Properties

Label 3042.2.a.g
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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Newspace parameters

Level: \( N \) \(=\) \( 3042 = 2 \cdot 3^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3042.a (trivial)

Newform invariants

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

\(f(q)\) \(=\) \( q - q^{2} + q^{4} + 3q^{5} - 3q^{7} - q^{8} + O(q^{10}) \) \( q - q^{2} + q^{4} + 3q^{5} - 3q^{7} - q^{8} - 3q^{10} + 3q^{14} + q^{16} + 3q^{17} - 6q^{19} + 3q^{20} - 6q^{23} + 4q^{25} - 3q^{28} - q^{32} - 3q^{34} - 9q^{35} - 3q^{37} + 6q^{38} - 3q^{40} + q^{43} + 6q^{46} + 3q^{47} + 2q^{49} - 4q^{50} + 6q^{53} + 3q^{56} - 6q^{59} - 8q^{61} + q^{64} - 12q^{67} + 3q^{68} + 9q^{70} - 15q^{71} - 6q^{73} + 3q^{74} - 6q^{76} + 10q^{79} + 3q^{80} - 6q^{83} + 9q^{85} - q^{86} - 6q^{89} - 6q^{92} - 3q^{94} - 18q^{95} - 12q^{97} - 2q^{98} + O(q^{100}) \)

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.

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 3.00000 0 −3.00000 −1.00000 0 −3.00000
\(n\): 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.g 1
3.b odd 2 1 338.2.a.d 1
12.b even 2 1 2704.2.a.j 1
13.b even 2 1 3042.2.a.j 1
13.d odd 4 2 234.2.b.b 2
15.d odd 2 1 8450.2.a.h 1
39.d odd 2 1 338.2.a.b 1
39.f even 4 2 26.2.b.a 2
39.h odd 6 2 338.2.c.f 2
39.i odd 6 2 338.2.c.b 2
39.k even 12 4 338.2.e.c 4
52.f even 4 2 1872.2.c.f 2
156.h even 2 1 2704.2.a.k 1
156.l odd 4 2 208.2.f.a 2
195.e odd 2 1 8450.2.a.u 1
195.j odd 4 2 650.2.c.d 2
195.n even 4 2 650.2.d.b 2
195.u odd 4 2 650.2.c.a 2
273.o odd 4 2 1274.2.d.c 2
273.cb odd 12 4 1274.2.n.c 4
273.cd even 12 4 1274.2.n.d 4
312.w odd 4 2 832.2.f.b 2
312.y even 4 2 832.2.f.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.2.b.a 2 39.f even 4 2
208.2.f.a 2 156.l odd 4 2
234.2.b.b 2 13.d odd 4 2
338.2.a.b 1 39.d odd 2 1
338.2.a.d 1 3.b odd 2 1
338.2.c.b 2 39.i odd 6 2
338.2.c.f 2 39.h odd 6 2
338.2.e.c 4 39.k even 12 4
650.2.c.a 2 195.u odd 4 2
650.2.c.d 2 195.j odd 4 2
650.2.d.b 2 195.n even 4 2
832.2.f.b 2 312.w odd 4 2
832.2.f.d 2 312.y even 4 2
1274.2.d.c 2 273.o odd 4 2
1274.2.n.c 4 273.cb odd 12 4
1274.2.n.d 4 273.cd even 12 4
1872.2.c.f 2 52.f even 4 2
2704.2.a.j 1 12.b even 2 1
2704.2.a.k 1 156.h even 2 1
3042.2.a.g 1 1.a even 1 1 trivial
3042.2.a.j 1 13.b even 2 1
8450.2.a.h 1 15.d odd 2 1
8450.2.a.u 1 195.e odd 2 1

Hecke kernels

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

\( T_{5} - 3 \)
\( T_{7} + 3 \)
\( T_{11} \)
\( T_{17} - 3 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T \)
$3$ \( T \)
$5$ \( -3 + T \)
$7$ \( 3 + T \)
$11$ \( T \)
$13$ \( T \)
$17$ \( -3 + T \)
$19$ \( 6 + T \)
$23$ \( 6 + T \)
$29$ \( T \)
$31$ \( T \)
$37$ \( 3 + T \)
$41$ \( T \)
$43$ \( -1 + T \)
$47$ \( -3 + T \)
$53$ \( -6 + T \)
$59$ \( 6 + T \)
$61$ \( 8 + T \)
$67$ \( 12 + T \)
$71$ \( 15 + T \)
$73$ \( 6 + T \)
$79$ \( -10 + T \)
$83$ \( 6 + T \)
$89$ \( 6 + T \)
$97$ \( 12 + T \)
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