Properties

Label 1575.4.a.f
Level 1575
Weight 4
Character orbit 1575.a
Self dual yes
Analytic conductor 92.928
Analytic rank 1
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1575.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(92.9280082590\)
Analytic rank: \(1\)
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

\(f(q)\) \(=\) \( q - 8q^{4} - 7q^{7} + O(q^{10}) \) \( q - 8q^{4} - 7q^{7} - 42q^{11} - 20q^{13} + 64q^{16} + 66q^{17} + 38q^{19} + 12q^{23} + 56q^{28} + 258q^{29} + 146q^{31} - 434q^{37} + 282q^{41} - 20q^{43} + 336q^{44} - 72q^{47} + 49q^{49} + 160q^{52} + 336q^{53} + 360q^{59} - 682q^{61} - 512q^{64} - 812q^{67} - 528q^{68} - 810q^{71} + 124q^{73} - 304q^{76} + 294q^{77} + 1136q^{79} + 156q^{83} + 1038q^{89} + 140q^{91} - 96q^{92} - 1208q^{97} + O(q^{100}) \)

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
0 0 −8.00000 0 0 −7.00000 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 1575.4.a.f 1
3.b odd 2 1 525.4.a.e 1
5.b even 2 1 315.4.a.d 1
15.d odd 2 1 105.4.a.a 1
15.e even 4 2 525.4.d.f 2
35.c odd 2 1 2205.4.a.o 1
60.h even 2 1 1680.4.a.s 1
105.g even 2 1 735.4.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.4.a.a 1 15.d odd 2 1
315.4.a.d 1 5.b even 2 1
525.4.a.e 1 3.b odd 2 1
525.4.d.f 2 15.e even 4 2
735.4.a.c 1 105.g even 2 1
1575.4.a.f 1 1.a even 1 1 trivial
1680.4.a.s 1 60.h even 2 1
2205.4.a.o 1 35.c odd 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1575))\):

\( T_{2} \)
\( T_{11} + 42 \)
\( T_{13} + 20 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 8 T^{2} \)
$3$ 1
$5$ 1
$7$ \( 1 + 7 T \)
$11$ \( 1 + 42 T + 1331 T^{2} \)
$13$ \( 1 + 20 T + 2197 T^{2} \)
$17$ \( 1 - 66 T + 4913 T^{2} \)
$19$ \( 1 - 38 T + 6859 T^{2} \)
$23$ \( 1 - 12 T + 12167 T^{2} \)
$29$ \( 1 - 258 T + 24389 T^{2} \)
$31$ \( 1 - 146 T + 29791 T^{2} \)
$37$ \( 1 + 434 T + 50653 T^{2} \)
$41$ \( 1 - 282 T + 68921 T^{2} \)
$43$ \( 1 + 20 T + 79507 T^{2} \)
$47$ \( 1 + 72 T + 103823 T^{2} \)
$53$ \( 1 - 336 T + 148877 T^{2} \)
$59$ \( 1 - 360 T + 205379 T^{2} \)
$61$ \( 1 + 682 T + 226981 T^{2} \)
$67$ \( 1 + 812 T + 300763 T^{2} \)
$71$ \( 1 + 810 T + 357911 T^{2} \)
$73$ \( 1 - 124 T + 389017 T^{2} \)
$79$ \( 1 - 1136 T + 493039 T^{2} \)
$83$ \( 1 - 156 T + 571787 T^{2} \)
$89$ \( 1 - 1038 T + 704969 T^{2} \)
$97$ \( 1 + 1208 T + 912673 T^{2} \)
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