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Properties

Label 550.2.be.a
Level $550$
Weight $2$
Character orbit 550.be
Analytic conductor $4.392$
Analytic rank $0$
Dimension $240$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [550,2,Mod(17,550)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(550, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([13, 18]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("550.17");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 550 = 2 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 550.be (of order \(20\), 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: \(4.39177211117\)
Analytic rank: \(0\)
Dimension: \(240\)
Relative dimension: \(30\) over \(\Q(\zeta_{20})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 240 q - 4 q^{3} + 12 q^{5}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 240 q - 4 q^{3} + 12 q^{5} + 20 q^{10} + 4 q^{12} + 4 q^{15} + 60 q^{16} - 20 q^{17} - 40 q^{19} - 4 q^{20} - 4 q^{22} + 12 q^{23} + 8 q^{27} - 20 q^{28} - 4 q^{33} + 60 q^{36} - 40 q^{37} - 36 q^{38} + 200 q^{39} - 40 q^{42} - 44 q^{45} + 4 q^{48} + 120 q^{49} + 100 q^{53} + 60 q^{54} + 16 q^{55} - 80 q^{57} - 32 q^{58} - 80 q^{59} - 20 q^{60} - 40 q^{62} + 60 q^{67} - 20 q^{68} - 60 q^{69} - 16 q^{70} + 60 q^{71} + 80 q^{73} - 76 q^{75} - 136 q^{77} + 16 q^{78} - 80 q^{79} + 8 q^{80} - 280 q^{81} - 40 q^{82} + 100 q^{83} - 40 q^{85} + 36 q^{88} + 8 q^{92} + 16 q^{93} - 160 q^{95} - 68 q^{97} - 80 q^{98} - 260 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1 −0.156434 + 0.987688i −2.25472 + 2.25472i −0.951057 − 0.309017i −0.0142489 + 2.23602i −1.87425 − 2.57968i 0.590571 − 3.72872i 0.453990 − 0.891007i − 7.16755i −2.20626 − 0.363864i
17.2 −0.156434 + 0.987688i −1.94873 + 1.94873i −0.951057 − 0.309017i 0.196081 − 2.22745i −1.61989 − 2.22958i −0.0699365 + 0.441562i 0.453990 − 0.891007i − 4.59508i 2.16936 + 0.542118i
17.3 −0.156434 + 0.987688i −1.50285 + 1.50285i −0.951057 − 0.309017i 2.03900 − 0.917869i −1.24925 − 1.71945i 0.177874 − 1.12305i 0.453990 − 0.891007i − 1.51713i 0.587599 + 2.15748i
17.4 −0.156434 + 0.987688i −1.44750 + 1.44750i −0.951057 − 0.309017i 0.569141 + 2.16242i −1.20324 − 1.65612i −0.351597 + 2.21990i 0.453990 − 0.891007i − 1.19051i −2.22483 + 0.223856i
17.5 −0.156434 + 0.987688i −0.740127 + 0.740127i −0.951057 − 0.309017i −2.21008 + 0.339905i −0.615234 − 0.846796i −0.122254 + 0.771880i 0.453990 − 0.891007i 1.90442i 0.0100130 − 2.23605i
17.6 −0.156434 + 0.987688i −0.610128 + 0.610128i −0.951057 − 0.309017i −0.571911 − 2.16169i −0.507171 − 0.698061i −0.170927 + 1.07919i 0.453990 − 0.891007i 2.25549i 2.22455 − 0.226707i
17.7 −0.156434 + 0.987688i −0.404187 + 0.404187i −0.951057 − 0.309017i −2.12951 − 0.682043i −0.335982 − 0.462439i 0.675669 − 4.26601i 0.453990 − 0.891007i 2.67327i 1.00678 − 1.99660i
17.8 −0.156434 + 0.987688i −0.303962 + 0.303962i −0.951057 − 0.309017i 2.18459 − 0.477032i −0.252670 − 0.347770i 0.680127 − 4.29415i 0.453990 − 0.891007i 2.81521i 0.129414 + 2.23232i
17.9 −0.156434 + 0.987688i −0.262093 + 0.262093i −0.951057 − 0.309017i 1.89405 + 1.18852i −0.217866 − 0.299867i −0.565455 + 3.57014i 0.453990 − 0.891007i 2.86261i −1.47018 + 1.68480i
17.10 −0.156434 + 0.987688i 0.611827 − 0.611827i −0.951057 − 0.309017i 0.0747434 + 2.23482i 0.508584 + 0.700006i 0.249049 − 1.57244i 0.453990 − 0.891007i 2.25133i −2.21900 − 0.275779i
17.11 −0.156434 + 0.987688i 1.05009 − 1.05009i −0.951057 − 0.309017i −1.56197 − 1.60008i 0.872888 + 1.20143i −0.583619 + 3.68482i 0.453990 − 0.891007i 0.794637i 1.82472 − 1.29244i
17.12 −0.156434 + 0.987688i 1.35692 − 1.35692i −0.951057 − 0.309017i −1.64976 + 1.50940i 1.12795 + 1.55249i −0.430834 + 2.72018i 0.453990 − 0.891007i − 0.682481i −1.23273 − 1.86557i
17.13 −0.156434 + 0.987688i 1.46950 − 1.46950i −0.951057 − 0.309017i 0.972993 − 2.01328i 1.22153 + 1.68129i 0.161998 − 1.02281i 0.453990 − 0.891007i − 1.31888i 1.83628 + 1.27596i
17.14 −0.156434 + 0.987688i 2.07180 − 2.07180i −0.951057 − 0.309017i 2.00608 − 0.987754i 1.72219 + 2.37039i −0.386494 + 2.44023i 0.453990 − 0.891007i − 5.58468i 0.661774 + 2.13590i
17.15 −0.156434 + 0.987688i 2.27212 − 2.27212i −0.951057 − 0.309017i 1.29215 + 1.82493i 1.88871 + 2.59959i 0.566635 − 3.57759i 0.453990 − 0.891007i − 7.32508i −2.00460 + 0.990756i
17.16 0.156434 − 0.987688i −2.36429 + 2.36429i −0.951057 − 0.309017i 2.05111 + 0.890472i 1.96532 + 2.70503i 0.210090 − 1.32645i −0.453990 + 0.891007i − 8.17969i 1.20037 − 1.88656i
17.17 0.156434 − 0.987688i −2.04084 + 2.04084i −0.951057 − 0.309017i −2.18439 + 0.477953i 1.69646 + 2.33498i 0.0496649 − 0.313572i −0.453990 + 0.891007i − 5.33010i 0.130355 + 2.23227i
17.18 0.156434 − 0.987688i −1.86653 + 1.86653i −0.951057 − 0.309017i −0.0904226 − 2.23424i 1.55156 + 2.13554i −0.728263 + 4.59807i −0.453990 + 0.891007i − 3.96788i −2.22088 − 0.260203i
17.19 0.156434 − 0.987688i −0.978652 + 0.978652i −0.951057 − 0.309017i 1.62188 + 1.53932i 0.813508 + 1.11970i −0.225196 + 1.42183i −0.453990 + 0.891007i 1.08448i 1.77409 − 1.36111i
17.20 0.156434 − 0.987688i −0.914565 + 0.914565i −0.951057 − 0.309017i −1.45376 − 1.69900i 0.760236 + 1.04637i −0.0490764 + 0.309856i −0.453990 + 0.891007i 1.32714i −1.90550 + 1.17008i
See next 80 embeddings (of 240 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.30
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
275.bl even 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 550.2.be.a ✓ 240
11.d odd 10 1 550.2.bp.a yes 240
25.f odd 20 1 550.2.bp.a yes 240
275.bl even 20 1 inner 550.2.be.a ✓ 240
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
550.2.be.a ✓ 240 1.a even 1 1 trivial
550.2.be.a ✓ 240 275.bl even 20 1 inner
550.2.bp.a yes 240 11.d odd 10 1
550.2.bp.a yes 240 25.f odd 20 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(550, [\chi])\).