Properties

Label 4005.2.a.l
Level 4005
Weight 2
Character orbit 4005.a
Self dual Yes
Analytic conductor 31.980
Analytic rank 0
Dimension 4
CM No
Inner twists 1

Related objects

Downloads

Learn more about

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: \(4\)
Coefficient field: 4.4.725.1
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.

\(f(q)\) \(=\) \( q\) \( + ( -1 + \beta_{1} + \beta_{2} ) q^{2} \) \( + ( \beta_{1} + \beta_{3} ) q^{4} \) \(+ q^{5}\) \( + ( 1 + 2 \beta_{1} - 2 \beta_{3} ) q^{7} \) \( + ( 2 + \beta_{1} ) q^{8} \) \(+O(q^{10})\) \( q\) \( + ( -1 + \beta_{1} + \beta_{2} ) q^{2} \) \( + ( \beta_{1} + \beta_{3} ) q^{4} \) \(+ q^{5}\) \( + ( 1 + 2 \beta_{1} - 2 \beta_{3} ) q^{7} \) \( + ( 2 + \beta_{1} ) q^{8} \) \( + ( -1 + \beta_{1} + \beta_{2} ) q^{10} \) \( + ( 3 + \beta_{2} ) q^{11} \) \( + ( -3 - \beta_{1} + \beta_{2} + 3 \beta_{3} ) q^{13} \) \( + ( -1 - \beta_{1} + \beta_{2} + 4 \beta_{3} ) q^{14} \) \( + ( -2 + \beta_{1} + 3 \beta_{2} - \beta_{3} ) q^{16} \) \( + ( 2 - \beta_{1} + \beta_{2} - 3 \beta_{3} ) q^{17} \) \( + ( -1 - 3 \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{19} \) \( + ( \beta_{1} + \beta_{3} ) q^{20} \) \( + ( -2 + 4 \beta_{1} + 3 \beta_{2} ) q^{22} \) \( + ( -\beta_{1} - \beta_{2} + 3 \beta_{3} ) q^{23} \) \(+ q^{25}\) \( + ( 4 + 3 \beta_{1} - \beta_{2} - 4 \beta_{3} ) q^{26} \) \( + ( 3 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{28} \) \( + ( 2 + 4 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{29} \) \( + ( -3 + 5 \beta_{1} - 2 \beta_{3} ) q^{31} \) \( + ( 1 - 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{32} \) \( + ( -1 - 4 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{34} \) \( + ( 1 + 2 \beta_{1} - 2 \beta_{3} ) q^{35} \) \( + ( 2 - 3 \beta_{1} - 4 \beta_{2} + 3 \beta_{3} ) q^{37} \) \( + ( 3 - 3 \beta_{2} - 4 \beta_{3} ) q^{38} \) \( + ( 2 + \beta_{1} ) q^{40} \) \( + ( -3 \beta_{1} - 2 \beta_{2} + 5 \beta_{3} ) q^{41} \) \( + ( 3 - 3 \beta_{1} + \beta_{2} - \beta_{3} ) q^{43} \) \( + ( -1 + 5 \beta_{1} + 4 \beta_{3} ) q^{44} \) \( + ( -1 + 4 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} ) q^{46} \) \( + ( 6 + 4 \beta_{1} + \beta_{2} - 3 \beta_{3} ) q^{47} \) \( + ( 2 - 4 \beta_{2} ) q^{49} \) \( + ( -1 + \beta_{1} + \beta_{2} ) q^{50} \) \( + ( 1 + \beta_{2} + \beta_{3} ) q^{52} \) \( + ( -5 + \beta_{1} + 3 \beta_{2} + 5 \beta_{3} ) q^{53} \) \( + ( 3 + \beta_{2} ) q^{55} \) \( + ( 4 + 5 \beta_{1} - 4 \beta_{3} ) q^{56} \) \( + ( -1 + 3 \beta_{1} + 4 \beta_{2} + 6 \beta_{3} ) q^{58} \) \( + ( 7 + 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{59} \) \( + ( 3 + 3 \beta_{1} + \beta_{2} - 10 \beta_{3} ) q^{61} \) \( + ( 3 - 2 \beta_{1} + 7 \beta_{3} ) q^{62} \) \( + ( 1 - \beta_{1} - 5 \beta_{2} - 2 \beta_{3} ) q^{64} \) \( + ( -3 - \beta_{1} + \beta_{2} + 3 \beta_{3} ) q^{65} \) \( + ( -\beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{67} \) \( + ( -5 - \beta_{1} - 5 \beta_{2} ) q^{68} \) \( + ( -1 - \beta_{1} + \beta_{2} + 4 \beta_{3} ) q^{70} \) \( + ( -1 - 2 \beta_{2} - 4 \beta_{3} ) q^{71} \) \( + ( 3 + 5 \beta_{1} - \beta_{2} - 6 \beta_{3} ) q^{73} \) \( + ( -6 + \beta_{1} + 2 \beta_{2} - 6 \beta_{3} ) q^{74} \) \( + ( -4 - 2 \beta_{1} - 5 \beta_{2} + 2 \beta_{3} ) q^{76} \) \( + ( 1 + 6 \beta_{1} + \beta_{2} - 4 \beta_{3} ) q^{77} \) \( + ( -7 - 5 \beta_{1} - 3 \beta_{2} + 6 \beta_{3} ) q^{79} \) \( + ( -2 + \beta_{1} + 3 \beta_{2} - \beta_{3} ) q^{80} \) \( + ( -2 + 5 \beta_{1} + 2 \beta_{2} - 8 \beta_{3} ) q^{82} \) \( + ( 2 - 4 \beta_{1} - 6 \beta_{2} + \beta_{3} ) q^{83} \) \( + ( 2 - \beta_{1} + \beta_{2} - 3 \beta_{3} ) q^{85} \) \( + ( -2 - \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{86} \) \( + ( 5 + 4 \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{88} \) \(+ q^{89}\) \( + ( -13 - \beta_{1} + 7 \beta_{2} + 5 \beta_{3} ) q^{91} \) \( + ( 3 - \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{92} \) \( + ( -5 + 5 \beta_{1} + 7 \beta_{2} + 7 \beta_{3} ) q^{94} \) \( + ( -1 - 3 \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{95} \) \( + ( 9 - 5 \beta_{1} + 2 \beta_{2} + 3 \beta_{3} ) q^{97} \) \( + ( -6 - 2 \beta_{1} + 2 \beta_{2} ) q^{98} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(4q \) \(\mathstrut -\mathstrut q^{2} \) \(\mathstrut +\mathstrut 3q^{4} \) \(\mathstrut +\mathstrut 4q^{5} \) \(\mathstrut +\mathstrut 2q^{7} \) \(\mathstrut +\mathstrut 9q^{8} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(4q \) \(\mathstrut -\mathstrut q^{2} \) \(\mathstrut +\mathstrut 3q^{4} \) \(\mathstrut +\mathstrut 4q^{5} \) \(\mathstrut +\mathstrut 2q^{7} \) \(\mathstrut +\mathstrut 9q^{8} \) \(\mathstrut -\mathstrut q^{10} \) \(\mathstrut +\mathstrut 14q^{11} \) \(\mathstrut -\mathstrut 5q^{13} \) \(\mathstrut +\mathstrut 5q^{14} \) \(\mathstrut -\mathstrut 3q^{16} \) \(\mathstrut +\mathstrut 3q^{17} \) \(\mathstrut -\mathstrut q^{19} \) \(\mathstrut +\mathstrut 3q^{20} \) \(\mathstrut +\mathstrut 2q^{22} \) \(\mathstrut +\mathstrut 3q^{23} \) \(\mathstrut +\mathstrut 4q^{25} \) \(\mathstrut +\mathstrut 9q^{26} \) \(\mathstrut +\mathstrut 5q^{28} \) \(\mathstrut +\mathstrut 10q^{29} \) \(\mathstrut -\mathstrut 11q^{31} \) \(\mathstrut +\mathstrut 2q^{32} \) \(\mathstrut -\mathstrut 8q^{34} \) \(\mathstrut +\mathstrut 2q^{35} \) \(\mathstrut +\mathstrut 3q^{37} \) \(\mathstrut -\mathstrut 2q^{38} \) \(\mathstrut +\mathstrut 9q^{40} \) \(\mathstrut +\mathstrut 3q^{41} \) \(\mathstrut +\mathstrut 9q^{43} \) \(\mathstrut +\mathstrut 9q^{44} \) \(\mathstrut -\mathstrut 4q^{46} \) \(\mathstrut +\mathstrut 24q^{47} \) \(\mathstrut -\mathstrut q^{50} \) \(\mathstrut +\mathstrut 8q^{52} \) \(\mathstrut -\mathstrut 3q^{53} \) \(\mathstrut +\mathstrut 14q^{55} \) \(\mathstrut +\mathstrut 13q^{56} \) \(\mathstrut +\mathstrut 19q^{58} \) \(\mathstrut +\mathstrut 22q^{59} \) \(\mathstrut -\mathstrut 3q^{61} \) \(\mathstrut +\mathstrut 24q^{62} \) \(\mathstrut -\mathstrut 11q^{64} \) \(\mathstrut -\mathstrut 5q^{65} \) \(\mathstrut -\mathstrut 9q^{67} \) \(\mathstrut -\mathstrut 31q^{68} \) \(\mathstrut +\mathstrut 5q^{70} \) \(\mathstrut -\mathstrut 16q^{71} \) \(\mathstrut +\mathstrut 3q^{73} \) \(\mathstrut -\mathstrut 31q^{74} \) \(\mathstrut -\mathstrut 24q^{76} \) \(\mathstrut +\mathstrut 4q^{77} \) \(\mathstrut -\mathstrut 27q^{79} \) \(\mathstrut -\mathstrut 3q^{80} \) \(\mathstrut -\mathstrut 15q^{82} \) \(\mathstrut -\mathstrut 6q^{83} \) \(\mathstrut +\mathstrut 3q^{85} \) \(\mathstrut -\mathstrut 15q^{86} \) \(\mathstrut +\mathstrut 30q^{88} \) \(\mathstrut +\mathstrut 4q^{89} \) \(\mathstrut -\mathstrut 29q^{91} \) \(\mathstrut +\mathstrut 17q^{92} \) \(\mathstrut +\mathstrut 13q^{94} \) \(\mathstrut -\mathstrut q^{95} \) \(\mathstrut +\mathstrut 41q^{97} \) \(\mathstrut -\mathstrut 22q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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

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

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
−0.477260
0.737640
−1.35567
2.09529
−1.77222 0 1.14077 1.00000 0 −3.19059 1.52274 0 −1.77222
1.2 −1.45589 0 0.119606 1.00000 0 3.71135 2.73764 0 −1.45589
1.3 −0.162147 0 −1.97371 1.00000 0 −0.475281 0.644326 0 −0.162147
1.4 2.39026 0 3.71333 1.00000 0 1.95452 4.09529 0 2.39026
\(n\): 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4005))\):

\(T_{2}^{4} \) \(\mathstrut +\mathstrut T_{2}^{3} \) \(\mathstrut -\mathstrut 5 T_{2}^{2} \) \(\mathstrut -\mathstrut 7 T_{2} \) \(\mathstrut -\mathstrut 1 \)
\(T_{7}^{4} \) \(\mathstrut -\mathstrut 2 T_{7}^{3} \) \(\mathstrut -\mathstrut 12 T_{7}^{2} \) \(\mathstrut +\mathstrut 18 T_{7} \) \(\mathstrut +\mathstrut 11 \)
\(T_{11}^{4} \) \(\mathstrut -\mathstrut 14 T_{11}^{3} \) \(\mathstrut +\mathstrut 70 T_{11}^{2} \) \(\mathstrut -\mathstrut 147 T_{11} \) \(\mathstrut +\mathstrut 109 \)