Properties

Label 4005.2.a.f
Level 4005
Weight 2
Character orbit 4005.a
Self dual Yes
Analytic conductor 31.980
Analytic rank 1
Dimension 2
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: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
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 \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

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

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
−1.73205
1.73205
−1.73205 0 1.00000 −1.00000 0 −2.73205 1.73205 0 1.73205
1.2 1.73205 0 1.00000 −1.00000 0 0.732051 −1.73205 0 −1.73205
\(n\): 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4005))\):

\(T_{2}^{2} \) \(\mathstrut -\mathstrut 3 \)
\(T_{7}^{2} \) \(\mathstrut +\mathstrut 2 T_{7} \) \(\mathstrut -\mathstrut 2 \)
\(T_{11} \)