Properties

Label 4004.2.a.f
Level 4004
Weight 2
Character orbit 4004.a
Self dual Yes
Analytic conductor 31.972
Analytic rank 1
Dimension 5
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4004 = 2^{2} \cdot 7 \cdot 11 \cdot 13 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4004.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(31.9721009693\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: 5.5.463341.1
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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)  \(=\)  \( q\) \( + ( 1 - \beta_{1} ) q^{3} \) \( - \beta_{2} q^{5} \) \(- q^{7}\) \( + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + ( 1 - \beta_{1} ) q^{3} \) \( - \beta_{2} q^{5} \) \(- q^{7}\) \( + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{9} \) \(- q^{11}\) \(+ q^{13}\) \( + ( -2 + 2 \beta_{1} + \beta_{3} + \beta_{4} ) q^{15} \) \( + ( -3 + \beta_{1} + 2 \beta_{3} + \beta_{4} ) q^{17} \) \( + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} ) q^{19} \) \( + ( -1 + \beta_{1} ) q^{21} \) \( + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} ) q^{23} \) \( + ( 1 - \beta_{2} - 2 \beta_{3} - \beta_{4} ) q^{25} \) \( + ( 2 - \beta_{1} + \beta_{2} - 3 \beta_{3} - \beta_{4} ) q^{27} \) \( + ( -1 + 2 \beta_{2} - \beta_{3} - \beta_{4} ) q^{29} \) \( + ( 1 + 2 \beta_{1} - 2 \beta_{3} ) q^{31} \) \( + ( -1 + \beta_{1} ) q^{33} \) \( + \beta_{2} q^{35} \) \( + ( -3 + 2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{37} \) \( + ( 1 - \beta_{1} ) q^{39} \) \( + ( -4 + \beta_{4} ) q^{41} \) \( + ( -5 - \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{43} \) \( + ( -7 + 2 \beta_{1} + 3 \beta_{3} + \beta_{4} ) q^{45} \) \( + ( 3 - \beta_{1} + \beta_{2} - 3 \beta_{3} + \beta_{4} ) q^{47} \) \(+ q^{49}\) \( + ( -4 + 4 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} + \beta_{4} ) q^{51} \) \( + ( 2 + \beta_{1} + 2 \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{53} \) \( + \beta_{2} q^{55} \) \( + ( -1 + \beta_{1} + \beta_{3} - 2 \beta_{4} ) q^{57} \) \( + ( 2 \beta_{1} + \beta_{3} - 3 \beta_{4} ) q^{59} \) \( + ( -5 - \beta_{1} + 3 \beta_{3} - \beta_{4} ) q^{61} \) \( + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{63} \) \( - \beta_{2} q^{65} \) \( + ( -4 + 3 \beta_{1} + \beta_{2} + 4 \beta_{3} + \beta_{4} ) q^{67} \) \( + ( -5 + 5 \beta_{1} + 3 \beta_{3} ) q^{69} \) \( + ( 2 + 4 \beta_{4} ) q^{71} \) \( + ( -8 + 4 \beta_{1} + \beta_{2} + 3 \beta_{3} + 2 \beta_{4} ) q^{73} \) \( + ( -3 + \beta_{2} - \beta_{3} ) q^{75} \) \(+ q^{77}\) \( + ( -5 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 3 \beta_{4} ) q^{79} \) \( + ( 1 - 3 \beta_{1} - \beta_{2} - 2 \beta_{3} - 2 \beta_{4} ) q^{81} \) \( + ( 2 + 2 \beta_{1} - \beta_{2} - 3 \beta_{3} ) q^{83} \) \( + ( -1 - 3 \beta_{1} + 4 \beta_{2} + 2 \beta_{4} ) q^{85} \) \( + ( 2 - 3 \beta_{1} + \beta_{2} - 3 \beta_{3} - 3 \beta_{4} ) q^{87} \) \( + ( -5 + \beta_{1} + \beta_{2} - \beta_{3} + 5 \beta_{4} ) q^{89} \) \(- q^{91}\) \( + ( -7 - 3 \beta_{1} - 2 \beta_{2} ) q^{93} \) \( + ( -4 - \beta_{1} + 2 \beta_{2} ) q^{95} \) \( + ( -5 - 2 \beta_{1} - 2 \beta_{3} - \beta_{4} ) q^{97} \) \( + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)  \(=\)  \(5q \) \(\mathstrut +\mathstrut 3q^{3} \) \(\mathstrut -\mathstrut 5q^{7} \) \(\mathstrut +\mathstrut 2q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(5q \) \(\mathstrut +\mathstrut 3q^{3} \) \(\mathstrut -\mathstrut 5q^{7} \) \(\mathstrut +\mathstrut 2q^{9} \) \(\mathstrut -\mathstrut 5q^{11} \) \(\mathstrut +\mathstrut 5q^{13} \) \(\mathstrut -\mathstrut 3q^{15} \) \(\mathstrut -\mathstrut 9q^{17} \) \(\mathstrut -\mathstrut 4q^{19} \) \(\mathstrut -\mathstrut 3q^{21} \) \(\mathstrut -\mathstrut 4q^{23} \) \(\mathstrut +\mathstrut q^{25} \) \(\mathstrut +\mathstrut 3q^{27} \) \(\mathstrut -\mathstrut 8q^{29} \) \(\mathstrut +\mathstrut 7q^{31} \) \(\mathstrut -\mathstrut 3q^{33} \) \(\mathstrut -\mathstrut 10q^{37} \) \(\mathstrut +\mathstrut 3q^{39} \) \(\mathstrut -\mathstrut 18q^{41} \) \(\mathstrut -\mathstrut 22q^{43} \) \(\mathstrut -\mathstrut 26q^{45} \) \(\mathstrut +\mathstrut 12q^{47} \) \(\mathstrut +\mathstrut 5q^{49} \) \(\mathstrut -\mathstrut 7q^{51} \) \(\mathstrut +\mathstrut 7q^{53} \) \(\mathstrut -\mathstrut 6q^{57} \) \(\mathstrut -\mathstrut q^{59} \) \(\mathstrut -\mathstrut 26q^{61} \) \(\mathstrut -\mathstrut 2q^{63} \) \(\mathstrut -\mathstrut 8q^{67} \) \(\mathstrut -\mathstrut 12q^{69} \) \(\mathstrut +\mathstrut 18q^{71} \) \(\mathstrut -\mathstrut 25q^{73} \) \(\mathstrut -\mathstrut 16q^{75} \) \(\mathstrut +\mathstrut 5q^{77} \) \(\mathstrut -\mathstrut 6q^{79} \) \(\mathstrut -\mathstrut 7q^{81} \) \(\mathstrut +\mathstrut 11q^{83} \) \(\mathstrut -\mathstrut 7q^{85} \) \(\mathstrut -\mathstrut 5q^{87} \) \(\mathstrut -\mathstrut 14q^{89} \) \(\mathstrut -\mathstrut 5q^{91} \) \(\mathstrut -\mathstrut 41q^{93} \) \(\mathstrut -\mathstrut 22q^{95} \) \(\mathstrut -\mathstrut 33q^{97} \) \(\mathstrut -\mathstrut 2q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5}\mathstrut -\mathstrut \) \(2\) \(x^{4}\mathstrut -\mathstrut \) \(6\) \(x^{3}\mathstrut +\mathstrut \) \(6\) \(x^{2}\mathstrut +\mathstrut \) \(6\) \(x\mathstrut +\mathstrut \) \(1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{4} - 2 \nu^{3} - 5 \nu^{2} + 5 \nu + 2 \)
\(\beta_{3}\)\(=\)\( \nu^{4} - 2 \nu^{3} - 6 \nu^{2} + 6 \nu + 5 \)
\(\beta_{4}\)\(=\)\( -2 \nu^{4} + 5 \nu^{3} + 10 \nu^{2} - 17 \nu - 6 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\)\(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(3\)
\(\nu^{3}\)\(=\)\(\beta_{4}\mathstrut +\mathstrut \) \(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(7\) \(\beta_{1}\mathstrut +\mathstrut \) \(2\)
\(\nu^{4}\)\(=\)\(2\) \(\beta_{4}\mathstrut -\mathstrut \) \(5\) \(\beta_{3}\mathstrut +\mathstrut \) \(10\) \(\beta_{2}\mathstrut +\mathstrut \) \(14\) \(\beta_{1}\mathstrut +\mathstrut \) \(17\)

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
3.09892
1.53838
−0.231747
−0.466886
−1.93866
0 −2.09892 0 −2.18167 0 −1.00000 0 1.40545 0
1.2 0 −0.538379 0 3.82180 0 −1.00000 0 −2.71015 0
1.3 0 1.23175 0 −0.600512 0 −1.00000 0 −1.48280 0
1.4 0 1.46689 0 1.17328 0 −1.00000 0 −0.848245 0
1.5 0 2.93866 0 −2.21290 0 −1.00000 0 5.63574 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
\(2\) \(-1\)
\(7\) \(1\)
\(11\) \(1\)
\(13\) \(-1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{3}^{5} \) \(\mathstrut -\mathstrut 3 T_{3}^{4} \) \(\mathstrut -\mathstrut 4 T_{3}^{3} \) \(\mathstrut +\mathstrut 14 T_{3}^{2} \) \(\mathstrut -\mathstrut 3 T_{3} \) \(\mathstrut -\mathstrut 6 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4004))\).