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Properties

Label 450.2.v.a
Level $450$
Weight $2$
Character orbit 450.v
Analytic conductor $3.593$
Analytic rank $0$
Dimension $240$
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] = [450,2,Mod(79,450)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(450, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([20, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("450.79");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 450.v (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: \(3.59326809096\)
Analytic rank: \(0\)
Dimension: \(240\)
Relative dimension: \(30\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 240 q - 30 q^{4} - 8 q^{5} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 240 q - 30 q^{4} - 8 q^{5} + 4 q^{9} - 4 q^{11} + 10 q^{12} + 8 q^{14} - 20 q^{15} + 30 q^{16} - 2 q^{20} + 24 q^{21} + 24 q^{25} - 96 q^{26} + 30 q^{27} + 12 q^{29} - 22 q^{30} + 12 q^{31} + 50 q^{33} - 32 q^{35} + 8 q^{36} - 52 q^{39} - 16 q^{41} - 8 q^{44} - 108 q^{45} - 50 q^{47} - 20 q^{48} + 120 q^{49} - 4 q^{50} - 32 q^{51} - 24 q^{54} + 24 q^{55} - 8 q^{56} + 18 q^{59} + 6 q^{60} - 60 q^{62} - 70 q^{63} + 60 q^{64} - 64 q^{65} - 16 q^{66} - 30 q^{67} - 8 q^{69} + 24 q^{70} + 76 q^{71} - 80 q^{74} - 6 q^{75} + 80 q^{77} - 20 q^{78} + 12 q^{79} - 4 q^{80} - 36 q^{81} - 140 q^{83} - 18 q^{84} + 12 q^{85} - 20 q^{86} - 150 q^{87} - 28 q^{89} + 62 q^{90} - 40 q^{92} + 36 q^{95} - 28 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
79.1 −0.743145 + 0.669131i −1.69642 − 0.349507i 0.104528 − 0.994522i 0.950392 − 2.02404i 1.49455 − 0.875393i −0.590039 − 0.340659i 0.587785 + 0.809017i 2.75569 + 1.18582i 0.648071 + 2.14009i
79.2 −0.743145 + 0.669131i −1.53471 − 0.802914i 0.104528 − 0.994522i 1.99429 + 1.01134i 1.67776 − 0.430239i 1.01614 + 0.586669i 0.587785 + 0.809017i 1.71066 + 2.46448i −2.15876 + 0.582871i
79.3 −0.743145 + 0.669131i −1.44495 + 0.955053i 0.104528 − 0.994522i −1.82705 − 1.28914i 0.434750 − 1.67660i 2.39808 + 1.38453i 0.587785 + 0.809017i 1.17575 − 2.76000i 2.22037 − 0.264522i
79.4 −0.743145 + 0.669131i −1.41885 + 0.993404i 0.104528 − 0.994522i −1.15383 + 1.91538i 0.389698 − 1.68764i −2.23155 − 1.28839i 0.587785 + 0.809017i 1.02630 − 2.81899i −0.424180 − 2.19547i
79.5 −0.743145 + 0.669131i −0.818331 − 1.52654i 0.104528 − 0.994522i −2.19550 − 0.423984i 1.62960 + 0.586873i 4.25551 + 2.45692i 0.587785 + 0.809017i −1.66067 + 2.49844i 1.91528 − 1.15400i
79.6 −0.743145 + 0.669131i −0.429668 + 1.67791i 0.104528 − 0.994522i 0.680748 − 2.12993i −0.803436 − 1.53444i −0.636266 − 0.367348i 0.587785 + 0.809017i −2.63077 − 1.44189i 0.919304 + 2.03835i
79.7 −0.743145 + 0.669131i −0.417886 − 1.68088i 0.104528 − 0.994522i 0.153377 + 2.23080i 1.43528 + 0.969520i −1.71704 − 0.991331i 0.587785 + 0.809017i −2.65074 + 1.40484i −1.60668 − 1.55518i
79.8 −0.743145 + 0.669131i 0.284495 − 1.70853i 0.104528 − 0.994522i 2.09970 − 0.768942i 0.931807 + 1.46005i 1.37548 + 0.794136i 0.587785 + 0.809017i −2.83813 − 0.972133i −1.04586 + 1.97641i
79.9 −0.743145 + 0.669131i 0.547614 + 1.64320i 0.104528 − 0.994522i −0.449052 + 2.19051i −1.50647 − 0.854713i 2.70306 + 1.56061i 0.587785 + 0.809017i −2.40024 + 1.79968i −1.13203 − 1.92834i
79.10 −0.743145 + 0.669131i 1.27970 − 1.16721i 0.104528 − 0.994522i −2.12793 + 0.686960i −0.169983 + 1.72369i −0.590637 − 0.341005i 0.587785 + 0.809017i 0.275240 − 2.98735i 1.12169 − 1.93437i
79.11 −0.743145 + 0.669131i 1.31955 + 1.12196i 0.104528 − 0.994522i 2.22159 − 0.254058i −1.73135 + 0.0491696i 1.56138 + 0.901461i 0.587785 + 0.809017i 0.482407 + 2.96096i −1.48096 + 1.67533i
79.12 −0.743145 + 0.669131i 1.33209 + 1.10704i 0.104528 − 0.994522i −0.575676 − 2.16069i −1.73069 + 0.0686527i −4.13897 − 2.38963i 0.587785 + 0.809017i 0.548931 + 2.94935i 1.87360 + 1.22051i
79.13 −0.743145 + 0.669131i 1.55685 + 0.759080i 0.104528 − 0.994522i −2.15809 + 0.585360i −1.66489 + 0.477633i 0.358219 + 0.206818i 0.587785 + 0.809017i 1.84759 + 2.36356i 1.21209 − 1.87905i
79.14 −0.743145 + 0.669131i 1.61003 − 0.638583i 0.104528 − 0.994522i 0.490186 − 2.18168i −0.769194 + 1.55188i 3.15543 + 1.82179i 0.587785 + 0.809017i 2.18442 − 2.05628i 1.09555 + 1.94930i
79.15 −0.743145 + 0.669131i 1.72218 − 0.184659i 0.104528 − 0.994522i 1.39685 + 1.74608i −1.15627 + 1.28959i −3.45470 − 1.99457i 0.587785 + 0.809017i 2.93180 − 0.636031i −2.20642 − 0.362915i
79.16 0.743145 − 0.669131i −1.71340 + 0.253483i 0.104528 − 0.994522i −2.04514 + 0.904112i −1.10369 + 1.33486i 0.528158 + 0.304932i −0.587785 − 0.809017i 2.87149 − 0.868637i −0.914863 + 2.04035i
79.17 0.743145 − 0.669131i −1.69433 − 0.359506i 0.104528 − 0.994522i 1.60067 + 1.56136i −1.49969 + 0.866564i −3.59465 − 2.07537i −0.587785 − 0.809017i 2.74151 + 1.21824i 2.23429 + 0.0892630i
79.18 0.743145 − 0.669131i −1.32725 − 1.11283i 0.104528 − 0.994522i −1.71721 − 1.43220i −1.73097 + 0.0611078i 0.546948 + 0.315781i −0.587785 − 0.809017i 0.523198 + 2.95403i −2.23446 + 0.0847027i
79.19 0.743145 − 0.669131i −1.20035 + 1.24866i 0.104528 − 0.994522i −0.514427 − 2.17609i −0.0565200 + 1.73113i 0.879689 + 0.507889i −0.587785 − 0.809017i −0.118303 − 2.99767i −1.83838 − 1.27293i
79.20 0.743145 − 0.669131i −0.817375 + 1.52706i 0.104528 − 0.994522i 2.01342 + 0.972707i 0.414372 + 1.68175i −0.0644733 − 0.0372237i −0.587785 − 0.809017i −1.66380 − 2.49635i 2.14713 − 0.624376i
See next 80 embeddings (of 240 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 79.30
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner
25.e even 10 1 inner
225.u even 30 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 450.2.v.a ✓ 240
9.c even 3 1 inner 450.2.v.a ✓ 240
25.e even 10 1 inner 450.2.v.a ✓ 240
225.u even 30 1 inner 450.2.v.a ✓ 240
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
450.2.v.a ✓ 240 1.a even 1 1 trivial
450.2.v.a ✓ 240 9.c even 3 1 inner
450.2.v.a ✓ 240 25.e even 10 1 inner
450.2.v.a ✓ 240 225.u even 30 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(450, [\chi])\).