Properties

Label 8001.2.a.l
Level 8001
Weight 2
Character orbit 8001.a
Self dual Yes
Analytic conductor 63.888
Analytic rank 1
Dimension 7
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8001.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(63.8883066572\)
Analytic rank: \(1\)
Dimension: \(7\)
Coefficient field: 7.7.118870813.1
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 \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( -\beta_{4} q^{2} \) \( + ( \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} ) q^{4} \) \( + ( 1 + \beta_{3} - \beta_{5} ) q^{5} \) \(+ q^{7}\) \( + ( 1 + \beta_{5} + \beta_{6} ) q^{8} \) \(+O(q^{10})\) \( q\) \( -\beta_{4} q^{2} \) \( + ( \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} ) q^{4} \) \( + ( 1 + \beta_{3} - \beta_{5} ) q^{5} \) \(+ q^{7}\) \( + ( 1 + \beta_{5} + \beta_{6} ) q^{8} \) \( + ( -\beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} ) q^{10} \) \( + ( 1 + \beta_{1} - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} ) q^{11} \) \( + ( -3 - \beta_{2} + \beta_{4} - \beta_{5} + \beta_{6} ) q^{13} \) \( -\beta_{4} q^{14} \) \( + ( -\beta_{1} + 2 \beta_{5} ) q^{16} \) \( + ( \beta_{1} - \beta_{3} + \beta_{5} - 2 \beta_{6} ) q^{17} \) \( + ( -1 - 2 \beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{5} - 2 \beta_{6} ) q^{19} \) \( + ( 1 + 2 \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} ) q^{20} \) \( + ( -2 - 3 \beta_{1} - \beta_{2} - 3 \beta_{3} - \beta_{4} - \beta_{6} ) q^{22} \) \( + ( -1 - 3 \beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - \beta_{6} ) q^{23} \) \( + ( 1 + 2 \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{25} \) \( + ( -1 - \beta_{1} - 2 \beta_{3} + 3 \beta_{4} - \beta_{6} ) q^{26} \) \( + ( \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} ) q^{28} \) \( + ( 1 - \beta_{2} + \beta_{3} + 2 \beta_{4} + 3 \beta_{5} + \beta_{6} ) q^{29} \) \( + ( -5 + \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} ) q^{31} \) \( + ( -2 + 2 \beta_{1} + 2 \beta_{2} + 3 \beta_{3} - 2 \beta_{5} ) q^{32} \) \( + ( -\beta_{2} + 2 \beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{6} ) q^{34} \) \( + ( 1 + \beta_{3} - \beta_{5} ) q^{35} \) \( + ( -5 - 3 \beta_{1} - \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{37} \) \( + ( 1 + 2 \beta_{1} + \beta_{2} + 2 \beta_{3} + 3 \beta_{4} - 2 \beta_{5} ) q^{38} \) \( + ( -2 - 2 \beta_{1} + \beta_{3} + \beta_{5} + 2 \beta_{6} ) q^{40} \) \( + ( 2 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} + 2 \beta_{6} ) q^{41} \) \( + ( -1 - \beta_{1} + 2 \beta_{2} + \beta_{5} - 3 \beta_{6} ) q^{43} \) \( + ( 1 - 3 \beta_{1} + \beta_{2} - 2 \beta_{3} - 3 \beta_{5} - 2 \beta_{6} ) q^{44} \) \( + ( -3 - \beta_{3} + 4 \beta_{4} - \beta_{6} ) q^{46} \) \( + ( -3 - \beta_{1} + 5 \beta_{2} + 3 \beta_{3} - 2 \beta_{4} ) q^{47} \) \(+ q^{49}\) \( + ( -3 - 2 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} + \beta_{4} + 2 \beta_{5} - 4 \beta_{6} ) q^{50} \) \( + ( -5 \beta_{1} - 2 \beta_{2} - 6 \beta_{3} + 3 \beta_{4} + 3 \beta_{5} - 6 \beta_{6} ) q^{52} \) \( + ( \beta_{1} - 3 \beta_{2} + \beta_{3} - 3 \beta_{5} ) q^{53} \) \( + ( -3 - \beta_{1} + 3 \beta_{2} + 3 \beta_{3} - 3 \beta_{5} + 2 \beta_{6} ) q^{55} \) \( + ( 1 + \beta_{5} + \beta_{6} ) q^{56} \) \( + ( -3 + 3 \beta_{1} + 3 \beta_{2} + 6 \beta_{3} + 2 \beta_{5} + 2 \beta_{6} ) q^{58} \) \( + ( 6 - \beta_{1} - 2 \beta_{2} - 4 \beta_{3} - \beta_{5} - 3 \beta_{6} ) q^{59} \) \( + ( -3 + \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + 3 \beta_{5} - \beta_{6} ) q^{61} \) \( + ( 1 - 2 \beta_{2} + 5 \beta_{4} + \beta_{5} - \beta_{6} ) q^{62} \) \( + ( -2 + \beta_{1} - 4 \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{5} - 2 \beta_{6} ) q^{64} \) \( + ( -1 - \beta_{1} - 2 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} + 3 \beta_{5} + 2 \beta_{6} ) q^{65} \) \( + ( -3 - \beta_{1} + \beta_{2} - 3 \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{67} \) \( + ( -3 - 2 \beta_{1} - 3 \beta_{2} + \beta_{3} + 3 \beta_{4} + \beta_{5} ) q^{68} \) \( + ( -\beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} ) q^{70} \) \( + ( -2 - \beta_{1} + 5 \beta_{2} + \beta_{3} - \beta_{4} + 3 \beta_{5} - \beta_{6} ) q^{71} \) \( + ( -3 + 5 \beta_{1} - \beta_{5} + 2 \beta_{6} ) q^{73} \) \( + ( 2 + 2 \beta_{1} + 3 \beta_{2} + \beta_{3} + 4 \beta_{4} - 2 \beta_{5} + 3 \beta_{6} ) q^{74} \) \( + ( -5 - 4 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - \beta_{6} ) q^{76} \) \( + ( 1 + \beta_{1} - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} ) q^{77} \) \( + ( 1 + \beta_{2} - 4 \beta_{4} - 5 \beta_{5} - 2 \beta_{6} ) q^{79} \) \( + ( -2 - 2 \beta_{1} - \beta_{2} - \beta_{3} + 3 \beta_{5} + \beta_{6} ) q^{80} \) \( + ( -\beta_{1} + 2 \beta_{2} - 3 \beta_{4} + 4 \beta_{6} ) q^{82} \) \( + ( 5 - 3 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - \beta_{4} + 3 \beta_{5} - 2 \beta_{6} ) q^{83} \) \( + ( -5 + 2 \beta_{2} + \beta_{3} - 6 \beta_{4} - 2 \beta_{5} - 5 \beta_{6} ) q^{85} \) \( + ( -2 - \beta_{1} - 4 \beta_{2} - \beta_{3} + 4 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} ) q^{86} \) \( + ( 3 - 4 \beta_{2} - 6 \beta_{3} + 3 \beta_{4} - \beta_{5} - 3 \beta_{6} ) q^{88} \) \( + ( \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{5} + 7 \beta_{6} ) q^{89} \) \( + ( -3 - \beta_{2} + \beta_{4} - \beta_{5} + \beta_{6} ) q^{91} \) \( + ( -6 + \beta_{1} - 3 \beta_{2} - 3 \beta_{3} + 4 \beta_{4} - \beta_{5} - 3 \beta_{6} ) q^{92} \) \( + ( -1 - 3 \beta_{2} - \beta_{3} + \beta_{4} + 6 \beta_{5} + 2 \beta_{6} ) q^{94} \) \( + ( -1 - 2 \beta_{1} - 4 \beta_{2} - 4 \beta_{3} - \beta_{4} + 4 \beta_{5} - 3 \beta_{6} ) q^{95} \) \( + ( 4 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} + 5 \beta_{6} ) q^{97} \) \( -\beta_{4} q^{98} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(7q \) \(\mathstrut +\mathstrut 2q^{2} \) \(\mathstrut +\mathstrut 4q^{4} \) \(\mathstrut +\mathstrut 8q^{5} \) \(\mathstrut +\mathstrut 7q^{7} \) \(\mathstrut +\mathstrut 9q^{8} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(7q \) \(\mathstrut +\mathstrut 2q^{2} \) \(\mathstrut +\mathstrut 4q^{4} \) \(\mathstrut +\mathstrut 8q^{5} \) \(\mathstrut +\mathstrut 7q^{7} \) \(\mathstrut +\mathstrut 9q^{8} \) \(\mathstrut +\mathstrut 3q^{11} \) \(\mathstrut -\mathstrut 23q^{13} \) \(\mathstrut +\mathstrut 2q^{14} \) \(\mathstrut +\mathstrut 2q^{16} \) \(\mathstrut -\mathstrut 3q^{17} \) \(\mathstrut -\mathstrut 9q^{19} \) \(\mathstrut +\mathstrut 9q^{20} \) \(\mathstrut -\mathstrut 19q^{22} \) \(\mathstrut -\mathstrut 12q^{23} \) \(\mathstrut +\mathstrut 3q^{25} \) \(\mathstrut -\mathstrut 18q^{26} \) \(\mathstrut +\mathstrut 4q^{28} \) \(\mathstrut +\mathstrut 9q^{29} \) \(\mathstrut -\mathstrut 33q^{31} \) \(\mathstrut -\mathstrut 10q^{32} \) \(\mathstrut -\mathstrut 2q^{34} \) \(\mathstrut +\mathstrut 8q^{35} \) \(\mathstrut -\mathstrut 33q^{37} \) \(\mathstrut +\mathstrut 3q^{38} \) \(\mathstrut -\mathstrut 9q^{40} \) \(\mathstrut +\mathstrut 3q^{41} \) \(\mathstrut -\mathstrut 9q^{43} \) \(\mathstrut -\mathstrut 2q^{44} \) \(\mathstrut -\mathstrut 32q^{46} \) \(\mathstrut -\mathstrut 11q^{47} \) \(\mathstrut +\mathstrut 7q^{49} \) \(\mathstrut -\mathstrut 29q^{50} \) \(\mathstrut -\mathstrut 21q^{52} \) \(\mathstrut -\mathstrut q^{53} \) \(\mathstrut -\mathstrut 16q^{55} \) \(\mathstrut +\mathstrut 9q^{56} \) \(\mathstrut -\mathstrut 5q^{58} \) \(\mathstrut +\mathstrut 30q^{59} \) \(\mathstrut -\mathstrut 19q^{61} \) \(\mathstrut -\mathstrut 3q^{62} \) \(\mathstrut -\mathstrut 21q^{64} \) \(\mathstrut -\mathstrut 14q^{65} \) \(\mathstrut -\mathstrut 30q^{67} \) \(\mathstrut -\mathstrut 24q^{68} \) \(\mathstrut -\mathstrut 8q^{71} \) \(\mathstrut -\mathstrut 20q^{73} \) \(\mathstrut +\mathstrut 9q^{74} \) \(\mathstrut -\mathstrut 42q^{76} \) \(\mathstrut +\mathstrut 3q^{77} \) \(\mathstrut +\mathstrut 8q^{79} \) \(\mathstrut -\mathstrut 12q^{80} \) \(\mathstrut +\mathstrut 10q^{82} \) \(\mathstrut +\mathstrut 34q^{83} \) \(\mathstrut -\mathstrut 28q^{85} \) \(\mathstrut -\mathstrut 24q^{86} \) \(\mathstrut -\mathstrut q^{88} \) \(\mathstrut +\mathstrut 12q^{89} \) \(\mathstrut -\mathstrut 23q^{91} \) \(\mathstrut -\mathstrut 60q^{92} \) \(\mathstrut -\mathstrut 3q^{94} \) \(\mathstrut -\mathstrut 12q^{95} \) \(\mathstrut +\mathstrut 7q^{97} \) \(\mathstrut +\mathstrut 2q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{7}\mathstrut -\mathstrut \) \(9\) \(x^{5}\mathstrut -\mathstrut \) \(3\) \(x^{4}\mathstrut +\mathstrut \) \(20\) \(x^{3}\mathstrut +\mathstrut \) \(7\) \(x^{2}\mathstrut -\mathstrut \) \(13\) \(x\mathstrut -\mathstrut \) \(4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{5} - 7 \nu^{3} - 4 \nu^{2} + 7 \nu + 5 \)
\(\beta_{3}\)\(=\)\( \nu^{5} - 8 \nu^{3} - 3 \nu^{2} + 11 \nu + 4 \)
\(\beta_{4}\)\(=\)\( -\nu^{6} + 8 \nu^{4} + 3 \nu^{3} - 12 \nu^{2} - 4 \nu + 2 \)
\(\beta_{5}\)\(=\)\( \nu^{6} + 2 \nu^{5} - 8 \nu^{4} - 18 \nu^{3} + 6 \nu^{2} + 22 \nu + 4 \)
\(\beta_{6}\)\(=\)\( 2 \nu^{6} + \nu^{5} - 15 \nu^{4} - 14 \nu^{3} + 15 \nu^{2} + 16 \nu + 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut -\mathstrut \) \(\beta_{3}\mathstrut -\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(3\)
\(\nu^{3}\)\(=\)\(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut -\mathstrut \) \(2\) \(\beta_{3}\mathstrut +\mathstrut \) \(4\) \(\beta_{1}\mathstrut +\mathstrut \) \(2\)
\(\nu^{4}\)\(=\)\(\beta_{6}\mathstrut +\mathstrut \) \(6\) \(\beta_{5}\mathstrut +\mathstrut \) \(8\) \(\beta_{4}\mathstrut -\mathstrut \) \(7\) \(\beta_{3}\mathstrut -\mathstrut \) \(6\) \(\beta_{2}\mathstrut +\mathstrut \) \(3\) \(\beta_{1}\mathstrut +\mathstrut \) \(15\)
\(\nu^{5}\)\(=\)\(11\) \(\beta_{5}\mathstrut +\mathstrut \) \(11\) \(\beta_{4}\mathstrut -\mathstrut \) \(18\) \(\beta_{3}\mathstrut -\mathstrut \) \(3\) \(\beta_{2}\mathstrut +\mathstrut \) \(21\) \(\beta_{1}\mathstrut +\mathstrut \) \(21\)
\(\nu^{6}\)\(=\)\(8\) \(\beta_{6}\mathstrut +\mathstrut \) \(39\) \(\beta_{5}\mathstrut +\mathstrut \) \(54\) \(\beta_{4}\mathstrut -\mathstrut \) \(50\) \(\beta_{3}\mathstrut -\mathstrut \) \(36\) \(\beta_{2}\mathstrut +\mathstrut \) \(32\) \(\beta_{1}\mathstrut +\mathstrut \) \(92\)

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
−0.301070
2.69855
−2.06168
−1.52532
1.20244
−1.14753
1.13462
−2.09968 0 2.40865 3.29054 0 1.00000 −0.858029 0 −6.90907
1.2 −0.840819 0 −1.29302 −2.74724 0 1.00000 2.76884 0 2.30993
1.3 −0.692358 0 −1.52064 2.78145 0 1.00000 2.43754 0 −1.92576
1.4 −0.246202 0 −1.93938 0.318209 0 1.00000 0.969884 0 −0.0783436
1.5 1.24280 0 −0.455452 3.33400 0 1.00000 −3.05163 0 4.14349
1.6 2.15625 0 2.64943 0.239094 0 1.00000 1.40032 0 0.515547
1.7 2.48001 0 4.15043 0.783950 0 1.00000 5.33307 0 1.94420
\(n\): 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
\(3\) \(-1\)
\(7\) \(-1\)
\(127\) \(1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8001))\):

\(T_{2}^{7} \) \(\mathstrut -\mathstrut 2 T_{2}^{6} \) \(\mathstrut -\mathstrut 7 T_{2}^{5} \) \(\mathstrut +\mathstrut 11 T_{2}^{4} \) \(\mathstrut +\mathstrut 14 T_{2}^{3} \) \(\mathstrut -\mathstrut 9 T_{2}^{2} \) \(\mathstrut -\mathstrut 11 T_{2} \) \(\mathstrut -\mathstrut 2 \)
\(T_{5}^{7} \) \(\mathstrut -\mathstrut 8 T_{5}^{6} \) \(\mathstrut +\mathstrut 13 T_{5}^{5} \) \(\mathstrut +\mathstrut 42 T_{5}^{4} \) \(\mathstrut -\mathstrut 149 T_{5}^{3} \) \(\mathstrut +\mathstrut 138 T_{5}^{2} \) \(\mathstrut -\mathstrut 46 T_{5} \) \(\mathstrut +\mathstrut 5 \)