# Properties

 Label 725.2.l.d Level $725$ Weight $2$ Character orbit 725.l Analytic conductor $5.789$ Analytic rank $0$ Dimension $24$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [725,2,Mod(226,725)]

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([0, 10]))

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

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

chi := DirichletCharacter("725.226");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$725 = 5^{2} \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 725.l (of order $$7$$, degree $$6$$, 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: $$24$$ Relative dimension: $$4$$ over $$\Q(\zeta_{7})$$ Twist minimal: no (minimal twist has level 145) Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$24 q + 2 q^{2} - 2 q^{3} - 4 q^{4} - 4 q^{6} - 4 q^{7} - 9 q^{8}+O(q^{10})$$ 24 * q + 2 * q^2 - 2 * q^3 - 4 * q^4 - 4 * q^6 - 4 * q^7 - 9 * q^8 $$\operatorname{Tr}(f)(q) =$$ $$24 q + 2 q^{2} - 2 q^{3} - 4 q^{4} - 4 q^{6} - 4 q^{7} - 9 q^{8} + 12 q^{11} - 2 q^{12} - 4 q^{13} - 32 q^{14} + 10 q^{16} + 40 q^{17} - 10 q^{18} + 2 q^{19} - 35 q^{21} - 13 q^{22} - 12 q^{23} - 22 q^{24} - 8 q^{26} + 13 q^{27} - 20 q^{28} + 5 q^{29} - 7 q^{31} - 22 q^{32} + 12 q^{33} + 12 q^{34} - 16 q^{36} + 14 q^{37} - 14 q^{38} - 9 q^{39} + 8 q^{41} - 19 q^{42} + 47 q^{43} + 17 q^{44} - 48 q^{46} - 22 q^{47} - 31 q^{48} - 10 q^{49} + 34 q^{51} - 13 q^{52} + 53 q^{53} + 23 q^{54} + 10 q^{56} - 20 q^{57} - 10 q^{58} - 38 q^{59} + 19 q^{62} - 11 q^{63} + 37 q^{64} + 2 q^{66} - 43 q^{67} + 50 q^{68} + 48 q^{69} - 61 q^{71} + 37 q^{72} - 24 q^{73} + 109 q^{74} + 24 q^{76} - 41 q^{77} + 42 q^{78} + 2 q^{79} - 15 q^{81} + 40 q^{82} - 37 q^{83} - 6 q^{84} - 18 q^{86} - 9 q^{87} + 20 q^{88} + 16 q^{89} - q^{91} - 29 q^{92} + 26 q^{93} - 54 q^{94} + 15 q^{96} + 57 q^{97} + 26 q^{98} + 88 q^{99}+O(q^{100})$$ 24 * q + 2 * q^2 - 2 * q^3 - 4 * q^4 - 4 * q^6 - 4 * q^7 - 9 * q^8 + 12 * q^11 - 2 * q^12 - 4 * q^13 - 32 * q^14 + 10 * q^16 + 40 * q^17 - 10 * q^18 + 2 * q^19 - 35 * q^21 - 13 * q^22 - 12 * q^23 - 22 * q^24 - 8 * q^26 + 13 * q^27 - 20 * q^28 + 5 * q^29 - 7 * q^31 - 22 * q^32 + 12 * q^33 + 12 * q^34 - 16 * q^36 + 14 * q^37 - 14 * q^38 - 9 * q^39 + 8 * q^41 - 19 * q^42 + 47 * q^43 + 17 * q^44 - 48 * q^46 - 22 * q^47 - 31 * q^48 - 10 * q^49 + 34 * q^51 - 13 * q^52 + 53 * q^53 + 23 * q^54 + 10 * q^56 - 20 * q^57 - 10 * q^58 - 38 * q^59 + 19 * q^62 - 11 * q^63 + 37 * q^64 + 2 * q^66 - 43 * q^67 + 50 * q^68 + 48 * q^69 - 61 * q^71 + 37 * q^72 - 24 * q^73 + 109 * q^74 + 24 * q^76 - 41 * q^77 + 42 * q^78 + 2 * q^79 - 15 * q^81 + 40 * q^82 - 37 * q^83 - 6 * q^84 - 18 * q^86 - 9 * q^87 + 20 * q^88 + 16 * q^89 - q^91 - 29 * q^92 + 26 * q^93 - 54 * q^94 + 15 * q^96 + 57 * q^97 + 26 * q^98 + 88 * 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
226.1 −0.926597 + 1.16192i 0.508780 2.22911i −0.0464250 0.203401i 0 2.11861 + 2.65665i −0.153120 + 0.670862i −2.39859 1.15510i −2.00718 0.966605i 0
226.2 −0.552702 + 0.693066i −0.484696 + 2.12359i 0.270180 + 1.18374i 0 −1.20390 1.50964i −0.932055 + 4.08360i −2.56709 1.23625i −1.57180 0.756939i 0
226.3 0.388637 0.487336i −0.153777 + 0.673740i 0.358585 + 1.57106i 0 0.268574 + 0.336781i 0.617778 2.70666i 2.02819 + 0.976724i 2.27263 + 1.09444i 0
226.4 1.36814 1.71559i 0.530661 2.32498i −0.626413 2.74450i 0 −3.26270 4.09130i 0.423324 1.85470i −1.61142 0.776019i −2.42102 1.16590i 0
326.1 −0.535053 2.34422i −0.390955 0.188274i −3.40714 + 1.64079i 0 −0.232174 + 1.01722i −1.10203 0.530707i 2.67101 + 3.34934i −1.75307 2.19828i 0
326.2 −0.263278 1.15350i 2.67743 + 1.28938i 0.540697 0.260386i 0 0.782391 3.42788i −3.34803 1.61232i −1.91809 2.40520i 3.63566 + 4.55898i 0
326.3 −0.0486603 0.213195i −1.07523 0.517805i 1.75885 0.847019i 0 −0.0580721 + 0.254431i 1.52525 + 0.734524i −0.538852 0.675700i −0.982464 1.23197i 0
326.4 0.446022 + 1.95415i −2.33473 1.12435i −1.81783 + 0.875423i 0 1.15580 5.06391i −0.000629566 0 0.000303183i −0.0220508 0.0276508i 2.31635 + 2.90461i 0
401.1 −0.926597 1.16192i 0.508780 + 2.22911i −0.0464250 + 0.203401i 0 2.11861 2.65665i −0.153120 0.670862i −2.39859 + 1.15510i −2.00718 + 0.966605i 0
401.2 −0.552702 0.693066i −0.484696 2.12359i 0.270180 1.18374i 0 −1.20390 + 1.50964i −0.932055 4.08360i −2.56709 + 1.23625i −1.57180 + 0.756939i 0
401.3 0.388637 + 0.487336i −0.153777 0.673740i 0.358585 1.57106i 0 0.268574 0.336781i 0.617778 + 2.70666i 2.02819 0.976724i 2.27263 1.09444i 0
401.4 1.36814 + 1.71559i 0.530661 + 2.32498i −0.626413 + 2.74450i 0 −3.26270 + 4.09130i 0.423324 + 1.85470i −1.61142 + 0.776019i −2.42102 + 1.16590i 0
426.1 −1.72674 + 0.831553i 0.700245 + 0.878079i 1.04316 1.30808i 0 −1.93931 0.933921i 1.05684 + 1.32524i 0.139412 0.610805i 0.386882 1.69504i 0
426.2 −0.0878682 + 0.0423151i −1.48831 1.86628i −1.24105 + 1.55623i 0 0.209747 + 0.101009i 1.00514 + 1.26040i 0.0866002 0.379420i −0.600378 + 2.63043i 0
426.3 0.994366 0.478861i 0.939621 + 1.17825i −0.487524 + 0.611336i 0 1.49854 + 0.721661i 0.880793 + 1.10448i −0.683208 + 2.99333i 0.162183 0.710572i 0
426.4 1.94373 0.936051i −0.429035 0.537992i 1.65491 2.07520i 0 −1.33752 0.644113i −1.97327 2.47441i 0.314092 1.37613i 0.562198 2.46315i 0
451.1 −1.72674 0.831553i 0.700245 0.878079i 1.04316 + 1.30808i 0 −1.93931 + 0.933921i 1.05684 1.32524i 0.139412 + 0.610805i 0.386882 + 1.69504i 0
451.2 −0.0878682 0.0423151i −1.48831 + 1.86628i −1.24105 1.55623i 0 0.209747 0.101009i 1.00514 1.26040i 0.0866002 + 0.379420i −0.600378 2.63043i 0
451.3 0.994366 + 0.478861i 0.939621 1.17825i −0.487524 0.611336i 0 1.49854 0.721661i 0.880793 1.10448i −0.683208 2.99333i 0.162183 + 0.710572i 0
451.4 1.94373 + 0.936051i −0.429035 + 0.537992i 1.65491 + 2.07520i 0 −1.33752 + 0.644113i −1.97327 + 2.47441i 0.314092 + 1.37613i 0.562198 + 2.46315i 0
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 226.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
29.d even 7 1 inner

## Twists

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

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{24} - 2 T_{2}^{23} + 8 T_{2}^{22} - 13 T_{2}^{21} + 31 T_{2}^{20} - 16 T_{2}^{19} + 64 T_{2}^{18} + \cdots + 1$$ acting on $$S_{2}^{\mathrm{new}}(725, [\chi])$$.