LMFDB - Newform orbit 538.4.a.d

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [538,4,Mod(1,538)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(538, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("538.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$N$	$=$	$538 = 2 \cdot 269$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	538.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:	$31.7430275831$
Analytic rank:	$0$
Dimension:	$21$
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$ -expansion

The algebraic $q$ -expansion of this newform has not been computed, but we have computed the trace expansion.

\operatorname{Tr}(f)(q) =

21 q + 42 q^{2} + 6 q^{3} + 84 q^{4} + 54 q^{5} + 12 q^{6} + 52 q^{7} + 168 q^{8} + 309 q^{9} + 108 q^{10} + 99 q^{11} + 24 q^{12} + 81 q^{13} + 104 q^{14} + 277 q^{15} + 336 q^{16} + 228 q^{17} + 618 q^{18}+ \cdots - 4470 q^{99}+O(q^{100})

21 * q + 42 * q^2 + 6 * q^3 + 84 * q^4 + 54 * q^5 + 12 * q^6 + 52 * q^7 + 168 * q^8 + 309 * q^9 + 108 * q^10 + 99 * q^11 + 24 * q^12 + 81 * q^13 + 104 * q^14 + 277 * q^15 + 336 * q^16 + 228 * q^17 + 618 * q^18 + 51 * q^19 + 216 * q^20 + 235 * q^21 + 198 * q^22 + 691 * q^23 + 48 * q^24 + 1035 * q^25 + 162 * q^26 + 291 * q^27 + 208 * q^28 + 540 * q^29 + 554 * q^30 + 312 * q^31 + 672 * q^32 + 513 * q^33 + 456 * q^34 + 410 * q^35 + 1236 * q^36 + 1120 * q^37 + 102 * q^38 + 904 * q^39 + 432 * q^40 + 731 * q^41 + 470 * q^42 + 788 * q^43 + 396 * q^44 + 1771 * q^45 + 1382 * q^46 + 1570 * q^47 + 96 * q^48 + 2337 * q^49 + 2070 * q^50 + 973 * q^51 + 324 * q^52 + 1801 * q^53 + 582 * q^54 + 1191 * q^55 + 416 * q^56 + 2122 * q^57 + 1080 * q^58 + 1352 * q^59 + 1108 * q^60 + 1868 * q^61 + 624 * q^62 + 2302 * q^63 + 1344 * q^64 + 2552 * q^65 + 1026 * q^66 + 1968 * q^67 + 912 * q^68 + 823 * q^69 + 820 * q^70 + 868 * q^71 + 2472 * q^72 + 1587 * q^73 + 2240 * q^74 - 1978 * q^75 + 204 * q^76 - 2020 * q^77 + 1808 * q^78 + 1168 * q^79 + 864 * q^80 + 3069 * q^81 + 1462 * q^82 - 1506 * q^83 + 940 * q^84 + 871 * q^85 + 1576 * q^86 - 6155 * q^87 + 792 * q^88 + 1471 * q^89 + 3542 * q^90 - 2157 * q^91 + 2764 * q^92 - 677 * q^93 + 3140 * q^94 + 544 * q^95 + 192 * q^96 - 801 * q^97 + 4674 * q^98 - 4470 * q^99

Embeddings

For each embedding $\iota_m$ of the coefficient field, the values $\iota_m(a_n)$ are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label	$a_{2}$	$a_{3}$	$a_{4}$	$a_{5}$	$a_{6}$	$a_{7}$	$a_{8}$	$a_{9}$	$a_{10}$
1.1	2.00000	−10.0400	4.00000	−10.3142	−20.0800	30.9953	8.00000	73.8018	−20.6284
1.2	2.00000	−8.94694	4.00000	17.4912	−17.8939	−29.4022	8.00000	53.0478	34.9825
1.3	2.00000	−8.51755	4.00000	16.7909	−17.0351	25.1792	8.00000	45.5487	33.5818
1.4	2.00000	−6.54123	4.00000	−13.1398	−13.0825	−12.7222	8.00000	15.7877	−26.2795
1.5	2.00000	−6.41803	4.00000	6.01917	−12.8361	26.2838	8.00000	14.1911	12.0383
1.6	2.00000	−5.47352	4.00000	−17.6842	−10.9470	−34.0172	8.00000	2.95944	−35.3683
1.7	2.00000	−4.26201	4.00000	6.83518	−8.52401	−31.0166	8.00000	−8.83529	13.6704
1.8	2.00000	−3.71008	4.00000	−7.25350	−7.42015	0.635451	8.00000	−13.2353	−14.5070
1.9	2.00000	−1.64279	4.00000	−16.1874	−3.28558	5.23738	8.00000	−24.3012	−32.3748
1.10	2.00000	−1.07079	4.00000	3.57975	−2.14158	21.8456	8.00000	−25.8534	7.15950
1.11	2.00000	−0.0924306	4.00000	22.1534	−0.184861	−8.74881	8.00000	−26.9915	44.3068
1.12	2.00000	0.291894	4.00000	10.4463	0.583789	19.7604	8.00000	−26.9148	20.8926
1.13	2.00000	2.97379	4.00000	−18.4835	5.94759	−11.2005	8.00000	−18.1566	−36.9670
1.14	2.00000	5.07974	4.00000	10.3107	10.1595	24.1687	8.00000	−1.19626	20.6214
1.15	2.00000	5.22913	4.00000	19.5728	10.4583	−1.73635	8.00000	0.343768	39.1455
1.16	2.00000	6.73834	4.00000	5.62452	13.4767	−24.9667	8.00000	18.4053	11.2490
1.17	2.00000	6.96110	4.00000	11.9308	13.9222	13.1346	8.00000	21.4569	23.8617
1.18	2.00000	7.70804	4.00000	−10.8854	15.4161	35.5612	8.00000	32.4139	−21.7708
1.19	2.00000	8.48473	4.00000	−6.56280	16.9695	3.72047	8.00000	44.9906	−13.1256
1.20	2.00000	9.24827	4.00000	15.1406	18.4965	12.8672	8.00000	58.5304	30.2812

See all 21 embeddings

Atkin-Lehner signs

$p$	Sign
$2$	$-1$
$269$	$-1$

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	538.4.a.d	✓	21

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
538.4.a.d	✓	21	1.a	even	1	1	trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{3}^{21} - 6 T_{3}^{20} - 420 T_{3}^{19} + 2387 T_{3}^{18} + 74643 T_{3}^{17} - 394163 T_{3}^{16} + \cdots + 2956968188160$ T3^21 - 6*T3^20 - 420*T3^19 + 2387*T3^18 + 74643*T3^17 - 394163*T3^16 - 7355254*T3^15 + 35092295*T3^14 + 442438137*T3^13 - 1829290752*T3^12 - 16839146512*T3^11 + 56565429119*T3^10 + 405014946478*T3^9 - 992444379138*T3^8 - 5913361981855*T3^7 + 8534459083285*T3^6 + 47066408368105*T3^5 - 19396880148429*T3^4 - 152266177851700*T3^3 - 78849350772176*T3^2 + 25985316207112*T3 + 2956968188160 acting on $S_{4}^{\mathrm{new}}(\Gamma_0(538))$ .

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Classical	Maass
Hilbert	Bianchi

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

$n$ :		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 1.21
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