Properties

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

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Newspace parameters

Level: \( N \) = \( 28 = 2^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 28.e (of order \(3\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(0.22358112566\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q\) \( -\zeta_{6} q^{3} \) \( + ( -3 + 3 \zeta_{6} ) q^{5} \) \( + ( -1 - 2 \zeta_{6} ) q^{7} \) \( + ( 2 - 2 \zeta_{6} ) q^{9} \) \(+O(q^{10})\) \( q\) \( -\zeta_{6} q^{3} \) \( + ( -3 + 3 \zeta_{6} ) q^{5} \) \( + ( -1 - 2 \zeta_{6} ) q^{7} \) \( + ( 2 - 2 \zeta_{6} ) q^{9} \) \( + 3 \zeta_{6} q^{11} \) \( + 2 q^{13} \) \( + 3 q^{15} \) \( -3 \zeta_{6} q^{17} \) \( + ( 1 - \zeta_{6} ) q^{19} \) \( + ( -2 + 3 \zeta_{6} ) q^{21} \) \( + ( -3 + 3 \zeta_{6} ) q^{23} \) \( -4 \zeta_{6} q^{25} \) \( -5 q^{27} \) \( -6 q^{29} \) \( + 7 \zeta_{6} q^{31} \) \( + ( 3 - 3 \zeta_{6} ) q^{33} \) \( + ( 9 - 3 \zeta_{6} ) q^{35} \) \( + ( 1 - \zeta_{6} ) q^{37} \) \( -2 \zeta_{6} q^{39} \) \( + 6 q^{41} \) \( -4 q^{43} \) \( + 6 \zeta_{6} q^{45} \) \( + ( 9 - 9 \zeta_{6} ) q^{47} \) \( + ( -3 + 8 \zeta_{6} ) q^{49} \) \( + ( -3 + 3 \zeta_{6} ) q^{51} \) \( -3 \zeta_{6} q^{53} \) \( -9 q^{55} \) \(- q^{57}\) \( -9 \zeta_{6} q^{59} \) \( + ( 1 - \zeta_{6} ) q^{61} \) \( + ( -6 + 2 \zeta_{6} ) q^{63} \) \( + ( -6 + 6 \zeta_{6} ) q^{65} \) \( + 7 \zeta_{6} q^{67} \) \( + 3 q^{69} \) \( + \zeta_{6} q^{73} \) \( + ( -4 + 4 \zeta_{6} ) q^{75} \) \( + ( 6 - 9 \zeta_{6} ) q^{77} \) \( + ( 13 - 13 \zeta_{6} ) q^{79} \) \( -\zeta_{6} q^{81} \) \( + 12 q^{83} \) \( + 9 q^{85} \) \( + 6 \zeta_{6} q^{87} \) \( + ( -15 + 15 \zeta_{6} ) q^{89} \) \( + ( -2 - 4 \zeta_{6} ) q^{91} \) \( + ( 7 - 7 \zeta_{6} ) q^{93} \) \( + 3 \zeta_{6} q^{95} \) \( -10 q^{97} \) \( + 6 q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut -\mathstrut q^{3} \) \(\mathstrut -\mathstrut 3q^{5} \) \(\mathstrut -\mathstrut 4q^{7} \) \(\mathstrut +\mathstrut 2q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut -\mathstrut q^{3} \) \(\mathstrut -\mathstrut 3q^{5} \) \(\mathstrut -\mathstrut 4q^{7} \) \(\mathstrut +\mathstrut 2q^{9} \) \(\mathstrut +\mathstrut 3q^{11} \) \(\mathstrut +\mathstrut 4q^{13} \) \(\mathstrut +\mathstrut 6q^{15} \) \(\mathstrut -\mathstrut 3q^{17} \) \(\mathstrut +\mathstrut q^{19} \) \(\mathstrut -\mathstrut q^{21} \) \(\mathstrut -\mathstrut 3q^{23} \) \(\mathstrut -\mathstrut 4q^{25} \) \(\mathstrut -\mathstrut 10q^{27} \) \(\mathstrut -\mathstrut 12q^{29} \) \(\mathstrut +\mathstrut 7q^{31} \) \(\mathstrut +\mathstrut 3q^{33} \) \(\mathstrut +\mathstrut 15q^{35} \) \(\mathstrut +\mathstrut q^{37} \) \(\mathstrut -\mathstrut 2q^{39} \) \(\mathstrut +\mathstrut 12q^{41} \) \(\mathstrut -\mathstrut 8q^{43} \) \(\mathstrut +\mathstrut 6q^{45} \) \(\mathstrut +\mathstrut 9q^{47} \) \(\mathstrut +\mathstrut 2q^{49} \) \(\mathstrut -\mathstrut 3q^{51} \) \(\mathstrut -\mathstrut 3q^{53} \) \(\mathstrut -\mathstrut 18q^{55} \) \(\mathstrut -\mathstrut 2q^{57} \) \(\mathstrut -\mathstrut 9q^{59} \) \(\mathstrut +\mathstrut q^{61} \) \(\mathstrut -\mathstrut 10q^{63} \) \(\mathstrut -\mathstrut 6q^{65} \) \(\mathstrut +\mathstrut 7q^{67} \) \(\mathstrut +\mathstrut 6q^{69} \) \(\mathstrut +\mathstrut q^{73} \) \(\mathstrut -\mathstrut 4q^{75} \) \(\mathstrut +\mathstrut 3q^{77} \) \(\mathstrut +\mathstrut 13q^{79} \) \(\mathstrut -\mathstrut q^{81} \) \(\mathstrut +\mathstrut 24q^{83} \) \(\mathstrut +\mathstrut 18q^{85} \) \(\mathstrut +\mathstrut 6q^{87} \) \(\mathstrut -\mathstrut 15q^{89} \) \(\mathstrut -\mathstrut 8q^{91} \) \(\mathstrut +\mathstrut 7q^{93} \) \(\mathstrut +\mathstrut 3q^{95} \) \(\mathstrut -\mathstrut 20q^{97} \) \(\mathstrut +\mathstrut 12q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

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

\(n\) \(15\) \(17\)
\(\chi(n)\) \(1\) \(-1 + \zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1
0.500000 + 0.866025i
0.500000 0.866025i
0 −0.500000 0.866025i 0 −1.50000 + 2.59808i 0 −2.00000 1.73205i 0 1.00000 1.73205i 0
25.1 0 −0.500000 + 0.866025i 0 −1.50000 2.59808i 0 −2.00000 + 1.73205i 0 1.00000 + 1.73205i 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
7.c Even 1 yes

Hecke kernels

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