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Properties

Label 27.7.d.a
Level $27$
Weight $7$
Character orbit 27.d
Analytic conductor $6.211$
Analytic rank $0$
Dimension $10$
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] = [27,7,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, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("27.8");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 27 = 3^{3} \)
Weight: \( k \) \(=\) \( 7 \)
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: \(6.21146025774\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - x^{9} + 75 x^{8} - 2 x^{7} + 4610 x^{6} - 2412 x^{5} + 66932 x^{4} - 174032 x^{3} + \cdots + 1982464 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{14} \)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{5} - 25 \beta_{4} + \cdots + \beta_1) q^{4}+ \cdots + (2 \beta_{9} - \beta_{8} - 2 \beta_{7} + \cdots - 35) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{5} - 25 \beta_{4} + \cdots + \beta_1) q^{4}+ \cdots + (866 \beta_{9} - 433 \beta_{8} + \cdots + 220807) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 3 q^{2} + 127 q^{4} + 219 q^{5} - 121 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 3 q^{2} + 127 q^{4} + 219 q^{5} - 121 q^{7} - 132 q^{10} - 483 q^{11} - 841 q^{13} - 12174 q^{14} - 1985 q^{16} + 6176 q^{19} + 63186 q^{20} + 3471 q^{22} - 53565 q^{23} + 8452 q^{25} - 22660 q^{28} + 80679 q^{29} - 24601 q^{31} - 218295 q^{32} + 7425 q^{34} + 12764 q^{37} + 371877 q^{38} + 54150 q^{40} - 232251 q^{41} - 93271 q^{43} + 112512 q^{46} + 142887 q^{47} + 86238 q^{49} - 318459 q^{50} + 186920 q^{52} - 419982 q^{55} - 342546 q^{56} - 380658 q^{58} + 995061 q^{59} - 59305 q^{61} + 403066 q^{64} - 1642029 q^{65} + 158513 q^{67} + 1693791 q^{68} - 304788 q^{70} + 933896 q^{73} - 595182 q^{74} + 666641 q^{76} + 2198883 q^{77} + 468707 q^{79} - 2038470 q^{82} - 3008337 q^{83} - 1189944 q^{85} - 1905549 q^{86} - 349773 q^{88} - 211778 q^{91} + 973788 q^{92} + 809124 q^{94} + 2562954 q^{95} + 336029 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - x^{9} + 75 x^{8} - 2 x^{7} + 4610 x^{6} - 2412 x^{5} + 66932 x^{4} - 174032 x^{3} + \cdots + 1982464 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 5928919 \nu^{9} + 4480309 \nu^{8} - 11157967 \nu^{7} + 1123354358 \nu^{6} + \cdots + 38990423194624 ) / 25215567395904 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 5928919 \nu^{9} - 4480309 \nu^{8} + 11157967 \nu^{7} - 1123354358 \nu^{6} + \cdots - 38990423194624 ) / 12607783697952 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 9684923 \nu^{9} + 688220647 \nu^{8} - 1713976261 \nu^{7} + 42631129082 \nu^{6} + \cdots + 11\!\cdots\!76 ) / 12607783697952 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 3461507741 \nu^{9} + 4504997485 \nu^{8} - 258824546191 \nu^{7} + 4959213290 \nu^{6} + \cdots - 112526238150656 ) / 44\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 301534626821 \nu^{9} + 161056711573 \nu^{8} - 22437956373703 \nu^{7} + \cdots - 93\!\cdots\!08 ) / 44\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 228970339285 \nu^{9} - 1071624020857 \nu^{8} - 17892439228289 \nu^{7} + \cdots - 27\!\cdots\!72 ) / 554742482709888 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 680904382841 \nu^{9} + 1735301772337 \nu^{8} - 45444196783291 \nu^{7} + \cdots + 93\!\cdots\!96 ) / 11\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 357064666657 \nu^{9} + 1468396024739 \nu^{8} - 24218221634837 \nu^{7} + \cdots + 76\!\cdots\!00 ) / 554742482709888 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 1183320135805 \nu^{9} + 2091348988481 \nu^{8} - 82073042998211 \nu^{7} + \cdots + 14\!\cdots\!40 ) / 11\!\cdots\!76 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{2} + 2\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} - 89\beta_{4} + \beta_{3} + 2\beta_{2} + \beta _1 - 89 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3\beta_{8} - 4\beta_{7} + \beta_{6} + 6\beta_{3} + 153\beta_{2} - 147\beta _1 - 107 ) / 9 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 5\beta_{9} - 5\beta_{8} + \beta_{7} + 2\beta_{6} - 204\beta_{5} + 13276\beta_{4} - 389\beta_{2} - 574\beta _1 - 1 ) / 9 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 217 \beta_{9} + 367 \beta_{7} + 75 \beta_{6} - 534 \beta_{5} + 23350 \beta_{4} - 534 \beta_{3} + \cdots + 23642 ) / 9 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 89\beta_{8} - 52\beta_{7} - 37\beta_{6} - 4230\beta_{3} - 11123\beta_{2} + 6893\beta _1 + 253495 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 4423 \beta_{9} - 4423 \beta_{8} - 1573 \beta_{7} - 3146 \beta_{6} + 14418 \beta_{5} - 755890 \beta_{4} + \cdots + 1573 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 1997 \beta_{9} - 5061 \beta_{7} - 3529 \beta_{6} + 262018 \beta_{5} - 15267234 \beta_{4} + \cdots - 15268766 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 782853 \beta_{8} - 1072036 \beta_{7} + 289183 \beta_{6} + 3309126 \beta_{3} + 33892269 \beta_{2} + \cdots - 183654569 ) / 9 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(-\beta_{4}\)

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
−3.54866 − 6.14646i
−2.32209 − 4.02197i
1.07323 + 1.85889i
1.22025 + 2.11353i
4.07727 + 7.06203i
−3.54866 + 6.14646i
−2.32209 + 4.02197i
1.07323 − 1.85889i
1.22025 − 2.11353i
4.07727 − 7.06203i
−10.6460 − 6.14646i 0 43.5579 + 75.4444i 157.562 − 90.9682i 0 83.3541 − 144.373i − 284.159i 0 −2236.53
8.2 −6.96626 − 4.02197i 0 0.352523 + 0.610587i −80.3236 + 46.3749i 0 60.0074 − 103.936i 509.141i 0 746.074
8.3 3.21969 + 1.85889i 0 −25.0891 − 43.4555i −136.563 + 78.8448i 0 −256.037 + 443.470i − 424.489i 0 −586.255
8.4 3.66074 + 2.11353i 0 −23.0660 − 39.9514i 64.9866 − 37.5201i 0 181.066 − 313.616i − 465.535i 0 317.199
8.5 12.2318 + 7.06203i 0 67.7447 + 117.337i 103.839 − 59.9512i 0 −128.891 + 223.245i 1009.72i 0 1693.51
17.1 −10.6460 + 6.14646i 0 43.5579 − 75.4444i 157.562 + 90.9682i 0 83.3541 + 144.373i 284.159i 0 −2236.53
17.2 −6.96626 + 4.02197i 0 0.352523 − 0.610587i −80.3236 − 46.3749i 0 60.0074 + 103.936i − 509.141i 0 746.074
17.3 3.21969 − 1.85889i 0 −25.0891 + 43.4555i −136.563 − 78.8448i 0 −256.037 − 443.470i 424.489i 0 −586.255
17.4 3.66074 − 2.11353i 0 −23.0660 + 39.9514i 64.9866 + 37.5201i 0 181.066 + 313.616i 465.535i 0 317.199
17.5 12.2318 − 7.06203i 0 67.7447 − 117.337i 103.839 + 59.9512i 0 −128.891 − 223.245i − 1009.72i 0 1693.51
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 8.5
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 27.7.d.a 10
3.b odd 2 1 9.7.d.a ✓ 10
4.b odd 2 1 432.7.q.a 10
9.c even 3 1 9.7.d.a ✓ 10
9.c even 3 1 81.7.b.a 10
9.d odd 6 1 inner 27.7.d.a 10
9.d odd 6 1 81.7.b.a 10
12.b even 2 1 144.7.q.a 10
36.f odd 6 1 144.7.q.a 10
36.h even 6 1 432.7.q.a 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9.7.d.a ✓ 10 3.b odd 2 1
9.7.d.a ✓ 10 9.c even 3 1
27.7.d.a 10 1.a even 1 1 trivial
27.7.d.a 10 9.d odd 6 1 inner
81.7.b.a 10 9.c even 3 1
81.7.b.a 10 9.d odd 6 1
144.7.q.a 10 12.b even 2 1
144.7.q.a 10 36.f odd 6 1
432.7.q.a 10 4.b odd 2 1
432.7.q.a 10 36.h even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(27, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + \cdots + 481738752 \) Copy content Toggle raw display
$3$ \( T^{10} \) Copy content Toggle raw display
$5$ \( T^{10} + \cdots + 57\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{10} + \cdots + 91\!\cdots\!96 \) Copy content Toggle raw display
$11$ \( T^{10} + \cdots + 96\!\cdots\!27 \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots + 11\!\cdots\!64 \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( (T^{5} + \cdots - 51\!\cdots\!36)^{2} \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 17\!\cdots\!92 \) Copy content Toggle raw display
$29$ \( T^{10} + \cdots + 24\!\cdots\!72 \) Copy content Toggle raw display
$31$ \( T^{10} + \cdots + 38\!\cdots\!56 \) Copy content Toggle raw display
$37$ \( (T^{5} + \cdots + 12\!\cdots\!72)^{2} \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots + 44\!\cdots\!63 \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots + 48\!\cdots\!01 \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 10\!\cdots\!28 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 41\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 73\!\cdots\!43 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots + 52\!\cdots\!44 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 73\!\cdots\!41 \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{5} + \cdots - 36\!\cdots\!80)^{2} \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 34\!\cdots\!96 \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 32\!\cdots\!88 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 14\!\cdots\!25 \) Copy content Toggle raw display
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