Properties

Label 13.3.f.a
Level 13
Weight 3
Character orbit 13.f
Analytic conductor 0.354
Analytic rank 0
Dimension 4
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 13 \)
Weight: \( k \) = \( 3 \)
Character orbit: \([\chi]\) = 13.f (of order \(12\) and degree \(4\))

Newform invariants

Self dual: No
Analytic conductor: \(0.354224343668\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Coefficient ring: \(\Z[a_1, a_2]\)
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.

\(f(q)\) \(=\) \( q\) \( + ( -1 + \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{2} \) \( + ( \zeta_{12} - \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{3} \) \( + ( -1 - 2 \zeta_{12} - \zeta_{12}^{2} ) q^{4} \) \( + ( -3 + \zeta_{12} - \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{5} \) \( + ( 3 - 3 \zeta_{12} - \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{6} \) \( + ( 2 + 2 \zeta_{12} + 4 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{7} \) \( + ( -4 + 5 \zeta_{12} + 5 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{8} \) \( + ( 5 + 2 \zeta_{12} - 5 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + ( -1 + \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{2} \) \( + ( \zeta_{12} - \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{3} \) \( + ( -1 - 2 \zeta_{12} - \zeta_{12}^{2} ) q^{4} \) \( + ( -3 + \zeta_{12} - \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{5} \) \( + ( 3 - 3 \zeta_{12} - \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{6} \) \( + ( 2 + 2 \zeta_{12} + 4 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{7} \) \( + ( -4 + 5 \zeta_{12} + 5 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{8} \) \( + ( 5 + 2 \zeta_{12} - 5 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{9} \) \( + ( 8 - 5 \zeta_{12} - 4 \zeta_{12}^{2} + 5 \zeta_{12}^{3} ) q^{10} \) \( + ( -4 - 6 \zeta_{12} + 10 \zeta_{12}^{2} + 10 \zeta_{12}^{3} ) q^{11} \) \( + ( 1 - 2 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{12} \) \( -13 \zeta_{12}^{2} q^{13} \) \( + ( -10 + 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{14} \) \( + ( -8 + \zeta_{12} + 9 \zeta_{12}^{2} - 9 \zeta_{12}^{3} ) q^{15} \) \( + ( 8 \zeta_{12} - \zeta_{12}^{2} + 8 \zeta_{12}^{3} ) q^{16} \) \( + ( -2 - 15 \zeta_{12} - 2 \zeta_{12}^{2} ) q^{17} \) \( + ( -2 - 3 \zeta_{12} + 3 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{18} \) \( + ( 1 - \zeta_{12} + 3 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{19} \) \( + ( 8 + 8 \zeta_{12} - 3 \zeta_{12}^{2} - 5 \zeta_{12}^{3} ) q^{20} \) \( + ( 14 - 8 \zeta_{12} - 8 \zeta_{12}^{2} + 14 \zeta_{12}^{3} ) q^{21} \) \( + ( -2 + 6 \zeta_{12} + 2 \zeta_{12}^{2} - 12 \zeta_{12}^{3} ) q^{22} \) \( + ( 6 + 15 \zeta_{12} - 3 \zeta_{12}^{2} - 15 \zeta_{12}^{3} ) q^{23} \) \( + ( 8 - 13 \zeta_{12} + 5 \zeta_{12}^{2} + 5 \zeta_{12}^{3} ) q^{24} \) \( + ( -7 + 14 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{25} \) \( + ( 13 - 13 \zeta_{12} + 13 \zeta_{12}^{3} ) q^{26} \) \( + ( -8 + 24 \zeta_{12} - 12 \zeta_{12}^{3} ) q^{27} \) \( + ( -10 - 12 \zeta_{12} - 2 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{28} \) \( + ( -6 \zeta_{12} + \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{29} \) \( + ( -9 + 17 \zeta_{12} - 9 \zeta_{12}^{2} ) q^{30} \) \( + ( -9 - 8 \zeta_{12} + 8 \zeta_{12}^{2} + 9 \zeta_{12}^{3} ) q^{31} \) \( + ( 1 - \zeta_{12} - 12 \zeta_{12}^{2} - 11 \zeta_{12}^{3} ) q^{32} \) \( + ( -4 - 4 \zeta_{12} - 8 \zeta_{12}^{2} + 12 \zeta_{12}^{3} ) q^{33} \) \( + ( -11 + 13 \zeta_{12} + 13 \zeta_{12}^{2} - 11 \zeta_{12}^{3} ) q^{34} \) \( + ( 16 - 22 \zeta_{12} - 16 \zeta_{12}^{2} + 44 \zeta_{12}^{3} ) q^{35} \) \( + ( -18 - 16 \zeta_{12} + 9 \zeta_{12}^{2} + 16 \zeta_{12}^{3} ) q^{36} \) \( + ( -5 + 29 \zeta_{12} - 24 \zeta_{12}^{2} - 24 \zeta_{12}^{3} ) q^{37} \) \( + ( -1 + 2 \zeta_{12}^{2} - 5 \zeta_{12}^{3} ) q^{38} \) \( + ( -13 + 13 \zeta_{12} + 13 \zeta_{12}^{2} - 26 \zeta_{12}^{3} ) q^{39} \) \( + ( 18 - 38 \zeta_{12} + 19 \zeta_{12}^{3} ) q^{40} \) \( + ( 27 + 23 \zeta_{12} - 4 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{41} \) \( + ( -14 \zeta_{12} + 22 \zeta_{12}^{2} - 14 \zeta_{12}^{3} ) q^{42} \) \( + ( 30 + 30 \zeta_{12}^{2} ) q^{43} \) \( + ( 34 + 24 \zeta_{12} - 24 \zeta_{12}^{2} - 34 \zeta_{12}^{3} ) q^{44} \) \( + ( -10 + 10 \zeta_{12} + 19 \zeta_{12}^{2} + 9 \zeta_{12}^{3} ) q^{45} \) \( + ( -3 - 3 \zeta_{12} - 9 \zeta_{12}^{2} + 12 \zeta_{12}^{3} ) q^{46} \) \( + ( -7 - 20 \zeta_{12} - 20 \zeta_{12}^{2} - 7 \zeta_{12}^{3} ) q^{47} \) \( + ( -25 + 9 \zeta_{12} + 25 \zeta_{12}^{2} - 18 \zeta_{12}^{3} ) q^{48} \) \( + ( -24 + 7 \zeta_{12} + 12 \zeta_{12}^{2} - 7 \zeta_{12}^{3} ) q^{49} \) \( + ( -8 + 15 \zeta_{12} - 7 \zeta_{12}^{2} - 7 \zeta_{12}^{3} ) q^{50} \) \( + ( 13 - 26 \zeta_{12}^{2} + 9 \zeta_{12}^{3} ) q^{51} \) \( + ( -13 + 26 \zeta_{12}^{2} + 26 \zeta_{12}^{3} ) q^{52} \) \( + ( 32 - 54 \zeta_{12} + 27 \zeta_{12}^{3} ) q^{53} \) \( + ( 20 - 12 \zeta_{12} - 32 \zeta_{12}^{2} + 32 \zeta_{12}^{3} ) q^{54} \) \( + ( -6 \zeta_{12} - 50 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{55} \) \( + ( -14 + 48 \zeta_{12} - 14 \zeta_{12}^{2} ) q^{56} \) \( + ( -4 + 2 \zeta_{12} - 2 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{57} \) \( + ( -13 + 13 \zeta_{12} + 6 \zeta_{12}^{2} - 7 \zeta_{12}^{3} ) q^{58} \) \( + ( -60 - 60 \zeta_{12} + 38 \zeta_{12}^{2} + 22 \zeta_{12}^{3} ) q^{59} \) \( + ( -1 + 6 \zeta_{12} + 6 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{60} \) \( + ( -62 + 21 \zeta_{12} + 62 \zeta_{12}^{2} - 42 \zeta_{12}^{3} ) q^{61} \) \( + ( 2 + 7 \zeta_{12} - \zeta_{12}^{2} - 7 \zeta_{12}^{3} ) q^{62} \) \( + ( 26 - 26 \zeta_{12}^{2} - 26 \zeta_{12}^{3} ) q^{63} \) \( + ( 31 - 62 \zeta_{12}^{2} + 14 \zeta_{12}^{3} ) q^{64} \) \( + ( -13 + 39 \zeta_{12} + 52 \zeta_{12}^{2} - 52 \zeta_{12}^{3} ) q^{65} \) \( + ( 20 - 16 \zeta_{12} + 8 \zeta_{12}^{3} ) q^{66} \) \( + ( 48 + 11 \zeta_{12} - 37 \zeta_{12}^{2} + 37 \zeta_{12}^{3} ) q^{67} \) \( + ( 19 \zeta_{12} + 36 \zeta_{12}^{2} + 19 \zeta_{12}^{3} ) q^{68} \) \( + ( 12 - 6 \zeta_{12} + 12 \zeta_{12}^{2} ) q^{69} \) \( + ( 22 - 38 \zeta_{12} + 38 \zeta_{12}^{2} - 22 \zeta_{12}^{3} ) q^{70} \) \( + ( -11 + 11 \zeta_{12} - 21 \zeta_{12}^{2} - 32 \zeta_{12}^{3} ) q^{71} \) \( + ( 17 + 17 \zeta_{12} + 2 \zeta_{12}^{2} - 19 \zeta_{12}^{3} ) q^{72} \) \( + ( 34 - 39 \zeta_{12} - 39 \zeta_{12}^{2} + 34 \zeta_{12}^{3} ) q^{73} \) \( + ( 34 - 29 \zeta_{12} - 34 \zeta_{12}^{2} + 58 \zeta_{12}^{3} ) q^{74} \) \( + ( 16 - 22 \zeta_{12} - 8 \zeta_{12}^{2} + 22 \zeta_{12}^{3} ) q^{75} \) \( + ( 10 + 3 \zeta_{12} - 13 \zeta_{12}^{2} - 13 \zeta_{12}^{3} ) q^{76} \) \( + ( -44 + 88 \zeta_{12}^{2} + 20 \zeta_{12}^{3} ) q^{77} \) \( + ( -13 + 26 \zeta_{12} - 26 \zeta_{12}^{2} + 13 \zeta_{12}^{3} ) q^{78} \) \( + ( -10 + 84 \zeta_{12} - 42 \zeta_{12}^{3} ) q^{79} \) \( + ( -57 - 13 \zeta_{12} + 44 \zeta_{12}^{2} - 44 \zeta_{12}^{3} ) q^{80} \) \( + ( -38 \zeta_{12} - \zeta_{12}^{2} - 38 \zeta_{12}^{3} ) q^{81} \) \( + ( 4 - 31 \zeta_{12} + 4 \zeta_{12}^{2} ) q^{82} \) \( + ( -34 + 26 \zeta_{12} - 26 \zeta_{12}^{2} + 34 \zeta_{12}^{3} ) q^{83} \) \( + ( 6 - 6 \zeta_{12} - 10 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{84} \) \( + ( 49 + 49 \zeta_{12} - 50 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{85} \) \( + ( -60 + 30 \zeta_{12} + 30 \zeta_{12}^{2} - 60 \zeta_{12}^{3} ) q^{86} \) \( + ( 19 - 7 \zeta_{12} - 19 \zeta_{12}^{2} + 14 \zeta_{12}^{3} ) q^{87} \) \( + ( -68 - 6 \zeta_{12} + 34 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{88} \) \( + ( -50 + 5 \zeta_{12} + 45 \zeta_{12}^{2} + 45 \zeta_{12}^{3} ) q^{89} \) \( + ( 10 - 20 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{90} \) \( + ( 52 - 78 \zeta_{12} - 78 \zeta_{12}^{2} + 52 \zeta_{12}^{3} ) q^{91} \) \( + ( -39 - 42 \zeta_{12} + 21 \zeta_{12}^{3} ) q^{92} \) \( + ( -2 - 8 \zeta_{12} - 6 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{93} \) \( + ( 7 \zeta_{12} + 13 \zeta_{12}^{2} + 7 \zeta_{12}^{3} ) q^{94} \) \( + ( -13 - \zeta_{12} - 13 \zeta_{12}^{2} ) q^{95} \) \( + ( 11 + 2 \zeta_{12} - 2 \zeta_{12}^{2} - 11 \zeta_{12}^{3} ) q^{96} \) \( + ( 35 - 35 \zeta_{12} + 19 \zeta_{12}^{2} + 54 \zeta_{12}^{3} ) q^{97} \) \( + ( 12 + 12 \zeta_{12} - 31 \zeta_{12}^{2} + 19 \zeta_{12}^{3} ) q^{98} \) \( + ( 26 + 52 \zeta_{12} + 52 \zeta_{12}^{2} + 26 \zeta_{12}^{3} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(4q \) \(\mathstrut -\mathstrut 2q^{2} \) \(\mathstrut -\mathstrut 2q^{3} \) \(\mathstrut -\mathstrut 6q^{4} \) \(\mathstrut -\mathstrut 14q^{5} \) \(\mathstrut +\mathstrut 10q^{6} \) \(\mathstrut +\mathstrut 16q^{7} \) \(\mathstrut -\mathstrut 6q^{8} \) \(\mathstrut +\mathstrut 10q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(4q \) \(\mathstrut -\mathstrut 2q^{2} \) \(\mathstrut -\mathstrut 2q^{3} \) \(\mathstrut -\mathstrut 6q^{4} \) \(\mathstrut -\mathstrut 14q^{5} \) \(\mathstrut +\mathstrut 10q^{6} \) \(\mathstrut +\mathstrut 16q^{7} \) \(\mathstrut -\mathstrut 6q^{8} \) \(\mathstrut +\mathstrut 10q^{9} \) \(\mathstrut +\mathstrut 24q^{10} \) \(\mathstrut +\mathstrut 4q^{11} \) \(\mathstrut -\mathstrut 26q^{13} \) \(\mathstrut -\mathstrut 40q^{14} \) \(\mathstrut -\mathstrut 14q^{15} \) \(\mathstrut -\mathstrut 2q^{16} \) \(\mathstrut -\mathstrut 12q^{17} \) \(\mathstrut -\mathstrut 2q^{18} \) \(\mathstrut +\mathstrut 10q^{19} \) \(\mathstrut +\mathstrut 26q^{20} \) \(\mathstrut +\mathstrut 40q^{21} \) \(\mathstrut -\mathstrut 4q^{22} \) \(\mathstrut +\mathstrut 18q^{23} \) \(\mathstrut +\mathstrut 42q^{24} \) \(\mathstrut +\mathstrut 52q^{26} \) \(\mathstrut -\mathstrut 32q^{27} \) \(\mathstrut -\mathstrut 44q^{28} \) \(\mathstrut +\mathstrut 2q^{29} \) \(\mathstrut -\mathstrut 54q^{30} \) \(\mathstrut -\mathstrut 20q^{31} \) \(\mathstrut -\mathstrut 20q^{32} \) \(\mathstrut -\mathstrut 32q^{33} \) \(\mathstrut -\mathstrut 18q^{34} \) \(\mathstrut +\mathstrut 32q^{35} \) \(\mathstrut -\mathstrut 54q^{36} \) \(\mathstrut -\mathstrut 68q^{37} \) \(\mathstrut -\mathstrut 26q^{39} \) \(\mathstrut +\mathstrut 72q^{40} \) \(\mathstrut +\mathstrut 100q^{41} \) \(\mathstrut +\mathstrut 44q^{42} \) \(\mathstrut +\mathstrut 180q^{43} \) \(\mathstrut +\mathstrut 88q^{44} \) \(\mathstrut -\mathstrut 2q^{45} \) \(\mathstrut -\mathstrut 30q^{46} \) \(\mathstrut -\mathstrut 68q^{47} \) \(\mathstrut -\mathstrut 50q^{48} \) \(\mathstrut -\mathstrut 72q^{49} \) \(\mathstrut -\mathstrut 46q^{50} \) \(\mathstrut +\mathstrut 128q^{53} \) \(\mathstrut +\mathstrut 16q^{54} \) \(\mathstrut -\mathstrut 100q^{55} \) \(\mathstrut -\mathstrut 84q^{56} \) \(\mathstrut -\mathstrut 20q^{57} \) \(\mathstrut -\mathstrut 40q^{58} \) \(\mathstrut -\mathstrut 164q^{59} \) \(\mathstrut +\mathstrut 8q^{60} \) \(\mathstrut -\mathstrut 124q^{61} \) \(\mathstrut +\mathstrut 6q^{62} \) \(\mathstrut +\mathstrut 52q^{63} \) \(\mathstrut +\mathstrut 52q^{65} \) \(\mathstrut +\mathstrut 80q^{66} \) \(\mathstrut +\mathstrut 118q^{67} \) \(\mathstrut +\mathstrut 72q^{68} \) \(\mathstrut +\mathstrut 72q^{69} \) \(\mathstrut +\mathstrut 164q^{70} \) \(\mathstrut -\mathstrut 86q^{71} \) \(\mathstrut +\mathstrut 72q^{72} \) \(\mathstrut +\mathstrut 58q^{73} \) \(\mathstrut +\mathstrut 68q^{74} \) \(\mathstrut +\mathstrut 48q^{75} \) \(\mathstrut +\mathstrut 14q^{76} \) \(\mathstrut -\mathstrut 104q^{78} \) \(\mathstrut -\mathstrut 40q^{79} \) \(\mathstrut -\mathstrut 140q^{80} \) \(\mathstrut -\mathstrut 2q^{81} \) \(\mathstrut +\mathstrut 24q^{82} \) \(\mathstrut -\mathstrut 188q^{83} \) \(\mathstrut +\mathstrut 4q^{84} \) \(\mathstrut +\mathstrut 96q^{85} \) \(\mathstrut -\mathstrut 180q^{86} \) \(\mathstrut +\mathstrut 38q^{87} \) \(\mathstrut -\mathstrut 204q^{88} \) \(\mathstrut -\mathstrut 110q^{89} \) \(\mathstrut +\mathstrut 52q^{91} \) \(\mathstrut -\mathstrut 156q^{92} \) \(\mathstrut -\mathstrut 20q^{93} \) \(\mathstrut +\mathstrut 26q^{94} \) \(\mathstrut -\mathstrut 78q^{95} \) \(\mathstrut +\mathstrut 40q^{96} \) \(\mathstrut +\mathstrut 178q^{97} \) \(\mathstrut -\mathstrut 14q^{98} \) \(\mathstrut +\mathstrut 208q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

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

\(n\) \(2\)
\(\chi(n)\) \(\zeta_{12}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1
0.866025 + 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 0.500000i
−0.500000 0.133975i 0.366025 + 0.633975i −3.23205 1.86603i −2.63397 + 2.63397i −0.0980762 0.366025i 5.73205 1.53590i 2.83013 + 2.83013i 4.23205 7.33013i 1.66987 0.964102i
6.1 −0.500000 1.86603i −1.36603 + 2.36603i 0.232051 0.133975i −4.36603 + 4.36603i 5.09808 + 1.36603i 2.26795 8.46410i −5.83013 5.83013i 0.767949 + 1.33013i 10.3301 + 5.96410i
7.1 −0.500000 + 0.133975i 0.366025 0.633975i −3.23205 + 1.86603i −2.63397 2.63397i −0.0980762 + 0.366025i 5.73205 + 1.53590i 2.83013 2.83013i 4.23205 + 7.33013i 1.66987 + 0.964102i
11.1 −0.500000 + 1.86603i −1.36603 2.36603i 0.232051 + 0.133975i −4.36603 4.36603i 5.09808 1.36603i 2.26795 + 8.46410i −5.83013 + 5.83013i 0.767949 1.33013i 10.3301 5.96410i
\(n\): 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.f Odd 1 yes

Hecke kernels

There are no other newforms in \(S_{3}^{\mathrm{new}}(13, [\chi])\).