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Properties

Label 861.2.bk.a
Level $861$
Weight $2$
Character orbit 861.bk
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(16,861)]
 
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, 10, 18]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("861.16");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 861 = 3 \cdot 7 \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 861.bk (of order \(15\), 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: \(224\)
Relative dimension: \(28\) over \(\Q(\zeta_{15})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 224 q - 112 q^{3} + 28 q^{4} + 24 q^{8} - 112 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 224 q - 112 q^{3} + 28 q^{4} + 24 q^{8} - 112 q^{9} + 17 q^{10} + 28 q^{12} + 12 q^{14} + 28 q^{16} + 10 q^{17} - 10 q^{19} + 56 q^{20} + 32 q^{22} - 4 q^{23} - 12 q^{24} + 48 q^{25} - 26 q^{26} + 224 q^{27} + 15 q^{28} - 44 q^{29} + 17 q^{30} - 10 q^{31} + 66 q^{32} + 76 q^{34} - 7 q^{35} - 56 q^{36} + 19 q^{37} + 4 q^{38} - 70 q^{40} + 10 q^{41} + 6 q^{42} + 4 q^{43} - 2 q^{44} + 32 q^{46} - 44 q^{47} - 56 q^{48} - 16 q^{49} - 60 q^{50} + 10 q^{51} - 46 q^{52} + 53 q^{53} + 28 q^{55} - 13 q^{56} + 20 q^{57} - 32 q^{58} - 6 q^{59} - 28 q^{60} - 4 q^{61} + 18 q^{62} - 188 q^{64} + 46 q^{65} - 16 q^{66} - 6 q^{67} + 36 q^{68} + 8 q^{69} + 189 q^{70} - 34 q^{71} - 12 q^{72} - 102 q^{73} - 118 q^{74} + 48 q^{75} + 70 q^{76} - 18 q^{77} + 52 q^{78} - 66 q^{79} - 44 q^{80} - 112 q^{81} - 24 q^{82} + 52 q^{83} + 6 q^{84} + 128 q^{85} + 5 q^{86} + 22 q^{87} + 59 q^{88} + 52 q^{89} - 34 q^{90} + 126 q^{91} - 30 q^{92} - 10 q^{93} - 97 q^{94} + 53 q^{95} + 66 q^{96} + 112 q^{97} + 30 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
16.1 −1.86678 − 2.07327i −0.500000 − 0.866025i −0.604524 + 5.75166i −3.20438 + 1.42668i −0.862115 + 2.65332i 2.48664 + 0.903676i 8.53918 − 6.20408i −0.500000 + 0.866025i 8.93979 + 3.98025i
16.2 −1.74316 − 1.93597i −0.500000 − 0.866025i −0.500333 + 4.76035i 2.49594 − 1.11126i −0.805022 + 2.47760i −2.53897 + 0.744073i 5.87292 − 4.26692i −0.500000 + 0.866025i −6.50218 − 2.89496i
16.3 −1.65398 − 1.83693i −0.500000 − 0.866025i −0.429607 + 4.08743i −1.63051 + 0.725949i −0.763838 + 2.35085i −0.962759 − 2.46437i 4.21938 − 3.06556i −0.500000 + 0.866025i 4.03034 + 1.79442i
16.4 −1.62445 − 1.80413i −0.500000 − 0.866025i −0.407005 + 3.87239i 0.561384 − 0.249944i −0.750200 + 2.30888i 0.911895 + 2.48364i 3.71936 − 2.70227i −0.500000 + 0.866025i −1.36287 − 0.606789i
16.5 −1.37188 − 1.52363i −0.500000 − 0.866025i −0.230330 + 2.19144i 2.59322 − 1.15458i −0.633561 + 1.94990i 0.560879 − 2.58562i 0.337565 − 0.245255i −0.500000 + 0.866025i −5.31674 − 2.36717i
16.6 −1.23957 − 1.37668i −0.500000 − 0.866025i −0.149662 + 1.42394i −1.12598 + 0.501318i −0.572456 + 1.76184i 0.716161 − 2.54698i −0.851592 + 0.618718i −0.500000 + 0.866025i 2.08589 + 0.928696i
16.7 −1.07412 − 1.19293i −0.500000 − 0.866025i −0.0602948 + 0.573666i −1.90350 + 0.847493i −0.496049 + 1.52668i 2.63290 − 0.260479i −1.84824 + 1.34282i −0.500000 + 0.866025i 3.05559 + 1.36044i
16.8 −0.986134 − 1.09521i −0.500000 − 0.866025i −0.0179738 + 0.171009i 3.34713 − 1.49024i −0.455415 + 1.40162i 1.59840 + 2.10835i −2.17957 + 1.58355i −0.500000 + 0.866025i −4.93284 − 2.19624i
16.9 −0.844253 − 0.937638i −0.500000 − 0.866025i 0.0426551 − 0.405836i 0.472555 − 0.210395i −0.389892 + 1.19996i −2.33449 + 1.24504i −2.45804 + 1.78587i −0.500000 + 0.866025i −0.596230 − 0.265459i
16.10 −0.730878 − 0.811723i −0.500000 − 0.866025i 0.0843465 − 0.802503i −3.96642 + 1.76596i −0.337533 + 1.03882i −0.624182 + 2.57107i −2.48040 + 1.80212i −0.500000 + 0.866025i 4.33244 + 1.92893i
16.11 −0.587555 − 0.652546i −0.500000 − 0.866025i 0.128462 − 1.22223i 0.576981 − 0.256888i −0.271344 + 0.835110i −2.42819 − 1.05066i −2.29381 + 1.66655i −0.500000 + 0.866025i −0.506639 − 0.225570i
16.12 −0.306305 − 0.340186i −0.500000 − 0.866025i 0.187153 − 1.78064i 2.78312 − 1.23913i −0.141457 + 0.435361i 1.36398 − 2.26706i −1.40375 + 1.01989i −0.500000 + 0.866025i −1.27402 − 0.567229i
16.13 −0.204292 − 0.226890i −0.500000 − 0.866025i 0.199313 − 1.89634i −0.674704 + 0.300397i −0.0943460 + 0.290367i −1.87874 − 1.86288i −0.964980 + 0.701099i −0.500000 + 0.866025i 0.205994 + 0.0917143i
16.14 −0.0476784 − 0.0529522i −0.500000 − 0.866025i 0.208526 − 1.98399i −0.353167 + 0.157240i −0.0220188 + 0.0677668i 1.68784 + 2.03745i −0.230291 + 0.167316i −0.500000 + 0.866025i 0.0251646 + 0.0112040i
16.15 0.0561408 + 0.0623506i −0.500000 − 0.866025i 0.208321 − 1.98204i −2.03186 + 0.904643i 0.0259269 − 0.0797947i 2.60898 + 0.439580i 0.271032 − 0.196916i −0.500000 + 0.866025i −0.170475 − 0.0759005i
16.16 0.208902 + 0.232009i −0.500000 − 0.866025i 0.198869 − 1.89211i −1.25831 + 0.560237i 0.0964749 − 0.296919i −0.434816 + 2.60978i 0.985680 − 0.716138i −0.500000 + 0.866025i −0.392844 − 0.174905i
16.17 0.595593 + 0.661473i −0.500000 − 0.866025i 0.126241 − 1.20111i 2.29769 − 1.02300i 0.275056 − 0.846535i −1.89993 + 1.84127i 2.30990 − 1.67824i −0.500000 + 0.866025i 2.04517 + 0.910569i
16.18 0.617066 + 0.685322i −0.500000 − 0.866025i 0.120162 − 1.14327i 0.0876093 − 0.0390062i 0.284973 − 0.877056i −1.43762 − 2.22109i 2.34979 − 1.70722i −0.500000 + 0.866025i 0.0807926 + 0.0359712i
16.19 0.701997 + 0.779646i −0.500000 − 0.866025i 0.0940079 − 0.894426i 1.81473 − 0.807971i 0.324195 − 0.997770i 1.90225 − 1.83887i 2.46084 − 1.78790i −0.500000 + 0.866025i 1.90387 + 0.847656i
16.20 0.781465 + 0.867905i −0.500000 − 0.866025i 0.0664857 − 0.632569i −3.75592 + 1.67224i 0.360895 − 1.11072i −2.59424 − 0.519545i 2.49064 − 1.80955i −0.500000 + 0.866025i −4.38647 − 1.95298i
See next 80 embeddings (of 224 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 16.28
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
41.d even 5 1 inner
287.s even 15 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 861.2.bk.a ✓ 224
7.c even 3 1 inner 861.2.bk.a ✓ 224
41.d even 5 1 inner 861.2.bk.a ✓ 224
287.s even 15 1 inner 861.2.bk.a ✓ 224
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
861.2.bk.a ✓ 224 1.a even 1 1 trivial
861.2.bk.a ✓ 224 7.c even 3 1 inner
861.2.bk.a ✓ 224 41.d even 5 1 inner
861.2.bk.a ✓ 224 287.s even 15 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{224} - 42 T_{2}^{222} - 8 T_{2}^{221} + 777 T_{2}^{220} + 294 T_{2}^{219} - 7072 T_{2}^{218} + \cdots + 21407287493601 \) acting on \(S_{2}^{\mathrm{new}}(861, [\chi])\). Copy content Toggle raw display