LMFDB - Embedded newform 475.2.a.j.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [475,2,Mod(1,475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(475, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("475.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 475 = 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	475.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:	$3.79289409601$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.66064384.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 9x^{4} + 13x^{2} - 1 $ x^6 - 9x^4 + 13x^2 - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 95)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.68667$ of defining polynomial
Character	$\chi$	$=$	475.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.82254 q^{2} -2.31446 q^{3} +1.32164 q^{4} +4.21819 q^{6} -1.45033 q^{7} +1.23634 q^{8} +2.35673 q^{9} +O(q^{10})$	\(q-1.82254 q^{2} -2.31446 q^{3} +1.32164 q^{4} +4.21819 q^{6} -1.45033 q^{7} +1.23634 q^{8} +2.35673 q^{9} -3.89655 q^{11} -3.05888 q^{12} -3.05888 q^{13} +2.64327 q^{14} -4.89655 q^{16} -3.92301 q^{17} -4.29522 q^{18} -1.00000 q^{19} +3.35673 q^{21} +7.10160 q^{22} +5.37334 q^{23} -2.86146 q^{24} +5.57491 q^{26} +1.48883 q^{27} -1.91681 q^{28} +6.00000 q^{29} -8.43637 q^{31} +6.45146 q^{32} +9.01841 q^{33} +7.14982 q^{34} +3.11474 q^{36} +5.95953 q^{37} +1.82254 q^{38} +7.07965 q^{39} +10.4364 q^{41} -6.11775 q^{42} +1.45033 q^{43} -5.14982 q^{44} -9.79310 q^{46} -4.90686 q^{47} +11.3329 q^{48} -4.89655 q^{49} +9.07965 q^{51} -4.04272 q^{52} +4.23127 q^{53} -2.71345 q^{54} -1.79310 q^{56} +2.31446 q^{57} -10.9352 q^{58} +3.35673 q^{59} +10.3329 q^{61} +15.3756 q^{62} -3.41802 q^{63} -1.96491 q^{64} -16.4364 q^{66} +9.84404 q^{67} -5.18479 q^{68} -12.4364 q^{69} +8.64327 q^{71} +2.91372 q^{72} +2.43418 q^{73} -10.8615 q^{74} -1.32164 q^{76} +5.65127 q^{77} -12.9029 q^{78} -12.4364 q^{79} -10.5160 q^{81} -19.0207 q^{82} -12.6635 q^{83} +4.43637 q^{84} -2.64327 q^{86} -13.8868 q^{87} -4.81746 q^{88} +12.3662 q^{89} +4.43637 q^{91} +7.10160 q^{92} +19.5256 q^{93} +8.94292 q^{94} -14.9316 q^{96} -3.05888 q^{97} +8.92414 q^{98} -9.18310 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 8 q^{4} + 14 q^{9}+O(q^{10}) $ 6 * q + 8 * q^4 + 14 * q^9	$ 6 q + 8 q^{4} + 14 q^{9} + 2 q^{11} + 16 q^{14} - 4 q^{16} - 6 q^{19} + 20 q^{21} + 8 q^{24} + 8 q^{26} + 36 q^{29} - 8 q^{34} - 32 q^{36} - 8 q^{39} + 12 q^{41} + 20 q^{44} - 8 q^{46} - 4 q^{49} + 4 q^{51} - 16 q^{54} + 40 q^{56} + 20 q^{59} - 14 q^{61} - 12 q^{64} - 48 q^{66} - 24 q^{69} + 52 q^{71} - 40 q^{74} - 8 q^{76} - 24 q^{79} + 38 q^{81} - 24 q^{84} - 16 q^{86} + 24 q^{89} - 24 q^{91} - 48 q^{94} - 64 q^{96} - 30 q^{99}+O(q^{100}) $ 6 * q + 8 * q^4 + 14 * q^9 + 2 * q^11 + 16 * q^14 - 4 * q^16 - 6 * q^19 + 20 * q^21 + 8 * q^24 + 8 * q^26 + 36 * q^29 - 8 * q^34 - 32 * q^36 - 8 * q^39 + 12 * q^41 + 20 * q^44 - 8 * q^46 - 4 * q^49 + 4 * q^51 - 16 * q^54 + 40 * q^56 + 20 * q^59 - 14 * q^61 - 12 * q^64 - 48 * q^66 - 24 * q^69 + 52 * q^71 - 40 * q^74 - 8 * q^76 - 24 * q^79 + 38 * q^81 - 24 * q^84 - 16 * q^86 + 24 * q^89 - 24 * q^91 - 48 * q^94 - 64 * q^96 - 30 * 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 $

$2$	−1.82254	−1.28873	−0.644364	−	0.764719i	$-0.722878\pi$
$2$	−1.82254	−1.28873	−0.644364	+	0.764719i	$0.722878\pi$
$3$	−2.31446	−1.33625	−0.668127	−	0.744047i	$-0.732904\pi$
$3$	−2.31446	−1.33625	−0.668127	+	0.744047i	$0.732904\pi$
$4$	1.32164	0.660819
$5$	0	0
$5$	0	0
$6$	4.21819	1.72207
$7$	−1.45033	−0.548172	−0.274086	−	0.961705i	$-0.588375\pi$
$7$	−1.45033	−0.548172	−0.274086	+	0.961705i	$0.588375\pi$
$8$	1.23634	0.437112
$9$	2.35673	0.785575
$10$	0	0
$11$	−3.89655	−1.17485	−0.587427	−	0.809277i	$-0.699859\pi$
$11$	−3.89655	−1.17485	−0.587427	+	0.809277i	$0.699859\pi$
$12$	−3.05888	−0.883022
$13$	−3.05888	−0.848380	−0.424190	−	0.905573i	$-0.639441\pi$
$13$	−3.05888	−0.848380	−0.424190	+	0.905573i	$0.639441\pi$
$14$	2.64327	0.706445
$15$	0	0
$16$	−4.89655	−1.22414
$17$	−3.92301	−0.951469	−0.475735	−	0.879589i	$-0.657818\pi$
$17$	−3.92301	−0.951469	−0.475735	+	0.879589i	$0.657818\pi$
$18$	−4.29522	−1.01239
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	3.35673	0.732498
$22$	7.10160	1.51407
$23$	5.37334	1.12042	0.560209	−	0.828351i	$-0.310721\pi$
$23$	5.37334	1.12042	0.560209	+	0.828351i	$0.310721\pi$
$24$	−2.86146	−0.584093
$25$	0	0
$26$	5.57491	1.09333
$27$	1.48883	0.286526
$28$	−1.91681	−0.362242
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	−8.43637	−1.51522	−0.757609	−	0.652709i	$-0.773632\pi$
$31$	−8.43637	−1.51522	−0.757609	+	0.652709i	$0.773632\pi$
$32$	6.45146	1.14047
$33$	9.01841	1.56990
$34$	7.14982	1.22618
$35$	0	0
$36$	3.11474	0.519123
$37$	5.95953	0.979741	0.489871	−	0.871795i	$-0.337044\pi$
$37$	5.95953	0.979741	0.489871	+	0.871795i	$0.337044\pi$
$38$	1.82254	0.295654
$39$	7.07965	1.13365
$40$	0	0
$41$	10.4364	1.62989	0.814944	−	0.579540i	$-0.196768\pi$
$41$	10.4364	1.62989	0.814944	+	0.579540i	$0.196768\pi$
$42$	−6.11775	−0.943990
$43$	1.45033	0.221173	0.110586	−	0.993867i	$-0.464727\pi$
$43$	1.45033	0.221173	0.110586	+	0.993867i	$0.464727\pi$
$44$	−5.14982	−0.776365
$45$	0	0
$46$	−9.79310	−1.44391
$47$	−4.90686	−0.715739	−0.357869	−	0.933772i	$-0.616497\pi$
$47$	−4.90686	−0.715739	−0.357869	+	0.933772i	$0.616497\pi$
$48$	11.3329	1.63576
$49$	−4.89655	−0.699507
$50$	0	0
$51$	9.07965	1.27140
$52$	−4.04272	−0.560625
$53$	4.23127	0.581209	0.290605	−	0.956843i	$-0.406144\pi$
$53$	4.23127	0.581209	0.290605	+	0.956843i	$0.406144\pi$
$54$	−2.71345	−0.369254
$55$	0	0
$56$	−1.79310	−0.239613
$57$	2.31446	0.306558
$58$	−10.9352	−1.43586
$59$	3.35673	0.437008	0.218504	−	0.975836i	$-0.429882\pi$
$59$	3.35673	0.437008	0.218504	+	0.975836i	$0.429882\pi$
$60$	0	0
$61$	10.3329	1.32300	0.661498	−	0.749947i	$-0.269921\pi$
$61$	10.3329	1.32300	0.661498	+	0.749947i	$0.269921\pi$
$62$	15.3756	1.95270
$63$	−3.41802	−0.430631
$64$	−1.96491	−0.245614
$65$	0	0
$66$	−16.4364	−2.02318
$67$	9.84404	1.20264	0.601320	−	0.799008i	$-0.294642\pi$
$67$	9.84404	1.20264	0.601320	+	0.799008i	$0.294642\pi$
$68$	−5.18479	−0.628749
$69$	−12.4364	−1.49716
$70$	0	0
$71$	8.64327	1.02577	0.512884	−	0.858458i	$-0.328577\pi$
$71$	8.64327	1.02577	0.512884	+	0.858458i	$0.328577\pi$
$72$	2.91372	0.343385
$73$	2.43418	0.284899	0.142449	−	0.989802i	$-0.454502\pi$
$73$	2.43418	0.284899	0.142449	+	0.989802i	$0.454502\pi$
$74$	−10.8615	−1.26262
$75$	0	0
$76$	−1.32164	−0.151602
$77$	5.65127	0.644022
$78$	−12.9029	−1.46097
$79$	−12.4364	−1.39920	−0.699601	−	0.714534i	$-0.746639\pi$
$79$	−12.4364	−1.39920	−0.699601	+	0.714534i	$0.746639\pi$
$80$	0	0
$81$	−10.5160	−1.16845
$82$	−19.0207	−2.10048
$83$	−12.6635	−1.39000	−0.694999	−	0.719011i	$-0.744595\pi$
$83$	−12.6635	−1.39000	−0.694999	+	0.719011i	$0.744595\pi$
$84$	4.43637	0.484048
$85$	0	0
$86$	−2.64327	−0.285032
$87$	−13.8868	−1.48882
$88$	−4.81746	−0.513543
$89$	12.3662	1.31081	0.655407	−	0.755276i	$-0.272497\pi$
$89$	12.3662	1.31081	0.655407	+	0.755276i	$0.272497\pi$
$90$	0	0
$91$	4.43637	0.465058
$92$	7.10160	0.740393
$93$	19.5256	2.02472
$94$	8.94292	0.922392
$95$	0	0
$96$	−14.9316	−1.52395
$97$	−3.05888	−0.310582	−0.155291	−	0.987869i	$-0.549631\pi$
$97$	−3.05888	−0.310582	−0.155291	+	0.987869i	$0.549631\pi$
$98$	8.92414	0.901474
$99$	−9.18310	−0.922936
$100$	0	0
$101$	3.35673	0.334007	0.167003	−	0.985956i	$-0.446591\pi$
$101$	3.35673	0.334007	0.167003	+	0.985956i	$0.446591\pi$
$102$	−16.5480	−1.63849
$103$	−13.0611	−1.28695	−0.643476	−	0.765466i	$-0.722508\pi$
$103$	−13.0611	−1.28695	−0.643476	+	0.765466i	$0.722508\pi$
$104$	−3.78181	−0.370837
$105$	0	0
$106$	−7.71164	−0.749020
$107$	5.77099	0.557903	0.278951	−	0.960305i	$-0.410013\pi$
$107$	5.77099	0.557903	0.278951	+	0.960305i	$0.410013\pi$
$108$	1.96770	0.189342
$109$	6.64327	0.636310	0.318155	−	0.948039i	$-0.396937\pi$
$109$	6.64327	0.636310	0.318155	+	0.948039i	$0.396937\pi$
$110$	0	0
$111$	−13.7931	−1.30918
$112$	7.10160	0.671038
$113$	9.41606	0.885789	0.442894	−	0.896574i	$-0.353952\pi$
$113$	9.41606	0.885789	0.442894	+	0.896574i	$0.353952\pi$
$114$	−4.21819	−0.395069
$115$	0	0
$116$	7.92982	0.736266
$117$	−7.20893	−0.666466
$118$	−6.11775	−0.563185
$119$	5.68965	0.521569
$120$	0	0
$121$	4.18310	0.380282
$122$	−18.8321	−1.70498
$123$	−24.1546	−2.17794
$124$	−11.1498	−1.00128
$125$	0	0
$126$	6.22947	0.554966
$127$	−11.0934	−0.984383	−0.492192	−	0.870487i	$-0.663804\pi$
$127$	−11.0934	−0.984383	−0.492192	+	0.870487i	$0.663804\pi$
$128$	−9.32179	−0.823938
$129$	−3.35673	−0.295543
$130$	0	0
$131$	4.61000	0.402778	0.201389	−	0.979511i	$-0.435455\pi$
$131$	4.61000	0.402778	0.201389	+	0.979511i	$0.435455\pi$
$132$	11.9191	1.03742
$133$	1.45033	0.125759
$134$	−17.9411	−1.54988
$135$	0	0
$136$	−4.85018	−0.415899
$137$	13.1808	1.12612	0.563058	−	0.826417i	$-0.309625\pi$
$137$	13.1808	1.12612	0.563058	+	0.826417i	$0.309625\pi$
$138$	22.6657	1.92944
$139$	1.18310	0.100349	0.0501745	−	0.998740i	$-0.484022\pi$
$139$	1.18310	0.100349	0.0501745	+	0.998740i	$0.484022\pi$
$140$	0	0
$141$	11.3567	0.956409
$142$	−15.7527	−1.32194
$143$	11.9191	0.996722
$144$	−11.5398	−0.961652
$145$	0	0
$146$	−4.43637	−0.367157
$147$	11.3329	0.934719
$148$	7.87634	0.647431
$149$	5.46018	0.447315	0.223658	−	0.974668i	$-0.428200\pi$
$149$	5.46018	0.447315	0.223658	+	0.974668i	$0.428200\pi$
$150$	0	0
$151$	−5.07965	−0.413376	−0.206688	−	0.978407i	$-0.566268\pi$
$151$	−5.07965	−0.413376	−0.206688	+	0.978407i	$0.566268\pi$
$152$	−1.23634	−0.100280
$153$	−9.24546	−0.747451
$154$	−10.2996	−0.829969
$155$	0	0
$156$	9.35673	0.749138
$157$	−6.11775	−0.488250	−0.244125	−	0.969744i	$-0.578501\pi$
$157$	−6.11775	−0.488250	−0.244125	+	0.969744i	$0.578501\pi$
$158$	22.6657	1.80319
$159$	−9.79310	−0.776643
$160$	0	0
$161$	−7.79310	−0.614182
$162$	19.1658	1.50581
$163$	16.4365	1.28740	0.643701	−	0.765277i	$-0.277398\pi$
$163$	16.4365	1.28740	0.643701	+	0.765277i	$0.277398\pi$
$164$	13.7931	1.07706
$165$	0	0
$166$	23.0796	1.79133
$167$	3.80329	0.294308	0.147154	−	0.989114i	$-0.452989\pi$
$167$	3.80329	0.294308	0.147154	+	0.989114i	$0.452989\pi$
$168$	4.15006	0.320184
$169$	−3.64327	−0.280252
$170$	0	0
$171$	−2.35673	−0.180223
$172$	1.91681	0.146155
$173$	11.3838	0.865491	0.432746	−	0.901516i	$-0.357545\pi$
$173$	11.3838	0.865491	0.432746	+	0.901516i	$0.357545\pi$
$174$	25.3091	1.91868
$175$	0	0
$176$	19.0796	1.43818
$177$	−7.76901	−0.583954
$178$	−22.5378	−1.68928
$179$	10.0702	0.752680	0.376340	−	0.926482i	$-0.377182\pi$
$179$	10.0702	0.752680	0.376340	+	0.926482i	$0.377182\pi$
$180$	0	0
$181$	0.573097	0.0425980	0.0212990	−	0.999773i	$-0.493220\pi$
$181$	0.573097	0.0425980	0.0212990	+	0.999773i	$0.493220\pi$
$182$	−8.08545	−0.599333
$183$	−23.9151	−1.76786
$184$	6.64327	0.489749
$185$	0	0
$186$	−35.5862	−2.60931
$187$	15.2862	1.11784
$188$	−6.48509	−0.472973
$189$	−2.15930	−0.157066
$190$	0	0
$191$	−3.18310	−0.230321	−0.115160	−	0.993347i	$-0.536738\pi$
$191$	−3.18310	−0.230321	−0.115160	+	0.993347i	$0.536738\pi$
$192$	4.54771	0.328203
$193$	−3.05888	−0.220183	−0.110091	−	0.993921i	$-0.535114\pi$
$193$	−3.05888	−0.220183	−0.110091	+	0.993921i	$0.535114\pi$
$194$	5.57491	0.400255
$195$	0	0
$196$	−6.47146	−0.462247
$197$	−21.4933	−1.53134	−0.765669	−	0.643235i	$-0.777592\pi$
$197$	−21.4933	−1.53134	−0.765669	+	0.643235i	$0.777592\pi$
$198$	16.7365	1.18941
$199$	−4.81690	−0.341461	−0.170731	−	0.985318i	$-0.554613\pi$
$199$	−4.81690	−0.341461	−0.170731	+	0.985318i	$0.554613\pi$
$200$	0	0
$201$	−22.7836	−1.60703
$202$	−6.11775	−0.430444
$203$	−8.70197	−0.610758
$204$	12.0000	0.840168
$205$	0	0
$206$	23.8044	1.65853
$207$	12.6635	0.880173
$208$	14.9779	1.03853
$209$	3.89655	0.269530
$210$	0	0
$211$	10.5066	0.723301	0.361650	−	0.932314i	$-0.382213\pi$
$211$	10.5066	0.723301	0.361650	+	0.932314i	$0.382213\pi$
$212$	5.59220	0.384074
$213$	−20.0045	−1.37069
$214$	−10.5178	−0.718984
$215$	0	0
$216$	1.84070	0.125244
$217$	12.2355	0.830600
$218$	−12.1076	−0.820031
$219$	−5.63380	−0.380697
$220$	0	0
$221$	12.0000	0.807207
$222$	25.1384	1.68718
$223$	16.8947	1.13136	0.565678	−	0.824626i	$-0.308615\pi$
$223$	16.8947	1.13136	0.565678	+	0.824626i	$0.308615\pi$
$224$	−9.35673	−0.625173
$225$	0	0
$226$	−17.1611	−1.14154
$227$	−17.1342	−1.13724	−0.568618	−	0.822602i	$-0.692522\pi$
$227$	−17.1342	−1.13724	−0.568618	+	0.822602i	$0.692522\pi$
$228$	3.05888	0.202579
$229$	25.0464	1.65511	0.827555	−	0.561384i	$-0.189731\pi$
$229$	25.0464	1.65511	0.827555	+	0.561384i	$0.189731\pi$
$230$	0	0
$231$	−13.0796	−0.860578
$232$	7.41804	0.487018
$233$	−19.2986	−1.26429	−0.632147	−	0.774849i	$-0.717826\pi$
$233$	−19.2986	−1.26429	−0.632147	+	0.774849i	$0.717826\pi$
$234$	13.1385	0.858893
$235$	0	0
$236$	4.43637	0.288783
$237$	28.7835	1.86969
$238$	−10.3696	−0.672161
$239$	18.7693	1.21408	0.607042	−	0.794669i	$-0.292356\pi$
$239$	18.7693	1.21408	0.607042	+	0.794669i	$0.292356\pi$
$240$	0	0
$241$	−14.4364	−0.929929	−0.464964	−	0.885329i	$-0.653933\pi$
$241$	−14.4364	−0.929929	−0.464964	+	0.885329i	$0.653933\pi$
$242$	−7.62385	−0.490079
$243$	19.8724	1.27482
$244$	13.6564	0.874260
$245$	0	0
$246$	44.0226	2.80678
$247$	3.05888	0.194632
$248$	−10.4302	−0.662320
$249$	29.3091	1.85739
$250$	0	0
$251$	−10.9762	−0.692811	−0.346406	−	0.938085i	$-0.612598\pi$
$251$	−10.9762	−0.692811	−0.346406	+	0.938085i	$0.612598\pi$
$252$	−4.51739	−0.284569
$253$	−20.9375	−1.31633
$254$	20.2182	1.26860
$255$	0	0
$256$	20.9191	1.30745
$257$	17.6392	1.10030	0.550150	−	0.835066i	$-0.314570\pi$
$257$	17.6392	1.10030	0.550150	+	0.835066i	$0.314570\pi$
$258$	6.11775	0.380875
$259$	−8.64327	−0.537067
$260$	0	0
$261$	14.1404	0.875266
$262$	−8.40189	−0.519071
$263$	1.68976	0.104195	0.0520975	−	0.998642i	$-0.483409\pi$
$263$	1.68976	0.104195	0.0520975	+	0.998642i	$0.483409\pi$
$264$	11.1498	0.686224
$265$	0	0
$266$	−2.64327	−0.162070
$267$	−28.6211	−1.75158
$268$	13.0102	0.794727
$269$	−27.1022	−1.65245	−0.826226	−	0.563339i	$-0.809516\pi$
$269$	−27.1022	−1.65245	−0.826226	+	0.563339i	$0.809516\pi$
$270$	0	0
$271$	23.9524	1.45500	0.727502	−	0.686105i	$-0.240681\pi$
$271$	23.9524	1.45500	0.727502	+	0.686105i	$0.240681\pi$
$272$	19.2092	1.16473
$273$	−10.2678	−0.621436
$274$	−24.0226	−1.45126
$275$	0	0
$276$	−16.4364	−0.989353
$277$	−8.23549	−0.494822	−0.247411	−	0.968911i	$-0.579580\pi$
$277$	−8.23549	−0.494822	−0.247411	+	0.968911i	$0.579580\pi$
$278$	−2.15624	−0.129323
$279$	−19.8822	−1.19032
$280$	0	0
$281$	−10.4364	−0.622582	−0.311291	−	0.950315i	$-0.600761\pi$
$281$	−10.4364	−0.622582	−0.311291	+	0.950315i	$0.600761\pi$
$282$	−20.6980	−1.23255
$283$	10.4687	0.622302	0.311151	−	0.950361i	$-0.399286\pi$
$283$	10.4687	0.622302	0.311151	+	0.950361i	$0.399286\pi$
$284$	11.4233	0.677847
$285$	0	0
$286$	−21.7229	−1.28450
$287$	−15.1362	−0.893459
$288$	15.2043	0.895923
$289$	−1.61000	−0.0947059
$290$	0	0
$291$	7.07965	0.415016
$292$	3.21710	0.188266
$293$	16.4668	0.961999	0.481000	−	0.876721i	$-0.340274\pi$
$293$	16.4668	0.961999	0.481000	+	0.876721i	$0.340274\pi$
$294$	−20.6546	−1.20460
$295$	0	0
$296$	7.36801	0.428257
$297$	−5.80131	−0.336626
$298$	−9.95137	−0.576467
$299$	−16.4364	−0.950540
$300$	0	0
$301$	−2.10345	−0.121241
$302$	9.25784	0.532729
$303$	−7.76901	−0.446318
$304$	4.89655	0.280836
$305$	0	0
$306$	16.8502	0.963260
$307$	−7.87634	−0.449526	−0.224763	−	0.974413i	$-0.572161\pi$
$307$	−7.87634	−0.449526	−0.224763	+	0.974413i	$0.572161\pi$
$308$	7.46893	0.425582
$309$	30.2295	1.71969
$310$	0	0
$311$	3.89655	0.220953	0.110477	−	0.993879i	$-0.464762\pi$
$311$	3.89655	0.220953	0.110477	+	0.993879i	$0.464762\pi$
$312$	8.75286	0.495533
$313$	−7.76901	−0.439130	−0.219565	−	0.975598i	$-0.570464\pi$
$313$	−7.76901	−0.439130	−0.219565	+	0.975598i	$0.570464\pi$
$314$	11.1498	0.629221
$315$	0	0
$316$	−16.4364	−0.924618
$317$	−19.0510	−1.07001	−0.535005	−	0.844849i	$-0.679690\pi$
$317$	−19.0510	−1.07001	−0.535005	+	0.844849i	$0.679690\pi$
$318$	17.8483	1.00088
$319$	−23.3793	−1.30899
$320$	0	0
$321$	−13.3567	−0.745500
$322$	14.2032	0.791514
$323$	3.92301	0.218282
$324$	−13.8984	−0.772131
$325$	0	0
$326$	−29.9560	−1.65911
$327$	−15.3756	−0.850272
$328$	12.9029	0.712444
$329$	7.11655	0.392348
$330$	0	0
$331$	8.00000	0.439720	0.219860	−	0.975531i	$-0.429440\pi$
$331$	8.00000	0.439720	0.219860	+	0.975531i	$0.429440\pi$
$332$	−16.7365	−0.918536
$333$	14.0450	0.769660
$334$	−6.93164	−0.379282
$335$	0	0
$336$	−16.4364	−0.896678
$337$	6.89249	0.375458	0.187729	−	0.982221i	$-0.439887\pi$
$337$	6.89249	0.375458	0.187729	+	0.982221i	$0.439887\pi$
$338$	6.64000	0.361168
$339$	−21.7931	−1.18364
$340$	0	0
$341$	32.8727	1.78016
$342$	4.29522	0.232259
$343$	17.2539	0.931623
$344$	1.79310	0.0966774
$345$	0	0
$346$	−20.7473	−1.11538
$347$	30.5503	1.64002	0.820012	−	0.572346i	$-0.193967\pi$
$347$	30.5503	1.64002	0.820012	+	0.572346i	$0.193967\pi$
$348$	−18.3533	−0.983838
$349$	16.7693	0.897640	0.448820	−	0.893622i	$-0.351844\pi$
$349$	16.7693	0.897640	0.448820	+	0.893622i	$0.351844\pi$
$350$	0	0
$351$	−4.55416	−0.243083
$352$	−25.1384	−1.33988
$353$	−29.0999	−1.54883	−0.774417	−	0.632676i	$-0.781956\pi$
$353$	−29.0999	−1.54883	−0.774417	+	0.632676i	$0.781956\pi$
$354$	14.1593	0.752558
$355$	0	0
$356$	16.3436	0.866210
$357$	−13.1685	−0.696949
$358$	−18.3533	−0.970000
$359$	−11.6896	−0.616956	−0.308478	−	0.951231i	$-0.599820\pi$
$359$	−11.6896	−0.616956	−0.308478	+	0.951231i	$0.599820\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−1.04449	−0.0548972
$363$	−9.68161	−0.508153
$364$	5.86328	0.307319
$365$	0	0
$366$	43.5862	2.27829
$367$	4.20095	0.219288	0.109644	−	0.993971i	$-0.465029\pi$
$367$	4.20095	0.219288	0.109644	+	0.993971i	$0.465029\pi$
$368$	−26.3108	−1.37155
$369$	24.5957	1.28040
$370$	0	0
$371$	−6.13672	−0.318603
$372$	25.8058	1.33797
$373$	16.9456	0.877412	0.438706	−	0.898631i	$-0.355437\pi$
$373$	16.9456	0.877412	0.438706	+	0.898631i	$0.355437\pi$
$374$	−27.8596	−1.44059
$375$	0	0
$376$	−6.06655	−0.312858
$377$	−18.3533	−0.945241
$378$	3.93539	0.202415
$379$	10.3662	0.532476	0.266238	−	0.963907i	$-0.414219\pi$
$379$	10.3662	0.532476	0.266238	+	0.963907i	$0.414219\pi$
$380$	0	0
$381$	25.6753	1.31539
$382$	5.80131	0.296821
$383$	−20.5907	−1.05214	−0.526068	−	0.850442i	$-0.676334\pi$
$383$	−20.5907	−1.05214	−0.526068	+	0.850442i	$0.676334\pi$
$384$	21.5749	1.10099
$385$	0	0
$386$	5.57491	0.283756
$387$	3.41802	0.173748
$388$	−4.04272	−0.205238
$389$	8.10345	0.410861	0.205431	−	0.978672i	$-0.434141\pi$
$389$	8.10345	0.410861	0.205431	+	0.978672i	$0.434141\pi$
$390$	0	0
$391$	−21.0796	−1.06604
$392$	−6.05380	−0.305763
$393$	−10.6697	−0.538213
$394$	39.1724	1.97348
$395$	0	0
$396$	−12.1367	−0.609893
$397$	−3.46891	−0.174100	−0.0870499	−	0.996204i	$-0.527744\pi$
$397$	−3.46891	−0.174100	−0.0870499	+	0.996204i	$0.527744\pi$
$398$	8.77898	0.440050
$399$	−3.35673	−0.168046
$400$	0	0
$401$	−15.9524	−0.796625	−0.398312	−	0.917250i	$-0.630404\pi$
$401$	−15.9524	−0.796625	−0.398312	+	0.917250i	$0.630404\pi$
$402$	41.5240	2.07103
$403$	25.8058	1.28548
$404$	4.43637	0.220718
$405$	0	0
$406$	15.8596	0.787101
$407$	−23.2216	−1.15105
$408$	11.2255	0.555747
$409$	−3.92982	−0.194317	−0.0971586	−	0.995269i	$-0.530975\pi$
$409$	−3.92982	−0.194317	−0.0971586	+	0.995269i	$0.530975\pi$
$410$	0	0
$411$	−30.5066	−1.50478
$412$	−17.2621	−0.850442
$413$	−4.86835	−0.239556
$414$	−23.0796	−1.13430
$415$	0	0
$416$	−19.7342	−0.967549
$417$	−2.73823	−0.134092
$418$	−7.10160	−0.347351
$419$	−34.2996	−1.67565	−0.837824	−	0.545941i	$-0.816172\pi$
$419$	−34.2996	−1.67565	−0.837824	+	0.545941i	$0.816172\pi$
$420$	0	0
$421$	−26.0891	−1.27151	−0.635753	−	0.771893i	$-0.719310\pi$
$421$	−26.0891	−1.27151	−0.635753	+	0.771893i	$0.719310\pi$
$422$	−19.1486	−0.932138
$423$	−11.5641	−0.562267
$424$	5.23129	0.254054
$425$	0	0
$426$	36.4589	1.76644
$427$	−14.9861	−0.725229
$428$	7.62715	0.368672
$429$	−27.5862	−1.33187
$430$	0	0
$431$	10.2996	0.496117	0.248058	−	0.968745i	$-0.420208\pi$
$431$	10.2996	0.496117	0.248058	+	0.968745i	$0.420208\pi$
$432$	−7.29014	−0.350747
$433$	36.2319	1.74119	0.870596	−	0.491999i	$-0.163734\pi$
$433$	36.2319	1.74119	0.870596	+	0.491999i	$0.163734\pi$
$434$	−22.2996	−1.07042
$435$	0	0
$436$	8.78000	0.420486
$437$	−5.37334	−0.257042
$438$	10.2678	0.490615
$439$	−24.0891	−1.14971	−0.574855	−	0.818255i	$-0.694942\pi$
$439$	−24.0891	−1.14971	−0.574855	+	0.818255i	$0.694942\pi$
$440$	0	0
$441$	−11.5398	−0.549515
$442$	−21.8704	−1.04027
$443$	5.52337	0.262423	0.131212	−	0.991354i	$-0.458113\pi$
$443$	5.52337	0.262423	0.131212	+	0.991354i	$0.458113\pi$
$444$	−18.2295	−0.865132
$445$	0	0
$446$	−30.7913	−1.45801
$447$	−12.6374	−0.597727
$448$	2.84977	0.134639
$449$	7.92982	0.374231	0.187116	−	0.982338i	$-0.440086\pi$
$449$	7.92982	0.374231	0.187116	+	0.982338i	$0.440086\pi$
$450$	0	0
$451$	−40.6658	−1.91488
$452$	12.4446	0.585346
$453$	11.7566	0.552375
$454$	31.2277	1.46559
$455$	0	0
$456$	2.86146	0.134000
$457$	22.6534	1.05968	0.529840	−	0.848098i	$-0.322252\pi$
$457$	22.6534	1.05968	0.529840	+	0.848098i	$0.322252\pi$
$458$	−45.6479	−2.13299
$459$	−5.84070	−0.272621
$460$	0	0
$461$	13.3900	0.623634	0.311817	−	0.950142i	$-0.399062\pi$
$461$	13.3900	0.623634	0.311817	+	0.950142i	$0.399062\pi$
$462$	23.8381	1.10905
$463$	16.9029	0.785546	0.392773	−	0.919635i	$-0.371516\pi$
$463$	16.9029	0.785546	0.392773	+	0.919635i	$0.371516\pi$
$464$	−29.3793	−1.36390
$465$	0	0
$466$	35.1724	1.62933
$467$	22.2501	1.02961	0.514807	−	0.857306i	$-0.327864\pi$
$467$	22.2501	1.02961	0.514807	+	0.857306i	$0.327864\pi$
$468$	−9.52759	−0.440413
$469$	−14.2771	−0.659254
$470$	0	0
$471$	14.1593	0.652426
$472$	4.15006	0.191022
$473$	−5.65127	−0.259846
$474$	−52.4589	−2.40952
$475$	0	0
$476$	7.51965	0.344663
$477$	9.97194	0.456584
$478$	−34.2077	−1.56462
$479$	0.366196	0.0167319	0.00836597	−	0.999965i	$-0.497337\pi$
$479$	0.366196	0.0167319	0.00836597	+	0.999965i	$0.497337\pi$
$480$	0	0
$481$	−18.2295	−0.831192
$482$	26.3108	1.19842
$483$	18.0368	0.820704
$484$	5.52854	0.251297
$485$	0	0
$486$	−36.2182	−1.64289
$487$	26.8461	1.21651	0.608257	−	0.793740i	$-0.291869\pi$
$487$	26.8461	1.21651	0.608257	+	0.793740i	$0.291869\pi$
$488$	12.7750	0.578298
$489$	−38.0415	−1.72030
$490$	0	0
$491$	−23.7266	−1.07076	−0.535382	−	0.844610i	$-0.679832\pi$
$491$	−23.7266	−1.07076	−0.535382	+	0.844610i	$0.679832\pi$
$492$	−31.9236	−1.43923
$493$	−23.5381	−1.06010
$494$	−5.57491	−0.250827
$495$	0	0
$496$	41.3091	1.85483
$497$	−12.5356	−0.562298
$498$	−53.4169	−2.39367
$499$	6.81690	0.305166	0.152583	−	0.988291i	$-0.451241\pi$
$499$	6.81690	0.305166	0.152583	+	0.988291i	$0.451241\pi$
$500$	0	0
$501$	−8.80257	−0.393270
$502$	20.0045	0.892845
$503$	23.4102	1.04381	0.521904	−	0.853004i	$-0.325222\pi$
$503$	23.4102	1.04381	0.521904	+	0.853004i	$0.325222\pi$
$504$	−4.22584	−0.188234
$505$	0	0
$506$	38.1593	1.69639
$507$	8.43221	0.374488
$508$	−14.6615	−0.650499
$509$	16.9204	0.749981	0.374991	−	0.927029i	$-0.377646\pi$
$509$	16.9204	0.749981	0.374991	+	0.927029i	$0.377646\pi$
$510$	0	0
$511$	−3.53035	−0.156174
$512$	−19.4823	−0.861003
$513$	−1.48883	−0.0657336
$514$	−32.1480	−1.41799
$515$	0	0
$516$	−4.43637	−0.195300
$517$	19.1198	0.840888
$518$	15.7527	0.692133
$519$	−26.3473	−1.15652
$520$	0	0
$521$	−3.49345	−0.153051	−0.0765254	−	0.997068i	$-0.524383\pi$
$521$	−3.49345	−0.153051	−0.0765254	+	0.997068i	$0.524383\pi$
$522$	−25.7713	−1.12798
$523$	14.9271	0.652714	0.326357	−	0.945247i	$-0.394179\pi$
$523$	14.9271	0.652714	0.326357	+	0.945247i	$0.394179\pi$
$524$	6.09275	0.266163
$525$	0	0
$526$	−3.07965	−0.134279
$527$	33.0960	1.44168
$528$	−44.1591	−1.92178
$529$	5.87275	0.255337
$530$	0	0
$531$	7.91088	0.343303
$532$	1.91681	0.0831041
$533$	−31.9236	−1.38276
$534$	52.1629	2.25731
$535$	0	0
$536$	12.1706	0.525689
$537$	−23.3070	−1.00577
$538$	49.3948	2.12956
$539$	19.0796	0.821819
$540$	0	0
$541$	−12.6991	−0.545978	−0.272989	−	0.962017i	$-0.588012\pi$
$541$	−12.6991	−0.545978	−0.272989	+	0.962017i	$0.588012\pi$
$542$	−43.6541	−1.87510
$543$	−1.32641	−0.0569217
$544$	−25.3091	−1.08512
$545$	0	0
$546$	18.7135	0.800862
$547$	31.8162	1.36036	0.680182	−	0.733043i	$-0.261901\pi$
$547$	31.8162	1.36036	0.680182	+	0.733043i	$0.261901\pi$
$548$	17.4203	0.744158
$549$	24.3519	1.03931
$550$	0	0
$551$	−6.00000	−0.255609
$552$	−15.3756	−0.654429
$553$	18.0368	0.767003
$554$	15.0095	0.637691
$555$	0	0
$556$	1.56363	0.0663125
$557$	9.64731	0.408770	0.204385	−	0.978891i	$-0.434481\pi$
$557$	9.64731	0.408770	0.204385	+	0.978891i	$0.434481\pi$
$558$	36.2361	1.53399
$559$	−4.43637	−0.187639
$560$	0	0
$561$	−35.3793	−1.49372
$562$	19.0207	0.802338
$563$	4.28216	0.180471	0.0902357	−	0.995920i	$-0.471238\pi$
$563$	4.28216	0.180471	0.0902357	+	0.995920i	$0.471238\pi$
$564$	15.0095	0.632013
$565$	0	0
$566$	−19.0796	−0.801977
$567$	15.2517	0.640510
$568$	10.6860	0.448376
$569$	42.2295	1.77035	0.885176	−	0.465257i	$-0.154038\pi$
$569$	42.2295	1.77035	0.885176	+	0.465257i	$0.154038\pi$
$570$	0	0
$571$	−19.2200	−0.804332	−0.402166	−	0.915567i	$-0.631743\pi$
$571$	−19.2200	−0.804332	−0.402166	+	0.915567i	$0.631743\pi$
$572$	15.7527	0.658653
$573$	7.36715	0.307767
$574$	27.5862	1.15143
$575$	0	0
$576$	−4.63076	−0.192948
$577$	40.7919	1.69819	0.849096	−	0.528239i	$-0.177148\pi$
$577$	40.7919	1.69819	0.849096	+	0.528239i	$0.177148\pi$
$578$	2.93428	0.122050
$579$	7.07965	0.294220
$580$	0	0
$581$	18.3662	0.761958
$582$	−12.9029	−0.534843
$583$	−16.4873	−0.682836
$584$	3.00947	0.124533
$585$	0	0
$586$	−30.0113	−1.23975
$587$	31.8851	1.31604	0.658019	−	0.753001i	$-0.271395\pi$
$587$	31.8851	1.31604	0.658019	+	0.753001i	$0.271395\pi$
$588$	14.9779	0.617680
$589$	8.43637	0.347615
$590$	0	0
$591$	49.7455	2.04626
$592$	−29.1811	−1.19934
$593$	38.8973	1.59732	0.798660	−	0.601783i	$-0.205543\pi$
$593$	38.8973	1.59732	0.798660	+	0.601783i	$0.205543\pi$
$594$	10.5731	0.433819
$595$	0	0
$596$	7.21637	0.295594
$597$	11.1485	0.456279
$598$	29.9559	1.22499
$599$	28.1629	1.15071	0.575353	−	0.817905i	$-0.304865\pi$
$599$	28.1629	1.15071	0.575353	+	0.817905i	$0.304865\pi$
$600$	0	0
$601$	−5.56363	−0.226945	−0.113473	−	0.993541i	$-0.536197\pi$
$601$	−5.56363	−0.226945	−0.113473	+	0.993541i	$0.536197\pi$
$602$	3.83361	0.156246
$603$	23.1997	0.944764
$604$	−6.71345	−0.273166
$605$	0	0
$606$	14.1593	0.575182
$607$	−33.9986	−1.37996	−0.689980	−	0.723828i	$-0.742381\pi$
$607$	−33.9986	−1.37996	−0.689980	+	0.723828i	$0.742381\pi$
$608$	−6.45146	−0.261641
$609$	20.1404	0.816128
$610$	0	0
$611$	15.0095	0.607218
$612$	−12.2191	−0.493929
$613$	−17.5703	−0.709659	−0.354830	−	0.934931i	$-0.615461\pi$
$613$	−17.5703	−0.709659	−0.354830	+	0.934931i	$0.615461\pi$
$614$	14.3549	0.579317
$615$	0	0
$616$	6.98690	0.281510
$617$	13.0791	0.526544	0.263272	−	0.964722i	$-0.415198\pi$
$617$	13.0791	0.526544	0.263272	+	0.964722i	$0.415198\pi$
$618$	−55.0943	−2.21622
$619$	−18.9393	−0.761234	−0.380617	−	0.924733i	$-0.624288\pi$
$619$	−18.9393	−0.761234	−0.380617	+	0.924733i	$0.624288\pi$
$620$	0	0
$621$	8.00000	0.321029
$622$	−7.10160	−0.284748
$623$	−17.9350	−0.718552
$624$	−34.6658	−1.38774
$625$	0	0
$626$	14.1593	0.565919
$627$	−9.01841	−0.360161
$628$	−8.08545	−0.322645
$629$	−23.3793	−0.932194
$630$	0	0
$631$	31.6896	1.26154	0.630772	−	0.775968i	$-0.282738\pi$
$631$	31.6896	1.26154	0.630772	+	0.775968i	$0.282738\pi$
$632$	−15.3756	−0.611608
$633$	−24.3170	−0.966514
$634$	34.7211	1.37895
$635$	0	0
$636$	−12.9429	−0.513220
$637$	14.9779	0.593448
$638$	42.6096	1.68693
$639$	20.3698	0.805818
$640$	0	0
$641$	47.9750	1.89490	0.947449	−	0.319908i	$-0.103652\pi$
$641$	47.9750	1.89490	0.947449	+	0.319908i	$0.103652\pi$
$642$	24.3431	0.960746
$643$	0.200927	0.00792378	0.00396189	−	0.999992i	$-0.498739\pi$
$643$	0.200927	0.00792378	0.00396189	+	0.999992i	$0.498739\pi$
$644$	−10.2996	−0.405863
$645$	0	0
$646$	−7.14982	−0.281306
$647$	−1.58798	−0.0624299	−0.0312150	−	0.999513i	$-0.509938\pi$
$647$	−1.58798	−0.0624299	−0.0312150	+	0.999513i	$0.509938\pi$
$648$	−13.0014	−0.510743
$649$	−13.0796	−0.513421
$650$	0	0
$651$	−28.3186	−1.10989
$652$	21.7230	0.850739
$653$	33.5624	1.31340	0.656700	−	0.754152i	$-0.271952\pi$
$653$	33.5624	1.31340	0.656700	+	0.754152i	$0.271952\pi$
$654$	28.0226	1.09577
$655$	0	0
$656$	−51.1022	−1.99521
$657$	5.73669	0.223809
$658$	−12.9702	−0.505630
$659$	15.3567	0.598213	0.299107	−	0.954220i	$-0.403311\pi$
$659$	15.3567	0.598213	0.299107	+	0.954220i	$0.403311\pi$
$660$	0	0
$661$	12.8026	0.497962	0.248981	−	0.968508i	$-0.419904\pi$
$661$	12.8026	0.497962	0.248981	+	0.968508i	$0.419904\pi$
$662$	−14.5803	−0.566679
$663$	−27.7735	−1.07863
$664$	−15.6564	−0.607585
$665$	0	0
$666$	−25.5975	−0.991882
$667$	32.2400	1.24834
$668$	5.02657	0.194484
$669$	−39.1022	−1.51178
$670$	0	0
$671$	−40.2627	−1.55433
$672$	21.6558	0.835389
$673$	−7.82545	−0.301649	−0.150824	−	0.988561i	$-0.548193\pi$
$673$	−7.82545	−0.301649	−0.150824	+	0.988561i	$0.548193\pi$
$674$	−12.5618	−0.483863
$675$	0	0
$676$	−4.81509	−0.185196
$677$	−21.6516	−0.832137	−0.416069	−	0.909333i	$-0.636592\pi$
$677$	−21.6516	−0.832137	−0.416069	+	0.909333i	$0.636592\pi$
$678$	39.7187	1.52539
$679$	4.43637	0.170252
$680$	0	0
$681$	39.6564	1.51964
$682$	−59.9118	−2.29414
$683$	−6.38751	−0.244411	−0.122206	−	0.992505i	$-0.538997\pi$
$683$	−6.38751	−0.244411	−0.122206	+	0.992505i	$0.538997\pi$
$684$	−3.11474	−0.119095
$685$	0	0
$686$	−31.4458	−1.20061
$687$	−57.9688	−2.21165
$688$	−7.10160	−0.270746
$689$	−12.9429	−0.493086
$690$	0	0
$691$	44.4958	1.69270	0.846351	−	0.532626i	$-0.178795\pi$
$691$	44.4958	1.69270	0.846351	+	0.532626i	$0.178795\pi$
$692$	15.0452	0.571933
$693$	13.3185	0.505928
$694$	−55.6789	−2.11354
$695$	0	0
$696$	−17.1688	−0.650780
$697$	−40.9420	−1.55079
$698$	−30.5626	−1.15681
$699$	44.6658	1.68942
$700$	0	0
$701$	17.5160	0.661571	0.330785	−	0.943706i	$-0.392686\pi$
$701$	17.5160	0.661571	0.330785	+	0.943706i	$0.392686\pi$
$702$	8.30011	0.313268
$703$	−5.95953	−0.224768
$704$	7.65638	0.288560
$705$	0	0
$706$	53.0357	1.99602
$707$	−4.86835	−0.183093
$708$	−10.2678	−0.385888
$709$	11.4269	0.429146	0.214573	−	0.976708i	$-0.431164\pi$
$709$	11.4269	0.429146	0.214573	+	0.976708i	$0.431164\pi$
$710$	0	0
$711$	−29.3091	−1.09918
$712$	15.2888	0.572973
$713$	−45.3315	−1.69768
$714$	24.0000	0.898177
$715$	0	0
$716$	13.3091	0.497385
$717$	−43.4408	−1.62233
$718$	21.3048	0.795088
$719$	15.8965	0.592841	0.296421	−	0.955057i	$-0.404207\pi$
$719$	15.8965	0.592841	0.296421	+	0.955057i	$0.404207\pi$
$720$	0	0
$721$	18.9429	0.705471
$722$	−1.82254	−0.0678278
$723$	33.4124	1.24262
$724$	0.757427	0.0281495
$725$	0	0
$726$	17.6451	0.654871
$727$	−41.9905	−1.55734	−0.778670	−	0.627434i	$-0.784105\pi$
$727$	−41.9905	−1.55734	−0.778670	+	0.627434i	$0.784105\pi$
$728$	5.48487	0.203283
$729$	−14.4458	−0.535031
$730$	0	0
$731$	−5.68965	−0.210439
$732$	−31.6071	−1.16823
$733$	0.632884	0.0233761	0.0116881	−	0.999932i	$-0.496279\pi$
$733$	0.632884	0.0233761	0.0116881	+	0.999932i	$0.496279\pi$
$734$	−7.65638	−0.282602
$735$	0	0
$736$	34.6658	1.27780
$737$	−38.3578	−1.41293
$738$	−44.8265	−1.65009
$739$	−29.5493	−1.08699	−0.543494	−	0.839413i	$-0.682899\pi$
$739$	−29.5493	−1.08699	−0.543494	+	0.839413i	$0.682899\pi$
$740$	0	0
$741$	−7.07965	−0.260077
$742$	11.1844	0.410592
$743$	31.3374	1.14966	0.574829	−	0.818274i	$-0.305069\pi$
$743$	31.3374	1.14966	0.574829	+	0.818274i	$0.305069\pi$
$744$	24.1404	0.885028
$745$	0	0
$746$	−30.8840	−1.13074
$747$	−29.8443	−1.09195
$748$	20.2028	0.738688
$749$	−8.36983	−0.305827
$750$	0	0
$751$	25.0131	0.912741	0.456371	−	0.889790i	$-0.349149\pi$
$751$	25.0131	0.912741	0.456371	+	0.889790i	$0.349149\pi$
$752$	24.0267	0.876163
$753$	25.4040	0.925772
$754$	33.4495	1.21816
$755$	0	0
$756$	−2.85381	−0.103792
$757$	32.0900	1.16633	0.583165	−	0.812354i	$-0.301814\pi$
$757$	32.0900	1.16633	0.583165	+	0.812354i	$0.301814\pi$
$758$	−18.8928	−0.686216
$759$	48.4589	1.75895
$760$	0	0
$761$	−40.4922	−1.46784	−0.733921	−	0.679235i	$-0.762312\pi$
$761$	−40.4922	−1.46784	−0.733921	+	0.679235i	$0.762312\pi$
$762$	−46.7942	−1.69517
$763$	−9.63492	−0.348808
$764$	−4.20690	−0.152200
$765$	0	0
$766$	37.5273	1.35592
$767$	−10.2678	−0.370749
$768$	−48.4165	−1.74708
$769$	3.09398	0.111572	0.0557859	−	0.998443i	$-0.482234\pi$
$769$	3.09398	0.111572	0.0557859	+	0.998443i	$0.482234\pi$
$770$	0	0
$771$	−40.8251	−1.47028
$772$	−4.04272	−0.145501
$773$	1.96350	0.0706220	0.0353110	−	0.999376i	$-0.488758\pi$
$773$	1.96350	0.0706220	0.0353110	+	0.999376i	$0.488758\pi$
$774$	−6.22947	−0.223914
$775$	0	0
$776$	−3.78181	−0.135759
$777$	20.0045	0.717658
$778$	−14.7688	−0.529488
$779$	−10.4364	−0.373922
$780$	0	0
$781$	−33.6789	−1.20513
$782$	38.4184	1.37384
$783$	8.93300	0.319239
$784$	23.9762	0.856293
$785$	0	0
$786$	19.4458	0.693610
$787$	0.107331	0.00382595	0.00191297	−	0.999998i	$-0.499391\pi$
$787$	0.107331	0.00382595	0.00191297	+	0.999998i	$0.499391\pi$
$788$	−28.4064	−1.01194
$789$	−3.91088	−0.139231
$790$	0	0
$791$	−13.6564	−0.485565
$792$	−11.3534	−0.403427
$793$	−31.6071	−1.12240
$794$	6.32222	0.224367
$795$	0	0
$796$	−6.36620	−0.225644
$797$	8.32068	0.294734	0.147367	−	0.989082i	$-0.452920\pi$
$797$	8.32068	0.294734	0.147367	+	0.989082i	$0.452920\pi$
$798$	6.11775	0.216566
$799$	19.2496	0.681003
$800$	0	0
$801$	29.1437	1.02974
$802$	29.0738	1.02663
$803$	−9.48489	−0.334714
$804$	−30.1117	−1.06196
$805$	0	0
$806$	−47.0320	−1.65663
$807$	62.7270	2.20810
$808$	4.15006	0.145998
$809$	−27.6231	−0.971177	−0.485588	−	0.874188i	$-0.661395\pi$
$809$	−27.6231	−0.971177	−0.485588	+	0.874188i	$0.661395\pi$
$810$	0	0
$811$	23.0095	0.807972	0.403986	−	0.914765i	$-0.367624\pi$
$811$	23.0095	0.807972	0.403986	+	0.914765i	$0.367624\pi$
$812$	−11.5008	−0.403600
$813$	−55.4369	−1.94426
$814$	42.3222	1.48339
$815$	0	0
$816$	−44.4589	−1.55637
$817$	−1.45033	−0.0507405
$818$	7.16224	0.250422
$819$	10.4553	0.365338
$820$	0	0
$821$	31.1355	1.08664	0.543318	−	0.839527i	$-0.317168\pi$
$821$	31.1355	1.08664	0.543318	+	0.839527i	$0.317168\pi$
$822$	55.5993	1.93925
$823$	−20.4201	−0.711800	−0.355900	−	0.934524i	$-0.615826\pi$
$823$	−20.4201	−0.711800	−0.355900	+	0.934524i	$0.615826\pi$
$824$	−16.1480	−0.562543
$825$	0	0
$826$	8.87275	0.308722
$827$	0.902638	0.0313878	0.0156939	−	0.999877i	$-0.495004\pi$
$827$	0.902638	0.0313878	0.0156939	+	0.999877i	$0.495004\pi$
$828$	16.7365	0.581634
$829$	−13.4971	−0.468773	−0.234386	−	0.972143i	$-0.575308\pi$
$829$	−13.4971	−0.468773	−0.234386	+	0.972143i	$0.575308\pi$
$830$	0	0
$831$	19.0607	0.661209
$832$	6.01042	0.208374
$833$	19.2092	0.665560
$834$	4.99053	0.172808
$835$	0	0
$836$	5.14982	0.178110
$837$	−12.5603	−0.434149
$838$	62.5123	2.15945
$839$	33.1022	1.14282	0.571408	−	0.820666i	$-0.306397\pi$
$839$	33.1022	1.14282	0.571408	+	0.820666i	$0.306397\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	47.5484	1.63862
$843$	24.1546	0.831928
$844$	13.8858	0.477971
$845$	0	0
$846$	21.0760	0.724608
$847$	−6.06686	−0.208460
$848$	−20.7186	−0.711480
$849$	−24.2295	−0.831553
$850$	0	0
$851$	32.0226	1.09772
$852$	−26.4387	−0.905775
$853$	−50.9097	−1.74312	−0.871558	−	0.490293i	$-0.836890\pi$
$853$	−50.9097	−1.74312	−0.871558	+	0.490293i	$0.836890\pi$
$854$	27.3128	0.934623
$855$	0	0
$856$	7.13491	0.243866
$857$	−21.2333	−0.725317	−0.362659	−	0.931922i	$-0.618131\pi$
$857$	−21.2333	−0.725317	−0.362659	+	0.931922i	$0.618131\pi$
$858$	50.2768	1.71642
$859$	29.4827	1.00594	0.502969	−	0.864304i	$-0.332241\pi$
$859$	29.4827	1.00594	0.502969	+	0.864304i	$0.332241\pi$
$860$	0	0
$861$	35.0320	1.19389
$862$	−18.7715	−0.639359
$863$	−25.7755	−0.877408	−0.438704	−	0.898632i	$-0.644562\pi$
$863$	−25.7755	−0.877408	−0.438704	+	0.898632i	$0.644562\pi$
$864$	9.60514	0.326773
$865$	0	0
$866$	−66.0339	−2.24392
$867$	3.72628	0.126551
$868$	16.1709	0.548876
$869$	48.4589	1.64386
$870$	0	0
$871$	−30.1117	−1.02030
$872$	8.21335	0.278139
$873$	−7.20893	−0.243985
$874$	9.79310	0.331257
$875$	0	0
$876$	−7.44584	−0.251572
$877$	−54.2687	−1.83252	−0.916261	−	0.400581i	$-0.868808\pi$
$877$	−54.2687	−1.83252	−0.916261	+	0.400581i	$0.868808\pi$
$878$	43.9033	1.48166
$879$	−38.1117	−1.28548
$880$	0	0
$881$	−28.4922	−0.959927	−0.479964	−	0.877288i	$-0.659350\pi$
$881$	−28.4922	−0.959927	−0.479964	+	0.877288i	$0.659350\pi$
$882$	21.0317	0.708176
$883$	−40.5264	−1.36382	−0.681911	−	0.731435i	$-0.738851\pi$
$883$	−40.5264	−1.36382	−0.681911	+	0.731435i	$0.738851\pi$
$884$	15.8596	0.533418
$885$	0	0
$886$	−10.0665	−0.338192
$887$	11.4705	0.385142	0.192571	−	0.981283i	$-0.438317\pi$
$887$	11.4705	0.385142	0.192571	+	0.981283i	$0.438317\pi$
$888$	−17.0530	−0.572260
$889$	16.0891	0.539612
$890$	0	0
$891$	40.9762	1.37275
$892$	22.3287	0.747621
$893$	4.90686	0.164202
$894$	23.0320	0.770307
$895$	0	0
$896$	13.5197	0.451660
$897$	38.0413	1.27016
$898$	−14.4524	−0.482282
$899$	−50.6182	−1.68821
$900$	0	0
$901$	−16.5993	−0.553003
$902$	74.1150	2.46776
$903$	4.86835	0.162009
$904$	11.6415	0.387189
$905$	0	0
$906$	−21.4269	−0.711861
$907$	48.1000	1.59714	0.798568	−	0.601905i	$-0.205591\pi$
$907$	48.1000	1.59714	0.798568	+	0.601905i	$0.205591\pi$
$908$	−22.6452	−0.751506
$909$	7.91088	0.262387
$910$	0	0
$911$	−12.8062	−0.424288	−0.212144	−	0.977238i	$-0.568045\pi$
$911$	−12.8062	−0.424288	−0.212144	+	0.977238i	$0.568045\pi$
$912$	−11.3329	−0.375269
$913$	49.3439	1.63304
$914$	−41.2865	−1.36564
$915$	0	0
$916$	33.1022	1.09373
$917$	−6.68601	−0.220792
$918$	10.6449	0.351334
$919$	−38.7135	−1.27704	−0.638519	−	0.769606i	$-0.720453\pi$
$919$	−38.7135	−1.27704	−0.638519	+	0.769606i	$0.720453\pi$
$920$	0	0
$921$	18.2295	0.600682
$922$	−24.4038	−0.803695
$923$	−26.4387	−0.870241
$924$	−17.2865	−0.568686
$925$	0	0
$926$	−30.8062	−1.01235
$927$	−30.7815	−1.01100
$928$	38.7087	1.27068
$929$	−36.0189	−1.18174	−0.590872	−	0.806766i	$-0.701216\pi$
$929$	−36.0189	−1.18174	−0.590872	+	0.806766i	$0.701216\pi$
$930$	0	0
$931$	4.89655	0.160478
$932$	−25.5058	−0.835469
$933$	−9.01841	−0.295249
$934$	−40.5517	−1.32689
$935$	0	0
$936$	−8.91270	−0.291321
$937$	−45.2421	−1.47799	−0.738997	−	0.673709i	$-0.764700\pi$
$937$	−45.2421	−1.47799	−0.738997	+	0.673709i	$0.764700\pi$
$938$	26.0205	0.849599
$939$	17.9811	0.586790
$940$	0	0
$941$	−11.7455	−0.382892	−0.191446	−	0.981503i	$-0.561318\pi$
$941$	−11.7455	−0.382892	−0.191446	+	0.981503i	$0.561318\pi$
$942$	−25.8058	−0.840799
$943$	56.0781	1.82616
$944$	−16.4364	−0.534958
$945$	0	0
$946$	10.2996	0.334870
$947$	−13.7752	−0.447635	−0.223817	−	0.974631i	$-0.571852\pi$
$947$	−13.7752	−0.447635	−0.223817	+	0.974631i	$0.571852\pi$
$948$	38.0413	1.23552
$949$	−7.44584	−0.241702
$950$	0	0
$951$	44.0927	1.42981
$952$	7.03434	0.227984
$953$	9.01421	0.291999	0.145999	−	0.989285i	$-0.453360\pi$
$953$	9.01421	0.291999	0.145999	+	0.989285i	$0.453360\pi$
$954$	−18.1742	−0.588412
$955$	0	0
$956$	24.8062	0.802290
$957$	54.1105	1.74914
$958$	−0.667406	−0.0215629
$959$	−19.1166	−0.617306
$960$	0	0
$961$	40.1724	1.29588
$962$	33.2239	1.07118
$963$	13.6006	0.438274
$964$	−19.0796	−0.614514
$965$	0	0
$966$	−32.8727	−1.05766
$967$	30.3232	0.975129	0.487564	−	0.873087i	$-0.337885\pi$
$967$	30.3232	0.975129	0.487564	+	0.873087i	$0.337885\pi$
$968$	5.17173	0.166226
$969$	−9.07965	−0.291680
$970$	0	0
$971$	−31.5636	−1.01292	−0.506462	−	0.862262i	$-0.669047\pi$
$971$	−31.5636	−1.01292	−0.506462	+	0.862262i	$0.669047\pi$
$972$	26.2641	0.842422
$973$	−1.71588	−0.0550086
$974$	−48.9280	−1.56775
$975$	0	0
$976$	−50.5957	−1.61953
$977$	43.1285	1.37980	0.689902	−	0.723903i	$-0.257654\pi$
$977$	43.1285	1.37980	0.689902	+	0.723903i	$0.257654\pi$
$978$	69.3320	2.21699
$979$	−48.1855	−1.54002
$980$	0	0
$981$	15.6564	0.499870
$982$	43.2425	1.37992
$983$	7.81570	0.249282	0.124641	−	0.992202i	$-0.460222\pi$
$983$	7.81570	0.249282	0.124641	+	0.992202i	$0.460222\pi$
$984$	−29.8633	−0.952006
$985$	0	0
$986$	42.8989	1.36618
$987$	−16.4710	−0.524277
$988$	4.04272	0.128616
$989$	7.79310	0.247806
$990$	0	0
$991$	−23.5197	−0.747126	−0.373563	−	0.927605i	$-0.621864\pi$
$991$	−23.5197	−0.747126	−0.373563	+	0.927605i	$0.621864\pi$
$992$	−54.4269	−1.72806
$993$	−18.5157	−0.587577
$994$	22.8465	0.724648
$995$	0	0
$996$	38.7360	1.22740
$997$	−33.6395	−1.06537	−0.532686	−	0.846313i	$-0.678817\pi$
$997$	−33.6395	−1.06537	−0.532686	+	0.846313i	$0.678817\pi$
$998$	−12.4240	−0.393276
$999$	8.87275	0.280721

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	475.2.a.j.1.2		6
3.2	odd	2		4275.2.a.br.1.5		6
4.3	odd	2		7600.2.a.ck.1.5		6
5.2	odd	4		95.2.b.b.39.2	✓	6
5.3	odd	4		95.2.b.b.39.5	yes	6
5.4	even	2	inner	475.2.a.j.1.5		6
15.2	even	4		855.2.c.d.514.5		6
15.8	even	4		855.2.c.d.514.2		6
15.14	odd	2		4275.2.a.br.1.2		6
19.18	odd	2		9025.2.a.bx.1.5		6
20.3	even	4		1520.2.d.h.609.5		6
20.7	even	4		1520.2.d.h.609.2		6
20.19	odd	2		7600.2.a.ck.1.2		6
95.18	even	4		1805.2.b.e.1084.2		6
95.37	even	4		1805.2.b.e.1084.5		6
95.94	odd	2		9025.2.a.bx.1.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
95.2.b.b.39.2	✓	6	5.2	odd	4
95.2.b.b.39.5	yes	6	5.3	odd	4
475.2.a.j.1.2		6	1.1	even	1	trivial
475.2.a.j.1.5		6	5.4	even	2	inner
855.2.c.d.514.2		6	15.8	even	4
855.2.c.d.514.5		6	15.2	even	4
1520.2.d.h.609.2		6	20.7	even	4
1520.2.d.h.609.5		6	20.3	even	4
1805.2.b.e.1084.2		6	95.18	even	4
1805.2.b.e.1084.5		6	95.37	even	4
4275.2.a.br.1.2		6	15.14	odd	2
4275.2.a.br.1.5		6	3.2	odd	2
7600.2.a.ck.1.2		6	20.19	odd	2
7600.2.a.ck.1.5		6	4.3	odd	2
9025.2.a.bx.1.2		6	95.94	odd	2
9025.2.a.bx.1.5		6	19.18	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:	\( N \)	\(=\)	\( 475 = 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	475.a (trivial)

\(f(q)\)	\(=\)	\(q-1.82254 q^{2} -2.31446 q^{3} +1.32164 q^{4} +4.21819 q^{6} -1.45033 q^{7} +1.23634 q^{8} +2.35673 q^{9} +O(q^{10})\)	\(q-1.82254 q^{2} -2.31446 q^{3} +1.32164 q^{4} +4.21819 q^{6} -1.45033 q^{7} +1.23634 q^{8} +2.35673 q^{9} -3.89655 q^{11} -3.05888 q^{12} -3.05888 q^{13} +2.64327 q^{14} -4.89655 q^{16} -3.92301 q^{17} -4.29522 q^{18} -1.00000 q^{19} +3.35673 q^{21} +7.10160 q^{22} +5.37334 q^{23} -2.86146 q^{24} +5.57491 q^{26} +1.48883 q^{27} -1.91681 q^{28} +6.00000 q^{29} -8.43637 q^{31} +6.45146 q^{32} +9.01841 q^{33} +7.14982 q^{34} +3.11474 q^{36} +5.95953 q^{37} +1.82254 q^{38} +7.07965 q^{39} +10.4364 q^{41} -6.11775 q^{42} +1.45033 q^{43} -5.14982 q^{44} -9.79310 q^{46} -4.90686 q^{47} +11.3329 q^{48} -4.89655 q^{49} +9.07965 q^{51} -4.04272 q^{52} +4.23127 q^{53} -2.71345 q^{54} -1.79310 q^{56} +2.31446 q^{57} -10.9352 q^{58} +3.35673 q^{59} +10.3329 q^{61} +15.3756 q^{62} -3.41802 q^{63} -1.96491 q^{64} -16.4364 q^{66} +9.84404 q^{67} -5.18479 q^{68} -12.4364 q^{69} +8.64327 q^{71} +2.91372 q^{72} +2.43418 q^{73} -10.8615 q^{74} -1.32164 q^{76} +5.65127 q^{77} -12.9029 q^{78} -12.4364 q^{79} -10.5160 q^{81} -19.0207 q^{82} -12.6635 q^{83} +4.43637 q^{84} -2.64327 q^{86} -13.8868 q^{87} -4.81746 q^{88} +12.3662 q^{89} +4.43637 q^{91} +7.10160 q^{92} +19.5256 q^{93} +8.94292 q^{94} -14.9316 q^{96} -3.05888 q^{97} +8.92414 q^{98} -9.18310 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 8 q^{4} + 14 q^{9}+O(q^{10}) \) 6 * q + 8 * q^4 + 14 * q^9	\( 6 q + 8 q^{4} + 14 q^{9} + 2 q^{11} + 16 q^{14} - 4 q^{16} - 6 q^{19} + 20 q^{21} + 8 q^{24} + 8 q^{26} + 36 q^{29} - 8 q^{34} - 32 q^{36} - 8 q^{39} + 12 q^{41} + 20 q^{44} - 8 q^{46} - 4 q^{49} + 4 q^{51} - 16 q^{54} + 40 q^{56} + 20 q^{59} - 14 q^{61} - 12 q^{64} - 48 q^{66} - 24 q^{69} + 52 q^{71} - 40 q^{74} - 8 q^{76} - 24 q^{79} + 38 q^{81} - 24 q^{84} - 16 q^{86} + 24 q^{89} - 24 q^{91} - 48 q^{94} - 64 q^{96} - 30 q^{99}+O(q^{100}) \) 6 * q + 8 * q^4 + 14 * q^9 + 2 * q^11 + 16 * q^14 - 4 * q^16 - 6 * q^19 + 20 * q^21 + 8 * q^24 + 8 * q^26 + 36 * q^29 - 8 * q^34 - 32 * q^36 - 8 * q^39 + 12 * q^41 + 20 * q^44 - 8 * q^46 - 4 * q^49 + 4 * q^51 - 16 * q^54 + 40 * q^56 + 20 * q^59 - 14 * q^61 - 12 * q^64 - 48 * q^66 - 24 * q^69 + 52 * q^71 - 40 * q^74 - 8 * q^76 - 24 * q^79 + 38 * q^81 - 24 * q^84 - 16 * q^86 + 24 * q^89 - 24 * q^91 - 48 * q^94 - 64 * q^96 - 30 * q^99

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