LMFDB - Newform orbit 371.6.a.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [371,6,Mod(1,371)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("371.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(371, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 371 = 7 \cdot 53 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	371.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$59.5023971506$
Analytic rank:	$2$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

$ q - 7 q^{2} + 4 q^{3} + 17 q^{4} - 47 q^{5} - 28 q^{6} - 49 q^{7} + 105 q^{8} - 227 q^{9} + 329 q^{10} - 641 q^{11} + 68 q^{12} + 340 q^{13} + 343 q^{14} - 188 q^{15} - 1279 q^{16} - 2271 q^{17} + 1589 q^{18}+ \cdots + 145507 q^{99}+O(q^{100}) $

q - 7 * q^2 + 4 * q^3 + 17 * q^4 - 47 * q^5 - 28 * q^6 - 49 * q^7 + 105 * q^8 - 227 * q^9 + 329 * q^10 - 641 * q^11 + 68 * q^12 + 340 * q^13 + 343 * q^14 - 188 * q^15 - 1279 * q^16 - 2271 * q^17 + 1589 * q^18 + 2032 * q^19 - 799 * q^20 - 196 * q^21 + 4487 * q^22 - 4047 * q^23 + 420 * q^24 - 916 * q^25 - 2380 * q^26 - 1880 * q^27 - 833 * q^28 - 2384 * q^29 + 1316 * q^30 - 303 * q^31 + 5593 * q^32 - 2564 * q^33 + 15897 * q^34 + 2303 * q^35 - 3859 * q^36 - 2832 * q^37 - 14224 * q^38 + 1360 * q^39 - 4935 * q^40 - 11902 * q^41 + 1372 * q^42 - 6775 * q^43 - 10897 * q^44 + 10669 * q^45 + 28329 * q^46 + 6048 * q^47 - 5116 * q^48 + 2401 * q^49 + 6412 * q^50 - 9084 * q^51 + 5780 * q^52 + 2809 * q^53 + 13160 * q^54 + 30127 * q^55 - 5145 * q^56 + 8128 * q^57 + 16688 * q^58 - 49977 * q^59 - 3196 * q^60 - 55910 * q^61 + 2121 * q^62 + 11123 * q^63 + 1777 * q^64 - 15980 * q^65 + 17948 * q^66 + 2956 * q^67 - 38607 * q^68 - 16188 * q^69 - 16121 * q^70 - 35988 * q^71 - 23835 * q^72 - 65346 * q^73 + 19824 * q^74 - 3664 * q^75 + 34544 * q^76 + 31409 * q^77 - 9520 * q^78 + 79133 * q^79 + 60113 * q^80 + 47641 * q^81 + 83314 * q^82 - 16656 * q^83 - 3332 * q^84 + 106737 * q^85 + 47425 * q^86 - 9536 * q^87 - 67305 * q^88 - 61457 * q^89 - 74683 * q^90 - 16660 * q^91 - 68799 * q^92 - 1212 * q^93 - 42336 * q^94 - 95504 * q^95 + 22372 * q^96 + 52725 * q^97 - 16807 * q^98 + 145507 * 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

$\iota_m(\nu)$

$ a_{2} $

$ a_{3} $

$ a_{4} $

$ a_{5} $

$ a_{6} $

$ a_{7} $

$ a_{8} $

$ a_{9} $

$ a_{10} $

1.1

−7.00000

4.00000

17.0000

−47.0000

−28.0000

−49.0000

105.000

−227.000

329.000

Atkin-Lehner signs

$ p $	Sign
$7$	$ +1 $
$53$	$ -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	371.6.a.a	✓	1

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{2} + 7 $ T2 + 7 acting on $S_{6}^{\mathrm{new}}(\Gamma_0(371))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T + 7 $ T + 7
$3$	$ T - 4 $ T - 4
$5$	$ T + 47 $ T + 47
$7$	$ T + 49 $ T + 49
$11$	$ T + 641 $ T + 641
$13$	$ T - 340 $ T - 340
$17$	$ T + 2271 $ T + 2271
$19$	$ T - 2032 $ T - 2032
$23$	$ T + 4047 $ T + 4047
$29$	$ T + 2384 $ T + 2384
$31$	$ T + 303 $ T + 303
$37$	$ T + 2832 $ T + 2832
$41$	$ T + 11902 $ T + 11902
$43$	$ T + 6775 $ T + 6775
$47$	$ T - 6048 $ T - 6048
$53$	$ T - 2809 $ T - 2809
$59$	$ T + 49977 $ T + 49977
$61$	$ T + 55910 $ T + 55910
$67$	$ T - 2956 $ T - 2956
$71$	$ T + 35988 $ T + 35988
$73$	$ T + 65346 $ T + 65346
$79$	$ T - 79133 $ T - 79133
$83$	$ T + 16656 $ T + 16656
$89$	$ T + 61457 $ T + 61457
$97$	$ T - 52725 $ T - 52725
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Level:	\( N \)	\(=\)	\( 371 = 7 \cdot 53 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	371.a (trivial)

\(n\):		e.g. 2-40 or 80-90
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