Properties

Label 75.3.c.a
Level $75$
Weight $3$
Character orbit 75.c
Self dual yes
Analytic conductor $2.044$
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: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 75.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(2.04360198270\)
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

\(f(q)\) \(=\) \( q - 3q^{3} + 4q^{4} + 11q^{7} + 9q^{9} + O(q^{10}) \) \( q - 3q^{3} + 4q^{4} + 11q^{7} + 9q^{9} - 12q^{12} - q^{13} + 16q^{16} - 37q^{19} - 33q^{21} - 27q^{27} + 44q^{28} - 13q^{31} + 36q^{36} + 26q^{37} + 3q^{39} - 61q^{43} - 48q^{48} + 72q^{49} - 4q^{52} + 111q^{57} + 47q^{61} + 99q^{63} + 64q^{64} - 109q^{67} - 46q^{73} - 148q^{76} - 142q^{79} + 81q^{81} - 132q^{84} - 11q^{91} + 39q^{93} - 169q^{97} + O(q^{100}) \)

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 −3.00000 4.00000 0 0 11.0000 0 9.00000 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 75.3.c.a 1
3.b odd 2 1 CM 75.3.c.a 1
4.b odd 2 1 1200.3.l.d 1
5.b even 2 1 75.3.c.b yes 1
5.c odd 4 2 75.3.d.a 2
12.b even 2 1 1200.3.l.d 1
15.d odd 2 1 75.3.c.b yes 1
15.e even 4 2 75.3.d.a 2
20.d odd 2 1 1200.3.l.c 1
20.e even 4 2 1200.3.c.a 2
60.h even 2 1 1200.3.l.c 1
60.l odd 4 2 1200.3.c.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.3.c.a 1 1.a even 1 1 trivial
75.3.c.a 1 3.b odd 2 1 CM
75.3.c.b yes 1 5.b even 2 1
75.3.c.b yes 1 15.d odd 2 1
75.3.d.a 2 5.c odd 4 2
75.3.d.a 2 15.e even 4 2
1200.3.c.a 2 20.e even 4 2
1200.3.c.a 2 60.l odd 4 2
1200.3.l.c 1 20.d odd 2 1
1200.3.l.c 1 60.h even 2 1
1200.3.l.d 1 4.b odd 2 1
1200.3.l.d 1 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(75, [\chi])\):

\( T_{2} \)
\( T_{7} - 11 \)
\( T_{13} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( 3 + T \)
$5$ \( T \)
$7$ \( -11 + T \)
$11$ \( T \)
$13$ \( 1 + T \)
$17$ \( T \)
$19$ \( 37 + T \)
$23$ \( T \)
$29$ \( T \)
$31$ \( 13 + T \)
$37$ \( -26 + T \)
$41$ \( T \)
$43$ \( 61 + T \)
$47$ \( T \)
$53$ \( T \)
$59$ \( T \)
$61$ \( -47 + T \)
$67$ \( 109 + T \)
$71$ \( T \)
$73$ \( 46 + T \)
$79$ \( 142 + T \)
$83$ \( T \)
$89$ \( T \)
$97$ \( 169 + T \)
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