Properties

Label 4005.2.a.q
Level 4005
Weight 2
Character orbit 4005.a
Self dual Yes
Analytic conductor 31.980
Analytic rank 1
Dimension 9
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: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)  \(=\)  \( q\) \( + ( -1 + \beta_{1} ) q^{2} \) \( + ( 2 - \beta_{1} + \beta_{2} ) q^{4} \) \(+ q^{5}\) \( + ( - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{8} ) q^{7} \) \( + ( -3 + 2 \beta_{1} - \beta_{2} + \beta_{4} + \beta_{5} ) q^{8} \) \(+O(q^{10})\) \( q\) \( + ( -1 + \beta_{1} ) q^{2} \) \( + ( 2 - \beta_{1} + \beta_{2} ) q^{4} \) \(+ q^{5}\) \( + ( - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{8} ) q^{7} \) \( + ( -3 + 2 \beta_{1} - \beta_{2} + \beta_{4} + \beta_{5} ) q^{8} \) \( + ( -1 + \beta_{1} ) q^{10} \) \( + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} ) q^{11} \) \( + ( 1 - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} ) q^{13} \) \( + ( 1 - \beta_{2} - \beta_{4} + 2 \beta_{6} - \beta_{7} - 2 \beta_{8} ) q^{14} \) \( + ( 3 - 4 \beta_{1} + \beta_{3} - 2 \beta_{4} - 2 \beta_{5} ) q^{16} \) \( + ( -3 - \beta_{1} - \beta_{2} - \beta_{4} + \beta_{6} + \beta_{7} + \beta_{8} ) q^{17} \) \( + ( -1 - \beta_{3} + 2 \beta_{4} + \beta_{5} - \beta_{8} ) q^{19} \) \( + ( 2 - \beta_{1} + \beta_{2} ) q^{20} \) \( + ( 1 - \beta_{1} + \beta_{3} + \beta_{4} + 2 \beta_{7} ) q^{22} \) \( + ( -3 + \beta_{1} + \beta_{3} - \beta_{4} - 2 \beta_{6} ) q^{23} \) \(+ q^{25}\) \( + ( -2 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{5} - 2 \beta_{6} + \beta_{8} ) q^{26} \) \( + ( -2 + \beta_{3} + \beta_{4} - \beta_{5} - 4 \beta_{6} + 2 \beta_{7} + 4 \beta_{8} ) q^{28} \) \( + ( -1 + 2 \beta_{1} - \beta_{2} - \beta_{5} - \beta_{6} + \beta_{7} - 2 \beta_{8} ) q^{29} \) \( + ( 1 - 3 \beta_{1} - \beta_{3} - \beta_{4} - 2 \beta_{5} - 2 \beta_{7} ) q^{31} \) \( + ( -7 + 3 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{32} \) \( + ( 4 - 5 \beta_{1} - \beta_{3} - 2 \beta_{5} - \beta_{7} ) q^{34} \) \( + ( - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{8} ) q^{35} \) \( + ( \beta_{1} + \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{7} ) q^{37} \) \( + ( -2 - 5 \beta_{1} + 2 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} + 3 \beta_{8} ) q^{38} \) \( + ( -3 + 2 \beta_{1} - \beta_{2} + \beta_{4} + \beta_{5} ) q^{40} \) \( + ( 2 - 3 \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} + \beta_{8} ) q^{41} \) \( + ( -2 + \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} + 3 \beta_{7} ) q^{43} \) \( + ( -5 + \beta_{2} - \beta_{3} + \beta_{6} + 3 \beta_{8} ) q^{44} \) \( + ( 3 + 2 \beta_{1} - \beta_{2} - \beta_{3} + 4 \beta_{5} + \beta_{6} - 2 \beta_{7} - 5 \beta_{8} ) q^{46} \) \( + ( -1 - \beta_{1} - \beta_{2} - \beta_{5} + 2 \beta_{6} - 2 \beta_{7} - 2 \beta_{8} ) q^{47} \) \( + ( 3 - 3 \beta_{1} - 2 \beta_{2} + \beta_{3} - 2 \beta_{4} - 3 \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} ) q^{49} \) \( + ( -1 + \beta_{1} ) q^{50} \) \( + ( 4 - \beta_{2} - 2 \beta_{4} + 2 \beta_{6} - 2 \beta_{7} - 5 \beta_{8} ) q^{52} \) \( + ( -5 + \beta_{1} + \beta_{2} + \beta_{5} ) q^{53} \) \( + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} ) q^{55} \) \( + ( 2 + 3 \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} + 3 \beta_{6} - \beta_{7} - 8 \beta_{8} ) q^{56} \) \( + ( 3 - 2 \beta_{1} + \beta_{2} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} + 4 \beta_{8} ) q^{58} \) \( + ( -2 - \beta_{2} + 3 \beta_{3} + \beta_{4} - 3 \beta_{5} - 2 \beta_{6} + 3 \beta_{7} + 2 \beta_{8} ) q^{59} \) \( + ( -4 + 2 \beta_{1} + \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + \beta_{5} + \beta_{6} + 2 \beta_{8} ) q^{61} \) \( + ( -5 - 3 \beta_{2} - \beta_{3} + 2 \beta_{5} + \beta_{6} + 2 \beta_{7} - \beta_{8} ) q^{62} \) \( + ( 10 - 8 \beta_{1} + 2 \beta_{2} - 4 \beta_{4} - 2 \beta_{5} + 3 \beta_{6} - 3 \beta_{7} ) q^{64} \) \( + ( 1 - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} ) q^{65} \) \( + ( -1 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} ) q^{67} \) \( + ( -9 + 4 \beta_{1} - 3 \beta_{2} + 2 \beta_{4} - \beta_{6} - \beta_{8} ) q^{68} \) \( + ( 1 - \beta_{2} - \beta_{4} + 2 \beta_{6} - \beta_{7} - 2 \beta_{8} ) q^{70} \) \( + ( 2 - \beta_{1} - \beta_{2} - \beta_{4} + 2 \beta_{5} + \beta_{6} - 2 \beta_{7} - 3 \beta_{8} ) q^{71} \) \( + ( 2 + \beta_{1} - 3 \beta_{2} - 3 \beta_{4} + 3 \beta_{6} - 2 \beta_{7} - \beta_{8} ) q^{73} \) \( + ( 3 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{5} - \beta_{6} - 3 \beta_{7} - 2 \beta_{8} ) q^{74} \) \( + ( -3 + 5 \beta_{1} - 5 \beta_{2} - \beta_{3} + 3 \beta_{4} + 3 \beta_{5} + \beta_{6} + 2 \beta_{7} - 4 \beta_{8} ) q^{76} \) \( + ( -3 \beta_{1} + 4 \beta_{2} - \beta_{4} + 3 \beta_{5} + 4 \beta_{6} - 2 \beta_{7} - \beta_{8} ) q^{77} \) \( + ( -1 - 3 \beta_{1} + 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + \beta_{6} + 2 \beta_{7} + 4 \beta_{8} ) q^{79} \) \( + ( 3 - 4 \beta_{1} + \beta_{3} - 2 \beta_{4} - 2 \beta_{5} ) q^{80} \) \( + ( -9 + 7 \beta_{1} - 2 \beta_{2} + 6 \beta_{4} + 2 \beta_{5} - \beta_{6} + 3 \beta_{7} + \beta_{8} ) q^{82} \) \( + ( -2 + \beta_{1} + \beta_{2} + 2 \beta_{3} + 3 \beta_{4} - \beta_{6} + 2 \beta_{7} - \beta_{8} ) q^{83} \) \( + ( -3 - \beta_{1} - \beta_{2} - \beta_{4} + \beta_{6} + \beta_{7} + \beta_{8} ) q^{85} \) \( + ( -5 \beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{4} - 5 \beta_{5} - \beta_{6} - 3 \beta_{7} + 2 \beta_{8} ) q^{86} \) \( + ( 12 - 4 \beta_{1} + \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - \beta_{5} + 3 \beta_{6} - 4 \beta_{7} - 4 \beta_{8} ) q^{88} \) \(- q^{89}\) \( + ( -9 + 5 \beta_{1} - 2 \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} - 3 \beta_{6} + \beta_{7} + \beta_{8} ) q^{91} \) \( + ( -2 - 4 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} - 4 \beta_{4} - \beta_{5} - \beta_{6} - 2 \beta_{7} + 6 \beta_{8} ) q^{92} \) \( + ( -3 - 5 \beta_{1} + \beta_{2} + 2 \beta_{4} - \beta_{5} - 4 \beta_{6} + 5 \beta_{7} + 7 \beta_{8} ) q^{94} \) \( + ( -1 - \beta_{3} + 2 \beta_{4} + \beta_{5} - \beta_{8} ) q^{95} \) \( + ( 3 + \beta_{1} + 4 \beta_{2} - 3 \beta_{3} + \beta_{4} + 2 \beta_{6} - 4 \beta_{7} - 2 \beta_{8} ) q^{97} \) \( + ( -12 + 4 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + \beta_{5} - \beta_{6} + 2 \beta_{8} ) q^{98} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)  \(=\)  \(9q \) \(\mathstrut -\mathstrut 5q^{2} \) \(\mathstrut +\mathstrut 11q^{4} \) \(\mathstrut +\mathstrut 9q^{5} \) \(\mathstrut -\mathstrut 3q^{7} \) \(\mathstrut -\mathstrut 18q^{8} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(9q \) \(\mathstrut -\mathstrut 5q^{2} \) \(\mathstrut +\mathstrut 11q^{4} \) \(\mathstrut +\mathstrut 9q^{5} \) \(\mathstrut -\mathstrut 3q^{7} \) \(\mathstrut -\mathstrut 18q^{8} \) \(\mathstrut -\mathstrut 5q^{10} \) \(\mathstrut -\mathstrut 4q^{11} \) \(\mathstrut +\mathstrut 5q^{13} \) \(\mathstrut +\mathstrut q^{14} \) \(\mathstrut +\mathstrut 15q^{16} \) \(\mathstrut -\mathstrut 21q^{17} \) \(\mathstrut -\mathstrut 18q^{19} \) \(\mathstrut +\mathstrut 11q^{20} \) \(\mathstrut +\mathstrut 6q^{22} \) \(\mathstrut -\mathstrut 16q^{23} \) \(\mathstrut +\mathstrut 9q^{25} \) \(\mathstrut -\mathstrut 8q^{26} \) \(\mathstrut +\mathstrut 6q^{28} \) \(\mathstrut -\mathstrut 3q^{29} \) \(\mathstrut -\mathstrut 6q^{31} \) \(\mathstrut -\mathstrut 46q^{32} \) \(\mathstrut +\mathstrut 12q^{34} \) \(\mathstrut -\mathstrut 3q^{35} \) \(\mathstrut +\mathstrut 11q^{37} \) \(\mathstrut -\mathstrut 20q^{38} \) \(\mathstrut -\mathstrut 18q^{40} \) \(\mathstrut +\mathstrut q^{41} \) \(\mathstrut -\mathstrut 3q^{43} \) \(\mathstrut -\mathstrut 38q^{44} \) \(\mathstrut +\mathstrut 16q^{46} \) \(\mathstrut -\mathstrut 27q^{47} \) \(\mathstrut +\mathstrut 24q^{49} \) \(\mathstrut -\mathstrut 5q^{50} \) \(\mathstrut +\mathstrut 17q^{52} \) \(\mathstrut -\mathstrut 43q^{53} \) \(\mathstrut -\mathstrut 4q^{55} \) \(\mathstrut -\mathstrut 5q^{56} \) \(\mathstrut +\mathstrut 34q^{58} \) \(\mathstrut -\mathstrut 3q^{59} \) \(\mathstrut -\mathstrut 30q^{61} \) \(\mathstrut -\mathstrut 36q^{62} \) \(\mathstrut +\mathstrut 50q^{64} \) \(\mathstrut +\mathstrut 5q^{65} \) \(\mathstrut -\mathstrut 12q^{67} \) \(\mathstrut -\mathstrut 64q^{68} \) \(\mathstrut +\mathstrut q^{70} \) \(\mathstrut +\mathstrut 4q^{71} \) \(\mathstrut +\mathstrut 26q^{73} \) \(\mathstrut -\mathstrut 2q^{74} \) \(\mathstrut -\mathstrut 12q^{76} \) \(\mathstrut -\mathstrut 34q^{77} \) \(\mathstrut +\mathstrut q^{79} \) \(\mathstrut +\mathstrut 15q^{80} \) \(\mathstrut -\mathstrut 51q^{82} \) \(\mathstrut -\mathstrut 24q^{83} \) \(\mathstrut -\mathstrut 21q^{85} \) \(\mathstrut -\mathstrut 18q^{86} \) \(\mathstrut +\mathstrut 64q^{88} \) \(\mathstrut -\mathstrut 9q^{89} \) \(\mathstrut -\mathstrut 50q^{91} \) \(\mathstrut -\mathstrut 10q^{92} \) \(\mathstrut -\mathstrut 11q^{94} \) \(\mathstrut -\mathstrut 18q^{95} \) \(\mathstrut -\mathstrut 4q^{97} \) \(\mathstrut -\mathstrut 75q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{9}\mathstrut -\mathstrut \) \(4\) \(x^{8}\mathstrut -\mathstrut \) \(6\) \(x^{7}\mathstrut +\mathstrut \) \(31\) \(x^{6}\mathstrut +\mathstrut \) \(13\) \(x^{5}\mathstrut -\mathstrut \) \(75\) \(x^{4}\mathstrut -\mathstrut \) \(17\) \(x^{3}\mathstrut +\mathstrut \) \(52\) \(x^{2}\mathstrut +\mathstrut \) \(11\) \(x\mathstrut -\mathstrut \) \(1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{4} - 2 \nu^{3} - 4 \nu^{2} + 4 \nu + 2 \)
\(\beta_{4}\)\(=\)\( \nu^{6} - 3 \nu^{5} - 4 \nu^{4} + 13 \nu^{3} + 3 \nu^{2} - 12 \nu \)
\(\beta_{5}\)\(=\)\( -\nu^{6} + 3 \nu^{5} + 4 \nu^{4} - 12 \nu^{3} - 5 \nu^{2} + 8 \nu + 3 \)
\(\beta_{6}\)\(=\)\( \nu^{8} - 5 \nu^{7} + \nu^{6} + 23 \nu^{5} - 16 \nu^{4} - 28 \nu^{3} + 12 \nu^{2} + 10 \nu + 1 \)
\(\beta_{7}\)\(=\)\( \nu^{8} - 5 \nu^{7} + 27 \nu^{5} - 15 \nu^{4} - 44 \nu^{3} + 19 \nu^{2} + 22 \nu - 2 \)
\(\beta_{8}\)\(=\)\( -\nu^{8} + 6 \nu^{7} - 5 \nu^{6} - 25 \nu^{5} + 35 \nu^{4} + 24 \nu^{3} - 35 \nu^{2} - 5 \nu + 4 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(3\)
\(\nu^{3}\)\(=\)\(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(6\) \(\beta_{1}\mathstrut +\mathstrut \) \(3\)
\(\nu^{4}\)\(=\)\(2\) \(\beta_{5}\mathstrut +\mathstrut \) \(2\) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(8\) \(\beta_{2}\mathstrut +\mathstrut \) \(12\) \(\beta_{1}\mathstrut +\mathstrut \) \(16\)
\(\nu^{5}\)\(=\)\(\beta_{7}\mathstrut -\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(9\) \(\beta_{5}\mathstrut +\mathstrut \) \(10\) \(\beta_{4}\mathstrut +\mathstrut \) \(3\) \(\beta_{3}\mathstrut +\mathstrut \) \(20\) \(\beta_{2}\mathstrut +\mathstrut \) \(44\) \(\beta_{1}\mathstrut +\mathstrut \) \(30\)
\(\nu^{6}\)\(=\)\(3\) \(\beta_{7}\mathstrut -\mathstrut \) \(3\) \(\beta_{6}\mathstrut +\mathstrut \) \(22\) \(\beta_{5}\mathstrut +\mathstrut \) \(26\) \(\beta_{4}\mathstrut +\mathstrut \) \(13\) \(\beta_{3}\mathstrut +\mathstrut \) \(63\) \(\beta_{2}\mathstrut +\mathstrut \) \(111\) \(\beta_{1}\mathstrut +\mathstrut \) \(106\)
\(\nu^{7}\)\(=\)\(\beta_{8}\mathstrut +\mathstrut \) \(14\) \(\beta_{7}\mathstrut -\mathstrut \) \(13\) \(\beta_{6}\mathstrut +\mathstrut \) \(72\) \(\beta_{5}\mathstrut +\mathstrut \) \(90\) \(\beta_{4}\mathstrut +\mathstrut \) \(39\) \(\beta_{3}\mathstrut +\mathstrut \) \(171\) \(\beta_{2}\mathstrut +\mathstrut \) \(346\) \(\beta_{1}\mathstrut +\mathstrut \) \(256\)
\(\nu^{8}\)\(=\)\(5\) \(\beta_{8}\mathstrut +\mathstrut \) \(44\) \(\beta_{7}\mathstrut -\mathstrut \) \(38\) \(\beta_{6}\mathstrut +\mathstrut \) \(191\) \(\beta_{5}\mathstrut +\mathstrut \) \(254\) \(\beta_{4}\mathstrut +\mathstrut \) \(129\) \(\beta_{3}\mathstrut +\mathstrut \) \(504\) \(\beta_{2}\mathstrut +\mathstrut \) \(945\) \(\beta_{1}\mathstrut +\mathstrut \) \(787\)

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
−1.72546
−1.70432
−1.02867
−0.286244
0.0690386
1.09195
1.89144
2.81142
2.88084
−2.72546 0 5.42813 1.00000 0 −4.89499 −9.34325 0 −2.72546
1.2 −2.70432 0 5.31333 1.00000 0 3.91511 −8.96030 0 −2.70432
1.3 −2.02867 0 2.11549 1.00000 0 2.64395 −0.234298 0 −2.02867
1.4 −1.28624 0 −0.345575 1.00000 0 −0.646623 3.01698 0 −1.28624
1.5 −0.930961 0 −1.13331 1.00000 0 −1.89313 2.91699 0 −0.930961
1.6 0.0919478 0 −1.99155 1.00000 0 −4.79397 −0.367014 0 0.0919478
1.7 0.891444 0 −1.20533 1.00000 0 3.64096 −2.85737 0 0.891444
1.8 1.81142 0 1.28123 1.00000 0 −0.549078 −1.30199 0 1.81142
1.9 1.88084 0 1.53757 1.00000 0 −0.422231 −0.869761 0 1.88084
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4005))\):

\(T_{2}^{9} + \cdots\)
\(T_{7}^{9} + \cdots\)
\(T_{11}^{9} + \cdots\)