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

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

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

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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8040))\):

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