Properties

Label 4864.2.a.bm
Level $4864$
Weight $2$
Character orbit 4864.a
Self dual yes
Analytic conductor $38.839$
Analytic rank $1$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 4864 = 2^{8} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4864.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(38.8392355432\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: 8.8.34309996544.1
Defining polynomial: \(x^{8} - 4 x^{7} - 8 x^{6} + 28 x^{5} + 31 x^{4} - 36 x^{3} - 22 x^{2} + 12 x + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 2432)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} + \beta_{6} q^{5} + \beta_{4} q^{7} + ( -\beta_{1} + \beta_{7} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} + \beta_{6} q^{5} + \beta_{4} q^{7} + ( -\beta_{1} + \beta_{7} ) q^{9} + ( -2 + \beta_{1} - \beta_{7} ) q^{11} + ( -\beta_{2} + \beta_{4} + \beta_{5} + \beta_{6} ) q^{13} + ( -\beta_{4} - \beta_{5} - 2 \beta_{6} ) q^{15} + ( -2 - \beta_{1} - \beta_{3} ) q^{17} + q^{19} + ( \beta_{2} - 2 \beta_{4} - \beta_{6} ) q^{21} + ( \beta_{2} - \beta_{4} - \beta_{5} - \beta_{6} ) q^{23} + ( 3 - \beta_{1} + 2 \beta_{3} - \beta_{7} ) q^{25} + ( \beta_{3} - \beta_{7} ) q^{27} + ( -2 \beta_{4} - \beta_{5} - \beta_{6} ) q^{29} + ( \beta_{4} + \beta_{5} ) q^{31} + ( -5 \beta_{1} - \beta_{3} + \beta_{7} ) q^{33} + ( -4 - 3 \beta_{1} - 2 \beta_{3} + 3 \beta_{7} ) q^{35} + ( \beta_{2} - \beta_{4} + 2 \beta_{5} ) q^{37} + ( -\beta_{4} + 2 \beta_{5} - \beta_{6} ) q^{39} + ( 2 + \beta_{1} + \beta_{3} + \beta_{7} ) q^{41} + ( -2 + \beta_{1} + 2 \beta_{3} - 3 \beta_{7} ) q^{43} + ( 2 \beta_{4} + \beta_{6} ) q^{45} + ( \beta_{2} - 2 \beta_{4} - \beta_{5} - 3 \beta_{6} ) q^{47} + ( -1 - 3 \beta_{1} - \beta_{3} ) q^{49} + ( -1 + \beta_{3} - 2 \beta_{7} ) q^{51} + ( -\beta_{2} - 2 \beta_{5} - \beta_{6} ) q^{53} + ( -2 \beta_{4} - 3 \beta_{6} ) q^{55} + \beta_{1} q^{57} + ( -3 + 3 \beta_{1} - \beta_{7} ) q^{59} + ( -2 \beta_{4} - 2 \beta_{5} - \beta_{6} ) q^{61} + ( -2 \beta_{2} + 2 \beta_{4} + 3 \beta_{6} ) q^{63} + ( -2 - 4 \beta_{1} - 2 \beta_{3} ) q^{65} + ( -5 - 2 \beta_{1} - \beta_{3} + 2 \beta_{7} ) q^{67} + ( \beta_{4} - 2 \beta_{5} + \beta_{6} ) q^{69} + ( -2 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} ) q^{71} + ( 1 + 2 \beta_{1} + 2 \beta_{3} - 2 \beta_{7} ) q^{73} + ( -10 - 3 \beta_{3} + \beta_{7} ) q^{75} + ( 2 \beta_{2} - 4 \beta_{4} - 3 \beta_{6} ) q^{77} + ( 2 \beta_{2} + \beta_{4} + 3 \beta_{5} ) q^{79} + ( -5 - 2 \beta_{3} - 2 \beta_{7} ) q^{81} + ( -6 - 2 \beta_{3} + 2 \beta_{7} ) q^{83} + ( -2 \beta_{2} + 2 \beta_{4} - \beta_{6} ) q^{85} + ( -\beta_{2} + 3 \beta_{4} - \beta_{5} + 3 \beta_{6} ) q^{87} + ( -\beta_{1} + 3 \beta_{3} - \beta_{7} ) q^{89} + ( -1 - 5 \beta_{1} + 3 \beta_{7} ) q^{91} + 2 \beta_{5} q^{93} + \beta_{6} q^{95} + ( 4 - 2 \beta_{1} ) q^{97} + ( -4 + 5 \beta_{1} + 2 \beta_{3} - 3 \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 4q^{3} + 4q^{9} + O(q^{10}) \) \( 8q - 4q^{3} + 4q^{9} - 20q^{11} - 8q^{17} + 8q^{19} + 20q^{25} - 4q^{27} + 24q^{33} - 12q^{35} + 8q^{41} - 28q^{43} + 8q^{49} - 12q^{51} - 4q^{57} - 36q^{59} + 8q^{65} - 28q^{67} - 8q^{73} - 68q^{75} - 32q^{81} - 40q^{83} - 8q^{89} + 12q^{91} + 40q^{97} - 60q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 4 x^{7} - 8 x^{6} + 28 x^{5} + 31 x^{4} - 36 x^{3} - 22 x^{2} + 12 x + 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -5 \nu^{7} + 37 \nu^{6} - 39 \nu^{5} - 205 \nu^{4} + 256 \nu^{3} + 412 \nu^{2} - 126 \nu - 162 \)\()/52\)
\(\beta_{2}\)\(=\)\((\)\( -5 \nu^{7} + 37 \nu^{6} - 39 \nu^{5} - 205 \nu^{4} + 308 \nu^{3} + 308 \nu^{2} - 438 \nu - 58 \)\()/52\)
\(\beta_{3}\)\(=\)\((\)\( -9 \nu^{7} + 25 \nu^{6} + 117 \nu^{5} - 161 \nu^{4} - 600 \nu^{3} - 28 \nu^{2} + 522 \nu + 10 \)\()/52\)
\(\beta_{4}\)\(=\)\((\)\( -5 \nu^{7} + 24 \nu^{6} + 13 \nu^{5} - 114 \nu^{4} - 30 \nu^{3} - 4 \nu^{2} - 22 \nu + 98 \)\()/26\)
\(\beta_{5}\)\(=\)\((\)\( 15 \nu^{7} - 59 \nu^{6} - 91 \nu^{5} + 251 \nu^{4} + 428 \nu^{3} + 220 \nu^{2} - 142 \nu - 138 \)\()/52\)
\(\beta_{6}\)\(=\)\((\)\( 10 \nu^{7} - 35 \nu^{6} - 104 \nu^{5} + 267 \nu^{4} + 424 \nu^{3} - 304 \nu^{2} - 268 \nu + 38 \)\()/26\)
\(\beta_{7}\)\(=\)\((\)\( 23 \nu^{7} - 87 \nu^{6} - 195 \nu^{5} + 579 \nu^{4} + 736 \nu^{3} - 460 \nu^{2} - 294 \nu + 38 \)\()/52\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-2 \beta_{7} + 2 \beta_{6} + \beta_{5} + \beta_{3} - \beta_{2} + \beta_{1} + 3\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(-2 \beta_{7} + 2 \beta_{6} - 2 \beta_{4} + \beta_{3} + \beta_{1} + 9\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(-10 \beta_{7} + 10 \beta_{6} + 3 \beta_{5} - 4 \beta_{4} + 5 \beta_{3} - \beta_{2} + 3 \beta_{1} + 23\)\()/2\)
\(\nu^{4}\)\(=\)\(-15 \beta_{7} + 16 \beta_{6} + 3 \beta_{5} - 10 \beta_{4} + 10 \beta_{3} + \beta_{2} + 5 \beta_{1} + 48\)
\(\nu^{5}\)\(=\)\((\)\(-116 \beta_{7} + 124 \beta_{6} + 35 \beta_{5} - 62 \beta_{4} + 83 \beta_{3} + 11 \beta_{2} + 31 \beta_{1} + 323\)\()/2\)
\(\nu^{6}\)\(=\)\(-199 \beta_{7} + 222 \beta_{6} + 60 \beta_{5} - 120 \beta_{4} + 167 \beta_{3} + 38 \beta_{2} + 54 \beta_{1} + 611\)
\(\nu^{7}\)\(=\)\(-731 \beta_{7} + 829 \beta_{6} + 255 \beta_{5} - 421 \beta_{4} + 665 \beta_{3} + 178 \beta_{2} + 175 \beta_{1} + 2202\)

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
−1.58857
−0.139869
0.545005
3.79159
−0.836706
3.15093
0.904430
−1.82681
0 −2.63640 0 −2.63403 0 −3.88603 0 3.95063 0
1.2 0 −2.63640 0 2.63403 0 3.88603 0 3.95063 0
1.3 0 −1.65222 0 −4.30533 0 2.73423 0 −0.270160 0
1.4 0 −1.65222 0 4.30533 0 −2.73423 0 −0.270160 0
1.5 0 0.222191 0 −1.55081 0 3.06334 0 −2.95063 0
1.6 0 0.222191 0 1.55081 0 −3.06334 0 −2.95063 0
1.7 0 2.06644 0 −1.45635 0 −0.196723 0 1.27016 0
1.8 0 2.06644 0 1.45635 0 0.196723 0 1.27016 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4864.2.a.bm 8
4.b odd 2 1 4864.2.a.br 8
8.b even 2 1 4864.2.a.br 8
8.d odd 2 1 inner 4864.2.a.bm 8
16.e even 4 2 2432.2.c.i 16
16.f odd 4 2 2432.2.c.i 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2432.2.c.i 16 16.e even 4 2
2432.2.c.i 16 16.f odd 4 2
4864.2.a.bm 8 1.a even 1 1 trivial
4864.2.a.bm 8 8.d odd 2 1 inner
4864.2.a.br 8 4.b odd 2 1
4864.2.a.br 8 8.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4864))\):

\( T_{3}^{4} + 2 T_{3}^{3} - 5 T_{3}^{2} - 8 T_{3} + 2 \)
\( T_{5}^{8} - 30 T_{5}^{6} + 249 T_{5}^{4} - 712 T_{5}^{2} + 656 \)
\( T_{7}^{8} - 32 T_{7}^{6} + 326 T_{7}^{4} - 1072 T_{7}^{2} + 41 \)
\( T_{11}^{4} + 10 T_{11}^{3} + 25 T_{11}^{2} - 32 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( ( 2 - 8 T - 5 T^{2} + 2 T^{3} + T^{4} )^{2} \)
$5$ \( 656 - 712 T^{2} + 249 T^{4} - 30 T^{6} + T^{8} \)
$7$ \( 41 - 1072 T^{2} + 326 T^{4} - 32 T^{6} + T^{8} \)
$11$ \( ( -32 + 25 T^{2} + 10 T^{3} + T^{4} )^{2} \)
$13$ \( 2624 - 9232 T^{2} + 1505 T^{4} - 74 T^{6} + T^{8} \)
$17$ \( ( -7 + 2 T + T^{2} )^{4} \)
$19$ \( ( -1 + T )^{8} \)
$23$ \( 2624 - 9232 T^{2} + 1505 T^{4} - 74 T^{6} + T^{8} \)
$29$ \( 164 - 30124 T^{2} + 3325 T^{4} - 110 T^{6} + T^{8} \)
$31$ \( 2624 - 2400 T^{2} + 660 T^{4} - 52 T^{6} + T^{8} \)
$37$ \( 4410944 - 673056 T^{2} + 21460 T^{4} - 252 T^{6} + T^{8} \)
$41$ \( ( 32 + 48 T - 30 T^{2} - 4 T^{3} + T^{4} )^{2} \)
$43$ \( ( -1756 - 1172 T - 59 T^{2} + 14 T^{3} + T^{4} )^{2} \)
$47$ \( 758336 - 154704 T^{2} + 10233 T^{4} - 230 T^{6} + T^{8} \)
$53$ \( 393764 - 178164 T^{2} + 11485 T^{4} - 198 T^{6} + T^{8} \)
$59$ \( ( -4 - 20 T + 69 T^{2} + 18 T^{3} + T^{4} )^{2} \)
$61$ \( 189584 - 124040 T^{2} + 8265 T^{4} - 174 T^{6} + T^{8} \)
$67$ \( ( -562 - 260 T + 23 T^{2} + 14 T^{3} + T^{4} )^{2} \)
$71$ \( 52910336 - 3418112 T^{2} + 64864 T^{4} - 448 T^{6} + T^{8} \)
$73$ \( ( -191 - 252 T - 74 T^{2} + 4 T^{3} + T^{4} )^{2} \)
$79$ \( 13983296 - 2761568 T^{2} + 66900 T^{4} - 468 T^{6} + T^{8} \)
$83$ \( ( -64 - 32 T + 52 T^{2} + 20 T^{3} + T^{4} )^{2} \)
$89$ \( ( -1864 - 1192 T - 158 T^{2} + 4 T^{3} + T^{4} )^{2} \)
$97$ \( ( -32 - 224 T + 124 T^{2} - 20 T^{3} + T^{4} )^{2} \)
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