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Properties

Label 847.2.f.y
Level $847$
Weight $2$
Character orbit 847.f
Analytic conductor $6.763$
Analytic rank $0$
Dimension $24$
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(148,847)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(847, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("847.148");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 847 = 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 847.f (of order \(5\), degree \(4\), 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: \(6\) over \(\Q(\zeta_{5})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 4 q^{2} + 2 q^{3} - 4 q^{4} + 4 q^{5} + 6 q^{6} - 6 q^{7} - 12 q^{8} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 4 q^{2} + 2 q^{3} - 4 q^{4} + 4 q^{5} + 6 q^{6} - 6 q^{7} - 12 q^{8} - 8 q^{9} - 32 q^{10} - 56 q^{12} - 4 q^{13} - 4 q^{14} - 2 q^{15} - 8 q^{16} - 22 q^{17} - 24 q^{18} - 6 q^{19} - 2 q^{20} - 8 q^{21} + 8 q^{23} + 20 q^{24} - 4 q^{25} - 6 q^{26} + 2 q^{27} - 4 q^{28} - 12 q^{29} - 20 q^{30} + 2 q^{31} + 32 q^{32} + 96 q^{34} + 4 q^{35} - 18 q^{36} - 14 q^{37} + 22 q^{38} - 20 q^{39} - 18 q^{40} - 26 q^{41} + 6 q^{42} - 16 q^{43} - 144 q^{45} - 12 q^{46} + 16 q^{47} + 24 q^{48} - 6 q^{49} + 4 q^{50} + 4 q^{51} - 12 q^{52} - 4 q^{53} - 128 q^{54} + 48 q^{56} - 20 q^{57} + 2 q^{58} + 4 q^{59} - 24 q^{60} + 8 q^{61} - 20 q^{62} - 8 q^{63} - 26 q^{64} + 96 q^{65} + 24 q^{67} - 12 q^{68} + 14 q^{69} + 8 q^{70} - 22 q^{71} - 16 q^{72} - 14 q^{73} - 44 q^{74} + 20 q^{75} - 120 q^{76} + 128 q^{78} + 28 q^{79} + 4 q^{80} + 6 q^{81} + 4 q^{82} - 22 q^{83} + 14 q^{84} + 24 q^{85} + 30 q^{86} + 88 q^{87} + 22 q^{90} - 4 q^{91} - 10 q^{92} + 50 q^{93} + 38 q^{94} + 24 q^{95} + 62 q^{96} + 4 q^{97} + 16 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
148.1 −0.428283 − 1.31812i 0.0990877 + 0.0719914i 0.0640220 − 0.0465147i −0.0411006 + 0.126495i 0.0524557 − 0.161442i −0.809017 + 0.587785i −2.33125 − 1.69375i −0.922415 − 2.83890i 0.184338
148.2 −0.254493 − 0.783248i 0.777181 + 0.564655i 1.06932 − 0.776908i 0.922616 − 2.83952i 0.244478 − 0.752427i −0.809017 + 0.587785i −2.21319 − 1.60798i −0.641876 − 1.97549i −2.45885
148.3 0.0371933 + 0.114469i −2.23923 − 1.62690i 1.60631 − 1.16706i −0.867882 + 2.67107i 0.102945 − 0.316832i −0.809017 + 0.587785i 0.388082 + 0.281958i 1.44031 + 4.43281i −0.338034
148.4 0.394480 + 1.21408i 2.08406 + 1.51415i 0.299647 − 0.217706i −1.26432 + 3.89119i −1.01619 + 3.12752i −0.809017 + 0.587785i 2.44804 + 1.77861i 1.12357 + 3.45799i −5.22298
148.5 0.651837 + 2.00615i −1.37413 − 0.998361i −1.98170 + 1.43979i 0.152157 − 0.468292i 1.10715 − 3.40747i −0.809017 + 0.587785i −0.767119 − 0.557345i −0.0355535 − 0.109422i 1.03864
148.6 0.835334 + 2.57089i 2.27106 + 1.65003i −4.29367 + 3.11953i −0.137535 + 0.423289i −2.34494 + 7.21698i −0.809017 + 0.587785i −7.23277 − 5.25492i 1.50810 + 4.64146i −1.20312
323.1 −2.18693 − 1.58890i −0.867470 + 2.66980i 1.64004 + 5.04751i 0.360071 − 0.261607i 6.13913 − 4.46034i 0.309017 + 0.951057i 2.76267 − 8.50263i −3.94826 − 2.86858i −1.20312
323.2 −1.70653 − 1.23987i 0.524869 − 1.61538i 0.756943 + 2.32963i −0.398353 + 0.289420i −2.89857 + 2.10593i 0.309017 + 0.951057i 0.293014 − 0.901803i 0.0930802 + 0.0676268i 1.03864
323.3 −1.03276 − 0.750346i −0.796038 + 2.44995i −0.114455 − 0.352256i 3.31004 − 2.40489i 2.66043 − 1.93292i 0.309017 + 0.951057i −0.935069 + 2.87785i −2.94155 − 2.13716i −5.22298
323.4 −0.0973732 − 0.0707458i 0.855309 − 2.63237i −0.613557 − 1.88834i 2.27214 − 1.65081i −0.269513 + 0.195813i 0.309017 + 0.951057i −0.148234 + 0.456218i −3.77078 − 2.73963i −0.338034
323.5 0.666271 + 0.484074i −0.296857 + 0.913631i −0.408445 − 1.25706i −2.41544 + 1.75492i −0.640052 + 0.465025i 0.309017 + 0.951057i 0.845363 − 2.60176i 1.68045 + 1.22092i −2.45885
323.6 1.12126 + 0.814642i −0.0378481 + 0.116485i −0.0244542 − 0.0752624i 0.107603 − 0.0781781i −0.137331 + 0.0997767i 0.309017 + 0.951057i 0.890458 − 2.74055i 2.41491 + 1.75454i 0.184338
372.1 −0.428283 + 1.31812i 0.0990877 − 0.0719914i 0.0640220 + 0.0465147i −0.0411006 − 0.126495i 0.0524557 + 0.161442i −0.809017 − 0.587785i −2.33125 + 1.69375i −0.922415 + 2.83890i 0.184338
372.2 −0.254493 + 0.783248i 0.777181 − 0.564655i 1.06932 + 0.776908i 0.922616 + 2.83952i 0.244478 + 0.752427i −0.809017 − 0.587785i −2.21319 + 1.60798i −0.641876 + 1.97549i −2.45885
372.3 0.0371933 − 0.114469i −2.23923 + 1.62690i 1.60631 + 1.16706i −0.867882 − 2.67107i 0.102945 + 0.316832i −0.809017 − 0.587785i 0.388082 − 0.281958i 1.44031 − 4.43281i −0.338034
372.4 0.394480 − 1.21408i 2.08406 − 1.51415i 0.299647 + 0.217706i −1.26432 − 3.89119i −1.01619 − 3.12752i −0.809017 − 0.587785i 2.44804 − 1.77861i 1.12357 − 3.45799i −5.22298
372.5 0.651837 − 2.00615i −1.37413 + 0.998361i −1.98170 − 1.43979i 0.152157 + 0.468292i 1.10715 + 3.40747i −0.809017 − 0.587785i −0.767119 + 0.557345i −0.0355535 + 0.109422i 1.03864
372.6 0.835334 − 2.57089i 2.27106 − 1.65003i −4.29367 − 3.11953i −0.137535 − 0.423289i −2.34494 − 7.21698i −0.809017 − 0.587785i −7.23277 + 5.25492i 1.50810 − 4.64146i −1.20312
729.1 −2.18693 + 1.58890i −0.867470 − 2.66980i 1.64004 − 5.04751i 0.360071 + 0.261607i 6.13913 + 4.46034i 0.309017 − 0.951057i 2.76267 + 8.50263i −3.94826 + 2.86858i −1.20312
729.2 −1.70653 + 1.23987i 0.524869 + 1.61538i 0.756943 − 2.32963i −0.398353 − 0.289420i −2.89857 − 2.10593i 0.309017 − 0.951057i 0.293014 + 0.901803i 0.0930802 − 0.0676268i 1.03864
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 148.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 3 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 847.2.f.y 24
11.b odd 2 1 847.2.f.z 24
11.c even 5 1 847.2.a.n yes 6
11.c even 5 3 inner 847.2.f.y 24
11.d odd 10 1 847.2.a.m ✓ 6
11.d odd 10 3 847.2.f.z 24
33.f even 10 1 7623.2.a.cs 6
33.h odd 10 1 7623.2.a.cp 6
77.j odd 10 1 5929.2.a.bm 6
77.l even 10 1 5929.2.a.bj 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
847.2.a.m ✓ 6 11.d odd 10 1
847.2.a.n yes 6 11.c even 5 1
847.2.f.y 24 1.a even 1 1 trivial
847.2.f.y 24 11.c even 5 3 inner
847.2.f.z 24 11.b odd 2 1
847.2.f.z 24 11.d odd 10 3
5929.2.a.bj 6 77.l even 10 1
5929.2.a.bm 6 77.j odd 10 1
7623.2.a.cp 6 33.h odd 10 1
7623.2.a.cs 6 33.f even 10 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(847, [\chi])\):

\( T_{2}^{24} + 4 T_{2}^{23} + 16 T_{2}^{22} + 52 T_{2}^{21} + 164 T_{2}^{20} + 304 T_{2}^{19} + 687 T_{2}^{18} + \cdots + 1 \) Copy content Toggle raw display
\( T_{3}^{24} - 2 T_{3}^{23} + 15 T_{3}^{22} - 32 T_{3}^{21} + 164 T_{3}^{20} - 254 T_{3}^{19} + 1365 T_{3}^{18} + \cdots + 256 \) Copy content Toggle raw display
\( T_{13}^{24} + 4 T_{13}^{23} + 35 T_{13}^{22} + 144 T_{13}^{21} + 989 T_{13}^{20} + 1744 T_{13}^{19} + \cdots + 65536 \) Copy content Toggle raw display