Properties

Label 1792.2.m.b
Level $1792$
Weight $2$
Character orbit 1792.m
Analytic conductor $14.309$
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] = [1792,2,Mod(449,1792)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1792, base_ring=CyclotomicField(4))
 
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("1792.449");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1792 = 2^{8} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1792.m (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: \(14.3091920422\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
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: \( 2^{4} \)
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.

\(f(q)\) \(=\) \( q - \beta_{3} q^{3} + ( - \beta_{7} - \beta_{6} + \beta_{4} + 1) q^{5} + \beta_{6} q^{7} + ( - \beta_{7} - \beta_{6} + \beta_{3} + \beta_{2} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{3} + ( - \beta_{7} - \beta_{6} + \beta_{4} + 1) q^{5} + \beta_{6} q^{7} + ( - \beta_{7} - \beta_{6} + \beta_{3} + \beta_{2} - 1) q^{9} + (\beta_{7} - 2 \beta_{4} + \beta_1) q^{11} + ( - \beta_{7} - \beta_{6} - \beta_{3} - \beta_1 - 1) q^{13} + \beta_1 q^{15} + (\beta_{3} - \beta_{2} + 1) q^{17} + ( - \beta_{7} - 2 \beta_{6} - 3 \beta_{5} - \beta_1 + 1) q^{19} + ( - \beta_{2} + 1) q^{21} + ( - \beta_{7} + 2 \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} - \beta_1 + 2) q^{23} + ( - 5 \beta_{7} - 3 \beta_{6} - 2 \beta_{5} + 2 \beta_{4} + \beta_{3} + \beta_{2} - 2 \beta_1 + 1) q^{25} + (\beta_{7} - 2 \beta_{6} + \beta_{4} + \beta_{2} - 2 \beta_1 + 1) q^{27} + ( - 2 \beta_{7} + \beta_{6} - 2 \beta_{3} - 2 \beta_1 + 1) q^{29} + ( - \beta_{5} - \beta_{4} - 2 \beta_{3} + 2 \beta_{2} + \beta_1 - 1) q^{31} + (\beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} - 3 \beta_1 - 2) q^{33} + (\beta_{7} + \beta_{6} + \beta_{5} + \beta_1) q^{35} + ( - \beta_{7} - \beta_{6} - 2 \beta_{4} + 2 \beta_{2} + 3 \beta_1 - 1) q^{37} + ( - \beta_{7} + 2 \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} - \beta_1) q^{39} + (4 \beta_{6} + 2 \beta_{5} - 2 \beta_{4} + \beta_{3} + \beta_{2} + 2 \beta_1 - 3) q^{41} + ( - 2 \beta_{7} - 4 \beta_{6} + 2 \beta_{2} + 2 \beta_1 + 2) q^{43} + ( - 3 \beta_{7} - 3 \beta_{6} - 2 \beta_{5} + \beta_{3} - 3 \beta_1 - 1) q^{45} + ( - \beta_{3} + \beta_{2} - 5) q^{47} - q^{49} + (\beta_{7} - 2 \beta_{6} - 2 \beta_{3} + \beta_1 - 2) q^{51} + (\beta_{7} + 3 \beta_{6} - 2 \beta_{2} - \beta_1 - 1) q^{53} + (4 \beta_{7} + 8 \beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} + \beta_1) q^{55} + ( - 3 \beta_{7} + 2 \beta_{5} - 2 \beta_{4} + \beta_{3} + \beta_{2} + 2 \beta_1 - 3) q^{57} + ( - \beta_{7} - 6 \beta_{6} + \beta_{4} - 2 \beta_{2} + 8) q^{59} + ( - \beta_{7} + 5 \beta_{6} + 3 \beta_{5} - 2 \beta_{3} - \beta_1 + 2) q^{61} + ( - \beta_{3} + \beta_{2} + \beta_1) q^{63} + ( - 2 \beta_{5} - 2 \beta_{4} + \beta_{3} - \beta_{2} - \beta_1 - 3) q^{65} + ( - 2 \beta_{7} - 6 \beta_{6} + 4 \beta_{3} - 2 \beta_1 - 6) q^{67} + ( - 2 \beta_{7} + 2 \beta_{6} - \beta_{4} - \beta_{2} + 3 \beta_1 - 1) q^{69} + (6 \beta_{6} + 3 \beta_{5} - 3 \beta_{4} + \beta_{3} + \beta_{2} + 3 \beta_1 - 4) q^{71} + (2 \beta_{7} + 6 \beta_{6} - 4 \beta_{3} - 4 \beta_{2} + 4) q^{73} + ( - \beta_{7} - 2 \beta_{6} + \beta_{4} - 4 \beta_{2} + 6) q^{75} + ( - \beta_{7} - 2 \beta_{5} - \beta_1 + 2) q^{77} + ( - \beta_{5} - \beta_{4} + 5 \beta_{3} - 5 \beta_{2} + \beta_1) q^{79} + ( - 2 \beta_{5} - 2 \beta_{4} + \beta_{3} - \beta_{2} - \beta_1 + 2) q^{81} + (4 \beta_{6} + 2 \beta_{5} - 3 \beta_{3} + 2) q^{83} + (\beta_{7} - \beta_1) q^{85} + ( - 2 \beta_{7} + 4 \beta_{6} - 2 \beta_{5} + 2 \beta_{4} - \beta_{3} - \beta_{2} - 2 \beta_1 + 3) q^{87} + ( - 8 \beta_{7} - 2 \beta_{6} - 3 \beta_{5} + 3 \beta_{4} - 3 \beta_{3} - 3 \beta_{2} + \cdots + 6) q^{89}+ \cdots + (3 \beta_{7} + 2 \beta_{6} + 2 \beta_{5} + 3 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{3} + 4 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{3} + 4 q^{5} + 8 q^{11} - 12 q^{13} + 8 q^{17} - 4 q^{19} + 4 q^{21} + 8 q^{27} - 8 q^{31} - 16 q^{33} + 4 q^{35} + 8 q^{37} + 24 q^{43} - 12 q^{45} - 40 q^{47} - 8 q^{49} - 24 q^{51} - 16 q^{53} + 52 q^{59} + 20 q^{61} - 24 q^{65} - 32 q^{67} - 8 q^{69} + 28 q^{75} + 8 q^{77} + 16 q^{81} + 12 q^{83} + 12 q^{91} + 40 q^{93} - 80 q^{95} + 72 q^{97} + 8 q^{99}+O(q^{100}) \) 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

\(\beta_{1}\)\(=\) \( \nu^{6} - 3\nu^{5} + 10\nu^{4} - 15\nu^{3} + 19\nu^{2} - 12\nu + 4 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{7} - 3\nu^{6} + 11\nu^{5} - 17\nu^{4} + 26\nu^{3} - 19\nu^{2} + 13\nu - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{7} - 4\nu^{6} + 14\nu^{5} - 28\nu^{4} + 43\nu^{3} - 43\nu^{2} + 29\nu - 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( 5\nu^{7} - 17\nu^{6} + 59\nu^{5} - 102\nu^{4} + 146\nu^{3} - 120\nu^{2} + 65\nu - 14 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -5\nu^{7} + 17\nu^{6} - 59\nu^{5} + 103\nu^{4} - 148\nu^{3} + 127\nu^{2} - 71\nu + 19 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -8\nu^{7} + 28\nu^{6} - 98\nu^{5} + 175\nu^{4} - 256\nu^{3} + 223\nu^{2} - 126\nu + 31 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -10\nu^{7} + 35\nu^{6} - 123\nu^{5} + 220\nu^{4} - 325\nu^{3} + 285\nu^{2} - 166\nu + 42 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{7} - \beta_{6} + \beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} + 2\beta_{3} + \beta _1 - 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -4\beta_{7} + 5\beta_{6} + \beta_{5} + 2\beta_{4} - \beta_{3} - 4\beta_{2} + \beta _1 - 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -9\beta_{7} + 11\beta_{6} - 3\beta_{5} - \beta_{4} - 10\beta_{3} - 2\beta_{2} - 5\beta _1 + 14 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 9\beta_{7} - 13\beta_{6} - 5\beta_{5} - 10\beta_{4} - 7\beta_{3} + 18\beta_{2} - 10\beta _1 + 18 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 50\beta_{7} - 67\beta_{6} + 11\beta_{5} - 9\beta_{4} + 38\beta_{3} + 26\beta_{2} + 18\beta _1 - 48 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 8\beta_{7} - 7\beta_{6} + 30\beta_{5} + 33\beta_{4} + 72\beta_{3} - 61\beta_{2} + 72\beta _1 - 122 ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1792\mathbb{Z}\right)^\times\).

\(n\) \(1023\) \(1025\) \(1541\)
\(\chi(n)\) \(1\) \(1\) \(-\beta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
449.1
0.500000 0.691860i
0.500000 1.44392i
0.500000 + 0.0297061i
0.500000 + 2.10607i
0.500000 + 0.691860i
0.500000 + 1.44392i
0.500000 0.0297061i
0.500000 2.10607i
0 −1.89897 + 1.89897i 0 0.372364 + 0.372364i 0 1.00000i 0 4.21215i 0
449.2 0 −1.23681 + 1.23681i 0 −0.571717 0.571717i 0 1.00000i 0 0.0594122i 0
449.3 0 0.236813 0.236813i 0 2.98593 + 2.98593i 0 1.00000i 0 2.88784i 0
449.4 0 0.898966 0.898966i 0 −0.786578 0.786578i 0 1.00000i 0 1.38372i 0
1345.1 0 −1.89897 1.89897i 0 0.372364 0.372364i 0 1.00000i 0 4.21215i 0
1345.2 0 −1.23681 1.23681i 0 −0.571717 + 0.571717i 0 1.00000i 0 0.0594122i 0
1345.3 0 0.236813 + 0.236813i 0 2.98593 2.98593i 0 1.00000i 0 2.88784i 0
1345.4 0 0.898966 + 0.898966i 0 −0.786578 + 0.786578i 0 1.00000i 0 1.38372i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 449.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1792.2.m.b yes 8
4.b odd 2 1 1792.2.m.d yes 8
8.b even 2 1 1792.2.m.c yes 8
8.d odd 2 1 1792.2.m.a 8
16.e even 4 1 inner 1792.2.m.b yes 8
16.e even 4 1 1792.2.m.c yes 8
16.f odd 4 1 1792.2.m.a 8
16.f odd 4 1 1792.2.m.d yes 8
32.g even 8 1 7168.2.a.s 4
32.g even 8 1 7168.2.a.w 4
32.h odd 8 1 7168.2.a.t 4
32.h odd 8 1 7168.2.a.x 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1792.2.m.a 8 8.d odd 2 1
1792.2.m.a 8 16.f odd 4 1
1792.2.m.b yes 8 1.a even 1 1 trivial
1792.2.m.b yes 8 16.e even 4 1 inner
1792.2.m.c yes 8 8.b even 2 1
1792.2.m.c yes 8 16.e even 4 1
1792.2.m.d yes 8 4.b odd 2 1
1792.2.m.d yes 8 16.f odd 4 1
7168.2.a.s 4 32.g even 8 1
7168.2.a.t 4 32.h odd 8 1
7168.2.a.w 4 32.g even 8 1
7168.2.a.x 4 32.h odd 8 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1792, [\chi])\):

\( T_{3}^{8} + 4T_{3}^{7} + 8T_{3}^{6} + 8T_{3}^{3} + 32T_{3}^{2} - 16T_{3} + 4 \) Copy content Toggle raw display
\( T_{5}^{8} - 4T_{5}^{7} + 8T_{5}^{6} + 24T_{5}^{5} + 32T_{5}^{4} + 8T_{5}^{3} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + 4 T^{7} + 8 T^{6} + 8 T^{3} + \cdots + 4 \) Copy content Toggle raw display
$5$ \( T^{8} - 4 T^{7} + 8 T^{6} + 24 T^{5} + \cdots + 4 \) Copy content Toggle raw display
$7$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$11$ \( T^{8} - 8 T^{7} + 32 T^{6} + 32 T^{5} + \cdots + 256 \) Copy content Toggle raw display
$13$ \( T^{8} + 12 T^{7} + 72 T^{6} + \cdots + 3844 \) Copy content Toggle raw display
$17$ \( (T^{4} - 4 T^{3} - 4 T^{2} + 16 T + 8)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + 4 T^{7} + 8 T^{6} + \cdots + 31684 \) Copy content Toggle raw display
$23$ \( T^{8} + 56 T^{6} + 792 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$29$ \( T^{8} + 128 T^{5} + 2696 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$31$ \( (T^{4} + 4 T^{3} - 44 T^{2} + 32 T + 8)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} - 8 T^{7} + 32 T^{6} + \cdots + 258064 \) Copy content Toggle raw display
$41$ \( T^{8} + 184 T^{6} + 8736 T^{4} + \cdots + 61504 \) Copy content Toggle raw display
$43$ \( T^{8} - 24 T^{7} + 288 T^{6} + \cdots + 984064 \) Copy content Toggle raw display
$47$ \( (T^{4} + 20 T^{3} + 140 T^{2} + 400 T + 392)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} + 16 T^{7} + 128 T^{6} + \cdots + 80656 \) Copy content Toggle raw display
$59$ \( T^{8} - 52 T^{7} + 1352 T^{6} + \cdots + 21104836 \) Copy content Toggle raw display
$61$ \( T^{8} - 20 T^{7} + 200 T^{6} + \cdots + 40934404 \) Copy content Toggle raw display
$67$ \( T^{8} + 32 T^{7} + 512 T^{6} + \cdots + 20647936 \) Copy content Toggle raw display
$71$ \( T^{8} + 384 T^{6} + 44320 T^{4} + \cdots + 5837056 \) Copy content Toggle raw display
$73$ \( T^{8} + 368 T^{6} + 44896 T^{4} + \cdots + 2027776 \) Copy content Toggle raw display
$79$ \( (T^{4} - 288 T^{2} + 128 T + 19088)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} - 12 T^{7} + 72 T^{6} + \cdots + 9604 \) Copy content Toggle raw display
$89$ \( T^{8} + 576 T^{6} + \cdots + 24760576 \) Copy content Toggle raw display
$97$ \( (T^{4} - 36 T^{3} + 388 T^{2} - 960 T - 3064)^{2} \) Copy content Toggle raw display
show more
show less