Properties

Label 216.4.i.a
Level $216$
Weight $4$
Character orbit 216.i
Analytic conductor $12.744$
Analytic rank $0$
Dimension $8$
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] = [216,4,Mod(73,216)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(216, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 4]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("216.73");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 216 = 2^{3} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 216.i (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: \(12.7444125612\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.5206055409.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{7} + x^{6} + 9x^{5} - 23x^{4} + 27x^{3} + 9x^{2} - 81x + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{6} \)
Twist minimal: no (minimal twist has level 72)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{3} + \beta_1 + 1) q^{5} + ( - \beta_{7} - \beta_{6} - \beta_1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} + \beta_1 + 1) q^{5} + ( - \beta_{7} - \beta_{6} - \beta_1) q^{7} + (\beta_{7} + \beta_{6} + \beta_{5} + 2 \beta_{4} - 2 \beta_{3} + \beta_{2} + 4 \beta_1) q^{11} + ( - 4 \beta_{7} - \beta_{2} + 6 \beta_1 + 6) q^{13} + (6 \beta_{6} + 5 \beta_{5} - 10 \beta_{4} - 1) q^{17} + ( - 2 \beta_{6} + 6 \beta_{5} + \beta_{4} - 28) q^{19} + ( - 7 \beta_{7} + 3 \beta_{3} + \beta_{2} - 10 \beta_1 - 10) q^{23} + ( - 2 \beta_{7} - 2 \beta_{6} + 4 \beta_{5} + 9 \beta_{4} - 9 \beta_{3} + \cdots - 26 \beta_1) q^{25}+ \cdots + (64 \beta_{7} + 64 \beta_{6} - 21 \beta_{5} - 39 \beta_{4} + 39 \beta_{3} + \cdots - 606 \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 5 q^{5} + 3 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 5 q^{5} + 3 q^{7} - 16 q^{11} + 29 q^{13} + 34 q^{17} - 218 q^{19} - 37 q^{23} + 97 q^{25} + 3 q^{29} + 331 q^{31} + 342 q^{35} - 732 q^{37} + 378 q^{41} + 506 q^{43} - 171 q^{47} + 829 q^{49} - 820 q^{53} - 2326 q^{55} - 616 q^{59} + 1331 q^{61} + 815 q^{65} + 1162 q^{67} + 688 q^{71} - 2614 q^{73} + 741 q^{77} + 1853 q^{79} - 1421 q^{83} + 2074 q^{85} - 1632 q^{89} - 3990 q^{91} - 1292 q^{95} + 2506 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{7} + x^{6} + 9x^{5} - 23x^{4} + 27x^{3} + 9x^{2} - 81x + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{7} + 3\nu^{6} - 8\nu^{5} + 15\nu^{4} - 14\nu^{3} - 30\nu^{2} + 99\nu - 135 ) / 81 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2\nu^{7} - 48\nu^{6} + 146\nu^{5} - 186\nu^{4} - 109\nu^{3} + 372\nu^{2} - 936\nu + 1107 ) / 81 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 11\nu^{7} + 15\nu^{6} - 34\nu^{5} - 15\nu^{4} + 89\nu^{3} - 78\nu^{2} - 45\nu + 297 ) / 81 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -2\nu^{7} + 9\nu^{6} - 2\nu^{5} - 42\nu^{4} + 55\nu^{3} + 12\nu^{2} - 36\nu + 189 ) / 27 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -10\nu^{7} + 15\nu^{6} + 26\nu^{5} - 78\nu^{4} + 113\nu^{3} + 48\nu^{2} - 396\nu + 486 ) / 27 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -10\nu^{7} + 21\nu^{6} + 8\nu^{5} - 72\nu^{4} + 167\nu^{3} - 144\nu^{2} - 72\nu + 513 ) / 27 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -37\nu^{7} + 69\nu^{6} - 82\nu^{5} - 51\nu^{4} + 275\nu^{3} - 600\nu^{2} + 225\nu + 135 ) / 81 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -2\beta_{7} + 2\beta_{6} - 2\beta_{5} + 4\beta_{4} - 5\beta_{3} - \beta_{2} + 7\beta _1 + 17 ) / 36 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -4\beta_{7} - 2\beta_{6} + 2\beta_{5} + 8\beta_{4} - 7\beta_{3} - 5\beta_{2} - 13\beta _1 + 25 ) / 36 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -4\beta_{7} + 4\beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} - 5\beta_{2} - 43\beta _1 - 44 ) / 18 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -10\beta_{7} + 4\beta_{6} + 11\beta_{5} - 25\beta_{4} - 4\beta_{3} - 14\beta_{2} + 2\beta _1 + 127 ) / 36 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -26\beta_{7} + 26\beta_{6} + 13\beta_{5} - 29\beta_{4} - 20\beta_{3} + 2\beta_{2} + 250\beta _1 - 19 ) / 36 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -4\beta_{7} - 2\beta_{6} + 14\beta_{5} - \beta_{4} + 2\beta_{3} + \beta_{2} + 182\beta _1 + 82 ) / 9 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -44\beta_{7} + 26\beta_{6} + \beta_{5} - 29\beta_{4} + 133\beta_{3} + 23\beta_{2} + 415\beta _1 - 346 ) / 36 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/216\mathbb{Z}\right)^\times\).

\(n\) \(55\) \(109\) \(137\)
\(\chi(n)\) \(1\) \(1\) \(-1 - \beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
73.1
1.35516 1.07868i
1.70133 + 0.324778i
−1.72895 0.103515i
0.172469 + 1.72344i
1.35516 + 1.07868i
1.70133 0.324778i
−1.72895 + 0.103515i
0.172469 1.72344i
0 0 0 −4.00813 6.94228i 0 −0.468615 + 0.811666i 0 0 0
73.2 0 0 0 −2.99723 5.19136i 0 −7.78882 + 13.4906i 0 0 0
73.3 0 0 0 0.845922 + 1.46518i 0 8.57067 14.8448i 0 0 0
73.4 0 0 0 8.65944 + 14.9986i 0 1.18676 2.05553i 0 0 0
145.1 0 0 0 −4.00813 + 6.94228i 0 −0.468615 0.811666i 0 0 0
145.2 0 0 0 −2.99723 + 5.19136i 0 −7.78882 13.4906i 0 0 0
145.3 0 0 0 0.845922 1.46518i 0 8.57067 + 14.8448i 0 0 0
145.4 0 0 0 8.65944 14.9986i 0 1.18676 + 2.05553i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 73.4
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 216.4.i.a 8
3.b odd 2 1 72.4.i.a 8
4.b odd 2 1 432.4.i.e 8
9.c even 3 1 inner 216.4.i.a 8
9.c even 3 1 648.4.a.h 4
9.d odd 6 1 72.4.i.a 8
9.d odd 6 1 648.4.a.i 4
12.b even 2 1 144.4.i.e 8
36.f odd 6 1 432.4.i.e 8
36.f odd 6 1 1296.4.a.y 4
36.h even 6 1 144.4.i.e 8
36.h even 6 1 1296.4.a.ba 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.4.i.a 8 3.b odd 2 1
72.4.i.a 8 9.d odd 6 1
144.4.i.e 8 12.b even 2 1
144.4.i.e 8 36.h even 6 1
216.4.i.a 8 1.a even 1 1 trivial
216.4.i.a 8 9.c even 3 1 inner
432.4.i.e 8 4.b odd 2 1
432.4.i.e 8 36.f odd 6 1
648.4.a.h 4 9.c even 3 1
648.4.a.i 4 9.d odd 6 1
1296.4.a.y 4 36.f odd 6 1
1296.4.a.ba 4 36.h even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} - 5T_{5}^{7} + 214T_{5}^{6} + 1951T_{5}^{5} + 31798T_{5}^{4} + 109147T_{5}^{3} + 519121T_{5}^{2} - 708224T_{5} + 1982464 \) acting on \(S_{4}^{\mathrm{new}}(216, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} - 5 T^{7} + 214 T^{6} + \cdots + 1982464 \) Copy content Toggle raw display
$7$ \( T^{8} - 3 T^{7} + 276 T^{6} + \cdots + 352836 \) Copy content Toggle raw display
$11$ \( T^{8} + 16 T^{7} + \cdots + 1515467041 \) Copy content Toggle raw display
$13$ \( T^{8} - 29 T^{7} + \cdots + 19954952644 \) Copy content Toggle raw display
$17$ \( (T^{4} - 17 T^{3} - 14562 T^{2} + \cdots + 2567224)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 109 T^{3} - 11772 T^{2} + \cdots - 5081408)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + 37 T^{7} + \cdots + 14\!\cdots\!84 \) Copy content Toggle raw display
$29$ \( T^{8} - 3 T^{7} + \cdots + 24\!\cdots\!84 \) Copy content Toggle raw display
$31$ \( T^{8} - 331 T^{7} + \cdots + 10\!\cdots\!16 \) Copy content Toggle raw display
$37$ \( (T^{4} + 366 T^{3} - 18492 T^{2} + \cdots - 215981856)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} - 378 T^{7} + \cdots + 13\!\cdots\!89 \) Copy content Toggle raw display
$43$ \( T^{8} - 506 T^{7} + \cdots + 32\!\cdots\!89 \) Copy content Toggle raw display
$47$ \( T^{8} + 171 T^{7} + \cdots + 96\!\cdots\!84 \) Copy content Toggle raw display
$53$ \( (T^{4} + 410 T^{3} + \cdots - 15963093536)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + 616 T^{7} + \cdots + 20\!\cdots\!89 \) Copy content Toggle raw display
$61$ \( T^{8} - 1331 T^{7} + \cdots + 23\!\cdots\!16 \) Copy content Toggle raw display
$67$ \( T^{8} - 1162 T^{7} + \cdots + 18\!\cdots\!41 \) Copy content Toggle raw display
$71$ \( (T^{4} - 344 T^{3} - 654384 T^{2} + \cdots + 9762389248)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 1307 T^{3} + \cdots - 88005243128)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} - 1853 T^{7} + \cdots + 97\!\cdots\!36 \) Copy content Toggle raw display
$83$ \( T^{8} + 1421 T^{7} + \cdots + 46\!\cdots\!84 \) Copy content Toggle raw display
$89$ \( (T^{4} + 816 T^{3} + \cdots + 22003976592)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} - 2506 T^{7} + \cdots + 67\!\cdots\!49 \) Copy content Toggle raw display
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