Properties

Label 8004.2.a.e
Level $8004$
Weight $2$
Character orbit 8004.a
Self dual yes
Analytic conductor $63.912$
Analytic rank $1$
Dimension $9$
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: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Defining polynomial: \(x^{9} - 3 x^{8} - 13 x^{7} + 32 x^{6} + 40 x^{5} - 79 x^{4} - 39 x^{3} + 58 x^{2} + 9 x - 11\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2 \)
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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{3} + \beta_{1} q^{5} + ( 1 - \beta_{3} ) q^{7} + q^{9} +O(q^{10})\) \( q - q^{3} + \beta_{1} q^{5} + ( 1 - \beta_{3} ) q^{7} + q^{9} + ( \beta_{3} + \beta_{5} + \beta_{6} + \beta_{8} ) q^{11} + ( -1 - \beta_{1} + \beta_{4} - \beta_{5} - \beta_{7} ) q^{13} -\beta_{1} q^{15} + ( \beta_{3} + \beta_{5} + \beta_{7} ) q^{17} + ( \beta_{1} - \beta_{3} - \beta_{4} - \beta_{7} ) q^{19} + ( -1 + \beta_{3} ) q^{21} - q^{23} + ( -\beta_{2} - \beta_{4} + \beta_{7} - \beta_{8} ) q^{25} - q^{27} + q^{29} + ( -2 \beta_{1} + 2 \beta_{4} - \beta_{5} - 2 \beta_{6} - \beta_{8} ) q^{31} + ( -\beta_{3} - \beta_{5} - \beta_{6} - \beta_{8} ) q^{33} + ( -1 + \beta_{2} + \beta_{3} - \beta_{6} + \beta_{7} ) q^{35} + ( -1 - \beta_{1} + \beta_{2} - \beta_{4} + \beta_{6} ) q^{37} + ( 1 + \beta_{1} - \beta_{4} + \beta_{5} + \beta_{7} ) q^{39} + ( -2 + \beta_{2} - 2 \beta_{3} - \beta_{5} - \beta_{6} - \beta_{8} ) q^{41} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{4} - 3 \beta_{5} + \beta_{7} ) q^{43} + \beta_{1} q^{45} + ( \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{6} - \beta_{8} ) q^{47} + ( -2 - 2 \beta_{3} - \beta_{6} ) q^{49} + ( -\beta_{3} - \beta_{5} - \beta_{7} ) q^{51} + ( -3 - \beta_{4} - \beta_{5} - \beta_{6} + 2 \beta_{8} ) q^{53} + ( 1 - 2 \beta_{1} - \beta_{2} + 2 \beta_{4} - \beta_{5} + \beta_{7} - \beta_{8} ) q^{55} + ( -\beta_{1} + \beta_{3} + \beta_{4} + \beta_{7} ) q^{57} + ( -1 + \beta_{3} - \beta_{4} + \beta_{6} + \beta_{7} - \beta_{8} ) q^{59} + ( -1 - \beta_{1} + \beta_{3} - \beta_{4} - \beta_{7} + \beta_{8} ) q^{61} + ( 1 - \beta_{3} ) q^{63} + ( -3 - 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{5} - \beta_{6} - \beta_{7} ) q^{65} + ( 2 + \beta_{1} - 2 \beta_{2} + \beta_{3} + 3 \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} ) q^{67} + q^{69} + ( -1 - \beta_{1} + \beta_{5} - \beta_{7} - \beta_{8} ) q^{71} + ( -\beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{73} + ( \beta_{2} + \beta_{4} - \beta_{7} + \beta_{8} ) q^{75} + ( \beta_{1} - \beta_{2} + 2 \beta_{3} + 4 \beta_{5} + 3 \beta_{6} + \beta_{7} ) q^{77} + ( -2 - \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - \beta_{5} - \beta_{6} - 2 \beta_{8} ) q^{79} + q^{81} + ( -1 - \beta_{1} + 2 \beta_{2} + 4 \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} ) q^{83} + ( 1 - 2 \beta_{2} - 3 \beta_{3} + 2 \beta_{4} - \beta_{5} + 2 \beta_{6} - \beta_{7} + \beta_{8} ) q^{85} - q^{87} + ( -3 \beta_{1} - 2 \beta_{2} + 2 \beta_{4} + \beta_{5} + 3 \beta_{6} - \beta_{7} + 2 \beta_{8} ) q^{89} + ( -3 - \beta_{1} + \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{8} ) q^{91} + ( 2 \beta_{1} - 2 \beta_{4} + \beta_{5} + 2 \beta_{6} + \beta_{8} ) q^{93} + ( 2 + \beta_{1} + 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} - 2 \beta_{8} ) q^{95} + ( -2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 3 \beta_{5} - 3 \beta_{7} - \beta_{8} ) q^{97} + ( \beta_{3} + \beta_{5} + \beta_{6} + \beta_{8} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9q - 9q^{3} - 3q^{5} + 7q^{7} + 9q^{9} + O(q^{10}) \) \( 9q - 9q^{3} - 3q^{5} + 7q^{7} + 9q^{9} - 2q^{11} - 7q^{13} + 3q^{15} - q^{19} - 7q^{21} - 9q^{23} + 2q^{25} - 9q^{27} + 9q^{29} + 8q^{31} + 2q^{33} - 5q^{35} - 8q^{37} + 7q^{39} - 19q^{41} - 3q^{43} - 3q^{45} - 3q^{47} - 18q^{49} - 17q^{53} + 9q^{55} + q^{57} - 10q^{59} + q^{61} + 7q^{63} - 16q^{65} + 12q^{67} + 9q^{69} - 7q^{71} + 13q^{73} - 2q^{75} - 15q^{77} - 10q^{79} + 9q^{81} + 9q^{83} - 6q^{85} - 9q^{87} - 5q^{89} - 18q^{91} - 8q^{93} + 31q^{95} - 7q^{97} - 2q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{9} - 3 x^{8} - 13 x^{7} + 32 x^{6} + 40 x^{5} - 79 x^{4} - 39 x^{3} + 58 x^{2} + 9 x - 11\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 93 \nu^{8} - 7 \nu^{7} - 1649 \nu^{6} - 1742 \nu^{5} + 8484 \nu^{4} + 13848 \nu^{3} - 21492 \nu^{2} - 13414 \nu + 16720 \)\()/4877\)
\(\beta_{2}\)\(=\)\((\)\( -676 \nu^{8} - 54 \nu^{7} + 14451 \nu^{6} + 5373 \nu^{5} - 80862 \nu^{4} - 26245 \nu^{3} + 118464 \nu^{2} + 29593 \nu - 29868 \)\()/4877\)
\(\beta_{3}\)\(=\)\((\)\( -1248 \nu^{8} + 4027 \nu^{7} + 15049 \nu^{6} - 42227 \nu^{5} - 37863 \nu^{4} + 93731 \nu^{3} + 26624 \nu^{2} - 41031 \nu - 4120 \)\()/4877\)
\(\beta_{4}\)\(=\)\((\)\( 1613 \nu^{8} - 4474 \nu^{7} - 21416 \nu^{6} + 45249 \nu^{5} + 67542 \nu^{4} - 97748 \nu^{3} - 66924 \nu^{2} + 53254 \nu + 16986 \)\()/4877\)
\(\beta_{5}\)\(=\)\((\)\( 1706 \nu^{8} - 4481 \nu^{7} - 23065 \nu^{6} + 43507 \nu^{5} + 76026 \nu^{4} - 83900 \nu^{3} - 83539 \nu^{2} + 34963 \nu + 19075 \)\()/4877\)
\(\beta_{6}\)\(=\)\((\)\( -2082 \nu^{8} + 5663 \nu^{7} + 27005 \nu^{6} - 53822 \nu^{5} - 79649 \nu^{4} + 92100 \nu^{3} + 68801 \nu^{2} - 18907 \nu - 12313 \)\()/4877\)
\(\beta_{7}\)\(=\)\((\)\( 2159 \nu^{8} - 5931 \nu^{7} - 29524 \nu^{6} + 60980 \nu^{5} + 102091 \nu^{4} - 136012 \nu^{3} - 122465 \nu^{2} + 64804 \nu + 35858 \)\()/4877\)
\(\beta_{8}\)\(=\)\((\)\( 2535 \nu^{8} - 7113 \nu^{7} - 33464 \nu^{6} + 71295 \nu^{5} + 105714 \nu^{4} - 144212 \nu^{3} - 107727 \nu^{2} + 58502 \nu + 24219 \)\()/4877\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{8} - \beta_{7} + \beta_{6} + \beta_{5} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{8} - \beta_{7} + \beta_{6} + 3 \beta_{5} - 2 \beta_{4} - 2 \beta_{1} + 7\)\()/2\)
\(\nu^{3}\)\(=\)\(4 \beta_{8} - 5 \beta_{7} + 5 \beta_{6} + 6 \beta_{5} - 3 \beta_{4} - 4 \beta_{3} - \beta_{2} + 7\)
\(\nu^{4}\)\(=\)\((\)\(13 \beta_{8} - 19 \beta_{7} + 23 \beta_{6} + 53 \beta_{5} - 32 \beta_{4} - 14 \beta_{3} - 2 \beta_{2} - 18 \beta_{1} + 75\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(83 \beta_{8} - 129 \beta_{7} + 135 \beta_{6} + 193 \beta_{5} - 100 \beta_{4} - 130 \beta_{3} - 30 \beta_{2} - 14 \beta_{1} + 225\)\()/2\)
\(\nu^{6}\)\(=\)\(96 \beta_{8} - 174 \beta_{7} + 201 \beta_{6} + 413 \beta_{5} - 231 \beta_{4} - 167 \beta_{3} - 29 \beta_{2} - 99 \beta_{1} + 520\)
\(\nu^{7}\)\(=\)\((\)\(1033 \beta_{8} - 1823 \beta_{7} + 1943 \beta_{6} + 3085 \beta_{5} - 1588 \beta_{4} - 1928 \beta_{3} - 424 \beta_{2} - 342 \beta_{1} + 3591\)\()/2\)
\(\nu^{8}\)\(=\)\((\)\(3035 \beta_{8} - 5877 \beta_{7} + 6591 \beta_{6} + 12709 \beta_{5} - 6834 \beta_{4} - 6034 \beta_{3} - 1142 \beta_{2} - 2514 \beta_{1} + 15401\)\()/2\)

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
−2.86352
−1.47135
3.93556
2.25739
0.761439
−0.975163
1.39711
0.511914
−0.553378
0 −1.00000 0 −3.70748 0 −1.59469 0 1.00000 0
1.2 0 −1.00000 0 −3.48372 0 2.18462 0 1.00000 0
1.3 0 −1.00000 0 −2.36120 0 2.82157 0 1.00000 0
1.4 0 −1.00000 0 −0.650765 0 −1.33403 0 1.00000 0
1.5 0 −1.00000 0 0.461945 0 0.732054 0 1.00000 0
1.6 0 −1.00000 0 0.900888 0 3.90501 0 1.00000 0
1.7 0 −1.00000 0 1.20115 0 −1.61263 0 1.00000 0
1.8 0 −1.00000 0 1.34732 0 2.91839 0 1.00000 0
1.9 0 −1.00000 0 3.29187 0 −1.02028 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
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.e 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8004.2.a.e 9 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8004))\):

\(T_{5}^{9} + \cdots\)
\(T_{7}^{9} - \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{9} \)
$3$ \( ( 1 + T )^{9} \)
$5$ \( -44 + 116 T + 66 T^{2} - 314 T^{3} + 117 T^{4} + 117 T^{5} - 44 T^{6} - 19 T^{7} + 3 T^{8} + T^{9} \)
$7$ \( -180 - 36 T + 418 T^{2} + 138 T^{3} - 245 T^{4} - 53 T^{5} + 65 T^{6} + 2 T^{7} - 7 T^{8} + T^{9} \)
$11$ \( -496 + 2768 T - 2876 T^{2} - 3252 T^{3} + 879 T^{4} + 738 T^{5} - 81 T^{6} - 50 T^{7} + 2 T^{8} + T^{9} \)
$13$ \( 11717 - 3991 T - 9799 T^{2} + 1086 T^{3} + 2742 T^{4} + 126 T^{5} - 259 T^{6} - 30 T^{7} + 7 T^{8} + T^{9} \)
$17$ \( 107 + 1235 T + 3855 T^{2} + 5125 T^{3} + 3129 T^{4} + 666 T^{5} - 115 T^{6} - 55 T^{7} + T^{9} \)
$19$ \( 2047 + 3651 T - 7106 T^{2} - 8782 T^{3} + 360 T^{4} + 1620 T^{5} - 38 T^{6} - 74 T^{7} + T^{8} + T^{9} \)
$23$ \( ( 1 + T )^{9} \)
$29$ \( ( -1 + T )^{9} \)
$31$ \( 333252 + 677160 T + 227862 T^{2} - 83112 T^{3} - 28327 T^{4} + 5072 T^{5} + 887 T^{6} - 126 T^{7} - 8 T^{8} + T^{9} \)
$37$ \( 236939 + 17654 T - 103193 T^{2} - 10160 T^{3} + 13397 T^{4} + 1657 T^{5} - 603 T^{6} - 80 T^{7} + 8 T^{8} + T^{9} \)
$41$ \( -47916 + 137244 T + 194794 T^{2} + 58442 T^{3} - 12795 T^{4} - 9313 T^{5} - 1344 T^{6} + 31 T^{7} + 19 T^{8} + T^{9} \)
$43$ \( -2401487 + 3031193 T + 1931464 T^{2} - 795889 T^{3} - 25400 T^{4} + 24434 T^{5} - 232 T^{6} - 267 T^{7} + 3 T^{8} + T^{9} \)
$47$ \( -24692 + 51072 T + 11386 T^{2} - 43380 T^{3} + 1481 T^{4} + 5176 T^{5} - 151 T^{6} - 145 T^{7} + 3 T^{8} + T^{9} \)
$53$ \( 3308752 + 4727792 T + 2570424 T^{2} + 594496 T^{3} + 14717 T^{4} - 19079 T^{5} - 3156 T^{6} - 79 T^{7} + 17 T^{8} + T^{9} \)
$59$ \( 1772000 + 414448 T - 428480 T^{2} - 62852 T^{3} + 33402 T^{4} + 3755 T^{5} - 1016 T^{6} - 104 T^{7} + 10 T^{8} + T^{9} \)
$61$ \( 576 + 7488 T + 21424 T^{2} - 18480 T^{3} - 3467 T^{4} + 3253 T^{5} + 207 T^{6} - 134 T^{7} - T^{8} + T^{9} \)
$67$ \( 99286848 + 52981632 T + 2512680 T^{2} - 2390880 T^{3} - 223495 T^{4} + 39839 T^{5} + 3197 T^{6} - 329 T^{7} - 12 T^{8} + T^{9} \)
$71$ \( -3875 - 39919 T - 85437 T^{2} - 21210 T^{3} + 16544 T^{4} + 3278 T^{5} - 647 T^{6} - 106 T^{7} + 7 T^{8} + T^{9} \)
$73$ \( -18180 - 114408 T + 231246 T^{2} + 108816 T^{3} - 41263 T^{4} - 7516 T^{5} + 2845 T^{6} - 151 T^{7} - 13 T^{8} + T^{9} \)
$79$ \( 21425 + 510976 T + 1684309 T^{2} + 1191902 T^{3} + 283161 T^{4} + 10439 T^{5} - 3605 T^{6} - 286 T^{7} + 10 T^{8} + T^{9} \)
$83$ \( 69516 - 588396 T - 1219862 T^{2} - 609114 T^{3} - 18339 T^{4} + 27593 T^{5} + 1634 T^{6} - 325 T^{7} - 9 T^{8} + T^{9} \)
$89$ \( -61987357 + 61570299 T + 4599889 T^{2} - 4809342 T^{3} + 21438 T^{4} + 84560 T^{5} - 1243 T^{6} - 510 T^{7} + 5 T^{8} + T^{9} \)
$97$ \( -15236672 + 27880896 T + 15146896 T^{2} - 6719040 T^{3} + 126147 T^{4} + 109753 T^{5} - 2879 T^{6} - 604 T^{7} + 7 T^{8} + T^{9} \)
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