Properties

Label 2005.2.a.c
Level 2005
Weight 2
Character orbit 2005.a
Self dual Yes
Analytic conductor 16.010
Analytic rank 1
Dimension 3
CM No
Inner twists 1

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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: \(3\)
Coefficient field: \(\Q(\zeta_{18})^+\)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + \beta_{1} ) q^{2} + ( -1 - 2 \beta_{1} + \beta_{2} ) q^{3} + ( 1 - 2 \beta_{1} + \beta_{2} ) q^{4} + q^{5} + ( -2 + 2 \beta_{1} - 3 \beta_{2} ) q^{6} + ( -1 + \beta_{1} - 3 \beta_{2} ) q^{7} + ( -2 + 2 \beta_{1} - 3 \beta_{2} ) q^{8} + ( 4 + \beta_{1} + \beta_{2} ) q^{9} +O(q^{10})\) \( q + ( -1 + \beta_{1} ) q^{2} + ( -1 - 2 \beta_{1} + \beta_{2} ) q^{3} + ( 1 - 2 \beta_{1} + \beta_{2} ) q^{4} + q^{5} + ( -2 + 2 \beta_{1} - 3 \beta_{2} ) q^{6} + ( -1 + \beta_{1} - 3 \beta_{2} ) q^{7} + ( -2 + 2 \beta_{1} - 3 \beta_{2} ) q^{8} + ( 4 + \beta_{1} + \beta_{2} ) q^{9} + ( -1 + \beta_{1} ) q^{10} + ( -\beta_{1} + 2 \beta_{2} ) q^{11} + ( 5 - 3 \beta_{1} + 3 \beta_{2} ) q^{12} + ( -1 + 3 \beta_{1} - 2 \beta_{2} ) q^{13} + ( -5 \beta_{1} + 4 \beta_{2} ) q^{14} + ( -1 - 2 \beta_{1} + \beta_{2} ) q^{15} + ( 1 - 3 \beta_{1} + 3 \beta_{2} ) q^{16} + ( -2 + 2 \beta_{1} + \beta_{2} ) q^{17} + ( -1 + 4 \beta_{1} ) q^{18} + 3 q^{19} + ( 1 - 2 \beta_{1} + \beta_{2} ) q^{20} + ( -2 + 5 \beta_{1} + 3 \beta_{2} ) q^{21} + ( 3 \beta_{1} - 3 \beta_{2} ) q^{22} + ( -3 - \beta_{1} + 2 \beta_{2} ) q^{23} + ( -4 + 7 \beta_{1} ) q^{24} + q^{25} + ( 5 - 6 \beta_{1} + 5 \beta_{2} ) q^{26} + ( -4 - 3 \beta_{1} - 3 \beta_{2} ) q^{27} + ( -4 + 7 \beta_{1} - 3 \beta_{2} ) q^{28} + ( -4 \beta_{1} + \beta_{2} ) q^{29} + ( -2 + 2 \beta_{1} - 3 \beta_{2} ) q^{30} + ( 2 - \beta_{1} + 3 \beta_{2} ) q^{31} + 3 \beta_{1} q^{32} + ( 3 - 2 \beta_{1} - 2 \beta_{2} ) q^{33} + ( 7 - 3 \beta_{1} + \beta_{2} ) q^{34} + ( -1 + \beta_{1} - 3 \beta_{2} ) q^{35} + ( 1 - 7 \beta_{1} + 2 \beta_{2} ) q^{36} + ( 5 + 3 \beta_{2} ) q^{37} + ( -3 + 3 \beta_{1} ) q^{38} + ( -8 + 4 \beta_{1} - 3 \beta_{2} ) q^{39} + ( -2 + 2 \beta_{1} - 3 \beta_{2} ) q^{40} + ( -6 + 3 \beta_{1} - 2 \beta_{2} ) q^{41} + ( 15 - 4 \beta_{1} + 2 \beta_{2} ) q^{42} + ( -2 + 3 \beta_{1} - 3 \beta_{2} ) q^{43} + ( 3 - 4 \beta_{1} + 2 \beta_{2} ) q^{44} + ( 4 + \beta_{1} + \beta_{2} ) q^{45} + ( 3 - 3 \beta_{2} ) q^{46} + ( -5 \beta_{1} + 2 \beta_{2} ) q^{47} + ( 8 - 5 \beta_{1} + \beta_{2} ) q^{48} + ( 8 + \beta_{1} - 2 \beta_{2} ) q^{49} + ( -1 + \beta_{1} ) q^{50} + ( -4 + 3 \beta_{1} - 8 \beta_{2} ) q^{51} + ( -10 + 10 \beta_{1} - 7 \beta_{2} ) q^{52} + ( -5 - \beta_{2} ) q^{53} + ( -5 - 4 \beta_{1} ) q^{54} + ( -\beta_{1} + 2 \beta_{2} ) q^{55} + ( 15 - 4 \beta_{1} + 2 \beta_{2} ) q^{56} + ( -3 - 6 \beta_{1} + 3 \beta_{2} ) q^{57} + ( -7 + 5 \beta_{1} - 5 \beta_{2} ) q^{58} + ( -2 + 6 \beta_{1} - \beta_{2} ) q^{59} + ( 5 - 3 \beta_{1} + 3 \beta_{2} ) q^{60} + ( -1 - \beta_{1} - 4 \beta_{2} ) q^{61} + ( -1 + 6 \beta_{1} - 4 \beta_{2} ) q^{62} + ( -10 - 2 \beta_{1} - 9 \beta_{2} ) q^{63} + ( 4 + 3 \beta_{1} - 3 \beta_{2} ) q^{64} + ( -1 + 3 \beta_{1} - 2 \beta_{2} ) q^{65} + ( -9 + 3 \beta_{1} ) q^{66} + ( -9 - 4 \beta_{1} + 2 \beta_{2} ) q^{67} + ( -8 + 7 \beta_{1} - 6 \beta_{2} ) q^{68} + ( 6 + 4 \beta_{1} - 5 \beta_{2} ) q^{69} + ( -5 \beta_{1} + 4 \beta_{2} ) q^{70} + ( 5 \beta_{1} - 7 \beta_{2} ) q^{71} + ( -11 + 2 \beta_{1} - 9 \beta_{2} ) q^{72} + ( 1 + \beta_{1} - 4 \beta_{2} ) q^{73} + ( -2 + 8 \beta_{1} - 3 \beta_{2} ) q^{74} + ( -1 - 2 \beta_{1} + \beta_{2} ) q^{75} + ( 3 - 6 \beta_{1} + 3 \beta_{2} ) q^{76} + ( -9 + 3 \beta_{2} ) q^{77} + ( 13 - 15 \beta_{1} + 7 \beta_{2} ) q^{78} + ( 1 + 10 \beta_{1} - 4 \beta_{2} ) q^{79} + ( 1 - 3 \beta_{1} + 3 \beta_{2} ) q^{80} + ( 1 + 8 \beta_{1} + 5 \beta_{2} ) q^{81} + ( 10 - 11 \beta_{1} + 5 \beta_{2} ) q^{82} + ( -13 + \beta_{1} - 2 \beta_{2} ) q^{83} + ( -17 + 11 \beta_{1} - 12 \beta_{2} ) q^{84} + ( -2 + 2 \beta_{1} + \beta_{2} ) q^{85} + ( 5 - 8 \beta_{1} + 6 \beta_{2} ) q^{86} + ( 12 - \beta_{1} + 6 \beta_{2} ) q^{87} + ( -9 + 3 \beta_{1} ) q^{88} + ( -1 + 5 \beta_{1} - 7 \beta_{2} ) q^{89} + ( -1 + 4 \beta_{1} ) q^{90} + ( 8 - 9 \beta_{1} + 2 \beta_{2} ) q^{91} + ( 2 \beta_{1} - \beta_{2} ) q^{92} + ( 1 - 7 \beta_{1} - 2 \beta_{2} ) q^{93} + ( -8 + 7 \beta_{1} - 7 \beta_{2} ) q^{94} + 3 q^{95} + ( -9 - 6 \beta_{2} ) q^{96} + ( -4 - \beta_{1} + 7 \beta_{2} ) q^{97} + ( -8 + 5 \beta_{1} + 3 \beta_{2} ) q^{98} + ( 3 - \beta_{1} + 5 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 3q^{2} - 3q^{3} + 3q^{4} + 3q^{5} - 6q^{6} - 3q^{7} - 6q^{8} + 12q^{9} + O(q^{10}) \) \( 3q - 3q^{2} - 3q^{3} + 3q^{4} + 3q^{5} - 6q^{6} - 3q^{7} - 6q^{8} + 12q^{9} - 3q^{10} + 15q^{12} - 3q^{13} - 3q^{15} + 3q^{16} - 6q^{17} - 3q^{18} + 9q^{19} + 3q^{20} - 6q^{21} - 9q^{23} - 12q^{24} + 3q^{25} + 15q^{26} - 12q^{27} - 12q^{28} - 6q^{30} + 6q^{31} + 9q^{33} + 21q^{34} - 3q^{35} + 3q^{36} + 15q^{37} - 9q^{38} - 24q^{39} - 6q^{40} - 18q^{41} + 45q^{42} - 6q^{43} + 9q^{44} + 12q^{45} + 9q^{46} + 24q^{48} + 24q^{49} - 3q^{50} - 12q^{51} - 30q^{52} - 15q^{53} - 15q^{54} + 45q^{56} - 9q^{57} - 21q^{58} - 6q^{59} + 15q^{60} - 3q^{61} - 3q^{62} - 30q^{63} + 12q^{64} - 3q^{65} - 27q^{66} - 27q^{67} - 24q^{68} + 18q^{69} - 33q^{72} + 3q^{73} - 6q^{74} - 3q^{75} + 9q^{76} - 27q^{77} + 39q^{78} + 3q^{79} + 3q^{80} + 3q^{81} + 30q^{82} - 39q^{83} - 51q^{84} - 6q^{85} + 15q^{86} + 36q^{87} - 27q^{88} - 3q^{89} - 3q^{90} + 24q^{91} + 3q^{93} - 24q^{94} + 9q^{95} - 27q^{96} - 12q^{97} - 24q^{98} + 9q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of \(\nu = \zeta_{18} + \zeta_{18}^{-1}\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 2 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 2\)

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
−1.53209
−0.347296
1.87939
−2.53209 2.41147 4.41147 1.00000 −6.10607 −3.57398 −6.10607 2.81521 −2.53209
1.2 −1.34730 −2.18479 −0.184793 1.00000 2.94356 4.29086 2.94356 1.77332 −1.34730
1.3 0.879385 −3.22668 −1.22668 1.00000 −2.83750 −3.71688 −2.83750 7.41147 0.879385
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2005))\):

\( T_{2}^{3} + 3 T_{2}^{2} - 3 \)
\( T_{11}^{3} - 9 T_{11} + 9 \)