Properties

Label 576.7.g.k
Level $576$
Weight $7$
Character orbit 576.g
Analytic conductor $132.511$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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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: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 96)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 5 \beta_{2} - 50) q^{5} + (5 \beta_{3} + 73 \beta_1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 5 \beta_{2} - 50) q^{5} + (5 \beta_{3} + 73 \beta_1) q^{7} + ( - 19 \beta_{3} + 94 \beta_1) q^{11} + ( - 86 \beta_{2} - 2458) q^{13} + (38 \beta_{2} + 198) q^{17} + (221 \beta_{3} + 1146 \beta_1) q^{19} + (768 \beta_{3} + 430 \beta_1) q^{23} + (500 \beta_{2} - 2325) q^{25} + (629 \beta_{2} - 20698) q^{29} + ( - 169 \beta_{3} + 5209 \beta_1) q^{31} + ( - 1710 \beta_{3} - 6350 \beta_1) q^{35} + ( - 3092 \beta_{2} - 834) q^{37} + ( - 2246 \beta_{2} - 45962) q^{41} + (5087 \beta_{3} - 6058 \beta_1) q^{43} + (2026 \beta_{3} - 2846 \beta_1) q^{47} + ( - 2920 \beta_{2} + 21585) q^{49} + (10853 \beta_{2} + 26070) q^{53} + ( - 930 \beta_{3} + 5560 \beta_1) q^{55} + (6223 \beta_{3} + 33400 \beta_1) q^{59} + ( - 8448 \beta_{2} - 163714) q^{61} + (16590 \beta_{2} + 308660) q^{65} + ( - 277 \beta_{3} - 22264 \beta_1) q^{67} + ( - 444 \beta_{3} + 76838 \beta_1) q^{71} + ( - 18664 \beta_{2} + 360066) q^{73} + (3668 \beta_{2} - 68752) q^{77} + (18783 \beta_{3} + 69509 \beta_1) q^{79} + (2275 \beta_{3} - 178546 \beta_1) q^{83} + ( - 2890 \beta_{2} - 91980) q^{85} + (51188 \beta_{2} - 18066) q^{89} + ( - 37402 \beta_{3} - 225874 \beta_1) q^{91} + ( - 33970 \beta_{3} - 176640 \beta_1) q^{95} + ( - 42852 \beta_{2} + 488770) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 200 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 200 q^{5} - 9832 q^{13} + 792 q^{17} - 9300 q^{25} - 82792 q^{29} - 3336 q^{37} - 183848 q^{41} + 86340 q^{49} + 104280 q^{53} - 654856 q^{61} + 1234640 q^{65} + 1440264 q^{73} - 275008 q^{77} - 367920 q^{85} - 72264 q^{89} + 1955080 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( 4\zeta_{12}^{3} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -12\zeta_{12}^{3} + 24\zeta_{12} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 24\zeta_{12}^{2} - 12 \) Copy content Toggle raw display
\(\zeta_{12}\)\(=\) \( ( \beta_{2} + 3\beta_1 ) / 24 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( ( \beta_{3} + 12 ) / 24 \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( ( \beta_1 ) / 4 \) 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
0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 0.500000i
−0.866025 + 0.500000i
0 0 0 −153.923 0 395.923i 0 0 0
127.2 0 0 0 −153.923 0 395.923i 0 0 0
127.3 0 0 0 53.9230 0 188.077i 0 0 0
127.4 0 0 0 53.9230 0 188.077i 0 0 0
\(n\): e.g. 2-40 or 990-1000
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.k 4
3.b odd 2 1 192.7.g.d 4
4.b odd 2 1 inner 576.7.g.k 4
8.b even 2 1 288.7.g.c 4
8.d odd 2 1 288.7.g.c 4
12.b even 2 1 192.7.g.d 4
24.f even 2 1 96.7.g.a 4
24.h odd 2 1 96.7.g.a 4
48.i odd 4 1 768.7.b.a 4
48.i odd 4 1 768.7.b.e 4
48.k even 4 1 768.7.b.a 4
48.k even 4 1 768.7.b.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
96.7.g.a 4 24.f even 2 1
96.7.g.a 4 24.h odd 2 1
192.7.g.d 4 3.b odd 2 1
192.7.g.d 4 12.b even 2 1
288.7.g.c 4 8.b even 2 1
288.7.g.c 4 8.d odd 2 1
576.7.g.k 4 1.a even 1 1 trivial
576.7.g.k 4 4.b odd 2 1 inner
768.7.b.a 4 48.i odd 4 1
768.7.b.a 4 48.k even 4 1
768.7.b.e 4 48.i odd 4 1
768.7.b.e 4 48.k even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 100T_{5} - 8300 \) acting on \(S_{7}^{\mathrm{new}}(576, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} + 100 T - 8300)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + 192128 T^{2} + \cdots + 5544887296 \) Copy content Toggle raw display
$11$ \( T^{4} + 594656 T^{2} + \cdots + 212459776 \) Copy content Toggle raw display
$13$ \( (T^{2} + 4916 T + 2846692)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 396 T - 584604)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 84224736 T^{2} + \cdots + 7440097536 \) Copy content Toggle raw display
$23$ \( T^{4} + 515524736 T^{2} + \cdots + 63\!\cdots\!24 \) Copy content Toggle raw display
$29$ \( (T^{2} + 41396 T + 257490292)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 892954496 T^{2} + \cdots + 17\!\cdots\!36 \) Copy content Toggle raw display
$37$ \( (T^{2} + 1668 T - 4129424892)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 91924 T - 66725468)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 23532599264 T^{2} + \cdots + 11\!\cdots\!56 \) Copy content Toggle raw display
$47$ \( T^{4} + 3805630976 T^{2} + \cdots + 27\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( (T^{2} - 52140 T - 50204602188)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 69156949856 T^{2} + \cdots + 12\!\cdots\!84 \) Copy content Toggle raw display
$61$ \( (T^{2} + 327428 T - 4029006332)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 15928236128 T^{2} + \cdots + 62\!\cdots\!64 \) Copy content Toggle raw display
$71$ \( T^{4} + 189100829312 T^{2} + \cdots + 89\!\cdots\!04 \) Copy content Toggle raw display
$73$ \( (T^{2} - 720132 T - 20837470716)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} + 459428175488 T^{2} + \cdots + 56\!\cdots\!04 \) Copy content Toggle raw display
$83$ \( T^{4} + 1024589311712 T^{2} + \cdots + 25\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( (T^{2} + 36132 T - 1131604920252)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 977540 T - 554382853628)^{2} \) Copy content Toggle raw display
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