LMFDB - Embedded newform 625.2.a.f.1.3

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			1.3
Root			$-1.13370$ 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)

$f(q)$	$=$	$q-1.13370 q^{2} +2.60278 q^{3} -0.714715 q^{4} -2.95078 q^{6} -0.407162 q^{7} +3.07768 q^{8} +3.77447 q^{9} +O(q^{10})$	\(q-1.13370 q^{2} +2.60278 q^{3} -0.714715 q^{4} -2.95078 q^{6} -0.407162 q^{7} +3.07768 q^{8} +3.77447 q^{9} +2.00000 q^{11} -1.86025 q^{12} +0.700668 q^{13} +0.461601 q^{14} -2.05975 q^{16} -1.58273 q^{17} -4.27913 q^{18} +4.95078 q^{19} -1.05975 q^{21} -2.26741 q^{22} -1.20145 q^{23} +8.01054 q^{24} -0.794350 q^{26} +2.01577 q^{27} +0.291004 q^{28} +5.50906 q^{29} +8.20390 q^{31} -3.82022 q^{32} +5.20556 q^{33} +1.79435 q^{34} -2.69767 q^{36} +5.13532 q^{37} -5.61272 q^{38} +1.82368 q^{39} +7.21619 q^{41} +1.20145 q^{42} -9.16531 q^{43} -1.42943 q^{44} +1.36208 q^{46} +1.27323 q^{47} -5.36108 q^{48} -6.83422 q^{49} -4.11950 q^{51} -0.500778 q^{52} +5.07996 q^{53} -2.28528 q^{54} -1.25311 q^{56} +12.8858 q^{57} -6.24565 q^{58} -6.49972 q^{59} -9.42008 q^{61} -9.30079 q^{62} -1.53682 q^{63} +8.45050 q^{64} -5.90157 q^{66} +3.08173 q^{67} +1.13120 q^{68} -3.12710 q^{69} +6.86639 q^{71} +11.6166 q^{72} -0.545146 q^{73} -5.82193 q^{74} -3.53840 q^{76} -0.814323 q^{77} -2.06752 q^{78} +5.48159 q^{79} -6.07680 q^{81} -8.18102 q^{82} -0.974135 q^{83} +0.757421 q^{84} +10.3907 q^{86} +14.3389 q^{87} +6.15537 q^{88} -2.26649 q^{89} -0.285285 q^{91} +0.858691 q^{92} +21.3529 q^{93} -1.44347 q^{94} -9.94319 q^{96} -15.2185 q^{97} +7.74798 q^{98} +7.54893 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.13370	−0.801650	−0.400825	−	0.916155i	$-0.631276\pi$
$2$	−1.13370	−0.801650	−0.400825	+	0.916155i	$0.631276\pi$
$3$	2.60278	1.50272	0.751358	−	0.659895i	$-0.229399\pi$
$3$	2.60278	1.50272	0.751358	+	0.659895i	$0.229399\pi$
$4$	−0.714715	−0.357358
$5$	0	0
$5$	0	0
$6$	−2.95078	−1.20465
$7$	−0.407162	−0.153893	−0.0769463	−	0.997035i	$-0.524517\pi$
$7$	−0.407162	−0.153893	−0.0769463	+	0.997035i	$0.524517\pi$
$8$	3.07768	1.08813
$9$	3.77447	1.25816
$10$	0	0
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	−1.86025	−0.537007
$13$	0.700668	0.194330	0.0971651	−	0.995268i	$-0.469023\pi$
$13$	0.700668	0.194330	0.0971651	+	0.995268i	$0.469023\pi$
$14$	0.461601	0.123368
$15$	0	0
$16$	−2.05975	−0.514938
$17$	−1.58273	−0.383869	−0.191934	−	0.981408i	$-0.561476\pi$
$17$	−1.58273	−0.383869	−0.191934	+	0.981408i	$0.561476\pi$
$18$	−4.27913	−1.00860
$19$	4.95078	1.13579	0.567894	−	0.823102i	$-0.307758\pi$
$19$	4.95078	1.13579	0.567894	+	0.823102i	$0.307758\pi$
$20$	0	0
$21$	−1.05975	−0.231257
$22$	−2.26741	−0.483413
$23$	−1.20145	−0.250519	−0.125259	−	0.992124i	$-0.539976\pi$
$23$	−1.20145	−0.250519	−0.125259	+	0.992124i	$0.539976\pi$
$24$	8.01054	1.63514
$25$	0	0
$26$	−0.794350	−0.155785
$27$	2.01577	0.387935
$28$	0.291004	0.0549947
$29$	5.50906	1.02301	0.511504	−	0.859281i	$-0.329089\pi$
$29$	5.50906	1.02301	0.511504	+	0.859281i	$0.329089\pi$
$30$	0	0
$31$	8.20390	1.47346	0.736732	−	0.676185i	$-0.236368\pi$
$31$	8.20390	1.47346	0.736732	+	0.676185i	$0.236368\pi$
$32$	−3.82022	−0.675325
$33$	5.20556	0.906172
$34$	1.79435	0.307728
$35$	0	0
$36$	−2.69767	−0.449611
$37$	5.13532	0.844241	0.422121	−	0.906540i	$-0.361286\pi$
$37$	5.13532	0.844241	0.422121	+	0.906540i	$0.361286\pi$
$38$	−5.61272	−0.910504
$39$	1.82368	0.292023
$40$	0	0
$41$	7.21619	1.12698	0.563489	−	0.826123i	$-0.309459\pi$
$41$	7.21619	1.12698	0.563489	+	0.826123i	$0.309459\pi$
$42$	1.20145	0.185387
$43$	−9.16531	−1.39770	−0.698848	−	0.715270i	$-0.746304\pi$
$43$	−9.16531	−1.39770	−0.698848	+	0.715270i	$0.746304\pi$
$44$	−1.42943	−0.215495
$45$	0	0
$46$	1.36208	0.200828
$47$	1.27323	0.185720	0.0928602	−	0.995679i	$-0.470399\pi$
$47$	1.27323	0.185720	0.0928602	+	0.995679i	$0.470399\pi$
$48$	−5.36108	−0.773806
$49$	−6.83422	−0.976317
$50$	0	0
$51$	−4.11950	−0.576846
$52$	−0.500778	−0.0694454
$53$	5.07996	0.697786	0.348893	−	0.937162i	$-0.386558\pi$
$53$	5.07996	0.697786	0.348893	+	0.937162i	$0.386558\pi$
$54$	−2.28528	−0.310988
$55$	0	0
$56$	−1.25311	−0.167454
$57$	12.8858	1.70677
$58$	−6.24565	−0.820094
$59$	−6.49972	−0.846191	−0.423096	−	0.906085i	$-0.639057\pi$
$59$	−6.49972	−0.846191	−0.423096	+	0.906085i	$0.639057\pi$
$60$	0	0
$61$	−9.42008	−1.20612	−0.603059	−	0.797697i	$-0.706052\pi$
$61$	−9.42008	−1.20612	−0.603059	+	0.797697i	$0.706052\pi$
$62$	−9.30079	−1.18120
$63$	−1.53682	−0.193621
$64$	8.45050	1.05631
$65$	0	0
$66$	−5.90157	−0.726433
$67$	3.08173	0.376493	0.188247	−	0.982122i	$-0.439720\pi$
$67$	3.08173	0.376493	0.188247	+	0.982122i	$0.439720\pi$
$68$	1.13120	0.137178
$69$	−3.12710	−0.376458
$70$	0	0
$71$	6.86639	0.814891	0.407445	−	0.913230i	$-0.366420\pi$
$71$	6.86639	0.814891	0.407445	+	0.913230i	$0.366420\pi$
$72$	11.6166	1.36903
$73$	−0.545146	−0.0638045	−0.0319022	−	0.999491i	$-0.510157\pi$
$73$	−0.545146	−0.0638045	−0.0319022	+	0.999491i	$0.510157\pi$
$74$	−5.82193	−0.676786
$75$	0	0
$76$	−3.53840	−0.405882
$77$	−0.814323	−0.0928007
$78$	−2.06752	−0.234100
$79$	5.48159	0.616727	0.308363	−	0.951269i	$-0.400219\pi$
$79$	5.48159	0.616727	0.308363	+	0.951269i	$0.400219\pi$
$80$	0	0
$81$	−6.07680	−0.675200
$82$	−8.18102	−0.903442
$83$	−0.974135	−0.106925	−0.0534626	−	0.998570i	$-0.517026\pi$
$83$	−0.974135	−0.106925	−0.0534626	+	0.998570i	$0.517026\pi$
$84$	0.757421	0.0826414
$85$	0	0
$86$	10.3907	1.12046
$87$	14.3389	1.53729
$88$	6.15537	0.656164
$89$	−2.26649	−0.240247	−0.120124	−	0.992759i	$-0.538329\pi$
$89$	−2.26649	−0.240247	−0.120124	+	0.992759i	$0.538329\pi$
$90$	0	0
$91$	−0.285285	−0.0299060
$92$	0.858691	0.0895247
$93$	21.3529	2.21420
$94$	−1.44347	−0.148883
$95$	0	0
$96$	−9.94319	−1.01482
$97$	−15.2185	−1.54520	−0.772600	−	0.634893i	$-0.781044\pi$
$97$	−15.2185	−1.54520	−0.772600	+	0.634893i	$0.781044\pi$
$98$	7.74798	0.782664
$99$	7.54893	0.758696
$100$	0	0
$101$	18.3965	1.83052	0.915261	−	0.402861i	$-0.131984\pi$
$101$	18.3965	1.83052	0.915261	+	0.402861i	$0.131984\pi$
$102$	4.67030	0.462428
$103$	−11.8842	−1.17099	−0.585495	−	0.810676i	$-0.699100\pi$
$103$	−11.8842	−1.17099	−0.585495	+	0.810676i	$0.699100\pi$
$104$	2.15643	0.211456
$105$	0	0
$106$	−5.75917	−0.559380
$107$	−0.754919	−0.0729808	−0.0364904	−	0.999334i	$-0.511618\pi$
$107$	−0.754919	−0.0729808	−0.0364904	+	0.999334i	$0.511618\pi$
$108$	−1.44070	−0.138631
$109$	−9.15752	−0.877131	−0.438566	−	0.898699i	$-0.644513\pi$
$109$	−9.15752	−0.877131	−0.438566	+	0.898699i	$0.644513\pi$
$110$	0	0
$111$	13.3661	1.26865
$112$	0.838652	0.0792452
$113$	12.8399	1.20788	0.603938	−	0.797032i	$-0.293598\pi$
$113$	12.8399	1.20788	0.603938	+	0.797032i	$0.293598\pi$
$114$	−14.6087	−1.36823
$115$	0	0
$116$	−3.93741	−0.365579
$117$	2.64465	0.244498
$118$	7.36876	0.678349
$119$	0.644428	0.0590746
$120$	0	0
$121$	−7.00000	−0.636364
$122$	10.6796	0.966884
$123$	18.7821	1.69353
$124$	−5.86345	−0.526553
$125$	0	0
$126$	1.74230	0.155216
$127$	−7.10517	−0.630482	−0.315241	−	0.949012i	$-0.602085\pi$
$127$	−7.10517	−0.630482	−0.315241	+	0.949012i	$0.602085\pi$
$128$	−1.93993	−0.171467
$129$	−23.8553	−2.10034
$130$	0	0
$131$	2.39425	0.209187	0.104593	−	0.994515i	$-0.466646\pi$
$131$	2.39425	0.209187	0.104593	+	0.994515i	$0.466646\pi$
$132$	−3.72049	−0.323827
$133$	−2.01577	−0.174789
$134$	−3.49377	−0.301816
$135$	0	0
$136$	−4.87115	−0.417698
$137$	−19.0193	−1.62493	−0.812466	−	0.583009i	$-0.801875\pi$
$137$	−19.0193	−1.62493	−0.812466	+	0.583009i	$0.801875\pi$
$138$	3.54520	0.301788
$139$	−6.73470	−0.571230	−0.285615	−	0.958344i	$-0.592198\pi$
$139$	−6.73470	−0.571230	−0.285615	+	0.958344i	$0.592198\pi$
$140$	0	0
$141$	3.31395	0.279085
$142$	−7.78445	−0.653257
$143$	1.40134	0.117186
$144$	−7.77447	−0.647872
$145$	0	0
$146$	0.618034	0.0511489
$147$	−17.7880	−1.46713
$148$	−3.67029	−0.301696
$149$	−0.720492	−0.0590250	−0.0295125	−	0.999564i	$-0.509395\pi$
$149$	−0.720492	−0.0590250	−0.0295125	+	0.999564i	$0.509395\pi$
$150$	0	0
$151$	−15.5178	−1.26282	−0.631412	−	0.775447i	$-0.717524\pi$
$151$	−15.5178	−1.26282	−0.631412	+	0.775447i	$0.717524\pi$
$152$	15.2369	1.23588
$153$	−5.97397	−0.482967
$154$	0.923201	0.0743937
$155$	0	0
$156$	−1.30341	−0.104357
$157$	2.78418	0.222202	0.111101	−	0.993809i	$-0.464562\pi$
$157$	2.78418	0.222202	0.111101	+	0.993809i	$0.464562\pi$
$158$	−6.21450	−0.494399
$159$	13.2220	1.04857
$160$	0	0
$161$	0.489182	0.0385530
$162$	6.88929	0.541274
$163$	−24.2074	−1.89607	−0.948034	−	0.318170i	$-0.896932\pi$
$163$	−24.2074	−1.89607	−0.948034	+	0.318170i	$0.896932\pi$
$164$	−5.15752	−0.402734
$165$	0	0
$166$	1.10438	0.0857165
$167$	−19.0462	−1.47384	−0.736919	−	0.675982i	$-0.763720\pi$
$167$	−19.0462	−1.47384	−0.736919	+	0.675982i	$0.763720\pi$
$168$	−3.26158	−0.251637
$169$	−12.5091	−0.962236
$170$	0	0
$171$	18.6866	1.42900
$172$	6.55058	0.499477
$173$	−13.2470	−1.00715	−0.503577	−	0.863950i	$-0.667983\pi$
$173$	−13.2470	−1.00715	−0.503577	+	0.863950i	$0.667983\pi$
$174$	−16.2561	−1.23237
$175$	0	0
$176$	−4.11950	−0.310519
$177$	−16.9173	−1.27159
$178$	2.56952	0.192594
$179$	−12.2250	−0.913737	−0.456869	−	0.889534i	$-0.651029\pi$
$179$	−12.2250	−0.913737	−0.456869	+	0.889534i	$0.651029\pi$
$180$	0	0
$181$	10.9168	0.811439	0.405719	−	0.913998i	$-0.367021\pi$
$181$	10.9168	0.811439	0.405719	+	0.913998i	$0.367021\pi$
$182$	0.323429	0.0239741
$183$	−24.5184	−1.81245
$184$	−3.69767	−0.272596
$185$	0	0
$186$	−24.2079	−1.77501
$187$	−3.16546	−0.231482
$188$	−0.910000	−0.0663686
$189$	−0.820743	−0.0597003
$190$	0	0
$191$	1.56763	0.113430	0.0567148	−	0.998390i	$-0.481937\pi$
$191$	1.56763	0.113430	0.0567148	+	0.998390i	$0.481937\pi$
$192$	21.9948	1.58734
$193$	1.65786	0.119335	0.0596675	−	0.998218i	$-0.480996\pi$
$193$	1.65786	0.119335	0.0596675	+	0.998218i	$0.480996\pi$
$194$	17.2532	1.23871
$195$	0	0
$196$	4.88452	0.348894
$197$	13.2831	0.946380	0.473190	−	0.880960i	$-0.343102\pi$
$197$	13.2831	0.946380	0.473190	+	0.880960i	$0.343102\pi$
$198$	−8.55826	−0.608209
$199$	12.1025	0.857921	0.428960	−	0.903323i	$-0.358880\pi$
$199$	12.1025	0.857921	0.428960	+	0.903323i	$0.358880\pi$
$200$	0	0
$201$	8.02107	0.565763
$202$	−20.8562	−1.46744
$203$	−2.24308	−0.157433
$204$	2.94427	0.206140
$205$	0	0
$206$	13.4732	0.938724
$207$	−4.53482	−0.315192
$208$	−1.44320	−0.100068
$209$	9.90157	0.684906
$210$	0	0
$211$	−6.70061	−0.461289	−0.230644	−	0.973038i	$-0.574083\pi$
$211$	−6.70061	−0.461289	−0.230644	+	0.973038i	$0.574083\pi$
$212$	−3.63073	−0.249359
$213$	17.8717	1.22455
$214$	0.855855	0.0585050
$215$	0	0
$216$	6.20390	0.422122
$217$	−3.34031	−0.226755
$218$	10.3819	0.703152
$219$	−1.41889	−0.0958800
$220$	0	0
$221$	−1.10897	−0.0745973
$222$	−15.1532	−1.01702
$223$	−10.5917	−0.709272	−0.354636	−	0.935004i	$-0.615395\pi$
$223$	−10.5917	−0.709272	−0.354636	+	0.935004i	$0.615395\pi$
$224$	1.55545	0.103928
$225$	0	0
$226$	−14.5566	−0.968293
$227$	19.6112	1.30164	0.650822	−	0.759231i	$-0.274425\pi$
$227$	19.6112	1.30164	0.650822	+	0.759231i	$0.274425\pi$
$228$	−9.20968	−0.609926
$229$	8.83898	0.584096	0.292048	−	0.956404i	$-0.405663\pi$
$229$	8.83898	0.584096	0.292048	+	0.956404i	$0.405663\pi$
$230$	0	0
$231$	−2.11950	−0.139453
$232$	16.9552	1.11316
$233$	11.0624	0.724723	0.362362	−	0.932038i	$-0.381971\pi$
$233$	11.0624	0.724723	0.362362	+	0.932038i	$0.381971\pi$
$234$	−2.99825	−0.196002
$235$	0	0
$236$	4.64545	0.302393
$237$	14.2674	0.926765
$238$	−0.730590	−0.0473571
$239$	16.1055	1.04178	0.520888	−	0.853625i	$-0.325601\pi$
$239$	16.1055	1.04178	0.520888	+	0.853625i	$0.325601\pi$
$240$	0	0
$241$	5.98873	0.385768	0.192884	−	0.981222i	$-0.438216\pi$
$241$	5.98873	0.385768	0.192884	+	0.981222i	$0.438216\pi$
$242$	7.93593	0.510141
$243$	−21.8639	−1.40257
$244$	6.73268	0.431015
$245$	0	0
$246$	−21.2934	−1.35762
$247$	3.46885	0.220718
$248$	25.2490	1.60331
$249$	−2.53546	−0.160678
$250$	0	0
$251$	3.73176	0.235547	0.117773	−	0.993041i	$-0.462424\pi$
$251$	3.73176	0.235547	0.117773	+	0.993041i	$0.462424\pi$
$252$	1.09839	0.0691919
$253$	−2.40289	−0.151068
$254$	8.05516	0.505426
$255$	0	0
$256$	−14.7017	−0.918856
$257$	−23.5935	−1.47172	−0.735862	−	0.677132i	$-0.763223\pi$
$257$	−23.5935	−1.47172	−0.735862	+	0.677132i	$0.763223\pi$
$258$	27.0448	1.68374
$259$	−2.09090	−0.129922
$260$	0	0
$261$	20.7938	1.28710
$262$	−2.71438	−0.167695
$263$	−6.64126	−0.409518	−0.204759	−	0.978812i	$-0.565641\pi$
$263$	−6.64126	−0.409518	−0.204759	+	0.978812i	$0.565641\pi$
$264$	16.0211	0.986029
$265$	0	0
$266$	2.28528	0.140120
$267$	−5.89917	−0.361023
$268$	−2.20256	−0.134543
$269$	−13.0439	−0.795300	−0.397650	−	0.917537i	$-0.630174\pi$
$269$	−13.0439	−0.795300	−0.397650	+	0.917537i	$0.630174\pi$
$270$	0	0
$271$	−11.8488	−0.719762	−0.359881	−	0.932998i	$-0.617183\pi$
$271$	−11.8488	−0.719762	−0.359881	+	0.932998i	$0.617183\pi$
$272$	3.26004	0.197669
$273$	−0.742534	−0.0449402
$274$	21.5623	1.30263
$275$	0	0
$276$	2.23498	0.134530
$277$	16.9559	1.01878	0.509390	−	0.860536i	$-0.329871\pi$
$277$	16.9559	1.01878	0.509390	+	0.860536i	$0.329871\pi$
$278$	7.63516	0.457926
$279$	30.9653	1.85385
$280$	0	0
$281$	−5.07498	−0.302748	−0.151374	−	0.988477i	$-0.548370\pi$
$281$	−5.07498	−0.302748	−0.151374	+	0.988477i	$0.548370\pi$
$282$	−3.75704	−0.223728
$283$	11.8248	0.702914	0.351457	−	0.936204i	$-0.385686\pi$
$283$	11.8248	0.702914	0.351457	+	0.936204i	$0.385686\pi$
$284$	−4.90751	−0.291207
$285$	0	0
$286$	−1.58870	−0.0939418
$287$	−2.93815	−0.173434
$288$	−14.4193	−0.849665
$289$	−14.4950	−0.852645
$290$	0	0
$291$	−39.6103	−2.32200
$292$	0.389624	0.0228010
$293$	−19.4348	−1.13540	−0.567698	−	0.823237i	$-0.692166\pi$
$293$	−19.4348	−1.13540	−0.567698	+	0.823237i	$0.692166\pi$
$294$	20.1663	1.17612
$295$	0	0
$296$	15.8049	0.918640
$297$	4.03154	0.233934
$298$	0.816825	0.0473174
$299$	−0.841814	−0.0486834
$300$	0	0
$301$	3.73176	0.215095
$302$	17.5926	1.01234
$303$	47.8821	2.75076
$304$	−10.1974	−0.584860
$305$	0	0
$306$	6.77271	0.387170
$307$	25.4169	1.45062	0.725310	−	0.688423i	$-0.241697\pi$
$307$	25.4169	1.45062	0.725310	+	0.688423i	$0.241697\pi$
$308$	0.582009	0.0331630
$309$	−30.9321	−1.75967
$310$	0	0
$311$	15.8578	0.899212	0.449606	−	0.893227i	$-0.351564\pi$
$311$	15.8578	0.899212	0.449606	+	0.893227i	$0.351564\pi$
$312$	5.61272	0.317758
$313$	7.01644	0.396592	0.198296	−	0.980142i	$-0.436459\pi$
$313$	7.01644	0.396592	0.198296	+	0.980142i	$0.436459\pi$
$314$	−3.15643	−0.178128
$315$	0	0
$316$	−3.91777	−0.220392
$317$	13.0337	0.732044	0.366022	−	0.930606i	$-0.380720\pi$
$317$	13.0337	0.732044	0.366022	+	0.930606i	$0.380720\pi$
$318$	−14.9899	−0.840590
$319$	11.0181	0.616897
$320$	0	0
$321$	−1.96489	−0.109669
$322$	−0.554588	−0.0309060
$323$	−7.83576	−0.435994
$324$	4.34318	0.241288
$325$	0	0
$326$	27.4440	1.51998
$327$	−23.8350	−1.31808
$328$	22.2091	1.22629
$329$	−0.518412	−0.0285810
$330$	0	0
$331$	−17.6439	−0.969794	−0.484897	−	0.874571i	$-0.661143\pi$
$331$	−17.6439	−0.969794	−0.484897	+	0.874571i	$0.661143\pi$
$332$	0.696229	0.0382105
$333$	19.3831	1.06219
$334$	21.5927	1.18150
$335$	0	0
$336$	2.18283	0.119083
$337$	24.8495	1.35364	0.676819	−	0.736149i	$-0.263358\pi$
$337$	24.8495	1.35364	0.676819	+	0.736149i	$0.263358\pi$
$338$	14.1816	0.771376
$339$	33.4194	1.81509
$340$	0	0
$341$	16.4078	0.888532
$342$	−21.1850	−1.14556
$343$	5.63276	0.304141
$344$	−28.2079	−1.52087
$345$	0	0
$346$	15.0182	0.807385
$347$	13.2775	0.712773	0.356386	−	0.934339i	$-0.384009\pi$
$347$	13.2775	0.712773	0.356386	+	0.934339i	$0.384009\pi$
$348$	−10.2482	−0.549362
$349$	−18.1283	−0.970385	−0.485192	−	0.874407i	$-0.661250\pi$
$349$	−18.1283	−0.970385	−0.485192	+	0.874407i	$0.661250\pi$
$350$	0	0
$351$	1.41238	0.0753875
$352$	−7.64044	−0.407237
$353$	−17.7042	−0.942302	−0.471151	−	0.882053i	$-0.656161\pi$
$353$	−17.7042	−0.942302	−0.471151	+	0.882053i	$0.656161\pi$
$354$	19.1793	1.01937
$355$	0	0
$356$	1.61989	0.0858541
$357$	1.67730	0.0887723
$358$	13.8595	0.732497
$359$	32.5443	1.71762	0.858812	−	0.512290i	$-0.171203\pi$
$359$	32.5443	1.71762	0.858812	+	0.512290i	$0.171203\pi$
$360$	0	0
$361$	5.51025	0.290013
$362$	−12.3764	−0.650490
$363$	−18.2195	−0.956274
$364$	0.203897	0.0106871
$365$	0	0
$366$	27.7966	1.45295
$367$	32.9481	1.71988	0.859939	−	0.510397i	$-0.170501\pi$
$367$	32.9481	1.71988	0.859939	+	0.510397i	$0.170501\pi$
$368$	2.47468	0.129002
$369$	27.2373	1.41791
$370$	0	0
$371$	−2.06837	−0.107384
$372$	−15.2613	−0.791260
$373$	3.46648	0.179487	0.0897436	−	0.995965i	$-0.471395\pi$
$373$	3.46648	0.179487	0.0897436	+	0.995965i	$0.471395\pi$
$374$	3.58870	0.185567
$375$	0	0
$376$	3.91861	0.202087
$377$	3.86002	0.198801
$378$	0.930480	0.0478587
$379$	−1.92863	−0.0990670	−0.0495335	−	0.998772i	$-0.515773\pi$
$379$	−1.92863	−0.0990670	−0.0495335	+	0.998772i	$0.515773\pi$
$380$	0	0
$381$	−18.4932	−0.947436
$382$	−1.77723	−0.0909309
$383$	−4.93003	−0.251913	−0.125956	−	0.992036i	$-0.540200\pi$
$383$	−4.93003	−0.251913	−0.125956	+	0.992036i	$0.540200\pi$
$384$	−5.04922	−0.257667
$385$	0	0
$386$	−1.87952	−0.0956649
$387$	−34.5942	−1.75852
$388$	10.8769	0.552189
$389$	13.3860	0.678697	0.339348	−	0.940661i	$-0.389793\pi$
$389$	13.3860	0.678697	0.339348	+	0.940661i	$0.389793\pi$
$390$	0	0
$391$	1.90157	0.0961663
$392$	−21.0336	−1.06236
$393$	6.23172	0.314349
$394$	−15.0591	−0.758666
$395$	0	0
$396$	−5.39534	−0.271126
$397$	2.13887	0.107347	0.0536733	−	0.998559i	$-0.482907\pi$
$397$	2.13887	0.107347	0.0536733	+	0.998559i	$0.482907\pi$
$398$	−13.7206	−0.687752
$399$	−5.24660	−0.262659
$400$	0	0
$401$	−26.8213	−1.33939	−0.669696	−	0.742635i	$-0.733576\pi$
$401$	−26.8213	−1.33939	−0.669696	+	0.742635i	$0.733576\pi$
$402$	−9.09352	−0.453544
$403$	5.74821	0.286339
$404$	−13.1483	−0.654151
$405$	0	0
$406$	2.54299	0.126206
$407$	10.2706	0.509097
$408$	−12.6785	−0.627681
$409$	13.9982	0.692169	0.346084	−	0.938203i	$-0.387511\pi$
$409$	13.9982	0.692169	0.346084	+	0.938203i	$0.387511\pi$
$410$	0	0
$411$	−49.5032	−2.44181
$412$	8.49385	0.418462
$413$	2.64644	0.130223
$414$	5.14114	0.252673
$415$	0	0
$416$	−2.67670	−0.131236
$417$	−17.5290	−0.858397
$418$	−11.2254	−0.549055
$419$	30.9020	1.50966	0.754831	−	0.655919i	$-0.227719\pi$
$419$	30.9020	1.50966	0.754831	+	0.655919i	$0.227719\pi$
$420$	0	0
$421$	−8.23974	−0.401581	−0.200790	−	0.979634i	$-0.564351\pi$
$421$	−8.23974	−0.401581	−0.200790	+	0.979634i	$0.564351\pi$
$422$	7.59651	0.369792
$423$	4.80578	0.233665
$424$	15.6345	0.759279
$425$	0	0
$426$	−20.2612	−0.981660
$427$	3.83550	0.185613
$428$	0.539552	0.0260802
$429$	3.64737	0.176097
$430$	0	0
$431$	27.1482	1.30768	0.653841	−	0.756632i	$-0.273157\pi$
$431$	27.1482	1.30768	0.653841	+	0.756632i	$0.273157\pi$
$432$	−4.15198	−0.199762
$433$	−21.4905	−1.03277	−0.516383	−	0.856357i	$-0.672722\pi$
$433$	−21.4905	−1.03277	−0.516383	+	0.856357i	$0.672722\pi$
$434$	3.78692	0.181778
$435$	0	0
$436$	6.54502	0.313449
$437$	−5.94810	−0.284536
$438$	1.60861	0.0768622
$439$	−25.7956	−1.23116	−0.615578	−	0.788076i	$-0.711078\pi$
$439$	−25.7956	−1.23116	−0.615578	+	0.788076i	$0.711078\pi$
$440$	0	0
$441$	−25.7955	−1.22836
$442$	1.25724	0.0598009
$443$	−3.18479	−0.151314	−0.0756570	−	0.997134i	$-0.524105\pi$
$443$	−3.18479	−0.151314	−0.0756570	+	0.997134i	$0.524105\pi$
$444$	−9.55296	−0.453363
$445$	0	0
$446$	12.0078	0.568588
$447$	−1.87528	−0.0886978
$448$	−3.44072	−0.162559
$449$	36.0785	1.70265	0.851325	−	0.524639i	$-0.175800\pi$
$449$	36.0785	1.70265	0.851325	+	0.524639i	$0.175800\pi$
$450$	0	0
$451$	14.4324	0.679594
$452$	−9.17686	−0.431643
$453$	−40.3896	−1.89767
$454$	−22.2333	−1.04346
$455$	0	0
$456$	39.6584	1.85718
$457$	−25.5245	−1.19399	−0.596994	−	0.802246i	$-0.703638\pi$
$457$	−25.5245	−1.19399	−0.596994	+	0.802246i	$0.703638\pi$
$458$	−10.0208	−0.468241
$459$	−3.19042	−0.148916
$460$	0	0
$461$	−16.6392	−0.774963	−0.387482	−	0.921877i	$-0.626655\pi$
$461$	−16.6392	−0.774963	−0.387482	+	0.921877i	$0.626655\pi$
$462$	2.40289	0.111793
$463$	−17.8178	−0.828062	−0.414031	−	0.910263i	$-0.635880\pi$
$463$	−17.8178	−0.828062	−0.414031	+	0.910263i	$0.635880\pi$
$464$	−11.3473	−0.526786
$465$	0	0
$466$	−12.5415	−0.580974
$467$	14.2480	0.659321	0.329660	−	0.944100i	$-0.393066\pi$
$467$	14.2480	0.659321	0.329660	+	0.944100i	$0.393066\pi$
$468$	−1.89017	−0.0873731
$469$	−1.25476	−0.0579395
$470$	0	0
$471$	7.24660	0.333906
$472$	−20.0041	−0.920762
$473$	−18.3306	−0.842843
$474$	−16.1750	−0.742941
$475$	0	0
$476$	−0.460582	−0.0211107
$477$	19.1742	0.877924
$478$	−18.2588	−0.835139
$479$	2.90864	0.132899	0.0664496	−	0.997790i	$-0.478833\pi$
$479$	2.90864	0.132899	0.0664496	+	0.997790i	$0.478833\pi$
$480$	0	0
$481$	3.59815	0.164062
$482$	−6.78945	−0.309251
$483$	1.27323	0.0579342
$484$	5.00301	0.227409
$485$	0	0
$486$	24.7872	1.12437
$487$	17.0652	0.773298	0.386649	−	0.922227i	$-0.373632\pi$
$487$	17.0652	0.773298	0.386649	+	0.922227i	$0.373632\pi$
$488$	−28.9920	−1.31241
$489$	−63.0065	−2.84925
$490$	0	0
$491$	30.7768	1.38894	0.694470	−	0.719522i	$-0.255639\pi$
$491$	30.7768	1.38894	0.694470	+	0.719522i	$0.255639\pi$
$492$	−13.4239	−0.605195
$493$	−8.71937	−0.392701
$494$	−3.93265	−0.176938
$495$	0	0
$496$	−16.8980	−0.758742
$497$	−2.79573	−0.125406
$498$	2.87446	0.128808
$499$	−11.8824	−0.531927	−0.265964	−	0.963983i	$-0.585690\pi$
$499$	−11.8824	−0.531927	−0.265964	+	0.963983i	$0.585690\pi$
$500$	0	0
$501$	−49.5730	−2.21476
$502$	−4.23071	−0.188826
$503$	2.97823	0.132793	0.0663964	−	0.997793i	$-0.478850\pi$
$503$	2.97823	0.132793	0.0663964	+	0.997793i	$0.478850\pi$
$504$	−4.72984	−0.210684
$505$	0	0
$506$	2.72417	0.121104
$507$	−32.5584	−1.44597
$508$	5.07817	0.225308
$509$	−36.7159	−1.62740	−0.813702	−	0.581282i	$-0.802551\pi$
$509$	−36.7159	−1.62740	−0.813702	+	0.581282i	$0.802551\pi$
$510$	0	0
$511$	0.221962	0.00981904
$512$	20.5472	0.908068
$513$	9.97963	0.440612
$514$	26.7481	1.17981
$515$	0	0
$516$	17.0497	0.750573
$517$	2.54647	0.111994
$518$	2.37047	0.104152
$519$	−34.4792	−1.51347
$520$	0	0
$521$	2.50039	0.109544	0.0547720	−	0.998499i	$-0.482557\pi$
$521$	2.50039	0.109544	0.0547720	+	0.998499i	$0.482557\pi$
$522$	−23.5740	−1.03181
$523$	42.1258	1.84203	0.921017	−	0.389523i	$-0.127360\pi$
$523$	42.1258	1.84203	0.921017	+	0.389523i	$0.127360\pi$
$524$	−1.71121	−0.0747545
$525$	0	0
$526$	7.52922	0.328290
$527$	−12.9846	−0.565617
$528$	−10.7222	−0.466622
$529$	−21.5565	−0.937240
$530$	0	0
$531$	−24.5330	−1.06464
$532$	1.44070	0.0624623
$533$	5.05615	0.219006
$534$	6.68791	0.289414
$535$	0	0
$536$	9.48459	0.409672
$537$	−31.8189	−1.37309
$538$	14.7879	0.637552
$539$	−13.6684	−0.588741
$540$	0	0
$541$	−27.3492	−1.17583	−0.587916	−	0.808922i	$-0.700052\pi$
$541$	−27.3492	−1.17583	−0.587916	+	0.808922i	$0.700052\pi$
$542$	13.4330	0.576997
$543$	28.4140	1.21936
$544$	6.04638	0.259236
$545$	0	0
$546$	0.841814	0.0360263
$547$	20.3392	0.869640	0.434820	−	0.900517i	$-0.356812\pi$
$547$	20.3392	0.869640	0.434820	+	0.900517i	$0.356812\pi$
$548$	13.5934	0.580682
$549$	−35.5558	−1.51748
$550$	0	0
$551$	27.2742	1.16192
$552$	−9.62422	−0.409634
$553$	−2.23189	−0.0949097
$554$	−19.2229	−0.816705
$555$	0	0
$556$	4.81339	0.204133
$557$	28.2605	1.19744	0.598718	−	0.800960i	$-0.295677\pi$
$557$	28.2605	1.19744	0.598718	+	0.800960i	$0.295677\pi$
$558$	−35.1055	−1.48614
$559$	−6.42184	−0.271615
$560$	0	0
$561$	−8.23901	−0.347851
$562$	5.75353	0.242698
$563$	12.9254	0.544742	0.272371	−	0.962192i	$-0.412192\pi$
$563$	12.9254	0.544742	0.272371	+	0.962192i	$0.412192\pi$
$564$	−2.36853	−0.0997331
$565$	0	0
$566$	−13.4059	−0.563491
$567$	2.47424	0.103908
$568$	21.1326	0.886703
$569$	−3.08443	−0.129306	−0.0646531	−	0.997908i	$-0.520594\pi$
$569$	−3.08443	−0.129306	−0.0646531	+	0.997908i	$0.520594\pi$
$570$	0	0
$571$	−4.12766	−0.172737	−0.0863687	−	0.996263i	$-0.527526\pi$
$571$	−4.12766	−0.172737	−0.0863687	+	0.996263i	$0.527526\pi$
$572$	−1.00156	−0.0418771
$573$	4.08019	0.170453
$574$	3.33100	0.139033
$575$	0	0
$576$	31.8961	1.32901
$577$	6.35570	0.264591	0.132296	−	0.991210i	$-0.457765\pi$
$577$	6.35570	0.264591	0.132296	+	0.991210i	$0.457765\pi$
$578$	16.4330	0.683522
$579$	4.31503	0.179327
$580$	0	0
$581$	0.396630	0.0164550
$582$	44.9063	1.86143
$583$	10.1599	0.420781
$584$	−1.67779	−0.0694273
$585$	0	0
$586$	22.0334	0.910190
$587$	−21.1040	−0.871054	−0.435527	−	0.900176i	$-0.643438\pi$
$587$	−21.1040	−0.871054	−0.435527	+	0.900176i	$0.643438\pi$
$588$	12.7133	0.524289
$589$	40.6157	1.67354
$590$	0	0
$591$	34.5729	1.42214
$592$	−10.5775	−0.434732
$593$	21.6529	0.889177	0.444589	−	0.895735i	$-0.353350\pi$
$593$	21.6529	0.889177	0.444589	+	0.895735i	$0.353350\pi$
$594$	−4.57057	−0.187533
$595$	0	0
$596$	0.514946	0.0210930
$597$	31.5000	1.28921
$598$	0.954368	0.0390270
$599$	−3.38501	−0.138308	−0.0691539	−	0.997606i	$-0.522030\pi$
$599$	−3.38501	−0.138308	−0.0691539	+	0.997606i	$0.522030\pi$
$600$	0	0
$601$	28.8265	1.17586	0.587928	−	0.808913i	$-0.299944\pi$
$601$	28.8265	1.17586	0.587928	+	0.808913i	$0.299944\pi$
$602$	−4.23071	−0.172431
$603$	11.6319	0.473687
$604$	11.0908	0.451280
$605$	0	0
$606$	−54.2842	−2.20514
$607$	−15.6708	−0.636059	−0.318029	−	0.948081i	$-0.603021\pi$
$607$	−15.6708	−0.636059	−0.318029	+	0.948081i	$0.603021\pi$
$608$	−18.9131	−0.767026
$609$	−5.83824	−0.236578
$610$	0	0
$611$	0.892114	0.0360911
$612$	4.26969	0.172592
$613$	38.2005	1.54290	0.771451	−	0.636288i	$-0.219531\pi$
$613$	38.2005	1.54290	0.771451	+	0.636288i	$0.219531\pi$
$614$	−28.8153	−1.16289
$615$	0	0
$616$	−2.50623	−0.100979
$617$	−13.2318	−0.532691	−0.266346	−	0.963878i	$-0.585816\pi$
$617$	−13.2318	−0.532691	−0.266346	+	0.963878i	$0.585816\pi$
$618$	35.0678	1.41064
$619$	6.04463	0.242954	0.121477	−	0.992594i	$-0.461237\pi$
$619$	6.04463	0.242954	0.121477	+	0.992594i	$0.461237\pi$
$620$	0	0
$621$	−2.42184	−0.0971849
$622$	−17.9780	−0.720853
$623$	0.922826	0.0369722
$624$	−3.75634	−0.150374
$625$	0	0
$626$	−7.95456	−0.317928
$627$	25.7716	1.02922
$628$	−1.98989	−0.0794054
$629$	−8.12783	−0.324078
$630$	0	0
$631$	−37.7855	−1.50422	−0.752108	−	0.659040i	$-0.770963\pi$
$631$	−37.7855	−1.50422	−0.752108	+	0.659040i	$0.770963\pi$
$632$	16.8706	0.671076
$633$	−17.4402	−0.693186
$634$	−14.7763	−0.586843
$635$	0	0
$636$	−9.44998	−0.374716
$637$	−4.78852	−0.189728
$638$	−12.4913	−0.494535
$639$	25.9170	1.02526
$640$	0	0
$641$	−20.8570	−0.823802	−0.411901	−	0.911229i	$-0.635135\pi$
$641$	−20.8570	−0.823802	−0.411901	+	0.911229i	$0.635135\pi$
$642$	2.22760	0.0879164
$643$	37.5552	1.48103	0.740516	−	0.672039i	$-0.234581\pi$
$643$	37.5552	1.48103	0.740516	+	0.672039i	$0.234581\pi$
$644$	−0.349626	−0.0137772
$645$	0	0
$646$	8.88344	0.349514
$647$	27.4003	1.07722	0.538608	−	0.842557i	$-0.318951\pi$
$647$	27.4003	1.07722	0.538608	+	0.842557i	$0.318951\pi$
$648$	−18.7025	−0.734702
$649$	−12.9994	−0.510272
$650$	0	0
$651$	−8.69410	−0.340749
$652$	17.3014	0.677574
$653$	−23.0461	−0.901864	−0.450932	−	0.892558i	$-0.648908\pi$
$653$	−23.0461	−0.901864	−0.450932	+	0.892558i	$0.648908\pi$
$654$	27.0218	1.05664
$655$	0	0
$656$	−14.8636	−0.580324
$657$	−2.05763	−0.0802760
$658$	0.587726	0.0229119
$659$	20.4837	0.797931	0.398966	−	0.916966i	$-0.369369\pi$
$659$	20.4837	0.797931	0.398966	+	0.916966i	$0.369369\pi$
$660$	0	0
$661$	13.9900	0.544147	0.272074	−	0.962276i	$-0.412291\pi$
$661$	13.9900	0.544147	0.272074	+	0.962276i	$0.412291\pi$
$662$	20.0029	0.777436
$663$	−2.88640	−0.112099
$664$	−2.99808	−0.116348
$665$	0	0
$666$	−21.9747	−0.851502
$667$	−6.61884	−0.256283
$668$	13.6126	0.526687
$669$	−27.5678	−1.06583
$670$	0	0
$671$	−18.8402	−0.727317
$672$	4.04848	0.156174
$673$	−7.84572	−0.302430	−0.151215	−	0.988501i	$-0.548319\pi$
$673$	−7.84572	−0.302430	−0.151215	+	0.988501i	$0.548319\pi$
$674$	−28.1720	−1.08514
$675$	0	0
$676$	8.94042	0.343862
$677$	−14.3983	−0.553371	−0.276686	−	0.960960i	$-0.589236\pi$
$677$	−14.3983	−0.553371	−0.276686	+	0.960960i	$0.589236\pi$
$678$	−37.8877	−1.45507
$679$	6.19637	0.237795
$680$	0	0
$681$	51.0437	1.95600
$682$	−18.6016	−0.712291
$683$	12.6740	0.484956	0.242478	−	0.970157i	$-0.422040\pi$
$683$	12.6740	0.484956	0.242478	+	0.970157i	$0.422040\pi$
$684$	−13.3556	−0.510663
$685$	0	0
$686$	−6.38589	−0.243814
$687$	23.0059	0.877731
$688$	18.8783	0.719727
$689$	3.55937	0.135601
$690$	0	0
$691$	5.74530	0.218562	0.109281	−	0.994011i	$-0.465145\pi$
$691$	5.74530	0.218562	0.109281	+	0.994011i	$0.465145\pi$
$692$	9.46787	0.359914
$693$	−3.07364	−0.116758
$694$	−15.0527	−0.571394
$695$	0	0
$696$	44.1306	1.67276
$697$	−11.4213	−0.432612
$698$	20.5521	0.777909
$699$	28.7931	1.08905
$700$	0	0
$701$	−30.5834	−1.15512	−0.577560	−	0.816348i	$-0.695995\pi$
$701$	−30.5834	−1.15512	−0.577560	+	0.816348i	$0.695995\pi$
$702$	−1.60123	−0.0604344
$703$	25.4238	0.958879
$704$	16.9010	0.636980
$705$	0	0
$706$	20.0714	0.755396
$707$	−7.49036	−0.281704
$708$	12.0911	0.454411
$709$	−27.0696	−1.01662	−0.508310	−	0.861174i	$-0.669730\pi$
$709$	−27.0696	−1.01662	−0.508310	+	0.861174i	$0.669730\pi$
$710$	0	0
$711$	20.6901	0.775938
$712$	−6.97553	−0.261419
$713$	−9.85653	−0.369130
$714$	−1.90157	−0.0711643
$715$	0	0
$716$	8.73737	0.326531
$717$	41.9190	1.56549
$718$	−36.8957	−1.37693
$719$	−16.7066	−0.623052	−0.311526	−	0.950238i	$-0.600840\pi$
$719$	−16.7066	−0.623052	−0.311526	+	0.950238i	$0.600840\pi$
$720$	0	0
$721$	4.83881	0.180207
$722$	−6.24700	−0.232489
$723$	15.5874	0.579700
$724$	−7.80240	−0.289974
$725$	0	0
$726$	20.6555	0.766597
$727$	−11.7006	−0.433951	−0.216975	−	0.976177i	$-0.569619\pi$
$727$	−11.7006	−0.433951	−0.216975	+	0.976177i	$0.569619\pi$
$728$	−0.878017	−0.0325415
$729$	−38.6765	−1.43246
$730$	0	0
$731$	14.5062	0.536532
$732$	17.5237	0.647694
$733$	−34.1078	−1.25980	−0.629901	−	0.776676i	$-0.716904\pi$
$733$	−34.1078	−1.25980	−0.629901	+	0.776676i	$0.716904\pi$
$734$	−37.3534	−1.37874
$735$	0	0
$736$	4.58978	0.169182
$737$	6.16346	0.227034
$738$	−30.8790	−1.13667
$739$	5.74517	0.211340	0.105670	−	0.994401i	$-0.466301\pi$
$739$	5.74517	0.211340	0.105670	+	0.994401i	$0.466301\pi$
$740$	0	0
$741$	9.02866	0.331676
$742$	2.34491	0.0860845
$743$	−36.4348	−1.33666	−0.668332	−	0.743863i	$-0.732991\pi$
$743$	−36.4348	−1.33666	−0.668332	+	0.743863i	$0.732991\pi$
$744$	65.7176	2.40932
$745$	0	0
$746$	−3.92996	−0.143886
$747$	−3.67684	−0.134528
$748$	2.26240	0.0827217
$749$	0.307374	0.0112312
$750$	0	0
$751$	1.48912	0.0543387	0.0271693	−	0.999631i	$-0.491351\pi$
$751$	1.48912	0.0543387	0.0271693	+	0.999631i	$0.491351\pi$
$752$	−2.62255	−0.0956345
$753$	9.71296	0.353960
$754$	−4.37612	−0.159369
$755$	0	0
$756$	0.586598	0.0213344
$757$	5.53316	0.201106	0.100553	−	0.994932i	$-0.467939\pi$
$757$	5.53316	0.201106	0.100553	+	0.994932i	$0.467939\pi$
$758$	2.18649	0.0794171
$759$	−6.25420	−0.227013
$760$	0	0
$761$	−18.6697	−0.676776	−0.338388	−	0.941007i	$-0.609882\pi$
$761$	−18.6697	−0.676776	−0.338388	+	0.941007i	$0.609882\pi$
$762$	20.9658	0.759512
$763$	3.72859	0.134984
$764$	−1.12041	−0.0405349
$765$	0	0
$766$	5.58920	0.201946
$767$	−4.55414	−0.164441
$768$	−38.2653	−1.38078
$769$	−13.1433	−0.473960	−0.236980	−	0.971515i	$-0.576158\pi$
$769$	−13.1433	−0.473960	−0.236980	+	0.971515i	$0.576158\pi$
$770$	0	0
$771$	−61.4088	−2.21158
$772$	−1.18489	−0.0426453
$773$	−27.9736	−1.00614	−0.503070	−	0.864246i	$-0.667796\pi$
$773$	−27.9736	−1.00614	−0.503070	+	0.864246i	$0.667796\pi$
$774$	39.2195	1.40972
$775$	0	0
$776$	−46.8376	−1.68137
$777$	−5.44217	−0.195237
$778$	−15.1758	−0.544077
$779$	35.7258	1.28001
$780$	0	0
$781$	13.7328	0.491398
$782$	−2.15581	−0.0770917
$783$	11.1050	0.396860
$784$	14.0768	0.502743
$785$	0	0
$786$	−7.06492	−0.251998
$787$	−33.3670	−1.18940	−0.594702	−	0.803946i	$-0.702730\pi$
$787$	−33.3670	−1.18940	−0.594702	+	0.803946i	$0.702730\pi$
$788$	−9.49362	−0.338196
$789$	−17.2857	−0.615389
$790$	0	0
$791$	−5.22791	−0.185883
$792$	23.2332	0.825557
$793$	−6.60035	−0.234385
$794$	−2.42484	−0.0860544
$795$	0	0
$796$	−8.64981	−0.306584
$797$	−37.3986	−1.32473	−0.662364	−	0.749183i	$-0.730447\pi$
$797$	−37.3986	−1.32473	−0.662364	+	0.749183i	$0.730447\pi$
$798$	5.94810	0.210560
$799$	−2.01519	−0.0712923
$800$	0	0
$801$	−8.55478	−0.302268
$802$	30.4074	1.07372
$803$	−1.09029	−0.0384756
$804$	−5.73278	−0.202180
$805$	0	0
$806$	−6.51676	−0.229543
$807$	−33.9504	−1.19511
$808$	56.6187	1.99184
$809$	33.6017	1.18137	0.590687	−	0.806901i	$-0.298857\pi$
$809$	33.6017	1.18137	0.590687	+	0.806901i	$0.298857\pi$
$810$	0	0
$811$	−38.0056	−1.33456	−0.667278	−	0.744809i	$-0.732541\pi$
$811$	−38.0056	−1.33456	−0.667278	+	0.744809i	$0.732541\pi$
$812$	1.60316	0.0562600
$813$	−30.8398	−1.08160
$814$	−11.6439	−0.408117
$815$	0	0
$816$	8.48516	0.297040
$817$	−45.3755	−1.58749
$818$	−15.8699	−0.554877
$819$	−1.07680	−0.0376264
$820$	0	0
$821$	24.9332	0.870176	0.435088	−	0.900388i	$-0.356717\pi$
$821$	24.9332	0.870176	0.435088	+	0.900388i	$0.356717\pi$
$822$	56.1220	1.95748
$823$	25.4606	0.887499	0.443750	−	0.896151i	$-0.353648\pi$
$823$	25.4606	0.887499	0.443750	+	0.896151i	$0.353648\pi$
$824$	−36.5760	−1.27418
$825$	0	0
$826$	−3.00027	−0.104393
$827$	32.3387	1.12453	0.562263	−	0.826958i	$-0.309931\pi$
$827$	32.3387	1.12453	0.562263	+	0.826958i	$0.309931\pi$
$828$	3.24110	0.112636
$829$	24.9812	0.867633	0.433816	−	0.901001i	$-0.357167\pi$
$829$	24.9812	0.867633	0.433816	+	0.901001i	$0.357167\pi$
$830$	0	0
$831$	44.1324	1.53094
$832$	5.92099	0.205273
$833$	10.8167	0.374778
$834$	19.8726	0.688133
$835$	0	0
$836$	−7.07680	−0.244756
$837$	16.5372	0.571608
$838$	−35.0337	−1.21022
$839$	19.3526	0.668125	0.334062	−	0.942551i	$-0.391580\pi$
$839$	19.3526	0.668125	0.334062	+	0.942551i	$0.391580\pi$
$840$	0	0
$841$	1.34980	0.0465447
$842$	9.34143	0.321927
$843$	−13.2091	−0.454944
$844$	4.78903	0.164845
$845$	0	0
$846$	−5.44833	−0.187318
$847$	2.85013	0.0979317
$848$	−10.4635	−0.359317
$849$	30.7775	1.05628
$850$	0	0
$851$	−6.16980	−0.211498
$852$	−12.7732	−0.437602
$853$	26.7876	0.917190	0.458595	−	0.888645i	$-0.348353\pi$
$853$	26.7876	0.917190	0.458595	+	0.888645i	$0.348353\pi$
$854$	−4.34832	−0.148796
$855$	0	0
$856$	−2.32340	−0.0794122
$857$	−47.5186	−1.62320	−0.811602	−	0.584210i	$-0.801404\pi$
$857$	−47.5186	−1.62320	−0.811602	+	0.584210i	$0.801404\pi$
$858$	−4.13504	−0.141168
$859$	11.5290	0.393365	0.196682	−	0.980467i	$-0.436983\pi$
$859$	11.5290	0.393365	0.196682	+	0.980467i	$0.436983\pi$
$860$	0	0
$861$	−7.64737	−0.260622
$862$	−30.7780	−1.04830
$863$	18.0057	0.612922	0.306461	−	0.951883i	$-0.400855\pi$
$863$	18.0057	0.612922	0.306461	+	0.951883i	$0.400855\pi$
$864$	−7.70067	−0.261982
$865$	0	0
$866$	24.3639	0.827917
$867$	−37.7272	−1.28128
$868$	2.38737	0.0810327
$869$	10.9632	0.371900
$870$	0	0
$871$	2.15927	0.0731641
$872$	−28.1839	−0.954429
$873$	−57.4415	−1.94410
$874$	6.74338	0.228098
$875$	0	0
$876$	1.01411	0.0342635
$877$	−17.0973	−0.577335	−0.288668	−	0.957429i	$-0.593212\pi$
$877$	−17.0973	−0.577335	−0.288668	+	0.957429i	$0.593212\pi$
$878$	29.2446	0.986957
$879$	−50.5846	−1.70618
$880$	0	0
$881$	−38.0291	−1.28123	−0.640617	−	0.767861i	$-0.721321\pi$
$881$	−38.0291	−1.28123	−0.640617	+	0.767861i	$0.721321\pi$
$882$	29.2445	0.984714
$883$	−36.8494	−1.24008	−0.620040	−	0.784570i	$-0.712884\pi$
$883$	−36.8494	−1.24008	−0.620040	+	0.784570i	$0.712884\pi$
$884$	0.792597	0.0266579
$885$	0	0
$886$	3.61061	0.121301
$887$	29.8738	1.00306	0.501532	−	0.865139i	$-0.332770\pi$
$887$	29.8738	1.00306	0.501532	+	0.865139i	$0.332770\pi$
$888$	41.1366	1.38046
$889$	2.89295	0.0970265
$890$	0	0
$891$	−12.1536	−0.407161
$892$	7.57004	0.253464
$893$	6.30351	0.210939
$894$	2.12602	0.0711046
$895$	0	0
$896$	0.789866	0.0263876
$897$	−2.19106	−0.0731573
$898$	−40.9023	−1.36493
$899$	45.1958	1.50736
$900$	0	0
$901$	−8.04022	−0.267859
$902$	−16.3620	−0.544796
$903$	9.71296	0.323227
$904$	39.5171	1.31432
$905$	0	0
$906$	45.7898	1.52126
$907$	43.4897	1.44405	0.722026	−	0.691866i	$-0.243211\pi$
$907$	43.4897	1.44405	0.722026	+	0.691866i	$0.243211\pi$
$908$	−14.0164	−0.465152
$909$	69.4371	2.30308
$910$	0	0
$911$	−24.3690	−0.807383	−0.403691	−	0.914895i	$-0.632273\pi$
$911$	−24.3690	−0.807383	−0.403691	+	0.914895i	$0.632273\pi$
$912$	−26.5416	−0.878879
$913$	−1.94827	−0.0644783
$914$	28.9373	0.957160
$915$	0	0
$916$	−6.31735	−0.208731
$917$	−0.974848	−0.0321923
$918$	3.61699	0.119379
$919$	45.5143	1.50138	0.750690	−	0.660655i	$-0.229721\pi$
$919$	45.5143	1.50138	0.750690	+	0.660655i	$0.229721\pi$
$920$	0	0
$921$	66.1546	2.17987
$922$	18.8639	0.621249
$923$	4.81106	0.158358
$924$	1.51484	0.0498346
$925$	0	0
$926$	20.2001	0.663816
$927$	−44.8567	−1.47329
$928$	−21.0458	−0.690863
$929$	9.03795	0.296525	0.148263	−	0.988948i	$-0.452632\pi$
$929$	9.03795	0.296525	0.148263	+	0.988948i	$0.452632\pi$
$930$	0	0
$931$	−33.8347	−1.10889
$932$	−7.90648	−0.258985
$933$	41.2743	1.35126
$934$	−16.1531	−0.528545
$935$	0	0
$936$	8.13939	0.266044
$937$	−33.3272	−1.08875	−0.544376	−	0.838841i	$-0.683234\pi$
$937$	−33.3272	−1.08875	−0.544376	+	0.838841i	$0.683234\pi$
$938$	1.42253	0.0464472
$939$	18.2622	0.595966
$940$	0	0
$941$	−29.3010	−0.955186	−0.477593	−	0.878581i	$-0.658491\pi$
$941$	−29.3010	−0.955186	−0.477593	+	0.878581i	$0.658491\pi$
$942$	−8.21550	−0.267676
$943$	−8.66985	−0.282329
$944$	13.3878	0.435736
$945$	0	0
$946$	20.7815	0.675665
$947$	−56.8456	−1.84723	−0.923617	−	0.383317i	$-0.874782\pi$
$947$	−56.8456	−1.84723	−0.923617	+	0.383317i	$0.874782\pi$
$948$	−10.1971	−0.331187
$949$	−0.381966	−0.0123991
$950$	0	0
$951$	33.9238	1.10005
$952$	1.98334	0.0642806
$953$	−19.0368	−0.616663	−0.308331	−	0.951279i	$-0.599771\pi$
$953$	−19.0368	−0.616663	−0.308331	+	0.951279i	$0.599771\pi$
$954$	−21.7378	−0.703788
$955$	0	0
$956$	−11.5108	−0.372286
$957$	28.6778	0.927021
$958$	−3.29754	−0.106539
$959$	7.74394	0.250065
$960$	0	0
$961$	36.3039	1.17109
$962$	−4.07924	−0.131520
$963$	−2.84942	−0.0918212
$964$	−4.28024	−0.137857
$965$	0	0
$966$	−1.44347	−0.0464429
$967$	14.7105	0.473059	0.236529	−	0.971624i	$-0.423990\pi$
$967$	14.7105	0.473059	0.236529	+	0.971624i	$0.423990\pi$
$968$	−21.5438	−0.692443
$969$	−20.3948	−0.655174
$970$	0	0
$971$	−38.8453	−1.24661	−0.623303	−	0.781980i	$-0.714210\pi$
$971$	−38.8453	−1.24661	−0.623303	+	0.781980i	$0.714210\pi$
$972$	15.6264	0.501218
$973$	2.74211	0.0879081
$974$	−19.3469	−0.619914
$975$	0	0
$976$	19.4030	0.621076
$977$	−58.4544	−1.87012	−0.935060	−	0.354489i	$-0.884655\pi$
$977$	−58.4544	−1.87012	−0.935060	+	0.354489i	$0.884655\pi$
$978$	71.4307	2.28410
$979$	−4.53297	−0.144874
$980$	0	0
$981$	−34.5647	−1.10357
$982$	−34.8918	−1.11344
$983$	28.1595	0.898147	0.449074	−	0.893495i	$-0.351754\pi$
$983$	28.1595	0.898147	0.449074	+	0.893495i	$0.351754\pi$
$984$	57.8055	1.84277
$985$	0	0
$986$	9.88519	0.314809
$987$	−1.34931	−0.0429491
$988$	−2.47924	−0.0788752
$989$	11.0116	0.350149
$990$	0	0
$991$	39.3437	1.24979	0.624897	−	0.780707i	$-0.285141\pi$
$991$	39.3437	1.24979	0.624897	+	0.780707i	$0.285141\pi$
$992$	−31.3407	−0.995067
$993$	−45.9231	−1.45733
$994$	3.16953	0.100531
$995$	0	0
$996$	1.81213	0.0574195
$997$	43.1178	1.36555	0.682777	−	0.730627i	$-0.260772\pi$
$997$	43.1178	1.36555	0.682777	+	0.730627i	$0.260772\pi$
$998$	13.4711	0.426419
$999$	10.3516	0.327511

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	625.2.a.f.1.3		8
3.2	odd	2		5625.2.a.x.1.6		8
4.3	odd	2		10000.2.a.bj.1.1		8
5.2	odd	4		625.2.b.c.624.3		8
5.3	odd	4		625.2.b.c.624.6		8
5.4	even	2	inner	625.2.a.f.1.6		8
15.14	odd	2		5625.2.a.x.1.3		8
20.19	odd	2		10000.2.a.bj.1.8		8
25.2	odd	20		625.2.e.a.124.1		8
25.3	odd	20		25.2.e.a.9.2	✓	8
25.4	even	10		125.2.d.b.76.2		16
25.6	even	5		125.2.d.b.51.3		16
25.8	odd	20		125.2.e.b.74.1		8
25.9	even	10		625.2.d.o.126.3		16
25.11	even	5		625.2.d.o.501.2		16
25.12	odd	20		625.2.e.i.499.2		8
25.13	odd	20		625.2.e.a.499.1		8
25.14	even	10		625.2.d.o.501.3		16
25.16	even	5		625.2.d.o.126.2		16
25.17	odd	20		25.2.e.a.14.2	yes	8
25.19	even	10		125.2.d.b.51.2		16
25.21	even	5		125.2.d.b.76.3		16
25.22	odd	20		125.2.e.b.49.1		8
25.23	odd	20		625.2.e.i.124.2		8
75.17	even	20		225.2.m.a.64.1		8
75.53	even	20		225.2.m.a.109.1		8
100.3	even	20		400.2.y.c.209.2		8
100.67	even	20		400.2.y.c.289.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.2.e.a.9.2	✓	8	25.3	odd	20
25.2.e.a.14.2	yes	8	25.17	odd	20
125.2.d.b.51.2		16	25.19	even	10
125.2.d.b.51.3		16	25.6	even	5
125.2.d.b.76.2		16	25.4	even	10
125.2.d.b.76.3		16	25.21	even	5
125.2.e.b.49.1		8	25.22	odd	20
125.2.e.b.74.1		8	25.8	odd	20
225.2.m.a.64.1		8	75.17	even	20
225.2.m.a.109.1		8	75.53	even	20
400.2.y.c.209.2		8	100.3	even	20
400.2.y.c.289.2		8	100.67	even	20
625.2.a.f.1.3		8	1.1	even	1	trivial
625.2.a.f.1.6		8	5.4	even	2	inner
625.2.b.c.624.3		8	5.2	odd	4
625.2.b.c.624.6		8	5.3	odd	4
625.2.d.o.126.2		16	25.16	even	5
625.2.d.o.126.3		16	25.9	even	10
625.2.d.o.501.2		16	25.11	even	5
625.2.d.o.501.3		16	25.14	even	10
625.2.e.a.124.1		8	25.2	odd	20
625.2.e.a.499.1		8	25.13	odd	20
625.2.e.i.124.2		8	25.23	odd	20
625.2.e.i.499.2		8	25.12	odd	20
5625.2.a.x.1.3		8	15.14	odd	2
5625.2.a.x.1.6		8	3.2	odd	2
10000.2.a.bj.1.1		8	4.3	odd	2
10000.2.a.bj.1.8		8	20.19	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)

\(f(q)\)	\(=\)	\(q-1.13370 q^{2} +2.60278 q^{3} -0.714715 q^{4} -2.95078 q^{6} -0.407162 q^{7} +3.07768 q^{8} +3.77447 q^{9} +O(q^{10})\)	\(q-1.13370 q^{2} +2.60278 q^{3} -0.714715 q^{4} -2.95078 q^{6} -0.407162 q^{7} +3.07768 q^{8} +3.77447 q^{9} +2.00000 q^{11} -1.86025 q^{12} +0.700668 q^{13} +0.461601 q^{14} -2.05975 q^{16} -1.58273 q^{17} -4.27913 q^{18} +4.95078 q^{19} -1.05975 q^{21} -2.26741 q^{22} -1.20145 q^{23} +8.01054 q^{24} -0.794350 q^{26} +2.01577 q^{27} +0.291004 q^{28} +5.50906 q^{29} +8.20390 q^{31} -3.82022 q^{32} +5.20556 q^{33} +1.79435 q^{34} -2.69767 q^{36} +5.13532 q^{37} -5.61272 q^{38} +1.82368 q^{39} +7.21619 q^{41} +1.20145 q^{42} -9.16531 q^{43} -1.42943 q^{44} +1.36208 q^{46} +1.27323 q^{47} -5.36108 q^{48} -6.83422 q^{49} -4.11950 q^{51} -0.500778 q^{52} +5.07996 q^{53} -2.28528 q^{54} -1.25311 q^{56} +12.8858 q^{57} -6.24565 q^{58} -6.49972 q^{59} -9.42008 q^{61} -9.30079 q^{62} -1.53682 q^{63} +8.45050 q^{64} -5.90157 q^{66} +3.08173 q^{67} +1.13120 q^{68} -3.12710 q^{69} +6.86639 q^{71} +11.6166 q^{72} -0.545146 q^{73} -5.82193 q^{74} -3.53840 q^{76} -0.814323 q^{77} -2.06752 q^{78} +5.48159 q^{79} -6.07680 q^{81} -8.18102 q^{82} -0.974135 q^{83} +0.757421 q^{84} +10.3907 q^{86} +14.3389 q^{87} +6.15537 q^{88} -2.26649 q^{89} -0.285285 q^{91} +0.858691 q^{92} +21.3529 q^{93} -1.44347 q^{94} -9.94319 q^{96} -15.2185 q^{97} +7.74798 q^{98} +7.54893 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 6 q^{4} + 6 q^{6} + 4 q^{9}+O(q^{10}) \) 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

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