↑

Properties

Label 608.2.y.b
Level $608$
Weight $2$
Character orbit 608.y
Analytic conductor $4.855$
Analytic rank $0$
Dimension $30$
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] = [608,2,Mod(161,608)]
 
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([0, 0, 8]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("608.161");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 608 = 2^{5} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 608.y (of order \(9\), degree \(6\), 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: \(30\)
Relative dimension: \(5\) over \(\Q(\zeta_{9})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 30 q - 3 q^{3} + 12 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 30 q - 3 q^{3} + 12 q^{7} + 3 q^{9} + 6 q^{11} - 6 q^{13} - 24 q^{15} - 6 q^{17} - 12 q^{19} - 6 q^{21} - 6 q^{23} - 12 q^{25} + 39 q^{27} + 12 q^{29} + 39 q^{33} - 12 q^{35} + 36 q^{37} - 24 q^{39} + 9 q^{41} + 18 q^{43} + 18 q^{45} + 12 q^{47} - 9 q^{49} + 51 q^{51} + 6 q^{53} + 24 q^{55} + 12 q^{57} - 9 q^{59} + 18 q^{61} - 132 q^{63} + 18 q^{65} - 51 q^{67} - 24 q^{69} + 36 q^{71} - 30 q^{73} + 48 q^{75} + 36 q^{77} + 60 q^{79} - 3 q^{81} + 30 q^{83} - 12 q^{85} - 24 q^{89} - 60 q^{91} + 30 q^{93} - 36 q^{95} - 9 q^{97} + 27 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
161.1 0 −2.43577 + 2.04385i 0 2.82460 + 1.02807i 0 1.78679 − 3.09482i 0 1.23469 − 7.00228i 0
161.2 0 −0.990242 + 0.830911i 0 −3.16371 − 1.15150i 0 1.52889 − 2.64812i 0 −0.230780 + 1.30882i 0
161.3 0 0.480615 − 0.403284i 0 2.54736 + 0.927162i 0 −0.710121 + 1.22997i 0 −0.452592 + 2.56677i 0
161.4 0 1.08700 − 0.912100i 0 −1.73765 − 0.632454i 0 −0.766882 + 1.32828i 0 −0.171305 + 0.971520i 0
161.5 0 2.12444 − 1.78262i 0 −0.470583 − 0.171278i 0 2.04070 − 3.53460i 0 0.814579 − 4.61970i 0
225.1 0 −2.43759 + 0.887211i 0 −0.259419 + 1.47124i 0 −1.90668 − 3.30247i 0 2.85657 − 2.39695i 0
225.2 0 −1.72205 + 0.626776i 0 0.763801 − 4.33173i 0 1.17816 + 2.04063i 0 0.274488 − 0.230323i 0
225.3 0 −0.797869 + 0.290401i 0 −0.644333 + 3.65419i 0 2.35875 + 4.08547i 0 −1.74587 + 1.46496i 0
225.4 0 0.858265 − 0.312383i 0 −0.0129515 + 0.0734518i 0 0.379354 + 0.657061i 0 −1.65910 + 1.39215i 0
225.5 0 2.65956 − 0.967999i 0 0.152903 − 0.867153i 0 −0.356875 − 0.618126i 0 3.83808 − 3.22053i 0
289.1 0 −0.577667 − 3.27611i 0 0.144355 − 0.121128i 0 1.31827 − 2.28331i 0 −7.58013 + 2.75894i 0
289.2 0 −0.227930 − 1.29266i 0 −0.860715 + 0.722225i 0 −1.57981 + 2.73632i 0 1.20007 − 0.436789i 0
289.3 0 −0.0859948 − 0.487701i 0 2.79500 − 2.34529i 0 0.305432 − 0.529024i 0 2.58862 − 0.942181i 0
289.4 0 0.237448 + 1.34664i 0 0.0695384 − 0.0583496i 0 1.31604 − 2.27944i 0 1.06203 − 0.386548i 0
289.5 0 0.327792 + 1.85900i 0 −2.14818 + 1.80254i 0 −0.892013 + 1.54501i 0 −0.529359 + 0.192671i 0
321.1 0 −2.43577 − 2.04385i 0 2.82460 − 1.02807i 0 1.78679 + 3.09482i 0 1.23469 + 7.00228i 0
321.2 0 −0.990242 − 0.830911i 0 −3.16371 + 1.15150i 0 1.52889 + 2.64812i 0 −0.230780 − 1.30882i 0
321.3 0 0.480615 + 0.403284i 0 2.54736 − 0.927162i 0 −0.710121 − 1.22997i 0 −0.452592 − 2.56677i 0
321.4 0 1.08700 + 0.912100i 0 −1.73765 + 0.632454i 0 −0.766882 − 1.32828i 0 −0.171305 − 0.971520i 0
321.5 0 2.12444 + 1.78262i 0 −0.470583 + 0.171278i 0 2.04070 + 3.53460i 0 0.814579 + 4.61970i 0
See all 30 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 161.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.e even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 608.2.y.b ✓ 30
4.b odd 2 1 608.2.y.c yes 30
19.e even 9 1 inner 608.2.y.b ✓ 30
76.l odd 18 1 608.2.y.c yes 30
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
608.2.y.b ✓ 30 1.a even 1 1 trivial
608.2.y.b ✓ 30 19.e even 9 1 inner
608.2.y.c yes 30 4.b odd 2 1
608.2.y.c yes 30 76.l odd 18 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{30} + 3 T_{3}^{29} + 3 T_{3}^{28} - 13 T_{3}^{27} - 27 T_{3}^{26} + 63 T_{3}^{25} + 938 T_{3}^{24} + \cdots + 349281 \) acting on \(S_{2}^{\mathrm{new}}(608, [\chi])\). Copy content Toggle raw display