Properties

Label 4005.2.a.n
Level 4005
Weight 2
Character orbit 4005.a
Self dual yes
Analytic conductor 31.980
Analytic rank 0
Dimension 6
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: \(6\)
Coefficient field: 6.6.10407557.1
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 1335)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 + \beta_{2} ) q^{2} + ( 2 + \beta_{2} + \beta_{3} ) q^{4} + q^{5} -\beta_{4} q^{7} + ( 2 + 2 \beta_{2} + \beta_{3} - \beta_{5} ) q^{8} +O(q^{10})\) \( q + ( 1 + \beta_{2} ) q^{2} + ( 2 + \beta_{2} + \beta_{3} ) q^{4} + q^{5} -\beta_{4} q^{7} + ( 2 + 2 \beta_{2} + \beta_{3} - \beta_{5} ) q^{8} + ( 1 + \beta_{2} ) q^{10} + \beta_{1} q^{13} + ( 1 - \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{14} + ( 2 - \beta_{1} + 3 \beta_{2} + \beta_{3} + \beta_{4} - 2 \beta_{5} ) q^{16} + ( 3 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} ) q^{17} + 2 q^{19} + ( 2 + \beta_{2} + \beta_{3} ) q^{20} + ( 3 - \beta_{2} + \beta_{3} - \beta_{4} ) q^{23} + q^{25} + \beta_{4} q^{26} + ( \beta_{1} - 3 \beta_{4} + 2 \beta_{5} ) q^{28} + ( \beta_{1} - \beta_{5} ) q^{29} + ( 2 + 2 \beta_{4} ) q^{31} + ( 3 - 2 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} + 3 \beta_{4} - 2 \beta_{5} ) q^{32} + ( 1 + \beta_{1} + 3 \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{34} -\beta_{4} q^{35} + ( -1 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} ) q^{37} + ( 2 + 2 \beta_{2} ) q^{38} + ( 2 + 2 \beta_{2} + \beta_{3} - \beta_{5} ) q^{40} + ( -1 - 2 \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{41} + ( -1 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} ) q^{43} + ( 4 \beta_{2} - 2 \beta_{4} ) q^{46} + ( 2 - 2 \beta_{3} + \beta_{4} ) q^{47} + ( 1 + \beta_{1} - 2 \beta_{4} + 2 \beta_{5} ) q^{49} + ( 1 + \beta_{2} ) q^{50} + ( -1 - 2 \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{52} + ( 4 - 2 \beta_{2} - 2 \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{53} + ( 3 + 2 \beta_{1} - 3 \beta_{2} - \beta_{3} - 3 \beta_{4} + 3 \beta_{5} ) q^{56} + ( -1 - \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{58} + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} ) q^{59} + ( 2 - 2 \beta_{4} + 2 \beta_{5} ) q^{61} + ( 4 \beta_{2} - 2 \beta_{3} + 4 \beta_{4} - 2 \beta_{5} ) q^{62} + ( 1 + 6 \beta_{2} + 4 \beta_{4} - 3 \beta_{5} ) q^{64} + \beta_{1} q^{65} + ( -1 + 2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - 2 \beta_{5} ) q^{67} + ( 8 - \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{68} + ( 1 - \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{70} + ( 2 - 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{71} + ( -1 - 4 \beta_{1} + \beta_{2} + 3 \beta_{3} + 3 \beta_{4} - 2 \beta_{5} ) q^{73} + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{74} + ( 4 + 2 \beta_{2} + 2 \beta_{3} ) q^{76} + ( 1 + 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{79} + ( 2 - \beta_{1} + 3 \beta_{2} + \beta_{3} + \beta_{4} - 2 \beta_{5} ) q^{80} + ( -1 - 2 \beta_{1} + \beta_{2} + \beta_{3} + 4 \beta_{4} - 3 \beta_{5} ) q^{82} + ( 3 - \beta_{2} + \beta_{3} + \beta_{4} ) q^{83} + ( 3 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} ) q^{85} + ( 3 + \beta_{1} - 3 \beta_{2} - \beta_{3} + \beta_{5} ) q^{86} + q^{89} + ( -3 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} ) q^{91} + ( 8 + 4 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{92} + ( 3 - \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{94} + 2 q^{95} + ( -5 + 2 \beta_{1} - 3 \beta_{2} - \beta_{3} + \beta_{4} ) q^{97} + ( 5 + 2 \beta_{1} - 3 \beta_{2} - 5 \beta_{4} + 4 \beta_{5} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 4q^{2} + 8q^{4} + 6q^{5} + q^{7} + 9q^{8} + O(q^{10}) \) \( 6q + 4q^{2} + 8q^{4} + 6q^{5} + q^{7} + 9q^{8} + 4q^{10} - 3q^{13} + 5q^{14} + 12q^{16} + 13q^{17} + 12q^{19} + 8q^{20} + 19q^{23} + 6q^{25} - q^{26} - 6q^{28} + 10q^{31} + 17q^{32} - 2q^{34} + q^{35} - q^{37} + 8q^{38} + 9q^{40} + 4q^{41} - 7q^{43} - 6q^{46} + 15q^{47} - q^{49} + 4q^{50} + q^{52} + 27q^{53} + 14q^{56} - 6q^{58} + 4q^{59} + 8q^{61} - 2q^{62} - q^{64} - 3q^{65} - 11q^{67} + 47q^{68} + 5q^{70} + 16q^{71} + q^{73} + 10q^{74} + 16q^{76} + 12q^{80} + q^{82} + 17q^{83} + 13q^{85} + 20q^{86} + 6q^{89} - 18q^{91} + 36q^{92} + 17q^{94} + 12q^{95} - 29q^{97} + 23q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - x^{5} - 8 x^{4} + 2 x^{3} + 18 x^{2} + 7 x - 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{3} - \nu^{2} - 5 \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{4} - 2 \nu^{3} - 4 \nu^{2} + 5 \nu + 3 \)
\(\beta_{4}\)\(=\)\( \nu^{5} - 2 \nu^{4} - 6 \nu^{3} + 7 \nu^{2} + 10 \nu \)
\(\beta_{5}\)\(=\)\( 2 \nu^{5} - 4 \nu^{4} - 11 \nu^{3} + 14 \nu^{2} + 18 \nu - 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{5} - 2 \beta_{4} - \beta_{2} - \beta_{1}\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{5} - 2 \beta_{4} + 3 \beta_{2} - \beta_{1} + 12\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(3 \beta_{5} - 6 \beta_{4} - \beta_{2} - \beta_{1} + 6\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(11 \beta_{5} - 22 \beta_{4} + 4 \beta_{3} + 13 \beta_{2} - 3 \beta_{1} + 60\)\()/4\)
\(\nu^{5}\)\(=\)\((\)\(41 \beta_{5} - 78 \beta_{4} + 8 \beta_{3} + 3 \beta_{2} - \beta_{1} + 108\)\()/4\)

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
0.191235
−0.763968
2.09854
−1.43605
2.63450
−1.72426
−2.15466 0 2.64258 1.00000 0 −2.12396 −1.38454 0 −2.15466
1.2 −0.652386 0 −1.57439 1.00000 0 1.82035 2.33188 0 −0.652386
1.3 0.305330 0 −1.90677 1.00000 0 1.72660 −1.19286 0 0.305330
1.4 1.49828 0 0.244835 1.00000 0 −3.23110 −2.62972 0 1.49828
1.5 2.30610 0 3.31809 1.00000 0 4.21574 3.03965 0 2.30610
1.6 2.69734 0 5.27566 1.00000 0 −1.40763 8.83558 0 2.69734
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4005.2.a.n 6
3.b odd 2 1 1335.2.a.g 6
15.d odd 2 1 6675.2.a.u 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1335.2.a.g 6 3.b odd 2 1
4005.2.a.n 6 1.a even 1 1 trivial
6675.2.a.u 6 15.d odd 2 1

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(-1\)
\(89\) \(-1\)

Hecke kernels

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

\( T_{2}^{6} - 4 T_{2}^{5} - 2 T_{2}^{4} + 21 T_{2}^{3} - 13 T_{2}^{2} - 11 T_{2} + 4 \)
\( T_{7}^{6} - T_{7}^{5} - 20 T_{7}^{4} + 7 T_{7}^{3} + 96 T_{7}^{2} - 16 T_{7} - 128 \)
\( T_{11} \)