Properties

Label 403.2.a.e
Level 403
Weight 2
Character orbit 403.a
Self dual Yes
Analytic conductor 3.218
Analytic rank 0
Dimension 8
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: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - x^{7} - 11 x^{6} + 10 x^{5} + 37 x^{4} - 33 x^{3} - 36 x^{2} + 33 x - 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - \nu^{2} - 4 \nu + 3 \)
\(\beta_{4}\)\(=\)\( \nu^{5} - \nu^{4} - 7 \nu^{3} + 5 \nu^{2} + 10 \nu - 5 \)
\(\beta_{5}\)\(=\)\( -\nu^{6} + \nu^{5} + 8 \nu^{4} - 6 \nu^{3} - 15 \nu^{2} + 9 \nu + 2 \)
\(\beta_{6}\)\(=\)\( \nu^{6} - 2 \nu^{5} - 6 \nu^{4} + 12 \nu^{3} + 5 \nu^{2} - 15 \nu + 5 \)
\(\beta_{7}\)\(=\)\( \nu^{7} - 10 \nu^{5} - \nu^{4} + 28 \nu^{3} + \nu^{2} - 21 \nu + 4 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{3} + \beta_{2} + 4 \beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} + 6 \beta_{2} + 13\)
\(\nu^{5}\)\(=\)\(\beta_{6} + \beta_{5} + 2 \beta_{4} + 8 \beta_{3} + 8 \beta_{2} + 18 \beta_{1} + 3\)
\(\nu^{6}\)\(=\)\(9 \beta_{6} + 8 \beta_{5} + 10 \beta_{4} + 10 \beta_{3} + 35 \beta_{2} + 3 \beta_{1} + 64\)
\(\nu^{7}\)\(=\)\(\beta_{7} + 11 \beta_{6} + 11 \beta_{5} + 21 \beta_{4} + 53 \beta_{3} + 57 \beta_{2} + 89 \beta_{1} + 36\)

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
2.53815
1.71590
1.53950
0.656875
0.147670
−1.38904
−2.06827
−2.14079
−2.53815 2.96968 4.44218 2.45818 −7.53747 −1.29216 −6.19862 5.81898 −6.23922
1.2 −1.71590 −1.27835 0.944315 −2.00831 2.19352 −3.76625 1.81145 −1.36583 3.44606
1.3 −1.53950 0.734778 0.370072 3.72666 −1.13119 3.41650 2.50928 −2.46010 −5.73720
1.4 −0.656875 2.67827 −1.56852 −1.55873 −1.75929 2.17607 2.34407 4.17313 1.02389
1.5 −0.147670 −2.43721 −1.97819 4.40045 0.359904 −2.90659 0.587461 2.94000 −0.649816
1.6 1.38904 1.62343 −0.0705747 3.01447 2.25501 2.19250 −2.87611 −0.364464 4.18722
1.7 2.06827 2.49221 2.27774 1.40222 5.15456 −5.08294 0.574435 3.21112 2.90017
1.8 2.14079 0.217190 2.58298 −0.434938 0.464957 3.26287 1.24803 −2.95283 −0.931109
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
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 \(T_{2}^{8} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(403))\).