LMFDB - Embedded newform 2550.2.d.p.2449.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2550, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2550.2449");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$20.3618525154$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

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

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -5.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -5.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} -1.00000 q^{11} +1.00000i q^{12} -2.00000i q^{13} +5.00000 q^{14} +1.00000 q^{16} -1.00000i q^{17} -1.00000i q^{18} +7.00000 q^{19} -5.00000 q^{21} -1.00000i q^{22} +2.00000i q^{23} -1.00000 q^{24} +2.00000 q^{26} +1.00000i q^{27} +5.00000i q^{28} +10.0000 q^{29} -5.00000 q^{31} +1.00000i q^{32} +1.00000i q^{33} +1.00000 q^{34} +1.00000 q^{36} -3.00000i q^{37} +7.00000i q^{38} -2.00000 q^{39} -10.0000 q^{41} -5.00000i q^{42} -9.00000i q^{43} +1.00000 q^{44} -2.00000 q^{46} +7.00000i q^{47} -1.00000i q^{48} -18.0000 q^{49} -1.00000 q^{51} +2.00000i q^{52} -11.0000i q^{53} -1.00000 q^{54} -5.00000 q^{56} -7.00000i q^{57} +10.0000i q^{58} +2.00000 q^{61} -5.00000i q^{62} +5.00000i q^{63} -1.00000 q^{64} -1.00000 q^{66} -13.0000i q^{67} +1.00000i q^{68} +2.00000 q^{69} -10.0000 q^{71} +1.00000i q^{72} +4.00000i q^{73} +3.00000 q^{74} -7.00000 q^{76} +5.00000i q^{77} -2.00000i q^{78} -3.00000 q^{79} +1.00000 q^{81} -10.0000i q^{82} +14.0000i q^{83} +5.00000 q^{84} +9.00000 q^{86} -10.0000i q^{87} +1.00000i q^{88} -4.00000 q^{89} -10.0000 q^{91} -2.00000i q^{92} +5.00000i q^{93} -7.00000 q^{94} +1.00000 q^{96} -6.00000i q^{97} -18.0000i q^{98} +1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} - 2 q^{11} + 10 q^{14} + 2 q^{16} + 14 q^{19} - 10 q^{21} - 2 q^{24} + 4 q^{26} + 20 q^{29} - 10 q^{31} + 2 q^{34} + 2 q^{36} - 4 q^{39} - 20 q^{41} + 2 q^{44} - 4 q^{46} - 36 q^{49} - 2 q^{51} - 2 q^{54} - 10 q^{56} + 4 q^{61} - 2 q^{64} - 2 q^{66} + 4 q^{69} - 20 q^{71} + 6 q^{74} - 14 q^{76} - 6 q^{79} + 2 q^{81} + 10 q^{84} + 18 q^{86} - 8 q^{89} - 20 q^{91} - 14 q^{94} + 2 q^{96} + 2 q^{99}+O(q^{100}) $ 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9 - 2 * q^11 + 10 * q^14 + 2 * q^16 + 14 * q^19 - 10 * q^21 - 2 * q^24 + 4 * q^26 + 20 * q^29 - 10 * q^31 + 2 * q^34 + 2 * q^36 - 4 * q^39 - 20 * q^41 + 2 * q^44 - 4 * q^46 - 36 * q^49 - 2 * q^51 - 2 * q^54 - 10 * q^56 + 4 * q^61 - 2 * q^64 - 2 * q^66 + 4 * q^69 - 20 * q^71 + 6 * q^74 - 14 * q^76 - 6 * q^79 + 2 * q^81 + 10 * q^84 + 18 * q^86 - 8 * q^89 - 20 * q^91 - 14 * q^94 + 2 * q^96 + 2 * 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
$7$	− 5.00000i	− 1.88982i	−0.327327	−	0.944911i	$-0.606148\pi$
$7$	− 5.00000i	− 1.88982i	0.327327	−	0.944911i	$-0.393852\pi$
$8$	− 1.00000i	− 0.353553i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−1.00000	−0.301511	−0.150756	−	0.988571i	$-0.548171\pi$
$11$	−1.00000	−0.301511	−0.150756	+	0.988571i	$0.548171\pi$
$12$	1.00000i	0.288675i
$13$	− 2.00000i	− 0.554700i	−0.960769	−	0.277350i	$-0.910544\pi$
$13$	− 2.00000i	− 0.554700i	0.960769	−	0.277350i	$-0.0894562\pi$
$14$	5.00000	1.33631
$15$	0	0
$16$	1.00000	0.250000
$17$	− 1.00000i	− 0.242536i
$17$	− 1.00000i	− 0.242536i
$18$	− 1.00000i	− 0.235702i
$19$	7.00000	1.60591	0.802955	−	0.596040i	$-0.203260\pi$
$19$	7.00000	1.60591	0.802955	+	0.596040i	$0.203260\pi$
$20$	0	0
$21$	−5.00000	−1.09109
$22$	− 1.00000i	− 0.213201i
$23$	2.00000i	0.417029i	0.978019	+	0.208514i	$0.0668628\pi$
$23$	2.00000i	0.417029i	−0.978019	+	0.208514i	$0.933137\pi$
$24$	−1.00000	−0.204124
$25$	0	0
$26$	2.00000	0.392232
$27$	1.00000i	0.192450i
$28$	5.00000i	0.944911i
$29$	10.0000	1.85695	0.928477	−	0.371391i	$-0.121119\pi$
$29$	10.0000	1.85695	0.928477	+	0.371391i	$0.121119\pi$
$30$	0	0
$31$	−5.00000	−0.898027	−0.449013	−	0.893525i	$-0.648224\pi$
$31$	−5.00000	−0.898027	−0.449013	+	0.893525i	$0.648224\pi$
$32$	1.00000i	0.176777i
$33$	1.00000i	0.174078i
$34$	1.00000	0.171499
$35$	0	0
$36$	1.00000	0.166667
$37$	− 3.00000i	− 0.493197i	−0.969118	−	0.246598i	$-0.920687\pi$
$37$	− 3.00000i	− 0.493197i	0.969118	−	0.246598i	$-0.0793129\pi$
$38$	7.00000i	1.13555i
$39$	−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	$0.785223\pi$
$42$	− 5.00000i	− 0.771517i
$43$	− 9.00000i	− 1.37249i	−0.727372	−	0.686244i	$-0.759258\pi$
$43$	− 9.00000i	− 1.37249i	0.727372	−	0.686244i	$-0.240742\pi$
$44$	1.00000	0.150756
$45$	0	0
$46$	−2.00000	−0.294884
$47$	7.00000i	1.02105i	0.859861	+	0.510527i	$0.170550\pi$
$47$	7.00000i	1.02105i	−0.859861	+	0.510527i	$0.829450\pi$
$48$	− 1.00000i	− 0.144338i
$49$	−18.0000	−2.57143
$50$	0	0
$51$	−1.00000	−0.140028
$52$	2.00000i	0.277350i
$53$	− 11.0000i	− 1.51097i	−0.655168	−	0.755483i	$-0.727402\pi$
$53$	− 11.0000i	− 1.51097i	0.655168	−	0.755483i	$-0.272598\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	−5.00000	−0.668153
$57$	− 7.00000i	− 0.927173i
$58$	10.0000i	1.31306i
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	− 5.00000i	− 0.635001i
$63$	5.00000i	0.629941i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	−1.00000	−0.123091
$67$	− 13.0000i	− 1.58820i	−0.607785	−	0.794101i	$-0.707942\pi$
$67$	− 13.0000i	− 1.58820i	0.607785	−	0.794101i	$-0.292058\pi$
$68$	1.00000i	0.121268i
$69$	2.00000	0.240772
$70$	0	0
$71$	−10.0000	−1.18678	−0.593391	−	0.804914i	$-0.702211\pi$
$71$	−10.0000	−1.18678	−0.593391	+	0.804914i	$0.702211\pi$
$72$	1.00000i	0.117851i
$73$	4.00000i	0.468165i	0.972217	+	0.234082i	$0.0752085\pi$
$73$	4.00000i	0.468165i	−0.972217	+	0.234082i	$0.924791\pi$
$74$	3.00000	0.348743
$75$	0	0
$76$	−7.00000	−0.802955
$77$	5.00000i	0.569803i
$78$	− 2.00000i	− 0.226455i
$79$	−3.00000	−0.337526	−0.168763	−	0.985657i	$-0.553977\pi$
$79$	−3.00000	−0.337526	−0.168763	+	0.985657i	$0.553977\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	− 10.0000i	− 1.10432i
$83$	14.0000i	1.53670i	0.640030	+	0.768350i	$0.278922\pi$
$83$	14.0000i	1.53670i	−0.640030	+	0.768350i	$0.721078\pi$
$84$	5.00000	0.545545
$85$	0	0
$86$	9.00000	0.970495
$87$	− 10.0000i	− 1.07211i
$88$	1.00000i	0.106600i
$89$	−4.00000	−0.423999	−0.212000	−	0.977270i	$-0.567998\pi$
$89$	−4.00000	−0.423999	−0.212000	+	0.977270i	$0.567998\pi$
$90$	0	0
$91$	−10.0000	−1.04828
$92$	− 2.00000i	− 0.208514i
$93$	5.00000i	0.518476i
$94$	−7.00000	−0.721995
$95$	0	0
$96$	1.00000	0.102062
$97$	− 6.00000i	− 0.609208i	−0.952479	−	0.304604i	$-0.901476\pi$
$97$	− 6.00000i	− 0.609208i	0.952479	−	0.304604i	$-0.0985241\pi$
$98$	− 18.0000i	− 1.81827i
$99$	1.00000	0.100504
$100$	0	0
$101$	−7.00000	−0.696526	−0.348263	−	0.937397i	$-0.613228\pi$
$101$	−7.00000	−0.696526	−0.348263	+	0.937397i	$0.613228\pi$
$102$	− 1.00000i	− 0.0990148i
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	−2.00000	−0.196116
$105$	0	0
$106$	11.0000	1.06841
$107$	9.00000i	0.870063i	0.900415	+	0.435031i	$0.143263\pi$
$107$	9.00000i	0.870063i	−0.900415	+	0.435031i	$0.856737\pi$
$108$	− 1.00000i	− 0.0962250i
$109$	−19.0000	−1.81987	−0.909935	−	0.414751i	$-0.863869\pi$
$109$	−19.0000	−1.81987	−0.909935	+	0.414751i	$0.863869\pi$
$110$	0	0
$111$	−3.00000	−0.284747
$112$	− 5.00000i	− 0.472456i
$113$	− 3.00000i	− 0.282216i	−0.989994	−	0.141108i	$-0.954933\pi$
$113$	− 3.00000i	− 0.282216i	0.989994	−	0.141108i	$-0.0450665\pi$
$114$	7.00000	0.655610
$115$	0	0
$116$	−10.0000	−0.928477
$117$	2.00000i	0.184900i
$118$	0	0
$119$	−5.00000	−0.458349
$120$	0	0
$121$	−10.0000	−0.909091
$122$	2.00000i	0.181071i
$123$	10.0000i	0.901670i
$124$	5.00000	0.449013
$125$	0	0
$126$	−5.00000	−0.445435
$127$	4.00000i	0.354943i	0.984126	+	0.177471i	$0.0567917\pi$
$127$	4.00000i	0.354943i	−0.984126	+	0.177471i	$0.943208\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	−9.00000	−0.792406
$130$	0	0
$131$	−4.00000	−0.349482	−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000	−0.349482	−0.174741	+	0.984614i	$0.555909\pi$
$132$	− 1.00000i	− 0.0870388i
$133$	− 35.0000i	− 3.03488i
$134$	13.0000	1.12303
$135$	0	0
$136$	−1.00000	−0.0857493
$137$	− 12.0000i	− 1.02523i	−0.858619	−	0.512615i	$-0.828677\pi$
$137$	− 12.0000i	− 1.02523i	0.858619	−	0.512615i	$-0.171323\pi$
$138$	2.00000i	0.170251i
$139$	20.0000	1.69638	0.848189	−	0.529694i	$-0.177693\pi$
$139$	20.0000	1.69638	0.848189	+	0.529694i	$0.177693\pi$
$140$	0	0
$141$	7.00000	0.589506
$142$	− 10.0000i	− 0.839181i
$143$	2.00000i	0.167248i
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	−4.00000	−0.331042
$147$	18.0000i	1.48461i
$148$	3.00000i	0.246598i
$149$	2.00000	0.163846	0.0819232	−	0.996639i	$-0.473894\pi$
$149$	2.00000	0.163846	0.0819232	+	0.996639i	$0.473894\pi$
$150$	0	0
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	− 7.00000i	− 0.567775i
$153$	1.00000i	0.0808452i
$154$	−5.00000	−0.402911
$155$	0	0
$156$	2.00000	0.160128
$157$	16.0000i	1.27694i	0.769647	+	0.638470i	$0.220432\pi$
$157$	16.0000i	1.27694i	−0.769647	+	0.638470i	$0.779568\pi$
$158$	− 3.00000i	− 0.238667i
$159$	−11.0000	−0.872357
$160$	0	0
$161$	10.0000	0.788110
$162$	1.00000i	0.0785674i
$163$	16.0000i	1.25322i	0.779334	+	0.626608i	$0.215557\pi$
$163$	16.0000i	1.25322i	−0.779334	+	0.626608i	$0.784443\pi$
$164$	10.0000	0.780869
$165$	0	0
$166$	−14.0000	−1.08661
$167$	6.00000i	0.464294i	0.972681	+	0.232147i	$0.0745750\pi$
$167$	6.00000i	0.464294i	−0.972681	+	0.232147i	$0.925425\pi$
$168$	5.00000i	0.385758i
$169$	9.00000	0.692308
$170$	0	0
$171$	−7.00000	−0.535303
$172$	9.00000i	0.686244i
$173$	− 20.0000i	− 1.52057i	−0.649589	−	0.760286i	$-0.725059\pi$
$173$	− 20.0000i	− 1.52057i	0.649589	−	0.760286i	$-0.274941\pi$
$174$	10.0000	0.758098
$175$	0	0
$176$	−1.00000	−0.0753778
$177$	0	0
$178$	− 4.00000i	− 0.299813i
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	0	0
$181$	17.0000	1.26360	0.631800	−	0.775131i	$-0.282316\pi$
$181$	17.0000	1.26360	0.631800	+	0.775131i	$0.282316\pi$
$182$	− 10.0000i	− 0.741249i
$183$	− 2.00000i	− 0.147844i
$184$	2.00000	0.147442
$185$	0	0
$186$	−5.00000	−0.366618
$187$	1.00000i	0.0731272i
$188$	− 7.00000i	− 0.510527i
$189$	5.00000	0.363696
$190$	0	0
$191$	25.0000	1.80894	0.904468	−	0.426541i	$-0.140268\pi$
$191$	25.0000	1.80894	0.904468	+	0.426541i	$0.140268\pi$
$192$	1.00000i	0.0721688i
$193$	− 24.0000i	− 1.72756i	−0.503871	−	0.863779i	$-0.668091\pi$
$193$	− 24.0000i	− 1.72756i	0.503871	−	0.863779i	$-0.331909\pi$
$194$	6.00000	0.430775
$195$	0	0
$196$	18.0000	1.28571
$197$	8.00000i	0.569976i	0.958531	+	0.284988i	$0.0919897\pi$
$197$	8.00000i	0.569976i	−0.958531	+	0.284988i	$0.908010\pi$
$198$	1.00000i	0.0710669i
$199$	−7.00000	−0.496217	−0.248108	−	0.968732i	$-0.579809\pi$
$199$	−7.00000	−0.496217	−0.248108	+	0.968732i	$0.579809\pi$
$200$	0	0
$201$	−13.0000	−0.916949
$202$	− 7.00000i	− 0.492518i
$203$	− 50.0000i	− 3.50931i
$204$	1.00000	0.0700140
$205$	0	0
$206$	0	0
$207$	− 2.00000i	− 0.139010i
$208$	− 2.00000i	− 0.138675i
$209$	−7.00000	−0.484200
$210$	0	0
$211$	−14.0000	−0.963800	−0.481900	−	0.876226i	$-0.660053\pi$
$211$	−14.0000	−0.963800	−0.481900	+	0.876226i	$0.660053\pi$
$212$	11.0000i	0.755483i
$213$	10.0000i	0.685189i
$214$	−9.00000	−0.615227
$215$	0	0
$216$	1.00000	0.0680414
$217$	25.0000i	1.69711i
$218$	− 19.0000i	− 1.28684i
$219$	4.00000	0.270295
$220$	0	0
$221$	−2.00000	−0.134535
$222$	− 3.00000i	− 0.201347i
$223$	10.0000i	0.669650i	0.942280	+	0.334825i	$0.108677\pi$
$223$	10.0000i	0.669650i	−0.942280	+	0.334825i	$0.891323\pi$
$224$	5.00000	0.334077
$225$	0	0
$226$	3.00000	0.199557
$227$	− 1.00000i	− 0.0663723i	−0.999449	−	0.0331862i	$-0.989435\pi$
$227$	− 1.00000i	− 0.0663723i	0.999449	−	0.0331862i	$-0.0105654\pi$
$228$	7.00000i	0.463586i
$229$	12.0000	0.792982	0.396491	−	0.918039i	$-0.370228\pi$
$229$	12.0000	0.792982	0.396491	+	0.918039i	$0.370228\pi$
$230$	0	0
$231$	5.00000	0.328976
$232$	− 10.0000i	− 0.656532i
$233$	− 18.0000i	− 1.17922i	−0.807688	−	0.589610i	$-0.799282\pi$
$233$	− 18.0000i	− 1.17922i	0.807688	−	0.589610i	$-0.200718\pi$
$234$	−2.00000	−0.130744
$235$	0	0
$236$	0	0
$237$	3.00000i	0.194871i
$238$	− 5.00000i	− 0.324102i
$239$	−15.0000	−0.970269	−0.485135	−	0.874439i	$-0.661229\pi$
$239$	−15.0000	−0.970269	−0.485135	+	0.874439i	$0.661229\pi$
$240$	0	0
$241$	−20.0000	−1.28831	−0.644157	−	0.764894i	$-0.722792\pi$
$241$	−20.0000	−1.28831	−0.644157	+	0.764894i	$0.722792\pi$
$242$	− 10.0000i	− 0.642824i
$243$	− 1.00000i	− 0.0641500i
$244$	−2.00000	−0.128037
$245$	0	0
$246$	−10.0000	−0.637577
$247$	− 14.0000i	− 0.890799i
$248$	5.00000i	0.317500i
$249$	14.0000	0.887214
$250$	0	0
$251$	2.00000	0.126239	0.0631194	−	0.998006i	$-0.479895\pi$
$251$	2.00000	0.126239	0.0631194	+	0.998006i	$0.479895\pi$
$252$	− 5.00000i	− 0.314970i
$253$	− 2.00000i	− 0.125739i
$254$	−4.00000	−0.250982
$255$	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	$-0.189739\pi$
$258$	− 9.00000i	− 0.560316i
$259$	−15.0000	−0.932055
$260$	0	0
$261$	−10.0000	−0.618984
$262$	− 4.00000i	− 0.247121i
$263$	17.0000i	1.04826i	0.851637	+	0.524132i	$0.175610\pi$
$263$	17.0000i	1.04826i	−0.851637	+	0.524132i	$0.824390\pi$
$264$	1.00000	0.0615457
$265$	0	0
$266$	35.0000	2.14599
$267$	4.00000i	0.244796i
$268$	13.0000i	0.794101i
$269$	−6.00000	−0.365826	−0.182913	−	0.983129i	$-0.558553\pi$
$269$	−6.00000	−0.365826	−0.182913	+	0.983129i	$0.558553\pi$
$270$	0	0
$271$	−6.00000	−0.364474	−0.182237	−	0.983255i	$-0.558334\pi$
$271$	−6.00000	−0.364474	−0.182237	+	0.983255i	$0.558334\pi$
$272$	− 1.00000i	− 0.0606339i
$273$	10.0000i	0.605228i
$274$	12.0000	0.724947
$275$	0	0
$276$	−2.00000	−0.120386
$277$	− 23.0000i	− 1.38194i	−0.722885	−	0.690968i	$-0.757185\pi$
$277$	− 23.0000i	− 1.38194i	0.722885	−	0.690968i	$-0.242815\pi$
$278$	20.0000i	1.19952i
$279$	5.00000	0.299342
$280$	0	0
$281$	22.0000	1.31241	0.656205	−	0.754583i	$-0.272161\pi$
$281$	22.0000	1.31241	0.656205	+	0.754583i	$0.272161\pi$
$282$	7.00000i	0.416844i
$283$	− 14.0000i	− 0.832214i	−0.909316	−	0.416107i	$-0.863394\pi$
$283$	− 14.0000i	− 0.832214i	0.909316	−	0.416107i	$-0.136606\pi$
$284$	10.0000	0.593391
$285$	0	0
$286$	−2.00000	−0.118262
$287$	50.0000i	2.95141i
$288$	− 1.00000i	− 0.0589256i
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	−6.00000	−0.351726
$292$	− 4.00000i	− 0.234082i
$293$	− 14.0000i	− 0.817889i	−0.912559	−	0.408944i	$-0.865897\pi$
$293$	− 14.0000i	− 0.817889i	0.912559	−	0.408944i	$-0.134103\pi$
$294$	−18.0000	−1.04978
$295$	0	0
$296$	−3.00000	−0.174371
$297$	− 1.00000i	− 0.0580259i
$298$	2.00000i	0.115857i
$299$	4.00000	0.231326
$300$	0	0
$301$	−45.0000	−2.59376
$302$	− 8.00000i	− 0.460348i
$303$	7.00000i	0.402139i
$304$	7.00000	0.401478
$305$	0	0
$306$	−1.00000	−0.0571662
$307$	4.00000i	0.228292i	0.993464	+	0.114146i	$0.0364132\pi$
$307$	4.00000i	0.228292i	−0.993464	+	0.114146i	$0.963587\pi$
$308$	− 5.00000i	− 0.284901i
$309$	0	0
$310$	0	0
$311$	26.0000	1.47432	0.737162	−	0.675716i	$-0.236165\pi$
$311$	26.0000	1.47432	0.737162	+	0.675716i	$0.236165\pi$
$312$	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	$-0.0180015\pi$
$314$	−16.0000	−0.902932
$315$	0	0
$316$	3.00000	0.168763
$317$	20.0000i	1.12331i	0.827371	+	0.561656i	$0.189836\pi$
$317$	20.0000i	1.12331i	−0.827371	+	0.561656i	$0.810164\pi$
$318$	− 11.0000i	− 0.616849i
$319$	−10.0000	−0.559893
$320$	0	0
$321$	9.00000	0.502331
$322$	10.0000i	0.557278i
$323$	− 7.00000i	− 0.389490i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	−16.0000	−0.886158
$327$	19.0000i	1.05070i
$328$	10.0000i	0.552158i
$329$	35.0000	1.92961
$330$	0	0
$331$	7.00000	0.384755	0.192377	−	0.981321i	$-0.438380\pi$
$331$	7.00000	0.384755	0.192377	+	0.981321i	$0.438380\pi$
$332$	− 14.0000i	− 0.768350i
$333$	3.00000i	0.164399i
$334$	−6.00000	−0.328305
$335$	0	0
$336$	−5.00000	−0.272772
$337$	− 8.00000i	− 0.435788i	−0.975972	−	0.217894i	$-0.930081\pi$
$337$	− 8.00000i	− 0.435788i	0.975972	−	0.217894i	$-0.0699187\pi$
$338$	9.00000i	0.489535i
$339$	−3.00000	−0.162938
$340$	0	0
$341$	5.00000	0.270765
$342$	− 7.00000i	− 0.378517i
$343$	55.0000i	2.96972i
$344$	−9.00000	−0.485247
$345$	0	0
$346$	20.0000	1.07521
$347$	− 5.00000i	− 0.268414i	−0.990953	−	0.134207i	$-0.957151\pi$
$347$	− 5.00000i	− 0.268414i	0.990953	−	0.134207i	$-0.0428487\pi$
$348$	10.0000i	0.536056i
$349$	2.00000	0.107058	0.0535288	−	0.998566i	$-0.482953\pi$
$349$	2.00000	0.107058	0.0535288	+	0.998566i	$0.482953\pi$
$350$	0	0
$351$	2.00000	0.106752
$352$	− 1.00000i	− 0.0533002i
$353$	2.00000i	0.106449i	0.998583	+	0.0532246i	$0.0169499\pi$
$353$	2.00000i	0.106449i	−0.998583	+	0.0532246i	$0.983050\pi$
$354$	0	0
$355$	0	0
$356$	4.00000	0.212000
$357$	5.00000i	0.264628i
$358$	12.0000i	0.634220i
$359$	3.00000	0.158334	0.0791670	−	0.996861i	$-0.474774\pi$
$359$	3.00000	0.158334	0.0791670	+	0.996861i	$0.474774\pi$
$360$	0	0
$361$	30.0000	1.57895
$362$	17.0000i	0.893500i
$363$	10.0000i	0.524864i
$364$	10.0000	0.524142
$365$	0	0
$366$	2.00000	0.104542
$367$	5.00000i	0.260998i	0.991448	+	0.130499i	$0.0416579\pi$
$367$	5.00000i	0.260998i	−0.991448	+	0.130499i	$0.958342\pi$
$368$	2.00000i	0.104257i
$369$	10.0000	0.520579
$370$	0	0
$371$	−55.0000	−2.85546
$372$	− 5.00000i	− 0.259238i
$373$	26.0000i	1.34623i	0.739538	+	0.673114i	$0.235044\pi$
$373$	26.0000i	1.34623i	−0.739538	+	0.673114i	$0.764956\pi$
$374$	−1.00000	−0.0517088
$375$	0	0
$376$	7.00000	0.360997
$377$	− 20.0000i	− 1.03005i
$378$	5.00000i	0.257172i
$379$	−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
$381$	4.00000	0.204926
$382$	25.0000i	1.27911i
$383$	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
$386$	24.0000	1.22157
$387$	9.00000i	0.457496i
$388$	6.00000i	0.304604i
$389$	15.0000	0.760530	0.380265	−	0.924878i	$-0.375833\pi$
$389$	15.0000	0.760530	0.380265	+	0.924878i	$0.375833\pi$
$390$	0	0
$391$	2.00000	0.101144
$392$	18.0000i	0.909137i
$393$	4.00000i	0.201773i
$394$	−8.00000	−0.403034
$395$	0	0
$396$	−1.00000	−0.0502519
$397$	− 3.00000i	− 0.150566i	−0.997162	−	0.0752828i	$-0.976014\pi$
$397$	− 3.00000i	− 0.150566i	0.997162	−	0.0752828i	$-0.0239860\pi$
$398$	− 7.00000i	− 0.350878i
$399$	−35.0000	−1.75219
$400$	0	0
$401$	30.0000	1.49813	0.749064	−	0.662497i	$-0.230503\pi$
$401$	30.0000	1.49813	0.749064	+	0.662497i	$0.230503\pi$
$402$	− 13.0000i	− 0.648381i
$403$	10.0000i	0.498135i
$404$	7.00000	0.348263
$405$	0	0
$406$	50.0000	2.48146
$407$	3.00000i	0.148704i
$408$	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
$411$	−12.0000	−0.591916
$412$	0	0
$413$	0	0
$414$	2.00000	0.0982946
$415$	0	0
$416$	2.00000	0.0980581
$417$	− 20.0000i	− 0.979404i
$418$	− 7.00000i	− 0.342381i
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	−30.0000	−1.46211	−0.731055	−	0.682318i	$-0.760972\pi$
$421$	−30.0000	−1.46211	−0.731055	+	0.682318i	$0.760972\pi$
$422$	− 14.0000i	− 0.681509i
$423$	− 7.00000i	− 0.340352i
$424$	−11.0000	−0.534207
$425$	0	0
$426$	−10.0000	−0.484502
$427$	− 10.0000i	− 0.483934i
$428$	− 9.00000i	− 0.435031i
$429$	2.00000	0.0965609
$430$	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
$433$	− 35.0000i	− 1.68199i	−0.541041	−	0.840996i	$-0.681970\pi$
$433$	− 35.0000i	− 1.68199i	0.541041	−	0.840996i	$-0.318030\pi$
$434$	−25.0000	−1.20004
$435$	0	0
$436$	19.0000	0.909935
$437$	14.0000i	0.669711i
$438$	4.00000i	0.191127i
$439$	24.0000	1.14546	0.572729	−	0.819745i	$-0.305885\pi$
$439$	24.0000	1.14546	0.572729	+	0.819745i	$0.305885\pi$
$440$	0	0
$441$	18.0000	0.857143
$442$	− 2.00000i	− 0.0951303i
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	3.00000	0.142374
$445$	0	0
$446$	−10.0000	−0.473514
$447$	− 2.00000i	− 0.0945968i
$448$	5.00000i	0.236228i
$449$	−9.00000	−0.424736	−0.212368	−	0.977190i	$-0.568118\pi$
$449$	−9.00000	−0.424736	−0.212368	+	0.977190i	$0.568118\pi$
$450$	0	0
$451$	10.0000	0.470882
$452$	3.00000i	0.141108i
$453$	8.00000i	0.375873i
$454$	1.00000	0.0469323
$455$	0	0
$456$	−7.00000	−0.327805
$457$	17.0000i	0.795226i	0.917553	+	0.397613i	$0.130161\pi$
$457$	17.0000i	0.795226i	−0.917553	+	0.397613i	$0.869839\pi$
$458$	12.0000i	0.560723i
$459$	1.00000	0.0466760
$460$	0	0
$461$	35.0000	1.63011	0.815056	−	0.579382i	$-0.196706\pi$
$461$	35.0000	1.63011	0.815056	+	0.579382i	$0.196706\pi$
$462$	5.00000i	0.232621i
$463$	16.0000i	0.743583i	0.928316	+	0.371792i	$0.121256\pi$
$463$	16.0000i	0.743583i	−0.928316	+	0.371792i	$0.878744\pi$
$464$	10.0000	0.464238
$465$	0	0
$466$	18.0000	0.833834
$467$	− 6.00000i	− 0.277647i	−0.990317	−	0.138823i	$-0.955668\pi$
$467$	− 6.00000i	− 0.277647i	0.990317	−	0.138823i	$-0.0443321\pi$
$468$	− 2.00000i	− 0.0924500i
$469$	−65.0000	−3.00142
$470$	0	0
$471$	16.0000	0.737241
$472$	0	0
$473$	9.00000i	0.413820i
$474$	−3.00000	−0.137795
$475$	0	0
$476$	5.00000	0.229175
$477$	11.0000i	0.503655i
$478$	− 15.0000i	− 0.686084i
$479$	22.0000	1.00521	0.502603	−	0.864517i	$-0.332376\pi$
$479$	22.0000	1.00521	0.502603	+	0.864517i	$0.332376\pi$
$480$	0	0
$481$	−6.00000	−0.273576
$482$	− 20.0000i	− 0.910975i
$483$	− 10.0000i	− 0.455016i
$484$	10.0000	0.454545
$485$	0	0
$486$	1.00000	0.0453609
$487$	− 8.00000i	− 0.362515i	−0.983436	−	0.181257i	$-0.941983\pi$
$487$	− 8.00000i	− 0.362515i	0.983436	−	0.181257i	$-0.0580167\pi$
$488$	− 2.00000i	− 0.0905357i
$489$	16.0000	0.723545
$490$	0	0
$491$	12.0000	0.541552	0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000	0.541552	0.270776	+	0.962642i	$0.412720\pi$
$492$	− 10.0000i	− 0.450835i
$493$	− 10.0000i	− 0.450377i
$494$	14.0000	0.629890
$495$	0	0
$496$	−5.00000	−0.224507
$497$	50.0000i	2.24281i
$498$	14.0000i	0.627355i
$499$	−22.0000	−0.984855	−0.492428	−	0.870353i	$-0.663890\pi$
$499$	−22.0000	−0.984855	−0.492428	+	0.870353i	$0.663890\pi$
$500$	0	0
$501$	6.00000	0.268060
$502$	2.00000i	0.0892644i
$503$	− 22.0000i	− 0.980932i	−0.871460	−	0.490466i	$-0.836827\pi$
$503$	− 22.0000i	− 0.980932i	0.871460	−	0.490466i	$-0.163173\pi$
$504$	5.00000	0.222718
$505$	0	0
$506$	2.00000	0.0889108
$507$	− 9.00000i	− 0.399704i
$508$	− 4.00000i	− 0.177471i
$509$	37.0000	1.64000	0.819998	−	0.572366i	$-0.193974\pi$
$509$	37.0000	1.64000	0.819998	+	0.572366i	$0.193974\pi$
$510$	0	0
$511$	20.0000	0.884748
$512$	1.00000i	0.0441942i
$513$	7.00000i	0.309058i
$514$	18.0000	0.793946
$515$	0	0
$516$	9.00000	0.396203
$517$	− 7.00000i	− 0.307860i
$518$	− 15.0000i	− 0.659062i
$519$	−20.0000	−0.877903
$520$	0	0
$521$	−27.0000	−1.18289	−0.591446	−	0.806345i	$-0.701443\pi$
$521$	−27.0000	−1.18289	−0.591446	+	0.806345i	$0.701443\pi$
$522$	− 10.0000i	− 0.437688i
$523$	4.00000i	0.174908i	0.996169	+	0.0874539i	$0.0278730\pi$
$523$	4.00000i	0.174908i	−0.996169	+	0.0874539i	$0.972127\pi$
$524$	4.00000	0.174741
$525$	0	0
$526$	−17.0000	−0.741235
$527$	5.00000i	0.217803i
$528$	1.00000i	0.0435194i
$529$	19.0000	0.826087
$530$	0	0
$531$	0	0
$532$	35.0000i	1.51744i
$533$	20.0000i	0.866296i
$534$	−4.00000	−0.173097
$535$	0	0
$536$	−13.0000	−0.561514
$537$	− 12.0000i	− 0.517838i
$538$	− 6.00000i	− 0.258678i
$539$	18.0000	0.775315
$540$	0	0
$541$	−39.0000	−1.67674	−0.838370	−	0.545101i	$-0.816491\pi$
$541$	−39.0000	−1.67674	−0.838370	+	0.545101i	$0.816491\pi$
$542$	− 6.00000i	− 0.257722i
$543$	− 17.0000i	− 0.729540i
$544$	1.00000	0.0428746
$545$	0	0
$546$	−10.0000	−0.427960
$547$	− 38.0000i	− 1.62476i	−0.583127	−	0.812381i	$-0.698171\pi$
$547$	− 38.0000i	− 1.62476i	0.583127	−	0.812381i	$-0.301829\pi$
$548$	12.0000i	0.512615i
$549$	−2.00000	−0.0853579
$550$	0	0
$551$	70.0000	2.98210
$552$	− 2.00000i	− 0.0851257i
$553$	15.0000i	0.637865i
$554$	23.0000	0.977176
$555$	0	0
$556$	−20.0000	−0.848189
$557$	29.0000i	1.22877i	0.789007	+	0.614385i	$0.210596\pi$
$557$	29.0000i	1.22877i	−0.789007	+	0.614385i	$0.789404\pi$
$558$	5.00000i	0.211667i
$559$	−18.0000	−0.761319
$560$	0	0
$561$	1.00000	0.0422200
$562$	22.0000i	0.928014i
$563$	12.0000i	0.505740i	0.967500	+	0.252870i	$0.0813744\pi$
$563$	12.0000i	0.505740i	−0.967500	+	0.252870i	$0.918626\pi$
$564$	−7.00000	−0.294753
$565$	0	0
$566$	14.0000	0.588464
$567$	− 5.00000i	− 0.209980i
$568$	10.0000i	0.419591i
$569$	36.0000	1.50920	0.754599	−	0.656186i	$-0.227831\pi$
$569$	36.0000	1.50920	0.754599	+	0.656186i	$0.227831\pi$
$570$	0	0
$571$	−20.0000	−0.836974	−0.418487	−	0.908223i	$-0.637439\pi$
$571$	−20.0000	−0.836974	−0.418487	+	0.908223i	$0.637439\pi$
$572$	− 2.00000i	− 0.0836242i
$573$	− 25.0000i	− 1.04439i
$574$	−50.0000	−2.08696
$575$	0	0
$576$	1.00000	0.0416667
$577$	39.0000i	1.62359i	0.583942	+	0.811796i	$0.301510\pi$
$577$	39.0000i	1.62359i	−0.583942	+	0.811796i	$0.698490\pi$
$578$	− 1.00000i	− 0.0415945i
$579$	−24.0000	−0.997406
$580$	0	0
$581$	70.0000	2.90409
$582$	− 6.00000i	− 0.248708i
$583$	11.0000i	0.455573i
$584$	4.00000	0.165521
$585$	0	0
$586$	14.0000	0.578335
$587$	− 26.0000i	− 1.07313i	−0.843857	−	0.536567i	$-0.819721\pi$
$587$	− 26.0000i	− 1.07313i	0.843857	−	0.536567i	$-0.180279\pi$
$588$	− 18.0000i	− 0.742307i
$589$	−35.0000	−1.44215
$590$	0	0
$591$	8.00000	0.329076
$592$	− 3.00000i	− 0.123299i
$593$	12.0000i	0.492781i	0.969171	+	0.246390i	$0.0792446\pi$
$593$	12.0000i	0.492781i	−0.969171	+	0.246390i	$0.920755\pi$
$594$	1.00000	0.0410305
$595$	0	0
$596$	−2.00000	−0.0819232
$597$	7.00000i	0.286491i
$598$	4.00000i	0.163572i
$599$	−33.0000	−1.34834	−0.674172	−	0.738575i	$-0.735499\pi$
$599$	−33.0000	−1.34834	−0.674172	+	0.738575i	$0.735499\pi$
$600$	0	0
$601$	18.0000	0.734235	0.367118	−	0.930175i	$-0.380345\pi$
$601$	18.0000	0.734235	0.367118	+	0.930175i	$0.380345\pi$
$602$	− 45.0000i	− 1.83406i
$603$	13.0000i	0.529401i
$604$	8.00000	0.325515
$605$	0	0
$606$	−7.00000	−0.284356
$607$	− 28.0000i	− 1.13648i	−0.822861	−	0.568242i	$-0.807624\pi$
$607$	− 28.0000i	− 1.13648i	0.822861	−	0.568242i	$-0.192376\pi$
$608$	7.00000i	0.283887i
$609$	−50.0000	−2.02610
$610$	0	0
$611$	14.0000	0.566379
$612$	− 1.00000i	− 0.0404226i
$613$	− 2.00000i	− 0.0807792i	−0.999184	−	0.0403896i	$-0.987140\pi$
$613$	− 2.00000i	− 0.0807792i	0.999184	−	0.0403896i	$-0.0128599\pi$
$614$	−4.00000	−0.161427
$615$	0	0
$616$	5.00000	0.201456
$617$	9.00000i	0.362326i	0.983453	+	0.181163i	$0.0579862\pi$
$617$	9.00000i	0.362326i	−0.983453	+	0.181163i	$0.942014\pi$
$618$	0	0
$619$	16.0000	0.643094	0.321547	−	0.946894i	$-0.395797\pi$
$619$	16.0000	0.643094	0.321547	+	0.946894i	$0.395797\pi$
$620$	0	0
$621$	−2.00000	−0.0802572
$622$	26.0000i	1.04251i
$623$	20.0000i	0.801283i
$624$	−2.00000	−0.0800641
$625$	0	0
$626$	2.00000	0.0799361
$627$	7.00000i	0.279553i
$628$	− 16.0000i	− 0.638470i
$629$	−3.00000	−0.119618
$630$	0	0
$631$	38.0000	1.51276	0.756378	−	0.654135i	$-0.226967\pi$
$631$	38.0000	1.51276	0.756378	+	0.654135i	$0.226967\pi$
$632$	3.00000i	0.119334i
$633$	14.0000i	0.556450i
$634$	−20.0000	−0.794301
$635$	0	0
$636$	11.0000	0.436178
$637$	36.0000i	1.42637i
$638$	− 10.0000i	− 0.395904i
$639$	10.0000	0.395594
$640$	0	0
$641$	18.0000	0.710957	0.355479	−	0.934684i	$-0.384318\pi$
$641$	18.0000	0.710957	0.355479	+	0.934684i	$0.384318\pi$
$642$	9.00000i	0.355202i
$643$	− 14.0000i	− 0.552106i	−0.961142	−	0.276053i	$-0.910973\pi$
$643$	− 14.0000i	− 0.552106i	0.961142	−	0.276053i	$-0.0890266\pi$
$644$	−10.0000	−0.394055
$645$	0	0
$646$	7.00000	0.275411
$647$	− 32.0000i	− 1.25805i	−0.777385	−	0.629025i	$-0.783454\pi$
$647$	− 32.0000i	− 1.25805i	0.777385	−	0.629025i	$-0.216546\pi$
$648$	− 1.00000i	− 0.0392837i
$649$	0	0
$650$	0	0
$651$	25.0000	0.979827
$652$	− 16.0000i	− 0.626608i
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	−19.0000	−0.742959
$655$	0	0
$656$	−10.0000	−0.390434
$657$	− 4.00000i	− 0.156055i
$658$	35.0000i	1.36444i
$659$	−44.0000	−1.71400	−0.856998	−	0.515319i	$-0.827673\pi$
$659$	−44.0000	−1.71400	−0.856998	+	0.515319i	$0.827673\pi$
$660$	0	0
$661$	−16.0000	−0.622328	−0.311164	−	0.950356i	$-0.600719\pi$
$661$	−16.0000	−0.622328	−0.311164	+	0.950356i	$0.600719\pi$
$662$	7.00000i	0.272063i
$663$	2.00000i	0.0776736i
$664$	14.0000	0.543305
$665$	0	0
$666$	−3.00000	−0.116248
$667$	20.0000i	0.774403i
$668$	− 6.00000i	− 0.232147i
$669$	10.0000	0.386622
$670$	0	0
$671$	−2.00000	−0.0772091
$672$	− 5.00000i	− 0.192879i
$673$	10.0000i	0.385472i	0.981251	+	0.192736i	$0.0617360\pi$
$673$	10.0000i	0.385472i	−0.981251	+	0.192736i	$0.938264\pi$
$674$	8.00000	0.308148
$675$	0	0
$676$	−9.00000	−0.346154
$677$	− 36.0000i	− 1.38359i	−0.722093	−	0.691796i	$-0.756820\pi$
$677$	− 36.0000i	− 1.38359i	0.722093	−	0.691796i	$-0.243180\pi$
$678$	− 3.00000i	− 0.115214i
$679$	−30.0000	−1.15129
$680$	0	0
$681$	−1.00000	−0.0383201
$682$	5.00000i	0.191460i
$683$	0	0	1.00000			$0$
$683$	0	0	−1.00000			$\pi$
$684$	7.00000	0.267652
$685$	0	0
$686$	−55.0000	−2.09991
$687$	− 12.0000i	− 0.457829i
$688$	− 9.00000i	− 0.343122i
$689$	−22.0000	−0.838133
$690$	0	0
$691$	4.00000	0.152167	0.0760836	−	0.997101i	$-0.475758\pi$
$691$	4.00000	0.152167	0.0760836	+	0.997101i	$0.475758\pi$
$692$	20.0000i	0.760286i
$693$	− 5.00000i	− 0.189934i
$694$	5.00000	0.189797
$695$	0	0
$696$	−10.0000	−0.379049
$697$	10.0000i	0.378777i
$698$	2.00000i	0.0757011i
$699$	−18.0000	−0.680823
$700$	0	0
$701$	−42.0000	−1.58632	−0.793159	−	0.609015i	$-0.791565\pi$
$701$	−42.0000	−1.58632	−0.793159	+	0.609015i	$0.791565\pi$
$702$	2.00000i	0.0754851i
$703$	− 21.0000i	− 0.792030i
$704$	1.00000	0.0376889
$705$	0	0
$706$	−2.00000	−0.0752710
$707$	35.0000i	1.31631i
$708$	0	0
$709$	17.0000	0.638448	0.319224	−	0.947679i	$-0.396578\pi$
$709$	17.0000	0.638448	0.319224	+	0.947679i	$0.396578\pi$
$710$	0	0
$711$	3.00000	0.112509
$712$	4.00000i	0.149906i
$713$	− 10.0000i	− 0.374503i
$714$	−5.00000	−0.187120
$715$	0	0
$716$	−12.0000	−0.448461
$717$	15.0000i	0.560185i
$718$	3.00000i	0.111959i
$719$	2.00000	0.0745874	0.0372937	−	0.999304i	$-0.488126\pi$
$719$	2.00000	0.0745874	0.0372937	+	0.999304i	$0.488126\pi$
$720$	0	0
$721$	0	0
$722$	30.0000i	1.11648i
$723$	20.0000i	0.743808i
$724$	−17.0000	−0.631800
$725$	0	0
$726$	−10.0000	−0.371135
$727$	− 26.0000i	− 0.964287i	−0.876092	−	0.482143i	$-0.839858\pi$
$727$	− 26.0000i	− 0.964287i	0.876092	−	0.482143i	$-0.160142\pi$
$728$	10.0000i	0.370625i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−9.00000	−0.332877
$732$	2.00000i	0.0739221i
$733$	36.0000i	1.32969i	0.746981	+	0.664845i	$0.231502\pi$
$733$	36.0000i	1.32969i	−0.746981	+	0.664845i	$0.768498\pi$
$734$	−5.00000	−0.184553
$735$	0	0
$736$	−2.00000	−0.0737210
$737$	13.0000i	0.478861i
$738$	10.0000i	0.368105i
$739$	51.0000	1.87607	0.938033	−	0.346547i	$-0.112646\pi$
$739$	51.0000	1.87607	0.938033	+	0.346547i	$0.112646\pi$
$740$	0	0
$741$	−14.0000	−0.514303
$742$	− 55.0000i	− 2.01911i
$743$	34.0000i	1.24734i	0.781688	+	0.623670i	$0.214359\pi$
$743$	34.0000i	1.24734i	−0.781688	+	0.623670i	$0.785641\pi$
$744$	5.00000	0.183309
$745$	0	0
$746$	−26.0000	−0.951928
$747$	− 14.0000i	− 0.512233i
$748$	− 1.00000i	− 0.0365636i
$749$	45.0000	1.64426
$750$	0	0
$751$	−24.0000	−0.875772	−0.437886	−	0.899030i	$-0.644273\pi$
$751$	−24.0000	−0.875772	−0.437886	+	0.899030i	$0.644273\pi$
$752$	7.00000i	0.255264i
$753$	− 2.00000i	− 0.0728841i
$754$	20.0000	0.728357
$755$	0	0
$756$	−5.00000	−0.181848
$757$	10.0000i	0.363456i	0.983349	+	0.181728i	$0.0581691\pi$
$757$	10.0000i	0.363456i	−0.983349	+	0.181728i	$0.941831\pi$
$758$	− 20.0000i	− 0.726433i
$759$	−2.00000	−0.0725954
$760$	0	0
$761$	−36.0000	−1.30500	−0.652499	−	0.757789i	$-0.726280\pi$
$761$	−36.0000	−1.30500	−0.652499	+	0.757789i	$0.726280\pi$
$762$	4.00000i	0.144905i
$763$	95.0000i	3.43923i
$764$	−25.0000	−0.904468
$765$	0	0
$766$	−8.00000	−0.289052
$767$	0	0
$768$	− 1.00000i	− 0.0360844i
$769$	13.0000	0.468792	0.234396	−	0.972141i	$-0.424689\pi$
$769$	13.0000	0.468792	0.234396	+	0.972141i	$0.424689\pi$
$770$	0	0
$771$	−18.0000	−0.648254
$772$	24.0000i	0.863779i
$773$	− 6.00000i	− 0.215805i	−0.994161	−	0.107903i	$-0.965587\pi$
$773$	− 6.00000i	− 0.215805i	0.994161	−	0.107903i	$-0.0344134\pi$
$774$	−9.00000	−0.323498
$775$	0	0
$776$	−6.00000	−0.215387
$777$	15.0000i	0.538122i
$778$	15.0000i	0.537776i
$779$	−70.0000	−2.50801
$780$	0	0
$781$	10.0000	0.357828
$782$	2.00000i	0.0715199i
$783$	10.0000i	0.357371i
$784$	−18.0000	−0.642857
$785$	0	0
$786$	−4.00000	−0.142675
$787$	− 18.0000i	− 0.641631i	−0.947142	−	0.320815i	$-0.896043\pi$
$787$	− 18.0000i	− 0.641631i	0.947142	−	0.320815i	$-0.103957\pi$
$788$	− 8.00000i	− 0.284988i
$789$	17.0000	0.605216
$790$	0	0
$791$	−15.0000	−0.533339
$792$	− 1.00000i	− 0.0355335i
$793$	− 4.00000i	− 0.142044i
$794$	3.00000	0.106466
$795$	0	0
$796$	7.00000	0.248108
$797$	41.0000i	1.45229i	0.687539	+	0.726147i	$0.258691\pi$
$797$	41.0000i	1.45229i	−0.687539	+	0.726147i	$0.741309\pi$
$798$	− 35.0000i	− 1.23899i
$799$	7.00000	0.247642
$800$	0	0
$801$	4.00000	0.141333
$802$	30.0000i	1.05934i
$803$	− 4.00000i	− 0.141157i
$804$	13.0000	0.458475
$805$	0	0
$806$	−10.0000	−0.352235
$807$	6.00000i	0.211210i
$808$	7.00000i	0.246259i
$809$	−45.0000	−1.58212	−0.791058	−	0.611741i	$-0.790469\pi$
$809$	−45.0000	−1.58212	−0.791058	+	0.611741i	$0.790469\pi$
$810$	0	0
$811$	8.00000	0.280918	0.140459	−	0.990086i	$-0.455142\pi$
$811$	8.00000	0.280918	0.140459	+	0.990086i	$0.455142\pi$
$812$	50.0000i	1.75466i
$813$	6.00000i	0.210429i
$814$	−3.00000	−0.105150
$815$	0	0
$816$	−1.00000	−0.0350070
$817$	− 63.0000i	− 2.20409i
$818$	26.0000i	0.909069i
$819$	10.0000	0.349428
$820$	0	0
$821$	18.0000	0.628204	0.314102	−	0.949389i	$-0.398297\pi$
$821$	18.0000	0.628204	0.314102	+	0.949389i	$0.398297\pi$
$822$	− 12.0000i	− 0.418548i
$823$	− 4.00000i	− 0.139431i	−0.997567	−	0.0697156i	$-0.977791\pi$
$823$	− 4.00000i	− 0.139431i	0.997567	−	0.0697156i	$-0.0222092\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 23.0000i	− 0.799788i	−0.916561	−	0.399894i	$-0.869047\pi$
$827$	− 23.0000i	− 0.799788i	0.916561	−	0.399894i	$-0.130953\pi$
$828$	2.00000i	0.0695048i
$829$	22.0000	0.764092	0.382046	−	0.924143i	$-0.375220\pi$
$829$	22.0000	0.764092	0.382046	+	0.924143i	$0.375220\pi$
$830$	0	0
$831$	−23.0000	−0.797861
$832$	2.00000i	0.0693375i
$833$	18.0000i	0.623663i
$834$	20.0000	0.692543
$835$	0	0
$836$	7.00000	0.242100
$837$	− 5.00000i	− 0.172825i
$838$	0	0
$839$	−36.0000	−1.24286	−0.621429	−	0.783470i	$-0.713448\pi$
$839$	−36.0000	−1.24286	−0.621429	+	0.783470i	$0.713448\pi$
$840$	0	0
$841$	71.0000	2.44828
$842$	− 30.0000i	− 1.03387i
$843$	− 22.0000i	− 0.757720i
$844$	14.0000	0.481900
$845$	0	0
$846$	7.00000	0.240665
$847$	50.0000i	1.71802i
$848$	− 11.0000i	− 0.377742i
$849$	−14.0000	−0.480479
$850$	0	0
$851$	6.00000	0.205677
$852$	− 10.0000i	− 0.342594i
$853$	5.00000i	0.171197i	0.996330	+	0.0855984i	$0.0272802\pi$
$853$	5.00000i	0.171197i	−0.996330	+	0.0855984i	$0.972720\pi$
$854$	10.0000	0.342193
$855$	0	0
$856$	9.00000	0.307614
$857$	− 23.0000i	− 0.785665i	−0.919610	−	0.392833i	$-0.871495\pi$
$857$	− 23.0000i	− 0.785665i	0.919610	−	0.392833i	$-0.128505\pi$
$858$	2.00000i	0.0682789i
$859$	47.0000	1.60362	0.801810	−	0.597580i	$-0.203871\pi$
$859$	47.0000	1.60362	0.801810	+	0.597580i	$0.203871\pi$
$860$	0	0
$861$	50.0000	1.70400
$862$	− 16.0000i	− 0.544962i
$863$	− 43.0000i	− 1.46374i	−0.681446	−	0.731869i	$-0.738649\pi$
$863$	− 43.0000i	− 1.46374i	0.681446	−	0.731869i	$-0.261351\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	35.0000	1.18935
$867$	1.00000i	0.0339618i
$868$	− 25.0000i	− 0.848555i
$869$	3.00000	0.101768
$870$	0	0
$871$	−26.0000	−0.880976
$872$	19.0000i	0.643421i
$873$	6.00000i	0.203069i
$874$	−14.0000	−0.473557
$875$	0	0
$876$	−4.00000	−0.135147
$877$	− 50.0000i	− 1.68838i	−0.536044	−	0.844190i	$-0.680082\pi$
$877$	− 50.0000i	− 1.68838i	0.536044	−	0.844190i	$-0.319918\pi$
$878$	24.0000i	0.809961i
$879$	−14.0000	−0.472208
$880$	0	0
$881$	3.00000	0.101073	0.0505363	−	0.998722i	$-0.483907\pi$
$881$	3.00000	0.101073	0.0505363	+	0.998722i	$0.483907\pi$
$882$	18.0000i	0.606092i
$883$	44.0000i	1.48072i	0.672212	+	0.740359i	$0.265344\pi$
$883$	44.0000i	1.48072i	−0.672212	+	0.740359i	$0.734656\pi$
$884$	2.00000	0.0672673
$885$	0	0
$886$	0	0
$887$	24.0000i	0.805841i	0.915235	+	0.402921i	$0.132005\pi$
$887$	24.0000i	0.805841i	−0.915235	+	0.402921i	$0.867995\pi$
$888$	3.00000i	0.100673i
$889$	20.0000	0.670778
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	− 10.0000i	− 0.334825i
$893$	49.0000i	1.63972i
$894$	2.00000	0.0668900
$895$	0	0
$896$	−5.00000	−0.167038
$897$	− 4.00000i	− 0.133556i
$898$	− 9.00000i	− 0.300334i
$899$	−50.0000	−1.66759
$900$	0	0
$901$	−11.0000	−0.366463
$902$	10.0000i	0.332964i
$903$	45.0000i	1.49751i
$904$	−3.00000	−0.0997785
$905$	0	0
$906$	−8.00000	−0.265782
$907$	38.0000i	1.26177i	0.775877	+	0.630885i	$0.217308\pi$
$907$	38.0000i	1.26177i	−0.775877	+	0.630885i	$0.782692\pi$
$908$	1.00000i	0.0331862i
$909$	7.00000	0.232175
$910$	0	0
$911$	8.00000	0.265052	0.132526	−	0.991180i	$-0.457691\pi$
$911$	8.00000	0.265052	0.132526	+	0.991180i	$0.457691\pi$
$912$	− 7.00000i	− 0.231793i
$913$	− 14.0000i	− 0.463332i
$914$	−17.0000	−0.562310
$915$	0	0
$916$	−12.0000	−0.396491
$917$	20.0000i	0.660458i
$918$	1.00000i	0.0330049i
$919$	−4.00000	−0.131948	−0.0659739	−	0.997821i	$-0.521015\pi$
$919$	−4.00000	−0.131948	−0.0659739	+	0.997821i	$0.521015\pi$
$920$	0	0
$921$	4.00000	0.131804
$922$	35.0000i	1.15266i
$923$	20.0000i	0.658308i
$924$	−5.00000	−0.164488
$925$	0	0
$926$	−16.0000	−0.525793
$927$	0	0
$928$	10.0000i	0.328266i
$929$	9.00000	0.295280	0.147640	−	0.989041i	$-0.452832\pi$
$929$	9.00000	0.295280	0.147640	+	0.989041i	$0.452832\pi$
$930$	0	0
$931$	−126.000	−4.12948
$932$	18.0000i	0.589610i
$933$	− 26.0000i	− 0.851202i
$934$	6.00000	0.196326
$935$	0	0
$936$	2.00000	0.0653720
$937$	− 26.0000i	− 0.849383i	−0.905338	−	0.424691i	$-0.860383\pi$
$937$	− 26.0000i	− 0.849383i	0.905338	−	0.424691i	$-0.139617\pi$
$938$	− 65.0000i	− 2.12233i
$939$	−2.00000	−0.0652675
$940$	0	0
$941$	10.0000	0.325991	0.162995	−	0.986627i	$-0.447884\pi$
$941$	10.0000	0.325991	0.162995	+	0.986627i	$0.447884\pi$
$942$	16.0000i	0.521308i
$943$	− 20.0000i	− 0.651290i
$944$	0	0
$945$	0	0
$946$	−9.00000	−0.292615
$947$	− 41.0000i	− 1.33232i	−0.745808	−	0.666160i	$-0.767937\pi$
$947$	− 41.0000i	− 1.33232i	0.745808	−	0.666160i	$-0.232063\pi$
$948$	− 3.00000i	− 0.0974355i
$949$	8.00000	0.259691
$950$	0	0
$951$	20.0000	0.648544
$952$	5.00000i	0.162051i
$953$	− 54.0000i	− 1.74923i	−0.484817	−	0.874616i	$-0.661114\pi$
$953$	− 54.0000i	− 1.74923i	0.484817	−	0.874616i	$-0.338886\pi$
$954$	−11.0000	−0.356138
$955$	0	0
$956$	15.0000	0.485135
$957$	10.0000i	0.323254i
$958$	22.0000i	0.710788i
$959$	−60.0000	−1.93750
$960$	0	0
$961$	−6.00000	−0.193548
$962$	− 6.00000i	− 0.193448i
$963$	− 9.00000i	− 0.290021i
$964$	20.0000	0.644157
$965$	0	0
$966$	10.0000	0.321745
$967$	16.0000i	0.514525i	0.966342	+	0.257263i	$0.0828206\pi$
$967$	16.0000i	0.514525i	−0.966342	+	0.257263i	$0.917179\pi$
$968$	10.0000i	0.321412i
$969$	−7.00000	−0.224872
$970$	0	0
$971$	22.0000	0.706014	0.353007	−	0.935621i	$-0.385159\pi$
$971$	22.0000	0.706014	0.353007	+	0.935621i	$0.385159\pi$
$972$	1.00000i	0.0320750i
$973$	− 100.000i	− 3.20585i
$974$	8.00000	0.256337
$975$	0	0
$976$	2.00000	0.0640184
$977$	− 18.0000i	− 0.575871i	−0.957650	−	0.287936i	$-0.907031\pi$
$977$	− 18.0000i	− 0.575871i	0.957650	−	0.287936i	$-0.0929689\pi$
$978$	16.0000i	0.511624i
$979$	4.00000	0.127841
$980$	0	0
$981$	19.0000	0.606623
$982$	12.0000i	0.382935i
$983$	− 10.0000i	− 0.318950i	−0.987202	−	0.159475i	$-0.949020\pi$
$983$	− 10.0000i	− 0.318950i	0.987202	−	0.159475i	$-0.0509802\pi$
$984$	10.0000	0.318788
$985$	0	0
$986$	10.0000	0.318465
$987$	− 35.0000i	− 1.11406i
$988$	14.0000i	0.445399i
$989$	18.0000	0.572367
$990$	0	0
$991$	−4.00000	−0.127064	−0.0635321	−	0.997980i	$-0.520237\pi$
$991$	−4.00000	−0.127064	−0.0635321	+	0.997980i	$0.520237\pi$
$992$	− 5.00000i	− 0.158750i
$993$	− 7.00000i	− 0.222138i
$994$	−50.0000	−1.58590
$995$	0	0
$996$	−14.0000	−0.443607
$997$	− 43.0000i	− 1.36182i	−0.732365	−	0.680912i	$-0.761584\pi$
$997$	− 43.0000i	− 1.36182i	0.732365	−	0.680912i	$-0.238416\pi$
$998$	− 22.0000i	− 0.696398i
$999$	3.00000	0.0949158

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2550.2.d.p.2449.2		2
5.2	odd	4		2550.2.a.g.1.1	✓	1
5.3	odd	4		2550.2.a.ba.1.1	yes	1
5.4	even	2	inner	2550.2.d.p.2449.1		2
15.2	even	4		7650.2.a.cp.1.1		1
15.8	even	4		7650.2.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2550.2.a.g.1.1	✓	1	5.2	odd	4
2550.2.a.ba.1.1	yes	1	5.3	odd	4
2550.2.d.p.2449.1		2	5.4	even	2	inner
2550.2.d.p.2449.2		2	1.1	even	1	trivial
7650.2.a.b.1.1		1	15.8	even	4
7650.2.a.cp.1.1		1	15.2	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -5.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -5.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} -1.00000 q^{11} +1.00000i q^{12} -2.00000i q^{13} +5.00000 q^{14} +1.00000 q^{16} -1.00000i q^{17} -1.00000i q^{18} +7.00000 q^{19} -5.00000 q^{21} -1.00000i q^{22} +2.00000i q^{23} -1.00000 q^{24} +2.00000 q^{26} +1.00000i q^{27} +5.00000i q^{28} +10.0000 q^{29} -5.00000 q^{31} +1.00000i q^{32} +1.00000i q^{33} +1.00000 q^{34} +1.00000 q^{36} -3.00000i q^{37} +7.00000i q^{38} -2.00000 q^{39} -10.0000 q^{41} -5.00000i q^{42} -9.00000i q^{43} +1.00000 q^{44} -2.00000 q^{46} +7.00000i q^{47} -1.00000i q^{48} -18.0000 q^{49} -1.00000 q^{51} +2.00000i q^{52} -11.0000i q^{53} -1.00000 q^{54} -5.00000 q^{56} -7.00000i q^{57} +10.0000i q^{58} +2.00000 q^{61} -5.00000i q^{62} +5.00000i q^{63} -1.00000 q^{64} -1.00000 q^{66} -13.0000i q^{67} +1.00000i q^{68} +2.00000 q^{69} -10.0000 q^{71} +1.00000i q^{72} +4.00000i q^{73} +3.00000 q^{74} -7.00000 q^{76} +5.00000i q^{77} -2.00000i q^{78} -3.00000 q^{79} +1.00000 q^{81} -10.0000i q^{82} +14.0000i q^{83} +5.00000 q^{84} +9.00000 q^{86} -10.0000i q^{87} +1.00000i q^{88} -4.00000 q^{89} -10.0000 q^{91} -2.00000i q^{92} +5.00000i q^{93} -7.00000 q^{94} +1.00000 q^{96} -6.00000i q^{97} -18.0000i q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9	\( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} - 2 q^{11} + 10 q^{14} + 2 q^{16} + 14 q^{19} - 10 q^{21} - 2 q^{24} + 4 q^{26} + 20 q^{29} - 10 q^{31} + 2 q^{34} + 2 q^{36} - 4 q^{39} - 20 q^{41} + 2 q^{44} - 4 q^{46} - 36 q^{49} - 2 q^{51} - 2 q^{54} - 10 q^{56} + 4 q^{61} - 2 q^{64} - 2 q^{66} + 4 q^{69} - 20 q^{71} + 6 q^{74} - 14 q^{76} - 6 q^{79} + 2 q^{81} + 10 q^{84} + 18 q^{86} - 8 q^{89} - 20 q^{91} - 14 q^{94} + 2 q^{96} + 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9 - 2 * q^11 + 10 * q^14 + 2 * q^16 + 14 * q^19 - 10 * q^21 - 2 * q^24 + 4 * q^26 + 20 * q^29 - 10 * q^31 + 2 * q^34 + 2 * q^36 - 4 * q^39 - 20 * q^41 + 2 * q^44 - 4 * q^46 - 36 * q^49 - 2 * q^51 - 2 * q^54 - 10 * q^56 + 4 * q^61 - 2 * q^64 - 2 * q^66 + 4 * q^69 - 20 * q^71 + 6 * q^74 - 14 * q^76 - 6 * q^79 + 2 * q^81 + 10 * q^84 + 18 * q^86 - 8 * q^89 - 20 * q^91 - 14 * q^94 + 2 * q^96 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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