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Properties

Label 675.2.l.f
Level $675$
Weight $2$
Character orbit 675.l
Analytic conductor $5.390$
Analytic rank $0$
Dimension $66$
CM no
Inner twists $2$

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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: \(66\)
Relative dimension: \(11\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 66 q - 6 q^{2} - 6 q^{6} - 6 q^{7} - 12 q^{8} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 66 q - 6 q^{2} - 6 q^{6} - 6 q^{7} - 12 q^{8} - 6 q^{9} + 15 q^{11} - 18 q^{12} + 15 q^{14} + 18 q^{16} - 30 q^{17} + 12 q^{18} + 12 q^{19} + 12 q^{21} + 45 q^{22} - 36 q^{23} - 39 q^{24} + 6 q^{26} - 51 q^{27} + 36 q^{28} - 15 q^{29} + 3 q^{31} - 27 q^{32} + 3 q^{33} + 30 q^{36} - 6 q^{37} + 12 q^{38} - 15 q^{39} + 39 q^{41} - 48 q^{42} - 12 q^{43} + 51 q^{44} + 9 q^{46} - 30 q^{47} + 132 q^{48} - 6 q^{49} - 9 q^{52} + 24 q^{53} + 75 q^{54} + 144 q^{56} - 33 q^{57} - 27 q^{58} + 45 q^{59} - 54 q^{61} - 66 q^{62} + 120 q^{63} - 24 q^{64} + 48 q^{66} - 9 q^{67} + 69 q^{68} + 51 q^{69} - 15 q^{71} - 9 q^{72} + 15 q^{73} + 96 q^{74} - 48 q^{76} + 36 q^{77} + 18 q^{78} + 48 q^{79} - 54 q^{81} + 36 q^{82} - 30 q^{83} + 57 q^{84} - 111 q^{86} + 33 q^{87} - 36 q^{88} - 12 q^{89} + 9 q^{91} + 219 q^{92} - 63 q^{93} + 36 q^{94} - 249 q^{96} - 57 q^{97} - 75 q^{98} - 48 q^{99}+O(q^{100}) \) 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 −1.84958 + 1.55198i 1.53931 + 0.794060i 0.665000 − 3.77140i 0 −4.07944 + 0.920299i 0.0583477 + 0.330906i 2.20872 + 3.82562i 1.73894 + 2.44461i 0
76.2 −1.55121 + 1.30162i −1.45868 − 0.933937i 0.364742 − 2.06856i 0 3.47836 − 0.449921i 0.0652816 + 0.370230i 0.101721 + 0.176186i 1.25552 + 2.72464i 0
76.3 −1.23677 + 1.03778i 1.21085 − 1.23848i 0.105334 − 0.597379i 0 −0.212279 + 2.78832i 0.302296 + 1.71440i −1.12482 − 1.94825i −0.0676836 − 2.99924i 0
76.4 −1.01600 + 0.852522i 0.196018 + 1.72092i −0.0418420 + 0.237298i 0 −1.66628 − 1.58134i −0.769922 − 4.36644i −1.48608 − 2.57396i −2.92315 + 0.674663i 0
76.5 −0.594907 + 0.499186i −0.900618 + 1.47949i −0.242569 + 1.37568i 0 −0.202756 − 1.32973i 0.855088 + 4.84944i −1.31901 − 2.28459i −1.37777 − 2.66491i 0
76.6 0.194784 − 0.163444i −1.72511 + 0.154950i −0.336069 + 1.90594i 0 −0.310698 + 0.312139i −0.449901 − 2.55151i 0.500326 + 0.866590i 2.95198 − 0.534610i 0
76.7 0.211903 − 0.177808i −0.483303 − 1.66326i −0.334009 + 1.89426i 0 −0.398153 − 0.266514i 0.293371 + 1.66379i 0.542656 + 0.939908i −2.53284 + 1.60771i 0
76.8 0.672508 − 0.564301i 1.45527 − 0.939248i −0.213465 + 1.21062i 0 0.448662 − 1.45286i −0.0883715 − 0.501180i 1.41750 + 2.45517i 1.23563 − 2.73372i 0
76.9 1.02568 − 0.860644i 0.817526 + 1.52697i −0.0359939 + 0.204131i 0 2.15270 + 0.862581i 0.0601126 + 0.340915i 1.47769 + 2.55944i −1.66330 + 2.49668i 0
76.10 1.70314 − 1.42910i −0.320463 − 1.70215i 0.511048 − 2.89830i 0 −2.97833 − 2.44102i −0.653660 − 3.70709i −1.04829 − 1.81569i −2.79461 + 1.09095i 0
76.11 2.03285 − 1.70577i 1.72224 − 0.184133i 0.875557 − 4.96553i 0 3.18696 − 3.31205i 0.859447 + 4.87417i −4.03645 − 6.99134i 2.93219 − 0.634241i 0
151.1 −1.84958 − 1.55198i 1.53931 − 0.794060i 0.665000 + 3.77140i 0 −4.07944 − 0.920299i 0.0583477 − 0.330906i 2.20872 − 3.82562i 1.73894 − 2.44461i 0
151.2 −1.55121 − 1.30162i −1.45868 + 0.933937i 0.364742 + 2.06856i 0 3.47836 + 0.449921i 0.0652816 − 0.370230i 0.101721 − 0.176186i 1.25552 − 2.72464i 0
151.3 −1.23677 − 1.03778i 1.21085 + 1.23848i 0.105334 + 0.597379i 0 −0.212279 − 2.78832i 0.302296 − 1.71440i −1.12482 + 1.94825i −0.0676836 + 2.99924i 0
151.4 −1.01600 − 0.852522i 0.196018 − 1.72092i −0.0418420 − 0.237298i 0 −1.66628 + 1.58134i −0.769922 + 4.36644i −1.48608 + 2.57396i −2.92315 − 0.674663i 0
151.5 −0.594907 − 0.499186i −0.900618 − 1.47949i −0.242569 − 1.37568i 0 −0.202756 + 1.32973i 0.855088 − 4.84944i −1.31901 + 2.28459i −1.37777 + 2.66491i 0
151.6 0.194784 + 0.163444i −1.72511 − 0.154950i −0.336069 − 1.90594i 0 −0.310698 − 0.312139i −0.449901 + 2.55151i 0.500326 − 0.866590i 2.95198 + 0.534610i 0
151.7 0.211903 + 0.177808i −0.483303 + 1.66326i −0.334009 − 1.89426i 0 −0.398153 + 0.266514i 0.293371 − 1.66379i 0.542656 − 0.939908i −2.53284 − 1.60771i 0
151.8 0.672508 + 0.564301i 1.45527 + 0.939248i −0.213465 − 1.21062i 0 0.448662 + 1.45286i −0.0883715 + 0.501180i 1.41750 − 2.45517i 1.23563 + 2.73372i 0
151.9 1.02568 + 0.860644i 0.817526 − 1.52697i −0.0359939 − 0.204131i 0 2.15270 − 0.862581i 0.0601126 − 0.340915i 1.47769 − 2.55944i −1.66330 − 2.49668i 0
See all 66 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 76.11
Significant digits:
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Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 675.2.l.f ✓ 66
5.b even 2 1 675.2.l.g yes 66
5.c odd 4 2 675.2.u.e 132
27.e even 9 1 inner 675.2.l.f ✓ 66
135.p even 18 1 675.2.l.g yes 66
135.r odd 36 2 675.2.u.e 132
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
675.2.l.f ✓ 66 1.a even 1 1 trivial
675.2.l.f ✓ 66 27.e even 9 1 inner
675.2.l.g yes 66 5.b even 2 1
675.2.l.g yes 66 135.p even 18 1
675.2.u.e 132 5.c odd 4 2
675.2.u.e 132 135.r odd 36 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{66} + 6 T_{2}^{65} + 18 T_{2}^{64} + 44 T_{2}^{63} + 84 T_{2}^{62} + 99 T_{2}^{61} + 527 T_{2}^{60} + \cdots + 23409 \) acting on \(S_{2}^{\mathrm{new}}(675, [\chi])\). Copy content Toggle raw display