Properties

Label 1216.4.a.a
Level $1216$
Weight $4$
Character orbit 1216.a
Self dual yes
Analytic conductor $71.746$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1216 = 2^{6} \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1216.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(71.7463225670\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 19)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 5q^{3} + 12q^{5} - 11q^{7} - 2q^{9} + O(q^{10}) \) \( q - 5q^{3} + 12q^{5} - 11q^{7} - 2q^{9} - 54q^{11} - 11q^{13} - 60q^{15} - 93q^{17} + 19q^{19} + 55q^{21} - 183q^{23} + 19q^{25} + 145q^{27} + 249q^{29} - 56q^{31} + 270q^{33} - 132q^{35} + 250q^{37} + 55q^{39} + 240q^{41} - 196q^{43} - 24q^{45} + 168q^{47} - 222q^{49} + 465q^{51} - 435q^{53} - 648q^{55} - 95q^{57} + 195q^{59} + 358q^{61} + 22q^{63} - 132q^{65} - 961q^{67} + 915q^{69} + 246q^{71} + 353q^{73} - 95q^{75} + 594q^{77} + 34q^{79} - 671q^{81} + 234q^{83} - 1116q^{85} - 1245q^{87} - 168q^{89} + 121q^{91} + 280q^{93} + 228q^{95} + 758q^{97} + 108q^{99} + O(q^{100}) \)

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 −5.00000 0 12.0000 0 −11.0000 0 −2.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(19\) \(-1\)

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 1216.4.a.a 1
4.b odd 2 1 1216.4.a.f 1
8.b even 2 1 304.4.a.b 1
8.d odd 2 1 19.4.a.a 1
24.f even 2 1 171.4.a.d 1
40.e odd 2 1 475.4.a.e 1
40.k even 4 2 475.4.b.c 2
56.e even 2 1 931.4.a.a 1
88.g even 2 1 2299.4.a.b 1
152.b even 2 1 361.4.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.4.a.a 1 8.d odd 2 1
171.4.a.d 1 24.f even 2 1
304.4.a.b 1 8.b even 2 1
361.4.a.b 1 152.b even 2 1
475.4.a.e 1 40.e odd 2 1
475.4.b.c 2 40.k even 4 2
931.4.a.a 1 56.e even 2 1
1216.4.a.a 1 1.a even 1 1 trivial
1216.4.a.f 1 4.b odd 2 1
2299.4.a.b 1 88.g even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1216))\):

\( T_{3} + 5 \)
\( T_{5} - 12 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( 5 + T \)
$5$ \( -12 + T \)
$7$ \( 11 + T \)
$11$ \( 54 + T \)
$13$ \( 11 + T \)
$17$ \( 93 + T \)
$19$ \( -19 + T \)
$23$ \( 183 + T \)
$29$ \( -249 + T \)
$31$ \( 56 + T \)
$37$ \( -250 + T \)
$41$ \( -240 + T \)
$43$ \( 196 + T \)
$47$ \( -168 + T \)
$53$ \( 435 + T \)
$59$ \( -195 + T \)
$61$ \( -358 + T \)
$67$ \( 961 + T \)
$71$ \( -246 + T \)
$73$ \( -353 + T \)
$79$ \( -34 + T \)
$83$ \( -234 + T \)
$89$ \( 168 + T \)
$97$ \( -758 + T \)
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