Properties

Label 403.2.c.a
Level 403
Weight 2
Character orbit 403.c
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.c (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(3.21797120146\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 q^{3} + 2 q^{4} + 4 i q^{5} -2 i q^{7} + q^{9} +O(q^{10})\) \( q + 2 q^{3} + 2 q^{4} + 4 i q^{5} -2 i q^{7} + q^{9} + i q^{11} + 4 q^{12} + ( -2 - 3 i ) q^{13} + 8 i q^{15} + 4 q^{16} -2 q^{17} -2 i q^{19} + 8 i q^{20} -4 i q^{21} -4 q^{23} -11 q^{25} -4 q^{27} -4 i q^{28} + 8 q^{29} -i q^{31} + 2 i q^{33} + 8 q^{35} + 2 q^{36} -3 i q^{37} + ( -4 - 6 i ) q^{39} -4 i q^{41} + 4 q^{43} + 2 i q^{44} + 4 i q^{45} -2 i q^{47} + 8 q^{48} + 3 q^{49} -4 q^{51} + ( -4 - 6 i ) q^{52} -4 q^{53} -4 q^{55} -4 i q^{57} + 6 i q^{59} + 16 i q^{60} + 10 q^{61} -2 i q^{63} + 8 q^{64} + ( 12 - 8 i ) q^{65} + 16 i q^{67} -4 q^{68} -8 q^{69} -6 i q^{71} -7 i q^{73} -22 q^{75} -4 i q^{76} + 2 q^{77} -16 q^{79} + 16 i q^{80} -11 q^{81} + 12 i q^{83} -8 i q^{84} -8 i q^{85} + 16 q^{87} -15 i q^{89} + ( -6 + 4 i ) q^{91} -8 q^{92} -2 i q^{93} + 8 q^{95} + 2 i q^{97} + i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 4q^{3} + 4q^{4} + 2q^{9} + O(q^{10}) \) \( 2q + 4q^{3} + 4q^{4} + 2q^{9} + 8q^{12} - 4q^{13} + 8q^{16} - 4q^{17} - 8q^{23} - 22q^{25} - 8q^{27} + 16q^{29} + 16q^{35} + 4q^{36} - 8q^{39} + 8q^{43} + 16q^{48} + 6q^{49} - 8q^{51} - 8q^{52} - 8q^{53} - 8q^{55} + 20q^{61} + 16q^{64} + 24q^{65} - 8q^{68} - 16q^{69} - 44q^{75} + 4q^{77} - 32q^{79} - 22q^{81} + 32q^{87} - 12q^{91} - 16q^{92} + 16q^{95} + O(q^{100}) \)

Character Values

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

\(n\) \(249\) \(313\)
\(\chi(n)\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
311.1
1.00000i
1.00000i
0 2.00000 2.00000 4.00000i 0 2.00000i 0 1.00000 0
311.2 0 2.00000 2.00000 4.00000i 0 2.00000i 0 1.00000 0
\(n\): 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
13.b Even 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{2}^{\mathrm{new}}(403, [\chi])\).