LMFDB - Embedded newform 3150.2.g.t.2899.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3150,2,Mod(2899,3150)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3150, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("3150.2899");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3150.g (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:	$25.1528766367$
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 210)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2899.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	3150.2899
Dual form			3150.2.g.t.2899.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000 q^{4} -1.00000i q^{7} -1.00000i q^{8} +O(q^{10})$	$q+1.00000i q^{2} -1.00000 q^{4} -1.00000i q^{7} -1.00000i q^{8} +4.00000 q^{11} +2.00000i q^{13} +1.00000 q^{14} +1.00000 q^{16} +6.00000i q^{17} +4.00000i q^{22} -8.00000i q^{23} -2.00000 q^{26} +1.00000i q^{28} +10.0000 q^{29} -8.00000 q^{31} +1.00000i q^{32} -6.00000 q^{34} +2.00000i q^{37} +2.00000 q^{41} -8.00000i q^{43} -4.00000 q^{44} +8.00000 q^{46} -4.00000i q^{47} -1.00000 q^{49} -2.00000i q^{52} +10.0000i q^{53} -1.00000 q^{56} +10.0000i q^{58} +4.00000 q^{59} -6.00000 q^{61} -8.00000i q^{62} -1.00000 q^{64} -6.00000i q^{68} +12.0000 q^{71} +6.00000i q^{73} -2.00000 q^{74} -4.00000i q^{77} +8.00000 q^{79} +2.00000i q^{82} -4.00000i q^{83} +8.00000 q^{86} -4.00000i q^{88} +14.0000 q^{89} +2.00000 q^{91} +8.00000i q^{92} +4.00000 q^{94} +2.00000i q^{97} -1.00000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4}+O(q^{10}) $ 2 * q - 2 * q^4	$ 2 q - 2 q^{4} + 8 q^{11} + 2 q^{14} + 2 q^{16} - 4 q^{26} + 20 q^{29} - 16 q^{31} - 12 q^{34} + 4 q^{41} - 8 q^{44} + 16 q^{46} - 2 q^{49} - 2 q^{56} + 8 q^{59} - 12 q^{61} - 2 q^{64} + 24 q^{71} - 4 q^{74} + 16 q^{79} + 16 q^{86} + 28 q^{89} + 4 q^{91} + 8 q^{94}+O(q^{100}) $ 2 * q - 2 * q^4 + 8 * q^11 + 2 * q^14 + 2 * q^16 - 4 * q^26 + 20 * q^29 - 16 * q^31 - 12 * q^34 + 4 * q^41 - 8 * q^44 + 16 * q^46 - 2 * q^49 - 2 * q^56 + 8 * q^59 - 12 * q^61 - 2 * q^64 + 24 * q^71 - 4 * q^74 + 16 * q^79 + 16 * q^86 + 28 * q^89 + 4 * q^91 + 8 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3150\mathbb{Z}\right)^\times$.

$n$	$127$	$451$	$2801$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	0	0
$3$	0	0
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 1.00000i	− 0.377964i
$7$	− 1.00000i	− 0.377964i
$8$	− 1.00000i	− 0.353553i
$9$	0	0
$10$	0	0
$11$	4.00000	1.20605	0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000	1.20605	0.603023	+	0.797724i	$0.293963\pi$
$12$	0	0
$13$	2.00000i	0.554700i	0.960769	+	0.277350i	$0.0894562\pi$
$13$	2.00000i	0.554700i	−0.960769	+	0.277350i	$0.910544\pi$
$14$	1.00000	0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	6.00000i	1.45521i	0.685994	+	0.727607i	$0.259367\pi$
$17$	6.00000i	1.45521i	−0.685994	+	0.727607i	$0.740633\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	0	0
$22$	4.00000i	0.852803i
$23$	− 8.00000i	− 1.66812i	−0.551677	−	0.834058i	$-0.686012\pi$
$23$	− 8.00000i	− 1.66812i	0.551677	−	0.834058i	$-0.313988\pi$
$24$	0	0
$25$	0	0
$26$	−2.00000	−0.392232
$27$	0	0
$28$	1.00000i	0.188982i
$29$	10.0000	1.85695	0.928477	−	0.371391i	$-0.121119\pi$
$29$	10.0000	1.85695	0.928477	+	0.371391i	$0.121119\pi$
$30$	0	0
$31$	−8.00000	−1.43684	−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000	−1.43684	−0.718421	+	0.695608i	$0.755135\pi$
$32$	1.00000i	0.176777i
$33$	0	0
$34$	−6.00000	−1.02899
$35$	0	0
$36$	0	0
$37$	2.00000i	0.328798i	0.986394	+	0.164399i	$0.0525685\pi$
$37$	2.00000i	0.328798i	−0.986394	+	0.164399i	$0.947432\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	0	0
$43$	− 8.00000i	− 1.21999i	−0.792406	−	0.609994i	$-0.791172\pi$
$43$	− 8.00000i	− 1.21999i	0.792406	−	0.609994i	$-0.208828\pi$
$44$	−4.00000	−0.603023
$45$	0	0
$46$	8.00000	1.17954
$47$	− 4.00000i	− 0.583460i	−0.956501	−	0.291730i	$-0.905769\pi$
$47$	− 4.00000i	− 0.583460i	0.956501	−	0.291730i	$-0.0942309\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	− 2.00000i	− 0.277350i
$53$	10.0000i	1.37361i	0.726844	+	0.686803i	$0.240986\pi$
$53$	10.0000i	1.37361i	−0.726844	+	0.686803i	$0.759014\pi$
$54$	0	0
$55$	0	0
$56$	−1.00000	−0.133631
$57$	0	0
$58$	10.0000i	1.31306i
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	0	0
$61$	−6.00000	−0.768221	−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000	−0.768221	−0.384111	+	0.923287i	$0.625492\pi$
$62$	− 8.00000i	− 1.01600i
$63$	0	0
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	− 6.00000i	− 0.727607i
$69$	0	0
$70$	0	0
$71$	12.0000	1.42414	0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000	1.42414	0.712069	+	0.702109i	$0.247758\pi$
$72$	0	0
$73$	6.00000i	0.702247i	0.936329	+	0.351123i	$0.114200\pi$
$73$	6.00000i	0.702247i	−0.936329	+	0.351123i	$0.885800\pi$
$74$	−2.00000	−0.232495
$75$	0	0
$76$	0	0
$77$	− 4.00000i	− 0.455842i
$78$	0	0
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	0	0
$81$	0	0
$82$	2.00000i	0.220863i
$83$	− 4.00000i	− 0.439057i	−0.975606	−	0.219529i	$-0.929548\pi$
$83$	− 4.00000i	− 0.439057i	0.975606	−	0.219529i	$-0.0704519\pi$
$84$	0	0
$85$	0	0
$86$	8.00000	0.862662
$87$	0	0
$88$	− 4.00000i	− 0.426401i
$89$	14.0000	1.48400	0.741999	−	0.670402i	$-0.233878\pi$
$89$	14.0000	1.48400	0.741999	+	0.670402i	$0.233878\pi$
$90$	0	0
$91$	2.00000	0.209657
$92$	8.00000i	0.834058i
$93$	0	0
$94$	4.00000	0.412568
$95$	0	0
$96$	0	0
$97$	2.00000i	0.203069i	0.994832	+	0.101535i	$0.0323753\pi$
$97$	2.00000i	0.203069i	−0.994832	+	0.101535i	$0.967625\pi$
$98$	− 1.00000i	− 0.101015i
$99$	0	0
$100$	0	0
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	0	0
$103$	16.0000i	1.57653i	0.615338	+	0.788263i	$0.289020\pi$
$103$	16.0000i	1.57653i	−0.615338	+	0.788263i	$0.710980\pi$
$104$	2.00000	0.196116
$105$	0	0
$106$	−10.0000	−0.971286
$107$	4.00000i	0.386695i	0.981130	+	0.193347i	$0.0619344\pi$
$107$	4.00000i	0.386695i	−0.981130	+	0.193347i	$0.938066\pi$
$108$	0	0
$109$	18.0000	1.72409	0.862044	−	0.506834i	$-0.169184\pi$
$109$	18.0000	1.72409	0.862044	+	0.506834i	$0.169184\pi$
$110$	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
$116$	−10.0000	−0.928477
$117$	0	0
$118$	4.00000i	0.368230i
$119$	6.00000	0.550019
$120$	0	0
$121$	5.00000	0.454545
$122$	− 6.00000i	− 0.543214i
$123$	0	0
$124$	8.00000	0.718421
$125$	0	0
$126$	0	0
$127$	16.0000i	1.41977i	0.704317	+	0.709885i	$0.251253\pi$
$127$	16.0000i	1.41977i	−0.704317	+	0.709885i	$0.748747\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	0	0
$130$	0	0
$131$	−4.00000	−0.349482	−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000	−0.349482	−0.174741	+	0.984614i	$0.555909\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	6.00000	0.514496
$137$	− 6.00000i	− 0.512615i	−0.966595	−	0.256307i	$-0.917494\pi$
$137$	− 6.00000i	− 0.512615i	0.966595	−	0.256307i	$-0.0825059\pi$
$138$	0	0
$139$	16.0000	1.35710	0.678551	−	0.734553i	$-0.262608\pi$
$139$	16.0000	1.35710	0.678551	+	0.734553i	$0.262608\pi$
$140$	0	0
$141$	0	0
$142$	12.0000i	1.00702i
$143$	8.00000i	0.668994i
$144$	0	0
$145$	0	0
$146$	−6.00000	−0.496564
$147$	0	0
$148$	− 2.00000i	− 0.164399i
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	0	0
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	0	0
$153$	0	0
$154$	4.00000	0.322329
$155$	0	0
$156$	0	0
$157$	− 10.0000i	− 0.798087i	−0.916932	−	0.399043i	$-0.869342\pi$
$157$	− 10.0000i	− 0.798087i	0.916932	−	0.399043i	$-0.130658\pi$
$158$	8.00000i	0.636446i
$159$	0	0
$160$	0	0
$161$	−8.00000	−0.630488
$162$	0	0
$163$	− 16.0000i	− 1.25322i	−0.779334	−	0.626608i	$-0.784443\pi$
$163$	− 16.0000i	− 1.25322i	0.779334	−	0.626608i	$-0.215557\pi$
$164$	−2.00000	−0.156174
$165$	0	0
$166$	4.00000	0.310460
$167$	12.0000i	0.928588i	0.885681	+	0.464294i	$0.153692\pi$
$167$	12.0000i	0.928588i	−0.885681	+	0.464294i	$0.846308\pi$
$168$	0	0
$169$	9.00000	0.692308
$170$	0	0
$171$	0	0
$172$	8.00000i	0.609994i
$173$	18.0000i	1.36851i	0.729241	+	0.684257i	$0.239873\pi$
$173$	18.0000i	1.36851i	−0.729241	+	0.684257i	$0.760127\pi$
$174$	0	0
$175$	0	0
$176$	4.00000	0.301511
$177$	0	0
$178$	14.0000i	1.04934i
$179$	−4.00000	−0.298974	−0.149487	−	0.988764i	$-0.547762\pi$
$179$	−4.00000	−0.298974	−0.149487	+	0.988764i	$0.547762\pi$
$180$	0	0
$181$	18.0000	1.33793	0.668965	−	0.743294i	$-0.266738\pi$
$181$	18.0000	1.33793	0.668965	+	0.743294i	$0.266738\pi$
$182$	2.00000i	0.148250i
$183$	0	0
$184$	−8.00000	−0.589768
$185$	0	0
$186$	0	0
$187$	24.0000i	1.75505i
$188$	4.00000i	0.291730i
$189$	0	0
$190$	0	0
$191$	−12.0000	−0.868290	−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000	−0.868290	−0.434145	+	0.900843i	$0.642949\pi$
$192$	0	0
$193$	− 10.0000i	− 0.719816i	−0.932988	−	0.359908i	$-0.882808\pi$
$193$	− 10.0000i	− 0.719816i	0.932988	−	0.359908i	$-0.117192\pi$
$194$	−2.00000	−0.143592
$195$	0	0
$196$	1.00000	0.0714286
$197$	22.0000i	1.56744i	0.621117	+	0.783718i	$0.286679\pi$
$197$	22.0000i	1.56744i	−0.621117	+	0.783718i	$0.713321\pi$
$198$	0	0
$199$	−24.0000	−1.70131	−0.850657	−	0.525720i	$-0.823796\pi$
$199$	−24.0000	−1.70131	−0.850657	+	0.525720i	$0.823796\pi$
$200$	0	0
$201$	0	0
$202$	6.00000i	0.422159i
$203$	− 10.0000i	− 0.701862i
$204$	0	0
$205$	0	0
$206$	−16.0000	−1.11477
$207$	0	0
$208$	2.00000i	0.138675i
$209$	0	0
$210$	0	0
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	− 10.0000i	− 0.686803i
$213$	0	0
$214$	−4.00000	−0.273434
$215$	0	0
$216$	0	0
$217$	8.00000i	0.543075i
$218$	18.0000i	1.21911i
$219$	0	0
$220$	0	0
$221$	−12.0000	−0.807207
$222$	0	0
$223$	24.0000i	1.60716i	0.595198	+	0.803579i	$0.297074\pi$
$223$	24.0000i	1.60716i	−0.595198	+	0.803579i	$0.702926\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	−6.00000	−0.399114
$227$	− 20.0000i	− 1.32745i	−0.747978	−	0.663723i	$-0.768975\pi$
$227$	− 20.0000i	− 1.32745i	0.747978	−	0.663723i	$-0.231025\pi$
$228$	0	0
$229$	22.0000	1.45380	0.726900	−	0.686743i	$-0.240960\pi$
$229$	22.0000	1.45380	0.726900	+	0.686743i	$0.240960\pi$
$230$	0	0
$231$	0	0
$232$	− 10.0000i	− 0.656532i
$233$	− 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
$236$	−4.00000	−0.260378
$237$	0	0
$238$	6.00000i	0.388922i
$239$	−12.0000	−0.776215	−0.388108	−	0.921614i	$-0.626871\pi$
$239$	−12.0000	−0.776215	−0.388108	+	0.921614i	$0.626871\pi$
$240$	0	0
$241$	−14.0000	−0.901819	−0.450910	−	0.892570i	$-0.648900\pi$
$241$	−14.0000	−0.901819	−0.450910	+	0.892570i	$0.648900\pi$
$242$	5.00000i	0.321412i
$243$	0	0
$244$	6.00000	0.384111
$245$	0	0
$246$	0	0
$247$	0	0
$248$	8.00000i	0.508001i
$249$	0	0
$250$	0	0
$251$	20.0000	1.26239	0.631194	−	0.775625i	$-0.282565\pi$
$251$	20.0000	1.26239	0.631194	+	0.775625i	$0.282565\pi$
$252$	0	0
$253$	− 32.0000i	− 2.01182i
$254$	−16.0000	−1.00393
$255$	0	0
$256$	1.00000	0.0625000
$257$	14.0000i	0.873296i	0.899632	+	0.436648i	$0.143834\pi$
$257$	14.0000i	0.873296i	−0.899632	+	0.436648i	$0.856166\pi$
$258$	0	0
$259$	2.00000	0.124274
$260$	0	0
$261$	0	0
$262$	− 4.00000i	− 0.247121i
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	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
$271$	8.00000	0.485965	0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000	0.485965	0.242983	+	0.970031i	$0.421874\pi$
$272$	6.00000i	0.363803i
$273$	0	0
$274$	6.00000	0.362473
$275$	0	0
$276$	0	0
$277$	− 22.0000i	− 1.32185i	−0.750451	−	0.660926i	$-0.770164\pi$
$277$	− 22.0000i	− 1.32185i	0.750451	−	0.660926i	$-0.229836\pi$
$278$	16.0000i	0.959616i
$279$	0	0
$280$	0	0
$281$	22.0000	1.31241	0.656205	−	0.754583i	$-0.272161\pi$
$281$	22.0000	1.31241	0.656205	+	0.754583i	$0.272161\pi$
$282$	0	0
$283$	− 28.0000i	− 1.66443i	−0.554455	−	0.832214i	$-0.687073\pi$
$283$	− 28.0000i	− 1.66443i	0.554455	−	0.832214i	$-0.312927\pi$
$284$	−12.0000	−0.712069
$285$	0	0
$286$	−8.00000	−0.473050
$287$	− 2.00000i	− 0.118056i
$288$	0	0
$289$	−19.0000	−1.11765
$290$	0	0
$291$	0	0
$292$	− 6.00000i	− 0.351123i
$293$	− 30.0000i	− 1.75262i	−0.481749	−	0.876309i	$-0.659998\pi$
$293$	− 30.0000i	− 1.75262i	0.481749	−	0.876309i	$-0.340002\pi$
$294$	0	0
$295$	0	0
$296$	2.00000	0.116248
$297$	0	0
$298$	− 6.00000i	− 0.347571i
$299$	16.0000	0.925304
$300$	0	0
$301$	−8.00000	−0.461112
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	20.0000i	1.14146i	0.821138	+	0.570730i	$0.193340\pi$
$307$	20.0000i	1.14146i	−0.821138	+	0.570730i	$0.806660\pi$
$308$	4.00000i	0.227921i
$309$	0	0
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	6.00000i	0.339140i	0.985518	+	0.169570i	$0.0542379\pi$
$313$	6.00000i	0.339140i	−0.985518	+	0.169570i	$0.945762\pi$
$314$	10.0000	0.564333
$315$	0	0
$316$	−8.00000	−0.450035
$317$	30.0000i	1.68497i	0.538721	+	0.842484i	$0.318908\pi$
$317$	30.0000i	1.68497i	−0.538721	+	0.842484i	$0.681092\pi$
$318$	0	0
$319$	40.0000	2.23957
$320$	0	0
$321$	0	0
$322$	− 8.00000i	− 0.445823i
$323$	0	0
$324$	0	0
$325$	0	0
$326$	16.0000	0.886158
$327$	0	0
$328$	− 2.00000i	− 0.110432i
$329$	−4.00000	−0.220527
$330$	0	0
$331$	20.0000	1.09930	0.549650	−	0.835395i	$-0.314761\pi$
$331$	20.0000	1.09930	0.549650	+	0.835395i	$0.314761\pi$
$332$	4.00000i	0.219529i
$333$	0	0
$334$	−12.0000	−0.656611
$335$	0	0
$336$	0	0
$337$	− 6.00000i	− 0.326841i	−0.986557	−	0.163420i	$-0.947747\pi$
$337$	− 6.00000i	− 0.326841i	0.986557	−	0.163420i	$-0.0522527\pi$
$338$	9.00000i	0.489535i
$339$	0	0
$340$	0	0
$341$	−32.0000	−1.73290
$342$	0	0
$343$	1.00000i	0.0539949i
$344$	−8.00000	−0.431331
$345$	0	0
$346$	−18.0000	−0.967686
$347$	− 20.0000i	− 1.07366i	−0.843692	−	0.536828i	$-0.819622\pi$
$347$	− 20.0000i	− 1.07366i	0.843692	−	0.536828i	$-0.180378\pi$
$348$	0	0
$349$	−18.0000	−0.963518	−0.481759	−	0.876304i	$-0.660002\pi$
$349$	−18.0000	−0.963518	−0.481759	+	0.876304i	$0.660002\pi$
$350$	0	0
$351$	0	0
$352$	4.00000i	0.213201i
$353$	18.0000i	0.958043i	0.877803	+	0.479022i	$0.159008\pi$
$353$	18.0000i	0.958043i	−0.877803	+	0.479022i	$0.840992\pi$
$354$	0	0
$355$	0	0
$356$	−14.0000	−0.741999
$357$	0	0
$358$	− 4.00000i	− 0.211407i
$359$	−20.0000	−1.05556	−0.527780	−	0.849381i	$-0.676975\pi$
$359$	−20.0000	−1.05556	−0.527780	+	0.849381i	$0.676975\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	18.0000i	0.946059i
$363$	0	0
$364$	−2.00000	−0.104828
$365$	0	0
$366$	0	0
$367$	16.0000i	0.835193i	0.908633	+	0.417597i	$0.137127\pi$
$367$	16.0000i	0.835193i	−0.908633	+	0.417597i	$0.862873\pi$
$368$	− 8.00000i	− 0.417029i
$369$	0	0
$370$	0	0
$371$	10.0000	0.519174
$372$	0	0
$373$	30.0000i	1.55334i	0.629907	+	0.776671i	$0.283093\pi$
$373$	30.0000i	1.55334i	−0.629907	+	0.776671i	$0.716907\pi$
$374$	−24.0000	−1.24101
$375$	0	0
$376$	−4.00000	−0.206284
$377$	20.0000i	1.03005i
$378$	0	0
$379$	28.0000	1.43826	0.719132	−	0.694874i	$-0.244540\pi$
$379$	28.0000	1.43826	0.719132	+	0.694874i	$0.244540\pi$
$380$	0	0
$381$	0	0
$382$	− 12.0000i	− 0.613973i
$383$	− 28.0000i	− 1.43073i	−0.698749	−	0.715367i	$-0.746260\pi$
$383$	− 28.0000i	− 1.43073i	0.698749	−	0.715367i	$-0.253740\pi$
$384$	0	0
$385$	0	0
$386$	10.0000	0.508987
$387$	0	0
$388$	− 2.00000i	− 0.101535i
$389$	10.0000	0.507020	0.253510	−	0.967333i	$-0.418415\pi$
$389$	10.0000	0.507020	0.253510	+	0.967333i	$0.418415\pi$
$390$	0	0
$391$	48.0000	2.42746
$392$	1.00000i	0.0505076i
$393$	0	0
$394$	−22.0000	−1.10834
$395$	0	0
$396$	0	0
$397$	14.0000i	0.702640i	0.936255	+	0.351320i	$0.114267\pi$
$397$	14.0000i	0.702640i	−0.936255	+	0.351320i	$0.885733\pi$
$398$	− 24.0000i	− 1.20301i
$399$	0	0
$400$	0	0
$401$	−34.0000	−1.69788	−0.848939	−	0.528490i	$-0.822758\pi$
$401$	−34.0000	−1.69788	−0.848939	+	0.528490i	$0.822758\pi$
$402$	0	0
$403$	− 16.0000i	− 0.797017i
$404$	−6.00000	−0.298511
$405$	0	0
$406$	10.0000	0.496292
$407$	8.00000i	0.396545i
$408$	0	0
$409$	−26.0000	−1.28562	−0.642809	−	0.766027i	$-0.722231\pi$
$409$	−26.0000	−1.28562	−0.642809	+	0.766027i	$0.722231\pi$
$410$	0	0
$411$	0	0
$412$	− 16.0000i	− 0.788263i
$413$	− 4.00000i	− 0.196827i
$414$	0	0
$415$	0	0
$416$	−2.00000	−0.0980581
$417$	0	0
$418$	0	0
$419$	28.0000	1.36789	0.683945	−	0.729534i	$-0.260263\pi$
$419$	28.0000	1.36789	0.683945	+	0.729534i	$0.260263\pi$
$420$	0	0
$421$	−26.0000	−1.26716	−0.633581	−	0.773676i	$-0.718416\pi$
$421$	−26.0000	−1.26716	−0.633581	+	0.773676i	$0.718416\pi$
$422$	− 4.00000i	− 0.194717i
$423$	0	0
$424$	10.0000	0.485643
$425$	0	0
$426$	0	0
$427$	6.00000i	0.290360i
$428$	− 4.00000i	− 0.193347i
$429$	0	0
$430$	0	0
$431$	36.0000	1.73406	0.867029	−	0.498257i	$-0.166026\pi$
$431$	36.0000	1.73406	0.867029	+	0.498257i	$0.166026\pi$
$432$	0	0
$433$	− 34.0000i	− 1.63394i	−0.576683	−	0.816968i	$-0.695653\pi$
$433$	− 34.0000i	− 1.63394i	0.576683	−	0.816968i	$-0.304347\pi$
$434$	−8.00000	−0.384012
$435$	0	0
$436$	−18.0000	−0.862044
$437$	0	0
$438$	0	0
$439$	−32.0000	−1.52728	−0.763638	−	0.645644i	$-0.776589\pi$
$439$	−32.0000	−1.52728	−0.763638	+	0.645644i	$0.776589\pi$
$440$	0	0
$441$	0	0
$442$	− 12.0000i	− 0.570782i
$443$	4.00000i	0.190046i	0.995475	+	0.0950229i	$0.0302924\pi$
$443$	4.00000i	0.190046i	−0.995475	+	0.0950229i	$0.969708\pi$
$444$	0	0
$445$	0	0
$446$	−24.0000	−1.13643
$447$	0	0
$448$	1.00000i	0.0472456i
$449$	26.0000	1.22702	0.613508	−	0.789689i	$-0.289758\pi$
$449$	26.0000	1.22702	0.613508	+	0.789689i	$0.289758\pi$
$450$	0	0
$451$	8.00000	0.376705
$452$	− 6.00000i	− 0.282216i
$453$	0	0
$454$	20.0000	0.938647
$455$	0	0
$456$	0	0
$457$	− 6.00000i	− 0.280668i	−0.990104	−	0.140334i	$-0.955182\pi$
$457$	− 6.00000i	− 0.280668i	0.990104	−	0.140334i	$-0.0448177\pi$
$458$	22.0000i	1.02799i
$459$	0	0
$460$	0	0
$461$	−18.0000	−0.838344	−0.419172	−	0.907907i	$-0.637680\pi$
$461$	−18.0000	−0.838344	−0.419172	+	0.907907i	$0.637680\pi$
$462$	0	0
$463$	− 8.00000i	− 0.371792i	−0.982569	−	0.185896i	$-0.940481\pi$
$463$	− 8.00000i	− 0.371792i	0.982569	−	0.185896i	$-0.0595187\pi$
$464$	10.0000	0.464238
$465$	0	0
$466$	18.0000	0.833834
$467$	− 12.0000i	− 0.555294i	−0.960683	−	0.277647i	$-0.910445\pi$
$467$	− 12.0000i	− 0.555294i	0.960683	−	0.277647i	$-0.0895545\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	− 4.00000i	− 0.184115i
$473$	− 32.0000i	− 1.47136i
$474$	0	0
$475$	0	0
$476$	−6.00000	−0.275010
$477$	0	0
$478$	− 12.0000i	− 0.548867i
$479$	−8.00000	−0.365529	−0.182765	−	0.983157i	$-0.558505\pi$
$479$	−8.00000	−0.365529	−0.182765	+	0.983157i	$0.558505\pi$
$480$	0	0
$481$	−4.00000	−0.182384
$482$	− 14.0000i	− 0.637683i
$483$	0	0
$484$	−5.00000	−0.227273
$485$	0	0
$486$	0	0
$487$	8.00000i	0.362515i	0.983436	+	0.181257i	$0.0580167\pi$
$487$	8.00000i	0.362515i	−0.983436	+	0.181257i	$0.941983\pi$
$488$	6.00000i	0.271607i
$489$	0	0
$490$	0	0
$491$	−20.0000	−0.902587	−0.451294	−	0.892375i	$-0.649037\pi$
$491$	−20.0000	−0.902587	−0.451294	+	0.892375i	$0.649037\pi$
$492$	0	0
$493$	60.0000i	2.70226i
$494$	0	0
$495$	0	0
$496$	−8.00000	−0.359211
$497$	− 12.0000i	− 0.538274i
$498$	0	0
$499$	−28.0000	−1.25345	−0.626726	−	0.779240i	$-0.715605\pi$
$499$	−28.0000	−1.25345	−0.626726	+	0.779240i	$0.715605\pi$
$500$	0	0
$501$	0	0
$502$	20.0000i	0.892644i
$503$	− 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
$506$	32.0000	1.42257
$507$	0	0
$508$	− 16.0000i	− 0.709885i
$509$	2.00000	0.0886484	0.0443242	−	0.999017i	$-0.485887\pi$
$509$	2.00000	0.0886484	0.0443242	+	0.999017i	$0.485887\pi$
$510$	0	0
$511$	6.00000	0.265424
$512$	1.00000i	0.0441942i
$513$	0	0
$514$	−14.0000	−0.617514
$515$	0	0
$516$	0	0
$517$	− 16.0000i	− 0.703679i
$518$	2.00000i	0.0878750i
$519$	0	0
$520$	0	0
$521$	18.0000	0.788594	0.394297	−	0.918983i	$-0.370988\pi$
$521$	18.0000	0.788594	0.394297	+	0.918983i	$0.370988\pi$
$522$	0	0
$523$	28.0000i	1.22435i	0.790721	+	0.612177i	$0.209706\pi$
$523$	28.0000i	1.22435i	−0.790721	+	0.612177i	$0.790294\pi$
$524$	4.00000	0.174741
$525$	0	0
$526$	0	0
$527$	− 48.0000i	− 2.09091i
$528$	0	0
$529$	−41.0000	−1.78261
$530$	0	0
$531$	0	0
$532$	0	0
$533$	4.00000i	0.173259i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	2.00000i	0.0862261i
$539$	−4.00000	−0.172292
$540$	0	0
$541$	−10.0000	−0.429934	−0.214967	−	0.976621i	$-0.568964\pi$
$541$	−10.0000	−0.429934	−0.214967	+	0.976621i	$0.568964\pi$
$542$	8.00000i	0.343629i
$543$	0	0
$544$	−6.00000	−0.257248
$545$	0	0
$546$	0	0
$547$	− 32.0000i	− 1.36822i	−0.729378	−	0.684111i	$-0.760191\pi$
$547$	− 32.0000i	− 1.36822i	0.729378	−	0.684111i	$-0.239809\pi$
$548$	6.00000i	0.256307i
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	− 8.00000i	− 0.340195i
$554$	22.0000	0.934690
$555$	0	0
$556$	−16.0000	−0.678551
$557$	− 18.0000i	− 0.762684i	−0.924434	−	0.381342i	$-0.875462\pi$
$557$	− 18.0000i	− 0.762684i	0.924434	−	0.381342i	$-0.124538\pi$
$558$	0	0
$559$	16.0000	0.676728
$560$	0	0
$561$	0	0
$562$	22.0000i	0.928014i
$563$	4.00000i	0.168580i	0.996441	+	0.0842900i	$0.0268622\pi$
$563$	4.00000i	0.168580i	−0.996441	+	0.0842900i	$0.973138\pi$
$564$	0	0
$565$	0	0
$566$	28.0000	1.17693
$567$	0	0
$568$	− 12.0000i	− 0.503509i
$569$	−6.00000	−0.251533	−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000	−0.251533	−0.125767	+	0.992060i	$0.540139\pi$
$570$	0	0
$571$	−20.0000	−0.836974	−0.418487	−	0.908223i	$-0.637439\pi$
$571$	−20.0000	−0.836974	−0.418487	+	0.908223i	$0.637439\pi$
$572$	− 8.00000i	− 0.334497i
$573$	0	0
$574$	2.00000	0.0834784
$575$	0	0
$576$	0	0
$577$	− 14.0000i	− 0.582828i	−0.956597	−	0.291414i	$-0.905874\pi$
$577$	− 14.0000i	− 0.582828i	0.956597	−	0.291414i	$-0.0941257\pi$
$578$	− 19.0000i	− 0.790296i
$579$	0	0
$580$	0	0
$581$	−4.00000	−0.165948
$582$	0	0
$583$	40.0000i	1.65663i
$584$	6.00000	0.248282
$585$	0	0
$586$	30.0000	1.23929
$587$	− 12.0000i	− 0.495293i	−0.968850	−	0.247647i	$-0.920343\pi$
$587$	− 12.0000i	− 0.495293i	0.968850	−	0.247647i	$-0.0796572\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	2.00000i	0.0821995i
$593$	− 38.0000i	− 1.56047i	−0.625485	−	0.780236i	$-0.715099\pi$
$593$	− 38.0000i	− 1.56047i	0.625485	−	0.780236i	$-0.284901\pi$
$594$	0	0
$595$	0	0
$596$	6.00000	0.245770
$597$	0	0
$598$	16.0000i	0.654289i
$599$	20.0000	0.817178	0.408589	−	0.912719i	$-0.366021\pi$
$599$	20.0000	0.817178	0.408589	+	0.912719i	$0.366021\pi$
$600$	0	0
$601$	10.0000	0.407909	0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000	0.407909	0.203954	+	0.978980i	$0.434621\pi$
$602$	− 8.00000i	− 0.326056i
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 40.0000i	− 1.62355i	−0.583970	−	0.811775i	$-0.698502\pi$
$607$	− 40.0000i	− 1.62355i	0.583970	−	0.811775i	$-0.301498\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	8.00000	0.323645
$612$	0	0
$613$	14.0000i	0.565455i	0.959200	+	0.282727i	$0.0912392\pi$
$613$	14.0000i	0.565455i	−0.959200	+	0.282727i	$0.908761\pi$
$614$	−20.0000	−0.807134
$615$	0	0
$616$	−4.00000	−0.161165
$617$	2.00000i	0.0805170i	0.999189	+	0.0402585i	$0.0128181\pi$
$617$	2.00000i	0.0805170i	−0.999189	+	0.0402585i	$0.987182\pi$
$618$	0	0
$619$	−8.00000	−0.321547	−0.160774	−	0.986991i	$-0.551399\pi$
$619$	−8.00000	−0.321547	−0.160774	+	0.986991i	$0.551399\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 14.0000i	− 0.560898i
$624$	0	0
$625$	0	0
$626$	−6.00000	−0.239808
$627$	0	0
$628$	10.0000i	0.399043i
$629$	−12.0000	−0.478471
$630$	0	0
$631$	−8.00000	−0.318475	−0.159237	−	0.987240i	$-0.550904\pi$
$631$	−8.00000	−0.318475	−0.159237	+	0.987240i	$0.550904\pi$
$632$	− 8.00000i	− 0.318223i
$633$	0	0
$634$	−30.0000	−1.19145
$635$	0	0
$636$	0	0
$637$	− 2.00000i	− 0.0792429i
$638$	40.0000i	1.58362i
$639$	0	0
$640$	0	0
$641$	6.00000	0.236986	0.118493	−	0.992955i	$-0.462194\pi$
$641$	6.00000	0.236986	0.118493	+	0.992955i	$0.462194\pi$
$642$	0	0
$643$	4.00000i	0.157745i	0.996885	+	0.0788723i	$0.0251319\pi$
$643$	4.00000i	0.157745i	−0.996885	+	0.0788723i	$0.974868\pi$
$644$	8.00000	0.315244
$645$	0	0
$646$	0	0
$647$	− 28.0000i	− 1.10079i	−0.834903	−	0.550397i	$-0.814476\pi$
$647$	− 28.0000i	− 1.10079i	0.834903	−	0.550397i	$-0.185524\pi$
$648$	0	0
$649$	16.0000	0.628055
$650$	0	0
$651$	0	0
$652$	16.0000i	0.626608i
$653$	− 14.0000i	− 0.547862i	−0.961749	−	0.273931i	$-0.911676\pi$
$653$	− 14.0000i	− 0.547862i	0.961749	−	0.273931i	$-0.0883240\pi$
$654$	0	0
$655$	0	0
$656$	2.00000	0.0780869
$657$	0	0
$658$	− 4.00000i	− 0.155936i
$659$	−12.0000	−0.467454	−0.233727	−	0.972302i	$-0.575092\pi$
$659$	−12.0000	−0.467454	−0.233727	+	0.972302i	$0.575092\pi$
$660$	0	0
$661$	−30.0000	−1.16686	−0.583432	−	0.812162i	$-0.698291\pi$
$661$	−30.0000	−1.16686	−0.583432	+	0.812162i	$0.698291\pi$
$662$	20.0000i	0.777322i
$663$	0	0
$664$	−4.00000	−0.155230
$665$	0	0
$666$	0	0
$667$	− 80.0000i	− 3.09761i
$668$	− 12.0000i	− 0.464294i
$669$	0	0
$670$	0	0
$671$	−24.0000	−0.926510
$672$	0	0
$673$	− 10.0000i	− 0.385472i	−0.981251	−	0.192736i	$-0.938264\pi$
$673$	− 10.0000i	− 0.385472i	0.981251	−	0.192736i	$-0.0617360\pi$
$674$	6.00000	0.231111
$675$	0	0
$676$	−9.00000	−0.346154
$677$	22.0000i	0.845529i	0.906240	+	0.422764i	$0.138940\pi$
$677$	22.0000i	0.845529i	−0.906240	+	0.422764i	$0.861060\pi$
$678$	0	0
$679$	2.00000	0.0767530
$680$	0	0
$681$	0	0
$682$	− 32.0000i	− 1.22534i
$683$	28.0000i	1.07139i	0.844411	+	0.535695i	$0.179950\pi$
$683$	28.0000i	1.07139i	−0.844411	+	0.535695i	$0.820050\pi$
$684$	0	0
$685$	0	0
$686$	−1.00000	−0.0381802
$687$	0	0
$688$	− 8.00000i	− 0.304997i
$689$	−20.0000	−0.761939
$690$	0	0
$691$	0	0		−	1.00000i	$-0.5\pi$
$691$	0	0			1.00000i	$0.5\pi$
$692$	− 18.0000i	− 0.684257i
$693$	0	0
$694$	20.0000	0.759190
$695$	0	0
$696$	0	0
$697$	12.0000i	0.454532i
$698$	− 18.0000i	− 0.681310i
$699$	0	0
$700$	0	0
$701$	−42.0000	−1.58632	−0.793159	−	0.609015i	$-0.791565\pi$
$701$	−42.0000	−1.58632	−0.793159	+	0.609015i	$0.791565\pi$
$702$	0	0
$703$	0	0
$704$	−4.00000	−0.150756
$705$	0	0
$706$	−18.0000	−0.677439
$707$	− 6.00000i	− 0.225653i
$708$	0	0
$709$	−14.0000	−0.525781	−0.262891	−	0.964826i	$-0.584676\pi$
$709$	−14.0000	−0.525781	−0.262891	+	0.964826i	$0.584676\pi$
$710$	0	0
$711$	0	0
$712$	− 14.0000i	− 0.524672i
$713$	64.0000i	2.39682i
$714$	0	0
$715$	0	0
$716$	4.00000	0.149487
$717$	0	0
$718$	− 20.0000i	− 0.746393i
$719$	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
$721$	16.0000	0.595871
$722$	− 19.0000i	− 0.707107i
$723$	0	0
$724$	−18.0000	−0.668965
$725$	0	0
$726$	0	0
$727$	− 16.0000i	− 0.593407i	−0.954970	−	0.296704i	$-0.904113\pi$
$727$	− 16.0000i	− 0.593407i	0.954970	−	0.296704i	$-0.0958873\pi$
$728$	− 2.00000i	− 0.0741249i
$729$	0	0
$730$	0	0
$731$	48.0000	1.77534
$732$	0	0
$733$	− 14.0000i	− 0.517102i	−0.965998	−	0.258551i	$-0.916755\pi$
$733$	− 14.0000i	− 0.517102i	0.965998	−	0.258551i	$-0.0832450\pi$
$734$	−16.0000	−0.590571
$735$	0	0
$736$	8.00000	0.294884
$737$	0	0
$738$	0	0
$739$	4.00000	0.147142	0.0735712	−	0.997290i	$-0.476560\pi$
$739$	4.00000	0.147142	0.0735712	+	0.997290i	$0.476560\pi$
$740$	0	0
$741$	0	0
$742$	10.0000i	0.367112i
$743$	− 8.00000i	− 0.293492i	−0.989174	−	0.146746i	$-0.953120\pi$
$743$	− 8.00000i	− 0.293492i	0.989174	−	0.146746i	$-0.0468799\pi$
$744$	0	0
$745$	0	0
$746$	−30.0000	−1.09838
$747$	0	0
$748$	− 24.0000i	− 0.877527i
$749$	4.00000	0.146157
$750$	0	0
$751$	−16.0000	−0.583848	−0.291924	−	0.956441i	$-0.594295\pi$
$751$	−16.0000	−0.583848	−0.291924	+	0.956441i	$0.594295\pi$
$752$	− 4.00000i	− 0.145865i
$753$	0	0
$754$	−20.0000	−0.728357
$755$	0	0
$756$	0	0
$757$	2.00000i	0.0726912i	0.999339	+	0.0363456i	$0.0115717\pi$
$757$	2.00000i	0.0726912i	−0.999339	+	0.0363456i	$0.988428\pi$
$758$	28.0000i	1.01701i
$759$	0	0
$760$	0	0
$761$	42.0000	1.52250	0.761249	−	0.648459i	$-0.224586\pi$
$761$	42.0000	1.52250	0.761249	+	0.648459i	$0.224586\pi$
$762$	0	0
$763$	− 18.0000i	− 0.651644i
$764$	12.0000	0.434145
$765$	0	0
$766$	28.0000	1.01168
$767$	8.00000i	0.288863i
$768$	0	0
$769$	−10.0000	−0.360609	−0.180305	−	0.983611i	$-0.557708\pi$
$769$	−10.0000	−0.360609	−0.180305	+	0.983611i	$0.557708\pi$
$770$	0	0
$771$	0	0
$772$	10.0000i	0.359908i
$773$	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
$776$	2.00000	0.0717958
$777$	0	0
$778$	10.0000i	0.358517i
$779$	0	0
$780$	0	0
$781$	48.0000	1.71758
$782$	48.0000i	1.71648i
$783$	0	0
$784$	−1.00000	−0.0357143
$785$	0	0
$786$	0	0
$787$	4.00000i	0.142585i	0.997455	+	0.0712923i	$0.0227123\pi$
$787$	4.00000i	0.142585i	−0.997455	+	0.0712923i	$0.977288\pi$
$788$	− 22.0000i	− 0.783718i
$789$	0	0
$790$	0	0
$791$	6.00000	0.213335
$792$	0	0
$793$	− 12.0000i	− 0.426132i
$794$	−14.0000	−0.496841
$795$	0	0
$796$	24.0000	0.850657
$797$	− 18.0000i	− 0.637593i	−0.947823	−	0.318796i	$-0.896721\pi$
$797$	− 18.0000i	− 0.637593i	0.947823	−	0.318796i	$-0.103279\pi$
$798$	0	0
$799$	24.0000	0.849059
$800$	0	0
$801$	0	0
$802$	− 34.0000i	− 1.20058i
$803$	24.0000i	0.846942i
$804$	0	0
$805$	0	0
$806$	16.0000	0.563576
$807$	0	0
$808$	− 6.00000i	− 0.211079i
$809$	42.0000	1.47664	0.738321	−	0.674450i	$-0.235619\pi$
$809$	42.0000	1.47664	0.738321	+	0.674450i	$0.235619\pi$
$810$	0	0
$811$	0	0		−	1.00000i	$-0.5\pi$
$811$	0	0			1.00000i	$0.5\pi$
$812$	10.0000i	0.350931i
$813$	0	0
$814$	−8.00000	−0.280400
$815$	0	0
$816$	0	0
$817$	0	0
$818$	− 26.0000i	− 0.909069i
$819$	0	0
$820$	0	0
$821$	−18.0000	−0.628204	−0.314102	−	0.949389i	$-0.601703\pi$
$821$	−18.0000	−0.628204	−0.314102	+	0.949389i	$0.601703\pi$
$822$	0	0
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	16.0000	0.557386
$825$	0	0
$826$	4.00000	0.139178
$827$	28.0000i	0.973655i	0.873498	+	0.486828i	$0.161846\pi$
$827$	28.0000i	0.973655i	−0.873498	+	0.486828i	$0.838154\pi$
$828$	0	0
$829$	−34.0000	−1.18087	−0.590434	−	0.807086i	$-0.701044\pi$
$829$	−34.0000	−1.18087	−0.590434	+	0.807086i	$0.701044\pi$
$830$	0	0
$831$	0	0
$832$	− 2.00000i	− 0.0693375i
$833$	− 6.00000i	− 0.207888i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	28.0000i	0.967244i
$839$	8.00000	0.276191	0.138095	−	0.990419i	$-0.455902\pi$
$839$	8.00000	0.276191	0.138095	+	0.990419i	$0.455902\pi$
$840$	0	0
$841$	71.0000	2.44828
$842$	− 26.0000i	− 0.896019i
$843$	0	0
$844$	4.00000	0.137686
$845$	0	0
$846$	0	0
$847$	− 5.00000i	− 0.171802i
$848$	10.0000i	0.343401i
$849$	0	0
$850$	0	0
$851$	16.0000	0.548473
$852$	0	0
$853$	− 22.0000i	− 0.753266i	−0.926363	−	0.376633i	$-0.877082\pi$
$853$	− 22.0000i	− 0.753266i	0.926363	−	0.376633i	$-0.122918\pi$
$854$	−6.00000	−0.205316
$855$	0	0
$856$	4.00000	0.136717
$857$	38.0000i	1.29806i	0.760765	+	0.649028i	$0.224824\pi$
$857$	38.0000i	1.29806i	−0.760765	+	0.649028i	$0.775176\pi$
$858$	0	0
$859$	−24.0000	−0.818869	−0.409435	−	0.912339i	$-0.634274\pi$
$859$	−24.0000	−0.818869	−0.409435	+	0.912339i	$0.634274\pi$
$860$	0	0
$861$	0	0
$862$	36.0000i	1.22616i
$863$	− 8.00000i	− 0.272323i	−0.990687	−	0.136162i	$-0.956523\pi$
$863$	− 8.00000i	− 0.272323i	0.990687	−	0.136162i	$-0.0434766\pi$
$864$	0	0
$865$	0	0
$866$	34.0000	1.15537
$867$	0	0
$868$	− 8.00000i	− 0.271538i
$869$	32.0000	1.08553
$870$	0	0
$871$	0	0
$872$	− 18.0000i	− 0.609557i
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	2.00000i	0.0675352i	0.999430	+	0.0337676i	$0.0107506\pi$
$877$	2.00000i	0.0675352i	−0.999430	+	0.0337676i	$0.989249\pi$
$878$	− 32.0000i	− 1.07995i
$879$	0	0
$880$	0	0
$881$	2.00000	0.0673817	0.0336909	−	0.999432i	$-0.489274\pi$
$881$	2.00000	0.0673817	0.0336909	+	0.999432i	$0.489274\pi$
$882$	0	0
$883$	− 48.0000i	− 1.61533i	−0.589643	−	0.807664i	$-0.700731\pi$
$883$	− 48.0000i	− 1.61533i	0.589643	−	0.807664i	$-0.299269\pi$
$884$	12.0000	0.403604
$885$	0	0
$886$	−4.00000	−0.134383
$887$	− 20.0000i	− 0.671534i	−0.941945	−	0.335767i	$-0.891004\pi$
$887$	− 20.0000i	− 0.671534i	0.941945	−	0.335767i	$-0.108996\pi$
$888$	0	0
$889$	16.0000	0.536623
$890$	0	0
$891$	0	0
$892$	− 24.0000i	− 0.803579i
$893$	0	0
$894$	0	0
$895$	0	0
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	26.0000i	0.867631i
$899$	−80.0000	−2.66815
$900$	0	0
$901$	−60.0000	−1.99889
$902$	8.00000i	0.266371i
$903$	0	0
$904$	6.00000	0.199557
$905$	0	0
$906$	0	0
$907$	8.00000i	0.265636i	0.991140	+	0.132818i	$0.0424025\pi$
$907$	8.00000i	0.265636i	−0.991140	+	0.132818i	$0.957597\pi$
$908$	20.0000i	0.663723i
$909$	0	0
$910$	0	0
$911$	20.0000	0.662630	0.331315	−	0.943520i	$-0.392508\pi$
$911$	20.0000	0.662630	0.331315	+	0.943520i	$0.392508\pi$
$912$	0	0
$913$	− 16.0000i	− 0.529523i
$914$	6.00000	0.198462
$915$	0	0
$916$	−22.0000	−0.726900
$917$	4.00000i	0.132092i
$918$	0	0
$919$	24.0000	0.791687	0.395843	−	0.918318i	$-0.370452\pi$
$919$	24.0000	0.791687	0.395843	+	0.918318i	$0.370452\pi$
$920$	0	0
$921$	0	0
$922$	− 18.0000i	− 0.592798i
$923$	24.0000i	0.789970i
$924$	0	0
$925$	0	0
$926$	8.00000	0.262896
$927$	0	0
$928$	10.0000i	0.328266i
$929$	−18.0000	−0.590561	−0.295280	−	0.955411i	$-0.595413\pi$
$929$	−18.0000	−0.590561	−0.295280	+	0.955411i	$0.595413\pi$
$930$	0	0
$931$	0	0
$932$	18.0000i	0.589610i
$933$	0	0
$934$	12.0000	0.392652
$935$	0	0
$936$	0	0
$937$	42.0000i	1.37208i	0.727564	+	0.686040i	$0.240653\pi$
$937$	42.0000i	1.37208i	−0.727564	+	0.686040i	$0.759347\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−18.0000	−0.586783	−0.293392	−	0.955992i	$-0.594784\pi$
$941$	−18.0000	−0.586783	−0.293392	+	0.955992i	$0.594784\pi$
$942$	0	0
$943$	− 16.0000i	− 0.521032i
$944$	4.00000	0.130189
$945$	0	0
$946$	32.0000	1.04041
$947$	− 52.0000i	− 1.68977i	−0.534946	−	0.844886i	$-0.679668\pi$
$947$	− 52.0000i	− 1.68977i	0.534946	−	0.844886i	$-0.320332\pi$
$948$	0	0
$949$	−12.0000	−0.389536
$950$	0	0
$951$	0	0
$952$	− 6.00000i	− 0.194461i
$953$	− 50.0000i	− 1.61966i	−0.586665	−	0.809829i	$-0.699560\pi$
$953$	− 50.0000i	− 1.61966i	0.586665	−	0.809829i	$-0.300440\pi$
$954$	0	0
$955$	0	0
$956$	12.0000	0.388108
$957$	0	0
$958$	− 8.00000i	− 0.258468i
$959$	−6.00000	−0.193750
$960$	0	0
$961$	33.0000	1.06452
$962$	− 4.00000i	− 0.128965i
$963$	0	0
$964$	14.0000	0.450910
$965$	0	0
$966$	0	0
$967$	− 8.00000i	− 0.257263i	−0.991692	−	0.128631i	$-0.958942\pi$
$967$	− 8.00000i	− 0.257263i	0.991692	−	0.128631i	$-0.0410584\pi$
$968$	− 5.00000i	− 0.160706i
$969$	0	0
$970$	0	0
$971$	36.0000	1.15529	0.577647	−	0.816286i	$-0.303971\pi$
$971$	36.0000	1.15529	0.577647	+	0.816286i	$0.303971\pi$
$972$	0	0
$973$	− 16.0000i	− 0.512936i
$974$	−8.00000	−0.256337
$975$	0	0
$976$	−6.00000	−0.192055
$977$	− 6.00000i	− 0.191957i	−0.995383	−	0.0959785i	$-0.969402\pi$
$977$	− 6.00000i	− 0.191957i	0.995383	−	0.0959785i	$-0.0305980\pi$
$978$	0	0
$979$	56.0000	1.78977
$980$	0	0
$981$	0	0
$982$	− 20.0000i	− 0.638226i
$983$	28.0000i	0.893061i	0.894768	+	0.446531i	$0.147341\pi$
$983$	28.0000i	0.893061i	−0.894768	+	0.446531i	$0.852659\pi$
$984$	0	0
$985$	0	0
$986$	−60.0000	−1.91079
$987$	0	0
$988$	0	0
$989$	−64.0000	−2.03508
$990$	0	0
$991$	16.0000	0.508257	0.254128	−	0.967170i	$-0.418211\pi$
$991$	16.0000	0.508257	0.254128	+	0.967170i	$0.418211\pi$
$992$	− 8.00000i	− 0.254000i
$993$	0	0
$994$	12.0000	0.380617
$995$	0	0
$996$	0	0
$997$	− 10.0000i	− 0.316703i	−0.987383	−	0.158352i	$-0.949382\pi$
$997$	− 10.0000i	− 0.316703i	0.987383	−	0.158352i	$-0.0506179\pi$
$998$	− 28.0000i	− 0.886325i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3150.2.g.t.2899.2		2
3.2	odd	2		1050.2.g.f.799.1		2
5.2	odd	4		3150.2.a.t.1.1		1
5.3	odd	4		630.2.a.i.1.1		1
5.4	even	2	inner	3150.2.g.t.2899.1		2
15.2	even	4		1050.2.a.q.1.1		1
15.8	even	4		210.2.a.a.1.1	✓	1
15.14	odd	2		1050.2.g.f.799.2		2
20.3	even	4		5040.2.a.bg.1.1		1
35.13	even	4		4410.2.a.bc.1.1		1
60.23	odd	4		1680.2.a.o.1.1		1
60.47	odd	4		8400.2.a.m.1.1		1
105.23	even	12		1470.2.i.t.361.1		2
105.38	odd	12		1470.2.i.n.961.1		2
105.53	even	12		1470.2.i.t.961.1		2
105.62	odd	4		7350.2.a.bo.1.1		1
105.68	odd	12		1470.2.i.n.361.1		2
105.83	odd	4		1470.2.a.g.1.1		1
120.53	even	4		6720.2.a.cg.1.1		1
120.83	odd	4		6720.2.a.z.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.2.a.a.1.1	✓	1	15.8	even	4
630.2.a.i.1.1		1	5.3	odd	4
1050.2.a.q.1.1		1	15.2	even	4
1050.2.g.f.799.1		2	3.2	odd	2
1050.2.g.f.799.2		2	15.14	odd	2
1470.2.a.g.1.1		1	105.83	odd	4
1470.2.i.n.361.1		2	105.68	odd	12
1470.2.i.n.961.1		2	105.38	odd	12
1470.2.i.t.361.1		2	105.23	even	12
1470.2.i.t.961.1		2	105.53	even	12
1680.2.a.o.1.1		1	60.23	odd	4
3150.2.a.t.1.1		1	5.2	odd	4
3150.2.g.t.2899.1		2	5.4	even	2	inner
3150.2.g.t.2899.2		2	1.1	even	1	trivial
4410.2.a.bc.1.1		1	35.13	even	4
5040.2.a.bg.1.1		1	20.3	even	4
6720.2.a.z.1.1		1	120.83	odd	4
6720.2.a.cg.1.1		1	120.53	even	4
7350.2.a.bo.1.1		1	105.62	odd	4
8400.2.a.m.1.1		1	60.47	odd	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3150.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{4} -1.00000i q^{7} -1.00000i q^{8} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000 q^{4} -1.00000i q^{7} -1.00000i q^{8} +4.00000 q^{11} +2.00000i q^{13} +1.00000 q^{14} +1.00000 q^{16} +6.00000i q^{17} +4.00000i q^{22} -8.00000i q^{23} -2.00000 q^{26} +1.00000i q^{28} +10.0000 q^{29} -8.00000 q^{31} +1.00000i q^{32} -6.00000 q^{34} +2.00000i q^{37} +2.00000 q^{41} -8.00000i q^{43} -4.00000 q^{44} +8.00000 q^{46} -4.00000i q^{47} -1.00000 q^{49} -2.00000i q^{52} +10.0000i q^{53} -1.00000 q^{56} +10.0000i q^{58} +4.00000 q^{59} -6.00000 q^{61} -8.00000i q^{62} -1.00000 q^{64} -6.00000i q^{68} +12.0000 q^{71} +6.00000i q^{73} -2.00000 q^{74} -4.00000i q^{77} +8.00000 q^{79} +2.00000i q^{82} -4.00000i q^{83} +8.00000 q^{86} -4.00000i q^{88} +14.0000 q^{89} +2.00000 q^{91} +8.00000i q^{92} +4.00000 q^{94} +2.00000i q^{97} -1.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4}+O(q^{10}) \) 2 * q - 2 * q^4	\( 2 q - 2 q^{4} + 8 q^{11} + 2 q^{14} + 2 q^{16} - 4 q^{26} + 20 q^{29} - 16 q^{31} - 12 q^{34} + 4 q^{41} - 8 q^{44} + 16 q^{46} - 2 q^{49} - 2 q^{56} + 8 q^{59} - 12 q^{61} - 2 q^{64} + 24 q^{71} - 4 q^{74} + 16 q^{79} + 16 q^{86} + 28 q^{89} + 4 q^{91} + 8 q^{94}+O(q^{100}) \) 2 * q - 2 * q^4 + 8 * q^11 + 2 * q^14 + 2 * q^16 - 4 * q^26 + 20 * q^29 - 16 * q^31 - 12 * q^34 + 4 * q^41 - 8 * q^44 + 16 * q^46 - 2 * q^49 - 2 * q^56 + 8 * q^59 - 12 * q^61 - 2 * q^64 + 24 * q^71 - 4 * q^74 + 16 * q^79 + 16 * q^86 + 28 * q^89 + 4 * q^91 + 8 * q^94

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