LMFDB - Embedded newform 3450.2.d.o.2899.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3450,2,Mod(2899,3450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3450, 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("3450.2899");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3450 = 2 \cdot 3 \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3450.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:	$27.5483886973$
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			2899.2
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	3450.2899
Dual form			3450.2.d.o.2899.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -3.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -3.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} -1.00000 q^{11} +1.00000i q^{12} +3.00000i q^{13} +3.00000 q^{14} +1.00000 q^{16} -1.00000i q^{18} +5.00000 q^{19} -3.00000 q^{21} -1.00000i q^{22} -1.00000i q^{23} -1.00000 q^{24} -3.00000 q^{26} +1.00000i q^{27} +3.00000i q^{28} +9.00000 q^{29} -2.00000 q^{31} +1.00000i q^{32} +1.00000i q^{33} +1.00000 q^{36} +5.00000i q^{38} +3.00000 q^{39} -5.00000 q^{41} -3.00000i q^{42} -7.00000i q^{43} +1.00000 q^{44} +1.00000 q^{46} -2.00000i q^{47} -1.00000i q^{48} -2.00000 q^{49} -3.00000i q^{52} +2.00000i q^{53} -1.00000 q^{54} -3.00000 q^{56} -5.00000i q^{57} +9.00000i q^{58} +12.0000 q^{59} +4.00000 q^{61} -2.00000i q^{62} +3.00000i q^{63} -1.00000 q^{64} -1.00000 q^{66} -8.00000i q^{67} -1.00000 q^{69} -10.0000 q^{71} +1.00000i q^{72} -5.00000i q^{73} -5.00000 q^{76} +3.00000i q^{77} +3.00000i q^{78} -9.00000 q^{79} +1.00000 q^{81} -5.00000i q^{82} -11.0000i q^{83} +3.00000 q^{84} +7.00000 q^{86} -9.00000i q^{87} +1.00000i q^{88} -10.0000 q^{89} +9.00000 q^{91} +1.00000i q^{92} +2.00000i q^{93} +2.00000 q^{94} +1.00000 q^{96} -14.0000i q^{97} -2.00000i q^{98} +1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 6 q^{14} + 2 q^{16} + 10 q^{19} - 6 q^{21} - 2 q^{24} - 6 q^{26} + 18 q^{29} - 4 q^{31} + 2 q^{36} + 6 q^{39} - 10 q^{41} + 2 q^{44} + 2 q^{46} - 4 q^{49} - 2 q^{54} - 6 q^{56} + 24 q^{59} + 8 q^{61} - 2 q^{64} - 2 q^{66} - 2 q^{69} - 20 q^{71} - 10 q^{76} - 18 q^{79} + 2 q^{81} + 6 q^{84} + 14 q^{86} - 20 q^{89} + 18 q^{91} + 4 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 + 6 * q^14 + 2 * q^16 + 10 * q^19 - 6 * q^21 - 2 * q^24 - 6 * q^26 + 18 * q^29 - 4 * q^31 + 2 * q^36 + 6 * q^39 - 10 * q^41 + 2 * q^44 + 2 * q^46 - 4 * q^49 - 2 * q^54 - 6 * q^56 + 24 * q^59 + 8 * q^61 - 2 * q^64 - 2 * q^66 - 2 * q^69 - 20 * q^71 - 10 * q^76 - 18 * q^79 + 2 * q^81 + 6 * q^84 + 14 * q^86 - 20 * q^89 + 18 * q^91 + 4 * q^94 + 2 * q^96 + 2 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3450\mathbb{Z}\right)^\times$.

$n$	$277$	$1151$	$1201$
$\chi(n)$	$-1$	$1$	$1$

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
$7$	− 3.00000i	− 1.13389i	−0.823754	−	0.566947i	$-0.808125\pi$
$7$	− 3.00000i	− 1.13389i	0.823754	−	0.566947i	$-0.191875\pi$
$8$	− 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
$13$	3.00000i	0.832050i	0.909353	+	0.416025i	$0.136577\pi$
$13$	3.00000i	0.832050i	−0.909353	+	0.416025i	$0.863423\pi$
$14$	3.00000	0.801784
$15$	0	0
$16$	1.00000	0.250000
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	− 1.00000i	− 0.235702i
$19$	5.00000	1.14708	0.573539	−	0.819178i	$-0.305570\pi$
$19$	5.00000	1.14708	0.573539	+	0.819178i	$0.305570\pi$
$20$	0	0
$21$	−3.00000	−0.654654
$22$	− 1.00000i	− 0.213201i
$23$	− 1.00000i	− 0.208514i
$23$	− 1.00000i	− 0.208514i
$24$	−1.00000	−0.204124
$25$	0	0
$26$	−3.00000	−0.588348
$27$	1.00000i	0.192450i
$28$	3.00000i	0.566947i
$29$	9.00000	1.67126	0.835629	−	0.549294i	$-0.185103\pi$
$29$	9.00000	1.67126	0.835629	+	0.549294i	$0.185103\pi$
$30$	0	0
$31$	−2.00000	−0.359211	−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000	−0.359211	−0.179605	+	0.983739i	$0.557482\pi$
$32$	1.00000i	0.176777i
$33$	1.00000i	0.174078i
$34$	0	0
$35$	0	0
$36$	1.00000	0.166667
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	5.00000i	0.811107i
$39$	3.00000	0.480384
$40$	0	0
$41$	−5.00000	−0.780869	−0.390434	−	0.920631i	$-0.627675\pi$
$41$	−5.00000	−0.780869	−0.390434	+	0.920631i	$0.627675\pi$
$42$	− 3.00000i	− 0.462910i
$43$	− 7.00000i	− 1.06749i	−0.845645	−	0.533745i	$-0.820784\pi$
$43$	− 7.00000i	− 1.06749i	0.845645	−	0.533745i	$-0.179216\pi$
$44$	1.00000	0.150756
$45$	0	0
$46$	1.00000	0.147442
$47$	− 2.00000i	− 0.291730i	−0.989305	−	0.145865i	$-0.953403\pi$
$47$	− 2.00000i	− 0.291730i	0.989305	−	0.145865i	$-0.0465965\pi$
$48$	− 1.00000i	− 0.144338i
$49$	−2.00000	−0.285714
$50$	0	0
$51$	0	0
$52$	− 3.00000i	− 0.416025i
$53$	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
$56$	−3.00000	−0.400892
$57$	− 5.00000i	− 0.662266i
$58$	9.00000i	1.18176i
$59$	12.0000	1.56227	0.781133	−	0.624364i	$-0.214642\pi$
$59$	12.0000	1.56227	0.781133	+	0.624364i	$0.214642\pi$
$60$	0	0
$61$	4.00000	0.512148	0.256074	−	0.966657i	$-0.417571\pi$
$61$	4.00000	0.512148	0.256074	+	0.966657i	$0.417571\pi$
$62$	− 2.00000i	− 0.254000i
$63$	3.00000i	0.377964i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	−1.00000	−0.123091
$67$	− 8.00000i	− 0.977356i	−0.872464	−	0.488678i	$-0.837479\pi$
$67$	− 8.00000i	− 0.977356i	0.872464	−	0.488678i	$-0.162521\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	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
$73$	− 5.00000i	− 0.585206i	−0.956234	−	0.292603i	$-0.905479\pi$
$73$	− 5.00000i	− 0.585206i	0.956234	−	0.292603i	$-0.0945214\pi$
$74$	0	0
$75$	0	0
$76$	−5.00000	−0.573539
$77$	3.00000i	0.341882i
$78$	3.00000i	0.339683i
$79$	−9.00000	−1.01258	−0.506290	−	0.862364i	$-0.668983\pi$
$79$	−9.00000	−1.01258	−0.506290	+	0.862364i	$0.668983\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	− 5.00000i	− 0.552158i
$83$	− 11.0000i	− 1.20741i	−0.797209	−	0.603703i	$-0.793691\pi$
$83$	− 11.0000i	− 1.20741i	0.797209	−	0.603703i	$-0.206309\pi$
$84$	3.00000	0.327327
$85$	0	0
$86$	7.00000	0.754829
$87$	− 9.00000i	− 0.964901i
$88$	1.00000i	0.106600i
$89$	−10.0000	−1.06000	−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000	−1.06000	−0.529999	+	0.847998i	$0.677808\pi$
$90$	0	0
$91$	9.00000	0.943456
$92$	1.00000i	0.104257i
$93$	2.00000i	0.207390i
$94$	2.00000	0.206284
$95$	0	0
$96$	1.00000	0.102062
$97$	− 14.0000i	− 1.42148i	−0.703452	−	0.710742i	$-0.748359\pi$
$97$	− 14.0000i	− 1.42148i	0.703452	−	0.710742i	$-0.251641\pi$
$98$	− 2.00000i	− 0.202031i
$99$	1.00000	0.100504
$100$	0	0
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	0	0
$103$	− 11.0000i	− 1.08386i	−0.840423	−	0.541931i	$-0.817693\pi$
$103$	− 11.0000i	− 1.08386i	0.840423	−	0.541931i	$-0.182307\pi$
$104$	3.00000	0.294174
$105$	0	0
$106$	−2.00000	−0.194257
$107$	− 4.00000i	− 0.386695i	−0.981130	−	0.193347i	$-0.938066\pi$
$107$	− 4.00000i	− 0.386695i	0.981130	−	0.193347i	$-0.0619344\pi$
$108$	− 1.00000i	− 0.0962250i
$109$	16.0000	1.53252	0.766261	−	0.642529i	$-0.222115\pi$
$109$	16.0000	1.53252	0.766261	+	0.642529i	$0.222115\pi$
$110$	0	0
$111$	0	0
$112$	− 3.00000i	− 0.283473i
$113$	2.00000i	0.188144i	0.995565	+	0.0940721i	$0.0299884\pi$
$113$	2.00000i	0.188144i	−0.995565	+	0.0940721i	$0.970012\pi$
$114$	5.00000	0.468293
$115$	0	0
$116$	−9.00000	−0.835629
$117$	− 3.00000i	− 0.277350i
$118$	12.0000i	1.10469i
$119$	0	0
$120$	0	0
$121$	−10.0000	−0.909091
$122$	4.00000i	0.362143i
$123$	5.00000i	0.450835i
$124$	2.00000	0.179605
$125$	0	0
$126$	−3.00000	−0.267261
$127$	− 12.0000i	− 1.06483i	−0.846484	−	0.532414i	$-0.821285\pi$
$127$	− 12.0000i	− 1.06483i	0.846484	−	0.532414i	$-0.178715\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	−7.00000	−0.616316
$130$	0	0
$131$	2.00000	0.174741	0.0873704	−	0.996176i	$-0.472154\pi$
$131$	2.00000	0.174741	0.0873704	+	0.996176i	$0.472154\pi$
$132$	− 1.00000i	− 0.0870388i
$133$	− 15.0000i	− 1.30066i
$134$	8.00000	0.691095
$135$	0	0
$136$	0	0
$137$	2.00000i	0.170872i	0.996344	+	0.0854358i	$0.0272282\pi$
$137$	2.00000i	0.170872i	−0.996344	+	0.0854358i	$0.972772\pi$
$138$	− 1.00000i	− 0.0851257i
$139$	−12.0000	−1.01783	−0.508913	−	0.860818i	$-0.669953\pi$
$139$	−12.0000	−1.01783	−0.508913	+	0.860818i	$0.669953\pi$
$140$	0	0
$141$	−2.00000	−0.168430
$142$	− 10.0000i	− 0.839181i
$143$	− 3.00000i	− 0.250873i
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	5.00000	0.413803
$147$	2.00000i	0.164957i
$148$	0	0
$149$	−22.0000	−1.80231	−0.901155	−	0.433497i	$-0.857280\pi$
$149$	−22.0000	−1.80231	−0.901155	+	0.433497i	$0.857280\pi$
$150$	0	0
$151$	−2.00000	−0.162758	−0.0813788	−	0.996683i	$-0.525932\pi$
$151$	−2.00000	−0.162758	−0.0813788	+	0.996683i	$0.525932\pi$
$152$	− 5.00000i	− 0.405554i
$153$	0	0
$154$	−3.00000	−0.241747
$155$	0	0
$156$	−3.00000	−0.240192
$157$	− 6.00000i	− 0.478852i	−0.970915	−	0.239426i	$-0.923041\pi$
$157$	− 6.00000i	− 0.478852i	0.970915	−	0.239426i	$-0.0769593\pi$
$158$	− 9.00000i	− 0.716002i
$159$	2.00000	0.158610
$160$	0	0
$161$	−3.00000	−0.236433
$162$	1.00000i	0.0785674i
$163$	− 22.0000i	− 1.72317i	−0.507611	−	0.861586i	$-0.669471\pi$
$163$	− 22.0000i	− 1.72317i	0.507611	−	0.861586i	$-0.330529\pi$
$164$	5.00000	0.390434
$165$	0	0
$166$	11.0000	0.853766
$167$	2.00000i	0.154765i	0.997001	+	0.0773823i	$0.0246562\pi$
$167$	2.00000i	0.154765i	−0.997001	+	0.0773823i	$0.975344\pi$
$168$	3.00000i	0.231455i
$169$	4.00000	0.307692
$170$	0	0
$171$	−5.00000	−0.382360
$172$	7.00000i	0.533745i
$173$	− 3.00000i	− 0.228086i	−0.993476	−	0.114043i	$-0.963620\pi$
$173$	− 3.00000i	− 0.228086i	0.993476	−	0.114043i	$-0.0363801\pi$
$174$	9.00000	0.682288
$175$	0	0
$176$	−1.00000	−0.0753778
$177$	− 12.0000i	− 0.901975i
$178$	− 10.0000i	− 0.749532i
$179$	6.00000	0.448461	0.224231	−	0.974536i	$-0.428013\pi$
$179$	6.00000	0.448461	0.224231	+	0.974536i	$0.428013\pi$
$180$	0	0
$181$	−12.0000	−0.891953	−0.445976	−	0.895045i	$-0.647144\pi$
$181$	−12.0000	−0.891953	−0.445976	+	0.895045i	$0.647144\pi$
$182$	9.00000i	0.667124i
$183$	− 4.00000i	− 0.295689i
$184$	−1.00000	−0.0737210
$185$	0	0
$186$	−2.00000	−0.146647
$187$	0	0
$188$	2.00000i	0.145865i
$189$	3.00000	0.218218
$190$	0	0
$191$	−5.00000	−0.361787	−0.180894	−	0.983503i	$-0.557899\pi$
$191$	−5.00000	−0.361787	−0.180894	+	0.983503i	$0.557899\pi$
$192$	1.00000i	0.0721688i
$193$	14.0000i	1.00774i	0.863779	+	0.503871i	$0.168091\pi$
$193$	14.0000i	1.00774i	−0.863779	+	0.503871i	$0.831909\pi$
$194$	14.0000	1.00514
$195$	0	0
$196$	2.00000	0.142857
$197$	11.0000i	0.783718i	0.920025	+	0.391859i	$0.128168\pi$
$197$	11.0000i	0.783718i	−0.920025	+	0.391859i	$0.871832\pi$
$198$	1.00000i	0.0710669i
$199$	3.00000	0.212664	0.106332	−	0.994331i	$-0.466089\pi$
$199$	3.00000	0.212664	0.106332	+	0.994331i	$0.466089\pi$
$200$	0	0
$201$	−8.00000	−0.564276
$202$	− 6.00000i	− 0.422159i
$203$	− 27.0000i	− 1.89503i
$204$	0	0
$205$	0	0
$206$	11.0000	0.766406
$207$	1.00000i	0.0695048i
$208$	3.00000i	0.208013i
$209$	−5.00000	−0.345857
$210$	0	0
$211$	10.0000	0.688428	0.344214	−	0.938891i	$-0.388145\pi$
$211$	10.0000	0.688428	0.344214	+	0.938891i	$0.388145\pi$
$212$	− 2.00000i	− 0.137361i
$213$	10.0000i	0.685189i
$214$	4.00000	0.273434
$215$	0	0
$216$	1.00000	0.0680414
$217$	6.00000i	0.407307i
$218$	16.0000i	1.08366i
$219$	−5.00000	−0.337869
$220$	0	0
$221$	0	0
$222$	0	0
$223$	2.00000i	0.133930i	0.997755	+	0.0669650i	$0.0213316\pi$
$223$	2.00000i	0.133930i	−0.997755	+	0.0669650i	$0.978668\pi$
$224$	3.00000	0.200446
$225$	0	0
$226$	−2.00000	−0.133038
$227$	20.0000i	1.32745i	0.747978	+	0.663723i	$0.231025\pi$
$227$	20.0000i	1.32745i	−0.747978	+	0.663723i	$0.768975\pi$
$228$	5.00000i	0.331133i
$229$	14.0000	0.925146	0.462573	−	0.886581i	$-0.346926\pi$
$229$	14.0000	0.925146	0.462573	+	0.886581i	$0.346926\pi$
$230$	0	0
$231$	3.00000	0.197386
$232$	− 9.00000i	− 0.590879i
$233$	5.00000i	0.327561i	0.986497	+	0.163780i	$0.0523689\pi$
$233$	5.00000i	0.327561i	−0.986497	+	0.163780i	$0.947631\pi$
$234$	3.00000	0.196116
$235$	0	0
$236$	−12.0000	−0.781133
$237$	9.00000i	0.584613i
$238$	0	0
$239$	26.0000	1.68180	0.840900	−	0.541190i	$-0.182026\pi$
$239$	26.0000	1.68180	0.840900	+	0.541190i	$0.182026\pi$
$240$	0	0
$241$	14.0000	0.901819	0.450910	−	0.892570i	$-0.351100\pi$
$241$	14.0000	0.901819	0.450910	+	0.892570i	$0.351100\pi$
$242$	− 10.0000i	− 0.642824i
$243$	− 1.00000i	− 0.0641500i
$244$	−4.00000	−0.256074
$245$	0	0
$246$	−5.00000	−0.318788
$247$	15.0000i	0.954427i
$248$	2.00000i	0.127000i
$249$	−11.0000	−0.697097
$250$	0	0
$251$	24.0000	1.51487	0.757433	−	0.652913i	$-0.226453\pi$
$251$	24.0000	1.51487	0.757433	+	0.652913i	$0.226453\pi$
$252$	− 3.00000i	− 0.188982i
$253$	1.00000i	0.0628695i
$254$	12.0000	0.752947
$255$	0	0
$256$	1.00000	0.0625000
$257$	6.00000i	0.374270i	0.982334	+	0.187135i	$0.0599201\pi$
$257$	6.00000i	0.374270i	−0.982334	+	0.187135i	$0.940080\pi$
$258$	− 7.00000i	− 0.435801i
$259$	0	0
$260$	0	0
$261$	−9.00000	−0.557086
$262$	2.00000i	0.123560i
$263$	− 16.0000i	− 0.986602i	−0.869859	−	0.493301i	$-0.835790\pi$
$263$	− 16.0000i	− 0.986602i	0.869859	−	0.493301i	$-0.164210\pi$
$264$	1.00000	0.0615457
$265$	0	0
$266$	15.0000	0.919709
$267$	10.0000i	0.611990i
$268$	8.00000i	0.488678i
$269$	17.0000	1.03651	0.518254	−	0.855227i	$-0.326582\pi$
$269$	17.0000	1.03651	0.518254	+	0.855227i	$0.326582\pi$
$270$	0	0
$271$	−2.00000	−0.121491	−0.0607457	−	0.998153i	$-0.519348\pi$
$271$	−2.00000	−0.121491	−0.0607457	+	0.998153i	$0.519348\pi$
$272$	0	0
$273$	− 9.00000i	− 0.544705i
$274$	−2.00000	−0.120824
$275$	0	0
$276$	1.00000	0.0601929
$277$	− 21.0000i	− 1.26177i	−0.775877	−	0.630884i	$-0.782692\pi$
$277$	− 21.0000i	− 1.26177i	0.775877	−	0.630884i	$-0.217308\pi$
$278$	− 12.0000i	− 0.719712i
$279$	2.00000	0.119737
$280$	0	0
$281$	6.00000	0.357930	0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000	0.357930	0.178965	+	0.983855i	$0.442725\pi$
$282$	− 2.00000i	− 0.119098i
$283$	− 16.0000i	− 0.951101i	−0.879688	−	0.475551i	$-0.842249\pi$
$283$	− 16.0000i	− 0.951101i	0.879688	−	0.475551i	$-0.157751\pi$
$284$	10.0000	0.593391
$285$	0	0
$286$	3.00000	0.177394
$287$	15.0000i	0.885422i
$288$	− 1.00000i	− 0.0589256i
$289$	17.0000	1.00000
$290$	0	0
$291$	−14.0000	−0.820695
$292$	5.00000i	0.292603i
$293$	− 16.0000i	− 0.934730i	−0.884064	−	0.467365i	$-0.845203\pi$
$293$	− 16.0000i	− 0.934730i	0.884064	−	0.467365i	$-0.154797\pi$
$294$	−2.00000	−0.116642
$295$	0	0
$296$	0	0
$297$	− 1.00000i	− 0.0580259i
$298$	− 22.0000i	− 1.27443i
$299$	3.00000	0.173494
$300$	0	0
$301$	−21.0000	−1.21042
$302$	− 2.00000i	− 0.115087i
$303$	6.00000i	0.344691i
$304$	5.00000	0.286770
$305$	0	0
$306$	0	0
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	− 3.00000i	− 0.170941i
$309$	−11.0000	−0.625768
$310$	0	0
$311$	16.0000	0.907277	0.453638	−	0.891186i	$-0.350126\pi$
$311$	16.0000	0.907277	0.453638	+	0.891186i	$0.350126\pi$
$312$	− 3.00000i	− 0.169842i
$313$	− 16.0000i	− 0.904373i	−0.891923	−	0.452187i	$-0.850644\pi$
$313$	− 16.0000i	− 0.904373i	0.891923	−	0.452187i	$-0.149356\pi$
$314$	6.00000	0.338600
$315$	0	0
$316$	9.00000	0.506290
$317$	3.00000i	0.168497i	0.996445	+	0.0842484i	$0.0268489\pi$
$317$	3.00000i	0.168497i	−0.996445	+	0.0842484i	$0.973151\pi$
$318$	2.00000i	0.112154i
$319$	−9.00000	−0.503903
$320$	0	0
$321$	−4.00000	−0.223258
$322$	− 3.00000i	− 0.167183i
$323$	0	0
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	22.0000	1.21847
$327$	− 16.0000i	− 0.884802i
$328$	5.00000i	0.276079i
$329$	−6.00000	−0.330791
$330$	0	0
$331$	−26.0000	−1.42909	−0.714545	−	0.699590i	$-0.753366\pi$
$331$	−26.0000	−1.42909	−0.714545	+	0.699590i	$0.753366\pi$
$332$	11.0000i	0.603703i
$333$	0	0
$334$	−2.00000	−0.109435
$335$	0	0
$336$	−3.00000	−0.163663
$337$	10.0000i	0.544735i	0.962193	+	0.272367i	$0.0878066\pi$
$337$	10.0000i	0.544735i	−0.962193	+	0.272367i	$0.912193\pi$
$338$	4.00000i	0.217571i
$339$	2.00000	0.108625
$340$	0	0
$341$	2.00000	0.108306
$342$	− 5.00000i	− 0.270369i
$343$	− 15.0000i	− 0.809924i
$344$	−7.00000	−0.377415
$345$	0	0
$346$	3.00000	0.161281
$347$	8.00000i	0.429463i	0.976673	+	0.214731i	$0.0688876\pi$
$347$	8.00000i	0.429463i	−0.976673	+	0.214731i	$0.931112\pi$
$348$	9.00000i	0.482451i
$349$	−11.0000	−0.588817	−0.294408	−	0.955680i	$-0.595123\pi$
$349$	−11.0000	−0.588817	−0.294408	+	0.955680i	$0.595123\pi$
$350$	0	0
$351$	−3.00000	−0.160128
$352$	− 1.00000i	− 0.0533002i
$353$	3.00000i	0.159674i	0.996808	+	0.0798369i	$0.0254400\pi$
$353$	3.00000i	0.159674i	−0.996808	+	0.0798369i	$0.974560\pi$
$354$	12.0000	0.637793
$355$	0	0
$356$	10.0000	0.529999
$357$	0	0
$358$	6.00000i	0.317110i
$359$	11.0000	0.580558	0.290279	−	0.956942i	$-0.406252\pi$
$359$	11.0000	0.580558	0.290279	+	0.956942i	$0.406252\pi$
$360$	0	0
$361$	6.00000	0.315789
$362$	− 12.0000i	− 0.630706i
$363$	10.0000i	0.524864i
$364$	−9.00000	−0.471728
$365$	0	0
$366$	4.00000	0.209083
$367$	− 11.0000i	− 0.574195i	−0.957901	−	0.287098i	$-0.907310\pi$
$367$	− 11.0000i	− 0.574195i	0.957901	−	0.287098i	$-0.0926904\pi$
$368$	− 1.00000i	− 0.0521286i
$369$	5.00000	0.260290
$370$	0	0
$371$	6.00000	0.311504
$372$	− 2.00000i	− 0.103695i
$373$	− 12.0000i	− 0.621336i	−0.950518	−	0.310668i	$-0.899447\pi$
$373$	− 12.0000i	− 0.621336i	0.950518	−	0.310668i	$-0.100553\pi$
$374$	0	0
$375$	0	0
$376$	−2.00000	−0.103142
$377$	27.0000i	1.39057i
$378$	3.00000i	0.154303i
$379$	20.0000	1.02733	0.513665	−	0.857991i	$-0.328287\pi$
$379$	20.0000	1.02733	0.513665	+	0.857991i	$0.328287\pi$
$380$	0	0
$381$	−12.0000	−0.614779
$382$	− 5.00000i	− 0.255822i
$383$	− 1.00000i	− 0.0510976i	−0.999674	−	0.0255488i	$-0.991867\pi$
$383$	− 1.00000i	− 0.0510976i	0.999674	−	0.0255488i	$-0.00813332\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	−14.0000	−0.712581
$387$	7.00000i	0.355830i
$388$	14.0000i	0.710742i
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	0	0
$391$	0	0
$392$	2.00000i	0.101015i
$393$	− 2.00000i	− 0.100887i
$394$	−11.0000	−0.554172
$395$	0	0
$396$	−1.00000	−0.0502519
$397$	14.0000i	0.702640i	0.936255	+	0.351320i	$0.114267\pi$
$397$	14.0000i	0.702640i	−0.936255	+	0.351320i	$0.885733\pi$
$398$	3.00000i	0.150376i
$399$	−15.0000	−0.750939
$400$	0	0
$401$	−12.0000	−0.599251	−0.299626	−	0.954057i	$-0.596862\pi$
$401$	−12.0000	−0.599251	−0.299626	+	0.954057i	$0.596862\pi$
$402$	− 8.00000i	− 0.399004i
$403$	− 6.00000i	− 0.298881i
$404$	6.00000	0.298511
$405$	0	0
$406$	27.0000	1.33999
$407$	0	0
$408$	0	0
$409$	−25.0000	−1.23617	−0.618085	−	0.786111i	$-0.712091\pi$
$409$	−25.0000	−1.23617	−0.618085	+	0.786111i	$0.712091\pi$
$410$	0	0
$411$	2.00000	0.0986527
$412$	11.0000i	0.541931i
$413$	− 36.0000i	− 1.77144i
$414$	−1.00000	−0.0491473
$415$	0	0
$416$	−3.00000	−0.147087
$417$	12.0000i	0.587643i
$418$	− 5.00000i	− 0.244558i
$419$	−9.00000	−0.439679	−0.219839	−	0.975536i	$-0.570553\pi$
$419$	−9.00000	−0.439679	−0.219839	+	0.975536i	$0.570553\pi$
$420$	0	0
$421$	−26.0000	−1.26716	−0.633581	−	0.773676i	$-0.718416\pi$
$421$	−26.0000	−1.26716	−0.633581	+	0.773676i	$0.718416\pi$
$422$	10.0000i	0.486792i
$423$	2.00000i	0.0972433i
$424$	2.00000	0.0971286
$425$	0	0
$426$	−10.0000	−0.484502
$427$	− 12.0000i	− 0.580721i
$428$	4.00000i	0.193347i
$429$	−3.00000	−0.144841
$430$	0	0
$431$	8.00000	0.385346	0.192673	−	0.981263i	$-0.438284\pi$
$431$	8.00000	0.385346	0.192673	+	0.981263i	$0.438284\pi$
$432$	1.00000i	0.0481125i
$433$	28.0000i	1.34559i	0.739827	+	0.672797i	$0.234907\pi$
$433$	28.0000i	1.34559i	−0.739827	+	0.672797i	$0.765093\pi$
$434$	−6.00000	−0.288009
$435$	0	0
$436$	−16.0000	−0.766261
$437$	− 5.00000i	− 0.239182i
$438$	− 5.00000i	− 0.238909i
$439$	36.0000	1.71819	0.859093	−	0.511819i	$-0.171028\pi$
$439$	36.0000	1.71819	0.859093	+	0.511819i	$0.171028\pi$
$440$	0	0
$441$	2.00000	0.0952381
$442$	0	0
$443$	6.00000i	0.285069i	0.989790	+	0.142534i	$0.0455251\pi$
$443$	6.00000i	0.285069i	−0.989790	+	0.142534i	$0.954475\pi$
$444$	0	0
$445$	0	0
$446$	−2.00000	−0.0947027
$447$	22.0000i	1.04056i
$448$	3.00000i	0.141737i
$449$	−22.0000	−1.03824	−0.519122	−	0.854700i	$-0.673741\pi$
$449$	−22.0000	−1.03824	−0.519122	+	0.854700i	$0.673741\pi$
$450$	0	0
$451$	5.00000	0.235441
$452$	− 2.00000i	− 0.0940721i
$453$	2.00000i	0.0939682i
$454$	−20.0000	−0.938647
$455$	0	0
$456$	−5.00000	−0.234146
$457$	− 26.0000i	− 1.21623i	−0.793849	−	0.608114i	$-0.791926\pi$
$457$	− 26.0000i	− 1.21623i	0.793849	−	0.608114i	$-0.208074\pi$
$458$	14.0000i	0.654177i
$459$	0	0
$460$	0	0
$461$	35.0000	1.63011	0.815056	−	0.579382i	$-0.196706\pi$
$461$	35.0000	1.63011	0.815056	+	0.579382i	$0.196706\pi$
$462$	3.00000i	0.139573i
$463$	20.0000i	0.929479i	0.885448	+	0.464739i	$0.153852\pi$
$463$	20.0000i	0.929479i	−0.885448	+	0.464739i	$0.846148\pi$
$464$	9.00000	0.417815
$465$	0	0
$466$	−5.00000	−0.231621
$467$	17.0000i	0.786666i	0.919396	+	0.393333i	$0.128678\pi$
$467$	17.0000i	0.786666i	−0.919396	+	0.393333i	$0.871322\pi$
$468$	3.00000i	0.138675i
$469$	−24.0000	−1.10822
$470$	0	0
$471$	−6.00000	−0.276465
$472$	− 12.0000i	− 0.552345i
$473$	7.00000i	0.321860i
$474$	−9.00000	−0.413384
$475$	0	0
$476$	0	0
$477$	− 2.00000i	− 0.0915737i
$478$	26.0000i	1.18921i
$479$	21.0000	0.959514	0.479757	−	0.877401i	$-0.340725\pi$
$479$	21.0000	0.959514	0.479757	+	0.877401i	$0.340725\pi$
$480$	0	0
$481$	0	0
$482$	14.0000i	0.637683i
$483$	3.00000i	0.136505i
$484$	10.0000	0.454545
$485$	0	0
$486$	1.00000	0.0453609
$487$	2.00000i	0.0906287i	0.998973	+	0.0453143i	$0.0144289\pi$
$487$	2.00000i	0.0906287i	−0.998973	+	0.0453143i	$0.985571\pi$
$488$	− 4.00000i	− 0.181071i
$489$	−22.0000	−0.994874
$490$	0	0
$491$	−12.0000	−0.541552	−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000	−0.541552	−0.270776	+	0.962642i	$0.587280\pi$
$492$	− 5.00000i	− 0.225417i
$493$	0	0
$494$	−15.0000	−0.674882
$495$	0	0
$496$	−2.00000	−0.0898027
$497$	30.0000i	1.34568i
$498$	− 11.0000i	− 0.492922i
$499$	−30.0000	−1.34298	−0.671492	−	0.741012i	$-0.734346\pi$
$499$	−30.0000	−1.34298	−0.671492	+	0.741012i	$0.734346\pi$
$500$	0	0
$501$	2.00000	0.0893534
$502$	24.0000i	1.07117i
$503$	33.0000i	1.47140i	0.677309	+	0.735699i	$0.263146\pi$
$503$	33.0000i	1.47140i	−0.677309	+	0.735699i	$0.736854\pi$
$504$	3.00000	0.133631
$505$	0	0
$506$	−1.00000	−0.0444554
$507$	− 4.00000i	− 0.177646i
$508$	12.0000i	0.532414i
$509$	30.0000	1.32973	0.664863	−	0.746965i	$-0.268490\pi$
$509$	30.0000	1.32973	0.664863	+	0.746965i	$0.268490\pi$
$510$	0	0
$511$	−15.0000	−0.663561
$512$	1.00000i	0.0441942i
$513$	5.00000i	0.220755i
$514$	−6.00000	−0.264649
$515$	0	0
$516$	7.00000	0.308158
$517$	2.00000i	0.0879599i
$518$	0	0
$519$	−3.00000	−0.131685
$520$	0	0
$521$	−36.0000	−1.57719	−0.788594	−	0.614914i	$-0.789191\pi$
$521$	−36.0000	−1.57719	−0.788594	+	0.614914i	$0.789191\pi$
$522$	− 9.00000i	− 0.393919i
$523$	11.0000i	0.480996i	0.970650	+	0.240498i	$0.0773108\pi$
$523$	11.0000i	0.480996i	−0.970650	+	0.240498i	$0.922689\pi$
$524$	−2.00000	−0.0873704
$525$	0	0
$526$	16.0000	0.697633
$527$	0	0
$528$	1.00000i	0.0435194i
$529$	−1.00000	−0.0434783
$530$	0	0
$531$	−12.0000	−0.520756
$532$	15.0000i	0.650332i
$533$	− 15.0000i	− 0.649722i
$534$	−10.0000	−0.432742
$535$	0	0
$536$	−8.00000	−0.345547
$537$	− 6.00000i	− 0.258919i
$538$	17.0000i	0.732922i
$539$	2.00000	0.0861461
$540$	0	0
$541$	11.0000	0.472927	0.236463	−	0.971640i	$-0.424012\pi$
$541$	11.0000	0.472927	0.236463	+	0.971640i	$0.424012\pi$
$542$	− 2.00000i	− 0.0859074i
$543$	12.0000i	0.514969i
$544$	0	0
$545$	0	0
$546$	9.00000	0.385164
$547$	2.00000i	0.0855138i	0.999086	+	0.0427569i	$0.0136141\pi$
$547$	2.00000i	0.0855138i	−0.999086	+	0.0427569i	$0.986386\pi$
$548$	− 2.00000i	− 0.0854358i
$549$	−4.00000	−0.170716
$550$	0	0
$551$	45.0000	1.91706
$552$	1.00000i	0.0425628i
$553$	27.0000i	1.14816i
$554$	21.0000	0.892205
$555$	0	0
$556$	12.0000	0.508913
$557$	− 36.0000i	− 1.52537i	−0.646771	−	0.762684i	$-0.723881\pi$
$557$	− 36.0000i	− 1.52537i	0.646771	−	0.762684i	$-0.276119\pi$
$558$	2.00000i	0.0846668i
$559$	21.0000	0.888205
$560$	0	0
$561$	0	0
$562$	6.00000i	0.253095i
$563$	− 31.0000i	− 1.30649i	−0.757145	−	0.653247i	$-0.773406\pi$
$563$	− 31.0000i	− 1.30649i	0.757145	−	0.653247i	$-0.226594\pi$
$564$	2.00000	0.0842152
$565$	0	0
$566$	16.0000	0.672530
$567$	− 3.00000i	− 0.125988i
$568$	10.0000i	0.419591i
$569$	24.0000	1.00613	0.503066	−	0.864248i	$-0.332205\pi$
$569$	24.0000	1.00613	0.503066	+	0.864248i	$0.332205\pi$
$570$	0	0
$571$	44.0000	1.84134	0.920671	−	0.390339i	$-0.127642\pi$
$571$	44.0000	1.84134	0.920671	+	0.390339i	$0.127642\pi$
$572$	3.00000i	0.125436i
$573$	5.00000i	0.208878i
$574$	−15.0000	−0.626088
$575$	0	0
$576$	1.00000	0.0416667
$577$	− 47.0000i	− 1.95664i	−0.207109	−	0.978318i	$-0.566406\pi$
$577$	− 47.0000i	− 1.95664i	0.207109	−	0.978318i	$-0.433594\pi$
$578$	17.0000i	0.707107i
$579$	14.0000	0.581820
$580$	0	0
$581$	−33.0000	−1.36907
$582$	− 14.0000i	− 0.580319i
$583$	− 2.00000i	− 0.0828315i
$584$	−5.00000	−0.206901
$585$	0	0
$586$	16.0000	0.660954
$587$	44.0000i	1.81607i	0.418890	+	0.908037i	$0.362419\pi$
$587$	44.0000i	1.81607i	−0.418890	+	0.908037i	$0.637581\pi$
$588$	− 2.00000i	− 0.0824786i
$589$	−10.0000	−0.412043
$590$	0	0
$591$	11.0000	0.452480
$592$	0	0
$593$	9.00000i	0.369586i	0.982777	+	0.184793i	$0.0591614\pi$
$593$	9.00000i	0.369586i	−0.982777	+	0.184793i	$0.940839\pi$
$594$	1.00000	0.0410305
$595$	0	0
$596$	22.0000	0.901155
$597$	− 3.00000i	− 0.122782i
$598$	3.00000i	0.122679i
$599$	−38.0000	−1.55264	−0.776319	−	0.630340i	$-0.782915\pi$
$599$	−38.0000	−1.55264	−0.776319	+	0.630340i	$0.782915\pi$
$600$	0	0
$601$	10.0000	0.407909	0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000	0.407909	0.203954	+	0.978980i	$0.434621\pi$
$602$	− 21.0000i	− 0.855896i
$603$	8.00000i	0.325785i
$604$	2.00000	0.0813788
$605$	0	0
$606$	−6.00000	−0.243733
$607$	14.0000i	0.568242i	0.958788	+	0.284121i	$0.0917018\pi$
$607$	14.0000i	0.568242i	−0.958788	+	0.284121i	$0.908298\pi$
$608$	5.00000i	0.202777i
$609$	−27.0000	−1.09410
$610$	0	0
$611$	6.00000	0.242734
$612$	0	0
$613$	16.0000i	0.646234i	0.946359	+	0.323117i	$0.104731\pi$
$613$	16.0000i	0.646234i	−0.946359	+	0.323117i	$0.895269\pi$
$614$	0	0
$615$	0	0
$616$	3.00000	0.120873
$617$	12.0000i	0.483102i	0.970388	+	0.241551i	$0.0776561\pi$
$617$	12.0000i	0.483102i	−0.970388	+	0.241551i	$0.922344\pi$
$618$	− 11.0000i	− 0.442485i
$619$	−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
$621$	1.00000	0.0401286
$622$	16.0000i	0.641542i
$623$	30.0000i	1.20192i
$624$	3.00000	0.120096
$625$	0	0
$626$	16.0000	0.639489
$627$	5.00000i	0.199681i
$628$	6.00000i	0.239426i
$629$	0	0
$630$	0	0
$631$	−37.0000	−1.47295	−0.736473	−	0.676467i	$-0.763510\pi$
$631$	−37.0000	−1.47295	−0.736473	+	0.676467i	$0.763510\pi$
$632$	9.00000i	0.358001i
$633$	− 10.0000i	− 0.397464i
$634$	−3.00000	−0.119145
$635$	0	0
$636$	−2.00000	−0.0793052
$637$	− 6.00000i	− 0.237729i
$638$	− 9.00000i	− 0.356313i
$639$	10.0000	0.395594
$640$	0	0
$641$	16.0000	0.631962	0.315981	−	0.948766i	$-0.397666\pi$
$641$	16.0000	0.631962	0.315981	+	0.948766i	$0.397666\pi$
$642$	− 4.00000i	− 0.157867i
$643$	− 31.0000i	− 1.22252i	−0.791430	−	0.611260i	$-0.790663\pi$
$643$	− 31.0000i	− 1.22252i	0.791430	−	0.611260i	$-0.209337\pi$
$644$	3.00000	0.118217
$645$	0	0
$646$	0	0
$647$	30.0000i	1.17942i	0.807614	+	0.589711i	$0.200758\pi$
$647$	30.0000i	1.17942i	−0.807614	+	0.589711i	$0.799242\pi$
$648$	− 1.00000i	− 0.0392837i
$649$	−12.0000	−0.471041
$650$	0	0
$651$	6.00000	0.235159
$652$	22.0000i	0.861586i
$653$	23.0000i	0.900060i	0.893014	+	0.450030i	$0.148587\pi$
$653$	23.0000i	0.900060i	−0.893014	+	0.450030i	$0.851413\pi$
$654$	16.0000	0.625650
$655$	0	0
$656$	−5.00000	−0.195217
$657$	5.00000i	0.195069i
$658$	− 6.00000i	− 0.233904i
$659$	3.00000	0.116863	0.0584317	−	0.998291i	$-0.481390\pi$
$659$	3.00000	0.116863	0.0584317	+	0.998291i	$0.481390\pi$
$660$	0	0
$661$	−14.0000	−0.544537	−0.272268	−	0.962221i	$-0.587774\pi$
$661$	−14.0000	−0.544537	−0.272268	+	0.962221i	$0.587774\pi$
$662$	− 26.0000i	− 1.01052i
$663$	0	0
$664$	−11.0000	−0.426883
$665$	0	0
$666$	0	0
$667$	− 9.00000i	− 0.348481i
$668$	− 2.00000i	− 0.0773823i
$669$	2.00000	0.0773245
$670$	0	0
$671$	−4.00000	−0.154418
$672$	− 3.00000i	− 0.115728i
$673$	− 11.0000i	− 0.424019i	−0.977268	−	0.212009i	$-0.931999\pi$
$673$	− 11.0000i	− 0.424019i	0.977268	−	0.212009i	$-0.0680008\pi$
$674$	−10.0000	−0.385186
$675$	0	0
$676$	−4.00000	−0.153846
$677$	24.0000i	0.922395i	0.887298	+	0.461197i	$0.152580\pi$
$677$	24.0000i	0.922395i	−0.887298	+	0.461197i	$0.847420\pi$
$678$	2.00000i	0.0768095i
$679$	−42.0000	−1.61181
$680$	0	0
$681$	20.0000	0.766402
$682$	2.00000i	0.0765840i
$683$	46.0000i	1.76014i	0.474843	+	0.880071i	$0.342505\pi$
$683$	46.0000i	1.76014i	−0.474843	+	0.880071i	$0.657495\pi$
$684$	5.00000	0.191180
$685$	0	0
$686$	15.0000	0.572703
$687$	− 14.0000i	− 0.534133i
$688$	− 7.00000i	− 0.266872i
$689$	−6.00000	−0.228582
$690$	0	0
$691$	−4.00000	−0.152167	−0.0760836	−	0.997101i	$-0.524242\pi$
$691$	−4.00000	−0.152167	−0.0760836	+	0.997101i	$0.524242\pi$
$692$	3.00000i	0.114043i
$693$	− 3.00000i	− 0.113961i
$694$	−8.00000	−0.303676
$695$	0	0
$696$	−9.00000	−0.341144
$697$	0	0
$698$	− 11.0000i	− 0.416356i
$699$	5.00000	0.189117
$700$	0	0
$701$	12.0000	0.453234	0.226617	−	0.973984i	$-0.427233\pi$
$701$	12.0000	0.453234	0.226617	+	0.973984i	$0.427233\pi$
$702$	− 3.00000i	− 0.113228i
$703$	0	0
$704$	1.00000	0.0376889
$705$	0	0
$706$	−3.00000	−0.112906
$707$	18.0000i	0.676960i
$708$	12.0000i	0.450988i
$709$	26.0000	0.976450	0.488225	−	0.872718i	$-0.337644\pi$
$709$	26.0000	0.976450	0.488225	+	0.872718i	$0.337644\pi$
$710$	0	0
$711$	9.00000	0.337526
$712$	10.0000i	0.374766i
$713$	2.00000i	0.0749006i
$714$	0	0
$715$	0	0
$716$	−6.00000	−0.224231
$717$	− 26.0000i	− 0.970988i
$718$	11.0000i	0.410516i
$719$	28.0000	1.04422	0.522112	−	0.852877i	$-0.325144\pi$
$719$	28.0000	1.04422	0.522112	+	0.852877i	$0.325144\pi$
$720$	0	0
$721$	−33.0000	−1.22898
$722$	6.00000i	0.223297i
$723$	− 14.0000i	− 0.520666i
$724$	12.0000	0.445976
$725$	0	0
$726$	−10.0000	−0.371135
$727$	8.00000i	0.296704i	0.988935	+	0.148352i	$0.0473968\pi$
$727$	8.00000i	0.296704i	−0.988935	+	0.148352i	$0.952603\pi$
$728$	− 9.00000i	− 0.333562i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	0	0
$732$	4.00000i	0.147844i
$733$	6.00000i	0.221615i	0.993842	+	0.110808i	$0.0353437\pi$
$733$	6.00000i	0.221615i	−0.993842	+	0.110808i	$0.964656\pi$
$734$	11.0000	0.406017
$735$	0	0
$736$	1.00000	0.0368605
$737$	8.00000i	0.294684i
$738$	5.00000i	0.184053i
$739$	30.0000	1.10357	0.551784	−	0.833987i	$-0.313947\pi$
$739$	30.0000	1.10357	0.551784	+	0.833987i	$0.313947\pi$
$740$	0	0
$741$	15.0000	0.551039
$742$	6.00000i	0.220267i
$743$	39.0000i	1.43077i	0.698730	+	0.715386i	$0.253749\pi$
$743$	39.0000i	1.43077i	−0.698730	+	0.715386i	$0.746251\pi$
$744$	2.00000	0.0733236
$745$	0	0
$746$	12.0000	0.439351
$747$	11.0000i	0.402469i
$748$	0	0
$749$	−12.0000	−0.438470
$750$	0	0
$751$	−13.0000	−0.474377	−0.237188	−	0.971464i	$-0.576226\pi$
$751$	−13.0000	−0.474377	−0.237188	+	0.971464i	$0.576226\pi$
$752$	− 2.00000i	− 0.0729325i
$753$	− 24.0000i	− 0.874609i
$754$	−27.0000	−0.983282
$755$	0	0
$756$	−3.00000	−0.109109
$757$	12.0000i	0.436147i	0.975932	+	0.218074i	$0.0699773\pi$
$757$	12.0000i	0.436147i	−0.975932	+	0.218074i	$0.930023\pi$
$758$	20.0000i	0.726433i
$759$	1.00000	0.0362977
$760$	0	0
$761$	25.0000	0.906249	0.453125	−	0.891447i	$-0.350309\pi$
$761$	25.0000	0.906249	0.453125	+	0.891447i	$0.350309\pi$
$762$	− 12.0000i	− 0.434714i
$763$	− 48.0000i	− 1.73772i
$764$	5.00000	0.180894
$765$	0	0
$766$	1.00000	0.0361315
$767$	36.0000i	1.29988i
$768$	− 1.00000i	− 0.0360844i
$769$	−24.0000	−0.865462	−0.432731	−	0.901523i	$-0.642450\pi$
$769$	−24.0000	−0.865462	−0.432731	+	0.901523i	$0.642450\pi$
$770$	0	0
$771$	6.00000	0.216085
$772$	− 14.0000i	− 0.503871i
$773$	40.0000i	1.43870i	0.694648	+	0.719350i	$0.255560\pi$
$773$	40.0000i	1.43870i	−0.694648	+	0.719350i	$0.744440\pi$
$774$	−7.00000	−0.251610
$775$	0	0
$776$	−14.0000	−0.502571
$777$	0	0
$778$	− 6.00000i	− 0.215110i
$779$	−25.0000	−0.895718
$780$	0	0
$781$	10.0000	0.357828
$782$	0	0
$783$	9.00000i	0.321634i
$784$	−2.00000	−0.0714286
$785$	0	0
$786$	2.00000	0.0713376
$787$	− 27.0000i	− 0.962446i	−0.876598	−	0.481223i	$-0.840193\pi$
$787$	− 27.0000i	− 0.962446i	0.876598	−	0.481223i	$-0.159807\pi$
$788$	− 11.0000i	− 0.391859i
$789$	−16.0000	−0.569615
$790$	0	0
$791$	6.00000	0.213335
$792$	− 1.00000i	− 0.0355335i
$793$	12.0000i	0.426132i
$794$	−14.0000	−0.496841
$795$	0	0
$796$	−3.00000	−0.106332
$797$	− 12.0000i	− 0.425062i	−0.977154	−	0.212531i	$-0.931829\pi$
$797$	− 12.0000i	− 0.425062i	0.977154	−	0.212531i	$-0.0681706\pi$
$798$	− 15.0000i	− 0.530994i
$799$	0	0
$800$	0	0
$801$	10.0000	0.353333
$802$	− 12.0000i	− 0.423735i
$803$	5.00000i	0.176446i
$804$	8.00000	0.282138
$805$	0	0
$806$	6.00000	0.211341
$807$	− 17.0000i	− 0.598428i
$808$	6.00000i	0.211079i
$809$	−33.0000	−1.16022	−0.580109	−	0.814539i	$-0.696990\pi$
$809$	−33.0000	−1.16022	−0.580109	+	0.814539i	$0.696990\pi$
$810$	0	0
$811$	−52.0000	−1.82597	−0.912983	−	0.407997i	$-0.866228\pi$
$811$	−52.0000	−1.82597	−0.912983	+	0.407997i	$0.866228\pi$
$812$	27.0000i	0.947514i
$813$	2.00000i	0.0701431i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 35.0000i	− 1.22449i
$818$	− 25.0000i	− 0.874105i
$819$	−9.00000	−0.314485
$820$	0	0
$821$	29.0000	1.01211	0.506053	−	0.862502i	$-0.331104\pi$
$821$	29.0000	1.01211	0.506053	+	0.862502i	$0.331104\pi$
$822$	2.00000i	0.0697580i
$823$	− 44.0000i	− 1.53374i	−0.641800	−	0.766872i	$-0.721812\pi$
$823$	− 44.0000i	− 1.53374i	0.641800	−	0.766872i	$-0.278188\pi$
$824$	−11.0000	−0.383203
$825$	0	0
$826$	36.0000	1.25260
$827$	37.0000i	1.28662i	0.765607	+	0.643308i	$0.222439\pi$
$827$	37.0000i	1.28662i	−0.765607	+	0.643308i	$0.777561\pi$
$828$	− 1.00000i	− 0.0347524i
$829$	−27.0000	−0.937749	−0.468874	−	0.883265i	$-0.655340\pi$
$829$	−27.0000	−0.937749	−0.468874	+	0.883265i	$0.655340\pi$
$830$	0	0
$831$	−21.0000	−0.728482
$832$	− 3.00000i	− 0.104006i
$833$	0	0
$834$	−12.0000	−0.415526
$835$	0	0
$836$	5.00000	0.172929
$837$	− 2.00000i	− 0.0691301i
$838$	− 9.00000i	− 0.310900i
$839$	−3.00000	−0.103572	−0.0517858	−	0.998658i	$-0.516491\pi$
$839$	−3.00000	−0.103572	−0.0517858	+	0.998658i	$0.516491\pi$
$840$	0	0
$841$	52.0000	1.79310
$842$	− 26.0000i	− 0.896019i
$843$	− 6.00000i	− 0.206651i
$844$	−10.0000	−0.344214
$845$	0	0
$846$	−2.00000	−0.0687614
$847$	30.0000i	1.03081i
$848$	2.00000i	0.0686803i
$849$	−16.0000	−0.549119
$850$	0	0
$851$	0	0
$852$	− 10.0000i	− 0.342594i
$853$	31.0000i	1.06142i	0.847554	+	0.530710i	$0.178075\pi$
$853$	31.0000i	1.06142i	−0.847554	+	0.530710i	$0.821925\pi$
$854$	12.0000	0.410632
$855$	0	0
$856$	−4.00000	−0.136717
$857$	− 42.0000i	− 1.43469i	−0.696717	−	0.717346i	$-0.745357\pi$
$857$	− 42.0000i	− 1.43469i	0.696717	−	0.717346i	$-0.254643\pi$
$858$	− 3.00000i	− 0.102418i
$859$	−16.0000	−0.545913	−0.272956	−	0.962026i	$-0.588002\pi$
$859$	−16.0000	−0.545913	−0.272956	+	0.962026i	$0.588002\pi$
$860$	0	0
$861$	15.0000	0.511199
$862$	8.00000i	0.272481i
$863$	− 18.0000i	− 0.612727i	−0.951915	−	0.306364i	$-0.900888\pi$
$863$	− 18.0000i	− 0.612727i	0.951915	−	0.306364i	$-0.0991123\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	−28.0000	−0.951479
$867$	− 17.0000i	− 0.577350i
$868$	− 6.00000i	− 0.203653i
$869$	9.00000	0.305304
$870$	0	0
$871$	24.0000	0.813209
$872$	− 16.0000i	− 0.541828i
$873$	14.0000i	0.473828i
$874$	5.00000	0.169128
$875$	0	0
$876$	5.00000	0.168934
$877$	50.0000i	1.68838i	0.536044	+	0.844190i	$0.319918\pi$
$877$	50.0000i	1.68838i	−0.536044	+	0.844190i	$0.680082\pi$
$878$	36.0000i	1.21494i
$879$	−16.0000	−0.539667
$880$	0	0
$881$	48.0000	1.61716	0.808581	−	0.588386i	$-0.200236\pi$
$881$	48.0000	1.61716	0.808581	+	0.588386i	$0.200236\pi$
$882$	2.00000i	0.0673435i
$883$	16.0000i	0.538443i	0.963078	+	0.269221i	$0.0867663\pi$
$883$	16.0000i	0.538443i	−0.963078	+	0.269221i	$0.913234\pi$
$884$	0	0
$885$	0	0
$886$	−6.00000	−0.201574
$887$	− 4.00000i	− 0.134307i	−0.997743	−	0.0671534i	$-0.978608\pi$
$887$	− 4.00000i	− 0.134307i	0.997743	−	0.0671534i	$-0.0213917\pi$
$888$	0	0
$889$	−36.0000	−1.20740
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	− 2.00000i	− 0.0669650i
$893$	− 10.0000i	− 0.334637i
$894$	−22.0000	−0.735790
$895$	0	0
$896$	−3.00000	−0.100223
$897$	− 3.00000i	− 0.100167i
$898$	− 22.0000i	− 0.734150i
$899$	−18.0000	−0.600334
$900$	0	0
$901$	0	0
$902$	5.00000i	0.166482i
$903$	21.0000i	0.698836i
$904$	2.00000	0.0665190
$905$	0	0
$906$	−2.00000	−0.0664455
$907$	− 31.0000i	− 1.02934i	−0.857389	−	0.514669i	$-0.827915\pi$
$907$	− 31.0000i	− 1.02934i	0.857389	−	0.514669i	$-0.172085\pi$
$908$	− 20.0000i	− 0.663723i
$909$	6.00000	0.199007
$910$	0	0
$911$	5.00000	0.165657	0.0828287	−	0.996564i	$-0.473605\pi$
$911$	5.00000	0.165657	0.0828287	+	0.996564i	$0.473605\pi$
$912$	− 5.00000i	− 0.165567i
$913$	11.0000i	0.364047i
$914$	26.0000	0.860004
$915$	0	0
$916$	−14.0000	−0.462573
$917$	− 6.00000i	− 0.198137i
$918$	0	0
$919$	0	0		−	1.00000i	$-0.5\pi$
$919$	0	0			1.00000i	$0.5\pi$
$920$	0	0
$921$	0	0
$922$	35.0000i	1.15266i
$923$	− 30.0000i	− 0.987462i
$924$	−3.00000	−0.0986928
$925$	0	0
$926$	−20.0000	−0.657241
$927$	11.0000i	0.361287i
$928$	9.00000i	0.295439i
$929$	−51.0000	−1.67326	−0.836628	−	0.547772i	$-0.815476\pi$
$929$	−51.0000	−1.67326	−0.836628	+	0.547772i	$0.815476\pi$
$930$	0	0
$931$	−10.0000	−0.327737
$932$	− 5.00000i	− 0.163780i
$933$	− 16.0000i	− 0.523816i
$934$	−17.0000	−0.556257
$935$	0	0
$936$	−3.00000	−0.0980581
$937$	− 2.00000i	− 0.0653372i	−0.999466	−	0.0326686i	$-0.989599\pi$
$937$	− 2.00000i	− 0.0653372i	0.999466	−	0.0326686i	$-0.0104006\pi$
$938$	− 24.0000i	− 0.783628i
$939$	−16.0000	−0.522140
$940$	0	0
$941$	−26.0000	−0.847576	−0.423788	−	0.905761i	$-0.639300\pi$
$941$	−26.0000	−0.847576	−0.423788	+	0.905761i	$0.639300\pi$
$942$	− 6.00000i	− 0.195491i
$943$	5.00000i	0.162822i
$944$	12.0000	0.390567
$945$	0	0
$946$	−7.00000	−0.227590
$947$	− 38.0000i	− 1.23483i	−0.786636	−	0.617417i	$-0.788179\pi$
$947$	− 38.0000i	− 1.23483i	0.786636	−	0.617417i	$-0.211821\pi$
$948$	− 9.00000i	− 0.292306i
$949$	15.0000	0.486921
$950$	0	0
$951$	3.00000	0.0972817
$952$	0	0
$953$	16.0000i	0.518291i	0.965838	+	0.259145i	$0.0834409\pi$
$953$	16.0000i	0.518291i	−0.965838	+	0.259145i	$0.916559\pi$
$954$	2.00000	0.0647524
$955$	0	0
$956$	−26.0000	−0.840900
$957$	9.00000i	0.290929i
$958$	21.0000i	0.678479i
$959$	6.00000	0.193750
$960$	0	0
$961$	−27.0000	−0.870968
$962$	0	0
$963$	4.00000i	0.128898i
$964$	−14.0000	−0.450910
$965$	0	0
$966$	−3.00000	−0.0965234
$967$	28.0000i	0.900419i	0.892923	+	0.450210i	$0.148651\pi$
$967$	28.0000i	0.900419i	−0.892923	+	0.450210i	$0.851349\pi$
$968$	10.0000i	0.321412i
$969$	0	0
$970$	0	0
$971$	−15.0000	−0.481373	−0.240686	−	0.970603i	$-0.577373\pi$
$971$	−15.0000	−0.481373	−0.240686	+	0.970603i	$0.577373\pi$
$972$	1.00000i	0.0320750i
$973$	36.0000i	1.15411i
$974$	−2.00000	−0.0640841
$975$	0	0
$976$	4.00000	0.128037
$977$	− 22.0000i	− 0.703842i	−0.936030	−	0.351921i	$-0.885529\pi$
$977$	− 22.0000i	− 0.703842i	0.936030	−	0.351921i	$-0.114471\pi$
$978$	− 22.0000i	− 0.703482i
$979$	10.0000	0.319601
$980$	0	0
$981$	−16.0000	−0.510841
$982$	− 12.0000i	− 0.382935i
$983$	− 3.00000i	− 0.0956851i	−0.998855	−	0.0478426i	$-0.984765\pi$
$983$	− 3.00000i	− 0.0956851i	0.998855	−	0.0478426i	$-0.0152346\pi$
$984$	5.00000	0.159394
$985$	0	0
$986$	0	0
$987$	6.00000i	0.190982i
$988$	− 15.0000i	− 0.477214i
$989$	−7.00000	−0.222587
$990$	0	0
$991$	54.0000	1.71537	0.857683	−	0.514178i	$-0.171903\pi$
$991$	54.0000	1.71537	0.857683	+	0.514178i	$0.171903\pi$
$992$	− 2.00000i	− 0.0635001i
$993$	26.0000i	0.825085i
$994$	−30.0000	−0.951542
$995$	0	0
$996$	11.0000	0.348548
$997$	− 25.0000i	− 0.791758i	−0.918303	−	0.395879i	$-0.870440\pi$
$997$	− 25.0000i	− 0.791758i	0.918303	−	0.395879i	$-0.129560\pi$
$998$	− 30.0000i	− 0.949633i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3450.2.d.o.2899.2		2
5.2	odd	4		3450.2.a.g.1.1	✓	1
5.3	odd	4		3450.2.a.v.1.1	yes	1
5.4	even	2	inner	3450.2.d.o.2899.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3450.2.a.g.1.1	✓	1	5.2	odd	4
3450.2.a.v.1.1	yes	1	5.3	odd	4
3450.2.d.o.2899.1		2	5.4	even	2	inner
3450.2.d.o.2899.2		2	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 3450 = 2 \cdot 3 \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3450.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -3.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -3.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} -1.00000 q^{11} +1.00000i q^{12} +3.00000i q^{13} +3.00000 q^{14} +1.00000 q^{16} -1.00000i q^{18} +5.00000 q^{19} -3.00000 q^{21} -1.00000i q^{22} -1.00000i q^{23} -1.00000 q^{24} -3.00000 q^{26} +1.00000i q^{27} +3.00000i q^{28} +9.00000 q^{29} -2.00000 q^{31} +1.00000i q^{32} +1.00000i q^{33} +1.00000 q^{36} +5.00000i q^{38} +3.00000 q^{39} -5.00000 q^{41} -3.00000i q^{42} -7.00000i q^{43} +1.00000 q^{44} +1.00000 q^{46} -2.00000i q^{47} -1.00000i q^{48} -2.00000 q^{49} -3.00000i q^{52} +2.00000i q^{53} -1.00000 q^{54} -3.00000 q^{56} -5.00000i q^{57} +9.00000i q^{58} +12.0000 q^{59} +4.00000 q^{61} -2.00000i q^{62} +3.00000i q^{63} -1.00000 q^{64} -1.00000 q^{66} -8.00000i q^{67} -1.00000 q^{69} -10.0000 q^{71} +1.00000i q^{72} -5.00000i q^{73} -5.00000 q^{76} +3.00000i q^{77} +3.00000i q^{78} -9.00000 q^{79} +1.00000 q^{81} -5.00000i q^{82} -11.0000i q^{83} +3.00000 q^{84} +7.00000 q^{86} -9.00000i q^{87} +1.00000i q^{88} -10.0000 q^{89} +9.00000 q^{91} +1.00000i q^{92} +2.00000i q^{93} +2.00000 q^{94} +1.00000 q^{96} -14.0000i q^{97} -2.00000i q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9	\( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} - 2 q^{11} + 6 q^{14} + 2 q^{16} + 10 q^{19} - 6 q^{21} - 2 q^{24} - 6 q^{26} + 18 q^{29} - 4 q^{31} + 2 q^{36} + 6 q^{39} - 10 q^{41} + 2 q^{44} + 2 q^{46} - 4 q^{49} - 2 q^{54} - 6 q^{56} + 24 q^{59} + 8 q^{61} - 2 q^{64} - 2 q^{66} - 2 q^{69} - 20 q^{71} - 10 q^{76} - 18 q^{79} + 2 q^{81} + 6 q^{84} + 14 q^{86} - 20 q^{89} + 18 q^{91} + 4 q^{94} + 2 q^{96} + 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9 - 2 * q^11 + 6 * q^14 + 2 * q^16 + 10 * q^19 - 6 * q^21 - 2 * q^24 - 6 * q^26 + 18 * q^29 - 4 * q^31 + 2 * q^36 + 6 * q^39 - 10 * q^41 + 2 * q^44 + 2 * q^46 - 4 * q^49 - 2 * q^54 - 6 * q^56 + 24 * q^59 + 8 * q^61 - 2 * q^64 - 2 * q^66 - 2 * q^69 - 20 * q^71 - 10 * q^76 - 18 * q^79 + 2 * q^81 + 6 * q^84 + 14 * q^86 - 20 * q^89 + 18 * q^91 + 4 * q^94 + 2 * q^96 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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