Properties

Label 975.2.a.c
Level $975$
Weight $2$
Character orbit 975.a
Self dual yes
Analytic conductor $7.785$
Analytic rank $1$
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: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 195)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 2 q^{2} + q^{3} + 2 q^{4} - 2 q^{6} - 3 q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{2} + q^{3} + 2 q^{4} - 2 q^{6} - 3 q^{7} + q^{9} - q^{11} + 2 q^{12} + q^{13} + 6 q^{14} - 4 q^{16} + q^{17} - 2 q^{18} - 2 q^{19} - 3 q^{21} + 2 q^{22} + 3 q^{23} - 2 q^{26} + q^{27} - 6 q^{28} - 2 q^{29} - 6 q^{31} + 8 q^{32} - q^{33} - 2 q^{34} + 2 q^{36} - 11 q^{37} + 4 q^{38} + q^{39} - 5 q^{41} + 6 q^{42} - 4 q^{43} - 2 q^{44} - 6 q^{46} + 10 q^{47} - 4 q^{48} + 2 q^{49} + q^{51} + 2 q^{52} - 11 q^{53} - 2 q^{54} - 2 q^{57} + 4 q^{58} + 8 q^{59} + 13 q^{61} + 12 q^{62} - 3 q^{63} - 8 q^{64} + 2 q^{66} - 12 q^{67} + 2 q^{68} + 3 q^{69} - 5 q^{71} - 10 q^{73} + 22 q^{74} - 4 q^{76} + 3 q^{77} - 2 q^{78} - 3 q^{79} + q^{81} + 10 q^{82} + 12 q^{83} - 6 q^{84} + 8 q^{86} - 2 q^{87} - 15 q^{89} - 3 q^{91} + 6 q^{92} - 6 q^{93} - 20 q^{94} + 8 q^{96} - 17 q^{97} - 4 q^{98} - q^{99}+O(q^{100}) \) 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
−2.00000 1.00000 2.00000 0 −2.00000 −3.00000 0 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.c 1
3.b odd 2 1 2925.2.a.q 1
5.b even 2 1 195.2.a.b 1
5.c odd 4 2 975.2.c.a 2
15.d odd 2 1 585.2.a.b 1
15.e even 4 2 2925.2.c.c 2
20.d odd 2 1 3120.2.a.u 1
35.c odd 2 1 9555.2.a.v 1
60.h even 2 1 9360.2.a.d 1
65.d even 2 1 2535.2.a.a 1
195.e odd 2 1 7605.2.a.u 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.2.a.b 1 5.b even 2 1
585.2.a.b 1 15.d odd 2 1
975.2.a.c 1 1.a even 1 1 trivial
975.2.c.a 2 5.c odd 4 2
2535.2.a.a 1 65.d even 2 1
2925.2.a.q 1 3.b odd 2 1
2925.2.c.c 2 15.e even 4 2
3120.2.a.u 1 20.d odd 2 1
7605.2.a.u 1 195.e odd 2 1
9360.2.a.d 1 60.h even 2 1
9555.2.a.v 1 35.c odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(975))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 2 \) 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
$17$ \( T - 1 \) Copy content Toggle raw display
$19$ \( T + 2 \) Copy content Toggle raw display
$23$ \( T - 3 \) Copy content Toggle raw display
$29$ \( T + 2 \) Copy content Toggle raw display
$31$ \( T + 6 \) Copy content Toggle raw display
$37$ \( T + 11 \) Copy content Toggle raw display
$41$ \( T + 5 \) Copy content Toggle raw display
$43$ \( T + 4 \) Copy content Toggle raw display
$47$ \( T - 10 \) Copy content Toggle raw display
$53$ \( T + 11 \) Copy content Toggle raw display
$59$ \( T - 8 \) Copy content Toggle raw display
$61$ \( T - 13 \) Copy content Toggle raw display
$67$ \( T + 12 \) Copy content Toggle raw display
$71$ \( T + 5 \) Copy content Toggle raw display
$73$ \( T + 10 \) Copy content Toggle raw display
$79$ \( T + 3 \) Copy content Toggle raw display
$83$ \( T - 12 \) Copy content Toggle raw display
$89$ \( T + 15 \) Copy content Toggle raw display
$97$ \( T + 17 \) Copy content Toggle raw display
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