Properties

Label 27.6.a.c
Level 27
Weight 6
Character orbit 27.a
Self dual Yes
Analytic conductor 4.330
Analytic rank 0
Dimension 2
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 27 = 3^{3} \)
Weight: \( k \) = \( 6 \)
Character orbit: \([\chi]\) = 27.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(4.33036313495\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{6}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 3 \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

\(f(q)\) \(=\) \( q\) \( + \beta q^{2} \) \( + 22 q^{4} \) \( + 8 \beta q^{5} \) \( + 167 q^{7} \) \( -10 \beta q^{8} \) \(+O(q^{10})\) \( q\) \( + \beta q^{2} \) \( + 22 q^{4} \) \( + 8 \beta q^{5} \) \( + 167 q^{7} \) \( -10 \beta q^{8} \) \( + 432 q^{10} \) \( -104 \beta q^{11} \) \( -235 q^{13} \) \( + 167 \beta q^{14} \) \( -1244 q^{16} \) \( + 24 \beta q^{17} \) \( + 1361 q^{19} \) \( + 176 \beta q^{20} \) \( -5616 q^{22} \) \( -328 \beta q^{23} \) \( + 331 q^{25} \) \( -235 \beta q^{26} \) \( + 3674 q^{28} \) \( + 64 \beta q^{29} \) \( + 3500 q^{31} \) \( -924 \beta q^{32} \) \( + 1296 q^{34} \) \( + 1336 \beta q^{35} \) \( + 13115 q^{37} \) \( + 1361 \beta q^{38} \) \( -4320 q^{40} \) \( + 1280 \beta q^{41} \) \( + 104 q^{43} \) \( -2288 \beta q^{44} \) \( -17712 q^{46} \) \( -2792 \beta q^{47} \) \( + 11082 q^{49} \) \( + 331 \beta q^{50} \) \( -5170 q^{52} \) \( -144 \beta q^{53} \) \( -44928 q^{55} \) \( -1670 \beta q^{56} \) \( + 3456 q^{58} \) \( + 4184 \beta q^{59} \) \( -7393 q^{61} \) \( + 3500 \beta q^{62} \) \( -10088 q^{64} \) \( -1880 \beta q^{65} \) \( + 38861 q^{67} \) \( + 528 \beta q^{68} \) \( + 72144 q^{70} \) \( -336 \beta q^{71} \) \( + 5465 q^{73} \) \( + 13115 \beta q^{74} \) \( + 29942 q^{76} \) \( -17368 \beta q^{77} \) \( -82903 q^{79} \) \( -9952 \beta q^{80} \) \( + 69120 q^{82} \) \( -1808 \beta q^{83} \) \( + 10368 q^{85} \) \( + 104 \beta q^{86} \) \( + 56160 q^{88} \) \( + 12216 \beta q^{89} \) \( -39245 q^{91} \) \( -7216 \beta q^{92} \) \( -150768 q^{94} \) \( + 10888 \beta q^{95} \) \( -49603 q^{97} \) \( + 11082 \beta q^{98} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut +\mathstrut 44q^{4} \) \(\mathstrut +\mathstrut 334q^{7} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut +\mathstrut 44q^{4} \) \(\mathstrut +\mathstrut 334q^{7} \) \(\mathstrut +\mathstrut 864q^{10} \) \(\mathstrut -\mathstrut 470q^{13} \) \(\mathstrut -\mathstrut 2488q^{16} \) \(\mathstrut +\mathstrut 2722q^{19} \) \(\mathstrut -\mathstrut 11232q^{22} \) \(\mathstrut +\mathstrut 662q^{25} \) \(\mathstrut +\mathstrut 7348q^{28} \) \(\mathstrut +\mathstrut 7000q^{31} \) \(\mathstrut +\mathstrut 2592q^{34} \) \(\mathstrut +\mathstrut 26230q^{37} \) \(\mathstrut -\mathstrut 8640q^{40} \) \(\mathstrut +\mathstrut 208q^{43} \) \(\mathstrut -\mathstrut 35424q^{46} \) \(\mathstrut +\mathstrut 22164q^{49} \) \(\mathstrut -\mathstrut 10340q^{52} \) \(\mathstrut -\mathstrut 89856q^{55} \) \(\mathstrut +\mathstrut 6912q^{58} \) \(\mathstrut -\mathstrut 14786q^{61} \) \(\mathstrut -\mathstrut 20176q^{64} \) \(\mathstrut +\mathstrut 77722q^{67} \) \(\mathstrut +\mathstrut 144288q^{70} \) \(\mathstrut +\mathstrut 10930q^{73} \) \(\mathstrut +\mathstrut 59884q^{76} \) \(\mathstrut -\mathstrut 165806q^{79} \) \(\mathstrut +\mathstrut 138240q^{82} \) \(\mathstrut +\mathstrut 20736q^{85} \) \(\mathstrut +\mathstrut 112320q^{88} \) \(\mathstrut -\mathstrut 78490q^{91} \) \(\mathstrut -\mathstrut 301536q^{94} \) \(\mathstrut -\mathstrut 99206q^{97} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−2.44949
2.44949
−7.34847 0 22.0000 −58.7878 0 167.000 73.4847 0 432.000
1.2 7.34847 0 22.0000 58.7878 0 167.000 −73.4847 0 432.000
\(n\): 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

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{2} \) \(\mathstrut -\mathstrut 54 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(27))\).