Properties

Label 864.3.b.a
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: \( x^{16} + x^{14} - 8x^{12} + 4x^{10} + 160x^{8} + 64x^{6} - 2048x^{4} + 4096x^{2} + 65536 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{30}\cdot 3^{10} \)
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.

\(f(q)\) \(=\) \( q + \beta_{4} q^{5} + \beta_{3} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{5} + \beta_{3} q^{7} - \beta_{9} q^{11} + \beta_{2} q^{13} + (\beta_{7} - \beta_1) q^{17} + (\beta_{5} - 2) q^{19} + \beta_{10} q^{23} + (\beta_{6} - 5) q^{25} + (\beta_{12} - \beta_{10}) q^{29} + \beta_{8} q^{31} + (\beta_{13} - 2 \beta_{7}) q^{35} + ( - \beta_{11} + \beta_{8}) q^{37} + ( - \beta_{13} + 2 \beta_{9} + \beta_{7} + 2 \beta_1) q^{41} + ( - \beta_{15} + \beta_{6} - 8) q^{43} + ( - \beta_{14} - 2 \beta_{12} - 3 \beta_{4}) q^{47} + ( - \beta_{15} + 2 \beta_{5} - 5) q^{49} + (\beta_{14} - \beta_{12} - \beta_{10}) q^{53} + ( - \beta_{11} + \beta_{8} + \beta_{3} + 3 \beta_{2}) q^{55} + ( - \beta_{13} - 2 \beta_{7} + 3 \beta_1) q^{59} + (2 \beta_{11} + 2 \beta_{8} + 4 \beta_{3} + 2 \beta_{2}) q^{61} + (4 \beta_{9} - \beta_{7} - 3 \beta_1) q^{65} + (\beta_{15} + 3 \beta_{6} + 2 \beta_{5} - 8) q^{67} + (3 \beta_{14} + 2 \beta_{12} + \beta_{10} - 7 \beta_{4}) q^{71} + (\beta_{6} + 4 \beta_{5} - 10) q^{73} + ( - 2 \beta_{14} + \beta_{4}) q^{77} + ( - \beta_{11} + 5 \beta_{3} + 7 \beta_{2}) q^{79} + ( - \beta_{13} - \beta_{9} + 2 \beta_{7} + \beta_1) q^{83} + ( - 4 \beta_{11} + 12 \beta_{3} + \beta_{2}) q^{85} + (3 \beta_{13} - 2 \beta_{9} + 3 \beta_1) q^{89} + ( - \beta_{15} + \beta_{6} + \beta_{5} + 6) q^{91} + ( - 4 \beta_{14} - 3 \beta_{10} + 4 \beta_{4}) q^{95} + (\beta_{15} + 3 \beta_{6} + 2 \beta_{5} - 13) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 32 q^{19} - 80 q^{25} - 128 q^{43} - 80 q^{49} - 128 q^{67} - 160 q^{73} + 96 q^{91} - 208 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( ( -\nu^{15} - 9\nu^{13} - 196\nu^{9} + 576\nu^{7} + 1728\nu^{5} + 512\nu^{3} ) / 32768 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{12} - 3\nu^{10} + 4\nu^{8} - 12\nu^{6} + 208\nu^{4} + 256\nu^{2} - 1024 ) / 512 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{14} + \nu^{12} + 24\nu^{10} + 164\nu^{8} + 544\nu^{6} - 320\nu^{4} + 3584\nu^{2} + 16384 ) / 8192 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{15} - 7\nu^{13} + 48\nu^{11} + 132\nu^{9} - 384\nu^{7} + 3136\nu^{5} + 3584\nu^{3} ) / 32768 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{12} + \nu^{10} - 8\nu^{8} + 4\nu^{6} - 96\nu^{4} - 192\nu^{2} - 1024 ) / 256 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{14} - 7\nu^{12} + 48\nu^{10} + 132\nu^{8} - 1408\nu^{6} + 64\nu^{4} + 13824\nu^{2} + 12288 ) / 4096 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3\nu^{15} - 37\nu^{13} - 192\nu^{11} + 460\nu^{9} + 576\nu^{7} - 1600\nu^{5} + 29184\nu^{3} + 49152\nu ) / 32768 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -7\nu^{14} - 7\nu^{12} - 40\nu^{10} + 516\nu^{8} - 1248\nu^{6} - 7488\nu^{4} - 14848\nu^{2} + 16384 ) / 8192 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -5\nu^{15} + 19\nu^{13} - 64\nu^{11} - 84\nu^{9} - 448\nu^{7} + 8128\nu^{5} - 15872\nu^{3} - 49152\nu ) / 32768 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -\nu^{15} + 3\nu^{13} + 12\nu^{11} + 92\nu^{9} + 496\nu^{7} + 64\nu^{5} + 2816\nu^{3} - 2048\nu ) / 4096 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -7\nu^{14} + 9\nu^{12} + 168\nu^{10} - 188\nu^{8} - 416\nu^{6} + 9152\nu^{4} - 6656\nu^{2} - 65536 ) / 8192 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( -\nu^{15} + 7\nu^{13} + 16\nu^{11} - 68\nu^{9} - 128\nu^{7} + 1216\nu^{5} + 10752\nu^{3} - 28672\nu ) / 4096 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 15\nu^{15} + 7\nu^{13} - 128\nu^{11} - 132\nu^{9} + 3136\nu^{7} + 2752\nu^{5} - 32256\nu^{3} - 196608\nu ) / 32768 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 17\nu^{15} + 9\nu^{13} - 80\nu^{11} + 196\nu^{9} + 2176\nu^{7} + 4160\nu^{5} - 29184\nu^{3} + 327680\nu ) / 32768 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( -9\nu^{14} - \nu^{12} + 16\nu^{10} - 164\nu^{8} + 128\nu^{6} - 576\nu^{4} + 35328\nu^{2} - 36864 ) / 4096 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{14} - \beta_{13} - \beta_{4} + \beta_1 ) / 16 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{15} - \beta_{11} - \beta_{8} + \beta_{6} + 2\beta_{3} + \beta_{2} - 2 ) / 16 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{14} - \beta_{13} + 4\beta_{12} - 4\beta_{9} + 4\beta_{7} - \beta_{4} + \beta_1 ) / 16 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -\beta_{15} + 5\beta_{11} - 3\beta_{8} - \beta_{6} - 8\beta_{5} - 2\beta_{3} + 11\beta_{2} + 34 ) / 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 7\beta_{14} + \beta_{13} + 4\beta_{12} + 28\beta_{9} + 4\beta_{7} + 57\beta_{4} + 31\beta_1 ) / 16 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 5\beta_{15} - \beta_{11} - 9\beta_{8} - 27\beta_{6} - 8\beta_{5} + 74\beta_{3} - 15\beta_{2} - 74 ) / 16 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 13\beta_{14} + 27\beta_{13} - 4\beta_{12} + 64\beta_{10} - 28\beta_{9} - 4\beta_{7} - 77\beta_{4} + 197\beta_1 ) / 16 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -17\beta_{15} + 13\beta_{11} + 85\beta_{8} + 15\beta_{6} - 56\beta_{5} + 350\beta_{3} + 131\beta_{2} - 926 ) / 16 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 57 \beta_{14} + 17 \beta_{13} - 108 \beta_{12} + 192 \beta_{10} + 140 \beta_{9} + 148 \beta_{7} + 633 \beta_{4} - 1009 \beta_1 ) / 16 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( -35\beta_{15} + 311\beta_{11} - 49\beta_{8} + 189\beta_{6} + 280\beta_{5} + 826\beta_{3} - 647\beta_{2} - 122 ) / 16 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 91 \beta_{14} - 93 \beta_{13} - 68 \beta_{12} + 320 \beta_{10} - 1372 \beta_{9} - 1348 \beta_{7} + 2843 \beta_{4} + 701 \beta_1 ) / 16 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 25 \beta_{15} + 85 \beta_{11} + 285 \beta_{8} + 135 \beta_{6} + 2632 \beta_{5} + 1870 \beta_{3} + 3003 \beta_{2} + 12274 ) / 16 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 2495 \beta_{14} + 25 \beta_{13} + 2420 \beta_{12} + 960 \beta_{10} + 2476 \beta_{9} - 2444 \beta_{7} - 10623 \beta_{4} - 14585 \beta_1 ) / 16 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 2971 \beta_{15} - 3953 \beta_{11} - 5529 \beta_{8} + 3653 \beta_{6} + 1624 \beta_{5} + 3914 \beta_{3} - 863 \beta_{2} - 61322 ) / 16 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 8813 \beta_{14} + 13211 \beta_{13} + 6044 \beta_{12} - 9408 \beta_{10} - 19516 \beta_{9} - 356 \beta_{7} + 25171 \beta_{4} - 27707 \beta_1 ) / 16 \) Copy content Toggle raw display

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
−0.941181 + 1.76470i
0.941181 + 1.76470i
1.95589 0.417734i
−1.95589 0.417734i
0.316912 1.97473i
−0.316912 1.97473i
1.71413 + 1.03041i
−1.71413 + 1.03041i
−1.71413 1.03041i
1.71413 1.03041i
−0.316912 + 1.97473i
0.316912 + 1.97473i
−1.95589 + 0.417734i
1.95589 + 0.417734i
0.941181 1.76470i
−0.941181 1.76470i
0 0 0 8.60639i 0 4.81948i 0 0 0
271.2 0 0 0 8.60639i 0 4.81948i 0 0 0
271.3 0 0 0 5.14075i 0 12.6525i 0 0 0
271.4 0 0 0 5.14075i 0 12.6525i 0 0 0
271.5 0 0 0 4.41296i 0 3.59985i 0 0 0
271.6 0 0 0 4.41296i 0 3.59985i 0 0 0
271.7 0 0 0 0.169019i 0 4.44165i 0 0 0
271.8 0 0 0 0.169019i 0 4.44165i 0 0 0
271.9 0 0 0 0.169019i 0 4.44165i 0 0 0
271.10 0 0 0 0.169019i 0 4.44165i 0 0 0
271.11 0 0 0 4.41296i 0 3.59985i 0 0 0
271.12 0 0 0 4.41296i 0 3.59985i 0 0 0
271.13 0 0 0 5.14075i 0 12.6525i 0 0 0
271.14 0 0 0 5.14075i 0 12.6525i 0 0 0
271.15 0 0 0 8.60639i 0 4.81948i 0 0 0
271.16 0 0 0 8.60639i 0 4.81948i 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.a 16
3.b odd 2 1 inner 864.3.b.a 16
4.b odd 2 1 216.3.b.a 16
8.b even 2 1 216.3.b.a 16
8.d odd 2 1 inner 864.3.b.a 16
12.b even 2 1 216.3.b.a 16
24.f even 2 1 inner 864.3.b.a 16
24.h odd 2 1 216.3.b.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
216.3.b.a 16 4.b odd 2 1
216.3.b.a 16 8.b even 2 1
216.3.b.a 16 12.b even 2 1
216.3.b.a 16 24.h odd 2 1
864.3.b.a 16 1.a even 1 1 trivial
864.3.b.a 16 3.b odd 2 1 inner
864.3.b.a 16 8.d odd 2 1 inner
864.3.b.a 16 24.f even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 120T_{5}^{6} + 3918T_{5}^{4} + 38232T_{5}^{2} + 1089 \) 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
$5$ \( (T^{8} + 120 T^{6} + 3918 T^{4} + \cdots + 1089)^{2} \) Copy content Toggle raw display
$7$ \( (T^{8} + 216 T^{6} + 9966 T^{4} + \cdots + 950625)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} - 508 T^{6} + 60486 T^{4} + \cdots + 4060225)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + 600 T^{6} + 90768 T^{4} + \cdots + 58982400)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} - 1336 T^{6} + \cdots + 1435500544)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 8 T^{3} - 756 T^{2} + \cdots + 129088)^{4} \) 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
$37$ \( (T^{8} + 7128 T^{6} + \cdots + 4052652134400)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} - 5368 T^{6} + \cdots + 84239257600)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 32 T^{3} - 6096 T^{2} + \cdots + 5478400)^{4} \) Copy content Toggle raw display
$47$ \( (T^{8} + 10176 T^{6} + \cdots + 7614420811776)^{2} \) 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
$61$ \( (T^{8} + 25728 T^{6} + \cdots + 872632396283904)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 32 T^{3} - 13440 T^{2} + \cdots + 8139520)^{4} \) Copy content Toggle raw display
$71$ \( (T^{8} + 25416 T^{6} + \cdots + 90326016000000)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 40 T^{3} - 10962 T^{2} + \cdots + 15407905)^{4} \) Copy content Toggle raw display
$79$ \( (T^{8} + 39000 T^{6} + \cdots + 62\!\cdots\!44)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} - 6940 T^{6} + \cdots + 5715511201)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} - 22240 T^{6} + \cdots + 338984068710400)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 52 T^{3} - 12810 T^{2} + \cdots + 4535185)^{4} \) Copy content Toggle raw display
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