−33.3769	−1.29528
$665$	0	0
$666$	0	0
$667$	22.9729	0.889514
$668$	1.31086	0.0507187
$669$	0	0
$670$	0	0
$671$	24.3873	0.941462
$672$	0	0
$673$	−2.88826	−0.111334	−0.0556672	−	0.998449i	$-0.517729\pi$
$673$	−2.88826	−0.111334	−0.0556672	+	0.998449i	$0.517729\pi$
$674$	−31.7878	−1.22442
$675$	0	0
$676$	−2.24601	−0.0863851
$677$	13.2556	0.509453	0.254727	−	0.967013i	$-0.418015\pi$
$677$	13.2556	0.509453	0.254727	+	0.967013i	$0.418015\pi$
$678$	0	0
$679$	−31.1861	−1.19681
$680$	0	0
$681$	0	0
$682$	69.8457	2.67453
$683$	−35.5937	−1.36196	−0.680978	−	0.732304i	$-0.738445\pi$
$683$	−35.5937	−1.36196	−0.680978	+	0.732304i	$0.738445\pi$
$684$	0	0
$685$	0	0
$686$	−46.6913	−1.78268
$687$	0	0
$688$	10.7742	0.410763
$689$	0.651005	0.0248013
$690$	0	0
$691$	−32.1060	−1.22137	−0.610685	−	0.791874i	$-0.709106\pi$
$691$	−32.1060	−1.22137	−0.610685	+	0.791874i	$0.709106\pi$
$692$	−1.87768	−0.0713785
$693$	0	0
$694$	−13.6816	−0.519348
$695$	0	0
$696$	0	0
$697$	−16.0826	−0.609172
$698$	46.6921	1.76732
$699$	0	0
$700$	0	0
$701$	32.6062	1.23152	0.615760	−	0.787933i	$-0.288849\pi$
$701$	32.6062	1.23152	0.615760	+	0.787933i	$0.288849\pi$
$702$	0	0
$703$	−3.39627	−0.128093
$704$	40.1207	1.51211
$705$	0	0
$706$	−15.6216	−0.587928
$707$	−0.233089	−0.00876623
$708$	0	0
$709$	−19.8281	−0.744661	−0.372331	−	0.928100i	$-0.621441\pi$
$709$	−19.8281	−0.744661	−0.372331	+	0.928100i	$0.621441\pi$
$710$	0	0
$711$	0	0
$712$	−13.9760	−0.523772
$713$	−52.1663	−1.95364
$714$	0	0
$715$	0	0
$716$	−2.72507	−0.101841
$717$	0	0
$718$	41.3830	1.54440
$719$	16.3835	0.611003	0.305501	−	0.952192i	$-0.401176\pi$
$719$	16.3835	0.611003	0.305501	+	0.952192i	$0.401176\pi$
$720$	0	0
$721$	61.9194	2.30600
$722$	−20.1013	−0.748093
$723$	0	0
$724$	−3.19334	−0.118680
$725$	0	0
$726$	0	0
$727$	−36.7604	−1.36337	−0.681685	−	0.731646i	$-0.738752\pi$
$727$	−36.7604	−1.36337	−0.681685	+	0.731646i	$0.738752\pi$
$728$	−1.25901	−0.0466622
$729$	0	0
$730$	0	0
$731$	−14.5587	−0.538471
$732$	0	0
$733$	−21.3173	−0.787372	−0.393686	−	0.919245i	$-0.628800\pi$
$733$	−21.3173	−0.787372	−0.393686	+	0.919245i	$0.628800\pi$
$734$	30.6135	1.12996
$735$	0	0
$736$	5.98935	0.220770
$737$	−44.6931	−1.64629
$738$	0	0
$739$	−11.4938	−0.422805	−0.211403	−	0.977399i	$-0.567803\pi$
$739$	−11.4938	−0.422805	−0.211403	+	0.977399i	$0.567803\pi$
$740$	0	0
$741$	0	0
$742$	42.9547	1.57692
$743$	−43.7222	−1.60401	−0.802006	−	0.597316i	$-0.796234\pi$
$743$	−43.7222	−1.60401	−0.802006	+	0.597316i	$0.796234\pi$
$744$	0	0
$745$	0	0
$746$	51.7817	1.89586
$747$	0	0
$748$	5.62379	0.205626
$749$	12.2583	0.447909
$750$	0	0
$751$	−18.5913	−0.678406	−0.339203	−	0.940713i	$-0.610157\pi$
$751$	−18.5913	−0.678406	−0.339203	+	0.940713i	$0.610157\pi$
$752$	37.1661	1.35531
$753$	0	0
$754$	0.563620	0.0205258
$755$	0	0
$756$	0	0
$757$	7.64333	0.277802	0.138901	−	0.990306i	$-0.455643\pi$
$757$	7.64333	0.277802	0.138901	+	0.990306i	$0.455643\pi$
$758$	−43.7746	−1.58996
$759$	0	0
$760$	0	0
$761$	−44.8988	−1.62758	−0.813789	−	0.581160i	$-0.802599\pi$
$761$	−44.8988	−1.62758	−0.813789	+	0.581160i	$0.802599\pi$
$762$	0	0
$763$	−1.87080	−0.0677274
$764$	−2.68070	−0.0969842
$765$	0	0
$766$	49.8361	1.80065
$767$	0.707694	0.0255533
$768$	0	0
$769$	−29.9521	−1.08010	−0.540051	−	0.841632i	$-0.681595\pi$
$769$	−29.9521	−1.08010	−0.540051	+	0.841632i	$0.681595\pi$
$770$	0	0
$771$	0	0
$772$	−1.30685	−0.0470347
$773$	−36.8137	−1.32410	−0.662049	−	0.749461i	$-0.730313\pi$
$773$	−36.8137	−1.32410	−0.662049	+	0.749461i	$0.730313\pi$
$774$	0	0
$775$	0	0
$776$	−18.3617	−0.659148
$777$	0	0
$778$	−9.02436	−0.323539
$779$	−6.38662	−0.228824
$780$	0	0
$781$	−82.2690	−2.94381
$782$	−52.7841	−1.88755
$783$	0	0
$784$	−60.0970	−2.14632
$785$	0	0
$786$	0	0
$787$	−35.8705	−1.27865	−0.639323	−	0.768938i	$-0.720785\pi$
$787$	−35.8705	−1.27865	−0.639323	+	0.768938i	$0.720785\pi$
$788$	−0.152080	−0.00541763
$789$	0	0
$790$	0	0
$791$	33.3583	1.18608
$792$	0	0
$793$	−0.446874	−0.0158689
$794$	49.6111	1.76063
$795$	0	0
$796$	2.36426	0.0837990
$797$	−45.2506	−1.60286	−0.801430	−	0.598089i	$-0.795927\pi$
$797$	−45.2506	−1.60286	−0.801430	+	0.598089i	$0.795927\pi$
$798$	0	0
$799$	−50.2207	−1.77668
$800$	0	0
$801$	0	0
$802$	−16.0729	−0.567554
$803$	−60.9106	−2.14949
$804$	0	0
$805$	0	0
$806$	−1.27985	−0.0450809
$807$	0	0
$808$	−0.137239	−0.00482803
$809$	19.4022	0.682146	0.341073	−	0.940037i	$-0.389210\pi$
$809$	19.4022	0.682146	0.341073	+	0.940037i	$0.389210\pi$
$810$	0	0
$811$	−13.2251	−0.464396	−0.232198	−	0.972669i	$-0.574592\pi$
$811$	−13.2251	−0.464396	−0.232198	+	0.972669i	$0.574592\pi$
$812$	2.95930	0.103851
$813$	0	0
$814$	−12.0559	−0.422560
$815$	0	0
$816$	0	0
$817$	−5.78143	−0.202267
$818$	40.9055	1.43023
$819$	0	0
$820$	0	0
$821$	−22.4781	−0.784493	−0.392246	−	0.919860i	$-0.628302\pi$
$821$	−22.4781	−0.784493	−0.392246	+	0.919860i	$0.628302\pi$
$822$	0	0
$823$	13.9707	0.486987	0.243494	−	0.969902i	$-0.421707\pi$
$823$	13.9707	0.486987	0.243494	+	0.969902i	$0.421707\pi$
$824$	36.4569	1.27004
$825$	0	0
$826$	46.6951	1.62473
$827$	−14.5443	−0.505756	−0.252878	−	0.967498i	$-0.581377\pi$
$827$	−14.5443	−0.505756	−0.252878	+	0.967498i	$0.581377\pi$
$828$	0	0
$829$	39.2519	1.36327	0.681637	−	0.731690i	$-0.261268\pi$
$829$	39.2519	1.36327	0.681637	+	0.731690i	$0.261268\pi$
$830$	0	0
$831$	0	0
$832$	−0.735172	−0.0254875
$833$	81.2061	2.81362
$834$	0	0
$835$	0	0
$836$	2.23328	0.0772396
$837$	0	0
$838$	−21.6600	−0.748231
$839$	5.21412	0.180011	0.0900057	−	0.995941i	$-0.471311\pi$
$839$	5.21412	0.180011	0.0900057	+	0.995941i	$0.471311\pi$
$840$	0	0
$841$	−15.0013	−0.517287
$842$	25.5435	0.880288
$843$	0	0
$844$	2.80583	0.0965807
$845$	0	0
$846$	0	0
$847$	−91.9579	−3.15971
$848$	27.4938	0.944143
$849$	0	0
$850$	0	0
$851$	9.00433	0.308664
$852$	0	0
$853$	43.8238	1.50050	0.750249	−	0.661155i	$-0.229934\pi$
$853$	43.8238	1.50050	0.750249	+	0.661155i	$0.229934\pi$
$854$	−29.4857	−1.00898
$855$	0	0
$856$	7.21746	0.246688
$857$	26.5704	0.907628	0.453814	−	0.891096i	$-0.350063\pi$
$857$	26.5704	0.907628	0.453814	+	0.891096i	$0.350063\pi$
$858$	0	0
$859$	−1.57966	−0.0538973	−0.0269487	−	0.999637i	$-0.508579\pi$
$859$	−1.57966	−0.0538973	−0.0269487	+	0.999637i	$0.508579\pi$
$860$	0	0
$861$	0	0
$862$	12.1952	0.415370
$863$	−12.7587	−0.434311	−0.217155	−	0.976137i	$-0.569678\pi$
$863$	−12.7587	−0.434311	−0.217155	+	0.976137i	$0.569678\pi$
$864$	0	0
$865$	0	0
$866$	37.4876	1.27388
$867$	0	0
$868$	−6.71990	−0.228088
$869$	−49.7826	−1.68876
$870$	0	0
$871$	0.818957	0.0277493
$872$	−1.10149	−0.0373011
$873$	0	0
$874$	−20.9612	−0.709024
$875$	0	0
$876$	0	0
$877$	11.2864	0.381115	0.190558	−	0.981676i	$-0.438970\pi$
$877$	11.2864	0.381115	0.190558	+	0.981676i	$0.438970\pi$
$878$	−2.07897	−0.0701619
$879$	0	0
$880$	0	0
$881$	47.0578	1.58542	0.792708	−	0.609601i	$-0.208670\pi$
$881$	47.0578	1.58542	0.792708	+	0.609601i	$0.208670\pi$
$882$	0	0
$883$	18.8049	0.632834	0.316417	−	0.948620i	$-0.397520\pi$
$883$	18.8049	0.632834	0.316417	+	0.948620i	$0.397520\pi$
$884$	−0.103050	−0.00346596
$885$	0	0
$886$	−2.48993	−0.0836509
$887$	−44.6699	−1.49987	−0.749933	−	0.661513i	$-0.769915\pi$
$887$	−44.6699	−1.49987	−0.749933	+	0.661513i	$0.769915\pi$
$888$	0	0
$889$	49.1055	1.64694
$890$	0	0
$891$	0	0
$892$	−1.20492	−0.0403437
$893$	−19.9433	−0.667377
$894$	0	0
$895$	0	0
$896$	−57.4323	−1.91868
$897$	0	0
$898$	−31.2847	−1.04398
$899$	−31.7878	−1.06018
$900$	0	0
$901$	−37.1511	−1.23768
$902$	−22.6709	−0.754859
$903$	0	0
$904$	19.6407	0.653241
$905$	0	0
$906$	0	0
$907$	−10.3156	−0.342525	−0.171262	−	0.985225i	$-0.554785\pi$
$907$	−10.3156	−0.342525	−0.171262	+	0.985225i	$0.554785\pi$
$908$	2.91859	0.0968569
$909$	0	0
$910$	0	0
$911$	31.3633	1.03911	0.519557	−	0.854436i	$-0.326097\pi$
$911$	31.3633	1.03911	0.519557	+	0.854436i	$0.326097\pi$
$912$	0	0
$913$	69.1141	2.28734
$914$	−40.0262	−1.32395
$915$	0	0
$916$	−0.805088	−0.0266009
$917$	−58.1294	−1.91960
$918$	0	0
$919$	1.17198	0.0386599	0.0193300	−	0.999813i	$-0.493847\pi$
$919$	1.17198	0.0386599	0.0193300	+	0.999813i	$0.493847\pi$
$920$	0	0
$921$	0	0
$922$	−37.4277	−1.23262
$923$	1.50750	0.0496199
$924$	0	0
$925$	0	0
$926$	−32.7648	−1.07672
$927$	0	0
$928$	3.64965	0.119806
$929$	2.81931	0.0924985	0.0462493	−	0.998930i	$-0.485273\pi$
$929$	2.81931	0.0924985	0.0462493	+	0.998930i	$0.485273\pi$
$930$	0	0
$931$	32.2480	1.05689
$932$	−1.11496	−0.0365216
$933$	0	0
$934$	−34.2410	−1.12040
$935$	0	0
$936$	0	0
$937$	−24.3041	−0.793982	−0.396991	−	0.917823i	$-0.629946\pi$
$937$	−24.3041	−0.793982	−0.396991	+	0.917823i	$0.629946\pi$
$938$	54.0365	1.76435
$939$	0	0
$940$	0	0
$941$	−17.2668	−0.562881	−0.281440	−	0.959579i	$-0.590812\pi$
$941$	−17.2668	−0.562881	−0.281440	+	0.959579i	$0.590812\pi$
$942$	0	0
$943$	16.9324	0.551396
$944$	29.8880	0.972771
$945$	0	0
$946$	−20.5227	−0.667250
$947$	26.2414	0.852729	0.426365	−	0.904551i	$-0.359794\pi$
$947$	26.2414	0.852729	0.426365	+	0.904551i	$0.359794\pi$
$948$	0	0
$949$	1.11613	0.0362310
$950$	0	0
$951$	0	0
$952$	71.8485	2.32862
$953$	−37.4166	−1.21204	−0.606021	−	0.795449i	$-0.707235\pi$
$953$	−37.4166	−1.21204	−0.606021	+	0.795449i	$0.707235\pi$
$954$	0	0
$955$	0	0
$956$	−3.73267	−0.120723
$957$	0	0
$958$	15.1864	0.490649
$959$	−50.8781	−1.64294
$960$	0	0
$961$	41.1830	1.32848
$962$	0.220913	0.00712253
$963$	0	0
$964$	−0.103239	−0.00332509
$965$	0	0
$966$	0	0
$967$	14.2911	0.459572	0.229786	−	0.973241i	$-0.426197\pi$
$967$	14.2911	0.459572	0.229786	+	0.973241i	$0.426197\pi$
$968$	−54.1431	−1.74022
$969$	0	0
$970$	0	0
$971$	9.60912	0.308371	0.154186	−	0.988042i	$-0.450725\pi$
$971$	9.60912	0.308371	0.154186	+	0.988042i	$0.450725\pi$
$972$	0	0
$973$	106.998	3.43020
$974$	−1.86925	−0.0598946
$975$	0	0
$976$	−18.8728	−0.604103
$977$	22.8916	0.732368	0.366184	−	0.930542i	$-0.380664\pi$
$977$	22.8916	0.732368	0.366184	+	0.930542i	$0.380664\pi$
$978$	0	0
$979$	28.9403	0.924935
$980$	0	0
$981$	0	0
$982$	28.6945	0.915678
$983$	17.6300	0.562311	0.281155	−	0.959662i	$-0.409282\pi$
$983$	17.6300	0.562311	0.281155	+	0.959662i	$0.409282\pi$
$984$	0	0
$985$	0	0
$986$	−32.1643	−1.02432
$987$	0	0
$988$	−0.0409226	−0.00130192
$989$	15.3280	0.487401
$990$	0	0
$991$	−3.26738	−0.103792	−0.0518959	−	0.998652i	$-0.516526\pi$
$991$	−3.26738	−0.103792	−0.0518959	+	0.998652i	$0.516526\pi$
$992$	−8.28752	−0.263129
$993$	0	0
$994$	99.4678	3.15493
$995$	0	0
$996$	0	0
$997$	33.7052	1.06745	0.533727	−	0.845657i	$-0.320791\pi$
$997$	33.7052	1.06745	0.533727	+	0.845657i	$0.320791\pi$
$998$	−30.0865	−0.952371
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5625.2.a.y.1.5	yes	8
3.2	odd	2	inner	5625.2.a.y.1.4	✓	8
5.4	even	2		5625.2.a.ba.1.4	yes	8
15.14	odd	2		5625.2.a.ba.1.5	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5625.2.a.y.1.4	✓	8	3.2	odd	2	inner
5625.2.a.y.1.5	yes	8	1.1	even	1	trivial
5625.2.a.ba.1.4	yes	8	5.4	even	2
5625.2.a.ba.1.5	yes	8	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+1.47408 q^{2} +0.172909 q^{4} -4.57433 q^{7} -2.69328 q^{8} +O(q^{10})\)	\(q+1.47408 q^{2} +0.172909 q^{4} -4.57433 q^{7} -2.69328 q^{8} +5.57701 q^{11} -0.102193 q^{13} -6.74292 q^{14} -4.31592 q^{16} +5.83189 q^{17} +2.31592 q^{19} +8.22095 q^{22} -6.14006 q^{23} -0.150641 q^{26} -0.790943 q^{28} -3.74148 q^{29} +8.49606 q^{31} -0.975455 q^{32} +8.59667 q^{34} -1.46649 q^{37} +3.41385 q^{38} -2.75770 q^{41} -2.49639 q^{43} +0.964316 q^{44} -9.05093 q^{46} -8.61138 q^{47} +13.9245 q^{49} -0.0176701 q^{52} -6.37033 q^{53} +12.3199 q^{56} -5.51524 q^{58} -6.92506 q^{59} +4.37283 q^{61} +12.5239 q^{62} +7.19394 q^{64} -8.01381 q^{67} +1.00839 q^{68} -14.7514 q^{71} -10.9217 q^{73} -2.16172 q^{74} +0.400444 q^{76} -25.5111 q^{77} -8.92640 q^{79} -4.06507 q^{82} +12.3927 q^{83} -3.67987 q^{86} -15.0204 q^{88} +5.18921 q^{89} +0.467465 q^{91} -1.06167 q^{92} -12.6939 q^{94} +6.81762 q^{97} +20.5258 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 14 q^{4} - 10 q^{7}+O(q^{10}) \) 8 * q + 14 * q^4 - 10 * q^7	\( 8 q + 14 q^{4} - 10 q^{7} - 10 q^{13} - 18 q^{16} + 2 q^{19} - 20 q^{22} - 10 q^{28} + 4 q^{31} + 50 q^{34} - 50 q^{43} - 30 q^{46} - 6 q^{49} - 30 q^{52} - 60 q^{58} + 46 q^{61} - 14 q^{64} - 40 q^{67} - 50 q^{73} - 34 q^{76} - 12 q^{79} - 60 q^{82} - 70 q^{88} - 10 q^{91} - 20 q^{94} - 50 q^{97}+O(q^{100}) \) 8 * q + 14 * q^4 - 10 * q^7 - 10 * q^13 - 18 * q^16 + 2 * q^19 - 20 * q^22 - 10 * q^28 + 4 * q^31 + 50 * q^34 - 50 * q^43 - 30 * q^46 - 6 * q^49 - 30 * q^52 - 60 * q^58 + 46 * q^61 - 14 * q^64 - 40 * q^67 - 50 * q^73 - 34 * q^76 - 12 * q^79 - 60 * q^82 - 70 * q^88 - 10 * q^91 - 20 * q^94 - 50 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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