LMFDB - Embedded newform 625.2.a.f.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [625,2,Mod(1,625)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(625, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("625.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 625 = 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	625.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:	$4.99065012633$
Analytic rank:	$0$
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	$\chi$	$=$	625.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.30927 q^{2} -0.474903 q^{3} +3.33275 q^{4} +1.09668 q^{6} -3.03582 q^{7} -3.07768 q^{8} -2.77447 q^{9} +O(q^{10})$	\(q-2.30927 q^{2} -0.474903 q^{3} +3.33275 q^{4} +1.09668 q^{6} -3.03582 q^{7} -3.07768 q^{8} -2.77447 q^{9} +2.00000 q^{11} -1.58273 q^{12} +1.42721 q^{13} +7.01054 q^{14} +0.441718 q^{16} -1.86025 q^{17} +6.40701 q^{18} +0.903319 q^{19} +1.44172 q^{21} -4.61855 q^{22} +3.32932 q^{23} +1.46160 q^{24} -3.29582 q^{26} +2.74231 q^{27} -10.1176 q^{28} +3.96307 q^{29} -6.43997 q^{31} +5.13532 q^{32} -0.949806 q^{33} +4.29582 q^{34} -9.24660 q^{36} -3.82022 q^{37} -2.08601 q^{38} -0.677786 q^{39} -1.83422 q^{41} -3.32932 q^{42} +3.59445 q^{43} +6.66550 q^{44} -7.68832 q^{46} +4.79995 q^{47} -0.209773 q^{48} +2.21619 q^{49} +0.883436 q^{51} +4.75653 q^{52} +9.50473 q^{53} -6.33275 q^{54} +9.34328 q^{56} -0.428989 q^{57} -9.15182 q^{58} +10.6456 q^{59} +14.2742 q^{61} +14.8716 q^{62} +8.42278 q^{63} -12.7423 q^{64} +2.19336 q^{66} +10.6902 q^{67} -6.19974 q^{68} -1.58111 q^{69} +12.4598 q^{71} +8.53893 q^{72} -0.267631 q^{73} +8.82193 q^{74} +3.01054 q^{76} -6.07163 q^{77} +1.56519 q^{78} -8.57176 q^{79} +7.02107 q^{81} +4.23572 q^{82} +12.6182 q^{83} +4.80489 q^{84} -8.30058 q^{86} -1.88207 q^{87} -6.15537 q^{88} -4.76796 q^{89} -4.33275 q^{91} +11.0958 q^{92} +3.05836 q^{93} -11.0844 q^{94} -2.43878 q^{96} +9.95805 q^{97} -5.11778 q^{98} -5.54893 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 6 q^{4} + 6 q^{6} + 4 q^{9}+O(q^{10}) $ 8 * q + 6 * q^4 + 6 * q^6 + 4 * q^9	$ 8 q + 6 q^{4} + 6 q^{6} + 4 q^{9} + 16 q^{11} + 12 q^{14} - 2 q^{16} + 10 q^{19} + 6 q^{21} + 20 q^{24} + 6 q^{26} + 20 q^{29} + 16 q^{31} + 2 q^{34} - 12 q^{36} + 18 q^{39} + 26 q^{41} + 12 q^{44} + 6 q^{46} - 14 q^{49} - 4 q^{51} - 30 q^{54} + 10 q^{56} + 30 q^{59} + 6 q^{61} - 44 q^{64} + 12 q^{66} + 8 q^{69} + 46 q^{71} + 12 q^{74} - 20 q^{76} + 10 q^{79} - 32 q^{81} - 18 q^{84} - 14 q^{86} + 30 q^{89} - 14 q^{91} - 68 q^{94} - 54 q^{96} + 8 q^{99}+O(q^{100}) $ 8 * q + 6 * q^4 + 6 * q^6 + 4 * q^9 + 16 * q^11 + 12 * q^14 - 2 * q^16 + 10 * q^19 + 6 * q^21 + 20 * q^24 + 6 * q^26 + 20 * q^29 + 16 * q^31 + 2 * q^34 - 12 * q^36 + 18 * q^39 + 26 * q^41 + 12 * q^44 + 6 * q^46 - 14 * q^49 - 4 * q^51 - 30 * q^54 + 10 * q^56 + 30 * q^59 + 6 * q^61 - 44 * q^64 + 12 * q^66 + 8 * q^69 + 46 * q^71 + 12 * q^74 - 20 * q^76 + 10 * q^79 - 32 * q^81 - 18 * q^84 - 14 * q^86 + 30 * q^89 - 14 * q^91 - 68 * q^94 - 54 * q^96 + 8 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−2.30927	−1.63290	−0.816452	−	0.577414i	$-0.804062\pi$
$2$	−2.30927	−1.63290	−0.816452	+	0.577414i	$0.804062\pi$
$3$	−0.474903	−0.274185	−0.137093	−	0.990558i	$-0.543776\pi$
$3$	−0.474903	−0.274185	−0.137093	+	0.990558i	$0.543776\pi$
$4$	3.33275	1.66637
$5$	0	0
$5$	0	0
$6$	1.09668	0.447718
$7$	−3.03582	−1.14743	−0.573716	−	0.819055i	$-0.694498\pi$
$7$	−3.03582	−1.14743	−0.573716	+	0.819055i	$0.694498\pi$
$8$	−3.07768	−1.08813
$9$	−2.77447	−0.924822
$10$	0	0
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	−1.58273	−0.456895
$13$	1.42721	0.395837	0.197918	−	0.980218i	$-0.436582\pi$
$13$	1.42721	0.395837	0.197918	+	0.980218i	$0.436582\pi$
$14$	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	$0.572430\pi$
$18$	6.40701	1.51015
$19$	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
$21$	1.44172	0.314609
$22$	−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	$0.387164\pi$
$24$	1.46160	0.298348
$25$	0	0
$26$	−3.29582	−0.646364
$27$	2.74231	0.527758
$28$	−10.1176	−1.91205
$29$	3.96307	0.735924	0.367962	−	0.929841i	$-0.380056\pi$
$29$	3.96307	0.735924	0.367962	+	0.929841i	$0.380056\pi$
$30$	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
$33$	−0.949806	−0.165340
$34$	4.29582	0.736727
$35$	0	0
$36$	−9.24660	−1.54110
$37$	−3.82022	−0.628040	−0.314020	−	0.949416i	$-0.601676\pi$
$37$	−3.82022	−0.628040	−0.314020	+	0.949416i	$0.601676\pi$
$38$	−2.08601	−0.338396
$39$	−0.677786	−0.108533
$40$	0	0
$41$	−1.83422	−0.286457	−0.143228	−	0.989690i	$-0.545748\pi$
$41$	−1.83422	−0.286457	−0.143228	+	0.989690i	$0.545748\pi$
$42$	−3.32932	−0.513726
$43$	3.59445	0.548149	0.274074	−	0.961708i	$-0.411629\pi$
$43$	3.59445	0.548149	0.274074	+	0.961708i	$0.411629\pi$
$44$	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	$0.386157\pi$
$48$	−0.209773	−0.0302782
$49$	2.21619	0.316598
$50$	0	0
$51$	0.883436	0.123706
$52$	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	$0.273599\pi$
$54$	−6.33275	−0.861778
$55$	0	0
$56$	9.34328	1.24855
$57$	−0.428989	−0.0568209
$58$	−9.15182	−1.20169
$59$	10.6456	1.38594	0.692971	−	0.720966i	$-0.256302\pi$
$59$	10.6456	1.38594	0.692971	+	0.720966i	$0.256302\pi$
$60$	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
$63$	8.42278	1.06117
$64$	−12.7423	−1.59279
$65$	0	0
$66$	2.19336	0.269984
$67$	10.6902	1.30601	0.653007	−	0.757352i	$-0.273507\pi$
$67$	10.6902	1.30601	0.653007	+	0.757352i	$0.273507\pi$
$68$	−6.19974	−0.751828
$69$	−1.58111	−0.190343
$70$	0	0
$71$	12.4598	1.47871	0.739356	−	0.673315i	$-0.235130\pi$
$71$	12.4598	1.47871	0.739356	+	0.673315i	$0.235130\pi$
$72$	8.53893	1.00632
$73$	−0.267631	−0.0313239	−0.0156619	−	0.999877i	$-0.504986\pi$
$73$	−0.267631	−0.0313239	−0.0156619	+	0.999877i	$0.504986\pi$
$74$	8.82193	1.02553
$75$	0	0
$76$	3.01054	0.345332
$77$	−6.07163	−0.691927
$78$	1.56519	0.177223
$79$	−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
$81$	7.02107	0.780119
$82$	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	$0.256504\pi$
$84$	4.80489	0.524256
$85$	0	0
$86$	−8.30058	−0.895074
$87$	−1.88207	−0.201779
$88$	−6.15537	−0.656164
$89$	−4.76796	−0.505402	−0.252701	−	0.967544i	$-0.581319\pi$
$89$	−4.76796	−0.505402	−0.252701	+	0.967544i	$0.581319\pi$
$90$	0	0
$91$	−4.33275	−0.454196
$92$	11.0958	1.15682
$93$	3.05836	0.317137
$94$	−11.0844	−1.14327
$95$	0	0
$96$	−2.43878	−0.248907
$97$	9.95805	1.01109	0.505543	−	0.862801i	$-0.331292\pi$
$97$	9.95805	1.01109	0.505543	+	0.862801i	$0.331292\pi$
$98$	−5.11778	−0.516974
$99$	−5.54893	−0.557689
$100$	0	0
$101$	9.34612	0.929974	0.464987	−	0.885318i	$-0.346059\pi$
$101$	9.34612	0.929974	0.464987	+	0.885318i	$0.346059\pi$
$102$	−2.04010	−0.202000
$103$	−9.08408	−0.895081	−0.447540	−	0.894264i	$-0.647700\pi$
$103$	−9.08408	−0.895081	−0.447540	+	0.894264i	$0.647700\pi$
$104$	−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	$0.587710\pi$
$108$	9.13943	0.879442
$109$	−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
$111$	1.81423	0.172199
$112$	−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	$0.668084\pi$
$114$	0.990653	0.0927831
$115$	0	0
$116$	13.2079	1.22632
$117$	−3.95975	−0.366079
$118$	−24.5836	−2.26311
$119$	5.64737	0.517693
$120$	0	0
$121$	−7.00000	−0.636364
$122$	−32.9630	−2.98433
$123$	0.871076	0.0785423
$124$	−21.4628	−1.92742
$125$	0	0
$126$	−19.4505	−1.73279
$127$	11.3609	1.00812	0.504060	−	0.863669i	$-0.331839\pi$
$127$	11.3609	1.00812	0.504060	+	0.863669i	$0.331839\pi$
$128$	19.1548	1.69306
$129$	−1.70702	−0.150294
$130$	0	0
$131$	7.98771	0.697890	0.348945	−	0.937143i	$-0.386540\pi$
$131$	7.98771	0.697890	0.348945	+	0.937143i	$0.386540\pi$
$132$	−3.16546	−0.275518
$133$	−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	$0.630597\pi$
$138$	3.65121	0.310811
$139$	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
$141$	−2.27951	−0.191969
$142$	−28.7732	−2.41459
$143$	2.85442	0.238699
$144$	−1.22553	−0.102128
$145$	0	0
$146$	0.618034	0.0511489
$147$	−1.05247	−0.0868065
$148$	−12.7318	−1.04655
$149$	−6.31395	−0.517259	−0.258629	−	0.965977i	$-0.583271\pi$
$149$	−6.31395	−0.517259	−0.258629	+	0.965977i	$0.583271\pi$
$150$	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
$153$	5.16119	0.417258
$154$	14.0211	1.12985
$155$	0	0
$156$	−2.25889	−0.180856
$157$	−1.46908	−0.117245	−0.0586225	−	0.998280i	$-0.518671\pi$
$157$	−1.46908	−0.117245	−0.0586225	+	0.998280i	$0.518671\pi$
$158$	19.7945	1.57477
$159$	−4.51382	−0.357969
$160$	0	0
$161$	−10.1072	−0.796560
$162$	−16.2136	−1.27386
$163$	−4.45969	−0.349310	−0.174655	−	0.984630i	$-0.555881\pi$
$163$	−4.45969	−0.349310	−0.174655	+	0.984630i	$0.555881\pi$
$164$	−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	$0.632273\pi$
$168$	−4.43715	−0.342334
$169$	−10.9631	−0.843313
$170$	0	0
$171$	−2.50623	−0.191656
$172$	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	$0.405744\pi$
$174$	4.34623	0.329486
$175$	0	0
$176$	0.883436	0.0665915
$177$	−5.05563	−0.380005
$178$	11.0105	0.825273
$179$	15.5168	1.15978	0.579889	−	0.814696i	$-0.303096\pi$
$179$	15.5168	1.15978	0.579889	+	0.814696i	$0.303096\pi$
$180$	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
$183$	−6.77885	−0.501107
$184$	−10.2466	−0.755390
$185$	0	0
$186$	−7.06259	−0.517854
$187$	−3.72049	−0.272069
$188$	15.9970	1.16670
$189$	−8.32515	−0.605566
$190$	0	0
$191$	19.6684	1.42316	0.711579	−	0.702606i	$-0.247980\pi$
$191$	19.6684	1.42316	0.711579	+	0.702606i	$0.247980\pi$
$192$	6.05135	0.436719
$193$	−13.1100	−0.943680	−0.471840	−	0.881684i	$-0.656410\pi$
$193$	−13.1100	−0.943680	−0.471840	+	0.881684i	$0.656410\pi$
$194$	−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	$0.461015\pi$
$198$	12.8140	0.910652
$199$	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
$201$	−5.07680	−0.358090
$202$	−21.5828	−1.51856
$203$	−12.0312	−0.844422
$204$	2.94427	0.206140
$205$	0	0
$206$	20.9776	1.46158
$207$	−9.23710	−0.642023
$208$	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
$213$	−5.91722	−0.405441
$214$	12.9982	0.888542
$215$	0	0
$216$	−8.43997	−0.574267
$217$	19.5506	1.32718
$218$	23.3537	1.58171
$219$	0.127099	0.00858854
$220$	0	0
$221$	−2.65496	−0.178592
$222$	−4.18956	−0.281185
$223$	−28.7148	−1.92288	−0.961441	−	0.275010i	$-0.911319\pi$
$223$	−28.7148	−1.92288	−0.961441	+	0.275010i	$0.911319\pi$
$224$	−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	$0.627167\pi$
$228$	−1.42971	−0.0946850
$229$	−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
$231$	2.88344	0.189716
$232$	−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	$0.763989\pi$
$234$	9.14415	0.597771
$235$	0	0
$236$	35.4792	2.30950
$237$	4.07075	0.264424
$238$	−13.0413	−0.845344
$239$	−6.63333	−0.429074	−0.214537	−	0.976716i	$-0.568824\pi$
$239$	−6.63333	−0.429074	−0.214537	+	0.976716i	$0.568824\pi$
$240$	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
$243$	−11.5613	−0.741655
$244$	47.5723	3.04550
$245$	0	0
$246$	−2.01155	−0.128252
$247$	1.28923	0.0820315
$248$	19.8202	1.25858
$249$	−5.99241	−0.379753
$250$	0	0
$251$	−10.9121	−0.688766	−0.344383	−	0.938829i	$-0.611912\pi$
$251$	−10.9121	−0.688766	−0.344383	+	0.938829i	$0.611912\pi$
$252$	28.0710	1.76831
$253$	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	$0.565798\pi$
$258$	3.94197	0.245416
$259$	11.5975	0.720632
$260$	0	0
$261$	−10.9954	−0.680599
$262$	−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	$0.185032\pi$
$264$	2.92320	0.179911
$265$	0	0
$266$	6.33275	0.388286
$267$	2.26432	0.138574
$268$	35.6277	2.17631
$269$	1.00945	0.0615474	0.0307737	−	0.999526i	$-0.490203\pi$
$269$	1.00945	0.0615474	0.0307737	+	0.999526i	$0.490203\pi$
$270$	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
$273$	2.05763	0.124534
$274$	21.5623	1.30263
$275$	0	0
$276$	−5.26943	−0.317182
$277$	24.6703	1.48230	0.741148	−	0.671342i	$-0.234282\pi$
$277$	24.6703	1.48230	0.741148	+	0.671342i	$0.234282\pi$
$278$	−41.3708	−2.48125
$279$	17.8675	1.06970
$280$	0	0
$281$	1.83891	0.109700	0.0548502	−	0.998495i	$-0.482532\pi$
$281$	1.83891	0.109700	0.0548502	+	0.998495i	$0.482532\pi$
$282$	5.26401	0.313467
$283$	8.64116	0.513664	0.256832	−	0.966456i	$-0.417321\pi$
$283$	8.64116	0.513664	0.256832	+	0.966456i	$0.417321\pi$
$284$	41.5255	2.46409
$285$	0	0
$286$	−6.59164	−0.389772
$287$	5.56835	0.328690
$288$	−14.2478	−0.839558
$289$	−13.5395	−0.796440
$290$	0	0
$291$	−4.72910	−0.277225
$292$	−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	$0.558833\pi$
$294$	2.43045	0.141747
$295$	0	0
$296$	11.7574	0.683386
$297$	5.48462	0.318250
$298$	14.5806	0.844634
$299$	4.75164	0.274795
$300$	0	0
$301$	−10.9121	−0.628963
$302$	−10.8986	−0.627142
$303$	−4.43850	−0.254985
$304$	0.399012	0.0228849
$305$	0	0
$306$	−11.9186	−0.681342
$307$	28.6661	1.63606	0.818030	−	0.575175i	$-0.195066\pi$
$307$	28.6661	1.63606	0.818030	+	0.575175i	$0.195066\pi$
$308$	−20.2352	−1.15301
$309$	4.31405	0.245418
$310$	0	0
$311$	−7.83649	−0.444367	−0.222183	−	0.975005i	$-0.571318\pi$
$311$	−7.83649	−0.444367	−0.222183	+	0.975005i	$0.571318\pi$
$312$	2.08601	0.118097
$313$	−21.4093	−1.21012	−0.605061	−	0.796179i	$-0.706851\pi$
$313$	−21.4093	−1.21012	−0.605061	+	0.796179i	$0.706851\pi$
$314$	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	$0.536010\pi$
$318$	10.4237	0.584530
$319$	7.92614	0.443779
$320$	0	0
$321$	2.67309	0.149197
$322$	23.3403	1.30071
$323$	−1.68040	−0.0934997
$324$	23.3995	1.29997
$325$	0	0
$326$	10.2987	0.570390
$327$	4.80269	0.265589
$328$	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
$333$	10.5991	0.580825
$334$	24.0942	1.31838
$335$	0	0
$336$	0.636833	0.0347421
$337$	21.5348	1.17307	0.586537	−	0.809922i	$-0.300491\pi$
$337$	21.5348	1.17307	0.586537	+	0.809922i	$0.300491\pi$
$338$	25.3167	1.37705
$339$	5.08716	0.276297
$340$	0	0
$341$	−12.8799	−0.697487
$342$	5.78757	0.312956
$343$	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	$0.637497\pi$
$348$	−6.27248	−0.336240
$349$	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
$351$	3.91385	0.208906
$352$	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	$0.568483\pi$
$354$	11.6748	0.620511
$355$	0	0
$356$	−15.8904	−0.842190
$357$	−2.68195	−0.141944
$358$	−35.8325	−1.89380
$359$	−12.3427	−0.651424	−0.325712	−	0.945469i	$-0.605604\pi$
$359$	−12.3427	−0.651424	−0.325712	+	0.945469i	$0.605604\pi$
$360$	0	0
$361$	−18.1840	−0.957053
$362$	3.67303	0.193050
$363$	3.32432	0.174482
$364$	−14.4400	−0.756860
$365$	0	0
$366$	15.6542	0.818260
$367$	−26.8749	−1.40286	−0.701430	−	0.712738i	$-0.747455\pi$
$367$	−26.8749	−1.40286	−0.701430	+	0.712738i	$0.747455\pi$
$368$	1.47062	0.0766615
$369$	5.08898	0.264922
$370$	0	0
$371$	−28.8546	−1.49806
$372$	10.1927	0.528469
$373$	27.6389	1.43109	0.715544	−	0.698567i	$-0.246179\pi$
$373$	27.6389	1.43109	0.715544	+	0.698567i	$0.246179\pi$
$374$	8.59164	0.444263
$375$	0	0
$376$	−14.7727	−0.761845
$377$	5.65614	0.291306
$378$	19.2251	0.988831
$379$	−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
$381$	−5.39534	−0.276412
$382$	−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	$0.746513\pi$
$384$	−9.09668	−0.464213
$385$	0	0
$386$	30.2746	1.54094
$387$	−9.97269	−0.506940
$388$	33.1877	1.68485
$389$	10.8845	0.551867	0.275934	−	0.961177i	$-0.411013\pi$
$389$	10.8845	0.551867	0.275934	+	0.961177i	$0.411013\pi$
$390$	0	0
$391$	−6.19336	−0.313212
$392$	−6.82072	−0.344498
$393$	−3.79339	−0.191351
$394$	−7.91963	−0.398985
$395$	0	0
$396$	−18.4932	−0.929319
$397$	−16.2212	−0.814120	−0.407060	−	0.913401i	$-0.633446\pi$
$397$	−16.2212	−0.814120	−0.407060	+	0.913401i	$0.633446\pi$
$398$	−40.8647	−2.04836
$399$	1.30233	0.0651981
$400$	0	0
$401$	3.78686	0.189107	0.0945534	−	0.995520i	$-0.469858\pi$
$401$	3.78686	0.189107	0.0945534	+	0.995520i	$0.469858\pi$
$402$	11.7237	0.584726
$403$	−9.19118	−0.457845
$404$	31.1483	1.54968
$405$	0	0
$406$	27.7832	1.37886
$407$	−7.64044	−0.378722
$408$	−2.71894	−0.134607
$409$	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
$411$	4.43429	0.218728
$412$	−30.2750	−1.49154
$413$	−32.3181	−1.59027
$414$	21.3310	1.04836
$415$	0	0
$416$	7.32918	0.359343
$417$	−8.50790	−0.416634
$418$	−4.17202	−0.204060
$419$	14.3472	0.700907	0.350453	−	0.936580i	$-0.386028\pi$
$419$	14.3472	0.700907	0.350453	+	0.936580i	$0.386028\pi$
$420$	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
$423$	−13.3173	−0.647509
$424$	−29.2525	−1.42063
$425$	0	0
$426$	13.6645	0.662046
$427$	−43.3338	−2.09707
$428$	−18.7591	−0.906755
$429$	−1.35557	−0.0654477
$430$	0	0
$431$	12.5043	0.602311	0.301156	−	0.953575i	$-0.402628\pi$
$431$	12.5043	0.602311	0.301156	+	0.953575i	$0.402628\pi$
$432$	1.21133	0.0582801
$433$	22.4951	1.08105	0.540524	−	0.841329i	$-0.318226\pi$
$433$	22.4951	1.08105	0.540524	+	0.841329i	$0.318226\pi$
$434$	−45.1476	−2.16715
$435$	0	0
$436$	−33.7041	−1.61413
$437$	3.00744	0.143865
$438$	−0.293506	−0.0140243
$439$	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
$441$	−6.14873	−0.292797
$442$	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	$0.336272\pi$
$444$	6.04638	0.286949
$445$	0	0
$446$	66.3103	3.13988
$447$	2.99851	0.141825
$448$	38.6833	1.82761
$449$	−25.9539	−1.22484	−0.612420	−	0.790533i	$-0.709804\pi$
$449$	−25.9539	−1.22484	−0.612420	+	0.790533i	$0.709804\pi$
$450$	0	0
$451$	−3.66844	−0.172740
$452$	−35.7004	−1.67921
$453$	−2.24129	−0.105305
$454$	27.0662	1.27028
$455$	0	0
$456$	1.32029	0.0618283
$457$	8.50150	0.397684	0.198842	−	0.980032i	$-0.436282\pi$
$457$	8.50150	0.397684	0.198842	+	0.980032i	$0.436282\pi$
$458$	37.8751	1.76979
$459$	−5.10137	−0.238112
$460$	0	0
$461$	14.7851	0.688609	0.344305	−	0.938858i	$-0.388115\pi$
$461$	14.7851	0.688609	0.344305	+	0.938858i	$0.388115\pi$
$462$	−6.65865	−0.309788
$463$	22.1921	1.03135	0.515677	−	0.856783i	$-0.327540\pi$
$463$	22.1921	1.03135	0.515677	+	0.856783i	$0.327540\pi$
$464$	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	$0.270792\pi$
$468$	−13.1968	−0.610024
$469$	−32.4534	−1.49856
$470$	0	0
$471$	0.697669	0.0321469
$472$	−32.7638	−1.50808
$473$	7.18891	0.330546
$474$	−9.40048	−0.431779
$475$	0	0
$476$	18.8213	0.862671
$477$	−26.3706	−1.20743
$478$	15.3182	0.700637
$479$	25.0569	1.14488	0.572440	−	0.819947i	$-0.305997\pi$
$479$	25.0569	1.14488	0.572440	+	0.819947i	$0.305997\pi$
$480$	0	0
$481$	−5.45225	−0.248601
$482$	−60.5632	−2.75858
$483$	4.79995	0.218405
$484$	−23.3292	−1.06042
$485$	0	0
$486$	26.6981	1.21105
$487$	1.46479	0.0663758	0.0331879	−	0.999449i	$-0.489434\pi$
$487$	1.46479	0.0663758	0.0331879	+	0.999449i	$0.489434\pi$
$488$	−43.9314	−1.98868
$489$	2.11792	0.0957757
$490$	0	0
$491$	−20.0686	−0.905685	−0.452843	−	0.891591i	$-0.649590\pi$
$491$	−20.0686	−0.905685	−0.452843	+	0.891591i	$0.649590\pi$
$492$	2.90308	0.130881
$493$	−7.37229	−0.332031
$494$	−2.97718	−0.133950
$495$	0	0
$496$	−2.84465	−0.127729
$497$	−37.8258	−1.69672
$498$	13.8381	0.620101
$499$	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
$501$	4.95498	0.221372
$502$	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	$0.358377\pi$
$504$	−25.9226	−1.15469
$505$	0	0
$506$	−15.3766	−0.683575
$507$	5.20639	0.231224
$508$	37.8631	1.67990
$509$	−10.5202	−0.466298	−0.233149	−	0.972441i	$-0.574903\pi$
$509$	−10.5202	−0.466298	−0.233149	+	0.972441i	$0.574903\pi$
$510$	0	0
$511$	0.812479	0.0359420
$512$	4.98730	0.220410
$513$	2.47718	0.109370
$514$	15.1962	0.670276
$515$	0	0
$516$	−5.68906	−0.250447
$517$	9.59989	0.422203
$518$	−26.7818	−1.17672
$519$	−3.64545	−0.160017
$520$	0	0
$521$	−10.0070	−0.438413	−0.219207	−	0.975678i	$-0.570347\pi$
$521$	−10.0070	−0.438413	−0.219207	+	0.975678i	$0.570347\pi$
$522$	25.3914	1.11135
$523$	−22.7830	−0.996233	−0.498117	−	0.867110i	$-0.665975\pi$
$523$	−22.7830	−0.996233	−0.498117	+	0.867110i	$0.665975\pi$
$524$	26.6210	1.16295
$525$	0	0
$526$	−62.5981	−2.72941
$527$	11.9799	0.521854
$528$	−0.419546	−0.0182584
$529$	−11.9156	−0.518070
$530$	0	0
$531$	−29.5359	−1.28175
$532$	−9.13943	−0.396245
$533$	−2.61782	−0.113390
$534$	−5.22893	−0.226278
$535$	0	0
$536$	−32.9010	−1.42111
$537$	−7.36896	−0.317994
$538$	−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
$543$	0.755360	0.0324156
$544$	−9.55296	−0.409580
$545$	0	0
$546$	−4.75164	−0.203352
$547$	13.5883	0.580993	0.290497	−	0.956876i	$-0.406179\pi$
$547$	13.5883	0.580993	0.290497	+	0.956876i	$0.406179\pi$
$548$	−31.1188	−1.32933
$549$	−39.6033	−1.69023
$550$	0	0
$551$	3.57992	0.152510
$552$	4.86614	0.207117
$553$	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	$0.300870\pi$
$558$	−41.2609	−1.74671
$559$	5.13004	0.216978
$560$	0	0
$561$	1.76687	0.0745974
$562$	−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	$0.488868\pi$
$564$	−7.59703	−0.319893
$565$	0	0
$566$	−19.9548	−0.838763
$567$	−21.3147	−0.895133
$568$	−38.3475	−1.60902
$569$	17.8828	0.749686	0.374843	−	0.927088i	$-0.377697\pi$
$569$	17.8828	0.749686	0.374843	+	0.927088i	$0.377697\pi$
$570$	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
$573$	−9.34060	−0.390209
$574$	−12.8589	−0.536718
$575$	0	0
$576$	35.3531	1.47305
$577$	22.8137	0.949746	0.474873	−	0.880054i	$-0.342494\pi$
$577$	22.8137	0.949746	0.474873	+	0.880054i	$0.342494\pi$
$578$	31.2664	1.30051
$579$	6.22599	0.258743
$580$	0	0
$581$	−38.3065	−1.58922
$582$	10.9208	0.452682
$583$	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	$0.426529\pi$
$588$	−3.50763	−0.144652
$589$	−5.81734	−0.239699
$590$	0	0
$591$	−1.62867	−0.0669947
$592$	−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	$0.573405\pi$
$594$	−12.6655	−0.519672
$595$	0	0
$596$	−21.0428	−0.861947
$597$	−8.40384	−0.343946
$598$	−10.9729	−0.448713
$599$	36.2736	1.48210	0.741049	−	0.671451i	$-0.234329\pi$
$599$	36.2736	1.48210	0.741049	+	0.671451i	$0.234329\pi$
$600$	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
$603$	−29.6596	−1.20783
$604$	15.7288	0.639997
$605$	0	0
$606$	10.2497	0.416366
$607$	33.5066	1.35999	0.679996	−	0.733216i	$-0.261982\pi$
$607$	33.5066	1.35999	0.679996	+	0.733216i	$0.261982\pi$
$608$	4.63883	0.188129
$609$	5.71363	0.231528
$610$	0	0
$611$	6.85053	0.277143
$612$	17.2010	0.695308
$613$	28.0289	1.13208	0.566038	−	0.824379i	$-0.308476\pi$
$613$	28.0289	1.13208	0.566038	+	0.824379i	$0.308476\pi$
$614$	−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	$0.710598\pi$
$618$	−9.96234	−0.400744
$619$	−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
$621$	9.13004	0.366376
$622$	18.0966	0.725608
$623$	14.4746	0.579914
$624$	−0.299391	−0.0119852
$625$	0	0
$626$	49.4399	1.97601
$627$	−0.857977	−0.0342643
$628$	−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
$633$	1.54042	0.0612264
$634$	9.28290	0.368671
$635$	0	0
$636$	−15.0434	−0.596511
$637$	3.16296	0.125321
$638$	−18.3036	−0.724648
$639$	−34.5694	−1.36755
$640$	0	0
$641$	−22.1774	−0.875956	−0.437978	−	0.898986i	$-0.644305\pi$
$641$	−22.1774	−0.875956	−0.437978	+	0.898986i	$0.644305\pi$
$642$	−6.17290	−0.243625
$643$	13.2767	0.523583	0.261792	−	0.965124i	$-0.415687\pi$
$643$	13.2767	0.523583	0.261792	+	0.965124i	$0.415687\pi$
$644$	−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	$0.428795\pi$
$648$	−21.6086	−0.848867
$649$	21.2912	0.835754
$650$	0	0
$651$	−9.28462	−0.363893
$652$	−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	$0.252852\pi$
$654$	−11.0907	−0.433682
$655$	0	0
$656$	−0.810208	−0.0316333
$657$	0.742534	0.0289690
$658$	33.6502	1.31182
$659$	39.7655	1.54905	0.774523	−	0.632546i	$-0.217990\pi$
$659$	39.7655	1.54905	0.774523	+	0.632546i	$0.217990\pi$
$660$	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
$663$	1.26085	0.0489673
$664$	−38.8347	−1.50708
$665$	0	0
$666$	−24.4762	−0.948432
$667$	13.1943	0.510887
$668$	−34.7728	−1.34540
$669$	13.6367	0.527226
$670$	0	0
$671$	28.5484	1.10210
$672$	7.40368	0.285603
$673$	41.4627	1.59827	0.799135	−	0.601151i	$-0.205291\pi$
$673$	41.4627	1.59827	0.799135	+	0.601151i	$0.205291\pi$
$674$	−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	$0.491196\pi$
$678$	−11.7477	−0.451166
$679$	−30.2308	−1.16015
$680$	0	0
$681$	5.56616	0.213296
$682$	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	$0.448090\pi$
$684$	−8.35263	−0.319371
$685$	0	0
$686$	−33.5371	−1.28045
$687$	7.78902	0.297170
$688$	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
$693$	16.8456	0.639910
$694$	36.0183	1.36724
$695$	0	0
$696$	5.79243	0.219561
$697$	3.41210	0.129243
$698$	−12.8534	−0.486508
$699$	10.6922	0.404418
$700$	0	0
$701$	0.840795	0.0317564	0.0158782	−	0.999874i	$-0.494946\pi$
$701$	0.840795	0.0317564	0.0158782	+	0.999874i	$0.494946\pi$
$702$	−9.03816	−0.341124
$703$	−3.45087	−0.130152
$704$	−25.4846	−0.960487
$705$	0	0
$706$	18.5254	0.697212
$707$	−28.3731	−1.06708
$708$	−16.8492	−0.633230
$709$	−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
$711$	23.7821	0.891897
$712$	14.6743	0.549941
$713$	−21.4407	−0.802962
$714$	6.19336	0.231781
$715$	0	0
$716$	51.7135	1.93262
$717$	3.15019	0.117646
$718$	28.5027	1.06371
$719$	43.4148	1.61910	0.809550	−	0.587051i	$-0.199711\pi$
$719$	43.4148	1.61910	0.809550	+	0.587051i	$0.199711\pi$
$720$	0	0
$721$	27.5776	1.02704
$722$	41.9919	1.56278
$723$	−12.4548	−0.463200
$724$	−5.30093	−0.197007
$725$	0	0
$726$	−7.67677	−0.284912
$727$	32.0480	1.18859	0.594297	−	0.804245i	$-0.297430\pi$
$727$	32.0480	1.18859	0.594297	+	0.804245i	$0.297430\pi$
$728$	13.3348	0.494222
$729$	−15.5727	−0.576768
$730$	0	0
$731$	−6.68657	−0.247312
$732$	−22.5922	−0.835032
$733$	−8.13928	−0.300631	−0.150316	−	0.988638i	$-0.548029\pi$
$733$	−8.13928	−0.300631	−0.150316	+	0.988638i	$0.548029\pi$
$734$	62.0616	2.29074
$735$	0	0
$736$	17.0971	0.630209
$737$	21.3804	0.787556
$738$	−11.7519	−0.432592
$739$	−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
$741$	−0.612257	−0.0224918
$742$	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	$0.631612\pi$
$744$	−9.41266	−0.345085
$745$	0	0
$746$	−63.8258	−2.33683
$747$	−35.0087	−1.28090
$748$	−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
$753$	5.18219	0.188849
$754$	−13.0616	−0.475674
$755$	0	0
$756$	−27.7457	−1.00910
$757$	−45.6524	−1.65926	−0.829632	−	0.558311i	$-0.811450\pi$
$757$	−45.6524	−1.65926	−0.829632	+	0.558311i	$0.811450\pi$
$758$	8.02386	0.291440
$759$	−3.16221	−0.114781
$760$	0	0
$761$	39.9058	1.44658	0.723291	−	0.690543i	$-0.242628\pi$
$761$	39.9058	1.44658	0.723291	+	0.690543i	$0.242628\pi$
$762$	12.4593	0.451353
$763$	30.7012	1.11146
$764$	65.5500	2.37151
$765$	0	0
$766$	63.2092	2.28384
$767$	15.1935	0.548607
$768$	8.90403	0.321296
$769$	−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
$771$	3.12510	0.112548
$772$	−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	$0.256609\pi$
$774$	23.0297	0.827785
$775$	0	0
$776$	−30.6477	−1.10019
$777$	−5.50768	−0.197587
$778$	−25.1353	−0.901146
$779$	−1.65689	−0.0593641
$780$	0	0
$781$	24.9197	0.891697
$782$	14.3022	0.511445
$783$	10.8680	0.388390
$784$	0.978929	0.0349618
$785$	0	0
$786$	8.75997	0.312458
$787$	53.0202	1.88997	0.944983	−	0.327120i	$-0.106078\pi$
$787$	53.0202	1.88997	0.944983	+	0.327120i	$0.106078\pi$
$788$	11.4296	0.407164
$789$	−12.8733	−0.458302
$790$	0	0
$791$	32.5197	1.15627
$792$	17.0779	0.606835
$793$	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	$0.427919\pi$
$798$	−3.00744	−0.106462
$799$	−8.92908	−0.315888
$800$	0	0
$801$	13.2285	0.467407
$802$	−8.74490	−0.308793
$803$	−0.535262	−0.0188890
$804$	−16.9197	−0.596712
$805$	0	0
$806$	21.2250	0.747618
$807$	−0.479392	−0.0168754
$808$	−28.7644	−1.01193
$809$	−41.8935	−1.47290	−0.736449	−	0.676493i	$-0.763499\pi$
$809$	−41.8935	−1.47290	−0.736449	+	0.676493i	$0.763499\pi$
$810$	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
$813$	−2.96911	−0.104131
$814$	17.6439	0.618417
$815$	0	0
$816$	0.390230	0.0136608
$817$	3.24694	0.113596
$818$	−4.28568	−0.149845
$819$	12.0211	0.420050
$820$	0	0
$821$	−21.1349	−0.737612	−0.368806	−	0.929506i	$-0.620233\pi$
$821$	−21.1349	−0.737612	−0.368806	+	0.929506i	$0.620233\pi$
$822$	−10.2400	−0.357161
$823$	−3.17714	−0.110748	−0.0553740	−	0.998466i	$-0.517635\pi$
$823$	−3.17714	−0.110748	−0.0553740	+	0.998466i	$0.517635\pi$
$824$	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	$0.554193\pi$
$828$	−30.7849	−1.06985
$829$	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
$831$	−11.7160	−0.406424
$832$	−18.1859	−0.630484
$833$	−4.12265	−0.142841
$834$	19.6471	0.680323
$835$	0	0
$836$	6.02107	0.208243
$837$	−17.6604	−0.610432
$838$	−33.1316	−1.14451
$839$	42.6819	1.47354	0.736771	−	0.676142i	$-0.236350\pi$
$839$	42.6819	1.47354	0.736771	+	0.676142i	$0.236350\pi$
$840$	0	0
$841$	−13.2941	−0.458416
$842$	−35.6887	−1.22992
$843$	−0.873305	−0.0300782
$844$	−10.8103	−0.372106
$845$	0	0
$846$	30.7533	1.05732
$847$	21.2507	0.730183
$848$	4.19841	0.144174
$849$	−4.10371	−0.140839
$850$	0	0
$851$	−12.7187	−0.435993
$852$	−19.7206	−0.675617
$853$	−16.9610	−0.580733	−0.290367	−	0.956915i	$-0.593777\pi$
$853$	−16.9610	−0.580733	−0.290367	+	0.956915i	$0.593777\pi$
$854$	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	$0.265637\pi$
$858$	3.13039	0.106870
$859$	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
$861$	−2.64443	−0.0901219
$862$	−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	$0.504927\pi$
$864$	14.0826	0.479101
$865$	0	0
$866$	−51.9474	−1.76525
$867$	6.42994	0.218372
$868$	65.1571	2.21158
$869$	−17.1435	−0.581554
$870$	0	0
$871$	15.2571	0.516968
$872$	31.1246	1.05401
$873$	−27.6283	−0.935075
$874$	−6.94501	−0.234918
$875$	0	0
$876$	0.423589	0.0143117
$877$	−33.6613	−1.13666	−0.568331	−	0.822800i	$-0.692411\pi$
$877$	−33.6613	−1.13666	−0.568331	+	0.822800i	$0.692411\pi$
$878$	−28.1213	−0.949050
$879$	2.98788	0.100779
$880$	0	0
$881$	−19.9283	−0.671402	−0.335701	−	0.941969i	$-0.608973\pi$
$881$	−19.9283	−0.671402	−0.335701	+	0.941969i	$0.608973\pi$
$882$	14.1991	0.478109
$883$	−15.6081	−0.525255	−0.262627	−	0.964897i	$-0.584589\pi$
$883$	−15.6081	−0.525255	−0.262627	+	0.964897i	$0.584589\pi$
$884$	−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	$0.788532\pi$
$888$	−5.58363	−0.187374
$889$	−34.4897	−1.15675
$890$	0	0
$891$	14.0421	0.470429
$892$	−95.6991	−3.20424
$893$	4.33588	0.145095
$894$	−6.92439	−0.231586
$895$	0	0
$896$	−58.1505	−1.94267
$897$	−2.25657	−0.0753447
$898$	59.9347	2.00005
$899$	−25.5220	−0.851208
$900$	0	0
$901$	−17.6811	−0.589044
$902$	8.47143	0.282068
$903$	5.18219	0.172452
$904$	32.9682	1.09651
$905$	0	0
$906$	5.17576	0.171953
$907$	−1.43447	−0.0476308	−0.0238154	−	0.999716i	$-0.507581\pi$
$907$	−1.43447	−0.0476308	−0.0238154	+	0.999716i	$0.507581\pi$
$908$	−39.0619	−1.29632
$909$	−25.9305	−0.860061
$910$	0	0
$911$	−2.81129	−0.0931422	−0.0465711	−	0.998915i	$-0.514829\pi$
$911$	−2.81129	−0.0931422	−0.0465711	+	0.998915i	$0.514829\pi$
$912$	−0.189492	−0.00627471
$913$	25.2363	0.835201
$914$	−19.6323	−0.649379
$915$	0	0
$916$	−54.6614	−1.80606
$917$	−24.2492	−0.800780
$918$	11.7805	0.388814
$919$	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
$921$	−13.6136	−0.448584
$922$	−34.1428	−1.12443
$923$	17.7828	0.585329
$924$	9.60977	0.316138
$925$	0	0
$926$	−51.2477	−1.68410
$927$	25.2035	0.827791
$928$	20.3516	0.668075
$929$	33.3227	1.09328	0.546642	−	0.837367i	$-0.315906\pi$
$929$	33.3227	1.09328	0.546642	+	0.837367i	$0.315906\pi$
$930$	0	0
$931$	2.00192	0.0656104
$932$	−75.0355	−2.45787
$933$	3.72157	0.121839
$934$	−65.8175	−2.15361
$935$	0	0
$936$	12.1869	0.398340
$937$	−13.3675	−0.436698	−0.218349	−	0.975871i	$-0.570067\pi$
$937$	−13.3675	−0.436698	−0.218349	+	0.975871i	$0.570067\pi$
$938$	74.9439	2.44701
$939$	10.1673	0.331798
$940$	0	0
$941$	−2.14982	−0.0700820	−0.0350410	−	0.999386i	$-0.511156\pi$
$941$	−2.14982	−0.0700820	−0.0350410	+	0.999386i	$0.511156\pi$
$942$	−1.61111	−0.0524928
$943$	−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	$0.692030\pi$
$948$	13.5668	0.440629
$949$	−0.381966	−0.0123991
$950$	0	0
$951$	1.90903	0.0619046
$952$	−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	$0.457189\pi$
$954$	60.8969	1.97161
$955$	0	0
$956$	−22.1072	−0.714998
$957$	−3.76415	−0.121678
$958$	−57.8633	−1.86948
$959$	28.3462	0.915347
$960$	0	0
$961$	10.4732	0.337844
$962$	12.5908	0.405942
$963$	15.6167	0.503241
$964$	87.4048	2.81512
$965$	0	0
$966$	−11.0844	−0.356635
$967$	−29.4871	−0.948241	−0.474121	−	0.880460i	$-0.657234\pi$
$967$	−29.4871	−0.948241	−0.474121	+	0.880460i	$0.657234\pi$
$968$	21.5438	0.692443
$969$	0.798025	0.0256363
$970$	0	0
$971$	21.8666	0.701733	0.350867	−	0.936425i	$-0.385887\pi$
$971$	21.8666	0.701733	0.350867	+	0.936425i	$0.385887\pi$
$972$	−38.5308	−1.23588
$973$	−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	$0.571102\pi$
$978$	−4.89086	−0.156392
$979$	−9.53591	−0.304769
$980$	0	0
$981$	28.0582	0.895828
$982$	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	$0.514906\pi$
$984$	−2.68090	−0.0854639
$985$	0	0
$986$	17.0246	0.542175
$987$	6.92017	0.220271
$988$	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
$993$	−5.52970	−0.175480
$994$	87.3502	2.77058
$995$	0	0
$996$	−19.9712	−0.632811
$997$	12.0885	0.382845	0.191423	−	0.981508i	$-0.438690\pi$
$997$	12.0885	0.382845	0.191423	+	0.981508i	$0.438690\pi$
$998$	−1.44329	−0.0456867
$999$	−10.4762	−0.331453

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	625.2.a.f.1.1		8
3.2	odd	2		5625.2.a.x.1.8		8
4.3	odd	2		10000.2.a.bj.1.5		8
5.2	odd	4		625.2.b.c.624.1		8
5.3	odd	4		625.2.b.c.624.8		8
5.4	even	2	inner	625.2.a.f.1.8		8
15.14	odd	2		5625.2.a.x.1.1		8
20.19	odd	2		10000.2.a.bj.1.4		8
25.2	odd	20		625.2.e.i.124.1		8
25.3	odd	20		125.2.e.b.49.2		8
25.4	even	10		125.2.d.b.76.1		16
25.6	even	5		125.2.d.b.51.4		16
25.8	odd	20		25.2.e.a.14.1	yes	8
25.9	even	10		625.2.d.o.126.4		16
25.11	even	5		625.2.d.o.501.1		16
25.12	odd	20		625.2.e.a.499.2		8
25.13	odd	20		625.2.e.i.499.1		8
25.14	even	10		625.2.d.o.501.4		16
25.16	even	5		625.2.d.o.126.1		16
25.17	odd	20		125.2.e.b.74.2		8
25.19	even	10		125.2.d.b.51.1		16
25.21	even	5		125.2.d.b.76.4		16
25.22	odd	20		25.2.e.a.9.1	✓	8
25.23	odd	20		625.2.e.a.124.2		8
75.8	even	20		225.2.m.a.64.2		8
75.47	even	20		225.2.m.a.109.2		8
100.47	even	20		400.2.y.c.209.1		8
100.83	even	20		400.2.y.c.289.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.2.e.a.9.1	✓	8	25.22	odd	20
25.2.e.a.14.1	yes	8	25.8	odd	20
125.2.d.b.51.1		16	25.19	even	10
125.2.d.b.51.4		16	25.6	even	5
125.2.d.b.76.1		16	25.4	even	10
125.2.d.b.76.4		16	25.21	even	5
125.2.e.b.49.2		8	25.3	odd	20
125.2.e.b.74.2		8	25.17	odd	20
225.2.m.a.64.2		8	75.8	even	20
225.2.m.a.109.2		8	75.47	even	20
400.2.y.c.209.1		8	100.47	even	20
400.2.y.c.289.1		8	100.83	even	20
625.2.a.f.1.1		8	1.1	even	1	trivial
625.2.a.f.1.8		8	5.4	even	2	inner
625.2.b.c.624.1		8	5.2	odd	4
625.2.b.c.624.8		8	5.3	odd	4
625.2.d.o.126.1		16	25.16	even	5
625.2.d.o.126.4		16	25.9	even	10
625.2.d.o.501.1		16	25.11	even	5
625.2.d.o.501.4		16	25.14	even	10
625.2.e.a.124.2		8	25.23	odd	20
625.2.e.a.499.2		8	25.12	odd	20
625.2.e.i.124.1		8	25.2	odd	20
625.2.e.i.499.1		8	25.13	odd	20
5625.2.a.x.1.1		8	15.14	odd	2
5625.2.a.x.1.8		8	3.2	odd	2
10000.2.a.bj.1.4		8	20.19	odd	2
10000.2.a.bj.1.5		8	4.3	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:	\( N \)	\(=\)	\( 625 = 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	625.a (trivial)

\(f(q)\)	\(=\)	\(q-2.30927 q^{2} -0.474903 q^{3} +3.33275 q^{4} +1.09668 q^{6} -3.03582 q^{7} -3.07768 q^{8} -2.77447 q^{9} +O(q^{10})\)	\(q-2.30927 q^{2} -0.474903 q^{3} +3.33275 q^{4} +1.09668 q^{6} -3.03582 q^{7} -3.07768 q^{8} -2.77447 q^{9} +2.00000 q^{11} -1.58273 q^{12} +1.42721 q^{13} +7.01054 q^{14} +0.441718 q^{16} -1.86025 q^{17} +6.40701 q^{18} +0.903319 q^{19} +1.44172 q^{21} -4.61855 q^{22} +3.32932 q^{23} +1.46160 q^{24} -3.29582 q^{26} +2.74231 q^{27} -10.1176 q^{28} +3.96307 q^{29} -6.43997 q^{31} +5.13532 q^{32} -0.949806 q^{33} +4.29582 q^{34} -9.24660 q^{36} -3.82022 q^{37} -2.08601 q^{38} -0.677786 q^{39} -1.83422 q^{41} -3.32932 q^{42} +3.59445 q^{43} +6.66550 q^{44} -7.68832 q^{46} +4.79995 q^{47} -0.209773 q^{48} +2.21619 q^{49} +0.883436 q^{51} +4.75653 q^{52} +9.50473 q^{53} -6.33275 q^{54} +9.34328 q^{56} -0.428989 q^{57} -9.15182 q^{58} +10.6456 q^{59} +14.2742 q^{61} +14.8716 q^{62} +8.42278 q^{63} -12.7423 q^{64} +2.19336 q^{66} +10.6902 q^{67} -6.19974 q^{68} -1.58111 q^{69} +12.4598 q^{71} +8.53893 q^{72} -0.267631 q^{73} +8.82193 q^{74} +3.01054 q^{76} -6.07163 q^{77} +1.56519 q^{78} -8.57176 q^{79} +7.02107 q^{81} +4.23572 q^{82} +12.6182 q^{83} +4.80489 q^{84} -8.30058 q^{86} -1.88207 q^{87} -6.15537 q^{88} -4.76796 q^{89} -4.33275 q^{91} +11.0958 q^{92} +3.05836 q^{93} -11.0844 q^{94} -2.43878 q^{96} +9.95805 q^{97} -5.11778 q^{98} -5.54893 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 6 q^{4} + 6 q^{6} + 4 q^{9}+O(q^{10}) \) 8 * q + 6 * q^4 + 6 * q^6 + 4 * q^9	\( 8 q + 6 q^{4} + 6 q^{6} + 4 q^{9} + 16 q^{11} + 12 q^{14} - 2 q^{16} + 10 q^{19} + 6 q^{21} + 20 q^{24} + 6 q^{26} + 20 q^{29} + 16 q^{31} + 2 q^{34} - 12 q^{36} + 18 q^{39} + 26 q^{41} + 12 q^{44} + 6 q^{46} - 14 q^{49} - 4 q^{51} - 30 q^{54} + 10 q^{56} + 30 q^{59} + 6 q^{61} - 44 q^{64} + 12 q^{66} + 8 q^{69} + 46 q^{71} + 12 q^{74} - 20 q^{76} + 10 q^{79} - 32 q^{81} - 18 q^{84} - 14 q^{86} + 30 q^{89} - 14 q^{91} - 68 q^{94} - 54 q^{96} + 8 q^{99}+O(q^{100}) \) 8 * q + 6 * q^4 + 6 * q^6 + 4 * q^9 + 16 * q^11 + 12 * q^14 - 2 * q^16 + 10 * q^19 + 6 * q^21 + 20 * q^24 + 6 * q^26 + 20 * q^29 + 16 * q^31 + 2 * q^34 - 12 * q^36 + 18 * q^39 + 26 * q^41 + 12 * q^44 + 6 * q^46 - 14 * q^49 - 4 * q^51 - 30 * q^54 + 10 * q^56 + 30 * q^59 + 6 * q^61 - 44 * q^64 + 12 * q^66 + 8 * q^69 + 46 * q^71 + 12 * q^74 - 20 * q^76 + 10 * q^79 - 32 * q^81 - 18 * q^84 - 14 * q^86 + 30 * q^89 - 14 * q^91 - 68 * q^94 - 54 * q^96 + 8 * q^99

Introduction

L-functions

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Database

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