LMFDB - Embedded newform 950.2.a.f.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [950,2,Mod(1,950)] 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("950.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(950, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 950 = 2 \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	950.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,-2,2,2,0,-2,6,-2,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	yes
Analytic conductor:	$7.58578819202$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 190)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	950.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000 q^{2} -0.414214 q^{3} +1.00000 q^{4} +0.414214 q^{6} +4.41421 q^{7} -1.00000 q^{8} -2.82843 q^{9} -1.41421 q^{11} -0.414214 q^{12} +5.82843 q^{13} -4.41421 q^{14} +1.00000 q^{16} +1.00000 q^{17} +2.82843 q^{18} -1.00000 q^{19} -1.82843 q^{21} +1.41421 q^{22} +0.757359 q^{23} +0.414214 q^{24} -5.82843 q^{26} +2.41421 q^{27} +4.41421 q^{28} -0.171573 q^{29} +6.24264 q^{31} -1.00000 q^{32} +0.585786 q^{33} -1.00000 q^{34} -2.82843 q^{36} -8.48528 q^{37} +1.00000 q^{38} -2.41421 q^{39} -4.24264 q^{41} +1.82843 q^{42} +1.75736 q^{43} -1.41421 q^{44} -0.757359 q^{46} -0.414214 q^{48} +12.4853 q^{49} -0.414214 q^{51} +5.82843 q^{52} +5.48528 q^{53} -2.41421 q^{54} -4.41421 q^{56} +0.414214 q^{57} +0.171573 q^{58} +6.89949 q^{59} +14.2426 q^{61} -6.24264 q^{62} -12.4853 q^{63} +1.00000 q^{64} -0.585786 q^{66} +4.75736 q^{67} +1.00000 q^{68} -0.313708 q^{69} -13.4142 q^{71} +2.82843 q^{72} +11.4853 q^{73} +8.48528 q^{74} -1.00000 q^{76} -6.24264 q^{77} +2.41421 q^{78} -6.48528 q^{79} +7.48528 q^{81} +4.24264 q^{82} +14.4853 q^{83} -1.82843 q^{84} -1.75736 q^{86} +0.0710678 q^{87} +1.41421 q^{88} +7.07107 q^{89} +25.7279 q^{91} +0.757359 q^{92} -2.58579 q^{93} +0.414214 q^{96} -0.343146 q^{97} -12.4853 q^{98} +4.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} + 6 q^{7} - 2 q^{8} + 2 q^{12} + 6 q^{13} - 6 q^{14} + 2 q^{16} + 2 q^{17} - 2 q^{19} + 2 q^{21} + 10 q^{23} - 2 q^{24} - 6 q^{26} + 2 q^{27} + 6 q^{28} - 6 q^{29}+ \cdots + 8 q^{99}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^6 + 6 * q^7 - 2 * q^8 + 2 * q^12 + 6 * q^13 - 6 * q^14 + 2 * q^16 + 2 * q^17 - 2 * q^19 + 2 * q^21 + 10 * q^23 - 2 * q^24 - 6 * q^26 + 2 * q^27 + 6 * q^28 - 6 * q^29 + 4 * q^31 - 2 * q^32 + 4 * q^33 - 2 * q^34 + 2 * q^38 - 2 * q^39 - 2 * q^42 + 12 * q^43 - 10 * q^46 + 2 * q^48 + 8 * q^49 + 2 * q^51 + 6 * q^52 - 6 * q^53 - 2 * q^54 - 6 * q^56 - 2 * q^57 + 6 * q^58 - 6 * q^59 + 20 * q^61 - 4 * q^62 - 8 * q^63 + 2 * q^64 - 4 * q^66 + 18 * q^67 + 2 * q^68 + 22 * q^69 - 24 * q^71 + 6 * q^73 - 2 * q^76 - 4 * q^77 + 2 * q^78 + 4 * q^79 - 2 * q^81 + 12 * q^83 + 2 * q^84 - 12 * q^86 - 14 * q^87 + 26 * q^91 + 10 * q^92 - 8 * q^93 - 2 * q^96 - 12 * q^97 - 8 * q^98 + 8 * 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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−0.414214	−0.239146	−0.119573	−	0.992825i	$-0.538153\pi$
$3$	−0.414214	−0.239146	−0.119573	+	0.992825i	$0.538153\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	0.414214	0.169102
$7$	4.41421	1.66842	0.834208	−	0.551450i	$-0.185925\pi$
$7$	4.41421	1.66842	0.834208	+	0.551450i	$0.185925\pi$
$8$	−1.00000	−0.353553
$9$	−2.82843	−0.942809
$10$	0	0
$11$	−1.41421	−0.426401	−0.213201	−	0.977008i	$-0.568389\pi$
$11$	−1.41421	−0.426401	−0.213201	+	0.977008i	$0.568389\pi$
$12$	−0.414214	−0.119573
$13$	5.82843	1.61651	0.808257	−	0.588829i	$-0.200411\pi$
$13$	5.82843	1.61651	0.808257	+	0.588829i	$0.200411\pi$
$14$	−4.41421	−1.17975
$15$	0	0
$16$	1.00000	0.250000
$17$	1.00000	0.242536	0.121268	−	0.992620i	$-0.461304\pi$
$17$	1.00000	0.242536	0.121268	+	0.992620i	$0.461304\pi$
$18$	2.82843	0.666667
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−1.82843	−0.398996
$22$	1.41421	0.301511
$23$	0.757359	0.157920	0.0789602	−	0.996878i	$-0.474840\pi$
$23$	0.757359	0.157920	0.0789602	+	0.996878i	$0.474840\pi$
$24$	0.414214	0.0845510
$25$	0	0
$26$	−5.82843	−1.14305
$27$	2.41421	0.464616
$28$	4.41421	0.834208
$29$	−0.171573	−0.0318603	−0.0159301	−	0.999873i	$-0.505071\pi$
$29$	−0.171573	−0.0318603	−0.0159301	+	0.999873i	$0.505071\pi$
$30$	0	0
$31$	6.24264	1.12121	0.560606	−	0.828083i	$-0.310568\pi$
$31$	6.24264	1.12121	0.560606	+	0.828083i	$0.310568\pi$
$32$	−1.00000	−0.176777
$33$	0.585786	0.101972
$34$	−1.00000	−0.171499
$35$	0	0
$36$	−2.82843	−0.471405
$37$	−8.48528	−1.39497	−0.697486	−	0.716599i	$-0.745698\pi$
$37$	−8.48528	−1.39497	−0.697486	+	0.716599i	$0.745698\pi$
$38$	1.00000	0.162221
$39$	−2.41421	−0.386584
$40$	0	0
$41$	−4.24264	−0.662589	−0.331295	−	0.943527i	$-0.607485\pi$
$41$	−4.24264	−0.662589	−0.331295	+	0.943527i	$0.607485\pi$
$42$	1.82843	0.282132
$43$	1.75736	0.267995	0.133997	−	0.990982i	$-0.457219\pi$
$43$	1.75736	0.267995	0.133997	+	0.990982i	$0.457219\pi$
$44$	−1.41421	−0.213201
$45$	0	0
$46$	−0.757359	−0.111667
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	−0.414214	−0.0597866
$49$	12.4853	1.78361
$50$	0	0
$51$	−0.414214	−0.0580015
$52$	5.82843	0.808257
$53$	5.48528	0.753461	0.376731	−	0.926323i	$-0.377048\pi$
$53$	5.48528	0.753461	0.376731	+	0.926323i	$0.377048\pi$
$54$	−2.41421	−0.328533
$55$	0	0
$56$	−4.41421	−0.589874
$57$	0.414214	0.0548639
$58$	0.171573	0.0225286
$59$	6.89949	0.898238	0.449119	−	0.893472i	$-0.351738\pi$
$59$	6.89949	0.898238	0.449119	+	0.893472i	$0.351738\pi$
$60$	0	0
$61$	14.2426	1.82358	0.911792	−	0.410653i	$-0.134699\pi$
$61$	14.2426	1.82358	0.911792	+	0.410653i	$0.134699\pi$
$62$	−6.24264	−0.792816
$63$	−12.4853	−1.57300
$64$	1.00000	0.125000
$65$	0	0
$66$	−0.585786	−0.0721053
$67$	4.75736	0.581204	0.290602	−	0.956844i	$-0.406144\pi$
$67$	4.75736	0.581204	0.290602	+	0.956844i	$0.406144\pi$
$68$	1.00000	0.121268
$69$	−0.313708	−0.0377661
$70$	0	0
$71$	−13.4142	−1.59197	−0.795987	−	0.605314i	$-0.793048\pi$
$71$	−13.4142	−1.59197	−0.795987	+	0.605314i	$0.793048\pi$
$72$	2.82843	0.333333
$73$	11.4853	1.34425	0.672125	−	0.740437i	$-0.265382\pi$
$73$	11.4853	1.34425	0.672125	+	0.740437i	$0.265382\pi$
$74$	8.48528	0.986394
$75$	0	0
$76$	−1.00000	−0.114708
$77$	−6.24264	−0.711415
$78$	2.41421	0.273356
$79$	−6.48528	−0.729651	−0.364826	−	0.931076i	$-0.618871\pi$
$79$	−6.48528	−0.729651	−0.364826	+	0.931076i	$0.618871\pi$
$80$	0	0
$81$	7.48528	0.831698
$82$	4.24264	0.468521
$83$	14.4853	1.58997	0.794983	−	0.606632i	$-0.207480\pi$
$83$	14.4853	1.58997	0.794983	+	0.606632i	$0.207480\pi$
$84$	−1.82843	−0.199498
$85$	0	0
$86$	−1.75736	−0.189501
$87$	0.0710678	0.00761927
$88$	1.41421	0.150756
$89$	7.07107	0.749532	0.374766	−	0.927119i	$-0.377723\pi$
$89$	7.07107	0.749532	0.374766	+	0.927119i	$0.377723\pi$
$90$	0	0
$91$	25.7279	2.69702
$92$	0.757359	0.0789602
$93$	−2.58579	−0.268134
$94$	0	0
$95$	0	0
$96$	0.414214	0.0422755
$97$	−0.343146	−0.0348412	−0.0174206	−	0.999848i	$-0.505545\pi$
$97$	−0.343146	−0.0348412	−0.0174206	+	0.999848i	$0.505545\pi$
$98$	−12.4853	−1.26120
$99$	4.00000	0.402015
$100$	0	0
$101$	13.0711	1.30062	0.650310	−	0.759669i	$-0.274639\pi$
$101$	13.0711	1.30062	0.650310	+	0.759669i	$0.274639\pi$
$102$	0.414214	0.0410133
$103$	4.24264	0.418040	0.209020	−	0.977911i	$-0.432973\pi$
$103$	4.24264	0.418040	0.209020	+	0.977911i	$0.432973\pi$
$104$	−5.82843	−0.571524
$105$	0	0
$106$	−5.48528	−0.532778
$107$	19.7279	1.90717	0.953585	−	0.301124i	$-0.0973617\pi$
$107$	19.7279	1.90717	0.953585	+	0.301124i	$0.0973617\pi$
$108$	2.41421	0.232308
$109$	−17.9706	−1.72127	−0.860634	−	0.509224i	$-0.829932\pi$
$109$	−17.9706	−1.72127	−0.860634	+	0.509224i	$0.829932\pi$
$110$	0	0
$111$	3.51472	0.333602
$112$	4.41421	0.417104
$113$	−10.2426	−0.963547	−0.481773	−	0.876296i	$-0.660007\pi$
$113$	−10.2426	−0.963547	−0.481773	+	0.876296i	$0.660007\pi$
$114$	−0.414214	−0.0387947
$115$	0	0
$116$	−0.171573	−0.0159301
$117$	−16.4853	−1.52406
$118$	−6.89949	−0.635150
$119$	4.41421	0.404650
$120$	0	0
$121$	−9.00000	−0.818182
$122$	−14.2426	−1.28947
$123$	1.75736	0.158456
$124$	6.24264	0.560606
$125$	0	0
$126$	12.4853	1.11228
$127$	−2.48528	−0.220533	−0.110267	−	0.993902i	$-0.535170\pi$
$127$	−2.48528	−0.220533	−0.110267	+	0.993902i	$0.535170\pi$
$128$	−1.00000	−0.0883883
$129$	−0.727922	−0.0640900
$130$	0	0
$131$	−16.9706	−1.48272	−0.741362	−	0.671105i	$-0.765820\pi$
$131$	−16.9706	−1.48272	−0.741362	+	0.671105i	$0.765820\pi$
$132$	0.585786	0.0509862
$133$	−4.41421	−0.382761
$134$	−4.75736	−0.410973
$135$	0	0
$136$	−1.00000	−0.0857493
$137$	13.0000	1.11066	0.555332	−	0.831628i	$-0.312591\pi$
$137$	13.0000	1.11066	0.555332	+	0.831628i	$0.312591\pi$
$138$	0.313708	0.0267046
$139$	−12.0000	−1.01783	−0.508913	−	0.860818i	$-0.669953\pi$
$139$	−12.0000	−1.01783	−0.508913	+	0.860818i	$0.669953\pi$
$140$	0	0
$141$	0	0
$142$	13.4142	1.12570
$143$	−8.24264	−0.689284
$144$	−2.82843	−0.235702
$145$	0	0
$146$	−11.4853	−0.950529
$147$	−5.17157	−0.426544
$148$	−8.48528	−0.697486
$149$	17.6569	1.44651	0.723253	−	0.690583i	$-0.242646\pi$
$149$	17.6569	1.44651	0.723253	+	0.690583i	$0.242646\pi$
$150$	0	0
$151$	−10.4853	−0.853280	−0.426640	−	0.904422i	$-0.640303\pi$
$151$	−10.4853	−0.853280	−0.426640	+	0.904422i	$0.640303\pi$
$152$	1.00000	0.0811107
$153$	−2.82843	−0.228665
$154$	6.24264	0.503046
$155$	0	0
$156$	−2.41421	−0.193292
$157$	11.6569	0.930318	0.465159	−	0.885227i	$-0.345997\pi$
$157$	11.6569	0.930318	0.465159	+	0.885227i	$0.345997\pi$
$158$	6.48528	0.515941
$159$	−2.27208	−0.180188
$160$	0	0
$161$	3.34315	0.263477
$162$	−7.48528	−0.588099
$163$	−10.2426	−0.802266	−0.401133	−	0.916020i	$-0.631383\pi$
$163$	−10.2426	−0.802266	−0.401133	+	0.916020i	$0.631383\pi$
$164$	−4.24264	−0.331295
$165$	0	0
$166$	−14.4853	−1.12428
$167$	−18.2426	−1.41166	−0.705829	−	0.708382i	$-0.749425\pi$
$167$	−18.2426	−1.41166	−0.705829	+	0.708382i	$0.749425\pi$
$168$	1.82843	0.141066
$169$	20.9706	1.61312
$170$	0	0
$171$	2.82843	0.216295
$172$	1.75736	0.133997
$173$	0.485281	0.0368953	0.0184476	−	0.999830i	$-0.494128\pi$
$173$	0.485281	0.0368953	0.0184476	+	0.999830i	$0.494128\pi$
$174$	−0.0710678	−0.00538764
$175$	0	0
$176$	−1.41421	−0.106600
$177$	−2.85786	−0.214810
$178$	−7.07107	−0.529999
$179$	−11.6569	−0.871274	−0.435637	−	0.900122i	$-0.643477\pi$
$179$	−11.6569	−0.871274	−0.435637	+	0.900122i	$0.643477\pi$
$180$	0	0
$181$	−8.48528	−0.630706	−0.315353	−	0.948974i	$-0.602123\pi$
$181$	−8.48528	−0.630706	−0.315353	+	0.948974i	$0.602123\pi$
$182$	−25.7279	−1.90708
$183$	−5.89949	−0.436103
$184$	−0.757359	−0.0558333
$185$	0	0
$186$	2.58579	0.189599
$187$	−1.41421	−0.103418
$188$	0	0
$189$	10.6569	0.775172
$190$	0	0
$191$	12.5563	0.908546	0.454273	−	0.890863i	$-0.349899\pi$
$191$	12.5563	0.908546	0.454273	+	0.890863i	$0.349899\pi$
$192$	−0.414214	−0.0298933
$193$	−0.343146	−0.0247002	−0.0123501	−	0.999924i	$-0.503931\pi$
$193$	−0.343146	−0.0247002	−0.0123501	+	0.999924i	$0.503931\pi$
$194$	0.343146	0.0246364
$195$	0	0
$196$	12.4853	0.891806
$197$	−11.7574	−0.837677	−0.418839	−	0.908061i	$-0.637563\pi$
$197$	−11.7574	−0.837677	−0.418839	+	0.908061i	$0.637563\pi$
$198$	−4.00000	−0.284268
$199$	9.24264	0.655193	0.327597	−	0.944818i	$-0.393761\pi$
$199$	9.24264	0.655193	0.327597	+	0.944818i	$0.393761\pi$
$200$	0	0
$201$	−1.97056	−0.138993
$202$	−13.0711	−0.919677
$203$	−0.757359	−0.0531562
$204$	−0.414214	−0.0290008
$205$	0	0
$206$	−4.24264	−0.295599
$207$	−2.14214	−0.148889
$208$	5.82843	0.404129
$209$	1.41421	0.0978232
$210$	0	0
$211$	−19.7279	−1.35813	−0.679063	−	0.734080i	$-0.737614\pi$
$211$	−19.7279	−1.35813	−0.679063	+	0.734080i	$0.737614\pi$
$212$	5.48528	0.376731
$213$	5.55635	0.380715
$214$	−19.7279	−1.34857
$215$	0	0
$216$	−2.41421	−0.164266
$217$	27.5563	1.87065
$218$	17.9706	1.21712
$219$	−4.75736	−0.321473
$220$	0	0
$221$	5.82843	0.392062
$222$	−3.51472	−0.235892
$223$	−15.1716	−1.01596	−0.507982	−	0.861368i	$-0.669608\pi$
$223$	−15.1716	−1.01596	−0.507982	+	0.861368i	$0.669608\pi$
$224$	−4.41421	−0.294937
$225$	0	0
$226$	10.2426	0.681330
$227$	−16.7574	−1.11223	−0.556113	−	0.831107i	$-0.687708\pi$
$227$	−16.7574	−1.11223	−0.556113	+	0.831107i	$0.687708\pi$
$228$	0.414214	0.0274320
$229$	14.9706	0.989283	0.494641	−	0.869097i	$-0.335299\pi$
$229$	14.9706	0.989283	0.494641	+	0.869097i	$0.335299\pi$
$230$	0	0
$231$	2.58579	0.170132
$232$	0.171573	0.0112643
$233$	−24.9706	−1.63588	−0.817938	−	0.575306i	$-0.804883\pi$
$233$	−24.9706	−1.63588	−0.817938	+	0.575306i	$0.804883\pi$
$234$	16.4853	1.07768
$235$	0	0
$236$	6.89949	0.449119
$237$	2.68629	0.174493
$238$	−4.41421	−0.286131
$239$	−6.89949	−0.446291	−0.223146	−	0.974785i	$-0.571633\pi$
$239$	−6.89949	−0.446291	−0.223146	+	0.974785i	$0.571633\pi$
$240$	0	0
$241$	8.97056	0.577845	0.288922	−	0.957353i	$-0.406703\pi$
$241$	8.97056	0.577845	0.288922	+	0.957353i	$0.406703\pi$
$242$	9.00000	0.578542
$243$	−10.3431	−0.663513
$244$	14.2426	0.911792
$245$	0	0
$246$	−1.75736	−0.112045
$247$	−5.82843	−0.370854
$248$	−6.24264	−0.396408
$249$	−6.00000	−0.380235
$250$	0	0
$251$	3.55635	0.224475	0.112237	−	0.993681i	$-0.464198\pi$
$251$	3.55635	0.224475	0.112237	+	0.993681i	$0.464198\pi$
$252$	−12.4853	−0.786499
$253$	−1.07107	−0.0673375
$254$	2.48528	0.155940
$255$	0	0
$256$	1.00000	0.0625000
$257$	20.7279	1.29297	0.646486	−	0.762926i	$-0.276238\pi$
$257$	20.7279	1.29297	0.646486	+	0.762926i	$0.276238\pi$
$258$	0.727922	0.0453184
$259$	−37.4558	−2.32739
$260$	0	0
$261$	0.485281	0.0300382
$262$	16.9706	1.04844
$263$	−26.9706	−1.66308	−0.831538	−	0.555468i	$-0.812539\pi$
$263$	−26.9706	−1.66308	−0.831538	+	0.555468i	$0.812539\pi$
$264$	−0.585786	−0.0360527
$265$	0	0
$266$	4.41421	0.270653
$267$	−2.92893	−0.179248
$268$	4.75736	0.290602
$269$	−16.6274	−1.01379	−0.506896	−	0.862007i	$-0.669207\pi$
$269$	−16.6274	−1.01379	−0.506896	+	0.862007i	$0.669207\pi$
$270$	0	0
$271$	−27.2426	−1.65487	−0.827436	−	0.561560i	$-0.810202\pi$
$271$	−27.2426	−1.65487	−0.827436	+	0.561560i	$0.810202\pi$
$272$	1.00000	0.0606339
$273$	−10.6569	−0.644982
$274$	−13.0000	−0.785359
$275$	0	0
$276$	−0.313708	−0.0188830
$277$	0	0		−	1.00000i	$-0.5\pi$
$277$	0	0			1.00000i	$0.5\pi$
$278$	12.0000	0.719712
$279$	−17.6569	−1.05709
$280$	0	0
$281$	4.24264	0.253095	0.126547	−	0.991961i	$-0.459610\pi$
$281$	4.24264	0.253095	0.126547	+	0.991961i	$0.459610\pi$
$282$	0	0
$283$	32.1421	1.91065	0.955326	−	0.295555i	$-0.0955045\pi$
$283$	32.1421	1.91065	0.955326	+	0.295555i	$0.0955045\pi$
$284$	−13.4142	−0.795987
$285$	0	0
$286$	8.24264	0.487398
$287$	−18.7279	−1.10547
$288$	2.82843	0.166667
$289$	−16.0000	−0.941176
$290$	0	0
$291$	0.142136	0.00833214
$292$	11.4853	0.672125
$293$	−5.48528	−0.320454	−0.160227	−	0.987080i	$-0.551223\pi$
$293$	−5.48528	−0.320454	−0.160227	+	0.987080i	$0.551223\pi$
$294$	5.17157	0.301612
$295$	0	0
$296$	8.48528	0.493197
$297$	−3.41421	−0.198113
$298$	−17.6569	−1.02283
$299$	4.41421	0.255281
$300$	0	0
$301$	7.75736	0.447127
$302$	10.4853	0.603360
$303$	−5.41421	−0.311038
$304$	−1.00000	−0.0573539
$305$	0	0
$306$	2.82843	0.161690
$307$	−17.6569	−1.00773	−0.503865	−	0.863782i	$-0.668089\pi$
$307$	−17.6569	−1.00773	−0.503865	+	0.863782i	$0.668089\pi$
$308$	−6.24264	−0.355707
$309$	−1.75736	−0.0999727
$310$	0	0
$311$	−4.75736	−0.269765	−0.134883	−	0.990862i	$-0.543066\pi$
$311$	−4.75736	−0.269765	−0.134883	+	0.990862i	$0.543066\pi$
$312$	2.41421	0.136678
$313$	−7.97056	−0.450523	−0.225261	−	0.974298i	$-0.572324\pi$
$313$	−7.97056	−0.450523	−0.225261	+	0.974298i	$0.572324\pi$
$314$	−11.6569	−0.657834
$315$	0	0
$316$	−6.48528	−0.364826
$317$	−9.48528	−0.532746	−0.266373	−	0.963870i	$-0.585825\pi$
$317$	−9.48528	−0.532746	−0.266373	+	0.963870i	$0.585825\pi$
$318$	2.27208	0.127412
$319$	0.242641	0.0135853
$320$	0	0
$321$	−8.17157	−0.456093
$322$	−3.34315	−0.186306
$323$	−1.00000	−0.0556415
$324$	7.48528	0.415849
$325$	0	0
$326$	10.2426	0.567287
$327$	7.44365	0.411635
$328$	4.24264	0.234261
$329$	0	0
$330$	0	0
$331$	−19.2426	−1.05767	−0.528836	−	0.848724i	$-0.677371\pi$
$331$	−19.2426	−1.05767	−0.528836	+	0.848724i	$0.677371\pi$
$332$	14.4853	0.794983
$333$	24.0000	1.31519
$334$	18.2426	0.998193
$335$	0	0
$336$	−1.82843	−0.0997489
$337$	14.1005	0.768103	0.384052	−	0.923312i	$-0.374528\pi$
$337$	14.1005	0.768103	0.384052	+	0.923312i	$0.374528\pi$
$338$	−20.9706	−1.14065
$339$	4.24264	0.230429
$340$	0	0
$341$	−8.82843	−0.478086
$342$	−2.82843	−0.152944
$343$	24.2132	1.30739
$344$	−1.75736	−0.0947505
$345$	0	0
$346$	−0.485281	−0.0260889
$347$	−5.51472	−0.296046	−0.148023	−	0.988984i	$-0.547291\pi$
$347$	−5.51472	−0.296046	−0.148023	+	0.988984i	$0.547291\pi$
$348$	0.0710678	0.00380963
$349$	6.00000	0.321173	0.160586	−	0.987022i	$-0.448662\pi$
$349$	6.00000	0.321173	0.160586	+	0.987022i	$0.448662\pi$
$350$	0	0
$351$	14.0711	0.751058
$352$	1.41421	0.0753778
$353$	−2.51472	−0.133845	−0.0669225	−	0.997758i	$-0.521318\pi$
$353$	−2.51472	−0.133845	−0.0669225	+	0.997758i	$0.521318\pi$
$354$	2.85786	0.151894
$355$	0	0
$356$	7.07107	0.374766
$357$	−1.82843	−0.0967706
$358$	11.6569	0.616084
$359$	−22.7574	−1.20109	−0.600544	−	0.799592i	$-0.705049\pi$
$359$	−22.7574	−1.20109	−0.600544	+	0.799592i	$0.705049\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	8.48528	0.445976
$363$	3.72792	0.195665
$364$	25.7279	1.34851
$365$	0	0
$366$	5.89949	0.308372
$367$	−25.4558	−1.32878	−0.664392	−	0.747384i	$-0.731309\pi$
$367$	−25.4558	−1.32878	−0.664392	+	0.747384i	$0.731309\pi$
$368$	0.757359	0.0394801
$369$	12.0000	0.624695
$370$	0	0
$371$	24.2132	1.25709
$372$	−2.58579	−0.134067
$373$	9.00000	0.466002	0.233001	−	0.972476i	$-0.425145\pi$
$373$	9.00000	0.466002	0.233001	+	0.972476i	$0.425145\pi$
$374$	1.41421	0.0731272
$375$	0	0
$376$	0	0
$377$	−1.00000	−0.0515026
$378$	−10.6569	−0.548129
$379$	11.2426	0.577496	0.288748	−	0.957405i	$-0.406761\pi$
$379$	11.2426	0.577496	0.288748	+	0.957405i	$0.406761\pi$
$380$	0	0
$381$	1.02944	0.0527397
$382$	−12.5563	−0.642439
$383$	12.2426	0.625570	0.312785	−	0.949824i	$-0.398738\pi$
$383$	12.2426	0.625570	0.312785	+	0.949824i	$0.398738\pi$
$384$	0.414214	0.0211377
$385$	0	0
$386$	0.343146	0.0174657
$387$	−4.97056	−0.252668
$388$	−0.343146	−0.0174206
$389$	22.9289	1.16254	0.581272	−	0.813710i	$-0.302555\pi$
$389$	22.9289	1.16254	0.581272	+	0.813710i	$0.302555\pi$
$390$	0	0
$391$	0.757359	0.0383013
$392$	−12.4853	−0.630602
$393$	7.02944	0.354588
$394$	11.7574	0.592327
$395$	0	0
$396$	4.00000	0.201008
$397$	24.0000	1.20453	0.602263	−	0.798298i	$-0.294266\pi$
$397$	24.0000	1.20453	0.602263	+	0.798298i	$0.294266\pi$
$398$	−9.24264	−0.463292
$399$	1.82843	0.0915358
$400$	0	0
$401$	−25.4142	−1.26913	−0.634563	−	0.772871i	$-0.718820\pi$
$401$	−25.4142	−1.26913	−0.634563	+	0.772871i	$0.718820\pi$
$402$	1.97056	0.0982827
$403$	36.3848	1.81245
$404$	13.0711	0.650310
$405$	0	0
$406$	0.757359	0.0375871
$407$	12.0000	0.594818
$408$	0.414214	0.0205066
$409$	25.2132	1.24671	0.623356	−	0.781938i	$-0.285769\pi$
$409$	25.2132	1.24671	0.623356	+	0.781938i	$0.285769\pi$
$410$	0	0
$411$	−5.38478	−0.265611
$412$	4.24264	0.209020
$413$	30.4558	1.49863
$414$	2.14214	0.105280
$415$	0	0
$416$	−5.82843	−0.285762
$417$	4.97056	0.243410
$418$	−1.41421	−0.0691714
$419$	19.4142	0.948446	0.474223	−	0.880405i	$-0.342729\pi$
$419$	19.4142	0.948446	0.474223	+	0.880405i	$0.342729\pi$
$420$	0	0
$421$	19.4853	0.949655	0.474827	−	0.880079i	$-0.342511\pi$
$421$	19.4853	0.949655	0.474827	+	0.880079i	$0.342511\pi$
$422$	19.7279	0.960340
$423$	0	0
$424$	−5.48528	−0.266389
$425$	0	0
$426$	−5.55635	−0.269206
$427$	62.8701	3.04250
$428$	19.7279	0.953585
$429$	3.41421	0.164840
$430$	0	0
$431$	−6.38478	−0.307544	−0.153772	−	0.988106i	$-0.549142\pi$
$431$	−6.38478	−0.307544	−0.153772	+	0.988106i	$0.549142\pi$
$432$	2.41421	0.116154
$433$	−9.55635	−0.459249	−0.229624	−	0.973279i	$-0.573750\pi$
$433$	−9.55635	−0.459249	−0.229624	+	0.973279i	$0.573750\pi$
$434$	−27.5563	−1.32275
$435$	0	0
$436$	−17.9706	−0.860634
$437$	−0.757359	−0.0362294
$438$	4.75736	0.227315
$439$	−5.75736	−0.274784	−0.137392	−	0.990517i	$-0.543872\pi$
$439$	−5.75736	−0.274784	−0.137392	+	0.990517i	$0.543872\pi$
$440$	0	0
$441$	−35.3137	−1.68161
$442$	−5.82843	−0.277230
$443$	4.24264	0.201574	0.100787	−	0.994908i	$-0.467864\pi$
$443$	4.24264	0.201574	0.100787	+	0.994908i	$0.467864\pi$
$444$	3.51472	0.166801
$445$	0	0
$446$	15.1716	0.718395
$447$	−7.31371	−0.345927
$448$	4.41421	0.208552
$449$	17.3137	0.817084	0.408542	−	0.912739i	$-0.366037\pi$
$449$	17.3137	0.817084	0.408542	+	0.912739i	$0.366037\pi$
$450$	0	0
$451$	6.00000	0.282529
$452$	−10.2426	−0.481773
$453$	4.34315	0.204059
$454$	16.7574	0.786462
$455$	0	0
$456$	−0.414214	−0.0193973
$457$	−3.00000	−0.140334	−0.0701670	−	0.997535i	$-0.522353\pi$
$457$	−3.00000	−0.140334	−0.0701670	+	0.997535i	$0.522353\pi$
$458$	−14.9706	−0.699528
$459$	2.41421	0.112686
$460$	0	0
$461$	−3.55635	−0.165636	−0.0828178	−	0.996565i	$-0.526392\pi$
$461$	−3.55635	−0.165636	−0.0828178	+	0.996565i	$0.526392\pi$
$462$	−2.58579	−0.120302
$463$	14.1421	0.657241	0.328620	−	0.944462i	$-0.393416\pi$
$463$	14.1421	0.657241	0.328620	+	0.944462i	$0.393416\pi$
$464$	−0.171573	−0.00796507
$465$	0	0
$466$	24.9706	1.15674
$467$	0.727922	0.0336842	0.0168421	−	0.999858i	$-0.494639\pi$
$467$	0.727922	0.0336842	0.0168421	+	0.999858i	$0.494639\pi$
$468$	−16.4853	−0.762032
$469$	21.0000	0.969690
$470$	0	0
$471$	−4.82843	−0.222482
$472$	−6.89949	−0.317575
$473$	−2.48528	−0.114273
$474$	−2.68629	−0.123385
$475$	0	0
$476$	4.41421	0.202325
$477$	−15.5147	−0.710370
$478$	6.89949	0.315576
$479$	−31.1127	−1.42158	−0.710788	−	0.703407i	$-0.751661\pi$
$479$	−31.1127	−1.42158	−0.710788	+	0.703407i	$0.751661\pi$
$480$	0	0
$481$	−49.4558	−2.25499
$482$	−8.97056	−0.408598
$483$	−1.38478	−0.0630095
$484$	−9.00000	−0.409091
$485$	0	0
$486$	10.3431	0.469175
$487$	37.7990	1.71284	0.856418	−	0.516283i	$-0.172685\pi$
$487$	37.7990	1.71284	0.856418	+	0.516283i	$0.172685\pi$
$488$	−14.2426	−0.644734
$489$	4.24264	0.191859
$490$	0	0
$491$	−33.5563	−1.51438	−0.757188	−	0.653197i	$-0.773428\pi$
$491$	−33.5563	−1.51438	−0.757188	+	0.653197i	$0.773428\pi$
$492$	1.75736	0.0792279
$493$	−0.171573	−0.00772725
$494$	5.82843	0.262233
$495$	0	0
$496$	6.24264	0.280303
$497$	−59.2132	−2.65608
$498$	6.00000	0.268866
$499$	−25.7574	−1.15306	−0.576529	−	0.817077i	$-0.695593\pi$
$499$	−25.7574	−1.15306	−0.576529	+	0.817077i	$0.695593\pi$
$500$	0	0
$501$	7.55635	0.337593
$502$	−3.55635	−0.158728
$503$	14.2721	0.636361	0.318180	−	0.948030i	$-0.396928\pi$
$503$	14.2721	0.636361	0.318180	+	0.948030i	$0.396928\pi$
$504$	12.4853	0.556139
$505$	0	0
$506$	1.07107	0.0476148
$507$	−8.68629	−0.385772
$508$	−2.48528	−0.110267
$509$	28.9706	1.28410	0.642049	−	0.766664i	$-0.278085\pi$
$509$	28.9706	1.28410	0.642049	+	0.766664i	$0.278085\pi$
$510$	0	0
$511$	50.6985	2.24277
$512$	−1.00000	−0.0441942
$513$	−2.41421	−0.106590
$514$	−20.7279	−0.914269
$515$	0	0
$516$	−0.727922	−0.0320450
$517$	0	0
$518$	37.4558	1.64572
$519$	−0.201010	−0.00882337
$520$	0	0
$521$	23.3137	1.02139	0.510696	−	0.859761i	$-0.329388\pi$
$521$	23.3137	1.02139	0.510696	+	0.859761i	$0.329388\pi$
$522$	−0.485281	−0.0212402
$523$	−2.27208	−0.0993510	−0.0496755	−	0.998765i	$-0.515819\pi$
$523$	−2.27208	−0.0993510	−0.0496755	+	0.998765i	$0.515819\pi$
$524$	−16.9706	−0.741362
$525$	0	0
$526$	26.9706	1.17597
$527$	6.24264	0.271934
$528$	0.585786	0.0254931
$529$	−22.4264	−0.975061
$530$	0	0
$531$	−19.5147	−0.846867
$532$	−4.41421	−0.191380
$533$	−24.7279	−1.07109
$534$	2.92893	0.126747
$535$	0	0
$536$	−4.75736	−0.205487
$537$	4.82843	0.208362
$538$	16.6274	0.716859
$539$	−17.6569	−0.760535
$540$	0	0
$541$	9.75736	0.419502	0.209751	−	0.977755i	$-0.432735\pi$
$541$	9.75736	0.419502	0.209751	+	0.977755i	$0.432735\pi$
$542$	27.2426	1.17017
$543$	3.51472	0.150831
$544$	−1.00000	−0.0428746
$545$	0	0
$546$	10.6569	0.456071
$547$	−17.3137	−0.740281	−0.370140	−	0.928976i	$-0.620690\pi$
$547$	−17.3137	−0.740281	−0.370140	+	0.928976i	$0.620690\pi$
$548$	13.0000	0.555332
$549$	−40.2843	−1.71929
$550$	0	0
$551$	0.171573	0.00730925
$552$	0.313708	0.0133523
$553$	−28.6274	−1.21736
$554$	0	0
$555$	0	0
$556$	−12.0000	−0.508913
$557$	16.0000	0.677942	0.338971	−	0.940797i	$-0.389921\pi$
$557$	16.0000	0.677942	0.338971	+	0.940797i	$0.389921\pi$
$558$	17.6569	0.747474
$559$	10.2426	0.433218
$560$	0	0
$561$	0.585786	0.0247319
$562$	−4.24264	−0.178965
$563$	4.97056	0.209484	0.104742	−	0.994499i	$-0.466598\pi$
$563$	4.97056	0.209484	0.104742	+	0.994499i	$0.466598\pi$
$564$	0	0
$565$	0	0
$566$	−32.1421	−1.35103
$567$	33.0416	1.38762
$568$	13.4142	0.562848
$569$	−28.2843	−1.18574	−0.592869	−	0.805299i	$-0.702005\pi$
$569$	−28.2843	−1.18574	−0.592869	+	0.805299i	$0.702005\pi$
$570$	0	0
$571$	−2.24264	−0.0938516	−0.0469258	−	0.998898i	$-0.514942\pi$
$571$	−2.24264	−0.0938516	−0.0469258	+	0.998898i	$0.514942\pi$
$572$	−8.24264	−0.344642
$573$	−5.20101	−0.217275
$574$	18.7279	0.781688
$575$	0	0
$576$	−2.82843	−0.117851
$577$	−20.3137	−0.845671	−0.422835	−	0.906207i	$-0.638965\pi$
$577$	−20.3137	−0.845671	−0.422835	+	0.906207i	$0.638965\pi$
$578$	16.0000	0.665512
$579$	0.142136	0.00590695
$580$	0	0
$581$	63.9411	2.65272
$582$	−0.142136	−0.00589171
$583$	−7.75736	−0.321277
$584$	−11.4853	−0.475264
$585$	0	0
$586$	5.48528	0.226595
$587$	30.2426	1.24825	0.624124	−	0.781326i	$-0.285456\pi$
$587$	30.2426	1.24825	0.624124	+	0.781326i	$0.285456\pi$
$588$	−5.17157	−0.213272
$589$	−6.24264	−0.257224
$590$	0	0
$591$	4.87006	0.200327
$592$	−8.48528	−0.348743
$593$	22.0000	0.903432	0.451716	−	0.892162i	$-0.350812\pi$
$593$	22.0000	0.903432	0.451716	+	0.892162i	$0.350812\pi$
$594$	3.41421	0.140087
$595$	0	0
$596$	17.6569	0.723253
$597$	−3.82843	−0.156687
$598$	−4.41421	−0.180511
$599$	−15.2132	−0.621595	−0.310797	−	0.950476i	$-0.600596\pi$
$599$	−15.2132	−0.621595	−0.310797	+	0.950476i	$0.600596\pi$
$600$	0	0
$601$	−20.2426	−0.825715	−0.412857	−	0.910796i	$-0.635469\pi$
$601$	−20.2426	−0.825715	−0.412857	+	0.910796i	$0.635469\pi$
$602$	−7.75736	−0.316166
$603$	−13.4558	−0.547964
$604$	−10.4853	−0.426640
$605$	0	0
$606$	5.41421	0.219937
$607$	3.17157	0.128730	0.0643651	−	0.997926i	$-0.479498\pi$
$607$	3.17157	0.128730	0.0643651	+	0.997926i	$0.479498\pi$
$608$	1.00000	0.0405554
$609$	0.313708	0.0127121
$610$	0	0
$611$	0	0
$612$	−2.82843	−0.114332
$613$	−11.6985	−0.472497	−0.236249	−	0.971693i	$-0.575918\pi$
$613$	−11.6985	−0.472497	−0.236249	+	0.971693i	$0.575918\pi$
$614$	17.6569	0.712573
$615$	0	0
$616$	6.24264	0.251523
$617$	−4.48528	−0.180571	−0.0902853	−	0.995916i	$-0.528778\pi$
$617$	−4.48528	−0.180571	−0.0902853	+	0.995916i	$0.528778\pi$
$618$	1.75736	0.0706914
$619$	7.75736	0.311795	0.155897	−	0.987773i	$-0.450173\pi$
$619$	7.75736	0.311795	0.155897	+	0.987773i	$0.450173\pi$
$620$	0	0
$621$	1.82843	0.0733723
$622$	4.75736	0.190753
$623$	31.2132	1.25053
$624$	−2.41421	−0.0966459
$625$	0	0
$626$	7.97056	0.318568
$627$	−0.585786	−0.0233941
$628$	11.6569	0.465159
$629$	−8.48528	−0.338330
$630$	0	0
$631$	−38.9706	−1.55139	−0.775697	−	0.631106i	$-0.782601\pi$
$631$	−38.9706	−1.55139	−0.775697	+	0.631106i	$0.782601\pi$
$632$	6.48528	0.257971
$633$	8.17157	0.324791
$634$	9.48528	0.376709
$635$	0	0
$636$	−2.27208	−0.0900938
$637$	72.7696	2.88323
$638$	−0.242641	−0.00960624
$639$	37.9411	1.50093
$640$	0	0
$641$	−18.0416	−0.712602	−0.356301	−	0.934371i	$-0.615962\pi$
$641$	−18.0416	−0.712602	−0.356301	+	0.934371i	$0.615962\pi$
$642$	8.17157	0.322506
$643$	−14.4853	−0.571244	−0.285622	−	0.958342i	$-0.592200\pi$
$643$	−14.4853	−0.571244	−0.285622	+	0.958342i	$0.592200\pi$
$644$	3.34315	0.131738
$645$	0	0
$646$	1.00000	0.0393445
$647$	−18.7574	−0.737428	−0.368714	−	0.929543i	$-0.620202\pi$
$647$	−18.7574	−0.737428	−0.368714	+	0.929543i	$0.620202\pi$
$648$	−7.48528	−0.294050
$649$	−9.75736	−0.383010
$650$	0	0
$651$	−11.4142	−0.447358
$652$	−10.2426	−0.401133
$653$	−28.9706	−1.13371	−0.566853	−	0.823819i	$-0.691839\pi$
$653$	−28.9706	−1.13371	−0.566853	+	0.823819i	$0.691839\pi$
$654$	−7.44365	−0.291070
$655$	0	0
$656$	−4.24264	−0.165647
$657$	−32.4853	−1.26737
$658$	0	0
$659$	18.8995	0.736220	0.368110	−	0.929782i	$-0.380005\pi$
$659$	18.8995	0.736220	0.368110	+	0.929782i	$0.380005\pi$
$660$	0	0
$661$	−18.4558	−0.717849	−0.358925	−	0.933367i	$-0.616856\pi$
$661$	−18.4558	−0.717849	−0.358925	+	0.933367i	$0.616856\pi$
$662$	19.2426	0.747886
$663$	−2.41421	−0.0937603
$664$	−14.4853	−0.562138
$665$	0	0
$666$	−24.0000	−0.929981
$667$	−0.129942	−0.00503139
$668$	−18.2426	−0.705829
$669$	6.28427	0.242964
$670$	0	0
$671$	−20.1421	−0.777579
$672$	1.82843	0.0705331
$673$	−12.0000	−0.462566	−0.231283	−	0.972887i	$-0.574292\pi$
$673$	−12.0000	−0.462566	−0.231283	+	0.972887i	$0.574292\pi$
$674$	−14.1005	−0.543131
$675$	0	0
$676$	20.9706	0.806560
$677$	−40.9411	−1.57350	−0.786748	−	0.617275i	$-0.788237\pi$
$677$	−40.9411	−1.57350	−0.786748	+	0.617275i	$0.788237\pi$
$678$	−4.24264	−0.162938
$679$	−1.51472	−0.0581296
$680$	0	0
$681$	6.94113	0.265985
$682$	8.82843	0.338058
$683$	12.0000	0.459167	0.229584	−	0.973289i	$-0.426264\pi$
$683$	12.0000	0.459167	0.229584	+	0.973289i	$0.426264\pi$
$684$	2.82843	0.108148
$685$	0	0
$686$	−24.2132	−0.924464
$687$	−6.20101	−0.236583
$688$	1.75736	0.0669987
$689$	31.9706	1.21798
$690$	0	0
$691$	22.4853	0.855380	0.427690	−	0.903925i	$-0.359327\pi$
$691$	22.4853	0.855380	0.427690	+	0.903925i	$0.359327\pi$
$692$	0.485281	0.0184476
$693$	17.6569	0.670728
$694$	5.51472	0.209336
$695$	0	0
$696$	−0.0710678	−0.00269382
$697$	−4.24264	−0.160701
$698$	−6.00000	−0.227103
$699$	10.3431	0.391214
$700$	0	0
$701$	16.9706	0.640969	0.320485	−	0.947254i	$-0.396154\pi$
$701$	16.9706	0.640969	0.320485	+	0.947254i	$0.396154\pi$
$702$	−14.0711	−0.531078
$703$	8.48528	0.320028
$704$	−1.41421	−0.0533002
$705$	0	0
$706$	2.51472	0.0946427
$707$	57.6985	2.16997
$708$	−2.85786	−0.107405
$709$	−34.0000	−1.27690	−0.638448	−	0.769665i	$-0.720423\pi$
$709$	−34.0000	−1.27690	−0.638448	+	0.769665i	$0.720423\pi$
$710$	0	0
$711$	18.3431	0.687922
$712$	−7.07107	−0.264999
$713$	4.72792	0.177062
$714$	1.82843	0.0684272
$715$	0	0
$716$	−11.6569	−0.435637
$717$	2.85786	0.106729
$718$	22.7574	0.849297
$719$	−5.10051	−0.190217	−0.0951084	−	0.995467i	$-0.530320\pi$
$719$	−5.10051	−0.190217	−0.0951084	+	0.995467i	$0.530320\pi$
$720$	0	0
$721$	18.7279	0.697464
$722$	−1.00000	−0.0372161
$723$	−3.71573	−0.138189
$724$	−8.48528	−0.315353
$725$	0	0
$726$	−3.72792	−0.138356
$727$	9.72792	0.360789	0.180394	−	0.983594i	$-0.442263\pi$
$727$	9.72792	0.360789	0.180394	+	0.983594i	$0.442263\pi$
$728$	−25.7279	−0.953540
$729$	−18.1716	−0.673021
$730$	0	0
$731$	1.75736	0.0649983
$732$	−5.89949	−0.218052
$733$	12.0000	0.443230	0.221615	−	0.975134i	$-0.428867\pi$
$733$	12.0000	0.443230	0.221615	+	0.975134i	$0.428867\pi$
$734$	25.4558	0.939592
$735$	0	0
$736$	−0.757359	−0.0279166
$737$	−6.72792	−0.247826
$738$	−12.0000	−0.441726
$739$	1.27208	0.0467941	0.0233971	−	0.999726i	$-0.492552\pi$
$739$	1.27208	0.0467941	0.0233971	+	0.999726i	$0.492552\pi$
$740$	0	0
$741$	2.41421	0.0886884
$742$	−24.2132	−0.888895
$743$	6.72792	0.246824	0.123412	−	0.992356i	$-0.460616\pi$
$743$	6.72792	0.246824	0.123412	+	0.992356i	$0.460616\pi$
$744$	2.58579	0.0947995
$745$	0	0
$746$	−9.00000	−0.329513
$747$	−40.9706	−1.49903
$748$	−1.41421	−0.0517088
$749$	87.0833	3.18195
$750$	0	0
$751$	−26.7279	−0.975316	−0.487658	−	0.873035i	$-0.662149\pi$
$751$	−26.7279	−0.975316	−0.487658	+	0.873035i	$0.662149\pi$
$752$	0	0
$753$	−1.47309	−0.0536823
$754$	1.00000	0.0364179
$755$	0	0
$756$	10.6569	0.387586
$757$	12.3431	0.448619	0.224310	−	0.974518i	$-0.427987\pi$
$757$	12.3431	0.448619	0.224310	+	0.974518i	$0.427987\pi$
$758$	−11.2426	−0.408351
$759$	0.443651	0.0161035
$760$	0	0
$761$	43.9706	1.59393	0.796966	−	0.604024i	$-0.206437\pi$
$761$	43.9706	1.59393	0.796966	+	0.604024i	$0.206437\pi$
$762$	−1.02944	−0.0372926
$763$	−79.3259	−2.87179
$764$	12.5563	0.454273
$765$	0	0
$766$	−12.2426	−0.442345
$767$	40.2132	1.45201
$768$	−0.414214	−0.0149466
$769$	−36.4558	−1.31463	−0.657316	−	0.753615i	$-0.728308\pi$
$769$	−36.4558	−1.31463	−0.657316	+	0.753615i	$0.728308\pi$
$770$	0	0
$771$	−8.58579	−0.309210
$772$	−0.343146	−0.0123501
$773$	13.9706	0.502486	0.251243	−	0.967924i	$-0.419161\pi$
$773$	13.9706	0.502486	0.251243	+	0.967924i	$0.419161\pi$
$774$	4.97056	0.178663
$775$	0	0
$776$	0.343146	0.0123182
$777$	15.5147	0.556587
$778$	−22.9289	−0.822042
$779$	4.24264	0.152008
$780$	0	0
$781$	18.9706	0.678820
$782$	−0.757359	−0.0270831
$783$	−0.414214	−0.0148028
$784$	12.4853	0.445903
$785$	0	0
$786$	−7.02944	−0.250732
$787$	−41.1838	−1.46804	−0.734021	−	0.679126i	$-0.762359\pi$
$787$	−41.1838	−1.46804	−0.734021	+	0.679126i	$0.762359\pi$
$788$	−11.7574	−0.418839
$789$	11.1716	0.397719
$790$	0	0
$791$	−45.2132	−1.60760
$792$	−4.00000	−0.142134
$793$	83.0122	2.94785
$794$	−24.0000	−0.851728
$795$	0	0
$796$	9.24264	0.327597
$797$	−47.4853	−1.68201	−0.841007	−	0.541023i	$-0.818037\pi$
$797$	−47.4853	−1.68201	−0.841007	+	0.541023i	$0.818037\pi$
$798$	−1.82843	−0.0647256
$799$	0	0
$800$	0	0
$801$	−20.0000	−0.706665
$802$	25.4142	0.897407
$803$	−16.2426	−0.573190
$804$	−1.97056	−0.0694964
$805$	0	0
$806$	−36.3848	−1.28160
$807$	6.88730	0.242444
$808$	−13.0711	−0.459839
$809$	−28.7990	−1.01252	−0.506259	−	0.862381i	$-0.668972\pi$
$809$	−28.7990	−1.01252	−0.506259	+	0.862381i	$0.668972\pi$
$810$	0	0
$811$	52.6985	1.85049	0.925247	−	0.379365i	$-0.123858\pi$
$811$	52.6985	1.85049	0.925247	+	0.379365i	$0.123858\pi$
$812$	−0.757359	−0.0265781
$813$	11.2843	0.395757
$814$	−12.0000	−0.420600
$815$	0	0
$816$	−0.414214	−0.0145004
$817$	−1.75736	−0.0614822
$818$	−25.2132	−0.881559
$819$	−72.7696	−2.54277
$820$	0	0
$821$	6.34315	0.221377	0.110689	−	0.993855i	$-0.464694\pi$
$821$	6.34315	0.221377	0.110689	+	0.993855i	$0.464694\pi$
$822$	5.38478	0.187816
$823$	−15.7279	−0.548241	−0.274120	−	0.961695i	$-0.588387\pi$
$823$	−15.7279	−0.548241	−0.274120	+	0.961695i	$0.588387\pi$
$824$	−4.24264	−0.147799
$825$	0	0
$826$	−30.4558	−1.05969
$827$	−20.6985	−0.719757	−0.359878	−	0.932999i	$-0.617182\pi$
$827$	−20.6985	−0.719757	−0.359878	+	0.932999i	$0.617182\pi$
$828$	−2.14214	−0.0744444
$829$	10.4558	0.363146	0.181573	−	0.983377i	$-0.441881\pi$
$829$	10.4558	0.363146	0.181573	+	0.983377i	$0.441881\pi$
$830$	0	0
$831$	0	0
$832$	5.82843	0.202064
$833$	12.4853	0.432589
$834$	−4.97056	−0.172117
$835$	0	0
$836$	1.41421	0.0489116
$837$	15.0711	0.520932
$838$	−19.4142	−0.670653
$839$	22.6274	0.781185	0.390593	−	0.920564i	$-0.372270\pi$
$839$	22.6274	0.781185	0.390593	+	0.920564i	$0.372270\pi$
$840$	0	0
$841$	−28.9706	−0.998985
$842$	−19.4853	−0.671507
$843$	−1.75736	−0.0605267
$844$	−19.7279	−0.679063
$845$	0	0
$846$	0	0
$847$	−39.7279	−1.36507
$848$	5.48528	0.188365
$849$	−13.3137	−0.456925
$850$	0	0
$851$	−6.42641	−0.220294
$852$	5.55635	0.190357
$853$	−27.9411	−0.956686	−0.478343	−	0.878173i	$-0.658762\pi$
$853$	−27.9411	−0.956686	−0.478343	+	0.878173i	$0.658762\pi$
$854$	−62.8701	−2.15137
$855$	0	0
$856$	−19.7279	−0.674286
$857$	0	0		−	1.00000i	$-0.5\pi$
$857$	0	0			1.00000i	$0.5\pi$
$858$	−3.41421	−0.116559
$859$	4.72792	0.161315	0.0806573	−	0.996742i	$-0.474298\pi$
$859$	4.72792	0.161315	0.0806573	+	0.996742i	$0.474298\pi$
$860$	0	0
$861$	7.75736	0.264370
$862$	6.38478	0.217466
$863$	0.727922	0.0247788	0.0123894	−	0.999923i	$-0.496056\pi$
$863$	0.727922	0.0247788	0.0123894	+	0.999923i	$0.496056\pi$
$864$	−2.41421	−0.0821332
$865$	0	0
$866$	9.55635	0.324738
$867$	6.62742	0.225079
$868$	27.5563	0.935323
$869$	9.17157	0.311124
$870$	0	0
$871$	27.7279	0.939525
$872$	17.9706	0.608560
$873$	0.970563	0.0328486
$874$	0.757359	0.0256181
$875$	0	0
$876$	−4.75736	−0.160736
$877$	−18.8579	−0.636785	−0.318392	−	0.947959i	$-0.603143\pi$
$877$	−18.8579	−0.636785	−0.318392	+	0.947959i	$0.603143\pi$
$878$	5.75736	0.194301
$879$	2.27208	0.0766353
$880$	0	0
$881$	−39.5980	−1.33409	−0.667045	−	0.745018i	$-0.732441\pi$
$881$	−39.5980	−1.33409	−0.667045	+	0.745018i	$0.732441\pi$
$882$	35.3137	1.18907
$883$	14.4853	0.487469	0.243734	−	0.969842i	$-0.421628\pi$
$883$	14.4853	0.487469	0.243734	+	0.969842i	$0.421628\pi$
$884$	5.82843	0.196031
$885$	0	0
$886$	−4.24264	−0.142534
$887$	−45.2132	−1.51811	−0.759055	−	0.651026i	$-0.774339\pi$
$887$	−45.2132	−1.51811	−0.759055	+	0.651026i	$0.774339\pi$
$888$	−3.51472	−0.117946
$889$	−10.9706	−0.367941
$890$	0	0
$891$	−10.5858	−0.354637
$892$	−15.1716	−0.507982
$893$	0	0
$894$	7.31371	0.244607
$895$	0	0
$896$	−4.41421	−0.147469
$897$	−1.82843	−0.0610494
$898$	−17.3137	−0.577766
$899$	−1.07107	−0.0357221
$900$	0	0
$901$	5.48528	0.182741
$902$	−6.00000	−0.199778
$903$	−3.21320	−0.106929
$904$	10.2426	0.340665
$905$	0	0
$906$	−4.34315	−0.144291
$907$	38.6985	1.28496	0.642481	−	0.766302i	$-0.277905\pi$
$907$	38.6985	1.28496	0.642481	+	0.766302i	$0.277905\pi$
$908$	−16.7574	−0.556113
$909$	−36.9706	−1.22624
$910$	0	0
$911$	40.2843	1.33468	0.667339	−	0.744754i	$-0.267433\pi$
$911$	40.2843	1.33468	0.667339	+	0.744754i	$0.267433\pi$
$912$	0.414214	0.0137160
$913$	−20.4853	−0.677964
$914$	3.00000	0.0992312
$915$	0	0
$916$	14.9706	0.494641
$917$	−74.9117	−2.47380
$918$	−2.41421	−0.0796809
$919$	−6.21320	−0.204955	−0.102477	−	0.994735i	$-0.532677\pi$
$919$	−6.21320	−0.204955	−0.102477	+	0.994735i	$0.532677\pi$
$920$	0	0
$921$	7.31371	0.240995
$922$	3.55635	0.117122
$923$	−78.1838	−2.57345
$924$	2.58579	0.0850661
$925$	0	0
$926$	−14.1421	−0.464739
$927$	−12.0000	−0.394132
$928$	0.171573	0.00563216
$929$	−12.1716	−0.399336	−0.199668	−	0.979864i	$-0.563986\pi$
$929$	−12.1716	−0.399336	−0.199668	+	0.979864i	$0.563986\pi$
$930$	0	0
$931$	−12.4853	−0.409189
$932$	−24.9706	−0.817938
$933$	1.97056	0.0645133
$934$	−0.727922	−0.0238183
$935$	0	0
$936$	16.4853	0.538838
$937$	27.0000	0.882052	0.441026	−	0.897494i	$-0.354615\pi$
$937$	27.0000	0.882052	0.441026	+	0.897494i	$0.354615\pi$
$938$	−21.0000	−0.685674
$939$	3.30152	0.107741
$940$	0	0
$941$	53.1421	1.73238	0.866192	−	0.499711i	$-0.166561\pi$
$941$	53.1421	1.73238	0.866192	+	0.499711i	$0.166561\pi$
$942$	4.82843	0.157319
$943$	−3.21320	−0.104636
$944$	6.89949	0.224559
$945$	0	0
$946$	2.48528	0.0808035
$947$	−12.0000	−0.389948	−0.194974	−	0.980808i	$-0.562462\pi$
$947$	−12.0000	−0.389948	−0.194974	+	0.980808i	$0.562462\pi$
$948$	2.68629	0.0872467
$949$	66.9411	2.17300
$950$	0	0
$951$	3.92893	0.127404
$952$	−4.41421	−0.143065
$953$	18.7279	0.606657	0.303328	−	0.952886i	$-0.401902\pi$
$953$	18.7279	0.606657	0.303328	+	0.952886i	$0.401902\pi$
$954$	15.5147	0.502308
$955$	0	0
$956$	−6.89949	−0.223146
$957$	−0.100505	−0.00324887
$958$	31.1127	1.00521
$959$	57.3848	1.85305
$960$	0	0
$961$	7.97056	0.257115
$962$	49.4558	1.59452
$963$	−55.7990	−1.79810
$964$	8.97056	0.288922
$965$	0	0
$966$	1.38478	0.0445544
$967$	43.1127	1.38641	0.693205	−	0.720740i	$-0.256198\pi$
$967$	43.1127	1.38641	0.693205	+	0.720740i	$0.256198\pi$
$968$	9.00000	0.289271
$969$	0.414214	0.0133065
$970$	0	0
$971$	41.6569	1.33683	0.668416	−	0.743788i	$-0.266973\pi$
$971$	41.6569	1.33683	0.668416	+	0.743788i	$0.266973\pi$
$972$	−10.3431	−0.331757
$973$	−52.9706	−1.69816
$974$	−37.7990	−1.21116
$975$	0	0
$976$	14.2426	0.455896
$977$	0.242641	0.00776276	0.00388138	−	0.999992i	$-0.498765\pi$
$977$	0.242641	0.00776276	0.00388138	+	0.999992i	$0.498765\pi$
$978$	−4.24264	−0.135665
$979$	−10.0000	−0.319601
$980$	0	0
$981$	50.8284	1.62283
$982$	33.5563	1.07083
$983$	−51.4558	−1.64119	−0.820593	−	0.571513i	$-0.806357\pi$
$983$	−51.4558	−1.64119	−0.820593	+	0.571513i	$0.806357\pi$
$984$	−1.75736	−0.0560226
$985$	0	0
$986$	0.171573	0.00546399
$987$	0	0
$988$	−5.82843	−0.185427
$989$	1.33095	0.0423218
$990$	0	0
$991$	13.7574	0.437017	0.218508	−	0.975835i	$-0.429881\pi$
$991$	13.7574	0.437017	0.218508	+	0.975835i	$0.429881\pi$
$992$	−6.24264	−0.198204
$993$	7.97056	0.252938
$994$	59.2132	1.87813
$995$	0	0
$996$	−6.00000	−0.190117
$997$	48.7279	1.54323	0.771614	−	0.636091i	$-0.219450\pi$
$997$	48.7279	1.54323	0.771614	+	0.636091i	$0.219450\pi$
$998$	25.7574	0.815335
$999$	−20.4853	−0.648126

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	950.2.a.f.1.1		2
3.2	odd	2		8550.2.a.cb.1.2		2
4.3	odd	2		7600.2.a.v.1.2		2
5.2	odd	4		190.2.b.a.39.2	✓	4
5.3	odd	4		190.2.b.a.39.3	yes	4
5.4	even	2		950.2.a.g.1.2		2
15.2	even	4		1710.2.d.c.1369.3		4
15.8	even	4		1710.2.d.c.1369.1		4
15.14	odd	2		8550.2.a.bn.1.1		2
20.3	even	4		1520.2.d.e.609.3		4
20.7	even	4		1520.2.d.e.609.2		4
20.19	odd	2		7600.2.a.bg.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
190.2.b.a.39.2	✓	4	5.2	odd	4
190.2.b.a.39.3	yes	4	5.3	odd	4
950.2.a.f.1.1		2	1.1	even	1	trivial
950.2.a.g.1.2		2	5.4	even	2
1520.2.d.e.609.2		4	20.7	even	4
1520.2.d.e.609.3		4	20.3	even	4
1710.2.d.c.1369.1		4	15.8	even	4
1710.2.d.c.1369.3		4	15.2	even	4
7600.2.a.v.1.2		2	4.3	odd	2
7600.2.a.bg.1.1		2	20.19	odd	2
8550.2.a.bn.1.1		2	15.14	odd	2
8550.2.a.cb.1.2		2	3.2	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:	\( N \)	\(=\)	\( 950 = 2 \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	950.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -0.414214 q^{3} +1.00000 q^{4} +0.414214 q^{6} +4.41421 q^{7} -1.00000 q^{8} -2.82843 q^{9} -1.41421 q^{11} -0.414214 q^{12} +5.82843 q^{13} -4.41421 q^{14} +1.00000 q^{16} +1.00000 q^{17} +2.82843 q^{18} -1.00000 q^{19} -1.82843 q^{21} +1.41421 q^{22} +0.757359 q^{23} +0.414214 q^{24} -5.82843 q^{26} +2.41421 q^{27} +4.41421 q^{28} -0.171573 q^{29} +6.24264 q^{31} -1.00000 q^{32} +0.585786 q^{33} -1.00000 q^{34} -2.82843 q^{36} -8.48528 q^{37} +1.00000 q^{38} -2.41421 q^{39} -4.24264 q^{41} +1.82843 q^{42} +1.75736 q^{43} -1.41421 q^{44} -0.757359 q^{46} -0.414214 q^{48} +12.4853 q^{49} -0.414214 q^{51} +5.82843 q^{52} +5.48528 q^{53} -2.41421 q^{54} -4.41421 q^{56} +0.414214 q^{57} +0.171573 q^{58} +6.89949 q^{59} +14.2426 q^{61} -6.24264 q^{62} -12.4853 q^{63} +1.00000 q^{64} -0.585786 q^{66} +4.75736 q^{67} +1.00000 q^{68} -0.313708 q^{69} -13.4142 q^{71} +2.82843 q^{72} +11.4853 q^{73} +8.48528 q^{74} -1.00000 q^{76} -6.24264 q^{77} +2.41421 q^{78} -6.48528 q^{79} +7.48528 q^{81} +4.24264 q^{82} +14.4853 q^{83} -1.82843 q^{84} -1.75736 q^{86} +0.0710678 q^{87} +1.41421 q^{88} +7.07107 q^{89} +25.7279 q^{91} +0.757359 q^{92} -2.58579 q^{93} +0.414214 q^{96} -0.343146 q^{97} -12.4853 q^{98} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} + 6 q^{7} - 2 q^{8} + 2 q^{12} + 6 q^{13} - 6 q^{14} + 2 q^{16} + 2 q^{17} - 2 q^{19} + 2 q^{21} + 10 q^{23} - 2 q^{24} - 6 q^{26} + 2 q^{27} + 6 q^{28} - 6 q^{29}+ \cdots + 8 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^6 + 6 * q^7 - 2 * q^8 + 2 * q^12 + 6 * q^13 - 6 * q^14 + 2 * q^16 + 2 * q^17 - 2 * q^19 + 2 * q^21 + 10 * q^23 - 2 * q^24 - 6 * q^26 + 2 * q^27 + 6 * q^28 - 6 * q^29 + 4 * q^31 - 2 * q^32 + 4 * q^33 - 2 * q^34 + 2 * q^38 - 2 * q^39 - 2 * q^42 + 12 * q^43 - 10 * q^46 + 2 * q^48 + 8 * q^49 + 2 * q^51 + 6 * q^52 - 6 * q^53 - 2 * q^54 - 6 * q^56 - 2 * q^57 + 6 * q^58 - 6 * q^59 + 20 * q^61 - 4 * q^62 - 8 * q^63 + 2 * q^64 - 4 * q^66 + 18 * q^67 + 2 * q^68 + 22 * q^69 - 24 * q^71 + 6 * q^73 - 2 * q^76 - 4 * q^77 + 2 * q^78 + 4 * q^79 - 2 * q^81 + 12 * q^83 + 2 * q^84 - 12 * q^86 - 14 * q^87 + 26 * q^91 + 10 * q^92 - 8 * q^93 - 2 * q^96 - 12 * q^97 - 8 * q^98 + 8 * q^99

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