Properties

Label 4005.2.a.s
Level 4005
Weight 2
Character orbit 4005.a
Self dual Yes
Analytic conductor 31.980
Analytic rank 1
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: \(1\)
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.

\(f(q)\) \(=\) \( q\) \( -\beta_{1} q^{2} \) \( + ( 1 + \beta_{2} ) q^{4} \) \(- q^{5}\) \( + \beta_{6} q^{7} \) \( + ( -\beta_{1} + \beta_{4} - \beta_{5} ) q^{8} \) \(+O(q^{10})\) \( q\) \( -\beta_{1} q^{2} \) \( + ( 1 + \beta_{2} ) q^{4} \) \(- q^{5}\) \( + \beta_{6} q^{7} \) \( + ( -\beta_{1} + \beta_{4} - \beta_{5} ) q^{8} \) \( + \beta_{1} q^{10} \) \( + ( -1 - \beta_{4} - \beta_{9} ) q^{11} \) \( + ( \beta_{1} + \beta_{5} + \beta_{7} - \beta_{9} ) q^{13} \) \( + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{8} ) q^{14} \) \( + ( 1 + \beta_{2} - \beta_{7} + \beta_{8} ) q^{16} \) \( + ( -1 - \beta_{3} + \beta_{5} - \beta_{6} + \beta_{8} + \beta_{9} ) q^{17} \) \( + ( \beta_{1} - \beta_{2} + \beta_{3} - \beta_{6} - \beta_{8} + \beta_{9} ) q^{19} \) \( + ( -1 - \beta_{2} ) q^{20} \) \( + ( 1 - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{7} + \beta_{9} ) q^{22} \) \( + ( 1 - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} ) q^{23} \) \(+ q^{25}\) \( + ( -2 - \beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - \beta_{7} + \beta_{8} + 2 \beta_{9} ) q^{26} \) \( + ( -1 + 2 \beta_{1} - \beta_{2} - \beta_{4} + \beta_{5} - \beta_{6} - \beta_{8} + \beta_{9} ) q^{28} \) \( + ( -4 + \beta_{1} - \beta_{3} + \beta_{4} + \beta_{6} ) q^{29} \) \( + ( 1 - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{7} + \beta_{9} ) q^{31} \) \( + ( -1 - \beta_{1} - \beta_{3} - \beta_{4} + \beta_{7} - \beta_{9} ) q^{32} \) \( + ( -2 + 2 \beta_{1} - \beta_{2} + \beta_{4} + \beta_{5} - \beta_{6} + 2 \beta_{7} + \beta_{8} - 2 \beta_{9} ) q^{34} \) \( -\beta_{6} q^{35} \) \( + ( 1 + \beta_{1} - \beta_{2} - \beta_{3} - \beta_{7} ) q^{37} \) \( + ( -1 + \beta_{1} - \beta_{2} - \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} ) q^{38} \) \( + ( \beta_{1} - \beta_{4} + \beta_{5} ) q^{40} \) \( + ( -2 - 2 \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{9} ) q^{41} \) \( + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{5} - \beta_{7} ) q^{43} \) \( + ( \beta_{1} - \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - 3 \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} ) q^{44} \) \( + ( 2 - \beta_{1} + \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} ) q^{46} \) \( + ( 2 - \beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{7} - \beta_{8} + \beta_{9} ) q^{47} \) \( + ( 2 - \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{4} - 2 \beta_{7} ) q^{49} \) \( -\beta_{1} q^{50} \) \( + ( -1 + 5 \beta_{1} - \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - \beta_{6} + 3 \beta_{7} + \beta_{8} - 2 \beta_{9} ) q^{52} \) \( + ( -4 + \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} - 2 \beta_{7} - \beta_{8} + \beta_{9} ) q^{53} \) \( + ( 1 + \beta_{4} + \beta_{9} ) q^{55} \) \( + ( -3 + \beta_{1} - 2 \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} + 2 \beta_{7} + \beta_{8} - \beta_{9} ) q^{56} \) \( + ( -5 + 6 \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{5} - \beta_{6} + \beta_{8} - \beta_{9} ) q^{58} \) \( + ( -4 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{4} + \beta_{7} + \beta_{8} - \beta_{9} ) q^{59} \) \( + ( 2 - \beta_{3} + \beta_{5} - \beta_{7} + 2 \beta_{8} ) q^{61} \) \( + ( -2 + \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} ) q^{62} \) \( + ( 1 + \beta_{1} - \beta_{2} - \beta_{4} - \beta_{6} - \beta_{8} + \beta_{9} ) q^{64} \) \( + ( -\beta_{1} - \beta_{5} - \beta_{7} + \beta_{9} ) q^{65} \) \( + ( -3 + 2 \beta_{1} - \beta_{2} + 2 \beta_{3} + 3 \beta_{4} - \beta_{6} - \beta_{7} + 2 \beta_{8} ) q^{67} \) \( + ( -2 + \beta_{1} + \beta_{2} - 2 \beta_{3} + 4 \beta_{4} + 3 \beta_{5} + 2 \beta_{6} - 3 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} ) q^{68} \) \( + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{8} ) q^{70} \) \( + ( -3 - \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - 2 \beta_{5} - \beta_{6} + \beta_{8} ) q^{71} \) \( + ( -1 - \beta_{1} - \beta_{3} + \beta_{4} + \beta_{7} - \beta_{8} + \beta_{9} ) q^{73} \) \( + ( -4 + 2 \beta_{1} - \beta_{2} + \beta_{3} - 3 \beta_{4} - \beta_{6} + \beta_{7} + \beta_{8} - 2 \beta_{9} ) q^{74} \) \( + ( -3 + 2 \beta_{1} - 2 \beta_{2} - \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - \beta_{9} ) q^{76} \) \( + ( -5 + \beta_{1} + \beta_{2} + 3 \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} - 2 \beta_{9} ) q^{77} \) \( + ( 4 - 3 \beta_{1} - 2 \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{6} + 2 \beta_{7} - \beta_{8} ) q^{79} \) \( + ( -1 - \beta_{2} + \beta_{7} - \beta_{8} ) q^{80} \) \( + ( 4 + 6 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 3 \beta_{4} - \beta_{6} + 2 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} ) q^{82} \) \( + ( 1 + 3 \beta_{1} + \beta_{3} - \beta_{4} + 3 \beta_{7} - 3 \beta_{8} - \beta_{9} ) q^{83} \) \( + ( 1 + \beta_{3} - \beta_{5} + \beta_{6} - \beta_{8} - \beta_{9} ) q^{85} \) \( + ( -2 - 2 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - 4 \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} ) q^{86} \) \( + ( -2 \beta_{1} + 2 \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{5} + 2 \beta_{6} - 3 \beta_{7} - 2 \beta_{8} + 2 \beta_{9} ) q^{88} \) \(+ q^{89}\) \( + ( -\beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{5} + \beta_{6} - \beta_{7} - 2 \beta_{8} - \beta_{9} ) q^{91} \) \( + ( -2 + 3 \beta_{3} - 4 \beta_{4} - 3 \beta_{5} + \beta_{7} - 2 \beta_{8} - 4 \beta_{9} ) q^{92} \) \( + ( 4 - 4 \beta_{1} + \beta_{3} - \beta_{4} - 3 \beta_{5} + \beta_{6} + 3 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} ) q^{94} \) \( + ( -\beta_{1} + \beta_{2} - \beta_{3} + \beta_{6} + \beta_{8} - \beta_{9} ) q^{95} \) \( + ( 1 + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} - 2 \beta_{7} - 2 \beta_{8} ) q^{97} \) \( + ( 4 + \beta_{1} + 2 \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{6} + 2 \beta_{7} - \beta_{9} ) q^{98} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(10q \) \(\mathstrut -\mathstrut q^{2} \) \(\mathstrut +\mathstrut 13q^{4} \) \(\mathstrut -\mathstrut 10q^{5} \) \(\mathstrut -\mathstrut q^{7} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(10q \) \(\mathstrut -\mathstrut q^{2} \) \(\mathstrut +\mathstrut 13q^{4} \) \(\mathstrut -\mathstrut 10q^{5} \) \(\mathstrut -\mathstrut q^{7} \) \(\mathstrut +\mathstrut q^{10} \) \(\mathstrut -\mathstrut 10q^{11} \) \(\mathstrut +\mathstrut 5q^{13} \) \(\mathstrut -\mathstrut 13q^{14} \) \(\mathstrut +\mathstrut 19q^{16} \) \(\mathstrut -\mathstrut 9q^{17} \) \(\mathstrut -\mathstrut 6q^{19} \) \(\mathstrut -\mathstrut 13q^{20} \) \(\mathstrut +\mathstrut 3q^{23} \) \(\mathstrut +\mathstrut 10q^{25} \) \(\mathstrut -\mathstrut 14q^{26} \) \(\mathstrut -\mathstrut 18q^{28} \) \(\mathstrut -\mathstrut 38q^{29} \) \(\mathstrut +\mathstrut 2q^{31} \) \(\mathstrut -\mathstrut 16q^{32} \) \(\mathstrut -\mathstrut 8q^{34} \) \(\mathstrut +\mathstrut q^{35} \) \(\mathstrut +\mathstrut 9q^{37} \) \(\mathstrut -\mathstrut 20q^{38} \) \(\mathstrut -\mathstrut 36q^{41} \) \(\mathstrut -\mathstrut 7q^{43} \) \(\mathstrut -\mathstrut 16q^{44} \) \(\mathstrut +\mathstrut 2q^{46} \) \(\mathstrut +\mathstrut 23q^{47} \) \(\mathstrut +\mathstrut 25q^{49} \) \(\mathstrut -\mathstrut q^{50} \) \(\mathstrut +\mathstrut 13q^{52} \) \(\mathstrut -\mathstrut 27q^{53} \) \(\mathstrut +\mathstrut 10q^{55} \) \(\mathstrut -\mathstrut 41q^{56} \) \(\mathstrut -\mathstrut 32q^{58} \) \(\mathstrut -\mathstrut 20q^{59} \) \(\mathstrut +\mathstrut 30q^{61} \) \(\mathstrut +\mathstrut 2q^{62} \) \(\mathstrut -\mathstrut 2q^{64} \) \(\mathstrut -\mathstrut 5q^{65} \) \(\mathstrut -\mathstrut 5q^{67} \) \(\mathstrut +\mathstrut 10q^{68} \) \(\mathstrut +\mathstrut 13q^{70} \) \(\mathstrut -\mathstrut 24q^{71} \) \(\mathstrut -\mathstrut 19q^{73} \) \(\mathstrut -\mathstrut 42q^{74} \) \(\mathstrut -\mathstrut 30q^{76} \) \(\mathstrut -\mathstrut 18q^{77} \) \(\mathstrut +\mathstrut 12q^{79} \) \(\mathstrut -\mathstrut 19q^{80} \) \(\mathstrut +\mathstrut 29q^{82} \) \(\mathstrut -\mathstrut 3q^{83} \) \(\mathstrut +\mathstrut 9q^{85} \) \(\mathstrut -\mathstrut 38q^{86} \) \(\mathstrut -\mathstrut 16q^{88} \) \(\mathstrut +\mathstrut 10q^{89} \) \(\mathstrut -\mathstrut 6q^{91} \) \(\mathstrut -\mathstrut 32q^{92} \) \(\mathstrut +\mathstrut 17q^{94} \) \(\mathstrut +\mathstrut 6q^{95} \) \(\mathstrut +\mathstrut 3q^{97} \) \(\mathstrut +\mathstrut 37q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10}\mathstrut -\mathstrut \) \(x^{9}\mathstrut -\mathstrut \) \(16\) \(x^{8}\mathstrut +\mathstrut \) \(15\) \(x^{7}\mathstrut +\mathstrut \) \(85\) \(x^{6}\mathstrut -\mathstrut \) \(75\) \(x^{5}\mathstrut -\mathstrut \) \(163\) \(x^{4}\mathstrut +\mathstrut \) \(138\) \(x^{3}\mathstrut +\mathstrut \) \(78\) \(x^{2}\mathstrut -\mathstrut \) \(67\) \(x\mathstrut +\mathstrut \) \(4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\((\)\( 5 \nu^{9} - 8 \nu^{8} - 106 \nu^{7} + 77 \nu^{6} + 733 \nu^{5} - 168 \nu^{4} - 1823 \nu^{3} + 59 \nu^{2} + 1140 \nu - 95 \)\()/77\)
\(\beta_{4}\)\(=\)\((\)\( -9 \nu^{9} - \nu^{8} + 160 \nu^{7} - 919 \nu^{5} + 56 \nu^{4} + 1803 \nu^{3} - 137 \nu^{2} - 743 \nu + 94 \)\()/77\)
\(\beta_{5}\)\(=\)\((\)\( -9 \nu^{9} - \nu^{8} + 160 \nu^{7} - 919 \nu^{5} + 56 \nu^{4} + 1880 \nu^{3} - 137 \nu^{2} - 1128 \nu + 94 \)\()/77\)
\(\beta_{6}\)\(=\)\((\)\( 13 \nu^{9} + 10 \nu^{8} - 214 \nu^{7} - 154 \nu^{6} + 1182 \nu^{5} + 749 \nu^{4} - 2399 \nu^{3} - 1171 \nu^{2} + 1270 \nu + 292 \)\()/77\)
\(\beta_{7}\)\(=\)\((\)\( 4 \nu^{9} - 2 \nu^{8} - 65 \nu^{7} + 22 \nu^{6} + 351 \nu^{5} - 75 \nu^{4} - 684 \nu^{3} + 111 \nu^{2} + 340 \nu - 65 \)\()/11\)
\(\beta_{8}\)\(=\)\((\)\( 4 \nu^{9} - 2 \nu^{8} - 65 \nu^{7} + 22 \nu^{6} + 351 \nu^{5} - 64 \nu^{4} - 684 \nu^{3} + 34 \nu^{2} + 340 \nu + 1 \)\()/11\)
\(\beta_{9}\)\(=\)\((\)\( 32 \nu^{9} - 5 \nu^{8} - 509 \nu^{7} + 77 \nu^{6} + 2720 \nu^{5} - 413 \nu^{4} - 5384 \nu^{3} + 855 \nu^{2} + 2830 \nu - 531 \)\()/77\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(3\)
\(\nu^{3}\)\(=\)\(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(5\) \(\beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{8}\mathstrut -\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(7\) \(\beta_{2}\mathstrut +\mathstrut \) \(15\)
\(\nu^{5}\)\(=\)\(\beta_{9}\mathstrut -\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(8\) \(\beta_{5}\mathstrut -\mathstrut \) \(7\) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(29\) \(\beta_{1}\mathstrut +\mathstrut \) \(1\)
\(\nu^{6}\)\(=\)\(\beta_{9}\mathstrut +\mathstrut \) \(9\) \(\beta_{8}\mathstrut -\mathstrut \) \(10\) \(\beta_{7}\mathstrut -\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(45\) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(87\)
\(\nu^{7}\)\(=\)\(11\) \(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{8}\mathstrut -\mathstrut \) \(11\) \(\beta_{7}\mathstrut +\mathstrut \) \(56\) \(\beta_{5}\mathstrut -\mathstrut \) \(43\) \(\beta_{4}\mathstrut +\mathstrut \) \(9\) \(\beta_{3}\mathstrut +\mathstrut \) \(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(177\) \(\beta_{1}\mathstrut +\mathstrut \) \(12\)
\(\nu^{8}\)\(=\)\(13\) \(\beta_{9}\mathstrut +\mathstrut \) \(63\) \(\beta_{8}\mathstrut -\mathstrut \) \(78\) \(\beta_{7}\mathstrut -\mathstrut \) \(9\) \(\beta_{6}\mathstrut -\mathstrut \) \(14\) \(\beta_{4}\mathstrut -\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(287\) \(\beta_{2}\mathstrut +\mathstrut \) \(14\) \(\beta_{1}\mathstrut +\mathstrut \) \(534\)
\(\nu^{9}\)\(=\)\(92\) \(\beta_{9}\mathstrut +\mathstrut \) \(17\) \(\beta_{8}\mathstrut -\mathstrut \) \(91\) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(379\) \(\beta_{5}\mathstrut -\mathstrut \) \(257\) \(\beta_{4}\mathstrut +\mathstrut \) \(58\) \(\beta_{3}\mathstrut +\mathstrut \) \(32\) \(\beta_{2}\mathstrut +\mathstrut \) \(1103\) \(\beta_{1}\mathstrut +\mathstrut \) \(110\)

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
2.58455
2.52467
1.69046
0.970031
0.644507
0.0651711
−0.875863
−1.67217
−2.41366
−2.51771
−2.58455 0 4.67989 −1.00000 0 1.54056 −6.92630 0 2.58455
1.2 −2.52467 0 4.37398 −1.00000 0 −0.151868 −5.99353 0 2.52467
1.3 −1.69046 0 0.857665 −1.00000 0 0.437617 1.93108 0 1.69046
1.4 −0.970031 0 −1.05904 −1.00000 0 −4.84165 2.96736 0 0.970031
1.5 −0.644507 0 −1.58461 −1.00000 0 2.88508 2.31031 0 0.644507
1.6 −0.0651711 0 −1.99575 −1.00000 0 4.79409 0.260408 0 0.0651711
1.7 0.875863 0 −1.23286 −1.00000 0 −3.38461 −2.83155 0 −0.875863
1.8 1.67217 0 0.796137 −1.00000 0 3.27977 −2.01306 0 −1.67217
1.9 2.41366 0 3.82573 −1.00000 0 −2.78117 4.40669 0 −2.41366
1.10 2.51771 0 4.33887 −1.00000 0 −2.77781 5.88859 0 −2.51771
\(n\): 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4005))\):

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