Properties

Label 819.2.ep.a
Level $819$
Weight $2$
Character orbit 819.ep
Analytic conductor $6.540$
Analytic rank $0$
Dimension $432$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [819,2,Mod(229,819)]
 
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([4, 10, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("819.229");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 819 = 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 819.ep (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: \(432\)
Relative dimension: \(108\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 432 q - 4 q^{2} - 12 q^{3} - 6 q^{5} - 12 q^{6} - 8 q^{8} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 432 q - 4 q^{2} - 12 q^{3} - 6 q^{5} - 12 q^{6} - 8 q^{8} - 4 q^{9} + 6 q^{11} - 4 q^{14} - 14 q^{15} - 392 q^{16} - 10 q^{18} - 12 q^{19} + 60 q^{20} - 18 q^{21} - 8 q^{22} - 12 q^{26} + 36 q^{27} + 4 q^{28} - 24 q^{29} - 20 q^{32} - 6 q^{33} + 12 q^{34} - 60 q^{35} - 2 q^{37} - 28 q^{39} - 12 q^{40} + 68 q^{42} + 32 q^{44} - 6 q^{45} + 4 q^{46} + 12 q^{48} + 22 q^{50} + 66 q^{52} + 8 q^{53} + 18 q^{54} - 44 q^{57} - 6 q^{58} + 50 q^{60} + 50 q^{63} - 32 q^{65} - 36 q^{66} - 32 q^{67} - 12 q^{68} + 2 q^{70} + 48 q^{71} + 22 q^{72} - 12 q^{73} - 52 q^{74} + 48 q^{76} + 2 q^{78} - 56 q^{79} - 6 q^{80} + 4 q^{81} - 84 q^{83} - 68 q^{84} - 6 q^{85} - 42 q^{86} - 108 q^{87} + 18 q^{89} + 52 q^{92} + 90 q^{93} - 6 q^{96} - 6 q^{97} - 112 q^{98} - 50 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
229.1 −1.96596 1.96596i −1.25489 + 1.19384i 5.72998i −0.404127 0.108286i 4.81410 + 0.120037i −1.02182 + 2.44047i 7.33299 7.33299i 0.149514 2.99627i 0.581612 + 1.00738i
229.2 −1.93236 1.93236i 1.39105 1.03198i 5.46804i −3.14351 0.842300i −4.68217 0.693857i 0.961386 2.46490i 6.70151 6.70151i 0.870042 2.87107i 4.44676 + 7.70202i
229.3 −1.89917 1.89917i −0.785142 1.54388i 5.21370i −2.25967 0.605476i −1.44096 + 4.42320i −2.56472 + 0.649790i 6.10335 6.10335i −1.76710 + 2.42432i 3.14159 + 5.44139i
229.4 −1.89057 1.89057i 1.73151 + 0.0432369i 5.14849i 3.71009 + 0.994115i −3.19180 3.35528i −2.52357 0.794740i 5.95244 5.95244i 2.99626 + 0.149731i −5.13473 8.89361i
229.5 −1.88944 1.88944i 0.160590 + 1.72459i 5.13995i 0.658675 + 0.176491i 2.95508 3.56193i −1.14253 2.38634i 5.93274 5.93274i −2.94842 + 0.553903i −0.911055 1.57799i
229.6 −1.88804 1.88804i −1.68739 0.390799i 5.12938i 2.57278 + 0.689373i 2.44801 + 3.92370i 1.85236 1.88912i 5.90840 5.90840i 2.69455 + 1.31886i −3.55594 6.15907i
229.7 −1.75840 1.75840i 0.504792 + 1.65686i 4.18391i −3.80271 1.01893i 2.02579 3.80104i 2.37359 + 1.16879i 3.84019 3.84019i −2.49037 + 1.67274i 4.89498 + 8.47835i
229.8 −1.74030 1.74030i 1.46132 + 0.929814i 4.05728i 0.578919 + 0.155121i −0.924973 4.16128i 2.60158 0.481414i 3.58028 3.58028i 1.27089 + 2.71751i −0.737536 1.27745i
229.9 −1.66329 1.66329i 0.126557 1.72742i 3.53307i 0.103056 + 0.0276137i −3.08370 + 2.66270i 1.55851 + 2.13800i 2.54993 2.54993i −2.96797 0.437234i −0.125482 0.217342i
229.10 −1.64531 1.64531i −1.67759 + 0.430934i 3.41408i −3.69224 0.989333i 3.46917 + 2.05113i 0.423123 2.61170i 2.32660 2.32660i 2.62859 1.44586i 4.44712 + 7.70264i
229.11 −1.63578 1.63578i −1.32238 1.11862i 3.35153i −1.82513 0.489041i 0.333304 + 3.99293i 2.40039 + 1.11271i 2.21080 2.21080i 0.497375 + 2.95848i 2.18554 + 3.78546i
229.12 −1.60413 1.60413i −1.72570 + 0.148152i 3.14644i 1.50819 + 0.404118i 3.00590 + 2.53059i 0.912160 + 2.48354i 1.83903 1.83903i 2.95610 0.511333i −1.77107 3.06758i
229.13 −1.60407 1.60407i 0.794613 + 1.53902i 3.14605i 2.37922 + 0.637509i 1.19408 3.74330i −0.371203 + 2.61958i 1.83834 1.83834i −1.73718 + 2.44585i −2.79381 4.83902i
229.14 −1.60074 1.60074i 1.07753 1.35607i 3.12472i 0.414662 + 0.111108i −3.89556 + 0.445865i −2.57658 + 0.601027i 1.80038 1.80038i −0.677849 2.92242i −0.485909 0.841619i
229.15 −1.58613 1.58613i 1.55738 0.758001i 3.03162i −1.33707 0.358267i −3.67250 1.26792i −0.831100 + 2.51183i 1.63628 1.63628i 1.85087 2.36099i 1.55251 + 2.68902i
229.16 −1.56728 1.56728i −1.50556 0.856331i 2.91271i 2.16047 + 0.578896i 1.01751 + 3.70173i −2.49862 0.869986i 1.43046 1.43046i 1.53340 + 2.57851i −2.47876 4.29334i
229.17 −1.55336 1.55336i −0.812740 + 1.52953i 2.82587i 3.56579 + 0.955449i 3.63839 1.11343i 2.64211 0.138853i 1.28288 1.28288i −1.67891 2.48621i −4.05480 7.02312i
229.18 −1.48199 1.48199i 1.56721 + 0.737461i 2.39258i −2.00664 0.537677i −1.22968 3.41550i −2.24356 1.40230i 0.581795 0.581795i 1.91230 + 2.31151i 2.17698 + 3.77065i
229.19 −1.43877 1.43877i −0.688101 + 1.58950i 2.14012i −0.107608 0.0288335i 3.27695 1.29691i 0.792554 2.52425i 0.201598 0.201598i −2.05303 2.18748i 0.113339 + 0.196308i
229.20 −1.43568 1.43568i 0.573580 1.63432i 2.12234i 2.32171 + 0.622101i −3.16984 + 1.52288i −0.895150 2.48972i 0.175648 0.175648i −2.34201 1.87483i −2.44009 4.22637i
See next 80 embeddings (of 432 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 229.108
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.d odd 4 1 inner
63.t odd 6 1 inner
819.ep 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.ep.a 432
7.d odd 6 1 819.2.fk.a yes 432
9.c even 3 1 819.2.fk.a yes 432
13.d odd 4 1 inner 819.2.ep.a 432
63.t odd 6 1 inner 819.2.ep.a 432
91.bb even 12 1 819.2.fk.a yes 432
117.y odd 12 1 819.2.fk.a yes 432
819.ep even 12 1 inner 819.2.ep.a 432
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
819.2.ep.a 432 1.a even 1 1 trivial
819.2.ep.a 432 13.d odd 4 1 inner
819.2.ep.a 432 63.t odd 6 1 inner
819.2.ep.a 432 819.ep even 12 1 inner
819.2.fk.a yes 432 7.d odd 6 1
819.2.fk.a yes 432 9.c even 3 1
819.2.fk.a yes 432 91.bb even 12 1
819.2.fk.a yes 432 117.y odd 12 1

Hecke kernels

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