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Properties

Label 576.8.f.a
Level $576$
Weight $8$
Character orbit 576.f
Analytic conductor $179.934$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [576,8,Mod(287,576)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(576, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("576.287");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 576 = 2^{6} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 576.f (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: \(179.933774679\)
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: \( x^{16} + 403381798 x^{12} + \cdots + 47\!\cdots\!25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{48}\cdot 3^{24} \)
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.

\(f(q)\) \(=\) \( q + \beta_{8} q^{5} + ( - \beta_{5} + 17 \beta_{4}) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{8} q^{5} + ( - \beta_{5} + 17 \beta_{4}) q^{7} - \beta_{11} q^{11} - \beta_{3} q^{13} + (\beta_{15} + 21 \beta_{10}) q^{17} + ( - 2 \beta_{7} + \beta_{6}) q^{19} + (11 \beta_{14} + 12 \beta_{13}) q^{23} + ( - 13 \beta_1 - 14351) q^{25} + ( - 17 \beta_{12} + 277 \beta_{8}) q^{29} + (145 \beta_{5} - 2522 \beta_{4}) q^{31} + (35 \beta_{11} + 69 \beta_{9}) q^{35} + ( - 13 \beta_{3} - 25 \beta_{2}) q^{37} + (75 \beta_{15} + 1517 \beta_{10}) q^{41} + ( - 7 \beta_{7} - 43 \beta_{6}) q^{43} + (197 \beta_{14} + 234 \beta_{13}) q^{47} + ( - 140 \beta_1 - 137901) q^{49} + (163 \beta_{12} + 2531 \beta_{8}) q^{53} + (586 \beta_{5} - 12460 \beta_{4}) q^{55} + ( - 236 \beta_{11} + 535 \beta_{9}) q^{59} + (99 \beta_{3} - 197 \beta_{2}) q^{61} + ( - 89 \beta_{15} - 3346 \beta_{10}) q^{65} + ( - 11 \beta_{7} - 234 \beta_{6}) q^{67} + ( - 345 \beta_{14} - 53 \beta_{13}) q^{71} + (166 \beta_1 + 433780) q^{73} + ( - 556 \beta_{12} + 8788 \beta_{8}) q^{77} + ( - 2745 \beta_{5} + 21002 \beta_{4}) q^{79} + (557 \beta_{11} + 931 \beta_{9}) q^{83} + (210 \beta_{3} - 131 \beta_{2}) q^{85} + ( - 350 \beta_{15} - 17957 \beta_{10}) q^{89} + ( - 252 \beta_{7} + 391 \beta_{6}) q^{91} + ( - 1714 \beta_{14} + 145 \beta_{13}) q^{95} + (2357 \beta_1 + 4085000) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 229616 q^{25} - 2206416 q^{49} + 6940480 q^{73} + 65360000 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 403381798 x^{12} + \cdots + 47\!\cdots\!25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 15942 \nu^{12} - 4822578573724 \nu^{8} + \cdots + 29\!\cdots\!50 ) / 78\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 54712573367103 \nu^{12} + \cdots + 66\!\cdots\!25 ) / 63\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 83\!\cdots\!96 \nu^{12} + \cdots + 11\!\cdots\!50 ) / 57\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1430676885136 \nu^{14} + \cdots + 60\!\cdots\!56 \nu^{2} ) / 46\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 27\!\cdots\!64 \nu^{14} + \cdots - 12\!\cdots\!69 \nu^{2} ) / 15\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 21\!\cdots\!02 \nu^{14} + \cdots - 28\!\cdots\!37 \nu^{2} ) / 17\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 10\!\cdots\!98 \nu^{14} + \cdots + 13\!\cdots\!83 \nu^{2} ) / 52\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 63\!\cdots\!47 \nu^{15} + \cdots + 27\!\cdots\!25 \nu ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 12\!\cdots\!94 \nu^{15} + \cdots - 54\!\cdots\!50 \nu ) / 26\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 17\!\cdots\!07 \nu^{15} + \cdots + 45\!\cdots\!25 \nu ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 14\!\cdots\!92 \nu^{15} + \cdots + 62\!\cdots\!75 \nu ) / 48\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 27\!\cdots\!53 \nu^{15} + \cdots - 11\!\cdots\!25 \nu ) / 48\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 35\!\cdots\!14 \nu^{15} + \cdots + 90\!\cdots\!50 \nu ) / 26\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 42\!\cdots\!44 \nu^{15} + \cdots + 10\!\cdots\!75 \nu ) / 69\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 39\!\cdots\!53 \nu^{15} + \cdots + 10\!\cdots\!50 \nu ) / 34\!\cdots\!75 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{13} - 8\beta_{10} - 81\beta_{9} + 648\beta_{8} ) / 2592 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{7} + 6\beta_{6} + 5616\beta_{5} + 433269\beta_{4} ) / 864 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 2106 \beta_{15} + 4212 \beta_{14} - 24851 \beta_{13} - 73710 \beta_{12} - 147420 \beta_{11} + \cdots - 8227332 \beta_{8} ) / 2592 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -12285\beta_{3} + 27214\beta_{2} + 2797821\beta _1 - 10891308546 ) / 108 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 27979965 \beta_{15} - 55959930 \beta_{14} + 264569093 \beta_{13} - 587505555 \beta_{12} + \cdots - 57078002214 \beta_{8} ) / 1296 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -288623492\beta_{7} - 474159156\beta_{6} - 213309922176\beta_{5} - 12692809671939\beta_{4} ) / 144 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 1120072953636 \beta_{15} - 2240145907272 \beta_{14} + 10288806606161 \beta_{13} + \cdots + 16\!\cdots\!12 \beta_{8} ) / 2592 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 5600168908425\beta_{3} - 10732586126024\beta_{2} - 564135013914966\beta _1 + 2121778169178175386 ) / 108 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 20\!\cdots\!10 \beta_{15} + \cdots + 22\!\cdots\!08 \beta_{8} ) / 2592 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 57\!\cdots\!02 \beta_{7} + \cdots + 15\!\cdots\!49 \beta_{4} ) / 864 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 17\!\cdots\!33 \beta_{15} + \cdots - 15\!\cdots\!46 \beta_{8} ) / 1296 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 13\!\cdots\!45 \beta_{3} + \cdots - 35\!\cdots\!73 ) / 9 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 58\!\cdots\!40 \beta_{15} + \cdots - 44\!\cdots\!88 \beta_{8} ) / 2592 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 15\!\cdots\!02 \beta_{7} + \cdots - 29\!\cdots\!89 \beta_{4} ) / 864 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 94\!\cdots\!46 \beta_{15} + \cdots + 62\!\cdots\!72 \beta_{8} ) / 2592 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/576\mathbb{Z}\right)^\times\).

\(n\) \(65\) \(127\) \(325\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
287.1
−83.5506 − 84.2577i
−84.2577 − 83.5506i
−84.2577 + 83.5506i
−83.5506 + 84.2577i
−30.1724 + 30.8795i
−30.8795 + 30.1724i
−30.8795 − 30.1724i
−30.1724 − 30.8795i
30.1724 − 30.8795i
30.8795 − 30.1724i
30.8795 + 30.1724i
30.1724 + 30.8795i
83.5506 + 84.2577i
84.2577 + 83.5506i
84.2577 − 83.5506i
83.5506 − 84.2577i
0 0 0 −335.617 0 − 659.704i 0 0 0
287.2 0 0 0 −335.617 0 − 659.704i 0 0 0
287.3 0 0 0 −335.617 0 659.704i 0 0 0
287.4 0 0 0 −335.617 0 659.704i 0 0 0
287.5 0 0 0 −122.104 0 − 1219.70i 0 0 0
287.6 0 0 0 −122.104 0 − 1219.70i 0 0 0
287.7 0 0 0 −122.104 0 1219.70i 0 0 0
287.8 0 0 0 −122.104 0 1219.70i 0 0 0
287.9 0 0 0 122.104 0 − 1219.70i 0 0 0
287.10 0 0 0 122.104 0 − 1219.70i 0 0 0
287.11 0 0 0 122.104 0 1219.70i 0 0 0
287.12 0 0 0 122.104 0 1219.70i 0 0 0
287.13 0 0 0 335.617 0 − 659.704i 0 0 0
287.14 0 0 0 335.617 0 − 659.704i 0 0 0
287.15 0 0 0 335.617 0 659.704i 0 0 0
287.16 0 0 0 335.617 0 659.704i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 287.16
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 576.8.f.a ✓ 16
3.b odd 2 1 inner 576.8.f.a ✓ 16
4.b odd 2 1 inner 576.8.f.a ✓ 16
8.b even 2 1 inner 576.8.f.a ✓ 16
8.d odd 2 1 inner 576.8.f.a ✓ 16
12.b even 2 1 inner 576.8.f.a ✓ 16
24.f even 2 1 inner 576.8.f.a ✓ 16
24.h odd 2 1 inner 576.8.f.a ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
576.8.f.a ✓ 16 1.a even 1 1 trivial
576.8.f.a ✓ 16 3.b odd 2 1 inner
576.8.f.a ✓ 16 4.b odd 2 1 inner
576.8.f.a ✓ 16 8.b even 2 1 inner
576.8.f.a ✓ 16 8.d odd 2 1 inner
576.8.f.a ✓ 16 12.b even 2 1 inner
576.8.f.a ✓ 16 24.f even 2 1 inner
576.8.f.a ✓ 16 24.h odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 127548T_{5}^{2} + 1679372100 \) acting on \(S_{8}^{\mathrm{new}}(576, [\chi])\). 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
$5$ \( (T^{4} - 127548 T^{2} + 1679372100)^{4} \) Copy content Toggle raw display
$7$ \( (T^{4} + 1922888 T^{2} + 647451966736)^{4} \) Copy content Toggle raw display
$11$ \( (T^{4} + \cdots + 40749523942464)^{4} \) Copy content Toggle raw display
$13$ \( (T^{4} + \cdots + 77237456792976)^{4} \) Copy content Toggle raw display
$17$ \( (T^{4} + \cdots + 33\!\cdots\!24)^{4} \) Copy content Toggle raw display
$19$ \( (T^{4} + \cdots + 12\!\cdots\!96)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} + \cdots + 30\!\cdots\!24)^{4} \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots + 29\!\cdots\!04)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} + \cdots + 28\!\cdots\!36)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} + \cdots + 51\!\cdots\!36)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots + 10\!\cdots\!64)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots + 70\!\cdots\!56)^{4} \) Copy content Toggle raw display
$47$ \( (T^{4} + \cdots + 30\!\cdots\!84)^{4} \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots + 57\!\cdots\!84)^{4} \) Copy content Toggle raw display
$59$ \( (T^{4} + \cdots + 55\!\cdots\!44)^{4} \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots + 89\!\cdots\!76)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} + \cdots + 74\!\cdots\!36)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots + 35\!\cdots\!04)^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} - 867560 T - 201165479024)^{8} \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots + 42\!\cdots\!96)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + \cdots + 60\!\cdots\!04)^{4} \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots + 13\!\cdots\!84)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} + \cdots - 61804069508096)^{8} \) Copy content Toggle raw display
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