# Properties

 Label 725.2.e.c Level $725$ Weight $2$ Character orbit 725.e Analytic conductor $5.789$ Analytic rank $0$ Dimension $26$ Inner twists $2$

# 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: $$26$$ Relative dimension: $$13$$ over $$\Q(i)$$ Twist minimal: no (minimal twist has level 145) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$26 q + 4 q^{3} - 22 q^{4} + 4 q^{7} + 10 q^{9}+O(q^{10})$$ 26 * q + 4 * q^3 - 22 * q^4 + 4 * q^7 + 10 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$26 q + 4 q^{3} - 22 q^{4} + 4 q^{7} + 10 q^{9} - 8 q^{11} + 8 q^{12} + 14 q^{13} + 4 q^{14} + 6 q^{16} + 16 q^{21} - 8 q^{22} + 4 q^{23} + 6 q^{26} + 4 q^{27} - 8 q^{28} + 8 q^{31} + 32 q^{34} - 22 q^{36} - 16 q^{37} - 8 q^{38} + 16 q^{39} - 6 q^{41} - 4 q^{42} - 12 q^{43} - 32 q^{46} + 36 q^{47} - 4 q^{48} - 26 q^{52} - 14 q^{53} - 32 q^{56} - 12 q^{57} - 58 q^{58} + 18 q^{61} + 28 q^{62} + 60 q^{63} - 30 q^{64} + 20 q^{66} + 32 q^{67} - 12 q^{69} + 20 q^{76} - 56 q^{78} - 4 q^{79} - 86 q^{81} + 58 q^{82} + 60 q^{83} - 76 q^{84} + 12 q^{87} + 68 q^{88} + 46 q^{89} - 28 q^{92} - 8 q^{93} + 8 q^{97} - 34 q^{98} - 36 q^{99}+O(q^{100})$$ 26 * q + 4 * q^3 - 22 * q^4 + 4 * q^7 + 10 * q^9 - 8 * q^11 + 8 * q^12 + 14 * q^13 + 4 * q^14 + 6 * q^16 + 16 * q^21 - 8 * q^22 + 4 * q^23 + 6 * q^26 + 4 * q^27 - 8 * q^28 + 8 * q^31 + 32 * q^34 - 22 * q^36 - 16 * q^37 - 8 * q^38 + 16 * q^39 - 6 * q^41 - 4 * q^42 - 12 * q^43 - 32 * q^46 + 36 * q^47 - 4 * q^48 - 26 * q^52 - 14 * q^53 - 32 * q^56 - 12 * q^57 - 58 * q^58 + 18 * q^61 + 28 * q^62 + 60 * q^63 - 30 * q^64 + 20 * q^66 + 32 * q^67 - 12 * q^69 + 20 * q^76 - 56 * q^78 - 4 * q^79 - 86 * q^81 + 58 * q^82 + 60 * q^83 - 76 * q^84 + 12 * q^87 + 68 * q^88 + 46 * q^89 - 28 * q^92 - 8 * q^93 + 8 * q^97 - 34 * q^98 - 36 * 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 2.26693i 2.65401 −3.13899 0 6.01647i 2.59753 + 2.59753i 2.58201i 4.04378 0
157.2 2.23019i −1.25170 −2.97373 0 2.79153i −0.483409 0.483409i 2.17160i −1.43324 0
157.3 1.41066i −2.58872 0.0100302 0 3.65181i 0.510628 + 0.510628i 2.83547i 3.70148 0
157.4 1.26373i 0.913274 0.402981 0 1.15413i −2.03055 2.03055i 3.03672i −2.16593 0
157.5 0.895351i 2.11245 1.19835 0 1.89138i −1.38465 1.38465i 2.86364i 1.46244 0
157.6 0.222351i −1.02589 1.95056 0 0.228107i 2.35964 + 2.35964i 0.878412i −1.94756 0
157.7 0.342532i 2.64611 1.88267 0 0.906377i 1.55474 + 1.55474i 1.32994i 4.00189 0
157.8 0.839004i −0.711801 1.29607 0 0.597203i −1.13987 1.13987i 2.76542i −2.49334 0
157.9 1.36192i 0.228160 0.145179 0 0.310736i −3.45046 3.45046i 2.92156i −2.94794 0
157.10 1.77873i 1.38965 −1.16389 0 2.47181i 3.41296 + 3.41296i 1.48721i −1.06888 0
157.11 1.82099i −2.59340 −1.31599 0 4.72256i 0.820621 + 0.820621i 1.24557i 3.72575 0
157.12 2.41122i 1.85973 −3.81399 0 4.48422i 0.291676 + 0.291676i 4.37394i 0.458586 0
157.13 2.73482i −1.63186 −5.47925 0 4.46285i −1.05887 1.05887i 9.51511i −0.337028 0
568.1 2.73482i −1.63186 −5.47925 0 4.46285i −1.05887 + 1.05887i 9.51511i −0.337028 0
568.2 2.41122i 1.85973 −3.81399 0 4.48422i 0.291676 0.291676i 4.37394i 0.458586 0
568.3 1.82099i −2.59340 −1.31599 0 4.72256i 0.820621 0.820621i 1.24557i 3.72575 0
568.4 1.77873i 1.38965 −1.16389 0 2.47181i 3.41296 3.41296i 1.48721i −1.06888 0
568.5 1.36192i 0.228160 0.145179 0 0.310736i −3.45046 + 3.45046i 2.92156i −2.94794 0
568.6 0.839004i −0.711801 1.29607 0 0.597203i −1.13987 + 1.13987i 2.76542i −2.49334 0
568.7 0.342532i 2.64611 1.88267 0 0.906377i 1.55474 1.55474i 1.32994i 4.00189 0
See all 26 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 157.13 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
145.e 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.c 26
5.b even 2 1 145.2.e.a 26
5.c odd 4 1 145.2.j.a yes 26
5.c odd 4 1 725.2.j.c 26
29.c odd 4 1 725.2.j.c 26
145.e even 4 1 inner 725.2.e.c 26
145.f odd 4 1 145.2.j.a yes 26
145.j even 4 1 145.2.e.a 26

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
145.2.e.a 26 5.b even 2 1
145.2.e.a 26 145.j even 4 1
145.2.j.a yes 26 5.c odd 4 1
145.2.j.a yes 26 145.f odd 4 1
725.2.e.c 26 1.a even 1 1 trivial
725.2.e.c 26 145.e even 4 1 inner
725.2.j.c 26 5.c odd 4 1
725.2.j.c 26 29.c odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{26} + 37 T_{2}^{24} + 598 T_{2}^{22} + 5562 T_{2}^{20} + 33015 T_{2}^{18} + 131099 T_{2}^{16} + \cdots + 225$$ acting on $$S_{2}^{\mathrm{new}}(725, [\chi])$$.