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 \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.59305157015\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-11})\)
Defining polynomial: \(x^{4} - x^{3} - 2 x^{2} - 3 x + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 9)
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})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 3 q^{2} - 5 q^{4} + 15 q^{5} - 7 q^{7} - 66 q^{8} + O(q^{10}) \) \( 4 q + 3 q^{2} - 5 q^{4} + 15 q^{5} - 7 q^{7} - 66 q^{8} + 12 q^{10} + 66 q^{11} + 11 q^{13} + 60 q^{14} + 7 q^{16} - 198 q^{17} - 154 q^{19} - 12 q^{20} + 33 q^{22} + 33 q^{23} + 121 q^{25} + 528 q^{26} + 332 q^{28} - 51 q^{29} - 43 q^{31} - 423 q^{32} - 297 q^{34} - 6 q^{35} - 100 q^{37} - 561 q^{38} - 264 q^{40} + 132 q^{41} - 88 q^{43} + 462 q^{44} - 528 q^{46} + 399 q^{47} + 513 q^{49} - 429 q^{50} + 770 q^{52} - 108 q^{53} + 1254 q^{55} + 66 q^{56} + 60 q^{58} + 798 q^{59} - 439 q^{61} - 228 q^{62} - 1454 q^{64} + 165 q^{65} - 988 q^{67} + 693 q^{68} - 318 q^{70} - 2736 q^{71} - 910 q^{73} + 816 q^{74} + 1529 q^{76} - 165 q^{77} + 803 q^{79} - 192 q^{80} + 3630 q^{82} + 813 q^{83} - 594 q^{85} + 33 q^{86} - 1221 q^{88} + 792 q^{89} - 1562 q^{91} - 858 q^{92} - 2100 q^{94} - 132 q^{95} - 736 q^{97} + 846 q^{98} + O(q^{100}) \)

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 Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 27.4.c.a 4
3.b odd 2 1 9.4.c.a 4
4.b odd 2 1 432.4.i.c 4
9.c even 3 1 inner 27.4.c.a 4
9.c even 3 1 81.4.a.a 2
9.d odd 6 1 9.4.c.a 4
9.d odd 6 1 81.4.a.d 2
12.b even 2 1 144.4.i.c 4
15.d odd 2 1 225.4.e.b 4
15.e even 4 2 225.4.k.b 8
36.f odd 6 1 432.4.i.c 4
36.f odd 6 1 1296.4.a.i 2
36.h even 6 1 144.4.i.c 4
36.h even 6 1 1296.4.a.u 2
45.h odd 6 1 225.4.e.b 4
45.h odd 6 1 2025.4.a.g 2
45.j even 6 1 2025.4.a.n 2
45.l even 12 2 225.4.k.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9.4.c.a 4 3.b odd 2 1
9.4.c.a 4 9.d odd 6 1
27.4.c.a 4 1.a even 1 1 trivial
27.4.c.a 4 9.c even 3 1 inner
81.4.a.a 2 9.c even 3 1
81.4.a.d 2 9.d odd 6 1
144.4.i.c 4 12.b even 2 1
144.4.i.c 4 36.h even 6 1
225.4.e.b 4 15.d odd 2 1
225.4.e.b 4 45.h odd 6 1
225.4.k.b 8 15.e even 4 2
225.4.k.b 8 45.l even 12 2
432.4.i.c 4 4.b odd 2 1
432.4.i.c 4 36.f odd 6 1
1296.4.a.i 2 36.f odd 6 1
1296.4.a.u 2 36.h even 6 1
2025.4.a.g 2 45.h odd 6 1
2025.4.a.n 2 45.j even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(27, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 36 + 18 T + 15 T^{2} - 3 T^{3} + T^{4} \)
$3$ \( T^{4} \)
$5$ \( 2304 - 720 T + 177 T^{2} - 15 T^{3} + T^{4} \)
$7$ \( 3844 - 434 T + 111 T^{2} + 7 T^{3} + T^{4} \)
$11$ \( 314721 - 37026 T + 3795 T^{2} - 66 T^{3} + T^{4} \)
$13$ \( 3334276 + 20086 T + 1947 T^{2} - 11 T^{3} + T^{4} \)
$17$ \( ( 1782 + 99 T + T^{2} )^{2} \)
$19$ \( ( -4532 + 77 T + T^{2} )^{2} \)
$23$ \( 7322436 + 89298 T + 3795 T^{2} - 33 T^{3} + T^{4} \)
$29$ \( 412164 + 32742 T + 1959 T^{2} + 51 T^{3} + T^{4} \)
$31$ \( 150544 + 16684 T + 1461 T^{2} + 43 T^{3} + T^{4} \)
$37$ \( ( -23432 + 50 T + T^{2} )^{2} \)
$41$ \( 5606565129 + 9883764 T + 92301 T^{2} - 132 T^{3} + T^{4} \)
$43$ \( 2686321 + 144232 T + 6105 T^{2} + 88 T^{3} + T^{4} \)
$47$ \( 813276324 + 11378682 T + 187719 T^{2} - 399 T^{3} + T^{4} \)
$53$ \( ( -215784 + 54 T + T^{2} )^{2} \)
$59$ \( 43678881 - 5273982 T + 630195 T^{2} - 798 T^{3} + T^{4} \)
$61$ \( 1829786176 - 18778664 T + 235497 T^{2} + 439 T^{3} + T^{4} \)
$67$ \( 43305193801 + 205601812 T + 768045 T^{2} + 988 T^{3} + T^{4} \)
$71$ \( ( 296784 + 1368 T + T^{2} )^{2} \)
$73$ \( ( -435398 + 455 T + T^{2} )^{2} \)
$79$ \( 392522298256 + 503092348 T + 1271325 T^{2} - 803 T^{3} + T^{4} \)
$83$ \( 5010940944 - 57550644 T + 590181 T^{2} - 813 T^{3} + T^{4} \)
$89$ \( ( -3564 - 396 T + T^{2} )^{2} \)
$97$ \( 2459267281 + 36498976 T + 492105 T^{2} + 736 T^{3} + T^{4} \)
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