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Properties

Label 675.2.u.e
Level $675$
Weight $2$
Character orbit 675.u
Analytic conductor $5.390$
Analytic rank $0$
Dimension $132$
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(49,675)]
 
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, 9]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("675.49");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 675 = 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 675.u (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: \(5.38990213644\)
Analytic rank: \(0\)
Dimension: \(132\)
Relative dimension: \(22\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 132 q - 12 q^{6} + 12 q^{9} + 30 q^{11} - 30 q^{14} + 36 q^{16} - 24 q^{19} + 24 q^{21} + 78 q^{24} + 12 q^{26} + 30 q^{29} + 6 q^{31} + 60 q^{36} + 30 q^{39} + 78 q^{41} - 102 q^{44} + 18 q^{46} + 12 q^{49}+ \cdots + 96 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1 −1.70577 − 2.03285i 0.184133 + 1.72224i −0.875557 + 4.96553i 0 3.18696 − 3.31205i 4.87417 − 0.859447i 6.99134 − 4.03645i −2.93219 + 0.634241i 0
49.2 −1.55198 − 1.84958i 0.794060 − 1.53931i −0.665000 + 3.77140i 0 −4.07944 + 0.920299i −0.330906 + 0.0583477i 3.82562 − 2.20872i −1.73894 − 2.44461i 0
49.3 −1.42910 − 1.70314i 1.70215 − 0.320463i −0.511048 + 2.89830i 0 −2.97833 − 2.44102i −3.70709 + 0.653660i 1.81569 − 1.04829i 2.79461 − 1.09095i 0
49.4 −1.30162 − 1.55121i −0.933937 + 1.45868i −0.364742 + 2.06856i 0 3.47836 − 0.449921i −0.370230 + 0.0652816i 0.176186 − 0.101721i −1.25552 − 2.72464i 0
49.5 −1.03778 − 1.23677i −1.23848 − 1.21085i −0.105334 + 0.597379i 0 −0.212279 + 2.78832i −1.71440 + 0.302296i −1.94825 + 1.12482i 0.0676836 + 2.99924i 0
49.6 −0.860644 − 1.02568i −1.52697 + 0.817526i 0.0359939 − 0.204131i 0 2.15270 + 0.862581i 0.340915 − 0.0601126i −2.55944 + 1.47769i 1.66330 − 2.49668i 0
49.7 −0.852522 − 1.01600i 1.72092 − 0.196018i 0.0418420 − 0.237298i 0 −1.66628 − 1.58134i 4.36644 − 0.769922i −2.57396 + 1.48608i 2.92315 − 0.674663i 0
49.8 −0.564301 − 0.672508i 0.939248 + 1.45527i 0.213465 − 1.21062i 0 0.448662 − 1.45286i −0.501180 + 0.0883715i −2.45517 + 1.41750i −1.23563 + 2.73372i 0
49.9 −0.499186 − 0.594907i 1.47949 + 0.900618i 0.242569 − 1.37568i 0 −0.202756 − 1.32973i −4.84944 + 0.855088i −2.28459 + 1.31901i 1.37777 + 2.66491i 0
49.10 −0.177808 − 0.211903i 1.66326 − 0.483303i 0.334009 − 1.89426i 0 −0.398153 − 0.266514i 1.66379 − 0.293371i −0.939908 + 0.542656i 2.53284 − 1.60771i 0
49.11 −0.163444 − 0.194784i −0.154950 − 1.72511i 0.336069 − 1.90594i 0 −0.310698 + 0.312139i −2.55151 + 0.449901i −0.866590 + 0.500326i −2.95198 + 0.534610i 0
49.12 0.163444 + 0.194784i 0.154950 + 1.72511i 0.336069 − 1.90594i 0 −0.310698 + 0.312139i 2.55151 − 0.449901i 0.866590 − 0.500326i −2.95198 + 0.534610i 0
49.13 0.177808 + 0.211903i −1.66326 + 0.483303i 0.334009 − 1.89426i 0 −0.398153 − 0.266514i −1.66379 + 0.293371i 0.939908 − 0.542656i 2.53284 − 1.60771i 0
49.14 0.499186 + 0.594907i −1.47949 − 0.900618i 0.242569 − 1.37568i 0 −0.202756 − 1.32973i 4.84944 − 0.855088i 2.28459 − 1.31901i 1.37777 + 2.66491i 0
49.15 0.564301 + 0.672508i −0.939248 − 1.45527i 0.213465 − 1.21062i 0 0.448662 − 1.45286i 0.501180 − 0.0883715i 2.45517 − 1.41750i −1.23563 + 2.73372i 0
49.16 0.852522 + 1.01600i −1.72092 + 0.196018i 0.0418420 − 0.237298i 0 −1.66628 − 1.58134i −4.36644 + 0.769922i 2.57396 − 1.48608i 2.92315 − 0.674663i 0
49.17 0.860644 + 1.02568i 1.52697 − 0.817526i 0.0359939 − 0.204131i 0 2.15270 + 0.862581i −0.340915 + 0.0601126i 2.55944 − 1.47769i 1.66330 − 2.49668i 0
49.18 1.03778 + 1.23677i 1.23848 + 1.21085i −0.105334 + 0.597379i 0 −0.212279 + 2.78832i 1.71440 − 0.302296i 1.94825 − 1.12482i 0.0676836 + 2.99924i 0
49.19 1.30162 + 1.55121i 0.933937 − 1.45868i −0.364742 + 2.06856i 0 3.47836 − 0.449921i 0.370230 − 0.0652816i −0.176186 + 0.101721i −1.25552 − 2.72464i 0
49.20 1.42910 + 1.70314i −1.70215 + 0.320463i −0.511048 + 2.89830i 0 −2.97833 − 2.44102i 3.70709 − 0.653660i −1.81569 + 1.04829i 2.79461 − 1.09095i 0
See next 80 embeddings (of 132 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.22
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.u.e 132
5.b even 2 1 inner 675.2.u.e 132
5.c odd 4 1 675.2.l.f ✓ 66
5.c odd 4 1 675.2.l.g yes 66
27.e even 9 1 inner 675.2.u.e 132
135.p even 18 1 inner 675.2.u.e 132
135.r odd 36 1 675.2.l.f ✓ 66
135.r odd 36 1 675.2.l.g yes 66
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
675.2.l.f ✓ 66 5.c odd 4 1
675.2.l.f ✓ 66 135.r odd 36 1
675.2.l.g yes 66 5.c odd 4 1
675.2.l.g yes 66 135.r odd 36 1
675.2.u.e 132 1.a even 1 1 trivial
675.2.u.e 132 5.b even 2 1 inner
675.2.u.e 132 27.e even 9 1 inner
675.2.u.e 132 135.p even 18 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{132} - 36 T_{2}^{128} - 954 T_{2}^{126} + 1530 T_{2}^{124} + 28719 T_{2}^{122} + 547425 T_{2}^{120} + \cdots + 547981281 \) acting on \(S_{2}^{\mathrm{new}}(675, [\chi])\). Copy content Toggle raw display