Properties

Label 975.2.a.d
Level $975$
Weight $2$
Character orbit 975.a
Self dual yes
Analytic conductor $7.785$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 975.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(7.78541419707\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{2} - q^{3} - q^{4} + q^{6} + 3 q^{7} + 3 q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} - q^{3} - q^{4} + q^{6} + 3 q^{7} + 3 q^{8} + q^{9} - q^{11} + q^{12} - q^{13} - 3 q^{14} - q^{16} + 5 q^{17} - q^{18} - 8 q^{19} - 3 q^{21} + q^{22} - 3 q^{24} + q^{26} - q^{27} - 3 q^{28} + q^{29} + 3 q^{31} - 5 q^{32} + q^{33} - 5 q^{34} - q^{36} + 8 q^{37} + 8 q^{38} + q^{39} - 2 q^{41} + 3 q^{42} - 8 q^{43} + q^{44} + 11 q^{47} + q^{48} + 2 q^{49} - 5 q^{51} + q^{52} + 11 q^{53} + q^{54} + 9 q^{56} + 8 q^{57} - q^{58} + 5 q^{59} + q^{61} - 3 q^{62} + 3 q^{63} + 7 q^{64} - q^{66} - 3 q^{67} - 5 q^{68} + 16 q^{71} + 3 q^{72} + 4 q^{73} - 8 q^{74} + 8 q^{76} - 3 q^{77} - q^{78} + 12 q^{79} + q^{81} + 2 q^{82} + 3 q^{83} + 3 q^{84} + 8 q^{86} - q^{87} - 3 q^{88} - 3 q^{91} - 3 q^{93} - 11 q^{94} + 5 q^{96} + 2 q^{97} - 2 q^{98} - q^{99}+O(q^{100}) \) Copy content Toggle raw display

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 −1.00000 −1.00000 0 1.00000 3.00000 3.00000 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(5\) \(-1\)
\(13\) \(1\)

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 975.2.a.d 1
3.b odd 2 1 2925.2.a.o 1
5.b even 2 1 975.2.a.k yes 1
5.c odd 4 2 975.2.c.g 2
15.d odd 2 1 2925.2.a.b 1
15.e even 4 2 2925.2.c.i 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
975.2.a.d 1 1.a even 1 1 trivial
975.2.a.k yes 1 5.b even 2 1
975.2.c.g 2 5.c odd 4 2
2925.2.a.b 1 15.d odd 2 1
2925.2.a.o 1 3.b odd 2 1
2925.2.c.i 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(975))\):

\( T_{2} + 1 \) Copy content Toggle raw display
\( T_{7} - 3 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 1 \) Copy content Toggle raw display
$3$ \( T + 1 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T - 3 \) Copy content Toggle raw display
$11$ \( T + 1 \) Copy content Toggle raw display
$13$ \( T + 1 \) Copy content Toggle raw display
$17$ \( T - 5 \) Copy content Toggle raw display
$19$ \( T + 8 \) Copy content Toggle raw display
$23$ \( T \) Copy content Toggle raw display
$29$ \( T - 1 \) Copy content Toggle raw display
$31$ \( T - 3 \) Copy content Toggle raw display
$37$ \( T - 8 \) Copy content Toggle raw display
$41$ \( T + 2 \) Copy content Toggle raw display
$43$ \( T + 8 \) Copy content Toggle raw display
$47$ \( T - 11 \) Copy content Toggle raw display
$53$ \( T - 11 \) Copy content Toggle raw display
$59$ \( T - 5 \) Copy content Toggle raw display
$61$ \( T - 1 \) Copy content Toggle raw display
$67$ \( T + 3 \) Copy content Toggle raw display
$71$ \( T - 16 \) Copy content Toggle raw display
$73$ \( T - 4 \) Copy content Toggle raw display
$79$ \( T - 12 \) Copy content Toggle raw display
$83$ \( T - 3 \) Copy content Toggle raw display
$89$ \( T \) Copy content Toggle raw display
$97$ \( T - 2 \) Copy content Toggle raw display
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