LMFDB - Embedded newform 1350.2.c.h.649.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1350,2,Mod(649,1350)]

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

lf = mfeigenbasis(mf)

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")

//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);

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

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:	yes
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.h.649.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{4} -4.00000i q^{7} +1.00000i q^{8} +O(q^{10})$	$q-1.00000i q^{2} -1.00000 q^{4} -4.00000i q^{7} +1.00000i q^{8} +3.00000 q^{11} +1.00000i q^{13} -4.00000 q^{14} +1.00000 q^{16} -2.00000 q^{19} -3.00000i q^{22} -9.00000i q^{23} +1.00000 q^{26} +4.00000i q^{28} +6.00000 q^{29} -10.0000 q^{31} -1.00000i q^{32} -7.00000i q^{37} +2.00000i q^{38} +6.00000 q^{41} +10.0000i q^{43} -3.00000 q^{44} -9.00000 q^{46} -9.00000i q^{47} -9.00000 q^{49} -1.00000i q^{52} +6.00000i q^{53} +4.00000 q^{56} -6.00000i q^{58} -3.00000 q^{59} -7.00000 q^{61} +10.0000i q^{62} -1.00000 q^{64} -4.00000i q^{67} -15.0000 q^{71} -2.00000i q^{73} -7.00000 q^{74} +2.00000 q^{76} -12.0000i q^{77} -8.00000 q^{79} -6.00000i q^{82} -12.0000i q^{83} +10.0000 q^{86} +3.00000i q^{88} +12.0000 q^{89} +4.00000 q^{91} +9.00000i q^{92} -9.00000 q^{94} -19.0000i q^{97} +9.00000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4}+O(q^{10}) $ 2 * q - 2 * q^4	$ 2 q - 2 q^{4} + 6 q^{11} - 8 q^{14} + 2 q^{16} - 4 q^{19} + 2 q^{26} + 12 q^{29} - 20 q^{31} + 12 q^{41} - 6 q^{44} - 18 q^{46} - 18 q^{49} + 8 q^{56} - 6 q^{59} - 14 q^{61} - 2 q^{64} - 30 q^{71} - 14 q^{74} + 4 q^{76} - 16 q^{79} + 20 q^{86} + 24 q^{89} + 8 q^{91} - 18 q^{94}+O(q^{100}) $ 2 * q - 2 * q^4 + 6 * q^11 - 8 * q^14 + 2 * q^16 - 4 * q^19 + 2 * q^26 + 12 * q^29 - 20 * q^31 + 12 * q^41 - 6 * q^44 - 18 * q^46 - 18 * q^49 + 8 * q^56 - 6 * q^59 - 14 * q^61 - 2 * q^64 - 30 * q^71 - 14 * q^74 + 4 * q^76 - 16 * q^79 + 20 * q^86 + 24 * q^89 + 8 * q^91 - 18 * q^94

Display 100 coefficients 10 coefficients

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
$7$	− 4.00000i	− 1.51186i	−0.654654	−	0.755929i	$-0.727186\pi$
$7$	− 4.00000i	− 1.51186i	0.654654	−	0.755929i	$-0.272814\pi$
$8$	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
$13$	1.00000i	0.277350i	0.990338	+	0.138675i	$0.0442844\pi$
$13$	1.00000i	0.277350i	−0.990338	+	0.138675i	$0.955716\pi$
$14$	−4.00000	−1.06904
$15$	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
$23$	− 9.00000i	− 1.87663i	−0.345782	−	0.938315i	$-0.612386\pi$
$23$	− 9.00000i	− 1.87663i	0.345782	−	0.938315i	$-0.387614\pi$
$24$	0	0
$25$	0	0
$26$	1.00000	0.196116
$27$	0	0
$28$	4.00000i	0.755929i
$29$	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
$31$	−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
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 7.00000i	− 1.15079i	−0.817875	−	0.575396i	$-0.804848\pi$
$37$	− 7.00000i	− 1.15079i	0.817875	−	0.575396i	$-0.195152\pi$
$38$	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	$0.276025\pi$
$43$	10.0000i	1.52499i	−0.646997	+	0.762493i	$0.723975\pi$
$44$	−3.00000	−0.452267
$45$	0	0
$46$	−9.00000	−1.32698
$47$	− 9.00000i	− 1.31278i	−0.754420	−	0.656392i	$-0.772082\pi$
$47$	− 9.00000i	− 1.31278i	0.754420	−	0.656392i	$-0.227918\pi$
$48$	0	0
$49$	−9.00000	−1.28571
$50$	0	0
$51$	0	0
$52$	− 1.00000i	− 0.138675i
$53$	6.00000i	0.824163i	0.911147	+	0.412082i	$0.135198\pi$
$53$	6.00000i	0.824163i	−0.911147	+	0.412082i	$0.864802\pi$
$54$	0	0
$55$	0	0
$56$	4.00000	0.534522
$57$	0	0
$58$	− 6.00000i	− 0.787839i
$59$	−3.00000	−0.390567	−0.195283	−	0.980747i	$-0.562563\pi$
$59$	−3.00000	−0.390567	−0.195283	+	0.980747i	$0.562563\pi$
$60$	0	0
$61$	−7.00000	−0.896258	−0.448129	−	0.893969i	$-0.647910\pi$
$61$	−7.00000	−0.896258	−0.448129	+	0.893969i	$0.647910\pi$
$62$	10.0000i	1.27000i
$63$	0	0
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	− 4.00000i	− 0.488678i	−0.969690	−	0.244339i	$-0.921429\pi$
$67$	− 4.00000i	− 0.488678i	0.969690	−	0.244339i	$-0.0785709\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−15.0000	−1.78017	−0.890086	−	0.455792i	$-0.849356\pi$
$71$	−15.0000	−1.78017	−0.890086	+	0.455792i	$0.849356\pi$
$72$	0	0
$73$	− 2.00000i	− 0.234082i	−0.993127	−	0.117041i	$-0.962659\pi$
$73$	− 2.00000i	− 0.234082i	0.993127	−	0.117041i	$-0.0373409\pi$
$74$	−7.00000	−0.813733
$75$	0	0
$76$	2.00000	0.229416
$77$	− 12.0000i	− 1.36753i
$78$	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
$83$	− 12.0000i	− 1.31717i	−0.752506	−	0.658586i	$-0.771155\pi$
$83$	− 12.0000i	− 1.31717i	0.752506	−	0.658586i	$-0.228845\pi$
$84$	0	0
$85$	0	0
$86$	10.0000	1.07833
$87$	0	0
$88$	3.00000i	0.319801i
$89$	12.0000	1.27200	0.635999	−	0.771690i	$-0.280588\pi$
$89$	12.0000	1.27200	0.635999	+	0.771690i	$0.280588\pi$
$90$	0	0
$91$	4.00000	0.419314
$92$	9.00000i	0.938315i
$93$	0	0
$94$	−9.00000	−0.928279
$95$	0	0
$96$	0	0
$97$	− 19.0000i	− 1.92916i	−0.263795	−	0.964579i	$-0.584974\pi$
$97$	− 19.0000i	− 1.92916i	0.263795	−	0.964579i	$-0.415026\pi$
$98$	9.00000i	0.909137i
$99$	0	0
$100$	0	0
$101$	−12.0000	−1.19404	−0.597022	−	0.802225i	$-0.703650\pi$
$101$	−12.0000	−1.19404	−0.597022	+	0.802225i	$0.703650\pi$
$102$	0	0
$103$	− 14.0000i	− 1.37946i	−0.724066	−	0.689730i	$-0.757729\pi$
$103$	− 14.0000i	− 1.37946i	0.724066	−	0.689730i	$-0.242271\pi$
$104$	−1.00000	−0.0980581
$105$	0	0
$106$	6.00000	0.582772
$107$	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
$112$	− 4.00000i	− 0.377964i
$113$	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
$118$	3.00000i	0.276172i
$119$	0	0
$120$	0	0
$121$	−2.00000	−0.181818
$122$	7.00000i	0.633750i
$123$	0	0
$124$	10.0000	0.898027
$125$	0	0
$126$	0	0
$127$	2.00000i	0.177471i	0.996055	+	0.0887357i	$0.0282826\pi$
$127$	2.00000i	0.177471i	−0.996055	+	0.0887357i	$0.971717\pi$
$128$	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
$133$	8.00000i	0.693688i
$134$	−4.00000	−0.345547
$135$	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
$142$	15.0000i	1.25877i
$143$	3.00000i	0.250873i
$144$	0	0
$145$	0	0
$146$	−2.00000	−0.165521
$147$	0	0
$148$	7.00000i	0.575396i
$149$	18.0000	1.47462	0.737309	−	0.675556i	$-0.236096\pi$
$149$	18.0000	1.47462	0.737309	+	0.675556i	$0.236096\pi$
$150$	0	0
$151$	2.00000	0.162758	0.0813788	−	0.996683i	$-0.474068\pi$
$151$	2.00000	0.162758	0.0813788	+	0.996683i	$0.474068\pi$
$152$	− 2.00000i	− 0.162221i
$153$	0	0
$154$	−12.0000	−0.966988
$155$	0	0
$156$	0	0
$157$	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
$161$	−36.0000	−2.83720
$162$	0	0
$163$	10.0000i	0.783260i	0.920123	+	0.391630i	$0.128089\pi$
$163$	10.0000i	0.783260i	−0.920123	+	0.391630i	$0.871911\pi$
$164$	−6.00000	−0.468521
$165$	0	0
$166$	−12.0000	−0.931381
$167$	− 21.0000i	− 1.62503i	−0.582941	−	0.812514i	$-0.698098\pi$
$167$	− 21.0000i	− 1.62503i	0.582941	−	0.812514i	$-0.301902\pi$
$168$	0	0
$169$	12.0000	0.923077
$170$	0	0
$171$	0	0
$172$	− 10.0000i	− 0.762493i
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	0	0
$175$	0	0
$176$	3.00000	0.226134
$177$	0	0
$178$	− 12.0000i	− 0.899438i
$179$	21.0000	1.56961	0.784807	−	0.619740i	$-0.212762\pi$
$179$	21.0000	1.56961	0.784807	+	0.619740i	$0.212762\pi$
$180$	0	0
$181$	−1.00000	−0.0743294	−0.0371647	−	0.999309i	$-0.511833\pi$
$181$	−1.00000	−0.0743294	−0.0371647	+	0.999309i	$0.511833\pi$
$182$	− 4.00000i	− 0.296500i
$183$	0	0
$184$	9.00000	0.663489
$185$	0	0
$186$	0	0
$187$	0	0
$188$	9.00000i	0.656392i
$189$	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
$193$	10.0000i	0.719816i	0.932988	+	0.359908i	$0.117192\pi$
$193$	10.0000i	0.719816i	−0.932988	+	0.359908i	$0.882808\pi$
$194$	−19.0000	−1.36412
$195$	0	0
$196$	9.00000	0.642857
$197$	24.0000i	1.70993i	0.518686	+	0.854965i	$0.326421\pi$
$197$	24.0000i	1.70993i	−0.518686	+	0.854965i	$0.673579\pi$
$198$	0	0
$199$	22.0000	1.55954	0.779769	−	0.626067i	$-0.215336\pi$
$199$	22.0000	1.55954	0.779769	+	0.626067i	$0.215336\pi$
$200$	0	0
$201$	0	0
$202$	12.0000i	0.844317i
$203$	− 24.0000i	− 1.68447i
$204$	0	0
$205$	0	0
$206$	−14.0000	−0.975426
$207$	0	0
$208$	1.00000i	0.0693375i
$209$	−6.00000	−0.415029
$210$	0	0
$211$	8.00000	0.550743	0.275371	−	0.961338i	$-0.411199\pi$
$211$	8.00000	0.550743	0.275371	+	0.961338i	$0.411199\pi$
$212$	− 6.00000i	− 0.412082i
$213$	0	0
$214$	9.00000	0.615227
$215$	0	0
$216$	0	0
$217$	40.0000i	2.71538i
$218$	2.00000i	0.135457i
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	− 2.00000i	− 0.133930i	−0.997755	−	0.0669650i	$-0.978668\pi$
$223$	− 2.00000i	− 0.133930i	0.997755	−	0.0669650i	$-0.0213316\pi$
$224$	−4.00000	−0.267261
$225$	0	0
$226$	6.00000	0.399114
$227$	3.00000i	0.199117i	0.995032	+	0.0995585i	$0.0317430\pi$
$227$	3.00000i	0.199117i	−0.995032	+	0.0995585i	$0.968257\pi$
$228$	0	0
$229$	1.00000	0.0660819	0.0330409	−	0.999454i	$-0.489481\pi$
$229$	1.00000	0.0660819	0.0330409	+	0.999454i	$0.489481\pi$
$230$	0	0
$231$	0	0
$232$	6.00000i	0.393919i
$233$	12.0000i	0.786146i	0.919507	+	0.393073i	$0.128588\pi$
$233$	12.0000i	0.786146i	−0.919507	+	0.393073i	$0.871412\pi$
$234$	0	0
$235$	0	0
$236$	3.00000	0.195283
$237$	0	0
$238$	0	0
$239$	−15.0000	−0.970269	−0.485135	−	0.874439i	$-0.661229\pi$
$239$	−15.0000	−0.970269	−0.485135	+	0.874439i	$0.661229\pi$
$240$	0	0
$241$	5.00000	0.322078	0.161039	−	0.986948i	$-0.448515\pi$
$241$	5.00000	0.322078	0.161039	+	0.986948i	$0.448515\pi$
$242$	2.00000i	0.128565i
$243$	0	0
$244$	7.00000	0.448129
$245$	0	0
$246$	0	0
$247$	− 2.00000i	− 0.127257i
$248$	− 10.0000i	− 0.635001i
$249$	0	0
$250$	0	0
$251$	15.0000	0.946792	0.473396	−	0.880850i	$-0.343028\pi$
$251$	15.0000	0.946792	0.473396	+	0.880850i	$0.343028\pi$
$252$	0	0
$253$	− 27.0000i	− 1.69748i
$254$	2.00000	0.125491
$255$	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
$259$	−28.0000	−1.73984
$260$	0	0
$261$	0	0
$262$	− 15.0000i	− 0.926703i
$263$	15.0000i	0.924940i	0.886635	+	0.462470i	$0.153037\pi$
$263$	15.0000i	0.924940i	−0.886635	+	0.462470i	$0.846963\pi$
$264$	0	0
$265$	0	0
$266$	8.00000	0.490511
$267$	0	0
$268$	4.00000i	0.244339i
$269$	12.0000	0.731653	0.365826	−	0.930683i	$-0.380786\pi$
$269$	12.0000	0.731653	0.365826	+	0.930683i	$0.380786\pi$
$270$	0	0
$271$	20.0000	1.21491	0.607457	−	0.794353i	$-0.292190\pi$
$271$	20.0000	1.21491	0.607457	+	0.794353i	$0.292190\pi$
$272$	0	0
$273$	0	0
$274$	6.00000	0.362473
$275$	0	0
$276$	0	0
$277$	2.00000i	0.120168i	0.998193	+	0.0600842i	$0.0191369\pi$
$277$	2.00000i	0.120168i	−0.998193	+	0.0600842i	$0.980863\pi$
$278$	− 4.00000i	− 0.239904i
$279$	0	0
$280$	0	0
$281$	6.00000	0.357930	0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000	0.357930	0.178965	+	0.983855i	$0.442725\pi$
$282$	0	0
$283$	− 26.0000i	− 1.54554i	−0.634686	−	0.772770i	$-0.718871\pi$
$283$	− 26.0000i	− 1.54554i	0.634686	−	0.772770i	$-0.281129\pi$
$284$	15.0000	0.890086
$285$	0	0
$286$	3.00000	0.177394
$287$	− 24.0000i	− 1.41668i
$288$	0	0
$289$	17.0000	1.00000
$290$	0	0
$291$	0	0
$292$	2.00000i	0.117041i
$293$	− 24.0000i	− 1.40209i	−0.713115	−	0.701047i	$-0.752716\pi$
$293$	− 24.0000i	− 1.40209i	0.713115	−	0.701047i	$-0.247284\pi$
$294$	0	0
$295$	0	0
$296$	7.00000	0.406867
$297$	0	0
$298$	− 18.0000i	− 1.04271i
$299$	9.00000	0.520483
$300$	0	0
$301$	40.0000	2.30556
$302$	− 2.00000i	− 0.115087i
$303$	0	0
$304$	−2.00000	−0.114708
$305$	0	0
$306$	0	0
$307$	26.0000i	1.48390i	0.670456	+	0.741949i	$0.266098\pi$
$307$	26.0000i	1.48390i	−0.670456	+	0.741949i	$0.733902\pi$
$308$	12.0000i	0.683763i
$309$	0	0
$310$	0	0
$311$	21.0000	1.19080	0.595400	−	0.803429i	$-0.296993\pi$
$311$	21.0000	1.19080	0.595400	+	0.803429i	$0.296993\pi$
$312$	0	0
$313$	− 14.0000i	− 0.791327i	−0.918396	−	0.395663i	$-0.870515\pi$
$313$	− 14.0000i	− 0.791327i	0.918396	−	0.395663i	$-0.129485\pi$
$314$	14.0000	0.790066
$315$	0	0
$316$	8.00000	0.450035
$317$	− 18.0000i	− 1.01098i	−0.862832	−	0.505490i	$-0.831312\pi$
$317$	− 18.0000i	− 1.01098i	0.862832	−	0.505490i	$-0.168688\pi$
$318$	0	0
$319$	18.0000	1.00781
$320$	0	0
$321$	0	0
$322$	36.0000i	2.00620i
$323$	0	0
$324$	0	0
$325$	0	0
$326$	10.0000	0.553849
$327$	0	0
$328$	6.00000i	0.331295i
$329$	−36.0000	−1.98474
$330$	0	0
$331$	−10.0000	−0.549650	−0.274825	−	0.961494i	$-0.588620\pi$
$331$	−10.0000	−0.549650	−0.274825	+	0.961494i	$0.588620\pi$
$332$	12.0000i	0.658586i
$333$	0	0
$334$	−21.0000	−1.14907
$335$	0	0
$336$	0	0
$337$	2.00000i	0.108947i	0.998515	+	0.0544735i	$0.0173480\pi$
$337$	2.00000i	0.108947i	−0.998515	+	0.0544735i	$0.982652\pi$
$338$	− 12.0000i	− 0.652714i
$339$	0	0
$340$	0	0
$341$	−30.0000	−1.62459
$342$	0	0
$343$	8.00000i	0.431959i
$344$	−10.0000	−0.539164
$345$	0	0
$346$	0	0
$347$	− 12.0000i	− 0.644194i	−0.946707	−	0.322097i	$-0.895612\pi$
$347$	− 12.0000i	− 0.644194i	0.946707	−	0.322097i	$-0.104388\pi$
$348$	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
$356$	−12.0000	−0.635999
$357$	0	0
$358$	− 21.0000i	− 1.10988i
$359$	−27.0000	−1.42501	−0.712503	−	0.701669i	$-0.752438\pi$
$359$	−27.0000	−1.42501	−0.712503	+	0.701669i	$0.752438\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	1.00000i	0.0525588i
$363$	0	0
$364$	−4.00000	−0.209657
$365$	0	0
$366$	0	0
$367$	8.00000i	0.417597i	0.977959	+	0.208798i	$0.0669552\pi$
$367$	8.00000i	0.417597i	−0.977959	+	0.208798i	$0.933045\pi$
$368$	− 9.00000i	− 0.469157i
$369$	0	0
$370$	0	0
$371$	24.0000	1.24602
$372$	0	0
$373$	22.0000i	1.13912i	0.821951	+	0.569558i	$0.192886\pi$
$373$	22.0000i	1.13912i	−0.821951	+	0.569558i	$0.807114\pi$
$374$	0	0
$375$	0	0
$376$	9.00000	0.464140
$377$	6.00000i	0.309016i
$378$	0	0
$379$	10.0000	0.513665	0.256833	−	0.966456i	$-0.417321\pi$
$379$	10.0000	0.513665	0.256833	+	0.966456i	$0.417321\pi$
$380$	0	0
$381$	0	0
$382$	− 12.0000i	− 0.613973i
$383$	9.00000i	0.459879i	0.973205	+	0.229939i	$0.0738528\pi$
$383$	9.00000i	0.459879i	−0.973205	+	0.229939i	$0.926147\pi$
$384$	0	0
$385$	0	0
$386$	10.0000	0.508987
$387$	0	0
$388$	19.0000i	0.964579i
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	0	0
$391$	0	0
$392$	− 9.00000i	− 0.454569i
$393$	0	0
$394$	24.0000	1.20910
$395$	0	0
$396$	0	0
$397$	5.00000i	0.250943i	0.992097	+	0.125471i	$0.0400443\pi$
$397$	5.00000i	0.250943i	−0.992097	+	0.125471i	$0.959956\pi$
$398$	− 22.0000i	− 1.10276i
$399$	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
$403$	− 10.0000i	− 0.498135i
$404$	12.0000	0.597022
$405$	0	0
$406$	−24.0000	−1.19110
$407$	− 21.0000i	− 1.04093i
$408$	0	0
$409$	7.00000	0.346128	0.173064	−	0.984911i	$-0.444633\pi$
$409$	7.00000	0.346128	0.173064	+	0.984911i	$0.444633\pi$
$410$	0	0
$411$	0	0
$412$	14.0000i	0.689730i
$413$	12.0000i	0.590481i
$414$	0	0
$415$	0	0
$416$	1.00000	0.0490290
$417$	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
$421$	−37.0000	−1.80327	−0.901635	−	0.432498i	$-0.857632\pi$
$421$	−37.0000	−1.80327	−0.901635	+	0.432498i	$0.857632\pi$
$422$	− 8.00000i	− 0.389434i
$423$	0	0
$424$	−6.00000	−0.291386
$425$	0	0
$426$	0	0
$427$	28.0000i	1.35501i
$428$	− 9.00000i	− 0.435031i
$429$	0	0
$430$	0	0
$431$	27.0000	1.30054	0.650272	−	0.759701i	$-0.274655\pi$
$431$	27.0000	1.30054	0.650272	+	0.759701i	$0.274655\pi$
$432$	0	0
$433$	− 11.0000i	− 0.528626i	−0.964437	−	0.264313i	$-0.914855\pi$
$433$	− 11.0000i	− 0.528626i	0.964437	−	0.264313i	$-0.0851452\pi$
$434$	40.0000	1.92006
$435$	0	0
$436$	2.00000	0.0957826
$437$	18.0000i	0.861057i
$438$	0	0
$439$	4.00000	0.190910	0.0954548	−	0.995434i	$-0.469569\pi$
$439$	4.00000	0.190910	0.0954548	+	0.995434i	$0.469569\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 3.00000i	− 0.142534i	−0.997457	−	0.0712672i	$-0.977296\pi$
$443$	− 3.00000i	− 0.142534i	0.997457	−	0.0712672i	$-0.0227043\pi$
$444$	0	0
$445$	0	0
$446$	−2.00000	−0.0947027
$447$	0	0
$448$	4.00000i	0.188982i
$449$	−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
$454$	3.00000	0.140797
$455$	0	0
$456$	0	0
$457$	− 19.0000i	− 0.888783i	−0.895833	−	0.444391i	$-0.853420\pi$
$457$	− 19.0000i	− 0.888783i	0.895833	−	0.444391i	$-0.146580\pi$
$458$	− 1.00000i	− 0.0467269i
$459$	0	0
$460$	0	0
$461$	−24.0000	−1.11779	−0.558896	−	0.829238i	$-0.688775\pi$
$461$	−24.0000	−1.11779	−0.558896	+	0.829238i	$0.688775\pi$
$462$	0	0
$463$	22.0000i	1.02243i	0.859454	+	0.511213i	$0.170804\pi$
$463$	22.0000i	1.02243i	−0.859454	+	0.511213i	$0.829196\pi$
$464$	6.00000	0.278543
$465$	0	0
$466$	12.0000	0.555889
$467$	3.00000i	0.138823i	0.997588	+	0.0694117i	$0.0221122\pi$
$467$	3.00000i	0.138823i	−0.997588	+	0.0694117i	$0.977888\pi$
$468$	0	0
$469$	−16.0000	−0.738811
$470$	0	0
$471$	0	0
$472$	− 3.00000i	− 0.138086i
$473$	30.0000i	1.37940i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	15.0000i	0.686084i
$479$	−24.0000	−1.09659	−0.548294	−	0.836286i	$-0.684723\pi$
$479$	−24.0000	−1.09659	−0.548294	+	0.836286i	$0.684723\pi$
$480$	0	0
$481$	7.00000	0.319173
$482$	− 5.00000i	− 0.227744i
$483$	0	0
$484$	2.00000	0.0909091
$485$	0	0
$486$	0	0
$487$	− 4.00000i	− 0.181257i	−0.995885	−	0.0906287i	$-0.971112\pi$
$487$	− 4.00000i	− 0.181257i	0.995885	−	0.0906287i	$-0.0288876\pi$
$488$	− 7.00000i	− 0.316875i
$489$	0	0
$490$	0	0
$491$	36.0000	1.62466	0.812329	−	0.583200i	$-0.198200\pi$
$491$	36.0000	1.62466	0.812329	+	0.583200i	$0.198200\pi$
$492$	0	0
$493$	0	0
$494$	−2.00000	−0.0899843
$495$	0	0
$496$	−10.0000	−0.449013
$497$	60.0000i	2.69137i
$498$	0	0
$499$	16.0000	0.716258	0.358129	−	0.933672i	$-0.383415\pi$
$499$	16.0000	0.716258	0.358129	+	0.933672i	$0.383415\pi$
$500$	0	0
$501$	0	0
$502$	− 15.0000i	− 0.669483i
$503$	− 12.0000i	− 0.535054i	−0.963550	−	0.267527i	$-0.913794\pi$
$503$	− 12.0000i	− 0.535054i	0.963550	−	0.267527i	$-0.0862064\pi$
$504$	0	0
$505$	0	0
$506$	−27.0000	−1.20030
$507$	0	0
$508$	− 2.00000i	− 0.0887357i
$509$	30.0000	1.32973	0.664863	−	0.746965i	$-0.268490\pi$
$509$	30.0000	1.32973	0.664863	+	0.746965i	$0.268490\pi$
$510$	0	0
$511$	−8.00000	−0.353899
$512$	− 1.00000i	− 0.0441942i
$513$	0	0
$514$	12.0000	0.529297
$515$	0	0
$516$	0	0
$517$	− 27.0000i	− 1.18746i
$518$	28.0000i	1.23025i
$519$	0	0
$520$	0	0
$521$	24.0000	1.05146	0.525730	−	0.850652i	$-0.323792\pi$
$521$	24.0000	1.05146	0.525730	+	0.850652i	$0.323792\pi$
$522$	0	0
$523$	4.00000i	0.174908i	0.996169	+	0.0874539i	$0.0278730\pi$
$523$	4.00000i	0.174908i	−0.996169	+	0.0874539i	$0.972127\pi$
$524$	−15.0000	−0.655278
$525$	0	0
$526$	15.0000	0.654031
$527$	0	0
$528$	0	0
$529$	−58.0000	−2.52174
$530$	0	0
$531$	0	0
$532$	− 8.00000i	− 0.346844i
$533$	6.00000i	0.259889i
$534$	0	0
$535$	0	0
$536$	4.00000	0.172774
$537$	0	0
$538$	− 12.0000i	− 0.517357i
$539$	−27.0000	−1.16297
$540$	0	0
$541$	35.0000	1.50477	0.752384	−	0.658725i	$-0.228904\pi$
$541$	35.0000	1.50477	0.752384	+	0.658725i	$0.228904\pi$
$542$	− 20.0000i	− 0.859074i
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 28.0000i	− 1.19719i	−0.801050	−	0.598597i	$-0.795725\pi$
$547$	− 28.0000i	− 1.19719i	0.801050	−	0.598597i	$-0.204275\pi$
$548$	− 6.00000i	− 0.256307i
$549$	0	0
$550$	0	0
$551$	−12.0000	−0.511217
$552$	0	0
$553$	32.0000i	1.36078i
$554$	2.00000	0.0849719
$555$	0	0
$556$	−4.00000	−0.169638
$557$	12.0000i	0.508456i	0.967144	+	0.254228i	$0.0818214\pi$
$557$	12.0000i	0.508456i	−0.967144	+	0.254228i	$0.918179\pi$
$558$	0	0
$559$	−10.0000	−0.422955
$560$	0	0
$561$	0	0
$562$	− 6.00000i	− 0.253095i
$563$	27.0000i	1.13791i	0.822367	+	0.568957i	$0.192653\pi$
$563$	27.0000i	1.13791i	−0.822367	+	0.568957i	$0.807347\pi$
$564$	0	0
$565$	0	0
$566$	−26.0000	−1.09286
$567$	0	0
$568$	− 15.0000i	− 0.629386i
$569$	18.0000	0.754599	0.377300	−	0.926091i	$-0.376853\pi$
$569$	18.0000	0.754599	0.377300	+	0.926091i	$0.376853\pi$
$570$	0	0
$571$	−34.0000	−1.42286	−0.711428	−	0.702759i	$-0.751951\pi$
$571$	−34.0000	−1.42286	−0.711428	+	0.702759i	$0.751951\pi$
$572$	− 3.00000i	− 0.125436i
$573$	0	0
$574$	−24.0000	−1.00174
$575$	0	0
$576$	0	0
$577$	− 13.0000i	− 0.541197i	−0.962692	−	0.270599i	$-0.912778\pi$
$577$	− 13.0000i	− 0.541197i	0.962692	−	0.270599i	$-0.0872216\pi$
$578$	− 17.0000i	− 0.707107i
$579$	0	0
$580$	0	0
$581$	−48.0000	−1.99138
$582$	0	0
$583$	18.0000i	0.745484i
$584$	2.00000	0.0827606
$585$	0	0
$586$	−24.0000	−0.991431
$587$	12.0000i	0.495293i	0.968850	+	0.247647i	$0.0796572\pi$
$587$	12.0000i	0.495293i	−0.968850	+	0.247647i	$0.920343\pi$
$588$	0	0
$589$	20.0000	0.824086
$590$	0	0
$591$	0	0
$592$	− 7.00000i	− 0.287698i
$593$	12.0000i	0.492781i	0.969171	+	0.246390i	$0.0792446\pi$
$593$	12.0000i	0.492781i	−0.969171	+	0.246390i	$0.920755\pi$
$594$	0	0
$595$	0	0
$596$	−18.0000	−0.737309
$597$	0	0
$598$	− 9.00000i	− 0.368037i
$599$	−12.0000	−0.490307	−0.245153	−	0.969484i	$-0.578838\pi$
$599$	−12.0000	−0.490307	−0.245153	+	0.969484i	$0.578838\pi$
$600$	0	0
$601$	35.0000	1.42768	0.713840	−	0.700309i	$-0.246954\pi$
$601$	35.0000	1.42768	0.713840	+	0.700309i	$0.246954\pi$
$602$	− 40.0000i	− 1.63028i
$603$	0	0
$604$	−2.00000	−0.0813788
$605$	0	0
$606$	0	0
$607$	− 22.0000i	− 0.892952i	−0.894795	−	0.446476i	$-0.852679\pi$
$607$	− 22.0000i	− 0.892952i	0.894795	−	0.446476i	$-0.147321\pi$
$608$	2.00000i	0.0811107i
$609$	0	0
$610$	0	0
$611$	9.00000	0.364101
$612$	0	0
$613$	− 29.0000i	− 1.17130i	−0.810564	−	0.585649i	$-0.800840\pi$
$613$	− 29.0000i	− 1.17130i	0.810564	−	0.585649i	$-0.199160\pi$
$614$	26.0000	1.04927
$615$	0	0
$616$	12.0000	0.483494
$617$	18.0000i	0.724653i	0.932051	+	0.362326i	$0.118017\pi$
$617$	18.0000i	0.724653i	−0.932051	+	0.362326i	$0.881983\pi$
$618$	0	0
$619$	−2.00000	−0.0803868	−0.0401934	−	0.999192i	$-0.512797\pi$
$619$	−2.00000	−0.0803868	−0.0401934	+	0.999192i	$0.512797\pi$
$620$	0	0
$621$	0	0
$622$	− 21.0000i	− 0.842023i
$623$	− 48.0000i	− 1.92308i
$624$	0	0
$625$	0	0
$626$	−14.0000	−0.559553
$627$	0	0
$628$	− 14.0000i	− 0.558661i
$629$	0	0
$630$	0	0
$631$	−10.0000	−0.398094	−0.199047	−	0.979990i	$-0.563785\pi$
$631$	−10.0000	−0.398094	−0.199047	+	0.979990i	$0.563785\pi$
$632$	− 8.00000i	− 0.318223i
$633$	0	0
$634$	−18.0000	−0.714871
$635$	0	0
$636$	0	0
$637$	− 9.00000i	− 0.356593i
$638$	− 18.0000i	− 0.712627i
$639$	0	0
$640$	0	0
$641$	−12.0000	−0.473972	−0.236986	−	0.971513i	$-0.576159\pi$
$641$	−12.0000	−0.473972	−0.236986	+	0.971513i	$0.576159\pi$
$642$	0	0
$643$	− 14.0000i	− 0.552106i	−0.961142	−	0.276053i	$-0.910973\pi$
$643$	− 14.0000i	− 0.552106i	0.961142	−	0.276053i	$-0.0890266\pi$
$644$	36.0000	1.41860
$645$	0	0
$646$	0	0
$647$	15.0000i	0.589711i	0.955542	+	0.294855i	$0.0952715\pi$
$647$	15.0000i	0.589711i	−0.955542	+	0.294855i	$0.904729\pi$
$648$	0	0
$649$	−9.00000	−0.353281
$650$	0	0
$651$	0	0
$652$	− 10.0000i	− 0.391630i
$653$	36.0000i	1.40879i	0.709809	+	0.704394i	$0.248781\pi$
$653$	36.0000i	1.40879i	−0.709809	+	0.704394i	$0.751219\pi$
$654$	0	0
$655$	0	0
$656$	6.00000	0.234261
$657$	0	0
$658$	36.0000i	1.40343i
$659$	−36.0000	−1.40236	−0.701180	−	0.712984i	$-0.747343\pi$
$659$	−36.0000	−1.40236	−0.701180	+	0.712984i	$0.747343\pi$
$660$	0	0
$661$	−31.0000	−1.20576	−0.602880	−	0.797832i	$-0.705980\pi$
$661$	−31.0000	−1.20576	−0.602880	+	0.797832i	$0.705980\pi$
$662$	10.0000i	0.388661i
$663$	0	0
$664$	12.0000	0.465690
$665$	0	0
$666$	0	0
$667$	− 54.0000i	− 2.09089i
$668$	21.0000i	0.812514i
$669$	0	0
$670$	0	0
$671$	−21.0000	−0.810696
$672$	0	0
$673$	1.00000i	0.0385472i	0.999814	+	0.0192736i	$0.00613535\pi$
$673$	1.00000i	0.0385472i	−0.999814	+	0.0192736i	$0.993865\pi$
$674$	2.00000	0.0770371
$675$	0	0
$676$	−12.0000	−0.461538
$677$	− 18.0000i	− 0.691796i	−0.938272	−	0.345898i	$-0.887574\pi$
$677$	− 18.0000i	− 0.691796i	0.938272	−	0.345898i	$-0.112426\pi$
$678$	0	0
$679$	−76.0000	−2.91661
$680$	0	0
$681$	0	0
$682$	30.0000i	1.14876i
$683$	− 9.00000i	− 0.344375i	−0.985064	−	0.172188i	$-0.944916\pi$
$683$	− 9.00000i	− 0.344375i	0.985064	−	0.172188i	$-0.0550836\pi$
$684$	0	0
$685$	0	0
$686$	8.00000	0.305441
$687$	0	0
$688$	10.0000i	0.381246i
$689$	−6.00000	−0.228582
$690$	0	0
$691$	−46.0000	−1.74992	−0.874961	−	0.484193i	$-0.839113\pi$
$691$	−46.0000	−1.74992	−0.874961	+	0.484193i	$0.839113\pi$
$692$	0	0
$693$	0	0
$694$	−12.0000	−0.455514
$695$	0	0
$696$	0	0
$697$	0	0
$698$	− 10.0000i	− 0.378506i
$699$	0	0
$700$	0	0
$701$	36.0000	1.35970	0.679851	−	0.733351i	$-0.262045\pi$
$701$	36.0000	1.35970	0.679851	+	0.733351i	$0.262045\pi$
$702$	0	0
$703$	14.0000i	0.528020i
$704$	−3.00000	−0.113067
$705$	0	0
$706$	−6.00000	−0.225813
$707$	48.0000i	1.80523i
$708$	0	0
$709$	1.00000	0.0375558	0.0187779	−	0.999824i	$-0.494022\pi$
$709$	1.00000	0.0375558	0.0187779	+	0.999824i	$0.494022\pi$
$710$	0	0
$711$	0	0
$712$	12.0000i	0.449719i
$713$	90.0000i	3.37053i
$714$	0	0
$715$	0	0
$716$	−21.0000	−0.784807
$717$	0	0
$718$	27.0000i	1.00763i
$719$	21.0000	0.783168	0.391584	−	0.920142i	$-0.371927\pi$
$719$	21.0000	0.783168	0.391584	+	0.920142i	$0.371927\pi$
$720$	0	0
$721$	−56.0000	−2.08555
$722$	15.0000i	0.558242i
$723$	0	0
$724$	1.00000	0.0371647
$725$	0	0
$726$	0	0
$727$	− 4.00000i	− 0.148352i	−0.997245	−	0.0741759i	$-0.976367\pi$
$727$	− 4.00000i	− 0.148352i	0.997245	−	0.0741759i	$-0.0236326\pi$
$728$	4.00000i	0.148250i
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	13.0000i	0.480166i	0.970752	+	0.240083i	$0.0771747\pi$
$733$	13.0000i	0.480166i	−0.970752	+	0.240083i	$0.922825\pi$
$734$	8.00000	0.295285
$735$	0	0
$736$	−9.00000	−0.331744
$737$	− 12.0000i	− 0.442026i
$738$	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
$742$	− 24.0000i	− 0.881068i
$743$	3.00000i	0.110059i	0.998485	+	0.0550297i	$0.0175253\pi$
$743$	3.00000i	0.110059i	−0.998485	+	0.0550297i	$0.982475\pi$
$744$	0	0
$745$	0	0
$746$	22.0000	0.805477
$747$	0	0
$748$	0	0
$749$	36.0000	1.31541
$750$	0	0
$751$	−4.00000	−0.145962	−0.0729810	−	0.997333i	$-0.523251\pi$
$751$	−4.00000	−0.145962	−0.0729810	+	0.997333i	$0.523251\pi$
$752$	− 9.00000i	− 0.328196i
$753$	0	0
$754$	6.00000	0.218507
$755$	0	0
$756$	0	0
$757$	23.0000i	0.835949i	0.908459	+	0.417975i	$0.137260\pi$
$757$	23.0000i	0.835949i	−0.908459	+	0.417975i	$0.862740\pi$
$758$	− 10.0000i	− 0.363216i
$759$	0	0
$760$	0	0
$761$	−18.0000	−0.652499	−0.326250	−	0.945284i	$-0.605785\pi$
$761$	−18.0000	−0.652499	−0.326250	+	0.945284i	$0.605785\pi$
$762$	0	0
$763$	8.00000i	0.289619i
$764$	−12.0000	−0.434145
$765$	0	0
$766$	9.00000	0.325183
$767$	− 3.00000i	− 0.108324i
$768$	0	0
$769$	46.0000	1.65880	0.829401	−	0.558653i	$-0.188682\pi$
$769$	46.0000	1.65880	0.829401	+	0.558653i	$0.188682\pi$
$770$	0	0
$771$	0	0
$772$	− 10.0000i	− 0.359908i
$773$	− 42.0000i	− 1.51064i	−0.655359	−	0.755318i	$-0.727483\pi$
$773$	− 42.0000i	− 1.51064i	0.655359	−	0.755318i	$-0.272517\pi$
$774$	0	0
$775$	0	0
$776$	19.0000	0.682060
$777$	0	0
$778$	6.00000i	0.215110i
$779$	−12.0000	−0.429945
$780$	0	0
$781$	−45.0000	−1.61023
$782$	0	0
$783$	0	0
$784$	−9.00000	−0.321429
$785$	0	0
$786$	0	0
$787$	8.00000i	0.285169i	0.989783	+	0.142585i	$0.0455413\pi$
$787$	8.00000i	0.285169i	−0.989783	+	0.142585i	$0.954459\pi$
$788$	− 24.0000i	− 0.854965i
$789$	0	0
$790$	0	0
$791$	24.0000	0.853342
$792$	0	0
$793$	− 7.00000i	− 0.248577i
$794$	5.00000	0.177443
$795$	0	0
$796$	−22.0000	−0.779769
$797$	− 6.00000i	− 0.212531i	−0.994338	−	0.106265i	$-0.966111\pi$
$797$	− 6.00000i	− 0.212531i	0.994338	−	0.106265i	$-0.0338893\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	12.0000i	0.423735i
$803$	− 6.00000i	− 0.211735i
$804$	0	0
$805$	0	0
$806$	−10.0000	−0.352235
$807$	0	0
$808$	− 12.0000i	− 0.422159i
$809$	−42.0000	−1.47664	−0.738321	−	0.674450i	$-0.764381\pi$
$809$	−42.0000	−1.47664	−0.738321	+	0.674450i	$0.764381\pi$
$810$	0	0
$811$	32.0000	1.12367	0.561836	−	0.827249i	$-0.310095\pi$
$811$	32.0000	1.12367	0.561836	+	0.827249i	$0.310095\pi$
$812$	24.0000i	0.842235i
$813$	0	0
$814$	−21.0000	−0.736050
$815$	0	0
$816$	0	0
$817$	− 20.0000i	− 0.699711i
$818$	− 7.00000i	− 0.244749i
$819$	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
$823$	− 26.0000i	− 0.906303i	−0.891434	−	0.453152i	$-0.850300\pi$
$823$	− 26.0000i	− 0.906303i	0.891434	−	0.453152i	$-0.149700\pi$
$824$	14.0000	0.487713
$825$	0	0
$826$	12.0000	0.417533
$827$	− 45.0000i	− 1.56480i	−0.622774	−	0.782402i	$-0.713994\pi$
$827$	− 45.0000i	− 1.56480i	0.622774	−	0.782402i	$-0.286006\pi$
$828$	0	0
$829$	−47.0000	−1.63238	−0.816189	−	0.577785i	$-0.803917\pi$
$829$	−47.0000	−1.63238	−0.816189	+	0.577785i	$0.803917\pi$
$830$	0	0
$831$	0	0
$832$	− 1.00000i	− 0.0346688i
$833$	0	0
$834$	0	0
$835$	0	0
$836$	6.00000	0.207514
$837$	0	0
$838$	− 12.0000i	− 0.414533i
$839$	9.00000	0.310715	0.155357	−	0.987858i	$-0.450347\pi$
$839$	9.00000	0.310715	0.155357	+	0.987858i	$0.450347\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	37.0000i	1.27510i
$843$	0	0
$844$	−8.00000	−0.275371
$845$	0	0
$846$	0	0
$847$	8.00000i	0.274883i
$848$	6.00000i	0.206041i
$849$	0	0
$850$	0	0
$851$	−63.0000	−2.15961
$852$	0	0
$853$	55.0000i	1.88316i	0.336784	+	0.941582i	$0.390661\pi$
$853$	55.0000i	1.88316i	−0.336784	+	0.941582i	$0.609339\pi$
$854$	28.0000	0.958140
$855$	0	0
$856$	−9.00000	−0.307614
$857$	12.0000i	0.409912i	0.978771	+	0.204956i	$0.0657052\pi$
$857$	12.0000i	0.409912i	−0.978771	+	0.204956i	$0.934295\pi$
$858$	0	0
$859$	−50.0000	−1.70598	−0.852989	−	0.521929i	$-0.825213\pi$
$859$	−50.0000	−1.70598	−0.852989	+	0.521929i	$0.825213\pi$
$860$	0	0
$861$	0	0
$862$	− 27.0000i	− 0.919624i
$863$	− 24.0000i	− 0.816970i	−0.912765	−	0.408485i	$-0.866057\pi$
$863$	− 24.0000i	− 0.816970i	0.912765	−	0.408485i	$-0.133943\pi$
$864$	0	0
$865$	0	0
$866$	−11.0000	−0.373795
$867$	0	0
$868$	− 40.0000i	− 1.35769i
$869$	−24.0000	−0.814144
$870$	0	0
$871$	4.00000	0.135535
$872$	− 2.00000i	− 0.0677285i
$873$	0	0
$874$	18.0000	0.608859
$875$	0	0
$876$	0	0
$877$	− 43.0000i	− 1.45201i	−0.687691	−	0.726003i	$-0.741376\pi$
$877$	− 43.0000i	− 1.45201i	0.687691	−	0.726003i	$-0.258624\pi$
$878$	− 4.00000i	− 0.134993i
$879$	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
$883$	52.0000i	1.74994i	0.484178	+	0.874970i	$0.339119\pi$
$883$	52.0000i	1.74994i	−0.484178	+	0.874970i	$0.660881\pi$
$884$	0	0
$885$	0	0
$886$	−3.00000	−0.100787
$887$	− 3.00000i	− 0.100730i	−0.998731	−	0.0503651i	$-0.983962\pi$
$887$	− 3.00000i	− 0.100730i	0.998731	−	0.0503651i	$-0.0160385\pi$
$888$	0	0
$889$	8.00000	0.268311
$890$	0	0
$891$	0	0
$892$	2.00000i	0.0669650i
$893$	18.0000i	0.602347i
$894$	0	0
$895$	0	0
$896$	4.00000	0.133631
$897$	0	0
$898$	18.0000i	0.600668i
$899$	−60.0000	−2.00111
$900$	0	0
$901$	0	0
$902$	− 18.0000i	− 0.599334i
$903$	0	0
$904$	−6.00000	−0.199557
$905$	0	0
$906$	0	0
$907$	− 46.0000i	− 1.52740i	−0.645568	−	0.763702i	$-0.723379\pi$
$907$	− 46.0000i	− 1.52740i	0.645568	−	0.763702i	$-0.276621\pi$
$908$	− 3.00000i	− 0.0995585i
$909$	0	0
$910$	0	0
$911$	39.0000	1.29213	0.646064	−	0.763283i	$-0.276414\pi$
$911$	39.0000	1.29213	0.646064	+	0.763283i	$0.276414\pi$
$912$	0	0
$913$	− 36.0000i	− 1.19143i
$914$	−19.0000	−0.628464
$915$	0	0
$916$	−1.00000	−0.0330409
$917$	− 60.0000i	− 1.98137i
$918$	0	0
$919$	46.0000	1.51740	0.758700	−	0.651440i	$-0.225835\pi$
$919$	46.0000	1.51740	0.758700	+	0.651440i	$0.225835\pi$
$920$	0	0
$921$	0	0
$922$	24.0000i	0.790398i
$923$	− 15.0000i	− 0.493731i
$924$	0	0
$925$	0	0
$926$	22.0000	0.722965
$927$	0	0
$928$	− 6.00000i	− 0.196960i
$929$	−42.0000	−1.37798	−0.688988	−	0.724773i	$-0.741945\pi$
$929$	−42.0000	−1.37798	−0.688988	+	0.724773i	$0.741945\pi$
$930$	0	0
$931$	18.0000	0.589926
$932$	− 12.0000i	− 0.393073i
$933$	0	0
$934$	3.00000	0.0981630
$935$	0	0
$936$	0	0
$937$	− 1.00000i	− 0.0326686i	−0.999867	−	0.0163343i	$-0.994800\pi$
$937$	− 1.00000i	− 0.0326686i	0.999867	−	0.0163343i	$-0.00519960\pi$
$938$	16.0000i	0.522419i
$939$	0	0
$940$	0	0
$941$	−12.0000	−0.391189	−0.195594	−	0.980685i	$-0.562664\pi$
$941$	−12.0000	−0.391189	−0.195594	+	0.980685i	$0.562664\pi$
$942$	0	0
$943$	− 54.0000i	− 1.75848i
$944$	−3.00000	−0.0976417
$945$	0	0
$946$	30.0000	0.975384
$947$	− 24.0000i	− 0.779895i	−0.920837	−	0.389948i	$-0.872493\pi$
$947$	− 24.0000i	− 0.779895i	0.920837	−	0.389948i	$-0.127507\pi$
$948$	0	0
$949$	2.00000	0.0649227
$950$	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
$956$	15.0000	0.485135
$957$	0	0
$958$	24.0000i	0.775405i
$959$	24.0000	0.775000
$960$	0	0
$961$	69.0000	2.22581
$962$	− 7.00000i	− 0.225689i
$963$	0	0
$964$	−5.00000	−0.161039
$965$	0	0
$966$	0	0
$967$	32.0000i	1.02905i	0.857475	+	0.514525i	$0.172032\pi$
$967$	32.0000i	1.02905i	−0.857475	+	0.514525i	$0.827968\pi$
$968$	− 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
$973$	− 16.0000i	− 0.512936i
$974$	−4.00000	−0.128168
$975$	0	0
$976$	−7.00000	−0.224065
$977$	48.0000i	1.53566i	0.640656	+	0.767828i	$0.278662\pi$
$977$	48.0000i	1.53566i	−0.640656	+	0.767828i	$0.721338\pi$
$978$	0	0
$979$	36.0000	1.15056
$980$	0	0
$981$	0	0
$982$	− 36.0000i	− 1.14881i
$983$	9.00000i	0.287055i	0.989646	+	0.143528i	$0.0458446\pi$
$983$	9.00000i	0.287055i	−0.989646	+	0.143528i	$0.954155\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	2.00000i	0.0636285i
$989$	90.0000	2.86183
$990$	0	0
$991$	32.0000	1.01651	0.508257	−	0.861206i	$-0.330290\pi$
$991$	32.0000	1.01651	0.508257	+	0.861206i	$0.330290\pi$
$992$	10.0000i	0.317500i
$993$	0	0
$994$	60.0000	1.90308
$995$	0	0
$996$	0	0
$997$	53.0000i	1.67853i	0.543725	+	0.839263i	$0.317013\pi$
$997$	53.0000i	1.67853i	−0.543725	+	0.839263i	$0.682987\pi$
$998$	− 16.0000i	− 0.506471i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1350.2.c.h.649.1		2
3.2	odd	2		1350.2.c.e.649.2		2
5.2	odd	4		1350.2.a.u.1.1	yes	1
5.3	odd	4		1350.2.a.a.1.1	✓	1
5.4	even	2	inner	1350.2.c.h.649.2		2
15.2	even	4		1350.2.a.k.1.1	yes	1
15.8	even	4		1350.2.a.m.1.1	yes	1
15.14	odd	2		1350.2.c.e.649.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1350.2.a.a.1.1	✓	1	5.3	odd	4
1350.2.a.k.1.1	yes	1	15.2	even	4
1350.2.a.m.1.1	yes	1	15.8	even	4
1350.2.a.u.1.1	yes	1	5.2	odd	4
1350.2.c.e.649.1		2	15.14	odd	2
1350.2.c.e.649.2		2	3.2	odd	2
1350.2.c.h.649.1		2	1.1	even	1	trivial
1350.2.c.h.649.2		2	5.4	even	2	inner

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

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} -4.00000i q^{7} +1.00000i q^{8} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000 q^{4} -4.00000i q^{7} +1.00000i q^{8} +3.00000 q^{11} +1.00000i q^{13} -4.00000 q^{14} +1.00000 q^{16} -2.00000 q^{19} -3.00000i q^{22} -9.00000i q^{23} +1.00000 q^{26} +4.00000i q^{28} +6.00000 q^{29} -10.0000 q^{31} -1.00000i q^{32} -7.00000i q^{37} +2.00000i q^{38} +6.00000 q^{41} +10.0000i q^{43} -3.00000 q^{44} -9.00000 q^{46} -9.00000i q^{47} -9.00000 q^{49} -1.00000i q^{52} +6.00000i q^{53} +4.00000 q^{56} -6.00000i q^{58} -3.00000 q^{59} -7.00000 q^{61} +10.0000i q^{62} -1.00000 q^{64} -4.00000i q^{67} -15.0000 q^{71} -2.00000i q^{73} -7.00000 q^{74} +2.00000 q^{76} -12.0000i q^{77} -8.00000 q^{79} -6.00000i q^{82} -12.0000i q^{83} +10.0000 q^{86} +3.00000i q^{88} +12.0000 q^{89} +4.00000 q^{91} +9.00000i q^{92} -9.00000 q^{94} -19.0000i q^{97} +9.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4}+O(q^{10}) \) 2 * q - 2 * q^4	\( 2 q - 2 q^{4} + 6 q^{11} - 8 q^{14} + 2 q^{16} - 4 q^{19} + 2 q^{26} + 12 q^{29} - 20 q^{31} + 12 q^{41} - 6 q^{44} - 18 q^{46} - 18 q^{49} + 8 q^{56} - 6 q^{59} - 14 q^{61} - 2 q^{64} - 30 q^{71} - 14 q^{74} + 4 q^{76} - 16 q^{79} + 20 q^{86} + 24 q^{89} + 8 q^{91} - 18 q^{94}+O(q^{100}) \) 2 * q - 2 * q^4 + 6 * q^11 - 8 * q^14 + 2 * q^16 - 4 * q^19 + 2 * q^26 + 12 * q^29 - 20 * q^31 + 12 * q^41 - 6 * q^44 - 18 * q^46 - 18 * q^49 + 8 * q^56 - 6 * q^59 - 14 * q^61 - 2 * q^64 - 30 * q^71 - 14 * q^74 + 4 * q^76 - 16 * q^79 + 20 * q^86 + 24 * q^89 + 8 * q^91 - 18 * q^94

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