Properties

Label 13.2.e.a
Level 13
Weight 2
Character orbit 13.e
Analytic conductor 0.104
Analytic rank 0
Dimension 2
CM No
Inner twists 2

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Newspace parameters

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

Newform invariants

Self dual: No
Analytic conductor: \(0.103805522628\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
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.

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

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/13\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1
0.500000 + 0.866025i
0.500000 0.866025i
−1.50000 0.866025i −1.00000 + 1.73205i 0.500000 + 0.866025i 1.73205i 3.00000 1.73205i 0 1.73205i −0.500000 0.866025i −1.50000 + 2.59808i
10.1 −1.50000 + 0.866025i −1.00000 1.73205i 0.500000 0.866025i 1.73205i 3.00000 + 1.73205i 0 1.73205i −0.500000 + 0.866025i −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
13.e Even 1 yes

Hecke kernels

There are no other newforms in \(S_{2}^{\mathrm{new}}(13, [\chi])\).