Properties

Label 8028.2.a.j
Level 8028
Weight 2
Character orbit 8028.a
Self dual Yes
Analytic conductor 64.104
Analytic rank 0
Dimension 7
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: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
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 \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( 1 + \beta_{1} + \beta_{6} ) q^{5} \) \( + ( \beta_{3} + \beta_{6} ) q^{7} \) \(+O(q^{10})\) \( q\) \( + ( 1 + \beta_{1} + \beta_{6} ) q^{5} \) \( + ( \beta_{3} + \beta_{6} ) q^{7} \) \( + ( 1 + \beta_{1} - \beta_{2} + \beta_{4} + \beta_{5} - \beta_{6} ) q^{11} \) \( + ( -1 - \beta_{2} - \beta_{3} + \beta_{4} ) q^{13} \) \( + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} + \beta_{6} ) q^{17} \) \( + ( -1 + \beta_{2} + \beta_{6} ) q^{19} \) \( + ( 2 + \beta_{2} - \beta_{4} - \beta_{5} + \beta_{6} ) q^{23} \) \( + ( 1 + \beta_{1} + \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{6} ) q^{25} \) \( + ( 3 - \beta_{2} - \beta_{5} ) q^{29} \) \( + ( 2 \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{5} + \beta_{6} ) q^{31} \) \( + ( 3 - 2 \beta_{1} + \beta_{3} - \beta_{5} + \beta_{6} ) q^{35} \) \( + ( -2 + \beta_{3} + 2 \beta_{4} - \beta_{6} ) q^{37} \) \( + ( -1 + \beta_{1} + \beta_{3} + \beta_{6} ) q^{41} \) \( + ( -3 + 2 \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{5} - \beta_{6} ) q^{43} \) \( + ( 3 - \beta_{2} + 2 \beta_{5} ) q^{47} \) \( + ( 1 - \beta_{3} + \beta_{4} - \beta_{5} ) q^{49} \) \( + ( 6 - \beta_{1} + 2 \beta_{2} - \beta_{4} - \beta_{5} + \beta_{6} ) q^{53} \) \( + ( 4 \beta_{1} - \beta_{2} - 2 \beta_{3} + 3 \beta_{4} + 3 \beta_{5} ) q^{55} \) \( + ( 4 - 2 \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} ) q^{59} \) \( + ( -3 + \beta_{1} - 3 \beta_{2} - \beta_{4} + \beta_{5} + 2 \beta_{6} ) q^{61} \) \( + ( -4 \beta_{1} - 3 \beta_{2} + \beta_{4} + \beta_{5} - 3 \beta_{6} ) q^{65} \) \( + ( -3 + \beta_{3} - \beta_{5} ) q^{67} \) \( + ( 1 - \beta_{1} - \beta_{2} + \beta_{4} - 3 \beta_{5} + \beta_{6} ) q^{71} \) \( + ( 4 + \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + \beta_{4} - 2 \beta_{5} ) q^{73} \) \( + ( 1 + \beta_{2} + \beta_{3} + 3 \beta_{4} + 4 \beta_{5} ) q^{77} \) \( + ( -1 + 2 \beta_{1} - 3 \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{5} + \beta_{6} ) q^{79} \) \( + ( 6 + 2 \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} + 3 \beta_{6} ) q^{83} \) \( + ( 2 + \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{5} - \beta_{6} ) q^{85} \) \( + ( 4 \beta_{1} - 3 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{89} \) \( + ( \beta_{1} - 2 \beta_{2} - \beta_{3} + 3 \beta_{5} ) q^{91} \) \( + ( 1 - 3 \beta_{1} + 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{6} ) q^{95} \) \( + ( 5 + \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{97} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(7q \) \(\mathstrut +\mathstrut 7q^{5} \) \(\mathstrut -\mathstrut q^{7} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(7q \) \(\mathstrut +\mathstrut 7q^{5} \) \(\mathstrut -\mathstrut q^{7} \) \(\mathstrut +\mathstrut 12q^{11} \) \(\mathstrut -\mathstrut 9q^{13} \) \(\mathstrut -\mathstrut 8q^{17} \) \(\mathstrut -\mathstrut 12q^{19} \) \(\mathstrut +\mathstrut 12q^{23} \) \(\mathstrut +\mathstrut 12q^{25} \) \(\mathstrut +\mathstrut 24q^{29} \) \(\mathstrut +\mathstrut q^{31} \) \(\mathstrut +\mathstrut 15q^{35} \) \(\mathstrut -\mathstrut 13q^{37} \) \(\mathstrut -\mathstrut 5q^{41} \) \(\mathstrut -\mathstrut 13q^{43} \) \(\mathstrut +\mathstrut 21q^{47} \) \(\mathstrut +\mathstrut 4q^{49} \) \(\mathstrut +\mathstrut 35q^{53} \) \(\mathstrut +\mathstrut q^{55} \) \(\mathstrut +\mathstrut 23q^{59} \) \(\mathstrut -\mathstrut 17q^{61} \) \(\mathstrut -\mathstrut 18q^{67} \) \(\mathstrut +\mathstrut 4q^{71} \) \(\mathstrut +\mathstrut 23q^{73} \) \(\mathstrut -\mathstrut 3q^{77} \) \(\mathstrut +\mathstrut 4q^{79} \) \(\mathstrut +\mathstrut 44q^{83} \) \(\mathstrut +\mathstrut 20q^{85} \) \(\mathstrut +\mathstrut 2q^{91} \) \(\mathstrut +\mathstrut 12q^{95} \) \(\mathstrut +\mathstrut 32q^{97} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{7}\mathstrut -\mathstrut \) \(3\) \(x^{6}\mathstrut -\mathstrut \) \(11\) \(x^{5}\mathstrut +\mathstrut \) \(24\) \(x^{4}\mathstrut +\mathstrut \) \(38\) \(x^{3}\mathstrut -\mathstrut \) \(46\) \(x^{2}\mathstrut -\mathstrut \) \(36\) \(x\mathstrut +\mathstrut \) \(9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{6} - 4 \nu^{5} - 3 \nu^{4} + 15 \nu^{3} - 5 \nu^{2} + 11 \nu - 3 \)\()/12\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{6} - 2 \nu^{5} - 13 \nu^{4} + 15 \nu^{3} + 33 \nu^{2} - 13 \nu + 15 \)\()/12\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{5} - 3 \nu^{4} - 10 \nu^{3} + 19 \nu^{2} + 26 \nu - 15 \)\()/6\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{6} - 2 \nu^{5} - 13 \nu^{4} + 15 \nu^{3} + 45 \nu^{2} - 37 \nu - 33 \)\()/12\)
\(\beta_{6}\)\(=\)\((\)\( -\nu^{6} + 4 \nu^{5} + 7 \nu^{4} - 29 \nu^{3} - 13 \nu^{2} + 41 \nu + 3 \)\()/6\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(2\) \(\beta_{1}\mathstrut +\mathstrut \) \(4\)
\(\nu^{3}\)\(=\)\(\beta_{6}\mathstrut +\mathstrut \) \(3\) \(\beta_{5}\mathstrut -\mathstrut \) \(2\) \(\beta_{4}\mathstrut -\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(10\) \(\beta_{1}\mathstrut +\mathstrut \) \(4\)
\(\nu^{4}\)\(=\)\(5\) \(\beta_{6}\mathstrut +\mathstrut \) \(15\) \(\beta_{5}\mathstrut -\mathstrut \) \(7\) \(\beta_{4}\mathstrut -\mathstrut \) \(8\) \(\beta_{3}\mathstrut +\mathstrut \) \(3\) \(\beta_{2}\mathstrut +\mathstrut \) \(31\) \(\beta_{1}\mathstrut +\mathstrut \) \(32\)
\(\nu^{5}\)\(=\)\(25\) \(\beta_{6}\mathstrut +\mathstrut \) \(56\) \(\beta_{5}\mathstrut -\mathstrut \) \(35\) \(\beta_{4}\mathstrut -\mathstrut \) \(15\) \(\beta_{3}\mathstrut +\mathstrut \) \(9\) \(\beta_{2}\mathstrut +\mathstrut \) \(129\) \(\beta_{1}\mathstrut +\mathstrut \) \(75\)
\(\nu^{6}\)\(=\)\(100\) \(\beta_{6}\mathstrut +\mathstrut \) \(229\) \(\beta_{5}\mathstrut -\mathstrut \) \(131\) \(\beta_{4}\mathstrut -\mathstrut \) \(74\) \(\beta_{3}\mathstrut +\mathstrut \) \(57\) \(\beta_{2}\mathstrut +\mathstrut \) \(458\) \(\beta_{1}\mathstrut +\mathstrut \) \(359\)

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
−0.828592
−2.20362
2.33527
1.49109
−1.80533
0.206062
3.80512
0 0 0 −3.49299 0 −0.758389 0 0 0
1.2 0 0 0 −0.778137 0 −3.30215 0 0 0
1.3 0 0 0 0.389930 0 −3.58741 0 0 0
1.4 0 0 0 1.18873 0 2.92270 0 0 0
1.5 0 0 0 2.57376 0 2.76633 0 0 0
1.6 0 0 0 2.98220 0 2.92860 0 0 0
1.7 0 0 0 4.13650 0 −1.96968 0 0 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
\(2\) \(-1\)
\(3\) \(-1\)
\(223\) \(-1\)

Hecke kernels

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

\(T_{5}^{7} \) \(\mathstrut -\mathstrut 7 T_{5}^{6} \) \(\mathstrut +\mathstrut T_{5}^{5} \) \(\mathstrut +\mathstrut 83 T_{5}^{4} \) \(\mathstrut -\mathstrut 171 T_{5}^{3} \) \(\mathstrut +\mathstrut 30 T_{5}^{2} \) \(\mathstrut +\mathstrut 112 T_{5} \) \(\mathstrut -\mathstrut 40 \)
\(T_{7}^{7} \) \(\mathstrut +\mathstrut T_{7}^{6} \) \(\mathstrut -\mathstrut 26 T_{7}^{5} \) \(\mathstrut -\mathstrut 20 T_{7}^{4} \) \(\mathstrut +\mathstrut 218 T_{7}^{3} \) \(\mathstrut +\mathstrut 141 T_{7}^{2} \) \(\mathstrut -\mathstrut 571 T_{7} \) \(\mathstrut -\mathstrut 419 \)