Properties

Label 8037.2.a.i
Level 8037
Weight 2
Character orbit 8037.a
Self dual Yes
Analytic conductor 64.176
Analytic rank 1
Dimension 6
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 8037 = 3^{2} \cdot 19 \cdot 47 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8037.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(64.1757681045\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.5476681.1
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( -1 - \beta_{4} ) q^{2} \) \( + ( 1 + \beta_{2} + 2 \beta_{4} ) q^{4} \) \( + ( -\beta_{4} - \beta_{5} ) q^{5} \) \( + \beta_{1} q^{7} \) \( + ( -2 - 2 \beta_{2} - \beta_{4} ) q^{8} \) \(+O(q^{10})\) \( q\) \( + ( -1 - \beta_{4} ) q^{2} \) \( + ( 1 + \beta_{2} + 2 \beta_{4} ) q^{4} \) \( + ( -\beta_{4} - \beta_{5} ) q^{5} \) \( + \beta_{1} q^{7} \) \( + ( -2 - 2 \beta_{2} - \beta_{4} ) q^{8} \) \( + ( 1 - \beta_{1} + \beta_{5} ) q^{10} \) \( + ( 2 + \beta_{4} ) q^{11} \) \( + ( -3 - 2 \beta_{2} - 2 \beta_{4} - \beta_{5} ) q^{13} \) \( + ( 1 - \beta_{1} + \beta_{3} + \beta_{4} + \beta_{5} ) q^{14} \) \( + ( -\beta_{2} - \beta_{4} ) q^{16} \) \( + ( 1 - \beta_{3} + \beta_{5} ) q^{17} \) \(- q^{19}\) \( + ( -1 + 2 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} ) q^{20} \) \( + ( -4 - \beta_{2} - 3 \beta_{4} ) q^{22} \) \( + ( 1 + \beta_{1} + 2 \beta_{2} + 2 \beta_{4} ) q^{23} \) \( + ( -2 - \beta_{1} - 3 \beta_{2} - 3 \beta_{4} - \beta_{5} ) q^{25} \) \( + ( 4 - \beta_{1} + \beta_{2} + 4 \beta_{4} + \beta_{5} ) q^{26} \) \( + ( -2 + \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{5} ) q^{28} \) \( + ( \beta_{1} + \beta_{2} ) q^{29} \) \( + ( 1 - \beta_{1} + 2 \beta_{2} + \beta_{5} ) q^{31} \) \( + ( 5 + 5 \beta_{2} + 3 \beta_{4} ) q^{32} \) \( + ( -1 - \beta_{4} - \beta_{5} ) q^{34} \) \( + ( \beta_{2} + \beta_{3} + 3 \beta_{4} ) q^{35} \) \( + ( -\beta_{1} + 3 \beta_{2} + \beta_{3} + \beta_{5} ) q^{37} \) \( + ( 1 + \beta_{4} ) q^{38} \) \( + ( -1 - \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + \beta_{4} ) q^{40} \) \( + ( -1 - 2 \beta_{1} + 2 \beta_{3} + 3 \beta_{4} + 2 \beta_{5} ) q^{41} \) \( + ( 2 + \beta_{1} + 4 \beta_{2} - \beta_{3} - 2 \beta_{5} ) q^{43} \) \( + ( 5 + 3 \beta_{2} + 5 \beta_{4} ) q^{44} \) \( + ( -2 - \beta_{1} - 2 \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{46} \) \(+ q^{47}\) \( + ( -2 + \beta_{2} + \beta_{5} ) q^{49} \) \( + ( 3 + 2 \beta_{2} - \beta_{3} + 3 \beta_{4} ) q^{50} \) \( + ( -5 + 2 \beta_{1} + \beta_{2} - \beta_{3} - 4 \beta_{4} ) q^{52} \) \( + ( -2 - 3 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 3 \beta_{4} + \beta_{5} ) q^{53} \) \( + ( -1 + \beta_{1} - \beta_{4} - 2 \beta_{5} ) q^{55} \) \( + ( 1 - 2 \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} ) q^{56} \) \( + ( 2 - \beta_{1} + \beta_{3} + \beta_{4} + \beta_{5} ) q^{58} \) \( + ( -2 - 3 \beta_{3} + \beta_{4} + \beta_{5} ) q^{59} \) \( + ( 2 - 2 \beta_{1} - \beta_{2} + 2 \beta_{4} + 3 \beta_{5} ) q^{61} \) \( + ( 1 + 2 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{5} ) q^{62} \) \( + ( -6 - \beta_{2} - 6 \beta_{4} ) q^{64} \) \( + ( 2 - 2 \beta_{1} - 5 \beta_{2} + 2 \beta_{3} - \beta_{4} ) q^{65} \) \( + ( -1 + \beta_{1} - \beta_{3} + 4 \beta_{4} - \beta_{5} ) q^{67} \) \( + ( -\beta_{1} + 2 \beta_{3} + \beta_{4} - \beta_{5} ) q^{68} \) \( + ( -4 + \beta_{1} - 2 \beta_{2} - 2 \beta_{4} ) q^{70} \) \( + ( -2 + 5 \beta_{1} + \beta_{5} ) q^{71} \) \( + ( 2 - 2 \beta_{1} + 6 \beta_{2} - \beta_{3} + \beta_{5} ) q^{73} \) \( + ( 4 + 3 \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{5} ) q^{74} \) \( + ( -1 - \beta_{2} - 2 \beta_{4} ) q^{76} \) \( + ( -1 + 2 \beta_{1} - \beta_{3} - \beta_{4} - \beta_{5} ) q^{77} \) \( + ( -2 - 4 \beta_{2} + 2 \beta_{4} + 4 \beta_{5} ) q^{79} \) \( + ( -\beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} ) q^{80} \) \( + ( -3 + 6 \beta_{1} + \beta_{2} - 2 \beta_{3} - 4 \beta_{5} ) q^{82} \) \( + ( 3 \beta_{1} + 5 \beta_{2} - \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{83} \) \( + ( -1 + 7 \beta_{2} - \beta_{3} + 2 \beta_{4} ) q^{85} \) \( + ( -4 \beta_{1} - 3 \beta_{2} + \beta_{3} - 4 \beta_{4} + 3 \beta_{5} ) q^{86} \) \( + ( -4 - 3 \beta_{2} - 4 \beta_{4} ) q^{88} \) \( + ( 5 - 2 \beta_{1} - 5 \beta_{2} - \beta_{3} + \beta_{5} ) q^{89} \) \( + ( 1 - 3 \beta_{1} - \beta_{2} + 2 \beta_{4} + \beta_{5} ) q^{91} \) \( + ( 3 + \beta_{1} - \beta_{3} + \beta_{4} - 2 \beta_{5} ) q^{92} \) \( + ( -1 - \beta_{4} ) q^{94} \) \( + ( \beta_{4} + \beta_{5} ) q^{95} \) \( + ( -8 - \beta_{1} - 5 \beta_{2} - 2 \beta_{3} - 5 \beta_{4} + \beta_{5} ) q^{97} \) \( + ( 4 + \beta_{1} + \beta_{2} + 3 \beta_{4} - \beta_{5} ) q^{98} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(6q \) \(\mathstrut -\mathstrut 4q^{2} \) \(\mathstrut +\mathstrut 4q^{5} \) \(\mathstrut +\mathstrut 2q^{7} \) \(\mathstrut -\mathstrut 6q^{8} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(6q \) \(\mathstrut -\mathstrut 4q^{2} \) \(\mathstrut +\mathstrut 4q^{5} \) \(\mathstrut +\mathstrut 2q^{7} \) \(\mathstrut -\mathstrut 6q^{8} \) \(\mathstrut +\mathstrut 2q^{10} \) \(\mathstrut +\mathstrut 10q^{11} \) \(\mathstrut -\mathstrut 8q^{13} \) \(\mathstrut +\mathstrut q^{14} \) \(\mathstrut +\mathstrut 4q^{16} \) \(\mathstrut +\mathstrut 3q^{17} \) \(\mathstrut -\mathstrut 6q^{19} \) \(\mathstrut -\mathstrut 7q^{20} \) \(\mathstrut -\mathstrut 16q^{22} \) \(\mathstrut +\mathstrut 10q^{26} \) \(\mathstrut -\mathstrut 7q^{28} \) \(\mathstrut -\mathstrut 2q^{31} \) \(\mathstrut +\mathstrut 14q^{32} \) \(\mathstrut -\mathstrut 2q^{34} \) \(\mathstrut -\mathstrut 7q^{35} \) \(\mathstrut -\mathstrut 9q^{37} \) \(\mathstrut +\mathstrut 4q^{38} \) \(\mathstrut -\mathstrut 4q^{40} \) \(\mathstrut -\mathstrut 18q^{41} \) \(\mathstrut +\mathstrut 9q^{43} \) \(\mathstrut +\mathstrut 14q^{44} \) \(\mathstrut -\mathstrut 7q^{46} \) \(\mathstrut +\mathstrut 6q^{47} \) \(\mathstrut -\mathstrut 16q^{49} \) \(\mathstrut +\mathstrut 7q^{50} \) \(\mathstrut -\mathstrut 21q^{52} \) \(\mathstrut -\mathstrut 20q^{53} \) \(\mathstrut +\mathstrut 2q^{55} \) \(\mathstrut +\mathstrut 5q^{56} \) \(\mathstrut +\mathstrut 7q^{58} \) \(\mathstrut -\mathstrut 19q^{59} \) \(\mathstrut +\mathstrut 13q^{62} \) \(\mathstrut -\mathstrut 22q^{64} \) \(\mathstrut +\mathstrut 22q^{65} \) \(\mathstrut -\mathstrut 11q^{67} \) \(\mathstrut -\mathstrut 14q^{70} \) \(\mathstrut -\mathstrut 4q^{71} \) \(\mathstrut -\mathstrut 7q^{73} \) \(\mathstrut +\mathstrut 27q^{74} \) \(\mathstrut +\mathstrut q^{77} \) \(\mathstrut -\mathstrut 16q^{79} \) \(\mathstrut +\mathstrut 5q^{80} \) \(\mathstrut -\mathstrut 2q^{82} \) \(\mathstrut -\mathstrut 7q^{83} \) \(\mathstrut -\mathstrut 25q^{85} \) \(\mathstrut +\mathstrut q^{86} \) \(\mathstrut -\mathstrut 10q^{88} \) \(\mathstrut +\mathstrut 33q^{89} \) \(\mathstrut -\mathstrut 4q^{91} \) \(\mathstrut +\mathstrut 21q^{92} \) \(\mathstrut -\mathstrut 4q^{94} \) \(\mathstrut -\mathstrut 4q^{95} \) \(\mathstrut -\mathstrut 34q^{97} \) \(\mathstrut +\mathstrut 20q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6}\mathstrut -\mathstrut \) \(2\) \(x^{5}\mathstrut -\mathstrut \) \(11\) \(x^{4}\mathstrut +\mathstrut \) \(19\) \(x^{3}\mathstrut +\mathstrut \) \(32\) \(x^{2}\mathstrut -\mathstrut \) \(44\) \(x\mathstrut -\mathstrut \) \(8\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{4} - 2 \nu^{3} - 5 \nu^{2} + 9 \nu - 2 \)\()/2\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{5} - 2 \nu^{4} - 3 \nu^{3} + 7 \nu^{2} - 12 \nu + 8 \)\()/4\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{5} - 4 \nu^{4} - 3 \nu^{3} + 21 \nu^{2} - 10 \nu - 4 \)\()/4\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{4} + 2 \nu^{3} + 7 \nu^{2} - 9 \nu - 8 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(5\)
\(\nu^{3}\)\(=\)\(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(5\) \(\beta_{1}\mathstrut +\mathstrut \) \(1\)
\(\nu^{4}\)\(=\)\(7\) \(\beta_{5}\mathstrut -\mathstrut \) \(2\) \(\beta_{4}\mathstrut +\mathstrut \) \(2\) \(\beta_{3}\mathstrut +\mathstrut \) \(7\) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(29\)
\(\nu^{5}\)\(=\)\(10\) \(\beta_{5}\mathstrut -\mathstrut \) \(7\) \(\beta_{4}\mathstrut +\mathstrut \) \(11\) \(\beta_{3}\mathstrut +\mathstrut \) \(7\) \(\beta_{2}\mathstrut +\mathstrut \) \(29\) \(\beta_{1}\mathstrut +\mathstrut \) \(18\)

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
−2.20947
1.40754
2.41126
−0.164281
2.84087
−2.28591
−2.24698 0 3.04892 −1.57380 0 −2.20947 −2.35690 0 3.53629
1.2 −2.24698 0 3.04892 1.32682 0 1.40754 −2.35690 0 −2.98134
1.3 −0.554958 0 −1.69202 −2.17107 0 2.41126 2.04892 0 1.20486
1.4 −0.554958 0 −1.69202 3.61612 0 −0.164281 2.04892 0 −2.00679
1.5 0.801938 0 −1.35690 −0.0216037 0 2.84087 −2.69202 0 −0.0173248
1.6 0.801938 0 −1.35690 2.82354 0 −2.28591 −2.69202 0 2.26430
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(19\) \(1\)
\(47\) \(-1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8037))\):

\(T_{2}^{3} \) \(\mathstrut +\mathstrut 2 T_{2}^{2} \) \(\mathstrut -\mathstrut T_{2} \) \(\mathstrut -\mathstrut 1 \)
\(T_{5}^{6} \) \(\mathstrut -\mathstrut 4 T_{5}^{5} \) \(\mathstrut -\mathstrut 7 T_{5}^{4} \) \(\mathstrut +\mathstrut 30 T_{5}^{3} \) \(\mathstrut +\mathstrut 14 T_{5}^{2} \) \(\mathstrut -\mathstrut 46 T_{5} \) \(\mathstrut -\mathstrut 1 \)