LMFDB - Embedded newform 4650.2.d.d.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:	$1$
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.d.3349.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +1.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} -3.00000 q^{11} -1.00000i q^{12} +2.00000i q^{13} -1.00000 q^{14} +1.00000 q^{16} -1.00000i q^{18} -5.00000 q^{19} -1.00000 q^{21} -3.00000i q^{22} +9.00000i q^{23} +1.00000 q^{24} -2.00000 q^{26} -1.00000i q^{27} -1.00000i q^{28} +1.00000 q^{31} +1.00000i q^{32} -3.00000i q^{33} +1.00000 q^{36} -8.00000i q^{37} -5.00000i q^{38} -2.00000 q^{39} -1.00000i q^{42} +11.0000i q^{43} +3.00000 q^{44} -9.00000 q^{46} +6.00000i q^{47} +1.00000i q^{48} +6.00000 q^{49} -2.00000i q^{52} -9.00000i q^{53} +1.00000 q^{54} +1.00000 q^{56} -5.00000i q^{57} -6.00000 q^{59} -10.0000 q^{61} +1.00000i q^{62} -1.00000i q^{63} -1.00000 q^{64} +3.00000 q^{66} -14.0000i q^{67} -9.00000 q^{69} +9.00000 q^{71} +1.00000i q^{72} -1.00000i q^{73} +8.00000 q^{74} +5.00000 q^{76} -3.00000i q^{77} -2.00000i q^{78} +1.00000 q^{79} +1.00000 q^{81} -12.0000i q^{83} +1.00000 q^{84} -11.0000 q^{86} +3.00000i q^{88} +9.00000 q^{89} -2.00000 q^{91} -9.00000i q^{92} +1.00000i q^{93} -6.00000 q^{94} -1.00000 q^{96} -14.0000i q^{97} +6.00000i q^{98} +3.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9} - 6 q^{11} - 2 q^{14} + 2 q^{16} - 10 q^{19} - 2 q^{21} + 2 q^{24} - 4 q^{26} + 2 q^{31} + 2 q^{36} - 4 q^{39} + 6 q^{44} - 18 q^{46} + 12 q^{49} + 2 q^{54} + 2 q^{56}+ \cdots + 6 q^{99}+O(q^{100}) $ 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9 - 6 * q^11 - 2 * q^14 + 2 * q^16 - 10 * q^19 - 2 * q^21 + 2 * q^24 - 4 * q^26 + 2 * q^31 + 2 * q^36 - 4 * q^39 + 6 * q^44 - 18 * q^46 + 12 * q^49 + 2 * q^54 + 2 * q^56 - 12 * q^59 - 20 * q^61 - 2 * q^64 + 6 * q^66 - 18 * q^69 + 18 * q^71 + 16 * q^74 + 10 * q^76 + 2 * q^79 + 2 * q^81 + 2 * q^84 - 22 * q^86 + 18 * q^89 - 4 * q^91 - 12 * q^94 - 2 * q^96 + 6 * q^99

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
$7$	1.00000i	0.377964i	0.981981	+	0.188982i	$0.0605189\pi$
$7$	1.00000i	0.377964i	−0.981981	+	0.188982i	$0.939481\pi$
$8$	− 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
$13$	2.00000i	0.554700i	0.960769	+	0.277350i	$0.0894562\pi$
$13$	2.00000i	0.554700i	−0.960769	+	0.277350i	$0.910544\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	− 1.00000i	− 0.235702i
$19$	−5.00000	−1.14708	−0.573539	−	0.819178i	$-0.694430\pi$
$19$	−5.00000	−1.14708	−0.573539	+	0.819178i	$0.694430\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	− 3.00000i	− 0.639602i
$23$	9.00000i	1.87663i	0.345782	+	0.938315i	$0.387614\pi$
$23$	9.00000i	1.87663i	−0.345782	+	0.938315i	$0.612386\pi$
$24$	1.00000	0.204124
$25$	0	0
$26$	−2.00000	−0.392232
$27$	− 1.00000i	− 0.192450i
$28$	− 1.00000i	− 0.188982i
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	1.00000i	0.176777i
$33$	− 3.00000i	− 0.522233i
$34$	0	0
$35$	0	0
$36$	1.00000	0.166667
$37$	− 8.00000i	− 1.31519i	−0.753371	−	0.657596i	$-0.771573\pi$
$37$	− 8.00000i	− 1.31519i	0.753371	−	0.657596i	$-0.228427\pi$
$38$	− 5.00000i	− 0.811107i
$39$	−2.00000	−0.320256
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	− 1.00000i	− 0.154303i
$43$	11.0000i	1.67748i	0.544529	+	0.838742i	$0.316708\pi$
$43$	11.0000i	1.67748i	−0.544529	+	0.838742i	$0.683292\pi$
$44$	3.00000	0.452267
$45$	0	0
$46$	−9.00000	−1.32698
$47$	6.00000i	0.875190i	0.899172	+	0.437595i	$0.144170\pi$
$47$	6.00000i	0.875190i	−0.899172	+	0.437595i	$0.855830\pi$
$48$	1.00000i	0.144338i
$49$	6.00000	0.857143
$50$	0	0
$51$	0	0
$52$	− 2.00000i	− 0.277350i
$53$	− 9.00000i	− 1.23625i	−0.786082	−	0.618123i	$-0.787894\pi$
$53$	− 9.00000i	− 1.23625i	0.786082	−	0.618123i	$-0.212106\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	1.00000	0.133631
$57$	− 5.00000i	− 0.662266i
$58$	0	0
$59$	−6.00000	−0.781133	−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−6.00000	−0.781133	−0.390567	+	0.920575i	$0.627721\pi$
$60$	0	0
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	1.00000i	0.127000i
$63$	− 1.00000i	− 0.125988i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	3.00000	0.369274
$67$	− 14.0000i	− 1.71037i	−0.518321	−	0.855186i	$-0.673443\pi$
$67$	− 14.0000i	− 1.71037i	0.518321	−	0.855186i	$-0.326557\pi$
$68$	0	0
$69$	−9.00000	−1.08347
$70$	0	0
$71$	9.00000	1.06810	0.534052	−	0.845452i	$-0.320669\pi$
$71$	9.00000	1.06810	0.534052	+	0.845452i	$0.320669\pi$
$72$	1.00000i	0.117851i
$73$	− 1.00000i	− 0.117041i	−0.998286	−	0.0585206i	$-0.981362\pi$
$73$	− 1.00000i	− 0.117041i	0.998286	−	0.0585206i	$-0.0186383\pi$
$74$	8.00000	0.929981
$75$	0	0
$76$	5.00000	0.573539
$77$	− 3.00000i	− 0.341882i
$78$	− 2.00000i	− 0.226455i
$79$	1.00000	0.112509	0.0562544	−	0.998416i	$-0.482084\pi$
$79$	1.00000	0.112509	0.0562544	+	0.998416i	$0.482084\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 12.0000i	− 1.31717i	−0.752506	−	0.658586i	$-0.771155\pi$
$83$	− 12.0000i	− 1.31717i	0.752506	−	0.658586i	$-0.228845\pi$
$84$	1.00000	0.109109
$85$	0	0
$86$	−11.0000	−1.18616
$87$	0	0
$88$	3.00000i	0.319801i
$89$	9.00000	0.953998	0.476999	−	0.878904i	$-0.341725\pi$
$89$	9.00000	0.953998	0.476999	+	0.878904i	$0.341725\pi$
$90$	0	0
$91$	−2.00000	−0.209657
$92$	− 9.00000i	− 0.938315i
$93$	1.00000i	0.103695i
$94$	−6.00000	−0.618853
$95$	0	0
$96$	−1.00000	−0.102062
$97$	− 14.0000i	− 1.42148i	−0.703452	−	0.710742i	$-0.748359\pi$
$97$	− 14.0000i	− 1.42148i	0.703452	−	0.710742i	$-0.251641\pi$
$98$	6.00000i	0.606092i
$99$	3.00000	0.301511
$100$	0	0
$101$	−15.0000	−1.49256	−0.746278	−	0.665635i	$-0.768161\pi$
$101$	−15.0000	−1.49256	−0.746278	+	0.665635i	$0.768161\pi$
$102$	0	0
$103$	8.00000i	0.788263i	0.919054	+	0.394132i	$0.128955\pi$
$103$	8.00000i	0.788263i	−0.919054	+	0.394132i	$0.871045\pi$
$104$	2.00000	0.196116
$105$	0	0
$106$	9.00000	0.874157
$107$	3.00000i	0.290021i	0.989430	+	0.145010i	$0.0463216\pi$
$107$	3.00000i	0.290021i	−0.989430	+	0.145010i	$0.953678\pi$
$108$	1.00000i	0.0962250i
$109$	−8.00000	−0.766261	−0.383131	−	0.923694i	$-0.625154\pi$
$109$	−8.00000	−0.766261	−0.383131	+	0.923694i	$0.625154\pi$
$110$	0	0
$111$	8.00000	0.759326
$112$	1.00000i	0.0944911i
$113$	− 3.00000i	− 0.282216i	−0.989994	−	0.141108i	$-0.954933\pi$
$113$	− 3.00000i	− 0.282216i	0.989994	−	0.141108i	$-0.0450665\pi$
$114$	5.00000	0.468293
$115$	0	0
$116$	0	0
$117$	− 2.00000i	− 0.184900i
$118$	− 6.00000i	− 0.552345i
$119$	0	0
$120$	0	0
$121$	−2.00000	−0.181818
$122$	− 10.0000i	− 0.905357i
$123$	0	0
$124$	−1.00000	−0.0898027
$125$	0	0
$126$	1.00000	0.0890871
$127$	− 8.00000i	− 0.709885i	−0.934888	−	0.354943i	$-0.884500\pi$
$127$	− 8.00000i	− 0.709885i	0.934888	−	0.354943i	$-0.115500\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	−11.0000	−0.968496
$130$	0	0
$131$	−6.00000	−0.524222	−0.262111	−	0.965038i	$-0.584419\pi$
$131$	−6.00000	−0.524222	−0.262111	+	0.965038i	$0.584419\pi$
$132$	3.00000i	0.261116i
$133$	− 5.00000i	− 0.433555i
$134$	14.0000	1.20942
$135$	0	0
$136$	0	0
$137$	− 18.0000i	− 1.53784i	−0.639343	−	0.768922i	$-0.720793\pi$
$137$	− 18.0000i	− 1.53784i	0.639343	−	0.768922i	$-0.279207\pi$
$138$	− 9.00000i	− 0.766131i
$139$	−14.0000	−1.18746	−0.593732	−	0.804663i	$-0.702346\pi$
$139$	−14.0000	−1.18746	−0.593732	+	0.804663i	$0.702346\pi$
$140$	0	0
$141$	−6.00000	−0.505291
$142$	9.00000i	0.755263i
$143$	− 6.00000i	− 0.501745i
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	1.00000	0.0827606
$147$	6.00000i	0.494872i
$148$	8.00000i	0.657596i
$149$	−3.00000	−0.245770	−0.122885	−	0.992421i	$-0.539215\pi$
$149$	−3.00000	−0.245770	−0.122885	+	0.992421i	$0.539215\pi$
$150$	0	0
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	5.00000i	0.405554i
$153$	0	0
$154$	3.00000	0.241747
$155$	0	0
$156$	2.00000	0.160128
$157$	7.00000i	0.558661i	0.960195	+	0.279330i	$0.0901125\pi$
$157$	7.00000i	0.558661i	−0.960195	+	0.279330i	$0.909888\pi$
$158$	1.00000i	0.0795557i
$159$	9.00000	0.713746
$160$	0	0
$161$	−9.00000	−0.709299
$162$	1.00000i	0.0785674i
$163$	− 22.0000i	− 1.72317i	−0.507611	−	0.861586i	$-0.669471\pi$
$163$	− 22.0000i	− 1.72317i	0.507611	−	0.861586i	$-0.330529\pi$
$164$	0	0
$165$	0	0
$166$	12.0000	0.931381
$167$	− 9.00000i	− 0.696441i	−0.937413	−	0.348220i	$-0.886786\pi$
$167$	− 9.00000i	− 0.696441i	0.937413	−	0.348220i	$-0.113214\pi$
$168$	1.00000i	0.0771517i
$169$	9.00000	0.692308
$170$	0	0
$171$	5.00000	0.382360
$172$	− 11.0000i	− 0.838742i
$173$	6.00000i	0.456172i	0.973641	+	0.228086i	$0.0732467\pi$
$173$	6.00000i	0.456172i	−0.973641	+	0.228086i	$0.926753\pi$
$174$	0	0
$175$	0	0
$176$	−3.00000	−0.226134
$177$	− 6.00000i	− 0.450988i
$178$	9.00000i	0.674579i
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	0	0
$181$	5.00000	0.371647	0.185824	−	0.982583i	$-0.440505\pi$
$181$	5.00000	0.371647	0.185824	+	0.982583i	$0.440505\pi$
$182$	− 2.00000i	− 0.148250i
$183$	− 10.0000i	− 0.739221i
$184$	9.00000	0.663489
$185$	0	0
$186$	−1.00000	−0.0733236
$187$	0	0
$188$	− 6.00000i	− 0.437595i
$189$	1.00000	0.0727393
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	− 1.00000i	− 0.0721688i
$193$	− 10.0000i	− 0.719816i	−0.932988	−	0.359908i	$-0.882808\pi$
$193$	− 10.0000i	− 0.719816i	0.932988	−	0.359908i	$-0.117192\pi$
$194$	14.0000	1.00514
$195$	0	0
$196$	−6.00000	−0.428571
$197$	6.00000i	0.427482i	0.976890	+	0.213741i	$0.0685649\pi$
$197$	6.00000i	0.427482i	−0.976890	+	0.213741i	$0.931435\pi$
$198$	3.00000i	0.213201i
$199$	25.0000	1.77220	0.886102	−	0.463491i	$-0.153403\pi$
$199$	25.0000	1.77220	0.886102	+	0.463491i	$0.153403\pi$
$200$	0	0
$201$	14.0000	0.987484
$202$	− 15.0000i	− 1.05540i
$203$	0	0
$204$	0	0
$205$	0	0
$206$	−8.00000	−0.557386
$207$	− 9.00000i	− 0.625543i
$208$	2.00000i	0.138675i
$209$	15.0000	1.03757
$210$	0	0
$211$	−19.0000	−1.30801	−0.654007	−	0.756489i	$-0.726913\pi$
$211$	−19.0000	−1.30801	−0.654007	+	0.756489i	$0.726913\pi$
$212$	9.00000i	0.618123i
$213$	9.00000i	0.616670i
$214$	−3.00000	−0.205076
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	1.00000i	0.0678844i
$218$	− 8.00000i	− 0.541828i
$219$	1.00000	0.0675737
$220$	0	0
$221$	0	0
$222$	8.00000i	0.536925i
$223$	− 10.0000i	− 0.669650i	−0.942280	−	0.334825i	$-0.891323\pi$
$223$	− 10.0000i	− 0.669650i	0.942280	−	0.334825i	$-0.108677\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	3.00000	0.199557
$227$	15.0000i	0.995585i	0.867296	+	0.497792i	$0.165856\pi$
$227$	15.0000i	0.995585i	−0.867296	+	0.497792i	$0.834144\pi$
$228$	5.00000i	0.331133i
$229$	−5.00000	−0.330409	−0.165205	−	0.986259i	$-0.552828\pi$
$229$	−5.00000	−0.330409	−0.165205	+	0.986259i	$0.552828\pi$
$230$	0	0
$231$	3.00000	0.197386
$232$	0	0
$233$	3.00000i	0.196537i	0.995160	+	0.0982683i	$0.0313303\pi$
$233$	3.00000i	0.196537i	−0.995160	+	0.0982683i	$0.968670\pi$
$234$	2.00000	0.130744
$235$	0	0
$236$	6.00000	0.390567
$237$	1.00000i	0.0649570i
$238$	0	0
$239$	−12.0000	−0.776215	−0.388108	−	0.921614i	$-0.626871\pi$
$239$	−12.0000	−0.776215	−0.388108	+	0.921614i	$0.626871\pi$
$240$	0	0
$241$	−22.0000	−1.41714	−0.708572	−	0.705638i	$-0.750660\pi$
$241$	−22.0000	−1.41714	−0.708572	+	0.705638i	$0.750660\pi$
$242$	− 2.00000i	− 0.128565i
$243$	1.00000i	0.0641500i
$244$	10.0000	0.640184
$245$	0	0
$246$	0	0
$247$	− 10.0000i	− 0.636285i
$248$	− 1.00000i	− 0.0635001i
$249$	12.0000	0.760469
$250$	0	0
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	1.00000i	0.0629941i
$253$	− 27.0000i	− 1.69748i
$254$	8.00000	0.501965
$255$	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	$0.681315\pi$
$258$	− 11.0000i	− 0.684830i
$259$	8.00000	0.497096
$260$	0	0
$261$	0	0
$262$	− 6.00000i	− 0.370681i
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	−3.00000	−0.184637
$265$	0	0
$266$	5.00000	0.306570
$267$	9.00000i	0.550791i
$268$	14.0000i	0.855186i
$269$	−6.00000	−0.365826	−0.182913	−	0.983129i	$-0.558553\pi$
$269$	−6.00000	−0.365826	−0.182913	+	0.983129i	$0.558553\pi$
$270$	0	0
$271$	5.00000	0.303728	0.151864	−	0.988401i	$-0.451472\pi$
$271$	5.00000	0.303728	0.151864	+	0.988401i	$0.451472\pi$
$272$	0	0
$273$	− 2.00000i	− 0.121046i
$274$	18.0000	1.08742
$275$	0	0
$276$	9.00000	0.541736
$277$	− 8.00000i	− 0.480673i	−0.970690	−	0.240337i	$-0.922742\pi$
$277$	− 8.00000i	− 0.480673i	0.970690	−	0.240337i	$-0.0772579\pi$
$278$	− 14.0000i	− 0.839664i
$279$	−1.00000	−0.0598684
$280$	0	0
$281$	−24.0000	−1.43172	−0.715860	−	0.698244i	$-0.753965\pi$
$281$	−24.0000	−1.43172	−0.715860	+	0.698244i	$0.753965\pi$
$282$	− 6.00000i	− 0.357295i
$283$	− 4.00000i	− 0.237775i	−0.992908	−	0.118888i	$-0.962067\pi$
$283$	− 4.00000i	− 0.237775i	0.992908	−	0.118888i	$-0.0379328\pi$
$284$	−9.00000	−0.534052
$285$	0	0
$286$	6.00000	0.354787
$287$	0	0
$288$	− 1.00000i	− 0.0589256i
$289$	17.0000	1.00000
$290$	0	0
$291$	14.0000	0.820695
$292$	1.00000i	0.0585206i
$293$	18.0000i	1.05157i	0.850617	+	0.525786i	$0.176229\pi$
$293$	18.0000i	1.05157i	−0.850617	+	0.525786i	$0.823771\pi$
$294$	−6.00000	−0.349927
$295$	0	0
$296$	−8.00000	−0.464991
$297$	3.00000i	0.174078i
$298$	− 3.00000i	− 0.173785i
$299$	−18.0000	−1.04097
$300$	0	0
$301$	−11.0000	−0.634029
$302$	− 16.0000i	− 0.920697i
$303$	− 15.0000i	− 0.861727i
$304$	−5.00000	−0.286770
$305$	0	0
$306$	0	0
$307$	− 32.0000i	− 1.82634i	−0.407583	−	0.913168i	$-0.633628\pi$
$307$	− 32.0000i	− 1.82634i	0.407583	−	0.913168i	$-0.366372\pi$
$308$	3.00000i	0.170941i
$309$	−8.00000	−0.455104
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	2.00000i	0.113228i
$313$	− 22.0000i	− 1.24351i	−0.783210	−	0.621757i	$-0.786419\pi$
$313$	− 22.0000i	− 1.24351i	0.783210	−	0.621757i	$-0.213581\pi$
$314$	−7.00000	−0.395033
$315$	0	0
$316$	−1.00000	−0.0562544
$317$	− 24.0000i	− 1.34797i	−0.738743	−	0.673987i	$-0.764580\pi$
$317$	− 24.0000i	− 1.34797i	0.738743	−	0.673987i	$-0.235420\pi$
$318$	9.00000i	0.504695i
$319$	0	0
$320$	0	0
$321$	−3.00000	−0.167444
$322$	− 9.00000i	− 0.501550i
$323$	0	0
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	22.0000	1.21847
$327$	− 8.00000i	− 0.442401i
$328$	0	0
$329$	−6.00000	−0.330791
$330$	0	0
$331$	8.00000	0.439720	0.219860	−	0.975531i	$-0.429440\pi$
$331$	8.00000	0.439720	0.219860	+	0.975531i	$0.429440\pi$
$332$	12.0000i	0.658586i
$333$	8.00000i	0.438397i
$334$	9.00000	0.492458
$335$	0	0
$336$	−1.00000	−0.0545545
$337$	22.0000i	1.19842i	0.800593	+	0.599208i	$0.204518\pi$
$337$	22.0000i	1.19842i	−0.800593	+	0.599208i	$0.795482\pi$
$338$	9.00000i	0.489535i
$339$	3.00000	0.162938
$340$	0	0
$341$	−3.00000	−0.162459
$342$	5.00000i	0.270369i
$343$	13.0000i	0.701934i
$344$	11.0000	0.593080
$345$	0	0
$346$	−6.00000	−0.322562
$347$	36.0000i	1.93258i	0.257454	+	0.966291i	$0.417117\pi$
$347$	36.0000i	1.93258i	−0.257454	+	0.966291i	$0.582883\pi$
$348$	0	0
$349$	4.00000	0.214115	0.107058	−	0.994253i	$-0.465857\pi$
$349$	4.00000	0.214115	0.107058	+	0.994253i	$0.465857\pi$
$350$	0	0
$351$	2.00000	0.106752
$352$	− 3.00000i	− 0.159901i
$353$	6.00000i	0.319348i	0.987170	+	0.159674i	$0.0510443\pi$
$353$	6.00000i	0.319348i	−0.987170	+	0.159674i	$0.948956\pi$
$354$	6.00000	0.318896
$355$	0	0
$356$	−9.00000	−0.476999
$357$	0	0
$358$	12.0000i	0.634220i
$359$	−9.00000	−0.475002	−0.237501	−	0.971387i	$-0.576328\pi$
$359$	−9.00000	−0.475002	−0.237501	+	0.971387i	$0.576328\pi$
$360$	0	0
$361$	6.00000	0.315789
$362$	5.00000i	0.262794i
$363$	− 2.00000i	− 0.104973i
$364$	2.00000	0.104828
$365$	0	0
$366$	10.0000	0.522708
$367$	− 26.0000i	− 1.35719i	−0.734513	−	0.678594i	$-0.762589\pi$
$367$	− 26.0000i	− 1.35719i	0.734513	−	0.678594i	$-0.237411\pi$
$368$	9.00000i	0.469157i
$369$	0	0
$370$	0	0
$371$	9.00000	0.467257
$372$	− 1.00000i	− 0.0518476i
$373$	− 19.0000i	− 0.983783i	−0.870657	−	0.491891i	$-0.836306\pi$
$373$	− 19.0000i	− 0.983783i	0.870657	−	0.491891i	$-0.163694\pi$
$374$	0	0
$375$	0	0
$376$	6.00000	0.309426
$377$	0	0
$378$	1.00000i	0.0514344i
$379$	1.00000	0.0513665	0.0256833	−	0.999670i	$-0.491824\pi$
$379$	1.00000	0.0513665	0.0256833	+	0.999670i	$0.491824\pi$
$380$	0	0
$381$	8.00000	0.409852
$382$	0	0
$383$	36.0000i	1.83951i	0.392488	+	0.919757i	$0.371614\pi$
$383$	36.0000i	1.83951i	−0.392488	+	0.919757i	$0.628386\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	10.0000	0.508987
$387$	− 11.0000i	− 0.559161i
$388$	14.0000i	0.710742i
$389$	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
$391$	0	0
$392$	− 6.00000i	− 0.303046i
$393$	− 6.00000i	− 0.302660i
$394$	−6.00000	−0.302276
$395$	0	0
$396$	−3.00000	−0.150756
$397$	7.00000i	0.351320i	0.984451	+	0.175660i	$0.0562059\pi$
$397$	7.00000i	0.351320i	−0.984451	+	0.175660i	$0.943794\pi$
$398$	25.0000i	1.25314i
$399$	5.00000	0.250313
$400$	0	0
$401$	15.0000	0.749064	0.374532	−	0.927214i	$-0.377803\pi$
$401$	15.0000	0.749064	0.374532	+	0.927214i	$0.377803\pi$
$402$	14.0000i	0.698257i
$403$	2.00000i	0.0996271i
$404$	15.0000	0.746278
$405$	0	0
$406$	0	0
$407$	24.0000i	1.18964i
$408$	0	0
$409$	28.0000	1.38451	0.692255	−	0.721653i	$-0.256617\pi$
$409$	28.0000	1.38451	0.692255	+	0.721653i	$0.256617\pi$
$410$	0	0
$411$	18.0000	0.887875
$412$	− 8.00000i	− 0.394132i
$413$	− 6.00000i	− 0.295241i
$414$	9.00000	0.442326
$415$	0	0
$416$	−2.00000	−0.0980581
$417$	− 14.0000i	− 0.685583i
$418$	15.0000i	0.733674i
$419$	24.0000	1.17248	0.586238	−	0.810139i	$-0.300608\pi$
$419$	24.0000	1.17248	0.586238	+	0.810139i	$0.300608\pi$
$420$	0	0
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	− 19.0000i	− 0.924906i
$423$	− 6.00000i	− 0.291730i
$424$	−9.00000	−0.437079
$425$	0	0
$426$	−9.00000	−0.436051
$427$	− 10.0000i	− 0.483934i
$428$	− 3.00000i	− 0.145010i
$429$	6.00000	0.289683
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	− 1.00000i	− 0.0481125i
$433$	5.00000i	0.240285i	0.992757	+	0.120142i	$0.0383351\pi$
$433$	5.00000i	0.240285i	−0.992757	+	0.120142i	$0.961665\pi$
$434$	−1.00000	−0.0480015
$435$	0	0
$436$	8.00000	0.383131
$437$	− 45.0000i	− 2.15264i
$438$	1.00000i	0.0477818i
$439$	−14.0000	−0.668184	−0.334092	−	0.942541i	$-0.608430\pi$
$439$	−14.0000	−0.668184	−0.334092	+	0.942541i	$0.608430\pi$
$440$	0	0
$441$	−6.00000	−0.285714
$442$	0	0
$443$	27.0000i	1.28281i	0.767203	+	0.641404i	$0.221648\pi$
$443$	27.0000i	1.28281i	−0.767203	+	0.641404i	$0.778352\pi$
$444$	−8.00000	−0.379663
$445$	0	0
$446$	10.0000	0.473514
$447$	− 3.00000i	− 0.141895i
$448$	− 1.00000i	− 0.0472456i
$449$	−6.00000	−0.283158	−0.141579	−	0.989927i	$-0.545218\pi$
$449$	−6.00000	−0.283158	−0.141579	+	0.989927i	$0.545218\pi$
$450$	0	0
$451$	0	0
$452$	3.00000i	0.141108i
$453$	− 16.0000i	− 0.751746i
$454$	−15.0000	−0.703985
$455$	0	0
$456$	−5.00000	−0.234146
$457$	− 38.0000i	− 1.77757i	−0.458329	−	0.888783i	$-0.651552\pi$
$457$	− 38.0000i	− 1.77757i	0.458329	−	0.888783i	$-0.348448\pi$
$458$	− 5.00000i	− 0.233635i
$459$	0	0
$460$	0	0
$461$	0	0		−	1.00000i	$-0.5\pi$
$461$	0	0			1.00000i	$0.5\pi$
$462$	3.00000i	0.139573i
$463$	32.0000i	1.48717i	0.668644	+	0.743583i	$0.266875\pi$
$463$	32.0000i	1.48717i	−0.668644	+	0.743583i	$0.733125\pi$
$464$	0	0
$465$	0	0
$466$	−3.00000	−0.138972
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	2.00000i	0.0924500i
$469$	14.0000	0.646460
$470$	0	0
$471$	−7.00000	−0.322543
$472$	6.00000i	0.276172i
$473$	− 33.0000i	− 1.51734i
$474$	−1.00000	−0.0459315
$475$	0	0
$476$	0	0
$477$	9.00000i	0.412082i
$478$	− 12.0000i	− 0.548867i
$479$	−21.0000	−0.959514	−0.479757	−	0.877401i	$-0.659275\pi$
$479$	−21.0000	−0.959514	−0.479757	+	0.877401i	$0.659275\pi$
$480$	0	0
$481$	16.0000	0.729537
$482$	− 22.0000i	− 1.00207i
$483$	− 9.00000i	− 0.409514i
$484$	2.00000	0.0909091
$485$	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	$0.638905\pi$
$488$	10.0000i	0.452679i
$489$	22.0000	0.994874
$490$	0	0
$491$	21.0000	0.947717	0.473858	−	0.880601i	$-0.342861\pi$
$491$	21.0000	0.947717	0.473858	+	0.880601i	$0.342861\pi$
$492$	0	0
$493$	0	0
$494$	10.0000	0.449921
$495$	0	0
$496$	1.00000	0.0449013
$497$	9.00000i	0.403705i
$498$	12.0000i	0.537733i
$499$	−32.0000	−1.43252	−0.716258	−	0.697835i	$-0.754147\pi$
$499$	−32.0000	−1.43252	−0.716258	+	0.697835i	$0.754147\pi$
$500$	0	0
$501$	9.00000	0.402090
$502$	12.0000i	0.535586i
$503$	− 24.0000i	− 1.07011i	−0.844818	−	0.535054i	$-0.820291\pi$
$503$	− 24.0000i	− 1.07011i	0.844818	−	0.535054i	$-0.179709\pi$
$504$	−1.00000	−0.0445435
$505$	0	0
$506$	27.0000	1.20030
$507$	9.00000i	0.399704i
$508$	8.00000i	0.354943i
$509$	−36.0000	−1.59567	−0.797836	−	0.602875i	$-0.794022\pi$
$509$	−36.0000	−1.59567	−0.797836	+	0.602875i	$0.794022\pi$
$510$	0	0
$511$	1.00000	0.0442374
$512$	1.00000i	0.0441942i
$513$	5.00000i	0.220755i
$514$	−27.0000	−1.19092
$515$	0	0
$516$	11.0000	0.484248
$517$	− 18.0000i	− 0.791639i
$518$	8.00000i	0.351500i
$519$	−6.00000	−0.263371
$520$	0	0
$521$	18.0000	0.788594	0.394297	−	0.918983i	$-0.370988\pi$
$521$	18.0000	0.788594	0.394297	+	0.918983i	$0.370988\pi$
$522$	0	0
$523$	− 25.0000i	− 1.09317i	−0.837402	−	0.546587i	$-0.815927\pi$
$523$	− 25.0000i	− 1.09317i	0.837402	−	0.546587i	$-0.184073\pi$
$524$	6.00000	0.262111
$525$	0	0
$526$	0	0
$527$	0	0
$528$	− 3.00000i	− 0.130558i
$529$	−58.0000	−2.52174
$530$	0	0
$531$	6.00000	0.260378
$532$	5.00000i	0.216777i
$533$	0	0
$534$	−9.00000	−0.389468
$535$	0	0
$536$	−14.0000	−0.604708
$537$	12.0000i	0.517838i
$538$	− 6.00000i	− 0.258678i
$539$	−18.0000	−0.775315
$540$	0	0
$541$	−34.0000	−1.46177	−0.730887	−	0.682498i	$-0.760893\pi$
$541$	−34.0000	−1.46177	−0.730887	+	0.682498i	$0.760893\pi$
$542$	5.00000i	0.214768i
$543$	5.00000i	0.214571i
$544$	0	0
$545$	0	0
$546$	2.00000	0.0855921
$547$	34.0000i	1.45374i	0.686778	+	0.726868i	$0.259025\pi$
$547$	34.0000i	1.45374i	−0.686778	+	0.726868i	$0.740975\pi$
$548$	18.0000i	0.768922i
$549$	10.0000	0.426790
$550$	0	0
$551$	0	0
$552$	9.00000i	0.383065i
$553$	1.00000i	0.0425243i
$554$	8.00000	0.339887
$555$	0	0
$556$	14.0000	0.593732
$557$	− 3.00000i	− 0.127114i	−0.997978	−	0.0635570i	$-0.979756\pi$
$557$	− 3.00000i	− 0.127114i	0.997978	−	0.0635570i	$-0.0202445\pi$
$558$	− 1.00000i	− 0.0423334i
$559$	−22.0000	−0.930501
$560$	0	0
$561$	0	0
$562$	− 24.0000i	− 1.01238i
$563$	− 12.0000i	− 0.505740i	−0.967500	−	0.252870i	$-0.918626\pi$
$563$	− 12.0000i	− 0.505740i	0.967500	−	0.252870i	$-0.0813744\pi$
$564$	6.00000	0.252646
$565$	0	0
$566$	4.00000	0.168133
$567$	1.00000i	0.0419961i
$568$	− 9.00000i	− 0.377632i
$569$	−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
$571$	14.0000	0.585882	0.292941	−	0.956131i	$-0.405366\pi$
$571$	14.0000	0.585882	0.292941	+	0.956131i	$0.405366\pi$
$572$	6.00000i	0.250873i
$573$	0	0
$574$	0	0
$575$	0	0
$576$	1.00000	0.0416667
$577$	16.0000i	0.666089i	0.942911	+	0.333044i	$0.108076\pi$
$577$	16.0000i	0.666089i	−0.942911	+	0.333044i	$0.891924\pi$
$578$	17.0000i	0.707107i
$579$	10.0000	0.415586
$580$	0	0
$581$	12.0000	0.497844
$582$	14.0000i	0.580319i
$583$	27.0000i	1.11823i
$584$	−1.00000	−0.0413803
$585$	0	0
$586$	−18.0000	−0.743573
$587$	24.0000i	0.990586i	0.868726	+	0.495293i	$0.164939\pi$
$587$	24.0000i	0.990586i	−0.868726	+	0.495293i	$0.835061\pi$
$588$	− 6.00000i	− 0.247436i
$589$	−5.00000	−0.206021
$590$	0	0
$591$	−6.00000	−0.246807
$592$	− 8.00000i	− 0.328798i
$593$	6.00000i	0.246390i	0.992382	+	0.123195i	$0.0393141\pi$
$593$	6.00000i	0.246390i	−0.992382	+	0.123195i	$0.960686\pi$
$594$	−3.00000	−0.123091
$595$	0	0
$596$	3.00000	0.122885
$597$	25.0000i	1.02318i
$598$	− 18.0000i	− 0.736075i
$599$	−27.0000	−1.10319	−0.551595	−	0.834112i	$-0.685981\pi$
$599$	−27.0000	−1.10319	−0.551595	+	0.834112i	$0.685981\pi$
$600$	0	0
$601$	−10.0000	−0.407909	−0.203954	−	0.978980i	$-0.565379\pi$
$601$	−10.0000	−0.407909	−0.203954	+	0.978980i	$0.565379\pi$
$602$	− 11.0000i	− 0.448327i
$603$	14.0000i	0.570124i
$604$	16.0000	0.651031
$605$	0	0
$606$	15.0000	0.609333
$607$	1.00000i	0.0405887i	0.999794	+	0.0202944i	$0.00646034\pi$
$607$	1.00000i	0.0405887i	−0.999794	+	0.0202944i	$0.993540\pi$
$608$	− 5.00000i	− 0.202777i
$609$	0	0
$610$	0	0
$611$	−12.0000	−0.485468
$612$	0	0
$613$	− 22.0000i	− 0.888572i	−0.895885	−	0.444286i	$-0.853457\pi$
$613$	− 22.0000i	− 0.888572i	0.895885	−	0.444286i	$-0.146543\pi$
$614$	32.0000	1.29141
$615$	0	0
$616$	−3.00000	−0.120873
$617$	15.0000i	0.603877i	0.953327	+	0.301939i	$0.0976338\pi$
$617$	15.0000i	0.603877i	−0.953327	+	0.301939i	$0.902366\pi$
$618$	− 8.00000i	− 0.321807i
$619$	−14.0000	−0.562708	−0.281354	−	0.959604i	$-0.590783\pi$
$619$	−14.0000	−0.562708	−0.281354	+	0.959604i	$0.590783\pi$
$620$	0	0
$621$	9.00000	0.361158
$622$	0	0
$623$	9.00000i	0.360577i
$624$	−2.00000	−0.0800641
$625$	0	0
$626$	22.0000	0.879297
$627$	15.0000i	0.599042i
$628$	− 7.00000i	− 0.279330i
$629$	0	0
$630$	0	0
$631$	−7.00000	−0.278666	−0.139333	−	0.990246i	$-0.544496\pi$
$631$	−7.00000	−0.278666	−0.139333	+	0.990246i	$0.544496\pi$
$632$	− 1.00000i	− 0.0397779i
$633$	− 19.0000i	− 0.755182i
$634$	24.0000	0.953162
$635$	0	0
$636$	−9.00000	−0.356873
$637$	12.0000i	0.475457i
$638$	0	0
$639$	−9.00000	−0.356034
$640$	0	0
$641$	30.0000	1.18493	0.592464	−	0.805597i	$-0.298155\pi$
$641$	30.0000	1.18493	0.592464	+	0.805597i	$0.298155\pi$
$642$	− 3.00000i	− 0.118401i
$643$	− 31.0000i	− 1.22252i	−0.791430	−	0.611260i	$-0.790663\pi$
$643$	− 31.0000i	− 1.22252i	0.791430	−	0.611260i	$-0.209337\pi$
$644$	9.00000	0.354650
$645$	0	0
$646$	0	0
$647$	33.0000i	1.29736i	0.761060	+	0.648682i	$0.224679\pi$
$647$	33.0000i	1.29736i	−0.761060	+	0.648682i	$0.775321\pi$
$648$	− 1.00000i	− 0.0392837i
$649$	18.0000	0.706562
$650$	0	0
$651$	−1.00000	−0.0391931
$652$	22.0000i	0.861586i
$653$	30.0000i	1.17399i	0.809590	+	0.586995i	$0.199689\pi$
$653$	30.0000i	1.17399i	−0.809590	+	0.586995i	$0.800311\pi$
$654$	8.00000	0.312825
$655$	0	0
$656$	0	0
$657$	1.00000i	0.0390137i
$658$	− 6.00000i	− 0.233904i
$659$	48.0000	1.86981	0.934907	−	0.354892i	$-0.115482\pi$
$659$	48.0000	1.86981	0.934907	+	0.354892i	$0.115482\pi$
$660$	0	0
$661$	−46.0000	−1.78919	−0.894596	−	0.446875i	$-0.852537\pi$
$661$	−46.0000	−1.78919	−0.894596	+	0.446875i	$0.852537\pi$
$662$	8.00000i	0.310929i
$663$	0	0
$664$	−12.0000	−0.465690
$665$	0	0
$666$	−8.00000	−0.309994
$667$	0	0
$668$	9.00000i	0.348220i
$669$	10.0000	0.386622
$670$	0	0
$671$	30.0000	1.15814
$672$	− 1.00000i	− 0.0385758i
$673$	− 10.0000i	− 0.385472i	−0.981251	−	0.192736i	$-0.938264\pi$
$673$	− 10.0000i	− 0.385472i	0.981251	−	0.192736i	$-0.0617360\pi$
$674$	−22.0000	−0.847408
$675$	0	0
$676$	−9.00000	−0.346154
$677$	33.0000i	1.26829i	0.773213	+	0.634147i	$0.218648\pi$
$677$	33.0000i	1.26829i	−0.773213	+	0.634147i	$0.781352\pi$
$678$	3.00000i	0.115214i
$679$	14.0000	0.537271
$680$	0	0
$681$	−15.0000	−0.574801
$682$	− 3.00000i	− 0.114876i
$683$	− 15.0000i	− 0.573959i	−0.957937	−	0.286980i	$-0.907349\pi$
$683$	− 15.0000i	− 0.573959i	0.957937	−	0.286980i	$-0.0926512\pi$
$684$	−5.00000	−0.191180
$685$	0	0
$686$	−13.0000	−0.496342
$687$	− 5.00000i	− 0.190762i
$688$	11.0000i	0.419371i
$689$	18.0000	0.685745
$690$	0	0
$691$	17.0000	0.646710	0.323355	−	0.946278i	$-0.395189\pi$
$691$	17.0000	0.646710	0.323355	+	0.946278i	$0.395189\pi$
$692$	− 6.00000i	− 0.228086i
$693$	3.00000i	0.113961i
$694$	−36.0000	−1.36654
$695$	0	0
$696$	0	0
$697$	0	0
$698$	4.00000i	0.151402i
$699$	−3.00000	−0.113470
$700$	0	0
$701$	−3.00000	−0.113308	−0.0566542	−	0.998394i	$-0.518043\pi$
$701$	−3.00000	−0.113308	−0.0566542	+	0.998394i	$0.518043\pi$
$702$	2.00000i	0.0754851i
$703$	40.0000i	1.50863i
$704$	3.00000	0.113067
$705$	0	0
$706$	−6.00000	−0.225813
$707$	− 15.0000i	− 0.564133i
$708$	6.00000i	0.225494i
$709$	−23.0000	−0.863783	−0.431892	−	0.901926i	$-0.642154\pi$
$709$	−23.0000	−0.863783	−0.431892	+	0.901926i	$0.642154\pi$
$710$	0	0
$711$	−1.00000	−0.0375029
$712$	− 9.00000i	− 0.337289i
$713$	9.00000i	0.337053i
$714$	0	0
$715$	0	0
$716$	−12.0000	−0.448461
$717$	− 12.0000i	− 0.448148i
$718$	− 9.00000i	− 0.335877i
$719$	−18.0000	−0.671287	−0.335643	−	0.941989i	$-0.608954\pi$
$719$	−18.0000	−0.671287	−0.335643	+	0.941989i	$0.608954\pi$
$720$	0	0
$721$	−8.00000	−0.297936
$722$	6.00000i	0.223297i
$723$	− 22.0000i	− 0.818189i
$724$	−5.00000	−0.185824
$725$	0	0
$726$	2.00000	0.0742270
$727$	− 23.0000i	− 0.853023i	−0.904482	−	0.426511i	$-0.859742\pi$
$727$	− 23.0000i	− 0.853023i	0.904482	−	0.426511i	$-0.140258\pi$
$728$	2.00000i	0.0741249i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	0	0
$732$	10.0000i	0.369611i
$733$	26.0000i	0.960332i	0.877178	+	0.480166i	$0.159424\pi$
$733$	26.0000i	0.960332i	−0.877178	+	0.480166i	$0.840576\pi$
$734$	26.0000	0.959678
$735$	0	0
$736$	−9.00000	−0.331744
$737$	42.0000i	1.54709i
$738$	0	0
$739$	−32.0000	−1.17714	−0.588570	−	0.808447i	$-0.700309\pi$
$739$	−32.0000	−1.17714	−0.588570	+	0.808447i	$0.700309\pi$
$740$	0	0
$741$	10.0000	0.367359
$742$	9.00000i	0.330400i
$743$	− 9.00000i	− 0.330178i	−0.986279	−	0.165089i	$-0.947209\pi$
$743$	− 9.00000i	− 0.330178i	0.986279	−	0.165089i	$-0.0527911\pi$
$744$	1.00000	0.0366618
$745$	0	0
$746$	19.0000	0.695639
$747$	12.0000i	0.439057i
$748$	0	0
$749$	−3.00000	−0.109618
$750$	0	0
$751$	26.0000	0.948753	0.474377	−	0.880322i	$-0.342673\pi$
$751$	26.0000	0.948753	0.474377	+	0.880322i	$0.342673\pi$
$752$	6.00000i	0.218797i
$753$	12.0000i	0.437304i
$754$	0	0
$755$	0	0
$756$	−1.00000	−0.0363696
$757$	− 2.00000i	− 0.0726912i	−0.999339	−	0.0363456i	$-0.988428\pi$
$757$	− 2.00000i	− 0.0726912i	0.999339	−	0.0363456i	$-0.0115717\pi$
$758$	1.00000i	0.0363216i
$759$	27.0000	0.980038
$760$	0	0
$761$	−21.0000	−0.761249	−0.380625	−	0.924730i	$-0.624291\pi$
$761$	−21.0000	−0.761249	−0.380625	+	0.924730i	$0.624291\pi$
$762$	8.00000i	0.289809i
$763$	− 8.00000i	− 0.289619i
$764$	0	0
$765$	0	0
$766$	−36.0000	−1.30073
$767$	− 12.0000i	− 0.433295i
$768$	1.00000i	0.0360844i
$769$	31.0000	1.11789	0.558944	−	0.829205i	$-0.311207\pi$
$769$	31.0000	1.11789	0.558944	+	0.829205i	$0.311207\pi$
$770$	0	0
$771$	−27.0000	−0.972381
$772$	10.0000i	0.359908i
$773$	− 27.0000i	− 0.971123i	−0.874203	−	0.485561i	$-0.838615\pi$
$773$	− 27.0000i	− 0.971123i	0.874203	−	0.485561i	$-0.161385\pi$
$774$	11.0000	0.395387
$775$	0	0
$776$	−14.0000	−0.502571
$777$	8.00000i	0.286998i
$778$	30.0000i	1.07555i
$779$	0	0
$780$	0	0
$781$	−27.0000	−0.966136
$782$	0	0
$783$	0	0
$784$	6.00000	0.214286
$785$	0	0
$786$	6.00000	0.214013
$787$	25.0000i	0.891154i	0.895244	+	0.445577i	$0.147001\pi$
$787$	25.0000i	0.891154i	−0.895244	+	0.445577i	$0.852999\pi$
$788$	− 6.00000i	− 0.213741i
$789$	0	0
$790$	0	0
$791$	3.00000	0.106668
$792$	− 3.00000i	− 0.106600i
$793$	− 20.0000i	− 0.710221i
$794$	−7.00000	−0.248421
$795$	0	0
$796$	−25.0000	−0.886102
$797$	− 18.0000i	− 0.637593i	−0.947823	−	0.318796i	$-0.896721\pi$
$797$	− 18.0000i	− 0.637593i	0.947823	−	0.318796i	$-0.103279\pi$
$798$	5.00000i	0.176998i
$799$	0	0
$800$	0	0
$801$	−9.00000	−0.317999
$802$	15.0000i	0.529668i
$803$	3.00000i	0.105868i
$804$	−14.0000	−0.493742
$805$	0	0
$806$	−2.00000	−0.0704470
$807$	− 6.00000i	− 0.211210i
$808$	15.0000i	0.527698i
$809$	−39.0000	−1.37117	−0.685583	−	0.727994i	$-0.740453\pi$
$809$	−39.0000	−1.37117	−0.685583	+	0.727994i	$0.740453\pi$
$810$	0	0
$811$	−43.0000	−1.50993	−0.754967	−	0.655763i	$-0.772347\pi$
$811$	−43.0000	−1.50993	−0.754967	+	0.655763i	$0.772347\pi$
$812$	0	0
$813$	5.00000i	0.175358i
$814$	−24.0000	−0.841200
$815$	0	0
$816$	0	0
$817$	− 55.0000i	− 1.92421i
$818$	28.0000i	0.978997i
$819$	2.00000	0.0698857
$820$	0	0
$821$	24.0000	0.837606	0.418803	−	0.908077i	$-0.362450\pi$
$821$	24.0000	0.837606	0.418803	+	0.908077i	$0.362450\pi$
$822$	18.0000i	0.627822i
$823$	− 22.0000i	− 0.766872i	−0.923567	−	0.383436i	$-0.874741\pi$
$823$	− 22.0000i	− 0.766872i	0.923567	−	0.383436i	$-0.125259\pi$
$824$	8.00000	0.278693
$825$	0	0
$826$	6.00000	0.208767
$827$	36.0000i	1.25184i	0.779886	+	0.625921i	$0.215277\pi$
$827$	36.0000i	1.25184i	−0.779886	+	0.625921i	$0.784723\pi$
$828$	9.00000i	0.312772i
$829$	−29.0000	−1.00721	−0.503606	−	0.863934i	$-0.667994\pi$
$829$	−29.0000	−1.00721	−0.503606	+	0.863934i	$0.667994\pi$
$830$	0	0
$831$	8.00000	0.277517
$832$	− 2.00000i	− 0.0693375i
$833$	0	0
$834$	14.0000	0.484780
$835$	0	0
$836$	−15.0000	−0.518786
$837$	− 1.00000i	− 0.0345651i
$838$	24.0000i	0.829066i
$839$	−21.0000	−0.725001	−0.362500	−	0.931984i	$-0.618077\pi$
$839$	−21.0000	−0.725001	−0.362500	+	0.931984i	$0.618077\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	− 10.0000i	− 0.344623i
$843$	− 24.0000i	− 0.826604i
$844$	19.0000	0.654007
$845$	0	0
$846$	6.00000	0.206284
$847$	− 2.00000i	− 0.0687208i
$848$	− 9.00000i	− 0.309061i
$849$	4.00000	0.137280
$850$	0	0
$851$	72.0000	2.46813
$852$	− 9.00000i	− 0.308335i
$853$	− 19.0000i	− 0.650548i	−0.945620	−	0.325274i	$-0.894544\pi$
$853$	− 19.0000i	− 0.650548i	0.945620	−	0.325274i	$-0.105456\pi$
$854$	10.0000	0.342193
$855$	0	0
$856$	3.00000	0.102538
$857$	6.00000i	0.204956i	0.994735	+	0.102478i	$0.0326771\pi$
$857$	6.00000i	0.204956i	−0.994735	+	0.102478i	$0.967323\pi$
$858$	6.00000i	0.204837i
$859$	−8.00000	−0.272956	−0.136478	−	0.990643i	$-0.543578\pi$
$859$	−8.00000	−0.272956	−0.136478	+	0.990643i	$0.543578\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	15.0000i	0.510606i	0.966861	+	0.255303i	$0.0821752\pi$
$863$	15.0000i	0.510606i	−0.966861	+	0.255303i	$0.917825\pi$
$864$	1.00000	0.0340207
$865$	0	0
$866$	−5.00000	−0.169907
$867$	17.0000i	0.577350i
$868$	− 1.00000i	− 0.0339422i
$869$	−3.00000	−0.101768
$870$	0	0
$871$	28.0000	0.948744
$872$	8.00000i	0.270914i
$873$	14.0000i	0.473828i
$874$	45.0000	1.52215
$875$	0	0
$876$	−1.00000	−0.0337869
$877$	22.0000i	0.742887i	0.928456	+	0.371444i	$0.121137\pi$
$877$	22.0000i	0.742887i	−0.928456	+	0.371444i	$0.878863\pi$
$878$	− 14.0000i	− 0.472477i
$879$	−18.0000	−0.607125
$880$	0	0
$881$	−18.0000	−0.606435	−0.303218	−	0.952921i	$-0.598061\pi$
$881$	−18.0000	−0.606435	−0.303218	+	0.952921i	$0.598061\pi$
$882$	− 6.00000i	− 0.202031i
$883$	− 31.0000i	− 1.04323i	−0.853180	−	0.521617i	$-0.825329\pi$
$883$	− 31.0000i	− 1.04323i	0.853180	−	0.521617i	$-0.174671\pi$
$884$	0	0
$885$	0	0
$886$	−27.0000	−0.907083
$887$	18.0000i	0.604381i	0.953248	+	0.302190i	$0.0977178\pi$
$887$	18.0000i	0.604381i	−0.953248	+	0.302190i	$0.902282\pi$
$888$	− 8.00000i	− 0.268462i
$889$	8.00000	0.268311
$890$	0	0
$891$	−3.00000	−0.100504
$892$	10.0000i	0.334825i
$893$	− 30.0000i	− 1.00391i
$894$	3.00000	0.100335
$895$	0	0
$896$	1.00000	0.0334077
$897$	− 18.0000i	− 0.601003i
$898$	− 6.00000i	− 0.200223i
$899$	0	0
$900$	0	0
$901$	0	0
$902$	0	0
$903$	− 11.0000i	− 0.366057i
$904$	−3.00000	−0.0997785
$905$	0	0
$906$	16.0000	0.531564
$907$	− 8.00000i	− 0.265636i	−0.991140	−	0.132818i	$-0.957597\pi$
$907$	− 8.00000i	− 0.265636i	0.991140	−	0.132818i	$-0.0424025\pi$
$908$	− 15.0000i	− 0.497792i
$909$	15.0000	0.497519
$910$	0	0
$911$	12.0000	0.397578	0.198789	−	0.980042i	$-0.436299\pi$
$911$	12.0000	0.397578	0.198789	+	0.980042i	$0.436299\pi$
$912$	− 5.00000i	− 0.165567i
$913$	36.0000i	1.19143i
$914$	38.0000	1.25693
$915$	0	0
$916$	5.00000	0.165205
$917$	− 6.00000i	− 0.198137i
$918$	0	0
$919$	−26.0000	−0.857661	−0.428830	−	0.903385i	$-0.641074\pi$
$919$	−26.0000	−0.857661	−0.428830	+	0.903385i	$0.641074\pi$
$920$	0	0
$921$	32.0000	1.05444
$922$	0	0
$923$	18.0000i	0.592477i
$924$	−3.00000	−0.0986928
$925$	0	0
$926$	−32.0000	−1.05159
$927$	− 8.00000i	− 0.262754i
$928$	0	0
$929$	15.0000	0.492134	0.246067	−	0.969253i	$-0.420862\pi$
$929$	15.0000	0.492134	0.246067	+	0.969253i	$0.420862\pi$
$930$	0	0
$931$	−30.0000	−0.983210
$932$	− 3.00000i	− 0.0982683i
$933$	0	0
$934$	0	0
$935$	0	0
$936$	−2.00000	−0.0653720
$937$	− 20.0000i	− 0.653372i	−0.945133	−	0.326686i	$-0.894068\pi$
$937$	− 20.0000i	− 0.653372i	0.945133	−	0.326686i	$-0.105932\pi$
$938$	14.0000i	0.457116i
$939$	22.0000	0.717943
$940$	0	0
$941$	−12.0000	−0.391189	−0.195594	−	0.980685i	$-0.562664\pi$
$941$	−12.0000	−0.391189	−0.195594	+	0.980685i	$0.562664\pi$
$942$	− 7.00000i	− 0.228072i
$943$	0	0
$944$	−6.00000	−0.195283
$945$	0	0
$946$	33.0000	1.07292
$947$	− 54.0000i	− 1.75476i	−0.479792	−	0.877382i	$-0.659288\pi$
$947$	− 54.0000i	− 1.75476i	0.479792	−	0.877382i	$-0.340712\pi$
$948$	− 1.00000i	− 0.0324785i
$949$	2.00000	0.0649227
$950$	0	0
$951$	24.0000	0.778253
$952$	0	0
$953$	− 12.0000i	− 0.388718i	−0.980930	−	0.194359i	$-0.937737\pi$
$953$	− 12.0000i	− 0.388718i	0.980930	−	0.194359i	$-0.0622627\pi$
$954$	−9.00000	−0.291386
$955$	0	0
$956$	12.0000	0.388108
$957$	0	0
$958$	− 21.0000i	− 0.678479i
$959$	18.0000	0.581250
$960$	0	0
$961$	1.00000	0.0322581
$962$	16.0000i	0.515861i
$963$	− 3.00000i	− 0.0966736i
$964$	22.0000	0.708572
$965$	0	0
$966$	9.00000	0.289570
$967$	4.00000i	0.128631i	0.997930	+	0.0643157i	$0.0204865\pi$
$967$	4.00000i	0.128631i	−0.997930	+	0.0643157i	$0.979514\pi$
$968$	2.00000i	0.0642824i
$969$	0	0
$970$	0	0
$971$	−42.0000	−1.34784	−0.673922	−	0.738802i	$-0.735392\pi$
$971$	−42.0000	−1.34784	−0.673922	+	0.738802i	$0.735392\pi$
$972$	− 1.00000i	− 0.0320750i
$973$	− 14.0000i	− 0.448819i
$974$	−40.0000	−1.28168
$975$	0	0
$976$	−10.0000	−0.320092
$977$	18.0000i	0.575871i	0.957650	+	0.287936i	$0.0929689\pi$
$977$	18.0000i	0.575871i	−0.957650	+	0.287936i	$0.907031\pi$
$978$	22.0000i	0.703482i
$979$	−27.0000	−0.862924
$980$	0	0
$981$	8.00000	0.255420
$982$	21.0000i	0.670137i
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 6.00000i	− 0.190982i
$988$	10.0000i	0.318142i
$989$	−99.0000	−3.14802
$990$	0	0
$991$	23.0000	0.730619	0.365310	−	0.930886i	$-0.380963\pi$
$991$	23.0000	0.730619	0.365310	+	0.930886i	$0.380963\pi$
$992$	1.00000i	0.0317500i
$993$	8.00000i	0.253872i
$994$	−9.00000	−0.285463
$995$	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	$-0.135070\pi$
$998$	− 32.0000i	− 1.01294i
$999$	−8.00000	−0.253109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4650.2.d.d.3349.2		2
5.2	odd	4		930.2.a.i.1.1	✓	1
5.3	odd	4		4650.2.a.bd.1.1		1
5.4	even	2	inner	4650.2.d.d.3349.1		2
15.2	even	4		2790.2.a.r.1.1		1
20.7	even	4		7440.2.a.k.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
930.2.a.i.1.1	✓	1	5.2	odd	4
2790.2.a.r.1.1		1	15.2	even	4
4650.2.a.bd.1.1		1	5.3	odd	4
4650.2.d.d.3349.1		2	5.4	even	2	inner
4650.2.d.d.3349.2		2	1.1	even	1	trivial
7440.2.a.k.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 $\F_{q}$
Belyi maps

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)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +1.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} -3.00000 q^{11} -1.00000i q^{12} +2.00000i q^{13} -1.00000 q^{14} +1.00000 q^{16} -1.00000i q^{18} -5.00000 q^{19} -1.00000 q^{21} -3.00000i q^{22} +9.00000i q^{23} +1.00000 q^{24} -2.00000 q^{26} -1.00000i q^{27} -1.00000i q^{28} +1.00000 q^{31} +1.00000i q^{32} -3.00000i q^{33} +1.00000 q^{36} -8.00000i q^{37} -5.00000i q^{38} -2.00000 q^{39} -1.00000i q^{42} +11.0000i q^{43} +3.00000 q^{44} -9.00000 q^{46} +6.00000i q^{47} +1.00000i q^{48} +6.00000 q^{49} -2.00000i q^{52} -9.00000i q^{53} +1.00000 q^{54} +1.00000 q^{56} -5.00000i q^{57} -6.00000 q^{59} -10.0000 q^{61} +1.00000i q^{62} -1.00000i q^{63} -1.00000 q^{64} +3.00000 q^{66} -14.0000i q^{67} -9.00000 q^{69} +9.00000 q^{71} +1.00000i q^{72} -1.00000i q^{73} +8.00000 q^{74} +5.00000 q^{76} -3.00000i q^{77} -2.00000i q^{78} +1.00000 q^{79} +1.00000 q^{81} -12.0000i q^{83} +1.00000 q^{84} -11.0000 q^{86} +3.00000i q^{88} +9.00000 q^{89} -2.00000 q^{91} -9.00000i q^{92} +1.00000i q^{93} -6.00000 q^{94} -1.00000 q^{96} -14.0000i q^{97} +6.00000i q^{98} +3.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9} - 6 q^{11} - 2 q^{14} + 2 q^{16} - 10 q^{19} - 2 q^{21} + 2 q^{24} - 4 q^{26} + 2 q^{31} + 2 q^{36} - 4 q^{39} + 6 q^{44} - 18 q^{46} + 12 q^{49} + 2 q^{54} + 2 q^{56}+ \cdots + 6 q^{99}+O(q^{100}) \) 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9 - 6 * q^11 - 2 * q^14 + 2 * q^16 - 10 * q^19 - 2 * q^21 + 2 * q^24 - 4 * q^26 + 2 * q^31 + 2 * q^36 - 4 * q^39 + 6 * q^44 - 18 * q^46 + 12 * q^49 + 2 * q^54 + 2 * q^56 - 12 * q^59 - 20 * q^61 - 2 * q^64 + 6 * q^66 - 18 * q^69 + 18 * q^71 + 16 * q^74 + 10 * q^76 + 2 * q^79 + 2 * q^81 + 2 * q^84 - 22 * q^86 + 18 * q^89 - 4 * q^91 - 12 * q^94 - 2 * q^96 + 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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