LMFDB - Embedded newform 2550.2.d.c.2449.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2550,2,Mod(2449,2550)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2550, 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("2550.2449");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2550 = 2 \cdot 3 \cdot 5^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2550.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:	$20.3618525154$
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			2449.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	2550.2449
Dual form			2550.2.d.c.2449.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +1.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +1.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} -3.00000 q^{11} -1.00000i q^{12} +2.00000i q^{13} -1.00000 q^{14} +1.00000 q^{16} +1.00000i q^{17} -1.00000i q^{18} +7.00000 q^{19} -1.00000 q^{21} -3.00000i q^{22} +6.00000i q^{23} +1.00000 q^{24} -2.00000 q^{26} -1.00000i q^{27} -1.00000i q^{28} -6.00000 q^{29} -7.00000 q^{31} +1.00000i q^{32} -3.00000i q^{33} -1.00000 q^{34} +1.00000 q^{36} +7.00000i q^{37} +7.00000i q^{38} -2.00000 q^{39} -6.00000 q^{41} -1.00000i q^{42} -1.00000i q^{43} +3.00000 q^{44} -6.00000 q^{46} -9.00000i q^{47} +1.00000i q^{48} +6.00000 q^{49} -1.00000 q^{51} -2.00000i q^{52} -3.00000i q^{53} +1.00000 q^{54} +1.00000 q^{56} +7.00000i q^{57} -6.00000i q^{58} -10.0000 q^{61} -7.00000i q^{62} -1.00000i q^{63} -1.00000 q^{64} +3.00000 q^{66} -5.00000i q^{67} -1.00000i q^{68} -6.00000 q^{69} -6.00000 q^{71} +1.00000i q^{72} -16.0000i q^{73} -7.00000 q^{74} -7.00000 q^{76} -3.00000i q^{77} -2.00000i q^{78} -17.0000 q^{79} +1.00000 q^{81} -6.00000i q^{82} -6.00000i q^{83} +1.00000 q^{84} +1.00000 q^{86} -6.00000i q^{87} +3.00000i q^{88} +12.0000 q^{89} -2.00000 q^{91} -6.00000i q^{92} -7.00000i q^{93} +9.00000 q^{94} -1.00000 q^{96} +10.0000i q^{97} +6.00000i q^{98} +3.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} - 2 q^{14} + 2 q^{16} + 14 q^{19} - 2 q^{21} + 2 q^{24} - 4 q^{26} - 12 q^{29} - 14 q^{31} - 2 q^{34} + 2 q^{36} - 4 q^{39} - 12 q^{41} + 6 q^{44} - 12 q^{46} + 12 q^{49} - 2 q^{51} + 2 q^{54} + 2 q^{56} - 20 q^{61} - 2 q^{64} + 6 q^{66} - 12 q^{69} - 12 q^{71} - 14 q^{74} - 14 q^{76} - 34 q^{79} + 2 q^{81} + 2 q^{84} + 2 q^{86} + 24 q^{89} - 4 q^{91} + 18 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 - 2 * q^14 + 2 * q^16 + 14 * q^19 - 2 * q^21 + 2 * q^24 - 4 * q^26 - 12 * q^29 - 14 * q^31 - 2 * q^34 + 2 * q^36 - 4 * q^39 - 12 * q^41 + 6 * q^44 - 12 * q^46 + 12 * q^49 - 2 * q^51 + 2 * q^54 + 2 * q^56 - 20 * q^61 - 2 * q^64 + 6 * q^66 - 12 * q^69 - 12 * q^71 - 14 * q^74 - 14 * q^76 - 34 * q^79 + 2 * q^81 + 2 * q^84 + 2 * q^86 + 24 * q^89 - 4 * q^91 + 18 * q^94 - 2 * q^96 + 6 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2550\mathbb{Z}\right)^\times$.

$n$	$751$	$851$	$1327$
$\chi(n)$	$1$	$1$	$-1$

Coefficient data

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

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

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

$2$	1.00000i	0.707107i
$2$	1.00000i	0.707107i
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	−1.00000	−0.408248
$7$	1.00000i	0.377964i	0.981981	+	0.188982i	$0.0605189\pi$
$7$	1.00000i	0.377964i	−0.981981	+	0.188982i	$0.939481\pi$
$8$	− 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
$13$	2.00000i	0.554700i	0.960769	+	0.277350i	$0.0894562\pi$
$13$	2.00000i	0.554700i	−0.960769	+	0.277350i	$0.910544\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	1.00000i	0.242536i
$17$	1.00000i	0.242536i
$18$	− 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
$21$	−1.00000	−0.218218
$22$	− 3.00000i	− 0.639602i
$23$	6.00000i	1.25109i	0.780189	+	0.625543i	$0.215123\pi$
$23$	6.00000i	1.25109i	−0.780189	+	0.625543i	$0.784877\pi$
$24$	1.00000	0.204124
$25$	0	0
$26$	−2.00000	−0.392232
$27$	− 1.00000i	− 0.192450i
$28$	− 1.00000i	− 0.188982i
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	0	0
$31$	−7.00000	−1.25724	−0.628619	−	0.777714i	$-0.716379\pi$
$31$	−7.00000	−1.25724	−0.628619	+	0.777714i	$0.716379\pi$
$32$	1.00000i	0.176777i
$33$	− 3.00000i	− 0.522233i
$34$	−1.00000	−0.171499
$35$	0	0
$36$	1.00000	0.166667
$37$	7.00000i	1.15079i	0.817875	+	0.575396i	$0.195152\pi$
$37$	7.00000i	1.15079i	−0.817875	+	0.575396i	$0.804848\pi$
$38$	7.00000i	1.13555i
$39$	−2.00000	−0.320256
$40$	0	0
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	− 1.00000i	− 0.154303i
$43$	− 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
$46$	−6.00000	−0.884652
$47$	− 9.00000i	− 1.31278i	−0.754420	−	0.656392i	$-0.772082\pi$
$47$	− 9.00000i	− 1.31278i	0.754420	−	0.656392i	$-0.227918\pi$
$48$	1.00000i	0.144338i
$49$	6.00000	0.857143
$50$	0	0
$51$	−1.00000	−0.140028
$52$	− 2.00000i	− 0.277350i
$53$	− 3.00000i	− 0.412082i	−0.978543	−	0.206041i	$-0.933942\pi$
$53$	− 3.00000i	− 0.412082i	0.978543	−	0.206041i	$-0.0660580\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	1.00000	0.133631
$57$	7.00000i	0.927173i
$58$	− 6.00000i	− 0.787839i
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	− 7.00000i	− 0.889001i
$63$	− 1.00000i	− 0.125988i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	3.00000	0.369274
$67$	− 5.00000i	− 0.610847i	−0.952217	−	0.305424i	$-0.901202\pi$
$67$	− 5.00000i	− 0.610847i	0.952217	−	0.305424i	$-0.0987981\pi$
$68$	− 1.00000i	− 0.121268i
$69$	−6.00000	−0.722315
$70$	0	0
$71$	−6.00000	−0.712069	−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000	−0.712069	−0.356034	+	0.934473i	$0.615871\pi$
$72$	1.00000i	0.117851i
$73$	− 16.0000i	− 1.87266i	−0.351123	−	0.936329i	$-0.614200\pi$
$73$	− 16.0000i	− 1.87266i	0.351123	−	0.936329i	$-0.385800\pi$
$74$	−7.00000	−0.813733
$75$	0	0
$76$	−7.00000	−0.802955
$77$	− 3.00000i	− 0.341882i
$78$	− 2.00000i	− 0.226455i
$79$	−17.0000	−1.91265	−0.956325	−	0.292306i	$-0.905577\pi$
$79$	−17.0000	−1.91265	−0.956325	+	0.292306i	$0.905577\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	− 6.00000i	− 0.662589i
$83$	− 6.00000i	− 0.658586i	−0.944228	−	0.329293i	$-0.893190\pi$
$83$	− 6.00000i	− 0.658586i	0.944228	−	0.329293i	$-0.106810\pi$
$84$	1.00000	0.109109
$85$	0	0
$86$	1.00000	0.107833
$87$	− 6.00000i	− 0.643268i
$88$	3.00000i	0.319801i
$89$	12.0000	1.27200	0.635999	−	0.771690i	$-0.280588\pi$
$89$	12.0000	1.27200	0.635999	+	0.771690i	$0.280588\pi$
$90$	0	0
$91$	−2.00000	−0.209657
$92$	− 6.00000i	− 0.625543i
$93$	− 7.00000i	− 0.725866i
$94$	9.00000	0.928279
$95$	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	$0.830506\pi$
$98$	6.00000i	0.606092i
$99$	3.00000	0.301511
$100$	0	0
$101$	−15.0000	−1.49256	−0.746278	−	0.665635i	$-0.768161\pi$
$101$	−15.0000	−1.49256	−0.746278	+	0.665635i	$0.768161\pi$
$102$	− 1.00000i	− 0.0990148i
$103$	8.00000i	0.788263i	0.919054	+	0.394132i	$0.128955\pi$
$103$	8.00000i	0.788263i	−0.919054	+	0.394132i	$0.871045\pi$
$104$	2.00000	0.196116
$105$	0	0
$106$	3.00000	0.291386
$107$	3.00000i	0.290021i	0.989430	+	0.145010i	$0.0463216\pi$
$107$	3.00000i	0.290021i	−0.989430	+	0.145010i	$0.953678\pi$
$108$	1.00000i	0.0962250i
$109$	7.00000	0.670478	0.335239	−	0.942133i	$-0.391183\pi$
$109$	7.00000	0.670478	0.335239	+	0.942133i	$0.391183\pi$
$110$	0	0
$111$	−7.00000	−0.664411
$112$	1.00000i	0.0944911i
$113$	15.0000i	1.41108i	0.708669	+	0.705541i	$0.249296\pi$
$113$	15.0000i	1.41108i	−0.708669	+	0.705541i	$0.750704\pi$
$114$	−7.00000	−0.655610
$115$	0	0
$116$	6.00000	0.557086
$117$	− 2.00000i	− 0.184900i
$118$	0	0
$119$	−1.00000	−0.0916698
$120$	0	0
$121$	−2.00000	−0.181818
$122$	− 10.0000i	− 0.905357i
$123$	− 6.00000i	− 0.541002i
$124$	7.00000	0.628619
$125$	0	0
$126$	1.00000	0.0890871
$127$	4.00000i	0.354943i	0.984126	+	0.177471i	$0.0567917\pi$
$127$	4.00000i	0.354943i	−0.984126	+	0.177471i	$0.943208\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	1.00000	0.0880451
$130$	0	0
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	3.00000i	0.261116i
$133$	7.00000i	0.606977i
$134$	5.00000	0.431934
$135$	0	0
$136$	1.00000	0.0857493
$137$	− 12.0000i	− 1.02523i	−0.858619	−	0.512615i	$-0.828677\pi$
$137$	− 12.0000i	− 1.02523i	0.858619	−	0.512615i	$-0.171323\pi$
$138$	− 6.00000i	− 0.510754i
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	0	0
$141$	9.00000	0.757937
$142$	− 6.00000i	− 0.503509i
$143$	− 6.00000i	− 0.501745i
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	16.0000	1.32417
$147$	6.00000i	0.494872i
$148$	− 7.00000i	− 0.575396i
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	0	0
$151$	−4.00000	−0.325515	−0.162758	−	0.986666i	$-0.552039\pi$
$151$	−4.00000	−0.325515	−0.162758	+	0.986666i	$0.552039\pi$
$152$	− 7.00000i	− 0.567775i
$153$	− 1.00000i	− 0.0808452i
$154$	3.00000	0.241747
$155$	0	0
$156$	2.00000	0.160128
$157$	4.00000i	0.319235i	0.987179	+	0.159617i	$0.0510260\pi$
$157$	4.00000i	0.319235i	−0.987179	+	0.159617i	$0.948974\pi$
$158$	− 17.0000i	− 1.35245i
$159$	3.00000	0.237915
$160$	0	0
$161$	−6.00000	−0.472866
$162$	1.00000i	0.0785674i
$163$	8.00000i	0.626608i	0.949653	+	0.313304i	$0.101436\pi$
$163$	8.00000i	0.626608i	−0.949653	+	0.313304i	$0.898564\pi$
$164$	6.00000	0.468521
$165$	0	0
$166$	6.00000	0.465690
$167$	− 6.00000i	− 0.464294i	−0.972681	−	0.232147i	$-0.925425\pi$
$167$	− 6.00000i	− 0.464294i	0.972681	−	0.232147i	$-0.0745750\pi$
$168$	1.00000i	0.0771517i
$169$	9.00000	0.692308
$170$	0	0
$171$	−7.00000	−0.535303
$172$	1.00000i	0.0762493i
$173$	− 12.0000i	− 0.912343i	−0.889892	−	0.456172i	$-0.849220\pi$
$173$	− 12.0000i	− 0.912343i	0.889892	−	0.456172i	$-0.150780\pi$
$174$	6.00000	0.454859
$175$	0	0
$176$	−3.00000	−0.226134
$177$	0	0
$178$	12.0000i	0.899438i
$179$	−24.0000	−1.79384	−0.896922	−	0.442189i	$-0.854202\pi$
$179$	−24.0000	−1.79384	−0.896922	+	0.442189i	$0.854202\pi$
$180$	0	0
$181$	11.0000	0.817624	0.408812	−	0.912619i	$-0.365943\pi$
$181$	11.0000	0.817624	0.408812	+	0.912619i	$0.365943\pi$
$182$	− 2.00000i	− 0.148250i
$183$	− 10.0000i	− 0.739221i
$184$	6.00000	0.442326
$185$	0	0
$186$	7.00000	0.513265
$187$	− 3.00000i	− 0.219382i
$188$	9.00000i	0.656392i
$189$	1.00000	0.0727393
$190$	0	0
$191$	−15.0000	−1.08536	−0.542681	−	0.839939i	$-0.682591\pi$
$191$	−15.0000	−1.08536	−0.542681	+	0.839939i	$0.682591\pi$
$192$	− 1.00000i	− 0.0721688i
$193$	− 16.0000i	− 1.15171i	−0.817554	−	0.575853i	$-0.804670\pi$
$193$	− 16.0000i	− 1.15171i	0.817554	−	0.575853i	$-0.195330\pi$
$194$	−10.0000	−0.717958
$195$	0	0
$196$	−6.00000	−0.428571
$197$	− 12.0000i	− 0.854965i	−0.904024	−	0.427482i	$-0.859401\pi$
$197$	− 12.0000i	− 0.854965i	0.904024	−	0.427482i	$-0.140599\pi$
$198$	3.00000i	0.213201i
$199$	−5.00000	−0.354441	−0.177220	−	0.984171i	$-0.556711\pi$
$199$	−5.00000	−0.354441	−0.177220	+	0.984171i	$0.556711\pi$
$200$	0	0
$201$	5.00000	0.352673
$202$	− 15.0000i	− 1.05540i
$203$	− 6.00000i	− 0.421117i
$204$	1.00000	0.0700140
$205$	0	0
$206$	−8.00000	−0.557386
$207$	− 6.00000i	− 0.417029i
$208$	2.00000i	0.138675i
$209$	−21.0000	−1.45260
$210$	0	0
$211$	14.0000	0.963800	0.481900	−	0.876226i	$-0.339947\pi$
$211$	14.0000	0.963800	0.481900	+	0.876226i	$0.339947\pi$
$212$	3.00000i	0.206041i
$213$	− 6.00000i	− 0.411113i
$214$	−3.00000	−0.205076
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	− 7.00000i	− 0.475191i
$218$	7.00000i	0.474100i
$219$	16.0000	1.08118
$220$	0	0
$221$	−2.00000	−0.134535
$222$	− 7.00000i	− 0.469809i
$223$	2.00000i	0.133930i	0.997755	+	0.0669650i	$0.0213316\pi$
$223$	2.00000i	0.133930i	−0.997755	+	0.0669650i	$0.978668\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	−15.0000	−0.997785
$227$	− 3.00000i	− 0.199117i	−0.995032	−	0.0995585i	$-0.968257\pi$
$227$	− 3.00000i	− 0.199117i	0.995032	−	0.0995585i	$-0.0317430\pi$
$228$	− 7.00000i	− 0.463586i
$229$	28.0000	1.85029	0.925146	−	0.379611i	$-0.123942\pi$
$229$	28.0000	1.85029	0.925146	+	0.379611i	$0.123942\pi$
$230$	0	0
$231$	3.00000	0.197386
$232$	6.00000i	0.393919i
$233$	18.0000i	1.17922i	0.807688	+	0.589610i	$0.200718\pi$
$233$	18.0000i	1.17922i	−0.807688	+	0.589610i	$0.799282\pi$
$234$	2.00000	0.130744
$235$	0	0
$236$	0	0
$237$	− 17.0000i	− 1.10427i
$238$	− 1.00000i	− 0.0648204i
$239$	9.00000	0.582162	0.291081	−	0.956698i	$-0.405985\pi$
$239$	9.00000	0.582162	0.291081	+	0.956698i	$0.405985\pi$
$240$	0	0
$241$	−4.00000	−0.257663	−0.128831	−	0.991667i	$-0.541123\pi$
$241$	−4.00000	−0.257663	−0.128831	+	0.991667i	$0.541123\pi$
$242$	− 2.00000i	− 0.128565i
$243$	1.00000i	0.0641500i
$244$	10.0000	0.640184
$245$	0	0
$246$	6.00000	0.382546
$247$	14.0000i	0.890799i
$248$	7.00000i	0.444500i
$249$	6.00000	0.380235
$250$	0	0
$251$	6.00000	0.378717	0.189358	−	0.981908i	$-0.439359\pi$
$251$	6.00000	0.378717	0.189358	+	0.981908i	$0.439359\pi$
$252$	1.00000i	0.0629941i
$253$	− 18.0000i	− 1.13165i
$254$	−4.00000	−0.250982
$255$	0	0
$256$	1.00000	0.0625000
$257$	18.0000i	1.12281i	0.827541	+	0.561405i	$0.189739\pi$
$257$	18.0000i	1.12281i	−0.827541	+	0.561405i	$0.810261\pi$
$258$	1.00000i	0.0622573i
$259$	−7.00000	−0.434959
$260$	0	0
$261$	6.00000	0.371391
$262$	− 12.0000i	− 0.741362i
$263$	9.00000i	0.554964i	0.960731	+	0.277482i	$0.0894999\pi$
$263$	9.00000i	0.554964i	−0.960731	+	0.277482i	$0.910500\pi$
$264$	−3.00000	−0.184637
$265$	0	0
$266$	−7.00000	−0.429198
$267$	12.0000i	0.734388i
$268$	5.00000i	0.305424i
$269$	6.00000	0.365826	0.182913	−	0.983129i	$-0.441447\pi$
$269$	6.00000	0.365826	0.182913	+	0.983129i	$0.441447\pi$
$270$	0	0
$271$	14.0000	0.850439	0.425220	−	0.905090i	$-0.360197\pi$
$271$	14.0000	0.850439	0.425220	+	0.905090i	$0.360197\pi$
$272$	1.00000i	0.0606339i
$273$	− 2.00000i	− 0.121046i
$274$	12.0000	0.724947
$275$	0	0
$276$	6.00000	0.361158
$277$	19.0000i	1.14160i	0.821089	+	0.570800i	$0.193367\pi$
$277$	19.0000i	1.14160i	−0.821089	+	0.570800i	$0.806633\pi$
$278$	4.00000i	0.239904i
$279$	7.00000	0.419079
$280$	0	0
$281$	18.0000	1.07379	0.536895	−	0.843649i	$-0.319597\pi$
$281$	18.0000	1.07379	0.536895	+	0.843649i	$0.319597\pi$
$282$	9.00000i	0.535942i
$283$	− 22.0000i	− 1.30776i	−0.756596	−	0.653882i	$-0.773139\pi$
$283$	− 22.0000i	− 1.30776i	0.756596	−	0.653882i	$-0.226861\pi$
$284$	6.00000	0.356034
$285$	0	0
$286$	6.00000	0.354787
$287$	− 6.00000i	− 0.354169i
$288$	− 1.00000i	− 0.0589256i
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	−10.0000	−0.586210
$292$	16.0000i	0.936329i
$293$	− 6.00000i	− 0.350524i	−0.984522	−	0.175262i	$-0.943923\pi$
$293$	− 6.00000i	− 0.350524i	0.984522	−	0.175262i	$-0.0560772\pi$
$294$	−6.00000	−0.349927
$295$	0	0
$296$	7.00000	0.406867
$297$	3.00000i	0.174078i
$298$	− 6.00000i	− 0.347571i
$299$	−12.0000	−0.693978
$300$	0	0
$301$	1.00000	0.0576390
$302$	− 4.00000i	− 0.230174i
$303$	− 15.0000i	− 0.861727i
$304$	7.00000	0.401478
$305$	0	0
$306$	1.00000	0.0571662
$307$	28.0000i	1.59804i	0.601302	+	0.799022i	$0.294649\pi$
$307$	28.0000i	1.59804i	−0.601302	+	0.799022i	$0.705351\pi$
$308$	3.00000i	0.170941i
$309$	−8.00000	−0.455104
$310$	0	0
$311$	−30.0000	−1.70114	−0.850572	−	0.525859i	$-0.823744\pi$
$311$	−30.0000	−1.70114	−0.850572	+	0.525859i	$0.823744\pi$
$312$	2.00000i	0.113228i
$313$	− 10.0000i	− 0.565233i	−0.959233	−	0.282617i	$-0.908798\pi$
$313$	− 10.0000i	− 0.565233i	0.959233	−	0.282617i	$-0.0912024\pi$
$314$	−4.00000	−0.225733
$315$	0	0
$316$	17.0000	0.956325
$317$	24.0000i	1.34797i	0.738743	+	0.673987i	$0.235420\pi$
$317$	24.0000i	1.34797i	−0.738743	+	0.673987i	$0.764580\pi$
$318$	3.00000i	0.168232i
$319$	18.0000	1.00781
$320$	0	0
$321$	−3.00000	−0.167444
$322$	− 6.00000i	− 0.334367i
$323$	7.00000i	0.389490i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	−8.00000	−0.443079
$327$	7.00000i	0.387101i
$328$	6.00000i	0.331295i
$329$	9.00000	0.496186
$330$	0	0
$331$	−25.0000	−1.37412	−0.687062	−	0.726599i	$-0.741100\pi$
$331$	−25.0000	−1.37412	−0.687062	+	0.726599i	$0.741100\pi$
$332$	6.00000i	0.329293i
$333$	− 7.00000i	− 0.383598i
$334$	6.00000	0.328305
$335$	0	0
$336$	−1.00000	−0.0545545
$337$	16.0000i	0.871576i	0.900049	+	0.435788i	$0.143530\pi$
$337$	16.0000i	0.871576i	−0.900049	+	0.435788i	$0.856470\pi$
$338$	9.00000i	0.489535i
$339$	−15.0000	−0.814688
$340$	0	0
$341$	21.0000	1.13721
$342$	− 7.00000i	− 0.378517i
$343$	13.0000i	0.701934i
$344$	−1.00000	−0.0539164
$345$	0	0
$346$	12.0000	0.645124
$347$	9.00000i	0.483145i	0.970383	+	0.241573i	$0.0776632\pi$
$347$	9.00000i	0.483145i	−0.970383	+	0.241573i	$0.922337\pi$
$348$	6.00000i	0.321634i
$349$	−26.0000	−1.39175	−0.695874	−	0.718164i	$-0.744983\pi$
$349$	−26.0000	−1.39175	−0.695874	+	0.718164i	$0.744983\pi$
$350$	0	0
$351$	2.00000	0.106752
$352$	− 3.00000i	− 0.159901i
$353$	6.00000i	0.319348i	0.987170	+	0.159674i	$0.0510443\pi$
$353$	6.00000i	0.319348i	−0.987170	+	0.159674i	$0.948956\pi$
$354$	0	0
$355$	0	0
$356$	−12.0000	−0.635999
$357$	− 1.00000i	− 0.0529256i
$358$	− 24.0000i	− 1.26844i
$359$	−21.0000	−1.10834	−0.554169	−	0.832404i	$-0.686964\pi$
$359$	−21.0000	−1.10834	−0.554169	+	0.832404i	$0.686964\pi$
$360$	0	0
$361$	30.0000	1.57895
$362$	11.0000i	0.578147i
$363$	− 2.00000i	− 0.104973i
$364$	2.00000	0.104828
$365$	0	0
$366$	10.0000	0.522708
$367$	− 17.0000i	− 0.887393i	−0.896177	−	0.443696i	$-0.853667\pi$
$367$	− 17.0000i	− 0.887393i	0.896177	−	0.443696i	$-0.146333\pi$
$368$	6.00000i	0.312772i
$369$	6.00000	0.312348
$370$	0	0
$371$	3.00000	0.155752
$372$	7.00000i	0.362933i
$373$	14.0000i	0.724893i	0.932005	+	0.362446i	$0.118058\pi$
$373$	14.0000i	0.724893i	−0.932005	+	0.362446i	$0.881942\pi$
$374$	3.00000	0.155126
$375$	0	0
$376$	−9.00000	−0.464140
$377$	− 12.0000i	− 0.618031i
$378$	1.00000i	0.0514344i
$379$	16.0000	0.821865	0.410932	−	0.911666i	$-0.365203\pi$
$379$	16.0000	0.821865	0.410932	+	0.911666i	$0.365203\pi$
$380$	0	0
$381$	−4.00000	−0.204926
$382$	− 15.0000i	− 0.767467i
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	16.0000	0.814379
$387$	1.00000i	0.0508329i
$388$	− 10.0000i	− 0.507673i
$389$	−9.00000	−0.456318	−0.228159	−	0.973624i	$-0.573271\pi$
$389$	−9.00000	−0.456318	−0.228159	+	0.973624i	$0.573271\pi$
$390$	0	0
$391$	−6.00000	−0.303433
$392$	− 6.00000i	− 0.303046i
$393$	− 12.0000i	− 0.605320i
$394$	12.0000	0.604551
$395$	0	0
$396$	−3.00000	−0.150756
$397$	31.0000i	1.55585i	0.628360	+	0.777923i	$0.283727\pi$
$397$	31.0000i	1.55585i	−0.628360	+	0.777923i	$0.716273\pi$
$398$	− 5.00000i	− 0.250627i
$399$	−7.00000	−0.350438
$400$	0	0
$401$	18.0000	0.898877	0.449439	−	0.893311i	$-0.351624\pi$
$401$	18.0000	0.898877	0.449439	+	0.893311i	$0.351624\pi$
$402$	5.00000i	0.249377i
$403$	− 14.0000i	− 0.697390i
$404$	15.0000	0.746278
$405$	0	0
$406$	6.00000	0.297775
$407$	− 21.0000i	− 1.04093i
$408$	1.00000i	0.0495074i
$409$	−14.0000	−0.692255	−0.346128	−	0.938187i	$-0.612504\pi$
$409$	−14.0000	−0.692255	−0.346128	+	0.938187i	$0.612504\pi$
$410$	0	0
$411$	12.0000	0.591916
$412$	− 8.00000i	− 0.394132i
$413$	0	0
$414$	6.00000	0.294884
$415$	0	0
$416$	−2.00000	−0.0980581
$417$	4.00000i	0.195881i
$418$	− 21.0000i	− 1.02714i
$419$	−24.0000	−1.17248	−0.586238	−	0.810139i	$-0.699392\pi$
$419$	−24.0000	−1.17248	−0.586238	+	0.810139i	$0.699392\pi$
$420$	0	0
$421$	26.0000	1.26716	0.633581	−	0.773676i	$-0.281584\pi$
$421$	26.0000	1.26716	0.633581	+	0.773676i	$0.281584\pi$
$422$	14.0000i	0.681509i
$423$	9.00000i	0.437595i
$424$	−3.00000	−0.145693
$425$	0	0
$426$	6.00000	0.290701
$427$	− 10.0000i	− 0.483934i
$428$	− 3.00000i	− 0.145010i
$429$	6.00000	0.289683
$430$	0	0
$431$	12.0000	0.578020	0.289010	−	0.957326i	$-0.406674\pi$
$431$	12.0000	0.578020	0.289010	+	0.957326i	$0.406674\pi$
$432$	− 1.00000i	− 0.0481125i
$433$	29.0000i	1.39365i	0.717241	+	0.696826i	$0.245405\pi$
$433$	29.0000i	1.39365i	−0.717241	+	0.696826i	$0.754595\pi$
$434$	7.00000	0.336011
$435$	0	0
$436$	−7.00000	−0.335239
$437$	42.0000i	2.00913i
$438$	16.0000i	0.764510i
$439$	−8.00000	−0.381819	−0.190910	−	0.981608i	$-0.561144\pi$
$439$	−8.00000	−0.381819	−0.190910	+	0.981608i	$0.561144\pi$
$440$	0	0
$441$	−6.00000	−0.285714
$442$	− 2.00000i	− 0.0951303i
$443$	36.0000i	1.71041i	0.518289	+	0.855206i	$0.326569\pi$
$443$	36.0000i	1.71041i	−0.518289	+	0.855206i	$0.673431\pi$
$444$	7.00000	0.332205
$445$	0	0
$446$	−2.00000	−0.0947027
$447$	− 6.00000i	− 0.283790i
$448$	− 1.00000i	− 0.0472456i
$449$	−3.00000	−0.141579	−0.0707894	−	0.997491i	$-0.522552\pi$
$449$	−3.00000	−0.141579	−0.0707894	+	0.997491i	$0.522552\pi$
$450$	0	0
$451$	18.0000	0.847587
$452$	− 15.0000i	− 0.705541i
$453$	− 4.00000i	− 0.187936i
$454$	3.00000	0.140797
$455$	0	0
$456$	7.00000	0.327805
$457$	1.00000i	0.0467780i	0.999726	+	0.0233890i	$0.00744563\pi$
$457$	1.00000i	0.0467780i	−0.999726	+	0.0233890i	$0.992554\pi$
$458$	28.0000i	1.30835i
$459$	1.00000	0.0466760
$460$	0	0
$461$	−21.0000	−0.978068	−0.489034	−	0.872265i	$-0.662651\pi$
$461$	−21.0000	−0.978068	−0.489034	+	0.872265i	$0.662651\pi$
$462$	3.00000i	0.139573i
$463$	20.0000i	0.929479i	0.885448	+	0.464739i	$0.153852\pi$
$463$	20.0000i	0.929479i	−0.885448	+	0.464739i	$0.846148\pi$
$464$	−6.00000	−0.278543
$465$	0	0
$466$	−18.0000	−0.833834
$467$	− 18.0000i	− 0.832941i	−0.909149	−	0.416470i	$-0.863267\pi$
$467$	− 18.0000i	− 0.832941i	0.909149	−	0.416470i	$-0.136733\pi$
$468$	2.00000i	0.0924500i
$469$	5.00000	0.230879
$470$	0	0
$471$	−4.00000	−0.184310
$472$	0	0
$473$	3.00000i	0.137940i
$474$	17.0000	0.780836
$475$	0	0
$476$	1.00000	0.0458349
$477$	3.00000i	0.137361i
$478$	9.00000i	0.411650i
$479$	6.00000	0.274147	0.137073	−	0.990561i	$-0.456230\pi$
$479$	6.00000	0.274147	0.137073	+	0.990561i	$0.456230\pi$
$480$	0	0
$481$	−14.0000	−0.638345
$482$	− 4.00000i	− 0.182195i
$483$	− 6.00000i	− 0.273009i
$484$	2.00000	0.0909091
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	− 8.00000i	− 0.362515i	−0.983436	−	0.181257i	$-0.941983\pi$
$487$	− 8.00000i	− 0.362515i	0.983436	−	0.181257i	$-0.0580167\pi$
$488$	10.0000i	0.452679i
$489$	−8.00000	−0.361773
$490$	0	0
$491$	−36.0000	−1.62466	−0.812329	−	0.583200i	$-0.801800\pi$
$491$	−36.0000	−1.62466	−0.812329	+	0.583200i	$0.801800\pi$
$492$	6.00000i	0.270501i
$493$	− 6.00000i	− 0.270226i
$494$	−14.0000	−0.629890
$495$	0	0
$496$	−7.00000	−0.314309
$497$	− 6.00000i	− 0.269137i
$498$	6.00000i	0.268866i
$499$	−14.0000	−0.626726	−0.313363	−	0.949633i	$-0.601456\pi$
$499$	−14.0000	−0.626726	−0.313363	+	0.949633i	$0.601456\pi$
$500$	0	0
$501$	6.00000	0.268060
$502$	6.00000i	0.267793i
$503$	− 6.00000i	− 0.267527i	−0.991013	−	0.133763i	$-0.957294\pi$
$503$	− 6.00000i	− 0.267527i	0.991013	−	0.133763i	$-0.0427062\pi$
$504$	−1.00000	−0.0445435
$505$	0	0
$506$	18.0000	0.800198
$507$	9.00000i	0.399704i
$508$	− 4.00000i	− 0.177471i
$509$	−27.0000	−1.19675	−0.598377	−	0.801215i	$-0.704187\pi$
$509$	−27.0000	−1.19675	−0.598377	+	0.801215i	$0.704187\pi$
$510$	0	0
$511$	16.0000	0.707798
$512$	1.00000i	0.0441942i
$513$	− 7.00000i	− 0.309058i
$514$	−18.0000	−0.793946
$515$	0	0
$516$	−1.00000	−0.0440225
$517$	27.0000i	1.18746i
$518$	− 7.00000i	− 0.307562i
$519$	12.0000	0.526742
$520$	0	0
$521$	−33.0000	−1.44576	−0.722878	−	0.690976i	$-0.757181\pi$
$521$	−33.0000	−1.44576	−0.722878	+	0.690976i	$0.757181\pi$
$522$	6.00000i	0.262613i
$523$	− 4.00000i	− 0.174908i	−0.996169	−	0.0874539i	$-0.972127\pi$
$523$	− 4.00000i	− 0.174908i	0.996169	−	0.0874539i	$-0.0278730\pi$
$524$	12.0000	0.524222
$525$	0	0
$526$	−9.00000	−0.392419
$527$	− 7.00000i	− 0.304925i
$528$	− 3.00000i	− 0.130558i
$529$	−13.0000	−0.565217
$530$	0	0
$531$	0	0
$532$	− 7.00000i	− 0.303488i
$533$	− 12.0000i	− 0.519778i
$534$	−12.0000	−0.519291
$535$	0	0
$536$	−5.00000	−0.215967
$537$	− 24.0000i	− 1.03568i
$538$	6.00000i	0.258678i
$539$	−18.0000	−0.775315
$540$	0	0
$541$	35.0000	1.50477	0.752384	−	0.658725i	$-0.228904\pi$
$541$	35.0000	1.50477	0.752384	+	0.658725i	$0.228904\pi$
$542$	14.0000i	0.601351i
$543$	11.0000i	0.472055i
$544$	−1.00000	−0.0428746
$545$	0	0
$546$	2.00000	0.0855921
$547$	22.0000i	0.940652i	0.882493	+	0.470326i	$0.155864\pi$
$547$	22.0000i	0.940652i	−0.882493	+	0.470326i	$0.844136\pi$
$548$	12.0000i	0.512615i
$549$	10.0000	0.426790
$550$	0	0
$551$	−42.0000	−1.78926
$552$	6.00000i	0.255377i
$553$	− 17.0000i	− 0.722914i
$554$	−19.0000	−0.807233
$555$	0	0
$556$	−4.00000	−0.169638
$557$	21.0000i	0.889799i	0.895581	+	0.444899i	$0.146761\pi$
$557$	21.0000i	0.889799i	−0.895581	+	0.444899i	$0.853239\pi$
$558$	7.00000i	0.296334i
$559$	2.00000	0.0845910
$560$	0	0
$561$	3.00000	0.126660
$562$	18.0000i	0.759284i
$563$	0	0	1.00000			$0$
$563$	0	0	−1.00000			$\pi$
$564$	−9.00000	−0.378968
$565$	0	0
$566$	22.0000	0.924729
$567$	1.00000i	0.0419961i
$568$	6.00000i	0.251754i
$569$	24.0000	1.00613	0.503066	−	0.864248i	$-0.332205\pi$
$569$	24.0000	1.00613	0.503066	+	0.864248i	$0.332205\pi$
$570$	0	0
$571$	44.0000	1.84134	0.920671	−	0.390339i	$-0.127642\pi$
$571$	44.0000	1.84134	0.920671	+	0.390339i	$0.127642\pi$
$572$	6.00000i	0.250873i
$573$	− 15.0000i	− 0.626634i
$574$	6.00000	0.250435
$575$	0	0
$576$	1.00000	0.0416667
$577$	− 17.0000i	− 0.707719i	−0.935299	−	0.353860i	$-0.884869\pi$
$577$	− 17.0000i	− 0.707719i	0.935299	−	0.353860i	$-0.115131\pi$
$578$	− 1.00000i	− 0.0415945i
$579$	16.0000	0.664937
$580$	0	0
$581$	6.00000	0.248922
$582$	− 10.0000i	− 0.414513i
$583$	9.00000i	0.372742i
$584$	−16.0000	−0.662085
$585$	0	0
$586$	6.00000	0.247858
$587$	30.0000i	1.23823i	0.785299	+	0.619116i	$0.212509\pi$
$587$	30.0000i	1.23823i	−0.785299	+	0.619116i	$0.787491\pi$
$588$	− 6.00000i	− 0.247436i
$589$	−49.0000	−2.01901
$590$	0	0
$591$	12.0000	0.493614
$592$	7.00000i	0.287698i
$593$	48.0000i	1.97112i	0.169316	+	0.985562i	$0.445844\pi$
$593$	48.0000i	1.97112i	−0.169316	+	0.985562i	$0.554156\pi$
$594$	−3.00000	−0.123091
$595$	0	0
$596$	6.00000	0.245770
$597$	− 5.00000i	− 0.204636i
$598$	− 12.0000i	− 0.490716i
$599$	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
$601$	26.0000	1.06056	0.530281	−	0.847822i	$-0.322086\pi$
$601$	26.0000	1.06056	0.530281	+	0.847822i	$0.322086\pi$
$602$	1.00000i	0.0407570i
$603$	5.00000i	0.203616i
$604$	4.00000	0.162758
$605$	0	0
$606$	15.0000	0.609333
$607$	4.00000i	0.162355i	0.996700	+	0.0811775i	$0.0258681\pi$
$607$	4.00000i	0.162355i	−0.996700	+	0.0811775i	$0.974132\pi$
$608$	7.00000i	0.283887i
$609$	6.00000	0.243132
$610$	0	0
$611$	18.0000	0.728202
$612$	1.00000i	0.0404226i
$613$	26.0000i	1.05013i	0.851062	+	0.525065i	$0.175959\pi$
$613$	26.0000i	1.05013i	−0.851062	+	0.525065i	$0.824041\pi$
$614$	−28.0000	−1.12999
$615$	0	0
$616$	−3.00000	−0.120873
$617$	− 21.0000i	− 0.845428i	−0.906263	−	0.422714i	$-0.861077\pi$
$617$	− 21.0000i	− 0.845428i	0.906263	−	0.422714i	$-0.138923\pi$
$618$	− 8.00000i	− 0.321807i
$619$	−20.0000	−0.803868	−0.401934	−	0.915669i	$-0.631662\pi$
$619$	−20.0000	−0.803868	−0.401934	+	0.915669i	$0.631662\pi$
$620$	0	0
$621$	6.00000	0.240772
$622$	− 30.0000i	− 1.20289i
$623$	12.0000i	0.480770i
$624$	−2.00000	−0.0800641
$625$	0	0
$626$	10.0000	0.399680
$627$	− 21.0000i	− 0.838659i
$628$	− 4.00000i	− 0.159617i
$629$	−7.00000	−0.279108
$630$	0	0
$631$	−22.0000	−0.875806	−0.437903	−	0.899022i	$-0.644279\pi$
$631$	−22.0000	−0.875806	−0.437903	+	0.899022i	$0.644279\pi$
$632$	17.0000i	0.676224i
$633$	14.0000i	0.556450i
$634$	−24.0000	−0.953162
$635$	0	0
$636$	−3.00000	−0.118958
$637$	12.0000i	0.475457i
$638$	18.0000i	0.712627i
$639$	6.00000	0.237356
$640$	0	0
$641$	30.0000	1.18493	0.592464	−	0.805597i	$-0.298155\pi$
$641$	30.0000	1.18493	0.592464	+	0.805597i	$0.298155\pi$
$642$	− 3.00000i	− 0.118401i
$643$	14.0000i	0.552106i	0.961142	+	0.276053i	$0.0890266\pi$
$643$	14.0000i	0.552106i	−0.961142	+	0.276053i	$0.910973\pi$
$644$	6.00000	0.236433
$645$	0	0
$646$	−7.00000	−0.275411
$647$	48.0000i	1.88707i	0.331266	+	0.943537i	$0.392524\pi$
$647$	48.0000i	1.88707i	−0.331266	+	0.943537i	$0.607476\pi$
$648$	− 1.00000i	− 0.0392837i
$649$	0	0
$650$	0	0
$651$	7.00000	0.274352
$652$	− 8.00000i	− 0.313304i
$653$	− 12.0000i	− 0.469596i	−0.972044	−	0.234798i	$-0.924557\pi$
$653$	− 12.0000i	− 0.469596i	0.972044	−	0.234798i	$-0.0754429\pi$
$654$	−7.00000	−0.273722
$655$	0	0
$656$	−6.00000	−0.234261
$657$	16.0000i	0.624219i
$658$	9.00000i	0.350857i
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	0	0
$661$	20.0000	0.777910	0.388955	−	0.921257i	$-0.372836\pi$
$661$	20.0000	0.777910	0.388955	+	0.921257i	$0.372836\pi$
$662$	− 25.0000i	− 0.971653i
$663$	− 2.00000i	− 0.0776736i
$664$	−6.00000	−0.232845
$665$	0	0
$666$	7.00000	0.271244
$667$	− 36.0000i	− 1.39393i
$668$	6.00000i	0.232147i
$669$	−2.00000	−0.0773245
$670$	0	0
$671$	30.0000	1.15814
$672$	− 1.00000i	− 0.0385758i
$673$	2.00000i	0.0770943i	0.999257	+	0.0385472i	$0.0122730\pi$
$673$	2.00000i	0.0770943i	−0.999257	+	0.0385472i	$0.987727\pi$
$674$	−16.0000	−0.616297
$675$	0	0
$676$	−9.00000	−0.346154
$677$	− 48.0000i	− 1.84479i	−0.386248	−	0.922395i	$-0.626229\pi$
$677$	− 48.0000i	− 1.84479i	0.386248	−	0.922395i	$-0.373771\pi$
$678$	− 15.0000i	− 0.576072i
$679$	−10.0000	−0.383765
$680$	0	0
$681$	3.00000	0.114960
$682$	21.0000i	0.804132i
$683$	24.0000i	0.918334i	0.888350	+	0.459167i	$0.151852\pi$
$683$	24.0000i	0.918334i	−0.888350	+	0.459167i	$0.848148\pi$
$684$	7.00000	0.267652
$685$	0	0
$686$	−13.0000	−0.496342
$687$	28.0000i	1.06827i
$688$	− 1.00000i	− 0.0381246i
$689$	6.00000	0.228582
$690$	0	0
$691$	−16.0000	−0.608669	−0.304334	−	0.952565i	$-0.598434\pi$
$691$	−16.0000	−0.608669	−0.304334	+	0.952565i	$0.598434\pi$
$692$	12.0000i	0.456172i
$693$	3.00000i	0.113961i
$694$	−9.00000	−0.341635
$695$	0	0
$696$	−6.00000	−0.227429
$697$	− 6.00000i	− 0.227266i
$698$	− 26.0000i	− 0.984115i
$699$	−18.0000	−0.680823
$700$	0	0
$701$	−18.0000	−0.679851	−0.339925	−	0.940452i	$-0.610402\pi$
$701$	−18.0000	−0.679851	−0.339925	+	0.940452i	$0.610402\pi$
$702$	2.00000i	0.0754851i
$703$	49.0000i	1.84807i
$704$	3.00000	0.113067
$705$	0	0
$706$	−6.00000	−0.225813
$707$	− 15.0000i	− 0.564133i
$708$	0	0
$709$	19.0000	0.713560	0.356780	−	0.934188i	$-0.383875\pi$
$709$	19.0000	0.713560	0.356780	+	0.934188i	$0.383875\pi$
$710$	0	0
$711$	17.0000	0.637550
$712$	− 12.0000i	− 0.449719i
$713$	− 42.0000i	− 1.57291i
$714$	1.00000	0.0374241
$715$	0	0
$716$	24.0000	0.896922
$717$	9.00000i	0.336111i
$718$	− 21.0000i	− 0.783713i
$719$	18.0000	0.671287	0.335643	−	0.941989i	$-0.391046\pi$
$719$	18.0000	0.671287	0.335643	+	0.941989i	$0.391046\pi$
$720$	0	0
$721$	−8.00000	−0.297936
$722$	30.0000i	1.11648i
$723$	− 4.00000i	− 0.148762i
$724$	−11.0000	−0.408812
$725$	0	0
$726$	2.00000	0.0742270
$727$	− 26.0000i	− 0.964287i	−0.876092	−	0.482143i	$-0.839858\pi$
$727$	− 26.0000i	− 0.964287i	0.876092	−	0.482143i	$-0.160142\pi$
$728$	2.00000i	0.0741249i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	1.00000	0.0369863
$732$	10.0000i	0.369611i
$733$	20.0000i	0.738717i	0.929287	+	0.369358i	$0.120423\pi$
$733$	20.0000i	0.738717i	−0.929287	+	0.369358i	$0.879577\pi$
$734$	17.0000	0.627481
$735$	0	0
$736$	−6.00000	−0.221163
$737$	15.0000i	0.552532i
$738$	6.00000i	0.220863i
$739$	−53.0000	−1.94964	−0.974818	−	0.223001i	$-0.928415\pi$
$739$	−53.0000	−1.94964	−0.974818	+	0.223001i	$0.928415\pi$
$740$	0	0
$741$	−14.0000	−0.514303
$742$	3.00000i	0.110133i
$743$	42.0000i	1.54083i	0.637542	+	0.770415i	$0.279951\pi$
$743$	42.0000i	1.54083i	−0.637542	+	0.770415i	$0.720049\pi$
$744$	−7.00000	−0.256632
$745$	0	0
$746$	−14.0000	−0.512576
$747$	6.00000i	0.219529i
$748$	3.00000i	0.109691i
$749$	−3.00000	−0.109618
$750$	0	0
$751$	32.0000	1.16770	0.583848	−	0.811863i	$-0.301546\pi$
$751$	32.0000	1.16770	0.583848	+	0.811863i	$0.301546\pi$
$752$	− 9.00000i	− 0.328196i
$753$	6.00000i	0.218652i
$754$	12.0000	0.437014
$755$	0	0
$756$	−1.00000	−0.0363696
$757$	− 38.0000i	− 1.38113i	−0.723269	−	0.690567i	$-0.757361\pi$
$757$	− 38.0000i	− 1.38113i	0.723269	−	0.690567i	$-0.242639\pi$
$758$	16.0000i	0.581146i
$759$	18.0000	0.653359
$760$	0	0
$761$	36.0000	1.30500	0.652499	−	0.757789i	$-0.273720\pi$
$761$	36.0000	1.30500	0.652499	+	0.757789i	$0.273720\pi$
$762$	− 4.00000i	− 0.144905i
$763$	7.00000i	0.253417i
$764$	15.0000	0.542681
$765$	0	0
$766$	0	0
$767$	0	0
$768$	1.00000i	0.0360844i
$769$	37.0000	1.33425	0.667127	−	0.744944i	$-0.267524\pi$
$769$	37.0000	1.33425	0.667127	+	0.744944i	$0.267524\pi$
$770$	0	0
$771$	−18.0000	−0.648254
$772$	16.0000i	0.575853i
$773$	− 30.0000i	− 1.07903i	−0.841978	−	0.539513i	$-0.818609\pi$
$773$	− 30.0000i	− 1.07903i	0.841978	−	0.539513i	$-0.181391\pi$
$774$	−1.00000	−0.0359443
$775$	0	0
$776$	10.0000	0.358979
$777$	− 7.00000i	− 0.251124i
$778$	− 9.00000i	− 0.322666i
$779$	−42.0000	−1.50481
$780$	0	0
$781$	18.0000	0.644091
$782$	− 6.00000i	− 0.214560i
$783$	6.00000i	0.214423i
$784$	6.00000	0.214286
$785$	0	0
$786$	12.0000	0.428026
$787$	− 38.0000i	− 1.35455i	−0.735728	−	0.677277i	$-0.763160\pi$
$787$	− 38.0000i	− 1.35455i	0.735728	−	0.677277i	$-0.236840\pi$
$788$	12.0000i	0.427482i
$789$	−9.00000	−0.320408
$790$	0	0
$791$	−15.0000	−0.533339
$792$	− 3.00000i	− 0.106600i
$793$	− 20.0000i	− 0.710221i
$794$	−31.0000	−1.10015
$795$	0	0
$796$	5.00000	0.177220
$797$	− 39.0000i	− 1.38145i	−0.723117	−	0.690725i	$-0.757291\pi$
$797$	− 39.0000i	− 1.38145i	0.723117	−	0.690725i	$-0.242709\pi$
$798$	− 7.00000i	− 0.247797i
$799$	9.00000	0.318397
$800$	0	0
$801$	−12.0000	−0.423999
$802$	18.0000i	0.635602i
$803$	48.0000i	1.69388i
$804$	−5.00000	−0.176336
$805$	0	0
$806$	14.0000	0.493129
$807$	6.00000i	0.211210i
$808$	15.0000i	0.527698i
$809$	33.0000	1.16022	0.580109	−	0.814539i	$-0.303010\pi$
$809$	33.0000	1.16022	0.580109	+	0.814539i	$0.303010\pi$
$810$	0	0
$811$	−28.0000	−0.983213	−0.491606	−	0.870817i	$-0.663590\pi$
$811$	−28.0000	−0.983213	−0.491606	+	0.870817i	$0.663590\pi$
$812$	6.00000i	0.210559i
$813$	14.0000i	0.491001i
$814$	21.0000	0.736050
$815$	0	0
$816$	−1.00000	−0.0350070
$817$	− 7.00000i	− 0.244899i
$818$	− 14.0000i	− 0.489499i
$819$	2.00000	0.0698857
$820$	0	0
$821$	−18.0000	−0.628204	−0.314102	−	0.949389i	$-0.601703\pi$
$821$	−18.0000	−0.628204	−0.314102	+	0.949389i	$0.601703\pi$
$822$	12.0000i	0.418548i
$823$	− 52.0000i	− 1.81261i	−0.422628	−	0.906303i	$-0.638892\pi$
$823$	− 52.0000i	− 1.81261i	0.422628	−	0.906303i	$-0.361108\pi$
$824$	8.00000	0.278693
$825$	0	0
$826$	0	0
$827$	3.00000i	0.104320i	0.998639	+	0.0521601i	$0.0166106\pi$
$827$	3.00000i	0.104320i	−0.998639	+	0.0521601i	$0.983389\pi$
$828$	6.00000i	0.208514i
$829$	46.0000	1.59765	0.798823	−	0.601566i	$-0.205456\pi$
$829$	46.0000	1.59765	0.798823	+	0.601566i	$0.205456\pi$
$830$	0	0
$831$	−19.0000	−0.659103
$832$	− 2.00000i	− 0.0693375i
$833$	6.00000i	0.207888i
$834$	−4.00000	−0.138509
$835$	0	0
$836$	21.0000	0.726300
$837$	7.00000i	0.241955i
$838$	− 24.0000i	− 0.829066i
$839$	48.0000	1.65714	0.828572	−	0.559883i	$-0.189154\pi$
$839$	48.0000	1.65714	0.828572	+	0.559883i	$0.189154\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	26.0000i	0.896019i
$843$	18.0000i	0.619953i
$844$	−14.0000	−0.481900
$845$	0	0
$846$	−9.00000	−0.309426
$847$	− 2.00000i	− 0.0687208i
$848$	− 3.00000i	− 0.103020i
$849$	22.0000	0.755038
$850$	0	0
$851$	−42.0000	−1.43974
$852$	6.00000i	0.205557i
$853$	− 1.00000i	− 0.0342393i	−0.999853	−	0.0171197i	$-0.994550\pi$
$853$	− 1.00000i	− 0.0342393i	0.999853	−	0.0171197i	$-0.00544963\pi$
$854$	10.0000	0.342193
$855$	0	0
$856$	3.00000	0.102538
$857$	3.00000i	0.102478i	0.998686	+	0.0512390i	$0.0163170\pi$
$857$	3.00000i	0.102478i	−0.998686	+	0.0512390i	$0.983683\pi$
$858$	6.00000i	0.204837i
$859$	−41.0000	−1.39890	−0.699451	−	0.714681i	$-0.746572\pi$
$859$	−41.0000	−1.39890	−0.699451	+	0.714681i	$0.746572\pi$
$860$	0	0
$861$	6.00000	0.204479
$862$	12.0000i	0.408722i
$863$	45.0000i	1.53182i	0.642949	+	0.765909i	$0.277711\pi$
$863$	45.0000i	1.53182i	−0.642949	+	0.765909i	$0.722289\pi$
$864$	1.00000	0.0340207
$865$	0	0
$866$	−29.0000	−0.985460
$867$	− 1.00000i	− 0.0339618i
$868$	7.00000i	0.237595i
$869$	51.0000	1.73006
$870$	0	0
$871$	10.0000	0.338837
$872$	− 7.00000i	− 0.237050i
$873$	− 10.0000i	− 0.338449i
$874$	−42.0000	−1.42067
$875$	0	0
$876$	−16.0000	−0.540590
$877$	34.0000i	1.14810i	0.818821	+	0.574049i	$0.194628\pi$
$877$	34.0000i	1.14810i	−0.818821	+	0.574049i	$0.805372\pi$
$878$	− 8.00000i	− 0.269987i
$879$	6.00000	0.202375
$880$	0	0
$881$	33.0000	1.11180	0.555899	−	0.831250i	$-0.312374\pi$
$881$	33.0000	1.11180	0.555899	+	0.831250i	$0.312374\pi$
$882$	− 6.00000i	− 0.202031i
$883$	20.0000i	0.673054i	0.941674	+	0.336527i	$0.109252\pi$
$883$	20.0000i	0.673054i	−0.941674	+	0.336527i	$0.890748\pi$
$884$	2.00000	0.0672673
$885$	0	0
$886$	−36.0000	−1.20944
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	7.00000i	0.234905i
$889$	−4.00000	−0.134156
$890$	0	0
$891$	−3.00000	−0.100504
$892$	− 2.00000i	− 0.0669650i
$893$	− 63.0000i	− 2.10821i
$894$	6.00000	0.200670
$895$	0	0
$896$	1.00000	0.0334077
$897$	− 12.0000i	− 0.400668i
$898$	− 3.00000i	− 0.100111i
$899$	42.0000	1.40078
$900$	0	0
$901$	3.00000	0.0999445
$902$	18.0000i	0.599334i
$903$	1.00000i	0.0332779i
$904$	15.0000	0.498893
$905$	0	0
$906$	4.00000	0.132891
$907$	22.0000i	0.730498i	0.930910	+	0.365249i	$0.119016\pi$
$907$	22.0000i	0.730498i	−0.930910	+	0.365249i	$0.880984\pi$
$908$	3.00000i	0.0995585i
$909$	15.0000	0.497519
$910$	0	0
$911$	36.0000	1.19273	0.596367	−	0.802712i	$-0.296610\pi$
$911$	36.0000	1.19273	0.596367	+	0.802712i	$0.296610\pi$
$912$	7.00000i	0.231793i
$913$	18.0000i	0.595713i
$914$	−1.00000	−0.0330771
$915$	0	0
$916$	−28.0000	−0.925146
$917$	− 12.0000i	− 0.396275i
$918$	1.00000i	0.0330049i
$919$	−8.00000	−0.263896	−0.131948	−	0.991257i	$-0.542123\pi$
$919$	−8.00000	−0.263896	−0.131948	+	0.991257i	$0.542123\pi$
$920$	0	0
$921$	−28.0000	−0.922631
$922$	− 21.0000i	− 0.691598i
$923$	− 12.0000i	− 0.394985i
$924$	−3.00000	−0.0986928
$925$	0	0
$926$	−20.0000	−0.657241
$927$	− 8.00000i	− 0.262754i
$928$	− 6.00000i	− 0.196960i
$929$	−45.0000	−1.47640	−0.738201	−	0.674581i	$-0.764324\pi$
$929$	−45.0000	−1.47640	−0.738201	+	0.674581i	$0.764324\pi$
$930$	0	0
$931$	42.0000	1.37649
$932$	− 18.0000i	− 0.589610i
$933$	− 30.0000i	− 0.982156i
$934$	18.0000	0.588978
$935$	0	0
$936$	−2.00000	−0.0653720
$937$	22.0000i	0.718709i	0.933201	+	0.359354i	$0.117003\pi$
$937$	22.0000i	0.718709i	−0.933201	+	0.359354i	$0.882997\pi$
$938$	5.00000i	0.163256i
$939$	10.0000	0.326338
$940$	0	0
$941$	−18.0000	−0.586783	−0.293392	−	0.955992i	$-0.594784\pi$
$941$	−18.0000	−0.586783	−0.293392	+	0.955992i	$0.594784\pi$
$942$	− 4.00000i	− 0.130327i
$943$	− 36.0000i	− 1.17232i
$944$	0	0
$945$	0	0
$946$	−3.00000	−0.0975384
$947$	− 51.0000i	− 1.65728i	−0.559784	−	0.828639i	$-0.689116\pi$
$947$	− 51.0000i	− 1.65728i	0.559784	−	0.828639i	$-0.310884\pi$
$948$	17.0000i	0.552134i
$949$	32.0000	1.03876
$950$	0	0
$951$	−24.0000	−0.778253
$952$	1.00000i	0.0324102i
$953$	− 18.0000i	− 0.583077i	−0.956559	−	0.291539i	$-0.905833\pi$
$953$	− 18.0000i	− 0.583077i	0.956559	−	0.291539i	$-0.0941672\pi$
$954$	−3.00000	−0.0971286
$955$	0	0
$956$	−9.00000	−0.291081
$957$	18.0000i	0.581857i
$958$	6.00000i	0.193851i
$959$	12.0000	0.387500
$960$	0	0
$961$	18.0000	0.580645
$962$	− 14.0000i	− 0.451378i
$963$	− 3.00000i	− 0.0966736i
$964$	4.00000	0.128831
$965$	0	0
$966$	6.00000	0.193047
$967$	4.00000i	0.128631i	0.997930	+	0.0643157i	$0.0204865\pi$
$967$	4.00000i	0.128631i	−0.997930	+	0.0643157i	$0.979514\pi$
$968$	2.00000i	0.0642824i
$969$	−7.00000	−0.224872
$970$	0	0
$971$	30.0000	0.962746	0.481373	−	0.876516i	$-0.340138\pi$
$971$	30.0000	0.962746	0.481373	+	0.876516i	$0.340138\pi$
$972$	− 1.00000i	− 0.0320750i
$973$	4.00000i	0.128234i
$974$	8.00000	0.256337
$975$	0	0
$976$	−10.0000	−0.320092
$977$	− 42.0000i	− 1.34370i	−0.740688	−	0.671850i	$-0.765500\pi$
$977$	− 42.0000i	− 1.34370i	0.740688	−	0.671850i	$-0.234500\pi$
$978$	− 8.00000i	− 0.255812i
$979$	−36.0000	−1.15056
$980$	0	0
$981$	−7.00000	−0.223493
$982$	− 36.0000i	− 1.14881i
$983$	42.0000i	1.33959i	0.742545	+	0.669796i	$0.233618\pi$
$983$	42.0000i	1.33959i	−0.742545	+	0.669796i	$0.766382\pi$
$984$	−6.00000	−0.191273
$985$	0	0
$986$	6.00000	0.191079
$987$	9.00000i	0.286473i
$988$	− 14.0000i	− 0.445399i
$989$	6.00000	0.190789
$990$	0	0
$991$	−4.00000	−0.127064	−0.0635321	−	0.997980i	$-0.520237\pi$
$991$	−4.00000	−0.127064	−0.0635321	+	0.997980i	$0.520237\pi$
$992$	− 7.00000i	− 0.222250i
$993$	− 25.0000i	− 0.793351i
$994$	6.00000	0.190308
$995$	0	0
$996$	−6.00000	−0.190117
$997$	55.0000i	1.74187i	0.491400	+	0.870934i	$0.336485\pi$
$997$	55.0000i	1.74187i	−0.491400	+	0.870934i	$0.663515\pi$
$998$	− 14.0000i	− 0.443162i
$999$	7.00000	0.221470

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2550.2.d.c.2449.2		2
5.2	odd	4		2550.2.a.j.1.1	✓	1
5.3	odd	4		2550.2.a.w.1.1	yes	1
5.4	even	2	inner	2550.2.d.c.2449.1		2
15.2	even	4		7650.2.a.bt.1.1		1
15.8	even	4		7650.2.a.v.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2550.2.a.j.1.1	✓	1	5.2	odd	4
2550.2.a.w.1.1	yes	1	5.3	odd	4
2550.2.d.c.2449.1		2	5.4	even	2	inner
2550.2.d.c.2449.2		2	1.1	even	1	trivial
7650.2.a.v.1.1		1	15.8	even	4
7650.2.a.bt.1.1		1	15.2	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2550 = 2 \cdot 3 \cdot 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2550.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +1.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +1.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} -3.00000 q^{11} -1.00000i q^{12} +2.00000i q^{13} -1.00000 q^{14} +1.00000 q^{16} +1.00000i q^{17} -1.00000i q^{18} +7.00000 q^{19} -1.00000 q^{21} -3.00000i q^{22} +6.00000i q^{23} +1.00000 q^{24} -2.00000 q^{26} -1.00000i q^{27} -1.00000i q^{28} -6.00000 q^{29} -7.00000 q^{31} +1.00000i q^{32} -3.00000i q^{33} -1.00000 q^{34} +1.00000 q^{36} +7.00000i q^{37} +7.00000i q^{38} -2.00000 q^{39} -6.00000 q^{41} -1.00000i q^{42} -1.00000i q^{43} +3.00000 q^{44} -6.00000 q^{46} -9.00000i q^{47} +1.00000i q^{48} +6.00000 q^{49} -1.00000 q^{51} -2.00000i q^{52} -3.00000i q^{53} +1.00000 q^{54} +1.00000 q^{56} +7.00000i q^{57} -6.00000i q^{58} -10.0000 q^{61} -7.00000i q^{62} -1.00000i q^{63} -1.00000 q^{64} +3.00000 q^{66} -5.00000i q^{67} -1.00000i q^{68} -6.00000 q^{69} -6.00000 q^{71} +1.00000i q^{72} -16.0000i q^{73} -7.00000 q^{74} -7.00000 q^{76} -3.00000i q^{77} -2.00000i q^{78} -17.0000 q^{79} +1.00000 q^{81} -6.00000i q^{82} -6.00000i q^{83} +1.00000 q^{84} +1.00000 q^{86} -6.00000i q^{87} +3.00000i q^{88} +12.0000 q^{89} -2.00000 q^{91} -6.00000i q^{92} -7.00000i q^{93} +9.00000 q^{94} -1.00000 q^{96} +10.0000i q^{97} +6.00000i q^{98} +3.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9	\( 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9} - 6 q^{11} - 2 q^{14} + 2 q^{16} + 14 q^{19} - 2 q^{21} + 2 q^{24} - 4 q^{26} - 12 q^{29} - 14 q^{31} - 2 q^{34} + 2 q^{36} - 4 q^{39} - 12 q^{41} + 6 q^{44} - 12 q^{46} + 12 q^{49} - 2 q^{51} + 2 q^{54} + 2 q^{56} - 20 q^{61} - 2 q^{64} + 6 q^{66} - 12 q^{69} - 12 q^{71} - 14 q^{74} - 14 q^{76} - 34 q^{79} + 2 q^{81} + 2 q^{84} + 2 q^{86} + 24 q^{89} - 4 q^{91} + 18 q^{94} - 2 q^{96} + 6 q^{99}+O(q^{100}) \) 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9 - 6 * q^11 - 2 * q^14 + 2 * q^16 + 14 * q^19 - 2 * q^21 + 2 * q^24 - 4 * q^26 - 12 * q^29 - 14 * q^31 - 2 * q^34 + 2 * q^36 - 4 * q^39 - 12 * q^41 + 6 * q^44 - 12 * q^46 + 12 * q^49 - 2 * q^51 + 2 * q^54 + 2 * q^56 - 20 * q^61 - 2 * q^64 + 6 * q^66 - 12 * q^69 - 12 * q^71 - 14 * q^74 - 14 * q^76 - 34 * q^79 + 2 * q^81 + 2 * q^84 + 2 * q^86 + 24 * q^89 - 4 * q^91 + 18 * q^94 - 2 * q^96 + 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more