↑

Properties

Label 608.2.bh.b
Level $608$
Weight $2$
Character orbit 608.bh
Analytic conductor $4.855$
Analytic rank $0$
Dimension $96$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [608,2,Mod(15,608)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(608, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([9, 9, 11]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("608.15");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 608 = 2^{5} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 608.bh (of order \(18\), degree \(6\), 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: \(4.85490444289\)
Analytic rank: \(0\)
Dimension: \(96\)
Relative dimension: \(16\) over \(\Q(\zeta_{18})\)
Twist minimal: no (minimal twist has level 152)
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 96 q + 6 q^{3} - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 96 q + 6 q^{3} - 18 q^{9} + 6 q^{11} - 12 q^{17} + 12 q^{19} - 12 q^{25} - 72 q^{27} - 24 q^{33} + 42 q^{35} - 42 q^{41} + 12 q^{43} + 54 q^{49} + 78 q^{51} - 12 q^{57} - 6 q^{59} - 18 q^{65} + 54 q^{67} - 24 q^{73} - 96 q^{81} + 6 q^{83} - 12 q^{89} + 54 q^{91} + 18 q^{97} - 72 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
15.1 0 −2.53648 + 0.447249i 0 −2.37173 − 2.82652i 0 −3.89125 − 2.24662i 0 3.41460 − 1.24281i 0
15.2 0 −2.53648 + 0.447249i 0 2.37173 + 2.82652i 0 3.89125 + 2.24662i 0 3.41460 − 1.24281i 0
15.3 0 −1.96968 + 0.347307i 0 −0.894548 − 1.06608i 0 1.03814 + 0.599372i 0 0.939926 − 0.342105i 0
15.4 0 −1.96968 + 0.347307i 0 0.894548 + 1.06608i 0 −1.03814 − 0.599372i 0 0.939926 − 0.342105i 0
15.5 0 −0.751131 + 0.132445i 0 −1.20402 − 1.43489i 0 1.81443 + 1.04756i 0 −2.27242 + 0.827094i 0
15.6 0 −0.751131 + 0.132445i 0 1.20402 + 1.43489i 0 −1.81443 − 1.04756i 0 −2.27242 + 0.827094i 0
15.7 0 0.180694 − 0.0318612i 0 −2.49583 − 2.97441i 0 −0.331565 − 0.191429i 0 −2.78744 + 1.01455i 0
15.8 0 0.180694 − 0.0318612i 0 2.49583 + 2.97441i 0 0.331565 + 0.191429i 0 −2.78744 + 1.01455i 0
15.9 0 0.352643 − 0.0621805i 0 −1.07506 − 1.28121i 0 2.83308 + 1.63568i 0 −2.69859 + 0.982205i 0
15.10 0 0.352643 − 0.0621805i 0 1.07506 + 1.28121i 0 −2.83308 − 1.63568i 0 −2.69859 + 0.982205i 0
15.11 0 1.27093 − 0.224098i 0 −0.462300 − 0.550948i 0 −4.05999 − 2.34404i 0 −1.25405 + 0.456436i 0
15.12 0 1.27093 − 0.224098i 0 0.462300 + 0.550948i 0 4.05999 + 2.34404i 0 −1.25405 + 0.456436i 0
15.13 0 2.18275 − 0.384878i 0 −1.74321 − 2.07748i 0 −2.89817 − 1.67326i 0 1.79719 − 0.654124i 0
15.14 0 2.18275 − 0.384878i 0 1.74321 + 2.07748i 0 2.89817 + 1.67326i 0 1.79719 − 0.654124i 0
15.15 0 2.53632 − 0.447221i 0 −1.51324 − 1.80341i 0 1.36279 + 0.786807i 0 3.41381 − 1.24253i 0
15.16 0 2.53632 − 0.447221i 0 1.51324 + 1.80341i 0 −1.36279 − 0.786807i 0 3.41381 − 1.24253i 0
79.1 0 −0.750355 − 2.06158i 0 −1.96163 − 0.345888i 0 3.50473 − 2.02345i 0 −1.38896 + 1.16547i 0
79.2 0 −0.750355 − 2.06158i 0 1.96163 + 0.345888i 0 −3.50473 + 2.02345i 0 −1.38896 + 1.16547i 0
79.3 0 −0.735462 − 2.02067i 0 −2.19693 − 0.387378i 0 −3.18527 + 1.83902i 0 −1.24405 + 1.04389i 0
79.4 0 −0.735462 − 2.02067i 0 2.19693 + 0.387378i 0 3.18527 − 1.83902i 0 −1.24405 + 1.04389i 0
See all 96 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 15.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner
19.f odd 18 1 inner
152.v even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 608.2.bh.b 96
4.b odd 2 1 152.2.v.b ✓ 96
8.b even 2 1 152.2.v.b ✓ 96
8.d odd 2 1 inner 608.2.bh.b 96
19.f odd 18 1 inner 608.2.bh.b 96
76.k even 18 1 152.2.v.b ✓ 96
152.s odd 18 1 152.2.v.b ✓ 96
152.v even 18 1 inner 608.2.bh.b 96
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.2.v.b ✓ 96 4.b odd 2 1
152.2.v.b ✓ 96 8.b even 2 1
152.2.v.b ✓ 96 76.k even 18 1
152.2.v.b ✓ 96 152.s odd 18 1
608.2.bh.b 96 1.a even 1 1 trivial
608.2.bh.b 96 8.d odd 2 1 inner
608.2.bh.b 96 19.f odd 18 1 inner
608.2.bh.b 96 152.v even 18 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{48} - 3 T_{3}^{47} + 9 T_{3}^{46} - 6 T_{3}^{45} + 30 T_{3}^{44} - 138 T_{3}^{43} - 101 T_{3}^{42} + \cdots + 3249 \) acting on \(S_{2}^{\mathrm{new}}(608, [\chi])\). Copy content Toggle raw display