LMFDB - Embedded newform 4650.2.d.h.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.h.3349.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -4.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -4.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +2.00000 q^{11} -1.00000i q^{12} +2.00000i q^{13} +4.00000 q^{14} +1.00000 q^{16} -1.00000i q^{18} +4.00000 q^{21} +2.00000i q^{22} -6.00000i q^{23} +1.00000 q^{24} -2.00000 q^{26} -1.00000i q^{27} +4.00000i q^{28} +1.00000 q^{31} +1.00000i q^{32} +2.00000i q^{33} +1.00000 q^{36} +2.00000i q^{37} -2.00000 q^{39} -10.0000 q^{41} +4.00000i q^{42} -4.00000i q^{43} -2.00000 q^{44} +6.00000 q^{46} -4.00000i q^{47} +1.00000i q^{48} -9.00000 q^{49} -2.00000i q^{52} +6.00000i q^{53} +1.00000 q^{54} -4.00000 q^{56} +4.00000 q^{59} +1.00000i q^{62} +4.00000i q^{63} -1.00000 q^{64} -2.00000 q^{66} -4.00000i q^{67} +6.00000 q^{69} -16.0000 q^{71} +1.00000i q^{72} +4.00000i q^{73} -2.00000 q^{74} -8.00000i q^{77} -2.00000i q^{78} -4.00000 q^{79} +1.00000 q^{81} -10.0000i q^{82} +8.00000i q^{83} -4.00000 q^{84} +4.00000 q^{86} -2.00000i q^{88} -6.00000 q^{89} +8.00000 q^{91} +6.00000i q^{92} +1.00000i q^{93} +4.00000 q^{94} -1.00000 q^{96} -14.0000i q^{97} -9.00000i q^{98} -2.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 4 q^{11} + 8 q^{14} + 2 q^{16} + 8 q^{21} + 2 q^{24} - 4 q^{26} + 2 q^{31} + 2 q^{36} - 4 q^{39} - 20 q^{41} - 4 q^{44} + 12 q^{46} - 18 q^{49} + 2 q^{54} - 8 q^{56} + 8 q^{59} - 2 q^{64} - 4 q^{66} + 12 q^{69} - 32 q^{71} - 4 q^{74} - 8 q^{79} + 2 q^{81} - 8 q^{84} + 8 q^{86} - 12 q^{89} + 16 q^{91} + 8 q^{94} - 2 q^{96} - 4 q^{99}+O(q^{100}) $ 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9 + 4 * q^11 + 8 * q^14 + 2 * q^16 + 8 * q^21 + 2 * q^24 - 4 * q^26 + 2 * q^31 + 2 * q^36 - 4 * q^39 - 20 * q^41 - 4 * q^44 + 12 * q^46 - 18 * q^49 + 2 * q^54 - 8 * q^56 + 8 * q^59 - 2 * q^64 - 4 * q^66 + 12 * q^69 - 32 * q^71 - 4 * q^74 - 8 * q^79 + 2 * q^81 - 8 * q^84 + 8 * q^86 - 12 * q^89 + 16 * q^91 + 8 * q^94 - 2 * q^96 - 4 * 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
$7$	− 4.00000i	− 1.51186i	−0.654654	−	0.755929i	$-0.727186\pi$
$7$	− 4.00000i	− 1.51186i	0.654654	−	0.755929i	$-0.272814\pi$
$8$	− 1.00000i	− 0.353553i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	− 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	$0.910544\pi$
$14$	4.00000	1.06904
$15$	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
$21$	4.00000	0.872872
$22$	2.00000i	0.426401i
$23$	− 6.00000i	− 1.25109i	−0.780189	−	0.625543i	$-0.784877\pi$
$23$	− 6.00000i	− 1.25109i	0.780189	−	0.625543i	$-0.215123\pi$
$24$	1.00000	0.204124
$25$	0	0
$26$	−2.00000	−0.392232
$27$	− 1.00000i	− 0.192450i
$28$	4.00000i	0.755929i
$29$	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
$33$	2.00000i	0.348155i
$34$	0	0
$35$	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	$0.947432\pi$
$38$	0	0
$39$	−2.00000	−0.320256
$40$	0	0
$41$	−10.0000	−1.56174	−0.780869	−	0.624695i	$-0.785223\pi$
$41$	−10.0000	−1.56174	−0.780869	+	0.624695i	$0.785223\pi$
$42$	4.00000i	0.617213i
$43$	− 4.00000i	− 0.609994i	−0.952353	−	0.304997i	$-0.901344\pi$
$43$	− 4.00000i	− 0.609994i	0.952353	−	0.304997i	$-0.0986555\pi$
$44$	−2.00000	−0.301511
$45$	0	0
$46$	6.00000	0.884652
$47$	− 4.00000i	− 0.583460i	−0.956501	−	0.291730i	$-0.905769\pi$
$47$	− 4.00000i	− 0.583460i	0.956501	−	0.291730i	$-0.0942309\pi$
$48$	1.00000i	0.144338i
$49$	−9.00000	−1.28571
$50$	0	0
$51$	0	0
$52$	− 2.00000i	− 0.277350i
$53$	6.00000i	0.824163i	0.911147	+	0.412082i	$0.135198\pi$
$53$	6.00000i	0.824163i	−0.911147	+	0.412082i	$0.864802\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	−4.00000	−0.534522
$57$	0	0
$58$	0	0
$59$	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
$61$	0	0		−	1.00000i	$-0.5\pi$
$61$	0	0			1.00000i	$0.5\pi$
$62$	1.00000i	0.127000i
$63$	4.00000i	0.503953i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	−2.00000	−0.246183
$67$	− 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
$71$	−16.0000	−1.89885	−0.949425	−	0.313993i	$-0.898333\pi$
$71$	−16.0000	−1.89885	−0.949425	+	0.313993i	$0.898333\pi$
$72$	1.00000i	0.117851i
$73$	4.00000i	0.468165i	0.972217	+	0.234082i	$0.0752085\pi$
$73$	4.00000i	0.468165i	−0.972217	+	0.234082i	$0.924791\pi$
$74$	−2.00000	−0.232495
$75$	0	0
$76$	0	0
$77$	− 8.00000i	− 0.911685i
$78$	− 2.00000i	− 0.226455i
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	− 10.0000i	− 1.10432i
$83$	8.00000i	0.878114i	0.898459	+	0.439057i	$0.144687\pi$
$83$	8.00000i	0.878114i	−0.898459	+	0.439057i	$0.855313\pi$
$84$	−4.00000	−0.436436
$85$	0	0
$86$	4.00000	0.431331
$87$	0	0
$88$	− 2.00000i	− 0.213201i
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	0	0
$91$	8.00000	0.838628
$92$	6.00000i	0.625543i
$93$	1.00000i	0.103695i
$94$	4.00000	0.412568
$95$	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	$-0.251641\pi$
$98$	− 9.00000i	− 0.909137i
$99$	−2.00000	−0.201008
$100$	0	0
$101$	−10.0000	−0.995037	−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000	−0.995037	−0.497519	+	0.867453i	$0.665755\pi$
$102$	0	0
$103$	− 12.0000i	− 1.18240i	−0.806527	−	0.591198i	$-0.798655\pi$
$103$	− 12.0000i	− 1.18240i	0.806527	−	0.591198i	$-0.201345\pi$
$104$	2.00000	0.196116
$105$	0	0
$106$	−6.00000	−0.582772
$107$	− 12.0000i	− 1.16008i	−0.814587	−	0.580042i	$-0.803036\pi$
$107$	− 12.0000i	− 1.16008i	0.814587	−	0.580042i	$-0.196964\pi$
$108$	1.00000i	0.0962250i
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	0	0
$111$	−2.00000	−0.189832
$112$	− 4.00000i	− 0.377964i
$113$	2.00000i	0.188144i	0.995565	+	0.0940721i	$0.0299884\pi$
$113$	2.00000i	0.188144i	−0.995565	+	0.0940721i	$0.970012\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 2.00000i	− 0.184900i
$118$	4.00000i	0.368230i
$119$	0	0
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	− 10.0000i	− 0.901670i
$124$	−1.00000	−0.0898027
$125$	0	0
$126$	−4.00000	−0.356348
$127$	− 18.0000i	− 1.59724i	−0.601834	−	0.798621i	$-0.705563\pi$
$127$	− 18.0000i	− 1.59724i	0.601834	−	0.798621i	$-0.294437\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	4.00000	0.352180
$130$	0	0
$131$	4.00000	0.349482	0.174741	−	0.984614i	$-0.444091\pi$
$131$	4.00000	0.349482	0.174741	+	0.984614i	$0.444091\pi$
$132$	− 2.00000i	− 0.174078i
$133$	0	0
$134$	4.00000	0.345547
$135$	0	0
$136$	0	0
$137$	12.0000i	1.02523i	0.858619	+	0.512615i	$0.171323\pi$
$137$	12.0000i	1.02523i	−0.858619	+	0.512615i	$0.828677\pi$
$138$	6.00000i	0.510754i
$139$	6.00000	0.508913	0.254457	−	0.967084i	$-0.418103\pi$
$139$	6.00000	0.508913	0.254457	+	0.967084i	$0.418103\pi$
$140$	0	0
$141$	4.00000	0.336861
$142$	− 16.0000i	− 1.34269i
$143$	4.00000i	0.334497i
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	−4.00000	−0.331042
$147$	− 9.00000i	− 0.742307i
$148$	− 2.00000i	− 0.164399i
$149$	2.00000	0.163846	0.0819232	−	0.996639i	$-0.473894\pi$
$149$	2.00000	0.163846	0.0819232	+	0.996639i	$0.473894\pi$
$150$	0	0
$151$	4.00000	0.325515	0.162758	−	0.986666i	$-0.447961\pi$
$151$	4.00000	0.325515	0.162758	+	0.986666i	$0.447961\pi$
$152$	0	0
$153$	0	0
$154$	8.00000	0.644658
$155$	0	0
$156$	2.00000	0.160128
$157$	− 18.0000i	− 1.43656i	−0.695756	−	0.718278i	$-0.744931\pi$
$157$	− 18.0000i	− 1.43656i	0.695756	−	0.718278i	$-0.255069\pi$
$158$	− 4.00000i	− 0.318223i
$159$	−6.00000	−0.475831
$160$	0	0
$161$	−24.0000	−1.89146
$162$	1.00000i	0.0785674i
$163$	− 12.0000i	− 0.939913i	−0.882690	−	0.469956i	$-0.844270\pi$
$163$	− 12.0000i	− 0.939913i	0.882690	−	0.469956i	$-0.155730\pi$
$164$	10.0000	0.780869
$165$	0	0
$166$	−8.00000	−0.620920
$167$	− 14.0000i	− 1.08335i	−0.840587	−	0.541676i	$-0.817790\pi$
$167$	− 14.0000i	− 1.08335i	0.840587	−	0.541676i	$-0.182210\pi$
$168$	− 4.00000i	− 0.308607i
$169$	9.00000	0.692308
$170$	0	0
$171$	0	0
$172$	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	$-0.178635\pi$
$174$	0	0
$175$	0	0
$176$	2.00000	0.150756
$177$	4.00000i	0.300658i
$178$	− 6.00000i	− 0.449719i
$179$	2.00000	0.149487	0.0747435	−	0.997203i	$-0.476186\pi$
$179$	2.00000	0.149487	0.0747435	+	0.997203i	$0.476186\pi$
$180$	0	0
$181$	−20.0000	−1.48659	−0.743294	−	0.668965i	$-0.766738\pi$
$181$	−20.0000	−1.48659	−0.743294	+	0.668965i	$0.766738\pi$
$182$	8.00000i	0.592999i
$183$	0	0
$184$	−6.00000	−0.442326
$185$	0	0
$186$	−1.00000	−0.0733236
$187$	0	0
$188$	4.00000i	0.291730i
$189$	−4.00000	−0.290957
$190$	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
$196$	9.00000	0.642857
$197$	6.00000i	0.427482i	0.976890	+	0.213741i	$0.0685649\pi$
$197$	6.00000i	0.427482i	−0.976890	+	0.213741i	$0.931435\pi$
$198$	− 2.00000i	− 0.142134i
$199$	−20.0000	−1.41776	−0.708881	−	0.705328i	$-0.750800\pi$
$199$	−20.0000	−1.41776	−0.708881	+	0.705328i	$0.750800\pi$
$200$	0	0
$201$	4.00000	0.282138
$202$	− 10.0000i	− 0.703598i
$203$	0	0
$204$	0	0
$205$	0	0
$206$	12.0000	0.836080
$207$	6.00000i	0.417029i
$208$	2.00000i	0.138675i
$209$	0	0
$210$	0	0
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	− 6.00000i	− 0.412082i
$213$	− 16.0000i	− 1.09630i
$214$	12.0000	0.820303
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	− 4.00000i	− 0.271538i
$218$	2.00000i	0.135457i
$219$	−4.00000	−0.270295
$220$	0	0
$221$	0	0
$222$	− 2.00000i	− 0.134231i
$223$	− 10.0000i	− 0.669650i	−0.942280	−	0.334825i	$-0.891323\pi$
$223$	− 10.0000i	− 0.669650i	0.942280	−	0.334825i	$-0.108677\pi$
$224$	4.00000	0.267261
$225$	0	0
$226$	−2.00000	−0.133038
$227$	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
$229$	20.0000	1.32164	0.660819	−	0.750546i	$-0.270209\pi$
$229$	20.0000	1.32164	0.660819	+	0.750546i	$0.270209\pi$
$230$	0	0
$231$	8.00000	0.526361
$232$	0	0
$233$	− 22.0000i	− 1.44127i	−0.693316	−	0.720634i	$-0.743851\pi$
$233$	− 22.0000i	− 1.44127i	0.693316	−	0.720634i	$-0.256149\pi$
$234$	2.00000	0.130744
$235$	0	0
$236$	−4.00000	−0.260378
$237$	− 4.00000i	− 0.259828i
$238$	0	0
$239$	8.00000	0.517477	0.258738	−	0.965947i	$-0.416693\pi$
$239$	8.00000	0.517477	0.258738	+	0.965947i	$0.416693\pi$
$240$	0	0
$241$	−22.0000	−1.41714	−0.708572	−	0.705638i	$-0.750660\pi$
$241$	−22.0000	−1.41714	−0.708572	+	0.705638i	$0.750660\pi$
$242$	− 7.00000i	− 0.449977i
$243$	1.00000i	0.0641500i
$244$	0	0
$245$	0	0
$246$	10.0000	0.637577
$247$	0	0
$248$	− 1.00000i	− 0.0635001i
$249$	−8.00000	−0.506979
$250$	0	0
$251$	2.00000	0.126239	0.0631194	−	0.998006i	$-0.479895\pi$
$251$	2.00000	0.126239	0.0631194	+	0.998006i	$0.479895\pi$
$252$	− 4.00000i	− 0.251976i
$253$	− 12.0000i	− 0.754434i
$254$	18.0000	1.12942
$255$	0	0
$256$	1.00000	0.0625000
$257$	− 18.0000i	− 1.12281i	−0.827541	−	0.561405i	$-0.810261\pi$
$257$	− 18.0000i	− 1.12281i	0.827541	−	0.561405i	$-0.189739\pi$
$258$	4.00000i	0.249029i
$259$	8.00000	0.497096
$260$	0	0
$261$	0	0
$262$	4.00000i	0.247121i
$263$	10.0000i	0.616626i	0.951285	+	0.308313i	$0.0997645\pi$
$263$	10.0000i	0.616626i	−0.951285	+	0.308313i	$0.900236\pi$
$264$	2.00000	0.123091
$265$	0	0
$266$	0	0
$267$	− 6.00000i	− 0.367194i
$268$	4.00000i	0.244339i
$269$	24.0000	1.46331	0.731653	−	0.681677i	$-0.238749\pi$
$269$	24.0000	1.46331	0.731653	+	0.681677i	$0.238749\pi$
$270$	0	0
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	0	0
$273$	8.00000i	0.484182i
$274$	−12.0000	−0.724947
$275$	0	0
$276$	−6.00000	−0.361158
$277$	2.00000i	0.120168i	0.998193	+	0.0600842i	$0.0191369\pi$
$277$	2.00000i	0.120168i	−0.998193	+	0.0600842i	$0.980863\pi$
$278$	6.00000i	0.359856i
$279$	−1.00000	−0.0598684
$280$	0	0
$281$	6.00000	0.357930	0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000	0.357930	0.178965	+	0.983855i	$0.442725\pi$
$282$	4.00000i	0.238197i
$283$	− 4.00000i	− 0.237775i	−0.992908	−	0.118888i	$-0.962067\pi$
$283$	− 4.00000i	− 0.237775i	0.992908	−	0.118888i	$-0.0379328\pi$
$284$	16.0000	0.949425
$285$	0	0
$286$	−4.00000	−0.236525
$287$	40.0000i	2.36113i
$288$	− 1.00000i	− 0.0589256i
$289$	17.0000	1.00000
$290$	0	0
$291$	14.0000	0.820695
$292$	− 4.00000i	− 0.234082i
$293$	− 2.00000i	− 0.116841i	−0.998292	−	0.0584206i	$-0.981394\pi$
$293$	− 2.00000i	− 0.116841i	0.998292	−	0.0584206i	$-0.0186065\pi$
$294$	9.00000	0.524891
$295$	0	0
$296$	2.00000	0.116248
$297$	− 2.00000i	− 0.116052i
$298$	2.00000i	0.115857i
$299$	12.0000	0.693978
$300$	0	0
$301$	−16.0000	−0.922225
$302$	4.00000i	0.230174i
$303$	− 10.0000i	− 0.574485i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	28.0000i	1.59804i	0.601302	+	0.799022i	$0.294649\pi$
$307$	28.0000i	1.59804i	−0.601302	+	0.799022i	$0.705351\pi$
$308$	8.00000i	0.455842i
$309$	12.0000	0.682656
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	2.00000i	0.113228i
$313$	28.0000i	1.58265i	0.611393	+	0.791327i	$0.290609\pi$
$313$	28.0000i	1.58265i	−0.611393	+	0.791327i	$0.709391\pi$
$314$	18.0000	1.01580
$315$	0	0
$316$	4.00000	0.225018
$317$	6.00000i	0.336994i	0.985702	+	0.168497i	$0.0538913\pi$
$317$	6.00000i	0.336994i	−0.985702	+	0.168497i	$0.946109\pi$
$318$	− 6.00000i	− 0.336463i
$319$	0	0
$320$	0	0
$321$	12.0000	0.669775
$322$	− 24.0000i	− 1.33747i
$323$	0	0
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	12.0000	0.664619
$327$	2.00000i	0.110600i
$328$	10.0000i	0.552158i
$329$	−16.0000	−0.882109
$330$	0	0
$331$	18.0000	0.989369	0.494685	−	0.869072i	$-0.335284\pi$
$331$	18.0000	0.989369	0.494685	+	0.869072i	$0.335284\pi$
$332$	− 8.00000i	− 0.439057i
$333$	− 2.00000i	− 0.109599i
$334$	14.0000	0.766046
$335$	0	0
$336$	4.00000	0.218218
$337$	32.0000i	1.74315i	0.490261	+	0.871576i	$0.336901\pi$
$337$	32.0000i	1.74315i	−0.490261	+	0.871576i	$0.663099\pi$
$338$	9.00000i	0.489535i
$339$	−2.00000	−0.108625
$340$	0	0
$341$	2.00000	0.108306
$342$	0	0
$343$	8.00000i	0.431959i
$344$	−4.00000	−0.215666
$345$	0	0
$346$	14.0000	0.752645
$347$	− 24.0000i	− 1.28839i	−0.764862	−	0.644194i	$-0.777193\pi$
$347$	− 24.0000i	− 1.28839i	0.764862	−	0.644194i	$-0.222807\pi$
$348$	0	0
$349$	14.0000	0.749403	0.374701	−	0.927146i	$-0.377745\pi$
$349$	14.0000	0.749403	0.374701	+	0.927146i	$0.377745\pi$
$350$	0	0
$351$	2.00000	0.106752
$352$	2.00000i	0.106600i
$353$	36.0000i	1.91609i	0.286623	+	0.958043i	$0.407467\pi$
$353$	36.0000i	1.91609i	−0.286623	+	0.958043i	$0.592533\pi$
$354$	−4.00000	−0.212598
$355$	0	0
$356$	6.00000	0.317999
$357$	0	0
$358$	2.00000i	0.105703i
$359$	−24.0000	−1.26667	−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000	−1.26667	−0.633336	+	0.773877i	$0.718315\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	− 20.0000i	− 1.05118i
$363$	− 7.00000i	− 0.367405i
$364$	−8.00000	−0.419314
$365$	0	0
$366$	0	0
$367$	− 26.0000i	− 1.35719i	−0.734513	−	0.678594i	$-0.762589\pi$
$367$	− 26.0000i	− 1.35719i	0.734513	−	0.678594i	$-0.237411\pi$
$368$	− 6.00000i	− 0.312772i
$369$	10.0000	0.520579
$370$	0	0
$371$	24.0000	1.24602
$372$	− 1.00000i	− 0.0518476i
$373$	6.00000i	0.310668i	0.987862	+	0.155334i	$0.0496454\pi$
$373$	6.00000i	0.310668i	−0.987862	+	0.155334i	$0.950355\pi$
$374$	0	0
$375$	0	0
$376$	−4.00000	−0.206284
$377$	0	0
$378$	− 4.00000i	− 0.205738i
$379$	16.0000	0.821865	0.410932	−	0.911666i	$-0.365203\pi$
$379$	16.0000	0.821865	0.410932	+	0.911666i	$0.365203\pi$
$380$	0	0
$381$	18.0000	0.922168
$382$	0	0
$383$	6.00000i	0.306586i	0.988181	+	0.153293i	$0.0489878\pi$
$383$	6.00000i	0.306586i	−0.988181	+	0.153293i	$0.951012\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	10.0000	0.508987
$387$	4.00000i	0.203331i
$388$	14.0000i	0.710742i
$389$	−20.0000	−1.01404	−0.507020	−	0.861934i	$-0.669253\pi$
$389$	−20.0000	−1.01404	−0.507020	+	0.861934i	$0.669253\pi$
$390$	0	0
$391$	0	0
$392$	9.00000i	0.454569i
$393$	4.00000i	0.201773i
$394$	−6.00000	−0.302276
$395$	0	0
$396$	2.00000	0.100504
$397$	− 18.0000i	− 0.903394i	−0.892171	−	0.451697i	$-0.850819\pi$
$397$	− 18.0000i	− 0.903394i	0.892171	−	0.451697i	$-0.149181\pi$
$398$	− 20.0000i	− 1.00251i
$399$	0	0
$400$	0	0
$401$	30.0000	1.49813	0.749064	−	0.662497i	$-0.230503\pi$
$401$	30.0000	1.49813	0.749064	+	0.662497i	$0.230503\pi$
$402$	4.00000i	0.199502i
$403$	2.00000i	0.0996271i
$404$	10.0000	0.497519
$405$	0	0
$406$	0	0
$407$	4.00000i	0.198273i
$408$	0	0
$409$	18.0000	0.890043	0.445021	−	0.895520i	$-0.353196\pi$
$409$	18.0000	0.890043	0.445021	+	0.895520i	$0.353196\pi$
$410$	0	0
$411$	−12.0000	−0.591916
$412$	12.0000i	0.591198i
$413$	− 16.0000i	− 0.787309i
$414$	−6.00000	−0.294884
$415$	0	0
$416$	−2.00000	−0.0980581
$417$	6.00000i	0.293821i
$418$	0	0
$419$	−36.0000	−1.75872	−0.879358	−	0.476162i	$-0.842028\pi$
$419$	−36.0000	−1.75872	−0.879358	+	0.476162i	$0.842028\pi$
$420$	0	0
$421$	−30.0000	−1.46211	−0.731055	−	0.682318i	$-0.760972\pi$
$421$	−30.0000	−1.46211	−0.731055	+	0.682318i	$0.760972\pi$
$422$	− 4.00000i	− 0.194717i
$423$	4.00000i	0.194487i
$424$	6.00000	0.291386
$425$	0	0
$426$	16.0000	0.775203
$427$	0	0
$428$	12.0000i	0.580042i
$429$	−4.00000	−0.193122
$430$	0	0
$431$	40.0000	1.92673	0.963366	−	0.268190i	$-0.0864254\pi$
$431$	40.0000	1.92673	0.963366	+	0.268190i	$0.0864254\pi$
$432$	− 1.00000i	− 0.0481125i
$433$	− 20.0000i	− 0.961139i	−0.876957	−	0.480569i	$-0.840430\pi$
$433$	− 20.0000i	− 0.961139i	0.876957	−	0.480569i	$-0.159570\pi$
$434$	4.00000	0.192006
$435$	0	0
$436$	−2.00000	−0.0957826
$437$	0	0
$438$	− 4.00000i	− 0.191127i
$439$	−24.0000	−1.14546	−0.572729	−	0.819745i	$-0.694115\pi$
$439$	−24.0000	−1.14546	−0.572729	+	0.819745i	$0.694115\pi$
$440$	0	0
$441$	9.00000	0.428571
$442$	0	0
$443$	12.0000i	0.570137i	0.958507	+	0.285069i	$0.0920164\pi$
$443$	12.0000i	0.570137i	−0.958507	+	0.285069i	$0.907984\pi$
$444$	2.00000	0.0949158
$445$	0	0
$446$	10.0000	0.473514
$447$	2.00000i	0.0945968i
$448$	4.00000i	0.188982i
$449$	−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
$451$	−20.0000	−0.941763
$452$	− 2.00000i	− 0.0940721i
$453$	4.00000i	0.187936i
$454$	−20.0000	−0.938647
$455$	0	0
$456$	0	0
$457$	− 8.00000i	− 0.374224i	−0.982339	−	0.187112i	$-0.940087\pi$
$457$	− 8.00000i	− 0.374224i	0.982339	−	0.187112i	$-0.0599128\pi$
$458$	20.0000i	0.934539i
$459$	0	0
$460$	0	0
$461$	0	0		−	1.00000i	$-0.5\pi$
$461$	0	0			1.00000i	$0.5\pi$
$462$	8.00000i	0.372194i
$463$	− 18.0000i	− 0.836531i	−0.908325	−	0.418265i	$-0.862638\pi$
$463$	− 18.0000i	− 0.836531i	0.908325	−	0.418265i	$-0.137362\pi$
$464$	0	0
$465$	0	0
$466$	22.0000	1.01913
$467$	20.0000i	0.925490i	0.886492	+	0.462745i	$0.153135\pi$
$467$	20.0000i	0.925490i	−0.886492	+	0.462745i	$0.846865\pi$
$468$	2.00000i	0.0924500i
$469$	−16.0000	−0.738811
$470$	0	0
$471$	18.0000	0.829396
$472$	− 4.00000i	− 0.184115i
$473$	− 8.00000i	− 0.367840i
$474$	4.00000	0.183726
$475$	0	0
$476$	0	0
$477$	− 6.00000i	− 0.274721i
$478$	8.00000i	0.365911i
$479$	−16.0000	−0.731059	−0.365529	−	0.930800i	$-0.619112\pi$
$479$	−16.0000	−0.731059	−0.365529	+	0.930800i	$0.619112\pi$
$480$	0	0
$481$	−4.00000	−0.182384
$482$	− 22.0000i	− 1.00207i
$483$	− 24.0000i	− 1.09204i
$484$	7.00000	0.318182
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	− 10.0000i	− 0.453143i	−0.973995	−	0.226572i	$-0.927248\pi$
$487$	− 10.0000i	− 0.453143i	0.973995	−	0.226572i	$-0.0727517\pi$
$488$	0	0
$489$	12.0000	0.542659
$490$	0	0
$491$	6.00000	0.270776	0.135388	−	0.990793i	$-0.456772\pi$
$491$	6.00000	0.270776	0.135388	+	0.990793i	$0.456772\pi$
$492$	10.0000i	0.450835i
$493$	0	0
$494$	0	0
$495$	0	0
$496$	1.00000	0.0449013
$497$	64.0000i	2.87079i
$498$	− 8.00000i	− 0.358489i
$499$	−22.0000	−0.984855	−0.492428	−	0.870353i	$-0.663890\pi$
$499$	−22.0000	−0.984855	−0.492428	+	0.870353i	$0.663890\pi$
$500$	0	0
$501$	14.0000	0.625474
$502$	2.00000i	0.0892644i
$503$	16.0000i	0.713405i	0.934218	+	0.356702i	$0.116099\pi$
$503$	16.0000i	0.713405i	−0.934218	+	0.356702i	$0.883901\pi$
$504$	4.00000	0.178174
$505$	0	0
$506$	12.0000	0.533465
$507$	9.00000i	0.399704i
$508$	18.0000i	0.798621i
$509$	4.00000	0.177297	0.0886484	−	0.996063i	$-0.471745\pi$
$509$	4.00000	0.177297	0.0886484	+	0.996063i	$0.471745\pi$
$510$	0	0
$511$	16.0000	0.707798
$512$	1.00000i	0.0441942i
$513$	0	0
$514$	18.0000	0.793946
$515$	0	0
$516$	−4.00000	−0.176090
$517$	− 8.00000i	− 0.351840i
$518$	8.00000i	0.351500i
$519$	14.0000	0.614532
$520$	0	0
$521$	−42.0000	−1.84005	−0.920027	−	0.391856i	$-0.871833\pi$
$521$	−42.0000	−1.84005	−0.920027	+	0.391856i	$0.871833\pi$
$522$	0	0
$523$	− 20.0000i	− 0.874539i	−0.899331	−	0.437269i	$-0.855946\pi$
$523$	− 20.0000i	− 0.874539i	0.899331	−	0.437269i	$-0.144054\pi$
$524$	−4.00000	−0.174741
$525$	0	0
$526$	−10.0000	−0.436021
$527$	0	0
$528$	2.00000i	0.0870388i
$529$	−13.0000	−0.565217
$530$	0	0
$531$	−4.00000	−0.173585
$532$	0	0
$533$	− 20.0000i	− 0.866296i
$534$	6.00000	0.259645
$535$	0	0
$536$	−4.00000	−0.172774
$537$	2.00000i	0.0863064i
$538$	24.0000i	1.03471i
$539$	−18.0000	−0.775315
$540$	0	0
$541$	6.00000	0.257960	0.128980	−	0.991647i	$-0.458830\pi$
$541$	6.00000	0.257960	0.128980	+	0.991647i	$0.458830\pi$
$542$	0	0
$543$	− 20.0000i	− 0.858282i
$544$	0	0
$545$	0	0
$546$	−8.00000	−0.342368
$547$	− 36.0000i	− 1.53925i	−0.638497	−	0.769624i	$-0.720443\pi$
$547$	− 36.0000i	− 1.53925i	0.638497	−	0.769624i	$-0.279557\pi$
$548$	− 12.0000i	− 0.512615i
$549$	0	0
$550$	0	0
$551$	0	0
$552$	− 6.00000i	− 0.255377i
$553$	16.0000i	0.680389i
$554$	−2.00000	−0.0849719
$555$	0	0
$556$	−6.00000	−0.254457
$557$	− 18.0000i	− 0.762684i	−0.924434	−	0.381342i	$-0.875462\pi$
$557$	− 18.0000i	− 0.762684i	0.924434	−	0.381342i	$-0.124538\pi$
$558$	− 1.00000i	− 0.0423334i
$559$	8.00000	0.338364
$560$	0	0
$561$	0	0
$562$	6.00000i	0.253095i
$563$	− 12.0000i	− 0.505740i	−0.967500	−	0.252870i	$-0.918626\pi$
$563$	− 12.0000i	− 0.505740i	0.967500	−	0.252870i	$-0.0813744\pi$
$564$	−4.00000	−0.168430
$565$	0	0
$566$	4.00000	0.168133
$567$	− 4.00000i	− 0.167984i
$568$	16.0000i	0.671345i
$569$	−30.0000	−1.25767	−0.628833	−	0.777541i	$-0.716467\pi$
$569$	−30.0000	−1.25767	−0.628833	+	0.777541i	$0.716467\pi$
$570$	0	0
$571$	−26.0000	−1.08807	−0.544033	−	0.839064i	$-0.683103\pi$
$571$	−26.0000	−1.08807	−0.544033	+	0.839064i	$0.683103\pi$
$572$	− 4.00000i	− 0.167248i
$573$	0	0
$574$	−40.0000	−1.66957
$575$	0	0
$576$	1.00000	0.0416667
$577$	6.00000i	0.249783i	0.992170	+	0.124892i	$0.0398583\pi$
$577$	6.00000i	0.249783i	−0.992170	+	0.124892i	$0.960142\pi$
$578$	17.0000i	0.707107i
$579$	10.0000	0.415586
$580$	0	0
$581$	32.0000	1.32758
$582$	14.0000i	0.580319i
$583$	12.0000i	0.496989i
$584$	4.00000	0.165521
$585$	0	0
$586$	2.00000	0.0826192
$587$	24.0000i	0.990586i	0.868726	+	0.495293i	$0.164939\pi$
$587$	24.0000i	0.990586i	−0.868726	+	0.495293i	$0.835061\pi$
$588$	9.00000i	0.371154i
$589$	0	0
$590$	0	0
$591$	−6.00000	−0.246807
$592$	2.00000i	0.0821995i
$593$	− 34.0000i	− 1.39621i	−0.715994	−	0.698106i	$-0.754026\pi$
$593$	− 34.0000i	− 1.39621i	0.715994	−	0.698106i	$-0.245974\pi$
$594$	2.00000	0.0820610
$595$	0	0
$596$	−2.00000	−0.0819232
$597$	− 20.0000i	− 0.818546i
$598$	12.0000i	0.490716i
$599$	48.0000	1.96123	0.980613	−	0.195952i	$-0.0627798\pi$
$599$	48.0000	1.96123	0.980613	+	0.195952i	$0.0627798\pi$
$600$	0	0
$601$	−10.0000	−0.407909	−0.203954	−	0.978980i	$-0.565379\pi$
$601$	−10.0000	−0.407909	−0.203954	+	0.978980i	$0.565379\pi$
$602$	− 16.0000i	− 0.652111i
$603$	4.00000i	0.162893i
$604$	−4.00000	−0.162758
$605$	0	0
$606$	10.0000	0.406222
$607$	16.0000i	0.649420i	0.945814	+	0.324710i	$0.105267\pi$
$607$	16.0000i	0.649420i	−0.945814	+	0.324710i	$0.894733\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	8.00000	0.323645
$612$	0	0
$613$	− 42.0000i	− 1.69636i	−0.529705	−	0.848182i	$-0.677697\pi$
$613$	− 42.0000i	− 1.69636i	0.529705	−	0.848182i	$-0.322303\pi$
$614$	−28.0000	−1.12999
$615$	0	0
$616$	−8.00000	−0.322329
$617$	− 30.0000i	− 1.20775i	−0.797077	−	0.603877i	$-0.793622\pi$
$617$	− 30.0000i	− 1.20775i	0.797077	−	0.603877i	$-0.206378\pi$
$618$	12.0000i	0.482711i
$619$	−34.0000	−1.36658	−0.683288	−	0.730149i	$-0.739451\pi$
$619$	−34.0000	−1.36658	−0.683288	+	0.730149i	$0.739451\pi$
$620$	0	0
$621$	−6.00000	−0.240772
$622$	0	0
$623$	24.0000i	0.961540i
$624$	−2.00000	−0.0800641
$625$	0	0
$626$	−28.0000	−1.11911
$627$	0	0
$628$	18.0000i	0.718278i
$629$	0	0
$630$	0	0
$631$	8.00000	0.318475	0.159237	−	0.987240i	$-0.449096\pi$
$631$	8.00000	0.318475	0.159237	+	0.987240i	$0.449096\pi$
$632$	4.00000i	0.159111i
$633$	− 4.00000i	− 0.158986i
$634$	−6.00000	−0.238290
$635$	0	0
$636$	6.00000	0.237915
$637$	− 18.0000i	− 0.713186i
$638$	0	0
$639$	16.0000	0.632950
$640$	0	0
$641$	10.0000	0.394976	0.197488	−	0.980305i	$-0.436722\pi$
$641$	10.0000	0.394976	0.197488	+	0.980305i	$0.436722\pi$
$642$	12.0000i	0.473602i
$643$	44.0000i	1.73519i	0.497271	+	0.867595i	$0.334335\pi$
$643$	44.0000i	1.73519i	−0.497271	+	0.867595i	$0.665665\pi$
$644$	24.0000	0.945732
$645$	0	0
$646$	0	0
$647$	18.0000i	0.707653i	0.935311	+	0.353827i	$0.115120\pi$
$647$	18.0000i	0.707653i	−0.935311	+	0.353827i	$0.884880\pi$
$648$	− 1.00000i	− 0.0392837i
$649$	8.00000	0.314027
$650$	0	0
$651$	4.00000	0.156772
$652$	12.0000i	0.469956i
$653$	30.0000i	1.17399i	0.809590	+	0.586995i	$0.199689\pi$
$653$	30.0000i	1.17399i	−0.809590	+	0.586995i	$0.800311\pi$
$654$	−2.00000	−0.0782062
$655$	0	0
$656$	−10.0000	−0.390434
$657$	− 4.00000i	− 0.156055i
$658$	− 16.0000i	− 0.623745i
$659$	28.0000	1.09073	0.545363	−	0.838200i	$-0.316392\pi$
$659$	28.0000	1.09073	0.545363	+	0.838200i	$0.316392\pi$
$660$	0	0
$661$	14.0000	0.544537	0.272268	−	0.962221i	$-0.412226\pi$
$661$	14.0000	0.544537	0.272268	+	0.962221i	$0.412226\pi$
$662$	18.0000i	0.699590i
$663$	0	0
$664$	8.00000	0.310460
$665$	0	0
$666$	2.00000	0.0774984
$667$	0	0
$668$	14.0000i	0.541676i
$669$	10.0000	0.386622
$670$	0	0
$671$	0	0
$672$	4.00000i	0.154303i
$673$	− 40.0000i	− 1.54189i	−0.636904	−	0.770943i	$-0.719785\pi$
$673$	− 40.0000i	− 1.54189i	0.636904	−	0.770943i	$-0.280215\pi$
$674$	−32.0000	−1.23259
$675$	0	0
$676$	−9.00000	−0.346154
$677$	− 42.0000i	− 1.61419i	−0.590421	−	0.807096i	$-0.701038\pi$
$677$	− 42.0000i	− 1.61419i	0.590421	−	0.807096i	$-0.298962\pi$
$678$	− 2.00000i	− 0.0768095i
$679$	−56.0000	−2.14908
$680$	0	0
$681$	−20.0000	−0.766402
$682$	2.00000i	0.0765840i
$683$	− 20.0000i	− 0.765279i	−0.923898	−	0.382639i	$-0.875015\pi$
$683$	− 20.0000i	− 0.765279i	0.923898	−	0.382639i	$-0.124985\pi$
$684$	0	0
$685$	0	0
$686$	−8.00000	−0.305441
$687$	20.0000i	0.763048i
$688$	− 4.00000i	− 0.152499i
$689$	−12.0000	−0.457164
$690$	0	0
$691$	−28.0000	−1.06517	−0.532585	−	0.846376i	$-0.678779\pi$
$691$	−28.0000	−1.06517	−0.532585	+	0.846376i	$0.678779\pi$
$692$	14.0000i	0.532200i
$693$	8.00000i	0.303895i
$694$	24.0000	0.911028
$695$	0	0
$696$	0	0
$697$	0	0
$698$	14.0000i	0.529908i
$699$	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	$0.754769\pi$
$702$	2.00000i	0.0754851i
$703$	0	0
$704$	−2.00000	−0.0753778
$705$	0	0
$706$	−36.0000	−1.35488
$707$	40.0000i	1.50435i
$708$	− 4.00000i	− 0.150329i
$709$	32.0000	1.20179	0.600893	−	0.799330i	$-0.294812\pi$
$709$	32.0000	1.20179	0.600893	+	0.799330i	$0.294812\pi$
$710$	0	0
$711$	4.00000	0.150012
$712$	6.00000i	0.224860i
$713$	− 6.00000i	− 0.224702i
$714$	0	0
$715$	0	0
$716$	−2.00000	−0.0747435
$717$	8.00000i	0.298765i
$718$	− 24.0000i	− 0.895672i
$719$	12.0000	0.447524	0.223762	−	0.974644i	$-0.428166\pi$
$719$	12.0000	0.447524	0.223762	+	0.974644i	$0.428166\pi$
$720$	0	0
$721$	−48.0000	−1.78761
$722$	− 19.0000i	− 0.707107i
$723$	− 22.0000i	− 0.818189i
$724$	20.0000	0.743294
$725$	0	0
$726$	7.00000	0.259794
$727$	− 48.0000i	− 1.78022i	−0.455744	−	0.890111i	$-0.650627\pi$
$727$	− 48.0000i	− 1.78022i	0.455744	−	0.890111i	$-0.349373\pi$
$728$	− 8.00000i	− 0.296500i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	0	0
$732$	0	0
$733$	− 14.0000i	− 0.517102i	−0.965998	−	0.258551i	$-0.916755\pi$
$733$	− 14.0000i	− 0.517102i	0.965998	−	0.258551i	$-0.0832450\pi$
$734$	26.0000	0.959678
$735$	0	0
$736$	6.00000	0.221163
$737$	− 8.00000i	− 0.294684i
$738$	10.0000i	0.368105i
$739$	−2.00000	−0.0735712	−0.0367856	−	0.999323i	$-0.511712\pi$
$739$	−2.00000	−0.0735712	−0.0367856	+	0.999323i	$0.511712\pi$
$740$	0	0
$741$	0	0
$742$	24.0000i	0.881068i
$743$	26.0000i	0.953847i	0.878945	+	0.476924i	$0.158248\pi$
$743$	26.0000i	0.953847i	−0.878945	+	0.476924i	$0.841752\pi$
$744$	1.00000	0.0366618
$745$	0	0
$746$	−6.00000	−0.219676
$747$	− 8.00000i	− 0.292705i
$748$	0	0
$749$	−48.0000	−1.75388
$750$	0	0
$751$	16.0000	0.583848	0.291924	−	0.956441i	$-0.405705\pi$
$751$	16.0000	0.583848	0.291924	+	0.956441i	$0.405705\pi$
$752$	− 4.00000i	− 0.145865i
$753$	2.00000i	0.0728841i
$754$	0	0
$755$	0	0
$756$	4.00000	0.145479
$757$	18.0000i	0.654221i	0.944986	+	0.327111i	$0.106075\pi$
$757$	18.0000i	0.654221i	−0.944986	+	0.327111i	$0.893925\pi$
$758$	16.0000i	0.581146i
$759$	12.0000	0.435572
$760$	0	0
$761$	−26.0000	−0.942499	−0.471250	−	0.882000i	$-0.656197\pi$
$761$	−26.0000	−0.942499	−0.471250	+	0.882000i	$0.656197\pi$
$762$	18.0000i	0.652071i
$763$	− 8.00000i	− 0.289619i
$764$	0	0
$765$	0	0
$766$	−6.00000	−0.216789
$767$	8.00000i	0.288863i
$768$	1.00000i	0.0360844i
$769$	−14.0000	−0.504853	−0.252426	−	0.967616i	$-0.581229\pi$
$769$	−14.0000	−0.504853	−0.252426	+	0.967616i	$0.581229\pi$
$770$	0	0
$771$	18.0000	0.648254
$772$	10.0000i	0.359908i
$773$	− 2.00000i	− 0.0719350i	−0.999353	−	0.0359675i	$-0.988549\pi$
$773$	− 2.00000i	− 0.0719350i	0.999353	−	0.0359675i	$-0.0114513\pi$
$774$	−4.00000	−0.143777
$775$	0	0
$776$	−14.0000	−0.502571
$777$	8.00000i	0.286998i
$778$	− 20.0000i	− 0.717035i
$779$	0	0
$780$	0	0
$781$	−32.0000	−1.14505
$782$	0	0
$783$	0	0
$784$	−9.00000	−0.321429
$785$	0	0
$786$	−4.00000	−0.142675
$787$	40.0000i	1.42585i	0.701242	+	0.712923i	$0.252629\pi$
$787$	40.0000i	1.42585i	−0.701242	+	0.712923i	$0.747371\pi$
$788$	− 6.00000i	− 0.213741i
$789$	−10.0000	−0.356009
$790$	0	0
$791$	8.00000	0.284447
$792$	2.00000i	0.0710669i
$793$	0	0
$794$	18.0000	0.638796
$795$	0	0
$796$	20.0000	0.708881
$797$	− 38.0000i	− 1.34603i	−0.739629	−	0.673015i	$-0.764999\pi$
$797$	− 38.0000i	− 1.34603i	0.739629	−	0.673015i	$-0.235001\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	6.00000	0.212000
$802$	30.0000i	1.05934i
$803$	8.00000i	0.282314i
$804$	−4.00000	−0.141069
$805$	0	0
$806$	−2.00000	−0.0704470
$807$	24.0000i	0.844840i
$808$	10.0000i	0.351799i
$809$	6.00000	0.210949	0.105474	−	0.994422i	$-0.466364\pi$
$809$	6.00000	0.210949	0.105474	+	0.994422i	$0.466364\pi$
$810$	0	0
$811$	32.0000	1.12367	0.561836	−	0.827249i	$-0.310095\pi$
$811$	32.0000	1.12367	0.561836	+	0.827249i	$0.310095\pi$
$812$	0	0
$813$	0	0
$814$	−4.00000	−0.140200
$815$	0	0
$816$	0	0
$817$	0	0
$818$	18.0000i	0.629355i
$819$	−8.00000	−0.279543
$820$	0	0
$821$	24.0000	0.837606	0.418803	−	0.908077i	$-0.362450\pi$
$821$	24.0000	0.837606	0.418803	+	0.908077i	$0.362450\pi$
$822$	− 12.0000i	− 0.418548i
$823$	18.0000i	0.627441i	0.949515	+	0.313720i	$0.101575\pi$
$823$	18.0000i	0.627441i	−0.949515	+	0.313720i	$0.898425\pi$
$824$	−12.0000	−0.418040
$825$	0	0
$826$	16.0000	0.556711
$827$	56.0000i	1.94731i	0.228024	+	0.973655i	$0.426773\pi$
$827$	56.0000i	1.94731i	−0.228024	+	0.973655i	$0.573227\pi$
$828$	− 6.00000i	− 0.208514i
$829$	−4.00000	−0.138926	−0.0694629	−	0.997585i	$-0.522129\pi$
$829$	−4.00000	−0.138926	−0.0694629	+	0.997585i	$0.522129\pi$
$830$	0	0
$831$	−2.00000	−0.0693792
$832$	− 2.00000i	− 0.0693375i
$833$	0	0
$834$	−6.00000	−0.207763
$835$	0	0
$836$	0	0
$837$	− 1.00000i	− 0.0345651i
$838$	− 36.0000i	− 1.24360i
$839$	−16.0000	−0.552381	−0.276191	−	0.961103i	$-0.589072\pi$
$839$	−16.0000	−0.552381	−0.276191	+	0.961103i	$0.589072\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	− 30.0000i	− 1.03387i
$843$	6.00000i	0.206651i
$844$	4.00000	0.137686
$845$	0	0
$846$	−4.00000	−0.137523
$847$	28.0000i	0.962091i
$848$	6.00000i	0.206041i
$849$	4.00000	0.137280
$850$	0	0
$851$	12.0000	0.411355
$852$	16.0000i	0.548151i
$853$	6.00000i	0.205436i	0.994711	+	0.102718i	$0.0327539\pi$
$853$	6.00000i	0.205436i	−0.994711	+	0.102718i	$0.967246\pi$
$854$	0	0
$855$	0	0
$856$	−12.0000	−0.410152
$857$	46.0000i	1.57133i	0.618652	+	0.785665i	$0.287679\pi$
$857$	46.0000i	1.57133i	−0.618652	+	0.785665i	$0.712321\pi$
$858$	− 4.00000i	− 0.136558i
$859$	2.00000	0.0682391	0.0341196	−	0.999418i	$-0.489137\pi$
$859$	2.00000	0.0682391	0.0341196	+	0.999418i	$0.489137\pi$
$860$	0	0
$861$	−40.0000	−1.36320
$862$	40.0000i	1.36241i
$863$	30.0000i	1.02121i	0.859815	+	0.510606i	$0.170579\pi$
$863$	30.0000i	1.02121i	−0.859815	+	0.510606i	$0.829421\pi$
$864$	1.00000	0.0340207
$865$	0	0
$866$	20.0000	0.679628
$867$	17.0000i	0.577350i
$868$	4.00000i	0.135769i
$869$	−8.00000	−0.271381
$870$	0	0
$871$	8.00000	0.271070
$872$	− 2.00000i	− 0.0677285i
$873$	14.0000i	0.473828i
$874$	0	0
$875$	0	0
$876$	4.00000	0.135147
$877$	− 18.0000i	− 0.607817i	−0.952701	−	0.303908i	$-0.901708\pi$
$877$	− 18.0000i	− 0.607817i	0.952701	−	0.303908i	$-0.0982917\pi$
$878$	− 24.0000i	− 0.809961i
$879$	2.00000	0.0674583
$880$	0	0
$881$	22.0000	0.741199	0.370599	−	0.928793i	$-0.379152\pi$
$881$	22.0000	0.741199	0.370599	+	0.928793i	$0.379152\pi$
$882$	9.00000i	0.303046i
$883$	− 36.0000i	− 1.21150i	−0.795656	−	0.605748i	$-0.792874\pi$
$883$	− 36.0000i	− 1.21150i	0.795656	−	0.605748i	$-0.207126\pi$
$884$	0	0
$885$	0	0
$886$	−12.0000	−0.403148
$887$	28.0000i	0.940148i	0.882627	+	0.470074i	$0.155773\pi$
$887$	28.0000i	0.940148i	−0.882627	+	0.470074i	$0.844227\pi$
$888$	2.00000i	0.0671156i
$889$	−72.0000	−2.41480
$890$	0	0
$891$	2.00000	0.0670025
$892$	10.0000i	0.334825i
$893$	0	0
$894$	−2.00000	−0.0668900
$895$	0	0
$896$	−4.00000	−0.133631
$897$	12.0000i	0.400668i
$898$	− 6.00000i	− 0.200223i
$899$	0	0
$900$	0	0
$901$	0	0
$902$	− 20.0000i	− 0.665927i
$903$	− 16.0000i	− 0.532447i
$904$	2.00000	0.0665190
$905$	0	0
$906$	−4.00000	−0.132891
$907$	− 28.0000i	− 0.929725i	−0.885383	−	0.464862i	$-0.846104\pi$
$907$	− 28.0000i	− 0.929725i	0.885383	−	0.464862i	$-0.153896\pi$
$908$	− 20.0000i	− 0.663723i
$909$	10.0000	0.331679
$910$	0	0
$911$	32.0000	1.06021	0.530104	−	0.847933i	$-0.322153\pi$
$911$	32.0000	1.06021	0.530104	+	0.847933i	$0.322153\pi$
$912$	0	0
$913$	16.0000i	0.529523i
$914$	8.00000	0.264616
$915$	0	0
$916$	−20.0000	−0.660819
$917$	− 16.0000i	− 0.528367i
$918$	0	0
$919$	−16.0000	−0.527791	−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000	−0.527791	−0.263896	+	0.964551i	$0.585007\pi$
$920$	0	0
$921$	−28.0000	−0.922631
$922$	0	0
$923$	− 32.0000i	− 1.05329i
$924$	−8.00000	−0.263181
$925$	0	0
$926$	18.0000	0.591517
$927$	12.0000i	0.394132i
$928$	0	0
$929$	−30.0000	−0.984268	−0.492134	−	0.870519i	$-0.663783\pi$
$929$	−30.0000	−0.984268	−0.492134	+	0.870519i	$0.663783\pi$
$930$	0	0
$931$	0	0
$932$	22.0000i	0.720634i
$933$	0	0
$934$	−20.0000	−0.654420
$935$	0	0
$936$	−2.00000	−0.0653720
$937$	− 30.0000i	− 0.980057i	−0.871706	−	0.490029i	$-0.836986\pi$
$937$	− 30.0000i	− 0.980057i	0.871706	−	0.490029i	$-0.163014\pi$
$938$	− 16.0000i	− 0.522419i
$939$	−28.0000	−0.913745
$940$	0	0
$941$	28.0000	0.912774	0.456387	−	0.889781i	$-0.349143\pi$
$941$	28.0000	0.912774	0.456387	+	0.889781i	$0.349143\pi$
$942$	18.0000i	0.586472i
$943$	60.0000i	1.95387i
$944$	4.00000	0.130189
$945$	0	0
$946$	8.00000	0.260102
$947$	36.0000i	1.16984i	0.811090	+	0.584921i	$0.198875\pi$
$947$	36.0000i	1.16984i	−0.811090	+	0.584921i	$0.801125\pi$
$948$	4.00000i	0.129914i
$949$	−8.00000	−0.259691
$950$	0	0
$951$	−6.00000	−0.194563
$952$	0	0
$953$	− 12.0000i	− 0.388718i	−0.980930	−	0.194359i	$-0.937737\pi$
$953$	− 12.0000i	− 0.388718i	0.980930	−	0.194359i	$-0.0622627\pi$
$954$	6.00000	0.194257
$955$	0	0
$956$	−8.00000	−0.258738
$957$	0	0
$958$	− 16.0000i	− 0.516937i
$959$	48.0000	1.55000
$960$	0	0
$961$	1.00000	0.0322581
$962$	− 4.00000i	− 0.128965i
$963$	12.0000i	0.386695i
$964$	22.0000	0.708572
$965$	0	0
$966$	24.0000	0.772187
$967$	− 26.0000i	− 0.836104i	−0.908423	−	0.418052i	$-0.862713\pi$
$967$	− 26.0000i	− 0.836104i	0.908423	−	0.418052i	$-0.137287\pi$
$968$	7.00000i	0.224989i
$969$	0	0
$970$	0	0
$971$	48.0000	1.54039	0.770197	−	0.637806i	$-0.220158\pi$
$971$	48.0000	1.54039	0.770197	+	0.637806i	$0.220158\pi$
$972$	− 1.00000i	− 0.0320750i
$973$	− 24.0000i	− 0.769405i
$974$	10.0000	0.320421
$975$	0	0
$976$	0	0
$977$	18.0000i	0.575871i	0.957650	+	0.287936i	$0.0929689\pi$
$977$	18.0000i	0.575871i	−0.957650	+	0.287936i	$0.907031\pi$
$978$	12.0000i	0.383718i
$979$	−12.0000	−0.383522
$980$	0	0
$981$	−2.00000	−0.0638551
$982$	6.00000i	0.191468i
$983$	10.0000i	0.318950i	0.987202	+	0.159475i	$0.0509802\pi$
$983$	10.0000i	0.318950i	−0.987202	+	0.159475i	$0.949020\pi$
$984$	−10.0000	−0.318788
$985$	0	0
$986$	0	0
$987$	− 16.0000i	− 0.509286i
$988$	0	0
$989$	−24.0000	−0.763156
$990$	0	0
$991$	28.0000	0.889449	0.444725	−	0.895667i	$-0.353302\pi$
$991$	28.0000	0.889449	0.444725	+	0.895667i	$0.353302\pi$
$992$	1.00000i	0.0317500i
$993$	18.0000i	0.571213i
$994$	−64.0000	−2.02996
$995$	0	0
$996$	8.00000	0.253490
$997$	− 6.00000i	− 0.190022i	−0.995476	−	0.0950110i	$-0.969711\pi$
$997$	− 6.00000i	− 0.190022i	0.995476	−	0.0950110i	$-0.0302886\pi$
$998$	− 22.0000i	− 0.696398i
$999$	2.00000	0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4650.2.d.h.3349.2		2
5.2	odd	4		930.2.a.j.1.1	✓	1
5.3	odd	4		4650.2.a.x.1.1		1
5.4	even	2	inner	4650.2.d.h.3349.1		2
15.2	even	4		2790.2.a.v.1.1		1
20.7	even	4		7440.2.a.g.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
930.2.a.j.1.1	✓	1	5.2	odd	4
2790.2.a.v.1.1		1	15.2	even	4
4650.2.a.x.1.1		1	5.3	odd	4
4650.2.d.h.3349.1		2	5.4	even	2	inner
4650.2.d.h.3349.2		2	1.1	even	1	trivial
7440.2.a.g.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)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -4.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -4.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +2.00000 q^{11} -1.00000i q^{12} +2.00000i q^{13} +4.00000 q^{14} +1.00000 q^{16} -1.00000i q^{18} +4.00000 q^{21} +2.00000i q^{22} -6.00000i q^{23} +1.00000 q^{24} -2.00000 q^{26} -1.00000i q^{27} +4.00000i q^{28} +1.00000 q^{31} +1.00000i q^{32} +2.00000i q^{33} +1.00000 q^{36} +2.00000i q^{37} -2.00000 q^{39} -10.0000 q^{41} +4.00000i q^{42} -4.00000i q^{43} -2.00000 q^{44} +6.00000 q^{46} -4.00000i q^{47} +1.00000i q^{48} -9.00000 q^{49} -2.00000i q^{52} +6.00000i q^{53} +1.00000 q^{54} -4.00000 q^{56} +4.00000 q^{59} +1.00000i q^{62} +4.00000i q^{63} -1.00000 q^{64} -2.00000 q^{66} -4.00000i q^{67} +6.00000 q^{69} -16.0000 q^{71} +1.00000i q^{72} +4.00000i q^{73} -2.00000 q^{74} -8.00000i q^{77} -2.00000i q^{78} -4.00000 q^{79} +1.00000 q^{81} -10.0000i q^{82} +8.00000i q^{83} -4.00000 q^{84} +4.00000 q^{86} -2.00000i q^{88} -6.00000 q^{89} +8.00000 q^{91} +6.00000i q^{92} +1.00000i q^{93} +4.00000 q^{94} -1.00000 q^{96} -14.0000i q^{97} -9.00000i q^{98} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9	\( 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9} + 4 q^{11} + 8 q^{14} + 2 q^{16} + 8 q^{21} + 2 q^{24} - 4 q^{26} + 2 q^{31} + 2 q^{36} - 4 q^{39} - 20 q^{41} - 4 q^{44} + 12 q^{46} - 18 q^{49} + 2 q^{54} - 8 q^{56} + 8 q^{59} - 2 q^{64} - 4 q^{66} + 12 q^{69} - 32 q^{71} - 4 q^{74} - 8 q^{79} + 2 q^{81} - 8 q^{84} + 8 q^{86} - 12 q^{89} + 16 q^{91} + 8 q^{94} - 2 q^{96} - 4 q^{99}+O(q^{100}) \) 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9 + 4 * q^11 + 8 * q^14 + 2 * q^16 + 8 * q^21 + 2 * q^24 - 4 * q^26 + 2 * q^31 + 2 * q^36 - 4 * q^39 - 20 * q^41 - 4 * q^44 + 12 * q^46 - 18 * q^49 + 2 * q^54 - 8 * q^56 + 8 * q^59 - 2 * q^64 - 4 * q^66 + 12 * q^69 - 32 * q^71 - 4 * q^74 - 8 * q^79 + 2 * q^81 - 8 * q^84 + 8 * q^86 - 12 * q^89 + 16 * q^91 + 8 * q^94 - 2 * q^96 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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