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Properties

Label 608.2.y.d
Level $608$
Weight $2$
Character orbit 608.y
Analytic conductor $4.855$
Analytic rank $0$
Dimension $36$
Inner twists $4$

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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: \(36\)
Relative dimension: \(6\) over \(\Q(\zeta_{9})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 36 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q - 18 q^{13} + 6 q^{17} - 6 q^{21} + 60 q^{25} - 6 q^{29} - 54 q^{33} - 72 q^{37} - 36 q^{41} - 48 q^{45} - 36 q^{49} + 66 q^{53} + 24 q^{57} + 36 q^{61} - 90 q^{65} + 42 q^{69} + 24 q^{73} + 72 q^{77} - 108 q^{81} + 24 q^{85} - 60 q^{89} - 96 q^{93} + 42 q^{97}+O(q^{100}) \) 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 −2.55256 + 2.14185i 0 −3.31341 − 1.20598i 0 0.597615 − 1.03510i 0 1.40708 − 7.97997i 0
161.2 0 −1.45449 + 1.22046i 0 3.37154 + 1.22714i 0 −1.74603 + 3.02421i 0 0.105071 − 0.595885i 0
161.3 0 −0.413837 + 0.347251i 0 −0.997819 − 0.363176i 0 −0.759121 + 1.31484i 0 −0.470266 + 2.66701i 0
161.4 0 0.413837 − 0.347251i 0 −0.997819 − 0.363176i 0 0.759121 − 1.31484i 0 −0.470266 + 2.66701i 0
161.5 0 1.45449 − 1.22046i 0 3.37154 + 1.22714i 0 1.74603 − 3.02421i 0 0.105071 − 0.595885i 0
161.6 0 2.55256 − 2.14185i 0 −3.31341 − 1.20598i 0 −0.597615 + 1.03510i 0 1.40708 − 7.97997i 0
225.1 0 −3.08612 + 1.12325i 0 −0.134353 + 0.761956i 0 2.31054 + 4.00198i 0 5.96428 − 5.00463i 0
225.2 0 −1.80058 + 0.655358i 0 0.506610 − 2.87313i 0 −1.71295 − 2.96691i 0 0.514464 − 0.431687i 0
225.3 0 −0.692190 + 0.251937i 0 −0.198608 + 1.12636i 0 −1.17605 − 2.03699i 0 −1.88248 + 1.57959i 0
225.4 0 0.692190 − 0.251937i 0 −0.198608 + 1.12636i 0 1.17605 + 2.03699i 0 −1.88248 + 1.57959i 0
225.5 0 1.80058 − 0.655358i 0 0.506610 − 2.87313i 0 1.71295 + 2.96691i 0 0.514464 − 0.431687i 0
225.6 0 3.08612 − 1.12325i 0 −0.134353 + 0.761956i 0 −2.31054 − 4.00198i 0 5.96428 − 5.00463i 0
289.1 0 −0.482395 − 2.73580i 0 2.69160 − 2.25852i 0 −1.46473 + 2.53699i 0 −4.43281 + 1.61341i 0
289.2 0 −0.443050 − 2.51266i 0 −2.72993 + 2.29068i 0 −0.0450625 + 0.0780505i 0 −3.29810 + 1.20041i 0
289.3 0 −0.152666 − 0.865812i 0 0.804374 − 0.674950i 0 2.11296 − 3.65976i 0 2.09275 − 0.761700i 0
289.4 0 0.152666 + 0.865812i 0 0.804374 − 0.674950i 0 −2.11296 + 3.65976i 0 2.09275 − 0.761700i 0
289.5 0 0.443050 + 2.51266i 0 −2.72993 + 2.29068i 0 0.0450625 − 0.0780505i 0 −3.29810 + 1.20041i 0
289.6 0 0.482395 + 2.73580i 0 2.69160 − 2.25852i 0 1.46473 − 2.53699i 0 −4.43281 + 1.61341i 0
321.1 0 −2.55256 − 2.14185i 0 −3.31341 + 1.20598i 0 0.597615 + 1.03510i 0 1.40708 + 7.97997i 0
321.2 0 −1.45449 − 1.22046i 0 3.37154 − 1.22714i 0 −1.74603 − 3.02421i 0 0.105071 + 0.595885i 0
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 161.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
19.e even 9 1 inner
76.l odd 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 608.2.y.d ✓ 36
4.b odd 2 1 inner 608.2.y.d ✓ 36
19.e even 9 1 inner 608.2.y.d ✓ 36
76.l odd 18 1 inner 608.2.y.d ✓ 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
608.2.y.d ✓ 36 1.a even 1 1 trivial
608.2.y.d ✓ 36 4.b odd 2 1 inner
608.2.y.d ✓ 36 19.e even 9 1 inner
608.2.y.d ✓ 36 76.l odd 18 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{36} + 27 T_{3}^{32} + 1152 T_{3}^{30} + 459 T_{3}^{28} + 31617 T_{3}^{26} + 1292706 T_{3}^{24} + \cdots + 95004009 \) acting on \(S_{2}^{\mathrm{new}}(608, [\chi])\). Copy content Toggle raw display