Properties

Label 1216.2.a.x
Level $1216$
Weight $2$
Character orbit 1216.a
Self dual yes
Analytic conductor $9.710$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

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

Newform invariants

Self dual: yes
Analytic conductor: \(9.70980888579\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.15317.1
Defining polynomial: \(x^{4} - 2 x^{3} - 4 x^{2} + 5 x + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 608)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 + \beta_{2} ) q^{3} + ( -\beta_{1} + \beta_{2} ) q^{5} -\beta_{3} q^{7} + ( 2 + \beta_{2} ) q^{9} +O(q^{10})\) \( q + ( 1 + \beta_{2} ) q^{3} + ( -\beta_{1} + \beta_{2} ) q^{5} -\beta_{3} q^{7} + ( 2 + \beta_{2} ) q^{9} + ( 2 - \beta_{1} + \beta_{2} ) q^{11} + ( -2 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{13} + ( 2 - 2 \beta_{1} + 2 \beta_{3} ) q^{15} + ( 1 + \beta_{1} - 2 \beta_{3} ) q^{17} + q^{19} + ( -2 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{21} + ( 2 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{23} + ( 5 - \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{25} + ( 3 - \beta_{2} ) q^{27} + ( -2 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{29} + ( 2 + 2 \beta_{2} - 2 \beta_{3} ) q^{31} + ( 4 - 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{33} + ( \beta_{1} - 3 \beta_{2} ) q^{35} + ( -4 - 2 \beta_{2} + 2 \beta_{3} ) q^{37} + ( 2 + 3 \beta_{1} - 3 \beta_{2} - \beta_{3} ) q^{39} + 2 \beta_{1} q^{41} + ( -\beta_{1} - \beta_{2} ) q^{43} + ( 2 - 3 \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{45} + ( 2 + \beta_{1} + 3 \beta_{2} - 2 \beta_{3} ) q^{47} + ( -\beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{49} + ( -1 + 4 \beta_{1} - \beta_{2} ) q^{51} + ( -2 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{53} + ( 10 - 3 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} ) q^{55} + ( 1 + \beta_{2} ) q^{57} + ( 5 + 2 \beta_{1} - \beta_{2} ) q^{59} + ( 2 - \beta_{1} + 3 \beta_{2} ) q^{61} + ( -2 + \beta_{1} - \beta_{2} ) q^{63} + ( -6 + 2 \beta_{1} - 8 \beta_{2} + 2 \beta_{3} ) q^{65} + ( 1 - 2 \beta_{1} - \beta_{2} ) q^{67} + ( 2 + \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{69} + ( 8 + 2 \beta_{3} ) q^{71} + ( 3 + \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{73} + ( 11 - 4 \beta_{1} + 7 \beta_{2} ) q^{75} + ( \beta_{1} - 3 \beta_{2} - 2 \beta_{3} ) q^{77} + ( -4 - 4 \beta_{2} + 2 \beta_{3} ) q^{79} -7 q^{81} + ( 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{83} + ( -8 + \beta_{1} - 7 \beta_{2} ) q^{85} + ( 2 + 3 \beta_{1} - 3 \beta_{2} - \beta_{3} ) q^{87} + ( -6 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{89} + ( 3 - \beta_{2} + 4 \beta_{3} ) q^{91} + ( 6 + 2 \beta_{1} + 2 \beta_{3} ) q^{93} + ( -\beta_{1} + \beta_{2} ) q^{95} + ( -2 + 2 \beta_{1} - 2 \beta_{2} ) q^{97} + ( 6 - 3 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{3} - q^{5} + q^{7} + 6q^{9} + O(q^{10}) \) \( 4q + 2q^{3} - q^{5} + q^{7} + 6q^{9} + 7q^{11} - 10q^{13} + 8q^{15} + 5q^{17} + 4q^{19} - 8q^{21} + 8q^{23} + 17q^{25} + 14q^{27} - 10q^{29} + 6q^{31} + 12q^{33} + 5q^{35} - 14q^{37} + 12q^{39} - 2q^{41} + 3q^{43} + 7q^{45} + 3q^{47} + 7q^{49} - 6q^{51} - 4q^{53} + 35q^{55} + 2q^{57} + 20q^{59} + 3q^{61} - 7q^{63} - 12q^{65} + 8q^{67} + 4q^{69} + 30q^{71} + 9q^{73} + 34q^{75} + 7q^{77} - 10q^{79} - 28q^{81} + 4q^{83} - 19q^{85} + 12q^{87} - 16q^{89} + 10q^{91} + 20q^{93} - q^{95} - 6q^{97} + 19q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 2 x^{3} - 4 x^{2} + 5 x + 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{3} - \nu^{2} - 3 \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - \nu^{2} - 5 \nu + 1 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{3} + \beta_{1} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{3} + 2 \beta_{2} + \beta_{1} + 7\)\()/2\)
\(\nu^{3}\)\(=\)\(-2 \beta_{3} + \beta_{2} + 3 \beta_{1} + 5\)

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
−0.329727
1.32973
2.69353
−1.69353
0 −1.56155 0 −3.40617 0 −2.50407 0 −0.561553 0
1.2 0 −1.56155 0 0.844614 0 5.06562 0 −0.561553 0
1.3 0 2.56155 0 −2.64453 0 0.180969 0 3.56155 0
1.4 0 2.56155 0 4.20608 0 −1.74252 0 3.56155 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 1216.2.a.x 4
4.b odd 2 1 1216.2.a.w 4
8.b even 2 1 608.2.a.i 4
8.d odd 2 1 608.2.a.j yes 4
24.f even 2 1 5472.2.a.bs 4
24.h odd 2 1 5472.2.a.bt 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
608.2.a.i 4 8.b even 2 1
608.2.a.j yes 4 8.d odd 2 1
1216.2.a.w 4 4.b odd 2 1
1216.2.a.x 4 1.a even 1 1 trivial
5472.2.a.bs 4 24.f even 2 1
5472.2.a.bt 4 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1216))\):

\( T_{3}^{2} - T_{3} - 4 \)
\( T_{5}^{4} + T_{5}^{3} - 18 T_{5}^{2} - 24 T_{5} + 32 \)
\( T_{7}^{4} - T_{7}^{3} - 17 T_{7}^{2} - 19 T_{7} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( -4 - T + T^{2} )^{2} \)
$5$ \( 32 - 24 T - 18 T^{2} + T^{3} + T^{4} \)
$7$ \( 4 - 19 T - 17 T^{2} - T^{3} + T^{4} \)
$11$ \( 16 + 28 T - 7 T^{3} + T^{4} \)
$13$ \( -344 - 158 T + 7 T^{2} + 10 T^{3} + T^{4} \)
$17$ \( 698 + 141 T - 51 T^{2} - 5 T^{3} + T^{4} \)
$19$ \( ( -1 + T )^{4} \)
$23$ \( 64 + 132 T - 17 T^{2} - 8 T^{3} + T^{4} \)
$29$ \( -344 - 158 T + 7 T^{2} + 10 T^{3} + T^{4} \)
$31$ \( -512 + 416 T - 56 T^{2} - 6 T^{3} + T^{4} \)
$37$ \( -1504 - 536 T + 4 T^{2} + 14 T^{3} + T^{4} \)
$41$ \( 512 - 192 T - 72 T^{2} + 2 T^{3} + T^{4} \)
$43$ \( -64 + 96 T - 32 T^{2} - 3 T^{3} + T^{4} \)
$47$ \( -64 + 176 T - 108 T^{2} - 3 T^{3} + T^{4} \)
$53$ \( -508 - 384 T - 69 T^{2} + 4 T^{3} + T^{4} \)
$59$ \( -152 + 150 T + 85 T^{2} - 20 T^{3} + T^{4} \)
$61$ \( 344 + 28 T - 66 T^{2} - 3 T^{3} + T^{4} \)
$67$ \( -824 + 602 T - 75 T^{2} - 8 T^{3} + T^{4} \)
$71$ \( -2432 - 424 T + 268 T^{2} - 30 T^{3} + T^{4} \)
$73$ \( 2 + 309 T - 47 T^{2} - 9 T^{3} + T^{4} \)
$79$ \( -3136 - 1288 T - 100 T^{2} + 10 T^{3} + T^{4} \)
$83$ \( -1024 + 784 T - 116 T^{2} - 4 T^{3} + T^{4} \)
$89$ \( -15296 - 4048 T - 204 T^{2} + 16 T^{3} + T^{4} \)
$97$ \( 608 - 88 T - 60 T^{2} + 6 T^{3} + T^{4} \)
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