Properties

Label 819.2.fm.e
Level $819$
Weight $2$
Character orbit 819.fm
Analytic conductor $6.540$
Analytic rank $0$
Dimension $32$
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(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 273)
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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 2 q^{2} + 6 q^{4} - 2 q^{5} + 2 q^{7} - 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 2 q^{2} + 6 q^{4} - 2 q^{5} + 2 q^{7} - 2 q^{8} + 2 q^{10} + 4 q^{11} + 6 q^{13} - 34 q^{14} + 14 q^{16} + 8 q^{17} + 2 q^{19} - 44 q^{20} - 4 q^{22} + 18 q^{23} + 28 q^{26} - 32 q^{28} + 18 q^{29} - 14 q^{31} + 8 q^{32} - 66 q^{34} - 22 q^{35} - 24 q^{37} - 24 q^{38} - 6 q^{43} + 20 q^{44} - 58 q^{46} + 28 q^{47} + 8 q^{49} - 70 q^{50} + 28 q^{52} + 80 q^{53} + 60 q^{55} + 54 q^{56} - 4 q^{58} + 42 q^{59} + 36 q^{61} - 52 q^{62} - 14 q^{65} + 26 q^{67} + 72 q^{68} - 116 q^{70} + 4 q^{71} + 12 q^{73} + 18 q^{74} - 48 q^{76} - 28 q^{77} - 4 q^{79} + 98 q^{80} + 20 q^{82} + 36 q^{83} - 10 q^{85} + 40 q^{86} + 96 q^{88} + 54 q^{89} + 148 q^{91} + 4 q^{92} - 60 q^{95} - 40 q^{97} - 36 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
370.1 −0.687394 2.56539i 0 −4.37666 + 2.52687i 1.17771 + 1.17771i 0 −1.53750 2.15316i 5.73490 + 5.73490i 0 2.21174 3.83085i
370.2 −0.385482 1.43864i 0 −0.189037 + 0.109141i 1.07205 + 1.07205i 0 1.81743 1.92274i −1.87643 1.87643i 0 1.12904 1.95556i
370.3 −0.0578232 0.215799i 0 1.68883 0.975044i −1.08760 1.08760i 0 −0.725943 + 2.54421i −0.624019 0.624019i 0 −0.171815 + 0.297592i
370.4 −0.0473445 0.176692i 0 1.70307 0.983269i −2.80040 2.80040i 0 −0.467560 2.60411i −0.513062 0.513062i 0 −0.362225 + 0.627392i
370.5 0.189683 + 0.707908i 0 1.26690 0.731443i 1.23329 + 1.23329i 0 −2.64473 0.0736014i 1.79455 + 1.79455i 0 −0.639122 + 1.10699i
370.6 0.281068 + 1.04896i 0 0.710736 0.410344i 1.02734 + 1.02734i 0 2.23270 1.41953i 2.16598 + 2.16598i 0 −0.788887 + 1.36639i
370.7 0.500642 + 1.86842i 0 −1.50830 + 0.870817i −2.44068 2.44068i 0 2.02526 + 1.70244i 0.353388 + 0.353388i 0 3.33831 5.78213i
370.8 0.706652 + 2.63726i 0 −4.72373 + 2.72725i 2.18431 + 2.18431i 0 0.666364 + 2.56046i −6.66928 6.66928i 0 −4.21705 + 7.30414i
496.1 −2.37607 0.636667i 0 3.50833 + 2.02554i 0.498430 + 0.498430i 0 −2.62820 0.304236i −3.56765 3.56765i 0 −0.866973 1.50164i
496.2 −1.96303 0.525993i 0 1.84478 + 1.06508i 1.56698 + 1.56698i 0 2.60496 + 0.462799i −0.187059 0.187059i 0 −2.25182 3.90026i
496.3 −1.34112 0.359352i 0 −0.0625832 0.0361324i −2.52867 2.52867i 0 −0.324044 + 2.62583i 2.03448 + 2.03448i 0 2.48257 + 4.29993i
496.4 −0.604240 0.161906i 0 −1.39316 0.804341i −0.965431 0.965431i 0 2.12303 1.57884i 1.59624 + 1.59624i 0 0.427043 + 0.739660i
496.5 1.11595 + 0.299019i 0 −0.576113 0.332619i −0.549341 0.549341i 0 1.61214 + 2.09786i −2.17732 2.17732i 0 −0.448776 0.777302i
496.6 1.38083 + 0.369991i 0 0.0377371 + 0.0217876i −0.512287 0.512287i 0 −2.41005 1.09162i −1.97762 1.97762i 0 −0.517838 0.896921i
496.7 2.11902 + 0.567791i 0 2.43582 + 1.40632i 3.00219 + 3.00219i 0 −2.60148 + 0.481982i 1.26060 + 1.26060i 0 4.65710 + 8.06633i
496.8 2.16866 + 0.581092i 0 2.63339 + 1.52039i −1.87790 1.87790i 0 1.25762 2.32774i 1.65230 + 1.65230i 0 −2.98130 5.16377i
622.1 −0.687394 + 2.56539i 0 −4.37666 2.52687i 1.17771 1.17771i 0 −1.53750 + 2.15316i 5.73490 5.73490i 0 2.21174 + 3.83085i
622.2 −0.385482 + 1.43864i 0 −0.189037 0.109141i 1.07205 1.07205i 0 1.81743 + 1.92274i −1.87643 + 1.87643i 0 1.12904 + 1.95556i
622.3 −0.0578232 + 0.215799i 0 1.68883 + 0.975044i −1.08760 + 1.08760i 0 −0.725943 2.54421i −0.624019 + 0.624019i 0 −0.171815 0.297592i
622.4 −0.0473445 + 0.176692i 0 1.70307 + 0.983269i −2.80040 + 2.80040i 0 −0.467560 + 2.60411i −0.513062 + 0.513062i 0 −0.362225 0.627392i
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
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.e 32
3.b odd 2 1 273.2.by.d yes 32
7.b odd 2 1 819.2.fm.f 32
13.f odd 12 1 819.2.fm.f 32
21.c even 2 1 273.2.by.c 32
39.k even 12 1 273.2.by.c 32
91.bc even 12 1 inner 819.2.fm.e 32
273.ca odd 12 1 273.2.by.d yes 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.by.c 32 21.c even 2 1
273.2.by.c 32 39.k even 12 1
273.2.by.d yes 32 3.b odd 2 1
273.2.by.d yes 32 273.ca odd 12 1
819.2.fm.e 32 1.a even 1 1 trivial
819.2.fm.e 32 91.bc even 12 1 inner
819.2.fm.f 32 7.b odd 2 1
819.2.fm.f 32 13.f odd 12 1

Hecke kernels

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

\( T_{2}^{32} - 2 T_{2}^{31} - T_{2}^{30} + 8 T_{2}^{29} - 67 T_{2}^{28} + 96 T_{2}^{27} + 112 T_{2}^{26} - 404 T_{2}^{25} + 3019 T_{2}^{24} - 4794 T_{2}^{23} - 8633 T_{2}^{22} + 26164 T_{2}^{21} - 40469 T_{2}^{20} + 3296 T_{2}^{19} + 182806 T_{2}^{18} + \cdots + 576 \) Copy content Toggle raw display
\( T_{5}^{32} + 2 T_{5}^{31} + 2 T_{5}^{30} - 18 T_{5}^{29} + 501 T_{5}^{28} + 916 T_{5}^{27} + 992 T_{5}^{26} - 9260 T_{5}^{25} + 71882 T_{5}^{24} + 85244 T_{5}^{23} + 115940 T_{5}^{22} - 1087128 T_{5}^{21} + 4494824 T_{5}^{20} + \cdots + 404653456 \) Copy content Toggle raw display
\( T_{19}^{32} - 2 T_{19}^{31} - 10 T_{19}^{30} - 116 T_{19}^{29} - 1607 T_{19}^{28} + 8838 T_{19}^{27} + 18222 T_{19}^{26} + 79068 T_{19}^{25} + 1914778 T_{19}^{24} - 12478412 T_{19}^{23} + 16914440 T_{19}^{22} + \cdots + 12\!\cdots\!36 \) Copy content Toggle raw display