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Properties

Label 819.2.m.f
Level $819$
Weight $2$
Character orbit 819.m
Analytic conductor $6.540$
Analytic rank $0$
Dimension $36$
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(274,819)]
 
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([2, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("819.274");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 819 = 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 819.m (of order \(3\), 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: \(36\)
Relative dimension: \(18\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 36 q - 2 q^{2} + 2 q^{3} - 18 q^{4} - 7 q^{5} - 4 q^{6} + 18 q^{7} + 12 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q - 2 q^{2} + 2 q^{3} - 18 q^{4} - 7 q^{5} - 4 q^{6} + 18 q^{7} + 12 q^{8} + 2 q^{9} - 12 q^{10} - 7 q^{11} + 3 q^{12} - 18 q^{13} + 2 q^{14} - 12 q^{15} - 18 q^{16} + 34 q^{17} - 6 q^{18} + 6 q^{19} - 14 q^{20} + q^{21} - 15 q^{22} - 14 q^{23} - 39 q^{24} - 21 q^{25} + 4 q^{26} - 4 q^{27} - 36 q^{28} - 36 q^{29} + 4 q^{30} - 3 q^{31} - 14 q^{32} - 17 q^{33} + 12 q^{34} - 14 q^{35} + 58 q^{36} + 6 q^{37} + 2 q^{38} - q^{39} + 33 q^{40} - 31 q^{41} - 8 q^{42} - 15 q^{43} + 36 q^{44} + 35 q^{45} - 12 q^{46} + 4 q^{47} - 26 q^{48} - 18 q^{49} - 21 q^{50} + 17 q^{51} - 18 q^{52} + 104 q^{53} + 70 q^{54} + 18 q^{55} + 6 q^{56} - 4 q^{57} - 12 q^{58} - 4 q^{59} - 73 q^{60} + 9 q^{61} + 58 q^{62} + 7 q^{63} - 36 q^{64} - 7 q^{65} + 18 q^{66} + 6 q^{67} - 17 q^{68} + 15 q^{69} - 6 q^{70} + 30 q^{71} + 109 q^{72} + 18 q^{73} - 27 q^{74} - 83 q^{75} - 3 q^{76} + 7 q^{77} - 4 q^{78} + 12 q^{79} - 26 q^{80} + 54 q^{81} + 6 q^{82} + 15 q^{83} - 12 q^{84} - 41 q^{86} - 13 q^{87} + 27 q^{88} - 10 q^{89} + 7 q^{90} - 36 q^{91} - 81 q^{92} - 80 q^{93} + 6 q^{94} - 54 q^{95} + 85 q^{96} - 30 q^{97} + 4 q^{98} + 58 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
274.1 −1.33261 − 2.30815i −1.73203 + 0.00771438i −2.55169 + 4.41966i 1.63903 − 2.83889i 2.32593 + 3.98751i 0.500000 + 0.866025i 8.27118 2.99988 − 0.0267231i −8.73676
274.2 −1.28139 − 2.21943i −0.0332475 − 1.73173i −2.28392 + 3.95586i −0.934061 + 1.61784i −3.80086 + 2.29281i 0.500000 + 0.866025i 6.58079 −2.99779 + 0.115152i 4.78758
274.3 −1.14388 − 1.98126i 1.71280 + 0.257527i −1.61693 + 2.80061i −1.31551 + 2.27853i −1.44901 − 3.68808i 0.500000 + 0.866025i 2.82279 2.86736 + 0.882184i 6.01916
274.4 −1.01710 − 1.76167i 1.15148 − 1.29387i −1.06899 + 1.85154i 1.28679 − 2.22878i −3.45054 − 0.712538i 0.500000 + 0.866025i 0.280670 −0.348182 − 2.97973i −5.23517
274.5 −0.924832 − 1.60186i −0.856254 + 1.50560i −0.710627 + 1.23084i −1.83762 + 3.18285i 3.20364 − 0.0208315i 0.500000 + 0.866025i −1.07048 −1.53366 − 2.57835i 6.79795
274.6 −0.688507 − 1.19253i 1.40384 + 1.01451i 0.0519166 − 0.0899222i 0.149565 − 0.259054i 0.243280 − 2.37262i 0.500000 + 0.866025i −2.89701 0.941534 + 2.84842i −0.411905
274.7 −0.350507 − 0.607096i −1.71493 + 0.242903i 0.754289 − 1.30647i −0.338522 + 0.586338i 0.748562 + 0.955991i 0.500000 + 0.866025i −2.45956 2.88200 − 0.833124i 0.474618
274.8 −0.319536 − 0.553453i −0.0362785 + 1.73167i 0.795794 − 1.37835i −0.640782 + 1.10987i 0.969990 − 0.533253i 0.500000 + 0.866025i −2.29528 −2.99737 − 0.125645i 0.819012
274.9 −0.276821 − 0.479468i 0.409538 − 1.68294i 0.846740 − 1.46660i 1.43150 − 2.47943i −0.920283 + 0.269512i 0.500000 + 0.866025i −2.04486 −2.66456 − 1.37846i −1.58508
274.10 −0.0278051 − 0.0481598i −1.31692 − 1.12504i 0.998454 − 1.72937i −1.73113 + 2.99840i −0.0175644 + 0.0947046i 0.500000 + 0.866025i −0.222269 0.468580 + 2.96318i 0.192537
274.11 0.250511 + 0.433898i 1.55930 + 0.754045i 0.874489 − 1.51466i 1.34303 − 2.32620i 0.0634430 + 0.865473i 0.500000 + 0.866025i 1.87832 1.86283 + 2.35157i 1.34578
274.12 0.388771 + 0.673371i 1.71552 − 0.238730i 0.697715 − 1.20848i −1.60620 + 2.78201i 0.827697 + 1.06237i 0.500000 + 0.866025i 2.64009 2.88602 − 0.819091i −2.49777
274.13 0.560564 + 0.970925i −1.42007 − 0.991668i 0.371536 − 0.643519i 1.65705 − 2.87009i 0.166796 − 1.93467i 0.500000 + 0.866025i 3.07533 1.03319 + 2.81647i 3.71553
274.14 0.650988 + 1.12754i −1.57445 + 0.721870i 0.152429 − 0.264015i −0.255938 + 0.443298i −1.83889 − 1.30534i 0.500000 + 0.866025i 3.00087 1.95781 − 2.27310i −0.666450
274.15 0.917211 + 1.58866i −0.510282 + 1.65518i −0.682553 + 1.18222i −0.0639793 + 0.110815i −3.09754 + 0.707485i 0.500000 + 0.866025i 1.16466 −2.47923 − 1.68921i −0.234730
274.16 1.16212 + 2.01284i 0.393500 − 1.68676i −1.70103 + 2.94627i −2.09366 + 3.62633i 3.85248 − 1.16816i 0.500000 + 0.866025i −3.25871 −2.69032 − 1.32748i −9.73232
274.17 1.16904 + 2.02483i 1.51479 − 0.839887i −1.73329 + 3.00215i 0.247196 − 0.428156i 3.47147 + 2.08533i 0.500000 + 0.866025i −3.42898 1.58918 − 2.54451i 1.15592
274.18 1.26379 + 2.18895i 0.333707 + 1.69960i −2.19433 + 3.80070i −0.436766 + 0.756501i −3.29860 + 2.87841i 0.500000 + 0.866025i −6.03755 −2.77728 + 1.13434i −2.20792
547.1 −1.33261 + 2.30815i −1.73203 − 0.00771438i −2.55169 − 4.41966i 1.63903 + 2.83889i 2.32593 − 3.98751i 0.500000 − 0.866025i 8.27118 2.99988 + 0.0267231i −8.73676
547.2 −1.28139 + 2.21943i −0.0332475 + 1.73173i −2.28392 − 3.95586i −0.934061 − 1.61784i −3.80086 − 2.29281i 0.500000 − 0.866025i 6.58079 −2.99779 − 0.115152i 4.78758
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 274.18
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 819.2.m.f ✓ 36
9.c even 3 1 inner 819.2.m.f ✓ 36
9.c even 3 1 7371.2.a.bb 18
9.d odd 6 1 7371.2.a.ba 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
819.2.m.f ✓ 36 1.a even 1 1 trivial
819.2.m.f ✓ 36 9.c even 3 1 inner
7371.2.a.ba 18 9.d odd 6 1
7371.2.a.bb 18 9.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{36} + 2 T_{2}^{35} + 29 T_{2}^{34} + 46 T_{2}^{33} + 468 T_{2}^{32} + 650 T_{2}^{31} + 5043 T_{2}^{30} + \cdots + 256 \) acting on \(S_{2}^{\mathrm{new}}(819, [\chi])\). Copy content Toggle raw display