Properties

Label 1003.2.a.f
Level 1003
Weight 2
Character orbit 1003.a
Self dual Yes
Analytic conductor 8.009
Analytic rank 1
Dimension 4
CM No
Inner twists 1

Related objects

Downloads

Learn more about

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: \(4\)
Coefficient field: 4.4.2225.1
Coefficient ring: \(\Z[a_1, a_2]\)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4}\mathstrut -\mathstrut \) \(x^{3}\mathstrut -\mathstrut \) \(5\) \(x^{2}\mathstrut +\mathstrut \) \(2\) \(x\mathstrut +\mathstrut \) \(4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - \nu^{2} - 3 \nu + 1 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(3\)
\(\nu^{3}\)\(=\)\(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(4\) \(\beta_{1}\mathstrut +\mathstrut \) \(2\)

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
−1.75660
−0.820249
1.13856
2.43828
−2.75660 1.84224 5.59883 −0.0856374 −5.07830 −0.236068 −9.92054 0.393832 0.236068
1.2 −1.82025 −1.50694 1.31331 2.32719 2.74301 4.23607 1.24995 −0.729126 −4.23607
1.3 0.138564 −2.84224 −1.98080 1.70367 −0.393832 −0.236068 −0.551597 5.07830 0.236068
1.4 1.43828 0.506942 0.0686587 −2.94523 0.729126 4.23607 −2.77782 −2.74301 −4.23607
\(n\): 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1003))\):

\(T_{2}^{4} \) \(\mathstrut +\mathstrut 3 T_{2}^{3} \) \(\mathstrut -\mathstrut 2 T_{2}^{2} \) \(\mathstrut -\mathstrut 7 T_{2} \) \(\mathstrut +\mathstrut 1 \)
\(T_{3}^{4} \) \(\mathstrut +\mathstrut 2 T_{3}^{3} \) \(\mathstrut -\mathstrut 5 T_{3}^{2} \) \(\mathstrut -\mathstrut 6 T_{3} \) \(\mathstrut +\mathstrut 4 \)