LMFDB - Embedded newform 4732.2.a.r.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4732,2,Mod(1,4732)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4732, 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("4732.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4732 = 2^{2} \cdot 7 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4732.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:	$37.7852102365$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.2854789.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{5} - 6x^{4} + 17x^{3} + 11x^{2} - 20x - 13 $ x^6 - 3x^5 - 6x^4 + 17x^3 + 11x^2 - 20*x - 13
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.00976$ of defining polynomial
Character	$\chi$	$=$	4732.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.85869 q^{3} +0.131645 q^{5} +1.00000 q^{7} +0.454712 q^{9} -2.36231 q^{11} -0.244686 q^{15} +2.27305 q^{17} +3.58645 q^{19} -1.85869 q^{21} -2.09259 q^{23} -4.98267 q^{25} +4.73089 q^{27} -0.321152 q^{29} -5.39622 q^{31} +4.39080 q^{33} +0.131645 q^{35} +0.682469 q^{37} +1.02616 q^{41} +3.11857 q^{43} +0.0598603 q^{45} +1.21943 q^{47} +1.00000 q^{49} -4.22489 q^{51} -1.92141 q^{53} -0.310986 q^{55} -6.66608 q^{57} +7.56302 q^{59} -10.4314 q^{61} +0.454712 q^{63} -0.338568 q^{67} +3.88946 q^{69} +9.67585 q^{71} -6.00107 q^{73} +9.26122 q^{75} -2.36231 q^{77} +16.2814 q^{79} -10.1574 q^{81} -11.3399 q^{83} +0.299235 q^{85} +0.596921 q^{87} -4.69788 q^{89} +10.0299 q^{93} +0.472137 q^{95} +13.4540 q^{97} -1.07417 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{3} + 6 q^{7} + 20 q^{9} + 4 q^{11} + 3 q^{15} - 3 q^{17} + 5 q^{19} + 6 q^{21} + 18 q^{23} + 2 q^{25} + 24 q^{27} + 17 q^{29} - 17 q^{31} + 21 q^{33} + q^{37} + 14 q^{41} + 14 q^{43} - 29 q^{45}+ \cdots + 3 q^{99}+O(q^{100}) $ 6 * q + 6 * q^3 + 6 * q^7 + 20 * q^9 + 4 * q^11 + 3 * q^15 - 3 * q^17 + 5 * q^19 + 6 * q^21 + 18 * q^23 + 2 * q^25 + 24 * q^27 + 17 * q^29 - 17 * q^31 + 21 * q^33 + q^37 + 14 * q^41 + 14 * q^43 - 29 * q^45 - 6 * q^47 + 6 * q^49 + 2 * q^51 + 3 * q^53 + 18 * q^55 - 14 * q^59 - 4 * q^61 + 20 * q^63 + 4 * q^67 + 39 * q^69 + 3 * q^71 - 31 * q^73 + 37 * q^75 + 4 * q^77 + 22 * q^79 + 6 * q^81 - 15 * q^83 + 51 * q^85 + 51 * q^87 - 20 * q^89 + 13 * q^93 + 19 * q^95 - 29 * q^97 + 3 * q^99

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
$3$	−1.85869	−1.07311	−0.536556	−	0.843865i	$-0.680275\pi$
$3$	−1.85869	−1.07311	−0.536556	+	0.843865i	$0.680275\pi$
$4$	0	0
$5$	0.131645	0.0588732	0.0294366	−	0.999567i	$-0.490629\pi$
$5$	0.131645	0.0588732	0.0294366	+	0.999567i	$0.490629\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	0.454712	0.151571
$10$	0	0
$11$	−2.36231	−0.712265	−0.356132	−	0.934436i	$-0.615905\pi$
$11$	−2.36231	−0.712265	−0.356132	+	0.934436i	$0.615905\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	−0.244686	−0.0631776
$16$	0	0
$17$	2.27305	0.551296	0.275648	−	0.961259i	$-0.411108\pi$
$17$	2.27305	0.551296	0.275648	+	0.961259i	$0.411108\pi$
$18$	0	0
$19$	3.58645	0.822788	0.411394	−	0.911458i	$-0.365042\pi$
$19$	3.58645	0.822788	0.411394	+	0.911458i	$0.365042\pi$
$20$	0	0
$21$	−1.85869	−0.405598
$22$	0	0
$23$	−2.09259	−0.436334	−0.218167	−	0.975911i	$-0.570008\pi$
$23$	−2.09259	−0.436334	−0.218167	+	0.975911i	$0.570008\pi$
$24$	0	0
$25$	−4.98267	−0.996534
$26$	0	0
$27$	4.73089	0.910460
$28$	0	0
$29$	−0.321152	−0.0596364	−0.0298182	−	0.999555i	$-0.509493\pi$
$29$	−0.321152	−0.0596364	−0.0298182	+	0.999555i	$0.509493\pi$
$30$	0	0
$31$	−5.39622	−0.969190	−0.484595	−	0.874739i	$-0.661033\pi$
$31$	−5.39622	−0.969190	−0.484595	+	0.874739i	$0.661033\pi$
$32$	0	0
$33$	4.39080	0.764340
$34$	0	0
$35$	0.131645	0.0222520
$36$	0	0
$37$	0.682469	0.112197	0.0560986	−	0.998425i	$-0.482134\pi$
$37$	0.682469	0.112197	0.0560986	+	0.998425i	$0.482134\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	1.02616	0.160259	0.0801295	−	0.996784i	$-0.474467\pi$
$41$	1.02616	0.160259	0.0801295	+	0.996784i	$0.474467\pi$
$42$	0	0
$43$	3.11857	0.475577	0.237788	−	0.971317i	$-0.423578\pi$
$43$	3.11857	0.475577	0.237788	+	0.971317i	$0.423578\pi$
$44$	0	0
$45$	0.0598603	0.00892345
$46$	0	0
$47$	1.21943	0.177872	0.0889361	−	0.996037i	$-0.471653\pi$
$47$	1.21943	0.177872	0.0889361	+	0.996037i	$0.471653\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−4.22489	−0.591603
$52$	0	0
$53$	−1.92141	−0.263925	−0.131963	−	0.991255i	$-0.542128\pi$
$53$	−1.92141	−0.263925	−0.131963	+	0.991255i	$0.542128\pi$
$54$	0	0
$55$	−0.310986	−0.0419333
$56$	0	0
$57$	−6.66608	−0.882944
$58$	0	0
$59$	7.56302	0.984621	0.492311	−	0.870420i	$-0.336152\pi$
$59$	7.56302	0.984621	0.492311	+	0.870420i	$0.336152\pi$
$60$	0	0
$61$	−10.4314	−1.33560	−0.667800	−	0.744340i	$-0.732764\pi$
$61$	−10.4314	−1.33560	−0.667800	+	0.744340i	$0.732764\pi$
$62$	0	0
$63$	0.454712	0.0572883
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−0.338568	−0.0413627	−0.0206813	−	0.999786i	$-0.506584\pi$
$67$	−0.338568	−0.0413627	−0.0206813	+	0.999786i	$0.506584\pi$
$68$	0	0
$69$	3.88946	0.468236
$70$	0	0
$71$	9.67585	1.14831	0.574156	−	0.818746i	$-0.305330\pi$
$71$	9.67585	1.14831	0.574156	+	0.818746i	$0.305330\pi$
$72$	0	0
$73$	−6.00107	−0.702372	−0.351186	−	0.936306i	$-0.614222\pi$
$73$	−6.00107	−0.702372	−0.351186	+	0.936306i	$0.614222\pi$
$74$	0	0
$75$	9.26122	1.06939
$76$	0	0
$77$	−2.36231	−0.269211
$78$	0	0
$79$	16.2814	1.83180	0.915902	−	0.401401i	$-0.131477\pi$
$79$	16.2814	1.83180	0.915902	+	0.401401i	$0.131477\pi$
$80$	0	0
$81$	−10.1574	−1.12860
$82$	0	0
$83$	−11.3399	−1.24471	−0.622357	−	0.782733i	$-0.713825\pi$
$83$	−11.3399	−1.24471	−0.622357	+	0.782733i	$0.713825\pi$
$84$	0	0
$85$	0.299235	0.0324566
$86$	0	0
$87$	0.596921	0.0639966
$88$	0	0
$89$	−4.69788	−0.497975	−0.248987	−	0.968507i	$-0.580098\pi$
$89$	−4.69788	−0.497975	−0.248987	+	0.968507i	$0.580098\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	10.0299	1.04005
$94$	0	0
$95$	0.472137	0.0484402
$96$	0	0
$97$	13.4540	1.36605	0.683024	−	0.730396i	$-0.260664\pi$
$97$	13.4540	1.36605	0.683024	+	0.730396i	$0.260664\pi$
$98$	0	0
$99$	−1.07417	−0.107958
$100$	0	0
$101$	6.04879	0.601877	0.300939	−	0.953644i	$-0.402700\pi$
$101$	6.04879	0.601877	0.300939	+	0.953644i	$0.402700\pi$
$102$	0	0
$103$	0.536101	0.0528236	0.0264118	−	0.999651i	$-0.491592\pi$
$103$	0.536101	0.0528236	0.0264118	+	0.999651i	$0.491592\pi$
$104$	0	0
$105$	−0.244686	−0.0238789
$106$	0	0
$107$	−12.2807	−1.18722	−0.593609	−	0.804754i	$-0.702297\pi$
$107$	−12.2807	−1.18722	−0.593609	+	0.804754i	$0.702297\pi$
$108$	0	0
$109$	−9.45733	−0.905848	−0.452924	−	0.891549i	$-0.649619\pi$
$109$	−9.45733	−0.905848	−0.452924	+	0.891549i	$0.649619\pi$
$110$	0	0
$111$	−1.26850	−0.120400
$112$	0	0
$113$	−0.510557	−0.0480291	−0.0240146	−	0.999712i	$-0.507645\pi$
$113$	−0.510557	−0.0480291	−0.0240146	+	0.999712i	$0.507645\pi$
$114$	0	0
$115$	−0.275477	−0.0256884
$116$	0	0
$117$	0	0
$118$	0	0
$119$	2.27305	0.208370
$120$	0	0
$121$	−5.41947	−0.492679
$122$	0	0
$123$	−1.90731	−0.171976
$124$	0	0
$125$	−1.31416	−0.117542
$126$	0	0
$127$	6.53782	0.580138	0.290069	−	0.957006i	$-0.406322\pi$
$127$	6.53782	0.580138	0.290069	+	0.957006i	$0.406322\pi$
$128$	0	0
$129$	−5.79643	−0.510347
$130$	0	0
$131$	6.25397	0.546412	0.273206	−	0.961956i	$-0.411916\pi$
$131$	6.25397	0.546412	0.273206	+	0.961956i	$0.411916\pi$
$132$	0	0
$133$	3.58645	0.310985
$134$	0	0
$135$	0.622796	0.0536017
$136$	0	0
$137$	3.29480	0.281494	0.140747	−	0.990046i	$-0.455050\pi$
$137$	3.29480	0.281494	0.140747	+	0.990046i	$0.455050\pi$
$138$	0	0
$139$	12.0396	1.02118	0.510591	−	0.859824i	$-0.329427\pi$
$139$	12.0396	1.02118	0.510591	+	0.859824i	$0.329427\pi$
$140$	0	0
$141$	−2.26654	−0.190877
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−0.0422779	−0.00351099
$146$	0	0
$147$	−1.85869	−0.153302
$148$	0	0
$149$	18.6364	1.52676	0.763378	−	0.645952i	$-0.223539\pi$
$149$	18.6364	1.52676	0.763378	+	0.645952i	$0.223539\pi$
$150$	0	0
$151$	−5.47950	−0.445915	−0.222958	−	0.974828i	$-0.571571\pi$
$151$	−5.47950	−0.445915	−0.222958	+	0.974828i	$0.571571\pi$
$152$	0	0
$153$	1.03358	0.0835603
$154$	0	0
$155$	−0.710383	−0.0570593
$156$	0	0
$157$	−0.911538	−0.0727486	−0.0363743	−	0.999338i	$-0.511581\pi$
$157$	−0.911538	−0.0727486	−0.0363743	+	0.999338i	$0.511581\pi$
$158$	0	0
$159$	3.57129	0.283222
$160$	0	0
$161$	−2.09259	−0.164919
$162$	0	0
$163$	−4.79431	−0.375519	−0.187760	−	0.982215i	$-0.560123\pi$
$163$	−4.79431	−0.375519	−0.187760	+	0.982215i	$0.560123\pi$
$164$	0	0
$165$	0.578025	0.0449992
$166$	0	0
$167$	−1.86827	−0.144571	−0.0722856	−	0.997384i	$-0.523029\pi$
$167$	−1.86827	−0.144571	−0.0722856	+	0.997384i	$0.523029\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	1.63080	0.124711
$172$	0	0
$173$	17.6075	1.33867	0.669337	−	0.742959i	$-0.266578\pi$
$173$	17.6075	1.33867	0.669337	+	0.742959i	$0.266578\pi$
$174$	0	0
$175$	−4.98267	−0.376654
$176$	0	0
$177$	−14.0573	−1.05661
$178$	0	0
$179$	23.2059	1.73449	0.867245	−	0.497881i	$-0.165888\pi$
$179$	23.2059	1.73449	0.867245	+	0.497881i	$0.165888\pi$
$180$	0	0
$181$	5.38803	0.400489	0.200245	−	0.979746i	$-0.435826\pi$
$181$	5.38803	0.400489	0.200245	+	0.979746i	$0.435826\pi$
$182$	0	0
$183$	19.3886	1.43325
$184$	0	0
$185$	0.0898433	0.00660541
$186$	0	0
$187$	−5.36967	−0.392669
$188$	0	0
$189$	4.73089	0.344122
$190$	0	0
$191$	13.9086	1.00639	0.503194	−	0.864174i	$-0.332158\pi$
$191$	13.9086	1.00639	0.503194	+	0.864174i	$0.332158\pi$
$192$	0	0
$193$	−3.90267	−0.280920	−0.140460	−	0.990086i	$-0.544858\pi$
$193$	−3.90267	−0.280920	−0.140460	+	0.990086i	$0.544858\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	15.2429	1.08601	0.543006	−	0.839729i	$-0.317286\pi$
$197$	15.2429	1.08601	0.543006	+	0.839729i	$0.317286\pi$
$198$	0	0
$199$	9.92221	0.703367	0.351684	−	0.936119i	$-0.385609\pi$
$199$	9.92221	0.703367	0.351684	+	0.936119i	$0.385609\pi$
$200$	0	0
$201$	0.629292	0.0443868
$202$	0	0
$203$	−0.321152	−0.0225405
$204$	0	0
$205$	0.135088	0.00943496
$206$	0	0
$207$	−0.951524	−0.0661355
$208$	0	0
$209$	−8.47233	−0.586043
$210$	0	0
$211$	1.91955	0.132147	0.0660737	−	0.997815i	$-0.478953\pi$
$211$	1.91955	0.132147	0.0660737	+	0.997815i	$0.478953\pi$
$212$	0	0
$213$	−17.9844	−1.23227
$214$	0	0
$215$	0.410542	0.0279987
$216$	0	0
$217$	−5.39622	−0.366319
$218$	0	0
$219$	11.1541	0.753725
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−4.10432	−0.274845	−0.137423	−	0.990512i	$-0.543882\pi$
$223$	−4.10432	−0.274845	−0.137423	+	0.990512i	$0.543882\pi$
$224$	0	0
$225$	−2.26568	−0.151045
$226$	0	0
$227$	−18.0147	−1.19568	−0.597840	−	0.801615i	$-0.703974\pi$
$227$	−18.0147	−1.19568	−0.597840	+	0.801615i	$0.703974\pi$
$228$	0	0
$229$	−22.6010	−1.49352	−0.746759	−	0.665094i	$-0.768391\pi$
$229$	−22.6010	−1.49352	−0.746759	+	0.665094i	$0.768391\pi$
$230$	0	0
$231$	4.39080	0.288893
$232$	0	0
$233$	16.3437	1.07071	0.535357	−	0.844626i	$-0.320177\pi$
$233$	16.3437	1.07071	0.535357	+	0.844626i	$0.320177\pi$
$234$	0	0
$235$	0.160531	0.0104719
$236$	0	0
$237$	−30.2621	−1.96573
$238$	0	0
$239$	26.7717	1.73172	0.865859	−	0.500288i	$-0.166773\pi$
$239$	26.7717	1.73172	0.865859	+	0.500288i	$0.166773\pi$
$240$	0	0
$241$	−6.33239	−0.407905	−0.203953	−	0.978981i	$-0.565379\pi$
$241$	−6.33239	−0.407905	−0.203953	+	0.978981i	$0.565379\pi$
$242$	0	0
$243$	4.68669	0.300651
$244$	0	0
$245$	0.131645	0.00841046
$246$	0	0
$247$	0	0
$248$	0	0
$249$	21.0773	1.33572
$250$	0	0
$251$	19.9584	1.25976	0.629882	−	0.776691i	$-0.283103\pi$
$251$	19.9584	1.25976	0.629882	+	0.776691i	$0.283103\pi$
$252$	0	0
$253$	4.94335	0.310786
$254$	0	0
$255$	−0.556184	−0.0348296
$256$	0	0
$257$	17.9829	1.12174	0.560872	−	0.827902i	$-0.310466\pi$
$257$	17.9829	1.12174	0.560872	+	0.827902i	$0.310466\pi$
$258$	0	0
$259$	0.682469	0.0424066
$260$	0	0
$261$	−0.146032	−0.00903913
$262$	0	0
$263$	8.77880	0.541324	0.270662	−	0.962674i	$-0.412757\pi$
$263$	8.77880	0.541324	0.270662	+	0.962674i	$0.412757\pi$
$264$	0	0
$265$	−0.252943	−0.0155381
$266$	0	0
$267$	8.73189	0.534383
$268$	0	0
$269$	−11.5933	−0.706857	−0.353428	−	0.935462i	$-0.614984\pi$
$269$	−11.5933	−0.706857	−0.353428	+	0.935462i	$0.614984\pi$
$270$	0	0
$271$	−16.0772	−0.976618	−0.488309	−	0.872671i	$-0.662386\pi$
$271$	−16.0772	−0.976618	−0.488309	+	0.872671i	$0.662386\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	11.7706	0.709796
$276$	0	0
$277$	15.6459	0.940072	0.470036	−	0.882647i	$-0.344241\pi$
$277$	15.6459	0.940072	0.470036	+	0.882647i	$0.344241\pi$
$278$	0	0
$279$	−2.45372	−0.146901
$280$	0	0
$281$	−25.0545	−1.49462	−0.747312	−	0.664473i	$-0.768656\pi$
$281$	−25.0545	−1.49462	−0.747312	+	0.664473i	$0.768656\pi$
$282$	0	0
$283$	32.0525	1.90532	0.952661	−	0.304034i	$-0.0983338\pi$
$283$	32.0525	1.90532	0.952661	+	0.304034i	$0.0983338\pi$
$284$	0	0
$285$	−0.877553	−0.0519818
$286$	0	0
$287$	1.02616	0.0605722
$288$	0	0
$289$	−11.8332	−0.696072
$290$	0	0
$291$	−25.0068	−1.46592
$292$	0	0
$293$	−27.6751	−1.61680	−0.808398	−	0.588636i	$-0.799665\pi$
$293$	−27.6751	−1.61680	−0.808398	+	0.588636i	$0.799665\pi$
$294$	0	0
$295$	0.995630	0.0579678
$296$	0	0
$297$	−11.1759	−0.648489
$298$	0	0
$299$	0	0
$300$	0	0
$301$	3.11857	0.179751
$302$	0	0
$303$	−11.2428	−0.645882
$304$	0	0
$305$	−1.37323	−0.0786311
$306$	0	0
$307$	15.9781	0.911918	0.455959	−	0.890001i	$-0.349296\pi$
$307$	15.9781	0.911918	0.455959	+	0.890001i	$0.349296\pi$
$308$	0	0
$309$	−0.996443	−0.0566856
$310$	0	0
$311$	13.2374	0.750622	0.375311	−	0.926899i	$-0.377536\pi$
$311$	13.2374	0.750622	0.375311	+	0.926899i	$0.377536\pi$
$312$	0	0
$313$	−1.87122	−0.105767	−0.0528837	−	0.998601i	$-0.516841\pi$
$313$	−1.87122	−0.105767	−0.0528837	+	0.998601i	$0.516841\pi$
$314$	0	0
$315$	0.0598603	0.00337275
$316$	0	0
$317$	30.3503	1.70464	0.852321	−	0.523019i	$-0.175194\pi$
$317$	30.3503	1.70464	0.852321	+	0.523019i	$0.175194\pi$
$318$	0	0
$319$	0.758662	0.0424769
$320$	0	0
$321$	22.8259	1.27402
$322$	0	0
$323$	8.15219	0.453600
$324$	0	0
$325$	0	0
$326$	0	0
$327$	17.5782	0.972077
$328$	0	0
$329$	1.21943	0.0672294
$330$	0	0
$331$	33.0288	1.81542	0.907712	−	0.419593i	$-0.137827\pi$
$331$	33.0288	1.81542	0.907712	+	0.419593i	$0.137827\pi$
$332$	0	0
$333$	0.310327	0.0170058
$334$	0	0
$335$	−0.0445706	−0.00243515
$336$	0	0
$337$	11.7594	0.640577	0.320288	−	0.947320i	$-0.396220\pi$
$337$	11.7594	0.640577	0.320288	+	0.947320i	$0.396220\pi$
$338$	0	0
$339$	0.948964	0.0515406
$340$	0	0
$341$	12.7476	0.690319
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0.512026	0.0275666
$346$	0	0
$347$	13.8295	0.742409	0.371205	−	0.928551i	$-0.378945\pi$
$347$	13.8295	0.742409	0.371205	+	0.928551i	$0.378945\pi$
$348$	0	0
$349$	−0.576073	−0.0308365	−0.0154182	−	0.999881i	$-0.504908\pi$
$349$	−0.576073	−0.0308365	−0.0154182	+	0.999881i	$0.504908\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	9.77058	0.520035	0.260018	−	0.965604i	$-0.416272\pi$
$353$	9.77058	0.520035	0.260018	+	0.965604i	$0.416272\pi$
$354$	0	0
$355$	1.27377	0.0676048
$356$	0	0
$357$	−4.22489	−0.223605
$358$	0	0
$359$	12.2276	0.645347	0.322674	−	0.946510i	$-0.395418\pi$
$359$	12.2276	0.645347	0.322674	+	0.946510i	$0.395418\pi$
$360$	0	0
$361$	−6.13737	−0.323020
$362$	0	0
$363$	10.0731	0.528700
$364$	0	0
$365$	−0.790008	−0.0413509
$366$	0	0
$367$	9.83874	0.513578	0.256789	−	0.966467i	$-0.417335\pi$
$367$	9.83874	0.513578	0.256789	+	0.966467i	$0.417335\pi$
$368$	0	0
$369$	0.466606	0.0242906
$370$	0	0
$371$	−1.92141	−0.0997544
$372$	0	0
$373$	13.3518	0.691330	0.345665	−	0.938358i	$-0.387653\pi$
$373$	13.3518	0.691330	0.345665	+	0.938358i	$0.387653\pi$
$374$	0	0
$375$	2.44262	0.126136
$376$	0	0
$377$	0	0
$378$	0	0
$379$	12.8560	0.660369	0.330184	−	0.943916i	$-0.392889\pi$
$379$	12.8560	0.660369	0.330184	+	0.943916i	$0.392889\pi$
$380$	0	0
$381$	−12.1517	−0.622553
$382$	0	0
$383$	−34.0832	−1.74157	−0.870785	−	0.491664i	$-0.836389\pi$
$383$	−34.0832	−1.74157	−0.870785	+	0.491664i	$0.836389\pi$
$384$	0	0
$385$	−0.310986	−0.0158493
$386$	0	0
$387$	1.41805	0.0720835
$388$	0	0
$389$	33.7101	1.70917	0.854586	−	0.519310i	$-0.173811\pi$
$389$	33.7101	1.70917	0.854586	+	0.519310i	$0.173811\pi$
$390$	0	0
$391$	−4.75656	−0.240550
$392$	0	0
$393$	−11.6242	−0.586361
$394$	0	0
$395$	2.14336	0.107844
$396$	0	0
$397$	17.5776	0.882195	0.441098	−	0.897459i	$-0.354589\pi$
$397$	17.5776	0.882195	0.441098	+	0.897459i	$0.354589\pi$
$398$	0	0
$399$	−6.66608	−0.333722
$400$	0	0
$401$	35.0416	1.74989	0.874947	−	0.484218i	$-0.160896\pi$
$401$	35.0416	1.74989	0.874947	+	0.484218i	$0.160896\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−1.33716	−0.0664441
$406$	0	0
$407$	−1.61221	−0.0799141
$408$	0	0
$409$	0.339940	0.0168089	0.00840447	−	0.999965i	$-0.497325\pi$
$409$	0.339940	0.0168089	0.00840447	+	0.999965i	$0.497325\pi$
$410$	0	0
$411$	−6.12401	−0.302075
$412$	0	0
$413$	7.56302	0.372152
$414$	0	0
$415$	−1.49283	−0.0732804
$416$	0	0
$417$	−22.3777	−1.09584
$418$	0	0
$419$	−9.42519	−0.460451	−0.230225	−	0.973137i	$-0.573946\pi$
$419$	−9.42519	−0.460451	−0.230225	+	0.973137i	$0.573946\pi$
$420$	0	0
$421$	13.0726	0.637121	0.318561	−	0.947902i	$-0.396801\pi$
$421$	13.0726	0.637121	0.318561	+	0.947902i	$0.396801\pi$
$422$	0	0
$423$	0.554490	0.0269602
$424$	0	0
$425$	−11.3259	−0.549386
$426$	0	0
$427$	−10.4314	−0.504810
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−20.4012	−0.982692	−0.491346	−	0.870964i	$-0.663495\pi$
$431$	−20.4012	−0.982692	−0.491346	+	0.870964i	$0.663495\pi$
$432$	0	0
$433$	14.6979	0.706337	0.353168	−	0.935560i	$-0.385104\pi$
$433$	14.6979	0.706337	0.353168	+	0.935560i	$0.385104\pi$
$434$	0	0
$435$	0.0785813	0.00376769
$436$	0	0
$437$	−7.50496	−0.359011
$438$	0	0
$439$	3.70299	0.176734	0.0883671	−	0.996088i	$-0.471835\pi$
$439$	3.70299	0.176734	0.0883671	+	0.996088i	$0.471835\pi$
$440$	0	0
$441$	0.454712	0.0216529
$442$	0	0
$443$	41.7855	1.98529	0.992644	−	0.121072i	$-0.0386332\pi$
$443$	41.7855	1.98529	0.992644	+	0.121072i	$0.0386332\pi$
$444$	0	0
$445$	−0.618451	−0.0293174
$446$	0	0
$447$	−34.6393	−1.63838
$448$	0	0
$449$	−3.68838	−0.174065	−0.0870327	−	0.996205i	$-0.527738\pi$
$449$	−3.68838	−0.174065	−0.0870327	+	0.996205i	$0.527738\pi$
$450$	0	0
$451$	−2.42411	−0.114147
$452$	0	0
$453$	10.1847	0.478517
$454$	0	0
$455$	0	0
$456$	0	0
$457$	6.43519	0.301025	0.150513	−	0.988608i	$-0.451908\pi$
$457$	6.43519	0.301025	0.150513	+	0.988608i	$0.451908\pi$
$458$	0	0
$459$	10.7536	0.501933
$460$	0	0
$461$	13.5422	0.630724	0.315362	−	0.948971i	$-0.397874\pi$
$461$	13.5422	0.630724	0.315362	+	0.948971i	$0.397874\pi$
$462$	0	0
$463$	16.9260	0.786620	0.393310	−	0.919406i	$-0.371330\pi$
$463$	16.9260	0.786620	0.393310	+	0.919406i	$0.371330\pi$
$464$	0	0
$465$	1.32038	0.0612311
$466$	0	0
$467$	7.69644	0.356149	0.178074	−	0.984017i	$-0.443013\pi$
$467$	7.69644	0.356149	0.178074	+	0.984017i	$0.443013\pi$
$468$	0	0
$469$	−0.338568	−0.0156336
$470$	0	0
$471$	1.69426	0.0780675
$472$	0	0
$473$	−7.36703	−0.338737
$474$	0	0
$475$	−17.8701	−0.819936
$476$	0	0
$477$	−0.873686	−0.0400033
$478$	0	0
$479$	−15.2407	−0.696363	−0.348182	−	0.937427i	$-0.613201\pi$
$479$	−15.2407	−0.696363	−0.348182	+	0.937427i	$0.613201\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	3.88946	0.176977
$484$	0	0
$485$	1.77115	0.0804237
$486$	0	0
$487$	−21.7251	−0.984460	−0.492230	−	0.870465i	$-0.663818\pi$
$487$	−21.7251	−0.984460	−0.492230	+	0.870465i	$0.663818\pi$
$488$	0	0
$489$	8.91112	0.402975
$490$	0	0
$491$	27.8105	1.25507	0.627535	−	0.778589i	$-0.284064\pi$
$491$	27.8105	1.25507	0.627535	+	0.778589i	$0.284064\pi$
$492$	0	0
$493$	−0.729996	−0.0328774
$494$	0	0
$495$	−0.141409	−0.00635586
$496$	0	0
$497$	9.67585	0.434021
$498$	0	0
$499$	34.8170	1.55862	0.779312	−	0.626636i	$-0.215569\pi$
$499$	34.8170	1.55862	0.779312	+	0.626636i	$0.215569\pi$
$500$	0	0
$501$	3.47253	0.155141
$502$	0	0
$503$	31.7059	1.41370	0.706849	−	0.707365i	$-0.250116\pi$
$503$	31.7059	1.41370	0.706849	+	0.707365i	$0.250116\pi$
$504$	0	0
$505$	0.796290	0.0354344
$506$	0	0
$507$	0	0
$508$	0	0
$509$	0.320953	0.0142260	0.00711300	−	0.999975i	$-0.497736\pi$
$509$	0.320953	0.0142260	0.00711300	+	0.999975i	$0.497736\pi$
$510$	0	0
$511$	−6.00107	−0.265472
$512$	0	0
$513$	16.9671	0.749116
$514$	0	0
$515$	0.0705747	0.00310989
$516$	0	0
$517$	−2.88068	−0.126692
$518$	0	0
$519$	−32.7268	−1.43655
$520$	0	0
$521$	−40.8872	−1.79130	−0.895650	−	0.444759i	$-0.853289\pi$
$521$	−40.8872	−1.79130	−0.895650	+	0.444759i	$0.853289\pi$
$522$	0	0
$523$	13.0414	0.570261	0.285131	−	0.958489i	$-0.407963\pi$
$523$	13.0414	0.570261	0.285131	+	0.958489i	$0.407963\pi$
$524$	0	0
$525$	9.26122	0.404193
$526$	0	0
$527$	−12.2659	−0.534311
$528$	0	0
$529$	−18.6211	−0.809612
$530$	0	0
$531$	3.43899	0.149240
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−1.61668	−0.0698953
$536$	0	0
$537$	−43.1325	−1.86130
$538$	0	0
$539$	−2.36231	−0.101752
$540$	0	0
$541$	−17.2705	−0.742516	−0.371258	−	0.928530i	$-0.621073\pi$
$541$	−17.2705	−0.742516	−0.371258	+	0.928530i	$0.621073\pi$
$542$	0	0
$543$	−10.0147	−0.429770
$544$	0	0
$545$	−1.24501	−0.0533302
$546$	0	0
$547$	17.2989	0.739648	0.369824	−	0.929102i	$-0.379418\pi$
$547$	17.2989	0.739648	0.369824	+	0.929102i	$0.379418\pi$
$548$	0	0
$549$	−4.74327	−0.202438
$550$	0	0
$551$	−1.15180	−0.0490682
$552$	0	0
$553$	16.2814	0.692357
$554$	0	0
$555$	−0.166990	−0.00708835
$556$	0	0
$557$	−23.4693	−0.994425	−0.497213	−	0.867629i	$-0.665643\pi$
$557$	−23.4693	−0.994425	−0.497213	+	0.867629i	$0.665643\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	9.98052	0.421378
$562$	0	0
$563$	35.5176	1.49689	0.748444	−	0.663198i	$-0.230801\pi$
$563$	35.5176	1.49689	0.748444	+	0.663198i	$0.230801\pi$
$564$	0	0
$565$	−0.0672120	−0.00282763
$566$	0	0
$567$	−10.1574	−0.426570
$568$	0	0
$569$	−11.7437	−0.492323	−0.246161	−	0.969229i	$-0.579169\pi$
$569$	−11.7437	−0.492323	−0.246161	+	0.969229i	$0.579169\pi$
$570$	0	0
$571$	−8.81763	−0.369006	−0.184503	−	0.982832i	$-0.559068\pi$
$571$	−8.81763	−0.369006	−0.184503	+	0.982832i	$0.559068\pi$
$572$	0	0
$573$	−25.8516	−1.07997
$574$	0	0
$575$	10.4267	0.434822
$576$	0	0
$577$	−29.4694	−1.22683	−0.613415	−	0.789761i	$-0.710205\pi$
$577$	−29.4694	−1.22683	−0.613415	+	0.789761i	$0.710205\pi$
$578$	0	0
$579$	7.25384	0.301459
$580$	0	0
$581$	−11.3399	−0.470458
$582$	0	0
$583$	4.53897	0.187985
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−9.93307	−0.409982	−0.204991	−	0.978764i	$-0.565716\pi$
$587$	−9.93307	−0.409982	−0.204991	+	0.978764i	$0.565716\pi$
$588$	0	0
$589$	−19.3533	−0.797438
$590$	0	0
$591$	−28.3318	−1.16541
$592$	0	0
$593$	−9.21272	−0.378321	−0.189161	−	0.981946i	$-0.560577\pi$
$593$	−9.21272	−0.378321	−0.189161	+	0.981946i	$0.560577\pi$
$594$	0	0
$595$	0.299235	0.0122674
$596$	0	0
$597$	−18.4423	−0.754792
$598$	0	0
$599$	−10.9241	−0.446347	−0.223174	−	0.974779i	$-0.571642\pi$
$599$	−10.9241	−0.446347	−0.223174	+	0.974779i	$0.571642\pi$
$600$	0	0
$601$	−35.3665	−1.44263	−0.721314	−	0.692608i	$-0.756462\pi$
$601$	−35.3665	−1.44263	−0.721314	+	0.692608i	$0.756462\pi$
$602$	0	0
$603$	−0.153951	−0.00626937
$604$	0	0
$605$	−0.713443	−0.0290056
$606$	0	0
$607$	−23.8814	−0.969317	−0.484659	−	0.874703i	$-0.661056\pi$
$607$	−23.8814	−0.969317	−0.484659	+	0.874703i	$0.661056\pi$
$608$	0	0
$609$	0.596921	0.0241884
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−41.9592	−1.69472	−0.847359	−	0.531021i	$-0.821809\pi$
$613$	−41.9592	−1.69472	−0.847359	+	0.531021i	$0.821809\pi$
$614$	0	0
$615$	−0.251086	−0.0101248
$616$	0	0
$617$	6.39341	0.257389	0.128695	−	0.991684i	$-0.458921\pi$
$617$	6.39341	0.257389	0.128695	+	0.991684i	$0.458921\pi$
$618$	0	0
$619$	13.7701	0.553467	0.276733	−	0.960947i	$-0.410748\pi$
$619$	13.7701	0.553467	0.276733	+	0.960947i	$0.410748\pi$
$620$	0	0
$621$	−9.89979	−0.397265
$622$	0	0
$623$	−4.69788	−0.188217
$624$	0	0
$625$	24.7403	0.989614
$626$	0	0
$627$	15.7474	0.628890
$628$	0	0
$629$	1.55129	0.0618539
$630$	0	0
$631$	−4.23366	−0.168539	−0.0842697	−	0.996443i	$-0.526856\pi$
$631$	−4.23366	−0.168539	−0.0842697	+	0.996443i	$0.526856\pi$
$632$	0	0
$633$	−3.56784	−0.141809
$634$	0	0
$635$	0.860668	0.0341546
$636$	0	0
$637$	0	0
$638$	0	0
$639$	4.39972	0.174050
$640$	0	0
$641$	−4.30218	−0.169926	−0.0849630	−	0.996384i	$-0.527077\pi$
$641$	−4.30218	−0.169926	−0.0849630	+	0.996384i	$0.527077\pi$
$642$	0	0
$643$	14.7113	0.580157	0.290078	−	0.957003i	$-0.406319\pi$
$643$	14.7113	0.580157	0.290078	+	0.957003i	$0.406319\pi$
$644$	0	0
$645$	−0.763068	−0.0300458
$646$	0	0
$647$	−24.8806	−0.978156	−0.489078	−	0.872240i	$-0.662667\pi$
$647$	−24.8806	−0.978156	−0.489078	+	0.872240i	$0.662667\pi$
$648$	0	0
$649$	−17.8662	−0.701311
$650$	0	0
$651$	10.0299	0.393102
$652$	0	0
$653$	17.9954	0.704215	0.352107	−	0.935960i	$-0.385465\pi$
$653$	17.9954	0.704215	0.352107	+	0.935960i	$0.385465\pi$
$654$	0	0
$655$	0.823301	0.0321690
$656$	0	0
$657$	−2.72876	−0.106459
$658$	0	0
$659$	0.397705	0.0154924	0.00774618	−	0.999970i	$-0.497534\pi$
$659$	0.397705	0.0154924	0.00774618	+	0.999970i	$0.497534\pi$
$660$	0	0
$661$	−39.7965	−1.54790	−0.773951	−	0.633245i	$-0.781723\pi$
$661$	−39.7965	−1.54790	−0.773951	+	0.633245i	$0.781723\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0.472137	0.0183087
$666$	0	0
$667$	0.672038	0.0260214
$668$	0	0
$669$	7.62863	0.294940
$670$	0	0
$671$	24.6422	0.951301
$672$	0	0
$673$	−49.7077	−1.91609	−0.958045	−	0.286619i	$-0.907469\pi$
$673$	−49.7077	−1.91609	−0.958045	+	0.286619i	$0.907469\pi$
$674$	0	0
$675$	−23.5725	−0.907305
$676$	0	0
$677$	−21.5739	−0.829151	−0.414576	−	0.910015i	$-0.636070\pi$
$677$	−21.5739	−0.829151	−0.414576	+	0.910015i	$0.636070\pi$
$678$	0	0
$679$	13.4540	0.516318
$680$	0	0
$681$	33.4838	1.28310
$682$	0	0
$683$	−29.8563	−1.14242	−0.571210	−	0.820804i	$-0.693526\pi$
$683$	−29.8563	−1.14242	−0.571210	+	0.820804i	$0.693526\pi$
$684$	0	0
$685$	0.433743	0.0165725
$686$	0	0
$687$	42.0082	1.60271
$688$	0	0
$689$	0	0
$690$	0	0
$691$	8.56216	0.325720	0.162860	−	0.986649i	$-0.447928\pi$
$691$	8.56216	0.325720	0.162860	+	0.986649i	$0.447928\pi$
$692$	0	0
$693$	−1.07417	−0.0408044
$694$	0	0
$695$	1.58494	0.0601202
$696$	0	0
$697$	2.33251	0.0883502
$698$	0	0
$699$	−30.3779	−1.14900
$700$	0	0
$701$	34.4649	1.30172	0.650861	−	0.759197i	$-0.274408\pi$
$701$	34.4649	1.30172	0.650861	+	0.759197i	$0.274408\pi$
$702$	0	0
$703$	2.44764	0.0923146
$704$	0	0
$705$	−0.298377	−0.0112375
$706$	0	0
$707$	6.04879	0.227488
$708$	0	0
$709$	5.24226	0.196877	0.0984385	−	0.995143i	$-0.468615\pi$
$709$	5.24226	0.196877	0.0984385	+	0.995143i	$0.468615\pi$
$710$	0	0
$711$	7.40336	0.277648
$712$	0	0
$713$	11.2921	0.422891
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−49.7602	−1.85833
$718$	0	0
$719$	−25.2552	−0.941859	−0.470930	−	0.882171i	$-0.656081\pi$
$719$	−25.2552	−0.941859	−0.470930	+	0.882171i	$0.656081\pi$
$720$	0	0
$721$	0.536101	0.0199654
$722$	0	0
$723$	11.7699	0.437728
$724$	0	0
$725$	1.60019	0.0594297
$726$	0	0
$727$	−5.96955	−0.221399	−0.110699	−	0.993854i	$-0.535309\pi$
$727$	−5.96955	−0.221399	−0.110699	+	0.993854i	$0.535309\pi$
$728$	0	0
$729$	21.7610	0.805964
$730$	0	0
$731$	7.08867	0.262184
$732$	0	0
$733$	6.28559	0.232164	0.116082	−	0.993240i	$-0.462967\pi$
$733$	6.28559	0.232164	0.116082	+	0.993240i	$0.462967\pi$
$734$	0	0
$735$	−0.244686	−0.00902537
$736$	0	0
$737$	0.799804	0.0294612
$738$	0	0
$739$	−8.81639	−0.324316	−0.162158	−	0.986765i	$-0.551845\pi$
$739$	−8.81639	−0.324316	−0.162158	+	0.986765i	$0.551845\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−21.5988	−0.792383	−0.396192	−	0.918168i	$-0.629668\pi$
$743$	−21.5988	−0.792383	−0.396192	+	0.918168i	$0.629668\pi$
$744$	0	0
$745$	2.45339	0.0898851
$746$	0	0
$747$	−5.15638	−0.188662
$748$	0	0
$749$	−12.2807	−0.448726
$750$	0	0
$751$	−47.1739	−1.72140	−0.860699	−	0.509114i	$-0.829973\pi$
$751$	−47.1739	−1.72140	−0.860699	+	0.509114i	$0.829973\pi$
$752$	0	0
$753$	−37.0964	−1.35187
$754$	0	0
$755$	−0.721346	−0.0262525
$756$	0	0
$757$	17.4772	0.635221	0.317610	−	0.948221i	$-0.397120\pi$
$757$	17.4772	0.635221	0.317610	+	0.948221i	$0.397120\pi$
$758$	0	0
$759$	−9.18813	−0.333508
$760$	0	0
$761$	34.2234	1.24060	0.620299	−	0.784366i	$-0.287011\pi$
$761$	34.2234	1.24060	0.620299	+	0.784366i	$0.287011\pi$
$762$	0	0
$763$	−9.45733	−0.342378
$764$	0	0
$765$	0.136066	0.00491947
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−49.2744	−1.77688	−0.888440	−	0.458992i	$-0.848211\pi$
$769$	−49.2744	−1.77688	−0.888440	+	0.458992i	$0.848211\pi$
$770$	0	0
$771$	−33.4246	−1.20376
$772$	0	0
$773$	1.07412	0.0386333	0.0193167	−	0.999813i	$-0.493851\pi$
$773$	1.07412	0.0386333	0.0193167	+	0.999813i	$0.493851\pi$
$774$	0	0
$775$	26.8876	0.965830
$776$	0	0
$777$	−1.26850	−0.0455070
$778$	0	0
$779$	3.68027	0.131859
$780$	0	0
$781$	−22.8574	−0.817902
$782$	0	0
$783$	−1.51934	−0.0542966
$784$	0	0
$785$	−0.119999	−0.00428295
$786$	0	0
$787$	32.8350	1.17044	0.585221	−	0.810874i	$-0.301008\pi$
$787$	32.8350	1.17044	0.585221	+	0.810874i	$0.301008\pi$
$788$	0	0
$789$	−16.3170	−0.580902
$790$	0	0
$791$	−0.510557	−0.0181533
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0.470141	0.0166742
$796$	0	0
$797$	15.1418	0.536350	0.268175	−	0.963370i	$-0.413579\pi$
$797$	15.1418	0.536350	0.268175	+	0.963370i	$0.413579\pi$
$798$	0	0
$799$	2.77183	0.0980603
$800$	0	0
$801$	−2.13618	−0.0754783
$802$	0	0
$803$	14.1764	0.500275
$804$	0	0
$805$	−0.275477	−0.00970930
$806$	0	0
$807$	21.5483	0.758537
$808$	0	0
$809$	−29.9721	−1.05376	−0.526882	−	0.849939i	$-0.676639\pi$
$809$	−29.9721	−1.05376	−0.526882	+	0.849939i	$0.676639\pi$
$810$	0	0
$811$	−47.0471	−1.65205	−0.826024	−	0.563635i	$-0.809403\pi$
$811$	−47.0471	−1.65205	−0.826024	+	0.563635i	$0.809403\pi$
$812$	0	0
$813$	29.8824	1.04802
$814$	0	0
$815$	−0.631145	−0.0221080
$816$	0	0
$817$	11.1846	0.391299
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−7.12942	−0.248819	−0.124409	−	0.992231i	$-0.539704\pi$
$821$	−7.12942	−0.248819	−0.124409	+	0.992231i	$0.539704\pi$
$822$	0	0
$823$	34.7593	1.21163	0.605817	−	0.795604i	$-0.292846\pi$
$823$	34.7593	1.21163	0.605817	+	0.795604i	$0.292846\pi$
$824$	0	0
$825$	−21.8779	−0.761691
$826$	0	0
$827$	−16.2065	−0.563557	−0.281778	−	0.959480i	$-0.590924\pi$
$827$	−16.2065	−0.563557	−0.281778	+	0.959480i	$0.590924\pi$
$828$	0	0
$829$	−2.04708	−0.0710981	−0.0355491	−	0.999368i	$-0.511318\pi$
$829$	−2.04708	−0.0710981	−0.0355491	+	0.999368i	$0.511318\pi$
$830$	0	0
$831$	−29.0808	−1.00880
$832$	0	0
$833$	2.27305	0.0787566
$834$	0	0
$835$	−0.245948	−0.00851137
$836$	0	0
$837$	−25.5289	−0.882409
$838$	0	0
$839$	−56.4277	−1.94810	−0.974050	−	0.226332i	$-0.927327\pi$
$839$	−56.4277	−1.94810	−0.974050	+	0.226332i	$0.927327\pi$
$840$	0	0
$841$	−28.8969	−0.996443
$842$	0	0
$843$	46.5684	1.60390
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−5.41947	−0.186215
$848$	0	0
$849$	−59.5755	−2.04463
$850$	0	0
$851$	−1.42813	−0.0489555
$852$	0	0
$853$	43.8562	1.50161	0.750804	−	0.660525i	$-0.229666\pi$
$853$	43.8562	1.50161	0.750804	+	0.660525i	$0.229666\pi$
$854$	0	0
$855$	0.214686	0.00734211
$856$	0	0
$857$	23.9447	0.817934	0.408967	−	0.912549i	$-0.365889\pi$
$857$	23.9447	0.817934	0.408967	+	0.912549i	$0.365889\pi$
$858$	0	0
$859$	38.9353	1.32846	0.664228	−	0.747530i	$-0.268760\pi$
$859$	38.9353	1.32846	0.664228	+	0.747530i	$0.268760\pi$
$860$	0	0
$861$	−1.90731	−0.0650008
$862$	0	0
$863$	48.5181	1.65158	0.825788	−	0.563981i	$-0.190731\pi$
$863$	48.5181	1.65158	0.825788	+	0.563981i	$0.190731\pi$
$864$	0	0
$865$	2.31793	0.0788120
$866$	0	0
$867$	21.9942	0.746964
$868$	0	0
$869$	−38.4619	−1.30473
$870$	0	0
$871$	0	0
$872$	0	0
$873$	6.11770	0.207053
$874$	0	0
$875$	−1.31416	−0.0444268
$876$	0	0
$877$	43.2601	1.46079	0.730395	−	0.683025i	$-0.239336\pi$
$877$	43.2601	1.46079	0.730395	+	0.683025i	$0.239336\pi$
$878$	0	0
$879$	51.4393	1.73500
$880$	0	0
$881$	−27.4066	−0.923351	−0.461675	−	0.887049i	$-0.652752\pi$
$881$	−27.4066	−0.923351	−0.461675	+	0.887049i	$0.652752\pi$
$882$	0	0
$883$	−38.6691	−1.30132	−0.650659	−	0.759370i	$-0.725507\pi$
$883$	−38.6691	−1.30132	−0.650659	+	0.759370i	$0.725507\pi$
$884$	0	0
$885$	−1.85056	−0.0622060
$886$	0	0
$887$	29.1524	0.978841	0.489420	−	0.872048i	$-0.337208\pi$
$887$	29.1524	0.978841	0.489420	+	0.872048i	$0.337208\pi$
$888$	0	0
$889$	6.53782	0.219271
$890$	0	0
$891$	23.9949	0.803860
$892$	0	0
$893$	4.37343	0.146351
$894$	0	0
$895$	3.05493	0.102115
$896$	0	0
$897$	0	0
$898$	0	0
$899$	1.73301	0.0577990
$900$	0	0
$901$	−4.36746	−0.145501
$902$	0	0
$903$	−5.79643	−0.192893
$904$	0	0
$905$	0.709304	0.0235781
$906$	0	0
$907$	−10.9871	−0.364821	−0.182410	−	0.983222i	$-0.558390\pi$
$907$	−10.9871	−0.364821	−0.182410	+	0.983222i	$0.558390\pi$
$908$	0	0
$909$	2.75046	0.0912269
$910$	0	0
$911$	−34.5446	−1.14451	−0.572257	−	0.820074i	$-0.693932\pi$
$911$	−34.5446	−1.14451	−0.572257	+	0.820074i	$0.693932\pi$
$912$	0	0
$913$	26.7884	0.886566
$914$	0	0
$915$	2.55241	0.0843800
$916$	0	0
$917$	6.25397	0.206524
$918$	0	0
$919$	−42.9584	−1.41707	−0.708533	−	0.705678i	$-0.750643\pi$
$919$	−42.9584	−1.41707	−0.708533	+	0.705678i	$0.750643\pi$
$920$	0	0
$921$	−29.6983	−0.978591
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−3.40052	−0.111808
$926$	0	0
$927$	0.243771	0.00800650
$928$	0	0
$929$	−22.3855	−0.734444	−0.367222	−	0.930133i	$-0.619691\pi$
$929$	−22.3855	−0.734444	−0.367222	+	0.930133i	$0.619691\pi$
$930$	0	0
$931$	3.58645	0.117541
$932$	0	0
$933$	−24.6041	−0.805502
$934$	0	0
$935$	−0.706887	−0.0231177
$936$	0	0
$937$	−56.3514	−1.84092	−0.920461	−	0.390835i	$-0.872186\pi$
$937$	−56.3514	−1.84092	−0.920461	+	0.390835i	$0.872186\pi$
$938$	0	0
$939$	3.47800	0.113500
$940$	0	0
$941$	22.3189	0.727576	0.363788	−	0.931482i	$-0.381483\pi$
$941$	22.3189	0.727576	0.363788	+	0.931482i	$0.381483\pi$
$942$	0	0
$943$	−2.14732	−0.0699265
$944$	0	0
$945$	0.622796	0.0202595
$946$	0	0
$947$	−10.2411	−0.332790	−0.166395	−	0.986059i	$-0.553213\pi$
$947$	−10.2411	−0.332790	−0.166395	+	0.986059i	$0.553213\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−56.4116	−1.82927
$952$	0	0
$953$	−33.5047	−1.08532	−0.542662	−	0.839951i	$-0.682583\pi$
$953$	−33.5047	−1.08532	−0.542662	+	0.839951i	$0.682583\pi$
$954$	0	0
$955$	1.83099	0.0592493
$956$	0	0
$957$	−1.41011	−0.0455825
$958$	0	0
$959$	3.29480	0.106395
$960$	0	0
$961$	−1.88082	−0.0606716
$962$	0	0
$963$	−5.58417	−0.179947
$964$	0	0
$965$	−0.513765	−0.0165387
$966$	0	0
$967$	47.4995	1.52748	0.763741	−	0.645523i	$-0.223361\pi$
$967$	47.4995	1.52748	0.763741	+	0.645523i	$0.223361\pi$
$968$	0	0
$969$	−15.1524	−0.486764
$970$	0	0
$971$	−40.6859	−1.30567	−0.652837	−	0.757499i	$-0.726421\pi$
$971$	−40.6859	−1.30567	−0.652837	+	0.757499i	$0.726421\pi$
$972$	0	0
$973$	12.0396	0.385970
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−15.2640	−0.488337	−0.244169	−	0.969733i	$-0.578515\pi$
$977$	−15.2640	−0.488337	−0.244169	+	0.969733i	$0.578515\pi$
$978$	0	0
$979$	11.0979	0.354690
$980$	0	0
$981$	−4.30036	−0.137300
$982$	0	0
$983$	46.3542	1.47847	0.739235	−	0.673448i	$-0.235187\pi$
$983$	46.3542	1.47847	0.739235	+	0.673448i	$0.235187\pi$
$984$	0	0
$985$	2.00664	0.0639370
$986$	0	0
$987$	−2.26654	−0.0721447
$988$	0	0
$989$	−6.52587	−0.207510
$990$	0	0
$991$	−16.0194	−0.508873	−0.254437	−	0.967089i	$-0.581890\pi$
$991$	−16.0194	−0.508873	−0.254437	+	0.967089i	$0.581890\pi$
$992$	0	0
$993$	−61.3901	−1.94816
$994$	0	0
$995$	1.30620	0.0414095
$996$	0	0
$997$	−15.2746	−0.483753	−0.241876	−	0.970307i	$-0.577763\pi$
$997$	−15.2746	−0.483753	−0.241876	+	0.970307i	$0.577763\pi$
$998$	0	0
$999$	3.22869	0.102151

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4732.2.a.r.1.2	yes	6
13.5	odd	4		4732.2.g.j.337.3		12
13.8	odd	4		4732.2.g.j.337.4		12
13.12	even	2		4732.2.a.q.1.2	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4732.2.a.q.1.2	✓	6	13.12	even	2
4732.2.a.r.1.2	yes	6	1.1	even	1	trivial
4732.2.g.j.337.3		12	13.5	odd	4
4732.2.g.j.337.4		12	13.8	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:	\( N \)	\(=\)	\( 4732 = 2^{2} \cdot 7 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4732.a (trivial)

\(f(q)\)	\(=\)	\(q-1.85869 q^{3} +0.131645 q^{5} +1.00000 q^{7} +0.454712 q^{9} -2.36231 q^{11} -0.244686 q^{15} +2.27305 q^{17} +3.58645 q^{19} -1.85869 q^{21} -2.09259 q^{23} -4.98267 q^{25} +4.73089 q^{27} -0.321152 q^{29} -5.39622 q^{31} +4.39080 q^{33} +0.131645 q^{35} +0.682469 q^{37} +1.02616 q^{41} +3.11857 q^{43} +0.0598603 q^{45} +1.21943 q^{47} +1.00000 q^{49} -4.22489 q^{51} -1.92141 q^{53} -0.310986 q^{55} -6.66608 q^{57} +7.56302 q^{59} -10.4314 q^{61} +0.454712 q^{63} -0.338568 q^{67} +3.88946 q^{69} +9.67585 q^{71} -6.00107 q^{73} +9.26122 q^{75} -2.36231 q^{77} +16.2814 q^{79} -10.1574 q^{81} -11.3399 q^{83} +0.299235 q^{85} +0.596921 q^{87} -4.69788 q^{89} +10.0299 q^{93} +0.472137 q^{95} +13.4540 q^{97} -1.07417 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{3} + 6 q^{7} + 20 q^{9} + 4 q^{11} + 3 q^{15} - 3 q^{17} + 5 q^{19} + 6 q^{21} + 18 q^{23} + 2 q^{25} + 24 q^{27} + 17 q^{29} - 17 q^{31} + 21 q^{33} + q^{37} + 14 q^{41} + 14 q^{43} - 29 q^{45}+ \cdots + 3 q^{99}+O(q^{100}) \) 6 * q + 6 * q^3 + 6 * q^7 + 20 * q^9 + 4 * q^11 + 3 * q^15 - 3 * q^17 + 5 * q^19 + 6 * q^21 + 18 * q^23 + 2 * q^25 + 24 * q^27 + 17 * q^29 - 17 * q^31 + 21 * q^33 + q^37 + 14 * q^41 + 14 * q^43 - 29 * q^45 - 6 * q^47 + 6 * q^49 + 2 * q^51 + 3 * q^53 + 18 * q^55 - 14 * q^59 - 4 * q^61 + 20 * q^63 + 4 * q^67 + 39 * q^69 + 3 * q^71 - 31 * q^73 + 37 * q^75 + 4 * q^77 + 22 * q^79 + 6 * q^81 - 15 * q^83 + 51 * q^85 + 51 * q^87 - 20 * q^89 + 13 * q^93 + 19 * q^95 - 29 * q^97 + 3 * q^99

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