LMFDB - Newform orbit 975.4.a.h

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [975,4,Mod(1,975)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(975, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("975.1"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 975 = 3 \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	975.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,3,3,1,0,9,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$57.5268622556$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 195)
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 + 3 q^{2} + 3 q^{3} + q^{4} + 9 q^{6} - 2 q^{7} - 21 q^{8} + 9 q^{9} + 24 q^{11} + 3 q^{12} - 13 q^{13} - 6 q^{14} - 71 q^{16} - 24 q^{17} + 27 q^{18} - 70 q^{19} - 6 q^{21} + 72 q^{22} - 90 q^{23}+ \cdots + 216 q^{99}+O(q^{100}) $

q + 3 * q^2 + 3 * q^3 + q^4 + 9 * q^6 - 2 * q^7 - 21 * q^8 + 9 * q^9 + 24 * q^11 + 3 * q^12 - 13 * q^13 - 6 * q^14 - 71 * q^16 - 24 * q^17 + 27 * q^18 - 70 * q^19 - 6 * q^21 + 72 * q^22 - 90 * q^23 - 63 * q^24 - 39 * q^26 + 27 * q^27 - 2 * q^28 - 120 * q^29 - 196 * q^31 - 45 * q^32 + 72 * q^33 - 72 * q^34 + 9 * q^36 + 214 * q^37 - 210 * q^38 - 39 * q^39 - 54 * q^41 - 18 * q^42 + 196 * q^43 + 24 * q^44 - 270 * q^46 - 120 * q^47 - 213 * q^48 - 339 * q^49 - 72 * q^51 - 13 * q^52 - 18 * q^53 + 81 * q^54 + 42 * q^56 - 210 * q^57 - 360 * q^58 - 312 * q^59 - 322 * q^61 - 588 * q^62 - 18 * q^63 + 433 * q^64 + 216 * q^66 + 376 * q^67 - 24 * q^68 - 270 * q^69 + 240 * q^71 - 189 * q^72 - 1136 * q^73 + 642 * q^74 - 70 * q^76 - 48 * q^77 - 117 * q^78 - 808 * q^79 + 81 * q^81 - 162 * q^82 - 1092 * q^83 - 6 * q^84 + 588 * q^86 - 360 * q^87 - 504 * q^88 - 618 * q^89 + 26 * q^91 - 90 * q^92 - 588 * q^93 - 360 * q^94 - 135 * q^96 + 880 * q^97 - 1017 * q^98 + 216 * 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

3.00000

1.00000

9.00000

−2.00000

−21.0000

9.00000

Atkin-Lehner signs

$ p $	Sign
$3$	$ -1 $
$5$	$ +1 $
$13$	$ +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	975.4.a.h		1
5.b	even	2	1		195.4.a.b	✓	1
15.d	odd	2	1		585.4.a.d		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
195.4.a.b	✓	1	5.b	even	2	1
585.4.a.d		1	15.d	odd	2	1
975.4.a.h		1	1.a	even	1	1	trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(\Gamma_0(975))$:

$ T_{2} - 3 $

T2 - 3

$ T_{7} + 2 $

T7 + 2

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T - 3 $ T - 3
$3$	$ T - 3 $ T - 3
$5$	$ T $ T
$7$	$ T + 2 $ T + 2
$11$	$ T - 24 $ T - 24
$13$	$ T + 13 $ T + 13
$17$	$ T + 24 $ T + 24
$19$	$ T + 70 $ T + 70
$23$	$ T + 90 $ T + 90
$29$	$ T + 120 $ T + 120
$31$	$ T + 196 $ T + 196
$37$	$ T - 214 $ T - 214
$41$	$ T + 54 $ T + 54
$43$	$ T - 196 $ T - 196
$47$	$ T + 120 $ T + 120
$53$	$ T + 18 $ T + 18
$59$	$ T + 312 $ T + 312
$61$	$ T + 322 $ T + 322
$67$	$ T - 376 $ T - 376
$71$	$ T - 240 $ T - 240
$73$	$ T + 1136 $ T + 1136
$79$	$ T + 808 $ T + 808
$83$	$ T + 1092 $ T + 1092
$89$	$ T + 618 $ T + 618
$97$	$ T - 880 $ T - 880
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Classical	Maass
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Elliptic curves over $\Q$
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Level:	\( N \)	\(=\)	\( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	975.a (trivial)

\(n\):		e.g. 2-40 or 990-1000
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