Properties

Label 28.3.h.a
Level 28
Weight 3
Character orbit 28.h
Analytic conductor 0.763
Analytic rank 0
Dimension 2
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 28 = 2^{2} \cdot 7 \)
Weight: \( k \) = \( 3 \)
Character orbit: \([\chi]\) = 28.h (of order \(6\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(0.762944740209\)
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: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q\) \( + ( 1 + \zeta_{6} ) q^{3} \) \( + ( 2 - \zeta_{6} ) q^{5} \) \( -7 q^{7} \) \( -6 \zeta_{6} q^{9} \) \(+O(q^{10})\) \( q\) \( + ( 1 + \zeta_{6} ) q^{3} \) \( + ( 2 - \zeta_{6} ) q^{5} \) \( -7 q^{7} \) \( -6 \zeta_{6} q^{9} \) \( + ( -15 + 15 \zeta_{6} ) q^{11} \) \( + ( 8 - 16 \zeta_{6} ) q^{13} \) \( + 3 q^{15} \) \( + ( 17 + 17 \zeta_{6} ) q^{17} \) \( + ( 18 - 9 \zeta_{6} ) q^{19} \) \( + ( -7 - 7 \zeta_{6} ) q^{21} \) \( + 9 \zeta_{6} q^{23} \) \( + ( -22 + 22 \zeta_{6} ) q^{25} \) \( + ( 15 - 30 \zeta_{6} ) q^{27} \) \( -6 q^{29} \) \( + ( -7 - 7 \zeta_{6} ) q^{31} \) \( + ( -30 + 15 \zeta_{6} ) q^{33} \) \( + ( -14 + 7 \zeta_{6} ) q^{35} \) \( -31 \zeta_{6} q^{37} \) \( + ( 24 - 24 \zeta_{6} ) q^{39} \) \( + ( -32 + 64 \zeta_{6} ) q^{41} \) \( + 10 q^{43} \) \( + ( -6 - 6 \zeta_{6} ) q^{45} \) \( + ( 50 - 25 \zeta_{6} ) q^{47} \) \( + 49 q^{49} \) \( + 51 \zeta_{6} q^{51} \) \( + ( 57 - 57 \zeta_{6} ) q^{53} \) \( + ( -15 + 30 \zeta_{6} ) q^{55} \) \( + 27 q^{57} \) \( + ( -47 - 47 \zeta_{6} ) q^{59} \) \( + ( -94 + 47 \zeta_{6} ) q^{61} \) \( + 42 \zeta_{6} q^{63} \) \( -24 \zeta_{6} q^{65} \) \( + ( 49 - 49 \zeta_{6} ) q^{67} \) \( + ( -9 + 18 \zeta_{6} ) q^{69} \) \( -126 q^{71} \) \( + ( -15 - 15 \zeta_{6} ) q^{73} \) \( + ( -44 + 22 \zeta_{6} ) q^{75} \) \( + ( 105 - 105 \zeta_{6} ) q^{77} \) \( + 73 \zeta_{6} q^{79} \) \( + ( -9 + 9 \zeta_{6} ) q^{81} \) \( + ( -8 + 16 \zeta_{6} ) q^{83} \) \( + 51 q^{85} \) \( + ( -6 - 6 \zeta_{6} ) q^{87} \) \( + ( 66 - 33 \zeta_{6} ) q^{89} \) \( + ( -56 + 112 \zeta_{6} ) q^{91} \) \( -21 \zeta_{6} q^{93} \) \( + ( 27 - 27 \zeta_{6} ) q^{95} \) \( + ( 16 - 32 \zeta_{6} ) q^{97} \) \( + 90 q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut +\mathstrut 3q^{3} \) \(\mathstrut +\mathstrut 3q^{5} \) \(\mathstrut -\mathstrut 14q^{7} \) \(\mathstrut -\mathstrut 6q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut +\mathstrut 3q^{3} \) \(\mathstrut +\mathstrut 3q^{5} \) \(\mathstrut -\mathstrut 14q^{7} \) \(\mathstrut -\mathstrut 6q^{9} \) \(\mathstrut -\mathstrut 15q^{11} \) \(\mathstrut +\mathstrut 6q^{15} \) \(\mathstrut +\mathstrut 51q^{17} \) \(\mathstrut +\mathstrut 27q^{19} \) \(\mathstrut -\mathstrut 21q^{21} \) \(\mathstrut +\mathstrut 9q^{23} \) \(\mathstrut -\mathstrut 22q^{25} \) \(\mathstrut -\mathstrut 12q^{29} \) \(\mathstrut -\mathstrut 21q^{31} \) \(\mathstrut -\mathstrut 45q^{33} \) \(\mathstrut -\mathstrut 21q^{35} \) \(\mathstrut -\mathstrut 31q^{37} \) \(\mathstrut +\mathstrut 24q^{39} \) \(\mathstrut +\mathstrut 20q^{43} \) \(\mathstrut -\mathstrut 18q^{45} \) \(\mathstrut +\mathstrut 75q^{47} \) \(\mathstrut +\mathstrut 98q^{49} \) \(\mathstrut +\mathstrut 51q^{51} \) \(\mathstrut +\mathstrut 57q^{53} \) \(\mathstrut +\mathstrut 54q^{57} \) \(\mathstrut -\mathstrut 141q^{59} \) \(\mathstrut -\mathstrut 141q^{61} \) \(\mathstrut +\mathstrut 42q^{63} \) \(\mathstrut -\mathstrut 24q^{65} \) \(\mathstrut +\mathstrut 49q^{67} \) \(\mathstrut -\mathstrut 252q^{71} \) \(\mathstrut -\mathstrut 45q^{73} \) \(\mathstrut -\mathstrut 66q^{75} \) \(\mathstrut +\mathstrut 105q^{77} \) \(\mathstrut +\mathstrut 73q^{79} \) \(\mathstrut -\mathstrut 9q^{81} \) \(\mathstrut +\mathstrut 102q^{85} \) \(\mathstrut -\mathstrut 18q^{87} \) \(\mathstrut +\mathstrut 99q^{89} \) \(\mathstrut -\mathstrut 21q^{93} \) \(\mathstrut +\mathstrut 27q^{95} \) \(\mathstrut +\mathstrut 180q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
0.500000 0.866025i
0.500000 + 0.866025i
0 1.50000 0.866025i 0 1.50000 + 0.866025i 0 −7.00000 0 −3.00000 + 5.19615i 0
17.1 0 1.50000 + 0.866025i 0 1.50000 0.866025i 0 −7.00000 0 −3.00000 5.19615i 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.d Odd 1 yes

Hecke kernels

There are no other newforms in \(S_{3}^{\mathrm{new}}(28, [\chi])\).