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Properties

Label 285.2.p.b
Level $285$
Weight $2$
Character orbit 285.p
Analytic conductor $2.276$
Analytic rank $0$
Dimension $52$
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] = [285,2,Mod(221,285)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(285, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("285.221");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 285 = 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 285.p (of order \(6\), degree \(2\), 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: \(2.27573645761\)
Analytic rank: \(0\)
Dimension: \(52\)
Relative dimension: \(26\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 52 q - 12 q^{3} - 32 q^{4} - 2 q^{6} + 12 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 52 q - 12 q^{3} - 32 q^{4} - 2 q^{6} + 12 q^{7} + 12 q^{10} - 30 q^{13} - 52 q^{16} + 16 q^{19} - 6 q^{21} - 36 q^{22} + 2 q^{24} + 26 q^{25} - 26 q^{28} + 28 q^{30} + 18 q^{34} - 4 q^{36} - 40 q^{39} - 6 q^{40} + 14 q^{42} + 58 q^{43} - 8 q^{45} - 54 q^{48} + 72 q^{49} + 12 q^{51} + 54 q^{52} - 2 q^{54} + 30 q^{57} - 16 q^{58} - 70 q^{61} - 30 q^{63} + 84 q^{64} - 14 q^{66} - 6 q^{67} - 30 q^{70} + 102 q^{72} - 38 q^{73} - 2 q^{76} - 24 q^{78} - 18 q^{79} + 36 q^{81} + 16 q^{82} - 20 q^{85} + 12 q^{87} + 48 q^{90} - 30 q^{91} - 26 q^{93} - 160 q^{96} - 108 q^{97} + 60 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
221.1 −1.36812 − 2.36966i 1.72171 − 0.188945i −2.74351 + 4.75190i −0.866025 + 0.500000i −2.80325 − 3.82137i 2.39037 9.54134 2.92860 − 0.650619i 2.36966 + 1.36812i
221.2 −1.29839 − 2.24887i −0.221535 + 1.71782i −2.37163 + 4.10778i 0.866025 − 0.500000i 4.15081 − 1.73220i −1.78544 7.12361 −2.90184 − 0.761118i −2.24887 − 1.29839i
221.3 −1.26670 − 2.19399i −1.64962 − 0.527985i −2.20907 + 3.82623i 0.866025 − 0.500000i 0.931178 + 4.28805i 4.60478 6.12615 2.44246 + 1.74194i −2.19399 − 1.26670i
221.4 −1.19211 − 2.06479i −1.72787 + 0.120203i −1.84224 + 3.19085i −0.866025 + 0.500000i 2.30801 + 3.42440i −4.13275 4.01614 2.97110 − 0.415393i 2.06479 + 1.19211i
221.5 −1.10826 − 1.91956i −0.142214 − 1.72620i −1.45646 + 2.52267i −0.866025 + 0.500000i −3.15593 + 2.18606i −0.689836 2.02351 −2.95955 + 0.490982i 1.91956 + 1.10826i
221.6 −0.993785 − 1.72129i 0.391868 + 1.68714i −0.975218 + 1.68913i −0.866025 + 0.500000i 2.51462 − 2.35117i 2.20393 −0.0985113 −2.69288 + 1.32227i 1.72129 + 0.993785i
221.7 −0.792294 − 1.37229i −0.932493 − 1.45961i −0.255461 + 0.442471i 0.866025 − 0.500000i −1.26420 + 2.43609i −1.64134 −2.35958 −1.26091 + 2.72215i −1.37229 − 0.792294i
221.8 −0.612592 − 1.06104i −1.71390 + 0.250109i 0.249462 − 0.432081i −0.866025 + 0.500000i 1.31530 + 1.66530i 3.95218 −3.06164 2.87489 − 0.857322i 1.06104 + 0.612592i
221.9 −0.594742 − 1.03012i −0.816211 + 1.52768i 0.292565 − 0.506737i 0.866025 − 0.500000i 2.05913 − 0.0677763i 1.51467 −3.07497 −1.66760 − 2.49381i −1.03012 − 0.594742i
221.10 −0.392604 − 0.680010i 1.13787 − 1.30586i 0.691724 − 1.19810i −0.866025 + 0.500000i −1.33473 − 0.261077i −3.79523 −2.65671 −0.410515 − 2.97178i 0.680010 + 0.392604i
221.11 −0.324144 − 0.561434i 1.28418 + 1.16228i 0.789862 − 1.36808i 0.866025 − 0.500000i 0.236281 − 1.09773i 3.01197 −2.32069 0.298234 + 2.98514i −0.561434 − 0.324144i
221.12 −0.286447 − 0.496140i −1.00111 + 1.41343i 0.835897 − 1.44782i −0.866025 + 0.500000i 0.988023 + 0.0918163i −3.42979 −2.10355 −0.995574 − 2.82999i 0.496140 + 0.286447i
221.13 −0.0544110 − 0.0942426i 1.59742 + 0.669509i 0.994079 − 1.72180i −0.866025 + 0.500000i −0.0238211 − 0.186974i 0.796510 −0.433999 2.10352 + 2.13898i 0.0942426 + 0.0544110i
221.14 0.0544110 + 0.0942426i 0.218900 − 1.71816i 0.994079 − 1.72180i 0.866025 − 0.500000i 0.173835 − 0.0728572i 0.796510 0.433999 −2.90417 − 0.752210i 0.0942426 + 0.0544110i
221.15 0.286447 + 0.496140i −1.72462 + 0.160268i 0.835897 − 1.44782i 0.866025 − 0.500000i −0.573527 − 0.809745i −3.42979 2.10355 2.94863 − 0.552802i 0.496140 + 0.286447i
221.16 0.324144 + 0.561434i −0.364470 − 1.69327i 0.789862 − 1.36808i −0.866025 + 0.500000i 0.832518 − 0.753489i 3.01197 2.32069 −2.73432 + 1.23429i −0.561434 − 0.324144i
221.17 0.392604 + 0.680010i 1.69984 − 0.332495i 0.691724 − 1.19810i 0.866025 − 0.500000i 0.893463 + 1.02537i −3.79523 2.65671 2.77889 − 1.13037i 0.680010 + 0.392604i
221.18 0.594742 + 1.03012i −1.73111 − 0.0569796i 0.292565 − 0.506737i −0.866025 + 0.500000i −0.970869 − 1.81715i 1.51467 3.07497 2.99351 + 0.197276i −1.03012 − 0.594742i
221.19 0.612592 + 1.06104i −1.07355 + 1.35922i 0.249462 − 0.432081i 0.866025 − 0.500000i −2.09984 − 0.306430i 3.95218 3.06164 −0.694983 − 2.91839i 1.06104 + 0.612592i
221.20 0.792294 + 1.37229i 0.797812 + 1.53737i −0.255461 + 0.442471i −0.866025 + 0.500000i −1.47762 + 2.31288i −1.64134 2.35958 −1.72699 + 2.45306i −1.37229 − 0.792294i
See all 52 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 221.26
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
19.d odd 6 1 inner
57.f even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 285.2.p.b ✓ 52
3.b odd 2 1 inner 285.2.p.b ✓ 52
19.d odd 6 1 inner 285.2.p.b ✓ 52
57.f even 6 1 inner 285.2.p.b ✓ 52
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
285.2.p.b ✓ 52 1.a even 1 1 trivial
285.2.p.b ✓ 52 3.b odd 2 1 inner
285.2.p.b ✓ 52 19.d odd 6 1 inner
285.2.p.b ✓ 52 57.f even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{52} + 42 T_{2}^{50} + 995 T_{2}^{48} + 16176 T_{2}^{46} + 199355 T_{2}^{44} + 1943476 T_{2}^{42} + \cdots + 36864 \) acting on \(S_{2}^{\mathrm{new}}(285, [\chi])\). Copy content Toggle raw display