Properties

Label 575.2.a.a
Level $575$
Weight $2$
Character orbit 575.a
Self dual yes
Analytic conductor $4.591$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 575 = 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 575.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.59139811622\)
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

\(f(q)\) \(=\) \( q - 2q^{2} - 2q^{3} + 2q^{4} + 4q^{6} - q^{7} + q^{9} + O(q^{10}) \) \( q - 2q^{2} - 2q^{3} + 2q^{4} + 4q^{6} - q^{7} + q^{9} - 4q^{12} - 2q^{13} + 2q^{14} - 4q^{16} + 5q^{17} - 2q^{18} + 8q^{19} + 2q^{21} + q^{23} + 4q^{26} + 4q^{27} - 2q^{28} - 5q^{29} - 5q^{31} + 8q^{32} - 10q^{34} + 2q^{36} - 7q^{37} - 16q^{38} + 4q^{39} - 7q^{41} - 4q^{42} - 4q^{43} - 2q^{46} + 2q^{47} + 8q^{48} - 6q^{49} - 10q^{51} - 4q^{52} + q^{53} - 8q^{54} - 16q^{57} + 10q^{58} + 3q^{59} - 6q^{61} + 10q^{62} - q^{63} - 8q^{64} - 13q^{67} + 10q^{68} - 2q^{69} + 13q^{71} - 8q^{73} + 14q^{74} + 16q^{76} - 8q^{78} - 14q^{79} - 11q^{81} + 14q^{82} + 3q^{83} + 4q^{84} + 8q^{86} + 10q^{87} - 14q^{89} + 2q^{91} + 2q^{92} + 10q^{93} - 4q^{94} - 16q^{96} - 14q^{97} + 12q^{98} + O(q^{100}) \)

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 −2.00000 2.00000 0 4.00000 −1.00000 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 575.2.a.a 1
3.b odd 2 1 5175.2.a.z 1
4.b odd 2 1 9200.2.a.bg 1
5.b even 2 1 575.2.a.e 1
5.c odd 4 2 115.2.b.a 2
15.d odd 2 1 5175.2.a.a 1
15.e even 4 2 1035.2.b.a 2
20.d odd 2 1 9200.2.a.g 1
20.e even 4 2 1840.2.e.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
115.2.b.a 2 5.c odd 4 2
575.2.a.a 1 1.a even 1 1 trivial
575.2.a.e 1 5.b even 2 1
1035.2.b.a 2 15.e even 4 2
1840.2.e.b 2 20.e even 4 2
5175.2.a.a 1 15.d odd 2 1
5175.2.a.z 1 3.b odd 2 1
9200.2.a.g 1 20.d odd 2 1
9200.2.a.bg 1 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(575))\):

\( T_{2} + 2 \)
\( T_{3} + 2 \)

Hecke characteristic polynomials

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