# Properties

 Label 675.2.u.e Level $675$ Weight $2$ Character orbit 675.u Analytic conductor $5.390$ Analytic rank $0$ Dimension $132$ CM no Inner twists $4$

# Related objects

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

 $$\operatorname{Tr}(f)(q) =$$ $$132 q - 12 q^{6} + 12 q^{9}+O(q^{10})$$ 132 * q - 12 * q^6 + 12 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$132 q - 12 q^{6} + 12 q^{9} + 30 q^{11} - 30 q^{14} + 36 q^{16} - 24 q^{19} + 24 q^{21} + 78 q^{24} + 12 q^{26} + 30 q^{29} + 6 q^{31} + 60 q^{36} + 30 q^{39} + 78 q^{41} - 102 q^{44} + 18 q^{46} + 12 q^{49} - 150 q^{54} + 288 q^{56} - 90 q^{59} - 108 q^{61} + 48 q^{64} + 96 q^{66} - 102 q^{69} - 30 q^{71} - 192 q^{74} - 96 q^{76} - 96 q^{79} - 108 q^{81} - 114 q^{84} - 222 q^{86} + 24 q^{89} + 18 q^{91} - 72 q^{94} - 498 q^{96} + 96 q^{99}+O(q^{100})$$ 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 - 150 * q^54 + 288 * q^56 - 90 * q^59 - 108 * q^61 + 48 * q^64 + 96 * q^66 - 102 * q^69 - 30 * q^71 - 192 * q^74 - 96 * q^76 - 96 * q^79 - 108 * q^81 - 114 * q^84 - 222 * q^86 + 24 * q^89 + 18 * q^91 - 72 * q^94 - 498 * q^96 + 96 * q^99

## 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: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 $$T_{2}^{132} - 36 T_{2}^{128} - 954 T_{2}^{126} + 1530 T_{2}^{124} + 28719 T_{2}^{122} + 547425 T_{2}^{120} - 1023489 T_{2}^{118} - 15099327 T_{2}^{116} - 187694634 T_{2}^{114} + 458980893 T_{2}^{112} + \cdots + 547981281$$ acting on $$S_{2}^{\mathrm{new}}(675, [\chi])$$.