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Properties

Label 855.2.j.b
Level $855$
Weight $2$
Character orbit 855.j
Analytic conductor $6.827$
Analytic rank $0$
Dimension $80$
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] = [855,2,Mod(106,855)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(855, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("855.106");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 855 = 3^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 855.j (of order \(3\), degree \(2\), 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: \(6.82720937282\)
Analytic rank: \(0\)
Dimension: \(80\)
Relative dimension: \(40\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 80 q - 2 q^{2} + q^{3} - 40 q^{4} + 80 q^{5} + 9 q^{6} + q^{7} + 24 q^{8} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 80 q - 2 q^{2} + q^{3} - 40 q^{4} + 80 q^{5} + 9 q^{6} + q^{7} + 24 q^{8} + 5 q^{9} - 2 q^{10} - q^{11} - 6 q^{12} - 7 q^{13} + q^{15} - 36 q^{16} + 6 q^{17} + 11 q^{18} + q^{19} - 40 q^{20} + 4 q^{21} + 30 q^{22} - 29 q^{24} + 80 q^{25} - 8 q^{26} - 2 q^{27} - 5 q^{28} + 6 q^{29} + 9 q^{30} - 16 q^{31} - 26 q^{32} + 4 q^{33} - 34 q^{34} + q^{35} + 29 q^{36} + 36 q^{37} - 22 q^{38} - 37 q^{39} + 24 q^{40} + 6 q^{41} - 22 q^{42} + 5 q^{43} - 10 q^{44} + 5 q^{45} + 34 q^{46} - 36 q^{47} + 94 q^{48} - 45 q^{49} - 2 q^{50} + 12 q^{51} - 20 q^{52} - 16 q^{53} - 54 q^{54} - q^{55} + 20 q^{56} - 11 q^{57} - 4 q^{58} + 4 q^{59} - 6 q^{60} - 14 q^{61} - 92 q^{62} + 17 q^{63} + 4 q^{64} - 7 q^{65} - 33 q^{66} - 43 q^{67} + 17 q^{68} + 4 q^{69} + 2 q^{71} - 29 q^{72} + 4 q^{73} + 12 q^{74} + q^{75} + 10 q^{76} + 10 q^{77} + 70 q^{78} - 19 q^{79} - 36 q^{80} + 21 q^{81} - 11 q^{82} + 3 q^{83} + 133 q^{84} + 6 q^{85} - 7 q^{86} - 18 q^{87} - 64 q^{88} - 18 q^{89} + 11 q^{90} - 44 q^{91} + 69 q^{92} - 30 q^{93} - 42 q^{94} + q^{95} + 52 q^{96} - 43 q^{97} + 5 q^{98} - 88 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
106.1 −1.37674 − 2.38459i −1.63198 − 0.580220i −2.79084 + 4.83387i 1.00000 0.863224 + 4.69040i −0.469184 + 0.812650i 9.86208 2.32669 + 1.89381i −1.37674 − 2.38459i
106.2 −1.35649 − 2.34951i 0.0589336 + 1.73105i −2.68012 + 4.64211i 1.00000 3.98717 − 2.48661i −2.27615 + 3.94240i 9.11628 −2.99305 + 0.204034i −1.35649 − 2.34951i
106.3 −1.29334 − 2.24012i 1.63474 − 0.572389i −2.34543 + 4.06241i 1.00000 −3.39649 − 2.92173i 2.15876 − 3.73909i 6.96039 2.34474 − 1.87141i −1.29334 − 2.24012i
106.4 −1.19404 − 2.06814i 0.00721012 − 1.73204i −1.85148 + 3.20686i 1.00000 −3.59071 + 2.05321i 1.33736 − 2.31638i 4.06682 −2.99990 − 0.0249764i −1.19404 − 2.06814i
106.5 −1.15785 − 2.00546i −1.61983 + 0.613316i −1.68125 + 2.91201i 1.00000 3.10551 + 2.53837i 1.06860 − 1.85088i 3.15516 2.24769 − 1.98693i −1.15785 − 2.00546i
106.6 −1.15126 − 1.99405i 0.108267 + 1.72866i −1.65081 + 2.85929i 1.00000 3.32239 − 2.20604i 1.99949 − 3.46323i 2.99703 −2.97656 + 0.374316i −1.15126 − 1.99405i
106.7 −1.10932 − 1.92140i 1.71023 + 0.274039i −1.46119 + 2.53085i 1.00000 −1.37066 − 3.59004i −1.00295 + 1.73716i 2.04642 2.84981 + 0.937342i −1.10932 − 1.92140i
106.8 −1.05655 − 1.83000i −0.991124 − 1.42045i −1.23261 + 2.13494i 1.00000 −1.55225 + 3.31454i −1.03622 + 1.79479i 0.983041 −1.03535 + 2.81568i −1.05655 − 1.83000i
106.9 −0.930646 − 1.61193i −0.615340 + 1.61906i −0.732204 + 1.26821i 1.00000 3.18247 − 0.514889i −0.798852 + 1.38365i −0.996894 −2.24271 − 1.99255i −0.930646 − 1.61193i
106.10 −0.864578 − 1.49749i 1.29553 − 1.14961i −0.494992 + 0.857351i 1.00000 −2.84162 − 0.946128i −1.73579 + 3.00647i −1.74648 0.356812 − 2.97871i −0.864578 − 1.49749i
106.11 −0.763234 − 1.32196i 0.113271 − 1.72834i −0.165051 + 0.285877i 1.00000 −2.37125 + 1.16939i −0.343527 + 0.595006i −2.54904 −2.97434 − 0.391543i −0.763234 − 1.32196i
106.12 −0.730029 − 1.26445i −1.42212 + 0.988719i −0.0658852 + 0.114116i 1.00000 2.28838 + 1.07641i −0.822725 + 1.42500i −2.72772 1.04487 − 2.81216i −0.730029 − 1.26445i
106.13 −0.605513 − 1.04878i 1.23428 + 1.21513i 0.266708 − 0.461952i 1.00000 0.527031 − 2.03027i 1.00958 − 1.74864i −3.06803 0.0469096 + 2.99963i −0.605513 − 1.04878i
106.14 −0.588598 − 1.01948i 1.72441 − 0.162534i 0.307105 − 0.531921i 1.00000 −1.18068 − 1.66234i 1.93837 − 3.35736i −3.07744 2.94717 − 0.560549i −0.588598 − 1.01948i
106.15 −0.486991 − 0.843493i 1.21019 + 1.23913i 0.525679 − 0.910503i 1.00000 0.455846 − 1.62423i −2.30991 + 4.00088i −2.97197 −0.0708839 + 2.99916i −0.486991 − 0.843493i
106.16 −0.439666 − 0.761524i −1.68909 − 0.383362i 0.613387 − 1.06242i 1.00000 0.450698 + 1.45484i 0.556752 − 0.964323i −2.83741 2.70607 + 1.29507i −0.439666 − 0.761524i
106.17 −0.288146 − 0.499083i −0.696653 + 1.58577i 0.833944 − 1.44443i 1.00000 0.992170 − 0.109246i 0.884301 − 1.53165i −2.11377 −2.02935 − 2.20947i −0.288146 − 0.499083i
106.18 −0.203503 − 0.352477i 0.817108 − 1.52720i 0.917173 − 1.58859i 1.00000 −0.704587 + 0.0227773i 0.599682 − 1.03868i −1.56060 −1.66467 − 2.49577i −0.203503 − 0.352477i
106.19 −0.0568503 − 0.0984675i −1.70654 − 0.296182i 0.993536 − 1.72085i 1.00000 0.0678529 + 0.184877i −1.28211 + 2.22069i −0.453332 2.82455 + 1.01089i −0.0568503 − 0.0984675i
106.20 −0.0453751 − 0.0785920i 1.67443 − 0.443056i 0.995882 − 1.72492i 1.00000 −0.110798 − 0.111493i −1.56479 + 2.71029i −0.362254 2.60740 − 1.48373i −0.0453751 − 0.0785920i
See all 80 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 106.40
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
171.g even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 855.2.j.b ✓ 80
9.c even 3 1 855.2.l.a yes 80
19.c even 3 1 855.2.l.a yes 80
171.g even 3 1 inner 855.2.j.b ✓ 80
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
855.2.j.b ✓ 80 1.a even 1 1 trivial
855.2.j.b ✓ 80 171.g even 3 1 inner
855.2.l.a yes 80 9.c even 3 1
855.2.l.a yes 80 19.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{80} + 2 T_{2}^{79} + 62 T_{2}^{78} + 108 T_{2}^{77} + 2043 T_{2}^{76} + 3184 T_{2}^{75} + \cdots + 729 \) acting on \(S_{2}^{\mathrm{new}}(855, [\chi])\). Copy content Toggle raw display