Properties

Label 3.25.b.a
Level $3$
Weight $25$
Character orbit 3.b
Self dual yes
Analytic conductor $10.949$
Analytic rank $0$
Dimension $1$
CM discriminant -3
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 3 \)
Weight: \( k \) \(=\) \( 25 \)
Character orbit: \([\chi]\) \(=\) 3.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(10.9490145677\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

\(f(q)\) \(=\) \( q + 531441 q^{3} + 16777216 q^{4} - 4119710398 q^{7} + 282429536481 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 531441 q^{3} + 16777216 q^{4} - 4119710398 q^{7} + 282429536481 q^{9} + 8916100448256 q^{12} + 41718377201762 q^{13} + 281474976710656 q^{16} - 41\!\cdots\!78 q^{19}+ \cdots + 13\!\cdots\!82 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1
0
0 531441. 1.67772e7 0 0 −4.11971e9 0 2.82430e11 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3.25.b.a 1
3.b odd 2 1 CM 3.25.b.a 1
4.b odd 2 1 48.25.e.a 1
12.b even 2 1 48.25.e.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.25.b.a 1 1.a even 1 1 trivial
3.25.b.a 1 3.b odd 2 1 CM
48.25.e.a 1 4.b odd 2 1
48.25.e.a 1 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{25}^{\mathrm{new}}(3, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T - 531441 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T + 4119710398 \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T - 41718377201762 \) Copy content Toggle raw display
$17$ \( T \) Copy content Toggle raw display
$19$ \( T + 4106686428542878 \) Copy content Toggle raw display
$23$ \( T \) Copy content Toggle raw display
$29$ \( T \) Copy content Toggle raw display
$31$ \( T + 12\!\cdots\!78 \) Copy content Toggle raw display
$37$ \( T + 51\!\cdots\!38 \) Copy content Toggle raw display
$41$ \( T \) Copy content Toggle raw display
$43$ \( T + 79\!\cdots\!98 \) Copy content Toggle raw display
$47$ \( T \) Copy content Toggle raw display
$53$ \( T \) Copy content Toggle raw display
$59$ \( T \) Copy content Toggle raw display
$61$ \( T - 17\!\cdots\!42 \) Copy content Toggle raw display
$67$ \( T - 64\!\cdots\!22 \) Copy content Toggle raw display
$71$ \( T \) Copy content Toggle raw display
$73$ \( T + 34\!\cdots\!58 \) Copy content Toggle raw display
$79$ \( T - 79\!\cdots\!82 \) Copy content Toggle raw display
$83$ \( T \) Copy content Toggle raw display
$89$ \( T \) Copy content Toggle raw display
$97$ \( T - 13\!\cdots\!82 \) Copy content Toggle raw display
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