Properties

Label 403.2.a.c
Level 403
Weight 2
Character orbit 403.a
Self dual Yes
Analytic conductor 3.218
Analytic rank 0
Dimension 7
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: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
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 \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{7} - 2 x^{6} - 9 x^{5} + 12 x^{4} + 22 x^{3} - 18 x^{2} - 13 x - 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{5} - \nu^{4} - 9 \nu^{3} + 2 \nu^{2} + 15 \nu + 1 \)
\(\beta_{3}\)\(=\)\( \nu^{5} - \nu^{4} - 9 \nu^{3} + \nu^{2} + 16 \nu + 4 \)
\(\beta_{4}\)\(=\)\( -\nu^{6} + \nu^{5} + 9 \nu^{4} - \nu^{3} - 17 \nu^{2} - 4 \nu + 3 \)
\(\beta_{5}\)\(=\)\( -\nu^{6} + \nu^{5} + 10 \nu^{4} - 4 \nu^{3} - 21 \nu^{2} + 7 \nu + 3 \)
\(\beta_{6}\)\(=\)\( 2 \nu^{6} - \nu^{5} - 19 \nu^{4} - 6 \nu^{3} + 33 \nu^{2} + 19 \nu + 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{3} + \beta_{2} + \beta_{1} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{6} + 2 \beta_{4} - 3 \beta_{3} + 2 \beta_{2} + 7 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(3 \beta_{6} + \beta_{5} + 5 \beta_{4} - 13 \beta_{3} + 10 \beta_{2} + 14 \beta_{1} + 15\)
\(\nu^{5}\)\(=\)\(12 \beta_{6} + \beta_{5} + 23 \beta_{4} - 38 \beta_{3} + 27 \beta_{2} + 60 \beta_{1} + 17\)
\(\nu^{6}\)\(=\)\(38 \beta_{6} + 10 \beta_{5} + 65 \beta_{4} - 135 \beta_{3} + 98 \beta_{2} + 158 \beta_{1} + 103\)

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
3.17535
−0.434125
−2.01774
1.56423
−1.70914
1.51046
−0.0890306
−2.10645 −2.17535 2.43713 2.39413 4.58226 1.34409 −0.920801 1.73214 −5.04311
1.2 −2.07211 1.43412 2.29366 0.0880198 −2.97167 0.184197 −0.608494 −0.943287 −0.182387
1.3 −0.299542 3.01774 −1.91027 3.67569 −0.903940 −0.783156 1.17129 6.10678 −1.10102
1.4 0.406064 −0.564229 −1.83511 2.08199 −0.229113 4.31735 −1.55730 −2.68165 0.845422
1.5 1.39140 2.70914 −0.0639949 −0.456939 3.76951 1.81179 −2.87185 4.33945 −0.635787
1.6 2.09092 −0.510463 2.37194 4.43975 −1.06734 −2.30124 0.777706 −2.73943 9.28317
1.7 2.58972 1.08903 4.70665 −1.22264 2.82028 −0.573038 7.00945 −1.81401 −3.16629
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
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 \( T_{2}^{7} - 2 T_{2}^{6} - 9 T_{2}^{5} + 17 T_{2}^{4} + 20 T_{2}^{3} - 37 T_{2}^{2} + T_{2} + 4 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(403))\).