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Properties

Label 1045.2.b.d
Level $1045$
Weight $2$
Character orbit 1045.b
Analytic conductor $8.344$
Analytic rank $0$
Dimension $22$
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] = [1045,2,Mod(419,1045)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1045, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1045.419");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1045 = 5 \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1045.b (of order \(2\), degree \(1\), 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: \(8.34436701122\)
Analytic rank: \(0\)
Dimension: \(22\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 22 q - 32 q^{4} + 7 q^{5} - 12 q^{6} - 34 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 22 q - 32 q^{4} + 7 q^{5} - 12 q^{6} - 34 q^{9} + 2 q^{10} - 22 q^{11} + 8 q^{14} - 23 q^{15} + 40 q^{16} + 22 q^{19} - 22 q^{20} - 22 q^{21} + 22 q^{24} + 13 q^{25} + 16 q^{26} + 10 q^{29} - 22 q^{30} + 76 q^{31} - 56 q^{34} - 2 q^{35} + 104 q^{36} + 8 q^{39} - 20 q^{40} + 6 q^{41} + 32 q^{44} - 12 q^{45} + 88 q^{46} - 28 q^{49} - 20 q^{50} + 8 q^{51} - 38 q^{54} - 7 q^{55} + 44 q^{56} - 40 q^{59} + 78 q^{60} - 6 q^{61} - 140 q^{64} - 22 q^{65} + 12 q^{66} - 74 q^{69} - 24 q^{70} + 62 q^{71} + 26 q^{74} + 13 q^{75} - 32 q^{76} - 102 q^{79} + 142 q^{80} + 94 q^{81} + 38 q^{84} + 26 q^{85} + 28 q^{86} - 54 q^{89} + 118 q^{90} + 88 q^{91} - 36 q^{94} + 7 q^{95} + 2 q^{96} + 34 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
419.1 − 2.74370i − 2.61424i −5.52788 1.89425 − 1.18821i −7.17269 − 3.68386i 9.67944i −3.83426 −3.26008 − 5.19724i
419.2 − 2.71685i 2.46712i −5.38125 2.08881 + 0.798049i 6.70278 − 0.267393i 9.18634i −3.08666 2.16818 − 5.67497i
419.3 − 2.58163i − 0.787674i −4.66484 −1.04139 + 1.97876i −2.03349 0.906789i 6.87964i 2.37957 5.10844 + 2.68850i
419.4 − 2.06343i − 3.28666i −2.25776 0.951609 − 2.02347i −6.78181 3.51972i 0.531875i −7.80213 −4.17530 − 1.96358i
419.5 − 1.84475i 3.33093i −1.40312 −2.19819 + 0.409812i 6.14475 3.04962i − 1.10110i −8.09513 0.756002 + 4.05513i
419.6 − 1.66420i − 1.15649i −0.769547 1.63516 + 1.52521i −1.92462 3.92678i − 2.04771i 1.66254 2.53825 − 2.72122i
419.7 − 1.60597i − 0.776012i −0.579148 −1.81070 − 1.31201i −1.24625 0.953282i − 2.28185i 2.39780 −2.10704 + 2.90794i
419.8 − 1.28593i 0.496540i 0.346393 −0.328055 + 2.21187i 0.638514 − 3.51780i − 3.01729i 2.75345 2.84431 + 0.421855i
419.9 − 1.06190i − 2.29282i 0.872373 −1.55246 − 1.60931i −2.43474 − 3.97400i − 3.05017i −2.25702 −1.70892 + 1.64855i
419.10 − 0.790175i 2.57534i 1.37562 1.76786 − 1.36919i 2.03497 − 0.0669205i − 2.66733i −3.63236 −1.08190 − 1.39692i
419.11 − 0.104144i 0.696995i 1.98915 2.09312 − 0.786662i 0.0725876 3.21657i − 0.415445i 2.51420 −0.0819259 − 0.217986i
419.12 0.104144i − 0.696995i 1.98915 2.09312 + 0.786662i 0.0725876 − 3.21657i 0.415445i 2.51420 −0.0819259 + 0.217986i
419.13 0.790175i − 2.57534i 1.37562 1.76786 + 1.36919i 2.03497 0.0669205i 2.66733i −3.63236 −1.08190 + 1.39692i
419.14 1.06190i 2.29282i 0.872373 −1.55246 + 1.60931i −2.43474 3.97400i 3.05017i −2.25702 −1.70892 − 1.64855i
419.15 1.28593i − 0.496540i 0.346393 −0.328055 − 2.21187i 0.638514 3.51780i 3.01729i 2.75345 2.84431 − 0.421855i
419.16 1.60597i 0.776012i −0.579148 −1.81070 + 1.31201i −1.24625 − 0.953282i 2.28185i 2.39780 −2.10704 − 2.90794i
419.17 1.66420i 1.15649i −0.769547 1.63516 − 1.52521i −1.92462 − 3.92678i 2.04771i 1.66254 2.53825 + 2.72122i
419.18 1.84475i − 3.33093i −1.40312 −2.19819 − 0.409812i 6.14475 − 3.04962i 1.10110i −8.09513 0.756002 − 4.05513i
419.19 2.06343i 3.28666i −2.25776 0.951609 + 2.02347i −6.78181 − 3.51972i − 0.531875i −7.80213 −4.17530 + 1.96358i
419.20 2.58163i 0.787674i −4.66484 −1.04139 − 1.97876i −2.03349 − 0.906789i − 6.87964i 2.37957 5.10844 − 2.68850i
See all 22 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 419.22
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1045.2.b.d ✓ 22
5.b even 2 1 inner 1045.2.b.d ✓ 22
5.c odd 4 2 5225.2.a.bb 22
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1045.2.b.d ✓ 22 1.a even 1 1 trivial
1045.2.b.d ✓ 22 5.b even 2 1 inner
5225.2.a.bb 22 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{22} + 38 T_{2}^{20} + 620 T_{2}^{18} + 5698 T_{2}^{16} + 32546 T_{2}^{14} + 120279 T_{2}^{12} + \cdots + 484 \) acting on \(S_{2}^{\mathrm{new}}(1045, [\chi])\). Copy content Toggle raw display