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Properties

Label 27.10.c.a
Level $27$
Weight $10$
Character orbit 27.c
Analytic conductor $13.906$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [27,10,Mod(10,27)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(27, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("27.10");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 27 = 3^{3} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 27.c (of order \(3\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(13.9059675764\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 8 x^{15} + 1984 x^{14} - 13748 x^{13} + 1552498 x^{12} - 9136628 x^{11} + 609566956 x^{10} + \cdots + 13\!\cdots\!25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{14}\cdot 3^{40}\cdot 17^{2} \)
Twist minimal: no (minimal twist has level 9)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (2 \beta_{9} + \beta_{8} - 2) q^{2} + ( - \beta_{10} - 224 \beta_{9} + \cdots + 2 \beta_1) q^{4}+ \cdots + ( - \beta_{5} - \beta_{4} - \beta_{3} + \cdots + 930) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (2 \beta_{9} + \beta_{8} - 2) q^{2} + ( - \beta_{10} - 224 \beta_{9} + \cdots + 2 \beta_1) q^{4}+ \cdots + ( - 18719 \beta_{7} - 35179 \beta_{6} + \cdots - 302145943) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 15 q^{2} - 1793 q^{4} - 453 q^{5} - 343 q^{7} + 14478 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 15 q^{2} - 1793 q^{4} - 453 q^{5} - 343 q^{7} + 14478 q^{8} + 1020 q^{10} - 99150 q^{11} + 32435 q^{13} - 394824 q^{14} - 328193 q^{16} + 831078 q^{17} - 170554 q^{19} - 1855164 q^{20} + 529359 q^{22} - 1064559 q^{23} - 2293229 q^{25} - 2436312 q^{26} + 1225724 q^{28} + 1309053 q^{29} - 2359819 q^{31} - 5760063 q^{32} + 981801 q^{34} + 31066554 q^{35} + 16391516 q^{37} - 39490203 q^{38} - 16760496 q^{40} - 54747318 q^{41} + 15249608 q^{43} + 332509926 q^{44} + 2390520 q^{46} - 156295545 q^{47} + 15239583 q^{49} - 315590163 q^{50} - 19773358 q^{52} + 525516228 q^{53} - 7579770 q^{55} - 470339790 q^{56} + 55408560 q^{58} - 307774074 q^{59} + 69192125 q^{61} + 914436924 q^{62} - 403588478 q^{64} - 482470359 q^{65} + 14328044 q^{67} - 915409575 q^{68} - 229271934 q^{70} + 1239601392 q^{71} + 598613198 q^{73} - 1022736000 q^{74} + 119954093 q^{76} - 717995541 q^{77} + 30257531 q^{79} + 2927826528 q^{80} - 202376022 q^{82} - 1176168291 q^{83} + 4818366 q^{85} - 1426944009 q^{86} + 911312427 q^{88} + 3317041296 q^{89} - 739230122 q^{91} + 76813998 q^{92} - 1954316784 q^{94} + 391400652 q^{95} - 267311278 q^{97} - 4827300318 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 8 x^{15} + 1984 x^{14} - 13748 x^{13} + 1552498 x^{12} - 9136628 x^{11} + 609566956 x^{10} + \cdots + 13\!\cdots\!25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 5356486831063 \nu^{14} + 37495407817441 \nu^{13} + \cdots + 13\!\cdots\!25 ) / 31\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 5356486831063 \nu^{14} + 37495407817441 \nu^{13} + \cdots + 23\!\cdots\!25 ) / 31\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 31\!\cdots\!83 \nu^{14} + \cdots + 22\!\cdots\!75 ) / 37\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 17\!\cdots\!57 \nu^{14} + \cdots - 36\!\cdots\!25 ) / 15\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 13\!\cdots\!53 \nu^{14} + \cdots - 20\!\cdots\!75 ) / 55\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 45\!\cdots\!87 \nu^{14} + \cdots - 18\!\cdots\!75 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 34\!\cdots\!21 \nu^{14} + \cdots + 30\!\cdots\!75 ) / 55\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 96\!\cdots\!97 \nu^{15} + \cdots + 19\!\cdots\!75 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 19\!\cdots\!94 \nu^{15} + \cdots - 44\!\cdots\!25 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 14\!\cdots\!69 \nu^{15} + \cdots + 32\!\cdots\!00 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 16\!\cdots\!35 \nu^{15} + \cdots - 44\!\cdots\!75 ) / 41\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 25\!\cdots\!04 \nu^{15} + \cdots - 68\!\cdots\!25 ) / 41\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 35\!\cdots\!88 \nu^{15} + \cdots + 91\!\cdots\!25 ) / 41\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 42\!\cdots\!51 \nu^{15} + \cdots - 94\!\cdots\!25 ) / 41\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 11\!\cdots\!21 \nu^{15} + \cdots + 25\!\cdots\!75 ) / 41\!\cdots\!00 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{9} + 2\beta_{8} - \beta _1 + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{9} + 2\beta_{8} + \beta_{2} - 2\beta _1 - 732 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2 \beta_{14} - 2 \beta_{12} - 2 \beta_{11} + 3 \beta_{10} + 650 \beta_{9} - 2459 \beta_{8} + \beta_{5} + \cdots - 3619 ) / 9 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 4 \beta_{14} - 4 \beta_{12} - 4 \beta_{11} + 6 \beta_{10} + 1297 \beta_{9} - 4924 \beta_{8} + \cdots + 897597 ) / 9 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 260 \beta_{15} - 4609 \beta_{14} + 20 \beta_{13} + 3301 \beta_{12} + 5177 \beta_{11} - 18299 \beta_{10} + \cdots + 8283966 ) / 27 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 260 \beta_{15} - 4619 \beta_{14} + 20 \beta_{13} + 3311 \beta_{12} + 5187 \beta_{11} - 18314 \beta_{10} + \cdots - 416029317 ) / 9 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 217192 \beta_{15} + 2838597 \beta_{14} - 60904 \beta_{13} - 1616397 \beta_{12} - 3534069 \beta_{11} + \cdots - 6240494943 ) / 27 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 872408 \beta_{15} + 11419082 \beta_{14} - 243896 \beta_{13} - 6511970 \beta_{12} + \cdots + 611798904726 ) / 27 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 139152644 \beta_{15} - 1628842083 \beta_{14} + 62926964 \beta_{13} + 803003703 \beta_{12} + \cdots + 4412210621253 ) / 27 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 702311740 \beta_{15} - 8229950589 \beta_{14} + 316464460 \beta_{13} + 4063927881 \beta_{12} + \cdots - 308392716149673 ) / 27 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 81064649544 \beta_{15} + 907087393079 \beta_{14} - 52356195816 \beta_{13} - 413813942399 \beta_{12} + \cdots - 29\!\cdots\!04 ) / 27 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 494127738296 \beta_{15} + 5533242534858 \beta_{14} - 317622309560 \beta_{13} - 2527694527314 \beta_{12} + \cdots + 15\!\cdots\!63 ) / 27 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 14959139996928 \beta_{15} - 166029476861446 \beta_{14} + 13187437396064 \beta_{13} + \cdots + 63\!\cdots\!95 ) / 9 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 321659706889952 \beta_{15} + \cdots - 81\!\cdots\!52 ) / 27 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 80\!\cdots\!36 \beta_{15} + \cdots - 39\!\cdots\!67 ) / 9 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(-\beta_{9}\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
10.1
0.500000 − 22.2094i
0.500000 − 19.2639i
0.500000 − 9.36376i
0.500000 − 3.55897i
0.500000 − 0.103648i
0.500000 + 13.2694i
0.500000 + 15.6774i
0.500000 + 23.8209i
0.500000 + 22.2094i
0.500000 + 19.2639i
0.500000 + 9.36376i
0.500000 + 3.55897i
0.500000 + 0.103648i
0.500000 − 13.2694i
0.500000 − 15.6774i
0.500000 − 23.8209i
−19.9839 − 34.6131i 0 −542.712 + 940.005i −1103.58 + 1911.45i 0 −2172.76 − 3763.33i 22918.5 0 88215.0
10.2 −17.4330 − 30.1949i 0 −351.819 + 609.369i 1003.26 − 1737.70i 0 −2433.94 − 4215.71i 6681.68 0 −69959.4
10.3 −8.85925 − 15.3447i 0 99.0272 − 171.520i −369.939 + 640.753i 0 3013.67 + 5219.83i −12581.1 0 13109.5
10.4 −3.83216 − 6.63749i 0 226.629 − 392.533i 1105.48 − 1914.74i 0 3116.58 + 5398.07i −7398.05 0 −16945.4
10.5 −0.839762 − 1.45451i 0 254.590 − 440.962i −769.303 + 1332.47i 0 −2828.47 − 4899.05i −1715.10 0 2584.13
10.6 10.7416 + 18.6050i 0 25.2357 − 43.7094i −22.4201 + 38.8327i 0 −4672.78 − 8093.49i 12083.7 0 −963.310
10.7 12.8270 + 22.2170i 0 −73.0637 + 126.550i 353.226 − 611.805i 0 2284.21 + 3956.36i 9386.09 0 18123.3
10.8 19.8795 + 34.4322i 0 −534.387 + 925.585i −423.223 + 733.045i 0 3521.99 + 6100.27i −22136.7 0 −33653.8
19.1 −19.9839 + 34.6131i 0 −542.712 − 940.005i −1103.58 − 1911.45i 0 −2172.76 + 3763.33i 22918.5 0 88215.0
19.2 −17.4330 + 30.1949i 0 −351.819 − 609.369i 1003.26 + 1737.70i 0 −2433.94 + 4215.71i 6681.68 0 −69959.4
19.3 −8.85925 + 15.3447i 0 99.0272 + 171.520i −369.939 − 640.753i 0 3013.67 − 5219.83i −12581.1 0 13109.5
19.4 −3.83216 + 6.63749i 0 226.629 + 392.533i 1105.48 + 1914.74i 0 3116.58 − 5398.07i −7398.05 0 −16945.4
19.5 −0.839762 + 1.45451i 0 254.590 + 440.962i −769.303 − 1332.47i 0 −2828.47 + 4899.05i −1715.10 0 2584.13
19.6 10.7416 − 18.6050i 0 25.2357 + 43.7094i −22.4201 − 38.8327i 0 −4672.78 + 8093.49i 12083.7 0 −963.310
19.7 12.8270 − 22.2170i 0 −73.0637 − 126.550i 353.226 + 611.805i 0 2284.21 − 3956.36i 9386.09 0 18123.3
19.8 19.8795 − 34.4322i 0 −534.387 − 925.585i −423.223 − 733.045i 0 3521.99 − 6100.27i −22136.7 0 −33653.8
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 10.8
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 27.10.c.a 16
3.b odd 2 1 9.10.c.a ✓ 16
9.c even 3 1 inner 27.10.c.a 16
9.c even 3 1 81.10.a.d 8
9.d odd 6 1 9.10.c.a ✓ 16
9.d odd 6 1 81.10.a.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9.10.c.a ✓ 16 3.b odd 2 1
9.10.c.a ✓ 16 9.d odd 6 1
27.10.c.a 16 1.a even 1 1 trivial
27.10.c.a 16 9.c even 3 1 inner
81.10.a.c 8 9.d odd 6 1
81.10.a.d 8 9.c even 3 1

Hecke kernels

This newform subspace is the entire newspace \(S_{10}^{\mathrm{new}}(27, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + \cdots + 48\!\cdots\!16 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 89\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 18\!\cdots\!96 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 21\!\cdots\!41 \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 27\!\cdots\!36 \) Copy content Toggle raw display
$17$ \( (T^{8} + \cdots - 82\!\cdots\!84)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + \cdots - 88\!\cdots\!00)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 10\!\cdots\!24 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 75\!\cdots\!56 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 69\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots + 99\!\cdots\!68)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 66\!\cdots\!25 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 14\!\cdots\!21 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 22\!\cdots\!44 \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots + 11\!\cdots\!88)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 15\!\cdots\!09 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 77\!\cdots\!04 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 88\!\cdots\!69 \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots - 16\!\cdots\!84)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots - 10\!\cdots\!04)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 38\!\cdots\!84 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 79\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots - 31\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 20\!\cdots\!01 \) Copy content Toggle raw display
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