# Properties

 Label 675.2.u.d Level $675$ Weight $2$ Character orbit 675.u Analytic conductor $5.390$ Analytic rank $0$ Dimension $84$ 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: $$84$$ Relative dimension: $$14$$ over $$\Q(\zeta_{18})$$ Twist minimal: no (minimal twist has level 135) 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.

 $$\operatorname{Tr}(f)(q) =$$ $$84 q - 12 q^{6} + 6 q^{9}+O(q^{10})$$ 84 * q - 12 * q^6 + 6 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$84 q - 12 q^{6} + 6 q^{9} - 12 q^{11} + 54 q^{14} - 24 q^{16} + 48 q^{19} + 24 q^{21} + 6 q^{24} - 36 q^{26} - 24 q^{31} - 102 q^{34} - 72 q^{36} + 60 q^{39} + 66 q^{41} - 24 q^{44} - 60 q^{46} - 36 q^{49} - 78 q^{51} - 60 q^{54} - 48 q^{56} + 54 q^{59} - 12 q^{61} + 126 q^{64} - 60 q^{66} - 102 q^{69} + 24 q^{71} + 144 q^{74} + 276 q^{76} + 36 q^{79} + 90 q^{81} + 102 q^{84} + 36 q^{86} - 18 q^{89} - 138 q^{91} + 66 q^{94} + 306 q^{96} + 18 q^{99}+O(q^{100})$$ 84 * q - 12 * q^6 + 6 * q^9 - 12 * q^11 + 54 * q^14 - 24 * q^16 + 48 * q^19 + 24 * q^21 + 6 * q^24 - 36 * q^26 - 24 * q^31 - 102 * q^34 - 72 * q^36 + 60 * q^39 + 66 * q^41 - 24 * q^44 - 60 * q^46 - 36 * q^49 - 78 * q^51 - 60 * q^54 - 48 * q^56 + 54 * q^59 - 12 * q^61 + 126 * q^64 - 60 * q^66 - 102 * q^69 + 24 * q^71 + 144 * q^74 + 276 * q^76 + 36 * q^79 + 90 * q^81 + 102 * q^84 + 36 * q^86 - 18 * q^89 - 138 * q^91 + 66 * q^94 + 306 * q^96 + 18 * 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 −1.75919 2.09652i −1.40194 + 1.01713i −0.953350 + 5.40672i 0 4.59871 + 1.14987i −2.48743 + 0.438601i 8.27211 4.77590i 0.930888 2.85192i 0
49.2 −1.69563 2.02078i 0.112798 1.72837i −0.861072 + 4.88338i 0 −3.68392 + 2.70275i −0.0335646 + 0.00591834i 6.75926 3.90246i −2.97455 0.389915i 0
49.3 −1.19967 1.42971i 1.58534 + 0.697646i −0.257569 + 1.46075i 0 −0.904448 3.10352i 3.42349 0.603654i −0.835177 + 0.482190i 2.02658 + 2.21201i 0
49.4 −0.949646 1.13174i −0.0588713 1.73105i −0.0317206 + 0.179896i 0 −1.90320 + 1.71051i −3.64163 + 0.642118i −2.32519 + 1.34245i −2.99307 + 0.203818i 0
49.5 −0.708387 0.844223i −0.290600 + 1.70750i 0.136396 0.773542i 0 1.64737 0.964239i 2.94069 0.518523i −2.65848 + 1.53487i −2.83110 0.992398i 0
49.6 −0.659865 0.786397i −0.795743 + 1.53844i 0.164299 0.931783i 0 1.73491 0.389393i −4.04068 + 0.712481i −2.61923 + 1.51222i −1.73359 2.44840i 0
49.7 −0.504838 0.601643i −1.44576 0.953823i 0.240184 1.36215i 0 0.156014 + 1.35136i 1.40161 0.247142i −2.30112 + 1.32855i 1.18044 + 2.75800i 0
49.8 0.504838 + 0.601643i 1.44576 + 0.953823i 0.240184 1.36215i 0 0.156014 + 1.35136i −1.40161 + 0.247142i 2.30112 1.32855i 1.18044 + 2.75800i 0
49.9 0.659865 + 0.786397i 0.795743 1.53844i 0.164299 0.931783i 0 1.73491 0.389393i 4.04068 0.712481i 2.61923 1.51222i −1.73359 2.44840i 0
49.10 0.708387 + 0.844223i 0.290600 1.70750i 0.136396 0.773542i 0 1.64737 0.964239i −2.94069 + 0.518523i 2.65848 1.53487i −2.83110 0.992398i 0
49.11 0.949646 + 1.13174i 0.0588713 + 1.73105i −0.0317206 + 0.179896i 0 −1.90320 + 1.71051i 3.64163 0.642118i 2.32519 1.34245i −2.99307 + 0.203818i 0
49.12 1.19967 + 1.42971i −1.58534 0.697646i −0.257569 + 1.46075i 0 −0.904448 3.10352i −3.42349 + 0.603654i 0.835177 0.482190i 2.02658 + 2.21201i 0
49.13 1.69563 + 2.02078i −0.112798 + 1.72837i −0.861072 + 4.88338i 0 −3.68392 + 2.70275i 0.0335646 0.00591834i −6.75926 + 3.90246i −2.97455 0.389915i 0
49.14 1.75919 + 2.09652i 1.40194 1.01713i −0.953350 + 5.40672i 0 4.59871 + 1.14987i 2.48743 0.438601i −8.27211 + 4.77590i 0.930888 2.85192i 0
124.1 −1.75919 + 2.09652i −1.40194 1.01713i −0.953350 5.40672i 0 4.59871 1.14987i −2.48743 0.438601i 8.27211 + 4.77590i 0.930888 + 2.85192i 0
124.2 −1.69563 + 2.02078i 0.112798 + 1.72837i −0.861072 4.88338i 0 −3.68392 2.70275i −0.0335646 0.00591834i 6.75926 + 3.90246i −2.97455 + 0.389915i 0
124.3 −1.19967 + 1.42971i 1.58534 0.697646i −0.257569 1.46075i 0 −0.904448 + 3.10352i 3.42349 + 0.603654i −0.835177 0.482190i 2.02658 2.21201i 0
124.4 −0.949646 + 1.13174i −0.0588713 + 1.73105i −0.0317206 0.179896i 0 −1.90320 1.71051i −3.64163 0.642118i −2.32519 1.34245i −2.99307 0.203818i 0
124.5 −0.708387 + 0.844223i −0.290600 1.70750i 0.136396 + 0.773542i 0 1.64737 + 0.964239i 2.94069 + 0.518523i −2.65848 1.53487i −2.83110 + 0.992398i 0
124.6 −0.659865 + 0.786397i −0.795743 1.53844i 0.164299 + 0.931783i 0 1.73491 + 0.389393i −4.04068 0.712481i −2.61923 1.51222i −1.73359 + 2.44840i 0
See all 84 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 49.14 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.d 84
5.b even 2 1 inner 675.2.u.d 84
5.c odd 4 1 135.2.k.b 42
5.c odd 4 1 675.2.l.e 42
15.e even 4 1 405.2.k.b 42
27.e even 9 1 inner 675.2.u.d 84
135.p even 18 1 inner 675.2.u.d 84
135.q even 36 1 405.2.k.b 42
135.q even 36 1 3645.2.a.k 21
135.r odd 36 1 135.2.k.b 42
135.r odd 36 1 675.2.l.e 42
135.r odd 36 1 3645.2.a.l 21

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
135.2.k.b 42 5.c odd 4 1
135.2.k.b 42 135.r odd 36 1
405.2.k.b 42 15.e even 4 1
405.2.k.b 42 135.q even 36 1
675.2.l.e 42 5.c odd 4 1
675.2.l.e 42 135.r odd 36 1
675.2.u.d 84 1.a even 1 1 trivial
675.2.u.d 84 5.b even 2 1 inner
675.2.u.d 84 27.e even 9 1 inner
675.2.u.d 84 135.p even 18 1 inner
3645.2.a.k 21 135.q even 36 1
3645.2.a.l 21 135.r odd 36 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{84} + 6 T_{2}^{80} - 755 T_{2}^{78} - 261 T_{2}^{76} - 1530 T_{2}^{74} + 397962 T_{2}^{72} + 21933 T_{2}^{70} + 395559 T_{2}^{68} - 105797931 T_{2}^{66} + 57036870 T_{2}^{64} + 15621750 T_{2}^{62} + \cdots + 855036081$$ acting on $$S_{2}^{\mathrm{new}}(675, [\chi])$$.