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Properties

Label 819.2.fk.a
Level $819$
Weight $2$
Character orbit 819.fk
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(31,819)]
 
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, 2, 9]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("819.31");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 819 = 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 819.fk (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.

\(\operatorname{Tr}(f)(q) = \) \( 432 q + 2 q^{2} - 12 q^{3} + 12 q^{6} - 8 q^{8} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 432 q + 2 q^{2} - 12 q^{3} + 12 q^{6} - 8 q^{8} - 4 q^{9} - 12 q^{11} - 4 q^{14} - 14 q^{15} + 196 q^{16} + 14 q^{18} - 12 q^{19} - 60 q^{20} - 8 q^{22} - 24 q^{24} - 12 q^{26} - 36 q^{27} + 4 q^{28} - 24 q^{29} - 30 q^{31} + 10 q^{32} - 60 q^{33} - 12 q^{34} - 60 q^{35} - 2 q^{37} - 22 q^{39} - 100 q^{42} + 32 q^{44} - 6 q^{45} + 4 q^{46} + 30 q^{47} - 12 q^{48} + 22 q^{50} + 8 q^{53} + 30 q^{54} - 44 q^{57} + 12 q^{58} - 6 q^{59} - 64 q^{60} - 12 q^{61} + 14 q^{63} + 16 q^{65} - 12 q^{66} + 16 q^{67} + 32 q^{70} + 48 q^{71} + 34 q^{72} - 12 q^{73} + 104 q^{74} - 48 q^{76} + 2 q^{78} + 28 q^{79} - 6 q^{80} - 44 q^{81} + 84 q^{83} - 38 q^{84} - 6 q^{85} + 84 q^{86} + 24 q^{87} + 18 q^{89} + 52 q^{92} - 108 q^{93} - 12 q^{94} + 42 q^{96} + 6 q^{97} - 112 q^{98} - 50 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1 −0.719591 − 2.68555i −1.66134 − 0.489852i −4.96231 + 2.86499i 0.295842 + 0.295842i −0.120037 + 4.81410i 2.62442 + 0.335307i 7.33299 + 7.33299i 2.52009 + 1.62762i 0.581612 − 1.00738i
31.2 −0.707293 − 2.63965i 1.58924 + 0.688696i −4.73546 + 2.73402i 2.30121 + 2.30121i 0.693857 − 4.68217i −2.61536 − 0.399865i 6.70151 + 6.70151i 2.05140 + 2.18901i 4.44676 − 7.70202i
31.3 −0.695145 − 2.59431i 0.944464 − 1.45189i −4.51519 + 2.60685i 1.65419 + 1.65419i −4.42320 − 1.44096i 1.84509 − 1.89622i 6.10335 + 6.10335i −1.21597 − 2.74252i 3.14159 − 5.44139i
31.4 −0.691996 − 2.58256i 0.828311 + 1.52115i −4.45873 + 2.57425i −2.71597 − 2.71597i 3.35528 − 3.19180i 0.573519 − 2.58284i 5.95244 + 5.95244i −1.62780 + 2.51997i −5.13473 + 8.89361i
31.5 −0.691582 − 2.58102i −1.41324 + 1.00137i −4.45133 + 2.56997i −0.482183 − 0.482183i 3.56193 + 2.95508i −1.49536 − 2.18263i 5.93274 + 5.93274i 0.994516 − 2.83036i −0.911055 + 1.57799i
31.6 −0.691070 − 2.57911i −0.505252 − 1.65672i −4.44218 + 2.56469i −1.88340 − 1.88340i −3.92370 + 2.44801i −2.56220 + 0.659627i 5.90840 + 5.90840i −2.48944 + 1.67412i −3.55594 + 6.15907i
31.7 −0.643618 − 2.40201i −1.18249 + 1.26559i −3.62338 + 2.09196i 2.78377 + 2.78377i 3.80104 + 2.02579i −0.174594 + 2.63998i 3.84019 + 3.84019i −0.203450 − 2.99309i 4.89498 − 8.47835i
31.8 −0.636993 − 2.37729i −0.0745843 + 1.73044i −3.51371 + 2.02864i −0.423798 − 0.423798i 4.16128 − 0.924973i −1.71771 + 2.01233i 3.58028 + 3.58028i −2.98887 − 0.258128i −0.737536 + 1.27745i
31.9 −0.608806 − 2.27210i 1.55927 − 0.754109i −3.05972 + 1.76653i −0.0754422 − 0.0754422i −2.66270 − 3.08370i 1.07231 + 2.41871i 2.54993 + 2.54993i 1.86264 − 2.35172i −0.125482 + 0.217342i
31.10 −0.602225 − 2.24753i −1.21199 − 1.23737i −2.95668 + 1.70704i 2.70291 + 2.70291i −2.05113 + 3.46917i −2.47336 − 0.939414i 2.32660 + 2.32660i −0.0621446 + 2.99936i 4.44712 − 7.70264i
31.11 −0.598736 − 2.23451i 0.307564 − 1.70452i −2.90251 + 1.67577i 1.33609 + 1.33609i −3.99293 + 0.333304i −0.236565 + 2.63515i 2.21080 + 2.21080i −2.81081 − 1.04850i 2.18554 − 3.78546i
31.12 −0.587151 − 2.19128i −0.991155 − 1.42043i −2.72489 + 1.57322i −1.10407 − 1.10407i −2.53059 + 3.00590i 1.69473 + 2.03172i 1.83903 + 1.83903i −1.03522 + 2.81573i −1.77107 + 3.06758i
31.13 −0.587129 − 2.19119i −0.935526 + 1.45767i −2.72456 + 1.57302i −1.74171 − 1.74171i 3.74330 + 1.19408i 2.45423 + 0.988320i 1.83834 + 1.83834i −1.24958 − 2.72737i −2.79381 + 4.83902i
31.14 −0.585910 − 2.18665i 1.71316 + 0.255135i −2.70609 + 1.56236i −0.303553 − 0.303553i −0.445865 − 3.89556i 1.80880 − 1.93087i 1.80038 + 1.80038i 2.86981 + 0.874174i −0.485909 + 0.841619i
31.15 −0.580564 − 2.16669i 1.43514 + 0.969731i −2.62546 + 1.51581i 0.978803 + 0.978803i 1.26792 − 3.67250i 2.59086 + 0.536160i 1.63628 + 1.63628i 1.11924 + 2.78340i 1.55251 − 2.68902i
31.16 −0.573663 − 2.14094i −0.0111737 − 1.73201i −2.52248 + 1.45635i −1.58157 − 1.58157i −3.70173 + 1.01751i 0.495882 − 2.59887i 1.43046 + 1.43046i −2.99975 + 0.0387061i −2.47876 + 4.29334i
31.17 −0.568570 − 2.12193i −1.73098 + 0.0609107i −2.44728 + 1.41294i −2.61034 − 2.61034i 1.11343 + 3.63839i −1.44130 + 2.21870i 1.28288 + 1.28288i 2.99258 − 0.210870i −4.05480 + 7.02312i
31.18 −0.542445 − 2.02443i 0.144946 + 1.72598i −2.07203 + 1.19629i 1.46896 + 1.46896i 3.41550 − 1.22968i −0.0926511 − 2.64413i 0.581795 + 0.581795i −2.95798 + 0.500347i 2.17698 − 3.77065i
31.19 −0.526626 − 1.96540i −1.72060 + 0.198838i −1.85340 + 1.07006i 0.0787747 + 0.0787747i 1.29691 + 3.27695i −2.58235 − 0.575755i 0.201598 + 0.201598i 2.92093 − 0.684240i 0.113339 − 0.196308i
31.20 −0.525495 − 1.96117i 1.70215 − 0.320426i −1.83800 + 1.06117i −1.69961 − 1.69961i −1.52288 − 3.16984i −1.70859 − 2.02008i 0.175648 + 0.175648i 2.79465 − 1.09083i −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 31.108
Significant digits:
Format:

Inner twists

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

Hecke kernels

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