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Properties

Label 500.2.o.a
Level $500$
Weight $2$
Character orbit 500.o
Analytic conductor $3.993$
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] = [500,2,Mod(9,500)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(500, base_ring=CyclotomicField(50))
 
chi = DirichletCharacter(H, H._module([0, 7]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("500.9");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 500 = 2^{2} \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 500.o (of order \(50\), degree \(20\), 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: \(3.99252010106\)
Analytic rank: \(0\)
Dimension: \(240\)
Relative dimension: \(12\) over \(\Q(\zeta_{50})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{50}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 240 q - 5 q^{5}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 240 q - 5 q^{5} - 5 q^{11} + 20 q^{15} + 5 q^{17} + 10 q^{19} - 30 q^{23} + 55 q^{25} + 10 q^{29} + 10 q^{31} - 15 q^{33} + 5 q^{35} + 10 q^{37} - 20 q^{39} - 15 q^{41} - 160 q^{45} + 80 q^{47} + 40 q^{49} + 20 q^{51} - 30 q^{53} - 35 q^{55} - 15 q^{59} - 20 q^{61} - 20 q^{63} + 25 q^{65} + 5 q^{67} + 10 q^{69} - 95 q^{71} + 60 q^{73} + 5 q^{75} + 20 q^{77} + 20 q^{79} + 5 q^{81} + 10 q^{83} + 5 q^{85} + 55 q^{87} + 35 q^{89} + 120 q^{91} - 100 q^{93} + 20 q^{95} - 25 q^{97} - 30 q^{99}+O(q^{100}) \) 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 −2.95034 + 0.185620i 0 1.43603 − 1.71401i 0 0.497527 − 0.161656i 0 5.69372 − 0.719284i 0
9.2 0 −2.61567 + 0.164564i 0 −2.16514 + 0.558727i 0 −3.53759 + 1.14943i 0 3.83833 − 0.484893i 0
9.3 0 −1.64478 + 0.103480i 0 2.00455 + 0.990849i 0 1.32769 − 0.431392i 0 −0.281767 + 0.0355955i 0
9.4 0 −0.819141 + 0.0515360i 0 −1.26404 − 1.84450i 0 0.129594 − 0.0421077i 0 −2.30801 + 0.291569i 0
9.5 0 −0.559565 + 0.0352049i 0 1.64320 − 1.51654i 0 −3.69411 + 1.20029i 0 −2.66447 + 0.336601i 0
9.6 0 −0.430418 + 0.0270796i 0 −0.0493601 + 2.23552i 0 −2.08239 + 0.676608i 0 −2.79182 + 0.352689i 0
9.7 0 −0.411644 + 0.0258984i 0 −2.23089 + 0.152123i 0 4.62661 − 1.50328i 0 −2.80756 + 0.354678i 0
9.8 0 1.05034 − 0.0660820i 0 −1.15636 + 1.91385i 0 −1.41344 + 0.459253i 0 −1.87749 + 0.237182i 0
9.9 0 1.81284 − 0.114054i 0 0.738418 − 2.11063i 0 1.36282 − 0.442808i 0 0.297029 − 0.0375235i 0
9.10 0 1.88201 − 0.118406i 0 1.90434 + 1.17196i 0 3.66797 − 1.19180i 0 0.551590 − 0.0696820i 0
9.11 0 2.79853 − 0.176068i 0 1.92750 + 1.13346i 0 −3.81176 + 1.23852i 0 4.82442 − 0.609466i 0
9.12 0 3.25423 − 0.204739i 0 −2.05345 + 0.885059i 0 1.74259 − 0.566202i 0 7.57177 − 0.956537i 0
29.1 0 −1.24397 − 2.26278i 0 0.0332163 + 2.23582i 0 −0.565731 + 0.778662i 0 −1.96522 + 3.09669i 0
29.2 0 −1.23002 − 2.23741i 0 0.473807 − 2.18529i 0 −2.10010 + 2.89054i 0 −1.88554 + 2.97114i 0
29.3 0 −1.14184 − 2.07700i 0 2.23531 + 0.0582132i 0 2.53657 − 3.49129i 0 −1.40265 + 2.21023i 0
29.4 0 −0.863284 − 1.57031i 0 −0.636583 − 2.14354i 0 1.24978 − 1.72018i 0 −0.113128 + 0.178261i 0
29.5 0 −0.452598 − 0.823273i 0 −1.59833 + 1.56376i 0 0.150710 − 0.207435i 0 1.13455 − 1.78776i 0
29.6 0 −0.0430389 − 0.0782874i 0 −2.15680 − 0.590110i 0 −1.77232 + 2.43939i 0 1.60320 − 2.52624i 0
29.7 0 0.329750 + 0.599812i 0 1.87991 − 1.21076i 0 −0.896123 + 1.23341i 0 1.35644 − 2.13741i 0
29.8 0 0.527135 + 0.958854i 0 0.463389 + 2.18753i 0 2.53572 − 3.49011i 0 0.965950 − 1.52209i 0
See next 80 embeddings (of 240 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 9.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
125.h even 50 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 500.2.o.a ✓ 240
125.h even 50 1 inner 500.2.o.a ✓ 240
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
500.2.o.a ✓ 240 1.a even 1 1 trivial
500.2.o.a ✓ 240 125.h even 50 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(500, [\chi])\).