Properties

Label 28.2.f.a
Level 28
Weight 2
Character orbit 28.f
Analytic conductor 0.224
Analytic rank 0
Dimension 4
CM No
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 28 = 2^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 28.f (of order \(6\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(0.22358112566\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

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

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/28\mathbb{Z}\right)^\times\).

\(n\) \(15\) \(17\)
\(\chi(n)\) \(-1\) \(\zeta_{12}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
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 −1.50000 + 0.866025i −1.73205 + 1.73205i −1.73205 + 2.00000i −2.00000 2.00000i 0 2.36603 0.633975i
3.2 0.366025 1.36603i −0.866025 + 1.50000i −1.73205 1.00000i −1.50000 + 0.866025i 1.73205 + 1.73205i 1.73205 2.00000i −2.00000 + 2.00000i 0 0.633975 + 2.36603i
19.1 −1.36603 + 0.366025i 0.866025 + 1.50000i 1.73205 1.00000i −1.50000 0.866025i −1.73205 1.73205i −1.73205 2.00000i −2.00000 + 2.00000i 0 2.36603 + 0.633975i
19.2 0.366025 + 1.36603i −0.866025 1.50000i −1.73205 + 1.00000i −1.50000 0.866025i 1.73205 1.73205i 1.73205 + 2.00000i −2.00000 2.00000i 0 0.633975 2.36603i
\(n\): 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
4.b Odd 1 yes
7.d Odd 1 yes
28.f Even 1 yes

Hecke kernels

There are no other newforms in \(S_{2}^{\mathrm{new}}(28, [\chi])\).