LMFDB - Embedded newform 1950.2.e.f.1249.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1950,2,Mod(1249,1950)]

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, 1, 0]))

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

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

chi := DirichletCharacter("1950.1249");

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.e (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$15.5708283941$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 390)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1249.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	1950.1249
Dual form			1950.2.e.f.1249.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +4.00000 q^{11} +1.00000i q^{12} -1.00000i q^{13} -2.00000 q^{14} +1.00000 q^{16} -8.00000i q^{17} +1.00000i q^{18} +6.00000 q^{19} -2.00000 q^{21} -4.00000i q^{22} +6.00000i q^{23} +1.00000 q^{24} -1.00000 q^{26} +1.00000i q^{27} +2.00000i q^{28} +4.00000 q^{29} -1.00000i q^{32} -4.00000i q^{33} -8.00000 q^{34} +1.00000 q^{36} +2.00000i q^{37} -6.00000i q^{38} -1.00000 q^{39} -2.00000 q^{41} +2.00000i q^{42} -4.00000i q^{43} -4.00000 q^{44} +6.00000 q^{46} -1.00000i q^{48} +3.00000 q^{49} -8.00000 q^{51} +1.00000i q^{52} -10.0000i q^{53} +1.00000 q^{54} +2.00000 q^{56} -6.00000i q^{57} -4.00000i q^{58} -4.00000 q^{59} -10.0000 q^{61} +2.00000i q^{63} -1.00000 q^{64} -4.00000 q^{66} -12.0000i q^{67} +8.00000i q^{68} +6.00000 q^{69} -8.00000 q^{71} -1.00000i q^{72} -8.00000i q^{73} +2.00000 q^{74} -6.00000 q^{76} -8.00000i q^{77} +1.00000i q^{78} -8.00000 q^{79} +1.00000 q^{81} +2.00000i q^{82} +12.0000i q^{83} +2.00000 q^{84} -4.00000 q^{86} -4.00000i q^{87} +4.00000i q^{88} +14.0000 q^{89} -2.00000 q^{91} -6.00000i q^{92} -1.00000 q^{96} +16.0000i q^{97} -3.00000i q^{98} -4.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9	$ 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9} + 8 q^{11} - 4 q^{14} + 2 q^{16} + 12 q^{19} - 4 q^{21} + 2 q^{24} - 2 q^{26} + 8 q^{29} - 16 q^{34} + 2 q^{36} - 2 q^{39} - 4 q^{41} - 8 q^{44} + 12 q^{46} + 6 q^{49} - 16 q^{51} + 2 q^{54} + 4 q^{56} - 8 q^{59} - 20 q^{61} - 2 q^{64} - 8 q^{66} + 12 q^{69} - 16 q^{71} + 4 q^{74} - 12 q^{76} - 16 q^{79} + 2 q^{81} + 4 q^{84} - 8 q^{86} + 28 q^{89} - 4 q^{91} - 2 q^{96} - 8 q^{99}+O(q^{100}) $ 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9 + 8 * q^11 - 4 * q^14 + 2 * q^16 + 12 * q^19 - 4 * q^21 + 2 * q^24 - 2 * q^26 + 8 * q^29 - 16 * q^34 + 2 * q^36 - 2 * q^39 - 4 * q^41 - 8 * q^44 + 12 * q^46 + 6 * q^49 - 16 * q^51 + 2 * q^54 + 4 * q^56 - 8 * q^59 - 20 * q^61 - 2 * q^64 - 8 * q^66 + 12 * q^69 - 16 * q^71 + 4 * q^74 - 12 * q^76 - 16 * q^79 + 2 * q^81 + 4 * q^84 - 8 * q^86 + 28 * q^89 - 4 * q^91 - 2 * q^96 - 8 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1950\mathbb{Z}\right)^\times$.

$n$	$301$	$1301$	$1327$
$\chi(n)$	$1$	$1$	$-1$

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 $

$2$	− 1.00000i	− 0.707107i
$2$	− 1.00000i	− 0.707107i
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	−1.00000	−0.408248
$7$	− 2.00000i	− 0.755929i	−0.925820	−	0.377964i	$-0.876624\pi$
$7$	− 2.00000i	− 0.755929i	0.925820	−	0.377964i	$-0.123376\pi$
$8$	1.00000i	0.353553i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	4.00000	1.20605	0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000	1.20605	0.603023	+	0.797724i	$0.293963\pi$
$12$	1.00000i	0.288675i
$13$	− 1.00000i	− 0.277350i
$13$	− 1.00000i	− 0.277350i
$14$	−2.00000	−0.534522
$15$	0	0
$16$	1.00000	0.250000
$17$	− 8.00000i	− 1.94029i	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	− 8.00000i	− 1.94029i	0.242536	−	0.970143i	$-0.422021\pi$
$18$	1.00000i	0.235702i
$19$	6.00000	1.37649	0.688247	−	0.725476i	$-0.258380\pi$
$19$	6.00000	1.37649	0.688247	+	0.725476i	$0.258380\pi$
$20$	0	0
$21$	−2.00000	−0.436436
$22$	− 4.00000i	− 0.852803i
$23$	6.00000i	1.25109i	0.780189	+	0.625543i	$0.215123\pi$
$23$	6.00000i	1.25109i	−0.780189	+	0.625543i	$0.784877\pi$
$24$	1.00000	0.204124
$25$	0	0
$26$	−1.00000	−0.196116
$27$	1.00000i	0.192450i
$28$	2.00000i	0.377964i
$29$	4.00000	0.742781	0.371391	−	0.928477i	$-0.378881\pi$
$29$	4.00000	0.742781	0.371391	+	0.928477i	$0.378881\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	− 1.00000i	− 0.176777i
$33$	− 4.00000i	− 0.696311i
$34$	−8.00000	−1.37199
$35$	0	0
$36$	1.00000	0.166667
$37$	2.00000i	0.328798i	0.986394	+	0.164399i	$0.0525685\pi$
$37$	2.00000i	0.328798i	−0.986394	+	0.164399i	$0.947432\pi$
$38$	− 6.00000i	− 0.973329i
$39$	−1.00000	−0.160128
$40$	0	0
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	2.00000i	0.308607i
$43$	− 4.00000i	− 0.609994i	−0.952353	−	0.304997i	$-0.901344\pi$
$43$	− 4.00000i	− 0.609994i	0.952353	−	0.304997i	$-0.0986555\pi$
$44$	−4.00000	−0.603023
$45$	0	0
$46$	6.00000	0.884652
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	− 1.00000i	− 0.144338i
$49$	3.00000	0.428571
$50$	0	0
$51$	−8.00000	−1.12022
$52$	1.00000i	0.138675i
$53$	− 10.0000i	− 1.37361i	−0.726844	−	0.686803i	$-0.759014\pi$
$53$	− 10.0000i	− 1.37361i	0.726844	−	0.686803i	$-0.240986\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	2.00000	0.267261
$57$	− 6.00000i	− 0.794719i
$58$	− 4.00000i	− 0.525226i
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	0	0
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	0	0
$63$	2.00000i	0.251976i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	−4.00000	−0.492366
$67$	− 12.0000i	− 1.46603i	−0.680211	−	0.733017i	$-0.738112\pi$
$67$	− 12.0000i	− 1.46603i	0.680211	−	0.733017i	$-0.261888\pi$
$68$	8.00000i	0.970143i
$69$	6.00000	0.722315
$70$	0	0
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	− 1.00000i	− 0.117851i
$73$	− 8.00000i	− 0.936329i	−0.883641	−	0.468165i	$-0.844915\pi$
$73$	− 8.00000i	− 0.936329i	0.883641	−	0.468165i	$-0.155085\pi$
$74$	2.00000	0.232495
$75$	0	0
$76$	−6.00000	−0.688247
$77$	− 8.00000i	− 0.911685i
$78$	1.00000i	0.113228i
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	2.00000i	0.220863i
$83$	12.0000i	1.31717i	0.752506	+	0.658586i	$0.228845\pi$
$83$	12.0000i	1.31717i	−0.752506	+	0.658586i	$0.771155\pi$
$84$	2.00000	0.218218
$85$	0	0
$86$	−4.00000	−0.431331
$87$	− 4.00000i	− 0.428845i
$88$	4.00000i	0.426401i
$89$	14.0000	1.48400	0.741999	−	0.670402i	$-0.233878\pi$
$89$	14.0000	1.48400	0.741999	+	0.670402i	$0.233878\pi$
$90$	0	0
$91$	−2.00000	−0.209657
$92$	− 6.00000i	− 0.625543i
$93$	0	0
$94$	0	0
$95$	0	0
$96$	−1.00000	−0.102062
$97$	16.0000i	1.62455i	0.583272	+	0.812277i	$0.301772\pi$
$97$	16.0000i	1.62455i	−0.583272	+	0.812277i	$0.698228\pi$
$98$	− 3.00000i	− 0.303046i
$99$	−4.00000	−0.402015
$100$	0	0
$101$	−16.0000	−1.59206	−0.796030	−	0.605257i	$-0.793070\pi$
$101$	−16.0000	−1.59206	−0.796030	+	0.605257i	$0.793070\pi$
$102$	8.00000i	0.792118i
$103$	− 12.0000i	− 1.18240i	−0.806527	−	0.591198i	$-0.798655\pi$
$103$	− 12.0000i	− 1.18240i	0.806527	−	0.591198i	$-0.201345\pi$
$104$	1.00000	0.0980581
$105$	0	0
$106$	−10.0000	−0.971286
$107$	− 12.0000i	− 1.16008i	−0.814587	−	0.580042i	$-0.803036\pi$
$107$	− 12.0000i	− 1.16008i	0.814587	−	0.580042i	$-0.196964\pi$
$108$	− 1.00000i	− 0.0962250i
$109$	−12.0000	−1.14939	−0.574696	−	0.818367i	$-0.694880\pi$
$109$	−12.0000	−1.14939	−0.574696	+	0.818367i	$0.694880\pi$
$110$	0	0
$111$	2.00000	0.189832
$112$	− 2.00000i	− 0.188982i
$113$	20.0000i	1.88144i	0.339182	+	0.940721i	$0.389850\pi$
$113$	20.0000i	1.88144i	−0.339182	+	0.940721i	$0.610150\pi$
$114$	−6.00000	−0.561951
$115$	0	0
$116$	−4.00000	−0.371391
$117$	1.00000i	0.0924500i
$118$	4.00000i	0.368230i
$119$	−16.0000	−1.46672
$120$	0	0
$121$	5.00000	0.454545
$122$	10.0000i	0.905357i
$123$	2.00000i	0.180334i
$124$	0	0
$125$	0	0
$126$	2.00000	0.178174
$127$	− 4.00000i	− 0.354943i	−0.984126	−	0.177471i	$-0.943208\pi$
$127$	− 4.00000i	− 0.354943i	0.984126	−	0.177471i	$-0.0567917\pi$
$128$	1.00000i	0.0883883i
$129$	−4.00000	−0.352180
$130$	0	0
$131$	10.0000	0.873704	0.436852	−	0.899533i	$-0.356093\pi$
$131$	10.0000	0.873704	0.436852	+	0.899533i	$0.356093\pi$
$132$	4.00000i	0.348155i
$133$	− 12.0000i	− 1.04053i
$134$	−12.0000	−1.03664
$135$	0	0
$136$	8.00000	0.685994
$137$	6.00000i	0.512615i	0.966595	+	0.256307i	$0.0825059\pi$
$137$	6.00000i	0.512615i	−0.966595	+	0.256307i	$0.917494\pi$
$138$	− 6.00000i	− 0.510754i
$139$	8.00000	0.678551	0.339276	−	0.940687i	$-0.389818\pi$
$139$	8.00000	0.678551	0.339276	+	0.940687i	$0.389818\pi$
$140$	0	0
$141$	0	0
$142$	8.00000i	0.671345i
$143$	− 4.00000i	− 0.334497i
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	−8.00000	−0.662085
$147$	− 3.00000i	− 0.247436i
$148$	− 2.00000i	− 0.164399i
$149$	10.0000	0.819232	0.409616	−	0.912258i	$-0.365663\pi$
$149$	10.0000	0.819232	0.409616	+	0.912258i	$0.365663\pi$
$150$	0	0
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	6.00000i	0.486664i
$153$	8.00000i	0.646762i
$154$	−8.00000	−0.644658
$155$	0	0
$156$	1.00000	0.0800641
$157$	− 22.0000i	− 1.75579i	−0.478852	−	0.877896i	$-0.658947\pi$
$157$	− 22.0000i	− 1.75579i	0.478852	−	0.877896i	$-0.341053\pi$
$158$	8.00000i	0.636446i
$159$	−10.0000	−0.793052
$160$	0	0
$161$	12.0000	0.945732
$162$	− 1.00000i	− 0.0785674i
$163$	− 16.0000i	− 1.25322i	−0.779334	−	0.626608i	$-0.784443\pi$
$163$	− 16.0000i	− 1.25322i	0.779334	−	0.626608i	$-0.215557\pi$
$164$	2.00000	0.156174
$165$	0	0
$166$	12.0000	0.931381
$167$	4.00000i	0.309529i	0.987951	+	0.154765i	$0.0494619\pi$
$167$	4.00000i	0.309529i	−0.987951	+	0.154765i	$0.950538\pi$
$168$	− 2.00000i	− 0.154303i
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	−6.00000	−0.458831
$172$	4.00000i	0.304997i
$173$	22.0000i	1.67263i	0.548250	+	0.836315i	$0.315294\pi$
$173$	22.0000i	1.67263i	−0.548250	+	0.836315i	$0.684706\pi$
$174$	−4.00000	−0.303239
$175$	0	0
$176$	4.00000	0.301511
$177$	4.00000i	0.300658i
$178$	− 14.0000i	− 1.04934i
$179$	10.0000	0.747435	0.373718	−	0.927543i	$-0.378083\pi$
$179$	10.0000	0.747435	0.373718	+	0.927543i	$0.378083\pi$
$180$	0	0
$181$	10.0000	0.743294	0.371647	−	0.928374i	$-0.378793\pi$
$181$	10.0000	0.743294	0.371647	+	0.928374i	$0.378793\pi$
$182$	2.00000i	0.148250i
$183$	10.0000i	0.739221i
$184$	−6.00000	−0.442326
$185$	0	0
$186$	0	0
$187$	− 32.0000i	− 2.34007i
$188$	0	0
$189$	2.00000	0.145479
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	1.00000i	0.0721688i
$193$	− 4.00000i	− 0.287926i	−0.989583	−	0.143963i	$-0.954015\pi$
$193$	− 4.00000i	− 0.287926i	0.989583	−	0.143963i	$-0.0459847\pi$
$194$	16.0000	1.14873
$195$	0	0
$196$	−3.00000	−0.214286
$197$	18.0000i	1.28245i	0.767354	+	0.641223i	$0.221573\pi$
$197$	18.0000i	1.28245i	−0.767354	+	0.641223i	$0.778427\pi$
$198$	4.00000i	0.284268i
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	0	0
$201$	−12.0000	−0.846415
$202$	16.0000i	1.12576i
$203$	− 8.00000i	− 0.561490i
$204$	8.00000	0.560112
$205$	0	0
$206$	−12.0000	−0.836080
$207$	− 6.00000i	− 0.417029i
$208$	− 1.00000i	− 0.0693375i
$209$	24.0000	1.66011
$210$	0	0
$211$	−20.0000	−1.37686	−0.688428	−	0.725304i	$-0.741699\pi$
$211$	−20.0000	−1.37686	−0.688428	+	0.725304i	$0.741699\pi$
$212$	10.0000i	0.686803i
$213$	8.00000i	0.548151i
$214$	−12.0000	−0.820303
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	0	0
$218$	12.0000i	0.812743i
$219$	−8.00000	−0.540590
$220$	0	0
$221$	−8.00000	−0.538138
$222$	− 2.00000i	− 0.134231i
$223$	− 2.00000i	− 0.133930i	−0.997755	−	0.0669650i	$-0.978668\pi$
$223$	− 2.00000i	− 0.133930i	0.997755	−	0.0669650i	$-0.0213316\pi$
$224$	−2.00000	−0.133631
$225$	0	0
$226$	20.0000	1.33038
$227$	4.00000i	0.265489i	0.991150	+	0.132745i	$0.0423790\pi$
$227$	4.00000i	0.265489i	−0.991150	+	0.132745i	$0.957621\pi$
$228$	6.00000i	0.397360i
$229$	−4.00000	−0.264327	−0.132164	−	0.991228i	$-0.542192\pi$
$229$	−4.00000	−0.264327	−0.132164	+	0.991228i	$0.542192\pi$
$230$	0	0
$231$	−8.00000	−0.526361
$232$	4.00000i	0.262613i
$233$	24.0000i	1.57229i	0.618041	+	0.786146i	$0.287927\pi$
$233$	24.0000i	1.57229i	−0.618041	+	0.786146i	$0.712073\pi$
$234$	1.00000	0.0653720
$235$	0	0
$236$	4.00000	0.260378
$237$	8.00000i	0.519656i
$238$	16.0000i	1.03713i
$239$	−16.0000	−1.03495	−0.517477	−	0.855697i	$-0.673129\pi$
$239$	−16.0000	−1.03495	−0.517477	+	0.855697i	$0.673129\pi$
$240$	0	0
$241$	2.00000	0.128831	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000	0.128831	0.0644157	+	0.997923i	$0.479482\pi$
$242$	− 5.00000i	− 0.321412i
$243$	− 1.00000i	− 0.0641500i
$244$	10.0000	0.640184
$245$	0	0
$246$	2.00000	0.127515
$247$	− 6.00000i	− 0.381771i
$248$	0	0
$249$	12.0000	0.760469
$250$	0	0
$251$	6.00000	0.378717	0.189358	−	0.981908i	$-0.439359\pi$
$251$	6.00000	0.378717	0.189358	+	0.981908i	$0.439359\pi$
$252$	− 2.00000i	− 0.125988i
$253$	24.0000i	1.50887i
$254$	−4.00000	−0.250982
$255$	0	0
$256$	1.00000	0.0625000
$257$	− 12.0000i	− 0.748539i	−0.927320	−	0.374270i	$-0.877893\pi$
$257$	− 12.0000i	− 0.748539i	0.927320	−	0.374270i	$-0.122107\pi$
$258$	4.00000i	0.249029i
$259$	4.00000	0.248548
$260$	0	0
$261$	−4.00000	−0.247594
$262$	− 10.0000i	− 0.617802i
$263$	2.00000i	0.123325i	0.998097	+	0.0616626i	$0.0196403\pi$
$263$	2.00000i	0.123325i	−0.998097	+	0.0616626i	$0.980360\pi$
$264$	4.00000	0.246183
$265$	0	0
$266$	−12.0000	−0.735767
$267$	− 14.0000i	− 0.856786i
$268$	12.0000i	0.733017i
$269$	24.0000	1.46331	0.731653	−	0.681677i	$-0.238749\pi$
$269$	24.0000	1.46331	0.731653	+	0.681677i	$0.238749\pi$
$270$	0	0
$271$	16.0000	0.971931	0.485965	−	0.873978i	$-0.338468\pi$
$271$	16.0000	0.971931	0.485965	+	0.873978i	$0.338468\pi$
$272$	− 8.00000i	− 0.485071i
$273$	2.00000i	0.121046i
$274$	6.00000	0.362473
$275$	0	0
$276$	−6.00000	−0.361158
$277$	10.0000i	0.600842i	0.953807	+	0.300421i	$0.0971271\pi$
$277$	10.0000i	0.600842i	−0.953807	+	0.300421i	$0.902873\pi$
$278$	− 8.00000i	− 0.479808i
$279$	0	0
$280$	0	0
$281$	−6.00000	−0.357930	−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000	−0.357930	−0.178965	+	0.983855i	$0.557275\pi$
$282$	0	0
$283$	− 4.00000i	− 0.237775i	−0.992908	−	0.118888i	$-0.962067\pi$
$283$	− 4.00000i	− 0.237775i	0.992908	−	0.118888i	$-0.0379328\pi$
$284$	8.00000	0.474713
$285$	0	0
$286$	−4.00000	−0.236525
$287$	4.00000i	0.236113i
$288$	1.00000i	0.0589256i
$289$	−47.0000	−2.76471
$290$	0	0
$291$	16.0000	0.937937
$292$	8.00000i	0.468165i
$293$	− 26.0000i	− 1.51894i	−0.650545	−	0.759468i	$-0.725459\pi$
$293$	− 26.0000i	− 1.51894i	0.650545	−	0.759468i	$-0.274541\pi$
$294$	−3.00000	−0.174964
$295$	0	0
$296$	−2.00000	−0.116248
$297$	4.00000i	0.232104i
$298$	− 10.0000i	− 0.579284i
$299$	6.00000	0.346989
$300$	0	0
$301$	−8.00000	−0.461112
$302$	0	0
$303$	16.0000i	0.919176i
$304$	6.00000	0.344124
$305$	0	0
$306$	8.00000	0.457330
$307$	− 12.0000i	− 0.684876i	−0.939540	−	0.342438i	$-0.888747\pi$
$307$	− 12.0000i	− 0.684876i	0.939540	−	0.342438i	$-0.111253\pi$
$308$	8.00000i	0.455842i
$309$	−12.0000	−0.682656
$310$	0	0
$311$	24.0000	1.36092	0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000	1.36092	0.680458	+	0.732787i	$0.261781\pi$
$312$	− 1.00000i	− 0.0566139i
$313$	6.00000i	0.339140i	0.985518	+	0.169570i	$0.0542379\pi$
$313$	6.00000i	0.339140i	−0.985518	+	0.169570i	$0.945762\pi$
$314$	−22.0000	−1.24153
$315$	0	0
$316$	8.00000	0.450035
$317$	30.0000i	1.68497i	0.538721	+	0.842484i	$0.318908\pi$
$317$	30.0000i	1.68497i	−0.538721	+	0.842484i	$0.681092\pi$
$318$	10.0000i	0.560772i
$319$	16.0000	0.895828
$320$	0	0
$321$	−12.0000	−0.669775
$322$	− 12.0000i	− 0.668734i
$323$	− 48.0000i	− 2.67079i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	−16.0000	−0.886158
$327$	12.0000i	0.663602i
$328$	− 2.00000i	− 0.110432i
$329$	0	0
$330$	0	0
$331$	10.0000	0.549650	0.274825	−	0.961494i	$-0.411380\pi$
$331$	10.0000	0.549650	0.274825	+	0.961494i	$0.411380\pi$
$332$	− 12.0000i	− 0.658586i
$333$	− 2.00000i	− 0.109599i
$334$	4.00000	0.218870
$335$	0	0
$336$	−2.00000	−0.109109
$337$	− 14.0000i	− 0.762629i	−0.924445	−	0.381314i	$-0.875472\pi$
$337$	− 14.0000i	− 0.762629i	0.924445	−	0.381314i	$-0.124528\pi$
$338$	1.00000i	0.0543928i
$339$	20.0000	1.08625
$340$	0	0
$341$	0	0
$342$	6.00000i	0.324443i
$343$	− 20.0000i	− 1.07990i
$344$	4.00000	0.215666
$345$	0	0
$346$	22.0000	1.18273
$347$	24.0000i	1.28839i	0.764862	+	0.644194i	$0.222807\pi$
$347$	24.0000i	1.28839i	−0.764862	+	0.644194i	$0.777193\pi$
$348$	4.00000i	0.214423i
$349$	28.0000	1.49881	0.749403	−	0.662114i	$-0.230341\pi$
$349$	28.0000	1.49881	0.749403	+	0.662114i	$0.230341\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	− 4.00000i	− 0.213201i
$353$	− 14.0000i	− 0.745145i	−0.928003	−	0.372572i	$-0.878476\pi$
$353$	− 14.0000i	− 0.745145i	0.928003	−	0.372572i	$-0.121524\pi$
$354$	4.00000	0.212598
$355$	0	0
$356$	−14.0000	−0.741999
$357$	16.0000i	0.846810i
$358$	− 10.0000i	− 0.528516i
$359$	24.0000	1.26667	0.633336	−	0.773877i	$-0.281685\pi$
$359$	24.0000	1.26667	0.633336	+	0.773877i	$0.281685\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	− 10.0000i	− 0.525588i
$363$	− 5.00000i	− 0.262432i
$364$	2.00000	0.104828
$365$	0	0
$366$	10.0000	0.522708
$367$	36.0000i	1.87918i	0.342296	+	0.939592i	$0.388796\pi$
$367$	36.0000i	1.87918i	−0.342296	+	0.939592i	$0.611204\pi$
$368$	6.00000i	0.312772i
$369$	2.00000	0.104116
$370$	0	0
$371$	−20.0000	−1.03835
$372$	0	0
$373$	− 2.00000i	− 0.103556i	−0.998659	−	0.0517780i	$-0.983511\pi$
$373$	− 2.00000i	− 0.103556i	0.998659	−	0.0517780i	$-0.0164888\pi$
$374$	−32.0000	−1.65468
$375$	0	0
$376$	0	0
$377$	− 4.00000i	− 0.206010i
$378$	− 2.00000i	− 0.102869i
$379$	18.0000	0.924598	0.462299	−	0.886724i	$-0.347025\pi$
$379$	18.0000	0.924598	0.462299	+	0.886724i	$0.347025\pi$
$380$	0	0
$381$	−4.00000	−0.204926
$382$	0	0
$383$	− 12.0000i	− 0.613171i	−0.951843	−	0.306586i	$-0.900813\pi$
$383$	− 12.0000i	− 0.613171i	0.951843	−	0.306586i	$-0.0991866\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	−4.00000	−0.203595
$387$	4.00000i	0.203331i
$388$	− 16.0000i	− 0.812277i
$389$	−8.00000	−0.405616	−0.202808	−	0.979219i	$-0.565007\pi$
$389$	−8.00000	−0.405616	−0.202808	+	0.979219i	$0.565007\pi$
$390$	0	0
$391$	48.0000	2.42746
$392$	3.00000i	0.151523i
$393$	− 10.0000i	− 0.504433i
$394$	18.0000	0.906827
$395$	0	0
$396$	4.00000	0.201008
$397$	− 14.0000i	− 0.702640i	−0.936255	−	0.351320i	$-0.885733\pi$
$397$	− 14.0000i	− 0.702640i	0.936255	−	0.351320i	$-0.114267\pi$
$398$	0	0
$399$	−12.0000	−0.600751
$400$	0	0
$401$	30.0000	1.49813	0.749064	−	0.662497i	$-0.230503\pi$
$401$	30.0000	1.49813	0.749064	+	0.662497i	$0.230503\pi$
$402$	12.0000i	0.598506i
$403$	0	0
$404$	16.0000	0.796030
$405$	0	0
$406$	−8.00000	−0.397033
$407$	8.00000i	0.396545i
$408$	− 8.00000i	− 0.396059i
$409$	26.0000	1.28562	0.642809	−	0.766027i	$-0.277769\pi$
$409$	26.0000	1.28562	0.642809	+	0.766027i	$0.277769\pi$
$410$	0	0
$411$	6.00000	0.295958
$412$	12.0000i	0.591198i
$413$	8.00000i	0.393654i
$414$	−6.00000	−0.294884
$415$	0	0
$416$	−1.00000	−0.0490290
$417$	− 8.00000i	− 0.391762i
$418$	− 24.0000i	− 1.17388i
$419$	−10.0000	−0.488532	−0.244266	−	0.969708i	$-0.578547\pi$
$419$	−10.0000	−0.488532	−0.244266	+	0.969708i	$0.578547\pi$
$420$	0	0
$421$	−8.00000	−0.389896	−0.194948	−	0.980814i	$-0.562454\pi$
$421$	−8.00000	−0.389896	−0.194948	+	0.980814i	$0.562454\pi$
$422$	20.0000i	0.973585i
$423$	0	0
$424$	10.0000	0.485643
$425$	0	0
$426$	8.00000	0.387601
$427$	20.0000i	0.967868i
$428$	12.0000i	0.580042i
$429$	−4.00000	−0.193122
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	1.00000i	0.0481125i
$433$	− 2.00000i	− 0.0961139i	−0.998845	−	0.0480569i	$-0.984697\pi$
$433$	− 2.00000i	− 0.0961139i	0.998845	−	0.0480569i	$-0.0153029\pi$
$434$	0	0
$435$	0	0
$436$	12.0000	0.574696
$437$	36.0000i	1.72211i
$438$	8.00000i	0.382255i
$439$	32.0000	1.52728	0.763638	−	0.645644i	$-0.223411\pi$
$439$	32.0000	1.52728	0.763638	+	0.645644i	$0.223411\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	8.00000i	0.380521i
$443$	16.0000i	0.760183i	0.924949	+	0.380091i	$0.124107\pi$
$443$	16.0000i	0.760183i	−0.924949	+	0.380091i	$0.875893\pi$
$444$	−2.00000	−0.0949158
$445$	0	0
$446$	−2.00000	−0.0947027
$447$	− 10.0000i	− 0.472984i
$448$	2.00000i	0.0944911i
$449$	6.00000	0.283158	0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000	0.283158	0.141579	+	0.989927i	$0.454782\pi$
$450$	0	0
$451$	−8.00000	−0.376705
$452$	− 20.0000i	− 0.940721i
$453$	0	0
$454$	4.00000	0.187729
$455$	0	0
$456$	6.00000	0.280976
$457$	− 8.00000i	− 0.374224i	−0.982339	−	0.187112i	$-0.940087\pi$
$457$	− 8.00000i	− 0.374224i	0.982339	−	0.187112i	$-0.0599128\pi$
$458$	4.00000i	0.186908i
$459$	8.00000	0.373408
$460$	0	0
$461$	−6.00000	−0.279448	−0.139724	−	0.990190i	$-0.544622\pi$
$461$	−6.00000	−0.279448	−0.139724	+	0.990190i	$0.544622\pi$
$462$	8.00000i	0.372194i
$463$	− 26.0000i	− 1.20832i	−0.796862	−	0.604161i	$-0.793508\pi$
$463$	− 26.0000i	− 1.20832i	0.796862	−	0.604161i	$-0.206492\pi$
$464$	4.00000	0.185695
$465$	0	0
$466$	24.0000	1.11178
$467$	− 28.0000i	− 1.29569i	−0.761774	−	0.647843i	$-0.775671\pi$
$467$	− 28.0000i	− 1.29569i	0.761774	−	0.647843i	$-0.224329\pi$
$468$	− 1.00000i	− 0.0462250i
$469$	−24.0000	−1.10822
$470$	0	0
$471$	−22.0000	−1.01371
$472$	− 4.00000i	− 0.184115i
$473$	− 16.0000i	− 0.735681i
$474$	8.00000	0.367452
$475$	0	0
$476$	16.0000	0.733359
$477$	10.0000i	0.457869i
$478$	16.0000i	0.731823i
$479$	−32.0000	−1.46212	−0.731059	−	0.682315i	$-0.760973\pi$
$479$	−32.0000	−1.46212	−0.731059	+	0.682315i	$0.760973\pi$
$480$	0	0
$481$	2.00000	0.0911922
$482$	− 2.00000i	− 0.0910975i
$483$	− 12.0000i	− 0.546019i
$484$	−5.00000	−0.227273
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	− 26.0000i	− 1.17817i	−0.808070	−	0.589086i	$-0.799488\pi$
$487$	− 26.0000i	− 1.17817i	0.808070	−	0.589086i	$-0.200512\pi$
$488$	− 10.0000i	− 0.452679i
$489$	−16.0000	−0.723545
$490$	0	0
$491$	42.0000	1.89543	0.947717	−	0.319113i	$-0.103385\pi$
$491$	42.0000	1.89543	0.947717	+	0.319113i	$0.103385\pi$
$492$	− 2.00000i	− 0.0901670i
$493$	− 32.0000i	− 1.44121i
$494$	−6.00000	−0.269953
$495$	0	0
$496$	0	0
$497$	16.0000i	0.717698i
$498$	− 12.0000i	− 0.537733i
$499$	−38.0000	−1.70111	−0.850557	−	0.525883i	$-0.823735\pi$
$499$	−38.0000	−1.70111	−0.850557	+	0.525883i	$0.823735\pi$
$500$	0	0
$501$	4.00000	0.178707
$502$	− 6.00000i	− 0.267793i
$503$	10.0000i	0.445878i	0.974832	+	0.222939i	$0.0715651\pi$
$503$	10.0000i	0.445878i	−0.974832	+	0.222939i	$0.928435\pi$
$504$	−2.00000	−0.0890871
$505$	0	0
$506$	24.0000	1.06693
$507$	1.00000i	0.0444116i
$508$	4.00000i	0.177471i
$509$	−18.0000	−0.797836	−0.398918	−	0.916987i	$-0.630614\pi$
$509$	−18.0000	−0.797836	−0.398918	+	0.916987i	$0.630614\pi$
$510$	0	0
$511$	−16.0000	−0.707798
$512$	− 1.00000i	− 0.0441942i
$513$	6.00000i	0.264906i
$514$	−12.0000	−0.529297
$515$	0	0
$516$	4.00000	0.176090
$517$	0	0
$518$	− 4.00000i	− 0.175750i
$519$	22.0000	0.965693
$520$	0	0
$521$	−26.0000	−1.13908	−0.569540	−	0.821963i	$-0.692879\pi$
$521$	−26.0000	−1.13908	−0.569540	+	0.821963i	$0.692879\pi$
$522$	4.00000i	0.175075i
$523$	36.0000i	1.57417i	0.616844	+	0.787085i	$0.288411\pi$
$523$	36.0000i	1.57417i	−0.616844	+	0.787085i	$0.711589\pi$
$524$	−10.0000	−0.436852
$525$	0	0
$526$	2.00000	0.0872041
$527$	0	0
$528$	− 4.00000i	− 0.174078i
$529$	−13.0000	−0.565217
$530$	0	0
$531$	4.00000	0.173585
$532$	12.0000i	0.520266i
$533$	2.00000i	0.0866296i
$534$	−14.0000	−0.605839
$535$	0	0
$536$	12.0000	0.518321
$537$	− 10.0000i	− 0.431532i
$538$	− 24.0000i	− 1.03471i
$539$	12.0000	0.516877
$540$	0	0
$541$	−8.00000	−0.343947	−0.171973	−	0.985102i	$-0.555014\pi$
$541$	−8.00000	−0.343947	−0.171973	+	0.985102i	$0.555014\pi$
$542$	− 16.0000i	− 0.687259i
$543$	− 10.0000i	− 0.429141i
$544$	−8.00000	−0.342997
$545$	0	0
$546$	2.00000	0.0855921
$547$	20.0000i	0.855138i	0.903983	+	0.427569i	$0.140630\pi$
$547$	20.0000i	0.855138i	−0.903983	+	0.427569i	$0.859370\pi$
$548$	− 6.00000i	− 0.256307i
$549$	10.0000	0.426790
$550$	0	0
$551$	24.0000	1.02243
$552$	6.00000i	0.255377i
$553$	16.0000i	0.680389i
$554$	10.0000	0.424859
$555$	0	0
$556$	−8.00000	−0.339276
$557$	− 2.00000i	− 0.0847427i	−0.999102	−	0.0423714i	$-0.986509\pi$
$557$	− 2.00000i	− 0.0847427i	0.999102	−	0.0423714i	$-0.0134913\pi$
$558$	0	0
$559$	−4.00000	−0.169182
$560$	0	0
$561$	−32.0000	−1.35104
$562$	6.00000i	0.253095i
$563$	24.0000i	1.01148i	0.862686	+	0.505740i	$0.168780\pi$
$563$	24.0000i	1.01148i	−0.862686	+	0.505740i	$0.831220\pi$
$564$	0	0
$565$	0	0
$566$	−4.00000	−0.168133
$567$	− 2.00000i	− 0.0839921i
$568$	− 8.00000i	− 0.335673i
$569$	−6.00000	−0.251533	−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000	−0.251533	−0.125767	+	0.992060i	$0.540139\pi$
$570$	0	0
$571$	40.0000	1.67395	0.836974	−	0.547243i	$-0.184323\pi$
$571$	40.0000	1.67395	0.836974	+	0.547243i	$0.184323\pi$
$572$	4.00000i	0.167248i
$573$	0	0
$574$	4.00000	0.166957
$575$	0	0
$576$	1.00000	0.0416667
$577$	− 12.0000i	− 0.499567i	−0.968302	−	0.249783i	$-0.919641\pi$
$577$	− 12.0000i	− 0.499567i	0.968302	−	0.249783i	$-0.0803594\pi$
$578$	47.0000i	1.95494i
$579$	−4.00000	−0.166234
$580$	0	0
$581$	24.0000	0.995688
$582$	− 16.0000i	− 0.663221i
$583$	− 40.0000i	− 1.65663i
$584$	8.00000	0.331042
$585$	0	0
$586$	−26.0000	−1.07405
$587$	− 36.0000i	− 1.48588i	−0.669359	−	0.742940i	$-0.733431\pi$
$587$	− 36.0000i	− 1.48588i	0.669359	−	0.742940i	$-0.266569\pi$
$588$	3.00000i	0.123718i
$589$	0	0
$590$	0	0
$591$	18.0000	0.740421
$592$	2.00000i	0.0821995i
$593$	22.0000i	0.903432i	0.892162	+	0.451716i	$0.149188\pi$
$593$	22.0000i	0.903432i	−0.892162	+	0.451716i	$0.850812\pi$
$594$	4.00000	0.164122
$595$	0	0
$596$	−10.0000	−0.409616
$597$	0	0
$598$	− 6.00000i	− 0.245358i
$599$	−24.0000	−0.980613	−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000	−0.980613	−0.490307	+	0.871550i	$0.663115\pi$
$600$	0	0
$601$	22.0000	0.897399	0.448699	−	0.893683i	$-0.351887\pi$
$601$	22.0000	0.897399	0.448699	+	0.893683i	$0.351887\pi$
$602$	8.00000i	0.326056i
$603$	12.0000i	0.488678i
$604$	0	0
$605$	0	0
$606$	16.0000	0.649956
$607$	28.0000i	1.13648i	0.822861	+	0.568242i	$0.192376\pi$
$607$	28.0000i	1.13648i	−0.822861	+	0.568242i	$0.807624\pi$
$608$	− 6.00000i	− 0.243332i
$609$	−8.00000	−0.324176
$610$	0	0
$611$	0	0
$612$	− 8.00000i	− 0.323381i
$613$	− 10.0000i	− 0.403896i	−0.979396	−	0.201948i	$-0.935273\pi$
$613$	− 10.0000i	− 0.403896i	0.979396	−	0.201948i	$-0.0647272\pi$
$614$	−12.0000	−0.484281
$615$	0	0
$616$	8.00000	0.322329
$617$	6.00000i	0.241551i	0.992680	+	0.120775i	$0.0385381\pi$
$617$	6.00000i	0.241551i	−0.992680	+	0.120775i	$0.961462\pi$
$618$	12.0000i	0.482711i
$619$	10.0000	0.401934	0.200967	−	0.979598i	$-0.435592\pi$
$619$	10.0000	0.401934	0.200967	+	0.979598i	$0.435592\pi$
$620$	0	0
$621$	−6.00000	−0.240772
$622$	− 24.0000i	− 0.962312i
$623$	− 28.0000i	− 1.12180i
$624$	−1.00000	−0.0400320
$625$	0	0
$626$	6.00000	0.239808
$627$	− 24.0000i	− 0.958468i
$628$	22.0000i	0.877896i
$629$	16.0000	0.637962
$630$	0	0
$631$	12.0000	0.477712	0.238856	−	0.971055i	$-0.423228\pi$
$631$	12.0000	0.477712	0.238856	+	0.971055i	$0.423228\pi$
$632$	− 8.00000i	− 0.318223i
$633$	20.0000i	0.794929i
$634$	30.0000	1.19145
$635$	0	0
$636$	10.0000	0.396526
$637$	− 3.00000i	− 0.118864i
$638$	− 16.0000i	− 0.633446i
$639$	8.00000	0.316475
$640$	0	0
$641$	18.0000	0.710957	0.355479	−	0.934684i	$-0.384318\pi$
$641$	18.0000	0.710957	0.355479	+	0.934684i	$0.384318\pi$
$642$	12.0000i	0.473602i
$643$	44.0000i	1.73519i	0.497271	+	0.867595i	$0.334335\pi$
$643$	44.0000i	1.73519i	−0.497271	+	0.867595i	$0.665665\pi$
$644$	−12.0000	−0.472866
$645$	0	0
$646$	−48.0000	−1.88853
$647$	30.0000i	1.17942i	0.807614	+	0.589711i	$0.200758\pi$
$647$	30.0000i	1.17942i	−0.807614	+	0.589711i	$0.799242\pi$
$648$	1.00000i	0.0392837i
$649$	−16.0000	−0.628055
$650$	0	0
$651$	0	0
$652$	16.0000i	0.626608i
$653$	42.0000i	1.64359i	0.569785	+	0.821794i	$0.307026\pi$
$653$	42.0000i	1.64359i	−0.569785	+	0.821794i	$0.692974\pi$
$654$	12.0000	0.469237
$655$	0	0
$656$	−2.00000	−0.0780869
$657$	8.00000i	0.312110i
$658$	0	0
$659$	−2.00000	−0.0779089	−0.0389545	−	0.999241i	$-0.512403\pi$
$659$	−2.00000	−0.0779089	−0.0389545	+	0.999241i	$0.512403\pi$
$660$	0	0
$661$	48.0000	1.86698	0.933492	−	0.358599i	$-0.116745\pi$
$661$	48.0000	1.86698	0.933492	+	0.358599i	$0.116745\pi$
$662$	− 10.0000i	− 0.388661i
$663$	8.00000i	0.310694i
$664$	−12.0000	−0.465690
$665$	0	0
$666$	−2.00000	−0.0774984
$667$	24.0000i	0.929284i
$668$	− 4.00000i	− 0.154765i
$669$	−2.00000	−0.0773245
$670$	0	0
$671$	−40.0000	−1.54418
$672$	2.00000i	0.0771517i
$673$	26.0000i	1.00223i	0.865382	+	0.501113i	$0.167076\pi$
$673$	26.0000i	1.00223i	−0.865382	+	0.501113i	$0.832924\pi$
$674$	−14.0000	−0.539260
$675$	0	0
$676$	1.00000	0.0384615
$677$	30.0000i	1.15299i	0.817099	+	0.576497i	$0.195581\pi$
$677$	30.0000i	1.15299i	−0.817099	+	0.576497i	$0.804419\pi$
$678$	− 20.0000i	− 0.768095i
$679$	32.0000	1.22805
$680$	0	0
$681$	4.00000	0.153280
$682$	0	0
$683$	− 44.0000i	− 1.68361i	−0.539779	−	0.841807i	$-0.681492\pi$
$683$	− 44.0000i	− 1.68361i	0.539779	−	0.841807i	$-0.318508\pi$
$684$	6.00000	0.229416
$685$	0	0
$686$	−20.0000	−0.763604
$687$	4.00000i	0.152610i
$688$	− 4.00000i	− 0.152499i
$689$	−10.0000	−0.380970
$690$	0	0
$691$	−14.0000	−0.532585	−0.266293	−	0.963892i	$-0.585799\pi$
$691$	−14.0000	−0.532585	−0.266293	+	0.963892i	$0.585799\pi$
$692$	− 22.0000i	− 0.836315i
$693$	8.00000i	0.303895i
$694$	24.0000	0.911028
$695$	0	0
$696$	4.00000	0.151620
$697$	16.0000i	0.606043i
$698$	− 28.0000i	− 1.05982i
$699$	24.0000	0.907763
$700$	0	0
$701$	32.0000	1.20862	0.604312	−	0.796748i	$-0.293448\pi$
$701$	32.0000	1.20862	0.604312	+	0.796748i	$0.293448\pi$
$702$	− 1.00000i	− 0.0377426i
$703$	12.0000i	0.452589i
$704$	−4.00000	−0.150756
$705$	0	0
$706$	−14.0000	−0.526897
$707$	32.0000i	1.20348i
$708$	− 4.00000i	− 0.150329i
$709$	4.00000	0.150223	0.0751116	−	0.997175i	$-0.476069\pi$
$709$	4.00000	0.150223	0.0751116	+	0.997175i	$0.476069\pi$
$710$	0	0
$711$	8.00000	0.300023
$712$	14.0000i	0.524672i
$713$	0	0
$714$	16.0000	0.598785
$715$	0	0
$716$	−10.0000	−0.373718
$717$	16.0000i	0.597531i
$718$	− 24.0000i	− 0.895672i
$719$	−36.0000	−1.34257	−0.671287	−	0.741198i	$-0.734258\pi$
$719$	−36.0000	−1.34257	−0.671287	+	0.741198i	$0.734258\pi$
$720$	0	0
$721$	−24.0000	−0.893807
$722$	− 17.0000i	− 0.632674i
$723$	− 2.00000i	− 0.0743808i
$724$	−10.0000	−0.371647
$725$	0	0
$726$	−5.00000	−0.185567
$727$	− 16.0000i	− 0.593407i	−0.954970	−	0.296704i	$-0.904113\pi$
$727$	− 16.0000i	− 0.593407i	0.954970	−	0.296704i	$-0.0958873\pi$
$728$	− 2.00000i	− 0.0741249i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−32.0000	−1.18356
$732$	− 10.0000i	− 0.369611i
$733$	38.0000i	1.40356i	0.712393	+	0.701781i	$0.247612\pi$
$733$	38.0000i	1.40356i	−0.712393	+	0.701781i	$0.752388\pi$
$734$	36.0000	1.32878
$735$	0	0
$736$	6.00000	0.221163
$737$	− 48.0000i	− 1.76810i
$738$	− 2.00000i	− 0.0736210i
$739$	−30.0000	−1.10357	−0.551784	−	0.833987i	$-0.686053\pi$
$739$	−30.0000	−1.10357	−0.551784	+	0.833987i	$0.686053\pi$
$740$	0	0
$741$	−6.00000	−0.220416
$742$	20.0000i	0.734223i
$743$	12.0000i	0.440237i	0.975473	+	0.220119i	$0.0706445\pi$
$743$	12.0000i	0.440237i	−0.975473	+	0.220119i	$0.929356\pi$
$744$	0	0
$745$	0	0
$746$	−2.00000	−0.0732252
$747$	− 12.0000i	− 0.439057i
$748$	32.0000i	1.17004i
$749$	−24.0000	−0.876941
$750$	0	0
$751$	−16.0000	−0.583848	−0.291924	−	0.956441i	$-0.594295\pi$
$751$	−16.0000	−0.583848	−0.291924	+	0.956441i	$0.594295\pi$
$752$	0	0
$753$	− 6.00000i	− 0.218652i
$754$	−4.00000	−0.145671
$755$	0	0
$756$	−2.00000	−0.0727393
$757$	6.00000i	0.218074i	0.994038	+	0.109037i	$0.0347767\pi$
$757$	6.00000i	0.218074i	−0.994038	+	0.109037i	$0.965223\pi$
$758$	− 18.0000i	− 0.653789i
$759$	24.0000	0.871145
$760$	0	0
$761$	22.0000	0.797499	0.398750	−	0.917060i	$-0.369444\pi$
$761$	22.0000	0.797499	0.398750	+	0.917060i	$0.369444\pi$
$762$	4.00000i	0.144905i
$763$	24.0000i	0.868858i
$764$	0	0
$765$	0	0
$766$	−12.0000	−0.433578
$767$	4.00000i	0.144432i
$768$	− 1.00000i	− 0.0360844i
$769$	−10.0000	−0.360609	−0.180305	−	0.983611i	$-0.557708\pi$
$769$	−10.0000	−0.360609	−0.180305	+	0.983611i	$0.557708\pi$
$770$	0	0
$771$	−12.0000	−0.432169
$772$	4.00000i	0.143963i
$773$	38.0000i	1.36677i	0.730061	+	0.683383i	$0.239492\pi$
$773$	38.0000i	1.36677i	−0.730061	+	0.683383i	$0.760508\pi$
$774$	4.00000	0.143777
$775$	0	0
$776$	−16.0000	−0.574367
$777$	− 4.00000i	− 0.143499i
$778$	8.00000i	0.286814i
$779$	−12.0000	−0.429945
$780$	0	0
$781$	−32.0000	−1.14505
$782$	− 48.0000i	− 1.71648i
$783$	4.00000i	0.142948i
$784$	3.00000	0.107143
$785$	0	0
$786$	−10.0000	−0.356688
$787$	32.0000i	1.14068i	0.821410	+	0.570338i	$0.193188\pi$
$787$	32.0000i	1.14068i	−0.821410	+	0.570338i	$0.806812\pi$
$788$	− 18.0000i	− 0.641223i
$789$	2.00000	0.0712019
$790$	0	0
$791$	40.0000	1.42224
$792$	− 4.00000i	− 0.142134i
$793$	10.0000i	0.355110i
$794$	−14.0000	−0.496841
$795$	0	0
$796$	0	0
$797$	10.0000i	0.354218i	0.984191	+	0.177109i	$0.0566745\pi$
$797$	10.0000i	0.354218i	−0.984191	+	0.177109i	$0.943325\pi$
$798$	12.0000i	0.424795i
$799$	0	0
$800$	0	0
$801$	−14.0000	−0.494666
$802$	− 30.0000i	− 1.05934i
$803$	− 32.0000i	− 1.12926i
$804$	12.0000	0.423207
$805$	0	0
$806$	0	0
$807$	− 24.0000i	− 0.844840i
$808$	− 16.0000i	− 0.562878i
$809$	−2.00000	−0.0703163	−0.0351581	−	0.999382i	$-0.511193\pi$
$809$	−2.00000	−0.0703163	−0.0351581	+	0.999382i	$0.511193\pi$
$810$	0	0
$811$	−38.0000	−1.33436	−0.667180	−	0.744896i	$-0.732499\pi$
$811$	−38.0000	−1.33436	−0.667180	+	0.744896i	$0.732499\pi$
$812$	8.00000i	0.280745i
$813$	− 16.0000i	− 0.561144i
$814$	8.00000	0.280400
$815$	0	0
$816$	−8.00000	−0.280056
$817$	− 24.0000i	− 0.839654i
$818$	− 26.0000i	− 0.909069i
$819$	2.00000	0.0698857
$820$	0	0
$821$	−10.0000	−0.349002	−0.174501	−	0.984657i	$-0.555831\pi$
$821$	−10.0000	−0.349002	−0.174501	+	0.984657i	$0.555831\pi$
$822$	− 6.00000i	− 0.209274i
$823$	− 52.0000i	− 1.81261i	−0.422628	−	0.906303i	$-0.638892\pi$
$823$	− 52.0000i	− 1.81261i	0.422628	−	0.906303i	$-0.361108\pi$
$824$	12.0000	0.418040
$825$	0	0
$826$	8.00000	0.278356
$827$	12.0000i	0.417281i	0.977992	+	0.208640i	$0.0669038\pi$
$827$	12.0000i	0.417281i	−0.977992	+	0.208640i	$0.933096\pi$
$828$	6.00000i	0.208514i
$829$	10.0000	0.347314	0.173657	−	0.984806i	$-0.444442\pi$
$829$	10.0000	0.347314	0.173657	+	0.984806i	$0.444442\pi$
$830$	0	0
$831$	10.0000	0.346896
$832$	1.00000i	0.0346688i
$833$	− 24.0000i	− 0.831551i
$834$	−8.00000	−0.277017
$835$	0	0
$836$	−24.0000	−0.830057
$837$	0	0
$838$	10.0000i	0.345444i
$839$	40.0000	1.38095	0.690477	−	0.723355i	$-0.257401\pi$
$839$	40.0000	1.38095	0.690477	+	0.723355i	$0.257401\pi$
$840$	0	0
$841$	−13.0000	−0.448276
$842$	8.00000i	0.275698i
$843$	6.00000i	0.206651i
$844$	20.0000	0.688428
$845$	0	0
$846$	0	0
$847$	− 10.0000i	− 0.343604i
$848$	− 10.0000i	− 0.343401i
$849$	−4.00000	−0.137280
$850$	0	0
$851$	−12.0000	−0.411355
$852$	− 8.00000i	− 0.274075i
$853$	46.0000i	1.57501i	0.616308	+	0.787505i	$0.288628\pi$
$853$	46.0000i	1.57501i	−0.616308	+	0.787505i	$0.711372\pi$
$854$	20.0000	0.684386
$855$	0	0
$856$	12.0000	0.410152
$857$	− 4.00000i	− 0.136637i	−0.997664	−	0.0683187i	$-0.978237\pi$
$857$	− 4.00000i	− 0.136637i	0.997664	−	0.0683187i	$-0.0217635\pi$
$858$	4.00000i	0.136558i
$859$	−36.0000	−1.22830	−0.614152	−	0.789188i	$-0.710502\pi$
$859$	−36.0000	−1.22830	−0.614152	+	0.789188i	$0.710502\pi$
$860$	0	0
$861$	4.00000	0.136320
$862$	0	0
$863$	− 12.0000i	− 0.408485i	−0.978920	−	0.204242i	$-0.934527\pi$
$863$	− 12.0000i	− 0.408485i	0.978920	−	0.204242i	$-0.0654731\pi$
$864$	1.00000	0.0340207
$865$	0	0
$866$	−2.00000	−0.0679628
$867$	47.0000i	1.59620i
$868$	0	0
$869$	−32.0000	−1.08553
$870$	0	0
$871$	−12.0000	−0.406604
$872$	− 12.0000i	− 0.406371i
$873$	− 16.0000i	− 0.541518i
$874$	36.0000	1.21772
$875$	0	0
$876$	8.00000	0.270295
$877$	14.0000i	0.472746i	0.971662	+	0.236373i	$0.0759588\pi$
$877$	14.0000i	0.472746i	−0.971662	+	0.236373i	$0.924041\pi$
$878$	− 32.0000i	− 1.07995i
$879$	−26.0000	−0.876958
$880$	0	0
$881$	−18.0000	−0.606435	−0.303218	−	0.952921i	$-0.598061\pi$
$881$	−18.0000	−0.606435	−0.303218	+	0.952921i	$0.598061\pi$
$882$	3.00000i	0.101015i
$883$	− 28.0000i	− 0.942275i	−0.882060	−	0.471138i	$-0.843844\pi$
$883$	− 28.0000i	− 0.942275i	0.882060	−	0.471138i	$-0.156156\pi$
$884$	8.00000	0.269069
$885$	0	0
$886$	16.0000	0.537531
$887$	2.00000i	0.0671534i	0.999436	+	0.0335767i	$0.0106898\pi$
$887$	2.00000i	0.0671534i	−0.999436	+	0.0335767i	$0.989310\pi$
$888$	2.00000i	0.0671156i
$889$	−8.00000	−0.268311
$890$	0	0
$891$	4.00000	0.134005
$892$	2.00000i	0.0669650i
$893$	0	0
$894$	−10.0000	−0.334450
$895$	0	0
$896$	2.00000	0.0668153
$897$	− 6.00000i	− 0.200334i
$898$	− 6.00000i	− 0.200223i
$899$	0	0
$900$	0	0
$901$	−80.0000	−2.66519
$902$	8.00000i	0.266371i
$903$	8.00000i	0.266223i
$904$	−20.0000	−0.665190
$905$	0	0
$906$	0	0
$907$	− 52.0000i	− 1.72663i	−0.504664	−	0.863316i	$-0.668384\pi$
$907$	− 52.0000i	− 1.72663i	0.504664	−	0.863316i	$-0.331616\pi$
$908$	− 4.00000i	− 0.132745i
$909$	16.0000	0.530687
$910$	0	0
$911$	−20.0000	−0.662630	−0.331315	−	0.943520i	$-0.607492\pi$
$911$	−20.0000	−0.662630	−0.331315	+	0.943520i	$0.607492\pi$
$912$	− 6.00000i	− 0.198680i
$913$	48.0000i	1.58857i
$914$	−8.00000	−0.264616
$915$	0	0
$916$	4.00000	0.132164
$917$	− 20.0000i	− 0.660458i
$918$	− 8.00000i	− 0.264039i
$919$	−40.0000	−1.31948	−0.659739	−	0.751495i	$-0.729333\pi$
$919$	−40.0000	−1.31948	−0.659739	+	0.751495i	$0.729333\pi$
$920$	0	0
$921$	−12.0000	−0.395413
$922$	6.00000i	0.197599i
$923$	8.00000i	0.263323i
$924$	8.00000	0.263181
$925$	0	0
$926$	−26.0000	−0.854413
$927$	12.0000i	0.394132i
$928$	− 4.00000i	− 0.131306i
$929$	30.0000	0.984268	0.492134	−	0.870519i	$-0.336217\pi$
$929$	30.0000	0.984268	0.492134	+	0.870519i	$0.336217\pi$
$930$	0	0
$931$	18.0000	0.589926
$932$	− 24.0000i	− 0.786146i
$933$	− 24.0000i	− 0.785725i
$934$	−28.0000	−0.916188
$935$	0	0
$936$	−1.00000	−0.0326860
$937$	− 18.0000i	− 0.588034i	−0.955800	−	0.294017i	$-0.905008\pi$
$937$	− 18.0000i	− 0.588034i	0.955800	−	0.294017i	$-0.0949923\pi$
$938$	24.0000i	0.783628i
$939$	6.00000	0.195803
$940$	0	0
$941$	−34.0000	−1.10837	−0.554184	−	0.832394i	$-0.686970\pi$
$941$	−34.0000	−1.10837	−0.554184	+	0.832394i	$0.686970\pi$
$942$	22.0000i	0.716799i
$943$	− 12.0000i	− 0.390774i
$944$	−4.00000	−0.130189
$945$	0	0
$946$	−16.0000	−0.520205
$947$	− 4.00000i	− 0.129983i	−0.997886	−	0.0649913i	$-0.979298\pi$
$947$	− 4.00000i	− 0.129983i	0.997886	−	0.0649913i	$-0.0207020\pi$
$948$	− 8.00000i	− 0.259828i
$949$	−8.00000	−0.259691
$950$	0	0
$951$	30.0000	0.972817
$952$	− 16.0000i	− 0.518563i
$953$	24.0000i	0.777436i	0.921357	+	0.388718i	$0.127082\pi$
$953$	24.0000i	0.777436i	−0.921357	+	0.388718i	$0.872918\pi$
$954$	10.0000	0.323762
$955$	0	0
$956$	16.0000	0.517477
$957$	− 16.0000i	− 0.517207i
$958$	32.0000i	1.03387i
$959$	12.0000	0.387500
$960$	0	0
$961$	−31.0000	−1.00000
$962$	− 2.00000i	− 0.0644826i
$963$	12.0000i	0.386695i
$964$	−2.00000	−0.0644157
$965$	0	0
$966$	−12.0000	−0.386094
$967$	14.0000i	0.450210i	0.974335	+	0.225105i	$0.0722725\pi$
$967$	14.0000i	0.450210i	−0.974335	+	0.225105i	$0.927728\pi$
$968$	5.00000i	0.160706i
$969$	−48.0000	−1.54198
$970$	0	0
$971$	38.0000	1.21948	0.609739	−	0.792602i	$-0.291274\pi$
$971$	38.0000	1.21948	0.609739	+	0.792602i	$0.291274\pi$
$972$	1.00000i	0.0320750i
$973$	− 16.0000i	− 0.512936i
$974$	−26.0000	−0.833094
$975$	0	0
$976$	−10.0000	−0.320092
$977$	− 30.0000i	− 0.959785i	−0.877327	−	0.479893i	$-0.840676\pi$
$977$	− 30.0000i	− 0.959785i	0.877327	−	0.479893i	$-0.159324\pi$
$978$	16.0000i	0.511624i
$979$	56.0000	1.78977
$980$	0	0
$981$	12.0000	0.383131
$982$	− 42.0000i	− 1.34027i
$983$	− 52.0000i	− 1.65854i	−0.558846	−	0.829271i	$-0.688756\pi$
$983$	− 52.0000i	− 1.65854i	0.558846	−	0.829271i	$-0.311244\pi$
$984$	−2.00000	−0.0637577
$985$	0	0
$986$	−32.0000	−1.01909
$987$	0	0
$988$	6.00000i	0.190885i
$989$	24.0000	0.763156
$990$	0	0
$991$	0	0		−	1.00000i	$-0.5\pi$
$991$	0	0			1.00000i	$0.5\pi$
$992$	0	0
$993$	− 10.0000i	− 0.317340i
$994$	16.0000	0.507489
$995$	0	0
$996$	−12.0000	−0.380235
$997$	26.0000i	0.823428i	0.911313	+	0.411714i	$0.135070\pi$
$997$	26.0000i	0.823428i	−0.911313	+	0.411714i	$0.864930\pi$
$998$	38.0000i	1.20287i
$999$	−2.00000	−0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1950.2.e.f.1249.1		2
3.2	odd	2		5850.2.e.i.5149.2		2
5.2	odd	4		390.2.a.e.1.1	✓	1
5.3	odd	4		1950.2.a.h.1.1		1
5.4	even	2	inner	1950.2.e.f.1249.2		2
15.2	even	4		1170.2.a.e.1.1		1
15.8	even	4		5850.2.a.bi.1.1		1
15.14	odd	2		5850.2.e.i.5149.1		2
20.7	even	4		3120.2.a.o.1.1		1
60.47	odd	4		9360.2.a.bh.1.1		1
65.12	odd	4		5070.2.a.e.1.1		1
65.47	even	4		5070.2.b.e.1351.2		2
65.57	even	4		5070.2.b.e.1351.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
390.2.a.e.1.1	✓	1	5.2	odd	4
1170.2.a.e.1.1		1	15.2	even	4
1950.2.a.h.1.1		1	5.3	odd	4
1950.2.e.f.1249.1		2	1.1	even	1	trivial
1950.2.e.f.1249.2		2	5.4	even	2	inner
3120.2.a.o.1.1		1	20.7	even	4
5070.2.a.e.1.1		1	65.12	odd	4
5070.2.b.e.1351.1		2	65.57	even	4
5070.2.b.e.1351.2		2	65.47	even	4
5850.2.a.bi.1.1		1	15.8	even	4
5850.2.e.i.5149.1		2	15.14	odd	2
5850.2.e.i.5149.2		2	3.2	odd	2
9360.2.a.bh.1.1		1	60.47	odd	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.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +4.00000 q^{11} +1.00000i q^{12} -1.00000i q^{13} -2.00000 q^{14} +1.00000 q^{16} -8.00000i q^{17} +1.00000i q^{18} +6.00000 q^{19} -2.00000 q^{21} -4.00000i q^{22} +6.00000i q^{23} +1.00000 q^{24} -1.00000 q^{26} +1.00000i q^{27} +2.00000i q^{28} +4.00000 q^{29} -1.00000i q^{32} -4.00000i q^{33} -8.00000 q^{34} +1.00000 q^{36} +2.00000i q^{37} -6.00000i q^{38} -1.00000 q^{39} -2.00000 q^{41} +2.00000i q^{42} -4.00000i q^{43} -4.00000 q^{44} +6.00000 q^{46} -1.00000i q^{48} +3.00000 q^{49} -8.00000 q^{51} +1.00000i q^{52} -10.0000i q^{53} +1.00000 q^{54} +2.00000 q^{56} -6.00000i q^{57} -4.00000i q^{58} -4.00000 q^{59} -10.0000 q^{61} +2.00000i q^{63} -1.00000 q^{64} -4.00000 q^{66} -12.0000i q^{67} +8.00000i q^{68} +6.00000 q^{69} -8.00000 q^{71} -1.00000i q^{72} -8.00000i q^{73} +2.00000 q^{74} -6.00000 q^{76} -8.00000i q^{77} +1.00000i q^{78} -8.00000 q^{79} +1.00000 q^{81} +2.00000i q^{82} +12.0000i q^{83} +2.00000 q^{84} -4.00000 q^{86} -4.00000i q^{87} +4.00000i q^{88} +14.0000 q^{89} -2.00000 q^{91} -6.00000i q^{92} -1.00000 q^{96} +16.0000i q^{97} -3.00000i q^{98} -4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9	\( 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9} + 8 q^{11} - 4 q^{14} + 2 q^{16} + 12 q^{19} - 4 q^{21} + 2 q^{24} - 2 q^{26} + 8 q^{29} - 16 q^{34} + 2 q^{36} - 2 q^{39} - 4 q^{41} - 8 q^{44} + 12 q^{46} + 6 q^{49} - 16 q^{51} + 2 q^{54} + 4 q^{56} - 8 q^{59} - 20 q^{61} - 2 q^{64} - 8 q^{66} + 12 q^{69} - 16 q^{71} + 4 q^{74} - 12 q^{76} - 16 q^{79} + 2 q^{81} + 4 q^{84} - 8 q^{86} + 28 q^{89} - 4 q^{91} - 2 q^{96} - 8 q^{99}+O(q^{100}) \) 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9 + 8 * q^11 - 4 * q^14 + 2 * q^16 + 12 * q^19 - 4 * q^21 + 2 * q^24 - 2 * q^26 + 8 * q^29 - 16 * q^34 + 2 * q^36 - 2 * q^39 - 4 * q^41 - 8 * q^44 + 12 * q^46 + 6 * q^49 - 16 * q^51 + 2 * q^54 + 4 * q^56 - 8 * q^59 - 20 * q^61 - 2 * q^64 - 8 * q^66 + 12 * q^69 - 16 * q^71 + 4 * q^74 - 12 * q^76 - 16 * q^79 + 2 * q^81 + 4 * q^84 - 8 * q^86 + 28 * q^89 - 4 * q^91 - 2 * q^96 - 8 * q^99

Introduction

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