Properties

Label 1638.4.a.d
Level $1638$
Weight $4$
Character orbit 1638.a
Self dual yes
Analytic conductor $96.645$
Analytic rank $1$
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: \(1\)
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

\(f(q)\) \(=\) \( q - 2q^{2} + 4q^{4} - 3q^{5} + 7q^{7} - 8q^{8} + O(q^{10}) \) \( q - 2q^{2} + 4q^{4} - 3q^{5} + 7q^{7} - 8q^{8} + 6q^{10} + 54q^{11} + 13q^{13} - 14q^{14} + 16q^{16} + 96q^{17} - 151q^{19} - 12q^{20} - 108q^{22} - 33q^{23} - 116q^{25} - 26q^{26} + 28q^{28} - 183q^{29} - 331q^{31} - 32q^{32} - 192q^{34} - 21q^{35} - 88q^{37} + 302q^{38} + 24q^{40} + 42q^{41} + 353q^{43} + 216q^{44} + 66q^{46} + 465q^{47} + 49q^{49} + 232q^{50} + 52q^{52} - 195q^{53} - 162q^{55} - 56q^{56} + 366q^{58} - 552q^{59} + 470q^{61} + 662q^{62} + 64q^{64} - 39q^{65} + 254q^{67} + 384q^{68} + 42q^{70} - 132q^{71} - 943q^{73} + 176q^{74} - 604q^{76} + 378q^{77} - 727q^{79} - 48q^{80} - 84q^{82} + 1197q^{83} - 288q^{85} - 706q^{86} - 432q^{88} - 753q^{89} + 91q^{91} - 132q^{92} - 930q^{94} + 453q^{95} + 1037q^{97} - 98q^{98} + O(q^{100}) \)

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 −3.00000 0 7.00000 −8.00000 0 6.00000
\(n\): 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.d 1
3.b odd 2 1 182.4.a.b 1
12.b even 2 1 1456.4.a.g 1
21.c even 2 1 1274.4.a.i 1
39.d odd 2 1 2366.4.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
182.4.a.b 1 3.b odd 2 1
1274.4.a.i 1 21.c even 2 1
1456.4.a.g 1 12.b even 2 1
1638.4.a.d 1 1.a even 1 1 trivial
2366.4.a.a 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1638))\):

\( T_{5} + 3 \)
\( T_{11} - 54 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 2 + T \)
$3$ \( T \)
$5$ \( 3 + T \)
$7$ \( -7 + T \)
$11$ \( -54 + T \)
$13$ \( -13 + T \)
$17$ \( -96 + T \)
$19$ \( 151 + T \)
$23$ \( 33 + T \)
$29$ \( 183 + T \)
$31$ \( 331 + T \)
$37$ \( 88 + T \)
$41$ \( -42 + T \)
$43$ \( -353 + T \)
$47$ \( -465 + T \)
$53$ \( 195 + T \)
$59$ \( 552 + T \)
$61$ \( -470 + T \)
$67$ \( -254 + T \)
$71$ \( 132 + T \)
$73$ \( 943 + T \)
$79$ \( 727 + T \)
$83$ \( -1197 + T \)
$89$ \( 753 + T \)
$97$ \( -1037 + T \)
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