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Properties

Label 650.2.bg.a
Level $650$
Weight $2$
Character orbit 650.bg
Analytic conductor $5.190$
Analytic rank $0$
Dimension $272$
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] = [650,2,Mod(9,650)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(650, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([21, 20]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("650.9");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 650 = 2 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 650.bg (of order \(30\), 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: \(272\)
Relative dimension: \(34\) over \(\Q(\zeta_{30})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{30}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 272 q - 34 q^{4} - 30 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 272 q - 34 q^{4} - 30 q^{9} + 2 q^{10} - 16 q^{14} + 36 q^{15} + 34 q^{16} - 24 q^{19} - 20 q^{23} + 36 q^{25} + 4 q^{26} + 4 q^{29} + 16 q^{30} + 120 q^{33} + 6 q^{35} + 30 q^{36} - 10 q^{37} - 52 q^{39} + 4 q^{40} - 22 q^{41} + 48 q^{45} + 100 q^{47} + 104 q^{49} + 8 q^{50} - 80 q^{51} - 80 q^{53} - 12 q^{54} - 42 q^{55} - 8 q^{56} - 12 q^{59} - 8 q^{60} - 12 q^{61} - 60 q^{62} + 70 q^{63} + 68 q^{64} - 40 q^{65} + 32 q^{66} + 50 q^{69} - 104 q^{70} - 32 q^{71} - 160 q^{73} - 84 q^{74} - 142 q^{75} - 16 q^{76} + 240 q^{77} - 110 q^{78} - 16 q^{79} - 26 q^{81} - 140 q^{83} + 36 q^{85} - 24 q^{86} - 80 q^{87} + 42 q^{89} + 40 q^{90} - 100 q^{91} + 40 q^{92} + 8 q^{95} - 40 q^{98} + 40 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1 −0.406737 − 0.913545i −0.632629 + 2.97629i −0.669131 + 0.743145i −1.85490 − 1.24874i 2.97629 − 0.632629i 2.35997 − 1.36253i 0.951057 + 0.309017i −5.71742 − 2.54556i −0.386322 + 2.20244i
9.2 −0.406737 − 0.913545i −0.544582 + 2.56206i −0.669131 + 0.743145i −2.19029 + 0.450132i 2.56206 − 0.544582i −3.74928 + 2.16465i 0.951057 + 0.309017i −3.52693 − 1.57029i 1.30209 + 1.81785i
9.3 −0.406737 − 0.913545i −0.503547 + 2.36900i −0.669131 + 0.743145i 1.95673 + 1.08222i 2.36900 − 0.503547i −1.00962 + 0.582904i 0.951057 + 0.309017i −2.61799 − 1.16560i 0.192782 − 2.22774i
9.4 −0.406737 − 0.913545i −0.471013 + 2.21594i −0.669131 + 0.743145i 1.15380 − 1.91540i 2.21594 − 0.471013i −2.15473 + 1.24404i 0.951057 + 0.309017i −1.94790 − 0.867261i −2.21910 − 0.274981i
9.5 −0.406737 − 0.913545i −0.394787 + 1.85733i −0.669131 + 0.743145i −0.211378 + 2.22605i 1.85733 − 0.394787i 3.45186 − 1.99293i 0.951057 + 0.309017i −0.553166 − 0.246286i 2.11958 − 0.712315i
9.6 −0.406737 − 0.913545i −0.234529 + 1.10337i −0.669131 + 0.743145i −1.30839 + 1.81331i 1.10337 − 0.234529i 1.25066 − 0.722067i 0.951057 + 0.309017i 1.57821 + 0.702662i 2.18872 + 0.457734i
9.7 −0.406737 − 0.913545i −0.181439 + 0.853604i −0.669131 + 0.743145i 2.14829 − 0.620362i 0.853604 − 0.181439i 1.67000 − 0.964175i 0.951057 + 0.309017i 2.04492 + 0.910455i −1.44052 − 1.71024i
9.8 −0.406737 − 0.913545i 0.0255953 − 0.120416i −0.669131 + 0.743145i 1.72446 + 1.42346i −0.120416 + 0.0255953i 0.0894711 − 0.0516562i 0.951057 + 0.309017i 2.72679 + 1.21405i 0.598998 − 2.15434i
9.9 −0.406737 − 0.913545i 0.0543711 − 0.255796i −0.669131 + 0.743145i −2.00428 − 0.991394i −0.255796 + 0.0543711i 2.27285 − 1.31223i 0.951057 + 0.309017i 2.67816 + 1.19239i −0.0904694 + 2.23424i
9.10 −0.406737 − 0.913545i 0.106075 − 0.499044i −0.669131 + 0.743145i 0.136414 − 2.23190i −0.499044 + 0.106075i 4.25550 − 2.45691i 0.951057 + 0.309017i 2.50284 + 1.11434i −2.09443 + 0.783176i
9.11 −0.406737 − 0.913545i 0.114099 − 0.536795i −0.669131 + 0.743145i 2.03288 − 0.931338i −0.536795 + 0.114099i −3.77562 + 2.17986i 0.951057 + 0.309017i 2.46551 + 1.09771i −1.67767 − 1.47832i
9.12 −0.406737 − 0.913545i 0.183330 − 0.862500i −0.669131 + 0.743145i −1.68669 − 1.46802i −0.862500 + 0.183330i −2.54180 + 1.46751i 0.951057 + 0.309017i 2.03034 + 0.903966i −0.655059 + 2.13797i
9.13 −0.406737 − 0.913545i 0.344024 − 1.61851i −0.669131 + 0.743145i −1.75542 + 1.38510i −1.61851 + 0.344024i 0.496665 − 0.286750i 0.951057 + 0.309017i 0.239422 + 0.106598i 1.97934 + 1.04029i
9.14 −0.406737 − 0.913545i 0.447030 − 2.10311i −0.669131 + 0.743145i 1.16329 + 1.90965i −2.10311 + 0.447030i 0.884920 − 0.510909i 0.951057 + 0.309017i −1.48259 − 0.660093i 1.27139 − 1.83945i
9.15 −0.406737 − 0.913545i 0.526212 − 2.47563i −0.669131 + 0.743145i 0.184535 − 2.22844i −2.47563 + 0.526212i −1.11487 + 0.643673i 0.951057 + 0.309017i −3.11122 − 1.38520i −2.11084 + 0.737807i
9.16 −0.406737 − 0.913545i 0.536590 − 2.52446i −0.669131 + 0.743145i −0.726192 + 2.11486i −2.52446 + 0.536590i −2.11766 + 1.22263i 0.951057 + 0.309017i −3.34432 − 1.48899i 2.22739 − 0.196783i
9.17 −0.406737 − 0.913545i 0.625200 − 2.94134i −0.669131 + 0.743145i 2.18820 − 0.460184i −2.94134 + 0.625200i 3.19580 − 1.84510i 0.951057 + 0.309017i −5.51995 − 2.45764i −1.31042 − 1.81185i
9.18 0.406737 + 0.913545i −0.683079 + 3.21363i −0.669131 + 0.743145i −0.977235 − 2.01122i −3.21363 + 0.683079i −0.650236 + 0.375414i −0.951057 − 0.309017i −7.12020 − 3.17012i 1.43987 − 1.71079i
9.19 0.406737 + 0.913545i −0.621151 + 2.92229i −0.669131 + 0.743145i 2.02804 + 0.941834i −2.92229 + 0.621151i 1.57843 − 0.911308i −0.951057 − 0.309017i −5.41329 − 2.41015i −0.0355297 + 2.23579i
9.20 0.406737 + 0.913545i −0.542647 + 2.55296i −0.669131 + 0.743145i −1.16114 + 1.91096i −2.55296 + 0.542647i −2.08252 + 1.20235i −0.951057 − 0.309017i −3.48248 − 1.55050i −2.21802 − 0.283496i
See next 80 embeddings (of 272 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 9.34
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner
25.e even 10 1 inner
325.bf even 30 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 650.2.bg.a ✓ 272
13.c even 3 1 inner 650.2.bg.a ✓ 272
25.e even 10 1 inner 650.2.bg.a ✓ 272
325.bf even 30 1 inner 650.2.bg.a ✓ 272
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
650.2.bg.a ✓ 272 1.a even 1 1 trivial
650.2.bg.a ✓ 272 13.c even 3 1 inner
650.2.bg.a ✓ 272 25.e even 10 1 inner
650.2.bg.a ✓ 272 325.bf even 30 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(650, [\chi])\).