LMFDB - Embedded newform 2550.2.d.s.2449.1

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:	no (minimal twist has level 510)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2449.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	2550.2449
Dual form			2550.2.d.s.2449.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} +4.00000 q^{11} -1.00000i q^{12} +2.00000i q^{13} +1.00000 q^{16} -1.00000i q^{17} +1.00000i q^{18} +4.00000 q^{19} -4.00000i q^{22} +4.00000i q^{23} -1.00000 q^{24} +2.00000 q^{26} -1.00000i q^{27} -2.00000 q^{29} -4.00000 q^{31} -1.00000i q^{32} +4.00000i q^{33} -1.00000 q^{34} +1.00000 q^{36} +6.00000i q^{37} -4.00000i q^{38} -2.00000 q^{39} -10.0000 q^{41} -8.00000i q^{43} -4.00000 q^{44} +4.00000 q^{46} +1.00000i q^{48} +7.00000 q^{49} +1.00000 q^{51} -2.00000i q^{52} +6.00000i q^{53} -1.00000 q^{54} +4.00000i q^{57} +2.00000i q^{58} +8.00000 q^{59} +10.0000 q^{61} +4.00000i q^{62} -1.00000 q^{64} +4.00000 q^{66} +8.00000i q^{67} +1.00000i q^{68} -4.00000 q^{69} +8.00000 q^{71} -1.00000i q^{72} -2.00000i q^{73} +6.00000 q^{74} -4.00000 q^{76} +2.00000i q^{78} -4.00000 q^{79} +1.00000 q^{81} +10.0000i q^{82} +4.00000i q^{83} -8.00000 q^{86} -2.00000i q^{87} +4.00000i q^{88} +14.0000 q^{89} -4.00000i q^{92} -4.00000i q^{93} +1.00000 q^{96} +10.0000i q^{97} -7.00000i q^{98} -4.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 8 q^{11} + 2 q^{16} + 8 q^{19} - 2 q^{24} + 4 q^{26} - 4 q^{29} - 8 q^{31} - 2 q^{34} + 2 q^{36} - 4 q^{39} - 20 q^{41} - 8 q^{44} + 8 q^{46} + 14 q^{49} + 2 q^{51} - 2 q^{54} + 16 q^{59} + 20 q^{61} - 2 q^{64} + 8 q^{66} - 8 q^{69} + 16 q^{71} + 12 q^{74} - 8 q^{76} - 8 q^{79} + 2 q^{81} - 16 q^{86} + 28 q^{89} + 2 q^{96} - 8 q^{99}+O(q^{100}) $ 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9 + 8 * q^11 + 2 * q^16 + 8 * q^19 - 2 * q^24 + 4 * q^26 - 4 * q^29 - 8 * q^31 - 2 * q^34 + 2 * q^36 - 4 * q^39 - 20 * q^41 - 8 * q^44 + 8 * q^46 + 14 * q^49 + 2 * q^51 - 2 * q^54 + 16 * q^59 + 20 * q^61 - 2 * q^64 + 8 * q^66 - 8 * q^69 + 16 * q^71 + 12 * q^74 - 8 * q^76 - 8 * q^79 + 2 * q^81 - 16 * q^86 + 28 * q^89 + 2 * q^96 - 8 * 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
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	1.00000i	0.353553i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	4.00000	1.20605	0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000	1.20605	0.603023	+	0.797724i	$0.293963\pi$
$12$	− 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	$0.910544\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	− 1.00000i	− 0.242536i
$17$	− 1.00000i	− 0.242536i
$18$	1.00000i	0.235702i
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	0	0
$21$	0	0
$22$	− 4.00000i	− 0.852803i
$23$	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
$26$	2.00000	0.392232
$27$	− 1.00000i	− 0.192450i
$28$	0	0
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	0	0
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	− 1.00000i	− 0.176777i
$33$	4.00000i	0.696311i
$34$	−1.00000	−0.171499
$35$	0	0
$36$	1.00000	0.166667
$37$	6.00000i	0.986394i	0.869918	+	0.493197i	$0.164172\pi$
$37$	6.00000i	0.986394i	−0.869918	+	0.493197i	$0.835828\pi$
$38$	− 4.00000i	− 0.648886i
$39$	−2.00000	−0.320256
$40$	0	0
$41$	−10.0000	−1.56174	−0.780869	−	0.624695i	$-0.785223\pi$
$41$	−10.0000	−1.56174	−0.780869	+	0.624695i	$0.785223\pi$
$42$	0	0
$43$	− 8.00000i	− 1.21999i	−0.792406	−	0.609994i	$-0.791172\pi$
$43$	− 8.00000i	− 1.21999i	0.792406	−	0.609994i	$-0.208828\pi$
$44$	−4.00000	−0.603023
$45$	0	0
$46$	4.00000	0.589768
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	1.00000i	0.144338i
$49$	7.00000	1.00000
$50$	0	0
$51$	1.00000	0.140028
$52$	− 2.00000i	− 0.277350i
$53$	6.00000i	0.824163i	0.911147	+	0.412082i	$0.135198\pi$
$53$	6.00000i	0.824163i	−0.911147	+	0.412082i	$0.864802\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	0	0
$57$	4.00000i	0.529813i
$58$	2.00000i	0.262613i
$59$	8.00000	1.04151	0.520756	−	0.853706i	$-0.325650\pi$
$59$	8.00000	1.04151	0.520756	+	0.853706i	$0.325650\pi$
$60$	0	0
$61$	10.0000	1.28037	0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000	1.28037	0.640184	+	0.768221i	$0.278858\pi$
$62$	4.00000i	0.508001i
$63$	0	0
$64$	−1.00000	−0.125000
$65$	0	0
$66$	4.00000	0.492366
$67$	8.00000i	0.977356i	0.872464	+	0.488678i	$0.162521\pi$
$67$	8.00000i	0.977356i	−0.872464	+	0.488678i	$0.837479\pi$
$68$	1.00000i	0.121268i
$69$	−4.00000	−0.481543
$70$	0	0
$71$	8.00000	0.949425	0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	− 1.00000i	− 0.117851i
$73$	− 2.00000i	− 0.234082i	−0.993127	−	0.117041i	$-0.962659\pi$
$73$	− 2.00000i	− 0.234082i	0.993127	−	0.117041i	$-0.0373409\pi$
$74$	6.00000	0.697486
$75$	0	0
$76$	−4.00000	−0.458831
$77$	0	0
$78$	2.00000i	0.226455i
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	10.0000i	1.10432i
$83$	4.00000i	0.439057i	0.975606	+	0.219529i	$0.0704519\pi$
$83$	4.00000i	0.439057i	−0.975606	+	0.219529i	$0.929548\pi$
$84$	0	0
$85$	0	0
$86$	−8.00000	−0.862662
$87$	− 2.00000i	− 0.214423i
$88$	4.00000i	0.426401i
$89$	14.0000	1.48400	0.741999	−	0.670402i	$-0.233878\pi$
$89$	14.0000	1.48400	0.741999	+	0.670402i	$0.233878\pi$
$90$	0	0
$91$	0	0
$92$	− 4.00000i	− 0.417029i
$93$	− 4.00000i	− 0.414781i
$94$	0	0
$95$	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	$0.830506\pi$
$98$	− 7.00000i	− 0.707107i
$99$	−4.00000	−0.402015
$100$	0	0
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	− 1.00000i	− 0.0990148i
$103$	16.0000i	1.57653i	0.615338	+	0.788263i	$0.289020\pi$
$103$	16.0000i	1.57653i	−0.615338	+	0.788263i	$0.710980\pi$
$104$	−2.00000	−0.196116
$105$	0	0
$106$	6.00000	0.582772
$107$	− 12.0000i	− 1.16008i	−0.814587	−	0.580042i	$-0.803036\pi$
$107$	− 12.0000i	− 1.16008i	0.814587	−	0.580042i	$-0.196964\pi$
$108$	1.00000i	0.0962250i
$109$	14.0000	1.34096	0.670478	−	0.741929i	$-0.266089\pi$
$109$	14.0000	1.34096	0.670478	+	0.741929i	$0.266089\pi$
$110$	0	0
$111$	−6.00000	−0.569495
$112$	0	0
$113$	18.0000i	1.69330i	0.532152	+	0.846649i	$0.321383\pi$
$113$	18.0000i	1.69330i	−0.532152	+	0.846649i	$0.678617\pi$
$114$	4.00000	0.374634
$115$	0	0
$116$	2.00000	0.185695
$117$	− 2.00000i	− 0.184900i
$118$	− 8.00000i	− 0.736460i
$119$	0	0
$120$	0	0
$121$	5.00000	0.454545
$122$	− 10.0000i	− 0.905357i
$123$	− 10.0000i	− 0.901670i
$124$	4.00000	0.359211
$125$	0	0
$126$	0	0
$127$	16.0000i	1.41977i	0.704317	+	0.709885i	$0.251253\pi$
$127$	16.0000i	1.41977i	−0.704317	+	0.709885i	$0.748747\pi$
$128$	1.00000i	0.0883883i
$129$	8.00000	0.704361
$130$	0	0
$131$	4.00000	0.349482	0.174741	−	0.984614i	$-0.444091\pi$
$131$	4.00000	0.349482	0.174741	+	0.984614i	$0.444091\pi$
$132$	− 4.00000i	− 0.348155i
$133$	0	0
$134$	8.00000	0.691095
$135$	0	0
$136$	1.00000	0.0857493
$137$	6.00000i	0.512615i	0.966595	+	0.256307i	$0.0825059\pi$
$137$	6.00000i	0.512615i	−0.966595	+	0.256307i	$0.917494\pi$
$138$	4.00000i	0.340503i
$139$	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
$141$	0	0
$142$	− 8.00000i	− 0.671345i
$143$	8.00000i	0.668994i
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	−2.00000	−0.165521
$147$	7.00000i	0.577350i
$148$	− 6.00000i	− 0.493197i
$149$	−2.00000	−0.163846	−0.0819232	−	0.996639i	$-0.526106\pi$
$149$	−2.00000	−0.163846	−0.0819232	+	0.996639i	$0.526106\pi$
$150$	0	0
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	4.00000i	0.324443i
$153$	1.00000i	0.0808452i
$154$	0	0
$155$	0	0
$156$	2.00000	0.160128
$157$	6.00000i	0.478852i	0.970915	+	0.239426i	$0.0769593\pi$
$157$	6.00000i	0.478852i	−0.970915	+	0.239426i	$0.923041\pi$
$158$	4.00000i	0.318223i
$159$	−6.00000	−0.475831
$160$	0	0
$161$	0	0
$162$	− 1.00000i	− 0.0785674i
$163$	12.0000i	0.939913i	0.882690	+	0.469956i	$0.155730\pi$
$163$	12.0000i	0.939913i	−0.882690	+	0.469956i	$0.844270\pi$
$164$	10.0000	0.780869
$165$	0	0
$166$	4.00000	0.310460
$167$	− 12.0000i	− 0.928588i	−0.885681	−	0.464294i	$-0.846308\pi$
$167$	− 12.0000i	− 0.928588i	0.885681	−	0.464294i	$-0.153692\pi$
$168$	0	0
$169$	9.00000	0.692308
$170$	0	0
$171$	−4.00000	−0.305888
$172$	8.00000i	0.609994i
$173$	2.00000i	0.152057i	0.997106	+	0.0760286i	$0.0242240\pi$
$173$	2.00000i	0.152057i	−0.997106	+	0.0760286i	$0.975776\pi$
$174$	−2.00000	−0.151620
$175$	0	0
$176$	4.00000	0.301511
$177$	8.00000i	0.601317i
$178$	− 14.0000i	− 1.04934i
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	−14.0000	−1.04061	−0.520306	−	0.853980i	$-0.674182\pi$
$181$	−14.0000	−1.04061	−0.520306	+	0.853980i	$0.674182\pi$
$182$	0	0
$183$	10.0000i	0.739221i
$184$	−4.00000	−0.294884
$185$	0	0
$186$	−4.00000	−0.293294
$187$	− 4.00000i	− 0.292509i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	− 1.00000i	− 0.0721688i
$193$	6.00000i	0.431889i	0.976406	+	0.215945i	$0.0692831\pi$
$193$	6.00000i	0.431889i	−0.976406	+	0.215945i	$0.930717\pi$
$194$	10.0000	0.717958
$195$	0	0
$196$	−7.00000	−0.500000
$197$	− 18.0000i	− 1.28245i	−0.767354	−	0.641223i	$-0.778427\pi$
$197$	− 18.0000i	− 1.28245i	0.767354	−	0.641223i	$-0.221573\pi$
$198$	4.00000i	0.284268i
$199$	4.00000	0.283552	0.141776	−	0.989899i	$-0.454719\pi$
$199$	4.00000	0.283552	0.141776	+	0.989899i	$0.454719\pi$
$200$	0	0
$201$	−8.00000	−0.564276
$202$	6.00000i	0.422159i
$203$	0	0
$204$	−1.00000	−0.0700140
$205$	0	0
$206$	16.0000	1.11477
$207$	− 4.00000i	− 0.278019i
$208$	2.00000i	0.138675i
$209$	16.0000	1.10674
$210$	0	0
$211$	12.0000	0.826114	0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000	0.826114	0.413057	+	0.910705i	$0.364461\pi$
$212$	− 6.00000i	− 0.412082i
$213$	8.00000i	0.548151i
$214$	−12.0000	−0.820303
$215$	0	0
$216$	1.00000	0.0680414
$217$	0	0
$218$	− 14.0000i	− 0.948200i
$219$	2.00000	0.135147
$220$	0	0
$221$	2.00000	0.134535
$222$	6.00000i	0.402694i
$223$	24.0000i	1.60716i	0.595198	+	0.803579i	$0.297074\pi$
$223$	24.0000i	1.60716i	−0.595198	+	0.803579i	$0.702926\pi$
$224$	0	0
$225$	0	0
$226$	18.0000	1.19734
$227$	− 12.0000i	− 0.796468i	−0.917284	−	0.398234i	$-0.869623\pi$
$227$	− 12.0000i	− 0.796468i	0.917284	−	0.398234i	$-0.130377\pi$
$228$	− 4.00000i	− 0.264906i
$229$	−6.00000	−0.396491	−0.198246	−	0.980152i	$-0.563524\pi$
$229$	−6.00000	−0.396491	−0.198246	+	0.980152i	$0.563524\pi$
$230$	0	0
$231$	0	0
$232$	− 2.00000i	− 0.131306i
$233$	− 30.0000i	− 1.96537i	−0.185296	−	0.982683i	$-0.559325\pi$
$233$	− 30.0000i	− 1.96537i	0.185296	−	0.982683i	$-0.440675\pi$
$234$	−2.00000	−0.130744
$235$	0	0
$236$	−8.00000	−0.520756
$237$	− 4.00000i	− 0.259828i
$238$	0	0
$239$	−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
$241$	10.0000	0.644157	0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000	0.644157	0.322078	+	0.946713i	$0.395619\pi$
$242$	− 5.00000i	− 0.321412i
$243$	1.00000i	0.0641500i
$244$	−10.0000	−0.640184
$245$	0	0
$246$	−10.0000	−0.637577
$247$	8.00000i	0.509028i
$248$	− 4.00000i	− 0.254000i
$249$	−4.00000	−0.253490
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	16.0000i	1.00591i
$254$	16.0000	1.00393
$255$	0	0
$256$	1.00000	0.0625000
$257$	− 18.0000i	− 1.12281i	−0.827541	−	0.561405i	$-0.810261\pi$
$257$	− 18.0000i	− 1.12281i	0.827541	−	0.561405i	$-0.189739\pi$
$258$	− 8.00000i	− 0.498058i
$259$	0	0
$260$	0	0
$261$	2.00000	0.123797
$262$	− 4.00000i	− 0.247121i
$263$	16.0000i	0.986602i	0.869859	+	0.493301i	$0.164210\pi$
$263$	16.0000i	0.986602i	−0.869859	+	0.493301i	$0.835790\pi$
$264$	−4.00000	−0.246183
$265$	0	0
$266$	0	0
$267$	14.0000i	0.856786i
$268$	− 8.00000i	− 0.488678i
$269$	14.0000	0.853595	0.426798	−	0.904347i	$-0.359642\pi$
$269$	14.0000	0.853595	0.426798	+	0.904347i	$0.359642\pi$
$270$	0	0
$271$	−8.00000	−0.485965	−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000	−0.485965	−0.242983	+	0.970031i	$0.578126\pi$
$272$	− 1.00000i	− 0.0606339i
$273$	0	0
$274$	6.00000	0.362473
$275$	0	0
$276$	4.00000	0.240772
$277$	− 10.0000i	− 0.600842i	−0.953807	−	0.300421i	$-0.902873\pi$
$277$	− 10.0000i	− 0.600842i	0.953807	−	0.300421i	$-0.0971271\pi$
$278$	− 4.00000i	− 0.239904i
$279$	4.00000	0.239474
$280$	0	0
$281$	−30.0000	−1.78965	−0.894825	−	0.446417i	$-0.852700\pi$
$281$	−30.0000	−1.78965	−0.894825	+	0.446417i	$0.852700\pi$
$282$	0	0
$283$	20.0000i	1.18888i	0.804141	+	0.594438i	$0.202626\pi$
$283$	20.0000i	1.18888i	−0.804141	+	0.594438i	$0.797374\pi$
$284$	−8.00000	−0.474713
$285$	0	0
$286$	8.00000	0.473050
$287$	0	0
$288$	1.00000i	0.0589256i
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	−10.0000	−0.586210
$292$	2.00000i	0.117041i
$293$	6.00000i	0.350524i	0.984522	+	0.175262i	$0.0560772\pi$
$293$	6.00000i	0.350524i	−0.984522	+	0.175262i	$0.943923\pi$
$294$	7.00000	0.408248
$295$	0	0
$296$	−6.00000	−0.348743
$297$	− 4.00000i	− 0.232104i
$298$	2.00000i	0.115857i
$299$	−8.00000	−0.462652
$300$	0	0
$301$	0	0
$302$	0	0
$303$	− 6.00000i	− 0.344691i
$304$	4.00000	0.229416
$305$	0	0
$306$	1.00000	0.0571662
$307$	− 16.0000i	− 0.913168i	−0.889680	−	0.456584i	$-0.849073\pi$
$307$	− 16.0000i	− 0.913168i	0.889680	−	0.456584i	$-0.150927\pi$
$308$	0	0
$309$	−16.0000	−0.910208
$310$	0	0
$311$	−8.00000	−0.453638	−0.226819	−	0.973937i	$-0.572833\pi$
$311$	−8.00000	−0.453638	−0.226819	+	0.973937i	$0.572833\pi$
$312$	− 2.00000i	− 0.113228i
$313$	− 2.00000i	− 0.113047i	−0.998401	−	0.0565233i	$-0.981998\pi$
$313$	− 2.00000i	− 0.113047i	0.998401	−	0.0565233i	$-0.0180015\pi$
$314$	6.00000	0.338600
$315$	0	0
$316$	4.00000	0.225018
$317$	− 26.0000i	− 1.46031i	−0.683284	−	0.730153i	$-0.739449\pi$
$317$	− 26.0000i	− 1.46031i	0.683284	−	0.730153i	$-0.260551\pi$
$318$	6.00000i	0.336463i
$319$	−8.00000	−0.447914
$320$	0	0
$321$	12.0000	0.669775
$322$	0	0
$323$	− 4.00000i	− 0.222566i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	12.0000	0.664619
$327$	14.0000i	0.774202i
$328$	− 10.0000i	− 0.552158i
$329$	0	0
$330$	0	0
$331$	20.0000	1.09930	0.549650	−	0.835395i	$-0.314761\pi$
$331$	20.0000	1.09930	0.549650	+	0.835395i	$0.314761\pi$
$332$	− 4.00000i	− 0.219529i
$333$	− 6.00000i	− 0.328798i
$334$	−12.0000	−0.656611
$335$	0	0
$336$	0	0
$337$	− 6.00000i	− 0.326841i	−0.986557	−	0.163420i	$-0.947747\pi$
$337$	− 6.00000i	− 0.326841i	0.986557	−	0.163420i	$-0.0522527\pi$
$338$	− 9.00000i	− 0.489535i
$339$	−18.0000	−0.977626
$340$	0	0
$341$	−16.0000	−0.866449
$342$	4.00000i	0.216295i
$343$	0	0
$344$	8.00000	0.431331
$345$	0	0
$346$	2.00000	0.107521
$347$	− 4.00000i	− 0.214731i	−0.994220	−	0.107366i	$-0.965758\pi$
$347$	− 4.00000i	− 0.214731i	0.994220	−	0.107366i	$-0.0342415\pi$
$348$	2.00000i	0.107211i
$349$	34.0000	1.81998	0.909989	−	0.414632i	$-0.136090\pi$
$349$	34.0000	1.81998	0.909989	+	0.414632i	$0.136090\pi$
$350$	0	0
$351$	2.00000	0.106752
$352$	− 4.00000i	− 0.213201i
$353$	− 30.0000i	− 1.59674i	−0.602168	−	0.798369i	$-0.705696\pi$
$353$	− 30.0000i	− 1.59674i	0.602168	−	0.798369i	$-0.294304\pi$
$354$	8.00000	0.425195
$355$	0	0
$356$	−14.0000	−0.741999
$357$	0	0
$358$	0	0
$359$	−32.0000	−1.68890	−0.844448	−	0.535638i	$-0.820071\pi$
$359$	−32.0000	−1.68890	−0.844448	+	0.535638i	$0.820071\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	14.0000i	0.735824i
$363$	5.00000i	0.262432i
$364$	0	0
$365$	0	0
$366$	10.0000	0.522708
$367$	− 8.00000i	− 0.417597i	−0.977959	−	0.208798i	$-0.933045\pi$
$367$	− 8.00000i	− 0.417597i	0.977959	−	0.208798i	$-0.0669552\pi$
$368$	4.00000i	0.208514i
$369$	10.0000	0.520579
$370$	0	0
$371$	0	0
$372$	4.00000i	0.207390i
$373$	18.0000i	0.932005i	0.884783	+	0.466002i	$0.154306\pi$
$373$	18.0000i	0.932005i	−0.884783	+	0.466002i	$0.845694\pi$
$374$	−4.00000	−0.206835
$375$	0	0
$376$	0	0
$377$	− 4.00000i	− 0.206010i
$378$	0	0
$379$	−20.0000	−1.02733	−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000	−1.02733	−0.513665	+	0.857991i	$0.671713\pi$
$380$	0	0
$381$	−16.0000	−0.819705
$382$	0	0
$383$	− 24.0000i	− 1.22634i	−0.789950	−	0.613171i	$-0.789894\pi$
$383$	− 24.0000i	− 1.22634i	0.789950	−	0.613171i	$-0.210106\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	6.00000	0.305392
$387$	8.00000i	0.406663i
$388$	− 10.0000i	− 0.507673i
$389$	30.0000	1.52106	0.760530	−	0.649303i	$-0.224939\pi$
$389$	30.0000	1.52106	0.760530	+	0.649303i	$0.224939\pi$
$390$	0	0
$391$	4.00000	0.202289
$392$	7.00000i	0.353553i
$393$	4.00000i	0.201773i
$394$	−18.0000	−0.906827
$395$	0	0
$396$	4.00000	0.201008
$397$	− 18.0000i	− 0.903394i	−0.892171	−	0.451697i	$-0.850819\pi$
$397$	− 18.0000i	− 0.903394i	0.892171	−	0.451697i	$-0.149181\pi$
$398$	− 4.00000i	− 0.200502i
$399$	0	0
$400$	0	0
$401$	6.00000	0.299626	0.149813	−	0.988714i	$-0.452133\pi$
$401$	6.00000	0.299626	0.149813	+	0.988714i	$0.452133\pi$
$402$	8.00000i	0.399004i
$403$	− 8.00000i	− 0.398508i
$404$	6.00000	0.298511
$405$	0	0
$406$	0	0
$407$	24.0000i	1.18964i
$408$	1.00000i	0.0495074i
$409$	−10.0000	−0.494468	−0.247234	−	0.968956i	$-0.579522\pi$
$409$	−10.0000	−0.494468	−0.247234	+	0.968956i	$0.579522\pi$
$410$	0	0
$411$	−6.00000	−0.295958
$412$	− 16.0000i	− 0.788263i
$413$	0	0
$414$	−4.00000	−0.196589
$415$	0	0
$416$	2.00000	0.0980581
$417$	4.00000i	0.195881i
$418$	− 16.0000i	− 0.782586i
$419$	−28.0000	−1.36789	−0.683945	−	0.729534i	$-0.739737\pi$
$419$	−28.0000	−1.36789	−0.683945	+	0.729534i	$0.739737\pi$
$420$	0	0
$421$	30.0000	1.46211	0.731055	−	0.682318i	$-0.239028\pi$
$421$	30.0000	1.46211	0.731055	+	0.682318i	$0.239028\pi$
$422$	− 12.0000i	− 0.584151i
$423$	0	0
$424$	−6.00000	−0.291386
$425$	0	0
$426$	8.00000	0.387601
$427$	0	0
$428$	12.0000i	0.580042i
$429$	−8.00000	−0.386244
$430$	0	0
$431$	−16.0000	−0.770693	−0.385346	−	0.922772i	$-0.625918\pi$
$431$	−16.0000	−0.770693	−0.385346	+	0.922772i	$0.625918\pi$
$432$	− 1.00000i	− 0.0481125i
$433$	− 6.00000i	− 0.288342i	−0.989553	−	0.144171i	$-0.953949\pi$
$433$	− 6.00000i	− 0.288342i	0.989553	−	0.144171i	$-0.0460515\pi$
$434$	0	0
$435$	0	0
$436$	−14.0000	−0.670478
$437$	16.0000i	0.765384i
$438$	− 2.00000i	− 0.0955637i
$439$	−20.0000	−0.954548	−0.477274	−	0.878755i	$-0.658375\pi$
$439$	−20.0000	−0.954548	−0.477274	+	0.878755i	$0.658375\pi$
$440$	0	0
$441$	−7.00000	−0.333333
$442$	− 2.00000i	− 0.0951303i
$443$	− 28.0000i	− 1.33032i	−0.746701	−	0.665160i	$-0.768363\pi$
$443$	− 28.0000i	− 1.33032i	0.746701	−	0.665160i	$-0.231637\pi$
$444$	6.00000	0.284747
$445$	0	0
$446$	24.0000	1.13643
$447$	− 2.00000i	− 0.0945968i
$448$	0	0
$449$	26.0000	1.22702	0.613508	−	0.789689i	$-0.289758\pi$
$449$	26.0000	1.22702	0.613508	+	0.789689i	$0.289758\pi$
$450$	0	0
$451$	−40.0000	−1.88353
$452$	− 18.0000i	− 0.846649i
$453$	0	0
$454$	−12.0000	−0.563188
$455$	0	0
$456$	−4.00000	−0.187317
$457$	− 2.00000i	− 0.0935561i	−0.998905	−	0.0467780i	$-0.985105\pi$
$457$	− 2.00000i	− 0.0935561i	0.998905	−	0.0467780i	$-0.0148953\pi$
$458$	6.00000i	0.280362i
$459$	−1.00000	−0.0466760
$460$	0	0
$461$	−38.0000	−1.76984	−0.884918	−	0.465746i	$-0.845786\pi$
$461$	−38.0000	−1.76984	−0.884918	+	0.465746i	$0.845786\pi$
$462$	0	0
$463$	− 24.0000i	− 1.11537i	−0.830051	−	0.557687i	$-0.811689\pi$
$463$	− 24.0000i	− 1.11537i	0.830051	−	0.557687i	$-0.188311\pi$
$464$	−2.00000	−0.0928477
$465$	0	0
$466$	−30.0000	−1.38972
$467$	12.0000i	0.555294i	0.960683	+	0.277647i	$0.0895545\pi$
$467$	12.0000i	0.555294i	−0.960683	+	0.277647i	$0.910445\pi$
$468$	2.00000i	0.0924500i
$469$	0	0
$470$	0	0
$471$	−6.00000	−0.276465
$472$	8.00000i	0.368230i
$473$	− 32.0000i	− 1.47136i
$474$	−4.00000	−0.183726
$475$	0	0
$476$	0	0
$477$	− 6.00000i	− 0.274721i
$478$	8.00000i	0.365911i
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	−12.0000	−0.547153
$482$	− 10.0000i	− 0.455488i
$483$	0	0
$484$	−5.00000	−0.227273
$485$	0	0
$486$	1.00000	0.0453609
$487$	40.0000i	1.81257i	0.422664	+	0.906287i	$0.361095\pi$
$487$	40.0000i	1.81257i	−0.422664	+	0.906287i	$0.638905\pi$
$488$	10.0000i	0.452679i
$489$	−12.0000	−0.542659
$490$	0	0
$491$	24.0000	1.08310	0.541552	−	0.840667i	$-0.317837\pi$
$491$	24.0000	1.08310	0.541552	+	0.840667i	$0.317837\pi$
$492$	10.0000i	0.450835i
$493$	2.00000i	0.0900755i
$494$	8.00000	0.359937
$495$	0	0
$496$	−4.00000	−0.179605
$497$	0	0
$498$	4.00000i	0.179244i
$499$	−28.0000	−1.25345	−0.626726	−	0.779240i	$-0.715605\pi$
$499$	−28.0000	−1.25345	−0.626726	+	0.779240i	$0.715605\pi$
$500$	0	0
$501$	12.0000	0.536120
$502$	0	0
$503$	− 28.0000i	− 1.24846i	−0.781241	−	0.624229i	$-0.785413\pi$
$503$	− 28.0000i	− 1.24846i	0.781241	−	0.624229i	$-0.214587\pi$
$504$	0	0
$505$	0	0
$506$	16.0000	0.711287
$507$	9.00000i	0.399704i
$508$	− 16.0000i	− 0.709885i
$509$	6.00000	0.265945	0.132973	−	0.991120i	$-0.457548\pi$
$509$	6.00000	0.265945	0.132973	+	0.991120i	$0.457548\pi$
$510$	0	0
$511$	0	0
$512$	− 1.00000i	− 0.0441942i
$513$	− 4.00000i	− 0.176604i
$514$	−18.0000	−0.793946
$515$	0	0
$516$	−8.00000	−0.352180
$517$	0	0
$518$	0	0
$519$	−2.00000	−0.0877903
$520$	0	0
$521$	30.0000	1.31432	0.657162	−	0.753749i	$-0.271757\pi$
$521$	30.0000	1.31432	0.657162	+	0.753749i	$0.271757\pi$
$522$	− 2.00000i	− 0.0875376i
$523$	− 40.0000i	− 1.74908i	−0.484955	−	0.874539i	$-0.661164\pi$
$523$	− 40.0000i	− 1.74908i	0.484955	−	0.874539i	$-0.338836\pi$
$524$	−4.00000	−0.174741
$525$	0	0
$526$	16.0000	0.697633
$527$	4.00000i	0.174243i
$528$	4.00000i	0.174078i
$529$	7.00000	0.304348
$530$	0	0
$531$	−8.00000	−0.347170
$532$	0	0
$533$	− 20.0000i	− 0.866296i
$534$	14.0000	0.605839
$535$	0	0
$536$	−8.00000	−0.345547
$537$	0	0
$538$	− 14.0000i	− 0.603583i
$539$	28.0000	1.20605
$540$	0	0
$541$	−22.0000	−0.945854	−0.472927	−	0.881102i	$-0.656803\pi$
$541$	−22.0000	−0.945854	−0.472927	+	0.881102i	$0.656803\pi$
$542$	8.00000i	0.343629i
$543$	− 14.0000i	− 0.600798i
$544$	−1.00000	−0.0428746
$545$	0	0
$546$	0	0
$547$	20.0000i	0.855138i	0.903983	+	0.427569i	$0.140630\pi$
$547$	20.0000i	0.855138i	−0.903983	+	0.427569i	$0.859370\pi$
$548$	− 6.00000i	− 0.256307i
$549$	−10.0000	−0.426790
$550$	0	0
$551$	−8.00000	−0.340811
$552$	− 4.00000i	− 0.170251i
$553$	0	0
$554$	−10.0000	−0.424859
$555$	0	0
$556$	−4.00000	−0.169638
$557$	− 30.0000i	− 1.27114i	−0.772043	−	0.635570i	$-0.780765\pi$
$557$	− 30.0000i	− 1.27114i	0.772043	−	0.635570i	$-0.219235\pi$
$558$	− 4.00000i	− 0.169334i
$559$	16.0000	0.676728
$560$	0	0
$561$	4.00000	0.168880
$562$	30.0000i	1.26547i
$563$	20.0000i	0.842900i	0.906852	+	0.421450i	$0.138479\pi$
$563$	20.0000i	0.842900i	−0.906852	+	0.421450i	$0.861521\pi$
$564$	0	0
$565$	0	0
$566$	20.0000	0.840663
$567$	0	0
$568$	8.00000i	0.335673i
$569$	−26.0000	−1.08998	−0.544988	−	0.838444i	$-0.683466\pi$
$569$	−26.0000	−1.08998	−0.544988	+	0.838444i	$0.683466\pi$
$570$	0	0
$571$	−44.0000	−1.84134	−0.920671	−	0.390339i	$-0.872358\pi$
$571$	−44.0000	−1.84134	−0.920671	+	0.390339i	$0.872358\pi$
$572$	− 8.00000i	− 0.334497i
$573$	0	0
$574$	0	0
$575$	0	0
$576$	1.00000	0.0416667
$577$	6.00000i	0.249783i	0.992170	+	0.124892i	$0.0398583\pi$
$577$	6.00000i	0.249783i	−0.992170	+	0.124892i	$0.960142\pi$
$578$	1.00000i	0.0415945i
$579$	−6.00000	−0.249351
$580$	0	0
$581$	0	0
$582$	10.0000i	0.414513i
$583$	24.0000i	0.993978i
$584$	2.00000	0.0827606
$585$	0	0
$586$	6.00000	0.247858
$587$	12.0000i	0.495293i	0.968850	+	0.247647i	$0.0796572\pi$
$587$	12.0000i	0.495293i	−0.968850	+	0.247647i	$0.920343\pi$
$588$	− 7.00000i	− 0.288675i
$589$	−16.0000	−0.659269
$590$	0	0
$591$	18.0000	0.740421
$592$	6.00000i	0.246598i
$593$	− 30.0000i	− 1.23195i	−0.787765	−	0.615976i	$-0.788762\pi$
$593$	− 30.0000i	− 1.23195i	0.787765	−	0.615976i	$-0.211238\pi$
$594$	−4.00000	−0.164122
$595$	0	0
$596$	2.00000	0.0819232
$597$	4.00000i	0.163709i
$598$	8.00000i	0.327144i
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	34.0000	1.38689	0.693444	−	0.720510i	$-0.256092\pi$
$601$	34.0000	1.38689	0.693444	+	0.720510i	$0.256092\pi$
$602$	0	0
$603$	− 8.00000i	− 0.325785i
$604$	0	0
$605$	0	0
$606$	−6.00000	−0.243733
$607$	32.0000i	1.29884i	0.760430	+	0.649420i	$0.224988\pi$
$607$	32.0000i	1.29884i	−0.760430	+	0.649420i	$0.775012\pi$
$608$	− 4.00000i	− 0.162221i
$609$	0	0
$610$	0	0
$611$	0	0
$612$	− 1.00000i	− 0.0404226i
$613$	10.0000i	0.403896i	0.979396	+	0.201948i	$0.0647272\pi$
$613$	10.0000i	0.403896i	−0.979396	+	0.201948i	$0.935273\pi$
$614$	−16.0000	−0.645707
$615$	0	0
$616$	0	0
$617$	− 2.00000i	− 0.0805170i	−0.999189	−	0.0402585i	$-0.987182\pi$
$617$	− 2.00000i	− 0.0805170i	0.999189	−	0.0402585i	$-0.0128181\pi$
$618$	16.0000i	0.643614i
$619$	−44.0000	−1.76851	−0.884255	−	0.467005i	$-0.845333\pi$
$619$	−44.0000	−1.76851	−0.884255	+	0.467005i	$0.845333\pi$
$620$	0	0
$621$	4.00000	0.160514
$622$	8.00000i	0.320771i
$623$	0	0
$624$	−2.00000	−0.0800641
$625$	0	0
$626$	−2.00000	−0.0799361
$627$	16.0000i	0.638978i
$628$	− 6.00000i	− 0.239426i
$629$	6.00000	0.239236
$630$	0	0
$631$	32.0000	1.27390	0.636950	−	0.770905i	$-0.280196\pi$
$631$	32.0000	1.27390	0.636950	+	0.770905i	$0.280196\pi$
$632$	− 4.00000i	− 0.159111i
$633$	12.0000i	0.476957i
$634$	−26.0000	−1.03259
$635$	0	0
$636$	6.00000	0.237915
$637$	14.0000i	0.554700i
$638$	8.00000i	0.316723i
$639$	−8.00000	−0.316475
$640$	0	0
$641$	−42.0000	−1.65890	−0.829450	−	0.558581i	$-0.811346\pi$
$641$	−42.0000	−1.65890	−0.829450	+	0.558581i	$0.811346\pi$
$642$	− 12.0000i	− 0.473602i
$643$	− 20.0000i	− 0.788723i	−0.918955	−	0.394362i	$-0.870966\pi$
$643$	− 20.0000i	− 0.788723i	0.918955	−	0.394362i	$-0.129034\pi$
$644$	0	0
$645$	0	0
$646$	−4.00000	−0.157378
$647$	32.0000i	1.25805i	0.777385	+	0.629025i	$0.216546\pi$
$647$	32.0000i	1.25805i	−0.777385	+	0.629025i	$0.783454\pi$
$648$	1.00000i	0.0392837i
$649$	32.0000	1.25611
$650$	0	0
$651$	0	0
$652$	− 12.0000i	− 0.469956i
$653$	34.0000i	1.33052i	0.746611	+	0.665261i	$0.231680\pi$
$653$	34.0000i	1.33052i	−0.746611	+	0.665261i	$0.768320\pi$
$654$	14.0000	0.547443
$655$	0	0
$656$	−10.0000	−0.390434
$657$	2.00000i	0.0780274i
$658$	0	0
$659$	−24.0000	−0.934907	−0.467454	−	0.884018i	$-0.654829\pi$
$659$	−24.0000	−0.934907	−0.467454	+	0.884018i	$0.654829\pi$
$660$	0	0
$661$	22.0000	0.855701	0.427850	−	0.903850i	$-0.359271\pi$
$661$	22.0000	0.855701	0.427850	+	0.903850i	$0.359271\pi$
$662$	− 20.0000i	− 0.777322i
$663$	2.00000i	0.0776736i
$664$	−4.00000	−0.155230
$665$	0	0
$666$	−6.00000	−0.232495
$667$	− 8.00000i	− 0.309761i
$668$	12.0000i	0.464294i
$669$	−24.0000	−0.927894
$670$	0	0
$671$	40.0000	1.54418
$672$	0	0
$673$	− 10.0000i	− 0.385472i	−0.981251	−	0.192736i	$-0.938264\pi$
$673$	− 10.0000i	− 0.385472i	0.981251	−	0.192736i	$-0.0617360\pi$
$674$	−6.00000	−0.231111
$675$	0	0
$676$	−9.00000	−0.346154
$677$	− 18.0000i	− 0.691796i	−0.938272	−	0.345898i	$-0.887574\pi$
$677$	− 18.0000i	− 0.691796i	0.938272	−	0.345898i	$-0.112426\pi$
$678$	18.0000i	0.691286i
$679$	0	0
$680$	0	0
$681$	12.0000	0.459841
$682$	16.0000i	0.612672i
$683$	44.0000i	1.68361i	0.539779	+	0.841807i	$0.318508\pi$
$683$	44.0000i	1.68361i	−0.539779	+	0.841807i	$0.681492\pi$
$684$	4.00000	0.152944
$685$	0	0
$686$	0	0
$687$	− 6.00000i	− 0.228914i
$688$	− 8.00000i	− 0.304997i
$689$	−12.0000	−0.457164
$690$	0	0
$691$	−36.0000	−1.36950	−0.684752	−	0.728776i	$-0.740090\pi$
$691$	−36.0000	−1.36950	−0.684752	+	0.728776i	$0.740090\pi$
$692$	− 2.00000i	− 0.0760286i
$693$	0	0
$694$	−4.00000	−0.151838
$695$	0	0
$696$	2.00000	0.0758098
$697$	10.0000i	0.378777i
$698$	− 34.0000i	− 1.28692i
$699$	30.0000	1.13470
$700$	0	0
$701$	10.0000	0.377695	0.188847	−	0.982006i	$-0.439525\pi$
$701$	10.0000	0.377695	0.188847	+	0.982006i	$0.439525\pi$
$702$	− 2.00000i	− 0.0754851i
$703$	24.0000i	0.905177i
$704$	−4.00000	−0.150756
$705$	0	0
$706$	−30.0000	−1.12906
$707$	0	0
$708$	− 8.00000i	− 0.300658i
$709$	−10.0000	−0.375558	−0.187779	−	0.982211i	$-0.560129\pi$
$709$	−10.0000	−0.375558	−0.187779	+	0.982211i	$0.560129\pi$
$710$	0	0
$711$	4.00000	0.150012
$712$	14.0000i	0.524672i
$713$	− 16.0000i	− 0.599205i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 8.00000i	− 0.298765i
$718$	32.0000i	1.19423i
$719$	48.0000	1.79010	0.895049	−	0.445968i	$-0.147140\pi$
$719$	48.0000	1.79010	0.895049	+	0.445968i	$0.147140\pi$
$720$	0	0
$721$	0	0
$722$	3.00000i	0.111648i
$723$	10.0000i	0.371904i
$724$	14.0000	0.520306
$725$	0	0
$726$	5.00000	0.185567
$727$	− 40.0000i	− 1.48352i	−0.670667	−	0.741759i	$-0.733992\pi$
$727$	− 40.0000i	− 1.48352i	0.670667	−	0.741759i	$-0.266008\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−8.00000	−0.295891
$732$	− 10.0000i	− 0.369611i
$733$	26.0000i	0.960332i	0.877178	+	0.480166i	$0.159424\pi$
$733$	26.0000i	0.960332i	−0.877178	+	0.480166i	$0.840576\pi$
$734$	−8.00000	−0.295285
$735$	0	0
$736$	4.00000	0.147442
$737$	32.0000i	1.17874i
$738$	− 10.0000i	− 0.368105i
$739$	44.0000	1.61857	0.809283	−	0.587419i	$-0.199856\pi$
$739$	44.0000	1.61857	0.809283	+	0.587419i	$0.199856\pi$
$740$	0	0
$741$	−8.00000	−0.293887
$742$	0	0
$743$	28.0000i	1.02722i	0.858024	+	0.513610i	$0.171692\pi$
$743$	28.0000i	1.02722i	−0.858024	+	0.513610i	$0.828308\pi$
$744$	4.00000	0.146647
$745$	0	0
$746$	18.0000	0.659027
$747$	− 4.00000i	− 0.146352i
$748$	4.00000i	0.146254i
$749$	0	0
$750$	0	0
$751$	−4.00000	−0.145962	−0.0729810	−	0.997333i	$-0.523251\pi$
$751$	−4.00000	−0.145962	−0.0729810	+	0.997333i	$0.523251\pi$
$752$	0	0
$753$	0	0
$754$	−4.00000	−0.145671
$755$	0	0
$756$	0	0
$757$	14.0000i	0.508839i	0.967094	+	0.254419i	$0.0818843\pi$
$757$	14.0000i	0.508839i	−0.967094	+	0.254419i	$0.918116\pi$
$758$	20.0000i	0.726433i
$759$	−16.0000	−0.580763
$760$	0	0
$761$	42.0000	1.52250	0.761249	−	0.648459i	$-0.224586\pi$
$761$	42.0000	1.52250	0.761249	+	0.648459i	$0.224586\pi$
$762$	16.0000i	0.579619i
$763$	0	0
$764$	0	0
$765$	0	0
$766$	−24.0000	−0.867155
$767$	16.0000i	0.577727i
$768$	1.00000i	0.0360844i
$769$	−2.00000	−0.0721218	−0.0360609	−	0.999350i	$-0.511481\pi$
$769$	−2.00000	−0.0721218	−0.0360609	+	0.999350i	$0.511481\pi$
$770$	0	0
$771$	18.0000	0.648254
$772$	− 6.00000i	− 0.215945i
$773$	14.0000i	0.503545i	0.967786	+	0.251773i	$0.0810135\pi$
$773$	14.0000i	0.503545i	−0.967786	+	0.251773i	$0.918987\pi$
$774$	8.00000	0.287554
$775$	0	0
$776$	−10.0000	−0.358979
$777$	0	0
$778$	− 30.0000i	− 1.07555i
$779$	−40.0000	−1.43315
$780$	0	0
$781$	32.0000	1.14505
$782$	− 4.00000i	− 0.143040i
$783$	2.00000i	0.0714742i
$784$	7.00000	0.250000
$785$	0	0
$786$	4.00000	0.142675
$787$	− 4.00000i	− 0.142585i	−0.997455	−	0.0712923i	$-0.977288\pi$
$787$	− 4.00000i	− 0.142585i	0.997455	−	0.0712923i	$-0.0227123\pi$
$788$	18.0000i	0.641223i
$789$	−16.0000	−0.569615
$790$	0	0
$791$	0	0
$792$	− 4.00000i	− 0.142134i
$793$	20.0000i	0.710221i
$794$	−18.0000	−0.638796
$795$	0	0
$796$	−4.00000	−0.141776
$797$	− 6.00000i	− 0.212531i	−0.994338	−	0.106265i	$-0.966111\pi$
$797$	− 6.00000i	− 0.212531i	0.994338	−	0.106265i	$-0.0338893\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−14.0000	−0.494666
$802$	− 6.00000i	− 0.211867i
$803$	− 8.00000i	− 0.282314i
$804$	8.00000	0.282138
$805$	0	0
$806$	−8.00000	−0.281788
$807$	14.0000i	0.492823i
$808$	− 6.00000i	− 0.211079i
$809$	−54.0000	−1.89854	−0.949269	−	0.314464i	$-0.898175\pi$
$809$	−54.0000	−1.89854	−0.949269	+	0.314464i	$0.898175\pi$
$810$	0	0
$811$	−28.0000	−0.983213	−0.491606	−	0.870817i	$-0.663590\pi$
$811$	−28.0000	−0.983213	−0.491606	+	0.870817i	$0.663590\pi$
$812$	0	0
$813$	− 8.00000i	− 0.280572i
$814$	24.0000	0.841200
$815$	0	0
$816$	1.00000	0.0350070
$817$	− 32.0000i	− 1.11954i
$818$	10.0000i	0.349642i
$819$	0	0
$820$	0	0
$821$	−6.00000	−0.209401	−0.104701	−	0.994504i	$-0.533388\pi$
$821$	−6.00000	−0.209401	−0.104701	+	0.994504i	$0.533388\pi$
$822$	6.00000i	0.209274i
$823$	16.0000i	0.557725i	0.960331	+	0.278862i	$0.0899574\pi$
$823$	16.0000i	0.557725i	−0.960331	+	0.278862i	$0.910043\pi$
$824$	−16.0000	−0.557386
$825$	0	0
$826$	0	0
$827$	− 36.0000i	− 1.25184i	−0.779886	−	0.625921i	$-0.784723\pi$
$827$	− 36.0000i	− 1.25184i	0.779886	−	0.625921i	$-0.215277\pi$
$828$	4.00000i	0.139010i
$829$	−54.0000	−1.87550	−0.937749	−	0.347314i	$-0.887094\pi$
$829$	−54.0000	−1.87550	−0.937749	+	0.347314i	$0.887094\pi$
$830$	0	0
$831$	10.0000	0.346896
$832$	− 2.00000i	− 0.0693375i
$833$	− 7.00000i	− 0.242536i
$834$	4.00000	0.138509
$835$	0	0
$836$	−16.0000	−0.553372
$837$	4.00000i	0.138260i
$838$	28.0000i	0.967244i
$839$	−16.0000	−0.552381	−0.276191	−	0.961103i	$-0.589072\pi$
$839$	−16.0000	−0.552381	−0.276191	+	0.961103i	$0.589072\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	− 30.0000i	− 1.03387i
$843$	− 30.0000i	− 1.03325i
$844$	−12.0000	−0.413057
$845$	0	0
$846$	0	0
$847$	0	0
$848$	6.00000i	0.206041i
$849$	−20.0000	−0.686398
$850$	0	0
$851$	−24.0000	−0.822709
$852$	− 8.00000i	− 0.274075i
$853$	− 54.0000i	− 1.84892i	−0.381273	−	0.924462i	$-0.624514\pi$
$853$	− 54.0000i	− 1.84892i	0.381273	−	0.924462i	$-0.375486\pi$
$854$	0	0
$855$	0	0
$856$	12.0000	0.410152
$857$	− 26.0000i	− 0.888143i	−0.895991	−	0.444072i	$-0.853534\pi$
$857$	− 26.0000i	− 0.888143i	0.895991	−	0.444072i	$-0.146466\pi$
$858$	8.00000i	0.273115i
$859$	−20.0000	−0.682391	−0.341196	−	0.939992i	$-0.610832\pi$
$859$	−20.0000	−0.682391	−0.341196	+	0.939992i	$0.610832\pi$
$860$	0	0
$861$	0	0
$862$	16.0000i	0.544962i
$863$	− 8.00000i	− 0.272323i	−0.990687	−	0.136162i	$-0.956523\pi$
$863$	− 8.00000i	− 0.272323i	0.990687	−	0.136162i	$-0.0434766\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	−6.00000	−0.203888
$867$	− 1.00000i	− 0.0339618i
$868$	0	0
$869$	−16.0000	−0.542763
$870$	0	0
$871$	−16.0000	−0.542139
$872$	14.0000i	0.474100i
$873$	− 10.0000i	− 0.338449i
$874$	16.0000	0.541208
$875$	0	0
$876$	−2.00000	−0.0675737
$877$	− 50.0000i	− 1.68838i	−0.536044	−	0.844190i	$-0.680082\pi$
$877$	− 50.0000i	− 1.68838i	0.536044	−	0.844190i	$-0.319918\pi$
$878$	20.0000i	0.674967i
$879$	−6.00000	−0.202375
$880$	0	0
$881$	−2.00000	−0.0673817	−0.0336909	−	0.999432i	$-0.510726\pi$
$881$	−2.00000	−0.0673817	−0.0336909	+	0.999432i	$0.510726\pi$
$882$	7.00000i	0.235702i
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	−2.00000	−0.0672673
$885$	0	0
$886$	−28.0000	−0.940678
$887$	− 12.0000i	− 0.402921i	−0.979497	−	0.201460i	$-0.935431\pi$
$887$	− 12.0000i	− 0.402921i	0.979497	−	0.201460i	$-0.0645687\pi$
$888$	− 6.00000i	− 0.201347i
$889$	0	0
$890$	0	0
$891$	4.00000	0.134005
$892$	− 24.0000i	− 0.803579i
$893$	0	0
$894$	−2.00000	−0.0668900
$895$	0	0
$896$	0	0
$897$	− 8.00000i	− 0.267112i
$898$	− 26.0000i	− 0.867631i
$899$	8.00000	0.266815
$900$	0	0
$901$	6.00000	0.199889
$902$	40.0000i	1.33185i
$903$	0	0
$904$	−18.0000	−0.598671
$905$	0	0
$906$	0	0
$907$	36.0000i	1.19536i	0.801735	+	0.597680i	$0.203911\pi$
$907$	36.0000i	1.19536i	−0.801735	+	0.597680i	$0.796089\pi$
$908$	12.0000i	0.398234i
$909$	6.00000	0.199007
$910$	0	0
$911$	48.0000	1.59031	0.795155	−	0.606406i	$-0.207389\pi$
$911$	48.0000	1.59031	0.795155	+	0.606406i	$0.207389\pi$
$912$	4.00000i	0.132453i
$913$	16.0000i	0.529523i
$914$	−2.00000	−0.0661541
$915$	0	0
$916$	6.00000	0.198246
$917$	0	0
$918$	1.00000i	0.0330049i
$919$	−16.0000	−0.527791	−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000	−0.527791	−0.263896	+	0.964551i	$0.585007\pi$
$920$	0	0
$921$	16.0000	0.527218
$922$	38.0000i	1.25146i
$923$	16.0000i	0.526646i
$924$	0	0
$925$	0	0
$926$	−24.0000	−0.788689
$927$	− 16.0000i	− 0.525509i
$928$	2.00000i	0.0656532i
$929$	50.0000	1.64045	0.820223	−	0.572043i	$-0.193849\pi$
$929$	50.0000	1.64045	0.820223	+	0.572043i	$0.193849\pi$
$930$	0	0
$931$	28.0000	0.917663
$932$	30.0000i	0.982683i
$933$	− 8.00000i	− 0.261908i
$934$	12.0000	0.392652
$935$	0	0
$936$	2.00000	0.0653720
$937$	− 2.00000i	− 0.0653372i	−0.999466	−	0.0326686i	$-0.989599\pi$
$937$	− 2.00000i	− 0.0653372i	0.999466	−	0.0326686i	$-0.0104006\pi$
$938$	0	0
$939$	2.00000	0.0652675
$940$	0	0
$941$	−30.0000	−0.977972	−0.488986	−	0.872292i	$-0.662633\pi$
$941$	−30.0000	−0.977972	−0.488986	+	0.872292i	$0.662633\pi$
$942$	6.00000i	0.195491i
$943$	− 40.0000i	− 1.30258i
$944$	8.00000	0.260378
$945$	0	0
$946$	−32.0000	−1.04041
$947$	28.0000i	0.909878i	0.890523	+	0.454939i	$0.150339\pi$
$947$	28.0000i	0.909878i	−0.890523	+	0.454939i	$0.849661\pi$
$948$	4.00000i	0.129914i
$949$	4.00000	0.129845
$950$	0	0
$951$	26.0000	0.843108
$952$	0	0
$953$	− 6.00000i	− 0.194359i	−0.995267	−	0.0971795i	$-0.969018\pi$
$953$	− 6.00000i	− 0.194359i	0.995267	−	0.0971795i	$-0.0309821\pi$
$954$	−6.00000	−0.194257
$955$	0	0
$956$	8.00000	0.258738
$957$	− 8.00000i	− 0.258603i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−15.0000	−0.483871
$962$	12.0000i	0.386896i
$963$	12.0000i	0.386695i
$964$	−10.0000	−0.322078
$965$	0	0
$966$	0	0
$967$	− 24.0000i	− 0.771788i	−0.922543	−	0.385894i	$-0.873893\pi$
$967$	− 24.0000i	− 0.771788i	0.922543	−	0.385894i	$-0.126107\pi$
$968$	5.00000i	0.160706i
$969$	4.00000	0.128499
$970$	0	0
$971$	32.0000	1.02693	0.513464	−	0.858111i	$-0.328362\pi$
$971$	32.0000	1.02693	0.513464	+	0.858111i	$0.328362\pi$
$972$	− 1.00000i	− 0.0320750i
$973$	0	0
$974$	40.0000	1.28168
$975$	0	0
$976$	10.0000	0.320092
$977$	− 18.0000i	− 0.575871i	−0.957650	−	0.287936i	$-0.907031\pi$
$977$	− 18.0000i	− 0.575871i	0.957650	−	0.287936i	$-0.0929689\pi$
$978$	12.0000i	0.383718i
$979$	56.0000	1.78977
$980$	0	0
$981$	−14.0000	−0.446986
$982$	− 24.0000i	− 0.765871i
$983$	− 44.0000i	− 1.40338i	−0.712481	−	0.701691i	$-0.752429\pi$
$983$	− 44.0000i	− 1.40338i	0.712481	−	0.701691i	$-0.247571\pi$
$984$	10.0000	0.318788
$985$	0	0
$986$	2.00000	0.0636930
$987$	0	0
$988$	− 8.00000i	− 0.254514i
$989$	32.0000	1.01754
$990$	0	0
$991$	−44.0000	−1.39771	−0.698853	−	0.715265i	$-0.746306\pi$
$991$	−44.0000	−1.39771	−0.698853	+	0.715265i	$0.746306\pi$
$992$	4.00000i	0.127000i
$993$	20.0000i	0.634681i
$994$	0	0
$995$	0	0
$996$	4.00000	0.126745
$997$	− 10.0000i	− 0.316703i	−0.987383	−	0.158352i	$-0.949382\pi$
$997$	− 10.0000i	− 0.316703i	0.987383	−	0.158352i	$-0.0506179\pi$
$998$	28.0000i	0.886325i
$999$	6.00000	0.189832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2550.2.d.s.2449.1		2
5.2	odd	4		510.2.a.f.1.1	✓	1
5.3	odd	4		2550.2.a.d.1.1		1
5.4	even	2	inner	2550.2.d.s.2449.2		2
15.2	even	4		1530.2.a.f.1.1		1
15.8	even	4		7650.2.a.bw.1.1		1
20.7	even	4		4080.2.a.c.1.1		1
85.67	odd	4		8670.2.a.r.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
510.2.a.f.1.1	✓	1	5.2	odd	4
1530.2.a.f.1.1		1	15.2	even	4
2550.2.a.d.1.1		1	5.3	odd	4
2550.2.d.s.2449.1		2	1.1	even	1	trivial
2550.2.d.s.2449.2		2	5.4	even	2	inner
4080.2.a.c.1.1		1	20.7	even	4
7650.2.a.bw.1.1		1	15.8	even	4
8670.2.a.r.1.1		1	85.67	odd	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)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} +4.00000 q^{11} -1.00000i q^{12} +2.00000i q^{13} +1.00000 q^{16} -1.00000i q^{17} +1.00000i q^{18} +4.00000 q^{19} -4.00000i q^{22} +4.00000i q^{23} -1.00000 q^{24} +2.00000 q^{26} -1.00000i q^{27} -2.00000 q^{29} -4.00000 q^{31} -1.00000i q^{32} +4.00000i q^{33} -1.00000 q^{34} +1.00000 q^{36} +6.00000i q^{37} -4.00000i q^{38} -2.00000 q^{39} -10.0000 q^{41} -8.00000i q^{43} -4.00000 q^{44} +4.00000 q^{46} +1.00000i q^{48} +7.00000 q^{49} +1.00000 q^{51} -2.00000i q^{52} +6.00000i q^{53} -1.00000 q^{54} +4.00000i q^{57} +2.00000i q^{58} +8.00000 q^{59} +10.0000 q^{61} +4.00000i q^{62} -1.00000 q^{64} +4.00000 q^{66} +8.00000i q^{67} +1.00000i q^{68} -4.00000 q^{69} +8.00000 q^{71} -1.00000i q^{72} -2.00000i q^{73} +6.00000 q^{74} -4.00000 q^{76} +2.00000i q^{78} -4.00000 q^{79} +1.00000 q^{81} +10.0000i q^{82} +4.00000i q^{83} -8.00000 q^{86} -2.00000i q^{87} +4.00000i q^{88} +14.0000 q^{89} -4.00000i q^{92} -4.00000i q^{93} +1.00000 q^{96} +10.0000i q^{97} -7.00000i q^{98} -4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9	\( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} + 8 q^{11} + 2 q^{16} + 8 q^{19} - 2 q^{24} + 4 q^{26} - 4 q^{29} - 8 q^{31} - 2 q^{34} + 2 q^{36} - 4 q^{39} - 20 q^{41} - 8 q^{44} + 8 q^{46} + 14 q^{49} + 2 q^{51} - 2 q^{54} + 16 q^{59} + 20 q^{61} - 2 q^{64} + 8 q^{66} - 8 q^{69} + 16 q^{71} + 12 q^{74} - 8 q^{76} - 8 q^{79} + 2 q^{81} - 16 q^{86} + 28 q^{89} + 2 q^{96} - 8 q^{99}+O(q^{100}) \) 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9 + 8 * q^11 + 2 * q^16 + 8 * q^19 - 2 * q^24 + 4 * q^26 - 4 * q^29 - 8 * q^31 - 2 * q^34 + 2 * q^36 - 4 * q^39 - 20 * q^41 - 8 * q^44 + 8 * q^46 + 14 * q^49 + 2 * q^51 - 2 * q^54 + 16 * q^59 + 20 * q^61 - 2 * q^64 + 8 * q^66 - 8 * q^69 + 16 * q^71 + 12 * q^74 - 8 * q^76 - 8 * q^79 + 2 * q^81 - 16 * q^86 + 28 * q^89 + 2 * q^96 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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