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Properties

Label 819.2.et.b
Level $819$
Weight $2$
Character orbit 819.et
Analytic conductor $6.540$
Analytic rank $0$
Dimension $28$
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] = [819,2,Mod(136,819)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(819, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 2, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("819.136");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 819 = 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 819.et (of order \(12\), 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: \(6.53974792554\)
Analytic rank: \(0\)
Dimension: \(28\)
Relative dimension: \(7\) over \(\Q(\zeta_{12})\)
Twist minimal: no (minimal twist has level 91)
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 28 q + 2 q^{2} + 6 q^{5} - 6 q^{7} + 4 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q + 2 q^{2} + 6 q^{5} - 6 q^{7} + 4 q^{8} - 6 q^{10} - 2 q^{11} + 20 q^{14} + 4 q^{16} + 12 q^{17} + 14 q^{19} - 36 q^{20} - 8 q^{22} - 24 q^{26} + 2 q^{28} + 8 q^{29} - 4 q^{31} - 10 q^{32} - 12 q^{34} + 20 q^{35} - 10 q^{37} + 48 q^{40} + 18 q^{41} + 48 q^{43} + 6 q^{44} + 24 q^{46} + 6 q^{47} - 50 q^{49} - 10 q^{50} - 26 q^{52} - 12 q^{53} + 6 q^{55} - 54 q^{56} - 46 q^{58} - 42 q^{59} + 30 q^{61} - 36 q^{62} - 28 q^{65} - 10 q^{67} - 88 q^{70} + 42 q^{71} + 40 q^{73} - 12 q^{74} - 52 q^{76} + 4 q^{79} - 30 q^{80} - 54 q^{82} - 66 q^{83} - 54 q^{85} + 18 q^{86} - 6 q^{88} + 26 q^{91} + 156 q^{92} - 18 q^{94} - 62 q^{97} + 56 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
136.1 −1.74384 + 1.74384i 0 − 4.08193i 0.638637 − 2.38343i 0 −2.04541 + 1.67818i 3.63054 + 3.63054i 0 3.04263 + 5.26998i
136.2 −1.14693 + 1.14693i 0 − 0.630890i −0.395109 + 1.47457i 0 0.0531605 − 2.64522i −1.57027 − 1.57027i 0 −1.23806 − 2.14439i
136.3 −0.490988 + 0.490988i 0 1.51786i 0.00962681 − 0.0359277i 0 −0.176775 + 2.63984i −1.72723 − 1.72723i 0 0.0129135 + 0.0223668i
136.4 0.193244 − 0.193244i 0 1.92531i 0.383199 − 1.43012i 0 −2.15474 − 1.53528i 0.758543 + 0.758543i 0 −0.202311 − 0.350412i
136.5 0.270646 − 0.270646i 0 1.85350i −0.959617 + 3.58134i 0 1.30385 + 2.30217i 1.04293 + 1.04293i 0 0.709559 + 1.22899i
136.6 0.984398 − 0.984398i 0 0.0619199i 0.172312 − 0.643078i 0 −2.46519 − 0.960657i 2.02975 + 2.02975i 0 −0.463421 − 0.802669i
136.7 1.56744 − 1.56744i 0 − 2.91373i 0.784926 − 2.92938i 0 2.25305 + 1.38700i −1.43221 − 1.43221i 0 −3.36131 − 5.82195i
145.1 −1.51485 + 1.51485i 0 − 2.58954i 1.34505 − 0.360406i 0 −0.246373 + 2.63426i 0.893066 + 0.893066i 0 −1.49159 + 2.58351i
145.2 −1.28453 + 1.28453i 0 − 1.30006i −1.07541 + 0.288156i 0 −0.978346 − 2.45822i −0.899098 − 0.899098i 0 1.01126 − 1.75155i
145.3 −0.203761 + 0.203761i 0 1.91696i −0.499383 + 0.133809i 0 −2.60732 − 0.449339i −0.798123 − 0.798123i 0 0.0744896 − 0.129020i
145.4 0.347096 − 0.347096i 0 1.75905i 3.47544 − 0.931242i 0 0.701045 + 2.55118i 1.30475 + 1.30475i 0 0.883082 − 1.52954i
145.5 0.876516 − 0.876516i 0 0.463441i 2.51660 − 0.674321i 0 2.20101 − 1.46818i 2.15924 + 2.15924i 0 1.61479 − 2.79689i
145.6 1.42500 − 1.42500i 0 − 2.06123i −3.16920 + 0.849184i 0 −0.111396 + 2.64341i −0.0872533 − 0.0872533i 0 −3.30601 + 5.72618i
145.7 1.72056 − 1.72056i 0 − 3.92067i −0.227080 + 0.0608458i 0 1.27342 − 2.31914i −3.30464 − 3.30464i 0 −0.286016 + 0.495394i
271.1 −1.74384 − 1.74384i 0 4.08193i 0.638637 + 2.38343i 0 −2.04541 − 1.67818i 3.63054 − 3.63054i 0 3.04263 − 5.26998i
271.2 −1.14693 − 1.14693i 0 0.630890i −0.395109 − 1.47457i 0 0.0531605 + 2.64522i −1.57027 + 1.57027i 0 −1.23806 + 2.14439i
271.3 −0.490988 − 0.490988i 0 − 1.51786i 0.00962681 + 0.0359277i 0 −0.176775 − 2.63984i −1.72723 + 1.72723i 0 0.0129135 − 0.0223668i
271.4 0.193244 + 0.193244i 0 − 1.92531i 0.383199 + 1.43012i 0 −2.15474 + 1.53528i 0.758543 − 0.758543i 0 −0.202311 + 0.350412i
271.5 0.270646 + 0.270646i 0 − 1.85350i −0.959617 − 3.58134i 0 1.30385 − 2.30217i 1.04293 − 1.04293i 0 0.709559 − 1.22899i
271.6 0.984398 + 0.984398i 0 − 0.0619199i 0.172312 + 0.643078i 0 −2.46519 + 0.960657i 2.02975 − 2.02975i 0 −0.463421 + 0.802669i
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 136.7
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
91.ba even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 819.2.et.b 28
3.b odd 2 1 91.2.ba.a yes 28
7.d odd 6 1 819.2.gh.b 28
13.f odd 12 1 819.2.gh.b 28
21.c even 2 1 637.2.bb.a 28
21.g even 6 1 91.2.w.a ✓ 28
21.g even 6 1 637.2.bd.a 28
21.h odd 6 1 637.2.x.a 28
21.h odd 6 1 637.2.bd.b 28
39.k even 12 1 91.2.w.a ✓ 28
91.ba even 12 1 inner 819.2.et.b 28
273.bs odd 12 1 91.2.ba.a yes 28
273.bv even 12 1 637.2.bb.a 28
273.bw even 12 1 637.2.bd.a 28
273.ca odd 12 1 637.2.x.a 28
273.ch odd 12 1 637.2.bd.b 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.w.a ✓ 28 21.g even 6 1
91.2.w.a ✓ 28 39.k even 12 1
91.2.ba.a yes 28 3.b odd 2 1
91.2.ba.a yes 28 273.bs odd 12 1
637.2.x.a 28 21.h odd 6 1
637.2.x.a 28 273.ca odd 12 1
637.2.bb.a 28 21.c even 2 1
637.2.bb.a 28 273.bv even 12 1
637.2.bd.a 28 21.g even 6 1
637.2.bd.a 28 273.bw even 12 1
637.2.bd.b 28 21.h odd 6 1
637.2.bd.b 28 273.ch odd 12 1
819.2.et.b 28 1.a even 1 1 trivial
819.2.et.b 28 91.ba even 12 1 inner
819.2.gh.b 28 7.d odd 6 1
819.2.gh.b 28 13.f odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{28} - 2 T_{2}^{27} + 2 T_{2}^{26} + 77 T_{2}^{24} - 152 T_{2}^{23} + 150 T_{2}^{22} - 6 T_{2}^{21} + \cdots + 9 \) acting on \(S_{2}^{\mathrm{new}}(819, [\chi])\). Copy content Toggle raw display