Properties

Label 1575.2.a.h
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 105)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{2} - q^{4} - q^{7} - 3q^{8} + O(q^{10}) \) \( q + q^{2} - q^{4} - q^{7} - 3q^{8} + 6q^{13} - q^{14} - q^{16} + 2q^{17} - 8q^{19} + 8q^{23} + 6q^{26} + q^{28} + 2q^{29} + 4q^{31} + 5q^{32} + 2q^{34} + 2q^{37} - 8q^{38} + 6q^{41} - 4q^{43} + 8q^{46} + 8q^{47} + q^{49} - 6q^{52} + 10q^{53} + 3q^{56} + 2q^{58} - 4q^{59} - 2q^{61} + 4q^{62} + 7q^{64} - 4q^{67} - 2q^{68} + 12q^{71} + 2q^{73} + 2q^{74} + 8q^{76} + 8q^{79} + 6q^{82} - 4q^{83} - 4q^{86} + 6q^{89} - 6q^{91} - 8q^{92} + 8q^{94} + 18q^{97} + q^{98} + O(q^{100}) \)

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.h 1
3.b odd 2 1 525.2.a.a 1
5.b even 2 1 315.2.a.a 1
5.c odd 4 2 1575.2.d.b 2
12.b even 2 1 8400.2.a.co 1
15.d odd 2 1 105.2.a.a 1
15.e even 4 2 525.2.d.b 2
20.d odd 2 1 5040.2.a.d 1
21.c even 2 1 3675.2.a.f 1
35.c odd 2 1 2205.2.a.b 1
60.h even 2 1 1680.2.a.f 1
105.g even 2 1 735.2.a.f 1
105.o odd 6 2 735.2.i.a 2
105.p even 6 2 735.2.i.b 2
120.i odd 2 1 6720.2.a.p 1
120.m even 2 1 6720.2.a.bk 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.a.a 1 15.d odd 2 1
315.2.a.a 1 5.b even 2 1
525.2.a.a 1 3.b odd 2 1
525.2.d.b 2 15.e even 4 2
735.2.a.f 1 105.g even 2 1
735.2.i.a 2 105.o odd 6 2
735.2.i.b 2 105.p even 6 2
1575.2.a.h 1 1.a even 1 1 trivial
1575.2.d.b 2 5.c odd 4 2
1680.2.a.f 1 60.h even 2 1
2205.2.a.b 1 35.c odd 2 1
3675.2.a.f 1 21.c even 2 1
5040.2.a.d 1 20.d odd 2 1
6720.2.a.p 1 120.i odd 2 1
6720.2.a.bk 1 120.m even 2 1
8400.2.a.co 1 12.b even 2 1

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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1575))\):

\( T_{2} - 1 \)
\( T_{11} \)
\( T_{13} - 6 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - T + 2 T^{2} \)
$3$ 1
$5$ 1
$7$ \( 1 + T \)
$11$ \( 1 + 11 T^{2} \)
$13$ \( 1 - 6 T + 13 T^{2} \)
$17$ \( 1 - 2 T + 17 T^{2} \)
$19$ \( 1 + 8 T + 19 T^{2} \)
$23$ \( 1 - 8 T + 23 T^{2} \)
$29$ \( 1 - 2 T + 29 T^{2} \)
$31$ \( 1 - 4 T + 31 T^{2} \)
$37$ \( 1 - 2 T + 37 T^{2} \)
$41$ \( 1 - 6 T + 41 T^{2} \)
$43$ \( 1 + 4 T + 43 T^{2} \)
$47$ \( 1 - 8 T + 47 T^{2} \)
$53$ \( 1 - 10 T + 53 T^{2} \)
$59$ \( 1 + 4 T + 59 T^{2} \)
$61$ \( 1 + 2 T + 61 T^{2} \)
$67$ \( 1 + 4 T + 67 T^{2} \)
$71$ \( 1 - 12 T + 71 T^{2} \)
$73$ \( 1 - 2 T + 73 T^{2} \)
$79$ \( 1 - 8 T + 79 T^{2} \)
$83$ \( 1 + 4 T + 83 T^{2} \)
$89$ \( 1 - 6 T + 89 T^{2} \)
$97$ \( 1 - 18 T + 97 T^{2} \)
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