LMFDB - Embedded newform 1650.2.c.f.199.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1650,2,Mod(199,1650)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1650, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1650.199"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 1650 = 2 \cdot 3 \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1650.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,-2,0,-2,0,0,-2,0,2,0,0,-6,0,2,0,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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

Self dual:	no
Analytic conductor:	$13.1753163335$
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			199.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	1650.199
Dual form			1650.2.c.f.199.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -3.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +1.00000 q^{11} +1.00000i q^{12} -3.00000 q^{14} +1.00000 q^{16} -5.00000i q^{17} +1.00000i q^{18} +1.00000 q^{19} -3.00000 q^{21} -1.00000i q^{22} -3.00000i q^{23} +1.00000 q^{24} +1.00000i q^{27} +3.00000i q^{28} -2.00000 q^{29} -6.00000 q^{31} -1.00000i q^{32} -1.00000i q^{33} -5.00000 q^{34} +1.00000 q^{36} -3.00000i q^{37} -1.00000i q^{38} -1.00000 q^{41} +3.00000i q^{42} +4.00000i q^{43} -1.00000 q^{44} -3.00000 q^{46} -7.00000i q^{47} -1.00000i q^{48} -2.00000 q^{49} -5.00000 q^{51} +8.00000i q^{53} +1.00000 q^{54} +3.00000 q^{56} -1.00000i q^{57} +2.00000i q^{58} -11.0000 q^{59} -8.00000 q^{61} +6.00000i q^{62} +3.00000i q^{63} -1.00000 q^{64} -1.00000 q^{66} -8.00000i q^{67} +5.00000i q^{68} -3.00000 q^{69} +1.00000 q^{71} -1.00000i q^{72} +8.00000i q^{73} -3.00000 q^{74} -1.00000 q^{76} -3.00000i q^{77} +17.0000 q^{79} +1.00000 q^{81} +1.00000i q^{82} +6.00000i q^{83} +3.00000 q^{84} +4.00000 q^{86} +2.00000i q^{87} +1.00000i q^{88} -12.0000 q^{89} +3.00000i q^{92} +6.00000i q^{93} -7.00000 q^{94} -1.00000 q^{96} +9.00000i q^{97} +2.00000i q^{98} -1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9} + 2 q^{11} - 6 q^{14} + 2 q^{16} + 2 q^{19} - 6 q^{21} + 2 q^{24} - 4 q^{29} - 12 q^{31} - 10 q^{34} + 2 q^{36} - 2 q^{41} - 2 q^{44} - 6 q^{46} - 4 q^{49} - 10 q^{51}+ \cdots - 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 + 2 * q^19 - 6 * q^21 + 2 * q^24 - 4 * q^29 - 12 * q^31 - 10 * q^34 + 2 * q^36 - 2 * q^41 - 2 * q^44 - 6 * q^46 - 4 * q^49 - 10 * q^51 + 2 * q^54 + 6 * q^56 - 22 * q^59 - 16 * q^61 - 2 * q^64 - 2 * q^66 - 6 * q^69 + 2 * q^71 - 6 * q^74 - 2 * q^76 + 34 * q^79 + 2 * q^81 + 6 * q^84 + 8 * q^86 - 24 * q^89 - 14 * q^94 - 2 * q^96 - 2 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1650\mathbb{Z}\right)^\times$.

$n$	$551$	$727$	$1201$
$\chi(n)$	$1$	$-1$	$1$

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
$11$	1.00000	0.301511
$12$	1.00000i	0.288675i
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	−3.00000	−0.801784
$15$	0	0
$16$	1.00000	0.250000
$17$	− 5.00000i	− 1.21268i	−0.795206	−	0.606339i	$-0.792637\pi$
$17$	− 5.00000i	− 1.21268i	0.795206	−	0.606339i	$-0.207363\pi$
$18$	1.00000i	0.235702i
$19$	1.00000	0.229416	0.114708	−	0.993399i	$-0.463407\pi$
$19$	1.00000	0.229416	0.114708	+	0.993399i	$0.463407\pi$
$20$	0	0
$21$	−3.00000	−0.654654
$22$	− 1.00000i	− 0.213201i
$23$	− 3.00000i	− 0.625543i	−0.949828	−	0.312772i	$-0.898743\pi$
$23$	− 3.00000i	− 0.625543i	0.949828	−	0.312772i	$-0.101257\pi$
$24$	1.00000	0.204124
$25$	0	0
$26$	0	0
$27$	1.00000i	0.192450i
$28$	3.00000i	0.566947i
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	0	0
$31$	−6.00000	−1.07763	−0.538816	−	0.842424i	$-0.681128\pi$
$31$	−6.00000	−1.07763	−0.538816	+	0.842424i	$0.681128\pi$
$32$	− 1.00000i	− 0.176777i
$33$	− 1.00000i	− 0.174078i
$34$	−5.00000	−0.857493
$35$	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	$-0.0793129\pi$
$38$	− 1.00000i	− 0.162221i
$39$	0	0
$40$	0	0
$41$	−1.00000	−0.156174	−0.0780869	−	0.996947i	$-0.524881\pi$
$41$	−1.00000	−0.156174	−0.0780869	+	0.996947i	$0.524881\pi$
$42$	3.00000i	0.462910i
$43$	4.00000i	0.609994i	0.952353	+	0.304997i	$0.0986555\pi$
$43$	4.00000i	0.609994i	−0.952353	+	0.304997i	$0.901344\pi$
$44$	−1.00000	−0.150756
$45$	0	0
$46$	−3.00000	−0.442326
$47$	− 7.00000i	− 1.02105i	−0.859861	−	0.510527i	$-0.829450\pi$
$47$	− 7.00000i	− 1.02105i	0.859861	−	0.510527i	$-0.170550\pi$
$48$	− 1.00000i	− 0.144338i
$49$	−2.00000	−0.285714
$50$	0	0
$51$	−5.00000	−0.700140
$52$	0	0
$53$	8.00000i	1.09888i	0.835532	+	0.549442i	$0.185160\pi$
$53$	8.00000i	1.09888i	−0.835532	+	0.549442i	$0.814840\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	3.00000	0.400892
$57$	− 1.00000i	− 0.132453i
$58$	2.00000i	0.262613i
$59$	−11.0000	−1.43208	−0.716039	−	0.698060i	$-0.754047\pi$
$59$	−11.0000	−1.43208	−0.716039	+	0.698060i	$0.754047\pi$
$60$	0	0
$61$	−8.00000	−1.02430	−0.512148	−	0.858898i	$-0.671150\pi$
$61$	−8.00000	−1.02430	−0.512148	+	0.858898i	$0.671150\pi$
$62$	6.00000i	0.762001i
$63$	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	$-0.162521\pi$
$68$	5.00000i	0.606339i
$69$	−3.00000	−0.361158
$70$	0	0
$71$	1.00000	0.118678	0.0593391	−	0.998238i	$-0.481101\pi$
$71$	1.00000	0.118678	0.0593391	+	0.998238i	$0.481101\pi$
$72$	− 1.00000i	− 0.117851i
$73$	8.00000i	0.936329i	0.883641	+	0.468165i	$0.155085\pi$
$73$	8.00000i	0.936329i	−0.883641	+	0.468165i	$0.844915\pi$
$74$	−3.00000	−0.348743
$75$	0	0
$76$	−1.00000	−0.114708
$77$	− 3.00000i	− 0.341882i
$78$	0	0
$79$	17.0000	1.91265	0.956325	−	0.292306i	$-0.0944227\pi$
$79$	17.0000	1.91265	0.956325	+	0.292306i	$0.0944227\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	1.00000i	0.110432i
$83$	6.00000i	0.658586i	0.944228	+	0.329293i	$0.106810\pi$
$83$	6.00000i	0.658586i	−0.944228	+	0.329293i	$0.893190\pi$
$84$	3.00000	0.327327
$85$	0	0
$86$	4.00000	0.431331
$87$	2.00000i	0.214423i
$88$	1.00000i	0.106600i
$89$	−12.0000	−1.27200	−0.635999	−	0.771690i	$-0.719412\pi$
$89$	−12.0000	−1.27200	−0.635999	+	0.771690i	$0.719412\pi$
$90$	0	0
$91$	0	0
$92$	3.00000i	0.312772i
$93$	6.00000i	0.622171i
$94$	−7.00000	−0.721995
$95$	0	0
$96$	−1.00000	−0.102062
$97$	9.00000i	0.913812i	0.889515	+	0.456906i	$0.151042\pi$
$97$	9.00000i	0.913812i	−0.889515	+	0.456906i	$0.848958\pi$
$98$	2.00000i	0.202031i
$99$	−1.00000	−0.100504
$100$	0	0
$101$	−13.0000	−1.29355	−0.646774	−	0.762682i	$-0.723882\pi$
$101$	−13.0000	−1.29355	−0.646774	+	0.762682i	$0.723882\pi$
$102$	5.00000i	0.495074i
$103$	− 2.00000i	− 0.197066i	−0.995134	−	0.0985329i	$-0.968585\pi$
$103$	− 2.00000i	− 0.197066i	0.995134	−	0.0985329i	$-0.0314150\pi$
$104$	0	0
$105$	0	0
$106$	8.00000	0.777029
$107$	− 2.00000i	− 0.193347i	−0.995316	−	0.0966736i	$-0.969180\pi$
$107$	− 2.00000i	− 0.193347i	0.995316	−	0.0966736i	$-0.0308203\pi$
$108$	− 1.00000i	− 0.0962250i
$109$	−8.00000	−0.766261	−0.383131	−	0.923694i	$-0.625154\pi$
$109$	−8.00000	−0.766261	−0.383131	+	0.923694i	$0.625154\pi$
$110$	0	0
$111$	−3.00000	−0.284747
$112$	− 3.00000i	− 0.283473i
$113$	− 16.0000i	− 1.50515i	−0.658505	−	0.752577i	$-0.728811\pi$
$113$	− 16.0000i	− 1.50515i	0.658505	−	0.752577i	$-0.271189\pi$
$114$	−1.00000	−0.0936586
$115$	0	0
$116$	2.00000	0.185695
$117$	0	0
$118$	11.0000i	1.01263i
$119$	−15.0000	−1.37505
$120$	0	0
$121$	1.00000	0.0909091
$122$	8.00000i	0.724286i
$123$	1.00000i	0.0901670i
$124$	6.00000	0.538816
$125$	0	0
$126$	3.00000	0.267261
$127$	− 5.00000i	− 0.443678i	−0.975083	−	0.221839i	$-0.928794\pi$
$127$	− 5.00000i	− 0.443678i	0.975083	−	0.221839i	$-0.0712060\pi$
$128$	1.00000i	0.0883883i
$129$	4.00000	0.352180
$130$	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
$133$	− 3.00000i	− 0.260133i
$134$	−8.00000	−0.691095
$135$	0	0
$136$	5.00000	0.428746
$137$	− 6.00000i	− 0.512615i	−0.966595	−	0.256307i	$-0.917494\pi$
$137$	− 6.00000i	− 0.512615i	0.966595	−	0.256307i	$-0.0825059\pi$
$138$	3.00000i	0.255377i
$139$	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
$142$	− 1.00000i	− 0.0839181i
$143$	0	0
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	8.00000	0.662085
$147$	2.00000i	0.164957i
$148$	3.00000i	0.246598i
$149$	−21.0000	−1.72039	−0.860194	−	0.509968i	$-0.829657\pi$
$149$	−21.0000	−1.72039	−0.860194	+	0.509968i	$0.829657\pi$
$150$	0	0
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	1.00000i	0.0811107i
$153$	5.00000i	0.404226i
$154$	−3.00000	−0.241747
$155$	0	0
$156$	0	0
$157$	− 6.00000i	− 0.478852i	−0.970915	−	0.239426i	$-0.923041\pi$
$157$	− 6.00000i	− 0.478852i	0.970915	−	0.239426i	$-0.0769593\pi$
$158$	− 17.0000i	− 1.35245i
$159$	8.00000	0.634441
$160$	0	0
$161$	−9.00000	−0.709299
$162$	− 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	$0.784443\pi$
$164$	1.00000	0.0780869
$165$	0	0
$166$	6.00000	0.465690
$167$	12.0000i	0.928588i	0.885681	+	0.464294i	$0.153692\pi$
$167$	12.0000i	0.928588i	−0.885681	+	0.464294i	$0.846308\pi$
$168$	− 3.00000i	− 0.231455i
$169$	13.0000	1.00000
$170$	0	0
$171$	−1.00000	−0.0764719
$172$	− 4.00000i	− 0.304997i
$173$	− 13.0000i	− 0.988372i	−0.869356	−	0.494186i	$-0.835466\pi$
$173$	− 13.0000i	− 0.988372i	0.869356	−	0.494186i	$-0.164534\pi$
$174$	2.00000	0.151620
$175$	0	0
$176$	1.00000	0.0753778
$177$	11.0000i	0.826811i
$178$	12.0000i	0.899438i
$179$	9.00000	0.672692	0.336346	−	0.941739i	$-0.390809\pi$
$179$	9.00000	0.672692	0.336346	+	0.941739i	$0.390809\pi$
$180$	0	0
$181$	5.00000	0.371647	0.185824	−	0.982583i	$-0.440505\pi$
$181$	5.00000	0.371647	0.185824	+	0.982583i	$0.440505\pi$
$182$	0	0
$183$	8.00000i	0.591377i
$184$	3.00000	0.221163
$185$	0	0
$186$	6.00000	0.439941
$187$	− 5.00000i	− 0.365636i
$188$	7.00000i	0.510527i
$189$	3.00000	0.218218
$190$	0	0
$191$	−15.0000	−1.08536	−0.542681	−	0.839939i	$-0.682591\pi$
$191$	−15.0000	−1.08536	−0.542681	+	0.839939i	$0.682591\pi$
$192$	1.00000i	0.0721688i
$193$	4.00000i	0.287926i	0.989583	+	0.143963i	$0.0459847\pi$
$193$	4.00000i	0.287926i	−0.989583	+	0.143963i	$0.954015\pi$
$194$	9.00000	0.646162
$195$	0	0
$196$	2.00000	0.142857
$197$	− 15.0000i	− 1.06871i	−0.845262	−	0.534353i	$-0.820555\pi$
$197$	− 15.0000i	− 1.06871i	0.845262	−	0.534353i	$-0.179445\pi$
$198$	1.00000i	0.0710669i
$199$	28.0000	1.98487	0.992434	−	0.122782i	$-0.0391815\pi$
$199$	28.0000	1.98487	0.992434	+	0.122782i	$0.0391815\pi$
$200$	0	0
$201$	−8.00000	−0.564276
$202$	13.0000i	0.914677i
$203$	6.00000i	0.421117i
$204$	5.00000	0.350070
$205$	0	0
$206$	−2.00000	−0.139347
$207$	3.00000i	0.208514i
$208$	0	0
$209$	1.00000	0.0691714
$210$	0	0
$211$	−20.0000	−1.37686	−0.688428	−	0.725304i	$-0.741699\pi$
$211$	−20.0000	−1.37686	−0.688428	+	0.725304i	$0.741699\pi$
$212$	− 8.00000i	− 0.549442i
$213$	− 1.00000i	− 0.0685189i
$214$	−2.00000	−0.136717
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	18.0000i	1.22192i
$218$	8.00000i	0.541828i
$219$	8.00000	0.540590
$220$	0	0
$221$	0	0
$222$	3.00000i	0.201347i
$223$	− 24.0000i	− 1.60716i	−0.595198	−	0.803579i	$-0.702926\pi$
$223$	− 24.0000i	− 1.60716i	0.595198	−	0.803579i	$-0.297074\pi$
$224$	−3.00000	−0.200446
$225$	0	0
$226$	−16.0000	−1.06430
$227$	14.0000i	0.929213i	0.885517	+	0.464606i	$0.153804\pi$
$227$	14.0000i	0.929213i	−0.885517	+	0.464606i	$0.846196\pi$
$228$	1.00000i	0.0662266i
$229$	−1.00000	−0.0660819	−0.0330409	−	0.999454i	$-0.510519\pi$
$229$	−1.00000	−0.0660819	−0.0330409	+	0.999454i	$0.510519\pi$
$230$	0	0
$231$	−3.00000	−0.197386
$232$	− 2.00000i	− 0.131306i
$233$	− 9.00000i	− 0.589610i	−0.955557	−	0.294805i	$-0.904745\pi$
$233$	− 9.00000i	− 0.589610i	0.955557	−	0.294805i	$-0.0952546\pi$
$234$	0	0
$235$	0	0
$236$	11.0000	0.716039
$237$	− 17.0000i	− 1.10427i
$238$	15.0000i	0.972306i
$239$	20.0000	1.29369	0.646846	−	0.762620i	$-0.276088\pi$
$239$	20.0000	1.29369	0.646846	+	0.762620i	$0.276088\pi$
$240$	0	0
$241$	18.0000	1.15948	0.579741	−	0.814801i	$-0.303154\pi$
$241$	18.0000	1.15948	0.579741	+	0.814801i	$0.303154\pi$
$242$	− 1.00000i	− 0.0642824i
$243$	− 1.00000i	− 0.0641500i
$244$	8.00000	0.512148
$245$	0	0
$246$	1.00000	0.0637577
$247$	0	0
$248$	− 6.00000i	− 0.381000i
$249$	6.00000	0.380235
$250$	0	0
$251$	8.00000	0.504956	0.252478	−	0.967603i	$-0.418755\pi$
$251$	8.00000	0.504956	0.252478	+	0.967603i	$0.418755\pi$
$252$	− 3.00000i	− 0.188982i
$253$	− 3.00000i	− 0.188608i
$254$	−5.00000	−0.313728
$255$	0	0
$256$	1.00000	0.0625000
$257$	− 10.0000i	− 0.623783i	−0.950118	−	0.311891i	$-0.899037\pi$
$257$	− 10.0000i	− 0.623783i	0.950118	−	0.311891i	$-0.100963\pi$
$258$	− 4.00000i	− 0.249029i
$259$	−9.00000	−0.559233
$260$	0	0
$261$	2.00000	0.123797
$262$	− 2.00000i	− 0.123560i
$263$	− 18.0000i	− 1.10993i	−0.831875	−	0.554964i	$-0.812732\pi$
$263$	− 18.0000i	− 1.10993i	0.831875	−	0.554964i	$-0.187268\pi$
$264$	1.00000	0.0615457
$265$	0	0
$266$	−3.00000	−0.183942
$267$	12.0000i	0.734388i
$268$	8.00000i	0.488678i
$269$	2.00000	0.121942	0.0609711	−	0.998140i	$-0.480580\pi$
$269$	2.00000	0.121942	0.0609711	+	0.998140i	$0.480580\pi$
$270$	0	0
$271$	−5.00000	−0.303728	−0.151864	−	0.988401i	$-0.548528\pi$
$271$	−5.00000	−0.303728	−0.151864	+	0.988401i	$0.548528\pi$
$272$	− 5.00000i	− 0.303170i
$273$	0	0
$274$	−6.00000	−0.362473
$275$	0	0
$276$	3.00000	0.180579
$277$	12.0000i	0.721010i	0.932757	+	0.360505i	$0.117396\pi$
$277$	12.0000i	0.721010i	−0.932757	+	0.360505i	$0.882604\pi$
$278$	− 20.0000i	− 1.19952i
$279$	6.00000	0.359211
$280$	0	0
$281$	3.00000	0.178965	0.0894825	−	0.995988i	$-0.471479\pi$
$281$	3.00000	0.178965	0.0894825	+	0.995988i	$0.471479\pi$
$282$	7.00000i	0.416844i
$283$	− 27.0000i	− 1.60498i	−0.596663	−	0.802492i	$-0.703507\pi$
$283$	− 27.0000i	− 1.60498i	0.596663	−	0.802492i	$-0.296493\pi$
$284$	−1.00000	−0.0593391
$285$	0	0
$286$	0	0
$287$	3.00000i	0.177084i
$288$	1.00000i	0.0589256i
$289$	−8.00000	−0.470588
$290$	0	0
$291$	9.00000	0.527589
$292$	− 8.00000i	− 0.468165i
$293$	9.00000i	0.525786i	0.964825	+	0.262893i	$0.0846766\pi$
$293$	9.00000i	0.525786i	−0.964825	+	0.262893i	$0.915323\pi$
$294$	2.00000	0.116642
$295$	0	0
$296$	3.00000	0.174371
$297$	1.00000i	0.0580259i
$298$	21.0000i	1.21650i
$299$	0	0
$300$	0	0
$301$	12.0000	0.691669
$302$	− 8.00000i	− 0.460348i
$303$	13.0000i	0.746830i
$304$	1.00000	0.0573539
$305$	0	0
$306$	5.00000	0.285831
$307$	20.0000i	1.14146i	0.821138	+	0.570730i	$0.193340\pi$
$307$	20.0000i	1.14146i	−0.821138	+	0.570730i	$0.806660\pi$
$308$	3.00000i	0.170941i
$309$	−2.00000	−0.113776
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	− 33.0000i	− 1.86527i	−0.360821	−	0.932635i	$-0.617503\pi$
$313$	− 33.0000i	− 1.86527i	0.360821	−	0.932635i	$-0.382497\pi$
$314$	−6.00000	−0.338600
$315$	0	0
$316$	−17.0000	−0.956325
$317$	− 32.0000i	− 1.79730i	−0.438667	−	0.898650i	$-0.644549\pi$
$317$	− 32.0000i	− 1.79730i	0.438667	−	0.898650i	$-0.355451\pi$
$318$	− 8.00000i	− 0.448618i
$319$	−2.00000	−0.111979
$320$	0	0
$321$	−2.00000	−0.111629
$322$	9.00000i	0.501550i
$323$	− 5.00000i	− 0.278207i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	16.0000	0.886158
$327$	8.00000i	0.442401i
$328$	− 1.00000i	− 0.0552158i
$329$	−21.0000	−1.15777
$330$	0	0
$331$	20.0000	1.09930	0.549650	−	0.835395i	$-0.314761\pi$
$331$	20.0000	1.09930	0.549650	+	0.835395i	$0.314761\pi$
$332$	− 6.00000i	− 0.329293i
$333$	3.00000i	0.164399i
$334$	12.0000	0.656611
$335$	0	0
$336$	−3.00000	−0.163663
$337$	− 28.0000i	− 1.52526i	−0.646837	−	0.762629i	$-0.723908\pi$
$337$	− 28.0000i	− 1.52526i	0.646837	−	0.762629i	$-0.276092\pi$
$338$	− 13.0000i	− 0.707107i
$339$	−16.0000	−0.869001
$340$	0	0
$341$	−6.00000	−0.324918
$342$	1.00000i	0.0540738i
$343$	− 15.0000i	− 0.809924i
$344$	−4.00000	−0.215666
$345$	0	0
$346$	−13.0000	−0.698884
$347$	− 12.0000i	− 0.644194i	−0.946707	−	0.322097i	$-0.895612\pi$
$347$	− 12.0000i	− 0.644194i	0.946707	−	0.322097i	$-0.104388\pi$
$348$	− 2.00000i	− 0.107211i
$349$	6.00000	0.321173	0.160586	−	0.987022i	$-0.448662\pi$
$349$	6.00000	0.321173	0.160586	+	0.987022i	$0.448662\pi$
$350$	0	0
$351$	0	0
$352$	− 1.00000i	− 0.0533002i
$353$	− 32.0000i	− 1.70319i	−0.524202	−	0.851594i	$-0.675636\pi$
$353$	− 32.0000i	− 1.70319i	0.524202	−	0.851594i	$-0.324364\pi$
$354$	11.0000	0.584643
$355$	0	0
$356$	12.0000	0.635999
$357$	15.0000i	0.793884i
$358$	− 9.00000i	− 0.475665i
$359$	2.00000	0.105556	0.0527780	−	0.998606i	$-0.483192\pi$
$359$	2.00000	0.105556	0.0527780	+	0.998606i	$0.483192\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	− 5.00000i	− 0.262794i
$363$	− 1.00000i	− 0.0524864i
$364$	0	0
$365$	0	0
$366$	8.00000	0.418167
$367$	26.0000i	1.35719i	0.734513	+	0.678594i	$0.237411\pi$
$367$	26.0000i	1.35719i	−0.734513	+	0.678594i	$0.762589\pi$
$368$	− 3.00000i	− 0.156386i
$369$	1.00000	0.0520579
$370$	0	0
$371$	24.0000	1.24602
$372$	− 6.00000i	− 0.311086i
$373$	− 18.0000i	− 0.932005i	−0.884783	−	0.466002i	$-0.845694\pi$
$373$	− 18.0000i	− 0.932005i	0.884783	−	0.466002i	$-0.154306\pi$
$374$	−5.00000	−0.258544
$375$	0	0
$376$	7.00000	0.360997
$377$	0	0
$378$	− 3.00000i	− 0.154303i
$379$	18.0000	0.924598	0.462299	−	0.886724i	$-0.347025\pi$
$379$	18.0000	0.924598	0.462299	+	0.886724i	$0.347025\pi$
$380$	0	0
$381$	−5.00000	−0.256158
$382$	15.0000i	0.767467i
$383$	− 8.00000i	− 0.408781i	−0.978889	−	0.204390i	$-0.934479\pi$
$383$	− 8.00000i	− 0.408781i	0.978889	−	0.204390i	$-0.0655212\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	4.00000	0.203595
$387$	− 4.00000i	− 0.203331i
$388$	− 9.00000i	− 0.456906i
$389$	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
$391$	−15.0000	−0.758583
$392$	− 2.00000i	− 0.101015i
$393$	− 2.00000i	− 0.100887i
$394$	−15.0000	−0.755689
$395$	0	0
$396$	1.00000	0.0502519
$397$	2.00000i	0.100377i	0.998740	+	0.0501886i	$0.0159822\pi$
$397$	2.00000i	0.100377i	−0.998740	+	0.0501886i	$0.984018\pi$
$398$	− 28.0000i	− 1.40351i
$399$	−3.00000	−0.150188
$400$	0	0
$401$	32.0000	1.59800	0.799002	−	0.601329i	$-0.205362\pi$
$401$	32.0000	1.59800	0.799002	+	0.601329i	$0.205362\pi$
$402$	8.00000i	0.399004i
$403$	0	0
$404$	13.0000	0.646774
$405$	0	0
$406$	6.00000	0.297775
$407$	− 3.00000i	− 0.148704i
$408$	− 5.00000i	− 0.247537i
$409$	−16.0000	−0.791149	−0.395575	−	0.918434i	$-0.629455\pi$
$409$	−16.0000	−0.791149	−0.395575	+	0.918434i	$0.629455\pi$
$410$	0	0
$411$	−6.00000	−0.295958
$412$	2.00000i	0.0985329i
$413$	33.0000i	1.62382i
$414$	3.00000	0.147442
$415$	0	0
$416$	0	0
$417$	− 20.0000i	− 0.979404i
$418$	− 1.00000i	− 0.0489116i
$419$	−21.0000	−1.02592	−0.512959	−	0.858413i	$-0.671451\pi$
$419$	−21.0000	−1.02592	−0.512959	+	0.858413i	$0.671451\pi$
$420$	0	0
$421$	23.0000	1.12095	0.560476	−	0.828171i	$-0.310618\pi$
$421$	23.0000	1.12095	0.560476	+	0.828171i	$0.310618\pi$
$422$	20.0000i	0.973585i
$423$	7.00000i	0.340352i
$424$	−8.00000	−0.388514
$425$	0	0
$426$	−1.00000	−0.0484502
$427$	24.0000i	1.16144i
$428$	2.00000i	0.0966736i
$429$	0	0
$430$	0	0
$431$	−14.0000	−0.674356	−0.337178	−	0.941441i	$-0.609472\pi$
$431$	−14.0000	−0.674356	−0.337178	+	0.941441i	$0.609472\pi$
$432$	1.00000i	0.0481125i
$433$	2.00000i	0.0961139i	0.998845	+	0.0480569i	$0.0153029\pi$
$433$	2.00000i	0.0961139i	−0.998845	+	0.0480569i	$0.984697\pi$
$434$	18.0000	0.864028
$435$	0	0
$436$	8.00000	0.383131
$437$	− 3.00000i	− 0.143509i
$438$	− 8.00000i	− 0.382255i
$439$	−11.0000	−0.525001	−0.262501	−	0.964932i	$-0.584547\pi$
$439$	−11.0000	−0.525001	−0.262501	+	0.964932i	$0.584547\pi$
$440$	0	0
$441$	2.00000	0.0952381
$442$	0	0
$443$	− 29.0000i	− 1.37783i	−0.724841	−	0.688916i	$-0.758087\pi$
$443$	− 29.0000i	− 1.37783i	0.724841	−	0.688916i	$-0.241913\pi$
$444$	3.00000	0.142374
$445$	0	0
$446$	−24.0000	−1.13643
$447$	21.0000i	0.993266i
$448$	3.00000i	0.141737i
$449$	−36.0000	−1.69895	−0.849473	−	0.527633i	$-0.823080\pi$
$449$	−36.0000	−1.69895	−0.849473	+	0.527633i	$0.823080\pi$
$450$	0	0
$451$	−1.00000	−0.0470882
$452$	16.0000i	0.752577i
$453$	− 8.00000i	− 0.375873i
$454$	14.0000	0.657053
$455$	0	0
$456$	1.00000	0.0468293
$457$	18.0000i	0.842004i	0.907060	+	0.421002i	$0.138322\pi$
$457$	18.0000i	0.842004i	−0.907060	+	0.421002i	$0.861678\pi$
$458$	1.00000i	0.0467269i
$459$	5.00000	0.233380
$460$	0	0
$461$	30.0000	1.39724	0.698620	−	0.715493i	$-0.253798\pi$
$461$	30.0000	1.39724	0.698620	+	0.715493i	$0.253798\pi$
$462$	3.00000i	0.139573i
$463$	10.0000i	0.464739i	0.972628	+	0.232370i	$0.0746479\pi$
$463$	10.0000i	0.464739i	−0.972628	+	0.232370i	$0.925352\pi$
$464$	−2.00000	−0.0928477
$465$	0	0
$466$	−9.00000	−0.416917
$467$	4.00000i	0.185098i	0.995708	+	0.0925490i	$0.0295015\pi$
$467$	4.00000i	0.185098i	−0.995708	+	0.0925490i	$0.970499\pi$
$468$	0	0
$469$	−24.0000	−1.10822
$470$	0	0
$471$	−6.00000	−0.276465
$472$	− 11.0000i	− 0.506316i
$473$	4.00000i	0.183920i
$474$	−17.0000	−0.780836
$475$	0	0
$476$	15.0000	0.687524
$477$	− 8.00000i	− 0.366295i
$478$	− 20.0000i	− 0.914779i
$479$	12.0000	0.548294	0.274147	−	0.961688i	$-0.411605\pi$
$479$	12.0000	0.548294	0.274147	+	0.961688i	$0.411605\pi$
$480$	0	0
$481$	0	0
$482$	− 18.0000i	− 0.819878i
$483$	9.00000i	0.409514i
$484$	−1.00000	−0.0454545
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	− 30.0000i	− 1.35943i	−0.733476	−	0.679715i	$-0.762104\pi$
$487$	− 30.0000i	− 1.35943i	0.733476	−	0.679715i	$-0.237896\pi$
$488$	− 8.00000i	− 0.362143i
$489$	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	$0.736695\pi$
$492$	− 1.00000i	− 0.0450835i
$493$	10.0000i	0.450377i
$494$	0	0
$495$	0	0
$496$	−6.00000	−0.269408
$497$	− 3.00000i	− 0.134568i
$498$	− 6.00000i	− 0.268866i
$499$	28.0000	1.25345	0.626726	−	0.779240i	$-0.284395\pi$
$499$	28.0000	1.25345	0.626726	+	0.779240i	$0.284395\pi$
$500$	0	0
$501$	12.0000	0.536120
$502$	− 8.00000i	− 0.357057i
$503$	12.0000i	0.535054i	0.963550	+	0.267527i	$0.0862064\pi$
$503$	12.0000i	0.535054i	−0.963550	+	0.267527i	$0.913794\pi$
$504$	−3.00000	−0.133631
$505$	0	0
$506$	−3.00000	−0.133366
$507$	− 13.0000i	− 0.577350i
$508$	5.00000i	0.221839i
$509$	−8.00000	−0.354594	−0.177297	−	0.984157i	$-0.556735\pi$
$509$	−8.00000	−0.354594	−0.177297	+	0.984157i	$0.556735\pi$
$510$	0	0
$511$	24.0000	1.06170
$512$	− 1.00000i	− 0.0441942i
$513$	1.00000i	0.0441511i
$514$	−10.0000	−0.441081
$515$	0	0
$516$	−4.00000	−0.176090
$517$	− 7.00000i	− 0.307860i
$518$	9.00000i	0.395437i
$519$	−13.0000	−0.570637
$520$	0	0
$521$	22.0000	0.963837	0.481919	−	0.876216i	$-0.339940\pi$
$521$	22.0000	0.963837	0.481919	+	0.876216i	$0.339940\pi$
$522$	− 2.00000i	− 0.0875376i
$523$	35.0000i	1.53044i	0.643767	+	0.765222i	$0.277371\pi$
$523$	35.0000i	1.53044i	−0.643767	+	0.765222i	$0.722629\pi$
$524$	−2.00000	−0.0873704
$525$	0	0
$526$	−18.0000	−0.784837
$527$	30.0000i	1.30682i
$528$	− 1.00000i	− 0.0435194i
$529$	14.0000	0.608696
$530$	0	0
$531$	11.0000	0.477359
$532$	3.00000i	0.130066i
$533$	0	0
$534$	12.0000	0.519291
$535$	0	0
$536$	8.00000	0.345547
$537$	− 9.00000i	− 0.388379i
$538$	− 2.00000i	− 0.0862261i
$539$	−2.00000	−0.0861461
$540$	0	0
$541$	16.0000	0.687894	0.343947	−	0.938989i	$-0.388236\pi$
$541$	16.0000	0.687894	0.343947	+	0.938989i	$0.388236\pi$
$542$	5.00000i	0.214768i
$543$	− 5.00000i	− 0.214571i
$544$	−5.00000	−0.214373
$545$	0	0
$546$	0	0
$547$	37.0000i	1.58201i	0.611812	+	0.791003i	$0.290441\pi$
$547$	37.0000i	1.58201i	−0.611812	+	0.791003i	$0.709559\pi$
$548$	6.00000i	0.256307i
$549$	8.00000	0.341432
$550$	0	0
$551$	−2.00000	−0.0852029
$552$	− 3.00000i	− 0.127688i
$553$	− 51.0000i	− 2.16874i
$554$	12.0000	0.509831
$555$	0	0
$556$	−20.0000	−0.848189
$557$	− 30.0000i	− 1.27114i	−0.772043	−	0.635570i	$-0.780765\pi$
$557$	− 30.0000i	− 1.27114i	0.772043	−	0.635570i	$-0.219235\pi$
$558$	− 6.00000i	− 0.254000i
$559$	0	0
$560$	0	0
$561$	−5.00000	−0.211100
$562$	− 3.00000i	− 0.126547i
$563$	42.0000i	1.77009i	0.465506	+	0.885044i	$0.345872\pi$
$563$	42.0000i	1.77009i	−0.465506	+	0.885044i	$0.654128\pi$
$564$	7.00000	0.294753
$565$	0	0
$566$	−27.0000	−1.13489
$567$	− 3.00000i	− 0.125988i
$568$	1.00000i	0.0419591i
$569$	15.0000	0.628833	0.314416	−	0.949285i	$-0.398191\pi$
$569$	15.0000	0.628833	0.314416	+	0.949285i	$0.398191\pi$
$570$	0	0
$571$	24.0000	1.00437	0.502184	−	0.864761i	$-0.332530\pi$
$571$	24.0000	1.00437	0.502184	+	0.864761i	$0.332530\pi$
$572$	0	0
$573$	15.0000i	0.626634i
$574$	3.00000	0.125218
$575$	0	0
$576$	1.00000	0.0416667
$577$	43.0000i	1.79011i	0.445952	+	0.895057i	$0.352865\pi$
$577$	43.0000i	1.79011i	−0.445952	+	0.895057i	$0.647135\pi$
$578$	8.00000i	0.332756i
$579$	4.00000	0.166234
$580$	0	0
$581$	18.0000	0.746766
$582$	− 9.00000i	− 0.373062i
$583$	8.00000i	0.331326i
$584$	−8.00000	−0.331042
$585$	0	0
$586$	9.00000	0.371787
$587$	9.00000i	0.371470i	0.982600	+	0.185735i	$0.0594666\pi$
$587$	9.00000i	0.371470i	−0.982600	+	0.185735i	$0.940533\pi$
$588$	− 2.00000i	− 0.0824786i
$589$	−6.00000	−0.247226
$590$	0	0
$591$	−15.0000	−0.617018
$592$	− 3.00000i	− 0.123299i
$593$	− 10.0000i	− 0.410651i	−0.978694	−	0.205325i	$-0.934175\pi$
$593$	− 10.0000i	− 0.410651i	0.978694	−	0.205325i	$-0.0658253\pi$
$594$	1.00000	0.0410305
$595$	0	0
$596$	21.0000	0.860194
$597$	− 28.0000i	− 1.14596i
$598$	0	0
$599$	21.0000	0.858037	0.429018	−	0.903296i	$-0.358860\pi$
$599$	21.0000	0.858037	0.429018	+	0.903296i	$0.358860\pi$
$600$	0	0
$601$	0	0		−	1.00000i	$-0.5\pi$
$601$	0	0			1.00000i	$0.5\pi$
$602$	− 12.0000i	− 0.489083i
$603$	8.00000i	0.325785i
$604$	−8.00000	−0.325515
$605$	0	0
$606$	13.0000	0.528089
$607$	− 16.0000i	− 0.649420i	−0.945814	−	0.324710i	$-0.894733\pi$
$607$	− 16.0000i	− 0.649420i	0.945814	−	0.324710i	$-0.105267\pi$
$608$	− 1.00000i	− 0.0405554i
$609$	6.00000	0.243132
$610$	0	0
$611$	0	0
$612$	− 5.00000i	− 0.202113i
$613$	4.00000i	0.161558i	0.996732	+	0.0807792i	$0.0257409\pi$
$613$	4.00000i	0.161558i	−0.996732	+	0.0807792i	$0.974259\pi$
$614$	20.0000	0.807134
$615$	0	0
$616$	3.00000	0.120873
$617$	10.0000i	0.402585i	0.979531	+	0.201292i	$0.0645141\pi$
$617$	10.0000i	0.402585i	−0.979531	+	0.201292i	$0.935486\pi$
$618$	2.00000i	0.0804518i
$619$	10.0000	0.401934	0.200967	−	0.979598i	$-0.435592\pi$
$619$	10.0000	0.401934	0.200967	+	0.979598i	$0.435592\pi$
$620$	0	0
$621$	3.00000	0.120386
$622$	0	0
$623$	36.0000i	1.44231i
$624$	0	0
$625$	0	0
$626$	−33.0000	−1.31895
$627$	− 1.00000i	− 0.0399362i
$628$	6.00000i	0.239426i
$629$	−15.0000	−0.598089
$630$	0	0
$631$	18.0000	0.716569	0.358284	−	0.933613i	$-0.383362\pi$
$631$	18.0000	0.716569	0.358284	+	0.933613i	$0.383362\pi$
$632$	17.0000i	0.676224i
$633$	20.0000i	0.794929i
$634$	−32.0000	−1.27088
$635$	0	0
$636$	−8.00000	−0.317221
$637$	0	0
$638$	2.00000i	0.0791808i
$639$	−1.00000	−0.0395594
$640$	0	0
$641$	12.0000	0.473972	0.236986	−	0.971513i	$-0.423841\pi$
$641$	12.0000	0.473972	0.236986	+	0.971513i	$0.423841\pi$
$642$	2.00000i	0.0789337i
$643$	− 14.0000i	− 0.552106i	−0.961142	−	0.276053i	$-0.910973\pi$
$643$	− 14.0000i	− 0.552106i	0.961142	−	0.276053i	$-0.0890266\pi$
$644$	9.00000	0.354650
$645$	0	0
$646$	−5.00000	−0.196722
$647$	− 27.0000i	− 1.06148i	−0.847535	−	0.530740i	$-0.821914\pi$
$647$	− 27.0000i	− 1.06148i	0.847535	−	0.530740i	$-0.178086\pi$
$648$	1.00000i	0.0392837i
$649$	−11.0000	−0.431788
$650$	0	0
$651$	18.0000	0.705476
$652$	− 16.0000i	− 0.626608i
$653$	14.0000i	0.547862i	0.961749	+	0.273931i	$0.0883240\pi$
$653$	14.0000i	0.547862i	−0.961749	+	0.273931i	$0.911676\pi$
$654$	8.00000	0.312825
$655$	0	0
$656$	−1.00000	−0.0390434
$657$	− 8.00000i	− 0.312110i
$658$	21.0000i	0.818665i
$659$	−24.0000	−0.934907	−0.467454	−	0.884018i	$-0.654829\pi$
$659$	−24.0000	−0.934907	−0.467454	+	0.884018i	$0.654829\pi$
$660$	0	0
$661$	−43.0000	−1.67251	−0.836253	−	0.548344i	$-0.815259\pi$
$661$	−43.0000	−1.67251	−0.836253	+	0.548344i	$0.815259\pi$
$662$	− 20.0000i	− 0.777322i
$663$	0	0
$664$	−6.00000	−0.232845
$665$	0	0
$666$	3.00000	0.116248
$667$	6.00000i	0.232321i
$668$	− 12.0000i	− 0.464294i
$669$	−24.0000	−0.927894
$670$	0	0
$671$	−8.00000	−0.308837
$672$	3.00000i	0.115728i
$673$	8.00000i	0.308377i	0.988041	+	0.154189i	$0.0492764\pi$
$673$	8.00000i	0.308377i	−0.988041	+	0.154189i	$0.950724\pi$
$674$	−28.0000	−1.07852
$675$	0	0
$676$	−13.0000	−0.500000
$677$	26.0000i	0.999261i	0.866239	+	0.499631i	$0.166531\pi$
$677$	26.0000i	0.999261i	−0.866239	+	0.499631i	$0.833469\pi$
$678$	16.0000i	0.614476i
$679$	27.0000	1.03616
$680$	0	0
$681$	14.0000	0.536481
$682$	6.00000i	0.229752i
$683$	5.00000i	0.191320i	0.995414	+	0.0956598i	$0.0304961\pi$
$683$	5.00000i	0.191320i	−0.995414	+	0.0956598i	$0.969504\pi$
$684$	1.00000	0.0382360
$685$	0	0
$686$	−15.0000	−0.572703
$687$	1.00000i	0.0381524i
$688$	4.00000i	0.152499i
$689$	0	0
$690$	0	0
$691$	−8.00000	−0.304334	−0.152167	−	0.988355i	$-0.548625\pi$
$691$	−8.00000	−0.304334	−0.152167	+	0.988355i	$0.548625\pi$
$692$	13.0000i	0.494186i
$693$	3.00000i	0.113961i
$694$	−12.0000	−0.455514
$695$	0	0
$696$	−2.00000	−0.0758098
$697$	5.00000i	0.189389i
$698$	− 6.00000i	− 0.227103i
$699$	−9.00000	−0.340411
$700$	0	0
$701$	13.0000	0.491003	0.245502	−	0.969396i	$-0.421047\pi$
$701$	13.0000	0.491003	0.245502	+	0.969396i	$0.421047\pi$
$702$	0	0
$703$	− 3.00000i	− 0.113147i
$704$	−1.00000	−0.0376889
$705$	0	0
$706$	−32.0000	−1.20434
$707$	39.0000i	1.46675i
$708$	− 11.0000i	− 0.413405i
$709$	−41.0000	−1.53979	−0.769894	−	0.638172i	$-0.779691\pi$
$709$	−41.0000	−1.53979	−0.769894	+	0.638172i	$0.779691\pi$
$710$	0	0
$711$	−17.0000	−0.637550
$712$	− 12.0000i	− 0.449719i
$713$	18.0000i	0.674105i
$714$	15.0000	0.561361
$715$	0	0
$716$	−9.00000	−0.336346
$717$	− 20.0000i	− 0.746914i
$718$	− 2.00000i	− 0.0746393i
$719$	8.00000	0.298350	0.149175	−	0.988811i	$-0.452338\pi$
$719$	8.00000	0.298350	0.149175	+	0.988811i	$0.452338\pi$
$720$	0	0
$721$	−6.00000	−0.223452
$722$	18.0000i	0.669891i
$723$	− 18.0000i	− 0.669427i
$724$	−5.00000	−0.185824
$725$	0	0
$726$	−1.00000	−0.0371135
$727$	− 32.0000i	− 1.18681i	−0.804902	−	0.593407i	$-0.797782\pi$
$727$	− 32.0000i	− 1.18681i	0.804902	−	0.593407i	$-0.202218\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	20.0000	0.739727
$732$	− 8.00000i	− 0.295689i
$733$	− 36.0000i	− 1.32969i	−0.746981	−	0.664845i	$-0.768498\pi$
$733$	− 36.0000i	− 1.32969i	0.746981	−	0.664845i	$-0.231502\pi$
$734$	26.0000	0.959678
$735$	0	0
$736$	−3.00000	−0.110581
$737$	− 8.00000i	− 0.294684i
$738$	− 1.00000i	− 0.0368105i
$739$	35.0000	1.28750	0.643748	−	0.765238i	$-0.277379\pi$
$739$	35.0000	1.28750	0.643748	+	0.765238i	$0.277379\pi$
$740$	0	0
$741$	0	0
$742$	− 24.0000i	− 0.881068i
$743$	48.0000i	1.76095i	0.474093	+	0.880475i	$0.342776\pi$
$743$	48.0000i	1.76095i	−0.474093	+	0.880475i	$0.657224\pi$
$744$	−6.00000	−0.219971
$745$	0	0
$746$	−18.0000	−0.659027
$747$	− 6.00000i	− 0.219529i
$748$	5.00000i	0.182818i
$749$	−6.00000	−0.219235
$750$	0	0
$751$	−4.00000	−0.145962	−0.0729810	−	0.997333i	$-0.523251\pi$
$751$	−4.00000	−0.145962	−0.0729810	+	0.997333i	$0.523251\pi$
$752$	− 7.00000i	− 0.255264i
$753$	− 8.00000i	− 0.291536i
$754$	0	0
$755$	0	0
$756$	−3.00000	−0.109109
$757$	− 2.00000i	− 0.0726912i	−0.999339	−	0.0363456i	$-0.988428\pi$
$757$	− 2.00000i	− 0.0726912i	0.999339	−	0.0363456i	$-0.0115717\pi$
$758$	− 18.0000i	− 0.653789i
$759$	−3.00000	−0.108893
$760$	0	0
$761$	−6.00000	−0.217500	−0.108750	−	0.994069i	$-0.534685\pi$
$761$	−6.00000	−0.217500	−0.108750	+	0.994069i	$0.534685\pi$
$762$	5.00000i	0.181131i
$763$	24.0000i	0.868858i
$764$	15.0000	0.542681
$765$	0	0
$766$	−8.00000	−0.289052
$767$	0	0
$768$	− 1.00000i	− 0.0360844i
$769$	−32.0000	−1.15395	−0.576975	−	0.816762i	$-0.695767\pi$
$769$	−32.0000	−1.15395	−0.576975	+	0.816762i	$0.695767\pi$
$770$	0	0
$771$	−10.0000	−0.360141
$772$	− 4.00000i	− 0.143963i
$773$	− 8.00000i	− 0.287740i	−0.989597	−	0.143870i	$-0.954045\pi$
$773$	− 8.00000i	− 0.287740i	0.989597	−	0.143870i	$-0.0459547\pi$
$774$	−4.00000	−0.143777
$775$	0	0
$776$	−9.00000	−0.323081
$777$	9.00000i	0.322873i
$778$	− 30.0000i	− 1.07555i
$779$	−1.00000	−0.0358287
$780$	0	0
$781$	1.00000	0.0357828
$782$	15.0000i	0.536399i
$783$	− 2.00000i	− 0.0714742i
$784$	−2.00000	−0.0714286
$785$	0	0
$786$	−2.00000	−0.0713376
$787$	− 19.0000i	− 0.677277i	−0.940917	−	0.338638i	$-0.890034\pi$
$787$	− 19.0000i	− 0.677277i	0.940917	−	0.338638i	$-0.109966\pi$
$788$	15.0000i	0.534353i
$789$	−18.0000	−0.640817
$790$	0	0
$791$	−48.0000	−1.70668
$792$	− 1.00000i	− 0.0355335i
$793$	0	0
$794$	2.00000	0.0709773
$795$	0	0
$796$	−28.0000	−0.992434
$797$	18.0000i	0.637593i	0.947823	+	0.318796i	$0.103279\pi$
$797$	18.0000i	0.637593i	−0.947823	+	0.318796i	$0.896721\pi$
$798$	3.00000i	0.106199i
$799$	−35.0000	−1.23821
$800$	0	0
$801$	12.0000	0.423999
$802$	− 32.0000i	− 1.12996i
$803$	8.00000i	0.282314i
$804$	8.00000	0.282138
$805$	0	0
$806$	0	0
$807$	− 2.00000i	− 0.0704033i
$808$	− 13.0000i	− 0.457338i
$809$	−9.00000	−0.316423	−0.158212	−	0.987405i	$-0.550573\pi$
$809$	−9.00000	−0.316423	−0.158212	+	0.987405i	$0.550573\pi$
$810$	0	0
$811$	7.00000	0.245803	0.122902	−	0.992419i	$-0.460780\pi$
$811$	7.00000	0.245803	0.122902	+	0.992419i	$0.460780\pi$
$812$	− 6.00000i	− 0.210559i
$813$	5.00000i	0.175358i
$814$	−3.00000	−0.105150
$815$	0	0
$816$	−5.00000	−0.175035
$817$	4.00000i	0.139942i
$818$	16.0000i	0.559427i
$819$	0	0
$820$	0	0
$821$	−30.0000	−1.04701	−0.523504	−	0.852023i	$-0.675375\pi$
$821$	−30.0000	−1.04701	−0.523504	+	0.852023i	$0.675375\pi$
$822$	6.00000i	0.209274i
$823$	44.0000i	1.53374i	0.641800	+	0.766872i	$0.278188\pi$
$823$	44.0000i	1.53374i	−0.641800	+	0.766872i	$0.721812\pi$
$824$	2.00000	0.0696733
$825$	0	0
$826$	33.0000	1.14822
$827$	16.0000i	0.556375i	0.960527	+	0.278187i	$0.0897336\pi$
$827$	16.0000i	0.556375i	−0.960527	+	0.278187i	$0.910266\pi$
$828$	− 3.00000i	− 0.104257i
$829$	38.0000	1.31979	0.659897	−	0.751356i	$-0.270600\pi$
$829$	38.0000	1.31979	0.659897	+	0.751356i	$0.270600\pi$
$830$	0	0
$831$	12.0000	0.416275
$832$	0	0
$833$	10.0000i	0.346479i
$834$	−20.0000	−0.692543
$835$	0	0
$836$	−1.00000	−0.0345857
$837$	− 6.00000i	− 0.207390i
$838$	21.0000i	0.725433i
$839$	48.0000	1.65714	0.828572	−	0.559883i	$-0.189154\pi$
$839$	48.0000	1.65714	0.828572	+	0.559883i	$0.189154\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	− 23.0000i	− 0.792632i
$843$	− 3.00000i	− 0.103325i
$844$	20.0000	0.688428
$845$	0	0
$846$	7.00000	0.240665
$847$	− 3.00000i	− 0.103081i
$848$	8.00000i	0.274721i
$849$	−27.0000	−0.926638
$850$	0	0
$851$	−9.00000	−0.308516
$852$	1.00000i	0.0342594i
$853$	46.0000i	1.57501i	0.616308	+	0.787505i	$0.288628\pi$
$853$	46.0000i	1.57501i	−0.616308	+	0.787505i	$0.711372\pi$
$854$	24.0000	0.821263
$855$	0	0
$856$	2.00000	0.0683586
$857$	43.0000i	1.46885i	0.678689	+	0.734426i	$0.262549\pi$
$857$	43.0000i	1.46885i	−0.678689	+	0.734426i	$0.737451\pi$
$858$	0	0
$859$	36.0000	1.22830	0.614152	−	0.789188i	$-0.289498\pi$
$859$	36.0000	1.22830	0.614152	+	0.789188i	$0.289498\pi$
$860$	0	0
$861$	3.00000	0.102240
$862$	14.0000i	0.476842i
$863$	24.0000i	0.816970i	0.912765	+	0.408485i	$0.133943\pi$
$863$	24.0000i	0.816970i	−0.912765	+	0.408485i	$0.866057\pi$
$864$	1.00000	0.0340207
$865$	0	0
$866$	2.00000	0.0679628
$867$	8.00000i	0.271694i
$868$	− 18.0000i	− 0.610960i
$869$	17.0000	0.576686
$870$	0	0
$871$	0	0
$872$	− 8.00000i	− 0.270914i
$873$	− 9.00000i	− 0.304604i
$874$	−3.00000	−0.101477
$875$	0	0
$876$	−8.00000	−0.270295
$877$	22.0000i	0.742887i	0.928456	+	0.371444i	$0.121137\pi$
$877$	22.0000i	0.742887i	−0.928456	+	0.371444i	$0.878863\pi$
$878$	11.0000i	0.371232i
$879$	9.00000	0.303562
$880$	0	0
$881$	−28.0000	−0.943344	−0.471672	−	0.881774i	$-0.656349\pi$
$881$	−28.0000	−0.943344	−0.471672	+	0.881774i	$0.656349\pi$
$882$	− 2.00000i	− 0.0673435i
$883$	6.00000i	0.201916i	0.994891	+	0.100958i	$0.0321908\pi$
$883$	6.00000i	0.201916i	−0.994891	+	0.100958i	$0.967809\pi$
$884$	0	0
$885$	0	0
$886$	−29.0000	−0.974274
$887$	4.00000i	0.134307i	0.997743	+	0.0671534i	$0.0213917\pi$
$887$	4.00000i	0.134307i	−0.997743	+	0.0671534i	$0.978608\pi$
$888$	− 3.00000i	− 0.100673i
$889$	−15.0000	−0.503084
$890$	0	0
$891$	1.00000	0.0335013
$892$	24.0000i	0.803579i
$893$	− 7.00000i	− 0.234246i
$894$	21.0000	0.702345
$895$	0	0
$896$	3.00000	0.100223
$897$	0	0
$898$	36.0000i	1.20134i
$899$	12.0000	0.400222
$900$	0	0
$901$	40.0000	1.33259
$902$	1.00000i	0.0332964i
$903$	− 12.0000i	− 0.399335i
$904$	16.0000	0.532152
$905$	0	0
$906$	−8.00000	−0.265782
$907$	− 30.0000i	− 0.996134i	−0.867139	−	0.498067i	$-0.834043\pi$
$907$	− 30.0000i	− 0.996134i	0.867139	−	0.498067i	$-0.165957\pi$
$908$	− 14.0000i	− 0.464606i
$909$	13.0000	0.431183
$910$	0	0
$911$	−39.0000	−1.29213	−0.646064	−	0.763283i	$-0.723586\pi$
$911$	−39.0000	−1.29213	−0.646064	+	0.763283i	$0.723586\pi$
$912$	− 1.00000i	− 0.0331133i
$913$	6.00000i	0.198571i
$914$	18.0000	0.595387
$915$	0	0
$916$	1.00000	0.0330409
$917$	− 6.00000i	− 0.198137i
$918$	− 5.00000i	− 0.165025i
$919$	−31.0000	−1.02260	−0.511298	−	0.859404i	$-0.670835\pi$
$919$	−31.0000	−1.02260	−0.511298	+	0.859404i	$0.670835\pi$
$920$	0	0
$921$	20.0000	0.659022
$922$	− 30.0000i	− 0.987997i
$923$	0	0
$924$	3.00000	0.0986928
$925$	0	0
$926$	10.0000	0.328620
$927$	2.00000i	0.0656886i
$928$	2.00000i	0.0656532i
$929$	4.00000	0.131236	0.0656179	−	0.997845i	$-0.479098\pi$
$929$	4.00000	0.131236	0.0656179	+	0.997845i	$0.479098\pi$
$930$	0	0
$931$	−2.00000	−0.0655474
$932$	9.00000i	0.294805i
$933$	0	0
$934$	4.00000	0.130884
$935$	0	0
$936$	0	0
$937$	34.0000i	1.11073i	0.831606	+	0.555366i	$0.187422\pi$
$937$	34.0000i	1.11073i	−0.831606	+	0.555366i	$0.812578\pi$
$938$	24.0000i	0.783628i
$939$	−33.0000	−1.07691
$940$	0	0
$941$	−49.0000	−1.59735	−0.798677	−	0.601760i	$-0.794466\pi$
$941$	−49.0000	−1.59735	−0.798677	+	0.601760i	$0.794466\pi$
$942$	6.00000i	0.195491i
$943$	3.00000i	0.0976934i
$944$	−11.0000	−0.358020
$945$	0	0
$946$	4.00000	0.130051
$947$	− 27.0000i	− 0.877382i	−0.898638	−	0.438691i	$-0.855442\pi$
$947$	− 27.0000i	− 0.877382i	0.898638	−	0.438691i	$-0.144558\pi$
$948$	17.0000i	0.552134i
$949$	0	0
$950$	0	0
$951$	−32.0000	−1.03767
$952$	− 15.0000i	− 0.486153i
$953$	45.0000i	1.45769i	0.684677	+	0.728846i	$0.259943\pi$
$953$	45.0000i	1.45769i	−0.684677	+	0.728846i	$0.740057\pi$
$954$	−8.00000	−0.259010
$955$	0	0
$956$	−20.0000	−0.646846
$957$	2.00000i	0.0646508i
$958$	− 12.0000i	− 0.387702i
$959$	−18.0000	−0.581250
$960$	0	0
$961$	5.00000	0.161290
$962$	0	0
$963$	2.00000i	0.0644491i
$964$	−18.0000	−0.579741
$965$	0	0
$966$	9.00000	0.289570
$967$	− 24.0000i	− 0.771788i	−0.922543	−	0.385894i	$-0.873893\pi$
$967$	− 24.0000i	− 0.771788i	0.922543	−	0.385894i	$-0.126107\pi$
$968$	1.00000i	0.0321412i
$969$	−5.00000	−0.160623
$970$	0	0
$971$	45.0000	1.44412	0.722059	−	0.691831i	$-0.243196\pi$
$971$	45.0000	1.44412	0.722059	+	0.691831i	$0.243196\pi$
$972$	1.00000i	0.0320750i
$973$	− 60.0000i	− 1.92351i
$974$	−30.0000	−0.961262
$975$	0	0
$976$	−8.00000	−0.256074
$977$	− 30.0000i	− 0.959785i	−0.877327	−	0.479893i	$-0.840676\pi$
$977$	− 30.0000i	− 0.959785i	0.877327	−	0.479893i	$-0.159324\pi$
$978$	− 16.0000i	− 0.511624i
$979$	−12.0000	−0.383522
$980$	0	0
$981$	8.00000	0.255420
$982$	30.0000i	0.957338i
$983$	− 45.0000i	− 1.43528i	−0.696416	−	0.717639i	$-0.745223\pi$
$983$	− 45.0000i	− 1.43528i	0.696416	−	0.717639i	$-0.254777\pi$
$984$	−1.00000	−0.0318788
$985$	0	0
$986$	10.0000	0.318465
$987$	21.0000i	0.668437i
$988$	0	0
$989$	12.0000	0.381578
$990$	0	0
$991$	−44.0000	−1.39771	−0.698853	−	0.715265i	$-0.746306\pi$
$991$	−44.0000	−1.39771	−0.698853	+	0.715265i	$0.746306\pi$
$992$	6.00000i	0.190500i
$993$	− 20.0000i	− 0.634681i
$994$	−3.00000	−0.0951542
$995$	0	0
$996$	−6.00000	−0.190117
$997$	14.0000i	0.443384i	0.975117	+	0.221692i	$0.0711580\pi$
$997$	14.0000i	0.443384i	−0.975117	+	0.221692i	$0.928842\pi$
$998$	− 28.0000i	− 0.886325i
$999$	3.00000	0.0949158

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1650.2.c.f.199.1		2
3.2	odd	2		4950.2.c.n.199.2		2
5.2	odd	4		1650.2.a.p.1.1	yes	1
5.3	odd	4		1650.2.a.f.1.1	✓	1
5.4	even	2	inner	1650.2.c.f.199.2		2
15.2	even	4		4950.2.a.p.1.1		1
15.8	even	4		4950.2.a.bb.1.1		1
15.14	odd	2		4950.2.c.n.199.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1650.2.a.f.1.1	✓	1	5.3	odd	4
1650.2.a.p.1.1	yes	1	5.2	odd	4
1650.2.c.f.199.1		2	1.1	even	1	trivial
1650.2.c.f.199.2		2	5.4	even	2	inner
4950.2.a.p.1.1		1	15.2	even	4
4950.2.a.bb.1.1		1	15.8	even	4
4950.2.c.n.199.1		2	15.14	odd	2
4950.2.c.n.199.2		2	3.2	odd	2

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1650 = 2 \cdot 3 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1650.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -3.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +1.00000 q^{11} +1.00000i q^{12} -3.00000 q^{14} +1.00000 q^{16} -5.00000i q^{17} +1.00000i q^{18} +1.00000 q^{19} -3.00000 q^{21} -1.00000i q^{22} -3.00000i q^{23} +1.00000 q^{24} +1.00000i q^{27} +3.00000i q^{28} -2.00000 q^{29} -6.00000 q^{31} -1.00000i q^{32} -1.00000i q^{33} -5.00000 q^{34} +1.00000 q^{36} -3.00000i q^{37} -1.00000i q^{38} -1.00000 q^{41} +3.00000i q^{42} +4.00000i q^{43} -1.00000 q^{44} -3.00000 q^{46} -7.00000i q^{47} -1.00000i q^{48} -2.00000 q^{49} -5.00000 q^{51} +8.00000i q^{53} +1.00000 q^{54} +3.00000 q^{56} -1.00000i q^{57} +2.00000i q^{58} -11.0000 q^{59} -8.00000 q^{61} +6.00000i q^{62} +3.00000i q^{63} -1.00000 q^{64} -1.00000 q^{66} -8.00000i q^{67} +5.00000i q^{68} -3.00000 q^{69} +1.00000 q^{71} -1.00000i q^{72} +8.00000i q^{73} -3.00000 q^{74} -1.00000 q^{76} -3.00000i q^{77} +17.0000 q^{79} +1.00000 q^{81} +1.00000i q^{82} +6.00000i q^{83} +3.00000 q^{84} +4.00000 q^{86} +2.00000i q^{87} +1.00000i q^{88} -12.0000 q^{89} +3.00000i q^{92} +6.00000i q^{93} -7.00000 q^{94} -1.00000 q^{96} +9.00000i q^{97} +2.00000i q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9} + 2 q^{11} - 6 q^{14} + 2 q^{16} + 2 q^{19} - 6 q^{21} + 2 q^{24} - 4 q^{29} - 12 q^{31} - 10 q^{34} + 2 q^{36} - 2 q^{41} - 2 q^{44} - 6 q^{46} - 4 q^{49} - 10 q^{51}+ \cdots - 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9 + 2 * q^11 - 6 * q^14 + 2 * q^16 + 2 * q^19 - 6 * q^21 + 2 * q^24 - 4 * q^29 - 12 * q^31 - 10 * q^34 + 2 * q^36 - 2 * q^41 - 2 * q^44 - 6 * q^46 - 4 * q^49 - 10 * q^51 + 2 * q^54 + 6 * q^56 - 22 * q^59 - 16 * q^61 - 2 * q^64 - 2 * q^66 - 6 * q^69 + 2 * q^71 - 6 * q^74 - 2 * q^76 + 34 * q^79 + 2 * q^81 + 6 * q^84 + 8 * q^86 - 24 * q^89 - 14 * q^94 - 2 * q^96 - 2 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

Properties

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