Properties

Label 3042.2.a.l
Level $3042$
Weight $2$
Character orbit 3042.a
Self dual yes
Analytic conductor $24.290$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3042,2,Mod(1,3042)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3042, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("3042.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3042 = 2 \cdot 3^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3042.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: \(24.2904922949\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 26)
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)
 
\(f(q)\) \(=\) \( q + q^{2} + q^{4} - q^{5} - q^{7} + q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + q^{4} - q^{5} - q^{7} + q^{8} - q^{10} - 2 q^{11} - q^{14} + q^{16} + 3 q^{17} - 6 q^{19} - q^{20} - 2 q^{22} + 4 q^{23} - 4 q^{25} - q^{28} - 2 q^{29} - 4 q^{31} + q^{32} + 3 q^{34} + q^{35} - 3 q^{37} - 6 q^{38} - q^{40} - 5 q^{43} - 2 q^{44} + 4 q^{46} + 13 q^{47} - 6 q^{49} - 4 q^{50} - 12 q^{53} + 2 q^{55} - q^{56} - 2 q^{58} - 10 q^{59} - 8 q^{61} - 4 q^{62} + q^{64} + 2 q^{67} + 3 q^{68} + q^{70} - 5 q^{71} + 10 q^{73} - 3 q^{74} - 6 q^{76} + 2 q^{77} - 4 q^{79} - q^{80} - 3 q^{85} - 5 q^{86} - 2 q^{88} + 6 q^{89} + 4 q^{92} + 13 q^{94} + 6 q^{95} - 14 q^{97} - 6 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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
0
1.00000 0 1.00000 −1.00000 0 −1.00000 1.00000 0 −1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(3\) \( -1 \)
\(13\) \( +1 \)

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 3042.2.a.l 1
3.b odd 2 1 338.2.a.a 1
12.b even 2 1 2704.2.a.n 1
13.b even 2 1 234.2.a.b 1
13.d odd 4 2 3042.2.b.f 2
15.d odd 2 1 8450.2.a.y 1
39.d odd 2 1 26.2.a.b 1
39.f even 4 2 338.2.b.a 2
39.h odd 6 2 338.2.c.c 2
39.i odd 6 2 338.2.c.g 2
39.k even 12 4 338.2.e.d 4
52.b odd 2 1 1872.2.a.m 1
65.d even 2 1 5850.2.a.bn 1
65.h odd 4 2 5850.2.e.v 2
104.e even 2 1 7488.2.a.w 1
104.h odd 2 1 7488.2.a.v 1
117.n odd 6 2 2106.2.e.h 2
117.t even 6 2 2106.2.e.t 2
156.h even 2 1 208.2.a.d 1
156.l odd 4 2 2704.2.f.j 2
195.e odd 2 1 650.2.a.g 1
195.s even 4 2 650.2.b.a 2
273.g even 2 1 1274.2.a.o 1
273.w odd 6 2 1274.2.f.l 2
273.ba even 6 2 1274.2.f.a 2
312.b odd 2 1 832.2.a.j 1
312.h even 2 1 832.2.a.a 1
429.e even 2 1 3146.2.a.a 1
624.v even 4 2 3328.2.b.k 2
624.bi odd 4 2 3328.2.b.g 2
663.g odd 2 1 7514.2.a.i 1
741.d even 2 1 9386.2.a.f 1
780.d even 2 1 5200.2.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.2.a.b 1 39.d odd 2 1
208.2.a.d 1 156.h even 2 1
234.2.a.b 1 13.b even 2 1
338.2.a.a 1 3.b odd 2 1
338.2.b.a 2 39.f even 4 2
338.2.c.c 2 39.h odd 6 2
338.2.c.g 2 39.i odd 6 2
338.2.e.d 4 39.k even 12 4
650.2.a.g 1 195.e odd 2 1
650.2.b.a 2 195.s even 4 2
832.2.a.a 1 312.h even 2 1
832.2.a.j 1 312.b odd 2 1
1274.2.a.o 1 273.g even 2 1
1274.2.f.a 2 273.ba even 6 2
1274.2.f.l 2 273.w odd 6 2
1872.2.a.m 1 52.b odd 2 1
2106.2.e.h 2 117.n odd 6 2
2106.2.e.t 2 117.t even 6 2
2704.2.a.n 1 12.b even 2 1
2704.2.f.j 2 156.l odd 4 2
3042.2.a.l 1 1.a even 1 1 trivial
3042.2.b.f 2 13.d odd 4 2
3146.2.a.a 1 429.e even 2 1
3328.2.b.g 2 624.bi odd 4 2
3328.2.b.k 2 624.v even 4 2
5200.2.a.c 1 780.d even 2 1
5850.2.a.bn 1 65.d even 2 1
5850.2.e.v 2 65.h odd 4 2
7488.2.a.v 1 104.h odd 2 1
7488.2.a.w 1 104.e even 2 1
7514.2.a.i 1 663.g odd 2 1
8450.2.a.y 1 15.d odd 2 1
9386.2.a.f 1 741.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3042))\):

\( T_{5} + 1 \) Copy content Toggle raw display
\( T_{7} + 1 \) Copy content Toggle raw display
\( T_{11} + 2 \) Copy content Toggle raw display
\( T_{17} - 3 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 1 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T + 1 \) Copy content Toggle raw display
$7$ \( T + 1 \) Copy content Toggle raw display
$11$ \( T + 2 \) Copy content Toggle raw display
$13$ \( T \) Copy content Toggle raw display
$17$ \( T - 3 \) Copy content Toggle raw display
$19$ \( T + 6 \) Copy content Toggle raw display
$23$ \( T - 4 \) Copy content Toggle raw display
$29$ \( T + 2 \) Copy content Toggle raw display
$31$ \( T + 4 \) Copy content Toggle raw display
$37$ \( T + 3 \) Copy content Toggle raw display
$41$ \( T \) Copy content Toggle raw display
$43$ \( T + 5 \) Copy content Toggle raw display
$47$ \( T - 13 \) Copy content Toggle raw display
$53$ \( T + 12 \) Copy content Toggle raw display
$59$ \( T + 10 \) Copy content Toggle raw display
$61$ \( T + 8 \) Copy content Toggle raw display
$67$ \( T - 2 \) Copy content Toggle raw display
$71$ \( T + 5 \) Copy content Toggle raw display
$73$ \( T - 10 \) Copy content Toggle raw display
$79$ \( T + 4 \) Copy content Toggle raw display
$83$ \( T \) Copy content Toggle raw display
$89$ \( T - 6 \) Copy content Toggle raw display
$97$ \( T + 14 \) Copy content Toggle raw display
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