LMFDB - Embedded newform 2550.2.d.u.2449.4

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:	$4$
Coefficient field:	$\Q(i, \sqrt{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9 $ x^4 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 510)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2449.4
Root			$-1.22474 + 1.22474i$ of defining polynomial
Character	$\chi$	$=$	2550.2449
Dual form			2550.2.d.u.2449.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +4.89898i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +4.89898i q^{7} -1.00000i q^{8} -1.00000 q^{9} +1.00000i q^{12} +6.89898i q^{13} -4.89898 q^{14} +1.00000 q^{16} +1.00000i q^{17} -1.00000i q^{18} -4.00000 q^{19} +4.89898 q^{21} -4.00000i q^{23} -1.00000 q^{24} -6.89898 q^{26} +1.00000i q^{27} -4.89898i q^{28} -6.00000 q^{29} +4.00000 q^{31} +1.00000i q^{32} -1.00000 q^{34} +1.00000 q^{36} -6.00000i q^{37} -4.00000i q^{38} +6.89898 q^{39} -2.89898 q^{41} +4.89898i q^{42} +8.89898i q^{43} +4.00000 q^{46} -9.79796i q^{47} -1.00000i q^{48} -17.0000 q^{49} +1.00000 q^{51} -6.89898i q^{52} -7.79796i q^{53} -1.00000 q^{54} +4.89898 q^{56} +4.00000i q^{57} -6.00000i q^{58} -4.89898 q^{59} +11.7980 q^{61} +4.00000i q^{62} -4.89898i q^{63} -1.00000 q^{64} -0.898979i q^{67} -1.00000i q^{68} -4.00000 q^{69} -8.89898 q^{71} +1.00000i q^{72} -10.8990i q^{73} +6.00000 q^{74} +4.00000 q^{76} +6.89898i q^{78} +5.79796 q^{79} +1.00000 q^{81} -2.89898i q^{82} +13.7980i q^{83} -4.89898 q^{84} -8.89898 q^{86} +6.00000i q^{87} +7.79796 q^{89} -33.7980 q^{91} +4.00000i q^{92} -4.00000i q^{93} +9.79796 q^{94} +1.00000 q^{96} +12.6969i q^{97} -17.0000i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4} + 4 q^{6} - 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^4 + 4 * q^6 - 4 * q^9	$ 4 q - 4 q^{4} + 4 q^{6} - 4 q^{9} + 4 q^{16} - 16 q^{19} - 4 q^{24} - 8 q^{26} - 24 q^{29} + 16 q^{31} - 4 q^{34} + 4 q^{36} + 8 q^{39} + 8 q^{41} + 16 q^{46} - 68 q^{49} + 4 q^{51} - 4 q^{54} + 8 q^{61} - 4 q^{64} - 16 q^{69} - 16 q^{71} + 24 q^{74} + 16 q^{76} - 16 q^{79} + 4 q^{81} - 16 q^{86} - 8 q^{89} - 96 q^{91} + 4 q^{96}+O(q^{100}) $ 4 * q - 4 * q^4 + 4 * q^6 - 4 * q^9 + 4 * q^16 - 16 * q^19 - 4 * q^24 - 8 * q^26 - 24 * q^29 + 16 * q^31 - 4 * q^34 + 4 * q^36 + 8 * q^39 + 8 * q^41 + 16 * q^46 - 68 * q^49 + 4 * q^51 - 4 * q^54 + 8 * q^61 - 4 * q^64 - 16 * q^69 - 16 * q^71 + 24 * q^74 + 16 * q^76 - 16 * q^79 + 4 * q^81 - 16 * q^86 - 8 * q^89 - 96 * q^91 + 4 * q^96

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
$7$	4.89898i	1.85164i	0.377964	+	0.925820i	$0.376624\pi$
$7$	4.89898i	1.85164i	−0.377964	+	0.925820i	$0.623376\pi$
$8$	− 1.00000i	− 0.353553i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	1.00000i	0.288675i
$13$	6.89898i	1.91343i	0.291022	+	0.956716i	$0.406005\pi$
$13$	6.89898i	1.91343i	−0.291022	+	0.956716i	$0.593995\pi$
$14$	−4.89898	−1.30931
$15$	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	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	0	0
$21$	4.89898	1.06904
$22$	0	0
$23$	− 4.00000i	− 0.834058i	−0.908893	−	0.417029i	$-0.863071\pi$
$23$	− 4.00000i	− 0.834058i	0.908893	−	0.417029i	$-0.136929\pi$
$24$	−1.00000	−0.204124
$25$	0	0
$26$	−6.89898	−1.35300
$27$	1.00000i	0.192450i
$28$	− 4.89898i	− 0.925820i
$29$	−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
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	1.00000i	0.176777i
$33$	0	0
$34$	−1.00000	−0.171499
$35$	0	0
$36$	1.00000	0.166667
$37$	− 6.00000i	− 0.986394i	−0.869918	−	0.493197i	$-0.835828\pi$
$37$	− 6.00000i	− 0.986394i	0.869918	−	0.493197i	$-0.164172\pi$
$38$	− 4.00000i	− 0.648886i
$39$	6.89898	1.10472
$40$	0	0
$41$	−2.89898	−0.452745	−0.226372	−	0.974041i	$-0.572687\pi$
$41$	−2.89898	−0.452745	−0.226372	+	0.974041i	$0.572687\pi$
$42$	4.89898i	0.755929i
$43$	8.89898i	1.35708i	0.734563	+	0.678541i	$0.237387\pi$
$43$	8.89898i	1.35708i	−0.734563	+	0.678541i	$0.762613\pi$
$44$	0	0
$45$	0	0
$46$	4.00000	0.589768
$47$	− 9.79796i	− 1.42918i	−0.699544	−	0.714590i	$-0.746613\pi$
$47$	− 9.79796i	− 1.42918i	0.699544	−	0.714590i	$-0.253387\pi$
$48$	− 1.00000i	− 0.144338i
$49$	−17.0000	−2.42857
$50$	0	0
$51$	1.00000	0.140028
$52$	− 6.89898i	− 0.956716i
$53$	− 7.79796i	− 1.07113i	−0.844493	−	0.535566i	$-0.820098\pi$
$53$	− 7.79796i	− 1.07113i	0.844493	−	0.535566i	$-0.179902\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	4.89898	0.654654
$57$	4.00000i	0.529813i
$58$	− 6.00000i	− 0.787839i
$59$	−4.89898	−0.637793	−0.318896	−	0.947790i	$-0.603312\pi$
$59$	−4.89898	−0.637793	−0.318896	+	0.947790i	$0.603312\pi$
$60$	0	0
$61$	11.7980	1.51057	0.755287	−	0.655394i	$-0.227498\pi$
$61$	11.7980	1.51057	0.755287	+	0.655394i	$0.227498\pi$
$62$	4.00000i	0.508001i
$63$	− 4.89898i	− 0.617213i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	− 0.898979i	− 0.109828i	−0.998491	−	0.0549139i	$-0.982512\pi$
$67$	− 0.898979i	− 0.109828i	0.998491	−	0.0549139i	$-0.0174884\pi$
$68$	− 1.00000i	− 0.121268i
$69$	−4.00000	−0.481543
$70$	0	0
$71$	−8.89898	−1.05611	−0.528057	−	0.849209i	$-0.677079\pi$
$71$	−8.89898	−1.05611	−0.528057	+	0.849209i	$0.677079\pi$
$72$	1.00000i	0.117851i
$73$	− 10.8990i	− 1.27563i	−0.770190	−	0.637815i	$-0.779839\pi$
$73$	− 10.8990i	− 1.27563i	0.770190	−	0.637815i	$-0.220161\pi$
$74$	6.00000	0.697486
$75$	0	0
$76$	4.00000	0.458831
$77$	0	0
$78$	6.89898i	0.781156i
$79$	5.79796	0.652321	0.326161	−	0.945314i	$-0.394245\pi$
$79$	5.79796	0.652321	0.326161	+	0.945314i	$0.394245\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	− 2.89898i	− 0.320139i
$83$	13.7980i	1.51452i	0.653112	+	0.757261i	$0.273463\pi$
$83$	13.7980i	1.51452i	−0.653112	+	0.757261i	$0.726537\pi$
$84$	−4.89898	−0.534522
$85$	0	0
$86$	−8.89898	−0.959602
$87$	6.00000i	0.643268i
$88$	0	0
$89$	7.79796	0.826582	0.413291	−	0.910599i	$-0.364379\pi$
$89$	7.79796	0.826582	0.413291	+	0.910599i	$0.364379\pi$
$90$	0	0
$91$	−33.7980	−3.54299
$92$	4.00000i	0.417029i
$93$	− 4.00000i	− 0.414781i
$94$	9.79796	1.01058
$95$	0	0
$96$	1.00000	0.102062
$97$	12.6969i	1.28918i	0.764529	+	0.644589i	$0.222972\pi$
$97$	12.6969i	1.28918i	−0.764529	+	0.644589i	$0.777028\pi$
$98$	− 17.0000i	− 1.71726i
$99$	0	0
$100$	0	0
$101$	−18.8990	−1.88052	−0.940259	−	0.340459i	$-0.889418\pi$
$101$	−18.8990	−1.88052	−0.940259	+	0.340459i	$0.889418\pi$
$102$	1.00000i	0.0990148i
$103$	4.00000i	0.394132i	0.980390	+	0.197066i	$0.0631413\pi$
$103$	4.00000i	0.394132i	−0.980390	+	0.197066i	$0.936859\pi$
$104$	6.89898	0.676501
$105$	0	0
$106$	7.79796	0.757405
$107$	− 5.79796i	− 0.560510i	−0.959926	−	0.280255i	$-0.909581\pi$
$107$	− 5.79796i	− 0.560510i	0.959926	−	0.280255i	$-0.0904190\pi$
$108$	− 1.00000i	− 0.0962250i
$109$	−11.7980	−1.13004	−0.565020	−	0.825077i	$-0.691131\pi$
$109$	−11.7980	−1.13004	−0.565020	+	0.825077i	$0.691131\pi$
$110$	0	0
$111$	−6.00000	−0.569495
$112$	4.89898i	0.462910i
$113$	7.79796i	0.733570i	0.930306	+	0.366785i	$0.119542\pi$
$113$	7.79796i	0.733570i	−0.930306	+	0.366785i	$0.880458\pi$
$114$	−4.00000	−0.374634
$115$	0	0
$116$	6.00000	0.557086
$117$	− 6.89898i	− 0.637811i
$118$	− 4.89898i	− 0.450988i
$119$	−4.89898	−0.449089
$120$	0	0
$121$	−11.0000	−1.00000
$122$	11.7980i	1.06814i
$123$	2.89898i	0.261392i
$124$	−4.00000	−0.359211
$125$	0	0
$126$	4.89898	0.436436
$127$	12.0000i	1.06483i	0.846484	+	0.532414i	$0.178715\pi$
$127$	12.0000i	1.06483i	−0.846484	+	0.532414i	$0.821285\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	8.89898	0.783511
$130$	0	0
$131$	9.79796	0.856052	0.428026	−	0.903767i	$-0.359209\pi$
$131$	9.79796	0.856052	0.428026	+	0.903767i	$0.359209\pi$
$132$	0	0
$133$	− 19.5959i	− 1.69918i
$134$	0.898979	0.0776600
$135$	0	0
$136$	1.00000	0.0857493
$137$	− 14.0000i	− 1.19610i	−0.801459	−	0.598050i	$-0.795942\pi$
$137$	− 14.0000i	− 1.19610i	0.801459	−	0.598050i	$-0.204058\pi$
$138$	− 4.00000i	− 0.340503i
$139$	−13.7980	−1.17033	−0.585164	−	0.810915i	$-0.698970\pi$
$139$	−13.7980	−1.17033	−0.585164	+	0.810915i	$0.698970\pi$
$140$	0	0
$141$	−9.79796	−0.825137
$142$	− 8.89898i	− 0.746786i
$143$	0	0
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	10.8990	0.902006
$147$	17.0000i	1.40214i
$148$	6.00000i	0.493197i
$149$	18.8990	1.54826	0.774132	−	0.633024i	$-0.218186\pi$
$149$	18.8990	1.54826	0.774132	+	0.633024i	$0.218186\pi$
$150$	0	0
$151$	−9.79796	−0.797347	−0.398673	−	0.917093i	$-0.630529\pi$
$151$	−9.79796	−0.797347	−0.398673	+	0.917093i	$0.630529\pi$
$152$	4.00000i	0.324443i
$153$	− 1.00000i	− 0.0808452i
$154$	0	0
$155$	0	0
$156$	−6.89898	−0.552360
$157$	1.10102i	0.0878710i	0.999034	+	0.0439355i	$0.0139896\pi$
$157$	1.10102i	0.0878710i	−0.999034	+	0.0439355i	$0.986010\pi$
$158$	5.79796i	0.461261i
$159$	−7.79796	−0.618418
$160$	0	0
$161$	19.5959	1.54437
$162$	1.00000i	0.0785674i
$163$	− 2.20204i	− 0.172477i	−0.996275	−	0.0862386i	$-0.972515\pi$
$163$	− 2.20204i	− 0.172477i	0.996275	−	0.0862386i	$-0.0274847\pi$
$164$	2.89898	0.226372
$165$	0	0
$166$	−13.7980	−1.07093
$167$	− 2.20204i	− 0.170399i	−0.996364	−	0.0851995i	$-0.972847\pi$
$167$	− 2.20204i	− 0.170399i	0.996364	−	0.0851995i	$-0.0271528\pi$
$168$	− 4.89898i	− 0.377964i
$169$	−34.5959	−2.66122
$170$	0	0
$171$	4.00000	0.305888
$172$	− 8.89898i	− 0.678541i
$173$	6.00000i	0.456172i	0.973641	+	0.228086i	$0.0732467\pi$
$173$	6.00000i	0.456172i	−0.973641	+	0.228086i	$0.926753\pi$
$174$	−6.00000	−0.454859
$175$	0	0
$176$	0	0
$177$	4.89898i	0.368230i
$178$	7.79796i	0.584482i
$179$	4.89898	0.366167	0.183083	−	0.983097i	$-0.441392\pi$
$179$	4.89898	0.366167	0.183083	+	0.983097i	$0.441392\pi$
$180$	0	0
$181$	−4.20204	−0.312335	−0.156168	−	0.987731i	$-0.549914\pi$
$181$	−4.20204	−0.312335	−0.156168	+	0.987731i	$0.549914\pi$
$182$	− 33.7980i	− 2.50527i
$183$	− 11.7980i	− 0.872130i
$184$	−4.00000	−0.294884
$185$	0	0
$186$	4.00000	0.293294
$187$	0	0
$188$	9.79796i	0.714590i
$189$	−4.89898	−0.356348
$190$	0	0
$191$	−9.79796	−0.708955	−0.354478	−	0.935064i	$-0.615341\pi$
$191$	−9.79796	−0.708955	−0.354478	+	0.935064i	$0.615341\pi$
$192$	1.00000i	0.0721688i
$193$	− 1.10102i	− 0.0792532i	−0.999215	−	0.0396266i	$-0.987383\pi$
$193$	− 1.10102i	− 0.0792532i	0.999215	−	0.0396266i	$-0.0126168\pi$
$194$	−12.6969	−0.911587
$195$	0	0
$196$	17.0000	1.21429
$197$	13.5959i	0.968669i	0.874883	+	0.484335i	$0.160938\pi$
$197$	13.5959i	0.968669i	−0.874883	+	0.484335i	$0.839062\pi$
$198$	0	0
$199$	−15.5959	−1.10557	−0.552783	−	0.833325i	$-0.686434\pi$
$199$	−15.5959	−1.10557	−0.552783	+	0.833325i	$0.686434\pi$
$200$	0	0
$201$	−0.898979	−0.0634091
$202$	− 18.8990i	− 1.32973i
$203$	− 29.3939i	− 2.06305i
$204$	−1.00000	−0.0700140
$205$	0	0
$206$	−4.00000	−0.278693
$207$	4.00000i	0.278019i
$208$	6.89898i	0.478358i
$209$	0	0
$210$	0	0
$211$	−12.0000	−0.826114	−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000	−0.826114	−0.413057	+	0.910705i	$0.635539\pi$
$212$	7.79796i	0.535566i
$213$	8.89898i	0.609748i
$214$	5.79796	0.396340
$215$	0	0
$216$	1.00000	0.0680414
$217$	19.5959i	1.33026i
$218$	− 11.7980i	− 0.799059i
$219$	−10.8990	−0.736485
$220$	0	0
$221$	−6.89898	−0.464076
$222$	− 6.00000i	− 0.402694i
$223$	− 4.00000i	− 0.267860i	−0.990991	−	0.133930i	$-0.957240\pi$
$223$	− 4.00000i	− 0.267860i	0.990991	−	0.133930i	$-0.0427597\pi$
$224$	−4.89898	−0.327327
$225$	0	0
$226$	−7.79796	−0.518713
$227$	21.7980i	1.44678i	0.690439	+	0.723391i	$0.257417\pi$
$227$	21.7980i	1.44678i	−0.690439	+	0.723391i	$0.742583\pi$
$228$	− 4.00000i	− 0.264906i
$229$	13.5959	0.898444	0.449222	−	0.893420i	$-0.351701\pi$
$229$	13.5959	0.898444	0.449222	+	0.893420i	$0.351701\pi$
$230$	0	0
$231$	0	0
$232$	6.00000i	0.393919i
$233$	14.0000i	0.917170i	0.888650	+	0.458585i	$0.151644\pi$
$233$	14.0000i	0.917170i	−0.888650	+	0.458585i	$0.848356\pi$
$234$	6.89898	0.451000
$235$	0	0
$236$	4.89898	0.318896
$237$	− 5.79796i	− 0.376618i
$238$	− 4.89898i	− 0.317554i
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	13.5959	0.875790	0.437895	−	0.899026i	$-0.355724\pi$
$241$	13.5959	0.875790	0.437895	+	0.899026i	$0.355724\pi$
$242$	− 11.0000i	− 0.707107i
$243$	− 1.00000i	− 0.0641500i
$244$	−11.7980	−0.755287
$245$	0	0
$246$	−2.89898	−0.184832
$247$	− 27.5959i	− 1.75589i
$248$	− 4.00000i	− 0.254000i
$249$	13.7980	0.874410
$250$	0	0
$251$	4.89898	0.309221	0.154610	−	0.987976i	$-0.450588\pi$
$251$	4.89898	0.309221	0.154610	+	0.987976i	$0.450588\pi$
$252$	4.89898i	0.308607i
$253$	0	0
$254$	−12.0000	−0.752947
$255$	0	0
$256$	1.00000	0.0625000
$257$	2.00000i	0.124757i	0.998053	+	0.0623783i	$0.0198685\pi$
$257$	2.00000i	0.124757i	−0.998053	+	0.0623783i	$0.980131\pi$
$258$	8.89898i	0.554026i
$259$	29.3939	1.82645
$260$	0	0
$261$	6.00000	0.371391
$262$	9.79796i	0.605320i
$263$	8.00000i	0.493301i	0.969104	+	0.246651i	$0.0793300\pi$
$263$	8.00000i	0.493301i	−0.969104	+	0.246651i	$0.920670\pi$
$264$	0	0
$265$	0	0
$266$	19.5959	1.20150
$267$	− 7.79796i	− 0.477227i
$268$	0.898979i	0.0549139i
$269$	−9.59592	−0.585073	−0.292537	−	0.956254i	$-0.594499\pi$
$269$	−9.59592	−0.585073	−0.292537	+	0.956254i	$0.594499\pi$
$270$	0	0
$271$	−1.79796	−0.109218	−0.0546091	−	0.998508i	$-0.517391\pi$
$271$	−1.79796	−0.109218	−0.0546091	+	0.998508i	$0.517391\pi$
$272$	1.00000i	0.0606339i
$273$	33.7980i	2.04555i
$274$	14.0000	0.845771
$275$	0	0
$276$	4.00000	0.240772
$277$	− 7.79796i	− 0.468534i	−0.972172	−	0.234267i	$-0.924731\pi$
$277$	− 7.79796i	− 0.468534i	0.972172	−	0.234267i	$-0.0752690\pi$
$278$	− 13.7980i	− 0.827547i
$279$	−4.00000	−0.239474
$280$	0	0
$281$	27.7980	1.65829	0.829144	−	0.559036i	$-0.188829\pi$
$281$	27.7980	1.65829	0.829144	+	0.559036i	$0.188829\pi$
$282$	− 9.79796i	− 0.583460i
$283$	23.5959i	1.40263i	0.712851	+	0.701316i	$0.247404\pi$
$283$	23.5959i	1.40263i	−0.712851	+	0.701316i	$0.752596\pi$
$284$	8.89898	0.528057
$285$	0	0
$286$	0	0
$287$	− 14.2020i	− 0.838320i
$288$	− 1.00000i	− 0.0589256i
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	12.6969	0.744308
$292$	10.8990i	0.637815i
$293$	13.5959i	0.794282i	0.917758	+	0.397141i	$0.129998\pi$
$293$	13.5959i	0.794282i	−0.917758	+	0.397141i	$0.870002\pi$
$294$	−17.0000	−0.991460
$295$	0	0
$296$	−6.00000	−0.348743
$297$	0	0
$298$	18.8990i	1.09479i
$299$	27.5959	1.59591
$300$	0	0
$301$	−43.5959	−2.51283
$302$	− 9.79796i	− 0.563809i
$303$	18.8990i	1.08572i
$304$	−4.00000	−0.229416
$305$	0	0
$306$	1.00000	0.0571662
$307$	− 0.898979i	− 0.0513075i	−0.999671	−	0.0256537i	$-0.991833\pi$
$307$	− 0.898979i	− 0.0513075i	0.999671	−	0.0256537i	$-0.00816673\pi$
$308$	0	0
$309$	4.00000	0.227552
$310$	0	0
$311$	−7.10102	−0.402662	−0.201331	−	0.979523i	$-0.564527\pi$
$311$	−7.10102	−0.402662	−0.201331	+	0.979523i	$0.564527\pi$
$312$	− 6.89898i	− 0.390578i
$313$	6.89898i	0.389953i	0.980808	+	0.194977i	$0.0624631\pi$
$313$	6.89898i	0.389953i	−0.980808	+	0.194977i	$0.937537\pi$
$314$	−1.10102	−0.0621342
$315$	0	0
$316$	−5.79796	−0.326161
$317$	− 14.0000i	− 0.786318i	−0.919470	−	0.393159i	$-0.871382\pi$
$317$	− 14.0000i	− 0.786318i	0.919470	−	0.393159i	$-0.128618\pi$
$318$	− 7.79796i	− 0.437288i
$319$	0	0
$320$	0	0
$321$	−5.79796	−0.323611
$322$	19.5959i	1.09204i
$323$	− 4.00000i	− 0.222566i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	2.20204	0.121960
$327$	11.7980i	0.652429i
$328$	2.89898i	0.160069i
$329$	48.0000	2.64633
$330$	0	0
$331$	−5.79796	−0.318685	−0.159342	−	0.987223i	$-0.550937\pi$
$331$	−5.79796	−0.318685	−0.159342	+	0.987223i	$0.550937\pi$
$332$	− 13.7980i	− 0.757261i
$333$	6.00000i	0.328798i
$334$	2.20204	0.120490
$335$	0	0
$336$	4.89898	0.267261
$337$	− 32.6969i	− 1.78112i	−0.454870	−	0.890558i	$-0.650314\pi$
$337$	− 32.6969i	− 1.78112i	0.454870	−	0.890558i	$-0.349686\pi$
$338$	− 34.5959i	− 1.88177i
$339$	7.79796	0.423527
$340$	0	0
$341$	0	0
$342$	4.00000i	0.216295i
$343$	− 48.9898i	− 2.64520i
$344$	8.89898	0.479801
$345$	0	0
$346$	−6.00000	−0.322562
$347$	− 21.7980i	− 1.17018i	−0.810970	−	0.585088i	$-0.801060\pi$
$347$	− 21.7980i	− 1.17018i	0.810970	−	0.585088i	$-0.198940\pi$
$348$	− 6.00000i	− 0.321634i
$349$	−4.20204	−0.224930	−0.112465	−	0.993656i	$-0.535875\pi$
$349$	−4.20204	−0.224930	−0.112465	+	0.993656i	$0.535875\pi$
$350$	0	0
$351$	−6.89898	−0.368240
$352$	0	0
$353$	− 26.0000i	− 1.38384i	−0.721974	−	0.691920i	$-0.756765\pi$
$353$	− 26.0000i	− 1.38384i	0.721974	−	0.691920i	$-0.243235\pi$
$354$	−4.89898	−0.260378
$355$	0	0
$356$	−7.79796	−0.413291
$357$	4.89898i	0.259281i
$358$	4.89898i	0.258919i
$359$	−37.3939	−1.97357	−0.986787	−	0.162025i	$-0.948198\pi$
$359$	−37.3939	−1.97357	−0.986787	+	0.162025i	$0.948198\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	− 4.20204i	− 0.220854i
$363$	11.0000i	0.577350i
$364$	33.7980	1.77149
$365$	0	0
$366$	11.7980	0.616689
$367$	27.1010i	1.41466i	0.706883	+	0.707331i	$0.250101\pi$
$367$	27.1010i	1.41466i	−0.706883	+	0.707331i	$0.749899\pi$
$368$	− 4.00000i	− 0.208514i
$369$	2.89898	0.150915
$370$	0	0
$371$	38.2020	1.98335
$372$	4.00000i	0.207390i
$373$	24.6969i	1.27876i	0.768891	+	0.639380i	$0.220809\pi$
$373$	24.6969i	1.27876i	−0.768891	+	0.639380i	$0.779191\pi$
$374$	0	0
$375$	0	0
$376$	−9.79796	−0.505291
$377$	− 41.3939i	− 2.13189i
$378$	− 4.89898i	− 0.251976i
$379$	29.7980	1.53062	0.765309	−	0.643663i	$-0.222586\pi$
$379$	29.7980	1.53062	0.765309	+	0.643663i	$0.222586\pi$
$380$	0	0
$381$	12.0000	0.614779
$382$	− 9.79796i	− 0.501307i
$383$	8.00000i	0.408781i	0.978889	+	0.204390i	$0.0655212\pi$
$383$	8.00000i	0.408781i	−0.978889	+	0.204390i	$0.934479\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	1.10102	0.0560405
$387$	− 8.89898i	− 0.452361i
$388$	− 12.6969i	− 0.644589i
$389$	38.4949	1.95177	0.975884	−	0.218288i	$-0.0700472\pi$
$389$	38.4949	1.95177	0.975884	+	0.218288i	$0.0700472\pi$
$390$	0	0
$391$	4.00000	0.202289
$392$	17.0000i	0.858630i
$393$	− 9.79796i	− 0.494242i
$394$	−13.5959	−0.684952
$395$	0	0
$396$	0	0
$397$	− 1.59592i	− 0.0800968i	−0.999198	−	0.0400484i	$-0.987249\pi$
$397$	− 1.59592i	− 0.0800968i	0.999198	−	0.0400484i	$-0.0127512\pi$
$398$	− 15.5959i	− 0.781753i
$399$	−19.5959	−0.981023
$400$	0	0
$401$	22.8990	1.14352	0.571760	−	0.820421i	$-0.306261\pi$
$401$	22.8990	1.14352	0.571760	+	0.820421i	$0.306261\pi$
$402$	− 0.898979i	− 0.0448370i
$403$	27.5959i	1.37465i
$404$	18.8990	0.940259
$405$	0	0
$406$	29.3939	1.45879
$407$	0	0
$408$	− 1.00000i	− 0.0495074i
$409$	17.5959	0.870062	0.435031	−	0.900415i	$-0.356737\pi$
$409$	17.5959	0.870062	0.435031	+	0.900415i	$0.356737\pi$
$410$	0	0
$411$	−14.0000	−0.690569
$412$	− 4.00000i	− 0.197066i
$413$	− 24.0000i	− 1.18096i
$414$	−4.00000	−0.196589
$415$	0	0
$416$	−6.89898	−0.338250
$417$	13.7980i	0.675689i
$418$	0	0
$419$	17.7980	0.869487	0.434744	−	0.900554i	$-0.356839\pi$
$419$	17.7980	0.869487	0.434744	+	0.900554i	$0.356839\pi$
$420$	0	0
$421$	14.0000	0.682318	0.341159	−	0.940006i	$-0.389181\pi$
$421$	14.0000	0.682318	0.341159	+	0.940006i	$0.389181\pi$
$422$	− 12.0000i	− 0.584151i
$423$	9.79796i	0.476393i
$424$	−7.79796	−0.378702
$425$	0	0
$426$	−8.89898	−0.431157
$427$	57.7980i	2.79704i
$428$	5.79796i	0.280255i
$429$	0	0
$430$	0	0
$431$	−23.1010	−1.11274	−0.556369	−	0.830936i	$-0.687806\pi$
$431$	−23.1010	−1.11274	−0.556369	+	0.830936i	$0.687806\pi$
$432$	1.00000i	0.0481125i
$433$	− 35.3939i	− 1.70092i	−0.526039	−	0.850461i	$-0.676323\pi$
$433$	− 35.3939i	− 1.70092i	0.526039	−	0.850461i	$-0.323677\pi$
$434$	−19.5959	−0.940634
$435$	0	0
$436$	11.7980	0.565020
$437$	16.0000i	0.765384i
$438$	− 10.8990i	− 0.520773i
$439$	−5.79796	−0.276721	−0.138361	−	0.990382i	$-0.544183\pi$
$439$	−5.79796	−0.276721	−0.138361	+	0.990382i	$0.544183\pi$
$440$	0	0
$441$	17.0000	0.809524
$442$	− 6.89898i	− 0.328151i
$443$	37.7980i	1.79584i	0.440164	+	0.897918i	$0.354920\pi$
$443$	37.7980i	1.79584i	−0.440164	+	0.897918i	$0.645080\pi$
$444$	6.00000	0.284747
$445$	0	0
$446$	4.00000	0.189405
$447$	− 18.8990i	− 0.893891i
$448$	− 4.89898i	− 0.231455i
$449$	−6.89898	−0.325583	−0.162791	−	0.986660i	$-0.552050\pi$
$449$	−6.89898	−0.325583	−0.162791	+	0.986660i	$0.552050\pi$
$450$	0	0
$451$	0	0
$452$	− 7.79796i	− 0.366785i
$453$	9.79796i	0.460348i
$454$	−21.7980	−1.02303
$455$	0	0
$456$	4.00000	0.187317
$457$	− 16.2020i	− 0.757900i	−0.925417	−	0.378950i	$-0.876285\pi$
$457$	− 16.2020i	− 0.757900i	0.925417	−	0.378950i	$-0.123715\pi$
$458$	13.5959i	0.635296i
$459$	−1.00000	−0.0466760
$460$	0	0
$461$	−28.6969	−1.33655	−0.668275	−	0.743914i	$-0.732967\pi$
$461$	−28.6969	−1.33655	−0.668275	+	0.743914i	$0.732967\pi$
$462$	0	0
$463$	7.59592i	0.353012i	0.984300	+	0.176506i	$0.0564795\pi$
$463$	7.59592i	0.353012i	−0.984300	+	0.176506i	$0.943520\pi$
$464$	−6.00000	−0.278543
$465$	0	0
$466$	−14.0000	−0.648537
$467$	7.59592i	0.351497i	0.984435	+	0.175749i	$0.0562346\pi$
$467$	7.59592i	0.351497i	−0.984435	+	0.175749i	$0.943765\pi$
$468$	6.89898i	0.318905i
$469$	4.40408	0.203362
$470$	0	0
$471$	1.10102	0.0507323
$472$	4.89898i	0.225494i
$473$	0	0
$474$	5.79796	0.266309
$475$	0	0
$476$	4.89898	0.224544
$477$	7.79796i	0.357044i
$478$	0	0
$479$	32.8990	1.50319	0.751596	−	0.659623i	$-0.229284\pi$
$479$	32.8990	1.50319	0.751596	+	0.659623i	$0.229284\pi$
$480$	0	0
$481$	41.3939	1.88740
$482$	13.5959i	0.619277i
$483$	− 19.5959i	− 0.891645i
$484$	11.0000	0.500000
$485$	0	0
$486$	1.00000	0.0453609
$487$	20.8990i	0.947023i	0.880788	+	0.473512i	$0.157014\pi$
$487$	20.8990i	0.947023i	−0.880788	+	0.473512i	$0.842986\pi$
$488$	− 11.7980i	− 0.534069i
$489$	−2.20204	−0.0995797
$490$	0	0
$491$	−1.30306	−0.0588063	−0.0294032	−	0.999568i	$-0.509361\pi$
$491$	−1.30306	−0.0588063	−0.0294032	+	0.999568i	$0.509361\pi$
$492$	− 2.89898i	− 0.130696i
$493$	− 6.00000i	− 0.270226i
$494$	27.5959	1.24160
$495$	0	0
$496$	4.00000	0.179605
$497$	− 43.5959i	− 1.95554i
$498$	13.7980i	0.618301i
$499$	39.5959	1.77256	0.886278	−	0.463153i	$-0.153282\pi$
$499$	39.5959	1.77256	0.886278	+	0.463153i	$0.153282\pi$
$500$	0	0
$501$	−2.20204	−0.0983799
$502$	4.89898i	0.218652i
$503$	4.00000i	0.178351i	0.996016	+	0.0891756i	$0.0284232\pi$
$503$	4.00000i	0.178351i	−0.996016	+	0.0891756i	$0.971577\pi$
$504$	−4.89898	−0.218218
$505$	0	0
$506$	0	0
$507$	34.5959i	1.53646i
$508$	− 12.0000i	− 0.532414i
$509$	15.3031	0.678296	0.339148	−	0.940733i	$-0.389861\pi$
$509$	15.3031	0.678296	0.339148	+	0.940733i	$0.389861\pi$
$510$	0	0
$511$	53.3939	2.36201
$512$	1.00000i	0.0441942i
$513$	− 4.00000i	− 0.176604i
$514$	−2.00000	−0.0882162
$515$	0	0
$516$	−8.89898	−0.391756
$517$	0	0
$518$	29.3939i	1.29149i
$519$	6.00000	0.263371
$520$	0	0
$521$	−40.2929	−1.76526	−0.882631	−	0.470066i	$-0.844230\pi$
$521$	−40.2929	−1.76526	−0.882631	+	0.470066i	$0.844230\pi$
$522$	6.00000i	0.262613i
$523$	18.6969i	0.817560i	0.912633	+	0.408780i	$0.134046\pi$
$523$	18.6969i	0.817560i	−0.912633	+	0.408780i	$0.865954\pi$
$524$	−9.79796	−0.428026
$525$	0	0
$526$	−8.00000	−0.348817
$527$	4.00000i	0.174243i
$528$	0	0
$529$	7.00000	0.304348
$530$	0	0
$531$	4.89898	0.212598
$532$	19.5959i	0.849591i
$533$	− 20.0000i	− 0.866296i
$534$	7.79796	0.337451
$535$	0	0
$536$	−0.898979	−0.0388300
$537$	− 4.89898i	− 0.211407i
$538$	− 9.59592i	− 0.413709i
$539$	0	0
$540$	0	0
$541$	−7.79796	−0.335260	−0.167630	−	0.985850i	$-0.553611\pi$
$541$	−7.79796	−0.335260	−0.167630	+	0.985850i	$0.553611\pi$
$542$	− 1.79796i	− 0.0772290i
$543$	4.20204i	0.180327i
$544$	−1.00000	−0.0428746
$545$	0	0
$546$	−33.7980	−1.44642
$547$	− 0.404082i	− 0.0172773i	−0.999963	−	0.00863865i	$-0.997250\pi$
$547$	− 0.404082i	− 0.0172773i	0.999963	−	0.00863865i	$-0.00274980\pi$
$548$	14.0000i	0.598050i
$549$	−11.7980	−0.503525
$550$	0	0
$551$	24.0000	1.02243
$552$	4.00000i	0.170251i
$553$	28.4041i	1.20786i
$554$	7.79796	0.331304
$555$	0	0
$556$	13.7980	0.585164
$557$	− 19.7980i	− 0.838866i	−0.907786	−	0.419433i	$-0.862229\pi$
$557$	− 19.7980i	− 0.838866i	0.907786	−	0.419433i	$-0.137771\pi$
$558$	− 4.00000i	− 0.169334i
$559$	−61.3939	−2.59668
$560$	0	0
$561$	0	0
$562$	27.7980i	1.17259i
$563$	21.7980i	0.918674i	0.888262	+	0.459337i	$0.151913\pi$
$563$	21.7980i	0.918674i	−0.888262	+	0.459337i	$0.848087\pi$
$564$	9.79796	0.412568
$565$	0	0
$566$	−23.5959	−0.991810
$567$	4.89898i	0.205738i
$568$	8.89898i	0.373393i
$569$	−34.0000	−1.42535	−0.712677	−	0.701492i	$-0.752517\pi$
$569$	−34.0000	−1.42535	−0.712677	+	0.701492i	$0.752517\pi$
$570$	0	0
$571$	29.7980	1.24701	0.623503	−	0.781821i	$-0.285709\pi$
$571$	29.7980	1.24701	0.623503	+	0.781821i	$0.285709\pi$
$572$	0	0
$573$	9.79796i	0.409316i
$574$	14.2020	0.592782
$575$	0	0
$576$	1.00000	0.0416667
$577$	27.3939i	1.14042i	0.821498	+	0.570211i	$0.193139\pi$
$577$	27.3939i	1.14042i	−0.821498	+	0.570211i	$0.806861\pi$
$578$	− 1.00000i	− 0.0415945i
$579$	−1.10102	−0.0457569
$580$	0	0
$581$	−67.5959	−2.80435
$582$	12.6969i	0.526305i
$583$	0	0
$584$	−10.8990	−0.451003
$585$	0	0
$586$	−13.5959	−0.561642
$587$	41.3939i	1.70851i	0.519856	+	0.854254i	$0.325986\pi$
$587$	41.3939i	1.70851i	−0.519856	+	0.854254i	$0.674014\pi$
$588$	− 17.0000i	− 0.701068i
$589$	−16.0000	−0.659269
$590$	0	0
$591$	13.5959	0.559261
$592$	− 6.00000i	− 0.246598i
$593$	1.59592i	0.0655365i	0.999463	+	0.0327682i	$0.0104323\pi$
$593$	1.59592i	0.0655365i	−0.999463	+	0.0327682i	$0.989568\pi$
$594$	0	0
$595$	0	0
$596$	−18.8990	−0.774132
$597$	15.5959i	0.638298i
$598$	27.5959i	1.12848i
$599$	−21.3939	−0.874130	−0.437065	−	0.899430i	$-0.643982\pi$
$599$	−21.3939	−0.874130	−0.437065	+	0.899430i	$0.643982\pi$
$600$	0	0
$601$	16.2020	0.660895	0.330448	−	0.943824i	$-0.392800\pi$
$601$	16.2020	0.660895	0.330448	+	0.943824i	$0.392800\pi$
$602$	− 43.5959i	− 1.77684i
$603$	0.898979i	0.0366093i
$604$	9.79796	0.398673
$605$	0	0
$606$	−18.8990	−0.767719
$607$	32.4949i	1.31893i	0.751736	+	0.659464i	$0.229217\pi$
$607$	32.4949i	1.31893i	−0.751736	+	0.659464i	$0.770783\pi$
$608$	− 4.00000i	− 0.162221i
$609$	−29.3939	−1.19110
$610$	0	0
$611$	67.5959	2.73464
$612$	1.00000i	0.0404226i
$613$	− 14.4949i	− 0.585443i	−0.956198	−	0.292722i	$-0.905439\pi$
$613$	− 14.4949i	− 0.585443i	0.956198	−	0.292722i	$-0.0945609\pi$
$614$	0.898979	0.0362799
$615$	0	0
$616$	0	0
$617$	37.5959i	1.51355i	0.653673	+	0.756777i	$0.273227\pi$
$617$	37.5959i	1.51355i	−0.653673	+	0.756777i	$0.726773\pi$
$618$	4.00000i	0.160904i
$619$	−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
$621$	4.00000	0.160514
$622$	− 7.10102i	− 0.284725i
$623$	38.2020i	1.53053i
$624$	6.89898	0.276180
$625$	0	0
$626$	−6.89898	−0.275739
$627$	0	0
$628$	− 1.10102i	− 0.0439355i
$629$	6.00000	0.239236
$630$	0	0
$631$	−40.0000	−1.59237	−0.796187	−	0.605050i	$-0.793153\pi$
$631$	−40.0000	−1.59237	−0.796187	+	0.605050i	$0.793153\pi$
$632$	− 5.79796i	− 0.230630i
$633$	12.0000i	0.476957i
$634$	14.0000	0.556011
$635$	0	0
$636$	7.79796	0.309209
$637$	− 117.283i	− 4.64691i
$638$	0	0
$639$	8.89898	0.352038
$640$	0	0
$641$	−20.6969	−0.817480	−0.408740	−	0.912651i	$-0.634032\pi$
$641$	−20.6969	−0.817480	−0.408740	+	0.912651i	$0.634032\pi$
$642$	− 5.79796i	− 0.228827i
$643$	29.7980i	1.17512i	0.809182	+	0.587558i	$0.199911\pi$
$643$	29.7980i	1.17512i	−0.809182	+	0.587558i	$0.800089\pi$
$644$	−19.5959	−0.772187
$645$	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
$649$	0	0
$650$	0	0
$651$	19.5959	0.768025
$652$	2.20204i	0.0862386i
$653$	30.0000i	1.17399i	0.809590	+	0.586995i	$0.199689\pi$
$653$	30.0000i	1.17399i	−0.809590	+	0.586995i	$0.800311\pi$
$654$	−11.7980	−0.461337
$655$	0	0
$656$	−2.89898	−0.113186
$657$	10.8990i	0.425210i
$658$	48.0000i	1.87123i
$659$	−30.6969	−1.19578	−0.597891	−	0.801577i	$-0.703995\pi$
$659$	−30.6969	−1.19578	−0.597891	+	0.801577i	$0.703995\pi$
$660$	0	0
$661$	−19.7980	−0.770051	−0.385026	−	0.922906i	$-0.625807\pi$
$661$	−19.7980	−0.770051	−0.385026	+	0.922906i	$0.625807\pi$
$662$	− 5.79796i	− 0.225344i
$663$	6.89898i	0.267934i
$664$	13.7980	0.535465
$665$	0	0
$666$	−6.00000	−0.232495
$667$	24.0000i	0.929284i
$668$	2.20204i	0.0851995i
$669$	−4.00000	−0.154649
$670$	0	0
$671$	0	0
$672$	4.89898i	0.188982i
$673$	− 33.1010i	− 1.27595i	−0.770057	−	0.637975i	$-0.779772\pi$
$673$	− 33.1010i	− 1.27595i	0.770057	−	0.637975i	$-0.220228\pi$
$674$	32.6969	1.25944
$675$	0	0
$676$	34.5959	1.33061
$677$	13.5959i	0.522534i	0.965267	+	0.261267i	$0.0841402\pi$
$677$	13.5959i	0.522534i	−0.965267	+	0.261267i	$0.915860\pi$
$678$	7.79796i	0.299479i
$679$	−62.2020	−2.38710
$680$	0	0
$681$	21.7980	0.835300
$682$	0	0
$683$	12.0000i	0.459167i	0.973289	+	0.229584i	$0.0737364\pi$
$683$	12.0000i	0.459167i	−0.973289	+	0.229584i	$0.926264\pi$
$684$	−4.00000	−0.152944
$685$	0	0
$686$	48.9898	1.87044
$687$	− 13.5959i	− 0.518717i
$688$	8.89898i	0.339270i
$689$	53.7980	2.04954
$690$	0	0
$691$	−27.1918	−1.03443	−0.517213	−	0.855857i	$-0.673031\pi$
$691$	−27.1918	−1.03443	−0.517213	+	0.855857i	$0.673031\pi$
$692$	− 6.00000i	− 0.228086i
$693$	0	0
$694$	21.7980	0.827439
$695$	0	0
$696$	6.00000	0.227429
$697$	− 2.89898i	− 0.109807i
$698$	− 4.20204i	− 0.159050i
$699$	14.0000	0.529529
$700$	0	0
$701$	14.8990	0.562727	0.281363	−	0.959601i	$-0.409213\pi$
$701$	14.8990	0.562727	0.281363	+	0.959601i	$0.409213\pi$
$702$	− 6.89898i	− 0.260385i
$703$	24.0000i	0.905177i
$704$	0	0
$705$	0	0
$706$	26.0000	0.978523
$707$	− 92.5857i	− 3.48204i
$708$	− 4.89898i	− 0.184115i
$709$	31.7980	1.19420	0.597099	−	0.802168i	$-0.296320\pi$
$709$	31.7980	1.19420	0.597099	+	0.802168i	$0.296320\pi$
$710$	0	0
$711$	−5.79796	−0.217440
$712$	− 7.79796i	− 0.292241i
$713$	− 16.0000i	− 0.599205i
$714$	−4.89898	−0.183340
$715$	0	0
$716$	−4.89898	−0.183083
$717$	0	0
$718$	− 37.3939i	− 1.39553i
$719$	12.4949	0.465981	0.232991	−	0.972479i	$-0.425149\pi$
$719$	12.4949	0.465981	0.232991	+	0.972479i	$0.425149\pi$
$720$	0	0
$721$	−19.5959	−0.729790
$722$	− 3.00000i	− 0.111648i
$723$	− 13.5959i	− 0.505638i
$724$	4.20204	0.156168
$725$	0	0
$726$	−11.0000	−0.408248
$727$	− 0.404082i	− 0.0149866i	−0.999972	−	0.00749329i	$-0.997615\pi$
$727$	− 0.404082i	− 0.0149866i	0.999972	−	0.00749329i	$-0.00238521\pi$
$728$	33.7980i	1.25264i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−8.89898	−0.329141
$732$	11.7980i	0.436065i
$733$	− 46.4949i	− 1.71733i	−0.512539	−	0.858664i	$-0.671295\pi$
$733$	− 46.4949i	− 1.71733i	0.512539	−	0.858664i	$-0.328705\pi$
$734$	−27.1010	−1.00032
$735$	0	0
$736$	4.00000	0.147442
$737$	0	0
$738$	2.89898i	0.106713i
$739$	9.39388	0.345559	0.172780	−	0.984960i	$-0.444725\pi$
$739$	9.39388	0.345559	0.172780	+	0.984960i	$0.444725\pi$
$740$	0	0
$741$	−27.5959	−1.01376
$742$	38.2020i	1.40244i
$743$	− 9.39388i	− 0.344628i	−0.985042	−	0.172314i	$-0.944876\pi$
$743$	− 9.39388i	− 0.344628i	0.985042	−	0.172314i	$-0.0551244\pi$
$744$	−4.00000	−0.146647
$745$	0	0
$746$	−24.6969	−0.904219
$747$	− 13.7980i	− 0.504841i
$748$	0	0
$749$	28.4041	1.03786
$750$	0	0
$751$	21.7980	0.795419	0.397709	−	0.917511i	$-0.369805\pi$
$751$	21.7980	0.795419	0.397709	+	0.917511i	$0.369805\pi$
$752$	− 9.79796i	− 0.357295i
$753$	− 4.89898i	− 0.178529i
$754$	41.3939	1.50748
$755$	0	0
$756$	4.89898	0.178174
$757$	4.69694i	0.170713i	0.996350	+	0.0853566i	$0.0272029\pi$
$757$	4.69694i	0.170713i	−0.996350	+	0.0853566i	$0.972797\pi$
$758$	29.7980i	1.08231i
$759$	0	0
$760$	0	0
$761$	−1.59592	−0.0578520	−0.0289260	−	0.999582i	$-0.509209\pi$
$761$	−1.59592	−0.0578520	−0.0289260	+	0.999582i	$0.509209\pi$
$762$	12.0000i	0.434714i
$763$	− 57.7980i	− 2.09243i
$764$	9.79796	0.354478
$765$	0	0
$766$	−8.00000	−0.289052
$767$	− 33.7980i	− 1.22037i
$768$	− 1.00000i	− 0.0360844i
$769$	−26.0000	−0.937584	−0.468792	−	0.883309i	$-0.655311\pi$
$769$	−26.0000	−0.937584	−0.468792	+	0.883309i	$0.655311\pi$
$770$	0	0
$771$	2.00000	0.0720282
$772$	1.10102i	0.0396266i
$773$	− 35.3939i	− 1.27303i	−0.771265	−	0.636515i	$-0.780375\pi$
$773$	− 35.3939i	− 1.27303i	0.771265	−	0.636515i	$-0.219625\pi$
$774$	8.89898	0.319867
$775$	0	0
$776$	12.6969	0.455794
$777$	− 29.3939i	− 1.05450i
$778$	38.4949i	1.38011i
$779$	11.5959	0.415467
$780$	0	0
$781$	0	0
$782$	4.00000i	0.143040i
$783$	− 6.00000i	− 0.214423i
$784$	−17.0000	−0.607143
$785$	0	0
$786$	9.79796	0.349482
$787$	45.7980i	1.63252i	0.577684	+	0.816260i	$0.303957\pi$
$787$	45.7980i	1.63252i	−0.577684	+	0.816260i	$0.696043\pi$
$788$	− 13.5959i	− 0.484335i
$789$	8.00000	0.284808
$790$	0	0
$791$	−38.2020	−1.35831
$792$	0	0
$793$	81.3939i	2.89038i
$794$	1.59592	0.0566370
$795$	0	0
$796$	15.5959	0.552783
$797$	− 26.0000i	− 0.920967i	−0.887668	−	0.460484i	$-0.847676\pi$
$797$	− 26.0000i	− 0.920967i	0.887668	−	0.460484i	$-0.152324\pi$
$798$	− 19.5959i	− 0.693688i
$799$	9.79796	0.346627
$800$	0	0
$801$	−7.79796	−0.275527
$802$	22.8990i	0.808591i
$803$	0	0
$804$	0.898979	0.0317046
$805$	0	0
$806$	−27.5959	−0.972025
$807$	9.59592i	0.337792i
$808$	18.8990i	0.664864i
$809$	−32.6969	−1.14956	−0.574782	−	0.818307i	$-0.694913\pi$
$809$	−32.6969	−1.14956	−0.574782	+	0.818307i	$0.694913\pi$
$810$	0	0
$811$	25.3939	0.891700	0.445850	−	0.895108i	$-0.352902\pi$
$811$	25.3939	0.891700	0.445850	+	0.895108i	$0.352902\pi$
$812$	29.3939i	1.03152i
$813$	1.79796i	0.0630572i
$814$	0	0
$815$	0	0
$816$	1.00000	0.0350070
$817$	− 35.5959i	− 1.24534i
$818$	17.5959i	0.615227i
$819$	33.7980	1.18100
$820$	0	0
$821$	15.7980	0.551353	0.275676	−	0.961251i	$-0.411098\pi$
$821$	15.7980	0.551353	0.275676	+	0.961251i	$0.411098\pi$
$822$	− 14.0000i	− 0.488306i
$823$	20.8990i	0.728493i	0.931303	+	0.364246i	$0.118673\pi$
$823$	20.8990i	0.728493i	−0.931303	+	0.364246i	$0.881327\pi$
$824$	4.00000	0.139347
$825$	0	0
$826$	24.0000	0.835067
$827$	4.00000i	0.139094i	0.997579	+	0.0695468i	$0.0221553\pi$
$827$	4.00000i	0.139094i	−0.997579	+	0.0695468i	$0.977845\pi$
$828$	− 4.00000i	− 0.139010i
$829$	−3.39388	−0.117874	−0.0589371	−	0.998262i	$-0.518771\pi$
$829$	−3.39388	−0.117874	−0.0589371	+	0.998262i	$0.518771\pi$
$830$	0	0
$831$	−7.79796	−0.270508
$832$	− 6.89898i	− 0.239179i
$833$	− 17.0000i	− 0.589015i
$834$	−13.7980	−0.477784
$835$	0	0
$836$	0	0
$837$	4.00000i	0.138260i
$838$	17.7980i	0.614820i
$839$	7.10102	0.245154	0.122577	−	0.992459i	$-0.460884\pi$
$839$	7.10102	0.245154	0.122577	+	0.992459i	$0.460884\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	14.0000i	0.482472i
$843$	− 27.7980i	− 0.957413i
$844$	12.0000	0.413057
$845$	0	0
$846$	−9.79796	−0.336861
$847$	− 53.8888i	− 1.85164i
$848$	− 7.79796i	− 0.267783i
$849$	23.5959	0.809810
$850$	0	0
$851$	−24.0000	−0.822709
$852$	− 8.89898i	− 0.304874i
$853$	11.3939i	0.390119i	0.980791	+	0.195059i	$0.0624900\pi$
$853$	11.3939i	0.390119i	−0.980791	+	0.195059i	$0.937510\pi$
$854$	−57.7980	−1.97781
$855$	0	0
$856$	−5.79796	−0.198170
$857$	10.0000i	0.341593i	0.985306	+	0.170797i	$0.0546341\pi$
$857$	10.0000i	0.341593i	−0.985306	+	0.170797i	$0.945366\pi$
$858$	0	0
$859$	−5.79796	−0.197824	−0.0989119	−	0.995096i	$-0.531536\pi$
$859$	−5.79796	−0.197824	−0.0989119	+	0.995096i	$0.531536\pi$
$860$	0	0
$861$	−14.2020	−0.484004
$862$	− 23.1010i	− 0.786824i
$863$	− 45.3939i	− 1.54523i	−0.634878	−	0.772613i	$-0.718949\pi$
$863$	− 45.3939i	− 1.54523i	0.634878	−	0.772613i	$-0.281051\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	35.3939	1.20273
$867$	1.00000i	0.0339618i
$868$	− 19.5959i	− 0.665129i
$869$	0	0
$870$	0	0
$871$	6.20204	0.210148
$872$	11.7980i	0.399529i
$873$	− 12.6969i	− 0.429726i
$874$	−16.0000	−0.541208
$875$	0	0
$876$	10.8990	0.368242
$877$	31.3939i	1.06010i	0.847968	+	0.530048i	$0.177826\pi$
$877$	31.3939i	1.06010i	−0.847968	+	0.530048i	$0.822174\pi$
$878$	− 5.79796i	− 0.195672i
$879$	13.5959	0.458579
$880$	0	0
$881$	34.4949	1.16216	0.581081	−	0.813846i	$-0.302630\pi$
$881$	34.4949	1.16216	0.581081	+	0.813846i	$0.302630\pi$
$882$	17.0000i	0.572420i
$883$	26.6969i	0.898424i	0.893425	+	0.449212i	$0.148295\pi$
$883$	26.6969i	0.898424i	−0.893425	+	0.449212i	$0.851705\pi$
$884$	6.89898	0.232038
$885$	0	0
$886$	−37.7980	−1.26985
$887$	12.0000i	0.402921i	0.979497	+	0.201460i	$0.0645687\pi$
$887$	12.0000i	0.402921i	−0.979497	+	0.201460i	$0.935431\pi$
$888$	6.00000i	0.201347i
$889$	−58.7878	−1.97168
$890$	0	0
$891$	0	0
$892$	4.00000i	0.133930i
$893$	39.1918i	1.31150i
$894$	18.8990	0.632076
$895$	0	0
$896$	4.89898	0.163663
$897$	− 27.5959i	− 0.921401i
$898$	− 6.89898i	− 0.230222i
$899$	−24.0000	−0.800445
$900$	0	0
$901$	7.79796	0.259788
$902$	0	0
$903$	43.5959i	1.45078i
$904$	7.79796	0.259356
$905$	0	0
$906$	−9.79796	−0.325515
$907$	33.3939i	1.10883i	0.832242	+	0.554413i	$0.187057\pi$
$907$	33.3939i	1.10883i	−0.832242	+	0.554413i	$0.812943\pi$
$908$	− 21.7980i	− 0.723391i
$909$	18.8990	0.626840
$910$	0	0
$911$	23.1010	0.765371	0.382685	−	0.923879i	$-0.374999\pi$
$911$	23.1010	0.765371	0.382685	+	0.923879i	$0.374999\pi$
$912$	4.00000i	0.132453i
$913$	0	0
$914$	16.2020	0.535916
$915$	0	0
$916$	−13.5959	−0.449222
$917$	48.0000i	1.58510i
$918$	− 1.00000i	− 0.0330049i
$919$	−1.79796	−0.0593092	−0.0296546	−	0.999560i	$-0.509441\pi$
$919$	−1.79796	−0.0593092	−0.0296546	+	0.999560i	$0.509441\pi$
$920$	0	0
$921$	−0.898979	−0.0296224
$922$	− 28.6969i	− 0.945083i
$923$	− 61.3939i	− 2.02080i
$924$	0	0
$925$	0	0
$926$	−7.59592	−0.249617
$927$	− 4.00000i	− 0.131377i
$928$	− 6.00000i	− 0.196960i
$929$	−36.2929	−1.19073	−0.595365	−	0.803455i	$-0.702993\pi$
$929$	−36.2929	−1.19073	−0.595365	+	0.803455i	$0.702993\pi$
$930$	0	0
$931$	68.0000	2.22861
$932$	− 14.0000i	− 0.458585i
$933$	7.10102i	0.232477i
$934$	−7.59592	−0.248546
$935$	0	0
$936$	−6.89898	−0.225500
$937$	− 39.3939i	− 1.28694i	−0.765471	−	0.643471i	$-0.777494\pi$
$937$	− 39.3939i	− 1.28694i	0.765471	−	0.643471i	$-0.222506\pi$
$938$	4.40408i	0.143798i
$939$	6.89898	0.225140
$940$	0	0
$941$	−16.2020	−0.528171	−0.264086	−	0.964499i	$-0.585070\pi$
$941$	−16.2020	−0.528171	−0.264086	+	0.964499i	$0.585070\pi$
$942$	1.10102i	0.0358732i
$943$	11.5959i	0.377615i
$944$	−4.89898	−0.159448
$945$	0	0
$946$	0	0
$947$	− 21.7980i	− 0.708338i	−0.935181	−	0.354169i	$-0.884764\pi$
$947$	− 21.7980i	− 0.708338i	0.935181	−	0.354169i	$-0.115236\pi$
$948$	5.79796i	0.188309i
$949$	75.1918	2.44083
$950$	0	0
$951$	−14.0000	−0.453981
$952$	4.89898i	0.158777i
$953$	− 41.1918i	− 1.33433i	−0.744908	−	0.667167i	$-0.767507\pi$
$953$	− 41.1918i	− 1.33433i	0.744908	−	0.667167i	$-0.232493\pi$
$954$	−7.79796	−0.252468
$955$	0	0
$956$	0	0
$957$	0	0
$958$	32.8990i	1.06292i
$959$	68.5857	2.21475
$960$	0	0
$961$	−15.0000	−0.483871
$962$	41.3939i	1.33459i
$963$	5.79796i	0.186837i
$964$	−13.5959	−0.437895
$965$	0	0
$966$	19.5959	0.630488
$967$	41.3939i	1.33114i	0.746337	+	0.665569i	$0.231811\pi$
$967$	41.3939i	1.33114i	−0.746337	+	0.665569i	$0.768189\pi$
$968$	11.0000i	0.353553i
$969$	−4.00000	−0.128499
$970$	0	0
$971$	38.6969	1.24184	0.620922	−	0.783872i	$-0.286758\pi$
$971$	38.6969	1.24184	0.620922	+	0.783872i	$0.286758\pi$
$972$	1.00000i	0.0320750i
$973$	− 67.5959i	− 2.16703i
$974$	−20.8990	−0.669646
$975$	0	0
$976$	11.7980	0.377643
$977$	29.5959i	0.946857i	0.880832	+	0.473429i	$0.156984\pi$
$977$	29.5959i	0.946857i	−0.880832	+	0.473429i	$0.843016\pi$
$978$	− 2.20204i	− 0.0704135i
$979$	0	0
$980$	0	0
$981$	11.7980	0.376680
$982$	− 1.30306i	− 0.0415824i
$983$	9.39388i	0.299618i	0.988715	+	0.149809i	$0.0478659\pi$
$983$	9.39388i	0.299618i	−0.988715	+	0.149809i	$0.952134\pi$
$984$	2.89898	0.0924161
$985$	0	0
$986$	6.00000	0.191079
$987$	− 48.0000i	− 1.52786i
$988$	27.5959i	0.877943i
$989$	35.5959	1.13188
$990$	0	0
$991$	−15.5959	−0.495421	−0.247710	−	0.968834i	$-0.579678\pi$
$991$	−15.5959	−0.495421	−0.247710	+	0.968834i	$0.579678\pi$
$992$	4.00000i	0.127000i
$993$	5.79796i	0.183993i
$994$	43.5959	1.38278
$995$	0	0
$996$	−13.7980	−0.437205
$997$	21.5959i	0.683950i	0.939709	+	0.341975i	$0.111096\pi$
$997$	21.5959i	0.683950i	−0.939709	+	0.341975i	$0.888904\pi$
$998$	39.5959i	1.25339i
$999$	6.00000	0.189832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2550.2.d.u.2449.4		4
5.2	odd	4		510.2.a.h.1.1	✓	2
5.3	odd	4		2550.2.a.bl.1.2		2
5.4	even	2	inner	2550.2.d.u.2449.1		4
15.2	even	4		1530.2.a.s.1.1		2
15.8	even	4		7650.2.a.cu.1.2		2
20.7	even	4		4080.2.a.bq.1.2		2
85.67	odd	4		8670.2.a.be.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
510.2.a.h.1.1	✓	2	5.2	odd	4
1530.2.a.s.1.1		2	15.2	even	4
2550.2.a.bl.1.2		2	5.3	odd	4
2550.2.d.u.2449.1		4	5.4	even	2	inner
2550.2.d.u.2449.4		4	1.1	even	1	trivial
4080.2.a.bq.1.2		2	20.7	even	4
7650.2.a.cu.1.2		2	15.8	even	4
8670.2.a.be.1.2		2	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)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +4.89898i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +4.89898i q^{7} -1.00000i q^{8} -1.00000 q^{9} +1.00000i q^{12} +6.89898i q^{13} -4.89898 q^{14} +1.00000 q^{16} +1.00000i q^{17} -1.00000i q^{18} -4.00000 q^{19} +4.89898 q^{21} -4.00000i q^{23} -1.00000 q^{24} -6.89898 q^{26} +1.00000i q^{27} -4.89898i q^{28} -6.00000 q^{29} +4.00000 q^{31} +1.00000i q^{32} -1.00000 q^{34} +1.00000 q^{36} -6.00000i q^{37} -4.00000i q^{38} +6.89898 q^{39} -2.89898 q^{41} +4.89898i q^{42} +8.89898i q^{43} +4.00000 q^{46} -9.79796i q^{47} -1.00000i q^{48} -17.0000 q^{49} +1.00000 q^{51} -6.89898i q^{52} -7.79796i q^{53} -1.00000 q^{54} +4.89898 q^{56} +4.00000i q^{57} -6.00000i q^{58} -4.89898 q^{59} +11.7980 q^{61} +4.00000i q^{62} -4.89898i q^{63} -1.00000 q^{64} -0.898979i q^{67} -1.00000i q^{68} -4.00000 q^{69} -8.89898 q^{71} +1.00000i q^{72} -10.8990i q^{73} +6.00000 q^{74} +4.00000 q^{76} +6.89898i q^{78} +5.79796 q^{79} +1.00000 q^{81} -2.89898i q^{82} +13.7980i q^{83} -4.89898 q^{84} -8.89898 q^{86} +6.00000i q^{87} +7.79796 q^{89} -33.7980 q^{91} +4.00000i q^{92} -4.00000i q^{93} +9.79796 q^{94} +1.00000 q^{96} +12.6969i q^{97} -17.0000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4} + 4 q^{6} - 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^4 + 4 * q^6 - 4 * q^9	\( 4 q - 4 q^{4} + 4 q^{6} - 4 q^{9} + 4 q^{16} - 16 q^{19} - 4 q^{24} - 8 q^{26} - 24 q^{29} + 16 q^{31} - 4 q^{34} + 4 q^{36} + 8 q^{39} + 8 q^{41} + 16 q^{46} - 68 q^{49} + 4 q^{51} - 4 q^{54} + 8 q^{61} - 4 q^{64} - 16 q^{69} - 16 q^{71} + 24 q^{74} + 16 q^{76} - 16 q^{79} + 4 q^{81} - 16 q^{86} - 8 q^{89} - 96 q^{91} + 4 q^{96}+O(q^{100}) \) 4 * q - 4 * q^4 + 4 * q^6 - 4 * q^9 + 4 * q^16 - 16 * q^19 - 4 * q^24 - 8 * q^26 - 24 * q^29 + 16 * q^31 - 4 * q^34 + 4 * q^36 + 8 * q^39 + 8 * q^41 + 16 * q^46 - 68 * q^49 + 4 * q^51 - 4 * q^54 + 8 * q^61 - 4 * q^64 - 16 * q^69 - 16 * q^71 + 24 * q^74 + 16 * q^76 - 16 * q^79 + 4 * q^81 - 16 * q^86 - 8 * q^89 - 96 * q^91 + 4 * q^96

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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