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Properties

Label 896.2.ba.f
Level $896$
Weight $2$
Character orbit 896.ba
Analytic conductor $7.155$
Analytic rank $0$
Dimension $48$
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] = [896,2,Mod(289,896)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(896, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 9, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("896.289");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 896 = 2^{7} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 896.ba (of order \(12\), 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: \(7.15459602111\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(12\) over \(\Q(\zeta_{12})\)
Twist minimal: no (minimal twist has level 112)
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 4 q^{5}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 4 q^{5} + 4 q^{11} + 24 q^{13} - 40 q^{15} + 8 q^{17} + 4 q^{19} + 8 q^{21} + 24 q^{27} - 24 q^{29} + 28 q^{31} + 16 q^{33} - 28 q^{35} + 24 q^{37} + 40 q^{43} + 28 q^{45} - 20 q^{47} - 24 q^{51} + 16 q^{53} + 20 q^{59} - 8 q^{61} - 16 q^{63} + 8 q^{65} - 48 q^{67} + 40 q^{69} + 4 q^{75} + 20 q^{77} - 36 q^{79} + 8 q^{83} - 64 q^{91} - 8 q^{93} - 4 q^{95} - 48 q^{97} + 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
289.1 0 −0.814813 + 3.04092i 0 −0.501787 − 1.87270i 0 1.89831 − 1.84294i 0 −5.98523 − 3.45557i 0
289.2 0 −0.659252 + 2.46036i 0 0.502490 + 1.87532i 0 −0.364450 + 2.62053i 0 −3.02070 − 1.74400i 0
289.3 0 −0.589510 + 2.20008i 0 0.622192 + 2.32205i 0 −2.41058 − 1.09047i 0 −1.89476 − 1.09394i 0
289.4 0 −0.430984 + 1.60845i 0 0.522232 + 1.94900i 0 1.89075 + 1.85069i 0 0.196696 + 0.113563i 0
289.5 0 −0.224854 + 0.839165i 0 −0.847576 − 3.16320i 0 −0.654939 − 2.56341i 0 1.94444 + 1.12262i 0
289.6 0 −0.0145698 + 0.0543752i 0 0.337028 + 1.25781i 0 −0.230738 − 2.63567i 0 2.59533 + 1.49842i 0
289.7 0 0.221819 − 0.827840i 0 1.10095 + 4.10878i 0 2.50325 − 0.856573i 0 1.96196 + 1.13274i 0
289.8 0 0.222846 − 0.831674i 0 −0.543265 − 2.02749i 0 2.63544 − 0.233350i 0 1.95606 + 1.12933i 0
289.9 0 0.312249 − 1.16533i 0 0.262843 + 0.980942i 0 −2.60251 − 0.476386i 0 1.33758 + 0.772254i 0
289.10 0 0.523249 − 1.95279i 0 0.256983 + 0.959072i 0 −0.292831 + 2.62950i 0 −0.941530 − 0.543593i 0
289.11 0 0.672759 − 2.51077i 0 −0.780276 − 2.91203i 0 −1.41955 − 2.23268i 0 −3.25328 − 1.87828i 0
289.12 0 0.781062 − 2.91496i 0 −0.199758 − 0.745506i 0 2.51194 + 0.830765i 0 −5.28887 − 3.05353i 0
417.1 0 −2.91496 + 0.781062i 0 0.745506 + 0.199758i 0 −2.51194 + 0.830765i 0 5.28887 − 3.05353i 0
417.2 0 −2.51077 + 0.672759i 0 2.91203 + 0.780276i 0 1.41955 − 2.23268i 0 3.25328 − 1.87828i 0
417.3 0 −1.95279 + 0.523249i 0 −0.959072 − 0.256983i 0 0.292831 + 2.62950i 0 0.941530 − 0.543593i 0
417.4 0 −1.16533 + 0.312249i 0 −0.980942 − 0.262843i 0 2.60251 − 0.476386i 0 −1.33758 + 0.772254i 0
417.5 0 −0.831674 + 0.222846i 0 2.02749 + 0.543265i 0 −2.63544 − 0.233350i 0 −1.95606 + 1.12933i 0
417.6 0 −0.827840 + 0.221819i 0 −4.10878 − 1.10095i 0 −2.50325 − 0.856573i 0 −1.96196 + 1.13274i 0
417.7 0 0.0543752 − 0.0145698i 0 −1.25781 − 0.337028i 0 0.230738 − 2.63567i 0 −2.59533 + 1.49842i 0
417.8 0 0.839165 − 0.224854i 0 3.16320 + 0.847576i 0 0.654939 − 2.56341i 0 −1.94444 + 1.12262i 0
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 289.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
16.e even 4 1 inner
112.w even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 896.2.ba.f 48
4.b odd 2 1 896.2.ba.e 48
7.c even 3 1 inner 896.2.ba.f 48
8.b even 2 1 112.2.w.c ✓ 48
8.d odd 2 1 448.2.ba.c 48
16.e even 4 1 112.2.w.c ✓ 48
16.e even 4 1 inner 896.2.ba.f 48
16.f odd 4 1 448.2.ba.c 48
16.f odd 4 1 896.2.ba.e 48
28.g odd 6 1 896.2.ba.e 48
56.h odd 2 1 784.2.x.o 48
56.j odd 6 1 784.2.m.k 24
56.j odd 6 1 784.2.x.o 48
56.k odd 6 1 448.2.ba.c 48
56.p even 6 1 112.2.w.c ✓ 48
56.p even 6 1 784.2.m.j 24
112.l odd 4 1 784.2.x.o 48
112.u odd 12 1 448.2.ba.c 48
112.u odd 12 1 896.2.ba.e 48
112.w even 12 1 112.2.w.c ✓ 48
112.w even 12 1 784.2.m.j 24
112.w even 12 1 inner 896.2.ba.f 48
112.x odd 12 1 784.2.m.k 24
112.x odd 12 1 784.2.x.o 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.2.w.c ✓ 48 8.b even 2 1
112.2.w.c ✓ 48 16.e even 4 1
112.2.w.c ✓ 48 56.p even 6 1
112.2.w.c ✓ 48 112.w even 12 1
448.2.ba.c 48 8.d odd 2 1
448.2.ba.c 48 16.f odd 4 1
448.2.ba.c 48 56.k odd 6 1
448.2.ba.c 48 112.u odd 12 1
784.2.m.j 24 56.p even 6 1
784.2.m.j 24 112.w even 12 1
784.2.m.k 24 56.j odd 6 1
784.2.m.k 24 112.x odd 12 1
784.2.x.o 48 56.h odd 2 1
784.2.x.o 48 56.j odd 6 1
784.2.x.o 48 112.l odd 4 1
784.2.x.o 48 112.x odd 12 1
896.2.ba.e 48 4.b odd 2 1
896.2.ba.e 48 16.f odd 4 1
896.2.ba.e 48 28.g odd 6 1
896.2.ba.e 48 112.u odd 12 1
896.2.ba.f 48 1.a even 1 1 trivial
896.2.ba.f 48 7.c even 3 1 inner
896.2.ba.f 48 16.e even 4 1 inner
896.2.ba.f 48 112.w even 12 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{48} - 8 T_{3}^{45} - 162 T_{3}^{44} + 24 T_{3}^{43} + 32 T_{3}^{42} + 1116 T_{3}^{41} + \cdots + 194481 \) acting on \(S_{2}^{\mathrm{new}}(896, [\chi])\). Copy content Toggle raw display