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Properties

Label 847.2.i.b
Level $847$
Weight $2$
Character orbit 847.i
Analytic conductor $6.763$
Analytic rank $0$
Dimension $48$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [847,2,Mod(241,847)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(847, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([1, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("847.241");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 847 = 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 847.i (of order \(6\), 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: \(6.76332905120\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(24\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 77)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 6 q^{3} + 16 q^{4} + 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q + 6 q^{3} + 16 q^{4} + 14 q^{9} - 12 q^{12} + 32 q^{14} + 20 q^{15} + 8 q^{16} + 10 q^{23} - 24 q^{25} - 78 q^{26} - 6 q^{31} + 72 q^{36} - 36 q^{37} - 102 q^{38} + 44 q^{42} - 84 q^{45} - 12 q^{47} - 46 q^{49} + 12 q^{53} - 8 q^{56} - 74 q^{58} + 132 q^{59} + 20 q^{60} + 12 q^{64} + 44 q^{67} + 38 q^{70} - 100 q^{71} - 42 q^{75} + 92 q^{78} + 90 q^{80} - 4 q^{81} + 66 q^{82} - 22 q^{86} + 6 q^{89} + 58 q^{91} + 80 q^{92} + 68 q^{93}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
241.1 −2.28265 + 1.31789i −1.38566 − 0.800011i 2.47367 − 4.28452i −1.85953 + 1.07360i 4.21731 2.40915 − 1.09362i 7.76855i −0.219966 − 0.380993i 2.82977 − 4.90131i
241.2 −2.04120 + 1.17849i 1.74857 + 1.00954i 1.77766 − 3.07900i 0.0257408 − 0.0148614i −4.75890 −2.64277 − 0.125567i 3.66586i 0.538327 + 0.932409i −0.0350280 + 0.0606703i
241.3 −1.79811 + 1.03814i −0.161824 − 0.0934290i 1.15547 − 2.00133i 2.22195 − 1.28284i 0.387969 −1.08720 + 2.41205i 0.645585i −1.48254 − 2.56784i −2.66354 + 4.61339i
241.4 −1.79299 + 1.03518i −2.72051 − 1.57069i 1.14321 − 1.98009i −1.02883 + 0.593996i 6.50379 −0.998358 − 2.45016i 0.592974i 3.43411 + 5.94805i 1.22979 − 2.13006i
241.5 −1.55412 + 0.897271i 2.50180 + 1.44441i 0.610191 − 1.05688i −1.74224 + 1.00588i −5.18412 −0.444263 + 2.60819i − 1.39906i 2.67266 + 4.62919i 1.80510 − 3.12652i
241.6 −1.29666 + 0.748626i −0.522378 − 0.301595i 0.120882 − 0.209374i 2.04664 − 1.18163i 0.903128 −2.63692 + 0.216008i − 2.63252i −1.31808 − 2.28298i −1.76920 + 3.06434i
241.7 −1.21617 + 0.702153i 0.177127 + 0.102264i −0.0139610 + 0.0241811i −1.31641 + 0.760032i −0.287221 1.53513 + 2.15485i − 2.84782i −1.47908 − 2.56185i 1.06732 − 1.84865i
241.8 −1.13445 + 0.654973i 2.36543 + 1.36568i −0.142020 + 0.245986i −0.354392 + 0.204609i −3.57794 1.58052 − 2.12178i − 2.99197i 2.23017 + 3.86277i 0.268026 − 0.464235i
241.9 −0.967760 + 0.558737i 1.35168 + 0.780393i −0.375626 + 0.650604i 2.66674 − 1.53964i −1.74414 −1.90251 − 1.83861i − 3.07445i −0.281972 − 0.488390i −1.72051 + 2.98001i
241.10 −0.450429 + 0.260055i −0.981461 − 0.566647i −0.864743 + 1.49778i −2.82673 + 1.63201i 0.589437 −0.556745 − 2.58651i − 1.93974i −0.857823 − 1.48579i 0.848825 − 1.47021i
241.11 −0.415153 + 0.239689i −1.93896 − 1.11946i −0.885099 + 1.53304i 1.16505 − 0.672641i 1.07328 −1.47018 − 2.19967i − 1.80735i 1.00637 + 1.74308i −0.322449 + 0.558497i
241.12 −0.0235483 + 0.0135956i 1.06618 + 0.615561i −0.999630 + 1.73141i 1.00201 − 0.578511i −0.0334757 1.78515 − 1.95275i − 0.108745i −0.742170 − 1.28548i −0.0157304 + 0.0272459i
241.13 0.0235483 − 0.0135956i 1.06618 + 0.615561i −0.999630 + 1.73141i 1.00201 − 0.578511i 0.0334757 −1.78515 + 1.95275i 0.108745i −0.742170 − 1.28548i 0.0157304 − 0.0272459i
241.14 0.415153 − 0.239689i −1.93896 − 1.11946i −0.885099 + 1.53304i 1.16505 − 0.672641i −1.07328 1.47018 + 2.19967i 1.80735i 1.00637 + 1.74308i 0.322449 − 0.558497i
241.15 0.450429 − 0.260055i −0.981461 − 0.566647i −0.864743 + 1.49778i −2.82673 + 1.63201i −0.589437 0.556745 + 2.58651i 1.93974i −0.857823 − 1.48579i −0.848825 + 1.47021i
241.16 0.967760 − 0.558737i 1.35168 + 0.780393i −0.375626 + 0.650604i 2.66674 − 1.53964i 1.74414 1.90251 + 1.83861i 3.07445i −0.281972 − 0.488390i 1.72051 − 2.98001i
241.17 1.13445 − 0.654973i 2.36543 + 1.36568i −0.142020 + 0.245986i −0.354392 + 0.204609i 3.57794 −1.58052 + 2.12178i 2.99197i 2.23017 + 3.86277i −0.268026 + 0.464235i
241.18 1.21617 − 0.702153i 0.177127 + 0.102264i −0.0139610 + 0.0241811i −1.31641 + 0.760032i 0.287221 −1.53513 − 2.15485i 2.84782i −1.47908 − 2.56185i −1.06732 + 1.84865i
241.19 1.29666 − 0.748626i −0.522378 − 0.301595i 0.120882 − 0.209374i 2.04664 − 1.18163i −0.903128 2.63692 − 0.216008i 2.63252i −1.31808 − 2.28298i 1.76920 − 3.06434i
241.20 1.55412 − 0.897271i 2.50180 + 1.44441i 0.610191 − 1.05688i −1.74224 + 1.00588i 5.18412 0.444263 − 2.60819i 1.39906i 2.67266 + 4.62919i −1.80510 + 3.12652i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 241.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner
11.b odd 2 1 inner
77.i even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 847.2.i.b 48
7.d odd 6 1 inner 847.2.i.b 48
11.b odd 2 1 inner 847.2.i.b 48
11.c even 5 1 77.2.n.a ✓ 48
11.c even 5 1 847.2.r.a 48
11.c even 5 1 847.2.r.c 48
11.c even 5 1 847.2.r.d 48
11.d odd 10 1 77.2.n.a ✓ 48
11.d odd 10 1 847.2.r.a 48
11.d odd 10 1 847.2.r.c 48
11.d odd 10 1 847.2.r.d 48
33.f even 10 1 693.2.cg.a 48
33.h odd 10 1 693.2.cg.a 48
77.i even 6 1 inner 847.2.i.b 48
77.j odd 10 1 539.2.s.d 48
77.l even 10 1 539.2.s.d 48
77.m even 15 1 539.2.m.a 48
77.m even 15 1 539.2.s.d 48
77.n even 30 1 77.2.n.a ✓ 48
77.n even 30 1 539.2.m.a 48
77.n even 30 1 847.2.r.a 48
77.n even 30 1 847.2.r.c 48
77.n even 30 1 847.2.r.d 48
77.o odd 30 1 539.2.m.a 48
77.o odd 30 1 539.2.s.d 48
77.p odd 30 1 77.2.n.a ✓ 48
77.p odd 30 1 539.2.m.a 48
77.p odd 30 1 847.2.r.a 48
77.p odd 30 1 847.2.r.c 48
77.p odd 30 1 847.2.r.d 48
231.bc even 30 1 693.2.cg.a 48
231.bf odd 30 1 693.2.cg.a 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.n.a ✓ 48 11.c even 5 1
77.2.n.a ✓ 48 11.d odd 10 1
77.2.n.a ✓ 48 77.n even 30 1
77.2.n.a ✓ 48 77.p odd 30 1
539.2.m.a 48 77.m even 15 1
539.2.m.a 48 77.n even 30 1
539.2.m.a 48 77.o odd 30 1
539.2.m.a 48 77.p odd 30 1
539.2.s.d 48 77.j odd 10 1
539.2.s.d 48 77.l even 10 1
539.2.s.d 48 77.m even 15 1
539.2.s.d 48 77.o odd 30 1
693.2.cg.a 48 33.f even 10 1
693.2.cg.a 48 33.h odd 10 1
693.2.cg.a 48 231.bc even 30 1
693.2.cg.a 48 231.bf odd 30 1
847.2.i.b 48 1.a even 1 1 trivial
847.2.i.b 48 7.d odd 6 1 inner
847.2.i.b 48 11.b odd 2 1 inner
847.2.i.b 48 77.i even 6 1 inner
847.2.r.a 48 11.c even 5 1
847.2.r.a 48 11.d odd 10 1
847.2.r.a 48 77.n even 30 1
847.2.r.a 48 77.p odd 30 1
847.2.r.c 48 11.c even 5 1
847.2.r.c 48 11.d odd 10 1
847.2.r.c 48 77.n even 30 1
847.2.r.c 48 77.p odd 30 1
847.2.r.d 48 11.c even 5 1
847.2.r.d 48 11.d odd 10 1
847.2.r.d 48 77.n even 30 1
847.2.r.d 48 77.p odd 30 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{48} - 32 T_{2}^{46} + 582 T_{2}^{44} - 7218 T_{2}^{42} + 67545 T_{2}^{40} - 496978 T_{2}^{38} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(847, [\chi])\). Copy content Toggle raw display