Properties

Label 8034.2.a.r
Level $8034$
Weight $2$
Character orbit 8034.a
Self dual yes
Analytic conductor $64.152$
Analytic rank $1$
Dimension $9$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8034 = 2 \cdot 3 \cdot 13 \cdot 103 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8034.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.1518129839\)
Analytic rank: \(1\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Defining polynomial: \(x^{9} - 4 x^{8} - 9 x^{7} + 45 x^{6} + 7 x^{5} - 123 x^{4} + 37 x^{3} + 87 x^{2} - 54 x + 8\)
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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} - q^{3} + q^{4} -\beta_{1} q^{5} - q^{6} -\beta_{3} q^{7} + q^{8} + q^{9} +O(q^{10})\) \( q + q^{2} - q^{3} + q^{4} -\beta_{1} q^{5} - q^{6} -\beta_{3} q^{7} + q^{8} + q^{9} -\beta_{1} q^{10} + ( -1 - \beta_{7} ) q^{11} - q^{12} + q^{13} -\beta_{3} q^{14} + \beta_{1} q^{15} + q^{16} + ( -1 + 2 \beta_{1} - \beta_{2} - \beta_{4} + \beta_{5} + \beta_{6} ) q^{17} + q^{18} + ( -1 - \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{19} -\beta_{1} q^{20} + \beta_{3} q^{21} + ( -1 - \beta_{7} ) q^{22} + ( 1 - \beta_{2} + \beta_{4} + \beta_{5} - \beta_{6} + 3 \beta_{7} + \beta_{8} ) q^{23} - q^{24} + ( -2 + \beta_{2} + \beta_{3} - \beta_{5} ) q^{25} + q^{26} - q^{27} -\beta_{3} q^{28} + ( -4 + \beta_{1} + \beta_{2} - \beta_{4} - \beta_{5} - 2 \beta_{7} - \beta_{8} ) q^{29} + \beta_{1} q^{30} + ( -1 + \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} ) q^{31} + q^{32} + ( 1 + \beta_{7} ) q^{33} + ( -1 + 2 \beta_{1} - \beta_{2} - \beta_{4} + \beta_{5} + \beta_{6} ) q^{34} + ( 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{35} + q^{36} + ( -1 + \beta_{1} - \beta_{3} - \beta_{6} + 2 \beta_{7} - \beta_{8} ) q^{37} + ( -1 - \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{38} - q^{39} -\beta_{1} q^{40} + ( -2 - \beta_{1} + \beta_{3} + \beta_{4} ) q^{41} + \beta_{3} q^{42} + ( -4 + \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - \beta_{4} - 2 \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{43} + ( -1 - \beta_{7} ) q^{44} -\beta_{1} q^{45} + ( 1 - \beta_{2} + \beta_{4} + \beta_{5} - \beta_{6} + 3 \beta_{7} + \beta_{8} ) q^{46} + ( 2 + \beta_{1} - 3 \beta_{6} + 3 \beta_{7} + \beta_{8} ) q^{47} - q^{48} + ( -1 + \beta_{1} + \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} ) q^{49} + ( -2 + \beta_{2} + \beta_{3} - \beta_{5} ) q^{50} + ( 1 - 2 \beta_{1} + \beta_{2} + \beta_{4} - \beta_{5} - \beta_{6} ) q^{51} + q^{52} + ( -1 - \beta_{1} + \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - \beta_{6} + \beta_{7} ) q^{53} - q^{54} + ( -1 + \beta_{1} - \beta_{5} ) q^{55} -\beta_{3} q^{56} + ( 1 + \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} ) q^{57} + ( -4 + \beta_{1} + \beta_{2} - \beta_{4} - \beta_{5} - 2 \beta_{7} - \beta_{8} ) q^{58} + ( -2 + \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{5} + \beta_{6} - 3 \beta_{8} ) q^{59} + \beta_{1} q^{60} + ( -2 + 2 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{7} + \beta_{8} ) q^{61} + ( -1 + \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} ) q^{62} -\beta_{3} q^{63} + q^{64} -\beta_{1} q^{65} + ( 1 + \beta_{7} ) q^{66} + ( -1 + \beta_{2} + \beta_{3} + 3 \beta_{5} + \beta_{6} - 2 \beta_{8} ) q^{67} + ( -1 + 2 \beta_{1} - \beta_{2} - \beta_{4} + \beta_{5} + \beta_{6} ) q^{68} + ( -1 + \beta_{2} - \beta_{4} - \beta_{5} + \beta_{6} - 3 \beta_{7} - \beta_{8} ) q^{69} + ( 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{70} + ( -3 + 2 \beta_{1} + 3 \beta_{2} + 3 \beta_{6} - 2 \beta_{7} - \beta_{8} ) q^{71} + q^{72} + ( -3 + \beta_{2} - \beta_{5} - \beta_{6} - \beta_{7} + 3 \beta_{8} ) q^{73} + ( -1 + \beta_{1} - \beta_{3} - \beta_{6} + 2 \beta_{7} - \beta_{8} ) q^{74} + ( 2 - \beta_{2} - \beta_{3} + \beta_{5} ) q^{75} + ( -1 - \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{76} + ( -\beta_{1} + \beta_{2} - \beta_{4} - \beta_{5} + \beta_{6} - 2 \beta_{7} + \beta_{8} ) q^{77} - q^{78} + ( 1 - \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{7} + 4 \beta_{8} ) q^{79} -\beta_{1} q^{80} + q^{81} + ( -2 - \beta_{1} + \beta_{3} + \beta_{4} ) q^{82} + ( 2 - 2 \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} + 3 \beta_{6} - 3 \beta_{7} + \beta_{8} ) q^{83} + \beta_{3} q^{84} + ( -2 + 2 \beta_{1} - 3 \beta_{2} - \beta_{3} + \beta_{5} + \beta_{6} + \beta_{7} + 2 \beta_{8} ) q^{85} + ( -4 + \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - \beta_{4} - 2 \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{86} + ( 4 - \beta_{1} - \beta_{2} + \beta_{4} + \beta_{5} + 2 \beta_{7} + \beta_{8} ) q^{87} + ( -1 - \beta_{7} ) q^{88} + ( 2 + 2 \beta_{1} - 2 \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{6} + 2 \beta_{7} + 3 \beta_{8} ) q^{89} -\beta_{1} q^{90} -\beta_{3} q^{91} + ( 1 - \beta_{2} + \beta_{4} + \beta_{5} - \beta_{6} + 3 \beta_{7} + \beta_{8} ) q^{92} + ( 1 - \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} ) q^{93} + ( 2 + \beta_{1} - 3 \beta_{6} + 3 \beta_{7} + \beta_{8} ) q^{94} + ( -3 \beta_{1} + 3 \beta_{2} - \beta_{3} + 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{8} ) q^{95} - q^{96} + ( -4 + \beta_{2} + \beta_{5} - 3 \beta_{7} - 2 \beta_{8} ) q^{97} + ( -1 + \beta_{1} + \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} ) q^{98} + ( -1 - \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9q + 9q^{2} - 9q^{3} + 9q^{4} - 4q^{5} - 9q^{6} - 4q^{7} + 9q^{8} + 9q^{9} + O(q^{10}) \) \( 9q + 9q^{2} - 9q^{3} + 9q^{4} - 4q^{5} - 9q^{6} - 4q^{7} + 9q^{8} + 9q^{9} - 4q^{10} - 5q^{11} - 9q^{12} + 9q^{13} - 4q^{14} + 4q^{15} + 9q^{16} - 6q^{17} + 9q^{18} - 4q^{19} - 4q^{20} + 4q^{21} - 5q^{22} - 6q^{23} - 9q^{24} - 11q^{25} + 9q^{26} - 9q^{27} - 4q^{28} - 19q^{29} + 4q^{30} - 6q^{31} + 9q^{32} + 5q^{33} - 6q^{34} + 10q^{35} + 9q^{36} - 13q^{37} - 4q^{38} - 9q^{39} - 4q^{40} - 18q^{41} + 4q^{42} - 20q^{43} - 5q^{44} - 4q^{45} - 6q^{46} + 14q^{47} - 9q^{48} - 3q^{49} - 11q^{50} + 6q^{51} + 9q^{52} - 3q^{53} - 9q^{54} - 4q^{55} - 4q^{56} + 4q^{57} - 19q^{58} - 9q^{59} + 4q^{60} - 24q^{61} - 6q^{62} - 4q^{63} + 9q^{64} - 4q^{65} + 5q^{66} - 4q^{67} - 6q^{68} + 6q^{69} + 10q^{70} - 9q^{71} + 9q^{72} - 24q^{73} - 13q^{74} + 11q^{75} - 4q^{76} + 3q^{77} - 9q^{78} - 15q^{79} - 4q^{80} + 9q^{81} - 18q^{82} + 20q^{83} + 4q^{84} - 31q^{85} - 20q^{86} + 19q^{87} - 5q^{88} + 3q^{89} - 4q^{90} - 4q^{91} - 6q^{92} + 6q^{93} + 14q^{94} - 4q^{95} - 9q^{96} - 19q^{97} - 3q^{98} - 5q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{9} - 4 x^{8} - 9 x^{7} + 45 x^{6} + 7 x^{5} - 123 x^{4} + 37 x^{3} + 87 x^{2} - 54 x + 8\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 55 \nu^{8} - 272 \nu^{7} - 307 \nu^{6} + 2869 \nu^{5} - 1267 \nu^{4} - 6789 \nu^{3} + 3935 \nu^{2} + 4811 \nu - 2412 \)\()/634\)
\(\beta_{3}\)\(=\)\((\)\( 129 \nu^{8} - 442 \nu^{7} - 1331 \nu^{6} + 4781 \nu^{5} + 2815 \nu^{4} - 11785 \nu^{3} - 223 \nu^{2} + 7065 \nu - 3444 \)\()/634\)
\(\beta_{4}\)\(=\)\((\)\( -131 \nu^{8} + 498 \nu^{7} + 1273 \nu^{6} - 5381 \nu^{5} - 2377 \nu^{4} + 12931 \nu^{3} + 1175 \nu^{2} - 5995 \nu + 1964 \)\()/634\)
\(\beta_{5}\)\(=\)\((\)\( 92 \nu^{8} - 357 \nu^{7} - 819 \nu^{6} + 3825 \nu^{5} + 774 \nu^{4} - 9287 \nu^{3} + 1539 \nu^{2} + 5938 \nu - 1977 \)\()/317\)
\(\beta_{6}\)\(=\)\((\)\( -359 \nu^{8} + 1176 \nu^{7} + 4171 \nu^{6} - 13551 \nu^{5} - 12675 \nu^{4} + 38965 \nu^{3} + 11433 \nu^{2} - 29201 \nu + 5692 \)\()/634\)
\(\beta_{7}\)\(=\)\((\)\( -415 \nu^{8} + 1476 \nu^{7} + 4449 \nu^{6} - 17037 \nu^{5} - 10555 \nu^{4} + 49497 \nu^{3} + 3219 \nu^{2} - 39183 \nu + 10534 \)\()/634\)
\(\beta_{8}\)\(=\)\((\)\( 261 \nu^{8} - 968 \nu^{7} - 2575 \nu^{6} + 10779 \nu^{5} + 4656 \nu^{4} - 28776 \nu^{3} + 28 \nu^{2} + 20133 \nu - 5619 \)\()/317\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{5} + \beta_{3} + \beta_{2} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{8} + \beta_{6} - 2 \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} + 7 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(2 \beta_{8} + 2 \beta_{7} - \beta_{6} - 11 \beta_{5} + \beta_{4} + 8 \beta_{3} + 10 \beta_{2} + \beta_{1} + 21\)
\(\nu^{5}\)\(=\)\(14 \beta_{8} + 2 \beta_{7} + 10 \beta_{6} - 27 \beta_{5} - 7 \beta_{4} + 12 \beta_{3} - 7 \beta_{2} + 54 \beta_{1} + 17\)
\(\nu^{6}\)\(=\)\(34 \beta_{8} + 27 \beta_{7} - 7 \beta_{6} - 114 \beta_{5} + 17 \beta_{4} + 73 \beta_{3} + 86 \beta_{2} + 17 \beta_{1} + 177\)
\(\nu^{7}\)\(=\)\(165 \beta_{8} + 45 \beta_{7} + 86 \beta_{6} - 305 \beta_{5} - 25 \beta_{4} + 141 \beta_{3} - 35 \beta_{2} + 434 \beta_{1} + 213\)
\(\nu^{8}\)\(=\)\(445 \beta_{8} + 315 \beta_{7} - 35 \beta_{6} - 1165 \beta_{5} + 236 \beta_{4} + 715 \beta_{3} + 719 \beta_{2} + 224 \beta_{1} + 1591\)

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
3.20969
2.80812
1.90297
0.840786
0.435443
0.254984
−1.25914
−1.44298
−2.74988
1.00000 −1.00000 1.00000 −3.20969 −1.00000 −2.02210 1.00000 1.00000 −3.20969
1.2 1.00000 −1.00000 1.00000 −2.80812 −1.00000 −1.72587 1.00000 1.00000 −2.80812
1.3 1.00000 −1.00000 1.00000 −1.90297 −1.00000 −5.10863 1.00000 1.00000 −1.90297
1.4 1.00000 −1.00000 1.00000 −0.840786 −1.00000 2.87036 1.00000 1.00000 −0.840786
1.5 1.00000 −1.00000 1.00000 −0.435443 −1.00000 1.91967 1.00000 1.00000 −0.435443
1.6 1.00000 −1.00000 1.00000 −0.254984 −1.00000 2.89551 1.00000 1.00000 −0.254984
1.7 1.00000 −1.00000 1.00000 1.25914 −1.00000 −0.797445 1.00000 1.00000 1.25914
1.8 1.00000 −1.00000 1.00000 1.44298 −1.00000 0.367530 1.00000 1.00000 1.44298
1.9 1.00000 −1.00000 1.00000 2.74988 −1.00000 −2.39902 1.00000 1.00000 2.74988
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(13\) \(-1\)
\(103\) \(1\)

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 8034.2.a.r 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8034.2.a.r 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8034))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{9} \)
$3$ \( ( 1 + T )^{9} \)
$5$ \( -8 - 54 T - 87 T^{2} + 37 T^{3} + 123 T^{4} + 7 T^{5} - 45 T^{6} - 9 T^{7} + 4 T^{8} + T^{9} \)
$7$ \( 200 - 199 T - 929 T^{2} - 210 T^{3} + 456 T^{4} + 139 T^{5} - 74 T^{6} - 22 T^{7} + 4 T^{8} + T^{9} \)
$11$ \( 62 - 63 T - 189 T^{2} + 96 T^{3} + 179 T^{4} - 20 T^{5} - 60 T^{6} - 7 T^{7} + 5 T^{8} + T^{9} \)
$13$ \( ( -1 + T )^{9} \)
$17$ \( -4562 - 15355 T - 13631 T^{2} + 1041 T^{3} + 3975 T^{4} + 538 T^{5} - 281 T^{6} - 48 T^{7} + 6 T^{8} + T^{9} \)
$19$ \( -25904 - 39618 T + 4909 T^{2} + 29658 T^{3} + 13546 T^{4} + 1083 T^{5} - 492 T^{6} - 82 T^{7} + 4 T^{8} + T^{9} \)
$23$ \( 32768 + 52976 T - 30965 T^{2} - 21438 T^{3} + 7780 T^{4} + 2149 T^{5} - 441 T^{6} - 85 T^{7} + 6 T^{8} + T^{9} \)
$29$ \( 18820 + 11736 T - 50163 T^{2} - 75589 T^{3} - 43209 T^{4} - 11671 T^{5} - 1256 T^{6} + 45 T^{7} + 19 T^{8} + T^{9} \)
$31$ \( 199652 - 19580 T - 114211 T^{2} - 8810 T^{3} + 15724 T^{4} + 2071 T^{5} - 676 T^{6} - 106 T^{7} + 6 T^{8} + T^{9} \)
$37$ \( 370136 + 657914 T + 424407 T^{2} + 105345 T^{3} - 5527 T^{4} - 8223 T^{5} - 1478 T^{6} - 41 T^{7} + 13 T^{8} + T^{9} \)
$41$ \( 1900 + 4348 T + 1137 T^{2} - 3853 T^{3} - 3626 T^{4} - 1007 T^{5} + 94 T^{6} + 101 T^{7} + 18 T^{8} + T^{9} \)
$43$ \( 2944376 - 1382426 T - 713157 T^{2} + 255353 T^{3} + 69686 T^{4} - 10096 T^{5} - 2856 T^{6} - 46 T^{7} + 20 T^{8} + T^{9} \)
$47$ \( 11242472 - 6529462 T - 238831 T^{2} + 796600 T^{3} - 139834 T^{4} - 6547 T^{5} + 3278 T^{6} - 154 T^{7} - 14 T^{8} + T^{9} \)
$53$ \( 48526 + 249663 T - 202045 T^{2} - 60115 T^{3} + 22892 T^{4} + 5148 T^{5} - 707 T^{6} - 158 T^{7} + 3 T^{8} + T^{9} \)
$59$ \( 6686528 + 5480180 T - 1441719 T^{2} - 679502 T^{3} + 79124 T^{4} + 21592 T^{5} - 1533 T^{6} - 251 T^{7} + 9 T^{8} + T^{9} \)
$61$ \( -8840 - 3162 T + 15929 T^{2} + 5663 T^{3} - 6170 T^{4} - 3144 T^{5} - 82 T^{6} + 154 T^{7} + 24 T^{8} + T^{9} \)
$67$ \( -39059282 + 20828449 T + 789840 T^{2} - 1570415 T^{3} + 50889 T^{4} + 38455 T^{5} - 1046 T^{6} - 351 T^{7} + 4 T^{8} + T^{9} \)
$71$ \( -7122160 - 10689486 T - 4246533 T^{2} + 61734 T^{3} + 239501 T^{4} + 15255 T^{5} - 3475 T^{6} - 309 T^{7} + 9 T^{8} + T^{9} \)
$73$ \( 42267172 + 53504045 T + 25054390 T^{2} + 5171458 T^{3} + 321771 T^{4} - 43861 T^{5} - 6980 T^{6} - 112 T^{7} + 24 T^{8} + T^{9} \)
$79$ \( 13472944 - 12965946 T - 3709561 T^{2} + 1885625 T^{3} + 563173 T^{4} + 16251 T^{5} - 5698 T^{6} - 329 T^{7} + 15 T^{8} + T^{9} \)
$83$ \( -123824 + 2009058 T - 2820857 T^{2} + 996862 T^{3} - 51248 T^{4} - 31407 T^{5} + 5325 T^{6} - 143 T^{7} - 20 T^{8} + T^{9} \)
$89$ \( -18020120 + 11645106 T + 1176121 T^{2} - 1468401 T^{3} - 7449 T^{4} + 54156 T^{5} + 873 T^{6} - 451 T^{7} - 3 T^{8} + T^{9} \)
$97$ \( -110764 + 189712 T + 43221 T^{2} - 181553 T^{3} + 62498 T^{4} + 3197 T^{5} - 2224 T^{6} - 80 T^{7} + 19 T^{8} + T^{9} \)
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