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Properties

Label 1536.2.d.g
Level $1536$
Weight $2$
Character orbit 1536.d
Analytic conductor $12.265$
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] = [1536,2,Mod(769,1536)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1536, 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("1536.769");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1536 = 2^{9} \cdot 3 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1536.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: \(12.2650217505\)
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: \( 2^{10} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q - \beta_{4} q^{3} + \beta_{5} q^{5} - \beta_{2} q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{3} + \beta_{5} q^{5} - \beta_{2} q^{7} - q^{9} + ( - \beta_{6} + 2 \beta_{4}) q^{11} + (\beta_{7} - \beta_{5}) q^{13} + \beta_1 q^{15} + (\beta_{3} + 2) q^{17} - \beta_{6} q^{19} + \beta_{7} q^{21} + 2 \beta_1 q^{23} + (2 \beta_{3} - 5) q^{25} + \beta_{4} q^{27} + ( - 2 \beta_{7} + \beta_{5}) q^{29} - \beta_{2} q^{31} + ( - \beta_{3} + 2) q^{33} + ( - 3 \beta_{6} - 2 \beta_{4}) q^{35} + ( - \beta_{7} - \beta_{5}) q^{37} + (\beta_{2} - \beta_1) q^{39} + ( - \beta_{3} - 6) q^{41} + ( - 3 \beta_{6} - 4 \beta_{4}) q^{43} - \beta_{5} q^{45} - 2 \beta_1 q^{47} + (4 \beta_{3} + 7) q^{49} + ( - \beta_{6} - 2 \beta_{4}) q^{51} + ( - 2 \beta_{7} - \beta_{5}) q^{53} + (2 \beta_{2} - 4 \beta_1) q^{55} - \beta_{3} q^{57} + ( - 2 \beta_{6} + 4 \beta_{4}) q^{59} + (\beta_{7} + \beta_{5}) q^{61} + \beta_{2} q^{63} + ( - 5 \beta_{3} + 8) q^{65} + ( - 2 \beta_{6} + 8 \beta_{4}) q^{67} - 2 \beta_{5} q^{69} + 2 \beta_{2} q^{71} - 4 q^{73} + ( - 2 \beta_{6} + 5 \beta_{4}) q^{75} + 2 \beta_{5} q^{77} + \beta_{2} q^{79} + q^{81} + (\beta_{6} - 6 \beta_{4}) q^{83} + 2 \beta_{7} q^{85} + ( - 2 \beta_{2} + \beta_1) q^{87} - 10 q^{89} + ( - \beta_{6} - 12 \beta_{4}) q^{91} + \beta_{7} q^{93} + (2 \beta_{2} - 2 \beta_1) q^{95} + ( - 2 \beta_{3} + 6) q^{97} + (\beta_{6} - 2 \beta_{4}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{9} + 16 q^{17} - 40 q^{25} + 16 q^{33} - 48 q^{41} + 56 q^{49} + 64 q^{65} - 32 q^{73} + 8 q^{81} - 80 q^{89} + 48 q^{97}+O(q^{100}) \) 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

\(\beta_{1}\)\(=\) \( \nu^{6} - 3\nu^{5} + 10\nu^{4} - 15\nu^{3} + 17\nu^{2} - 10\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\nu^{6} + 3\nu^{5} - 12\nu^{4} + 19\nu^{3} - 31\nu^{2} + 22\nu - 10 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\nu^{6} - 6\nu^{5} + 20\nu^{4} - 30\nu^{3} + 38\nu^{2} - 24\nu + 8 \) 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
\(\beta_{5}\)\(=\) \( 18\nu^{7} - 63\nu^{6} + 219\nu^{5} - 390\nu^{4} + 565\nu^{3} - 489\nu^{2} + 272\nu - 66 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -20\nu^{7} + 70\nu^{6} - 246\nu^{5} + 440\nu^{4} - 650\nu^{3} + 570\nu^{2} - 332\nu + 84 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( 38\nu^{7} - 133\nu^{6} + 465\nu^{5} - 830\nu^{4} + 1215\nu^{3} - 1059\nu^{2} + 608\nu - 152 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} + \beta_{6} - \beta_{5} + 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} + \beta_{6} - \beta_{5} + \beta_{3} - 2\beta _1 - 6 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -4\beta_{7} - 7\beta_{6} + 6\beta_{5} + 12\beta_{4} + 3\beta_{3} - 6\beta _1 - 20 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -5\beta_{7} - 8\beta_{6} + 7\beta_{5} + 12\beta_{4} - 4\beta_{3} - 2\beta_{2} + 6\beta _1 + 14 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 6\beta_{7} + 13\beta_{6} - 16\beta_{5} - 40\beta_{4} - 25\beta_{3} - 10\beta_{2} + 40\beta _1 + 104 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 22\beta_{7} + 40\beta_{6} - 42\beta_{5} - 90\beta_{4} + 8\beta_{3} + 5\beta_{2} - 7\beta _1 - 12 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 14\beta_{7} + 19\beta_{6} - 8\beta_{5} + 147\beta_{3} + 70\beta_{2} - 196\beta _1 - 472 ) / 8 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1536\mathbb{Z}\right)^\times\).

\(n\) \(511\) \(517\) \(1025\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
769.1
0.500000 + 0.691860i
0.500000 − 0.0297061i
0.500000 + 1.44392i
0.500000 − 2.10607i
0.500000 + 2.10607i
0.500000 − 1.44392i
0.500000 + 0.0297061i
0.500000 − 0.691860i
0 − 1.00000i 0 − 3.95687i 0 −1.63899 0 −1.00000 0
769.2 0 − 1.00000i 0 − 2.08402i 0 5.03127 0 −1.00000 0
769.3 0 − 1.00000i 0 2.08402i 0 −5.03127 0 −1.00000 0
769.4 0 − 1.00000i 0 3.95687i 0 1.63899 0 −1.00000 0
769.5 0 1.00000i 0 − 3.95687i 0 1.63899 0 −1.00000 0
769.6 0 1.00000i 0 − 2.08402i 0 −5.03127 0 −1.00000 0
769.7 0 1.00000i 0 2.08402i 0 5.03127 0 −1.00000 0
769.8 0 1.00000i 0 3.95687i 0 −1.63899 0 −1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 769.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1536.2.d.g 8
3.b odd 2 1 4608.2.d.p 8
4.b odd 2 1 inner 1536.2.d.g 8
8.b even 2 1 inner 1536.2.d.g 8
8.d odd 2 1 inner 1536.2.d.g 8
12.b even 2 1 4608.2.d.p 8
16.e even 4 1 1536.2.a.m ✓ 4
16.e even 4 1 1536.2.a.n yes 4
16.f odd 4 1 1536.2.a.m ✓ 4
16.f odd 4 1 1536.2.a.n yes 4
24.f even 2 1 4608.2.d.p 8
24.h odd 2 1 4608.2.d.p 8
48.i odd 4 1 4608.2.a.t 4
48.i odd 4 1 4608.2.a.ba 4
48.k even 4 1 4608.2.a.t 4
48.k even 4 1 4608.2.a.ba 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1536.2.a.m ✓ 4 16.e even 4 1
1536.2.a.m ✓ 4 16.f odd 4 1
1536.2.a.n yes 4 16.e even 4 1
1536.2.a.n yes 4 16.f odd 4 1
1536.2.d.g 8 1.a even 1 1 trivial
1536.2.d.g 8 4.b odd 2 1 inner
1536.2.d.g 8 8.b even 2 1 inner
1536.2.d.g 8 8.d odd 2 1 inner
4608.2.a.t 4 48.i odd 4 1
4608.2.a.t 4 48.k even 4 1
4608.2.a.ba 4 48.i odd 4 1
4608.2.a.ba 4 48.k even 4 1
4608.2.d.p 8 3.b odd 2 1
4608.2.d.p 8 12.b even 2 1
4608.2.d.p 8 24.f even 2 1
4608.2.d.p 8 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1536, [\chi])\):

\( T_{5}^{4} + 20T_{5}^{2} + 68 \) Copy content Toggle raw display
\( T_{7}^{4} - 28T_{7}^{2} + 68 \) Copy content Toggle raw display
\( T_{23}^{4} - 80T_{23}^{2} + 1088 \) 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
$5$ \( (T^{4} + 20 T^{2} + 68)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} - 28 T^{2} + 68)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 24 T^{2} + 16)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 40 T^{2} + 272)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 4 T - 4)^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} + 8)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} - 80 T^{2} + 1088)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 116 T^{2} + 3332)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 28 T^{2} + 68)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 56 T^{2} + 272)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 12 T + 28)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} + 176 T^{2} + 3136)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - 80 T^{2} + 1088)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 148 T^{2} + 68)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 96 T^{2} + 256)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 56 T^{2} + 272)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 192 T^{2} + 1024)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} - 112 T^{2} + 1088)^{2} \) Copy content Toggle raw display
$73$ \( (T + 4)^{8} \) Copy content Toggle raw display
$79$ \( (T^{4} - 28 T^{2} + 68)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 88 T^{2} + 784)^{2} \) Copy content Toggle raw display
$89$ \( (T + 10)^{8} \) Copy content Toggle raw display
$97$ \( (T^{2} - 12 T + 4)^{4} \) Copy content Toggle raw display
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