Properties

Label 1575.4.a.d
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 525)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 2q^{2} - 4q^{4} + 7q^{7} + 24q^{8} + O(q^{10}) \) \( q - 2q^{2} - 4q^{4} + 7q^{7} + 24q^{8} + 21q^{11} - 24q^{13} - 14q^{14} - 16q^{16} - 22q^{17} + 16q^{19} - 42q^{22} - 25q^{23} + 48q^{26} - 28q^{28} - 167q^{29} + 10q^{31} - 160q^{32} + 44q^{34} + 133q^{37} - 32q^{38} + 168q^{41} + 97q^{43} - 84q^{44} + 50q^{46} - 400q^{47} + 49q^{49} + 96q^{52} - 182q^{53} + 168q^{56} + 334q^{58} - 488q^{59} + 28q^{61} - 20q^{62} + 448q^{64} + 967q^{67} + 88q^{68} + 285q^{71} + 838q^{73} - 266q^{74} - 64q^{76} + 147q^{77} - 469q^{79} - 336q^{82} - 406q^{83} - 194q^{86} + 504q^{88} - 324q^{89} - 168q^{91} + 100q^{92} + 800q^{94} + 114q^{97} - 98q^{98} + O(q^{100}) \)

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 −4.00000 0 0 7.00000 24.0000 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 1575.4.a.d 1
3.b odd 2 1 525.4.a.f yes 1
5.b even 2 1 1575.4.a.h 1
15.d odd 2 1 525.4.a.d 1
15.e even 4 2 525.4.d.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.4.a.d 1 15.d odd 2 1
525.4.a.f yes 1 3.b odd 2 1
525.4.d.e 2 15.e even 4 2
1575.4.a.d 1 1.a even 1 1 trivial
1575.4.a.h 1 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1575))\):

\( T_{2} + 2 \)
\( T_{11} - 21 \)
\( T_{13} + 24 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 2 + T \)
$3$ \( T \)
$5$ \( T \)
$7$ \( -7 + T \)
$11$ \( -21 + T \)
$13$ \( 24 + T \)
$17$ \( 22 + T \)
$19$ \( -16 + T \)
$23$ \( 25 + T \)
$29$ \( 167 + T \)
$31$ \( -10 + T \)
$37$ \( -133 + T \)
$41$ \( -168 + T \)
$43$ \( -97 + T \)
$47$ \( 400 + T \)
$53$ \( 182 + T \)
$59$ \( 488 + T \)
$61$ \( -28 + T \)
$67$ \( -967 + T \)
$71$ \( -285 + T \)
$73$ \( -838 + T \)
$79$ \( 469 + T \)
$83$ \( 406 + T \)
$89$ \( 324 + T \)
$97$ \( -114 + T \)
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