LMFDB - Embedded newform 5148.2.a.p.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5148,2,Mod(1,5148)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5148, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("5148.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5148 = 2^{2} \cdot 3^{2} \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5148.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:	$41.1069869606$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.82128.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 12x^{2} - 10x + 3 $ x^4 - 12x^2 - 10x + 3
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			1.3
Root			$0.234381$ of defining polynomial
Character	$\chi$	$=$	5148.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.765619 q^{5} +1.10827 q^{7} +O(q^{10})$	$q-0.765619 q^{5} +1.10827 q^{7} -1.00000 q^{11} +1.00000 q^{13} -3.49414 q^{17} -1.57703 q^{19} +2.72852 q^{23} -4.41383 q^{25} -4.34265 q^{29} +8.22266 q^{31} -0.848516 q^{35} +3.83679 q^{37} +0.891727 q^{41} +5.49414 q^{43} +3.62025 q^{47} -5.77173 q^{49} -2.68531 q^{53} +0.765619 q^{55} -14.1423 q^{59} -7.62025 q^{61} -0.765619 q^{65} -11.3767 q^{67} -6.93494 q^{71} +4.04580 q^{73} -1.10827 q^{77} +0.0371003 q^{79} -11.4570 q^{83} +2.67518 q^{85} -15.2135 q^{89} +1.10827 q^{91} +1.20741 q^{95} +3.83679 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{5} + 2 q^{7}+O(q^{10}) $ 4 * q - 4 * q^5 + 2 * q^7	$ 4 q - 4 q^{5} + 2 q^{7} - 4 q^{11} + 4 q^{13} + 4 q^{17} - 2 q^{19} - 8 q^{23} + 8 q^{25} - 14 q^{29} - 4 q^{31} - 18 q^{35} - 6 q^{37} + 6 q^{41} + 4 q^{43} - 2 q^{47} + 8 q^{49} - 4 q^{53} + 4 q^{55} - 12 q^{59} - 14 q^{61} - 4 q^{65} - 18 q^{71} + 10 q^{73} - 2 q^{77} + 20 q^{79} - 8 q^{83} + 18 q^{85} + 6 q^{89} + 2 q^{91} - 30 q^{95} - 6 q^{97}+O(q^{100}) $ 4 * q - 4 * q^5 + 2 * q^7 - 4 * q^11 + 4 * q^13 + 4 * q^17 - 2 * q^19 - 8 * q^23 + 8 * q^25 - 14 * q^29 - 4 * q^31 - 18 * q^35 - 6 * q^37 + 6 * q^41 + 4 * q^43 - 2 * q^47 + 8 * q^49 - 4 * q^53 + 4 * q^55 - 12 * q^59 - 14 * q^61 - 4 * q^65 - 18 * q^71 + 10 * q^73 - 2 * q^77 + 20 * q^79 - 8 * q^83 + 18 * q^85 + 6 * q^89 + 2 * q^91 - 30 * q^95 - 6 * 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
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−0.765619	−0.342395	−0.171198	−	0.985237i	$-0.554764\pi$
$5$	−0.765619	−0.342395	−0.171198	+	0.985237i	$0.554764\pi$
$6$	0	0
$7$	1.10827	0.418888	0.209444	−	0.977821i	$-0.432835\pi$
$7$	1.10827	0.418888	0.209444	+	0.977821i	$0.432835\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−3.49414	−0.847453	−0.423727	−	0.905790i	$-0.639278\pi$
$17$	−3.49414	−0.847453	−0.423727	+	0.905790i	$0.639278\pi$
$18$	0	0
$19$	−1.57703	−0.361797	−0.180898	−	0.983502i	$-0.557900\pi$
$19$	−1.57703	−0.361797	−0.180898	+	0.983502i	$0.557900\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	2.72852	0.568936	0.284468	−	0.958686i	$-0.408183\pi$
$23$	2.72852	0.568936	0.284468	+	0.958686i	$0.408183\pi$
$24$	0	0
$25$	−4.41383	−0.882765
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−4.34265	−0.806411	−0.403205	−	0.915110i	$-0.632104\pi$
$29$	−4.34265	−0.806411	−0.403205	+	0.915110i	$0.632104\pi$
$30$	0	0
$31$	8.22266	1.47683	0.738416	−	0.674345i	$-0.235574\pi$
$31$	8.22266	1.47683	0.738416	+	0.674345i	$0.235574\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−0.848516	−0.143425
$36$	0	0
$37$	3.83679	0.630765	0.315382	−	0.948965i	$-0.397867\pi$
$37$	3.83679	0.630765	0.315382	+	0.948965i	$0.397867\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	0.891727	0.139264	0.0696322	−	0.997573i	$-0.477817\pi$
$41$	0.891727	0.139264	0.0696322	+	0.997573i	$0.477817\pi$
$42$	0	0
$43$	5.49414	0.837848	0.418924	−	0.908021i	$-0.362407\pi$
$43$	5.49414	0.837848	0.418924	+	0.908021i	$0.362407\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	3.62025	0.528067	0.264034	−	0.964513i	$-0.414947\pi$
$47$	3.62025	0.528067	0.264034	+	0.964513i	$0.414947\pi$
$48$	0	0
$49$	−5.77173	−0.824533
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−2.68531	−0.368855	−0.184428	−	0.982846i	$-0.559043\pi$
$53$	−2.68531	−0.368855	−0.184428	+	0.982846i	$0.559043\pi$
$54$	0	0
$55$	0.765619	0.103236
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−14.1423	−1.84118	−0.920588	−	0.390534i	$-0.872290\pi$
$59$	−14.1423	−1.84118	−0.920588	+	0.390534i	$0.872290\pi$
$60$	0	0
$61$	−7.62025	−0.975672	−0.487836	−	0.872935i	$-0.662214\pi$
$61$	−7.62025	−0.975672	−0.487836	+	0.872935i	$0.662214\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−0.765619	−0.0949634
$66$	0	0
$67$	−11.3767	−1.38989	−0.694944	−	0.719064i	$-0.744571\pi$
$67$	−11.3767	−1.38989	−0.694944	+	0.719064i	$0.744571\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−6.93494	−0.823026	−0.411513	−	0.911404i	$-0.634999\pi$
$71$	−6.93494	−0.823026	−0.411513	+	0.911404i	$0.634999\pi$
$72$	0	0
$73$	4.04580	0.473525	0.236762	−	0.971568i	$-0.423914\pi$
$73$	4.04580	0.473525	0.236762	+	0.971568i	$0.423914\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−1.10827	−0.126299
$78$	0	0
$79$	0.0371003	0.00417411	0.00208705	−	0.999998i	$-0.499336\pi$
$79$	0.0371003	0.00417411	0.00208705	+	0.999998i	$0.499336\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−11.4570	−1.25757	−0.628787	−	0.777578i	$-0.716448\pi$
$83$	−11.4570	−1.25757	−0.628787	+	0.777578i	$0.716448\pi$
$84$	0	0
$85$	2.67518	0.290164
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−15.2135	−1.61263	−0.806315	−	0.591487i	$-0.798541\pi$
$89$	−15.2135	−1.61263	−0.806315	+	0.591487i	$0.798541\pi$
$90$	0	0
$91$	1.10827	0.116179
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1.20741	0.123877
$96$	0	0
$97$	3.83679	0.389567	0.194784	−	0.980846i	$-0.437600\pi$
$97$	3.83679	0.389567	0.194784	+	0.980846i	$0.437600\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	8.95118	0.890675	0.445338	−	0.895363i	$-0.353084\pi$
$101$	8.95118	0.890675	0.445338	+	0.895363i	$0.353084\pi$
$102$	0	0
$103$	12.8277	1.26395	0.631973	−	0.774990i	$-0.282245\pi$
$103$	12.8277	1.26395	0.631973	+	0.774990i	$0.282245\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	10.8277	1.04675	0.523374	−	0.852103i	$-0.324673\pi$
$107$	10.8277	1.04675	0.523374	+	0.852103i	$0.324673\pi$
$108$	0	0
$109$	−10.2623	−0.982954	−0.491477	−	0.870891i	$-0.663543\pi$
$109$	−10.2623	−0.982954	−0.491477	+	0.870891i	$0.663543\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	6.35889	0.598194	0.299097	−	0.954223i	$-0.403315\pi$
$113$	6.35889	0.598194	0.299097	+	0.954223i	$0.403315\pi$
$114$	0	0
$115$	−2.08901	−0.194801
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−3.87246	−0.354988
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	7.20741	0.644650
$126$	0	0
$127$	−0.559201	−0.0496210	−0.0248105	−	0.999692i	$-0.507898\pi$
$127$	−0.559201	−0.0496210	−0.0248105	+	0.999692i	$0.507898\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−6.90185	−0.603018	−0.301509	−	0.953463i	$-0.597490\pi$
$131$	−6.90185	−0.603018	−0.301509	+	0.953463i	$0.597490\pi$
$132$	0	0
$133$	−1.74779	−0.151552
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−9.50427	−0.812004	−0.406002	−	0.913872i	$-0.633077\pi$
$137$	−9.50427	−0.812004	−0.406002	+	0.913872i	$0.633077\pi$
$138$	0	0
$139$	1.58315	0.134281	0.0671403	−	0.997744i	$-0.478612\pi$
$139$	1.58315	0.134281	0.0671403	+	0.997744i	$0.478612\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	3.32482	0.276111
$146$	0	0
$147$	0	0
$148$	0	0
$149$	21.6452	1.77324	0.886621	−	0.462496i	$-0.153046\pi$
$149$	21.6452	1.77324	0.886621	+	0.462496i	$0.153046\pi$
$150$	0	0
$151$	−10.0966	−0.821646	−0.410823	−	0.911715i	$-0.634759\pi$
$151$	−10.0966	−0.821646	−0.410823	+	0.911715i	$0.634759\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−6.29543	−0.505661
$156$	0	0
$157$	8.65432	0.690690	0.345345	−	0.938476i	$-0.387762\pi$
$157$	8.65432	0.690690	0.345345	+	0.938476i	$0.387762\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	3.02394	0.238320
$162$	0	0
$163$	−16.7804	−1.31434	−0.657172	−	0.753740i	$-0.728248\pi$
$163$	−16.7804	−1.31434	−0.657172	+	0.753740i	$0.728248\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−8.93752	−0.691606	−0.345803	−	0.938307i	$-0.612393\pi$
$167$	−8.93752	−0.691606	−0.345803	+	0.938307i	$0.612393\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−19.6722	−1.49565	−0.747823	−	0.663898i	$-0.768901\pi$
$173$	−19.6722	−1.49565	−0.747823	+	0.663898i	$0.768901\pi$
$174$	0	0
$175$	−4.89173	−0.369780
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−24.6543	−1.84275	−0.921375	−	0.388674i	$-0.872933\pi$
$179$	−24.6543	−1.84275	−0.921375	+	0.388674i	$0.872933\pi$
$180$	0	0
$181$	5.08741	0.378144	0.189072	−	0.981963i	$-0.439452\pi$
$181$	5.08741	0.378144	0.189072	+	0.981963i	$0.439452\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−2.93752	−0.215971
$186$	0	0
$187$	3.49414	0.255517
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−2.68531	−0.194302	−0.0971510	−	0.995270i	$-0.530973\pi$
$191$	−2.68531	−0.194302	−0.0971510	+	0.995270i	$0.530973\pi$
$192$	0	0
$193$	11.2863	0.812405	0.406202	−	0.913783i	$-0.366853\pi$
$193$	11.2863	0.812405	0.406202	+	0.913783i	$0.366853\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−11.4469	−0.815559	−0.407779	−	0.913081i	$-0.633697\pi$
$197$	−11.4469	−0.815559	−0.407779	+	0.913081i	$0.633697\pi$
$198$	0	0
$199$	−5.15407	−0.365362	−0.182681	−	0.983172i	$-0.558478\pi$
$199$	−5.15407	−0.365362	−0.182681	+	0.983172i	$0.558478\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−4.81285	−0.337796
$204$	0	0
$205$	−0.682723	−0.0476835
$206$	0	0
$207$	0	0
$208$	0	0
$209$	1.57703	0.109086
$210$	0	0
$211$	11.1170	0.765324	0.382662	−	0.923888i	$-0.375007\pi$
$211$	11.1170	0.765324	0.382662	+	0.923888i	$0.375007\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−4.20642	−0.286875
$216$	0	0
$217$	9.11295	0.618627
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−3.49414	−0.235041
$222$	0	0
$223$	0.875488	0.0586270	0.0293135	−	0.999570i	$-0.490668\pi$
$223$	0.875488	0.0586270	0.0293135	+	0.999570i	$0.490668\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−10.3131	−0.684504	−0.342252	−	0.939608i	$-0.611190\pi$
$227$	−10.3131	−0.684504	−0.342252	+	0.939608i	$0.611190\pi$
$228$	0	0
$229$	−21.8875	−1.44637	−0.723185	−	0.690654i	$-0.757323\pi$
$229$	−21.8875	−1.44637	−0.723185	+	0.690654i	$0.757323\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	2.30699	0.151136	0.0755678	−	0.997141i	$-0.475923\pi$
$233$	2.30699	0.151136	0.0755678	+	0.997141i	$0.475923\pi$
$234$	0	0
$235$	−2.77173	−0.180808
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−17.4672	−1.12986	−0.564929	−	0.825140i	$-0.691096\pi$
$239$	−17.4672	−1.12986	−0.564929	+	0.825140i	$0.691096\pi$
$240$	0	0
$241$	−6.39456	−0.411910	−0.205955	−	0.978561i	$-0.566030\pi$
$241$	−6.39456	−0.411910	−0.205955	+	0.978561i	$0.566030\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	4.41895	0.282316
$246$	0	0
$247$	−1.57703	−0.100344
$248$	0	0
$249$	0	0
$250$	0	0
$251$	16.6619	1.05169	0.525844	−	0.850581i	$-0.323750\pi$
$251$	16.6619	1.05169	0.525844	+	0.850581i	$0.323750\pi$
$252$	0	0
$253$	−2.72852	−0.171541
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−27.7417	−1.73048	−0.865241	−	0.501357i	$-0.832835\pi$
$257$	−27.7417	−1.73048	−0.865241	+	0.501357i	$0.832835\pi$
$258$	0	0
$259$	4.25221	0.264220
$260$	0	0
$261$	0	0
$262$	0	0
$263$	7.51296	0.463269	0.231635	−	0.972803i	$-0.425593\pi$
$263$	7.51296	0.463269	0.231635	+	0.972803i	$0.425593\pi$
$264$	0	0
$265$	2.05592	0.126294
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−20.7717	−1.26647	−0.633237	−	0.773958i	$-0.718274\pi$
$269$	−20.7717	−1.26647	−0.633237	+	0.773958i	$0.718274\pi$
$270$	0	0
$271$	11.6635	0.708505	0.354252	−	0.935150i	$-0.384735\pi$
$271$	11.6635	0.708505	0.354252	+	0.935150i	$0.384735\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	4.41383	0.266164
$276$	0	0
$277$	21.2607	1.27743	0.638717	−	0.769442i	$-0.279465\pi$
$277$	21.2607	1.27743	0.638717	+	0.769442i	$0.279465\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−15.9359	−0.950658	−0.475329	−	0.879808i	$-0.657671\pi$
$281$	−15.9359	−0.950658	−0.475329	+	0.879808i	$0.657671\pi$
$282$	0	0
$283$	−21.4174	−1.27313	−0.636565	−	0.771223i	$-0.719645\pi$
$283$	−21.4174	−1.27313	−0.636565	+	0.771223i	$0.719645\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0.988277	0.0583361
$288$	0	0
$289$	−4.79100	−0.281823
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−0.0284047	−0.00165942	−0.000829710	−	1.00000i	$-0.500264\pi$
$293$	−0.0284047	−0.00165942	−0.000829710		1.00000i	$0.500264\pi$
$294$	0	0
$295$	10.8277	0.630410
$296$	0	0
$297$	0	0
$298$	0	0
$299$	2.72852	0.157794
$300$	0	0
$301$	6.08901	0.350965
$302$	0	0
$303$	0	0
$304$	0	0
$305$	5.83421	0.334066
$306$	0	0
$307$	13.4286	0.766413	0.383206	−	0.923663i	$-0.374820\pi$
$307$	13.4286	0.766413	0.383206	+	0.923663i	$0.374820\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−7.57445	−0.429508	−0.214754	−	0.976668i	$-0.568895\pi$
$311$	−7.57445	−0.429508	−0.214754	+	0.976668i	$0.568895\pi$
$312$	0	0
$313$	−10.8201	−0.611589	−0.305794	−	0.952098i	$-0.598922\pi$
$313$	−10.8201	−0.611589	−0.305794	+	0.952098i	$0.598922\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	9.84290	0.552833	0.276416	−	0.961038i	$-0.410853\pi$
$317$	9.84290	0.552833	0.276416	+	0.961038i	$0.410853\pi$
$318$	0	0
$319$	4.34265	0.243142
$320$	0	0
$321$	0	0
$322$	0	0
$323$	5.51038	0.306606
$324$	0	0
$325$	−4.41383	−0.244835
$326$	0	0
$327$	0	0
$328$	0	0
$329$	4.01222	0.221201
$330$	0	0
$331$	−9.28772	−0.510499	−0.255250	−	0.966875i	$-0.582158\pi$
$331$	−9.28772	−0.510499	−0.255250	+	0.966875i	$0.582158\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	8.71024	0.475891
$336$	0	0
$337$	−33.1865	−1.80779	−0.903893	−	0.427758i	$-0.859303\pi$
$337$	−33.1865	−1.80779	−0.903893	+	0.427758i	$0.859303\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−8.22266	−0.445282
$342$	0	0
$343$	−14.1546	−0.764275
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−36.0351	−1.93446	−0.967232	−	0.253895i	$-0.918288\pi$
$347$	−36.0351	−1.93446	−0.967232	+	0.253895i	$0.918288\pi$
$348$	0	0
$349$	−11.1541	−0.597064	−0.298532	−	0.954400i	$-0.596497\pi$
$349$	−11.1541	−0.597064	−0.298532	+	0.954400i	$0.596497\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	32.2786	1.71802	0.859008	−	0.511963i	$-0.171081\pi$
$353$	32.2786	1.71802	0.859008	+	0.511963i	$0.171081\pi$
$354$	0	0
$355$	5.30952	0.281800
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−16.9425	−0.894190	−0.447095	−	0.894486i	$-0.647541\pi$
$359$	−16.9425	−0.894190	−0.447095	+	0.894486i	$0.647541\pi$
$360$	0	0
$361$	−16.5130	−0.869103
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−3.09754	−0.162133
$366$	0	0
$367$	−22.9497	−1.19797	−0.598983	−	0.800761i	$-0.704428\pi$
$367$	−22.9497	−1.19797	−0.598983	+	0.800761i	$0.704428\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−2.97606	−0.154509
$372$	0	0
$373$	−29.1814	−1.51095	−0.755477	−	0.655175i	$-0.772595\pi$
$373$	−29.1814	−1.51095	−0.755477	+	0.655175i	$0.772595\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−4.34265	−0.223658
$378$	0	0
$379$	34.6908	1.78195	0.890974	−	0.454055i	$-0.150023\pi$
$379$	34.6908	1.78195	0.890974	+	0.454055i	$0.150023\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−6.06815	−0.310068	−0.155034	−	0.987909i	$-0.549549\pi$
$383$	−6.06815	−0.310068	−0.155034	+	0.987909i	$0.549549\pi$
$384$	0	0
$385$	0.848516	0.0432444
$386$	0	0
$387$	0	0
$388$	0	0
$389$	13.5312	0.686061	0.343031	−	0.939324i	$-0.388547\pi$
$389$	13.5312	0.686061	0.343031	+	0.939324i	$0.388547\pi$
$390$	0	0
$391$	−9.53382	−0.482146
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−0.0284047	−0.00142920
$396$	0	0
$397$	30.4245	1.52696	0.763480	−	0.645832i	$-0.223489\pi$
$397$	30.4245	1.52696	0.763480	+	0.645832i	$0.223489\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	11.7591	0.587220	0.293610	−	0.955925i	$-0.405143\pi$
$401$	11.7591	0.587220	0.293610	+	0.955925i	$0.405143\pi$
$402$	0	0
$403$	8.22266	0.409600
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−3.83679	−0.190183
$408$	0	0
$409$	31.3014	1.54775	0.773877	−	0.633336i	$-0.218315\pi$
$409$	31.3014	1.54775	0.773877	+	0.633336i	$0.218315\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−15.6736	−0.771247
$414$	0	0
$415$	8.77173	0.430587
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−33.6121	−1.64206	−0.821029	−	0.570886i	$-0.806600\pi$
$419$	−33.6121	−1.64206	−0.821029	+	0.570886i	$0.806600\pi$
$420$	0	0
$421$	33.9380	1.65404	0.827019	−	0.562174i	$-0.190035\pi$
$421$	33.9380	1.65404	0.827019	+	0.562174i	$0.190035\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	15.4225	0.748102
$426$	0	0
$427$	−8.44532	−0.408697
$428$	0	0
$429$	0	0
$430$	0	0
$431$	3.06248	0.147514	0.0737572	−	0.997276i	$-0.476501\pi$
$431$	3.06248	0.147514	0.0737572	+	0.997276i	$0.476501\pi$
$432$	0	0
$433$	−17.8516	−0.857893	−0.428947	−	0.903330i	$-0.641115\pi$
$433$	−17.8516	−0.857893	−0.428947	+	0.903330i	$0.641115\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−4.30297	−0.205839
$438$	0	0
$439$	16.4082	0.783121	0.391561	−	0.920152i	$-0.371935\pi$
$439$	16.4082	0.783121	0.391561	+	0.920152i	$0.371935\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−8.60594	−0.408880	−0.204440	−	0.978879i	$-0.565537\pi$
$443$	−8.60594	−0.408880	−0.204440	+	0.978879i	$0.565537\pi$
$444$	0	0
$445$	11.6478	0.552157
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−4.24610	−0.200386	−0.100193	−	0.994968i	$-0.531946\pi$
$449$	−4.24610	−0.200386	−0.100193	+	0.994968i	$0.531946\pi$
$450$	0	0
$451$	−0.891727	−0.0419898
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−0.848516	−0.0397790
$456$	0	0
$457$	11.7985	0.551912	0.275956	−	0.961170i	$-0.411006\pi$
$457$	11.7985	0.551912	0.275956	+	0.961170i	$0.411006\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−11.8261	−0.550794	−0.275397	−	0.961331i	$-0.588809\pi$
$461$	−11.8261	−0.550794	−0.275397	+	0.961331i	$0.588809\pi$
$462$	0	0
$463$	−5.00303	−0.232510	−0.116255	−	0.993219i	$-0.537089\pi$
$463$	−5.00303	−0.232510	−0.116255	+	0.993219i	$0.537089\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	5.12308	0.237068	0.118534	−	0.992950i	$-0.462181\pi$
$467$	5.12308	0.237068	0.118534	+	0.992950i	$0.462181\pi$
$468$	0	0
$469$	−12.6085	−0.582207
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−5.49414	−0.252621
$474$	0	0
$475$	6.96076	0.319381
$476$	0	0
$477$	0	0
$478$	0	0
$479$	23.0106	1.05138	0.525691	−	0.850675i	$-0.323807\pi$
$479$	23.0106	1.05138	0.525691	+	0.850675i	$0.323807\pi$
$480$	0	0
$481$	3.83679	0.174943
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−2.93752	−0.133386
$486$	0	0
$487$	−6.14587	−0.278496	−0.139248	−	0.990258i	$-0.544469\pi$
$487$	−6.14587	−0.278496	−0.139248	+	0.990258i	$0.544469\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−26.3206	−1.18783	−0.593917	−	0.804526i	$-0.702419\pi$
$491$	−26.3206	−1.18783	−0.593917	+	0.804526i	$0.702419\pi$
$492$	0	0
$493$	15.1738	0.683395
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−7.68581	−0.344756
$498$	0	0
$499$	0.875488	0.0391922	0.0195961	−	0.999808i	$-0.493762\pi$
$499$	0.875488	0.0391922	0.0195961	+	0.999808i	$0.493762\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−17.7093	−0.789617	−0.394808	−	0.918764i	$-0.629189\pi$
$503$	−17.7093	−0.789617	−0.394808	+	0.918764i	$0.629189\pi$
$504$	0	0
$505$	−6.85319	−0.304963
$506$	0	0
$507$	0	0
$508$	0	0
$509$	14.0980	0.624882	0.312441	−	0.949937i	$-0.398853\pi$
$509$	14.0980	0.624882	0.312441	+	0.949937i	$0.398853\pi$
$510$	0	0
$511$	4.48385	0.198354
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−9.82110	−0.432769
$516$	0	0
$517$	−3.62025	−0.159218
$518$	0	0
$519$	0	0
$520$	0	0
$521$	29.5435	1.29432	0.647161	−	0.762353i	$-0.275956\pi$
$521$	29.5435	1.29432	0.647161	+	0.762353i	$0.275956\pi$
$522$	0	0
$523$	−3.04563	−0.133176	−0.0665881	−	0.997781i	$-0.521211\pi$
$523$	−3.04563	−0.133176	−0.0665881	+	0.997781i	$0.521211\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−28.7311	−1.25155
$528$	0	0
$529$	−15.5552	−0.676312
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0.891727	0.0386250
$534$	0	0
$535$	−8.28986	−0.358402
$536$	0	0
$537$	0	0
$538$	0	0
$539$	5.77173	0.248606
$540$	0	0
$541$	3.78345	0.162663	0.0813317	−	0.996687i	$-0.474083\pi$
$541$	3.78345	0.162663	0.0813317	+	0.996687i	$0.474083\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	7.85705	0.336559
$546$	0	0
$547$	42.2267	1.80548	0.902742	−	0.430182i	$-0.141551\pi$
$547$	42.2267	1.80548	0.902742	+	0.430182i	$0.141551\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	6.84852	0.291757
$552$	0	0
$553$	0.0411173	0.00174848
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−25.9664	−1.10023	−0.550116	−	0.835088i	$-0.685417\pi$
$557$	−25.9664	−1.10023	−0.550116	+	0.835088i	$0.685417\pi$
$558$	0	0
$559$	5.49414	0.232377
$560$	0	0
$561$	0	0
$562$	0	0
$563$	3.58716	0.151181	0.0755904	−	0.997139i	$-0.475916\pi$
$563$	3.58716	0.151181	0.0755904	+	0.997139i	$0.475916\pi$
$564$	0	0
$565$	−4.86849	−0.204819
$566$	0	0
$567$	0	0
$568$	0	0
$569$	40.6370	1.70359	0.851795	−	0.523874i	$-0.175514\pi$
$569$	40.6370	1.70359	0.851795	+	0.523874i	$0.175514\pi$
$570$	0	0
$571$	−39.5571	−1.65541	−0.827707	−	0.561161i	$-0.810355\pi$
$571$	−39.5571	−1.65541	−0.827707	+	0.561161i	$0.810355\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−12.0432	−0.502237
$576$	0	0
$577$	−7.29642	−0.303754	−0.151877	−	0.988399i	$-0.548532\pi$
$577$	−7.29642	−0.303754	−0.151877	+	0.988399i	$0.548532\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−12.6975	−0.526782
$582$	0	0
$583$	2.68531	0.111214
$584$	0	0
$585$	0	0
$586$	0	0
$587$	33.3237	1.37542	0.687709	−	0.725987i	$-0.258617\pi$
$587$	33.3237	1.37542	0.687709	+	0.725987i	$0.258617\pi$
$588$	0	0
$589$	−12.9674	−0.534313
$590$	0	0
$591$	0	0
$592$	0	0
$593$	26.9477	1.10661	0.553304	−	0.832980i	$-0.313367\pi$
$593$	26.9477	1.10661	0.553304	+	0.832980i	$0.313367\pi$
$594$	0	0
$595$	2.96483	0.121546
$596$	0	0
$597$	0	0
$598$	0	0
$599$	20.7332	0.847135	0.423568	−	0.905864i	$-0.360778\pi$
$599$	20.7332	0.847135	0.423568	+	0.905864i	$0.360778\pi$
$600$	0	0
$601$	0.555184	0.0226464	0.0113232	−	0.999936i	$-0.496396\pi$
$601$	0.555184	0.0226464	0.0113232	+	0.999936i	$0.496396\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−0.765619	−0.0311269
$606$	0	0
$607$	30.0336	1.21903	0.609514	−	0.792776i	$-0.291365\pi$
$607$	30.0336	1.21903	0.609514	+	0.792776i	$0.291365\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	3.62025	0.146459
$612$	0	0
$613$	−15.8261	−0.639208	−0.319604	−	0.947551i	$-0.603550\pi$
$613$	−15.8261	−0.639208	−0.319604	+	0.947551i	$0.603550\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	37.5698	1.51250	0.756252	−	0.654280i	$-0.227028\pi$
$617$	37.5698	1.51250	0.756252	+	0.654280i	$0.227028\pi$
$618$	0	0
$619$	1.18104	0.0474701	0.0237350	−	0.999718i	$-0.492444\pi$
$619$	1.18104	0.0474701	0.0237350	+	0.999718i	$0.492444\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−16.8607	−0.675511
$624$	0	0
$625$	16.5510	0.662040
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−13.4063	−0.534544
$630$	0	0
$631$	11.5907	0.461418	0.230709	−	0.973023i	$-0.425895\pi$
$631$	11.5907	0.461418	0.230709	+	0.973023i	$0.425895\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0.428135	0.0169900
$636$	0	0
$637$	−5.77173	−0.228684
$638$	0	0
$639$	0	0
$640$	0	0
$641$	4.75951	0.187989	0.0939946	−	0.995573i	$-0.470036\pi$
$641$	4.75951	0.187989	0.0939946	+	0.995573i	$0.470036\pi$
$642$	0	0
$643$	−18.0061	−0.710092	−0.355046	−	0.934849i	$-0.615535\pi$
$643$	−18.0061	−0.710092	−0.355046	+	0.934849i	$0.615535\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−6.33864	−0.249198	−0.124599	−	0.992207i	$-0.539764\pi$
$647$	−6.33864	−0.249198	−0.124599	+	0.992207i	$0.539764\pi$
$648$	0	0
$649$	14.1423	0.555136
$650$	0	0
$651$	0	0
$652$	0	0
$653$	19.3857	0.758621	0.379311	−	0.925269i	$-0.376161\pi$
$653$	19.3857	0.758621	0.379311	+	0.925269i	$0.376161\pi$
$654$	0	0
$655$	5.28419	0.206471
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−23.3446	−0.909376	−0.454688	−	0.890651i	$-0.650249\pi$
$659$	−23.3446	−0.909376	−0.454688	+	0.890651i	$0.650249\pi$
$660$	0	0
$661$	−25.7600	−1.00195	−0.500974	−	0.865462i	$-0.667025\pi$
$661$	−25.7600	−1.00195	−0.500974	+	0.865462i	$0.667025\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	1.33814	0.0518908
$666$	0	0
$667$	−11.8490	−0.458796
$668$	0	0
$669$	0	0
$670$	0	0
$671$	7.62025	0.294176
$672$	0	0
$673$	34.0299	1.31176	0.655878	−	0.754867i	$-0.272299\pi$
$673$	34.0299	1.31176	0.655878	+	0.754867i	$0.272299\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	21.9655	0.844202	0.422101	−	0.906549i	$-0.361293\pi$
$677$	21.9655	0.844202	0.422101	+	0.906549i	$0.361293\pi$
$678$	0	0
$679$	4.25221	0.163185
$680$	0	0
$681$	0	0
$682$	0	0
$683$	14.1163	0.540146	0.270073	−	0.962840i	$-0.412952\pi$
$683$	14.1163	0.540146	0.270073	+	0.962840i	$0.412952\pi$
$684$	0	0
$685$	7.27665	0.278027
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−2.68531	−0.102302
$690$	0	0
$691$	−29.0326	−1.10445	−0.552227	−	0.833694i	$-0.686222\pi$
$691$	−29.0326	−1.10445	−0.552227	+	0.833694i	$0.686222\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−1.21209	−0.0459771
$696$	0	0
$697$	−3.11582	−0.118020
$698$	0	0
$699$	0	0
$700$	0	0
$701$	18.3908	0.694612	0.347306	−	0.937752i	$-0.387097\pi$
$701$	18.3908	0.694612	0.347306	+	0.937752i	$0.387097\pi$
$702$	0	0
$703$	−6.05075	−0.228209
$704$	0	0
$705$	0	0
$706$	0	0
$707$	9.92035	0.373093
$708$	0	0
$709$	−20.2995	−0.762364	−0.381182	−	0.924500i	$-0.624483\pi$
$709$	−20.2995	−0.762364	−0.381182	+	0.924500i	$0.624483\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	22.4357	0.840223
$714$	0	0
$715$	0.765619	0.0286325
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−51.7032	−1.92820	−0.964102	−	0.265532i	$-0.914452\pi$
$719$	−51.7032	−1.92820	−0.964102	+	0.265532i	$0.914452\pi$
$720$	0	0
$721$	14.2165	0.529452
$722$	0	0
$723$	0	0
$724$	0	0
$725$	19.1677	0.711871
$726$	0	0
$727$	−29.6379	−1.09921	−0.549605	−	0.835425i	$-0.685222\pi$
$727$	−29.6379	−1.09921	−0.549605	+	0.835425i	$0.685222\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−19.1973	−0.710037
$732$	0	0
$733$	−11.6075	−0.428734	−0.214367	−	0.976753i	$-0.568769\pi$
$733$	−11.6075	−0.428734	−0.214367	+	0.976753i	$0.568769\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	11.3767	0.419067
$738$	0	0
$739$	2.28062	0.0838939	0.0419470	−	0.999120i	$-0.486644\pi$
$739$	2.28062	0.0838939	0.0419470	+	0.999120i	$0.486644\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	7.27902	0.267041	0.133521	−	0.991046i	$-0.457372\pi$
$743$	7.27902	0.267041	0.133521	+	0.991046i	$0.457372\pi$
$744$	0	0
$745$	−16.5720	−0.607150
$746$	0	0
$747$	0	0
$748$	0	0
$749$	12.0000	0.438470
$750$	0	0
$751$	45.0179	1.64273	0.821363	−	0.570406i	$-0.193214\pi$
$751$	45.0179	1.64273	0.821363	+	0.570406i	$0.193214\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	7.73011	0.281328
$756$	0	0
$757$	−27.5388	−1.00091	−0.500457	−	0.865761i	$-0.666835\pi$
$757$	−27.5388	−1.00091	−0.500457	+	0.865761i	$0.666835\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	8.12517	0.294537	0.147269	−	0.989097i	$-0.452952\pi$
$761$	8.12517	0.294537	0.147269	+	0.989097i	$0.452952\pi$
$762$	0	0
$763$	−11.3735	−0.411748
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−14.1423	−0.510651
$768$	0	0
$769$	25.6116	0.923578	0.461789	−	0.886990i	$-0.347208\pi$
$769$	25.6116	0.923578	0.461789	+	0.886990i	$0.347208\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	9.39759	0.338008	0.169004	−	0.985615i	$-0.445945\pi$
$773$	9.39759	0.338008	0.169004	+	0.985615i	$0.445945\pi$
$774$	0	0
$775$	−36.2934	−1.30370
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−1.40628	−0.0503853
$780$	0	0
$781$	6.93494	0.248152
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−6.62591	−0.236489
$786$	0	0
$787$	39.6959	1.41501	0.707504	−	0.706710i	$-0.249821\pi$
$787$	39.6959	1.41501	0.707504	+	0.706710i	$0.249821\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	7.04739	0.250576
$792$	0	0
$793$	−7.62025	−0.270603
$794$	0	0
$795$	0	0
$796$	0	0
$797$	4.82765	0.171004	0.0855021	−	0.996338i	$-0.472751\pi$
$797$	4.82765	0.171004	0.0855021	+	0.996338i	$0.472751\pi$
$798$	0	0
$799$	−12.6496	−0.447512
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−4.04580	−0.142773
$804$	0	0
$805$	−2.31519	−0.0815998
$806$	0	0
$807$	0	0
$808$	0	0
$809$	35.5980	1.25156	0.625779	−	0.780001i	$-0.284781\pi$
$809$	35.5980	1.25156	0.625779	+	0.780001i	$0.284781\pi$
$810$	0	0
$811$	8.07827	0.283667	0.141833	−	0.989891i	$-0.454700\pi$
$811$	8.07827	0.283667	0.141833	+	0.989891i	$0.454700\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	12.8474	0.450025
$816$	0	0
$817$	−8.66445	−0.303131
$818$	0	0
$819$	0	0
$820$	0	0
$821$	49.5710	1.73004	0.865020	−	0.501738i	$-0.167306\pi$
$821$	49.5710	1.73004	0.865020	+	0.501738i	$0.167306\pi$
$822$	0	0
$823$	33.5811	1.17056	0.585282	−	0.810830i	$-0.300984\pi$
$823$	33.5811	1.17056	0.585282	+	0.810830i	$0.300984\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−3.07261	−0.106845	−0.0534225	−	0.998572i	$-0.517013\pi$
$827$	−3.07261	−0.106845	−0.0534225	+	0.998572i	$0.517013\pi$
$828$	0	0
$829$	26.7178	0.927947	0.463974	−	0.885849i	$-0.346423\pi$
$829$	26.7178	0.927947	0.463974	+	0.885849i	$0.346423\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	20.1672	0.698753
$834$	0	0
$835$	6.84274	0.236803
$836$	0	0
$837$	0	0
$838$	0	0
$839$	6.70875	0.231612	0.115806	−	0.993272i	$-0.463055\pi$
$839$	6.70875	0.231612	0.115806	+	0.993272i	$0.463055\pi$
$840$	0	0
$841$	−10.1414	−0.349702
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−0.765619	−0.0263381
$846$	0	0
$847$	1.10827	0.0380807
$848$	0	0
$849$	0	0
$850$	0	0
$851$	10.4688	0.358865
$852$	0	0
$853$	50.6610	1.73460	0.867299	−	0.497787i	$-0.165854\pi$
$853$	50.6610	1.73460	0.867299	+	0.497787i	$0.165854\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	20.7067	0.707328	0.353664	−	0.935373i	$-0.384936\pi$
$857$	20.7067	0.707328	0.353664	+	0.935373i	$0.384936\pi$
$858$	0	0
$859$	−46.0090	−1.56981	−0.784904	−	0.619617i	$-0.787288\pi$
$859$	−46.0090	−1.56981	−0.784904	+	0.619617i	$0.787288\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−44.0447	−1.49930	−0.749650	−	0.661835i	$-0.769778\pi$
$863$	−44.0447	−1.49930	−0.749650	+	0.661835i	$0.769778\pi$
$864$	0	0
$865$	15.0614	0.512102
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−0.0371003	−0.00125854
$870$	0	0
$871$	−11.3767	−0.385486
$872$	0	0
$873$	0	0
$874$	0	0
$875$	7.98778	0.270036
$876$	0	0
$877$	22.0630	0.745014	0.372507	−	0.928029i	$-0.378498\pi$
$877$	22.0630	0.745014	0.372507	+	0.928029i	$0.378498\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−16.7891	−0.565640	−0.282820	−	0.959173i	$-0.591270\pi$
$881$	−16.7891	−0.565640	−0.282820	+	0.959173i	$0.591270\pi$
$882$	0	0
$883$	−1.67358	−0.0563206	−0.0281603	−	0.999603i	$-0.508965\pi$
$883$	−1.67358	−0.0563206	−0.0281603	+	0.999603i	$0.508965\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	15.4159	0.517616	0.258808	−	0.965929i	$-0.416670\pi$
$887$	15.4159	0.517616	0.258808	+	0.965929i	$0.416670\pi$
$888$	0	0
$889$	−0.619747	−0.0207857
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−5.70925	−0.191053
$894$	0	0
$895$	18.8758	0.630949
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−35.7082	−1.19093
$900$	0	0
$901$	9.38284	0.312588
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−3.89502	−0.129475
$906$	0	0
$907$	−42.4604	−1.40987	−0.704937	−	0.709270i	$-0.749025\pi$
$907$	−42.4604	−1.40987	−0.704937	+	0.709270i	$0.749025\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	35.7295	1.18377	0.591886	−	0.806022i	$-0.298384\pi$
$911$	35.7295	1.18377	0.591886	+	0.806022i	$0.298384\pi$
$912$	0	0
$913$	11.4570	0.379173
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−7.64914	−0.252597
$918$	0	0
$919$	−28.7645	−0.948854	−0.474427	−	0.880295i	$-0.657345\pi$
$919$	−28.7645	−0.948854	−0.474427	+	0.880295i	$0.657345\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−6.93494	−0.228266
$924$	0	0
$925$	−16.9349	−0.556817
$926$	0	0
$927$	0	0
$928$	0	0
$929$	51.4657	1.68854	0.844268	−	0.535921i	$-0.180036\pi$
$929$	51.4657	1.68854	0.844268	+	0.535921i	$0.180036\pi$
$930$	0	0
$931$	9.10222	0.298313
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−2.67518	−0.0874877
$936$	0	0
$937$	2.13073	0.0696079	0.0348040	−	0.999394i	$-0.488919\pi$
$937$	2.13073	0.0696079	0.0348040	+	0.999394i	$0.488919\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−0.911982	−0.0297298	−0.0148649	−	0.999890i	$-0.504732\pi$
$941$	−0.911982	−0.0297298	−0.0148649	+	0.999890i	$0.504732\pi$
$942$	0	0
$943$	2.43309	0.0792324
$944$	0	0
$945$	0	0
$946$	0	0
$947$	8.79776	0.285889	0.142944	−	0.989731i	$-0.454343\pi$
$947$	8.79776	0.285889	0.142944	+	0.989731i	$0.454343\pi$
$948$	0	0
$949$	4.04580	0.131332
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−19.6776	−0.637420	−0.318710	−	0.947852i	$-0.603250\pi$
$953$	−19.6776	−0.637420	−0.318710	+	0.947852i	$0.603250\pi$
$954$	0	0
$955$	2.05592	0.0665281
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−10.5333	−0.340139
$960$	0	0
$961$	36.6121	1.18104
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−8.64100	−0.278164
$966$	0	0
$967$	−34.4757	−1.10866	−0.554332	−	0.832296i	$-0.687026\pi$
$967$	−34.4757	−1.10866	−0.554332	+	0.832296i	$0.687026\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−27.5031	−0.882617	−0.441308	−	0.897355i	$-0.645486\pi$
$971$	−27.5031	−0.882617	−0.441308	+	0.897355i	$0.645486\pi$
$972$	0	0
$973$	1.75456	0.0562485
$974$	0	0
$975$	0	0
$976$	0	0
$977$	40.6381	1.30013	0.650064	−	0.759880i	$-0.274742\pi$
$977$	40.6381	1.30013	0.650064	+	0.759880i	$0.274742\pi$
$978$	0	0
$979$	15.2135	0.486226
$980$	0	0
$981$	0	0
$982$	0	0
$983$	29.7957	0.950334	0.475167	−	0.879896i	$-0.342388\pi$
$983$	29.7957	0.950334	0.475167	+	0.879896i	$0.342388\pi$
$984$	0	0
$985$	8.76398	0.279243
$986$	0	0
$987$	0	0
$988$	0	0
$989$	14.9909	0.476682
$990$	0	0
$991$	−31.4214	−0.998133	−0.499066	−	0.866564i	$-0.666324\pi$
$991$	−31.4214	−0.998133	−0.499066	+	0.866564i	$0.666324\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	3.94606	0.125098
$996$	0	0
$997$	14.6975	0.465475	0.232738	−	0.972540i	$-0.425232\pi$
$997$	14.6975	0.465475	0.232738	+	0.972540i	$0.425232\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5148.2.a.p.1.3	✓	4
3.2	odd	2		5148.2.a.r.1.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5148.2.a.p.1.3	✓	4	1.1	even	1	trivial
5148.2.a.r.1.2	yes	4	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 5148 = 2^{2} \cdot 3^{2} \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5148.a (trivial)

\(f(q)\)	\(=\)	\(q-0.765619 q^{5} +1.10827 q^{7} +O(q^{10})\)	\(q-0.765619 q^{5} +1.10827 q^{7} -1.00000 q^{11} +1.00000 q^{13} -3.49414 q^{17} -1.57703 q^{19} +2.72852 q^{23} -4.41383 q^{25} -4.34265 q^{29} +8.22266 q^{31} -0.848516 q^{35} +3.83679 q^{37} +0.891727 q^{41} +5.49414 q^{43} +3.62025 q^{47} -5.77173 q^{49} -2.68531 q^{53} +0.765619 q^{55} -14.1423 q^{59} -7.62025 q^{61} -0.765619 q^{65} -11.3767 q^{67} -6.93494 q^{71} +4.04580 q^{73} -1.10827 q^{77} +0.0371003 q^{79} -11.4570 q^{83} +2.67518 q^{85} -15.2135 q^{89} +1.10827 q^{91} +1.20741 q^{95} +3.83679 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{5} + 2 q^{7}+O(q^{10}) \) 4 * q - 4 * q^5 + 2 * q^7	\( 4 q - 4 q^{5} + 2 q^{7} - 4 q^{11} + 4 q^{13} + 4 q^{17} - 2 q^{19} - 8 q^{23} + 8 q^{25} - 14 q^{29} - 4 q^{31} - 18 q^{35} - 6 q^{37} + 6 q^{41} + 4 q^{43} - 2 q^{47} + 8 q^{49} - 4 q^{53} + 4 q^{55} - 12 q^{59} - 14 q^{61} - 4 q^{65} - 18 q^{71} + 10 q^{73} - 2 q^{77} + 20 q^{79} - 8 q^{83} + 18 q^{85} + 6 q^{89} + 2 q^{91} - 30 q^{95} - 6 q^{97}+O(q^{100}) \) 4 * q - 4 * q^5 + 2 * q^7 - 4 * q^11 + 4 * q^13 + 4 * q^17 - 2 * q^19 - 8 * q^23 + 8 * q^25 - 14 * q^29 - 4 * q^31 - 18 * q^35 - 6 * q^37 + 6 * q^41 + 4 * q^43 - 2 * q^47 + 8 * q^49 - 4 * q^53 + 4 * q^55 - 12 * q^59 - 14 * q^61 - 4 * q^65 - 18 * q^71 + 10 * q^73 - 2 * q^77 + 20 * q^79 - 8 * q^83 + 18 * q^85 + 6 * q^89 + 2 * q^91 - 30 * q^95 - 6 * q^97

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