Properties

Label 403.2.a.b
Level 403
Weight 2
Character orbit 403.a
Self dual Yes
Analytic conductor 3.218
Analytic rank 1
Dimension 6
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 403 = 13 \cdot 31 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 403.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(3.21797120146\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.5748973.1
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - x^{5} - 6 x^{4} + 4 x^{3} + 8 x^{2} - 4 x - 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 2 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - \nu^{2} - 3 \nu + 2 \)
\(\beta_{4}\)\(=\)\( \nu^{5} - \nu^{4} - 5 \nu^{3} + 3 \nu^{2} + 4 \nu - 2 \)
\(\beta_{5}\)\(=\)\( \nu^{5} - 6 \nu^{3} - 2 \nu^{2} + 6 \nu + 2 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 2\)
\(\nu^{3}\)\(=\)\(\beta_{3} + \beta_{2} + 3 \beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{5} - \beta_{4} + \beta_{3} + 6 \beta_{2} + \beta_{1} + 6\)
\(\nu^{5}\)\(=\)\(\beta_{5} + 6 \beta_{3} + 8 \beta_{2} + 12 \beta_{1} + 2\)

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
−0.187851
1.31089
−1.85232
0.699790
2.35805
−1.32857
−2.61388 −1.18785 4.83234 1.45573 3.10489 −1.87247 −7.40339 −1.58901 −3.80510
1.2 −1.94648 0.310892 1.78879 −3.27007 −0.605146 3.06335 0.411120 −2.90335 6.36512
1.3 −0.917109 −2.85232 −1.15891 −0.732299 2.61588 4.57774 2.89707 5.13571 0.671598
1.4 0.482827 −0.300210 −1.76688 0.848172 −0.144950 −1.74674 −1.81875 −2.90987 0.409520
1.5 0.543189 1.35805 −1.70495 −3.27954 0.737678 0.540055 −2.01249 −1.15570 −1.78141
1.6 2.45145 −2.32857 4.00960 −4.02200 −5.70836 −4.56194 4.92644 2.42222 −9.85973
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(13\) \(-1\)
\(31\) \(-1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \( T_{2}^{6} + 2 T_{2}^{5} - 7 T_{2}^{4} - 13 T_{2}^{3} + 6 T_{2}^{2} + 7 T_{2} - 3 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(403))\).