Properties

Label 27.3.b.b
Level 27
Weight 3
Character orbit 27.b
Analytic conductor 0.736
Analytic rank 0
Dimension 2
CM No
Inner twists 2

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 27 = 3^{3} \)
Weight: \( k \) = \( 3 \)
Character orbit: \([\chi]\) = 27.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(0.735696713773\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 3 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + \beta q^{2} \) \( -5 q^{4} \) \( -\beta q^{5} \) \( + 5 q^{7} \) \( -\beta q^{8} \) \(+O(q^{10})\) \( q\) \( + \beta q^{2} \) \( -5 q^{4} \) \( -\beta q^{5} \) \( + 5 q^{7} \) \( -\beta q^{8} \) \( + 9 q^{10} \) \( -5 \beta q^{11} \) \( -10 q^{13} \) \( + 5 \beta q^{14} \) \( -11 q^{16} \) \( + 6 \beta q^{17} \) \( -16 q^{19} \) \( + 5 \beta q^{20} \) \( + 45 q^{22} \) \( -4 \beta q^{23} \) \( + 16 q^{25} \) \( -10 \beta q^{26} \) \( -25 q^{28} \) \( + 10 \beta q^{29} \) \(- q^{31}\) \( -15 \beta q^{32} \) \( -54 q^{34} \) \( -5 \beta q^{35} \) \( + 20 q^{37} \) \( -16 \beta q^{38} \) \( -9 q^{40} \) \( + 20 \beta q^{41} \) \( + 50 q^{43} \) \( + 25 \beta q^{44} \) \( + 36 q^{46} \) \( -2 \beta q^{47} \) \( -24 q^{49} \) \( + 16 \beta q^{50} \) \( + 50 q^{52} \) \( -9 \beta q^{53} \) \( -45 q^{55} \) \( -5 \beta q^{56} \) \( -90 q^{58} \) \( -10 \beta q^{59} \) \( -76 q^{61} \) \( -\beta q^{62} \) \( + 91 q^{64} \) \( + 10 \beta q^{65} \) \( -10 q^{67} \) \( -30 \beta q^{68} \) \( + 45 q^{70} \) \( -30 \beta q^{71} \) \( + 65 q^{73} \) \( + 20 \beta q^{74} \) \( + 80 q^{76} \) \( -25 \beta q^{77} \) \( + 14 q^{79} \) \( + 11 \beta q^{80} \) \( -180 q^{82} \) \( + \beta q^{83} \) \( + 54 q^{85} \) \( + 50 \beta q^{86} \) \( -45 q^{88} \) \( + 30 \beta q^{89} \) \( -50 q^{91} \) \( + 20 \beta q^{92} \) \( + 18 q^{94} \) \( + 16 \beta q^{95} \) \( -85 q^{97} \) \( -24 \beta q^{98} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut -\mathstrut 10q^{4} \) \(\mathstrut +\mathstrut 10q^{7} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut -\mathstrut 10q^{4} \) \(\mathstrut +\mathstrut 10q^{7} \) \(\mathstrut +\mathstrut 18q^{10} \) \(\mathstrut -\mathstrut 20q^{13} \) \(\mathstrut -\mathstrut 22q^{16} \) \(\mathstrut -\mathstrut 32q^{19} \) \(\mathstrut +\mathstrut 90q^{22} \) \(\mathstrut +\mathstrut 32q^{25} \) \(\mathstrut -\mathstrut 50q^{28} \) \(\mathstrut -\mathstrut 2q^{31} \) \(\mathstrut -\mathstrut 108q^{34} \) \(\mathstrut +\mathstrut 40q^{37} \) \(\mathstrut -\mathstrut 18q^{40} \) \(\mathstrut +\mathstrut 100q^{43} \) \(\mathstrut +\mathstrut 72q^{46} \) \(\mathstrut -\mathstrut 48q^{49} \) \(\mathstrut +\mathstrut 100q^{52} \) \(\mathstrut -\mathstrut 90q^{55} \) \(\mathstrut -\mathstrut 180q^{58} \) \(\mathstrut -\mathstrut 152q^{61} \) \(\mathstrut +\mathstrut 182q^{64} \) \(\mathstrut -\mathstrut 20q^{67} \) \(\mathstrut +\mathstrut 90q^{70} \) \(\mathstrut +\mathstrut 130q^{73} \) \(\mathstrut +\mathstrut 160q^{76} \) \(\mathstrut +\mathstrut 28q^{79} \) \(\mathstrut -\mathstrut 360q^{82} \) \(\mathstrut +\mathstrut 108q^{85} \) \(\mathstrut -\mathstrut 90q^{88} \) \(\mathstrut -\mathstrut 100q^{91} \) \(\mathstrut +\mathstrut 36q^{94} \) \(\mathstrut -\mathstrut 170q^{97} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/27\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
26.1
1.00000i
1.00000i
3.00000i 0 −5.00000 3.00000i 0 5.00000 3.00000i 0 9.00000
26.2 3.00000i 0 −5.00000 3.00000i 0 5.00000 3.00000i 0 9.00000
\(n\): 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
3.b Odd 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{2} \) \(\mathstrut +\mathstrut 9 \) acting on \(S_{3}^{\mathrm{new}}(27, [\chi])\).