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Properties

Label 4005.2.a.g
Level 4005
Weight 2
Character orbit 4005.a
Self dual Yes
Analytic conductor 31.980
Analytic rank 0
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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
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 \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + ( 2 + \beta ) q^{4} - q^{5} + ( 4 + \beta ) q^{8} +O(q^{10})\) \( q + \beta q^{2} + ( 2 + \beta ) q^{4} - q^{5} + ( 4 + \beta ) q^{8} -\beta q^{10} -4 q^{11} + 2 q^{13} + 3 \beta q^{16} + ( -4 + 2 \beta ) q^{17} + ( -2 + 2 \beta ) q^{19} + ( -2 - \beta ) q^{20} -4 \beta q^{22} + ( 3 + \beta ) q^{23} + q^{25} + 2 \beta q^{26} + ( -5 + 3 \beta ) q^{29} + ( 2 + 2 \beta ) q^{31} + ( 4 + \beta ) q^{32} + ( 8 - 2 \beta ) q^{34} + 10 q^{37} + 8 q^{38} + ( -4 - \beta ) q^{40} + ( 5 - 3 \beta ) q^{41} + 4 \beta q^{43} + ( -8 - 4 \beta ) q^{44} + ( 4 + 4 \beta ) q^{46} + 8 q^{47} -7 q^{49} + \beta q^{50} + ( 4 + 2 \beta ) q^{52} + ( -2 + 4 \beta ) q^{53} + 4 q^{55} + ( 12 - 2 \beta ) q^{58} + ( -5 + \beta ) q^{59} + ( 8 - 2 \beta ) q^{61} + ( 8 + 4 \beta ) q^{62} + ( 4 - \beta ) q^{64} -2 q^{65} + ( -3 - 5 \beta ) q^{67} + 2 \beta q^{68} + ( 4 - 4 \beta ) q^{71} + ( -5 + 3 \beta ) q^{73} + 10 \beta q^{74} + ( 4 + 4 \beta ) q^{76} + ( -5 + \beta ) q^{79} -3 \beta q^{80} + ( -12 + 2 \beta ) q^{82} + ( 3 - 3 \beta ) q^{83} + ( 4 - 2 \beta ) q^{85} + ( 16 + 4 \beta ) q^{86} + ( -16 - 4 \beta ) q^{88} - q^{89} + ( 10 + 6 \beta ) q^{92} + 8 \beta q^{94} + ( 2 - 2 \beta ) q^{95} + ( -11 + \beta ) q^{97} -7 \beta q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{2} + 5q^{4} - 2q^{5} + 9q^{8} + O(q^{10}) \) \( 2q + q^{2} + 5q^{4} - 2q^{5} + 9q^{8} - q^{10} - 8q^{11} + 4q^{13} + 3q^{16} - 6q^{17} - 2q^{19} - 5q^{20} - 4q^{22} + 7q^{23} + 2q^{25} + 2q^{26} - 7q^{29} + 6q^{31} + 9q^{32} + 14q^{34} + 20q^{37} + 16q^{38} - 9q^{40} + 7q^{41} + 4q^{43} - 20q^{44} + 12q^{46} + 16q^{47} - 14q^{49} + q^{50} + 10q^{52} + 8q^{55} + 22q^{58} - 9q^{59} + 14q^{61} + 20q^{62} + 7q^{64} - 4q^{65} - 11q^{67} + 2q^{68} + 4q^{71} - 7q^{73} + 10q^{74} + 12q^{76} - 9q^{79} - 3q^{80} - 22q^{82} + 3q^{83} + 6q^{85} + 36q^{86} - 36q^{88} - 2q^{89} + 26q^{92} + 8q^{94} + 2q^{95} - 21q^{97} - 7q^{98} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

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
−1.56155
2.56155
−1.56155 0 0.438447 −1.00000 0 0 2.43845 0 1.56155
1.2 2.56155 0 4.56155 −1.00000 0 0 6.56155 0 −2.56155
\(n\): 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4005))\):

\( T_{2}^{2} - T_{2} - 4 \)
\( T_{7} \)
\( T_{11} + 4 \)