LMFDB - Embedded newform 4650.2.d.q.3349.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4650,2,Mod(3349,4650)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4650, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4650.3349");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4650 = 2 \cdot 3 \cdot 5^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4650.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$37.1304369399$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

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

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -3.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -3.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} -3.00000 q^{11} -1.00000i q^{12} -1.00000i q^{13} -3.00000 q^{14} +1.00000 q^{16} +2.00000i q^{17} +1.00000i q^{18} +3.00000 q^{21} +3.00000i q^{22} +4.00000i q^{23} -1.00000 q^{24} -1.00000 q^{26} -1.00000i q^{27} +3.00000i q^{28} -10.0000 q^{29} +1.00000 q^{31} -1.00000i q^{32} -3.00000i q^{33} +2.00000 q^{34} +1.00000 q^{36} -3.00000i q^{37} +1.00000 q^{39} +7.00000 q^{41} -3.00000i q^{42} -1.00000i q^{43} +3.00000 q^{44} +4.00000 q^{46} +7.00000i q^{47} +1.00000i q^{48} -2.00000 q^{49} -2.00000 q^{51} +1.00000i q^{52} +9.00000i q^{53} -1.00000 q^{54} +3.00000 q^{56} +10.0000i q^{58} +7.00000 q^{61} -1.00000i q^{62} +3.00000i q^{63} -1.00000 q^{64} -3.00000 q^{66} +2.00000i q^{67} -2.00000i q^{68} -4.00000 q^{69} +7.00000 q^{71} -1.00000i q^{72} +4.00000i q^{73} -3.00000 q^{74} +9.00000i q^{77} -1.00000i q^{78} +10.0000 q^{79} +1.00000 q^{81} -7.00000i q^{82} +9.00000i q^{83} -3.00000 q^{84} -1.00000 q^{86} -10.0000i q^{87} -3.00000i q^{88} -10.0000 q^{89} -3.00000 q^{91} -4.00000i q^{92} +1.00000i q^{93} +7.00000 q^{94} +1.00000 q^{96} +2.00000i q^{97} +2.00000i q^{98} +3.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9}+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} - 6 q^{11} - 6 q^{14} + 2 q^{16} + 6 q^{21} - 2 q^{24} - 2 q^{26} - 20 q^{29} + 2 q^{31} + 4 q^{34} + 2 q^{36} + 2 q^{39} + 14 q^{41} + 6 q^{44} + 8 q^{46} - 4 q^{49} - 4 q^{51} - 2 q^{54} + 6 q^{56} + 14 q^{61} - 2 q^{64} - 6 q^{66} - 8 q^{69} + 14 q^{71} - 6 q^{74} + 20 q^{79} + 2 q^{81} - 6 q^{84} - 2 q^{86} - 20 q^{89} - 6 q^{91} + 14 q^{94} + 2 q^{96} + 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^21 - 2 * q^24 - 2 * q^26 - 20 * q^29 + 2 * q^31 + 4 * q^34 + 2 * q^36 + 2 * q^39 + 14 * q^41 + 6 * q^44 + 8 * q^46 - 4 * q^49 - 4 * q^51 - 2 * q^54 + 6 * q^56 + 14 * q^61 - 2 * q^64 - 6 * q^66 - 8 * q^69 + 14 * q^71 - 6 * q^74 + 20 * q^79 + 2 * q^81 - 6 * q^84 - 2 * q^86 - 20 * q^89 - 6 * q^91 + 14 * q^94 + 2 * q^96 + 6 * 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
$7$	− 3.00000i	− 1.13389i	−0.823754	−	0.566947i	$-0.808125\pi$
$7$	− 3.00000i	− 1.13389i	0.823754	−	0.566947i	$-0.191875\pi$
$8$	1.00000i	0.353553i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−3.00000	−0.904534	−0.452267	−	0.891883i	$-0.649385\pi$
$11$	−3.00000	−0.904534	−0.452267	+	0.891883i	$0.649385\pi$
$12$	− 1.00000i	− 0.288675i
$13$	− 1.00000i	− 0.277350i	−0.990338	−	0.138675i	$-0.955716\pi$
$13$	− 1.00000i	− 0.277350i	0.990338	−	0.138675i	$-0.0442844\pi$
$14$	−3.00000	−0.801784
$15$	0	0
$16$	1.00000	0.250000
$17$	2.00000i	0.485071i	0.970143	+	0.242536i	$0.0779791\pi$
$17$	2.00000i	0.485071i	−0.970143	+	0.242536i	$0.922021\pi$
$18$	1.00000i	0.235702i
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	3.00000	0.654654
$22$	3.00000i	0.639602i
$23$	4.00000i	0.834058i	0.908893	+	0.417029i	$0.136929\pi$
$23$	4.00000i	0.834058i	−0.908893	+	0.417029i	$0.863071\pi$
$24$	−1.00000	−0.204124
$25$	0	0
$26$	−1.00000	−0.196116
$27$	− 1.00000i	− 0.192450i
$28$	3.00000i	0.566947i
$29$	−10.0000	−1.85695	−0.928477	−	0.371391i	$-0.878881\pi$
$29$	−10.0000	−1.85695	−0.928477	+	0.371391i	$0.878881\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	− 1.00000i	− 0.176777i
$33$	− 3.00000i	− 0.522233i
$34$	2.00000	0.342997
$35$	0	0
$36$	1.00000	0.166667
$37$	− 3.00000i	− 0.493197i	−0.969118	−	0.246598i	$-0.920687\pi$
$37$	− 3.00000i	− 0.493197i	0.969118	−	0.246598i	$-0.0793129\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	7.00000	1.09322	0.546608	−	0.837389i	$-0.315919\pi$
$41$	7.00000	1.09322	0.546608	+	0.837389i	$0.315919\pi$
$42$	− 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
$46$	4.00000	0.589768
$47$	7.00000i	1.02105i	0.859861	+	0.510527i	$0.170550\pi$
$47$	7.00000i	1.02105i	−0.859861	+	0.510527i	$0.829450\pi$
$48$	1.00000i	0.144338i
$49$	−2.00000	−0.285714
$50$	0	0
$51$	−2.00000	−0.280056
$52$	1.00000i	0.138675i
$53$	9.00000i	1.23625i	0.786082	+	0.618123i	$0.212106\pi$
$53$	9.00000i	1.23625i	−0.786082	+	0.618123i	$0.787894\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	3.00000	0.400892
$57$	0	0
$58$	10.0000i	1.31306i
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	7.00000	0.896258	0.448129	−	0.893969i	$-0.352090\pi$
$61$	7.00000	0.896258	0.448129	+	0.893969i	$0.352090\pi$
$62$	− 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	$0.0389851\pi$
$67$	2.00000i	0.244339i	−0.992509	+	0.122169i	$0.961015\pi$
$68$	− 2.00000i	− 0.242536i
$69$	−4.00000	−0.481543
$70$	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
$73$	4.00000i	0.468165i	0.972217	+	0.234082i	$0.0752085\pi$
$73$	4.00000i	0.468165i	−0.972217	+	0.234082i	$0.924791\pi$
$74$	−3.00000	−0.348743
$75$	0	0
$76$	0	0
$77$	9.00000i	1.02565i
$78$	− 1.00000i	− 0.113228i
$79$	10.0000	1.12509	0.562544	−	0.826767i	$-0.309823\pi$
$79$	10.0000	1.12509	0.562544	+	0.826767i	$0.309823\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	− 7.00000i	− 0.773021i
$83$	9.00000i	0.987878i	0.869496	+	0.493939i	$0.164443\pi$
$83$	9.00000i	0.987878i	−0.869496	+	0.493939i	$0.835557\pi$
$84$	−3.00000	−0.327327
$85$	0	0
$86$	−1.00000	−0.107833
$87$	− 10.0000i	− 1.07211i
$88$	− 3.00000i	− 0.319801i
$89$	−10.0000	−1.06000	−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000	−1.06000	−0.529999	+	0.847998i	$0.677808\pi$
$90$	0	0
$91$	−3.00000	−0.314485
$92$	− 4.00000i	− 0.417029i
$93$	1.00000i	0.103695i
$94$	7.00000	0.721995
$95$	0	0
$96$	1.00000	0.102062
$97$	2.00000i	0.203069i	0.994832	+	0.101535i	$0.0323753\pi$
$97$	2.00000i	0.203069i	−0.994832	+	0.101535i	$0.967625\pi$
$98$	2.00000i	0.202031i
$99$	3.00000	0.301511
$100$	0	0
$101$	2.00000	0.199007	0.0995037	−	0.995037i	$-0.468274\pi$
$101$	2.00000	0.199007	0.0995037	+	0.995037i	$0.468274\pi$
$102$	2.00000i	0.198030i
$103$	− 1.00000i	− 0.0985329i	−0.998786	−	0.0492665i	$-0.984312\pi$
$103$	− 1.00000i	− 0.0985329i	0.998786	−	0.0492665i	$-0.0156884\pi$
$104$	1.00000	0.0980581
$105$	0	0
$106$	9.00000	0.874157
$107$	− 18.0000i	− 1.74013i	−0.492941	−	0.870063i	$-0.664078\pi$
$107$	− 18.0000i	− 1.74013i	0.492941	−	0.870063i	$-0.335922\pi$
$108$	1.00000i	0.0962250i
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	0	0
$111$	3.00000	0.284747
$112$	− 3.00000i	− 0.283473i
$113$	− 6.00000i	− 0.564433i	−0.959351	−	0.282216i	$-0.908930\pi$
$113$	− 6.00000i	− 0.564433i	0.959351	−	0.282216i	$-0.0910696\pi$
$114$	0	0
$115$	0	0
$116$	10.0000	0.928477
$117$	1.00000i	0.0924500i
$118$	0	0
$119$	6.00000	0.550019
$120$	0	0
$121$	−2.00000	−0.181818
$122$	− 7.00000i	− 0.633750i
$123$	7.00000i	0.631169i
$124$	−1.00000	−0.0898027
$125$	0	0
$126$	3.00000	0.267261
$127$	2.00000i	0.177471i	0.996055	+	0.0887357i	$0.0282826\pi$
$127$	2.00000i	0.177471i	−0.996055	+	0.0887357i	$0.971717\pi$
$128$	1.00000i	0.0883883i
$129$	1.00000	0.0880451
$130$	0	0
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	3.00000i	0.261116i
$133$	0	0
$134$	2.00000	0.172774
$135$	0	0
$136$	−2.00000	−0.171499
$137$	2.00000i	0.170872i	0.996344	+	0.0854358i	$0.0272282\pi$
$137$	2.00000i	0.170872i	−0.996344	+	0.0854358i	$0.972772\pi$
$138$	4.00000i	0.340503i
$139$	5.00000	0.424094	0.212047	−	0.977259i	$-0.431987\pi$
$139$	5.00000	0.424094	0.212047	+	0.977259i	$0.431987\pi$
$140$	0	0
$141$	−7.00000	−0.589506
$142$	− 7.00000i	− 0.587427i
$143$	3.00000i	0.250873i
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	4.00000	0.331042
$147$	− 2.00000i	− 0.164957i
$148$	3.00000i	0.246598i
$149$	20.0000	1.63846	0.819232	−	0.573462i	$-0.194400\pi$
$149$	20.0000	1.63846	0.819232	+	0.573462i	$0.194400\pi$
$150$	0	0
$151$	−18.0000	−1.46482	−0.732410	−	0.680864i	$-0.761604\pi$
$151$	−18.0000	−1.46482	−0.732410	+	0.680864i	$0.761604\pi$
$152$	0	0
$153$	− 2.00000i	− 0.161690i
$154$	9.00000	0.725241
$155$	0	0
$156$	−1.00000	−0.0800641
$157$	22.0000i	1.75579i	0.478852	+	0.877896i	$0.341053\pi$
$157$	22.0000i	1.75579i	−0.478852	+	0.877896i	$0.658947\pi$
$158$	− 10.0000i	− 0.795557i
$159$	−9.00000	−0.713746
$160$	0	0
$161$	12.0000	0.945732
$162$	− 1.00000i	− 0.0785674i
$163$	4.00000i	0.313304i	0.987654	+	0.156652i	$0.0500701\pi$
$163$	4.00000i	0.313304i	−0.987654	+	0.156652i	$0.949930\pi$
$164$	−7.00000	−0.546608
$165$	0	0
$166$	9.00000	0.698535
$167$	2.00000i	0.154765i	0.997001	+	0.0773823i	$0.0246562\pi$
$167$	2.00000i	0.154765i	−0.997001	+	0.0773823i	$0.975344\pi$
$168$	3.00000i	0.231455i
$169$	12.0000	0.923077
$170$	0	0
$171$	0	0
$172$	1.00000i	0.0762493i
$173$	4.00000i	0.304114i	0.988372	+	0.152057i	$0.0485898\pi$
$173$	4.00000i	0.304114i	−0.988372	+	0.152057i	$0.951410\pi$
$174$	−10.0000	−0.758098
$175$	0	0
$176$	−3.00000	−0.226134
$177$	0	0
$178$	10.0000i	0.749532i
$179$	5.00000	0.373718	0.186859	−	0.982387i	$-0.440169\pi$
$179$	5.00000	0.373718	0.186859	+	0.982387i	$0.440169\pi$
$180$	0	0
$181$	−13.0000	−0.966282	−0.483141	−	0.875542i	$-0.660504\pi$
$181$	−13.0000	−0.966282	−0.483141	+	0.875542i	$0.660504\pi$
$182$	3.00000i	0.222375i
$183$	7.00000i	0.517455i
$184$	−4.00000	−0.294884
$185$	0	0
$186$	1.00000	0.0733236
$187$	− 6.00000i	− 0.438763i
$188$	− 7.00000i	− 0.510527i
$189$	−3.00000	−0.218218
$190$	0	0
$191$	−8.00000	−0.578860	−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000	−0.578860	−0.289430	+	0.957199i	$0.593466\pi$
$192$	− 1.00000i	− 0.0721688i
$193$	9.00000i	0.647834i	0.946085	+	0.323917i	$0.105000\pi$
$193$	9.00000i	0.647834i	−0.946085	+	0.323917i	$0.895000\pi$
$194$	2.00000	0.143592
$195$	0	0
$196$	2.00000	0.142857
$197$	27.0000i	1.92367i	0.273629	+	0.961835i	$0.411776\pi$
$197$	27.0000i	1.92367i	−0.273629	+	0.961835i	$0.588224\pi$
$198$	− 3.00000i	− 0.213201i
$199$	20.0000	1.41776	0.708881	−	0.705328i	$-0.249200\pi$
$199$	20.0000	1.41776	0.708881	+	0.705328i	$0.249200\pi$
$200$	0	0
$201$	−2.00000	−0.141069
$202$	− 2.00000i	− 0.140720i
$203$	30.0000i	2.10559i
$204$	2.00000	0.140028
$205$	0	0
$206$	−1.00000	−0.0696733
$207$	− 4.00000i	− 0.278019i
$208$	− 1.00000i	− 0.0693375i
$209$	0	0
$210$	0	0
$211$	2.00000	0.137686	0.0688428	−	0.997628i	$-0.478069\pi$
$211$	2.00000	0.137686	0.0688428	+	0.997628i	$0.478069\pi$
$212$	− 9.00000i	− 0.618123i
$213$	7.00000i	0.479632i
$214$	−18.0000	−1.23045
$215$	0	0
$216$	1.00000	0.0680414
$217$	− 3.00000i	− 0.203653i
$218$	− 10.0000i	− 0.677285i
$219$	−4.00000	−0.270295
$220$	0	0
$221$	2.00000	0.134535
$222$	− 3.00000i	− 0.201347i
$223$	− 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
$226$	−6.00000	−0.399114
$227$	12.0000i	0.796468i	0.917284	+	0.398234i	$0.130377\pi$
$227$	12.0000i	0.796468i	−0.917284	+	0.398234i	$0.869623\pi$
$228$	0	0
$229$	−10.0000	−0.660819	−0.330409	−	0.943838i	$-0.607187\pi$
$229$	−10.0000	−0.660819	−0.330409	+	0.943838i	$0.607187\pi$
$230$	0	0
$231$	−9.00000	−0.592157
$232$	− 10.0000i	− 0.656532i
$233$	9.00000i	0.589610i	0.955557	+	0.294805i	$0.0952546\pi$
$233$	9.00000i	0.589610i	−0.955557	+	0.294805i	$0.904745\pi$
$234$	1.00000	0.0653720
$235$	0	0
$236$	0	0
$237$	10.0000i	0.649570i
$238$	− 6.00000i	− 0.388922i
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	12.0000	0.772988	0.386494	−	0.922292i	$-0.373686\pi$
$241$	12.0000	0.772988	0.386494	+	0.922292i	$0.373686\pi$
$242$	2.00000i	0.128565i
$243$	1.00000i	0.0641500i
$244$	−7.00000	−0.448129
$245$	0	0
$246$	7.00000	0.446304
$247$	0	0
$248$	1.00000i	0.0635001i
$249$	−9.00000	−0.570352
$250$	0	0
$251$	−3.00000	−0.189358	−0.0946792	−	0.995508i	$-0.530183\pi$
$251$	−3.00000	−0.189358	−0.0946792	+	0.995508i	$0.530183\pi$
$252$	− 3.00000i	− 0.188982i
$253$	− 12.0000i	− 0.754434i
$254$	2.00000	0.125491
$255$	0	0
$256$	1.00000	0.0625000
$257$	7.00000i	0.436648i	0.975876	+	0.218324i	$0.0700590\pi$
$257$	7.00000i	0.436648i	−0.975876	+	0.218324i	$0.929941\pi$
$258$	− 1.00000i	− 0.0622573i
$259$	−9.00000	−0.559233
$260$	0	0
$261$	10.0000	0.618984
$262$	− 12.0000i	− 0.741362i
$263$	− 6.00000i	− 0.369976i	−0.982741	−	0.184988i	$-0.940775\pi$
$263$	− 6.00000i	− 0.369976i	0.982741	−	0.184988i	$-0.0592246\pi$
$264$	3.00000	0.184637
$265$	0	0
$266$	0	0
$267$	− 10.0000i	− 0.611990i
$268$	− 2.00000i	− 0.122169i
$269$	10.0000	0.609711	0.304855	−	0.952399i	$-0.401392\pi$
$269$	10.0000	0.609711	0.304855	+	0.952399i	$0.401392\pi$
$270$	0	0
$271$	−8.00000	−0.485965	−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000	−0.485965	−0.242983	+	0.970031i	$0.578126\pi$
$272$	2.00000i	0.121268i
$273$	− 3.00000i	− 0.181568i
$274$	2.00000	0.120824
$275$	0	0
$276$	4.00000	0.240772
$277$	− 18.0000i	− 1.08152i	−0.841178	−	0.540758i	$-0.818138\pi$
$277$	− 18.0000i	− 1.08152i	0.841178	−	0.540758i	$-0.181862\pi$
$278$	− 5.00000i	− 0.299880i
$279$	−1.00000	−0.0598684
$280$	0	0
$281$	7.00000	0.417585	0.208792	−	0.977960i	$-0.433047\pi$
$281$	7.00000	0.417585	0.208792	+	0.977960i	$0.433047\pi$
$282$	7.00000i	0.416844i
$283$	24.0000i	1.42665i	0.700832	+	0.713326i	$0.252812\pi$
$283$	24.0000i	1.42665i	−0.700832	+	0.713326i	$0.747188\pi$
$284$	−7.00000	−0.415374
$285$	0	0
$286$	3.00000	0.177394
$287$	− 21.0000i	− 1.23959i
$288$	1.00000i	0.0589256i
$289$	13.0000	0.764706
$290$	0	0
$291$	−2.00000	−0.117242
$292$	− 4.00000i	− 0.234082i
$293$	24.0000i	1.40209i	0.713115	+	0.701047i	$0.247284\pi$
$293$	24.0000i	1.40209i	−0.713115	+	0.701047i	$0.752716\pi$
$294$	−2.00000	−0.116642
$295$	0	0
$296$	3.00000	0.174371
$297$	3.00000i	0.174078i
$298$	− 20.0000i	− 1.15857i
$299$	4.00000	0.231326
$300$	0	0
$301$	−3.00000	−0.172917
$302$	18.0000i	1.03578i
$303$	2.00000i	0.114897i
$304$	0	0
$305$	0	0
$306$	−2.00000	−0.114332
$307$	22.0000i	1.25561i	0.778372	+	0.627803i	$0.216046\pi$
$307$	22.0000i	1.25561i	−0.778372	+	0.627803i	$0.783954\pi$
$308$	− 9.00000i	− 0.512823i
$309$	1.00000	0.0568880
$310$	0	0
$311$	27.0000	1.53103	0.765515	−	0.643418i	$-0.222484\pi$
$311$	27.0000	1.53103	0.765515	+	0.643418i	$0.222484\pi$
$312$	1.00000i	0.0566139i
$313$	24.0000i	1.35656i	0.734803	+	0.678280i	$0.237274\pi$
$313$	24.0000i	1.35656i	−0.734803	+	0.678280i	$0.762726\pi$
$314$	22.0000	1.24153
$315$	0	0
$316$	−10.0000	−0.562544
$317$	22.0000i	1.23564i	0.786318	+	0.617822i	$0.211985\pi$
$317$	22.0000i	1.23564i	−0.786318	+	0.617822i	$0.788015\pi$
$318$	9.00000i	0.504695i
$319$	30.0000	1.67968
$320$	0	0
$321$	18.0000	1.00466
$322$	− 12.0000i	− 0.668734i
$323$	0	0
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	4.00000	0.221540
$327$	10.0000i	0.553001i
$328$	7.00000i	0.386510i
$329$	21.0000	1.15777
$330$	0	0
$331$	−3.00000	−0.164895	−0.0824475	−	0.996595i	$-0.526274\pi$
$331$	−3.00000	−0.164895	−0.0824475	+	0.996595i	$0.526274\pi$
$332$	− 9.00000i	− 0.493939i
$333$	3.00000i	0.164399i
$334$	2.00000	0.109435
$335$	0	0
$336$	3.00000	0.163663
$337$	− 8.00000i	− 0.435788i	−0.975972	−	0.217894i	$-0.930081\pi$
$337$	− 8.00000i	− 0.435788i	0.975972	−	0.217894i	$-0.0699187\pi$
$338$	− 12.0000i	− 0.652714i
$339$	6.00000	0.325875
$340$	0	0
$341$	−3.00000	−0.162459
$342$	0	0
$343$	− 15.0000i	− 0.809924i
$344$	1.00000	0.0539164
$345$	0	0
$346$	4.00000	0.215041
$347$	7.00000i	0.375780i	0.982190	+	0.187890i	$0.0601648\pi$
$347$	7.00000i	0.375780i	−0.982190	+	0.187890i	$0.939835\pi$
$348$	10.0000i	0.536056i
$349$	0	0		−	1.00000i	$-0.5\pi$
$349$	0	0			1.00000i	$0.5\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	3.00000i	0.159901i
$353$	24.0000i	1.27739i	0.769460	+	0.638696i	$0.220526\pi$
$353$	24.0000i	1.27739i	−0.769460	+	0.638696i	$0.779474\pi$
$354$	0	0
$355$	0	0
$356$	10.0000	0.529999
$357$	6.00000i	0.317554i
$358$	− 5.00000i	− 0.264258i
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	13.0000i	0.683265i
$363$	− 2.00000i	− 0.104973i
$364$	3.00000	0.157243
$365$	0	0
$366$	7.00000	0.365896
$367$	− 28.0000i	− 1.46159i	−0.682598	−	0.730794i	$-0.739150\pi$
$367$	− 28.0000i	− 1.46159i	0.682598	−	0.730794i	$-0.260850\pi$
$368$	4.00000i	0.208514i
$369$	−7.00000	−0.364405
$370$	0	0
$371$	27.0000	1.40177
$372$	− 1.00000i	− 0.0518476i
$373$	24.0000i	1.24267i	0.783544	+	0.621336i	$0.213410\pi$
$373$	24.0000i	1.24267i	−0.783544	+	0.621336i	$0.786590\pi$
$374$	−6.00000	−0.310253
$375$	0	0
$376$	−7.00000	−0.360997
$377$	10.0000i	0.515026i
$378$	3.00000i	0.154303i
$379$	0	0		−	1.00000i	$-0.5\pi$
$379$	0	0			1.00000i	$0.5\pi$
$380$	0	0
$381$	−2.00000	−0.102463
$382$	8.00000i	0.409316i
$383$	24.0000i	1.22634i	0.789950	+	0.613171i	$0.210106\pi$
$383$	24.0000i	1.22634i	−0.789950	+	0.613171i	$0.789894\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	9.00000	0.458088
$387$	1.00000i	0.0508329i
$388$	− 2.00000i	− 0.101535i
$389$	−10.0000	−0.507020	−0.253510	−	0.967333i	$-0.581585\pi$
$389$	−10.0000	−0.507020	−0.253510	+	0.967333i	$0.581585\pi$
$390$	0	0
$391$	−8.00000	−0.404577
$392$	− 2.00000i	− 0.101015i
$393$	12.0000i	0.605320i
$394$	27.0000	1.36024
$395$	0	0
$396$	−3.00000	−0.150756
$397$	− 28.0000i	− 1.40528i	−0.711546	−	0.702640i	$-0.752005\pi$
$397$	− 28.0000i	− 1.40528i	0.711546	−	0.702640i	$-0.247995\pi$
$398$	− 20.0000i	− 1.00251i
$399$	0	0
$400$	0	0
$401$	−8.00000	−0.399501	−0.199750	−	0.979847i	$-0.564013\pi$
$401$	−8.00000	−0.399501	−0.199750	+	0.979847i	$0.564013\pi$
$402$	2.00000i	0.0997509i
$403$	− 1.00000i	− 0.0498135i
$404$	−2.00000	−0.0995037
$405$	0	0
$406$	30.0000	1.48888
$407$	9.00000i	0.446113i
$408$	− 2.00000i	− 0.0990148i
$409$	10.0000	0.494468	0.247234	−	0.968956i	$-0.420478\pi$
$409$	10.0000	0.494468	0.247234	+	0.968956i	$0.420478\pi$
$410$	0	0
$411$	−2.00000	−0.0986527
$412$	1.00000i	0.0492665i
$413$	0	0
$414$	−4.00000	−0.196589
$415$	0	0
$416$	−1.00000	−0.0490290
$417$	5.00000i	0.244851i
$418$	0	0
$419$	30.0000	1.46560	0.732798	−	0.680446i	$-0.238214\pi$
$419$	30.0000	1.46560	0.732798	+	0.680446i	$0.238214\pi$
$420$	0	0
$421$	−28.0000	−1.36464	−0.682318	−	0.731055i	$-0.739028\pi$
$421$	−28.0000	−1.36464	−0.682318	+	0.731055i	$0.739028\pi$
$422$	− 2.00000i	− 0.0973585i
$423$	− 7.00000i	− 0.340352i
$424$	−9.00000	−0.437079
$425$	0	0
$426$	7.00000	0.339151
$427$	− 21.0000i	− 1.01626i
$428$	18.0000i	0.870063i
$429$	−3.00000	−0.144841
$430$	0	0
$431$	7.00000	0.337178	0.168589	−	0.985686i	$-0.446079\pi$
$431$	7.00000	0.337178	0.168589	+	0.985686i	$0.446079\pi$
$432$	− 1.00000i	− 0.0481125i
$433$	4.00000i	0.192228i	0.995370	+	0.0961139i	$0.0306413\pi$
$433$	4.00000i	0.192228i	−0.995370	+	0.0961139i	$0.969359\pi$
$434$	−3.00000	−0.144005
$435$	0	0
$436$	−10.0000	−0.478913
$437$	0	0
$438$	4.00000i	0.191127i
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	0	0
$441$	2.00000	0.0952381
$442$	− 2.00000i	− 0.0951303i
$443$	− 36.0000i	− 1.71041i	−0.518289	−	0.855206i	$-0.673431\pi$
$443$	− 36.0000i	− 1.71041i	0.518289	−	0.855206i	$-0.326569\pi$
$444$	−3.00000	−0.142374
$445$	0	0
$446$	−6.00000	−0.284108
$447$	20.0000i	0.945968i
$448$	3.00000i	0.141737i
$449$	−20.0000	−0.943858	−0.471929	−	0.881636i	$-0.656442\pi$
$449$	−20.0000	−0.943858	−0.471929	+	0.881636i	$0.656442\pi$
$450$	0	0
$451$	−21.0000	−0.988851
$452$	6.00000i	0.282216i
$453$	− 18.0000i	− 0.845714i
$454$	12.0000	0.563188
$455$	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	$-0.0599128\pi$
$458$	10.0000i	0.467269i
$459$	2.00000	0.0933520
$460$	0	0
$461$	−3.00000	−0.139724	−0.0698620	−	0.997557i	$-0.522256\pi$
$461$	−3.00000	−0.139724	−0.0698620	+	0.997557i	$0.522256\pi$
$462$	9.00000i	0.418718i
$463$	14.0000i	0.650635i	0.945605	+	0.325318i	$0.105471\pi$
$463$	14.0000i	0.650635i	−0.945605	+	0.325318i	$0.894529\pi$
$464$	−10.0000	−0.464238
$465$	0	0
$466$	9.00000	0.416917
$467$	− 18.0000i	− 0.832941i	−0.909149	−	0.416470i	$-0.863267\pi$
$467$	− 18.0000i	− 0.832941i	0.909149	−	0.416470i	$-0.136733\pi$
$468$	− 1.00000i	− 0.0462250i
$469$	6.00000	0.277054
$470$	0	0
$471$	−22.0000	−1.01371
$472$	0	0
$473$	3.00000i	0.137940i
$474$	10.0000	0.459315
$475$	0	0
$476$	−6.00000	−0.275010
$477$	− 9.00000i	− 0.412082i
$478$	0	0
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	−3.00000	−0.136788
$482$	− 12.0000i	− 0.546585i
$483$	12.0000i	0.546019i
$484$	2.00000	0.0909091
$485$	0	0
$486$	1.00000	0.0453609
$487$	− 28.0000i	− 1.26880i	−0.773004	−	0.634401i	$-0.781247\pi$
$487$	− 28.0000i	− 1.26880i	0.773004	−	0.634401i	$-0.218753\pi$
$488$	7.00000i	0.316875i
$489$	−4.00000	−0.180886
$490$	0	0
$491$	−28.0000	−1.26362	−0.631811	−	0.775122i	$-0.717688\pi$
$491$	−28.0000	−1.26362	−0.631811	+	0.775122i	$0.717688\pi$
$492$	− 7.00000i	− 0.315584i
$493$	− 20.0000i	− 0.900755i
$494$	0	0
$495$	0	0
$496$	1.00000	0.0449013
$497$	− 21.0000i	− 0.941979i
$498$	9.00000i	0.403300i
$499$	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
$501$	−2.00000	−0.0893534
$502$	3.00000i	0.133897i
$503$	− 11.0000i	− 0.490466i	−0.969464	−	0.245233i	$-0.921136\pi$
$503$	− 11.0000i	− 0.490466i	0.969464	−	0.245233i	$-0.0788644\pi$
$504$	−3.00000	−0.133631
$505$	0	0
$506$	−12.0000	−0.533465
$507$	12.0000i	0.532939i
$508$	− 2.00000i	− 0.0887357i
$509$	15.0000	0.664863	0.332432	−	0.943127i	$-0.392131\pi$
$509$	15.0000	0.664863	0.332432	+	0.943127i	$0.392131\pi$
$510$	0	0
$511$	12.0000	0.530849
$512$	− 1.00000i	− 0.0441942i
$513$	0	0
$514$	7.00000	0.308757
$515$	0	0
$516$	−1.00000	−0.0440225
$517$	− 21.0000i	− 0.923579i
$518$	9.00000i	0.395437i
$519$	−4.00000	−0.175581
$520$	0	0
$521$	−33.0000	−1.44576	−0.722878	−	0.690976i	$-0.757181\pi$
$521$	−33.0000	−1.44576	−0.722878	+	0.690976i	$0.757181\pi$
$522$	− 10.0000i	− 0.437688i
$523$	4.00000i	0.174908i	0.996169	+	0.0874539i	$0.0278730\pi$
$523$	4.00000i	0.174908i	−0.996169	+	0.0874539i	$0.972127\pi$
$524$	−12.0000	−0.524222
$525$	0	0
$526$	−6.00000	−0.261612
$527$	2.00000i	0.0871214i
$528$	− 3.00000i	− 0.130558i
$529$	7.00000	0.304348
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 7.00000i	− 0.303204i
$534$	−10.0000	−0.432742
$535$	0	0
$536$	−2.00000	−0.0863868
$537$	5.00000i	0.215766i
$538$	− 10.0000i	− 0.431131i
$539$	6.00000	0.258438
$540$	0	0
$541$	−8.00000	−0.343947	−0.171973	−	0.985102i	$-0.555014\pi$
$541$	−8.00000	−0.343947	−0.171973	+	0.985102i	$0.555014\pi$
$542$	8.00000i	0.343629i
$543$	− 13.0000i	− 0.557883i
$544$	2.00000	0.0857493
$545$	0	0
$546$	−3.00000	−0.128388
$547$	22.0000i	0.940652i	0.882493	+	0.470326i	$0.155864\pi$
$547$	22.0000i	0.940652i	−0.882493	+	0.470326i	$0.844136\pi$
$548$	− 2.00000i	− 0.0854358i
$549$	−7.00000	−0.298753
$550$	0	0
$551$	0	0
$552$	− 4.00000i	− 0.170251i
$553$	− 30.0000i	− 1.27573i
$554$	−18.0000	−0.764747
$555$	0	0
$556$	−5.00000	−0.212047
$557$	2.00000i	0.0847427i	0.999102	+	0.0423714i	$0.0134913\pi$
$557$	2.00000i	0.0847427i	−0.999102	+	0.0423714i	$0.986509\pi$
$558$	1.00000i	0.0423334i
$559$	−1.00000	−0.0422955
$560$	0	0
$561$	6.00000	0.253320
$562$	− 7.00000i	− 0.295277i
$563$	34.0000i	1.43293i	0.697623	+	0.716465i	$0.254241\pi$
$563$	34.0000i	1.43293i	−0.697623	+	0.716465i	$0.745759\pi$
$564$	7.00000	0.294753
$565$	0	0
$566$	24.0000	1.00880
$567$	− 3.00000i	− 0.125988i
$568$	7.00000i	0.293713i
$569$	−10.0000	−0.419222	−0.209611	−	0.977785i	$-0.567220\pi$
$569$	−10.0000	−0.419222	−0.209611	+	0.977785i	$0.567220\pi$
$570$	0	0
$571$	−23.0000	−0.962520	−0.481260	−	0.876578i	$-0.659821\pi$
$571$	−23.0000	−0.962520	−0.481260	+	0.876578i	$0.659821\pi$
$572$	− 3.00000i	− 0.125436i
$573$	− 8.00000i	− 0.334205i
$574$	−21.0000	−0.876523
$575$	0	0
$576$	1.00000	0.0416667
$577$	22.0000i	0.915872i	0.888985	+	0.457936i	$0.151411\pi$
$577$	22.0000i	0.915872i	−0.888985	+	0.457936i	$0.848589\pi$
$578$	− 13.0000i	− 0.540729i
$579$	−9.00000	−0.374027
$580$	0	0
$581$	27.0000	1.12015
$582$	2.00000i	0.0829027i
$583$	− 27.0000i	− 1.11823i
$584$	−4.00000	−0.165521
$585$	0	0
$586$	24.0000	0.991431
$587$	17.0000i	0.701665i	0.936438	+	0.350833i	$0.114101\pi$
$587$	17.0000i	0.701665i	−0.936438	+	0.350833i	$0.885899\pi$
$588$	2.00000i	0.0824786i
$589$	0	0
$590$	0	0
$591$	−27.0000	−1.11063
$592$	− 3.00000i	− 0.123299i
$593$	− 1.00000i	− 0.0410651i	−0.999789	−	0.0205325i	$-0.993464\pi$
$593$	− 1.00000i	− 0.0410651i	0.999789	−	0.0205325i	$-0.00653617\pi$
$594$	3.00000	0.123091
$595$	0	0
$596$	−20.0000	−0.819232
$597$	20.0000i	0.818546i
$598$	− 4.00000i	− 0.163572i
$599$	15.0000	0.612883	0.306442	−	0.951889i	$-0.400862\pi$
$599$	15.0000	0.612883	0.306442	+	0.951889i	$0.400862\pi$
$600$	0	0
$601$	12.0000	0.489490	0.244745	−	0.969587i	$-0.421296\pi$
$601$	12.0000	0.489490	0.244745	+	0.969587i	$0.421296\pi$
$602$	3.00000i	0.122271i
$603$	− 2.00000i	− 0.0814463i
$604$	18.0000	0.732410
$605$	0	0
$606$	2.00000	0.0812444
$607$	− 43.0000i	− 1.74532i	−0.488332	−	0.872658i	$-0.662394\pi$
$607$	− 43.0000i	− 1.74532i	0.488332	−	0.872658i	$-0.337606\pi$
$608$	0	0
$609$	−30.0000	−1.21566
$610$	0	0
$611$	7.00000	0.283190
$612$	2.00000i	0.0808452i
$613$	− 26.0000i	− 1.05013i	−0.851062	−	0.525065i	$-0.824041\pi$
$613$	− 26.0000i	− 1.05013i	0.851062	−	0.525065i	$-0.175959\pi$
$614$	22.0000	0.887848
$615$	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	$-0.0192337\pi$
$618$	− 1.00000i	− 0.0402259i
$619$	5.00000	0.200967	0.100483	−	0.994939i	$-0.467961\pi$
$619$	5.00000	0.200967	0.100483	+	0.994939i	$0.467961\pi$
$620$	0	0
$621$	4.00000	0.160514
$622$	− 27.0000i	− 1.08260i
$623$	30.0000i	1.20192i
$624$	1.00000	0.0400320
$625$	0	0
$626$	24.0000	0.959233
$627$	0	0
$628$	− 22.0000i	− 0.877896i
$629$	6.00000	0.239236
$630$	0	0
$631$	2.00000	0.0796187	0.0398094	−	0.999207i	$-0.487325\pi$
$631$	2.00000	0.0796187	0.0398094	+	0.999207i	$0.487325\pi$
$632$	10.0000i	0.397779i
$633$	2.00000i	0.0794929i
$634$	22.0000	0.873732
$635$	0	0
$636$	9.00000	0.356873
$637$	2.00000i	0.0792429i
$638$	− 30.0000i	− 1.18771i
$639$	−7.00000	−0.276916
$640$	0	0
$641$	2.00000	0.0789953	0.0394976	−	0.999220i	$-0.487424\pi$
$641$	2.00000	0.0789953	0.0394976	+	0.999220i	$0.487424\pi$
$642$	− 18.0000i	− 0.710403i
$643$	− 31.0000i	− 1.22252i	−0.791430	−	0.611260i	$-0.790663\pi$
$643$	− 31.0000i	− 1.22252i	0.791430	−	0.611260i	$-0.209337\pi$
$644$	−12.0000	−0.472866
$645$	0	0
$646$	0	0
$647$	42.0000i	1.65119i	0.564263	+	0.825595i	$0.309160\pi$
$647$	42.0000i	1.65119i	−0.564263	+	0.825595i	$0.690840\pi$
$648$	1.00000i	0.0392837i
$649$	0	0
$650$	0	0
$651$	3.00000	0.117579
$652$	− 4.00000i	− 0.156652i
$653$	− 16.0000i	− 0.626128i	−0.949732	−	0.313064i	$-0.898644\pi$
$653$	− 16.0000i	− 0.626128i	0.949732	−	0.313064i	$-0.101356\pi$
$654$	10.0000	0.391031
$655$	0	0
$656$	7.00000	0.273304
$657$	− 4.00000i	− 0.156055i
$658$	− 21.0000i	− 0.818665i
$659$	40.0000	1.55818	0.779089	−	0.626913i	$-0.215682\pi$
$659$	40.0000	1.55818	0.779089	+	0.626913i	$0.215682\pi$
$660$	0	0
$661$	−38.0000	−1.47803	−0.739014	−	0.673690i	$-0.764708\pi$
$661$	−38.0000	−1.47803	−0.739014	+	0.673690i	$0.764708\pi$
$662$	3.00000i	0.116598i
$663$	2.00000i	0.0776736i
$664$	−9.00000	−0.349268
$665$	0	0
$666$	3.00000	0.116248
$667$	− 40.0000i	− 1.54881i
$668$	− 2.00000i	− 0.0773823i
$669$	6.00000	0.231973
$670$	0	0
$671$	−21.0000	−0.810696
$672$	− 3.00000i	− 0.115728i
$673$	− 26.0000i	− 1.00223i	−0.865382	−	0.501113i	$-0.832924\pi$
$673$	− 26.0000i	− 1.00223i	0.865382	−	0.501113i	$-0.167076\pi$
$674$	−8.00000	−0.308148
$675$	0	0
$676$	−12.0000	−0.461538
$677$	27.0000i	1.03769i	0.854867	+	0.518847i	$0.173639\pi$
$677$	27.0000i	1.03769i	−0.854867	+	0.518847i	$0.826361\pi$
$678$	− 6.00000i	− 0.230429i
$679$	6.00000	0.230259
$680$	0	0
$681$	−12.0000	−0.459841
$682$	3.00000i	0.114876i
$683$	− 26.0000i	− 0.994862i	−0.867503	−	0.497431i	$-0.834277\pi$
$683$	− 26.0000i	− 0.994862i	0.867503	−	0.497431i	$-0.165723\pi$
$684$	0	0
$685$	0	0
$686$	−15.0000	−0.572703
$687$	− 10.0000i	− 0.381524i
$688$	− 1.00000i	− 0.0381246i
$689$	9.00000	0.342873
$690$	0	0
$691$	−28.0000	−1.06517	−0.532585	−	0.846376i	$-0.678779\pi$
$691$	−28.0000	−1.06517	−0.532585	+	0.846376i	$0.678779\pi$
$692$	− 4.00000i	− 0.152057i
$693$	− 9.00000i	− 0.341882i
$694$	7.00000	0.265716
$695$	0	0
$696$	10.0000	0.379049
$697$	14.0000i	0.530288i
$698$	0	0
$699$	−9.00000	−0.340411
$700$	0	0
$701$	−48.0000	−1.81293	−0.906467	−	0.422276i	$-0.861231\pi$
$701$	−48.0000	−1.81293	−0.906467	+	0.422276i	$0.861231\pi$
$702$	1.00000i	0.0377426i
$703$	0	0
$704$	3.00000	0.113067
$705$	0	0
$706$	24.0000	0.903252
$707$	− 6.00000i	− 0.225653i
$708$	0	0
$709$	−50.0000	−1.87779	−0.938895	−	0.344204i	$-0.888149\pi$
$709$	−50.0000	−1.87779	−0.938895	+	0.344204i	$0.888149\pi$
$710$	0	0
$711$	−10.0000	−0.375029
$712$	− 10.0000i	− 0.374766i
$713$	4.00000i	0.149801i
$714$	6.00000	0.224544
$715$	0	0
$716$	−5.00000	−0.186859
$717$	0	0
$718$	0	0
$719$	−50.0000	−1.86469	−0.932343	−	0.361576i	$-0.882239\pi$
$719$	−50.0000	−1.86469	−0.932343	+	0.361576i	$0.882239\pi$
$720$	0	0
$721$	−3.00000	−0.111726
$722$	19.0000i	0.707107i
$723$	12.0000i	0.446285i
$724$	13.0000	0.483141
$725$	0	0
$726$	−2.00000	−0.0742270
$727$	27.0000i	1.00137i	0.865628	+	0.500687i	$0.166919\pi$
$727$	27.0000i	1.00137i	−0.865628	+	0.500687i	$0.833081\pi$
$728$	− 3.00000i	− 0.111187i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	2.00000	0.0739727
$732$	− 7.00000i	− 0.258727i
$733$	− 36.0000i	− 1.32969i	−0.746981	−	0.664845i	$-0.768498\pi$
$733$	− 36.0000i	− 1.32969i	0.746981	−	0.664845i	$-0.231502\pi$
$734$	−28.0000	−1.03350
$735$	0	0
$736$	4.00000	0.147442
$737$	− 6.00000i	− 0.221013i
$738$	7.00000i	0.257674i
$739$	20.0000	0.735712	0.367856	−	0.929883i	$-0.380092\pi$
$739$	20.0000	0.735712	0.367856	+	0.929883i	$0.380092\pi$
$740$	0	0
$741$	0	0
$742$	− 27.0000i	− 0.991201i
$743$	4.00000i	0.146746i	0.997305	+	0.0733729i	$0.0233763\pi$
$743$	4.00000i	0.146746i	−0.997305	+	0.0733729i	$0.976624\pi$
$744$	−1.00000	−0.0366618
$745$	0	0
$746$	24.0000	0.878702
$747$	− 9.00000i	− 0.329293i
$748$	6.00000i	0.219382i
$749$	−54.0000	−1.97312
$750$	0	0
$751$	7.00000	0.255434	0.127717	−	0.991811i	$-0.459235\pi$
$751$	7.00000	0.255434	0.127717	+	0.991811i	$0.459235\pi$
$752$	7.00000i	0.255264i
$753$	− 3.00000i	− 0.109326i
$754$	10.0000	0.364179
$755$	0	0
$756$	3.00000	0.109109
$757$	− 13.0000i	− 0.472493i	−0.971693	−	0.236247i	$-0.924083\pi$
$757$	− 13.0000i	− 0.472493i	0.971693	−	0.236247i	$-0.0759173\pi$
$758$	0	0
$759$	12.0000	0.435572
$760$	0	0
$761$	12.0000	0.435000	0.217500	−	0.976060i	$-0.430210\pi$
$761$	12.0000	0.435000	0.217500	+	0.976060i	$0.430210\pi$
$762$	2.00000i	0.0724524i
$763$	− 30.0000i	− 1.08607i
$764$	8.00000	0.289430
$765$	0	0
$766$	24.0000	0.867155
$767$	0	0
$768$	1.00000i	0.0360844i
$769$	45.0000	1.62274	0.811371	−	0.584532i	$-0.198722\pi$
$769$	45.0000	1.62274	0.811371	+	0.584532i	$0.198722\pi$
$770$	0	0
$771$	−7.00000	−0.252099
$772$	− 9.00000i	− 0.323917i
$773$	34.0000i	1.22290i	0.791285	+	0.611448i	$0.209412\pi$
$773$	34.0000i	1.22290i	−0.791285	+	0.611448i	$0.790588\pi$
$774$	1.00000	0.0359443
$775$	0	0
$776$	−2.00000	−0.0717958
$777$	− 9.00000i	− 0.322873i
$778$	10.0000i	0.358517i
$779$	0	0
$780$	0	0
$781$	−21.0000	−0.751439
$782$	8.00000i	0.286079i
$783$	10.0000i	0.357371i
$784$	−2.00000	−0.0714286
$785$	0	0
$786$	12.0000	0.428026
$787$	7.00000i	0.249523i	0.992187	+	0.124762i	$0.0398166\pi$
$787$	7.00000i	0.249523i	−0.992187	+	0.124762i	$0.960183\pi$
$788$	− 27.0000i	− 0.961835i
$789$	6.00000	0.213606
$790$	0	0
$791$	−18.0000	−0.640006
$792$	3.00000i	0.106600i
$793$	− 7.00000i	− 0.248577i
$794$	−28.0000	−0.993683
$795$	0	0
$796$	−20.0000	−0.708881
$797$	2.00000i	0.0708436i	0.999372	+	0.0354218i	$0.0112775\pi$
$797$	2.00000i	0.0708436i	−0.999372	+	0.0354218i	$0.988723\pi$
$798$	0	0
$799$	−14.0000	−0.495284
$800$	0	0
$801$	10.0000	0.353333
$802$	8.00000i	0.282490i
$803$	− 12.0000i	− 0.423471i
$804$	2.00000	0.0705346
$805$	0	0
$806$	−1.00000	−0.0352235
$807$	10.0000i	0.352017i
$808$	2.00000i	0.0703598i
$809$	10.0000	0.351581	0.175791	−	0.984428i	$-0.443752\pi$
$809$	10.0000	0.351581	0.175791	+	0.984428i	$0.443752\pi$
$810$	0	0
$811$	12.0000	0.421377	0.210688	−	0.977553i	$-0.432429\pi$
$811$	12.0000	0.421377	0.210688	+	0.977553i	$0.432429\pi$
$812$	− 30.0000i	− 1.05279i
$813$	− 8.00000i	− 0.280572i
$814$	9.00000	0.315450
$815$	0	0
$816$	−2.00000	−0.0700140
$817$	0	0
$818$	− 10.0000i	− 0.349642i
$819$	3.00000	0.104828
$820$	0	0
$821$	−53.0000	−1.84971	−0.924856	−	0.380317i	$-0.875815\pi$
$821$	−53.0000	−1.84971	−0.924856	+	0.380317i	$0.875815\pi$
$822$	2.00000i	0.0697580i
$823$	24.0000i	0.836587i	0.908312	+	0.418294i	$0.137372\pi$
$823$	24.0000i	0.836587i	−0.908312	+	0.418294i	$0.862628\pi$
$824$	1.00000	0.0348367
$825$	0	0
$826$	0	0
$827$	− 28.0000i	− 0.973655i	−0.873498	−	0.486828i	$-0.838154\pi$
$827$	− 28.0000i	− 0.973655i	0.873498	−	0.486828i	$-0.161846\pi$
$828$	4.00000i	0.139010i
$829$	−55.0000	−1.91023	−0.955114	−	0.296237i	$-0.904268\pi$
$829$	−55.0000	−1.91023	−0.955114	+	0.296237i	$0.904268\pi$
$830$	0	0
$831$	18.0000	0.624413
$832$	1.00000i	0.0346688i
$833$	− 4.00000i	− 0.138592i
$834$	5.00000	0.173136
$835$	0	0
$836$	0	0
$837$	− 1.00000i	− 0.0345651i
$838$	− 30.0000i	− 1.03633i
$839$	−15.0000	−0.517858	−0.258929	−	0.965896i	$-0.583369\pi$
$839$	−15.0000	−0.517858	−0.258929	+	0.965896i	$0.583369\pi$
$840$	0	0
$841$	71.0000	2.44828
$842$	28.0000i	0.964944i
$843$	7.00000i	0.241093i
$844$	−2.00000	−0.0688428
$845$	0	0
$846$	−7.00000	−0.240665
$847$	6.00000i	0.206162i
$848$	9.00000i	0.309061i
$849$	−24.0000	−0.823678
$850$	0	0
$851$	12.0000	0.411355
$852$	− 7.00000i	− 0.239816i
$853$	− 16.0000i	− 0.547830i	−0.961754	−	0.273915i	$-0.911681\pi$
$853$	− 16.0000i	− 0.547830i	0.961754	−	0.273915i	$-0.0883186\pi$
$854$	−21.0000	−0.718605
$855$	0	0
$856$	18.0000	0.615227
$857$	− 3.00000i	− 0.102478i	−0.998686	−	0.0512390i	$-0.983683\pi$
$857$	− 3.00000i	− 0.102478i	0.998686	−	0.0512390i	$-0.0163170\pi$
$858$	3.00000i	0.102418i
$859$	25.0000	0.852989	0.426494	−	0.904490i	$-0.359748\pi$
$859$	25.0000	0.852989	0.426494	+	0.904490i	$0.359748\pi$
$860$	0	0
$861$	21.0000	0.715678
$862$	− 7.00000i	− 0.238421i
$863$	− 46.0000i	− 1.56586i	−0.622111	−	0.782929i	$-0.713725\pi$
$863$	− 46.0000i	− 1.56586i	0.622111	−	0.782929i	$-0.286275\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	4.00000	0.135926
$867$	13.0000i	0.441503i
$868$	3.00000i	0.101827i
$869$	−30.0000	−1.01768
$870$	0	0
$871$	2.00000	0.0677674
$872$	10.0000i	0.338643i
$873$	− 2.00000i	− 0.0676897i
$874$	0	0
$875$	0	0
$876$	4.00000	0.135147
$877$	− 8.00000i	− 0.270141i	−0.990836	−	0.135070i	$-0.956874\pi$
$877$	− 8.00000i	− 0.270141i	0.990836	−	0.135070i	$-0.0431261\pi$
$878$	0	0
$879$	−24.0000	−0.809500
$880$	0	0
$881$	32.0000	1.07811	0.539054	−	0.842271i	$-0.318782\pi$
$881$	32.0000	1.07811	0.539054	+	0.842271i	$0.318782\pi$
$882$	− 2.00000i	− 0.0673435i
$883$	− 11.0000i	− 0.370179i	−0.982722	−	0.185090i	$-0.940742\pi$
$883$	− 11.0000i	− 0.370179i	0.982722	−	0.185090i	$-0.0592576\pi$
$884$	−2.00000	−0.0672673
$885$	0	0
$886$	−36.0000	−1.20944
$887$	− 13.0000i	− 0.436497i	−0.975893	−	0.218249i	$-0.929966\pi$
$887$	− 13.0000i	− 0.436497i	0.975893	−	0.218249i	$-0.0700344\pi$
$888$	3.00000i	0.100673i
$889$	6.00000	0.201234
$890$	0	0
$891$	−3.00000	−0.100504
$892$	6.00000i	0.200895i
$893$	0	0
$894$	20.0000	0.668900
$895$	0	0
$896$	3.00000	0.100223
$897$	4.00000i	0.133556i
$898$	20.0000i	0.667409i
$899$	−10.0000	−0.333519
$900$	0	0
$901$	−18.0000	−0.599667
$902$	21.0000i	0.699224i
$903$	− 3.00000i	− 0.0998337i
$904$	6.00000	0.199557
$905$	0	0
$906$	−18.0000	−0.598010
$907$	32.0000i	1.06254i	0.847202	+	0.531271i	$0.178286\pi$
$907$	32.0000i	1.06254i	−0.847202	+	0.531271i	$0.821714\pi$
$908$	− 12.0000i	− 0.398234i
$909$	−2.00000	−0.0663358
$910$	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
$913$	− 27.0000i	− 0.893570i
$914$	−8.00000	−0.264616
$915$	0	0
$916$	10.0000	0.330409
$917$	− 36.0000i	− 1.18882i
$918$	− 2.00000i	− 0.0660098i
$919$	−25.0000	−0.824674	−0.412337	−	0.911031i	$-0.635287\pi$
$919$	−25.0000	−0.824674	−0.412337	+	0.911031i	$0.635287\pi$
$920$	0	0
$921$	−22.0000	−0.724925
$922$	3.00000i	0.0987997i
$923$	− 7.00000i	− 0.230408i
$924$	9.00000	0.296078
$925$	0	0
$926$	14.0000	0.460069
$927$	1.00000i	0.0328443i
$928$	10.0000i	0.328266i
$929$	−40.0000	−1.31236	−0.656179	−	0.754606i	$-0.727828\pi$
$929$	−40.0000	−1.31236	−0.656179	+	0.754606i	$0.727828\pi$
$930$	0	0
$931$	0	0
$932$	− 9.00000i	− 0.294805i
$933$	27.0000i	0.883940i
$934$	−18.0000	−0.588978
$935$	0	0
$936$	−1.00000	−0.0326860
$937$	2.00000i	0.0653372i	0.999466	+	0.0326686i	$0.0104006\pi$
$937$	2.00000i	0.0653372i	−0.999466	+	0.0326686i	$0.989599\pi$
$938$	− 6.00000i	− 0.195907i
$939$	−24.0000	−0.783210
$940$	0	0
$941$	−38.0000	−1.23876	−0.619382	−	0.785090i	$-0.712617\pi$
$941$	−38.0000	−1.23876	−0.619382	+	0.785090i	$0.712617\pi$
$942$	22.0000i	0.716799i
$943$	28.0000i	0.911805i
$944$	0	0
$945$	0	0
$946$	3.00000	0.0975384
$947$	− 3.00000i	− 0.0974869i	−0.998811	−	0.0487435i	$-0.984478\pi$
$947$	− 3.00000i	− 0.0974869i	0.998811	−	0.0487435i	$-0.0155217\pi$
$948$	− 10.0000i	− 0.324785i
$949$	4.00000	0.129845
$950$	0	0
$951$	−22.0000	−0.713399
$952$	6.00000i	0.194461i
$953$	34.0000i	1.10137i	0.834714	+	0.550684i	$0.185633\pi$
$953$	34.0000i	1.10137i	−0.834714	+	0.550684i	$0.814367\pi$
$954$	−9.00000	−0.291386
$955$	0	0
$956$	0	0
$957$	30.0000i	0.969762i
$958$	0	0
$959$	6.00000	0.193750
$960$	0	0
$961$	1.00000	0.0322581
$962$	3.00000i	0.0967239i
$963$	18.0000i	0.580042i
$964$	−12.0000	−0.386494
$965$	0	0
$966$	12.0000	0.386094
$967$	− 18.0000i	− 0.578841i	−0.957202	−	0.289420i	$-0.906537\pi$
$967$	− 18.0000i	− 0.578841i	0.957202	−	0.289420i	$-0.0934626\pi$
$968$	− 2.00000i	− 0.0642824i
$969$	0	0
$970$	0	0
$971$	12.0000	0.385098	0.192549	−	0.981287i	$-0.438325\pi$
$971$	12.0000	0.385098	0.192549	+	0.981287i	$0.438325\pi$
$972$	− 1.00000i	− 0.0320750i
$973$	− 15.0000i	− 0.480878i
$974$	−28.0000	−0.897178
$975$	0	0
$976$	7.00000	0.224065
$977$	− 13.0000i	− 0.415907i	−0.978139	−	0.207953i	$-0.933320\pi$
$977$	− 13.0000i	− 0.415907i	0.978139	−	0.207953i	$-0.0666802\pi$
$978$	4.00000i	0.127906i
$979$	30.0000	0.958804
$980$	0	0
$981$	−10.0000	−0.319275
$982$	28.0000i	0.893516i
$983$	− 16.0000i	− 0.510321i	−0.966899	−	0.255160i	$-0.917872\pi$
$983$	− 16.0000i	− 0.510321i	0.966899	−	0.255160i	$-0.0821283\pi$
$984$	−7.00000	−0.223152
$985$	0	0
$986$	−20.0000	−0.636930
$987$	21.0000i	0.668437i
$988$	0	0
$989$	4.00000	0.127193
$990$	0	0
$991$	2.00000	0.0635321	0.0317660	−	0.999495i	$-0.489887\pi$
$991$	2.00000	0.0635321	0.0317660	+	0.999495i	$0.489887\pi$
$992$	− 1.00000i	− 0.0317500i
$993$	− 3.00000i	− 0.0952021i
$994$	−21.0000	−0.666080
$995$	0	0
$996$	9.00000	0.285176
$997$	42.0000i	1.33015i	0.746775	+	0.665077i	$0.231601\pi$
$997$	42.0000i	1.33015i	−0.746775	+	0.665077i	$0.768399\pi$
$998$	− 20.0000i	− 0.633089i
$999$	−3.00000	−0.0949158

Twists

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

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4650 = 2 \cdot 3 \cdot 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4650.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -3.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -3.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} -3.00000 q^{11} -1.00000i q^{12} -1.00000i q^{13} -3.00000 q^{14} +1.00000 q^{16} +2.00000i q^{17} +1.00000i q^{18} +3.00000 q^{21} +3.00000i q^{22} +4.00000i q^{23} -1.00000 q^{24} -1.00000 q^{26} -1.00000i q^{27} +3.00000i q^{28} -10.0000 q^{29} +1.00000 q^{31} -1.00000i q^{32} -3.00000i q^{33} +2.00000 q^{34} +1.00000 q^{36} -3.00000i q^{37} +1.00000 q^{39} +7.00000 q^{41} -3.00000i q^{42} -1.00000i q^{43} +3.00000 q^{44} +4.00000 q^{46} +7.00000i q^{47} +1.00000i q^{48} -2.00000 q^{49} -2.00000 q^{51} +1.00000i q^{52} +9.00000i q^{53} -1.00000 q^{54} +3.00000 q^{56} +10.0000i q^{58} +7.00000 q^{61} -1.00000i q^{62} +3.00000i q^{63} -1.00000 q^{64} -3.00000 q^{66} +2.00000i q^{67} -2.00000i q^{68} -4.00000 q^{69} +7.00000 q^{71} -1.00000i q^{72} +4.00000i q^{73} -3.00000 q^{74} +9.00000i q^{77} -1.00000i q^{78} +10.0000 q^{79} +1.00000 q^{81} -7.00000i q^{82} +9.00000i q^{83} -3.00000 q^{84} -1.00000 q^{86} -10.0000i q^{87} -3.00000i q^{88} -10.0000 q^{89} -3.00000 q^{91} -4.00000i q^{92} +1.00000i q^{93} +7.00000 q^{94} +1.00000 q^{96} +2.00000i q^{97} +2.00000i q^{98} +3.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9	\( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} - 6 q^{11} - 6 q^{14} + 2 q^{16} + 6 q^{21} - 2 q^{24} - 2 q^{26} - 20 q^{29} + 2 q^{31} + 4 q^{34} + 2 q^{36} + 2 q^{39} + 14 q^{41} + 6 q^{44} + 8 q^{46} - 4 q^{49} - 4 q^{51} - 2 q^{54} + 6 q^{56} + 14 q^{61} - 2 q^{64} - 6 q^{66} - 8 q^{69} + 14 q^{71} - 6 q^{74} + 20 q^{79} + 2 q^{81} - 6 q^{84} - 2 q^{86} - 20 q^{89} - 6 q^{91} + 14 q^{94} + 2 q^{96} + 6 q^{99}+O(q^{100}) \) 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9 - 6 * q^11 - 6 * q^14 + 2 * q^16 + 6 * q^21 - 2 * q^24 - 2 * q^26 - 20 * q^29 + 2 * q^31 + 4 * q^34 + 2 * q^36 + 2 * q^39 + 14 * q^41 + 6 * q^44 + 8 * q^46 - 4 * q^49 - 4 * q^51 - 2 * q^54 + 6 * q^56 + 14 * q^61 - 2 * q^64 - 6 * q^66 - 8 * q^69 + 14 * q^71 - 6 * q^74 + 20 * q^79 + 2 * q^81 - 6 * q^84 - 2 * q^86 - 20 * q^89 - 6 * q^91 + 14 * q^94 + 2 * q^96 + 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more