LMFDB - Embedded newform 6272.2.a.bt.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6272,2,Mod(1,6272)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6272, 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("6272.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6272 = 2^{7} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6272.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:	$50.0821721477$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.11344.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 4x^{2} + 4x + 3 $ x^4 - 2x^3 - 4x^2 + 4*x + 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 896)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.78165$ of defining polynomial
Character	$\chi$	$=$	6272.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.95594 q^{3} +0.703158 q^{5} +0.825711 q^{9} +O(q^{10})$	$q-1.95594 q^{3} +0.703158 q^{5} +0.825711 q^{9} -1.90421 q^{11} -2.73760 q^{13} -1.37534 q^{15} -5.38902 q^{17} +3.31052 q^{19} -6.39670 q^{23} -4.50557 q^{25} +4.25278 q^{27} -2.23203 q^{29} +4.78165 q^{31} +3.72452 q^{33} -0.819709 q^{37} +5.35458 q^{39} -8.14391 q^{41} +0.593684 q^{43} +0.580605 q^{45} +1.56692 q^{47} +10.5406 q^{51} -12.3223 q^{53} -1.33896 q^{55} -6.47519 q^{57} -1.36226 q^{59} +10.5400 q^{61} -1.92496 q^{65} +12.9256 q^{67} +12.5116 q^{69} -6.00000 q^{71} +10.0385 q^{73} +8.81263 q^{75} -9.33896 q^{79} -10.7953 q^{81} +12.6626 q^{83} -3.78933 q^{85} +4.36572 q^{87} -2.75489 q^{89} -9.35264 q^{93} +2.32782 q^{95} -14.4044 q^{97} -1.57232 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{3} + 2 q^{5} + 4 q^{9}+O(q^{10}) $ 4 * q + 2 * q^3 + 2 * q^5 + 4 * q^9	$ 4 q + 2 q^{3} + 2 q^{5} + 4 q^{9} + 8 q^{13} + 2 q^{15} - 4 q^{17} + 4 q^{19} + 4 q^{23} + 8 q^{27} - 8 q^{29} + 10 q^{31} - 8 q^{33} - 2 q^{37} + 22 q^{39} - 12 q^{41} + 4 q^{43} + 14 q^{47} - 8 q^{51} + 10 q^{53} + 24 q^{55} + 12 q^{57} + 6 q^{59} + 6 q^{61} + 8 q^{65} + 22 q^{67} + 34 q^{69} - 24 q^{71} - 16 q^{73} + 32 q^{75} - 8 q^{79} - 24 q^{81} + 16 q^{83} + 6 q^{85} - 18 q^{87} - 8 q^{89} + 2 q^{93} + 16 q^{95} - 16 q^{97} - 30 q^{99}+O(q^{100}) $ 4 * q + 2 * q^3 + 2 * q^5 + 4 * q^9 + 8 * q^13 + 2 * q^15 - 4 * q^17 + 4 * q^19 + 4 * q^23 + 8 * q^27 - 8 * q^29 + 10 * q^31 - 8 * q^33 - 2 * q^37 + 22 * q^39 - 12 * q^41 + 4 * q^43 + 14 * q^47 - 8 * q^51 + 10 * q^53 + 24 * q^55 + 12 * q^57 + 6 * q^59 + 6 * q^61 + 8 * q^65 + 22 * q^67 + 34 * q^69 - 24 * q^71 - 16 * q^73 + 32 * q^75 - 8 * q^79 - 24 * q^81 + 16 * q^83 + 6 * q^85 - 18 * q^87 - 8 * q^89 + 2 * q^93 + 16 * q^95 - 16 * q^97 - 30 * q^99

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
$3$	−1.95594	−1.12926	−0.564632	−	0.825343i	$-0.690982\pi$
$3$	−1.95594	−1.12926	−0.564632	+	0.825343i	$0.690982\pi$
$4$	0	0
$5$	0.703158	0.314462	0.157231	−	0.987562i	$-0.449743\pi$
$5$	0.703158	0.314462	0.157231	+	0.987562i	$0.449743\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	0.825711	0.275237
$10$	0	0
$11$	−1.90421	−0.574140	−0.287070	−	0.957910i	$-0.592681\pi$
$11$	−1.90421	−0.574140	−0.287070	+	0.957910i	$0.592681\pi$
$12$	0	0
$13$	−2.73760	−0.759272	−0.379636	−	0.925136i	$-0.623951\pi$
$13$	−2.73760	−0.759272	−0.379636	+	0.925136i	$0.623951\pi$
$14$	0	0
$15$	−1.37534	−0.355110
$16$	0	0
$17$	−5.38902	−1.30703	−0.653514	−	0.756914i	$-0.726706\pi$
$17$	−5.38902	−1.30703	−0.653514	+	0.756914i	$0.726706\pi$
$18$	0	0
$19$	3.31052	0.759486	0.379743	−	0.925092i	$-0.376012\pi$
$19$	3.31052	0.759486	0.379743	+	0.925092i	$0.376012\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−6.39670	−1.33380	−0.666902	−	0.745146i	$-0.732380\pi$
$23$	−6.39670	−1.33380	−0.666902	+	0.745146i	$0.732380\pi$
$24$	0	0
$25$	−4.50557	−0.901114
$26$	0	0
$27$	4.25278	0.818449
$28$	0	0
$29$	−2.23203	−0.414477	−0.207239	−	0.978290i	$-0.566448\pi$
$29$	−2.23203	−0.414477	−0.207239	+	0.978290i	$0.566448\pi$
$30$	0	0
$31$	4.78165	0.858810	0.429405	−	0.903112i	$-0.358723\pi$
$31$	4.78165	0.858810	0.429405	+	0.903112i	$0.358723\pi$
$32$	0	0
$33$	3.72452	0.648355
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−0.819709	−0.134759	−0.0673797	−	0.997727i	$-0.521464\pi$
$37$	−0.819709	−0.134759	−0.0673797	+	0.997727i	$0.521464\pi$
$38$	0	0
$39$	5.35458	0.857419
$40$	0	0
$41$	−8.14391	−1.27187	−0.635933	−	0.771745i	$-0.719384\pi$
$41$	−8.14391	−1.27187	−0.635933	+	0.771745i	$0.719384\pi$
$42$	0	0
$43$	0.593684	0.0905359	0.0452679	−	0.998975i	$-0.485586\pi$
$43$	0.593684	0.0905359	0.0452679	+	0.998975i	$0.485586\pi$
$44$	0	0
$45$	0.580605	0.0865516
$46$	0	0
$47$	1.56692	0.228559	0.114280	−	0.993449i	$-0.463544\pi$
$47$	1.56692	0.228559	0.114280	+	0.993449i	$0.463544\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	10.5406	1.47598
$52$	0	0
$53$	−12.3223	−1.69259	−0.846296	−	0.532712i	$-0.821173\pi$
$53$	−12.3223	−1.69259	−0.846296	+	0.532712i	$0.821173\pi$
$54$	0	0
$55$	−1.33896	−0.180545
$56$	0	0
$57$	−6.47519	−0.857660
$58$	0	0
$59$	−1.36226	−0.177351	−0.0886755	−	0.996061i	$-0.528263\pi$
$59$	−1.36226	−0.177351	−0.0886755	+	0.996061i	$0.528263\pi$
$60$	0	0
$61$	10.5400	1.34951	0.674755	−	0.738042i	$-0.264249\pi$
$61$	10.5400	1.34951	0.674755	+	0.738042i	$0.264249\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−1.92496	−0.238762
$66$	0	0
$67$	12.9256	1.57911	0.789555	−	0.613680i	$-0.210312\pi$
$67$	12.9256	1.57911	0.789555	+	0.613680i	$0.210312\pi$
$68$	0	0
$69$	12.5116	1.50622
$70$	0	0
$71$	−6.00000	−0.712069	−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000	−0.712069	−0.356034	+	0.934473i	$0.615871\pi$
$72$	0	0
$73$	10.0385	1.17492	0.587459	−	0.809254i	$-0.300128\pi$
$73$	10.0385	1.17492	0.587459	+	0.809254i	$0.300128\pi$
$74$	0	0
$75$	8.81263	1.01760
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−9.33896	−1.05071	−0.525357	−	0.850882i	$-0.676068\pi$
$79$	−9.33896	−1.05071	−0.525357	+	0.850882i	$0.676068\pi$
$80$	0	0
$81$	−10.7953	−1.19948
$82$	0	0
$83$	12.6626	1.38990	0.694948	−	0.719060i	$-0.255427\pi$
$83$	12.6626	1.38990	0.694948	+	0.719060i	$0.255427\pi$
$84$	0	0
$85$	−3.78933	−0.411011
$86$	0	0
$87$	4.36572	0.468054
$88$	0	0
$89$	−2.75489	−0.292018	−0.146009	−	0.989283i	$-0.546643\pi$
$89$	−2.75489	−0.292018	−0.146009	+	0.989283i	$0.546643\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−9.35264	−0.969823
$94$	0	0
$95$	2.32782	0.238829
$96$	0	0
$97$	−14.4044	−1.46254	−0.731271	−	0.682087i	$-0.761073\pi$
$97$	−14.4044	−1.46254	−0.731271	+	0.682087i	$0.761073\pi$
$98$	0	0
$99$	−1.57232	−0.158025
$100$	0	0
$101$	0.239103	0.0237917	0.0118958	−	0.999929i	$-0.496213\pi$
$101$	0.239103	0.0237917	0.0118958	+	0.999929i	$0.496213\pi$
$102$	0	0
$103$	3.89113	0.383404	0.191702	−	0.981453i	$-0.438599\pi$
$103$	3.89113	0.383404	0.191702	+	0.981453i	$0.438599\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	3.72798	0.360397	0.180199	−	0.983630i	$-0.442326\pi$
$107$	3.72798	0.360397	0.180199	+	0.983630i	$0.442326\pi$
$108$	0	0
$109$	19.7417	1.89091	0.945454	−	0.325756i	$-0.105619\pi$
$109$	19.7417	1.89091	0.945454	+	0.325756i	$0.105619\pi$
$110$	0	0
$111$	1.60330	0.152179
$112$	0	0
$113$	19.7761	1.86038	0.930189	−	0.367080i	$-0.119643\pi$
$113$	19.7761	1.86038	0.930189	+	0.367080i	$0.119643\pi$
$114$	0	0
$115$	−4.49789	−0.419430
$116$	0	0
$117$	−2.26046	−0.208980
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−7.37400	−0.670364
$122$	0	0
$123$	15.9290	1.43627
$124$	0	0
$125$	−6.68392	−0.597828
$126$	0	0
$127$	14.7780	1.31134	0.655669	−	0.755048i	$-0.272387\pi$
$127$	14.7780	1.31134	0.655669	+	0.755048i	$0.272387\pi$
$128$	0	0
$129$	−1.16121	−0.102239
$130$	0	0
$131$	−19.4585	−1.70010	−0.850048	−	0.526705i	$-0.823427\pi$
$131$	−19.4585	−1.70010	−0.850048	+	0.526705i	$0.823427\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	2.99038	0.257371
$136$	0	0
$137$	−9.02844	−0.771351	−0.385676	−	0.922634i	$-0.626032\pi$
$137$	−9.02844	−0.771351	−0.385676	+	0.922634i	$0.626032\pi$
$138$	0	0
$139$	9.50979	0.806610	0.403305	−	0.915066i	$-0.367861\pi$
$139$	9.50979	0.806610	0.403305	+	0.915066i	$0.367861\pi$
$140$	0	0
$141$	−3.06481	−0.258104
$142$	0	0
$143$	5.21295	0.435929
$144$	0	0
$145$	−1.56947	−0.130337
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−21.5501	−1.76545	−0.882725	−	0.469890i	$-0.844294\pi$
$149$	−21.5501	−1.76545	−0.882725	+	0.469890i	$0.844294\pi$
$150$	0	0
$151$	−7.22647	−0.588082	−0.294041	−	0.955793i	$-0.595000\pi$
$151$	−7.22647	−0.588082	−0.294041	+	0.955793i	$0.595000\pi$
$152$	0	0
$153$	−4.44977	−0.359743
$154$	0	0
$155$	3.36226	0.270063
$156$	0	0
$157$	14.3069	1.14182	0.570908	−	0.821014i	$-0.306591\pi$
$157$	14.3069	1.14182	0.570908	+	0.821014i	$0.306591\pi$
$158$	0	0
$159$	24.1016	1.91138
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−18.5162	−1.45030	−0.725152	−	0.688589i	$-0.758230\pi$
$163$	−18.5162	−1.45030	−0.725152	+	0.688589i	$0.758230\pi$
$164$	0	0
$165$	2.61893	0.203883
$166$	0	0
$167$	10.5573	0.816949	0.408474	−	0.912770i	$-0.366061\pi$
$167$	10.5573	0.816949	0.408474	+	0.912770i	$0.366061\pi$
$168$	0	0
$169$	−5.50557	−0.423505
$170$	0	0
$171$	2.73353	0.209039
$172$	0	0
$173$	10.1379	0.770771	0.385386	−	0.922756i	$-0.374068\pi$
$173$	10.1379	0.770771	0.385386	+	0.922756i	$0.374068\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	2.66450	0.200276
$178$	0	0
$179$	−0.437136	−0.0326731	−0.0163366	−	0.999867i	$-0.505200\pi$
$179$	−0.437136	−0.0326731	−0.0163366	+	0.999867i	$0.505200\pi$
$180$	0	0
$181$	−3.45983	−0.257167	−0.128584	−	0.991699i	$-0.541043\pi$
$181$	−3.45983	−0.257167	−0.128584	+	0.991699i	$0.541043\pi$
$182$	0	0
$183$	−20.6156	−1.52395
$184$	0	0
$185$	−0.576385	−0.0423767
$186$	0	0
$187$	10.2618	0.750417
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−7.62044	−0.551396	−0.275698	−	0.961244i	$-0.588909\pi$
$191$	−7.62044	−0.551396	−0.275698	+	0.961244i	$0.588909\pi$
$192$	0	0
$193$	13.9373	1.00323	0.501615	−	0.865091i	$-0.332740\pi$
$193$	13.9373	1.00323	0.501615	+	0.865091i	$0.332740\pi$
$194$	0	0
$195$	3.76512	0.269626
$196$	0	0
$197$	−7.52287	−0.535982	−0.267991	−	0.963421i	$-0.586360\pi$
$197$	−7.52287	−0.535982	−0.267991	+	0.963421i	$0.586360\pi$
$198$	0	0
$199$	−21.2265	−1.50470	−0.752352	−	0.658761i	$-0.771081\pi$
$199$	−21.2265	−1.50470	−0.752352	+	0.658761i	$0.771081\pi$
$200$	0	0
$201$	−25.2817	−1.78323
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−5.72646	−0.399953
$206$	0	0
$207$	−5.28182	−0.367112
$208$	0	0
$209$	−6.30392	−0.436051
$210$	0	0
$211$	−18.2878	−1.25899	−0.629493	−	0.777006i	$-0.716737\pi$
$211$	−18.2878	−1.25899	−0.629493	+	0.777006i	$0.716737\pi$
$212$	0	0
$213$	11.7357	0.804114
$214$	0	0
$215$	0.417453	0.0284701
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−19.6347	−1.32679
$220$	0	0
$221$	14.7530	0.992391
$222$	0	0
$223$	15.1874	1.01702	0.508511	−	0.861056i	$-0.330196\pi$
$223$	15.1874	1.01702	0.508511	+	0.861056i	$0.330196\pi$
$224$	0	0
$225$	−3.72030	−0.248020
$226$	0	0
$227$	16.0004	1.06199	0.530993	−	0.847376i	$-0.321819\pi$
$227$	16.0004	1.06199	0.530993	+	0.847376i	$0.321819\pi$
$228$	0	0
$229$	2.85203	0.188467	0.0942336	−	0.995550i	$-0.469960\pi$
$229$	2.85203	0.188467	0.0942336	+	0.995550i	$0.469960\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1.92918	0.126385	0.0631925	−	0.998001i	$-0.479872\pi$
$233$	1.92918	0.126385	0.0631925	+	0.998001i	$0.479872\pi$
$234$	0	0
$235$	1.10180	0.0718732
$236$	0	0
$237$	18.2665	1.18653
$238$	0	0
$239$	−6.63818	−0.429388	−0.214694	−	0.976681i	$-0.568875\pi$
$239$	−6.63818	−0.429388	−0.214694	+	0.976681i	$0.568875\pi$
$240$	0	0
$241$	8.24209	0.530920	0.265460	−	0.964122i	$-0.414476\pi$
$241$	8.24209	0.530920	0.265460	+	0.964122i	$0.414476\pi$
$242$	0	0
$243$	8.35670	0.536083
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−9.06287	−0.576657
$248$	0	0
$249$	−24.7672	−1.56956
$250$	0	0
$251$	18.4812	1.16652	0.583261	−	0.812285i	$-0.301776\pi$
$251$	18.4812	1.16652	0.583261	+	0.812285i	$0.301776\pi$
$252$	0	0
$253$	12.1806	0.765790
$254$	0	0
$255$	7.41172	0.464140
$256$	0	0
$257$	−1.56525	−0.0976375	−0.0488187	−	0.998808i	$-0.515546\pi$
$257$	−1.56525	−0.0976375	−0.0488187	+	0.998808i	$0.515546\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−1.84301	−0.114079
$262$	0	0
$263$	29.2316	1.80250	0.901249	−	0.433302i	$-0.142652\pi$
$263$	29.2316	1.80250	0.901249	+	0.433302i	$0.142652\pi$
$264$	0	0
$265$	−8.66450	−0.532256
$266$	0	0
$267$	5.38842	0.329766
$268$	0	0
$269$	−13.4681	−0.821166	−0.410583	−	0.911823i	$-0.634675\pi$
$269$	−13.4681	−0.821166	−0.410583	+	0.911823i	$0.634675\pi$
$270$	0	0
$271$	22.3432	1.35725	0.678626	−	0.734484i	$-0.262576\pi$
$271$	22.3432	1.35725	0.678626	+	0.734484i	$0.262576\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	8.57953	0.517365
$276$	0	0
$277$	14.1926	0.852753	0.426376	−	0.904546i	$-0.359790\pi$
$277$	14.1926	0.852753	0.426376	+	0.904546i	$0.359790\pi$
$278$	0	0
$279$	3.94826	0.236376
$280$	0	0
$281$	1.59562	0.0951870	0.0475935	−	0.998867i	$-0.484845\pi$
$281$	1.59562	0.0951870	0.0475935	+	0.998867i	$0.484845\pi$
$282$	0	0
$283$	32.3490	1.92295	0.961475	−	0.274893i	$-0.0886424\pi$
$283$	32.3490	1.92295	0.961475	+	0.274893i	$0.0886424\pi$
$284$	0	0
$285$	−4.55308	−0.269701
$286$	0	0
$287$	0	0
$288$	0	0
$289$	12.0415	0.708324
$290$	0	0
$291$	28.1741	1.65160
$292$	0	0
$293$	28.6364	1.67296	0.836478	−	0.548000i	$-0.184611\pi$
$293$	28.6364	1.67296	0.836478	+	0.548000i	$0.184611\pi$
$294$	0	0
$295$	−0.957884	−0.0557701
$296$	0	0
$297$	−8.09818	−0.469904
$298$	0	0
$299$	17.5116	1.01272
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−0.467673	−0.0268671
$304$	0	0
$305$	7.41129	0.424369
$306$	0	0
$307$	−10.8208	−0.617573	−0.308787	−	0.951131i	$-0.599923\pi$
$307$	−10.8208	−0.617573	−0.308787	+	0.951131i	$0.599923\pi$
$308$	0	0
$309$	−7.61082	−0.432965
$310$	0	0
$311$	−1.45323	−0.0824051	−0.0412025	−	0.999151i	$-0.513119\pi$
$311$	−1.45323	−0.0824051	−0.0412025	+	0.999151i	$0.513119\pi$
$312$	0	0
$313$	−19.6210	−1.10905	−0.554524	−	0.832168i	$-0.687099\pi$
$313$	−19.6210	−1.10905	−0.554524	+	0.832168i	$0.687099\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	23.1003	1.29744	0.648721	−	0.761026i	$-0.275304\pi$
$317$	23.1003	1.29744	0.648721	+	0.761026i	$0.275304\pi$
$318$	0	0
$319$	4.25024	0.237968
$320$	0	0
$321$	−7.29171	−0.406983
$322$	0	0
$323$	−17.8405	−0.992670
$324$	0	0
$325$	12.3344	0.684191
$326$	0	0
$327$	−38.6135	−2.13533
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−9.15867	−0.503406	−0.251703	−	0.967805i	$-0.580991\pi$
$331$	−9.15867	−0.503406	−0.251703	+	0.967805i	$0.580991\pi$
$332$	0	0
$333$	−0.676843	−0.0370907
$334$	0	0
$335$	9.08872	0.496570
$336$	0	0
$337$	−24.0952	−1.31255	−0.656274	−	0.754523i	$-0.727868\pi$
$337$	−24.0952	−1.31255	−0.656274	+	0.754523i	$0.727868\pi$
$338$	0	0
$339$	−38.6809	−2.10086
$340$	0	0
$341$	−9.10525	−0.493077
$342$	0	0
$343$	0	0
$344$	0	0
$345$	8.79761	0.473648
$346$	0	0
$347$	−10.8375	−0.581785	−0.290892	−	0.956756i	$-0.593952\pi$
$347$	−10.8375	−0.581785	−0.290892	+	0.956756i	$0.593952\pi$
$348$	0	0
$349$	20.5114	1.09795	0.548975	−	0.835839i	$-0.315018\pi$
$349$	20.5114	1.09795	0.548975	+	0.835839i	$0.315018\pi$
$350$	0	0
$351$	−11.6424	−0.621426
$352$	0	0
$353$	−7.54601	−0.401633	−0.200817	−	0.979629i	$-0.564360\pi$
$353$	−7.54601	−0.401633	−0.200817	+	0.979629i	$0.564360\pi$
$354$	0	0
$355$	−4.21895	−0.223919
$356$	0	0
$357$	0	0
$358$	0	0
$359$	36.4100	1.92164	0.960822	−	0.277166i	$-0.0893953\pi$
$359$	36.4100	1.92164	0.960822	+	0.277166i	$0.0893953\pi$
$360$	0	0
$361$	−8.04044	−0.423181
$362$	0	0
$363$	14.4231	0.757017
$364$	0	0
$365$	7.05865	0.369467
$366$	0	0
$367$	14.3540	0.749272	0.374636	−	0.927172i	$-0.377768\pi$
$367$	14.3540	0.749272	0.374636	+	0.927172i	$0.377768\pi$
$368$	0	0
$369$	−6.72452	−0.350064
$370$	0	0
$371$	0	0
$372$	0	0
$373$	27.2299	1.40991	0.704956	−	0.709251i	$-0.250967\pi$
$373$	27.2299	1.40991	0.704956	+	0.709251i	$0.250967\pi$
$374$	0	0
$375$	13.0734	0.675105
$376$	0	0
$377$	6.11039	0.314701
$378$	0	0
$379$	3.63428	0.186681	0.0933403	−	0.995634i	$-0.470246\pi$
$379$	3.63428	0.186681	0.0933403	+	0.995634i	$0.470246\pi$
$380$	0	0
$381$	−28.9050	−1.48085
$382$	0	0
$383$	−4.54467	−0.232222	−0.116111	−	0.993236i	$-0.537043\pi$
$383$	−4.54467	−0.232222	−0.116111	+	0.993236i	$0.537043\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0.490211	0.0249188
$388$	0	0
$389$	31.7832	1.61147	0.805735	−	0.592276i	$-0.201770\pi$
$389$	31.7832	1.61147	0.805735	+	0.592276i	$0.201770\pi$
$390$	0	0
$391$	34.4719	1.74332
$392$	0	0
$393$	38.0597	1.91986
$394$	0	0
$395$	−6.56677	−0.330410
$396$	0	0
$397$	25.9868	1.30424	0.652119	−	0.758117i	$-0.273880\pi$
$397$	25.9868	1.30424	0.652119	+	0.758117i	$0.273880\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	9.66484	0.482639	0.241319	−	0.970446i	$-0.422420\pi$
$401$	9.66484	0.482639	0.241319	+	0.970446i	$0.422420\pi$
$402$	0	0
$403$	−13.0902	−0.652071
$404$	0	0
$405$	−7.59083	−0.377191
$406$	0	0
$407$	1.56089	0.0773707
$408$	0	0
$409$	−21.7803	−1.07697	−0.538484	−	0.842636i	$-0.681003\pi$
$409$	−21.7803	−1.07697	−0.538484	+	0.842636i	$0.681003\pi$
$410$	0	0
$411$	17.6591	0.871059
$412$	0	0
$413$	0	0
$414$	0	0
$415$	8.90378	0.437069
$416$	0	0
$417$	−18.6006	−0.910875
$418$	0	0
$419$	10.5329	0.514567	0.257284	−	0.966336i	$-0.417173\pi$
$419$	10.5329	0.514567	0.257284	+	0.966336i	$0.417173\pi$
$420$	0	0
$421$	−24.9908	−1.21798	−0.608989	−	0.793179i	$-0.708425\pi$
$421$	−24.9908	−1.21798	−0.608989	+	0.793179i	$0.708425\pi$
$422$	0	0
$423$	1.29383	0.0629080
$424$	0	0
$425$	24.2806	1.17778
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−10.1962	−0.492278
$430$	0	0
$431$	15.0695	0.725871	0.362936	−	0.931814i	$-0.381774\pi$
$431$	15.0695	0.725871	0.362936	+	0.931814i	$0.381774\pi$
$432$	0	0
$433$	−24.8338	−1.19344	−0.596719	−	0.802450i	$-0.703529\pi$
$433$	−24.8338	−1.19344	−0.596719	+	0.802450i	$0.703529\pi$
$434$	0	0
$435$	3.06979	0.147185
$436$	0	0
$437$	−21.1764	−1.01301
$438$	0	0
$439$	12.0653	0.575843	0.287922	−	0.957654i	$-0.407036\pi$
$439$	12.0653	0.575843	0.287922	+	0.957654i	$0.407036\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−19.5244	−0.927631	−0.463816	−	0.885932i	$-0.653520\pi$
$443$	−19.5244	−0.927631	−0.463816	+	0.885932i	$0.653520\pi$
$444$	0	0
$445$	−1.93713	−0.0918286
$446$	0	0
$447$	42.1507	1.99366
$448$	0	0
$449$	33.7773	1.59405	0.797025	−	0.603947i	$-0.206406\pi$
$449$	33.7773	1.59405	0.797025	+	0.603947i	$0.206406\pi$
$450$	0	0
$451$	15.5077	0.730229
$452$	0	0
$453$	14.1346	0.664099
$454$	0	0
$455$	0	0
$456$	0	0
$457$	25.1651	1.17717	0.588587	−	0.808434i	$-0.299684\pi$
$457$	25.1651	1.17717	0.588587	+	0.808434i	$0.299684\pi$
$458$	0	0
$459$	−22.9183	−1.06974
$460$	0	0
$461$	18.8826	0.879450	0.439725	−	0.898133i	$-0.355076\pi$
$461$	18.8826	0.879450	0.439725	+	0.898133i	$0.355076\pi$
$462$	0	0
$463$	33.6129	1.56213	0.781063	−	0.624452i	$-0.214678\pi$
$463$	33.6129	1.56213	0.781063	+	0.624452i	$0.214678\pi$
$464$	0	0
$465$	−6.57639	−0.304973
$466$	0	0
$467$	5.41698	0.250668	0.125334	−	0.992115i	$-0.460000\pi$
$467$	5.41698	0.250668	0.125334	+	0.992115i	$0.460000\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−27.9835	−1.28941
$472$	0	0
$473$	−1.13050	−0.0519803
$474$	0	0
$475$	−14.9158	−0.684383
$476$	0	0
$477$	−10.1746	−0.465864
$478$	0	0
$479$	−2.71791	−0.124185	−0.0620923	−	0.998070i	$-0.519777\pi$
$479$	−2.71791	−0.124185	−0.0620923	+	0.998070i	$0.519777\pi$
$480$	0	0
$481$	2.24403	0.102319
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−10.1286	−0.459914
$486$	0	0
$487$	23.2662	1.05429	0.527146	−	0.849775i	$-0.323262\pi$
$487$	23.2662	1.05429	0.527146	+	0.849775i	$0.323262\pi$
$488$	0	0
$489$	36.2167	1.63778
$490$	0	0
$491$	14.9678	0.675489	0.337745	−	0.941238i	$-0.390336\pi$
$491$	14.9678	0.675489	0.337745	+	0.941238i	$0.390336\pi$
$492$	0	0
$493$	12.0284	0.541733
$494$	0	0
$495$	−1.10559	−0.0496927
$496$	0	0
$497$	0	0
$498$	0	0
$499$	0.515029	0.0230559	0.0115279	−	0.999934i	$-0.496330\pi$
$499$	0.515029	0.0230559	0.0115279	+	0.999934i	$0.496330\pi$
$500$	0	0
$501$	−20.6495	−0.922551
$502$	0	0
$503$	−41.8400	−1.86555	−0.932777	−	0.360454i	$-0.882622\pi$
$503$	−41.8400	−1.86555	−0.932777	+	0.360454i	$0.882622\pi$
$504$	0	0
$505$	0.168128	0.00748158
$506$	0	0
$507$	10.7686	0.478249
$508$	0	0
$509$	8.01412	0.355220	0.177610	−	0.984101i	$-0.443163\pi$
$509$	8.01412	0.355220	0.177610	+	0.984101i	$0.443163\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	14.0789	0.621600
$514$	0	0
$515$	2.73608	0.120566
$516$	0	0
$517$	−2.98375	−0.131225
$518$	0	0
$519$	−19.8292	−0.870404
$520$	0	0
$521$	−18.4024	−0.806225	−0.403113	−	0.915150i	$-0.632072\pi$
$521$	−18.4024	−0.806225	−0.403113	+	0.915150i	$0.632072\pi$
$522$	0	0
$523$	35.2378	1.54084	0.770420	−	0.637537i	$-0.220047\pi$
$523$	35.2378	1.54084	0.770420	+	0.637537i	$0.220047\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−25.7684	−1.12249
$528$	0	0
$529$	17.9177	0.779032
$530$	0	0
$531$	−1.12483	−0.0488136
$532$	0	0
$533$	22.2947	0.965692
$534$	0	0
$535$	2.62136	0.113331
$536$	0	0
$537$	0.855014	0.0368966
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−0.251108	−0.0107960	−0.00539798	−	0.999985i	$-0.501718\pi$
$541$	−0.251108	−0.0107960	−0.00539798	+	0.999985i	$0.501718\pi$
$542$	0	0
$543$	6.76724	0.290410
$544$	0	0
$545$	13.8815	0.594618
$546$	0	0
$547$	−25.7113	−1.09933	−0.549667	−	0.835384i	$-0.685246\pi$
$547$	−25.7113	−1.09933	−0.549667	+	0.835384i	$0.685246\pi$
$548$	0	0
$549$	8.70300	0.371435
$550$	0	0
$551$	−7.38918	−0.314790
$552$	0	0
$553$	0	0
$554$	0	0
$555$	1.12738	0.0478544
$556$	0	0
$557$	0.301062	0.0127564	0.00637821	−	0.999980i	$-0.497970\pi$
$557$	0.301062	0.0127564	0.00637821	+	0.999980i	$0.497970\pi$
$558$	0	0
$559$	−1.62527	−0.0687414
$560$	0	0
$561$	−20.0715	−0.847419
$562$	0	0
$563$	−17.7917	−0.749831	−0.374916	−	0.927059i	$-0.622328\pi$
$563$	−17.7917	−0.749831	−0.374916	+	0.927059i	$0.622328\pi$
$564$	0	0
$565$	13.9057	0.585018
$566$	0	0
$567$	0	0
$568$	0	0
$569$	27.9777	1.17289	0.586444	−	0.809990i	$-0.300527\pi$
$569$	27.9777	1.17289	0.586444	+	0.809990i	$0.300527\pi$
$570$	0	0
$571$	−11.1484	−0.466548	−0.233274	−	0.972411i	$-0.574944\pi$
$571$	−11.1484	−0.466548	−0.233274	+	0.972411i	$0.574944\pi$
$572$	0	0
$573$	14.9051	0.622671
$574$	0	0
$575$	28.8208	1.20191
$576$	0	0
$577$	−14.1874	−0.590628	−0.295314	−	0.955400i	$-0.595424\pi$
$577$	−14.1874	−0.590628	−0.295314	+	0.955400i	$0.595424\pi$
$578$	0	0
$579$	−27.2606	−1.13291
$580$	0	0
$581$	0	0
$582$	0	0
$583$	23.4641	0.971785
$584$	0	0
$585$	−1.58946	−0.0657162
$586$	0	0
$587$	−28.2391	−1.16555	−0.582776	−	0.812633i	$-0.698033\pi$
$587$	−28.2391	−1.16555	−0.582776	+	0.812633i	$0.698033\pi$
$588$	0	0
$589$	15.8298	0.652254
$590$	0	0
$591$	14.7143	0.605265
$592$	0	0
$593$	−19.2663	−0.791172	−0.395586	−	0.918429i	$-0.629459\pi$
$593$	−19.2663	−0.791172	−0.395586	+	0.918429i	$0.629459\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	41.5178	1.69921
$598$	0	0
$599$	7.52719	0.307553	0.153776	−	0.988106i	$-0.450856\pi$
$599$	7.52719	0.307553	0.153776	+	0.988106i	$0.450856\pi$
$600$	0	0
$601$	−12.1854	−0.497054	−0.248527	−	0.968625i	$-0.579946\pi$
$601$	−12.1854	−0.497054	−0.248527	+	0.968625i	$0.579946\pi$
$602$	0	0
$603$	10.6728	0.434629
$604$	0	0
$605$	−5.18509	−0.210804
$606$	0	0
$607$	30.6674	1.24475	0.622375	−	0.782719i	$-0.286168\pi$
$607$	30.6674	1.24475	0.622375	+	0.782719i	$0.286168\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−4.28961	−0.173539
$612$	0	0
$613$	7.45027	0.300914	0.150457	−	0.988617i	$-0.451926\pi$
$613$	7.45027	0.300914	0.150457	+	0.988617i	$0.451926\pi$
$614$	0	0
$615$	11.2006	0.451653
$616$	0	0
$617$	−22.6076	−0.910150	−0.455075	−	0.890453i	$-0.650387\pi$
$617$	−22.6076	−0.910150	−0.455075	+	0.890453i	$0.650387\pi$
$618$	0	0
$619$	24.8790	0.999970	0.499985	−	0.866034i	$-0.333339\pi$
$619$	24.8790	0.999970	0.499985	+	0.866034i	$0.333339\pi$
$620$	0	0
$621$	−27.2038	−1.09165
$622$	0	0
$623$	0	0
$624$	0	0
$625$	17.8280	0.713120
$626$	0	0
$627$	12.3301	0.492417
$628$	0	0
$629$	4.41743	0.176134
$630$	0	0
$631$	−20.0770	−0.799253	−0.399626	−	0.916678i	$-0.630860\pi$
$631$	−20.0770	−0.799253	−0.399626	+	0.916678i	$0.630860\pi$
$632$	0	0
$633$	35.7699	1.42173
$634$	0	0
$635$	10.3913	0.412366
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−4.95427	−0.195988
$640$	0	0
$641$	−5.03428	−0.198842	−0.0994210	−	0.995045i	$-0.531699\pi$
$641$	−5.03428	−0.198842	−0.0994210	+	0.995045i	$0.531699\pi$
$642$	0	0
$643$	0.881508	0.0347633	0.0173816	−	0.999849i	$-0.494467\pi$
$643$	0.881508	0.0347633	0.0173816	+	0.999849i	$0.494467\pi$
$644$	0	0
$645$	−0.816515	−0.0321502
$646$	0	0
$647$	−13.7718	−0.541424	−0.270712	−	0.962660i	$-0.587259\pi$
$647$	−13.7718	−0.541424	−0.270712	+	0.962660i	$0.587259\pi$
$648$	0	0
$649$	2.59402	0.101824
$650$	0	0
$651$	0	0
$652$	0	0
$653$	8.82663	0.345413	0.172706	−	0.984973i	$-0.444749\pi$
$653$	8.82663	0.345413	0.172706	+	0.984973i	$0.444749\pi$
$654$	0	0
$655$	−13.6824	−0.534616
$656$	0	0
$657$	8.28890	0.323381
$658$	0	0
$659$	−21.1835	−0.825191	−0.412596	−	0.910914i	$-0.635378\pi$
$659$	−21.1835	−0.825191	−0.412596	+	0.910914i	$0.635378\pi$
$660$	0	0
$661$	−6.19143	−0.240819	−0.120409	−	0.992724i	$-0.538421\pi$
$661$	−6.19143	−0.240819	−0.120409	+	0.992724i	$0.538421\pi$
$662$	0	0
$663$	−28.8559	−1.12067
$664$	0	0
$665$	0	0
$666$	0	0
$667$	14.2776	0.552831
$668$	0	0
$669$	−29.7056	−1.14849
$670$	0	0
$671$	−20.0703	−0.774807
$672$	0	0
$673$	2.31626	0.0892853	0.0446426	−	0.999003i	$-0.485785\pi$
$673$	2.31626	0.0892853	0.0446426	+	0.999003i	$0.485785\pi$
$674$	0	0
$675$	−19.1612	−0.737515
$676$	0	0
$677$	17.0325	0.654612	0.327306	−	0.944918i	$-0.393859\pi$
$677$	17.0325	0.654612	0.327306	+	0.944918i	$0.393859\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−31.2959	−1.19926
$682$	0	0
$683$	20.2110	0.773351	0.386675	−	0.922216i	$-0.373623\pi$
$683$	20.2110	0.773351	0.386675	+	0.922216i	$0.373623\pi$
$684$	0	0
$685$	−6.34842	−0.242561
$686$	0	0
$687$	−5.57840	−0.212829
$688$	0	0
$689$	33.7334	1.28514
$690$	0	0
$691$	−17.1929	−0.654050	−0.327025	−	0.945016i	$-0.606046\pi$
$691$	−17.1929	−0.654050	−0.327025	+	0.945016i	$0.606046\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	6.68689	0.253648
$696$	0	0
$697$	43.8877	1.66236
$698$	0	0
$699$	−3.77337	−0.142722
$700$	0	0
$701$	−9.40632	−0.355272	−0.177636	−	0.984096i	$-0.556845\pi$
$701$	−9.40632	−0.355272	−0.177636	+	0.984096i	$0.556845\pi$
$702$	0	0
$703$	−2.71366	−0.102348
$704$	0	0
$705$	−2.15505	−0.0811639
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−44.0010	−1.65249	−0.826247	−	0.563308i	$-0.809528\pi$
$709$	−44.0010	−1.65249	−0.826247	+	0.563308i	$0.809528\pi$
$710$	0	0
$711$	−7.71128	−0.289196
$712$	0	0
$713$	−30.5868	−1.14548
$714$	0	0
$715$	3.66553	0.137083
$716$	0	0
$717$	12.9839	0.484893
$718$	0	0
$719$	36.0370	1.34395	0.671977	−	0.740572i	$-0.265446\pi$
$719$	36.0370	1.34395	0.671977	+	0.740572i	$0.265446\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−16.1211	−0.599548
$724$	0	0
$725$	10.0566	0.373491
$726$	0	0
$727$	2.02401	0.0750663	0.0375332	−	0.999295i	$-0.488050\pi$
$727$	2.02401	0.0750663	0.0375332	+	0.999295i	$0.488050\pi$
$728$	0	0
$729$	16.0408	0.594103
$730$	0	0
$731$	−3.19937	−0.118333
$732$	0	0
$733$	−51.4651	−1.90091	−0.950454	−	0.310866i	$-0.899381\pi$
$733$	−51.4651	−1.90091	−0.950454	+	0.310866i	$0.899381\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−24.6129	−0.906629
$738$	0	0
$739$	4.89999	0.180249	0.0901244	−	0.995931i	$-0.471274\pi$
$739$	4.89999	0.180249	0.0901244	+	0.995931i	$0.471274\pi$
$740$	0	0
$741$	17.7265	0.651198
$742$	0	0
$743$	−9.94313	−0.364778	−0.182389	−	0.983226i	$-0.558383\pi$
$743$	−9.94313	−0.364778	−0.182389	+	0.983226i	$0.558383\pi$
$744$	0	0
$745$	−15.1531	−0.555167
$746$	0	0
$747$	10.4556	0.382551
$748$	0	0
$749$	0	0
$750$	0	0
$751$	0.662823	0.0241868	0.0120934	−	0.999927i	$-0.496150\pi$
$751$	0.662823	0.0241868	0.0120934	+	0.999927i	$0.496150\pi$
$752$	0	0
$753$	−36.1482	−1.31731
$754$	0	0
$755$	−5.08135	−0.184929
$756$	0	0
$757$	21.7680	0.791170	0.395585	−	0.918429i	$-0.370542\pi$
$757$	21.7680	0.791170	0.395585	+	0.918429i	$0.370542\pi$
$758$	0	0
$759$	−23.8246	−0.864779
$760$	0	0
$761$	26.4916	0.960321	0.480160	−	0.877181i	$-0.340578\pi$
$761$	26.4916	0.960321	0.480160	+	0.877181i	$0.340578\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−3.12889	−0.113125
$766$	0	0
$767$	3.72931	0.134658
$768$	0	0
$769$	13.5156	0.487386	0.243693	−	0.969852i	$-0.421641\pi$
$769$	13.5156	0.487386	0.243693	+	0.969852i	$0.421641\pi$
$770$	0	0
$771$	3.06153	0.110258
$772$	0	0
$773$	−4.61625	−0.166035	−0.0830175	−	0.996548i	$-0.526456\pi$
$773$	−4.61625	−0.166035	−0.0830175	+	0.996548i	$0.526456\pi$
$774$	0	0
$775$	−21.5441	−0.773886
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−26.9606	−0.965964
$780$	0	0
$781$	11.4252	0.408827
$782$	0	0
$783$	−9.49233	−0.339228
$784$	0	0
$785$	10.0600	0.359057
$786$	0	0
$787$	2.76522	0.0985695	0.0492848	−	0.998785i	$-0.484306\pi$
$787$	2.76522	0.0985695	0.0492848	+	0.998785i	$0.484306\pi$
$788$	0	0
$789$	−57.1753	−2.03550
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−28.8543	−1.02465
$794$	0	0
$795$	16.9473	0.601058
$796$	0	0
$797$	−55.1217	−1.95251	−0.976255	−	0.216626i	$-0.930495\pi$
$797$	−55.1217	−1.95251	−0.976255	+	0.216626i	$0.930495\pi$
$798$	0	0
$799$	−8.44418	−0.298734
$800$	0	0
$801$	−2.27475	−0.0803742
$802$	0	0
$803$	−19.1154	−0.674567
$804$	0	0
$805$	0	0
$806$	0	0
$807$	26.3429	0.927313
$808$	0	0
$809$	21.2686	0.747764	0.373882	−	0.927476i	$-0.378027\pi$
$809$	21.2686	0.747764	0.373882	+	0.927476i	$0.378027\pi$
$810$	0	0
$811$	32.9239	1.15611	0.578057	−	0.815996i	$-0.303811\pi$
$811$	32.9239	1.15611	0.578057	+	0.815996i	$0.303811\pi$
$812$	0	0
$813$	−43.7020	−1.53270
$814$	0	0
$815$	−13.0198	−0.456065
$816$	0	0
$817$	1.96540	0.0687607
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−3.55342	−0.124015	−0.0620077	−	0.998076i	$-0.519750\pi$
$821$	−3.55342	−0.124015	−0.0620077	+	0.998076i	$0.519750\pi$
$822$	0	0
$823$	−25.2770	−0.881101	−0.440550	−	0.897728i	$-0.645217\pi$
$823$	−25.2770	−0.881101	−0.440550	+	0.897728i	$0.645217\pi$
$824$	0	0
$825$	−16.7811	−0.584242
$826$	0	0
$827$	20.6746	0.718925	0.359463	−	0.933160i	$-0.382960\pi$
$827$	20.6746	0.718925	0.359463	+	0.933160i	$0.382960\pi$
$828$	0	0
$829$	20.8563	0.724368	0.362184	−	0.932107i	$-0.382031\pi$
$829$	20.8563	0.724368	0.362184	+	0.932107i	$0.382031\pi$
$830$	0	0
$831$	−27.7600	−0.962983
$832$	0	0
$833$	0	0
$834$	0	0
$835$	7.42346	0.256899
$836$	0	0
$837$	20.3353	0.702892
$838$	0	0
$839$	18.5672	0.641011	0.320506	−	0.947247i	$-0.396147\pi$
$839$	18.5672	0.641011	0.320506	+	0.947247i	$0.396147\pi$
$840$	0	0
$841$	−24.0181	−0.828209
$842$	0	0
$843$	−3.12095	−0.107491
$844$	0	0
$845$	−3.87129	−0.133176
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−63.2728	−2.17152
$850$	0	0
$851$	5.24343	0.179742
$852$	0	0
$853$	−35.8522	−1.22756	−0.613778	−	0.789479i	$-0.710351\pi$
$853$	−35.8522	−1.22756	−0.613778	+	0.789479i	$0.710351\pi$
$854$	0	0
$855$	1.92211	0.0657347
$856$	0	0
$857$	11.4045	0.389571	0.194785	−	0.980846i	$-0.437599\pi$
$857$	11.4045	0.389571	0.194785	+	0.980846i	$0.437599\pi$
$858$	0	0
$859$	−24.8781	−0.848829	−0.424414	−	0.905468i	$-0.639520\pi$
$859$	−24.8781	−0.848829	−0.424414	+	0.905468i	$0.639520\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	51.9455	1.76825	0.884123	−	0.467254i	$-0.154757\pi$
$863$	51.9455	1.76825	0.884123	+	0.467254i	$0.154757\pi$
$864$	0	0
$865$	7.12855	0.242378
$866$	0	0
$867$	−23.5525	−0.799885
$868$	0	0
$869$	17.7833	0.603257
$870$	0	0
$871$	−35.3850	−1.19897
$872$	0	0
$873$	−11.8939	−0.402546
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−19.8259	−0.669473	−0.334736	−	0.942312i	$-0.608647\pi$
$877$	−19.8259	−0.669473	−0.334736	+	0.942312i	$0.608647\pi$
$878$	0	0
$879$	−56.0112	−1.88921
$880$	0	0
$881$	45.5107	1.53329	0.766647	−	0.642069i	$-0.221924\pi$
$881$	45.5107	1.53329	0.766647	+	0.642069i	$0.221924\pi$
$882$	0	0
$883$	15.4490	0.519901	0.259951	−	0.965622i	$-0.416294\pi$
$883$	15.4490	0.519901	0.259951	+	0.965622i	$0.416294\pi$
$884$	0	0
$885$	1.87357	0.0629792
$886$	0	0
$887$	4.38768	0.147324	0.0736619	−	0.997283i	$-0.476531\pi$
$887$	4.38768	0.147324	0.0736619	+	0.997283i	$0.476531\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	20.5565	0.688670
$892$	0	0
$893$	5.18734	0.173588
$894$	0	0
$895$	−0.307376	−0.0102744
$896$	0	0
$897$	−34.2516	−1.14363
$898$	0	0
$899$	−10.6728	−0.355957
$900$	0	0
$901$	66.4049	2.21227
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−2.43281	−0.0808694
$906$	0	0
$907$	−18.0287	−0.598633	−0.299317	−	0.954154i	$-0.596759\pi$
$907$	−18.0287	−0.598633	−0.299317	+	0.954154i	$0.596759\pi$
$908$	0	0
$909$	0.197430	0.00654835
$910$	0	0
$911$	5.65142	0.187240	0.0936200	−	0.995608i	$-0.470156\pi$
$911$	5.65142	0.187240	0.0936200	+	0.995608i	$0.470156\pi$
$912$	0	0
$913$	−24.1121	−0.797995
$914$	0	0
$915$	−14.4961	−0.479225
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−35.2298	−1.16212	−0.581062	−	0.813859i	$-0.697363\pi$
$919$	−35.2298	−1.16212	−0.581062	+	0.813859i	$0.697363\pi$
$920$	0	0
$921$	21.1648	0.697403
$922$	0	0
$923$	16.4256	0.540654
$924$	0	0
$925$	3.69325	0.121433
$926$	0	0
$927$	3.21295	0.105527
$928$	0	0
$929$	−20.6403	−0.677186	−0.338593	−	0.940933i	$-0.609951\pi$
$929$	−20.6403	−0.677186	−0.338593	+	0.940933i	$0.609951\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	2.84243	0.0930571
$934$	0	0
$935$	7.21567	0.235978
$936$	0	0
$937$	−16.1498	−0.527590	−0.263795	−	0.964579i	$-0.584974\pi$
$937$	−16.1498	−0.527590	−0.263795	+	0.964579i	$0.584974\pi$
$938$	0	0
$939$	38.3776	1.25241
$940$	0	0
$941$	24.5448	0.800137	0.400068	−	0.916485i	$-0.368986\pi$
$941$	24.5448	0.800137	0.400068	+	0.916485i	$0.368986\pi$
$942$	0	0
$943$	52.0941	1.69642
$944$	0	0
$945$	0	0
$946$	0	0
$947$	48.3006	1.56956	0.784780	−	0.619775i	$-0.212776\pi$
$947$	48.3006	1.56956	0.784780	+	0.619775i	$0.212776\pi$
$948$	0	0
$949$	−27.4814	−0.892082
$950$	0	0
$951$	−45.1829	−1.46515
$952$	0	0
$953$	11.1097	0.359877	0.179938	−	0.983678i	$-0.442410\pi$
$953$	11.1097	0.359877	0.179938	+	0.983678i	$0.442410\pi$
$954$	0	0
$955$	−5.35838	−0.173393
$956$	0	0
$957$	−8.31323	−0.268728
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−8.13579	−0.262445
$962$	0	0
$963$	3.07823	0.0991946
$964$	0	0
$965$	9.80013	0.315477
$966$	0	0
$967$	52.4719	1.68738	0.843691	−	0.536830i	$-0.180378\pi$
$967$	52.4719	1.68738	0.843691	+	0.536830i	$0.180378\pi$
$968$	0	0
$969$	34.8949	1.12099
$970$	0	0
$971$	−42.4555	−1.36246	−0.681231	−	0.732069i	$-0.738555\pi$
$971$	−42.4555	−1.36246	−0.681231	+	0.732069i	$0.738555\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−24.1254	−0.772632
$976$	0	0
$977$	−27.3517	−0.875060	−0.437530	−	0.899204i	$-0.644147\pi$
$977$	−27.3517	−0.875060	−0.437530	+	0.899204i	$0.644147\pi$
$978$	0	0
$979$	5.24589	0.167659
$980$	0	0
$981$	16.3009	0.520448
$982$	0	0
$983$	38.0340	1.21310	0.606548	−	0.795047i	$-0.292554\pi$
$983$	38.0340	1.21310	0.606548	+	0.795047i	$0.292554\pi$
$984$	0	0
$985$	−5.28977	−0.168546
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−3.79761	−0.120757
$990$	0	0
$991$	−35.1820	−1.11759	−0.558796	−	0.829305i	$-0.688737\pi$
$991$	−35.1820	−1.11759	−0.558796	+	0.829305i	$0.688737\pi$
$992$	0	0
$993$	17.9138	0.568478
$994$	0	0
$995$	−14.9256	−0.473172
$996$	0	0
$997$	−13.1418	−0.416205	−0.208103	−	0.978107i	$-0.566729\pi$
$997$	−13.1418	−0.416205	−0.208103	+	0.978107i	$0.566729\pi$
$998$	0	0
$999$	−3.48604	−0.110294

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6272.2.a.bt.1.1		4
4.3	odd	2		6272.2.a.bd.1.4		4
7.2	even	3		896.2.i.b.641.4	yes	8
7.4	even	3		896.2.i.b.513.4	✓	8
7.6	odd	2		6272.2.a.ba.1.4		4
8.3	odd	2		6272.2.a.bp.1.1		4
8.5	even	2		6272.2.a.z.1.4		4
28.11	odd	6		896.2.i.e.513.1	yes	8
28.23	odd	6		896.2.i.e.641.1	yes	8
28.27	even	2		6272.2.a.bq.1.1		4
56.11	odd	6		896.2.i.d.513.4	yes	8
56.13	odd	2		6272.2.a.bu.1.1		4
56.27	even	2		6272.2.a.be.1.4		4
56.37	even	6		896.2.i.g.641.1	yes	8
56.51	odd	6		896.2.i.d.641.4	yes	8
56.53	even	6		896.2.i.g.513.1	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
896.2.i.b.513.4	✓	8	7.4	even	3
896.2.i.b.641.4	yes	8	7.2	even	3
896.2.i.d.513.4	yes	8	56.11	odd	6
896.2.i.d.641.4	yes	8	56.51	odd	6
896.2.i.e.513.1	yes	8	28.11	odd	6
896.2.i.e.641.1	yes	8	28.23	odd	6
896.2.i.g.513.1	yes	8	56.53	even	6
896.2.i.g.641.1	yes	8	56.37	even	6
6272.2.a.z.1.4		4	8.5	even	2
6272.2.a.ba.1.4		4	7.6	odd	2
6272.2.a.bd.1.4		4	4.3	odd	2
6272.2.a.be.1.4		4	56.27	even	2
6272.2.a.bp.1.1		4	8.3	odd	2
6272.2.a.bq.1.1		4	28.27	even	2
6272.2.a.bt.1.1		4	1.1	even	1	trivial
6272.2.a.bu.1.1		4	56.13	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:	\( N \)	\(=\)	\( 6272 = 2^{7} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6272.a (trivial)

\(f(q)\)	\(=\)	\(q-1.95594 q^{3} +0.703158 q^{5} +0.825711 q^{9} +O(q^{10})\)	\(q-1.95594 q^{3} +0.703158 q^{5} +0.825711 q^{9} -1.90421 q^{11} -2.73760 q^{13} -1.37534 q^{15} -5.38902 q^{17} +3.31052 q^{19} -6.39670 q^{23} -4.50557 q^{25} +4.25278 q^{27} -2.23203 q^{29} +4.78165 q^{31} +3.72452 q^{33} -0.819709 q^{37} +5.35458 q^{39} -8.14391 q^{41} +0.593684 q^{43} +0.580605 q^{45} +1.56692 q^{47} +10.5406 q^{51} -12.3223 q^{53} -1.33896 q^{55} -6.47519 q^{57} -1.36226 q^{59} +10.5400 q^{61} -1.92496 q^{65} +12.9256 q^{67} +12.5116 q^{69} -6.00000 q^{71} +10.0385 q^{73} +8.81263 q^{75} -9.33896 q^{79} -10.7953 q^{81} +12.6626 q^{83} -3.78933 q^{85} +4.36572 q^{87} -2.75489 q^{89} -9.35264 q^{93} +2.32782 q^{95} -14.4044 q^{97} -1.57232 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{3} + 2 q^{5} + 4 q^{9}+O(q^{10}) \) 4 * q + 2 * q^3 + 2 * q^5 + 4 * q^9	\( 4 q + 2 q^{3} + 2 q^{5} + 4 q^{9} + 8 q^{13} + 2 q^{15} - 4 q^{17} + 4 q^{19} + 4 q^{23} + 8 q^{27} - 8 q^{29} + 10 q^{31} - 8 q^{33} - 2 q^{37} + 22 q^{39} - 12 q^{41} + 4 q^{43} + 14 q^{47} - 8 q^{51} + 10 q^{53} + 24 q^{55} + 12 q^{57} + 6 q^{59} + 6 q^{61} + 8 q^{65} + 22 q^{67} + 34 q^{69} - 24 q^{71} - 16 q^{73} + 32 q^{75} - 8 q^{79} - 24 q^{81} + 16 q^{83} + 6 q^{85} - 18 q^{87} - 8 q^{89} + 2 q^{93} + 16 q^{95} - 16 q^{97} - 30 q^{99}+O(q^{100}) \) 4 * q + 2 * q^3 + 2 * q^5 + 4 * q^9 + 8 * q^13 + 2 * q^15 - 4 * q^17 + 4 * q^19 + 4 * q^23 + 8 * q^27 - 8 * q^29 + 10 * q^31 - 8 * q^33 - 2 * q^37 + 22 * q^39 - 12 * q^41 + 4 * q^43 + 14 * q^47 - 8 * q^51 + 10 * q^53 + 24 * q^55 + 12 * q^57 + 6 * q^59 + 6 * q^61 + 8 * q^65 + 22 * q^67 + 34 * q^69 - 24 * q^71 - 16 * q^73 + 32 * q^75 - 8 * q^79 - 24 * q^81 + 16 * q^83 + 6 * q^85 - 18 * q^87 - 8 * q^89 + 2 * q^93 + 16 * q^95 - 16 * q^97 - 30 * q^99

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