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.

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

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 \(S_{3}^{\mathrm{new}}(28, [\chi])\).