Properties

Label 4002.2.a.bc
Level 4002
Weight 2
Character orbit 4002.a
Self dual Yes
Analytic conductor 31.956
Analytic rank 1
Dimension 4
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4002 = 2 \cdot 3 \cdot 23 \cdot 29 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4002.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(31.9561308889\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.11324.1
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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.

\(f(q)\) \(=\) \( q\) \(+ q^{2}\) \(+ q^{3}\) \(+ q^{4}\) \( + ( -1 + \beta_{1} ) q^{5} \) \(+ q^{6}\) \( + ( -1 - \beta_{1} ) q^{7} \) \(+ q^{8}\) \(+ q^{9}\) \(+O(q^{10})\) \( q\) \(+ q^{2}\) \(+ q^{3}\) \(+ q^{4}\) \( + ( -1 + \beta_{1} ) q^{5} \) \(+ q^{6}\) \( + ( -1 - \beta_{1} ) q^{7} \) \(+ q^{8}\) \(+ q^{9}\) \( + ( -1 + \beta_{1} ) q^{10} \) \( + ( -3 + \beta_{1} - \beta_{2} ) q^{11} \) \(+ q^{12}\) \( + ( -1 - \beta_{1} + \beta_{3} ) q^{13} \) \( + ( -1 - \beta_{1} ) q^{14} \) \( + ( -1 + \beta_{1} ) q^{15} \) \(+ q^{16}\) \( + ( -1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{17} \) \(+ q^{18}\) \( + ( -3 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{19} \) \( + ( -1 + \beta_{1} ) q^{20} \) \( + ( -1 - \beta_{1} ) q^{21} \) \( + ( -3 + \beta_{1} - \beta_{2} ) q^{22} \) \(- q^{23}\) \(+ q^{24}\) \( + ( -3 \beta_{1} + \beta_{2} - \beta_{3} ) q^{25} \) \( + ( -1 - \beta_{1} + \beta_{3} ) q^{26} \) \(+ q^{27}\) \( + ( -1 - \beta_{1} ) q^{28} \) \(- q^{29}\) \( + ( -1 + \beta_{1} ) q^{30} \) \( + ( -1 + \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{31} \) \(+ q^{32}\) \( + ( -3 + \beta_{1} - \beta_{2} ) q^{33} \) \( + ( -1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{34} \) \( + ( -3 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{35} \) \(+ q^{36}\) \( + ( -2 - 2 \beta_{2} + \beta_{3} ) q^{37} \) \( + ( -3 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{38} \) \( + ( -1 - \beta_{1} + \beta_{3} ) q^{39} \) \( + ( -1 + \beta_{1} ) q^{40} \) \( + ( - \beta_{2} + 2 \beta_{3} ) q^{41} \) \( + ( -1 - \beta_{1} ) q^{42} \) \( + ( -3 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{43} \) \( + ( -3 + \beta_{1} - \beta_{2} ) q^{44} \) \( + ( -1 + \beta_{1} ) q^{45} \) \(- q^{46}\) \( + ( 1 + \beta_{1} - \beta_{2} - \beta_{3} ) q^{47} \) \(+ q^{48}\) \( + ( -2 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{49} \) \( + ( -3 \beta_{1} + \beta_{2} - \beta_{3} ) q^{50} \) \( + ( -1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{51} \) \( + ( -1 - \beta_{1} + \beta_{3} ) q^{52} \) \( + ( -8 + 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{53} \) \(+ q^{54}\) \( + ( 5 - 5 \beta_{1} - \beta_{3} ) q^{55} \) \( + ( -1 - \beta_{1} ) q^{56} \) \( + ( -3 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{57} \) \(- q^{58}\) \( + ( -1 + 3 \beta_{1} - \beta_{2} - \beta_{3} ) q^{59} \) \( + ( -1 + \beta_{1} ) q^{60} \) \( + ( -2 + 2 \beta_{1} - \beta_{2} - 3 \beta_{3} ) q^{61} \) \( + ( -1 + \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{62} \) \( + ( -1 - \beta_{1} ) q^{63} \) \(+ q^{64}\) \( + ( -3 - \beta_{1} - \beta_{3} ) q^{65} \) \( + ( -3 + \beta_{1} - \beta_{2} ) q^{66} \) \( + ( - \beta_{2} + \beta_{3} ) q^{67} \) \( + ( -1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{68} \) \(- q^{69}\) \( + ( -3 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{70} \) \( + ( 1 - \beta_{1} - 2 \beta_{2} - 3 \beta_{3} ) q^{71} \) \(+ q^{72}\) \( + ( -2 - 2 \beta_{2} + 2 \beta_{3} ) q^{73} \) \( + ( -2 - 2 \beta_{2} + \beta_{3} ) q^{74} \) \( + ( -3 \beta_{1} + \beta_{2} - \beta_{3} ) q^{75} \) \( + ( -3 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{76} \) \( + ( 1 + 3 \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{77} \) \( + ( -1 - \beta_{1} + \beta_{3} ) q^{78} \) \( + ( -5 - \beta_{1} - \beta_{3} ) q^{79} \) \( + ( -1 + \beta_{1} ) q^{80} \) \(+ q^{81}\) \( + ( - \beta_{2} + 2 \beta_{3} ) q^{82} \) \( + ( -4 + 2 \beta_{3} ) q^{83} \) \( + ( -1 - \beta_{1} ) q^{84} \) \( + ( -1 + 3 \beta_{1} - \beta_{2} + 3 \beta_{3} ) q^{85} \) \( + ( -3 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{86} \) \(- q^{87}\) \( + ( -3 + \beta_{1} - \beta_{2} ) q^{88} \) \( + ( -2 + 2 \beta_{1} + 2 \beta_{2} ) q^{89} \) \( + ( -1 + \beta_{1} ) q^{90} \) \( + ( 5 + 3 \beta_{1} - \beta_{3} ) q^{91} \) \(- q^{92}\) \( + ( -1 + \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{93} \) \( + ( 1 + \beta_{1} - \beta_{2} - \beta_{3} ) q^{94} \) \( + ( 1 + \beta_{1} - \beta_{2} + 3 \beta_{3} ) q^{95} \) \(+ q^{96}\) \( + ( 2 + 2 \beta_{1} - 5 \beta_{2} + 3 \beta_{3} ) q^{97} \) \( + ( -2 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{98} \) \( + ( -3 + \beta_{1} - \beta_{2} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(4q \) \(\mathstrut +\mathstrut 4q^{2} \) \(\mathstrut +\mathstrut 4q^{3} \) \(\mathstrut +\mathstrut 4q^{4} \) \(\mathstrut -\mathstrut 4q^{5} \) \(\mathstrut +\mathstrut 4q^{6} \) \(\mathstrut -\mathstrut 4q^{7} \) \(\mathstrut +\mathstrut 4q^{8} \) \(\mathstrut +\mathstrut 4q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(4q \) \(\mathstrut +\mathstrut 4q^{2} \) \(\mathstrut +\mathstrut 4q^{3} \) \(\mathstrut +\mathstrut 4q^{4} \) \(\mathstrut -\mathstrut 4q^{5} \) \(\mathstrut +\mathstrut 4q^{6} \) \(\mathstrut -\mathstrut 4q^{7} \) \(\mathstrut +\mathstrut 4q^{8} \) \(\mathstrut +\mathstrut 4q^{9} \) \(\mathstrut -\mathstrut 4q^{10} \) \(\mathstrut -\mathstrut 12q^{11} \) \(\mathstrut +\mathstrut 4q^{12} \) \(\mathstrut -\mathstrut 6q^{13} \) \(\mathstrut -\mathstrut 4q^{14} \) \(\mathstrut -\mathstrut 4q^{15} \) \(\mathstrut +\mathstrut 4q^{16} \) \(\mathstrut -\mathstrut 2q^{17} \) \(\mathstrut +\mathstrut 4q^{18} \) \(\mathstrut -\mathstrut 10q^{19} \) \(\mathstrut -\mathstrut 4q^{20} \) \(\mathstrut -\mathstrut 4q^{21} \) \(\mathstrut -\mathstrut 12q^{22} \) \(\mathstrut -\mathstrut 4q^{23} \) \(\mathstrut +\mathstrut 4q^{24} \) \(\mathstrut +\mathstrut 2q^{25} \) \(\mathstrut -\mathstrut 6q^{26} \) \(\mathstrut +\mathstrut 4q^{27} \) \(\mathstrut -\mathstrut 4q^{28} \) \(\mathstrut -\mathstrut 4q^{29} \) \(\mathstrut -\mathstrut 4q^{30} \) \(\mathstrut -\mathstrut 6q^{31} \) \(\mathstrut +\mathstrut 4q^{32} \) \(\mathstrut -\mathstrut 12q^{33} \) \(\mathstrut -\mathstrut 2q^{34} \) \(\mathstrut -\mathstrut 14q^{35} \) \(\mathstrut +\mathstrut 4q^{36} \) \(\mathstrut -\mathstrut 10q^{37} \) \(\mathstrut -\mathstrut 10q^{38} \) \(\mathstrut -\mathstrut 6q^{39} \) \(\mathstrut -\mathstrut 4q^{40} \) \(\mathstrut -\mathstrut 4q^{41} \) \(\mathstrut -\mathstrut 4q^{42} \) \(\mathstrut -\mathstrut 14q^{43} \) \(\mathstrut -\mathstrut 12q^{44} \) \(\mathstrut -\mathstrut 4q^{45} \) \(\mathstrut -\mathstrut 4q^{46} \) \(\mathstrut +\mathstrut 6q^{47} \) \(\mathstrut +\mathstrut 4q^{48} \) \(\mathstrut -\mathstrut 6q^{49} \) \(\mathstrut +\mathstrut 2q^{50} \) \(\mathstrut -\mathstrut 2q^{51} \) \(\mathstrut -\mathstrut 6q^{52} \) \(\mathstrut -\mathstrut 30q^{53} \) \(\mathstrut +\mathstrut 4q^{54} \) \(\mathstrut +\mathstrut 22q^{55} \) \(\mathstrut -\mathstrut 4q^{56} \) \(\mathstrut -\mathstrut 10q^{57} \) \(\mathstrut -\mathstrut 4q^{58} \) \(\mathstrut -\mathstrut 2q^{59} \) \(\mathstrut -\mathstrut 4q^{60} \) \(\mathstrut -\mathstrut 2q^{61} \) \(\mathstrut -\mathstrut 6q^{62} \) \(\mathstrut -\mathstrut 4q^{63} \) \(\mathstrut +\mathstrut 4q^{64} \) \(\mathstrut -\mathstrut 10q^{65} \) \(\mathstrut -\mathstrut 12q^{66} \) \(\mathstrut -\mathstrut 2q^{67} \) \(\mathstrut -\mathstrut 2q^{68} \) \(\mathstrut -\mathstrut 4q^{69} \) \(\mathstrut -\mathstrut 14q^{70} \) \(\mathstrut +\mathstrut 10q^{71} \) \(\mathstrut +\mathstrut 4q^{72} \) \(\mathstrut -\mathstrut 12q^{73} \) \(\mathstrut -\mathstrut 10q^{74} \) \(\mathstrut +\mathstrut 2q^{75} \) \(\mathstrut -\mathstrut 10q^{76} \) \(\mathstrut +\mathstrut 2q^{77} \) \(\mathstrut -\mathstrut 6q^{78} \) \(\mathstrut -\mathstrut 18q^{79} \) \(\mathstrut -\mathstrut 4q^{80} \) \(\mathstrut +\mathstrut 4q^{81} \) \(\mathstrut -\mathstrut 4q^{82} \) \(\mathstrut -\mathstrut 20q^{83} \) \(\mathstrut -\mathstrut 4q^{84} \) \(\mathstrut -\mathstrut 10q^{85} \) \(\mathstrut -\mathstrut 14q^{86} \) \(\mathstrut -\mathstrut 4q^{87} \) \(\mathstrut -\mathstrut 12q^{88} \) \(\mathstrut -\mathstrut 8q^{89} \) \(\mathstrut -\mathstrut 4q^{90} \) \(\mathstrut +\mathstrut 22q^{91} \) \(\mathstrut -\mathstrut 4q^{92} \) \(\mathstrut -\mathstrut 6q^{93} \) \(\mathstrut +\mathstrut 6q^{94} \) \(\mathstrut -\mathstrut 2q^{95} \) \(\mathstrut +\mathstrut 4q^{96} \) \(\mathstrut +\mathstrut 2q^{97} \) \(\mathstrut -\mathstrut 6q^{98} \) \(\mathstrut -\mathstrut 12q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{3} - 4 \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} + \nu - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{2} - \nu - 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\)\(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(6\)\()/2\)
\(\nu^{3}\)\(=\)\(-\)\(2\) \(\beta_{3}\mathstrut +\mathstrut \) \(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\)

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
1.18994
−2.08963
−0.356500
2.25619
1.00000 1.00000 1.00000 −4.07486 1.00000 2.07486 1.00000 1.00000 −4.07486
1.2 1.00000 1.00000 1.00000 −1.76597 1.00000 −0.234033 1.00000 1.00000 −1.76597
1.3 1.00000 1.00000 1.00000 0.380691 1.00000 −2.38069 1.00000 1.00000 0.380691
1.4 1.00000 1.00000 1.00000 1.46013 1.00000 −3.46013 1.00000 1.00000 1.46013
\(n\): 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
\(2\) \(-1\)
\(3\) \(-1\)
\(23\) \(1\)
\(29\) \(1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4002))\):

\(T_{5}^{4} \) \(\mathstrut +\mathstrut 4 T_{5}^{3} \) \(\mathstrut -\mathstrut 3 T_{5}^{2} \) \(\mathstrut -\mathstrut 10 T_{5} \) \(\mathstrut +\mathstrut 4 \)
\(T_{7}^{4} \) \(\mathstrut +\mathstrut 4 T_{7}^{3} \) \(\mathstrut -\mathstrut 3 T_{7}^{2} \) \(\mathstrut -\mathstrut 18 T_{7} \) \(\mathstrut -\mathstrut 4 \)
\(T_{11}^{4} \) \(\mathstrut +\mathstrut 12 T_{11}^{3} \) \(\mathstrut +\mathstrut 38 T_{11}^{2} \) \(\mathstrut -\mathstrut 12 T_{11} \) \(\mathstrut -\mathstrut 136 \)