Properties

Label 26.3.f.a
Level 26
Weight 3
Character orbit 26.f
Analytic conductor 0.708
Analytic rank 0
Dimension 4
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 26 = 2 \cdot 13 \)
Weight: \( k \) = \( 3 \)
Character orbit: \([\chi]\) = 26.f (of order \(12\) and degree \(4\))

Newform invariants

Self dual: No
Analytic conductor: \(0.708448687337\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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.

\(f(q)\) \(=\) \( q\) \( + ( \zeta_{12} + \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{2} \) \( + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{3} \) \( + 2 \zeta_{12} q^{4} \) \( + ( -1 - 2 \zeta_{12} + 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{5} \) \( + ( -1 + \zeta_{12} - \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{6} \) \( + ( -4 - 4 \zeta_{12} - 3 \zeta_{12}^{2} + 7 \zeta_{12}^{3} ) q^{7} \) \( + ( 2 + 2 \zeta_{12}^{3} ) q^{8} \) \( + ( 6 - 6 \zeta_{12}^{2} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + ( \zeta_{12} + \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{2} \) \( + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{3} \) \( + 2 \zeta_{12} q^{4} \) \( + ( -1 - 2 \zeta_{12} + 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{5} \) \( + ( -1 + \zeta_{12} - \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{6} \) \( + ( -4 - 4 \zeta_{12} - 3 \zeta_{12}^{2} + 7 \zeta_{12}^{3} ) q^{7} \) \( + ( 2 + 2 \zeta_{12}^{3} ) q^{8} \) \( + ( 6 - 6 \zeta_{12}^{2} ) q^{9} \) \( + ( -4 + 2 \zeta_{12}^{2} ) q^{10} \) \( + ( 1 + 4 \zeta_{12} - 5 \zeta_{12}^{2} - 5 \zeta_{12}^{3} ) q^{11} \) \( + ( 2 - 4 \zeta_{12}^{2} ) q^{12} \) \( + ( -9 + 8 \zeta_{12} + 12 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{13} \) \( + ( -1 - 14 \zeta_{12} + 7 \zeta_{12}^{3} ) q^{14} \) \( + ( 3 \zeta_{12} + 3 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{15} \) \( + 4 \zeta_{12}^{2} q^{16} \) \( + ( 13 - 6 \zeta_{12} + 13 \zeta_{12}^{2} ) q^{17} \) \( + ( 6 - 6 \zeta_{12}^{3} ) q^{18} \) \( + ( -12 + 12 \zeta_{12} - 5 \zeta_{12}^{2} - 17 \zeta_{12}^{3} ) q^{19} \) \( + ( -2 - 2 \zeta_{12} - 2 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{20} \) \( + ( 10 + \zeta_{12} + \zeta_{12}^{2} + 10 \zeta_{12}^{3} ) q^{21} \) \( + ( 9 + \zeta_{12} - 9 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{22} \) \( + ( 4 + 21 \zeta_{12} - 2 \zeta_{12}^{2} - 21 \zeta_{12}^{3} ) q^{23} \) \( + ( 4 - 2 \zeta_{12} - 2 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{24} \) \( + 19 \zeta_{12}^{3} q^{25} \) \( + ( -4 + 5 \zeta_{12} + \zeta_{12}^{2} + 15 \zeta_{12}^{3} ) q^{26} \) \( + ( -30 \zeta_{12} + 15 \zeta_{12}^{3} ) q^{27} \) \( + ( -14 - 8 \zeta_{12} + 6 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{28} \) \( + ( -16 \zeta_{12} - 27 \zeta_{12}^{2} - 16 \zeta_{12}^{3} ) q^{29} \) \( + 6 \zeta_{12} q^{30} \) \( + ( -37 - 10 \zeta_{12} + 10 \zeta_{12}^{2} + 37 \zeta_{12}^{3} ) q^{31} \) \( + ( -4 + 4 \zeta_{12} + 4 \zeta_{12}^{2} ) q^{32} \) \( + ( -6 - 6 \zeta_{12} - 3 \zeta_{12}^{2} + 9 \zeta_{12}^{3} ) q^{33} \) \( + ( -19 + 26 \zeta_{12} + 26 \zeta_{12}^{2} - 19 \zeta_{12}^{3} ) q^{34} \) \( + ( 21 + \zeta_{12} - 21 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{35} \) \( + ( 12 \zeta_{12} - 12 \zeta_{12}^{3} ) q^{36} \) \( + ( 7 - 13 \zeta_{12} + 6 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{37} \) \( + ( 17 - 34 \zeta_{12}^{2} + 7 \zeta_{12}^{3} ) q^{38} \) \( + ( 4 + 21 \zeta_{12} - 14 \zeta_{12}^{2} - 15 \zeta_{12}^{3} ) q^{39} \) \( + ( -8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{40} \) \( + ( -13 + 13 \zeta_{12} + 26 \zeta_{12}^{2} - 26 \zeta_{12}^{3} ) q^{41} \) \( + ( \zeta_{12} + 21 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{42} \) \( + ( 20 - 39 \zeta_{12} + 20 \zeta_{12}^{2} ) q^{43} \) \( + ( 10 + 2 \zeta_{12} - 2 \zeta_{12}^{2} - 10 \zeta_{12}^{3} ) q^{44} \) \( + ( 6 - 6 \zeta_{12} + 6 \zeta_{12}^{2} + 12 \zeta_{12}^{3} ) q^{45} \) \( + ( 23 + 23 \zeta_{12} - 19 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{46} \) \( + ( 33 + 33 \zeta_{12}^{3} ) q^{47} \) \( + ( 4 \zeta_{12} - 8 \zeta_{12}^{3} ) q^{48} \) \( + ( 14 + 25 \zeta_{12} - 7 \zeta_{12}^{2} - 25 \zeta_{12}^{3} ) q^{49} \) \( + ( -19 \zeta_{12} + 19 \zeta_{12}^{2} + 19 \zeta_{12}^{3} ) q^{50} \) \( + ( -6 + 12 \zeta_{12}^{2} - 39 \zeta_{12}^{3} ) q^{51} \) \( + ( 4 - 18 \zeta_{12} + 12 \zeta_{12}^{2} + 24 \zeta_{12}^{3} ) q^{52} \) \( + ( -42 + 44 \zeta_{12} - 22 \zeta_{12}^{3} ) q^{53} \) \( + ( -30 - 15 \zeta_{12} + 15 \zeta_{12}^{2} - 15 \zeta_{12}^{3} ) q^{54} \) \( + ( 9 \zeta_{12} + 3 \zeta_{12}^{2} + 9 \zeta_{12}^{3} ) q^{55} \) \( + ( -14 - 2 \zeta_{12} - 14 \zeta_{12}^{2} ) q^{56} \) \( + ( -22 + 7 \zeta_{12} - 7 \zeta_{12}^{2} + 22 \zeta_{12}^{3} ) q^{57} \) \( + ( 11 - 11 \zeta_{12} - 43 \zeta_{12}^{2} - 32 \zeta_{12}^{3} ) q^{58} \) \( + ( -10 - 10 \zeta_{12} + 23 \zeta_{12}^{2} - 13 \zeta_{12}^{3} ) q^{59} \) \( + ( 6 + 6 \zeta_{12}^{3} ) q^{60} \) \( + ( 39 + 6 \zeta_{12} - 39 \zeta_{12}^{2} - 12 \zeta_{12}^{3} ) q^{61} \) \( + ( -20 - 64 \zeta_{12} + 10 \zeta_{12}^{2} + 64 \zeta_{12}^{3} ) q^{62} \) \( + ( -42 + 18 \zeta_{12} + 24 \zeta_{12}^{2} + 24 \zeta_{12}^{3} ) q^{63} \) \( + 8 \zeta_{12}^{3} q^{64} \) \( + ( -25 + 2 \zeta_{12} - 10 \zeta_{12}^{2} - 7 \zeta_{12}^{3} ) q^{65} \) \( + ( -3 - 18 \zeta_{12} + 9 \zeta_{12}^{3} ) q^{66} \) \( + ( 33 + 10 \zeta_{12} - 23 \zeta_{12}^{2} + 23 \zeta_{12}^{3} ) q^{67} \) \( + ( 26 \zeta_{12} - 12 \zeta_{12}^{2} + 26 \zeta_{12}^{3} ) q^{68} \) \( + ( -21 - 6 \zeta_{12} - 21 \zeta_{12}^{2} ) q^{69} \) \( + ( 22 + 2 \zeta_{12} - 2 \zeta_{12}^{2} - 22 \zeta_{12}^{3} ) q^{70} \) \( + ( -4 + 4 \zeta_{12} + 29 \zeta_{12}^{2} + 25 \zeta_{12}^{3} ) q^{71} \) \( + ( 12 + 12 \zeta_{12} - 12 \zeta_{12}^{2} ) q^{72} \) \( + ( -7 - 54 \zeta_{12} - 54 \zeta_{12}^{2} - 7 \zeta_{12}^{3} ) q^{73} \) \( + ( -19 + 7 \zeta_{12} + 19 \zeta_{12}^{2} - 14 \zeta_{12}^{3} ) q^{74} \) \( + ( 38 - 19 \zeta_{12}^{2} ) q^{75} \) \( + ( 34 - 24 \zeta_{12} - 10 \zeta_{12}^{2} - 10 \zeta_{12}^{3} ) q^{76} \) \( + ( -32 + 64 \zeta_{12}^{2} + 15 \zeta_{12}^{3} ) q^{77} \) \( + ( 35 + 5 \zeta_{12} - 25 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{78} \) \( + ( 48 + 72 \zeta_{12} - 36 \zeta_{12}^{3} ) q^{79} \) \( + ( -8 - 4 \zeta_{12} + 4 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{80} \) \( -9 \zeta_{12}^{2} q^{81} \) \( + ( -13 + 39 \zeta_{12} - 13 \zeta_{12}^{2} ) q^{82} \) \( + ( 25 - 46 \zeta_{12} + 46 \zeta_{12}^{2} - 25 \zeta_{12}^{3} ) q^{83} \) \( + ( -20 + 20 \zeta_{12} + 22 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{84} \) \( + ( -33 - 33 \zeta_{12} + 45 \zeta_{12}^{2} - 12 \zeta_{12}^{3} ) q^{85} \) \( + ( -59 + 40 \zeta_{12} + 40 \zeta_{12}^{2} - 59 \zeta_{12}^{3} ) q^{86} \) \( + ( -48 - 27 \zeta_{12} + 48 \zeta_{12}^{2} + 54 \zeta_{12}^{3} ) q^{87} \) \( + ( 4 + 18 \zeta_{12} - 2 \zeta_{12}^{2} - 18 \zeta_{12}^{3} ) q^{88} \) \( + ( 13 + 43 \zeta_{12} - 56 \zeta_{12}^{2} - 56 \zeta_{12}^{3} ) q^{89} \) \( + ( -12 + 24 \zeta_{12}^{2} ) q^{90} \) \( + ( 22 - 86 \zeta_{12} - 25 \zeta_{12}^{2} - 37 \zeta_{12}^{3} ) q^{91} \) \( + ( 42 + 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{92} \) \( + ( 64 + 47 \zeta_{12} - 17 \zeta_{12}^{2} + 17 \zeta_{12}^{3} ) q^{93} \) \( + 66 \zeta_{12}^{2} q^{94} \) \( + ( -7 + 51 \zeta_{12} - 7 \zeta_{12}^{2} ) q^{95} \) \( + ( 4 + 8 \zeta_{12} - 8 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{96} \) \( + ( 71 - 71 \zeta_{12} - 2 \zeta_{12}^{2} + 69 \zeta_{12}^{3} ) q^{97} \) \( + ( 32 + 32 \zeta_{12} - 18 \zeta_{12}^{2} - 14 \zeta_{12}^{3} ) q^{98} \) \( + ( -24 - 6 \zeta_{12} - 6 \zeta_{12}^{2} - 24 \zeta_{12}^{3} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(4q \) \(\mathstrut +\mathstrut 2q^{2} \) \(\mathstrut -\mathstrut 6q^{6} \) \(\mathstrut -\mathstrut 22q^{7} \) \(\mathstrut +\mathstrut 8q^{8} \) \(\mathstrut +\mathstrut 12q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(4q \) \(\mathstrut +\mathstrut 2q^{2} \) \(\mathstrut -\mathstrut 6q^{6} \) \(\mathstrut -\mathstrut 22q^{7} \) \(\mathstrut +\mathstrut 8q^{8} \) \(\mathstrut +\mathstrut 12q^{9} \) \(\mathstrut -\mathstrut 12q^{10} \) \(\mathstrut -\mathstrut 6q^{11} \) \(\mathstrut -\mathstrut 12q^{13} \) \(\mathstrut -\mathstrut 4q^{14} \) \(\mathstrut +\mathstrut 6q^{15} \) \(\mathstrut +\mathstrut 8q^{16} \) \(\mathstrut +\mathstrut 78q^{17} \) \(\mathstrut +\mathstrut 24q^{18} \) \(\mathstrut -\mathstrut 58q^{19} \) \(\mathstrut -\mathstrut 12q^{20} \) \(\mathstrut +\mathstrut 42q^{21} \) \(\mathstrut +\mathstrut 18q^{22} \) \(\mathstrut +\mathstrut 12q^{23} \) \(\mathstrut +\mathstrut 12q^{24} \) \(\mathstrut -\mathstrut 14q^{26} \) \(\mathstrut -\mathstrut 44q^{28} \) \(\mathstrut -\mathstrut 54q^{29} \) \(\mathstrut -\mathstrut 128q^{31} \) \(\mathstrut -\mathstrut 8q^{32} \) \(\mathstrut -\mathstrut 30q^{33} \) \(\mathstrut -\mathstrut 24q^{34} \) \(\mathstrut +\mathstrut 42q^{35} \) \(\mathstrut +\mathstrut 40q^{37} \) \(\mathstrut -\mathstrut 12q^{39} \) \(\mathstrut +\mathstrut 42q^{42} \) \(\mathstrut +\mathstrut 120q^{43} \) \(\mathstrut +\mathstrut 36q^{44} \) \(\mathstrut +\mathstrut 36q^{45} \) \(\mathstrut +\mathstrut 54q^{46} \) \(\mathstrut +\mathstrut 132q^{47} \) \(\mathstrut +\mathstrut 42q^{49} \) \(\mathstrut +\mathstrut 38q^{50} \) \(\mathstrut +\mathstrut 40q^{52} \) \(\mathstrut -\mathstrut 168q^{53} \) \(\mathstrut -\mathstrut 90q^{54} \) \(\mathstrut +\mathstrut 6q^{55} \) \(\mathstrut -\mathstrut 84q^{56} \) \(\mathstrut -\mathstrut 102q^{57} \) \(\mathstrut -\mathstrut 42q^{58} \) \(\mathstrut +\mathstrut 6q^{59} \) \(\mathstrut +\mathstrut 24q^{60} \) \(\mathstrut +\mathstrut 78q^{61} \) \(\mathstrut -\mathstrut 60q^{62} \) \(\mathstrut -\mathstrut 120q^{63} \) \(\mathstrut -\mathstrut 120q^{65} \) \(\mathstrut -\mathstrut 12q^{66} \) \(\mathstrut +\mathstrut 86q^{67} \) \(\mathstrut -\mathstrut 24q^{68} \) \(\mathstrut -\mathstrut 126q^{69} \) \(\mathstrut +\mathstrut 84q^{70} \) \(\mathstrut +\mathstrut 42q^{71} \) \(\mathstrut +\mathstrut 24q^{72} \) \(\mathstrut -\mathstrut 136q^{73} \) \(\mathstrut -\mathstrut 38q^{74} \) \(\mathstrut +\mathstrut 114q^{75} \) \(\mathstrut +\mathstrut 116q^{76} \) \(\mathstrut +\mathstrut 90q^{78} \) \(\mathstrut +\mathstrut 192q^{79} \) \(\mathstrut -\mathstrut 24q^{80} \) \(\mathstrut -\mathstrut 18q^{81} \) \(\mathstrut -\mathstrut 78q^{82} \) \(\mathstrut +\mathstrut 192q^{83} \) \(\mathstrut -\mathstrut 36q^{84} \) \(\mathstrut -\mathstrut 42q^{85} \) \(\mathstrut -\mathstrut 156q^{86} \) \(\mathstrut -\mathstrut 96q^{87} \) \(\mathstrut +\mathstrut 12q^{88} \) \(\mathstrut -\mathstrut 60q^{89} \) \(\mathstrut +\mathstrut 38q^{91} \) \(\mathstrut +\mathstrut 168q^{92} \) \(\mathstrut +\mathstrut 222q^{93} \) \(\mathstrut +\mathstrut 132q^{94} \) \(\mathstrut -\mathstrut 42q^{95} \) \(\mathstrut +\mathstrut 280q^{97} \) \(\mathstrut +\mathstrut 92q^{98} \) \(\mathstrut -\mathstrut 108q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/26\mathbb{Z}\right)^\times\).

\(n\) \(15\)
\(\chi(n)\) \(\zeta_{12}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
0.866025 0.500000i
−0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
1.36603 0.366025i −0.866025 + 1.50000i 1.73205 1.00000i −1.73205 1.73205i −0.633975 + 2.36603i −8.96410 2.40192i 2.00000 2.00000i 3.00000 + 5.19615i −3.00000 1.73205i
11.1 −0.366025 + 1.36603i 0.866025 + 1.50000i −1.73205 1.00000i 1.73205 + 1.73205i −2.36603 + 0.633975i −2.03590 7.59808i 2.00000 2.00000i 3.00000 5.19615i −3.00000 + 1.73205i
15.1 1.36603 + 0.366025i −0.866025 1.50000i 1.73205 + 1.00000i −1.73205 + 1.73205i −0.633975 2.36603i −8.96410 + 2.40192i 2.00000 + 2.00000i 3.00000 5.19615i −3.00000 + 1.73205i
19.1 −0.366025 1.36603i 0.866025 1.50000i −1.73205 + 1.00000i 1.73205 1.73205i −2.36603 0.633975i −2.03590 + 7.59808i 2.00000 + 2.00000i 3.00000 + 5.19615i −3.00000 1.73205i
\(n\): 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

This newform can be constructed as the kernel of the linear operator \(T_{3}^{4} \) \(\mathstrut +\mathstrut 3 T_{3}^{2} \) \(\mathstrut +\mathstrut 9 \) acting on \(S_{3}^{\mathrm{new}}(26, [\chi])\).