Properties

Label 1575.2.a.k
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 35)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 2q^{2} + 2q^{4} + q^{7} + O(q^{10}) \) \( q + 2q^{2} + 2q^{4} + q^{7} + 3q^{11} - q^{13} + 2q^{14} - 4q^{16} + 7q^{17} + 6q^{22} + 6q^{23} - 2q^{26} + 2q^{28} + 5q^{29} + 2q^{31} - 8q^{32} + 14q^{34} - 2q^{37} - 2q^{41} + 4q^{43} + 6q^{44} + 12q^{46} - 3q^{47} + q^{49} - 2q^{52} + 6q^{53} + 10q^{58} - 10q^{59} - 8q^{61} + 4q^{62} - 8q^{64} - 2q^{67} + 14q^{68} + 8q^{71} - 6q^{73} - 4q^{74} + 3q^{77} - 5q^{79} - 4q^{82} - 4q^{83} + 8q^{86} - q^{91} + 12q^{92} - 6q^{94} - 7q^{97} + 2q^{98} + O(q^{100}) \)

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:

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.k 1
3.b odd 2 1 175.2.a.a 1
5.b even 2 1 1575.2.a.a 1
5.c odd 4 2 315.2.d.a 2
12.b even 2 1 2800.2.a.w 1
15.d odd 2 1 175.2.a.c 1
15.e even 4 2 35.2.b.a 2
20.e even 4 2 5040.2.t.p 2
21.c even 2 1 1225.2.a.a 1
35.f even 4 2 2205.2.d.b 2
60.h even 2 1 2800.2.a.l 1
60.l odd 4 2 560.2.g.b 2
105.g even 2 1 1225.2.a.i 1
105.k odd 4 2 245.2.b.a 2
105.w odd 12 4 245.2.j.d 4
105.x even 12 4 245.2.j.e 4
120.q odd 4 2 2240.2.g.g 2
120.w even 4 2 2240.2.g.h 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.2.b.a 2 15.e even 4 2
175.2.a.a 1 3.b odd 2 1
175.2.a.c 1 15.d odd 2 1
245.2.b.a 2 105.k odd 4 2
245.2.j.d 4 105.w odd 12 4
245.2.j.e 4 105.x even 12 4
315.2.d.a 2 5.c odd 4 2
560.2.g.b 2 60.l odd 4 2
1225.2.a.a 1 21.c even 2 1
1225.2.a.i 1 105.g even 2 1
1575.2.a.a 1 5.b even 2 1
1575.2.a.k 1 1.a even 1 1 trivial
2205.2.d.b 2 35.f even 4 2
2240.2.g.g 2 120.q odd 4 2
2240.2.g.h 2 120.w even 4 2
2800.2.a.l 1 60.h even 2 1
2800.2.a.w 1 12.b even 2 1
5040.2.t.p 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1575))\):

\( T_{2} - 2 \)
\( T_{11} - 3 \)
\( T_{13} + 1 \)

Hecke characteristic polynomials

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