Properties

Label 1216.4.a.c
Level $1216$
Weight $4$
Character orbit 1216.a
Self dual yes
Analytic conductor $71.746$
Analytic rank $1$
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: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 608)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{3} + 8q^{5} + 17q^{7} - 26q^{9} + O(q^{10}) \) \( q - q^{3} + 8q^{5} + 17q^{7} - 26q^{9} - 70q^{11} + 61q^{13} - 8q^{15} + 83q^{17} + 19q^{19} - 17q^{21} - 115q^{23} - 61q^{25} + 53q^{27} - 279q^{29} + 72q^{31} + 70q^{33} + 136q^{35} + 34q^{37} - 61q^{39} + 108q^{41} - 192q^{43} - 208q^{45} + 392q^{47} - 54q^{49} - 83q^{51} - 131q^{53} - 560q^{55} - 19q^{57} - 609q^{59} - 338q^{61} - 442q^{63} + 488q^{65} - 461q^{67} + 115q^{69} - 750q^{71} + 1177q^{73} + 61q^{75} - 1190q^{77} + 22q^{79} + 649q^{81} - 810q^{83} + 664q^{85} + 279q^{87} - 476q^{89} + 1037q^{91} - 72q^{93} + 152q^{95} + 1426q^{97} + 1820q^{99} + O(q^{100}) \)

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 −1.00000 0 8.00000 0 17.0000 0 −26.0000 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.c 1
4.b odd 2 1 1216.4.a.d 1
8.b even 2 1 608.4.a.b yes 1
8.d odd 2 1 608.4.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
608.4.a.a 1 8.d odd 2 1
608.4.a.b yes 1 8.b even 2 1
1216.4.a.c 1 1.a even 1 1 trivial
1216.4.a.d 1 4.b odd 2 1

Hecke kernels

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

\( T_{3} + 1 \)
\( T_{5} - 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( 1 + T \)
$5$ \( -8 + T \)
$7$ \( -17 + T \)
$11$ \( 70 + T \)
$13$ \( -61 + T \)
$17$ \( -83 + T \)
$19$ \( -19 + T \)
$23$ \( 115 + T \)
$29$ \( 279 + T \)
$31$ \( -72 + T \)
$37$ \( -34 + T \)
$41$ \( -108 + T \)
$43$ \( 192 + T \)
$47$ \( -392 + T \)
$53$ \( 131 + T \)
$59$ \( 609 + T \)
$61$ \( 338 + T \)
$67$ \( 461 + T \)
$71$ \( 750 + T \)
$73$ \( -1177 + T \)
$79$ \( -22 + T \)
$83$ \( 810 + T \)
$89$ \( 476 + T \)
$97$ \( -1426 + T \)
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