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Properties

Label 806.2.bk.c
Level $806$
Weight $2$
Character orbit 806.bk
Analytic conductor $6.436$
Analytic rank $0$
Dimension $72$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [806,2,Mod(131,806)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(806, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([0, 28]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("806.131");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 806 = 2 \cdot 13 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 806.bk (of order \(15\), degree \(8\), 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: \(6.43594240292\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(9\) over \(\Q(\zeta_{15})\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 72 q + 18 q^{2} - q^{3} - 18 q^{4} + 5 q^{5} + 6 q^{6} + 17 q^{7} + 18 q^{8} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 72 q + 18 q^{2} - q^{3} - 18 q^{4} + 5 q^{5} + 6 q^{6} + 17 q^{7} + 18 q^{8} + 6 q^{9} - 10 q^{10} - 13 q^{11} - q^{12} + 9 q^{13} + 13 q^{14} - 11 q^{15} - 18 q^{16} + 5 q^{17} - 31 q^{18} + q^{19} - 5 q^{20} + 30 q^{21} - 2 q^{22} - 27 q^{23} - 4 q^{24} - 67 q^{25} + 36 q^{26} + 26 q^{27} + 2 q^{28} - 2 q^{29} + 6 q^{30} - 15 q^{31} - 72 q^{32} - 9 q^{33} - 6 q^{35} - 44 q^{36} + 5 q^{37} + 44 q^{38} - 8 q^{39} + 5 q^{40} - 22 q^{41} - 15 q^{42} - 11 q^{43} + 22 q^{44} + 17 q^{45} - 33 q^{46} - 12 q^{47} + 4 q^{48} - 8 q^{50} + 87 q^{51} + 9 q^{52} - 23 q^{53} - 26 q^{54} - 22 q^{55} + 8 q^{56} + 8 q^{57} + 17 q^{58} + 31 q^{59} - 11 q^{60} + 26 q^{61} - 10 q^{62} + 52 q^{63} - 18 q^{64} + 10 q^{65} - 21 q^{66} - 56 q^{67} - 10 q^{68} + 152 q^{69} + 6 q^{70} - 4 q^{71} - q^{72} - 50 q^{73} + 5 q^{74} + 119 q^{75} + 46 q^{76} - 16 q^{77} - 2 q^{78} - 16 q^{79} - 5 q^{80} + 8 q^{81} - 8 q^{82} + 77 q^{83} - 3 q^{85} - 4 q^{86} - 64 q^{87} - 12 q^{88} + 36 q^{89} - 72 q^{90} - 4 q^{91} - 12 q^{92} - 118 q^{93} + 62 q^{94} - 37 q^{95} + q^{96} - 19 q^{97} + 20 q^{98} + 51 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
131.1 −0.309017 − 0.951057i −2.70533 + 0.575036i −0.809017 + 0.587785i 1.84399 − 3.19389i 1.38289 + 2.39523i −3.76511 − 1.67633i 0.809017 + 0.587785i 4.24752 − 1.89112i −3.60739 − 0.766774i
131.2 −0.309017 − 0.951057i −2.19505 + 0.466573i −0.809017 + 0.587785i −1.31993 + 2.28619i 1.12205 + 1.94344i 4.77414 + 2.12558i 0.809017 + 0.587785i 1.85993 − 0.828093i 2.58218 + 0.548860i
131.3 −0.309017 − 0.951057i −2.07913 + 0.441934i −0.809017 + 0.587785i 1.81133 − 3.13732i 1.06279 + 1.84081i 1.59299 + 0.709246i 0.809017 + 0.587785i 1.38686 − 0.617469i −3.54350 − 0.753194i
131.4 −0.309017 − 0.951057i −2.00044 + 0.425207i −0.809017 + 0.587785i −1.35368 + 2.34465i 1.02257 + 1.77114i −2.51087 − 1.11791i 0.809017 + 0.587785i 1.08033 − 0.480992i 2.64820 + 0.562893i
131.5 −0.309017 − 0.951057i −0.422776 + 0.0898637i −0.809017 + 0.587785i 0.809306 − 1.40176i 0.216110 + 0.374314i 1.11167 + 0.494949i 0.809017 + 0.587785i −2.56997 + 1.14423i −1.58324 − 0.336528i
131.6 −0.309017 − 0.951057i 0.0818419 − 0.0173960i −0.809017 + 0.587785i −0.597246 + 1.03446i −0.0418351 − 0.0724606i 1.57189 + 0.699853i 0.809017 + 0.587785i −2.73424 + 1.21736i 1.16839 + 0.248349i
131.7 −0.309017 − 0.951057i 1.36026 − 0.289132i −0.809017 + 0.587785i −0.788259 + 1.36530i −0.695323 − 1.20433i −1.72127 − 0.766357i 0.809017 + 0.587785i −0.973934 + 0.433623i 1.54207 + 0.327776i
131.8 −0.309017 − 0.951057i 2.31668 − 0.492426i −0.809017 + 0.587785i 0.966494 − 1.67402i −1.18422 − 2.05113i 2.06438 + 0.919120i 0.809017 + 0.587785i 2.38389 − 1.06137i −1.89075 − 0.401891i
131.9 −0.309017 − 0.951057i 3.08313 − 0.655340i −0.809017 + 0.587785i 0.915163 − 1.58511i −1.57601 − 2.72972i 0.569376 + 0.253503i 0.809017 + 0.587785i 6.33560 − 2.82079i −1.79033 − 0.380546i
183.1 0.809017 − 0.587785i −0.336673 + 3.20323i 0.309017 − 0.951057i 1.91379 − 3.31478i 1.61044 + 2.78936i 2.31181 − 0.491391i −0.309017 − 0.951057i −7.21289 − 1.53315i −0.400091 − 3.80661i
183.2 0.809017 − 0.587785i −0.329786 + 3.13770i 0.309017 − 0.951057i −1.49555 + 2.59038i 1.57749 + 2.73230i −0.915206 + 0.194533i −0.309017 − 0.951057i −6.80197 − 1.44580i 0.312656 + 2.97472i
183.3 0.809017 − 0.587785i −0.181168 + 1.72370i 0.309017 − 0.951057i −0.179115 + 0.310237i 0.866595 + 1.50099i 0.455036 − 0.0967210i −0.309017 − 0.951057i −0.00386310 0.000821127i 0.0374453 + 0.356268i
183.4 0.809017 − 0.587785i −0.0863538 + 0.821601i 0.309017 − 0.951057i 0.242565 − 0.420135i 0.413063 + 0.715447i 0.661165 − 0.140535i −0.309017 − 0.951057i 2.26687 + 0.481838i −0.0507099 − 0.482473i
183.5 0.809017 − 0.587785i 0.0634124 − 0.603328i 0.309017 − 0.951057i 2.08161 − 3.60546i −0.303326 − 0.525376i −4.02740 + 0.856050i −0.309017 − 0.951057i 2.57446 + 0.547218i −0.435176 − 4.14042i
183.6 0.809017 − 0.587785i 0.100008 − 0.951509i 0.309017 − 0.951057i 0.0166005 − 0.0287530i −0.478375 − 0.828570i 2.71024 − 0.576079i −0.309017 − 0.951057i 2.03907 + 0.433419i −0.00347046 − 0.0330192i
183.7 0.809017 − 0.587785i 0.175936 − 1.67392i 0.309017 − 0.951057i −2.05161 + 3.55349i −0.841571 − 1.45764i −0.131468 + 0.0279445i −0.309017 − 0.951057i 0.163386 + 0.0347288i 0.428903 + 4.08074i
183.8 0.809017 − 0.587785i 0.259078 − 2.46496i 0.309017 − 0.951057i 0.743630 − 1.28801i −1.23927 − 2.14647i 3.59055 − 0.763196i −0.309017 − 0.951057i −3.07445 − 0.653495i −0.155461 − 1.47911i
183.9 0.809017 − 0.587785i 0.295620 − 2.81264i 0.309017 − 0.951057i −0.976410 + 1.69119i −1.41407 − 2.44923i −3.38847 + 0.720241i −0.309017 − 0.951057i −4.88910 − 1.03921i 0.204125 + 1.94212i
235.1 0.809017 − 0.587785i −2.49143 + 1.10926i 0.309017 − 0.951057i 0.435014 + 0.753466i −1.36361 + 2.36184i −1.12464 − 1.24904i −0.309017 − 0.951057i 2.96940 − 3.29785i 0.794810 + 0.353872i
235.2 0.809017 − 0.587785i −1.83768 + 0.818189i 0.309017 − 0.951057i 0.910956 + 1.57782i −1.00580 + 1.74209i 2.95535 + 3.28225i −0.309017 − 0.951057i 0.700253 − 0.777710i 1.66440 + 0.741039i
See all 72 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 131.9
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.g even 15 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 806.2.bk.c ✓ 72
31.g even 15 1 inner 806.2.bk.c ✓ 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
806.2.bk.c ✓ 72 1.a even 1 1 trivial
806.2.bk.c ✓ 72 31.g even 15 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{72} + T_{3}^{71} - 16 T_{3}^{70} - 9 T_{3}^{69} + 23 T_{3}^{68} + 44 T_{3}^{67} + 1971 T_{3}^{66} + \cdots + 38950081 \) acting on \(S_{2}^{\mathrm{new}}(806, [\chi])\). Copy content Toggle raw display