↑

Properties

Label 847.2.n.j
Level $847$
Weight $2$
Character orbit 847.n
Analytic conductor $6.763$
Analytic rank $0$
Dimension $40$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [847,2,Mod(9,847)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(847, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([10, 18]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("847.9");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 847 = 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 847.n (of order \(15\), degree \(8\), 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: \(6.76332905120\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(5\) over \(\Q(\zeta_{15})\)
Twist minimal: no (minimal twist has level 77)
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 40 q + 3 q^{2} - 4 q^{3} - 3 q^{4} + 4 q^{5} + 16 q^{6} + 2 q^{7} + 38 q^{8} + 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q + 3 q^{2} - 4 q^{3} - 3 q^{4} + 4 q^{5} + 16 q^{6} + 2 q^{7} + 38 q^{8} + 7 q^{9} - 14 q^{10} - 18 q^{12} - 6 q^{13} - 3 q^{14} - 14 q^{15} - 5 q^{16} + 7 q^{17} - 24 q^{18} + 4 q^{19} - 30 q^{20} + 2 q^{21} - 14 q^{23} + 12 q^{24} + 21 q^{25} - 16 q^{27} - 16 q^{28} - 16 q^{30} - 17 q^{31} + 30 q^{32} + 48 q^{34} + 14 q^{35} + 14 q^{36} + 24 q^{37} + 12 q^{38} - 28 q^{39} - 10 q^{40} - 60 q^{41} - 70 q^{42} + 72 q^{43} - 16 q^{45} - 8 q^{46} + 13 q^{47} + 128 q^{48} - 10 q^{49} - 6 q^{50} + 7 q^{51} - 2 q^{52} + 33 q^{53} - 34 q^{54} + 24 q^{56} - 44 q^{57} - 17 q^{58} + 21 q^{59} - 48 q^{60} + 52 q^{62} - 24 q^{63} + 94 q^{64} + 40 q^{65} - 38 q^{67} + 23 q^{68} - 124 q^{69} - 3 q^{70} + 20 q^{71} + 38 q^{72} - 11 q^{73} + 41 q^{74} - 11 q^{75} + 96 q^{76} - 100 q^{78} - 21 q^{79} + 12 q^{80} - 58 q^{81} + 6 q^{82} + 46 q^{83} + 29 q^{84} + 78 q^{85} + 7 q^{86} - 48 q^{87} - 10 q^{89} + 18 q^{90} - 14 q^{91} - 110 q^{92} + 12 q^{93} - 37 q^{94} - 7 q^{95} + 53 q^{96} - 54 q^{97} - 116 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1 −1.76087 − 1.95564i −2.02209 − 0.900292i −0.514820 + 4.89819i 1.15065 + 0.244578i 1.79998 + 5.53977i −0.510555 + 2.59602i 6.22764 − 4.52465i 1.27093 + 1.41151i −1.54783 − 2.68093i
9.2 −0.391628 − 0.434946i −1.26312 − 0.562375i 0.173251 − 1.64837i 3.87755 + 0.824199i 0.250068 + 0.769629i 1.49228 − 2.18474i −1.73180 + 1.25823i −0.728198 − 0.808746i −1.16007 − 2.00931i
9.3 0.0508685 + 0.0564952i −2.15146 − 0.957893i 0.208453 − 1.98330i −2.52173 − 0.536011i −0.0553254 − 0.170274i −2.37978 + 1.15613i 0.245656 − 0.178480i 1.70384 + 1.89231i −0.0979948 − 0.169732i
9.4 0.450970 + 0.500853i 1.96516 + 0.874946i 0.161577 − 1.53730i −1.65218 − 0.351182i 0.448009 + 1.37883i −1.55484 − 2.14067i 1.93333 − 1.40464i 1.08893 + 1.20938i −0.569194 − 0.985873i
9.5 1.23711 + 1.37395i 1.07981 + 0.480764i −0.148239 + 1.41040i 2.31107 + 0.491232i 0.675302 + 2.07837i 0.122985 + 2.64289i 0.870261 − 0.632281i −1.07253 − 1.19116i 2.18411 + 3.78299i
81.1 −0.255843 + 2.43419i 1.95053 + 2.16628i −3.90352 − 0.829718i −0.303226 − 0.135005i −5.77217 + 4.19373i 1.95978 + 1.77744i 1.50568 − 4.63401i −0.574630 + 5.46724i 0.406206 − 0.703570i
81.2 −0.234707 + 2.23309i −1.27743 − 1.41873i −2.97532 − 0.632424i 2.25193 + 1.00262i 3.46797 − 2.51963i 1.07525 − 2.41740i 0.722861 − 2.22474i −0.0673794 + 0.641072i −2.76749 + 4.79344i
81.3 −0.0457018 + 0.434823i −1.72872 − 1.91994i 1.76931 + 0.376079i −1.11068 − 0.494505i 0.913838 − 0.663942i 0.0819410 + 2.64448i −0.514604 + 1.58379i −0.384102 + 3.65449i 0.265782 − 0.460348i
81.4 0.133281 − 1.26809i −0.0932166 − 0.103528i 0.366016 + 0.0777992i 1.08262 + 0.482012i −0.143706 + 0.104408i 2.63367 − 0.252511i 0.935477 − 2.87910i 0.311557 − 2.96426i 0.755525 − 1.30861i
81.5 0.233841 − 2.22485i 0.893246 + 0.992050i −2.93896 − 0.624696i −0.791435 − 0.352369i 2.41604 − 1.75535i −2.03186 + 1.69457i −0.694498 + 2.13745i 0.127310 − 1.21128i −0.969038 + 1.67842i
130.1 −1.80843 + 0.384393i −0.123553 − 1.17553i 1.29556 − 0.576822i −1.58095 + 1.75583i 0.675302 + 2.07837i 2.55154 − 0.699733i 0.870261 − 0.632281i 1.56784 − 0.333255i 2.18411 − 3.78299i
130.2 −0.659236 + 0.140125i −0.224855 − 2.13935i −1.41213 + 0.628722i 1.13022 − 1.25524i 0.448009 + 1.37883i −2.51637 − 0.817238i 1.93333 − 1.40464i −1.59182 + 0.338352i −0.569194 + 0.985873i
130.3 −0.0743606 + 0.0158058i 0.246172 + 2.34217i −1.82181 + 0.811123i 1.72507 − 1.91588i −0.0553254 − 0.170274i 0.364157 − 2.62057i 0.245656 − 0.178480i −2.49071 + 0.529416i −0.0979948 + 0.169732i
130.4 0.572488 − 0.121686i 0.144526 + 1.37508i −1.51416 + 0.674145i −2.65255 + 2.94596i 0.250068 + 0.769629i −1.61668 + 2.09436i −1.73180 + 1.25823i 1.06449 − 0.226265i −1.16007 + 2.00931i
130.5 2.57407 − 0.547134i 0.231369 + 2.20133i 4.49937 − 2.00325i −0.787136 + 0.874203i 1.79998 + 5.53977i 2.31119 − 1.28778i 6.22764 − 4.52465i −1.85787 + 0.394902i −1.54783 + 2.68093i
366.1 −0.255843 − 2.43419i 1.95053 − 2.16628i −3.90352 + 0.829718i −0.303226 + 0.135005i −5.77217 − 4.19373i 1.95978 − 1.77744i 1.50568 + 4.63401i −0.574630 − 5.46724i 0.406206 + 0.703570i
366.2 −0.234707 − 2.23309i −1.27743 + 1.41873i −2.97532 + 0.632424i 2.25193 − 1.00262i 3.46797 + 2.51963i 1.07525 + 2.41740i 0.722861 + 2.22474i −0.0673794 − 0.641072i −2.76749 − 4.79344i
366.3 −0.0457018 − 0.434823i −1.72872 + 1.91994i 1.76931 − 0.376079i −1.11068 + 0.494505i 0.913838 + 0.663942i 0.0819410 − 2.64448i −0.514604 − 1.58379i −0.384102 − 3.65449i 0.265782 + 0.460348i
366.4 0.133281 + 1.26809i −0.0932166 + 0.103528i 0.366016 − 0.0777992i 1.08262 − 0.482012i −0.143706 − 0.104408i 2.63367 + 0.252511i 0.935477 + 2.87910i 0.311557 + 2.96426i 0.755525 + 1.30861i
366.5 0.233841 + 2.22485i 0.893246 − 0.992050i −2.93896 + 0.624696i −0.791435 + 0.352369i 2.41604 + 1.75535i −2.03186 − 1.69457i −0.694498 − 2.13745i 0.127310 + 1.21128i −0.969038 − 1.67842i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 9.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
11.c even 5 1 inner
77.m even 15 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 847.2.n.j 40
7.c even 3 1 inner 847.2.n.j 40
11.b odd 2 1 77.2.m.b ✓ 40
11.c even 5 1 847.2.e.h 20
11.c even 5 2 847.2.n.h 40
11.c even 5 1 inner 847.2.n.j 40
11.d odd 10 1 77.2.m.b ✓ 40
11.d odd 10 1 847.2.e.i 20
11.d odd 10 2 847.2.n.i 40
33.d even 2 1 693.2.by.b 40
33.f even 10 1 693.2.by.b 40
77.b even 2 1 539.2.q.h 40
77.h odd 6 1 77.2.m.b ✓ 40
77.h odd 6 1 539.2.f.h 20
77.i even 6 1 539.2.f.g 20
77.i even 6 1 539.2.q.h 40
77.l even 10 1 539.2.q.h 40
77.m even 15 1 847.2.e.h 20
77.m even 15 2 847.2.n.h 40
77.m even 15 1 inner 847.2.n.j 40
77.m even 15 1 5929.2.a.by 10
77.n even 30 1 539.2.f.g 20
77.n even 30 1 539.2.q.h 40
77.n even 30 1 5929.2.a.bx 10
77.o odd 30 1 77.2.m.b ✓ 40
77.o odd 30 1 539.2.f.h 20
77.o odd 30 1 847.2.e.i 20
77.o odd 30 2 847.2.n.i 40
77.o odd 30 1 5929.2.a.bw 10
77.p odd 30 1 5929.2.a.bz 10
231.l even 6 1 693.2.by.b 40
231.be even 30 1 693.2.by.b 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.m.b ✓ 40 11.b odd 2 1
77.2.m.b ✓ 40 11.d odd 10 1
77.2.m.b ✓ 40 77.h odd 6 1
77.2.m.b ✓ 40 77.o odd 30 1
539.2.f.g 20 77.i even 6 1
539.2.f.g 20 77.n even 30 1
539.2.f.h 20 77.h odd 6 1
539.2.f.h 20 77.o odd 30 1
539.2.q.h 40 77.b even 2 1
539.2.q.h 40 77.i even 6 1
539.2.q.h 40 77.l even 10 1
539.2.q.h 40 77.n even 30 1
693.2.by.b 40 33.d even 2 1
693.2.by.b 40 33.f even 10 1
693.2.by.b 40 231.l even 6 1
693.2.by.b 40 231.be even 30 1
847.2.e.h 20 11.c even 5 1
847.2.e.h 20 77.m even 15 1
847.2.e.i 20 11.d odd 10 1
847.2.e.i 20 77.o odd 30 1
847.2.n.h 40 11.c even 5 2
847.2.n.h 40 77.m even 15 2
847.2.n.i 40 11.d odd 10 2
847.2.n.i 40 77.o odd 30 2
847.2.n.j 40 1.a even 1 1 trivial
847.2.n.j 40 7.c even 3 1 inner
847.2.n.j 40 11.c even 5 1 inner
847.2.n.j 40 77.m even 15 1 inner
5929.2.a.bw 10 77.o odd 30 1
5929.2.a.bx 10 77.n even 30 1
5929.2.a.by 10 77.m even 15 1
5929.2.a.bz 10 77.p odd 30 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{40} - 3 T_{2}^{39} + T_{2}^{38} - 12 T_{2}^{37} + 21 T_{2}^{36} + 50 T_{2}^{35} + 238 T_{2}^{34} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(847, [\chi])\). Copy content Toggle raw display