Properties

Label 216.3.b.b
Level $216$
Weight $3$
Character orbit 216.b
Analytic conductor $5.886$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [216,3,Mod(163,216)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(216, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("216.163");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 216 = 2^{3} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 216.b (of order \(2\), degree \(1\), 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: \(5.88557371018\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 2x^{14} - 2x^{12} - 56x^{10} + 400x^{8} - 896x^{6} - 512x^{4} - 8192x^{2} + 65536 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{8} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{2} q^{4} - \beta_{6} q^{5} - \beta_{14} q^{7} + \beta_{3} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{2} q^{4} - \beta_{6} q^{5} - \beta_{14} q^{7} + \beta_{3} q^{8} + (\beta_{8} + 1) q^{10} + ( - \beta_{13} + \beta_1) q^{11} + ( - \beta_{12} + \beta_{8}) q^{13} + ( - \beta_{13} - \beta_{10}) q^{14} + ( - \beta_{14} + \beta_{12} - \beta_{7} - \beta_{5} + 1) q^{16} + ( - \beta_{13} - \beta_{11} - \beta_{9} + \beta_{3}) q^{17} + ( - \beta_{15} - \beta_{8} + \beta_{2} - 4) q^{19} + ( - \beta_{13} - \beta_{11} + \beta_{10} - 2 \beta_{6} - \beta_{4} + 2 \beta_1) q^{20} + (\beta_{15} + 2 \beta_{14} - \beta_{7} + \beta_{2} + 5) q^{22} + ( - \beta_{11} + \beta_{9} - \beta_{6} - \beta_{4} + \beta_{3} + 3 \beta_1) q^{23} + ( - \beta_{15} - \beta_{8} + \beta_{7} + 2 \beta_{5} - 3 \beta_{2} - 5) q^{25} + ( - \beta_{13} - \beta_{11} + \beta_{10} - 2 \beta_{9} - 2 \beta_{6} + \beta_1) q^{26} + (\beta_{15} + \beta_{14} + \beta_{12} - \beta_{8} + 2 \beta_{7} + \beta_{5} - 1) q^{28} + (2 \beta_{11} - 2 \beta_{10} + 2 \beta_1) q^{29} + (\beta_{15} - 2 \beta_{12} - \beta_{8} + \beta_{7} - \beta_{2} + 2) q^{31} + ( - \beta_{13} + \beta_{11} - \beta_{10} + 2 \beta_{9} + \beta_{4}) q^{32} + (2 \beta_{14} + 2 \beta_{12} - \beta_{8} + 2 \beta_{7} - 2 \beta_{5} + 1) q^{34} + (\beta_{13} + 2 \beta_{11} + 2 \beta_{10} + 2 \beta_{6} + 4 \beta_{3} + \beta_1) q^{35} + (\beta_{15} - 2 \beta_{14} - \beta_{12} - 2 \beta_{8} + \beta_{7} - 2 \beta_{5} - 5 \beta_{2} + 4) q^{37} + (2 \beta_{13} + \beta_{11} - 2 \beta_{10} - 4 \beta_{6} + 2 \beta_{4} - 2 \beta_1) q^{38} + (4 \beta_{14} + 2 \beta_{8} - 4 \beta_{7} + 2 \beta_{2} - 6) q^{40} + ( - 2 \beta_{11} - 2 \beta_{10} - 2 \beta_{6} + 2 \beta_{4} - 4 \beta_{3} + 6 \beta_1) q^{41} + ( - \beta_{15} - \beta_{8} + \beta_{7} + 4 \beta_{5} - 7 \beta_{2} - 4) q^{43} + (\beta_{13} + \beta_{11} + 3 \beta_{10} + 6 \beta_{6} - \beta_{4} + 2 \beta_{3} + 2 \beta_1) q^{44} + ( - \beta_{15} + 2 \beta_{12} - 5 \beta_{7} + 2 \beta_{5} + 3 \beta_{2} - 15) q^{46} + ( - 3 \beta_{11} + 2 \beta_{10} + \beta_{9} + \beta_{6} + \beta_{4} + \beta_{3} - 7 \beta_1) q^{47} + ( - 2 \beta_{15} - 2 \beta_{8} - 2 \beta_{7} + 2 \beta_{2} - 8) q^{49} + (2 \beta_{13} - 2 \beta_{10} - 4 \beta_{6} - 2 \beta_{4} - 4 \beta_{3} - \beta_1) q^{50} + (4 \beta_{14} + 2 \beta_{8} + 4 \beta_{7} - 4 \beta_{5} + \beta_{2} + 6) q^{52} + (2 \beta_{11} + 2 \beta_{10} - 4 \beta_{9} - 4 \beta_{3} + 2 \beta_1) q^{53} + ( - \beta_{15} - 2 \beta_{14} - 2 \beta_{12} + 5 \beta_{8} - \beta_{7} - 4 \beta_{5} - 7 \beta_{2} + \cdots + 2) q^{55}+ \cdots + (4 \beta_{13} + 4 \beta_{11} - 4 \beta_{10} - 8 \beta_{6} + 4 \beta_{4} - 4 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 4 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 4 q^{4} + 24 q^{10} + 16 q^{16} - 64 q^{19} + 80 q^{22} - 80 q^{25} - 12 q^{28} + 8 q^{34} - 72 q^{40} - 64 q^{43} - 192 q^{46} - 128 q^{49} + 84 q^{52} - 96 q^{58} + 376 q^{64} + 128 q^{67} + 192 q^{70} + 80 q^{73} + 308 q^{76} + 272 q^{82} - 136 q^{88} + 192 q^{91} + 336 q^{94} + 80 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 2x^{14} - 2x^{12} - 56x^{10} + 400x^{8} - 896x^{6} - 512x^{4} - 8192x^{2} + 65536 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{15} + 2\nu^{13} + 2\nu^{11} + 56\nu^{9} - 400\nu^{7} + 896\nu^{5} + 512\nu^{3} + 8192\nu ) / 4096 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{14} - 2\nu^{12} - 2\nu^{10} - 56\nu^{8} + 400\nu^{6} - 896\nu^{4} - 512\nu^{2} - 6144 ) / 2048 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{15} - 10\nu^{13} + 78\nu^{11} - 168\nu^{9} + 720\nu^{7} - 3584\nu^{5} + 15872\nu^{3} - 36864\nu ) / 8192 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3\nu^{14} - 22\nu^{12} + 90\nu^{10} - 264\nu^{8} + 944\nu^{6} - 6528\nu^{4} + 19968\nu^{2} - 28672 ) / 4096 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( \nu^{14} - 10\nu^{12} + 14\nu^{10} - 40\nu^{8} + 336\nu^{6} - 2048\nu^{4} + 3584\nu^{2} + 7168 ) / 1024 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -3\nu^{15} - 10\nu^{13} + 166\nu^{11} - 568\nu^{9} + 464\nu^{7} - 1664\nu^{5} + 46592\nu^{3} - 139264\nu ) / 8192 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -3\nu^{15} - 6\nu^{13} + 62\nu^{11} - 128\nu^{9} - 80\nu^{7} - 1344\nu^{5} + 18944\nu^{3} - 38912\nu ) / 4096 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -3\nu^{15} + 22\nu^{13} - 90\nu^{11} + 264\nu^{9} - 944\nu^{7} + 6528\nu^{5} - 19968\nu^{3} + 28672\nu ) / 4096 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( -\nu^{14} - 6\nu^{12} + 82\nu^{10} - 312\nu^{8} + 432\nu^{6} - 1280\nu^{4} + 23040\nu^{2} - 73728 ) / 2048 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( -\nu^{15} + 5\nu^{13} + 4\nu^{11} - 30\nu^{9} - 200\nu^{7} + 752\nu^{5} + 2048\nu^{3} - 15872\nu ) / 1024 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( -7\nu^{14} + 14\nu^{12} + 78\nu^{10} - 248\nu^{8} - 880\nu^{6} + 1664\nu^{4} + 27136\nu^{2} - 102400 ) / 4096 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 5\nu^{14} - 58\nu^{12} + 278\nu^{10} - 568\nu^{8} + 3280\nu^{6} - 16000\nu^{4} + 61952\nu^{2} - 106496 ) / 4096 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{14} + \beta_{12} - \beta_{7} - \beta_{5} + 1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{13} + \beta_{11} - \beta_{10} + 2\beta_{9} + \beta_{4} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2\beta_{15} - 6\beta_{7} + 4\beta_{5} + 22 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -2\beta_{13} + 6\beta_{11} + 2\beta_{10} + 12\beta_{6} - 10\beta_{4} + 2\beta_{3} + 20\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 8\beta_{15} - 2\beta_{14} - 6\beta_{12} - 4\beta_{8} - 10\beta_{7} - 10\beta_{5} + 20\beta_{2} - 130 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -6\beta_{13} + 14\beta_{11} + 2\beta_{10} - 12\beta_{9} + 56\beta_{6} + 22\beta_{4} + 28\beta_{3} - 168\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 20\beta_{15} - 28\beta_{14} + 12\beta_{12} - 40\beta_{8} + 8\beta_{7} - 68\beta_{5} - 168\beta_{2} + 384 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -8\beta_{13} + 32\beta_{11} - 48\beta_{10} + 24\beta_{9} + 200\beta_{6} + 144\beta_{4} - 148\beta_{3} + 216\beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 40\beta_{15} + 84\beta_{14} - 132\beta_{12} - 216\beta_{8} + 188\beta_{7} + 212\beta_{5} + 216\beta_{2} + 1852 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 260 \beta_{13} + 28 \beta_{11} - 92 \beta_{10} - 264 \beta_{9} + 672 \beta_{6} - 116 \beta_{4} + 256 \beta_{3} + 1728 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 232 \beta_{15} - 896 \beta_{14} + 320 \beta_{12} - 736 \beta_{8} + 1336 \beta_{7} - 720 \beta_{5} + 1728 \beta_{2} - 4568 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 72 \beta_{13} - 600 \beta_{11} - 1864 \beta_{10} + 640 \beta_{9} + 80 \beta_{6} + 2088 \beta_{4} + 1496 \beta_{3} - 5328 \beta_1 \) Copy content Toggle raw display

Character values

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

\(n\) \(55\) \(109\) \(137\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
163.1
−1.98451 0.248465i
−1.98451 + 0.248465i
−1.77866 0.914534i
−1.77866 + 0.914534i
−0.862987 1.80423i
−0.862987 + 1.80423i
−0.808307 1.82938i
−0.808307 + 1.82938i
0.808307 1.82938i
0.808307 + 1.82938i
0.862987 1.80423i
0.862987 + 1.80423i
1.77866 0.914534i
1.77866 + 0.914534i
1.98451 0.248465i
1.98451 + 0.248465i
−1.98451 0.248465i 0 3.87653 + 0.986159i 8.47512i 0 7.86423i −7.44797 2.92022i 0 2.10577 16.8189i
163.2 −1.98451 + 0.248465i 0 3.87653 0.986159i 8.47512i 0 7.86423i −7.44797 + 2.92022i 0 2.10577 + 16.8189i
163.3 −1.77866 0.914534i 0 2.32725 + 3.25329i 3.96500i 0 3.99887i −1.16415 7.91484i 0 −3.62613 + 7.05238i
163.4 −1.77866 + 0.914534i 0 2.32725 3.25329i 3.96500i 0 3.99887i −1.16415 + 7.91484i 0 −3.62613 7.05238i
163.5 −0.862987 1.80423i 0 −2.51051 + 3.11406i 1.42436i 0 4.94379i 7.78502 + 1.84214i 0 −2.56987 + 1.22920i
163.6 −0.862987 + 1.80423i 0 −2.51051 3.11406i 1.42436i 0 4.94379i 7.78502 1.84214i 0 −2.56987 1.22920i
163.7 −0.808307 1.82938i 0 −2.69328 + 2.95740i 5.51565i 0 11.2126i 7.58722 + 2.53655i 0 10.0902 4.45834i
163.8 −0.808307 + 1.82938i 0 −2.69328 2.95740i 5.51565i 0 11.2126i 7.58722 2.53655i 0 10.0902 + 4.45834i
163.9 0.808307 1.82938i 0 −2.69328 2.95740i 5.51565i 0 11.2126i −7.58722 + 2.53655i 0 10.0902 + 4.45834i
163.10 0.808307 + 1.82938i 0 −2.69328 + 2.95740i 5.51565i 0 11.2126i −7.58722 2.53655i 0 10.0902 4.45834i
163.11 0.862987 1.80423i 0 −2.51051 3.11406i 1.42436i 0 4.94379i −7.78502 + 1.84214i 0 −2.56987 1.22920i
163.12 0.862987 + 1.80423i 0 −2.51051 + 3.11406i 1.42436i 0 4.94379i −7.78502 1.84214i 0 −2.56987 + 1.22920i
163.13 1.77866 0.914534i 0 2.32725 3.25329i 3.96500i 0 3.99887i 1.16415 7.91484i 0 −3.62613 7.05238i
163.14 1.77866 + 0.914534i 0 2.32725 + 3.25329i 3.96500i 0 3.99887i 1.16415 + 7.91484i 0 −3.62613 + 7.05238i
163.15 1.98451 0.248465i 0 3.87653 0.986159i 8.47512i 0 7.86423i 7.44797 2.92022i 0 2.10577 + 16.8189i
163.16 1.98451 + 0.248465i 0 3.87653 + 0.986159i 8.47512i 0 7.86423i 7.44797 + 2.92022i 0 2.10577 16.8189i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 163.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
8.d odd 2 1 inner
24.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 216.3.b.b 16
3.b odd 2 1 inner 216.3.b.b 16
4.b odd 2 1 864.3.b.b 16
8.b even 2 1 864.3.b.b 16
8.d odd 2 1 inner 216.3.b.b 16
12.b even 2 1 864.3.b.b 16
24.f even 2 1 inner 216.3.b.b 16
24.h odd 2 1 864.3.b.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
216.3.b.b 16 1.a even 1 1 trivial
216.3.b.b 16 3.b odd 2 1 inner
216.3.b.b 16 8.d odd 2 1 inner
216.3.b.b 16 24.f even 2 1 inner
864.3.b.b 16 4.b odd 2 1
864.3.b.b 16 8.b even 2 1
864.3.b.b 16 12.b even 2 1
864.3.b.b 16 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 120T_{5}^{6} + 4032T_{5}^{4} + 42048T_{5}^{2} + 69696 \) acting on \(S_{3}^{\mathrm{new}}(216, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - 2 T^{14} - 2 T^{12} + \cdots + 65536 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{8} + 120 T^{6} + 4032 T^{4} + \cdots + 69696)^{2} \) Copy content Toggle raw display
$7$ \( (T^{8} + 228 T^{6} + 15750 T^{4} + \cdots + 3038913)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} - 568 T^{6} + 94272 T^{4} + \cdots + 31294528)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + 732 T^{6} + 172134 T^{4} + \cdots + 300808737)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} - 1000 T^{6} + 141504 T^{4} + \cdots + 33082432)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 16 T^{3} - 750 T^{2} + \cdots + 59233)^{4} \) Copy content Toggle raw display
$23$ \( (T^{8} + 1704 T^{6} + 643392 T^{4} + \cdots + 655769664)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + 3360 T^{6} + \cdots + 89942409216)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + 4080 T^{6} + \cdots + 64927957248)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + 7596 T^{6} + \cdots + 225812702097)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} - 8896 T^{6} + \cdots + 11517064118272)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 16 T^{3} - 3936 T^{2} + \cdots + 417856)^{4} \) Copy content Toggle raw display
$47$ \( (T^{8} + 7464 T^{6} + \cdots + 54006041664)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + 17568 T^{6} + \cdots + 13649064247296)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} - 21496 T^{6} + \cdots + 12544953873472)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + 11388 T^{6} + \cdots + 199870871553)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 16 T + 37)^{8} \) Copy content Toggle raw display
$71$ \( (T^{8} + 19584 T^{6} + \cdots + 516742447104)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} - 20 T^{3} - 4458 T^{2} + \cdots + 3184753)^{4} \) Copy content Toggle raw display
$79$ \( (T^{8} + 33780 T^{6} + \cdots + 593133122547153)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} - 38464 T^{6} + \cdots + 4297326592)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} - 35368 T^{6} + \cdots + 44\!\cdots\!12)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 20 T^{3} - 23394 T^{2} + \cdots + 34042441)^{4} \) Copy content Toggle raw display
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