# Properties

 Label 725.2.r.c Level $725$ Weight $2$ Character orbit 725.r Analytic conductor $5.789$ Analytic rank $0$ Dimension $48$ Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [725,2,Mod(24,725)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(725, base_ring=CyclotomicField(14))

chi = DirichletCharacter(H, H._module([7, 4]))

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

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

chi := DirichletCharacter("725.24");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$725 = 5^{2} \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 725.r (of order $$14$$, 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: $$5.78915414654$$ Analytic rank: $$0$$ Dimension: $$48$$ Relative dimension: $$8$$ over $$\Q(\zeta_{14})$$ Twist minimal: no (minimal twist has level 145) Sato-Tate group: $\mathrm{SU}(2)[C_{14}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$48 q + 8 q^{4} - 8 q^{6}+O(q^{10})$$ 48 * q + 8 * q^4 - 8 * q^6 $$\operatorname{Tr}(f)(q) =$$ $$48 q + 8 q^{4} - 8 q^{6} + 24 q^{11} + 64 q^{14} + 20 q^{16} - 4 q^{19} - 70 q^{21} + 44 q^{24} - 16 q^{26} - 10 q^{29} - 14 q^{31} - 24 q^{34} - 32 q^{36} + 18 q^{39} + 16 q^{41} - 34 q^{44} - 96 q^{46} + 20 q^{49} + 68 q^{51} - 46 q^{54} + 20 q^{56} + 76 q^{59} - 74 q^{64} + 4 q^{66} - 96 q^{69} - 122 q^{71} - 218 q^{74} + 48 q^{76} - 4 q^{79} - 30 q^{81} + 12 q^{84} - 36 q^{86} - 32 q^{89} - 2 q^{91} + 108 q^{94} + 30 q^{96} - 176 q^{99}+O(q^{100})$$ 48 * q + 8 * q^4 - 8 * q^6 + 24 * q^11 + 64 * q^14 + 20 * q^16 - 4 * q^19 - 70 * q^21 + 44 * q^24 - 16 * q^26 - 10 * q^29 - 14 * q^31 - 24 * q^34 - 32 * q^36 + 18 * q^39 + 16 * q^41 - 34 * q^44 - 96 * q^46 + 20 * q^49 + 68 * q^51 - 46 * q^54 + 20 * q^56 + 76 * q^59 - 74 * q^64 + 4 * q^66 - 96 * q^69 - 122 * q^71 - 218 * q^74 + 48 * q^76 - 4 * q^79 - 30 * q^81 + 12 * q^84 - 36 * q^86 - 32 * q^89 - 2 * q^91 + 108 * q^94 + 30 * q^96 - 176 * 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
24.1 −1.71559 + 1.36814i 2.32498 0.530661i 0.626413 2.74450i 0 −3.26270 + 4.09130i −1.85470 + 0.423324i 0.776019 + 1.61142i 2.42102 1.16590i 0
24.2 −1.16192 + 0.926597i −2.22911 + 0.508780i 0.0464250 0.203401i 0 2.11861 2.65665i −0.670862 + 0.153120i −1.15510 2.39859i 2.00718 0.966605i 0
24.3 −0.693066 + 0.552702i 2.12359 0.484696i −0.270180 + 1.18374i 0 −1.20390 + 1.50964i −4.08360 + 0.932055i −1.23625 2.56709i 1.57180 0.756939i 0
24.4 −0.487336 + 0.388637i −0.673740 + 0.153777i −0.358585 + 1.57106i 0 0.268574 0.336781i −2.70666 + 0.617778i −0.976724 2.02819i −2.27263 + 1.09444i 0
24.5 0.487336 0.388637i 0.673740 0.153777i −0.358585 + 1.57106i 0 0.268574 0.336781i 2.70666 0.617778i 0.976724 + 2.02819i −2.27263 + 1.09444i 0
24.6 0.693066 0.552702i −2.12359 + 0.484696i −0.270180 + 1.18374i 0 −1.20390 + 1.50964i 4.08360 0.932055i 1.23625 + 2.56709i 1.57180 0.756939i 0
24.7 1.16192 0.926597i 2.22911 0.508780i 0.0464250 0.203401i 0 2.11861 2.65665i 0.670862 0.153120i 1.15510 + 2.39859i 2.00718 0.966605i 0
24.8 1.71559 1.36814i −2.32498 + 0.530661i 0.626413 2.74450i 0 −3.26270 + 4.09130i 1.85470 0.423324i −0.776019 1.61142i 2.42102 1.16590i 0
49.1 −0.936051 1.94373i 0.537992 0.429035i −1.65491 + 2.07520i 0 −1.33752 0.644113i −2.47441 + 1.97327i 1.37613 + 0.314092i −0.562198 + 2.46315i 0
49.2 −0.831553 1.72674i 0.878079 0.700245i −1.04316 + 1.30808i 0 −1.93931 0.933921i −1.32524 + 1.05684i −0.610805 0.139412i −0.386882 + 1.69504i 0
49.3 −0.478861 0.994366i −1.17825 + 0.939621i 0.487524 0.611336i 0 1.49854 + 0.721661i 1.10448 0.880793i −2.99333 0.683208i −0.162183 + 0.710572i 0
49.4 −0.0423151 0.0878682i −1.86628 + 1.48831i 1.24105 1.55623i 0 0.209747 + 0.101009i −1.26040 + 1.00514i −0.379420 0.0866002i 0.600378 2.63043i 0
49.5 0.0423151 + 0.0878682i 1.86628 1.48831i 1.24105 1.55623i 0 0.209747 + 0.101009i 1.26040 1.00514i 0.379420 + 0.0866002i 0.600378 2.63043i 0
49.6 0.478861 + 0.994366i 1.17825 0.939621i 0.487524 0.611336i 0 1.49854 + 0.721661i −1.10448 + 0.880793i 2.99333 + 0.683208i −0.162183 + 0.710572i 0
49.7 0.831553 + 1.72674i −0.878079 + 0.700245i −1.04316 + 1.30808i 0 −1.93931 0.933921i 1.32524 1.05684i 0.610805 + 0.139412i −0.386882 + 1.69504i 0
49.8 0.936051 + 1.94373i −0.537992 + 0.429035i −1.65491 + 2.07520i 0 −1.33752 0.644113i 2.47441 1.97327i −1.37613 0.314092i −0.562198 + 2.46315i 0
74.1 −0.936051 + 1.94373i 0.537992 + 0.429035i −1.65491 2.07520i 0 −1.33752 + 0.644113i −2.47441 1.97327i 1.37613 0.314092i −0.562198 2.46315i 0
74.2 −0.831553 + 1.72674i 0.878079 + 0.700245i −1.04316 1.30808i 0 −1.93931 + 0.933921i −1.32524 1.05684i −0.610805 + 0.139412i −0.386882 1.69504i 0
74.3 −0.478861 + 0.994366i −1.17825 0.939621i 0.487524 + 0.611336i 0 1.49854 0.721661i 1.10448 + 0.880793i −2.99333 + 0.683208i −0.162183 0.710572i 0
74.4 −0.0423151 + 0.0878682i −1.86628 1.48831i 1.24105 + 1.55623i 0 0.209747 0.101009i −1.26040 1.00514i −0.379420 + 0.0866002i 0.600378 + 2.63043i 0
See all 48 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 24.8 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
29.d even 7 1 inner
145.n even 14 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 725.2.r.c 48
5.b even 2 1 inner 725.2.r.c 48
5.c odd 4 1 145.2.k.a 24
5.c odd 4 1 725.2.l.d 24
29.d even 7 1 inner 725.2.r.c 48
145.n even 14 1 inner 725.2.r.c 48
145.p odd 28 1 145.2.k.a 24
145.p odd 28 1 725.2.l.d 24
145.p odd 28 1 4205.2.a.q 12
145.q odd 28 1 4205.2.a.r 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
145.2.k.a 24 5.c odd 4 1
145.2.k.a 24 145.p odd 28 1
725.2.l.d 24 5.c odd 4 1
725.2.l.d 24 145.p odd 28 1
725.2.r.c 48 1.a even 1 1 trivial
725.2.r.c 48 5.b even 2 1 inner
725.2.r.c 48 29.d even 7 1 inner
725.2.r.c 48 145.n even 14 1 inner
4205.2.a.q 12 145.p odd 28 1
4205.2.a.r 12 145.q odd 28 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{48} - 12 T_{2}^{46} + 74 T_{2}^{44} - 391 T_{2}^{42} + 2231 T_{2}^{40} - 11128 T_{2}^{38} + \cdots + 1$$ acting on $$S_{2}^{\mathrm{new}}(725, [\chi])$$.