Properties

Label 4624.2.a.bm
Level $4624$
Weight $2$
Character orbit 4624.a
Self dual yes
Analytic conductor $36.923$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 4624 = 2^{4} \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4624.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(36.9228258946\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{16})^+\)
Defining polynomial: \(x^{4} - 4 x^{2} + 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 136)
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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} -\beta_{3} q^{5} + ( -2 \beta_{1} - \beta_{3} ) q^{7} + ( -1 + \beta_{2} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} -\beta_{3} q^{5} + ( -2 \beta_{1} - \beta_{3} ) q^{7} + ( -1 + \beta_{2} ) q^{9} + \beta_{1} q^{11} -\beta_{2} q^{15} + ( 2 + 3 \beta_{2} ) q^{19} + ( -4 - 3 \beta_{2} ) q^{21} + ( -\beta_{1} - 4 \beta_{3} ) q^{23} + ( -3 - \beta_{2} ) q^{25} + ( -3 \beta_{1} + \beta_{3} ) q^{27} -3 \beta_{3} q^{29} + 3 \beta_{1} q^{31} + ( 2 + \beta_{2} ) q^{33} + ( 2 + \beta_{2} ) q^{35} + \beta_{1} q^{37} + ( 4 \beta_{1} + 3 \beta_{3} ) q^{41} + ( 6 - 3 \beta_{2} ) q^{43} + ( -\beta_{1} + 2 \beta_{3} ) q^{45} + ( 10 - 2 \beta_{2} ) q^{47} + ( 3 + 7 \beta_{2} ) q^{49} + ( 4 + 5 \beta_{2} ) q^{53} -\beta_{2} q^{55} + ( 5 \beta_{1} + 3 \beta_{3} ) q^{57} + ( 2 + 7 \beta_{2} ) q^{59} + ( -4 \beta_{1} - 7 \beta_{3} ) q^{61} -\beta_{1} q^{63} + ( 4 - 4 \beta_{2} ) q^{67} + ( -2 - 5 \beta_{2} ) q^{69} + ( -3 \beta_{1} + 4 \beta_{3} ) q^{71} + ( 8 \beta_{1} - \beta_{3} ) q^{73} + ( -4 \beta_{1} - \beta_{3} ) q^{75} + ( -4 - 3 \beta_{2} ) q^{77} + ( 9 \beta_{1} - 4 \beta_{3} ) q^{79} + ( -3 - 5 \beta_{2} ) q^{81} + ( 6 - 3 \beta_{2} ) q^{83} -3 \beta_{2} q^{87} + ( -4 - 4 \beta_{2} ) q^{89} + ( 6 + 3 \beta_{2} ) q^{93} + ( -3 \beta_{1} + \beta_{3} ) q^{95} + ( 4 \beta_{1} - 7 \beta_{3} ) q^{97} + \beta_{3} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{9} + O(q^{10}) \) \( 4 q - 4 q^{9} + 8 q^{19} - 16 q^{21} - 12 q^{25} + 8 q^{33} + 8 q^{35} + 24 q^{43} + 40 q^{47} + 12 q^{49} + 16 q^{53} + 8 q^{59} + 16 q^{67} - 8 q^{69} - 16 q^{77} - 12 q^{81} + 24 q^{83} - 16 q^{89} + 24 q^{93} + O(q^{100}) \)

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
−1.84776
−0.765367
0.765367
1.84776
0 −1.84776 0 0.765367 0 4.46088 0 0.414214 0
1.2 0 −0.765367 0 −1.84776 0 −0.317025 0 −2.41421 0
1.3 0 0.765367 0 1.84776 0 0.317025 0 −2.41421 0
1.4 0 1.84776 0 −0.765367 0 −4.46088 0 0.414214 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(17\) \(-1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4624.2.a.bm 4
4.b odd 2 1 2312.2.a.s 4
17.b even 2 1 inner 4624.2.a.bm 4
17.e odd 16 2 272.2.v.e 4
68.d odd 2 1 2312.2.a.s 4
68.f odd 4 2 2312.2.b.j 4
68.i even 16 2 136.2.n.a 4
204.t odd 16 2 1224.2.bq.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
136.2.n.a 4 68.i even 16 2
272.2.v.e 4 17.e odd 16 2
1224.2.bq.a 4 204.t odd 16 2
2312.2.a.s 4 4.b odd 2 1
2312.2.a.s 4 68.d odd 2 1
2312.2.b.j 4 68.f odd 4 2
4624.2.a.bm 4 1.a even 1 1 trivial
4624.2.a.bm 4 17.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4624))\):

\( T_{3}^{4} - 4 T_{3}^{2} + 2 \)
\( T_{5}^{4} - 4 T_{5}^{2} + 2 \)
\( T_{7}^{4} - 20 T_{7}^{2} + 2 \)
\( T_{13} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 2 - 4 T^{2} + T^{4} \)
$5$ \( 2 - 4 T^{2} + T^{4} \)
$7$ \( 2 - 20 T^{2} + T^{4} \)
$11$ \( 2 - 4 T^{2} + T^{4} \)
$13$ \( T^{4} \)
$17$ \( T^{4} \)
$19$ \( ( -14 - 4 T + T^{2} )^{2} \)
$23$ \( 1058 - 68 T^{2} + T^{4} \)
$29$ \( 162 - 36 T^{2} + T^{4} \)
$31$ \( 162 - 36 T^{2} + T^{4} \)
$37$ \( 2 - 4 T^{2} + T^{4} \)
$41$ \( 578 - 100 T^{2} + T^{4} \)
$43$ \( ( 18 - 12 T + T^{2} )^{2} \)
$47$ \( ( 92 - 20 T + T^{2} )^{2} \)
$53$ \( ( -34 - 8 T + T^{2} )^{2} \)
$59$ \( ( -94 - 4 T + T^{2} )^{2} \)
$61$ \( 15842 - 260 T^{2} + T^{4} \)
$67$ \( ( -16 - 8 T + T^{2} )^{2} \)
$71$ \( 578 - 100 T^{2} + T^{4} \)
$73$ \( 12482 - 260 T^{2} + T^{4} \)
$79$ \( 37538 - 388 T^{2} + T^{4} \)
$83$ \( ( 18 - 12 T + T^{2} )^{2} \)
$89$ \( ( -16 + 8 T + T^{2} )^{2} \)
$97$ \( 1058 - 260 T^{2} + T^{4} \)
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