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Properties

Label 861.2.n.e
Level $861$
Weight $2$
Character orbit 861.n
Analytic conductor $6.875$
Analytic rank $0$
Dimension $40$
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] = [861,2,Mod(379,861)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(861, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("861.379");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 861 = 3 \cdot 7 \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 861.n (of order \(5\), 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: \(6.87511961403\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(10\) over \(\Q(\zeta_{5})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 3 q^{2} - 40 q^{3} - 19 q^{4} + 6 q^{5} + 3 q^{6} - 10 q^{7} - 5 q^{8} + 40 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q - 3 q^{2} - 40 q^{3} - 19 q^{4} + 6 q^{5} + 3 q^{6} - 10 q^{7} - 5 q^{8} + 40 q^{9} - 12 q^{10} + q^{11} + 19 q^{12} + 22 q^{13} + 2 q^{14} - 6 q^{15} + 5 q^{16} - 4 q^{17} - 3 q^{18} + 7 q^{19} - 39 q^{20} + 10 q^{21} + 17 q^{22} + 10 q^{23} + 5 q^{24} - 22 q^{25} - 20 q^{26} - 40 q^{27} - 4 q^{28} + 14 q^{29} + 12 q^{30} - 17 q^{31} - 28 q^{32} - q^{33} - 26 q^{34} - 9 q^{35} - 19 q^{36} - 6 q^{37} - 28 q^{38} - 22 q^{39} - 98 q^{40} - 6 q^{41} - 2 q^{42} - 15 q^{43} - 14 q^{44} + 6 q^{45} + 35 q^{46} - 13 q^{47} - 5 q^{48} - 10 q^{49} - 54 q^{50} + 4 q^{51} - 12 q^{52} + 36 q^{53} + 3 q^{54} - 92 q^{55} - 5 q^{56} - 7 q^{57} + 39 q^{58} + 10 q^{59} + 39 q^{60} + 23 q^{62} - 10 q^{63} + 35 q^{64} - 31 q^{65} - 17 q^{66} + 37 q^{67} + 246 q^{68} - 10 q^{69} - 12 q^{70} - 29 q^{71} - 5 q^{72} + 48 q^{73} - 101 q^{74} + 22 q^{75} + 8 q^{76} + q^{77} + 20 q^{78} - 112 q^{79} - 120 q^{80} + 40 q^{81} - 35 q^{82} + 152 q^{83} + 4 q^{84} - 86 q^{85} - 96 q^{86} - 14 q^{87} + 82 q^{88} + 13 q^{89} - 12 q^{90} - 28 q^{91} + 2 q^{92} + 17 q^{93} + 18 q^{95} + 28 q^{96} - 4 q^{97} - 3 q^{98} + q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
379.1 −2.05092 − 1.49008i −1.00000 1.36789 + 4.20994i 0.749014 + 2.30523i 2.05092 + 1.49008i −0.809017 + 0.587785i 1.90094 − 5.85050i 1.00000 1.89881 − 5.84392i
379.2 −1.75068 − 1.27194i −1.00000 0.829009 + 2.55143i −0.874705 − 2.69207i 1.75068 + 1.27194i −0.809017 + 0.587785i 0.456542 − 1.40509i 1.00000 −1.89283 + 5.82553i
379.3 −1.25194 − 0.909585i −1.00000 0.121966 + 0.375373i −0.134104 − 0.412730i 1.25194 + 0.909585i −0.809017 + 0.587785i −0.767655 + 2.36260i 1.00000 −0.207523 + 0.638691i
379.4 −0.789928 − 0.573916i −1.00000 −0.323428 − 0.995408i 0.165515 + 0.509404i 0.789928 + 0.573916i −0.809017 + 0.587785i −0.919248 + 2.82915i 1.00000 0.161610 − 0.497385i
379.5 −0.748328 − 0.543692i −1.00000 −0.353640 − 1.08839i 1.24839 + 3.84215i 0.748328 + 0.543692i −0.809017 + 0.587785i −0.898783 + 2.76617i 1.00000 1.15474 − 3.55393i
379.6 −0.0474178 − 0.0344510i −1.00000 −0.616972 − 1.89885i 0.137025 + 0.421721i 0.0474178 + 0.0344510i −0.809017 + 0.587785i −0.0723857 + 0.222780i 1.00000 0.00803128 − 0.0247177i
379.7 0.578443 + 0.420264i −1.00000 −0.460059 − 1.41592i −1.04672 − 3.22147i −0.578443 − 0.420264i −0.809017 + 0.587785i 0.770831 − 2.37237i 1.00000 0.748399 − 2.30333i
379.8 1.04389 + 0.758430i −1.00000 −0.103545 − 0.318680i 0.753340 + 2.31854i −1.04389 − 0.758430i −0.809017 + 0.587785i 0.931067 − 2.86553i 1.00000 −0.972048 + 2.99166i
379.9 1.63189 + 1.18564i −1.00000 0.639302 + 1.96757i −0.682008 − 2.09901i −1.63189 − 1.18564i −0.809017 + 0.587785i −0.0429009 + 0.132035i 1.00000 1.37570 − 4.23397i
379.10 2.07596 + 1.50828i −1.00000 1.41670 + 4.36014i 1.18425 + 3.64474i −2.07596 − 1.50828i −0.809017 + 0.587785i −2.04939 + 6.30739i 1.00000 −3.03882 + 9.35253i
631.1 −0.871631 + 2.68260i −1.00000 −4.81859 − 3.50091i 1.18938 + 0.864139i 0.871631 − 2.68260i 0.309017 + 0.951057i 9.02766 − 6.55898i 1.00000 −3.35485 + 2.43744i
631.2 −0.644982 + 1.98505i −1.00000 −1.90639 − 1.38507i 0.173347 + 0.125944i 0.644982 − 1.98505i 0.309017 + 0.951057i 0.601856 − 0.437274i 1.00000 −0.361810 + 0.262870i
631.3 −0.520464 + 1.60182i −1.00000 −0.676923 − 0.491813i −0.682871 − 0.496135i 0.520464 − 1.60182i 0.309017 + 0.951057i −1.58507 + 1.15162i 1.00000 1.15013 − 0.835619i
631.4 −0.460795 + 1.41818i −1.00000 −0.180874 − 0.131413i 3.42821 + 2.49074i 0.460795 − 1.41818i 0.309017 + 0.951057i −2.14304 + 1.55701i 1.00000 −5.11202 + 3.71410i
631.5 −0.138407 + 0.425972i −1.00000 1.45574 + 1.05766i −3.36253 − 2.44302i 0.138407 − 0.425972i 0.309017 + 0.951057i −1.37672 + 1.00025i 1.00000 1.50606 − 1.09421i
631.6 −0.0301873 + 0.0929069i −1.00000 1.61031 + 1.16996i 1.86597 + 1.35571i 0.0301873 − 0.0929069i 0.309017 + 0.951057i −0.315371 + 0.229130i 1.00000 −0.182283 + 0.132436i
631.7 0.410384 − 1.26303i −1.00000 0.191201 + 0.138916i −1.52489 − 1.10790i −0.410384 + 1.26303i 0.309017 + 0.951057i 2.40272 − 1.74568i 1.00000 −2.02510 + 1.47132i
631.8 0.492891 − 1.51696i −1.00000 −0.440198 − 0.319822i −1.96225 − 1.42566i −0.492891 + 1.51696i 0.309017 + 0.951057i 1.87868 − 1.36494i 1.00000 −3.12985 + 2.27397i
631.9 0.754720 − 2.32279i −1.00000 −3.20772 − 2.33054i 0.0540505 + 0.0392700i −0.754720 + 2.32279i 0.309017 + 0.951057i −3.88253 + 2.82082i 1.00000 0.132009 − 0.0959102i
631.10 0.817488 − 2.51597i −1.00000 −4.04378 − 2.93798i 2.32159 + 1.68673i −0.817488 + 2.51597i 0.309017 + 0.951057i −6.41719 + 4.66236i 1.00000 6.14164 − 4.46217i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 379.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
41.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 861.2.n.e ✓ 40
41.d even 5 1 inner 861.2.n.e ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
861.2.n.e ✓ 40 1.a even 1 1 trivial
861.2.n.e ✓ 40 41.d even 5 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{40} + 3 T_{2}^{39} + 24 T_{2}^{38} + 62 T_{2}^{37} + 303 T_{2}^{36} + 708 T_{2}^{35} + 2759 T_{2}^{34} + \cdots + 361 \) acting on \(S_{2}^{\mathrm{new}}(861, [\chi])\). Copy content Toggle raw display