Properties

Label 8036.2.a.h
Level 8036
Weight 2
Character orbit 8036.a
Self dual Yes
Analytic conductor 64.168
Analytic rank 0
Dimension 3
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 8036 = 2^{2} \cdot 7^{2} \cdot 41 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8036.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(64.1677830643\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{18})^+\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 - \beta_{1} ) q^{3} + ( 2 \beta_{1} - 2 \beta_{2} ) q^{5} + ( -2 \beta_{1} + \beta_{2} ) q^{9} +O(q^{10})\) \( q + ( 1 - \beta_{1} ) q^{3} + ( 2 \beta_{1} - 2 \beta_{2} ) q^{5} + ( -2 \beta_{1} + \beta_{2} ) q^{9} + ( -2 + 2 \beta_{1} - 4 \beta_{2} ) q^{11} + ( 1 - \beta_{1} + 3 \beta_{2} ) q^{13} + ( -2 + 4 \beta_{1} - 4 \beta_{2} ) q^{15} + ( -1 - \beta_{2} ) q^{17} + ( 3 - 2 \beta_{1} - \beta_{2} ) q^{19} + ( 1 - \beta_{1} - 2 \beta_{2} ) q^{23} + ( 3 - 4 \beta_{1} ) q^{25} + 3 \beta_{2} q^{27} + ( -4 + 2 \beta_{1} ) q^{29} + ( 4 + 4 \beta_{1} - 4 \beta_{2} ) q^{31} + ( -2 + 8 \beta_{1} - 6 \beta_{2} ) q^{33} + ( -7 - 3 \beta_{1} + \beta_{2} ) q^{37} + ( -5 \beta_{1} + 4 \beta_{2} ) q^{39} + q^{41} + ( -1 + 4 \beta_{1} - \beta_{2} ) q^{43} + ( -6 + 4 \beta_{1} - 2 \beta_{2} ) q^{45} + ( 1 - 5 \beta_{1} - \beta_{2} ) q^{47} + ( 2 \beta_{1} - \beta_{2} ) q^{51} + ( 6 + 2 \beta_{1} + 2 \beta_{2} ) q^{53} + ( 12 - 8 \beta_{1} ) q^{55} + ( 8 - 4 \beta_{1} + \beta_{2} ) q^{57} + ( 2 + 4 \beta_{1} - 4 \beta_{2} ) q^{59} + ( 10 - 2 \beta_{1} ) q^{61} + ( -8 + 4 \beta_{1} + 2 \beta_{2} ) q^{65} + ( -6 - 4 \beta_{1} + 4 \beta_{2} ) q^{67} + ( 5 - \beta_{2} ) q^{69} + ( 4 + 2 \beta_{1} - 6 \beta_{2} ) q^{71} + ( 4 - 4 \beta_{1} + 4 \beta_{2} ) q^{73} + ( 11 - 7 \beta_{1} + 4 \beta_{2} ) q^{75} + ( 4 - 6 \beta_{1} + 6 \beta_{2} ) q^{79} + ( -3 + 3 \beta_{1} ) q^{81} + ( 6 - 4 \beta_{1} ) q^{83} + ( 2 - 2 \beta_{1} ) q^{85} + ( -8 + 6 \beta_{1} - 2 \beta_{2} ) q^{87} + ( 1 - 5 \beta_{1} + 7 \beta_{2} ) q^{89} + ( 4 \beta_{1} - 8 \beta_{2} ) q^{93} + ( -2 + 10 \beta_{1} - 12 \beta_{2} ) q^{95} + ( -1 - 5 \beta_{1} + 4 \beta_{2} ) q^{97} + ( -6 + 10 \beta_{1} - 2 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{3} + O(q^{10}) \) \( 3q + 3q^{3} - 6q^{11} + 3q^{13} - 6q^{15} - 3q^{17} + 9q^{19} + 3q^{23} + 9q^{25} - 12q^{29} + 12q^{31} - 6q^{33} - 21q^{37} + 3q^{41} - 3q^{43} - 18q^{45} + 3q^{47} + 18q^{53} + 36q^{55} + 24q^{57} + 6q^{59} + 30q^{61} - 24q^{65} - 18q^{67} + 15q^{69} + 12q^{71} + 12q^{73} + 33q^{75} + 12q^{79} - 9q^{81} + 18q^{83} + 6q^{85} - 24q^{87} + 3q^{89} - 6q^{95} - 3q^{97} - 18q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of \(\nu = \zeta_{18} + \zeta_{18}^{-1}\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 2 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 2\)

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
1.87939
−0.347296
−1.53209
0 −0.879385 0 0.694593 0 0 0 −2.22668 0
1.2 0 1.34730 0 3.06418 0 0 0 −1.18479 0
1.3 0 2.53209 0 −3.75877 0 0 0 3.41147 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(7\) \(-1\)
\(41\) \(-1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8036))\):

\( T_{3}^{3} - 3 T_{3}^{2} + 3 \)
\( T_{5}^{3} - 12 T_{5} + 8 \)
\( T_{11}^{3} + 6 T_{11}^{2} - 24 T_{11} - 136 \)