# Properties

 Label 675.2.u.b Level $675$ Weight $2$ Character orbit 675.u Analytic conductor $5.390$ Analytic rank $0$ Dimension $24$ 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: $$24$$ Relative dimension: $$4$$ over $$\Q(\zeta_{18})$$ Twist minimal: no (minimal twist has level 27) 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.

 $$\operatorname{Tr}(f)(q) =$$ $$24 q + 12 q^{4}+O(q^{10})$$ 24 * q + 12 * q^4 $$\operatorname{Tr}(f)(q) =$$ $$24 q + 12 q^{4} + 6 q^{11} - 30 q^{14} + 6 q^{19} - 24 q^{21} + 36 q^{24} - 60 q^{26} + 12 q^{29} + 6 q^{31} - 18 q^{34} + 36 q^{36} - 66 q^{39} + 30 q^{41} - 6 q^{44} - 6 q^{46} - 24 q^{49} - 36 q^{51} + 108 q^{54} - 66 q^{56} + 24 q^{59} + 24 q^{61} - 24 q^{64} - 18 q^{66} - 18 q^{69} + 54 q^{71} - 66 q^{74} - 96 q^{76} + 84 q^{79} + 72 q^{81} - 12 q^{84} + 102 q^{86} - 18 q^{89} + 12 q^{91} + 30 q^{94} + 54 q^{99}+O(q^{100})$$ 24 * q + 12 * q^4 + 6 * q^11 - 30 * q^14 + 6 * q^19 - 24 * q^21 + 36 * q^24 - 60 * q^26 + 12 * q^29 + 6 * q^31 - 18 * q^34 + 36 * q^36 - 66 * q^39 + 30 * q^41 - 6 * q^44 - 6 * q^46 - 24 * q^49 - 36 * q^51 + 108 * q^54 - 66 * q^56 + 24 * q^59 + 24 * q^61 - 24 * q^64 - 18 * q^66 - 18 * q^69 + 54 * q^71 - 66 * q^74 - 96 * q^76 + 84 * q^79 + 72 * q^81 - 12 * q^84 + 102 * q^86 - 18 * q^89 + 12 * q^91 + 30 * q^94 + 54 * 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 −1.36054 1.62143i 1.42389 + 0.986166i −0.430663 + 2.44241i 0 −0.338267 3.65046i −0.957561 + 0.168844i 0.880031 0.508086i 1.05495 + 2.80839i 0
49.2 −0.267057 0.318266i −1.72466 0.159815i 0.317323 1.79963i 0 0.409719 + 0.591580i 1.29958 0.229151i −1.37711 + 0.795075i 2.94892 + 0.551252i 0
49.3 0.267057 + 0.318266i 1.72466 + 0.159815i 0.317323 1.79963i 0 0.409719 + 0.591580i −1.29958 + 0.229151i 1.37711 0.795075i 2.94892 + 0.551252i 0
49.4 1.36054 + 1.62143i −1.42389 0.986166i −0.430663 + 2.44241i 0 −0.338267 3.65046i 0.957561 0.168844i −0.880031 + 0.508086i 1.05495 + 2.80839i 0
124.1 −1.36054 + 1.62143i 1.42389 0.986166i −0.430663 2.44241i 0 −0.338267 + 3.65046i −0.957561 0.168844i 0.880031 + 0.508086i 1.05495 2.80839i 0
124.2 −0.267057 + 0.318266i −1.72466 + 0.159815i 0.317323 + 1.79963i 0 0.409719 0.591580i 1.29958 + 0.229151i −1.37711 0.795075i 2.94892 0.551252i 0
124.3 0.267057 0.318266i 1.72466 0.159815i 0.317323 + 1.79963i 0 0.409719 0.591580i −1.29958 0.229151i 1.37711 + 0.795075i 2.94892 0.551252i 0
124.4 1.36054 1.62143i −1.42389 + 0.986166i −0.430663 2.44241i 0 −0.338267 + 3.65046i 0.957561 + 0.168844i −0.880031 0.508086i 1.05495 2.80839i 0
274.1 −0.574906 + 1.57954i 0.940501 + 1.45446i −0.632343 0.530599i 0 −2.83808 + 0.649381i 2.51261 + 2.99441i −1.70978 + 0.987144i −1.23092 + 2.73584i 0
274.2 −0.274138 + 0.753189i −0.386327 + 1.68842i 1.03995 + 0.872619i 0 −1.16579 0.753837i 1.52780 + 1.82076i −2.33062 + 1.34559i −2.70150 1.30456i 0
274.3 0.274138 0.753189i 0.386327 1.68842i 1.03995 + 0.872619i 0 −1.16579 0.753837i −1.52780 1.82076i 2.33062 1.34559i −2.70150 1.30456i 0
274.4 0.574906 1.57954i −0.940501 1.45446i −0.632343 0.530599i 0 −2.83808 + 0.649381i −2.51261 2.99441i 1.70978 0.987144i −1.23092 + 2.73584i 0
349.1 −2.36514 0.417037i −1.71926 + 0.210069i 3.54056 + 1.28866i 0 4.15390 + 0.220155i 0.198324 + 0.544891i −3.67675 2.12277i 2.91174 0.722330i 0
349.2 −1.03831 0.183082i −0.0916693 1.72962i −0.834822 0.303850i 0 −0.221481 + 1.81266i 0.841112 + 2.31094i 2.63732 + 1.52266i −2.98319 + 0.317107i 0
349.3 1.03831 + 0.183082i 0.0916693 + 1.72962i −0.834822 0.303850i 0 −0.221481 + 1.81266i −0.841112 2.31094i −2.63732 1.52266i −2.98319 + 0.317107i 0
349.4 2.36514 + 0.417037i 1.71926 0.210069i 3.54056 + 1.28866i 0 4.15390 + 0.220155i −0.198324 0.544891i 3.67675 + 2.12277i 2.91174 0.722330i 0
499.1 −2.36514 + 0.417037i −1.71926 0.210069i 3.54056 1.28866i 0 4.15390 0.220155i 0.198324 0.544891i −3.67675 + 2.12277i 2.91174 + 0.722330i 0
499.2 −1.03831 + 0.183082i −0.0916693 + 1.72962i −0.834822 + 0.303850i 0 −0.221481 1.81266i 0.841112 2.31094i 2.63732 1.52266i −2.98319 0.317107i 0
499.3 1.03831 0.183082i 0.0916693 1.72962i −0.834822 + 0.303850i 0 −0.221481 1.81266i −0.841112 + 2.31094i −2.63732 + 1.52266i −2.98319 0.317107i 0
499.4 2.36514 0.417037i 1.71926 + 0.210069i 3.54056 1.28866i 0 4.15390 0.220155i −0.198324 + 0.544891i 3.67675 2.12277i 2.91174 + 0.722330i 0
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 49.4 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.b 24
5.b even 2 1 inner 675.2.u.b 24
5.c odd 4 1 27.2.e.a 12
5.c odd 4 1 675.2.l.c 12
15.e even 4 1 81.2.e.a 12
20.e even 4 1 432.2.u.c 12
27.e even 9 1 inner 675.2.u.b 24
45.k odd 12 1 243.2.e.c 12
45.k odd 12 1 243.2.e.d 12
45.l even 12 1 243.2.e.a 12
45.l even 12 1 243.2.e.b 12
135.p even 18 1 inner 675.2.u.b 24
135.q even 36 1 81.2.e.a 12
135.q even 36 1 243.2.e.a 12
135.q even 36 1 243.2.e.b 12
135.q even 36 1 729.2.a.d 6
135.q even 36 2 729.2.c.b 12
135.r odd 36 1 27.2.e.a 12
135.r odd 36 1 243.2.e.c 12
135.r odd 36 1 243.2.e.d 12
135.r odd 36 1 675.2.l.c 12
135.r odd 36 1 729.2.a.a 6
135.r odd 36 2 729.2.c.e 12
540.bh even 36 1 432.2.u.c 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
27.2.e.a 12 5.c odd 4 1
27.2.e.a 12 135.r odd 36 1
81.2.e.a 12 15.e even 4 1
81.2.e.a 12 135.q even 36 1
243.2.e.a 12 45.l even 12 1
243.2.e.a 12 135.q even 36 1
243.2.e.b 12 45.l even 12 1
243.2.e.b 12 135.q even 36 1
243.2.e.c 12 45.k odd 12 1
243.2.e.c 12 135.r odd 36 1
243.2.e.d 12 45.k odd 12 1
243.2.e.d 12 135.r odd 36 1
432.2.u.c 12 20.e even 4 1
432.2.u.c 12 540.bh even 36 1
675.2.l.c 12 5.c odd 4 1
675.2.l.c 12 135.r odd 36 1
675.2.u.b 24 1.a even 1 1 trivial
675.2.u.b 24 5.b even 2 1 inner
675.2.u.b 24 27.e even 9 1 inner
675.2.u.b 24 135.p even 18 1 inner
729.2.a.a 6 135.r odd 36 1
729.2.a.d 6 135.q even 36 1
729.2.c.b 12 135.q even 36 2
729.2.c.e 12 135.r odd 36 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{24} - 6 T_{2}^{22} + 9 T_{2}^{20} - 84 T_{2}^{18} + 324 T_{2}^{16} + 1350 T_{2}^{14} + 3564 T_{2}^{12} - 6237 T_{2}^{10} - 1782 T_{2}^{8} + 2403 T_{2}^{6} + 2835 T_{2}^{4} + 243 T_{2}^{2} + 81$$ acting on $$S_{2}^{\mathrm{new}}(675, [\chi])$$.