Properties

Label 576.7.g.p
Level $576$
Weight $7$
Character orbit 576.g
Analytic conductor $132.511$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [576,7,Mod(127,576)]
 
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, 0, 0]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("576.127");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 576 = 2^{6} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 576.g (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: \(132.511152165\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.50898483.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 8x^{4} - 10x^{3} + 64x^{2} - 40x + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{30}\cdot 3^{5} \)
Twist minimal: no (minimal twist has level 12)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 - 7) q^{5} + ( - \beta_{3} + 2 \beta_{2}) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 - 7) q^{5} + ( - \beta_{3} + 2 \beta_{2}) q^{7} + ( - \beta_{4} - 2 \beta_{3}) q^{11} + ( - \beta_{5} + 2 \beta_1 + 557) q^{13} + (2 \beta_{5} + 6 \beta_1 - 2038) q^{17} + ( - 18 \beta_{3} - 29 \beta_{2}) q^{19} + ( - 4 \beta_{4} - 14 \beta_{3} + 98 \beta_{2}) q^{23} + (10 \beta_{5} + 52 \beta_1 + 9389) q^{25} + ( - 6 \beta_{5} + 65 \beta_1 - 14167) q^{29} + ( - 18 \beta_{4} + 7 \beta_{3} + 14 \beta_{2}) q^{31} + (19 \beta_{4} - 118 \beta_{3} + 409 \beta_{2}) q^{35} + ( - 23 \beta_{5} + 316 \beta_1 - 3579) q^{37} + (2 \beta_{5} + 186 \beta_1 + 10814) q^{41} + (18 \beta_{4} - 38 \beta_{3} - 713 \beta_{2}) q^{43} + (22 \beta_{4} - 106 \beta_{3} + 1262 \beta_{2}) q^{47} + ( - 22 \beta_{5} + 728 \beta_1 - 18841) q^{49} + (2 \beta_{5} + 337 \beta_1 + 78273) q^{53} + (126 \beta_{4} - 32 \beta_{3} - 2624 \beta_{2}) q^{55} + ( - 98 \beta_{4} + 464 \beta_{3} + 2309 \beta_{2}) q^{59} + (125 \beta_{5} + 416 \beta_1 - 2251) q^{61} + ( - 50 \beta_{5} + 462 \beta_1 - 53756) q^{65} + ( - 54 \beta_{4} + 464 \beta_{3} - 3691 \beta_{2}) q^{67} + ( - 30 \beta_{4} + 738 \beta_{3} + 3256 \beta_{2}) q^{71} + ( - 106 \beta_{5} - 616 \beta_1 - 213732) q^{73} + (176 \beta_{5} - 460 \beta_1 - 247676) q^{77} + ( - 306 \beta_{4} - 277 \beta_{3} - 6894 \beta_{2}) q^{79} + (209 \beta_{4} - 122 \beta_{3} + 7836 \beta_{2}) q^{83} + ( - 450 \beta_1 - 135670) q^{85} + (68 \beta_{5} - 2412 \beta_1 - 120994) q^{89} + ( - 252 \beta_{4} - 262 \beta_{3} - 8256 \beta_{2}) q^{91} + ( - 48 \beta_{4} - 1344 \beta_{3} + 7622 \beta_{2}) q^{95} + (176 \beta_{5} - 1828 \beta_1 + 542238) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 44 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 44 q^{5} + 3348 q^{13} - 12220 q^{17} + 56418 q^{25} - 84860 q^{29} - 20796 q^{37} + 65252 q^{41} - 111546 q^{49} + 470308 q^{53} - 12924 q^{61} - 321512 q^{65} - 1283412 q^{73} - 1487328 q^{77} - 814920 q^{85} - 730924 q^{89} + 3249420 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 8x^{4} - 10x^{3} + 64x^{2} - 40x + 25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( -2\nu^{4} - 10\nu^{3} - 16\nu^{2} + 10\nu - 35 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 9\nu^{5} + 72\nu^{3} - 45\nu^{2} + 576\nu - 180 ) / 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 67\nu^{5} + 30\nu^{4} + 546\nu^{3} - 575\nu^{2} + 4298\nu - 1390 ) / 5 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 257\nu^{5} + 120\nu^{4} + 1776\nu^{3} - 2245\nu^{2} + 11368\nu - 3740 ) / 5 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -64\nu^{4} - 32\nu^{3} - 512\nu^{2} + 320\nu - 2571 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -2\beta_{5} - 9\beta_{4} + 36\beta_{3} - 11\beta_{2} + 64\beta _1 - 22 ) / 9216 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -10\beta_{5} - 3\beta_{4} - 84\beta_{3} + 711\beta_{2} + 32\beta _1 - 24590 ) / 9216 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{5} - 32\beta _1 + 1451 ) / 288 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -90\beta_{5} - 21\beta_{4} + 852\beta_{3} - 5743\beta_{2} + 576\beta _1 - 196830 ) / 9216 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -178\beta_{5} + 561\beta_{4} - 2724\beta_{3} + 9379\beta_{2} + 4256\beta _1 - 308678 ) / 9216 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
127.1
1.21966 2.11251i
1.21966 + 2.11251i
0.330560 + 0.572547i
0.330560 0.572547i
−1.55022 + 2.68505i
−1.55022 2.68505i
0 0 0 −212.349 0 87.6116i 0 0 0
127.2 0 0 0 −212.349 0 87.6116i 0 0 0
127.3 0 0 0 18.1171 0 321.465i 0 0 0
127.4 0 0 0 18.1171 0 321.465i 0 0 0
127.5 0 0 0 172.232 0 545.623i 0 0 0
127.6 0 0 0 172.232 0 545.623i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 127.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 576.7.g.p 6
3.b odd 2 1 192.7.g.e 6
4.b odd 2 1 inner 576.7.g.p 6
8.b even 2 1 36.7.d.e 6
8.d odd 2 1 36.7.d.e 6
12.b even 2 1 192.7.g.e 6
24.f even 2 1 12.7.d.a 6
24.h odd 2 1 12.7.d.a 6
48.i odd 4 2 768.7.b.h 12
48.k even 4 2 768.7.b.h 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.7.d.a 6 24.f even 2 1
12.7.d.a 6 24.h odd 2 1
36.7.d.e 6 8.b even 2 1
36.7.d.e 6 8.d odd 2 1
192.7.g.e 6 3.b odd 2 1
192.7.g.e 6 12.b even 2 1
576.7.g.p 6 1.a even 1 1 trivial
576.7.g.p 6 4.b odd 2 1 inner
768.7.b.h 12 48.i odd 4 2
768.7.b.h 12 48.k even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{3} + 22T_{5}^{2} - 37300T_{5} + 662600 \) acting on \(S_{7}^{\mathrm{new}}(576, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( (T^{3} + 22 T^{2} - 37300 T + 662600)^{2} \) Copy content Toggle raw display
$7$ \( T^{6} + \cdots + 236143965745152 \) Copy content Toggle raw display
$11$ \( T^{6} + 8147856 T^{4} + \cdots + 36\!\cdots\!68 \) Copy content Toggle raw display
$13$ \( (T^{3} - 1674 T^{2} - 3363636 T + 94471112)^{2} \) Copy content Toggle raw display
$17$ \( (T^{3} + 6110 T^{2} + \cdots - 2355174680)^{2} \) Copy content Toggle raw display
$19$ \( T^{6} + 136216080 T^{4} + \cdots + 19\!\cdots\!48 \) Copy content Toggle raw display
$23$ \( T^{6} + 280862976 T^{4} + \cdots + 26\!\cdots\!92 \) Copy content Toggle raw display
$29$ \( (T^{3} + 42430 T^{2} + \cdots + 391390375016)^{2} \) Copy content Toggle raw display
$31$ \( T^{6} + 2161233552 T^{4} + \cdots + 15\!\cdots\!28 \) Copy content Toggle raw display
$37$ \( (T^{3} + 10398 T^{2} + \cdots + 154889066690024)^{2} \) Copy content Toggle raw display
$41$ \( (T^{3} - 32626 T^{2} + \cdots + 1472139052456)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + 8785307280 T^{4} + \cdots + 13\!\cdots\!68 \) Copy content Toggle raw display
$47$ \( T^{6} + 25020342336 T^{4} + \cdots + 54\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{3} - 235154 T^{2} + \cdots - 201035875990744)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} + 214124946960 T^{4} + \cdots + 17\!\cdots\!28 \) Copy content Toggle raw display
$61$ \( (T^{3} + 6462 T^{2} + \cdots + 59\!\cdots\!84)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + 246351150480 T^{4} + \cdots + 16\!\cdots\!12 \) Copy content Toggle raw display
$71$ \( T^{6} + 353413380096 T^{4} + \cdots + 40\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{3} + 641706 T^{2} + \cdots - 54\!\cdots\!28)^{2} \) Copy content Toggle raw display
$79$ \( T^{6} + 1223799739536 T^{4} + \cdots + 12\!\cdots\!72 \) Copy content Toggle raw display
$83$ \( T^{6} + 1007078587536 T^{4} + \cdots + 55\!\cdots\!52 \) Copy content Toggle raw display
$89$ \( (T^{3} + 365462 T^{2} + \cdots - 49\!\cdots\!24)^{2} \) Copy content Toggle raw display
$97$ \( (T^{3} - 1624710 T^{2} + \cdots - 65\!\cdots\!16)^{2} \) Copy content Toggle raw display
show more
show less