Properties

Label 27.5.b.b
Level 27
Weight 5
Character orbit 27.b
Analytic conductor 2.791
Analytic rank 0
Dimension 2
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 27 = 3^{3} \)
Weight: \( k \) = \( 5 \)
Character orbit: \([\chi]\) = 27.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(2.79098900326\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-6}) \)
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 \(\beta = 3\sqrt{-6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + \beta q^{2} \) \( -38 q^{4} \) \( + 2 \beta q^{5} \) \( + 17 q^{7} \) \( -22 \beta q^{8} \) \(+O(q^{10})\) \( q\) \( + \beta q^{2} \) \( -38 q^{4} \) \( + 2 \beta q^{5} \) \( + 17 q^{7} \) \( -22 \beta q^{8} \) \( -108 q^{10} \) \( + 22 \beta q^{11} \) \( + 95 q^{13} \) \( + 17 \beta q^{14} \) \( + 580 q^{16} \) \( + 42 \beta q^{17} \) \( + 209 q^{19} \) \( -76 \beta q^{20} \) \( -1188 q^{22} \) \( -118 \beta q^{23} \) \( + 409 q^{25} \) \( + 95 \beta q^{26} \) \( -646 q^{28} \) \( -44 \beta q^{29} \) \( + 950 q^{31} \) \( + 228 \beta q^{32} \) \( -2268 q^{34} \) \( + 34 \beta q^{35} \) \( -1177 q^{37} \) \( + 209 \beta q^{38} \) \( + 2376 q^{40} \) \( -292 \beta q^{41} \) \( + 1430 q^{43} \) \( -836 \beta q^{44} \) \( + 6372 q^{46} \) \( + 214 \beta q^{47} \) \( -2112 q^{49} \) \( + 409 \beta q^{50} \) \( -3610 q^{52} \) \( + 396 \beta q^{53} \) \( -2376 q^{55} \) \( -374 \beta q^{56} \) \( + 2376 q^{58} \) \( + 290 \beta q^{59} \) \( -1441 q^{61} \) \( + 950 \beta q^{62} \) \( -3032 q^{64} \) \( + 190 \beta q^{65} \) \( + 3497 q^{67} \) \( -1596 \beta q^{68} \) \( -1836 q^{70} \) \( -264 \beta q^{71} \) \( -9025 q^{73} \) \( -1177 \beta q^{74} \) \( -7942 q^{76} \) \( + 374 \beta q^{77} \) \( + 5273 q^{79} \) \( + 1160 \beta q^{80} \) \( + 15768 q^{82} \) \( -836 \beta q^{83} \) \( -4536 q^{85} \) \( + 1430 \beta q^{86} \) \( + 26136 q^{88} \) \( -1518 \beta q^{89} \) \( + 1615 q^{91} \) \( + 4484 \beta q^{92} \) \( -11556 q^{94} \) \( + 418 \beta q^{95} \) \( -2809 q^{97} \) \( -2112 \beta q^{98} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut -\mathstrut 76q^{4} \) \(\mathstrut +\mathstrut 34q^{7} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut -\mathstrut 76q^{4} \) \(\mathstrut +\mathstrut 34q^{7} \) \(\mathstrut -\mathstrut 216q^{10} \) \(\mathstrut +\mathstrut 190q^{13} \) \(\mathstrut +\mathstrut 1160q^{16} \) \(\mathstrut +\mathstrut 418q^{19} \) \(\mathstrut -\mathstrut 2376q^{22} \) \(\mathstrut +\mathstrut 818q^{25} \) \(\mathstrut -\mathstrut 1292q^{28} \) \(\mathstrut +\mathstrut 1900q^{31} \) \(\mathstrut -\mathstrut 4536q^{34} \) \(\mathstrut -\mathstrut 2354q^{37} \) \(\mathstrut +\mathstrut 4752q^{40} \) \(\mathstrut +\mathstrut 2860q^{43} \) \(\mathstrut +\mathstrut 12744q^{46} \) \(\mathstrut -\mathstrut 4224q^{49} \) \(\mathstrut -\mathstrut 7220q^{52} \) \(\mathstrut -\mathstrut 4752q^{55} \) \(\mathstrut +\mathstrut 4752q^{58} \) \(\mathstrut -\mathstrut 2882q^{61} \) \(\mathstrut -\mathstrut 6064q^{64} \) \(\mathstrut +\mathstrut 6994q^{67} \) \(\mathstrut -\mathstrut 3672q^{70} \) \(\mathstrut -\mathstrut 18050q^{73} \) \(\mathstrut -\mathstrut 15884q^{76} \) \(\mathstrut +\mathstrut 10546q^{79} \) \(\mathstrut +\mathstrut 31536q^{82} \) \(\mathstrut -\mathstrut 9072q^{85} \) \(\mathstrut +\mathstrut 52272q^{88} \) \(\mathstrut +\mathstrut 3230q^{91} \) \(\mathstrut -\mathstrut 23112q^{94} \) \(\mathstrut -\mathstrut 5618q^{97} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
2.44949i
2.44949i
7.34847i 0 −38.0000 14.6969i 0 17.0000 161.666i 0 −108.000
26.2 7.34847i 0 −38.0000 14.6969i 0 17.0000 161.666i 0 −108.000
\(n\): 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 \(T_{2}^{2} \) \(\mathstrut +\mathstrut 54 \) acting on \(S_{5}^{\mathrm{new}}(27, [\chi])\).