Properties

Label 1638.4.a.a
Level $1638$
Weight $4$
Character orbit 1638.a
Self dual yes
Analytic conductor $96.645$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1638.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(96.6451285894\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 182)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 2 q^{2} + 4 q^{4} - 16 q^{5} + 7 q^{7} - 8 q^{8} + O(q^{10}) \) \( q - 2 q^{2} + 4 q^{4} - 16 q^{5} + 7 q^{7} - 8 q^{8} + 32 q^{10} + 15 q^{11} - 13 q^{13} - 14 q^{14} + 16 q^{16} + 44 q^{17} - 138 q^{19} - 64 q^{20} - 30 q^{22} - 111 q^{23} + 131 q^{25} + 26 q^{26} + 28 q^{28} + 12 q^{29} + 215 q^{31} - 32 q^{32} - 88 q^{34} - 112 q^{35} + 55 q^{37} + 276 q^{38} + 128 q^{40} + 133 q^{41} - 180 q^{43} + 60 q^{44} + 222 q^{46} - 471 q^{47} + 49 q^{49} - 262 q^{50} - 52 q^{52} + 260 q^{53} - 240 q^{55} - 56 q^{56} - 24 q^{58} - 110 q^{59} - 271 q^{61} - 430 q^{62} + 64 q^{64} + 208 q^{65} - 799 q^{67} + 176 q^{68} + 224 q^{70} - 912 q^{71} + 747 q^{73} - 110 q^{74} - 552 q^{76} + 105 q^{77} - 883 q^{79} - 256 q^{80} - 266 q^{82} + 924 q^{83} - 704 q^{85} + 360 q^{86} - 120 q^{88} - 142 q^{89} - 91 q^{91} - 444 q^{92} + 942 q^{94} + 2208 q^{95} - 1407 q^{97} - 98 q^{98} + O(q^{100}) \)

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
−2.00000 0 4.00000 −16.0000 0 7.00000 −8.00000 0 32.0000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(7\) \(-1\)
\(13\) \(1\)

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 1638.4.a.a 1
3.b odd 2 1 182.4.a.d 1
12.b even 2 1 1456.4.a.c 1
21.c even 2 1 1274.4.a.c 1
39.d odd 2 1 2366.4.a.e 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
182.4.a.d 1 3.b odd 2 1
1274.4.a.c 1 21.c even 2 1
1456.4.a.c 1 12.b even 2 1
1638.4.a.a 1 1.a even 1 1 trivial
2366.4.a.e 1 39.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1638))\):

\( T_{5} + 16 \)
\( T_{11} - 15 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 2 + T \)
$3$ \( T \)
$5$ \( 16 + T \)
$7$ \( -7 + T \)
$11$ \( -15 + T \)
$13$ \( 13 + T \)
$17$ \( -44 + T \)
$19$ \( 138 + T \)
$23$ \( 111 + T \)
$29$ \( -12 + T \)
$31$ \( -215 + T \)
$37$ \( -55 + T \)
$41$ \( -133 + T \)
$43$ \( 180 + T \)
$47$ \( 471 + T \)
$53$ \( -260 + T \)
$59$ \( 110 + T \)
$61$ \( 271 + T \)
$67$ \( 799 + T \)
$71$ \( 912 + T \)
$73$ \( -747 + T \)
$79$ \( 883 + T \)
$83$ \( -924 + T \)
$89$ \( 142 + T \)
$97$ \( 1407 + T \)
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