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Properties

Label 450.3.m.a
Level $450$
Weight $3$
Character orbit 450.m
Analytic conductor $12.262$
Analytic rank $0$
Dimension $80$
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] = [450,3,Mod(89,450)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(450, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([5, 3]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("450.89");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 450.m (of order \(10\), degree \(4\), 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: \(12.2616118962\)
Analytic rank: \(0\)
Dimension: \(80\)
Relative dimension: \(20\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 80 q - 40 q^{4}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 80 q - 40 q^{4} - 20 q^{10} - 80 q^{16} - 120 q^{19} - 80 q^{22} - 280 q^{25} + 80 q^{28} - 120 q^{31} + 60 q^{34} + 20 q^{37} - 40 q^{40} - 240 q^{46} - 560 q^{49} + 160 q^{55} + 280 q^{61} - 160 q^{64} + 560 q^{67} + 400 q^{70} - 240 q^{73} + 160 q^{76} - 320 q^{79} - 1560 q^{85} - 320 q^{88} + 680 q^{91} - 320 q^{94} - 40 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
89.1 −1.14412 + 0.831254i 0 0.618034 − 1.90211i −3.52834 + 3.54271i 0 − 9.26862i 0.874032 + 2.68999i 0 1.09196 − 6.98625i
89.2 −1.14412 + 0.831254i 0 0.618034 − 1.90211i −2.86045 − 4.10096i 0 − 9.76646i 0.874032 + 2.68999i 0 6.68164 + 2.31424i
89.3 −1.14412 + 0.831254i 0 0.618034 − 1.90211i 4.37273 + 2.42471i 0 9.18094i 0.874032 + 2.68999i 0 −7.01849 + 0.860681i
89.4 −1.14412 + 0.831254i 0 0.618034 − 1.90211i 3.67212 − 3.39345i 0 2.00115i 0.874032 + 2.68999i 0 −1.38054 + 6.93499i
89.5 −1.14412 + 0.831254i 0 0.618034 − 1.90211i 0.171218 + 4.99707i 0 − 8.65476i 0.874032 + 2.68999i 0 −4.34973 − 5.57493i
89.6 −1.14412 + 0.831254i 0 0.618034 − 1.90211i 3.23870 − 3.80931i 0 − 3.30438i 0.874032 + 2.68999i 0 −0.538964 + 7.05050i
89.7 −1.14412 + 0.831254i 0 0.618034 − 1.90211i −3.05794 − 3.95588i 0 6.39767i 0.874032 + 2.68999i 0 6.78700 + 1.98409i
89.8 −1.14412 + 0.831254i 0 0.618034 − 1.90211i 0.724575 + 4.94722i 0 11.3836i 0.874032 + 2.68999i 0 −4.94140 − 5.05792i
89.9 −1.14412 + 0.831254i 0 0.618034 − 1.90211i −4.96345 − 0.603474i 0 8.84070i 0.874032 + 2.68999i 0 6.18043 − 3.43544i
89.10 −1.14412 + 0.831254i 0 0.618034 − 1.90211i −1.90864 + 4.62137i 0 0.798619i 0.874032 + 2.68999i 0 −1.65781 − 6.87398i
89.11 1.14412 − 0.831254i 0 0.618034 − 1.90211i −0.724575 − 4.94722i 0 11.3836i −0.874032 − 2.68999i 0 −4.94140 − 5.05792i
89.12 1.14412 − 0.831254i 0 0.618034 − 1.90211i 1.90864 − 4.62137i 0 0.798619i −0.874032 − 2.68999i 0 −1.65781 − 6.87398i
89.13 1.14412 − 0.831254i 0 0.618034 − 1.90211i −3.67212 + 3.39345i 0 2.00115i −0.874032 − 2.68999i 0 −1.38054 + 6.93499i
89.14 1.14412 − 0.831254i 0 0.618034 − 1.90211i 3.52834 − 3.54271i 0 − 9.26862i −0.874032 − 2.68999i 0 1.09196 − 6.98625i
89.15 1.14412 − 0.831254i 0 0.618034 − 1.90211i −4.37273 − 2.42471i 0 9.18094i −0.874032 − 2.68999i 0 −7.01849 + 0.860681i
89.16 1.14412 − 0.831254i 0 0.618034 − 1.90211i −3.23870 + 3.80931i 0 − 3.30438i −0.874032 − 2.68999i 0 −0.538964 + 7.05050i
89.17 1.14412 − 0.831254i 0 0.618034 − 1.90211i 4.96345 + 0.603474i 0 8.84070i −0.874032 − 2.68999i 0 6.18043 − 3.43544i
89.18 1.14412 − 0.831254i 0 0.618034 − 1.90211i 2.86045 + 4.10096i 0 − 9.76646i −0.874032 − 2.68999i 0 6.68164 + 2.31424i
89.19 1.14412 − 0.831254i 0 0.618034 − 1.90211i 3.05794 + 3.95588i 0 6.39767i −0.874032 − 2.68999i 0 6.78700 + 1.98409i
89.20 1.14412 − 0.831254i 0 0.618034 − 1.90211i −0.171218 − 4.99707i 0 − 8.65476i −0.874032 − 2.68999i 0 −4.34973 − 5.57493i
See all 80 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 89.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
25.e even 10 1 inner
75.h odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 450.3.m.a ✓ 80
3.b odd 2 1 inner 450.3.m.a ✓ 80
25.e even 10 1 inner 450.3.m.a ✓ 80
75.h odd 10 1 inner 450.3.m.a ✓ 80
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
450.3.m.a ✓ 80 1.a even 1 1 trivial
450.3.m.a ✓ 80 3.b odd 2 1 inner
450.3.m.a ✓ 80 25.e even 10 1 inner
450.3.m.a ✓ 80 75.h odd 10 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(450, [\chi])\).