Properties

Label 27.4.a.c
Level 27
Weight 4
Character orbit 27.a
Self dual Yes
Analytic conductor 1.593
Analytic rank 0
Dimension 2
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 27 = 3^{3} \)
Weight: \( k \) = \( 4 \)
Character orbit: \([\chi]\) = 27.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(1.59305157016\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
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 \(\beta = 3\sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + \beta q^{2} \) \( + 10 q^{4} \) \( -4 \beta q^{5} \) \( + 11 q^{7} \) \( + 2 \beta q^{8} \) \(+O(q^{10})\) \( q\) \( + \beta q^{2} \) \( + 10 q^{4} \) \( -4 \beta q^{5} \) \( + 11 q^{7} \) \( + 2 \beta q^{8} \) \( -72 q^{10} \) \( + 4 \beta q^{11} \) \( + 29 q^{13} \) \( + 11 \beta q^{14} \) \( -44 q^{16} \) \( -12 \beta q^{17} \) \( + 29 q^{19} \) \( -40 \beta q^{20} \) \( + 72 q^{22} \) \( + 20 \beta q^{23} \) \( + 163 q^{25} \) \( + 29 \beta q^{26} \) \( + 110 q^{28} \) \( + 64 \beta q^{29} \) \( -268 q^{31} \) \( -60 \beta q^{32} \) \( -216 q^{34} \) \( -44 \beta q^{35} \) \( + 83 q^{37} \) \( + 29 \beta q^{38} \) \( -144 q^{40} \) \( -64 \beta q^{41} \) \( -232 q^{43} \) \( + 40 \beta q^{44} \) \( + 360 q^{46} \) \( -92 \beta q^{47} \) \( -222 q^{49} \) \( + 163 \beta q^{50} \) \( + 290 q^{52} \) \( + 72 \beta q^{53} \) \( -288 q^{55} \) \( + 22 \beta q^{56} \) \( + 1152 q^{58} \) \( + 68 \beta q^{59} \) \( + 767 q^{61} \) \( -268 \beta q^{62} \) \( -728 q^{64} \) \( -116 \beta q^{65} \) \( -511 q^{67} \) \( -120 \beta q^{68} \) \( -792 q^{70} \) \( + 168 \beta q^{71} \) \( + 137 q^{73} \) \( + 83 \beta q^{74} \) \( + 290 q^{76} \) \( + 44 \beta q^{77} \) \( -475 q^{79} \) \( + 176 \beta q^{80} \) \( -1152 q^{82} \) \( + 136 \beta q^{83} \) \( + 864 q^{85} \) \( -232 \beta q^{86} \) \( + 144 q^{88} \) \( -60 \beta q^{89} \) \( + 319 q^{91} \) \( + 200 \beta q^{92} \) \( -1656 q^{94} \) \( -116 \beta q^{95} \) \( + 821 q^{97} \) \( -222 \beta q^{98} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut +\mathstrut 20q^{4} \) \(\mathstrut +\mathstrut 22q^{7} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut +\mathstrut 20q^{4} \) \(\mathstrut +\mathstrut 22q^{7} \) \(\mathstrut -\mathstrut 144q^{10} \) \(\mathstrut +\mathstrut 58q^{13} \) \(\mathstrut -\mathstrut 88q^{16} \) \(\mathstrut +\mathstrut 58q^{19} \) \(\mathstrut +\mathstrut 144q^{22} \) \(\mathstrut +\mathstrut 326q^{25} \) \(\mathstrut +\mathstrut 220q^{28} \) \(\mathstrut -\mathstrut 536q^{31} \) \(\mathstrut -\mathstrut 432q^{34} \) \(\mathstrut +\mathstrut 166q^{37} \) \(\mathstrut -\mathstrut 288q^{40} \) \(\mathstrut -\mathstrut 464q^{43} \) \(\mathstrut +\mathstrut 720q^{46} \) \(\mathstrut -\mathstrut 444q^{49} \) \(\mathstrut +\mathstrut 580q^{52} \) \(\mathstrut -\mathstrut 576q^{55} \) \(\mathstrut +\mathstrut 2304q^{58} \) \(\mathstrut +\mathstrut 1534q^{61} \) \(\mathstrut -\mathstrut 1456q^{64} \) \(\mathstrut -\mathstrut 1022q^{67} \) \(\mathstrut -\mathstrut 1584q^{70} \) \(\mathstrut +\mathstrut 274q^{73} \) \(\mathstrut +\mathstrut 580q^{76} \) \(\mathstrut -\mathstrut 950q^{79} \) \(\mathstrut -\mathstrut 2304q^{82} \) \(\mathstrut +\mathstrut 1728q^{85} \) \(\mathstrut +\mathstrut 288q^{88} \) \(\mathstrut +\mathstrut 638q^{91} \) \(\mathstrut -\mathstrut 3312q^{94} \) \(\mathstrut +\mathstrut 1642q^{97} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
−1.41421
1.41421
−4.24264 0 10.0000 16.9706 0 11.0000 −8.48528 0 −72.0000
1.2 4.24264 0 10.0000 −16.9706 0 11.0000 8.48528 0 −72.0000
\(n\): 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

Atkin-Lehner signs

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

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{2} \) \(\mathstrut -\mathstrut 18 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(27))\).