LMFDB - Embedded newform 9075.2.a.bf.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9075,2,Mod(1,9075)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9075, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("9075.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9075 = 3 \cdot 5^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9075.a (trivial)

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:	yes
Analytic conductor:	$72.4642398343$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{15}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 15 $ x^2 - 15
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.87298$ of defining polynomial
Character	$\chi$	$=$	9075.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -2.00000 q^{4} +3.87298 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -2.00000 q^{4} +3.87298 q^{7} +1.00000 q^{9} +2.00000 q^{12} -3.87298 q^{13} +4.00000 q^{16} -7.74597 q^{17} -3.87298 q^{19} -3.87298 q^{21} -6.00000 q^{23} -1.00000 q^{27} -7.74597 q^{28} -7.74597 q^{29} -5.00000 q^{31} -2.00000 q^{36} -2.00000 q^{37} +3.87298 q^{39} +7.74597 q^{41} +3.87298 q^{43} +12.0000 q^{47} -4.00000 q^{48} +8.00000 q^{49} +7.74597 q^{51} +7.74597 q^{52} -6.00000 q^{53} +3.87298 q^{57} -11.6190 q^{61} +3.87298 q^{63} -8.00000 q^{64} +7.00000 q^{67} +15.4919 q^{68} +6.00000 q^{69} +12.0000 q^{71} +7.74597 q^{76} -7.74597 q^{79} +1.00000 q^{81} +7.74597 q^{83} +7.74597 q^{84} +7.74597 q^{87} +6.00000 q^{89} -15.0000 q^{91} +12.0000 q^{92} +5.00000 q^{93} -7.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 4 q^{4} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 - 4 * q^4 + 2 * q^9	$ 2 q - 2 q^{3} - 4 q^{4} + 2 q^{9} + 4 q^{12} + 8 q^{16} - 12 q^{23} - 2 q^{27} - 10 q^{31} - 4 q^{36} - 4 q^{37} + 24 q^{47} - 8 q^{48} + 16 q^{49} - 12 q^{53} - 16 q^{64} + 14 q^{67} + 12 q^{69} + 24 q^{71} + 2 q^{81} + 12 q^{89} - 30 q^{91} + 24 q^{92} + 10 q^{93} - 14 q^{97}+O(q^{100}) $ 2 * q - 2 * q^3 - 4 * q^4 + 2 * q^9 + 4 * q^12 + 8 * q^16 - 12 * q^23 - 2 * q^27 - 10 * q^31 - 4 * q^36 - 4 * q^37 + 24 * q^47 - 8 * q^48 + 16 * q^49 - 12 * q^53 - 16 * q^64 + 14 * q^67 + 12 * q^69 + 24 * q^71 + 2 * q^81 + 12 * q^89 - 30 * q^91 + 24 * q^92 + 10 * q^93 - 14 * q^97

Display 100 coefficients 10 coefficients

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 $

$2$	0	0		−	1.00000i	$-0.5\pi$
$2$	0	0			1.00000i	$0.5\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−2.00000	−1.00000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	3.87298	1.46385	0.731925	−	0.681385i	$-0.238622\pi$
$7$	3.87298	1.46385	0.731925	+	0.681385i	$0.238622\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	0	0
$11$	0	0
$12$	2.00000	0.577350
$13$	−3.87298	−1.07417	−0.537086	−	0.843527i	$-0.680475\pi$
$13$	−3.87298	−1.07417	−0.537086	+	0.843527i	$0.680475\pi$
$14$	0	0
$15$	0	0
$16$	4.00000	1.00000
$17$	−7.74597	−1.87867	−0.939336	−	0.342997i	$-0.888558\pi$
$17$	−7.74597	−1.87867	−0.939336	+	0.342997i	$0.888558\pi$
$18$	0	0
$19$	−3.87298	−0.888523	−0.444262	−	0.895897i	$-0.646534\pi$
$19$	−3.87298	−0.888523	−0.444262	+	0.895897i	$0.646534\pi$
$20$	0	0
$21$	−3.87298	−0.845154
$22$	0	0
$23$	−6.00000	−1.25109	−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−6.00000	−1.25109	−0.625543	+	0.780189i	$0.715123\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−1.00000	−0.192450
$28$	−7.74597	−1.46385
$29$	−7.74597	−1.43839	−0.719195	−	0.694808i	$-0.755489\pi$
$29$	−7.74597	−1.43839	−0.719195	+	0.694808i	$0.755489\pi$
$30$	0	0
$31$	−5.00000	−0.898027	−0.449013	−	0.893525i	$-0.648224\pi$
$31$	−5.00000	−0.898027	−0.449013	+	0.893525i	$0.648224\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	−2.00000	−0.333333
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	0	0
$39$	3.87298	0.620174
$40$	0	0
$41$	7.74597	1.20972	0.604858	−	0.796333i	$-0.293230\pi$
$41$	7.74597	1.20972	0.604858	+	0.796333i	$0.293230\pi$
$42$	0	0
$43$	3.87298	0.590624	0.295312	−	0.955401i	$-0.404576\pi$
$43$	3.87298	0.590624	0.295312	+	0.955401i	$0.404576\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	12.0000	1.75038	0.875190	−	0.483779i	$-0.160736\pi$
$47$	12.0000	1.75038	0.875190	+	0.483779i	$0.160736\pi$
$48$	−4.00000	−0.577350
$49$	8.00000	1.14286
$50$	0	0
$51$	7.74597	1.08465
$52$	7.74597	1.07417
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	3.87298	0.512989
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	−11.6190	−1.48765	−0.743827	−	0.668372i	$-0.766991\pi$
$61$	−11.6190	−1.48765	−0.743827	+	0.668372i	$0.766991\pi$
$62$	0	0
$63$	3.87298	0.487950
$64$	−8.00000	−1.00000
$65$	0	0
$66$	0	0
$67$	7.00000	0.855186	0.427593	−	0.903971i	$-0.359362\pi$
$67$	7.00000	0.855186	0.427593	+	0.903971i	$0.359362\pi$
$68$	15.4919	1.87867
$69$	6.00000	0.722315
$70$	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
$73$	0	0		−	1.00000i	$-0.5\pi$
$73$	0	0			1.00000i	$0.5\pi$
$74$	0	0
$75$	0	0
$76$	7.74597	0.888523
$77$	0	0
$78$	0	0
$79$	−7.74597	−0.871489	−0.435745	−	0.900070i	$-0.643515\pi$
$79$	−7.74597	−0.871489	−0.435745	+	0.900070i	$0.643515\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	7.74597	0.850230	0.425115	−	0.905139i	$-0.360234\pi$
$83$	7.74597	0.850230	0.425115	+	0.905139i	$0.360234\pi$
$84$	7.74597	0.845154
$85$	0	0
$86$	0	0
$87$	7.74597	0.830455
$88$	0	0
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	−15.0000	−1.57243
$92$	12.0000	1.25109
$93$	5.00000	0.518476
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−7.00000	−0.710742	−0.355371	−	0.934725i	$-0.615646\pi$
$97$	−7.00000	−0.710742	−0.355371	+	0.934725i	$0.615646\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	15.4919	1.54150	0.770752	−	0.637135i	$-0.219880\pi$
$101$	15.4919	1.54150	0.770752	+	0.637135i	$0.219880\pi$
$102$	0	0
$103$	−16.0000	−1.57653	−0.788263	−	0.615338i	$-0.789020\pi$
$103$	−16.0000	−1.57653	−0.788263	+	0.615338i	$0.789020\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	7.74597	0.748831	0.374415	−	0.927261i	$-0.377843\pi$
$107$	7.74597	0.748831	0.374415	+	0.927261i	$0.377843\pi$
$108$	2.00000	0.192450
$109$	3.87298	0.370965	0.185482	−	0.982648i	$-0.440615\pi$
$109$	3.87298	0.370965	0.185482	+	0.982648i	$0.440615\pi$
$110$	0	0
$111$	2.00000	0.189832
$112$	15.4919	1.46385
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	0	0
$115$	0	0
$116$	15.4919	1.43839
$117$	−3.87298	−0.358057
$118$	0	0
$119$	−30.0000	−2.75010
$120$	0	0
$121$	0	0
$122$	0	0
$123$	−7.74597	−0.698430
$124$	10.0000	0.898027
$125$	0	0
$126$	0	0
$127$	7.74597	0.687343	0.343672	−	0.939090i	$-0.388329\pi$
$127$	7.74597	0.687343	0.343672	+	0.939090i	$0.388329\pi$
$128$	0	0
$129$	−3.87298	−0.340997
$130$	0	0
$131$	7.74597	0.676768	0.338384	−	0.941008i	$-0.390120\pi$
$131$	7.74597	0.676768	0.338384	+	0.941008i	$0.390120\pi$
$132$	0	0
$133$	−15.0000	−1.30066
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−12.0000	−1.02523	−0.512615	−	0.858619i	$-0.671323\pi$
$137$	−12.0000	−1.02523	−0.512615	+	0.858619i	$0.671323\pi$
$138$	0	0
$139$	7.74597	0.657004	0.328502	−	0.944503i	$-0.393456\pi$
$139$	7.74597	0.657004	0.328502	+	0.944503i	$0.393456\pi$
$140$	0	0
$141$	−12.0000	−1.01058
$142$	0	0
$143$	0	0
$144$	4.00000	0.333333
$145$	0	0
$146$	0	0
$147$	−8.00000	−0.659829
$148$	4.00000	0.328798
$149$	7.74597	0.634574	0.317287	−	0.948330i	$-0.397228\pi$
$149$	7.74597	0.634574	0.317287	+	0.948330i	$0.397228\pi$
$150$	0	0
$151$	−11.6190	−0.945537	−0.472768	−	0.881187i	$-0.656745\pi$
$151$	−11.6190	−0.945537	−0.472768	+	0.881187i	$0.656745\pi$
$152$	0	0
$153$	−7.74597	−0.626224
$154$	0	0
$155$	0	0
$156$	−7.74597	−0.620174
$157$	13.0000	1.03751	0.518756	−	0.854922i	$-0.326395\pi$
$157$	13.0000	1.03751	0.518756	+	0.854922i	$0.326395\pi$
$158$	0	0
$159$	6.00000	0.475831
$160$	0	0
$161$	−23.2379	−1.83140
$162$	0	0
$163$	11.0000	0.861586	0.430793	−	0.902451i	$-0.358234\pi$
$163$	11.0000	0.861586	0.430793	+	0.902451i	$0.358234\pi$
$164$	−15.4919	−1.20972
$165$	0	0
$166$	0	0
$167$	−15.4919	−1.19880	−0.599401	−	0.800449i	$-0.704594\pi$
$167$	−15.4919	−1.19880	−0.599401	+	0.800449i	$0.704594\pi$
$168$	0	0
$169$	2.00000	0.153846
$170$	0	0
$171$	−3.87298	−0.296174
$172$	−7.74597	−0.590624
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	6.00000	0.448461	0.224231	−	0.974536i	$-0.428013\pi$
$179$	6.00000	0.448461	0.224231	+	0.974536i	$0.428013\pi$
$180$	0	0
$181$	5.00000	0.371647	0.185824	−	0.982583i	$-0.440505\pi$
$181$	5.00000	0.371647	0.185824	+	0.982583i	$0.440505\pi$
$182$	0	0
$183$	11.6190	0.858898
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	−24.0000	−1.75038
$189$	−3.87298	−0.281718
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	8.00000	0.577350
$193$	11.6190	0.836350	0.418175	−	0.908366i	$-0.362670\pi$
$193$	11.6190	0.836350	0.418175	+	0.908366i	$0.362670\pi$
$194$	0	0
$195$	0	0
$196$	−16.0000	−1.14286
$197$	−15.4919	−1.10375	−0.551877	−	0.833925i	$-0.686088\pi$
$197$	−15.4919	−1.10375	−0.551877	+	0.833925i	$0.686088\pi$
$198$	0	0
$199$	25.0000	1.77220	0.886102	−	0.463491i	$-0.153403\pi$
$199$	25.0000	1.77220	0.886102	+	0.463491i	$0.153403\pi$
$200$	0	0
$201$	−7.00000	−0.493742
$202$	0	0
$203$	−30.0000	−2.10559
$204$	−15.4919	−1.08465
$205$	0	0
$206$	0	0
$207$	−6.00000	−0.417029
$208$	−15.4919	−1.07417
$209$	0	0
$210$	0	0
$211$	−19.3649	−1.33314	−0.666568	−	0.745444i	$-0.732237\pi$
$211$	−19.3649	−1.33314	−0.666568	+	0.745444i	$0.732237\pi$
$212$	12.0000	0.824163
$213$	−12.0000	−0.822226
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−19.3649	−1.31458
$218$	0	0
$219$	0	0
$220$	0	0
$221$	30.0000	2.01802
$222$	0	0
$223$	−19.0000	−1.27233	−0.636167	−	0.771551i	$-0.719481\pi$
$223$	−19.0000	−1.27233	−0.636167	+	0.771551i	$0.719481\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	15.4919	1.02824	0.514118	−	0.857720i	$-0.328119\pi$
$227$	15.4919	1.02824	0.514118	+	0.857720i	$0.328119\pi$
$228$	−7.74597	−0.512989
$229$	−5.00000	−0.330409	−0.165205	−	0.986259i	$-0.552828\pi$
$229$	−5.00000	−0.330409	−0.165205	+	0.986259i	$0.552828\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−15.4919	−1.01491	−0.507455	−	0.861678i	$-0.669414\pi$
$233$	−15.4919	−1.01491	−0.507455	+	0.861678i	$0.669414\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	7.74597	0.503155
$238$	0	0
$239$	23.2379	1.50313	0.751567	−	0.659656i	$-0.229298\pi$
$239$	23.2379	1.50313	0.751567	+	0.659656i	$0.229298\pi$
$240$	0	0
$241$	−27.1109	−1.74637	−0.873183	−	0.487393i	$-0.837948\pi$
$241$	−27.1109	−1.74637	−0.873183	+	0.487393i	$0.837948\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	23.2379	1.48765
$245$	0	0
$246$	0	0
$247$	15.0000	0.954427
$248$	0	0
$249$	−7.74597	−0.490881
$250$	0	0
$251$	18.0000	1.13615	0.568075	−	0.822977i	$-0.307688\pi$
$251$	18.0000	1.13615	0.568075	+	0.822977i	$0.307688\pi$
$252$	−7.74597	−0.487950
$253$	0	0
$254$	0	0
$255$	0	0
$256$	16.0000	1.00000
$257$	12.0000	0.748539	0.374270	−	0.927320i	$-0.377893\pi$
$257$	12.0000	0.748539	0.374270	+	0.927320i	$0.377893\pi$
$258$	0	0
$259$	−7.74597	−0.481311
$260$	0	0
$261$	−7.74597	−0.479463
$262$	0	0
$263$	23.2379	1.43291	0.716455	−	0.697633i	$-0.245763\pi$
$263$	23.2379	1.43291	0.716455	+	0.697633i	$0.245763\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−6.00000	−0.367194
$268$	−14.0000	−0.855186
$269$	30.0000	1.82913	0.914566	−	0.404436i	$-0.132532\pi$
$269$	30.0000	1.82913	0.914566	+	0.404436i	$0.132532\pi$
$270$	0	0
$271$	23.2379	1.41160	0.705801	−	0.708410i	$-0.250587\pi$
$271$	23.2379	1.41160	0.705801	+	0.708410i	$0.250587\pi$
$272$	−30.9839	−1.87867
$273$	15.0000	0.907841
$274$	0	0
$275$	0	0
$276$	−12.0000	−0.722315
$277$	−27.1109	−1.62894	−0.814468	−	0.580209i	$-0.802971\pi$
$277$	−27.1109	−1.62894	−0.814468	+	0.580209i	$0.802971\pi$
$278$	0	0
$279$	−5.00000	−0.299342
$280$	0	0
$281$	0	0		−	1.00000i	$-0.5\pi$
$281$	0	0			1.00000i	$0.5\pi$
$282$	0	0
$283$	3.87298	0.230225	0.115112	−	0.993352i	$-0.463277\pi$
$283$	3.87298	0.230225	0.115112	+	0.993352i	$0.463277\pi$
$284$	−24.0000	−1.42414
$285$	0	0
$286$	0	0
$287$	30.0000	1.77084
$288$	0	0
$289$	43.0000	2.52941
$290$	0	0
$291$	7.00000	0.410347
$292$	0	0
$293$	−23.2379	−1.35757	−0.678786	−	0.734336i	$-0.737494\pi$
$293$	−23.2379	−1.35757	−0.678786	+	0.734336i	$0.737494\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	23.2379	1.34388
$300$	0	0
$301$	15.0000	0.864586
$302$	0	0
$303$	−15.4919	−0.889988
$304$	−15.4919	−0.888523
$305$	0	0
$306$	0	0
$307$	11.6190	0.663129	0.331564	−	0.943433i	$-0.392424\pi$
$307$	11.6190	0.663129	0.331564	+	0.943433i	$0.392424\pi$
$308$	0	0
$309$	16.0000	0.910208
$310$	0	0
$311$	18.0000	1.02069	0.510343	−	0.859971i	$-0.329518\pi$
$311$	18.0000	1.02069	0.510343	+	0.859971i	$0.329518\pi$
$312$	0	0
$313$	1.00000	0.0565233	0.0282617	−	0.999601i	$-0.491003\pi$
$313$	1.00000	0.0565233	0.0282617	+	0.999601i	$0.491003\pi$
$314$	0	0
$315$	0	0
$316$	15.4919	0.871489
$317$	−18.0000	−1.01098	−0.505490	−	0.862832i	$-0.668688\pi$
$317$	−18.0000	−1.01098	−0.505490	+	0.862832i	$0.668688\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−7.74597	−0.432338
$322$	0	0
$323$	30.0000	1.66924
$324$	−2.00000	−0.111111
$325$	0	0
$326$	0	0
$327$	−3.87298	−0.214176
$328$	0	0
$329$	46.4758	2.56229
$330$	0	0
$331$	20.0000	1.09930	0.549650	−	0.835395i	$-0.314761\pi$
$331$	20.0000	1.09930	0.549650	+	0.835395i	$0.314761\pi$
$332$	−15.4919	−0.850230
$333$	−2.00000	−0.109599
$334$	0	0
$335$	0	0
$336$	−15.4919	−0.845154
$337$	−19.3649	−1.05487	−0.527437	−	0.849594i	$-0.676847\pi$
$337$	−19.3649	−1.05487	−0.527437	+	0.849594i	$0.676847\pi$
$338$	0	0
$339$	6.00000	0.325875
$340$	0	0
$341$	0	0
$342$	0	0
$343$	3.87298	0.209121
$344$	0	0
$345$	0	0
$346$	0	0
$347$	15.4919	0.831651	0.415825	−	0.909445i	$-0.363493\pi$
$347$	15.4919	0.831651	0.415825	+	0.909445i	$0.363493\pi$
$348$	−15.4919	−0.830455
$349$	15.4919	0.829264	0.414632	−	0.909989i	$-0.363910\pi$
$349$	15.4919	0.829264	0.414632	+	0.909989i	$0.363910\pi$
$350$	0	0
$351$	3.87298	0.206725
$352$	0	0
$353$	−6.00000	−0.319348	−0.159674	−	0.987170i	$-0.551044\pi$
$353$	−6.00000	−0.319348	−0.159674	+	0.987170i	$0.551044\pi$
$354$	0	0
$355$	0	0
$356$	−12.0000	−0.635999
$357$	30.0000	1.58777
$358$	0	0
$359$	15.4919	0.817633	0.408816	−	0.912617i	$-0.365942\pi$
$359$	15.4919	0.817633	0.408816	+	0.912617i	$0.365942\pi$
$360$	0	0
$361$	−4.00000	−0.210526
$362$	0	0
$363$	0	0
$364$	30.0000	1.57243
$365$	0	0
$366$	0	0
$367$	−13.0000	−0.678594	−0.339297	−	0.940679i	$-0.610189\pi$
$367$	−13.0000	−0.678594	−0.339297	+	0.940679i	$0.610189\pi$
$368$	−24.0000	−1.25109
$369$	7.74597	0.403239
$370$	0	0
$371$	−23.2379	−1.20645
$372$	−10.0000	−0.518476
$373$	3.87298	0.200535	0.100268	−	0.994960i	$-0.468030\pi$
$373$	3.87298	0.200535	0.100268	+	0.994960i	$0.468030\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	30.0000	1.54508
$378$	0	0
$379$	5.00000	0.256833	0.128416	−	0.991720i	$-0.459011\pi$
$379$	5.00000	0.256833	0.128416	+	0.991720i	$0.459011\pi$
$380$	0	0
$381$	−7.74597	−0.396838
$382$	0	0
$383$	6.00000	0.306586	0.153293	−	0.988181i	$-0.451012\pi$
$383$	6.00000	0.306586	0.153293	+	0.988181i	$0.451012\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	3.87298	0.196875
$388$	14.0000	0.710742
$389$	30.0000	1.52106	0.760530	−	0.649303i	$-0.224939\pi$
$389$	30.0000	1.52106	0.760530	+	0.649303i	$0.224939\pi$
$390$	0	0
$391$	46.4758	2.35038
$392$	0	0
$393$	−7.74597	−0.390732
$394$	0	0
$395$	0	0
$396$	0	0
$397$	23.0000	1.15434	0.577168	−	0.816625i	$-0.304158\pi$
$397$	23.0000	1.15434	0.577168	+	0.816625i	$0.304158\pi$
$398$	0	0
$399$	15.0000	0.750939
$400$	0	0
$401$	−30.0000	−1.49813	−0.749064	−	0.662497i	$-0.769497\pi$
$401$	−30.0000	−1.49813	−0.749064	+	0.662497i	$0.769497\pi$
$402$	0	0
$403$	19.3649	0.964635
$404$	−30.9839	−1.54150
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−27.1109	−1.34055	−0.670273	−	0.742114i	$-0.733823\pi$
$409$	−27.1109	−1.34055	−0.670273	+	0.742114i	$0.733823\pi$
$410$	0	0
$411$	12.0000	0.591916
$412$	32.0000	1.57653
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−7.74597	−0.379322
$418$	0	0
$419$	24.0000	1.17248	0.586238	−	0.810139i	$-0.300608\pi$
$419$	24.0000	1.17248	0.586238	+	0.810139i	$0.300608\pi$
$420$	0	0
$421$	10.0000	0.487370	0.243685	−	0.969854i	$-0.421644\pi$
$421$	10.0000	0.487370	0.243685	+	0.969854i	$0.421644\pi$
$422$	0	0
$423$	12.0000	0.583460
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−45.0000	−2.17770
$428$	−15.4919	−0.748831
$429$	0	0
$430$	0	0
$431$	7.74597	0.373110	0.186555	−	0.982445i	$-0.440268\pi$
$431$	7.74597	0.373110	0.186555	+	0.982445i	$0.440268\pi$
$432$	−4.00000	−0.192450
$433$	11.0000	0.528626	0.264313	−	0.964437i	$-0.414855\pi$
$433$	11.0000	0.528626	0.264313	+	0.964437i	$0.414855\pi$
$434$	0	0
$435$	0	0
$436$	−7.74597	−0.370965
$437$	23.2379	1.11162
$438$	0	0
$439$	−34.8569	−1.66363	−0.831813	−	0.555055i	$-0.812697\pi$
$439$	−34.8569	−1.66363	−0.831813	+	0.555055i	$0.812697\pi$
$440$	0	0
$441$	8.00000	0.380952
$442$	0	0
$443$	−6.00000	−0.285069	−0.142534	−	0.989790i	$-0.545525\pi$
$443$	−6.00000	−0.285069	−0.142534	+	0.989790i	$0.545525\pi$
$444$	−4.00000	−0.189832
$445$	0	0
$446$	0	0
$447$	−7.74597	−0.366372
$448$	−30.9839	−1.46385
$449$	−36.0000	−1.69895	−0.849473	−	0.527633i	$-0.823080\pi$
$449$	−36.0000	−1.69895	−0.849473	+	0.527633i	$0.823080\pi$
$450$	0	0
$451$	0	0
$452$	12.0000	0.564433
$453$	11.6190	0.545906
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−15.4919	−0.724682	−0.362341	−	0.932046i	$-0.618022\pi$
$457$	−15.4919	−0.724682	−0.362341	+	0.932046i	$0.618022\pi$
$458$	0	0
$459$	7.74597	0.361551
$460$	0	0
$461$	7.74597	0.360766	0.180383	−	0.983596i	$-0.442266\pi$
$461$	7.74597	0.360766	0.180383	+	0.983596i	$0.442266\pi$
$462$	0	0
$463$	−4.00000	−0.185896	−0.0929479	−	0.995671i	$-0.529629\pi$
$463$	−4.00000	−0.185896	−0.0929479	+	0.995671i	$0.529629\pi$
$464$	−30.9839	−1.43839
$465$	0	0
$466$	0	0
$467$	−42.0000	−1.94353	−0.971764	−	0.235954i	$-0.924178\pi$
$467$	−42.0000	−1.94353	−0.971764	+	0.235954i	$0.924178\pi$
$468$	7.74597	0.358057
$469$	27.1109	1.25186
$470$	0	0
$471$	−13.0000	−0.599008
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	60.0000	2.75010
$477$	−6.00000	−0.274721
$478$	0	0
$479$	−30.9839	−1.41569	−0.707845	−	0.706368i	$-0.750332\pi$
$479$	−30.9839	−1.41569	−0.707845	+	0.706368i	$0.750332\pi$
$480$	0	0
$481$	7.74597	0.353186
$482$	0	0
$483$	23.2379	1.05736
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−23.0000	−1.04223	−0.521115	−	0.853487i	$-0.674484\pi$
$487$	−23.0000	−1.04223	−0.521115	+	0.853487i	$0.674484\pi$
$488$	0	0
$489$	−11.0000	−0.497437
$490$	0	0
$491$	7.74597	0.349571	0.174785	−	0.984607i	$-0.444077\pi$
$491$	7.74597	0.349571	0.174785	+	0.984607i	$0.444077\pi$
$492$	15.4919	0.698430
$493$	60.0000	2.70226
$494$	0	0
$495$	0	0
$496$	−20.0000	−0.898027
$497$	46.4758	2.08472
$498$	0	0
$499$	25.0000	1.11915	0.559577	−	0.828778i	$-0.310964\pi$
$499$	25.0000	1.11915	0.559577	+	0.828778i	$0.310964\pi$
$500$	0	0
$501$	15.4919	0.692129
$502$	0	0
$503$	7.74597	0.345376	0.172688	−	0.984977i	$-0.444755\pi$
$503$	7.74597	0.345376	0.172688	+	0.984977i	$0.444755\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−2.00000	−0.0888231
$508$	−15.4919	−0.687343
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	3.87298	0.170996
$514$	0	0
$515$	0	0
$516$	7.74597	0.340997
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	30.0000	1.31432	0.657162	−	0.753749i	$-0.271757\pi$
$521$	30.0000	1.31432	0.657162	+	0.753749i	$0.271757\pi$
$522$	0	0
$523$	−11.6190	−0.508061	−0.254031	−	0.967196i	$-0.581756\pi$
$523$	−11.6190	−0.508061	−0.254031	+	0.967196i	$0.581756\pi$
$524$	−15.4919	−0.676768
$525$	0	0
$526$	0	0
$527$	38.7298	1.68710
$528$	0	0
$529$	13.0000	0.565217
$530$	0	0
$531$	0	0
$532$	30.0000	1.30066
$533$	−30.0000	−1.29944
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−6.00000	−0.258919
$538$	0	0
$539$	0	0
$540$	0	0
$541$	42.6028	1.83164	0.915819	−	0.401591i	$-0.131543\pi$
$541$	42.6028	1.83164	0.915819	+	0.401591i	$0.131543\pi$
$542$	0	0
$543$	−5.00000	−0.214571
$544$	0	0
$545$	0	0
$546$	0	0
$547$	23.2379	0.993581	0.496790	−	0.867871i	$-0.334512\pi$
$547$	23.2379	0.993581	0.496790	+	0.867871i	$0.334512\pi$
$548$	24.0000	1.02523
$549$	−11.6190	−0.495885
$550$	0	0
$551$	30.0000	1.27804
$552$	0	0
$553$	−30.0000	−1.27573
$554$	0	0
$555$	0	0
$556$	−15.4919	−0.657004
$557$	−15.4919	−0.656414	−0.328207	−	0.944606i	$-0.606444\pi$
$557$	−15.4919	−0.656414	−0.328207	+	0.944606i	$0.606444\pi$
$558$	0	0
$559$	−15.0000	−0.634432
$560$	0	0
$561$	0	0
$562$	0	0
$563$	7.74597	0.326454	0.163227	−	0.986589i	$-0.447810\pi$
$563$	7.74597	0.326454	0.163227	+	0.986589i	$0.447810\pi$
$564$	24.0000	1.01058
$565$	0	0
$566$	0	0
$567$	3.87298	0.162650
$568$	0	0
$569$	−23.2379	−0.974183	−0.487092	−	0.873351i	$-0.661942\pi$
$569$	−23.2379	−0.974183	−0.487092	+	0.873351i	$0.661942\pi$
$570$	0	0
$571$	19.3649	0.810397	0.405198	−	0.914229i	$-0.367202\pi$
$571$	19.3649	0.810397	0.405198	+	0.914229i	$0.367202\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	−8.00000	−0.333333
$577$	37.0000	1.54033	0.770165	−	0.637845i	$-0.220174\pi$
$577$	37.0000	1.54033	0.770165	+	0.637845i	$0.220174\pi$
$578$	0	0
$579$	−11.6190	−0.482867
$580$	0	0
$581$	30.0000	1.24461
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	18.0000	0.742940	0.371470	−	0.928445i	$-0.378854\pi$
$587$	18.0000	0.742940	0.371470	+	0.928445i	$0.378854\pi$
$588$	16.0000	0.659829
$589$	19.3649	0.797917
$590$	0	0
$591$	15.4919	0.637253
$592$	−8.00000	−0.328798
$593$	−23.2379	−0.954266	−0.477133	−	0.878831i	$-0.658324\pi$
$593$	−23.2379	−0.954266	−0.477133	+	0.878831i	$0.658324\pi$
$594$	0	0
$595$	0	0
$596$	−15.4919	−0.634574
$597$	−25.0000	−1.02318
$598$	0	0
$599$	24.0000	0.980613	0.490307	−	0.871550i	$-0.336885\pi$
$599$	24.0000	0.980613	0.490307	+	0.871550i	$0.336885\pi$
$600$	0	0
$601$	−27.1109	−1.10588	−0.552938	−	0.833222i	$-0.686493\pi$
$601$	−27.1109	−1.10588	−0.552938	+	0.833222i	$0.686493\pi$
$602$	0	0
$603$	7.00000	0.285062
$604$	23.2379	0.945537
$605$	0	0
$606$	0	0
$607$	38.7298	1.57200	0.785998	−	0.618229i	$-0.212150\pi$
$607$	38.7298	1.57200	0.785998	+	0.618229i	$0.212150\pi$
$608$	0	0
$609$	30.0000	1.21566
$610$	0	0
$611$	−46.4758	−1.88021
$612$	15.4919	0.626224
$613$	46.4758	1.87714	0.938570	−	0.345089i	$-0.112151\pi$
$613$	46.4758	1.87714	0.938570	+	0.345089i	$0.112151\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	12.0000	0.483102	0.241551	−	0.970388i	$-0.422344\pi$
$617$	12.0000	0.483102	0.241551	+	0.970388i	$0.422344\pi$
$618$	0	0
$619$	−1.00000	−0.0401934	−0.0200967	−	0.999798i	$-0.506397\pi$
$619$	−1.00000	−0.0401934	−0.0200967	+	0.999798i	$0.506397\pi$
$620$	0	0
$621$	6.00000	0.240772
$622$	0	0
$623$	23.2379	0.931007
$624$	15.4919	0.620174
$625$	0	0
$626$	0	0
$627$	0	0
$628$	−26.0000	−1.03751
$629$	15.4919	0.617704
$630$	0	0
$631$	7.00000	0.278666	0.139333	−	0.990246i	$-0.455504\pi$
$631$	7.00000	0.278666	0.139333	+	0.990246i	$0.455504\pi$
$632$	0	0
$633$	19.3649	0.769686
$634$	0	0
$635$	0	0
$636$	−12.0000	−0.475831
$637$	−30.9839	−1.22763
$638$	0	0
$639$	12.0000	0.474713
$640$	0	0
$641$	30.0000	1.18493	0.592464	−	0.805597i	$-0.298155\pi$
$641$	30.0000	1.18493	0.592464	+	0.805597i	$0.298155\pi$
$642$	0	0
$643$	−16.0000	−0.630978	−0.315489	−	0.948929i	$-0.602169\pi$
$643$	−16.0000	−0.630978	−0.315489	+	0.948929i	$0.602169\pi$
$644$	46.4758	1.83140
$645$	0	0
$646$	0	0
$647$	12.0000	0.471769	0.235884	−	0.971781i	$-0.424201\pi$
$647$	12.0000	0.471769	0.235884	+	0.971781i	$0.424201\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	19.3649	0.758971
$652$	−22.0000	−0.861586
$653$	24.0000	0.939193	0.469596	−	0.882881i	$-0.344399\pi$
$653$	24.0000	0.939193	0.469596	+	0.882881i	$0.344399\pi$
$654$	0	0
$655$	0	0
$656$	30.9839	1.20972
$657$	0	0
$658$	0	0
$659$	−46.4758	−1.81044	−0.905220	−	0.424943i	$-0.860294\pi$
$659$	−46.4758	−1.81044	−0.905220	+	0.424943i	$0.860294\pi$
$660$	0	0
$661$	−2.00000	−0.0777910	−0.0388955	−	0.999243i	$-0.512384\pi$
$661$	−2.00000	−0.0777910	−0.0388955	+	0.999243i	$0.512384\pi$
$662$	0	0
$663$	−30.0000	−1.16510
$664$	0	0
$665$	0	0
$666$	0	0
$667$	46.4758	1.79955
$668$	30.9839	1.19880
$669$	19.0000	0.734582
$670$	0	0
$671$	0	0
$672$	0	0
$673$	30.9839	1.19434	0.597170	−	0.802115i	$-0.296292\pi$
$673$	30.9839	1.19434	0.597170	+	0.802115i	$0.296292\pi$
$674$	0	0
$675$	0	0
$676$	−4.00000	−0.153846
$677$	−23.2379	−0.893105	−0.446553	−	0.894757i	$-0.647348\pi$
$677$	−23.2379	−0.893105	−0.446553	+	0.894757i	$0.647348\pi$
$678$	0	0
$679$	−27.1109	−1.04042
$680$	0	0
$681$	−15.4919	−0.593652
$682$	0	0
$683$	6.00000	0.229584	0.114792	−	0.993390i	$-0.463380\pi$
$683$	6.00000	0.229584	0.114792	+	0.993390i	$0.463380\pi$
$684$	7.74597	0.296174
$685$	0	0
$686$	0	0
$687$	5.00000	0.190762
$688$	15.4919	0.590624
$689$	23.2379	0.885293
$690$	0	0
$691$	−28.0000	−1.06517	−0.532585	−	0.846376i	$-0.678779\pi$
$691$	−28.0000	−1.06517	−0.532585	+	0.846376i	$0.678779\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−60.0000	−2.27266
$698$	0	0
$699$	15.4919	0.585959
$700$	0	0
$701$	30.9839	1.17024	0.585122	−	0.810945i	$-0.301047\pi$
$701$	30.9839	1.17024	0.585122	+	0.810945i	$0.301047\pi$
$702$	0	0
$703$	7.74597	0.292145
$704$	0	0
$705$	0	0
$706$	0	0
$707$	60.0000	2.25653
$708$	0	0
$709$	19.0000	0.713560	0.356780	−	0.934188i	$-0.383875\pi$
$709$	19.0000	0.713560	0.356780	+	0.934188i	$0.383875\pi$
$710$	0	0
$711$	−7.74597	−0.290496
$712$	0	0
$713$	30.0000	1.12351
$714$	0	0
$715$	0	0
$716$	−12.0000	−0.448461
$717$	−23.2379	−0.867835
$718$	0	0
$719$	6.00000	0.223762	0.111881	−	0.993722i	$-0.464312\pi$
$719$	6.00000	0.223762	0.111881	+	0.993722i	$0.464312\pi$
$720$	0	0
$721$	−61.9677	−2.30780
$722$	0	0
$723$	27.1109	1.00826
$724$	−10.0000	−0.371647
$725$	0	0
$726$	0	0
$727$	−17.0000	−0.630495	−0.315248	−	0.949009i	$-0.602088\pi$
$727$	−17.0000	−0.630495	−0.315248	+	0.949009i	$0.602088\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−30.0000	−1.10959
$732$	−23.2379	−0.858898
$733$	−30.9839	−1.14442	−0.572208	−	0.820109i	$-0.693913\pi$
$733$	−30.9839	−1.14442	−0.572208	+	0.820109i	$0.693913\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	7.74597	0.284940	0.142470	−	0.989799i	$-0.454496\pi$
$739$	7.74597	0.284940	0.142470	+	0.989799i	$0.454496\pi$
$740$	0	0
$741$	−15.0000	−0.551039
$742$	0	0
$743$	−23.2379	−0.852516	−0.426258	−	0.904602i	$-0.640168\pi$
$743$	−23.2379	−0.852516	−0.426258	+	0.904602i	$0.640168\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	7.74597	0.283410
$748$	0	0
$749$	30.0000	1.09618
$750$	0	0
$751$	−8.00000	−0.291924	−0.145962	−	0.989290i	$-0.546628\pi$
$751$	−8.00000	−0.291924	−0.145962	+	0.989290i	$0.546628\pi$
$752$	48.0000	1.75038
$753$	−18.0000	−0.655956
$754$	0	0
$755$	0	0
$756$	7.74597	0.281718
$757$	−17.0000	−0.617876	−0.308938	−	0.951082i	$-0.599973\pi$
$757$	−17.0000	−0.617876	−0.308938	+	0.951082i	$0.599973\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−54.2218	−1.96554	−0.982769	−	0.184839i	$-0.940824\pi$
$761$	−54.2218	−1.96554	−0.982769	+	0.184839i	$0.940824\pi$
$762$	0	0
$763$	15.0000	0.543036
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	−16.0000	−0.577350
$769$	19.3649	0.698317	0.349158	−	0.937064i	$-0.386468\pi$
$769$	19.3649	0.698317	0.349158	+	0.937064i	$0.386468\pi$
$770$	0	0
$771$	−12.0000	−0.432169
$772$	−23.2379	−0.836350
$773$	6.00000	0.215805	0.107903	−	0.994161i	$-0.465587\pi$
$773$	6.00000	0.215805	0.107903	+	0.994161i	$0.465587\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	7.74597	0.277885
$778$	0	0
$779$	−30.0000	−1.07486
$780$	0	0
$781$	0	0
$782$	0	0
$783$	7.74597	0.276818
$784$	32.0000	1.14286
$785$	0	0
$786$	0	0
$787$	−11.6190	−0.414171	−0.207085	−	0.978323i	$-0.566398\pi$
$787$	−11.6190	−0.414171	−0.207085	+	0.978323i	$0.566398\pi$
$788$	30.9839	1.10375
$789$	−23.2379	−0.827291
$790$	0	0
$791$	−23.2379	−0.826245
$792$	0	0
$793$	45.0000	1.59800
$794$	0	0
$795$	0	0
$796$	−50.0000	−1.77220
$797$	−12.0000	−0.425062	−0.212531	−	0.977154i	$-0.568171\pi$
$797$	−12.0000	−0.425062	−0.212531	+	0.977154i	$0.568171\pi$
$798$	0	0
$799$	−92.9516	−3.28839
$800$	0	0
$801$	6.00000	0.212000
$802$	0	0
$803$	0	0
$804$	14.0000	0.493742
$805$	0	0
$806$	0	0
$807$	−30.0000	−1.05605
$808$	0	0
$809$	−15.4919	−0.544667	−0.272334	−	0.962203i	$-0.587795\pi$
$809$	−15.4919	−0.544667	−0.272334	+	0.962203i	$0.587795\pi$
$810$	0	0
$811$	−3.87298	−0.135999	−0.0679994	−	0.997685i	$-0.521662\pi$
$811$	−3.87298	−0.135999	−0.0679994	+	0.997685i	$0.521662\pi$
$812$	60.0000	2.10559
$813$	−23.2379	−0.814989
$814$	0	0
$815$	0	0
$816$	30.9839	1.08465
$817$	−15.0000	−0.524784
$818$	0	0
$819$	−15.0000	−0.524142
$820$	0	0
$821$	46.4758	1.62202	0.811008	−	0.585035i	$-0.198919\pi$
$821$	46.4758	1.62202	0.811008	+	0.585035i	$0.198919\pi$
$822$	0	0
$823$	−31.0000	−1.08059	−0.540296	−	0.841475i	$-0.681688\pi$
$823$	−31.0000	−1.08059	−0.540296	+	0.841475i	$0.681688\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	23.2379	0.808061	0.404030	−	0.914746i	$-0.367609\pi$
$827$	23.2379	0.808061	0.404030	+	0.914746i	$0.367609\pi$
$828$	12.0000	0.417029
$829$	14.0000	0.486240	0.243120	−	0.969996i	$-0.421829\pi$
$829$	14.0000	0.486240	0.243120	+	0.969996i	$0.421829\pi$
$830$	0	0
$831$	27.1109	0.940466
$832$	30.9839	1.07417
$833$	−61.9677	−2.14705
$834$	0	0
$835$	0	0
$836$	0	0
$837$	5.00000	0.172825
$838$	0	0
$839$	−30.0000	−1.03572	−0.517858	−	0.855467i	$-0.673270\pi$
$839$	−30.0000	−1.03572	−0.517858	+	0.855467i	$0.673270\pi$
$840$	0	0
$841$	31.0000	1.06897
$842$	0	0
$843$	0	0
$844$	38.7298	1.33314
$845$	0	0
$846$	0	0
$847$	0	0
$848$	−24.0000	−0.824163
$849$	−3.87298	−0.132920
$850$	0	0
$851$	12.0000	0.411355
$852$	24.0000	0.822226
$853$	−11.6190	−0.397825	−0.198913	−	0.980017i	$-0.563741\pi$
$853$	−11.6190	−0.397825	−0.198913	+	0.980017i	$0.563741\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−23.2379	−0.793792	−0.396896	−	0.917864i	$-0.629913\pi$
$857$	−23.2379	−0.793792	−0.396896	+	0.917864i	$0.629913\pi$
$858$	0	0
$859$	44.0000	1.50126	0.750630	−	0.660722i	$-0.229750\pi$
$859$	44.0000	1.50126	0.750630	+	0.660722i	$0.229750\pi$
$860$	0	0
$861$	−30.0000	−1.02240
$862$	0	0
$863$	36.0000	1.22545	0.612727	−	0.790295i	$-0.290072\pi$
$863$	36.0000	1.22545	0.612727	+	0.790295i	$0.290072\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−43.0000	−1.46036
$868$	38.7298	1.31458
$869$	0	0
$870$	0	0
$871$	−27.1109	−0.918617
$872$	0	0
$873$	−7.00000	−0.236914
$874$	0	0
$875$	0	0
$876$	0	0
$877$	19.3649	0.653907	0.326953	−	0.945040i	$-0.393978\pi$
$877$	19.3649	0.653907	0.326953	+	0.945040i	$0.393978\pi$
$878$	0	0
$879$	23.2379	0.783795
$880$	0	0
$881$	−18.0000	−0.606435	−0.303218	−	0.952921i	$-0.598061\pi$
$881$	−18.0000	−0.606435	−0.303218	+	0.952921i	$0.598061\pi$
$882$	0	0
$883$	−29.0000	−0.975928	−0.487964	−	0.872864i	$-0.662260\pi$
$883$	−29.0000	−0.975928	−0.487964	+	0.872864i	$0.662260\pi$
$884$	−60.0000	−2.01802
$885$	0	0
$886$	0	0
$887$	7.74597	0.260084	0.130042	−	0.991508i	$-0.458489\pi$
$887$	7.74597	0.260084	0.130042	+	0.991508i	$0.458489\pi$
$888$	0	0
$889$	30.0000	1.00617
$890$	0	0
$891$	0	0
$892$	38.0000	1.27233
$893$	−46.4758	−1.55525
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−23.2379	−0.775891
$898$	0	0
$899$	38.7298	1.29171
$900$	0	0
$901$	46.4758	1.54833
$902$	0	0
$903$	−15.0000	−0.499169
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−28.0000	−0.929725	−0.464862	−	0.885383i	$-0.653896\pi$
$907$	−28.0000	−0.929725	−0.464862	+	0.885383i	$0.653896\pi$
$908$	−30.9839	−1.02824
$909$	15.4919	0.513835
$910$	0	0
$911$	30.0000	0.993944	0.496972	−	0.867766i	$-0.334445\pi$
$911$	30.0000	0.993944	0.496972	+	0.867766i	$0.334445\pi$
$912$	15.4919	0.512989
$913$	0	0
$914$	0	0
$915$	0	0
$916$	10.0000	0.330409
$917$	30.0000	0.990687
$918$	0	0
$919$	11.6190	0.383274	0.191637	−	0.981466i	$-0.438620\pi$
$919$	11.6190	0.383274	0.191637	+	0.981466i	$0.438620\pi$
$920$	0	0
$921$	−11.6190	−0.382857
$922$	0	0
$923$	−46.4758	−1.52977
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−16.0000	−0.525509
$928$	0	0
$929$	60.0000	1.96854	0.984268	−	0.176682i	$-0.0565363\pi$
$929$	60.0000	1.96854	0.984268	+	0.176682i	$0.0565363\pi$
$930$	0	0
$931$	−30.9839	−1.01546
$932$	30.9839	1.01491
$933$	−18.0000	−0.589294
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−27.1109	−0.885674	−0.442837	−	0.896602i	$-0.646028\pi$
$937$	−27.1109	−0.885674	−0.442837	+	0.896602i	$0.646028\pi$
$938$	0	0
$939$	−1.00000	−0.0326338
$940$	0	0
$941$	30.9839	1.01005	0.505023	−	0.863106i	$-0.331484\pi$
$941$	30.9839	1.01005	0.505023	+	0.863106i	$0.331484\pi$
$942$	0	0
$943$	−46.4758	−1.51346
$944$	0	0
$945$	0	0
$946$	0	0
$947$	18.0000	0.584921	0.292461	−	0.956278i	$-0.405526\pi$
$947$	18.0000	0.584921	0.292461	+	0.956278i	$0.405526\pi$
$948$	−15.4919	−0.503155
$949$	0	0
$950$	0	0
$951$	18.0000	0.583690
$952$	0	0
$953$	7.74597	0.250916	0.125458	−	0.992099i	$-0.459960\pi$
$953$	7.74597	0.250916	0.125458	+	0.992099i	$0.459960\pi$
$954$	0	0
$955$	0	0
$956$	−46.4758	−1.50313
$957$	0	0
$958$	0	0
$959$	−46.4758	−1.50078
$960$	0	0
$961$	−6.00000	−0.193548
$962$	0	0
$963$	7.74597	0.249610
$964$	54.2218	1.74637
$965$	0	0
$966$	0	0
$967$	23.2379	0.747280	0.373640	−	0.927574i	$-0.378109\pi$
$967$	23.2379	0.747280	0.373640	+	0.927574i	$0.378109\pi$
$968$	0	0
$969$	−30.0000	−0.963739
$970$	0	0
$971$	−30.0000	−0.962746	−0.481373	−	0.876516i	$-0.659862\pi$
$971$	−30.0000	−0.962746	−0.481373	+	0.876516i	$0.659862\pi$
$972$	2.00000	0.0641500
$973$	30.0000	0.961756
$974$	0	0
$975$	0	0
$976$	−46.4758	−1.48765
$977$	42.0000	1.34370	0.671850	−	0.740688i	$-0.265500\pi$
$977$	42.0000	1.34370	0.671850	+	0.740688i	$0.265500\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	3.87298	0.123655
$982$	0	0
$983$	24.0000	0.765481	0.382741	−	0.923856i	$-0.374980\pi$
$983$	24.0000	0.765481	0.382741	+	0.923856i	$0.374980\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−46.4758	−1.47934
$988$	−30.0000	−0.954427
$989$	−23.2379	−0.738922
$990$	0	0
$991$	55.0000	1.74713	0.873566	−	0.486705i	$-0.161801\pi$
$991$	55.0000	1.74713	0.873566	+	0.486705i	$0.161801\pi$
$992$	0	0
$993$	−20.0000	−0.634681
$994$	0	0
$995$	0	0
$996$	15.4919	0.490881
$997$	−46.4758	−1.47190	−0.735952	−	0.677034i	$-0.763265\pi$
$997$	−46.4758	−1.47190	−0.735952	+	0.677034i	$0.763265\pi$
$998$	0	0
$999$	2.00000	0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9075.2.a.bf.1.2	yes	2
5.4	even	2		9075.2.a.bm.1.1	yes	2
11.10	odd	2	inner	9075.2.a.bf.1.1	✓	2
55.54	odd	2		9075.2.a.bm.1.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9075.2.a.bf.1.1	✓	2	11.10	odd	2	inner
9075.2.a.bf.1.2	yes	2	1.1	even	1	trivial
9075.2.a.bm.1.1	yes	2	5.4	even	2
9075.2.a.bm.1.2	yes	2	55.54	odd	2

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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:	\( N \)	\(=\)	\( 9075 = 3 \cdot 5^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9075.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -2.00000 q^{4} +3.87298 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -2.00000 q^{4} +3.87298 q^{7} +1.00000 q^{9} +2.00000 q^{12} -3.87298 q^{13} +4.00000 q^{16} -7.74597 q^{17} -3.87298 q^{19} -3.87298 q^{21} -6.00000 q^{23} -1.00000 q^{27} -7.74597 q^{28} -7.74597 q^{29} -5.00000 q^{31} -2.00000 q^{36} -2.00000 q^{37} +3.87298 q^{39} +7.74597 q^{41} +3.87298 q^{43} +12.0000 q^{47} -4.00000 q^{48} +8.00000 q^{49} +7.74597 q^{51} +7.74597 q^{52} -6.00000 q^{53} +3.87298 q^{57} -11.6190 q^{61} +3.87298 q^{63} -8.00000 q^{64} +7.00000 q^{67} +15.4919 q^{68} +6.00000 q^{69} +12.0000 q^{71} +7.74597 q^{76} -7.74597 q^{79} +1.00000 q^{81} +7.74597 q^{83} +7.74597 q^{84} +7.74597 q^{87} +6.00000 q^{89} -15.0000 q^{91} +12.0000 q^{92} +5.00000 q^{93} -7.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 4 q^{4} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 - 4 * q^4 + 2 * q^9	\( 2 q - 2 q^{3} - 4 q^{4} + 2 q^{9} + 4 q^{12} + 8 q^{16} - 12 q^{23} - 2 q^{27} - 10 q^{31} - 4 q^{36} - 4 q^{37} + 24 q^{47} - 8 q^{48} + 16 q^{49} - 12 q^{53} - 16 q^{64} + 14 q^{67} + 12 q^{69} + 24 q^{71} + 2 q^{81} + 12 q^{89} - 30 q^{91} + 24 q^{92} + 10 q^{93} - 14 q^{97}+O(q^{100}) \) 2 * q - 2 * q^3 - 4 * q^4 + 2 * q^9 + 4 * q^12 + 8 * q^16 - 12 * q^23 - 2 * q^27 - 10 * q^31 - 4 * q^36 - 4 * q^37 + 24 * q^47 - 8 * q^48 + 16 * q^49 - 12 * q^53 - 16 * q^64 + 14 * q^67 + 12 * q^69 + 24 * q^71 + 2 * q^81 + 12 * q^89 - 30 * q^91 + 24 * q^92 + 10 * q^93 - 14 * q^97

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