LMFDB - Embedded newform 1950.2.e.j.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:	yes
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.j.1249.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -3.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -3.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} -3.00000 q^{11} -1.00000i q^{12} +1.00000i q^{13} -3.00000 q^{14} +1.00000 q^{16} -3.00000i q^{17} +1.00000i q^{18} +3.00000 q^{21} +3.00000i q^{22} +4.00000i q^{23} -1.00000 q^{24} +1.00000 q^{26} -1.00000i q^{27} +3.00000i q^{28} -5.00000 q^{29} -3.00000 q^{31} -1.00000i q^{32} -3.00000i q^{33} -3.00000 q^{34} +1.00000 q^{36} +12.0000i q^{37} -1.00000 q^{39} +2.00000 q^{41} -3.00000i q^{42} +4.00000i q^{43} +3.00000 q^{44} +4.00000 q^{46} -3.00000i q^{47} +1.00000i q^{48} -2.00000 q^{49} +3.00000 q^{51} -1.00000i q^{52} +9.00000i q^{53} -1.00000 q^{54} +3.00000 q^{56} +5.00000i q^{58} -15.0000 q^{59} -3.00000 q^{61} +3.00000i q^{62} +3.00000i q^{63} -1.00000 q^{64} -3.00000 q^{66} +7.00000i q^{67} +3.00000i q^{68} -4.00000 q^{69} -8.00000 q^{71} -1.00000i q^{72} -16.0000i q^{73} +12.0000 q^{74} +9.00000i q^{77} +1.00000i q^{78} +1.00000 q^{81} -2.00000i q^{82} -1.00000i q^{83} -3.00000 q^{84} +4.00000 q^{86} -5.00000i q^{87} -3.00000i q^{88} +3.00000 q^{91} -4.00000i q^{92} -3.00000i q^{93} -3.00000 q^{94} +1.00000 q^{96} +2.00000i q^{97} +2.00000i q^{98} +3.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} - 6 q^{11} - 6 q^{14} + 2 q^{16} + 6 q^{21} - 2 q^{24} + 2 q^{26} - 10 q^{29} - 6 q^{31} - 6 q^{34} + 2 q^{36} - 2 q^{39} + 4 q^{41} + 6 q^{44} + 8 q^{46} - 4 q^{49} + 6 q^{51} - 2 q^{54} + 6 q^{56} - 30 q^{59} - 6 q^{61} - 2 q^{64} - 6 q^{66} - 8 q^{69} - 16 q^{71} + 24 q^{74} + 2 q^{81} - 6 q^{84} + 8 q^{86} + 6 q^{91} - 6 q^{94} + 2 q^{96} + 6 q^{99}+O(q^{100}) $ 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9 - 6 * q^11 - 6 * q^14 + 2 * q^16 + 6 * q^21 - 2 * q^24 + 2 * q^26 - 10 * q^29 - 6 * q^31 - 6 * q^34 + 2 * q^36 - 2 * q^39 + 4 * q^41 + 6 * q^44 + 8 * q^46 - 4 * q^49 + 6 * q^51 - 2 * q^54 + 6 * q^56 - 30 * q^59 - 6 * q^61 - 2 * q^64 - 6 * q^66 - 8 * q^69 - 16 * q^71 + 24 * q^74 + 2 * q^81 - 6 * q^84 + 8 * q^86 + 6 * q^91 - 6 * q^94 + 2 * q^96 + 6 * 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
$7$	− 3.00000i	− 1.13389i	−0.823754	−	0.566947i	$-0.808125\pi$
$7$	− 3.00000i	− 1.13389i	0.823754	−	0.566947i	$-0.191875\pi$
$8$	1.00000i	0.353553i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−3.00000	−0.904534	−0.452267	−	0.891883i	$-0.649385\pi$
$11$	−3.00000	−0.904534	−0.452267	+	0.891883i	$0.649385\pi$
$12$	− 1.00000i	− 0.288675i
$13$	1.00000i	0.277350i
$13$	1.00000i	0.277350i
$14$	−3.00000	−0.801784
$15$	0	0
$16$	1.00000	0.250000
$17$	− 3.00000i	− 0.727607i	−0.931476	−	0.363803i	$-0.881478\pi$
$17$	− 3.00000i	− 0.727607i	0.931476	−	0.363803i	$-0.118522\pi$
$18$	1.00000i	0.235702i
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	3.00000	0.654654
$22$	3.00000i	0.639602i
$23$	4.00000i	0.834058i	0.908893	+	0.417029i	$0.136929\pi$
$23$	4.00000i	0.834058i	−0.908893	+	0.417029i	$0.863071\pi$
$24$	−1.00000	−0.204124
$25$	0	0
$26$	1.00000	0.196116
$27$	− 1.00000i	− 0.192450i
$28$	3.00000i	0.566947i
$29$	−5.00000	−0.928477	−0.464238	−	0.885710i	$-0.653672\pi$
$29$	−5.00000	−0.928477	−0.464238	+	0.885710i	$0.653672\pi$
$30$	0	0
$31$	−3.00000	−0.538816	−0.269408	−	0.963026i	$-0.586828\pi$
$31$	−3.00000	−0.538816	−0.269408	+	0.963026i	$0.586828\pi$
$32$	− 1.00000i	− 0.176777i
$33$	− 3.00000i	− 0.522233i
$34$	−3.00000	−0.514496
$35$	0	0
$36$	1.00000	0.166667
$37$	12.0000i	1.97279i	0.164399	+	0.986394i	$0.447432\pi$
$37$	12.0000i	1.97279i	−0.164399	+	0.986394i	$0.552568\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	− 3.00000i	− 0.462910i
$43$	4.00000i	0.609994i	0.952353	+	0.304997i	$0.0986555\pi$
$43$	4.00000i	0.609994i	−0.952353	+	0.304997i	$0.901344\pi$
$44$	3.00000	0.452267
$45$	0	0
$46$	4.00000	0.589768
$47$	− 3.00000i	− 0.437595i	−0.975770	−	0.218797i	$-0.929787\pi$
$47$	− 3.00000i	− 0.437595i	0.975770	−	0.218797i	$-0.0702134\pi$
$48$	1.00000i	0.144338i
$49$	−2.00000	−0.285714
$50$	0	0
$51$	3.00000	0.420084
$52$	− 1.00000i	− 0.138675i
$53$	9.00000i	1.23625i	0.786082	+	0.618123i	$0.212106\pi$
$53$	9.00000i	1.23625i	−0.786082	+	0.618123i	$0.787894\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	3.00000	0.400892
$57$	0	0
$58$	5.00000i	0.656532i
$59$	−15.0000	−1.95283	−0.976417	−	0.215894i	$-0.930733\pi$
$59$	−15.0000	−1.95283	−0.976417	+	0.215894i	$0.930733\pi$
$60$	0	0
$61$	−3.00000	−0.384111	−0.192055	−	0.981384i	$-0.561515\pi$
$61$	−3.00000	−0.384111	−0.192055	+	0.981384i	$0.561515\pi$
$62$	3.00000i	0.381000i
$63$	3.00000i	0.377964i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	−3.00000	−0.369274
$67$	7.00000i	0.855186i	0.903971	+	0.427593i	$0.140638\pi$
$67$	7.00000i	0.855186i	−0.903971	+	0.427593i	$0.859362\pi$
$68$	3.00000i	0.363803i
$69$	−4.00000	−0.481543
$70$	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
$73$	− 16.0000i	− 1.87266i	−0.351123	−	0.936329i	$-0.614200\pi$
$73$	− 16.0000i	− 1.87266i	0.351123	−	0.936329i	$-0.385800\pi$
$74$	12.0000	1.39497
$75$	0	0
$76$	0	0
$77$	9.00000i	1.02565i
$78$	1.00000i	0.113228i
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	− 2.00000i	− 0.220863i
$83$	− 1.00000i	− 0.109764i	−0.998493	−	0.0548821i	$-0.982522\pi$
$83$	− 1.00000i	− 0.109764i	0.998493	−	0.0548821i	$-0.0174783\pi$
$84$	−3.00000	−0.327327
$85$	0	0
$86$	4.00000	0.431331
$87$	− 5.00000i	− 0.536056i
$88$	− 3.00000i	− 0.319801i
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	3.00000	0.314485
$92$	− 4.00000i	− 0.417029i
$93$	− 3.00000i	− 0.311086i
$94$	−3.00000	−0.309426
$95$	0	0
$96$	1.00000	0.102062
$97$	2.00000i	0.203069i	0.994832	+	0.101535i	$0.0323753\pi$
$97$	2.00000i	0.203069i	−0.994832	+	0.101535i	$0.967625\pi$
$98$	2.00000i	0.202031i
$99$	3.00000	0.301511
$100$	0	0
$101$	−3.00000	−0.298511	−0.149256	−	0.988799i	$-0.547688\pi$
$101$	−3.00000	−0.298511	−0.149256	+	0.988799i	$0.547688\pi$
$102$	− 3.00000i	− 0.297044i
$103$	− 6.00000i	− 0.591198i	−0.955312	−	0.295599i	$-0.904481\pi$
$103$	− 6.00000i	− 0.591198i	0.955312	−	0.295599i	$-0.0955191\pi$
$104$	−1.00000	−0.0980581
$105$	0	0
$106$	9.00000	0.874157
$107$	2.00000i	0.193347i	0.995316	+	0.0966736i	$0.0308203\pi$
$107$	2.00000i	0.193347i	−0.995316	+	0.0966736i	$0.969180\pi$
$108$	1.00000i	0.0962250i
$109$	−10.0000	−0.957826	−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000	−0.957826	−0.478913	+	0.877862i	$0.658969\pi$
$110$	0	0
$111$	−12.0000	−1.13899
$112$	− 3.00000i	− 0.283473i
$113$	14.0000i	1.31701i	0.752577	+	0.658505i	$0.228811\pi$
$113$	14.0000i	1.31701i	−0.752577	+	0.658505i	$0.771189\pi$
$114$	0	0
$115$	0	0
$116$	5.00000	0.464238
$117$	− 1.00000i	− 0.0924500i
$118$	15.0000i	1.38086i
$119$	−9.00000	−0.825029
$120$	0	0
$121$	−2.00000	−0.181818
$122$	3.00000i	0.271607i
$123$	2.00000i	0.180334i
$124$	3.00000	0.269408
$125$	0	0
$126$	3.00000	0.267261
$127$	12.0000i	1.06483i	0.846484	+	0.532414i	$0.178715\pi$
$127$	12.0000i	1.06483i	−0.846484	+	0.532414i	$0.821285\pi$
$128$	1.00000i	0.0883883i
$129$	−4.00000	−0.352180
$130$	0	0
$131$	−18.0000	−1.57267	−0.786334	−	0.617802i	$-0.788023\pi$
$131$	−18.0000	−1.57267	−0.786334	+	0.617802i	$0.788023\pi$
$132$	3.00000i	0.261116i
$133$	0	0
$134$	7.00000	0.604708
$135$	0	0
$136$	3.00000	0.257248
$137$	12.0000i	1.02523i	0.858619	+	0.512615i	$0.171323\pi$
$137$	12.0000i	1.02523i	−0.858619	+	0.512615i	$0.828677\pi$
$138$	4.00000i	0.340503i
$139$	−10.0000	−0.848189	−0.424094	−	0.905618i	$-0.639408\pi$
$139$	−10.0000	−0.848189	−0.424094	+	0.905618i	$0.639408\pi$
$140$	0	0
$141$	3.00000	0.252646
$142$	8.00000i	0.671345i
$143$	− 3.00000i	− 0.250873i
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	−16.0000	−1.32417
$147$	− 2.00000i	− 0.164957i
$148$	− 12.0000i	− 0.986394i
$149$	−10.0000	−0.819232	−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000	−0.819232	−0.409616	+	0.912258i	$0.634337\pi$
$150$	0	0
$151$	17.0000	1.38344	0.691720	−	0.722166i	$-0.256853\pi$
$151$	17.0000	1.38344	0.691720	+	0.722166i	$0.256853\pi$
$152$	0	0
$153$	3.00000i	0.242536i
$154$	9.00000	0.725241
$155$	0	0
$156$	1.00000	0.0800641
$157$	7.00000i	0.558661i	0.960195	+	0.279330i	$0.0901125\pi$
$157$	7.00000i	0.558661i	−0.960195	+	0.279330i	$0.909888\pi$
$158$	0	0
$159$	−9.00000	−0.713746
$160$	0	0
$161$	12.0000	0.945732
$162$	− 1.00000i	− 0.0785674i
$163$	4.00000i	0.313304i	0.987654	+	0.156652i	$0.0500701\pi$
$163$	4.00000i	0.313304i	−0.987654	+	0.156652i	$0.949930\pi$
$164$	−2.00000	−0.156174
$165$	0	0
$166$	−1.00000	−0.0776151
$167$	12.0000i	0.928588i	0.885681	+	0.464294i	$0.153692\pi$
$167$	12.0000i	0.928588i	−0.885681	+	0.464294i	$0.846308\pi$
$168$	3.00000i	0.231455i
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	0	0
$172$	− 4.00000i	− 0.304997i
$173$	− 1.00000i	− 0.0760286i	−0.999277	−	0.0380143i	$-0.987897\pi$
$173$	− 1.00000i	− 0.0760286i	0.999277	−	0.0380143i	$-0.0121032\pi$
$174$	−5.00000	−0.379049
$175$	0	0
$176$	−3.00000	−0.226134
$177$	− 15.0000i	− 1.12747i
$178$	0	0
$179$	−10.0000	−0.747435	−0.373718	−	0.927543i	$-0.621917\pi$
$179$	−10.0000	−0.747435	−0.373718	+	0.927543i	$0.621917\pi$
$180$	0	0
$181$	17.0000	1.26360	0.631800	−	0.775131i	$-0.282316\pi$
$181$	17.0000	1.26360	0.631800	+	0.775131i	$0.282316\pi$
$182$	− 3.00000i	− 0.222375i
$183$	− 3.00000i	− 0.221766i
$184$	−4.00000	−0.294884
$185$	0	0
$186$	−3.00000	−0.219971
$187$	9.00000i	0.658145i
$188$	3.00000i	0.218797i
$189$	−3.00000	−0.218218
$190$	0	0
$191$	−18.0000	−1.30243	−0.651217	−	0.758891i	$-0.725741\pi$
$191$	−18.0000	−1.30243	−0.651217	+	0.758891i	$0.725741\pi$
$192$	− 1.00000i	− 0.0721688i
$193$	− 26.0000i	− 1.87152i	−0.352636	−	0.935760i	$-0.614715\pi$
$193$	− 26.0000i	− 1.87152i	0.352636	−	0.935760i	$-0.385285\pi$
$194$	2.00000	0.143592
$195$	0	0
$196$	2.00000	0.142857
$197$	2.00000i	0.142494i	0.997459	+	0.0712470i	$0.0226979\pi$
$197$	2.00000i	0.142494i	−0.997459	+	0.0712470i	$0.977302\pi$
$198$	− 3.00000i	− 0.213201i
$199$	10.0000	0.708881	0.354441	−	0.935079i	$-0.384671\pi$
$199$	10.0000	0.708881	0.354441	+	0.935079i	$0.384671\pi$
$200$	0	0
$201$	−7.00000	−0.493742
$202$	3.00000i	0.211079i
$203$	15.0000i	1.05279i
$204$	−3.00000	−0.210042
$205$	0	0
$206$	−6.00000	−0.418040
$207$	− 4.00000i	− 0.278019i
$208$	1.00000i	0.0693375i
$209$	0	0
$210$	0	0
$211$	12.0000	0.826114	0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000	0.826114	0.413057	+	0.910705i	$0.364461\pi$
$212$	− 9.00000i	− 0.618123i
$213$	− 8.00000i	− 0.548151i
$214$	2.00000	0.136717
$215$	0	0
$216$	1.00000	0.0680414
$217$	9.00000i	0.610960i
$218$	10.0000i	0.677285i
$219$	16.0000	1.08118
$220$	0	0
$221$	3.00000	0.201802
$222$	12.0000i	0.805387i
$223$	− 16.0000i	− 1.07144i	−0.844396	−	0.535720i	$-0.820040\pi$
$223$	− 16.0000i	− 1.07144i	0.844396	−	0.535720i	$-0.179960\pi$
$224$	−3.00000	−0.200446
$225$	0	0
$226$	14.0000	0.931266
$227$	− 13.0000i	− 0.862840i	−0.902151	−	0.431420i	$-0.858013\pi$
$227$	− 13.0000i	− 0.862840i	0.902151	−	0.431420i	$-0.141987\pi$
$228$	0	0
$229$	10.0000	0.660819	0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000	0.660819	0.330409	+	0.943838i	$0.392813\pi$
$230$	0	0
$231$	−9.00000	−0.592157
$232$	− 5.00000i	− 0.328266i
$233$	− 6.00000i	− 0.393073i	−0.980497	−	0.196537i	$-0.937031\pi$
$233$	− 6.00000i	− 0.393073i	0.980497	−	0.196537i	$-0.0629694\pi$
$234$	−1.00000	−0.0653720
$235$	0	0
$236$	15.0000	0.976417
$237$	0	0
$238$	9.00000i	0.583383i
$239$	5.00000	0.323423	0.161712	−	0.986838i	$-0.448299\pi$
$239$	5.00000	0.323423	0.161712	+	0.986838i	$0.448299\pi$
$240$	0	0
$241$	22.0000	1.41714	0.708572	−	0.705638i	$-0.249340\pi$
$241$	22.0000	1.41714	0.708572	+	0.705638i	$0.249340\pi$
$242$	2.00000i	0.128565i
$243$	1.00000i	0.0641500i
$244$	3.00000	0.192055
$245$	0	0
$246$	2.00000	0.127515
$247$	0	0
$248$	− 3.00000i	− 0.190500i
$249$	1.00000	0.0633724
$250$	0	0
$251$	2.00000	0.126239	0.0631194	−	0.998006i	$-0.479895\pi$
$251$	2.00000	0.126239	0.0631194	+	0.998006i	$0.479895\pi$
$252$	− 3.00000i	− 0.188982i
$253$	− 12.0000i	− 0.754434i
$254$	12.0000	0.752947
$255$	0	0
$256$	1.00000	0.0625000
$257$	27.0000i	1.68421i	0.539311	+	0.842107i	$0.318685\pi$
$257$	27.0000i	1.68421i	−0.539311	+	0.842107i	$0.681315\pi$
$258$	4.00000i	0.249029i
$259$	36.0000	2.23693
$260$	0	0
$261$	5.00000	0.309492
$262$	18.0000i	1.11204i
$263$	− 26.0000i	− 1.60323i	−0.597841	−	0.801614i	$-0.703975\pi$
$263$	− 26.0000i	− 1.60323i	0.597841	−	0.801614i	$-0.296025\pi$
$264$	3.00000	0.184637
$265$	0	0
$266$	0	0
$267$	0	0
$268$	− 7.00000i	− 0.427593i
$269$	−5.00000	−0.304855	−0.152428	−	0.988315i	$-0.548709\pi$
$269$	−5.00000	−0.304855	−0.152428	+	0.988315i	$0.548709\pi$
$270$	0	0
$271$	−3.00000	−0.182237	−0.0911185	−	0.995840i	$-0.529044\pi$
$271$	−3.00000	−0.182237	−0.0911185	+	0.995840i	$0.529044\pi$
$272$	− 3.00000i	− 0.181902i
$273$	3.00000i	0.181568i
$274$	12.0000	0.724947
$275$	0	0
$276$	4.00000	0.240772
$277$	2.00000i	0.120168i	0.998193	+	0.0600842i	$0.0191369\pi$
$277$	2.00000i	0.120168i	−0.998193	+	0.0600842i	$0.980863\pi$
$278$	10.0000i	0.599760i
$279$	3.00000	0.179605
$280$	0	0
$281$	−18.0000	−1.07379	−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000	−1.07379	−0.536895	+	0.843649i	$0.680403\pi$
$282$	− 3.00000i	− 0.178647i
$283$	− 6.00000i	− 0.356663i	−0.983970	−	0.178331i	$-0.942930\pi$
$283$	− 6.00000i	− 0.356663i	0.983970	−	0.178331i	$-0.0570699\pi$
$284$	8.00000	0.474713
$285$	0	0
$286$	−3.00000	−0.177394
$287$	− 6.00000i	− 0.354169i
$288$	1.00000i	0.0589256i
$289$	8.00000	0.470588
$290$	0	0
$291$	−2.00000	−0.117242
$292$	16.0000i	0.936329i
$293$	− 26.0000i	− 1.51894i	−0.650545	−	0.759468i	$-0.725459\pi$
$293$	− 26.0000i	− 1.51894i	0.650545	−	0.759468i	$-0.274541\pi$
$294$	−2.00000	−0.116642
$295$	0	0
$296$	−12.0000	−0.697486
$297$	3.00000i	0.174078i
$298$	10.0000i	0.579284i
$299$	−4.00000	−0.231326
$300$	0	0
$301$	12.0000	0.691669
$302$	− 17.0000i	− 0.978240i
$303$	− 3.00000i	− 0.172345i
$304$	0	0
$305$	0	0
$306$	3.00000	0.171499
$307$	− 28.0000i	− 1.59804i	−0.601302	−	0.799022i	$-0.705351\pi$
$307$	− 28.0000i	− 1.59804i	0.601302	−	0.799022i	$-0.294649\pi$
$308$	− 9.00000i	− 0.512823i
$309$	6.00000	0.341328
$310$	0	0
$311$	2.00000	0.113410	0.0567048	−	0.998391i	$-0.481941\pi$
$311$	2.00000	0.113410	0.0567048	+	0.998391i	$0.481941\pi$
$312$	− 1.00000i	− 0.0566139i
$313$	19.0000i	1.07394i	0.843600	+	0.536972i	$0.180432\pi$
$313$	19.0000i	1.07394i	−0.843600	+	0.536972i	$0.819568\pi$
$314$	7.00000	0.395033
$315$	0	0
$316$	0	0
$317$	− 28.0000i	− 1.57264i	−0.617822	−	0.786318i	$-0.711985\pi$
$317$	− 28.0000i	− 1.57264i	0.617822	−	0.786318i	$-0.288015\pi$
$318$	9.00000i	0.504695i
$319$	15.0000	0.839839
$320$	0	0
$321$	−2.00000	−0.111629
$322$	− 12.0000i	− 0.668734i
$323$	0	0
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	4.00000	0.221540
$327$	− 10.0000i	− 0.553001i
$328$	2.00000i	0.110432i
$329$	−9.00000	−0.496186
$330$	0	0
$331$	−28.0000	−1.53902	−0.769510	−	0.638635i	$-0.779499\pi$
$331$	−28.0000	−1.53902	−0.769510	+	0.638635i	$0.779499\pi$
$332$	1.00000i	0.0548821i
$333$	− 12.0000i	− 0.657596i
$334$	12.0000	0.656611
$335$	0	0
$336$	3.00000	0.163663
$337$	27.0000i	1.47078i	0.677642	+	0.735392i	$0.263002\pi$
$337$	27.0000i	1.47078i	−0.677642	+	0.735392i	$0.736998\pi$
$338$	1.00000i	0.0543928i
$339$	−14.0000	−0.760376
$340$	0	0
$341$	9.00000	0.487377
$342$	0	0
$343$	− 15.0000i	− 0.809924i
$344$	−4.00000	−0.215666
$345$	0	0
$346$	−1.00000	−0.0537603
$347$	− 18.0000i	− 0.966291i	−0.875540	−	0.483145i	$-0.839494\pi$
$347$	− 18.0000i	− 0.966291i	0.875540	−	0.483145i	$-0.160506\pi$
$348$	5.00000i	0.268028i
$349$	20.0000	1.07058	0.535288	−	0.844670i	$-0.320203\pi$
$349$	20.0000	1.07058	0.535288	+	0.844670i	$0.320203\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	3.00000i	0.159901i
$353$	14.0000i	0.745145i	0.928003	+	0.372572i	$0.121524\pi$
$353$	14.0000i	0.745145i	−0.928003	+	0.372572i	$0.878476\pi$
$354$	−15.0000	−0.797241
$355$	0	0
$356$	0	0
$357$	− 9.00000i	− 0.476331i
$358$	10.0000i	0.528516i
$359$	15.0000	0.791670	0.395835	−	0.918322i	$-0.370455\pi$
$359$	15.0000	0.791670	0.395835	+	0.918322i	$0.370455\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	− 17.0000i	− 0.893500i
$363$	− 2.00000i	− 0.104973i
$364$	−3.00000	−0.157243
$365$	0	0
$366$	−3.00000	−0.156813
$367$	− 28.0000i	− 1.46159i	−0.682598	−	0.730794i	$-0.739150\pi$
$367$	− 28.0000i	− 1.46159i	0.682598	−	0.730794i	$-0.260850\pi$
$368$	4.00000i	0.208514i
$369$	−2.00000	−0.104116
$370$	0	0
$371$	27.0000	1.40177
$372$	3.00000i	0.155543i
$373$	29.0000i	1.50156i	0.660551	+	0.750782i	$0.270323\pi$
$373$	29.0000i	1.50156i	−0.660551	+	0.750782i	$0.729677\pi$
$374$	9.00000	0.465379
$375$	0	0
$376$	3.00000	0.154713
$377$	− 5.00000i	− 0.257513i
$378$	3.00000i	0.154303i
$379$	−15.0000	−0.770498	−0.385249	−	0.922813i	$-0.625884\pi$
$379$	−15.0000	−0.770498	−0.385249	+	0.922813i	$0.625884\pi$
$380$	0	0
$381$	−12.0000	−0.614779
$382$	18.0000i	0.920960i
$383$	− 16.0000i	− 0.817562i	−0.912633	−	0.408781i	$-0.865954\pi$
$383$	− 16.0000i	− 0.817562i	0.912633	−	0.408781i	$-0.134046\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	−26.0000	−1.32337
$387$	− 4.00000i	− 0.203331i
$388$	− 2.00000i	− 0.101535i
$389$	30.0000	1.52106	0.760530	−	0.649303i	$-0.224939\pi$
$389$	30.0000	1.52106	0.760530	+	0.649303i	$0.224939\pi$
$390$	0	0
$391$	12.0000	0.606866
$392$	− 2.00000i	− 0.101015i
$393$	− 18.0000i	− 0.907980i
$394$	2.00000	0.100759
$395$	0	0
$396$	−3.00000	−0.150756
$397$	22.0000i	1.10415i	0.833795	+	0.552074i	$0.186163\pi$
$397$	22.0000i	1.10415i	−0.833795	+	0.552074i	$0.813837\pi$
$398$	− 10.0000i	− 0.501255i
$399$	0	0
$400$	0	0
$401$	12.0000	0.599251	0.299626	−	0.954057i	$-0.403138\pi$
$401$	12.0000	0.599251	0.299626	+	0.954057i	$0.403138\pi$
$402$	7.00000i	0.349128i
$403$	− 3.00000i	− 0.149441i
$404$	3.00000	0.149256
$405$	0	0
$406$	15.0000	0.744438
$407$	− 36.0000i	− 1.78445i
$408$	3.00000i	0.148522i
$409$	10.0000	0.494468	0.247234	−	0.968956i	$-0.420478\pi$
$409$	10.0000	0.494468	0.247234	+	0.968956i	$0.420478\pi$
$410$	0	0
$411$	−12.0000	−0.591916
$412$	6.00000i	0.295599i
$413$	45.0000i	2.21431i
$414$	−4.00000	−0.196589
$415$	0	0
$416$	1.00000	0.0490290
$417$	− 10.0000i	− 0.489702i
$418$	0	0
$419$	−30.0000	−1.46560	−0.732798	−	0.680446i	$-0.761786\pi$
$419$	−30.0000	−1.46560	−0.732798	+	0.680446i	$0.761786\pi$
$420$	0	0
$421$	−28.0000	−1.36464	−0.682318	−	0.731055i	$-0.739028\pi$
$421$	−28.0000	−1.36464	−0.682318	+	0.731055i	$0.739028\pi$
$422$	− 12.0000i	− 0.584151i
$423$	3.00000i	0.145865i
$424$	−9.00000	−0.437079
$425$	0	0
$426$	−8.00000	−0.387601
$427$	9.00000i	0.435541i
$428$	− 2.00000i	− 0.0966736i
$429$	3.00000	0.144841
$430$	0	0
$431$	12.0000	0.578020	0.289010	−	0.957326i	$-0.406674\pi$
$431$	12.0000	0.578020	0.289010	+	0.957326i	$0.406674\pi$
$432$	− 1.00000i	− 0.0481125i
$433$	− 26.0000i	− 1.24948i	−0.780833	−	0.624740i	$-0.785205\pi$
$433$	− 26.0000i	− 1.24948i	0.780833	−	0.624740i	$-0.214795\pi$
$434$	9.00000	0.432014
$435$	0	0
$436$	10.0000	0.478913
$437$	0	0
$438$	− 16.0000i	− 0.764510i
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	0	0
$441$	2.00000	0.0952381
$442$	− 3.00000i	− 0.142695i
$443$	4.00000i	0.190046i	0.995475	+	0.0950229i	$0.0302924\pi$
$443$	4.00000i	0.190046i	−0.995475	+	0.0950229i	$0.969708\pi$
$444$	12.0000	0.569495
$445$	0	0
$446$	−16.0000	−0.757622
$447$	− 10.0000i	− 0.472984i
$448$	3.00000i	0.141737i
$449$	20.0000	0.943858	0.471929	−	0.881636i	$-0.343558\pi$
$449$	20.0000	0.943858	0.471929	+	0.881636i	$0.343558\pi$
$450$	0	0
$451$	−6.00000	−0.282529
$452$	− 14.0000i	− 0.658505i
$453$	17.0000i	0.798730i
$454$	−13.0000	−0.610120
$455$	0	0
$456$	0	0
$457$	− 18.0000i	− 0.842004i	−0.907060	−	0.421002i	$-0.861678\pi$
$457$	− 18.0000i	− 0.842004i	0.907060	−	0.421002i	$-0.138322\pi$
$458$	− 10.0000i	− 0.467269i
$459$	−3.00000	−0.140028
$460$	0	0
$461$	−38.0000	−1.76984	−0.884918	−	0.465746i	$-0.845786\pi$
$461$	−38.0000	−1.76984	−0.884918	+	0.465746i	$0.845786\pi$
$462$	9.00000i	0.418718i
$463$	9.00000i	0.418265i	0.977887	+	0.209133i	$0.0670641\pi$
$463$	9.00000i	0.418265i	−0.977887	+	0.209133i	$0.932936\pi$
$464$	−5.00000	−0.232119
$465$	0	0
$466$	−6.00000	−0.277945
$467$	32.0000i	1.48078i	0.672176	+	0.740392i	$0.265360\pi$
$467$	32.0000i	1.48078i	−0.672176	+	0.740392i	$0.734640\pi$
$468$	1.00000i	0.0462250i
$469$	21.0000	0.969690
$470$	0	0
$471$	−7.00000	−0.322543
$472$	− 15.0000i	− 0.690431i
$473$	− 12.0000i	− 0.551761i
$474$	0	0
$475$	0	0
$476$	9.00000	0.412514
$477$	− 9.00000i	− 0.412082i
$478$	− 5.00000i	− 0.228695i
$479$	−15.0000	−0.685367	−0.342684	−	0.939451i	$-0.611336\pi$
$479$	−15.0000	−0.685367	−0.342684	+	0.939451i	$0.611336\pi$
$480$	0	0
$481$	−12.0000	−0.547153
$482$	− 22.0000i	− 1.00207i
$483$	12.0000i	0.546019i
$484$	2.00000	0.0909091
$485$	0	0
$486$	1.00000	0.0453609
$487$	− 43.0000i	− 1.94852i	−0.225436	−	0.974258i	$-0.572381\pi$
$487$	− 43.0000i	− 1.94852i	0.225436	−	0.974258i	$-0.427619\pi$
$488$	− 3.00000i	− 0.135804i
$489$	−4.00000	−0.180886
$490$	0	0
$491$	2.00000	0.0902587	0.0451294	−	0.998981i	$-0.485630\pi$
$491$	2.00000	0.0902587	0.0451294	+	0.998981i	$0.485630\pi$
$492$	− 2.00000i	− 0.0901670i
$493$	15.0000i	0.675566i
$494$	0	0
$495$	0	0
$496$	−3.00000	−0.134704
$497$	24.0000i	1.07655i
$498$	− 1.00000i	− 0.0448111i
$499$	−5.00000	−0.223831	−0.111915	−	0.993718i	$-0.535699\pi$
$499$	−5.00000	−0.223831	−0.111915	+	0.993718i	$0.535699\pi$
$500$	0	0
$501$	−12.0000	−0.536120
$502$	− 2.00000i	− 0.0892644i
$503$	− 6.00000i	− 0.267527i	−0.991013	−	0.133763i	$-0.957294\pi$
$503$	− 6.00000i	− 0.267527i	0.991013	−	0.133763i	$-0.0427062\pi$
$504$	−3.00000	−0.133631
$505$	0	0
$506$	−12.0000	−0.533465
$507$	− 1.00000i	− 0.0444116i
$508$	− 12.0000i	− 0.532414i
$509$	−10.0000	−0.443242	−0.221621	−	0.975133i	$-0.571135\pi$
$509$	−10.0000	−0.443242	−0.221621	+	0.975133i	$0.571135\pi$
$510$	0	0
$511$	−48.0000	−2.12339
$512$	− 1.00000i	− 0.0441942i
$513$	0	0
$514$	27.0000	1.19092
$515$	0	0
$516$	4.00000	0.176090
$517$	9.00000i	0.395820i
$518$	− 36.0000i	− 1.58175i
$519$	1.00000	0.0438951
$520$	0	0
$521$	22.0000	0.963837	0.481919	−	0.876216i	$-0.339940\pi$
$521$	22.0000	0.963837	0.481919	+	0.876216i	$0.339940\pi$
$522$	− 5.00000i	− 0.218844i
$523$	14.0000i	0.612177i	0.952003	+	0.306089i	$0.0990204\pi$
$523$	14.0000i	0.612177i	−0.952003	+	0.306089i	$0.900980\pi$
$524$	18.0000	0.786334
$525$	0	0
$526$	−26.0000	−1.13365
$527$	9.00000i	0.392046i
$528$	− 3.00000i	− 0.130558i
$529$	7.00000	0.304348
$530$	0	0
$531$	15.0000	0.650945
$532$	0	0
$533$	2.00000i	0.0866296i
$534$	0	0
$535$	0	0
$536$	−7.00000	−0.302354
$537$	− 10.0000i	− 0.431532i
$538$	5.00000i	0.215565i
$539$	6.00000	0.258438
$540$	0	0
$541$	32.0000	1.37579	0.687894	−	0.725811i	$-0.258536\pi$
$541$	32.0000	1.37579	0.687894	+	0.725811i	$0.258536\pi$
$542$	3.00000i	0.128861i
$543$	17.0000i	0.729540i
$544$	−3.00000	−0.128624
$545$	0	0
$546$	3.00000	0.128388
$547$	22.0000i	0.940652i	0.882493	+	0.470326i	$0.155864\pi$
$547$	22.0000i	0.940652i	−0.882493	+	0.470326i	$0.844136\pi$
$548$	− 12.0000i	− 0.512615i
$549$	3.00000	0.128037
$550$	0	0
$551$	0	0
$552$	− 4.00000i	− 0.170251i
$553$	0	0
$554$	2.00000	0.0849719
$555$	0	0
$556$	10.0000	0.424094
$557$	− 18.0000i	− 0.762684i	−0.924434	−	0.381342i	$-0.875462\pi$
$557$	− 18.0000i	− 0.762684i	0.924434	−	0.381342i	$-0.124538\pi$
$558$	− 3.00000i	− 0.127000i
$559$	−4.00000	−0.169182
$560$	0	0
$561$	−9.00000	−0.379980
$562$	18.0000i	0.759284i
$563$	4.00000i	0.168580i	0.996441	+	0.0842900i	$0.0268622\pi$
$563$	4.00000i	0.168580i	−0.996441	+	0.0842900i	$0.973138\pi$
$564$	−3.00000	−0.126323
$565$	0	0
$566$	−6.00000	−0.252199
$567$	− 3.00000i	− 0.125988i
$568$	− 8.00000i	− 0.335673i
$569$	−15.0000	−0.628833	−0.314416	−	0.949285i	$-0.601809\pi$
$569$	−15.0000	−0.628833	−0.314416	+	0.949285i	$0.601809\pi$
$570$	0	0
$571$	−28.0000	−1.17176	−0.585882	−	0.810397i	$-0.699252\pi$
$571$	−28.0000	−1.17176	−0.585882	+	0.810397i	$0.699252\pi$
$572$	3.00000i	0.125436i
$573$	− 18.0000i	− 0.751961i
$574$	−6.00000	−0.250435
$575$	0	0
$576$	1.00000	0.0416667
$577$	22.0000i	0.915872i	0.888985	+	0.457936i	$0.151411\pi$
$577$	22.0000i	0.915872i	−0.888985	+	0.457936i	$0.848589\pi$
$578$	− 8.00000i	− 0.332756i
$579$	26.0000	1.08052
$580$	0	0
$581$	−3.00000	−0.124461
$582$	2.00000i	0.0829027i
$583$	− 27.0000i	− 1.11823i
$584$	16.0000	0.662085
$585$	0	0
$586$	−26.0000	−1.07405
$587$	− 33.0000i	− 1.36206i	−0.732257	−	0.681028i	$-0.761533\pi$
$587$	− 33.0000i	− 1.36206i	0.732257	−	0.681028i	$-0.238467\pi$
$588$	2.00000i	0.0824786i
$589$	0	0
$590$	0	0
$591$	−2.00000	−0.0822690
$592$	12.0000i	0.493197i
$593$	− 6.00000i	− 0.246390i	−0.992382	−	0.123195i	$-0.960686\pi$
$593$	− 6.00000i	− 0.246390i	0.992382	−	0.123195i	$-0.0393141\pi$
$594$	3.00000	0.123091
$595$	0	0
$596$	10.0000	0.409616
$597$	10.0000i	0.409273i
$598$	4.00000i	0.163572i
$599$	−30.0000	−1.22577	−0.612883	−	0.790173i	$-0.709990\pi$
$599$	−30.0000	−1.22577	−0.612883	+	0.790173i	$0.709990\pi$
$600$	0	0
$601$	−3.00000	−0.122373	−0.0611863	−	0.998126i	$-0.519488\pi$
$601$	−3.00000	−0.122373	−0.0611863	+	0.998126i	$0.519488\pi$
$602$	− 12.0000i	− 0.489083i
$603$	− 7.00000i	− 0.285062i
$604$	−17.0000	−0.691720
$605$	0	0
$606$	−3.00000	−0.121867
$607$	− 18.0000i	− 0.730597i	−0.930890	−	0.365299i	$-0.880967\pi$
$607$	− 18.0000i	− 0.730597i	0.930890	−	0.365299i	$-0.119033\pi$
$608$	0	0
$609$	−15.0000	−0.607831
$610$	0	0
$611$	3.00000	0.121367
$612$	− 3.00000i	− 0.121268i
$613$	− 26.0000i	− 1.05013i	−0.851062	−	0.525065i	$-0.824041\pi$
$613$	− 26.0000i	− 1.05013i	0.851062	−	0.525065i	$-0.175959\pi$
$614$	−28.0000	−1.12999
$615$	0	0
$616$	−9.00000	−0.362620
$617$	42.0000i	1.69086i	0.534089	+	0.845428i	$0.320655\pi$
$617$	42.0000i	1.69086i	−0.534089	+	0.845428i	$0.679345\pi$
$618$	− 6.00000i	− 0.241355i
$619$	20.0000	0.803868	0.401934	−	0.915669i	$-0.368338\pi$
$619$	20.0000	0.803868	0.401934	+	0.915669i	$0.368338\pi$
$620$	0	0
$621$	4.00000	0.160514
$622$	− 2.00000i	− 0.0801927i
$623$	0	0
$624$	−1.00000	−0.0400320
$625$	0	0
$626$	19.0000	0.759393
$627$	0	0
$628$	− 7.00000i	− 0.279330i
$629$	36.0000	1.43541
$630$	0	0
$631$	12.0000	0.477712	0.238856	−	0.971055i	$-0.423228\pi$
$631$	12.0000	0.477712	0.238856	+	0.971055i	$0.423228\pi$
$632$	0	0
$633$	12.0000i	0.476957i
$634$	−28.0000	−1.11202
$635$	0	0
$636$	9.00000	0.356873
$637$	− 2.00000i	− 0.0792429i
$638$	− 15.0000i	− 0.593856i
$639$	8.00000	0.316475
$640$	0	0
$641$	17.0000	0.671460	0.335730	−	0.941958i	$-0.391017\pi$
$641$	17.0000	0.671460	0.335730	+	0.941958i	$0.391017\pi$
$642$	2.00000i	0.0789337i
$643$	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
$646$	0	0
$647$	− 48.0000i	− 1.88707i	−0.331266	−	0.943537i	$-0.607476\pi$
$647$	− 48.0000i	− 1.88707i	0.331266	−	0.943537i	$-0.392524\pi$
$648$	1.00000i	0.0392837i
$649$	45.0000	1.76640
$650$	0	0
$651$	−9.00000	−0.352738
$652$	− 4.00000i	− 0.156652i
$653$	39.0000i	1.52619i	0.646288	+	0.763094i	$0.276321\pi$
$653$	39.0000i	1.52619i	−0.646288	+	0.763094i	$0.723679\pi$
$654$	−10.0000	−0.391031
$655$	0	0
$656$	2.00000	0.0780869
$657$	16.0000i	0.624219i
$658$	9.00000i	0.350857i
$659$	−40.0000	−1.55818	−0.779089	−	0.626913i	$-0.784318\pi$
$659$	−40.0000	−1.55818	−0.779089	+	0.626913i	$0.784318\pi$
$660$	0	0
$661$	42.0000	1.63361	0.816805	−	0.576913i	$-0.195743\pi$
$661$	42.0000	1.63361	0.816805	+	0.576913i	$0.195743\pi$
$662$	28.0000i	1.08825i
$663$	3.00000i	0.116510i
$664$	1.00000	0.0388075
$665$	0	0
$666$	−12.0000	−0.464991
$667$	− 20.0000i	− 0.774403i
$668$	− 12.0000i	− 0.464294i
$669$	16.0000	0.618596
$670$	0	0
$671$	9.00000	0.347441
$672$	− 3.00000i	− 0.115728i
$673$	− 51.0000i	− 1.96591i	−0.183858	−	0.982953i	$-0.558859\pi$
$673$	− 51.0000i	− 1.96591i	0.183858	−	0.982953i	$-0.441141\pi$
$674$	27.0000	1.04000
$675$	0	0
$676$	1.00000	0.0384615
$677$	42.0000i	1.61419i	0.590421	+	0.807096i	$0.298962\pi$
$677$	42.0000i	1.61419i	−0.590421	+	0.807096i	$0.701038\pi$
$678$	14.0000i	0.537667i
$679$	6.00000	0.230259
$680$	0	0
$681$	13.0000	0.498161
$682$	− 9.00000i	− 0.344628i
$683$	− 21.0000i	− 0.803543i	−0.915740	−	0.401771i	$-0.868395\pi$
$683$	− 21.0000i	− 0.803543i	0.915740	−	0.401771i	$-0.131605\pi$
$684$	0	0
$685$	0	0
$686$	−15.0000	−0.572703
$687$	10.0000i	0.381524i
$688$	4.00000i	0.152499i
$689$	−9.00000	−0.342873
$690$	0	0
$691$	17.0000	0.646710	0.323355	−	0.946278i	$-0.395189\pi$
$691$	17.0000	0.646710	0.323355	+	0.946278i	$0.395189\pi$
$692$	1.00000i	0.0380143i
$693$	− 9.00000i	− 0.341882i
$694$	−18.0000	−0.683271
$695$	0	0
$696$	5.00000	0.189525
$697$	− 6.00000i	− 0.227266i
$698$	− 20.0000i	− 0.757011i
$699$	6.00000	0.226941
$700$	0	0
$701$	27.0000	1.01978	0.509888	−	0.860241i	$-0.329687\pi$
$701$	27.0000	1.01978	0.509888	+	0.860241i	$0.329687\pi$
$702$	− 1.00000i	− 0.0377426i
$703$	0	0
$704$	3.00000	0.113067
$705$	0	0
$706$	14.0000	0.526897
$707$	9.00000i	0.338480i
$708$	15.0000i	0.563735i
$709$	10.0000	0.375558	0.187779	−	0.982211i	$-0.439871\pi$
$709$	10.0000	0.375558	0.187779	+	0.982211i	$0.439871\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 12.0000i	− 0.449404i
$714$	−9.00000	−0.336817
$715$	0	0
$716$	10.0000	0.373718
$717$	5.00000i	0.186728i
$718$	− 15.0000i	− 0.559795i
$719$	−30.0000	−1.11881	−0.559406	−	0.828894i	$-0.688971\pi$
$719$	−30.0000	−1.11881	−0.559406	+	0.828894i	$0.688971\pi$
$720$	0	0
$721$	−18.0000	−0.670355
$722$	19.0000i	0.707107i
$723$	22.0000i	0.818189i
$724$	−17.0000	−0.631800
$725$	0	0
$726$	−2.00000	−0.0742270
$727$	42.0000i	1.55769i	0.627214	+	0.778847i	$0.284195\pi$
$727$	42.0000i	1.55769i	−0.627214	+	0.778847i	$0.715805\pi$
$728$	3.00000i	0.111187i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	12.0000	0.443836
$732$	3.00000i	0.110883i
$733$	4.00000i	0.147743i	0.997268	+	0.0738717i	$0.0235355\pi$
$733$	4.00000i	0.147743i	−0.997268	+	0.0738717i	$0.976464\pi$
$734$	−28.0000	−1.03350
$735$	0	0
$736$	4.00000	0.147442
$737$	− 21.0000i	− 0.773545i
$738$	2.00000i	0.0736210i
$739$	−5.00000	−0.183928	−0.0919640	−	0.995762i	$-0.529314\pi$
$739$	−5.00000	−0.183928	−0.0919640	+	0.995762i	$0.529314\pi$
$740$	0	0
$741$	0	0
$742$	− 27.0000i	− 0.991201i
$743$	− 21.0000i	− 0.770415i	−0.922830	−	0.385208i	$-0.874130\pi$
$743$	− 21.0000i	− 0.770415i	0.922830	−	0.385208i	$-0.125870\pi$
$744$	3.00000	0.109985
$745$	0	0
$746$	29.0000	1.06177
$747$	1.00000i	0.0365881i
$748$	− 9.00000i	− 0.329073i
$749$	6.00000	0.219235
$750$	0	0
$751$	12.0000	0.437886	0.218943	−	0.975738i	$-0.429739\pi$
$751$	12.0000	0.437886	0.218943	+	0.975738i	$0.429739\pi$
$752$	− 3.00000i	− 0.109399i
$753$	2.00000i	0.0728841i
$754$	−5.00000	−0.182089
$755$	0	0
$756$	3.00000	0.109109
$757$	− 43.0000i	− 1.56286i	−0.623992	−	0.781431i	$-0.714490\pi$
$757$	− 43.0000i	− 1.56286i	0.623992	−	0.781431i	$-0.285510\pi$
$758$	15.0000i	0.544825i
$759$	12.0000	0.435572
$760$	0	0
$761$	−28.0000	−1.01500	−0.507500	−	0.861652i	$-0.669430\pi$
$761$	−28.0000	−1.01500	−0.507500	+	0.861652i	$0.669430\pi$
$762$	12.0000i	0.434714i
$763$	30.0000i	1.08607i
$764$	18.0000	0.651217
$765$	0	0
$766$	−16.0000	−0.578103
$767$	− 15.0000i	− 0.541619i
$768$	1.00000i	0.0360844i
$769$	−40.0000	−1.44244	−0.721218	−	0.692708i	$-0.756418\pi$
$769$	−40.0000	−1.44244	−0.721218	+	0.692708i	$0.756418\pi$
$770$	0	0
$771$	−27.0000	−0.972381
$772$	26.0000i	0.935760i
$773$	34.0000i	1.22290i	0.791285	+	0.611448i	$0.209412\pi$
$773$	34.0000i	1.22290i	−0.791285	+	0.611448i	$0.790588\pi$
$774$	−4.00000	−0.143777
$775$	0	0
$776$	−2.00000	−0.0717958
$777$	36.0000i	1.29149i
$778$	− 30.0000i	− 1.07555i
$779$	0	0
$780$	0	0
$781$	24.0000	0.858788
$782$	− 12.0000i	− 0.429119i
$783$	5.00000i	0.178685i
$784$	−2.00000	−0.0714286
$785$	0	0
$786$	−18.0000	−0.642039
$787$	− 3.00000i	− 0.106938i	−0.998569	−	0.0534692i	$-0.982972\pi$
$787$	− 3.00000i	− 0.106938i	0.998569	−	0.0534692i	$-0.0170279\pi$
$788$	− 2.00000i	− 0.0712470i
$789$	26.0000	0.925625
$790$	0	0
$791$	42.0000	1.49335
$792$	3.00000i	0.106600i
$793$	− 3.00000i	− 0.106533i
$794$	22.0000	0.780751
$795$	0	0
$796$	−10.0000	−0.354441
$797$	− 33.0000i	− 1.16892i	−0.811423	−	0.584460i	$-0.801306\pi$
$797$	− 33.0000i	− 1.16892i	0.811423	−	0.584460i	$-0.198694\pi$
$798$	0	0
$799$	−9.00000	−0.318397
$800$	0	0
$801$	0	0
$802$	− 12.0000i	− 0.423735i
$803$	48.0000i	1.69388i
$804$	7.00000	0.246871
$805$	0	0
$806$	−3.00000	−0.105670
$807$	− 5.00000i	− 0.176008i
$808$	− 3.00000i	− 0.105540i
$809$	10.0000	0.351581	0.175791	−	0.984428i	$-0.443752\pi$
$809$	10.0000	0.351581	0.175791	+	0.984428i	$0.443752\pi$
$810$	0	0
$811$	−43.0000	−1.50993	−0.754967	−	0.655763i	$-0.772347\pi$
$811$	−43.0000	−1.50993	−0.754967	+	0.655763i	$0.772347\pi$
$812$	− 15.0000i	− 0.526397i
$813$	− 3.00000i	− 0.105215i
$814$	−36.0000	−1.26180
$815$	0	0
$816$	3.00000	0.105021
$817$	0	0
$818$	− 10.0000i	− 0.349642i
$819$	−3.00000	−0.104828
$820$	0	0
$821$	−28.0000	−0.977207	−0.488603	−	0.872506i	$-0.662493\pi$
$821$	−28.0000	−0.977207	−0.488603	+	0.872506i	$0.662493\pi$
$822$	12.0000i	0.418548i
$823$	44.0000i	1.53374i	0.641800	+	0.766872i	$0.278188\pi$
$823$	44.0000i	1.53374i	−0.641800	+	0.766872i	$0.721812\pi$
$824$	6.00000	0.209020
$825$	0	0
$826$	45.0000	1.56575
$827$	− 3.00000i	− 0.104320i	−0.998639	−	0.0521601i	$-0.983389\pi$
$827$	− 3.00000i	− 0.104320i	0.998639	−	0.0521601i	$-0.0166106\pi$
$828$	4.00000i	0.139010i
$829$	−15.0000	−0.520972	−0.260486	−	0.965478i	$-0.583883\pi$
$829$	−15.0000	−0.520972	−0.260486	+	0.965478i	$0.583883\pi$
$830$	0	0
$831$	−2.00000	−0.0693792
$832$	− 1.00000i	− 0.0346688i
$833$	6.00000i	0.207888i
$834$	−10.0000	−0.346272
$835$	0	0
$836$	0	0
$837$	3.00000i	0.103695i
$838$	30.0000i	1.03633i
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	−4.00000	−0.137931
$842$	28.0000i	0.964944i
$843$	− 18.0000i	− 0.619953i
$844$	−12.0000	−0.413057
$845$	0	0
$846$	3.00000	0.103142
$847$	6.00000i	0.206162i
$848$	9.00000i	0.309061i
$849$	6.00000	0.205919
$850$	0	0
$851$	−48.0000	−1.64542
$852$	8.00000i	0.274075i
$853$	− 16.0000i	− 0.547830i	−0.961754	−	0.273915i	$-0.911681\pi$
$853$	− 16.0000i	− 0.547830i	0.961754	−	0.273915i	$-0.0883186\pi$
$854$	9.00000	0.307974
$855$	0	0
$856$	−2.00000	−0.0683586
$857$	22.0000i	0.751506i	0.926720	+	0.375753i	$0.122616\pi$
$857$	22.0000i	0.751506i	−0.926720	+	0.375753i	$0.877384\pi$
$858$	− 3.00000i	− 0.102418i
$859$	−40.0000	−1.36478	−0.682391	−	0.730987i	$-0.739060\pi$
$859$	−40.0000	−1.36478	−0.682391	+	0.730987i	$0.739060\pi$
$860$	0	0
$861$	6.00000	0.204479
$862$	− 12.0000i	− 0.408722i
$863$	39.0000i	1.32758i	0.747921	+	0.663788i	$0.231052\pi$
$863$	39.0000i	1.32758i	−0.747921	+	0.663788i	$0.768948\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	−26.0000	−0.883516
$867$	8.00000i	0.271694i
$868$	− 9.00000i	− 0.305480i
$869$	0	0
$870$	0	0
$871$	−7.00000	−0.237186
$872$	− 10.0000i	− 0.338643i
$873$	− 2.00000i	− 0.0676897i
$874$	0	0
$875$	0	0
$876$	−16.0000	−0.540590
$877$	− 8.00000i	− 0.270141i	−0.990836	−	0.135070i	$-0.956874\pi$
$877$	− 8.00000i	− 0.270141i	0.990836	−	0.135070i	$-0.0431261\pi$
$878$	0	0
$879$	26.0000	0.876958
$880$	0	0
$881$	7.00000	0.235836	0.117918	−	0.993023i	$-0.462378\pi$
$881$	7.00000	0.235836	0.117918	+	0.993023i	$0.462378\pi$
$882$	− 2.00000i	− 0.0673435i
$883$	− 16.0000i	− 0.538443i	−0.963078	−	0.269221i	$-0.913234\pi$
$883$	− 16.0000i	− 0.538443i	0.963078	−	0.269221i	$-0.0867663\pi$
$884$	−3.00000	−0.100901
$885$	0	0
$886$	4.00000	0.134383
$887$	− 48.0000i	− 1.61168i	−0.592132	−	0.805841i	$-0.701714\pi$
$887$	− 48.0000i	− 1.61168i	0.592132	−	0.805841i	$-0.298286\pi$
$888$	− 12.0000i	− 0.402694i
$889$	36.0000	1.20740
$890$	0	0
$891$	−3.00000	−0.100504
$892$	16.0000i	0.535720i
$893$	0	0
$894$	−10.0000	−0.334450
$895$	0	0
$896$	3.00000	0.100223
$897$	− 4.00000i	− 0.133556i
$898$	− 20.0000i	− 0.667409i
$899$	15.0000	0.500278
$900$	0	0
$901$	27.0000	0.899500
$902$	6.00000i	0.199778i
$903$	12.0000i	0.399335i
$904$	−14.0000	−0.465633
$905$	0	0
$906$	17.0000	0.564787
$907$	− 8.00000i	− 0.265636i	−0.991140	−	0.132818i	$-0.957597\pi$
$907$	− 8.00000i	− 0.265636i	0.991140	−	0.132818i	$-0.0424025\pi$
$908$	13.0000i	0.431420i
$909$	3.00000	0.0995037
$910$	0	0
$911$	52.0000	1.72284	0.861418	−	0.507896i	$-0.169577\pi$
$911$	52.0000	1.72284	0.861418	+	0.507896i	$0.169577\pi$
$912$	0	0
$913$	3.00000i	0.0992855i
$914$	−18.0000	−0.595387
$915$	0	0
$916$	−10.0000	−0.330409
$917$	54.0000i	1.78324i
$918$	3.00000i	0.0990148i
$919$	−50.0000	−1.64935	−0.824674	−	0.565608i	$-0.808641\pi$
$919$	−50.0000	−1.64935	−0.824674	+	0.565608i	$0.808641\pi$
$920$	0	0
$921$	28.0000	0.922631
$922$	38.0000i	1.25146i
$923$	− 8.00000i	− 0.263323i
$924$	9.00000	0.296078
$925$	0	0
$926$	9.00000	0.295758
$927$	6.00000i	0.197066i
$928$	5.00000i	0.164133i
$929$	10.0000	0.328089	0.164045	−	0.986453i	$-0.447546\pi$
$929$	10.0000	0.328089	0.164045	+	0.986453i	$0.447546\pi$
$930$	0	0
$931$	0	0
$932$	6.00000i	0.196537i
$933$	2.00000i	0.0654771i
$934$	32.0000	1.04707
$935$	0	0
$936$	1.00000	0.0326860
$937$	− 13.0000i	− 0.424691i	−0.977195	−	0.212346i	$-0.931890\pi$
$937$	− 13.0000i	− 0.424691i	0.977195	−	0.212346i	$-0.0681103\pi$
$938$	− 21.0000i	− 0.685674i
$939$	−19.0000	−0.620042
$940$	0	0
$941$	52.0000	1.69515	0.847576	−	0.530674i	$-0.178061\pi$
$941$	52.0000	1.69515	0.847576	+	0.530674i	$0.178061\pi$
$942$	7.00000i	0.228072i
$943$	8.00000i	0.260516i
$944$	−15.0000	−0.488208
$945$	0	0
$946$	−12.0000	−0.390154
$947$	57.0000i	1.85225i	0.377215	+	0.926126i	$0.376882\pi$
$947$	57.0000i	1.85225i	−0.377215	+	0.926126i	$0.623118\pi$
$948$	0	0
$949$	16.0000	0.519382
$950$	0	0
$951$	28.0000	0.907962
$952$	− 9.00000i	− 0.291692i
$953$	29.0000i	0.939402i	0.882826	+	0.469701i	$0.155638\pi$
$953$	29.0000i	0.939402i	−0.882826	+	0.469701i	$0.844362\pi$
$954$	−9.00000	−0.291386
$955$	0	0
$956$	−5.00000	−0.161712
$957$	15.0000i	0.484881i
$958$	15.0000i	0.484628i
$959$	36.0000	1.16250
$960$	0	0
$961$	−22.0000	−0.709677
$962$	12.0000i	0.386896i
$963$	− 2.00000i	− 0.0644491i
$964$	−22.0000	−0.708572
$965$	0	0
$966$	12.0000	0.386094
$967$	37.0000i	1.18984i	0.803785	+	0.594920i	$0.202816\pi$
$967$	37.0000i	1.18984i	−0.803785	+	0.594920i	$0.797184\pi$
$968$	− 2.00000i	− 0.0642824i
$969$	0	0
$970$	0	0
$971$	42.0000	1.34784	0.673922	−	0.738802i	$-0.264608\pi$
$971$	42.0000	1.34784	0.673922	+	0.738802i	$0.264608\pi$
$972$	− 1.00000i	− 0.0320750i
$973$	30.0000i	0.961756i
$974$	−43.0000	−1.37781
$975$	0	0
$976$	−3.00000	−0.0960277
$977$	52.0000i	1.66363i	0.555055	+	0.831814i	$0.312697\pi$
$977$	52.0000i	1.66363i	−0.555055	+	0.831814i	$0.687303\pi$
$978$	4.00000i	0.127906i
$979$	0	0
$980$	0	0
$981$	10.0000	0.319275
$982$	− 2.00000i	− 0.0638226i
$983$	29.0000i	0.924956i	0.886631	+	0.462478i	$0.153040\pi$
$983$	29.0000i	0.924956i	−0.886631	+	0.462478i	$0.846960\pi$
$984$	−2.00000	−0.0637577
$985$	0	0
$986$	15.0000	0.477697
$987$	− 9.00000i	− 0.286473i
$988$	0	0
$989$	−16.0000	−0.508770
$990$	0	0
$991$	−8.00000	−0.254128	−0.127064	−	0.991894i	$-0.540555\pi$
$991$	−8.00000	−0.254128	−0.127064	+	0.991894i	$0.540555\pi$
$992$	3.00000i	0.0952501i
$993$	− 28.0000i	− 0.888553i
$994$	24.0000	0.761234
$995$	0	0
$996$	−1.00000	−0.0316862
$997$	17.0000i	0.538395i	0.963085	+	0.269198i	$0.0867585\pi$
$997$	17.0000i	0.538395i	−0.963085	+	0.269198i	$0.913241\pi$
$998$	5.00000i	0.158272i
$999$	12.0000	0.379663

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1950.2.e.j.1249.1		2
3.2	odd	2		5850.2.e.z.5149.2		2
5.2	odd	4		1950.2.a.bb.1.1	yes	1
5.3	odd	4		1950.2.a.a.1.1	✓	1
5.4	even	2	inner	1950.2.e.j.1249.2		2
15.2	even	4		5850.2.a.w.1.1		1
15.8	even	4		5850.2.a.bf.1.1		1
15.14	odd	2		5850.2.e.z.5149.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1950.2.a.a.1.1	✓	1	5.3	odd	4
1950.2.a.bb.1.1	yes	1	5.2	odd	4
1950.2.e.j.1249.1		2	1.1	even	1	trivial
1950.2.e.j.1249.2		2	5.4	even	2	inner
5850.2.a.w.1.1		1	15.2	even	4
5850.2.a.bf.1.1		1	15.8	even	4
5850.2.e.z.5149.1		2	15.14	odd	2
5850.2.e.z.5149.2		2	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -3.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -3.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} -3.00000 q^{11} -1.00000i q^{12} +1.00000i q^{13} -3.00000 q^{14} +1.00000 q^{16} -3.00000i q^{17} +1.00000i q^{18} +3.00000 q^{21} +3.00000i q^{22} +4.00000i q^{23} -1.00000 q^{24} +1.00000 q^{26} -1.00000i q^{27} +3.00000i q^{28} -5.00000 q^{29} -3.00000 q^{31} -1.00000i q^{32} -3.00000i q^{33} -3.00000 q^{34} +1.00000 q^{36} +12.0000i q^{37} -1.00000 q^{39} +2.00000 q^{41} -3.00000i q^{42} +4.00000i q^{43} +3.00000 q^{44} +4.00000 q^{46} -3.00000i q^{47} +1.00000i q^{48} -2.00000 q^{49} +3.00000 q^{51} -1.00000i q^{52} +9.00000i q^{53} -1.00000 q^{54} +3.00000 q^{56} +5.00000i q^{58} -15.0000 q^{59} -3.00000 q^{61} +3.00000i q^{62} +3.00000i q^{63} -1.00000 q^{64} -3.00000 q^{66} +7.00000i q^{67} +3.00000i q^{68} -4.00000 q^{69} -8.00000 q^{71} -1.00000i q^{72} -16.0000i q^{73} +12.0000 q^{74} +9.00000i q^{77} +1.00000i q^{78} +1.00000 q^{81} -2.00000i q^{82} -1.00000i q^{83} -3.00000 q^{84} +4.00000 q^{86} -5.00000i q^{87} -3.00000i q^{88} +3.00000 q^{91} -4.00000i q^{92} -3.00000i q^{93} -3.00000 q^{94} +1.00000 q^{96} +2.00000i q^{97} +2.00000i q^{98} +3.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9	\( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} - 6 q^{11} - 6 q^{14} + 2 q^{16} + 6 q^{21} - 2 q^{24} + 2 q^{26} - 10 q^{29} - 6 q^{31} - 6 q^{34} + 2 q^{36} - 2 q^{39} + 4 q^{41} + 6 q^{44} + 8 q^{46} - 4 q^{49} + 6 q^{51} - 2 q^{54} + 6 q^{56} - 30 q^{59} - 6 q^{61} - 2 q^{64} - 6 q^{66} - 8 q^{69} - 16 q^{71} + 24 q^{74} + 2 q^{81} - 6 q^{84} + 8 q^{86} + 6 q^{91} - 6 q^{94} + 2 q^{96} + 6 q^{99}+O(q^{100}) \) 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9 - 6 * q^11 - 6 * q^14 + 2 * q^16 + 6 * q^21 - 2 * q^24 + 2 * q^26 - 10 * q^29 - 6 * q^31 - 6 * q^34 + 2 * q^36 - 2 * q^39 + 4 * q^41 + 6 * q^44 + 8 * q^46 - 4 * q^49 + 6 * q^51 - 2 * q^54 + 6 * q^56 - 30 * q^59 - 6 * q^61 - 2 * q^64 - 6 * q^66 - 8 * q^69 - 16 * q^71 + 24 * q^74 + 2 * q^81 - 6 * q^84 + 8 * q^86 + 6 * q^91 - 6 * q^94 + 2 * q^96 + 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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