Properties

Label 8028.2.a.i
Level 8028
Weight 2
Character orbit 8028.a
Self dual Yes
Analytic conductor 64.104
Analytic rank 0
Dimension 5
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 8028 = 2^{2} \cdot 3^{2} \cdot 223 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8028.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(64.1039027427\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.1710888.1
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( 1 - \beta_{1} + \beta_{2} ) q^{5} \) \( + ( 1 - \beta_{4} ) q^{7} \) \(+O(q^{10})\) \( q\) \( + ( 1 - \beta_{1} + \beta_{2} ) q^{5} \) \( + ( 1 - \beta_{4} ) q^{7} \) \( + ( 3 + \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} ) q^{17} \) \( + ( -1 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} ) q^{19} \) \( -2 \beta_{4} q^{23} \) \( + ( -2 \beta_{1} + 3 \beta_{2} - \beta_{3} ) q^{25} \) \( + ( -2 + 3 \beta_{1} - 2 \beta_{2} ) q^{29} \) \( + ( \beta_{1} - 2 \beta_{2} ) q^{31} \) \( + ( 2 - 2 \beta_{3} - 2 \beta_{4} ) q^{35} \) \( + ( 2 - \beta_{2} - \beta_{3} - \beta_{4} ) q^{37} \) \( + ( 3 + \beta_{4} ) q^{41} \) \( + ( 2 - \beta_{1} + 4 \beta_{2} + 2 \beta_{3} ) q^{43} \) \( + ( 4 + 2 \beta_{2} + \beta_{3} + \beta_{4} ) q^{47} \) \( + ( -1 - \beta_{1} + 2 \beta_{2} ) q^{49} \) \( + ( 1 - 3 \beta_{1} + \beta_{2} + 3 \beta_{3} + \beta_{4} ) q^{53} \) \( + ( 4 - 2 \beta_{1} - 2 \beta_{2} ) q^{59} \) \( + ( -2 - 2 \beta_{1} + 2 \beta_{3} - 2 \beta_{4} ) q^{61} \) \( + ( 5 - 2 \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{67} \) \( + ( 2 - 4 \beta_{1} + 2 \beta_{2} + 2 \beta_{4} ) q^{71} \) \( + ( -3 - \beta_{1} - 3 \beta_{2} + \beta_{3} ) q^{73} \) \( + ( 3 + 2 \beta_{1} - 3 \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{79} \) \( + ( 1 + 4 \beta_{1} + \beta_{2} + 2 \beta_{4} ) q^{83} \) \( + ( 2 - 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 4 \beta_{4} ) q^{85} \) \( + ( 7 - 2 \beta_{1} + 2 \beta_{3} - 3 \beta_{4} ) q^{89} \) \( + ( 4 - 2 \beta_{1} + 4 \beta_{2} ) q^{95} \) \( + ( -4 + 2 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} ) q^{97} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(5q \) \(\mathstrut +\mathstrut 6q^{5} \) \(\mathstrut +\mathstrut 3q^{7} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(5q \) \(\mathstrut +\mathstrut 6q^{5} \) \(\mathstrut +\mathstrut 3q^{7} \) \(\mathstrut +\mathstrut 13q^{17} \) \(\mathstrut -\mathstrut q^{19} \) \(\mathstrut -\mathstrut 4q^{23} \) \(\mathstrut +\mathstrut 5q^{25} \) \(\mathstrut -\mathstrut 11q^{29} \) \(\mathstrut -\mathstrut 3q^{31} \) \(\mathstrut +\mathstrut 8q^{35} \) \(\mathstrut +\mathstrut 7q^{37} \) \(\mathstrut +\mathstrut 17q^{41} \) \(\mathstrut +\mathstrut 15q^{43} \) \(\mathstrut +\mathstrut 25q^{47} \) \(\mathstrut -\mathstrut 2q^{49} \) \(\mathstrut +\mathstrut 3q^{53} \) \(\mathstrut +\mathstrut 14q^{59} \) \(\mathstrut -\mathstrut 18q^{61} \) \(\mathstrut +\mathstrut 24q^{67} \) \(\mathstrut +\mathstrut 14q^{71} \) \(\mathstrut -\mathstrut 23q^{73} \) \(\mathstrut +\mathstrut 14q^{79} \) \(\mathstrut +\mathstrut 15q^{83} \) \(\mathstrut +\mathstrut 6q^{85} \) \(\mathstrut +\mathstrut 25q^{89} \) \(\mathstrut +\mathstrut 26q^{95} \) \(\mathstrut -\mathstrut 12q^{97} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5}\mathstrut -\mathstrut \) \(x^{4}\mathstrut -\mathstrut \) \(10\) \(x^{3}\mathstrut +\mathstrut \) \(3\) \(x^{2}\mathstrut +\mathstrut \) \(12\) \(x\mathstrut +\mathstrut \) \(4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{4} - \nu^{3} - 10 \nu^{2} + 5 \nu + 10 \)\()/2\)
\(\beta_{3}\)\(=\)\((\)\( -3 \nu^{4} + 5 \nu^{3} + 28 \nu^{2} - 27 \nu - 26 \)\()/2\)
\(\beta_{4}\)\(=\)\( -2 \nu^{4} + 3 \nu^{3} + 18 \nu^{2} - 15 \nu - 13 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\)\(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut -\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(5\)
\(\nu^{3}\)\(=\)\(-\)\(\beta_{4}\mathstrut +\mathstrut \) \(2\) \(\beta_{3}\mathstrut +\mathstrut \) \(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(7\) \(\beta_{1}\mathstrut +\mathstrut \) \(3\)
\(\nu^{4}\)\(=\)\(-\)\(11\) \(\beta_{4}\mathstrut +\mathstrut \) \(12\) \(\beta_{3}\mathstrut -\mathstrut \) \(6\) \(\beta_{2}\mathstrut +\mathstrut \) \(12\) \(\beta_{1}\mathstrut +\mathstrut \) \(43\)

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
1.42525
3.36194
−2.64396
−0.676546
−0.466688
0 0 0 −1.40326 0 −1.61825 0 0 0
1.2 0 0 0 −0.594895 0 2.48438 0 0 0
1.3 0 0 0 0.756442 0 1.69379 0 0 0
1.4 0 0 0 2.95619 0 −3.03905 0 0 0
1.5 0 0 0 4.28552 0 3.47913 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(223\) \(-1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8028))\):

\(T_{5}^{5} \) \(\mathstrut -\mathstrut 6 T_{5}^{4} \) \(\mathstrut +\mathstrut 3 T_{5}^{3} \) \(\mathstrut +\mathstrut 20 T_{5}^{2} \) \(\mathstrut -\mathstrut 4 T_{5} \) \(\mathstrut -\mathstrut 8 \)
\(T_{7}^{5} \) \(\mathstrut -\mathstrut 3 T_{7}^{4} \) \(\mathstrut -\mathstrut 12 T_{7}^{3} \) \(\mathstrut +\mathstrut 35 T_{7}^{2} \) \(\mathstrut +\mathstrut 24 T_{7} \) \(\mathstrut -\mathstrut 72 \)