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.

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

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

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

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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8036))\):

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