↑

Properties

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

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [819,2,Mod(76,819)]
 
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, 6, 7]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("819.76");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 819 = 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 819.fl (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.

\(\operatorname{Tr}(f)(q) = \) \( 432 q - 4 q^{2} - 12 q^{4} - 6 q^{7} - 8 q^{8} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 432 q - 4 q^{2} - 12 q^{4} - 6 q^{7} - 8 q^{8} - 4 q^{9} - 4 q^{14} + 28 q^{15} + 196 q^{16} - 40 q^{18} + 6 q^{21} + 4 q^{22} + 16 q^{28} + 12 q^{29} - 36 q^{30} - 20 q^{32} + 18 q^{35} - 12 q^{36} - 20 q^{37} + 8 q^{39} + 56 q^{42} - 40 q^{44} - 8 q^{46} - 32 q^{50} - 64 q^{53} - 44 q^{57} + 12 q^{58} + 32 q^{60} + 2 q^{63} + 52 q^{65} - 32 q^{67} + 2 q^{70} - 96 q^{71} - 68 q^{72} - 64 q^{74} - 72 q^{77} - 88 q^{78} - 20 q^{79} + 16 q^{81} - 146 q^{84} + 24 q^{85} + 132 q^{86} - 12 q^{88} + 40 q^{92} + 12 q^{93} + 36 q^{95} - 64 q^{98} - 92 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
76.1 −2.61602 + 0.700960i −1.71076 − 0.270735i 4.62016 − 2.66745i −1.07636 − 4.01704i 4.66516 − 0.490928i 1.57051 + 2.12920i −6.38654 + 6.38654i 2.85341 + 0.926325i 5.63158 + 9.75418i
76.2 −2.61602 + 0.700960i 1.71076 + 0.270735i 4.62016 − 2.66745i 1.07636 + 4.01704i −4.66516 + 0.490928i −2.12920 − 1.57051i −6.38654 + 6.38654i 2.85341 + 0.926325i −5.63158 − 9.75418i
76.3 −2.59133 + 0.694345i −0.770115 − 1.55143i 4.50084 − 2.59856i 0.313276 + 1.16916i 3.07285 + 3.48553i −2.64356 − 0.107742i −6.06489 + 6.06489i −1.81384 + 2.38955i −1.62360 − 2.81216i
76.4 −2.59133 + 0.694345i 0.770115 + 1.55143i 4.50084 − 2.59856i −0.313276 − 1.16916i −3.07285 − 3.48553i 0.107742 + 2.64356i −6.06489 + 6.06489i −1.81384 + 2.38955i 1.62360 + 2.81216i
76.5 −2.58412 + 0.692414i −1.29054 + 1.15521i 4.46620 − 2.57856i 0.359585 + 1.34199i 2.53503 − 3.87879i 2.13087 − 1.56824i −5.97237 + 5.97237i 0.330978 − 2.98169i −1.85842 − 3.21888i
76.6 −2.58412 + 0.692414i 1.29054 − 1.15521i 4.46620 − 2.57856i −0.359585 − 1.34199i −2.53503 + 3.87879i 1.56824 − 2.13087i −5.97237 + 5.97237i 0.330978 − 2.98169i 1.85842 + 3.21888i
76.7 −2.45952 + 0.659028i −0.878003 − 1.49302i 3.88289 − 2.24179i 0.799127 + 2.98238i 3.14341 + 3.09349i 2.61836 + 0.379744i −4.47167 + 4.47167i −1.45822 + 2.62175i −3.93094 − 6.80860i
76.8 −2.45952 + 0.659028i 0.878003 + 1.49302i 3.88289 − 2.24179i −0.799127 − 2.98238i −3.14341 − 3.09349i −0.379744 − 2.61836i −4.47167 + 4.47167i −1.45822 + 2.62175i 3.93094 + 6.80860i
76.9 −2.44167 + 0.654243i −0.663469 + 1.59994i 3.80167 − 2.19489i 0.0298378 + 0.111356i 0.573222 − 4.34060i −2.60786 − 0.446193i −4.27157 + 4.27157i −2.11962 − 2.12302i −0.145708 − 0.252374i
76.10 −2.44167 + 0.654243i 0.663469 − 1.59994i 3.80167 − 2.19489i −0.0298378 − 0.111356i −0.573222 + 4.34060i 0.446193 + 2.60786i −4.27157 + 4.27157i −2.11962 − 2.12302i 0.145708 + 0.252374i
76.11 −2.20363 + 0.590461i −1.41565 − 0.997968i 2.77530 − 1.60232i −0.404840 − 1.51088i 3.70883 + 1.36327i 0.793046 − 2.52410i −1.94328 + 1.94328i 1.00812 + 2.82554i 1.78424 + 3.09039i
76.12 −2.20363 + 0.590461i 1.41565 + 0.997968i 2.77530 − 1.60232i 0.404840 + 1.51088i −3.70883 − 1.36327i 2.52410 − 0.793046i −1.94328 + 1.94328i 1.00812 + 2.82554i −1.78424 − 3.09039i
76.13 −2.16672 + 0.580571i −1.70018 − 0.330739i 2.62557 − 1.51587i 0.174613 + 0.651666i 3.87583 − 0.270456i −2.25600 + 1.38220i −1.63650 + 1.63650i 2.78122 + 1.12463i −0.756677 − 1.31060i
76.14 −2.16672 + 0.580571i 1.70018 + 0.330739i 2.62557 − 1.51587i −0.174613 − 0.651666i −3.87583 + 0.270456i −1.38220 + 2.25600i −1.63650 + 1.63650i 2.78122 + 1.12463i 0.756677 + 1.31060i
76.15 −2.01562 + 0.540084i −1.32356 + 1.11722i 2.03899 − 1.17721i −0.964010 − 3.59773i 2.06441 − 2.96673i −2.64542 + 0.0418434i −0.522964 + 0.522964i 0.503644 − 2.95742i 3.88616 + 6.73103i
76.16 −2.01562 + 0.540084i 1.32356 − 1.11722i 2.03899 − 1.17721i 0.964010 + 3.59773i −2.06441 + 2.96673i −0.0418434 + 2.64542i −0.522964 + 0.522964i 0.503644 − 2.95742i −3.88616 − 6.73103i
76.17 −1.98798 + 0.532677i −1.22025 + 1.22922i 1.93627 − 1.11790i 0.0552041 + 0.206024i 1.77105 − 3.09367i 1.27825 + 2.31648i −0.343170 + 0.343170i −0.0219813 − 2.99992i −0.219489 − 0.380166i
76.18 −1.98798 + 0.532677i 1.22025 − 1.22922i 1.93627 − 1.11790i −0.0552041 − 0.206024i −1.77105 + 3.09367i −2.31648 − 1.27825i −0.343170 + 0.343170i −0.0219813 − 2.99992i 0.219489 + 0.380166i
76.19 −1.90643 + 0.510826i −1.72956 − 0.0928690i 1.64148 − 0.947709i 0.957845 + 3.57473i 3.34472 − 0.706456i −0.380634 − 2.61823i 0.145956 − 0.145956i 2.98275 + 0.321245i −3.65213 − 6.32567i
76.20 −1.90643 + 0.510826i 1.72956 + 0.0928690i 1.64148 − 0.947709i −0.957845 − 3.57473i −3.34472 + 0.706456i 2.61823 + 0.380634i 0.145956 − 0.145956i 2.98275 + 0.321245i 3.65213 + 6.32567i
See next 80 embeddings (of 432 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 76.108
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
117.bb odd 12 1 inner
819.fl 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.fl.a yes 432
7.b odd 2 1 inner 819.2.fl.a yes 432
9.c even 3 1 819.2.eq.a ✓ 432
13.f odd 12 1 819.2.eq.a ✓ 432
63.l odd 6 1 819.2.eq.a ✓ 432
91.bc even 12 1 819.2.eq.a ✓ 432
117.bb odd 12 1 inner 819.2.fl.a yes 432
819.fl even 12 1 inner 819.2.fl.a yes 432
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
819.2.eq.a ✓ 432 9.c even 3 1
819.2.eq.a ✓ 432 13.f odd 12 1
819.2.eq.a ✓ 432 63.l odd 6 1
819.2.eq.a ✓ 432 91.bc even 12 1
819.2.fl.a yes 432 1.a even 1 1 trivial
819.2.fl.a yes 432 7.b odd 2 1 inner
819.2.fl.a yes 432 117.bb odd 12 1 inner
819.2.fl.a yes 432 819.fl even 12 1 inner

Hecke kernels

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