Properties

Label 1575.2.a.g
Level 1575
Weight 2
Character orbit 1575.a
Self dual yes
Analytic conductor 12.576
Analytic rank 0
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1575.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(12.5764383184\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 315)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{2} - q^{4} - q^{7} - 3q^{8} + O(q^{10}) \) \( q + q^{2} - q^{4} - q^{7} - 3q^{8} + 4q^{13} - q^{14} - q^{16} + 2q^{17} + 4q^{26} + q^{28} + 8q^{29} - 4q^{31} + 5q^{32} + 2q^{34} + 8q^{37} + 4q^{41} + 8q^{43} - 12q^{47} + q^{49} - 4q^{52} + 6q^{53} + 3q^{56} + 8q^{58} + 8q^{59} + 10q^{61} - 4q^{62} + 7q^{64} + 8q^{67} - 2q^{68} - 16q^{71} + 12q^{73} + 8q^{74} - 8q^{79} + 4q^{82} + 16q^{83} + 8q^{86} - 12q^{89} - 4q^{91} - 12q^{94} + 4q^{97} + q^{98} + O(q^{100}) \)

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

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 1575.2.a.g 1
3.b odd 2 1 1575.2.a.b 1
5.b even 2 1 1575.2.a.d 1
5.c odd 4 2 315.2.d.d yes 2
15.d odd 2 1 1575.2.a.j 1
15.e even 4 2 315.2.d.b 2
20.e even 4 2 5040.2.t.r 2
35.f even 4 2 2205.2.d.c 2
60.l odd 4 2 5040.2.t.c 2
105.k odd 4 2 2205.2.d.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.2.d.b 2 15.e even 4 2
315.2.d.d yes 2 5.c odd 4 2
1575.2.a.b 1 3.b odd 2 1
1575.2.a.d 1 5.b even 2 1
1575.2.a.g 1 1.a even 1 1 trivial
1575.2.a.j 1 15.d odd 2 1
2205.2.d.c 2 35.f even 4 2
2205.2.d.g 2 105.k odd 4 2
5040.2.t.c 2 60.l odd 4 2
5040.2.t.r 2 20.e even 4 2

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(5\) \(-1\)
\(7\) \(1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1575))\):

\( T_{2} - 1 \)
\( T_{11} \)
\( T_{13} - 4 \)

Hecke characteristic polynomials

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