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Properties

Label 861.2.bz.a
Level $861$
Weight $2$
Character orbit 861.bz
Analytic conductor $6.875$
Analytic rank $0$
Dimension $448$
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] = [861,2,Mod(4,861)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(861, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([0, 20, 9]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("861.4");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 861 = 3 \cdot 7 \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 861.bz (of order \(30\), degree \(8\), 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: \(448\)
Relative dimension: \(56\) over \(\Q(\zeta_{30})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{30}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 448 q + 56 q^{4} - 24 q^{8} + 224 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 448 q + 56 q^{4} - 24 q^{8} + 224 q^{9} - 10 q^{10} + 56 q^{16} + 72 q^{20} - 40 q^{22} + 12 q^{23} + 40 q^{25} - 50 q^{28} + 40 q^{29} + 16 q^{31} + 4 q^{32} - 8 q^{33} - 120 q^{34} - 20 q^{35} + 112 q^{36} + 22 q^{37} - 52 q^{40} + 52 q^{41} - 8 q^{42} + 8 q^{43} - 12 q^{46} - 60 q^{47} - 56 q^{49} - 88 q^{50} - 20 q^{51} + 100 q^{52} - 120 q^{56} + 24 q^{57} + 20 q^{59} - 48 q^{61} + 36 q^{62} - 40 q^{64} - 90 q^{65} + 40 q^{66} - 20 q^{67} - 40 q^{69} + 10 q^{70} + 40 q^{71} - 12 q^{72} + 52 q^{73} - 4 q^{74} + 40 q^{75} - 60 q^{76} - 76 q^{77} - 172 q^{80} - 224 q^{81} + 88 q^{82} - 136 q^{83} - 72 q^{84} + 44 q^{86} - 24 q^{87} - 230 q^{88} + 60 q^{89} - 20 q^{90} - 124 q^{91} - 44 q^{92} + 60 q^{93} + 150 q^{94} - 40 q^{97} - 148 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 −2.50237 − 1.11413i −0.866025 − 0.500000i 3.68230 + 4.08961i 3.27023 − 0.695109i 1.61005 + 2.21604i −1.19685 − 2.35956i −2.96523 − 9.12602i 0.500000 + 0.866025i −8.95775 − 1.90403i
4.2 −2.49123 − 1.10917i 0.866025 + 0.500000i 3.63771 + 4.04009i −0.704765 + 0.149802i −1.60288 − 2.20618i 2.63512 − 0.236970i −2.89587 − 8.91257i 0.500000 + 0.866025i 1.92189 + 0.408510i
4.3 −2.35182 − 1.04710i 0.866025 + 0.500000i 3.09638 + 3.43888i 4.07438 − 0.866036i −1.51319 − 2.08272i −1.30140 + 2.30355i −2.09023 − 6.43307i 0.500000 + 0.866025i −10.4890 − 2.22951i
4.4 −2.31543 − 1.03089i −0.866025 − 0.500000i 2.96019 + 3.28762i 0.665188 − 0.141390i 1.48977 + 2.05049i 1.82050 + 1.91984i −1.89847 − 5.84289i 0.500000 + 0.866025i −1.68595 − 0.358360i
4.5 −2.22722 − 0.991620i −0.866025 − 0.500000i 2.63892 + 2.93081i −2.27705 + 0.484002i 1.43302 + 1.97238i −2.08770 + 1.62528i −1.46442 − 4.50703i 0.500000 + 0.866025i 5.55143 + 1.17999i
4.6 −2.22063 − 0.988686i 0.866025 + 0.500000i 2.61542 + 2.90471i 0.559286 − 0.118880i −1.42878 − 1.96654i −2.10920 − 1.59727i −1.43371 − 4.41250i 0.500000 + 0.866025i −1.35950 − 0.288970i
4.7 −2.22049 − 0.988626i −0.866025 − 0.500000i 2.61494 + 2.90418i −2.37463 + 0.504742i 1.42869 + 1.96642i 2.33152 − 1.25060i −1.43308 − 4.41056i 0.500000 + 0.866025i 5.77184 + 1.22684i
4.8 −2.12615 − 0.946623i 0.866025 + 0.500000i 2.28616 + 2.53903i 0.269170 − 0.0572139i −1.36799 − 1.88287i −1.40539 − 2.24163i −1.01882 − 3.13559i 0.500000 + 0.866025i −0.626456 − 0.133157i
4.9 −2.00458 − 0.892498i 0.866025 + 0.500000i 1.88354 + 2.09188i −3.19213 + 0.678507i −1.28977 − 1.77522i −1.47599 + 2.19578i −0.552564 − 1.70062i 0.500000 + 0.866025i 7.00444 + 1.48884i
4.10 −1.92977 − 0.859188i −0.866025 − 0.500000i 1.64754 + 1.82978i 1.61753 − 0.343817i 1.24163 + 1.70896i 1.98413 − 1.75020i −0.301713 − 0.928578i 0.500000 + 0.866025i −3.41687 − 0.726277i
4.11 −1.65371 − 0.736278i −0.866025 − 0.500000i 0.854382 + 0.948888i 3.68738 − 0.783776i 1.06401 + 1.46449i 0.122031 + 2.64294i 0.404517 + 1.24497i 0.500000 + 0.866025i −6.67492 − 1.41880i
4.12 −1.64992 − 0.734592i 0.866025 + 0.500000i 0.844351 + 0.937747i 2.62856 − 0.558719i −1.06158 − 1.46114i 2.63223 + 0.267176i 0.411957 + 1.26787i 0.500000 + 0.866025i −4.74735 − 1.00908i
4.13 −1.61875 − 0.720716i 0.866025 + 0.500000i 0.762671 + 0.847032i −2.13993 + 0.454857i −1.04152 − 1.43353i 1.02852 − 2.43765i 0.471015 + 1.44964i 0.500000 + 0.866025i 3.79185 + 0.805982i
4.14 −1.57542 − 0.701423i −0.866025 − 0.500000i 0.651696 + 0.723782i −2.95298 + 0.627675i 1.01364 + 1.39516i −2.42473 − 1.05862i 0.546789 + 1.68284i 0.500000 + 0.866025i 5.09245 + 1.08243i
4.15 −1.42632 − 0.635038i 0.866025 + 0.500000i 0.292849 + 0.325242i 0.387592 − 0.0823852i −0.917709 − 1.26312i 2.32033 + 1.27126i 0.753781 + 2.31990i 0.500000 + 0.866025i −0.605147 − 0.128628i
4.16 −1.41770 − 0.631202i −0.866025 − 0.500000i 0.273206 + 0.303426i 1.00143 − 0.212860i 0.912166 + 1.25549i 0.279657 + 2.63093i 0.763306 + 2.34922i 0.500000 + 0.866025i −1.55409 − 0.330331i
4.17 −1.13127 − 0.503673i −0.866025 − 0.500000i −0.312180 − 0.346711i 0.553673 − 0.117687i 0.727870 + 1.00183i −1.38069 − 2.25692i 0.943859 + 2.90490i 0.500000 + 0.866025i −0.685628 − 0.145735i
4.18 −1.08901 − 0.484858i 0.866025 + 0.500000i −0.387407 − 0.430259i −1.91218 + 0.406446i −0.700681 − 0.964405i −2.62403 + 0.338310i 0.950015 + 2.92384i 0.500000 + 0.866025i 2.27945 + 0.484512i
4.19 −1.05376 − 0.469165i −0.866025 − 0.500000i −0.447965 − 0.497515i −0.0948512 + 0.0201613i 0.678001 + 0.933189i 1.99461 − 1.73825i 0.951524 + 2.92849i 0.500000 + 0.866025i 0.109409 + 0.0232557i
4.20 −0.993017 − 0.442119i 0.866025 + 0.500000i −0.547649 − 0.608226i 2.51882 − 0.535391i −0.638918 − 0.879395i 0.142090 − 2.64193i 0.946714 + 2.91369i 0.500000 + 0.866025i −2.73793 − 0.581966i
See next 80 embeddings (of 448 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.56
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
41.f even 10 1 inner
287.z even 30 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 861.2.bz.a ✓ 448
7.c even 3 1 inner 861.2.bz.a ✓ 448
41.f even 10 1 inner 861.2.bz.a ✓ 448
287.z even 30 1 inner 861.2.bz.a ✓ 448
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
861.2.bz.a ✓ 448 1.a even 1 1 trivial
861.2.bz.a ✓ 448 7.c even 3 1 inner
861.2.bz.a ✓ 448 41.f even 10 1 inner
861.2.bz.a ✓ 448 287.z even 30 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(861, [\chi])\).