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Properties

Label 27.11.d.a
Level $27$
Weight $11$
Character orbit 27.d
Analytic conductor $17.155$
Analytic rank $0$
Dimension $18$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [27,11,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, 11, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("27.8");
 
S:= CuspForms(chi, 11);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 27 = 3^{3} \)
Weight: \( k \) \(=\) \( 11 \)
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: \(17.1546458222\)
Analytic rank: \(0\)
Dimension: \(18\)
Relative dimension: \(9\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - x^{17} + 2219 x^{16} + 4286 x^{15} + 3372866 x^{14} + 7237076 x^{13} + 2694115412 x^{12} + \cdots + 64\!\cdots\!96 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{14}\cdot 3^{52} \)
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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + (\beta_{6} - \beta_{4} + 2 \beta_{3} + \cdots + 454) q^{4} + ( - \beta_{10} - 184 \beta_1 - 183) q^{5} + (\beta_{11} - 2 \beta_{10} + \beta_{7} + \cdots + 1) q^{7} + (\beta_{12} - 2 \beta_{11} + \cdots + 2975) q^{8}+ \cdots + (188738 \beta_{17} - 94369 \beta_{16} + \cdots - 665280001) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 3 q^{2} + 4095 q^{4} - 4956 q^{5} - 6120 q^{7} - 2052 q^{10} - 969 q^{11} + 140274 q^{13} + 2134578 q^{14} - 1571841 q^{16} + 2771370 q^{19} - 14542734 q^{20} - 3475521 q^{22} + 9944382 q^{23} + 14726277 q^{25}+ \cdots - 14510723337 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - x^{17} + 2219 x^{16} + 4286 x^{15} + 3372866 x^{14} + 7237076 x^{13} + 2694115412 x^{12} + \cdots + 64\!\cdots\!96 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 31\!\cdots\!65 \nu^{17} + \cdots + 52\!\cdots\!16 ) / 53\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 15\!\cdots\!37 \nu^{17} + \cdots - 71\!\cdots\!80 ) / 19\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 15\!\cdots\!37 \nu^{17} + \cdots + 71\!\cdots\!80 ) / 19\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 10\!\cdots\!89 \nu^{17} + \cdots + 15\!\cdots\!08 ) / 10\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 10\!\cdots\!41 \nu^{17} + \cdots - 19\!\cdots\!88 ) / 55\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 23\!\cdots\!03 \nu^{17} + \cdots + 38\!\cdots\!04 ) / 26\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 49\!\cdots\!69 \nu^{17} + \cdots - 10\!\cdots\!96 ) / 78\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 27\!\cdots\!61 \nu^{17} + \cdots + 20\!\cdots\!04 ) / 27\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 11\!\cdots\!37 \nu^{17} + \cdots + 80\!\cdots\!04 ) / 97\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 97\!\cdots\!07 \nu^{17} + \cdots - 16\!\cdots\!28 ) / 78\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 49\!\cdots\!85 \nu^{17} + \cdots - 12\!\cdots\!24 ) / 29\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 70\!\cdots\!79 \nu^{17} + \cdots + 20\!\cdots\!92 ) / 39\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 95\!\cdots\!69 \nu^{17} + \cdots + 19\!\cdots\!48 ) / 19\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 51\!\cdots\!61 \nu^{17} + \cdots - 48\!\cdots\!60 ) / 87\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 35\!\cdots\!17 \nu^{17} + \cdots - 62\!\cdots\!24 ) / 39\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 25\!\cdots\!77 \nu^{17} + \cdots + 95\!\cdots\!40 ) / 19\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 15\!\cdots\!75 \nu^{17} + \cdots - 12\!\cdots\!96 ) / 43\!\cdots\!08 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{6} + 4\beta_{3} - 2\beta_{2} - 1478\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{12} - \beta_{9} + \beta_{5} + 13\beta_{4} + 2480\beta_{3} - 4959\beta_{2} - 8927 ) / 9 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 11 \beta_{17} - 2 \beta_{15} + 5 \beta_{14} - 15 \beta_{13} - 8 \beta_{12} + 6 \beta_{11} + \cdots - 3664401 ) / 9 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 87 \beta_{17} - 87 \beta_{16} - 1100 \beta_{15} - 561 \beta_{14} - 550 \beta_{13} - 9260 \beta_{12} + \cdots + 2902 ) / 27 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 19969 \beta_{16} - 790 \beta_{15} + 20759 \beta_{13} - 18076 \beta_{12} + 245600 \beta_{10} + \cdots + 3511431267 ) / 9 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 11655 \beta_{17} + 1017274 \beta_{15} + 1153695 \beta_{14} + 2046203 \beta_{13} + 5925528 \beta_{12} + \cdots + 89644548449 ) / 27 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 83122567 \beta_{17} - 83122567 \beta_{16} - 6347020 \beta_{15} - 17937921 \beta_{14} - 3173510 \beta_{13} + \cdots + 894279876 ) / 27 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 244256755 \beta_{16} + 1394383106 \beta_{15} - 1638639861 \beta_{13} + 7193004708 \beta_{12} + \cdots - 130673260056737 ) / 27 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 104350128023 \beta_{17} + 7063557862 \beta_{15} + 14066443089 \beta_{14} - 90223012299 \beta_{13} + \cdots - 11\!\cdots\!37 ) / 27 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 589801731731 \beta_{17} + 589801731731 \beta_{16} - 3423548897156 \beta_{15} - 2148260945739 \beta_{14} + \cdots - 8984501536884 ) / 27 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 41701120391141 \beta_{16} + 2812155327346 \beta_{15} + 38888965063795 \beta_{13} - 58743022150340 \beta_{12} + \cdots + 43\!\cdots\!27 ) / 9 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 10\!\cdots\!19 \beta_{17} + \cdots + 23\!\cdots\!29 ) / 27 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 14\!\cdots\!99 \beta_{17} + \cdots + 14\!\cdots\!64 ) / 27 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 16\!\cdots\!15 \beta_{16} + \cdots - 30\!\cdots\!65 ) / 27 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( - 17\!\cdots\!03 \beta_{17} + \cdots - 16\!\cdots\!85 ) / 27 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( - 23\!\cdots\!23 \beta_{17} + \cdots - 30\!\cdots\!28 ) / 27 \) Copy content Toggle raw display

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

Character values

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

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

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.

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
−15.7177 − 27.2239i
−13.3248 − 23.0793i
−9.48684 − 16.4317i
−2.95332 − 5.11529i
−2.54168 − 4.40231i
5.37603 + 9.31155i
8.49892 + 14.7206i
13.2461 + 22.9429i
17.4033 + 30.1435i
−15.7177 + 27.2239i
−13.3248 + 23.0793i
−9.48684 + 16.4317i
−2.95332 + 5.11529i
−2.54168 + 4.40231i
5.37603 − 9.31155i
8.49892 − 14.7206i
13.2461 − 22.9429i
17.4033 − 30.1435i
−47.1532 − 27.2239i 0 970.286 + 1680.58i −3579.43 + 2066.59i 0 −11845.8 + 20517.5i − 49905.4i 0 225043.
8.2 −39.9745 − 23.0793i 0 553.305 + 958.352i 794.787 − 458.870i 0 4431.20 − 7675.06i − 3813.13i 0 −42361.6
8.3 −28.4605 − 16.4317i 0 28.0010 + 48.4991i 4600.38 − 2656.03i 0 −1786.25 + 3093.87i 31811.7i 0 −174572.
8.4 −8.85995 − 5.11529i 0 −459.668 − 796.168i −5050.65 + 2916.00i 0 1337.45 − 2316.53i 19881.5i 0 59664.7
8.5 −7.62503 − 4.40231i 0 −473.239 − 819.674i 620.145 − 358.041i 0 9365.12 − 16220.9i 17349.3i 0 −6304.84
8.6 16.1281 + 9.31155i 0 −338.590 − 586.455i 748.776 − 432.306i 0 −13727.6 + 23777.0i − 31681.3i 0 16101.8
8.7 25.4968 + 14.7206i 0 −78.6099 − 136.156i −205.899 + 118.876i 0 −3544.69 + 6139.58i − 34776.4i 0 −6999.68
8.8 39.7383 + 22.9429i 0 540.755 + 936.616i 2605.51 − 1504.29i 0 13406.9 − 23221.4i 2638.94i 0 138051.
8.9 52.2100 + 30.1435i 0 1305.26 + 2260.78i −3011.61 + 1738.75i 0 −696.306 + 1206.04i 95646.4i 0 −209648.
17.1 −47.1532 + 27.2239i 0 970.286 − 1680.58i −3579.43 − 2066.59i 0 −11845.8 − 20517.5i 49905.4i 0 225043.
17.2 −39.9745 + 23.0793i 0 553.305 − 958.352i 794.787 + 458.870i 0 4431.20 + 7675.06i 3813.13i 0 −42361.6
17.3 −28.4605 + 16.4317i 0 28.0010 − 48.4991i 4600.38 + 2656.03i 0 −1786.25 − 3093.87i − 31811.7i 0 −174572.
17.4 −8.85995 + 5.11529i 0 −459.668 + 796.168i −5050.65 − 2916.00i 0 1337.45 + 2316.53i − 19881.5i 0 59664.7
17.5 −7.62503 + 4.40231i 0 −473.239 + 819.674i 620.145 + 358.041i 0 9365.12 + 16220.9i − 17349.3i 0 −6304.84
17.6 16.1281 − 9.31155i 0 −338.590 + 586.455i 748.776 + 432.306i 0 −13727.6 − 23777.0i 31681.3i 0 16101.8
17.7 25.4968 − 14.7206i 0 −78.6099 + 136.156i −205.899 − 118.876i 0 −3544.69 − 6139.58i 34776.4i 0 −6999.68
17.8 39.7383 − 22.9429i 0 540.755 − 936.616i 2605.51 + 1504.29i 0 13406.9 + 23221.4i − 2638.94i 0 138051.
17.9 52.2100 − 30.1435i 0 1305.26 − 2260.78i −3011.61 − 1738.75i 0 −696.306 − 1206.04i − 95646.4i 0 −209648.
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 8.9
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 27.11.d.a 18
3.b odd 2 1 9.11.d.a ✓ 18
9.c even 3 1 9.11.d.a ✓ 18
9.c even 3 1 81.11.b.a 18
9.d odd 6 1 inner 27.11.d.a 18
9.d odd 6 1 81.11.b.a 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9.11.d.a ✓ 18 3.b odd 2 1
9.11.d.a ✓ 18 9.c even 3 1
27.11.d.a 18 1.a even 1 1 trivial
27.11.d.a 18 9.d odd 6 1 inner
81.11.b.a 18 9.c even 3 1
81.11.b.a 18 9.d odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{11}^{\mathrm{new}}(27, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} + \cdots + 12\!\cdots\!68 \) Copy content Toggle raw display
$3$ \( T^{18} \) Copy content Toggle raw display
$5$ \( T^{18} + \cdots + 32\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{18} + \cdots + 74\!\cdots\!24 \) Copy content Toggle raw display
$11$ \( T^{18} + \cdots + 76\!\cdots\!43 \) Copy content Toggle raw display
$13$ \( T^{18} + \cdots + 29\!\cdots\!56 \) Copy content Toggle raw display
$17$ \( T^{18} + \cdots + 34\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( (T^{9} + \cdots - 17\!\cdots\!56)^{2} \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots + 56\!\cdots\!88 \) Copy content Toggle raw display
$29$ \( T^{18} + \cdots + 35\!\cdots\!08 \) Copy content Toggle raw display
$31$ \( T^{18} + \cdots + 51\!\cdots\!44 \) Copy content Toggle raw display
$37$ \( (T^{9} + \cdots - 92\!\cdots\!44)^{2} \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots + 24\!\cdots\!87 \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots + 28\!\cdots\!61 \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots + 15\!\cdots\!72 \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots + 65\!\cdots\!23 \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots + 17\!\cdots\!56 \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 58\!\cdots\!81 \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots + 46\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{9} + \cdots - 13\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( T^{18} + \cdots + 56\!\cdots\!84 \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots + 33\!\cdots\!72 \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots + 16\!\cdots\!25 \) Copy content Toggle raw display
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