Properties

Label 8004.2.a.d
Level $8004$
Weight $2$
Character orbit 8004.a
Self dual yes
Analytic conductor $63.912$
Analytic rank $1$
Dimension $8$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8004.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(63.9122617778\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - 3 x^{7} - 8 x^{6} + 19 x^{5} + 19 x^{4} - 35 x^{3} - 10 x^{2} + 18 x - 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + q^{3} + ( -1 - \beta_{7} ) q^{5} + ( \beta_{6} + \beta_{7} ) q^{7} + q^{9} +O(q^{10})\) \( q + q^{3} + ( -1 - \beta_{7} ) q^{5} + ( \beta_{6} + \beta_{7} ) q^{7} + q^{9} + ( -\beta_{3} + \beta_{7} ) q^{11} + ( -1 + \beta_{4} ) q^{13} + ( -1 - \beta_{7} ) q^{15} + ( -1 + 2 \beta_{1} + \beta_{3} - \beta_{4} - \beta_{6} ) q^{17} + ( -1 - \beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{19} + ( \beta_{6} + \beta_{7} ) q^{21} + q^{23} + ( 1 + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{7} ) q^{25} + q^{27} + q^{29} + ( -\beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{31} + ( -\beta_{3} + \beta_{7} ) q^{33} + ( -3 - \beta_{1} - \beta_{2} + \beta_{4} + \beta_{5} ) q^{35} + ( -2 - \beta_{2} - \beta_{3} + 2 \beta_{5} + \beta_{6} + \beta_{7} ) q^{37} + ( -1 + \beta_{4} ) q^{39} + ( -1 - 4 \beta_{1} + \beta_{3} + \beta_{5} - \beta_{7} ) q^{41} + ( -1 - \beta_{1} - 2 \beta_{4} + 2 \beta_{5} - \beta_{6} - \beta_{7} ) q^{43} + ( -1 - \beta_{7} ) q^{45} + ( -2 + 2 \beta_{2} - \beta_{4} + 2 \beta_{5} - 3 \beta_{6} - \beta_{7} ) q^{47} + ( -3 + \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - 3 \beta_{6} - 2 \beta_{7} ) q^{49} + ( -1 + 2 \beta_{1} + \beta_{3} - \beta_{4} - \beta_{6} ) q^{51} + ( -1 - 4 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} ) q^{53} + ( -3 + \beta_{2} + 2 \beta_{3} - \beta_{6} - 2 \beta_{7} ) q^{55} + ( -1 - \beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{57} + ( 2 - 3 \beta_{3} + 3 \beta_{4} - 2 \beta_{5} + 4 \beta_{6} + 2 \beta_{7} ) q^{59} + ( -\beta_{1} + \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - \beta_{6} - \beta_{7} ) q^{61} + ( \beta_{6} + \beta_{7} ) q^{63} + ( 2 + \beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} ) q^{65} + ( -1 - 2 \beta_{1} - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + 3 \beta_{6} ) q^{67} + q^{69} + ( 3 \beta_{1} + \beta_{2} + 3 \beta_{3} - 3 \beta_{4} - 3 \beta_{6} + \beta_{7} ) q^{71} + ( -5 + 5 \beta_{1} - 4 \beta_{2} - \beta_{3} + \beta_{6} + 3 \beta_{7} ) q^{73} + ( 1 + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{7} ) q^{75} + ( \beta_{1} - \beta_{2} - \beta_{4} - \beta_{5} - \beta_{6} ) q^{77} + ( -2 + 4 \beta_{1} - \beta_{2} - 3 \beta_{4} + \beta_{5} - 3 \beta_{6} + \beta_{7} ) q^{79} + q^{81} + ( 2 - 2 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} - 2 \beta_{5} + 7 \beta_{6} + 4 \beta_{7} ) q^{83} + ( -\beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} + \beta_{7} ) q^{85} + q^{87} + ( -2 + \beta_{1} + \beta_{2} + 6 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - \beta_{6} ) q^{89} + ( -2 + 2 \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} + \beta_{6} ) q^{91} + ( -\beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{93} + ( 1 + 2 \beta_{1} - 2 \beta_{5} + 3 \beta_{7} ) q^{95} + ( 3 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - \beta_{5} - 4 \beta_{6} - 3 \beta_{7} ) q^{97} + ( -\beta_{3} + \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 8q^{3} - 5q^{5} - 4q^{7} + 8q^{9} + O(q^{10}) \) \( 8q + 8q^{3} - 5q^{5} - 4q^{7} + 8q^{9} - 5q^{11} - 4q^{13} - 5q^{15} - 3q^{17} - 5q^{19} - 4q^{21} + 8q^{23} - 5q^{25} + 8q^{27} + 8q^{29} - 2q^{31} - 5q^{33} - 15q^{35} - 10q^{37} - 4q^{39} - 11q^{41} - 7q^{43} - 5q^{45} - 14q^{47} - 18q^{49} - 3q^{51} - 15q^{53} - 17q^{55} - 5q^{57} + 4q^{59} + q^{61} - 4q^{63} - 5q^{67} + 8q^{69} - q^{71} - 21q^{73} - 5q^{75} - 8q^{79} + 8q^{81} + 3q^{83} + 8q^{87} - 20q^{89} - 7q^{91} - 2q^{93} - 3q^{95} - 7q^{97} - 5q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 3 x^{7} - 8 x^{6} + 19 x^{5} + 19 x^{4} - 35 x^{3} - 10 x^{2} + 18 x - 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 5 \nu^{7} - 13 \nu^{6} - 46 \nu^{5} + 79 \nu^{4} + 131 \nu^{3} - 131 \nu^{2} - 108 \nu + 50 \)\()/2\)
\(\beta_{3}\)\(=\)\((\)\( -5 \nu^{7} + 13 \nu^{6} + 46 \nu^{5} - 79 \nu^{4} - 131 \nu^{3} + 133 \nu^{2} + 106 \nu - 56 \)\()/2\)
\(\beta_{4}\)\(=\)\( 5 \nu^{7} - 13 \nu^{6} - 45 \nu^{5} + 76 \nu^{4} + 126 \nu^{3} - 122 \nu^{2} - 102 \nu + 50 \)
\(\beta_{5}\)\(=\)\((\)\( -11 \nu^{7} + 29 \nu^{6} + 98 \nu^{5} - 171 \nu^{4} - 269 \nu^{3} + 273 \nu^{2} + 210 \nu - 104 \)\()/2\)
\(\beta_{6}\)\(=\)\((\)\( -13 \nu^{7} + 33 \nu^{6} + 120 \nu^{5} - 195 \nu^{4} - 339 \nu^{3} + 317 \nu^{2} + 272 \nu - 132 \)\()/2\)
\(\beta_{7}\)\(=\)\( 7 \nu^{7} - 18 \nu^{6} - 64 \nu^{5} + 107 \nu^{4} + 179 \nu^{3} - 175 \nu^{2} - 143 \nu + 71 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + \beta_{2} + \beta_{1} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + 6 \beta_{1} + 3\)
\(\nu^{4}\)\(=\)\(5 \beta_{7} + 4 \beta_{6} + 3 \beta_{5} + 2 \beta_{4} + 10 \beta_{3} + 9 \beta_{2} + 16 \beta_{1} + 20\)
\(\nu^{5}\)\(=\)\(20 \beta_{7} + 17 \beta_{6} + 14 \beta_{5} + 12 \beta_{4} + 31 \beta_{3} + 26 \beta_{2} + 63 \beta_{1} + 48\)
\(\nu^{6}\)\(=\)\(77 \beta_{7} + 60 \beta_{6} + 48 \beta_{5} + 34 \beta_{4} + 120 \beta_{3} + 98 \beta_{2} + 211 \beta_{1} + 199\)
\(\nu^{7}\)\(=\)\(279 \beta_{7} + 223 \beta_{6} + 180 \beta_{5} + 141 \beta_{4} + 413 \beta_{3} + 326 \beta_{2} + 766 \beta_{1} + 633\)

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
−1.85152
0.386133
−1.03502
3.55127
−1.58311
1.24887
0.411238
1.87214
0 1.00000 0 −3.41642 0 2.05098 0 1.00000 0
1.2 0 1.00000 0 −2.77471 0 0.556399 0 1.00000 0
1.3 0 1.00000 0 −1.84132 0 −2.79101 0 1.00000 0
1.4 0 1.00000 0 −1.29538 0 2.07218 0 1.00000 0
1.5 0 1.00000 0 −0.270840 0 0.409136 0 1.00000 0
1.6 0 1.00000 0 0.138622 0 −3.33662 0 1.00000 0
1.7 0 1.00000 0 1.71910 0 0.210007 0 1.00000 0
1.8 0 1.00000 0 2.74094 0 −3.17108 0 1.00000 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
\(2\) \(-1\)
\(3\) \(-1\)
\(23\) \(-1\)
\(29\) \(-1\)

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 8004.2.a.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8004.2.a.d 8 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8004))\):

\(T_{5}^{8} + \cdots\)
\(T_{7}^{8} + \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( ( -1 + T )^{8} \)
$5$ \( -4 + 10 T + 122 T^{2} + 107 T^{3} - 35 T^{4} - 52 T^{5} - 5 T^{6} + 5 T^{7} + T^{8} \)
$7$ \( -6 + 54 T - 145 T^{2} + 104 T^{3} + 56 T^{4} - 41 T^{5} - 11 T^{6} + 4 T^{7} + T^{8} \)
$11$ \( -96 - 264 T + 203 T^{2} + 273 T^{3} - 51 T^{4} - 91 T^{5} - 12 T^{6} + 5 T^{7} + T^{8} \)
$13$ \( -1 - 18 T + 9 T^{2} + 103 T^{3} + 57 T^{4} - 45 T^{5} - 14 T^{6} + 4 T^{7} + T^{8} \)
$17$ \( -461 + 76 T + 1521 T^{2} + 1318 T^{3} + 207 T^{4} - 127 T^{5} - 34 T^{6} + 3 T^{7} + T^{8} \)
$19$ \( -3418 - 5137 T - 311 T^{2} + 1799 T^{3} + 381 T^{4} - 187 T^{5} - 45 T^{6} + 5 T^{7} + T^{8} \)
$23$ \( ( -1 + T )^{8} \)
$29$ \( ( -1 + T )^{8} \)
$31$ \( 67344 + 56346 T - 50966 T^{2} - 6831 T^{3} + 5804 T^{4} - 5 T^{5} - 144 T^{6} + 2 T^{7} + T^{8} \)
$37$ \( -2944 - 12715 T + 8258 T^{2} + 5604 T^{3} - 940 T^{4} - 677 T^{5} - 45 T^{6} + 10 T^{7} + T^{8} \)
$41$ \( 969324 - 332022 T - 115226 T^{2} + 36043 T^{3} + 4993 T^{4} - 1142 T^{5} - 111 T^{6} + 11 T^{7} + T^{8} \)
$43$ \( 11216 - 48741 T + 2005 T^{2} + 11113 T^{3} + 534 T^{4} - 721 T^{5} - 86 T^{6} + 7 T^{7} + T^{8} \)
$47$ \( -54 - 1260 T - 5905 T^{2} - 9897 T^{3} - 5917 T^{4} - 1292 T^{5} - 46 T^{6} + 14 T^{7} + T^{8} \)
$53$ \( 43448 + 230316 T + 173102 T^{2} + 28941 T^{3} - 8435 T^{4} - 2774 T^{5} - 127 T^{6} + 15 T^{7} + T^{8} \)
$59$ \( 2324416 + 32512 T - 357392 T^{2} - 11272 T^{3} + 15760 T^{4} + 548 T^{5} - 227 T^{6} - 4 T^{7} + T^{8} \)
$61$ \( -712512 - 619824 T - 90868 T^{2} + 46313 T^{3} + 12123 T^{4} - 319 T^{5} - 206 T^{6} - T^{7} + T^{8} \)
$67$ \( -8 - 1988 T + 12017 T^{2} - 19118 T^{3} + 7472 T^{4} - 278 T^{5} - 169 T^{6} + 5 T^{7} + T^{8} \)
$71$ \( -5344884 - 3294291 T - 377525 T^{2} + 106124 T^{3} + 19819 T^{4} - 900 T^{5} - 259 T^{6} + T^{7} + T^{8} \)
$73$ \( 7960224 - 4849590 T + 42434 T^{2} + 237401 T^{3} + 126 T^{4} - 3903 T^{5} - 105 T^{6} + 21 T^{7} + T^{8} \)
$79$ \( 15902 - 10257 T - 17776 T^{2} + 5080 T^{3} + 3614 T^{4} - 569 T^{5} - 191 T^{6} + 8 T^{7} + T^{8} \)
$83$ \( -3977348 + 3091546 T - 275138 T^{2} - 176597 T^{3} + 28215 T^{4} + 1478 T^{5} - 325 T^{6} - 3 T^{7} + T^{8} \)
$89$ \( 194617089 - 61327428 T - 5179001 T^{2} + 1326091 T^{3} + 69665 T^{4} - 9161 T^{5} - 448 T^{6} + 20 T^{7} + T^{8} \)
$97$ \( -85056 - 245328 T - 45478 T^{2} + 40017 T^{3} + 8911 T^{4} - 1061 T^{5} - 184 T^{6} + 7 T^{7} + T^{8} \)
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