Properties

Label 3042.2.a.h
Level $3042$
Weight $2$
Character orbit 3042.a
Self dual yes
Analytic conductor $24.290$
Analytic rank $0$
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: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 78)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{2} + q^{4} + 3 q^{5} - 2 q^{7} - q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{4} + 3 q^{5} - 2 q^{7} - q^{8} - 3 q^{10} + 6 q^{11} + 2 q^{14} + q^{16} + 3 q^{17} - 2 q^{19} + 3 q^{20} - 6 q^{22} + 6 q^{23} + 4 q^{25} - 2 q^{28} - 3 q^{29} + 4 q^{31} - q^{32} - 3 q^{34} - 6 q^{35} + 7 q^{37} + 2 q^{38} - 3 q^{40} - 3 q^{41} - 10 q^{43} + 6 q^{44} - 6 q^{46} + 6 q^{47} - 3 q^{49} - 4 q^{50} - 3 q^{53} + 18 q^{55} + 2 q^{56} + 3 q^{58} - 7 q^{61} - 4 q^{62} + q^{64} + 10 q^{67} + 3 q^{68} + 6 q^{70} + 6 q^{71} + 13 q^{73} - 7 q^{74} - 2 q^{76} - 12 q^{77} - 4 q^{79} + 3 q^{80} + 3 q^{82} - 6 q^{83} + 9 q^{85} + 10 q^{86} - 6 q^{88} + 18 q^{89} + 6 q^{92} - 6 q^{94} - 6 q^{95} - 14 q^{97} + 3 q^{98}+O(q^{100}) \) 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.

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 −2.00000 −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.h 1
3.b odd 2 1 1014.2.a.f 1
12.b even 2 1 8112.2.a.c 1
13.b even 2 1 3042.2.a.i 1
13.d odd 4 2 3042.2.b.h 2
13.e even 6 2 234.2.h.a 2
39.d odd 2 1 1014.2.a.c 1
39.f even 4 2 1014.2.b.c 2
39.h odd 6 2 78.2.e.a 2
39.i odd 6 2 1014.2.e.a 2
39.k even 12 4 1014.2.i.b 4
52.i odd 6 2 1872.2.t.c 2
156.h even 2 1 8112.2.a.m 1
156.r even 6 2 624.2.q.g 2
195.y odd 6 2 1950.2.i.m 2
195.bf even 12 4 1950.2.z.g 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.2.e.a 2 39.h odd 6 2
234.2.h.a 2 13.e even 6 2
624.2.q.g 2 156.r even 6 2
1014.2.a.c 1 39.d odd 2 1
1014.2.a.f 1 3.b odd 2 1
1014.2.b.c 2 39.f even 4 2
1014.2.e.a 2 39.i odd 6 2
1014.2.i.b 4 39.k even 12 4
1872.2.t.c 2 52.i odd 6 2
1950.2.i.m 2 195.y odd 6 2
1950.2.z.g 4 195.bf even 12 4
3042.2.a.h 1 1.a even 1 1 trivial
3042.2.a.i 1 13.b even 2 1
3042.2.b.h 2 13.d odd 4 2
8112.2.a.c 1 12.b even 2 1
8112.2.a.m 1 156.h even 2 1

Hecke kernels

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

\( T_{5} - 3 \) Copy content Toggle raw display
\( T_{7} + 2 \) Copy content Toggle raw display
\( T_{11} - 6 \) 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
$5$ \( T - 3 \) Copy content Toggle raw display
$7$ \( T + 2 \) Copy content Toggle raw display
$11$ \( T - 6 \) Copy content Toggle raw display
$13$ \( T \) Copy content Toggle raw display
$17$ \( T - 3 \) Copy content Toggle raw display
$19$ \( T + 2 \) Copy content Toggle raw display
$23$ \( T - 6 \) Copy content Toggle raw display
$29$ \( T + 3 \) Copy content Toggle raw display
$31$ \( T - 4 \) Copy content Toggle raw display
$37$ \( T - 7 \) Copy content Toggle raw display
$41$ \( T + 3 \) Copy content Toggle raw display
$43$ \( T + 10 \) Copy content Toggle raw display
$47$ \( T - 6 \) Copy content Toggle raw display
$53$ \( T + 3 \) Copy content Toggle raw display
$59$ \( T \) Copy content Toggle raw display
$61$ \( T + 7 \) Copy content Toggle raw display
$67$ \( T - 10 \) Copy content Toggle raw display
$71$ \( T - 6 \) Copy content Toggle raw display
$73$ \( T - 13 \) Copy content Toggle raw display
$79$ \( T + 4 \) Copy content Toggle raw display
$83$ \( T + 6 \) Copy content Toggle raw display
$89$ \( T - 18 \) Copy content Toggle raw display
$97$ \( T + 14 \) Copy content Toggle raw display
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