Properties

Label 4005.2.a.k
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: 4.4.8069.1
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.

\(f(q)\) \(=\) \( q\) \( -\beta_{1} q^{2} \) \( + ( 1 + \beta_{2} ) q^{4} \) \(- q^{5}\) \( + 3 q^{7} \) \( + ( 1 - \beta_{3} ) q^{8} \) \(+O(q^{10})\) \( q\) \( -\beta_{1} q^{2} \) \( + ( 1 + \beta_{2} ) q^{4} \) \(- q^{5}\) \( + 3 q^{7} \) \( + ( 1 - \beta_{3} ) q^{8} \) \( + \beta_{1} q^{10} \) \( + ( -1 + \beta_{1} + \beta_{3} ) q^{11} \) \( + ( -2 + \beta_{1} - \beta_{2} - \beta_{3} ) q^{13} \) \( -3 \beta_{1} q^{14} \) \( + ( -1 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{16} \) \( + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{17} \) \( + ( 2 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{19} \) \( + ( -1 - \beta_{2} ) q^{20} \) \( + ( -4 + \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{22} \) \( + ( -3 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{23} \) \(+ q^{25}\) \( + ( -3 + 3 \beta_{1} + 2 \beta_{3} ) q^{26} \) \( + ( 3 + 3 \beta_{2} ) q^{28} \) \( + ( -2 - 3 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{29} \) \( + ( -4 - 2 \beta_{2} + 3 \beta_{3} ) q^{31} \) \( + ( -1 + 2 \beta_{1} + 2 \beta_{3} ) q^{32} \) \( + ( -5 + 2 \beta_{1} - 2 \beta_{2} ) q^{34} \) \( -3 q^{35} \) \( + ( -1 + 2 \beta_{1} + 3 \beta_{2} ) q^{37} \) \( + ( -3 - \beta_{1} + \beta_{3} ) q^{38} \) \( + ( -1 + \beta_{3} ) q^{40} \) \( + ( -3 + 2 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} ) q^{41} \) \( + ( -2 - 3 \beta_{1} - \beta_{2} + \beta_{3} ) q^{43} \) \( + ( -2 + 4 \beta_{1} + \beta_{3} ) q^{44} \) \( + ( 3 + 2 \beta_{1} - 2 \beta_{3} ) q^{46} \) \( + ( -2 - 3 \beta_{1} - \beta_{2} + 4 \beta_{3} ) q^{47} \) \( + 2 q^{49} \) \( -\beta_{1} q^{50} \) \( + ( -7 + \beta_{1} - 3 \beta_{2} ) q^{52} \) \( + ( -2 + 3 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} ) q^{53} \) \( + ( 1 - \beta_{1} - \beta_{3} ) q^{55} \) \( + ( 3 - 3 \beta_{3} ) q^{56} \) \( + ( 12 + 4 \beta_{2} - \beta_{3} ) q^{58} \) \( + ( 5 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{59} \) \( + ( -3 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{61} \) \( + ( -5 + 6 \beta_{1} - 3 \beta_{2} - \beta_{3} ) q^{62} \) \( + ( -6 + 3 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} ) q^{64} \) \( + ( 2 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{65} \) \( + ( 5 - 2 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} ) q^{67} \) \( + ( -6 + 5 \beta_{1} ) q^{68} \) \( + 3 \beta_{1} q^{70} \) \( + ( 7 - 2 \beta_{1} + 2 \beta_{3} ) q^{71} \) \( + ( -3 \beta_{1} - 2 \beta_{2} ) q^{73} \) \( + ( -3 - 2 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} ) q^{74} \) \( + ( 2 - \beta_{1} - 2 \beta_{2} + 3 \beta_{3} ) q^{76} \) \( + ( -3 + 3 \beta_{1} + 3 \beta_{3} ) q^{77} \) \( + ( -2 + \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{79} \) \( + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{80} \) \( + ( -7 + 6 \beta_{1} + 5 \beta_{3} ) q^{82} \) \( + ( -4 - 4 \beta_{1} + \beta_{2} + \beta_{3} ) q^{83} \) \( + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{85} \) \( + ( 7 + 3 \beta_{1} + 2 \beta_{2} ) q^{86} \) \( + ( -5 - \beta_{2} + \beta_{3} ) q^{88} \) \(+ q^{89}\) \( + ( -6 + 3 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} ) q^{91} \) \( + ( 2 - \beta_{1} - 2 \beta_{2} ) q^{92} \) \( + ( 4 + 3 \beta_{1} - \beta_{2} - 3 \beta_{3} ) q^{94} \) \( + ( -2 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{95} \) \( + ( -4 - 3 \beta_{2} ) q^{97} \) \( -2 \beta_{1} q^{98} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(4q \) \(\mathstrut -\mathstrut q^{2} \) \(\mathstrut +\mathstrut 3q^{4} \) \(\mathstrut -\mathstrut 4q^{5} \) \(\mathstrut +\mathstrut 12q^{7} \) \(\mathstrut +\mathstrut 3q^{8} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(4q \) \(\mathstrut -\mathstrut q^{2} \) \(\mathstrut +\mathstrut 3q^{4} \) \(\mathstrut -\mathstrut 4q^{5} \) \(\mathstrut +\mathstrut 12q^{7} \) \(\mathstrut +\mathstrut 3q^{8} \) \(\mathstrut +\mathstrut q^{10} \) \(\mathstrut -\mathstrut 2q^{11} \) \(\mathstrut -\mathstrut 7q^{13} \) \(\mathstrut -\mathstrut 3q^{14} \) \(\mathstrut -\mathstrut 3q^{16} \) \(\mathstrut -\mathstrut q^{17} \) \(\mathstrut -\mathstrut q^{19} \) \(\mathstrut -\mathstrut 3q^{20} \) \(\mathstrut -\mathstrut 14q^{22} \) \(\mathstrut -\mathstrut 13q^{23} \) \(\mathstrut +\mathstrut 4q^{25} \) \(\mathstrut -\mathstrut 7q^{26} \) \(\mathstrut +\mathstrut 9q^{28} \) \(\mathstrut -\mathstrut 14q^{29} \) \(\mathstrut -\mathstrut 11q^{31} \) \(\mathstrut -\mathstrut 16q^{34} \) \(\mathstrut -\mathstrut 12q^{35} \) \(\mathstrut -\mathstrut 5q^{37} \) \(\mathstrut -\mathstrut 12q^{38} \) \(\mathstrut -\mathstrut 3q^{40} \) \(\mathstrut -\mathstrut 9q^{41} \) \(\mathstrut -\mathstrut 9q^{43} \) \(\mathstrut -\mathstrut 3q^{44} \) \(\mathstrut +\mathstrut 12q^{46} \) \(\mathstrut -\mathstrut 6q^{47} \) \(\mathstrut +\mathstrut 8q^{49} \) \(\mathstrut -\mathstrut q^{50} \) \(\mathstrut -\mathstrut 24q^{52} \) \(\mathstrut -\mathstrut 5q^{53} \) \(\mathstrut +\mathstrut 2q^{55} \) \(\mathstrut +\mathstrut 9q^{56} \) \(\mathstrut +\mathstrut 43q^{58} \) \(\mathstrut +\mathstrut 18q^{59} \) \(\mathstrut -\mathstrut 3q^{61} \) \(\mathstrut -\mathstrut 12q^{62} \) \(\mathstrut -\mathstrut 23q^{64} \) \(\mathstrut +\mathstrut 7q^{65} \) \(\mathstrut +\mathstrut 13q^{67} \) \(\mathstrut -\mathstrut 19q^{68} \) \(\mathstrut +\mathstrut 3q^{70} \) \(\mathstrut +\mathstrut 28q^{71} \) \(\mathstrut -\mathstrut q^{73} \) \(\mathstrut -\mathstrut 15q^{74} \) \(\mathstrut +\mathstrut 12q^{76} \) \(\mathstrut -\mathstrut 6q^{77} \) \(\mathstrut -\mathstrut 7q^{79} \) \(\mathstrut +\mathstrut 3q^{80} \) \(\mathstrut -\mathstrut 17q^{82} \) \(\mathstrut -\mathstrut 20q^{83} \) \(\mathstrut +\mathstrut q^{85} \) \(\mathstrut +\mathstrut 29q^{86} \) \(\mathstrut -\mathstrut 18q^{88} \) \(\mathstrut +\mathstrut 4q^{89} \) \(\mathstrut -\mathstrut 21q^{91} \) \(\mathstrut +\mathstrut 9q^{92} \) \(\mathstrut +\mathstrut 17q^{94} \) \(\mathstrut +\mathstrut q^{95} \) \(\mathstrut -\mathstrut 13q^{97} \) \(\mathstrut -\mathstrut 2q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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

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

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
2.14281
1.23307
−0.171704
−2.20418
−2.14281 0 2.59164 −1.00000 0 3.00000 −1.26778 0 2.14281
1.2 −1.23307 0 −0.479533 −1.00000 0 3.00000 3.05744 0 1.23307
1.3 0.171704 0 −1.97052 −1.00000 0 3.00000 −0.681754 0 −0.171704
1.4 2.20418 0 2.85841 −1.00000 0 3.00000 1.89209 0 −2.20418
\(n\): 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4005))\):

\(T_{2}^{4} \) \(\mathstrut +\mathstrut T_{2}^{3} \) \(\mathstrut -\mathstrut 5 T_{2}^{2} \) \(\mathstrut -\mathstrut 5 T_{2} \) \(\mathstrut +\mathstrut 1 \)
\(T_{7} \) \(\mathstrut -\mathstrut 3 \)
\(T_{11}^{4} \) \(\mathstrut +\mathstrut 2 T_{11}^{3} \) \(\mathstrut -\mathstrut 14 T_{11}^{2} \) \(\mathstrut -\mathstrut 19 T_{11} \) \(\mathstrut +\mathstrut 13 \)