LMFDB - Embedded newform 4925.2.a.j.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4925,2,Mod(1,4925)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4925.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4925, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4925 = 5^{2} \cdot 197 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4925.a (trivial)

Newform invariants

comment:select newform

sage:traces = [10,2,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$39.3263229955$
Analytic rank:	$1$
Dimension:	$10$
Coefficient field:	10.10.21886214112361.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2x^{9} - 9x^{8} + 16x^{7} + 26x^{6} - 38x^{5} - 27x^{4} + 32x^{3} + 6x^{2} - 7x + 1 $ x^10 - 2x^9 - 9x^8 + 16x^7 + 26x^6 - 38x^5 - 27x^4 + 32x^3 + 6x^2 - 7*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 985)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.638386$ of defining polynomial
Character	$\chi$	$=$	4925.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.638386 q^{2} -1.45932 q^{3} -1.59246 q^{4} +0.931607 q^{6} +2.37873 q^{7} +2.29338 q^{8} -0.870397 q^{9} -1.81176 q^{11} +2.32391 q^{12} +4.54899 q^{13} -1.51855 q^{14} +1.72086 q^{16} -0.836504 q^{17} +0.555649 q^{18} -1.45674 q^{19} -3.47133 q^{21} +1.15660 q^{22} -1.66081 q^{23} -3.34676 q^{24} -2.90401 q^{26} +5.64813 q^{27} -3.78805 q^{28} -6.58183 q^{29} -7.71065 q^{31} -5.68533 q^{32} +2.64393 q^{33} +0.534013 q^{34} +1.38607 q^{36} +5.65226 q^{37} +0.929961 q^{38} -6.63841 q^{39} +10.8079 q^{41} +2.21605 q^{42} -7.93991 q^{43} +2.88516 q^{44} +1.06024 q^{46} -1.60862 q^{47} -2.51128 q^{48} -1.34162 q^{49} +1.22072 q^{51} -7.24409 q^{52} +3.16405 q^{53} -3.60569 q^{54} +5.45534 q^{56} +2.12584 q^{57} +4.20175 q^{58} +0.808859 q^{59} +6.67393 q^{61} +4.92237 q^{62} -2.07044 q^{63} +0.187711 q^{64} -1.68785 q^{66} -0.0994966 q^{67} +1.33210 q^{68} +2.42364 q^{69} +3.36616 q^{71} -1.99615 q^{72} +0.0821733 q^{73} -3.60833 q^{74} +2.31980 q^{76} -4.30969 q^{77} +4.23787 q^{78} -10.7394 q^{79} -5.63122 q^{81} -6.89963 q^{82} +3.37126 q^{83} +5.52796 q^{84} +5.06873 q^{86} +9.60497 q^{87} -4.15505 q^{88} +2.82869 q^{89} +10.8208 q^{91} +2.64477 q^{92} +11.2523 q^{93} +1.02692 q^{94} +8.29670 q^{96} +7.99399 q^{97} +0.856473 q^{98} +1.57695 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 2 q^{2} + 3 q^{3} + 2 q^{4} - 9 q^{6} + 6 q^{7} + 6 q^{8} - 5 q^{9} - 11 q^{11} + 5 q^{13} - 9 q^{14} - 2 q^{16} + 4 q^{17} - 15 q^{18} - 28 q^{19} - 7 q^{21} + 12 q^{22} + 24 q^{23} - 3 q^{24} + 7 q^{26}+ \cdots - 16 q^{99}+O(q^{100}) $ 10 * q + 2 * q^2 + 3 * q^3 + 2 * q^4 - 9 * q^6 + 6 * q^7 + 6 * q^8 - 5 * q^9 - 11 * q^11 + 5 * q^13 - 9 * q^14 - 2 * q^16 + 4 * q^17 - 15 * q^18 - 28 * q^19 - 7 * q^21 + 12 * q^22 + 24 * q^23 - 3 * q^24 + 7 * q^26 + 9 * q^27 - 3 * q^28 - 16 * q^29 - 34 * q^31 + 4 * q^32 + 10 * q^33 + q^34 - q^36 - 5 * q^37 - 13 * q^38 - 15 * q^39 - 15 * q^41 - 7 * q^42 + 5 * q^43 - 11 * q^44 + 6 * q^46 + 14 * q^47 - 12 * q^48 + 4 * q^49 - q^51 + 7 * q^52 + 15 * q^53 - 4 * q^54 - 5 * q^56 - 9 * q^57 - 28 * q^58 - 49 * q^59 - 16 * q^61 - 20 * q^62 - 8 * q^63 - 16 * q^64 + 40 * q^66 + 2 * q^67 - 19 * q^68 - 16 * q^69 - 64 * q^71 - q^72 - 18 * q^73 - 37 * q^74 - 19 * q^76 - 8 * q^77 - 4 * q^78 - 11 * q^79 - 14 * q^81 - 10 * q^82 - 4 * q^83 - 11 * q^84 + 5 * q^86 + q^87 - 19 * q^88 - 5 * q^89 - 30 * q^91 + 32 * q^92 - 41 * q^93 - 37 * q^94 + 26 * q^96 - 20 * q^97 + 37 * q^98 - 16 * q^99

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 $

$2$	−0.638386	−0.451407	−0.225704	−	0.974196i	$-0.572468\pi$
$2$	−0.638386	−0.451407	−0.225704	+	0.974196i	$0.572468\pi$
$3$	−1.45932	−0.842536	−0.421268	−	0.906936i	$-0.638415\pi$
$3$	−1.45932	−0.842536	−0.421268	+	0.906936i	$0.638415\pi$
$4$	−1.59246	−0.796231
$5$	0	0
$5$	0	0
$6$	0.931607	0.380327
$7$	2.37873	0.899077	0.449539	−	0.893261i	$-0.351588\pi$
$7$	2.37873	0.899077	0.449539	+	0.893261i	$0.351588\pi$
$8$	2.29338	0.810832
$9$	−0.870397	−0.290132
$10$	0	0
$11$	−1.81176	−0.546265	−0.273133	−	0.961976i	$-0.588060\pi$
$11$	−1.81176	−0.546265	−0.273133	+	0.961976i	$0.588060\pi$
$12$	2.32391	0.670854
$13$	4.54899	1.26166	0.630831	−	0.775920i	$-0.282714\pi$
$13$	4.54899	1.26166	0.630831	+	0.775920i	$0.282714\pi$
$14$	−1.51855	−0.405850
$15$	0	0
$16$	1.72086	0.430216
$17$	−0.836504	−0.202882	−0.101441	−	0.994842i	$-0.532345\pi$
$17$	−0.836504	−0.202882	−0.101441	+	0.994842i	$0.532345\pi$
$18$	0.555649	0.130968
$19$	−1.45674	−0.334199	−0.167099	−	0.985940i	$-0.553440\pi$
$19$	−1.45674	−0.334199	−0.167099	+	0.985940i	$0.553440\pi$
$20$	0	0
$21$	−3.47133	−0.757505
$22$	1.15660	0.246588
$23$	−1.66081	−0.346302	−0.173151	−	0.984895i	$-0.555395\pi$
$23$	−1.66081	−0.346302	−0.173151	+	0.984895i	$0.555395\pi$
$24$	−3.34676	−0.683155
$25$	0	0
$26$	−2.90401	−0.569523
$27$	5.64813	1.08698
$28$	−3.78805	−0.715874
$29$	−6.58183	−1.22222	−0.611108	−	0.791547i	$-0.709276\pi$
$29$	−6.58183	−1.22222	−0.611108	+	0.791547i	$0.709276\pi$
$30$	0	0
$31$	−7.71065	−1.38487	−0.692437	−	0.721479i	$-0.743463\pi$
$31$	−7.71065	−1.38487	−0.692437	+	0.721479i	$0.743463\pi$
$32$	−5.68533	−1.00503
$33$	2.64393	0.460248
$34$	0.534013	0.0915825
$35$	0	0
$36$	1.38607	0.231012
$37$	5.65226	0.929226	0.464613	−	0.885514i	$-0.346193\pi$
$37$	5.65226	0.929226	0.464613	+	0.885514i	$0.346193\pi$
$38$	0.929961	0.150860
$39$	−6.63841	−1.06300
$40$	0	0
$41$	10.8079	1.68791	0.843957	−	0.536411i	$-0.180220\pi$
$41$	10.8079	1.68791	0.843957	+	0.536411i	$0.180220\pi$
$42$	2.21605	0.341943
$43$	−7.93991	−1.21082	−0.605412	−	0.795912i	$-0.706992\pi$
$43$	−7.93991	−1.21082	−0.605412	+	0.795912i	$0.706992\pi$
$44$	2.88516	0.434954
$45$	0	0
$46$	1.06024	0.156323
$47$	−1.60862	−0.234641	−0.117320	−	0.993094i	$-0.537430\pi$
$47$	−1.60862	−0.234641	−0.117320	+	0.993094i	$0.537430\pi$
$48$	−2.51128	−0.362473
$49$	−1.34162	−0.191660
$50$	0	0
$51$	1.22072	0.170936
$52$	−7.24409	−1.00458
$53$	3.16405	0.434616	0.217308	−	0.976103i	$-0.430272\pi$
$53$	3.16405	0.434616	0.217308	+	0.976103i	$0.430272\pi$
$54$	−3.60569	−0.490672
$55$	0	0
$56$	5.45534	0.729001
$57$	2.12584	0.281574
$58$	4.20175	0.551717
$59$	0.808859	0.105304	0.0526522	−	0.998613i	$-0.483233\pi$
$59$	0.808859	0.105304	0.0526522	+	0.998613i	$0.483233\pi$
$60$	0	0
$61$	6.67393	0.854509	0.427255	−	0.904131i	$-0.359481\pi$
$61$	6.67393	0.854509	0.427255	+	0.904131i	$0.359481\pi$
$62$	4.92237	0.625142
$63$	−2.07044	−0.260851
$64$	0.187711	0.0234638
$65$	0	0
$66$	−1.68785	−0.207759
$67$	−0.0994966	−0.0121554	−0.00607772	−	0.999982i	$-0.501935\pi$
$67$	−0.0994966	−0.0121554	−0.00607772	+	0.999982i	$0.501935\pi$
$68$	1.33210	0.161541
$69$	2.42364	0.291772
$70$	0	0
$71$	3.36616	0.399490	0.199745	−	0.979848i	$-0.435989\pi$
$71$	3.36616	0.399490	0.199745	+	0.979848i	$0.435989\pi$
$72$	−1.99615	−0.235249
$73$	0.0821733	0.00961766	0.00480883	−	0.999988i	$-0.498469\pi$
$73$	0.0821733	0.00961766	0.00480883	+	0.999988i	$0.498469\pi$
$74$	−3.60833	−0.419460
$75$	0	0
$76$	2.31980	0.266099
$77$	−4.30969	−0.491135
$78$	4.23787	0.479844
$79$	−10.7394	−1.20828	−0.604141	−	0.796878i	$-0.706483\pi$
$79$	−10.7394	−1.20828	−0.604141	+	0.796878i	$0.706483\pi$
$80$	0	0
$81$	−5.63122	−0.625691
$82$	−6.89963	−0.761936
$83$	3.37126	0.370044	0.185022	−	0.982734i	$-0.440764\pi$
$83$	3.37126	0.370044	0.185022	+	0.982734i	$0.440764\pi$
$84$	5.52796	0.603150
$85$	0	0
$86$	5.06873	0.546575
$87$	9.60497	1.02976
$88$	−4.15505	−0.442929
$89$	2.82869	0.299840	0.149920	−	0.988698i	$-0.452098\pi$
$89$	2.82869	0.299840	0.149920	+	0.988698i	$0.452098\pi$
$90$	0	0
$91$	10.8208	1.13433
$92$	2.64477	0.275736
$93$	11.2523	1.16681
$94$	1.02692	0.105919
$95$	0	0
$96$	8.29670	0.846778
$97$	7.99399	0.811667	0.405834	−	0.913947i	$-0.366981\pi$
$97$	7.99399	0.811667	0.405834	+	0.913947i	$0.366981\pi$
$98$	0.856473	0.0865168
$99$	1.57695	0.158489
$100$	0	0
$101$	8.01271	0.797295	0.398647	−	0.917104i	$-0.369480\pi$
$101$	8.01271	0.797295	0.398647	+	0.917104i	$0.369480\pi$
$102$	−0.779294	−0.0771616
$103$	8.84323	0.871350	0.435675	−	0.900104i	$-0.356510\pi$
$103$	8.84323	0.871350	0.435675	+	0.900104i	$0.356510\pi$
$104$	10.4326	1.02300
$105$	0	0
$106$	−2.01989	−0.196189
$107$	5.17969	0.500739	0.250370	−	0.968150i	$-0.419448\pi$
$107$	5.17969	0.500739	0.250370	+	0.968150i	$0.419448\pi$
$108$	−8.99444	−0.865490
$109$	−3.07605	−0.294633	−0.147316	−	0.989089i	$-0.547064\pi$
$109$	−3.07605	−0.294633	−0.147316	+	0.989089i	$0.547064\pi$
$110$	0	0
$111$	−8.24844	−0.782907
$112$	4.09348	0.386797
$113$	−1.80734	−0.170020	−0.0850100	−	0.996380i	$-0.527092\pi$
$113$	−1.80734	−0.170020	−0.0850100	+	0.996380i	$0.527092\pi$
$114$	−1.35711	−0.127105
$115$	0	0
$116$	10.4813	0.973167
$117$	−3.95942	−0.366049
$118$	−0.516364	−0.0475352
$119$	−1.98982	−0.182407
$120$	0	0
$121$	−7.71754	−0.701594
$122$	−4.26054	−0.385732
$123$	−15.7722	−1.42213
$124$	12.2789	1.10268
$125$	0	0
$126$	1.32174	0.117750
$127$	−1.26176	−0.111963	−0.0559816	−	0.998432i	$-0.517829\pi$
$127$	−1.26176	−0.111963	−0.0559816	+	0.998432i	$0.517829\pi$
$128$	11.2508	0.994443
$129$	11.5868	1.02016
$130$	0	0
$131$	15.4514	1.34999	0.674996	−	0.737821i	$-0.264145\pi$
$131$	15.4514	1.34999	0.674996	+	0.737821i	$0.264145\pi$
$132$	−4.21035	−0.366464
$133$	−3.46519	−0.300470
$134$	0.0635173	0.00548706
$135$	0	0
$136$	−1.91842	−0.164503
$137$	−4.97641	−0.425163	−0.212582	−	0.977143i	$-0.568187\pi$
$137$	−4.97641	−0.425163	−0.212582	+	0.977143i	$0.568187\pi$
$138$	−1.54722	−0.131708
$139$	−18.4942	−1.56866	−0.784331	−	0.620343i	$-0.786993\pi$
$139$	−18.4942	−1.56866	−0.784331	+	0.620343i	$0.786993\pi$
$140$	0	0
$141$	2.34748	0.197693
$142$	−2.14891	−0.180333
$143$	−8.24166	−0.689202
$144$	−1.49783	−0.124820
$145$	0	0
$146$	−0.0524583	−0.00434148
$147$	1.95785	0.161481
$148$	−9.00102	−0.739879
$149$	−4.81035	−0.394079	−0.197039	−	0.980396i	$-0.563133\pi$
$149$	−4.81035	−0.394079	−0.197039	+	0.980396i	$0.563133\pi$
$150$	0	0
$151$	−6.81634	−0.554706	−0.277353	−	0.960768i	$-0.589457\pi$
$151$	−6.81634	−0.554706	−0.277353	+	0.960768i	$0.589457\pi$
$152$	−3.34085	−0.270979
$153$	0.728091	0.0588627
$154$	2.75125	0.221702
$155$	0	0
$156$	10.5714	0.846391
$157$	−11.1719	−0.891615	−0.445808	−	0.895129i	$-0.647083\pi$
$157$	−11.1719	−0.891615	−0.445808	+	0.895129i	$0.647083\pi$
$158$	6.85591	0.545427
$159$	−4.61735	−0.366180
$160$	0	0
$161$	−3.95061	−0.311352
$162$	3.59489	0.282441
$163$	−24.0764	−1.88581	−0.942906	−	0.333060i	$-0.891919\pi$
$163$	−24.0764	−1.88581	−0.942906	+	0.333060i	$0.891919\pi$
$164$	−17.2112	−1.34397
$165$	0	0
$166$	−2.15217	−0.167040
$167$	17.6344	1.36459	0.682295	−	0.731077i	$-0.260982\pi$
$167$	17.6344	1.36459	0.682295	+	0.731077i	$0.260982\pi$
$168$	−7.96107	−0.614210
$169$	7.69329	0.591791
$170$	0	0
$171$	1.26794	0.0969618
$172$	12.6440	0.964096
$173$	−0.983232	−0.0747537	−0.0373769	−	0.999301i	$-0.511900\pi$
$173$	−0.983232	−0.0747537	−0.0373769	+	0.999301i	$0.511900\pi$
$174$	−6.13168	−0.464842
$175$	0	0
$176$	−3.11779	−0.235012
$177$	−1.18038	−0.0887228
$178$	−1.80580	−0.135350
$179$	9.23360	0.690152	0.345076	−	0.938575i	$-0.387853\pi$
$179$	9.23360	0.690152	0.345076	+	0.938575i	$0.387853\pi$
$180$	0	0
$181$	2.78542	0.207039	0.103519	−	0.994627i	$-0.466990\pi$
$181$	2.78542	0.207039	0.103519	+	0.994627i	$0.466990\pi$
$182$	−6.90787	−0.512046
$183$	−9.73937	−0.719955
$184$	−3.80886	−0.280793
$185$	0	0
$186$	−7.18329	−0.526705
$187$	1.51554	0.110827
$188$	2.56166	0.186828
$189$	13.4354	0.977282
$190$	0	0
$191$	9.24802	0.669163	0.334582	−	0.942367i	$-0.391405\pi$
$191$	9.24802	0.669163	0.334582	+	0.942367i	$0.391405\pi$
$192$	−0.273929	−0.0197691
$193$	−5.72091	−0.411800	−0.205900	−	0.978573i	$-0.566012\pi$
$193$	−5.72091	−0.411800	−0.205900	+	0.978573i	$0.566012\pi$
$194$	−5.10326	−0.366392
$195$	0	0
$196$	2.13648	0.152606
$197$	−1.00000	−0.0712470
$197$	−1.00000	−0.0712470
$198$	−1.00670	−0.0715432
$199$	2.30305	0.163259	0.0816294	−	0.996663i	$-0.473988\pi$
$199$	2.30305	0.163259	0.0816294	+	0.996663i	$0.473988\pi$
$200$	0	0
$201$	0.145197	0.0102414
$202$	−5.11521	−0.359905
$203$	−15.6564	−1.09887
$204$	−1.94396	−0.136104
$205$	0	0
$206$	−5.64540	−0.393333
$207$	1.44556	0.100473
$208$	7.82819	0.542787
$209$	2.63926	0.182561
$210$	0	0
$211$	−26.3990	−1.81738	−0.908691	−	0.417470i	$-0.862917\pi$
$211$	−26.3990	−1.81738	−0.908691	+	0.417470i	$0.862917\pi$
$212$	−5.03864	−0.346055
$213$	−4.91229	−0.336585
$214$	−3.30664	−0.226037
$215$	0	0
$216$	12.9533	0.881361
$217$	−18.3416	−1.24511
$218$	1.96371	0.132999
$219$	−0.119917	−0.00810323
$220$	0	0
$221$	−3.80525	−0.255969
$222$	5.26569	0.353410
$223$	3.53923	0.237004	0.118502	−	0.992954i	$-0.462191\pi$
$223$	3.53923	0.237004	0.118502	+	0.992954i	$0.462191\pi$
$224$	−13.5239	−0.903604
$225$	0	0
$226$	1.15378	0.0767483
$227$	3.25072	0.215758	0.107879	−	0.994164i	$-0.465594\pi$
$227$	3.25072	0.215758	0.107879	+	0.994164i	$0.465594\pi$
$228$	−3.38532	−0.224198
$229$	15.6015	1.03098	0.515489	−	0.856896i	$-0.327610\pi$
$229$	15.6015	1.03098	0.515489	+	0.856896i	$0.327610\pi$
$230$	0	0
$231$	6.28920	0.413799
$232$	−15.0946	−0.991012
$233$	−7.40925	−0.485396	−0.242698	−	0.970102i	$-0.578032\pi$
$233$	−7.40925	−0.485396	−0.242698	+	0.970102i	$0.578032\pi$
$234$	2.52764	0.165237
$235$	0	0
$236$	−1.28808	−0.0838467
$237$	15.6722	1.01802
$238$	1.27028	0.0823397
$239$	−6.62976	−0.428843	−0.214422	−	0.976741i	$-0.568787\pi$
$239$	−6.62976	−0.428843	−0.214422	+	0.976741i	$0.568787\pi$
$240$	0	0
$241$	−4.19234	−0.270052	−0.135026	−	0.990842i	$-0.543112\pi$
$241$	−4.19234	−0.270052	−0.135026	+	0.990842i	$0.543112\pi$
$242$	4.92677	0.316705
$243$	−8.72667	−0.559816
$244$	−10.6280	−0.680387
$245$	0	0
$246$	10.0687	0.641959
$247$	−6.62668	−0.421646
$248$	−17.6834	−1.12290
$249$	−4.91973	−0.311775
$250$	0	0
$251$	−24.7951	−1.56505	−0.782527	−	0.622617i	$-0.786070\pi$
$251$	−24.7951	−1.56505	−0.782527	+	0.622617i	$0.786070\pi$
$252$	3.29710	0.207698
$253$	3.00898	0.189173
$254$	0.805490	0.0505410
$255$	0	0
$256$	−7.55780	−0.472363
$257$	4.95035	0.308794	0.154397	−	0.988009i	$-0.450656\pi$
$257$	4.95035	0.308794	0.154397	+	0.988009i	$0.450656\pi$
$258$	−7.39687	−0.460509
$259$	13.4452	0.835446
$260$	0	0
$261$	5.72881	0.354604
$262$	−9.86394	−0.609396
$263$	7.46313	0.460196	0.230098	−	0.973167i	$-0.426095\pi$
$263$	7.46313	0.460196	0.230098	+	0.973167i	$0.426095\pi$
$264$	6.06352	0.373184
$265$	0	0
$266$	2.21213	0.135634
$267$	−4.12795	−0.252627
$268$	0.158445	0.00967855
$269$	−18.1427	−1.10618	−0.553091	−	0.833121i	$-0.686552\pi$
$269$	−18.1427	−1.10618	−0.553091	+	0.833121i	$0.686552\pi$
$270$	0	0
$271$	−25.3166	−1.53787	−0.768935	−	0.639326i	$-0.779213\pi$
$271$	−25.3166	−1.53787	−0.768935	+	0.639326i	$0.779213\pi$
$272$	−1.43951	−0.0872832
$273$	−15.7910	−0.955716
$274$	3.17687	0.191922
$275$	0	0
$276$	−3.85956	−0.232318
$277$	−19.1476	−1.15047	−0.575233	−	0.817990i	$-0.695089\pi$
$277$	−19.1476	−1.15047	−0.575233	+	0.817990i	$0.695089\pi$
$278$	11.8065	0.708105
$279$	6.71132	0.401796
$280$	0	0
$281$	16.8848	1.00726	0.503630	−	0.863919i	$-0.331997\pi$
$281$	16.8848	1.00726	0.503630	+	0.863919i	$0.331997\pi$
$282$	−1.49860	−0.0892403
$283$	−5.06079	−0.300833	−0.150416	−	0.988623i	$-0.548061\pi$
$283$	−5.06079	−0.300833	−0.150416	+	0.988623i	$0.548061\pi$
$284$	−5.36048	−0.318086
$285$	0	0
$286$	5.26136	0.311111
$287$	25.7092	1.51756
$288$	4.94850	0.291593
$289$	−16.3003	−0.958839
$290$	0	0
$291$	−11.6658	−0.683859
$292$	−0.130858	−0.00765788
$293$	9.42268	0.550479	0.275240	−	0.961376i	$-0.411243\pi$
$293$	9.42268	0.550479	0.275240	+	0.961376i	$0.411243\pi$
$294$	−1.24986	−0.0728936
$295$	0	0
$296$	12.9628	0.753446
$297$	−10.2330	−0.593781
$298$	3.07086	0.177890
$299$	−7.55498	−0.436916
$300$	0	0
$301$	−18.8869	−1.08862
$302$	4.35146	0.250398
$303$	−11.6931	−0.671750
$304$	−2.50685	−0.143778
$305$	0	0
$306$	−0.464803	−0.0265710
$307$	19.7363	1.12641	0.563206	−	0.826316i	$-0.309568\pi$
$307$	19.7363	1.12641	0.563206	+	0.826316i	$0.309568\pi$
$308$	6.86302	0.391057
$309$	−12.9051	−0.734144
$310$	0	0
$311$	−18.2517	−1.03496	−0.517479	−	0.855696i	$-0.673130\pi$
$311$	−18.2517	−1.03496	−0.517479	+	0.855696i	$0.673130\pi$
$312$	−15.2244	−0.861911
$313$	16.3934	0.926611	0.463306	−	0.886199i	$-0.346663\pi$
$313$	16.3934	0.926611	0.463306	+	0.886199i	$0.346663\pi$
$314$	7.13199	0.402482
$315$	0	0
$316$	17.1022	0.962071
$317$	8.73465	0.490587	0.245294	−	0.969449i	$-0.421116\pi$
$317$	8.73465	0.490587	0.245294	+	0.969449i	$0.421116\pi$
$318$	2.94765	0.165296
$319$	11.9247	0.667654
$320$	0	0
$321$	−7.55880	−0.421891
$322$	2.52202	0.140547
$323$	1.21857	0.0678029
$324$	8.96751	0.498195
$325$	0	0
$326$	15.3701	0.851269
$327$	4.48894	0.248239
$328$	24.7867	1.36861
$329$	−3.82647	−0.210960
$330$	0	0
$331$	3.43899	0.189024	0.0945119	−	0.995524i	$-0.469871\pi$
$331$	3.43899	0.189024	0.0945119	+	0.995524i	$0.469871\pi$
$332$	−5.36861	−0.294641
$333$	−4.91971	−0.269599
$334$	−11.2576	−0.615986
$335$	0	0
$336$	−5.97368	−0.325891
$337$	−33.4243	−1.82074	−0.910368	−	0.413800i	$-0.864201\pi$
$337$	−33.4243	−1.82074	−0.910368	+	0.413800i	$0.864201\pi$
$338$	−4.91129	−0.267139
$339$	2.63748	0.143248
$340$	0	0
$341$	13.9698	0.756508
$342$	−0.809436	−0.0437693
$343$	−19.8425	−1.07139
$344$	−18.2092	−0.981775
$345$	0	0
$346$	0.627682	0.0337444
$347$	−14.0163	−0.752434	−0.376217	−	0.926532i	$-0.622775\pi$
$347$	−14.0163	−0.752434	−0.376217	+	0.926532i	$0.622775\pi$
$348$	−15.2956	−0.819928
$349$	22.8844	1.22498	0.612488	−	0.790480i	$-0.290169\pi$
$349$	22.8844	1.22498	0.612488	+	0.790480i	$0.290169\pi$
$350$	0	0
$351$	25.6933	1.37141
$352$	10.3004	0.549016
$353$	−33.2310	−1.76871	−0.884354	−	0.466816i	$-0.845401\pi$
$353$	−33.2310	−1.76871	−0.884354	+	0.466816i	$0.845401\pi$
$354$	0.753539	0.0400501
$355$	0	0
$356$	−4.50458	−0.238742
$357$	2.90378	0.153684
$358$	−5.89460	−0.311540
$359$	−8.96867	−0.473348	−0.236674	−	0.971589i	$-0.576057\pi$
$359$	−8.96867	−0.473348	−0.236674	+	0.971589i	$0.576057\pi$
$360$	0	0
$361$	−16.8779	−0.888311
$362$	−1.77817	−0.0934587
$363$	11.2623	0.591119
$364$	−17.2318	−0.903191
$365$	0	0
$366$	6.21748	0.324993
$367$	−29.2005	−1.52425	−0.762127	−	0.647428i	$-0.775845\pi$
$367$	−29.2005	−1.52425	−0.762127	+	0.647428i	$0.775845\pi$
$368$	−2.85802	−0.148985
$369$	−9.40718	−0.489718
$370$	0	0
$371$	7.52644	0.390753
$372$	−17.9188	−0.929048
$373$	−19.8718	−1.02892	−0.514462	−	0.857513i	$-0.672008\pi$
$373$	−19.8718	−1.02892	−0.514462	+	0.857513i	$0.672008\pi$
$374$	−0.967502	−0.0500283
$375$	0	0
$376$	−3.68917	−0.190254
$377$	−29.9407	−1.54202
$378$	−8.57698	−0.441152
$379$	13.8875	0.713355	0.356677	−	0.934228i	$-0.383910\pi$
$379$	13.8875	0.713355	0.356677	+	0.934228i	$0.383910\pi$
$380$	0	0
$381$	1.84131	0.0943330
$382$	−5.90381	−0.302065
$383$	6.02255	0.307738	0.153869	−	0.988091i	$-0.450827\pi$
$383$	6.02255	0.307738	0.153869	+	0.988091i	$0.450827\pi$
$384$	−16.4185	−0.837854
$385$	0	0
$386$	3.65215	0.185890
$387$	6.91087	0.351299
$388$	−12.7301	−0.646275
$389$	−27.4553	−1.39204	−0.696020	−	0.718022i	$-0.745048\pi$
$389$	−27.4553	−1.39204	−0.696020	+	0.718022i	$0.745048\pi$
$390$	0	0
$391$	1.38927	0.0702584
$392$	−3.07685	−0.155404
$393$	−22.5484	−1.13742
$394$	0.638386	0.0321614
$395$	0	0
$396$	−2.51123	−0.126194
$397$	−1.34398	−0.0674523	−0.0337262	−	0.999431i	$-0.510737\pi$
$397$	−1.34398	−0.0674523	−0.0337262	+	0.999431i	$0.510737\pi$
$398$	−1.47023	−0.0736962
$399$	5.05681	0.253157
$400$	0	0
$401$	−16.7130	−0.834605	−0.417303	−	0.908768i	$-0.637024\pi$
$401$	−16.7130	−0.834605	−0.417303	+	0.908768i	$0.637024\pi$
$402$	−0.0926918	−0.00462305
$403$	−35.0756	−1.74724
$404$	−12.7599	−0.634831
$405$	0	0
$406$	9.99485	0.496036
$407$	−10.2405	−0.507604
$408$	2.79958	0.138600
$409$	−1.84105	−0.0910340	−0.0455170	−	0.998964i	$-0.514494\pi$
$409$	−1.84105	−0.0910340	−0.0455170	+	0.998964i	$0.514494\pi$
$410$	0	0
$411$	7.26215	0.358216
$412$	−14.0825	−0.693796
$413$	1.92406	0.0946768
$414$	−0.922825	−0.0453544
$415$	0	0
$416$	−25.8625	−1.26801
$417$	26.9889	1.32165
$418$	−1.68486	−0.0824094
$419$	−21.5920	−1.05484	−0.527419	−	0.849605i	$-0.676840\pi$
$419$	−21.5920	−1.05484	−0.527419	+	0.849605i	$0.676840\pi$
$420$	0	0
$421$	−12.5755	−0.612894	−0.306447	−	0.951888i	$-0.599140\pi$
$421$	−12.5755	−0.612894	−0.306447	+	0.951888i	$0.599140\pi$
$422$	16.8527	0.820379
$423$	1.40013	0.0680769
$424$	7.25637	0.352401
$425$	0	0
$426$	3.13594	0.151937
$427$	15.8755	0.768270
$428$	−8.24846	−0.398704
$429$	12.0272	0.580678
$430$	0	0
$431$	−10.5575	−0.508538	−0.254269	−	0.967133i	$-0.581835\pi$
$431$	−10.5575	−0.508538	−0.254269	+	0.967133i	$0.581835\pi$
$432$	9.71967	0.467638
$433$	33.5743	1.61348	0.806738	−	0.590910i	$-0.201231\pi$
$433$	33.5743	1.61348	0.806738	+	0.590910i	$0.201231\pi$
$434$	11.7090	0.562051
$435$	0	0
$436$	4.89850	0.234596
$437$	2.41936	0.115734
$438$	0.0765533	0.00365786
$439$	3.35831	0.160283	0.0801416	−	0.996783i	$-0.474463\pi$
$439$	3.35831	0.160283	0.0801416	+	0.996783i	$0.474463\pi$
$440$	0	0
$441$	1.16774	0.0556068
$442$	2.42922	0.115546
$443$	22.8414	1.08523	0.542614	−	0.839982i	$-0.317435\pi$
$443$	22.8414	1.08523	0.542614	+	0.839982i	$0.317435\pi$
$444$	13.1353	0.623375
$445$	0	0
$446$	−2.25940	−0.106985
$447$	7.01982	0.332026
$448$	0.446514	0.0210958
$449$	−21.9195	−1.03445	−0.517223	−	0.855850i	$-0.673034\pi$
$449$	−21.9195	−1.03445	−0.517223	+	0.855850i	$0.673034\pi$
$450$	0	0
$451$	−19.5813	−0.922049
$452$	2.87812	0.135375
$453$	9.94720	0.467360
$454$	−2.07522	−0.0973947
$455$	0	0
$456$	4.87536	0.228310
$457$	28.2733	1.32257	0.661285	−	0.750135i	$-0.270011\pi$
$457$	28.2733	1.32257	0.661285	+	0.750135i	$0.270011\pi$
$458$	−9.95980	−0.465391
$459$	−4.72469	−0.220530
$460$	0	0
$461$	−10.9765	−0.511225	−0.255612	−	0.966779i	$-0.582277\pi$
$461$	−10.9765	−0.511225	−0.255612	+	0.966779i	$0.582277\pi$
$462$	−4.01494	−0.186792
$463$	−36.7887	−1.70972	−0.854858	−	0.518862i	$-0.826356\pi$
$463$	−36.7887	−1.70972	−0.854858	+	0.518862i	$0.826356\pi$
$464$	−11.3264	−0.525817
$465$	0	0
$466$	4.72997	0.219111
$467$	15.9738	0.739178	0.369589	−	0.929195i	$-0.379499\pi$
$467$	15.9738	0.739178	0.369589	+	0.929195i	$0.379499\pi$
$468$	6.30524	0.291460
$469$	−0.236676	−0.0109287
$470$	0	0
$471$	16.3033	0.751218
$472$	1.85502	0.0853842
$473$	14.3852	0.661431
$474$	−10.0049	−0.459542
$475$	0	0
$476$	3.16872	0.145238
$477$	−2.75398	−0.126096
$478$	4.23235	0.193583
$479$	−13.5586	−0.619509	−0.309755	−	0.950817i	$-0.600247\pi$
$479$	−13.5586	−0.619509	−0.309755	+	0.950817i	$0.600247\pi$
$480$	0	0
$481$	25.7121	1.17237
$482$	2.67633	0.121903
$483$	5.76520	0.262325
$484$	12.2899	0.558631
$485$	0	0
$486$	5.57099	0.252705
$487$	30.8938	1.39993	0.699965	−	0.714177i	$-0.253199\pi$
$487$	30.8938	1.39993	0.699965	+	0.714177i	$0.253199\pi$
$488$	15.3058	0.692863
$489$	35.1351	1.58886
$490$	0	0
$491$	−21.9059	−0.988598	−0.494299	−	0.869292i	$-0.664575\pi$
$491$	−21.9059	−0.988598	−0.494299	+	0.869292i	$0.664575\pi$
$492$	25.1166	1.13234
$493$	5.50573	0.247966
$494$	4.23038	0.190334
$495$	0	0
$496$	−13.2690	−0.595795
$497$	8.00720	0.359172
$498$	3.14069	0.140738
$499$	2.40142	0.107502	0.0537511	−	0.998554i	$-0.482882\pi$
$499$	2.40142	0.107502	0.0537511	+	0.998554i	$0.482882\pi$
$500$	0	0
$501$	−25.7342	−1.14972
$502$	15.8289	0.706477
$503$	1.79754	0.0801483	0.0400741	−	0.999197i	$-0.487241\pi$
$503$	1.79754	0.0801483	0.0400741	+	0.999197i	$0.487241\pi$
$504$	−4.74831	−0.211507
$505$	0	0
$506$	−1.92089	−0.0853939
$507$	−11.2269	−0.498606
$508$	2.00931	0.0891486
$509$	−36.0980	−1.60001	−0.800007	−	0.599990i	$-0.795171\pi$
$509$	−36.0980	−1.60001	−0.800007	+	0.599990i	$0.795171\pi$
$510$	0	0
$511$	0.195469	0.00864702
$512$	−17.6769	−0.781215
$513$	−8.22785	−0.363268
$514$	−3.16024	−0.139392
$515$	0	0
$516$	−18.4516	−0.812286
$517$	2.91442	0.128176
$518$	−8.58325	−0.377127
$519$	1.43485	0.0629828
$520$	0	0
$521$	−23.4982	−1.02948	−0.514738	−	0.857348i	$-0.672111\pi$
$521$	−23.4982	−1.02948	−0.514738	+	0.857348i	$0.672111\pi$
$522$	−3.65719	−0.160071
$523$	21.0190	0.919096	0.459548	−	0.888153i	$-0.348011\pi$
$523$	21.0190	0.919096	0.459548	+	0.888153i	$0.348011\pi$
$524$	−24.6057	−1.07491
$525$	0	0
$526$	−4.76436	−0.207736
$527$	6.44999	0.280966
$528$	4.54984	0.198006
$529$	−20.2417	−0.880075
$530$	0	0
$531$	−0.704028	−0.0305522
$532$	5.51819	0.239244
$533$	49.1651	2.12958
$534$	2.63523	0.114037
$535$	0	0
$536$	−0.228184	−0.00985603
$537$	−13.4747	−0.581478
$538$	11.5821	0.499339
$539$	2.43069	0.104697
$540$	0	0
$541$	31.3399	1.34741	0.673704	−	0.739002i	$-0.264702\pi$
$541$	31.3399	1.34741	0.673704	+	0.739002i	$0.264702\pi$
$542$	16.1617	0.694206
$543$	−4.06481	−0.174438
$544$	4.75581	0.203904
$545$	0	0
$546$	10.0808	0.431417
$547$	−39.8339	−1.70318	−0.851588	−	0.524212i	$-0.824360\pi$
$547$	−39.8339	−1.70318	−0.851588	+	0.524212i	$0.824360\pi$
$548$	7.92474	0.338528
$549$	−5.80897	−0.247921
$550$	0	0
$551$	9.58801	0.408463
$552$	5.55832	0.236578
$553$	−25.5463	−1.08634
$554$	12.2235	0.519329
$555$	0	0
$556$	29.4514	1.24902
$557$	−14.3352	−0.607400	−0.303700	−	0.952768i	$-0.598222\pi$
$557$	−14.3352	−0.607400	−0.303700	+	0.952768i	$0.598222\pi$
$558$	−4.28442	−0.181374
$559$	−36.1185	−1.52765
$560$	0	0
$561$	−2.21166	−0.0933762
$562$	−10.7790	−0.454685
$563$	40.0770	1.68904	0.844521	−	0.535522i	$-0.179885\pi$
$563$	40.0770	1.68904	0.844521	+	0.535522i	$0.179885\pi$
$564$	−3.73828	−0.157410
$565$	0	0
$566$	3.23074	0.135798
$567$	−13.3952	−0.562544
$568$	7.71988	0.323919
$569$	−8.23345	−0.345164	−0.172582	−	0.984995i	$-0.555211\pi$
$569$	−8.23345	−0.345164	−0.172582	+	0.984995i	$0.555211\pi$
$570$	0	0
$571$	−19.0529	−0.797341	−0.398671	−	0.917094i	$-0.630528\pi$
$571$	−19.0529	−0.797341	−0.398671	+	0.917094i	$0.630528\pi$
$572$	13.1245	0.548765
$573$	−13.4958	−0.563795
$574$	−16.4124	−0.685040
$575$	0	0
$576$	−0.163383	−0.00680762
$577$	−10.7142	−0.446038	−0.223019	−	0.974814i	$-0.571591\pi$
$577$	−10.7142	−0.446038	−0.223019	+	0.974814i	$0.571591\pi$
$578$	10.4059	0.432827
$579$	8.34861	0.346957
$580$	0	0
$581$	8.01933	0.332698
$582$	7.44726	0.308699
$583$	−5.73250	−0.237416
$584$	0.188455	0.00779830
$585$	0	0
$586$	−6.01531	−0.248490
$587$	−33.4292	−1.37977	−0.689886	−	0.723918i	$-0.742339\pi$
$587$	−33.4292	−1.37977	−0.689886	+	0.723918i	$0.742339\pi$
$588$	−3.11780	−0.128576
$589$	11.2324	0.462823
$590$	0	0
$591$	1.45932	0.0600282
$592$	9.72678	0.399768
$593$	−15.0267	−0.617071	−0.308535	−	0.951213i	$-0.599839\pi$
$593$	−15.0267	−0.617071	−0.308535	+	0.951213i	$0.599839\pi$
$594$	6.53263	0.268037
$595$	0	0
$596$	7.66030	0.313778
$597$	−3.36088	−0.137551
$598$	4.82300	0.197227
$599$	−24.7565	−1.01152	−0.505762	−	0.862673i	$-0.668789\pi$
$599$	−24.7565	−1.01152	−0.505762	+	0.862673i	$0.668789\pi$
$600$	0	0
$601$	11.0024	0.448798	0.224399	−	0.974497i	$-0.427958\pi$
$601$	11.0024	0.448798	0.224399	+	0.974497i	$0.427958\pi$
$602$	12.0572	0.491413
$603$	0.0866016	0.00352669
$604$	10.8548	0.441674
$605$	0	0
$606$	7.46470	0.303233
$607$	14.7004	0.596671	0.298336	−	0.954461i	$-0.403569\pi$
$607$	14.7004	0.596671	0.298336	+	0.954461i	$0.403569\pi$
$608$	8.28204	0.335881
$609$	22.8477	0.925835
$610$	0	0
$611$	−7.31758	−0.296037
$612$	−1.15946	−0.0468683
$613$	5.35706	0.216370	0.108185	−	0.994131i	$-0.465496\pi$
$613$	5.35706	0.216370	0.108185	+	0.994131i	$0.465496\pi$
$614$	−12.5994	−0.508471
$615$	0	0
$616$	−9.88375	−0.398228
$617$	−8.26597	−0.332775	−0.166388	−	0.986060i	$-0.553210\pi$
$617$	−8.26597	−0.332775	−0.166388	+	0.986060i	$0.553210\pi$
$618$	8.23842	0.331398
$619$	10.3460	0.415839	0.207920	−	0.978146i	$-0.433331\pi$
$619$	10.3460	0.415839	0.207920	+	0.978146i	$0.433331\pi$
$620$	0	0
$621$	−9.38045	−0.376424
$622$	11.6516	0.467188
$623$	6.72870	0.269580
$624$	−11.4238	−0.457318
$625$	0	0
$626$	−10.4653	−0.418279
$627$	−3.85151	−0.153814
$628$	17.7908	0.709932
$629$	−4.72814	−0.188523
$630$	0	0
$631$	6.37393	0.253742	0.126871	−	0.991919i	$-0.459507\pi$
$631$	6.37393	0.253742	0.126871	+	0.991919i	$0.459507\pi$
$632$	−24.6296	−0.979713
$633$	38.5245	1.53121
$634$	−5.57608	−0.221455
$635$	0	0
$636$	7.35296	0.291564
$637$	−6.10302	−0.241810
$638$	−7.61255	−0.301384
$639$	−2.92989	−0.115905
$640$	0	0
$641$	38.6778	1.52768	0.763841	−	0.645405i	$-0.223311\pi$
$641$	38.6778	1.52768	0.763841	+	0.645405i	$0.223311\pi$
$642$	4.82543	0.190445
$643$	14.1247	0.557025	0.278512	−	0.960433i	$-0.410159\pi$
$643$	14.1247	0.557025	0.278512	+	0.960433i	$0.410159\pi$
$644$	6.29121	0.247908
$645$	0	0
$646$	−0.777917	−0.0306067
$647$	10.2516	0.403033	0.201516	−	0.979485i	$-0.435413\pi$
$647$	10.2516	0.403033	0.201516	+	0.979485i	$0.435413\pi$
$648$	−12.9145	−0.507330
$649$	−1.46546	−0.0575242
$650$	0	0
$651$	26.7662	1.04905
$652$	38.3408	1.50154
$653$	28.5662	1.11788	0.558941	−	0.829207i	$-0.311208\pi$
$653$	28.5662	1.11788	0.558941	+	0.829207i	$0.311208\pi$
$654$	−2.86567	−0.112057
$655$	0	0
$656$	18.5990	0.726167
$657$	−0.0715234	−0.00279039
$658$	2.44277	0.0952290
$659$	20.6905	0.805986	0.402993	−	0.915203i	$-0.367970\pi$
$659$	20.6905	0.805986	0.402993	+	0.915203i	$0.367970\pi$
$660$	0	0
$661$	−19.9747	−0.776924	−0.388462	−	0.921465i	$-0.626994\pi$
$661$	−19.9747	−0.776924	−0.388462	+	0.921465i	$0.626994\pi$
$662$	−2.19540	−0.0853267
$663$	5.55306	0.215663
$664$	7.73158	0.300043
$665$	0	0
$666$	3.14068	0.121699
$667$	10.9311	0.423256
$668$	−28.0821	−1.08653
$669$	−5.16485	−0.199685
$670$	0	0
$671$	−12.0915	−0.466789
$672$	19.7356	0.761319
$673$	−24.2052	−0.933040	−0.466520	−	0.884511i	$-0.654492\pi$
$673$	−24.2052	−0.933040	−0.466520	+	0.884511i	$0.654492\pi$
$674$	21.3376	0.821893
$675$	0	0
$676$	−12.2513	−0.471203
$677$	−5.11565	−0.196610	−0.0983052	−	0.995156i	$-0.531342\pi$
$677$	−5.11565	−0.196610	−0.0983052	+	0.995156i	$0.531342\pi$
$678$	−1.68373	−0.0646632
$679$	19.0156	0.729751
$680$	0	0
$681$	−4.74383	−0.181784
$682$	−8.91814	−0.341493
$683$	−25.1783	−0.963420	−0.481710	−	0.876331i	$-0.659984\pi$
$683$	−25.1783	−0.963420	−0.481710	+	0.876331i	$0.659984\pi$
$684$	−2.01915	−0.0772040
$685$	0	0
$686$	12.6672	0.483635
$687$	−22.7676	−0.868637
$688$	−13.6635	−0.520916
$689$	14.3932	0.548339
$690$	0	0
$691$	−46.5869	−1.77225	−0.886124	−	0.463447i	$-0.846612\pi$
$691$	−46.5869	−1.77225	−0.886124	+	0.463447i	$0.846612\pi$
$692$	1.56576	0.0595213
$693$	3.75114	0.142494
$694$	8.94780	0.339654
$695$	0	0
$696$	22.0278	0.834963
$697$	−9.04087	−0.342447
$698$	−14.6091	−0.552963
$699$	10.8124	0.408964
$700$	0	0
$701$	−23.5751	−0.890419	−0.445210	−	0.895426i	$-0.646871\pi$
$701$	−23.5751	−0.890419	−0.445210	+	0.895426i	$0.646871\pi$
$702$	−16.4022	−0.619063
$703$	−8.23387	−0.310546
$704$	−0.340086	−0.0128175
$705$	0	0
$706$	21.2142	0.798408
$707$	19.0601	0.716829
$708$	1.87971	0.0706439
$709$	21.5784	0.810394	0.405197	−	0.914229i	$-0.367203\pi$
$709$	21.5784	0.810394	0.405197	+	0.914229i	$0.367203\pi$
$710$	0	0
$711$	9.34757	0.350561
$712$	6.48726	0.243120
$713$	12.8059	0.479584
$714$	−1.85373	−0.0693742
$715$	0	0
$716$	−14.7042	−0.549521
$717$	9.67491	0.361316
$718$	5.72547	0.213673
$719$	33.8809	1.26354	0.631772	−	0.775154i	$-0.282328\pi$
$719$	33.8809	1.26354	0.631772	+	0.775154i	$0.282328\pi$
$720$	0	0
$721$	21.0357	0.783410
$722$	10.7746	0.400990
$723$	6.11794	0.227529
$724$	−4.43568	−0.164851
$725$	0	0
$726$	−7.18971	−0.266835
$727$	15.2099	0.564105	0.282053	−	0.959399i	$-0.408985\pi$
$727$	15.2099	0.564105	0.282053	+	0.959399i	$0.408985\pi$
$728$	24.8163	0.919752
$729$	29.6286	1.09736
$730$	0	0
$731$	6.64177	0.245655
$732$	15.5096	0.573251
$733$	−31.9199	−1.17899	−0.589494	−	0.807773i	$-0.700673\pi$
$733$	−31.9199	−1.17899	−0.589494	+	0.807773i	$0.700673\pi$
$734$	18.6412	0.688059
$735$	0	0
$736$	9.44223	0.348045
$737$	0.180264	0.00664010
$738$	6.00541	0.221062
$739$	19.9347	0.733308	0.366654	−	0.930357i	$-0.380503\pi$
$739$	19.9347	0.733308	0.366654	+	0.930357i	$0.380503\pi$
$740$	0	0
$741$	9.67042	0.355252
$742$	−4.80478	−0.176389
$743$	50.1163	1.83859	0.919294	−	0.393572i	$-0.128761\pi$
$743$	50.1163	1.83859	0.919294	+	0.393572i	$0.128761\pi$
$744$	25.8057	0.946084
$745$	0	0
$746$	12.6859	0.464464
$747$	−2.93433	−0.107362
$748$	−2.41345	−0.0882443
$749$	12.3211	0.450203
$750$	0	0
$751$	−46.1121	−1.68266	−0.841328	−	0.540525i	$-0.818226\pi$
$751$	−46.1121	−1.68266	−0.841328	+	0.540525i	$0.818226\pi$
$752$	−2.76821	−0.100946
$753$	36.1839	1.31862
$754$	19.1137	0.696080
$755$	0	0
$756$	−21.3954	−0.778143
$757$	−9.69373	−0.352325	−0.176162	−	0.984361i	$-0.556368\pi$
$757$	−9.69373	−0.352325	−0.176162	+	0.984361i	$0.556368\pi$
$758$	−8.86561	−0.322014
$759$	−4.39105	−0.159385
$760$	0	0
$761$	46.7950	1.69632	0.848158	−	0.529743i	$-0.177712\pi$
$761$	46.7950	1.69632	0.848158	+	0.529743i	$0.177712\pi$
$762$	−1.17546	−0.0425826
$763$	−7.31712	−0.264897
$764$	−14.7271	−0.532809
$765$	0	0
$766$	−3.84471	−0.138915
$767$	3.67949	0.132859
$768$	11.0292	0.397983
$769$	52.0634	1.87745	0.938727	−	0.344662i	$-0.112007\pi$
$769$	52.0634	1.87745	0.938727	+	0.344662i	$0.112007\pi$
$770$	0	0
$771$	−7.22412	−0.260170
$772$	9.11034	0.327888
$773$	38.6344	1.38958	0.694791	−	0.719212i	$-0.255497\pi$
$773$	38.6344	1.38958	0.694791	+	0.719212i	$0.255497\pi$
$774$	−4.41180	−0.158579
$775$	0	0
$776$	18.3333	0.658126
$777$	−19.6208	−0.703894
$778$	17.5271	0.628377
$779$	−15.7443	−0.564098
$780$	0	0
$781$	−6.09866	−0.218227
$782$	−0.886891	−0.0317152
$783$	−37.1751	−1.32853
$784$	−2.30875	−0.0824553
$785$	0	0
$786$	14.3946	0.513439
$787$	−12.4300	−0.443080	−0.221540	−	0.975151i	$-0.571108\pi$
$787$	−12.4300	−0.443080	−0.221540	+	0.975151i	$0.571108\pi$
$788$	1.59246	0.0567291
$789$	−10.8911	−0.387732
$790$	0	0
$791$	−4.29918	−0.152861
$792$	3.61654	0.128508
$793$	30.3596	1.07810
$794$	0.857977	0.0304485
$795$	0	0
$796$	−3.66752	−0.129992
$797$	−34.3845	−1.21796	−0.608980	−	0.793186i	$-0.708421\pi$
$797$	−34.3845	−1.21796	−0.608980	+	0.793186i	$0.708421\pi$
$798$	−3.22820	−0.114277
$799$	1.34562	0.0476044
$800$	0	0
$801$	−2.46208	−0.0869934
$802$	10.6693	0.376747
$803$	−0.148878	−0.00525379
$804$	−0.231221	−0.00815453
$805$	0	0
$806$	22.3918	0.788718
$807$	26.4760	0.931999
$808$	18.3762	0.646472
$809$	41.9168	1.47372	0.736859	−	0.676047i	$-0.236308\pi$
$809$	41.9168	1.47372	0.736859	+	0.676047i	$0.236308\pi$
$810$	0	0
$811$	35.0481	1.23070	0.615352	−	0.788252i	$-0.289014\pi$
$811$	35.0481	1.23070	0.615352	+	0.788252i	$0.289014\pi$
$812$	24.9323	0.874952
$813$	36.9448	1.29571
$814$	6.53741	0.229136
$815$	0	0
$816$	2.10070	0.0735392
$817$	11.5664	0.404656
$818$	1.17530	0.0410934
$819$	−9.41842	−0.329106
$820$	0	0
$821$	−17.1696	−0.599225	−0.299612	−	0.954061i	$-0.596857\pi$
$821$	−17.1696	−0.599225	−0.299612	+	0.954061i	$0.596857\pi$
$822$	−4.63606	−0.161701
$823$	14.0943	0.491296	0.245648	−	0.969359i	$-0.420999\pi$
$823$	14.0943	0.491296	0.245648	+	0.969359i	$0.420999\pi$
$824$	20.2809	0.706518
$825$	0	0
$826$	−1.22829	−0.0427378
$827$	−35.3779	−1.23021	−0.615106	−	0.788445i	$-0.710887\pi$
$827$	−35.3779	−1.23021	−0.615106	+	0.788445i	$0.710887\pi$
$828$	−2.30200	−0.0800000
$829$	43.7694	1.52017	0.760087	−	0.649821i	$-0.225156\pi$
$829$	43.7694	1.52017	0.760087	+	0.649821i	$0.225156\pi$
$830$	0	0
$831$	27.9424	0.969310
$832$	0.853894	0.0296034
$833$	1.12227	0.0388844
$834$	−17.2294	−0.596604
$835$	0	0
$836$	−4.20292	−0.145361
$837$	−43.5507	−1.50533
$838$	13.7840	0.476162
$839$	−32.3344	−1.11631	−0.558153	−	0.829738i	$-0.688490\pi$
$839$	−32.3344	−1.11631	−0.558153	+	0.829738i	$0.688490\pi$
$840$	0	0
$841$	14.3205	0.493811
$842$	8.02805	0.276665
$843$	−24.6402	−0.848654
$844$	42.0394	1.44706
$845$	0	0
$846$	−0.893827	−0.0307304
$847$	−18.3580	−0.630787
$848$	5.44491	0.186979
$849$	7.38530	0.253463
$850$	0	0
$851$	−9.38731	−0.321793
$852$	7.82264	0.267999
$853$	−6.19646	−0.212163	−0.106081	−	0.994357i	$-0.533830\pi$
$853$	−6.19646	−0.212163	−0.106081	+	0.994357i	$0.533830\pi$
$854$	−10.1347	−0.346802
$855$	0	0
$856$	11.8790	0.406015
$857$	−30.3160	−1.03558	−0.517788	−	0.855509i	$-0.673244\pi$
$857$	−30.3160	−1.03558	−0.517788	+	0.855509i	$0.673244\pi$
$858$	−7.67799	−0.262122
$859$	−27.1160	−0.925185	−0.462593	−	0.886571i	$-0.653081\pi$
$859$	−27.1160	−0.925185	−0.462593	+	0.886571i	$0.653081\pi$
$860$	0	0
$861$	−37.5178	−1.27860
$862$	6.73978	0.229558
$863$	−27.7809	−0.945672	−0.472836	−	0.881150i	$-0.656770\pi$
$863$	−27.7809	−0.945672	−0.472836	+	0.881150i	$0.656770\pi$
$864$	−32.1115	−1.09246
$865$	0	0
$866$	−21.4333	−0.728335
$867$	23.7872	0.807857
$868$	29.2083	0.991394
$869$	19.4573	0.660042
$870$	0	0
$871$	−0.452609	−0.0153361
$872$	−7.05456	−0.238897
$873$	−6.95795	−0.235491
$874$	−1.54448	−0.0522430
$875$	0	0
$876$	0.190963	0.00645205
$877$	38.0939	1.28634	0.643170	−	0.765724i	$-0.277619\pi$
$877$	38.0939	1.28634	0.643170	+	0.765724i	$0.277619\pi$
$878$	−2.14390	−0.0723530
$879$	−13.7507	−0.463799
$880$	0	0
$881$	36.5035	1.22983	0.614917	−	0.788592i	$-0.289190\pi$
$881$	36.5035	1.22983	0.614917	+	0.788592i	$0.289190\pi$
$882$	−0.745471	−0.0251013
$883$	−40.4097	−1.35989	−0.679947	−	0.733261i	$-0.737997\pi$
$883$	−40.4097	−1.35989	−0.679947	+	0.733261i	$0.737997\pi$
$884$	6.05972	0.203810
$885$	0	0
$886$	−14.5816	−0.489880
$887$	15.6445	0.525289	0.262645	−	0.964893i	$-0.415405\pi$
$887$	15.6445	0.525289	0.262645	+	0.964893i	$0.415405\pi$
$888$	−18.9168	−0.634806
$889$	−3.00139	−0.100663
$890$	0	0
$891$	10.2024	0.341793
$892$	−5.63609	−0.188710
$893$	2.34333	0.0784166
$894$	−4.48135	−0.149879
$895$	0	0
$896$	26.7628	0.894081
$897$	11.0251	0.368118
$898$	13.9931	0.466957
$899$	50.7502	1.69261
$900$	0	0
$901$	−2.64674	−0.0881758
$902$	12.5004	0.416219
$903$	27.5620	0.917206
$904$	−4.14491	−0.137858
$905$	0	0
$906$	−6.35015	−0.210970
$907$	−42.4016	−1.40792	−0.703961	−	0.710239i	$-0.748587\pi$
$907$	−42.4016	−1.40792	−0.703961	+	0.710239i	$0.748587\pi$
$908$	−5.17665	−0.171793
$909$	−6.97424	−0.231321
$910$	0	0
$911$	22.2881	0.738437	0.369218	−	0.929343i	$-0.379625\pi$
$911$	22.2881	0.738437	0.369218	+	0.929343i	$0.379625\pi$
$912$	3.65828	0.121138
$913$	−6.10790	−0.202142
$914$	−18.0493	−0.597018
$915$	0	0
$916$	−24.8449	−0.820897
$917$	36.7547	1.21375
$918$	3.01618	0.0995486
$919$	−47.2107	−1.55734	−0.778669	−	0.627435i	$-0.784105\pi$
$919$	−47.2107	−1.55734	−0.778669	+	0.627435i	$0.784105\pi$
$920$	0	0
$921$	−28.8016	−0.949044
$922$	7.00722	0.230771
$923$	15.3126	0.504021
$924$	−10.0153	−0.329480
$925$	0	0
$926$	23.4854	0.771778
$927$	−7.69712	−0.252807
$928$	37.4199	1.22837
$929$	7.06514	0.231800	0.115900	−	0.993261i	$-0.463025\pi$
$929$	7.06514	0.231800	0.115900	+	0.993261i	$0.463025\pi$
$930$	0	0
$931$	1.95439	0.0640526
$932$	11.7990	0.386488
$933$	26.6350	0.871990
$934$	−10.1974	−0.333670
$935$	0	0
$936$	−9.08046	−0.296804
$937$	−41.1529	−1.34441	−0.672203	−	0.740367i	$-0.734652\pi$
$937$	−41.1529	−1.34441	−0.672203	+	0.740367i	$0.734652\pi$
$938$	0.151091	0.00493329
$939$	−23.9232	−0.780704
$940$	0	0
$941$	56.1332	1.82989	0.914944	−	0.403580i	$-0.132234\pi$
$941$	56.1332	1.82989	0.914944	+	0.403580i	$0.132234\pi$
$942$	−10.4078	−0.339105
$943$	−17.9498	−0.584527
$944$	1.39194	0.0453037
$945$	0	0
$946$	−9.18330	−0.298575
$947$	−1.33358	−0.0433354	−0.0216677	−	0.999765i	$-0.506898\pi$
$947$	−1.33358	−0.0433354	−0.0216677	+	0.999765i	$0.506898\pi$
$948$	−24.9575	−0.810580
$949$	0.373805	0.0121342
$950$	0	0
$951$	−12.7466	−0.413338
$952$	−4.56342	−0.147901
$953$	13.5724	0.439654	0.219827	−	0.975539i	$-0.429451\pi$
$953$	13.5724	0.439654	0.219827	+	0.975539i	$0.429451\pi$
$954$	1.75810	0.0569207
$955$	0	0
$956$	10.5576	0.341459
$957$	−17.4019	−0.562523
$958$	8.65564	0.279651
$959$	−11.8376	−0.382255
$960$	0	0
$961$	28.4541	0.917873
$962$	−16.4142	−0.529216
$963$	−4.50838	−0.145281
$964$	6.67614	0.215024
$965$	0	0
$966$	−3.68042	−0.118416
$967$	−31.1652	−1.00221	−0.501103	−	0.865388i	$-0.667072\pi$
$967$	−31.1652	−1.00221	−0.501103	+	0.865388i	$0.667072\pi$
$968$	−17.6992	−0.568875
$969$	−1.77828	−0.0571264
$970$	0	0
$971$	−52.1035	−1.67208	−0.836040	−	0.548669i	$-0.815135\pi$
$971$	−52.1035	−1.67208	−0.836040	+	0.548669i	$0.815135\pi$
$972$	13.8969	0.445743
$973$	−43.9929	−1.41035
$974$	−19.7222	−0.631939
$975$	0	0
$976$	11.4849	0.367624
$977$	18.5773	0.594339	0.297170	−	0.954825i	$-0.403957\pi$
$977$	18.5773	0.594339	0.297170	+	0.954825i	$0.403957\pi$
$978$	−22.4298	−0.717225
$979$	−5.12490	−0.163792
$980$	0	0
$981$	2.67739	0.0854824
$982$	13.9844	0.446260
$983$	59.8810	1.90991	0.954953	−	0.296757i	$-0.0959052\pi$
$983$	59.8810	1.90991	0.954953	+	0.296757i	$0.0959052\pi$
$984$	−36.1716	−1.15311
$985$	0	0
$986$	−3.51478	−0.111934
$987$	5.58403	0.177742
$988$	10.5527	0.335728
$989$	13.1866	0.419311
$990$	0	0
$991$	3.79908	0.120682	0.0603408	−	0.998178i	$-0.480781\pi$
$991$	3.79908	0.120682	0.0603408	+	0.998178i	$0.480781\pi$
$992$	43.8376	1.39185
$993$	−5.01857	−0.159260
$994$	−5.11169	−0.162133
$995$	0	0
$996$	7.83449	0.248245
$997$	−45.4216	−1.43852	−0.719258	−	0.694743i	$-0.755518\pi$
$997$	−45.4216	−1.43852	−0.719258	+	0.694743i	$0.755518\pi$
$998$	−1.53303	−0.0485273
$999$	31.9247	1.01005

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4925.2.a.j.1.4		10
5.4	even	2		985.2.a.e.1.7	✓	10
15.14	odd	2		8865.2.a.t.1.4		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
985.2.a.e.1.7	✓	10	5.4	even	2
4925.2.a.j.1.4		10	1.1	even	1	trivial
8865.2.a.t.1.4		10	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4925 = 5^{2} \cdot 197 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4925.a (trivial)

\(f(q)\)	\(=\)	\(q-0.638386 q^{2} -1.45932 q^{3} -1.59246 q^{4} +0.931607 q^{6} +2.37873 q^{7} +2.29338 q^{8} -0.870397 q^{9} -1.81176 q^{11} +2.32391 q^{12} +4.54899 q^{13} -1.51855 q^{14} +1.72086 q^{16} -0.836504 q^{17} +0.555649 q^{18} -1.45674 q^{19} -3.47133 q^{21} +1.15660 q^{22} -1.66081 q^{23} -3.34676 q^{24} -2.90401 q^{26} +5.64813 q^{27} -3.78805 q^{28} -6.58183 q^{29} -7.71065 q^{31} -5.68533 q^{32} +2.64393 q^{33} +0.534013 q^{34} +1.38607 q^{36} +5.65226 q^{37} +0.929961 q^{38} -6.63841 q^{39} +10.8079 q^{41} +2.21605 q^{42} -7.93991 q^{43} +2.88516 q^{44} +1.06024 q^{46} -1.60862 q^{47} -2.51128 q^{48} -1.34162 q^{49} +1.22072 q^{51} -7.24409 q^{52} +3.16405 q^{53} -3.60569 q^{54} +5.45534 q^{56} +2.12584 q^{57} +4.20175 q^{58} +0.808859 q^{59} +6.67393 q^{61} +4.92237 q^{62} -2.07044 q^{63} +0.187711 q^{64} -1.68785 q^{66} -0.0994966 q^{67} +1.33210 q^{68} +2.42364 q^{69} +3.36616 q^{71} -1.99615 q^{72} +0.0821733 q^{73} -3.60833 q^{74} +2.31980 q^{76} -4.30969 q^{77} +4.23787 q^{78} -10.7394 q^{79} -5.63122 q^{81} -6.89963 q^{82} +3.37126 q^{83} +5.52796 q^{84} +5.06873 q^{86} +9.60497 q^{87} -4.15505 q^{88} +2.82869 q^{89} +10.8208 q^{91} +2.64477 q^{92} +11.2523 q^{93} +1.02692 q^{94} +8.29670 q^{96} +7.99399 q^{97} +0.856473 q^{98} +1.57695 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 2 q^{2} + 3 q^{3} + 2 q^{4} - 9 q^{6} + 6 q^{7} + 6 q^{8} - 5 q^{9} - 11 q^{11} + 5 q^{13} - 9 q^{14} - 2 q^{16} + 4 q^{17} - 15 q^{18} - 28 q^{19} - 7 q^{21} + 12 q^{22} + 24 q^{23} - 3 q^{24} + 7 q^{26}+ \cdots - 16 q^{99}+O(q^{100}) \) 10 * q + 2 * q^2 + 3 * q^3 + 2 * q^4 - 9 * q^6 + 6 * q^7 + 6 * q^8 - 5 * q^9 - 11 * q^11 + 5 * q^13 - 9 * q^14 - 2 * q^16 + 4 * q^17 - 15 * q^18 - 28 * q^19 - 7 * q^21 + 12 * q^22 + 24 * q^23 - 3 * q^24 + 7 * q^26 + 9 * q^27 - 3 * q^28 - 16 * q^29 - 34 * q^31 + 4 * q^32 + 10 * q^33 + q^34 - q^36 - 5 * q^37 - 13 * q^38 - 15 * q^39 - 15 * q^41 - 7 * q^42 + 5 * q^43 - 11 * q^44 + 6 * q^46 + 14 * q^47 - 12 * q^48 + 4 * q^49 - q^51 + 7 * q^52 + 15 * q^53 - 4 * q^54 - 5 * q^56 - 9 * q^57 - 28 * q^58 - 49 * q^59 - 16 * q^61 - 20 * q^62 - 8 * q^63 - 16 * q^64 + 40 * q^66 + 2 * q^67 - 19 * q^68 - 16 * q^69 - 64 * q^71 - q^72 - 18 * q^73 - 37 * q^74 - 19 * q^76 - 8 * q^77 - 4 * q^78 - 11 * q^79 - 14 * q^81 - 10 * q^82 - 4 * q^83 - 11 * q^84 + 5 * q^86 + q^87 - 19 * q^88 - 5 * q^89 - 30 * q^91 + 32 * q^92 - 41 * q^93 - 37 * q^94 + 26 * q^96 - 20 * q^97 + 37 * q^98 - 16 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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