Properties

Label 847.2.r.a
Level $847$
Weight $2$
Character orbit 847.r
Analytic conductor $6.763$
Analytic rank $0$
Dimension $48$
CM no
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(40,847)]
 
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([25, 21]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("847.40");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 847 = 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 847.r (of order \(30\), 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: \(48\)
Relative dimension: \(6\) over \(\Q(\zeta_{30})\)
Twist minimal: no (minimal twist has level 77)
Sato-Tate group: $\mathrm{SU}(2)[C_{30}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 5 q^{2} + 6 q^{3} + q^{4} + 15 q^{5} + 10 q^{7} - 10 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 5 q^{2} + 6 q^{3} + q^{4} + 15 q^{5} + 10 q^{7} - 10 q^{8} + 4 q^{9} - 12 q^{12} - 8 q^{14} - 10 q^{15} + 23 q^{16} + 5 q^{18} + 30 q^{19} + 10 q^{23} + 60 q^{24} + 11 q^{25} + 12 q^{26} - 60 q^{28} - 20 q^{29} + 30 q^{30} - 6 q^{31} + 45 q^{35} + 2 q^{36} + 19 q^{37} + 18 q^{38} - 10 q^{39} - 75 q^{40} - 86 q^{42} - 84 q^{45} - 10 q^{46} + 3 q^{47} - 36 q^{49} + 10 q^{50} + 20 q^{51} - 60 q^{52} - 3 q^{53} - 8 q^{56} + 10 q^{57} - 9 q^{58} - 63 q^{59} + 5 q^{60} + 90 q^{61} - 70 q^{63} - 18 q^{64} + 44 q^{67} - 165 q^{68} + 8 q^{70} + 30 q^{71} + 50 q^{72} - 60 q^{74} + 78 q^{75} + 92 q^{78} - 70 q^{79} + 30 q^{80} + 31 q^{81} + 96 q^{82} + 65 q^{84} - 80 q^{85} + 73 q^{86} + 6 q^{89} - 17 q^{91} - 70 q^{92} + 58 q^{93} - 90 q^{94} - 50 q^{95} + 150 q^{96}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
40.1 −0.958667 2.15320i −0.419789 1.97495i −2.37898 + 2.64212i −0.0295601 + 0.00310689i −3.85003 + 2.79721i 0.936082 + 2.47462i 3.48644 + 1.13281i −0.983572 + 0.437914i 0.0350280 + 0.0606703i
40.2 −0.211548 0.475144i 0.235625 + 1.10853i 1.15725 1.28526i 3.24614 0.341183i 0.476865 0.346463i 2.63196 0.269780i −1.84481 0.599414i 1.56732 0.697816i −0.848825 1.47021i
40.3 0.0110597 + 0.0248404i −0.255965 1.20422i 1.33777 1.48574i −1.15068 + 0.120942i 0.0270824 0.0196765i −1.30554 + 2.30121i 0.103422 + 0.0336040i 1.35601 0.603736i −0.0157304 0.0272459i
40.4 0.608987 + 1.36781i 0.125410 + 0.590009i −0.161772 + 0.179666i −2.35031 + 0.247028i −0.730646 + 0.530845i −0.609417 2.57461i 2.50368 + 0.813494i 2.40825 1.07222i −1.76920 3.06434i
40.5 0.729906 + 1.63940i −0.600621 2.82570i −0.816596 + 0.906921i 2.00074 0.210287i 4.19404 3.04715i 2.34325 1.22849i 1.33058 + 0.432332i −4.88320 + 2.17414i 1.80510 + 3.12652i
40.6 1.07207 + 2.40791i 0.332663 + 1.56506i −3.31042 + 3.67659i 2.13544 0.224443i −3.41187 + 2.47887i −0.295626 + 2.62918i −7.38833 2.40061i 0.401899 0.178937i 2.82977 + 4.90131i
94.1 −1.53858 1.38534i 1.27771 + 2.86979i 0.238995 + 2.27389i 0.246997 + 1.16203i 2.00978 6.18547i 2.24786 + 1.39540i 0.348541 0.479726i −4.59574 + 5.10408i 1.22979 2.13006i
94.2 −1.04360 0.939665i −0.0831894 0.186846i −0.00291864 0.0277690i 0.316039 + 1.48685i −0.0887562 + 0.273164i −2.50854 0.840980i −1.67391 + 2.30394i 1.97940 2.19835i 1.06732 1.84865i
94.3 −0.830445 0.747736i −0.634829 1.42585i −0.0785273 0.747137i −0.640220 3.01200i −0.538968 + 1.65877i 2.61986 + 0.369199i −1.80712 + 2.48728i 0.377352 0.419092i −1.72051 + 2.98001i
94.4 0.356247 + 0.320766i 0.910648 + 2.04535i −0.185036 1.76050i −0.279700 1.31588i −0.331663 + 1.02075i −2.48234 0.915422i 1.06233 1.46217i −1.34678 + 1.49576i 0.322449 0.558497i
94.5 0.973480 + 0.876525i −1.11095 2.49522i −0.0296902 0.282484i 0.0850810 + 0.400275i 1.10564 3.40282i 0.0315105 2.64556i 1.75864 2.42055i −2.98455 + 3.31468i −0.268026 + 0.464235i
94.6 1.54298 + 1.38930i 0.0760020 + 0.170703i 0.241558 + 2.29827i −0.533436 2.50962i −0.119889 + 0.368981i 0.538211 + 2.59043i −0.379465 + 0.522289i 1.98403 2.20349i 2.66354 4.61339i
215.1 −0.430453 + 2.02512i 3.12416 + 0.328363i −2.08874 0.929967i −0.882850 + 0.794922i −2.00978 + 6.18547i 0.632479 + 2.56904i 0.348541 0.479726i 6.71813 + 1.42798i −1.22979 2.13006i
215.2 −0.291972 + 1.37362i −0.203408 0.0213791i 0.0255080 + 0.0113569i −1.12963 + 1.01712i 0.0887562 0.273164i −0.0246392 2.64564i −1.67391 + 2.30394i −2.89352 0.615038i −1.06732 1.84865i
215.3 −0.232336 + 1.09305i −1.55224 0.163147i 0.686304 + 0.305562i 2.28836 2.06045i 0.538968 1.65877i −0.458454 + 2.60573i −1.80712 + 2.48728i −0.551621 0.117251i 1.72051 + 2.98001i
215.4 0.0996681 0.468902i 2.22665 + 0.234030i 1.61716 + 0.720004i 0.999739 0.900169i 0.331663 1.02075i −0.103533 2.64372i 1.06233 1.46217i 1.96875 + 0.418472i −0.322449 0.558497i
215.5 0.272353 1.28132i −2.71640 0.285505i 0.259483 + 0.115529i −0.304108 + 0.273820i −1.10564 + 3.40282i −2.52582 0.787556i 1.75864 2.42055i 4.36287 + 0.927357i 0.268026 + 0.464235i
215.6 0.431683 2.03091i 0.185834 + 0.0195320i −2.11114 0.939941i 1.90668 1.71678i 0.119889 0.368981i 2.29733 + 1.31236i −0.379465 + 0.522289i −2.90029 0.616476i −2.66354 4.61339i
360.1 −0.958667 + 2.15320i −0.419789 + 1.97495i −2.37898 2.64212i −0.0295601 0.00310689i −3.85003 2.79721i 0.936082 2.47462i 3.48644 1.13281i −0.983572 0.437914i 0.0350280 0.0606703i
360.2 −0.211548 + 0.475144i 0.235625 1.10853i 1.15725 + 1.28526i 3.24614 + 0.341183i 0.476865 + 0.346463i 2.63196 + 0.269780i −1.84481 + 0.599414i 1.56732 + 0.697816i −0.848825 + 1.47021i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 40.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner
11.d odd 10 1 inner
77.n even 30 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 847.2.r.a 48
7.d odd 6 1 inner 847.2.r.a 48
11.b odd 2 1 847.2.r.d 48
11.c even 5 1 77.2.n.a 48
11.c even 5 1 847.2.i.b 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.i.b 48
11.d odd 10 1 inner 847.2.r.a 48
11.d odd 10 1 847.2.r.c 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 1 847.2.r.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.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.i.b 48
77.n even 30 1 inner 847.2.r.a 48
77.n even 30 1 847.2.r.c 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.i.b 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 11.c even 5 1
847.2.i.b 48 11.d odd 10 1
847.2.i.b 48 77.n even 30 1
847.2.i.b 48 77.p odd 30 1
847.2.r.a 48 1.a even 1 1 trivial
847.2.r.a 48 7.d odd 6 1 inner
847.2.r.a 48 11.d odd 10 1 inner
847.2.r.a 48 77.n even 30 1 inner
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.b odd 2 1
847.2.r.d 48 11.c even 5 1
847.2.r.d 48 77.i even 6 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 \( T_{2}^{48} + 5 T_{2}^{47} + 18 T_{2}^{46} + 65 T_{2}^{45} + 162 T_{2}^{44} + 360 T_{2}^{43} + 502 T_{2}^{42} - 50 T_{2}^{41} - 2375 T_{2}^{40} - 11210 T_{2}^{39} - 31718 T_{2}^{38} - 76060 T_{2}^{37} - 152303 T_{2}^{36} - 260405 T_{2}^{35} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(847, [\chi])\). Copy content Toggle raw display