LMFDB - Embedded newform 950.2.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [950,2,Mod(1,950)]

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

lf = mfeigenbasis(mf)

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")

//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);

Level:	$ N $	$=$	$ 950 = 2 \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	950.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:	$7.58578819202$
Analytic rank:	$1$
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)

$f(q)$	$=$	$q+1.00000 q^{2} -2.41421 q^{3} +1.00000 q^{4} -2.41421 q^{6} -1.58579 q^{7} +1.00000 q^{8} +2.82843 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -2.41421 q^{3} +1.00000 q^{4} -2.41421 q^{6} -1.58579 q^{7} +1.00000 q^{8} +2.82843 q^{9} +1.41421 q^{11} -2.41421 q^{12} -0.171573 q^{13} -1.58579 q^{14} +1.00000 q^{16} -1.00000 q^{17} +2.82843 q^{18} -1.00000 q^{19} +3.82843 q^{21} +1.41421 q^{22} -9.24264 q^{23} -2.41421 q^{24} -0.171573 q^{26} +0.414214 q^{27} -1.58579 q^{28} -5.82843 q^{29} -2.24264 q^{31} +1.00000 q^{32} -3.41421 q^{33} -1.00000 q^{34} +2.82843 q^{36} -8.48528 q^{37} -1.00000 q^{38} +0.414214 q^{39} +4.24264 q^{41} +3.82843 q^{42} -10.2426 q^{43} +1.41421 q^{44} -9.24264 q^{46} -2.41421 q^{48} -4.48528 q^{49} +2.41421 q^{51} -0.171573 q^{52} +11.4853 q^{53} +0.414214 q^{54} -1.58579 q^{56} +2.41421 q^{57} -5.82843 q^{58} -12.8995 q^{59} +5.75736 q^{61} -2.24264 q^{62} -4.48528 q^{63} +1.00000 q^{64} -3.41421 q^{66} -13.2426 q^{67} -1.00000 q^{68} +22.3137 q^{69} -10.5858 q^{71} +2.82843 q^{72} +5.48528 q^{73} -8.48528 q^{74} -1.00000 q^{76} -2.24264 q^{77} +0.414214 q^{78} +10.4853 q^{79} -9.48528 q^{81} +4.24264 q^{82} +2.48528 q^{83} +3.82843 q^{84} -10.2426 q^{86} +14.0711 q^{87} +1.41421 q^{88} -7.07107 q^{89} +0.272078 q^{91} -9.24264 q^{92} +5.41421 q^{93} -2.41421 q^{96} +11.6569 q^{97} -4.48528 q^{98} +4.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{6} - 6 q^{7} + 2 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 - 2 * q^6 - 6 * q^7 + 2 * q^8	$ 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}+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

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−2.41421	−1.39385	−0.696923	−	0.717146i	$-0.745448\pi$
$3$	−2.41421	−1.39385	−0.696923	+	0.717146i	$0.745448\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−2.41421	−0.985599
$7$	−1.58579	−0.599371	−0.299685	−	0.954038i	$-0.596882\pi$
$7$	−1.58579	−0.599371	−0.299685	+	0.954038i	$0.596882\pi$
$8$	1.00000	0.353553
$9$	2.82843	0.942809
$10$	0	0
$11$	1.41421	0.426401	0.213201	−	0.977008i	$-0.431611\pi$
$11$	1.41421	0.426401	0.213201	+	0.977008i	$0.431611\pi$
$12$	−2.41421	−0.696923
$13$	−0.171573	−0.0475858	−0.0237929	−	0.999717i	$-0.507574\pi$
$13$	−0.171573	−0.0475858	−0.0237929	+	0.999717i	$0.507574\pi$
$14$	−1.58579	−0.423819
$15$	0	0
$16$	1.00000	0.250000
$17$	−1.00000	−0.242536	−0.121268	−	0.992620i	$-0.538696\pi$
$17$	−1.00000	−0.242536	−0.121268	+	0.992620i	$0.538696\pi$
$18$	2.82843	0.666667
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	3.82843	0.835431
$22$	1.41421	0.301511
$23$	−9.24264	−1.92722	−0.963612	−	0.267305i	$-0.913867\pi$
$23$	−9.24264	−1.92722	−0.963612	+	0.267305i	$0.913867\pi$
$24$	−2.41421	−0.492799
$25$	0	0
$26$	−0.171573	−0.0336482
$27$	0.414214	0.0797154
$28$	−1.58579	−0.299685
$29$	−5.82843	−1.08231	−0.541156	−	0.840922i	$-0.682013\pi$
$29$	−5.82843	−1.08231	−0.541156	+	0.840922i	$0.682013\pi$
$30$	0	0
$31$	−2.24264	−0.402790	−0.201395	−	0.979510i	$-0.564548\pi$
$31$	−2.24264	−0.402790	−0.201395	+	0.979510i	$0.564548\pi$
$32$	1.00000	0.176777
$33$	−3.41421	−0.594338
$34$	−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
$39$	0.414214	0.0663273
$40$	0	0
$41$	4.24264	0.662589	0.331295	−	0.943527i	$-0.392515\pi$
$41$	4.24264	0.662589	0.331295	+	0.943527i	$0.392515\pi$
$42$	3.82843	0.590739
$43$	−10.2426	−1.56199	−0.780994	−	0.624538i	$-0.785287\pi$
$43$	−10.2426	−1.56199	−0.780994	+	0.624538i	$0.785287\pi$
$44$	1.41421	0.213201
$45$	0	0
$46$	−9.24264	−1.36275
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	−2.41421	−0.348462
$49$	−4.48528	−0.640754
$50$	0	0
$51$	2.41421	0.338058
$52$	−0.171573	−0.0237929
$53$	11.4853	1.57762	0.788812	−	0.614634i	$-0.210696\pi$
$53$	11.4853	1.57762	0.788812	+	0.614634i	$0.210696\pi$
$54$	0.414214	0.0563673
$55$	0	0
$56$	−1.58579	−0.211910
$57$	2.41421	0.319770
$58$	−5.82843	−0.765310
$59$	−12.8995	−1.67937	−0.839686	−	0.543073i	$-0.817261\pi$
$59$	−12.8995	−1.67937	−0.839686	+	0.543073i	$0.817261\pi$
$60$	0	0
$61$	5.75736	0.737154	0.368577	−	0.929597i	$-0.379845\pi$
$61$	5.75736	0.737154	0.368577	+	0.929597i	$0.379845\pi$
$62$	−2.24264	−0.284816
$63$	−4.48528	−0.565092
$64$	1.00000	0.125000
$65$	0	0
$66$	−3.41421	−0.420261
$67$	−13.2426	−1.61785	−0.808923	−	0.587915i	$-0.799949\pi$
$67$	−13.2426	−1.61785	−0.808923	+	0.587915i	$0.799949\pi$
$68$	−1.00000	−0.121268
$69$	22.3137	2.68625
$70$	0	0
$71$	−10.5858	−1.25630	−0.628151	−	0.778092i	$-0.716188\pi$
$71$	−10.5858	−1.25630	−0.628151	+	0.778092i	$0.716188\pi$
$72$	2.82843	0.333333
$73$	5.48528	0.642004	0.321002	−	0.947079i	$-0.395980\pi$
$73$	5.48528	0.642004	0.321002	+	0.947079i	$0.395980\pi$
$74$	−8.48528	−0.986394
$75$	0	0
$76$	−1.00000	−0.114708
$77$	−2.24264	−0.255573
$78$	0.414214	0.0469005
$79$	10.4853	1.17969	0.589843	−	0.807518i	$-0.299190\pi$
$79$	10.4853	1.17969	0.589843	+	0.807518i	$0.299190\pi$
$80$	0	0
$81$	−9.48528	−1.05392
$82$	4.24264	0.468521
$83$	2.48528	0.272795	0.136398	−	0.990654i	$-0.456448\pi$
$83$	2.48528	0.272795	0.136398	+	0.990654i	$0.456448\pi$
$84$	3.82843	0.417716
$85$	0	0
$86$	−10.2426	−1.10449
$87$	14.0711	1.50858
$88$	1.41421	0.150756
$89$	−7.07107	−0.749532	−0.374766	−	0.927119i	$-0.622277\pi$
$89$	−7.07107	−0.749532	−0.374766	+	0.927119i	$0.622277\pi$
$90$	0	0
$91$	0.272078	0.0285215
$92$	−9.24264	−0.963612
$93$	5.41421	0.561428
$94$	0	0
$95$	0	0
$96$	−2.41421	−0.246400
$97$	11.6569	1.18357	0.591787	−	0.806094i	$-0.298423\pi$
$97$	11.6569	1.18357	0.591787	+	0.806094i	$0.298423\pi$
$98$	−4.48528	−0.453082
$99$	4.00000	0.402015
$100$	0	0
$101$	−1.07107	−0.106575	−0.0532876	−	0.998579i	$-0.516970\pi$
$101$	−1.07107	−0.106575	−0.0532876	+	0.998579i	$0.516970\pi$
$102$	2.41421	0.239043
$103$	4.24264	0.418040	0.209020	−	0.977911i	$-0.432973\pi$
$103$	4.24264	0.418040	0.209020	+	0.977911i	$0.432973\pi$
$104$	−0.171573	−0.0168241
$105$	0	0
$106$	11.4853	1.11555
$107$	5.72792	0.553739	0.276870	−	0.960908i	$-0.410703\pi$
$107$	5.72792	0.553739	0.276870	+	0.960908i	$0.410703\pi$
$108$	0.414214	0.0398577
$109$	15.9706	1.52970	0.764851	−	0.644207i	$-0.222812\pi$
$109$	15.9706	1.52970	0.764851	+	0.644207i	$0.222812\pi$
$110$	0	0
$111$	20.4853	1.94438
$112$	−1.58579	−0.149843
$113$	1.75736	0.165318	0.0826592	−	0.996578i	$-0.473659\pi$
$113$	1.75736	0.165318	0.0826592	+	0.996578i	$0.473659\pi$
$114$	2.41421	0.226112
$115$	0	0
$116$	−5.82843	−0.541156
$117$	−0.485281	−0.0448643
$118$	−12.8995	−1.18749
$119$	1.58579	0.145369
$120$	0	0
$121$	−9.00000	−0.818182
$122$	5.75736	0.521247
$123$	−10.2426	−0.923548
$124$	−2.24264	−0.201395
$125$	0	0
$126$	−4.48528	−0.399581
$127$	−14.4853	−1.28536	−0.642680	−	0.766134i	$-0.722178\pi$
$127$	−14.4853	−1.28536	−0.642680	+	0.766134i	$0.722178\pi$
$128$	1.00000	0.0883883
$129$	24.7279	2.17717
$130$	0	0
$131$	16.9706	1.48272	0.741362	−	0.671105i	$-0.234180\pi$
$131$	16.9706	1.48272	0.741362	+	0.671105i	$0.234180\pi$
$132$	−3.41421	−0.297169
$133$	1.58579	0.137505
$134$	−13.2426	−1.14399
$135$	0	0
$136$	−1.00000	−0.0857493
$137$	−13.0000	−1.11066	−0.555332	−	0.831628i	$-0.687409\pi$
$137$	−13.0000	−1.11066	−0.555332	+	0.831628i	$0.687409\pi$
$138$	22.3137	1.89947
$139$	−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
$142$	−10.5858	−0.888339
$143$	−0.242641	−0.0202906
$144$	2.82843	0.235702
$145$	0	0
$146$	5.48528	0.453965
$147$	10.8284	0.893114
$148$	−8.48528	−0.697486
$149$	6.34315	0.519651	0.259825	−	0.965656i	$-0.416335\pi$
$149$	6.34315	0.519651	0.259825	+	0.965656i	$0.416335\pi$
$150$	0	0
$151$	6.48528	0.527765	0.263882	−	0.964555i	$-0.414997\pi$
$151$	6.48528	0.527765	0.263882	+	0.964555i	$0.414997\pi$
$152$	−1.00000	−0.0811107
$153$	−2.82843	−0.228665
$154$	−2.24264	−0.180717
$155$	0	0
$156$	0.414214	0.0331636
$157$	−0.343146	−0.0273860	−0.0136930	−	0.999906i	$-0.504359\pi$
$157$	−0.343146	−0.0273860	−0.0136930	+	0.999906i	$0.504359\pi$
$158$	10.4853	0.834164
$159$	−27.7279	−2.19897
$160$	0	0
$161$	14.6569	1.15512
$162$	−9.48528	−0.745234
$163$	1.75736	0.137647	0.0688235	−	0.997629i	$-0.478075\pi$
$163$	1.75736	0.137647	0.0688235	+	0.997629i	$0.478075\pi$
$164$	4.24264	0.331295
$165$	0	0
$166$	2.48528	0.192895
$167$	9.75736	0.755047	0.377524	−	0.926000i	$-0.376776\pi$
$167$	9.75736	0.755047	0.377524	+	0.926000i	$0.376776\pi$
$168$	3.82843	0.295370
$169$	−12.9706	−0.997736
$170$	0	0
$171$	−2.82843	−0.216295
$172$	−10.2426	−0.780994
$173$	16.4853	1.25335	0.626676	−	0.779280i	$-0.284415\pi$
$173$	16.4853	1.25335	0.626676	+	0.779280i	$0.284415\pi$
$174$	14.0711	1.06672
$175$	0	0
$176$	1.41421	0.106600
$177$	31.1421	2.34079
$178$	−7.07107	−0.529999
$179$	−0.343146	−0.0256479	−0.0128240	−	0.999918i	$-0.504082\pi$
$179$	−0.343146	−0.0256479	−0.0128240	+	0.999918i	$0.504082\pi$
$180$	0	0
$181$	8.48528	0.630706	0.315353	−	0.948974i	$-0.397877\pi$
$181$	8.48528	0.630706	0.315353	+	0.948974i	$0.397877\pi$
$182$	0.272078	0.0201678
$183$	−13.8995	−1.02748
$184$	−9.24264	−0.681377
$185$	0	0
$186$	5.41421	0.396989
$187$	−1.41421	−0.103418
$188$	0	0
$189$	−0.656854	−0.0477791
$190$	0	0
$191$	−18.5563	−1.34269	−0.671345	−	0.741145i	$-0.734283\pi$
$191$	−18.5563	−1.34269	−0.671345	+	0.741145i	$0.734283\pi$
$192$	−2.41421	−0.174231
$193$	11.6569	0.839079	0.419539	−	0.907737i	$-0.362192\pi$
$193$	11.6569	0.839079	0.419539	+	0.907737i	$0.362192\pi$
$194$	11.6569	0.836913
$195$	0	0
$196$	−4.48528	−0.320377
$197$	20.2426	1.44223	0.721114	−	0.692816i	$-0.243630\pi$
$197$	20.2426	1.44223	0.721114	+	0.692816i	$0.243630\pi$
$198$	4.00000	0.284268
$199$	0.757359	0.0536878	0.0268439	−	0.999640i	$-0.491454\pi$
$199$	0.757359	0.0536878	0.0268439	+	0.999640i	$0.491454\pi$
$200$	0	0
$201$	31.9706	2.25503
$202$	−1.07107	−0.0753601
$203$	9.24264	0.648706
$204$	2.41421	0.169029
$205$	0	0
$206$	4.24264	0.295599
$207$	−26.1421	−1.81700
$208$	−0.171573	−0.0118964
$209$	−1.41421	−0.0978232
$210$	0	0
$211$	5.72792	0.394326	0.197163	−	0.980371i	$-0.436827\pi$
$211$	5.72792	0.394326	0.197163	+	0.980371i	$0.436827\pi$
$212$	11.4853	0.788812
$213$	25.5563	1.75109
$214$	5.72792	0.391553
$215$	0	0
$216$	0.414214	0.0281837
$217$	3.55635	0.241421
$218$	15.9706	1.08166
$219$	−13.2426	−0.894855
$220$	0	0
$221$	0.171573	0.0115412
$222$	20.4853	1.37488
$223$	20.8284	1.39477	0.697387	−	0.716694i	$-0.254346\pi$
$223$	20.8284	1.39477	0.697387	+	0.716694i	$0.254346\pi$
$224$	−1.58579	−0.105955
$225$	0	0
$226$	1.75736	0.116898
$227$	25.2426	1.67541	0.837706	−	0.546121i	$-0.183896\pi$
$227$	25.2426	1.67541	0.837706	+	0.546121i	$0.183896\pi$
$228$	2.41421	0.159885
$229$	−18.9706	−1.25361	−0.626805	−	0.779176i	$-0.715638\pi$
$229$	−18.9706	−1.25361	−0.626805	+	0.779176i	$0.715638\pi$
$230$	0	0
$231$	5.41421	0.356229
$232$	−5.82843	−0.382655
$233$	−8.97056	−0.587681	−0.293841	−	0.955854i	$-0.594933\pi$
$233$	−8.97056	−0.587681	−0.293841	+	0.955854i	$0.594933\pi$
$234$	−0.485281	−0.0317238
$235$	0	0
$236$	−12.8995	−0.839686
$237$	−25.3137	−1.64430
$238$	1.58579	0.102791
$239$	12.8995	0.834399	0.417199	−	0.908815i	$-0.363012\pi$
$239$	12.8995	0.834399	0.417199	+	0.908815i	$0.363012\pi$
$240$	0	0
$241$	−24.9706	−1.60850	−0.804248	−	0.594294i	$-0.797431\pi$
$241$	−24.9706	−1.60850	−0.804248	+	0.594294i	$0.797431\pi$
$242$	−9.00000	−0.578542
$243$	21.6569	1.38929
$244$	5.75736	0.368577
$245$	0	0
$246$	−10.2426	−0.653047
$247$	0.171573	0.0109169
$248$	−2.24264	−0.142408
$249$	−6.00000	−0.380235
$250$	0	0
$251$	−27.5563	−1.73934	−0.869671	−	0.493632i	$-0.835669\pi$
$251$	−27.5563	−1.73934	−0.869671	+	0.493632i	$0.835669\pi$
$252$	−4.48528	−0.282546
$253$	−13.0711	−0.821771
$254$	−14.4853	−0.908887
$255$	0	0
$256$	1.00000	0.0625000
$257$	4.72792	0.294920	0.147460	−	0.989068i	$-0.452890\pi$
$257$	4.72792	0.294920	0.147460	+	0.989068i	$0.452890\pi$
$258$	24.7279	1.53949
$259$	13.4558	0.836105
$260$	0	0
$261$	−16.4853	−1.02041
$262$	16.9706	1.04844
$263$	−6.97056	−0.429823	−0.214912	−	0.976633i	$-0.568946\pi$
$263$	−6.97056	−0.429823	−0.214912	+	0.976633i	$0.568946\pi$
$264$	−3.41421	−0.210130
$265$	0	0
$266$	1.58579	0.0972308
$267$	17.0711	1.04473
$268$	−13.2426	−0.808923
$269$	28.6274	1.74544	0.872722	−	0.488217i	$-0.162353\pi$
$269$	28.6274	1.74544	0.872722	+	0.488217i	$0.162353\pi$
$270$	0	0
$271$	−18.7574	−1.13943	−0.569714	−	0.821843i	$-0.692946\pi$
$271$	−18.7574	−1.13943	−0.569714	+	0.821843i	$0.692946\pi$
$272$	−1.00000	−0.0606339
$273$	−0.656854	−0.0397546
$274$	−13.0000	−0.785359
$275$	0	0
$276$	22.3137	1.34313
$277$	0	0		−	1.00000i	$-0.5\pi$
$277$	0	0			1.00000i	$0.5\pi$
$278$	−12.0000	−0.719712
$279$	−6.34315	−0.379754
$280$	0	0
$281$	−4.24264	−0.253095	−0.126547	−	0.991961i	$-0.540390\pi$
$281$	−4.24264	−0.253095	−0.126547	+	0.991961i	$0.540390\pi$
$282$	0	0
$283$	−3.85786	−0.229326	−0.114663	−	0.993404i	$-0.536579\pi$
$283$	−3.85786	−0.229326	−0.114663	+	0.993404i	$0.536579\pi$
$284$	−10.5858	−0.628151
$285$	0	0
$286$	−0.242641	−0.0143476
$287$	−6.72792	−0.397137
$288$	2.82843	0.166667
$289$	−16.0000	−0.941176
$290$	0	0
$291$	−28.1421	−1.64972
$292$	5.48528	0.321002
$293$	−11.4853	−0.670977	−0.335489	−	0.942044i	$-0.608901\pi$
$293$	−11.4853	−0.670977	−0.335489	+	0.942044i	$0.608901\pi$
$294$	10.8284	0.631527
$295$	0	0
$296$	−8.48528	−0.493197
$297$	0.585786	0.0339908
$298$	6.34315	0.367449
$299$	1.58579	0.0917084
$300$	0	0
$301$	16.2426	0.936210
$302$	6.48528	0.373186
$303$	2.58579	0.148550
$304$	−1.00000	−0.0573539
$305$	0	0
$306$	−2.82843	−0.161690
$307$	6.34315	0.362022	0.181011	−	0.983481i	$-0.442063\pi$
$307$	6.34315	0.362022	0.181011	+	0.983481i	$0.442063\pi$
$308$	−2.24264	−0.127786
$309$	−10.2426	−0.582683
$310$	0	0
$311$	−13.2426	−0.750921	−0.375461	−	0.926838i	$-0.622515\pi$
$311$	−13.2426	−0.750921	−0.375461	+	0.926838i	$0.622515\pi$
$312$	0.414214	0.0234502
$313$	−25.9706	−1.46794	−0.733971	−	0.679180i	$-0.762335\pi$
$313$	−25.9706	−1.46794	−0.733971	+	0.679180i	$0.762335\pi$
$314$	−0.343146	−0.0193648
$315$	0	0
$316$	10.4853	0.589843
$317$	−7.48528	−0.420415	−0.210208	−	0.977657i	$-0.567414\pi$
$317$	−7.48528	−0.420415	−0.210208	+	0.977657i	$0.567414\pi$
$318$	−27.7279	−1.55490
$319$	−8.24264	−0.461499
$320$	0	0
$321$	−13.8284	−0.771828
$322$	14.6569	0.816795
$323$	1.00000	0.0556415
$324$	−9.48528	−0.526960
$325$	0	0
$326$	1.75736	0.0973311
$327$	−38.5563	−2.13217
$328$	4.24264	0.234261
$329$	0	0
$330$	0	0
$331$	−10.7574	−0.591278	−0.295639	−	0.955300i	$-0.595533\pi$
$331$	−10.7574	−0.591278	−0.295639	+	0.955300i	$0.595533\pi$
$332$	2.48528	0.136398
$333$	−24.0000	−1.31519
$334$	9.75736	0.533899
$335$	0	0
$336$	3.82843	0.208858
$337$	−33.8995	−1.84662	−0.923312	−	0.384052i	$-0.874528\pi$
$337$	−33.8995	−1.84662	−0.923312	+	0.384052i	$0.874528\pi$
$338$	−12.9706	−0.705506
$339$	−4.24264	−0.230429
$340$	0	0
$341$	−3.17157	−0.171750
$342$	−2.82843	−0.152944
$343$	18.2132	0.983421
$344$	−10.2426	−0.552246
$345$	0	0
$346$	16.4853	0.886254
$347$	22.4853	1.20707	0.603537	−	0.797335i	$-0.293758\pi$
$347$	22.4853	1.20707	0.603537	+	0.797335i	$0.293758\pi$
$348$	14.0711	0.754288
$349$	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
$351$	−0.0710678	−0.00379332
$352$	1.41421	0.0753778
$353$	19.4853	1.03710	0.518548	−	0.855048i	$-0.326473\pi$
$353$	19.4853	1.03710	0.518548	+	0.855048i	$0.326473\pi$
$354$	31.1421	1.65519
$355$	0	0
$356$	−7.07107	−0.374766
$357$	−3.82843	−0.202622
$358$	−0.343146	−0.0181358
$359$	−31.2426	−1.64892	−0.824462	−	0.565918i	$-0.808522\pi$
$359$	−31.2426	−1.64892	−0.824462	+	0.565918i	$0.808522\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	8.48528	0.445976
$363$	21.7279	1.14042
$364$	0.272078	0.0142608
$365$	0	0
$366$	−13.8995	−0.726538
$367$	−25.4558	−1.32878	−0.664392	−	0.747384i	$-0.731309\pi$
$367$	−25.4558	−1.32878	−0.664392	+	0.747384i	$0.731309\pi$
$368$	−9.24264	−0.481806
$369$	12.0000	0.624695
$370$	0	0
$371$	−18.2132	−0.945582
$372$	5.41421	0.280714
$373$	−9.00000	−0.466002	−0.233001	−	0.972476i	$-0.574855\pi$
$373$	−9.00000	−0.466002	−0.233001	+	0.972476i	$0.574855\pi$
$374$	−1.41421	−0.0731272
$375$	0	0
$376$	0	0
$377$	1.00000	0.0515026
$378$	−0.656854	−0.0337849
$379$	2.75736	0.141636	0.0708180	−	0.997489i	$-0.477439\pi$
$379$	2.75736	0.141636	0.0708180	+	0.997489i	$0.477439\pi$
$380$	0	0
$381$	34.9706	1.79160
$382$	−18.5563	−0.949425
$383$	−3.75736	−0.191992	−0.0959960	−	0.995382i	$-0.530604\pi$
$383$	−3.75736	−0.191992	−0.0959960	+	0.995382i	$0.530604\pi$
$384$	−2.41421	−0.123200
$385$	0	0
$386$	11.6569	0.593318
$387$	−28.9706	−1.47266
$388$	11.6569	0.591787
$389$	37.0711	1.87958	0.939789	−	0.341756i	$-0.111021\pi$
$389$	37.0711	1.87958	0.939789	+	0.341756i	$0.111021\pi$
$390$	0	0
$391$	9.24264	0.467420
$392$	−4.48528	−0.226541
$393$	−40.9706	−2.06669
$394$	20.2426	1.01981
$395$	0	0
$396$	4.00000	0.201008
$397$	−24.0000	−1.20453	−0.602263	−	0.798298i	$-0.705734\pi$
$397$	−24.0000	−1.20453	−0.602263	+	0.798298i	$0.705734\pi$
$398$	0.757359	0.0379630
$399$	−3.82843	−0.191661
$400$	0	0
$401$	−22.5858	−1.12788	−0.563940	−	0.825816i	$-0.690715\pi$
$401$	−22.5858	−1.12788	−0.563940	+	0.825816i	$0.690715\pi$
$402$	31.9706	1.59455
$403$	0.384776	0.0191671
$404$	−1.07107	−0.0532876
$405$	0	0
$406$	9.24264	0.458705
$407$	−12.0000	−0.594818
$408$	2.41421	0.119521
$409$	−17.2132	−0.851138	−0.425569	−	0.904926i	$-0.639926\pi$
$409$	−17.2132	−0.851138	−0.425569	+	0.904926i	$0.639926\pi$
$410$	0	0
$411$	31.3848	1.54810
$412$	4.24264	0.209020
$413$	20.4558	1.00657
$414$	−26.1421	−1.28482
$415$	0	0
$416$	−0.171573	−0.00841205
$417$	28.9706	1.41869
$418$	−1.41421	−0.0691714
$419$	16.5858	0.810269	0.405134	−	0.914257i	$-0.367225\pi$
$419$	16.5858	0.810269	0.405134	+	0.914257i	$0.367225\pi$
$420$	0	0
$421$	2.51472	0.122560	0.0612799	−	0.998121i	$-0.480482\pi$
$421$	2.51472	0.122560	0.0612799	+	0.998121i	$0.480482\pi$
$422$	5.72792	0.278831
$423$	0	0
$424$	11.4853	0.557775
$425$	0	0
$426$	25.5563	1.23821
$427$	−9.12994	−0.441829
$428$	5.72792	0.276870
$429$	0.585786	0.0282820
$430$	0	0
$431$	30.3848	1.46358	0.731792	−	0.681528i	$-0.238684\pi$
$431$	30.3848	1.46358	0.731792	+	0.681528i	$0.238684\pi$
$432$	0.414214	0.0199289
$433$	−21.5563	−1.03593	−0.517966	−	0.855401i	$-0.673311\pi$
$433$	−21.5563	−1.03593	−0.517966	+	0.855401i	$0.673311\pi$
$434$	3.55635	0.170710
$435$	0	0
$436$	15.9706	0.764851
$437$	9.24264	0.442135
$438$	−13.2426	−0.632758
$439$	−14.2426	−0.679764	−0.339882	−	0.940468i	$-0.610387\pi$
$439$	−14.2426	−0.679764	−0.339882	+	0.940468i	$0.610387\pi$
$440$	0	0
$441$	−12.6863	−0.604109
$442$	0.171573	0.00816089
$443$	4.24264	0.201574	0.100787	−	0.994908i	$-0.467864\pi$
$443$	4.24264	0.201574	0.100787	+	0.994908i	$0.467864\pi$
$444$	20.4853	0.972188
$445$	0	0
$446$	20.8284	0.986255
$447$	−15.3137	−0.724314
$448$	−1.58579	−0.0749214
$449$	−5.31371	−0.250769	−0.125385	−	0.992108i	$-0.540017\pi$
$449$	−5.31371	−0.250769	−0.125385	+	0.992108i	$0.540017\pi$
$450$	0	0
$451$	6.00000	0.282529
$452$	1.75736	0.0826592
$453$	−15.6569	−0.735623
$454$	25.2426	1.18470
$455$	0	0
$456$	2.41421	0.113056
$457$	3.00000	0.140334	0.0701670	−	0.997535i	$-0.477647\pi$
$457$	3.00000	0.140334	0.0701670	+	0.997535i	$0.477647\pi$
$458$	−18.9706	−0.886436
$459$	−0.414214	−0.0193338
$460$	0	0
$461$	27.5563	1.28343	0.641714	−	0.766944i	$-0.278224\pi$
$461$	27.5563	1.28343	0.641714	+	0.766944i	$0.278224\pi$
$462$	5.41421	0.251892
$463$	14.1421	0.657241	0.328620	−	0.944462i	$-0.393416\pi$
$463$	14.1421	0.657241	0.328620	+	0.944462i	$0.393416\pi$
$464$	−5.82843	−0.270578
$465$	0	0
$466$	−8.97056	−0.415553
$467$	24.7279	1.14427	0.572136	−	0.820159i	$-0.306115\pi$
$467$	24.7279	1.14427	0.572136	+	0.820159i	$0.306115\pi$
$468$	−0.485281	−0.0224321
$469$	21.0000	0.969690
$470$	0	0
$471$	0.828427	0.0381719
$472$	−12.8995	−0.593747
$473$	−14.4853	−0.666034
$474$	−25.3137	−1.16270
$475$	0	0
$476$	1.58579	0.0726844
$477$	32.4853	1.48740
$478$	12.8995	0.590009
$479$	31.1127	1.42158	0.710788	−	0.703407i	$-0.248339\pi$
$479$	31.1127	1.42158	0.710788	+	0.703407i	$0.248339\pi$
$480$	0	0
$481$	1.45584	0.0663808
$482$	−24.9706	−1.13738
$483$	−35.3848	−1.61006
$484$	−9.00000	−0.409091
$485$	0	0
$486$	21.6569	0.982375
$487$	1.79899	0.0815200	0.0407600	−	0.999169i	$-0.487022\pi$
$487$	1.79899	0.0815200	0.0407600	+	0.999169i	$0.487022\pi$
$488$	5.75736	0.260623
$489$	−4.24264	−0.191859
$490$	0	0
$491$	−2.44365	−0.110280	−0.0551402	−	0.998479i	$-0.517561\pi$
$491$	−2.44365	−0.110280	−0.0551402	+	0.998479i	$0.517561\pi$
$492$	−10.2426	−0.461774
$493$	5.82843	0.262499
$494$	0.171573	0.00771943
$495$	0	0
$496$	−2.24264	−0.100698
$497$	16.7868	0.752991
$498$	−6.00000	−0.268866
$499$	−34.2426	−1.53291	−0.766456	−	0.642297i	$-0.777981\pi$
$499$	−34.2426	−1.53291	−0.766456	+	0.642297i	$0.777981\pi$
$500$	0	0
$501$	−23.5563	−1.05242
$502$	−27.5563	−1.22990
$503$	−39.7279	−1.77138	−0.885690	−	0.464277i	$-0.846314\pi$
$503$	−39.7279	−1.77138	−0.885690	+	0.464277i	$0.846314\pi$
$504$	−4.48528	−0.199790
$505$	0	0
$506$	−13.0711	−0.581080
$507$	31.3137	1.39069
$508$	−14.4853	−0.642680
$509$	−4.97056	−0.220316	−0.110158	−	0.993914i	$-0.535136\pi$
$509$	−4.97056	−0.220316	−0.110158	+	0.993914i	$0.535136\pi$
$510$	0	0
$511$	−8.69848	−0.384798
$512$	1.00000	0.0441942
$513$	−0.414214	−0.0182880
$514$	4.72792	0.208540
$515$	0	0
$516$	24.7279	1.08859
$517$	0	0
$518$	13.4558	0.591216
$519$	−39.7990	−1.74698
$520$	0	0
$521$	0.686292	0.0300670	0.0150335	−	0.999887i	$-0.495215\pi$
$521$	0.686292	0.0300670	0.0150335	+	0.999887i	$0.495215\pi$
$522$	−16.4853	−0.721541
$523$	27.7279	1.21246	0.606229	−	0.795290i	$-0.292682\pi$
$523$	27.7279	1.21246	0.606229	+	0.795290i	$0.292682\pi$
$524$	16.9706	0.741362
$525$	0	0
$526$	−6.97056	−0.303931
$527$	2.24264	0.0976910
$528$	−3.41421	−0.148585
$529$	62.4264	2.71419
$530$	0	0
$531$	−36.4853	−1.58333
$532$	1.58579	0.0687526
$533$	−0.727922	−0.0315298
$534$	17.0711	0.738737
$535$	0	0
$536$	−13.2426	−0.571995
$537$	0.828427	0.0357493
$538$	28.6274	1.23422
$539$	−6.34315	−0.273219
$540$	0	0
$541$	18.2426	0.784312	0.392156	−	0.919899i	$-0.371729\pi$
$541$	18.2426	0.784312	0.392156	+	0.919899i	$0.371729\pi$
$542$	−18.7574	−0.805698
$543$	−20.4853	−0.879108
$544$	−1.00000	−0.0428746
$545$	0	0
$546$	−0.656854	−0.0281108
$547$	−5.31371	−0.227198	−0.113599	−	0.993527i	$-0.536238\pi$
$547$	−5.31371	−0.227198	−0.113599	+	0.993527i	$0.536238\pi$
$548$	−13.0000	−0.555332
$549$	16.2843	0.694996
$550$	0	0
$551$	5.82843	0.248299
$552$	22.3137	0.949735
$553$	−16.6274	−0.707070
$554$	0	0
$555$	0	0
$556$	−12.0000	−0.508913
$557$	−16.0000	−0.677942	−0.338971	−	0.940797i	$-0.610079\pi$
$557$	−16.0000	−0.677942	−0.338971	+	0.940797i	$0.610079\pi$
$558$	−6.34315	−0.268527
$559$	1.75736	0.0743284
$560$	0	0
$561$	3.41421	0.144148
$562$	−4.24264	−0.178965
$563$	28.9706	1.22096	0.610482	−	0.792030i	$-0.290976\pi$
$563$	28.9706	1.22096	0.610482	+	0.792030i	$0.290976\pi$
$564$	0	0
$565$	0	0
$566$	−3.85786	−0.162158
$567$	15.0416	0.631689
$568$	−10.5858	−0.444170
$569$	28.2843	1.18574	0.592869	−	0.805299i	$-0.297995\pi$
$569$	28.2843	1.18574	0.592869	+	0.805299i	$0.297995\pi$
$570$	0	0
$571$	6.24264	0.261246	0.130623	−	0.991432i	$-0.458302\pi$
$571$	6.24264	0.261246	0.130623	+	0.991432i	$0.458302\pi$
$572$	−0.242641	−0.0101453
$573$	44.7990	1.87150
$574$	−6.72792	−0.280818
$575$	0	0
$576$	2.82843	0.117851
$577$	−2.31371	−0.0963209	−0.0481605	−	0.998840i	$-0.515336\pi$
$577$	−2.31371	−0.0963209	−0.0481605	+	0.998840i	$0.515336\pi$
$578$	−16.0000	−0.665512
$579$	−28.1421	−1.16955
$580$	0	0
$581$	−3.94113	−0.163505
$582$	−28.1421	−1.16653
$583$	16.2426	0.672701
$584$	5.48528	0.226983
$585$	0	0
$586$	−11.4853	−0.474453
$587$	−21.7574	−0.898022	−0.449011	−	0.893526i	$-0.648224\pi$
$587$	−21.7574	−0.898022	−0.449011	+	0.893526i	$0.648224\pi$
$588$	10.8284	0.446557
$589$	2.24264	0.0924064
$590$	0	0
$591$	−48.8701	−2.01025
$592$	−8.48528	−0.348743
$593$	−22.0000	−0.903432	−0.451716	−	0.892162i	$-0.649188\pi$
$593$	−22.0000	−0.903432	−0.451716	+	0.892162i	$0.649188\pi$
$594$	0.585786	0.0240351
$595$	0	0
$596$	6.34315	0.259825
$597$	−1.82843	−0.0748325
$598$	1.58579	0.0648476
$599$	27.2132	1.11190	0.555951	−	0.831215i	$-0.312354\pi$
$599$	27.2132	1.11190	0.555951	+	0.831215i	$0.312354\pi$
$600$	0	0
$601$	−11.7574	−0.479593	−0.239796	−	0.970823i	$-0.577081\pi$
$601$	−11.7574	−0.479593	−0.239796	+	0.970823i	$0.577081\pi$
$602$	16.2426	0.662001
$603$	−37.4558	−1.52532
$604$	6.48528	0.263882
$605$	0	0
$606$	2.58579	0.105040
$607$	−8.82843	−0.358335	−0.179167	−	0.983819i	$-0.557340\pi$
$607$	−8.82843	−0.358335	−0.179167	+	0.983819i	$0.557340\pi$
$608$	−1.00000	−0.0405554
$609$	−22.3137	−0.904197
$610$	0	0
$611$	0	0
$612$	−2.82843	−0.114332
$613$	−47.6985	−1.92652	−0.963262	−	0.268564i	$-0.913451\pi$
$613$	−47.6985	−1.92652	−0.963262	+	0.268564i	$0.913451\pi$
$614$	6.34315	0.255989
$615$	0	0
$616$	−2.24264	−0.0903586
$617$	−12.4853	−0.502639	−0.251319	−	0.967904i	$-0.580864\pi$
$617$	−12.4853	−0.502639	−0.251319	+	0.967904i	$0.580864\pi$
$618$	−10.2426	−0.412019
$619$	16.2426	0.652847	0.326423	−	0.945224i	$-0.394156\pi$
$619$	16.2426	0.652847	0.326423	+	0.945224i	$0.394156\pi$
$620$	0	0
$621$	−3.82843	−0.153629
$622$	−13.2426	−0.530982
$623$	11.2132	0.449248
$624$	0.414214	0.0165818
$625$	0	0
$626$	−25.9706	−1.03799
$627$	3.41421	0.136351
$628$	−0.343146	−0.0136930
$629$	8.48528	0.338330
$630$	0	0
$631$	−5.02944	−0.200219	−0.100109	−	0.994976i	$-0.531919\pi$
$631$	−5.02944	−0.200219	−0.100109	+	0.994976i	$0.531919\pi$
$632$	10.4853	0.417082
$633$	−13.8284	−0.549631
$634$	−7.48528	−0.297279
$635$	0	0
$636$	−27.7279	−1.09948
$637$	0.769553	0.0304908
$638$	−8.24264	−0.326329
$639$	−29.9411	−1.18445
$640$	0	0
$641$	30.0416	1.18657	0.593287	−	0.804991i	$-0.297830\pi$
$641$	30.0416	1.18657	0.593287	+	0.804991i	$0.297830\pi$
$642$	−13.8284	−0.545764
$643$	−2.48528	−0.0980099	−0.0490050	−	0.998799i	$-0.515605\pi$
$643$	−2.48528	−0.0980099	−0.0490050	+	0.998799i	$0.515605\pi$
$644$	14.6569	0.577561
$645$	0	0
$646$	1.00000	0.0393445
$647$	27.2426	1.07102	0.535509	−	0.844529i	$-0.320120\pi$
$647$	27.2426	1.07102	0.535509	+	0.844529i	$0.320120\pi$
$648$	−9.48528	−0.372617
$649$	−18.2426	−0.716086
$650$	0	0
$651$	−8.58579	−0.336504
$652$	1.75736	0.0688235
$653$	−4.97056	−0.194513	−0.0972566	−	0.995259i	$-0.531007\pi$
$653$	−4.97056	−0.194513	−0.0972566	+	0.995259i	$0.531007\pi$
$654$	−38.5563	−1.50767
$655$	0	0
$656$	4.24264	0.165647
$657$	15.5147	0.605287
$658$	0	0
$659$	−0.899495	−0.0350393	−0.0175197	−	0.999847i	$-0.505577\pi$
$659$	−0.899495	−0.0350393	−0.0175197	+	0.999847i	$0.505577\pi$
$660$	0	0
$661$	32.4558	1.26239	0.631193	−	0.775626i	$-0.282566\pi$
$661$	32.4558	1.26239	0.631193	+	0.775626i	$0.282566\pi$
$662$	−10.7574	−0.418097
$663$	−0.414214	−0.0160867
$664$	2.48528	0.0964476
$665$	0	0
$666$	−24.0000	−0.929981
$667$	53.8701	2.08586
$668$	9.75736	0.377524
$669$	−50.2843	−1.94410
$670$	0	0
$671$	8.14214	0.314324
$672$	3.82843	0.147685
$673$	12.0000	0.462566	0.231283	−	0.972887i	$-0.425708\pi$
$673$	12.0000	0.462566	0.231283	+	0.972887i	$0.425708\pi$
$674$	−33.8995	−1.30576
$675$	0	0
$676$	−12.9706	−0.498868
$677$	−26.9411	−1.03543	−0.517716	−	0.855553i	$-0.673218\pi$
$677$	−26.9411	−1.03543	−0.517716	+	0.855553i	$0.673218\pi$
$678$	−4.24264	−0.162938
$679$	−18.4853	−0.709400
$680$	0	0
$681$	−60.9411	−2.33527
$682$	−3.17157	−0.121446
$683$	−12.0000	−0.459167	−0.229584	−	0.973289i	$-0.573736\pi$
$683$	−12.0000	−0.459167	−0.229584	+	0.973289i	$0.573736\pi$
$684$	−2.82843	−0.108148
$685$	0	0
$686$	18.2132	0.695383
$687$	45.7990	1.74734
$688$	−10.2426	−0.390497
$689$	−1.97056	−0.0750725
$690$	0	0
$691$	5.51472	0.209790	0.104895	−	0.994483i	$-0.466549\pi$
$691$	5.51472	0.209790	0.104895	+	0.994483i	$0.466549\pi$
$692$	16.4853	0.626676
$693$	−6.34315	−0.240956
$694$	22.4853	0.853530
$695$	0	0
$696$	14.0711	0.533362
$697$	−4.24264	−0.160701
$698$	6.00000	0.227103
$699$	21.6569	0.819137
$700$	0	0
$701$	−16.9706	−0.640969	−0.320485	−	0.947254i	$-0.603846\pi$
$701$	−16.9706	−0.640969	−0.320485	+	0.947254i	$0.603846\pi$
$702$	−0.0710678	−0.00268228
$703$	8.48528	0.320028
$704$	1.41421	0.0533002
$705$	0	0
$706$	19.4853	0.733338
$707$	1.69848	0.0638781
$708$	31.1421	1.17039
$709$	−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
$711$	29.6569	1.11222
$712$	−7.07107	−0.264999
$713$	20.7279	0.776267
$714$	−3.82843	−0.143275
$715$	0	0
$716$	−0.343146	−0.0128240
$717$	−31.1421	−1.16302
$718$	−31.2426	−1.16596
$719$	−24.8995	−0.928594	−0.464297	−	0.885679i	$-0.653693\pi$
$719$	−24.8995	−0.928594	−0.464297	+	0.885679i	$0.653693\pi$
$720$	0	0
$721$	−6.72792	−0.250561
$722$	1.00000	0.0372161
$723$	60.2843	2.24200
$724$	8.48528	0.315353
$725$	0	0
$726$	21.7279	0.806399
$727$	15.7279	0.583316	0.291658	−	0.956523i	$-0.405793\pi$
$727$	15.7279	0.583316	0.291658	+	0.956523i	$0.405793\pi$
$728$	0.272078	0.0100839
$729$	−23.8284	−0.882534
$730$	0	0
$731$	10.2426	0.378838
$732$	−13.8995	−0.513740
$733$	−12.0000	−0.443230	−0.221615	−	0.975134i	$-0.571133\pi$
$733$	−12.0000	−0.443230	−0.221615	+	0.975134i	$0.571133\pi$
$734$	−25.4558	−0.939592
$735$	0	0
$736$	−9.24264	−0.340688
$737$	−18.7279	−0.689852
$738$	12.0000	0.441726
$739$	26.7279	0.983203	0.491601	−	0.870820i	$-0.336412\pi$
$739$	26.7279	0.983203	0.491601	+	0.870820i	$0.336412\pi$
$740$	0	0
$741$	−0.414214	−0.0152165
$742$	−18.2132	−0.668628
$743$	18.7279	0.687061	0.343530	−	0.939142i	$-0.388377\pi$
$743$	18.7279	0.687061	0.343530	+	0.939142i	$0.388377\pi$
$744$	5.41421	0.198495
$745$	0	0
$746$	−9.00000	−0.329513
$747$	7.02944	0.257194
$748$	−1.41421	−0.0517088
$749$	−9.08326	−0.331895
$750$	0	0
$751$	−1.27208	−0.0464188	−0.0232094	−	0.999731i	$-0.507388\pi$
$751$	−1.27208	−0.0464188	−0.0232094	+	0.999731i	$0.507388\pi$
$752$	0	0
$753$	66.5269	2.42438
$754$	1.00000	0.0364179
$755$	0	0
$756$	−0.656854	−0.0238896
$757$	−23.6569	−0.859823	−0.429911	−	0.902871i	$-0.641455\pi$
$757$	−23.6569	−0.859823	−0.429911	+	0.902871i	$0.641455\pi$
$758$	2.75736	0.100152
$759$	31.5563	1.14542
$760$	0	0
$761$	10.0294	0.363567	0.181783	−	0.983339i	$-0.441813\pi$
$761$	10.0294	0.363567	0.181783	+	0.983339i	$0.441813\pi$
$762$	34.9706	1.26685
$763$	−25.3259	−0.916859
$764$	−18.5563	−0.671345
$765$	0	0
$766$	−3.75736	−0.135759
$767$	2.21320	0.0799141
$768$	−2.41421	−0.0871154
$769$	14.4558	0.521291	0.260646	−	0.965435i	$-0.416065\pi$
$769$	14.4558	0.521291	0.260646	+	0.965435i	$0.416065\pi$
$770$	0	0
$771$	−11.4142	−0.411073
$772$	11.6569	0.419539
$773$	19.9706	0.718291	0.359146	−	0.933282i	$-0.383068\pi$
$773$	19.9706	0.718291	0.359146	+	0.933282i	$0.383068\pi$
$774$	−28.9706	−1.04133
$775$	0	0
$776$	11.6569	0.418457
$777$	−32.4853	−1.16540
$778$	37.0711	1.32906
$779$	−4.24264	−0.152008
$780$	0	0
$781$	−14.9706	−0.535689
$782$	9.24264	0.330516
$783$	−2.41421	−0.0862770
$784$	−4.48528	−0.160189
$785$	0	0
$786$	−40.9706	−1.46137
$787$	−35.1838	−1.25417	−0.627083	−	0.778953i	$-0.715751\pi$
$787$	−35.1838	−1.25417	−0.627083	+	0.778953i	$0.715751\pi$
$788$	20.2426	0.721114
$789$	16.8284	0.599108
$790$	0	0
$791$	−2.78680	−0.0990871
$792$	4.00000	0.142134
$793$	−0.987807	−0.0350780
$794$	−24.0000	−0.851728
$795$	0	0
$796$	0.757359	0.0268439
$797$	30.5147	1.08089	0.540443	−	0.841380i	$-0.318257\pi$
$797$	30.5147	1.08089	0.540443	+	0.841380i	$0.318257\pi$
$798$	−3.82843	−0.135525
$799$	0	0
$800$	0	0
$801$	−20.0000	−0.706665
$802$	−22.5858	−0.797532
$803$	7.75736	0.273751
$804$	31.9706	1.12751
$805$	0	0
$806$	0.384776	0.0135532
$807$	−69.1127	−2.43288
$808$	−1.07107	−0.0376800
$809$	10.7990	0.379672	0.189836	−	0.981816i	$-0.439204\pi$
$809$	10.7990	0.379672	0.189836	+	0.981816i	$0.439204\pi$
$810$	0	0
$811$	−6.69848	−0.235216	−0.117608	−	0.993060i	$-0.537523\pi$
$811$	−6.69848	−0.235216	−0.117608	+	0.993060i	$0.537523\pi$
$812$	9.24264	0.324353
$813$	45.2843	1.58819
$814$	−12.0000	−0.420600
$815$	0	0
$816$	2.41421	0.0845144
$817$	10.2426	0.358345
$818$	−17.2132	−0.601846
$819$	0.769553	0.0268903
$820$	0	0
$821$	17.6569	0.616228	0.308114	−	0.951349i	$-0.400302\pi$
$821$	17.6569	0.616228	0.308114	+	0.951349i	$0.400302\pi$
$822$	31.3848	1.09467
$823$	−9.72792	−0.339094	−0.169547	−	0.985522i	$-0.554230\pi$
$823$	−9.72792	−0.339094	−0.169547	+	0.985522i	$0.554230\pi$
$824$	4.24264	0.147799
$825$	0	0
$826$	20.4558	0.711750
$827$	−38.6985	−1.34568	−0.672839	−	0.739789i	$-0.734925\pi$
$827$	−38.6985	−1.34568	−0.672839	+	0.739789i	$0.734925\pi$
$828$	−26.1421	−0.908502
$829$	−40.4558	−1.40509	−0.702545	−	0.711640i	$-0.747953\pi$
$829$	−40.4558	−1.40509	−0.702545	+	0.711640i	$0.747953\pi$
$830$	0	0
$831$	0	0
$832$	−0.171573	−0.00594822
$833$	4.48528	0.155406
$834$	28.9706	1.00317
$835$	0	0
$836$	−1.41421	−0.0489116
$837$	−0.928932	−0.0321086
$838$	16.5858	0.572946
$839$	−22.6274	−0.781185	−0.390593	−	0.920564i	$-0.627730\pi$
$839$	−22.6274	−0.781185	−0.390593	+	0.920564i	$0.627730\pi$
$840$	0	0
$841$	4.97056	0.171399
$842$	2.51472	0.0866629
$843$	10.2426	0.352775
$844$	5.72792	0.197163
$845$	0	0
$846$	0	0
$847$	14.2721	0.490394
$848$	11.4853	0.394406
$849$	9.31371	0.319646
$850$	0	0
$851$	78.4264	2.68842
$852$	25.5563	0.875546
$853$	−39.9411	−1.36756	−0.683779	−	0.729689i	$-0.739665\pi$
$853$	−39.9411	−1.36756	−0.683779	+	0.729689i	$0.739665\pi$
$854$	−9.12994	−0.312420
$855$	0	0
$856$	5.72792	0.195776
$857$	0	0		−	1.00000i	$-0.5\pi$
$857$	0	0			1.00000i	$0.5\pi$
$858$	0.585786	0.0199984
$859$	−20.7279	−0.707228	−0.353614	−	0.935392i	$-0.615047\pi$
$859$	−20.7279	−0.707228	−0.353614	+	0.935392i	$0.615047\pi$
$860$	0	0
$861$	16.2426	0.553548
$862$	30.3848	1.03491
$863$	24.7279	0.841748	0.420874	−	0.907119i	$-0.361723\pi$
$863$	24.7279	0.841748	0.420874	+	0.907119i	$0.361723\pi$
$864$	0.414214	0.0140918
$865$	0	0
$866$	−21.5563	−0.732515
$867$	38.6274	1.31186
$868$	3.55635	0.120710
$869$	14.8284	0.503020
$870$	0	0
$871$	2.27208	0.0769864
$872$	15.9706	0.540831
$873$	32.9706	1.11588
$874$	9.24264	0.312637
$875$	0	0
$876$	−13.2426	−0.447427
$877$	47.1421	1.59188	0.795938	−	0.605378i	$-0.206978\pi$
$877$	47.1421	1.59188	0.795938	+	0.605378i	$0.206978\pi$
$878$	−14.2426	−0.480666
$879$	27.7279	0.935240
$880$	0	0
$881$	39.5980	1.33409	0.667045	−	0.745018i	$-0.267559\pi$
$881$	39.5980	1.33409	0.667045	+	0.745018i	$0.267559\pi$
$882$	−12.6863	−0.427170
$883$	2.48528	0.0836364	0.0418182	−	0.999125i	$-0.486685\pi$
$883$	2.48528	0.0836364	0.0418182	+	0.999125i	$0.486685\pi$
$884$	0.171573	0.00577062
$885$	0	0
$886$	4.24264	0.142534
$887$	2.78680	0.0935715	0.0467857	−	0.998905i	$-0.485102\pi$
$887$	2.78680	0.0935715	0.0467857	+	0.998905i	$0.485102\pi$
$888$	20.4853	0.687441
$889$	22.9706	0.770408
$890$	0	0
$891$	−13.4142	−0.449393
$892$	20.8284	0.697387
$893$	0	0
$894$	−15.3137	−0.512167
$895$	0	0
$896$	−1.58579	−0.0529774
$897$	−3.82843	−0.127827
$898$	−5.31371	−0.177321
$899$	13.0711	0.435945
$900$	0	0
$901$	−11.4853	−0.382630
$902$	6.00000	0.199778
$903$	−39.2132	−1.30493
$904$	1.75736	0.0584489
$905$	0	0
$906$	−15.6569	−0.520164
$907$	20.6985	0.687282	0.343641	−	0.939101i	$-0.388340\pi$
$907$	20.6985	0.687282	0.343641	+	0.939101i	$0.388340\pi$
$908$	25.2426	0.837706
$909$	−3.02944	−0.100480
$910$	0	0
$911$	−16.2843	−0.539522	−0.269761	−	0.962927i	$-0.586945\pi$
$911$	−16.2843	−0.539522	−0.269761	+	0.962927i	$0.586945\pi$
$912$	2.41421	0.0799426
$913$	3.51472	0.116320
$914$	3.00000	0.0992312
$915$	0	0
$916$	−18.9706	−0.626805
$917$	−26.9117	−0.888702
$918$	−0.414214	−0.0136711
$919$	36.2132	1.19456	0.597282	−	0.802032i	$-0.296247\pi$
$919$	36.2132	1.19456	0.597282	+	0.802032i	$0.296247\pi$
$920$	0	0
$921$	−15.3137	−0.504604
$922$	27.5563	0.907520
$923$	1.81623	0.0597821
$924$	5.41421	0.178115
$925$	0	0
$926$	14.1421	0.464739
$927$	12.0000	0.394132
$928$	−5.82843	−0.191327
$929$	−17.8284	−0.584932	−0.292466	−	0.956276i	$-0.594476\pi$
$929$	−17.8284	−0.584932	−0.292466	+	0.956276i	$0.594476\pi$
$930$	0	0
$931$	4.48528	0.146999
$932$	−8.97056	−0.293841
$933$	31.9706	1.04667
$934$	24.7279	0.809122
$935$	0	0
$936$	−0.485281	−0.0158619
$937$	−27.0000	−0.882052	−0.441026	−	0.897494i	$-0.645385\pi$
$937$	−27.0000	−0.882052	−0.441026	+	0.897494i	$0.645385\pi$
$938$	21.0000	0.685674
$939$	62.6985	2.04609
$940$	0	0
$941$	24.8579	0.810343	0.405172	−	0.914241i	$-0.367212\pi$
$941$	24.8579	0.810343	0.405172	+	0.914241i	$0.367212\pi$
$942$	0.828427	0.0269916
$943$	−39.2132	−1.27696
$944$	−12.8995	−0.419843
$945$	0	0
$946$	−14.4853	−0.470957
$947$	12.0000	0.389948	0.194974	−	0.980808i	$-0.437538\pi$
$947$	12.0000	0.389948	0.194974	+	0.980808i	$0.437538\pi$
$948$	−25.3137	−0.822151
$949$	−0.941125	−0.0305502
$950$	0	0
$951$	18.0711	0.585995
$952$	1.58579	0.0513956
$953$	6.72792	0.217939	0.108969	−	0.994045i	$-0.465245\pi$
$953$	6.72792	0.217939	0.108969	+	0.994045i	$0.465245\pi$
$954$	32.4853	1.05175
$955$	0	0
$956$	12.8995	0.417199
$957$	19.8995	0.643259
$958$	31.1127	1.00521
$959$	20.6152	0.665700
$960$	0	0
$961$	−25.9706	−0.837760
$962$	1.45584	0.0469383
$963$	16.2010	0.522070
$964$	−24.9706	−0.804248
$965$	0	0
$966$	−35.3848	−1.13849
$967$	19.1127	0.614623	0.307311	−	0.951609i	$-0.400571\pi$
$967$	19.1127	0.614623	0.307311	+	0.951609i	$0.400571\pi$
$968$	−9.00000	−0.289271
$969$	−2.41421	−0.0775557
$970$	0	0
$971$	30.3431	0.973758	0.486879	−	0.873469i	$-0.338135\pi$
$971$	30.3431	0.973758	0.486879	+	0.873469i	$0.338135\pi$
$972$	21.6569	0.694644
$973$	19.0294	0.610056
$974$	1.79899	0.0576434
$975$	0	0
$976$	5.75736	0.184289
$977$	8.24264	0.263705	0.131853	−	0.991269i	$-0.457907\pi$
$977$	8.24264	0.263705	0.131853	+	0.991269i	$0.457907\pi$
$978$	−4.24264	−0.135665
$979$	−10.0000	−0.319601
$980$	0	0
$981$	45.1716	1.44222
$982$	−2.44365	−0.0779800
$983$	0.544156	0.0173559	0.00867794	−	0.999962i	$-0.497238\pi$
$983$	0.544156	0.0173559	0.00867794	+	0.999962i	$0.497238\pi$
$984$	−10.2426	−0.326523
$985$	0	0
$986$	5.82843	0.185615
$987$	0	0
$988$	0.171573	0.00545846
$989$	94.6690	3.01030
$990$	0	0
$991$	22.2426	0.706561	0.353280	−	0.935517i	$-0.385066\pi$
$991$	22.2426	0.706561	0.353280	+	0.935517i	$0.385066\pi$
$992$	−2.24264	−0.0712039
$993$	25.9706	0.824151
$994$	16.7868	0.532445
$995$	0	0
$996$	−6.00000	−0.190117
$997$	−23.2721	−0.737034	−0.368517	−	0.929621i	$-0.620134\pi$
$997$	−23.2721	−0.737034	−0.368517	+	0.929621i	$0.620134\pi$
$998$	−34.2426	−1.08393
$999$	−3.51472	−0.111201

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	950.2.a.g.1.1		2
3.2	odd	2		8550.2.a.bn.1.2		2
4.3	odd	2		7600.2.a.bg.1.2		2
5.2	odd	4		190.2.b.a.39.4	yes	4
5.3	odd	4		190.2.b.a.39.1	✓	4
5.4	even	2		950.2.a.f.1.2		2
15.2	even	4		1710.2.d.c.1369.2		4
15.8	even	4		1710.2.d.c.1369.4		4
15.14	odd	2		8550.2.a.cb.1.1		2
20.3	even	4		1520.2.d.e.609.4		4
20.7	even	4		1520.2.d.e.609.1		4
20.19	odd	2		7600.2.a.v.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
190.2.b.a.39.1	✓	4	5.3	odd	4
190.2.b.a.39.4	yes	4	5.2	odd	4
950.2.a.f.1.2		2	5.4	even	2
950.2.a.g.1.1		2	1.1	even	1	trivial
1520.2.d.e.609.1		4	20.7	even	4
1520.2.d.e.609.4		4	20.3	even	4
1710.2.d.c.1369.2		4	15.2	even	4
1710.2.d.c.1369.4		4	15.8	even	4
7600.2.a.v.1.1		2	20.19	odd	2
7600.2.a.bg.1.2		2	4.3	odd	2
8550.2.a.bn.1.2		2	3.2	odd	2
8550.2.a.cb.1.1		2	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -2.41421 q^{3} +1.00000 q^{4} -2.41421 q^{6} -1.58579 q^{7} +1.00000 q^{8} +2.82843 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -2.41421 q^{3} +1.00000 q^{4} -2.41421 q^{6} -1.58579 q^{7} +1.00000 q^{8} +2.82843 q^{9} +1.41421 q^{11} -2.41421 q^{12} -0.171573 q^{13} -1.58579 q^{14} +1.00000 q^{16} -1.00000 q^{17} +2.82843 q^{18} -1.00000 q^{19} +3.82843 q^{21} +1.41421 q^{22} -9.24264 q^{23} -2.41421 q^{24} -0.171573 q^{26} +0.414214 q^{27} -1.58579 q^{28} -5.82843 q^{29} -2.24264 q^{31} +1.00000 q^{32} -3.41421 q^{33} -1.00000 q^{34} +2.82843 q^{36} -8.48528 q^{37} -1.00000 q^{38} +0.414214 q^{39} +4.24264 q^{41} +3.82843 q^{42} -10.2426 q^{43} +1.41421 q^{44} -9.24264 q^{46} -2.41421 q^{48} -4.48528 q^{49} +2.41421 q^{51} -0.171573 q^{52} +11.4853 q^{53} +0.414214 q^{54} -1.58579 q^{56} +2.41421 q^{57} -5.82843 q^{58} -12.8995 q^{59} +5.75736 q^{61} -2.24264 q^{62} -4.48528 q^{63} +1.00000 q^{64} -3.41421 q^{66} -13.2426 q^{67} -1.00000 q^{68} +22.3137 q^{69} -10.5858 q^{71} +2.82843 q^{72} +5.48528 q^{73} -8.48528 q^{74} -1.00000 q^{76} -2.24264 q^{77} +0.414214 q^{78} +10.4853 q^{79} -9.48528 q^{81} +4.24264 q^{82} +2.48528 q^{83} +3.82843 q^{84} -10.2426 q^{86} +14.0711 q^{87} +1.41421 q^{88} -7.07107 q^{89} +0.272078 q^{91} -9.24264 q^{92} +5.41421 q^{93} -2.41421 q^{96} +11.6569 q^{97} -4.48528 q^{98} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{6} - 6 q^{7} + 2 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 - 2 * q^6 - 6 * q^7 + 2 * q^8	\( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{6} - 6 q^{7} + 2 q^{8} - 2 q^{12} - 6 q^{13} - 6 q^{14} + 2 q^{16} - 2 q^{17} - 2 q^{19} + 2 q^{21} - 10 q^{23} - 2 q^{24} - 6 q^{26} - 2 q^{27} - 6 q^{28} - 6 q^{29} + 4 q^{31} + 2 q^{32} - 4 q^{33} - 2 q^{34} - 2 q^{38} - 2 q^{39} + 2 q^{42} - 12 q^{43} - 10 q^{46} - 2 q^{48} + 8 q^{49} + 2 q^{51} - 6 q^{52} + 6 q^{53} - 2 q^{54} - 6 q^{56} + 2 q^{57} - 6 q^{58} - 6 q^{59} + 20 q^{61} + 4 q^{62} + 8 q^{63} + 2 q^{64} - 4 q^{66} - 18 q^{67} - 2 q^{68} + 22 q^{69} - 24 q^{71} - 6 q^{73} - 2 q^{76} + 4 q^{77} - 2 q^{78} + 4 q^{79} - 2 q^{81} - 12 q^{83} + 2 q^{84} - 12 q^{86} + 14 q^{87} + 26 q^{91} - 10 q^{92} + 8 q^{93} - 2 q^{96} + 12 q^{97} + 8 q^{98} + 8 q^{99}+O(q^{100}) \) 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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more