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Properties

Label 650.2.ba.b
Level $650$
Weight $2$
Character orbit 650.ba
Analytic conductor $5.190$
Analytic rank $0$
Dimension $144$
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] = [650,2,Mod(73,650)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(650, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([11, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("650.73");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 650 = 2 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 650.ba (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: \(5.19027613138\)
Analytic rank: \(0\)
Dimension: \(144\)
Relative dimension: \(18\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 144 q + 36 q^{2} - 36 q^{4} + 36 q^{8} - 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 144 q + 36 q^{2} - 36 q^{4} + 36 q^{8} - 20 q^{9} + 6 q^{13} + 8 q^{15} - 36 q^{16} - 6 q^{17} - 20 q^{19} - 6 q^{21} + 12 q^{23} + 2 q^{25} + 14 q^{26} - 24 q^{27} + 20 q^{29} + 12 q^{30} - 6 q^{31} - 144 q^{32} - 54 q^{33} + 6 q^{34} + 40 q^{35} + 20 q^{36} - 60 q^{37} + 20 q^{39} + 52 q^{41} + 6 q^{42} - 12 q^{43} - 6 q^{45} + 28 q^{46} + 50 q^{47} - 160 q^{49} - 22 q^{50} + 6 q^{52} + 20 q^{53} - 6 q^{54} - 32 q^{55} - 28 q^{57} + 20 q^{58} - 16 q^{59} - 12 q^{60} + 12 q^{61} + 6 q^{62} - 70 q^{63} - 36 q^{64} + 44 q^{65} - 16 q^{66} - 8 q^{67} - 6 q^{68} + 26 q^{69} + 10 q^{70} + 72 q^{71} - 52 q^{73} + 32 q^{75} - 56 q^{77} - 50 q^{78} - 20 q^{79} + 124 q^{81} - 22 q^{82} + 70 q^{83} - 6 q^{84} - 18 q^{85} - 8 q^{86} + 8 q^{87} + 28 q^{89} - 14 q^{90} - 26 q^{91} + 12 q^{92} + 50 q^{94} - 46 q^{95} + 28 q^{97} - 20 q^{98} + 40 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
73.1 0.809017 − 0.587785i −2.86287 + 1.45871i 0.309017 − 0.951057i −2.06627 + 0.854710i −1.45871 + 2.86287i − 2.35225i −0.309017 − 0.951057i 4.30486 − 5.92513i −1.16926 + 1.90600i
73.2 0.809017 − 0.587785i −2.34666 + 1.19569i 0.309017 − 0.951057i 0.373709 − 2.20462i −1.19569 + 2.34666i 2.39032i −0.309017 − 0.951057i 2.31381 − 3.18469i −0.993505 − 2.00323i
73.3 0.809017 − 0.587785i −1.92623 + 0.981464i 0.309017 − 0.951057i 0.916919 + 2.03943i −0.981464 + 1.92623i 0.691227i −0.309017 − 0.951057i 0.983741 − 1.35400i 1.94055 + 1.11098i
73.4 0.809017 − 0.587785i −1.88926 + 0.962626i 0.309017 − 0.951057i −0.207782 − 2.22639i −0.962626 + 1.88926i − 2.25263i −0.309017 − 0.951057i 0.879297 − 1.21025i −1.47674 − 1.67906i
73.5 0.809017 − 0.587785i −1.37577 + 0.700992i 0.309017 − 0.951057i −2.13959 + 0.649736i −0.700992 + 1.37577i 3.27030i −0.309017 − 0.951057i −0.361991 + 0.498237i −1.34906 + 1.78327i
73.6 0.809017 − 0.587785i −1.25413 + 0.639012i 0.309017 − 0.951057i 2.16343 − 0.565289i −0.639012 + 1.25413i 4.95684i −0.309017 − 0.951057i −0.598846 + 0.824240i 1.41799 − 1.72896i
73.7 0.809017 − 0.587785i −1.17085 + 0.596579i 0.309017 − 0.951057i −0.522021 + 2.17428i −0.596579 + 1.17085i − 4.83730i −0.309017 − 0.951057i −0.748366 + 1.03004i 0.855686 + 2.06587i
73.8 0.809017 − 0.587785i −0.772259 + 0.393485i 0.309017 − 0.951057i −2.17344 − 0.525498i −0.393485 + 0.772259i 0.519105i −0.309017 − 0.951057i −1.32180 + 1.81931i −2.06723 + 0.852381i
73.9 0.809017 − 0.587785i −0.269606 + 0.137371i 0.309017 − 0.951057i 1.66698 + 1.49036i −0.137371 + 0.269606i 0.151358i −0.309017 − 0.951057i −1.70954 + 2.35298i 2.22463 + 0.225904i
73.10 0.809017 − 0.587785i 0.485767 − 0.247511i 0.309017 − 0.951057i −2.07582 − 0.831238i 0.247511 − 0.485767i − 3.75726i −0.309017 − 0.951057i −1.58865 + 2.18659i −2.16797 + 0.547653i
73.11 0.809017 − 0.587785i 0.595251 − 0.303295i 0.309017 − 0.951057i 2.04549 + 0.903305i 0.303295 − 0.595251i − 2.88788i −0.309017 − 0.951057i −1.50102 + 2.06598i 2.18579 − 0.471522i
73.12 0.809017 − 0.587785i 0.845402 − 0.430754i 0.309017 − 0.951057i −1.66404 + 1.49364i 0.430754 − 0.845402i 3.62467i −0.309017 − 0.951057i −1.23420 + 1.69873i −0.468300 + 2.18648i
73.13 0.809017 − 0.587785i 0.932353 − 0.475057i 0.309017 − 0.951057i 0.203489 − 2.22679i 0.475057 − 0.932353i 1.00383i −0.309017 − 0.951057i −1.11975 + 1.54121i −1.14425 − 1.92112i
73.14 0.809017 − 0.587785i 1.17311 − 0.597729i 0.309017 − 0.951057i 1.70419 − 1.44767i 0.597729 − 1.17311i − 3.18119i −0.309017 − 0.951057i −0.744450 + 1.02465i 0.527798 − 2.17288i
73.15 0.809017 − 0.587785i 2.22141 − 1.13186i 0.309017 − 0.951057i 1.90207 − 1.17564i 1.13186 − 2.22141i 4.41958i −0.309017 − 0.951057i 1.89019 − 2.60162i 0.847781 − 2.06912i
73.16 0.809017 − 0.587785i 2.47541 − 1.26129i 0.309017 − 0.951057i −1.10470 + 1.94413i 1.26129 − 2.47541i − 1.46054i −0.309017 − 0.951057i 2.77347 − 3.81736i 0.249009 + 2.22216i
73.17 0.809017 − 0.587785i 2.50986 − 1.27884i 0.309017 − 0.951057i 1.84190 + 1.26784i 1.27884 − 2.50986i − 1.15940i −0.309017 − 0.951057i 2.90062 − 3.99237i 2.23534 − 0.0569404i
73.18 0.809017 − 0.587785i 2.62909 − 1.33959i 0.309017 − 0.951057i −1.81556 − 1.30527i 1.33959 − 2.62909i 0.861213i −0.309017 − 0.951057i 3.35425 − 4.61672i −2.23604 + 0.0111743i
187.1 0.809017 + 0.587785i −2.86287 − 1.45871i 0.309017 + 0.951057i −2.06627 − 0.854710i −1.45871 − 2.86287i 2.35225i −0.309017 + 0.951057i 4.30486 + 5.92513i −1.16926 − 1.90600i
187.2 0.809017 + 0.587785i −2.34666 − 1.19569i 0.309017 + 0.951057i 0.373709 + 2.20462i −1.19569 − 2.34666i − 2.39032i −0.309017 + 0.951057i 2.31381 + 3.18469i −0.993505 + 2.00323i
See next 80 embeddings (of 144 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 73.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
325.z even 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 650.2.ba.b ✓ 144
13.d odd 4 1 650.2.bd.b yes 144
25.f odd 20 1 650.2.bd.b yes 144
325.z even 20 1 inner 650.2.ba.b ✓ 144
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
650.2.ba.b ✓ 144 1.a even 1 1 trivial
650.2.ba.b ✓ 144 325.z even 20 1 inner
650.2.bd.b yes 144 13.d odd 4 1
650.2.bd.b yes 144 25.f odd 20 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{144} + 10 T_{3}^{142} + 8 T_{3}^{141} - 251 T_{3}^{140} + 136 T_{3}^{139} - 3048 T_{3}^{138} + \cdots + 60\!\cdots\!00 \) acting on \(S_{2}^{\mathrm{new}}(650, [\chi])\). Copy content Toggle raw display