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Properties

Label 861.2.ci.a
Level $861$
Weight $2$
Character orbit 861.ci
Analytic conductor $6.875$
Analytic rank $0$
Dimension $1792$
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(19,861)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(861, base_ring=CyclotomicField(120))
 
chi = DirichletCharacter(H, H._module([0, 100, 27]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("861.19");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 861 = 3 \cdot 7 \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 861.ci (of order \(120\), degree \(32\), 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: \(1792\)
Relative dimension: \(56\) over \(\Q(\zeta_{120})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{120}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 1792 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 1792 q + 8 q^{14} - 224 q^{16} + 48 q^{19} - 80 q^{22} - 72 q^{24} - 48 q^{26} + 64 q^{29} - 16 q^{30} - 40 q^{32} - 72 q^{37} - 72 q^{38} + 64 q^{42} - 64 q^{43} - 192 q^{44} - 24 q^{46} + 48 q^{47} - 160 q^{49} + 128 q^{50} + 72 q^{52} - 16 q^{53} + 112 q^{56} + 8 q^{58} - 64 q^{65} + 32 q^{67} - 744 q^{68} + 160 q^{70} + 32 q^{71} - 72 q^{73} + 56 q^{74} - 384 q^{75} - 32 q^{77} + 96 q^{78} - 8 q^{79} + 72 q^{80} - 576 q^{82} + 240 q^{84} + 32 q^{85} - 88 q^{88} - 384 q^{89} + 112 q^{91} + 216 q^{94} + 64 q^{95} - 480 q^{96} - 40 q^{98} - 32 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1 −0.146171 + 2.78911i −0.793353 − 0.608761i −5.76872 − 0.606317i −2.12837 − 0.817003i 1.81387 − 2.12377i −0.749695 − 2.53731i 1.66048 − 10.4839i 0.258819 + 0.965926i 2.58982 − 5.81682i
19.2 −0.143898 + 2.74574i −0.793353 − 0.608761i −5.52933 − 0.581157i 3.00858 + 1.15489i 1.78566 − 2.09074i −0.800122 + 2.52187i 1.53113 − 9.66717i 0.258819 + 0.965926i −3.60395 + 8.09460i
19.3 −0.136721 + 2.60879i 0.793353 + 0.608761i −4.79807 − 0.504298i −0.369414 − 0.141805i −1.69660 + 1.98647i −1.17330 + 2.37137i 1.15428 − 7.28781i 0.258819 + 0.965926i 0.420446 − 0.944336i
19.4 −0.132097 + 2.52057i 0.793353 + 0.608761i −4.34676 − 0.456863i −3.35114 − 1.28638i −1.63922 + 1.91928i 2.61716 − 0.387934i 0.936060 − 5.91005i 0.258819 + 0.965926i 3.68509 − 8.27686i
19.5 −0.131116 + 2.50184i 0.793353 + 0.608761i −4.25299 − 0.447007i 0.640190 + 0.245746i −1.62705 + 1.90503i 2.20989 − 1.45478i 0.892155 − 5.63284i 0.258819 + 0.965926i −0.698758 + 1.56944i
19.6 −0.121782 + 2.32374i −0.793353 − 0.608761i −3.39588 − 0.356922i −3.43338 − 1.31795i 1.51122 − 1.76941i 2.20419 + 1.46340i 0.514927 − 3.25112i 0.258819 + 0.965926i 3.48070 − 7.81778i
19.7 −0.121435 + 2.31712i −0.793353 − 0.608761i −3.36526 − 0.353703i −1.39955 − 0.537236i 1.50691 − 1.76437i −0.0317108 + 2.64556i 0.502283 − 3.17129i 0.258819 + 0.965926i 1.41479 − 3.17768i
19.8 −0.114313 + 2.18121i 0.793353 + 0.608761i −2.75559 − 0.289624i 3.90467 + 1.49886i −1.41853 + 1.66088i 1.20894 + 2.35339i 0.263359 − 1.66278i 0.258819 + 0.965926i −3.71569 + 8.34557i
19.9 −0.110336 + 2.10533i −0.793353 − 0.608761i −2.43119 − 0.255528i 0.0694164 + 0.0266465i 1.36918 − 1.60310i −2.64167 − 0.146905i 0.146622 − 0.925738i 0.258819 + 0.965926i −0.0637586 + 0.143204i
19.10 −0.107536 + 2.05190i 0.793353 + 0.608761i −2.20969 − 0.232248i 0.482441 + 0.185192i −1.33443 + 1.56242i −1.80042 − 1.93868i 0.0713137 − 0.450257i 0.258819 + 0.965926i −0.431875 + 0.970007i
19.11 −0.0990105 + 1.88923i −0.793353 − 0.608761i −1.57036 − 0.165051i 0.980356 + 0.376323i 1.22864 − 1.43856i −0.159200 − 2.64096i −0.124592 + 0.786641i 0.258819 + 0.965926i −0.808028 + 1.81486i
19.12 −0.0976441 + 1.86316i 0.793353 + 0.608761i −1.47279 − 0.154796i −0.260966 − 0.100176i −1.21169 + 1.41870i 0.614871 + 2.57331i −0.151505 + 0.956567i 0.258819 + 0.965926i 0.212125 − 0.476441i
19.13 −0.0904834 + 1.72653i 0.793353 + 0.608761i −0.983662 − 0.103387i −2.02542 − 0.777486i −1.12283 + 1.31466i −1.61038 − 2.09921i −0.273412 + 1.72626i 0.258819 + 0.965926i 1.52562 − 3.42659i
19.14 −0.0870888 + 1.66175i −0.793353 − 0.608761i −0.764799 − 0.0803836i 3.91509 + 1.50286i 1.08070 − 1.26534i −2.45902 + 0.976341i −0.320442 + 2.02319i 0.258819 + 0.965926i −2.83835 + 6.37503i
19.15 −0.0830929 + 1.58551i 0.793353 + 0.608761i −0.517887 − 0.0544321i 2.80671 + 1.07739i −1.03112 + 1.20728i −2.57849 − 0.592771i −0.367402 + 2.31968i 0.258819 + 0.965926i −1.94143 + 4.36053i
19.16 −0.0776716 + 1.48206i −0.793353 − 0.608761i −0.201434 − 0.0211715i −2.23543 − 0.858101i 0.963844 − 1.12852i 1.57288 − 2.12745i −0.417305 + 2.63476i 0.258819 + 0.965926i 1.44539 − 3.24640i
19.17 −0.0700176 + 1.33602i −0.793353 − 0.608761i 0.209010 + 0.0219678i 0.821301 + 0.315268i 0.868863 − 1.01731i 0.0333939 + 2.64554i −0.462555 + 2.92046i 0.258819 + 0.965926i −0.478709 + 1.07520i
19.18 −0.0666355 + 1.27148i −0.793353 − 0.608761i 0.376819 + 0.0396053i 2.66526 + 1.02310i 0.826894 − 0.968169i 2.64233 − 0.134433i −0.473820 + 2.99158i 0.258819 + 0.965926i −1.47845 + 3.32066i
19.19 −0.0589508 + 1.12485i 0.793353 + 0.608761i 0.727235 + 0.0764354i 2.18799 + 0.839891i −0.731533 + 0.856515i 2.59156 + 0.532756i −0.481263 + 3.03857i 0.258819 + 0.965926i −1.07373 + 2.41165i
19.20 −0.0520452 + 0.993082i 0.793353 + 0.608761i 1.00554 + 0.105687i −2.65786 − 1.02026i −0.645840 + 0.756182i 0.587556 − 2.57969i −0.468420 + 2.95749i 0.258819 + 0.965926i 1.15153 − 2.58638i
See next 80 embeddings (of 1792 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.56
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner
41.h odd 40 1 inner
287.be even 120 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 861.2.ci.a ✓ 1792
7.d odd 6 1 inner 861.2.ci.a ✓ 1792
41.h odd 40 1 inner 861.2.ci.a ✓ 1792
287.be even 120 1 inner 861.2.ci.a ✓ 1792
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
861.2.ci.a ✓ 1792 1.a even 1 1 trivial
861.2.ci.a ✓ 1792 7.d odd 6 1 inner
861.2.ci.a ✓ 1792 41.h odd 40 1 inner
861.2.ci.a ✓ 1792 287.be even 120 1 inner

Hecke kernels

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