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Properties

Label 861.2.ce.a
Level $861$
Weight $2$
Character orbit 861.ce
Analytic conductor $6.875$
Analytic rank $0$
Dimension $896$
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(46,861)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(861, base_ring=CyclotomicField(60))
 
chi = DirichletCharacter(H, H._module([0, 40, 33]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("861.46");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 861 = 3 \cdot 7 \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 861.ce (of order \(60\), degree \(16\), 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: \(896\)
Relative dimension: \(56\) over \(\Q(\zeta_{60})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{60}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 896 q - 112 q^{4}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 896 q - 112 q^{4} - 16 q^{10} + 8 q^{11} + 20 q^{14} + 112 q^{16} - 8 q^{19} - 240 q^{20} + 8 q^{22} - 12 q^{24} - 112 q^{25} + 12 q^{26} + 56 q^{28} + 8 q^{30} + 64 q^{31} + 32 q^{34} + 24 q^{35} + 12 q^{37} + 28 q^{38} - 192 q^{40} - 72 q^{41} - 72 q^{42} - 80 q^{43} + 96 q^{44} - 100 q^{46} - 80 q^{47} - 40 q^{49} + 28 q^{52} - 48 q^{53} - 8 q^{55} - 52 q^{56} - 48 q^{57} + 4 q^{58} - 24 q^{59} + 20 q^{60} + 80 q^{61} + 176 q^{64} - 12 q^{65} + 32 q^{66} + 8 q^{67} - 20 q^{68} - 80 q^{69} - 332 q^{70} - 8 q^{71} + 24 q^{72} + 64 q^{75} - 296 q^{76} - 200 q^{77} + 48 q^{78} - 28 q^{79} + 240 q^{80} + 448 q^{81} + 112 q^{82} - 224 q^{83} - 112 q^{85} + 16 q^{86} + 100 q^{88} + 8 q^{89} - 120 q^{90} + 216 q^{92} - 24 q^{93} - 4 q^{94} + 48 q^{95} - 16 q^{96} + 8 q^{97} - 236 q^{98} + 16 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
46.1 −1.09660 + 2.46300i −0.965926 + 0.258819i −3.52556 − 3.91554i −0.578784 − 2.72296i 0.421761 − 2.66289i −0.0393745 − 2.64546i 8.38182 − 2.72342i 0.866025 − 0.500000i 7.34134 + 1.56045i
46.2 −1.06114 + 2.38337i 0.965926 − 0.258819i −3.21615 − 3.57189i −0.861544 − 4.05325i −0.408125 + 2.57680i −1.43489 + 2.22285i 6.96345 − 2.26256i 0.866025 − 0.500000i 10.5746 + 2.24770i
46.3 −1.05000 + 2.35834i 0.965926 − 0.258819i −3.12102 − 3.46624i 0.0494545 + 0.232665i −0.403840 + 2.54975i 2.49139 + 0.890504i 6.54131 − 2.12540i 0.866025 − 0.500000i −0.600632 − 0.127668i
46.4 −1.02230 + 2.29613i 0.965926 − 0.258819i −2.88886 − 3.20841i 0.795530 + 3.74267i −0.393187 + 2.48249i −0.891136 + 2.49116i 5.53940 − 1.79986i 0.866025 − 0.500000i −9.40695 − 1.99951i
46.5 −1.00385 + 2.25468i −0.965926 + 0.258819i −2.73762 − 3.04043i −0.00519720 − 0.0244509i 0.386089 − 2.43767i −0.610568 + 2.57434i 4.90885 − 1.59498i 0.866025 − 0.500000i 0.0603463 + 0.0128270i
46.6 −0.979588 + 2.20019i −0.965926 + 0.258819i −2.54299 − 2.82427i 0.231403 + 1.08867i 0.376758 − 2.37876i −2.64407 + 0.0942094i 4.12395 − 1.33995i 0.866025 − 0.500000i −2.62195 − 0.557313i
46.7 −0.953842 + 2.14236i −0.965926 + 0.258819i −2.34165 − 2.60066i 0.510819 + 2.40322i 0.366856 − 2.31624i 1.70441 − 2.02361i 3.34447 − 1.08668i 0.866025 − 0.500000i −5.63580 − 1.19793i
46.8 −0.911586 + 2.04745i 0.965926 − 0.258819i −2.02282 − 2.24657i 0.544744 + 2.56282i −0.350604 + 2.21363i −1.13042 − 2.39210i 2.18068 − 0.708545i 0.866025 − 0.500000i −5.74383 − 1.22089i
46.9 −0.875053 + 1.96540i 0.965926 − 0.258819i −1.75882 − 1.95337i −0.0934928 − 0.439849i −0.336553 + 2.12491i −2.56869 − 0.633901i 1.28600 − 0.417848i 0.866025 − 0.500000i 0.946290 + 0.201140i
46.10 −0.755219 + 1.69625i −0.965926 + 0.258819i −0.968644 − 1.07579i 0.763902 + 3.59387i 0.290464 − 1.83392i 1.32888 + 2.28781i −0.975454 + 0.316944i 0.866025 − 0.500000i −6.67302 − 1.41839i
46.11 −0.738948 + 1.65971i 0.965926 − 0.258819i −0.870316 − 0.966584i −0.299780 − 1.41035i −0.284206 + 1.79441i 1.23755 − 2.33848i −1.20835 + 0.392615i 0.866025 − 0.500000i 2.56229 + 0.544632i
46.12 −0.727773 + 1.63460i 0.965926 − 0.258819i −0.804016 − 0.892950i −0.269547 − 1.26812i −0.279908 + 1.76727i 1.47959 + 2.19336i −1.35868 + 0.441463i 0.866025 − 0.500000i 2.26904 + 0.482299i
46.13 −0.708659 + 1.59168i −0.965926 + 0.258819i −0.692970 − 0.769621i −0.902681 − 4.24678i 0.272557 − 1.72085i −2.08850 − 1.62424i −1.59799 + 0.519220i 0.866025 − 0.500000i 7.39919 + 1.57275i
46.14 −0.696120 + 1.56351i −0.965926 + 0.258819i −0.621722 − 0.690492i −0.576505 − 2.71224i 0.267734 − 1.69040i 0.723572 + 2.54489i −1.74303 + 0.566346i 0.866025 − 0.500000i 4.64193 + 0.986674i
46.15 −0.622613 + 1.39841i −0.965926 + 0.258819i −0.229645 − 0.255046i −0.280820 − 1.32115i 0.239462 − 1.51190i 2.48432 − 0.910017i −2.41202 + 0.783714i 0.866025 − 0.500000i 2.02236 + 0.429865i
46.16 −0.620962 + 1.39470i 0.965926 − 0.258819i −0.221341 − 0.245824i −0.285253 − 1.34201i −0.238827 + 1.50790i −1.88604 + 1.85549i −2.42364 + 0.787490i 0.866025 − 0.500000i 2.04883 + 0.435493i
46.17 −0.548271 + 1.23144i −0.965926 + 0.258819i 0.122426 + 0.135968i 0.540658 + 2.54359i 0.210870 − 1.33138i −1.11412 − 2.39974i −2.79856 + 0.909307i 0.866025 − 0.500000i −3.42870 − 0.728793i
46.18 −0.480768 + 1.07982i 0.965926 − 0.258819i 0.403383 + 0.448002i 0.495260 + 2.33002i −0.184907 + 1.16746i 2.17481 + 1.50672i −2.92602 + 0.950721i 0.866025 − 0.500000i −2.75411 − 0.585404i
46.19 −0.478614 + 1.07499i −0.965926 + 0.258819i 0.411739 + 0.457282i −0.291035 − 1.36921i 0.184079 − 1.16223i −2.60093 − 0.484936i −2.92689 + 0.951003i 0.866025 − 0.500000i 1.61118 + 0.342466i
46.20 −0.392537 + 0.881652i 0.965926 − 0.258819i 0.715037 + 0.794129i 0.480664 + 2.26135i −0.150973 + 0.953206i 1.74872 − 1.98544i −2.81653 + 0.915146i 0.866025 − 0.500000i −2.18240 − 0.463883i
See next 80 embeddings (of 896 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 46.56
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
41.g even 20 1 inner
287.bc even 60 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 861.2.ce.a ✓ 896
7.c even 3 1 inner 861.2.ce.a ✓ 896
41.g even 20 1 inner 861.2.ce.a ✓ 896
287.bc even 60 1 inner 861.2.ce.a ✓ 896
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
861.2.ce.a ✓ 896 1.a even 1 1 trivial
861.2.ce.a ✓ 896 7.c even 3 1 inner
861.2.ce.a ✓ 896 41.g even 20 1 inner
861.2.ce.a ✓ 896 287.bc even 60 1 inner

Hecke kernels

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