LMFDB - Embedded newform 1950.2.a.bg.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1950,2,Mod(1,1950)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1950, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1950.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1950.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:	$15.5708283941$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{41}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 10 $ x^2 - x - 10
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.70156$ of defining polynomial
Character	$\chi$	$=$	1950.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +4.70156 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +4.70156 q^{7} +1.00000 q^{8} +1.00000 q^{9} +4.70156 q^{11} +1.00000 q^{12} -1.00000 q^{13} +4.70156 q^{14} +1.00000 q^{16} -0.701562 q^{17} +1.00000 q^{18} -1.70156 q^{19} +4.70156 q^{21} +4.70156 q^{22} +1.00000 q^{24} -1.00000 q^{26} +1.00000 q^{27} +4.70156 q^{28} -6.40312 q^{29} -10.1047 q^{31} +1.00000 q^{32} +4.70156 q^{33} -0.701562 q^{34} +1.00000 q^{36} -1.70156 q^{37} -1.70156 q^{38} -1.00000 q^{39} -3.70156 q^{41} +4.70156 q^{42} -11.4031 q^{43} +4.70156 q^{44} +7.00000 q^{47} +1.00000 q^{48} +15.1047 q^{49} -0.701562 q^{51} -1.00000 q^{52} +2.40312 q^{53} +1.00000 q^{54} +4.70156 q^{56} -1.70156 q^{57} -6.40312 q^{58} +2.70156 q^{59} +14.1047 q^{61} -10.1047 q^{62} +4.70156 q^{63} +1.00000 q^{64} +4.70156 q^{66} -6.40312 q^{67} -0.701562 q^{68} -1.70156 q^{71} +1.00000 q^{72} -12.0000 q^{73} -1.70156 q^{74} -1.70156 q^{76} +22.1047 q^{77} -1.00000 q^{78} -5.70156 q^{79} +1.00000 q^{81} -3.70156 q^{82} -10.7016 q^{83} +4.70156 q^{84} -11.4031 q^{86} -6.40312 q^{87} +4.70156 q^{88} +11.4031 q^{89} -4.70156 q^{91} -10.1047 q^{93} +7.00000 q^{94} +1.00000 q^{96} -2.59688 q^{97} +15.1047 q^{98} +4.70156 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{6} + 3 q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^6 + 3 * q^7 + 2 * q^8 + 2 * q^9	$ 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{6} + 3 q^{7} + 2 q^{8} + 2 q^{9} + 3 q^{11} + 2 q^{12} - 2 q^{13} + 3 q^{14} + 2 q^{16} + 5 q^{17} + 2 q^{18} + 3 q^{19} + 3 q^{21} + 3 q^{22} + 2 q^{24} - 2 q^{26} + 2 q^{27} + 3 q^{28} - q^{31} + 2 q^{32} + 3 q^{33} + 5 q^{34} + 2 q^{36} + 3 q^{37} + 3 q^{38} - 2 q^{39} - q^{41} + 3 q^{42} - 10 q^{43} + 3 q^{44} + 14 q^{47} + 2 q^{48} + 11 q^{49} + 5 q^{51} - 2 q^{52} - 8 q^{53} + 2 q^{54} + 3 q^{56} + 3 q^{57} - q^{59} + 9 q^{61} - q^{62} + 3 q^{63} + 2 q^{64} + 3 q^{66} + 5 q^{68} + 3 q^{71} + 2 q^{72} - 24 q^{73} + 3 q^{74} + 3 q^{76} + 25 q^{77} - 2 q^{78} - 5 q^{79} + 2 q^{81} - q^{82} - 15 q^{83} + 3 q^{84} - 10 q^{86} + 3 q^{88} + 10 q^{89} - 3 q^{91} - q^{93} + 14 q^{94} + 2 q^{96} - 18 q^{97} + 11 q^{98} + 3 q^{99}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^6 + 3 * q^7 + 2 * q^8 + 2 * q^9 + 3 * q^11 + 2 * q^12 - 2 * q^13 + 3 * q^14 + 2 * q^16 + 5 * q^17 + 2 * q^18 + 3 * q^19 + 3 * q^21 + 3 * q^22 + 2 * q^24 - 2 * q^26 + 2 * q^27 + 3 * q^28 - q^31 + 2 * q^32 + 3 * q^33 + 5 * q^34 + 2 * q^36 + 3 * q^37 + 3 * q^38 - 2 * q^39 - q^41 + 3 * q^42 - 10 * q^43 + 3 * q^44 + 14 * q^47 + 2 * q^48 + 11 * q^49 + 5 * q^51 - 2 * q^52 - 8 * q^53 + 2 * q^54 + 3 * q^56 + 3 * q^57 - q^59 + 9 * q^61 - q^62 + 3 * q^63 + 2 * q^64 + 3 * q^66 + 5 * q^68 + 3 * q^71 + 2 * q^72 - 24 * q^73 + 3 * q^74 + 3 * q^76 + 25 * q^77 - 2 * q^78 - 5 * q^79 + 2 * q^81 - q^82 - 15 * q^83 + 3 * q^84 - 10 * q^86 + 3 * q^88 + 10 * q^89 - 3 * q^91 - q^93 + 14 * q^94 + 2 * q^96 - 18 * q^97 + 11 * q^98 + 3 * 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	1.00000	0.408248
$7$	4.70156	1.77702	0.888512	−	0.458854i	$-0.151740\pi$
$7$	4.70156	1.77702	0.888512	+	0.458854i	$0.151740\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	4.70156	1.41757	0.708787	−	0.705422i	$-0.249243\pi$
$11$	4.70156	1.41757	0.708787	+	0.705422i	$0.249243\pi$
$12$	1.00000	0.288675
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	4.70156	1.25655
$15$	0	0
$16$	1.00000	0.250000
$17$	−0.701562	−0.170154	−0.0850769	−	0.996374i	$-0.527114\pi$
$17$	−0.701562	−0.170154	−0.0850769	+	0.996374i	$0.527114\pi$
$18$	1.00000	0.235702
$19$	−1.70156	−0.390365	−0.195183	−	0.980767i	$-0.562530\pi$
$19$	−1.70156	−0.390365	−0.195183	+	0.980767i	$0.562530\pi$
$20$	0	0
$21$	4.70156	1.02596
$22$	4.70156	1.00238
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	1.00000	0.204124
$25$	0	0
$26$	−1.00000	−0.196116
$27$	1.00000	0.192450
$28$	4.70156	0.888512
$29$	−6.40312	−1.18903	−0.594515	−	0.804084i	$-0.702656\pi$
$29$	−6.40312	−1.18903	−0.594515	+	0.804084i	$0.702656\pi$
$30$	0	0
$31$	−10.1047	−1.81486	−0.907428	−	0.420208i	$-0.861957\pi$
$31$	−10.1047	−1.81486	−0.907428	+	0.420208i	$0.861957\pi$
$32$	1.00000	0.176777
$33$	4.70156	0.818437
$34$	−0.701562	−0.120317
$35$	0	0
$36$	1.00000	0.166667
$37$	−1.70156	−0.279735	−0.139868	−	0.990170i	$-0.544668\pi$
$37$	−1.70156	−0.279735	−0.139868	+	0.990170i	$0.544668\pi$
$38$	−1.70156	−0.276030
$39$	−1.00000	−0.160128
$40$	0	0
$41$	−3.70156	−0.578087	−0.289043	−	0.957316i	$-0.593337\pi$
$41$	−3.70156	−0.578087	−0.289043	+	0.957316i	$0.593337\pi$
$42$	4.70156	0.725467
$43$	−11.4031	−1.73896	−0.869480	−	0.493968i	$-0.835546\pi$
$43$	−11.4031	−1.73896	−0.869480	+	0.493968i	$0.835546\pi$
$44$	4.70156	0.708787
$45$	0	0
$46$	0	0
$47$	7.00000	1.02105	0.510527	−	0.859861i	$-0.329450\pi$
$47$	7.00000	1.02105	0.510527	+	0.859861i	$0.329450\pi$
$48$	1.00000	0.144338
$49$	15.1047	2.15781
$50$	0	0
$51$	−0.701562	−0.0982383
$52$	−1.00000	−0.138675
$53$	2.40312	0.330095	0.165047	−	0.986286i	$-0.447222\pi$
$53$	2.40312	0.330095	0.165047	+	0.986286i	$0.447222\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	4.70156	0.628273
$57$	−1.70156	−0.225377
$58$	−6.40312	−0.840771
$59$	2.70156	0.351713	0.175857	−	0.984416i	$-0.443730\pi$
$59$	2.70156	0.351713	0.175857	+	0.984416i	$0.443730\pi$
$60$	0	0
$61$	14.1047	1.80592	0.902960	−	0.429725i	$-0.141389\pi$
$61$	14.1047	1.80592	0.902960	+	0.429725i	$0.141389\pi$
$62$	−10.1047	−1.28330
$63$	4.70156	0.592341
$64$	1.00000	0.125000
$65$	0	0
$66$	4.70156	0.578722
$67$	−6.40312	−0.782266	−0.391133	−	0.920334i	$-0.627917\pi$
$67$	−6.40312	−0.782266	−0.391133	+	0.920334i	$0.627917\pi$
$68$	−0.701562	−0.0850769
$69$	0	0
$70$	0	0
$71$	−1.70156	−0.201938	−0.100969	−	0.994890i	$-0.532194\pi$
$71$	−1.70156	−0.201938	−0.100969	+	0.994890i	$0.532194\pi$
$72$	1.00000	0.117851
$73$	−12.0000	−1.40449	−0.702247	−	0.711934i	$-0.747820\pi$
$73$	−12.0000	−1.40449	−0.702247	+	0.711934i	$0.747820\pi$
$74$	−1.70156	−0.197803
$75$	0	0
$76$	−1.70156	−0.195183
$77$	22.1047	2.51906
$78$	−1.00000	−0.113228
$79$	−5.70156	−0.641476	−0.320738	−	0.947168i	$-0.603931\pi$
$79$	−5.70156	−0.641476	−0.320738	+	0.947168i	$0.603931\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−3.70156	−0.408769
$83$	−10.7016	−1.17465	−0.587325	−	0.809352i	$-0.699819\pi$
$83$	−10.7016	−1.17465	−0.587325	+	0.809352i	$0.699819\pi$
$84$	4.70156	0.512982
$85$	0	0
$86$	−11.4031	−1.22963
$87$	−6.40312	−0.686487
$88$	4.70156	0.501188
$89$	11.4031	1.20873	0.604364	−	0.796708i	$-0.293427\pi$
$89$	11.4031	1.20873	0.604364	+	0.796708i	$0.293427\pi$
$90$	0	0
$91$	−4.70156	−0.492858
$92$	0	0
$93$	−10.1047	−1.04781
$94$	7.00000	0.721995
$95$	0	0
$96$	1.00000	0.102062
$97$	−2.59688	−0.263673	−0.131836	−	0.991271i	$-0.542087\pi$
$97$	−2.59688	−0.263673	−0.131836	+	0.991271i	$0.542087\pi$
$98$	15.1047	1.52580
$99$	4.70156	0.472525
$100$	0	0
$101$	6.70156	0.666830	0.333415	−	0.942780i	$-0.391799\pi$
$101$	6.70156	0.666830	0.333415	+	0.942780i	$0.391799\pi$
$102$	−0.701562	−0.0694650
$103$	1.40312	0.138254	0.0691270	−	0.997608i	$-0.477979\pi$
$103$	1.40312	0.138254	0.0691270	+	0.997608i	$0.477979\pi$
$104$	−1.00000	−0.0980581
$105$	0	0
$106$	2.40312	0.233412
$107$	19.1047	1.84692	0.923460	−	0.383695i	$-0.125349\pi$
$107$	19.1047	1.84692	0.923460	+	0.383695i	$0.125349\pi$
$108$	1.00000	0.0962250
$109$	4.29844	0.411716	0.205858	−	0.978582i	$-0.434002\pi$
$109$	4.29844	0.411716	0.205858	+	0.978582i	$0.434002\pi$
$110$	0	0
$111$	−1.70156	−0.161505
$112$	4.70156	0.444256
$113$	−14.0000	−1.31701	−0.658505	−	0.752577i	$-0.728811\pi$
$113$	−14.0000	−1.31701	−0.658505	+	0.752577i	$0.728811\pi$
$114$	−1.70156	−0.159366
$115$	0	0
$116$	−6.40312	−0.594515
$117$	−1.00000	−0.0924500
$118$	2.70156	0.248699
$119$	−3.29844	−0.302367
$120$	0	0
$121$	11.1047	1.00952
$122$	14.1047	1.27698
$123$	−3.70156	−0.333759
$124$	−10.1047	−0.907428
$125$	0	0
$126$	4.70156	0.418848
$127$	−6.29844	−0.558896	−0.279448	−	0.960161i	$-0.590151\pi$
$127$	−6.29844	−0.558896	−0.279448	+	0.960161i	$0.590151\pi$
$128$	1.00000	0.0883883
$129$	−11.4031	−1.00399
$130$	0	0
$131$	22.5078	1.96652	0.983258	−	0.182217i	$-0.0583275\pi$
$131$	22.5078	1.96652	0.983258	+	0.182217i	$0.0583275\pi$
$132$	4.70156	0.409218
$133$	−8.00000	−0.693688
$134$	−6.40312	−0.553146
$135$	0	0
$136$	−0.701562	−0.0601585
$137$	13.7016	1.17060	0.585302	−	0.810816i	$-0.300976\pi$
$137$	13.7016	1.17060	0.585302	+	0.810816i	$0.300976\pi$
$138$	0	0
$139$	9.40312	0.797563	0.398781	−	0.917046i	$-0.369433\pi$
$139$	9.40312	0.797563	0.398781	+	0.917046i	$0.369433\pi$
$140$	0	0
$141$	7.00000	0.589506
$142$	−1.70156	−0.142792
$143$	−4.70156	−0.393164
$144$	1.00000	0.0833333
$145$	0	0
$146$	−12.0000	−0.993127
$147$	15.1047	1.24581
$148$	−1.70156	−0.139868
$149$	6.59688	0.540437	0.270219	−	0.962799i	$-0.412904\pi$
$149$	6.59688	0.540437	0.270219	+	0.962799i	$0.412904\pi$
$150$	0	0
$151$	−14.1047	−1.14782	−0.573912	−	0.818917i	$-0.694575\pi$
$151$	−14.1047	−1.14782	−0.573912	+	0.818917i	$0.694575\pi$
$152$	−1.70156	−0.138015
$153$	−0.701562	−0.0567179
$154$	22.1047	1.78125
$155$	0	0
$156$	−1.00000	−0.0800641
$157$	−22.7016	−1.81178	−0.905891	−	0.423511i	$-0.860797\pi$
$157$	−22.7016	−1.81178	−0.905891	+	0.423511i	$0.860797\pi$
$158$	−5.70156	−0.453592
$159$	2.40312	0.190580
$160$	0	0
$161$	0	0
$162$	1.00000	0.0785674
$163$	−18.8062	−1.47302	−0.736510	−	0.676427i	$-0.763527\pi$
$163$	−18.8062	−1.47302	−0.736510	+	0.676427i	$0.763527\pi$
$164$	−3.70156	−0.289043
$165$	0	0
$166$	−10.7016	−0.830602
$167$	−1.10469	−0.0854832	−0.0427416	−	0.999086i	$-0.513609\pi$
$167$	−1.10469	−0.0854832	−0.0427416	+	0.999086i	$0.513609\pi$
$168$	4.70156	0.362733
$169$	1.00000	0.0769231
$170$	0	0
$171$	−1.70156	−0.130122
$172$	−11.4031	−0.869480
$173$	−0.193752	−0.0147307	−0.00736533	−	0.999973i	$-0.502344\pi$
$173$	−0.193752	−0.0147307	−0.00736533	+	0.999973i	$0.502344\pi$
$174$	−6.40312	−0.485420
$175$	0	0
$176$	4.70156	0.354394
$177$	2.70156	0.203062
$178$	11.4031	0.854700
$179$	−24.2094	−1.80949	−0.904747	−	0.425950i	$-0.859940\pi$
$179$	−24.2094	−1.80949	−0.904747	+	0.425950i	$0.859940\pi$
$180$	0	0
$181$	11.2984	0.839806	0.419903	−	0.907569i	$-0.362064\pi$
$181$	11.2984	0.839806	0.419903	+	0.907569i	$0.362064\pi$
$182$	−4.70156	−0.348503
$183$	14.1047	1.04265
$184$	0	0
$185$	0	0
$186$	−10.1047	−0.740912
$187$	−3.29844	−0.241206
$188$	7.00000	0.510527
$189$	4.70156	0.341988
$190$	0	0
$191$	−12.8062	−0.926628	−0.463314	−	0.886194i	$-0.653340\pi$
$191$	−12.8062	−0.926628	−0.463314	+	0.886194i	$0.653340\pi$
$192$	1.00000	0.0721688
$193$	16.2094	1.16678	0.583388	−	0.812194i	$-0.301727\pi$
$193$	16.2094	1.16678	0.583388	+	0.812194i	$0.301727\pi$
$194$	−2.59688	−0.186445
$195$	0	0
$196$	15.1047	1.07891
$197$	−5.40312	−0.384957	−0.192478	−	0.981301i	$-0.561653\pi$
$197$	−5.40312	−0.384957	−0.192478	+	0.981301i	$0.561653\pi$
$198$	4.70156	0.334125
$199$	8.29844	0.588261	0.294130	−	0.955765i	$-0.404970\pi$
$199$	8.29844	0.588261	0.294130	+	0.955765i	$0.404970\pi$
$200$	0	0
$201$	−6.40312	−0.451642
$202$	6.70156	0.471520
$203$	−30.1047	−2.11293
$204$	−0.701562	−0.0491192
$205$	0	0
$206$	1.40312	0.0977603
$207$	0	0
$208$	−1.00000	−0.0693375
$209$	−8.00000	−0.553372
$210$	0	0
$211$	6.80625	0.468561	0.234281	−	0.972169i	$-0.424727\pi$
$211$	6.80625	0.468561	0.234281	+	0.972169i	$0.424727\pi$
$212$	2.40312	0.165047
$213$	−1.70156	−0.116589
$214$	19.1047	1.30597
$215$	0	0
$216$	1.00000	0.0680414
$217$	−47.5078	−3.22504
$218$	4.29844	0.291127
$219$	−12.0000	−0.810885
$220$	0	0
$221$	0.701562	0.0471922
$222$	−1.70156	−0.114201
$223$	11.4031	0.763610	0.381805	−	0.924243i	$-0.375303\pi$
$223$	11.4031	0.763610	0.381805	+	0.924243i	$0.375303\pi$
$224$	4.70156	0.314136
$225$	0	0
$226$	−14.0000	−0.931266
$227$	−16.1047	−1.06891	−0.534453	−	0.845198i	$-0.679482\pi$
$227$	−16.1047	−1.06891	−0.534453	+	0.845198i	$0.679482\pi$
$228$	−1.70156	−0.112689
$229$	15.7016	1.03759	0.518794	−	0.854899i	$-0.326381\pi$
$229$	15.7016	1.03759	0.518794	+	0.854899i	$0.326381\pi$
$230$	0	0
$231$	22.1047	1.45438
$232$	−6.40312	−0.420386
$233$	−20.2094	−1.32396	−0.661980	−	0.749521i	$-0.730284\pi$
$233$	−20.2094	−1.32396	−0.661980	+	0.749521i	$0.730284\pi$
$234$	−1.00000	−0.0653720
$235$	0	0
$236$	2.70156	0.175857
$237$	−5.70156	−0.370356
$238$	−3.29844	−0.213806
$239$	6.10469	0.394879	0.197440	−	0.980315i	$-0.436737\pi$
$239$	6.10469	0.394879	0.197440	+	0.980315i	$0.436737\pi$
$240$	0	0
$241$	−2.59688	−0.167279	−0.0836397	−	0.996496i	$-0.526654\pi$
$241$	−2.59688	−0.167279	−0.0836397	+	0.996496i	$0.526654\pi$
$242$	11.1047	0.713836
$243$	1.00000	0.0641500
$244$	14.1047	0.902960
$245$	0	0
$246$	−3.70156	−0.236003
$247$	1.70156	0.108268
$248$	−10.1047	−0.641648
$249$	−10.7016	−0.678184
$250$	0	0
$251$	0.298438	0.0188372	0.00941862	−	0.999956i	$-0.497002\pi$
$251$	0.298438	0.0188372	0.00941862	+	0.999956i	$0.497002\pi$
$252$	4.70156	0.296171
$253$	0	0
$254$	−6.29844	−0.395199
$255$	0	0
$256$	1.00000	0.0625000
$257$	−15.2984	−0.954290	−0.477145	−	0.878824i	$-0.658328\pi$
$257$	−15.2984	−0.954290	−0.477145	+	0.878824i	$0.658328\pi$
$258$	−11.4031	−0.709928
$259$	−8.00000	−0.497096
$260$	0	0
$261$	−6.40312	−0.396343
$262$	22.5078	1.39054
$263$	2.00000	0.123325	0.0616626	−	0.998097i	$-0.480360\pi$
$263$	2.00000	0.123325	0.0616626	+	0.998097i	$0.480360\pi$
$264$	4.70156	0.289361
$265$	0	0
$266$	−8.00000	−0.490511
$267$	11.4031	0.697860
$268$	−6.40312	−0.391133
$269$	31.2094	1.90287	0.951435	−	0.307851i	$-0.0996099\pi$
$269$	31.2094	1.90287	0.951435	+	0.307851i	$0.0996099\pi$
$270$	0	0
$271$	19.5078	1.18502	0.592508	−	0.805565i	$-0.298138\pi$
$271$	19.5078	1.18502	0.592508	+	0.805565i	$0.298138\pi$
$272$	−0.701562	−0.0425385
$273$	−4.70156	−0.284551
$274$	13.7016	0.827742
$275$	0	0
$276$	0	0
$277$	−22.0000	−1.32185	−0.660926	−	0.750451i	$-0.729836\pi$
$277$	−22.0000	−1.32185	−0.660926	+	0.750451i	$0.729836\pi$
$278$	9.40312	0.563962
$279$	−10.1047	−0.604952
$280$	0	0
$281$	−21.9109	−1.30710	−0.653548	−	0.756885i	$-0.726720\pi$
$281$	−21.9109	−1.30710	−0.653548	+	0.756885i	$0.726720\pi$
$282$	7.00000	0.416844
$283$	21.4031	1.27228	0.636142	−	0.771572i	$-0.280529\pi$
$283$	21.4031	1.27228	0.636142	+	0.771572i	$0.280529\pi$
$284$	−1.70156	−0.100969
$285$	0	0
$286$	−4.70156	−0.278009
$287$	−17.4031	−1.02727
$288$	1.00000	0.0589256
$289$	−16.5078	−0.971048
$290$	0	0
$291$	−2.59688	−0.152232
$292$	−12.0000	−0.702247
$293$	21.4031	1.25038	0.625192	−	0.780471i	$-0.285021\pi$
$293$	21.4031	1.25038	0.625192	+	0.780471i	$0.285021\pi$
$294$	15.1047	0.880923
$295$	0	0
$296$	−1.70156	−0.0989013
$297$	4.70156	0.272812
$298$	6.59688	0.382147
$299$	0	0
$300$	0	0
$301$	−53.6125	−3.09017
$302$	−14.1047	−0.811633
$303$	6.70156	0.384995
$304$	−1.70156	−0.0975913
$305$	0	0
$306$	−0.701562	−0.0401056
$307$	−5.70156	−0.325405	−0.162703	−	0.986675i	$-0.552021\pi$
$307$	−5.70156	−0.325405	−0.162703	+	0.986675i	$0.552021\pi$
$308$	22.1047	1.25953
$309$	1.40312	0.0798209
$310$	0	0
$311$	30.0000	1.70114	0.850572	−	0.525859i	$-0.176256\pi$
$311$	30.0000	1.70114	0.850572	+	0.525859i	$0.176256\pi$
$312$	−1.00000	−0.0566139
$313$	−21.2094	−1.19882	−0.599412	−	0.800440i	$-0.704599\pi$
$313$	−21.2094	−1.19882	−0.599412	+	0.800440i	$0.704599\pi$
$314$	−22.7016	−1.28112
$315$	0	0
$316$	−5.70156	−0.320738
$317$	1.19375	0.0670478	0.0335239	−	0.999438i	$-0.489327\pi$
$317$	1.19375	0.0670478	0.0335239	+	0.999438i	$0.489327\pi$
$318$	2.40312	0.134761
$319$	−30.1047	−1.68554
$320$	0	0
$321$	19.1047	1.06632
$322$	0	0
$323$	1.19375	0.0664221
$324$	1.00000	0.0555556
$325$	0	0
$326$	−18.8062	−1.04158
$327$	4.29844	0.237704
$328$	−3.70156	−0.204385
$329$	32.9109	1.81444
$330$	0	0
$331$	−12.0000	−0.659580	−0.329790	−	0.944054i	$-0.606978\pi$
$331$	−12.0000	−0.659580	−0.329790	+	0.944054i	$0.606978\pi$
$332$	−10.7016	−0.587325
$333$	−1.70156	−0.0932450
$334$	−1.10469	−0.0604457
$335$	0	0
$336$	4.70156	0.256491
$337$	8.10469	0.441490	0.220745	−	0.975332i	$-0.429151\pi$
$337$	8.10469	0.441490	0.220745	+	0.975332i	$0.429151\pi$
$338$	1.00000	0.0543928
$339$	−14.0000	−0.760376
$340$	0	0
$341$	−47.5078	−2.57269
$342$	−1.70156	−0.0920099
$343$	38.1047	2.05746
$344$	−11.4031	−0.614815
$345$	0	0
$346$	−0.193752	−0.0104161
$347$	10.5078	0.564089	0.282044	−	0.959401i	$-0.408987\pi$
$347$	10.5078	0.564089	0.282044	+	0.959401i	$0.408987\pi$
$348$	−6.40312	−0.343243
$349$	−11.4031	−0.610395	−0.305198	−	0.952289i	$-0.598723\pi$
$349$	−11.4031	−0.610395	−0.305198	+	0.952289i	$0.598723\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	4.70156	0.250594
$353$	14.5078	0.772173	0.386086	−	0.922463i	$-0.373827\pi$
$353$	14.5078	0.772173	0.386086	+	0.922463i	$0.373827\pi$
$354$	2.70156	0.143586
$355$	0	0
$356$	11.4031	0.604364
$357$	−3.29844	−0.174572
$358$	−24.2094	−1.27951
$359$	−34.6125	−1.82678	−0.913389	−	0.407088i	$-0.866544\pi$
$359$	−34.6125	−1.82678	−0.913389	+	0.407088i	$0.866544\pi$
$360$	0	0
$361$	−16.1047	−0.847615
$362$	11.2984	0.593833
$363$	11.1047	0.582845
$364$	−4.70156	−0.246429
$365$	0	0
$366$	14.1047	0.737264
$367$	−2.29844	−0.119977	−0.0599887	−	0.998199i	$-0.519106\pi$
$367$	−2.29844	−0.119977	−0.0599887	+	0.998199i	$0.519106\pi$
$368$	0	0
$369$	−3.70156	−0.192696
$370$	0	0
$371$	11.2984	0.586586
$372$	−10.1047	−0.523904
$373$	5.29844	0.274343	0.137171	−	0.990547i	$-0.456199\pi$
$373$	5.29844	0.274343	0.137171	+	0.990547i	$0.456199\pi$
$374$	−3.29844	−0.170558
$375$	0	0
$376$	7.00000	0.360997
$377$	6.40312	0.329778
$378$	4.70156	0.241822
$379$	−13.8953	−0.713754	−0.356877	−	0.934151i	$-0.616159\pi$
$379$	−13.8953	−0.713754	−0.356877	+	0.934151i	$0.616159\pi$
$380$	0	0
$381$	−6.29844	−0.322679
$382$	−12.8062	−0.655225
$383$	32.5078	1.66107	0.830536	−	0.556965i	$-0.188034\pi$
$383$	32.5078	1.66107	0.830536	+	0.556965i	$0.188034\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	16.2094	0.825035
$387$	−11.4031	−0.579653
$388$	−2.59688	−0.131836
$389$	−27.1047	−1.37426	−0.687131	−	0.726533i	$-0.741130\pi$
$389$	−27.1047	−1.37426	−0.687131	+	0.726533i	$0.741130\pi$
$390$	0	0
$391$	0	0
$392$	15.1047	0.762902
$393$	22.5078	1.13537
$394$	−5.40312	−0.272205
$395$	0	0
$396$	4.70156	0.236262
$397$	−21.9109	−1.09968	−0.549839	−	0.835271i	$-0.685311\pi$
$397$	−21.9109	−1.09968	−0.549839	+	0.835271i	$0.685311\pi$
$398$	8.29844	0.415963
$399$	−8.00000	−0.400501
$400$	0	0
$401$	18.2094	0.909333	0.454666	−	0.890662i	$-0.349759\pi$
$401$	18.2094	0.909333	0.454666	+	0.890662i	$0.349759\pi$
$402$	−6.40312	−0.319359
$403$	10.1047	0.503350
$404$	6.70156	0.333415
$405$	0	0
$406$	−30.1047	−1.49407
$407$	−8.00000	−0.396545
$408$	−0.701562	−0.0347325
$409$	29.4031	1.45389	0.726945	−	0.686695i	$-0.240939\pi$
$409$	29.4031	1.45389	0.726945	+	0.686695i	$0.240939\pi$
$410$	0	0
$411$	13.7016	0.675848
$412$	1.40312	0.0691270
$413$	12.7016	0.625003
$414$	0	0
$415$	0	0
$416$	−1.00000	−0.0490290
$417$	9.40312	0.460473
$418$	−8.00000	−0.391293
$419$	9.91093	0.484181	0.242090	−	0.970254i	$-0.422167\pi$
$419$	9.91093	0.484181	0.242090	+	0.970254i	$0.422167\pi$
$420$	0	0
$421$	−0.596876	−0.0290899	−0.0145450	−	0.999894i	$-0.504630\pi$
$421$	−0.596876	−0.0290899	−0.0145450	+	0.999894i	$0.504630\pi$
$422$	6.80625	0.331323
$423$	7.00000	0.340352
$424$	2.40312	0.116706
$425$	0	0
$426$	−1.70156	−0.0824410
$427$	66.3141	3.20916
$428$	19.1047	0.923460
$429$	−4.70156	−0.226994
$430$	0	0
$431$	35.3141	1.70102	0.850509	−	0.525960i	$-0.176294\pi$
$431$	35.3141	1.70102	0.850509	+	0.525960i	$0.176294\pi$
$432$	1.00000	0.0481125
$433$	−11.1047	−0.533657	−0.266829	−	0.963744i	$-0.585976\pi$
$433$	−11.1047	−0.533657	−0.266829	+	0.963744i	$0.585976\pi$
$434$	−47.5078	−2.28045
$435$	0	0
$436$	4.29844	0.205858
$437$	0	0
$438$	−12.0000	−0.573382
$439$	−10.2984	−0.491518	−0.245759	−	0.969331i	$-0.579037\pi$
$439$	−10.2984	−0.491518	−0.245759	+	0.969331i	$0.579037\pi$
$440$	0	0
$441$	15.1047	0.719271
$442$	0.701562	0.0333699
$443$	32.5078	1.54449	0.772246	−	0.635323i	$-0.219133\pi$
$443$	32.5078	1.54449	0.772246	+	0.635323i	$0.219133\pi$
$444$	−1.70156	−0.0807526
$445$	0	0
$446$	11.4031	0.539954
$447$	6.59688	0.312022
$448$	4.70156	0.222128
$449$	2.29844	0.108470	0.0542350	−	0.998528i	$-0.482728\pi$
$449$	2.29844	0.108470	0.0542350	+	0.998528i	$0.482728\pi$
$450$	0	0
$451$	−17.4031	−0.819481
$452$	−14.0000	−0.658505
$453$	−14.1047	−0.662696
$454$	−16.1047	−0.755830
$455$	0	0
$456$	−1.70156	−0.0796829
$457$	−2.59688	−0.121477	−0.0607384	−	0.998154i	$-0.519346\pi$
$457$	−2.59688	−0.121477	−0.0607384	+	0.998154i	$0.519346\pi$
$458$	15.7016	0.733686
$459$	−0.701562	−0.0327461
$460$	0	0
$461$	36.2094	1.68644	0.843219	−	0.537570i	$-0.180658\pi$
$461$	36.2094	1.68644	0.843219	+	0.537570i	$0.180658\pi$
$462$	22.1047	1.02840
$463$	−21.2984	−0.989822	−0.494911	−	0.868944i	$-0.664799\pi$
$463$	−21.2984	−0.989822	−0.494911	+	0.868944i	$0.664799\pi$
$464$	−6.40312	−0.297258
$465$	0	0
$466$	−20.2094	−0.936181
$467$	30.2984	1.40204	0.701022	−	0.713139i	$-0.252727\pi$
$467$	30.2984	1.40204	0.701022	+	0.713139i	$0.252727\pi$
$468$	−1.00000	−0.0462250
$469$	−30.1047	−1.39011
$470$	0	0
$471$	−22.7016	−1.04603
$472$	2.70156	0.124349
$473$	−53.6125	−2.46511
$474$	−5.70156	−0.261881
$475$	0	0
$476$	−3.29844	−0.151184
$477$	2.40312	0.110032
$478$	6.10469	0.279222
$479$	−40.6125	−1.85563	−0.927816	−	0.373038i	$-0.878316\pi$
$479$	−40.6125	−1.85563	−0.927816	+	0.373038i	$0.878316\pi$
$480$	0	0
$481$	1.70156	0.0775846
$482$	−2.59688	−0.118284
$483$	0	0
$484$	11.1047	0.504758
$485$	0	0
$486$	1.00000	0.0453609
$487$	−7.89531	−0.357771	−0.178885	−	0.983870i	$-0.557249\pi$
$487$	−7.89531	−0.357771	−0.178885	+	0.983870i	$0.557249\pi$
$488$	14.1047	0.638489
$489$	−18.8062	−0.850448
$490$	0	0
$491$	33.4031	1.50746	0.753731	−	0.657183i	$-0.228252\pi$
$491$	33.4031	1.50746	0.753731	+	0.657183i	$0.228252\pi$
$492$	−3.70156	−0.166879
$493$	4.49219	0.202318
$494$	1.70156	0.0765569
$495$	0	0
$496$	−10.1047	−0.453714
$497$	−8.00000	−0.358849
$498$	−10.7016	−0.479548
$499$	17.2094	0.770397	0.385199	−	0.922834i	$-0.374133\pi$
$499$	17.2094	0.770397	0.385199	+	0.922834i	$0.374133\pi$
$500$	0	0
$501$	−1.10469	−0.0493537
$502$	0.298438	0.0133199
$503$	−12.2094	−0.544389	−0.272195	−	0.962242i	$-0.587749\pi$
$503$	−12.2094	−0.544389	−0.272195	+	0.962242i	$0.587749\pi$
$504$	4.70156	0.209424
$505$	0	0
$506$	0	0
$507$	1.00000	0.0444116
$508$	−6.29844	−0.279448
$509$	−5.40312	−0.239489	−0.119745	−	0.992805i	$-0.538208\pi$
$509$	−5.40312	−0.239489	−0.119745	+	0.992805i	$0.538208\pi$
$510$	0	0
$511$	−56.4187	−2.49582
$512$	1.00000	0.0441942
$513$	−1.70156	−0.0751258
$514$	−15.2984	−0.674785
$515$	0	0
$516$	−11.4031	−0.501995
$517$	32.9109	1.44742
$518$	−8.00000	−0.351500
$519$	−0.193752	−0.00850475
$520$	0	0
$521$	−36.2094	−1.58636	−0.793181	−	0.608986i	$-0.791576\pi$
$521$	−36.2094	−1.58636	−0.793181	+	0.608986i	$0.791576\pi$
$522$	−6.40312	−0.280257
$523$	−24.8062	−1.08470	−0.542351	−	0.840152i	$-0.682466\pi$
$523$	−24.8062	−1.08470	−0.542351	+	0.840152i	$0.682466\pi$
$524$	22.5078	0.983258
$525$	0	0
$526$	2.00000	0.0872041
$527$	7.08907	0.308805
$528$	4.70156	0.204609
$529$	−23.0000	−1.00000
$530$	0	0
$531$	2.70156	0.117238
$532$	−8.00000	−0.346844
$533$	3.70156	0.160332
$534$	11.4031	0.493461
$535$	0	0
$536$	−6.40312	−0.276573
$537$	−24.2094	−1.04471
$538$	31.2094	1.34553
$539$	71.0156	3.05886
$540$	0	0
$541$	−1.79063	−0.0769851	−0.0384925	−	0.999259i	$-0.512256\pi$
$541$	−1.79063	−0.0769851	−0.0384925	+	0.999259i	$0.512256\pi$
$542$	19.5078	0.837932
$543$	11.2984	0.484862
$544$	−0.701562	−0.0300792
$545$	0	0
$546$	−4.70156	−0.201208
$547$	−31.6125	−1.35165	−0.675826	−	0.737061i	$-0.736213\pi$
$547$	−31.6125	−1.35165	−0.675826	+	0.737061i	$0.736213\pi$
$548$	13.7016	0.585302
$549$	14.1047	0.601973
$550$	0	0
$551$	10.8953	0.464156
$552$	0	0
$553$	−26.8062	−1.13992
$554$	−22.0000	−0.934690
$555$	0	0
$556$	9.40312	0.398781
$557$	16.8062	0.712104	0.356052	−	0.934466i	$-0.384123\pi$
$557$	16.8062	0.712104	0.356052	+	0.934466i	$0.384123\pi$
$558$	−10.1047	−0.427765
$559$	11.4031	0.482301
$560$	0	0
$561$	−3.29844	−0.139260
$562$	−21.9109	−0.924257
$563$	−32.5078	−1.37004	−0.685020	−	0.728524i	$-0.740207\pi$
$563$	−32.5078	−1.37004	−0.685020	+	0.728524i	$0.740207\pi$
$564$	7.00000	0.294753
$565$	0	0
$566$	21.4031	0.899640
$567$	4.70156	0.197447
$568$	−1.70156	−0.0713960
$569$	−11.2984	−0.473655	−0.236828	−	0.971552i	$-0.576108\pi$
$569$	−11.2984	−0.473655	−0.236828	+	0.971552i	$0.576108\pi$
$570$	0	0
$571$	40.0000	1.67395	0.836974	−	0.547243i	$-0.184323\pi$
$571$	40.0000	1.67395	0.836974	+	0.547243i	$0.184323\pi$
$572$	−4.70156	−0.196582
$573$	−12.8062	−0.534989
$574$	−17.4031	−0.726392
$575$	0	0
$576$	1.00000	0.0416667
$577$	24.2094	1.00785	0.503925	−	0.863748i	$-0.331889\pi$
$577$	24.2094	1.00785	0.503925	+	0.863748i	$0.331889\pi$
$578$	−16.5078	−0.686634
$579$	16.2094	0.673639
$580$	0	0
$581$	−50.3141	−2.08738
$582$	−2.59688	−0.107644
$583$	11.2984	0.467933
$584$	−12.0000	−0.496564
$585$	0	0
$586$	21.4031	0.884155
$587$	−7.50781	−0.309881	−0.154940	−	0.987924i	$-0.549519\pi$
$587$	−7.50781	−0.309881	−0.154940	+	0.987924i	$0.549519\pi$
$588$	15.1047	0.622907
$589$	17.1938	0.708456
$590$	0	0
$591$	−5.40312	−0.222255
$592$	−1.70156	−0.0699338
$593$	17.9109	0.735514	0.367757	−	0.929922i	$-0.380126\pi$
$593$	17.9109	0.735514	0.367757	+	0.929922i	$0.380126\pi$
$594$	4.70156	0.192907
$595$	0	0
$596$	6.59688	0.270219
$597$	8.29844	0.339632
$598$	0	0
$599$	8.20937	0.335426	0.167713	−	0.985836i	$-0.446362\pi$
$599$	8.20937	0.335426	0.167713	+	0.985836i	$0.446362\pi$
$600$	0	0
$601$	10.6125	0.432893	0.216446	−	0.976295i	$-0.430553\pi$
$601$	10.6125	0.432893	0.216446	+	0.976295i	$0.430553\pi$
$602$	−53.6125	−2.18508
$603$	−6.40312	−0.260755
$604$	−14.1047	−0.573912
$605$	0	0
$606$	6.70156	0.272232
$607$	−35.1047	−1.42486	−0.712428	−	0.701746i	$-0.752404\pi$
$607$	−35.1047	−1.42486	−0.712428	+	0.701746i	$0.752404\pi$
$608$	−1.70156	−0.0690075
$609$	−30.1047	−1.21990
$610$	0	0
$611$	−7.00000	−0.283190
$612$	−0.701562	−0.0283590
$613$	−12.8062	−0.517240	−0.258620	−	0.965979i	$-0.583268\pi$
$613$	−12.8062	−0.517240	−0.258620	+	0.965979i	$0.583268\pi$
$614$	−5.70156	−0.230096
$615$	0	0
$616$	22.1047	0.890623
$617$	−27.1047	−1.09119	−0.545597	−	0.838048i	$-0.683697\pi$
$617$	−27.1047	−1.09119	−0.545597	+	0.838048i	$0.683697\pi$
$618$	1.40312	0.0564419
$619$	−13.1938	−0.530302	−0.265151	−	0.964207i	$-0.585422\pi$
$619$	−13.1938	−0.530302	−0.265151	+	0.964207i	$0.585422\pi$
$620$	0	0
$621$	0	0
$622$	30.0000	1.20289
$623$	53.6125	2.14794
$624$	−1.00000	−0.0400320
$625$	0	0
$626$	−21.2094	−0.847697
$627$	−8.00000	−0.319489
$628$	−22.7016	−0.905891
$629$	1.19375	0.0475980
$630$	0	0
$631$	33.6125	1.33809	0.669046	−	0.743221i	$-0.266703\pi$
$631$	33.6125	1.33809	0.669046	+	0.743221i	$0.266703\pi$
$632$	−5.70156	−0.226796
$633$	6.80625	0.270524
$634$	1.19375	0.0474099
$635$	0	0
$636$	2.40312	0.0952901
$637$	−15.1047	−0.598469
$638$	−30.1047	−1.19186
$639$	−1.70156	−0.0673128
$640$	0	0
$641$	5.50781	0.217545	0.108773	−	0.994067i	$-0.465308\pi$
$641$	5.50781	0.217545	0.108773	+	0.994067i	$0.465308\pi$
$642$	19.1047	0.754002
$643$	−10.2984	−0.406131	−0.203065	−	0.979165i	$-0.565090\pi$
$643$	−10.2984	−0.406131	−0.203065	+	0.979165i	$0.565090\pi$
$644$	0	0
$645$	0	0
$646$	1.19375	0.0469675
$647$	−24.0000	−0.943537	−0.471769	−	0.881722i	$-0.656384\pi$
$647$	−24.0000	−0.943537	−0.471769	+	0.881722i	$0.656384\pi$
$648$	1.00000	0.0392837
$649$	12.7016	0.498580
$650$	0	0
$651$	−47.5078	−1.86198
$652$	−18.8062	−0.736510
$653$	−21.2984	−0.833472	−0.416736	−	0.909027i	$-0.636826\pi$
$653$	−21.2984	−0.833472	−0.416736	+	0.909027i	$0.636826\pi$
$654$	4.29844	0.168082
$655$	0	0
$656$	−3.70156	−0.144522
$657$	−12.0000	−0.468165
$658$	32.9109	1.28300
$659$	−19.3141	−0.752369	−0.376184	−	0.926545i	$-0.622764\pi$
$659$	−19.3141	−0.752369	−0.376184	+	0.926545i	$0.622764\pi$
$660$	0	0
$661$	−40.1203	−1.56050	−0.780250	−	0.625468i	$-0.784908\pi$
$661$	−40.1203	−1.56050	−0.780250	+	0.625468i	$0.784908\pi$
$662$	−12.0000	−0.466393
$663$	0.701562	0.0272464
$664$	−10.7016	−0.415301
$665$	0	0
$666$	−1.70156	−0.0659342
$667$	0	0
$668$	−1.10469	−0.0427416
$669$	11.4031	0.440870
$670$	0	0
$671$	66.3141	2.56003
$672$	4.70156	0.181367
$673$	36.0156	1.38830	0.694150	−	0.719830i	$-0.255780\pi$
$673$	36.0156	1.38830	0.694150	+	0.719830i	$0.255780\pi$
$674$	8.10469	0.312181
$675$	0	0
$676$	1.00000	0.0384615
$677$	31.6125	1.21497	0.607483	−	0.794332i	$-0.292179\pi$
$677$	31.6125	1.21497	0.607483	+	0.794332i	$0.292179\pi$
$678$	−14.0000	−0.537667
$679$	−12.2094	−0.468553
$680$	0	0
$681$	−16.1047	−0.617133
$682$	−47.5078	−1.81917
$683$	44.7016	1.71046	0.855229	−	0.518251i	$-0.173417\pi$
$683$	44.7016	1.71046	0.855229	+	0.518251i	$0.173417\pi$
$684$	−1.70156	−0.0650609
$685$	0	0
$686$	38.1047	1.45484
$687$	15.7016	0.599052
$688$	−11.4031	−0.434740
$689$	−2.40312	−0.0915517
$690$	0	0
$691$	−22.1938	−0.844290	−0.422145	−	0.906528i	$-0.638723\pi$
$691$	−22.1938	−0.844290	−0.422145	+	0.906528i	$0.638723\pi$
$692$	−0.193752	−0.00736533
$693$	22.1047	0.839688
$694$	10.5078	0.398871
$695$	0	0
$696$	−6.40312	−0.242710
$697$	2.59688	0.0983637
$698$	−11.4031	−0.431615
$699$	−20.2094	−0.764389
$700$	0	0
$701$	30.9109	1.16749	0.583745	−	0.811937i	$-0.301587\pi$
$701$	30.9109	1.16749	0.583745	+	0.811937i	$0.301587\pi$
$702$	−1.00000	−0.0377426
$703$	2.89531	0.109199
$704$	4.70156	0.177197
$705$	0	0
$706$	14.5078	0.546009
$707$	31.5078	1.18497
$708$	2.70156	0.101531
$709$	15.6125	0.586340	0.293170	−	0.956060i	$-0.405290\pi$
$709$	15.6125	0.586340	0.293170	+	0.956060i	$0.405290\pi$
$710$	0	0
$711$	−5.70156	−0.213825
$712$	11.4031	0.427350
$713$	0	0
$714$	−3.29844	−0.123441
$715$	0	0
$716$	−24.2094	−0.904747
$717$	6.10469	0.227984
$718$	−34.6125	−1.29173
$719$	−11.0156	−0.410813	−0.205407	−	0.978677i	$-0.565852\pi$
$719$	−11.0156	−0.410813	−0.205407	+	0.978677i	$0.565852\pi$
$720$	0	0
$721$	6.59688	0.245680
$722$	−16.1047	−0.599354
$723$	−2.59688	−0.0965788
$724$	11.2984	0.419903
$725$	0	0
$726$	11.1047	0.412134
$727$	7.79063	0.288938	0.144469	−	0.989509i	$-0.453853\pi$
$727$	7.79063	0.288938	0.144469	+	0.989509i	$0.453853\pi$
$728$	−4.70156	−0.174251
$729$	1.00000	0.0370370
$730$	0	0
$731$	8.00000	0.295891
$732$	14.1047	0.521324
$733$	47.9109	1.76963	0.884815	−	0.465942i	$-0.154284\pi$
$733$	47.9109	1.76963	0.884815	+	0.465942i	$0.154284\pi$
$734$	−2.29844	−0.0848369
$735$	0	0
$736$	0	0
$737$	−30.1047	−1.10892
$738$	−3.70156	−0.136256
$739$	4.61250	0.169673	0.0848367	−	0.996395i	$-0.472963\pi$
$739$	4.61250	0.169673	0.0848367	+	0.996395i	$0.472963\pi$
$740$	0	0
$741$	1.70156	0.0625084
$742$	11.2984	0.414779
$743$	−30.6125	−1.12306	−0.561532	−	0.827455i	$-0.689788\pi$
$743$	−30.6125	−1.12306	−0.561532	+	0.827455i	$0.689788\pi$
$744$	−10.1047	−0.370456
$745$	0	0
$746$	5.29844	0.193990
$747$	−10.7016	−0.391550
$748$	−3.29844	−0.120603
$749$	89.8219	3.28202
$750$	0	0
$751$	8.50781	0.310454	0.155227	−	0.987879i	$-0.450389\pi$
$751$	8.50781	0.310454	0.155227	+	0.987879i	$0.450389\pi$
$752$	7.00000	0.255264
$753$	0.298438	0.0108757
$754$	6.40312	0.233188
$755$	0	0
$756$	4.70156	0.170994
$757$	−17.8953	−0.650416	−0.325208	−	0.945642i	$-0.605434\pi$
$757$	−17.8953	−0.650416	−0.325208	+	0.945642i	$0.605434\pi$
$758$	−13.8953	−0.504701
$759$	0	0
$760$	0	0
$761$	−26.7172	−0.968497	−0.484249	−	0.874930i	$-0.660907\pi$
$761$	−26.7172	−0.968497	−0.484249	+	0.874930i	$0.660907\pi$
$762$	−6.29844	−0.228168
$763$	20.2094	0.731628
$764$	−12.8062	−0.463314
$765$	0	0
$766$	32.5078	1.17455
$767$	−2.70156	−0.0975478
$768$	1.00000	0.0360844
$769$	−8.00000	−0.288487	−0.144244	−	0.989542i	$-0.546075\pi$
$769$	−8.00000	−0.288487	−0.144244	+	0.989542i	$0.546075\pi$
$770$	0	0
$771$	−15.2984	−0.550960
$772$	16.2094	0.583388
$773$	26.5969	0.956623	0.478312	−	0.878190i	$-0.341249\pi$
$773$	26.5969	0.956623	0.478312	+	0.878190i	$0.341249\pi$
$774$	−11.4031	−0.409877
$775$	0	0
$776$	−2.59688	−0.0932224
$777$	−8.00000	−0.286998
$778$	−27.1047	−0.971750
$779$	6.29844	0.225665
$780$	0	0
$781$	−8.00000	−0.286263
$782$	0	0
$783$	−6.40312	−0.228829
$784$	15.1047	0.539453
$785$	0	0
$786$	22.5078	0.802827
$787$	33.8953	1.20824	0.604119	−	0.796894i	$-0.293525\pi$
$787$	33.8953	1.20824	0.604119	+	0.796894i	$0.293525\pi$
$788$	−5.40312	−0.192478
$789$	2.00000	0.0712019
$790$	0	0
$791$	−65.8219	−2.34036
$792$	4.70156	0.167063
$793$	−14.1047	−0.500872
$794$	−21.9109	−0.777590
$795$	0	0
$796$	8.29844	0.294130
$797$	2.91093	0.103111	0.0515553	−	0.998670i	$-0.483582\pi$
$797$	2.91093	0.103111	0.0515553	+	0.998670i	$0.483582\pi$
$798$	−8.00000	−0.283197
$799$	−4.91093	−0.173736
$800$	0	0
$801$	11.4031	0.402910
$802$	18.2094	0.642995
$803$	−56.4187	−1.99097
$804$	−6.40312	−0.225821
$805$	0	0
$806$	10.1047	0.355922
$807$	31.2094	1.09862
$808$	6.70156	0.235760
$809$	7.19375	0.252919	0.126459	−	0.991972i	$-0.459639\pi$
$809$	7.19375	0.252919	0.126459	+	0.991972i	$0.459639\pi$
$810$	0	0
$811$	48.7016	1.71014	0.855072	−	0.518510i	$-0.173513\pi$
$811$	48.7016	1.71014	0.855072	+	0.518510i	$0.173513\pi$
$812$	−30.1047	−1.05647
$813$	19.5078	0.684169
$814$	−8.00000	−0.280400
$815$	0	0
$816$	−0.701562	−0.0245596
$817$	19.4031	0.678829
$818$	29.4031	1.02806
$819$	−4.70156	−0.164286
$820$	0	0
$821$	−20.5969	−0.718836	−0.359418	−	0.933177i	$-0.617025\pi$
$821$	−20.5969	−0.718836	−0.359418	+	0.933177i	$0.617025\pi$
$822$	13.7016	0.477897
$823$	17.1047	0.596232	0.298116	−	0.954530i	$-0.403642\pi$
$823$	17.1047	0.596232	0.298116	+	0.954530i	$0.403642\pi$
$824$	1.40312	0.0488801
$825$	0	0
$826$	12.7016	0.441944
$827$	20.7016	0.719864	0.359932	−	0.932979i	$-0.382800\pi$
$827$	20.7016	0.719864	0.359932	+	0.932979i	$0.382800\pi$
$828$	0	0
$829$	−5.50781	−0.191294	−0.0956471	−	0.995415i	$-0.530492\pi$
$829$	−5.50781	−0.191294	−0.0956471	+	0.995415i	$0.530492\pi$
$830$	0	0
$831$	−22.0000	−0.763172
$832$	−1.00000	−0.0346688
$833$	−10.5969	−0.367160
$834$	9.40312	0.325604
$835$	0	0
$836$	−8.00000	−0.276686
$837$	−10.1047	−0.349269
$838$	9.91093	0.342368
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	12.0000	0.413793
$842$	−0.596876	−0.0205697
$843$	−21.9109	−0.754653
$844$	6.80625	0.234281
$845$	0	0
$846$	7.00000	0.240665
$847$	52.2094	1.79394
$848$	2.40312	0.0825236
$849$	21.4031	0.734553
$850$	0	0
$851$	0	0
$852$	−1.70156	−0.0582946
$853$	0.507811	0.0173871	0.00869355	−	0.999962i	$-0.497233\pi$
$853$	0.507811	0.0173871	0.00869355	+	0.999962i	$0.497233\pi$
$854$	66.3141	2.26922
$855$	0	0
$856$	19.1047	0.652985
$857$	−7.61250	−0.260038	−0.130019	−	0.991512i	$-0.541504\pi$
$857$	−7.61250	−0.260038	−0.130019	+	0.991512i	$0.541504\pi$
$858$	−4.70156	−0.160509
$859$	−6.20937	−0.211861	−0.105931	−	0.994374i	$-0.533782\pi$
$859$	−6.20937	−0.211861	−0.105931	+	0.994374i	$0.533782\pi$
$860$	0	0
$861$	−17.4031	−0.593097
$862$	35.3141	1.20280
$863$	54.2250	1.84584	0.922920	−	0.384991i	$-0.125796\pi$
$863$	54.2250	1.84584	0.922920	+	0.384991i	$0.125796\pi$
$864$	1.00000	0.0340207
$865$	0	0
$866$	−11.1047	−0.377353
$867$	−16.5078	−0.560635
$868$	−47.5078	−1.61252
$869$	−26.8062	−0.909340
$870$	0	0
$871$	6.40312	0.216962
$872$	4.29844	0.145563
$873$	−2.59688	−0.0878909
$874$	0	0
$875$	0	0
$876$	−12.0000	−0.405442
$877$	−42.7172	−1.44246	−0.721228	−	0.692697i	$-0.756422\pi$
$877$	−42.7172	−1.44246	−0.721228	+	0.692697i	$0.756422\pi$
$878$	−10.2984	−0.347555
$879$	21.4031	0.721909
$880$	0	0
$881$	−39.7172	−1.33811	−0.669053	−	0.743215i	$-0.733300\pi$
$881$	−39.7172	−1.33811	−0.669053	+	0.743215i	$0.733300\pi$
$882$	15.1047	0.508601
$883$	45.6125	1.53498	0.767491	−	0.641059i	$-0.221505\pi$
$883$	45.6125	1.53498	0.767491	+	0.641059i	$0.221505\pi$
$884$	0.701562	0.0235961
$885$	0	0
$886$	32.5078	1.09212
$887$	−20.0000	−0.671534	−0.335767	−	0.941945i	$-0.608996\pi$
$887$	−20.0000	−0.671534	−0.335767	+	0.941945i	$0.608996\pi$
$888$	−1.70156	−0.0571007
$889$	−29.6125	−0.993171
$890$	0	0
$891$	4.70156	0.157508
$892$	11.4031	0.381805
$893$	−11.9109	−0.398584
$894$	6.59688	0.220633
$895$	0	0
$896$	4.70156	0.157068
$897$	0	0
$898$	2.29844	0.0766999
$899$	64.7016	2.15792
$900$	0	0
$901$	−1.68594	−0.0561668
$902$	−17.4031	−0.579461
$903$	−53.6125	−1.78411
$904$	−14.0000	−0.465633
$905$	0	0
$906$	−14.1047	−0.468597
$907$	−25.0156	−0.830630	−0.415315	−	0.909678i	$-0.636329\pi$
$907$	−25.0156	−0.830630	−0.415315	+	0.909678i	$0.636329\pi$
$908$	−16.1047	−0.534453
$909$	6.70156	0.222277
$910$	0	0
$911$	42.2094	1.39846	0.699229	−	0.714897i	$-0.253527\pi$
$911$	42.2094	1.39846	0.699229	+	0.714897i	$0.253527\pi$
$912$	−1.70156	−0.0563444
$913$	−50.3141	−1.66515
$914$	−2.59688	−0.0858970
$915$	0	0
$916$	15.7016	0.518794
$917$	105.822	3.49455
$918$	−0.701562	−0.0231550
$919$	−4.89531	−0.161481	−0.0807407	−	0.996735i	$-0.525729\pi$
$919$	−4.89531	−0.161481	−0.0807407	+	0.996735i	$0.525729\pi$
$920$	0	0
$921$	−5.70156	−0.187873
$922$	36.2094	1.19249
$923$	1.70156	0.0560076
$924$	22.1047	0.727191
$925$	0	0
$926$	−21.2984	−0.699910
$927$	1.40312	0.0460846
$928$	−6.40312	−0.210193
$929$	52.2984	1.71586	0.857928	−	0.513770i	$-0.171751\pi$
$929$	52.2984	1.71586	0.857928	+	0.513770i	$0.171751\pi$
$930$	0	0
$931$	−25.7016	−0.842335
$932$	−20.2094	−0.661980
$933$	30.0000	0.982156
$934$	30.2984	0.991395
$935$	0	0
$936$	−1.00000	−0.0326860
$937$	14.9109	0.487119	0.243560	−	0.969886i	$-0.421685\pi$
$937$	14.9109	0.487119	0.243560	+	0.969886i	$0.421685\pi$
$938$	−30.1047	−0.982953
$939$	−21.2094	−0.692142
$940$	0	0
$941$	28.4187	0.926425	0.463212	−	0.886247i	$-0.346697\pi$
$941$	28.4187	0.926425	0.463212	+	0.886247i	$0.346697\pi$
$942$	−22.7016	−0.739657
$943$	0	0
$944$	2.70156	0.0879284
$945$	0	0
$946$	−53.6125	−1.74309
$947$	47.5078	1.54380	0.771898	−	0.635746i	$-0.219307\pi$
$947$	47.5078	1.54380	0.771898	+	0.635746i	$0.219307\pi$
$948$	−5.70156	−0.185178
$949$	12.0000	0.389536
$950$	0	0
$951$	1.19375	0.0387100
$952$	−3.29844	−0.106903
$953$	−31.2984	−1.01386	−0.506928	−	0.861988i	$-0.669219\pi$
$953$	−31.2984	−1.01386	−0.506928	+	0.861988i	$0.669219\pi$
$954$	2.40312	0.0778040
$955$	0	0
$956$	6.10469	0.197440
$957$	−30.1047	−0.973146
$958$	−40.6125	−1.31213
$959$	64.4187	2.08019
$960$	0	0
$961$	71.1047	2.29370
$962$	1.70156	0.0548606
$963$	19.1047	0.615640
$964$	−2.59688	−0.0836397
$965$	0	0
$966$	0	0
$967$	29.8953	0.961368	0.480684	−	0.876894i	$-0.340388\pi$
$967$	29.8953	0.961368	0.480684	+	0.876894i	$0.340388\pi$
$968$	11.1047	0.356918
$969$	1.19375	0.0383488
$970$	0	0
$971$	−36.8953	−1.18403	−0.592013	−	0.805928i	$-0.701667\pi$
$971$	−36.8953	−1.18403	−0.592013	+	0.805928i	$0.701667\pi$
$972$	1.00000	0.0320750
$973$	44.2094	1.41729
$974$	−7.89531	−0.252982
$975$	0	0
$976$	14.1047	0.451480
$977$	−23.4031	−0.748732	−0.374366	−	0.927281i	$-0.622140\pi$
$977$	−23.4031	−0.748732	−0.374366	+	0.927281i	$0.622140\pi$
$978$	−18.8062	−0.601358
$979$	53.6125	1.71346
$980$	0	0
$981$	4.29844	0.137239
$982$	33.4031	1.06594
$983$	−7.50781	−0.239462	−0.119731	−	0.992806i	$-0.538203\pi$
$983$	−7.50781	−0.239462	−0.119731	+	0.992806i	$0.538203\pi$
$984$	−3.70156	−0.118001
$985$	0	0
$986$	4.49219	0.143060
$987$	32.9109	1.04757
$988$	1.70156	0.0541339
$989$	0	0
$990$	0	0
$991$	−9.10469	−0.289220	−0.144610	−	0.989489i	$-0.546193\pi$
$991$	−9.10469	−0.289220	−0.144610	+	0.989489i	$0.546193\pi$
$992$	−10.1047	−0.320824
$993$	−12.0000	−0.380808
$994$	−8.00000	−0.253745
$995$	0	0
$996$	−10.7016	−0.339092
$997$	−10.4922	−0.332291	−0.166145	−	0.986101i	$-0.553132\pi$
$997$	−10.4922	−0.332291	−0.166145	+	0.986101i	$0.553132\pi$
$998$	17.2094	0.544753
$999$	−1.70156	−0.0538350

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1950.2.a.bg.1.2	yes	2
3.2	odd	2		5850.2.a.cg.1.2		2
5.2	odd	4		1950.2.e.p.1249.4		4
5.3	odd	4		1950.2.e.p.1249.1		4
5.4	even	2		1950.2.a.bc.1.1	✓	2
15.2	even	4		5850.2.e.bi.5149.2		4
15.8	even	4		5850.2.e.bi.5149.3		4
15.14	odd	2		5850.2.a.cj.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1950.2.a.bc.1.1	✓	2	5.4	even	2
1950.2.a.bg.1.2	yes	2	1.1	even	1	trivial
1950.2.e.p.1249.1		4	5.3	odd	4
1950.2.e.p.1249.4		4	5.2	odd	4
5850.2.a.cg.1.2		2	3.2	odd	2
5850.2.a.cj.1.1		2	15.14	odd	2
5850.2.e.bi.5149.2		4	15.2	even	4
5850.2.e.bi.5149.3		4	15.8	even	4

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:	\( N \)	\(=\)	\( 1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1950.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +4.70156 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +4.70156 q^{7} +1.00000 q^{8} +1.00000 q^{9} +4.70156 q^{11} +1.00000 q^{12} -1.00000 q^{13} +4.70156 q^{14} +1.00000 q^{16} -0.701562 q^{17} +1.00000 q^{18} -1.70156 q^{19} +4.70156 q^{21} +4.70156 q^{22} +1.00000 q^{24} -1.00000 q^{26} +1.00000 q^{27} +4.70156 q^{28} -6.40312 q^{29} -10.1047 q^{31} +1.00000 q^{32} +4.70156 q^{33} -0.701562 q^{34} +1.00000 q^{36} -1.70156 q^{37} -1.70156 q^{38} -1.00000 q^{39} -3.70156 q^{41} +4.70156 q^{42} -11.4031 q^{43} +4.70156 q^{44} +7.00000 q^{47} +1.00000 q^{48} +15.1047 q^{49} -0.701562 q^{51} -1.00000 q^{52} +2.40312 q^{53} +1.00000 q^{54} +4.70156 q^{56} -1.70156 q^{57} -6.40312 q^{58} +2.70156 q^{59} +14.1047 q^{61} -10.1047 q^{62} +4.70156 q^{63} +1.00000 q^{64} +4.70156 q^{66} -6.40312 q^{67} -0.701562 q^{68} -1.70156 q^{71} +1.00000 q^{72} -12.0000 q^{73} -1.70156 q^{74} -1.70156 q^{76} +22.1047 q^{77} -1.00000 q^{78} -5.70156 q^{79} +1.00000 q^{81} -3.70156 q^{82} -10.7016 q^{83} +4.70156 q^{84} -11.4031 q^{86} -6.40312 q^{87} +4.70156 q^{88} +11.4031 q^{89} -4.70156 q^{91} -10.1047 q^{93} +7.00000 q^{94} +1.00000 q^{96} -2.59688 q^{97} +15.1047 q^{98} +4.70156 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{6} + 3 q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^6 + 3 * q^7 + 2 * q^8 + 2 * q^9	\( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{6} + 3 q^{7} + 2 q^{8} + 2 q^{9} + 3 q^{11} + 2 q^{12} - 2 q^{13} + 3 q^{14} + 2 q^{16} + 5 q^{17} + 2 q^{18} + 3 q^{19} + 3 q^{21} + 3 q^{22} + 2 q^{24} - 2 q^{26} + 2 q^{27} + 3 q^{28} - q^{31} + 2 q^{32} + 3 q^{33} + 5 q^{34} + 2 q^{36} + 3 q^{37} + 3 q^{38} - 2 q^{39} - q^{41} + 3 q^{42} - 10 q^{43} + 3 q^{44} + 14 q^{47} + 2 q^{48} + 11 q^{49} + 5 q^{51} - 2 q^{52} - 8 q^{53} + 2 q^{54} + 3 q^{56} + 3 q^{57} - q^{59} + 9 q^{61} - q^{62} + 3 q^{63} + 2 q^{64} + 3 q^{66} + 5 q^{68} + 3 q^{71} + 2 q^{72} - 24 q^{73} + 3 q^{74} + 3 q^{76} + 25 q^{77} - 2 q^{78} - 5 q^{79} + 2 q^{81} - q^{82} - 15 q^{83} + 3 q^{84} - 10 q^{86} + 3 q^{88} + 10 q^{89} - 3 q^{91} - q^{93} + 14 q^{94} + 2 q^{96} - 18 q^{97} + 11 q^{98} + 3 q^{99}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^6 + 3 * q^7 + 2 * q^8 + 2 * q^9 + 3 * q^11 + 2 * q^12 - 2 * q^13 + 3 * q^14 + 2 * q^16 + 5 * q^17 + 2 * q^18 + 3 * q^19 + 3 * q^21 + 3 * q^22 + 2 * q^24 - 2 * q^26 + 2 * q^27 + 3 * q^28 - q^31 + 2 * q^32 + 3 * q^33 + 5 * q^34 + 2 * q^36 + 3 * q^37 + 3 * q^38 - 2 * q^39 - q^41 + 3 * q^42 - 10 * q^43 + 3 * q^44 + 14 * q^47 + 2 * q^48 + 11 * q^49 + 5 * q^51 - 2 * q^52 - 8 * q^53 + 2 * q^54 + 3 * q^56 + 3 * q^57 - q^59 + 9 * q^61 - q^62 + 3 * q^63 + 2 * q^64 + 3 * q^66 + 5 * q^68 + 3 * q^71 + 2 * q^72 - 24 * q^73 + 3 * q^74 + 3 * q^76 + 25 * q^77 - 2 * q^78 - 5 * q^79 + 2 * q^81 - q^82 - 15 * q^83 + 3 * q^84 - 10 * q^86 + 3 * q^88 + 10 * q^89 - 3 * q^91 - q^93 + 14 * q^94 + 2 * q^96 - 18 * q^97 + 11 * q^98 + 3 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more