Properties

Label 403.2.a.d
Level 403
Weight 2
Character orbit 403.a
Self dual Yes
Analytic conductor 3.218
Analytic rank 1
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: \(1\)
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.

\(f(q)\) \(=\) \( q + ( -1 + \beta_{1} ) q^{2} -\beta_{6} q^{3} + ( 2 - \beta_{1} + \beta_{2} ) q^{4} + ( -2 - \beta_{2} - \beta_{5} ) q^{5} + ( -1 - \beta_{1} - \beta_{3} + \beta_{5} + 2 \beta_{6} ) q^{6} + ( \beta_{2} + \beta_{3} + \beta_{4} ) q^{7} + ( -3 + \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{8} + ( 1 - \beta_{1} - \beta_{4} + \beta_{5} + \beta_{6} ) q^{9} +O(q^{10})\) \( q + ( -1 + \beta_{1} ) q^{2} -\beta_{6} q^{3} + ( 2 - \beta_{1} + \beta_{2} ) q^{4} + ( -2 - \beta_{2} - \beta_{5} ) q^{5} + ( -1 - \beta_{1} - \beta_{3} + \beta_{5} + 2 \beta_{6} ) q^{6} + ( \beta_{2} + \beta_{3} + \beta_{4} ) q^{7} + ( -3 + \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{8} + ( 1 - \beta_{1} - \beta_{4} + \beta_{5} + \beta_{6} ) q^{9} + ( 2 - 3 \beta_{1} - \beta_{3} - \beta_{4} + \beta_{6} - \beta_{7} ) q^{10} + ( -2 - \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{11} + ( \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{5} - 3 \beta_{6} + \beta_{7} ) q^{12} - q^{13} + ( \beta_{1} + 2 \beta_{3} + \beta_{5} ) q^{14} + ( -\beta_{1} + 2 \beta_{2} + \beta_{4} + \beta_{5} + 3 \beta_{6} - \beta_{7} ) q^{15} + ( 2 - 3 \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} ) q^{16} + ( -2 + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} ) q^{17} + ( -3 + 2 \beta_{1} + \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - 3 \beta_{6} + \beta_{7} ) q^{18} + ( -1 - \beta_{1} - 2 \beta_{2} - \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + \beta_{6} - \beta_{7} ) q^{19} + ( -5 + 3 \beta_{1} - 2 \beta_{2} - \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{20} + ( -2 - \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{5} + \beta_{6} ) q^{21} + ( 2 - 2 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - 3 \beta_{6} ) q^{22} + ( 2 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{7} ) q^{23} + ( 1 - 2 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{5} + 4 \beta_{6} - 3 \beta_{7} ) q^{24} + ( 2 + 3 \beta_{2} + 4 \beta_{5} + \beta_{6} ) q^{25} + ( 1 - \beta_{1} ) q^{26} + ( -1 + 3 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - \beta_{4} - 2 \beta_{5} - 2 \beta_{6} ) q^{27} + ( 3 + 2 \beta_{2} + \beta_{4} - \beta_{6} + \beta_{7} ) q^{28} + ( 1 + 2 \beta_{2} + 2 \beta_{3} - \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{29} + ( 1 + 5 \beta_{1} - 2 \beta_{2} + 5 \beta_{3} - 3 \beta_{5} - 7 \beta_{6} + 2 \beta_{7} ) q^{30} - q^{31} + ( -5 + 2 \beta_{1} - 2 \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{32} + ( -2 + 3 \beta_{1} + 2 \beta_{2} + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} ) q^{33} + ( 4 - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - 3 \beta_{6} + 2 \beta_{7} ) q^{34} + ( -\beta_{1} - 2 \beta_{2} - 3 \beta_{3} - 2 \beta_{4} + \beta_{5} - \beta_{6} ) q^{35} + ( 3 - 4 \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{4} + 4 \beta_{5} + 6 \beta_{6} - 3 \beta_{7} ) q^{36} + ( -3 + 3 \beta_{1} + 2 \beta_{3} + \beta_{4} + 2 \beta_{5} + 2 \beta_{7} ) q^{37} + ( -2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - \beta_{4} - 4 \beta_{5} - \beta_{7} ) q^{38} + \beta_{6} q^{39} + ( 7 - 3 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} + \beta_{4} + 3 \beta_{5} + 3 \beta_{6} ) q^{40} + ( -2 - \beta_{1} - 2 \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{6} + \beta_{7} ) q^{41} + ( -2 \beta_{1} - 3 \beta_{2} - \beta_{3} - 3 \beta_{4} - \beta_{5} - 2 \beta_{7} ) q^{42} + ( -\beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{6} - 3 \beta_{7} ) q^{43} + ( -7 - 3 \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{5} + 5 \beta_{6} - 3 \beta_{7} ) q^{44} + ( -6 + 5 \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} - 3 \beta_{5} - 4 \beta_{6} + 3 \beta_{7} ) q^{45} + ( -1 + 2 \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{5} - \beta_{6} ) q^{46} + ( -3 + 3 \beta_{3} + 3 \beta_{4} + 2 \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{47} + ( 4 \beta_{1} + 2 \beta_{3} + 2 \beta_{4} - 4 \beta_{5} - 4 \beta_{6} + 3 \beta_{7} ) q^{48} + ( 3 \beta_{1} + 2 \beta_{2} + \beta_{3} + 2 \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{49} + ( -1 + 6 \beta_{1} + \beta_{2} + 4 \beta_{3} + 4 \beta_{4} - \beta_{5} - 6 \beta_{6} + 4 \beta_{7} ) q^{50} + ( -4 + 3 \beta_{1} - 2 \beta_{2} + \beta_{4} - \beta_{5} + \beta_{6} + 3 \beta_{7} ) q^{51} + ( -2 + \beta_{1} - \beta_{2} ) q^{52} + ( -1 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{53} + ( 8 - 5 \beta_{1} + 5 \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{5} + 6 \beta_{6} - 2 \beta_{7} ) q^{54} + ( 3 + 2 \beta_{1} + \beta_{2} + 5 \beta_{3} + \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{55} + ( -5 + 2 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} - \beta_{4} + \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{56} + ( \beta_{1} + \beta_{2} + 4 \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} ) q^{57} + ( -4 + \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} + 3 \beta_{5} + 5 \beta_{6} - 2 \beta_{7} ) q^{58} + ( -\beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} - 3 \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{59} + ( 5 - 6 \beta_{1} + 5 \beta_{2} - 4 \beta_{3} + 7 \beta_{5} + 11 \beta_{6} - 3 \beta_{7} ) q^{60} + ( 1 - \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} + 2 \beta_{5} - 2 \beta_{7} ) q^{61} + ( 1 - \beta_{1} ) q^{62} + ( -4 + \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{4} + 2 \beta_{6} ) q^{63} + ( 7 - \beta_{1} - \beta_{3} - 2 \beta_{5} - \beta_{6} + \beta_{7} ) q^{64} + ( 2 + \beta_{2} + \beta_{5} ) q^{65} + ( 11 - \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{66} + ( 2 - \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{67} + ( -5 - \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{5} + 5 \beta_{6} - \beta_{7} ) q^{68} + ( -3 + 2 \beta_{1} + \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - \beta_{5} + 2 \beta_{7} ) q^{69} + ( -4 - 3 \beta_{1} - \beta_{2} - 6 \beta_{3} - \beta_{5} + \beta_{6} + \beta_{7} ) q^{70} + ( -1 - \beta_{2} - 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{71} + ( 6 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} - \beta_{4} - 6 \beta_{5} - 10 \beta_{6} + 5 \beta_{7} ) q^{72} + ( 3 - 2 \beta_{1} - 3 \beta_{2} - \beta_{5} - \beta_{6} ) q^{73} + ( 10 - 3 \beta_{1} + 7 \beta_{2} + 2 \beta_{3} + 3 \beta_{4} + 3 \beta_{5} - 2 \beta_{6} ) q^{74} + ( -3 + 5 \beta_{1} - 7 \beta_{2} - 3 \beta_{4} - 4 \beta_{5} - 7 \beta_{6} + 3 \beta_{7} ) q^{75} + ( -3 - 2 \beta_{2} - 2 \beta_{3} - \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{76} + ( -6 - 3 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} + 2 \beta_{6} - 3 \beta_{7} ) q^{77} + ( 1 + \beta_{1} + \beta_{3} - \beta_{5} - 2 \beta_{6} ) q^{78} + ( 4 - 2 \beta_{2} - 2 \beta_{3} - 4 \beta_{5} + \beta_{7} ) q^{79} + ( -3 + 8 \beta_{1} - 2 \beta_{2} + 5 \beta_{3} - 5 \beta_{6} + \beta_{7} ) q^{80} + ( 3 - 6 \beta_{1} + \beta_{2} - 2 \beta_{3} + 2 \beta_{5} + 4 \beta_{6} - 4 \beta_{7} ) q^{81} + ( -3 - 3 \beta_{1} - 5 \beta_{2} - 3 \beta_{3} - 3 \beta_{4} + \beta_{5} + 4 \beta_{6} - 3 \beta_{7} ) q^{82} + ( -1 + \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{83} + ( -\beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} ) q^{84} + ( 3 - \beta_{2} + 3 \beta_{3} - \beta_{4} - 2 \beta_{5} - 4 \beta_{6} - \beta_{7} ) q^{85} + ( 5 + \beta_{1} - \beta_{2} - \beta_{3} - 7 \beta_{5} - 4 \beta_{6} + 3 \beta_{7} ) q^{86} + ( 4 - 5 \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{87} + ( 11 - \beta_{1} + 2 \beta_{2} + 3 \beta_{3} - 7 \beta_{5} - 6 \beta_{6} + 5 \beta_{7} ) q^{88} + ( -3 - 3 \beta_{1} + \beta_{2} - 4 \beta_{3} + \beta_{4} + 2 \beta_{6} - 2 \beta_{7} ) q^{89} + ( 14 - 11 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} - 4 \beta_{4} + 9 \beta_{5} + 11 \beta_{6} - 6 \beta_{7} ) q^{90} + ( -\beta_{2} - \beta_{3} - \beta_{4} ) q^{91} + ( 6 - 4 \beta_{1} + 3 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} ) q^{92} + \beta_{6} q^{93} + ( 6 - 2 \beta_{1} + 5 \beta_{2} + 4 \beta_{3} + 2 \beta_{4} - 4 \beta_{6} + 4 \beta_{7} ) q^{94} + ( 4 + 6 \beta_{1} + 5 \beta_{2} + \beta_{3} + 3 \beta_{4} + 4 \beta_{5} - 2 \beta_{6} + 3 \beta_{7} ) q^{95} + ( 3 - 2 \beta_{4} + 5 \beta_{5} + 4 \beta_{6} - \beta_{7} ) q^{96} + ( -3 - 2 \beta_{2} - 2 \beta_{3} - \beta_{6} - 2 \beta_{7} ) q^{97} + ( 6 + 4 \beta_{2} + \beta_{3} + 3 \beta_{4} + 3 \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{98} + ( 6 - 3 \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} + 3 \beta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 5q^{2} - 3q^{3} + 9q^{4} - 15q^{5} - 4q^{7} - 15q^{8} + 9q^{9} + O(q^{10}) \) \( 8q - 5q^{2} - 3q^{3} + 9q^{4} - 15q^{5} - 4q^{7} - 15q^{8} + 9q^{9} + 9q^{10} - 5q^{11} - 9q^{12} - 8q^{13} + 2q^{14} + 2q^{15} + 3q^{16} - 11q^{17} - 30q^{18} - 9q^{19} - 31q^{20} - 16q^{21} - 2q^{22} + 13q^{24} + 19q^{25} + 5q^{26} - 9q^{27} + 16q^{28} - 12q^{29} - 7q^{30} - 8q^{31} - 25q^{32} - 14q^{33} + 22q^{34} + 7q^{35} + 37q^{36} - 9q^{37} - 9q^{38} + 3q^{39} + 55q^{40} - 25q^{41} - 3q^{42} + 7q^{43} - 26q^{44} - 45q^{45} + 5q^{46} - 17q^{47} - 9q^{48} - 11q^{50} - 10q^{51} - 9q^{52} - 15q^{53} + 54q^{54} + 7q^{55} - 14q^{56} - 7q^{57} - 5q^{58} - 15q^{59} + 61q^{60} + 11q^{61} + 5q^{62} - 21q^{63} + 47q^{64} + 15q^{65} + 83q^{66} + 18q^{67} - 16q^{68} - 15q^{69} - 24q^{70} - 7q^{71} - 21q^{72} + 24q^{73} + 48q^{74} - 17q^{75} - 3q^{76} - 49q^{77} + 33q^{79} - 16q^{80} + 20q^{81} - q^{82} - 13q^{83} - 6q^{84} + q^{85} + 19q^{86} + 18q^{87} + 37q^{88} - 23q^{89} + 117q^{90} + 4q^{91} + 22q^{92} + 3q^{93} + 10q^{94} + 43q^{95} + 46q^{96} - 17q^{97} + 52q^{98} + 51q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 3 x^{7} - 7 x^{6} + 19 x^{5} + 21 x^{4} - 31 x^{3} - 29 x^{2} - x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - \nu^{2} - 4 \nu \)
\(\beta_{4}\)\(=\)\( \nu^{4} - 3 \nu^{3} - 3 \nu^{2} + 9 \nu + 3 \)
\(\beta_{5}\)\(=\)\( \nu^{5} - 3 \nu^{4} - 3 \nu^{3} + 9 \nu^{2} + 3 \nu \)
\(\beta_{6}\)\(=\)\( \nu^{7} - 4 \nu^{6} - 3 \nu^{5} + 21 \nu^{4} + 3 \nu^{3} - 30 \nu^{2} - 9 \nu + 2 \)
\(\beta_{7}\)\(=\)\( \nu^{7} - 3 \nu^{6} - 7 \nu^{5} + 20 \nu^{4} + 18 \nu^{3} - 34 \nu^{2} - 20 \nu + 2 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{3} + \beta_{2} + 5 \beta_{1} + 3\)
\(\nu^{4}\)\(=\)\(\beta_{4} + 3 \beta_{3} + 6 \beta_{2} + 9 \beta_{1} + 15\)
\(\nu^{5}\)\(=\)\(\beta_{5} + 3 \beta_{4} + 12 \beta_{3} + 12 \beta_{2} + 30 \beta_{1} + 27\)
\(\nu^{6}\)\(=\)\(\beta_{7} - \beta_{6} + 4 \beta_{5} + 13 \beta_{4} + 36 \beta_{3} + 43 \beta_{2} + 69 \beta_{1} + 90\)
\(\nu^{7}\)\(=\)\(4 \beta_{7} - 3 \beta_{6} + 19 \beta_{5} + 40 \beta_{4} + 114 \beta_{3} + 109 \beta_{2} + 201 \beta_{1} + 205\)

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
−1.66697
−1.57233
−0.759212
−0.247616
0.158875
1.93310
2.39479
2.75938
−2.66697 1.08287 5.11275 −1.31345 −2.88800 1.98184 −8.30164 −1.82738 3.50294
1.2 −2.57233 −3.30456 4.61688 −4.29347 8.50042 0.180190 −6.73148 7.92012 11.0442
1.3 −1.75921 2.94965 1.09483 −3.30948 −5.18907 −3.55859 1.59239 5.70046 5.82207
1.4 −1.24762 −2.42435 −0.443454 0.848757 3.02465 −1.14028 3.04849 2.87746 −1.05892
1.5 −0.841125 0.162061 −1.29251 0.443680 −0.136313 0.552392 2.76941 −2.97374 −0.373191
1.6 0.933096 1.42306 −1.12933 −3.66529 1.32785 −3.96149 −2.91997 −0.974910 −3.42006
1.7 1.39479 −1.21460 −0.0545724 −0.0316546 −1.69410 −2.20383 −2.86569 −1.52475 −0.0441514
1.8 1.75938 −1.67414 1.09540 −3.67909 −2.94544 4.14977 −1.59152 −0.197253 −6.47291
\(n\): 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 \( T_{2}^{8} + 5 T_{2}^{7} - 30 T_{2}^{5} - 24 T_{2}^{4} + 54 T_{2}^{3} + 54 T_{2}^{2} - 28 T_{2} - 29 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(403))\).