Properties

Label 946.2.l.a
Level $946$
Weight $2$
Character orbit 946.l
Analytic conductor $7.554$
Analytic rank $0$
Dimension $88$
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] = [946,2,Mod(85,946)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(946, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([3, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("946.85");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 946 = 2 \cdot 11 \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 946.l (of order \(10\), degree \(4\), 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: \(7.55384803121\)
Analytic rank: \(0\)
Dimension: \(88\)
Relative dimension: \(22\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 88 q - 22 q^{2} - 22 q^{4} + 5 q^{6} - 22 q^{8} + 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 88 q - 22 q^{2} - 22 q^{4} + 5 q^{6} - 22 q^{8} + 20 q^{9} - 7 q^{11} + 5 q^{13} - 8 q^{15} - 22 q^{16} - 10 q^{17} + 25 q^{18} - 13 q^{19} - 2 q^{22} + 10 q^{23} - 5 q^{24} + 6 q^{25} - 20 q^{26} + 15 q^{27} + 6 q^{29} - 8 q^{30} - 4 q^{31} + 88 q^{32} - 25 q^{33} + 60 q^{35} + 25 q^{36} + 20 q^{37} + 22 q^{38} - 36 q^{39} - 5 q^{43} + 13 q^{44} - 5 q^{46} - 17 q^{47} + 12 q^{49} + 36 q^{50} - 21 q^{51} + 20 q^{52} - 56 q^{55} - 5 q^{57} + 6 q^{58} + 15 q^{59} + 12 q^{60} - 24 q^{61} - 9 q^{62} - 8 q^{63} - 22 q^{64} + 52 q^{65} - 10 q^{66} + 12 q^{67} + 10 q^{68} + 30 q^{69} - 30 q^{71} + 20 q^{72} - 14 q^{73} - 20 q^{74} + 105 q^{75} - 18 q^{76} + 28 q^{77} + 4 q^{78} - 20 q^{79} - 25 q^{81} + 5 q^{82} + 5 q^{83} - 2 q^{85} + 30 q^{86} - 2 q^{88} - 30 q^{91} - 80 q^{93} + 18 q^{94} - 50 q^{95} + 28 q^{97} + 52 q^{98} + 92 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
85.1 −0.809017 + 0.587785i −2.99176 + 0.972082i 0.309017 0.951057i 0.0144910 0.0199451i 1.84901 2.54494i −0.635199 + 1.95494i 0.309017 + 0.951057i 5.57863 4.05311i 0.0246536i
85.2 −0.809017 + 0.587785i −2.64506 + 0.859431i 0.309017 0.951057i 2.19210 3.01717i 1.63473 2.25002i 0.978717 3.01218i 0.309017 + 0.951057i 3.83065 2.78313i 3.72943i
85.3 −0.809017 + 0.587785i −2.53894 + 0.824950i 0.309017 0.951057i −2.24874 + 3.09512i 1.56915 2.15975i 0.718873 2.21246i 0.309017 + 0.951057i 3.33860 2.42564i 3.82578i
85.4 −0.809017 + 0.587785i −2.22721 + 0.723664i 0.309017 0.951057i 0.743511 1.02336i 1.37649 1.89458i −0.765347 + 2.35549i 0.309017 + 0.951057i 2.00972 1.46015i 1.26494i
85.5 −0.809017 + 0.587785i −2.14623 + 0.697353i 0.309017 0.951057i −0.292520 + 0.402619i 1.32644 1.82569i −0.706871 + 2.17552i 0.309017 + 0.951057i 1.69296 1.23001i 0.497664i
85.6 −0.809017 + 0.587785i −2.00563 + 0.651669i 0.309017 0.951057i −1.09227 + 1.50338i 1.23955 1.70609i 0.593774 1.82745i 0.309017 + 0.951057i 1.17083 0.850656i 1.85828i
85.7 −0.809017 + 0.587785i −1.17692 + 0.382406i 0.309017 0.951057i 1.85246 2.54969i 0.727379 1.00115i 0.358232 1.10253i 0.309017 + 0.951057i −1.18814 + 0.863232i 3.15160i
85.8 −0.809017 + 0.587785i −0.875069 + 0.284327i 0.309017 0.951057i −2.08664 + 2.87201i 0.540822 0.744378i −0.991886 + 3.05271i 0.309017 + 0.951057i −1.74215 + 1.26574i 3.55000i
85.9 −0.809017 + 0.587785i −0.872172 + 0.283386i 0.309017 0.951057i 0.425915 0.586221i 0.539032 0.741914i 1.32297 4.07168i 0.309017 + 0.951057i −1.74667 + 1.26903i 0.724609i
85.10 −0.809017 + 0.587785i −0.441079 + 0.143315i 0.309017 0.951057i −0.893157 + 1.22933i 0.272602 0.375205i −1.18417 + 3.64449i 0.309017 + 0.951057i −2.25304 + 1.63693i 1.51953i
85.11 −0.809017 + 0.587785i −0.298204 + 0.0968924i 0.309017 0.951057i 1.80281 2.48135i 0.184300 0.253668i 0.489048 1.50513i 0.309017 + 0.951057i −2.34751 + 1.70557i 3.06712i
85.12 −0.809017 + 0.587785i −0.124065 + 0.0403112i 0.309017 0.951057i −0.207165 + 0.285138i 0.0766765 0.105536i 1.05333 3.24181i 0.309017 + 0.951057i −2.41328 + 1.75335i 0.352450i
85.13 −0.809017 + 0.587785i −0.0407586 + 0.0132433i 0.309017 0.951057i 2.35495 3.24131i 0.0251902 0.0346714i −1.44902 + 4.45963i 0.309017 + 0.951057i −2.42557 + 1.76228i 4.00648i
85.14 −0.809017 + 0.587785i 0.619989 0.201447i 0.309017 0.951057i −2.02952 + 2.79339i −0.383174 + 0.527394i 0.687411 2.11564i 0.309017 + 0.951057i −2.08325 + 1.51357i 3.45282i
85.15 −0.809017 + 0.587785i 0.917762 0.298199i 0.309017 0.951057i 1.35569 1.86595i −0.567208 + 0.780695i −0.732700 + 2.25502i 0.309017 + 0.951057i −1.67369 + 1.21600i 2.30644i
85.16 −0.809017 + 0.587785i 1.27926 0.415656i 0.309017 0.951057i −0.441745 + 0.608010i −0.790625 + 1.08820i 0.155387 0.478232i 0.309017 + 0.951057i −0.963319 + 0.699892i 0.751542i
85.17 −0.809017 + 0.587785i 1.34123 0.435794i 0.309017 0.951057i −0.610055 + 0.839668i −0.828929 + 1.14092i 0.0795635 0.244871i 0.309017 + 0.951057i −0.818056 + 0.594352i 1.03789i
85.18 −0.809017 + 0.587785i 2.15707 0.700873i 0.309017 0.951057i −1.57860 + 2.17275i −1.33314 + 1.83491i −1.12581 + 3.46488i 0.309017 + 0.951057i 1.73466 1.26030i 2.68567i
85.19 −0.809017 + 0.587785i 2.30619 0.749325i 0.309017 0.951057i 1.40834 1.93841i −1.42530 + 1.96176i −0.0303287 + 0.0933420i 0.309017 + 0.951057i 2.32996 1.69281i 2.39600i
85.20 −0.809017 + 0.587785i 2.43207 0.790226i 0.309017 0.951057i −0.118871 + 0.163612i −1.50310 + 2.06884i 0.609357 1.87541i 0.309017 + 0.951057i 2.86344 2.08041i 0.202236i
See all 88 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 85.22
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
473.k even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 946.2.l.a 88
11.d odd 10 1 946.2.l.b yes 88
43.b odd 2 1 946.2.l.b yes 88
473.k even 10 1 inner 946.2.l.a 88
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
946.2.l.a 88 1.a even 1 1 trivial
946.2.l.a 88 473.k even 10 1 inner
946.2.l.b yes 88 11.d odd 10 1
946.2.l.b yes 88 43.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{88} - 43 T_{3}^{86} + 5 T_{3}^{85} + 1086 T_{3}^{84} - 215 T_{3}^{83} - 21099 T_{3}^{82} + 5945 T_{3}^{81} + 349477 T_{3}^{80} - 119510 T_{3}^{79} - 4973995 T_{3}^{78} + 2030250 T_{3}^{77} + \cdots + 25825775616 \) acting on \(S_{2}^{\mathrm{new}}(946, [\chi])\). Copy content Toggle raw display