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 \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 3 \beta q^{2} + 10 q^{4} -12 \beta q^{5} + 11 q^{7} + 6 \beta q^{8} +O(q^{10})\) \( q + 3 \beta q^{2} + 10 q^{4} -12 \beta q^{5} + 11 q^{7} + 6 \beta q^{8} -72 q^{10} + 12 \beta q^{11} + 29 q^{13} + 33 \beta q^{14} -44 q^{16} -36 \beta q^{17} + 29 q^{19} -120 \beta q^{20} + 72 q^{22} + 60 \beta q^{23} + 163 q^{25} + 87 \beta q^{26} + 110 q^{28} + 192 \beta q^{29} -268 q^{31} -180 \beta q^{32} -216 q^{34} -132 \beta q^{35} + 83 q^{37} + 87 \beta q^{38} -144 q^{40} -192 \beta q^{41} -232 q^{43} + 120 \beta q^{44} + 360 q^{46} -276 \beta q^{47} -222 q^{49} + 489 \beta q^{50} + 290 q^{52} + 216 \beta q^{53} -288 q^{55} + 66 \beta q^{56} + 1152 q^{58} + 204 \beta q^{59} + 767 q^{61} -804 \beta q^{62} -728 q^{64} -348 \beta q^{65} -511 q^{67} -360 \beta q^{68} -792 q^{70} + 504 \beta q^{71} + 137 q^{73} + 249 \beta q^{74} + 290 q^{76} + 132 \beta q^{77} -475 q^{79} + 528 \beta q^{80} -1152 q^{82} + 408 \beta q^{83} + 864 q^{85} -696 \beta q^{86} + 144 q^{88} -180 \beta q^{89} + 319 q^{91} + 600 \beta q^{92} -1656 q^{94} -348 \beta q^{95} + 821 q^{97} -666 \beta q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 20q^{4} + 22q^{7} + O(q^{10}) \) \( 2q + 20q^{4} + 22q^{7} - 144q^{10} + 58q^{13} - 88q^{16} + 58q^{19} + 144q^{22} + 326q^{25} + 220q^{28} - 536q^{31} - 432q^{34} + 166q^{37} - 288q^{40} - 464q^{43} + 720q^{46} - 444q^{49} + 580q^{52} - 576q^{55} + 2304q^{58} + 1534q^{61} - 1456q^{64} - 1022q^{67} - 1584q^{70} + 274q^{73} + 580q^{76} - 950q^{79} - 2304q^{82} + 1728q^{85} + 288q^{88} + 638q^{91} - 3312q^{94} + 1642q^{97} + O(q^{100}) \)

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 \( T_{2}^{2} - 18 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(27))\).