# Properties

 Label 841.2.e.j Level $841$ Weight $2$ Character orbit 841.e Analytic conductor $6.715$ Analytic rank $0$ Dimension $24$ Inner twists $12$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [841,2,Mod(63,841)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(841, base_ring=CyclotomicField(14))

chi = DirichletCharacter(H, H._module([11]))

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

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

chi := DirichletCharacter("841.63");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$841 = 29^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 841.e (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: $$6.71541880999$$ Analytic rank: $$0$$ Dimension: $$24$$ Relative dimension: $$4$$ over $$\Q(\zeta_{14})$$ Twist minimal: yes 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.

 $$\operatorname{Tr}(f)(q) =$$ $$24 q - 2 q^{4} + 2 q^{5} - 6 q^{6} - 6 q^{9}+O(q^{10})$$ 24 * q - 2 * q^4 + 2 * q^5 - 6 * q^6 - 6 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$24 q - 2 q^{4} + 2 q^{5} - 6 q^{6} - 6 q^{9} + 8 q^{13} + 6 q^{16} + 16 q^{20} + 10 q^{22} - 4 q^{23} - 10 q^{24} - 26 q^{25} - 60 q^{28} + 72 q^{30} - 10 q^{33} + 16 q^{34} - 30 q^{35} - 8 q^{36} - 12 q^{38} - 10 q^{42} + 18 q^{45} + 8 q^{49} - 16 q^{51} - 6 q^{52} + 8 q^{53} - 22 q^{54} - 72 q^{57} + 12 q^{59} - 16 q^{62} + 10 q^{63} + 8 q^{64} + 26 q^{65} - 24 q^{67} + 24 q^{71} - 34 q^{74} + 22 q^{78} + 42 q^{80} + 4 q^{81} - 14 q^{82} + 4 q^{83} - 60 q^{88} + 20 q^{91} + 8 q^{92} + 16 q^{93} - 14 q^{94} + 4 q^{96}+O(q^{100})$$ 24 * q - 2 * q^4 + 2 * q^5 - 6 * q^6 - 6 * q^9 + 8 * q^13 + 6 * q^16 + 16 * q^20 + 10 * q^22 - 4 * q^23 - 10 * q^24 - 26 * q^25 - 60 * q^28 + 72 * q^30 - 10 * q^33 + 16 * q^34 - 30 * q^35 - 8 * q^36 - 12 * q^38 - 10 * q^42 + 18 * q^45 + 8 * q^49 - 16 * q^51 - 6 * q^52 + 8 * q^53 - 22 * q^54 - 72 * q^57 + 12 * q^59 - 16 * q^62 + 10 * q^63 + 8 * q^64 + 26 * q^65 - 24 * q^67 + 24 * q^71 - 34 * q^74 + 22 * q^78 + 42 * q^80 + 4 * q^81 - 14 * q^82 + 4 * q^83 - 60 * q^88 + 20 * q^91 + 8 * q^92 + 16 * q^93 - 14 * q^94 + 4 * q^96

## 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
63.1 −1.26503 + 1.00883i −1.57747 + 0.360046i 0.137526 0.602539i 1.77950 + 2.23143i 1.63232 2.04686i −0.497572 2.18001i −0.970194 2.01463i −0.344139 + 0.165729i −4.50225 1.02761i
63.2 −0.483198 + 0.385338i −0.602539 + 0.137526i −0.360046 + 1.57747i −2.40299 3.01326i 0.238152 0.298633i 0.497572 + 2.18001i −0.970194 2.01463i −2.35877 + 1.13592i 2.32225 + 0.530037i
63.3 0.483198 0.385338i 0.602539 0.137526i −0.360046 + 1.57747i −2.40299 3.01326i 0.238152 0.298633i 0.497572 + 2.18001i 0.970194 + 2.01463i −2.35877 + 1.13592i −2.32225 0.530037i
63.4 1.26503 1.00883i 1.57747 0.360046i 0.137526 0.602539i 1.77950 + 2.23143i 1.63232 2.04686i −0.497572 2.18001i 0.970194 + 2.01463i −0.344139 + 0.165729i 4.50225 + 1.02761i
196.1 −1.57747 + 0.360046i 0.702039 1.45780i 0.556829 0.268155i −0.635097 2.78254i −0.582567 + 2.55239i −2.01463 0.970194i 1.74823 1.39417i 0.238152 + 0.298633i 2.00369 + 4.16070i
196.2 −0.602539 + 0.137526i 0.268155 0.556829i −1.45780 + 0.702039i 0.857618 + 3.75747i −0.0849954 + 0.372389i 2.01463 + 0.970194i 1.74823 1.39417i 1.63232 + 2.04686i −1.03350 2.14608i
196.3 0.602539 0.137526i −0.268155 + 0.556829i −1.45780 + 0.702039i 0.857618 + 3.75747i −0.0849954 + 0.372389i 2.01463 + 0.970194i −1.74823 + 1.39417i 1.63232 + 2.04686i 1.03350 + 2.14608i
196.4 1.57747 0.360046i −0.702039 + 1.45780i 0.556829 0.268155i −0.635097 2.78254i −0.582567 + 2.55239i −2.01463 0.970194i −1.74823 + 1.39417i 0.238152 + 0.298633i −2.00369 4.16070i
236.1 −1.57747 0.360046i 0.702039 + 1.45780i 0.556829 + 0.268155i −0.635097 + 2.78254i −0.582567 2.55239i −2.01463 + 0.970194i 1.74823 + 1.39417i 0.238152 0.298633i 2.00369 4.16070i
236.2 −0.602539 0.137526i 0.268155 + 0.556829i −1.45780 0.702039i 0.857618 3.75747i −0.0849954 0.372389i 2.01463 0.970194i 1.74823 + 1.39417i 1.63232 2.04686i −1.03350 + 2.14608i
236.3 0.602539 + 0.137526i −0.268155 0.556829i −1.45780 0.702039i 0.857618 3.75747i −0.0849954 0.372389i 2.01463 0.970194i −1.74823 1.39417i 1.63232 2.04686i 1.03350 2.14608i
236.4 1.57747 + 0.360046i −0.702039 1.45780i 0.556829 + 0.268155i −0.635097 + 2.78254i −0.582567 2.55239i −2.01463 + 0.970194i −1.74823 1.39417i 0.238152 0.298633i −2.00369 + 4.16070i
267.1 −1.26503 1.00883i −1.57747 0.360046i 0.137526 + 0.602539i 1.77950 2.23143i 1.63232 + 2.04686i −0.497572 + 2.18001i −0.970194 + 2.01463i −0.344139 0.165729i −4.50225 + 1.02761i
267.2 −0.483198 0.385338i −0.602539 0.137526i −0.360046 1.57747i −2.40299 + 3.01326i 0.238152 + 0.298633i 0.497572 2.18001i −0.970194 + 2.01463i −2.35877 1.13592i 2.32225 0.530037i
267.3 0.483198 + 0.385338i 0.602539 + 0.137526i −0.360046 1.57747i −2.40299 + 3.01326i 0.238152 + 0.298633i 0.497572 2.18001i 0.970194 2.01463i −2.35877 1.13592i −2.32225 + 0.530037i
267.4 1.26503 + 1.00883i 1.57747 + 0.360046i 0.137526 + 0.602539i 1.77950 2.23143i 1.63232 + 2.04686i −0.497572 + 2.18001i 0.970194 2.01463i −0.344139 0.165729i 4.50225 1.02761i
270.1 −0.702039 1.45780i 1.26503 1.00883i −0.385338 + 0.483198i −2.57146 + 1.23835i −2.35877 1.13592i 1.39417 + 1.74823i −2.18001 0.497572i −0.0849954 + 0.372389i 3.61052 + 2.87930i
270.2 −0.268155 0.556829i 0.483198 0.385338i 1.00883 1.26503i 3.47243 1.67223i −0.344139 0.165729i −1.39417 1.74823i −2.18001 0.497572i −0.582567 + 2.55239i −1.86230 1.48513i
270.3 0.268155 + 0.556829i −0.483198 + 0.385338i 1.00883 1.26503i 3.47243 1.67223i −0.344139 0.165729i −1.39417 1.74823i 2.18001 + 0.497572i −0.582567 + 2.55239i 1.86230 + 1.48513i
270.4 0.702039 + 1.45780i −1.26503 + 1.00883i −0.385338 + 0.483198i −2.57146 + 1.23835i −2.35877 1.13592i 1.39417 + 1.74823i 2.18001 + 0.497572i −0.0849954 + 0.372389i −3.61052 2.87930i
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 63.4 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.b even 2 1 inner
29.d even 7 5 inner
29.e even 14 5 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 841.2.e.j 24
29.b even 2 1 inner 841.2.e.j 24
29.c odd 4 1 841.2.d.g 12
29.c odd 4 1 841.2.d.i 12
29.d even 7 1 841.2.b.b 4
29.d even 7 5 inner 841.2.e.j 24
29.e even 14 1 841.2.b.b 4
29.e even 14 5 inner 841.2.e.j 24
29.f odd 28 1 841.2.a.a 2
29.f odd 28 1 841.2.a.c yes 2
29.f odd 28 5 841.2.d.g 12
29.f odd 28 5 841.2.d.i 12
87.k even 28 1 7569.2.a.d 2
87.k even 28 1 7569.2.a.l 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
841.2.a.a 2 29.f odd 28 1
841.2.a.c yes 2 29.f odd 28 1
841.2.b.b 4 29.d even 7 1
841.2.b.b 4 29.e even 14 1
841.2.d.g 12 29.c odd 4 1
841.2.d.g 12 29.f odd 28 5
841.2.d.i 12 29.c odd 4 1
841.2.d.i 12 29.f odd 28 5
841.2.e.j 24 1.a even 1 1 trivial
841.2.e.j 24 29.b even 2 1 inner
841.2.e.j 24 29.d even 7 5 inner
841.2.e.j 24 29.e even 14 5 inner
7569.2.a.d 2 87.k even 28 1
7569.2.a.l 2 87.k even 28 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{24} - 3 T_{2}^{22} + 8 T_{2}^{20} - 21 T_{2}^{18} + 55 T_{2}^{16} - 144 T_{2}^{14} + 377 T_{2}^{12} + \cdots + 1$$ acting on $$S_{2}^{\mathrm{new}}(841, [\chi])$$.