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Properties

Label 450.3.n.b
Level $450$
Weight $3$
Character orbit 450.n
Analytic conductor $12.262$
Analytic rank $0$
Dimension $48$
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(71,450)]
 
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, 6]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("450.71");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 450.n (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: \(48\)
Relative dimension: \(12\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 24 q^{4} + 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q + 24 q^{4} + 8 q^{7} + 8 q^{10} - 16 q^{13} - 48 q^{16} - 108 q^{19} + 48 q^{22} + 236 q^{25} + 24 q^{28} + 132 q^{31} - 56 q^{34} + 28 q^{37} + 24 q^{40} + 200 q^{43} + 72 q^{46} + 976 q^{49} + 32 q^{52} - 464 q^{55} - 168 q^{58} + 96 q^{64} + 624 q^{67} + 48 q^{70} + 712 q^{73} - 144 q^{76} - 648 q^{79} - 328 q^{82} - 12 q^{85} + 184 q^{88} - 936 q^{91} - 480 q^{94} + 108 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
71.1 −1.34500 − 0.437016i 0 1.61803 + 1.17557i 4.89276 − 1.02998i 0 10.6899 −1.66251 − 2.28825i 0 −7.03087 − 0.752894i
71.2 −1.34500 − 0.437016i 0 1.61803 + 1.17557i −4.86266 − 1.16386i 0 13.6646 −1.66251 − 2.28825i 0 6.03163 + 3.69045i
71.3 −1.34500 − 0.437016i 0 1.61803 + 1.17557i −0.324944 + 4.98943i 0 −7.00769 −1.66251 − 2.28825i 0 2.61751 − 6.56876i
71.4 −1.34500 − 0.437016i 0 1.61803 + 1.17557i −4.99953 − 0.0683419i 0 −5.96107 −1.66251 − 2.28825i 0 6.69449 + 2.27680i
71.5 −1.34500 − 0.437016i 0 1.61803 + 1.17557i 4.08555 + 2.88241i 0 0.249737 −1.66251 − 2.28825i 0 −4.23539 − 5.66228i
71.6 −1.34500 − 0.437016i 0 1.61803 + 1.17557i 2.55382 − 4.29861i 0 −8.39938 −1.66251 − 2.28825i 0 −5.31344 + 4.66556i
71.7 1.34500 + 0.437016i 0 1.61803 + 1.17557i −4.08555 − 2.88241i 0 0.249737 1.66251 + 2.28825i 0 −4.23539 − 5.66228i
71.8 1.34500 + 0.437016i 0 1.61803 + 1.17557i −4.89276 + 1.02998i 0 10.6899 1.66251 + 2.28825i 0 −7.03087 − 0.752894i
71.9 1.34500 + 0.437016i 0 1.61803 + 1.17557i −2.55382 + 4.29861i 0 −8.39938 1.66251 + 2.28825i 0 −5.31344 + 4.66556i
71.10 1.34500 + 0.437016i 0 1.61803 + 1.17557i 4.99953 + 0.0683419i 0 −5.96107 1.66251 + 2.28825i 0 6.69449 + 2.27680i
71.11 1.34500 + 0.437016i 0 1.61803 + 1.17557i 4.86266 + 1.16386i 0 13.6646 1.66251 + 2.28825i 0 6.03163 + 3.69045i
71.12 1.34500 + 0.437016i 0 1.61803 + 1.17557i 0.324944 − 4.98943i 0 −7.00769 1.66251 + 2.28825i 0 2.61751 − 6.56876i
161.1 −0.831254 − 1.14412i 0 −0.618034 + 1.90211i −2.23153 + 4.47440i 0 12.3262 2.68999 − 0.874032i 0 6.97423 − 1.16622i
161.2 −0.831254 − 1.14412i 0 −0.618034 + 1.90211i 1.35057 + 4.81414i 0 −4.93601 2.68999 − 0.874032i 0 4.38530 − 5.54699i
161.3 −0.831254 − 1.14412i 0 −0.618034 + 1.90211i −2.73585 − 4.18511i 0 4.66866 2.68999 − 0.874032i 0 −2.51409 + 6.60904i
161.4 −0.831254 − 1.14412i 0 −0.618034 + 1.90211i 4.99754 + 0.156899i 0 3.81307 2.68999 − 0.874032i 0 −3.97471 − 5.84822i
161.5 −0.831254 − 1.14412i 0 −0.618034 + 1.90211i 4.42557 − 2.32687i 0 −5.65799 2.68999 − 0.874032i 0 −6.34100 − 3.12918i
161.6 −0.831254 − 1.14412i 0 −0.618034 + 1.90211i −4.97505 + 0.498900i 0 −11.4500 2.68999 − 0.874032i 0 4.70633 + 5.27735i
161.7 0.831254 + 1.14412i 0 −0.618034 + 1.90211i 4.97505 − 0.498900i 0 −11.4500 −2.68999 + 0.874032i 0 4.70633 + 5.27735i
161.8 0.831254 + 1.14412i 0 −0.618034 + 1.90211i −4.42557 + 2.32687i 0 −5.65799 −2.68999 + 0.874032i 0 −6.34100 − 3.12918i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 71.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
25.d even 5 1 inner
75.j 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.n.b ✓ 48
3.b odd 2 1 inner 450.3.n.b ✓ 48
25.d even 5 1 inner 450.3.n.b ✓ 48
75.j odd 10 1 inner 450.3.n.b ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
450.3.n.b ✓ 48 1.a even 1 1 trivial
450.3.n.b ✓ 48 3.b odd 2 1 inner
450.3.n.b ✓ 48 25.d even 5 1 inner
450.3.n.b ✓ 48 75.j odd 10 1 inner