Properties

Label 1003.2.a.e
Level 1003
Weight 2
Character orbit 1003.a
Self dual Yes
Analytic conductor 8.009
Analytic rank 1
Dimension 3
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 1003 = 17 \cdot 59 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 1003.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(8.00899532273\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.229.1
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3}\mathstrut -\mathstrut \) \(4\) \(x\mathstrut -\mathstrut \) \(1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{2} + \nu - 3 \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\)\(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(6\)\()/2\)

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
−0.254102
2.11491
−1.86081
−1.00000 −1.68133 −1.00000 −4.18953 1.68133 1.68133 3.00000 −0.173127 4.18953
1.2 −1.00000 0.357926 −1.00000 2.58774 −0.357926 −0.357926 3.00000 −2.87189 −2.58774
1.3 −1.00000 3.32340 −1.00000 −2.39821 −3.32340 −3.32340 3.00000 8.04502 2.39821
\(n\): 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
\(17\) \(1\)
\(59\) \(1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1003))\):

\(T_{2} \) \(\mathstrut +\mathstrut 1 \)
\(T_{3}^{3} \) \(\mathstrut -\mathstrut 2 T_{3}^{2} \) \(\mathstrut -\mathstrut 5 T_{3} \) \(\mathstrut +\mathstrut 2 \)