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Properties

Label 675.2.l.h
Level $675$
Weight $2$
Character orbit 675.l
Analytic conductor $5.390$
Analytic rank $0$
Dimension $96$
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] = [675,2,Mod(76,675)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(675, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([14, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("675.76");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 675 = 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 675.l (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: \(5.38990213644\)
Analytic rank: \(0\)
Dimension: \(96\)
Relative dimension: \(16\) over \(\Q(\zeta_{9})\)
Twist minimal: no (minimal twist has level 135)
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.

\(\operatorname{Tr}(f)(q) = \) \( 96 q + 12 q^{4} - 6 q^{6} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 96 q + 12 q^{4} - 6 q^{6} + 18 q^{9} - 6 q^{11} + 18 q^{14} - 24 q^{16} + 6 q^{19} + 24 q^{21} + 30 q^{24} + 48 q^{26} + 30 q^{29} - 30 q^{31} + 24 q^{34} + 54 q^{36} + 6 q^{39} - 12 q^{41} - 78 q^{44} - 6 q^{46} + 30 q^{49} - 90 q^{51} - 108 q^{54} - 96 q^{56} - 66 q^{59} + 6 q^{61} - 150 q^{66} - 24 q^{69} - 90 q^{71} - 66 q^{74} + 12 q^{76} - 24 q^{79} - 54 q^{81} - 198 q^{84} + 18 q^{86} - 96 q^{89} - 6 q^{91} - 24 q^{94} + 42 q^{96} - 36 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
76.1 −2.06784 + 1.73512i 0.239124 + 1.71546i 0.918006 − 5.20627i 0 −3.47101 − 3.13239i −0.175710 − 0.996499i 4.43585 + 7.68312i −2.88564 + 0.820419i 0
76.2 −1.72110 + 1.44418i −1.72712 + 0.130598i 0.529254 − 3.00155i 0 2.78395 − 2.71904i 0.538208 + 3.05233i 1.17713 + 2.03885i 2.96589 − 0.451117i 0
76.3 −1.70485 + 1.43054i −0.364996 − 1.69316i 0.512771 − 2.90807i 0 3.04438 + 2.36443i −0.641005 − 3.63532i 1.06038 + 1.83663i −2.73356 + 1.23599i 0
76.4 −1.26326 + 1.06000i 1.44261 + 0.958575i 0.124928 − 0.708504i 0 −2.83849 + 0.318243i −0.215456 − 1.22191i −1.05587 − 1.82882i 1.16227 + 2.76571i 0
76.5 −0.990387 + 0.831033i −1.55212 + 0.768718i −0.0570464 + 0.323526i 0 0.898367 − 2.05119i −0.134907 − 0.765094i −1.50522 − 2.60712i 1.81814 − 2.38628i 0
76.6 −0.948617 + 0.795984i 1.62630 − 0.595931i −0.0810128 + 0.459446i 0 −1.06839 + 1.85982i −0.395202 − 2.24130i −1.52719 − 2.64518i 2.28973 − 1.93833i 0
76.7 −0.232949 + 0.195467i 0.722492 − 1.57417i −0.331239 + 1.87855i 0 0.139394 + 0.507924i 0.592777 + 3.36181i −0.594126 − 1.02906i −1.95601 − 2.27465i 0
76.8 −0.158986 + 0.133405i −1.27412 − 1.17329i −0.339817 + 1.92720i 0 0.359090 + 0.0165625i −0.500029 − 2.83580i −0.410612 − 0.711201i 0.246774 + 2.98983i 0
76.9 0.158986 − 0.133405i 1.27412 + 1.17329i −0.339817 + 1.92720i 0 0.359090 + 0.0165625i 0.500029 + 2.83580i 0.410612 + 0.711201i 0.246774 + 2.98983i 0
76.10 0.232949 − 0.195467i −0.722492 + 1.57417i −0.331239 + 1.87855i 0 0.139394 + 0.507924i −0.592777 − 3.36181i 0.594126 + 1.02906i −1.95601 − 2.27465i 0
76.11 0.948617 − 0.795984i −1.62630 + 0.595931i −0.0810128 + 0.459446i 0 −1.06839 + 1.85982i 0.395202 + 2.24130i 1.52719 + 2.64518i 2.28973 − 1.93833i 0
76.12 0.990387 − 0.831033i 1.55212 − 0.768718i −0.0570464 + 0.323526i 0 0.898367 − 2.05119i 0.134907 + 0.765094i 1.50522 + 2.60712i 1.81814 − 2.38628i 0
76.13 1.26326 − 1.06000i −1.44261 − 0.958575i 0.124928 − 0.708504i 0 −2.83849 + 0.318243i 0.215456 + 1.22191i 1.05587 + 1.82882i 1.16227 + 2.76571i 0
76.14 1.70485 − 1.43054i 0.364996 + 1.69316i 0.512771 − 2.90807i 0 3.04438 + 2.36443i 0.641005 + 3.63532i −1.06038 − 1.83663i −2.73356 + 1.23599i 0
76.15 1.72110 − 1.44418i 1.72712 − 0.130598i 0.529254 − 3.00155i 0 2.78395 − 2.71904i −0.538208 − 3.05233i −1.17713 − 2.03885i 2.96589 − 0.451117i 0
76.16 2.06784 − 1.73512i −0.239124 − 1.71546i 0.918006 − 5.20627i 0 −3.47101 − 3.13239i 0.175710 + 0.996499i −4.43585 − 7.68312i −2.88564 + 0.820419i 0
151.1 −2.06784 − 1.73512i 0.239124 − 1.71546i 0.918006 + 5.20627i 0 −3.47101 + 3.13239i −0.175710 + 0.996499i 4.43585 − 7.68312i −2.88564 − 0.820419i 0
151.2 −1.72110 − 1.44418i −1.72712 − 0.130598i 0.529254 + 3.00155i 0 2.78395 + 2.71904i 0.538208 − 3.05233i 1.17713 − 2.03885i 2.96589 + 0.451117i 0
151.3 −1.70485 − 1.43054i −0.364996 + 1.69316i 0.512771 + 2.90807i 0 3.04438 − 2.36443i −0.641005 + 3.63532i 1.06038 − 1.83663i −2.73356 − 1.23599i 0
151.4 −1.26326 − 1.06000i 1.44261 − 0.958575i 0.124928 + 0.708504i 0 −2.83849 − 0.318243i −0.215456 + 1.22191i −1.05587 + 1.82882i 1.16227 − 2.76571i 0
See all 96 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 76.16
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
27.e even 9 1 inner
135.p even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 675.2.l.h 96
5.b even 2 1 inner 675.2.l.h 96
5.c odd 4 2 135.2.p.a ✓ 96
15.e even 4 2 405.2.p.a 96
27.e even 9 1 inner 675.2.l.h 96
135.p even 18 1 inner 675.2.l.h 96
135.q even 36 2 405.2.p.a 96
135.r odd 36 2 135.2.p.a ✓ 96
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
135.2.p.a ✓ 96 5.c odd 4 2
135.2.p.a ✓ 96 135.r odd 36 2
405.2.p.a 96 15.e even 4 2
405.2.p.a 96 135.q even 36 2
675.2.l.h 96 1.a even 1 1 trivial
675.2.l.h 96 5.b even 2 1 inner
675.2.l.h 96 27.e even 9 1 inner
675.2.l.h 96 135.p even 18 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{96} - 6 T_{2}^{94} + 33 T_{2}^{92} + 496 T_{2}^{90} - 3051 T_{2}^{88} + 19098 T_{2}^{86} + \cdots + 130321 \) acting on \(S_{2}^{\mathrm{new}}(675, [\chi])\). Copy content Toggle raw display