Properties

Label 4005.2.a.t
Level 4005
Weight 2
Character orbit 4005.a
Self dual Yes
Analytic conductor 31.980
Analytic rank 0
Dimension 10
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: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( -\beta_{1} q^{2} \) \( + ( 2 + \beta_{2} ) q^{4} \) \(+ q^{5}\) \( + ( 1 - \beta_{4} ) q^{7} \) \( + ( -\beta_{1} - \beta_{5} - \beta_{6} ) q^{8} \) \(+O(q^{10})\) \( q\) \( -\beta_{1} q^{2} \) \( + ( 2 + \beta_{2} ) q^{4} \) \(+ q^{5}\) \( + ( 1 - \beta_{4} ) q^{7} \) \( + ( -\beta_{1} - \beta_{5} - \beta_{6} ) q^{8} \) \( -\beta_{1} q^{10} \) \( + \beta_{8} q^{11} \) \( + ( 2 - \beta_{5} ) q^{13} \) \( + ( -2 \beta_{1} - \beta_{8} + \beta_{9} ) q^{14} \) \( + ( 3 + \beta_{2} + \beta_{3} + \beta_{7} + \beta_{9} ) q^{16} \) \( + ( -1 - \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{9} ) q^{17} \) \( + ( 1 - \beta_{7} ) q^{19} \) \( + ( 2 + \beta_{2} ) q^{20} \) \( + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} + 3 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{9} ) q^{22} \) \( + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{9} ) q^{23} \) \(+ q^{25}\) \( + ( 2 - 2 \beta_{1} + \beta_{2} + \beta_{7} ) q^{26} \) \( + ( 4 + \beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{5} - 2 \beta_{6} ) q^{28} \) \( + ( -1 + \beta_{3} - \beta_{7} ) q^{29} \) \( + ( -1 - \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{9} ) q^{31} \) \( + ( 1 - \beta_{2} - \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{32} \) \( + ( \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - 2 \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} ) q^{34} \) \( + ( 1 - \beta_{4} ) q^{35} \) \( + ( 3 + \beta_{3} + \beta_{7} + \beta_{8} ) q^{37} \) \( + ( -2 - 3 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} + 3 \beta_{5} + \beta_{6} - \beta_{7} + \beta_{9} ) q^{38} \) \( + ( -\beta_{1} - \beta_{5} - \beta_{6} ) q^{40} \) \( + ( -2 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{5} + \beta_{6} - \beta_{8} - \beta_{9} ) q^{41} \) \( + ( 1 + \beta_{3} - \beta_{7} - \beta_{8} - 2 \beta_{9} ) q^{43} \) \( + ( \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} - 3 \beta_{9} ) q^{44} \) \( + ( -2 - \beta_{1} - \beta_{2} - 3 \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} ) q^{46} \) \( + ( 2 \beta_{1} - \beta_{4} + \beta_{7} - 2 \beta_{9} ) q^{47} \) \( + ( 2 + \beta_{1} - \beta_{2} + \beta_{3} - 3 \beta_{4} - 2 \beta_{5} - \beta_{6} - 3 \beta_{9} ) q^{49} \) \( -\beta_{1} q^{50} \) \( + ( 6 - \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + \beta_{7} - \beta_{9} ) q^{52} \) \( + ( -2 - 2 \beta_{3} - \beta_{4} + 2 \beta_{5} - \beta_{7} ) q^{53} \) \( + \beta_{8} q^{55} \) \( + ( -3 \beta_{1} + 2 \beta_{2} - \beta_{5} - 2 \beta_{6} + \beta_{7} + 2 \beta_{9} ) q^{56} \) \( + ( -2 - \beta_{1} + \beta_{2} - 3 \beta_{3} + \beta_{4} + 3 \beta_{5} - \beta_{7} + \beta_{9} ) q^{58} \) \( + ( 2 + \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} ) q^{59} \) \( + ( 1 + \beta_{1} - 3 \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} + \beta_{8} - \beta_{9} ) q^{61} \) \( + ( 3 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} ) q^{62} \) \( + ( 2 + \beta_{2} + 3 \beta_{3} - \beta_{4} - 2 \beta_{5} + \beta_{6} + \beta_{7} - 2 \beta_{8} + \beta_{9} ) q^{64} \) \( + ( 2 - \beta_{5} ) q^{65} \) \( + ( 4 - 2 \beta_{1} + 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - \beta_{8} ) q^{67} \) \( + ( 2 - \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 3 \beta_{4} + 3 \beta_{5} + 3 \beta_{6} - \beta_{8} + 2 \beta_{9} ) q^{68} \) \( + ( -2 \beta_{1} - \beta_{8} + \beta_{9} ) q^{70} \) \( + ( 3 + 2 \beta_{2} + 2 \beta_{4} - \beta_{7} + \beta_{8} + 2 \beta_{9} ) q^{71} \) \( + ( 3 - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - \beta_{7} + \beta_{8} + 2 \beta_{9} ) q^{73} \) \( + ( 3 - 2 \beta_{1} - 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - \beta_{6} + \beta_{7} ) q^{74} \) \( + ( 2 \beta_{5} - 2 \beta_{7} + \beta_{8} ) q^{76} \) \( + ( 5 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - 3 \beta_{6} - \beta_{7} + 2 \beta_{8} + \beta_{9} ) q^{77} \) \( + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} + 3 \beta_{9} ) q^{79} \) \( + ( 3 + \beta_{2} + \beta_{3} + \beta_{7} + \beta_{9} ) q^{80} \) \( + ( 1 + 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - \beta_{7} - 3 \beta_{9} ) q^{82} \) \( + ( 1 + 2 \beta_{2} - 2 \beta_{6} - \beta_{7} + \beta_{8} ) q^{83} \) \( + ( -1 - \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{9} ) q^{85} \) \( + ( -1 - 2 \beta_{1} - 2 \beta_{3} + 2 \beta_{5} + \beta_{6} - \beta_{7} - 2 \beta_{9} ) q^{86} \) \( + ( -1 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} + 3 \beta_{5} + 3 \beta_{6} - \beta_{8} - \beta_{9} ) q^{88} \) \(+ q^{89}\) \( + ( 2 + \beta_{3} - 3 \beta_{4} - 2 \beta_{6} + \beta_{7} ) q^{91} \) \( + ( 1 + 5 \beta_{1} - \beta_{2} + 5 \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} + \beta_{8} - 3 \beta_{9} ) q^{92} \) \( + ( -4 + \beta_{1} - 3 \beta_{2} + \beta_{3} + \beta_{4} - 3 \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} - 2 \beta_{9} ) q^{94} \) \( + ( 1 - \beta_{7} ) q^{95} \) \( + ( 3 - 3 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} + \beta_{9} ) q^{97} \) \( + ( 4 - 4 \beta_{1} + 2 \beta_{2} - \beta_{3} + 3 \beta_{4} + \beta_{5} + 3 \beta_{6} + 2 \beta_{7} - 3 \beta_{8} + \beta_{9} ) q^{98} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(10q \) \(\mathstrut +\mathstrut 18q^{4} \) \(\mathstrut +\mathstrut 10q^{5} \) \(\mathstrut +\mathstrut 9q^{7} \) \(\mathstrut -\mathstrut 3q^{8} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(10q \) \(\mathstrut +\mathstrut 18q^{4} \) \(\mathstrut +\mathstrut 10q^{5} \) \(\mathstrut +\mathstrut 9q^{7} \) \(\mathstrut -\mathstrut 3q^{8} \) \(\mathstrut -\mathstrut 4q^{11} \) \(\mathstrut +\mathstrut 15q^{13} \) \(\mathstrut +\mathstrut q^{14} \) \(\mathstrut +\mathstrut 22q^{16} \) \(\mathstrut -\mathstrut 11q^{17} \) \(\mathstrut +\mathstrut 14q^{19} \) \(\mathstrut +\mathstrut 18q^{20} \) \(\mathstrut +\mathstrut 10q^{22} \) \(\mathstrut +\mathstrut 8q^{23} \) \(\mathstrut +\mathstrut 10q^{25} \) \(\mathstrut +\mathstrut 14q^{26} \) \(\mathstrut +\mathstrut 36q^{28} \) \(\mathstrut -\mathstrut 5q^{29} \) \(\mathstrut -\mathstrut 8q^{31} \) \(\mathstrut -\mathstrut q^{32} \) \(\mathstrut +\mathstrut 2q^{34} \) \(\mathstrut +\mathstrut 9q^{35} \) \(\mathstrut +\mathstrut 23q^{37} \) \(\mathstrut -\mathstrut 8q^{38} \) \(\mathstrut -\mathstrut 3q^{40} \) \(\mathstrut -\mathstrut 13q^{41} \) \(\mathstrut +\mathstrut 25q^{43} \) \(\mathstrut +\mathstrut 8q^{44} \) \(\mathstrut -\mathstrut 14q^{46} \) \(\mathstrut +\mathstrut q^{47} \) \(\mathstrut +\mathstrut 21q^{49} \) \(\mathstrut +\mathstrut 51q^{52} \) \(\mathstrut -\mathstrut 9q^{53} \) \(\mathstrut -\mathstrut 4q^{55} \) \(\mathstrut -\mathstrut 15q^{56} \) \(\mathstrut -\mathstrut 8q^{58} \) \(\mathstrut +\mathstrut 15q^{59} \) \(\mathstrut +\mathstrut 8q^{61} \) \(\mathstrut +\mathstrut 8q^{62} \) \(\mathstrut +\mathstrut 9q^{64} \) \(\mathstrut +\mathstrut 15q^{65} \) \(\mathstrut +\mathstrut 52q^{67} \) \(\mathstrut +\mathstrut 28q^{68} \) \(\mathstrut +\mathstrut q^{70} \) \(\mathstrut +\mathstrut 22q^{71} \) \(\mathstrut +\mathstrut 34q^{73} \) \(\mathstrut +\mathstrut 18q^{74} \) \(\mathstrut +\mathstrut 14q^{76} \) \(\mathstrut -\mathstrut 4q^{77} \) \(\mathstrut -\mathstrut 3q^{79} \) \(\mathstrut +\mathstrut 22q^{80} \) \(\mathstrut +\mathstrut 17q^{82} \) \(\mathstrut +\mathstrut 10q^{83} \) \(\mathstrut -\mathstrut 11q^{85} \) \(\mathstrut +\mathstrut 6q^{86} \) \(\mathstrut +\mathstrut 4q^{88} \) \(\mathstrut +\mathstrut 10q^{89} \) \(\mathstrut +\mathstrut 18q^{91} \) \(\mathstrut +\mathstrut 14q^{92} \) \(\mathstrut -\mathstrut 43q^{94} \) \(\mathstrut +\mathstrut 14q^{95} \) \(\mathstrut +\mathstrut 34q^{97} \) \(\mathstrut +\mathstrut 38q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10}\mathstrut -\mathstrut \) \(19\) \(x^{8}\mathstrut -\mathstrut \) \(x^{7}\mathstrut +\mathstrut \) \(128\) \(x^{6}\mathstrut +\mathstrut \) \(14\) \(x^{5}\mathstrut -\mathstrut \) \(358\) \(x^{4}\mathstrut -\mathstrut \) \(59\) \(x^{3}\mathstrut +\mathstrut \) \(344\) \(x^{2}\mathstrut +\mathstrut \) \(71\) \(x\mathstrut -\mathstrut \) \(21\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 4 \)
\(\beta_{3}\)\(=\)\((\)\( 14 \nu^{9} + 41 \nu^{8} - 212 \nu^{7} - 582 \nu^{6} + 1039 \nu^{5} + 2512 \nu^{4} - 1884 \nu^{3} - 3357 \nu^{2} + 1103 \nu + 696 \)\()/185\)
\(\beta_{4}\)\(=\)\((\)\( 7 \nu^{9} - 72 \nu^{8} - 106 \nu^{7} + 1004 \nu^{6} + 612 \nu^{5} - 4109 \nu^{4} - 1867 \nu^{3} + 4519 \nu^{2} + 2494 \nu + 163 \)\()/185\)
\(\beta_{5}\)\(=\)\((\)\( -13 \nu^{9} + 28 \nu^{8} + 144 \nu^{7} - 411 \nu^{6} - 238 \nu^{5} + 1896 \nu^{4} - 1052 \nu^{3} - 3001 \nu^{2} + 1579 \nu + 1098 \)\()/185\)
\(\beta_{6}\)\(=\)\((\)\( 13 \nu^{9} - 28 \nu^{8} - 144 \nu^{7} + 411 \nu^{6} + 238 \nu^{5} - 1896 \nu^{4} + 1237 \nu^{3} + 3001 \nu^{2} - 2504 \nu - 1098 \)\()/185\)
\(\beta_{7}\)\(=\)\((\)\( 28 \nu^{9} - 103 \nu^{8} - 424 \nu^{7} + 1426 \nu^{6} + 2078 \nu^{5} - 5706 \nu^{4} - 3768 \nu^{3} + 5866 \nu^{2} + 2021 \nu + 97 \)\()/185\)
\(\beta_{8}\)\(=\)\((\)\( 6 \nu^{9} + 7 \nu^{8} - 75 \nu^{7} - 112 \nu^{6} + 218 \nu^{5} + 548 \nu^{4} + 144 \nu^{3} - 778 \nu^{2} - 595 \nu - 3 \)\()/37\)
\(\beta_{9}\)\(=\)\((\)\( -42 \nu^{9} + 62 \nu^{8} + 636 \nu^{7} - 844 \nu^{6} - 3117 \nu^{5} + 3379 \nu^{4} + 5652 \nu^{3} - 3804 \nu^{2} - 3124 \nu + 132 \)\()/185\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(4\)
\(\nu^{3}\)\(=\)\(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(5\) \(\beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(7\) \(\beta_{2}\mathstrut +\mathstrut \) \(23\)
\(\nu^{5}\)\(=\)\(-\)\(\beta_{7}\mathstrut +\mathstrut \) \(10\) \(\beta_{6}\mathstrut +\mathstrut \) \(10\) \(\beta_{5}\mathstrut +\mathstrut \) \(2\) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(28\) \(\beta_{1}\mathstrut -\mathstrut \) \(1\)
\(\nu^{6}\)\(=\)\(11\) \(\beta_{9}\mathstrut -\mathstrut \) \(2\) \(\beta_{8}\mathstrut +\mathstrut \) \(11\) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(2\) \(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(13\) \(\beta_{3}\mathstrut +\mathstrut \) \(47\) \(\beta_{2}\mathstrut +\mathstrut \) \(144\)
\(\nu^{7}\)\(=\)\(2\) \(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{8}\mathstrut -\mathstrut \) \(13\) \(\beta_{7}\mathstrut +\mathstrut \) \(84\) \(\beta_{6}\mathstrut +\mathstrut \) \(82\) \(\beta_{5}\mathstrut +\mathstrut \) \(28\) \(\beta_{4}\mathstrut +\mathstrut \) \(14\) \(\beta_{3}\mathstrut +\mathstrut \) \(12\) \(\beta_{2}\mathstrut +\mathstrut \) \(168\) \(\beta_{1}\mathstrut -\mathstrut \) \(12\)
\(\nu^{8}\)\(=\)\(96\) \(\beta_{9}\mathstrut -\mathstrut \) \(28\) \(\beta_{8}\mathstrut +\mathstrut \) \(95\) \(\beta_{7}\mathstrut +\mathstrut \) \(14\) \(\beta_{6}\mathstrut -\mathstrut \) \(28\) \(\beta_{5}\mathstrut -\mathstrut \) \(14\) \(\beta_{4}\mathstrut +\mathstrut \) \(126\) \(\beta_{3}\mathstrut +\mathstrut \) \(320\) \(\beta_{2}\mathstrut -\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(947\)
\(\nu^{9}\)\(=\)\(27\) \(\beta_{9}\mathstrut +\mathstrut \) \(14\) \(\beta_{8}\mathstrut -\mathstrut \) \(123\) \(\beta_{7}\mathstrut +\mathstrut \) \(665\) \(\beta_{6}\mathstrut +\mathstrut \) \(633\) \(\beta_{5}\mathstrut +\mathstrut \) \(275\) \(\beta_{4}\mathstrut +\mathstrut \) \(143\) \(\beta_{3}\mathstrut +\mathstrut \) \(108\) \(\beta_{2}\mathstrut +\mathstrut \) \(1063\) \(\beta_{1}\mathstrut -\mathstrut \) \(112\)

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
2.72551
2.28323
2.20255
1.37466
0.167560
−0.395751
−1.33301
−2.11901
−2.19546
−2.71029
−2.72551 0 5.42842 1.00000 0 0.930190 −9.34419 0 −2.72551
1.2 −2.28323 0 3.21313 1.00000 0 0.569221 −2.76986 0 −2.28323
1.3 −2.20255 0 2.85124 1.00000 0 4.87392 −1.87489 0 −2.20255
1.4 −1.37466 0 −0.110304 1.00000 0 −2.28939 2.90096 0 −1.37466
1.5 −0.167560 0 −1.97192 1.00000 0 −2.76137 0.665537 0 −0.167560
1.6 0.395751 0 −1.84338 1.00000 0 1.55828 −1.52102 0 0.395751
1.7 1.33301 0 −0.223096 1.00000 0 4.48223 −2.96340 0 1.33301
1.8 2.11901 0 2.49018 1.00000 0 1.69488 1.03870 0 2.11901
1.9 2.19546 0 2.82005 1.00000 0 −3.75700 1.80038 0 2.19546
1.10 2.71029 0 5.34569 1.00000 0 3.69905 9.06779 0 2.71029
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4005))\):

\(T_{2}^{10} - \cdots\)
\(T_{7}^{10} - \cdots\)
\(T_{11}^{10} + \cdots\)