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Properties

Label 27.8.c.a
Level $27$
Weight $8$
Character orbit 27.c
Analytic conductor $8.434$
Analytic rank $0$
Dimension $12$
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] = [27,8,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, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("27.10");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 27 = 3^{3} \)
Weight: \( k \) \(=\) \( 8 \)
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: \(8.43439568807\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 6 x^{11} + 375 x^{10} - 1820 x^{9} + 50808 x^{8} - 192378 x^{7} + 3002887 x^{6} + \cdots + 754412211 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{21} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{7} + \beta_{6} - \beta_1) q^{2} + (\beta_{9} + 52 \beta_{7} + 3 \beta_{6} - 52) q^{4} + ( - \beta_{11} - 30 \beta_{7} + 30) q^{5} + ( - 3 \beta_{11} - \beta_{10} + \cdots + 16 \beta_1) q^{7}+ \cdots + (\beta_{5} - 2 \beta_{4} + \beta_{3} + \cdots - 472) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{7} + \beta_{6} - \beta_1) q^{2} + (\beta_{9} + 52 \beta_{7} + 3 \beta_{6} - 52) q^{4} + ( - \beta_{11} - 30 \beta_{7} + 30) q^{5} + ( - 3 \beta_{11} - \beta_{10} + \cdots + 16 \beta_1) q^{7}+ \cdots + (7147 \beta_{5} - 3458 \beta_{4} + \cdots + 8053744) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 9 q^{2} - 321 q^{4} + 180 q^{5} - 84 q^{7} - 5922 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 9 q^{2} - 321 q^{4} + 180 q^{5} - 84 q^{7} - 5922 q^{8} + 252 q^{10} + 8460 q^{11} - 1848 q^{13} + 16272 q^{14} - 12417 q^{16} - 30564 q^{17} + 24432 q^{19} + 40788 q^{20} - 35001 q^{22} + 51588 q^{23} + 4746 q^{25} - 536472 q^{26} + 75516 q^{28} + 414648 q^{29} + 8196 q^{31} + 1048977 q^{32} - 106623 q^{34} - 2210616 q^{35} + 139344 q^{37} + 1952685 q^{38} + 305496 q^{40} + 1731582 q^{41} + 408372 q^{43} - 5169114 q^{44} - 1684008 q^{46} + 1631484 q^{47} - 179010 q^{49} + 1654461 q^{50} + 681594 q^{52} - 2835648 q^{53} - 16056 q^{55} + 1784466 q^{56} - 948384 q^{58} + 2055636 q^{59} - 2723196 q^{61} + 1026828 q^{62} + 7178178 q^{64} + 1387620 q^{65} + 3806556 q^{67} - 2142639 q^{68} + 953442 q^{70} - 2408400 q^{71} - 10670052 q^{73} - 9846504 q^{74} - 6727827 q^{76} - 3478824 q^{77} + 6020916 q^{79} + 38072448 q^{80} + 9403002 q^{82} - 9605052 q^{83} - 1698624 q^{85} - 34278561 q^{86} - 16459029 q^{88} + 24630264 q^{89} + 13570104 q^{91} - 39143394 q^{92} + 12602808 q^{94} - 10422072 q^{95} + 9977226 q^{97} + 95833314 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 6 x^{11} + 375 x^{10} - 1820 x^{9} + 50808 x^{8} - 192378 x^{7} + 3002887 x^{6} + \cdots + 754412211 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{10} - 5 \nu^{9} + 631 \nu^{8} - 2494 \nu^{7} + 213005 \nu^{6} - 630307 \nu^{5} + \cdots + 3098815785 ) / 479986560 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 3\nu^{2} - 3\nu + 180 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 443 \nu^{10} - 2215 \nu^{9} + 183791 \nu^{8} - 721874 \nu^{7} + 26320567 \nu^{6} + \cdots + 18976459761 ) / 239993280 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 5029 \nu^{10} - 25145 \nu^{9} + 1832911 \nu^{8} - 7180774 \nu^{7} + 238629713 \nu^{6} + \cdots + 1000917153009 ) / 479986560 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2699 \nu^{10} - 13495 \nu^{9} + 873305 \nu^{8} - 3412250 \nu^{7} + 97211743 \nu^{6} + \cdots + 182728260423 ) / 159995520 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1958197 \nu^{11} - 9956688 \nu^{10} + 724465158 \nu^{9} - 2684366423 \nu^{8} + \cdots + 28\!\cdots\!61 ) / 780837815928960 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3916394 \nu^{11} - 21540167 \nu^{10} + 1457064271 \nu^{9} - 6395237967 \nu^{8} + \cdots - 87187467749373 ) / 780837815928960 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 499913231 \nu^{11} + 3836540593 \nu^{10} + 180673452721 \nu^{9} + 1190431529238 \nu^{8} + \cdots + 47\!\cdots\!40 ) / 780837815928960 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 77730382 \nu^{11} + 427517101 \nu^{10} - 28848101381 \nu^{9} + 126610077957 \nu^{8} + \cdots + 16\!\cdots\!71 ) / 86759757325440 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 868001755 \nu^{11} - 683443683 \nu^{10} + 299680949037 \nu^{9} + 86084626714 \nu^{8} + \cdots + 54\!\cdots\!20 ) / 780837815928960 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 674669385 \nu^{11} + 3350347411 \nu^{10} - 236441521415 \nu^{9} + 894769481542 \nu^{8} + \cdots + 27\!\cdots\!12 ) / 390418907964480 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} - 2\beta_{6} + \beta _1 + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} - 2\beta_{6} + \beta_{2} + \beta _1 - 179 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 2 \beta_{11} + 4 \beta_{10} + 11 \beta_{9} + 2 \beta_{8} - 155 \beta_{7} + 578 \beta_{6} - \beta_{5} + \cdots - 728 ) / 9 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 4 \beta_{11} + 8 \beta_{10} + 22 \beta_{9} + 4 \beta_{8} - 313 \beta_{7} + 1162 \beta_{6} + \cdots + 51556 ) / 9 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 1107 \beta_{11} - 1658 \beta_{10} - 6592 \beta_{9} - 1039 \beta_{8} - 203622 \beta_{7} - 184787 \beta_{6} + \cdots + 491337 ) / 27 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 1117 \beta_{11} - 1678 \beta_{10} - 6647 \beta_{9} - 1049 \beta_{8} - 202838 \beta_{7} - 187695 \beta_{6} + \cdots - 5339585 ) / 9 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 51457 \beta_{11} + 69106 \beta_{10} + 341729 \beta_{9} + 46673 \beta_{8} + 18594995 \beta_{7} + \cdots - 28796399 ) / 9 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 633150 \beta_{11} + 852820 \beta_{10} + 4193960 \beta_{9} + 574790 \beta_{8} + 225977475 \beta_{7} + \cdots + 1698678117 ) / 27 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 6852619 \beta_{11} - 8449306 \beta_{10} - 48596144 \beta_{9} - 5813303 \beta_{8} - 3307887749 \beta_{7} + \cdots + 4488984514 ) / 9 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 35853809 \beta_{11} - 44390366 \beta_{10} - 253512259 \beta_{9} - 30510853 \beta_{8} + \cdots - 60531727202 ) / 9 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 2716894845 \beta_{11} + 3079935034 \beta_{10} + 19761824426 \beta_{9} + 2103885437 \beta_{8} + \cdots - 1972568702727 ) / 27 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(-1 + \beta_{7}\)

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 + 9.80854i
0.500000 + 9.08282i
0.500000 + 1.48508i
0.500000 − 2.70685i
0.500000 − 6.17443i
0.500000 − 11.4952i
0.500000 − 9.80854i
0.500000 − 9.08282i
0.500000 − 1.48508i
0.500000 + 2.70685i
0.500000 + 6.17443i
0.500000 + 11.4952i
−7.74445 − 13.4138i 0 −55.9529 + 96.9133i −52.7641 + 91.3900i 0 761.419 + 1318.82i −249.280 0 1634.51
10.2 −7.11595 − 12.3252i 0 −37.2735 + 64.5595i 145.304 − 251.673i 0 −555.940 − 962.916i −760.739 0 −4135.89
10.3 −0.536120 − 0.928588i 0 63.4251 − 109.856i −47.9866 + 83.1153i 0 −189.000 − 327.358i −273.261 0 102.906
10.4 3.09420 + 5.35931i 0 44.8519 − 77.6857i −167.952 + 290.901i 0 442.025 + 765.610i 1347.24 0 −2078.70
10.5 6.09721 + 10.5607i 0 −10.3519 + 17.9301i 246.026 − 426.130i 0 −382.311 − 662.182i 1308.41 0 6000.29
10.6 10.7051 + 18.5418i 0 −165.199 + 286.133i −32.6274 + 56.5123i 0 −118.194 − 204.717i −4333.37 0 −1397.12
19.1 −7.74445 + 13.4138i 0 −55.9529 − 96.9133i −52.7641 − 91.3900i 0 761.419 − 1318.82i −249.280 0 1634.51
19.2 −7.11595 + 12.3252i 0 −37.2735 − 64.5595i 145.304 + 251.673i 0 −555.940 + 962.916i −760.739 0 −4135.89
19.3 −0.536120 + 0.928588i 0 63.4251 + 109.856i −47.9866 − 83.1153i 0 −189.000 + 327.358i −273.261 0 102.906
19.4 3.09420 − 5.35931i 0 44.8519 + 77.6857i −167.952 − 290.901i 0 442.025 − 765.610i 1347.24 0 −2078.70
19.5 6.09721 − 10.5607i 0 −10.3519 − 17.9301i 246.026 + 426.130i 0 −382.311 + 662.182i 1308.41 0 6000.29
19.6 10.7051 − 18.5418i 0 −165.199 − 286.133i −32.6274 − 56.5123i 0 −118.194 + 204.717i −4333.37 0 −1397.12
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 10.6
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 27.8.c.a 12
3.b odd 2 1 9.8.c.a ✓ 12
4.b odd 2 1 432.8.i.c 12
9.c even 3 1 inner 27.8.c.a 12
9.c even 3 1 81.8.a.c 6
9.d odd 6 1 9.8.c.a ✓ 12
9.d odd 6 1 81.8.a.e 6
12.b even 2 1 144.8.i.c 12
36.f odd 6 1 432.8.i.c 12
36.h even 6 1 144.8.i.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9.8.c.a ✓ 12 3.b odd 2 1
9.8.c.a ✓ 12 9.d odd 6 1
27.8.c.a 12 1.a even 1 1 trivial
27.8.c.a 12 9.c even 3 1 inner
81.8.a.c 6 9.c even 3 1
81.8.a.e 6 9.d odd 6 1
144.8.i.c 12 12.b even 2 1
144.8.i.c 12 36.h even 6 1
432.8.i.c 12 4.b odd 2 1
432.8.i.c 12 36.f odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{8}^{\mathrm{new}}(27, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + \cdots + 145838444544 \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 10\!\cdots\!04 \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 83\!\cdots\!81 \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 50\!\cdots\!64 \) Copy content Toggle raw display
$17$ \( (T^{6} + \cdots - 16\!\cdots\!76)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} + \cdots + 16\!\cdots\!00)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 73\!\cdots\!84 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 53\!\cdots\!44 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 23\!\cdots\!56 \) Copy content Toggle raw display
$37$ \( (T^{6} + \cdots + 17\!\cdots\!64)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 45\!\cdots\!61 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 27\!\cdots\!69 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 56\!\cdots\!04 \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots + 54\!\cdots\!44)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 74\!\cdots\!89 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 75\!\cdots\!96 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 21\!\cdots\!49 \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots + 81\!\cdots\!24)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + \cdots + 34\!\cdots\!16)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 11\!\cdots\!44 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 58\!\cdots\!64 \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots - 14\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 83\!\cdots\!69 \) Copy content Toggle raw display
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