LMFDB - Embedded newform 4650.2.d.bb.3349.1

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:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

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

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -1.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -1.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +5.00000 q^{11} -1.00000i q^{12} +1.00000i q^{13} -1.00000 q^{14} +1.00000 q^{16} +1.00000i q^{18} +1.00000 q^{21} -5.00000i q^{22} +6.00000i q^{23} -1.00000 q^{24} +1.00000 q^{26} -1.00000i q^{27} +1.00000i q^{28} +2.00000 q^{29} +1.00000 q^{31} -1.00000i q^{32} +5.00000i q^{33} +1.00000 q^{36} +7.00000i q^{37} -1.00000 q^{39} -9.00000 q^{41} -1.00000i q^{42} +11.0000i q^{43} -5.00000 q^{44} +6.00000 q^{46} -3.00000i q^{47} +1.00000i q^{48} +6.00000 q^{49} -1.00000i q^{52} -13.0000i q^{53} -1.00000 q^{54} +1.00000 q^{56} -2.00000i q^{58} +1.00000 q^{61} -1.00000i q^{62} +1.00000i q^{63} -1.00000 q^{64} +5.00000 q^{66} -4.00000i q^{67} -6.00000 q^{69} +1.00000 q^{71} -1.00000i q^{72} -4.00000i q^{73} +7.00000 q^{74} -5.00000i q^{77} +1.00000i q^{78} -10.0000 q^{79} +1.00000 q^{81} +9.00000i q^{82} +1.00000i q^{83} -1.00000 q^{84} +11.0000 q^{86} +2.00000i q^{87} +5.00000i q^{88} +2.00000 q^{89} +1.00000 q^{91} -6.00000i q^{92} +1.00000i q^{93} -3.00000 q^{94} +1.00000 q^{96} -2.00000i q^{97} -6.00000i q^{98} -5.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 10 q^{11} - 2 q^{14} + 2 q^{16} + 2 q^{21} - 2 q^{24} + 2 q^{26} + 4 q^{29} + 2 q^{31} + 2 q^{36} - 2 q^{39} - 18 q^{41} - 10 q^{44} + 12 q^{46} + 12 q^{49} - 2 q^{54} + 2 q^{56} + 2 q^{61} - 2 q^{64} + 10 q^{66} - 12 q^{69} + 2 q^{71} + 14 q^{74} - 20 q^{79} + 2 q^{81} - 2 q^{84} + 22 q^{86} + 4 q^{89} + 2 q^{91} - 6 q^{94} + 2 q^{96} - 10 q^{99}+O(q^{100}) $ 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9 + 10 * q^11 - 2 * q^14 + 2 * q^16 + 2 * q^21 - 2 * q^24 + 2 * q^26 + 4 * q^29 + 2 * q^31 + 2 * q^36 - 2 * q^39 - 18 * q^41 - 10 * q^44 + 12 * q^46 + 12 * q^49 - 2 * q^54 + 2 * q^56 + 2 * q^61 - 2 * q^64 + 10 * q^66 - 12 * q^69 + 2 * q^71 + 14 * q^74 - 20 * q^79 + 2 * q^81 - 2 * q^84 + 22 * q^86 + 4 * q^89 + 2 * q^91 - 6 * q^94 + 2 * q^96 - 10 * 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
$7$	− 1.00000i	− 0.377964i	−0.981981	−	0.188982i	$-0.939481\pi$
$7$	− 1.00000i	− 0.377964i	0.981981	−	0.188982i	$-0.0605189\pi$
$8$	1.00000i	0.353553i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	5.00000	1.50756	0.753778	−	0.657129i	$-0.228229\pi$
$11$	5.00000	1.50756	0.753778	+	0.657129i	$0.228229\pi$
$12$	− 1.00000i	− 0.288675i
$13$	1.00000i	0.277350i	0.990338	+	0.138675i	$0.0442844\pi$
$13$	1.00000i	0.277350i	−0.990338	+	0.138675i	$0.955716\pi$
$14$	−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
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	− 5.00000i	− 1.06600i
$23$	6.00000i	1.25109i	0.780189	+	0.625543i	$0.215123\pi$
$23$	6.00000i	1.25109i	−0.780189	+	0.625543i	$0.784877\pi$
$24$	−1.00000	−0.204124
$25$	0	0
$26$	1.00000	0.196116
$27$	− 1.00000i	− 0.192450i
$28$	1.00000i	0.188982i
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	− 1.00000i	− 0.176777i
$33$	5.00000i	0.870388i
$34$	0	0
$35$	0	0
$36$	1.00000	0.166667
$37$	7.00000i	1.15079i	0.817875	+	0.575396i	$0.195152\pi$
$37$	7.00000i	1.15079i	−0.817875	+	0.575396i	$0.804848\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	−9.00000	−1.40556	−0.702782	−	0.711405i	$-0.748059\pi$
$41$	−9.00000	−1.40556	−0.702782	+	0.711405i	$0.748059\pi$
$42$	− 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	$0.683292\pi$
$44$	−5.00000	−0.753778
$45$	0	0
$46$	6.00000	0.884652
$47$	− 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
$49$	6.00000	0.857143
$50$	0	0
$51$	0	0
$52$	− 1.00000i	− 0.138675i
$53$	− 13.0000i	− 1.78569i	−0.450367	−	0.892844i	$-0.648707\pi$
$53$	− 13.0000i	− 1.78569i	0.450367	−	0.892844i	$-0.351293\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	1.00000	0.133631
$57$	0	0
$58$	− 2.00000i	− 0.262613i
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	1.00000	0.128037	0.0640184	−	0.997949i	$-0.479608\pi$
$61$	1.00000	0.128037	0.0640184	+	0.997949i	$0.479608\pi$
$62$	− 1.00000i	− 0.127000i
$63$	1.00000i	0.125988i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	5.00000	0.615457
$67$	− 4.00000i	− 0.488678i	−0.969690	−	0.244339i	$-0.921429\pi$
$67$	− 4.00000i	− 0.488678i	0.969690	−	0.244339i	$-0.0785709\pi$
$68$	0	0
$69$	−6.00000	−0.722315
$70$	0	0
$71$	1.00000	0.118678	0.0593391	−	0.998238i	$-0.481101\pi$
$71$	1.00000	0.118678	0.0593391	+	0.998238i	$0.481101\pi$
$72$	− 1.00000i	− 0.117851i
$73$	− 4.00000i	− 0.468165i	−0.972217	−	0.234082i	$-0.924791\pi$
$73$	− 4.00000i	− 0.468165i	0.972217	−	0.234082i	$-0.0752085\pi$
$74$	7.00000	0.813733
$75$	0	0
$76$	0	0
$77$	− 5.00000i	− 0.569803i
$78$	1.00000i	0.113228i
$79$	−10.0000	−1.12509	−0.562544	−	0.826767i	$-0.690177\pi$
$79$	−10.0000	−1.12509	−0.562544	+	0.826767i	$0.690177\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	9.00000i	0.993884i
$83$	1.00000i	0.109764i	0.998493	+	0.0548821i	$0.0174783\pi$
$83$	1.00000i	0.109764i	−0.998493	+	0.0548821i	$0.982522\pi$
$84$	−1.00000	−0.109109
$85$	0	0
$86$	11.0000	1.18616
$87$	2.00000i	0.214423i
$88$	5.00000i	0.533002i
$89$	2.00000	0.212000	0.106000	−	0.994366i	$-0.466196\pi$
$89$	2.00000	0.212000	0.106000	+	0.994366i	$0.466196\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	− 6.00000i	− 0.625543i
$93$	1.00000i	0.103695i
$94$	−3.00000	−0.309426
$95$	0	0
$96$	1.00000	0.102062
$97$	− 2.00000i	− 0.203069i	−0.994832	−	0.101535i	$-0.967625\pi$
$97$	− 2.00000i	− 0.203069i	0.994832	−	0.101535i	$-0.0323753\pi$
$98$	− 6.00000i	− 0.606092i
$99$	−5.00000	−0.502519
$100$	0	0
$101$	8.00000	0.796030	0.398015	−	0.917379i	$-0.369699\pi$
$101$	8.00000	0.796030	0.398015	+	0.917379i	$0.369699\pi$
$102$	0	0
$103$	13.0000i	1.28093i	0.767988	+	0.640464i	$0.221258\pi$
$103$	13.0000i	1.28093i	−0.767988	+	0.640464i	$0.778742\pi$
$104$	−1.00000	−0.0980581
$105$	0	0
$106$	−13.0000	−1.26267
$107$	14.0000i	1.35343i	0.736245	+	0.676716i	$0.236597\pi$
$107$	14.0000i	1.35343i	−0.736245	+	0.676716i	$0.763403\pi$
$108$	1.00000i	0.0962250i
$109$	12.0000	1.14939	0.574696	−	0.818367i	$-0.305120\pi$
$109$	12.0000	1.14939	0.574696	+	0.818367i	$0.305120\pi$
$110$	0	0
$111$	−7.00000	−0.664411
$112$	− 1.00000i	− 0.0944911i
$113$	18.0000i	1.69330i	0.532152	+	0.846649i	$0.321383\pi$
$113$	18.0000i	1.69330i	−0.532152	+	0.846649i	$0.678617\pi$
$114$	0	0
$115$	0	0
$116$	−2.00000	−0.185695
$117$	− 1.00000i	− 0.0924500i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	14.0000	1.27273
$122$	− 1.00000i	− 0.0905357i
$123$	− 9.00000i	− 0.811503i
$124$	−1.00000	−0.0898027
$125$	0	0
$126$	1.00000	0.0890871
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	1.00000i	0.0883883i
$129$	−11.0000	−0.968496
$130$	0	0
$131$	14.0000	1.22319	0.611593	−	0.791173i	$-0.290529\pi$
$131$	14.0000	1.22319	0.611593	+	0.791173i	$0.290529\pi$
$132$	− 5.00000i	− 0.435194i
$133$	0	0
$134$	−4.00000	−0.345547
$135$	0	0
$136$	0	0
$137$	18.0000i	1.53784i	0.639343	+	0.768922i	$0.279207\pi$
$137$	18.0000i	1.53784i	−0.639343	+	0.768922i	$0.720793\pi$
$138$	6.00000i	0.510754i
$139$	9.00000	0.763370	0.381685	−	0.924292i	$-0.375344\pi$
$139$	9.00000	0.763370	0.381685	+	0.924292i	$0.375344\pi$
$140$	0	0
$141$	3.00000	0.252646
$142$	− 1.00000i	− 0.0839181i
$143$	5.00000i	0.418121i
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	−4.00000	−0.331042
$147$	6.00000i	0.494872i
$148$	− 7.00000i	− 0.575396i
$149$	−2.00000	−0.163846	−0.0819232	−	0.996639i	$-0.526106\pi$
$149$	−2.00000	−0.163846	−0.0819232	+	0.996639i	$0.526106\pi$
$150$	0	0
$151$	−10.0000	−0.813788	−0.406894	−	0.913475i	$-0.633388\pi$
$151$	−10.0000	−0.813788	−0.406894	+	0.913475i	$0.633388\pi$
$152$	0	0
$153$	0	0
$154$	−5.00000	−0.402911
$155$	0	0
$156$	1.00000	0.0800641
$157$	− 18.0000i	− 1.43656i	−0.695756	−	0.718278i	$-0.744931\pi$
$157$	− 18.0000i	− 1.43656i	0.695756	−	0.718278i	$-0.255069\pi$
$158$	10.0000i	0.795557i
$159$	13.0000	1.03097
$160$	0	0
$161$	6.00000	0.472866
$162$	− 1.00000i	− 0.0785674i
$163$	− 6.00000i	− 0.469956i	−0.972001	−	0.234978i	$-0.924498\pi$
$163$	− 6.00000i	− 0.469956i	0.972001	−	0.234978i	$-0.0755019\pi$
$164$	9.00000	0.702782
$165$	0	0
$166$	1.00000	0.0776151
$167$	− 6.00000i	− 0.464294i	−0.972681	−	0.232147i	$-0.925425\pi$
$167$	− 6.00000i	− 0.464294i	0.972681	−	0.232147i	$-0.0745750\pi$
$168$	1.00000i	0.0771517i
$169$	12.0000	0.923077
$170$	0	0
$171$	0	0
$172$	− 11.0000i	− 0.838742i
$173$	24.0000i	1.82469i	0.409426	+	0.912343i	$0.365729\pi$
$173$	24.0000i	1.82469i	−0.409426	+	0.912343i	$0.634271\pi$
$174$	2.00000	0.151620
$175$	0	0
$176$	5.00000	0.376889
$177$	0	0
$178$	− 2.00000i	− 0.149906i
$179$	1.00000	0.0747435	0.0373718	−	0.999301i	$-0.488101\pi$
$179$	1.00000	0.0747435	0.0373718	+	0.999301i	$0.488101\pi$
$180$	0	0
$181$	−3.00000	−0.222988	−0.111494	−	0.993765i	$-0.535564\pi$
$181$	−3.00000	−0.222988	−0.111494	+	0.993765i	$0.535564\pi$
$182$	− 1.00000i	− 0.0741249i
$183$	1.00000i	0.0739221i
$184$	−6.00000	−0.442326
$185$	0	0
$186$	1.00000	0.0733236
$187$	0	0
$188$	3.00000i	0.218797i
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	8.00000	0.578860	0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000	0.578860	0.289430	+	0.957199i	$0.406534\pi$
$192$	− 1.00000i	− 0.0721688i
$193$	− 7.00000i	− 0.503871i	−0.967744	−	0.251936i	$-0.918933\pi$
$193$	− 7.00000i	− 0.503871i	0.967744	−	0.251936i	$-0.0810671\pi$
$194$	−2.00000	−0.143592
$195$	0	0
$196$	−6.00000	−0.428571
$197$	5.00000i	0.356235i	0.984009	+	0.178118i	$0.0570008\pi$
$197$	5.00000i	0.356235i	−0.984009	+	0.178118i	$0.942999\pi$
$198$	5.00000i	0.355335i
$199$	6.00000	0.425329	0.212664	−	0.977125i	$-0.431786\pi$
$199$	6.00000	0.425329	0.212664	+	0.977125i	$0.431786\pi$
$200$	0	0
$201$	4.00000	0.282138
$202$	− 8.00000i	− 0.562878i
$203$	− 2.00000i	− 0.140372i
$204$	0	0
$205$	0	0
$206$	13.0000	0.905753
$207$	− 6.00000i	− 0.417029i
$208$	1.00000i	0.0693375i
$209$	0	0
$210$	0	0
$211$	−8.00000	−0.550743	−0.275371	−	0.961338i	$-0.588801\pi$
$211$	−8.00000	−0.550743	−0.275371	+	0.961338i	$0.588801\pi$
$212$	13.0000i	0.892844i
$213$	1.00000i	0.0685189i
$214$	14.0000	0.957020
$215$	0	0
$216$	1.00000	0.0680414
$217$	− 1.00000i	− 0.0678844i
$218$	− 12.0000i	− 0.812743i
$219$	4.00000	0.270295
$220$	0	0
$221$	0	0
$222$	7.00000i	0.469809i
$223$	26.0000i	1.74109i	0.492090	+	0.870544i	$0.336233\pi$
$223$	26.0000i	1.74109i	−0.492090	+	0.870544i	$0.663767\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	18.0000	1.19734
$227$	20.0000i	1.32745i	0.747978	+	0.663723i	$0.231025\pi$
$227$	20.0000i	1.32745i	−0.747978	+	0.663723i	$0.768975\pi$
$228$	0	0
$229$	14.0000	0.925146	0.462573	−	0.886581i	$-0.346926\pi$
$229$	14.0000	0.925146	0.462573	+	0.886581i	$0.346926\pi$
$230$	0	0
$231$	5.00000	0.328976
$232$	2.00000i	0.131306i
$233$	9.00000i	0.589610i	0.955557	+	0.294805i	$0.0952546\pi$
$233$	9.00000i	0.589610i	−0.955557	+	0.294805i	$0.904745\pi$
$234$	−1.00000	−0.0653720
$235$	0	0
$236$	0	0
$237$	− 10.0000i	− 0.649570i
$238$	0	0
$239$	16.0000	1.03495	0.517477	−	0.855697i	$-0.326871\pi$
$239$	16.0000	1.03495	0.517477	+	0.855697i	$0.326871\pi$
$240$	0	0
$241$	−26.0000	−1.67481	−0.837404	−	0.546585i	$-0.815928\pi$
$241$	−26.0000	−1.67481	−0.837404	+	0.546585i	$0.815928\pi$
$242$	− 14.0000i	− 0.899954i
$243$	1.00000i	0.0641500i
$244$	−1.00000	−0.0640184
$245$	0	0
$246$	−9.00000	−0.573819
$247$	0	0
$248$	1.00000i	0.0635001i
$249$	−1.00000	−0.0633724
$250$	0	0
$251$	17.0000	1.07303	0.536515	−	0.843891i	$-0.319740\pi$
$251$	17.0000	1.07303	0.536515	+	0.843891i	$0.319740\pi$
$252$	− 1.00000i	− 0.0629941i
$253$	30.0000i	1.88608i
$254$	0	0
$255$	0	0
$256$	1.00000	0.0625000
$257$	7.00000i	0.436648i	0.975876	+	0.218324i	$0.0700590\pi$
$257$	7.00000i	0.436648i	−0.975876	+	0.218324i	$0.929941\pi$
$258$	11.0000i	0.684830i
$259$	7.00000	0.434959
$260$	0	0
$261$	−2.00000	−0.123797
$262$	− 14.0000i	− 0.864923i
$263$	12.0000i	0.739952i	0.929041	+	0.369976i	$0.120634\pi$
$263$	12.0000i	0.739952i	−0.929041	+	0.369976i	$0.879366\pi$
$264$	−5.00000	−0.307729
$265$	0	0
$266$	0	0
$267$	2.00000i	0.122398i
$268$	4.00000i	0.244339i
$269$	−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
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	0	0
$273$	1.00000i	0.0605228i
$274$	18.0000	1.08742
$275$	0	0
$276$	6.00000	0.361158
$277$	22.0000i	1.32185i	0.750451	+	0.660926i	$0.229836\pi$
$277$	22.0000i	1.32185i	−0.750451	+	0.660926i	$0.770164\pi$
$278$	− 9.00000i	− 0.539784i
$279$	−1.00000	−0.0598684
$280$	0	0
$281$	27.0000	1.61068	0.805342	−	0.592810i	$-0.201981\pi$
$281$	27.0000	1.61068	0.805342	+	0.592810i	$0.201981\pi$
$282$	− 3.00000i	− 0.178647i
$283$	16.0000i	0.951101i	0.879688	+	0.475551i	$0.157751\pi$
$283$	16.0000i	0.951101i	−0.879688	+	0.475551i	$0.842249\pi$
$284$	−1.00000	−0.0593391
$285$	0	0
$286$	5.00000	0.295656
$287$	9.00000i	0.531253i
$288$	1.00000i	0.0589256i
$289$	17.0000	1.00000
$290$	0	0
$291$	2.00000	0.117242
$292$	4.00000i	0.234082i
$293$	− 14.0000i	− 0.817889i	−0.912559	−	0.408944i	$-0.865897\pi$
$293$	− 14.0000i	− 0.817889i	0.912559	−	0.408944i	$-0.134103\pi$
$294$	6.00000	0.349927
$295$	0	0
$296$	−7.00000	−0.406867
$297$	− 5.00000i	− 0.290129i
$298$	2.00000i	0.115857i
$299$	−6.00000	−0.346989
$300$	0	0
$301$	11.0000	0.634029
$302$	10.0000i	0.575435i
$303$	8.00000i	0.459588i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 22.0000i	− 1.25561i	−0.778372	−	0.627803i	$-0.783954\pi$
$307$	− 22.0000i	− 1.25561i	0.778372	−	0.627803i	$-0.216046\pi$
$308$	5.00000i	0.284901i
$309$	−13.0000	−0.739544
$310$	0	0
$311$	−3.00000	−0.170114	−0.0850572	−	0.996376i	$-0.527107\pi$
$311$	−3.00000	−0.170114	−0.0850572	+	0.996376i	$0.527107\pi$
$312$	− 1.00000i	− 0.0566139i
$313$	− 14.0000i	− 0.791327i	−0.918396	−	0.395663i	$-0.870515\pi$
$313$	− 14.0000i	− 0.791327i	0.918396	−	0.395663i	$-0.129485\pi$
$314$	−18.0000	−1.01580
$315$	0	0
$316$	10.0000	0.562544
$317$	− 4.00000i	− 0.224662i	−0.993671	−	0.112331i	$-0.964168\pi$
$317$	− 4.00000i	− 0.224662i	0.993671	−	0.112331i	$-0.0358318\pi$
$318$	− 13.0000i	− 0.729004i
$319$	10.0000	0.559893
$320$	0	0
$321$	−14.0000	−0.781404
$322$	− 6.00000i	− 0.334367i
$323$	0	0
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	−6.00000	−0.332309
$327$	12.0000i	0.663602i
$328$	− 9.00000i	− 0.496942i
$329$	−3.00000	−0.165395
$330$	0	0
$331$	−19.0000	−1.04433	−0.522167	−	0.852843i	$-0.674876\pi$
$331$	−19.0000	−1.04433	−0.522167	+	0.852843i	$0.674876\pi$
$332$	− 1.00000i	− 0.0548821i
$333$	− 7.00000i	− 0.383598i
$334$	−6.00000	−0.328305
$335$	0	0
$336$	1.00000	0.0545545
$337$	8.00000i	0.435788i	0.975972	+	0.217894i	$0.0699187\pi$
$337$	8.00000i	0.435788i	−0.975972	+	0.217894i	$0.930081\pi$
$338$	− 12.0000i	− 0.652714i
$339$	−18.0000	−0.977626
$340$	0	0
$341$	5.00000	0.270765
$342$	0	0
$343$	− 13.0000i	− 0.701934i
$344$	−11.0000	−0.593080
$345$	0	0
$346$	24.0000	1.29025
$347$	− 17.0000i	− 0.912608i	−0.889824	−	0.456304i	$-0.849173\pi$
$347$	− 17.0000i	− 0.912608i	0.889824	−	0.456304i	$-0.150827\pi$
$348$	− 2.00000i	− 0.107211i
$349$	16.0000	0.856460	0.428230	−	0.903670i	$-0.359137\pi$
$349$	16.0000	0.856460	0.428230	+	0.903670i	$0.359137\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	− 5.00000i	− 0.266501i
$353$	26.0000i	1.38384i	0.721974	+	0.691920i	$0.243235\pi$
$353$	26.0000i	1.38384i	−0.721974	+	0.691920i	$0.756765\pi$
$354$	0	0
$355$	0	0
$356$	−2.00000	−0.106000
$357$	0	0
$358$	− 1.00000i	− 0.0528516i
$359$	24.0000	1.26667	0.633336	−	0.773877i	$-0.281685\pi$
$359$	24.0000	1.26667	0.633336	+	0.773877i	$0.281685\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	3.00000i	0.157676i
$363$	14.0000i	0.734809i
$364$	−1.00000	−0.0524142
$365$	0	0
$366$	1.00000	0.0522708
$367$	− 18.0000i	− 0.939592i	−0.882775	−	0.469796i	$-0.844327\pi$
$367$	− 18.0000i	− 0.939592i	0.882775	−	0.469796i	$-0.155673\pi$
$368$	6.00000i	0.312772i
$369$	9.00000	0.468521
$370$	0	0
$371$	−13.0000	−0.674926
$372$	− 1.00000i	− 0.0518476i
$373$	− 24.0000i	− 1.24267i	−0.783544	−	0.621336i	$-0.786590\pi$
$373$	− 24.0000i	− 1.24267i	0.783544	−	0.621336i	$-0.213410\pi$
$374$	0	0
$375$	0	0
$376$	3.00000	0.154713
$377$	2.00000i	0.103005i
$378$	1.00000i	0.0514344i
$379$	26.0000	1.33553	0.667765	−	0.744372i	$-0.267251\pi$
$379$	26.0000	1.33553	0.667765	+	0.744372i	$0.267251\pi$
$380$	0	0
$381$	0	0
$382$	− 8.00000i	− 0.409316i
$383$	12.0000i	0.613171i	0.951843	+	0.306586i	$0.0991866\pi$
$383$	12.0000i	0.613171i	−0.951843	+	0.306586i	$0.900813\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	−7.00000	−0.356291
$387$	− 11.0000i	− 0.559161i
$388$	2.00000i	0.101535i
$389$	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
$391$	0	0
$392$	6.00000i	0.303046i
$393$	14.0000i	0.706207i
$394$	5.00000	0.251896
$395$	0	0
$396$	5.00000	0.251259
$397$	18.0000i	0.903394i	0.892171	+	0.451697i	$0.149181\pi$
$397$	18.0000i	0.903394i	−0.892171	+	0.451697i	$0.850819\pi$
$398$	− 6.00000i	− 0.300753i
$399$	0	0
$400$	0	0
$401$	−10.0000	−0.499376	−0.249688	−	0.968326i	$-0.580328\pi$
$401$	−10.0000	−0.499376	−0.249688	+	0.968326i	$0.580328\pi$
$402$	− 4.00000i	− 0.199502i
$403$	1.00000i	0.0498135i
$404$	−8.00000	−0.398015
$405$	0	0
$406$	−2.00000	−0.0992583
$407$	35.0000i	1.73489i
$408$	0	0
$409$	−12.0000	−0.593362	−0.296681	−	0.954977i	$-0.595880\pi$
$409$	−12.0000	−0.593362	−0.296681	+	0.954977i	$0.595880\pi$
$410$	0	0
$411$	−18.0000	−0.887875
$412$	− 13.0000i	− 0.640464i
$413$	0	0
$414$	−6.00000	−0.294884
$415$	0	0
$416$	1.00000	0.0490290
$417$	9.00000i	0.440732i
$418$	0	0
$419$	30.0000	1.46560	0.732798	−	0.680446i	$-0.238214\pi$
$419$	30.0000	1.46560	0.732798	+	0.680446i	$0.238214\pi$
$420$	0	0
$421$	−6.00000	−0.292422	−0.146211	−	0.989253i	$-0.546708\pi$
$421$	−6.00000	−0.292422	−0.146211	+	0.989253i	$0.546708\pi$
$422$	8.00000i	0.389434i
$423$	3.00000i	0.145865i
$424$	13.0000	0.631336
$425$	0	0
$426$	1.00000	0.0484502
$427$	− 1.00000i	− 0.0483934i
$428$	− 14.0000i	− 0.676716i
$429$	−5.00000	−0.241402
$430$	0	0
$431$	−7.00000	−0.337178	−0.168589	−	0.985686i	$-0.553921\pi$
$431$	−7.00000	−0.337178	−0.168589	+	0.985686i	$0.553921\pi$
$432$	− 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	$-0.214795\pi$
$434$	−1.00000	−0.0480015
$435$	0	0
$436$	−12.0000	−0.574696
$437$	0	0
$438$	− 4.00000i	− 0.191127i
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	0	0
$441$	−6.00000	−0.285714
$442$	0	0
$443$	18.0000i	0.855206i	0.903967	+	0.427603i	$0.140642\pi$
$443$	18.0000i	0.855206i	−0.903967	+	0.427603i	$0.859358\pi$
$444$	7.00000	0.332205
$445$	0	0
$446$	26.0000	1.23114
$447$	− 2.00000i	− 0.0945968i
$448$	1.00000i	0.0472456i
$449$	34.0000	1.60456	0.802280	−	0.596948i	$-0.203620\pi$
$449$	34.0000	1.60456	0.802280	+	0.596948i	$0.203620\pi$
$450$	0	0
$451$	−45.0000	−2.11897
$452$	− 18.0000i	− 0.846649i
$453$	− 10.0000i	− 0.469841i
$454$	20.0000	0.938647
$455$	0	0
$456$	0	0
$457$	2.00000i	0.0935561i	0.998905	+	0.0467780i	$0.0148953\pi$
$457$	2.00000i	0.0935561i	−0.998905	+	0.0467780i	$0.985105\pi$
$458$	− 14.0000i	− 0.654177i
$459$	0	0
$460$	0	0
$461$	−9.00000	−0.419172	−0.209586	−	0.977790i	$-0.567212\pi$
$461$	−9.00000	−0.419172	−0.209586	+	0.977790i	$0.567212\pi$
$462$	− 5.00000i	− 0.232621i
$463$	− 30.0000i	− 1.39422i	−0.716965	−	0.697109i	$-0.754469\pi$
$463$	− 30.0000i	− 1.39422i	0.716965	−	0.697109i	$-0.245531\pi$
$464$	2.00000	0.0928477
$465$	0	0
$466$	9.00000	0.416917
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	1.00000i	0.0462250i
$469$	−4.00000	−0.184703
$470$	0	0
$471$	18.0000	0.829396
$472$	0	0
$473$	55.0000i	2.52890i
$474$	−10.0000	−0.459315
$475$	0	0
$476$	0	0
$477$	13.0000i	0.595229i
$478$	− 16.0000i	− 0.731823i
$479$	−4.00000	−0.182765	−0.0913823	−	0.995816i	$-0.529129\pi$
$479$	−4.00000	−0.182765	−0.0913823	+	0.995816i	$0.529129\pi$
$480$	0	0
$481$	−7.00000	−0.319173
$482$	26.0000i	1.18427i
$483$	6.00000i	0.273009i
$484$	−14.0000	−0.636364
$485$	0	0
$486$	1.00000	0.0453609
$487$	− 34.0000i	− 1.54069i	−0.637629	−	0.770344i	$-0.720085\pi$
$487$	− 34.0000i	− 1.54069i	0.637629	−	0.770344i	$-0.279915\pi$
$488$	1.00000i	0.0452679i
$489$	6.00000	0.271329
$490$	0	0
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	9.00000i	0.405751i
$493$	0	0
$494$	0	0
$495$	0	0
$496$	1.00000	0.0449013
$497$	− 1.00000i	− 0.0448561i
$498$	1.00000i	0.0448111i
$499$	4.00000	0.179065	0.0895323	−	0.995984i	$-0.471463\pi$
$499$	4.00000	0.179065	0.0895323	+	0.995984i	$0.471463\pi$
$500$	0	0
$501$	6.00000	0.268060
$502$	− 17.0000i	− 0.758747i
$503$	27.0000i	1.20387i	0.798545	+	0.601935i	$0.205603\pi$
$503$	27.0000i	1.20387i	−0.798545	+	0.601935i	$0.794397\pi$
$504$	−1.00000	−0.0445435
$505$	0	0
$506$	30.0000	1.33366
$507$	12.0000i	0.532939i
$508$	0	0
$509$	−15.0000	−0.664863	−0.332432	−	0.943127i	$-0.607869\pi$
$509$	−15.0000	−0.664863	−0.332432	+	0.943127i	$0.607869\pi$
$510$	0	0
$511$	−4.00000	−0.176950
$512$	− 1.00000i	− 0.0441942i
$513$	0	0
$514$	7.00000	0.308757
$515$	0	0
$516$	11.0000	0.484248
$517$	− 15.0000i	− 0.659699i
$518$	− 7.00000i	− 0.307562i
$519$	−24.0000	−1.05348
$520$	0	0
$521$	−21.0000	−0.920027	−0.460013	−	0.887912i	$-0.652155\pi$
$521$	−21.0000	−0.920027	−0.460013	+	0.887912i	$0.652155\pi$
$522$	2.00000i	0.0875376i
$523$	12.0000i	0.524723i	0.964970	+	0.262362i	$0.0845013\pi$
$523$	12.0000i	0.524723i	−0.964970	+	0.262362i	$0.915499\pi$
$524$	−14.0000	−0.611593
$525$	0	0
$526$	12.0000	0.523225
$527$	0	0
$528$	5.00000i	0.217597i
$529$	−13.0000	−0.565217
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 9.00000i	− 0.389833i
$534$	2.00000	0.0865485
$535$	0	0
$536$	4.00000	0.172774
$537$	1.00000i	0.0431532i
$538$	18.0000i	0.776035i
$539$	30.0000	1.29219
$540$	0	0
$541$	−30.0000	−1.28980	−0.644900	−	0.764267i	$-0.723101\pi$
$541$	−30.0000	−1.28980	−0.644900	+	0.764267i	$0.723101\pi$
$542$	0	0
$543$	− 3.00000i	− 0.128742i
$544$	0	0
$545$	0	0
$546$	1.00000	0.0427960
$547$	− 26.0000i	− 1.11168i	−0.831289	−	0.555840i	$-0.812397\pi$
$547$	− 26.0000i	− 1.11168i	0.831289	−	0.555840i	$-0.187603\pi$
$548$	− 18.0000i	− 0.768922i
$549$	−1.00000	−0.0426790
$550$	0	0
$551$	0	0
$552$	− 6.00000i	− 0.255377i
$553$	10.0000i	0.425243i
$554$	22.0000	0.934690
$555$	0	0
$556$	−9.00000	−0.381685
$557$	18.0000i	0.762684i	0.924434	+	0.381342i	$0.124538\pi$
$557$	18.0000i	0.762684i	−0.924434	+	0.381342i	$0.875462\pi$
$558$	1.00000i	0.0423334i
$559$	−11.0000	−0.465250
$560$	0	0
$561$	0	0
$562$	− 27.0000i	− 1.13893i
$563$	− 44.0000i	− 1.85438i	−0.374593	−	0.927189i	$-0.622217\pi$
$563$	− 44.0000i	− 1.85438i	0.374593	−	0.927189i	$-0.377783\pi$
$564$	−3.00000	−0.126323
$565$	0	0
$566$	16.0000	0.672530
$567$	− 1.00000i	− 0.0419961i
$568$	1.00000i	0.0419591i
$569$	−34.0000	−1.42535	−0.712677	−	0.701492i	$-0.752517\pi$
$569$	−34.0000	−1.42535	−0.712677	+	0.701492i	$0.752517\pi$
$570$	0	0
$571$	5.00000	0.209243	0.104622	−	0.994512i	$-0.466637\pi$
$571$	5.00000	0.209243	0.104622	+	0.994512i	$0.466637\pi$
$572$	− 5.00000i	− 0.209061i
$573$	8.00000i	0.334205i
$574$	9.00000	0.375653
$575$	0	0
$576$	1.00000	0.0416667
$577$	− 18.0000i	− 0.749350i	−0.927156	−	0.374675i	$-0.877754\pi$
$577$	− 18.0000i	− 0.749350i	0.927156	−	0.374675i	$-0.122246\pi$
$578$	− 17.0000i	− 0.707107i
$579$	7.00000	0.290910
$580$	0	0
$581$	1.00000	0.0414870
$582$	− 2.00000i	− 0.0829027i
$583$	− 65.0000i	− 2.69202i
$584$	4.00000	0.165521
$585$	0	0
$586$	−14.0000	−0.578335
$587$	− 39.0000i	− 1.60970i	−0.593477	−	0.804851i	$-0.702245\pi$
$587$	− 39.0000i	− 1.60970i	0.593477	−	0.804851i	$-0.297755\pi$
$588$	− 6.00000i	− 0.247436i
$589$	0	0
$590$	0	0
$591$	−5.00000	−0.205673
$592$	7.00000i	0.287698i
$593$	− 5.00000i	− 0.205325i	−0.994716	−	0.102663i	$-0.967264\pi$
$593$	− 5.00000i	− 0.205325i	0.994716	−	0.102663i	$-0.0327362\pi$
$594$	−5.00000	−0.205152
$595$	0	0
$596$	2.00000	0.0819232
$597$	6.00000i	0.245564i
$598$	6.00000i	0.245358i
$599$	9.00000	0.367730	0.183865	−	0.982952i	$-0.441139\pi$
$599$	9.00000	0.367730	0.183865	+	0.982952i	$0.441139\pi$
$600$	0	0
$601$	−2.00000	−0.0815817	−0.0407909	−	0.999168i	$-0.512988\pi$
$601$	−2.00000	−0.0815817	−0.0407909	+	0.999168i	$0.512988\pi$
$602$	− 11.0000i	− 0.448327i
$603$	4.00000i	0.162893i
$604$	10.0000	0.406894
$605$	0	0
$606$	8.00000	0.324978
$607$	− 13.0000i	− 0.527654i	−0.964570	−	0.263827i	$-0.915015\pi$
$607$	− 13.0000i	− 0.527654i	0.964570	−	0.263827i	$-0.0849848\pi$
$608$	0	0
$609$	2.00000	0.0810441
$610$	0	0
$611$	3.00000	0.121367
$612$	0	0
$613$	− 6.00000i	− 0.242338i	−0.992632	−	0.121169i	$-0.961336\pi$
$613$	− 6.00000i	− 0.242338i	0.992632	−	0.121169i	$-0.0386643\pi$
$614$	−22.0000	−0.887848
$615$	0	0
$616$	5.00000	0.201456
$617$	41.0000i	1.65060i	0.564696	+	0.825299i	$0.308993\pi$
$617$	41.0000i	1.65060i	−0.564696	+	0.825299i	$0.691007\pi$
$618$	13.0000i	0.522937i
$619$	29.0000	1.16561	0.582804	−	0.812613i	$-0.301955\pi$
$619$	29.0000	1.16561	0.582804	+	0.812613i	$0.301955\pi$
$620$	0	0
$621$	6.00000	0.240772
$622$	3.00000i	0.120289i
$623$	− 2.00000i	− 0.0801283i
$624$	−1.00000	−0.0400320
$625$	0	0
$626$	−14.0000	−0.559553
$627$	0	0
$628$	18.0000i	0.718278i
$629$	0	0
$630$	0	0
$631$	−14.0000	−0.557331	−0.278666	−	0.960388i	$-0.589892\pi$
$631$	−14.0000	−0.557331	−0.278666	+	0.960388i	$0.589892\pi$
$632$	− 10.0000i	− 0.397779i
$633$	− 8.00000i	− 0.317971i
$634$	−4.00000	−0.158860
$635$	0	0
$636$	−13.0000	−0.515484
$637$	6.00000i	0.237729i
$638$	− 10.0000i	− 0.395904i
$639$	−1.00000	−0.0395594
$640$	0	0
$641$	−42.0000	−1.65890	−0.829450	−	0.558581i	$-0.811346\pi$
$641$	−42.0000	−1.65890	−0.829450	+	0.558581i	$0.811346\pi$
$642$	14.0000i	0.552536i
$643$	5.00000i	0.197181i	0.995128	+	0.0985904i	$0.0314334\pi$
$643$	5.00000i	0.197181i	−0.995128	+	0.0985904i	$0.968567\pi$
$644$	−6.00000	−0.236433
$645$	0	0
$646$	0	0
$647$	− 10.0000i	− 0.393141i	−0.980490	−	0.196570i	$-0.937020\pi$
$647$	− 10.0000i	− 0.393141i	0.980490	−	0.196570i	$-0.0629804\pi$
$648$	1.00000i	0.0392837i
$649$	0	0
$650$	0	0
$651$	1.00000	0.0391931
$652$	6.00000i	0.234978i
$653$	28.0000i	1.09572i	0.836569	+	0.547862i	$0.184558\pi$
$653$	28.0000i	1.09572i	−0.836569	+	0.547862i	$0.815442\pi$
$654$	12.0000	0.469237
$655$	0	0
$656$	−9.00000	−0.351391
$657$	4.00000i	0.156055i
$658$	3.00000i	0.116952i
$659$	10.0000	0.389545	0.194772	−	0.980848i	$-0.437603\pi$
$659$	10.0000	0.389545	0.194772	+	0.980848i	$0.437603\pi$
$660$	0	0
$661$	48.0000	1.86698	0.933492	−	0.358599i	$-0.116745\pi$
$661$	48.0000	1.86698	0.933492	+	0.358599i	$0.116745\pi$
$662$	19.0000i	0.738456i
$663$	0	0
$664$	−1.00000	−0.0388075
$665$	0	0
$666$	−7.00000	−0.271244
$667$	12.0000i	0.464642i
$668$	6.00000i	0.232147i
$669$	−26.0000	−1.00522
$670$	0	0
$671$	5.00000	0.193023
$672$	− 1.00000i	− 0.0385758i
$673$	− 2.00000i	− 0.0770943i	−0.999257	−	0.0385472i	$-0.987727\pi$
$673$	− 2.00000i	− 0.0770943i	0.999257	−	0.0385472i	$-0.0122730\pi$
$674$	8.00000	0.308148
$675$	0	0
$676$	−12.0000	−0.461538
$677$	− 31.0000i	− 1.19143i	−0.803197	−	0.595713i	$-0.796869\pi$
$677$	− 31.0000i	− 1.19143i	0.803197	−	0.595713i	$-0.203131\pi$
$678$	18.0000i	0.691286i
$679$	−2.00000	−0.0767530
$680$	0	0
$681$	−20.0000	−0.766402
$682$	− 5.00000i	− 0.191460i
$683$	8.00000i	0.306111i	0.988218	+	0.153056i	$0.0489114\pi$
$683$	8.00000i	0.306111i	−0.988218	+	0.153056i	$0.951089\pi$
$684$	0	0
$685$	0	0
$686$	−13.0000	−0.496342
$687$	14.0000i	0.534133i
$688$	11.0000i	0.419371i
$689$	13.0000	0.495261
$690$	0	0
$691$	−28.0000	−1.06517	−0.532585	−	0.846376i	$-0.678779\pi$
$691$	−28.0000	−1.06517	−0.532585	+	0.846376i	$0.678779\pi$
$692$	− 24.0000i	− 0.912343i
$693$	5.00000i	0.189934i
$694$	−17.0000	−0.645311
$695$	0	0
$696$	−2.00000	−0.0758098
$697$	0	0
$698$	− 16.0000i	− 0.605609i
$699$	−9.00000	−0.340411
$700$	0	0
$701$	44.0000	1.66186	0.830929	−	0.556379i	$-0.187810\pi$
$701$	44.0000	1.66186	0.830929	+	0.556379i	$0.187810\pi$
$702$	− 1.00000i	− 0.0377426i
$703$	0	0
$704$	−5.00000	−0.188445
$705$	0	0
$706$	26.0000	0.978523
$707$	− 8.00000i	− 0.300871i
$708$	0	0
$709$	−38.0000	−1.42712	−0.713560	−	0.700594i	$-0.752918\pi$
$709$	−38.0000	−1.42712	−0.713560	+	0.700594i	$0.752918\pi$
$710$	0	0
$711$	10.0000	0.375029
$712$	2.00000i	0.0749532i
$713$	6.00000i	0.224702i
$714$	0	0
$715$	0	0
$716$	−1.00000	−0.0373718
$717$	16.0000i	0.597531i
$718$	− 24.0000i	− 0.895672i
$719$	40.0000	1.49175	0.745874	−	0.666087i	$-0.232032\pi$
$719$	40.0000	1.49175	0.745874	+	0.666087i	$0.232032\pi$
$720$	0	0
$721$	13.0000	0.484145
$722$	19.0000i	0.707107i
$723$	− 26.0000i	− 0.966950i
$724$	3.00000	0.111494
$725$	0	0
$726$	14.0000	0.519589
$727$	9.00000i	0.333792i	0.985975	+	0.166896i	$0.0533743\pi$
$727$	9.00000i	0.333792i	−0.985975	+	0.166896i	$0.946626\pi$
$728$	1.00000i	0.0370625i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	0	0
$732$	− 1.00000i	− 0.0369611i
$733$	− 34.0000i	− 1.25582i	−0.778287	−	0.627909i	$-0.783911\pi$
$733$	− 34.0000i	− 1.25582i	0.778287	−	0.627909i	$-0.216089\pi$
$734$	−18.0000	−0.664392
$735$	0	0
$736$	6.00000	0.221163
$737$	− 20.0000i	− 0.736709i
$738$	− 9.00000i	− 0.331295i
$739$	20.0000	0.735712	0.367856	−	0.929883i	$-0.380092\pi$
$739$	20.0000	0.735712	0.367856	+	0.929883i	$0.380092\pi$
$740$	0	0
$741$	0	0
$742$	13.0000i	0.477245i
$743$	− 8.00000i	− 0.293492i	−0.989174	−	0.146746i	$-0.953120\pi$
$743$	− 8.00000i	− 0.293492i	0.989174	−	0.146746i	$-0.0468799\pi$
$744$	−1.00000	−0.0366618
$745$	0	0
$746$	−24.0000	−0.878702
$747$	− 1.00000i	− 0.0365881i
$748$	0	0
$749$	14.0000	0.511549
$750$	0	0
$751$	13.0000	0.474377	0.237188	−	0.971464i	$-0.423774\pi$
$751$	13.0000	0.474377	0.237188	+	0.971464i	$0.423774\pi$
$752$	− 3.00000i	− 0.109399i
$753$	17.0000i	0.619514i
$754$	2.00000	0.0728357
$755$	0	0
$756$	1.00000	0.0363696
$757$	− 15.0000i	− 0.545184i	−0.962130	−	0.272592i	$-0.912119\pi$
$757$	− 15.0000i	− 0.545184i	0.962130	−	0.272592i	$-0.0878810\pi$
$758$	− 26.0000i	− 0.944363i
$759$	−30.0000	−1.08893
$760$	0	0
$761$	0	0		−	1.00000i	$-0.5\pi$
$761$	0	0			1.00000i	$0.5\pi$
$762$	0	0
$763$	− 12.0000i	− 0.434429i
$764$	−8.00000	−0.289430
$765$	0	0
$766$	12.0000	0.433578
$767$	0	0
$768$	1.00000i	0.0360844i
$769$	−15.0000	−0.540914	−0.270457	−	0.962732i	$-0.587175\pi$
$769$	−15.0000	−0.540914	−0.270457	+	0.962732i	$0.587175\pi$
$770$	0	0
$771$	−7.00000	−0.252099
$772$	7.00000i	0.251936i
$773$	− 42.0000i	− 1.51064i	−0.655359	−	0.755318i	$-0.727483\pi$
$773$	− 42.0000i	− 1.51064i	0.655359	−	0.755318i	$-0.272517\pi$
$774$	−11.0000	−0.395387
$775$	0	0
$776$	2.00000	0.0717958
$777$	7.00000i	0.251124i
$778$	− 6.00000i	− 0.215110i
$779$	0	0
$780$	0	0
$781$	5.00000	0.178914
$782$	0	0
$783$	− 2.00000i	− 0.0714742i
$784$	6.00000	0.214286
$785$	0	0
$786$	14.0000	0.499363
$787$	− 37.0000i	− 1.31891i	−0.751745	−	0.659454i	$-0.770788\pi$
$787$	− 37.0000i	− 1.31891i	0.751745	−	0.659454i	$-0.229212\pi$
$788$	− 5.00000i	− 0.178118i
$789$	−12.0000	−0.427211
$790$	0	0
$791$	18.0000	0.640006
$792$	− 5.00000i	− 0.177667i
$793$	1.00000i	0.0355110i
$794$	18.0000	0.638796
$795$	0	0
$796$	−6.00000	−0.212664
$797$	− 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
$801$	−2.00000	−0.0706665
$802$	10.0000i	0.353112i
$803$	− 20.0000i	− 0.705785i
$804$	−4.00000	−0.141069
$805$	0	0
$806$	1.00000	0.0352235
$807$	− 18.0000i	− 0.633630i
$808$	8.00000i	0.281439i
$809$	−50.0000	−1.75791	−0.878953	−	0.476908i	$-0.841757\pi$
$809$	−50.0000	−1.75791	−0.878953	+	0.476908i	$0.841757\pi$
$810$	0	0
$811$	−40.0000	−1.40459	−0.702295	−	0.711886i	$-0.747841\pi$
$811$	−40.0000	−1.40459	−0.702295	+	0.711886i	$0.747841\pi$
$812$	2.00000i	0.0701862i
$813$	0	0
$814$	35.0000	1.22675
$815$	0	0
$816$	0	0
$817$	0	0
$818$	12.0000i	0.419570i
$819$	−1.00000	−0.0349428
$820$	0	0
$821$	−27.0000	−0.942306	−0.471153	−	0.882051i	$-0.656162\pi$
$821$	−27.0000	−0.942306	−0.471153	+	0.882051i	$0.656162\pi$
$822$	18.0000i	0.627822i
$823$	− 32.0000i	− 1.11545i	−0.830026	−	0.557725i	$-0.811674\pi$
$823$	− 32.0000i	− 1.11545i	0.830026	−	0.557725i	$-0.188326\pi$
$824$	−13.0000	−0.452876
$825$	0	0
$826$	0	0
$827$	− 4.00000i	− 0.139094i	−0.997579	−	0.0695468i	$-0.977845\pi$
$827$	− 4.00000i	− 0.139094i	0.997579	−	0.0695468i	$-0.0221553\pi$
$828$	6.00000i	0.208514i
$829$	−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
$831$	−22.0000	−0.763172
$832$	− 1.00000i	− 0.0346688i
$833$	0	0
$834$	9.00000	0.311645
$835$	0	0
$836$	0	0
$837$	− 1.00000i	− 0.0345651i
$838$	− 30.0000i	− 1.03633i
$839$	−29.0000	−1.00119	−0.500596	−	0.865681i	$-0.666886\pi$
$839$	−29.0000	−1.00119	−0.500596	+	0.865681i	$0.666886\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	6.00000i	0.206774i
$843$	27.0000i	0.929929i
$844$	8.00000	0.275371
$845$	0	0
$846$	3.00000	0.103142
$847$	− 14.0000i	− 0.481046i
$848$	− 13.0000i	− 0.446422i
$849$	−16.0000	−0.549119
$850$	0	0
$851$	−42.0000	−1.43974
$852$	− 1.00000i	− 0.0342594i
$853$	− 24.0000i	− 0.821744i	−0.911693	−	0.410872i	$-0.865224\pi$
$853$	− 24.0000i	− 0.821744i	0.911693	−	0.410872i	$-0.134776\pi$
$854$	−1.00000	−0.0342193
$855$	0	0
$856$	−14.0000	−0.478510
$857$	45.0000i	1.53717i	0.639747	+	0.768585i	$0.279039\pi$
$857$	45.0000i	1.53717i	−0.639747	+	0.768585i	$0.720961\pi$
$858$	5.00000i	0.170697i
$859$	5.00000	0.170598	0.0852989	−	0.996355i	$-0.472815\pi$
$859$	5.00000	0.170598	0.0852989	+	0.996355i	$0.472815\pi$
$860$	0	0
$861$	−9.00000	−0.306719
$862$	7.00000i	0.238421i
$863$	− 36.0000i	− 1.22545i	−0.790295	−	0.612727i	$-0.790072\pi$
$863$	− 36.0000i	− 1.22545i	0.790295	−	0.612727i	$-0.209928\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	−26.0000	−0.883516
$867$	17.0000i	0.577350i
$868$	1.00000i	0.0339422i
$869$	−50.0000	−1.69613
$870$	0	0
$871$	4.00000	0.135535
$872$	12.0000i	0.406371i
$873$	2.00000i	0.0676897i
$874$	0	0
$875$	0	0
$876$	−4.00000	−0.135147
$877$	36.0000i	1.21563i	0.794077	+	0.607817i	$0.207955\pi$
$877$	36.0000i	1.21563i	−0.794077	+	0.607817i	$0.792045\pi$
$878$	0	0
$879$	14.0000	0.472208
$880$	0	0
$881$	16.0000	0.539054	0.269527	−	0.962993i	$-0.413133\pi$
$881$	16.0000	0.539054	0.269527	+	0.962993i	$0.413133\pi$
$882$	6.00000i	0.202031i
$883$	− 7.00000i	− 0.235569i	−0.993039	−	0.117784i	$-0.962421\pi$
$883$	− 7.00000i	− 0.235569i	0.993039	−	0.117784i	$-0.0375792\pi$
$884$	0	0
$885$	0	0
$886$	18.0000	0.604722
$887$	− 3.00000i	− 0.100730i	−0.998731	−	0.0503651i	$-0.983962\pi$
$887$	− 3.00000i	− 0.100730i	0.998731	−	0.0503651i	$-0.0160385\pi$
$888$	− 7.00000i	− 0.234905i
$889$	0	0
$890$	0	0
$891$	5.00000	0.167506
$892$	− 26.0000i	− 0.870544i
$893$	0	0
$894$	−2.00000	−0.0668900
$895$	0	0
$896$	1.00000	0.0334077
$897$	− 6.00000i	− 0.200334i
$898$	− 34.0000i	− 1.13459i
$899$	2.00000	0.0667037
$900$	0	0
$901$	0	0
$902$	45.0000i	1.49834i
$903$	11.0000i	0.366057i
$904$	−18.0000	−0.598671
$905$	0	0
$906$	−10.0000	−0.332228
$907$	16.0000i	0.531271i	0.964073	+	0.265636i	$0.0855818\pi$
$907$	16.0000i	0.531271i	−0.964073	+	0.265636i	$0.914418\pi$
$908$	− 20.0000i	− 0.663723i
$909$	−8.00000	−0.265343
$910$	0	0
$911$	8.00000	0.265052	0.132526	−	0.991180i	$-0.457691\pi$
$911$	8.00000	0.265052	0.132526	+	0.991180i	$0.457691\pi$
$912$	0	0
$913$	5.00000i	0.165476i
$914$	2.00000	0.0661541
$915$	0	0
$916$	−14.0000	−0.462573
$917$	− 14.0000i	− 0.462321i
$918$	0	0
$919$	33.0000	1.08857	0.544285	−	0.838901i	$-0.316801\pi$
$919$	33.0000	1.08857	0.544285	+	0.838901i	$0.316801\pi$
$920$	0	0
$921$	22.0000	0.724925
$922$	9.00000i	0.296399i
$923$	1.00000i	0.0329154i
$924$	−5.00000	−0.164488
$925$	0	0
$926$	−30.0000	−0.985861
$927$	− 13.0000i	− 0.426976i
$928$	− 2.00000i	− 0.0656532i
$929$	−54.0000	−1.77168	−0.885841	−	0.463988i	$-0.846418\pi$
$929$	−54.0000	−1.77168	−0.885841	+	0.463988i	$0.846418\pi$
$930$	0	0
$931$	0	0
$932$	− 9.00000i	− 0.294805i
$933$	− 3.00000i	− 0.0982156i
$934$	0	0
$935$	0	0
$936$	1.00000	0.0326860
$937$	34.0000i	1.11073i	0.831606	+	0.555366i	$0.187422\pi$
$937$	34.0000i	1.11073i	−0.831606	+	0.555366i	$0.812578\pi$
$938$	4.00000i	0.130605i
$939$	14.0000	0.456873
$940$	0	0
$941$	−18.0000	−0.586783	−0.293392	−	0.955992i	$-0.594784\pi$
$941$	−18.0000	−0.586783	−0.293392	+	0.955992i	$0.594784\pi$
$942$	− 18.0000i	− 0.586472i
$943$	− 54.0000i	− 1.75848i
$944$	0	0
$945$	0	0
$946$	55.0000	1.78820
$947$	5.00000i	0.162478i	0.996695	+	0.0812391i	$0.0258877\pi$
$947$	5.00000i	0.162478i	−0.996695	+	0.0812391i	$0.974112\pi$
$948$	10.0000i	0.324785i
$949$	4.00000	0.129845
$950$	0	0
$951$	4.00000	0.129709
$952$	0	0
$953$	− 22.0000i	− 0.712650i	−0.934362	−	0.356325i	$-0.884030\pi$
$953$	− 22.0000i	− 0.712650i	0.934362	−	0.356325i	$-0.115970\pi$
$954$	13.0000	0.420891
$955$	0	0
$956$	−16.0000	−0.517477
$957$	10.0000i	0.323254i
$958$	4.00000i	0.129234i
$959$	18.0000	0.581250
$960$	0	0
$961$	1.00000	0.0322581
$962$	7.00000i	0.225689i
$963$	− 14.0000i	− 0.451144i
$964$	26.0000	0.837404
$965$	0	0
$966$	6.00000	0.193047
$967$	− 10.0000i	− 0.321578i	−0.986989	−	0.160789i	$-0.948596\pi$
$967$	− 10.0000i	− 0.321578i	0.986989	−	0.160789i	$-0.0514039\pi$
$968$	14.0000i	0.449977i
$969$	0	0
$970$	0	0
$971$	12.0000	0.385098	0.192549	−	0.981287i	$-0.438325\pi$
$971$	12.0000	0.385098	0.192549	+	0.981287i	$0.438325\pi$
$972$	− 1.00000i	− 0.0320750i
$973$	− 9.00000i	− 0.288527i
$974$	−34.0000	−1.08943
$975$	0	0
$976$	1.00000	0.0320092
$977$	3.00000i	0.0959785i	0.998848	+	0.0479893i	$0.0152813\pi$
$977$	3.00000i	0.0959785i	−0.998848	+	0.0479893i	$0.984719\pi$
$978$	− 6.00000i	− 0.191859i
$979$	10.0000	0.319601
$980$	0	0
$981$	−12.0000	−0.383131
$982$	0	0
$983$	18.0000i	0.574111i	0.957914	+	0.287055i	$0.0926764\pi$
$983$	18.0000i	0.574111i	−0.957914	+	0.287055i	$0.907324\pi$
$984$	9.00000	0.286910
$985$	0	0
$986$	0	0
$987$	− 3.00000i	− 0.0954911i
$988$	0	0
$989$	−66.0000	−2.09868
$990$	0	0
$991$	2.00000	0.0635321	0.0317660	−	0.999495i	$-0.489887\pi$
$991$	2.00000	0.0635321	0.0317660	+	0.999495i	$0.489887\pi$
$992$	− 1.00000i	− 0.0317500i
$993$	− 19.0000i	− 0.602947i
$994$	−1.00000	−0.0317181
$995$	0	0
$996$	1.00000	0.0316862
$997$	48.0000i	1.52018i	0.649821	+	0.760088i	$0.274844\pi$
$997$	48.0000i	1.52018i	−0.649821	+	0.760088i	$0.725156\pi$
$998$	− 4.00000i	− 0.126618i
$999$	7.00000	0.221470

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4650.2.d.bb.3349.1		2
5.2	odd	4		4650.2.a.br.1.1	yes	1
5.3	odd	4		4650.2.a.g.1.1	✓	1
5.4	even	2	inner	4650.2.d.bb.3349.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4650.2.a.g.1.1	✓	1	5.3	odd	4
4650.2.a.br.1.1	yes	1	5.2	odd	4
4650.2.d.bb.3349.1		2	1.1	even	1	trivial
4650.2.d.bb.3349.2		2	5.4	even	2	inner

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)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -1.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -1.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +5.00000 q^{11} -1.00000i q^{12} +1.00000i q^{13} -1.00000 q^{14} +1.00000 q^{16} +1.00000i q^{18} +1.00000 q^{21} -5.00000i q^{22} +6.00000i q^{23} -1.00000 q^{24} +1.00000 q^{26} -1.00000i q^{27} +1.00000i q^{28} +2.00000 q^{29} +1.00000 q^{31} -1.00000i q^{32} +5.00000i q^{33} +1.00000 q^{36} +7.00000i q^{37} -1.00000 q^{39} -9.00000 q^{41} -1.00000i q^{42} +11.0000i q^{43} -5.00000 q^{44} +6.00000 q^{46} -3.00000i q^{47} +1.00000i q^{48} +6.00000 q^{49} -1.00000i q^{52} -13.0000i q^{53} -1.00000 q^{54} +1.00000 q^{56} -2.00000i q^{58} +1.00000 q^{61} -1.00000i q^{62} +1.00000i q^{63} -1.00000 q^{64} +5.00000 q^{66} -4.00000i q^{67} -6.00000 q^{69} +1.00000 q^{71} -1.00000i q^{72} -4.00000i q^{73} +7.00000 q^{74} -5.00000i q^{77} +1.00000i q^{78} -10.0000 q^{79} +1.00000 q^{81} +9.00000i q^{82} +1.00000i q^{83} -1.00000 q^{84} +11.0000 q^{86} +2.00000i q^{87} +5.00000i q^{88} +2.00000 q^{89} +1.00000 q^{91} -6.00000i q^{92} +1.00000i q^{93} -3.00000 q^{94} +1.00000 q^{96} -2.00000i q^{97} -6.00000i q^{98} -5.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9	\( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} + 10 q^{11} - 2 q^{14} + 2 q^{16} + 2 q^{21} - 2 q^{24} + 2 q^{26} + 4 q^{29} + 2 q^{31} + 2 q^{36} - 2 q^{39} - 18 q^{41} - 10 q^{44} + 12 q^{46} + 12 q^{49} - 2 q^{54} + 2 q^{56} + 2 q^{61} - 2 q^{64} + 10 q^{66} - 12 q^{69} + 2 q^{71} + 14 q^{74} - 20 q^{79} + 2 q^{81} - 2 q^{84} + 22 q^{86} + 4 q^{89} + 2 q^{91} - 6 q^{94} + 2 q^{96} - 10 q^{99}+O(q^{100}) \) 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9 + 10 * q^11 - 2 * q^14 + 2 * q^16 + 2 * q^21 - 2 * q^24 + 2 * q^26 + 4 * q^29 + 2 * q^31 + 2 * q^36 - 2 * q^39 - 18 * q^41 - 10 * q^44 + 12 * q^46 + 12 * q^49 - 2 * q^54 + 2 * q^56 + 2 * q^61 - 2 * q^64 + 10 * q^66 - 12 * q^69 + 2 * q^71 + 14 * q^74 - 20 * q^79 + 2 * q^81 - 2 * q^84 + 22 * q^86 + 4 * q^89 + 2 * q^91 - 6 * q^94 + 2 * q^96 - 10 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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