LMFDB - Embedded newform 5625.2.a.x.1.1

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:	8.8.14884000000.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 11x^{6} + 36x^{4} - 31x^{2} + 1 $ x^8 - 11x^6 + 36x^4 - 31*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 25)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.30927$ 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)

$f(q)$	$=$	$q-2.30927 q^{2} +3.33275 q^{4} +3.03582 q^{7} -3.07768 q^{8} +O(q^{10})$	$q-2.30927 q^{2} +3.33275 q^{4} +3.03582 q^{7} -3.07768 q^{8} -2.00000 q^{11} -1.42721 q^{13} -7.01054 q^{14} +0.441718 q^{16} -1.86025 q^{17} +0.903319 q^{19} +4.61855 q^{22} +3.32932 q^{23} +3.29582 q^{26} +10.1176 q^{28} -3.96307 q^{29} -6.43997 q^{31} +5.13532 q^{32} +4.29582 q^{34} +3.82022 q^{37} -2.08601 q^{38} +1.83422 q^{41} -3.59445 q^{43} -6.66550 q^{44} -7.68832 q^{46} +4.79995 q^{47} +2.21619 q^{49} -4.75653 q^{52} +9.50473 q^{53} -9.34328 q^{56} +9.15182 q^{58} -10.6456 q^{59} +14.2742 q^{61} +14.8716 q^{62} -12.7423 q^{64} -10.6902 q^{67} -6.19974 q^{68} -12.4598 q^{71} +0.267631 q^{73} -8.82193 q^{74} +3.01054 q^{76} -6.07163 q^{77} -8.57176 q^{79} -4.23572 q^{82} +12.6182 q^{83} +8.30058 q^{86} +6.15537 q^{88} +4.76796 q^{89} -4.33275 q^{91} +11.0958 q^{92} -11.0844 q^{94} -9.95805 q^{97} -5.11778 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 6 q^{4}+O(q^{10}) $ 8 * q + 6 * q^4	$ 8 q + 6 q^{4} - 16 q^{11} - 12 q^{14} - 2 q^{16} + 10 q^{19} - 6 q^{26} - 20 q^{29} + 16 q^{31} + 2 q^{34} - 26 q^{41} - 12 q^{44} + 6 q^{46} - 14 q^{49} - 10 q^{56} - 30 q^{59} + 6 q^{61} - 44 q^{64} - 46 q^{71} - 12 q^{74} - 20 q^{76} + 10 q^{79} + 14 q^{86} - 30 q^{89} - 14 q^{91} - 68 q^{94}+O(q^{100}) $ 8 * q + 6 * q^4 - 16 * q^11 - 12 * q^14 - 2 * q^16 + 10 * q^19 - 6 * q^26 - 20 * q^29 + 16 * q^31 + 2 * q^34 - 26 * q^41 - 12 * q^44 + 6 * q^46 - 14 * q^49 - 10 * q^56 - 30 * q^59 + 6 * q^61 - 44 * q^64 - 46 * q^71 - 12 * q^74 - 20 * q^76 + 10 * q^79 + 14 * q^86 - 30 * q^89 - 14 * q^91 - 68 * q^94

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 $

$2$	−2.30927	−1.63290	−0.816452	−	0.577414i	$-0.804062\pi$
$2$	−2.30927	−1.63290	−0.816452	+	0.577414i	$0.804062\pi$
$3$	0	0
$3$	0	0
$4$	3.33275	1.66637
$5$	0	0
$5$	0	0
$6$	0	0
$7$	3.03582	1.14743	0.573716	−	0.819055i	$-0.305502\pi$
$7$	3.03582	1.14743	0.573716	+	0.819055i	$0.305502\pi$
$8$	−3.07768	−1.08813
$9$	0	0
$10$	0	0
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	0	0
$13$	−1.42721	−0.395837	−0.197918	−	0.980218i	$-0.563418\pi$
$13$	−1.42721	−0.395837	−0.197918	+	0.980218i	$0.563418\pi$
$14$	−7.01054	−1.87364
$15$	0	0
$16$	0.441718	0.110430
$17$	−1.86025	−0.451176	−0.225588	−	0.974223i	$-0.572430\pi$
$17$	−1.86025	−0.451176	−0.225588	+	0.974223i	$0.572430\pi$
$18$	0	0
$19$	0.903319	0.207236	0.103618	−	0.994617i	$-0.466958\pi$
$19$	0.903319	0.207236	0.103618	+	0.994617i	$0.466958\pi$
$20$	0	0
$21$	0	0
$22$	4.61855	0.984678
$23$	3.32932	0.694212	0.347106	−	0.937826i	$-0.387164\pi$
$23$	3.32932	0.694212	0.347106	+	0.937826i	$0.387164\pi$
$24$	0	0
$25$	0	0
$26$	3.29582	0.646364
$27$	0	0
$28$	10.1176	1.91205
$29$	−3.96307	−0.735924	−0.367962	−	0.929841i	$-0.619944\pi$
$29$	−3.96307	−0.735924	−0.367962	+	0.929841i	$0.619944\pi$
$30$	0	0
$31$	−6.43997	−1.15665	−0.578326	−	0.815806i	$-0.696294\pi$
$31$	−6.43997	−1.15665	−0.578326	+	0.815806i	$0.696294\pi$
$32$	5.13532	0.907805
$33$	0	0
$34$	4.29582	0.736727
$35$	0	0
$36$	0	0
$37$	3.82022	0.628040	0.314020	−	0.949416i	$-0.398324\pi$
$37$	3.82022	0.628040	0.314020	+	0.949416i	$0.398324\pi$
$38$	−2.08601	−0.338396
$39$	0	0
$40$	0	0
$41$	1.83422	0.286457	0.143228	−	0.989690i	$-0.454252\pi$
$41$	1.83422	0.286457	0.143228	+	0.989690i	$0.454252\pi$
$42$	0	0
$43$	−3.59445	−0.548149	−0.274074	−	0.961708i	$-0.588371\pi$
$43$	−3.59445	−0.548149	−0.274074	+	0.961708i	$0.588371\pi$
$44$	−6.66550	−1.00486
$45$	0	0
$46$	−7.68832	−1.13358
$47$	4.79995	0.700144	0.350072	−	0.936723i	$-0.386157\pi$
$47$	4.79995	0.700144	0.350072	+	0.936723i	$0.386157\pi$
$48$	0	0
$49$	2.21619	0.316598
$50$	0	0
$51$	0	0
$52$	−4.75653	−0.659613
$53$	9.50473	1.30558	0.652788	−	0.757541i	$-0.273599\pi$
$53$	9.50473	1.30558	0.652788	+	0.757541i	$0.273599\pi$
$54$	0	0
$55$	0	0
$56$	−9.34328	−1.24855
$57$	0	0
$58$	9.15182	1.20169
$59$	−10.6456	−1.38594	−0.692971	−	0.720966i	$-0.743698\pi$
$59$	−10.6456	−1.38594	−0.692971	+	0.720966i	$0.743698\pi$
$60$	0	0
$61$	14.2742	1.82762	0.913811	−	0.406140i	$-0.133125\pi$
$61$	14.2742	1.82762	0.913811	+	0.406140i	$0.133125\pi$
$62$	14.8716	1.88870
$63$	0	0
$64$	−12.7423	−1.59279
$65$	0	0
$66$	0	0
$67$	−10.6902	−1.30601	−0.653007	−	0.757352i	$-0.726493\pi$
$67$	−10.6902	−1.30601	−0.653007	+	0.757352i	$0.726493\pi$
$68$	−6.19974	−0.751828
$69$	0	0
$70$	0	0
$71$	−12.4598	−1.47871	−0.739356	−	0.673315i	$-0.764870\pi$
$71$	−12.4598	−1.47871	−0.739356	+	0.673315i	$0.764870\pi$
$72$	0	0
$73$	0.267631	0.0313239	0.0156619	−	0.999877i	$-0.495014\pi$
$73$	0.267631	0.0313239	0.0156619	+	0.999877i	$0.495014\pi$
$74$	−8.82193	−1.02553
$75$	0	0
$76$	3.01054	0.345332
$77$	−6.07163	−0.691927
$78$	0	0
$79$	−8.57176	−0.964398	−0.482199	−	0.876062i	$-0.660162\pi$
$79$	−8.57176	−0.964398	−0.482199	+	0.876062i	$0.660162\pi$
$80$	0	0
$81$	0	0
$82$	−4.23572	−0.467757
$83$	12.6182	1.38502	0.692512	−	0.721406i	$-0.256504\pi$
$83$	12.6182	1.38502	0.692512	+	0.721406i	$0.256504\pi$
$84$	0	0
$85$	0	0
$86$	8.30058	0.895074
$87$	0	0
$88$	6.15537	0.656164
$89$	4.76796	0.505402	0.252701	−	0.967544i	$-0.418681\pi$
$89$	4.76796	0.505402	0.252701	+	0.967544i	$0.418681\pi$
$90$	0	0
$91$	−4.33275	−0.454196
$92$	11.0958	1.15682
$93$	0	0
$94$	−11.0844	−1.14327
$95$	0	0
$96$	0	0
$97$	−9.95805	−1.01109	−0.505543	−	0.862801i	$-0.668708\pi$
$97$	−9.95805	−1.01109	−0.505543	+	0.862801i	$0.668708\pi$
$98$	−5.11778	−0.516974
$99$	0	0
$100$	0	0
$101$	−9.34612	−0.929974	−0.464987	−	0.885318i	$-0.653941\pi$
$101$	−9.34612	−0.929974	−0.464987	+	0.885318i	$0.653941\pi$
$102$	0	0
$103$	9.08408	0.895081	0.447540	−	0.894264i	$-0.352300\pi$
$103$	9.08408	0.895081	0.447540	+	0.894264i	$0.352300\pi$
$104$	4.39250	0.430720
$105$	0	0
$106$	−21.9490	−2.13188
$107$	−5.62871	−0.544148	−0.272074	−	0.962276i	$-0.587710\pi$
$107$	−5.62871	−0.544148	−0.272074	+	0.962276i	$0.587710\pi$
$108$	0	0
$109$	−10.1130	−0.968649	−0.484325	−	0.874888i	$-0.660935\pi$
$109$	−10.1130	−0.968649	−0.484325	+	0.874888i	$0.660935\pi$
$110$	0	0
$111$	0	0
$112$	1.34098	0.126710
$113$	−10.7120	−1.00770	−0.503851	−	0.863791i	$-0.668084\pi$
$113$	−10.7120	−1.00770	−0.503851	+	0.863791i	$0.668084\pi$
$114$	0	0
$115$	0	0
$116$	−13.2079	−1.22632
$117$	0	0
$118$	24.5836	2.26311
$119$	−5.64737	−0.517693
$120$	0	0
$121$	−7.00000	−0.636364
$122$	−32.9630	−2.98433
$123$	0	0
$124$	−21.4628	−1.92742
$125$	0	0
$126$	0	0
$127$	−11.3609	−1.00812	−0.504060	−	0.863669i	$-0.668161\pi$
$127$	−11.3609	−1.00812	−0.504060	+	0.863669i	$0.668161\pi$
$128$	19.1548	1.69306
$129$	0	0
$130$	0	0
$131$	−7.98771	−0.697890	−0.348945	−	0.937143i	$-0.613460\pi$
$131$	−7.98771	−0.697890	−0.348945	+	0.937143i	$0.613460\pi$
$132$	0	0
$133$	2.74231	0.237789
$134$	24.6866	2.13259
$135$	0	0
$136$	5.72525	0.490936
$137$	−9.33726	−0.797736	−0.398868	−	0.917008i	$-0.630597\pi$
$137$	−9.33726	−0.797736	−0.398868	+	0.917008i	$0.630597\pi$
$138$	0	0
$139$	17.9150	1.51953	0.759767	−	0.650195i	$-0.225313\pi$
$139$	17.9150	1.51953	0.759767	+	0.650195i	$0.225313\pi$
$140$	0	0
$141$	0	0
$142$	28.7732	2.41459
$143$	2.85442	0.238699
$144$	0	0
$145$	0	0
$146$	−0.618034	−0.0511489
$147$	0	0
$148$	12.7318	1.04655
$149$	6.31395	0.517259	0.258629	−	0.965977i	$-0.416729\pi$
$149$	6.31395	0.517259	0.258629	+	0.965977i	$0.416729\pi$
$150$	0	0
$151$	4.71947	0.384065	0.192033	−	0.981389i	$-0.438492\pi$
$151$	4.71947	0.384065	0.192033	+	0.981389i	$0.438492\pi$
$152$	−2.78013	−0.225498
$153$	0	0
$154$	14.0211	1.12985
$155$	0	0
$156$	0	0
$157$	1.46908	0.117245	0.0586225	−	0.998280i	$-0.481329\pi$
$157$	1.46908	0.117245	0.0586225	+	0.998280i	$0.481329\pi$
$158$	19.7945	1.57477
$159$	0	0
$160$	0	0
$161$	10.1072	0.796560
$162$	0	0
$163$	4.45969	0.349310	0.174655	−	0.984630i	$-0.444119\pi$
$163$	4.45969	0.349310	0.174655	+	0.984630i	$0.444119\pi$
$164$	6.11299	0.477345
$165$	0	0
$166$	−29.1388	−2.26161
$167$	−10.4337	−0.807381	−0.403691	−	0.914896i	$-0.632273\pi$
$167$	−10.4337	−0.807381	−0.403691	+	0.914896i	$0.632273\pi$
$168$	0	0
$169$	−10.9631	−0.843313
$170$	0	0
$171$	0	0
$172$	−11.9794	−0.913421
$173$	7.67619	0.583610	0.291805	−	0.956478i	$-0.405744\pi$
$173$	7.67619	0.583610	0.291805	+	0.956478i	$0.405744\pi$
$174$	0	0
$175$	0	0
$176$	−0.883436	−0.0665915
$177$	0	0
$178$	−11.0105	−0.825273
$179$	−15.5168	−1.15978	−0.579889	−	0.814696i	$-0.696904\pi$
$179$	−15.5168	−1.15978	−0.579889	+	0.814696i	$0.696904\pi$
$180$	0	0
$181$	−1.59056	−0.118225	−0.0591126	−	0.998251i	$-0.518827\pi$
$181$	−1.59056	−0.118225	−0.0591126	+	0.998251i	$0.518827\pi$
$182$	10.0055	0.741658
$183$	0	0
$184$	−10.2466	−0.755390
$185$	0	0
$186$	0	0
$187$	3.72049	0.272069
$188$	15.9970	1.16670
$189$	0	0
$190$	0	0
$191$	−19.6684	−1.42316	−0.711579	−	0.702606i	$-0.752020\pi$
$191$	−19.6684	−1.42316	−0.711579	+	0.702606i	$0.752020\pi$
$192$	0	0
$193$	13.1100	0.943680	0.471840	−	0.881684i	$-0.343590\pi$
$193$	13.1100	0.943680	0.471840	+	0.881684i	$0.343590\pi$
$194$	22.9959	1.65101
$195$	0	0
$196$	7.38599	0.527571
$197$	3.42949	0.244341	0.122170	−	0.992509i	$-0.461015\pi$
$197$	3.42949	0.244341	0.122170	+	0.992509i	$0.461015\pi$
$198$	0	0
$199$	17.6959	1.25443	0.627215	−	0.778846i	$-0.284195\pi$
$199$	17.6959	1.25443	0.627215	+	0.778846i	$0.284195\pi$
$200$	0	0
$201$	0	0
$202$	21.5828	1.51856
$203$	−12.0312	−0.844422
$204$	0	0
$205$	0	0
$206$	−20.9776	−1.46158
$207$	0	0
$208$	−0.630425	−0.0437121
$209$	−1.80664	−0.124968
$210$	0	0
$211$	−3.24366	−0.223303	−0.111651	−	0.993747i	$-0.535614\pi$
$211$	−3.24366	−0.223303	−0.111651	+	0.993747i	$0.535614\pi$
$212$	31.6769	2.17558
$213$	0	0
$214$	12.9982	0.888542
$215$	0	0
$216$	0	0
$217$	−19.5506	−1.32718
$218$	23.3537	1.58171
$219$	0	0
$220$	0	0
$221$	2.65496	0.178592
$222$	0	0
$223$	28.7148	1.92288	0.961441	−	0.275010i	$-0.0886812\pi$
$223$	28.7148	1.92288	0.961441	+	0.275010i	$0.0886812\pi$
$224$	15.5899	1.04164
$225$	0	0
$226$	24.7370	1.64548
$227$	−11.7206	−0.777926	−0.388963	−	0.921253i	$-0.627167\pi$
$227$	−11.7206	−0.777926	−0.388963	+	0.921253i	$0.627167\pi$
$228$	0	0
$229$	−16.4013	−1.08383	−0.541914	−	0.840434i	$-0.682300\pi$
$229$	−16.4013	−1.08383	−0.541914	+	0.840434i	$0.682300\pi$
$230$	0	0
$231$	0	0
$232$	12.1971	0.800777
$233$	−22.5146	−1.47498	−0.737490	−	0.675358i	$-0.763989\pi$
$233$	−22.5146	−1.47498	−0.737490	+	0.675358i	$0.763989\pi$
$234$	0	0
$235$	0	0
$236$	−35.4792	−2.30950
$237$	0	0
$238$	13.0413	0.845344
$239$	6.63333	0.429074	0.214537	−	0.976716i	$-0.431176\pi$
$239$	6.63333	0.429074	0.214537	+	0.976716i	$0.431176\pi$
$240$	0	0
$241$	26.2261	1.68937	0.844684	−	0.535265i	$-0.179788\pi$
$241$	26.2261	1.68937	0.844684	+	0.535265i	$0.179788\pi$
$242$	16.1649	1.03912
$243$	0	0
$244$	47.5723	3.04550
$245$	0	0
$246$	0	0
$247$	−1.28923	−0.0820315
$248$	19.8202	1.25858
$249$	0	0
$250$	0	0
$251$	10.9121	0.688766	0.344383	−	0.938829i	$-0.388088\pi$
$251$	10.9121	0.688766	0.344383	+	0.938829i	$0.388088\pi$
$252$	0	0
$253$	−6.65865	−0.418626
$254$	26.2355	1.64616
$255$	0	0
$256$	−18.7492	−1.17182
$257$	−6.58051	−0.410481	−0.205240	−	0.978712i	$-0.565798\pi$
$257$	−6.58051	−0.410481	−0.205240	+	0.978712i	$0.565798\pi$
$258$	0	0
$259$	11.5975	0.720632
$260$	0	0
$261$	0	0
$262$	18.4458	1.13959
$263$	27.1073	1.67151	0.835753	−	0.549106i	$-0.185032\pi$
$263$	27.1073	1.67151	0.835753	+	0.549106i	$0.185032\pi$
$264$	0	0
$265$	0	0
$266$	−6.33275	−0.388286
$267$	0	0
$268$	−35.6277	−2.17631
$269$	−1.00945	−0.0615474	−0.0307737	−	0.999526i	$-0.509797\pi$
$269$	−1.00945	−0.0615474	−0.0307737	+	0.999526i	$0.509797\pi$
$270$	0	0
$271$	6.25203	0.379784	0.189892	−	0.981805i	$-0.439186\pi$
$271$	6.25203	0.379784	0.189892	+	0.981805i	$0.439186\pi$
$272$	−0.821705	−0.0498232
$273$	0	0
$274$	21.5623	1.30263
$275$	0	0
$276$	0	0
$277$	−24.6703	−1.48230	−0.741148	−	0.671342i	$-0.765718\pi$
$277$	−24.6703	−1.48230	−0.741148	+	0.671342i	$0.765718\pi$
$278$	−41.3708	−2.48125
$279$	0	0
$280$	0	0
$281$	−1.83891	−0.109700	−0.0548502	−	0.998495i	$-0.517468\pi$
$281$	−1.83891	−0.109700	−0.0548502	+	0.998495i	$0.517468\pi$
$282$	0	0
$283$	−8.64116	−0.513664	−0.256832	−	0.966456i	$-0.582679\pi$
$283$	−8.64116	−0.513664	−0.256832	+	0.966456i	$0.582679\pi$
$284$	−41.5255	−2.46409
$285$	0	0
$286$	−6.59164	−0.389772
$287$	5.56835	0.328690
$288$	0	0
$289$	−13.5395	−0.796440
$290$	0	0
$291$	0	0
$292$	0.891948	0.0521973
$293$	−6.29156	−0.367557	−0.183779	−	0.982968i	$-0.558833\pi$
$293$	−6.29156	−0.367557	−0.183779	+	0.982968i	$0.558833\pi$
$294$	0	0
$295$	0	0
$296$	−11.7574	−0.683386
$297$	0	0
$298$	−14.5806	−0.844634
$299$	−4.75164	−0.274795
$300$	0	0
$301$	−10.9121	−0.628963
$302$	−10.8986	−0.627142
$303$	0	0
$304$	0.399012	0.0228849
$305$	0	0
$306$	0	0
$307$	−28.6661	−1.63606	−0.818030	−	0.575175i	$-0.804934\pi$
$307$	−28.6661	−1.63606	−0.818030	+	0.575175i	$0.804934\pi$
$308$	−20.2352	−1.15301
$309$	0	0
$310$	0	0
$311$	7.83649	0.444367	0.222183	−	0.975005i	$-0.428682\pi$
$311$	7.83649	0.444367	0.222183	+	0.975005i	$0.428682\pi$
$312$	0	0
$313$	21.4093	1.21012	0.605061	−	0.796179i	$-0.293149\pi$
$313$	21.4093	1.21012	0.605061	+	0.796179i	$0.293149\pi$
$314$	−3.39250	−0.191450
$315$	0	0
$316$	−28.5675	−1.60705
$317$	−4.01983	−0.225776	−0.112888	−	0.993608i	$-0.536010\pi$
$317$	−4.01983	−0.225776	−0.112888	+	0.993608i	$0.536010\pi$
$318$	0	0
$319$	7.92614	0.443779
$320$	0	0
$321$	0	0
$322$	−23.3403	−1.30071
$323$	−1.68040	−0.0934997
$324$	0	0
$325$	0	0
$326$	−10.2987	−0.570390
$327$	0	0
$328$	−5.64515	−0.311701
$329$	14.5718	0.803367
$330$	0	0
$331$	11.6439	0.640005	0.320002	−	0.947417i	$-0.396316\pi$
$331$	11.6439	0.640005	0.320002	+	0.947417i	$0.396316\pi$
$332$	42.0532	2.30797
$333$	0	0
$334$	24.0942	1.31838
$335$	0	0
$336$	0	0
$337$	−21.5348	−1.17307	−0.586537	−	0.809922i	$-0.699509\pi$
$337$	−21.5348	−1.17307	−0.586537	+	0.809922i	$0.699509\pi$
$338$	25.3167	1.37705
$339$	0	0
$340$	0	0
$341$	12.8799	0.697487
$342$	0	0
$343$	−14.5228	−0.784157
$344$	11.0626	0.596455
$345$	0	0
$346$	−17.7264	−0.952979
$347$	−15.5972	−0.837303	−0.418652	−	0.908147i	$-0.637497\pi$
$347$	−15.5972	−0.837303	−0.418652	+	0.908147i	$0.637497\pi$
$348$	0	0
$349$	5.56598	0.297940	0.148970	−	0.988842i	$-0.452404\pi$
$349$	5.56598	0.297940	0.148970	+	0.988842i	$0.452404\pi$
$350$	0	0
$351$	0	0
$352$	−10.2706	−0.547427
$353$	−8.02216	−0.426977	−0.213488	−	0.976946i	$-0.568483\pi$
$353$	−8.02216	−0.426977	−0.213488	+	0.976946i	$0.568483\pi$
$354$	0	0
$355$	0	0
$356$	15.8904	0.842190
$357$	0	0
$358$	35.8325	1.89380
$359$	12.3427	0.651424	0.325712	−	0.945469i	$-0.394396\pi$
$359$	12.3427	0.651424	0.325712	+	0.945469i	$0.394396\pi$
$360$	0	0
$361$	−18.1840	−0.957053
$362$	3.67303	0.193050
$363$	0	0
$364$	−14.4400	−0.756860
$365$	0	0
$366$	0	0
$367$	26.8749	1.40286	0.701430	−	0.712738i	$-0.252545\pi$
$367$	26.8749	1.40286	0.701430	+	0.712738i	$0.252545\pi$
$368$	1.47062	0.0766615
$369$	0	0
$370$	0	0
$371$	28.8546	1.49806
$372$	0	0
$373$	−27.6389	−1.43109	−0.715544	−	0.698567i	$-0.753821\pi$
$373$	−27.6389	−1.43109	−0.715544	+	0.698567i	$0.753821\pi$
$374$	−8.59164	−0.444263
$375$	0	0
$376$	−14.7727	−0.761845
$377$	5.65614	0.291306
$378$	0	0
$379$	−3.47462	−0.178479	−0.0892397	−	0.996010i	$-0.528444\pi$
$379$	−3.47462	−0.178479	−0.0892397	+	0.996010i	$0.528444\pi$
$380$	0	0
$381$	0	0
$382$	45.4198	2.32388
$383$	−27.3719	−1.39864	−0.699319	−	0.714810i	$-0.746513\pi$
$383$	−27.3719	−1.39864	−0.699319	+	0.714810i	$0.746513\pi$
$384$	0	0
$385$	0	0
$386$	−30.2746	−1.54094
$387$	0	0
$388$	−33.1877	−1.68485
$389$	−10.8845	−0.551867	−0.275934	−	0.961177i	$-0.588987\pi$
$389$	−10.8845	−0.551867	−0.275934	+	0.961177i	$0.588987\pi$
$390$	0	0
$391$	−6.19336	−0.313212
$392$	−6.82072	−0.344498
$393$	0	0
$394$	−7.91963	−0.398985
$395$	0	0
$396$	0	0
$397$	16.2212	0.814120	0.407060	−	0.913401i	$-0.366554\pi$
$397$	16.2212	0.814120	0.407060	+	0.913401i	$0.366554\pi$
$398$	−40.8647	−2.04836
$399$	0	0
$400$	0	0
$401$	−3.78686	−0.189107	−0.0945534	−	0.995520i	$-0.530142\pi$
$401$	−3.78686	−0.189107	−0.0945534	+	0.995520i	$0.530142\pi$
$402$	0	0
$403$	9.19118	0.457845
$404$	−31.1483	−1.54968
$405$	0	0
$406$	27.7832	1.37886
$407$	−7.64044	−0.378722
$408$	0	0
$409$	1.85585	0.0917661	0.0458831	−	0.998947i	$-0.485390\pi$
$409$	1.85585	0.0917661	0.0458831	+	0.998947i	$0.485390\pi$
$410$	0	0
$411$	0	0
$412$	30.2750	1.49154
$413$	−32.3181	−1.59027
$414$	0	0
$415$	0	0
$416$	−7.32918	−0.359343
$417$	0	0
$418$	4.17202	0.204060
$419$	−14.3472	−0.700907	−0.350453	−	0.936580i	$-0.613972\pi$
$419$	−14.3472	−0.700907	−0.350453	+	0.936580i	$0.613972\pi$
$420$	0	0
$421$	15.4545	0.753207	0.376604	−	0.926374i	$-0.377092\pi$
$421$	15.4545	0.753207	0.376604	+	0.926374i	$0.377092\pi$
$422$	7.49051	0.364632
$423$	0	0
$424$	−29.2525	−1.42063
$425$	0	0
$426$	0	0
$427$	43.3338	2.09707
$428$	−18.7591	−0.906755
$429$	0	0
$430$	0	0
$431$	−12.5043	−0.602311	−0.301156	−	0.953575i	$-0.597372\pi$
$431$	−12.5043	−0.602311	−0.301156	+	0.953575i	$0.597372\pi$
$432$	0	0
$433$	−22.4951	−1.08105	−0.540524	−	0.841329i	$-0.681774\pi$
$433$	−22.4951	−1.08105	−0.540524	+	0.841329i	$0.681774\pi$
$434$	45.1476	2.16715
$435$	0	0
$436$	−33.7041	−1.61413
$437$	3.00744	0.143865
$438$	0	0
$439$	12.1776	0.581204	0.290602	−	0.956844i	$-0.406145\pi$
$439$	12.1776	0.581204	0.290602	+	0.956844i	$0.406145\pi$
$440$	0	0
$441$	0	0
$442$	−6.13104	−0.291624
$443$	20.7101	0.983968	0.491984	−	0.870604i	$-0.336272\pi$
$443$	20.7101	0.983968	0.491984	+	0.870604i	$0.336272\pi$
$444$	0	0
$445$	0	0
$446$	−66.3103	−3.13988
$447$	0	0
$448$	−38.6833	−1.82761
$449$	25.9539	1.22484	0.612420	−	0.790533i	$-0.290196\pi$
$449$	25.9539	1.22484	0.612420	+	0.790533i	$0.290196\pi$
$450$	0	0
$451$	−3.66844	−0.172740
$452$	−35.7004	−1.67921
$453$	0	0
$454$	27.0662	1.27028
$455$	0	0
$456$	0	0
$457$	−8.50150	−0.397684	−0.198842	−	0.980032i	$-0.563718\pi$
$457$	−8.50150	−0.397684	−0.198842	+	0.980032i	$0.563718\pi$
$458$	37.8751	1.76979
$459$	0	0
$460$	0	0
$461$	−14.7851	−0.688609	−0.344305	−	0.938858i	$-0.611885\pi$
$461$	−14.7851	−0.688609	−0.344305	+	0.938858i	$0.611885\pi$
$462$	0	0
$463$	−22.1921	−1.03135	−0.515677	−	0.856783i	$-0.672460\pi$
$463$	−22.1921	−1.03135	−0.515677	+	0.856783i	$0.672460\pi$
$464$	−1.75056	−0.0812677
$465$	0	0
$466$	51.9924	2.40850
$467$	28.5014	1.31889	0.659443	−	0.751754i	$-0.270792\pi$
$467$	28.5014	1.31889	0.659443	+	0.751754i	$0.270792\pi$
$468$	0	0
$469$	−32.4534	−1.49856
$470$	0	0
$471$	0	0
$472$	32.7638	1.50808
$473$	7.18891	0.330546
$474$	0	0
$475$	0	0
$476$	−18.8213	−0.862671
$477$	0	0
$478$	−15.3182	−0.700637
$479$	−25.0569	−1.14488	−0.572440	−	0.819947i	$-0.694003\pi$
$479$	−25.0569	−1.14488	−0.572440	+	0.819947i	$0.694003\pi$
$480$	0	0
$481$	−5.45225	−0.248601
$482$	−60.5632	−2.75858
$483$	0	0
$484$	−23.3292	−1.06042
$485$	0	0
$486$	0	0
$487$	−1.46479	−0.0663758	−0.0331879	−	0.999449i	$-0.510566\pi$
$487$	−1.46479	−0.0663758	−0.0331879	+	0.999449i	$0.510566\pi$
$488$	−43.9314	−1.98868
$489$	0	0
$490$	0	0
$491$	20.0686	0.905685	0.452843	−	0.891591i	$-0.350410\pi$
$491$	20.0686	0.905685	0.452843	+	0.891591i	$0.350410\pi$
$492$	0	0
$493$	7.37229	0.332031
$494$	2.97718	0.133950
$495$	0	0
$496$	−2.84465	−0.127729
$497$	−37.8258	−1.69672
$498$	0	0
$499$	0.624999	0.0279788	0.0139894	−	0.999902i	$-0.495547\pi$
$499$	0.624999	0.0279788	0.0139894	+	0.999902i	$0.495547\pi$
$500$	0	0
$501$	0	0
$502$	−25.1990	−1.12469
$503$	19.3052	0.860776	0.430388	−	0.902644i	$-0.358377\pi$
$503$	19.3052	0.860776	0.430388	+	0.902644i	$0.358377\pi$
$504$	0	0
$505$	0	0
$506$	15.3766	0.683575
$507$	0	0
$508$	−37.8631	−1.67990
$509$	10.5202	0.466298	0.233149	−	0.972441i	$-0.425097\pi$
$509$	10.5202	0.466298	0.233149	+	0.972441i	$0.425097\pi$
$510$	0	0
$511$	0.812479	0.0359420
$512$	4.98730	0.220410
$513$	0	0
$514$	15.1962	0.670276
$515$	0	0
$516$	0	0
$517$	−9.59989	−0.422203
$518$	−26.7818	−1.17672
$519$	0	0
$520$	0	0
$521$	10.0070	0.438413	0.219207	−	0.975678i	$-0.429653\pi$
$521$	10.0070	0.438413	0.219207	+	0.975678i	$0.429653\pi$
$522$	0	0
$523$	22.7830	0.996233	0.498117	−	0.867110i	$-0.334025\pi$
$523$	22.7830	0.996233	0.498117	+	0.867110i	$0.334025\pi$
$524$	−26.6210	−1.16295
$525$	0	0
$526$	−62.5981	−2.72941
$527$	11.9799	0.521854
$528$	0	0
$529$	−11.9156	−0.518070
$530$	0	0
$531$	0	0
$532$	9.13943	0.396245
$533$	−2.61782	−0.113390
$534$	0	0
$535$	0	0
$536$	32.9010	1.42111
$537$	0	0
$538$	2.33110	0.100501
$539$	−4.43237	−0.190916
$540$	0	0
$541$	3.25900	0.140115	0.0700576	−	0.997543i	$-0.477682\pi$
$541$	3.25900	0.140115	0.0700576	+	0.997543i	$0.477682\pi$
$542$	−14.4377	−0.620150
$543$	0	0
$544$	−9.55296	−0.409580
$545$	0	0
$546$	0	0
$547$	−13.5883	−0.580993	−0.290497	−	0.956876i	$-0.593821\pi$
$547$	−13.5883	−0.580993	−0.290497	+	0.956876i	$0.593821\pi$
$548$	−31.1188	−1.32933
$549$	0	0
$550$	0	0
$551$	−3.57992	−0.152510
$552$	0	0
$553$	−26.0223	−1.10658
$554$	56.9706	2.42045
$555$	0	0
$556$	59.7063	2.53211
$557$	27.6399	1.17114	0.585571	−	0.810621i	$-0.300870\pi$
$557$	27.6399	1.17114	0.585571	+	0.810621i	$0.300870\pi$
$558$	0	0
$559$	5.13004	0.216978
$560$	0	0
$561$	0	0
$562$	4.24656	0.179130
$563$	1.65925	0.0699291	0.0349646	−	0.999389i	$-0.488868\pi$
$563$	1.65925	0.0699291	0.0349646	+	0.999389i	$0.488868\pi$
$564$	0	0
$565$	0	0
$566$	19.9548	0.838763
$567$	0	0
$568$	38.3475	1.60902
$569$	−17.8828	−0.749686	−0.374843	−	0.927088i	$-0.622303\pi$
$569$	−17.8828	−0.749686	−0.374843	+	0.927088i	$0.622303\pi$
$570$	0	0
$571$	−36.8723	−1.54306	−0.771530	−	0.636193i	$-0.780508\pi$
$571$	−36.8723	−1.54306	−0.771530	+	0.636193i	$0.780508\pi$
$572$	9.51307	0.397761
$573$	0	0
$574$	−12.8589	−0.536718
$575$	0	0
$576$	0	0
$577$	−22.8137	−0.949746	−0.474873	−	0.880054i	$-0.657506\pi$
$577$	−22.8137	−0.949746	−0.474873	+	0.880054i	$0.657506\pi$
$578$	31.2664	1.30051
$579$	0	0
$580$	0	0
$581$	38.3065	1.58922
$582$	0	0
$583$	−19.0095	−0.787291
$584$	−0.823684	−0.0340843
$585$	0	0
$586$	14.5289	0.600185
$587$	11.0855	0.457546	0.228773	−	0.973480i	$-0.426529\pi$
$587$	11.0855	0.457546	0.228773	+	0.973480i	$0.426529\pi$
$588$	0	0
$589$	−5.81734	−0.239699
$590$	0	0
$591$	0	0
$592$	1.68746	0.0693542
$593$	−11.1321	−0.457139	−0.228570	−	0.973528i	$-0.573405\pi$
$593$	−11.1321	−0.457139	−0.228570	+	0.973528i	$0.573405\pi$
$594$	0	0
$595$	0	0
$596$	21.0428	0.861947
$597$	0	0
$598$	10.9729	0.448713
$599$	−36.2736	−1.48210	−0.741049	−	0.671451i	$-0.765671\pi$
$599$	−36.2736	−1.48210	−0.741049	+	0.671451i	$0.765671\pi$
$600$	0	0
$601$	−15.1051	−0.616150	−0.308075	−	0.951362i	$-0.599685\pi$
$601$	−15.1051	−0.616150	−0.308075	+	0.951362i	$0.599685\pi$
$602$	25.1990	1.02704
$603$	0	0
$604$	15.7288	0.639997
$605$	0	0
$606$	0	0
$607$	−33.5066	−1.35999	−0.679996	−	0.733216i	$-0.738018\pi$
$607$	−33.5066	−1.35999	−0.679996	+	0.733216i	$0.738018\pi$
$608$	4.63883	0.188129
$609$	0	0
$610$	0	0
$611$	−6.85053	−0.277143
$612$	0	0
$613$	−28.0289	−1.13208	−0.566038	−	0.824379i	$-0.691524\pi$
$613$	−28.0289	−1.13208	−0.566038	+	0.824379i	$0.691524\pi$
$614$	66.1979	2.67153
$615$	0	0
$616$	18.6866	0.752903
$617$	−30.5223	−1.22878	−0.614391	−	0.789002i	$-0.710598\pi$
$617$	−30.5223	−1.22878	−0.614391	+	0.789002i	$0.710598\pi$
$618$	0	0
$619$	−21.6971	−0.872080	−0.436040	−	0.899927i	$-0.643619\pi$
$619$	−21.6971	−0.872080	−0.436040	+	0.899927i	$0.643619\pi$
$620$	0	0
$621$	0	0
$622$	−18.0966	−0.725608
$623$	14.4746	0.579914
$624$	0	0
$625$	0	0
$626$	−49.4399	−1.97601
$627$	0	0
$628$	4.89606	0.195374
$629$	−7.10655	−0.283357
$630$	0	0
$631$	−16.2277	−0.646015	−0.323007	−	0.946396i	$-0.604694\pi$
$631$	−16.2277	−0.646015	−0.323007	+	0.946396i	$0.604694\pi$
$632$	26.3812	1.04939
$633$	0	0
$634$	9.28290	0.368671
$635$	0	0
$636$	0	0
$637$	−3.16296	−0.125321
$638$	−18.3036	−0.724648
$639$	0	0
$640$	0	0
$641$	22.1774	0.875956	0.437978	−	0.898986i	$-0.355695\pi$
$641$	22.1774	0.875956	0.437978	+	0.898986i	$0.355695\pi$
$642$	0	0
$643$	−13.2767	−0.523583	−0.261792	−	0.965124i	$-0.584313\pi$
$643$	−13.2767	−0.523583	−0.261792	+	0.965124i	$0.584313\pi$
$644$	33.6848	1.32737
$645$	0	0
$646$	3.88050	0.152676
$647$	11.2853	0.443671	0.221835	−	0.975084i	$-0.428795\pi$
$647$	11.2853	0.443671	0.221835	+	0.975084i	$0.428795\pi$
$648$	0	0
$649$	21.2912	0.835754
$650$	0	0
$651$	0	0
$652$	14.8630	0.582082
$653$	35.8134	1.40149	0.700743	−	0.713414i	$-0.252852\pi$
$653$	35.8134	1.40149	0.700743	+	0.713414i	$0.252852\pi$
$654$	0	0
$655$	0	0
$656$	0.810208	0.0316333
$657$	0	0
$658$	−33.6502	−1.31182
$659$	−39.7655	−1.54905	−0.774523	−	0.632546i	$-0.782010\pi$
$659$	−39.7655	−1.54905	−0.774523	+	0.632546i	$0.782010\pi$
$660$	0	0
$661$	−6.24734	−0.242993	−0.121497	−	0.992592i	$-0.538769\pi$
$661$	−6.24734	−0.242993	−0.121497	+	0.992592i	$0.538769\pi$
$662$	−26.8889	−1.04507
$663$	0	0
$664$	−38.8347	−1.50708
$665$	0	0
$666$	0	0
$667$	−13.1943	−0.510887
$668$	−34.7728	−1.34540
$669$	0	0
$670$	0	0
$671$	−28.5484	−1.10210
$672$	0	0
$673$	−41.4627	−1.59827	−0.799135	−	0.601151i	$-0.794709\pi$
$673$	−41.4627	−1.59827	−0.799135	+	0.601151i	$0.794709\pi$
$674$	49.7297	1.91552
$675$	0	0
$676$	−36.5372	−1.40528
$677$	1.43915	0.0553112	0.0276556	−	0.999618i	$-0.491196\pi$
$677$	1.43915	0.0553112	0.0276556	+	0.999618i	$0.491196\pi$
$678$	0	0
$679$	−30.2308	−1.16015
$680$	0	0
$681$	0	0
$682$	−29.7433	−1.13893
$683$	8.48623	0.324716	0.162358	−	0.986732i	$-0.448090\pi$
$683$	8.48623	0.324716	0.162358	+	0.986732i	$0.448090\pi$
$684$	0	0
$685$	0	0
$686$	33.5371	1.28045
$687$	0	0
$688$	−1.58774	−0.0605318
$689$	−13.5652	−0.516795
$690$	0	0
$691$	−43.7797	−1.66546	−0.832730	−	0.553679i	$-0.813223\pi$
$691$	−43.7797	−1.66546	−0.832730	+	0.553679i	$0.813223\pi$
$692$	25.5828	0.972513
$693$	0	0
$694$	36.0183	1.36724
$695$	0	0
$696$	0	0
$697$	−3.41210	−0.129243
$698$	−12.8534	−0.486508
$699$	0	0
$700$	0	0
$701$	−0.840795	−0.0317564	−0.0158782	−	0.999874i	$-0.505054\pi$
$701$	−0.840795	−0.0317564	−0.0158782	+	0.999874i	$0.505054\pi$
$702$	0	0
$703$	3.45087	0.130152
$704$	25.4846	0.960487
$705$	0	0
$706$	18.5254	0.697212
$707$	−28.3731	−1.06708
$708$	0	0
$709$	−13.3812	−0.502543	−0.251271	−	0.967917i	$-0.580849\pi$
$709$	−13.3812	−0.502543	−0.251271	+	0.967917i	$0.580849\pi$
$710$	0	0
$711$	0	0
$712$	−14.6743	−0.549941
$713$	−21.4407	−0.802962
$714$	0	0
$715$	0	0
$716$	−51.7135	−1.93262
$717$	0	0
$718$	−28.5027	−1.06371
$719$	−43.4148	−1.61910	−0.809550	−	0.587051i	$-0.800289\pi$
$719$	−43.4148	−1.61910	−0.809550	+	0.587051i	$0.800289\pi$
$720$	0	0
$721$	27.5776	1.02704
$722$	41.9919	1.56278
$723$	0	0
$724$	−5.30093	−0.197007
$725$	0	0
$726$	0	0
$727$	−32.0480	−1.18859	−0.594297	−	0.804245i	$-0.702570\pi$
$727$	−32.0480	−1.18859	−0.594297	+	0.804245i	$0.702570\pi$
$728$	13.3348	0.494222
$729$	0	0
$730$	0	0
$731$	6.68657	0.247312
$732$	0	0
$733$	8.13928	0.300631	0.150316	−	0.988638i	$-0.451971\pi$
$733$	8.13928	0.300631	0.150316	+	0.988638i	$0.451971\pi$
$734$	−62.0616	−2.29074
$735$	0	0
$736$	17.0971	0.630209
$737$	21.3804	0.787556
$738$	0	0
$739$	−7.12714	−0.262176	−0.131088	−	0.991371i	$-0.541847\pi$
$739$	−7.12714	−0.262176	−0.131088	+	0.991371i	$0.541847\pi$
$740$	0	0
$741$	0	0
$742$	−66.6332	−2.44618
$743$	−21.9040	−0.803578	−0.401789	−	0.915732i	$-0.631612\pi$
$743$	−21.9040	−0.803578	−0.401789	+	0.915732i	$0.631612\pi$
$744$	0	0
$745$	0	0
$746$	63.8258	2.33683
$747$	0	0
$748$	12.3995	0.453370
$749$	−17.0877	−0.624373
$750$	0	0
$751$	9.21909	0.336409	0.168205	−	0.985752i	$-0.446203\pi$
$751$	9.21909	0.336409	0.168205	+	0.985752i	$0.446203\pi$
$752$	2.12022	0.0773166
$753$	0	0
$754$	−13.0616	−0.475674
$755$	0	0
$756$	0	0
$757$	45.6524	1.65926	0.829632	−	0.558311i	$-0.188550\pi$
$757$	45.6524	1.65926	0.829632	+	0.558311i	$0.188550\pi$
$758$	8.02386	0.291440
$759$	0	0
$760$	0	0
$761$	−39.9058	−1.44658	−0.723291	−	0.690543i	$-0.757372\pi$
$761$	−39.9058	−1.44658	−0.723291	+	0.690543i	$0.757372\pi$
$762$	0	0
$763$	−30.7012	−1.11146
$764$	−65.5500	−2.37151
$765$	0	0
$766$	63.2092	2.28384
$767$	15.1935	0.548607
$768$	0	0
$769$	−44.3420	−1.59901	−0.799506	−	0.600658i	$-0.794906\pi$
$769$	−44.3420	−1.59901	−0.799506	+	0.600658i	$0.794906\pi$
$770$	0	0
$771$	0	0
$772$	43.6924	1.57252
$773$	38.4944	1.38455	0.692273	−	0.721635i	$-0.256609\pi$
$773$	38.4944	1.38455	0.692273	+	0.721635i	$0.256609\pi$
$774$	0	0
$775$	0	0
$776$	30.6477	1.10019
$777$	0	0
$778$	25.1353	0.901146
$779$	1.65689	0.0593641
$780$	0	0
$781$	24.9197	0.891697
$782$	14.3022	0.511445
$783$	0	0
$784$	0.978929	0.0349618
$785$	0	0
$786$	0	0
$787$	−53.0202	−1.88997	−0.944983	−	0.327120i	$-0.893922\pi$
$787$	−53.0202	−1.88997	−0.944983	+	0.327120i	$0.893922\pi$
$788$	11.4296	0.407164
$789$	0	0
$790$	0	0
$791$	−32.5197	−1.15627
$792$	0	0
$793$	−20.3723	−0.723440
$794$	−37.4593	−1.32938
$795$	0	0
$796$	58.9760	2.09035
$797$	12.6769	0.449037	0.224519	−	0.974470i	$-0.427919\pi$
$797$	12.6769	0.449037	0.224519	+	0.974470i	$0.427919\pi$
$798$	0	0
$799$	−8.92908	−0.315888
$800$	0	0
$801$	0	0
$802$	8.74490	0.308793
$803$	−0.535262	−0.0188890
$804$	0	0
$805$	0	0
$806$	−21.2250	−0.747618
$807$	0	0
$808$	28.7644	1.01193
$809$	41.8935	1.47290	0.736449	−	0.676493i	$-0.236501\pi$
$809$	41.8935	1.47290	0.736449	+	0.676493i	$0.236501\pi$
$810$	0	0
$811$	−34.5486	−1.21317	−0.606583	−	0.795020i	$-0.707460\pi$
$811$	−34.5486	−1.21317	−0.606583	+	0.795020i	$0.707460\pi$
$812$	−40.0968	−1.40712
$813$	0	0
$814$	17.6439	0.618417
$815$	0	0
$816$	0	0
$817$	−3.24694	−0.113596
$818$	−4.28568	−0.149845
$819$	0	0
$820$	0	0
$821$	21.1349	0.737612	0.368806	−	0.929506i	$-0.379767\pi$
$821$	21.1349	0.737612	0.368806	+	0.929506i	$0.379767\pi$
$822$	0	0
$823$	3.17714	0.110748	0.0553740	−	0.998466i	$-0.482365\pi$
$823$	3.17714	0.110748	0.0553740	+	0.998466i	$0.482365\pi$
$824$	−27.9579	−0.973960
$825$	0	0
$826$	74.6315	2.59676
$827$	−9.74482	−0.338860	−0.169430	−	0.985542i	$-0.554193\pi$
$827$	−9.74482	−0.338860	−0.169430	+	0.985542i	$0.554193\pi$
$828$	0	0
$829$	23.4352	0.813938	0.406969	−	0.913442i	$-0.366586\pi$
$829$	23.4352	0.813938	0.406969	+	0.913442i	$0.366586\pi$
$830$	0	0
$831$	0	0
$832$	18.1859	0.630484
$833$	−4.12265	−0.142841
$834$	0	0
$835$	0	0
$836$	−6.02107	−0.208243
$837$	0	0
$838$	33.1316	1.14451
$839$	−42.6819	−1.47354	−0.736771	−	0.676142i	$-0.763650\pi$
$839$	−42.6819	−1.47354	−0.736771	+	0.676142i	$0.763650\pi$
$840$	0	0
$841$	−13.2941	−0.458416
$842$	−35.6887	−1.22992
$843$	0	0
$844$	−10.8103	−0.372106
$845$	0	0
$846$	0	0
$847$	−21.2507	−0.730183
$848$	4.19841	0.144174
$849$	0	0
$850$	0	0
$851$	12.7187	0.435993
$852$	0	0
$853$	16.9610	0.580733	0.290367	−	0.956915i	$-0.406223\pi$
$853$	16.9610	0.580733	0.290367	+	0.956915i	$0.406223\pi$
$854$	−100.070	−3.42431
$855$	0	0
$856$	17.3234	0.592102
$857$	39.3176	1.34306	0.671531	−	0.740976i	$-0.265637\pi$
$857$	39.3176	1.34306	0.671531	+	0.740976i	$0.265637\pi$
$858$	0	0
$859$	0.707056	0.0241244	0.0120622	−	0.999927i	$-0.496160\pi$
$859$	0.707056	0.0241244	0.0120622	+	0.999927i	$0.496160\pi$
$860$	0	0
$861$	0	0
$862$	28.8759	0.983516
$863$	−0.909409	−0.0309567	−0.0154783	−	0.999880i	$-0.504927\pi$
$863$	−0.909409	−0.0309567	−0.0154783	+	0.999880i	$0.504927\pi$
$864$	0	0
$865$	0	0
$866$	51.9474	1.76525
$867$	0	0
$868$	−65.1571	−2.21158
$869$	17.1435	0.581554
$870$	0	0
$871$	15.2571	0.516968
$872$	31.1246	1.05401
$873$	0	0
$874$	−6.94501	−0.234918
$875$	0	0
$876$	0	0
$877$	33.6613	1.13666	0.568331	−	0.822800i	$-0.307589\pi$
$877$	33.6613	1.13666	0.568331	+	0.822800i	$0.307589\pi$
$878$	−28.1213	−0.949050
$879$	0	0
$880$	0	0
$881$	19.9283	0.671402	0.335701	−	0.941969i	$-0.391027\pi$
$881$	19.9283	0.671402	0.335701	+	0.941969i	$0.391027\pi$
$882$	0	0
$883$	15.6081	0.525255	0.262627	−	0.964897i	$-0.415411\pi$
$883$	15.6081	0.525255	0.262627	+	0.964897i	$0.415411\pi$
$884$	8.84833	0.297601
$885$	0	0
$886$	−47.8254	−1.60673
$887$	−46.8968	−1.57464	−0.787320	−	0.616544i	$-0.788532\pi$
$887$	−46.8968	−1.57464	−0.787320	+	0.616544i	$0.788532\pi$
$888$	0	0
$889$	−34.4897	−1.15675
$890$	0	0
$891$	0	0
$892$	95.6991	3.20424
$893$	4.33588	0.145095
$894$	0	0
$895$	0	0
$896$	58.1505	1.94267
$897$	0	0
$898$	−59.9347	−2.00005
$899$	25.5220	0.851208
$900$	0	0
$901$	−17.6811	−0.589044
$902$	8.47143	0.282068
$903$	0	0
$904$	32.9682	1.09651
$905$	0	0
$906$	0	0
$907$	1.43447	0.0476308	0.0238154	−	0.999716i	$-0.492419\pi$
$907$	1.43447	0.0476308	0.0238154	+	0.999716i	$0.492419\pi$
$908$	−39.0619	−1.29632
$909$	0	0
$910$	0	0
$911$	2.81129	0.0931422	0.0465711	−	0.998915i	$-0.485171\pi$
$911$	2.81129	0.0931422	0.0465711	+	0.998915i	$0.485171\pi$
$912$	0	0
$913$	−25.2363	−0.835201
$914$	19.6323	0.649379
$915$	0	0
$916$	−54.6614	−1.80606
$917$	−24.2492	−0.800780
$918$	0	0
$919$	0.992236	0.0327308	0.0163654	−	0.999866i	$-0.494790\pi$
$919$	0.992236	0.0327308	0.0163654	+	0.999866i	$0.494790\pi$
$920$	0	0
$921$	0	0
$922$	34.1428	1.12443
$923$	17.7828	0.585329
$924$	0	0
$925$	0	0
$926$	51.2477	1.68410
$927$	0	0
$928$	−20.3516	−0.668075
$929$	−33.3227	−1.09328	−0.546642	−	0.837367i	$-0.684094\pi$
$929$	−33.3227	−1.09328	−0.546642	+	0.837367i	$0.684094\pi$
$930$	0	0
$931$	2.00192	0.0656104
$932$	−75.0355	−2.45787
$933$	0	0
$934$	−65.8175	−2.15361
$935$	0	0
$936$	0	0
$937$	13.3675	0.436698	0.218349	−	0.975871i	$-0.429933\pi$
$937$	13.3675	0.436698	0.218349	+	0.975871i	$0.429933\pi$
$938$	74.9439	2.44701
$939$	0	0
$940$	0	0
$941$	2.14982	0.0700820	0.0350410	−	0.999386i	$-0.488844\pi$
$941$	2.14982	0.0700820	0.0350410	+	0.999386i	$0.488844\pi$
$942$	0	0
$943$	6.10671	0.198862
$944$	−4.70236	−0.153049
$945$	0	0
$946$	−16.6012	−0.539750
$947$	−34.9183	−1.13469	−0.567346	−	0.823479i	$-0.692030\pi$
$947$	−34.9183	−1.13469	−0.567346	+	0.823479i	$0.692030\pi$
$948$	0	0
$949$	−0.381966	−0.0123991
$950$	0	0
$951$	0	0
$952$	17.3808	0.563315
$953$	8.27883	0.268178	0.134089	−	0.990969i	$-0.457189\pi$
$953$	8.27883	0.268178	0.134089	+	0.990969i	$0.457189\pi$
$954$	0	0
$955$	0	0
$956$	22.1072	0.714998
$957$	0	0
$958$	57.8633	1.86948
$959$	−28.3462	−0.915347
$960$	0	0
$961$	10.4732	0.337844
$962$	12.5908	0.405942
$963$	0	0
$964$	87.4048	2.81512
$965$	0	0
$966$	0	0
$967$	29.4871	0.948241	0.474121	−	0.880460i	$-0.342766\pi$
$967$	29.4871	0.948241	0.474121	+	0.880460i	$0.342766\pi$
$968$	21.5438	0.692443
$969$	0	0
$970$	0	0
$971$	−21.8666	−0.701733	−0.350867	−	0.936425i	$-0.614113\pi$
$971$	−21.8666	−0.701733	−0.350867	+	0.936425i	$0.614113\pi$
$972$	0	0
$973$	54.3868	1.74356
$974$	3.38260	0.108385
$975$	0	0
$976$	6.30517	0.201823
$977$	−13.8482	−0.443043	−0.221521	−	0.975155i	$-0.571102\pi$
$977$	−13.8482	−0.443043	−0.221521	+	0.975155i	$0.571102\pi$
$978$	0	0
$979$	−9.53591	−0.304769
$980$	0	0
$981$	0	0
$982$	−46.3440	−1.47890
$983$	−2.93538	−0.0936241	−0.0468120	−	0.998904i	$-0.514906\pi$
$983$	−2.93538	−0.0936241	−0.0468120	+	0.998904i	$0.514906\pi$
$984$	0	0
$985$	0	0
$986$	−17.0246	−0.542175
$987$	0	0
$988$	−4.29667	−0.136695
$989$	−11.9671	−0.380531
$990$	0	0
$991$	31.6137	1.00424	0.502122	−	0.864797i	$-0.332553\pi$
$991$	31.6137	1.00424	0.502122	+	0.864797i	$0.332553\pi$
$992$	−33.0713	−1.05001
$993$	0	0
$994$	87.3502	2.77058
$995$	0	0
$996$	0	0
$997$	−12.0885	−0.382845	−0.191423	−	0.981508i	$-0.561310\pi$
$997$	−12.0885	−0.382845	−0.191423	+	0.981508i	$0.561310\pi$
$998$	−1.44329	−0.0456867
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5625.2.a.x.1.1		8
3.2	odd	2		625.2.a.f.1.8		8
5.4	even	2	inner	5625.2.a.x.1.8		8
12.11	even	2		10000.2.a.bj.1.4		8
15.2	even	4		625.2.b.c.624.8		8
15.8	even	4		625.2.b.c.624.1		8
15.14	odd	2		625.2.a.f.1.1		8
25.3	odd	20		225.2.m.a.109.2		8
25.17	odd	20		225.2.m.a.64.2		8
60.59	even	2		10000.2.a.bj.1.5		8
75.2	even	20		625.2.e.a.124.2		8
75.8	even	20		125.2.e.b.74.2		8
75.11	odd	10		625.2.d.o.501.4		16
75.14	odd	10		625.2.d.o.501.1		16
75.17	even	20		25.2.e.a.14.1	yes	8
75.23	even	20		625.2.e.i.124.1		8
75.29	odd	10		125.2.d.b.76.4		16
75.38	even	20		625.2.e.a.499.2		8
75.41	odd	10		625.2.d.o.126.4		16
75.44	odd	10		125.2.d.b.51.4		16
75.47	even	20		125.2.e.b.49.2		8
75.53	even	20		25.2.e.a.9.1	✓	8
75.56	odd	10		125.2.d.b.51.1		16
75.59	odd	10		625.2.d.o.126.1		16
75.62	even	20		625.2.e.i.499.1		8
75.71	odd	10		125.2.d.b.76.1		16
300.167	odd	20		400.2.y.c.289.1		8
300.203	odd	20		400.2.y.c.209.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.2.e.a.9.1	✓	8	75.53	even	20
25.2.e.a.14.1	yes	8	75.17	even	20
125.2.d.b.51.1		16	75.56	odd	10
125.2.d.b.51.4		16	75.44	odd	10
125.2.d.b.76.1		16	75.71	odd	10
125.2.d.b.76.4		16	75.29	odd	10
125.2.e.b.49.2		8	75.47	even	20
125.2.e.b.74.2		8	75.8	even	20
225.2.m.a.64.2		8	25.17	odd	20
225.2.m.a.109.2		8	25.3	odd	20
400.2.y.c.209.1		8	300.203	odd	20
400.2.y.c.289.1		8	300.167	odd	20
625.2.a.f.1.1		8	15.14	odd	2
625.2.a.f.1.8		8	3.2	odd	2
625.2.b.c.624.1		8	15.8	even	4
625.2.b.c.624.8		8	15.2	even	4
625.2.d.o.126.1		16	75.59	odd	10
625.2.d.o.126.4		16	75.41	odd	10
625.2.d.o.501.1		16	75.14	odd	10
625.2.d.o.501.4		16	75.11	odd	10
625.2.e.a.124.2		8	75.2	even	20
625.2.e.a.499.2		8	75.38	even	20
625.2.e.i.124.1		8	75.23	even	20
625.2.e.i.499.1		8	75.62	even	20
5625.2.a.x.1.1		8	1.1	even	1	trivial
5625.2.a.x.1.8		8	5.4	even	2	inner
10000.2.a.bj.1.4		8	12.11	even	2
10000.2.a.bj.1.5		8	60.59	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)

\(f(q)\)	\(=\)	\(q-2.30927 q^{2} +3.33275 q^{4} +3.03582 q^{7} -3.07768 q^{8} +O(q^{10})\)	\(q-2.30927 q^{2} +3.33275 q^{4} +3.03582 q^{7} -3.07768 q^{8} -2.00000 q^{11} -1.42721 q^{13} -7.01054 q^{14} +0.441718 q^{16} -1.86025 q^{17} +0.903319 q^{19} +4.61855 q^{22} +3.32932 q^{23} +3.29582 q^{26} +10.1176 q^{28} -3.96307 q^{29} -6.43997 q^{31} +5.13532 q^{32} +4.29582 q^{34} +3.82022 q^{37} -2.08601 q^{38} +1.83422 q^{41} -3.59445 q^{43} -6.66550 q^{44} -7.68832 q^{46} +4.79995 q^{47} +2.21619 q^{49} -4.75653 q^{52} +9.50473 q^{53} -9.34328 q^{56} +9.15182 q^{58} -10.6456 q^{59} +14.2742 q^{61} +14.8716 q^{62} -12.7423 q^{64} -10.6902 q^{67} -6.19974 q^{68} -12.4598 q^{71} +0.267631 q^{73} -8.82193 q^{74} +3.01054 q^{76} -6.07163 q^{77} -8.57176 q^{79} -4.23572 q^{82} +12.6182 q^{83} +8.30058 q^{86} +6.15537 q^{88} +4.76796 q^{89} -4.33275 q^{91} +11.0958 q^{92} -11.0844 q^{94} -9.95805 q^{97} -5.11778 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 6 q^{4}+O(q^{10}) \) 8 * q + 6 * q^4	\( 8 q + 6 q^{4} - 16 q^{11} - 12 q^{14} - 2 q^{16} + 10 q^{19} - 6 q^{26} - 20 q^{29} + 16 q^{31} + 2 q^{34} - 26 q^{41} - 12 q^{44} + 6 q^{46} - 14 q^{49} - 10 q^{56} - 30 q^{59} + 6 q^{61} - 44 q^{64} - 46 q^{71} - 12 q^{74} - 20 q^{76} + 10 q^{79} + 14 q^{86} - 30 q^{89} - 14 q^{91} - 68 q^{94}+O(q^{100}) \) 8 * q + 6 * q^4 - 16 * q^11 - 12 * q^14 - 2 * q^16 + 10 * q^19 - 6 * q^26 - 20 * q^29 + 16 * q^31 + 2 * q^34 - 26 * q^41 - 12 * q^44 + 6 * q^46 - 14 * q^49 - 10 * q^56 - 30 * q^59 + 6 * q^61 - 44 * q^64 - 46 * q^71 - 12 * q^74 - 20 * q^76 + 10 * q^79 + 14 * q^86 - 30 * q^89 - 14 * q^91 - 68 * q^94

Introduction

L-functions

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