Properties

Label 2475.2.a.i
Level $2475$
Weight $2$
Character orbit 2475.a
Self dual yes
Analytic conductor $19.763$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2475.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(19.7629745003\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 55)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{2} - q^{4} - 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} - q^{4} - 3 q^{8} + q^{11} - 2 q^{13} - q^{16} + 6 q^{17} - 4 q^{19} + q^{22} + 4 q^{23} - 2 q^{26} - 6 q^{29} - 8 q^{31} + 5 q^{32} + 6 q^{34} + 2 q^{37} - 4 q^{38} - 2 q^{41} - 4 q^{43} - q^{44} + 4 q^{46} - 12 q^{47} - 7 q^{49} + 2 q^{52} - 2 q^{53} - 6 q^{58} - 4 q^{59} - 10 q^{61} - 8 q^{62} + 7 q^{64} + 16 q^{67} - 6 q^{68} - 8 q^{71} - 14 q^{73} + 2 q^{74} + 4 q^{76} + 8 q^{79} - 2 q^{82} - 4 q^{83} - 4 q^{86} - 3 q^{88} - 10 q^{89} - 4 q^{92} - 12 q^{94} - 10 q^{97} - 7 q^{98}+O(q^{100}) \) 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 0 −1.00000 0 0 0 −3.00000 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(1\)
\(11\) \(-1\)

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 2475.2.a.i 1
3.b odd 2 1 275.2.a.a 1
5.b even 2 1 495.2.a.a 1
5.c odd 4 2 2475.2.c.f 2
12.b even 2 1 4400.2.a.p 1
15.d odd 2 1 55.2.a.a 1
15.e even 4 2 275.2.b.b 2
20.d odd 2 1 7920.2.a.i 1
33.d even 2 1 3025.2.a.f 1
55.d odd 2 1 5445.2.a.i 1
60.h even 2 1 880.2.a.h 1
60.l odd 4 2 4400.2.b.n 2
105.g even 2 1 2695.2.a.c 1
120.i odd 2 1 3520.2.a.p 1
120.m even 2 1 3520.2.a.n 1
165.d even 2 1 605.2.a.b 1
165.o odd 10 4 605.2.g.a 4
165.r even 10 4 605.2.g.c 4
195.e odd 2 1 9295.2.a.b 1
660.g odd 2 1 9680.2.a.r 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.2.a.a 1 15.d odd 2 1
275.2.a.a 1 3.b odd 2 1
275.2.b.b 2 15.e even 4 2
495.2.a.a 1 5.b even 2 1
605.2.a.b 1 165.d even 2 1
605.2.g.a 4 165.o odd 10 4
605.2.g.c 4 165.r even 10 4
880.2.a.h 1 60.h even 2 1
2475.2.a.i 1 1.a even 1 1 trivial
2475.2.c.f 2 5.c odd 4 2
2695.2.a.c 1 105.g even 2 1
3025.2.a.f 1 33.d even 2 1
3520.2.a.n 1 120.m even 2 1
3520.2.a.p 1 120.i odd 2 1
4400.2.a.p 1 12.b even 2 1
4400.2.b.n 2 60.l odd 4 2
5445.2.a.i 1 55.d odd 2 1
7920.2.a.i 1 20.d odd 2 1
9295.2.a.b 1 195.e odd 2 1
9680.2.a.r 1 660.g odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2475))\):

\( T_{2} - 1 \) Copy content Toggle raw display
\( T_{7} \) Copy content Toggle raw display
\( T_{29} + 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 1 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T - 1 \) Copy content Toggle raw display
$13$ \( T + 2 \) Copy content Toggle raw display
$17$ \( T - 6 \) Copy content Toggle raw display
$19$ \( T + 4 \) Copy content Toggle raw display
$23$ \( T - 4 \) Copy content Toggle raw display
$29$ \( T + 6 \) Copy content Toggle raw display
$31$ \( T + 8 \) Copy content Toggle raw display
$37$ \( T - 2 \) Copy content Toggle raw display
$41$ \( T + 2 \) Copy content Toggle raw display
$43$ \( T + 4 \) Copy content Toggle raw display
$47$ \( T + 12 \) Copy content Toggle raw display
$53$ \( T + 2 \) Copy content Toggle raw display
$59$ \( T + 4 \) Copy content Toggle raw display
$61$ \( T + 10 \) Copy content Toggle raw display
$67$ \( T - 16 \) Copy content Toggle raw display
$71$ \( T + 8 \) Copy content Toggle raw display
$73$ \( T + 14 \) Copy content Toggle raw display
$79$ \( T - 8 \) Copy content Toggle raw display
$83$ \( T + 4 \) Copy content Toggle raw display
$89$ \( T + 10 \) Copy content Toggle raw display
$97$ \( T + 10 \) Copy content Toggle raw display
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