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Properties

Label 216.3.b.a
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: \( x^{16} + x^{14} - 8x^{12} + 4x^{10} + 160x^{8} + 64x^{6} - 2048x^{4} + 4096x^{2} + 65536 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{14}\cdot 3^{10} \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{2} q^{4} - \beta_{7} q^{5} - \beta_{6} q^{7} + \beta_{3} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{2} q^{4} - \beta_{7} q^{5} - \beta_{6} q^{7} + \beta_{3} q^{8} + ( - \beta_{10} - 1) q^{10} + ( - \beta_{13} + \beta_{3}) q^{11} + ( - \beta_{9} + \beta_{2}) q^{13} + (\beta_{15} - \beta_{11} + \cdots - \beta_1) q^{14}+ \cdots + ( - 7 \beta_{15} - 2 \beta_{13} + \cdots - 4 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 2 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 2 q^{4} - 18 q^{10} + 34 q^{16} + 32 q^{19} - 22 q^{22} - 80 q^{25} + 102 q^{28} + 68 q^{34} - 6 q^{40} + 128 q^{43} + 60 q^{46} - 80 q^{49} - 180 q^{52} - 156 q^{58} - 74 q^{64} + 128 q^{67} - 378 q^{70} - 160 q^{73} + 188 q^{76} - 508 q^{82} + 542 q^{88} - 96 q^{91} + 24 q^{94} - 208 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + x^{14} - 8x^{12} + 4x^{10} + 160x^{8} + 64x^{6} - 2048x^{4} + 4096x^{2} + 65536 \) : 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
\(\beta_{4}\)\(=\) \( ( -\nu^{15} - 9\nu^{13} - 196\nu^{9} + 576\nu^{7} + 1728\nu^{5} + 512\nu^{3} ) / 32768 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{12} + 3\nu^{10} - 4\nu^{8} + 12\nu^{6} + 304\nu^{4} + 256\nu^{2} ) / 512 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{14} + \nu^{12} + 24\nu^{10} + 164\nu^{8} + 544\nu^{6} - 320\nu^{4} + 3584\nu^{2} + 16384 ) / 8192 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{15} - 7\nu^{13} + 48\nu^{11} + 132\nu^{9} - 384\nu^{7} + 3136\nu^{5} + 3584\nu^{3} ) / 32768 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -\nu^{14} - \nu^{12} - 24\nu^{10} + 92\nu^{8} + 224\nu^{6} - 192\nu^{4} + 512\nu^{2} + 4096 ) / 4096 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -\nu^{12} + 3\nu^{10} - 4\nu^{8} + 12\nu^{6} - 208\nu^{4} + 256\nu^{2} + 1024 ) / 512 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -\nu^{14} + 7\nu^{12} + 16\nu^{10} - 68\nu^{8} + 384\nu^{6} + 704\nu^{4} - 512\nu^{2} - 12288 ) / 4096 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -3\nu^{15} - 3\nu^{13} + 56\nu^{11} + 148\nu^{9} + 928\nu^{7} - 1600\nu^{5} - 2560\nu^{3} ) / 16384 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 3\nu^{15} - 37\nu^{13} - 192\nu^{11} + 460\nu^{9} + 576\nu^{7} - 1600\nu^{5} - 3584\nu^{3} + 81920\nu ) / 32768 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( -5\nu^{15} + 19\nu^{13} - 64\nu^{11} - 84\nu^{9} - 448\nu^{7} + 8128\nu^{5} + 16896\nu^{3} - 49152\nu ) / 32768 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 5\nu^{14} - 11\nu^{12} - 88\nu^{10} - 12\nu^{8} + 1376\nu^{6} - 2880\nu^{4} - 14848\nu^{2} + 32768 ) / 8192 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( -\nu^{15} - \nu^{13} + 8\nu^{11} - 4\nu^{9} - 160\nu^{7} - 64\nu^{5} + 2048\nu^{3} - 4096\nu ) / 4096 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{9} + \beta_{5} + 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{15} + 2\beta_{13} + 4\beta_{7} + 2\beta_{4} - \beta_{3} + 2\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2\beta_{14} + 4\beta_{10} + \beta_{9} + 2\beta_{8} + 2\beta_{6} + \beta_{5} + 2\beta_{2} - 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -9\beta_{15} + 2\beta_{13} + 8\beta_{11} - 4\beta_{7} + 10\beta_{4} + 5\beta_{3} - 6\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -6\beta_{14} - 12\beta_{10} - 5\beta_{9} + 10\beta_{8} + 26\beta_{6} - \beta_{5} - 22\beta_{2} - 64 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -7\beta_{15} + 14\beta_{13} + 16\beta_{12} + 24\beta_{11} + 36\beta_{7} - 74\beta_{4} - \beta_{3} - 26\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -26\beta_{14} + 12\beta_{10} + 33\beta_{9} - 42\beta_{8} + 70\beta_{6} + 5\beta_{5} - 90\beta_{2} - 24 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 3\beta_{15} - 70\beta_{13} - 80\beta_{12} + 40\beta_{11} + 172\beta_{7} + 18\beta_{4} + 13\beta_{3} + 98\beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( -30\beta_{14} + 132\beta_{10} - 173\beta_{9} - 142\beta_{8} + 130\beta_{6} - 177\beta_{5} + 98\beta_{2} + 744 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 375 \beta_{15} + 366 \beta_{13} - 304 \beta_{12} + 120 \beta_{11} - 508 \beta_{7} - 970 \beta_{4} + \cdots + 934 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 550\beta_{14} - 628\beta_{10} - 663\beta_{9} - 1578\beta_{8} + 1030\beta_{6} - 3\beta_{5} + 998\beta_{2} - 3240 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 2165 \beta_{15} - 1430 \beta_{13} - 400 \beta_{12} - 1176 \beta_{11} + 2124 \beta_{7} + \cdots - 3310 \beta_1 \) Copy content Toggle raw display

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

Character values

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

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

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
−1.95589 − 0.417734i
−1.95589 + 0.417734i
−1.71413 − 1.03041i
−1.71413 + 1.03041i
−0.941181 − 1.76470i
−0.941181 + 1.76470i
−0.316912 − 1.97473i
−0.316912 + 1.97473i
0.316912 − 1.97473i
0.316912 + 1.97473i
0.941181 − 1.76470i
0.941181 + 1.76470i
1.71413 − 1.03041i
1.71413 + 1.03041i
1.95589 − 0.417734i
1.95589 + 0.417734i
−1.95589 − 0.417734i 0 3.65100 + 1.63408i 5.14075i 0 − 12.6525i −6.45833 − 4.72122i 0 2.14746 − 10.0547i
163.2 −1.95589 + 0.417734i 0 3.65100 − 1.63408i − 5.14075i 0 12.6525i −6.45833 + 4.72122i 0 2.14746 + 10.0547i
163.3 −1.71413 − 1.03041i 0 1.87649 + 3.53253i − 0.169019i 0 4.44165i 0.423412 − 7.98879i 0 −0.174160 + 0.289721i
163.4 −1.71413 + 1.03041i 0 1.87649 − 3.53253i 0.169019i 0 − 4.44165i 0.423412 + 7.98879i 0 −0.174160 − 0.289721i
163.5 −0.941181 − 1.76470i 0 −2.22836 + 3.32181i − 8.60639i 0 − 4.81948i 7.95930 + 0.805966i 0 −15.1877 + 8.10017i
163.6 −0.941181 + 1.76470i 0 −2.22836 − 3.32181i 8.60639i 0 4.81948i 7.95930 − 0.805966i 0 −15.1877 − 8.10017i
163.7 −0.316912 − 1.97473i 0 −3.79913 + 1.25163i 4.41296i 0 − 3.59985i 3.67563 + 7.10561i 0 8.71442 − 1.39852i
163.8 −0.316912 + 1.97473i 0 −3.79913 − 1.25163i − 4.41296i 0 3.59985i 3.67563 − 7.10561i 0 8.71442 + 1.39852i
163.9 0.316912 − 1.97473i 0 −3.79913 − 1.25163i 4.41296i 0 3.59985i −3.67563 + 7.10561i 0 8.71442 + 1.39852i
163.10 0.316912 + 1.97473i 0 −3.79913 + 1.25163i − 4.41296i 0 − 3.59985i −3.67563 − 7.10561i 0 8.71442 − 1.39852i
163.11 0.941181 − 1.76470i 0 −2.22836 − 3.32181i − 8.60639i 0 4.81948i −7.95930 + 0.805966i 0 −15.1877 − 8.10017i
163.12 0.941181 + 1.76470i 0 −2.22836 + 3.32181i 8.60639i 0 − 4.81948i −7.95930 − 0.805966i 0 −15.1877 + 8.10017i
163.13 1.71413 − 1.03041i 0 1.87649 − 3.53253i − 0.169019i 0 − 4.44165i −0.423412 − 7.98879i 0 −0.174160 − 0.289721i
163.14 1.71413 + 1.03041i 0 1.87649 + 3.53253i 0.169019i 0 4.44165i −0.423412 + 7.98879i 0 −0.174160 + 0.289721i
163.15 1.95589 − 0.417734i 0 3.65100 − 1.63408i 5.14075i 0 12.6525i 6.45833 − 4.72122i 0 2.14746 + 10.0547i
163.16 1.95589 + 0.417734i 0 3.65100 + 1.63408i − 5.14075i 0 − 12.6525i 6.45833 + 4.72122i 0 2.14746 − 10.0547i
\(n\): 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.a ✓ 16
3.b odd 2 1 inner 216.3.b.a ✓ 16
4.b odd 2 1 864.3.b.a 16
8.b even 2 1 864.3.b.a 16
8.d odd 2 1 inner 216.3.b.a ✓ 16
12.b even 2 1 864.3.b.a 16
24.f even 2 1 inner 216.3.b.a ✓ 16
24.h odd 2 1 864.3.b.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
216.3.b.a ✓ 16 1.a even 1 1 trivial
216.3.b.a ✓ 16 3.b odd 2 1 inner
216.3.b.a ✓ 16 8.d odd 2 1 inner
216.3.b.a ✓ 16 24.f even 2 1 inner
864.3.b.a 16 4.b odd 2 1
864.3.b.a 16 8.b even 2 1
864.3.b.a 16 12.b even 2 1
864.3.b.a 16 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 120T_{5}^{6} + 3918T_{5}^{4} + 38232T_{5}^{2} + 1089 \) acting on \(S_{3}^{\mathrm{new}}(216, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + T^{14} + \cdots + 65536 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{8} + 120 T^{6} + \cdots + 1089)^{2} \) Copy content Toggle raw display
$7$ \( (T^{8} + 216 T^{6} + \cdots + 950625)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} - 508 T^{6} + \cdots + 4060225)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + 600 T^{6} + \cdots + 58982400)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} - 1336 T^{6} + \cdots + 1435500544)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 8 T^{3} + \cdots + 129088)^{4} \) Copy content Toggle raw display
$23$ \( (T^{8} + 2376 T^{6} + \cdots + 49861103616)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + 3768 T^{6} + \cdots + 197136000000)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + 3144 T^{6} + \cdots + 69463346481)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots + 4052652134400)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} - 5368 T^{6} + \cdots + 84239257600)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 32 T^{3} + \cdots + 5478400)^{4} \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots + 7614420811776)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + 10296 T^{6} + \cdots + 269102600001)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} - 8416 T^{6} + \cdots + 12687769600)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots + 872632396283904)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 32 T^{3} + \cdots + 8139520)^{4} \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 90326016000000)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 40 T^{3} + \cdots + 15407905)^{4} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 62\!\cdots\!44)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} - 6940 T^{6} + \cdots + 5715511201)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 338984068710400)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 52 T^{3} + \cdots + 4535185)^{4} \) Copy content Toggle raw display
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