# Properties

 Label 375.2.i.d Level $375$ Weight $2$ Character orbit 375.i Analytic conductor $2.994$ 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] = [375,2,Mod(49,375)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(375, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([0, 7]))

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

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

chi := DirichletCharacter("375.49");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$375 = 3 \cdot 5^{3}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 375.i (of order $$10$$, degree $$4$$, 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: $$2.99439007580$$ Analytic rank: $$0$$ Dimension: $$24$$ Relative dimension: $$6$$ over $$\Q(\zeta_{10})$$ Twist minimal: no (minimal twist has level 75) Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$24 q + 20 q^{4} + 6 q^{9}+O(q^{10})$$ 24 * q + 20 * q^4 + 6 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$24 q + 20 q^{4} + 6 q^{9} - 8 q^{11} - 12 q^{14} + 32 q^{16} - 14 q^{19} - 6 q^{21} - 12 q^{24} - 112 q^{26} + 2 q^{29} + 26 q^{31} + 50 q^{34} - 4 q^{39} + 16 q^{41} - 66 q^{44} - 44 q^{46} + 56 q^{49} + 52 q^{51} + 90 q^{56} + 44 q^{59} - 16 q^{61} - 98 q^{64} - 6 q^{66} - 12 q^{69} - 42 q^{71} + 88 q^{74} - 104 q^{76} - 20 q^{79} - 6 q^{81} + 12 q^{84} + 112 q^{86} - 114 q^{89} - 14 q^{91} + 46 q^{94} - 46 q^{96} - 12 q^{99}+O(q^{100})$$ 24 * q + 20 * q^4 + 6 * q^9 - 8 * q^11 - 12 * q^14 + 32 * q^16 - 14 * q^19 - 6 * q^21 - 12 * q^24 - 112 * q^26 + 2 * q^29 + 26 * q^31 + 50 * q^34 - 4 * q^39 + 16 * q^41 - 66 * q^44 - 44 * q^46 + 56 * q^49 + 52 * q^51 + 90 * q^56 + 44 * q^59 - 16 * q^61 - 98 * q^64 - 6 * q^66 - 12 * q^69 - 42 * q^71 + 88 * q^74 - 104 * q^76 - 20 * q^79 - 6 * q^81 + 12 * q^84 + 112 * q^86 - 114 * q^89 - 14 * q^91 + 46 * q^94 - 46 * q^96 - 12 * 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 −1.26995 + 1.74793i 0.951057 0.309017i −0.824463 2.53744i 0 −0.667650 + 2.05481i 3.16056i 1.37265 + 0.446002i 0.809017 0.587785i 0
49.2 −1.18666 + 1.63330i −0.951057 + 0.309017i −0.641469 1.97424i 0 0.623865 1.92006i 1.01887i 0.145612 + 0.0473123i 0.809017 0.587785i 0
49.3 −0.0832830 + 0.114629i −0.951057 + 0.309017i 0.611830 + 1.88302i 0 0.0437845 0.134755i 0.858311i −0.536314 0.174259i 0.809017 0.587785i 0
49.4 0.0832830 0.114629i 0.951057 0.309017i 0.611830 + 1.88302i 0 0.0437845 0.134755i 0.858311i 0.536314 + 0.174259i 0.809017 0.587785i 0
49.5 1.18666 1.63330i 0.951057 0.309017i −0.641469 1.97424i 0 0.623865 1.92006i 1.01887i −0.145612 0.0473123i 0.809017 0.587785i 0
49.6 1.26995 1.74793i −0.951057 + 0.309017i −0.824463 2.53744i 0 −0.667650 + 2.05481i 3.16056i −1.37265 0.446002i 0.809017 0.587785i 0
199.1 −1.26995 1.74793i 0.951057 + 0.309017i −0.824463 + 2.53744i 0 −0.667650 2.05481i 3.16056i 1.37265 0.446002i 0.809017 + 0.587785i 0
199.2 −1.18666 1.63330i −0.951057 0.309017i −0.641469 + 1.97424i 0 0.623865 + 1.92006i 1.01887i 0.145612 0.0473123i 0.809017 + 0.587785i 0
199.3 −0.0832830 0.114629i −0.951057 0.309017i 0.611830 1.88302i 0 0.0437845 + 0.134755i 0.858311i −0.536314 + 0.174259i 0.809017 + 0.587785i 0
199.4 0.0832830 + 0.114629i 0.951057 + 0.309017i 0.611830 1.88302i 0 0.0437845 + 0.134755i 0.858311i 0.536314 0.174259i 0.809017 + 0.587785i 0
199.5 1.18666 + 1.63330i 0.951057 + 0.309017i −0.641469 + 1.97424i 0 0.623865 + 1.92006i 1.01887i −0.145612 + 0.0473123i 0.809017 + 0.587785i 0
199.6 1.26995 + 1.74793i −0.951057 0.309017i −0.824463 + 2.53744i 0 −0.667650 2.05481i 3.16056i −1.37265 + 0.446002i 0.809017 + 0.587785i 0
274.1 −2.55553 0.830342i 0.587785 0.809017i 4.22323 + 3.06835i 0 −2.17386 + 1.57940i 1.68704i −5.08599 7.00026i −0.309017 0.951057i 0
274.2 −2.32085 0.754089i −0.587785 + 0.809017i 3.19965 + 2.32468i 0 1.97423 1.43436i 3.44028i −2.80415 3.85959i −0.309017 0.951057i 0
274.3 −0.234682 0.0762527i −0.587785 + 0.809017i −1.56877 1.13978i 0 0.199632 0.145041i 1.24676i 0.571334 + 0.786373i −0.309017 0.951057i 0
274.4 0.234682 + 0.0762527i 0.587785 0.809017i −1.56877 1.13978i 0 0.199632 0.145041i 1.24676i −0.571334 0.786373i −0.309017 0.951057i 0
274.5 2.32085 + 0.754089i 0.587785 0.809017i 3.19965 + 2.32468i 0 1.97423 1.43436i 3.44028i 2.80415 + 3.85959i −0.309017 0.951057i 0
274.6 2.55553 + 0.830342i −0.587785 + 0.809017i 4.22323 + 3.06835i 0 −2.17386 + 1.57940i 1.68704i 5.08599 + 7.00026i −0.309017 0.951057i 0
349.1 −2.55553 + 0.830342i 0.587785 + 0.809017i 4.22323 3.06835i 0 −2.17386 1.57940i 1.68704i −5.08599 + 7.00026i −0.309017 + 0.951057i 0
349.2 −2.32085 + 0.754089i −0.587785 0.809017i 3.19965 2.32468i 0 1.97423 + 1.43436i 3.44028i −2.80415 + 3.85959i −0.309017 + 0.951057i 0
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 349.6 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
25.d even 5 1 inner
25.e even 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 375.2.i.d 24
5.b even 2 1 inner 375.2.i.d 24
5.c odd 4 1 75.2.g.c 12
5.c odd 4 1 375.2.g.c 12
15.e even 4 1 225.2.h.d 12
25.d even 5 1 inner 375.2.i.d 24
25.d even 5 1 1875.2.b.f 12
25.e even 10 1 inner 375.2.i.d 24
25.e even 10 1 1875.2.b.f 12
25.f odd 20 1 75.2.g.c 12
25.f odd 20 1 375.2.g.c 12
25.f odd 20 1 1875.2.a.j 6
25.f odd 20 1 1875.2.a.k 6
75.l even 20 1 225.2.h.d 12
75.l even 20 1 5625.2.a.p 6
75.l even 20 1 5625.2.a.q 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.2.g.c 12 5.c odd 4 1
75.2.g.c 12 25.f odd 20 1
225.2.h.d 12 15.e even 4 1
225.2.h.d 12 75.l even 20 1
375.2.g.c 12 5.c odd 4 1
375.2.g.c 12 25.f odd 20 1
375.2.i.d 24 1.a even 1 1 trivial
375.2.i.d 24 5.b even 2 1 inner
375.2.i.d 24 25.d even 5 1 inner
375.2.i.d 24 25.e even 10 1 inner
1875.2.a.j 6 25.f odd 20 1
1875.2.a.k 6 25.f odd 20 1
1875.2.b.f 12 25.d even 5 1
1875.2.b.f 12 25.e even 10 1
5625.2.a.p 6 75.l even 20 1
5625.2.a.q 6 75.l even 20 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{24} - 16 T_{2}^{22} + 132 T_{2}^{20} - 717 T_{2}^{18} + 4268 T_{2}^{16} - 19372 T_{2}^{14} + 64293 T_{2}^{12} - 147172 T_{2}^{10} + 681588 T_{2}^{8} - 58037 T_{2}^{6} + 1932 T_{2}^{4} + 4 T_{2}^{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(375, [\chi])$$.