# Properties

 Label 475.2.j.d Level $475$ Weight $2$ Character orbit 475.j Analytic conductor $3.793$ Analytic rank $0$ Dimension $24$ Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [475,2,Mod(49,475)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(475, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 4]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("475.49");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$475 = 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 475.j (of order $$6$$, degree $$2$$, 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: $$3.79289409601$$ Analytic rank: $$0$$ Dimension: $$24$$ Relative dimension: $$12$$ over $$\Q(\zeta_{6})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$24 q + 4 q^{4} + 2 q^{6} + 14 q^{9}+O(q^{10})$$ 24 * q + 4 * q^4 + 2 * q^6 + 14 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$24 q + 4 q^{4} + 2 q^{6} + 14 q^{9} - 4 q^{11} - 12 q^{14} + 12 q^{16} + 12 q^{19} - 6 q^{21} + 22 q^{24} + 76 q^{26} + 6 q^{29} - 12 q^{31} - 2 q^{34} - 26 q^{36} - 32 q^{39} - 22 q^{41} + 42 q^{44} - 48 q^{46} - 16 q^{49} + 34 q^{51} + 36 q^{54} + 16 q^{56} + 8 q^{59} - 50 q^{61} + 88 q^{64} - 68 q^{66} - 52 q^{69} - 36 q^{71} - 12 q^{74} + 48 q^{76} + 6 q^{79} - 4 q^{81} - 148 q^{84} - 18 q^{86} + 24 q^{89} + 22 q^{91} - 32 q^{94} - 52 q^{96} - 40 q^{99}+O(q^{100})$$ 24 * q + 4 * q^4 + 2 * q^6 + 14 * q^9 - 4 * q^11 - 12 * q^14 + 12 * q^16 + 12 * q^19 - 6 * q^21 + 22 * q^24 + 76 * q^26 + 6 * q^29 - 12 * q^31 - 2 * q^34 - 26 * q^36 - 32 * q^39 - 22 * q^41 + 42 * q^44 - 48 * q^46 - 16 * q^49 + 34 * q^51 + 36 * q^54 + 16 * q^56 + 8 * q^59 - 50 * q^61 + 88 * q^64 - 68 * q^66 - 52 * q^69 - 36 * q^71 - 12 * q^74 + 48 * q^76 + 6 * q^79 - 4 * q^81 - 148 * q^84 - 18 * q^86 + 24 * q^89 + 22 * q^91 - 32 * q^94 - 52 * q^96 - 40 * 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 −1.87935 1.08504i −1.22342 0.706345i 1.35464 + 2.34630i 0 1.53283 + 2.65494i 1.76171i 1.53919i −0.502155 0.869757i 0
49.2 −1.87935 1.08504i 2.55640 + 1.47594i 1.35464 + 2.34630i 0 −3.20292 5.54761i 0.591620i 1.53919i 2.85679 + 4.94811i 0
49.3 −1.28275 0.740597i −2.47401 1.42837i 0.0969683 + 0.167954i 0 2.11569 + 3.66449i 3.78541i 2.67513i 2.58048 + 4.46952i 0
49.4 −1.28275 0.740597i 0.157277 + 0.0908038i 0.0969683 + 0.167954i 0 −0.134498 0.232958i 1.30422i 2.67513i −1.48351 2.56951i 0
49.5 −0.269427 0.155554i −1.94349 1.12208i −0.951606 1.64823i 0 0.349087 + 0.604636i 3.96928i 1.21432i 1.01811 + 1.76343i 0
49.6 −0.269427 0.155554i 0.891863 + 0.514917i −0.951606 1.64823i 0 −0.160195 0.277466i 3.28038i 1.21432i −0.969720 1.67960i 0
49.7 0.269427 + 0.155554i −0.891863 0.514917i −0.951606 1.64823i 0 −0.160195 0.277466i 3.28038i 1.21432i −0.969720 1.67960i 0
49.8 0.269427 + 0.155554i 1.94349 + 1.12208i −0.951606 1.64823i 0 0.349087 + 0.604636i 3.96928i 1.21432i 1.01811 + 1.76343i 0
49.9 1.28275 + 0.740597i −0.157277 0.0908038i 0.0969683 + 0.167954i 0 −0.134498 0.232958i 1.30422i 2.67513i −1.48351 2.56951i 0
49.10 1.28275 + 0.740597i 2.47401 + 1.42837i 0.0969683 + 0.167954i 0 2.11569 + 3.66449i 3.78541i 2.67513i 2.58048 + 4.46952i 0
49.11 1.87935 + 1.08504i −2.55640 1.47594i 1.35464 + 2.34630i 0 −3.20292 5.54761i 0.591620i 1.53919i 2.85679 + 4.94811i 0
49.12 1.87935 + 1.08504i 1.22342 + 0.706345i 1.35464 + 2.34630i 0 1.53283 + 2.65494i 1.76171i 1.53919i −0.502155 0.869757i 0
349.1 −1.87935 + 1.08504i −1.22342 + 0.706345i 1.35464 2.34630i 0 1.53283 2.65494i 1.76171i 1.53919i −0.502155 + 0.869757i 0
349.2 −1.87935 + 1.08504i 2.55640 1.47594i 1.35464 2.34630i 0 −3.20292 + 5.54761i 0.591620i 1.53919i 2.85679 4.94811i 0
349.3 −1.28275 + 0.740597i −2.47401 + 1.42837i 0.0969683 0.167954i 0 2.11569 3.66449i 3.78541i 2.67513i 2.58048 4.46952i 0
349.4 −1.28275 + 0.740597i 0.157277 0.0908038i 0.0969683 0.167954i 0 −0.134498 + 0.232958i 1.30422i 2.67513i −1.48351 + 2.56951i 0
349.5 −0.269427 + 0.155554i −1.94349 + 1.12208i −0.951606 + 1.64823i 0 0.349087 0.604636i 3.96928i 1.21432i 1.01811 1.76343i 0
349.6 −0.269427 + 0.155554i 0.891863 0.514917i −0.951606 + 1.64823i 0 −0.160195 + 0.277466i 3.28038i 1.21432i −0.969720 + 1.67960i 0
349.7 0.269427 0.155554i −0.891863 + 0.514917i −0.951606 + 1.64823i 0 −0.160195 + 0.277466i 3.28038i 1.21432i −0.969720 + 1.67960i 0
349.8 0.269427 0.155554i 1.94349 1.12208i −0.951606 + 1.64823i 0 0.349087 0.604636i 3.96928i 1.21432i 1.01811 1.76343i 0
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 49.12 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
19.c even 3 1 inner
95.i even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 475.2.j.d 24
5.b even 2 1 inner 475.2.j.d 24
5.c odd 4 1 475.2.e.f 12
5.c odd 4 1 475.2.e.h yes 12
19.c even 3 1 inner 475.2.j.d 24
95.i even 6 1 inner 475.2.j.d 24
95.l even 12 1 9025.2.a.bs 6
95.l even 12 1 9025.2.a.by 6
95.m odd 12 1 475.2.e.f 12
95.m odd 12 1 475.2.e.h yes 12
95.m odd 12 1 9025.2.a.br 6
95.m odd 12 1 9025.2.a.bz 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
475.2.e.f 12 5.c odd 4 1
475.2.e.f 12 95.m odd 12 1
475.2.e.h yes 12 5.c odd 4 1
475.2.e.h yes 12 95.m odd 12 1
475.2.j.d 24 1.a even 1 1 trivial
475.2.j.d 24 5.b even 2 1 inner
475.2.j.d 24 19.c even 3 1 inner
475.2.j.d 24 95.i even 6 1 inner
9025.2.a.br 6 95.m odd 12 1
9025.2.a.bs 6 95.l even 12 1
9025.2.a.by 6 95.l even 12 1
9025.2.a.bz 6 95.m odd 12 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{12} - 7T_{2}^{10} + 38T_{2}^{8} - 75T_{2}^{6} + 114T_{2}^{4} - 11T_{2}^{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(475, [\chi])$$.