Properties

Label 512.2.e.i
Level $512$
Weight $2$
Character orbit 512.e
Analytic conductor $4.088$
Analytic rank $0$
Dimension $8$
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] = [512,2,Mod(129,512)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(512, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("512.129");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 512 = 2^{9} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 512.e (of order \(4\), degree \(2\), 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: \(4\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{16})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{7} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{6} q^{3} + ( - \beta_{4} + \beta_{2} - 1) q^{5} - \beta_{3} q^{7} + ( - \beta_{4} + \beta_{2} - \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{6} q^{3} + ( - \beta_{4} + \beta_{2} - 1) q^{5} - \beta_{3} q^{7} + ( - \beta_{4} + \beta_{2} - \beta_1) q^{9} + ( - \beta_{7} + \beta_{5} + \beta_{3}) q^{11} + (\beta_{2} + \beta_1 + 1) q^{13} + ( - \beta_{7} + 2 \beta_{6} + 2 \beta_{5}) q^{15} + (\beta_{4} - \beta_1) q^{17} + ( - \beta_{7} - \beta_{6} - \beta_{3}) q^{19} + 2 \beta_{4} q^{21} + ( - 2 \beta_{6} + 2 \beta_{5} - \beta_{3}) q^{23} + (2 \beta_{4} - \beta_{2} + 2 \beta_1) q^{25} + ( - \beta_{7} + \beta_{3}) q^{27} + ( - \beta_{2} - 3 \beta_1 - 1) q^{29} + (2 \beta_{6} + 2 \beta_{5}) q^{31} + ( - 3 \beta_{4} + 3 \beta_1 - 4) q^{33} - 2 \beta_{6} q^{35} + ( - \beta_{4} - 5 \beta_{2} + 5) q^{37} - \beta_{3} q^{39} - 4 \beta_{2} q^{41} + \beta_{5} q^{43} + ( - 5 \beta_{2} + 3 \beta_1 - 5) q^{45} + ( - 2 \beta_{6} - 2 \beta_{5}) q^{47} + (2 \beta_{4} - 2 \beta_1 - 1) q^{49} + (\beta_{7} - 2 \beta_{6} + \beta_{3}) q^{51} + (3 \beta_{4} - \beta_{2} + 1) q^{53} + (4 \beta_{6} - 4 \beta_{5} - \beta_{3}) q^{55} + (\beta_{4} + 4 \beta_{2} + \beta_1) q^{57} + \beta_{5} q^{59} + (5 \beta_{2} - 3 \beta_1 + 5) q^{61} + ( - \beta_{7} - 2 \beta_{6} - 2 \beta_{5}) q^{63} + 2 q^{65} + ( - \beta_{7} + 3 \beta_{6} - \beta_{3}) q^{67} + ( - 2 \beta_{4} + 8 \beta_{2} - 8) q^{69} + ( - 2 \beta_{6} + 2 \beta_{5} + 3 \beta_{3}) q^{71} + ( - 3 \beta_{4} + 2 \beta_{2} - 3 \beta_1) q^{73} + (2 \beta_{7} - 5 \beta_{5} - 2 \beta_{3}) q^{75} + (8 \beta_{2} + 2 \beta_1 + 8) q^{77} + 4 \beta_{7} q^{79} + (\beta_{4} - \beta_1 + 3) q^{81} + (2 \beta_{7} + 3 \beta_{6} + 2 \beta_{3}) q^{83} + ( - 2 \beta_{4} + 4 \beta_{2} - 4) q^{85} + ( - 2 \beta_{6} + 2 \beta_{5} + 3 \beta_{3}) q^{87} + ( - \beta_{4} - 2 \beta_{2} - \beta_1) q^{89} + (2 \beta_{7} + 2 \beta_{5} - 2 \beta_{3}) q^{91} + ( - 8 \beta_{2} + 4 \beta_1 - 8) q^{93} - \beta_{7} q^{95} + (\beta_{4} - \beta_1 + 8) q^{97} + 7 \beta_{6} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{5} + 8 q^{13} - 8 q^{29} - 32 q^{33} + 40 q^{37} - 40 q^{45} - 8 q^{49} + 8 q^{53} + 40 q^{61} + 16 q^{65} - 64 q^{69} + 64 q^{77} + 24 q^{81} - 32 q^{85} - 64 q^{93} + 64 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( 2\zeta_{16}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{16}^{4} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\zeta_{16}^{5} + 2\zeta_{16}^{3} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( 2\zeta_{16}^{6} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -\zeta_{16}^{7} - \zeta_{16}^{5} + \zeta_{16}^{3} - \zeta_{16} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \zeta_{16}^{7} - \zeta_{16}^{5} + \zeta_{16}^{3} + \zeta_{16} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -2\zeta_{16}^{7} + 2\zeta_{16} \) Copy content Toggle raw display
\(\zeta_{16}\)\(=\) \( ( \beta_{7} + \beta_{6} - \beta_{5} ) / 4 \) Copy content Toggle raw display
\(\zeta_{16}^{2}\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{16}^{3}\)\(=\) \( ( \beta_{6} + \beta_{5} + \beta_{3} ) / 4 \) Copy content Toggle raw display
\(\zeta_{16}^{4}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\zeta_{16}^{5}\)\(=\) \( ( -\beta_{6} - \beta_{5} + \beta_{3} ) / 4 \) Copy content Toggle raw display
\(\zeta_{16}^{6}\)\(=\) \( ( \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\zeta_{16}^{7}\)\(=\) \( ( -\beta_{7} + \beta_{6} - \beta_{5} ) / 4 \) Copy content Toggle raw display

Character values

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

\(n\) \(5\) \(511\)
\(\chi(n)\) \(-\beta_{2}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
129.1
−0.382683 + 0.923880i
0.923880 + 0.382683i
−0.923880 0.382683i
0.382683 0.923880i
−0.382683 0.923880i
0.923880 0.382683i
−0.923880 + 0.382683i
0.382683 + 0.923880i
0 −1.84776 1.84776i 0 −2.41421 + 2.41421i 0 1.53073i 0 3.82843i 0
129.2 0 −0.765367 0.765367i 0 0.414214 0.414214i 0 3.69552i 0 1.82843i 0
129.3 0 0.765367 + 0.765367i 0 0.414214 0.414214i 0 3.69552i 0 1.82843i 0
129.4 0 1.84776 + 1.84776i 0 −2.41421 + 2.41421i 0 1.53073i 0 3.82843i 0
385.1 0 −1.84776 + 1.84776i 0 −2.41421 2.41421i 0 1.53073i 0 3.82843i 0
385.2 0 −0.765367 + 0.765367i 0 0.414214 + 0.414214i 0 3.69552i 0 1.82843i 0
385.3 0 0.765367 0.765367i 0 0.414214 + 0.414214i 0 3.69552i 0 1.82843i 0
385.4 0 1.84776 1.84776i 0 −2.41421 2.41421i 0 1.53073i 0 3.82843i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 129.4
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 512.2.e.i 8
3.b odd 2 1 4608.2.k.bi 8
4.b odd 2 1 inner 512.2.e.i 8
8.b even 2 1 512.2.e.j yes 8
8.d odd 2 1 512.2.e.j yes 8
12.b even 2 1 4608.2.k.bi 8
16.e even 4 1 inner 512.2.e.i 8
16.e even 4 1 512.2.e.j yes 8
16.f odd 4 1 inner 512.2.e.i 8
16.f odd 4 1 512.2.e.j yes 8
24.f even 2 1 4608.2.k.bd 8
24.h odd 2 1 4608.2.k.bd 8
32.g even 8 1 1024.2.a.h 4
32.g even 8 1 1024.2.a.i 4
32.g even 8 2 1024.2.b.g 8
32.h odd 8 1 1024.2.a.h 4
32.h odd 8 1 1024.2.a.i 4
32.h odd 8 2 1024.2.b.g 8
48.i odd 4 1 4608.2.k.bd 8
48.i odd 4 1 4608.2.k.bi 8
48.k even 4 1 4608.2.k.bd 8
48.k even 4 1 4608.2.k.bi 8
96.o even 8 1 9216.2.a.w 4
96.o even 8 1 9216.2.a.bp 4
96.p odd 8 1 9216.2.a.w 4
96.p odd 8 1 9216.2.a.bp 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
512.2.e.i 8 1.a even 1 1 trivial
512.2.e.i 8 4.b odd 2 1 inner
512.2.e.i 8 16.e even 4 1 inner
512.2.e.i 8 16.f odd 4 1 inner
512.2.e.j yes 8 8.b even 2 1
512.2.e.j yes 8 8.d odd 2 1
512.2.e.j yes 8 16.e even 4 1
512.2.e.j yes 8 16.f odd 4 1
1024.2.a.h 4 32.g even 8 1
1024.2.a.h 4 32.h odd 8 1
1024.2.a.i 4 32.g even 8 1
1024.2.a.i 4 32.h odd 8 1
1024.2.b.g 8 32.g even 8 2
1024.2.b.g 8 32.h odd 8 2
4608.2.k.bd 8 24.f even 2 1
4608.2.k.bd 8 24.h odd 2 1
4608.2.k.bd 8 48.i odd 4 1
4608.2.k.bd 8 48.k even 4 1
4608.2.k.bi 8 3.b odd 2 1
4608.2.k.bi 8 12.b even 2 1
4608.2.k.bi 8 48.i odd 4 1
4608.2.k.bi 8 48.k even 4 1
9216.2.a.w 4 96.o even 8 1
9216.2.a.w 4 96.p odd 8 1
9216.2.a.bp 4 96.o even 8 1
9216.2.a.bp 4 96.p odd 8 1

Hecke kernels

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

\( T_{3}^{8} + 48T_{3}^{4} + 64 \) Copy content Toggle raw display
\( T_{5}^{4} + 4T_{5}^{3} + 8T_{5}^{2} - 8T_{5} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + 48T^{4} + 64 \) Copy content Toggle raw display
$5$ \( (T^{4} + 4 T^{3} + 8 T^{2} - 8 T + 4)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} + 16 T^{2} + 32)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} + 816 T^{4} + 153664 \) Copy content Toggle raw display
$13$ \( (T^{4} - 4 T^{3} + 8 T^{2} + 8 T + 4)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 8)^{4} \) Copy content Toggle raw display
$19$ \( T^{8} + 1584T^{4} + 64 \) Copy content Toggle raw display
$23$ \( (T^{4} + 80 T^{2} + 1568)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 4 T^{3} + 8 T^{2} - 136 T + 1156)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 64 T^{2} + 512)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 20 T^{3} + 200 T^{2} - 920 T + 2116)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 16)^{4} \) Copy content Toggle raw display
$43$ \( T^{8} + 48T^{4} + 64 \) Copy content Toggle raw display
$47$ \( (T^{4} - 64 T^{2} + 512)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 4 T^{3} + 8 T^{2} + 136 T + 1156)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + 48T^{4} + 64 \) Copy content Toggle raw display
$61$ \( (T^{4} - 20 T^{3} + 200 T^{2} - 280 T + 196)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + 10032 T^{4} + \cdots + 153664 \) Copy content Toggle raw display
$71$ \( (T^{4} + 208 T^{2} + 9248)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 152 T^{2} + 4624)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 256 T^{2} + 8192)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 35376 T^{4} + \cdots + 5345344 \) Copy content Toggle raw display
$89$ \( (T^{4} + 24 T^{2} + 16)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 16 T + 56)^{4} \) Copy content Toggle raw display
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