Properties

Label 819.2.be.a
Level $819$
Weight $2$
Character orbit 819.be
Analytic conductor $6.540$
Analytic rank $0$
Dimension $192$
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(131,819)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(819, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([5, 5, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("819.131");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 819 = 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 819.be (of order \(6\), degree \(2\), 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: \(192\)
Relative dimension: \(96\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 192 q - 192 q^{4} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 192 q - 192 q^{4} + 8 q^{9} + 30 q^{12} - 42 q^{14} - 6 q^{15} + 192 q^{16} - 18 q^{17} - 10 q^{18} + 4 q^{21} - 18 q^{23} - 96 q^{25} + 36 q^{27} + 36 q^{29} - 22 q^{30} - 24 q^{36} - 30 q^{38} - 12 q^{41} + 30 q^{42} + 66 q^{44} - 84 q^{45} + 12 q^{46} - 60 q^{48} - 12 q^{49} - 48 q^{50} - 10 q^{51} - 24 q^{53} - 84 q^{54} + 96 q^{56} + 10 q^{57} + 12 q^{58} + 60 q^{59} + 100 q^{60} + 36 q^{62} + 4 q^{63} - 192 q^{64} + 30 q^{66} + 42 q^{68} - 36 q^{70} + 2 q^{72} - 48 q^{75} - 6 q^{77} + 10 q^{78} + 24 q^{79} - 80 q^{81} - 60 q^{83} - 102 q^{84} - 60 q^{86} - 48 q^{87} + 84 q^{89} + 54 q^{90} + 42 q^{92} - 38 q^{93} - 84 q^{96} + 102 q^{98} + 10 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
131.1 2.75013i −0.525766 + 1.65032i −5.56324 −0.680031 1.17785i 4.53861 + 1.44593i 2.64066 0.164126i 9.79938i −2.44714 1.73537i −3.23924 + 1.87018i
131.2 2.74898i 1.63715 0.565445i −5.55690 −1.29542 2.24373i −1.55440 4.50050i −1.51202 2.17113i 9.77785i 2.36054 1.85144i −6.16798 + 3.56109i
131.3 2.73882i −1.09155 1.34481i −5.50114 1.60660 + 2.78271i −3.68320 + 2.98955i 0.236348 2.63517i 9.58898i −0.617044 + 2.93586i 7.62134 4.40018i
131.4 2.71616i −1.40234 1.01658i −5.37750 −0.333015 0.576798i −2.76118 + 3.80899i −0.171395 + 2.64019i 9.17383i 0.933138 + 2.85118i −1.56667 + 0.904520i
131.5 2.68351i 0.690276 + 1.58856i −5.20121 1.56627 + 2.71286i 4.26291 1.85236i −2.21901 1.44084i 8.59048i −2.04704 + 2.19309i 7.27998 4.20310i
131.6 2.54597i −1.52580 + 0.819716i −4.48196 0.609884 + 1.05635i 2.08697 + 3.88464i −1.82539 + 1.91519i 6.31900i 1.65613 2.50145i 2.68943 1.55275i
131.7 2.45977i 1.40603 1.01147i −4.05047 0.451648 + 0.782277i −2.48798 3.45852i 2.51399 0.824541i 5.04369i 0.953860 2.84432i 1.92422 1.11095i
131.8 2.45370i −1.69378 + 0.362108i −4.02064 −1.04334 1.80712i 0.888504 + 4.15602i −0.243984 2.63448i 4.95804i 2.73776 1.22666i −4.43414 + 2.56005i
131.9 2.42664i 1.65199 + 0.520509i −3.88858 0.293231 + 0.507891i 1.26309 4.00878i −0.999638 + 2.44964i 4.58290i 2.45814 + 1.71975i 1.23247 0.711565i
131.10 2.40217i 1.19422 + 1.25453i −3.77044 −1.77637 3.07677i 3.01360 2.86871i 1.13710 + 2.38893i 4.25291i −0.147698 + 2.99636i −7.39093 + 4.26715i
131.11 2.36062i 0.342821 1.69778i −3.57253 0.0272649 + 0.0472242i −4.00783 0.809270i −2.55198 0.698138i 3.71214i −2.76495 1.16407i 0.111478 0.0643620i
131.12 2.27732i 1.58919 + 0.688832i −3.18620 0.872025 + 1.51039i 1.56869 3.61909i 2.08014 1.63493i 2.70135i 2.05102 + 2.18936i 3.43965 1.98588i
131.13 2.20642i −0.106553 1.72877i −2.86827 0.566894 + 0.981889i −3.81438 + 0.235101i 2.63987 0.176277i 1.91576i −2.97729 + 0.368412i 2.16646 1.25080i
131.14 2.18169i −0.805241 + 1.53349i −2.75975 −2.02931 3.51487i 3.34559 + 1.75678i −2.62703 + 0.314218i 1.65754i −1.70317 2.46965i −7.66833 + 4.42731i
131.15 2.14803i −0.0525246 + 1.73125i −2.61402 1.40260 + 2.42937i 3.71878 + 0.112824i 1.44779 + 2.21447i 1.31894i −2.99448 0.181867i 5.21836 3.01282i
131.16 2.06287i −1.17177 1.27552i −2.25542 −1.56599 2.71237i −2.63123 + 2.41720i −2.52851 + 0.778876i 0.526900i −0.253916 + 2.98924i −5.59525 + 3.23042i
131.17 2.03908i −1.26173 + 1.18661i −2.15784 1.34835 + 2.33541i 2.41959 + 2.57276i −1.96236 1.77459i 0.321848i 0.183910 2.99436i 4.76208 2.74939i
131.18 1.94462i 1.40048 1.01915i −1.78156 −1.83006 3.16976i −1.98186 2.72340i 2.51888 + 0.809472i 0.424777i 0.922674 2.85459i −6.16400 + 3.55879i
131.19 1.93848i 0.0119177 + 1.73201i −1.75770 −0.441415 0.764554i 3.35746 0.0231021i −0.111737 2.64339i 0.469696i −2.99972 + 0.0412830i −1.48207 + 0.855674i
131.20 1.82356i −1.62712 0.593694i −1.32536 0.468964 + 0.812270i −1.08263 + 2.96715i 0.0300109 2.64558i 1.23025i 2.29505 + 1.93203i 1.48122 0.855183i
See next 80 embeddings (of 192 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 131.96
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.i even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 819.2.be.a 192
7.d odd 6 1 819.2.du.a yes 192
9.d odd 6 1 819.2.du.a yes 192
63.i even 6 1 inner 819.2.be.a 192
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
819.2.be.a 192 1.a even 1 1 trivial
819.2.be.a 192 63.i even 6 1 inner
819.2.du.a yes 192 7.d odd 6 1
819.2.du.a yes 192 9.d odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(819, [\chi])\).