LMFDB - Embedded newform 1350.2.c.k.649.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1350,2,Mod(649,1350)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1350.649"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1350, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1350 = 2 \cdot 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1350.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,-2,0,0,0,0,0,0,6,0,0,2,0,2,0,0,-4,0,0,0,0,0,0,-8,0,0,12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(29)] == traces)

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

Self dual:	no
Analytic conductor:	$10.7798042729$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 54)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	1350.649
Dual form			1350.2.c.k.649.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{7} +1.00000i q^{8} +3.00000 q^{11} -4.00000i q^{13} +1.00000 q^{14} +1.00000 q^{16} -2.00000 q^{19} -3.00000i q^{22} +6.00000i q^{23} -4.00000 q^{26} -1.00000i q^{28} +6.00000 q^{29} +5.00000 q^{31} -1.00000i q^{32} -2.00000i q^{37} +2.00000i q^{38} +6.00000 q^{41} -10.0000i q^{43} -3.00000 q^{44} +6.00000 q^{46} +6.00000i q^{47} +6.00000 q^{49} +4.00000i q^{52} -9.00000i q^{53} -1.00000 q^{56} -6.00000i q^{58} +12.0000 q^{59} +8.00000 q^{61} -5.00000i q^{62} -1.00000 q^{64} -14.0000i q^{67} -7.00000i q^{73} -2.00000 q^{74} +2.00000 q^{76} +3.00000i q^{77} -8.00000 q^{79} -6.00000i q^{82} +3.00000i q^{83} -10.0000 q^{86} +3.00000i q^{88} -18.0000 q^{89} +4.00000 q^{91} -6.00000i q^{92} +6.00000 q^{94} +1.00000i q^{97} -6.00000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} + 6 q^{11} + 2 q^{14} + 2 q^{16} - 4 q^{19} - 8 q^{26} + 12 q^{29} + 10 q^{31} + 12 q^{41} - 6 q^{44} + 12 q^{46} + 12 q^{49} - 2 q^{56} + 24 q^{59} + 16 q^{61} - 2 q^{64} - 4 q^{74} + 4 q^{76}+ \cdots + 12 q^{94}+O(q^{100}) $ 2 * q - 2 * q^4 + 6 * q^11 + 2 * q^14 + 2 * q^16 - 4 * q^19 - 8 * q^26 + 12 * q^29 + 10 * q^31 + 12 * q^41 - 6 * q^44 + 12 * q^46 + 12 * q^49 - 2 * q^56 + 24 * q^59 + 16 * q^61 - 2 * q^64 - 4 * q^74 + 4 * q^76 - 16 * q^79 - 20 * q^86 - 36 * q^89 + 8 * q^91 + 12 * q^94

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1350\mathbb{Z}\right)^\times$.

$n$	$1001$	$1027$
$\chi(n)$	$1$	$-1$

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
$3$	0	0
$3$	0	0
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.00000i	0.377964i	0.981981	+	0.188982i	$0.0605189\pi$
$7$	1.00000i	0.377964i	−0.981981	+	0.188982i	$0.939481\pi$
$8$	1.00000i	0.353553i
$9$	0	0
$10$	0	0
$11$	3.00000	0.904534	0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000	0.904534	0.452267	+	0.891883i	$0.350615\pi$
$12$	0	0
$13$	− 4.00000i	− 1.10940i	−0.832050	−	0.554700i	$-0.812833\pi$
$13$	− 4.00000i	− 1.10940i	0.832050	−	0.554700i	$-0.187167\pi$
$14$	1.00000	0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	0	0
$19$	−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
$21$	0	0
$22$	− 3.00000i	− 0.639602i
$23$	6.00000i	1.25109i	0.780189	+	0.625543i	$0.215123\pi$
$23$	6.00000i	1.25109i	−0.780189	+	0.625543i	$0.784877\pi$
$24$	0	0
$25$	0	0
$26$	−4.00000	−0.784465
$27$	0	0
$28$	− 1.00000i	− 0.188982i
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	5.00000	0.898027	0.449013	−	0.893525i	$-0.351776\pi$
$31$	5.00000	0.898027	0.449013	+	0.893525i	$0.351776\pi$
$32$	− 1.00000i	− 0.176777i
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 2.00000i	− 0.328798i	−0.986394	−	0.164399i	$-0.947432\pi$
$37$	− 2.00000i	− 0.328798i	0.986394	−	0.164399i	$-0.0525685\pi$
$38$	2.00000i	0.324443i
$39$	0	0
$40$	0	0
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	0	0
$43$	− 10.0000i	− 1.52499i	−0.646997	−	0.762493i	$-0.723975\pi$
$43$	− 10.0000i	− 1.52499i	0.646997	−	0.762493i	$-0.276025\pi$
$44$	−3.00000	−0.452267
$45$	0	0
$46$	6.00000	0.884652
$47$	6.00000i	0.875190i	0.899172	+	0.437595i	$0.144170\pi$
$47$	6.00000i	0.875190i	−0.899172	+	0.437595i	$0.855830\pi$
$48$	0	0
$49$	6.00000	0.857143
$50$	0	0
$51$	0	0
$52$	4.00000i	0.554700i
$53$	− 9.00000i	− 1.23625i	−0.786082	−	0.618123i	$-0.787894\pi$
$53$	− 9.00000i	− 1.23625i	0.786082	−	0.618123i	$-0.212106\pi$
$54$	0	0
$55$	0	0
$56$	−1.00000	−0.133631
$57$	0	0
$58$	− 6.00000i	− 0.787839i
$59$	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
$61$	8.00000	1.02430	0.512148	−	0.858898i	$-0.328850\pi$
$61$	8.00000	1.02430	0.512148	+	0.858898i	$0.328850\pi$
$62$	− 5.00000i	− 0.635001i
$63$	0	0
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	− 14.0000i	− 1.71037i	−0.518321	−	0.855186i	$-0.673443\pi$
$67$	− 14.0000i	− 1.71037i	0.518321	−	0.855186i	$-0.326557\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	− 7.00000i	− 0.819288i	−0.912245	−	0.409644i	$-0.865653\pi$
$73$	− 7.00000i	− 0.819288i	0.912245	−	0.409644i	$-0.134347\pi$
$74$	−2.00000	−0.232495
$75$	0	0
$76$	2.00000	0.229416
$77$	3.00000i	0.341882i
$78$	0	0
$79$	−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
$81$	0	0
$82$	− 6.00000i	− 0.662589i
$83$	3.00000i	0.329293i	0.986353	+	0.164646i	$0.0526483\pi$
$83$	3.00000i	0.329293i	−0.986353	+	0.164646i	$0.947352\pi$
$84$	0	0
$85$	0	0
$86$	−10.0000	−1.07833
$87$	0	0
$88$	3.00000i	0.319801i
$89$	−18.0000	−1.90800	−0.953998	−	0.299813i	$-0.903076\pi$
$89$	−18.0000	−1.90800	−0.953998	+	0.299813i	$0.903076\pi$
$90$	0	0
$91$	4.00000	0.419314
$92$	− 6.00000i	− 0.625543i
$93$	0	0
$94$	6.00000	0.618853
$95$	0	0
$96$	0	0
$97$	1.00000i	0.101535i	0.998711	+	0.0507673i	$0.0161667\pi$
$97$	1.00000i	0.101535i	−0.998711	+	0.0507673i	$0.983833\pi$
$98$	− 6.00000i	− 0.606092i
$99$	0	0
$100$	0	0
$101$	3.00000	0.298511	0.149256	−	0.988799i	$-0.452312\pi$
$101$	3.00000	0.298511	0.149256	+	0.988799i	$0.452312\pi$
$102$	0	0
$103$	− 4.00000i	− 0.394132i	−0.980390	−	0.197066i	$-0.936859\pi$
$103$	− 4.00000i	− 0.394132i	0.980390	−	0.197066i	$-0.0631413\pi$
$104$	4.00000	0.392232
$105$	0	0
$106$	−9.00000	−0.874157
$107$	9.00000i	0.870063i	0.900415	+	0.435031i	$0.143263\pi$
$107$	9.00000i	0.870063i	−0.900415	+	0.435031i	$0.856737\pi$
$108$	0	0
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	0	0
$111$	0	0
$112$	1.00000i	0.0944911i
$113$	6.00000i	0.564433i	0.959351	+	0.282216i	$0.0910696\pi$
$113$	6.00000i	0.564433i	−0.959351	+	0.282216i	$0.908930\pi$
$114$	0	0
$115$	0	0
$116$	−6.00000	−0.557086
$117$	0	0
$118$	− 12.0000i	− 1.10469i
$119$	0	0
$120$	0	0
$121$	−2.00000	−0.181818
$122$	− 8.00000i	− 0.724286i
$123$	0	0
$124$	−5.00000	−0.449013
$125$	0	0
$126$	0	0
$127$	7.00000i	0.621150i	0.950549	+	0.310575i	$0.100522\pi$
$127$	7.00000i	0.621150i	−0.950549	+	0.310575i	$0.899478\pi$
$128$	1.00000i	0.0883883i
$129$	0	0
$130$	0	0
$131$	15.0000	1.31056	0.655278	−	0.755388i	$-0.272551\pi$
$131$	15.0000	1.31056	0.655278	+	0.755388i	$0.272551\pi$
$132$	0	0
$133$	− 2.00000i	− 0.173422i
$134$	−14.0000	−1.20942
$135$	0	0
$136$	0	0
$137$	6.00000i	0.512615i	0.966595	+	0.256307i	$0.0825059\pi$
$137$	6.00000i	0.512615i	−0.966595	+	0.256307i	$0.917494\pi$
$138$	0	0
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 12.0000i	− 1.00349i
$144$	0	0
$145$	0	0
$146$	−7.00000	−0.579324
$147$	0	0
$148$	2.00000i	0.164399i
$149$	3.00000	0.245770	0.122885	−	0.992421i	$-0.460785\pi$
$149$	3.00000	0.245770	0.122885	+	0.992421i	$0.460785\pi$
$150$	0	0
$151$	17.0000	1.38344	0.691720	−	0.722166i	$-0.256853\pi$
$151$	17.0000	1.38344	0.691720	+	0.722166i	$0.256853\pi$
$152$	− 2.00000i	− 0.162221i
$153$	0	0
$154$	3.00000	0.241747
$155$	0	0
$156$	0	0
$157$	4.00000i	0.319235i	0.987179	+	0.159617i	$0.0510260\pi$
$157$	4.00000i	0.319235i	−0.987179	+	0.159617i	$0.948974\pi$
$158$	8.00000i	0.636446i
$159$	0	0
$160$	0	0
$161$	−6.00000	−0.472866
$162$	0	0
$163$	20.0000i	1.56652i	0.621694	+	0.783260i	$0.286445\pi$
$163$	20.0000i	1.56652i	−0.621694	+	0.783260i	$0.713555\pi$
$164$	−6.00000	−0.468521
$165$	0	0
$166$	3.00000	0.232845
$167$	− 6.00000i	− 0.464294i	−0.972681	−	0.232147i	$-0.925425\pi$
$167$	− 6.00000i	− 0.464294i	0.972681	−	0.232147i	$-0.0745750\pi$
$168$	0	0
$169$	−3.00000	−0.230769
$170$	0	0
$171$	0	0
$172$	10.0000i	0.762493i
$173$	− 15.0000i	− 1.14043i	−0.821496	−	0.570214i	$-0.806860\pi$
$173$	− 15.0000i	− 1.14043i	0.821496	−	0.570214i	$-0.193140\pi$
$174$	0	0
$175$	0	0
$176$	3.00000	0.226134
$177$	0	0
$178$	18.0000i	1.34916i
$179$	−9.00000	−0.672692	−0.336346	−	0.941739i	$-0.609191\pi$
$179$	−9.00000	−0.672692	−0.336346	+	0.941739i	$0.609191\pi$
$180$	0	0
$181$	−16.0000	−1.18927	−0.594635	−	0.803996i	$-0.702704\pi$
$181$	−16.0000	−1.18927	−0.594635	+	0.803996i	$0.702704\pi$
$182$	− 4.00000i	− 0.296500i
$183$	0	0
$184$	−6.00000	−0.442326
$185$	0	0
$186$	0	0
$187$	0	0
$188$	− 6.00000i	− 0.437595i
$189$	0	0
$190$	0	0
$191$	12.0000	0.868290	0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000	0.868290	0.434145	+	0.900843i	$0.357051\pi$
$192$	0	0
$193$	5.00000i	0.359908i	0.983675	+	0.179954i	$0.0575949\pi$
$193$	5.00000i	0.359908i	−0.983675	+	0.179954i	$0.942405\pi$
$194$	1.00000	0.0717958
$195$	0	0
$196$	−6.00000	−0.428571
$197$	9.00000i	0.641223i	0.947211	+	0.320612i	$0.103888\pi$
$197$	9.00000i	0.641223i	−0.947211	+	0.320612i	$0.896112\pi$
$198$	0	0
$199$	7.00000	0.496217	0.248108	−	0.968732i	$-0.420191\pi$
$199$	7.00000	0.496217	0.248108	+	0.968732i	$0.420191\pi$
$200$	0	0
$201$	0	0
$202$	− 3.00000i	− 0.211079i
$203$	6.00000i	0.421117i
$204$	0	0
$205$	0	0
$206$	−4.00000	−0.278693
$207$	0	0
$208$	− 4.00000i	− 0.277350i
$209$	−6.00000	−0.415029
$210$	0	0
$211$	−22.0000	−1.51454	−0.757271	−	0.653101i	$-0.773468\pi$
$211$	−22.0000	−1.51454	−0.757271	+	0.653101i	$0.773468\pi$
$212$	9.00000i	0.618123i
$213$	0	0
$214$	9.00000	0.615227
$215$	0	0
$216$	0	0
$217$	5.00000i	0.339422i
$218$	2.00000i	0.135457i
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	8.00000i	0.535720i	0.963458	+	0.267860i	$0.0863164\pi$
$223$	8.00000i	0.535720i	−0.963458	+	0.267860i	$0.913684\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	6.00000	0.399114
$227$	− 12.0000i	− 0.796468i	−0.917284	−	0.398234i	$-0.869623\pi$
$227$	− 12.0000i	− 0.796468i	0.917284	−	0.398234i	$-0.130377\pi$
$228$	0	0
$229$	−14.0000	−0.925146	−0.462573	−	0.886581i	$-0.653074\pi$
$229$	−14.0000	−0.925146	−0.462573	+	0.886581i	$0.653074\pi$
$230$	0	0
$231$	0	0
$232$	6.00000i	0.393919i
$233$	− 18.0000i	− 1.17922i	−0.807688	−	0.589610i	$-0.799282\pi$
$233$	− 18.0000i	− 1.17922i	0.807688	−	0.589610i	$-0.200718\pi$
$234$	0	0
$235$	0	0
$236$	−12.0000	−0.781133
$237$	0	0
$238$	0	0
$239$	30.0000	1.94054	0.970269	−	0.242028i	$-0.0778125\pi$
$239$	30.0000	1.94054	0.970269	+	0.242028i	$0.0778125\pi$
$240$	0	0
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	2.00000i	0.128565i
$243$	0	0
$244$	−8.00000	−0.512148
$245$	0	0
$246$	0	0
$247$	8.00000i	0.509028i
$248$	5.00000i	0.317500i
$249$	0	0
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	18.0000i	1.13165i
$254$	7.00000	0.439219
$255$	0	0
$256$	1.00000	0.0625000
$257$	12.0000i	0.748539i	0.927320	+	0.374270i	$0.122107\pi$
$257$	12.0000i	0.748539i	−0.927320	+	0.374270i	$0.877893\pi$
$258$	0	0
$259$	2.00000	0.124274
$260$	0	0
$261$	0	0
$262$	− 15.0000i	− 0.926703i
$263$	30.0000i	1.84988i	0.380114	+	0.924940i	$0.375885\pi$
$263$	30.0000i	1.84988i	−0.380114	+	0.924940i	$0.624115\pi$
$264$	0	0
$265$	0	0
$266$	−2.00000	−0.122628
$267$	0	0
$268$	14.0000i	0.855186i
$269$	−18.0000	−1.09748	−0.548740	−	0.835993i	$-0.684892\pi$
$269$	−18.0000	−1.09748	−0.548740	+	0.835993i	$0.684892\pi$
$270$	0	0
$271$	−25.0000	−1.51864	−0.759321	−	0.650716i	$-0.774469\pi$
$271$	−25.0000	−1.51864	−0.759321	+	0.650716i	$0.774469\pi$
$272$	0	0
$273$	0	0
$274$	6.00000	0.362473
$275$	0	0
$276$	0	0
$277$	− 8.00000i	− 0.480673i	−0.970690	−	0.240337i	$-0.922742\pi$
$277$	− 8.00000i	− 0.480673i	0.970690	−	0.240337i	$-0.0772579\pi$
$278$	− 4.00000i	− 0.239904i
$279$	0	0
$280$	0	0
$281$	−24.0000	−1.43172	−0.715860	−	0.698244i	$-0.753965\pi$
$281$	−24.0000	−1.43172	−0.715860	+	0.698244i	$0.753965\pi$
$282$	0	0
$283$	14.0000i	0.832214i	0.909316	+	0.416107i	$0.136606\pi$
$283$	14.0000i	0.832214i	−0.909316	+	0.416107i	$0.863394\pi$
$284$	0	0
$285$	0	0
$286$	−12.0000	−0.709575
$287$	6.00000i	0.354169i
$288$	0	0
$289$	17.0000	1.00000
$290$	0	0
$291$	0	0
$292$	7.00000i	0.409644i
$293$	6.00000i	0.350524i	0.984522	+	0.175262i	$0.0560772\pi$
$293$	6.00000i	0.350524i	−0.984522	+	0.175262i	$0.943923\pi$
$294$	0	0
$295$	0	0
$296$	2.00000	0.116248
$297$	0	0
$298$	− 3.00000i	− 0.173785i
$299$	24.0000	1.38796
$300$	0	0
$301$	10.0000	0.576390
$302$	− 17.0000i	− 0.978240i
$303$	0	0
$304$	−2.00000	−0.114708
$305$	0	0
$306$	0	0
$307$	16.0000i	0.913168i	0.889680	+	0.456584i	$0.150927\pi$
$307$	16.0000i	0.913168i	−0.889680	+	0.456584i	$0.849073\pi$
$308$	− 3.00000i	− 0.170941i
$309$	0	0
$310$	0	0
$311$	6.00000	0.340229	0.170114	−	0.985424i	$-0.445586\pi$
$311$	6.00000	0.340229	0.170114	+	0.985424i	$0.445586\pi$
$312$	0	0
$313$	− 19.0000i	− 1.07394i	−0.843600	−	0.536972i	$-0.819568\pi$
$313$	− 19.0000i	− 1.07394i	0.843600	−	0.536972i	$-0.180432\pi$
$314$	4.00000	0.225733
$315$	0	0
$316$	8.00000	0.450035
$317$	− 3.00000i	− 0.168497i	−0.996445	−	0.0842484i	$-0.973151\pi$
$317$	− 3.00000i	− 0.168497i	0.996445	−	0.0842484i	$-0.0268489\pi$
$318$	0	0
$319$	18.0000	1.00781
$320$	0	0
$321$	0	0
$322$	6.00000i	0.334367i
$323$	0	0
$324$	0	0
$325$	0	0
$326$	20.0000	1.10770
$327$	0	0
$328$	6.00000i	0.331295i
$329$	−6.00000	−0.330791
$330$	0	0
$331$	−10.0000	−0.549650	−0.274825	−	0.961494i	$-0.588620\pi$
$331$	−10.0000	−0.549650	−0.274825	+	0.961494i	$0.588620\pi$
$332$	− 3.00000i	− 0.164646i
$333$	0	0
$334$	−6.00000	−0.328305
$335$	0	0
$336$	0	0
$337$	22.0000i	1.19842i	0.800593	+	0.599208i	$0.204518\pi$
$337$	22.0000i	1.19842i	−0.800593	+	0.599208i	$0.795482\pi$
$338$	3.00000i	0.163178i
$339$	0	0
$340$	0	0
$341$	15.0000	0.812296
$342$	0	0
$343$	13.0000i	0.701934i
$344$	10.0000	0.539164
$345$	0	0
$346$	−15.0000	−0.806405
$347$	3.00000i	0.161048i	0.996753	+	0.0805242i	$0.0256594\pi$
$347$	3.00000i	0.161048i	−0.996753	+	0.0805242i	$0.974341\pi$
$348$	0	0
$349$	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
$351$	0	0
$352$	− 3.00000i	− 0.159901i
$353$	− 6.00000i	− 0.319348i	−0.987170	−	0.159674i	$-0.948956\pi$
$353$	− 6.00000i	− 0.319348i	0.987170	−	0.159674i	$-0.0510443\pi$
$354$	0	0
$355$	0	0
$356$	18.0000	0.953998
$357$	0	0
$358$	9.00000i	0.475665i
$359$	18.0000	0.950004	0.475002	−	0.879985i	$-0.342447\pi$
$359$	18.0000	0.950004	0.475002	+	0.879985i	$0.342447\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	16.0000i	0.840941i
$363$	0	0
$364$	−4.00000	−0.209657
$365$	0	0
$366$	0	0
$367$	− 17.0000i	− 0.887393i	−0.896177	−	0.443696i	$-0.853667\pi$
$367$	− 17.0000i	− 0.887393i	0.896177	−	0.443696i	$-0.146333\pi$
$368$	6.00000i	0.312772i
$369$	0	0
$370$	0	0
$371$	9.00000	0.467257
$372$	0	0
$373$	32.0000i	1.65690i	0.560065	+	0.828449i	$0.310776\pi$
$373$	32.0000i	1.65690i	−0.560065	+	0.828449i	$0.689224\pi$
$374$	0	0
$375$	0	0
$376$	−6.00000	−0.309426
$377$	− 24.0000i	− 1.23606i
$378$	0	0
$379$	−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
$381$	0	0
$382$	− 12.0000i	− 0.613973i
$383$	24.0000i	1.22634i	0.789950	+	0.613171i	$0.210106\pi$
$383$	24.0000i	1.22634i	−0.789950	+	0.613171i	$0.789894\pi$
$384$	0	0
$385$	0	0
$386$	5.00000	0.254493
$387$	0	0
$388$	− 1.00000i	− 0.0507673i
$389$	−21.0000	−1.06474	−0.532371	−	0.846511i	$-0.678699\pi$
$389$	−21.0000	−1.06474	−0.532371	+	0.846511i	$0.678699\pi$
$390$	0	0
$391$	0	0
$392$	6.00000i	0.303046i
$393$	0	0
$394$	9.00000	0.453413
$395$	0	0
$396$	0	0
$397$	− 20.0000i	− 1.00377i	−0.864934	−	0.501886i	$-0.832640\pi$
$397$	− 20.0000i	− 1.00377i	0.864934	−	0.501886i	$-0.167360\pi$
$398$	− 7.00000i	− 0.350878i
$399$	0	0
$400$	0	0
$401$	−12.0000	−0.599251	−0.299626	−	0.954057i	$-0.596862\pi$
$401$	−12.0000	−0.599251	−0.299626	+	0.954057i	$0.596862\pi$
$402$	0	0
$403$	− 20.0000i	− 0.996271i
$404$	−3.00000	−0.149256
$405$	0	0
$406$	6.00000	0.297775
$407$	− 6.00000i	− 0.297409i
$408$	0	0
$409$	−23.0000	−1.13728	−0.568638	−	0.822588i	$-0.692530\pi$
$409$	−23.0000	−1.13728	−0.568638	+	0.822588i	$0.692530\pi$
$410$	0	0
$411$	0	0
$412$	4.00000i	0.197066i
$413$	12.0000i	0.590481i
$414$	0	0
$415$	0	0
$416$	−4.00000	−0.196116
$417$	0	0
$418$	6.00000i	0.293470i
$419$	12.0000	0.586238	0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000	0.586238	0.293119	+	0.956076i	$0.405307\pi$
$420$	0	0
$421$	8.00000	0.389896	0.194948	−	0.980814i	$-0.437546\pi$
$421$	8.00000	0.389896	0.194948	+	0.980814i	$0.437546\pi$
$422$	22.0000i	1.07094i
$423$	0	0
$424$	9.00000	0.437079
$425$	0	0
$426$	0	0
$427$	8.00000i	0.387147i
$428$	− 9.00000i	− 0.435031i
$429$	0	0
$430$	0	0
$431$	−18.0000	−0.867029	−0.433515	−	0.901146i	$-0.642727\pi$
$431$	−18.0000	−0.867029	−0.433515	+	0.901146i	$0.642727\pi$
$432$	0	0
$433$	29.0000i	1.39365i	0.717241	+	0.696826i	$0.245405\pi$
$433$	29.0000i	1.39365i	−0.717241	+	0.696826i	$0.754595\pi$
$434$	5.00000	0.240008
$435$	0	0
$436$	2.00000	0.0957826
$437$	− 12.0000i	− 0.574038i
$438$	0	0
$439$	19.0000	0.906821	0.453410	−	0.891302i	$-0.350207\pi$
$439$	19.0000	0.906821	0.453410	+	0.891302i	$0.350207\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	12.0000i	0.570137i	0.958507	+	0.285069i	$0.0920164\pi$
$443$	12.0000i	0.570137i	−0.958507	+	0.285069i	$0.907984\pi$
$444$	0	0
$445$	0	0
$446$	8.00000	0.378811
$447$	0	0
$448$	− 1.00000i	− 0.0472456i
$449$	−18.0000	−0.849473	−0.424736	−	0.905317i	$-0.639633\pi$
$449$	−18.0000	−0.849473	−0.424736	+	0.905317i	$0.639633\pi$
$450$	0	0
$451$	18.0000	0.847587
$452$	− 6.00000i	− 0.282216i
$453$	0	0
$454$	−12.0000	−0.563188
$455$	0	0
$456$	0	0
$457$	1.00000i	0.0467780i	0.999726	+	0.0233890i	$0.00744563\pi$
$457$	1.00000i	0.0467780i	−0.999726	+	0.0233890i	$0.992554\pi$
$458$	14.0000i	0.654177i
$459$	0	0
$460$	0	0
$461$	21.0000	0.978068	0.489034	−	0.872265i	$-0.337349\pi$
$461$	21.0000	0.978068	0.489034	+	0.872265i	$0.337349\pi$
$462$	0	0
$463$	− 13.0000i	− 0.604161i	−0.953282	−	0.302081i	$-0.902319\pi$
$463$	− 13.0000i	− 0.604161i	0.953282	−	0.302081i	$-0.0976812\pi$
$464$	6.00000	0.278543
$465$	0	0
$466$	−18.0000	−0.833834
$467$	− 27.0000i	− 1.24941i	−0.780860	−	0.624705i	$-0.785219\pi$
$467$	− 27.0000i	− 1.24941i	0.780860	−	0.624705i	$-0.214781\pi$
$468$	0	0
$469$	14.0000	0.646460
$470$	0	0
$471$	0	0
$472$	12.0000i	0.552345i
$473$	− 30.0000i	− 1.37940i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	− 30.0000i	− 1.37217i
$479$	6.00000	0.274147	0.137073	−	0.990561i	$-0.456230\pi$
$479$	6.00000	0.274147	0.137073	+	0.990561i	$0.456230\pi$
$480$	0	0
$481$	−8.00000	−0.364769
$482$	10.0000i	0.455488i
$483$	0	0
$484$	2.00000	0.0909091
$485$	0	0
$486$	0	0
$487$	16.0000i	0.725029i	0.931978	+	0.362515i	$0.118082\pi$
$487$	16.0000i	0.725029i	−0.931978	+	0.362515i	$0.881918\pi$
$488$	8.00000i	0.362143i
$489$	0	0
$490$	0	0
$491$	−39.0000	−1.76005	−0.880023	−	0.474932i	$-0.842473\pi$
$491$	−39.0000	−1.76005	−0.880023	+	0.474932i	$0.842473\pi$
$492$	0	0
$493$	0	0
$494$	8.00000	0.359937
$495$	0	0
$496$	5.00000	0.224507
$497$	0	0
$498$	0	0
$499$	−14.0000	−0.626726	−0.313363	−	0.949633i	$-0.601456\pi$
$499$	−14.0000	−0.626726	−0.313363	+	0.949633i	$0.601456\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	18.0000i	0.802580i	0.915951	+	0.401290i	$0.131438\pi$
$503$	18.0000i	0.802580i	−0.915951	+	0.401290i	$0.868562\pi$
$504$	0	0
$505$	0	0
$506$	18.0000	0.800198
$507$	0	0
$508$	− 7.00000i	− 0.310575i
$509$	−15.0000	−0.664863	−0.332432	−	0.943127i	$-0.607869\pi$
$509$	−15.0000	−0.664863	−0.332432	+	0.943127i	$0.607869\pi$
$510$	0	0
$511$	7.00000	0.309662
$512$	− 1.00000i	− 0.0441942i
$513$	0	0
$514$	12.0000	0.529297
$515$	0	0
$516$	0	0
$517$	18.0000i	0.791639i
$518$	− 2.00000i	− 0.0878750i
$519$	0	0
$520$	0	0
$521$	−36.0000	−1.57719	−0.788594	−	0.614914i	$-0.789191\pi$
$521$	−36.0000	−1.57719	−0.788594	+	0.614914i	$0.789191\pi$
$522$	0	0
$523$	− 16.0000i	− 0.699631i	−0.936819	−	0.349816i	$-0.886244\pi$
$523$	− 16.0000i	− 0.699631i	0.936819	−	0.349816i	$-0.113756\pi$
$524$	−15.0000	−0.655278
$525$	0	0
$526$	30.0000	1.30806
$527$	0	0
$528$	0	0
$529$	−13.0000	−0.565217
$530$	0	0
$531$	0	0
$532$	2.00000i	0.0867110i
$533$	− 24.0000i	− 1.03956i
$534$	0	0
$535$	0	0
$536$	14.0000	0.604708
$537$	0	0
$538$	18.0000i	0.776035i
$539$	18.0000	0.775315
$540$	0	0
$541$	20.0000	0.859867	0.429934	−	0.902861i	$-0.358537\pi$
$541$	20.0000	0.859867	0.429934	+	0.902861i	$0.358537\pi$
$542$	25.0000i	1.07384i
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 8.00000i	− 0.342055i	−0.985266	−	0.171028i	$-0.945291\pi$
$547$	− 8.00000i	− 0.342055i	0.985266	−	0.171028i	$-0.0547087\pi$
$548$	− 6.00000i	− 0.256307i
$549$	0	0
$550$	0	0
$551$	−12.0000	−0.511217
$552$	0	0
$553$	− 8.00000i	− 0.340195i
$554$	−8.00000	−0.339887
$555$	0	0
$556$	−4.00000	−0.169638
$557$	27.0000i	1.14403i	0.820244	+	0.572013i	$0.193837\pi$
$557$	27.0000i	1.14403i	−0.820244	+	0.572013i	$0.806163\pi$
$558$	0	0
$559$	−40.0000	−1.69182
$560$	0	0
$561$	0	0
$562$	24.0000i	1.01238i
$563$	− 3.00000i	− 0.126435i	−0.998000	−	0.0632175i	$-0.979864\pi$
$563$	− 3.00000i	− 0.126435i	0.998000	−	0.0632175i	$-0.0201362\pi$
$564$	0	0
$565$	0	0
$566$	14.0000	0.588464
$567$	0	0
$568$	0	0
$569$	−12.0000	−0.503066	−0.251533	−	0.967849i	$-0.580935\pi$
$569$	−12.0000	−0.503066	−0.251533	+	0.967849i	$0.580935\pi$
$570$	0	0
$571$	−4.00000	−0.167395	−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000	−0.167395	−0.0836974	+	0.996491i	$0.526673\pi$
$572$	12.0000i	0.501745i
$573$	0	0
$574$	6.00000	0.250435
$575$	0	0
$576$	0	0
$577$	− 38.0000i	− 1.58196i	−0.611842	−	0.790980i	$-0.709571\pi$
$577$	− 38.0000i	− 1.58196i	0.611842	−	0.790980i	$-0.290429\pi$
$578$	− 17.0000i	− 0.707107i
$579$	0	0
$580$	0	0
$581$	−3.00000	−0.124461
$582$	0	0
$583$	− 27.0000i	− 1.11823i
$584$	7.00000	0.289662
$585$	0	0
$586$	6.00000	0.247858
$587$	− 3.00000i	− 0.123823i	−0.998082	−	0.0619116i	$-0.980280\pi$
$587$	− 3.00000i	− 0.123823i	0.998082	−	0.0619116i	$-0.0197197\pi$
$588$	0	0
$589$	−10.0000	−0.412043
$590$	0	0
$591$	0	0
$592$	− 2.00000i	− 0.0821995i
$593$	− 18.0000i	− 0.739171i	−0.929197	−	0.369586i	$-0.879500\pi$
$593$	− 18.0000i	− 0.739171i	0.929197	−	0.369586i	$-0.120500\pi$
$594$	0	0
$595$	0	0
$596$	−3.00000	−0.122885
$597$	0	0
$598$	− 24.0000i	− 0.981433i
$599$	−42.0000	−1.71607	−0.858037	−	0.513588i	$-0.828316\pi$
$599$	−42.0000	−1.71607	−0.858037	+	0.513588i	$0.828316\pi$
$600$	0	0
$601$	35.0000	1.42768	0.713840	−	0.700309i	$-0.246954\pi$
$601$	35.0000	1.42768	0.713840	+	0.700309i	$0.246954\pi$
$602$	− 10.0000i	− 0.407570i
$603$	0	0
$604$	−17.0000	−0.691720
$605$	0	0
$606$	0	0
$607$	− 32.0000i	− 1.29884i	−0.760430	−	0.649420i	$-0.775012\pi$
$607$	− 32.0000i	− 1.29884i	0.760430	−	0.649420i	$-0.224988\pi$
$608$	2.00000i	0.0811107i
$609$	0	0
$610$	0	0
$611$	24.0000	0.970936
$612$	0	0
$613$	− 34.0000i	− 1.37325i	−0.727013	−	0.686624i	$-0.759092\pi$
$613$	− 34.0000i	− 1.37325i	0.727013	−	0.686624i	$-0.240908\pi$
$614$	16.0000	0.645707
$615$	0	0
$616$	−3.00000	−0.120873
$617$	− 42.0000i	− 1.69086i	−0.534089	−	0.845428i	$-0.679345\pi$
$617$	− 42.0000i	− 1.69086i	0.534089	−	0.845428i	$-0.320655\pi$
$618$	0	0
$619$	28.0000	1.12542	0.562708	−	0.826656i	$-0.309760\pi$
$619$	28.0000	1.12542	0.562708	+	0.826656i	$0.309760\pi$
$620$	0	0
$621$	0	0
$622$	− 6.00000i	− 0.240578i
$623$	− 18.0000i	− 0.721155i
$624$	0	0
$625$	0	0
$626$	−19.0000	−0.759393
$627$	0	0
$628$	− 4.00000i	− 0.159617i
$629$	0	0
$630$	0	0
$631$	−25.0000	−0.995234	−0.497617	−	0.867397i	$-0.665792\pi$
$631$	−25.0000	−0.995234	−0.497617	+	0.867397i	$0.665792\pi$
$632$	− 8.00000i	− 0.318223i
$633$	0	0
$634$	−3.00000	−0.119145
$635$	0	0
$636$	0	0
$637$	− 24.0000i	− 0.950915i
$638$	− 18.0000i	− 0.712627i
$639$	0	0
$640$	0	0
$641$	−42.0000	−1.65890	−0.829450	−	0.558581i	$-0.811346\pi$
$641$	−42.0000	−1.65890	−0.829450	+	0.558581i	$0.811346\pi$
$642$	0	0
$643$	− 4.00000i	− 0.157745i	−0.996885	−	0.0788723i	$-0.974868\pi$
$643$	− 4.00000i	− 0.157745i	0.996885	−	0.0788723i	$-0.0251319\pi$
$644$	6.00000	0.236433
$645$	0	0
$646$	0	0
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	0	0
$649$	36.0000	1.41312
$650$	0	0
$651$	0	0
$652$	− 20.0000i	− 0.783260i
$653$	− 39.0000i	− 1.52619i	−0.646288	−	0.763094i	$-0.723679\pi$
$653$	− 39.0000i	− 1.52619i	0.646288	−	0.763094i	$-0.276321\pi$
$654$	0	0
$655$	0	0
$656$	6.00000	0.234261
$657$	0	0
$658$	6.00000i	0.233904i
$659$	−21.0000	−0.818044	−0.409022	−	0.912525i	$-0.634130\pi$
$659$	−21.0000	−0.818044	−0.409022	+	0.912525i	$0.634130\pi$
$660$	0	0
$661$	14.0000	0.544537	0.272268	−	0.962221i	$-0.412226\pi$
$661$	14.0000	0.544537	0.272268	+	0.962221i	$0.412226\pi$
$662$	10.0000i	0.388661i
$663$	0	0
$664$	−3.00000	−0.116423
$665$	0	0
$666$	0	0
$667$	36.0000i	1.39393i
$668$	6.00000i	0.232147i
$669$	0	0
$670$	0	0
$671$	24.0000	0.926510
$672$	0	0
$673$	− 19.0000i	− 0.732396i	−0.930537	−	0.366198i	$-0.880659\pi$
$673$	− 19.0000i	− 0.732396i	0.930537	−	0.366198i	$-0.119341\pi$
$674$	22.0000	0.847408
$675$	0	0
$676$	3.00000	0.115385
$677$	42.0000i	1.61419i	0.590421	+	0.807096i	$0.298962\pi$
$677$	42.0000i	1.61419i	−0.590421	+	0.807096i	$0.701038\pi$
$678$	0	0
$679$	−1.00000	−0.0383765
$680$	0	0
$681$	0	0
$682$	− 15.0000i	− 0.574380i
$683$	36.0000i	1.37750i	0.724998	+	0.688751i	$0.241841\pi$
$683$	36.0000i	1.37750i	−0.724998	+	0.688751i	$0.758159\pi$
$684$	0	0
$685$	0	0
$686$	13.0000	0.496342
$687$	0	0
$688$	− 10.0000i	− 0.381246i
$689$	−36.0000	−1.37149
$690$	0	0
$691$	44.0000	1.67384	0.836919	−	0.547326i	$-0.184354\pi$
$691$	44.0000	1.67384	0.836919	+	0.547326i	$0.184354\pi$
$692$	15.0000i	0.570214i
$693$	0	0
$694$	3.00000	0.113878
$695$	0	0
$696$	0	0
$697$	0	0
$698$	− 10.0000i	− 0.378506i
$699$	0	0
$700$	0	0
$701$	−9.00000	−0.339925	−0.169963	−	0.985451i	$-0.554365\pi$
$701$	−9.00000	−0.339925	−0.169963	+	0.985451i	$0.554365\pi$
$702$	0	0
$703$	4.00000i	0.150863i
$704$	−3.00000	−0.113067
$705$	0	0
$706$	−6.00000	−0.225813
$707$	3.00000i	0.112827i
$708$	0	0
$709$	−44.0000	−1.65245	−0.826227	−	0.563337i	$-0.809517\pi$
$709$	−44.0000	−1.65245	−0.826227	+	0.563337i	$0.809517\pi$
$710$	0	0
$711$	0	0
$712$	− 18.0000i	− 0.674579i
$713$	30.0000i	1.12351i
$714$	0	0
$715$	0	0
$716$	9.00000	0.336346
$717$	0	0
$718$	− 18.0000i	− 0.671754i
$719$	36.0000	1.34257	0.671287	−	0.741198i	$-0.265742\pi$
$719$	36.0000	1.34257	0.671287	+	0.741198i	$0.265742\pi$
$720$	0	0
$721$	4.00000	0.148968
$722$	15.0000i	0.558242i
$723$	0	0
$724$	16.0000	0.594635
$725$	0	0
$726$	0	0
$727$	1.00000i	0.0370879i	0.999828	+	0.0185440i	$0.00590307\pi$
$727$	1.00000i	0.0370879i	−0.999828	+	0.0185440i	$0.994097\pi$
$728$	4.00000i	0.148250i
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	− 22.0000i	− 0.812589i	−0.913742	−	0.406294i	$-0.866821\pi$
$733$	− 22.0000i	− 0.812589i	0.913742	−	0.406294i	$-0.133179\pi$
$734$	−17.0000	−0.627481
$735$	0	0
$736$	6.00000	0.221163
$737$	− 42.0000i	− 1.54709i
$738$	0	0
$739$	16.0000	0.588570	0.294285	−	0.955718i	$-0.404919\pi$
$739$	16.0000	0.588570	0.294285	+	0.955718i	$0.404919\pi$
$740$	0	0
$741$	0	0
$742$	− 9.00000i	− 0.330400i
$743$	− 12.0000i	− 0.440237i	−0.975473	−	0.220119i	$-0.929356\pi$
$743$	− 12.0000i	− 0.440237i	0.975473	−	0.220119i	$-0.0706445\pi$
$744$	0	0
$745$	0	0
$746$	32.0000	1.17160
$747$	0	0
$748$	0	0
$749$	−9.00000	−0.328853
$750$	0	0
$751$	41.0000	1.49611	0.748056	−	0.663636i	$-0.230988\pi$
$751$	41.0000	1.49611	0.748056	+	0.663636i	$0.230988\pi$
$752$	6.00000i	0.218797i
$753$	0	0
$754$	−24.0000	−0.874028
$755$	0	0
$756$	0	0
$757$	− 2.00000i	− 0.0726912i	−0.999339	−	0.0363456i	$-0.988428\pi$
$757$	− 2.00000i	− 0.0726912i	0.999339	−	0.0363456i	$-0.0115717\pi$
$758$	20.0000i	0.726433i
$759$	0	0
$760$	0	0
$761$	−48.0000	−1.74000	−0.869999	−	0.493053i	$-0.835881\pi$
$761$	−48.0000	−1.74000	−0.869999	+	0.493053i	$0.835881\pi$
$762$	0	0
$763$	− 2.00000i	− 0.0724049i
$764$	−12.0000	−0.434145
$765$	0	0
$766$	24.0000	0.867155
$767$	− 48.0000i	− 1.73318i
$768$	0	0
$769$	31.0000	1.11789	0.558944	−	0.829205i	$-0.311207\pi$
$769$	31.0000	1.11789	0.558944	+	0.829205i	$0.311207\pi$
$770$	0	0
$771$	0	0
$772$	− 5.00000i	− 0.179954i
$773$	18.0000i	0.647415i	0.946157	+	0.323708i	$0.104929\pi$
$773$	18.0000i	0.647415i	−0.946157	+	0.323708i	$0.895071\pi$
$774$	0	0
$775$	0	0
$776$	−1.00000	−0.0358979
$777$	0	0
$778$	21.0000i	0.752886i
$779$	−12.0000	−0.429945
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	6.00000	0.214286
$785$	0	0
$786$	0	0
$787$	− 32.0000i	− 1.14068i	−0.821410	−	0.570338i	$-0.806812\pi$
$787$	− 32.0000i	− 1.14068i	0.821410	−	0.570338i	$-0.193188\pi$
$788$	− 9.00000i	− 0.320612i
$789$	0	0
$790$	0	0
$791$	−6.00000	−0.213335
$792$	0	0
$793$	− 32.0000i	− 1.13635i
$794$	−20.0000	−0.709773
$795$	0	0
$796$	−7.00000	−0.248108
$797$	39.0000i	1.38145i	0.723117	+	0.690725i	$0.242709\pi$
$797$	39.0000i	1.38145i	−0.723117	+	0.690725i	$0.757291\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	12.0000i	0.423735i
$803$	− 21.0000i	− 0.741074i
$804$	0	0
$805$	0	0
$806$	−20.0000	−0.704470
$807$	0	0
$808$	3.00000i	0.105540i
$809$	18.0000	0.632846	0.316423	−	0.948618i	$-0.397518\pi$
$809$	18.0000	0.632846	0.316423	+	0.948618i	$0.397518\pi$
$810$	0	0
$811$	2.00000	0.0702295	0.0351147	−	0.999383i	$-0.488820\pi$
$811$	2.00000	0.0702295	0.0351147	+	0.999383i	$0.488820\pi$
$812$	− 6.00000i	− 0.210559i
$813$	0	0
$814$	−6.00000	−0.210300
$815$	0	0
$816$	0	0
$817$	20.0000i	0.699711i
$818$	23.0000i	0.804176i
$819$	0	0
$820$	0	0
$821$	−6.00000	−0.209401	−0.104701	−	0.994504i	$-0.533388\pi$
$821$	−6.00000	−0.209401	−0.104701	+	0.994504i	$0.533388\pi$
$822$	0	0
$823$	− 31.0000i	− 1.08059i	−0.841475	−	0.540296i	$-0.818312\pi$
$823$	− 31.0000i	− 1.08059i	0.841475	−	0.540296i	$-0.181688\pi$
$824$	4.00000	0.139347
$825$	0	0
$826$	12.0000	0.417533
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	0	0
$829$	−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
$831$	0	0
$832$	4.00000i	0.138675i
$833$	0	0
$834$	0	0
$835$	0	0
$836$	6.00000	0.207514
$837$	0	0
$838$	− 12.0000i	− 0.414533i
$839$	24.0000	0.828572	0.414286	−	0.910147i	$-0.364031\pi$
$839$	24.0000	0.828572	0.414286	+	0.910147i	$0.364031\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	− 8.00000i	− 0.275698i
$843$	0	0
$844$	22.0000	0.757271
$845$	0	0
$846$	0	0
$847$	− 2.00000i	− 0.0687208i
$848$	− 9.00000i	− 0.309061i
$849$	0	0
$850$	0	0
$851$	12.0000	0.411355
$852$	0	0
$853$	− 10.0000i	− 0.342393i	−0.985237	−	0.171197i	$-0.945237\pi$
$853$	− 10.0000i	− 0.342393i	0.985237	−	0.171197i	$-0.0547634\pi$
$854$	8.00000	0.273754
$855$	0	0
$856$	−9.00000	−0.307614
$857$	− 48.0000i	− 1.63965i	−0.572615	−	0.819824i	$-0.694071\pi$
$857$	− 48.0000i	− 1.63965i	0.572615	−	0.819824i	$-0.305929\pi$
$858$	0	0
$859$	40.0000	1.36478	0.682391	−	0.730987i	$-0.260940\pi$
$859$	40.0000	1.36478	0.682391	+	0.730987i	$0.260940\pi$
$860$	0	0
$861$	0	0
$862$	18.0000i	0.613082i
$863$	36.0000i	1.22545i	0.790295	+	0.612727i	$0.209928\pi$
$863$	36.0000i	1.22545i	−0.790295	+	0.612727i	$0.790072\pi$
$864$	0	0
$865$	0	0
$866$	29.0000	0.985460
$867$	0	0
$868$	− 5.00000i	− 0.169711i
$869$	−24.0000	−0.814144
$870$	0	0
$871$	−56.0000	−1.89749
$872$	− 2.00000i	− 0.0677285i
$873$	0	0
$874$	−12.0000	−0.405906
$875$	0	0
$876$	0	0
$877$	22.0000i	0.742887i	0.928456	+	0.371444i	$0.121137\pi$
$877$	22.0000i	0.742887i	−0.928456	+	0.371444i	$0.878863\pi$
$878$	− 19.0000i	− 0.641219i
$879$	0	0
$880$	0	0
$881$	36.0000	1.21287	0.606435	−	0.795133i	$-0.292599\pi$
$881$	36.0000	1.21287	0.606435	+	0.795133i	$0.292599\pi$
$882$	0	0
$883$	2.00000i	0.0673054i	0.999434	+	0.0336527i	$0.0107140\pi$
$883$	2.00000i	0.0673054i	−0.999434	+	0.0336527i	$0.989286\pi$
$884$	0	0
$885$	0	0
$886$	12.0000	0.403148
$887$	12.0000i	0.402921i	0.979497	+	0.201460i	$0.0645687\pi$
$887$	12.0000i	0.402921i	−0.979497	+	0.201460i	$0.935431\pi$
$888$	0	0
$889$	−7.00000	−0.234772
$890$	0	0
$891$	0	0
$892$	− 8.00000i	− 0.267860i
$893$	− 12.0000i	− 0.401565i
$894$	0	0
$895$	0	0
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	18.0000i	0.600668i
$899$	30.0000	1.00056
$900$	0	0
$901$	0	0
$902$	− 18.0000i	− 0.599334i
$903$	0	0
$904$	−6.00000	−0.199557
$905$	0	0
$906$	0	0
$907$	− 26.0000i	− 0.863316i	−0.902037	−	0.431658i	$-0.857929\pi$
$907$	− 26.0000i	− 0.863316i	0.902037	−	0.431658i	$-0.142071\pi$
$908$	12.0000i	0.398234i
$909$	0	0
$910$	0	0
$911$	−6.00000	−0.198789	−0.0993944	−	0.995048i	$-0.531691\pi$
$911$	−6.00000	−0.198789	−0.0993944	+	0.995048i	$0.531691\pi$
$912$	0	0
$913$	9.00000i	0.297857i
$914$	1.00000	0.0330771
$915$	0	0
$916$	14.0000	0.462573
$917$	15.0000i	0.495344i
$918$	0	0
$919$	−29.0000	−0.956622	−0.478311	−	0.878191i	$-0.658751\pi$
$919$	−29.0000	−0.956622	−0.478311	+	0.878191i	$0.658751\pi$
$920$	0	0
$921$	0	0
$922$	− 21.0000i	− 0.691598i
$923$	0	0
$924$	0	0
$925$	0	0
$926$	−13.0000	−0.427207
$927$	0	0
$928$	− 6.00000i	− 0.196960i
$929$	−12.0000	−0.393707	−0.196854	−	0.980433i	$-0.563072\pi$
$929$	−12.0000	−0.393707	−0.196854	+	0.980433i	$0.563072\pi$
$930$	0	0
$931$	−12.0000	−0.393284
$932$	18.0000i	0.589610i
$933$	0	0
$934$	−27.0000	−0.883467
$935$	0	0
$936$	0	0
$937$	− 11.0000i	− 0.359354i	−0.983726	−	0.179677i	$-0.942495\pi$
$937$	− 11.0000i	− 0.359354i	0.983726	−	0.179677i	$-0.0575053\pi$
$938$	− 14.0000i	− 0.457116i
$939$	0	0
$940$	0	0
$941$	33.0000	1.07577	0.537885	−	0.843018i	$-0.319224\pi$
$941$	33.0000	1.07577	0.537885	+	0.843018i	$0.319224\pi$
$942$	0	0
$943$	36.0000i	1.17232i
$944$	12.0000	0.390567
$945$	0	0
$946$	−30.0000	−0.975384
$947$	51.0000i	1.65728i	0.559784	+	0.828639i	$0.310884\pi$
$947$	51.0000i	1.65728i	−0.559784	+	0.828639i	$0.689116\pi$
$948$	0	0
$949$	−28.0000	−0.908918
$950$	0	0
$951$	0	0
$952$	0	0
$953$	36.0000i	1.16615i	0.812417	+	0.583077i	$0.198151\pi$
$953$	36.0000i	1.16615i	−0.812417	+	0.583077i	$0.801849\pi$
$954$	0	0
$955$	0	0
$956$	−30.0000	−0.970269
$957$	0	0
$958$	− 6.00000i	− 0.193851i
$959$	−6.00000	−0.193750
$960$	0	0
$961$	−6.00000	−0.193548
$962$	8.00000i	0.257930i
$963$	0	0
$964$	10.0000	0.322078
$965$	0	0
$966$	0	0
$967$	− 23.0000i	− 0.739630i	−0.929105	−	0.369815i	$-0.879421\pi$
$967$	− 23.0000i	− 0.739630i	0.929105	−	0.369815i	$-0.120579\pi$
$968$	− 2.00000i	− 0.0642824i
$969$	0	0
$970$	0	0
$971$	27.0000	0.866471	0.433236	−	0.901281i	$-0.357372\pi$
$971$	27.0000	0.866471	0.433236	+	0.901281i	$0.357372\pi$
$972$	0	0
$973$	4.00000i	0.128234i
$974$	16.0000	0.512673
$975$	0	0
$976$	8.00000	0.256074
$977$	− 42.0000i	− 1.34370i	−0.740688	−	0.671850i	$-0.765500\pi$
$977$	− 42.0000i	− 1.34370i	0.740688	−	0.671850i	$-0.234500\pi$
$978$	0	0
$979$	−54.0000	−1.72585
$980$	0	0
$981$	0	0
$982$	39.0000i	1.24454i
$983$	− 6.00000i	− 0.191370i	−0.995412	−	0.0956851i	$-0.969496\pi$
$983$	− 6.00000i	− 0.191370i	0.995412	−	0.0956851i	$-0.0305042\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	− 8.00000i	− 0.254514i
$989$	60.0000	1.90789
$990$	0	0
$991$	47.0000	1.49300	0.746502	−	0.665383i	$-0.231732\pi$
$991$	47.0000	1.49300	0.746502	+	0.665383i	$0.231732\pi$
$992$	− 5.00000i	− 0.158750i
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	28.0000i	0.886769i	0.896332	+	0.443384i	$0.146222\pi$
$997$	28.0000i	0.886769i	−0.896332	+	0.443384i	$0.853778\pi$
$998$	14.0000i	0.443162i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1350.2.c.k.649.1		2
3.2	odd	2		1350.2.c.b.649.2		2
5.2	odd	4		54.2.a.b.1.1	yes	1
5.3	odd	4		1350.2.a.h.1.1		1
5.4	even	2	inner	1350.2.c.k.649.2		2
15.2	even	4		54.2.a.a.1.1	✓	1
15.8	even	4		1350.2.a.r.1.1		1
15.14	odd	2		1350.2.c.b.649.1		2
20.7	even	4		432.2.a.b.1.1		1
35.27	even	4		2646.2.a.bd.1.1		1
40.27	even	4		1728.2.a.z.1.1		1
40.37	odd	4		1728.2.a.y.1.1		1
45.2	even	12		162.2.c.c.109.1		2
45.7	odd	12		162.2.c.b.109.1		2
45.22	odd	12		162.2.c.b.55.1		2
45.32	even	12		162.2.c.c.55.1		2
55.32	even	4		6534.2.a.b.1.1		1
60.47	odd	4		432.2.a.g.1.1		1
65.12	odd	4		9126.2.a.r.1.1		1
105.62	odd	4		2646.2.a.a.1.1		1
120.77	even	4		1728.2.a.c.1.1		1
120.107	odd	4		1728.2.a.d.1.1		1
165.32	odd	4		6534.2.a.bc.1.1		1
180.7	even	12		1296.2.i.o.433.1		2
180.47	odd	12		1296.2.i.c.433.1		2
180.67	even	12		1296.2.i.o.865.1		2
180.167	odd	12		1296.2.i.c.865.1		2
195.77	even	4		9126.2.a.u.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
54.2.a.a.1.1	✓	1	15.2	even	4
54.2.a.b.1.1	yes	1	5.2	odd	4
162.2.c.b.55.1		2	45.22	odd	12
162.2.c.b.109.1		2	45.7	odd	12
162.2.c.c.55.1		2	45.32	even	12
162.2.c.c.109.1		2	45.2	even	12
432.2.a.b.1.1		1	20.7	even	4
432.2.a.g.1.1		1	60.47	odd	4
1296.2.i.c.433.1		2	180.47	odd	12
1296.2.i.c.865.1		2	180.167	odd	12
1296.2.i.o.433.1		2	180.7	even	12
1296.2.i.o.865.1		2	180.67	even	12
1350.2.a.h.1.1		1	5.3	odd	4
1350.2.a.r.1.1		1	15.8	even	4
1350.2.c.b.649.1		2	15.14	odd	2
1350.2.c.b.649.2		2	3.2	odd	2
1350.2.c.k.649.1		2	1.1	even	1	trivial
1350.2.c.k.649.2		2	5.4	even	2	inner
1728.2.a.c.1.1		1	120.77	even	4
1728.2.a.d.1.1		1	120.107	odd	4
1728.2.a.y.1.1		1	40.37	odd	4
1728.2.a.z.1.1		1	40.27	even	4
2646.2.a.a.1.1		1	105.62	odd	4
2646.2.a.bd.1.1		1	35.27	even	4
6534.2.a.b.1.1		1	55.32	even	4
6534.2.a.bc.1.1		1	165.32	odd	4
9126.2.a.r.1.1		1	65.12	odd	4
9126.2.a.u.1.1		1	195.77	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1350 = 2 \cdot 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1350.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{7} +1.00000i q^{8} +3.00000 q^{11} -4.00000i q^{13} +1.00000 q^{14} +1.00000 q^{16} -2.00000 q^{19} -3.00000i q^{22} +6.00000i q^{23} -4.00000 q^{26} -1.00000i q^{28} +6.00000 q^{29} +5.00000 q^{31} -1.00000i q^{32} -2.00000i q^{37} +2.00000i q^{38} +6.00000 q^{41} -10.0000i q^{43} -3.00000 q^{44} +6.00000 q^{46} +6.00000i q^{47} +6.00000 q^{49} +4.00000i q^{52} -9.00000i q^{53} -1.00000 q^{56} -6.00000i q^{58} +12.0000 q^{59} +8.00000 q^{61} -5.00000i q^{62} -1.00000 q^{64} -14.0000i q^{67} -7.00000i q^{73} -2.00000 q^{74} +2.00000 q^{76} +3.00000i q^{77} -8.00000 q^{79} -6.00000i q^{82} +3.00000i q^{83} -10.0000 q^{86} +3.00000i q^{88} -18.0000 q^{89} +4.00000 q^{91} -6.00000i q^{92} +6.00000 q^{94} +1.00000i q^{97} -6.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} + 6 q^{11} + 2 q^{14} + 2 q^{16} - 4 q^{19} - 8 q^{26} + 12 q^{29} + 10 q^{31} + 12 q^{41} - 6 q^{44} + 12 q^{46} + 12 q^{49} - 2 q^{56} + 24 q^{59} + 16 q^{61} - 2 q^{64} - 4 q^{74} + 4 q^{76}+ \cdots + 12 q^{94}+O(q^{100}) \) 2 * q - 2 * q^4 + 6 * q^11 + 2 * q^14 + 2 * q^16 - 4 * q^19 - 8 * q^26 + 12 * q^29 + 10 * q^31 + 12 * q^41 - 6 * q^44 + 12 * q^46 + 12 * q^49 - 2 * q^56 + 24 * q^59 + 16 * q^61 - 2 * q^64 - 4 * q^74 + 4 * q^76 - 16 * q^79 - 20 * q^86 - 36 * q^89 + 8 * q^91 + 12 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more