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: \( 2^{2} \)
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.

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

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

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

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.

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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4005))\):

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