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Properties

Label 1152.2.v.b
Level $1152$
Weight $2$
Character orbit 1152.v
Analytic conductor $9.199$
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] = [1152,2,Mod(145,1152)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1152, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([0, 7, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1152.145");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1152 = 2^{7} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1152.v (of order \(8\), degree \(4\), 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: \(9.19876631285\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{8})\)
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: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 32)
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{7} + \beta_{4}) q^{5} + (\beta_{7} - \beta_{3} - \beta_1 + 1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{7} + \beta_{4}) q^{5} + (\beta_{7} - \beta_{3} - \beta_1 + 1) q^{7} + (2 \beta_{7} + \beta_{4} - \beta_{2} + 1) q^{11} + ( - \beta_{7} - \beta_{6} - 2 \beta_{5} + \cdots - 2) q^{13}+ \cdots + (3 \beta_{6} + \beta_{5} + 3 \beta_{4} + \cdots + 4) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{7} + 4 q^{11} - 8 q^{13} - 4 q^{19} - 8 q^{23} - 8 q^{25} - 32 q^{31} + 16 q^{35} - 8 q^{37} - 8 q^{41} + 12 q^{43} - 8 q^{53} + 16 q^{55} - 20 q^{59} + 24 q^{61} + 36 q^{67} - 24 q^{71} - 32 q^{73} - 16 q^{77} + 20 q^{83} + 8 q^{85} + 16 q^{89} - 40 q^{91} - 8 q^{95} + 32 q^{97}+O(q^{100}) \) 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

\(\beta_{1}\)\(=\) \( \nu^{7} - 4\nu^{6} + 14\nu^{5} - 27\nu^{4} + 41\nu^{3} - 37\nu^{2} + 24\nu - 5 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{7} - 4\nu^{6} + 14\nu^{5} - 28\nu^{4} + 43\nu^{3} - 44\nu^{2} + 30\nu - 10 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -5\nu^{7} + 17\nu^{6} - 59\nu^{5} + 102\nu^{4} - 146\nu^{3} + 121\nu^{2} - 66\nu + 15 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( 5\nu^{7} - 17\nu^{6} + 60\nu^{5} - 105\nu^{4} + 155\nu^{3} - 133\nu^{2} + 77\nu - 19 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -5\nu^{7} + 18\nu^{6} - 62\nu^{5} + 113\nu^{4} - 163\nu^{3} + 145\nu^{2} - 82\nu + 20 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -5\nu^{7} + 18\nu^{6} - 63\nu^{5} + 115\nu^{4} - 170\nu^{3} + 152\nu^{2} - 89\nu + 23 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -8\nu^{7} + 28\nu^{6} - 98\nu^{5} + 175\nu^{4} - 256\nu^{3} + 223\nu^{2} - 126\nu + 31 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{7} + 2\beta_{6} + \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{7} + 3\beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} + 2\beta _1 - 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 5\beta_{7} - 5\beta_{6} - 2\beta_{5} + 3\beta_{4} + \beta_{3} - 4\beta_{2} - \beta _1 - 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 11\beta_{7} - 19\beta_{6} + 3\beta_{5} - \beta_{4} - 5\beta_{3} - 4\beta_{2} - 8\beta _1 + 12 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -13\beta_{7} + 2\beta_{6} + 15\beta_{5} - 16\beta_{4} - 10\beta_{3} + 13\beta_{2} - 2\beta _1 + 23 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -67\beta_{7} + 90\beta_{6} + 4\beta_{5} - 10\beta_{4} + 16\beta_{3} + 31\beta_{2} + 33\beta _1 - 28 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -7\beta_{7} + 87\beta_{6} - 68\beta_{5} + 71\beta_{4} + 65\beta_{3} - 26\beta_{2} + 37\beta _1 - 125 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(641\) \(901\)
\(\chi(n)\) \(1\) \(1\) \(-\beta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
145.1
0.500000 − 0.691860i
0.500000 + 2.10607i
0.500000 − 0.0297061i
0.500000 + 1.44392i
0.500000 + 0.0297061i
0.500000 − 1.44392i
0.500000 + 0.691860i
0.500000 − 2.10607i
0 0 0 −0.707107 + 0.292893i 0 −1.68554 + 1.68554i 0 0 0
145.2 0 0 0 −0.707107 + 0.292893i 0 2.27133 − 2.27133i 0 0 0
433.1 0 0 0 0.707107 − 1.70711i 0 0.665096 + 0.665096i 0 0 0
433.2 0 0 0 0.707107 − 1.70711i 0 2.74912 + 2.74912i 0 0 0
721.1 0 0 0 0.707107 + 1.70711i 0 0.665096 − 0.665096i 0 0 0
721.2 0 0 0 0.707107 + 1.70711i 0 2.74912 − 2.74912i 0 0 0
1009.1 0 0 0 −0.707107 − 0.292893i 0 −1.68554 − 1.68554i 0 0 0
1009.2 0 0 0 −0.707107 − 0.292893i 0 2.27133 + 2.27133i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 145.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
32.g even 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1152.2.v.b 8
3.b odd 2 1 128.2.g.b 8
4.b odd 2 1 288.2.v.b 8
12.b even 2 1 32.2.g.b ✓ 8
24.f even 2 1 256.2.g.d 8
24.h odd 2 1 256.2.g.c 8
32.g even 8 1 inner 1152.2.v.b 8
32.h odd 8 1 288.2.v.b 8
48.i odd 4 1 512.2.g.f 8
48.i odd 4 1 512.2.g.g 8
48.k even 4 1 512.2.g.e 8
48.k even 4 1 512.2.g.h 8
60.h even 2 1 800.2.y.b 8
60.l odd 4 1 800.2.ba.c 8
60.l odd 4 1 800.2.ba.d 8
96.o even 8 1 32.2.g.b ✓ 8
96.o even 8 1 256.2.g.d 8
96.o even 8 1 512.2.g.e 8
96.o even 8 1 512.2.g.h 8
96.p odd 8 1 128.2.g.b 8
96.p odd 8 1 256.2.g.c 8
96.p odd 8 1 512.2.g.f 8
96.p odd 8 1 512.2.g.g 8
192.q odd 16 2 4096.2.a.q 8
192.s even 16 2 4096.2.a.k 8
480.bq odd 8 1 800.2.ba.d 8
480.bs even 8 1 800.2.y.b 8
480.ca odd 8 1 800.2.ba.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
32.2.g.b ✓ 8 12.b even 2 1
32.2.g.b ✓ 8 96.o even 8 1
128.2.g.b 8 3.b odd 2 1
128.2.g.b 8 96.p odd 8 1
256.2.g.c 8 24.h odd 2 1
256.2.g.c 8 96.p odd 8 1
256.2.g.d 8 24.f even 2 1
256.2.g.d 8 96.o even 8 1
288.2.v.b 8 4.b odd 2 1
288.2.v.b 8 32.h odd 8 1
512.2.g.e 8 48.k even 4 1
512.2.g.e 8 96.o even 8 1
512.2.g.f 8 48.i odd 4 1
512.2.g.f 8 96.p odd 8 1
512.2.g.g 8 48.i odd 4 1
512.2.g.g 8 96.p odd 8 1
512.2.g.h 8 48.k even 4 1
512.2.g.h 8 96.o even 8 1
800.2.y.b 8 60.h even 2 1
800.2.y.b 8 480.bs even 8 1
800.2.ba.c 8 60.l odd 4 1
800.2.ba.c 8 480.ca odd 8 1
800.2.ba.d 8 60.l odd 4 1
800.2.ba.d 8 480.bq odd 8 1
1152.2.v.b 8 1.a even 1 1 trivial
1152.2.v.b 8 32.g even 8 1 inner
4096.2.a.k 8 192.s even 16 2
4096.2.a.q 8 192.q odd 16 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 2T_{5}^{2} + 4T_{5} + 2 \) acting on \(S_{2}^{\mathrm{new}}(1152, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{4} + 2 T^{2} + 4 T + 2)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} - 8 T^{7} + \cdots + 784 \) Copy content Toggle raw display
$11$ \( T^{8} - 4 T^{7} + \cdots + 4 \) Copy content Toggle raw display
$13$ \( T^{8} + 8 T^{7} + \cdots + 6724 \) Copy content Toggle raw display
$17$ \( T^{8} + 64 T^{6} + \cdots + 256 \) Copy content Toggle raw display
$19$ \( T^{8} + 4 T^{7} + \cdots + 196 \) Copy content Toggle raw display
$23$ \( T^{8} + 8 T^{7} + \cdots + 16 \) Copy content Toggle raw display
$29$ \( T^{8} - 12 T^{6} + \cdots + 188356 \) Copy content Toggle raw display
$31$ \( (T^{2} + 8 T + 8)^{4} \) Copy content Toggle raw display
$37$ \( T^{8} + 8 T^{7} + \cdots + 64516 \) Copy content Toggle raw display
$41$ \( T^{8} + 8 T^{7} + \cdots + 26896 \) Copy content Toggle raw display
$43$ \( T^{8} - 12 T^{7} + \cdots + 31684 \) Copy content Toggle raw display
$47$ \( T^{8} + 64 T^{6} + \cdots + 256 \) Copy content Toggle raw display
$53$ \( T^{8} + 8 T^{7} + \cdots + 158404 \) Copy content Toggle raw display
$59$ \( T^{8} + 20 T^{7} + \cdots + 643204 \) Copy content Toggle raw display
$61$ \( T^{8} - 24 T^{7} + \cdots + 42436 \) Copy content Toggle raw display
$67$ \( T^{8} - 36 T^{7} + \cdots + 1285956 \) Copy content Toggle raw display
$71$ \( T^{8} + 24 T^{7} + \cdots + 21196816 \) Copy content Toggle raw display
$73$ \( T^{8} + 32 T^{7} + \cdots + 38416 \) Copy content Toggle raw display
$79$ \( T^{8} + 512 T^{6} + \cdots + 99361024 \) Copy content Toggle raw display
$83$ \( T^{8} - 20 T^{7} + \cdots + 138250564 \) Copy content Toggle raw display
$89$ \( T^{8} - 16 T^{7} + \cdots + 17007376 \) Copy content Toggle raw display
$97$ \( (T^{4} - 16 T^{3} + \cdots - 992)^{2} \) Copy content Toggle raw display
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