Properties

Label 4005.2.a.v
Level 4005
Weight 2
Character orbit 4005.a
Self dual Yes
Analytic conductor 31.980
Analytic rank 1
Dimension 12
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: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} - q^{5} + ( -1 - \beta_{2} + \beta_{5} ) q^{7} + ( 1 + \beta_{1} + \beta_{2} + \beta_{6} + \beta_{7} ) q^{8} +O(q^{10})\) \( q + \beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} - q^{5} + ( -1 - \beta_{2} + \beta_{5} ) q^{7} + ( 1 + \beta_{1} + \beta_{2} + \beta_{6} + \beta_{7} ) q^{8} -\beta_{1} q^{10} + ( -\beta_{2} + \beta_{9} ) q^{11} + ( -1 - \beta_{2} + \beta_{4} + \beta_{7} + \beta_{8} + \beta_{9} ) q^{13} + ( -3 \beta_{1} - \beta_{3} + \beta_{4} - \beta_{6} - \beta_{7} ) q^{14} + ( 2 \beta_{1} + 2 \beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{6} - 2 \beta_{9} + \beta_{11} ) q^{16} + ( -\beta_{1} - \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} ) q^{17} + ( -2 + \beta_{1} - \beta_{3} - \beta_{5} - \beta_{8} - \beta_{11} ) q^{19} + ( -1 - \beta_{2} ) q^{20} + ( -1 - \beta_{1} - 2 \beta_{2} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} + \beta_{9} - \beta_{10} - \beta_{11} ) q^{22} + ( 2 - \beta_{1} + \beta_{8} - \beta_{11} ) q^{23} + q^{25} + ( -\beta_{1} - \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} ) q^{26} + ( -4 - \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} ) q^{28} + ( -1 + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{7} + \beta_{10} + \beta_{11} ) q^{29} + ( 2 \beta_{2} + \beta_{3} - 2 \beta_{5} - 2 \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} ) q^{31} + ( 3 + 3 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} - 2 \beta_{9} + \beta_{10} + 2 \beta_{11} ) q^{32} + ( -2 - \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{8} + \beta_{9} + \beta_{10} ) q^{34} + ( 1 + \beta_{2} - \beta_{5} ) q^{35} + ( -1 + \beta_{2} + \beta_{3} - 3 \beta_{4} - \beta_{5} - \beta_{7} - \beta_{9} - \beta_{10} + \beta_{11} ) q^{37} + ( -3 \beta_{1} - \beta_{3} - 2 \beta_{6} - \beta_{7} - 2 \beta_{8} - \beta_{11} ) q^{38} + ( -1 - \beta_{1} - \beta_{2} - \beta_{6} - \beta_{7} ) q^{40} + ( 1 - 2 \beta_{1} + \beta_{3} + \beta_{5} + \beta_{8} - 2 \beta_{9} - \beta_{10} + \beta_{11} ) q^{41} + ( -4 - \beta_{1} + \beta_{2} - 2 \beta_{4} - \beta_{6} - 2 \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} ) q^{43} + ( -4 - 2 \beta_{1} - 5 \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{5} - 2 \beta_{6} + 3 \beta_{9} - 2 \beta_{11} ) q^{44} + ( -3 + 4 \beta_{1} - 3 \beta_{2} + \beta_{4} + \beta_{5} + \beta_{6} + 2 \beta_{7} + \beta_{8} + \beta_{9} - \beta_{11} ) q^{46} + ( 3 + \beta_{2} - 2 \beta_{5} + 2 \beta_{6} + \beta_{8} - \beta_{9} - \beta_{10} - \beta_{11} ) q^{47} + ( 1 - \beta_{1} + \beta_{2} + 2 \beta_{4} - \beta_{5} + \beta_{6} + 2 \beta_{7} + \beta_{8} - \beta_{10} - \beta_{11} ) q^{49} + \beta_{1} q^{50} + ( -2 + 2 \beta_{1} - 3 \beta_{2} + \beta_{4} - \beta_{5} + 2 \beta_{6} + 3 \beta_{7} + \beta_{8} + 2 \beta_{9} - \beta_{11} ) q^{52} + ( -2 \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} - 2 \beta_{8} + \beta_{11} ) q^{53} + ( \beta_{2} - \beta_{9} ) q^{55} + ( -2 - 4 \beta_{1} - 3 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} - 2 \beta_{6} - 3 \beta_{7} - 2 \beta_{8} + \beta_{9} - \beta_{11} ) q^{56} + ( 2 \beta_{2} + \beta_{3} - \beta_{4} - 3 \beta_{5} + \beta_{6} + \beta_{8} - 3 \beta_{9} - 2 \beta_{10} ) q^{58} + ( -3 \beta_{1} - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + \beta_{6} + 2 \beta_{8} + 2 \beta_{9} ) q^{59} + ( -5 + \beta_{1} - 3 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{6} + 2 \beta_{7} + \beta_{9} - 2 \beta_{10} - 2 \beta_{11} ) q^{61} + ( 2 - \beta_{1} + \beta_{2} + \beta_{5} + 2 \beta_{6} + 2 \beta_{9} + 3 \beta_{10} + 2 \beta_{11} ) q^{62} + ( 1 + 4 \beta_{1} + \beta_{2} + 3 \beta_{3} - \beta_{4} + 4 \beta_{6} + 4 \beta_{7} + 3 \beta_{8} - \beta_{9} + \beta_{11} ) q^{64} + ( 1 + \beta_{2} - \beta_{4} - \beta_{7} - \beta_{8} - \beta_{9} ) q^{65} + ( -3 - \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} ) q^{67} + ( -2 - 4 \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{5} - 2 \beta_{6} - 3 \beta_{7} - \beta_{8} - \beta_{10} - \beta_{11} ) q^{68} + ( 3 \beta_{1} + \beta_{3} - \beta_{4} + \beta_{6} + \beta_{7} ) q^{70} + ( -1 - 3 \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{6} + \beta_{7} + \beta_{9} - \beta_{11} ) q^{71} + ( -4 \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{4} - 2 \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} + 2 \beta_{11} ) q^{73} + ( -1 + 3 \beta_{1} - 2 \beta_{2} + \beta_{3} + 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + 5 \beta_{7} + \beta_{8} + 2 \beta_{9} - \beta_{10} - \beta_{11} ) q^{74} + ( -5 - \beta_{1} - 5 \beta_{2} - 2 \beta_{3} + 3 \beta_{4} + 2 \beta_{5} - 3 \beta_{6} - \beta_{8} + 2 \beta_{9} - \beta_{10} - \beta_{11} ) q^{76} + ( 2 + \beta_{1} + 4 \beta_{2} + \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + \beta_{7} - 2 \beta_{9} + \beta_{11} ) q^{77} + ( -1 - \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{5} + 2 \beta_{6} - \beta_{7} - \beta_{9} - \beta_{11} ) q^{79} + ( -2 \beta_{1} - 2 \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{5} - \beta_{6} + 2 \beta_{9} - \beta_{11} ) q^{80} + ( -3 + \beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} - \beta_{9} + 2 \beta_{10} + 3 \beta_{11} ) q^{82} + ( 2 + 3 \beta_{2} - \beta_{5} - \beta_{8} - \beta_{10} - \beta_{11} ) q^{83} + ( \beta_{1} + \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} ) q^{85} + ( -1 - 4 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 4 \beta_{4} + \beta_{7} - \beta_{8} + \beta_{9} - \beta_{11} ) q^{86} + ( -6 - 6 \beta_{1} - 7 \beta_{2} - 3 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - 4 \beta_{6} - 2 \beta_{7} - \beta_{8} + 2 \beta_{9} - \beta_{10} - 3 \beta_{11} ) q^{88} - q^{89} + ( -4 + \beta_{1} + \beta_{2} + 2 \beta_{3} - 5 \beta_{4} - 2 \beta_{6} - 3 \beta_{7} - 3 \beta_{9} + \beta_{11} ) q^{91} + ( 4 - 5 \beta_{1} + 3 \beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{5} - 2 \beta_{6} - 3 \beta_{7} - \beta_{8} - 2 \beta_{9} - \beta_{10} + 2 \beta_{11} ) q^{92} + ( -3 + 3 \beta_{1} + 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} + \beta_{8} + \beta_{9} + 3 \beta_{10} + 2 \beta_{11} ) q^{94} + ( 2 - \beta_{1} + \beta_{3} + \beta_{5} + \beta_{8} + \beta_{11} ) q^{95} + ( 1 + \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{5} + 3 \beta_{6} + 3 \beta_{7} + \beta_{9} + 3 \beta_{10} + \beta_{11} ) q^{97} + ( -4 + 4 \beta_{1} + 2 \beta_{2} + 3 \beta_{3} - 6 \beta_{4} + \beta_{5} + 2 \beta_{6} + \beta_{7} + \beta_{8} - 2 \beta_{9} + \beta_{10} + 2 \beta_{11} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + 3q^{2} + 11q^{4} - 12q^{5} - 8q^{7} + 9q^{8} + O(q^{10}) \) \( 12q + 3q^{2} + 11q^{4} - 12q^{5} - 8q^{7} + 9q^{8} - 3q^{10} - 12q^{13} - 4q^{14} + q^{16} - 24q^{19} - 11q^{20} - 16q^{22} + 24q^{23} + 12q^{25} + q^{26} - 44q^{28} - 8q^{29} - 12q^{31} + 31q^{32} - 18q^{34} + 8q^{35} - 10q^{37} - 2q^{38} - 9q^{40} + 10q^{41} - 42q^{43} - 42q^{44} - 24q^{46} + 22q^{47} - 4q^{49} + 3q^{50} - 30q^{52} - 8q^{53} - 27q^{56} - 12q^{58} - 4q^{59} - 52q^{61} + 14q^{62} + 7q^{64} + 12q^{65} - 40q^{67} - 23q^{68} + 4q^{70} - 2q^{71} - 8q^{73} - 26q^{74} - 46q^{76} + 12q^{77} - 26q^{79} - q^{80} - 26q^{82} + 14q^{83} - 32q^{86} - 60q^{88} - 12q^{89} - 24q^{91} + 38q^{92} - 26q^{94} + 24q^{95} - 6q^{97} - 30q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 3 x^{11} - 13 x^{10} + 41 x^{9} + 58 x^{8} - 202 x^{7} - 95 x^{6} + 432 x^{5} + 4 x^{4} - 368 x^{3} + 94 x^{2} + 77 x - 27\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\((\)\( -5 \nu^{11} + 8 \nu^{10} + 87 \nu^{9} - 121 \nu^{8} - 562 \nu^{7} + 666 \nu^{6} + 1618 \nu^{5} - 1612 \nu^{4} - 1915 \nu^{3} + 1616 \nu^{2} + 599 \nu - 432 \)\()/27\)
\(\beta_{4}\)\(=\)\((\)\( 2 \nu^{11} - 5 \nu^{10} - 33 \nu^{9} + 79 \nu^{8} + 196 \nu^{7} - 441 \nu^{6} - 496 \nu^{5} + 1021 \nu^{4} + 478 \nu^{3} - 866 \nu^{2} - 92 \nu + 153 \)\()/9\)
\(\beta_{5}\)\(=\)\((\)\( 31 \nu^{11} - 82 \nu^{10} - 426 \nu^{9} + 1085 \nu^{8} + 2156 \nu^{7} - 5112 \nu^{6} - 4934 \nu^{5} + 10286 \nu^{4} + 4799 \nu^{3} - 8059 \nu^{2} - 1273 \nu + 1512 \)\()/27\)
\(\beta_{6}\)\(=\)\((\)\( -35 \nu^{11} + 83 \nu^{10} + 501 \nu^{9} - 1117 \nu^{8} - 2638 \nu^{7} + 5391 \nu^{6} + 6196 \nu^{5} - 11203 \nu^{4} - 5953 \nu^{3} + 9125 \nu^{2} + 1331 \nu - 1728 \)\()/27\)
\(\beta_{7}\)\(=\)\((\)\( 35 \nu^{11} - 83 \nu^{10} - 501 \nu^{9} + 1117 \nu^{8} + 2638 \nu^{7} - 5391 \nu^{6} - 6196 \nu^{5} + 11203 \nu^{4} + 5980 \nu^{3} - 9152 \nu^{2} - 1466 \nu + 1782 \)\()/27\)
\(\beta_{8}\)\(=\)\((\)\( -43 \nu^{11} + 112 \nu^{10} + 597 \nu^{9} - 1505 \nu^{8} - 3035 \nu^{7} + 7245 \nu^{6} + 6884 \nu^{5} - 15008 \nu^{4} - 6506 \nu^{3} + 12229 \nu^{2} + 1690 \nu - 2403 \)\()/27\)
\(\beta_{9}\)\(=\)\((\)\( 67 \nu^{11} - 172 \nu^{10} - 939 \nu^{9} + 2318 \nu^{8} + 4847 \nu^{7} - 11214 \nu^{6} - 11270 \nu^{5} + 23399 \nu^{4} + 11054 \nu^{3} - 19192 \nu^{2} - 2902 \nu + 3726 \)\()/27\)
\(\beta_{10}\)\(=\)\((\)\( -68 \nu^{11} + 170 \nu^{10} + 969 \nu^{9} - 2308 \nu^{8} - 5089 \nu^{7} + 11241 \nu^{6} + 11995 \nu^{5} - 23554 \nu^{4} - 11806 \nu^{3} + 19346 \nu^{2} + 3056 \nu - 3789 \)\()/9\)
\(\beta_{11}\)\(=\)\((\)\( 217 \nu^{11} - 547 \nu^{10} - 3090 \nu^{9} + 7433 \nu^{8} + 16226 \nu^{7} - 36243 \nu^{6} - 38264 \nu^{5} + 76052 \nu^{4} + 37643 \nu^{3} - 62596 \nu^{2} - 9613 \nu + 12312 \)\()/27\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{7} + \beta_{6} + \beta_{2} + 5 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(\beta_{11} - 2 \beta_{9} + \beta_{6} - \beta_{5} - 2 \beta_{4} + \beta_{3} + 8 \beta_{2} + 2 \beta_{1} + 14\)
\(\nu^{5}\)\(=\)\(2 \beta_{11} + \beta_{10} - 2 \beta_{9} + \beta_{8} + 9 \beta_{7} + 9 \beta_{6} - \beta_{5} - 2 \beta_{4} + 2 \beta_{3} + 11 \beta_{2} + 28 \beta_{1} + 11\)
\(\nu^{6}\)\(=\)\(11 \beta_{11} - 21 \beta_{9} + 3 \beta_{8} + 4 \beta_{7} + 14 \beta_{6} - 10 \beta_{5} - 21 \beta_{4} + 13 \beta_{3} + 57 \beta_{2} + 24 \beta_{1} + 77\)
\(\nu^{7}\)\(=\)\(25 \beta_{11} + 10 \beta_{10} - 27 \beta_{9} + 16 \beta_{8} + 69 \beta_{7} + 73 \beta_{6} - 14 \beta_{5} - 29 \beta_{4} + 28 \beta_{3} + 97 \beta_{2} + 172 \beta_{1} + 94\)
\(\nu^{8}\)\(=\)\(96 \beta_{11} + 2 \beta_{10} - 172 \beta_{9} + 44 \beta_{8} + 62 \beta_{7} + 141 \beta_{6} - 83 \beta_{5} - 175 \beta_{4} + 124 \beta_{3} + 404 \beta_{2} + 222 \beta_{1} + 469\)
\(\nu^{9}\)\(=\)\(234 \beta_{11} + 76 \beta_{10} - 270 \beta_{9} + 168 \beta_{8} + 508 \beta_{7} + 572 \beta_{6} - 145 \beta_{5} - 300 \beta_{4} + 285 \beta_{3} + 795 \beta_{2} + 1130 \beta_{1} + 749\)
\(\nu^{10}\)\(=\)\(776 \beta_{11} + 34 \beta_{10} - 1311 \beta_{9} + 453 \beta_{8} + 666 \beta_{7} + 1248 \beta_{6} - 655 \beta_{5} - 1365 \beta_{4} + 1055 \beta_{3} + 2896 \beta_{2} + 1876 \beta_{1} + 3062\)
\(\nu^{11}\)\(=\)\(1970 \beta_{11} + 528 \beta_{10} - 2398 \beta_{9} + 1508 \beta_{8} + 3711 \beta_{7} + 4404 \beta_{6} - 1322 \beta_{5} - 2709 \beta_{4} + 2550 \beta_{3} + 6300 \beta_{2} + 7776 \beta_{1} + 5819\)

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
−2.29792
−1.85231
−1.76848
−1.26525
−0.551952
0.478750
0.491103
1.02694
1.62987
2.11454
2.24327
2.75145
−2.29792 0 3.28045 −1.00000 0 −2.76426 −2.94239 0 2.29792
1.2 −1.85231 0 1.43107 −1.00000 0 −0.0119316 1.05384 0 1.85231
1.3 −1.76848 0 1.12751 −1.00000 0 −4.90565 1.54298 0 1.76848
1.4 −1.26525 0 −0.399141 −1.00000 0 3.64120 3.03551 0 1.26525
1.5 −0.551952 0 −1.69535 −1.00000 0 1.42988 2.03966 0 0.551952
1.6 0.478750 0 −1.77080 −1.00000 0 −0.0239408 −1.80527 0 −0.478750
1.7 0.491103 0 −1.75882 −1.00000 0 −1.39011 −1.84597 0 −0.491103
1.8 1.02694 0 −0.945401 −1.00000 0 2.58018 −3.02474 0 −1.02694
1.9 1.62987 0 0.656467 −1.00000 0 −1.43414 −2.18978 0 −1.62987
1.10 2.11454 0 2.47129 −1.00000 0 −2.16289 0.996554 0 −2.11454
1.11 2.24327 0 3.03225 −1.00000 0 1.09855 2.31562 0 −2.24327
1.12 2.75145 0 5.57047 −1.00000 0 −4.05689 9.82397 0 −2.75145
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4005))\):

\(T_{2}^{12} - \cdots\)
\(T_{7}^{12} + \cdots\)
\(T_{11}^{12} - \cdots\)