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Properties

Label 27.9.d.a
Level $27$
Weight $9$
Character orbit 27.d
Analytic conductor $10.999$
Analytic rank $0$
Dimension $14$
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] = [27,9,Mod(8,27)]
 
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([1]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("27.8");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 27 = 3^{3} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 27.d (of order \(6\), 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: \(10.9992224717\)
Analytic rank: \(0\)
Dimension: \(14\)
Relative dimension: \(7\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - x^{13} + 427 x^{12} - 1362 x^{11} + 135762 x^{10} - 371244 x^{9} + 18261508 x^{8} + \cdots + 872385888256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{30} \)
Twist minimal: no (minimal twist has level 9)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + (\beta_{6} + \beta_{4} + \cdots - 2 \beta_1) q^{4}+ \cdots + ( - \beta_{13} + \beta_{11} + \cdots + 688) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + (\beta_{6} + \beta_{4} + \cdots - 2 \beta_1) q^{4}+ \cdots + (1715 \beta_{13} - 7861 \beta_{12} + \cdots + 14058781) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 3 q^{2} + 767 q^{4} - 438 q^{5} + 922 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 14 q + 3 q^{2} + 767 q^{4} - 438 q^{5} + 922 q^{7} - 516 q^{10} + 28677 q^{11} + 1684 q^{13} - 120966 q^{14} - 65281 q^{16} - 269630 q^{19} - 539454 q^{20} + 61311 q^{22} + 1000452 q^{23} + 65177 q^{25} + 1075708 q^{28} - 3797682 q^{29} - 164132 q^{31} + 8461881 q^{32} + 654993 q^{34} - 1671668 q^{37} - 10967691 q^{38} + 613326 q^{40} + 10239447 q^{41} + 791815 q^{43} + 1189536 q^{46} - 31148628 q^{47} - 4826637 q^{49} + 63849453 q^{50} - 5552720 q^{52} + 8107476 q^{55} - 116638674 q^{56} + 14211822 q^{58} + 83493795 q^{59} - 5255600 q^{61} - 26813830 q^{64} - 69232992 q^{65} - 8288855 q^{67} + 77746743 q^{68} + 27813756 q^{70} - 36721682 q^{73} + 10383450 q^{74} - 42822959 q^{76} - 56158710 q^{77} - 32771822 q^{79} + 236099418 q^{82} + 198915996 q^{83} + 97486146 q^{85} - 146190669 q^{86} + 24955827 q^{88} - 201514504 q^{91} + 295365804 q^{92} - 36698244 q^{94} - 386813838 q^{95} + 127049161 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - x^{13} + 427 x^{12} - 1362 x^{11} + 135762 x^{10} - 371244 x^{9} + 18261508 x^{8} + \cdots + 872385888256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 44\!\cdots\!83 \nu^{13} + \cdots + 86\!\cdots\!64 ) / 10\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 44\!\cdots\!83 \nu^{13} + \cdots - 86\!\cdots\!64 ) / 10\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 14\!\cdots\!61 \nu^{13} + \cdots - 10\!\cdots\!32 ) / 15\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 12\!\cdots\!85 \nu^{13} + \cdots - 67\!\cdots\!92 ) / 18\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 36\!\cdots\!33 \nu^{13} + \cdots - 26\!\cdots\!32 ) / 21\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 88\!\cdots\!19 \nu^{13} + \cdots + 10\!\cdots\!44 ) / 26\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 53\!\cdots\!05 \nu^{13} + \cdots + 12\!\cdots\!44 ) / 13\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 11\!\cdots\!37 \nu^{13} + \cdots - 79\!\cdots\!12 ) / 18\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 41\!\cdots\!81 \nu^{13} + \cdots - 12\!\cdots\!24 ) / 62\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 24\!\cdots\!79 \nu^{13} + \cdots + 28\!\cdots\!16 ) / 21\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 43\!\cdots\!09 \nu^{13} + \cdots + 86\!\cdots\!48 ) / 21\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 26\!\cdots\!37 \nu^{13} + \cdots - 29\!\cdots\!44 ) / 13\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 54\!\cdots\!03 \nu^{13} + \cdots + 38\!\cdots\!32 ) / 21\!\cdots\!12 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{2} + \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{6} + 366\beta_{3} + 2\beta_{2} - 4\beta _1 - 366 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{13} + \beta_{11} + 3\beta_{5} + 15\beta_{4} - 1238\beta_{2} + 619\beta _1 + 2062 ) / 9 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 50 \beta_{13} + 20 \beta_{12} + 25 \beta_{11} - 13 \beta_{10} - 20 \beta_{9} - 40 \beta_{8} + \cdots + 25 ) / 9 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 1339 \beta_{13} - 2114 \beta_{11} + 3387 \beta_{10} + 564 \beta_{9} + 4 \beta_{8} - 556 \beta_{7} + \cdots - 3324965 ) / 27 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 10271 \beta_{13} - 8044 \beta_{12} + 2227 \beta_{11} + 9212 \beta_{8} - 1168 \beta_{7} - 8044 \beta_{6} + \cdots + 54524090 ) / 9 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 311218 \beta_{13} + 89924 \beta_{12} + 155609 \beta_{11} - 345405 \beta_{10} - 89924 \beta_{9} + \cdots + 155609 ) / 9 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 3370043 \beta_{13} - 4167354 \beta_{11} + 2936991 \beta_{10} + 2572732 \beta_{9} + 2984108 \beta_{8} + \cdots - 14382888997 ) / 9 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 50363121 \beta_{13} - 32491604 \beta_{12} + 17871517 \beta_{11} + 7451492 \beta_{8} + \cdots + 145305653358 ) / 9 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 2072576502 \beta_{13} + 776340924 \beta_{12} + 1036288251 \beta_{11} - 1034796031 \beta_{10} + \cdots + 1036288251 ) / 9 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 47435231227 \beta_{13} - 62771371898 \beta_{11} + 86985137055 \beta_{10} + 32099090556 \beta_{9} + \cdots - 142787474627381 ) / 27 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 312030918355 \beta_{13} - 230751738140 \beta_{12} + 81279180215 \beta_{11} + 242454175948 \beta_{8} + \cdots + 11\!\cdots\!62 ) / 9 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 9782332803842 \beta_{13} + 3388187097460 \beta_{12} + 4891166401921 \beta_{11} - 8394482887053 \beta_{10} + \cdots + 4891166401921 ) / 9 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(\beta_{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
8.1
−8.68602 − 15.0446i
−5.49482 − 9.51731i
−2.00397 − 3.47098i
−0.447645 − 0.775344i
4.05115 + 7.01679i
5.69757 + 9.86849i
7.38374 + 12.7890i
−8.68602 + 15.0446i
−5.49482 + 9.51731i
−2.00397 + 3.47098i
−0.447645 + 0.775344i
4.05115 − 7.01679i
5.69757 − 9.86849i
7.38374 − 12.7890i
−26.0581 − 15.0446i 0 324.682 + 562.365i −189.131 + 109.195i 0 1404.67 − 2432.96i − 11836.0i 0 6571.17
8.2 −16.4845 − 9.51731i 0 53.1583 + 92.0729i 46.9888 − 27.1290i 0 −921.012 + 1595.24i 2849.17i 0 −1032.78
8.3 −6.01192 − 3.47098i 0 −103.905 − 179.968i −331.396 + 191.331i 0 467.516 − 809.762i 3219.75i 0 2656.43
8.4 −1.34294 − 0.775344i 0 −126.798 − 219.620i 604.549 − 349.037i 0 −1124.45 + 1947.61i 790.223i 0 −1082.49
8.5 12.1534 + 7.01679i 0 −29.5292 − 51.1461i −676.216 + 390.413i 0 2168.61 − 3756.14i − 4421.40i 0 −10957.8
8.6 17.0927 + 9.86849i 0 66.7741 + 115.656i 896.557 − 517.627i 0 −21.9132 + 37.9547i − 2416.83i 0 20432.8
8.7 22.1512 + 12.7890i 0 199.117 + 344.881i −570.352 + 329.293i 0 −1512.41 + 2619.58i 3638.08i 0 −16845.3
17.1 −26.0581 + 15.0446i 0 324.682 − 562.365i −189.131 − 109.195i 0 1404.67 + 2432.96i 11836.0i 0 6571.17
17.2 −16.4845 + 9.51731i 0 53.1583 − 92.0729i 46.9888 + 27.1290i 0 −921.012 − 1595.24i − 2849.17i 0 −1032.78
17.3 −6.01192 + 3.47098i 0 −103.905 + 179.968i −331.396 − 191.331i 0 467.516 + 809.762i − 3219.75i 0 2656.43
17.4 −1.34294 + 0.775344i 0 −126.798 + 219.620i 604.549 + 349.037i 0 −1124.45 − 1947.61i − 790.223i 0 −1082.49
17.5 12.1534 − 7.01679i 0 −29.5292 + 51.1461i −676.216 − 390.413i 0 2168.61 + 3756.14i 4421.40i 0 −10957.8
17.6 17.0927 − 9.86849i 0 66.7741 − 115.656i 896.557 + 517.627i 0 −21.9132 − 37.9547i 2416.83i 0 20432.8
17.7 22.1512 − 12.7890i 0 199.117 − 344.881i −570.352 − 329.293i 0 −1512.41 − 2619.58i − 3638.08i 0 −16845.3
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 8.7
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 27.9.d.a 14
3.b odd 2 1 9.9.d.a ✓ 14
4.b odd 2 1 432.9.q.a 14
9.c even 3 1 9.9.d.a ✓ 14
9.c even 3 1 81.9.b.a 14
9.d odd 6 1 inner 27.9.d.a 14
9.d odd 6 1 81.9.b.a 14
12.b even 2 1 144.9.q.a 14
36.f odd 6 1 144.9.q.a 14
36.h even 6 1 432.9.q.a 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9.9.d.a ✓ 14 3.b odd 2 1
9.9.d.a ✓ 14 9.c even 3 1
27.9.d.a 14 1.a even 1 1 trivial
27.9.d.a 14 9.d odd 6 1 inner
81.9.b.a 14 9.c even 3 1
81.9.b.a 14 9.d odd 6 1
144.9.q.a 14 12.b even 2 1
144.9.q.a 14 36.f odd 6 1
432.9.q.a 14 4.b odd 2 1
432.9.q.a 14 36.h even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{9}^{\mathrm{new}}(27, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} + \cdots + 19\!\cdots\!72 \) Copy content Toggle raw display
$3$ \( T^{14} \) Copy content Toggle raw display
$5$ \( T^{14} + \cdots + 28\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{14} + \cdots + 39\!\cdots\!36 \) Copy content Toggle raw display
$11$ \( T^{14} + \cdots + 15\!\cdots\!43 \) Copy content Toggle raw display
$13$ \( T^{14} + \cdots + 41\!\cdots\!96 \) Copy content Toggle raw display
$17$ \( T^{14} + \cdots + 19\!\cdots\!68 \) Copy content Toggle raw display
$19$ \( (T^{7} + \cdots + 18\!\cdots\!88)^{2} \) Copy content Toggle raw display
$23$ \( T^{14} + \cdots + 83\!\cdots\!52 \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots + 26\!\cdots\!88 \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots + 25\!\cdots\!84 \) Copy content Toggle raw display
$37$ \( (T^{7} + \cdots + 33\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots + 20\!\cdots\!23 \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots + 11\!\cdots\!69 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 35\!\cdots\!72 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 35\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 10\!\cdots\!67 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 26\!\cdots\!04 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 48\!\cdots\!49 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots + 37\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{7} + \cdots + 40\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 49\!\cdots\!84 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 55\!\cdots\!52 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 78\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 18\!\cdots\!25 \) Copy content Toggle raw display
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