Properties

Label 75.9.c.b
Level $75$
Weight $9$
Character orbit 75.c
Self dual yes
Analytic conductor $30.553$
Analytic rank $0$
Dimension $1$
CM discriminant -3
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 75.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(30.5533957546\)
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

\(f(q)\) \(=\) \( q + 81 q^{3} + 256 q^{4} - 4273 q^{7} + 6561 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 81 q^{3} + 256 q^{4} - 4273 q^{7} + 6561 q^{9} + 20736 q^{12} + 56447 q^{13} + 65536 q^{16} + 157967 q^{19} - 346113 q^{21} + 531441 q^{27} - 1093888 q^{28} + 1225967 q^{31} + 1679616 q^{36} + 503522 q^{37} + 4572207 q^{39} - 6837073 q^{43} + 5308416 q^{48} + 12493728 q^{49} + 14450432 q^{52} + 12795327 q^{57} - 307393 q^{61} - 28035153 q^{63} + 16777216 q^{64} - 31874833 q^{67} + 16169282 q^{73} + 40439552 q^{76} - 18887038 q^{79} + 43046721 q^{81} - 88604928 q^{84} - 241198031 q^{91} + 99303327 q^{93} - 82132513 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/75\mathbb{Z}\right)^\times\).

\(n\) \(26\) \(52\)
\(\chi(n)\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
26.1
0
0 81.0000 256.000 0 0 −4273.00 0 6561.00 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 75.9.c.b yes 1
3.b odd 2 1 CM 75.9.c.b yes 1
5.b even 2 1 75.9.c.a 1
5.c odd 4 2 75.9.d.a 2
15.d odd 2 1 75.9.c.a 1
15.e even 4 2 75.9.d.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.9.c.a 1 5.b even 2 1
75.9.c.a 1 15.d odd 2 1
75.9.c.b yes 1 1.a even 1 1 trivial
75.9.c.b yes 1 3.b odd 2 1 CM
75.9.d.a 2 5.c odd 4 2
75.9.d.a 2 15.e even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{9}^{\mathrm{new}}(75, [\chi])\):

\( T_{2} \) Copy content Toggle raw display
\( T_{7} + 4273 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T - 81 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T + 4273 \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T - 56447 \) Copy content Toggle raw display
$17$ \( T \) Copy content Toggle raw display
$19$ \( T - 157967 \) Copy content Toggle raw display
$23$ \( T \) Copy content Toggle raw display
$29$ \( T \) Copy content Toggle raw display
$31$ \( T - 1225967 \) Copy content Toggle raw display
$37$ \( T - 503522 \) Copy content Toggle raw display
$41$ \( T \) Copy content Toggle raw display
$43$ \( T + 6837073 \) 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
$61$ \( T + 307393 \) Copy content Toggle raw display
$67$ \( T + 31874833 \) Copy content Toggle raw display
$71$ \( T \) Copy content Toggle raw display
$73$ \( T - 16169282 \) Copy content Toggle raw display
$79$ \( T + 18887038 \) Copy content Toggle raw display
$83$ \( T \) Copy content Toggle raw display
$89$ \( T \) Copy content Toggle raw display
$97$ \( T + 82132513 \) Copy content Toggle raw display
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