Properties

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

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Newspace parameters

Level: \( N \) = \( 27 = 3^{3} \)
Weight: \( k \) = \( 4 \)
Character orbit: \([\chi]\) = 27.c (of order \(3\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(1.59305157016\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-11})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \beta_{1} + \beta_{3} ) q^{2} + ( -4 + \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{4} + ( 7 - 8 \beta_{1} - \beta_{2} + \beta_{3} ) q^{5} + ( -2 \beta_{1} - 3 \beta_{3} ) q^{7} + ( -16 + \beta_{2} ) q^{8} +O(q^{10})\) \( q + ( \beta_{1} + \beta_{3} ) q^{2} + ( -4 + \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{4} + ( 7 - 8 \beta_{1} - \beta_{2} + \beta_{3} ) q^{5} + ( -2 \beta_{1} - 3 \beta_{3} ) q^{7} + ( -16 + \beta_{2} ) q^{8} + ( 6 + 6 \beta_{2} ) q^{10} + ( 37 \beta_{1} - 8 \beta_{3} ) q^{11} + ( 13 + 2 \beta_{1} + 15 \beta_{2} - 15 \beta_{3} ) q^{13} + ( 34 - 26 \beta_{1} + 8 \beta_{2} - 8 \beta_{3} ) q^{14} + ( -\beta_{1} + 9 \beta_{3} ) q^{16} + ( -54 - 9 \beta_{2} ) q^{17} + ( -52 - 27 \beta_{2} ) q^{19} + ( -16 \beta_{1} + 20 \beta_{3} ) q^{20} + ( 6 - 27 \beta_{1} - 21 \beta_{2} + 21 \beta_{3} ) q^{22} + ( 7 - 26 \beta_{1} - 19 \beta_{2} + 19 \beta_{3} ) q^{23} + ( 53 \beta_{1} + 15 \beta_{3} ) q^{25} + ( 146 + 28 \beta_{2} ) q^{26} + ( 92 + 18 \beta_{2} ) q^{28} + ( -26 \beta_{1} + \beta_{3} ) q^{29} + ( -23 + 20 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{31} + ( -216 + 207 \beta_{1} - 9 \beta_{2} + 9 \beta_{3} ) q^{32} + ( -117 \beta_{1} - 63 \beta_{3} ) q^{34} + ( -11 - 19 \beta_{2} ) q^{35} + ( 2 + 54 \beta_{2} ) q^{37} + ( -241 \beta_{1} - 79 \beta_{3} ) q^{38} + ( -120 + 144 \beta_{1} + 24 \beta_{2} - 24 \beta_{3} ) q^{40} + ( 115 - 17 \beta_{1} + 98 \beta_{2} - 98 \beta_{3} ) q^{41} + ( -47 \beta_{1} + 6 \beta_{3} ) q^{43} + ( 76 - 79 \beta_{2} ) q^{44} + ( -138 - 12 \beta_{2} ) q^{46} + ( 154 \beta_{1} + 91 \beta_{3} ) q^{47} + ( 246 - 267 \beta_{1} - 21 \beta_{2} + 21 \beta_{3} ) q^{49} + ( -256 + 173 \beta_{1} - 83 \beta_{2} + 83 \beta_{3} ) q^{50} + ( 358 \beta_{1} + 54 \beta_{3} ) q^{52} + ( 54 + 162 \beta_{2} ) q^{53} + ( 267 - 93 \beta_{2} ) q^{55} + ( 10 \beta_{1} + 46 \beta_{3} ) q^{56} + ( 42 - 18 \beta_{1} + 24 \beta_{2} - 24 \beta_{3} ) q^{58} + ( 331 - 467 \beta_{1} - 136 \beta_{2} + 136 \beta_{3} ) q^{59} + ( -272 \beta_{1} + 105 \beta_{3} ) q^{61} + ( -70 - 26 \beta_{2} ) q^{62} + ( -440 - 153 \beta_{2} ) q^{64} + ( 136 \beta_{1} - 107 \beta_{3} ) q^{65} + ( -527 + 461 \beta_{1} - 66 \beta_{2} + 66 \beta_{3} ) q^{67} + ( 432 - 261 \beta_{1} + 171 \beta_{2} - 171 \beta_{3} ) q^{68} + ( -144 \beta_{1} - 30 \beta_{3} ) q^{70} + ( -756 - 144 \beta_{2} ) q^{71} + ( -106 + 243 \beta_{2} ) q^{73} + ( 380 \beta_{1} + 56 \beta_{3} ) q^{74} + ( 856 - 673 \beta_{1} + 183 \beta_{2} - 183 \beta_{3} ) q^{76} + ( -47 + 118 \beta_{1} + 71 \beta_{2} - 71 \beta_{3} ) q^{77} + ( 556 \beta_{1} - 309 \beta_{3} ) q^{79} + ( -16 + 64 \beta_{2} ) q^{80} + ( 1014 + 213 \beta_{2} ) q^{82} + ( 460 \beta_{1} - 107 \beta_{3} ) q^{83} + ( -306 + 288 \beta_{1} - 18 \beta_{2} + 18 \beta_{3} ) q^{85} + ( 34 + \beta_{1} + 35 \beta_{2} - 35 \beta_{3} ) q^{86} + ( -693 \beta_{1} + 165 \beta_{3} ) q^{88} + ( 162 - 72 \beta_{2} ) q^{89} + ( -425 - 69 \beta_{2} ) q^{91} + ( -430 \beta_{1} + 2 \beta_{3} ) q^{92} + ( -1218 + 882 \beta_{1} - 336 \beta_{2} + 336 \beta_{3} ) q^{94} + ( -148 - 16 \beta_{1} - 164 \beta_{2} + 164 \beta_{3} ) q^{95} + ( -317 \beta_{1} - 102 \beta_{3} ) q^{97} + ( 324 + 225 \beta_{2} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 3q^{2} - 5q^{4} + 15q^{5} - 7q^{7} - 66q^{8} + O(q^{10}) \) \( 4q + 3q^{2} - 5q^{4} + 15q^{5} - 7q^{7} - 66q^{8} + 12q^{10} + 66q^{11} + 11q^{13} + 60q^{14} + 7q^{16} - 198q^{17} - 154q^{19} - 12q^{20} + 33q^{22} + 33q^{23} + 121q^{25} + 528q^{26} + 332q^{28} - 51q^{29} - 43q^{31} - 423q^{32} - 297q^{34} - 6q^{35} - 100q^{37} - 561q^{38} - 264q^{40} + 132q^{41} - 88q^{43} + 462q^{44} - 528q^{46} + 399q^{47} + 513q^{49} - 429q^{50} + 770q^{52} - 108q^{53} + 1254q^{55} + 66q^{56} + 60q^{58} + 798q^{59} - 439q^{61} - 228q^{62} - 1454q^{64} + 165q^{65} - 988q^{67} + 693q^{68} - 318q^{70} - 2736q^{71} - 910q^{73} + 816q^{74} + 1529q^{76} - 165q^{77} + 803q^{79} - 192q^{80} + 3630q^{82} + 813q^{83} - 594q^{85} + 33q^{86} - 1221q^{88} + 792q^{89} - 1562q^{91} - 858q^{92} - 2100q^{94} - 132q^{95} - 736q^{97} + 846q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 2 x^{2} - 3 x + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} + 2 \nu^{2} - 2 \nu - 3 \)\()/6\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{3} + \nu^{2} + 5 \nu \)\()/3\)
\(\beta_{3}\)\(=\)\((\)\( 2 \nu^{3} + \nu^{2} + 2 \nu - 9 \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{2} - 2 \beta_{1} + 2\)\()/3\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{3} + 2 \beta_{2} + 8 \beta_{1} + 1\)\()/3\)
\(\nu^{3}\)\(=\)\((\)\(4 \beta_{3} - 2 \beta_{2} - 2 \beta_{1} + 11\)\()/3\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/27\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1 + \beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
10.1
−1.18614 1.26217i
1.68614 + 0.396143i
−1.18614 + 1.26217i
1.68614 0.396143i
−0.686141 1.18843i 0 3.05842 5.29734i 5.18614 8.98266i 0 2.55842 + 4.43132i −19.3723 0 −14.2337
10.2 2.18614 + 3.78651i 0 −5.55842 + 9.62747i 2.31386 4.00772i 0 −6.05842 10.4935i −13.6277 0 20.2337
19.1 −0.686141 + 1.18843i 0 3.05842 + 5.29734i 5.18614 + 8.98266i 0 2.55842 4.43132i −19.3723 0 −14.2337
19.2 2.18614 3.78651i 0 −5.55842 9.62747i 2.31386 + 4.00772i 0 −6.05842 + 10.4935i −13.6277 0 20.2337
\(n\): 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
9.c Even 1 yes

Hecke kernels

There are no other newforms in \(S_{4}^{\mathrm{new}}(27, [\chi])\).