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Properties

Label 847.2.n.d
Level $847$
Weight $2$
Character orbit 847.n
Analytic conductor $6.763$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $8$

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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: \(24\)
Relative dimension: \(3\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - q^{3} + 4 q^{4} - 2 q^{5} - 2 q^{6} + 2 q^{7} - 18 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - q^{3} + 4 q^{4} - 2 q^{5} - 2 q^{6} + 2 q^{7} - 18 q^{8} - 36 q^{10} + 36 q^{12} - 22 q^{13} - 12 q^{14} + 14 q^{15} + 2 q^{16} + 3 q^{17} - 10 q^{18} + 11 q^{19} - 28 q^{20} - 40 q^{21} - 48 q^{23} - 2 q^{24} + 3 q^{25} + q^{26} - 4 q^{27} + 13 q^{28} - 18 q^{29} - 2 q^{30} - 3 q^{31} - 12 q^{32} - 80 q^{34} + 9 q^{35} + 18 q^{36} - 4 q^{37} + 8 q^{38} + 5 q^{39} + 3 q^{40} - 10 q^{41} + 2 q^{42} - 16 q^{43} + 36 q^{45} + 10 q^{46} - 3 q^{47} - 20 q^{48} + 24 q^{49} - 6 q^{50} - 2 q^{51} + 7 q^{52} + 17 q^{53} - 32 q^{54} + 12 q^{56} + 40 q^{57} - 13 q^{58} + 8 q^{59} + 6 q^{60} + 24 q^{61} + 26 q^{62} + 12 q^{63} + 14 q^{64} + 60 q^{65} + 64 q^{67} - 5 q^{68} + 6 q^{69} + 27 q^{70} - 14 q^{71} - 10 q^{72} + 20 q^{73} - 22 q^{74} + 25 q^{75} + 312 q^{76} - 48 q^{78} - 3 q^{79} + 9 q^{80} - 17 q^{81} + 41 q^{82} - 22 q^{83} + 12 q^{84} + 22 q^{85} - 21 q^{86} + 120 q^{87} - 4 q^{89} + 20 q^{90} + 15 q^{91} - 50 q^{92} - 26 q^{93} + 10 q^{94} + 17 q^{95} - 27 q^{96} - 18 q^{97} + 96 q^{98}+O(q^{100}) \) 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 −1.22735 − 1.36311i −2.00860 − 0.894288i −0.142624 + 1.35697i 0.621664 + 0.132139i 1.24625 + 3.83555i 1.80096 − 1.93818i −0.943117 + 0.685215i 1.22735 + 1.36311i −0.582878 − 1.00958i
9.2 −0.439365 − 0.487964i 1.74691 + 0.777774i 0.163989 − 1.56026i 3.49086 + 0.742006i −0.388004 − 1.19415i 0.295442 + 2.62920i −1.89583 + 1.37740i 0.439365 + 0.487964i −1.17169 − 2.02943i
9.3 1.66671 + 1.85107i −0.651849 − 0.290222i −0.439480 + 4.18137i −2.15623 − 0.458321i −0.549224 − 1.69034i −2.30546 + 1.29802i −4.44220 + 3.22745i −1.66671 − 1.85107i −2.74543 − 4.75523i
81.1 −0.260366 + 2.47722i −0.477450 − 0.530262i −4.11253 − 0.874144i 2.01382 + 0.896611i 1.43789 − 1.04469i −1.94692 + 1.79151i 1.69677 − 5.22212i 0.260366 − 2.47722i −2.74543 + 4.75523i
81.2 0.0686355 − 0.653023i 1.27953 + 1.42106i 1.53457 + 0.326182i −3.26031 − 1.45158i 1.01581 − 0.738028i −2.40923 − 1.09345i 0.724144 − 2.22869i −0.0686355 + 0.653023i −1.17169 + 2.02943i
81.3 0.191731 − 1.82420i −1.47121 − 1.63395i −1.33463 − 0.283685i −0.580606 − 0.258502i −3.26271 + 2.37050i 2.39985 − 1.11389i 0.360239 − 1.10870i −0.191731 + 1.82420i −0.582878 + 1.00958i
130.1 −2.43643 + 0.517880i 0.0745850 + 0.709629i 3.84091 − 1.71008i 1.47503 − 1.63819i −0.549224 − 1.69034i 0.522062 − 2.59373i −4.44220 + 3.22745i 2.43643 − 0.517880i −2.74543 + 4.75523i
130.2 0.642272 − 0.136519i −0.199882 − 1.90175i −1.43322 + 0.638109i −2.38803 + 2.65217i −0.388004 − 1.19415i 2.59182 − 0.531487i −1.89583 + 1.37740i −0.642272 + 0.136519i −1.17169 + 2.02943i
130.3 1.79416 − 0.381361i 0.229826 + 2.18665i 1.24649 − 0.554971i −0.425267 + 0.472307i 1.24625 + 3.83555i −1.28679 + 2.31175i −0.943117 + 0.685215i −1.79416 + 0.381361i −0.582878 + 1.00958i
366.1 −0.260366 − 2.47722i −0.477450 + 0.530262i −4.11253 + 0.874144i 2.01382 − 0.896611i 1.43789 + 1.04469i −1.94692 − 1.79151i 1.69677 + 5.22212i 0.260366 + 2.47722i −2.74543 − 4.75523i
366.2 0.0686355 + 0.653023i 1.27953 − 1.42106i 1.53457 − 0.326182i −3.26031 + 1.45158i 1.01581 + 0.738028i −2.40923 + 1.09345i 0.724144 + 2.22869i −0.0686355 − 0.653023i −1.17169 − 2.02943i
366.3 0.191731 + 1.82420i −1.47121 + 1.63395i −1.33463 + 0.283685i −0.580606 + 0.258502i −3.26271 − 2.37050i 2.39985 + 1.11389i 0.360239 + 1.10870i −0.191731 − 1.82420i −0.582878 − 1.00958i
487.1 −1.67566 − 0.746054i 2.15064 + 0.457134i 0.912994 + 1.01398i 0.0664333 − 0.632070i −3.26271 − 2.37050i −2.59624 − 0.509441i 0.360239 + 1.10870i 1.67566 + 0.746054i −0.582878 + 1.00958i
487.2 −0.599853 − 0.267072i −1.87044 − 0.397575i −1.04977 − 1.16588i 0.373046 − 3.54930i 1.01581 + 0.738028i 1.30639 + 2.30073i 0.724144 + 2.22869i 0.599853 + 0.267072i −1.17169 + 2.02943i
487.3 2.27552 + 1.01313i 0.697945 + 0.148353i 2.81329 + 3.12448i −0.230423 + 2.19233i 1.43789 + 1.04469i 2.62811 − 0.304997i 1.69677 + 5.22212i −2.27552 − 1.01313i −2.74543 + 4.75523i
632.1 −2.43643 − 0.517880i 0.0745850 − 0.709629i 3.84091 + 1.71008i 1.47503 + 1.63819i −0.549224 + 1.69034i 0.522062 + 2.59373i −4.44220 − 3.22745i 2.43643 + 0.517880i −2.74543 − 4.75523i
632.2 0.642272 + 0.136519i −0.199882 + 1.90175i −1.43322 − 0.638109i −2.38803 − 2.65217i −0.388004 + 1.19415i 2.59182 + 0.531487i −1.89583 − 1.37740i −0.642272 − 0.136519i −1.17169 − 2.02943i
632.3 1.79416 + 0.381361i 0.229826 − 2.18665i 1.24649 + 0.554971i −0.425267 − 0.472307i 1.24625 − 3.83555i −1.28679 − 2.31175i −0.943117 − 0.685215i −1.79416 − 0.381361i −0.582878 − 1.00958i
753.1 −1.22735 + 1.36311i −2.00860 + 0.894288i −0.142624 − 1.35697i 0.621664 − 0.132139i 1.24625 − 3.83555i 1.80096 + 1.93818i −0.943117 − 0.685215i 1.22735 − 1.36311i −0.582878 + 1.00958i
753.2 −0.439365 + 0.487964i 1.74691 − 0.777774i 0.163989 + 1.56026i 3.49086 − 0.742006i −0.388004 + 1.19415i 0.295442 − 2.62920i −1.89583 − 1.37740i 0.439365 − 0.487964i −1.17169 + 2.02943i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 9.3
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 3 inner
77.m even 15 3 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 847.2.n.d 24
7.c even 3 1 inner 847.2.n.d 24
11.b odd 2 1 847.2.n.e 24
11.c even 5 1 847.2.e.d 6
11.c even 5 3 inner 847.2.n.d 24
11.d odd 10 1 77.2.e.b ✓ 6
11.d odd 10 3 847.2.n.e 24
33.f even 10 1 693.2.i.g 6
44.g even 10 1 1232.2.q.k 6
77.h odd 6 1 847.2.n.e 24
77.l even 10 1 539.2.e.l 6
77.m even 15 1 847.2.e.d 6
77.m even 15 3 inner 847.2.n.d 24
77.m even 15 1 5929.2.a.v 3
77.n even 30 1 539.2.a.i 3
77.n even 30 1 539.2.e.l 6
77.o odd 30 1 77.2.e.b ✓ 6
77.o odd 30 1 539.2.a.h 3
77.o odd 30 3 847.2.n.e 24
77.p odd 30 1 5929.2.a.w 3
231.be even 30 1 693.2.i.g 6
231.be even 30 1 4851.2.a.bo 3
231.bf odd 30 1 4851.2.a.bn 3
308.bc even 30 1 1232.2.q.k 6
308.bc even 30 1 8624.2.a.cl 3
308.bd odd 30 1 8624.2.a.ck 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.e.b ✓ 6 11.d odd 10 1
77.2.e.b ✓ 6 77.o odd 30 1
539.2.a.h 3 77.o odd 30 1
539.2.a.i 3 77.n even 30 1
539.2.e.l 6 77.l even 10 1
539.2.e.l 6 77.n even 30 1
693.2.i.g 6 33.f even 10 1
693.2.i.g 6 231.be even 30 1
847.2.e.d 6 11.c even 5 1
847.2.e.d 6 77.m even 15 1
847.2.n.d 24 1.a even 1 1 trivial
847.2.n.d 24 7.c even 3 1 inner
847.2.n.d 24 11.c even 5 3 inner
847.2.n.d 24 77.m even 15 3 inner
847.2.n.e 24 11.b odd 2 1
847.2.n.e 24 11.d odd 10 3
847.2.n.e 24 77.h odd 6 1
847.2.n.e 24 77.o odd 30 3
1232.2.q.k 6 44.g even 10 1
1232.2.q.k 6 308.bc even 30 1
4851.2.a.bn 3 231.bf odd 30 1
4851.2.a.bo 3 231.be even 30 1
5929.2.a.v 3 77.m even 15 1
5929.2.a.w 3 77.p odd 30 1
8624.2.a.ck 3 308.bd odd 30 1
8624.2.a.cl 3 308.bc even 30 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{24} - 5 T_{2}^{22} + 6 T_{2}^{21} + 30 T_{2}^{19} + 152 T_{2}^{18} - 375 T_{2}^{17} - 445 T_{2}^{16} + \cdots + 6561 \) acting on \(S_{2}^{\mathrm{new}}(847, [\chi])\). Copy content Toggle raw display