Properties

Label 5175.2.a.z
Level $5175$
Weight $2$
Character orbit 5175.a
Self dual yes
Analytic conductor $41.323$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5175 = 3^{2} \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5175.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(41.3225830460\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 115)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 2q^{2} + 2q^{4} - q^{7} + O(q^{10}) \) \( q + 2q^{2} + 2q^{4} - q^{7} - 2q^{13} - 2q^{14} - 4q^{16} - 5q^{17} + 8q^{19} - q^{23} - 4q^{26} - 2q^{28} + 5q^{29} - 5q^{31} - 8q^{32} - 10q^{34} - 7q^{37} + 16q^{38} + 7q^{41} - 4q^{43} - 2q^{46} - 2q^{47} - 6q^{49} - 4q^{52} - q^{53} + 10q^{58} - 3q^{59} - 6q^{61} - 10q^{62} - 8q^{64} - 13q^{67} - 10q^{68} - 13q^{71} - 8q^{73} - 14q^{74} + 16q^{76} - 14q^{79} + 14q^{82} - 3q^{83} - 8q^{86} + 14q^{89} + 2q^{91} - 2q^{92} - 4q^{94} - 14q^{97} - 12q^{98} + O(q^{100}) \)

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 0 2.00000 0 0 −1.00000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(-1\)
\(23\) \(1\)

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 5175.2.a.z 1
3.b odd 2 1 575.2.a.a 1
5.b even 2 1 5175.2.a.a 1
5.c odd 4 2 1035.2.b.a 2
12.b even 2 1 9200.2.a.bg 1
15.d odd 2 1 575.2.a.e 1
15.e even 4 2 115.2.b.a 2
60.h even 2 1 9200.2.a.g 1
60.l odd 4 2 1840.2.e.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
115.2.b.a 2 15.e even 4 2
575.2.a.a 1 3.b odd 2 1
575.2.a.e 1 15.d odd 2 1
1035.2.b.a 2 5.c odd 4 2
1840.2.e.b 2 60.l odd 4 2
5175.2.a.a 1 5.b even 2 1
5175.2.a.z 1 1.a even 1 1 trivial
9200.2.a.g 1 60.h even 2 1
9200.2.a.bg 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5175))\):

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

Hecke characteristic polynomials

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