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Properties

Label 819.2.fm.g
Level $819$
Weight $2$
Character orbit 819.fm
Analytic conductor $6.540$
Analytic rank $0$
Dimension $32$
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(370,819)]
 
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([0, 6, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("819.370");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 819 = 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 819.fm (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: \(32\)
Relative dimension: \(8\) over \(\Q(\zeta_{12})\)
Twist minimal: no (minimal twist has level 91)
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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 8 q^{2} - 12 q^{4} + 16 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 8 q^{2} - 12 q^{4} + 16 q^{8} + 4 q^{11} + 32 q^{14} + 12 q^{16} + 4 q^{22} + 12 q^{23} + 24 q^{28} - 4 q^{29} - 4 q^{32} + 20 q^{35} + 4 q^{37} - 48 q^{43} - 24 q^{44} + 84 q^{46} + 24 q^{49} + 44 q^{50} - 72 q^{53} - 60 q^{56} - 16 q^{58} - 4 q^{65} - 56 q^{67} + 56 q^{70} - 84 q^{71} + 24 q^{74} - 80 q^{79} + 36 q^{85} + 48 q^{86} - 228 q^{88} - 48 q^{91} - 24 q^{92} + 84 q^{95} + 32 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
370.1 −0.612835 − 2.28713i 0 −3.12335 + 1.80327i −2.25527 − 2.25527i 0 −2.60590 − 0.457468i 2.68981 + 2.68981i 0 −3.77599 + 6.54020i
370.2 −0.612835 − 2.28713i 0 −3.12335 + 1.80327i 2.25527 + 2.25527i 0 2.48551 + 0.906771i 2.68981 + 2.68981i 0 3.77599 − 6.54020i
370.3 −0.218036 − 0.813723i 0 1.11745 − 0.645157i −1.01306 − 1.01306i 0 2.43848 + 1.02656i −1.96000 − 1.96000i 0 −0.603466 + 1.04523i
370.4 −0.218036 − 0.813723i 0 1.11745 − 0.645157i 1.01306 + 1.01306i 0 −2.62506 − 0.330214i −1.96000 − 1.96000i 0 0.603466 − 1.04523i
370.5 0.357317 + 1.33353i 0 0.0814361 − 0.0470171i −2.02925 − 2.02925i 0 −2.32989 + 1.25364i 2.04421 + 2.04421i 0 1.98097 − 3.43114i
370.6 0.357317 + 1.33353i 0 0.0814361 − 0.0470171i 2.02925 + 2.02925i 0 1.39092 + 2.25063i 2.04421 + 2.04421i 0 −1.98097 + 3.43114i
370.7 0.607529 + 2.26733i 0 −3.03963 + 1.75493i −1.12678 − 1.12678i 0 −0.437995 − 2.60925i −2.50608 − 2.50608i 0 1.87023 − 3.23933i
370.8 0.607529 + 2.26733i 0 −3.03963 + 1.75493i 1.12678 + 1.12678i 0 1.68394 − 2.04067i −2.50608 − 2.50608i 0 −1.87023 + 3.23933i
496.1 −1.61897 − 0.433802i 0 0.700831 + 0.404625i −1.42145 − 1.42145i 0 −1.11484 − 2.39940i 1.41124 + 1.41124i 0 1.68466 + 2.91792i
496.2 −1.61897 − 0.433802i 0 0.700831 + 0.404625i 1.42145 + 1.42145i 0 0.234216 + 2.63536i 1.41124 + 1.41124i 0 −1.68466 − 2.91792i
496.3 0.112775 + 0.0302180i 0 −1.72025 − 0.993184i −1.24841 − 1.24841i 0 −2.63771 − 0.206123i −0.329103 − 0.329103i 0 −0.103066 − 0.178515i
496.4 0.112775 + 0.0302180i 0 −1.72025 − 0.993184i 1.24841 + 1.24841i 0 −2.18126 + 1.49736i −0.329103 − 0.329103i 0 0.103066 + 0.178515i
496.5 0.892257 + 0.239080i 0 −0.993087 − 0.573359i −2.80028 − 2.80028i 0 2.62900 + 0.297264i −2.05537 − 2.05537i 0 −1.82908 − 3.16806i
496.6 0.892257 + 0.239080i 0 −0.993087 − 0.573359i 2.80028 + 2.80028i 0 2.12815 − 1.57194i −2.05537 − 2.05537i 0 1.82908 + 3.16806i
496.7 2.47996 + 0.664504i 0 3.97660 + 2.29589i −0.281686 − 0.281686i 0 1.14426 + 2.38551i 4.70528 + 4.70528i 0 −0.511390 − 0.885754i
496.8 2.47996 + 0.664504i 0 3.97660 + 2.29589i 0.281686 + 0.281686i 0 −0.201802 − 2.63804i 4.70528 + 4.70528i 0 0.511390 + 0.885754i
622.1 −0.612835 + 2.28713i 0 −3.12335 − 1.80327i −2.25527 + 2.25527i 0 −2.60590 + 0.457468i 2.68981 − 2.68981i 0 −3.77599 − 6.54020i
622.2 −0.612835 + 2.28713i 0 −3.12335 − 1.80327i 2.25527 − 2.25527i 0 2.48551 − 0.906771i 2.68981 − 2.68981i 0 3.77599 + 6.54020i
622.3 −0.218036 + 0.813723i 0 1.11745 + 0.645157i −1.01306 + 1.01306i 0 2.43848 − 1.02656i −1.96000 + 1.96000i 0 −0.603466 − 1.04523i
622.4 −0.218036 + 0.813723i 0 1.11745 + 0.645157i 1.01306 − 1.01306i 0 −2.62506 + 0.330214i −1.96000 + 1.96000i 0 0.603466 + 1.04523i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 370.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
13.f odd 12 1 inner
91.bc 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.fm.g 32
3.b odd 2 1 91.2.bc.a ✓ 32
7.b odd 2 1 inner 819.2.fm.g 32
13.f odd 12 1 inner 819.2.fm.g 32
21.c even 2 1 91.2.bc.a ✓ 32
21.g even 6 1 637.2.x.b 32
21.g even 6 1 637.2.bb.b 32
21.h odd 6 1 637.2.x.b 32
21.h odd 6 1 637.2.bb.b 32
39.k even 12 1 91.2.bc.a ✓ 32
91.bc even 12 1 inner 819.2.fm.g 32
273.bs odd 12 1 637.2.x.b 32
273.bv even 12 1 637.2.x.b 32
273.bw even 12 1 637.2.bb.b 32
273.ca odd 12 1 91.2.bc.a ✓ 32
273.ch odd 12 1 637.2.bb.b 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.bc.a ✓ 32 3.b odd 2 1
91.2.bc.a ✓ 32 21.c even 2 1
91.2.bc.a ✓ 32 39.k even 12 1
91.2.bc.a ✓ 32 273.ca odd 12 1
637.2.x.b 32 21.g even 6 1
637.2.x.b 32 21.h odd 6 1
637.2.x.b 32 273.bs odd 12 1
637.2.x.b 32 273.bv even 12 1
637.2.bb.b 32 21.g even 6 1
637.2.bb.b 32 21.h odd 6 1
637.2.bb.b 32 273.bw even 12 1
637.2.bb.b 32 273.ch odd 12 1
819.2.fm.g 32 1.a even 1 1 trivial
819.2.fm.g 32 7.b odd 2 1 inner
819.2.fm.g 32 13.f odd 12 1 inner
819.2.fm.g 32 91.bc even 12 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(819, [\chi])\):

\( T_{2}^{16} - 4 T_{2}^{15} + 11 T_{2}^{14} - 28 T_{2}^{13} + 31 T_{2}^{12} - 10 T_{2}^{11} - 68 T_{2}^{10} + \cdots + 9 \) Copy content Toggle raw display
\( T_{5}^{32} + 454 T_{5}^{28} + 64945 T_{5}^{24} + 3728042 T_{5}^{20} + 87250612 T_{5}^{16} + \cdots + 187388721 \) Copy content Toggle raw display
\( T_{19}^{32} + 204 T_{19}^{30} + 17159 T_{19}^{28} + 670548 T_{19}^{26} + 7775888 T_{19}^{24} + \cdots + 55\!\cdots\!76 \) Copy content Toggle raw display