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Properties

Label 450.3.n.a
Level $450$
Weight $3$
Character orbit 450.n
Analytic conductor $12.262$
Analytic rank $0$
Dimension $32$
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: \(32\)
Relative dimension: \(8\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 16 q^{4} + 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 16 q^{4} + 8 q^{7} - 20 q^{10} - 16 q^{13} - 32 q^{16} - 12 q^{19} - 32 q^{22} - 20 q^{25} + 24 q^{28} - 12 q^{31} + 116 q^{34} + 88 q^{37} - 40 q^{43} + 168 q^{46} - 416 q^{49} + 32 q^{52} + 320 q^{55} + 192 q^{58} - 280 q^{61} + 64 q^{64} - 96 q^{67} + 160 q^{70} - 248 q^{73} - 16 q^{76} + 328 q^{79} - 688 q^{82} - 380 q^{85} - 56 q^{88} + 256 q^{91} + 160 q^{94} - 852 q^{97}+O(q^{100}) \) 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 −0.140171 + 4.99803i 0 9.93603 −1.66251 − 2.28825i 0 2.37275 − 6.66108i
71.2 −1.34500 − 0.437016i 0 1.61803 + 1.17557i 4.06913 + 2.90554i 0 −6.20054 −1.66251 − 2.28825i 0 −4.20320 − 5.68622i
71.3 −1.34500 − 0.437016i 0 1.61803 + 1.17557i −3.33518 − 3.72512i 0 2.11420 −1.66251 − 2.28825i 0 2.85787 + 6.46781i
71.4 −1.34500 − 0.437016i 0 1.61803 + 1.17557i 4.27247 − 2.59731i 0 −2.61362 −1.66251 − 2.28825i 0 −6.88152 + 1.62624i
71.5 1.34500 + 0.437016i 0 1.61803 + 1.17557i −4.27247 + 2.59731i 0 −2.61362 1.66251 + 2.28825i 0 −6.88152 + 1.62624i
71.6 1.34500 + 0.437016i 0 1.61803 + 1.17557i 0.140171 − 4.99803i 0 9.93603 1.66251 + 2.28825i 0 2.37275 − 6.66108i
71.7 1.34500 + 0.437016i 0 1.61803 + 1.17557i −4.06913 − 2.90554i 0 −6.20054 1.66251 + 2.28825i 0 −4.20320 − 5.68622i
71.8 1.34500 + 0.437016i 0 1.61803 + 1.17557i 3.33518 + 3.72512i 0 2.11420 1.66251 + 2.28825i 0 2.85787 + 6.46781i
161.1 −0.831254 − 1.14412i 0 −0.618034 + 1.90211i −4.91641 − 0.910429i 0 −1.49980 2.68999 − 0.874032i 0 3.04515 + 6.38178i
161.2 −0.831254 − 1.14412i 0 −0.618034 + 1.90211i 2.35177 − 4.41239i 0 −3.34126 2.68999 − 0.874032i 0 −7.00323 + 0.977095i
161.3 −0.831254 − 1.14412i 0 −0.618034 + 1.90211i 4.59060 + 1.98150i 0 9.53608 2.68999 − 0.874032i 0 −1.54887 − 6.89935i
161.4 −0.831254 − 1.14412i 0 −0.618034 + 1.90211i −0.877200 + 4.92245i 0 −5.93108 2.68999 − 0.874032i 0 6.36106 − 3.08818i
161.5 0.831254 + 1.14412i 0 −0.618034 + 1.90211i −2.35177 + 4.41239i 0 −3.34126 −2.68999 + 0.874032i 0 −7.00323 + 0.977095i
161.6 0.831254 + 1.14412i 0 −0.618034 + 1.90211i 0.877200 − 4.92245i 0 −5.93108 −2.68999 + 0.874032i 0 6.36106 − 3.08818i
161.7 0.831254 + 1.14412i 0 −0.618034 + 1.90211i −4.59060 − 1.98150i 0 9.53608 −2.68999 + 0.874032i 0 −1.54887 − 6.89935i
161.8 0.831254 + 1.14412i 0 −0.618034 + 1.90211i 4.91641 + 0.910429i 0 −1.49980 −2.68999 + 0.874032i 0 3.04515 + 6.38178i
341.1 −0.831254 + 1.14412i 0 −0.618034 − 1.90211i −4.91641 + 0.910429i 0 −1.49980 2.68999 + 0.874032i 0 3.04515 − 6.38178i
341.2 −0.831254 + 1.14412i 0 −0.618034 − 1.90211i 2.35177 + 4.41239i 0 −3.34126 2.68999 + 0.874032i 0 −7.00323 − 0.977095i
341.3 −0.831254 + 1.14412i 0 −0.618034 − 1.90211i −0.877200 − 4.92245i 0 −5.93108 2.68999 + 0.874032i 0 6.36106 + 3.08818i
341.4 −0.831254 + 1.14412i 0 −0.618034 − 1.90211i 4.59060 − 1.98150i 0 9.53608 2.68999 + 0.874032i 0 −1.54887 + 6.89935i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 71.8
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.a ✓ 32
3.b odd 2 1 inner 450.3.n.a ✓ 32
25.d even 5 1 inner 450.3.n.a ✓ 32
75.j odd 10 1 inner 450.3.n.a ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
450.3.n.a ✓ 32 1.a even 1 1 trivial
450.3.n.a ✓ 32 3.b odd 2 1 inner
450.3.n.a ✓ 32 25.d even 5 1 inner
450.3.n.a ✓ 32 75.j odd 10 1 inner