Properties

Label 8010.2.a.bn
Level 8010
Weight 2
Character orbit 8010.a
Self dual Yes
Analytic conductor 63.960
Analytic rank 0
Dimension 7
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 8010 = 2 \cdot 3^{2} \cdot 5 \cdot 89 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8010.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(63.960172019\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
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 \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \(+ q^{2}\) \(+ q^{4}\) \(+ q^{5}\) \( -\beta_{2} q^{7} \) \(+ q^{8}\) \(+O(q^{10})\) \( q\) \(+ q^{2}\) \(+ q^{4}\) \(+ q^{5}\) \( -\beta_{2} q^{7} \) \(+ q^{8}\) \(+ q^{10}\) \( + \beta_{3} q^{11} \) \( + ( -\beta_{2} - \beta_{5} ) q^{13} \) \( -\beta_{2} q^{14} \) \(+ q^{16}\) \( + ( 1 + \beta_{2} - \beta_{6} ) q^{17} \) \( + ( 1 - \beta_{4} ) q^{19} \) \(+ q^{20}\) \( + \beta_{3} q^{22} \) \( + ( 1 + \beta_{1} + \beta_{2} ) q^{23} \) \(+ q^{25}\) \( + ( -\beta_{2} - \beta_{5} ) q^{26} \) \( -\beta_{2} q^{28} \) \( + ( 1 - \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} ) q^{29} \) \( + ( 2 - \beta_{2} + \beta_{5} ) q^{31} \) \(+ q^{32}\) \( + ( 1 + \beta_{2} - \beta_{6} ) q^{34} \) \( -\beta_{2} q^{35} \) \( + ( -\beta_{3} + \beta_{4} - \beta_{6} ) q^{37} \) \( + ( 1 - \beta_{4} ) q^{38} \) \(+ q^{40}\) \( + ( -\beta_{1} + \beta_{2} - \beta_{3} - \beta_{6} ) q^{41} \) \( + ( -2 + 2 \beta_{1} ) q^{43} \) \( + \beta_{3} q^{44} \) \( + ( 1 + \beta_{1} + \beta_{2} ) q^{46} \) \( + ( 2 + \beta_{1} + \beta_{4} - \beta_{5} ) q^{47} \) \( + ( 5 - 2 \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} + \beta_{6} ) q^{49} \) \(+ q^{50}\) \( + ( -\beta_{2} - \beta_{5} ) q^{52} \) \( + ( 4 - \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} ) q^{53} \) \( + \beta_{3} q^{55} \) \( -\beta_{2} q^{56} \) \( + ( 1 - \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} ) q^{58} \) \( + ( 1 + \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} + \beta_{6} ) q^{59} \) \( + ( 4 - 2 \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} ) q^{61} \) \( + ( 2 - \beta_{2} + \beta_{5} ) q^{62} \) \(+ q^{64}\) \( + ( -\beta_{2} - \beta_{5} ) q^{65} \) \( + ( 1 - \beta_{3} + \beta_{4} - \beta_{5} ) q^{67} \) \( + ( 1 + \beta_{2} - \beta_{6} ) q^{68} \) \( -\beta_{2} q^{70} \) \( + ( 2 - 2 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} ) q^{71} \) \( + ( \beta_{1} + \beta_{2} + \beta_{6} ) q^{73} \) \( + ( -\beta_{3} + \beta_{4} - \beta_{6} ) q^{74} \) \( + ( 1 - \beta_{4} ) q^{76} \) \( + ( 1 + \beta_{4} + \beta_{5} ) q^{77} \) \( + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} ) q^{79} \) \(+ q^{80}\) \( + ( -\beta_{1} + \beta_{2} - \beta_{3} - \beta_{6} ) q^{82} \) \( + ( \beta_{1} - \beta_{2} - \beta_{3} - \beta_{6} ) q^{83} \) \( + ( 1 + \beta_{2} - \beta_{6} ) q^{85} \) \( + ( -2 + 2 \beta_{1} ) q^{86} \) \( + \beta_{3} q^{88} \) \(- q^{89}\) \( + ( 8 - 3 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + \beta_{6} ) q^{91} \) \( + ( 1 + \beta_{1} + \beta_{2} ) q^{92} \) \( + ( 2 + \beta_{1} + \beta_{4} - \beta_{5} ) q^{94} \) \( + ( 1 - \beta_{4} ) q^{95} \) \( + ( 4 + 2 \beta_{1} + 2 \beta_{2} ) q^{97} \) \( + ( 5 - 2 \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} + \beta_{6} ) q^{98} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(7q \) \(\mathstrut +\mathstrut 7q^{2} \) \(\mathstrut +\mathstrut 7q^{4} \) \(\mathstrut +\mathstrut 7q^{5} \) \(\mathstrut +\mathstrut q^{7} \) \(\mathstrut +\mathstrut 7q^{8} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(7q \) \(\mathstrut +\mathstrut 7q^{2} \) \(\mathstrut +\mathstrut 7q^{4} \) \(\mathstrut +\mathstrut 7q^{5} \) \(\mathstrut +\mathstrut q^{7} \) \(\mathstrut +\mathstrut 7q^{8} \) \(\mathstrut +\mathstrut 7q^{10} \) \(\mathstrut -\mathstrut q^{11} \) \(\mathstrut +\mathstrut q^{14} \) \(\mathstrut +\mathstrut 7q^{16} \) \(\mathstrut +\mathstrut 9q^{17} \) \(\mathstrut +\mathstrut 9q^{19} \) \(\mathstrut +\mathstrut 7q^{20} \) \(\mathstrut -\mathstrut q^{22} \) \(\mathstrut +\mathstrut 8q^{23} \) \(\mathstrut +\mathstrut 7q^{25} \) \(\mathstrut +\mathstrut q^{28} \) \(\mathstrut +\mathstrut 4q^{29} \) \(\mathstrut +\mathstrut 16q^{31} \) \(\mathstrut +\mathstrut 7q^{32} \) \(\mathstrut +\mathstrut 9q^{34} \) \(\mathstrut +\mathstrut q^{35} \) \(\mathstrut +\mathstrut 2q^{37} \) \(\mathstrut +\mathstrut 9q^{38} \) \(\mathstrut +\mathstrut 7q^{40} \) \(\mathstrut +\mathstrut q^{41} \) \(\mathstrut -\mathstrut 10q^{43} \) \(\mathstrut -\mathstrut q^{44} \) \(\mathstrut +\mathstrut 8q^{46} \) \(\mathstrut +\mathstrut 13q^{47} \) \(\mathstrut +\mathstrut 26q^{49} \) \(\mathstrut +\mathstrut 7q^{50} \) \(\mathstrut +\mathstrut 24q^{53} \) \(\mathstrut -\mathstrut q^{55} \) \(\mathstrut +\mathstrut q^{56} \) \(\mathstrut +\mathstrut 4q^{58} \) \(\mathstrut +\mathstrut 6q^{59} \) \(\mathstrut +\mathstrut 23q^{61} \) \(\mathstrut +\mathstrut 16q^{62} \) \(\mathstrut +\mathstrut 7q^{64} \) \(\mathstrut +\mathstrut 5q^{67} \) \(\mathstrut +\mathstrut 9q^{68} \) \(\mathstrut +\mathstrut q^{70} \) \(\mathstrut +\mathstrut 8q^{71} \) \(\mathstrut -\mathstrut 2q^{73} \) \(\mathstrut +\mathstrut 2q^{74} \) \(\mathstrut +\mathstrut 9q^{76} \) \(\mathstrut +\mathstrut 6q^{77} \) \(\mathstrut +\mathstrut 7q^{79} \) \(\mathstrut +\mathstrut 7q^{80} \) \(\mathstrut +\mathstrut q^{82} \) \(\mathstrut +\mathstrut 7q^{83} \) \(\mathstrut +\mathstrut 9q^{85} \) \(\mathstrut -\mathstrut 10q^{86} \) \(\mathstrut -\mathstrut q^{88} \) \(\mathstrut -\mathstrut 7q^{89} \) \(\mathstrut +\mathstrut 48q^{91} \) \(\mathstrut +\mathstrut 8q^{92} \) \(\mathstrut +\mathstrut 13q^{94} \) \(\mathstrut +\mathstrut 9q^{95} \) \(\mathstrut +\mathstrut 30q^{97} \) \(\mathstrut +\mathstrut 26q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{7}\mathstrut -\mathstrut \) \(3\) \(x^{6}\mathstrut -\mathstrut \) \(24\) \(x^{5}\mathstrut +\mathstrut \) \(34\) \(x^{4}\mathstrut +\mathstrut \) \(111\) \(x^{3}\mathstrut -\mathstrut \) \(127\) \(x^{2}\mathstrut -\mathstrut \) \(20\) \(x\mathstrut +\mathstrut \) \(16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 5 \nu^{6} + 17 \nu^{5} - 204 \nu^{4} - 750 \nu^{3} + 1539 \nu^{2} + 4973 \nu - 1820 \)\()/964\)
\(\beta_{2}\)\(=\)\((\)\( -11 \nu^{6} + 59 \nu^{5} + 256 \nu^{4} - 1242 \nu^{3} - 1265 \nu^{2} + 5351 \nu - 2744 \)\()/964\)
\(\beta_{3}\)\(=\)\((\)\( 27 \nu^{6} - 101 \nu^{5} - 716 \nu^{4} + 1734 \nu^{3} + 5033 \nu^{2} - 7657 \nu - 4044 \)\()/964\)
\(\beta_{4}\)\(=\)\((\)\( 27 \nu^{6} - 101 \nu^{5} - 716 \nu^{4} + 1734 \nu^{3} + 5033 \nu^{2} - 5729 \nu - 5008 \)\()/964\)
\(\beta_{5}\)\(=\)\((\)\( 47 \nu^{6} - 33 \nu^{5} - 1532 \nu^{4} - 302 \nu^{3} + 8297 \nu^{2} - 297 \nu - 2648 \)\()/964\)
\(\beta_{6}\)\(=\)\((\)\( 104 \nu^{6} - 273 \nu^{5} - 2508 \nu^{4} + 2234 \nu^{3} + 10996 \nu^{2} - 6313 \nu - 2670 \)\()/482\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{4}\mathstrut -\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(1\)\()/2\)
\(\nu^{2}\)\(=\)\(\beta_{4}\mathstrut +\mathstrut \) \(2\) \(\beta_{2}\mathstrut -\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(9\)
\(\nu^{3}\)\(=\)\((\)\(2\) \(\beta_{5}\mathstrut +\mathstrut \) \(19\) \(\beta_{4}\mathstrut -\mathstrut \) \(15\) \(\beta_{3}\mathstrut +\mathstrut \) \(12\) \(\beta_{2}\mathstrut -\mathstrut \) \(14\) \(\beta_{1}\mathstrut +\mathstrut \) \(49\)\()/2\)
\(\nu^{4}\)\(=\)\(\beta_{6}\mathstrut +\mathstrut \) \(2\) \(\beta_{5}\mathstrut +\mathstrut \) \(38\) \(\beta_{4}\mathstrut -\mathstrut \) \(19\) \(\beta_{3}\mathstrut +\mathstrut \) \(55\) \(\beta_{2}\mathstrut -\mathstrut \) \(42\) \(\beta_{1}\mathstrut +\mathstrut \) \(206\)
\(\nu^{5}\)\(=\)\((\)\(4\) \(\beta_{6}\mathstrut +\mathstrut \) \(68\) \(\beta_{5}\mathstrut +\mathstrut \) \(509\) \(\beta_{4}\mathstrut -\mathstrut \) \(357\) \(\beta_{3}\mathstrut +\mathstrut \) \(512\) \(\beta_{2}\mathstrut -\mathstrut \) \(500\) \(\beta_{1}\mathstrut +\mathstrut \) \(1869\)\()/2\)
\(\nu^{6}\)\(=\)\(34\) \(\beta_{6}\mathstrut +\mathstrut \) \(116\) \(\beta_{5}\mathstrut +\mathstrut \) \(1305\) \(\beta_{4}\mathstrut -\mathstrut \) \(796\) \(\beta_{3}\mathstrut +\mathstrut \) \(1658\) \(\beta_{2}\mathstrut -\mathstrut \) \(1413\) \(\beta_{1}\mathstrut +\mathstrut \) \(5999\)

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
5.68936
−3.29882
1.14258
−2.69745
0.327888
2.21696
−0.380516
1.00000 0 1.00000 1.00000 0 −5.07548 1.00000 0 1.00000
1.2 1.00000 0 1.00000 1.00000 0 −3.64705 1.00000 0 1.00000
1.3 1.00000 0 1.00000 1.00000 0 −0.407283 1.00000 0 1.00000
1.4 1.00000 0 1.00000 1.00000 0 1.15694 1.00000 0 1.00000
1.5 1.00000 0 1.00000 1.00000 0 1.20963 1.00000 0 1.00000
1.6 1.00000 0 1.00000 1.00000 0 2.69061 1.00000 0 1.00000
1.7 1.00000 0 1.00000 1.00000 0 5.07263 1.00000 0 1.00000
\(n\): 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
\(2\) \(-1\)
\(3\) \(-1\)
\(5\) \(-1\)
\(89\) \(1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8010))\):

\(T_{7}^{7} \) \(\mathstrut -\mathstrut T_{7}^{6} \) \(\mathstrut -\mathstrut 37 T_{7}^{5} \) \(\mathstrut +\mathstrut 46 T_{7}^{4} \) \(\mathstrut +\mathstrut 286 T_{7}^{3} \) \(\mathstrut -\mathstrut 526 T_{7}^{2} \) \(\mathstrut +\mathstrut 96 T_{7} \) \(\mathstrut +\mathstrut 144 \)
\(T_{11}^{7} \) \(\mathstrut +\mathstrut T_{11}^{6} \) \(\mathstrut -\mathstrut 65 T_{11}^{5} \) \(\mathstrut -\mathstrut 46 T_{11}^{4} \) \(\mathstrut +\mathstrut 1120 T_{11}^{3} \) \(\mathstrut +\mathstrut 220 T_{11}^{2} \) \(\mathstrut -\mathstrut 4544 T_{11} \) \(\mathstrut -\mathstrut 2304 \)
\(T_{13}^{7} \) \(\mathstrut -\mathstrut 86 T_{13}^{5} \) \(\mathstrut +\mathstrut 18 T_{13}^{4} \) \(\mathstrut +\mathstrut 2364 T_{13}^{3} \) \(\mathstrut -\mathstrut 1568 T_{13}^{2} \) \(\mathstrut -\mathstrut 20848 T_{13} \) \(\mathstrut +\mathstrut 29088 \)
\(T_{17}^{7} \) \(\mathstrut -\mathstrut 9 T_{17}^{6} \) \(\mathstrut -\mathstrut 53 T_{17}^{5} \) \(\mathstrut +\mathstrut 616 T_{17}^{4} \) \(\mathstrut -\mathstrut 20 T_{17}^{3} \) \(\mathstrut -\mathstrut 9804 T_{17}^{2} \) \(\mathstrut +\mathstrut 20464 T_{17} \) \(\mathstrut -\mathstrut 9392 \)