Properties

Label 8040.2.a.t
Level 8040
Weight 2
Character orbit 8040.a
Self dual Yes
Analytic conductor 64.200
Analytic rank 0
Dimension 7
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 8040 = 2^{3} \cdot 3 \cdot 5 \cdot 67 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8040.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(64.1997232251\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \(+ q^{3}\) \(- q^{5}\) \( + ( 1 + \beta_{3} ) q^{7} \) \(+ q^{9}\) \(+O(q^{10})\) \( q\) \(+ q^{3}\) \(- q^{5}\) \( + ( 1 + \beta_{3} ) q^{7} \) \(+ q^{9}\) \( + ( -\beta_{1} - \beta_{6} ) q^{11} \) \( + ( -\beta_{1} + \beta_{3} + \beta_{4} + \beta_{6} ) q^{13} \) \(- q^{15}\) \( -\beta_{2} q^{17} \) \( + ( 1 + \beta_{1} + \beta_{3} + \beta_{4} ) q^{19} \) \( + ( 1 + \beta_{3} ) q^{21} \) \( + ( 1 - \beta_{2} + \beta_{4} - \beta_{5} - \beta_{6} ) q^{23} \) \(+ q^{25}\) \(+ q^{27}\) \( + ( -\beta_{5} - \beta_{6} ) q^{29} \) \( + ( 1 - \beta_{4} ) q^{31} \) \( + ( -\beta_{1} - \beta_{6} ) q^{33} \) \( + ( -1 - \beta_{3} ) q^{35} \) \( + ( 2 + \beta_{1} + \beta_{2} - \beta_{4} + 2 \beta_{5} ) q^{37} \) \( + ( -\beta_{1} + \beta_{3} + \beta_{4} + \beta_{6} ) q^{39} \) \( + ( 1 + \beta_{4} ) q^{41} \) \( + ( -1 - \beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} + 2 \beta_{6} ) q^{43} \) \(- q^{45}\) \( + ( 2 - \beta_{1} - \beta_{3} - \beta_{4} + \beta_{6} ) q^{47} \) \( + ( 1 - \beta_{1} + 2 \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} ) q^{49} \) \( -\beta_{2} q^{51} \) \( + ( 3 - \beta_{1} - \beta_{3} + \beta_{4} - 2 \beta_{5} - 2 \beta_{6} ) q^{53} \) \( + ( \beta_{1} + \beta_{6} ) q^{55} \) \( + ( 1 + \beta_{1} + \beta_{3} + \beta_{4} ) q^{57} \) \( + ( -\beta_{1} - \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{6} ) q^{59} \) \( + ( 2 \beta_{1} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} ) q^{61} \) \( + ( 1 + \beta_{3} ) q^{63} \) \( + ( \beta_{1} - \beta_{3} - \beta_{4} - \beta_{6} ) q^{65} \) \(+ q^{67}\) \( + ( 1 - \beta_{2} + \beta_{4} - \beta_{5} - \beta_{6} ) q^{69} \) \( + ( 2 \beta_{1} + \beta_{3} - \beta_{5} + 2 \beta_{6} ) q^{71} \) \( + ( 1 + 3 \beta_{1} + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} ) q^{73} \) \(+ q^{75}\) \( + ( 1 - \beta_{1} + \beta_{2} - 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} ) q^{77} \) \( + ( 4 - 2 \beta_{1} - 2 \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{6} ) q^{79} \) \(+ q^{81}\) \( + ( -5 + \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{6} ) q^{83} \) \( + \beta_{2} q^{85} \) \( + ( -\beta_{5} - \beta_{6} ) q^{87} \) \( + ( 2 - 4 \beta_{1} - \beta_{3} + \beta_{5} + 2 \beta_{6} ) q^{89} \) \( + ( 5 - 3 \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} + 4 \beta_{6} ) q^{91} \) \( + ( 1 - \beta_{4} ) q^{93} \) \( + ( -1 - \beta_{1} - \beta_{3} - \beta_{4} ) q^{95} \) \( + ( 6 + \beta_{3} - 4 \beta_{5} - \beta_{6} ) q^{97} \) \( + ( -\beta_{1} - \beta_{6} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(7q \) \(\mathstrut +\mathstrut 7q^{3} \) \(\mathstrut -\mathstrut 7q^{5} \) \(\mathstrut +\mathstrut 10q^{7} \) \(\mathstrut +\mathstrut 7q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(7q \) \(\mathstrut +\mathstrut 7q^{3} \) \(\mathstrut -\mathstrut 7q^{5} \) \(\mathstrut +\mathstrut 10q^{7} \) \(\mathstrut +\mathstrut 7q^{9} \) \(\mathstrut -\mathstrut q^{13} \) \(\mathstrut -\mathstrut 7q^{15} \) \(\mathstrut -\mathstrut 2q^{17} \) \(\mathstrut +\mathstrut 9q^{19} \) \(\mathstrut +\mathstrut 10q^{21} \) \(\mathstrut +\mathstrut 2q^{23} \) \(\mathstrut +\mathstrut 7q^{25} \) \(\mathstrut +\mathstrut 7q^{27} \) \(\mathstrut -\mathstrut q^{29} \) \(\mathstrut +\mathstrut 9q^{31} \) \(\mathstrut -\mathstrut 10q^{35} \) \(\mathstrut +\mathstrut 23q^{37} \) \(\mathstrut -\mathstrut q^{39} \) \(\mathstrut +\mathstrut 5q^{41} \) \(\mathstrut -\mathstrut 3q^{43} \) \(\mathstrut -\mathstrut 7q^{45} \) \(\mathstrut +\mathstrut 11q^{47} \) \(\mathstrut +\mathstrut 13q^{49} \) \(\mathstrut -\mathstrut 2q^{51} \) \(\mathstrut +\mathstrut 13q^{53} \) \(\mathstrut +\mathstrut 9q^{57} \) \(\mathstrut +\mathstrut q^{59} \) \(\mathstrut +\mathstrut 4q^{61} \) \(\mathstrut +\mathstrut 10q^{63} \) \(\mathstrut +\mathstrut q^{65} \) \(\mathstrut +\mathstrut 7q^{67} \) \(\mathstrut +\mathstrut 2q^{69} \) \(\mathstrut +\mathstrut q^{71} \) \(\mathstrut +\mathstrut 14q^{73} \) \(\mathstrut +\mathstrut 7q^{75} \) \(\mathstrut +\mathstrut 18q^{77} \) \(\mathstrut +\mathstrut 25q^{79} \) \(\mathstrut +\mathstrut 7q^{81} \) \(\mathstrut -\mathstrut 29q^{83} \) \(\mathstrut +\mathstrut 2q^{85} \) \(\mathstrut -\mathstrut q^{87} \) \(\mathstrut +\mathstrut 7q^{89} \) \(\mathstrut +\mathstrut 27q^{91} \) \(\mathstrut +\mathstrut 9q^{93} \) \(\mathstrut -\mathstrut 9q^{95} \) \(\mathstrut +\mathstrut 38q^{97} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{7}\mathstrut -\mathstrut \) \(x^{6}\mathstrut -\mathstrut \) \(15\) \(x^{5}\mathstrut +\mathstrut \) \(3\) \(x^{4}\mathstrut +\mathstrut \) \(43\) \(x^{3}\mathstrut -\mathstrut \) \(6\) \(x^{2}\mathstrut -\mathstrut \) \(29\) \(x\mathstrut +\mathstrut \) \(6\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 4 \)
\(\beta_{3}\)\(=\)\((\)\( 6 \nu^{6} - 4 \nu^{5} - 73 \nu^{4} - 43 \nu^{3} + 42 \nu^{2} + 143 \nu + 112 \)\()/55\)
\(\beta_{4}\)\(=\)\((\)\( 7 \nu^{6} - 23 \nu^{5} - 76 \nu^{4} + 234 \nu^{3} + 104 \nu^{2} - 484 \nu + 39 \)\()/55\)
\(\beta_{5}\)\(=\)\((\)\( 16 \nu^{6} - 29 \nu^{5} - 213 \nu^{4} + 197 \nu^{3} + 497 \nu^{2} - 242 \nu - 123 \)\()/55\)
\(\beta_{6}\)\(=\)\((\)\( -24 \nu^{6} + 16 \nu^{5} + 347 \nu^{4} + 62 \nu^{3} - 773 \nu^{2} - 132 \nu + 267 \)\()/55\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(4\)
\(\nu^{3}\)\(=\)\(-\)\(\beta_{6}\mathstrut -\mathstrut \) \(2\) \(\beta_{5}\mathstrut +\mathstrut \) \(2\) \(\beta_{4}\mathstrut -\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(10\) \(\beta_{1}\mathstrut +\mathstrut \) \(5\)
\(\nu^{4}\)\(=\)\(-\)\(\beta_{6}\mathstrut -\mathstrut \) \(4\) \(\beta_{5}\mathstrut +\mathstrut \) \(4\) \(\beta_{4}\mathstrut +\mathstrut \) \(2\) \(\beta_{3}\mathstrut +\mathstrut \) \(13\) \(\beta_{2}\mathstrut +\mathstrut \) \(23\) \(\beta_{1}\mathstrut +\mathstrut \) \(41\)
\(\nu^{5}\)\(=\)\(-\)\(16\) \(\beta_{6}\mathstrut -\mathstrut \) \(33\) \(\beta_{5}\mathstrut +\mathstrut \) \(30\) \(\beta_{4}\mathstrut -\mathstrut \) \(11\) \(\beta_{3}\mathstrut +\mathstrut \) \(25\) \(\beta_{2}\mathstrut +\mathstrut \) \(134\) \(\beta_{1}\mathstrut +\mathstrut \) \(105\)
\(\nu^{6}\)\(=\)\(-\)\(30\) \(\beta_{6}\mathstrut -\mathstrut \) \(85\) \(\beta_{5}\mathstrut +\mathstrut \) \(83\) \(\beta_{4}\mathstrut +\mathstrut \) \(19\) \(\beta_{3}\mathstrut +\mathstrut \) \(175\) \(\beta_{2}\mathstrut +\mathstrut \) \(410\) \(\beta_{1}\mathstrut +\mathstrut \) \(558\)

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
−1.70642
1.51692
3.95767
−1.05266
0.211675
−2.84959
0.922407
0 1.00000 0 −1.00000 0 −2.80016 0 1.00000 0
1.2 0 1.00000 0 −1.00000 0 −0.274078 0 1.00000 0
1.3 0 1.00000 0 −1.00000 0 −0.212435 0 1.00000 0
1.4 0 1.00000 0 −1.00000 0 0.670274 0 1.00000 0
1.5 0 1.00000 0 −1.00000 0 3.61083 0 1.00000 0
1.6 0 1.00000 0 −1.00000 0 4.47701 0 1.00000 0
1.7 0 1.00000 0 −1.00000 0 4.52856 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(5\) \(1\)
\(67\) \(-1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8040))\):

\(T_{7}^{7} \) \(\mathstrut -\mathstrut 10 T_{7}^{6} \) \(\mathstrut +\mathstrut 19 T_{7}^{5} \) \(\mathstrut +\mathstrut 74 T_{7}^{4} \) \(\mathstrut -\mathstrut 223 T_{7}^{3} \) \(\mathstrut +\mathstrut 17 T_{7}^{2} \) \(\mathstrut +\mathstrut 52 T_{7} \) \(\mathstrut +\mathstrut 8 \)
\(T_{11}^{7} \) \(\mathstrut -\mathstrut 41 T_{11}^{5} \) \(\mathstrut +\mathstrut 18 T_{11}^{4} \) \(\mathstrut +\mathstrut 525 T_{11}^{3} \) \(\mathstrut -\mathstrut 377 T_{11}^{2} \) \(\mathstrut -\mathstrut 2000 T_{11} \) \(\mathstrut +\mathstrut 1530 \)
\(T_{13}^{7} \) \(\mathstrut +\mathstrut T_{13}^{6} \) \(\mathstrut -\mathstrut 55 T_{13}^{5} \) \(\mathstrut -\mathstrut 69 T_{13}^{4} \) \(\mathstrut +\mathstrut 971 T_{13}^{3} \) \(\mathstrut +\mathstrut 1494 T_{13}^{2} \) \(\mathstrut -\mathstrut 5580 T_{13} \) \(\mathstrut -\mathstrut 10152 \)
\(T_{17}^{7} \) \(\mathstrut +\mathstrut 2 T_{17}^{6} \) \(\mathstrut -\mathstrut 76 T_{17}^{5} \) \(\mathstrut +\mathstrut 51 T_{17}^{4} \) \(\mathstrut +\mathstrut 1653 T_{17}^{3} \) \(\mathstrut -\mathstrut 4766 T_{17}^{2} \) \(\mathstrut +\mathstrut 1834 T_{17} \) \(\mathstrut +\mathstrut 3332 \)