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Properties

Label 841.2.e.k
Level $841$
Weight $2$
Character orbit 841.e
Analytic conductor $6.715$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $12$

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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: no (minimal twist has level 29)
Sato-Tate group: $\mathrm{SU}(2)[C_{14}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 4 q^{4} - 4 q^{5} - 12 q^{6}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 4 q^{4} - 4 q^{5} - 12 q^{6} - 4 q^{13} - 12 q^{16} + 4 q^{20} + 4 q^{22} + 8 q^{23} + 20 q^{24} + 16 q^{25} + 192 q^{28} + 72 q^{30} - 4 q^{33} - 8 q^{34} - 32 q^{36} - 24 q^{38} + 32 q^{42} - 4 q^{49} + 8 q^{51} + 36 q^{52} - 4 q^{53} - 4 q^{54} - 144 q^{57} + 48 q^{59} - 52 q^{62} - 32 q^{63} - 28 q^{64} - 4 q^{65} - 24 q^{71} - 16 q^{74} - 44 q^{78} - 12 q^{80} + 4 q^{81} - 32 q^{82} - 8 q^{83} + 72 q^{86} - 24 q^{88} + 32 q^{91} + 56 q^{92} + 52 q^{93} - 20 q^{94} + 4 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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 −1.88751 + 1.50524i −2.35368 + 0.537213i 0.851905 − 3.73244i 0.623490 + 0.781831i 3.63396 − 4.55685i 0.629384 + 2.75751i 1.91526 + 3.97707i 2.54832 − 1.22721i −2.35368 − 0.537213i
63.2 −0.323845 + 0.258258i −0.403828 + 0.0921712i −0.406863 + 1.78258i 0.623490 + 0.781831i 0.106974 − 0.134141i −0.629384 − 2.75751i −0.688047 − 1.42874i −2.54832 + 1.22721i −0.403828 − 0.0921712i
63.3 0.323845 − 0.258258i 0.403828 − 0.0921712i −0.406863 + 1.78258i 0.623490 + 0.781831i 0.106974 − 0.134141i −0.629384 − 2.75751i 0.688047 + 1.42874i −2.54832 + 1.22721i 0.403828 + 0.0921712i
63.4 1.88751 − 1.50524i 2.35368 − 0.537213i 0.851905 − 3.73244i 0.623490 + 0.781831i 3.63396 − 4.55685i 0.629384 + 2.75751i −1.91526 − 3.97707i 2.54832 − 1.22721i 2.35368 + 0.537213i
196.1 −2.35368 + 0.537213i 1.04749 − 2.17513i 3.44929 − 1.66109i −0.222521 − 0.974928i −1.29695 + 5.68230i 2.54832 + 1.22721i −3.45117 + 2.75222i −1.76350 − 2.21135i 1.04749 + 2.17513i
196.2 −0.403828 + 0.0921712i 0.179721 − 0.373194i −1.64736 + 0.793325i −0.222521 − 0.974928i −0.0381786 + 0.167271i −2.54832 − 1.22721i 1.23982 − 0.988722i 1.76350 + 2.21135i 0.179721 + 0.373194i
196.3 0.403828 − 0.0921712i −0.179721 + 0.373194i −1.64736 + 0.793325i −0.222521 − 0.974928i −0.0381786 + 0.167271i −2.54832 − 1.22721i −1.23982 + 0.988722i 1.76350 + 2.21135i −0.179721 − 0.373194i
196.4 2.35368 − 0.537213i −1.04749 + 2.17513i 3.44929 − 1.66109i −0.222521 − 0.974928i −1.29695 + 5.68230i 2.54832 + 1.22721i 3.45117 − 2.75222i −1.76350 − 2.21135i −1.04749 − 2.17513i
236.1 −2.35368 − 0.537213i 1.04749 + 2.17513i 3.44929 + 1.66109i −0.222521 + 0.974928i −1.29695 − 5.68230i 2.54832 − 1.22721i −3.45117 − 2.75222i −1.76350 + 2.21135i 1.04749 − 2.17513i
236.2 −0.403828 − 0.0921712i 0.179721 + 0.373194i −1.64736 − 0.793325i −0.222521 + 0.974928i −0.0381786 − 0.167271i −2.54832 + 1.22721i 1.23982 + 0.988722i 1.76350 − 2.21135i 0.179721 − 0.373194i
236.3 0.403828 + 0.0921712i −0.179721 − 0.373194i −1.64736 − 0.793325i −0.222521 + 0.974928i −0.0381786 − 0.167271i −2.54832 + 1.22721i −1.23982 − 0.988722i 1.76350 − 2.21135i −0.179721 + 0.373194i
236.4 2.35368 + 0.537213i −1.04749 − 2.17513i 3.44929 + 1.66109i −0.222521 + 0.974928i −1.29695 − 5.68230i 2.54832 − 1.22721i 3.45117 + 2.75222i −1.76350 + 2.21135i −1.04749 + 2.17513i
267.1 −1.88751 − 1.50524i −2.35368 − 0.537213i 0.851905 + 3.73244i 0.623490 − 0.781831i 3.63396 + 4.55685i 0.629384 − 2.75751i 1.91526 − 3.97707i 2.54832 + 1.22721i −2.35368 + 0.537213i
267.2 −0.323845 − 0.258258i −0.403828 − 0.0921712i −0.406863 − 1.78258i 0.623490 − 0.781831i 0.106974 + 0.134141i −0.629384 + 2.75751i −0.688047 + 1.42874i −2.54832 − 1.22721i −0.403828 + 0.0921712i
267.3 0.323845 + 0.258258i 0.403828 + 0.0921712i −0.406863 − 1.78258i 0.623490 − 0.781831i 0.106974 + 0.134141i −0.629384 + 2.75751i 0.688047 − 1.42874i −2.54832 − 1.22721i 0.403828 − 0.0921712i
267.4 1.88751 + 1.50524i 2.35368 + 0.537213i 0.851905 + 3.73244i 0.623490 − 0.781831i 3.63396 + 4.55685i 0.629384 − 2.75751i −1.91526 + 3.97707i 2.54832 + 1.22721i 2.35368 − 0.537213i
270.1 −1.04749 − 2.17513i 1.88751 − 1.50524i −2.38699 + 2.99318i −0.900969 + 0.433884i −5.25123 − 2.52886i −1.76350 − 2.21135i 4.30354 + 0.982255i 0.629384 − 2.75751i 1.88751 + 1.50524i
270.2 −0.179721 − 0.373194i 0.323845 − 0.258258i 1.14001 − 1.42952i −0.900969 + 0.433884i −0.154582 − 0.0744427i 1.76350 + 2.21135i −1.54603 − 0.352871i −0.629384 + 2.75751i 0.323845 + 0.258258i
270.3 0.179721 + 0.373194i −0.323845 + 0.258258i 1.14001 − 1.42952i −0.900969 + 0.433884i −0.154582 − 0.0744427i 1.76350 + 2.21135i 1.54603 + 0.352871i −0.629384 + 2.75751i −0.323845 − 0.258258i
270.4 1.04749 + 2.17513i −1.88751 + 1.50524i −2.38699 + 2.99318i −0.900969 + 0.433884i −5.25123 − 2.52886i −1.76350 − 2.21135i −4.30354 − 0.982255i 0.629384 − 2.75751i −1.88751 − 1.50524i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 63.4
Significant digits:
Format:

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.k 24
29.b even 2 1 inner 841.2.e.k 24
29.c odd 4 1 841.2.d.f 12
29.c odd 4 1 841.2.d.j 12
29.d even 7 1 841.2.b.a 4
29.d even 7 5 inner 841.2.e.k 24
29.e even 14 1 841.2.b.a 4
29.e even 14 5 inner 841.2.e.k 24
29.f odd 28 1 29.2.a.a ✓ 2
29.f odd 28 1 841.2.a.d 2
29.f odd 28 5 841.2.d.f 12
29.f odd 28 5 841.2.d.j 12
87.k even 28 1 261.2.a.d 2
87.k even 28 1 7569.2.a.c 2
116.l even 28 1 464.2.a.h 2
145.o even 28 1 725.2.b.b 4
145.s odd 28 1 725.2.a.b 2
145.t even 28 1 725.2.b.b 4
203.r even 28 1 1421.2.a.j 2
232.u odd 28 1 1856.2.a.r 2
232.v even 28 1 1856.2.a.w 2
319.q even 28 1 3509.2.a.j 2
348.v odd 28 1 4176.2.a.bq 2
377.bb odd 28 1 4901.2.a.g 2
435.bk even 28 1 6525.2.a.o 2
493.y odd 28 1 8381.2.a.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
29.2.a.a ✓ 2 29.f odd 28 1
261.2.a.d 2 87.k even 28 1
464.2.a.h 2 116.l even 28 1
725.2.a.b 2 145.s odd 28 1
725.2.b.b 4 145.o even 28 1
725.2.b.b 4 145.t even 28 1
841.2.a.d 2 29.f odd 28 1
841.2.b.a 4 29.d even 7 1
841.2.b.a 4 29.e even 14 1
841.2.d.f 12 29.c odd 4 1
841.2.d.f 12 29.f odd 28 5
841.2.d.j 12 29.c odd 4 1
841.2.d.j 12 29.f odd 28 5
841.2.e.k 24 1.a even 1 1 trivial
841.2.e.k 24 29.b even 2 1 inner
841.2.e.k 24 29.d even 7 5 inner
841.2.e.k 24 29.e even 14 5 inner
1421.2.a.j 2 203.r even 28 1
1856.2.a.r 2 232.u odd 28 1
1856.2.a.w 2 232.v even 28 1
3509.2.a.j 2 319.q even 28 1
4176.2.a.bq 2 348.v odd 28 1
4901.2.a.g 2 377.bb odd 28 1
6525.2.a.o 2 435.bk even 28 1
7569.2.a.c 2 87.k even 28 1
8381.2.a.e 2 493.y odd 28 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{24} - 6 T_{2}^{22} + 35 T_{2}^{20} - 204 T_{2}^{18} + 1189 T_{2}^{16} - 6930 T_{2}^{14} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(841, [\chi])\). Copy content Toggle raw display