# Properties

 Label 725.2.e.d Level $725$ Weight $2$ Character orbit 725.e Analytic conductor $5.789$ Analytic rank $0$ Dimension $40$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [725,2,Mod(157,725)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(725, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([1, 1]))

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

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

chi := DirichletCharacter("725.157");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$725 = 5^{2} \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 725.e (of order $$4$$, degree $$2$$, 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: $$5.78915414654$$ Analytic rank: $$0$$ Dimension: $$40$$ Relative dimension: $$20$$ over $$\Q(i)$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$40 q - 40 q^{4} + 56 q^{9}+O(q^{10})$$ 40 * q - 40 * q^4 + 56 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$40 q - 40 q^{4} + 56 q^{9} + 12 q^{14} + 80 q^{16} - 4 q^{19} - 28 q^{21} - 12 q^{29} - 40 q^{31} - 8 q^{34} - 64 q^{36} - 68 q^{39} - 28 q^{41} + 40 q^{44} - 40 q^{46} - 40 q^{56} - 24 q^{61} - 80 q^{66} - 40 q^{69} - 108 q^{76} + 72 q^{79} + 200 q^{81} - 4 q^{84} - 40 q^{89} + 28 q^{99}+O(q^{100})$$ 40 * q - 40 * q^4 + 56 * q^9 + 12 * q^14 + 80 * q^16 - 4 * q^19 - 28 * q^21 - 12 * q^29 - 40 * q^31 - 8 * q^34 - 64 * q^36 - 68 * q^39 - 28 * q^41 + 40 * q^44 - 40 * q^46 - 40 * q^56 - 24 * q^61 - 80 * q^66 - 40 * q^69 - 108 * q^76 + 72 * q^79 + 200 * q^81 - 4 * q^84 - 40 * q^89 + 28 * 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
157.1 2.65926i −1.93483 −5.07167 0 5.14522i 3.36169 + 3.36169i 8.16838i 0.743572 0
157.2 2.52500i −3.42491 −4.37564 0 8.64790i −1.78637 1.78637i 5.99850i 8.72998 0
157.3 2.46344i 1.09135 −4.06854 0 2.68848i −0.369215 0.369215i 5.09573i −1.80895 0
157.4 2.15259i 0.669815 −2.63363 0 1.44183i −1.74569 1.74569i 1.36393i −2.55135 0
157.5 1.63047i −0.836393 −0.658440 0 1.36372i 1.64475 + 1.64475i 2.18738i −2.30045 0
157.6 1.23283i 3.00892 0.480125 0 3.70950i 1.53900 + 1.53900i 3.05758i 6.05362 0
157.7 1.10914i −2.35179 0.769812 0 2.60846i 0.825650 + 0.825650i 3.07210i 2.53090 0
157.8 0.624848i −1.25254 1.60957 0 0.782644i −2.62256 2.62256i 2.25543i −1.43115 0
157.9 0.204638i 3.05184 1.95812 0 0.624521i −3.34957 3.34957i 0.809982i 6.31371 0
157.10 0.0985170i 0.848593 1.99029 0 0.0836009i 0.686476 + 0.686476i 0.393112i −2.27989 0
157.11 0.0985170i −0.848593 1.99029 0 0.0836009i −0.686476 0.686476i 0.393112i −2.27989 0
157.12 0.204638i −3.05184 1.95812 0 0.624521i 3.34957 + 3.34957i 0.809982i 6.31371 0
157.13 0.624848i 1.25254 1.60957 0 0.782644i 2.62256 + 2.62256i 2.25543i −1.43115 0
157.14 1.10914i 2.35179 0.769812 0 2.60846i −0.825650 0.825650i 3.07210i 2.53090 0
157.15 1.23283i −3.00892 0.480125 0 3.70950i −1.53900 1.53900i 3.05758i 6.05362 0
157.16 1.63047i 0.836393 −0.658440 0 1.36372i −1.64475 1.64475i 2.18738i −2.30045 0
157.17 2.15259i −0.669815 −2.63363 0 1.44183i 1.74569 + 1.74569i 1.36393i −2.55135 0
157.18 2.46344i −1.09135 −4.06854 0 2.68848i 0.369215 + 0.369215i 5.09573i −1.80895 0
157.19 2.52500i 3.42491 −4.37564 0 8.64790i 1.78637 + 1.78637i 5.99850i 8.72998 0
157.20 2.65926i 1.93483 −5.07167 0 5.14522i −3.36169 3.36169i 8.16838i 0.743572 0
See all 40 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 157.20 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
145.e even 4 1 inner
145.j even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 725.2.e.d 40
5.b even 2 1 inner 725.2.e.d 40
5.c odd 4 2 725.2.j.d yes 40
29.c odd 4 1 725.2.j.d yes 40
145.e even 4 1 inner 725.2.e.d 40
145.f odd 4 1 725.2.j.d yes 40
145.j even 4 1 inner 725.2.e.d 40

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
725.2.e.d 40 1.a even 1 1 trivial
725.2.e.d 40 5.b even 2 1 inner
725.2.e.d 40 145.e even 4 1 inner
725.2.e.d 40 145.j even 4 1 inner
725.2.j.d yes 40 5.c odd 4 2
725.2.j.d yes 40 29.c odd 4 1
725.2.j.d yes 40 145.f odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{20} + 30 T_{2}^{18} + 370 T_{2}^{16} + 2420 T_{2}^{14} + 9043 T_{2}^{12} + 19436 T_{2}^{10} + 23083 T_{2}^{8} + 13664 T_{2}^{6} + 3115 T_{2}^{4} + 132 T_{2}^{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(725, [\chi])$$.