Properties

Label 3150.2.a.bq
Level $3150$
Weight $2$
Character orbit 3150.a
Self dual yes
Analytic conductor $25.153$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3150.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(25.1528766367\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 350)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{2} + q^{4} + q^{7} + q^{8} + O(q^{10}) \) \( q + q^{2} + q^{4} + q^{7} + q^{8} + 5q^{11} + 6q^{13} + q^{14} + q^{16} - q^{17} - 3q^{19} + 5q^{22} + 6q^{26} + q^{28} + 6q^{29} - 4q^{31} + q^{32} - q^{34} - 8q^{37} - 3q^{38} - 11q^{41} + 8q^{43} + 5q^{44} + 2q^{47} + q^{49} + 6q^{52} + 4q^{53} + q^{56} + 6q^{58} - 4q^{59} - 2q^{61} - 4q^{62} + q^{64} - 9q^{67} - q^{68} + 10q^{71} + 7q^{73} - 8q^{74} - 3q^{76} + 5q^{77} - 2q^{79} - 11q^{82} + 11q^{83} + 8q^{86} + 5q^{88} + 11q^{89} + 6q^{91} + 2q^{94} + 10q^{97} + q^{98} + O(q^{100}) \)

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

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(5\) \(1\)
\(7\) \(-1\)

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 3150.2.a.bq 1
3.b odd 2 1 350.2.a.c 1
5.b even 2 1 3150.2.a.j 1
5.c odd 4 2 3150.2.g.v 2
12.b even 2 1 2800.2.a.b 1
15.d odd 2 1 350.2.a.d yes 1
15.e even 4 2 350.2.c.a 2
21.c even 2 1 2450.2.a.a 1
60.h even 2 1 2800.2.a.bg 1
60.l odd 4 2 2800.2.g.a 2
105.g even 2 1 2450.2.a.bg 1
105.k odd 4 2 2450.2.c.r 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
350.2.a.c 1 3.b odd 2 1
350.2.a.d yes 1 15.d odd 2 1
350.2.c.a 2 15.e even 4 2
2450.2.a.a 1 21.c even 2 1
2450.2.a.bg 1 105.g even 2 1
2450.2.c.r 2 105.k odd 4 2
2800.2.a.b 1 12.b even 2 1
2800.2.a.bg 1 60.h even 2 1
2800.2.g.a 2 60.l odd 4 2
3150.2.a.j 1 5.b even 2 1
3150.2.a.bq 1 1.a even 1 1 trivial
3150.2.g.v 2 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3150))\):

\( T_{11} - 5 \)
\( T_{13} - 6 \)
\( T_{17} + 1 \)
\( T_{19} + 3 \)
\( T_{29} - 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 + T \)
$3$ \( T \)
$5$ \( T \)
$7$ \( -1 + T \)
$11$ \( -5 + T \)
$13$ \( -6 + T \)
$17$ \( 1 + T \)
$19$ \( 3 + T \)
$23$ \( T \)
$29$ \( -6 + T \)
$31$ \( 4 + T \)
$37$ \( 8 + T \)
$41$ \( 11 + T \)
$43$ \( -8 + T \)
$47$ \( -2 + T \)
$53$ \( -4 + T \)
$59$ \( 4 + T \)
$61$ \( 2 + T \)
$67$ \( 9 + T \)
$71$ \( -10 + T \)
$73$ \( -7 + T \)
$79$ \( 2 + T \)
$83$ \( -11 + T \)
$89$ \( -11 + T \)
$97$ \( -10 + T \)
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