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Properties

Label 220.2.r.d
Level $220$
Weight $2$
Character orbit 220.r
Analytic conductor $1.757$
Analytic rank $0$
Dimension $48$
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [220,2,Mod(51,220)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("220.51"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(220, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([5, 0, 7])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 220 = 2^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 220.r (of order \(10\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [48,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.75670884447\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(12\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 12 q^{5} + 20 q^{6} + 16 q^{9} + 14 q^{12} - 10 q^{13} - 21 q^{14} - 4 q^{16} - 25 q^{18} + 5 q^{20} + 15 q^{22} - 30 q^{24} - 12 q^{25} - 4 q^{26} - 60 q^{28} + 20 q^{29} - 10 q^{33} - 18 q^{34}+ \cdots + 38 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
51.1 −1.41421 + 0.00460387i −2.09469 − 0.680607i 1.99996 − 0.0130217i −0.809017 + 0.587785i 2.96546 + 0.952875i −0.548700 − 1.68872i −2.82829 + 0.0276229i 1.49746 + 1.08797i 1.14141 − 0.834974i
51.2 −1.30396 + 0.547426i 3.21575 + 1.04486i 1.40065 − 1.42765i −0.809017 + 0.587785i −4.76521 + 0.397924i −0.161812 − 0.498007i −1.04486 + 2.62836i 6.82226 + 4.95666i 0.733161 − 1.20933i
51.3 −1.01371 − 0.986100i −0.260392 − 0.0846065i 0.0552150 + 1.99924i −0.809017 + 0.587785i 0.180532 + 0.342539i −0.560317 − 1.72448i 1.91548 − 2.08109i −2.36641 − 1.71929i 1.39972 + 0.201928i
51.4 −1.00805 + 0.991888i 0.115659 + 0.0375799i 0.0323155 − 1.99974i −0.809017 + 0.587785i −0.153865 + 0.0768386i −1.38450 − 4.26105i 1.95094 + 2.04788i −2.41509 − 1.75466i 0.232509 − 1.39497i
51.5 −0.631838 + 1.26522i −0.115659 − 0.0375799i −1.20156 − 1.59883i −0.809017 + 0.587785i 0.120625 − 0.122590i 1.38450 + 4.26105i 2.78206 − 0.510037i −2.41509 − 1.75466i −0.232509 − 1.39497i
51.6 −0.385414 − 1.36068i −1.51690 − 0.492870i −1.70291 + 1.04885i −0.809017 + 0.587785i −0.0860056 + 2.25398i 1.09672 + 3.37535i 2.08348 + 1.91288i −0.368991 − 0.268088i 1.11160 + 0.874274i
51.7 −0.117686 + 1.40931i −3.21575 − 1.04486i −1.97230 − 0.331712i −0.809017 + 0.587785i 1.85098 − 4.40902i 0.161812 + 0.498007i 0.699596 − 2.74054i 6.82226 + 4.95666i −0.733161 − 1.20933i
51.8 0.432635 + 1.34641i 2.09469 + 0.680607i −1.62565 + 1.16501i −0.809017 + 0.587785i −0.0101400 + 3.11478i 0.548700 + 1.68872i −2.27190 − 1.68478i 1.49746 + 1.08797i −1.14141 − 0.834974i
51.9 0.766165 − 1.18869i −2.27074 − 0.737810i −0.825983 − 1.82147i −0.809017 + 0.587785i −2.61679 + 2.13393i −0.461886 − 1.42154i −2.79801 − 0.413706i 2.18487 + 1.58740i 0.0788559 + 1.41201i
51.10 0.893756 − 1.09599i 2.27074 + 0.737810i −0.402399 − 1.95910i −0.809017 + 0.587785i 2.83813 − 1.82930i 0.461886 + 1.42154i −2.50681 − 1.30993i 2.18487 + 1.58740i −0.0788559 + 1.41201i
51.11 1.25109 + 0.659374i 0.260392 + 0.0846065i 1.13045 + 1.64987i −0.809017 + 0.587785i 0.269987 + 0.277546i 0.560317 + 1.72448i 0.326416 + 2.80953i −2.36641 − 1.71929i −1.39972 + 0.201928i
51.12 1.41319 − 0.0539233i 1.51690 + 0.492870i 1.99418 − 0.152407i −0.809017 + 0.587785i 2.17024 + 0.614721i −1.09672 − 3.37535i 2.80993 − 0.322913i −0.368991 − 0.268088i −1.11160 + 0.874274i
151.1 −1.41421 − 0.00460387i −2.09469 + 0.680607i 1.99996 + 0.0130217i −0.809017 − 0.587785i 2.96546 − 0.952875i −0.548700 + 1.68872i −2.82829 − 0.0276229i 1.49746 − 1.08797i 1.14141 + 0.834974i
151.2 −1.30396 − 0.547426i 3.21575 − 1.04486i 1.40065 + 1.42765i −0.809017 − 0.587785i −4.76521 − 0.397924i −0.161812 + 0.498007i −1.04486 − 2.62836i 6.82226 − 4.95666i 0.733161 + 1.20933i
151.3 −1.01371 + 0.986100i −0.260392 + 0.0846065i 0.0552150 − 1.99924i −0.809017 − 0.587785i 0.180532 − 0.342539i −0.560317 + 1.72448i 1.91548 + 2.08109i −2.36641 + 1.71929i 1.39972 − 0.201928i
151.4 −1.00805 − 0.991888i 0.115659 − 0.0375799i 0.0323155 + 1.99974i −0.809017 − 0.587785i −0.153865 − 0.0768386i −1.38450 + 4.26105i 1.95094 − 2.04788i −2.41509 + 1.75466i 0.232509 + 1.39497i
151.5 −0.631838 − 1.26522i −0.115659 + 0.0375799i −1.20156 + 1.59883i −0.809017 − 0.587785i 0.120625 + 0.122590i 1.38450 − 4.26105i 2.78206 + 0.510037i −2.41509 + 1.75466i −0.232509 + 1.39497i
151.6 −0.385414 + 1.36068i −1.51690 + 0.492870i −1.70291 − 1.04885i −0.809017 − 0.587785i −0.0860056 − 2.25398i 1.09672 − 3.37535i 2.08348 − 1.91288i −0.368991 + 0.268088i 1.11160 − 0.874274i
151.7 −0.117686 − 1.40931i −3.21575 + 1.04486i −1.97230 + 0.331712i −0.809017 − 0.587785i 1.85098 + 4.40902i 0.161812 − 0.498007i 0.699596 + 2.74054i 6.82226 − 4.95666i −0.733161 + 1.20933i
151.8 0.432635 − 1.34641i 2.09469 − 0.680607i −1.62565 − 1.16501i −0.809017 − 0.587785i −0.0101400 − 3.11478i 0.548700 − 1.68872i −2.27190 + 1.68478i 1.49746 − 1.08797i −1.14141 + 0.834974i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 51.12
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
11.d odd 10 1 inner
44.g even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 220.2.r.d ✓ 48
4.b odd 2 1 inner 220.2.r.d ✓ 48
11.d odd 10 1 inner 220.2.r.d ✓ 48
44.g even 10 1 inner 220.2.r.d ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
220.2.r.d ✓ 48 1.a even 1 1 trivial
220.2.r.d ✓ 48 4.b odd 2 1 inner
220.2.r.d ✓ 48 11.d odd 10 1 inner
220.2.r.d ✓ 48 44.g even 10 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{48} - 26 T_{3}^{46} + 393 T_{3}^{44} - 4613 T_{3}^{42} + 51675 T_{3}^{40} - 431059 T_{3}^{38} + \cdots + 65536 \) acting on \(S_{2}^{\mathrm{new}}(220, [\chi])\). Copy content Toggle raw display