LMFDB - Embedded newform 6032.2.a.bd.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6032,2,Mod(1,6032)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6032, 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("6032.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6032 = 2^{4} \cdot 13 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6032.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:	$48.1657624992$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 6 x^{11} - 10 x^{10} + 98 x^{9} + 10 x^{8} - 585 x^{7} + 151 x^{6} + 1524 x^{5} - 445 x^{4} + \cdots + 68 $ x^12 - 6x^11 - 10x^10 + 98x^9 + 10x^8 - 585x^7 + 151x^6 + 1524x^5 - 445x^4 - 1567x^3 + 273x^2 + 424*x + 68
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 3016)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$0.848556$ of defining polynomial
Character	$\chi$	$=$	6032.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.151444 q^{3} +3.67772 q^{5} +0.901102 q^{7} -2.97706 q^{9} +O(q^{10})$	$q-0.151444 q^{3} +3.67772 q^{5} +0.901102 q^{7} -2.97706 q^{9} -1.75590 q^{11} +1.00000 q^{13} -0.556966 q^{15} -5.39861 q^{17} -2.14907 q^{19} -0.136466 q^{21} -4.92932 q^{23} +8.52559 q^{25} +0.905188 q^{27} +1.00000 q^{29} -3.97282 q^{31} +0.265919 q^{33} +3.31400 q^{35} +2.48072 q^{37} -0.151444 q^{39} +0.995414 q^{41} +2.34747 q^{43} -10.9488 q^{45} -6.61979 q^{47} -6.18801 q^{49} +0.817584 q^{51} +14.0893 q^{53} -6.45769 q^{55} +0.325463 q^{57} -10.1428 q^{59} -1.78100 q^{61} -2.68264 q^{63} +3.67772 q^{65} -14.3091 q^{67} +0.746513 q^{69} -13.2450 q^{71} +7.83764 q^{73} -1.29115 q^{75} -1.58224 q^{77} -6.15682 q^{79} +8.79411 q^{81} +14.3672 q^{83} -19.8545 q^{85} -0.151444 q^{87} -0.813326 q^{89} +0.901102 q^{91} +0.601658 q^{93} -7.90368 q^{95} -15.5159 q^{97} +5.22742 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 6 q^{3} + 3 q^{5} - 6 q^{7} + 20 q^{9}+O(q^{10}) $ 12 * q - 6 * q^3 + 3 * q^5 - 6 * q^7 + 20 * q^9	$ 12 q - 6 q^{3} + 3 q^{5} - 6 q^{7} + 20 q^{9} - 14 q^{11} + 12 q^{13} - 8 q^{15} + 4 q^{17} - 11 q^{19} - 5 q^{21} - 15 q^{23} + 5 q^{25} - 24 q^{27} + 12 q^{29} - 13 q^{31} - q^{33} - 18 q^{35} - 23 q^{37} - 6 q^{39} - 2 q^{41} - 26 q^{43} - 9 q^{45} - 15 q^{47} + 16 q^{49} - 21 q^{51} + 31 q^{53} - 10 q^{55} - 10 q^{57} - 7 q^{59} + 2 q^{61} + 25 q^{63} + 3 q^{65} - 47 q^{67} - 8 q^{69} - 32 q^{71} - 25 q^{73} - 31 q^{75} - 4 q^{77} - 7 q^{79} + 64 q^{81} - 12 q^{83} + 7 q^{85} - 6 q^{87} + 6 q^{89} - 6 q^{91} + 17 q^{93} - q^{95} - 7 q^{97} - 44 q^{99}+O(q^{100}) $ 12 * q - 6 * q^3 + 3 * q^5 - 6 * q^7 + 20 * q^9 - 14 * q^11 + 12 * q^13 - 8 * q^15 + 4 * q^17 - 11 * q^19 - 5 * q^21 - 15 * q^23 + 5 * q^25 - 24 * q^27 + 12 * q^29 - 13 * q^31 - q^33 - 18 * q^35 - 23 * q^37 - 6 * q^39 - 2 * q^41 - 26 * q^43 - 9 * q^45 - 15 * q^47 + 16 * q^49 - 21 * q^51 + 31 * q^53 - 10 * q^55 - 10 * q^57 - 7 * q^59 + 2 * q^61 + 25 * q^63 + 3 * q^65 - 47 * q^67 - 8 * q^69 - 32 * q^71 - 25 * q^73 - 31 * q^75 - 4 * q^77 - 7 * q^79 + 64 * q^81 - 12 * q^83 + 7 * q^85 - 6 * q^87 + 6 * q^89 - 6 * q^91 + 17 * q^93 - q^95 - 7 * q^97 - 44 * 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
$3$	−0.151444	−0.0874360	−0.0437180	−	0.999044i	$-0.513920\pi$
$3$	−0.151444	−0.0874360	−0.0437180	+	0.999044i	$0.513920\pi$
$4$	0	0
$5$	3.67772	1.64472	0.822362	−	0.568964i	$-0.192656\pi$
$5$	3.67772	1.64472	0.822362	+	0.568964i	$0.192656\pi$
$6$	0	0
$7$	0.901102	0.340585	0.170292	−	0.985394i	$-0.445529\pi$
$7$	0.901102	0.340585	0.170292	+	0.985394i	$0.445529\pi$
$8$	0	0
$9$	−2.97706	−0.992355
$10$	0	0
$11$	−1.75590	−0.529423	−0.264711	−	0.964328i	$-0.585277\pi$
$11$	−1.75590	−0.529423	−0.264711	+	0.964328i	$0.585277\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	−0.556966	−0.143808
$16$	0	0
$17$	−5.39861	−1.30935	−0.654677	−	0.755909i	$-0.727195\pi$
$17$	−5.39861	−1.30935	−0.654677	+	0.755909i	$0.727195\pi$
$18$	0	0
$19$	−2.14907	−0.493031	−0.246516	−	0.969139i	$-0.579286\pi$
$19$	−2.14907	−0.493031	−0.246516	+	0.969139i	$0.579286\pi$
$20$	0	0
$21$	−0.136466	−0.0297793
$22$	0	0
$23$	−4.92932	−1.02783	−0.513917	−	0.857840i	$-0.671806\pi$
$23$	−4.92932	−1.02783	−0.513917	+	0.857840i	$0.671806\pi$
$24$	0	0
$25$	8.52559	1.70512
$26$	0	0
$27$	0.905188	0.174203
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−3.97282	−0.713540	−0.356770	−	0.934192i	$-0.616122\pi$
$31$	−3.97282	−0.713540	−0.356770	+	0.934192i	$0.616122\pi$
$32$	0	0
$33$	0.265919	0.0462906
$34$	0	0
$35$	3.31400	0.560168
$36$	0	0
$37$	2.48072	0.407827	0.203914	−	0.978989i	$-0.434634\pi$
$37$	2.48072	0.407827	0.203914	+	0.978989i	$0.434634\pi$
$38$	0	0
$39$	−0.151444	−0.0242504
$40$	0	0
$41$	0.995414	0.155458	0.0777288	−	0.996975i	$-0.475233\pi$
$41$	0.995414	0.155458	0.0777288	+	0.996975i	$0.475233\pi$
$42$	0	0
$43$	2.34747	0.357986	0.178993	−	0.983850i	$-0.442716\pi$
$43$	2.34747	0.357986	0.178993	+	0.983850i	$0.442716\pi$
$44$	0	0
$45$	−10.9488	−1.63215
$46$	0	0
$47$	−6.61979	−0.965595	−0.482798	−	0.875732i	$-0.660379\pi$
$47$	−6.61979	−0.965595	−0.482798	+	0.875732i	$0.660379\pi$
$48$	0	0
$49$	−6.18801	−0.884002
$50$	0	0
$51$	0.817584	0.114485
$52$	0	0
$53$	14.0893	1.93531	0.967654	−	0.252279i	$-0.0811802\pi$
$53$	14.0893	1.93531	0.967654	+	0.252279i	$0.0811802\pi$
$54$	0	0
$55$	−6.45769	−0.870754
$56$	0	0
$57$	0.325463	0.0431087
$58$	0	0
$59$	−10.1428	−1.32048	−0.660238	−	0.751056i	$-0.729545\pi$
$59$	−10.1428	−1.32048	−0.660238	+	0.751056i	$0.729545\pi$
$60$	0	0
$61$	−1.78100	−0.228034	−0.114017	−	0.993479i	$-0.536372\pi$
$61$	−1.78100	−0.228034	−0.114017	+	0.993479i	$0.536372\pi$
$62$	0	0
$63$	−2.68264	−0.337981
$64$	0	0
$65$	3.67772	0.456164
$66$	0	0
$67$	−14.3091	−1.74813	−0.874065	−	0.485809i	$-0.838525\pi$
$67$	−14.3091	−1.74813	−0.874065	+	0.485809i	$0.838525\pi$
$68$	0	0
$69$	0.746513	0.0898696
$70$	0	0
$71$	−13.2450	−1.57189	−0.785947	−	0.618293i	$-0.787824\pi$
$71$	−13.2450	−1.57189	−0.785947	+	0.618293i	$0.787824\pi$
$72$	0	0
$73$	7.83764	0.917326	0.458663	−	0.888610i	$-0.348328\pi$
$73$	7.83764	0.917326	0.458663	+	0.888610i	$0.348328\pi$
$74$	0	0
$75$	−1.29115	−0.149089
$76$	0	0
$77$	−1.58224	−0.180313
$78$	0	0
$79$	−6.15682	−0.692696	−0.346348	−	0.938106i	$-0.612578\pi$
$79$	−6.15682	−0.692696	−0.346348	+	0.938106i	$0.612578\pi$
$80$	0	0
$81$	8.79411	0.977123
$82$	0	0
$83$	14.3672	1.57700	0.788502	−	0.615032i	$-0.210857\pi$
$83$	14.3672	1.57700	0.788502	+	0.615032i	$0.210857\pi$
$84$	0	0
$85$	−19.8545	−2.15353
$86$	0	0
$87$	−0.151444	−0.0162365
$88$	0	0
$89$	−0.813326	−0.0862124	−0.0431062	−	0.999070i	$-0.513725\pi$
$89$	−0.813326	−0.0862124	−0.0431062	+	0.999070i	$0.513725\pi$
$90$	0	0
$91$	0.901102	0.0944612
$92$	0	0
$93$	0.601658	0.0623891
$94$	0	0
$95$	−7.90368	−0.810901
$96$	0	0
$97$	−15.5159	−1.57541	−0.787703	−	0.616055i	$-0.788730\pi$
$97$	−15.5159	−1.57541	−0.787703	+	0.616055i	$0.788730\pi$
$98$	0	0
$99$	5.22742	0.525375
$100$	0	0
$101$	−19.4387	−1.93423	−0.967113	−	0.254346i	$-0.918140\pi$
$101$	−19.4387	−1.93423	−0.967113	+	0.254346i	$0.918140\pi$
$102$	0	0
$103$	−6.37882	−0.628523	−0.314262	−	0.949336i	$-0.601757\pi$
$103$	−6.37882	−0.628523	−0.314262	+	0.949336i	$0.601757\pi$
$104$	0	0
$105$	−0.501884	−0.0489788
$106$	0	0
$107$	7.54237	0.729148	0.364574	−	0.931174i	$-0.381215\pi$
$107$	7.54237	0.729148	0.364574	+	0.931174i	$0.381215\pi$
$108$	0	0
$109$	7.54868	0.723032	0.361516	−	0.932366i	$-0.382259\pi$
$109$	7.54868	0.723032	0.361516	+	0.932366i	$0.382259\pi$
$110$	0	0
$111$	−0.375688	−0.0356588
$112$	0	0
$113$	−16.1004	−1.51460	−0.757299	−	0.653068i	$-0.773482\pi$
$113$	−16.1004	−1.51460	−0.757299	+	0.653068i	$0.773482\pi$
$114$	0	0
$115$	−18.1286	−1.69050
$116$	0	0
$117$	−2.97706	−0.275230
$118$	0	0
$119$	−4.86470	−0.445946
$120$	0	0
$121$	−7.91683	−0.719712
$122$	0	0
$123$	−0.150749	−0.0135926
$124$	0	0
$125$	12.9661	1.15973
$126$	0	0
$127$	1.69500	0.150407	0.0752035	−	0.997168i	$-0.476039\pi$
$127$	1.69500	0.150407	0.0752035	+	0.997168i	$0.476039\pi$
$128$	0	0
$129$	−0.355509	−0.0313009
$130$	0	0
$131$	0.379139	0.0331255	0.0165628	−	0.999863i	$-0.494728\pi$
$131$	0.379139	0.0331255	0.0165628	+	0.999863i	$0.494728\pi$
$132$	0	0
$133$	−1.93653	−0.167919
$134$	0	0
$135$	3.32902	0.286517
$136$	0	0
$137$	18.5808	1.58746	0.793731	−	0.608269i	$-0.208136\pi$
$137$	18.5808	1.58746	0.793731	+	0.608269i	$0.208136\pi$
$138$	0	0
$139$	13.8534	1.17503	0.587514	−	0.809214i	$-0.300107\pi$
$139$	13.8534	1.17503	0.587514	+	0.809214i	$0.300107\pi$
$140$	0	0
$141$	1.00252	0.0844278
$142$	0	0
$143$	−1.75590	−0.146835
$144$	0	0
$145$	3.67772	0.305418
$146$	0	0
$147$	0.937135	0.0772936
$148$	0	0
$149$	−7.19281	−0.589258	−0.294629	−	0.955612i	$-0.595196\pi$
$149$	−7.19281	−0.589258	−0.294629	+	0.955612i	$0.595196\pi$
$150$	0	0
$151$	−19.8310	−1.61382	−0.806911	−	0.590673i	$-0.798862\pi$
$151$	−19.8310	−1.61382	−0.806911	+	0.590673i	$0.798862\pi$
$152$	0	0
$153$	16.0720	1.29934
$154$	0	0
$155$	−14.6109	−1.17358
$156$	0	0
$157$	−19.5102	−1.55708	−0.778540	−	0.627595i	$-0.784040\pi$
$157$	−19.5102	−1.55708	−0.778540	+	0.627595i	$0.784040\pi$
$158$	0	0
$159$	−2.13373	−0.169216
$160$	0	0
$161$	−4.44182	−0.350064
$162$	0	0
$163$	−5.83746	−0.457225	−0.228613	−	0.973517i	$-0.573419\pi$
$163$	−5.83746	−0.457225	−0.228613	+	0.973517i	$0.573419\pi$
$164$	0	0
$165$	0.977975	0.0761352
$166$	0	0
$167$	8.30449	0.642621	0.321310	−	0.946974i	$-0.395877\pi$
$167$	8.30449	0.642621	0.321310	+	0.946974i	$0.395877\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	6.39793	0.489262
$172$	0	0
$173$	7.75126	0.589317	0.294659	−	0.955603i	$-0.404794\pi$
$173$	7.75126	0.589317	0.294659	+	0.955603i	$0.404794\pi$
$174$	0	0
$175$	7.68243	0.580737
$176$	0	0
$177$	1.53606	0.115457
$178$	0	0
$179$	25.8853	1.93476	0.967379	−	0.253335i	$-0.0815275\pi$
$179$	25.8853	1.93476	0.967379	+	0.253335i	$0.0815275\pi$
$180$	0	0
$181$	7.35700	0.546842	0.273421	−	0.961894i	$-0.411845\pi$
$181$	7.35700	0.546842	0.273421	+	0.961894i	$0.411845\pi$
$182$	0	0
$183$	0.269722	0.0199384
$184$	0	0
$185$	9.12337	0.670763
$186$	0	0
$187$	9.47939	0.693202
$188$	0	0
$189$	0.815667	0.0593310
$190$	0	0
$191$	−18.0160	−1.30359	−0.651794	−	0.758396i	$-0.725983\pi$
$191$	−18.0160	−1.30359	−0.651794	+	0.758396i	$0.725983\pi$
$192$	0	0
$193$	16.5508	1.19135	0.595677	−	0.803224i	$-0.296884\pi$
$193$	16.5508	1.19135	0.595677	+	0.803224i	$0.296884\pi$
$194$	0	0
$195$	−0.556966	−0.0398852
$196$	0	0
$197$	−0.893253	−0.0636417	−0.0318208	−	0.999494i	$-0.510131\pi$
$197$	−0.893253	−0.0636417	−0.0318208	+	0.999494i	$0.510131\pi$
$198$	0	0
$199$	6.02017	0.426759	0.213379	−	0.976969i	$-0.431553\pi$
$199$	6.02017	0.426759	0.213379	+	0.976969i	$0.431553\pi$
$200$	0	0
$201$	2.16701	0.152849
$202$	0	0
$203$	0.901102	0.0632450
$204$	0	0
$205$	3.66085	0.255685
$206$	0	0
$207$	14.6749	1.01998
$208$	0	0
$209$	3.77355	0.261022
$210$	0	0
$211$	−21.5847	−1.48595	−0.742976	−	0.669318i	$-0.766587\pi$
$211$	−21.5847	−1.48595	−0.742976	+	0.669318i	$0.766587\pi$
$212$	0	0
$213$	2.00587	0.137440
$214$	0	0
$215$	8.63333	0.588788
$216$	0	0
$217$	−3.57992	−0.243021
$218$	0	0
$219$	−1.18696	−0.0802073
$220$	0	0
$221$	−5.39861	−0.363150
$222$	0	0
$223$	−23.9530	−1.60401	−0.802007	−	0.597315i	$-0.796234\pi$
$223$	−23.9530	−1.60401	−0.802007	+	0.597315i	$0.796234\pi$
$224$	0	0
$225$	−25.3812	−1.69208
$226$	0	0
$227$	15.0206	0.996949	0.498475	−	0.866904i	$-0.333894\pi$
$227$	15.0206	0.996949	0.498475	+	0.866904i	$0.333894\pi$
$228$	0	0
$229$	4.28623	0.283242	0.141621	−	0.989921i	$-0.454769\pi$
$229$	4.28623	0.283242	0.141621	+	0.989921i	$0.454769\pi$
$230$	0	0
$231$	0.239620	0.0157659
$232$	0	0
$233$	−14.4120	−0.944160	−0.472080	−	0.881556i	$-0.656497\pi$
$233$	−14.4120	−0.944160	−0.472080	+	0.881556i	$0.656497\pi$
$234$	0	0
$235$	−24.3457	−1.58814
$236$	0	0
$237$	0.932410	0.0605666
$238$	0	0
$239$	−1.47977	−0.0957187	−0.0478593	−	0.998854i	$-0.515240\pi$
$239$	−1.47977	−0.0957187	−0.0478593	+	0.998854i	$0.515240\pi$
$240$	0	0
$241$	6.59571	0.424867	0.212434	−	0.977176i	$-0.431861\pi$
$241$	6.59571	0.424867	0.212434	+	0.977176i	$0.431861\pi$
$242$	0	0
$243$	−4.04737	−0.259639
$244$	0	0
$245$	−22.7578	−1.45394
$246$	0	0
$247$	−2.14907	−0.136742
$248$	0	0
$249$	−2.17582	−0.137887
$250$	0	0
$251$	23.1573	1.46167	0.730837	−	0.682552i	$-0.239130\pi$
$251$	23.1573	1.46167	0.730837	+	0.682552i	$0.239130\pi$
$252$	0	0
$253$	8.65537	0.544158
$254$	0	0
$255$	3.00684	0.188296
$256$	0	0
$257$	−7.22022	−0.450385	−0.225192	−	0.974314i	$-0.572301\pi$
$257$	−7.22022	−0.450385	−0.225192	+	0.974314i	$0.572301\pi$
$258$	0	0
$259$	2.23538	0.138900
$260$	0	0
$261$	−2.97706	−0.184276
$262$	0	0
$263$	2.28307	0.140780	0.0703900	−	0.997520i	$-0.477576\pi$
$263$	2.28307	0.140780	0.0703900	+	0.997520i	$0.477576\pi$
$264$	0	0
$265$	51.8163	3.18305
$266$	0	0
$267$	0.123173	0.00753806
$268$	0	0
$269$	14.8718	0.906751	0.453376	−	0.891320i	$-0.350220\pi$
$269$	14.8718	0.906751	0.453376	+	0.891320i	$0.350220\pi$
$270$	0	0
$271$	8.23995	0.500541	0.250271	−	0.968176i	$-0.419480\pi$
$271$	8.23995	0.500541	0.250271	+	0.968176i	$0.419480\pi$
$272$	0	0
$273$	−0.136466	−0.00825930
$274$	0	0
$275$	−14.9701	−0.902728
$276$	0	0
$277$	−13.5877	−0.816405	−0.408203	−	0.912891i	$-0.633844\pi$
$277$	−13.5877	−0.816405	−0.408203	+	0.912891i	$0.633844\pi$
$278$	0	0
$279$	11.8274	0.708085
$280$	0	0
$281$	0.0997277	0.00594926	0.00297463	−	0.999996i	$-0.499053\pi$
$281$	0.0997277	0.00594926	0.00297463	+	0.999996i	$0.499053\pi$
$282$	0	0
$283$	16.7908	0.998111	0.499055	−	0.866570i	$-0.333680\pi$
$283$	16.7908	0.998111	0.499055	+	0.866570i	$0.333680\pi$
$284$	0	0
$285$	1.19696	0.0709019
$286$	0	0
$287$	0.896970	0.0529464
$288$	0	0
$289$	12.1449	0.714409
$290$	0	0
$291$	2.34979	0.137747
$292$	0	0
$293$	17.2159	1.00576	0.502882	−	0.864355i	$-0.332273\pi$
$293$	17.2159	1.00576	0.502882	+	0.864355i	$0.332273\pi$
$294$	0	0
$295$	−37.3022	−2.17182
$296$	0	0
$297$	−1.58942	−0.0922272
$298$	0	0
$299$	−4.92932	−0.285070
$300$	0	0
$301$	2.11531	0.121925
$302$	0	0
$303$	2.94387	0.169121
$304$	0	0
$305$	−6.55003	−0.375054
$306$	0	0
$307$	5.66389	0.323255	0.161628	−	0.986852i	$-0.448326\pi$
$307$	5.66389	0.323255	0.161628	+	0.986852i	$0.448326\pi$
$308$	0	0
$309$	0.966030	0.0549556
$310$	0	0
$311$	−9.16745	−0.519838	−0.259919	−	0.965630i	$-0.583696\pi$
$311$	−9.16745	−0.519838	−0.259919	+	0.965630i	$0.583696\pi$
$312$	0	0
$313$	−16.4627	−0.930524	−0.465262	−	0.885173i	$-0.654040\pi$
$313$	−16.4627	−0.930524	−0.465262	+	0.885173i	$0.654040\pi$
$314$	0	0
$315$	−9.86599	−0.555885
$316$	0	0
$317$	−16.1341	−0.906181	−0.453091	−	0.891464i	$-0.649679\pi$
$317$	−16.1341	−0.906181	−0.453091	+	0.891464i	$0.649679\pi$
$318$	0	0
$319$	−1.75590	−0.0983113
$320$	0	0
$321$	−1.14224	−0.0637538
$322$	0	0
$323$	11.6020	0.645553
$324$	0	0
$325$	8.52559	0.472915
$326$	0	0
$327$	−1.14320	−0.0632190
$328$	0	0
$329$	−5.96511	−0.328867
$330$	0	0
$331$	−2.64280	−0.145262	−0.0726308	−	0.997359i	$-0.523139\pi$
$331$	−2.64280	−0.145262	−0.0726308	+	0.997359i	$0.523139\pi$
$332$	0	0
$333$	−7.38525	−0.404709
$334$	0	0
$335$	−52.6246	−2.87519
$336$	0	0
$337$	5.89920	0.321350	0.160675	−	0.987007i	$-0.448633\pi$
$337$	5.89920	0.321350	0.160675	+	0.987007i	$0.448633\pi$
$338$	0	0
$339$	2.43830	0.132430
$340$	0	0
$341$	6.97586	0.377764
$342$	0	0
$343$	−11.8837	−0.641662
$344$	0	0
$345$	2.74546	0.147811
$346$	0	0
$347$	−9.99301	−0.536453	−0.268227	−	0.963356i	$-0.586438\pi$
$347$	−9.99301	−0.536453	−0.268227	+	0.963356i	$0.586438\pi$
$348$	0	0
$349$	7.52576	0.402845	0.201422	−	0.979504i	$-0.435444\pi$
$349$	7.52576	0.402845	0.201422	+	0.979504i	$0.435444\pi$
$350$	0	0
$351$	0.905188	0.0483154
$352$	0	0
$353$	−2.15035	−0.114452	−0.0572258	−	0.998361i	$-0.518225\pi$
$353$	−2.15035	−0.114452	−0.0572258	+	0.998361i	$0.518225\pi$
$354$	0	0
$355$	−48.7114	−2.58533
$356$	0	0
$357$	0.736727	0.0389917
$358$	0	0
$359$	20.1301	1.06242	0.531212	−	0.847239i	$-0.321737\pi$
$359$	20.1301	1.06242	0.531212	+	0.847239i	$0.321737\pi$
$360$	0	0
$361$	−14.3815	−0.756920
$362$	0	0
$363$	1.19895	0.0629287
$364$	0	0
$365$	28.8246	1.50875
$366$	0	0
$367$	27.4893	1.43493	0.717466	−	0.696594i	$-0.245302\pi$
$367$	27.4893	1.43493	0.717466	+	0.696594i	$0.245302\pi$
$368$	0	0
$369$	−2.96341	−0.154269
$370$	0	0
$371$	12.6959	0.659136
$372$	0	0
$373$	−12.5487	−0.649747	−0.324874	−	0.945757i	$-0.605322\pi$
$373$	−12.5487	−0.649747	−0.324874	+	0.945757i	$0.605322\pi$
$374$	0	0
$375$	−1.96364	−0.101402
$376$	0	0
$377$	1.00000	0.0515026
$378$	0	0
$379$	−17.6043	−0.904271	−0.452136	−	0.891949i	$-0.649338\pi$
$379$	−17.6043	−0.904271	−0.452136	+	0.891949i	$0.649338\pi$
$380$	0	0
$381$	−0.256697	−0.0131510
$382$	0	0
$383$	4.69708	0.240009	0.120005	−	0.992773i	$-0.461709\pi$
$383$	4.69708	0.240009	0.120005	+	0.992773i	$0.461709\pi$
$384$	0	0
$385$	−5.81903	−0.296565
$386$	0	0
$387$	−6.98858	−0.355249
$388$	0	0
$389$	−0.480660	−0.0243704	−0.0121852	−	0.999926i	$-0.503879\pi$
$389$	−0.480660	−0.0243704	−0.0121852	+	0.999926i	$0.503879\pi$
$390$	0	0
$391$	26.6114	1.34580
$392$	0	0
$393$	−0.0574182	−0.00289636
$394$	0	0
$395$	−22.6430	−1.13929
$396$	0	0
$397$	25.4080	1.27519	0.637596	−	0.770371i	$-0.279929\pi$
$397$	25.4080	1.27519	0.637596	+	0.770371i	$0.279929\pi$
$398$	0	0
$399$	0.293276	0.0146821
$400$	0	0
$401$	32.3342	1.61469	0.807347	−	0.590077i	$-0.200903\pi$
$401$	32.3342	1.61469	0.807347	+	0.590077i	$0.200903\pi$
$402$	0	0
$403$	−3.97282	−0.197900
$404$	0	0
$405$	32.3422	1.60710
$406$	0	0
$407$	−4.35588	−0.215913
$408$	0	0
$409$	−29.1569	−1.44172	−0.720859	−	0.693082i	$-0.756252\pi$
$409$	−29.1569	−1.44172	−0.720859	+	0.693082i	$0.756252\pi$
$410$	0	0
$411$	−2.81394	−0.138801
$412$	0	0
$413$	−9.13967	−0.449734
$414$	0	0
$415$	52.8385	2.59374
$416$	0	0
$417$	−2.09800	−0.102740
$418$	0	0
$419$	−29.0102	−1.41724	−0.708622	−	0.705589i	$-0.750683\pi$
$419$	−29.0102	−1.41724	−0.708622	+	0.705589i	$0.750683\pi$
$420$	0	0
$421$	27.9695	1.36315	0.681574	−	0.731749i	$-0.261296\pi$
$421$	27.9695	1.36315	0.681574	+	0.731749i	$0.261296\pi$
$422$	0	0
$423$	19.7075	0.958213
$424$	0	0
$425$	−46.0263	−2.23260
$426$	0	0
$427$	−1.60487	−0.0776650
$428$	0	0
$429$	0.265919	0.0128387
$430$	0	0
$431$	−34.2602	−1.65025	−0.825127	−	0.564948i	$-0.808896\pi$
$431$	−34.2602	−1.65025	−0.825127	+	0.564948i	$0.808896\pi$
$432$	0	0
$433$	−26.6551	−1.28096	−0.640482	−	0.767973i	$-0.721265\pi$
$433$	−26.6551	−1.28096	−0.640482	+	0.767973i	$0.721265\pi$
$434$	0	0
$435$	−0.556966	−0.0267045
$436$	0	0
$437$	10.5935	0.506754
$438$	0	0
$439$	11.8080	0.563567	0.281783	−	0.959478i	$-0.409074\pi$
$439$	11.8080	0.563567	0.281783	+	0.959478i	$0.409074\pi$
$440$	0	0
$441$	18.4221	0.877244
$442$	0	0
$443$	−29.9304	−1.42204	−0.711018	−	0.703174i	$-0.751765\pi$
$443$	−29.9304	−1.42204	−0.711018	+	0.703174i	$0.751765\pi$
$444$	0	0
$445$	−2.99118	−0.141796
$446$	0	0
$447$	1.08930	0.0515223
$448$	0	0
$449$	2.71866	0.128301	0.0641507	−	0.997940i	$-0.479566\pi$
$449$	2.71866	0.128301	0.0641507	+	0.997940i	$0.479566\pi$
$450$	0	0
$451$	−1.74784	−0.0823027
$452$	0	0
$453$	3.00327	0.141106
$454$	0	0
$455$	3.31400	0.155363
$456$	0	0
$457$	18.5587	0.868140	0.434070	−	0.900879i	$-0.357077\pi$
$457$	18.5587	0.868140	0.434070	+	0.900879i	$0.357077\pi$
$458$	0	0
$459$	−4.88675	−0.228094
$460$	0	0
$461$	−33.7482	−1.57181	−0.785905	−	0.618347i	$-0.787803\pi$
$461$	−33.7482	−1.57181	−0.785905	+	0.618347i	$0.787803\pi$
$462$	0	0
$463$	36.8991	1.71485	0.857423	−	0.514612i	$-0.172064\pi$
$463$	36.8991	1.71485	0.857423	+	0.514612i	$0.172064\pi$
$464$	0	0
$465$	2.21273	0.102613
$466$	0	0
$467$	35.7843	1.65590	0.827949	−	0.560804i	$-0.189508\pi$
$467$	35.7843	1.65590	0.827949	+	0.560804i	$0.189508\pi$
$468$	0	0
$469$	−12.8939	−0.595386
$470$	0	0
$471$	2.95469	0.136145
$472$	0	0
$473$	−4.12192	−0.189526
$474$	0	0
$475$	−18.3221	−0.840677
$476$	0	0
$477$	−41.9446	−1.92051
$478$	0	0
$479$	13.7132	0.626573	0.313287	−	0.949659i	$-0.398570\pi$
$479$	13.7132	0.626573	0.313287	+	0.949659i	$0.398570\pi$
$480$	0	0
$481$	2.48072	0.113111
$482$	0	0
$483$	0.672685	0.0306082
$484$	0	0
$485$	−57.0632	−2.59111
$486$	0	0
$487$	42.8351	1.94104	0.970521	−	0.241016i	$-0.0774807\pi$
$487$	42.8351	1.94104	0.970521	+	0.241016i	$0.0774807\pi$
$488$	0	0
$489$	0.884046	0.0399779
$490$	0	0
$491$	12.3967	0.559455	0.279727	−	0.960080i	$-0.409756\pi$
$491$	12.3967	0.559455	0.279727	+	0.960080i	$0.409756\pi$
$492$	0	0
$493$	−5.39861	−0.243141
$494$	0	0
$495$	19.2249	0.864097
$496$	0	0
$497$	−11.9351	−0.535363
$498$	0	0
$499$	−24.0055	−1.07463	−0.537316	−	0.843381i	$-0.680562\pi$
$499$	−24.0055	−1.07463	−0.537316	+	0.843381i	$0.680562\pi$
$500$	0	0
$501$	−1.25766	−0.0561882
$502$	0	0
$503$	6.88901	0.307166	0.153583	−	0.988136i	$-0.450919\pi$
$503$	6.88901	0.307166	0.153583	+	0.988136i	$0.450919\pi$
$504$	0	0
$505$	−71.4901	−3.18127
$506$	0	0
$507$	−0.151444	−0.00672584
$508$	0	0
$509$	−1.74513	−0.0773513	−0.0386757	−	0.999252i	$-0.512314\pi$
$509$	−1.74513	−0.0773513	−0.0386757	+	0.999252i	$0.512314\pi$
$510$	0	0
$511$	7.06251	0.312427
$512$	0	0
$513$	−1.94532	−0.0858878
$514$	0	0
$515$	−23.4595	−1.03375
$516$	0	0
$517$	11.6237	0.511208
$518$	0	0
$519$	−1.17388	−0.0515275
$520$	0	0
$521$	−20.3415	−0.891179	−0.445589	−	0.895237i	$-0.647006\pi$
$521$	−20.3415	−0.891179	−0.445589	+	0.895237i	$0.647006\pi$
$522$	0	0
$523$	−29.6117	−1.29483	−0.647415	−	0.762138i	$-0.724150\pi$
$523$	−29.6117	−1.29483	−0.647415	+	0.762138i	$0.724150\pi$
$524$	0	0
$525$	−1.16345	−0.0507773
$526$	0	0
$527$	21.4477	0.934277
$528$	0	0
$529$	1.29816	0.0564418
$530$	0	0
$531$	30.1957	1.31038
$532$	0	0
$533$	0.995414	0.0431162
$534$	0	0
$535$	27.7387	1.19925
$536$	0	0
$537$	−3.92016	−0.169167
$538$	0	0
$539$	10.8655	0.468011
$540$	0	0
$541$	−23.2754	−1.00069	−0.500345	−	0.865826i	$-0.666793\pi$
$541$	−23.2754	−1.00069	−0.500345	+	0.865826i	$0.666793\pi$
$542$	0	0
$543$	−1.11417	−0.0478136
$544$	0	0
$545$	27.7619	1.18919
$546$	0	0
$547$	41.8687	1.79017	0.895087	−	0.445891i	$-0.147113\pi$
$547$	41.8687	1.79017	0.895087	+	0.445891i	$0.147113\pi$
$548$	0	0
$549$	5.30217	0.226291
$550$	0	0
$551$	−2.14907	−0.0915536
$552$	0	0
$553$	−5.54792	−0.235922
$554$	0	0
$555$	−1.38167	−0.0586488
$556$	0	0
$557$	15.6468	0.662975	0.331488	−	0.943460i	$-0.392449\pi$
$557$	15.6468	0.662975	0.331488	+	0.943460i	$0.392449\pi$
$558$	0	0
$559$	2.34747	0.0992875
$560$	0	0
$561$	−1.43559	−0.0606108
$562$	0	0
$563$	−37.9092	−1.59768	−0.798842	−	0.601541i	$-0.794554\pi$
$563$	−37.9092	−1.59768	−0.798842	+	0.601541i	$0.794554\pi$
$564$	0	0
$565$	−59.2127	−2.49110
$566$	0	0
$567$	7.92439	0.332793
$568$	0	0
$569$	−19.2954	−0.808904	−0.404452	−	0.914559i	$-0.632538\pi$
$569$	−19.2954	−0.808904	−0.404452	+	0.914559i	$0.632538\pi$
$570$	0	0
$571$	27.1600	1.13661	0.568306	−	0.822817i	$-0.307599\pi$
$571$	27.1600	1.13661	0.568306	+	0.822817i	$0.307599\pi$
$572$	0	0
$573$	2.72840	0.113981
$574$	0	0
$575$	−42.0253	−1.75258
$576$	0	0
$577$	−15.9716	−0.664905	−0.332452	−	0.943120i	$-0.607876\pi$
$577$	−15.9716	−0.664905	−0.332452	+	0.943120i	$0.607876\pi$
$578$	0	0
$579$	−2.50651	−0.104167
$580$	0	0
$581$	12.9463	0.537103
$582$	0	0
$583$	−24.7393	−1.02460
$584$	0	0
$585$	−10.9488	−0.452677
$586$	0	0
$587$	−24.0642	−0.993235	−0.496618	−	0.867969i	$-0.665425\pi$
$587$	−24.0642	−0.993235	−0.496618	+	0.867969i	$0.665425\pi$
$588$	0	0
$589$	8.53789	0.351798
$590$	0	0
$591$	0.135277	0.00556457
$592$	0	0
$593$	12.4441	0.511018	0.255509	−	0.966807i	$-0.417757\pi$
$593$	12.4441	0.511018	0.255509	+	0.966807i	$0.417757\pi$
$594$	0	0
$595$	−17.8910	−0.733458
$596$	0	0
$597$	−0.911716	−0.0373141
$598$	0	0
$599$	−39.5216	−1.61481	−0.807404	−	0.589998i	$-0.799128\pi$
$599$	−39.5216	−1.61481	−0.807404	+	0.589998i	$0.799128\pi$
$600$	0	0
$601$	7.06213	0.288070	0.144035	−	0.989573i	$-0.453992\pi$
$601$	7.06213	0.288070	0.144035	+	0.989573i	$0.453992\pi$
$602$	0	0
$603$	42.5990	1.73477
$604$	0	0
$605$	−29.1158	−1.18373
$606$	0	0
$607$	−5.52134	−0.224104	−0.112052	−	0.993702i	$-0.535742\pi$
$607$	−5.52134	−0.224104	−0.112052	+	0.993702i	$0.535742\pi$
$608$	0	0
$609$	−0.136466	−0.00552989
$610$	0	0
$611$	−6.61979	−0.267808
$612$	0	0
$613$	−32.1248	−1.29751	−0.648753	−	0.760999i	$-0.724709\pi$
$613$	−32.1248	−1.29751	−0.648753	+	0.760999i	$0.724709\pi$
$614$	0	0
$615$	−0.554412	−0.0223560
$616$	0	0
$617$	−0.992815	−0.0399692	−0.0199846	−	0.999800i	$-0.506362\pi$
$617$	−0.992815	−0.0399692	−0.0199846	+	0.999800i	$0.506362\pi$
$618$	0	0
$619$	21.6626	0.870695	0.435348	−	0.900262i	$-0.356625\pi$
$619$	21.6626	0.870695	0.435348	+	0.900262i	$0.356625\pi$
$620$	0	0
$621$	−4.46196	−0.179052
$622$	0	0
$623$	−0.732890	−0.0293626
$624$	0	0
$625$	5.05777	0.202311
$626$	0	0
$627$	−0.571480	−0.0228227
$628$	0	0
$629$	−13.3924	−0.533990
$630$	0	0
$631$	4.52887	0.180291	0.0901457	−	0.995929i	$-0.471267\pi$
$631$	4.52887	0.180291	0.0901457	+	0.995929i	$0.471267\pi$
$632$	0	0
$633$	3.26887	0.129926
$634$	0	0
$635$	6.23373	0.247378
$636$	0	0
$637$	−6.18801	−0.245178
$638$	0	0
$639$	39.4313	1.55988
$640$	0	0
$641$	−16.6070	−0.655938	−0.327969	−	0.944688i	$-0.606364\pi$
$641$	−16.6070	−0.655938	−0.327969	+	0.944688i	$0.606364\pi$
$642$	0	0
$643$	−40.1928	−1.58505	−0.792525	−	0.609840i	$-0.791234\pi$
$643$	−40.1928	−1.58505	−0.792525	+	0.609840i	$0.791234\pi$
$644$	0	0
$645$	−1.30746	−0.0514813
$646$	0	0
$647$	−4.46501	−0.175537	−0.0877687	−	0.996141i	$-0.527974\pi$
$647$	−4.46501	−0.175537	−0.0877687	+	0.996141i	$0.527974\pi$
$648$	0	0
$649$	17.8096	0.699090
$650$	0	0
$651$	0.542156	0.0212488
$652$	0	0
$653$	25.2994	0.990043	0.495021	−	0.868881i	$-0.335160\pi$
$653$	25.2994	0.990043	0.495021	+	0.868881i	$0.335160\pi$
$654$	0	0
$655$	1.39437	0.0544824
$656$	0	0
$657$	−23.3332	−0.910313
$658$	0	0
$659$	27.4012	1.06740	0.533700	−	0.845674i	$-0.320801\pi$
$659$	27.4012	1.06740	0.533700	+	0.845674i	$0.320801\pi$
$660$	0	0
$661$	−46.9555	−1.82636	−0.913178	−	0.407561i	$-0.866379\pi$
$661$	−46.9555	−1.82636	−0.913178	+	0.407561i	$0.866379\pi$
$662$	0	0
$663$	0.817584	0.0317523
$664$	0	0
$665$	−7.12202	−0.276180
$666$	0	0
$667$	−4.92932	−0.190864
$668$	0	0
$669$	3.62753	0.140248
$670$	0	0
$671$	3.12726	0.120726
$672$	0	0
$673$	11.5114	0.443732	0.221866	−	0.975077i	$-0.428785\pi$
$673$	11.5114	0.443732	0.221866	+	0.975077i	$0.428785\pi$
$674$	0	0
$675$	7.71726	0.297038
$676$	0	0
$677$	−44.3924	−1.70614	−0.853070	−	0.521797i	$-0.825262\pi$
$677$	−44.3924	−1.70614	−0.853070	+	0.521797i	$0.825262\pi$
$678$	0	0
$679$	−13.9815	−0.536559
$680$	0	0
$681$	−2.27477	−0.0871692
$682$	0	0
$683$	5.39852	0.206569	0.103284	−	0.994652i	$-0.467065\pi$
$683$	5.39852	0.206569	0.103284	+	0.994652i	$0.467065\pi$
$684$	0	0
$685$	68.3348	2.61094
$686$	0	0
$687$	−0.649122	−0.0247655
$688$	0	0
$689$	14.0893	0.536758
$690$	0	0
$691$	24.0365	0.914390	0.457195	−	0.889367i	$-0.348854\pi$
$691$	24.0365	0.914390	0.457195	+	0.889367i	$0.348854\pi$
$692$	0	0
$693$	4.71044	0.178935
$694$	0	0
$695$	50.9487	1.93260
$696$	0	0
$697$	−5.37385	−0.203549
$698$	0	0
$699$	2.18260	0.0825535
$700$	0	0
$701$	11.2382	0.424460	0.212230	−	0.977220i	$-0.431927\pi$
$701$	11.2382	0.424460	0.212230	+	0.977220i	$0.431927\pi$
$702$	0	0
$703$	−5.33124	−0.201072
$704$	0	0
$705$	3.68700	0.138860
$706$	0	0
$707$	−17.5163	−0.658768
$708$	0	0
$709$	5.76759	0.216606	0.108303	−	0.994118i	$-0.465458\pi$
$709$	5.76759	0.216606	0.108303	+	0.994118i	$0.465458\pi$
$710$	0	0
$711$	18.3292	0.687400
$712$	0	0
$713$	19.5833	0.733400
$714$	0	0
$715$	−6.45769	−0.241504
$716$	0	0
$717$	0.224102	0.00836925
$718$	0	0
$719$	−23.6168	−0.880757	−0.440378	−	0.897812i	$-0.645156\pi$
$719$	−23.6168	−0.880757	−0.440378	+	0.897812i	$0.645156\pi$
$720$	0	0
$721$	−5.74797	−0.214065
$722$	0	0
$723$	−0.998878	−0.0371487
$724$	0	0
$725$	8.52559	0.316633
$726$	0	0
$727$	32.5806	1.20835	0.604174	−	0.796852i	$-0.293503\pi$
$727$	32.5806	1.20835	0.604174	+	0.796852i	$0.293503\pi$
$728$	0	0
$729$	−25.7694	−0.954421
$730$	0	0
$731$	−12.6731	−0.468731
$732$	0	0
$733$	−30.3059	−1.11937	−0.559687	−	0.828704i	$-0.689079\pi$
$733$	−30.3059	−1.11937	−0.559687	+	0.828704i	$0.689079\pi$
$734$	0	0
$735$	3.44652	0.127127
$736$	0	0
$737$	25.1252	0.925499
$738$	0	0
$739$	−11.2060	−0.412218	−0.206109	−	0.978529i	$-0.566080\pi$
$739$	−11.2060	−0.412218	−0.206109	+	0.978529i	$0.566080\pi$
$740$	0	0
$741$	0.325463	0.0119562
$742$	0	0
$743$	−35.7346	−1.31098	−0.655488	−	0.755206i	$-0.727537\pi$
$743$	−35.7346	−1.31098	−0.655488	+	0.755206i	$0.727537\pi$
$744$	0	0
$745$	−26.4531	−0.969167
$746$	0	0
$747$	−42.7721	−1.56495
$748$	0	0
$749$	6.79645	0.248337
$750$	0	0
$751$	13.9170	0.507837	0.253919	−	0.967226i	$-0.418280\pi$
$751$	13.9170	0.507837	0.253919	+	0.967226i	$0.418280\pi$
$752$	0	0
$753$	−3.50702	−0.127803
$754$	0	0
$755$	−72.9327	−2.65429
$756$	0	0
$757$	−33.2545	−1.20866	−0.604328	−	0.796736i	$-0.706558\pi$
$757$	−33.2545	−1.20866	−0.604328	+	0.796736i	$0.706558\pi$
$758$	0	0
$759$	−1.31080	−0.0475790
$760$	0	0
$761$	−10.8339	−0.392729	−0.196364	−	0.980531i	$-0.562914\pi$
$761$	−10.8339	−0.392729	−0.196364	+	0.980531i	$0.562914\pi$
$762$	0	0
$763$	6.80213	0.246254
$764$	0	0
$765$	59.1082	2.13706
$766$	0	0
$767$	−10.1428	−0.366234
$768$	0	0
$769$	−11.7373	−0.423258	−0.211629	−	0.977350i	$-0.567877\pi$
$769$	−11.7373	−0.423258	−0.211629	+	0.977350i	$0.567877\pi$
$770$	0	0
$771$	1.09346	0.0393798
$772$	0	0
$773$	−6.80724	−0.244839	−0.122420	−	0.992478i	$-0.539065\pi$
$773$	−6.80724	−0.244839	−0.122420	+	0.992478i	$0.539065\pi$
$774$	0	0
$775$	−33.8707	−1.21667
$776$	0	0
$777$	−0.338534	−0.0121448
$778$	0	0
$779$	−2.13922	−0.0766454
$780$	0	0
$781$	23.2569	0.832196
$782$	0	0
$783$	0.905188	0.0323488
$784$	0	0
$785$	−71.7528	−2.56097
$786$	0	0
$787$	−22.6210	−0.806351	−0.403175	−	0.915123i	$-0.632094\pi$
$787$	−22.6210	−0.806351	−0.403175	+	0.915123i	$0.632094\pi$
$788$	0	0
$789$	−0.345756	−0.0123092
$790$	0	0
$791$	−14.5081	−0.515849
$792$	0	0
$793$	−1.78100	−0.0632453
$794$	0	0
$795$	−7.84724	−0.278313
$796$	0	0
$797$	13.0778	0.463238	0.231619	−	0.972807i	$-0.425598\pi$
$797$	13.0778	0.463238	0.231619	+	0.972807i	$0.425598\pi$
$798$	0	0
$799$	35.7376	1.26431
$800$	0	0
$801$	2.42132	0.0855533
$802$	0	0
$803$	−13.7621	−0.485653
$804$	0	0
$805$	−16.3357	−0.575759
$806$	0	0
$807$	−2.25224	−0.0792827
$808$	0	0
$809$	−14.2905	−0.502426	−0.251213	−	0.967932i	$-0.580829\pi$
$809$	−14.2905	−0.502426	−0.251213	+	0.967932i	$0.580829\pi$
$810$	0	0
$811$	55.1553	1.93676	0.968382	−	0.249470i	$-0.0802565\pi$
$811$	55.1553	1.93676	0.968382	+	0.249470i	$0.0802565\pi$
$812$	0	0
$813$	−1.24789	−0.0437653
$814$	0	0
$815$	−21.4685	−0.752010
$816$	0	0
$817$	−5.04489	−0.176498
$818$	0	0
$819$	−2.68264	−0.0937390
$820$	0	0
$821$	−15.5822	−0.543823	−0.271912	−	0.962322i	$-0.587656\pi$
$821$	−15.5822	−0.543823	−0.271912	+	0.962322i	$0.587656\pi$
$822$	0	0
$823$	19.2627	0.671457	0.335729	−	0.941959i	$-0.391017\pi$
$823$	19.2627	0.671457	0.335729	+	0.941959i	$0.391017\pi$
$824$	0	0
$825$	2.26712	0.0789309
$826$	0	0
$827$	17.2620	0.600258	0.300129	−	0.953899i	$-0.402970\pi$
$827$	17.2620	0.600258	0.300129	+	0.953899i	$0.402970\pi$
$828$	0	0
$829$	40.1908	1.39589	0.697943	−	0.716154i	$-0.254099\pi$
$829$	40.1908	1.39589	0.697943	+	0.716154i	$0.254099\pi$
$830$	0	0
$831$	2.05777	0.0713832
$832$	0	0
$833$	33.4067	1.15747
$834$	0	0
$835$	30.5415	1.05693
$836$	0	0
$837$	−3.59615	−0.124301
$838$	0	0
$839$	18.9848	0.655430	0.327715	−	0.944777i	$-0.393721\pi$
$839$	18.9848	0.655430	0.327715	+	0.944777i	$0.393721\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	−0.0151031	−0.000520179 0
$844$	0	0
$845$	3.67772	0.126517
$846$	0	0
$847$	−7.13387	−0.245123
$848$	0	0
$849$	−2.54286	−0.0872708
$850$	0	0
$851$	−12.2282	−0.419178
$852$	0	0
$853$	−49.6950	−1.70153	−0.850763	−	0.525550i	$-0.823860\pi$
$853$	−49.6950	−1.70153	−0.850763	+	0.525550i	$0.823860\pi$
$854$	0	0
$855$	23.5298	0.804701
$856$	0	0
$857$	42.9392	1.46677	0.733387	−	0.679811i	$-0.237938\pi$
$857$	42.9392	1.46677	0.733387	+	0.679811i	$0.237938\pi$
$858$	0	0
$859$	−51.6962	−1.76385	−0.881925	−	0.471389i	$-0.843753\pi$
$859$	−51.6962	−1.76385	−0.881925	+	0.471389i	$0.843753\pi$
$860$	0	0
$861$	−0.135840	−0.00462942
$862$	0	0
$863$	39.3092	1.33810	0.669051	−	0.743217i	$-0.266701\pi$
$863$	39.3092	1.33810	0.669051	+	0.743217i	$0.266701\pi$
$864$	0	0
$865$	28.5069	0.969264
$866$	0	0
$867$	−1.83927	−0.0624650
$868$	0	0
$869$	10.8107	0.366729
$870$	0	0
$871$	−14.3091	−0.484844
$872$	0	0
$873$	46.1920	1.56336
$874$	0	0
$875$	11.6838	0.394985
$876$	0	0
$877$	57.6216	1.94574	0.972872	−	0.231343i	$-0.0743120\pi$
$877$	57.6216	1.94574	0.972872	+	0.231343i	$0.0743120\pi$
$878$	0	0
$879$	−2.60724	−0.0879400
$880$	0	0
$881$	0.800228	0.0269604	0.0134802	−	0.999909i	$-0.495709\pi$
$881$	0.800228	0.0269604	0.0134802	+	0.999909i	$0.495709\pi$
$882$	0	0
$883$	0.162596	0.00547178	0.00273589	−	0.999996i	$-0.499129\pi$
$883$	0.162596	0.00547178	0.00273589	+	0.999996i	$0.499129\pi$
$884$	0	0
$885$	5.64918	0.189895
$886$	0	0
$887$	57.1925	1.92034	0.960169	−	0.279420i	$-0.0901423\pi$
$887$	57.1925	1.92034	0.960169	+	0.279420i	$0.0901423\pi$
$888$	0	0
$889$	1.52737	0.0512263
$890$	0	0
$891$	−15.4415	−0.517311
$892$	0	0
$893$	14.2264	0.476069
$894$	0	0
$895$	95.1987	3.18214
$896$	0	0
$897$	0.746513	0.0249253
$898$	0	0
$899$	−3.97282	−0.132501
$900$	0	0
$901$	−76.0624	−2.53401
$902$	0	0
$903$	−0.320350	−0.0106606
$904$	0	0
$905$	27.0570	0.899404
$906$	0	0
$907$	−14.0921	−0.467922	−0.233961	−	0.972246i	$-0.575169\pi$
$907$	−14.0921	−0.467922	−0.233961	+	0.972246i	$0.575169\pi$
$908$	0	0
$909$	57.8704	1.91944
$910$	0	0
$911$	6.52045	0.216032	0.108016	−	0.994149i	$-0.465550\pi$
$911$	6.52045	0.216032	0.108016	+	0.994149i	$0.465550\pi$
$912$	0	0
$913$	−25.2273	−0.834902
$914$	0	0
$915$	0.991960	0.0327932
$916$	0	0
$917$	0.341643	0.0112820
$918$	0	0
$919$	19.2987	0.636607	0.318303	−	0.947989i	$-0.396887\pi$
$919$	19.2987	0.636607	0.318303	+	0.947989i	$0.396887\pi$
$920$	0	0
$921$	−0.857759	−0.0282641
$922$	0	0
$923$	−13.2450	−0.435965
$924$	0	0
$925$	21.1496	0.695394
$926$	0	0
$927$	18.9901	0.623718
$928$	0	0
$929$	6.92836	0.227312	0.113656	−	0.993520i	$-0.463744\pi$
$929$	6.92836	0.227312	0.113656	+	0.993520i	$0.463744\pi$
$930$	0	0
$931$	13.2985	0.435841
$932$	0	0
$933$	1.38835	0.0454526
$934$	0	0
$935$	34.8625	1.14013
$936$	0	0
$937$	20.6997	0.676230	0.338115	−	0.941105i	$-0.390211\pi$
$937$	20.6997	0.676230	0.338115	+	0.941105i	$0.390211\pi$
$938$	0	0
$939$	2.49316	0.0813613
$940$	0	0
$941$	49.6746	1.61934	0.809672	−	0.586883i	$-0.199645\pi$
$941$	49.6746	1.61934	0.809672	+	0.586883i	$0.199645\pi$
$942$	0	0
$943$	−4.90671	−0.159784
$944$	0	0
$945$	2.99979	0.0975832
$946$	0	0
$947$	32.9447	1.07056	0.535279	−	0.844675i	$-0.320206\pi$
$947$	32.9447	1.07056	0.535279	+	0.844675i	$0.320206\pi$
$948$	0	0
$949$	7.83764	0.254421
$950$	0	0
$951$	2.44341	0.0792329
$952$	0	0
$953$	17.5794	0.569452	0.284726	−	0.958609i	$-0.408097\pi$
$953$	17.5794	0.569452	0.284726	+	0.958609i	$0.408097\pi$
$954$	0	0
$955$	−66.2576	−2.14404
$956$	0	0
$957$	0.265919	0.00859594
$958$	0	0
$959$	16.7432	0.540665
$960$	0	0
$961$	−15.2167	−0.490861
$962$	0	0
$963$	−22.4541	−0.723574
$964$	0	0
$965$	60.8692	1.95945
$966$	0	0
$967$	−25.1577	−0.809017	−0.404508	−	0.914534i	$-0.632557\pi$
$967$	−25.1577	−0.809017	−0.404508	+	0.914534i	$0.632557\pi$
$968$	0	0
$969$	−1.75705	−0.0564445
$970$	0	0
$971$	41.4380	1.32981	0.664904	−	0.746929i	$-0.268473\pi$
$971$	41.4380	1.32981	0.664904	+	0.746929i	$0.268473\pi$
$972$	0	0
$973$	12.4833	0.400196
$974$	0	0
$975$	−1.29115	−0.0413498
$976$	0	0
$977$	11.0033	0.352026	0.176013	−	0.984388i	$-0.443680\pi$
$977$	11.0033	0.352026	0.176013	+	0.984388i	$0.443680\pi$
$978$	0	0
$979$	1.42812	0.0456428
$980$	0	0
$981$	−22.4729	−0.717505
$982$	0	0
$983$	41.5216	1.32433	0.662167	−	0.749357i	$-0.269637\pi$
$983$	41.5216	1.32433	0.662167	+	0.749357i	$0.269637\pi$
$984$	0	0
$985$	−3.28513	−0.104673
$986$	0	0
$987$	0.903377	0.0287548
$988$	0	0
$989$	−11.5714	−0.367950
$990$	0	0
$991$	17.9704	0.570850	0.285425	−	0.958401i	$-0.407865\pi$
$991$	17.9704	0.570850	0.285425	+	0.958401i	$0.407865\pi$
$992$	0	0
$993$	0.400235	0.0127011
$994$	0	0
$995$	22.1405	0.701900
$996$	0	0
$997$	−21.7607	−0.689168	−0.344584	−	0.938755i	$-0.611980\pi$
$997$	−21.7607	−0.689168	−0.344584	+	0.938755i	$0.611980\pi$
$998$	0	0
$999$	2.24551	0.0710449

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6032.2.a.bd.1.7		12
4.3	odd	2		3016.2.a.j.1.6	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3016.2.a.j.1.6	✓	12	4.3	odd	2
6032.2.a.bd.1.7		12	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 6032 = 2^{4} \cdot 13 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6032.a (trivial)

\(f(q)\)	\(=\)	\(q-0.151444 q^{3} +3.67772 q^{5} +0.901102 q^{7} -2.97706 q^{9} +O(q^{10})\)	\(q-0.151444 q^{3} +3.67772 q^{5} +0.901102 q^{7} -2.97706 q^{9} -1.75590 q^{11} +1.00000 q^{13} -0.556966 q^{15} -5.39861 q^{17} -2.14907 q^{19} -0.136466 q^{21} -4.92932 q^{23} +8.52559 q^{25} +0.905188 q^{27} +1.00000 q^{29} -3.97282 q^{31} +0.265919 q^{33} +3.31400 q^{35} +2.48072 q^{37} -0.151444 q^{39} +0.995414 q^{41} +2.34747 q^{43} -10.9488 q^{45} -6.61979 q^{47} -6.18801 q^{49} +0.817584 q^{51} +14.0893 q^{53} -6.45769 q^{55} +0.325463 q^{57} -10.1428 q^{59} -1.78100 q^{61} -2.68264 q^{63} +3.67772 q^{65} -14.3091 q^{67} +0.746513 q^{69} -13.2450 q^{71} +7.83764 q^{73} -1.29115 q^{75} -1.58224 q^{77} -6.15682 q^{79} +8.79411 q^{81} +14.3672 q^{83} -19.8545 q^{85} -0.151444 q^{87} -0.813326 q^{89} +0.901102 q^{91} +0.601658 q^{93} -7.90368 q^{95} -15.5159 q^{97} +5.22742 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 6 q^{3} + 3 q^{5} - 6 q^{7} + 20 q^{9}+O(q^{10}) \) 12 * q - 6 * q^3 + 3 * q^5 - 6 * q^7 + 20 * q^9	\( 12 q - 6 q^{3} + 3 q^{5} - 6 q^{7} + 20 q^{9} - 14 q^{11} + 12 q^{13} - 8 q^{15} + 4 q^{17} - 11 q^{19} - 5 q^{21} - 15 q^{23} + 5 q^{25} - 24 q^{27} + 12 q^{29} - 13 q^{31} - q^{33} - 18 q^{35} - 23 q^{37} - 6 q^{39} - 2 q^{41} - 26 q^{43} - 9 q^{45} - 15 q^{47} + 16 q^{49} - 21 q^{51} + 31 q^{53} - 10 q^{55} - 10 q^{57} - 7 q^{59} + 2 q^{61} + 25 q^{63} + 3 q^{65} - 47 q^{67} - 8 q^{69} - 32 q^{71} - 25 q^{73} - 31 q^{75} - 4 q^{77} - 7 q^{79} + 64 q^{81} - 12 q^{83} + 7 q^{85} - 6 q^{87} + 6 q^{89} - 6 q^{91} + 17 q^{93} - q^{95} - 7 q^{97} - 44 q^{99}+O(q^{100}) \) 12 * q - 6 * q^3 + 3 * q^5 - 6 * q^7 + 20 * q^9 - 14 * q^11 + 12 * q^13 - 8 * q^15 + 4 * q^17 - 11 * q^19 - 5 * q^21 - 15 * q^23 + 5 * q^25 - 24 * q^27 + 12 * q^29 - 13 * q^31 - q^33 - 18 * q^35 - 23 * q^37 - 6 * q^39 - 2 * q^41 - 26 * q^43 - 9 * q^45 - 15 * q^47 + 16 * q^49 - 21 * q^51 + 31 * q^53 - 10 * q^55 - 10 * q^57 - 7 * q^59 + 2 * q^61 + 25 * q^63 + 3 * q^65 - 47 * q^67 - 8 * q^69 - 32 * q^71 - 25 * q^73 - 31 * q^75 - 4 * q^77 - 7 * q^79 + 64 * q^81 - 12 * q^83 + 7 * q^85 - 6 * q^87 + 6 * q^89 - 6 * q^91 + 17 * q^93 - q^95 - 7 * q^97 - 44 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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