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Properties

Label 847.2.r.c
Level $847$
Weight $2$
Character orbit 847.r
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(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 algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 5 q^{2} - 9 q^{3} - 9 q^{4} - 15 q^{5} + 5 q^{7} - 11 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q + 5 q^{2} - 9 q^{3} - 9 q^{4} - 15 q^{5} + 5 q^{7} - 11 q^{9} - 12 q^{12} - 8 q^{14} - 27 q^{16} - 15 q^{17} - 20 q^{18} + 15 q^{19} + 10 q^{23} - 75 q^{24} + q^{25} + 27 q^{26} + 40 q^{28} + 40 q^{29} - 25 q^{30} + 9 q^{31} - 5 q^{35} - 38 q^{36} - q^{37} + 33 q^{38} + 45 q^{39} - 75 q^{40} + 64 q^{42} - 84 q^{45} + 20 q^{46} + 3 q^{47} + 59 q^{49} - 30 q^{50} - 55 q^{51} + 15 q^{52} - 3 q^{53} - 8 q^{56} - 60 q^{57} + 46 q^{58} - 3 q^{59} - 15 q^{60} + 30 q^{61} + 40 q^{63} + 12 q^{64} + 44 q^{67} + 75 q^{68} - 27 q^{70} + 20 q^{71} + 60 q^{72} + 60 q^{73} - 45 q^{74} - 57 q^{75} + 92 q^{78} + 70 q^{79} - 75 q^{80} - 29 q^{81} - 129 q^{82} + 125 q^{84} - 10 q^{85} - 62 q^{86} + 6 q^{89} - 12 q^{91} + 30 q^{92} - 92 q^{93} - 105 q^{94} - 30 q^{95} - 75 q^{96}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
40.1 −0.844499 − 1.89678i 0.0388500 + 0.182775i −1.54632 + 1.71736i −2.55163 + 0.268187i 0.313874 − 0.228043i −1.95804 + 1.77935i 0.613987 + 0.199497i 2.70874 − 1.20601i 2.66354 + 4.61339i
40.2 −0.532803 − 1.19670i −0.567882 − 2.67168i 0.190060 − 0.211083i 0.406975 − 0.0427748i −2.89461 + 2.10306i 1.52953 − 2.15883i −2.84553 − 0.924570i −4.07472 + 1.81418i −0.268026 − 0.464235i
40.3 −0.194980 − 0.437933i 0.465496 + 2.18999i 1.18449 − 1.31551i −1.33791 + 0.140620i 0.868306 − 0.630861i 2.54633 + 0.718490i −1.71889 − 0.558501i −1.83873 + 0.818654i 0.322449 + 0.558497i
40.4 0.454517 + 1.02086i −0.324506 − 1.52668i 0.502686 − 0.558290i −3.06242 + 0.321873i 1.41104 − 1.02518i −2.33652 − 1.24123i 2.92398 + 0.950058i 0.515189 − 0.229377i −1.72051 − 2.98001i
40.5 0.571183 + 1.28290i −0.0425239 − 0.200059i 0.0186834 − 0.0207501i 1.51174 − 0.158890i 0.232367 − 0.168824i 2.52376 + 0.794113i 2.70844 + 0.880026i 2.70242 − 1.20320i 1.06732 + 1.84865i
40.6 0.842093 + 1.89137i 0.653128 + 3.07273i −1.52991 + 1.69913i 1.18148 − 0.124179i −5.26167 + 3.82283i −2.63875 − 0.192354i −0.563952 − 0.183239i −6.27443 + 2.79356i 1.22979 + 2.13006i
94.1 −1.75157 − 1.57712i −0.821231 − 1.84451i 0.371633 + 3.53585i −0.00617973 − 0.0290734i −1.47058 + 4.52598i 2.21185 − 1.45180i 2.15474 − 2.96574i −0.720422 + 0.800109i −0.0350280 + 0.0606703i
94.2 −0.386517 − 0.348022i 0.460952 + 1.03532i −0.180780 − 1.72001i 0.678628 + 3.19269i 0.182146 − 0.560588i 1.97073 + 1.76528i −1.14015 + 1.56929i 1.14799 − 1.27497i 0.848825 − 1.47021i
94.3 0.0202070 + 0.0181945i −0.500742 − 1.12469i −0.208980 − 1.98831i −0.240558 − 1.13174i 0.0103446 − 0.0318373i 0.296416 − 2.62909i 0.0639186 − 0.0879764i 0.993218 − 1.10308i 0.0157304 − 0.0272459i
94.4 1.11268 + 1.00186i 0.245339 + 0.551041i 0.0252713 + 0.240440i −0.491349 − 2.31161i −0.279082 + 0.858925i −2.00635 + 1.72470i 1.54736 − 2.12976i 1.76394 − 1.95905i 1.76920 − 3.06434i
94.5 1.33360 + 1.20078i −1.17499 − 2.63908i 0.127565 + 1.21370i 0.418270 + 1.96780i 1.60198 − 4.93040i 1.17364 + 2.37120i 0.822345 − 1.13186i −3.57672 + 3.97235i −1.80510 + 3.12652i
94.6 1.95877 + 1.76368i 0.650787 + 1.46169i 0.517138 + 4.92024i 0.446428 + 2.10028i −1.30322 + 4.01090i 1.30623 − 2.30082i −4.56624 + 6.28489i 0.294372 − 0.326934i −2.82977 + 4.90131i
215.1 −0.490042 + 2.30547i −2.00801 − 0.211051i −3.24795 − 1.44608i 0.0220884 − 0.0198885i 1.47058 − 4.52598i −2.06424 + 1.65497i 2.15474 − 2.96574i 1.05313 + 0.223849i 0.0350280 + 0.0606703i
215.2 −0.108137 + 0.508745i 1.12709 + 0.118461i 1.57996 + 0.703445i −2.42564 + 2.18406i −0.182146 + 0.560588i 1.06990 + 2.41978i −1.14015 + 1.56929i −1.67816 − 0.356703i −0.848825 − 1.47021i
215.3 0.00565338 − 0.0265970i −1.22438 − 0.128687i 1.82642 + 0.813173i 0.859835 − 0.774198i −0.0103446 + 0.0318373i −2.59202 − 0.530526i 0.0639186 − 0.0879764i −1.45190 − 0.308612i −0.0157304 − 0.0272459i
215.4 0.311296 − 1.46453i 0.599886 + 0.0630505i −0.220863 − 0.0983344i 1.75624 − 1.58133i 0.279082 − 0.858925i 2.26028 − 1.37519i 1.54736 − 2.12976i −2.57856 − 0.548089i −1.76920 − 3.06434i
215.5 0.373106 − 1.75533i −2.87300 − 0.301965i −1.11488 − 0.496374i −1.49503 + 1.34613i −1.60198 + 4.93040i 1.89247 + 1.84893i 0.822345 − 1.13186i 5.22852 + 1.11136i 1.80510 + 3.12652i
215.6 0.548010 − 2.57818i 1.59126 + 0.167248i −4.51962 − 2.01227i −1.59568 + 1.43676i 1.30322 − 4.01090i −2.59186 + 0.531305i −4.56624 + 6.28489i −0.430319 − 0.0914672i 2.82977 + 4.90131i
360.1 −0.844499 + 1.89678i 0.0388500 − 0.182775i −1.54632 − 1.71736i −2.55163 − 0.268187i 0.313874 + 0.228043i −1.95804 − 1.77935i 0.613987 − 0.199497i 2.70874 + 1.20601i 2.66354 − 4.61339i
360.2 −0.532803 + 1.19670i −0.567882 + 2.67168i 0.190060 + 0.211083i 0.406975 + 0.0427748i −2.89461 − 2.10306i 1.52953 + 2.15883i −2.84553 + 0.924570i −4.07472 − 1.81418i −0.268026 + 0.464235i
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.c 48
7.d odd 6 1 inner 847.2.r.c 48
11.b odd 2 1 77.2.n.a ✓ 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.a 48
11.c even 5 1 847.2.r.d 48
11.d odd 10 1 847.2.i.b 48
11.d odd 10 1 847.2.r.a 48
11.d odd 10 1 inner 847.2.r.c 48
11.d odd 10 1 847.2.r.d 48
33.d even 2 1 693.2.cg.a 48
33.h odd 10 1 693.2.cg.a 48
77.b even 2 1 539.2.s.d 48
77.h odd 6 1 539.2.m.a 48
77.h odd 6 1 539.2.s.d 48
77.i even 6 1 77.2.n.a ✓ 48
77.i even 6 1 539.2.m.a 48
77.j odd 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 1 847.2.i.b 48
77.n even 30 1 847.2.r.a 48
77.n even 30 1 inner 847.2.r.c 48
77.n even 30 1 847.2.r.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.a 48
77.p odd 30 1 847.2.r.d 48
231.k odd 6 1 693.2.cg.a 48
231.bc even 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.b odd 2 1
77.2.n.a ✓ 48 11.c even 5 1
77.2.n.a ✓ 48 77.i even 6 1
77.2.n.a ✓ 48 77.p odd 30 1
539.2.m.a 48 77.h odd 6 1
539.2.m.a 48 77.i even 6 1
539.2.m.a 48 77.m even 15 1
539.2.m.a 48 77.p odd 30 1
539.2.s.d 48 77.b even 2 1
539.2.s.d 48 77.h odd 6 1
539.2.s.d 48 77.j odd 10 1
539.2.s.d 48 77.m even 15 1
693.2.cg.a 48 33.d even 2 1
693.2.cg.a 48 33.h odd 10 1
693.2.cg.a 48 231.k odd 6 1
693.2.cg.a 48 231.bc even 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 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 1.a even 1 1 trivial
847.2.r.c 48 7.d odd 6 1 inner
847.2.r.c 48 11.d odd 10 1 inner
847.2.r.c 48 77.n even 30 1 inner
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 \( T_{2}^{48} - 5 T_{2}^{47} + 23 T_{2}^{46} - 60 T_{2}^{45} + 142 T_{2}^{44} - 195 T_{2}^{43} + 222 T_{2}^{42} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(847, [\chi])\). Copy content Toggle raw display