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Properties

Label 861.2.bj.a
Level $861$
Weight $2$
Character orbit 861.bj
Analytic conductor $6.875$
Analytic rank $0$
Dimension $224$
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(214,861)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(861, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 8, 9]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("861.214");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 861 = 3 \cdot 7 \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 861.bj (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: \(6.87511961403\)
Analytic rank: \(0\)
Dimension: \(224\)
Relative dimension: \(56\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 224 q + 112 q^{4}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 224 q + 112 q^{4} + 16 q^{10} - 8 q^{11} - 20 q^{14} - 112 q^{16} + 8 q^{19} - 8 q^{22} + 12 q^{24} + 112 q^{25} - 12 q^{26} + 44 q^{28} - 8 q^{30} - 64 q^{31} - 32 q^{34} - 64 q^{35} + 8 q^{37} - 28 q^{38} - 48 q^{40} + 72 q^{41} + 72 q^{42} + 24 q^{44} - 40 q^{47} - 28 q^{52} + 48 q^{53} + 8 q^{55} + 52 q^{56} - 32 q^{57} - 4 q^{58} - 16 q^{59} - 20 q^{60} - 176 q^{64} + 12 q^{65} - 32 q^{66} - 8 q^{67} + 120 q^{68} - 8 q^{70} + 8 q^{71} - 24 q^{72} + 16 q^{75} + 176 q^{76} - 48 q^{78} + 28 q^{79} + 112 q^{81} + 8 q^{82} + 64 q^{83} + 112 q^{85} - 16 q^{86} + 20 q^{88} + 32 q^{89} + 144 q^{92} + 24 q^{93} - 56 q^{94} - 48 q^{95} - 4 q^{96} - 88 q^{97} + 136 q^{98} - 16 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
214.1 −2.37790 + 1.37288i −0.965926 + 0.258819i 2.76961 − 4.79710i −1.46552 + 0.846117i 1.94155 − 1.94155i −0.816473 + 2.51662i 9.71785i 0.866025 − 0.500000i 2.32324 − 4.02397i
214.2 −2.32444 + 1.34201i 0.965926 − 0.258819i 2.60201 − 4.50681i −2.47854 + 1.43098i −1.89790 + 1.89790i −1.76668 − 1.96948i 8.59966i 0.866025 − 0.500000i 3.84080 − 6.65247i
214.3 −2.20211 + 1.27139i 0.965926 − 0.258819i 2.23287 − 3.86744i 1.92787 − 1.11306i −1.79802 + 1.79802i 0.826816 − 2.51324i 6.26983i 0.866025 − 0.500000i −2.83026 + 4.90215i
214.4 −2.17278 + 1.25446i 0.965926 − 0.258819i 2.14732 − 3.71926i −0.188534 + 0.108850i −1.77407 + 1.77407i 2.40604 + 1.10045i 5.75702i 0.866025 − 0.500000i 0.273095 − 0.473014i
214.5 −2.12976 + 1.22962i −0.965926 + 0.258819i 2.02392 − 3.50553i −2.39895 + 1.38503i 1.73894 − 1.73894i 0.989702 − 2.45367i 5.03612i 0.866025 − 0.500000i 3.40612 − 5.89958i
214.6 −1.99156 + 1.14983i −0.965926 + 0.258819i 1.64421 − 2.84786i 0.452702 − 0.261367i 1.62610 − 1.62610i 2.11271 + 1.59262i 2.96293i 0.866025 − 0.500000i −0.601056 + 1.04106i
214.7 −1.93732 + 1.11851i −0.965926 + 0.258819i 1.50215 − 2.60179i 1.18561 − 0.684509i 1.58182 − 1.58182i −2.50692 − 0.845788i 2.24663i 0.866025 − 0.500000i −1.53127 + 2.65223i
214.8 −1.85701 + 1.07215i −0.965926 + 0.258819i 1.29899 − 2.24992i 3.80154 − 2.19482i 1.51624 − 1.51624i 1.58461 − 2.11873i 1.28226i 0.866025 − 0.500000i −4.70633 + 8.15161i
214.9 −1.83847 + 1.06144i −0.965926 + 0.258819i 1.25331 − 2.17080i 2.78416 − 1.60743i 1.50110 − 1.50110i −0.00946593 + 2.64573i 1.07549i 0.866025 − 0.500000i −3.41239 + 5.91043i
214.10 −1.68728 + 0.974153i 0.965926 − 0.258819i 0.897949 − 1.55529i −3.73766 + 2.15794i −1.37766 + 1.37766i 2.60065 + 0.486454i − 0.397653i 0.866025 − 0.500000i 4.20432 − 7.28210i
214.11 −1.66966 + 0.963976i 0.965926 − 0.258819i 0.858500 − 1.48697i 2.00889 − 1.15983i −1.36327 + 1.36327i −1.76834 − 1.96799i − 0.545611i 0.866025 − 0.500000i −2.23610 + 3.87304i
214.12 −1.66549 + 0.961572i 0.965926 − 0.258819i 0.849243 − 1.47093i −1.66904 + 0.963621i −1.35987 + 1.35987i −1.62356 + 2.08903i − 0.579856i 0.866025 − 0.500000i 1.85318 − 3.20981i
214.13 −1.56664 + 0.904501i 0.965926 − 0.258819i 0.636243 − 1.10201i 2.12301 − 1.22572i −1.27916 + 1.27916i −2.15887 + 1.52948i − 1.31607i 0.866025 − 0.500000i −2.21733 + 3.84053i
214.14 −1.47146 + 0.849550i −0.965926 + 0.258819i 0.443469 − 0.768110i −2.74350 + 1.58396i 1.20144 − 1.20144i 2.22860 + 1.42596i − 1.89120i 0.866025 − 0.500000i 2.69130 − 4.66147i
214.15 −1.39919 + 0.807825i −0.965926 + 0.258819i 0.305163 − 0.528558i −2.18800 + 1.26324i 1.14244 − 1.14244i −2.40342 + 1.10615i − 2.24523i 0.866025 − 0.500000i 2.04096 − 3.53504i
214.16 −1.12086 + 0.647128i 0.965926 − 0.258819i −0.162450 + 0.281372i 3.27353 − 1.88998i −0.915177 + 0.915177i 1.71041 + 2.01854i − 3.00902i 0.866025 − 0.500000i −2.44611 + 4.23679i
214.17 −1.10161 + 0.636015i −0.965926 + 0.258819i −0.190971 + 0.330771i −0.0615046 + 0.0355097i 0.899461 − 0.899461i −1.43724 − 2.22134i − 3.02990i 0.866025 − 0.500000i 0.0451694 − 0.0782357i
214.18 −0.989927 + 0.571535i 0.965926 − 0.258819i −0.346696 + 0.600495i −0.863260 + 0.498403i −0.808272 + 0.808272i 0.866481 + 2.49984i − 3.07873i 0.866025 − 0.500000i 0.569710 − 0.986766i
214.19 −0.970157 + 0.560120i 0.965926 − 0.258819i −0.372531 + 0.645242i 0.218664 − 0.126246i −0.792130 + 0.792130i 2.09260 − 1.61896i − 3.07513i 0.866025 − 0.500000i −0.141425 + 0.244956i
214.20 −0.849774 + 0.490617i 0.965926 − 0.258819i −0.518589 + 0.898223i −0.719645 + 0.415487i −0.693838 + 0.693838i 0.177871 − 2.63977i − 2.98018i 0.866025 − 0.500000i 0.407690 − 0.706140i
See next 80 embeddings (of 224 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 214.56
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
41.c even 4 1 inner
287.r even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 861.2.bj.a ✓ 224
7.c even 3 1 inner 861.2.bj.a ✓ 224
41.c even 4 1 inner 861.2.bj.a ✓ 224
287.r even 12 1 inner 861.2.bj.a ✓ 224
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
861.2.bj.a ✓ 224 1.a even 1 1 trivial
861.2.bj.a ✓ 224 7.c even 3 1 inner
861.2.bj.a ✓ 224 41.c even 4 1 inner
861.2.bj.a ✓ 224 287.r even 12 1 inner

Hecke kernels

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