# Properties

 Label 154.2.k.a Level $154$ Weight $2$ Character orbit 154.k Analytic conductor $1.230$ Analytic rank $0$ Dimension $32$ Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [154,2,Mod(13,154)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(154, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([5, 1]))

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

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

chi := DirichletCharacter("154.13");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$154 = 2 \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 154.k (of order $$10$$, degree $$4$$, 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: $$1.22969619113$$ Analytic rank: $$0$$ Dimension: $$32$$ Relative dimension: $$8$$ over $$\Q(\zeta_{10})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$32 q + 8 q^{4} + 10 q^{7} - 4 q^{9}+O(q^{10})$$ 32 * q + 8 * q^4 + 10 * q^7 - 4 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$32 q + 8 q^{4} + 10 q^{7} - 4 q^{9} - 4 q^{11} + 2 q^{14} - 24 q^{15} - 8 q^{16} + 4 q^{22} - 32 q^{23} + 20 q^{25} - 10 q^{28} - 20 q^{29} - 60 q^{35} - 16 q^{36} + 8 q^{37} - 20 q^{39} + 14 q^{42} - 16 q^{44} - 60 q^{51} + 16 q^{53} + 8 q^{56} + 80 q^{57} - 56 q^{58} - 16 q^{60} + 40 q^{63} + 8 q^{64} - 16 q^{67} + 16 q^{70} + 104 q^{71} + 40 q^{72} + 40 q^{74} - 4 q^{77} + 60 q^{79} + 60 q^{81} + 20 q^{84} - 80 q^{85} + 44 q^{86} + 16 q^{88} + 44 q^{91} - 28 q^{92} + 32 q^{93} + 100 q^{95} - 68 q^{99}+O(q^{100})$$ 32 * q + 8 * q^4 + 10 * q^7 - 4 * q^9 - 4 * q^11 + 2 * q^14 - 24 * q^15 - 8 * q^16 + 4 * q^22 - 32 * q^23 + 20 * q^25 - 10 * q^28 - 20 * q^29 - 60 * q^35 - 16 * q^36 + 8 * q^37 - 20 * q^39 + 14 * q^42 - 16 * q^44 - 60 * q^51 + 16 * q^53 + 8 * q^56 + 80 * q^57 - 56 * q^58 - 16 * q^60 + 40 * q^63 + 8 * q^64 - 16 * q^67 + 16 * q^70 + 104 * q^71 + 40 * q^72 + 40 * q^74 - 4 * q^77 + 60 * q^79 + 60 * q^81 + 20 * q^84 - 80 * q^85 + 44 * q^86 + 16 * q^88 + 44 * q^91 - 28 * q^92 + 32 * q^93 + 100 * q^95 - 68 * 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
13.1 −0.951057 0.309017i −1.71909 2.36612i 0.809017 + 0.587785i 2.56145 0.832265i 0.903778 + 2.78154i −1.23855 2.33795i −0.587785 0.809017i −1.71622 + 5.28197i −2.69327
13.2 −0.951057 0.309017i −1.15608 1.59121i 0.809017 + 0.587785i −3.16384 + 1.02799i 0.607787 + 1.87058i 0.184637 + 2.63930i −0.587785 0.809017i −0.268370 + 0.825959i 3.32666
13.3 −0.951057 0.309017i 1.15608 + 1.59121i 0.809017 + 0.587785i 3.16384 1.02799i −0.607787 1.87058i −2.56718 + 0.639989i −0.587785 0.809017i −0.268370 + 0.825959i −3.32666
13.4 −0.951057 0.309017i 1.71909 + 2.36612i 0.809017 + 0.587785i −2.56145 + 0.832265i −0.903778 2.78154i 2.60625 + 0.455461i −0.587785 0.809017i −1.71622 + 5.28197i 2.69327
13.5 0.951057 + 0.309017i −0.998766 1.37468i 0.809017 + 0.587785i 0.610430 0.198341i −0.525082 1.61604i 2.51989 0.806336i 0.587785 + 0.809017i 0.0348302 0.107196i 0.641844
13.6 0.951057 + 0.309017i −0.815841 1.12291i 0.809017 + 0.587785i 2.77340 0.901132i −0.428913 1.32006i −2.57794 + 0.595147i 0.587785 + 0.809017i 0.331723 1.02094i 2.91613
13.7 0.951057 + 0.309017i 0.815841 + 1.12291i 0.809017 + 0.587785i −2.77340 + 0.901132i 0.428913 + 1.32006i 0.230610 + 2.63568i 0.587785 + 0.809017i 0.331723 1.02094i −2.91613
13.8 0.951057 + 0.309017i 0.998766 + 1.37468i 0.809017 + 0.587785i −0.610430 + 0.198341i 0.525082 + 1.61604i −0.0118166 2.64572i 0.587785 + 0.809017i 0.0348302 0.107196i −0.641844
41.1 −0.587785 0.809017i −2.66489 + 0.865874i −0.309017 + 0.951057i 0.784057 1.07916i 2.26689 + 1.64699i 2.53536 + 0.756276i 0.951057 0.309017i 3.92483 2.85155i −1.33392
41.2 −0.587785 0.809017i −0.893059 + 0.290172i −0.309017 + 0.951057i −0.830043 + 1.14246i 0.759681 + 0.551941i −1.37230 + 2.26203i 0.951057 0.309017i −1.71370 + 1.24507i 1.41215
41.3 −0.587785 0.809017i 0.893059 0.290172i −0.309017 + 0.951057i 0.830043 1.14246i −0.759681 0.551941i 0.219373 2.63664i 0.951057 0.309017i −1.71370 + 1.24507i −1.41215
41.4 −0.587785 0.809017i 2.66489 0.865874i −0.309017 + 0.951057i −0.784057 + 1.07916i −2.26689 1.64699i 2.49568 + 0.878407i 0.951057 0.309017i 3.92483 2.85155i 1.33392
41.5 0.587785 + 0.809017i −1.88362 + 0.612024i −0.309017 + 0.951057i −0.402252 + 0.553653i −1.60230 1.16414i −1.89097 + 1.85047i −0.951057 + 0.309017i 0.746390 0.542284i −0.684352
41.6 0.587785 + 0.809017i −0.312890 + 0.101664i −0.309017 + 0.951057i 2.52967 3.48179i −0.266160 0.193377i 1.73223 + 1.99984i −0.951057 + 0.309017i −2.33949 + 1.69974i 4.30373
41.7 0.587785 + 0.809017i 0.312890 0.101664i −0.309017 + 0.951057i −2.52967 + 3.48179i 0.266160 + 0.193377i 2.57688 0.599728i −0.951057 + 0.309017i −2.33949 + 1.69974i −4.30373
41.8 0.587785 + 0.809017i 1.88362 0.612024i −0.309017 + 0.951057i 0.402252 0.553653i 1.60230 + 1.16414i −0.442149 2.60854i −0.951057 + 0.309017i 0.746390 0.542284i 0.684352
83.1 −0.951057 + 0.309017i −1.71909 + 2.36612i 0.809017 0.587785i 2.56145 + 0.832265i 0.903778 2.78154i −1.23855 + 2.33795i −0.587785 + 0.809017i −1.71622 5.28197i −2.69327
83.2 −0.951057 + 0.309017i −1.15608 + 1.59121i 0.809017 0.587785i −3.16384 1.02799i 0.607787 1.87058i 0.184637 2.63930i −0.587785 + 0.809017i −0.268370 0.825959i 3.32666
83.3 −0.951057 + 0.309017i 1.15608 1.59121i 0.809017 0.587785i 3.16384 + 1.02799i −0.607787 + 1.87058i −2.56718 0.639989i −0.587785 + 0.809017i −0.268370 0.825959i −3.32666
83.4 −0.951057 + 0.309017i 1.71909 2.36612i 0.809017 0.587785i −2.56145 0.832265i −0.903778 + 2.78154i 2.60625 0.455461i −0.587785 + 0.809017i −1.71622 5.28197i 2.69327
See all 32 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 13.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
11.d odd 10 1 inner
77.l even 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 154.2.k.a 32
7.b odd 2 1 inner 154.2.k.a 32
11.c even 5 1 1694.2.c.c 32
11.d odd 10 1 inner 154.2.k.a 32
11.d odd 10 1 1694.2.c.c 32
77.j odd 10 1 1694.2.c.c 32
77.l even 10 1 inner 154.2.k.a 32
77.l even 10 1 1694.2.c.c 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.2.k.a 32 1.a even 1 1 trivial
154.2.k.a 32 7.b odd 2 1 inner
154.2.k.a 32 11.d odd 10 1 inner
154.2.k.a 32 77.l even 10 1 inner
1694.2.c.c 32 11.c even 5 1
1694.2.c.c 32 11.d odd 10 1
1694.2.c.c 32 77.j odd 10 1
1694.2.c.c 32 77.l even 10 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(154, [\chi])$$.