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Properties

Label 950.2.bg.a
Level $950$
Weight $2$
Character orbit 950.bg
Analytic conductor $7.586$
Analytic rank $0$
Dimension $1200$
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] = [950,2,Mod(9,950)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(950, base_ring=CyclotomicField(90))
 
chi = DirichletCharacter(H, H._module([63, 40]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("950.9");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 950 = 2 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 950.bg (of order \(90\), degree \(24\), 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: \(7.58578819202\)
Analytic rank: \(0\)
Dimension: \(1200\)
Relative dimension: \(50\) over \(\Q(\zeta_{90})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{90}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 1200 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 1200 q + 18 q^{11} + 90 q^{15} + 12 q^{20} - 60 q^{22} - 60 q^{23} - 96 q^{25} - 36 q^{29} - 60 q^{33} + 30 q^{35} - 120 q^{45} - 60 q^{46} - 60 q^{47} + 660 q^{49} - 24 q^{50} + 120 q^{51} + 36 q^{54} + 36 q^{55} + 48 q^{56} - 36 q^{59} - 30 q^{60} - 144 q^{61} - 150 q^{64} + 120 q^{65} + 96 q^{66} + 48 q^{69} - 60 q^{70} - 120 q^{71} - 48 q^{79} - 96 q^{81} - 30 q^{83} + 78 q^{84} - 192 q^{85} + 60 q^{87} - 72 q^{89} - 36 q^{90} + 60 q^{92} - 96 q^{94} - 24 q^{95} - 180 q^{97} - 240 q^{98} - 192 q^{99}+O(q^{100}) \) 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 −0.898794 − 0.438371i −1.80064 + 2.88162i 0.615661 + 0.788011i −1.98988 − 1.01998i 2.88162 − 1.80064i −1.84599 − 1.06578i −0.207912 − 0.978148i −3.74633 − 7.68112i 1.34136 + 1.78906i
9.2 −0.898794 − 0.438371i −1.54722 + 2.47606i 0.615661 + 0.788011i 1.80261 + 1.32311i 2.47606 − 1.54722i 0.146440 + 0.0845473i −0.207912 − 0.978148i −2.42190 − 4.96562i −1.04016 − 1.97941i
9.3 −0.898794 − 0.438371i −1.48382 + 2.37460i 0.615661 + 0.788011i 1.05677 − 1.97059i 2.37460 − 1.48382i −0.246693 − 0.142428i −0.207912 − 0.978148i −2.12191 − 4.35057i −1.81367 + 1.30790i
9.4 −0.898794 − 0.438371i −1.22036 + 1.95299i 0.615661 + 0.788011i −0.755334 + 2.10463i 1.95299 − 1.22036i 1.23161 + 0.711072i −0.207912 − 0.978148i −1.00977 − 2.07033i 1.60150 − 1.56051i
9.5 −0.898794 − 0.438371i −1.21964 + 1.95183i 0.615661 + 0.788011i 0.512699 − 2.17650i 1.95183 − 1.21964i 1.49771 + 0.864704i −0.207912 − 0.978148i −1.00701 − 2.06468i −1.41492 + 1.73147i
9.6 −0.898794 − 0.438371i −0.891893 + 1.42733i 0.615661 + 0.788011i −0.847823 + 2.06911i 1.42733 − 0.891893i −4.31713 − 2.49250i −0.207912 − 0.978148i 0.0733240 + 0.150336i 1.66905 − 1.48804i
9.7 −0.898794 − 0.438371i −0.838664 + 1.34214i 0.615661 + 0.788011i 1.22007 + 1.87388i 1.34214 − 0.838664i 0.0845996 + 0.0488436i −0.207912 − 0.978148i 0.217124 + 0.445170i −0.275136 − 2.21908i
9.8 −0.898794 − 0.438371i −0.716423 + 1.14652i 0.615661 + 0.788011i −2.00197 − 0.996049i 1.14652 − 0.716423i −1.22683 − 0.708310i −0.207912 − 0.978148i 0.513875 + 1.05360i 1.36272 + 1.77285i
9.9 −0.898794 − 0.438371i −0.654190 + 1.04692i 0.615661 + 0.788011i 2.23588 − 0.0292423i 1.04692 − 0.654190i 3.55795 + 2.05418i −0.207912 − 0.978148i 0.647030 + 1.32661i −2.02241 − 0.953861i
9.10 −0.898794 − 0.438371i −0.443586 + 0.709885i 0.615661 + 0.788011i 0.161877 − 2.23020i 0.709885 − 0.443586i −3.38490 − 1.95427i −0.207912 − 0.978148i 1.00794 + 2.06659i −1.12315 + 1.93353i
9.11 −0.898794 − 0.438371i −0.401035 + 0.641790i 0.615661 + 0.788011i −2.22848 + 0.184025i 0.641790 − 0.401035i 1.40727 + 0.812487i −0.207912 − 0.978148i 1.06405 + 2.18162i 2.08362 + 0.811502i
9.12 −0.898794 − 0.438371i −0.0925144 + 0.148054i 0.615661 + 0.788011i 2.22820 + 0.187404i 0.148054 − 0.0925144i −3.43865 − 1.98531i −0.207912 − 0.978148i 1.30175 + 2.66899i −1.92054 − 1.14522i
9.13 −0.898794 − 0.438371i 0.123716 − 0.197987i 0.615661 + 0.788011i −2.12450 + 0.697499i −0.197987 + 0.123716i −0.938580 − 0.541889i −0.207912 − 0.978148i 1.29122 + 2.64739i 2.21525 + 0.304411i
9.14 −0.898794 − 0.438371i 0.145245 − 0.232441i 0.615661 + 0.788011i −1.06342 + 1.96701i −0.232441 + 0.145245i 2.84810 + 1.64435i −0.207912 − 0.978148i 1.28218 + 2.62886i 1.81808 − 1.30176i
9.15 −0.898794 − 0.438371i 0.281867 − 0.451082i 0.615661 + 0.788011i 1.08365 + 1.95594i −0.451082 + 0.281867i −1.73157 − 0.999724i −0.207912 − 0.978148i 1.19109 + 2.44209i −0.116554 − 2.23303i
9.16 −0.898794 − 0.438371i 0.324851 − 0.519870i 0.615661 + 0.788011i 1.89925 − 1.18020i −0.519870 + 0.324851i 2.84170 + 1.64066i −0.207912 − 0.978148i 1.15038 + 2.35862i −2.22440 + 0.228181i
9.17 −0.898794 − 0.438371i 0.543822 − 0.870297i 0.615661 + 0.788011i −0.314385 − 2.21386i −0.870297 + 0.543822i −0.435827 − 0.251625i −0.207912 − 0.978148i 0.853439 + 1.74981i −0.687924 + 2.12762i
9.18 −0.898794 − 0.438371i 0.613064 − 0.981108i 0.615661 + 0.788011i 1.45906 − 1.69444i −0.981108 + 0.613064i −2.05896 − 1.18874i −0.207912 − 0.978148i 0.728388 + 1.49342i −2.05419 + 0.883345i
9.19 −0.898794 − 0.438371i 0.750998 − 1.20185i 0.615661 + 0.788011i −0.805035 − 2.08613i −1.20185 + 0.750998i 2.85027 + 1.64561i −0.207912 − 0.978148i 0.434672 + 0.891210i −0.190936 + 2.22790i
9.20 −0.898794 − 0.438371i 1.25757 − 2.01253i 0.615661 + 0.788011i −0.556445 + 2.16573i −2.01253 + 1.25757i −1.28276 − 0.740599i −0.207912 − 0.978148i −1.15369 − 2.36542i 1.44952 − 1.70261i
See next 80 embeddings (of 1200 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 9.50
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.e even 9 1 inner
25.e even 10 1 inner
475.bg even 90 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 950.2.bg.a ✓ 1200
19.e even 9 1 inner 950.2.bg.a ✓ 1200
25.e even 10 1 inner 950.2.bg.a ✓ 1200
475.bg even 90 1 inner 950.2.bg.a ✓ 1200
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
950.2.bg.a ✓ 1200 1.a even 1 1 trivial
950.2.bg.a ✓ 1200 19.e even 9 1 inner
950.2.bg.a ✓ 1200 25.e even 10 1 inner
950.2.bg.a ✓ 1200 475.bg even 90 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(950, [\chi])\).