Properties

Label 4928.2.a.cj
Level 4928
Weight 2
Character orbit 4928.a
Self dual Yes
Analytic conductor 39.350
Analytic rank 0
Dimension 4
CM No
Inner twists 1

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 4928 = 2^{6} \cdot 7 \cdot 11 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4928.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(39.3502781161\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.8468.1
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( -\beta_{1} q^{3} \) \( + ( 1 + \beta_{3} ) q^{5} \) \(- q^{7}\) \( + \beta_{3} q^{9} \) \(+O(q^{10})\) \( q\) \( -\beta_{1} q^{3} \) \( + ( 1 + \beta_{3} ) q^{5} \) \(- q^{7}\) \( + \beta_{3} q^{9} \) \(- q^{11}\) \( + ( 2 + \beta_{2} ) q^{13} \) \( + ( -3 \beta_{1} - \beta_{2} ) q^{15} \) \( + ( 1 + \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{17} \) \( + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{19} \) \( + \beta_{1} q^{21} \) \( + ( 2 - \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{23} \) \( + ( 1 - \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{25} \) \( + ( \beta_{1} - \beta_{2} ) q^{27} \) \( + ( 2 + 2 \beta_{1} ) q^{29} \) \( + ( -3 - 2 \beta_{1} - \beta_{2} - \beta_{3} ) q^{31} \) \( + \beta_{1} q^{33} \) \( + ( -1 - \beta_{3} ) q^{35} \) \( + ( 3 - 4 \beta_{1} - \beta_{3} ) q^{37} \) \( + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{39} \) \( + ( -3 - \beta_{1} + \beta_{3} ) q^{41} \) \( + ( -1 + \beta_{1} + 3 \beta_{2} + \beta_{3} ) q^{43} \) \( + ( 5 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{45} \) \( + ( 1 - 3 \beta_{1} - \beta_{3} ) q^{47} \) \(+ q^{49}\) \( + ( -1 - \beta_{1} + \beta_{2} - 3 \beta_{3} ) q^{51} \) \( + ( 5 + 3 \beta_{1} - \beta_{2} + \beta_{3} ) q^{53} \) \( + ( -1 - \beta_{3} ) q^{55} \) \( + ( -4 - 2 \beta_{1} - 2 \beta_{2} ) q^{57} \) \( + ( 4 + \beta_{1} + 2 \beta_{2} ) q^{59} \) \( + ( 6 - 2 \beta_{1} + \beta_{2} ) q^{61} \) \( -\beta_{3} q^{63} \) \( + 2 \beta_{1} q^{65} \) \( + ( \beta_{1} - \beta_{2} ) q^{67} \) \( + ( 4 - 5 \beta_{1} - \beta_{2} ) q^{69} \) \( + ( 3 \beta_{1} + \beta_{2} ) q^{71} \) \( + ( 1 - \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{73} \) \( + ( 2 - 6 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} ) q^{75} \) \(+ q^{77}\) \( + ( -3 - \beta_{1} + \beta_{2} + 3 \beta_{3} ) q^{79} \) \( + ( -4 - \beta_{1} - \beta_{2} - 3 \beta_{3} ) q^{81} \) \( + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{83} \) \( + ( 2 + 6 \beta_{1} - 2 \beta_{3} ) q^{85} \) \( + ( -6 - 2 \beta_{1} - 2 \beta_{3} ) q^{87} \) \( + ( -2 + 5 \beta_{1} + \beta_{2} ) q^{89} \) \( + ( -2 - \beta_{2} ) q^{91} \) \( + ( 5 + 4 \beta_{1} + 3 \beta_{3} ) q^{93} \) \( + ( 6 + 2 \beta_{3} ) q^{95} \) \( + ( -7 \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{97} \) \( -\beta_{3} q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(4q \) \(\mathstrut +\mathstrut q^{3} \) \(\mathstrut +\mathstrut 5q^{5} \) \(\mathstrut -\mathstrut 4q^{7} \) \(\mathstrut +\mathstrut q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(4q \) \(\mathstrut +\mathstrut q^{3} \) \(\mathstrut +\mathstrut 5q^{5} \) \(\mathstrut -\mathstrut 4q^{7} \) \(\mathstrut +\mathstrut q^{9} \) \(\mathstrut -\mathstrut 4q^{11} \) \(\mathstrut +\mathstrut 6q^{13} \) \(\mathstrut +\mathstrut 5q^{15} \) \(\mathstrut -\mathstrut 2q^{19} \) \(\mathstrut -\mathstrut q^{21} \) \(\mathstrut +\mathstrut 9q^{23} \) \(\mathstrut +\mathstrut 9q^{25} \) \(\mathstrut +\mathstrut q^{27} \) \(\mathstrut +\mathstrut 6q^{29} \) \(\mathstrut -\mathstrut 9q^{31} \) \(\mathstrut -\mathstrut q^{33} \) \(\mathstrut -\mathstrut 5q^{35} \) \(\mathstrut +\mathstrut 15q^{37} \) \(\mathstrut +\mathstrut 2q^{39} \) \(\mathstrut -\mathstrut 10q^{41} \) \(\mathstrut -\mathstrut 10q^{43} \) \(\mathstrut +\mathstrut 24q^{45} \) \(\mathstrut +\mathstrut 6q^{47} \) \(\mathstrut +\mathstrut 4q^{49} \) \(\mathstrut -\mathstrut 8q^{51} \) \(\mathstrut +\mathstrut 20q^{53} \) \(\mathstrut -\mathstrut 5q^{55} \) \(\mathstrut -\mathstrut 10q^{57} \) \(\mathstrut +\mathstrut 11q^{59} \) \(\mathstrut +\mathstrut 24q^{61} \) \(\mathstrut -\mathstrut q^{63} \) \(\mathstrut -\mathstrut 2q^{65} \) \(\mathstrut +\mathstrut q^{67} \) \(\mathstrut +\mathstrut 23q^{69} \) \(\mathstrut -\mathstrut 5q^{71} \) \(\mathstrut +\mathstrut 10q^{73} \) \(\mathstrut +\mathstrut 22q^{75} \) \(\mathstrut +\mathstrut 4q^{77} \) \(\mathstrut -\mathstrut 10q^{79} \) \(\mathstrut -\mathstrut 16q^{81} \) \(\mathstrut +\mathstrut 2q^{83} \) \(\mathstrut -\mathstrut 24q^{87} \) \(\mathstrut -\mathstrut 15q^{89} \) \(\mathstrut -\mathstrut 6q^{91} \) \(\mathstrut +\mathstrut 19q^{93} \) \(\mathstrut +\mathstrut 26q^{95} \) \(\mathstrut +\mathstrut 7q^{97} \) \(\mathstrut -\mathstrut q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{2}\)\(=\)\( -\nu^{3} + 5 \nu \)
\(\beta_{3}\)\(=\)\( \nu^{3} - \nu^{2} - 3 \nu + 2 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(1\)\()/2\)
\(\nu^{2}\)\(=\)\(\beta_{1}\mathstrut +\mathstrut \) \(3\)
\(\nu^{3}\)\(=\)\((\)\(5\) \(\beta_{3}\mathstrut +\mathstrut \) \(3\) \(\beta_{2}\mathstrut +\mathstrut \) \(5\) \(\beta_{1}\mathstrut +\mathstrut \) \(5\)\()/2\)

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
2.27841
−1.89122
1.31743
−0.704624
0 −2.19117 0 2.80122 0 −1.00000 0 1.80122 0
1.2 0 −0.576713 0 −1.66740 0 −1.00000 0 −2.66740 0
1.3 0 1.26438 0 −0.401352 0 −1.00000 0 −1.40135 0
1.4 0 2.50350 0 4.26753 0 −1.00000 0 3.26753 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
\(2\) \(-1\)
\(7\) \(1\)
\(11\) \(1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4928))\):

\(T_{3}^{4} \) \(\mathstrut -\mathstrut T_{3}^{3} \) \(\mathstrut -\mathstrut 6 T_{3}^{2} \) \(\mathstrut +\mathstrut 4 T_{3} \) \(\mathstrut +\mathstrut 4 \)
\(T_{5}^{4} \) \(\mathstrut -\mathstrut 5 T_{5}^{3} \) \(\mathstrut -\mathstrut 2 T_{5}^{2} \) \(\mathstrut +\mathstrut 20 T_{5} \) \(\mathstrut +\mathstrut 8 \)
\(T_{13}^{4} \) \(\mathstrut -\mathstrut 6 T_{13}^{3} \) \(\mathstrut -\mathstrut 4 T_{13}^{2} \) \(\mathstrut +\mathstrut 12 T_{13} \) \(\mathstrut +\mathstrut 8 \)
\(T_{17}^{4} \) \(\mathstrut -\mathstrut 64 T_{17}^{2} \) \(\mathstrut -\mathstrut 12 T_{17} \) \(\mathstrut +\mathstrut 848 \)
\(T_{19}^{4} \) \(\mathstrut +\mathstrut 2 T_{19}^{3} \) \(\mathstrut -\mathstrut 40 T_{19}^{2} \) \(\mathstrut +\mathstrut 64 T_{19} \) \(\mathstrut +\mathstrut 32 \)
\(T_{23}^{4} \) \(\mathstrut -\mathstrut 9 T_{23}^{3} \) \(\mathstrut -\mathstrut 28 T_{23}^{2} \) \(\mathstrut +\mathstrut 384 T_{23} \) \(\mathstrut -\mathstrut 736 \)