Properties

Label 403.2.be.a
Level 403
Weight 2
Character orbit 403.be
Analytic conductor 3.218
Analytic rank 0
Dimension 4
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 403 = 13 \cdot 31 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 403.be (of order \(12\) and degree \(4\))

Newform invariants

Self dual: No
Analytic conductor: \(3.21797120146\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

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

Character Values

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

\(n\) \(249\) \(313\)
\(\chi(n)\) \(\zeta_{12}^{3}\) \(\zeta_{12}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
57.1
−0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
0.866025 + 0.500000i
−1.36603 + 1.36603i 0.633975 + 0.366025i 1.73205i −1.36603 0.366025i −1.36603 + 0.366025i −4.23205 + 1.13397i −0.366025 0.366025i −1.23205 2.13397i 2.36603 1.36603i
99.1 −1.36603 1.36603i 0.633975 0.366025i 1.73205i −1.36603 + 0.366025i −1.36603 0.366025i −4.23205 1.13397i −0.366025 + 0.366025i −1.23205 + 2.13397i 2.36603 + 1.36603i
161.1 0.366025 0.366025i 2.36603 1.36603i 1.73205i 0.366025 + 1.36603i 0.366025 1.36603i −0.767949 + 2.86603i 1.36603 + 1.36603i 2.23205 3.86603i 0.633975 + 0.366025i
398.1 0.366025 + 0.366025i 2.36603 + 1.36603i 1.73205i 0.366025 1.36603i 0.366025 + 1.36603i −0.767949 2.86603i 1.36603 1.36603i 2.23205 + 3.86603i 0.633975 0.366025i
\(n\): 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.be Even 1 yes

Hecke kernels

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

\( T_{2}^{4} + 2 T_{2}^{3} + 2 T_{2}^{2} - 2 T_{2} + 1 \)
\( T_{7}^{4} + 10 T_{7}^{3} + 41 T_{7}^{2} + 104 T_{7} + 169 \)