Properties

Label 4005.2.a.m
Level 4005
Weight 2
Character orbit 4005.a
Self dual Yes
Analytic conductor 31.980
Analytic rank 1
Dimension 4
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4005 = 3^{2} \cdot 5 \cdot 89 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4005.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(31.9800860095\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.2777.1
Coefficient ring: \(\Z[a_1, a_2]\)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{2} q^{2} + ( 1 - \beta_{3} ) q^{4} + q^{5} + ( -2 + \beta_{1} ) q^{7} + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{8} +O(q^{10})\) \( q -\beta_{2} q^{2} + ( 1 - \beta_{3} ) q^{4} + q^{5} + ( -2 + \beta_{1} ) q^{7} + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{8} -\beta_{2} q^{10} + 2 \beta_{2} q^{11} + ( -2 - \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{13} + ( 1 - \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{14} + ( -3 - \beta_{2} + \beta_{3} ) q^{16} + ( 3 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{17} + ( -2 - 2 \beta_{1} ) q^{19} + ( 1 - \beta_{3} ) q^{20} + ( -6 + 2 \beta_{3} ) q^{22} + ( -2 + 3 \beta_{1} + \beta_{3} ) q^{23} + q^{25} + ( -\beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{26} + ( -2 - \beta_{2} + 2 \beta_{3} ) q^{28} + ( 1 - 2 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{29} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{31} + ( -3 \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{32} + ( 1 - 3 \beta_{2} + \beta_{3} ) q^{34} + ( -2 + \beta_{1} ) q^{35} + ( -1 - 3 \beta_{1} + \beta_{2} - \beta_{3} ) q^{37} + ( -2 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{38} + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{40} -\beta_{3} q^{41} + ( -7 + 3 \beta_{1} + \beta_{2} + 3 \beta_{3} ) q^{43} + ( -2 - 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{44} + ( 2 - 4 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{46} + ( -\beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{47} + ( -1 - 3 \beta_{1} + \beta_{2} ) q^{49} -\beta_{2} q^{50} + ( -5 + \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{52} + ( -2 - \beta_{1} + 6 \beta_{3} ) q^{53} + 2 \beta_{2} q^{55} + ( -1 - 2 \beta_{2} + 3 \beta_{3} ) q^{56} + ( -3 + 4 \beta_{1} - \beta_{2} + \beta_{3} ) q^{58} + ( -2 - 3 \beta_{2} + 3 \beta_{3} ) q^{59} + ( 2 + 2 \beta_{2} - 6 \beta_{3} ) q^{61} + ( 4 - 4 \beta_{3} ) q^{62} + ( -2 + \beta_{1} + 2 \beta_{2} + 4 \beta_{3} ) q^{64} + ( -2 - \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{65} + ( -4 - \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{67} + ( 2 + \beta_{1} + \beta_{2} - 4 \beta_{3} ) q^{68} + ( 1 - \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{70} + ( 2 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{71} + ( -4 - 3 \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{73} + ( -5 + 4 \beta_{1} + \beta_{2} + 3 \beta_{3} ) q^{74} + ( -2 + 2 \beta_{2} + 2 \beta_{3} ) q^{76} + ( -2 + 2 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} ) q^{77} + ( -4 + 2 \beta_{1} + 4 \beta_{2} - 3 \beta_{3} ) q^{79} + ( -3 - \beta_{2} + \beta_{3} ) q^{80} + ( 1 + \beta_{1} - \beta_{3} ) q^{82} + ( 4 - 5 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{83} + ( 3 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{85} + ( -3 - 6 \beta_{1} + 7 \beta_{2} + \beta_{3} ) q^{86} + ( 2 + 2 \beta_{2} + 2 \beta_{3} ) q^{88} - q^{89} + ( 3 + 3 \beta_{2} - 5 \beta_{3} ) q^{91} + ( -4 - 2 \beta_{2} + 2 \beta_{3} ) q^{92} + ( 7 + 3 \beta_{1} - 3 \beta_{3} ) q^{94} + ( -2 - 2 \beta_{1} ) q^{95} + ( -4 + 5 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} ) q^{97} + ( -6 + 3 \beta_{1} + \beta_{2} + 4 \beta_{3} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{4} + 4q^{5} - 7q^{7} + 3q^{8} + O(q^{10}) \) \( 4q + 2q^{4} + 4q^{5} - 7q^{7} + 3q^{8} - 5q^{13} + q^{14} - 10q^{16} + 13q^{17} - 10q^{19} + 2q^{20} - 20q^{22} - 3q^{23} + 4q^{25} + 3q^{26} - 4q^{28} - 2q^{29} - 2q^{31} + q^{32} + 6q^{34} - 7q^{35} - 9q^{37} - 2q^{38} + 3q^{40} - 2q^{41} - 19q^{43} - 6q^{44} - 5q^{47} - 7q^{49} - 17q^{52} + 3q^{53} + 2q^{56} - 6q^{58} - 2q^{59} - 4q^{61} + 8q^{62} + q^{64} - 5q^{65} - 15q^{67} + q^{68} + q^{70} + 6q^{71} - 21q^{73} - 10q^{74} - 4q^{76} - 2q^{77} - 20q^{79} - 10q^{80} + 3q^{82} + 9q^{83} + 13q^{85} - 16q^{86} + 12q^{88} - 4q^{89} + 2q^{91} - 12q^{92} + 25q^{94} - 10q^{95} - 17q^{97} - 13q^{98} + O(q^{100}) \)

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

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

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
−1.50848
2.36234
−0.679643
0.825785
−1.78400 0 1.18264 1.00000 0 −3.50848 1.45816 0 −1.78400
1.2 −1.21831 0 −0.515722 1.00000 0 0.362340 3.06493 0 −1.21831
1.3 0.858442 0 −1.26308 1.00000 0 −2.67964 −2.80116 0 0.858442
1.4 2.14386 0 2.59615 1.00000 0 −1.17422 1.27807 0 2.14386
\(n\): 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
\(3\) \(-1\)
\(5\) \(-1\)
\(89\) \(1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4005))\):

\( T_{2}^{4} - 5 T_{2}^{2} - T_{2} + 4 \)
\( T_{7}^{4} + 7 T_{7}^{3} + 14 T_{7}^{2} + 5 T_{7} - 4 \)
\( T_{11}^{4} - 20 T_{11}^{2} + 8 T_{11} + 64 \)