Properties

Label 2005.2.a.d
Level 2005
Weight 2
Character orbit 2005.a
Self dual Yes
Analytic conductor 16.010
Analytic rank 1
Dimension 25
CM No

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Newspace parameters

Level: \( N \) = \( 2005 = 5 \cdot 401 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 2005.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(16.0100056053\)
Analytic rank: \(1\)
Dimension: \(25\)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 25q - 5q^{2} - 10q^{3} + 25q^{4} + 25q^{5} + 2q^{6} - 31q^{7} - 30q^{8} + 17q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 25q - 5q^{2} - 10q^{3} + 25q^{4} + 25q^{5} + 2q^{6} - 31q^{7} - 30q^{8} + 17q^{9} - 5q^{10} - 30q^{11} - 29q^{12} - 18q^{13} + 6q^{14} - 10q^{15} + 21q^{16} - 18q^{17} - 30q^{18} - 17q^{19} + 25q^{20} + 6q^{21} - 2q^{22} - 44q^{23} + 11q^{24} + 25q^{25} - 14q^{26} - 25q^{27} - 50q^{28} - 9q^{29} + 2q^{30} - 13q^{31} - 45q^{32} - 21q^{33} - 21q^{34} - 31q^{35} + 5q^{36} - 28q^{37} - 32q^{38} + 9q^{39} - 30q^{40} + 28q^{41} - 67q^{42} - 61q^{43} - 49q^{44} + 17q^{45} + 18q^{46} - 53q^{47} - 44q^{48} + 28q^{49} - 5q^{50} - 30q^{51} - 3q^{52} - 36q^{53} + 17q^{54} - 30q^{55} - 3q^{56} - 13q^{57} + 2q^{58} - 39q^{59} - 29q^{60} + 10q^{61} - 30q^{62} - 44q^{63} - 4q^{64} - 18q^{65} + 33q^{66} - 10q^{67} - 18q^{68} - 6q^{69} + 6q^{70} - 7q^{71} - q^{72} - 26q^{73} - 3q^{74} - 10q^{75} + 12q^{76} + 29q^{77} - 5q^{78} - 6q^{79} + 21q^{80} + 13q^{81} - 30q^{82} - 35q^{83} + 117q^{84} - 18q^{85} + 14q^{86} - 104q^{87} + 53q^{88} + 7q^{89} - 30q^{90} - 25q^{91} - 31q^{92} + 2q^{93} + 68q^{94} - 17q^{95} + 92q^{96} + 6q^{97} + 15q^{98} - 36q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −2.78880 −0.462551 5.77738 1.00000 1.28996 1.59492 −10.5343 −2.78605 −2.78880
1.2 −2.73364 −2.38520 5.47281 1.00000 6.52028 −4.95208 −9.49342 2.68916 −2.73364
1.3 −2.39330 −3.02965 3.72790 1.00000 7.25088 −0.331604 −4.13539 6.17880 −2.39330
1.4 −2.32427 0.0359659 3.40223 1.00000 −0.0835945 −0.619979 −3.25915 −2.99871 −2.32427
1.5 −2.32032 2.99682 3.38388 1.00000 −6.95357 −1.69858 −3.21105 5.98091 −2.32032
1.6 −2.11170 −1.19077 2.45926 1.00000 2.51454 −4.60904 −0.969827 −1.58207 −2.11170
1.7 −2.02550 1.24930 2.10265 1.00000 −2.53047 2.30907 −0.207923 −1.43924 −2.02550
1.8 −1.64150 1.09521 0.694512 1.00000 −1.79778 −0.515011 2.14295 −1.80052 −1.64150
1.9 −1.42545 −3.28864 0.0319048 1.00000 4.68779 −2.81826 2.80542 7.81515 −1.42545
1.10 −0.757050 −2.32990 −1.42688 1.00000 1.76385 −3.79091 2.59432 2.42842 −0.757050
1.11 −0.349213 −0.991069 −1.87805 1.00000 0.346094 −3.74674 1.35426 −2.01778 −0.349213
1.12 −0.280572 2.67538 −1.92128 1.00000 −0.750637 −1.90382 1.10020 4.15765 −0.280572
1.13 −0.181981 1.48815 −1.96688 1.00000 −0.270816 1.49043 0.721898 −0.785400 −0.181981
1.14 −0.0572643 1.42799 −1.99672 1.00000 −0.0817728 −3.04461 0.228869 −0.960848 −0.0572643
1.15 0.235509 −2.52549 −1.94454 1.00000 −0.594777 3.96079 −0.928975 3.37811 0.235509
1.16 0.641448 −2.54042 −1.58854 1.00000 −1.62955 1.24931 −2.30186 3.45375 0.641448
1.17 1.06984 1.18694 −0.855443 1.00000 1.26983 −0.905201 −3.05487 −1.59118 1.06984
1.18 1.18106 2.42838 −0.605102 1.00000 2.86806 −4.73589 −3.07678 2.89705 1.18106
1.19 1.31024 −1.68649 −0.283271 1.00000 −2.20970 0.0877260 −2.99163 −0.155757 1.31024
1.20 1.50483 0.906859 0.264506 1.00000 1.36467 −0.905011 −2.61162 −2.17761 1.50483
See all 25 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.25
Significant digits:
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Inner twists

This newform does not have CM; other inner twists have not been computed.

Atkin-Lehner signs

\( p \) Sign
\(5\) \(-1\)
\(401\) \(-1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2005))\):

\(T_{2}^{25} + \cdots\)
\(T_{11}^{25} + \cdots\)