LMFDB - Embedded newform 4650.2.d.y.3349.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4650,2,Mod(3349,4650)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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:traces = [2,0,0,-2,0,2,0,0,-2,0,6,0,0,-6,0,2,0,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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.y.3349.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +3.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +3.00000 q^{11} +1.00000i q^{12} -2.00000i q^{13} -3.00000 q^{14} +1.00000 q^{16} +4.00000i q^{17} -1.00000i q^{18} +3.00000 q^{19} +3.00000 q^{21} +3.00000i q^{22} +5.00000i q^{23} -1.00000 q^{24} +2.00000 q^{26} +1.00000i q^{27} -3.00000i q^{28} -4.00000 q^{29} +1.00000 q^{31} +1.00000i q^{32} -3.00000i q^{33} -4.00000 q^{34} +1.00000 q^{36} +3.00000i q^{38} -2.00000 q^{39} +4.00000 q^{41} +3.00000i q^{42} +1.00000i q^{43} -3.00000 q^{44} -5.00000 q^{46} -10.0000i q^{47} -1.00000i q^{48} -2.00000 q^{49} +4.00000 q^{51} +2.00000i q^{52} +3.00000i q^{53} -1.00000 q^{54} +3.00000 q^{56} -3.00000i q^{57} -4.00000i q^{58} -6.00000 q^{59} -2.00000 q^{61} +1.00000i q^{62} -3.00000i q^{63} -1.00000 q^{64} +3.00000 q^{66} -2.00000i q^{67} -4.00000i q^{68} +5.00000 q^{69} +7.00000 q^{71} +1.00000i q^{72} +5.00000i q^{73} -3.00000 q^{76} +9.00000i q^{77} -2.00000i q^{78} +1.00000 q^{79} +1.00000 q^{81} +4.00000i q^{82} +12.0000i q^{83} -3.00000 q^{84} -1.00000 q^{86} +4.00000i q^{87} -3.00000i q^{88} -1.00000 q^{89} +6.00000 q^{91} -5.00000i q^{92} -1.00000i q^{93} +10.0000 q^{94} +1.00000 q^{96} +10.0000i q^{97} -2.00000i q^{98} -3.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} + 6 q^{11} - 6 q^{14} + 2 q^{16} + 6 q^{19} + 6 q^{21} - 2 q^{24} + 4 q^{26} - 8 q^{29} + 2 q^{31} - 8 q^{34} + 2 q^{36} - 4 q^{39} + 8 q^{41} - 6 q^{44} - 10 q^{46} - 4 q^{49}+ \cdots - 6 q^{99}+O(q^{100}) $ 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9 + 6 * q^11 - 6 * q^14 + 2 * q^16 + 6 * q^19 + 6 * q^21 - 2 * q^24 + 4 * q^26 - 8 * q^29 + 2 * q^31 - 8 * q^34 + 2 * q^36 - 4 * q^39 + 8 * q^41 - 6 * q^44 - 10 * q^46 - 4 * q^49 + 8 * q^51 - 2 * q^54 + 6 * q^56 - 12 * q^59 - 4 * q^61 - 2 * q^64 + 6 * q^66 + 10 * q^69 + 14 * q^71 - 6 * q^76 + 2 * q^79 + 2 * q^81 - 6 * q^84 - 2 * q^86 - 2 * q^89 + 12 * q^91 + 20 * 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
$7$	3.00000i	1.13389i	0.823754	+	0.566947i	$0.191875\pi$
$7$	3.00000i	1.13389i	−0.823754	+	0.566947i	$0.808125\pi$
$8$	− 1.00000i	− 0.353553i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	3.00000	0.904534	0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000	0.904534	0.452267	+	0.891883i	$0.350615\pi$
$12$	1.00000i	0.288675i
$13$	− 2.00000i	− 0.554700i	−0.960769	−	0.277350i	$-0.910544\pi$
$13$	− 2.00000i	− 0.554700i	0.960769	−	0.277350i	$-0.0894562\pi$
$14$	−3.00000	−0.801784
$15$	0	0
$16$	1.00000	0.250000
$17$	4.00000i	0.970143i	0.874475	+	0.485071i	$0.161206\pi$
$17$	4.00000i	0.970143i	−0.874475	+	0.485071i	$0.838794\pi$
$18$	− 1.00000i	− 0.235702i
$19$	3.00000	0.688247	0.344124	−	0.938924i	$-0.388176\pi$
$19$	3.00000	0.688247	0.344124	+	0.938924i	$0.388176\pi$
$20$	0	0
$21$	3.00000	0.654654
$22$	3.00000i	0.639602i
$23$	5.00000i	1.04257i	0.853382	+	0.521286i	$0.174548\pi$
$23$	5.00000i	1.04257i	−0.853382	+	0.521286i	$0.825452\pi$
$24$	−1.00000	−0.204124
$25$	0	0
$26$	2.00000	0.392232
$27$	1.00000i	0.192450i
$28$	− 3.00000i	− 0.566947i
$29$	−4.00000	−0.742781	−0.371391	−	0.928477i	$-0.621119\pi$
$29$	−4.00000	−0.742781	−0.371391	+	0.928477i	$0.621119\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	1.00000i	0.176777i
$33$	− 3.00000i	− 0.522233i
$34$	−4.00000	−0.685994
$35$	0	0
$36$	1.00000	0.166667
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	3.00000i	0.486664i
$39$	−2.00000	−0.320256
$40$	0	0
$41$	4.00000	0.624695	0.312348	−	0.949968i	$-0.398885\pi$
$41$	4.00000	0.624695	0.312348	+	0.949968i	$0.398885\pi$
$42$	3.00000i	0.462910i
$43$	1.00000i	0.152499i	0.997089	+	0.0762493i	$0.0242945\pi$
$43$	1.00000i	0.152499i	−0.997089	+	0.0762493i	$0.975706\pi$
$44$	−3.00000	−0.452267
$45$	0	0
$46$	−5.00000	−0.737210
$47$	− 10.0000i	− 1.45865i	−0.684167	−	0.729325i	$-0.739834\pi$
$47$	− 10.0000i	− 1.45865i	0.684167	−	0.729325i	$-0.260166\pi$
$48$	− 1.00000i	− 0.144338i
$49$	−2.00000	−0.285714
$50$	0	0
$51$	4.00000	0.560112
$52$	2.00000i	0.277350i
$53$	3.00000i	0.412082i	0.978543	+	0.206041i	$0.0660580\pi$
$53$	3.00000i	0.412082i	−0.978543	+	0.206041i	$0.933942\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	3.00000	0.400892
$57$	− 3.00000i	− 0.397360i
$58$	− 4.00000i	− 0.525226i
$59$	−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
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	1.00000i	0.127000i
$63$	− 3.00000i	− 0.377964i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	3.00000	0.369274
$67$	− 2.00000i	− 0.244339i	−0.992509	−	0.122169i	$-0.961015\pi$
$67$	− 2.00000i	− 0.244339i	0.992509	−	0.122169i	$-0.0389851\pi$
$68$	− 4.00000i	− 0.485071i
$69$	5.00000	0.601929
$70$	0	0
$71$	7.00000	0.830747	0.415374	−	0.909651i	$-0.363651\pi$
$71$	7.00000	0.830747	0.415374	+	0.909651i	$0.363651\pi$
$72$	1.00000i	0.117851i
$73$	5.00000i	0.585206i	0.956234	+	0.292603i	$0.0945214\pi$
$73$	5.00000i	0.585206i	−0.956234	+	0.292603i	$0.905479\pi$
$74$	0	0
$75$	0	0
$76$	−3.00000	−0.344124
$77$	9.00000i	1.02565i
$78$	− 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
$82$	4.00000i	0.441726i
$83$	12.0000i	1.31717i	0.752506	+	0.658586i	$0.228845\pi$
$83$	12.0000i	1.31717i	−0.752506	+	0.658586i	$0.771155\pi$
$84$	−3.00000	−0.327327
$85$	0	0
$86$	−1.00000	−0.107833
$87$	4.00000i	0.428845i
$88$	− 3.00000i	− 0.319801i
$89$	−1.00000	−0.106000	−0.0529999	−	0.998595i	$-0.516878\pi$
$89$	−1.00000	−0.106000	−0.0529999	+	0.998595i	$0.516878\pi$
$90$	0	0
$91$	6.00000	0.628971
$92$	− 5.00000i	− 0.521286i
$93$	− 1.00000i	− 0.103695i
$94$	10.0000	1.03142
$95$	0	0
$96$	1.00000	0.102062
$97$	10.0000i	1.01535i	0.861550	+	0.507673i	$0.169494\pi$
$97$	10.0000i	1.01535i	−0.861550	+	0.507673i	$0.830506\pi$
$98$	− 2.00000i	− 0.202031i
$99$	−3.00000	−0.301511
$100$	0	0
$101$	−1.00000	−0.0995037	−0.0497519	−	0.998762i	$-0.515843\pi$
$101$	−1.00000	−0.0995037	−0.0497519	+	0.998762i	$0.515843\pi$
$102$	4.00000i	0.396059i
$103$	16.0000i	1.57653i	0.615338	+	0.788263i	$0.289020\pi$
$103$	16.0000i	1.57653i	−0.615338	+	0.788263i	$0.710980\pi$
$104$	−2.00000	−0.196116
$105$	0	0
$106$	−3.00000	−0.291386
$107$	− 9.00000i	− 0.870063i	−0.900415	−	0.435031i	$-0.856737\pi$
$107$	− 9.00000i	− 0.870063i	0.900415	−	0.435031i	$-0.143263\pi$
$108$	− 1.00000i	− 0.0962250i
$109$	−20.0000	−1.91565	−0.957826	−	0.287348i	$-0.907226\pi$
$109$	−20.0000	−1.91565	−0.957826	+	0.287348i	$0.907226\pi$
$110$	0	0
$111$	0	0
$112$	3.00000i	0.283473i
$113$	9.00000i	0.846649i	0.905978	+	0.423324i	$0.139137\pi$
$113$	9.00000i	0.846649i	−0.905978	+	0.423324i	$0.860863\pi$
$114$	3.00000	0.280976
$115$	0	0
$116$	4.00000	0.371391
$117$	2.00000i	0.184900i
$118$	− 6.00000i	− 0.552345i
$119$	−12.0000	−1.10004
$120$	0	0
$121$	−2.00000	−0.181818
$122$	− 2.00000i	− 0.181071i
$123$	− 4.00000i	− 0.360668i
$124$	−1.00000	−0.0898027
$125$	0	0
$126$	3.00000	0.267261
$127$	− 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
$129$	1.00000	0.0880451
$130$	0	0
$131$	6.00000	0.524222	0.262111	−	0.965038i	$-0.415581\pi$
$131$	6.00000	0.524222	0.262111	+	0.965038i	$0.415581\pi$
$132$	3.00000i	0.261116i
$133$	9.00000i	0.780399i
$134$	2.00000	0.172774
$135$	0	0
$136$	4.00000	0.342997
$137$	10.0000i	0.854358i	0.904167	+	0.427179i	$0.140493\pi$
$137$	10.0000i	0.854358i	−0.904167	+	0.427179i	$0.859507\pi$
$138$	5.00000i	0.425628i
$139$	14.0000	1.18746	0.593732	−	0.804663i	$-0.297654\pi$
$139$	14.0000	1.18746	0.593732	+	0.804663i	$0.297654\pi$
$140$	0	0
$141$	−10.0000	−0.842152
$142$	7.00000i	0.587427i
$143$	− 6.00000i	− 0.501745i
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	−5.00000	−0.413803
$147$	2.00000i	0.164957i
$148$	0	0
$149$	11.0000	0.901155	0.450578	−	0.892737i	$-0.351218\pi$
$149$	11.0000	0.901155	0.450578	+	0.892737i	$0.351218\pi$
$150$	0	0
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	− 3.00000i	− 0.243332i
$153$	− 4.00000i	− 0.323381i
$154$	−9.00000	−0.725241
$155$	0	0
$156$	2.00000	0.160128
$157$	5.00000i	0.399043i	0.979893	+	0.199522i	$0.0639388\pi$
$157$	5.00000i	0.399043i	−0.979893	+	0.199522i	$0.936061\pi$
$158$	1.00000i	0.0795557i
$159$	3.00000	0.237915
$160$	0	0
$161$	−15.0000	−1.18217
$162$	1.00000i	0.0785674i
$163$	14.0000i	1.09656i	0.836293	+	0.548282i	$0.184718\pi$
$163$	14.0000i	1.09656i	−0.836293	+	0.548282i	$0.815282\pi$
$164$	−4.00000	−0.312348
$165$	0	0
$166$	−12.0000	−0.931381
$167$	19.0000i	1.47026i	0.677924	+	0.735132i	$0.262880\pi$
$167$	19.0000i	1.47026i	−0.677924	+	0.735132i	$0.737120\pi$
$168$	− 3.00000i	− 0.231455i
$169$	9.00000	0.692308
$170$	0	0
$171$	−3.00000	−0.229416
$172$	− 1.00000i	− 0.0762493i
$173$	− 22.0000i	− 1.67263i	−0.548250	−	0.836315i	$-0.684706\pi$
$173$	− 22.0000i	− 1.67263i	0.548250	−	0.836315i	$-0.315294\pi$
$174$	−4.00000	−0.303239
$175$	0	0
$176$	3.00000	0.226134
$177$	6.00000i	0.450988i
$178$	− 1.00000i	− 0.0749532i
$179$	−4.00000	−0.298974	−0.149487	−	0.988764i	$-0.547762\pi$
$179$	−4.00000	−0.298974	−0.149487	+	0.988764i	$0.547762\pi$
$180$	0	0
$181$	5.00000	0.371647	0.185824	−	0.982583i	$-0.440505\pi$
$181$	5.00000	0.371647	0.185824	+	0.982583i	$0.440505\pi$
$182$	6.00000i	0.444750i
$183$	2.00000i	0.147844i
$184$	5.00000	0.368605
$185$	0	0
$186$	1.00000	0.0733236
$187$	12.0000i	0.877527i
$188$	10.0000i	0.729325i
$189$	−3.00000	−0.218218
$190$	0	0
$191$	16.0000	1.15772	0.578860	−	0.815427i	$-0.303498\pi$
$191$	16.0000	1.15772	0.578860	+	0.815427i	$0.303498\pi$
$192$	1.00000i	0.0721688i
$193$	− 6.00000i	− 0.431889i	−0.976406	−	0.215945i	$-0.930717\pi$
$193$	− 6.00000i	− 0.431889i	0.976406	−	0.215945i	$-0.0692831\pi$
$194$	−10.0000	−0.717958
$195$	0	0
$196$	2.00000	0.142857
$197$	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
$199$	−7.00000	−0.496217	−0.248108	−	0.968732i	$-0.579809\pi$
$199$	−7.00000	−0.496217	−0.248108	+	0.968732i	$0.579809\pi$
$200$	0	0
$201$	−2.00000	−0.141069
$202$	− 1.00000i	− 0.0703598i
$203$	− 12.0000i	− 0.842235i
$204$	−4.00000	−0.280056
$205$	0	0
$206$	−16.0000	−1.11477
$207$	− 5.00000i	− 0.347524i
$208$	− 2.00000i	− 0.138675i
$209$	9.00000	0.622543
$210$	0	0
$211$	−19.0000	−1.30801	−0.654007	−	0.756489i	$-0.726913\pi$
$211$	−19.0000	−1.30801	−0.654007	+	0.756489i	$0.726913\pi$
$212$	− 3.00000i	− 0.206041i
$213$	− 7.00000i	− 0.479632i
$214$	9.00000	0.615227
$215$	0	0
$216$	1.00000	0.0680414
$217$	3.00000i	0.203653i
$218$	− 20.0000i	− 1.35457i
$219$	5.00000	0.337869
$220$	0	0
$221$	8.00000	0.538138
$222$	0	0
$223$	− 6.00000i	− 0.401790i	−0.979613	−	0.200895i	$-0.935615\pi$
$223$	− 6.00000i	− 0.401790i	0.979613	−	0.200895i	$-0.0643850\pi$
$224$	−3.00000	−0.200446
$225$	0	0
$226$	−9.00000	−0.598671
$227$	27.0000i	1.79205i	0.444001	+	0.896026i	$0.353559\pi$
$227$	27.0000i	1.79205i	−0.444001	+	0.896026i	$0.646441\pi$
$228$	3.00000i	0.198680i
$229$	−13.0000	−0.859064	−0.429532	−	0.903052i	$-0.641321\pi$
$229$	−13.0000	−0.859064	−0.429532	+	0.903052i	$0.641321\pi$
$230$	0	0
$231$	9.00000	0.592157
$232$	4.00000i	0.262613i
$233$	15.0000i	0.982683i	0.870967	+	0.491341i	$0.163493\pi$
$233$	15.0000i	0.982683i	−0.870967	+	0.491341i	$0.836507\pi$
$234$	−2.00000	−0.130744
$235$	0	0
$236$	6.00000	0.390567
$237$	− 1.00000i	− 0.0649570i
$238$	− 12.0000i	− 0.777844i
$239$	−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
$241$	−18.0000	−1.15948	−0.579741	−	0.814801i	$-0.696846\pi$
$241$	−18.0000	−1.15948	−0.579741	+	0.814801i	$0.696846\pi$
$242$	− 2.00000i	− 0.128565i
$243$	− 1.00000i	− 0.0641500i
$244$	2.00000	0.128037
$245$	0	0
$246$	4.00000	0.255031
$247$	− 6.00000i	− 0.381771i
$248$	− 1.00000i	− 0.0635001i
$249$	12.0000	0.760469
$250$	0	0
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	3.00000i	0.188982i
$253$	15.0000i	0.943042i
$254$	8.00000	0.501965
$255$	0	0
$256$	1.00000	0.0625000
$257$	23.0000i	1.43470i	0.696713	+	0.717350i	$0.254645\pi$
$257$	23.0000i	1.43470i	−0.696713	+	0.717350i	$0.745355\pi$
$258$	1.00000i	0.0622573i
$259$	0	0
$260$	0	0
$261$	4.00000	0.247594
$262$	6.00000i	0.370681i
$263$	24.0000i	1.47990i	0.672660	+	0.739952i	$0.265152\pi$
$263$	24.0000i	1.47990i	−0.672660	+	0.739952i	$0.734848\pi$
$264$	−3.00000	−0.184637
$265$	0	0
$266$	−9.00000	−0.551825
$267$	1.00000i	0.0611990i
$268$	2.00000i	0.122169i
$269$	−2.00000	−0.121942	−0.0609711	−	0.998140i	$-0.519420\pi$
$269$	−2.00000	−0.121942	−0.0609711	+	0.998140i	$0.519420\pi$
$270$	0	0
$271$	13.0000	0.789694	0.394847	−	0.918747i	$-0.370798\pi$
$271$	13.0000	0.789694	0.394847	+	0.918747i	$0.370798\pi$
$272$	4.00000i	0.242536i
$273$	− 6.00000i	− 0.363137i
$274$	−10.0000	−0.604122
$275$	0	0
$276$	−5.00000	−0.300965
$277$	− 12.0000i	− 0.721010i	−0.932757	−	0.360505i	$-0.882604\pi$
$277$	− 12.0000i	− 0.721010i	0.932757	−	0.360505i	$-0.117396\pi$
$278$	14.0000i	0.839664i
$279$	−1.00000	−0.0598684
$280$	0	0
$281$	16.0000	0.954480	0.477240	−	0.878773i	$-0.341637\pi$
$281$	16.0000	0.954480	0.477240	+	0.878773i	$0.341637\pi$
$282$	− 10.0000i	− 0.595491i
$283$	− 24.0000i	− 1.42665i	−0.700832	−	0.713326i	$-0.747188\pi$
$283$	− 24.0000i	− 1.42665i	0.700832	−	0.713326i	$-0.252812\pi$
$284$	−7.00000	−0.415374
$285$	0	0
$286$	6.00000	0.354787
$287$	12.0000i	0.708338i
$288$	− 1.00000i	− 0.0589256i
$289$	1.00000	0.0588235
$290$	0	0
$291$	10.0000	0.586210
$292$	− 5.00000i	− 0.292603i
$293$	30.0000i	1.75262i	0.481749	+	0.876309i	$0.340002\pi$
$293$	30.0000i	1.75262i	−0.481749	+	0.876309i	$0.659998\pi$
$294$	−2.00000	−0.116642
$295$	0	0
$296$	0	0
$297$	3.00000i	0.174078i
$298$	11.0000i	0.637213i
$299$	10.0000	0.578315
$300$	0	0
$301$	−3.00000	−0.172917
$302$	0	0
$303$	1.00000i	0.0574485i
$304$	3.00000	0.172062
$305$	0	0
$306$	4.00000	0.228665
$307$	− 16.0000i	− 0.913168i	−0.889680	−	0.456584i	$-0.849073\pi$
$307$	− 16.0000i	− 0.913168i	0.889680	−	0.456584i	$-0.150927\pi$
$308$	− 9.00000i	− 0.512823i
$309$	16.0000	0.910208
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	2.00000i	0.113228i
$313$	30.0000i	1.69570i	0.530236	+	0.847850i	$0.322103\pi$
$313$	30.0000i	1.69570i	−0.530236	+	0.847850i	$0.677897\pi$
$314$	−5.00000	−0.282166
$315$	0	0
$316$	−1.00000	−0.0562544
$317$	8.00000i	0.449325i	0.974437	+	0.224662i	$0.0721279\pi$
$317$	8.00000i	0.449325i	−0.974437	+	0.224662i	$0.927872\pi$
$318$	3.00000i	0.168232i
$319$	−12.0000	−0.671871
$320$	0	0
$321$	−9.00000	−0.502331
$322$	− 15.0000i	− 0.835917i
$323$	12.0000i	0.667698i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	−14.0000	−0.775388
$327$	20.0000i	1.10600i
$328$	− 4.00000i	− 0.220863i
$329$	30.0000	1.65395
$330$	0	0
$331$	12.0000	0.659580	0.329790	−	0.944054i	$-0.393022\pi$
$331$	12.0000	0.659580	0.329790	+	0.944054i	$0.393022\pi$
$332$	− 12.0000i	− 0.658586i
$333$	0	0
$334$	−19.0000	−1.03963
$335$	0	0
$336$	3.00000	0.163663
$337$	2.00000i	0.108947i	0.998515	+	0.0544735i	$0.0173480\pi$
$337$	2.00000i	0.108947i	−0.998515	+	0.0544735i	$0.982652\pi$
$338$	9.00000i	0.489535i
$339$	9.00000	0.488813
$340$	0	0
$341$	3.00000	0.162459
$342$	− 3.00000i	− 0.162221i
$343$	15.0000i	0.809924i
$344$	1.00000	0.0539164
$345$	0	0
$346$	22.0000	1.18273
$347$	32.0000i	1.71785i	0.512101	+	0.858925i	$0.328867\pi$
$347$	32.0000i	1.71785i	−0.512101	+	0.858925i	$0.671133\pi$
$348$	− 4.00000i	− 0.214423i
$349$	24.0000	1.28469	0.642345	−	0.766415i	$-0.277962\pi$
$349$	24.0000	1.28469	0.642345	+	0.766415i	$0.277962\pi$
$350$	0	0
$351$	2.00000	0.106752
$352$	3.00000i	0.159901i
$353$	18.0000i	0.958043i	0.877803	+	0.479022i	$0.159008\pi$
$353$	18.0000i	0.958043i	−0.877803	+	0.479022i	$0.840992\pi$
$354$	−6.00000	−0.318896
$355$	0	0
$356$	1.00000	0.0529999
$357$	12.0000i	0.635107i
$358$	− 4.00000i	− 0.211407i
$359$	−15.0000	−0.791670	−0.395835	−	0.918322i	$-0.629545\pi$
$359$	−15.0000	−0.791670	−0.395835	+	0.918322i	$0.629545\pi$
$360$	0	0
$361$	−10.0000	−0.526316
$362$	5.00000i	0.262794i
$363$	2.00000i	0.104973i
$364$	−6.00000	−0.314485
$365$	0	0
$366$	−2.00000	−0.104542
$367$	− 2.00000i	− 0.104399i	−0.998637	−	0.0521996i	$-0.983377\pi$
$367$	− 2.00000i	− 0.104399i	0.998637	−	0.0521996i	$-0.0166232\pi$
$368$	5.00000i	0.260643i
$369$	−4.00000	−0.208232
$370$	0	0
$371$	−9.00000	−0.467257
$372$	1.00000i	0.0518476i
$373$	− 9.00000i	− 0.466002i	−0.972476	−	0.233001i	$-0.925145\pi$
$373$	− 9.00000i	− 0.466002i	0.972476	−	0.233001i	$-0.0748546\pi$
$374$	−12.0000	−0.620505
$375$	0	0
$376$	−10.0000	−0.515711
$377$	8.00000i	0.412021i
$378$	− 3.00000i	− 0.154303i
$379$	−15.0000	−0.770498	−0.385249	−	0.922813i	$-0.625884\pi$
$379$	−15.0000	−0.770498	−0.385249	+	0.922813i	$0.625884\pi$
$380$	0	0
$381$	−8.00000	−0.409852
$382$	16.0000i	0.818631i
$383$	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
$386$	6.00000	0.305392
$387$	− 1.00000i	− 0.0508329i
$388$	− 10.0000i	− 0.507673i
$389$	26.0000	1.31825	0.659126	−	0.752032i	$-0.270926\pi$
$389$	26.0000	1.31825	0.659126	+	0.752032i	$0.270926\pi$
$390$	0	0
$391$	−20.0000	−1.01144
$392$	2.00000i	0.101015i
$393$	− 6.00000i	− 0.302660i
$394$	−6.00000	−0.302276
$395$	0	0
$396$	3.00000	0.150756
$397$	− 35.0000i	− 1.75660i	−0.478110	−	0.878300i	$-0.658678\pi$
$397$	− 35.0000i	− 1.75660i	0.478110	−	0.878300i	$-0.341322\pi$
$398$	− 7.00000i	− 0.350878i
$399$	9.00000	0.450564
$400$	0	0
$401$	25.0000	1.24844	0.624220	−	0.781248i	$-0.285417\pi$
$401$	25.0000	1.24844	0.624220	+	0.781248i	$0.285417\pi$
$402$	− 2.00000i	− 0.0997509i
$403$	− 2.00000i	− 0.0996271i
$404$	1.00000	0.0497519
$405$	0	0
$406$	12.0000	0.595550
$407$	0	0
$408$	− 4.00000i	− 0.198030i
$409$	−8.00000	−0.395575	−0.197787	−	0.980245i	$-0.563376\pi$
$409$	−8.00000	−0.395575	−0.197787	+	0.980245i	$0.563376\pi$
$410$	0	0
$411$	10.0000	0.493264
$412$	− 16.0000i	− 0.788263i
$413$	− 18.0000i	− 0.885722i
$414$	5.00000	0.245737
$415$	0	0
$416$	2.00000	0.0980581
$417$	− 14.0000i	− 0.685583i
$418$	9.00000i	0.440204i
$419$	−12.0000	−0.586238	−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000	−0.586238	−0.293119	+	0.956076i	$0.594693\pi$
$420$	0	0
$421$	−22.0000	−1.07221	−0.536107	−	0.844150i	$-0.680106\pi$
$421$	−22.0000	−1.07221	−0.536107	+	0.844150i	$0.680106\pi$
$422$	− 19.0000i	− 0.924906i
$423$	10.0000i	0.486217i
$424$	3.00000	0.145693
$425$	0	0
$426$	7.00000	0.339151
$427$	− 6.00000i	− 0.290360i
$428$	9.00000i	0.435031i
$429$	−6.00000	−0.289683
$430$	0	0
$431$	16.0000	0.770693	0.385346	−	0.922772i	$-0.374082\pi$
$431$	16.0000	0.770693	0.385346	+	0.922772i	$0.374082\pi$
$432$	1.00000i	0.0481125i
$433$	− 1.00000i	− 0.0480569i	−0.999711	−	0.0240285i	$-0.992351\pi$
$433$	− 1.00000i	− 0.0480569i	0.999711	−	0.0240285i	$-0.00764923\pi$
$434$	−3.00000	−0.144005
$435$	0	0
$436$	20.0000	0.957826
$437$	15.0000i	0.717547i
$438$	5.00000i	0.238909i
$439$	−18.0000	−0.859093	−0.429547	−	0.903045i	$-0.641327\pi$
$439$	−18.0000	−0.859093	−0.429547	+	0.903045i	$0.641327\pi$
$440$	0	0
$441$	2.00000	0.0952381
$442$	8.00000i	0.380521i
$443$	15.0000i	0.712672i	0.934358	+	0.356336i	$0.115974\pi$
$443$	15.0000i	0.712672i	−0.934358	+	0.356336i	$0.884026\pi$
$444$	0	0
$445$	0	0
$446$	6.00000	0.284108
$447$	− 11.0000i	− 0.520282i
$448$	− 3.00000i	− 0.141737i
$449$	−2.00000	−0.0943858	−0.0471929	−	0.998886i	$-0.515028\pi$
$449$	−2.00000	−0.0943858	−0.0471929	+	0.998886i	$0.515028\pi$
$450$	0	0
$451$	12.0000	0.565058
$452$	− 9.00000i	− 0.423324i
$453$	0	0
$454$	−27.0000	−1.26717
$455$	0	0
$456$	−3.00000	−0.140488
$457$	− 10.0000i	− 0.467780i	−0.972263	−	0.233890i	$-0.924854\pi$
$457$	− 10.0000i	− 0.467780i	0.972263	−	0.233890i	$-0.0751456\pi$
$458$	− 13.0000i	− 0.607450i
$459$	−4.00000	−0.186704
$460$	0	0
$461$	0	0		−	1.00000i	$-0.5\pi$
$461$	0	0			1.00000i	$0.5\pi$
$462$	9.00000i	0.418718i
$463$	4.00000i	0.185896i	0.995671	+	0.0929479i	$0.0296290\pi$
$463$	4.00000i	0.185896i	−0.995671	+	0.0929479i	$0.970371\pi$
$464$	−4.00000	−0.185695
$465$	0	0
$466$	−15.0000	−0.694862
$467$	− 24.0000i	− 1.11059i	−0.831654	−	0.555294i	$-0.812606\pi$
$467$	− 24.0000i	− 1.11059i	0.831654	−	0.555294i	$-0.187394\pi$
$468$	− 2.00000i	− 0.0924500i
$469$	6.00000	0.277054
$470$	0	0
$471$	5.00000	0.230388
$472$	6.00000i	0.276172i
$473$	3.00000i	0.137940i
$474$	1.00000	0.0459315
$475$	0	0
$476$	12.0000	0.550019
$477$	− 3.00000i	− 0.137361i
$478$	− 12.0000i	− 0.548867i
$479$	−3.00000	−0.137073	−0.0685367	−	0.997649i	$-0.521833\pi$
$479$	−3.00000	−0.137073	−0.0685367	+	0.997649i	$0.521833\pi$
$480$	0	0
$481$	0	0
$482$	− 18.0000i	− 0.819878i
$483$	15.0000i	0.682524i
$484$	2.00000	0.0909091
$485$	0	0
$486$	1.00000	0.0453609
$487$	− 32.0000i	− 1.45006i	−0.688718	−	0.725029i	$-0.741826\pi$
$487$	− 32.0000i	− 1.45006i	0.688718	−	0.725029i	$-0.258174\pi$
$488$	2.00000i	0.0905357i
$489$	14.0000	0.633102
$490$	0	0
$491$	−37.0000	−1.66979	−0.834893	−	0.550412i	$-0.814471\pi$
$491$	−37.0000	−1.66979	−0.834893	+	0.550412i	$0.814471\pi$
$492$	4.00000i	0.180334i
$493$	− 16.0000i	− 0.720604i
$494$	6.00000	0.269953
$495$	0	0
$496$	1.00000	0.0449013
$497$	21.0000i	0.941979i
$498$	12.0000i	0.537733i
$499$	−4.00000	−0.179065	−0.0895323	−	0.995984i	$-0.528537\pi$
$499$	−4.00000	−0.179065	−0.0895323	+	0.995984i	$0.528537\pi$
$500$	0	0
$501$	19.0000	0.848857
$502$	− 12.0000i	− 0.535586i
$503$	20.0000i	0.891756i	0.895094	+	0.445878i	$0.147108\pi$
$503$	20.0000i	0.891756i	−0.895094	+	0.445878i	$0.852892\pi$
$504$	−3.00000	−0.133631
$505$	0	0
$506$	−15.0000	−0.666831
$507$	− 9.00000i	− 0.399704i
$508$	8.00000i	0.354943i
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	−15.0000	−0.663561
$512$	1.00000i	0.0441942i
$513$	3.00000i	0.132453i
$514$	−23.0000	−1.01449
$515$	0	0
$516$	−1.00000	−0.0440225
$517$	− 30.0000i	− 1.31940i
$518$	0	0
$519$	−22.0000	−0.965693
$520$	0	0
$521$	−18.0000	−0.788594	−0.394297	−	0.918983i	$-0.629012\pi$
$521$	−18.0000	−0.788594	−0.394297	+	0.918983i	$0.629012\pi$
$522$	4.00000i	0.175075i
$523$	− 43.0000i	− 1.88026i	−0.340818	−	0.940129i	$-0.610704\pi$
$523$	− 43.0000i	− 1.88026i	0.340818	−	0.940129i	$-0.389296\pi$
$524$	−6.00000	−0.262111
$525$	0	0
$526$	−24.0000	−1.04645
$527$	4.00000i	0.174243i
$528$	− 3.00000i	− 0.130558i
$529$	−2.00000	−0.0869565
$530$	0	0
$531$	6.00000	0.260378
$532$	− 9.00000i	− 0.390199i
$533$	− 8.00000i	− 0.346518i
$534$	−1.00000	−0.0432742
$535$	0	0
$536$	−2.00000	−0.0863868
$537$	4.00000i	0.172613i
$538$	− 2.00000i	− 0.0862261i
$539$	−6.00000	−0.258438
$540$	0	0
$541$	22.0000	0.945854	0.472927	−	0.881102i	$-0.343197\pi$
$541$	22.0000	0.945854	0.472927	+	0.881102i	$0.343197\pi$
$542$	13.0000i	0.558398i
$543$	− 5.00000i	− 0.214571i
$544$	−4.00000	−0.171499
$545$	0	0
$546$	6.00000	0.256776
$547$	− 22.0000i	− 0.940652i	−0.882493	−	0.470326i	$-0.844136\pi$
$547$	− 22.0000i	− 0.940652i	0.882493	−	0.470326i	$-0.155864\pi$
$548$	− 10.0000i	− 0.427179i
$549$	2.00000	0.0853579
$550$	0	0
$551$	−12.0000	−0.511217
$552$	− 5.00000i	− 0.212814i
$553$	3.00000i	0.127573i
$554$	12.0000	0.509831
$555$	0	0
$556$	−14.0000	−0.593732
$557$	1.00000i	0.0423714i	0.999776	+	0.0211857i	$0.00674412\pi$
$557$	1.00000i	0.0423714i	−0.999776	+	0.0211857i	$0.993256\pi$
$558$	− 1.00000i	− 0.0423334i
$559$	2.00000	0.0845910
$560$	0	0
$561$	12.0000	0.506640
$562$	16.0000i	0.674919i
$563$	− 4.00000i	− 0.168580i	−0.996441	−	0.0842900i	$-0.973138\pi$
$563$	− 4.00000i	− 0.168580i	0.996441	−	0.0842900i	$-0.0268622\pi$
$564$	10.0000	0.421076
$565$	0	0
$566$	24.0000	1.00880
$567$	3.00000i	0.125988i
$568$	− 7.00000i	− 0.293713i
$569$	23.0000	0.964210	0.482105	−	0.876113i	$-0.339872\pi$
$569$	23.0000	0.964210	0.482105	+	0.876113i	$0.339872\pi$
$570$	0	0
$571$	22.0000	0.920671	0.460336	−	0.887745i	$-0.347729\pi$
$571$	22.0000	0.920671	0.460336	+	0.887745i	$0.347729\pi$
$572$	6.00000i	0.250873i
$573$	− 16.0000i	− 0.668410i
$574$	−12.0000	−0.500870
$575$	0	0
$576$	1.00000	0.0416667
$577$	− 4.00000i	− 0.166522i	−0.996528	−	0.0832611i	$-0.973466\pi$
$577$	− 4.00000i	− 0.166522i	0.996528	−	0.0832611i	$-0.0265335\pi$
$578$	1.00000i	0.0415945i
$579$	−6.00000	−0.249351
$580$	0	0
$581$	−36.0000	−1.49353
$582$	10.0000i	0.414513i
$583$	9.00000i	0.372742i
$584$	5.00000	0.206901
$585$	0	0
$586$	−30.0000	−1.23929
$587$	− 44.0000i	− 1.81607i	−0.418890	−	0.908037i	$-0.637581\pi$
$587$	− 44.0000i	− 1.81607i	0.418890	−	0.908037i	$-0.362419\pi$
$588$	− 2.00000i	− 0.0824786i
$589$	3.00000	0.123613
$590$	0	0
$591$	6.00000	0.246807
$592$	0	0
$593$	22.0000i	0.903432i	0.892162	+	0.451716i	$0.149188\pi$
$593$	22.0000i	0.903432i	−0.892162	+	0.451716i	$0.850812\pi$
$594$	−3.00000	−0.123091
$595$	0	0
$596$	−11.0000	−0.450578
$597$	7.00000i	0.286491i
$598$	10.0000i	0.408930i
$599$	27.0000	1.10319	0.551595	−	0.834112i	$-0.314019\pi$
$599$	27.0000	1.10319	0.551595	+	0.834112i	$0.314019\pi$
$600$	0	0
$601$	42.0000	1.71322	0.856608	−	0.515968i	$-0.172568\pi$
$601$	42.0000	1.71322	0.856608	+	0.515968i	$0.172568\pi$
$602$	− 3.00000i	− 0.122271i
$603$	2.00000i	0.0814463i
$604$	0	0
$605$	0	0
$606$	−1.00000	−0.0406222
$607$	− 29.0000i	− 1.17707i	−0.808470	−	0.588537i	$-0.799704\pi$
$607$	− 29.0000i	− 1.17707i	0.808470	−	0.588537i	$-0.200296\pi$
$608$	3.00000i	0.121666i
$609$	−12.0000	−0.486265
$610$	0	0
$611$	−20.0000	−0.809113
$612$	4.00000i	0.161690i
$613$	− 22.0000i	− 0.888572i	−0.895885	−	0.444286i	$-0.853457\pi$
$613$	− 22.0000i	− 0.888572i	0.895885	−	0.444286i	$-0.146543\pi$
$614$	16.0000	0.645707
$615$	0	0
$616$	9.00000	0.362620
$617$	27.0000i	1.08698i	0.839416	+	0.543490i	$0.182897\pi$
$617$	27.0000i	1.08698i	−0.839416	+	0.543490i	$0.817103\pi$
$618$	16.0000i	0.643614i
$619$	38.0000	1.52735	0.763674	−	0.645601i	$-0.223393\pi$
$619$	38.0000	1.52735	0.763674	+	0.645601i	$0.223393\pi$
$620$	0	0
$621$	−5.00000	−0.200643
$622$	0	0
$623$	− 3.00000i	− 0.120192i
$624$	−2.00000	−0.0800641
$625$	0	0
$626$	−30.0000	−1.19904
$627$	− 9.00000i	− 0.359425i
$628$	− 5.00000i	− 0.199522i
$629$	0	0
$630$	0	0
$631$	17.0000	0.676759	0.338380	−	0.941010i	$-0.390121\pi$
$631$	17.0000	0.676759	0.338380	+	0.941010i	$0.390121\pi$
$632$	− 1.00000i	− 0.0397779i
$633$	19.0000i	0.755182i
$634$	−8.00000	−0.317721
$635$	0	0
$636$	−3.00000	−0.118958
$637$	4.00000i	0.158486i
$638$	− 12.0000i	− 0.475085i
$639$	−7.00000	−0.276916
$640$	0	0
$641$	26.0000	1.02694	0.513469	−	0.858108i	$-0.328360\pi$
$641$	26.0000	1.02694	0.513469	+	0.858108i	$0.328360\pi$
$642$	− 9.00000i	− 0.355202i
$643$	19.0000i	0.749287i	0.927169	+	0.374643i	$0.122235\pi$
$643$	19.0000i	0.749287i	−0.927169	+	0.374643i	$0.877765\pi$
$644$	15.0000	0.591083
$645$	0	0
$646$	−12.0000	−0.472134
$647$	− 3.00000i	− 0.117942i	−0.998260	−	0.0589711i	$-0.981218\pi$
$647$	− 3.00000i	− 0.117942i	0.998260	−	0.0589711i	$-0.0187820\pi$
$648$	− 1.00000i	− 0.0392837i
$649$	−18.0000	−0.706562
$650$	0	0
$651$	3.00000	0.117579
$652$	− 14.0000i	− 0.548282i
$653$	− 26.0000i	− 1.01746i	−0.860927	−	0.508729i	$-0.830115\pi$
$653$	− 26.0000i	− 1.01746i	0.860927	−	0.508729i	$-0.169885\pi$
$654$	−20.0000	−0.782062
$655$	0	0
$656$	4.00000	0.156174
$657$	− 5.00000i	− 0.195069i
$658$	30.0000i	1.16952i
$659$	4.00000	0.155818	0.0779089	−	0.996960i	$-0.475176\pi$
$659$	4.00000	0.155818	0.0779089	+	0.996960i	$0.475176\pi$
$660$	0	0
$661$	−14.0000	−0.544537	−0.272268	−	0.962221i	$-0.587774\pi$
$661$	−14.0000	−0.544537	−0.272268	+	0.962221i	$0.587774\pi$
$662$	12.0000i	0.466393i
$663$	− 8.00000i	− 0.310694i
$664$	12.0000	0.465690
$665$	0	0
$666$	0	0
$667$	− 20.0000i	− 0.774403i
$668$	− 19.0000i	− 0.735132i
$669$	−6.00000	−0.231973
$670$	0	0
$671$	−6.00000	−0.231627
$672$	3.00000i	0.115728i
$673$	26.0000i	1.00223i	0.865382	+	0.501113i	$0.167076\pi$
$673$	26.0000i	1.00223i	−0.865382	+	0.501113i	$0.832924\pi$
$674$	−2.00000	−0.0770371
$675$	0	0
$676$	−9.00000	−0.346154
$677$	− 27.0000i	− 1.03769i	−0.854867	−	0.518847i	$-0.826361\pi$
$677$	− 27.0000i	− 1.03769i	0.854867	−	0.518847i	$-0.173639\pi$
$678$	9.00000i	0.345643i
$679$	−30.0000	−1.15129
$680$	0	0
$681$	27.0000	1.03464
$682$	3.00000i	0.114876i
$683$	− 19.0000i	− 0.727015i	−0.931591	−	0.363507i	$-0.881579\pi$
$683$	− 19.0000i	− 0.727015i	0.931591	−	0.363507i	$-0.118421\pi$
$684$	3.00000	0.114708
$685$	0	0
$686$	−15.0000	−0.572703
$687$	13.0000i	0.495981i
$688$	1.00000i	0.0381246i
$689$	6.00000	0.228582
$690$	0	0
$691$	−31.0000	−1.17930	−0.589648	−	0.807661i	$-0.700733\pi$
$691$	−31.0000	−1.17930	−0.589648	+	0.807661i	$0.700733\pi$
$692$	22.0000i	0.836315i
$693$	− 9.00000i	− 0.341882i
$694$	−32.0000	−1.21470
$695$	0	0
$696$	4.00000	0.151620
$697$	16.0000i	0.606043i
$698$	24.0000i	0.908413i
$699$	15.0000	0.567352
$700$	0	0
$701$	−21.0000	−0.793159	−0.396580	−	0.918000i	$-0.629803\pi$
$701$	−21.0000	−0.793159	−0.396580	+	0.918000i	$0.629803\pi$
$702$	2.00000i	0.0754851i
$703$	0	0
$704$	−3.00000	−0.113067
$705$	0	0
$706$	−18.0000	−0.677439
$707$	− 3.00000i	− 0.112827i
$708$	− 6.00000i	− 0.225494i
$709$	25.0000	0.938895	0.469447	−	0.882960i	$-0.344453\pi$
$709$	25.0000	0.938895	0.469447	+	0.882960i	$0.344453\pi$
$710$	0	0
$711$	−1.00000	−0.0375029
$712$	1.00000i	0.0374766i
$713$	5.00000i	0.187251i
$714$	−12.0000	−0.449089
$715$	0	0
$716$	4.00000	0.149487
$717$	12.0000i	0.448148i
$718$	− 15.0000i	− 0.559795i
$719$	−26.0000	−0.969636	−0.484818	−	0.874615i	$-0.661114\pi$
$719$	−26.0000	−0.969636	−0.484818	+	0.874615i	$0.661114\pi$
$720$	0	0
$721$	−48.0000	−1.78761
$722$	− 10.0000i	− 0.372161i
$723$	18.0000i	0.669427i
$724$	−5.00000	−0.185824
$725$	0	0
$726$	−2.00000	−0.0742270
$727$	− 45.0000i	− 1.66896i	−0.551040	−	0.834479i	$-0.685769\pi$
$727$	− 45.0000i	− 1.66896i	0.551040	−	0.834479i	$-0.314231\pi$
$728$	− 6.00000i	− 0.222375i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−4.00000	−0.147945
$732$	− 2.00000i	− 0.0739221i
$733$	− 18.0000i	− 0.664845i	−0.943131	−	0.332423i	$-0.892134\pi$
$733$	− 18.0000i	− 0.664845i	0.943131	−	0.332423i	$-0.107866\pi$
$734$	2.00000	0.0738213
$735$	0	0
$736$	−5.00000	−0.184302
$737$	− 6.00000i	− 0.221013i
$738$	− 4.00000i	− 0.147242i
$739$	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
$741$	−6.00000	−0.220416
$742$	− 9.00000i	− 0.330400i
$743$	− 37.0000i	− 1.35740i	−0.734416	−	0.678699i	$-0.762544\pi$
$743$	− 37.0000i	− 1.35740i	0.734416	−	0.678699i	$-0.237456\pi$
$744$	−1.00000	−0.0366618
$745$	0	0
$746$	9.00000	0.329513
$747$	− 12.0000i	− 0.439057i
$748$	− 12.0000i	− 0.438763i
$749$	27.0000	0.986559
$750$	0	0
$751$	−50.0000	−1.82453	−0.912263	−	0.409605i	$-0.865667\pi$
$751$	−50.0000	−1.82453	−0.912263	+	0.409605i	$0.865667\pi$
$752$	− 10.0000i	− 0.364662i
$753$	12.0000i	0.437304i
$754$	−8.00000	−0.291343
$755$	0	0
$756$	3.00000	0.109109
$757$	− 50.0000i	− 1.81728i	−0.417579	−	0.908640i	$-0.637121\pi$
$757$	− 50.0000i	− 1.81728i	0.417579	−	0.908640i	$-0.362879\pi$
$758$	− 15.0000i	− 0.544825i
$759$	15.0000	0.544466
$760$	0	0
$761$	45.0000	1.63125	0.815624	−	0.578582i	$-0.196394\pi$
$761$	45.0000	1.63125	0.815624	+	0.578582i	$0.196394\pi$
$762$	− 8.00000i	− 0.289809i
$763$	− 60.0000i	− 2.17215i
$764$	−16.0000	−0.578860
$765$	0	0
$766$	−12.0000	−0.433578
$767$	12.0000i	0.433295i
$768$	− 1.00000i	− 0.0360844i
$769$	−9.00000	−0.324548	−0.162274	−	0.986746i	$-0.551883\pi$
$769$	−9.00000	−0.324548	−0.162274	+	0.986746i	$0.551883\pi$
$770$	0	0
$771$	23.0000	0.828325
$772$	6.00000i	0.215945i
$773$	− 31.0000i	− 1.11499i	−0.830179	−	0.557496i	$-0.811762\pi$
$773$	− 31.0000i	− 1.11499i	0.830179	−	0.557496i	$-0.188238\pi$
$774$	1.00000	0.0359443
$775$	0	0
$776$	10.0000	0.358979
$777$	0	0
$778$	26.0000i	0.932145i
$779$	12.0000	0.429945
$780$	0	0
$781$	21.0000	0.751439
$782$	− 20.0000i	− 0.715199i
$783$	− 4.00000i	− 0.142948i
$784$	−2.00000	−0.0714286
$785$	0	0
$786$	6.00000	0.214013
$787$	35.0000i	1.24762i	0.781578	+	0.623808i	$0.214415\pi$
$787$	35.0000i	1.24762i	−0.781578	+	0.623808i	$0.785585\pi$
$788$	− 6.00000i	− 0.213741i
$789$	24.0000	0.854423
$790$	0	0
$791$	−27.0000	−0.960009
$792$	3.00000i	0.106600i
$793$	4.00000i	0.142044i
$794$	35.0000	1.24210
$795$	0	0
$796$	7.00000	0.248108
$797$	− 2.00000i	− 0.0708436i	−0.999372	−	0.0354218i	$-0.988723\pi$
$797$	− 2.00000i	− 0.0708436i	0.999372	−	0.0354218i	$-0.0112775\pi$
$798$	9.00000i	0.318597i
$799$	40.0000	1.41510
$800$	0	0
$801$	1.00000	0.0353333
$802$	25.0000i	0.882781i
$803$	15.0000i	0.529339i
$804$	2.00000	0.0705346
$805$	0	0
$806$	2.00000	0.0704470
$807$	2.00000i	0.0704033i
$808$	1.00000i	0.0351799i
$809$	−17.0000	−0.597688	−0.298844	−	0.954302i	$-0.596601\pi$
$809$	−17.0000	−0.597688	−0.298844	+	0.954302i	$0.596601\pi$
$810$	0	0
$811$	−27.0000	−0.948098	−0.474049	−	0.880498i	$-0.657208\pi$
$811$	−27.0000	−0.948098	−0.474049	+	0.880498i	$0.657208\pi$
$812$	12.0000i	0.421117i
$813$	− 13.0000i	− 0.455930i
$814$	0	0
$815$	0	0
$816$	4.00000	0.140028
$817$	3.00000i	0.104957i
$818$	− 8.00000i	− 0.279713i
$819$	−6.00000	−0.209657
$820$	0	0
$821$	40.0000	1.39601	0.698005	−	0.716093i	$-0.254071\pi$
$821$	40.0000	1.39601	0.698005	+	0.716093i	$0.254071\pi$
$822$	10.0000i	0.348790i
$823$	− 18.0000i	− 0.627441i	−0.949515	−	0.313720i	$-0.898425\pi$
$823$	− 18.0000i	− 0.627441i	0.949515	−	0.313720i	$-0.101575\pi$
$824$	16.0000	0.557386
$825$	0	0
$826$	18.0000	0.626300
$827$	− 20.0000i	− 0.695468i	−0.937593	−	0.347734i	$-0.886951\pi$
$827$	− 20.0000i	− 0.695468i	0.937593	−	0.347734i	$-0.113049\pi$
$828$	5.00000i	0.173762i
$829$	35.0000	1.21560	0.607800	−	0.794090i	$-0.292052\pi$
$829$	35.0000	1.21560	0.607800	+	0.794090i	$0.292052\pi$
$830$	0	0
$831$	−12.0000	−0.416275
$832$	2.00000i	0.0693375i
$833$	− 8.00000i	− 0.277184i
$834$	14.0000	0.484780
$835$	0	0
$836$	−9.00000	−0.311272
$837$	1.00000i	0.0345651i
$838$	− 12.0000i	− 0.414533i
$839$	−51.0000	−1.76072	−0.880358	−	0.474310i	$-0.842698\pi$
$839$	−51.0000	−1.76072	−0.880358	+	0.474310i	$0.842698\pi$
$840$	0	0
$841$	−13.0000	−0.448276
$842$	− 22.0000i	− 0.758170i
$843$	− 16.0000i	− 0.551069i
$844$	19.0000	0.654007
$845$	0	0
$846$	−10.0000	−0.343807
$847$	− 6.00000i	− 0.206162i
$848$	3.00000i	0.103020i
$849$	−24.0000	−0.823678
$850$	0	0
$851$	0	0
$852$	7.00000i	0.239816i
$853$	− 41.0000i	− 1.40381i	−0.712269	−	0.701907i	$-0.752332\pi$
$853$	− 41.0000i	− 1.40381i	0.712269	−	0.701907i	$-0.247668\pi$
$854$	6.00000	0.205316
$855$	0	0
$856$	−9.00000	−0.307614
$857$	− 42.0000i	− 1.43469i	−0.696717	−	0.717346i	$-0.745357\pi$
$857$	− 42.0000i	− 1.43469i	0.696717	−	0.717346i	$-0.254643\pi$
$858$	− 6.00000i	− 0.204837i
$859$	−32.0000	−1.09183	−0.545913	−	0.837842i	$-0.683817\pi$
$859$	−32.0000	−1.09183	−0.545913	+	0.837842i	$0.683817\pi$
$860$	0	0
$861$	12.0000	0.408959
$862$	16.0000i	0.544962i
$863$	43.0000i	1.46374i	0.681446	+	0.731869i	$0.261351\pi$
$863$	43.0000i	1.46374i	−0.681446	+	0.731869i	$0.738649\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	1.00000	0.0339814
$867$	− 1.00000i	− 0.0339618i
$868$	− 3.00000i	− 0.101827i
$869$	3.00000	0.101768
$870$	0	0
$871$	−4.00000	−0.135535
$872$	20.0000i	0.677285i
$873$	− 10.0000i	− 0.338449i
$874$	−15.0000	−0.507383
$875$	0	0
$876$	−5.00000	−0.168934
$877$	− 46.0000i	− 1.55331i	−0.629926	−	0.776655i	$-0.716915\pi$
$877$	− 46.0000i	− 1.55331i	0.629926	−	0.776655i	$-0.283085\pi$
$878$	− 18.0000i	− 0.607471i
$879$	30.0000	1.01187
$880$	0	0
$881$	26.0000	0.875962	0.437981	−	0.898984i	$-0.355694\pi$
$881$	26.0000	0.875962	0.437981	+	0.898984i	$0.355694\pi$
$882$	2.00000i	0.0673435i
$883$	11.0000i	0.370179i	0.982722	+	0.185090i	$0.0592576\pi$
$883$	11.0000i	0.370179i	−0.982722	+	0.185090i	$0.940742\pi$
$884$	−8.00000	−0.269069
$885$	0	0
$886$	−15.0000	−0.503935
$887$	46.0000i	1.54453i	0.635301	+	0.772264i	$0.280876\pi$
$887$	46.0000i	1.54453i	−0.635301	+	0.772264i	$0.719124\pi$
$888$	0	0
$889$	24.0000	0.804934
$890$	0	0
$891$	3.00000	0.100504
$892$	6.00000i	0.200895i
$893$	− 30.0000i	− 1.00391i
$894$	11.0000	0.367895
$895$	0	0
$896$	3.00000	0.100223
$897$	− 10.0000i	− 0.333890i
$898$	− 2.00000i	− 0.0667409i
$899$	−4.00000	−0.133407
$900$	0	0
$901$	−12.0000	−0.399778
$902$	12.0000i	0.399556i
$903$	3.00000i	0.0998337i
$904$	9.00000	0.299336
$905$	0	0
$906$	0	0
$907$	52.0000i	1.72663i	0.504664	+	0.863316i	$0.331616\pi$
$907$	52.0000i	1.72663i	−0.504664	+	0.863316i	$0.668384\pi$
$908$	− 27.0000i	− 0.896026i
$909$	1.00000	0.0331679
$910$	0	0
$911$	20.0000	0.662630	0.331315	−	0.943520i	$-0.392508\pi$
$911$	20.0000	0.662630	0.331315	+	0.943520i	$0.392508\pi$
$912$	− 3.00000i	− 0.0993399i
$913$	36.0000i	1.19143i
$914$	10.0000	0.330771
$915$	0	0
$916$	13.0000	0.429532
$917$	18.0000i	0.594412i
$918$	− 4.00000i	− 0.132020i
$919$	38.0000	1.25350	0.626752	−	0.779219i	$-0.284384\pi$
$919$	38.0000	1.25350	0.626752	+	0.779219i	$0.284384\pi$
$920$	0	0
$921$	−16.0000	−0.527218
$922$	0	0
$923$	− 14.0000i	− 0.460816i
$924$	−9.00000	−0.296078
$925$	0	0
$926$	−4.00000	−0.131448
$927$	− 16.0000i	− 0.525509i
$928$	− 4.00000i	− 0.131306i
$929$	41.0000	1.34517	0.672583	−	0.740022i	$-0.265185\pi$
$929$	41.0000	1.34517	0.672583	+	0.740022i	$0.265185\pi$
$930$	0	0
$931$	−6.00000	−0.196642
$932$	− 15.0000i	− 0.491341i
$933$	0	0
$934$	24.0000	0.785304
$935$	0	0
$936$	2.00000	0.0653720
$937$	52.0000i	1.69877i	0.527777	+	0.849383i	$0.323026\pi$
$937$	52.0000i	1.69877i	−0.527777	+	0.849383i	$0.676974\pi$
$938$	6.00000i	0.195907i
$939$	30.0000	0.979013
$940$	0	0
$941$	40.0000	1.30396	0.651981	−	0.758235i	$-0.273938\pi$
$941$	40.0000	1.30396	0.651981	+	0.758235i	$0.273938\pi$
$942$	5.00000i	0.162909i
$943$	20.0000i	0.651290i
$944$	−6.00000	−0.195283
$945$	0	0
$946$	−3.00000	−0.0975384
$947$	− 30.0000i	− 0.974869i	−0.873160	−	0.487435i	$-0.837933\pi$
$947$	− 30.0000i	− 0.974869i	0.873160	−	0.487435i	$-0.162067\pi$
$948$	1.00000i	0.0324785i
$949$	10.0000	0.324614
$950$	0	0
$951$	8.00000	0.259418
$952$	12.0000i	0.388922i
$953$	− 52.0000i	− 1.68445i	−0.539130	−	0.842223i	$-0.681247\pi$
$953$	− 52.0000i	− 1.68445i	0.539130	−	0.842223i	$-0.318753\pi$
$954$	3.00000	0.0971286
$955$	0	0
$956$	12.0000	0.388108
$957$	12.0000i	0.387905i
$958$	− 3.00000i	− 0.0969256i
$959$	−30.0000	−0.968751
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	9.00000i	0.290021i
$964$	18.0000	0.579741
$965$	0	0
$966$	−15.0000	−0.482617
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	2.00000i	0.0642824i
$969$	12.0000	0.385496
$970$	0	0
$971$	−54.0000	−1.73294	−0.866471	−	0.499227i	$-0.833617\pi$
$971$	−54.0000	−1.73294	−0.866471	+	0.499227i	$0.833617\pi$
$972$	1.00000i	0.0320750i
$973$	42.0000i	1.34646i
$974$	32.0000	1.02535
$975$	0	0
$976$	−2.00000	−0.0640184
$977$	− 38.0000i	− 1.21573i	−0.794041	−	0.607864i	$-0.792027\pi$
$977$	− 38.0000i	− 1.21573i	0.794041	−	0.607864i	$-0.207973\pi$
$978$	14.0000i	0.447671i
$979$	−3.00000	−0.0958804
$980$	0	0
$981$	20.0000	0.638551
$982$	− 37.0000i	− 1.18072i
$983$	− 56.0000i	− 1.78612i	−0.449935	−	0.893061i	$-0.648553\pi$
$983$	− 56.0000i	− 1.78612i	0.449935	−	0.893061i	$-0.351447\pi$
$984$	−4.00000	−0.127515
$985$	0	0
$986$	16.0000	0.509544
$987$	− 30.0000i	− 0.954911i
$988$	6.00000i	0.190885i
$989$	−5.00000	−0.158991
$990$	0	0
$991$	−25.0000	−0.794151	−0.397076	−	0.917786i	$-0.629975\pi$
$991$	−25.0000	−0.794151	−0.397076	+	0.917786i	$0.629975\pi$
$992$	1.00000i	0.0317500i
$993$	− 12.0000i	− 0.380808i
$994$	−21.0000	−0.666080
$995$	0	0
$996$	−12.0000	−0.380235
$997$	42.0000i	1.33015i	0.746775	+	0.665077i	$0.231601\pi$
$997$	42.0000i	1.33015i	−0.746775	+	0.665077i	$0.768399\pi$
$998$	− 4.00000i	− 0.126618i
$999$	0	0

Twists

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

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

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +3.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +3.00000 q^{11} +1.00000i q^{12} -2.00000i q^{13} -3.00000 q^{14} +1.00000 q^{16} +4.00000i q^{17} -1.00000i q^{18} +3.00000 q^{19} +3.00000 q^{21} +3.00000i q^{22} +5.00000i q^{23} -1.00000 q^{24} +2.00000 q^{26} +1.00000i q^{27} -3.00000i q^{28} -4.00000 q^{29} +1.00000 q^{31} +1.00000i q^{32} -3.00000i q^{33} -4.00000 q^{34} +1.00000 q^{36} +3.00000i q^{38} -2.00000 q^{39} +4.00000 q^{41} +3.00000i q^{42} +1.00000i q^{43} -3.00000 q^{44} -5.00000 q^{46} -10.0000i q^{47} -1.00000i q^{48} -2.00000 q^{49} +4.00000 q^{51} +2.00000i q^{52} +3.00000i q^{53} -1.00000 q^{54} +3.00000 q^{56} -3.00000i q^{57} -4.00000i q^{58} -6.00000 q^{59} -2.00000 q^{61} +1.00000i q^{62} -3.00000i q^{63} -1.00000 q^{64} +3.00000 q^{66} -2.00000i q^{67} -4.00000i q^{68} +5.00000 q^{69} +7.00000 q^{71} +1.00000i q^{72} +5.00000i q^{73} -3.00000 q^{76} +9.00000i q^{77} -2.00000i q^{78} +1.00000 q^{79} +1.00000 q^{81} +4.00000i q^{82} +12.0000i q^{83} -3.00000 q^{84} -1.00000 q^{86} +4.00000i q^{87} -3.00000i q^{88} -1.00000 q^{89} +6.00000 q^{91} -5.00000i q^{92} -1.00000i q^{93} +10.0000 q^{94} +1.00000 q^{96} +10.0000i q^{97} -2.00000i q^{98} -3.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} + 6 q^{11} - 6 q^{14} + 2 q^{16} + 6 q^{19} + 6 q^{21} - 2 q^{24} + 4 q^{26} - 8 q^{29} + 2 q^{31} - 8 q^{34} + 2 q^{36} - 4 q^{39} + 8 q^{41} - 6 q^{44} - 10 q^{46} - 4 q^{49}+ \cdots - 6 q^{99}+O(q^{100}) \) 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9 + 6 * q^11 - 6 * q^14 + 2 * q^16 + 6 * q^19 + 6 * q^21 - 2 * q^24 + 4 * q^26 - 8 * q^29 + 2 * q^31 - 8 * q^34 + 2 * q^36 - 4 * q^39 + 8 * q^41 - 6 * q^44 - 10 * q^46 - 4 * q^49 + 8 * q^51 - 2 * q^54 + 6 * q^56 - 12 * q^59 - 4 * q^61 - 2 * q^64 + 6 * q^66 + 10 * q^69 + 14 * q^71 - 6 * q^76 + 2 * q^79 + 2 * q^81 - 6 * q^84 - 2 * q^86 - 2 * q^89 + 12 * q^91 + 20 * q^94 + 2 * q^96 - 6 * q^99

Introduction

L-functions

Modular forms

Varieties

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