Properties

Label 403.2.l.a
Level 403
Weight 2
Character orbit 403.l
Analytic conductor 3.218
Analytic rank 0
Dimension 2
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 403 = 13 \cdot 31 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 403.l (of order \(6\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(3.21797120146\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

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

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/403\mathbb{Z}\right)^\times\).

\(n\) \(249\) \(313\)
\(\chi(n)\) \(-1\) \(-1 + \zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
25.1
0.500000 + 0.866025i
0.500000 0.866025i
1.73205i 0.500000 + 0.866025i −1.00000 −1.50000 0.866025i 1.50000 0.866025i −1.50000 + 0.866025i 1.73205i 1.00000 1.73205i −1.50000 + 2.59808i
129.1 1.73205i 0.500000 0.866025i −1.00000 −1.50000 + 0.866025i 1.50000 + 0.866025i −1.50000 0.866025i 1.73205i 1.00000 + 1.73205i −1.50000 2.59808i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
403.l Even 1 yes

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(403, [\chi])\):

\( T_{2}^{2} + 3 \)
\( T_{5}^{2} + 3 T_{5} + 3 \)