Properties

Label 28.3.b.a
Level 28
Weight 3
Character orbit 28.b
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.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(0.762944740209\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-6}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{-6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + \beta q^{3} \) \( -\beta q^{5} \) \( + ( 5 - \beta ) q^{7} \) \( -15 q^{9} \) \(+O(q^{10})\) \( q\) \( + \beta q^{3} \) \( -\beta q^{5} \) \( + ( 5 - \beta ) q^{7} \) \( -15 q^{9} \) \( -6 q^{11} \) \( + \beta q^{13} \) \( + 24 q^{15} \) \( -4 \beta q^{17} \) \( + 5 \beta q^{19} \) \( + ( 24 + 5 \beta ) q^{21} \) \( -30 q^{23} \) \(+ q^{25}\) \( -6 \beta q^{27} \) \( -6 q^{29} \) \( -6 \beta q^{33} \) \( + ( -24 - 5 \beta ) q^{35} \) \( + 10 q^{37} \) \( -24 q^{39} \) \( + 10 \beta q^{41} \) \( + 10 q^{43} \) \( + 15 \beta q^{45} \) \( -4 \beta q^{47} \) \( + ( 1 - 10 \beta ) q^{49} \) \( + 96 q^{51} \) \( + 90 q^{53} \) \( + 6 \beta q^{55} \) \( -120 q^{57} \) \( -5 \beta q^{59} \) \( + 5 \beta q^{61} \) \( + ( -75 + 15 \beta ) q^{63} \) \( + 24 q^{65} \) \( -70 q^{67} \) \( -30 \beta q^{69} \) \( + 42 q^{71} \) \( -22 \beta q^{73} \) \( + \beta q^{75} \) \( + ( -30 + 6 \beta ) q^{77} \) \( + 74 q^{79} \) \( + 9 q^{81} \) \( + 13 \beta q^{83} \) \( -96 q^{85} \) \( -6 \beta q^{87} \) \( + 30 \beta q^{89} \) \( + ( 24 + 5 \beta ) q^{91} \) \( + 120 q^{95} \) \( + 16 \beta q^{97} \) \( + 90 q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut +\mathstrut 10q^{7} \) \(\mathstrut -\mathstrut 30q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut +\mathstrut 10q^{7} \) \(\mathstrut -\mathstrut 30q^{9} \) \(\mathstrut -\mathstrut 12q^{11} \) \(\mathstrut +\mathstrut 48q^{15} \) \(\mathstrut +\mathstrut 48q^{21} \) \(\mathstrut -\mathstrut 60q^{23} \) \(\mathstrut +\mathstrut 2q^{25} \) \(\mathstrut -\mathstrut 12q^{29} \) \(\mathstrut -\mathstrut 48q^{35} \) \(\mathstrut +\mathstrut 20q^{37} \) \(\mathstrut -\mathstrut 48q^{39} \) \(\mathstrut +\mathstrut 20q^{43} \) \(\mathstrut +\mathstrut 2q^{49} \) \(\mathstrut +\mathstrut 192q^{51} \) \(\mathstrut +\mathstrut 180q^{53} \) \(\mathstrut -\mathstrut 240q^{57} \) \(\mathstrut -\mathstrut 150q^{63} \) \(\mathstrut +\mathstrut 48q^{65} \) \(\mathstrut -\mathstrut 140q^{67} \) \(\mathstrut +\mathstrut 84q^{71} \) \(\mathstrut -\mathstrut 60q^{77} \) \(\mathstrut +\mathstrut 148q^{79} \) \(\mathstrut +\mathstrut 18q^{81} \) \(\mathstrut -\mathstrut 192q^{85} \) \(\mathstrut +\mathstrut 48q^{91} \) \(\mathstrut +\mathstrut 240q^{95} \) \(\mathstrut +\mathstrut 180q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

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

\(n\) \(15\) \(17\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1
2.44949i
2.44949i
0 4.89898i 0 4.89898i 0 5.00000 + 4.89898i 0 −15.0000 0
13.2 0 4.89898i 0 4.89898i 0 5.00000 4.89898i 0 −15.0000 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.b Odd 1 yes

Hecke kernels

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