Properties

Label 675.4.a.h
Level $675$
Weight $4$
Character orbit 675.a
Self dual yes
Analytic conductor $39.826$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 675 = 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 675.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(39.8262892539\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 135)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{2} - 7 q^{4} + 6 q^{7} - 15 q^{8} + O(q^{10}) \) \( q + q^{2} - 7 q^{4} + 6 q^{7} - 15 q^{8} + 47 q^{11} + 5 q^{13} + 6 q^{14} + 41 q^{16} - 131 q^{17} - 56 q^{19} + 47 q^{22} + 3 q^{23} + 5 q^{26} - 42 q^{28} + 157 q^{29} + 225 q^{31} + 161 q^{32} - 131 q^{34} + 70 q^{37} - 56 q^{38} - 140 q^{41} - 397 q^{43} - 329 q^{44} + 3 q^{46} - 347 q^{47} - 307 q^{49} - 35 q^{52} + 4 q^{53} - 90 q^{56} + 157 q^{58} - 748 q^{59} - 338 q^{61} + 225 q^{62} - 167 q^{64} - 492 q^{67} + 917 q^{68} - 32 q^{71} - 970 q^{73} + 70 q^{74} + 392 q^{76} + 282 q^{77} - 1257 q^{79} - 140 q^{82} - 102 q^{83} - 397 q^{86} - 705 q^{88} + 1488 q^{89} + 30 q^{91} - 21 q^{92} - 347 q^{94} - 974 q^{97} - 307 q^{98} + O(q^{100}) \)

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 −7.00000 0 0 6.00000 −15.0000 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 675.4.a.h 1
3.b odd 2 1 675.4.a.c 1
5.b even 2 1 135.4.a.b 1
5.c odd 4 2 675.4.b.f 2
15.d odd 2 1 135.4.a.c yes 1
15.e even 4 2 675.4.b.e 2
20.d odd 2 1 2160.4.a.f 1
45.h odd 6 2 405.4.e.f 2
45.j even 6 2 405.4.e.h 2
60.h even 2 1 2160.4.a.p 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
135.4.a.b 1 5.b even 2 1
135.4.a.c yes 1 15.d odd 2 1
405.4.e.f 2 45.h odd 6 2
405.4.e.h 2 45.j even 6 2
675.4.a.c 1 3.b odd 2 1
675.4.a.h 1 1.a even 1 1 trivial
675.4.b.e 2 15.e even 4 2
675.4.b.f 2 5.c odd 4 2
2160.4.a.f 1 20.d odd 2 1
2160.4.a.p 1 60.h even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(675))\):

\( T_{2} - 1 \)
\( T_{7} - 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 + T \)
$3$ \( T \)
$5$ \( T \)
$7$ \( -6 + T \)
$11$ \( -47 + T \)
$13$ \( -5 + T \)
$17$ \( 131 + T \)
$19$ \( 56 + T \)
$23$ \( -3 + T \)
$29$ \( -157 + T \)
$31$ \( -225 + T \)
$37$ \( -70 + T \)
$41$ \( 140 + T \)
$43$ \( 397 + T \)
$47$ \( 347 + T \)
$53$ \( -4 + T \)
$59$ \( 748 + T \)
$61$ \( 338 + T \)
$67$ \( 492 + T \)
$71$ \( 32 + T \)
$73$ \( 970 + T \)
$79$ \( 1257 + T \)
$83$ \( 102 + T \)
$89$ \( -1488 + T \)
$97$ \( 974 + T \)
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