Properties

Label 27.5.b.c
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: \(\Q(\sqrt{-1}) \)
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.

\(f(q)\) \(=\) \( q\) \( + \beta q^{2} \) \( + 7 q^{4} \) \( + 11 \beta q^{5} \) \( -19 q^{7} \) \( + 23 \beta q^{8} \) \(+O(q^{10})\) \( q\) \( + \beta q^{2} \) \( + 7 q^{4} \) \( + 11 \beta q^{5} \) \( -19 q^{7} \) \( + 23 \beta q^{8} \) \( -99 q^{10} \) \( -41 \beta q^{11} \) \( + 302 q^{13} \) \( -19 \beta q^{14} \) \( -95 q^{16} \) \( -138 \beta q^{17} \) \( -304 q^{19} \) \( + 77 \beta q^{20} \) \( + 369 q^{22} \) \( -100 \beta q^{23} \) \( -464 q^{25} \) \( + 302 \beta q^{26} \) \( -133 q^{28} \) \( + 226 \beta q^{29} \) \( + 239 q^{31} \) \( + 273 \beta q^{32} \) \( + 1242 q^{34} \) \( -209 \beta q^{35} \) \( + 740 q^{37} \) \( -304 \beta q^{38} \) \( -2277 q^{40} \) \( -76 \beta q^{41} \) \( -982 q^{43} \) \( -287 \beta q^{44} \) \( + 900 q^{46} \) \( -722 \beta q^{47} \) \( -2040 q^{49} \) \( -464 \beta q^{50} \) \( + 2114 q^{52} \) \( + 531 \beta q^{53} \) \( + 4059 q^{55} \) \( -437 \beta q^{56} \) \( -2034 q^{58} \) \( + 974 \beta q^{59} \) \( -316 q^{61} \) \( + 239 \beta q^{62} \) \( -3977 q^{64} \) \( + 3322 \beta q^{65} \) \( + 4622 q^{67} \) \( -966 \beta q^{68} \) \( + 1881 q^{70} \) \( -606 \beta q^{71} \) \( -3031 q^{73} \) \( + 740 \beta q^{74} \) \( -2128 q^{76} \) \( + 779 \beta q^{77} \) \( -10450 q^{79} \) \( -1045 \beta q^{80} \) \( + 684 q^{82} \) \( -4211 \beta q^{83} \) \( + 13662 q^{85} \) \( -982 \beta q^{86} \) \( + 8487 q^{88} \) \( + 2334 \beta q^{89} \) \( -5738 q^{91} \) \( -700 \beta q^{92} \) \( + 6498 q^{94} \) \( -3344 \beta q^{95} \) \( -6517 q^{97} \) \( -2040 \beta q^{98} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut +\mathstrut 14q^{4} \) \(\mathstrut -\mathstrut 38q^{7} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut +\mathstrut 14q^{4} \) \(\mathstrut -\mathstrut 38q^{7} \) \(\mathstrut -\mathstrut 198q^{10} \) \(\mathstrut +\mathstrut 604q^{13} \) \(\mathstrut -\mathstrut 190q^{16} \) \(\mathstrut -\mathstrut 608q^{19} \) \(\mathstrut +\mathstrut 738q^{22} \) \(\mathstrut -\mathstrut 928q^{25} \) \(\mathstrut -\mathstrut 266q^{28} \) \(\mathstrut +\mathstrut 478q^{31} \) \(\mathstrut +\mathstrut 2484q^{34} \) \(\mathstrut +\mathstrut 1480q^{37} \) \(\mathstrut -\mathstrut 4554q^{40} \) \(\mathstrut -\mathstrut 1964q^{43} \) \(\mathstrut +\mathstrut 1800q^{46} \) \(\mathstrut -\mathstrut 4080q^{49} \) \(\mathstrut +\mathstrut 4228q^{52} \) \(\mathstrut +\mathstrut 8118q^{55} \) \(\mathstrut -\mathstrut 4068q^{58} \) \(\mathstrut -\mathstrut 632q^{61} \) \(\mathstrut -\mathstrut 7954q^{64} \) \(\mathstrut +\mathstrut 9244q^{67} \) \(\mathstrut +\mathstrut 3762q^{70} \) \(\mathstrut -\mathstrut 6062q^{73} \) \(\mathstrut -\mathstrut 4256q^{76} \) \(\mathstrut -\mathstrut 20900q^{79} \) \(\mathstrut +\mathstrut 1368q^{82} \) \(\mathstrut +\mathstrut 27324q^{85} \) \(\mathstrut +\mathstrut 16974q^{88} \) \(\mathstrut -\mathstrut 11476q^{91} \) \(\mathstrut +\mathstrut 12996q^{94} \) \(\mathstrut -\mathstrut 13034q^{97} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 7.00000 33.0000i 0 −19.0000 69.0000i 0 −99.0000
26.2 3.00000i 0 7.00000 33.0000i 0 −19.0000 69.0000i 0 −99.0000
\(n\): 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 \(S_{5}^{\mathrm{new}}(27, [\chi])\).