Properties

Label 861.2.l.a
Level $861$
Weight $2$
Character orbit 861.l
Analytic conductor $6.875$
Analytic rank $0$
Dimension $216$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [861,2,Mod(419,861)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(861, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 2, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("861.419");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 861 = 3 \cdot 7 \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 861.l (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: \(6.87511961403\)
Analytic rank: \(0\)
Dimension: \(216\)
Relative dimension: \(108\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 216 q + 192 q^{4} - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 216 q + 192 q^{4} - 4 q^{7} + 20 q^{15} + 144 q^{16} - 24 q^{18} - 56 q^{22} - 200 q^{25} - 40 q^{28} + 32 q^{30} + 16 q^{37} + 4 q^{42} - 16 q^{51} - 64 q^{57} - 32 q^{58} + 40 q^{60} - 6 q^{63} + 48 q^{64} - 48 q^{67} + 48 q^{70} - 92 q^{72} + 28 q^{78} + 8 q^{79} - 120 q^{81} + 16 q^{85} - 144 q^{88} - 16 q^{93} + 56 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
419.1 −2.76490 −1.28288 + 1.16371i 5.64469 3.67998i 3.54703 3.21755i −1.30191 2.30327i −10.0772 0.291548 2.98580i 10.1748i
419.2 −2.76490 1.28288 1.16371i 5.64469 3.67998i −3.54703 + 3.21755i −2.30327 1.30191i −10.0772 0.291548 2.98580i 10.1748i
419.3 −2.54686 −1.08190 + 1.35258i 4.48648 3.20407i 2.75545 3.44484i 2.56048 0.666289i −6.33272 −0.658969 2.92673i 8.16030i
419.4 −2.54686 1.08190 1.35258i 4.48648 3.20407i −2.75545 + 3.44484i −0.666289 + 2.56048i −6.33272 −0.658969 2.92673i 8.16030i
419.5 −2.54534 −1.72582 0.146743i 4.47878 1.45380i 4.39282 + 0.373512i −2.11962 + 1.58342i −6.30935 2.95693 + 0.506505i 3.70042i
419.6 −2.54534 1.72582 + 0.146743i 4.47878 1.45380i −4.39282 0.373512i 1.58342 2.11962i −6.30935 2.95693 + 0.506505i 3.70042i
419.7 −2.51614 −0.724750 1.57313i 4.33098 1.53774i 1.82358 + 3.95822i −2.26225 1.37194i −5.86509 −1.94947 + 2.28025i 3.86917i
419.8 −2.51614 0.724750 + 1.57313i 4.33098 1.53774i −1.82358 3.95822i −1.37194 2.26225i −5.86509 −1.94947 + 2.28025i 3.86917i
419.9 −2.49211 −1.73194 0.0198358i 4.21059 0.195467i 4.31617 + 0.0494328i 2.64473 0.0733311i −5.50903 2.99921 + 0.0687086i 0.487125i
419.10 −2.49211 1.73194 + 0.0198358i 4.21059 0.195467i −4.31617 0.0494328i −0.0733311 + 2.64473i −5.50903 2.99921 + 0.0687086i 0.487125i
419.11 −2.39321 −1.20157 1.24749i 3.72747 2.99101i 2.87562 + 2.98550i 1.32481 + 2.29017i −4.13422 −0.112448 + 2.99789i 7.15812i
419.12 −2.39321 1.20157 + 1.24749i 3.72747 2.99101i −2.87562 2.98550i 2.29017 + 1.32481i −4.13422 −0.112448 + 2.99789i 7.15812i
419.13 −2.13419 −0.283105 + 1.70876i 2.55477 0.157669i 0.604199 3.64682i −2.21993 + 1.43942i −1.18399 −2.83970 0.967514i 0.336495i
419.14 −2.13419 0.283105 1.70876i 2.55477 0.157669i −0.604199 + 3.64682i 1.43942 2.21993i −1.18399 −2.83970 0.967514i 0.336495i
419.15 −2.09764 −0.839690 + 1.51490i 2.40008 2.02468i 1.76136 3.17771i 1.67336 + 2.04936i −0.839219 −1.58984 2.54409i 4.24705i
419.16 −2.09764 0.839690 1.51490i 2.40008 2.02468i −1.76136 + 3.17771i 2.04936 + 1.67336i −0.839219 −1.58984 2.54409i 4.24705i
419.17 −2.06720 −1.24187 1.20738i 2.27332 4.31335i 2.56719 + 2.49590i 0.905579 2.48595i −0.565003 0.0844609 + 2.99881i 8.91656i
419.18 −2.06720 1.24187 + 1.20738i 2.27332 4.31335i −2.56719 2.49590i −2.48595 + 0.905579i −0.565003 0.0844609 + 2.99881i 8.91656i
419.19 −2.04053 −0.410008 1.68282i 2.16376 2.10095i 0.836632 + 3.43385i −2.14224 + 1.55268i −0.334148 −2.66379 + 1.37994i 4.28704i
419.20 −2.04053 0.410008 + 1.68282i 2.16376 2.10095i −0.836632 3.43385i 1.55268 2.14224i −0.334148 −2.66379 + 1.37994i 4.28704i
See next 80 embeddings (of 216 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 419.108
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
21.c even 2 1 inner
41.c even 4 1 inner
123.f odd 4 1 inner
287.g odd 4 1 inner
861.l even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 861.2.l.a 216
3.b odd 2 1 inner 861.2.l.a 216
7.b odd 2 1 inner 861.2.l.a 216
21.c even 2 1 inner 861.2.l.a 216
41.c even 4 1 inner 861.2.l.a 216
123.f odd 4 1 inner 861.2.l.a 216
287.g odd 4 1 inner 861.2.l.a 216
861.l even 4 1 inner 861.2.l.a 216
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
861.2.l.a 216 1.a even 1 1 trivial
861.2.l.a 216 3.b odd 2 1 inner
861.2.l.a 216 7.b odd 2 1 inner
861.2.l.a 216 21.c even 2 1 inner
861.2.l.a 216 41.c even 4 1 inner
861.2.l.a 216 123.f odd 4 1 inner
861.2.l.a 216 287.g odd 4 1 inner
861.2.l.a 216 861.l even 4 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(861, [\chi])\).