LMFDB - Embedded newform 4650.2.d.o.3349.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4650,2,Mod(3349,4650)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4650, 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("4650.3349");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4650 = 2 \cdot 3 \cdot 5^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4650.d (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:	$37.1304369399$
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 930)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3349.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	4650.3349
Dual form			4650.2.d.o.3349.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -1.00000i q^{8} -1.00000 q^{9} -4.00000 q^{11} +1.00000i q^{12} +6.00000i q^{13} +1.00000 q^{16} -2.00000i q^{17} -1.00000i q^{18} +4.00000 q^{19} -4.00000i q^{22} -4.00000i q^{23} -1.00000 q^{24} -6.00000 q^{26} +1.00000i q^{27} -2.00000 q^{29} -1.00000 q^{31} +1.00000i q^{32} +4.00000i q^{33} +2.00000 q^{34} +1.00000 q^{36} +2.00000i q^{37} +4.00000i q^{38} +6.00000 q^{39} -6.00000 q^{41} -4.00000i q^{43} +4.00000 q^{44} +4.00000 q^{46} -1.00000i q^{48} +7.00000 q^{49} -2.00000 q^{51} -6.00000i q^{52} +2.00000i q^{53} -1.00000 q^{54} -4.00000i q^{57} -2.00000i q^{58} +4.00000 q^{59} -6.00000 q^{61} -1.00000i q^{62} -1.00000 q^{64} -4.00000 q^{66} -16.0000i q^{67} +2.00000i q^{68} -4.00000 q^{69} -12.0000 q^{71} +1.00000i q^{72} -6.00000i q^{73} -2.00000 q^{74} -4.00000 q^{76} +6.00000i q^{78} +16.0000 q^{79} +1.00000 q^{81} -6.00000i q^{82} -12.0000i q^{83} +4.00000 q^{86} +2.00000i q^{87} +4.00000i q^{88} +18.0000 q^{89} +4.00000i q^{92} +1.00000i q^{93} +1.00000 q^{96} +14.0000i q^{97} +7.00000i q^{98} +4.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 2 q^{16} + 8 q^{19} - 2 q^{24} - 12 q^{26} - 4 q^{29} - 2 q^{31} + 4 q^{34} + 2 q^{36} + 12 q^{39} - 12 q^{41} + 8 q^{44} + 8 q^{46} + 14 q^{49} - 4 q^{51} - 2 q^{54} + 8 q^{59} - 12 q^{61} - 2 q^{64} - 8 q^{66} - 8 q^{69} - 24 q^{71} - 4 q^{74} - 8 q^{76} + 32 q^{79} + 2 q^{81} + 8 q^{86} + 36 q^{89} + 2 q^{96} + 8 q^{99}+O(q^{100}) $ 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9 - 8 * q^11 + 2 * q^16 + 8 * q^19 - 2 * q^24 - 12 * q^26 - 4 * q^29 - 2 * q^31 + 4 * q^34 + 2 * q^36 + 12 * q^39 - 12 * q^41 + 8 * q^44 + 8 * q^46 + 14 * q^49 - 4 * q^51 - 2 * q^54 + 8 * q^59 - 12 * q^61 - 2 * q^64 - 8 * q^66 - 8 * q^69 - 24 * q^71 - 4 * q^74 - 8 * q^76 + 32 * q^79 + 2 * q^81 + 8 * q^86 + 36 * q^89 + 2 * q^96 + 8 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4650\mathbb{Z}\right)^\times$.

$n$	$1801$	$2977$	$3101$
$\chi(n)$	$1$	$-1$	$1$

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
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	− 1.00000i	− 0.353553i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	1.00000i	0.288675i
$13$	6.00000i	1.66410i	0.554700	+	0.832050i	$0.312833\pi$
$13$	6.00000i	1.66410i	−0.554700	+	0.832050i	$0.687167\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	− 2.00000i	− 0.485071i	−0.970143	−	0.242536i	$-0.922021\pi$
$17$	− 2.00000i	− 0.485071i	0.970143	−	0.242536i	$-0.0779791\pi$
$18$	− 1.00000i	− 0.235702i
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	0	0
$21$	0	0
$22$	− 4.00000i	− 0.852803i
$23$	− 4.00000i	− 0.834058i	−0.908893	−	0.417029i	$-0.863071\pi$
$23$	− 4.00000i	− 0.834058i	0.908893	−	0.417029i	$-0.136929\pi$
$24$	−1.00000	−0.204124
$25$	0	0
$26$	−6.00000	−1.17670
$27$	1.00000i	0.192450i
$28$	0	0
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	1.00000i	0.176777i
$33$	4.00000i	0.696311i
$34$	2.00000	0.342997
$35$	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	$0.947432\pi$
$38$	4.00000i	0.648886i
$39$	6.00000	0.960769
$40$	0	0
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	0	0
$43$	− 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
$46$	4.00000	0.589768
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	− 1.00000i	− 0.144338i
$49$	7.00000	1.00000
$50$	0	0
$51$	−2.00000	−0.280056
$52$	− 6.00000i	− 0.832050i
$53$	2.00000i	0.274721i	0.990521	+	0.137361i	$0.0438619\pi$
$53$	2.00000i	0.274721i	−0.990521	+	0.137361i	$0.956138\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	0	0
$57$	− 4.00000i	− 0.529813i
$58$	− 2.00000i	− 0.262613i
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	0	0
$61$	−6.00000	−0.768221	−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000	−0.768221	−0.384111	+	0.923287i	$0.625492\pi$
$62$	− 1.00000i	− 0.127000i
$63$	0	0
$64$	−1.00000	−0.125000
$65$	0	0
$66$	−4.00000	−0.492366
$67$	− 16.0000i	− 1.95471i	−0.211604	−	0.977356i	$-0.567869\pi$
$67$	− 16.0000i	− 1.95471i	0.211604	−	0.977356i	$-0.432131\pi$
$68$	2.00000i	0.242536i
$69$	−4.00000	−0.481543
$70$	0	0
$71$	−12.0000	−1.42414	−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000	−1.42414	−0.712069	+	0.702109i	$0.752242\pi$
$72$	1.00000i	0.117851i
$73$	− 6.00000i	− 0.702247i	−0.936329	−	0.351123i	$-0.885800\pi$
$73$	− 6.00000i	− 0.702247i	0.936329	−	0.351123i	$-0.114200\pi$
$74$	−2.00000	−0.232495
$75$	0	0
$76$	−4.00000	−0.458831
$77$	0	0
$78$	6.00000i	0.679366i
$79$	16.0000	1.80014	0.900070	−	0.435745i	$-0.143515\pi$
$79$	16.0000	1.80014	0.900070	+	0.435745i	$0.143515\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	− 6.00000i	− 0.662589i
$83$	− 12.0000i	− 1.31717i	−0.752506	−	0.658586i	$-0.771155\pi$
$83$	− 12.0000i	− 1.31717i	0.752506	−	0.658586i	$-0.228845\pi$
$84$	0	0
$85$	0	0
$86$	4.00000	0.431331
$87$	2.00000i	0.214423i
$88$	4.00000i	0.426401i
$89$	18.0000	1.90800	0.953998	−	0.299813i	$-0.0969242\pi$
$89$	18.0000	1.90800	0.953998	+	0.299813i	$0.0969242\pi$
$90$	0	0
$91$	0	0
$92$	4.00000i	0.417029i
$93$	1.00000i	0.103695i
$94$	0	0
$95$	0	0
$96$	1.00000	0.102062
$97$	14.0000i	1.42148i	0.703452	+	0.710742i	$0.251641\pi$
$97$	14.0000i	1.42148i	−0.703452	+	0.710742i	$0.748359\pi$
$98$	7.00000i	0.707107i
$99$	4.00000	0.402015
$100$	0	0
$101$	2.00000	0.199007	0.0995037	−	0.995037i	$-0.468274\pi$
$101$	2.00000	0.199007	0.0995037	+	0.995037i	$0.468274\pi$
$102$	− 2.00000i	− 0.198030i
$103$	8.00000i	0.788263i	0.919054	+	0.394132i	$0.128955\pi$
$103$	8.00000i	0.788263i	−0.919054	+	0.394132i	$0.871045\pi$
$104$	6.00000	0.588348
$105$	0	0
$106$	−2.00000	−0.194257
$107$	− 4.00000i	− 0.386695i	−0.981130	−	0.193347i	$-0.938066\pi$
$107$	− 4.00000i	− 0.386695i	0.981130	−	0.193347i	$-0.0619344\pi$
$108$	− 1.00000i	− 0.0962250i
$109$	−6.00000	−0.574696	−0.287348	−	0.957826i	$-0.592774\pi$
$109$	−6.00000	−0.574696	−0.287348	+	0.957826i	$0.592774\pi$
$110$	0	0
$111$	2.00000	0.189832
$112$	0	0
$113$	− 10.0000i	− 0.940721i	−0.882474	−	0.470360i	$-0.844124\pi$
$113$	− 10.0000i	− 0.940721i	0.882474	−	0.470360i	$-0.155876\pi$
$114$	4.00000	0.374634
$115$	0	0
$116$	2.00000	0.185695
$117$	− 6.00000i	− 0.554700i
$118$	4.00000i	0.368230i
$119$	0	0
$120$	0	0
$121$	5.00000	0.454545
$122$	− 6.00000i	− 0.543214i
$123$	6.00000i	0.541002i
$124$	1.00000	0.0898027
$125$	0	0
$126$	0	0
$127$	8.00000i	0.709885i	0.934888	+	0.354943i	$0.115500\pi$
$127$	8.00000i	0.709885i	−0.934888	+	0.354943i	$0.884500\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	−4.00000	−0.352180
$130$	0	0
$131$	−4.00000	−0.349482	−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000	−0.349482	−0.174741	+	0.984614i	$0.555909\pi$
$132$	− 4.00000i	− 0.348155i
$133$	0	0
$134$	16.0000	1.38219
$135$	0	0
$136$	−2.00000	−0.171499
$137$	− 18.0000i	− 1.53784i	−0.639343	−	0.768922i	$-0.720793\pi$
$137$	− 18.0000i	− 1.53784i	0.639343	−	0.768922i	$-0.279207\pi$
$138$	− 4.00000i	− 0.340503i
$139$	−8.00000	−0.678551	−0.339276	−	0.940687i	$-0.610182\pi$
$139$	−8.00000	−0.678551	−0.339276	+	0.940687i	$0.610182\pi$
$140$	0	0
$141$	0	0
$142$	− 12.0000i	− 1.00702i
$143$	− 24.0000i	− 2.00698i
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	6.00000	0.496564
$147$	− 7.00000i	− 0.577350i
$148$	− 2.00000i	− 0.164399i
$149$	−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
$151$	16.0000	1.30206	0.651031	−	0.759051i	$-0.274337\pi$
$151$	16.0000	1.30206	0.651031	+	0.759051i	$0.274337\pi$
$152$	− 4.00000i	− 0.324443i
$153$	2.00000i	0.161690i
$154$	0	0
$155$	0	0
$156$	−6.00000	−0.480384
$157$	− 18.0000i	− 1.43656i	−0.695756	−	0.718278i	$-0.744931\pi$
$157$	− 18.0000i	− 1.43656i	0.695756	−	0.718278i	$-0.255069\pi$
$158$	16.0000i	1.27289i
$159$	2.00000	0.158610
$160$	0	0
$161$	0	0
$162$	1.00000i	0.0785674i
$163$	− 24.0000i	− 1.87983i	−0.341415	−	0.939913i	$-0.610906\pi$
$163$	− 24.0000i	− 1.87983i	0.341415	−	0.939913i	$-0.389094\pi$
$164$	6.00000	0.468521
$165$	0	0
$166$	12.0000	0.931381
$167$	− 12.0000i	− 0.928588i	−0.885681	−	0.464294i	$-0.846308\pi$
$167$	− 12.0000i	− 0.928588i	0.885681	−	0.464294i	$-0.153692\pi$
$168$	0	0
$169$	−23.0000	−1.76923
$170$	0	0
$171$	−4.00000	−0.305888
$172$	4.00000i	0.304997i
$173$	− 14.0000i	− 1.06440i	−0.846619	−	0.532200i	$-0.821365\pi$
$173$	− 14.0000i	− 1.06440i	0.846619	−	0.532200i	$-0.178635\pi$
$174$	−2.00000	−0.151620
$175$	0	0
$176$	−4.00000	−0.301511
$177$	− 4.00000i	− 0.300658i
$178$	18.0000i	1.34916i
$179$	4.00000	0.298974	0.149487	−	0.988764i	$-0.452238\pi$
$179$	4.00000	0.298974	0.149487	+	0.988764i	$0.452238\pi$
$180$	0	0
$181$	−6.00000	−0.445976	−0.222988	−	0.974821i	$-0.571581\pi$
$181$	−6.00000	−0.445976	−0.222988	+	0.974821i	$0.571581\pi$
$182$	0	0
$183$	6.00000i	0.443533i
$184$	−4.00000	−0.294884
$185$	0	0
$186$	−1.00000	−0.0733236
$187$	8.00000i	0.585018i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	4.00000	0.289430	0.144715	−	0.989473i	$-0.453773\pi$
$191$	4.00000	0.289430	0.144715	+	0.989473i	$0.453773\pi$
$192$	1.00000i	0.0721688i
$193$	− 22.0000i	− 1.58359i	−0.610784	−	0.791797i	$-0.709146\pi$
$193$	− 22.0000i	− 1.58359i	0.610784	−	0.791797i	$-0.290854\pi$
$194$	−14.0000	−1.00514
$195$	0	0
$196$	−7.00000	−0.500000
$197$	− 2.00000i	− 0.142494i	−0.997459	−	0.0712470i	$-0.977302\pi$
$197$	− 2.00000i	− 0.142494i	0.997459	−	0.0712470i	$-0.0226979\pi$
$198$	4.00000i	0.284268i
$199$	−16.0000	−1.13421	−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000	−1.13421	−0.567105	+	0.823646i	$0.691937\pi$
$200$	0	0
$201$	−16.0000	−1.12855
$202$	2.00000i	0.140720i
$203$	0	0
$204$	2.00000	0.140028
$205$	0	0
$206$	−8.00000	−0.557386
$207$	4.00000i	0.278019i
$208$	6.00000i	0.416025i
$209$	−16.0000	−1.10674
$210$	0	0
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	− 2.00000i	− 0.137361i
$213$	12.0000i	0.822226i
$214$	4.00000	0.273434
$215$	0	0
$216$	1.00000	0.0680414
$217$	0	0
$218$	− 6.00000i	− 0.406371i
$219$	−6.00000	−0.405442
$220$	0	0
$221$	12.0000	0.807207
$222$	2.00000i	0.134231i
$223$	− 8.00000i	− 0.535720i	−0.963458	−	0.267860i	$-0.913684\pi$
$223$	− 8.00000i	− 0.535720i	0.963458	−	0.267860i	$-0.0863164\pi$
$224$	0	0
$225$	0	0
$226$	10.0000	0.665190
$227$	− 20.0000i	− 1.32745i	−0.747978	−	0.663723i	$-0.768975\pi$
$227$	− 20.0000i	− 1.32745i	0.747978	−	0.663723i	$-0.231025\pi$
$228$	4.00000i	0.264906i
$229$	−10.0000	−0.660819	−0.330409	−	0.943838i	$-0.607187\pi$
$229$	−10.0000	−0.660819	−0.330409	+	0.943838i	$0.607187\pi$
$230$	0	0
$231$	0	0
$232$	2.00000i	0.131306i
$233$	22.0000i	1.44127i	0.693316	+	0.720634i	$0.256149\pi$
$233$	22.0000i	1.44127i	−0.693316	+	0.720634i	$0.743851\pi$
$234$	6.00000	0.392232
$235$	0	0
$236$	−4.00000	−0.260378
$237$	− 16.0000i	− 1.03931i
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	−14.0000	−0.901819	−0.450910	−	0.892570i	$-0.648900\pi$
$241$	−14.0000	−0.901819	−0.450910	+	0.892570i	$0.648900\pi$
$242$	5.00000i	0.321412i
$243$	− 1.00000i	− 0.0641500i
$244$	6.00000	0.384111
$245$	0	0
$246$	−6.00000	−0.382546
$247$	24.0000i	1.52708i
$248$	1.00000i	0.0635001i
$249$	−12.0000	−0.760469
$250$	0	0
$251$	20.0000	1.26239	0.631194	−	0.775625i	$-0.282565\pi$
$251$	20.0000	1.26239	0.631194	+	0.775625i	$0.282565\pi$
$252$	0	0
$253$	16.0000i	1.00591i
$254$	−8.00000	−0.501965
$255$	0	0
$256$	1.00000	0.0625000
$257$	− 22.0000i	− 1.37232i	−0.727450	−	0.686161i	$-0.759294\pi$
$257$	− 22.0000i	− 1.37232i	0.727450	−	0.686161i	$-0.240706\pi$
$258$	− 4.00000i	− 0.249029i
$259$	0	0
$260$	0	0
$261$	2.00000	0.123797
$262$	− 4.00000i	− 0.247121i
$263$	− 12.0000i	− 0.739952i	−0.929041	−	0.369976i	$-0.879366\pi$
$263$	− 12.0000i	− 0.739952i	0.929041	−	0.369976i	$-0.120634\pi$
$264$	4.00000	0.246183
$265$	0	0
$266$	0	0
$267$	− 18.0000i	− 1.10158i
$268$	16.0000i	0.977356i
$269$	−18.0000	−1.09748	−0.548740	−	0.835993i	$-0.684892\pi$
$269$	−18.0000	−1.09748	−0.548740	+	0.835993i	$0.684892\pi$
$270$	0	0
$271$	−8.00000	−0.485965	−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000	−0.485965	−0.242983	+	0.970031i	$0.578126\pi$
$272$	− 2.00000i	− 0.121268i
$273$	0	0
$274$	18.0000	1.08742
$275$	0	0
$276$	4.00000	0.240772
$277$	− 6.00000i	− 0.360505i	−0.983620	−	0.180253i	$-0.942309\pi$
$277$	− 6.00000i	− 0.360505i	0.983620	−	0.180253i	$-0.0576915\pi$
$278$	− 8.00000i	− 0.479808i
$279$	1.00000	0.0598684
$280$	0	0
$281$	10.0000	0.596550	0.298275	−	0.954480i	$-0.403589\pi$
$281$	10.0000	0.596550	0.298275	+	0.954480i	$0.403589\pi$
$282$	0	0
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	12.0000	0.712069
$285$	0	0
$286$	24.0000	1.41915
$287$	0	0
$288$	− 1.00000i	− 0.0589256i
$289$	13.0000	0.764706
$290$	0	0
$291$	14.0000	0.820695
$292$	6.00000i	0.351123i
$293$	− 6.00000i	− 0.350524i	−0.984522	−	0.175262i	$-0.943923\pi$
$293$	− 6.00000i	− 0.350524i	0.984522	−	0.175262i	$-0.0560772\pi$
$294$	7.00000	0.408248
$295$	0	0
$296$	2.00000	0.116248
$297$	− 4.00000i	− 0.232104i
$298$	− 10.0000i	− 0.579284i
$299$	24.0000	1.38796
$300$	0	0
$301$	0	0
$302$	16.0000i	0.920697i
$303$	− 2.00000i	− 0.114897i
$304$	4.00000	0.229416
$305$	0	0
$306$	−2.00000	−0.114332
$307$	16.0000i	0.913168i	0.889680	+	0.456584i	$0.150927\pi$
$307$	16.0000i	0.913168i	−0.889680	+	0.456584i	$0.849073\pi$
$308$	0	0
$309$	8.00000	0.455104
$310$	0	0
$311$	−28.0000	−1.58773	−0.793867	−	0.608091i	$-0.791935\pi$
$311$	−28.0000	−1.58773	−0.793867	+	0.608091i	$0.791935\pi$
$312$	− 6.00000i	− 0.339683i
$313$	26.0000i	1.46961i	0.678280	+	0.734803i	$0.262726\pi$
$313$	26.0000i	1.46961i	−0.678280	+	0.734803i	$0.737274\pi$
$314$	18.0000	1.01580
$315$	0	0
$316$	−16.0000	−0.900070
$317$	− 2.00000i	− 0.112331i	−0.998421	−	0.0561656i	$-0.982113\pi$
$317$	− 2.00000i	− 0.112331i	0.998421	−	0.0561656i	$-0.0178875\pi$
$318$	2.00000i	0.112154i
$319$	8.00000	0.447914
$320$	0	0
$321$	−4.00000	−0.223258
$322$	0	0
$323$	− 8.00000i	− 0.445132i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	24.0000	1.32924
$327$	6.00000i	0.331801i
$328$	6.00000i	0.331295i
$329$	0	0
$330$	0	0
$331$	24.0000	1.31916	0.659580	−	0.751635i	$-0.270734\pi$
$331$	24.0000	1.31916	0.659580	+	0.751635i	$0.270734\pi$
$332$	12.0000i	0.658586i
$333$	− 2.00000i	− 0.109599i
$334$	12.0000	0.656611
$335$	0	0
$336$	0	0
$337$	− 2.00000i	− 0.108947i	−0.998515	−	0.0544735i	$-0.982652\pi$
$337$	− 2.00000i	− 0.108947i	0.998515	−	0.0544735i	$-0.0173480\pi$
$338$	− 23.0000i	− 1.25104i
$339$	−10.0000	−0.543125
$340$	0	0
$341$	4.00000	0.216612
$342$	− 4.00000i	− 0.216295i
$343$	0	0
$344$	−4.00000	−0.215666
$345$	0	0
$346$	14.0000	0.752645
$347$	4.00000i	0.214731i	0.994220	+	0.107366i	$0.0342415\pi$
$347$	4.00000i	0.214731i	−0.994220	+	0.107366i	$0.965758\pi$
$348$	− 2.00000i	− 0.107211i
$349$	2.00000	0.107058	0.0535288	−	0.998566i	$-0.482953\pi$
$349$	2.00000	0.107058	0.0535288	+	0.998566i	$0.482953\pi$
$350$	0	0
$351$	−6.00000	−0.320256
$352$	− 4.00000i	− 0.213201i
$353$	− 6.00000i	− 0.319348i	−0.987170	−	0.159674i	$-0.948956\pi$
$353$	− 6.00000i	− 0.319348i	0.987170	−	0.159674i	$-0.0510443\pi$
$354$	4.00000	0.212598
$355$	0	0
$356$	−18.0000	−0.953998
$357$	0	0
$358$	4.00000i	0.211407i
$359$	20.0000	1.05556	0.527780	−	0.849381i	$-0.323025\pi$
$359$	20.0000	1.05556	0.527780	+	0.849381i	$0.323025\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	− 6.00000i	− 0.315353i
$363$	− 5.00000i	− 0.262432i
$364$	0	0
$365$	0	0
$366$	−6.00000	−0.313625
$367$	− 16.0000i	− 0.835193i	−0.908633	−	0.417597i	$-0.862873\pi$
$367$	− 16.0000i	− 0.835193i	0.908633	−	0.417597i	$-0.137127\pi$
$368$	− 4.00000i	− 0.208514i
$369$	6.00000	0.312348
$370$	0	0
$371$	0	0
$372$	− 1.00000i	− 0.0518476i
$373$	− 38.0000i	− 1.96757i	−0.179364	−	0.983783i	$-0.557404\pi$
$373$	− 38.0000i	− 1.96757i	0.179364	−	0.983783i	$-0.442596\pi$
$374$	−8.00000	−0.413670
$375$	0	0
$376$	0	0
$377$	− 12.0000i	− 0.618031i
$378$	0	0
$379$	20.0000	1.02733	0.513665	−	0.857991i	$-0.328287\pi$
$379$	20.0000	1.02733	0.513665	+	0.857991i	$0.328287\pi$
$380$	0	0
$381$	8.00000	0.409852
$382$	4.00000i	0.204658i
$383$	4.00000i	0.204390i	0.994764	+	0.102195i	$0.0325866\pi$
$383$	4.00000i	0.204390i	−0.994764	+	0.102195i	$0.967413\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	22.0000	1.11977
$387$	4.00000i	0.203331i
$388$	− 14.0000i	− 0.710742i
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	0	0
$391$	−8.00000	−0.404577
$392$	− 7.00000i	− 0.353553i
$393$	4.00000i	0.201773i
$394$	2.00000	0.100759
$395$	0	0
$396$	−4.00000	−0.201008
$397$	14.0000i	0.702640i	0.936255	+	0.351320i	$0.114267\pi$
$397$	14.0000i	0.702640i	−0.936255	+	0.351320i	$0.885733\pi$
$398$	− 16.0000i	− 0.802008i
$399$	0	0
$400$	0	0
$401$	14.0000	0.699127	0.349563	−	0.936913i	$-0.386330\pi$
$401$	14.0000	0.699127	0.349563	+	0.936913i	$0.386330\pi$
$402$	− 16.0000i	− 0.798007i
$403$	− 6.00000i	− 0.298881i
$404$	−2.00000	−0.0995037
$405$	0	0
$406$	0	0
$407$	− 8.00000i	− 0.396545i
$408$	2.00000i	0.0990148i
$409$	6.00000	0.296681	0.148340	−	0.988936i	$-0.452607\pi$
$409$	6.00000	0.296681	0.148340	+	0.988936i	$0.452607\pi$
$410$	0	0
$411$	−18.0000	−0.887875
$412$	− 8.00000i	− 0.394132i
$413$	0	0
$414$	−4.00000	−0.196589
$415$	0	0
$416$	−6.00000	−0.294174
$417$	8.00000i	0.391762i
$418$	− 16.0000i	− 0.782586i
$419$	12.0000	0.586238	0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000	0.586238	0.293119	+	0.956076i	$0.405307\pi$
$420$	0	0
$421$	14.0000	0.682318	0.341159	−	0.940006i	$-0.389181\pi$
$421$	14.0000	0.682318	0.341159	+	0.940006i	$0.389181\pi$
$422$	− 4.00000i	− 0.194717i
$423$	0	0
$424$	2.00000	0.0971286
$425$	0	0
$426$	−12.0000	−0.581402
$427$	0	0
$428$	4.00000i	0.193347i
$429$	−24.0000	−1.15873
$430$	0	0
$431$	−12.0000	−0.578020	−0.289010	−	0.957326i	$-0.593326\pi$
$431$	−12.0000	−0.578020	−0.289010	+	0.957326i	$0.593326\pi$
$432$	1.00000i	0.0481125i
$433$	2.00000i	0.0961139i	0.998845	+	0.0480569i	$0.0153029\pi$
$433$	2.00000i	0.0961139i	−0.998845	+	0.0480569i	$0.984697\pi$
$434$	0	0
$435$	0	0
$436$	6.00000	0.287348
$437$	− 16.0000i	− 0.765384i
$438$	− 6.00000i	− 0.286691i
$439$	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
$441$	−7.00000	−0.333333
$442$	12.0000i	0.570782i
$443$	− 4.00000i	− 0.190046i	−0.995475	−	0.0950229i	$-0.969708\pi$
$443$	− 4.00000i	− 0.190046i	0.995475	−	0.0950229i	$-0.0302924\pi$
$444$	−2.00000	−0.0949158
$445$	0	0
$446$	8.00000	0.378811
$447$	10.0000i	0.472984i
$448$	0	0
$449$	18.0000	0.849473	0.424736	−	0.905317i	$-0.360367\pi$
$449$	18.0000	0.849473	0.424736	+	0.905317i	$0.360367\pi$
$450$	0	0
$451$	24.0000	1.13012
$452$	10.0000i	0.470360i
$453$	− 16.0000i	− 0.751746i
$454$	20.0000	0.938647
$455$	0	0
$456$	−4.00000	−0.187317
$457$	22.0000i	1.02912i	0.857455	+	0.514558i	$0.172044\pi$
$457$	22.0000i	1.02912i	−0.857455	+	0.514558i	$0.827956\pi$
$458$	− 10.0000i	− 0.467269i
$459$	2.00000	0.0933520
$460$	0	0
$461$	−14.0000	−0.652045	−0.326023	−	0.945362i	$-0.605709\pi$
$461$	−14.0000	−0.652045	−0.326023	+	0.945362i	$0.605709\pi$
$462$	0	0
$463$	− 24.0000i	− 1.11537i	−0.830051	−	0.557687i	$-0.811689\pi$
$463$	− 24.0000i	− 1.11537i	0.830051	−	0.557687i	$-0.188311\pi$
$464$	−2.00000	−0.0928477
$465$	0	0
$466$	−22.0000	−1.01913
$467$	4.00000i	0.185098i	0.995708	+	0.0925490i	$0.0295015\pi$
$467$	4.00000i	0.185098i	−0.995708	+	0.0925490i	$0.970499\pi$
$468$	6.00000i	0.277350i
$469$	0	0
$470$	0	0
$471$	−18.0000	−0.829396
$472$	− 4.00000i	− 0.184115i
$473$	16.0000i	0.735681i
$474$	16.0000	0.734904
$475$	0	0
$476$	0	0
$477$	− 2.00000i	− 0.0915737i
$478$	0	0
$479$	−36.0000	−1.64488	−0.822441	−	0.568850i	$-0.807388\pi$
$479$	−36.0000	−1.64488	−0.822441	+	0.568850i	$0.807388\pi$
$480$	0	0
$481$	−12.0000	−0.547153
$482$	− 14.0000i	− 0.637683i
$483$	0	0
$484$	−5.00000	−0.227273
$485$	0	0
$486$	1.00000	0.0453609
$487$	40.0000i	1.81257i	0.422664	+	0.906287i	$0.361095\pi$
$487$	40.0000i	1.81257i	−0.422664	+	0.906287i	$0.638905\pi$
$488$	6.00000i	0.271607i
$489$	−24.0000	−1.08532
$490$	0	0
$491$	20.0000	0.902587	0.451294	−	0.892375i	$-0.350963\pi$
$491$	20.0000	0.902587	0.451294	+	0.892375i	$0.350963\pi$
$492$	− 6.00000i	− 0.270501i
$493$	4.00000i	0.180151i
$494$	−24.0000	−1.07981
$495$	0	0
$496$	−1.00000	−0.0449013
$497$	0	0
$498$	− 12.0000i	− 0.537733i
$499$	−16.0000	−0.716258	−0.358129	−	0.933672i	$-0.616585\pi$
$499$	−16.0000	−0.716258	−0.358129	+	0.933672i	$0.616585\pi$
$500$	0	0
$501$	−12.0000	−0.536120
$502$	20.0000i	0.892644i
$503$	32.0000i	1.42681i	0.700752	+	0.713405i	$0.252848\pi$
$503$	32.0000i	1.42681i	−0.700752	+	0.713405i	$0.747152\pi$
$504$	0	0
$505$	0	0
$506$	−16.0000	−0.711287
$507$	23.0000i	1.02147i
$508$	− 8.00000i	− 0.354943i
$509$	−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
$511$	0	0
$512$	1.00000i	0.0441942i
$513$	4.00000i	0.176604i
$514$	22.0000	0.970378
$515$	0	0
$516$	4.00000	0.176090
$517$	0	0
$518$	0	0
$519$	−14.0000	−0.614532
$520$	0	0
$521$	−14.0000	−0.613351	−0.306676	−	0.951814i	$-0.599217\pi$
$521$	−14.0000	−0.613351	−0.306676	+	0.951814i	$0.599217\pi$
$522$	2.00000i	0.0875376i
$523$	20.0000i	0.874539i	0.899331	+	0.437269i	$0.144054\pi$
$523$	20.0000i	0.874539i	−0.899331	+	0.437269i	$0.855946\pi$
$524$	4.00000	0.174741
$525$	0	0
$526$	12.0000	0.523225
$527$	2.00000i	0.0871214i
$528$	4.00000i	0.174078i
$529$	7.00000	0.304348
$530$	0	0
$531$	−4.00000	−0.173585
$532$	0	0
$533$	− 36.0000i	− 1.55933i
$534$	18.0000	0.778936
$535$	0	0
$536$	−16.0000	−0.691095
$537$	− 4.00000i	− 0.172613i
$538$	− 18.0000i	− 0.776035i
$539$	−28.0000	−1.20605
$540$	0	0
$541$	−10.0000	−0.429934	−0.214967	−	0.976621i	$-0.568964\pi$
$541$	−10.0000	−0.429934	−0.214967	+	0.976621i	$0.568964\pi$
$542$	− 8.00000i	− 0.343629i
$543$	6.00000i	0.257485i
$544$	2.00000	0.0857493
$545$	0	0
$546$	0	0
$547$	− 16.0000i	− 0.684111i	−0.939680	−	0.342055i	$-0.888877\pi$
$547$	− 16.0000i	− 0.684111i	0.939680	−	0.342055i	$-0.111123\pi$
$548$	18.0000i	0.768922i
$549$	6.00000	0.256074
$550$	0	0
$551$	−8.00000	−0.340811
$552$	4.00000i	0.170251i
$553$	0	0
$554$	6.00000	0.254916
$555$	0	0
$556$	8.00000	0.339276
$557$	30.0000i	1.27114i	0.772043	+	0.635570i	$0.219235\pi$
$557$	30.0000i	1.27114i	−0.772043	+	0.635570i	$0.780765\pi$
$558$	1.00000i	0.0423334i
$559$	24.0000	1.01509
$560$	0	0
$561$	8.00000	0.337760
$562$	10.0000i	0.421825i
$563$	− 12.0000i	− 0.505740i	−0.967500	−	0.252870i	$-0.918626\pi$
$563$	− 12.0000i	− 0.505740i	0.967500	−	0.252870i	$-0.0813744\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	12.0000i	0.503509i
$569$	−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	$0.184323\pi$
$572$	24.0000i	1.00349i
$573$	− 4.00000i	− 0.167102i
$574$	0	0
$575$	0	0
$576$	1.00000	0.0416667
$577$	30.0000i	1.24892i	0.781058	+	0.624458i	$0.214680\pi$
$577$	30.0000i	1.24892i	−0.781058	+	0.624458i	$0.785320\pi$
$578$	13.0000i	0.540729i
$579$	−22.0000	−0.914289
$580$	0	0
$581$	0	0
$582$	14.0000i	0.580319i
$583$	− 8.00000i	− 0.331326i
$584$	−6.00000	−0.248282
$585$	0	0
$586$	6.00000	0.247858
$587$	− 4.00000i	− 0.165098i	−0.996587	−	0.0825488i	$-0.973694\pi$
$587$	− 4.00000i	− 0.165098i	0.996587	−	0.0825488i	$-0.0263060\pi$
$588$	7.00000i	0.288675i
$589$	−4.00000	−0.164817
$590$	0	0
$591$	−2.00000	−0.0822690
$592$	2.00000i	0.0821995i
$593$	14.0000i	0.574911i	0.957794	+	0.287456i	$0.0928094\pi$
$593$	14.0000i	0.574911i	−0.957794	+	0.287456i	$0.907191\pi$
$594$	4.00000	0.164122
$595$	0	0
$596$	10.0000	0.409616
$597$	16.0000i	0.654836i
$598$	24.0000i	0.981433i
$599$	4.00000	0.163436	0.0817178	−	0.996656i	$-0.473959\pi$
$599$	4.00000	0.163436	0.0817178	+	0.996656i	$0.473959\pi$
$600$	0	0
$601$	10.0000	0.407909	0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000	0.407909	0.203954	+	0.978980i	$0.434621\pi$
$602$	0	0
$603$	16.0000i	0.651570i
$604$	−16.0000	−0.651031
$605$	0	0
$606$	2.00000	0.0812444
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	4.00000i	0.162221i
$609$	0	0
$610$	0	0
$611$	0	0
$612$	− 2.00000i	− 0.0808452i
$613$	− 26.0000i	− 1.05013i	−0.851062	−	0.525065i	$-0.824041\pi$
$613$	− 26.0000i	− 1.05013i	0.851062	−	0.525065i	$-0.175959\pi$
$614$	−16.0000	−0.645707
$615$	0	0
$616$	0	0
$617$	− 46.0000i	− 1.85189i	−0.377658	−	0.925945i	$-0.623271\pi$
$617$	− 46.0000i	− 1.85189i	0.377658	−	0.925945i	$-0.376729\pi$
$618$	8.00000i	0.321807i
$619$	−32.0000	−1.28619	−0.643094	−	0.765787i	$-0.722350\pi$
$619$	−32.0000	−1.28619	−0.643094	+	0.765787i	$0.722350\pi$
$620$	0	0
$621$	4.00000	0.160514
$622$	− 28.0000i	− 1.12270i
$623$	0	0
$624$	6.00000	0.240192
$625$	0	0
$626$	−26.0000	−1.03917
$627$	16.0000i	0.638978i
$628$	18.0000i	0.718278i
$629$	4.00000	0.159490
$630$	0	0
$631$	8.00000	0.318475	0.159237	−	0.987240i	$-0.449096\pi$
$631$	8.00000	0.318475	0.159237	+	0.987240i	$0.449096\pi$
$632$	− 16.0000i	− 0.636446i
$633$	4.00000i	0.158986i
$634$	2.00000	0.0794301
$635$	0	0
$636$	−2.00000	−0.0793052
$637$	42.0000i	1.66410i
$638$	8.00000i	0.316723i
$639$	12.0000	0.474713
$640$	0	0
$641$	30.0000	1.18493	0.592464	−	0.805597i	$-0.298155\pi$
$641$	30.0000	1.18493	0.592464	+	0.805597i	$0.298155\pi$
$642$	− 4.00000i	− 0.157867i
$643$	− 20.0000i	− 0.788723i	−0.918955	−	0.394362i	$-0.870966\pi$
$643$	− 20.0000i	− 0.788723i	0.918955	−	0.394362i	$-0.129034\pi$
$644$	0	0
$645$	0	0
$646$	8.00000	0.314756
$647$	12.0000i	0.471769i	0.971781	+	0.235884i	$0.0757987\pi$
$647$	12.0000i	0.471769i	−0.971781	+	0.235884i	$0.924201\pi$
$648$	− 1.00000i	− 0.0392837i
$649$	−16.0000	−0.628055
$650$	0	0
$651$	0	0
$652$	24.0000i	0.939913i
$653$	18.0000i	0.704394i	0.935926	+	0.352197i	$0.114565\pi$
$653$	18.0000i	0.704394i	−0.935926	+	0.352197i	$0.885435\pi$
$654$	−6.00000	−0.234619
$655$	0	0
$656$	−6.00000	−0.234261
$657$	6.00000i	0.234082i
$658$	0	0
$659$	−36.0000	−1.40236	−0.701180	−	0.712984i	$-0.747343\pi$
$659$	−36.0000	−1.40236	−0.701180	+	0.712984i	$0.747343\pi$
$660$	0	0
$661$	−34.0000	−1.32245	−0.661223	−	0.750189i	$-0.729962\pi$
$661$	−34.0000	−1.32245	−0.661223	+	0.750189i	$0.729962\pi$
$662$	24.0000i	0.932786i
$663$	− 12.0000i	− 0.466041i
$664$	−12.0000	−0.465690
$665$	0	0
$666$	2.00000	0.0774984
$667$	8.00000i	0.309761i
$668$	12.0000i	0.464294i
$669$	−8.00000	−0.309298
$670$	0	0
$671$	24.0000	0.926510
$672$	0	0
$673$	34.0000i	1.31060i	0.755367	+	0.655302i	$0.227459\pi$
$673$	34.0000i	1.31060i	−0.755367	+	0.655302i	$0.772541\pi$
$674$	2.00000	0.0770371
$675$	0	0
$676$	23.0000	0.884615
$677$	− 42.0000i	− 1.61419i	−0.590421	−	0.807096i	$-0.701038\pi$
$677$	− 42.0000i	− 1.61419i	0.590421	−	0.807096i	$-0.298962\pi$
$678$	− 10.0000i	− 0.384048i
$679$	0	0
$680$	0	0
$681$	−20.0000	−0.766402
$682$	4.00000i	0.153168i
$683$	− 36.0000i	− 1.37750i	−0.724998	−	0.688751i	$-0.758159\pi$
$683$	− 36.0000i	− 1.37750i	0.724998	−	0.688751i	$-0.241841\pi$
$684$	4.00000	0.152944
$685$	0	0
$686$	0	0
$687$	10.0000i	0.381524i
$688$	− 4.00000i	− 0.152499i
$689$	−12.0000	−0.457164
$690$	0	0
$691$	28.0000	1.06517	0.532585	−	0.846376i	$-0.321221\pi$
$691$	28.0000	1.06517	0.532585	+	0.846376i	$0.321221\pi$
$692$	14.0000i	0.532200i
$693$	0	0
$694$	−4.00000	−0.151838
$695$	0	0
$696$	2.00000	0.0758098
$697$	12.0000i	0.454532i
$698$	2.00000i	0.0757011i
$699$	22.0000	0.832116
$700$	0	0
$701$	−38.0000	−1.43524	−0.717620	−	0.696435i	$-0.754769\pi$
$701$	−38.0000	−1.43524	−0.717620	+	0.696435i	$0.754769\pi$
$702$	− 6.00000i	− 0.226455i
$703$	8.00000i	0.301726i
$704$	4.00000	0.150756
$705$	0	0
$706$	6.00000	0.225813
$707$	0	0
$708$	4.00000i	0.150329i
$709$	46.0000	1.72757	0.863783	−	0.503864i	$-0.168089\pi$
$709$	46.0000	1.72757	0.863783	+	0.503864i	$0.168089\pi$
$710$	0	0
$711$	−16.0000	−0.600047
$712$	− 18.0000i	− 0.674579i
$713$	4.00000i	0.149801i
$714$	0	0
$715$	0	0
$716$	−4.00000	−0.149487
$717$	0	0
$718$	20.0000i	0.746393i
$719$	24.0000	0.895049	0.447524	−	0.894272i	$-0.352306\pi$
$719$	24.0000	0.895049	0.447524	+	0.894272i	$0.352306\pi$
$720$	0	0
$721$	0	0
$722$	− 3.00000i	− 0.111648i
$723$	14.0000i	0.520666i
$724$	6.00000	0.222988
$725$	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	$-0.0958873\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−8.00000	−0.295891
$732$	− 6.00000i	− 0.221766i
$733$	2.00000i	0.0738717i	0.999318	+	0.0369358i	$0.0117597\pi$
$733$	2.00000i	0.0738717i	−0.999318	+	0.0369358i	$0.988240\pi$
$734$	16.0000	0.590571
$735$	0	0
$736$	4.00000	0.147442
$737$	64.0000i	2.35747i
$738$	6.00000i	0.220863i
$739$	16.0000	0.588570	0.294285	−	0.955718i	$-0.404919\pi$
$739$	16.0000	0.588570	0.294285	+	0.955718i	$0.404919\pi$
$740$	0	0
$741$	24.0000	0.881662
$742$	0	0
$743$	12.0000i	0.440237i	0.975473	+	0.220119i	$0.0706445\pi$
$743$	12.0000i	0.440237i	−0.975473	+	0.220119i	$0.929356\pi$
$744$	1.00000	0.0366618
$745$	0	0
$746$	38.0000	1.39128
$747$	12.0000i	0.439057i
$748$	− 8.00000i	− 0.292509i
$749$	0	0
$750$	0	0
$751$	−24.0000	−0.875772	−0.437886	−	0.899030i	$-0.644273\pi$
$751$	−24.0000	−0.875772	−0.437886	+	0.899030i	$0.644273\pi$
$752$	0	0
$753$	− 20.0000i	− 0.728841i
$754$	12.0000	0.437014
$755$	0	0
$756$	0	0
$757$	18.0000i	0.654221i	0.944986	+	0.327111i	$0.106075\pi$
$757$	18.0000i	0.654221i	−0.944986	+	0.327111i	$0.893925\pi$
$758$	20.0000i	0.726433i
$759$	16.0000	0.580763
$760$	0	0
$761$	22.0000	0.797499	0.398750	−	0.917060i	$-0.369444\pi$
$761$	22.0000	0.797499	0.398750	+	0.917060i	$0.369444\pi$
$762$	8.00000i	0.289809i
$763$	0	0
$764$	−4.00000	−0.144715
$765$	0	0
$766$	−4.00000	−0.144526
$767$	24.0000i	0.866590i
$768$	− 1.00000i	− 0.0360844i
$769$	14.0000	0.504853	0.252426	−	0.967616i	$-0.418771\pi$
$769$	14.0000	0.504853	0.252426	+	0.967616i	$0.418771\pi$
$770$	0	0
$771$	−22.0000	−0.792311
$772$	22.0000i	0.791797i
$773$	50.0000i	1.79838i	0.437564	+	0.899188i	$0.355842\pi$
$773$	50.0000i	1.79838i	−0.437564	+	0.899188i	$0.644158\pi$
$774$	−4.00000	−0.143777
$775$	0	0
$776$	14.0000	0.502571
$777$	0	0
$778$	6.00000i	0.215110i
$779$	−24.0000	−0.859889
$780$	0	0
$781$	48.0000	1.71758
$782$	− 8.00000i	− 0.286079i
$783$	− 2.00000i	− 0.0714742i
$784$	7.00000	0.250000
$785$	0	0
$786$	−4.00000	−0.142675
$787$	− 28.0000i	− 0.998092i	−0.866575	−	0.499046i	$-0.833684\pi$
$787$	− 28.0000i	− 0.998092i	0.866575	−	0.499046i	$-0.166316\pi$
$788$	2.00000i	0.0712470i
$789$	−12.0000	−0.427211
$790$	0	0
$791$	0	0
$792$	− 4.00000i	− 0.142134i
$793$	− 36.0000i	− 1.27840i
$794$	−14.0000	−0.496841
$795$	0	0
$796$	16.0000	0.567105
$797$	− 18.0000i	− 0.637593i	−0.947823	−	0.318796i	$-0.896721\pi$
$797$	− 18.0000i	− 0.637593i	0.947823	−	0.318796i	$-0.103279\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−18.0000	−0.635999
$802$	14.0000i	0.494357i
$803$	24.0000i	0.846942i
$804$	16.0000	0.564276
$805$	0	0
$806$	6.00000	0.211341
$807$	18.0000i	0.633630i
$808$	− 2.00000i	− 0.0703598i
$809$	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
$811$	12.0000	0.421377	0.210688	−	0.977553i	$-0.432429\pi$
$811$	12.0000	0.421377	0.210688	+	0.977553i	$0.432429\pi$
$812$	0	0
$813$	8.00000i	0.280572i
$814$	8.00000	0.280400
$815$	0	0
$816$	−2.00000	−0.0700140
$817$	− 16.0000i	− 0.559769i
$818$	6.00000i	0.209785i
$819$	0	0
$820$	0	0
$821$	−22.0000	−0.767805	−0.383903	−	0.923374i	$-0.625420\pi$
$821$	−22.0000	−0.767805	−0.383903	+	0.923374i	$0.625420\pi$
$822$	− 18.0000i	− 0.627822i
$823$	40.0000i	1.39431i	0.716919	+	0.697156i	$0.245552\pi$
$823$	40.0000i	1.39431i	−0.716919	+	0.697156i	$0.754448\pi$
$824$	8.00000	0.278693
$825$	0	0
$826$	0	0
$827$	− 52.0000i	− 1.80822i	−0.427303	−	0.904109i	$-0.640536\pi$
$827$	− 52.0000i	− 1.80822i	0.427303	−	0.904109i	$-0.359464\pi$
$828$	− 4.00000i	− 0.139010i
$829$	−50.0000	−1.73657	−0.868286	−	0.496064i	$-0.834778\pi$
$829$	−50.0000	−1.73657	−0.868286	+	0.496064i	$0.834778\pi$
$830$	0	0
$831$	−6.00000	−0.208138
$832$	− 6.00000i	− 0.208013i
$833$	− 14.0000i	− 0.485071i
$834$	−8.00000	−0.277017
$835$	0	0
$836$	16.0000	0.553372
$837$	− 1.00000i	− 0.0345651i
$838$	12.0000i	0.414533i
$839$	−28.0000	−0.966667	−0.483334	−	0.875436i	$-0.660574\pi$
$839$	−28.0000	−0.966667	−0.483334	+	0.875436i	$0.660574\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	14.0000i	0.482472i
$843$	− 10.0000i	− 0.344418i
$844$	4.00000	0.137686
$845$	0	0
$846$	0	0
$847$	0	0
$848$	2.00000i	0.0686803i
$849$	0	0
$850$	0	0
$851$	8.00000	0.274236
$852$	− 12.0000i	− 0.411113i
$853$	58.0000i	1.98588i	0.118609	+	0.992941i	$0.462157\pi$
$853$	58.0000i	1.98588i	−0.118609	+	0.992941i	$0.537843\pi$
$854$	0	0
$855$	0	0
$856$	−4.00000	−0.136717
$857$	− 54.0000i	− 1.84460i	−0.386469	−	0.922302i	$-0.626305\pi$
$857$	− 54.0000i	− 1.84460i	0.386469	−	0.922302i	$-0.373695\pi$
$858$	− 24.0000i	− 0.819346i
$859$	−48.0000	−1.63774	−0.818869	−	0.573980i	$-0.805399\pi$
$859$	−48.0000	−1.63774	−0.818869	+	0.573980i	$0.805399\pi$
$860$	0	0
$861$	0	0
$862$	− 12.0000i	− 0.408722i
$863$	36.0000i	1.22545i	0.790295	+	0.612727i	$0.209928\pi$
$863$	36.0000i	1.22545i	−0.790295	+	0.612727i	$0.790072\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	−2.00000	−0.0679628
$867$	− 13.0000i	− 0.441503i
$868$	0	0
$869$	−64.0000	−2.17105
$870$	0	0
$871$	96.0000	3.25284
$872$	6.00000i	0.203186i
$873$	− 14.0000i	− 0.473828i
$874$	16.0000	0.541208
$875$	0	0
$876$	6.00000	0.202721
$877$	38.0000i	1.28317i	0.767052	+	0.641584i	$0.221723\pi$
$877$	38.0000i	1.28317i	−0.767052	+	0.641584i	$0.778277\pi$
$878$	32.0000i	1.07995i
$879$	−6.00000	−0.202375
$880$	0	0
$881$	−42.0000	−1.41502	−0.707508	−	0.706705i	$-0.750181\pi$
$881$	−42.0000	−1.41502	−0.707508	+	0.706705i	$0.750181\pi$
$882$	− 7.00000i	− 0.235702i
$883$	52.0000i	1.74994i	0.484178	+	0.874970i	$0.339119\pi$
$883$	52.0000i	1.74994i	−0.484178	+	0.874970i	$0.660881\pi$
$884$	−12.0000	−0.403604
$885$	0	0
$886$	4.00000	0.134383
$887$	24.0000i	0.805841i	0.915235	+	0.402921i	$0.132005\pi$
$887$	24.0000i	0.805841i	−0.915235	+	0.402921i	$0.867995\pi$
$888$	− 2.00000i	− 0.0671156i
$889$	0	0
$890$	0	0
$891$	−4.00000	−0.134005
$892$	8.00000i	0.267860i
$893$	0	0
$894$	−10.0000	−0.334450
$895$	0	0
$896$	0	0
$897$	− 24.0000i	− 0.801337i
$898$	18.0000i	0.600668i
$899$	2.00000	0.0667037
$900$	0	0
$901$	4.00000	0.133259
$902$	24.0000i	0.799113i
$903$	0	0
$904$	−10.0000	−0.332595
$905$	0	0
$906$	16.0000	0.531564
$907$	56.0000i	1.85945i	0.368255	+	0.929725i	$0.379955\pi$
$907$	56.0000i	1.85945i	−0.368255	+	0.929725i	$0.620045\pi$
$908$	20.0000i	0.663723i
$909$	−2.00000	−0.0663358
$910$	0	0
$911$	−48.0000	−1.59031	−0.795155	−	0.606406i	$-0.792611\pi$
$911$	−48.0000	−1.59031	−0.795155	+	0.606406i	$0.792611\pi$
$912$	− 4.00000i	− 0.132453i
$913$	48.0000i	1.58857i
$914$	−22.0000	−0.727695
$915$	0	0
$916$	10.0000	0.330409
$917$	0	0
$918$	2.00000i	0.0660098i
$919$	32.0000	1.05558	0.527791	−	0.849374i	$-0.323020\pi$
$919$	32.0000	1.05558	0.527791	+	0.849374i	$0.323020\pi$
$920$	0	0
$921$	16.0000	0.527218
$922$	− 14.0000i	− 0.461065i
$923$	− 72.0000i	− 2.36991i
$924$	0	0
$925$	0	0
$926$	24.0000	0.788689
$927$	− 8.00000i	− 0.262754i
$928$	− 2.00000i	− 0.0656532i
$929$	42.0000	1.37798	0.688988	−	0.724773i	$-0.258055\pi$
$929$	42.0000	1.37798	0.688988	+	0.724773i	$0.258055\pi$
$930$	0	0
$931$	28.0000	0.917663
$932$	− 22.0000i	− 0.720634i
$933$	28.0000i	0.916679i
$934$	−4.00000	−0.130884
$935$	0	0
$936$	−6.00000	−0.196116
$937$	− 10.0000i	− 0.326686i	−0.986569	−	0.163343i	$-0.947772\pi$
$937$	− 10.0000i	− 0.326686i	0.986569	−	0.163343i	$-0.0522277\pi$
$938$	0	0
$939$	26.0000	0.848478
$940$	0	0
$941$	18.0000	0.586783	0.293392	−	0.955992i	$-0.405216\pi$
$941$	18.0000	0.586783	0.293392	+	0.955992i	$0.405216\pi$
$942$	− 18.0000i	− 0.586472i
$943$	24.0000i	0.781548i
$944$	4.00000	0.130189
$945$	0	0
$946$	−16.0000	−0.520205
$947$	44.0000i	1.42981i	0.699223	+	0.714904i	$0.253530\pi$
$947$	44.0000i	1.42981i	−0.699223	+	0.714904i	$0.746470\pi$
$948$	16.0000i	0.519656i
$949$	36.0000	1.16861
$950$	0	0
$951$	−2.00000	−0.0648544
$952$	0	0
$953$	− 30.0000i	− 0.971795i	−0.874016	−	0.485898i	$-0.838493\pi$
$953$	− 30.0000i	− 0.971795i	0.874016	−	0.485898i	$-0.161507\pi$
$954$	2.00000	0.0647524
$955$	0	0
$956$	0	0
$957$	− 8.00000i	− 0.258603i
$958$	− 36.0000i	− 1.16311i
$959$	0	0
$960$	0	0
$961$	1.00000	0.0322581
$962$	− 12.0000i	− 0.386896i
$963$	4.00000i	0.128898i
$964$	14.0000	0.450910
$965$	0	0
$966$	0	0
$967$	8.00000i	0.257263i	0.991692	+	0.128631i	$0.0410584\pi$
$967$	8.00000i	0.257263i	−0.991692	+	0.128631i	$0.958942\pi$
$968$	− 5.00000i	− 0.160706i
$969$	−8.00000	−0.256997
$970$	0	0
$971$	−12.0000	−0.385098	−0.192549	−	0.981287i	$-0.561675\pi$
$971$	−12.0000	−0.385098	−0.192549	+	0.981287i	$0.561675\pi$
$972$	1.00000i	0.0320750i
$973$	0	0
$974$	−40.0000	−1.28168
$975$	0	0
$976$	−6.00000	−0.192055
$977$	− 22.0000i	− 0.703842i	−0.936030	−	0.351921i	$-0.885529\pi$
$977$	− 22.0000i	− 0.703842i	0.936030	−	0.351921i	$-0.114471\pi$
$978$	− 24.0000i	− 0.767435i
$979$	−72.0000	−2.30113
$980$	0	0
$981$	6.00000	0.191565
$982$	20.0000i	0.638226i
$983$	− 4.00000i	− 0.127580i	−0.997963	−	0.0637901i	$-0.979681\pi$
$983$	− 4.00000i	− 0.127580i	0.997963	−	0.0637901i	$-0.0203188\pi$
$984$	6.00000	0.191273
$985$	0	0
$986$	−4.00000	−0.127386
$987$	0	0
$988$	− 24.0000i	− 0.763542i
$989$	−16.0000	−0.508770
$990$	0	0
$991$	−16.0000	−0.508257	−0.254128	−	0.967170i	$-0.581789\pi$
$991$	−16.0000	−0.508257	−0.254128	+	0.967170i	$0.581789\pi$
$992$	− 1.00000i	− 0.0317500i
$993$	− 24.0000i	− 0.761617i
$994$	0	0
$995$	0	0
$996$	12.0000	0.380235
$997$	− 26.0000i	− 0.823428i	−0.911313	−	0.411714i	$-0.864930\pi$
$997$	− 26.0000i	− 0.823428i	0.911313	−	0.411714i	$-0.135070\pi$
$998$	− 16.0000i	− 0.506471i
$999$	−2.00000	−0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4650.2.d.o.3349.2		2
5.2	odd	4		930.2.a.b.1.1	✓	1
5.3	odd	4		4650.2.a.bp.1.1		1
5.4	even	2	inner	4650.2.d.o.3349.1		2
15.2	even	4		2790.2.a.ba.1.1		1
20.7	even	4		7440.2.a.q.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
930.2.a.b.1.1	✓	1	5.2	odd	4
2790.2.a.ba.1.1		1	15.2	even	4
4650.2.a.bp.1.1		1	5.3	odd	4
4650.2.d.o.3349.1		2	5.4	even	2	inner
4650.2.d.o.3349.2		2	1.1	even	1	trivial
7440.2.a.q.1.1		1	20.7	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:	\( N \)	\(=\)	\( 4650 = 2 \cdot 3 \cdot 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4650.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -1.00000i q^{8} -1.00000 q^{9} -4.00000 q^{11} +1.00000i q^{12} +6.00000i q^{13} +1.00000 q^{16} -2.00000i q^{17} -1.00000i q^{18} +4.00000 q^{19} -4.00000i q^{22} -4.00000i q^{23} -1.00000 q^{24} -6.00000 q^{26} +1.00000i q^{27} -2.00000 q^{29} -1.00000 q^{31} +1.00000i q^{32} +4.00000i q^{33} +2.00000 q^{34} +1.00000 q^{36} +2.00000i q^{37} +4.00000i q^{38} +6.00000 q^{39} -6.00000 q^{41} -4.00000i q^{43} +4.00000 q^{44} +4.00000 q^{46} -1.00000i q^{48} +7.00000 q^{49} -2.00000 q^{51} -6.00000i q^{52} +2.00000i q^{53} -1.00000 q^{54} -4.00000i q^{57} -2.00000i q^{58} +4.00000 q^{59} -6.00000 q^{61} -1.00000i q^{62} -1.00000 q^{64} -4.00000 q^{66} -16.0000i q^{67} +2.00000i q^{68} -4.00000 q^{69} -12.0000 q^{71} +1.00000i q^{72} -6.00000i q^{73} -2.00000 q^{74} -4.00000 q^{76} +6.00000i q^{78} +16.0000 q^{79} +1.00000 q^{81} -6.00000i q^{82} -12.0000i q^{83} +4.00000 q^{86} +2.00000i q^{87} +4.00000i q^{88} +18.0000 q^{89} +4.00000i q^{92} +1.00000i q^{93} +1.00000 q^{96} +14.0000i q^{97} +7.00000i q^{98} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9	\( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} - 8 q^{11} + 2 q^{16} + 8 q^{19} - 2 q^{24} - 12 q^{26} - 4 q^{29} - 2 q^{31} + 4 q^{34} + 2 q^{36} + 12 q^{39} - 12 q^{41} + 8 q^{44} + 8 q^{46} + 14 q^{49} - 4 q^{51} - 2 q^{54} + 8 q^{59} - 12 q^{61} - 2 q^{64} - 8 q^{66} - 8 q^{69} - 24 q^{71} - 4 q^{74} - 8 q^{76} + 32 q^{79} + 2 q^{81} + 8 q^{86} + 36 q^{89} + 2 q^{96} + 8 q^{99}+O(q^{100}) \) 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9 - 8 * q^11 + 2 * q^16 + 8 * q^19 - 2 * q^24 - 12 * q^26 - 4 * q^29 - 2 * q^31 + 4 * q^34 + 2 * q^36 + 12 * q^39 - 12 * q^41 + 8 * q^44 + 8 * q^46 + 14 * q^49 - 4 * q^51 - 2 * q^54 + 8 * q^59 - 12 * q^61 - 2 * q^64 - 8 * q^66 - 8 * q^69 - 24 * q^71 - 4 * q^74 - 8 * q^76 + 32 * q^79 + 2 * q^81 + 8 * q^86 + 36 * q^89 + 2 * q^96 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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