Properties

Label 4005.2.a.o
Level 4005
Weight 2
Character orbit 4005.a
Self dual Yes
Analytic conductor 31.980
Analytic rank 0
Dimension 7
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: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 - \beta_{1} ) q^{2} + ( 2 - \beta_{1} + \beta_{3} ) q^{4} - q^{5} + ( -2 - \beta_{6} ) q^{7} + ( 3 - \beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} - \beta_{6} ) q^{8} +O(q^{10})\) \( q + ( 1 - \beta_{1} ) q^{2} + ( 2 - \beta_{1} + \beta_{3} ) q^{4} - q^{5} + ( -2 - \beta_{6} ) q^{7} + ( 3 - \beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} - \beta_{6} ) q^{8} + ( -1 + \beta_{1} ) q^{10} + ( 1 - \beta_{4} - \beta_{5} + \beta_{6} ) q^{11} + ( -2 - \beta_{1} - \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{6} ) q^{13} + ( 3 \beta_{1} + 2 \beta_{3} + \beta_{4} - 2 \beta_{6} ) q^{14} + ( 4 - \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} - 3 \beta_{6} ) q^{16} + ( 2 + \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} ) q^{17} + ( -3 \beta_{1} - \beta_{2} + \beta_{6} ) q^{19} + ( -2 + \beta_{1} - \beta_{3} ) q^{20} + ( 1 - 2 \beta_{1} - \beta_{5} + \beta_{6} ) q^{22} + ( 1 - \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{5} ) q^{23} + q^{25} + ( -\beta_{1} - 2 \beta_{2} + \beta_{4} + 2 \beta_{5} + \beta_{6} ) q^{26} + ( -2 + \beta_{1} - \beta_{2} + \beta_{4} - 2 \beta_{5} - 3 \beta_{6} ) q^{28} + ( -1 - 2 \beta_{1} - 2 \beta_{3} - 2 \beta_{4} + \beta_{5} + 3 \beta_{6} ) q^{29} + ( -2 + \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{5} + 3 \beta_{6} ) q^{31} + ( 6 + \beta_{2} + 5 \beta_{3} + 2 \beta_{4} - 5 \beta_{6} ) q^{32} + ( 5 + \beta_{2} + 3 \beta_{3} + 2 \beta_{4} - 4 \beta_{5} - 4 \beta_{6} ) q^{34} + ( 2 + \beta_{6} ) q^{35} + ( -1 - \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{4} ) q^{37} + ( 6 - 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + 2 \beta_{6} ) q^{38} + ( -3 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{5} + \beta_{6} ) q^{40} + ( 2 - 3 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} ) q^{41} + ( -5 + \beta_{1} + \beta_{2} + \beta_{5} ) q^{43} + ( 4 - \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} ) q^{44} + ( 3 - 2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + 4 \beta_{5} + \beta_{6} ) q^{46} + ( 2 + 3 \beta_{2} - \beta_{4} - \beta_{6} ) q^{47} + ( 2 + \beta_{2} + \beta_{3} + 4 \beta_{6} ) q^{49} + ( 1 - \beta_{1} ) q^{50} + ( -2 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + \beta_{6} ) q^{52} + ( -\beta_{1} + 2 \beta_{2} - 3 \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{6} ) q^{53} + ( -1 + \beta_{4} + \beta_{5} - \beta_{6} ) q^{55} + ( 1 + \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - \beta_{5} - \beta_{6} ) q^{56} + ( -3 \beta_{1} - \beta_{2} - 3 \beta_{3} - \beta_{4} + 3 \beta_{5} + 6 \beta_{6} ) q^{58} + ( 2 - 2 \beta_{1} - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + \beta_{6} ) q^{59} + ( -1 - \beta_{1} - 3 \beta_{2} - 3 \beta_{3} - \beta_{4} + \beta_{6} ) q^{61} + ( -7 + \beta_{2} - 4 \beta_{3} - \beta_{4} - \beta_{5} + 5 \beta_{6} ) q^{62} + ( 7 - \beta_{1} + 4 \beta_{3} + \beta_{4} - 4 \beta_{5} - 7 \beta_{6} ) q^{64} + ( 2 + \beta_{1} + \beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{6} ) q^{65} + ( 1 - \beta_{1} + 2 \beta_{3} - \beta_{4} + \beta_{5} ) q^{67} + ( 12 + \beta_{1} + 8 \beta_{3} + 4 \beta_{4} - 6 \beta_{5} - 7 \beta_{6} ) q^{68} + ( -3 \beta_{1} - 2 \beta_{3} - \beta_{4} + 2 \beta_{6} ) q^{70} + ( 2 - 2 \beta_{1} + 2 \beta_{3} - 3 \beta_{4} ) q^{71} + ( 3 + \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - 4 \beta_{5} - 3 \beta_{6} ) q^{73} + ( 5 + 3 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} ) q^{74} + ( 7 - 6 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{5} ) q^{76} + ( -5 - 2 \beta_{2} + \beta_{4} + \beta_{5} - 3 \beta_{6} ) q^{77} + ( -1 - \beta_{1} + \beta_{2} - 3 \beta_{3} - \beta_{4} + 2 \beta_{5} + 3 \beta_{6} ) q^{79} + ( -4 + \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{5} + 3 \beta_{6} ) q^{80} + ( 6 - 5 \beta_{1} - 3 \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{82} + ( 8 - 4 \beta_{1} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} ) q^{83} + ( -2 - \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} ) q^{85} + ( -8 + 5 \beta_{1} - 2 \beta_{3} ) q^{86} + ( 4 - \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{6} ) q^{88} - q^{89} + ( 5 + 5 \beta_{1} + \beta_{2} + 2 \beta_{3} + 3 \beta_{4} - 5 \beta_{5} - \beta_{6} ) q^{91} + ( -1 - 7 \beta_{1} - 3 \beta_{2} - \beta_{3} + 2 \beta_{5} + 2 \beta_{6} ) q^{92} + ( 8 + \beta_{1} + 2 \beta_{2} + 3 \beta_{3} + 2 \beta_{4} - 3 \beta_{5} - 3 \beta_{6} ) q^{94} + ( 3 \beta_{1} + \beta_{2} - \beta_{6} ) q^{95} + ( 1 - \beta_{1} + \beta_{3} + 2 \beta_{4} - 2 \beta_{6} ) q^{97} + ( -5 - 6 \beta_{1} - 8 \beta_{3} - 4 \beta_{4} - 2 \beta_{5} + 7 \beta_{6} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7q + 4q^{2} + 8q^{4} - 7q^{5} - 16q^{7} + 12q^{8} + O(q^{10}) \) \( 7q + 4q^{2} + 8q^{4} - 7q^{5} - 16q^{7} + 12q^{8} - 4q^{10} + 10q^{11} - 7q^{13} - 3q^{14} + 10q^{16} + 13q^{17} - 7q^{19} - 8q^{20} + 2q^{22} + 13q^{23} + 7q^{25} - q^{26} - 21q^{28} + 4q^{29} + q^{31} + 13q^{32} + 10q^{34} + 16q^{35} - 5q^{37} + 40q^{38} - 12q^{40} - 5q^{41} - 31q^{43} + 21q^{44} + 16q^{46} + 14q^{47} + 19q^{49} + 4q^{50} + 13q^{53} - 10q^{55} + q^{56} + 17q^{58} + 14q^{59} + 3q^{61} - 26q^{62} + 14q^{64} + 7q^{65} + q^{67} + 35q^{68} + 3q^{70} + 8q^{71} + 9q^{73} + 35q^{74} + 40q^{76} - 42q^{77} + 9q^{79} - 10q^{80} + 29q^{82} + 42q^{83} - 13q^{85} - 35q^{86} + 30q^{88} - 7q^{89} + 31q^{91} - 19q^{92} + 37q^{94} + 7q^{95} - 7q^{97} - 9q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{7} - x^{6} - 8 x^{5} + 6 x^{4} + 19 x^{3} - 10 x^{2} - 12 x + 6\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{2} - 2 \)
\(\beta_{2}\)\(=\)\( \nu^{3} - \nu^{2} - 4 \nu + 2 \)
\(\beta_{3}\)\(=\)\( \nu^{4} - 5 \nu^{2} + 3 \)
\(\beta_{4}\)\(=\)\( -\nu^{4} + \nu^{3} + 4 \nu^{2} - 2 \nu - 2 \)
\(\beta_{5}\)\(=\)\( \nu^{5} - \nu^{4} - 6 \nu^{3} + 5 \nu^{2} + 8 \nu - 5 \)
\(\beta_{6}\)\(=\)\( \nu^{6} - \nu^{5} - 7 \nu^{4} + 5 \nu^{3} + 13 \nu^{2} - 4 \nu - 4 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{4} + \beta_{3} - \beta_{2} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\(\beta_{1} + 2\)
\(\nu^{3}\)\(=\)\(2 \beta_{4} + 2 \beta_{3} - \beta_{2} + \beta_{1} + 2\)
\(\nu^{4}\)\(=\)\(\beta_{3} + 5 \beta_{1} + 7\)
\(\nu^{5}\)\(=\)\(\beta_{5} + 8 \beta_{4} + 9 \beta_{3} - 2 \beta_{2} + 6 \beta_{1} + 10\)
\(\nu^{6}\)\(=\)\(\beta_{6} + \beta_{5} + 8 \beta_{3} + \beta_{2} + 23 \beta_{1} + 29\)

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
2.26266
−1.96388
1.89340
−1.49803
−1.07810
0.885013
0.498937
−2.11962 0 2.49279 −1.00000 0 −4.83304 −1.04452 0 2.11962
1.2 −0.856822 0 −1.26586 −1.00000 0 −0.580377 2.79826 0 0.856822
1.3 −0.584976 0 −1.65780 −1.00000 0 −2.74591 2.13973 0 0.584976
1.4 0.755898 0 −1.42862 −1.00000 0 0.0498231 −2.59169 0 −0.755898
1.5 1.83770 0 1.37716 −1.00000 0 −4.72699 −1.14460 0 −1.83770
1.6 2.21675 0 2.91399 −1.00000 0 −3.75132 2.02609 0 −2.21675
1.7 2.75106 0 5.56834 −1.00000 0 0.587818 9.81674 0 −2.75106
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4005))\):

\( T_{2}^{7} - 4 T_{2}^{6} - 3 T_{2}^{5} + 24 T_{2}^{4} - 8 T_{2}^{3} - 29 T_{2}^{2} + 6 T_{2} + 9 \)
\( T_{7}^{7} + 16 T_{7}^{6} + 94 T_{7}^{5} + 236 T_{7}^{4} + 189 T_{7}^{3} - 96 T_{7}^{2} - 76 T_{7} + 4 \)
\( T_{11}^{7} - 10 T_{11}^{6} + 14 T_{11}^{5} + 149 T_{11}^{4} - 639 T_{11}^{3} + 968 T_{11}^{2} - 608 T_{11} + 128 \)