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Properties

Label 3072.2.d.i
Level $3072$
Weight $2$
Character orbit 3072.d
Analytic conductor $24.530$
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] = [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: 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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 48)
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.

\(f(q)\) \(=\) \( q + \beta_{4} q^{3} + (\beta_{6} + \beta_{4}) q^{5} + ( - \beta_1 + 1) q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{3} + (\beta_{6} + \beta_{4}) q^{5} + ( - \beta_1 + 1) q^{7} - q^{9} + ( - \beta_{7} - \beta_{6} + \beta_{5}) q^{11} + (\beta_{7} + \beta_{6} - 2 \beta_{4}) q^{13} + ( - \beta_{3} - 1) q^{15} + (\beta_{3} + \beta_{2} + \beta_1) q^{17} + (\beta_{7} - \beta_{6} + \beta_{5}) q^{19} + (\beta_{7} + \beta_{4}) q^{21} + 2 \beta_{2} q^{23} + ( - 2 \beta_{3} + 2 \beta_{2} - 1) q^{25} - \beta_{4} q^{27} + (\beta_{6} + 2 \beta_{5} - 3 \beta_{4}) q^{29} + ( - \beta_1 - 3) q^{31} + (\beta_{3} + \beta_{2} - \beta_1) q^{33} + (\beta_{7} + \beta_{6} + 3 \beta_{5}) q^{35} + (\beta_{7} - \beta_{6} + \cdots + 4 \beta_{4}) q^{37}+ \cdots + (\beta_{7} + \beta_{6} - \beta_{5}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{7} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{7} - 8 q^{9} - 8 q^{15} - 8 q^{25} - 24 q^{31} + 16 q^{39} + 8 q^{49} + 32 q^{55} - 8 q^{63} - 16 q^{65} - 16 q^{71} + 16 q^{73} - 24 q^{79} + 8 q^{81} + 24 q^{87} + 16 q^{89} + 48 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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

\(\beta_{1}\)\(=\) \( -\nu^{4} + 2\nu^{3} - 7\nu^{2} + 6\nu - 5 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{6} - 3\nu^{5} + 10\nu^{4} - 15\nu^{3} + 19\nu^{2} - 12\nu + 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\nu^{6} + 3\nu^{5} - 11\nu^{4} + 17\nu^{3} - 24\nu^{2} + 16\nu - 5 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -8\nu^{7} + 28\nu^{6} - 98\nu^{5} + 175\nu^{4} - 256\nu^{3} + 223\nu^{2} - 126\nu + 31 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( 10\nu^{7} - 35\nu^{6} + 123\nu^{5} - 220\nu^{4} + 325\nu^{3} - 285\nu^{2} + 166\nu - 42 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( 10\nu^{7} - 35\nu^{6} + 123\nu^{5} - 220\nu^{4} + 325\nu^{3} - 285\nu^{2} + 168\nu - 43 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -28\nu^{7} + 98\nu^{6} - 342\nu^{5} + 610\nu^{4} - 890\nu^{3} + 774\nu^{2} - 440\nu + 109 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{6} - \beta_{5} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{6} - \beta_{5} + \beta_{3} + \beta_{2} - \beta _1 - 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{7} - 5\beta_{6} + 7\beta_{5} + 6\beta_{4} + 3\beta_{3} + 3\beta_{2} - 3\beta _1 - 10 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -\beta_{7} - 6\beta_{6} + 8\beta_{5} + 6\beta_{4} - 4\beta_{3} - 4\beta_{2} + 2\beta _1 + 7 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 5\beta_{7} + 11\beta_{6} - 13\beta_{5} - 20\beta_{4} - 25\beta_{3} - 25\beta_{2} + 15\beta _1 + 52 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 10\beta_{7} + 32\beta_{6} - 40\beta_{5} - 45\beta_{4} + 6\beta_{3} + 8\beta_{2} - \beta _1 - 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -3\beta_{7} + 11\beta_{6} - 19\beta_{5} + 133\beta_{3} + 147\beta_{2} - 63\beta _1 - 236 ) / 4 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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
0.500000 − 0.691860i
0.500000 − 1.44392i
0.500000 + 0.0297061i
0.500000 + 2.10607i
0.500000 − 2.10607i
0.500000 − 0.0297061i
0.500000 + 1.44392i
0.500000 + 0.691860i
0 − 1.00000i 0 − 3.79793i 0 2.15894 0 −1.00000 0
1537.2 0 − 1.00000i 0 − 2.47363i 0 −2.55765 0 −1.00000 0
1537.3 0 − 1.00000i 0 0.473626i 0 4.55765 0 −1.00000 0
1537.4 0 − 1.00000i 0 1.79793i 0 −0.158942 0 −1.00000 0
1537.5 0 1.00000i 0 − 1.79793i 0 −0.158942 0 −1.00000 0
1537.6 0 1.00000i 0 − 0.473626i 0 4.55765 0 −1.00000 0
1537.7 0 1.00000i 0 2.47363i 0 −2.55765 0 −1.00000 0
1537.8 0 1.00000i 0 3.79793i 0 2.15894 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.i 8
4.b odd 2 1 3072.2.d.f 8
8.b even 2 1 inner 3072.2.d.i 8
8.d odd 2 1 3072.2.d.f 8
16.e even 4 1 3072.2.a.n 4
16.e even 4 1 3072.2.a.o 4
16.f odd 4 1 3072.2.a.i 4
16.f odd 4 1 3072.2.a.t 4
32.g even 8 2 192.2.j.a 8
32.g even 8 2 384.2.j.a 8
32.h odd 8 2 48.2.j.a ✓ 8
32.h odd 8 2 384.2.j.b 8
48.i odd 4 1 9216.2.a.x 4
48.i odd 4 1 9216.2.a.bn 4
48.k even 4 1 9216.2.a.y 4
48.k even 4 1 9216.2.a.bo 4
96.o even 8 2 144.2.k.b 8
96.o even 8 2 1152.2.k.c 8
96.p odd 8 2 576.2.k.b 8
96.p odd 8 2 1152.2.k.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.2.j.a ✓ 8 32.h odd 8 2
144.2.k.b 8 96.o even 8 2
192.2.j.a 8 32.g even 8 2
384.2.j.a 8 32.g even 8 2
384.2.j.b 8 32.h odd 8 2
576.2.k.b 8 96.p odd 8 2
1152.2.k.c 8 96.o even 8 2
1152.2.k.f 8 96.p odd 8 2
3072.2.a.i 4 16.f odd 4 1
3072.2.a.n 4 16.e even 4 1
3072.2.a.o 4 16.e even 4 1
3072.2.a.t 4 16.f odd 4 1
3072.2.d.f 8 4.b odd 2 1
3072.2.d.f 8 8.d odd 2 1
3072.2.d.i 8 1.a even 1 1 trivial
3072.2.d.i 8 8.b even 2 1 inner
9216.2.a.x 4 48.i odd 4 1
9216.2.a.y 4 48.k even 4 1
9216.2.a.bn 4 48.i odd 4 1
9216.2.a.bo 4 48.k even 4 1

Hecke kernels

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

\( T_{5}^{8} + 24T_{5}^{6} + 160T_{5}^{4} + 320T_{5}^{2} + 64 \) Copy content Toggle raw display
\( T_{7}^{4} - 4T_{7}^{3} - 8T_{7}^{2} + 24T_{7} + 4 \) 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
$5$ \( T^{8} + 24 T^{6} + \cdots + 64 \) Copy content Toggle raw display
$7$ \( (T^{4} - 4 T^{3} - 8 T^{2} + \cdots + 4)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} + 48 T^{6} + \cdots + 1024 \) Copy content Toggle raw display
$13$ \( T^{8} + 56 T^{6} + \cdots + 16 \) Copy content Toggle raw display
$17$ \( (T^{4} - 32 T^{2} + \cdots + 16)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + 64 T^{6} + \cdots + 256 \) Copy content Toggle raw display
$23$ \( (T^{2} - 8)^{4} \) Copy content Toggle raw display
$29$ \( T^{8} + 88 T^{6} + \cdots + 61504 \) Copy content Toggle raw display
$31$ \( (T^{4} + 12 T^{3} + \cdots - 28)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + 152 T^{6} + \cdots + 1106704 \) Copy content Toggle raw display
$41$ \( (T^{4} - 64 T^{2} + \cdots - 112)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + 192 T^{6} + \cdots + 12544 \) Copy content Toggle raw display
$47$ \( (T^{2} - 8)^{4} \) Copy content Toggle raw display
$53$ \( T^{8} + 152 T^{6} + \cdots + 18496 \) Copy content Toggle raw display
$59$ \( (T^{2} + 32)^{4} \) Copy content Toggle raw display
$61$ \( T^{8} + 152 T^{6} + \cdots + 1106704 \) Copy content Toggle raw display
$67$ \( T^{8} + 256 T^{6} + \cdots + 65536 \) Copy content Toggle raw display
$71$ \( (T^{4} + 8 T^{3} - 32 T^{2} + \cdots + 64)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} - 8 T^{3} - 96 T^{2} + \cdots + 64)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 12 T^{3} + \cdots - 10108)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 432 T^{6} + \cdots + 1024 \) Copy content Toggle raw display
$89$ \( (T^{4} - 8 T^{3} + \cdots - 1904)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 224 T^{2} + \cdots + 512)^{2} \) Copy content Toggle raw display
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