Properties

Label 6004.2.a.d
Level $6004$
Weight $2$
Character orbit 6004.a
Self dual yes
Analytic conductor $47.942$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6004 = 2^{2} \cdot 19 \cdot 79 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6004.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(47.9421813736\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - 4 x^{7} - 15 x^{6} + 56 x^{5} + 87 x^{4} - 248 x^{3} - 241 x^{2} + 340 x + 248\)
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{5} -\beta_{4} q^{7} -3 q^{9} +O(q^{10})\) \( q + \beta_{1} q^{5} -\beta_{4} q^{7} -3 q^{9} + ( -1 + \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{7} ) q^{11} + ( -2 + \beta_{1} - \beta_{2} - \beta_{4} - \beta_{6} ) q^{13} + ( -1 - \beta_{5} + \beta_{6} ) q^{17} + q^{19} + ( 2 + \beta_{2} - \beta_{3} + \beta_{7} ) q^{23} + ( \beta_{1} + \beta_{2} - \beta_{5} + \beta_{6} ) q^{25} + ( 2 \beta_{2} + \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{29} + ( -1 + 2 \beta_{1} - 3 \beta_{2} - \beta_{4} + \beta_{5} - 2 \beta_{7} ) q^{31} + ( -2 + 2 \beta_{1} - 3 \beta_{2} - 2 \beta_{4} + 2 \beta_{5} ) q^{35} + ( -1 + \beta_{1} - \beta_{2} - \beta_{3} - \beta_{6} + 2 \beta_{7} ) q^{37} + ( -1 + \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{5} - \beta_{6} ) q^{41} + ( 1 - \beta_{2} + \beta_{3} + \beta_{4} ) q^{43} -3 \beta_{1} q^{45} + ( 2 - 2 \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} ) q^{47} + ( 4 + \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{49} + ( -\beta_{1} + 2 \beta_{2} + 3 \beta_{3} + \beta_{5} ) q^{53} + ( -1 - 3 \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{7} ) q^{55} + ( 2 \beta_{1} - 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{7} ) q^{59} + ( 1 + 2 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{5} + \beta_{6} - \beta_{7} ) q^{61} + 3 \beta_{4} q^{63} + ( 2 - 3 \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} + 2 \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{65} + ( 3 - \beta_{1} - \beta_{5} + \beta_{6} ) q^{67} + ( 6 - \beta_{1} + \beta_{3} - \beta_{4} + 2 \beta_{5} - \beta_{6} + \beta_{7} ) q^{71} + ( -3 - \beta_{1} + 2 \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{73} + ( 1 + 6 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} - \beta_{5} - 3 \beta_{6} + \beta_{7} ) q^{77} + q^{79} + 9 q^{81} + ( \beta_{2} - \beta_{3} - 2 \beta_{5} + \beta_{7} ) q^{83} + ( -3 + 2 \beta_{3} - \beta_{5} + 3 \beta_{6} - 2 \beta_{7} ) q^{85} + ( -2 - \beta_{1} + 3 \beta_{2} + 2 \beta_{3} + 3 \beta_{4} - 2 \beta_{5} + \beta_{6} ) q^{89} + ( 6 + 5 \beta_{1} - 2 \beta_{2} - 5 \beta_{3} - \beta_{4} + 2 \beta_{5} - \beta_{6} - \beta_{7} ) q^{91} + \beta_{1} q^{95} + ( -4 - 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{5} - 2 \beta_{7} ) q^{97} + ( 3 - 3 \beta_{1} + 6 \beta_{2} + 3 \beta_{3} + 3 \beta_{4} - 3 \beta_{5} + 3 \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 4q^{5} + q^{7} - 24q^{9} + O(q^{10}) \) \( 8q + 4q^{5} + q^{7} - 24q^{9} - 7q^{11} - 12q^{13} - 7q^{17} + 8q^{19} + 10q^{23} + 6q^{25} + 5q^{29} + 3q^{31} - 11q^{35} - 15q^{37} - 5q^{41} + 10q^{43} - 12q^{45} + 18q^{47} + 31q^{49} + 9q^{53} - 17q^{55} + 8q^{59} + 13q^{61} - 3q^{63} + 4q^{65} + 21q^{67} + 44q^{71} - 20q^{73} + 15q^{77} + 8q^{79} + 72q^{81} - 4q^{83} - 9q^{85} - 10q^{89} + 48q^{91} + 4q^{95} - 22q^{97} + 21q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 4 x^{7} - 15 x^{6} + 56 x^{5} + 87 x^{4} - 248 x^{3} - 241 x^{2} + 340 x + 248\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{6} - 4 \nu^{5} - 10 \nu^{4} + 36 \nu^{3} + 37 \nu^{2} - 68 \nu - 56 \)\()/8\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{7} + 3 \nu^{6} + 14 \nu^{5} - 26 \nu^{4} - 73 \nu^{3} + 31 \nu^{2} + 140 \nu + 72 \)\()/16\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{7} - 4 \nu^{6} - 10 \nu^{5} + 40 \nu^{4} + 25 \nu^{3} - 104 \nu^{2} + 4 \nu + 64 \)\()/8\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{7} - 5 \nu^{6} - 8 \nu^{5} + 56 \nu^{4} + 7 \nu^{3} - 171 \nu^{2} + 32 \nu + 112 \)\()/8\)
\(\beta_{6}\)\(=\)\((\)\( \nu^{7} - 6 \nu^{6} - 4 \nu^{5} + 66 \nu^{4} - 29 \nu^{3} - 200 \nu^{2} + 92 \nu + 128 \)\()/8\)
\(\beta_{7}\)\(=\)\((\)\( \nu^{7} - 7 \nu^{6} + 76 \nu^{4} - 69 \nu^{3} - 229 \nu^{2} + 188 \nu + 160 \)\()/8\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{6} - \beta_{5} + \beta_{2} + \beta_{1} + 5\)
\(\nu^{3}\)\(=\)\(-2 \beta_{7} + 4 \beta_{6} - 2 \beta_{5} + 9 \beta_{1} + 4\)
\(\nu^{4}\)\(=\)\(-6 \beta_{7} + 21 \beta_{6} - 15 \beta_{5} + 2 \beta_{4} + 4 \beta_{3} + 11 \beta_{2} + 17 \beta_{1} + 37\)
\(\nu^{5}\)\(=\)\(-36 \beta_{7} + 84 \beta_{6} - 52 \beta_{5} + 10 \beta_{4} + 12 \beta_{3} + 14 \beta_{2} + 97 \beta_{1} + 68\)
\(\nu^{6}\)\(=\)\(-132 \beta_{7} + 365 \beta_{6} - 249 \beta_{5} + 60 \beta_{4} + 88 \beta_{3} + 137 \beta_{2} + 265 \beta_{1} + 369\)
\(\nu^{7}\)\(=\)\(-598 \beta_{7} + 1464 \beta_{6} - 970 \beta_{5} + 268 \beta_{4} + 312 \beta_{3} + 352 \beta_{2} + 1225 \beta_{1} + 1032\)

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
−2.49694
−1.94482
−1.76571
−0.654712
1.60188
2.45095
2.82169
3.98766
0 0 0 −2.49694 0 4.08498 0 −3.00000 0
1.2 0 0 0 −1.94482 0 −0.969069 0 −3.00000 0
1.3 0 0 0 −1.76571 0 2.40269 0 −3.00000 0
1.4 0 0 0 −0.654712 0 −2.24646 0 −3.00000 0
1.5 0 0 0 1.60188 0 −2.96099 0 −3.00000 0
1.6 0 0 0 2.45095 0 −5.04197 0 −3.00000 0
1.7 0 0 0 2.82169 0 4.85844 0 −3.00000 0
1.8 0 0 0 3.98766 0 0.872374 0 −3.00000 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
\(2\) \(-1\)
\(19\) \(-1\)
\(79\) \(-1\)

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 6004.2.a.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6004.2.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6004))\):

\( T_{3} \)
\(T_{5}^{8} - \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( T^{8} \)
$5$ \( 248 + 340 T - 241 T^{2} - 248 T^{3} + 87 T^{4} + 56 T^{5} - 15 T^{6} - 4 T^{7} + T^{8} \)
$7$ \( 1352 - 2032 T^{2} - 166 T^{3} + 530 T^{4} + 33 T^{5} - 43 T^{6} - T^{7} + T^{8} \)
$11$ \( 1792 - 8768 T - 1192 T^{2} + 3468 T^{3} + 542 T^{4} - 299 T^{5} - 43 T^{6} + 7 T^{7} + T^{8} \)
$13$ \( -50176 - 19200 T + 12032 T^{2} + 4936 T^{3} - 800 T^{4} - 422 T^{5} + 12 T^{7} + T^{8} \)
$17$ \( -1664 - 2688 T + 2304 T^{2} + 2316 T^{3} - 20 T^{4} - 285 T^{5} - 33 T^{6} + 7 T^{7} + T^{8} \)
$19$ \( ( -1 + T )^{8} \)
$23$ \( 50944 + 32512 T - 8992 T^{2} - 6928 T^{3} + 616 T^{4} + 460 T^{5} - 36 T^{6} - 10 T^{7} + T^{8} \)
$29$ \( 15472 + 17432 T - 28172 T^{2} - 7562 T^{3} + 3494 T^{4} + 383 T^{5} - 109 T^{6} - 5 T^{7} + T^{8} \)
$31$ \( -283552 + 389328 T - 105280 T^{2} - 26496 T^{3} + 7742 T^{4} + 547 T^{5} - 163 T^{6} - 3 T^{7} + T^{8} \)
$37$ \( -151388 - 198996 T + 71034 T^{2} + 37066 T^{3} - 2112 T^{4} - 1583 T^{5} - 59 T^{6} + 15 T^{7} + T^{8} \)
$41$ \( 85172 - 78124 T - 46138 T^{2} + 12154 T^{3} + 4622 T^{4} - 569 T^{5} - 137 T^{6} + 5 T^{7} + T^{8} \)
$43$ \( 12334 + 17498 T - 1647 T^{2} - 6022 T^{3} + 581 T^{4} + 470 T^{5} - 49 T^{6} - 10 T^{7} + T^{8} \)
$47$ \( -634922 + 1951854 T + 291629 T^{2} - 143982 T^{3} - 4727 T^{4} + 3362 T^{5} - 113 T^{6} - 18 T^{7} + T^{8} \)
$53$ \( -2381252 - 224820 T + 843322 T^{2} - 223106 T^{3} + 8778 T^{4} + 2875 T^{5} - 243 T^{6} - 9 T^{7} + T^{8} \)
$59$ \( 143872 - 572160 T + 469440 T^{2} - 131328 T^{3} + 7376 T^{4} + 2064 T^{5} - 220 T^{6} - 8 T^{7} + T^{8} \)
$61$ \( -479744 + 233216 T + 83456 T^{2} - 41708 T^{3} - 500 T^{4} + 1429 T^{5} - 75 T^{6} - 13 T^{7} + T^{8} \)
$67$ \( -6184 + 352 T + 14670 T^{2} + 4150 T^{3} - 3790 T^{4} + 323 T^{5} + 111 T^{6} - 21 T^{7} + T^{8} \)
$71$ \( 307328 - 152880 T^{2} + 31640 T^{3} + 11660 T^{4} - 5070 T^{5} + 714 T^{6} - 44 T^{7} + T^{8} \)
$73$ \( -4804 + 23260 T - 35841 T^{2} + 17216 T^{3} + 347 T^{4} - 1184 T^{5} - 7 T^{6} + 20 T^{7} + T^{8} \)
$79$ \( ( -1 + T )^{8} \)
$83$ \( -1024 - 3072 T - 2624 T^{2} + 96 T^{3} + 832 T^{4} + 68 T^{5} - 88 T^{6} + 4 T^{7} + T^{8} \)
$89$ \( 2323328 - 1314176 T - 783232 T^{2} + 98632 T^{3} + 25624 T^{4} - 1854 T^{5} - 280 T^{6} + 10 T^{7} + T^{8} \)
$97$ \( -861184 + 829440 T + 454400 T^{2} + 544 T^{3} - 24912 T^{4} - 3488 T^{5} - 16 T^{6} + 22 T^{7} + T^{8} \)
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