LMFDB - Embedded newform 4650.2.d.ba.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:	no (minimal twist has level 930)
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.ba.3349.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +3.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +3.00000 q^{11} -1.00000i q^{12} +2.00000i q^{13} +3.00000 q^{14} +1.00000 q^{16} +8.00000i q^{17} +1.00000i q^{18} +7.00000 q^{19} -3.00000 q^{21} -3.00000i q^{22} -7.00000i q^{23} -1.00000 q^{24} +2.00000 q^{26} -1.00000i q^{27} -3.00000i q^{28} +8.00000 q^{29} -1.00000 q^{31} -1.00000i q^{32} +3.00000i q^{33} +8.00000 q^{34} +1.00000 q^{36} -4.00000i q^{37} -7.00000i q^{38} -2.00000 q^{39} +3.00000i q^{42} -1.00000i q^{43} -3.00000 q^{44} -7.00000 q^{46} +6.00000i q^{47} +1.00000i q^{48} -2.00000 q^{49} -8.00000 q^{51} -2.00000i q^{52} -5.00000i q^{53} -1.00000 q^{54} -3.00000 q^{56} +7.00000i q^{57} -8.00000i q^{58} -6.00000 q^{59} +2.00000 q^{61} +1.00000i q^{62} -3.00000i q^{63} -1.00000 q^{64} +3.00000 q^{66} +10.0000i q^{67} -8.00000i q^{68} +7.00000 q^{69} +9.00000 q^{71} -1.00000i q^{72} -1.00000i q^{73} -4.00000 q^{74} -7.00000 q^{76} +9.00000i q^{77} +2.00000i q^{78} -13.0000 q^{79} +1.00000 q^{81} +16.0000i q^{83} +3.00000 q^{84} -1.00000 q^{86} +8.00000i q^{87} +3.00000i q^{88} +3.00000 q^{89} -6.00000 q^{91} +7.00000i q^{92} -1.00000i q^{93} +6.00000 q^{94} +1.00000 q^{96} +6.00000i q^{97} +2.00000i q^{98} -3.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} + 6 q^{11} + 6 q^{14} + 2 q^{16} + 14 q^{19} - 6 q^{21} - 2 q^{24} + 4 q^{26} + 16 q^{29} - 2 q^{31} + 16 q^{34} + 2 q^{36} - 4 q^{39} - 6 q^{44} - 14 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 + 14 * q^19 - 6 * q^21 - 2 * q^24 + 4 * q^26 + 16 * q^29 - 2 * q^31 + 16 * q^34 + 2 * q^36 - 4 * q^39 - 6 * q^44 - 14 * q^46 - 4 * q^49 - 16 * q^51 - 2 * q^54 - 6 * q^56 - 12 * q^59 + 4 * q^61 - 2 * q^64 + 6 * q^66 + 14 * q^69 + 18 * q^71 - 8 * q^74 - 14 * q^76 - 26 * q^79 + 2 * q^81 + 6 * q^84 - 2 * q^86 + 6 * q^89 - 12 * q^91 + 12 * q^94 + 2 * q^96 - 6 * q^99

Character values

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

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

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	− 1.00000i	− 0.707107i
$2$	− 1.00000i	− 0.707107i
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	1.00000	0.408248
$7$	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	$0.0894562\pi$
$13$	2.00000i	0.554700i	−0.960769	+	0.277350i	$0.910544\pi$
$14$	3.00000	0.801784
$15$	0	0
$16$	1.00000	0.250000
$17$	8.00000i	1.94029i	0.242536	+	0.970143i	$0.422021\pi$
$17$	8.00000i	1.94029i	−0.242536	+	0.970143i	$0.577979\pi$
$18$	1.00000i	0.235702i
$19$	7.00000	1.60591	0.802955	−	0.596040i	$-0.203260\pi$
$19$	7.00000	1.60591	0.802955	+	0.596040i	$0.203260\pi$
$20$	0	0
$21$	−3.00000	−0.654654
$22$	− 3.00000i	− 0.639602i
$23$	− 7.00000i	− 1.45960i	−0.683660	−	0.729800i	$-0.739613\pi$
$23$	− 7.00000i	− 1.45960i	0.683660	−	0.729800i	$-0.260387\pi$
$24$	−1.00000	−0.204124
$25$	0	0
$26$	2.00000	0.392232
$27$	− 1.00000i	− 0.192450i
$28$	− 3.00000i	− 0.566947i
$29$	8.00000	1.48556	0.742781	−	0.669534i	$-0.233506\pi$
$29$	8.00000	1.48556	0.742781	+	0.669534i	$0.233506\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	− 1.00000i	− 0.176777i
$33$	3.00000i	0.522233i
$34$	8.00000	1.37199
$35$	0	0
$36$	1.00000	0.166667
$37$	− 4.00000i	− 0.657596i	−0.944400	−	0.328798i	$-0.893356\pi$
$37$	− 4.00000i	− 0.657596i	0.944400	−	0.328798i	$-0.106644\pi$
$38$	− 7.00000i	− 1.13555i
$39$	−2.00000	−0.320256
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	3.00000i	0.462910i
$43$	− 1.00000i	− 0.152499i	−0.997089	−	0.0762493i	$-0.975706\pi$
$43$	− 1.00000i	− 0.152499i	0.997089	−	0.0762493i	$-0.0242945\pi$
$44$	−3.00000	−0.452267
$45$	0	0
$46$	−7.00000	−1.03209
$47$	6.00000i	0.875190i	0.899172	+	0.437595i	$0.144170\pi$
$47$	6.00000i	0.875190i	−0.899172	+	0.437595i	$0.855830\pi$
$48$	1.00000i	0.144338i
$49$	−2.00000	−0.285714
$50$	0	0
$51$	−8.00000	−1.12022
$52$	− 2.00000i	− 0.277350i
$53$	− 5.00000i	− 0.686803i	−0.939189	−	0.343401i	$-0.888421\pi$
$53$	− 5.00000i	− 0.686803i	0.939189	−	0.343401i	$-0.111579\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	−3.00000	−0.400892
$57$	7.00000i	0.927173i
$58$	− 8.00000i	− 1.05045i
$59$	−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	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	1.00000i	0.127000i
$63$	− 3.00000i	− 0.377964i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	3.00000	0.369274
$67$	10.0000i	1.22169i	0.791748	+	0.610847i	$0.209171\pi$
$67$	10.0000i	1.22169i	−0.791748	+	0.610847i	$0.790829\pi$
$68$	− 8.00000i	− 0.970143i
$69$	7.00000	0.842701
$70$	0	0
$71$	9.00000	1.06810	0.534052	−	0.845452i	$-0.320669\pi$
$71$	9.00000	1.06810	0.534052	+	0.845452i	$0.320669\pi$
$72$	− 1.00000i	− 0.117851i
$73$	− 1.00000i	− 0.117041i	−0.998286	−	0.0585206i	$-0.981362\pi$
$73$	− 1.00000i	− 0.117041i	0.998286	−	0.0585206i	$-0.0186383\pi$
$74$	−4.00000	−0.464991
$75$	0	0
$76$	−7.00000	−0.802955
$77$	9.00000i	1.02565i
$78$	2.00000i	0.226455i
$79$	−13.0000	−1.46261	−0.731307	−	0.682048i	$-0.761089\pi$
$79$	−13.0000	−1.46261	−0.731307	+	0.682048i	$0.761089\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	16.0000i	1.75623i	0.478451	+	0.878114i	$0.341198\pi$
$83$	16.0000i	1.75623i	−0.478451	+	0.878114i	$0.658802\pi$
$84$	3.00000	0.327327
$85$	0	0
$86$	−1.00000	−0.107833
$87$	8.00000i	0.857690i
$88$	3.00000i	0.319801i
$89$	3.00000	0.317999	0.159000	−	0.987279i	$-0.449173\pi$
$89$	3.00000	0.317999	0.159000	+	0.987279i	$0.449173\pi$
$90$	0	0
$91$	−6.00000	−0.628971
$92$	7.00000i	0.729800i
$93$	− 1.00000i	− 0.103695i
$94$	6.00000	0.618853
$95$	0	0
$96$	1.00000	0.102062
$97$	6.00000i	0.609208i	0.952479	+	0.304604i	$0.0985241\pi$
$97$	6.00000i	0.609208i	−0.952479	+	0.304604i	$0.901476\pi$
$98$	2.00000i	0.202031i
$99$	−3.00000	−0.301511
$100$	0	0
$101$	5.00000	0.497519	0.248759	−	0.968565i	$-0.419977\pi$
$101$	5.00000	0.497519	0.248759	+	0.968565i	$0.419977\pi$
$102$	8.00000i	0.792118i
$103$	− 8.00000i	− 0.788263i	−0.919054	−	0.394132i	$-0.871045\pi$
$103$	− 8.00000i	− 0.788263i	0.919054	−	0.394132i	$-0.128955\pi$
$104$	−2.00000	−0.196116
$105$	0	0
$106$	−5.00000	−0.485643
$107$	− 3.00000i	− 0.290021i	−0.989430	−	0.145010i	$-0.953678\pi$
$107$	− 3.00000i	− 0.290021i	0.989430	−	0.145010i	$-0.0463216\pi$
$108$	1.00000i	0.0962250i
$109$	−4.00000	−0.383131	−0.191565	−	0.981480i	$-0.561356\pi$
$109$	−4.00000	−0.383131	−0.191565	+	0.981480i	$0.561356\pi$
$110$	0	0
$111$	4.00000	0.379663
$112$	3.00000i	0.283473i
$113$	− 1.00000i	− 0.0940721i	−0.998893	−	0.0470360i	$-0.985022\pi$
$113$	− 1.00000i	− 0.0940721i	0.998893	−	0.0470360i	$-0.0149776\pi$
$114$	7.00000	0.655610
$115$	0	0
$116$	−8.00000	−0.742781
$117$	− 2.00000i	− 0.184900i
$118$	6.00000i	0.552345i
$119$	−24.0000	−2.20008
$120$	0	0
$121$	−2.00000	−0.181818
$122$	− 2.00000i	− 0.181071i
$123$	0	0
$124$	1.00000	0.0898027
$125$	0	0
$126$	−3.00000	−0.267261
$127$	− 12.0000i	− 1.06483i	−0.846484	−	0.532414i	$-0.821285\pi$
$127$	− 12.0000i	− 1.06483i	0.846484	−	0.532414i	$-0.178715\pi$
$128$	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
$133$	21.0000i	1.82093i
$134$	10.0000	0.863868
$135$	0	0
$136$	−8.00000	−0.685994
$137$	10.0000i	0.854358i	0.904167	+	0.427179i	$0.140493\pi$
$137$	10.0000i	0.854358i	−0.904167	+	0.427179i	$0.859507\pi$
$138$	− 7.00000i	− 0.595880i
$139$	10.0000	0.848189	0.424094	−	0.905618i	$-0.360592\pi$
$139$	10.0000	0.848189	0.424094	+	0.905618i	$0.360592\pi$
$140$	0	0
$141$	−6.00000	−0.505291
$142$	− 9.00000i	− 0.755263i
$143$	6.00000i	0.501745i
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	−1.00000	−0.0827606
$147$	− 2.00000i	− 0.164957i
$148$	4.00000i	0.328798i
$149$	9.00000	0.737309	0.368654	−	0.929567i	$-0.379819\pi$
$149$	9.00000	0.737309	0.368654	+	0.929567i	$0.379819\pi$
$150$	0	0
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	7.00000i	0.567775i
$153$	− 8.00000i	− 0.646762i
$154$	9.00000	0.725241
$155$	0	0
$156$	2.00000	0.160128
$157$	− 7.00000i	− 0.558661i	−0.960195	−	0.279330i	$-0.909888\pi$
$157$	− 7.00000i	− 0.558661i	0.960195	−	0.279330i	$-0.0901125\pi$
$158$	13.0000i	1.03422i
$159$	5.00000	0.396526
$160$	0	0
$161$	21.0000	1.65503
$162$	− 1.00000i	− 0.0785674i
$163$	− 18.0000i	− 1.40987i	−0.709273	−	0.704934i	$-0.750976\pi$
$163$	− 18.0000i	− 1.40987i	0.709273	−	0.704934i	$-0.249024\pi$
$164$	0	0
$165$	0	0
$166$	16.0000	1.24184
$167$	15.0000i	1.16073i	0.814355	+	0.580367i	$0.197091\pi$
$167$	15.0000i	1.16073i	−0.814355	+	0.580367i	$0.802909\pi$
$168$	− 3.00000i	− 0.231455i
$169$	9.00000	0.692308
$170$	0	0
$171$	−7.00000	−0.535303
$172$	1.00000i	0.0762493i
$173$	6.00000i	0.456172i	0.973641	+	0.228086i	$0.0732467\pi$
$173$	6.00000i	0.456172i	−0.973641	+	0.228086i	$0.926753\pi$
$174$	8.00000	0.606478
$175$	0	0
$176$	3.00000	0.226134
$177$	− 6.00000i	− 0.450988i
$178$	− 3.00000i	− 0.224860i
$179$	−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
$181$	−21.0000	−1.56092	−0.780459	−	0.625207i	$-0.785014\pi$
$181$	−21.0000	−1.56092	−0.780459	+	0.625207i	$0.785014\pi$
$182$	6.00000i	0.444750i
$183$	2.00000i	0.147844i
$184$	7.00000	0.516047
$185$	0	0
$186$	−1.00000	−0.0733236
$187$	24.0000i	1.75505i
$188$	− 6.00000i	− 0.437595i
$189$	3.00000	0.218218
$190$	0	0
$191$	−24.0000	−1.73658	−0.868290	−	0.496058i	$-0.834780\pi$
$191$	−24.0000	−1.73658	−0.868290	+	0.496058i	$0.834780\pi$
$192$	− 1.00000i	− 0.0721688i
$193$	14.0000i	1.00774i	0.863779	+	0.503871i	$0.168091\pi$
$193$	14.0000i	1.00774i	−0.863779	+	0.503871i	$0.831909\pi$
$194$	6.00000	0.430775
$195$	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
$199$	3.00000	0.212664	0.106332	−	0.994331i	$-0.466089\pi$
$199$	3.00000	0.212664	0.106332	+	0.994331i	$0.466089\pi$
$200$	0	0
$201$	−10.0000	−0.705346
$202$	− 5.00000i	− 0.351799i
$203$	24.0000i	1.68447i
$204$	8.00000	0.560112
$205$	0	0
$206$	−8.00000	−0.557386
$207$	7.00000i	0.486534i
$208$	2.00000i	0.138675i
$209$	21.0000	1.45260
$210$	0	0
$211$	17.0000	1.17033	0.585164	−	0.810915i	$-0.301030\pi$
$211$	17.0000	1.17033	0.585164	+	0.810915i	$0.301030\pi$
$212$	5.00000i	0.343401i
$213$	9.00000i	0.616670i
$214$	−3.00000	−0.205076
$215$	0	0
$216$	1.00000	0.0680414
$217$	− 3.00000i	− 0.203653i
$218$	4.00000i	0.270914i
$219$	1.00000	0.0675737
$220$	0	0
$221$	−16.0000	−1.07628
$222$	− 4.00000i	− 0.268462i
$223$	10.0000i	0.669650i	0.942280	+	0.334825i	$0.108677\pi$
$223$	10.0000i	0.669650i	−0.942280	+	0.334825i	$0.891323\pi$
$224$	3.00000	0.200446
$225$	0	0
$226$	−1.00000	−0.0665190
$227$	25.0000i	1.65931i	0.558278	+	0.829654i	$0.311462\pi$
$227$	25.0000i	1.65931i	−0.558278	+	0.829654i	$0.688538\pi$
$228$	− 7.00000i	− 0.463586i
$229$	13.0000	0.859064	0.429532	−	0.903052i	$-0.358679\pi$
$229$	13.0000	0.859064	0.429532	+	0.903052i	$0.358679\pi$
$230$	0	0
$231$	−9.00000	−0.592157
$232$	8.00000i	0.525226i
$233$	25.0000i	1.63780i	0.573933	+	0.818902i	$0.305417\pi$
$233$	25.0000i	1.63780i	−0.573933	+	0.818902i	$0.694583\pi$
$234$	−2.00000	−0.130744
$235$	0	0
$236$	6.00000	0.390567
$237$	− 13.0000i	− 0.844441i
$238$	24.0000i	1.55569i
$239$	−8.00000	−0.517477	−0.258738	−	0.965947i	$-0.583307\pi$
$239$	−8.00000	−0.517477	−0.258738	+	0.965947i	$0.583307\pi$
$240$	0	0
$241$	−22.0000	−1.41714	−0.708572	−	0.705638i	$-0.750660\pi$
$241$	−22.0000	−1.41714	−0.708572	+	0.705638i	$0.750660\pi$
$242$	2.00000i	0.128565i
$243$	1.00000i	0.0641500i
$244$	−2.00000	−0.128037
$245$	0	0
$246$	0	0
$247$	14.0000i	0.890799i
$248$	− 1.00000i	− 0.0635001i
$249$	−16.0000	−1.01396
$250$	0	0
$251$	20.0000	1.26239	0.631194	−	0.775625i	$-0.282565\pi$
$251$	20.0000	1.26239	0.631194	+	0.775625i	$0.282565\pi$
$252$	3.00000i	0.188982i
$253$	− 21.0000i	− 1.32026i
$254$	−12.0000	−0.752947
$255$	0	0
$256$	1.00000	0.0625000
$257$	− 23.0000i	− 1.43470i	−0.696713	−	0.717350i	$-0.745355\pi$
$257$	− 23.0000i	− 1.43470i	0.696713	−	0.717350i	$-0.254645\pi$
$258$	− 1.00000i	− 0.0622573i
$259$	12.0000	0.745644
$260$	0	0
$261$	−8.00000	−0.495188
$262$	− 6.00000i	− 0.370681i
$263$	− 24.0000i	− 1.47990i	−0.672660	−	0.739952i	$-0.734848\pi$
$263$	− 24.0000i	− 1.47990i	0.672660	−	0.739952i	$-0.265152\pi$
$264$	−3.00000	−0.184637
$265$	0	0
$266$	21.0000	1.28759
$267$	3.00000i	0.183597i
$268$	− 10.0000i	− 0.610847i
$269$	18.0000	1.09748	0.548740	−	0.835993i	$-0.315108\pi$
$269$	18.0000	1.09748	0.548740	+	0.835993i	$0.315108\pi$
$270$	0	0
$271$	7.00000	0.425220	0.212610	−	0.977137i	$-0.431804\pi$
$271$	7.00000	0.425220	0.212610	+	0.977137i	$0.431804\pi$
$272$	8.00000i	0.485071i
$273$	− 6.00000i	− 0.363137i
$274$	10.0000	0.604122
$275$	0	0
$276$	−7.00000	−0.421350
$277$	− 8.00000i	− 0.480673i	−0.970690	−	0.240337i	$-0.922742\pi$
$277$	− 8.00000i	− 0.480673i	0.970690	−	0.240337i	$-0.0772579\pi$
$278$	− 10.0000i	− 0.599760i
$279$	1.00000	0.0598684
$280$	0	0
$281$	−28.0000	−1.67034	−0.835170	−	0.549992i	$-0.814631\pi$
$281$	−28.0000	−1.67034	−0.835170	+	0.549992i	$0.814631\pi$
$282$	6.00000i	0.357295i
$283$	16.0000i	0.951101i	0.879688	+	0.475551i	$0.157751\pi$
$283$	16.0000i	0.951101i	−0.879688	+	0.475551i	$0.842249\pi$
$284$	−9.00000	−0.534052
$285$	0	0
$286$	6.00000	0.354787
$287$	0	0
$288$	1.00000i	0.0589256i
$289$	−47.0000	−2.76471
$290$	0	0
$291$	−6.00000	−0.351726
$292$	1.00000i	0.0585206i
$293$	− 14.0000i	− 0.817889i	−0.912559	−	0.408944i	$-0.865897\pi$
$293$	− 14.0000i	− 0.817889i	0.912559	−	0.408944i	$-0.134103\pi$
$294$	−2.00000	−0.116642
$295$	0	0
$296$	4.00000	0.232495
$297$	− 3.00000i	− 0.174078i
$298$	− 9.00000i	− 0.521356i
$299$	14.0000	0.809641
$300$	0	0
$301$	3.00000	0.172917
$302$	8.00000i	0.460348i
$303$	5.00000i	0.287242i
$304$	7.00000	0.401478
$305$	0	0
$306$	−8.00000	−0.457330
$307$	− 12.0000i	− 0.684876i	−0.939540	−	0.342438i	$-0.888747\pi$
$307$	− 12.0000i	− 0.684876i	0.939540	−	0.342438i	$-0.111253\pi$
$308$	− 9.00000i	− 0.512823i
$309$	8.00000	0.455104
$310$	0	0
$311$	−16.0000	−0.907277	−0.453638	−	0.891186i	$-0.649874\pi$
$311$	−16.0000	−0.907277	−0.453638	+	0.891186i	$0.649874\pi$
$312$	− 2.00000i	− 0.113228i
$313$	10.0000i	0.565233i	0.959233	+	0.282617i	$0.0912024\pi$
$313$	10.0000i	0.565233i	−0.959233	+	0.282617i	$0.908798\pi$
$314$	−7.00000	−0.395033
$315$	0	0
$316$	13.0000	0.731307
$317$	24.0000i	1.34797i	0.738743	+	0.673987i	$0.235420\pi$
$317$	24.0000i	1.34797i	−0.738743	+	0.673987i	$0.764580\pi$
$318$	− 5.00000i	− 0.280386i
$319$	24.0000	1.34374
$320$	0	0
$321$	3.00000	0.167444
$322$	− 21.0000i	− 1.17028i
$323$	56.0000i	3.11592i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	−18.0000	−0.996928
$327$	− 4.00000i	− 0.221201i
$328$	0	0
$329$	−18.0000	−0.992372
$330$	0	0
$331$	32.0000	1.75888	0.879440	−	0.476011i	$-0.157918\pi$
$331$	32.0000	1.75888	0.879440	+	0.476011i	$0.157918\pi$
$332$	− 16.0000i	− 0.878114i
$333$	4.00000i	0.219199i
$334$	15.0000	0.820763
$335$	0	0
$336$	−3.00000	−0.163663
$337$	− 10.0000i	− 0.544735i	−0.962193	−	0.272367i	$-0.912193\pi$
$337$	− 10.0000i	− 0.544735i	0.962193	−	0.272367i	$-0.0878066\pi$
$338$	− 9.00000i	− 0.489535i
$339$	1.00000	0.0543125
$340$	0	0
$341$	−3.00000	−0.162459
$342$	7.00000i	0.378517i
$343$	15.0000i	0.809924i
$344$	1.00000	0.0539164
$345$	0	0
$346$	6.00000	0.322562
$347$	− 12.0000i	− 0.644194i	−0.946707	−	0.322097i	$-0.895612\pi$
$347$	− 12.0000i	− 0.644194i	0.946707	−	0.322097i	$-0.104388\pi$
$348$	− 8.00000i	− 0.428845i
$349$	−20.0000	−1.07058	−0.535288	−	0.844670i	$-0.679797\pi$
$349$	−20.0000	−1.07058	−0.535288	+	0.844670i	$0.679797\pi$
$350$	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
$356$	−3.00000	−0.159000
$357$	− 24.0000i	− 1.27021i
$358$	4.00000i	0.211407i
$359$	−1.00000	−0.0527780	−0.0263890	−	0.999652i	$-0.508401\pi$
$359$	−1.00000	−0.0527780	−0.0263890	+	0.999652i	$0.508401\pi$
$360$	0	0
$361$	30.0000	1.57895
$362$	21.0000i	1.10374i
$363$	− 2.00000i	− 0.104973i
$364$	6.00000	0.314485
$365$	0	0
$366$	2.00000	0.104542
$367$	− 18.0000i	− 0.939592i	−0.882775	−	0.469796i	$-0.844327\pi$
$367$	− 18.0000i	− 0.939592i	0.882775	−	0.469796i	$-0.155673\pi$
$368$	− 7.00000i	− 0.364900i
$369$	0	0
$370$	0	0
$371$	15.0000	0.778761
$372$	1.00000i	0.0518476i
$373$	− 13.0000i	− 0.673114i	−0.941663	−	0.336557i	$-0.890737\pi$
$373$	− 13.0000i	− 0.673114i	0.941663	−	0.336557i	$-0.109263\pi$
$374$	24.0000	1.24101
$375$	0	0
$376$	−6.00000	−0.309426
$377$	16.0000i	0.824042i
$378$	− 3.00000i	− 0.154303i
$379$	−19.0000	−0.975964	−0.487982	−	0.872854i	$-0.662267\pi$
$379$	−19.0000	−0.975964	−0.487982	+	0.872854i	$0.662267\pi$
$380$	0	0
$381$	12.0000	0.614779
$382$	24.0000i	1.22795i
$383$	36.0000i	1.83951i	0.392488	+	0.919757i	$0.371614\pi$
$383$	36.0000i	1.83951i	−0.392488	+	0.919757i	$0.628386\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	14.0000	0.712581
$387$	1.00000i	0.0508329i
$388$	− 6.00000i	− 0.304604i
$389$	22.0000	1.11544	0.557722	−	0.830028i	$-0.311675\pi$
$389$	22.0000	1.11544	0.557722	+	0.830028i	$0.311675\pi$
$390$	0	0
$391$	56.0000	2.83204
$392$	− 2.00000i	− 0.101015i
$393$	6.00000i	0.302660i
$394$	6.00000	0.302276
$395$	0	0
$396$	3.00000	0.150756
$397$	25.0000i	1.25471i	0.778732	+	0.627357i	$0.215863\pi$
$397$	25.0000i	1.25471i	−0.778732	+	0.627357i	$0.784137\pi$
$398$	− 3.00000i	− 0.150376i
$399$	−21.0000	−1.05131
$400$	0	0
$401$	−35.0000	−1.74782	−0.873908	−	0.486091i	$-0.838422\pi$
$401$	−35.0000	−1.74782	−0.873908	+	0.486091i	$0.838422\pi$
$402$	10.0000i	0.498755i
$403$	− 2.00000i	− 0.0996271i
$404$	−5.00000	−0.248759
$405$	0	0
$406$	24.0000	1.19110
$407$	− 12.0000i	− 0.594818i
$408$	− 8.00000i	− 0.396059i
$409$	−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
$411$	−10.0000	−0.493264
$412$	8.00000i	0.394132i
$413$	− 18.0000i	− 0.885722i
$414$	7.00000	0.344031
$415$	0	0
$416$	2.00000	0.0980581
$417$	10.0000i	0.489702i
$418$	− 21.0000i	− 1.02714i
$419$	−16.0000	−0.781651	−0.390826	−	0.920465i	$-0.627810\pi$
$419$	−16.0000	−0.781651	−0.390826	+	0.920465i	$0.627810\pi$
$420$	0	0
$421$	34.0000	1.65706	0.828529	−	0.559946i	$-0.189178\pi$
$421$	34.0000	1.65706	0.828529	+	0.559946i	$0.189178\pi$
$422$	− 17.0000i	− 0.827547i
$423$	− 6.00000i	− 0.291730i
$424$	5.00000	0.242821
$425$	0	0
$426$	9.00000	0.436051
$427$	6.00000i	0.290360i
$428$	3.00000i	0.145010i
$429$	−6.00000	−0.289683
$430$	0	0
$431$	32.0000	1.54139	0.770693	−	0.637207i	$-0.219910\pi$
$431$	32.0000	1.54139	0.770693	+	0.637207i	$0.219910\pi$
$432$	− 1.00000i	− 0.0481125i
$433$	− 19.0000i	− 0.913082i	−0.889702	−	0.456541i	$-0.849088\pi$
$433$	− 19.0000i	− 0.913082i	0.889702	−	0.456541i	$-0.150912\pi$
$434$	−3.00000	−0.144005
$435$	0	0
$436$	4.00000	0.191565
$437$	− 49.0000i	− 2.34399i
$438$	− 1.00000i	− 0.0477818i
$439$	26.0000	1.24091	0.620456	−	0.784241i	$-0.286947\pi$
$439$	26.0000	1.24091	0.620456	+	0.784241i	$0.286947\pi$
$440$	0	0
$441$	2.00000	0.0952381
$442$	16.0000i	0.761042i
$443$	− 19.0000i	− 0.902717i	−0.892343	−	0.451359i	$-0.850940\pi$
$443$	− 19.0000i	− 0.902717i	0.892343	−	0.451359i	$-0.149060\pi$
$444$	−4.00000	−0.189832
$445$	0	0
$446$	10.0000	0.473514
$447$	9.00000i	0.425685i
$448$	− 3.00000i	− 0.141737i
$449$	−34.0000	−1.60456	−0.802280	−	0.596948i	$-0.796380\pi$
$449$	−34.0000	−1.60456	−0.802280	+	0.596948i	$0.796380\pi$
$450$	0	0
$451$	0	0
$452$	1.00000i	0.0470360i
$453$	− 8.00000i	− 0.375873i
$454$	25.0000	1.17331
$455$	0	0
$456$	−7.00000	−0.327805
$457$	− 38.0000i	− 1.77757i	−0.458329	−	0.888783i	$-0.651552\pi$
$457$	− 38.0000i	− 1.77757i	0.458329	−	0.888783i	$-0.348448\pi$
$458$	− 13.0000i	− 0.607450i
$459$	8.00000	0.373408
$460$	0	0
$461$	−20.0000	−0.931493	−0.465746	−	0.884918i	$-0.654214\pi$
$461$	−20.0000	−0.931493	−0.465746	+	0.884918i	$0.654214\pi$
$462$	9.00000i	0.418718i
$463$	40.0000i	1.85896i	0.368875	+	0.929479i	$0.379743\pi$
$463$	40.0000i	1.85896i	−0.368875	+	0.929479i	$0.620257\pi$
$464$	8.00000	0.371391
$465$	0	0
$466$	25.0000	1.15810
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	2.00000i	0.0924500i
$469$	−30.0000	−1.38527
$470$	0	0
$471$	7.00000	0.322543
$472$	− 6.00000i	− 0.276172i
$473$	− 3.00000i	− 0.137940i
$474$	−13.0000	−0.597110
$475$	0	0
$476$	24.0000	1.10004
$477$	5.00000i	0.228934i
$478$	8.00000i	0.365911i
$479$	27.0000	1.23366	0.616831	−	0.787096i	$-0.288416\pi$
$479$	27.0000	1.23366	0.616831	+	0.787096i	$0.288416\pi$
$480$	0	0
$481$	8.00000	0.364769
$482$	22.0000i	1.00207i
$483$	21.0000i	0.955533i
$484$	2.00000	0.0909091
$485$	0	0
$486$	1.00000	0.0453609
$487$	− 20.0000i	− 0.906287i	−0.891438	−	0.453143i	$-0.850303\pi$
$487$	− 20.0000i	− 0.906287i	0.891438	−	0.453143i	$-0.149697\pi$
$488$	2.00000i	0.0905357i
$489$	18.0000	0.813988
$490$	0	0
$491$	−29.0000	−1.30875	−0.654376	−	0.756169i	$-0.727069\pi$
$491$	−29.0000	−1.30875	−0.654376	+	0.756169i	$0.727069\pi$
$492$	0	0
$493$	64.0000i	2.88242i
$494$	14.0000	0.629890
$495$	0	0
$496$	−1.00000	−0.0449013
$497$	27.0000i	1.21112i
$498$	16.0000i	0.716977i
$499$	20.0000	0.895323	0.447661	−	0.894203i	$-0.352257\pi$
$499$	20.0000	0.895323	0.447661	+	0.894203i	$0.352257\pi$
$500$	0	0
$501$	−15.0000	−0.670151
$502$	− 20.0000i	− 0.892644i
$503$	− 32.0000i	− 1.42681i	−0.700752	−	0.713405i	$-0.747152\pi$
$503$	− 32.0000i	− 1.42681i	0.700752	−	0.713405i	$-0.252848\pi$
$504$	3.00000	0.133631
$505$	0	0
$506$	−21.0000	−0.933564
$507$	9.00000i	0.399704i
$508$	12.0000i	0.532414i
$509$	24.0000	1.06378	0.531891	−	0.846813i	$-0.321482\pi$
$509$	24.0000	1.06378	0.531891	+	0.846813i	$0.321482\pi$
$510$	0	0
$511$	3.00000	0.132712
$512$	− 1.00000i	− 0.0441942i
$513$	− 7.00000i	− 0.309058i
$514$	−23.0000	−1.01449
$515$	0	0
$516$	−1.00000	−0.0440225
$517$	18.0000i	0.791639i
$518$	− 12.0000i	− 0.527250i
$519$	−6.00000	−0.263371
$520$	0	0
$521$	−18.0000	−0.788594	−0.394297	−	0.918983i	$-0.629012\pi$
$521$	−18.0000	−0.788594	−0.394297	+	0.918983i	$0.629012\pi$
$522$	8.00000i	0.350150i
$523$	− 29.0000i	− 1.26808i	−0.773300	−	0.634041i	$-0.781395\pi$
$523$	− 29.0000i	− 1.26808i	0.773300	−	0.634041i	$-0.218605\pi$
$524$	−6.00000	−0.262111
$525$	0	0
$526$	−24.0000	−1.04645
$527$	− 8.00000i	− 0.348485i
$528$	3.00000i	0.130558i
$529$	−26.0000	−1.13043
$530$	0	0
$531$	6.00000	0.260378
$532$	− 21.0000i	− 0.910465i
$533$	0	0
$534$	3.00000	0.129823
$535$	0	0
$536$	−10.0000	−0.431934
$537$	− 4.00000i	− 0.172613i
$538$	− 18.0000i	− 0.776035i
$539$	−6.00000	−0.258438
$540$	0	0
$541$	14.0000	0.601907	0.300954	−	0.953639i	$-0.402695\pi$
$541$	14.0000	0.601907	0.300954	+	0.953639i	$0.402695\pi$
$542$	− 7.00000i	− 0.300676i
$543$	− 21.0000i	− 0.901196i
$544$	8.00000	0.342997
$545$	0	0
$546$	−6.00000	−0.256776
$547$	− 14.0000i	− 0.598597i	−0.954160	−	0.299298i	$-0.903247\pi$
$547$	− 14.0000i	− 0.598597i	0.954160	−	0.299298i	$-0.0967526\pi$
$548$	− 10.0000i	− 0.427179i
$549$	−2.00000	−0.0853579
$550$	0	0
$551$	56.0000	2.38568
$552$	7.00000i	0.297940i
$553$	− 39.0000i	− 1.65845i
$554$	−8.00000	−0.339887
$555$	0	0
$556$	−10.0000	−0.424094
$557$	17.0000i	0.720313i	0.932892	+	0.360157i	$0.117277\pi$
$557$	17.0000i	0.720313i	−0.932892	+	0.360157i	$0.882723\pi$
$558$	− 1.00000i	− 0.0423334i
$559$	2.00000	0.0845910
$560$	0	0
$561$	−24.0000	−1.01328
$562$	28.0000i	1.18111i
$563$	36.0000i	1.51722i	0.651546	+	0.758610i	$0.274121\pi$
$563$	36.0000i	1.51722i	−0.651546	+	0.758610i	$0.725879\pi$
$564$	6.00000	0.252646
$565$	0	0
$566$	16.0000	0.672530
$567$	3.00000i	0.125988i
$568$	9.00000i	0.377632i
$569$	−37.0000	−1.55112	−0.775560	−	0.631273i	$-0.782533\pi$
$569$	−37.0000	−1.55112	−0.775560	+	0.631273i	$0.782533\pi$
$570$	0	0
$571$	−22.0000	−0.920671	−0.460336	−	0.887745i	$-0.652271\pi$
$571$	−22.0000	−0.920671	−0.460336	+	0.887745i	$0.652271\pi$
$572$	− 6.00000i	− 0.250873i
$573$	− 24.0000i	− 1.00261i
$574$	0	0
$575$	0	0
$576$	1.00000	0.0416667
$577$	− 8.00000i	− 0.333044i	−0.986038	−	0.166522i	$-0.946746\pi$
$577$	− 8.00000i	− 0.333044i	0.986038	−	0.166522i	$-0.0532537\pi$
$578$	47.0000i	1.95494i
$579$	−14.0000	−0.581820
$580$	0	0
$581$	−48.0000	−1.99138
$582$	6.00000i	0.248708i
$583$	− 15.0000i	− 0.621237i
$584$	1.00000	0.0413803
$585$	0	0
$586$	−14.0000	−0.578335
$587$	16.0000i	0.660391i	0.943913	+	0.330195i	$0.107115\pi$
$587$	16.0000i	0.660391i	−0.943913	+	0.330195i	$0.892885\pi$
$588$	2.00000i	0.0824786i
$589$	−7.00000	−0.288430
$590$	0	0
$591$	−6.00000	−0.246807
$592$	− 4.00000i	− 0.164399i
$593$	− 30.0000i	− 1.23195i	−0.787765	−	0.615976i	$-0.788762\pi$
$593$	− 30.0000i	− 1.23195i	0.787765	−	0.615976i	$-0.211238\pi$
$594$	−3.00000	−0.123091
$595$	0	0
$596$	−9.00000	−0.368654
$597$	3.00000i	0.122782i
$598$	− 14.0000i	− 0.572503i
$599$	−35.0000	−1.43006	−0.715031	−	0.699093i	$-0.753587\pi$
$599$	−35.0000	−1.43006	−0.715031	+	0.699093i	$0.753587\pi$
$600$	0	0
$601$	−30.0000	−1.22373	−0.611863	−	0.790964i	$-0.709580\pi$
$601$	−30.0000	−1.22373	−0.611863	+	0.790964i	$0.709580\pi$
$602$	− 3.00000i	− 0.122271i
$603$	− 10.0000i	− 0.407231i
$604$	8.00000	0.325515
$605$	0	0
$606$	5.00000	0.203111
$607$	− 13.0000i	− 0.527654i	−0.964570	−	0.263827i	$-0.915015\pi$
$607$	− 13.0000i	− 0.527654i	0.964570	−	0.263827i	$-0.0849848\pi$
$608$	− 7.00000i	− 0.283887i
$609$	−24.0000	−0.972529
$610$	0	0
$611$	−12.0000	−0.485468
$612$	8.00000i	0.323381i
$613$	− 26.0000i	− 1.05013i	−0.851062	−	0.525065i	$-0.824041\pi$
$613$	− 26.0000i	− 1.05013i	0.851062	−	0.525065i	$-0.175959\pi$
$614$	−12.0000	−0.484281
$615$	0	0
$616$	−9.00000	−0.362620
$617$	− 3.00000i	− 0.120775i	−0.998175	−	0.0603877i	$-0.980766\pi$
$617$	− 3.00000i	− 0.120775i	0.998175	−	0.0603877i	$-0.0192337\pi$
$618$	− 8.00000i	− 0.321807i
$619$	18.0000	0.723481	0.361741	−	0.932279i	$-0.382183\pi$
$619$	18.0000	0.723481	0.361741	+	0.932279i	$0.382183\pi$
$620$	0	0
$621$	−7.00000	−0.280900
$622$	16.0000i	0.641542i
$623$	9.00000i	0.360577i
$624$	−2.00000	−0.0800641
$625$	0	0
$626$	10.0000	0.399680
$627$	21.0000i	0.838659i
$628$	7.00000i	0.279330i
$629$	32.0000	1.27592
$630$	0	0
$631$	−5.00000	−0.199047	−0.0995234	−	0.995035i	$-0.531732\pi$
$631$	−5.00000	−0.199047	−0.0995234	+	0.995035i	$0.531732\pi$
$632$	− 13.0000i	− 0.517112i
$633$	17.0000i	0.675689i
$634$	24.0000	0.953162
$635$	0	0
$636$	−5.00000	−0.198263
$637$	− 4.00000i	− 0.158486i
$638$	− 24.0000i	− 0.950169i
$639$	−9.00000	−0.356034
$640$	0	0
$641$	−30.0000	−1.18493	−0.592464	−	0.805597i	$-0.701845\pi$
$641$	−30.0000	−1.18493	−0.592464	+	0.805597i	$0.701845\pi$
$642$	− 3.00000i	− 0.118401i
$643$	− 19.0000i	− 0.749287i	−0.927169	−	0.374643i	$-0.877765\pi$
$643$	− 19.0000i	− 0.749287i	0.927169	−	0.374643i	$-0.122235\pi$
$644$	−21.0000	−0.827516
$645$	0	0
$646$	56.0000	2.20329
$647$	49.0000i	1.92639i	0.268806	+	0.963194i	$0.413371\pi$
$647$	49.0000i	1.92639i	−0.268806	+	0.963194i	$0.586629\pi$
$648$	1.00000i	0.0392837i
$649$	−18.0000	−0.706562
$650$	0	0
$651$	3.00000	0.117579
$652$	18.0000i	0.704934i
$653$	6.00000i	0.234798i	0.993085	+	0.117399i	$0.0374557\pi$
$653$	6.00000i	0.234798i	−0.993085	+	0.117399i	$0.962544\pi$
$654$	−4.00000	−0.156412
$655$	0	0
$656$	0	0
$657$	1.00000i	0.0390137i
$658$	18.0000i	0.701713i
$659$	32.0000	1.24654	0.623272	−	0.782006i	$-0.285803\pi$
$659$	32.0000	1.24654	0.623272	+	0.782006i	$0.285803\pi$
$660$	0	0
$661$	−46.0000	−1.78919	−0.894596	−	0.446875i	$-0.852537\pi$
$661$	−46.0000	−1.78919	−0.894596	+	0.446875i	$0.852537\pi$
$662$	− 32.0000i	− 1.24372i
$663$	− 16.0000i	− 0.621389i
$664$	−16.0000	−0.620920
$665$	0	0
$666$	4.00000	0.154997
$667$	− 56.0000i	− 2.16833i
$668$	− 15.0000i	− 0.580367i
$669$	−10.0000	−0.386622
$670$	0	0
$671$	6.00000	0.231627
$672$	3.00000i	0.115728i
$673$	46.0000i	1.77317i	0.462566	+	0.886585i	$0.346929\pi$
$673$	46.0000i	1.77317i	−0.462566	+	0.886585i	$0.653071\pi$
$674$	−10.0000	−0.385186
$675$	0	0
$676$	−9.00000	−0.346154
$677$	− 43.0000i	− 1.65262i	−0.563212	−	0.826312i	$-0.690435\pi$
$677$	− 43.0000i	− 1.65262i	0.563212	−	0.826312i	$-0.309565\pi$
$678$	− 1.00000i	− 0.0384048i
$679$	−18.0000	−0.690777
$680$	0	0
$681$	−25.0000	−0.958002
$682$	3.00000i	0.114876i
$683$	15.0000i	0.573959i	0.957937	+	0.286980i	$0.0926512\pi$
$683$	15.0000i	0.573959i	−0.957937	+	0.286980i	$0.907349\pi$
$684$	7.00000	0.267652
$685$	0	0
$686$	15.0000	0.572703
$687$	13.0000i	0.495981i
$688$	− 1.00000i	− 0.0381246i
$689$	10.0000	0.380970
$690$	0	0
$691$	5.00000	0.190209	0.0951045	−	0.995467i	$-0.469681\pi$
$691$	5.00000	0.190209	0.0951045	+	0.995467i	$0.469681\pi$
$692$	− 6.00000i	− 0.228086i
$693$	− 9.00000i	− 0.341882i
$694$	−12.0000	−0.455514
$695$	0	0
$696$	−8.00000	−0.303239
$697$	0	0
$698$	20.0000i	0.757011i
$699$	−25.0000	−0.945587
$700$	0	0
$701$	33.0000	1.24639	0.623196	−	0.782065i	$-0.285834\pi$
$701$	33.0000	1.24639	0.623196	+	0.782065i	$0.285834\pi$
$702$	− 2.00000i	− 0.0754851i
$703$	− 28.0000i	− 1.05604i
$704$	−3.00000	−0.113067
$705$	0	0
$706$	18.0000	0.677439
$707$	15.0000i	0.564133i
$708$	6.00000i	0.225494i
$709$	15.0000	0.563337	0.281668	−	0.959512i	$-0.409112\pi$
$709$	15.0000	0.563337	0.281668	+	0.959512i	$0.409112\pi$
$710$	0	0
$711$	13.0000	0.487538
$712$	3.00000i	0.112430i
$713$	7.00000i	0.262152i
$714$	−24.0000	−0.898177
$715$	0	0
$716$	4.00000	0.149487
$717$	− 8.00000i	− 0.298765i
$718$	1.00000i	0.0373197i
$719$	30.0000	1.11881	0.559406	−	0.828894i	$-0.311029\pi$
$719$	30.0000	1.11881	0.559406	+	0.828894i	$0.311029\pi$
$720$	0	0
$721$	24.0000	0.893807
$722$	− 30.0000i	− 1.11648i
$723$	− 22.0000i	− 0.818189i
$724$	21.0000	0.780459
$725$	0	0
$726$	−2.00000	−0.0742270
$727$	− 29.0000i	− 1.07555i	−0.843088	−	0.537775i	$-0.819265\pi$
$727$	− 29.0000i	− 1.07555i	0.843088	−	0.537775i	$-0.180735\pi$
$728$	− 6.00000i	− 0.222375i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	8.00000	0.295891
$732$	− 2.00000i	− 0.0739221i
$733$	− 2.00000i	− 0.0738717i	−0.999318	−	0.0369358i	$-0.988240\pi$
$733$	− 2.00000i	− 0.0738717i	0.999318	−	0.0369358i	$-0.0117597\pi$
$734$	−18.0000	−0.664392
$735$	0	0
$736$	−7.00000	−0.258023
$737$	30.0000i	1.10506i
$738$	0	0
$739$	40.0000	1.47142	0.735712	−	0.677295i	$-0.236848\pi$
$739$	40.0000	1.47142	0.735712	+	0.677295i	$0.236848\pi$
$740$	0	0
$741$	−14.0000	−0.514303
$742$	− 15.0000i	− 0.550667i
$743$	− 1.00000i	− 0.0366864i	−0.999832	−	0.0183432i	$-0.994161\pi$
$743$	− 1.00000i	− 0.0366864i	0.999832	−	0.0183432i	$-0.00583916\pi$
$744$	1.00000	0.0366618
$745$	0	0
$746$	−13.0000	−0.475964
$747$	− 16.0000i	− 0.585409i
$748$	− 24.0000i	− 0.877527i
$749$	9.00000	0.328853
$750$	0	0
$751$	6.00000	0.218943	0.109472	−	0.993990i	$-0.465084\pi$
$751$	6.00000	0.218943	0.109472	+	0.993990i	$0.465084\pi$
$752$	6.00000i	0.218797i
$753$	20.0000i	0.728841i
$754$	16.0000	0.582686
$755$	0	0
$756$	−3.00000	−0.109109
$757$	− 2.00000i	− 0.0726912i	−0.999339	−	0.0363456i	$-0.988428\pi$
$757$	− 2.00000i	− 0.0726912i	0.999339	−	0.0363456i	$-0.0115717\pi$
$758$	19.0000i	0.690111i
$759$	21.0000	0.762252
$760$	0	0
$761$	49.0000	1.77625	0.888124	−	0.459603i	$-0.152008\pi$
$761$	49.0000	1.77625	0.888124	+	0.459603i	$0.152008\pi$
$762$	− 12.0000i	− 0.434714i
$763$	− 12.0000i	− 0.434429i
$764$	24.0000	0.868290
$765$	0	0
$766$	36.0000	1.30073
$767$	− 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
$772$	− 14.0000i	− 0.503871i
$773$	1.00000i	0.0359675i	0.999838	+	0.0179838i	$0.00572471\pi$
$773$	1.00000i	0.0359675i	−0.999838	+	0.0179838i	$0.994275\pi$
$774$	1.00000	0.0359443
$775$	0	0
$776$	−6.00000	−0.215387
$777$	12.0000i	0.430498i
$778$	− 22.0000i	− 0.788738i
$779$	0	0
$780$	0	0
$781$	27.0000	0.966136
$782$	− 56.0000i	− 2.00256i
$783$	− 8.00000i	− 0.285897i
$784$	−2.00000	−0.0714286
$785$	0	0
$786$	6.00000	0.214013
$787$	− 27.0000i	− 0.962446i	−0.876598	−	0.481223i	$-0.840193\pi$
$787$	− 27.0000i	− 0.962446i	0.876598	−	0.481223i	$-0.159807\pi$
$788$	− 6.00000i	− 0.213741i
$789$	24.0000	0.854423
$790$	0	0
$791$	3.00000	0.106668
$792$	− 3.00000i	− 0.106600i
$793$	4.00000i	0.142044i
$794$	25.0000	0.887217
$795$	0	0
$796$	−3.00000	−0.106332
$797$	− 2.00000i	− 0.0708436i	−0.999372	−	0.0354218i	$-0.988723\pi$
$797$	− 2.00000i	− 0.0708436i	0.999372	−	0.0354218i	$-0.0112775\pi$
$798$	21.0000i	0.743392i
$799$	−48.0000	−1.69812
$800$	0	0
$801$	−3.00000	−0.106000
$802$	35.0000i	1.23589i
$803$	− 3.00000i	− 0.105868i
$804$	10.0000	0.352673
$805$	0	0
$806$	−2.00000	−0.0704470
$807$	18.0000i	0.633630i
$808$	5.00000i	0.175899i
$809$	−13.0000	−0.457056	−0.228528	−	0.973537i	$-0.573391\pi$
$809$	−13.0000	−0.457056	−0.228528	+	0.973537i	$0.573391\pi$
$810$	0	0
$811$	−23.0000	−0.807639	−0.403820	−	0.914839i	$-0.632318\pi$
$811$	−23.0000	−0.807639	−0.403820	+	0.914839i	$0.632318\pi$
$812$	− 24.0000i	− 0.842235i
$813$	7.00000i	0.245501i
$814$	−12.0000	−0.420600
$815$	0	0
$816$	−8.00000	−0.280056
$817$	− 7.00000i	− 0.244899i
$818$	12.0000i	0.419570i
$819$	6.00000	0.209657
$820$	0	0
$821$	−24.0000	−0.837606	−0.418803	−	0.908077i	$-0.637550\pi$
$821$	−24.0000	−0.837606	−0.418803	+	0.908077i	$0.637550\pi$
$822$	10.0000i	0.348790i
$823$	− 14.0000i	− 0.488009i	−0.969774	−	0.244005i	$-0.921539\pi$
$823$	− 14.0000i	− 0.488009i	0.969774	−	0.244005i	$-0.0784612\pi$
$824$	8.00000	0.278693
$825$	0	0
$826$	−18.0000	−0.626300
$827$	− 44.0000i	− 1.53003i	−0.644013	−	0.765015i	$-0.722732\pi$
$827$	− 44.0000i	− 1.53003i	0.644013	−	0.765015i	$-0.277268\pi$
$828$	− 7.00000i	− 0.243267i
$829$	21.0000	0.729360	0.364680	−	0.931133i	$-0.381178\pi$
$829$	21.0000	0.729360	0.364680	+	0.931133i	$0.381178\pi$
$830$	0	0
$831$	8.00000	0.277517
$832$	− 2.00000i	− 0.0693375i
$833$	− 16.0000i	− 0.554367i
$834$	10.0000	0.346272
$835$	0	0
$836$	−21.0000	−0.726300
$837$	1.00000i	0.0345651i
$838$	16.0000i	0.552711i
$839$	27.0000	0.932144	0.466072	−	0.884747i	$-0.345669\pi$
$839$	27.0000	0.932144	0.466072	+	0.884747i	$0.345669\pi$
$840$	0	0
$841$	35.0000	1.20690
$842$	− 34.0000i	− 1.17172i
$843$	− 28.0000i	− 0.964371i
$844$	−17.0000	−0.585164
$845$	0	0
$846$	−6.00000	−0.206284
$847$	− 6.00000i	− 0.206162i
$848$	− 5.00000i	− 0.171701i
$849$	−16.0000	−0.549119
$850$	0	0
$851$	−28.0000	−0.959828
$852$	− 9.00000i	− 0.308335i
$853$	− 45.0000i	− 1.54077i	−0.637579	−	0.770385i	$-0.720064\pi$
$853$	− 45.0000i	− 1.54077i	0.637579	−	0.770385i	$-0.279936\pi$
$854$	6.00000	0.205316
$855$	0	0
$856$	3.00000	0.102538
$857$	− 22.0000i	− 0.751506i	−0.926720	−	0.375753i	$-0.877384\pi$
$857$	− 22.0000i	− 0.751506i	0.926720	−	0.375753i	$-0.122616\pi$
$858$	6.00000i	0.204837i
$859$	24.0000	0.818869	0.409435	−	0.912339i	$-0.365726\pi$
$859$	24.0000	0.818869	0.409435	+	0.912339i	$0.365726\pi$
$860$	0	0
$861$	0	0
$862$	− 32.0000i	− 1.08992i
$863$	31.0000i	1.05525i	0.849477	+	0.527626i	$0.176918\pi$
$863$	31.0000i	1.05525i	−0.849477	+	0.527626i	$0.823082\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	−19.0000	−0.645646
$867$	− 47.0000i	− 1.59620i
$868$	3.00000i	0.101827i
$869$	−39.0000	−1.32298
$870$	0	0
$871$	−20.0000	−0.677674
$872$	− 4.00000i	− 0.135457i
$873$	− 6.00000i	− 0.203069i
$874$	−49.0000	−1.65745
$875$	0	0
$876$	−1.00000	−0.0337869
$877$	2.00000i	0.0675352i	0.999430	+	0.0337676i	$0.0107506\pi$
$877$	2.00000i	0.0675352i	−0.999430	+	0.0337676i	$0.989249\pi$
$878$	− 26.0000i	− 0.877457i
$879$	14.0000	0.472208
$880$	0	0
$881$	18.0000	0.606435	0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000	0.606435	0.303218	+	0.952921i	$0.401939\pi$
$882$	− 2.00000i	− 0.0673435i
$883$	13.0000i	0.437485i	0.975783	+	0.218742i	$0.0701954\pi$
$883$	13.0000i	0.437485i	−0.975783	+	0.218742i	$0.929805\pi$
$884$	16.0000	0.538138
$885$	0	0
$886$	−19.0000	−0.638317
$887$	22.0000i	0.738688i	0.929293	+	0.369344i	$0.120418\pi$
$887$	22.0000i	0.738688i	−0.929293	+	0.369344i	$0.879582\pi$
$888$	4.00000i	0.134231i
$889$	36.0000	1.20740
$890$	0	0
$891$	3.00000	0.100504
$892$	− 10.0000i	− 0.334825i
$893$	42.0000i	1.40548i
$894$	9.00000	0.301005
$895$	0	0
$896$	−3.00000	−0.100223
$897$	14.0000i	0.467446i
$898$	34.0000i	1.13459i
$899$	−8.00000	−0.266815
$900$	0	0
$901$	40.0000	1.33259
$902$	0	0
$903$	3.00000i	0.0998337i
$904$	1.00000	0.0332595
$905$	0	0
$906$	−8.00000	−0.265782
$907$	− 40.0000i	− 1.32818i	−0.747653	−	0.664089i	$-0.768820\pi$
$907$	− 40.0000i	− 1.32818i	0.747653	−	0.664089i	$-0.231180\pi$
$908$	− 25.0000i	− 0.829654i
$909$	−5.00000	−0.165840
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	7.00000i	0.231793i
$913$	48.0000i	1.58857i
$914$	−38.0000	−1.25693
$915$	0	0
$916$	−13.0000	−0.429532
$917$	18.0000i	0.594412i
$918$	− 8.00000i	− 0.264039i
$919$	10.0000	0.329870	0.164935	−	0.986304i	$-0.447259\pi$
$919$	10.0000	0.329870	0.164935	+	0.986304i	$0.447259\pi$
$920$	0	0
$921$	12.0000	0.395413
$922$	20.0000i	0.658665i
$923$	18.0000i	0.592477i
$924$	9.00000	0.296078
$925$	0	0
$926$	40.0000	1.31448
$927$	8.00000i	0.262754i
$928$	− 8.00000i	− 0.262613i
$929$	−3.00000	−0.0984268	−0.0492134	−	0.998788i	$-0.515671\pi$
$929$	−3.00000	−0.0984268	−0.0492134	+	0.998788i	$0.515671\pi$
$930$	0	0
$931$	−14.0000	−0.458831
$932$	− 25.0000i	− 0.818902i
$933$	− 16.0000i	− 0.523816i
$934$	0	0
$935$	0	0
$936$	2.00000	0.0653720
$937$	− 16.0000i	− 0.522697i	−0.965244	−	0.261349i	$-0.915833\pi$
$937$	− 16.0000i	− 0.522697i	0.965244	−	0.261349i	$-0.0841672\pi$
$938$	30.0000i	0.979535i
$939$	−10.0000	−0.326338
$940$	0	0
$941$	−28.0000	−0.912774	−0.456387	−	0.889781i	$-0.650857\pi$
$941$	−28.0000	−0.912774	−0.456387	+	0.889781i	$0.650857\pi$
$942$	− 7.00000i	− 0.228072i
$943$	0	0
$944$	−6.00000	−0.195283
$945$	0	0
$946$	−3.00000	−0.0975384
$947$	38.0000i	1.23483i	0.786636	+	0.617417i	$0.211821\pi$
$947$	38.0000i	1.23483i	−0.786636	+	0.617417i	$0.788179\pi$
$948$	13.0000i	0.422220i
$949$	2.00000	0.0649227
$950$	0	0
$951$	−24.0000	−0.778253
$952$	− 24.0000i	− 0.777844i
$953$	− 12.0000i	− 0.388718i	−0.980930	−	0.194359i	$-0.937737\pi$
$953$	− 12.0000i	− 0.388718i	0.980930	−	0.194359i	$-0.0622627\pi$
$954$	5.00000	0.161881
$955$	0	0
$956$	8.00000	0.258738
$957$	24.0000i	0.775810i
$958$	− 27.0000i	− 0.872330i
$959$	−30.0000	−0.968751
$960$	0	0
$961$	1.00000	0.0322581
$962$	− 8.00000i	− 0.257930i
$963$	3.00000i	0.0966736i
$964$	22.0000	0.708572
$965$	0	0
$966$	21.0000	0.675664
$967$	− 32.0000i	− 1.02905i	−0.857475	−	0.514525i	$-0.827968\pi$
$967$	− 32.0000i	− 1.02905i	0.857475	−	0.514525i	$-0.172032\pi$
$968$	− 2.00000i	− 0.0642824i
$969$	−56.0000	−1.79898
$970$	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
$973$	30.0000i	0.961756i
$974$	−20.0000	−0.640841
$975$	0	0
$976$	2.00000	0.0640184
$977$	46.0000i	1.47167i	0.677161	+	0.735835i	$0.263210\pi$
$977$	46.0000i	1.47167i	−0.677161	+	0.735835i	$0.736790\pi$
$978$	− 18.0000i	− 0.575577i
$979$	9.00000	0.287641
$980$	0	0
$981$	4.00000	0.127710
$982$	29.0000i	0.925427i
$983$	− 48.0000i	− 1.53096i	−0.643458	−	0.765481i	$-0.722501\pi$
$983$	− 48.0000i	− 1.53096i	0.643458	−	0.765481i	$-0.277499\pi$
$984$	0	0
$985$	0	0
$986$	64.0000	2.03818
$987$	− 18.0000i	− 0.572946i
$988$	− 14.0000i	− 0.445399i
$989$	−7.00000	−0.222587
$990$	0	0
$991$	5.00000	0.158830	0.0794151	−	0.996842i	$-0.474695\pi$
$991$	5.00000	0.158830	0.0794151	+	0.996842i	$0.474695\pi$
$992$	1.00000i	0.0317500i
$993$	32.0000i	1.01549i
$994$	27.0000	0.856388
$995$	0	0
$996$	16.0000	0.506979
$997$	− 6.00000i	− 0.190022i	−0.995476	−	0.0950110i	$-0.969711\pi$
$997$	− 6.00000i	− 0.190022i	0.995476	−	0.0950110i	$-0.0302886\pi$
$998$	− 20.0000i	− 0.633089i
$999$	−4.00000	−0.126554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4650.2.d.ba.3349.1		2
5.2	odd	4		4650.2.a.bk.1.1		1
5.3	odd	4		930.2.a.e.1.1	✓	1
5.4	even	2	inner	4650.2.d.ba.3349.2		2
15.8	even	4		2790.2.a.u.1.1		1
20.3	even	4		7440.2.a.x.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
930.2.a.e.1.1	✓	1	5.3	odd	4
2790.2.a.u.1.1		1	15.8	even	4
4650.2.a.bk.1.1		1	5.2	odd	4
4650.2.d.ba.3349.1		2	1.1	even	1	trivial
4650.2.d.ba.3349.2		2	5.4	even	2	inner
7440.2.a.x.1.1		1	20.3	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)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +3.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +3.00000 q^{11} -1.00000i q^{12} +2.00000i q^{13} +3.00000 q^{14} +1.00000 q^{16} +8.00000i q^{17} +1.00000i q^{18} +7.00000 q^{19} -3.00000 q^{21} -3.00000i q^{22} -7.00000i q^{23} -1.00000 q^{24} +2.00000 q^{26} -1.00000i q^{27} -3.00000i q^{28} +8.00000 q^{29} -1.00000 q^{31} -1.00000i q^{32} +3.00000i q^{33} +8.00000 q^{34} +1.00000 q^{36} -4.00000i q^{37} -7.00000i q^{38} -2.00000 q^{39} +3.00000i q^{42} -1.00000i q^{43} -3.00000 q^{44} -7.00000 q^{46} +6.00000i q^{47} +1.00000i q^{48} -2.00000 q^{49} -8.00000 q^{51} -2.00000i q^{52} -5.00000i q^{53} -1.00000 q^{54} -3.00000 q^{56} +7.00000i q^{57} -8.00000i q^{58} -6.00000 q^{59} +2.00000 q^{61} +1.00000i q^{62} -3.00000i q^{63} -1.00000 q^{64} +3.00000 q^{66} +10.0000i q^{67} -8.00000i q^{68} +7.00000 q^{69} +9.00000 q^{71} -1.00000i q^{72} -1.00000i q^{73} -4.00000 q^{74} -7.00000 q^{76} +9.00000i q^{77} +2.00000i q^{78} -13.0000 q^{79} +1.00000 q^{81} +16.0000i q^{83} +3.00000 q^{84} -1.00000 q^{86} +8.00000i q^{87} +3.00000i q^{88} +3.00000 q^{89} -6.00000 q^{91} +7.00000i q^{92} -1.00000i q^{93} +6.00000 q^{94} +1.00000 q^{96} +6.00000i q^{97} +2.00000i q^{98} -3.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} + 6 q^{11} + 6 q^{14} + 2 q^{16} + 14 q^{19} - 6 q^{21} - 2 q^{24} + 4 q^{26} + 16 q^{29} - 2 q^{31} + 16 q^{34} + 2 q^{36} - 4 q^{39} - 6 q^{44} - 14 q^{46} - 4 q^{49}+ \cdots - 6 q^{99}+O(q^{100}) \) 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9 + 6 * q^11 + 6 * q^14 + 2 * q^16 + 14 * q^19 - 6 * q^21 - 2 * q^24 + 4 * q^26 + 16 * q^29 - 2 * q^31 + 16 * q^34 + 2 * q^36 - 4 * q^39 - 6 * q^44 - 14 * q^46 - 4 * q^49 - 16 * q^51 - 2 * q^54 - 6 * q^56 - 12 * q^59 + 4 * q^61 - 2 * q^64 + 6 * q^66 + 14 * q^69 + 18 * q^71 - 8 * q^74 - 14 * q^76 - 26 * q^79 + 2 * q^81 + 6 * q^84 - 2 * q^86 + 6 * q^89 - 12 * q^91 + 12 * q^94 + 2 * q^96 - 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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