Properties

Label 115.3.c.a
Level $115$
Weight $3$
Character orbit 115.c
Self dual yes
Analytic conductor $3.134$
Analytic rank $0$
Dimension $1$
CM discriminant -115
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 115 = 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 115.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(3.13352304014\)
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

\(f(q)\) \(=\) \( q + 4 q^{4} - 5 q^{5} + 9 q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 4 q^{4} - 5 q^{5} + 9 q^{7} + 9 q^{9} + 16 q^{16} - 11 q^{17} - 20 q^{20} + 23 q^{23} + 25 q^{25} + 36 q^{28} - 57 q^{29} - 53 q^{31} - 45 q^{35} + 36 q^{36} - 51 q^{37} - 33 q^{41} + 6 q^{43} - 45 q^{45} + 32 q^{49} + 101 q^{53} + 3 q^{59} + 81 q^{63} + 64 q^{64} - 111 q^{67} - 44 q^{68} + 27 q^{71} - 80 q^{80} + 81 q^{81} + 41 q^{83} + 55 q^{85} + 92 q^{92} + 174 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/115\mathbb{Z}\right)^\times\).

\(n\) \(47\) \(51\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
114.1
0
0 0 4.00000 −5.00000 0 9.00000 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
115.c odd 2 1 CM by \(\Q(\sqrt{-115}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 115.3.c.a 1
5.b even 2 1 115.3.c.b yes 1
5.c odd 4 2 575.3.d.a 2
23.b odd 2 1 115.3.c.b yes 1
115.c odd 2 1 CM 115.3.c.a 1
115.e even 4 2 575.3.d.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
115.3.c.a 1 1.a even 1 1 trivial
115.3.c.a 1 115.c odd 2 1 CM
115.3.c.b yes 1 5.b even 2 1
115.3.c.b yes 1 23.b odd 2 1
575.3.d.a 2 5.c odd 4 2
575.3.d.a 2 115.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 \(S_{3}^{\mathrm{new}}(115, [\chi])\):

\( T_{2} \) Copy content Toggle raw display
\( T_{7} - 9 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T + 5 \) Copy content Toggle raw display
$7$ \( T - 9 \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T \) Copy content Toggle raw display
$17$ \( T + 11 \) Copy content Toggle raw display
$19$ \( T \) Copy content Toggle raw display
$23$ \( T - 23 \) Copy content Toggle raw display
$29$ \( T + 57 \) Copy content Toggle raw display
$31$ \( T + 53 \) Copy content Toggle raw display
$37$ \( T + 51 \) Copy content Toggle raw display
$41$ \( T + 33 \) Copy content Toggle raw display
$43$ \( T - 6 \) Copy content Toggle raw display
$47$ \( T \) Copy content Toggle raw display
$53$ \( T - 101 \) Copy content Toggle raw display
$59$ \( T - 3 \) Copy content Toggle raw display
$61$ \( T \) Copy content Toggle raw display
$67$ \( T + 111 \) Copy content Toggle raw display
$71$ \( T - 27 \) Copy content Toggle raw display
$73$ \( T \) Copy content Toggle raw display
$79$ \( T \) Copy content Toggle raw display
$83$ \( T - 41 \) Copy content Toggle raw display
$89$ \( T \) Copy content Toggle raw display
$97$ \( T - 174 \) Copy content Toggle raw display
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