LMFDB - Embedded newform 2550.2.d.f.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.f.2449.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +1.00000i q^{12} +4.00000i q^{13} -2.00000 q^{14} +1.00000 q^{16} +1.00000i q^{17} +1.00000i q^{18} -4.00000 q^{19} -2.00000 q^{21} +4.00000i q^{23} +1.00000 q^{24} +4.00000 q^{26} +1.00000i q^{27} +2.00000i q^{28} -2.00000 q^{29} -1.00000i q^{32} +1.00000 q^{34} +1.00000 q^{36} +2.00000i q^{37} +4.00000i q^{38} +4.00000 q^{39} -4.00000 q^{41} +2.00000i q^{42} +10.0000i q^{43} +4.00000 q^{46} +8.00000i q^{47} -1.00000i q^{48} +3.00000 q^{49} +1.00000 q^{51} -4.00000i q^{52} +2.00000i q^{53} +1.00000 q^{54} +2.00000 q^{56} +4.00000i q^{57} +2.00000i q^{58} +2.00000 q^{59} -14.0000 q^{61} +2.00000i q^{63} -1.00000 q^{64} -2.00000i q^{67} -1.00000i q^{68} +4.00000 q^{69} -6.00000 q^{71} -1.00000i q^{72} -4.00000i q^{73} +2.00000 q^{74} +4.00000 q^{76} -4.00000i q^{78} +12.0000 q^{79} +1.00000 q^{81} +4.00000i q^{82} +8.00000i q^{83} +2.00000 q^{84} +10.0000 q^{86} +2.00000i q^{87} +10.0000 q^{89} +8.00000 q^{91} -4.00000i q^{92} +8.00000 q^{94} -1.00000 q^{96} -8.00000i q^{97} -3.00000i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} - 4 q^{14} + 2 q^{16} - 8 q^{19} - 4 q^{21} + 2 q^{24} + 8 q^{26} - 4 q^{29} + 2 q^{34} + 2 q^{36} + 8 q^{39} - 8 q^{41} + 8 q^{46} + 6 q^{49} + 2 q^{51} + 2 q^{54} + 4 q^{56} + 4 q^{59} - 28 q^{61} - 2 q^{64} + 8 q^{69} - 12 q^{71} + 4 q^{74} + 8 q^{76} + 24 q^{79} + 2 q^{81} + 4 q^{84} + 20 q^{86} + 20 q^{89} + 16 q^{91} + 16 q^{94} - 2 q^{96}+O(q^{100}) $ 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9 - 4 * q^14 + 2 * q^16 - 8 * q^19 - 4 * q^21 + 2 * q^24 + 8 * q^26 - 4 * q^29 + 2 * q^34 + 2 * q^36 + 8 * q^39 - 8 * q^41 + 8 * q^46 + 6 * q^49 + 2 * q^51 + 2 * q^54 + 4 * q^56 + 4 * q^59 - 28 * q^61 - 2 * q^64 + 8 * q^69 - 12 * q^71 + 4 * q^74 + 8 * q^76 + 24 * q^79 + 2 * q^81 + 4 * q^84 + 20 * q^86 + 20 * q^89 + 16 * q^91 + 16 * q^94 - 2 * 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
$7$	− 2.00000i	− 0.755929i	−0.925820	−	0.377964i	$-0.876624\pi$
$7$	− 2.00000i	− 0.755929i	0.925820	−	0.377964i	$-0.123376\pi$
$8$	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
$13$	4.00000i	1.10940i	0.832050	+	0.554700i	$0.187167\pi$
$13$	4.00000i	1.10940i	−0.832050	+	0.554700i	$0.812833\pi$
$14$	−2.00000	−0.534522
$15$	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
$21$	−2.00000	−0.436436
$22$	0	0
$23$	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
$26$	4.00000	0.784465
$27$	1.00000i	0.192450i
$28$	2.00000i	0.377964i
$29$	−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
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	− 1.00000i	− 0.176777i
$33$	0	0
$34$	1.00000	0.171499
$35$	0	0
$36$	1.00000	0.166667
$37$	2.00000i	0.328798i	0.986394	+	0.164399i	$0.0525685\pi$
$37$	2.00000i	0.328798i	−0.986394	+	0.164399i	$0.947432\pi$
$38$	4.00000i	0.648886i
$39$	4.00000	0.640513
$40$	0	0
$41$	−4.00000	−0.624695	−0.312348	−	0.949968i	$-0.601115\pi$
$41$	−4.00000	−0.624695	−0.312348	+	0.949968i	$0.601115\pi$
$42$	2.00000i	0.308607i
$43$	10.0000i	1.52499i	0.646997	+	0.762493i	$0.276025\pi$
$43$	10.0000i	1.52499i	−0.646997	+	0.762493i	$0.723975\pi$
$44$	0	0
$45$	0	0
$46$	4.00000	0.589768
$47$	8.00000i	1.16692i	0.812142	+	0.583460i	$0.198301\pi$
$47$	8.00000i	1.16692i	−0.812142	+	0.583460i	$0.801699\pi$
$48$	− 1.00000i	− 0.144338i
$49$	3.00000	0.428571
$50$	0	0
$51$	1.00000	0.140028
$52$	− 4.00000i	− 0.554700i
$53$	2.00000i	0.274721i	0.990521	+	0.137361i	$0.0438619\pi$
$53$	2.00000i	0.274721i	−0.990521	+	0.137361i	$0.956138\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	2.00000	0.267261
$57$	4.00000i	0.529813i
$58$	2.00000i	0.262613i
$59$	2.00000	0.260378	0.130189	−	0.991489i	$-0.458442\pi$
$59$	2.00000	0.260378	0.130189	+	0.991489i	$0.458442\pi$
$60$	0	0
$61$	−14.0000	−1.79252	−0.896258	−	0.443533i	$-0.853725\pi$
$61$	−14.0000	−1.79252	−0.896258	+	0.443533i	$0.853725\pi$
$62$	0	0
$63$	2.00000i	0.251976i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	− 2.00000i	− 0.244339i	−0.992509	−	0.122169i	$-0.961015\pi$
$67$	− 2.00000i	− 0.244339i	0.992509	−	0.122169i	$-0.0389851\pi$
$68$	− 1.00000i	− 0.121268i
$69$	4.00000	0.481543
$70$	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
$73$	− 4.00000i	− 0.468165i	−0.972217	−	0.234082i	$-0.924791\pi$
$73$	− 4.00000i	− 0.468165i	0.972217	−	0.234082i	$-0.0752085\pi$
$74$	2.00000	0.232495
$75$	0	0
$76$	4.00000	0.458831
$77$	0	0
$78$	− 4.00000i	− 0.452911i
$79$	12.0000	1.35011	0.675053	−	0.737769i	$-0.264121\pi$
$79$	12.0000	1.35011	0.675053	+	0.737769i	$0.264121\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	4.00000i	0.441726i
$83$	8.00000i	0.878114i	0.898459	+	0.439057i	$0.144687\pi$
$83$	8.00000i	0.878114i	−0.898459	+	0.439057i	$0.855313\pi$
$84$	2.00000	0.218218
$85$	0	0
$86$	10.0000	1.07833
$87$	2.00000i	0.214423i
$88$	0	0
$89$	10.0000	1.06000	0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000	1.06000	0.529999	+	0.847998i	$0.322192\pi$
$90$	0	0
$91$	8.00000	0.838628
$92$	− 4.00000i	− 0.417029i
$93$	0	0
$94$	8.00000	0.825137
$95$	0	0
$96$	−1.00000	−0.102062
$97$	− 8.00000i	− 0.812277i	−0.913812	−	0.406138i	$-0.866875\pi$
$97$	− 8.00000i	− 0.812277i	0.913812	−	0.406138i	$-0.133125\pi$
$98$	− 3.00000i	− 0.303046i
$99$	0	0
$100$	0	0
$101$	−4.00000	−0.398015	−0.199007	−	0.979998i	$-0.563772\pi$
$101$	−4.00000	−0.398015	−0.199007	+	0.979998i	$0.563772\pi$
$102$	− 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	$0.871045\pi$
$104$	−4.00000	−0.392232
$105$	0	0
$106$	2.00000	0.194257
$107$	12.0000i	1.16008i	0.814587	+	0.580042i	$0.196964\pi$
$107$	12.0000i	1.16008i	−0.814587	+	0.580042i	$0.803036\pi$
$108$	− 1.00000i	− 0.0962250i
$109$	−6.00000	−0.574696	−0.287348	−	0.957826i	$-0.592774\pi$
$109$	−6.00000	−0.574696	−0.287348	+	0.957826i	$0.592774\pi$
$110$	0	0
$111$	2.00000	0.189832
$112$	− 2.00000i	− 0.188982i
$113$	− 6.00000i	− 0.564433i	−0.959351	−	0.282216i	$-0.908930\pi$
$113$	− 6.00000i	− 0.564433i	0.959351	−	0.282216i	$-0.0910696\pi$
$114$	4.00000	0.374634
$115$	0	0
$116$	2.00000	0.185695
$117$	− 4.00000i	− 0.369800i
$118$	− 2.00000i	− 0.184115i
$119$	2.00000	0.183340
$120$	0	0
$121$	−11.0000	−1.00000
$122$	14.0000i	1.26750i
$123$	4.00000i	0.360668i
$124$	0	0
$125$	0	0
$126$	2.00000	0.178174
$127$	− 16.0000i	− 1.41977i	−0.704317	−	0.709885i	$-0.748747\pi$
$127$	− 16.0000i	− 1.41977i	0.704317	−	0.709885i	$-0.251253\pi$
$128$	1.00000i	0.0883883i
$129$	10.0000	0.880451
$130$	0	0
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	0	0
$133$	8.00000i	0.693688i
$134$	−2.00000	−0.172774
$135$	0	0
$136$	−1.00000	−0.0857493
$137$	18.0000i	1.53784i	0.639343	+	0.768922i	$0.279207\pi$
$137$	18.0000i	1.53784i	−0.639343	+	0.768922i	$0.720793\pi$
$138$	− 4.00000i	− 0.340503i
$139$	12.0000	1.01783	0.508913	−	0.860818i	$-0.330047\pi$
$139$	12.0000	1.01783	0.508913	+	0.860818i	$0.330047\pi$
$140$	0	0
$141$	8.00000	0.673722
$142$	6.00000i	0.503509i
$143$	0	0
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	−4.00000	−0.331042
$147$	− 3.00000i	− 0.247436i
$148$	− 2.00000i	− 0.164399i
$149$	12.0000	0.983078	0.491539	−	0.870855i	$-0.336434\pi$
$149$	12.0000	0.983078	0.491539	+	0.870855i	$0.336434\pi$
$150$	0	0
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	− 4.00000i	− 0.324443i
$153$	− 1.00000i	− 0.0808452i
$154$	0	0
$155$	0	0
$156$	−4.00000	−0.320256
$157$	− 4.00000i	− 0.319235i	−0.987179	−	0.159617i	$-0.948974\pi$
$157$	− 4.00000i	− 0.319235i	0.987179	−	0.159617i	$-0.0510260\pi$
$158$	− 12.0000i	− 0.954669i
$159$	2.00000	0.158610
$160$	0	0
$161$	8.00000	0.630488
$162$	− 1.00000i	− 0.0785674i
$163$	24.0000i	1.87983i	0.341415	+	0.939913i	$0.389094\pi$
$163$	24.0000i	1.87983i	−0.341415	+	0.939913i	$0.610906\pi$
$164$	4.00000	0.312348
$165$	0	0
$166$	8.00000	0.620920
$167$	8.00000i	0.619059i	0.950890	+	0.309529i	$0.100171\pi$
$167$	8.00000i	0.619059i	−0.950890	+	0.309529i	$0.899829\pi$
$168$	− 2.00000i	− 0.154303i
$169$	−3.00000	−0.230769
$170$	0	0
$171$	4.00000	0.305888
$172$	− 10.0000i	− 0.762493i
$173$	− 2.00000i	− 0.152057i	−0.997106	−	0.0760286i	$-0.975776\pi$
$173$	− 2.00000i	− 0.152057i	0.997106	−	0.0760286i	$-0.0242240\pi$
$174$	2.00000	0.151620
$175$	0	0
$176$	0	0
$177$	− 2.00000i	− 0.150329i
$178$	− 10.0000i	− 0.749532i
$179$	−18.0000	−1.34538	−0.672692	−	0.739923i	$-0.734862\pi$
$179$	−18.0000	−1.34538	−0.672692	+	0.739923i	$0.734862\pi$
$180$	0	0
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	− 8.00000i	− 0.592999i
$183$	14.0000i	1.03491i
$184$	−4.00000	−0.294884
$185$	0	0
$186$	0	0
$187$	0	0
$188$	− 8.00000i	− 0.583460i
$189$	2.00000	0.145479
$190$	0	0
$191$	12.0000	0.868290	0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000	0.868290	0.434145	+	0.900843i	$0.357051\pi$
$192$	1.00000i	0.0721688i
$193$	− 8.00000i	− 0.575853i	−0.957653	−	0.287926i	$-0.907034\pi$
$193$	− 8.00000i	− 0.575853i	0.957653	−	0.287926i	$-0.0929658\pi$
$194$	−8.00000	−0.574367
$195$	0	0
$196$	−3.00000	−0.214286
$197$	− 6.00000i	− 0.427482i	−0.976890	−	0.213741i	$-0.931435\pi$
$197$	− 6.00000i	− 0.427482i	0.976890	−	0.213741i	$-0.0685649\pi$
$198$	0	0
$199$	−16.0000	−1.13421	−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000	−1.13421	−0.567105	+	0.823646i	$0.691937\pi$
$200$	0	0
$201$	−2.00000	−0.141069
$202$	4.00000i	0.281439i
$203$	4.00000i	0.280745i
$204$	−1.00000	−0.0700140
$205$	0	0
$206$	8.00000	0.557386
$207$	− 4.00000i	− 0.278019i
$208$	4.00000i	0.277350i
$209$	0	0
$210$	0	0
$211$	12.0000	0.826114	0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000	0.826114	0.413057	+	0.910705i	$0.364461\pi$
$212$	− 2.00000i	− 0.137361i
$213$	6.00000i	0.411113i
$214$	12.0000	0.820303
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	0	0
$218$	6.00000i	0.406371i
$219$	−4.00000	−0.270295
$220$	0	0
$221$	−4.00000	−0.269069
$222$	− 2.00000i	− 0.134231i
$223$	− 16.0000i	− 1.07144i	−0.844396	−	0.535720i	$-0.820040\pi$
$223$	− 16.0000i	− 1.07144i	0.844396	−	0.535720i	$-0.179960\pi$
$224$	−2.00000	−0.133631
$225$	0	0
$226$	−6.00000	−0.399114
$227$	12.0000i	0.796468i	0.917284	+	0.398234i	$0.130377\pi$
$227$	12.0000i	0.796468i	−0.917284	+	0.398234i	$0.869623\pi$
$228$	− 4.00000i	− 0.264906i
$229$	2.00000	0.132164	0.0660819	−	0.997814i	$-0.478950\pi$
$229$	2.00000	0.132164	0.0660819	+	0.997814i	$0.478950\pi$
$230$	0	0
$231$	0	0
$232$	− 2.00000i	− 0.131306i
$233$	14.0000i	0.917170i	0.888650	+	0.458585i	$0.151644\pi$
$233$	14.0000i	0.917170i	−0.888650	+	0.458585i	$0.848356\pi$
$234$	−4.00000	−0.261488
$235$	0	0
$236$	−2.00000	−0.130189
$237$	− 12.0000i	− 0.779484i
$238$	− 2.00000i	− 0.129641i
$239$	24.0000	1.55243	0.776215	−	0.630468i	$-0.217137\pi$
$239$	24.0000	1.55243	0.776215	+	0.630468i	$0.217137\pi$
$240$	0	0
$241$	−30.0000	−1.93247	−0.966235	−	0.257663i	$-0.917048\pi$
$241$	−30.0000	−1.93247	−0.966235	+	0.257663i	$0.917048\pi$
$242$	11.0000i	0.707107i
$243$	− 1.00000i	− 0.0641500i
$244$	14.0000	0.896258
$245$	0	0
$246$	4.00000	0.255031
$247$	− 16.0000i	− 1.01806i
$248$	0	0
$249$	8.00000	0.506979
$250$	0	0
$251$	−2.00000	−0.126239	−0.0631194	−	0.998006i	$-0.520105\pi$
$251$	−2.00000	−0.126239	−0.0631194	+	0.998006i	$0.520105\pi$
$252$	− 2.00000i	− 0.125988i
$253$	0	0
$254$	−16.0000	−1.00393
$255$	0	0
$256$	1.00000	0.0625000
$257$	14.0000i	0.873296i	0.899632	+	0.436648i	$0.143834\pi$
$257$	14.0000i	0.873296i	−0.899632	+	0.436648i	$0.856166\pi$
$258$	− 10.0000i	− 0.622573i
$259$	4.00000	0.248548
$260$	0	0
$261$	2.00000	0.123797
$262$	− 12.0000i	− 0.741362i
$263$	24.0000i	1.47990i	0.672660	+	0.739952i	$0.265152\pi$
$263$	24.0000i	1.47990i	−0.672660	+	0.739952i	$0.734848\pi$
$264$	0	0
$265$	0	0
$266$	8.00000	0.490511
$267$	− 10.0000i	− 0.611990i
$268$	2.00000i	0.122169i
$269$	−10.0000	−0.609711	−0.304855	−	0.952399i	$-0.598608\pi$
$269$	−10.0000	−0.609711	−0.304855	+	0.952399i	$0.598608\pi$
$270$	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
$273$	− 8.00000i	− 0.484182i
$274$	18.0000	1.08742
$275$	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	$-0.0971271\pi$
$278$	− 12.0000i	− 0.719712i
$279$	0	0
$280$	0	0
$281$	22.0000	1.31241	0.656205	−	0.754583i	$-0.272161\pi$
$281$	22.0000	1.31241	0.656205	+	0.754583i	$0.272161\pi$
$282$	− 8.00000i	− 0.476393i
$283$	− 28.0000i	− 1.66443i	−0.554455	−	0.832214i	$-0.687073\pi$
$283$	− 28.0000i	− 1.66443i	0.554455	−	0.832214i	$-0.312927\pi$
$284$	6.00000	0.356034
$285$	0	0
$286$	0	0
$287$	8.00000i	0.472225i
$288$	1.00000i	0.0589256i
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	−8.00000	−0.468968
$292$	4.00000i	0.234082i
$293$	22.0000i	1.28525i	0.766179	+	0.642627i	$0.222155\pi$
$293$	22.0000i	1.28525i	−0.766179	+	0.642627i	$0.777845\pi$
$294$	−3.00000	−0.174964
$295$	0	0
$296$	−2.00000	−0.116248
$297$	0	0
$298$	− 12.0000i	− 0.695141i
$299$	−16.0000	−0.925304
$300$	0	0
$301$	20.0000	1.15278
$302$	− 8.00000i	− 0.460348i
$303$	4.00000i	0.229794i
$304$	−4.00000	−0.229416
$305$	0	0
$306$	−1.00000	−0.0571662
$307$	− 10.0000i	− 0.570730i	−0.958419	−	0.285365i	$-0.907885\pi$
$307$	− 10.0000i	− 0.570730i	0.958419	−	0.285365i	$-0.0921148\pi$
$308$	0	0
$309$	8.00000	0.455104
$310$	0	0
$311$	−10.0000	−0.567048	−0.283524	−	0.958965i	$-0.591504\pi$
$311$	−10.0000	−0.567048	−0.283524	+	0.958965i	$0.591504\pi$
$312$	4.00000i	0.226455i
$313$	− 16.0000i	− 0.904373i	−0.891923	−	0.452187i	$-0.850644\pi$
$313$	− 16.0000i	− 0.904373i	0.891923	−	0.452187i	$-0.149356\pi$
$314$	−4.00000	−0.225733
$315$	0	0
$316$	−12.0000	−0.675053
$317$	− 30.0000i	− 1.68497i	−0.538721	−	0.842484i	$-0.681092\pi$
$317$	− 30.0000i	− 1.68497i	0.538721	−	0.842484i	$-0.318908\pi$
$318$	− 2.00000i	− 0.112154i
$319$	0	0
$320$	0	0
$321$	12.0000	0.669775
$322$	− 8.00000i	− 0.445823i
$323$	− 4.00000i	− 0.222566i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	24.0000	1.32924
$327$	6.00000i	0.331801i
$328$	− 4.00000i	− 0.220863i
$329$	16.0000	0.882109
$330$	0	0
$331$	−16.0000	−0.879440	−0.439720	−	0.898135i	$-0.644922\pi$
$331$	−16.0000	−0.879440	−0.439720	+	0.898135i	$0.644922\pi$
$332$	− 8.00000i	− 0.439057i
$333$	− 2.00000i	− 0.109599i
$334$	8.00000	0.437741
$335$	0	0
$336$	−2.00000	−0.109109
$337$	− 12.0000i	− 0.653682i	−0.945079	−	0.326841i	$-0.894016\pi$
$337$	− 12.0000i	− 0.653682i	0.945079	−	0.326841i	$-0.105984\pi$
$338$	3.00000i	0.163178i
$339$	−6.00000	−0.325875
$340$	0	0
$341$	0	0
$342$	− 4.00000i	− 0.216295i
$343$	− 20.0000i	− 1.07990i
$344$	−10.0000	−0.539164
$345$	0	0
$346$	−2.00000	−0.107521
$347$	20.0000i	1.07366i	0.843692	+	0.536828i	$0.180378\pi$
$347$	20.0000i	1.07366i	−0.843692	+	0.536828i	$0.819622\pi$
$348$	− 2.00000i	− 0.107211i
$349$	−2.00000	−0.107058	−0.0535288	−	0.998566i	$-0.517047\pi$
$349$	−2.00000	−0.107058	−0.0535288	+	0.998566i	$0.517047\pi$
$350$	0	0
$351$	−4.00000	−0.213504
$352$	0	0
$353$	18.0000i	0.958043i	0.877803	+	0.479022i	$0.159008\pi$
$353$	18.0000i	0.958043i	−0.877803	+	0.479022i	$0.840992\pi$
$354$	−2.00000	−0.106299
$355$	0	0
$356$	−10.0000	−0.529999
$357$	− 2.00000i	− 0.105851i
$358$	18.0000i	0.951330i
$359$	−20.0000	−1.05556	−0.527780	−	0.849381i	$-0.676975\pi$
$359$	−20.0000	−1.05556	−0.527780	+	0.849381i	$0.676975\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	10.0000i	0.525588i
$363$	11.0000i	0.577350i
$364$	−8.00000	−0.419314
$365$	0	0
$366$	14.0000	0.731792
$367$	2.00000i	0.104399i	0.998637	+	0.0521996i	$0.0166232\pi$
$367$	2.00000i	0.104399i	−0.998637	+	0.0521996i	$0.983377\pi$
$368$	4.00000i	0.208514i
$369$	4.00000	0.208232
$370$	0	0
$371$	4.00000	0.207670
$372$	0	0
$373$	8.00000i	0.414224i	0.978317	+	0.207112i	$0.0664065\pi$
$373$	8.00000i	0.414224i	−0.978317	+	0.207112i	$0.933593\pi$
$374$	0	0
$375$	0	0
$376$	−8.00000	−0.412568
$377$	− 8.00000i	− 0.412021i
$378$	− 2.00000i	− 0.102869i
$379$	−28.0000	−1.43826	−0.719132	−	0.694874i	$-0.755460\pi$
$379$	−28.0000	−1.43826	−0.719132	+	0.694874i	$0.755460\pi$
$380$	0	0
$381$	−16.0000	−0.819705
$382$	− 12.0000i	− 0.613973i
$383$	16.0000i	0.817562i	0.912633	+	0.408781i	$0.134046\pi$
$383$	16.0000i	0.817562i	−0.912633	+	0.408781i	$0.865954\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	−8.00000	−0.407189
$387$	− 10.0000i	− 0.508329i
$388$	8.00000i	0.406138i
$389$	−12.0000	−0.608424	−0.304212	−	0.952604i	$-0.598393\pi$
$389$	−12.0000	−0.608424	−0.304212	+	0.952604i	$0.598393\pi$
$390$	0	0
$391$	−4.00000	−0.202289
$392$	3.00000i	0.151523i
$393$	− 12.0000i	− 0.605320i
$394$	−6.00000	−0.302276
$395$	0	0
$396$	0	0
$397$	10.0000i	0.501886i	0.968002	+	0.250943i	$0.0807406\pi$
$397$	10.0000i	0.501886i	−0.968002	+	0.250943i	$0.919259\pi$
$398$	16.0000i	0.802008i
$399$	8.00000	0.400501
$400$	0	0
$401$	−24.0000	−1.19850	−0.599251	−	0.800561i	$-0.704535\pi$
$401$	−24.0000	−1.19850	−0.599251	+	0.800561i	$0.704535\pi$
$402$	2.00000i	0.0997509i
$403$	0	0
$404$	4.00000	0.199007
$405$	0	0
$406$	4.00000	0.198517
$407$	0	0
$408$	1.00000i	0.0495074i
$409$	26.0000	1.28562	0.642809	−	0.766027i	$-0.277769\pi$
$409$	26.0000	1.28562	0.642809	+	0.766027i	$0.277769\pi$
$410$	0	0
$411$	18.0000	0.887875
$412$	− 8.00000i	− 0.394132i
$413$	− 4.00000i	− 0.196827i
$414$	−4.00000	−0.196589
$415$	0	0
$416$	4.00000	0.196116
$417$	− 12.0000i	− 0.587643i
$418$	0	0
$419$	4.00000	0.195413	0.0977064	−	0.995215i	$-0.468849\pi$
$419$	4.00000	0.195413	0.0977064	+	0.995215i	$0.468849\pi$
$420$	0	0
$421$	−34.0000	−1.65706	−0.828529	−	0.559946i	$-0.810822\pi$
$421$	−34.0000	−1.65706	−0.828529	+	0.559946i	$0.810822\pi$
$422$	− 12.0000i	− 0.584151i
$423$	− 8.00000i	− 0.388973i
$424$	−2.00000	−0.0971286
$425$	0	0
$426$	6.00000	0.290701
$427$	28.0000i	1.35501i
$428$	− 12.0000i	− 0.580042i
$429$	0	0
$430$	0	0
$431$	−34.0000	−1.63772	−0.818861	−	0.573992i	$-0.805394\pi$
$431$	−34.0000	−1.63772	−0.818861	+	0.573992i	$0.805394\pi$
$432$	1.00000i	0.0481125i
$433$	− 34.0000i	− 1.63394i	−0.576683	−	0.816968i	$-0.695653\pi$
$433$	− 34.0000i	− 1.63394i	0.576683	−	0.816968i	$-0.304347\pi$
$434$	0	0
$435$	0	0
$436$	6.00000	0.287348
$437$	− 16.0000i	− 0.765384i
$438$	4.00000i	0.191127i
$439$	12.0000	0.572729	0.286364	−	0.958121i	$-0.407553\pi$
$439$	12.0000	0.572729	0.286364	+	0.958121i	$0.407553\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	4.00000i	0.190261i
$443$	− 8.00000i	− 0.380091i	−0.981775	−	0.190046i	$-0.939136\pi$
$443$	− 8.00000i	− 0.380091i	0.981775	−	0.190046i	$-0.0608636\pi$
$444$	−2.00000	−0.0949158
$445$	0	0
$446$	−16.0000	−0.757622
$447$	− 12.0000i	− 0.567581i
$448$	2.00000i	0.0944911i
$449$	−40.0000	−1.88772	−0.943858	−	0.330350i	$-0.892833\pi$
$449$	−40.0000	−1.88772	−0.943858	+	0.330350i	$0.892833\pi$
$450$	0	0
$451$	0	0
$452$	6.00000i	0.282216i
$453$	− 8.00000i	− 0.375873i
$454$	12.0000	0.563188
$455$	0	0
$456$	−4.00000	−0.187317
$457$	34.0000i	1.59045i	0.606313	+	0.795226i	$0.292648\pi$
$457$	34.0000i	1.59045i	−0.606313	+	0.795226i	$0.707352\pi$
$458$	− 2.00000i	− 0.0934539i
$459$	−1.00000	−0.0466760
$460$	0	0
$461$	0	0		−	1.00000i	$-0.5\pi$
$461$	0	0			1.00000i	$0.5\pi$
$462$	0	0
$463$	16.0000i	0.743583i	0.928316	+	0.371792i	$0.121256\pi$
$463$	16.0000i	0.743583i	−0.928316	+	0.371792i	$0.878744\pi$
$464$	−2.00000	−0.0928477
$465$	0	0
$466$	14.0000	0.648537
$467$	36.0000i	1.66588i	0.553362	+	0.832941i	$0.313345\pi$
$467$	36.0000i	1.66588i	−0.553362	+	0.832941i	$0.686655\pi$
$468$	4.00000i	0.184900i
$469$	−4.00000	−0.184703
$470$	0	0
$471$	−4.00000	−0.184310
$472$	2.00000i	0.0920575i
$473$	0	0
$474$	−12.0000	−0.551178
$475$	0	0
$476$	−2.00000	−0.0916698
$477$	− 2.00000i	− 0.0915737i
$478$	− 24.0000i	− 1.09773i
$479$	−2.00000	−0.0913823	−0.0456912	−	0.998956i	$-0.514549\pi$
$479$	−2.00000	−0.0913823	−0.0456912	+	0.998956i	$0.514549\pi$
$480$	0	0
$481$	−8.00000	−0.364769
$482$	30.0000i	1.36646i
$483$	− 8.00000i	− 0.364013i
$484$	11.0000	0.500000
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	22.0000i	0.996915i	0.866914	+	0.498458i	$0.166100\pi$
$487$	22.0000i	0.996915i	−0.866914	+	0.498458i	$0.833900\pi$
$488$	− 14.0000i	− 0.633750i
$489$	24.0000	1.08532
$490$	0	0
$491$	−14.0000	−0.631811	−0.315906	−	0.948791i	$-0.602308\pi$
$491$	−14.0000	−0.631811	−0.315906	+	0.948791i	$0.602308\pi$
$492$	− 4.00000i	− 0.180334i
$493$	− 2.00000i	− 0.0900755i
$494$	−16.0000	−0.719874
$495$	0	0
$496$	0	0
$497$	12.0000i	0.538274i
$498$	− 8.00000i	− 0.358489i
$499$	12.0000	0.537194	0.268597	−	0.963253i	$-0.413440\pi$
$499$	12.0000	0.537194	0.268597	+	0.963253i	$0.413440\pi$
$500$	0	0
$501$	8.00000	0.357414
$502$	2.00000i	0.0892644i
$503$	28.0000i	1.24846i	0.781241	+	0.624229i	$0.214587\pi$
$503$	28.0000i	1.24846i	−0.781241	+	0.624229i	$0.785413\pi$
$504$	−2.00000	−0.0890871
$505$	0	0
$506$	0	0
$507$	3.00000i	0.133235i
$508$	16.0000i	0.709885i
$509$	−4.00000	−0.177297	−0.0886484	−	0.996063i	$-0.528255\pi$
$509$	−4.00000	−0.177297	−0.0886484	+	0.996063i	$0.528255\pi$
$510$	0	0
$511$	−8.00000	−0.353899
$512$	− 1.00000i	− 0.0441942i
$513$	− 4.00000i	− 0.176604i
$514$	14.0000	0.617514
$515$	0	0
$516$	−10.0000	−0.440225
$517$	0	0
$518$	− 4.00000i	− 0.175750i
$519$	−2.00000	−0.0877903
$520$	0	0
$521$	16.0000	0.700973	0.350486	−	0.936568i	$-0.386016\pi$
$521$	16.0000	0.700973	0.350486	+	0.936568i	$0.386016\pi$
$522$	− 2.00000i	− 0.0875376i
$523$	− 18.0000i	− 0.787085i	−0.919306	−	0.393543i	$-0.871249\pi$
$523$	− 18.0000i	− 0.787085i	0.919306	−	0.393543i	$-0.128751\pi$
$524$	−12.0000	−0.524222
$525$	0	0
$526$	24.0000	1.04645
$527$	0	0
$528$	0	0
$529$	7.00000	0.304348
$530$	0	0
$531$	−2.00000	−0.0867926
$532$	− 8.00000i	− 0.346844i
$533$	− 16.0000i	− 0.693037i
$534$	−10.0000	−0.432742
$535$	0	0
$536$	2.00000	0.0863868
$537$	18.0000i	0.776757i
$538$	10.0000i	0.431131i
$539$	0	0
$540$	0	0
$541$	2.00000	0.0859867	0.0429934	−	0.999075i	$-0.486311\pi$
$541$	2.00000	0.0859867	0.0429934	+	0.999075i	$0.486311\pi$
$542$	8.00000i	0.343629i
$543$	10.0000i	0.429141i
$544$	1.00000	0.0428746
$545$	0	0
$546$	−8.00000	−0.342368
$547$	36.0000i	1.53925i	0.638497	+	0.769624i	$0.279557\pi$
$547$	36.0000i	1.53925i	−0.638497	+	0.769624i	$0.720443\pi$
$548$	− 18.0000i	− 0.768922i
$549$	14.0000	0.597505
$550$	0	0
$551$	8.00000	0.340811
$552$	4.00000i	0.170251i
$553$	− 24.0000i	− 1.02058i
$554$	−10.0000	−0.424859
$555$	0	0
$556$	−12.0000	−0.508913
$557$	46.0000i	1.94908i	0.224208	+	0.974541i	$0.428020\pi$
$557$	46.0000i	1.94908i	−0.224208	+	0.974541i	$0.571980\pi$
$558$	0	0
$559$	−40.0000	−1.69182
$560$	0	0
$561$	0	0
$562$	− 22.0000i	− 0.928014i
$563$	− 32.0000i	− 1.34864i	−0.738440	−	0.674320i	$-0.764437\pi$
$563$	− 32.0000i	− 1.34864i	0.738440	−	0.674320i	$-0.235563\pi$
$564$	−8.00000	−0.336861
$565$	0	0
$566$	−28.0000	−1.17693
$567$	− 2.00000i	− 0.0839921i
$568$	− 6.00000i	− 0.251754i
$569$	−42.0000	−1.76073	−0.880366	−	0.474295i	$-0.842703\pi$
$569$	−42.0000	−1.76073	−0.880366	+	0.474295i	$0.842703\pi$
$570$	0	0
$571$	−12.0000	−0.502184	−0.251092	−	0.967963i	$-0.580790\pi$
$571$	−12.0000	−0.502184	−0.251092	+	0.967963i	$0.580790\pi$
$572$	0	0
$573$	− 12.0000i	− 0.501307i
$574$	8.00000	0.333914
$575$	0	0
$576$	1.00000	0.0416667
$577$	− 22.0000i	− 0.915872i	−0.888985	−	0.457936i	$-0.848589\pi$
$577$	− 22.0000i	− 0.915872i	0.888985	−	0.457936i	$-0.151411\pi$
$578$	1.00000i	0.0415945i
$579$	−8.00000	−0.332469
$580$	0	0
$581$	16.0000	0.663792
$582$	8.00000i	0.331611i
$583$	0	0
$584$	4.00000	0.165521
$585$	0	0
$586$	22.0000	0.908812
$587$	− 8.00000i	− 0.330195i	−0.986277	−	0.165098i	$-0.947206\pi$
$587$	− 8.00000i	− 0.330195i	0.986277	−	0.165098i	$-0.0527939\pi$
$588$	3.00000i	0.123718i
$589$	0	0
$590$	0	0
$591$	−6.00000	−0.246807
$592$	2.00000i	0.0821995i
$593$	− 30.0000i	− 1.23195i	−0.787765	−	0.615976i	$-0.788762\pi$
$593$	− 30.0000i	− 1.23195i	0.787765	−	0.615976i	$-0.211238\pi$
$594$	0	0
$595$	0	0
$596$	−12.0000	−0.491539
$597$	16.0000i	0.654836i
$598$	16.0000i	0.654289i
$599$	−12.0000	−0.490307	−0.245153	−	0.969484i	$-0.578838\pi$
$599$	−12.0000	−0.490307	−0.245153	+	0.969484i	$0.578838\pi$
$600$	0	0
$601$	38.0000	1.55005	0.775026	−	0.631929i	$-0.217737\pi$
$601$	38.0000	1.55005	0.775026	+	0.631929i	$0.217737\pi$
$602$	− 20.0000i	− 0.815139i
$603$	2.00000i	0.0814463i
$604$	−8.00000	−0.325515
$605$	0	0
$606$	4.00000	0.162489
$607$	14.0000i	0.568242i	0.958788	+	0.284121i	$0.0917018\pi$
$607$	14.0000i	0.568242i	−0.958788	+	0.284121i	$0.908298\pi$
$608$	4.00000i	0.162221i
$609$	4.00000	0.162088
$610$	0	0
$611$	−32.0000	−1.29458
$612$	1.00000i	0.0404226i
$613$	24.0000i	0.969351i	0.874694	+	0.484675i	$0.161062\pi$
$613$	24.0000i	0.969351i	−0.874694	+	0.484675i	$0.838938\pi$
$614$	−10.0000	−0.403567
$615$	0	0
$616$	0	0
$617$	− 6.00000i	− 0.241551i	−0.992680	−	0.120775i	$-0.961462\pi$
$617$	− 6.00000i	− 0.241551i	0.992680	−	0.120775i	$-0.0385381\pi$
$618$	− 8.00000i	− 0.321807i
$619$	−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
$622$	10.0000i	0.400963i
$623$	− 20.0000i	− 0.801283i
$624$	4.00000	0.160128
$625$	0	0
$626$	−16.0000	−0.639489
$627$	0	0
$628$	4.00000i	0.159617i
$629$	−2.00000	−0.0797452
$630$	0	0
$631$	32.0000	1.27390	0.636950	−	0.770905i	$-0.280196\pi$
$631$	32.0000	1.27390	0.636950	+	0.770905i	$0.280196\pi$
$632$	12.0000i	0.477334i
$633$	− 12.0000i	− 0.476957i
$634$	−30.0000	−1.19145
$635$	0	0
$636$	−2.00000	−0.0793052
$637$	12.0000i	0.475457i
$638$	0	0
$639$	6.00000	0.237356
$640$	0	0
$641$	40.0000	1.57991	0.789953	−	0.613168i	$-0.210105\pi$
$641$	40.0000	1.57991	0.789953	+	0.613168i	$0.210105\pi$
$642$	− 12.0000i	− 0.473602i
$643$	16.0000i	0.630978i	0.948929	+	0.315489i	$0.102169\pi$
$643$	16.0000i	0.630978i	−0.948929	+	0.315489i	$0.897831\pi$
$644$	−8.00000	−0.315244
$645$	0	0
$646$	−4.00000	−0.157378
$647$	8.00000i	0.314512i	0.987558	+	0.157256i	$0.0502649\pi$
$647$	8.00000i	0.314512i	−0.987558	+	0.157256i	$0.949735\pi$
$648$	1.00000i	0.0392837i
$649$	0	0
$650$	0	0
$651$	0	0
$652$	− 24.0000i	− 0.939913i
$653$	− 14.0000i	− 0.547862i	−0.961749	−	0.273931i	$-0.911676\pi$
$653$	− 14.0000i	− 0.547862i	0.961749	−	0.273931i	$-0.0883240\pi$
$654$	6.00000	0.234619
$655$	0	0
$656$	−4.00000	−0.156174
$657$	4.00000i	0.156055i
$658$	− 16.0000i	− 0.623745i
$659$	6.00000	0.233727	0.116863	−	0.993148i	$-0.462716\pi$
$659$	6.00000	0.233727	0.116863	+	0.993148i	$0.462716\pi$
$660$	0	0
$661$	−30.0000	−1.16686	−0.583432	−	0.812162i	$-0.698291\pi$
$661$	−30.0000	−1.16686	−0.583432	+	0.812162i	$0.698291\pi$
$662$	16.0000i	0.621858i
$663$	4.00000i	0.155347i
$664$	−8.00000	−0.310460
$665$	0	0
$666$	−2.00000	−0.0774984
$667$	− 8.00000i	− 0.309761i
$668$	− 8.00000i	− 0.309529i
$669$	−16.0000	−0.618596
$670$	0	0
$671$	0	0
$672$	2.00000i	0.0771517i
$673$	8.00000i	0.308377i	0.988041	+	0.154189i	$0.0492764\pi$
$673$	8.00000i	0.308377i	−0.988041	+	0.154189i	$0.950724\pi$
$674$	−12.0000	−0.462223
$675$	0	0
$676$	3.00000	0.115385
$677$	− 38.0000i	− 1.46046i	−0.683202	−	0.730229i	$-0.739413\pi$
$677$	− 38.0000i	− 1.46046i	0.683202	−	0.730229i	$-0.260587\pi$
$678$	6.00000i	0.230429i
$679$	−16.0000	−0.614024
$680$	0	0
$681$	12.0000	0.459841
$682$	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
$686$	−20.0000	−0.763604
$687$	− 2.00000i	− 0.0763048i
$688$	10.0000i	0.381246i
$689$	−8.00000	−0.304776
$690$	0	0
$691$	12.0000	0.456502	0.228251	−	0.973602i	$-0.426699\pi$
$691$	12.0000	0.456502	0.228251	+	0.973602i	$0.426699\pi$
$692$	2.00000i	0.0760286i
$693$	0	0
$694$	20.0000	0.759190
$695$	0	0
$696$	−2.00000	−0.0758098
$697$	− 4.00000i	− 0.151511i
$698$	2.00000i	0.0757011i
$699$	14.0000	0.529529
$700$	0	0
$701$	24.0000	0.906467	0.453234	−	0.891392i	$-0.350270\pi$
$701$	24.0000	0.906467	0.453234	+	0.891392i	$0.350270\pi$
$702$	4.00000i	0.150970i
$703$	− 8.00000i	− 0.301726i
$704$	0	0
$705$	0	0
$706$	18.0000	0.677439
$707$	8.00000i	0.300871i
$708$	2.00000i	0.0751646i
$709$	−26.0000	−0.976450	−0.488225	−	0.872718i	$-0.662356\pi$
$709$	−26.0000	−0.976450	−0.488225	+	0.872718i	$0.662356\pi$
$710$	0	0
$711$	−12.0000	−0.450035
$712$	10.0000i	0.374766i
$713$	0	0
$714$	−2.00000	−0.0748481
$715$	0	0
$716$	18.0000	0.672692
$717$	− 24.0000i	− 0.896296i
$718$	20.0000i	0.746393i
$719$	−26.0000	−0.969636	−0.484818	−	0.874615i	$-0.661114\pi$
$719$	−26.0000	−0.969636	−0.484818	+	0.874615i	$0.661114\pi$
$720$	0	0
$721$	16.0000	0.595871
$722$	3.00000i	0.111648i
$723$	30.0000i	1.11571i
$724$	10.0000	0.371647
$725$	0	0
$726$	11.0000	0.408248
$727$	24.0000i	0.890111i	0.895503	+	0.445055i	$0.146816\pi$
$727$	24.0000i	0.890111i	−0.895503	+	0.445055i	$0.853184\pi$
$728$	8.00000i	0.296500i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−10.0000	−0.369863
$732$	− 14.0000i	− 0.517455i
$733$	− 16.0000i	− 0.590973i	−0.955347	−	0.295487i	$-0.904518\pi$
$733$	− 16.0000i	− 0.590973i	0.955347	−	0.295487i	$-0.0954818\pi$
$734$	2.00000	0.0738213
$735$	0	0
$736$	4.00000	0.147442
$737$	0	0
$738$	− 4.00000i	− 0.147242i
$739$	16.0000	0.588570	0.294285	−	0.955718i	$-0.404919\pi$
$739$	16.0000	0.588570	0.294285	+	0.955718i	$0.404919\pi$
$740$	0	0
$741$	−16.0000	−0.587775
$742$	− 4.00000i	− 0.146845i
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	0	0
$745$	0	0
$746$	8.00000	0.292901
$747$	− 8.00000i	− 0.292705i
$748$	0	0
$749$	24.0000	0.876941
$750$	0	0
$751$	20.0000	0.729810	0.364905	−	0.931045i	$-0.381101\pi$
$751$	20.0000	0.729810	0.364905	+	0.931045i	$0.381101\pi$
$752$	8.00000i	0.291730i
$753$	2.00000i	0.0728841i
$754$	−8.00000	−0.291343
$755$	0	0
$756$	−2.00000	−0.0727393
$757$	− 52.0000i	− 1.88997i	−0.327111	−	0.944986i	$-0.606075\pi$
$757$	− 52.0000i	− 1.88997i	0.327111	−	0.944986i	$-0.393925\pi$
$758$	28.0000i	1.01701i
$759$	0	0
$760$	0	0
$761$	−54.0000	−1.95750	−0.978749	−	0.205061i	$-0.934261\pi$
$761$	−54.0000	−1.95750	−0.978749	+	0.205061i	$0.934261\pi$
$762$	16.0000i	0.579619i
$763$	12.0000i	0.434429i
$764$	−12.0000	−0.434145
$765$	0	0
$766$	16.0000	0.578103
$767$	8.00000i	0.288863i
$768$	− 1.00000i	− 0.0360844i
$769$	−14.0000	−0.504853	−0.252426	−	0.967616i	$-0.581229\pi$
$769$	−14.0000	−0.504853	−0.252426	+	0.967616i	$0.581229\pi$
$770$	0	0
$771$	14.0000	0.504198
$772$	8.00000i	0.287926i
$773$	42.0000i	1.51064i	0.655359	+	0.755318i	$0.272517\pi$
$773$	42.0000i	1.51064i	−0.655359	+	0.755318i	$0.727483\pi$
$774$	−10.0000	−0.359443
$775$	0	0
$776$	8.00000	0.287183
$777$	− 4.00000i	− 0.143499i
$778$	12.0000i	0.430221i
$779$	16.0000	0.573259
$780$	0	0
$781$	0	0
$782$	4.00000i	0.143040i
$783$	− 2.00000i	− 0.0714742i
$784$	3.00000	0.107143
$785$	0	0
$786$	−12.0000	−0.428026
$787$	− 8.00000i	− 0.285169i	−0.989783	−	0.142585i	$-0.954459\pi$
$787$	− 8.00000i	− 0.285169i	0.989783	−	0.142585i	$-0.0455413\pi$
$788$	6.00000i	0.213741i
$789$	24.0000	0.854423
$790$	0	0
$791$	−12.0000	−0.426671
$792$	0	0
$793$	− 56.0000i	− 1.98862i
$794$	10.0000	0.354887
$795$	0	0
$796$	16.0000	0.567105
$797$	− 6.00000i	− 0.212531i	−0.994338	−	0.106265i	$-0.966111\pi$
$797$	− 6.00000i	− 0.212531i	0.994338	−	0.106265i	$-0.0338893\pi$
$798$	− 8.00000i	− 0.283197i
$799$	−8.00000	−0.283020
$800$	0	0
$801$	−10.0000	−0.353333
$802$	24.0000i	0.847469i
$803$	0	0
$804$	2.00000	0.0705346
$805$	0	0
$806$	0	0
$807$	10.0000i	0.352017i
$808$	− 4.00000i	− 0.140720i
$809$	28.0000	0.984428	0.492214	−	0.870474i	$-0.336188\pi$
$809$	28.0000	0.984428	0.492214	+	0.870474i	$0.336188\pi$
$810$	0	0
$811$	52.0000	1.82597	0.912983	−	0.407997i	$-0.133772\pi$
$811$	52.0000	1.82597	0.912983	+	0.407997i	$0.133772\pi$
$812$	− 4.00000i	− 0.140372i
$813$	8.00000i	0.280572i
$814$	0	0
$815$	0	0
$816$	1.00000	0.0350070
$817$	− 40.0000i	− 1.39942i
$818$	− 26.0000i	− 0.909069i
$819$	−8.00000	−0.279543
$820$	0	0
$821$	30.0000	1.04701	0.523504	−	0.852023i	$-0.324625\pi$
$821$	30.0000	1.04701	0.523504	+	0.852023i	$0.324625\pi$
$822$	− 18.0000i	− 0.627822i
$823$	− 50.0000i	− 1.74289i	−0.490493	−	0.871445i	$-0.663183\pi$
$823$	− 50.0000i	− 1.74289i	0.490493	−	0.871445i	$-0.336817\pi$
$824$	−8.00000	−0.278693
$825$	0	0
$826$	−4.00000	−0.139178
$827$	− 52.0000i	− 1.80822i	−0.427303	−	0.904109i	$-0.640536\pi$
$827$	− 52.0000i	− 1.80822i	0.427303	−	0.904109i	$-0.359464\pi$
$828$	4.00000i	0.139010i
$829$	−2.00000	−0.0694629	−0.0347314	−	0.999397i	$-0.511058\pi$
$829$	−2.00000	−0.0694629	−0.0347314	+	0.999397i	$0.511058\pi$
$830$	0	0
$831$	−10.0000	−0.346896
$832$	− 4.00000i	− 0.138675i
$833$	3.00000i	0.103944i
$834$	−12.0000	−0.415526
$835$	0	0
$836$	0	0
$837$	0	0
$838$	− 4.00000i	− 0.138178i
$839$	−14.0000	−0.483334	−0.241667	−	0.970359i	$-0.577694\pi$
$839$	−14.0000	−0.483334	−0.241667	+	0.970359i	$0.577694\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	34.0000i	1.17172i
$843$	− 22.0000i	− 0.757720i
$844$	−12.0000	−0.413057
$845$	0	0
$846$	−8.00000	−0.275046
$847$	22.0000i	0.755929i
$848$	2.00000i	0.0686803i
$849$	−28.0000	−0.960958
$850$	0	0
$851$	−8.00000	−0.274236
$852$	− 6.00000i	− 0.205557i
$853$	42.0000i	1.43805i	0.694983	+	0.719026i	$0.255412\pi$
$853$	42.0000i	1.43805i	−0.694983	+	0.719026i	$0.744588\pi$
$854$	28.0000	0.958140
$855$	0	0
$856$	−12.0000	−0.410152
$857$	2.00000i	0.0683187i	0.999416	+	0.0341593i	$0.0108754\pi$
$857$	2.00000i	0.0683187i	−0.999416	+	0.0341593i	$0.989125\pi$
$858$	0	0
$859$	−24.0000	−0.818869	−0.409435	−	0.912339i	$-0.634274\pi$
$859$	−24.0000	−0.818869	−0.409435	+	0.912339i	$0.634274\pi$
$860$	0	0
$861$	8.00000	0.272639
$862$	34.0000i	1.15804i
$863$	32.0000i	1.08929i	0.838666	+	0.544646i	$0.183336\pi$
$863$	32.0000i	1.08929i	−0.838666	+	0.544646i	$0.816664\pi$
$864$	1.00000	0.0340207
$865$	0	0
$866$	−34.0000	−1.15537
$867$	1.00000i	0.0339618i
$868$	0	0
$869$	0	0
$870$	0	0
$871$	8.00000	0.271070
$872$	− 6.00000i	− 0.203186i
$873$	8.00000i	0.270759i
$874$	−16.0000	−0.541208
$875$	0	0
$876$	4.00000	0.135147
$877$	− 10.0000i	− 0.337676i	−0.985644	−	0.168838i	$-0.945999\pi$
$877$	− 10.0000i	− 0.337676i	0.985644	−	0.168838i	$-0.0540015\pi$
$878$	− 12.0000i	− 0.404980i
$879$	22.0000	0.742042
$880$	0	0
$881$	48.0000	1.61716	0.808581	−	0.588386i	$-0.200236\pi$
$881$	48.0000	1.61716	0.808581	+	0.588386i	$0.200236\pi$
$882$	3.00000i	0.101015i
$883$	− 2.00000i	− 0.0673054i	−0.999434	−	0.0336527i	$-0.989286\pi$
$883$	− 2.00000i	− 0.0673054i	0.999434	−	0.0336527i	$-0.0107140\pi$
$884$	4.00000	0.134535
$885$	0	0
$886$	−8.00000	−0.268765
$887$	− 20.0000i	− 0.671534i	−0.941945	−	0.335767i	$-0.891004\pi$
$887$	− 20.0000i	− 0.671534i	0.941945	−	0.335767i	$-0.108996\pi$
$888$	2.00000i	0.0671156i
$889$	−32.0000	−1.07325
$890$	0	0
$891$	0	0
$892$	16.0000i	0.535720i
$893$	− 32.0000i	− 1.07084i
$894$	−12.0000	−0.401340
$895$	0	0
$896$	2.00000	0.0668153
$897$	16.0000i	0.534224i
$898$	40.0000i	1.33482i
$899$	0	0
$900$	0	0
$901$	−2.00000	−0.0666297
$902$	0	0
$903$	− 20.0000i	− 0.665558i
$904$	6.00000	0.199557
$905$	0	0
$906$	−8.00000	−0.265782
$907$	8.00000i	0.265636i	0.991140	+	0.132818i	$0.0424025\pi$
$907$	8.00000i	0.265636i	−0.991140	+	0.132818i	$0.957597\pi$
$908$	− 12.0000i	− 0.398234i
$909$	4.00000	0.132672
$910$	0	0
$911$	18.0000	0.596367	0.298183	−	0.954509i	$-0.403619\pi$
$911$	18.0000	0.596367	0.298183	+	0.954509i	$0.403619\pi$
$912$	4.00000i	0.132453i
$913$	0	0
$914$	34.0000	1.12462
$915$	0	0
$916$	−2.00000	−0.0660819
$917$	− 24.0000i	− 0.792550i
$918$	1.00000i	0.0330049i
$919$	−32.0000	−1.05558	−0.527791	−	0.849374i	$-0.676980\pi$
$919$	−32.0000	−1.05558	−0.527791	+	0.849374i	$0.676980\pi$
$920$	0	0
$921$	−10.0000	−0.329511
$922$	0	0
$923$	− 24.0000i	− 0.789970i
$924$	0	0
$925$	0	0
$926$	16.0000	0.525793
$927$	− 8.00000i	− 0.262754i
$928$	2.00000i	0.0656532i
$929$	20.0000	0.656179	0.328089	−	0.944647i	$-0.393595\pi$
$929$	20.0000	0.656179	0.328089	+	0.944647i	$0.393595\pi$
$930$	0	0
$931$	−12.0000	−0.393284
$932$	− 14.0000i	− 0.458585i
$933$	10.0000i	0.327385i
$934$	36.0000	1.17796
$935$	0	0
$936$	4.00000	0.130744
$937$	− 54.0000i	− 1.76410i	−0.471153	−	0.882052i	$-0.656162\pi$
$937$	− 54.0000i	− 1.76410i	0.471153	−	0.882052i	$-0.343838\pi$
$938$	4.00000i	0.130605i
$939$	−16.0000	−0.522140
$940$	0	0
$941$	−10.0000	−0.325991	−0.162995	−	0.986627i	$-0.552116\pi$
$941$	−10.0000	−0.325991	−0.162995	+	0.986627i	$0.552116\pi$
$942$	4.00000i	0.130327i
$943$	− 16.0000i	− 0.521032i
$944$	2.00000	0.0650945
$945$	0	0
$946$	0	0
$947$	4.00000i	0.129983i	0.997886	+	0.0649913i	$0.0207020\pi$
$947$	4.00000i	0.129983i	−0.997886	+	0.0649913i	$0.979298\pi$
$948$	12.0000i	0.389742i
$949$	16.0000	0.519382
$950$	0	0
$951$	−30.0000	−0.972817
$952$	2.00000i	0.0648204i
$953$	− 34.0000i	− 1.10137i	−0.834714	−	0.550684i	$-0.814367\pi$
$953$	− 34.0000i	− 1.10137i	0.834714	−	0.550684i	$-0.185633\pi$
$954$	−2.00000	−0.0647524
$955$	0	0
$956$	−24.0000	−0.776215
$957$	0	0
$958$	2.00000i	0.0646171i
$959$	36.0000	1.16250
$960$	0	0
$961$	−31.0000	−1.00000
$962$	8.00000i	0.257930i
$963$	− 12.0000i	− 0.386695i
$964$	30.0000	0.966235
$965$	0	0
$966$	−8.00000	−0.257396
$967$	− 20.0000i	− 0.643157i	−0.946883	−	0.321578i	$-0.895787\pi$
$967$	− 20.0000i	− 0.643157i	0.946883	−	0.321578i	$-0.104213\pi$
$968$	− 11.0000i	− 0.353553i
$969$	−4.00000	−0.128499
$970$	0	0
$971$	−6.00000	−0.192549	−0.0962746	−	0.995355i	$-0.530693\pi$
$971$	−6.00000	−0.192549	−0.0962746	+	0.995355i	$0.530693\pi$
$972$	1.00000i	0.0320750i
$973$	− 24.0000i	− 0.769405i
$974$	22.0000	0.704925
$975$	0	0
$976$	−14.0000	−0.448129
$977$	− 18.0000i	− 0.575871i	−0.957650	−	0.287936i	$-0.907031\pi$
$977$	− 18.0000i	− 0.575871i	0.957650	−	0.287936i	$-0.0929689\pi$
$978$	− 24.0000i	− 0.767435i
$979$	0	0
$980$	0	0
$981$	6.00000	0.191565
$982$	14.0000i	0.446758i
$983$	56.0000i	1.78612i	0.449935	+	0.893061i	$0.351447\pi$
$983$	56.0000i	1.78612i	−0.449935	+	0.893061i	$0.648553\pi$
$984$	−4.00000	−0.127515
$985$	0	0
$986$	−2.00000	−0.0636930
$987$	− 16.0000i	− 0.509286i
$988$	16.0000i	0.509028i
$989$	−40.0000	−1.27193
$990$	0	0
$991$	24.0000	0.762385	0.381193	−	0.924496i	$-0.375513\pi$
$991$	24.0000	0.762385	0.381193	+	0.924496i	$0.375513\pi$
$992$	0	0
$993$	16.0000i	0.507745i
$994$	12.0000	0.380617
$995$	0	0
$996$	−8.00000	−0.253490
$997$	− 14.0000i	− 0.443384i	−0.975117	−	0.221692i	$-0.928842\pi$
$997$	− 14.0000i	− 0.443384i	0.975117	−	0.221692i	$-0.0711580\pi$
$998$	− 12.0000i	− 0.379853i
$999$	−2.00000	−0.0632772

Twists

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

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

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +1.00000i q^{12} +4.00000i q^{13} -2.00000 q^{14} +1.00000 q^{16} +1.00000i q^{17} +1.00000i q^{18} -4.00000 q^{19} -2.00000 q^{21} +4.00000i q^{23} +1.00000 q^{24} +4.00000 q^{26} +1.00000i q^{27} +2.00000i q^{28} -2.00000 q^{29} -1.00000i q^{32} +1.00000 q^{34} +1.00000 q^{36} +2.00000i q^{37} +4.00000i q^{38} +4.00000 q^{39} -4.00000 q^{41} +2.00000i q^{42} +10.0000i q^{43} +4.00000 q^{46} +8.00000i q^{47} -1.00000i q^{48} +3.00000 q^{49} +1.00000 q^{51} -4.00000i q^{52} +2.00000i q^{53} +1.00000 q^{54} +2.00000 q^{56} +4.00000i q^{57} +2.00000i q^{58} +2.00000 q^{59} -14.0000 q^{61} +2.00000i q^{63} -1.00000 q^{64} -2.00000i q^{67} -1.00000i q^{68} +4.00000 q^{69} -6.00000 q^{71} -1.00000i q^{72} -4.00000i q^{73} +2.00000 q^{74} +4.00000 q^{76} -4.00000i q^{78} +12.0000 q^{79} +1.00000 q^{81} +4.00000i q^{82} +8.00000i q^{83} +2.00000 q^{84} +10.0000 q^{86} +2.00000i q^{87} +10.0000 q^{89} +8.00000 q^{91} -4.00000i q^{92} +8.00000 q^{94} -1.00000 q^{96} -8.00000i q^{97} -3.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9	\( 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9} - 4 q^{14} + 2 q^{16} - 8 q^{19} - 4 q^{21} + 2 q^{24} + 8 q^{26} - 4 q^{29} + 2 q^{34} + 2 q^{36} + 8 q^{39} - 8 q^{41} + 8 q^{46} + 6 q^{49} + 2 q^{51} + 2 q^{54} + 4 q^{56} + 4 q^{59} - 28 q^{61} - 2 q^{64} + 8 q^{69} - 12 q^{71} + 4 q^{74} + 8 q^{76} + 24 q^{79} + 2 q^{81} + 4 q^{84} + 20 q^{86} + 20 q^{89} + 16 q^{91} + 16 q^{94} - 2 q^{96}+O(q^{100}) \) 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9 - 4 * q^14 + 2 * q^16 - 8 * q^19 - 4 * q^21 + 2 * q^24 + 8 * q^26 - 4 * q^29 + 2 * q^34 + 2 * q^36 + 8 * q^39 - 8 * q^41 + 8 * q^46 + 6 * q^49 + 2 * q^51 + 2 * q^54 + 4 * q^56 + 4 * q^59 - 28 * q^61 - 2 * q^64 + 8 * q^69 - 12 * q^71 + 4 * q^74 + 8 * q^76 + 24 * q^79 + 2 * q^81 + 4 * q^84 + 20 * q^86 + 20 * q^89 + 16 * q^91 + 16 * q^94 - 2 * q^96

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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