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.

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

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

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

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

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