Properties

Label 8008.2.a.j
Level 8008
Weight 2
Character orbit 8008.a
Self dual yes
Analytic conductor 63.944
Analytic rank 0
Dimension 5
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8008.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(63.9442019386\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.668973.1
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 3 \)
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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{1} q^{3} + \beta_{3} q^{5} + q^{7} + ( 1 + \beta_{2} - \beta_{4} ) q^{9} +O(q^{10})\) \( q -\beta_{1} q^{3} + \beta_{3} q^{5} + q^{7} + ( 1 + \beta_{2} - \beta_{4} ) q^{9} - q^{11} - q^{13} + ( -1 + \beta_{2} ) q^{15} + ( 4 + \beta_{3} ) q^{17} + ( -1 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{19} -\beta_{1} q^{21} + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{23} + ( 3 + \beta_{1} - 2 \beta_{2} + 2 \beta_{4} ) q^{25} + ( -2 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} ) q^{27} + ( -4 - 2 \beta_{1} - \beta_{2} - \beta_{4} ) q^{29} + ( 1 + 2 \beta_{1} - \beta_{4} ) q^{31} + \beta_{1} q^{33} + \beta_{3} q^{35} + ( 2 - \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{4} ) q^{37} + \beta_{1} q^{39} + ( 2 + \beta_{1} + \beta_{3} - 2 \beta_{4} ) q^{41} + ( -5 - 2 \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{43} + ( \beta_{2} - 2 \beta_{3} + \beta_{4} ) q^{45} + ( 1 + \beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{4} ) q^{47} + q^{49} + ( -1 - 4 \beta_{1} + \beta_{2} ) q^{51} + ( 4 + \beta_{2} - \beta_{3} - \beta_{4} ) q^{53} -\beta_{3} q^{55} + ( 3 + 2 \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{57} + ( 3 - 4 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{59} + ( 5 - \beta_{1} + \beta_{2} ) q^{61} + ( 1 + \beta_{2} - \beta_{4} ) q^{63} -\beta_{3} q^{65} + ( 5 - 2 \beta_{3} - \beta_{4} ) q^{67} + ( 3 - 2 \beta_{1} + 3 \beta_{2} + \beta_{3} ) q^{69} + ( 6 + 2 \beta_{4} ) q^{71} + ( -1 - \beta_{1} + 3 \beta_{2} ) q^{73} + ( 3 \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{4} ) q^{75} - q^{77} + ( -3 - \beta_{3} - 3 \beta_{4} ) q^{79} + ( 2 + 5 \beta_{1} - \beta_{3} + \beta_{4} ) q^{81} + ( -1 - 3 \beta_{1} + 3 \beta_{2} - 2 \beta_{4} ) q^{83} + ( 8 + \beta_{1} - 2 \beta_{2} + 4 \beta_{3} + 2 \beta_{4} ) q^{85} + ( 6 + 3 \beta_{1} - \beta_{2} - \beta_{3} - 3 \beta_{4} ) q^{87} + ( -7 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{89} - q^{91} + ( -10 - 3 \beta_{1} - 4 \beta_{2} + 2 \beta_{4} ) q^{93} + ( 7 + 3 \beta_{1} - 2 \beta_{2} + 3 \beta_{4} ) q^{95} + ( 3 - 3 \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{4} ) q^{97} + ( -1 - \beta_{2} + \beta_{4} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q + q^{3} - q^{5} + 5q^{7} + 6q^{9} + O(q^{10}) \) \( 5q + q^{3} - q^{5} + 5q^{7} + 6q^{9} - 5q^{11} - 5q^{13} - 5q^{15} + 19q^{17} - 5q^{19} + q^{21} + 5q^{23} + 12q^{25} - 11q^{27} - 17q^{29} + 4q^{31} - q^{33} - q^{35} + 10q^{37} - q^{39} + 10q^{41} - 25q^{43} + q^{45} + 3q^{47} + 5q^{49} - q^{51} + 22q^{53} + q^{55} + 16q^{57} + 21q^{59} + 26q^{61} + 6q^{63} + q^{65} + 28q^{67} + 16q^{69} + 28q^{71} - 4q^{73} - 5q^{77} - 11q^{79} + 5q^{81} + 33q^{85} + 31q^{87} - 37q^{89} - 5q^{91} - 49q^{93} + 29q^{95} + 18q^{97} - 6q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5} - x^{4} - 9 x^{3} - x^{2} + 7 x - 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{4} - \nu^{3} - 8 \nu^{2} - 3 \nu + 2 \)
\(\beta_{2}\)\(=\)\( \nu^{4} - 10 \nu^{2} - 10 \nu + 5 \)
\(\beta_{3}\)\(=\)\( -\nu^{4} + \nu^{3} + 9 \nu^{2} + 2 \nu - 6 \)
\(\beta_{4}\)\(=\)\( 3 \nu^{4} - \nu^{3} - 29 \nu^{2} - 19 \nu + 15 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{4} + \beta_{3} - 2 \beta_{2} + 1\)\()/3\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{4} + 4 \beta_{3} - 2 \beta_{2} + 3 \beta_{1} + 13\)\()/3\)
\(\nu^{3}\)\(=\)\(3 \beta_{4} + 5 \beta_{3} - 5 \beta_{2} + \beta_{1} + 8\)
\(\nu^{4}\)\(=\)\((\)\(20 \beta_{4} + 50 \beta_{3} - 37 \beta_{2} + 30 \beta_{1} + 125\)\()/3\)

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
−2.16366
3.50189
0.376114
0.553533
−1.26787
0 −3.08441 0 −0.239305 0 1.00000 0 6.51357 0
1.2 0 −0.830554 0 3.93077 0 1.00000 0 −2.31018 0
1.3 0 0.293231 0 −3.94142 0 1.00000 0 −2.91402 0
1.4 0 2.18752 0 −2.05962 0 1.00000 0 1.78523 0
1.5 0 2.43421 0 1.30958 0 1.00000 0 2.92540 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
Significant digits:
Format:

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 8008.2.a.j 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8008.2.a.j 5 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(7\) \(-1\)
\(11\) \(1\)
\(13\) \(1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8008))\):

\( T_{3}^{5} - T_{3}^{4} - 10 T_{3}^{3} + 12 T_{3}^{2} + 11 T_{3} - 4 \)
\( T_{5}^{5} + T_{5}^{4} - 18 T_{5}^{3} - 16 T_{5}^{2} + 39 T_{5} + 10 \)
\( T_{17}^{5} - 19 T_{17}^{4} + 126 T_{17}^{3} - 344 T_{17}^{2} + 327 T_{17} - 18 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( \)
$3$ \( 1 - T + 5 T^{2} + 11 T^{4} + 14 T^{5} + 33 T^{6} + 135 T^{8} - 81 T^{9} + 243 T^{10} \)
$5$ \( 1 + T + 7 T^{2} + 4 T^{3} + 19 T^{4} + 95 T^{6} + 100 T^{7} + 875 T^{8} + 625 T^{9} + 3125 T^{10} \)
$7$ \( ( 1 - T )^{5} \)
$11$ \( ( 1 + T )^{5} \)
$13$ \( ( 1 + T )^{5} \)
$17$ \( 1 - 19 T + 211 T^{2} - 1636 T^{3} + 9643 T^{4} - 44660 T^{5} + 163931 T^{6} - 472804 T^{7} + 1036643 T^{8} - 1586899 T^{9} + 1419857 T^{10} \)
$19$ \( 1 + 5 T + 65 T^{2} + 260 T^{3} + 2113 T^{4} + 6722 T^{5} + 40147 T^{6} + 93860 T^{7} + 445835 T^{8} + 651605 T^{9} + 2476099 T^{10} \)
$23$ \( 1 - 5 T + 79 T^{2} - 466 T^{3} + 2923 T^{4} - 16082 T^{5} + 67229 T^{6} - 246514 T^{7} + 961193 T^{8} - 1399205 T^{9} + 6436343 T^{10} \)
$29$ \( 1 + 17 T + 173 T^{2} + 1010 T^{3} + 4519 T^{4} + 18428 T^{5} + 131051 T^{6} + 849410 T^{7} + 4219297 T^{8} + 12023777 T^{9} + 20511149 T^{10} \)
$31$ \( 1 - 4 T + 77 T^{2} - 512 T^{3} + 2801 T^{4} - 24388 T^{5} + 86831 T^{6} - 492032 T^{7} + 2293907 T^{8} - 3694084 T^{9} + 28629151 T^{10} \)
$37$ \( 1 - 10 T + 125 T^{2} - 585 T^{3} + 4387 T^{4} - 14264 T^{5} + 162319 T^{6} - 800865 T^{7} + 6331625 T^{8} - 18741610 T^{9} + 69343957 T^{10} \)
$41$ \( 1 - 10 T + 93 T^{2} - 615 T^{3} + 4027 T^{4} - 29860 T^{5} + 165107 T^{6} - 1033815 T^{7} + 6409653 T^{8} - 28257610 T^{9} + 115856201 T^{10} \)
$43$ \( 1 + 25 T + 344 T^{2} + 3373 T^{3} + 27232 T^{4} + 190396 T^{5} + 1170976 T^{6} + 6236677 T^{7} + 27350408 T^{8} + 85470025 T^{9} + 147008443 T^{10} \)
$47$ \( 1 - 3 T + 137 T^{2} - 818 T^{3} + 8733 T^{4} - 62242 T^{5} + 410451 T^{6} - 1806962 T^{7} + 14223751 T^{8} - 14639043 T^{9} + 229345007 T^{10} \)
$53$ \( 1 - 22 T + 412 T^{2} - 4808 T^{3} + 49834 T^{4} - 382030 T^{5} + 2641202 T^{6} - 13505672 T^{7} + 61337324 T^{8} - 173590582 T^{9} + 418195493 T^{10} \)
$59$ \( 1 - 21 T + 186 T^{2} - 1247 T^{3} + 15310 T^{4} - 157544 T^{5} + 903290 T^{6} - 4340807 T^{7} + 38200494 T^{8} - 254464581 T^{9} + 714924299 T^{10} \)
$61$ \( 1 - 26 T + 547 T^{2} - 7381 T^{3} + 83563 T^{4} - 708520 T^{5} + 5097343 T^{6} - 27464701 T^{7} + 124158607 T^{8} - 359991866 T^{9} + 844596301 T^{10} \)
$67$ \( 1 - 28 T + 545 T^{2} - 7368 T^{3} + 82657 T^{4} - 735404 T^{5} + 5538019 T^{6} - 33074952 T^{7} + 163915835 T^{8} - 564231388 T^{9} + 1350125107 T^{10} \)
$71$ \( 1 - 28 T + 563 T^{2} - 7840 T^{3} + 91306 T^{4} - 835592 T^{5} + 6482726 T^{6} - 39521440 T^{7} + 201503893 T^{8} - 711527068 T^{9} + 1804229351 T^{10} \)
$73$ \( 1 + 4 T + 193 T^{2} + 29 T^{3} + 15799 T^{4} - 36904 T^{5} + 1153327 T^{6} + 154541 T^{7} + 75080281 T^{8} + 113592964 T^{9} + 2073071593 T^{10} \)
$79$ \( 1 + 11 T + 182 T^{2} + 1077 T^{3} + 22036 T^{4} + 150844 T^{5} + 1740844 T^{6} + 6721557 T^{7} + 89733098 T^{8} + 428450891 T^{9} + 3077056399 T^{10} \)
$83$ \( 1 + 193 T^{2} - 195 T^{3} + 20681 T^{4} - 45478 T^{5} + 1716523 T^{6} - 1343355 T^{7} + 110354891 T^{8} + 3939040643 T^{10} \)
$89$ \( 1 + 37 T + 945 T^{2} + 16086 T^{3} + 218857 T^{4} + 2276992 T^{5} + 19478273 T^{6} + 127417206 T^{7} + 666195705 T^{8} + 2321462917 T^{9} + 5584059449 T^{10} \)
$97$ \( 1 - 18 T + 469 T^{2} - 6621 T^{3} + 89573 T^{4} - 946052 T^{5} + 8688581 T^{6} - 62296989 T^{7} + 428043637 T^{8} - 1593527058 T^{9} + 8587340257 T^{10} \)
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