Properties

Label 8048.2.a.m
Level 8048
Weight 2
Character orbit 8048.a
Self dual Yes
Analytic conductor 64.264
Analytic rank 0
Dimension 3
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 8048 = 2^{4} \cdot 503 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8048.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(64.2636035467\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.257.1
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.

\(f(q)\) \(=\) \( q\) \( -\beta_{1} q^{3} \) \( + ( -\beta_{1} + \beta_{2} ) q^{7} \) \( + \beta_{2} q^{9} \) \(+O(q^{10})\) \( q\) \( -\beta_{1} q^{3} \) \( + ( -\beta_{1} + \beta_{2} ) q^{7} \) \( + \beta_{2} q^{9} \) \( + ( -4 + \beta_{1} ) q^{11} \) \( + ( 2 - \beta_{1} - \beta_{2} ) q^{13} \) \( + ( 4 + 2 \beta_{2} ) q^{17} \) \( -4 q^{19} \) \( + ( 3 - \beta_{1} ) q^{21} \) \( + 4 q^{23} \) \( -5 q^{25} \) \( + ( 2 \beta_{1} - \beta_{2} ) q^{27} \) \( + ( -2 \beta_{1} - 2 \beta_{2} ) q^{29} \) \( + ( 4 - 2 \beta_{1} + 2 \beta_{2} ) q^{31} \) \( + ( -3 + 4 \beta_{1} - \beta_{2} ) q^{33} \) \( + ( 2 - 4 \beta_{1} ) q^{37} \) \( + ( 3 - \beta_{1} + 2 \beta_{2} ) q^{39} \) \( + ( 2 \beta_{1} + 4 \beta_{2} ) q^{41} \) \( + ( -1 - 2 \beta_{1} + 3 \beta_{2} ) q^{43} \) \( + ( -7 + 2 \beta_{1} + \beta_{2} ) q^{47} \) \( + ( -1 - \beta_{1} - 2 \beta_{2} ) q^{49} \) \( + ( -6 \beta_{1} - 2 \beta_{2} ) q^{51} \) \( + ( 6 - 4 \beta_{1} - 2 \beta_{2} ) q^{53} \) \( + 4 \beta_{1} q^{57} \) \( + ( 1 + 2 \beta_{1} + \beta_{2} ) q^{59} \) \( + 2 q^{61} \) \( + ( 3 - 2 \beta_{2} ) q^{63} \) \( + ( -11 - 2 \beta_{1} + \beta_{2} ) q^{67} \) \( -4 \beta_{1} q^{69} \) \( + ( -4 + 2 \beta_{2} ) q^{71} \) \( + ( 12 - \beta_{1} + 2 \beta_{2} ) q^{73} \) \( + 5 \beta_{1} q^{75} \) \( + ( -3 + 5 \beta_{1} - 4 \beta_{2} ) q^{77} \) \( + ( 4 - 5 \beta_{1} - 3 \beta_{2} ) q^{79} \) \( + ( -6 + \beta_{1} - 4 \beta_{2} ) q^{81} \) \( + ( \beta_{1} - \beta_{2} ) q^{83} \) \( + ( 6 + 2 \beta_{1} + 4 \beta_{2} ) q^{87} \) \( + ( 4 \beta_{1} + 2 \beta_{2} ) q^{89} \) \( + ( -3 \beta_{1} + 4 \beta_{2} ) q^{91} \) \( + ( 6 - 6 \beta_{1} ) q^{93} \) \( + ( -6 - 5 \beta_{1} + 3 \beta_{2} ) q^{97} \) \( + ( \beta_{1} - 3 \beta_{2} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(3q \) \(\mathstrut -\mathstrut q^{3} \) \(\mathstrut -\mathstrut q^{7} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(3q \) \(\mathstrut -\mathstrut q^{3} \) \(\mathstrut -\mathstrut q^{7} \) \(\mathstrut -\mathstrut 11q^{11} \) \(\mathstrut +\mathstrut 5q^{13} \) \(\mathstrut +\mathstrut 12q^{17} \) \(\mathstrut -\mathstrut 12q^{19} \) \(\mathstrut +\mathstrut 8q^{21} \) \(\mathstrut +\mathstrut 12q^{23} \) \(\mathstrut -\mathstrut 15q^{25} \) \(\mathstrut +\mathstrut 2q^{27} \) \(\mathstrut -\mathstrut 2q^{29} \) \(\mathstrut +\mathstrut 10q^{31} \) \(\mathstrut -\mathstrut 5q^{33} \) \(\mathstrut +\mathstrut 2q^{37} \) \(\mathstrut +\mathstrut 8q^{39} \) \(\mathstrut +\mathstrut 2q^{41} \) \(\mathstrut -\mathstrut 5q^{43} \) \(\mathstrut -\mathstrut 19q^{47} \) \(\mathstrut -\mathstrut 4q^{49} \) \(\mathstrut -\mathstrut 6q^{51} \) \(\mathstrut +\mathstrut 14q^{53} \) \(\mathstrut +\mathstrut 4q^{57} \) \(\mathstrut +\mathstrut 5q^{59} \) \(\mathstrut +\mathstrut 6q^{61} \) \(\mathstrut +\mathstrut 9q^{63} \) \(\mathstrut -\mathstrut 35q^{67} \) \(\mathstrut -\mathstrut 4q^{69} \) \(\mathstrut -\mathstrut 12q^{71} \) \(\mathstrut +\mathstrut 35q^{73} \) \(\mathstrut +\mathstrut 5q^{75} \) \(\mathstrut -\mathstrut 4q^{77} \) \(\mathstrut +\mathstrut 7q^{79} \) \(\mathstrut -\mathstrut 17q^{81} \) \(\mathstrut +\mathstrut q^{83} \) \(\mathstrut +\mathstrut 20q^{87} \) \(\mathstrut +\mathstrut 4q^{89} \) \(\mathstrut -\mathstrut 3q^{91} \) \(\mathstrut +\mathstrut 12q^{93} \) \(\mathstrut -\mathstrut 23q^{97} \) \(\mathstrut +\mathstrut q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(3\)

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
2.19869
0.713538
−1.91223
0 −2.19869 0 0 0 −0.364448 0 1.83424 0
1.2 0 −0.713538 0 0 0 −3.20440 0 −2.49086 0
1.3 0 1.91223 0 0 0 2.56885 0 0.656620 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
\(2\) \(-1\)
\(503\) \(1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8048))\):

\(T_{3}^{3} \) \(\mathstrut +\mathstrut T_{3}^{2} \) \(\mathstrut -\mathstrut 4 T_{3} \) \(\mathstrut -\mathstrut 3 \)
\(T_{5} \)
\(T_{7}^{3} \) \(\mathstrut +\mathstrut T_{7}^{2} \) \(\mathstrut -\mathstrut 8 T_{7} \) \(\mathstrut -\mathstrut 3 \)
\(T_{13}^{3} \) \(\mathstrut -\mathstrut 5 T_{13}^{2} \) \(\mathstrut -\mathstrut 2 T_{13} \) \(\mathstrut +\mathstrut 25 \)