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

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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.

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

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 \( T_{2}^{2} + 9 \) acting on \(S_{3}^{\mathrm{new}}(27, [\chi])\).