Properties

Label 27.9.b.b
Level 27
Weight 9
Character orbit 27.b
Analytic conductor 10.999
Analytic rank 0
Dimension 2
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 27 = 3^{3} \)
Weight: \( k \) = \( 9 \)
Character orbit: \([\chi]\) = 27.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(10.9992224717\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-6}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 12\sqrt{-6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + \beta q^{2} \) \( -608 q^{4} \) \( -28 \beta q^{5} \) \( + 1967 q^{7} \) \( -352 \beta q^{8} \) \(+O(q^{10})\) \( q\) \( + \beta q^{2} \) \( -608 q^{4} \) \( -28 \beta q^{5} \) \( + 1967 q^{7} \) \( -352 \beta q^{8} \) \( + 24192 q^{10} \) \( -428 \beta q^{11} \) \( -45505 q^{13} \) \( + 1967 \beta q^{14} \) \( + 148480 q^{16} \) \( -2028 \beta q^{17} \) \( + 152399 q^{19} \) \( + 17024 \beta q^{20} \) \( + 369792 q^{22} \) \( -4468 \beta q^{23} \) \( -286751 q^{25} \) \( -45505 \beta q^{26} \) \( -1195936 q^{28} \) \( -20024 \beta q^{29} \) \( -164350 q^{31} \) \( + 58368 \beta q^{32} \) \( + 1752192 q^{34} \) \( -55076 \beta q^{35} \) \( -663937 q^{37} \) \( + 152399 \beta q^{38} \) \( -8515584 q^{40} \) \( -31912 \beta q^{41} \) \( + 575330 q^{43} \) \( + 260224 \beta q^{44} \) \( + 3860352 q^{46} \) \( -314156 \beta q^{47} \) \( -1895712 q^{49} \) \( -286751 \beta q^{50} \) \( + 27667040 q^{52} \) \( + 353016 \beta q^{53} \) \( -10354176 q^{55} \) \( -692384 \beta q^{56} \) \( + 17300736 q^{58} \) \( -171460 \beta q^{59} \) \( -19212961 q^{61} \) \( -164350 \beta q^{62} \) \( -12419072 q^{64} \) \( + 1274140 \beta q^{65} \) \( -598033 q^{67} \) \( + 1233024 \beta q^{68} \) \( + 47585664 q^{70} \) \( + 995856 \beta q^{71} \) \( + 12850175 q^{73} \) \( -663937 \beta q^{74} \) \( -92658592 q^{76} \) \( -841876 \beta q^{77} \) \( -23584657 q^{79} \) \( -4157440 \beta q^{80} \) \( + 27571968 q^{82} \) \( + 1138024 \beta q^{83} \) \( -49061376 q^{85} \) \( + 575330 \beta q^{86} \) \( -130166784 q^{88} \) \( -962268 \beta q^{89} \) \( -89508335 q^{91} \) \( + 2716544 \beta q^{92} \) \( + 271430784 q^{94} \) \( -4267172 \beta q^{95} \) \( + 136489631 q^{97} \) \( -1895712 \beta q^{98} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut -\mathstrut 1216q^{4} \) \(\mathstrut +\mathstrut 3934q^{7} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut -\mathstrut 1216q^{4} \) \(\mathstrut +\mathstrut 3934q^{7} \) \(\mathstrut +\mathstrut 48384q^{10} \) \(\mathstrut -\mathstrut 91010q^{13} \) \(\mathstrut +\mathstrut 296960q^{16} \) \(\mathstrut +\mathstrut 304798q^{19} \) \(\mathstrut +\mathstrut 739584q^{22} \) \(\mathstrut -\mathstrut 573502q^{25} \) \(\mathstrut -\mathstrut 2391872q^{28} \) \(\mathstrut -\mathstrut 328700q^{31} \) \(\mathstrut +\mathstrut 3504384q^{34} \) \(\mathstrut -\mathstrut 1327874q^{37} \) \(\mathstrut -\mathstrut 17031168q^{40} \) \(\mathstrut +\mathstrut 1150660q^{43} \) \(\mathstrut +\mathstrut 7720704q^{46} \) \(\mathstrut -\mathstrut 3791424q^{49} \) \(\mathstrut +\mathstrut 55334080q^{52} \) \(\mathstrut -\mathstrut 20708352q^{55} \) \(\mathstrut +\mathstrut 34601472q^{58} \) \(\mathstrut -\mathstrut 38425922q^{61} \) \(\mathstrut -\mathstrut 24838144q^{64} \) \(\mathstrut -\mathstrut 1196066q^{67} \) \(\mathstrut +\mathstrut 95171328q^{70} \) \(\mathstrut +\mathstrut 25700350q^{73} \) \(\mathstrut -\mathstrut 185317184q^{76} \) \(\mathstrut -\mathstrut 47169314q^{79} \) \(\mathstrut +\mathstrut 55143936q^{82} \) \(\mathstrut -\mathstrut 98122752q^{85} \) \(\mathstrut -\mathstrut 260333568q^{88} \) \(\mathstrut -\mathstrut 179016670q^{91} \) \(\mathstrut +\mathstrut 542861568q^{94} \) \(\mathstrut +\mathstrut 272979262q^{97} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
29.3939i 0 −608.000 823.029i 0 1967.00 10346.6i 0 24192.0
26.2 29.3939i 0 −608.000 823.029i 0 1967.00 10346.6i 0 24192.0
\(n\): 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
3.b Odd 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{2} \) \(\mathstrut +\mathstrut 864 \) acting on \(S_{9}^{\mathrm{new}}(27, [\chi])\).