Properties

Label 3072.2.d.e
Level $3072$
Weight $2$
Character orbit 3072.d
Analytic conductor $24.530$
Analytic rank $0$
Dimension $8$
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] = [3072,2,Mod(1537,3072)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3072, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3072.1537");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3072 = 2^{10} \cdot 3 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3072.d (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: \(24.5300435009\)
Analytic rank: \(0\)
Dimension: \(8\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: no (minimal twist has level 1536)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + ( - \beta_{3} + \beta_{2}) q^{5} + (\beta_{5} - 2) q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + ( - \beta_{3} + \beta_{2}) q^{5} + (\beta_{5} - 2) q^{7} - q^{9} + ( - \beta_{7} - \beta_{3}) q^{11} + (2 \beta_{7} + \beta_{2} + 2 \beta_1) q^{13} + (\beta_{5} - \beta_{4}) q^{15} + (\beta_{6} - \beta_{5} - 2 \beta_{4}) q^{17} + (\beta_{7} + \beta_{3} - 2 \beta_{2}) q^{19} + (\beta_{3} - 2 \beta_1) q^{21} - 4 q^{23} + (2 \beta_{6} + 2 \beta_{5} - 2 \beta_{4} - 1) q^{25} - \beta_1 q^{27} + ( - 2 \beta_{7} - \beta_{3} + \beta_{2}) q^{29} + (2 \beta_{6} + \beta_{5} + 2) q^{31} + ( - \beta_{6} + \beta_{5}) q^{33} + ( - \beta_{7} + 3 \beta_{3} - 4 \beta_{2} - 4 \beta_1) q^{35} + (2 \beta_{7} - 2 \beta_{3} - \beta_{2} - 2 \beta_1) q^{37} + (2 \beta_{6} - \beta_{4} - 2) q^{39} + ( - 3 \beta_{6} - \beta_{5} - 2 \beta_{4}) q^{41} + (3 \beta_{7} - \beta_{3} - 2 \beta_{2}) q^{43} + (\beta_{3} - \beta_{2}) q^{45} + (4 \beta_{4} + 4) q^{47} + ( - 4 \beta_{5} + 2 \beta_{4} + 1) q^{49} + ( - \beta_{7} - \beta_{3} - 2 \beta_{2}) q^{51} + ( - 2 \beta_{7} + 3 \beta_{3} + \beta_{2}) q^{53} + (2 \beta_{6} - 4) q^{55} + (\beta_{6} - \beta_{5} + 2 \beta_{4}) q^{57} + (2 \beta_{7} - 2 \beta_{3} - 4 \beta_{2} + 4 \beta_1) q^{59} + ( - 2 \beta_{7} - 2 \beta_{3} - \beta_{2} + 6 \beta_1) q^{61} + ( - \beta_{5} + 2) q^{63} + ( - \beta_{6} + 5 \beta_{5} - 6 \beta_{4} - 2) q^{65} + ( - 4 \beta_{3} - 4 \beta_{2} + 4 \beta_1) q^{67} - 4 \beta_1 q^{69} + ( - 2 \beta_{5} + 4 \beta_{4} - 4) q^{71} + ( - 4 \beta_{6} - 2 \beta_{4} + 2) q^{73} + ( - 2 \beta_{7} + 2 \beta_{3} - 2 \beta_{2} - \beta_1) q^{75} + (2 \beta_{7} + 2 \beta_{3} - 4 \beta_1) q^{77} + ( - 3 \beta_{5} - 4 \beta_{4} + 6) q^{79} + q^{81} + ( - \beta_{7} - \beta_{3} - 4 \beta_{2}) q^{83} + 2 \beta_{3} q^{85} + ( - 2 \beta_{6} + \beta_{5} - \beta_{4}) q^{87} + (4 \beta_{6} - 4 \beta_{4} + 2) q^{89} + ( - 5 \beta_{7} + 3 \beta_{3} - 6 \beta_{2} - 4 \beta_1) q^{91} + ( - 2 \beta_{7} + \beta_{3} + 2 \beta_1) q^{93} + ( - 4 \beta_{6} - 2 \beta_{5} + 8) q^{95} + ( - 2 \beta_{6} + 2 \beta_{5} + 4 \beta_{4} - 4) q^{97} + (\beta_{7} + \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 16 q^{7} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 16 q^{7} - 8 q^{9} - 32 q^{23} - 8 q^{25} + 16 q^{31} - 16 q^{39} + 32 q^{47} + 8 q^{49} - 32 q^{55} + 16 q^{63} - 16 q^{65} - 32 q^{71} + 16 q^{73} + 48 q^{79} + 8 q^{81} + 16 q^{89} + 64 q^{95} - 32 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3072\mathbb{Z}\right)^\times\).

\(n\) \(1025\) \(2047\) \(2053\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1537.1
−0.923880 + 0.382683i
0.382683 + 0.923880i
0.923880 0.382683i
−0.382683 0.923880i
−0.382683 + 0.923880i
0.923880 + 0.382683i
0.382683 0.923880i
−0.923880 0.382683i
0 1.00000i 0 4.02734i 0 −4.61313 0 −1.00000 0
1537.2 0 1.00000i 0 0.331821i 0 −3.08239 0 −1.00000 0
1537.3 0 1.00000i 0 1.19891i 0 0.613126 0 −1.00000 0
1537.4 0 1.00000i 0 2.49661i 0 −0.917608 0 −1.00000 0
1537.5 0 1.00000i 0 2.49661i 0 −0.917608 0 −1.00000 0
1537.6 0 1.00000i 0 1.19891i 0 0.613126 0 −1.00000 0
1537.7 0 1.00000i 0 0.331821i 0 −3.08239 0 −1.00000 0
1537.8 0 1.00000i 0 4.02734i 0 −4.61313 0 −1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1537.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3072.2.d.e 8
4.b odd 2 1 3072.2.d.j 8
8.b even 2 1 inner 3072.2.d.e 8
8.d odd 2 1 3072.2.d.j 8
16.e even 4 1 3072.2.a.m 4
16.e even 4 1 3072.2.a.s 4
16.f odd 4 1 3072.2.a.j 4
16.f odd 4 1 3072.2.a.p 4
32.g even 8 2 1536.2.j.e 8
32.g even 8 2 1536.2.j.j yes 8
32.h odd 8 2 1536.2.j.f yes 8
32.h odd 8 2 1536.2.j.i yes 8
48.i odd 4 1 9216.2.a.bl 4
48.i odd 4 1 9216.2.a.bm 4
48.k even 4 1 9216.2.a.z 4
48.k even 4 1 9216.2.a.ba 4
96.o even 8 2 4608.2.k.bc 8
96.o even 8 2 4608.2.k.bj 8
96.p odd 8 2 4608.2.k.be 8
96.p odd 8 2 4608.2.k.bh 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1536.2.j.e 8 32.g even 8 2
1536.2.j.f yes 8 32.h odd 8 2
1536.2.j.i yes 8 32.h odd 8 2
1536.2.j.j yes 8 32.g even 8 2
3072.2.a.j 4 16.f odd 4 1
3072.2.a.m 4 16.e even 4 1
3072.2.a.p 4 16.f odd 4 1
3072.2.a.s 4 16.e even 4 1
3072.2.d.e 8 1.a even 1 1 trivial
3072.2.d.e 8 8.b even 2 1 inner
3072.2.d.j 8 4.b odd 2 1
3072.2.d.j 8 8.d odd 2 1
4608.2.k.bc 8 96.o even 8 2
4608.2.k.be 8 96.p odd 8 2
4608.2.k.bh 8 96.p odd 8 2
4608.2.k.bj 8 96.o even 8 2
9216.2.a.z 4 48.k even 4 1
9216.2.a.ba 4 48.k even 4 1
9216.2.a.bl 4 48.i odd 4 1
9216.2.a.bm 4 48.i odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3072, [\chi])\):

\( T_{5}^{8} + 24T_{5}^{6} + 136T_{5}^{4} + 160T_{5}^{2} + 16 \) Copy content Toggle raw display
\( T_{7}^{4} + 8T_{7}^{3} + 16T_{7}^{2} - 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$5$ \( T^{8} + 24 T^{6} + 136 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$7$ \( (T^{4} + 8 T^{3} + 16 T^{2} - 8)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 16 T^{2} + 32)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + 88 T^{6} + 2584 T^{4} + \cdots + 35344 \) Copy content Toggle raw display
$17$ \( (T^{4} - 32 T^{2} - 64 T - 32)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + 64 T^{6} + 960 T^{4} + \cdots + 1024 \) Copy content Toggle raw display
$23$ \( (T + 4)^{8} \) Copy content Toggle raw display
$29$ \( T^{8} + 88 T^{6} + 1800 T^{4} + \cdots + 4624 \) Copy content Toggle raw display
$31$ \( (T^{4} - 8 T^{3} - 16 T^{2} + 128 T + 248)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + 152 T^{6} + 4504 T^{4} + \cdots + 150544 \) Copy content Toggle raw display
$41$ \( (T^{4} - 96 T^{2} - 64 T + 992)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + 192 T^{6} + 11200 T^{4} + \cdots + 984064 \) Copy content Toggle raw display
$47$ \( (T^{2} - 8 T - 16)^{4} \) Copy content Toggle raw display
$53$ \( T^{8} + 216 T^{6} + 12040 T^{4} + \cdots + 35344 \) Copy content Toggle raw display
$59$ \( T^{8} + 320 T^{6} + \cdots + 18939904 \) Copy content Toggle raw display
$61$ \( T^{8} + 280 T^{6} + 13464 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$67$ \( T^{8} + 448 T^{6} + \cdots + 62980096 \) Copy content Toggle raw display
$71$ \( (T^{4} + 16 T^{3} - 768 T - 2176)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} - 8 T^{3} - 120 T^{2} + 32 T + 1552)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 24 T^{3} + 80 T^{2} + 1344 T - 7688)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 160 T^{6} + 7488 T^{4} + \cdots + 295936 \) Copy content Toggle raw display
$89$ \( (T^{4} - 8 T^{3} - 168 T^{2} - 288 T + 272)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 16 T^{3} - 32 T^{2} - 256 T - 256)^{2} \) Copy content Toggle raw display
show more
show less