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Properties

Label 819.2.du.a
Level $819$
Weight $2$
Character orbit 819.du
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(248,819)]
 
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, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("819.248");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 819 = 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 819.du (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.

\(\operatorname{Tr}(f)(q) = \) \( 192 q + 96 q^{4} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 192 q + 96 q^{4} - 4 q^{9} - 12 q^{14} - 6 q^{15} - 96 q^{16} + 18 q^{17} + 20 q^{18} - 32 q^{21} - 36 q^{24} + 192 q^{25} - 36 q^{27} + 36 q^{29} + 26 q^{30} - 60 q^{32} - 24 q^{36} - 60 q^{38} + 12 q^{41} - 66 q^{44} - 18 q^{45} + 12 q^{46} + 60 q^{48} + 24 q^{49} - 48 q^{50} + 44 q^{51} + 24 q^{53} - 114 q^{54} + 144 q^{56} + 10 q^{57} - 24 q^{58} + 30 q^{59} + 16 q^{60} - 36 q^{61} - 36 q^{62} - 38 q^{63} - 192 q^{64} + 48 q^{66} + 84 q^{68} - 4 q^{72} - 30 q^{77} + 10 q^{78} - 12 q^{79} + 52 q^{81} + 60 q^{83} - 96 q^{84} - 102 q^{87} - 84 q^{89} - 54 q^{90} + 42 q^{92} + 28 q^{93} - 120 q^{95} + 18 q^{96} - 102 q^{98} + 10 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
248.1 −2.39384 − 1.38209i −1.46869 − 0.918125i 2.82033 + 4.88495i 3.74694 2.24688 + 4.22770i 2.32445 − 1.26369i − 10.0634i 1.31409 + 2.69688i −8.96959 − 5.17860i
248.2 −2.36436 − 1.36506i 0.175699 − 1.72312i 2.72679 + 4.72294i 0.765731 −2.76758 + 3.83422i −0.947751 + 2.47018i − 9.42871i −2.93826 − 0.605501i −1.81046 − 1.04527i
248.3 −2.33196 − 1.34636i −1.03553 + 1.38841i 2.62537 + 4.54728i −1.88040 4.28411 − 1.84353i 2.33867 − 1.23718i − 8.75334i −0.855361 − 2.87548i 4.38503 + 2.53170i
248.4 −2.27037 − 1.31080i 0.505836 + 1.65654i 2.43639 + 4.21996i −2.42432 1.02296 − 4.42402i −2.54119 + 0.736450i − 7.53130i −2.48826 + 1.67588i 5.50411 + 3.17780i
248.5 −2.24984 − 1.29895i 1.73201 + 0.0121846i 2.37452 + 4.11280i −1.88524 −3.88092 − 2.27720i 0.995403 + 2.45136i − 7.14173i 2.99970 + 0.0422076i 4.24148 + 2.44882i
248.6 −2.19910 − 1.26965i −0.324141 + 1.70145i 2.22403 + 3.85213i 4.04805 2.87307 − 3.33011i −2.41990 + 1.06961i − 6.21637i −2.78987 − 1.10302i −8.90206 − 5.13961i
248.7 −2.19585 − 1.26777i −1.54710 − 0.778770i 2.21449 + 3.83561i −3.88251 2.40989 + 3.67143i 0.608817 + 2.57475i − 6.15880i 1.78704 + 2.40967i 8.52540 + 4.92214i
248.8 −2.08994 − 1.20663i 1.26942 − 1.17838i 1.91190 + 3.31151i 1.19398 −4.07487 + 0.931029i −2.50633 − 0.847534i − 4.40130i 0.222842 − 2.99171i −2.49534 − 1.44069i
248.9 −2.05051 − 1.18386i −1.63258 + 0.578509i 1.80307 + 3.12300i 0.167993 4.03251 + 0.746516i −1.79191 − 1.94655i − 3.80288i 2.33065 − 1.88893i −0.344471 − 0.198880i
248.10 −2.01605 − 1.16397i −1.37406 + 1.05449i 1.70964 + 2.96118i 1.25946 3.99757 − 0.526551i 0.845263 + 2.50710i − 3.30399i 0.776085 − 2.89788i −2.53913 − 1.46597i
248.11 −1.99779 − 1.15342i −1.44823 − 0.950065i 1.66078 + 2.87655i −2.49909 1.79744 + 3.56846i −0.0985594 − 2.64391i − 3.04863i 1.19475 + 2.75183i 4.99266 + 2.88251i
248.12 −1.99641 − 1.15263i −0.135228 − 1.72676i 1.65710 + 2.87017i −0.0322327 −1.72034 + 3.60319i 1.11447 − 2.39957i − 3.02954i −2.96343 + 0.467015i 0.0643497 + 0.0371523i
248.13 −1.97727 − 1.14158i 1.01371 + 1.40442i 1.60641 + 2.78238i 2.22843 −0.401113 − 3.93415i 2.57100 + 0.624451i − 2.76904i −0.944803 + 2.84734i −4.40622 − 2.54393i
248.14 −1.92723 − 1.11269i 1.39230 − 1.03029i 1.47615 + 2.55677i 3.26876 −3.82968 + 0.436405i 1.99408 + 1.73886i − 2.11923i 0.877014 − 2.86895i −6.29967 − 3.63711i
248.15 −1.88458 − 1.08806i 0.165909 − 1.72409i 1.36777 + 2.36904i −2.09283 −2.18859 + 3.06866i 2.60415 + 0.467328i − 1.60061i −2.94495 − 0.572084i 3.94411 + 2.27713i
248.16 −1.86407 − 1.07622i 1.25555 − 1.19314i 1.31650 + 2.28025i −4.31441 −3.62452 + 0.872845i −2.42522 − 1.05750i − 1.36251i 0.152828 − 2.99610i 8.04236 + 4.64326i
248.17 −1.78273 − 1.02926i −1.18054 − 1.26741i 1.11876 + 1.93774i 2.56395 0.800083 + 3.47454i −2.63056 + 0.283152i − 0.488924i −0.212667 + 2.99245i −4.57084 − 2.63898i
248.18 −1.74017 − 1.00469i 0.892498 + 1.48440i 1.01880 + 1.76461i −3.88018 −0.0617378 − 3.47980i 1.70463 − 2.02342i − 0.0755449i −1.40690 + 2.64965i 6.75217 + 3.89837i
248.19 −1.65038 − 0.952850i −1.12199 − 1.31952i 0.815846 + 1.41309i 0.215736 0.594415 + 3.24681i −2.14734 + 1.54562i 0.701884i −0.482265 + 2.96098i −0.356047 − 0.205564i
248.20 −1.60546 − 0.926914i 1.72811 + 0.116830i 0.718340 + 1.24420i −1.95576 −2.66612 − 1.78937i 2.62025 − 0.366453i 1.04430i 2.97270 + 0.403789i 3.13991 + 1.81283i
See next 80 embeddings (of 192 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 248.96
Significant digits:
Format:

Inner twists

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

Hecke kernels

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