LMFDB - Embedded newform 3450.2.d.p.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.p.2899.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -1.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -1.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +1.00000i q^{12} -2.00000i q^{13} +1.00000 q^{14} +1.00000 q^{16} +3.00000i q^{17} -1.00000i q^{18} -2.00000 q^{19} -1.00000 q^{21} +1.00000i q^{23} -1.00000 q^{24} +2.00000 q^{26} +1.00000i q^{27} +1.00000i q^{28} -9.00000 q^{29} -10.0000 q^{31} +1.00000i q^{32} -3.00000 q^{34} +1.00000 q^{36} +11.0000i q^{37} -2.00000i q^{38} -2.00000 q^{39} +12.0000 q^{41} -1.00000i q^{42} +10.0000i q^{43} -1.00000 q^{46} -9.00000i q^{47} -1.00000i q^{48} +6.00000 q^{49} +3.00000 q^{51} +2.00000i q^{52} -1.00000 q^{54} -1.00000 q^{56} +2.00000i q^{57} -9.00000i q^{58} +6.00000 q^{59} +14.0000 q^{61} -10.0000i q^{62} +1.00000i q^{63} -1.00000 q^{64} +14.0000i q^{67} -3.00000i q^{68} +1.00000 q^{69} +3.00000 q^{71} +1.00000i q^{72} -11.0000i q^{73} -11.0000 q^{74} +2.00000 q^{76} -2.00000i q^{78} -8.00000 q^{79} +1.00000 q^{81} +12.0000i q^{82} +9.00000i q^{83} +1.00000 q^{84} -10.0000 q^{86} +9.00000i q^{87} -3.00000 q^{89} -2.00000 q^{91} -1.00000i q^{92} +10.0000i q^{93} +9.00000 q^{94} +1.00000 q^{96} -10.0000i q^{97} +6.00000i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 2 q^{14} + 2 q^{16} - 4 q^{19} - 2 q^{21} - 2 q^{24} + 4 q^{26} - 18 q^{29} - 20 q^{31} - 6 q^{34} + 2 q^{36} - 4 q^{39} + 24 q^{41} - 2 q^{46} + 12 q^{49} + 6 q^{51} - 2 q^{54} - 2 q^{56} + 12 q^{59} + 28 q^{61} - 2 q^{64} + 2 q^{69} + 6 q^{71} - 22 q^{74} + 4 q^{76} - 16 q^{79} + 2 q^{81} + 2 q^{84} - 20 q^{86} - 6 q^{89} - 4 q^{91} + 18 q^{94} + 2 q^{96}+O(q^{100}) $ 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9 + 2 * q^14 + 2 * q^16 - 4 * q^19 - 2 * q^21 - 2 * q^24 + 4 * q^26 - 18 * q^29 - 20 * q^31 - 6 * q^34 + 2 * q^36 - 4 * q^39 + 24 * q^41 - 2 * q^46 + 12 * q^49 + 6 * q^51 - 2 * q^54 - 2 * q^56 + 12 * q^59 + 28 * q^61 - 2 * q^64 + 2 * q^69 + 6 * q^71 - 22 * q^74 + 4 * q^76 - 16 * q^79 + 2 * q^81 + 2 * q^84 - 20 * q^86 - 6 * q^89 - 4 * q^91 + 18 * q^94 + 2 * q^96

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
$7$	− 1.00000i	− 0.377964i	−0.981981	−	0.188982i	$-0.939481\pi$
$7$	− 1.00000i	− 0.377964i	0.981981	−	0.188982i	$-0.0605189\pi$
$8$	− 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
$13$	− 2.00000i	− 0.554700i	−0.960769	−	0.277350i	$-0.910544\pi$
$13$	− 2.00000i	− 0.554700i	0.960769	−	0.277350i	$-0.0894562\pi$
$14$	1.00000	0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	3.00000i	0.727607i	0.931476	+	0.363803i	$0.118522\pi$
$17$	3.00000i	0.727607i	−0.931476	+	0.363803i	$0.881478\pi$
$18$	− 1.00000i	− 0.235702i
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	1.00000i	0.208514i
$23$	1.00000i	0.208514i
$24$	−1.00000	−0.204124
$25$	0	0
$26$	2.00000	0.392232
$27$	1.00000i	0.192450i
$28$	1.00000i	0.188982i
$29$	−9.00000	−1.67126	−0.835629	−	0.549294i	$-0.814897\pi$
$29$	−9.00000	−1.67126	−0.835629	+	0.549294i	$0.814897\pi$
$30$	0	0
$31$	−10.0000	−1.79605	−0.898027	−	0.439941i	$-0.854999\pi$
$31$	−10.0000	−1.79605	−0.898027	+	0.439941i	$0.854999\pi$
$32$	1.00000i	0.176777i
$33$	0	0
$34$	−3.00000	−0.514496
$35$	0	0
$36$	1.00000	0.166667
$37$	11.0000i	1.80839i	0.427121	+	0.904194i	$0.359528\pi$
$37$	11.0000i	1.80839i	−0.427121	+	0.904194i	$0.640472\pi$
$38$	− 2.00000i	− 0.324443i
$39$	−2.00000	−0.320256
$40$	0	0
$41$	12.0000	1.87409	0.937043	−	0.349215i	$-0.113552\pi$
$41$	12.0000	1.87409	0.937043	+	0.349215i	$0.113552\pi$
$42$	− 1.00000i	− 0.154303i
$43$	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
$46$	−1.00000	−0.147442
$47$	− 9.00000i	− 1.31278i	−0.754420	−	0.656392i	$-0.772082\pi$
$47$	− 9.00000i	− 1.31278i	0.754420	−	0.656392i	$-0.227918\pi$
$48$	− 1.00000i	− 0.144338i
$49$	6.00000	0.857143
$50$	0	0
$51$	3.00000	0.420084
$52$	2.00000i	0.277350i
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	−1.00000	−0.133631
$57$	2.00000i	0.264906i
$58$	− 9.00000i	− 1.18176i
$59$	6.00000	0.781133	0.390567	−	0.920575i	$-0.372279\pi$
$59$	6.00000	0.781133	0.390567	+	0.920575i	$0.372279\pi$
$60$	0	0
$61$	14.0000	1.79252	0.896258	−	0.443533i	$-0.146275\pi$
$61$	14.0000	1.79252	0.896258	+	0.443533i	$0.146275\pi$
$62$	− 10.0000i	− 1.27000i
$63$	1.00000i	0.125988i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	14.0000i	1.71037i	0.518321	+	0.855186i	$0.326557\pi$
$67$	14.0000i	1.71037i	−0.518321	+	0.855186i	$0.673443\pi$
$68$	− 3.00000i	− 0.363803i
$69$	1.00000	0.120386
$70$	0	0
$71$	3.00000	0.356034	0.178017	−	0.984027i	$-0.443032\pi$
$71$	3.00000	0.356034	0.178017	+	0.984027i	$0.443032\pi$
$72$	1.00000i	0.117851i
$73$	− 11.0000i	− 1.28745i	−0.765256	−	0.643726i	$-0.777388\pi$
$73$	− 11.0000i	− 1.28745i	0.765256	−	0.643726i	$-0.222612\pi$
$74$	−11.0000	−1.27872
$75$	0	0
$76$	2.00000	0.229416
$77$	0	0
$78$	− 2.00000i	− 0.226455i
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	12.0000i	1.32518i
$83$	9.00000i	0.987878i	0.869496	+	0.493939i	$0.164443\pi$
$83$	9.00000i	0.987878i	−0.869496	+	0.493939i	$0.835557\pi$
$84$	1.00000	0.109109
$85$	0	0
$86$	−10.0000	−1.07833
$87$	9.00000i	0.964901i
$88$	0	0
$89$	−3.00000	−0.317999	−0.159000	−	0.987279i	$-0.550827\pi$
$89$	−3.00000	−0.317999	−0.159000	+	0.987279i	$0.550827\pi$
$90$	0	0
$91$	−2.00000	−0.209657
$92$	− 1.00000i	− 0.104257i
$93$	10.0000i	1.03695i
$94$	9.00000	0.928279
$95$	0	0
$96$	1.00000	0.102062
$97$	− 10.0000i	− 1.01535i	−0.861550	−	0.507673i	$-0.830506\pi$
$97$	− 10.0000i	− 1.01535i	0.861550	−	0.507673i	$-0.169494\pi$
$98$	6.00000i	0.606092i
$99$	0	0
$100$	0	0
$101$	−9.00000	−0.895533	−0.447767	−	0.894150i	$-0.647781\pi$
$101$	−9.00000	−0.895533	−0.447767	+	0.894150i	$0.647781\pi$
$102$	3.00000i	0.297044i
$103$	7.00000i	0.689730i	0.938652	+	0.344865i	$0.112075\pi$
$103$	7.00000i	0.689730i	−0.938652	+	0.344865i	$0.887925\pi$
$104$	−2.00000	−0.196116
$105$	0	0
$106$	0	0
$107$	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
$109$	−11.0000	−1.05361	−0.526804	−	0.849987i	$-0.676610\pi$
$109$	−11.0000	−1.05361	−0.526804	+	0.849987i	$0.676610\pi$
$110$	0	0
$111$	11.0000	1.04407
$112$	− 1.00000i	− 0.0944911i
$113$	9.00000i	0.846649i	0.905978	+	0.423324i	$0.139137\pi$
$113$	9.00000i	0.846649i	−0.905978	+	0.423324i	$0.860863\pi$
$114$	−2.00000	−0.187317
$115$	0	0
$116$	9.00000	0.835629
$117$	2.00000i	0.184900i
$118$	6.00000i	0.552345i
$119$	3.00000	0.275010
$120$	0	0
$121$	−11.0000	−1.00000
$122$	14.0000i	1.26750i
$123$	− 12.0000i	− 1.08200i
$124$	10.0000	0.898027
$125$	0	0
$126$	−1.00000	−0.0890871
$127$	8.00000i	0.709885i	0.934888	+	0.354943i	$0.115500\pi$
$127$	8.00000i	0.709885i	−0.934888	+	0.354943i	$0.884500\pi$
$128$	− 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
$133$	2.00000i	0.173422i
$134$	−14.0000	−1.20942
$135$	0	0
$136$	3.00000	0.257248
$137$	9.00000i	0.768922i	0.923141	+	0.384461i	$0.125613\pi$
$137$	9.00000i	0.768922i	−0.923141	+	0.384461i	$0.874387\pi$
$138$	1.00000i	0.0851257i
$139$	−5.00000	−0.424094	−0.212047	−	0.977259i	$-0.568013\pi$
$139$	−5.00000	−0.424094	−0.212047	+	0.977259i	$0.568013\pi$
$140$	0	0
$141$	−9.00000	−0.757937
$142$	3.00000i	0.251754i
$143$	0	0
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	11.0000	0.910366
$147$	− 6.00000i	− 0.494872i
$148$	− 11.0000i	− 0.904194i
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	2.00000i	0.162221i
$153$	− 3.00000i	− 0.242536i
$154$	0	0
$155$	0	0
$156$	2.00000	0.160128
$157$	14.0000i	1.11732i	0.829396	+	0.558661i	$0.188685\pi$
$157$	14.0000i	1.11732i	−0.829396	+	0.558661i	$0.811315\pi$
$158$	− 8.00000i	− 0.636446i
$159$	0	0
$160$	0	0
$161$	1.00000	0.0788110
$162$	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	$0.784443\pi$
$164$	−12.0000	−0.937043
$165$	0	0
$166$	−9.00000	−0.698535
$167$	− 3.00000i	− 0.232147i	−0.993241	−	0.116073i	$-0.962969\pi$
$167$	− 3.00000i	− 0.232147i	0.993241	−	0.116073i	$-0.0370308\pi$
$168$	1.00000i	0.0771517i
$169$	9.00000	0.692308
$170$	0	0
$171$	2.00000	0.152944
$172$	− 10.0000i	− 0.762493i
$173$	6.00000i	0.456172i	0.973641	+	0.228086i	$0.0732467\pi$
$173$	6.00000i	0.456172i	−0.973641	+	0.228086i	$0.926753\pi$
$174$	−9.00000	−0.682288
$175$	0	0
$176$	0	0
$177$	− 6.00000i	− 0.450988i
$178$	− 3.00000i	− 0.224860i
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	−1.00000	−0.0743294	−0.0371647	−	0.999309i	$-0.511833\pi$
$181$	−1.00000	−0.0743294	−0.0371647	+	0.999309i	$0.511833\pi$
$182$	− 2.00000i	− 0.148250i
$183$	− 14.0000i	− 1.03491i
$184$	1.00000	0.0737210
$185$	0	0
$186$	−10.0000	−0.733236
$187$	0	0
$188$	9.00000i	0.656392i
$189$	1.00000	0.0727393
$190$	0	0
$191$	−18.0000	−1.30243	−0.651217	−	0.758891i	$-0.725741\pi$
$191$	−18.0000	−1.30243	−0.651217	+	0.758891i	$0.725741\pi$
$192$	1.00000i	0.0721688i
$193$	25.0000i	1.79954i	0.436365	+	0.899770i	$0.356266\pi$
$193$	25.0000i	1.79954i	−0.436365	+	0.899770i	$0.643734\pi$
$194$	10.0000	0.717958
$195$	0	0
$196$	−6.00000	−0.428571
$197$	− 15.0000i	− 1.06871i	−0.845262	−	0.534353i	$-0.820555\pi$
$197$	− 15.0000i	− 1.06871i	0.845262	−	0.534353i	$-0.179445\pi$
$198$	0	0
$199$	1.00000	0.0708881	0.0354441	−	0.999372i	$-0.488715\pi$
$199$	1.00000	0.0708881	0.0354441	+	0.999372i	$0.488715\pi$
$200$	0	0
$201$	14.0000	0.987484
$202$	− 9.00000i	− 0.633238i
$203$	9.00000i	0.631676i
$204$	−3.00000	−0.210042
$205$	0	0
$206$	−7.00000	−0.487713
$207$	− 1.00000i	− 0.0695048i
$208$	− 2.00000i	− 0.138675i
$209$	0	0
$210$	0	0
$211$	23.0000	1.58339	0.791693	−	0.610920i	$-0.209200\pi$
$211$	23.0000	1.58339	0.791693	+	0.610920i	$0.209200\pi$
$212$	0	0
$213$	− 3.00000i	− 0.205557i
$214$	−12.0000	−0.820303
$215$	0	0
$216$	1.00000	0.0680414
$217$	10.0000i	0.678844i
$218$	− 11.0000i	− 0.745014i
$219$	−11.0000	−0.743311
$220$	0	0
$221$	6.00000	0.403604
$222$	11.0000i	0.738272i
$223$	22.0000i	1.47323i	0.676313	+	0.736614i	$0.263577\pi$
$223$	22.0000i	1.47323i	−0.676313	+	0.736614i	$0.736423\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	−9.00000	−0.598671
$227$	− 21.0000i	− 1.39382i	−0.717159	−	0.696909i	$-0.754558\pi$
$227$	− 21.0000i	− 1.39382i	0.717159	−	0.696909i	$-0.245442\pi$
$228$	− 2.00000i	− 0.132453i
$229$	−2.00000	−0.132164	−0.0660819	−	0.997814i	$-0.521050\pi$
$229$	−2.00000	−0.132164	−0.0660819	+	0.997814i	$0.521050\pi$
$230$	0	0
$231$	0	0
$232$	9.00000i	0.590879i
$233$	24.0000i	1.57229i	0.618041	+	0.786146i	$0.287927\pi$
$233$	24.0000i	1.57229i	−0.618041	+	0.786146i	$0.712073\pi$
$234$	−2.00000	−0.130744
$235$	0	0
$236$	−6.00000	−0.390567
$237$	8.00000i	0.519656i
$238$	3.00000i	0.194461i
$239$	−27.0000	−1.74648	−0.873242	−	0.487286i	$-0.837987\pi$
$239$	−27.0000	−1.74648	−0.873242	+	0.487286i	$0.837987\pi$
$240$	0	0
$241$	8.00000	0.515325	0.257663	−	0.966235i	$-0.417048\pi$
$241$	8.00000	0.515325	0.257663	+	0.966235i	$0.417048\pi$
$242$	− 11.0000i	− 0.707107i
$243$	− 1.00000i	− 0.0641500i
$244$	−14.0000	−0.896258
$245$	0	0
$246$	12.0000	0.765092
$247$	4.00000i	0.254514i
$248$	10.0000i	0.635001i
$249$	9.00000	0.570352
$250$	0	0
$251$	3.00000	0.189358	0.0946792	−	0.995508i	$-0.469817\pi$
$251$	3.00000	0.189358	0.0946792	+	0.995508i	$0.469817\pi$
$252$	− 1.00000i	− 0.0629941i
$253$	0	0
$254$	−8.00000	−0.501965
$255$	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	$-0.189739\pi$
$258$	10.0000i	0.622573i
$259$	11.0000	0.683507
$260$	0	0
$261$	9.00000	0.557086
$262$	12.0000i	0.741362i
$263$	18.0000i	1.10993i	0.831875	+	0.554964i	$0.187268\pi$
$263$	18.0000i	1.10993i	−0.831875	+	0.554964i	$0.812732\pi$
$264$	0	0
$265$	0	0
$266$	−2.00000	−0.122628
$267$	3.00000i	0.183597i
$268$	− 14.0000i	− 0.855186i
$269$	−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
$271$	−28.0000	−1.70088	−0.850439	−	0.526073i	$-0.823664\pi$
$271$	−28.0000	−1.70088	−0.850439	+	0.526073i	$0.823664\pi$
$272$	3.00000i	0.181902i
$273$	2.00000i	0.121046i
$274$	−9.00000	−0.543710
$275$	0	0
$276$	−1.00000	−0.0601929
$277$	26.0000i	1.56219i	0.624413	+	0.781094i	$0.285338\pi$
$277$	26.0000i	1.56219i	−0.624413	+	0.781094i	$0.714662\pi$
$278$	− 5.00000i	− 0.299880i
$279$	10.0000	0.598684
$280$	0	0
$281$	3.00000	0.178965	0.0894825	−	0.995988i	$-0.471479\pi$
$281$	3.00000	0.178965	0.0894825	+	0.995988i	$0.471479\pi$
$282$	− 9.00000i	− 0.535942i
$283$	16.0000i	0.951101i	0.879688	+	0.475551i	$0.157751\pi$
$283$	16.0000i	0.951101i	−0.879688	+	0.475551i	$0.842249\pi$
$284$	−3.00000	−0.178017
$285$	0	0
$286$	0	0
$287$	− 12.0000i	− 0.708338i
$288$	− 1.00000i	− 0.0589256i
$289$	8.00000	0.470588
$290$	0	0
$291$	−10.0000	−0.586210
$292$	11.0000i	0.643726i
$293$	− 18.0000i	− 1.05157i	−0.850617	−	0.525786i	$-0.823771\pi$
$293$	− 18.0000i	− 1.05157i	0.850617	−	0.525786i	$-0.176229\pi$
$294$	6.00000	0.349927
$295$	0	0
$296$	11.0000	0.639362
$297$	0	0
$298$	0	0
$299$	2.00000	0.115663
$300$	0	0
$301$	10.0000	0.576390
$302$	8.00000i	0.460348i
$303$	9.00000i	0.517036i
$304$	−2.00000	−0.114708
$305$	0	0
$306$	3.00000	0.171499
$307$	5.00000i	0.285365i	0.989769	+	0.142683i	$0.0455728\pi$
$307$	5.00000i	0.285365i	−0.989769	+	0.142683i	$0.954427\pi$
$308$	0	0
$309$	7.00000	0.398216
$310$	0	0
$311$	−15.0000	−0.850572	−0.425286	−	0.905059i	$-0.639826\pi$
$311$	−15.0000	−0.850572	−0.425286	+	0.905059i	$0.639826\pi$
$312$	2.00000i	0.113228i
$313$	− 8.00000i	− 0.452187i	−0.974106	−	0.226093i	$-0.927405\pi$
$313$	− 8.00000i	− 0.452187i	0.974106	−	0.226093i	$-0.0725954\pi$
$314$	−14.0000	−0.790066
$315$	0	0
$316$	8.00000	0.450035
$317$	− 27.0000i	− 1.51647i	−0.651981	−	0.758236i	$-0.726062\pi$
$317$	− 27.0000i	− 1.51647i	0.651981	−	0.758236i	$-0.273938\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	12.0000	0.669775
$322$	1.00000i	0.0557278i
$323$	− 6.00000i	− 0.333849i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	−16.0000	−0.886158
$327$	11.0000i	0.608301i
$328$	− 12.0000i	− 0.662589i
$329$	−9.00000	−0.496186
$330$	0	0
$331$	5.00000	0.274825	0.137412	−	0.990514i	$-0.456121\pi$
$331$	5.00000	0.274825	0.137412	+	0.990514i	$0.456121\pi$
$332$	− 9.00000i	− 0.493939i
$333$	− 11.0000i	− 0.602796i
$334$	3.00000	0.164153
$335$	0	0
$336$	−1.00000	−0.0545545
$337$	− 28.0000i	− 1.52526i	−0.646837	−	0.762629i	$-0.723908\pi$
$337$	− 28.0000i	− 1.52526i	0.646837	−	0.762629i	$-0.276092\pi$
$338$	9.00000i	0.489535i
$339$	9.00000	0.488813
$340$	0	0
$341$	0	0
$342$	2.00000i	0.108148i
$343$	− 13.0000i	− 0.701934i
$344$	10.0000	0.539164
$345$	0	0
$346$	−6.00000	−0.322562
$347$	− 30.0000i	− 1.61048i	−0.592946	−	0.805242i	$-0.702035\pi$
$347$	− 30.0000i	− 1.61048i	0.592946	−	0.805242i	$-0.297965\pi$
$348$	− 9.00000i	− 0.482451i
$349$	10.0000	0.535288	0.267644	−	0.963518i	$-0.413755\pi$
$349$	10.0000	0.535288	0.267644	+	0.963518i	$0.413755\pi$
$350$	0	0
$351$	2.00000	0.106752
$352$	0	0
$353$	6.00000i	0.319348i	0.987170	+	0.159674i	$0.0510443\pi$
$353$	6.00000i	0.319348i	−0.987170	+	0.159674i	$0.948956\pi$
$354$	6.00000	0.318896
$355$	0	0
$356$	3.00000	0.159000
$357$	− 3.00000i	− 0.158777i
$358$	0	0
$359$	−36.0000	−1.90001	−0.950004	−	0.312239i	$-0.898921\pi$
$359$	−36.0000	−1.90001	−0.950004	+	0.312239i	$0.898921\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	− 1.00000i	− 0.0525588i
$363$	11.0000i	0.577350i
$364$	2.00000	0.104828
$365$	0	0
$366$	14.0000	0.731792
$367$	8.00000i	0.417597i	0.977959	+	0.208798i	$0.0669552\pi$
$367$	8.00000i	0.417597i	−0.977959	+	0.208798i	$0.933045\pi$
$368$	1.00000i	0.0521286i
$369$	−12.0000	−0.624695
$370$	0	0
$371$	0	0
$372$	− 10.0000i	− 0.518476i
$373$	13.0000i	0.673114i	0.941663	+	0.336557i	$0.109263\pi$
$373$	13.0000i	0.673114i	−0.941663	+	0.336557i	$0.890737\pi$
$374$	0	0
$375$	0	0
$376$	−9.00000	−0.464140
$377$	18.0000i	0.927047i
$378$	1.00000i	0.0514344i
$379$	−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
$381$	8.00000	0.409852
$382$	− 18.0000i	− 0.920960i
$383$	36.0000i	1.83951i	0.392488	+	0.919757i	$0.371614\pi$
$383$	36.0000i	1.83951i	−0.392488	+	0.919757i	$0.628386\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	−25.0000	−1.27247
$387$	− 10.0000i	− 0.508329i
$388$	10.0000i	0.507673i
$389$	30.0000	1.52106	0.760530	−	0.649303i	$-0.224939\pi$
$389$	30.0000	1.52106	0.760530	+	0.649303i	$0.224939\pi$
$390$	0	0
$391$	−3.00000	−0.151717
$392$	− 6.00000i	− 0.303046i
$393$	− 12.0000i	− 0.605320i
$394$	15.0000	0.755689
$395$	0	0
$396$	0	0
$397$	8.00000i	0.401508i	0.979642	+	0.200754i	$0.0643393\pi$
$397$	8.00000i	0.401508i	−0.979642	+	0.200754i	$0.935661\pi$
$398$	1.00000i	0.0501255i
$399$	2.00000	0.100125
$400$	0	0
$401$	−18.0000	−0.898877	−0.449439	−	0.893311i	$-0.648376\pi$
$401$	−18.0000	−0.898877	−0.449439	+	0.893311i	$0.648376\pi$
$402$	14.0000i	0.698257i
$403$	20.0000i	0.996271i
$404$	9.00000	0.447767
$405$	0	0
$406$	−9.00000	−0.446663
$407$	0	0
$408$	− 3.00000i	− 0.148522i
$409$	−17.0000	−0.840596	−0.420298	−	0.907386i	$-0.638074\pi$
$409$	−17.0000	−0.840596	−0.420298	+	0.907386i	$0.638074\pi$
$410$	0	0
$411$	9.00000	0.443937
$412$	− 7.00000i	− 0.344865i
$413$	− 6.00000i	− 0.295241i
$414$	1.00000	0.0491473
$415$	0	0
$416$	2.00000	0.0980581
$417$	5.00000i	0.244851i
$418$	0	0
$419$	−15.0000	−0.732798	−0.366399	−	0.930458i	$-0.619409\pi$
$419$	−15.0000	−0.732798	−0.366399	+	0.930458i	$0.619409\pi$
$420$	0	0
$421$	26.0000	1.26716	0.633581	−	0.773676i	$-0.281584\pi$
$421$	26.0000	1.26716	0.633581	+	0.773676i	$0.281584\pi$
$422$	23.0000i	1.11962i
$423$	9.00000i	0.437595i
$424$	0	0
$425$	0	0
$426$	3.00000	0.145350
$427$	− 14.0000i	− 0.677507i
$428$	− 12.0000i	− 0.580042i
$429$	0	0
$430$	0	0
$431$	30.0000	1.44505	0.722525	−	0.691345i	$-0.242982\pi$
$431$	30.0000	1.44505	0.722525	+	0.691345i	$0.242982\pi$
$432$	1.00000i	0.0481125i
$433$	− 2.00000i	− 0.0961139i	−0.998845	−	0.0480569i	$-0.984697\pi$
$433$	− 2.00000i	− 0.0961139i	0.998845	−	0.0480569i	$-0.0153029\pi$
$434$	−10.0000	−0.480015
$435$	0	0
$436$	11.0000	0.526804
$437$	− 2.00000i	− 0.0956730i
$438$	− 11.0000i	− 0.525600i
$439$	−20.0000	−0.954548	−0.477274	−	0.878755i	$-0.658375\pi$
$439$	−20.0000	−0.954548	−0.477274	+	0.878755i	$0.658375\pi$
$440$	0	0
$441$	−6.00000	−0.285714
$442$	6.00000i	0.285391i
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	−11.0000	−0.522037
$445$	0	0
$446$	−22.0000	−1.04173
$447$	0	0
$448$	1.00000i	0.0472456i
$449$	6.00000	0.283158	0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000	0.283158	0.141579	+	0.989927i	$0.454782\pi$
$450$	0	0
$451$	0	0
$452$	− 9.00000i	− 0.423324i
$453$	− 8.00000i	− 0.375873i
$454$	21.0000	0.985579
$455$	0	0
$456$	2.00000	0.0936586
$457$	32.0000i	1.49690i	0.663193	+	0.748448i	$0.269201\pi$
$457$	32.0000i	1.49690i	−0.663193	+	0.748448i	$0.730799\pi$
$458$	− 2.00000i	− 0.0934539i
$459$	−3.00000	−0.140028
$460$	0	0
$461$	3.00000	0.139724	0.0698620	−	0.997557i	$-0.477744\pi$
$461$	3.00000	0.139724	0.0698620	+	0.997557i	$0.477744\pi$
$462$	0	0
$463$	− 8.00000i	− 0.371792i	−0.982569	−	0.185896i	$-0.940481\pi$
$463$	− 8.00000i	− 0.371792i	0.982569	−	0.185896i	$-0.0595187\pi$
$464$	−9.00000	−0.417815
$465$	0	0
$466$	−24.0000	−1.11178
$467$	− 33.0000i	− 1.52706i	−0.645774	−	0.763529i	$-0.723465\pi$
$467$	− 33.0000i	− 1.52706i	0.645774	−	0.763529i	$-0.276535\pi$
$468$	− 2.00000i	− 0.0924500i
$469$	14.0000	0.646460
$470$	0	0
$471$	14.0000	0.645086
$472$	− 6.00000i	− 0.276172i
$473$	0	0
$474$	−8.00000	−0.367452
$475$	0	0
$476$	−3.00000	−0.137505
$477$	0	0
$478$	− 27.0000i	− 1.23495i
$479$	18.0000	0.822441	0.411220	−	0.911536i	$-0.365103\pi$
$479$	18.0000	0.822441	0.411220	+	0.911536i	$0.365103\pi$
$480$	0	0
$481$	22.0000	1.00311
$482$	8.00000i	0.364390i
$483$	− 1.00000i	− 0.0455016i
$484$	11.0000	0.500000
$485$	0	0
$486$	1.00000	0.0453609
$487$	− 28.0000i	− 1.26880i	−0.773004	−	0.634401i	$-0.781247\pi$
$487$	− 28.0000i	− 1.26880i	0.773004	−	0.634401i	$-0.218753\pi$
$488$	− 14.0000i	− 0.633750i
$489$	16.0000	0.723545
$490$	0	0
$491$	−30.0000	−1.35388	−0.676941	−	0.736038i	$-0.736695\pi$
$491$	−30.0000	−1.35388	−0.676941	+	0.736038i	$0.736695\pi$
$492$	12.0000i	0.541002i
$493$	− 27.0000i	− 1.21602i
$494$	−4.00000	−0.179969
$495$	0	0
$496$	−10.0000	−0.449013
$497$	− 3.00000i	− 0.134568i
$498$	9.00000i	0.403300i
$499$	19.0000	0.850557	0.425278	−	0.905063i	$-0.360176\pi$
$499$	19.0000	0.850557	0.425278	+	0.905063i	$0.360176\pi$
$500$	0	0
$501$	−3.00000	−0.134030
$502$	3.00000i	0.133897i
$503$	− 36.0000i	− 1.60516i	−0.596544	−	0.802580i	$-0.703460\pi$
$503$	− 36.0000i	− 1.60516i	0.596544	−	0.802580i	$-0.296540\pi$
$504$	1.00000	0.0445435
$505$	0	0
$506$	0	0
$507$	− 9.00000i	− 0.399704i
$508$	− 8.00000i	− 0.354943i
$509$	27.0000	1.19675	0.598377	−	0.801215i	$-0.295813\pi$
$509$	27.0000	1.19675	0.598377	+	0.801215i	$0.295813\pi$
$510$	0	0
$511$	−11.0000	−0.486611
$512$	1.00000i	0.0441942i
$513$	− 2.00000i	− 0.0883022i
$514$	18.0000	0.793946
$515$	0	0
$516$	−10.0000	−0.440225
$517$	0	0
$518$	11.0000i	0.483312i
$519$	6.00000	0.263371
$520$	0	0
$521$	−39.0000	−1.70862	−0.854311	−	0.519763i	$-0.826020\pi$
$521$	−39.0000	−1.70862	−0.854311	+	0.519763i	$0.826020\pi$
$522$	9.00000i	0.393919i
$523$	4.00000i	0.174908i	0.996169	+	0.0874539i	$0.0278730\pi$
$523$	4.00000i	0.174908i	−0.996169	+	0.0874539i	$0.972127\pi$
$524$	−12.0000	−0.524222
$525$	0	0
$526$	−18.0000	−0.784837
$527$	− 30.0000i	− 1.30682i
$528$	0	0
$529$	−1.00000	−0.0434783
$530$	0	0
$531$	−6.00000	−0.260378
$532$	− 2.00000i	− 0.0867110i
$533$	− 24.0000i	− 1.03956i
$534$	−3.00000	−0.129823
$535$	0	0
$536$	14.0000	0.604708
$537$	0	0
$538$	− 6.00000i	− 0.258678i
$539$	0	0
$540$	0	0
$541$	44.0000	1.89171	0.945854	−	0.324593i	$-0.105227\pi$
$541$	44.0000	1.89171	0.945854	+	0.324593i	$0.105227\pi$
$542$	− 28.0000i	− 1.20270i
$543$	1.00000i	0.0429141i
$544$	−3.00000	−0.128624
$545$	0	0
$546$	−2.00000	−0.0855921
$547$	− 1.00000i	− 0.0427569i	−0.999771	−	0.0213785i	$-0.993195\pi$
$547$	− 1.00000i	− 0.0427569i	0.999771	−	0.0213785i	$-0.00680549\pi$
$548$	− 9.00000i	− 0.384461i
$549$	−14.0000	−0.597505
$550$	0	0
$551$	18.0000	0.766826
$552$	− 1.00000i	− 0.0425628i
$553$	8.00000i	0.340195i
$554$	−26.0000	−1.10463
$555$	0	0
$556$	5.00000	0.212047
$557$	− 42.0000i	− 1.77960i	−0.456354	−	0.889799i	$-0.650845\pi$
$557$	− 42.0000i	− 1.77960i	0.456354	−	0.889799i	$-0.349155\pi$
$558$	10.0000i	0.423334i
$559$	20.0000	0.845910
$560$	0	0
$561$	0	0
$562$	3.00000i	0.126547i
$563$	− 24.0000i	− 1.01148i	−0.862686	−	0.505740i	$-0.831220\pi$
$563$	− 24.0000i	− 1.01148i	0.862686	−	0.505740i	$-0.168780\pi$
$564$	9.00000	0.378968
$565$	0	0
$566$	−16.0000	−0.672530
$567$	− 1.00000i	− 0.0419961i
$568$	− 3.00000i	− 0.125877i
$569$	42.0000	1.76073	0.880366	−	0.474295i	$-0.157297\pi$
$569$	42.0000	1.76073	0.880366	+	0.474295i	$0.157297\pi$
$570$	0	0
$571$	26.0000	1.08807	0.544033	−	0.839064i	$-0.316897\pi$
$571$	26.0000	1.08807	0.544033	+	0.839064i	$0.316897\pi$
$572$	0	0
$573$	18.0000i	0.751961i
$574$	12.0000	0.500870
$575$	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	$-0.151411\pi$
$578$	8.00000i	0.332756i
$579$	25.0000	1.03896
$580$	0	0
$581$	9.00000	0.373383
$582$	− 10.0000i	− 0.414513i
$583$	0	0
$584$	−11.0000	−0.455183
$585$	0	0
$586$	18.0000	0.743573
$587$	− 12.0000i	− 0.495293i	−0.968850	−	0.247647i	$-0.920343\pi$
$587$	− 12.0000i	− 0.495293i	0.968850	−	0.247647i	$-0.0796572\pi$
$588$	6.00000i	0.247436i
$589$	20.0000	0.824086
$590$	0	0
$591$	−15.0000	−0.617018
$592$	11.0000i	0.452097i
$593$	− 36.0000i	− 1.47834i	−0.673517	−	0.739171i	$-0.735217\pi$
$593$	− 36.0000i	− 1.47834i	0.673517	−	0.739171i	$-0.264783\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 1.00000i	− 0.0409273i
$598$	2.00000i	0.0817861i
$599$	−24.0000	−0.980613	−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000	−0.980613	−0.490307	+	0.871550i	$0.663115\pi$
$600$	0	0
$601$	−19.0000	−0.775026	−0.387513	−	0.921864i	$-0.626666\pi$
$601$	−19.0000	−0.775026	−0.387513	+	0.921864i	$0.626666\pi$
$602$	10.0000i	0.407570i
$603$	− 14.0000i	− 0.570124i
$604$	−8.00000	−0.325515
$605$	0	0
$606$	−9.00000	−0.365600
$607$	14.0000i	0.568242i	0.958788	+	0.284121i	$0.0917018\pi$
$607$	14.0000i	0.568242i	−0.958788	+	0.284121i	$0.908298\pi$
$608$	− 2.00000i	− 0.0811107i
$609$	9.00000	0.364698
$610$	0	0
$611$	−18.0000	−0.728202
$612$	3.00000i	0.121268i
$613$	19.0000i	0.767403i	0.923457	+	0.383701i	$0.125351\pi$
$613$	19.0000i	0.767403i	−0.923457	+	0.383701i	$0.874649\pi$
$614$	−5.00000	−0.201784
$615$	0	0
$616$	0	0
$617$	30.0000i	1.20775i	0.797077	+	0.603877i	$0.206378\pi$
$617$	30.0000i	1.20775i	−0.797077	+	0.603877i	$0.793622\pi$
$618$	7.00000i	0.281581i
$619$	−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
$621$	−1.00000	−0.0401286
$622$	− 15.0000i	− 0.601445i
$623$	3.00000i	0.120192i
$624$	−2.00000	−0.0800641
$625$	0	0
$626$	8.00000	0.319744
$627$	0	0
$628$	− 14.0000i	− 0.558661i
$629$	−33.0000	−1.31580
$630$	0	0
$631$	5.00000	0.199047	0.0995234	−	0.995035i	$-0.468268\pi$
$631$	5.00000	0.199047	0.0995234	+	0.995035i	$0.468268\pi$
$632$	8.00000i	0.318223i
$633$	− 23.0000i	− 0.914168i
$634$	27.0000	1.07231
$635$	0	0
$636$	0	0
$637$	− 12.0000i	− 0.475457i
$638$	0	0
$639$	−3.00000	−0.118678
$640$	0	0
$641$	−21.0000	−0.829450	−0.414725	−	0.909947i	$-0.636122\pi$
$641$	−21.0000	−0.829450	−0.414725	+	0.909947i	$0.636122\pi$
$642$	12.0000i	0.473602i
$643$	− 14.0000i	− 0.552106i	−0.961142	−	0.276053i	$-0.910973\pi$
$643$	− 14.0000i	− 0.552106i	0.961142	−	0.276053i	$-0.0890266\pi$
$644$	−1.00000	−0.0394055
$645$	0	0
$646$	6.00000	0.236067
$647$	21.0000i	0.825595i	0.910823	+	0.412798i	$0.135448\pi$
$647$	21.0000i	0.825595i	−0.910823	+	0.412798i	$0.864552\pi$
$648$	− 1.00000i	− 0.0392837i
$649$	0	0
$650$	0	0
$651$	10.0000	0.391931
$652$	− 16.0000i	− 0.626608i
$653$	− 9.00000i	− 0.352197i	−0.984373	−	0.176099i	$-0.943652\pi$
$653$	− 9.00000i	− 0.352197i	0.984373	−	0.176099i	$-0.0563478\pi$
$654$	−11.0000	−0.430134
$655$	0	0
$656$	12.0000	0.468521
$657$	11.0000i	0.429151i
$658$	− 9.00000i	− 0.350857i
$659$	−15.0000	−0.584317	−0.292159	−	0.956370i	$-0.594373\pi$
$659$	−15.0000	−0.584317	−0.292159	+	0.956370i	$0.594373\pi$
$660$	0	0
$661$	−31.0000	−1.20576	−0.602880	−	0.797832i	$-0.705980\pi$
$661$	−31.0000	−1.20576	−0.602880	+	0.797832i	$0.705980\pi$
$662$	5.00000i	0.194331i
$663$	− 6.00000i	− 0.233021i
$664$	9.00000	0.349268
$665$	0	0
$666$	11.0000	0.426241
$667$	− 9.00000i	− 0.348481i
$668$	3.00000i	0.116073i
$669$	22.0000	0.850569
$670$	0	0
$671$	0	0
$672$	− 1.00000i	− 0.0385758i
$673$	7.00000i	0.269830i	0.990857	+	0.134915i	$0.0430762\pi$
$673$	7.00000i	0.269830i	−0.990857	+	0.134915i	$0.956924\pi$
$674$	28.0000	1.07852
$675$	0	0
$676$	−9.00000	−0.346154
$677$	12.0000i	0.461197i	0.973049	+	0.230599i	$0.0740685\pi$
$677$	12.0000i	0.461197i	−0.973049	+	0.230599i	$0.925932\pi$
$678$	9.00000i	0.345643i
$679$	−10.0000	−0.383765
$680$	0	0
$681$	−21.0000	−0.804722
$682$	0	0
$683$	12.0000i	0.459167i	0.973289	+	0.229584i	$0.0737364\pi$
$683$	12.0000i	0.459167i	−0.973289	+	0.229584i	$0.926264\pi$
$684$	−2.00000	−0.0764719
$685$	0	0
$686$	13.0000	0.496342
$687$	2.00000i	0.0763048i
$688$	10.0000i	0.381246i
$689$	0	0
$690$	0	0
$691$	11.0000	0.418460	0.209230	−	0.977866i	$-0.432904\pi$
$691$	11.0000	0.418460	0.209230	+	0.977866i	$0.432904\pi$
$692$	− 6.00000i	− 0.228086i
$693$	0	0
$694$	30.0000	1.13878
$695$	0	0
$696$	9.00000	0.341144
$697$	36.0000i	1.36360i
$698$	10.0000i	0.378506i
$699$	24.0000	0.907763
$700$	0	0
$701$	−6.00000	−0.226617	−0.113308	−	0.993560i	$-0.536145\pi$
$701$	−6.00000	−0.226617	−0.113308	+	0.993560i	$0.536145\pi$
$702$	2.00000i	0.0754851i
$703$	− 22.0000i	− 0.829746i
$704$	0	0
$705$	0	0
$706$	−6.00000	−0.225813
$707$	9.00000i	0.338480i
$708$	6.00000i	0.225494i
$709$	19.0000	0.713560	0.356780	−	0.934188i	$-0.383875\pi$
$709$	19.0000	0.713560	0.356780	+	0.934188i	$0.383875\pi$
$710$	0	0
$711$	8.00000	0.300023
$712$	3.00000i	0.112430i
$713$	− 10.0000i	− 0.374503i
$714$	3.00000	0.112272
$715$	0	0
$716$	0	0
$717$	27.0000i	1.00833i
$718$	− 36.0000i	− 1.34351i
$719$	15.0000	0.559406	0.279703	−	0.960087i	$-0.409764\pi$
$719$	15.0000	0.559406	0.279703	+	0.960087i	$0.409764\pi$
$720$	0	0
$721$	7.00000	0.260694
$722$	− 15.0000i	− 0.558242i
$723$	− 8.00000i	− 0.297523i
$724$	1.00000	0.0371647
$725$	0	0
$726$	−11.0000	−0.408248
$727$	32.0000i	1.18681i	0.804902	+	0.593407i	$0.202218\pi$
$727$	32.0000i	1.18681i	−0.804902	+	0.593407i	$0.797782\pi$
$728$	2.00000i	0.0741249i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−30.0000	−1.10959
$732$	14.0000i	0.517455i
$733$	− 29.0000i	− 1.07114i	−0.844491	−	0.535570i	$-0.820097\pi$
$733$	− 29.0000i	− 1.07114i	0.844491	−	0.535570i	$-0.179903\pi$
$734$	−8.00000	−0.295285
$735$	0	0
$736$	−1.00000	−0.0368605
$737$	0	0
$738$	− 12.0000i	− 0.441726i
$739$	1.00000	0.0367856	0.0183928	−	0.999831i	$-0.494145\pi$
$739$	1.00000	0.0367856	0.0183928	+	0.999831i	$0.494145\pi$
$740$	0	0
$741$	4.00000	0.146944
$742$	0	0
$743$	30.0000i	1.10059i	0.834969	+	0.550297i	$0.185485\pi$
$743$	30.0000i	1.10059i	−0.834969	+	0.550297i	$0.814515\pi$
$744$	10.0000	0.366618
$745$	0	0
$746$	−13.0000	−0.475964
$747$	− 9.00000i	− 0.329293i
$748$	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	$0.576226\pi$
$752$	− 9.00000i	− 0.328196i
$753$	− 3.00000i	− 0.109326i
$754$	−18.0000	−0.655521
$755$	0	0
$756$	−1.00000	−0.0363696
$757$	− 13.0000i	− 0.472493i	−0.971693	−	0.236247i	$-0.924083\pi$
$757$	− 13.0000i	− 0.472493i	0.971693	−	0.236247i	$-0.0759173\pi$
$758$	− 20.0000i	− 0.726433i
$759$	0	0
$760$	0	0
$761$	−42.0000	−1.52250	−0.761249	−	0.648459i	$-0.775414\pi$
$761$	−42.0000	−1.52250	−0.761249	+	0.648459i	$0.775414\pi$
$762$	8.00000i	0.289809i
$763$	11.0000i	0.398227i
$764$	18.0000	0.651217
$765$	0	0
$766$	−36.0000	−1.30073
$767$	− 12.0000i	− 0.433295i
$768$	− 1.00000i	− 0.0360844i
$769$	34.0000	1.22607	0.613036	−	0.790055i	$-0.289948\pi$
$769$	34.0000	1.22607	0.613036	+	0.790055i	$0.289948\pi$
$770$	0	0
$771$	−18.0000	−0.648254
$772$	− 25.0000i	− 0.899770i
$773$	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
$776$	−10.0000	−0.358979
$777$	− 11.0000i	− 0.394623i
$778$	30.0000i	1.07555i
$779$	−24.0000	−0.859889
$780$	0	0
$781$	0	0
$782$	− 3.00000i	− 0.107280i
$783$	− 9.00000i	− 0.321634i
$784$	6.00000	0.214286
$785$	0	0
$786$	12.0000	0.428026
$787$	− 28.0000i	− 0.998092i	−0.866575	−	0.499046i	$-0.833684\pi$
$787$	− 28.0000i	− 0.998092i	0.866575	−	0.499046i	$-0.166316\pi$
$788$	15.0000i	0.534353i
$789$	18.0000	0.640817
$790$	0	0
$791$	9.00000	0.320003
$792$	0	0
$793$	− 28.0000i	− 0.994309i
$794$	−8.00000	−0.283909
$795$	0	0
$796$	−1.00000	−0.0354441
$797$	− 30.0000i	− 1.06265i	−0.847167	−	0.531327i	$-0.821693\pi$
$797$	− 30.0000i	− 1.06265i	0.847167	−	0.531327i	$-0.178307\pi$
$798$	2.00000i	0.0707992i
$799$	27.0000	0.955191
$800$	0	0
$801$	3.00000	0.106000
$802$	− 18.0000i	− 0.635602i
$803$	0	0
$804$	−14.0000	−0.493742
$805$	0	0
$806$	−20.0000	−0.704470
$807$	6.00000i	0.211210i
$808$	9.00000i	0.316619i
$809$	6.00000	0.210949	0.105474	−	0.994422i	$-0.466364\pi$
$809$	6.00000	0.210949	0.105474	+	0.994422i	$0.466364\pi$
$810$	0	0
$811$	−52.0000	−1.82597	−0.912983	−	0.407997i	$-0.866228\pi$
$811$	−52.0000	−1.82597	−0.912983	+	0.407997i	$0.866228\pi$
$812$	− 9.00000i	− 0.315838i
$813$	28.0000i	0.982003i
$814$	0	0
$815$	0	0
$816$	3.00000	0.105021
$817$	− 20.0000i	− 0.699711i
$818$	− 17.0000i	− 0.594391i
$819$	2.00000	0.0698857
$820$	0	0
$821$	18.0000	0.628204	0.314102	−	0.949389i	$-0.398297\pi$
$821$	18.0000	0.628204	0.314102	+	0.949389i	$0.398297\pi$
$822$	9.00000i	0.313911i
$823$	− 8.00000i	− 0.278862i	−0.990232	−	0.139431i	$-0.955473\pi$
$823$	− 8.00000i	− 0.278862i	0.990232	−	0.139431i	$-0.0445274\pi$
$824$	7.00000	0.243857
$825$	0	0
$826$	6.00000	0.208767
$827$	− 33.0000i	− 1.14752i	−0.819023	−	0.573761i	$-0.805484\pi$
$827$	− 33.0000i	− 1.14752i	0.819023	−	0.573761i	$-0.194516\pi$
$828$	1.00000i	0.0347524i
$829$	−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
$831$	26.0000	0.901930
$832$	2.00000i	0.0693375i
$833$	18.0000i	0.623663i
$834$	−5.00000	−0.173136
$835$	0	0
$836$	0	0
$837$	− 10.0000i	− 0.345651i
$838$	− 15.0000i	− 0.518166i
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	52.0000	1.79310
$842$	26.0000i	0.896019i
$843$	− 3.00000i	− 0.103325i
$844$	−23.0000	−0.791693
$845$	0	0
$846$	−9.00000	−0.309426
$847$	11.0000i	0.377964i
$848$	0	0
$849$	16.0000	0.549119
$850$	0	0
$851$	−11.0000	−0.377075
$852$	3.00000i	0.102778i
$853$	− 44.0000i	− 1.50653i	−0.657716	−	0.753266i	$-0.728477\pi$
$853$	− 44.0000i	− 1.50653i	0.657716	−	0.753266i	$-0.271523\pi$
$854$	14.0000	0.479070
$855$	0	0
$856$	12.0000	0.410152
$857$	− 6.00000i	− 0.204956i	−0.994735	−	0.102478i	$-0.967323\pi$
$857$	− 6.00000i	− 0.204956i	0.994735	−	0.102478i	$-0.0326771\pi$
$858$	0	0
$859$	4.00000	0.136478	0.0682391	−	0.997669i	$-0.478262\pi$
$859$	4.00000	0.136478	0.0682391	+	0.997669i	$0.478262\pi$
$860$	0	0
$861$	−12.0000	−0.408959
$862$	30.0000i	1.02180i
$863$	− 27.0000i	− 0.919091i	−0.888154	−	0.459545i	$-0.848012\pi$
$863$	− 27.0000i	− 0.919091i	0.888154	−	0.459545i	$-0.151988\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	2.00000	0.0679628
$867$	− 8.00000i	− 0.271694i
$868$	− 10.0000i	− 0.339422i
$869$	0	0
$870$	0	0
$871$	28.0000	0.948744
$872$	11.0000i	0.372507i
$873$	10.0000i	0.338449i
$874$	2.00000	0.0676510
$875$	0	0
$876$	11.0000	0.371656
$877$	38.0000i	1.28317i	0.767052	+	0.641584i	$0.221723\pi$
$877$	38.0000i	1.28317i	−0.767052	+	0.641584i	$0.778277\pi$
$878$	− 20.0000i	− 0.674967i
$879$	−18.0000	−0.607125
$880$	0	0
$881$	30.0000	1.01073	0.505363	−	0.862907i	$-0.331359\pi$
$881$	30.0000	1.01073	0.505363	+	0.862907i	$0.331359\pi$
$882$	− 6.00000i	− 0.202031i
$883$	− 11.0000i	− 0.370179i	−0.982722	−	0.185090i	$-0.940742\pi$
$883$	− 11.0000i	− 0.370179i	0.982722	−	0.185090i	$-0.0592576\pi$
$884$	−6.00000	−0.201802
$885$	0	0
$886$	0	0
$887$	39.0000i	1.30949i	0.755849	+	0.654746i	$0.227224\pi$
$887$	39.0000i	1.30949i	−0.755849	+	0.654746i	$0.772776\pi$
$888$	− 11.0000i	− 0.369136i
$889$	8.00000	0.268311
$890$	0	0
$891$	0	0
$892$	− 22.0000i	− 0.736614i
$893$	18.0000i	0.602347i
$894$	0	0
$895$	0	0
$896$	−1.00000	−0.0334077
$897$	− 2.00000i	− 0.0667781i
$898$	6.00000i	0.200223i
$899$	90.0000	3.00167
$900$	0	0
$901$	0	0
$902$	0	0
$903$	− 10.0000i	− 0.332779i
$904$	9.00000	0.299336
$905$	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	$0.782692\pi$
$908$	21.0000i	0.696909i
$909$	9.00000	0.298511
$910$	0	0
$911$	−24.0000	−0.795155	−0.397578	−	0.917568i	$-0.630149\pi$
$911$	−24.0000	−0.795155	−0.397578	+	0.917568i	$0.630149\pi$
$912$	2.00000i	0.0662266i
$913$	0	0
$914$	−32.0000	−1.05847
$915$	0	0
$916$	2.00000	0.0660819
$917$	− 12.0000i	− 0.396275i
$918$	− 3.00000i	− 0.0990148i
$919$	31.0000	1.02260	0.511298	−	0.859404i	$-0.329165\pi$
$919$	31.0000	1.02260	0.511298	+	0.859404i	$0.329165\pi$
$920$	0	0
$921$	5.00000	0.164756
$922$	3.00000i	0.0987997i
$923$	− 6.00000i	− 0.197492i
$924$	0	0
$925$	0	0
$926$	8.00000	0.262896
$927$	− 7.00000i	− 0.229910i
$928$	− 9.00000i	− 0.295439i
$929$	42.0000	1.37798	0.688988	−	0.724773i	$-0.258055\pi$
$929$	42.0000	1.37798	0.688988	+	0.724773i	$0.258055\pi$
$930$	0	0
$931$	−12.0000	−0.393284
$932$	− 24.0000i	− 0.786146i
$933$	15.0000i	0.491078i
$934$	33.0000	1.07979
$935$	0	0
$936$	2.00000	0.0653720
$937$	50.0000i	1.63343i	0.577042	+	0.816714i	$0.304207\pi$
$937$	50.0000i	1.63343i	−0.577042	+	0.816714i	$0.695793\pi$
$938$	14.0000i	0.457116i
$939$	−8.00000	−0.261070
$940$	0	0
$941$	42.0000	1.36916	0.684580	−	0.728937i	$-0.259985\pi$
$941$	42.0000	1.36916	0.684580	+	0.728937i	$0.259985\pi$
$942$	14.0000i	0.456145i
$943$	12.0000i	0.390774i
$944$	6.00000	0.195283
$945$	0	0
$946$	0	0
$947$	− 12.0000i	− 0.389948i	−0.980808	−	0.194974i	$-0.937538\pi$
$947$	− 12.0000i	− 0.389948i	0.980808	−	0.194974i	$-0.0624622\pi$
$948$	− 8.00000i	− 0.259828i
$949$	−22.0000	−0.714150
$950$	0	0
$951$	−27.0000	−0.875535
$952$	− 3.00000i	− 0.0972306i
$953$	− 15.0000i	− 0.485898i	−0.970039	−	0.242949i	$-0.921885\pi$
$953$	− 15.0000i	− 0.485898i	0.970039	−	0.242949i	$-0.0781147\pi$
$954$	0	0
$955$	0	0
$956$	27.0000	0.873242
$957$	0	0
$958$	18.0000i	0.581554i
$959$	9.00000	0.290625
$960$	0	0
$961$	69.0000	2.22581
$962$	22.0000i	0.709308i
$963$	− 12.0000i	− 0.386695i
$964$	−8.00000	−0.257663
$965$	0	0
$966$	1.00000	0.0321745
$967$	2.00000i	0.0643157i	0.999483	+	0.0321578i	$0.0102379\pi$
$967$	2.00000i	0.0643157i	−0.999483	+	0.0321578i	$0.989762\pi$
$968$	11.0000i	0.353553i
$969$	−6.00000	−0.192748
$970$	0	0
$971$	−21.0000	−0.673922	−0.336961	−	0.941519i	$-0.609399\pi$
$971$	−21.0000	−0.673922	−0.336961	+	0.941519i	$0.609399\pi$
$972$	1.00000i	0.0320750i
$973$	5.00000i	0.160293i
$974$	28.0000	0.897178
$975$	0	0
$976$	14.0000	0.448129
$977$	21.0000i	0.671850i	0.941889	+	0.335925i	$0.109049\pi$
$977$	21.0000i	0.671850i	−0.941889	+	0.335925i	$0.890951\pi$
$978$	16.0000i	0.511624i
$979$	0	0
$980$	0	0
$981$	11.0000	0.351203
$982$	− 30.0000i	− 0.957338i
$983$	− 36.0000i	− 1.14822i	−0.818778	−	0.574111i	$-0.805348\pi$
$983$	− 36.0000i	− 1.14822i	0.818778	−	0.574111i	$-0.194652\pi$
$984$	−12.0000	−0.382546
$985$	0	0
$986$	27.0000	0.859855
$987$	9.00000i	0.286473i
$988$	− 4.00000i	− 0.127257i
$989$	−10.0000	−0.317982
$990$	0	0
$991$	−40.0000	−1.27064	−0.635321	−	0.772248i	$-0.719132\pi$
$991$	−40.0000	−1.27064	−0.635321	+	0.772248i	$0.719132\pi$
$992$	− 10.0000i	− 0.317500i
$993$	− 5.00000i	− 0.158670i
$994$	3.00000	0.0951542
$995$	0	0
$996$	−9.00000	−0.285176
$997$	26.0000i	0.823428i	0.911313	+	0.411714i	$0.135070\pi$
$997$	26.0000i	0.823428i	−0.911313	+	0.411714i	$0.864930\pi$
$998$	19.0000i	0.601434i
$999$	−11.0000	−0.348025

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3450.2.a.f.1.1	✓	1	5.2	odd	4
3450.2.a.x.1.1	yes	1	5.3	odd	4
3450.2.d.p.2899.1		2	5.4	even	2	inner
3450.2.d.p.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)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -1.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -1.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +1.00000i q^{12} -2.00000i q^{13} +1.00000 q^{14} +1.00000 q^{16} +3.00000i q^{17} -1.00000i q^{18} -2.00000 q^{19} -1.00000 q^{21} +1.00000i q^{23} -1.00000 q^{24} +2.00000 q^{26} +1.00000i q^{27} +1.00000i q^{28} -9.00000 q^{29} -10.0000 q^{31} +1.00000i q^{32} -3.00000 q^{34} +1.00000 q^{36} +11.0000i q^{37} -2.00000i q^{38} -2.00000 q^{39} +12.0000 q^{41} -1.00000i q^{42} +10.0000i q^{43} -1.00000 q^{46} -9.00000i q^{47} -1.00000i q^{48} +6.00000 q^{49} +3.00000 q^{51} +2.00000i q^{52} -1.00000 q^{54} -1.00000 q^{56} +2.00000i q^{57} -9.00000i q^{58} +6.00000 q^{59} +14.0000 q^{61} -10.0000i q^{62} +1.00000i q^{63} -1.00000 q^{64} +14.0000i q^{67} -3.00000i q^{68} +1.00000 q^{69} +3.00000 q^{71} +1.00000i q^{72} -11.0000i q^{73} -11.0000 q^{74} +2.00000 q^{76} -2.00000i q^{78} -8.00000 q^{79} +1.00000 q^{81} +12.0000i q^{82} +9.00000i q^{83} +1.00000 q^{84} -10.0000 q^{86} +9.00000i q^{87} -3.00000 q^{89} -2.00000 q^{91} -1.00000i q^{92} +10.0000i q^{93} +9.00000 q^{94} +1.00000 q^{96} -10.0000i q^{97} +6.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9	\( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} + 2 q^{14} + 2 q^{16} - 4 q^{19} - 2 q^{21} - 2 q^{24} + 4 q^{26} - 18 q^{29} - 20 q^{31} - 6 q^{34} + 2 q^{36} - 4 q^{39} + 24 q^{41} - 2 q^{46} + 12 q^{49} + 6 q^{51} - 2 q^{54} - 2 q^{56} + 12 q^{59} + 28 q^{61} - 2 q^{64} + 2 q^{69} + 6 q^{71} - 22 q^{74} + 4 q^{76} - 16 q^{79} + 2 q^{81} + 2 q^{84} - 20 q^{86} - 6 q^{89} - 4 q^{91} + 18 q^{94} + 2 q^{96}+O(q^{100}) \) 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9 + 2 * q^14 + 2 * q^16 - 4 * q^19 - 2 * q^21 - 2 * q^24 + 4 * q^26 - 18 * q^29 - 20 * q^31 - 6 * q^34 + 2 * q^36 - 4 * q^39 + 24 * q^41 - 2 * q^46 + 12 * q^49 + 6 * q^51 - 2 * q^54 - 2 * q^56 + 12 * q^59 + 28 * q^61 - 2 * q^64 + 2 * q^69 + 6 * q^71 - 22 * q^74 + 4 * q^76 - 16 * q^79 + 2 * q^81 + 2 * q^84 - 20 * q^86 - 6 * q^89 - 4 * q^91 + 18 * q^94 + 2 * q^96

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more