Properties

Label 512.2.g.f
Level $512$
Weight $2$
Character orbit 512.g
Analytic conductor $4.088$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [512,2,Mod(65,512)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(512, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([0, 7]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("512.65");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 512 = 2^{9} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 512.g (of order \(8\), degree \(4\), 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: \(4.08834058349\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{8})\)
Coefficient field: 8.0.18939904.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} + 14x^{6} - 28x^{5} + 43x^{4} - 44x^{3} + 30x^{2} - 12x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 32)
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{5} + \beta_{4} - 1) q^{3} + (\beta_{6} + 1) q^{5} + ( - \beta_{7} - 2 \beta_{6} - \beta_{3} - \beta_1 - 1) q^{7} + ( - \beta_{7} + \beta_{6} + \beta_{5} - \beta_{3} + \beta_{2} + \beta_1 - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{5} + \beta_{4} - 1) q^{3} + (\beta_{6} + 1) q^{5} + ( - \beta_{7} - 2 \beta_{6} - \beta_{3} - \beta_1 - 1) q^{7} + ( - \beta_{7} + \beta_{6} + \beta_{5} - \beta_{3} + \beta_{2} + \beta_1 - 1) q^{9} + (\beta_{6} - \beta_{3} - 2) q^{11} + ( - \beta_{4} - 2 \beta_1 + 1) q^{13} + ( - \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} - 1) q^{15} + ( - 2 \beta_{7} - \beta_{6} - \beta_{5} + 3 \beta_{4} - \beta_{3} + \beta_{2} + \beta_1 - 2) q^{17} + ( - \beta_{6} - \beta_{3} - \beta_{2} - \beta_1 + 1) q^{19} + (\beta_{7} + 3 \beta_{6} - \beta_{4} + 2 \beta_{3} - 1) q^{21} + (\beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} - \beta_{2} - 1) q^{23} + (\beta_{7} - 3 \beta_{6} + 1) q^{25} + (\beta_{7} - 3 \beta_{6} - \beta_{5} + \beta_{4} - \beta_1 - 2) q^{27} + (2 \beta_{7} - \beta_{4} + 2 \beta_{3} - 2 \beta_{2} + 1) q^{29} + (2 \beta_{6} + 2 \beta_{4} - 4) q^{31} + ( - 2 \beta_{6} + \beta_{5} - 2 \beta_{4} - \beta_{3} - \beta_{2} + \beta_1 - 2) q^{33} + ( - 3 \beta_{7} - 2 \beta_{6} + 2 \beta_{4} - \beta_{3} + \beta_{2} - 2) q^{35} + (2 \beta_{7} - 5 \beta_{6} - 2 \beta_{5} - 2 \beta_{2} - 2 \beta_1 - 1) q^{37} + (\beta_{7} + 5 \beta_{6} - \beta_{5} + \beta_{4} + \beta_{2} + 1) q^{39} + (\beta_{7} - 2 \beta_{6} - 2 \beta_{5} + 2 \beta_{4} - 2 \beta_{2} - 1) q^{41} + ( - 2 \beta_{7} - 3 \beta_{6} + 2 \beta_{4} - \beta_{3} + 2) q^{43} + ( - \beta_{7} + \beta_{6} + 2 \beta_1 - 2) q^{45} + ( - 2 \beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} - \beta_1) q^{47} + (3 \beta_{7} + 2 \beta_{6} + 2 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} + 2) q^{49} + (2 \beta_{7} - 2 \beta_{6} - 4 \beta_{4} - 4) q^{51} + ( - 3 \beta_{6} + 2 \beta_{5} - 2 \beta_{3} - 2 \beta_1 + 3) q^{53} + (\beta_{7} - \beta_{3} + \beta_1 - 3) q^{55} + ( - 3 \beta_{7} + 3 \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + \beta_1 - 3) q^{57} + ( - 2 \beta_{7} + 2 \beta_{6} - \beta_{5} - 3 \beta_{4} - \beta_{2} - \beta_1 + 4) q^{59} + (2 \beta_{6} - 2 \beta_{5} + 3 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} - 3) q^{61} + ( - 4 \beta_{6} - 4 \beta_{4} + \beta_{2} - \beta_1 + 5) q^{63} + (\beta_{6} + \beta_{4} + 2 \beta_{2} - 2 \beta_1) q^{65} + (6 \beta_{7} + 6 \beta_{6} + 3 \beta_{5} - 3 \beta_{4} + 3) q^{67} + ( - 5 \beta_{7} + 3 \beta_{6} + 2 \beta_{5} - 3 \beta_{4} + 2 \beta_{2} + 2 \beta_1 - 1) q^{69} + ( - 3 \beta_{7} - 2 \beta_{6} + 3 \beta_{3} + 3 \beta_1 - 3) q^{71} + ( - 3 \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} - \beta_1 + 5) q^{73} + ( - \beta_{7} + 4 \beta_{6} - \beta_{5} + \beta_{4} + 3 \beta_{3} + \beta_1 - 2) q^{75} + (\beta_{7} + \beta_{6} + 3 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 1) q^{77} + (6 \beta_{7} - 3 \beta_{6} + 3 \beta_{5} + \beta_{4} + 3 \beta_{3} - \beta_{2} - \beta_1 + 4) q^{79} + (\beta_{7} + 2 \beta_{6} - \beta_{5} - 4 \beta_{4} - \beta_{3} - \beta_{2} - \beta_1) q^{81} + ( - 4 \beta_{7} + 7 \beta_{6} + 6 \beta_{4} + 3 \beta_{3} + 3 \beta_{2} + 3 \beta_1 + 3) q^{83} + ( - 4 \beta_{7} - 2 \beta_{6} - 2 \beta_{5} + 4 \beta_{4} - 2 \beta_{3} + 2 \beta_1 - 2) q^{85} + ( - 7 \beta_{7} + 3 \beta_{6} + 3 \beta_{5} + \beta_{4} + 3 \beta_{2} + 7) q^{87} + (3 \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} - 3 \beta_{3} - \beta_{2} - 3 \beta_1 + 3) q^{89} + (5 \beta_{7} + 3 \beta_{6} + 3 \beta_{5} + 5 \beta_{4} + 3 \beta_1) q^{91} + ( - 2 \beta_{7} + 4 \beta_{5} - 4 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} + 4) q^{93} + (\beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} + 1) q^{95} + (3 \beta_{6} - \beta_{5} + 3 \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 + 4) q^{97} + ( - \beta_{7} - 4 \beta_{6} - \beta_{5} - 3 \beta_{4} + 3 \beta_{3} - 3 \beta_{2} + 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{3} + 8 q^{5} - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{3} + 8 q^{5} - 8 q^{7} - 12 q^{11} + 4 q^{19} - 16 q^{21} - 8 q^{23} + 8 q^{25} - 16 q^{27} - 8 q^{29} - 32 q^{31} - 16 q^{33} - 8 q^{35} - 16 q^{37} + 16 q^{39} - 8 q^{41} + 20 q^{43} - 8 q^{45} - 32 q^{51} + 16 q^{53} - 16 q^{55} - 16 q^{57} + 28 q^{59} + 40 q^{63} + 12 q^{67} - 24 q^{71} + 32 q^{73} - 20 q^{75} + 16 q^{77} + 36 q^{83} + 8 q^{85} + 56 q^{87} + 16 q^{89} + 32 q^{93} + 8 q^{95} + 32 q^{97} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} + 14x^{6} - 28x^{5} + 43x^{4} - 44x^{3} + 30x^{2} - 12x + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{7} - 4\nu^{6} + 14\nu^{5} - 27\nu^{4} + 41\nu^{3} - 37\nu^{2} + 24\nu - 5 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{7} - 4\nu^{6} + 14\nu^{5} - 28\nu^{4} + 43\nu^{3} - 44\nu^{2} + 30\nu - 10 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -5\nu^{7} + 17\nu^{6} - 59\nu^{5} + 102\nu^{4} - 146\nu^{3} + 121\nu^{2} - 66\nu + 15 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( 5\nu^{7} - 17\nu^{6} + 60\nu^{5} - 105\nu^{4} + 155\nu^{3} - 133\nu^{2} + 77\nu - 19 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -5\nu^{7} + 18\nu^{6} - 62\nu^{5} + 113\nu^{4} - 163\nu^{3} + 145\nu^{2} - 82\nu + 20 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -5\nu^{7} + 18\nu^{6} - 63\nu^{5} + 115\nu^{4} - 170\nu^{3} + 152\nu^{2} - 89\nu + 23 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -8\nu^{7} + 28\nu^{6} - 98\nu^{5} + 175\nu^{4} - 256\nu^{3} + 223\nu^{2} - 126\nu + 31 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{7} + 2\beta_{6} + \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{7} + 3\beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} + 2\beta _1 - 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 5\beta_{7} - 5\beta_{6} - 2\beta_{5} + 3\beta_{4} + \beta_{3} - 4\beta_{2} - \beta _1 - 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 11\beta_{7} - 19\beta_{6} + 3\beta_{5} - \beta_{4} - 5\beta_{3} - 4\beta_{2} - 8\beta _1 + 12 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -13\beta_{7} + 2\beta_{6} + 15\beta_{5} - 16\beta_{4} - 10\beta_{3} + 13\beta_{2} - 2\beta _1 + 23 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -67\beta_{7} + 90\beta_{6} + 4\beta_{5} - 10\beta_{4} + 16\beta_{3} + 31\beta_{2} + 33\beta _1 - 28 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -7\beta_{7} + 87\beta_{6} - 68\beta_{5} + 71\beta_{4} + 65\beta_{3} - 26\beta_{2} + 37\beta _1 - 125 ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/512\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(511\)
\(\chi(n)\) \(\beta_{6}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
65.1
0.500000 + 0.0297061i
0.500000 1.44392i
0.500000 2.10607i
0.500000 + 0.691860i
0.500000 + 2.10607i
0.500000 0.691860i
0.500000 0.0297061i
0.500000 + 1.44392i
0 −0.529706 + 1.27882i 0 1.70711 0.707107i 0 −2.74912 + 2.74912i 0 0.766519 + 0.766519i 0
65.2 0 0.943920 2.27882i 0 1.70711 0.707107i 0 −0.665096 + 0.665096i 0 −2.18073 2.18073i 0
193.1 0 −2.60607 + 1.07947i 0 0.292893 0.707107i 0 1.68554 + 1.68554i 0 3.50504 3.50504i 0
193.2 0 0.191860 0.0794708i 0 0.292893 0.707107i 0 −2.27133 2.27133i 0 −2.09083 + 2.09083i 0
321.1 0 −2.60607 1.07947i 0 0.292893 + 0.707107i 0 1.68554 1.68554i 0 3.50504 + 3.50504i 0
321.2 0 0.191860 + 0.0794708i 0 0.292893 + 0.707107i 0 −2.27133 + 2.27133i 0 −2.09083 2.09083i 0
449.1 0 −0.529706 1.27882i 0 1.70711 + 0.707107i 0 −2.74912 2.74912i 0 0.766519 0.766519i 0
449.2 0 0.943920 + 2.27882i 0 1.70711 + 0.707107i 0 −0.665096 0.665096i 0 −2.18073 + 2.18073i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 65.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
32.g even 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 512.2.g.f 8
4.b odd 2 1 512.2.g.h 8
8.b even 2 1 512.2.g.g 8
8.d odd 2 1 512.2.g.e 8
16.e even 4 1 128.2.g.b 8
16.e even 4 1 256.2.g.c 8
16.f odd 4 1 32.2.g.b 8
16.f odd 4 1 256.2.g.d 8
32.g even 8 1 128.2.g.b 8
32.g even 8 1 256.2.g.c 8
32.g even 8 1 inner 512.2.g.f 8
32.g even 8 1 512.2.g.g 8
32.h odd 8 1 32.2.g.b 8
32.h odd 8 1 256.2.g.d 8
32.h odd 8 1 512.2.g.e 8
32.h odd 8 1 512.2.g.h 8
48.i odd 4 1 1152.2.v.b 8
48.k even 4 1 288.2.v.b 8
64.i even 16 2 4096.2.a.q 8
64.j odd 16 2 4096.2.a.k 8
80.j even 4 1 800.2.ba.d 8
80.k odd 4 1 800.2.y.b 8
80.s even 4 1 800.2.ba.c 8
96.o even 8 1 288.2.v.b 8
96.p odd 8 1 1152.2.v.b 8
160.u even 8 1 800.2.ba.c 8
160.y odd 8 1 800.2.y.b 8
160.ba even 8 1 800.2.ba.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
32.2.g.b 8 16.f odd 4 1
32.2.g.b 8 32.h odd 8 1
128.2.g.b 8 16.e even 4 1
128.2.g.b 8 32.g even 8 1
256.2.g.c 8 16.e even 4 1
256.2.g.c 8 32.g even 8 1
256.2.g.d 8 16.f odd 4 1
256.2.g.d 8 32.h odd 8 1
288.2.v.b 8 48.k even 4 1
288.2.v.b 8 96.o even 8 1
512.2.g.e 8 8.d odd 2 1
512.2.g.e 8 32.h odd 8 1
512.2.g.f 8 1.a even 1 1 trivial
512.2.g.f 8 32.g even 8 1 inner
512.2.g.g 8 8.b even 2 1
512.2.g.g 8 32.g even 8 1
512.2.g.h 8 4.b odd 2 1
512.2.g.h 8 32.h odd 8 1
800.2.y.b 8 80.k odd 4 1
800.2.y.b 8 160.y odd 8 1
800.2.ba.c 8 80.s even 4 1
800.2.ba.c 8 160.u even 8 1
800.2.ba.d 8 80.j even 4 1
800.2.ba.d 8 160.ba even 8 1
1152.2.v.b 8 48.i odd 4 1
1152.2.v.b 8 96.p odd 8 1
4096.2.a.k 8 64.j odd 16 2
4096.2.a.q 8 64.i even 16 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(512, [\chi])\):

\( T_{3}^{8} + 4T_{3}^{7} + 8T_{3}^{6} + 24T_{3}^{5} + 64T_{3}^{4} + 56T_{3}^{3} + 64T_{3}^{2} - 32T_{3} + 4 \) Copy content Toggle raw display
\( T_{5}^{4} - 4T_{5}^{3} + 6T_{5}^{2} - 4T_{5} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + 4 T^{7} + 8 T^{6} + 24 T^{5} + \cdots + 4 \) Copy content Toggle raw display
$5$ \( (T^{4} - 4 T^{3} + 6 T^{2} - 4 T + 2)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} + 8 T^{7} + 32 T^{6} + 48 T^{5} + \cdots + 784 \) Copy content Toggle raw display
$11$ \( T^{8} + 12 T^{7} + 72 T^{6} + 264 T^{5} + \cdots + 4 \) Copy content Toggle raw display
$13$ \( T^{8} - 4 T^{6} - 88 T^{5} + \cdots + 6724 \) Copy content Toggle raw display
$17$ \( T^{8} + 64 T^{6} + 1056 T^{4} + \cdots + 256 \) Copy content Toggle raw display
$19$ \( T^{8} - 4 T^{7} + 24 T^{6} - 104 T^{5} + \cdots + 196 \) Copy content Toggle raw display
$23$ \( T^{8} + 8 T^{7} + 32 T^{6} + 16 T^{5} + \cdots + 16 \) Copy content Toggle raw display
$29$ \( T^{8} + 8 T^{7} + 44 T^{6} + \cdots + 188356 \) Copy content Toggle raw display
$31$ \( (T^{2} + 8 T + 8)^{4} \) Copy content Toggle raw display
$37$ \( T^{8} + 16 T^{7} + 204 T^{6} + \cdots + 64516 \) Copy content Toggle raw display
$41$ \( T^{8} + 8 T^{7} + 32 T^{6} + \cdots + 26896 \) Copy content Toggle raw display
$43$ \( T^{8} - 20 T^{7} + 216 T^{6} + \cdots + 31684 \) Copy content Toggle raw display
$47$ \( T^{8} + 64 T^{6} + 544 T^{4} + \cdots + 256 \) Copy content Toggle raw display
$53$ \( T^{8} - 16 T^{7} + 60 T^{6} + \cdots + 158404 \) Copy content Toggle raw display
$59$ \( T^{8} - 28 T^{7} + 456 T^{6} + \cdots + 643204 \) Copy content Toggle raw display
$61$ \( T^{8} + 156 T^{6} + 1992 T^{5} + \cdots + 42436 \) Copy content Toggle raw display
$67$ \( T^{8} - 12 T^{7} + 216 T^{6} + \cdots + 1285956 \) Copy content Toggle raw display
$71$ \( T^{8} + 24 T^{7} + 288 T^{6} + \cdots + 21196816 \) Copy content Toggle raw display
$73$ \( T^{8} - 32 T^{7} + 512 T^{6} + \cdots + 38416 \) Copy content Toggle raw display
$79$ \( T^{8} + 512 T^{6} + \cdots + 99361024 \) Copy content Toggle raw display
$83$ \( T^{8} - 36 T^{7} + \cdots + 138250564 \) Copy content Toggle raw display
$89$ \( T^{8} - 16 T^{7} + 128 T^{6} + \cdots + 17007376 \) Copy content Toggle raw display
$97$ \( (T^{4} - 16 T^{3} + 40 T^{2} + 288 T - 992)^{2} \) Copy content Toggle raw display
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