Properties

Label 18.6.c.b
Level $18$
Weight $6$
Character orbit 18.c
Analytic conductor $2.887$
Analytic rank $0$
Dimension $6$
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] = [18,6,Mod(7,18)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(18, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("18.7");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 18 = 2 \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 18.c (of order \(3\), degree \(2\), 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: \(2.88690875663\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.47347183152.3
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{5} + 118x^{4} - 231x^{3} + 3700x^{2} - 3585x + 32331 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{4} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 4 \beta_1 q^{2} + ( - \beta_{3} + \beta_{2} + \beta_1 + 1) q^{3} + (16 \beta_1 - 16) q^{4} + (\beta_{5} + \beta_{3} + 2 \beta_{2} + \cdots - 20) q^{5}+ \cdots + (6 \beta_{5} - 3 \beta_{4} + \beta_{3} + \cdots - 50) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 4 \beta_1 q^{2} + ( - \beta_{3} + \beta_{2} + \beta_1 + 1) q^{3} + (16 \beta_1 - 16) q^{4} + (\beta_{5} + \beta_{3} + 2 \beta_{2} + \cdots - 20) q^{5}+ \cdots + ( - 504 \beta_{5} + 495 \beta_{4} + \cdots + 13191) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 12 q^{2} + 9 q^{3} - 48 q^{4} - 54 q^{5} - 12 q^{6} - 132 q^{7} + 384 q^{8} - 177 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 12 q^{2} + 9 q^{3} - 48 q^{4} - 54 q^{5} - 12 q^{6} - 132 q^{7} + 384 q^{8} - 177 q^{9} + 432 q^{10} - 315 q^{11} - 96 q^{12} - 744 q^{13} - 528 q^{14} + 2286 q^{15} - 768 q^{16} + 2898 q^{17} + 1056 q^{18} + 2262 q^{19} - 864 q^{20} - 11076 q^{21} - 1260 q^{22} - 3168 q^{23} + 576 q^{24} - 2883 q^{25} + 5952 q^{26} + 18144 q^{27} + 4224 q^{28} - 5148 q^{29} - 14400 q^{30} - 8610 q^{31} - 3072 q^{32} + 17469 q^{33} - 5796 q^{34} + 2700 q^{35} - 1392 q^{36} + 39936 q^{37} - 4524 q^{38} - 49026 q^{39} - 3456 q^{40} + 5049 q^{41} + 41352 q^{42} - 31389 q^{43} + 10080 q^{44} + 2538 q^{45} + 25344 q^{46} + 12924 q^{47} - 768 q^{48} - 52857 q^{49} - 11532 q^{50} + 36837 q^{51} - 11904 q^{52} - 96048 q^{53} - 71892 q^{54} + 126252 q^{55} - 8448 q^{56} - 17469 q^{57} - 20592 q^{58} + 62955 q^{59} + 21024 q^{60} - 75966 q^{61} + 68880 q^{62} + 49578 q^{63} + 24576 q^{64} + 108702 q^{65} + 2952 q^{66} - 32991 q^{67} - 23184 q^{68} - 29250 q^{69} - 5400 q^{70} - 129672 q^{71} - 11328 q^{72} - 8466 q^{73} - 79872 q^{74} - 105483 q^{75} - 18096 q^{76} + 88740 q^{77} + 171720 q^{78} + 89202 q^{79} + 27648 q^{80} + 123435 q^{81} - 40392 q^{82} + 32634 q^{83} + 11808 q^{84} + 71388 q^{85} - 125556 q^{86} - 151524 q^{87} - 20160 q^{88} + 66132 q^{89} - 263088 q^{90} - 301836 q^{91} - 50688 q^{92} + 57678 q^{93} + 51696 q^{94} - 82944 q^{95} - 6144 q^{96} + 46245 q^{97} + 422856 q^{98} + 282168 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 3x^{5} + 118x^{4} - 231x^{3} + 3700x^{2} - 3585x + 32331 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -2\nu^{5} + 5\nu^{4} + 176\nu^{3} - 269\nu^{2} + 16626\nu + 22035 ) / 60606 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 331\nu^{5} - 2875\nu^{4} + 35573\nu^{3} - 258101\nu^{2} + 670179\nu - 4096833 ) / 60606 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 331\nu^{5} - 418\nu^{4} + 30659\nu^{3} - 22229\nu^{2} + 527673\nu - 168909 ) / 30303 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 991\nu^{5} + 3665\nu^{4} + 82325\nu^{3} + 495697\nu^{2} + 769179\nu + 11037819 ) / 60606 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1655\nu^{5} - 9461\nu^{4} + 168037\nu^{3} - 636943\nu^{2} + 3065883\nu - 5840445 ) / 60606 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{5} - 2\beta_{4} + 4\beta_{3} - 7\beta_{2} + 2\beta _1 + 9 ) / 18 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} - \beta_{3} - 3\beta_{2} - 112 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -44\beta_{5} + 109\beta_{4} - 221\beta_{3} + 353\beta_{2} + 2870\beta _1 - 2505 ) / 18 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -101\beta_{5} + 17\beta_{4} + 98\beta_{3} + 264\beta_{2} + 976\beta _1 + 5208 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 2119\beta_{5} - 6779\beta_{4} + 16081\beta_{3} - 20746\beta_{2} - 261628\beta _1 + 221196 ) / 18 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/18\mathbb{Z}\right)^\times\).

\(n\) \(11\)
\(\chi(n)\) \(-1 + \beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
0.500000 4.03013i
0.500000 + 8.40123i
0.500000 5.23712i
0.500000 + 4.03013i
0.500000 8.40123i
0.500000 + 5.23712i
−2.00000 + 3.46410i −8.58066 + 13.0143i −8.00000 13.8564i −39.2420 67.9691i −27.9216 55.7529i −110.566 + 191.505i 64.0000 −95.7446 223.343i 313.936
7.2 −2.00000 + 3.46410i −2.43381 15.3973i −8.00000 13.8564i −20.8014 36.0292i 58.2054 + 22.3636i 101.661 176.082i 64.0000 −231.153 + 74.9483i 166.412
7.3 −2.00000 + 3.46410i 15.5145 + 1.51695i −8.00000 13.8564i 33.0434 + 57.2329i −36.2838 + 50.7098i −57.0952 + 98.8918i 64.0000 238.398 + 47.0695i −264.347
13.1 −2.00000 3.46410i −8.58066 13.0143i −8.00000 + 13.8564i −39.2420 + 67.9691i −27.9216 + 55.7529i −110.566 191.505i 64.0000 −95.7446 + 223.343i 313.936
13.2 −2.00000 3.46410i −2.43381 + 15.3973i −8.00000 + 13.8564i −20.8014 + 36.0292i 58.2054 22.3636i 101.661 + 176.082i 64.0000 −231.153 74.9483i 166.412
13.3 −2.00000 3.46410i 15.5145 1.51695i −8.00000 + 13.8564i 33.0434 57.2329i −36.2838 50.7098i −57.0952 98.8918i 64.0000 238.398 47.0695i −264.347
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.3
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 18.6.c.b 6
3.b odd 2 1 54.6.c.b 6
4.b odd 2 1 144.6.i.b 6
9.c even 3 1 inner 18.6.c.b 6
9.c even 3 1 162.6.a.j 3
9.d odd 6 1 54.6.c.b 6
9.d odd 6 1 162.6.a.i 3
12.b even 2 1 432.6.i.b 6
36.f odd 6 1 144.6.i.b 6
36.h even 6 1 432.6.i.b 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.6.c.b 6 1.a even 1 1 trivial
18.6.c.b 6 9.c even 3 1 inner
54.6.c.b 6 3.b odd 2 1
54.6.c.b 6 9.d odd 6 1
144.6.i.b 6 4.b odd 2 1
144.6.i.b 6 36.f odd 6 1
162.6.a.i 3 9.d odd 6 1
162.6.a.j 3 9.c even 3 1
432.6.i.b 6 12.b even 2 1
432.6.i.b 6 36.h even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} + 54T_{5}^{5} + 7587T_{5}^{4} + 179334T_{5}^{3} + 33470577T_{5}^{2} + 1007927064T_{5} + 46562734656 \) acting on \(S_{6}^{\mathrm{new}}(18, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 4 T + 16)^{3} \) Copy content Toggle raw display
$3$ \( T^{6} - 9 T^{5} + \cdots + 14348907 \) Copy content Toggle raw display
$5$ \( T^{6} + \cdots + 46562734656 \) Copy content Toggle raw display
$7$ \( T^{6} + \cdots + 26358756910084 \) Copy content Toggle raw display
$11$ \( T^{6} + \cdots + 17\!\cdots\!69 \) Copy content Toggle raw display
$13$ \( T^{6} + \cdots + 14\!\cdots\!84 \) Copy content Toggle raw display
$17$ \( (T^{3} - 1449 T^{2} + \cdots + 930192444)^{2} \) Copy content Toggle raw display
$19$ \( (T^{3} - 1131 T^{2} + \cdots - 352455920)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + \cdots + 18\!\cdots\!96 \) Copy content Toggle raw display
$29$ \( T^{6} + \cdots + 42\!\cdots\!16 \) Copy content Toggle raw display
$31$ \( T^{6} + \cdots + 35\!\cdots\!16 \) Copy content Toggle raw display
$37$ \( (T^{3} - 19968 T^{2} + \cdots - 188019064016)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} + \cdots + 46\!\cdots\!69 \) Copy content Toggle raw display
$43$ \( T^{6} + \cdots + 26\!\cdots\!21 \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots + 71\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{3} + \cdots + 1764512817552)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} + \cdots + 33\!\cdots\!49 \) Copy content Toggle raw display
$61$ \( T^{6} + \cdots + 28\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots + 92\!\cdots\!25 \) Copy content Toggle raw display
$71$ \( (T^{3} + 64836 T^{2} + \cdots - 139951336896)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} + \cdots + 14322358753732)^{2} \) Copy content Toggle raw display
$79$ \( T^{6} + \cdots + 13\!\cdots\!16 \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots + 10\!\cdots\!16 \) Copy content Toggle raw display
$89$ \( (T^{3} + \cdots + 9104584153608)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + \cdots + 85\!\cdots\!09 \) Copy content Toggle raw display
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