Properties

Label 1152.2.k.b
Level $1152$
Weight $2$
Character orbit 1152.k
Analytic conductor $9.199$
Analytic rank $0$
Dimension $2$
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(289,1152)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1152, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1152.289");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1152 = 2^{7} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1152.k (of order \(4\), degree \(2\), 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: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 16)
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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - i - 1) q^{5} - 2 i q^{7} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - i - 1) q^{5} - 2 i q^{7} + (i + 1) q^{11} + ( - i + 1) q^{13} + 2 q^{17} + (3 i - 3) q^{19} - 6 i q^{23} - 3 i q^{25} + ( - 3 i + 3) q^{29} - 8 q^{31} + (2 i - 2) q^{35} + ( - 3 i - 3) q^{37} + ( - 5 i - 5) q^{43} - 8 q^{47} + 3 q^{49} + ( - 5 i - 5) q^{53} - 2 i q^{55} + ( - 3 i - 3) q^{59} + ( - 9 i + 9) q^{61} - 2 q^{65} + ( - 5 i + 5) q^{67} + 10 i q^{71} - 4 i q^{73} + ( - 2 i + 2) q^{77} + (i - 1) q^{83} + ( - 2 i - 2) q^{85} - 4 i q^{89} + ( - 2 i - 2) q^{91} + 6 q^{95} - 2 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{5} + 2 q^{11} + 2 q^{13} + 4 q^{17} - 6 q^{19} + 6 q^{29} - 16 q^{31} - 4 q^{35} - 6 q^{37} - 10 q^{43} - 16 q^{47} + 6 q^{49} - 10 q^{53} - 6 q^{59} + 18 q^{61} - 4 q^{65} + 10 q^{67} + 4 q^{77} - 2 q^{83} - 4 q^{85} - 4 q^{91} + 12 q^{95} - 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(127\) \(641\) \(901\)
\(\chi(n)\) \(1\) \(1\) \(i\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
289.1
1.00000i
1.00000i
0 0 0 −1.00000 + 1.00000i 0 2.00000i 0 0 0
865.1 0 0 0 −1.00000 1.00000i 0 2.00000i 0 0 0
\(n\): e.g. 2-40 or 990-1000
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 1152.2.k.b 2
3.b odd 2 1 128.2.e.b 2
4.b odd 2 1 1152.2.k.a 2
8.b even 2 1 144.2.k.a 2
8.d odd 2 1 576.2.k.a 2
12.b even 2 1 128.2.e.a 2
16.e even 4 1 144.2.k.a 2
16.e even 4 1 inner 1152.2.k.b 2
16.f odd 4 1 576.2.k.a 2
16.f odd 4 1 1152.2.k.a 2
24.f even 2 1 64.2.e.a 2
24.h odd 2 1 16.2.e.a 2
32.g even 8 2 9216.2.a.d 2
32.h odd 8 2 9216.2.a.s 2
48.i odd 4 1 16.2.e.a 2
48.i odd 4 1 128.2.e.b 2
48.k even 4 1 64.2.e.a 2
48.k even 4 1 128.2.e.a 2
96.o even 8 2 1024.2.a.e 2
96.o even 8 2 1024.2.b.b 2
96.p odd 8 2 1024.2.a.b 2
96.p odd 8 2 1024.2.b.e 2
120.i odd 2 1 400.2.l.c 2
120.m even 2 1 1600.2.l.a 2
120.q odd 4 1 1600.2.q.a 2
120.q odd 4 1 1600.2.q.b 2
120.w even 4 1 400.2.q.a 2
120.w even 4 1 400.2.q.b 2
168.i even 2 1 784.2.m.b 2
168.s odd 6 2 784.2.x.f 4
168.ba even 6 2 784.2.x.c 4
240.t even 4 1 1600.2.l.a 2
240.z odd 4 1 1600.2.q.b 2
240.bb even 4 1 400.2.q.a 2
240.bd odd 4 1 1600.2.q.a 2
240.bf even 4 1 400.2.q.b 2
240.bm odd 4 1 400.2.l.c 2
336.y even 4 1 784.2.m.b 2
336.bo even 12 2 784.2.x.c 4
336.bt odd 12 2 784.2.x.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
16.2.e.a 2 24.h odd 2 1
16.2.e.a 2 48.i odd 4 1
64.2.e.a 2 24.f even 2 1
64.2.e.a 2 48.k even 4 1
128.2.e.a 2 12.b even 2 1
128.2.e.a 2 48.k even 4 1
128.2.e.b 2 3.b odd 2 1
128.2.e.b 2 48.i odd 4 1
144.2.k.a 2 8.b even 2 1
144.2.k.a 2 16.e even 4 1
400.2.l.c 2 120.i odd 2 1
400.2.l.c 2 240.bm odd 4 1
400.2.q.a 2 120.w even 4 1
400.2.q.a 2 240.bb even 4 1
400.2.q.b 2 120.w even 4 1
400.2.q.b 2 240.bf even 4 1
576.2.k.a 2 8.d odd 2 1
576.2.k.a 2 16.f odd 4 1
784.2.m.b 2 168.i even 2 1
784.2.m.b 2 336.y even 4 1
784.2.x.c 4 168.ba even 6 2
784.2.x.c 4 336.bo even 12 2
784.2.x.f 4 168.s odd 6 2
784.2.x.f 4 336.bt odd 12 2
1024.2.a.b 2 96.p odd 8 2
1024.2.a.e 2 96.o even 8 2
1024.2.b.b 2 96.o even 8 2
1024.2.b.e 2 96.p odd 8 2
1152.2.k.a 2 4.b odd 2 1
1152.2.k.a 2 16.f odd 4 1
1152.2.k.b 2 1.a even 1 1 trivial
1152.2.k.b 2 16.e even 4 1 inner
1600.2.l.a 2 120.m even 2 1
1600.2.l.a 2 240.t even 4 1
1600.2.q.a 2 120.q odd 4 1
1600.2.q.a 2 240.bd odd 4 1
1600.2.q.b 2 120.q odd 4 1
1600.2.q.b 2 240.z odd 4 1
9216.2.a.d 2 32.g even 8 2
9216.2.a.s 2 32.h odd 8 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1152, [\chi])\):

\( T_{5}^{2} + 2T_{5} + 2 \) Copy content Toggle raw display
\( T_{11}^{2} - 2T_{11} + 2 \) Copy content Toggle raw display
\( T_{19}^{2} + 6T_{19} + 18 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
$7$ \( T^{2} + 4 \) Copy content Toggle raw display
$11$ \( T^{2} - 2T + 2 \) Copy content Toggle raw display
$13$ \( T^{2} - 2T + 2 \) Copy content Toggle raw display
$17$ \( (T - 2)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 6T + 18 \) Copy content Toggle raw display
$23$ \( T^{2} + 36 \) Copy content Toggle raw display
$29$ \( T^{2} - 6T + 18 \) Copy content Toggle raw display
$31$ \( (T + 8)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 6T + 18 \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 10T + 50 \) Copy content Toggle raw display
$47$ \( (T + 8)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 10T + 50 \) Copy content Toggle raw display
$59$ \( T^{2} + 6T + 18 \) Copy content Toggle raw display
$61$ \( T^{2} - 18T + 162 \) Copy content Toggle raw display
$67$ \( T^{2} - 10T + 50 \) Copy content Toggle raw display
$71$ \( T^{2} + 100 \) Copy content Toggle raw display
$73$ \( T^{2} + 16 \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
$89$ \( T^{2} + 16 \) Copy content Toggle raw display
$97$ \( (T + 2)^{2} \) Copy content Toggle raw display
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