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Properties

Label 864.3.b.b
Level $864$
Weight $3$
Character orbit 864.b
Analytic conductor $23.542$
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] = [864,3,Mod(271,864)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(864, 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("864.271");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 864 = 2^{5} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 864.b (of order \(2\), degree \(1\), 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: \(23.5422948407\)
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: \( 2^{28}\cdot 3^{8} \)
Twist minimal: no (minimal twist has level 216)
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{5} - \beta_{9} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{5} - \beta_{9} q^{7} + \beta_{7} q^{11} + \beta_{12} q^{13} - \beta_{14} q^{17} + (\beta_{13} + 4) q^{19} + \beta_{6} q^{23} + ( - \beta_{13} + 2 \beta_{4} + \cdots - 5) q^{25}+ \cdots + (3 \beta_{13} - 4 \beta_{4} - 3 \beta_1 + 5) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 64 q^{19} - 80 q^{25} + 64 q^{43} - 128 q^{49} - 128 q^{67} + 80 q^{73} - 192 q^{91} + 80 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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

\(\beta_{1}\)\(=\) \( ( -\nu^{14} - 14\nu^{12} + 98\nu^{10} - 40\nu^{8} - 656\nu^{6} - 2944\nu^{4} + 38400\nu^{2} - 4096 ) / 4096 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{15} + 10\nu^{13} - 78\nu^{11} + 168\nu^{9} - 720\nu^{7} + 3584\nu^{5} - 15872\nu^{3} + 36864\nu ) / 8192 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -3\nu^{15} - 2\nu^{13} + 86\nu^{11} + 56\nu^{9} - 880\nu^{7} + 17920\nu^{3} - 69632\nu ) / 8192 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3\nu^{14} - 22\nu^{12} + 90\nu^{10} - 264\nu^{8} + 944\nu^{6} - 6528\nu^{4} + 19968\nu^{2} - 28672 ) / 4096 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{14} - 2\nu^{12} - 2\nu^{10} - 56\nu^{8} + 400\nu^{6} - 896\nu^{4} + 3584\nu^{2} - 8192 ) / 1024 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{15} + 12\nu^{13} - 66\nu^{11} + 260\nu^{9} - 608\nu^{7} + 3232\nu^{5} - 19456\nu^{3} + 37888\nu ) / 2048 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{15} + 5\nu^{13} + 4\nu^{11} - 30\nu^{9} - 200\nu^{7} + 752\nu^{5} + 2048\nu^{3} - 16896\nu ) / 1024 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 7\nu^{13} - 38\nu^{11} + 98\nu^{9} - 216\nu^{7} + 1968\nu^{5} - 9728\nu^{3} + 17920\nu ) / 512 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 7\nu^{14} - 14\nu^{12} - 78\nu^{10} + 248\nu^{8} + 880\nu^{6} - 1664\nu^{4} - 27136\nu^{2} + 102400 ) / 4096 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -3\nu^{15} + 12\nu^{13} + 10\nu^{11} - 4\nu^{9} - 800\nu^{7} + 2400\nu^{5} + 4608\nu^{3} - 9216\nu ) / 2048 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -\nu^{14} + 2\nu^{12} + 2\nu^{10} - 72\nu^{8} - 144\nu^{6} + 1152\nu^{4} + 3584\nu^{2} - 17408 ) / 512 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 3\nu^{14} - 14\nu^{12} - 54\nu^{10} + 232\nu^{8} + 240\nu^{6} - 2816\nu^{4} - 15872\nu^{2} + 88064 ) / 2048 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 9\nu^{14} - 98\nu^{12} + 334\nu^{10} - 728\nu^{8} + 4624\nu^{6} - 24192\nu^{4} + 72192\nu^{2} - 77824 ) / 4096 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( -17\nu^{15} + 74\nu^{13} + 18\nu^{11} - 280\nu^{9} - 3024\nu^{7} + 17408\nu^{5} + 14848\nu^{3} - 208896\nu ) / 8192 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( -15\nu^{15} + 42\nu^{13} + 38\nu^{11} - 16\nu^{9} - 2128\nu^{7} + 9792\nu^{5} + 38400\nu^{3} - 116736\nu ) / 4096 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{10} - \beta_{7} - \beta_{3} - \beta_{2} ) / 16 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} - \beta_{4} + \beta _1 + 2 ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{15} - 2\beta_{14} + \beta_{8} + \beta_{7} - 2\beta_{6} + \beta_{3} + \beta_{2} ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{13} - 2\beta_{12} + \beta_{11} + 4\beta_{9} - \beta_{5} - 4\beta_{4} + \beta _1 + 4 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 8\beta_{14} + \beta_{10} + 4\beta_{8} - 17\beta_{7} - 8\beta_{6} + 7\beta_{3} - \beta_{2} ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 6\beta_{13} - 4\beta_{12} - 2\beta_{11} + 5\beta_{5} - 23\beta_{4} - 3\beta _1 + 94 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 5\beta_{15} + 6\beta_{14} - 7\beta_{10} + \beta_{8} - 12\beta_{7} + 6\beta_{6} - 16\beta_{3} - 48\beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 7\beta_{13} - 6\beta_{12} - 9\beta_{11} + 4\beta_{9} + 2\beta_{5} - 41\beta_{4} + 12\beta _1 - 274 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 22\beta_{15} - 36\beta_{14} - 25\beta_{10} + 18\beta_{8} + 15\beta_{7} + 12\beta_{6} + 63\beta_{3} - 105\beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 16\beta_{13} - 72\beta_{12} - 24\beta_{11} + 56\beta_{9} - 71\beta_{5} - 5\beta_{4} - 3\beta _1 + 578 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 41 \beta_{15} + 90 \beta_{14} + 94 \beta_{10} + 7 \beta_{8} - 159 \beta_{7} + 42 \beta_{6} + \cdots - 411 \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( -57\beta_{13} - 62\beta_{12} - 97\beta_{11} - 84\beta_{9} + 85\beta_{5} + 112\beta_{4} - 21\beta _1 + 1828 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 48 \beta_{15} - 424 \beta_{14} + 187 \beta_{10} + 140 \beta_{8} + 709 \beta_{7} + 104 \beta_{6} + \cdots - 1499 \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( -278\beta_{13} - 412\beta_{12} - 46\beta_{11} + 896\beta_{9} + 7\beta_{5} + 923\beta_{4} + 367\beta _1 - 4646 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 121 \beta_{15} - 670 \beta_{14} + 231 \beta_{10} + 811 \beta_{8} + 632 \beta_{7} + \cdots - 164 \beta_{2} \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/864\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(353\) \(703\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
271.1
1.98451 + 0.248465i
−1.98451 + 0.248465i
−0.808307 + 1.82938i
0.808307 + 1.82938i
−1.77866 − 0.914534i
1.77866 − 0.914534i
−0.862987 − 1.80423i
0.862987 − 1.80423i
0.862987 + 1.80423i
−0.862987 + 1.80423i
1.77866 + 0.914534i
−1.77866 + 0.914534i
0.808307 − 1.82938i
−0.808307 − 1.82938i
−1.98451 − 0.248465i
1.98451 − 0.248465i
0 0 0 − 8.47512i 0 − 7.86423i 0 0 0
271.2 0 0 0 − 8.47512i 0 7.86423i 0 0 0
271.3 0 0 0 − 5.51565i 0 − 11.2126i 0 0 0
271.4 0 0 0 − 5.51565i 0 11.2126i 0 0 0
271.5 0 0 0 − 3.96500i 0 − 3.99887i 0 0 0
271.6 0 0 0 − 3.96500i 0 3.99887i 0 0 0
271.7 0 0 0 − 1.42436i 0 − 4.94379i 0 0 0
271.8 0 0 0 − 1.42436i 0 4.94379i 0 0 0
271.9 0 0 0 1.42436i 0 − 4.94379i 0 0 0
271.10 0 0 0 1.42436i 0 4.94379i 0 0 0
271.11 0 0 0 3.96500i 0 − 3.99887i 0 0 0
271.12 0 0 0 3.96500i 0 3.99887i 0 0 0
271.13 0 0 0 5.51565i 0 − 11.2126i 0 0 0
271.14 0 0 0 5.51565i 0 11.2126i 0 0 0
271.15 0 0 0 8.47512i 0 − 7.86423i 0 0 0
271.16 0 0 0 8.47512i 0 7.86423i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 271.16
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 864.3.b.b 16
3.b odd 2 1 inner 864.3.b.b 16
4.b odd 2 1 216.3.b.b ✓ 16
8.b even 2 1 216.3.b.b ✓ 16
8.d odd 2 1 inner 864.3.b.b 16
12.b even 2 1 216.3.b.b ✓ 16
24.f even 2 1 inner 864.3.b.b 16
24.h odd 2 1 216.3.b.b ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
216.3.b.b ✓ 16 4.b odd 2 1
216.3.b.b ✓ 16 8.b even 2 1
216.3.b.b ✓ 16 12.b even 2 1
216.3.b.b ✓ 16 24.h odd 2 1
864.3.b.b 16 1.a even 1 1 trivial
864.3.b.b 16 3.b odd 2 1 inner
864.3.b.b 16 8.d odd 2 1 inner
864.3.b.b 16 24.f even 2 1 inner

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 \(S_{3}^{\mathrm{new}}(864, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{8} + 120 T^{6} + \cdots + 69696)^{2} \) Copy content Toggle raw display
$7$ \( (T^{8} + 228 T^{6} + \cdots + 3038913)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} - 568 T^{6} + \cdots + 31294528)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + 732 T^{6} + \cdots + 300808737)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} - 1000 T^{6} + \cdots + 33082432)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 16 T^{3} + \cdots + 59233)^{4} \) Copy content Toggle raw display
$23$ \( (T^{8} + 1704 T^{6} + \cdots + 655769664)^{2} \) 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
$41$ \( (T^{8} + \cdots + 11517064118272)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 16 T^{3} + \cdots + 417856)^{4} \) Copy content Toggle raw display
$47$ \( (T^{8} + 7464 T^{6} + \cdots + 54006041664)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots + 13649064247296)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots + 12544953873472)^{2} \) 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
$73$ \( (T^{4} - 20 T^{3} + \cdots + 3184753)^{4} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 593133122547153)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} - 38464 T^{6} + \cdots + 4297326592)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 44\!\cdots\!12)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 20 T^{3} + \cdots + 34042441)^{4} \) Copy content Toggle raw display
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