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Properties

Label 630.4.be.a
Level $630$
Weight $4$
Character orbit 630.be
Analytic conductor $37.171$
Analytic rank $0$
Dimension $32$
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] = [630,4,Mod(341,630)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(630, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 5]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("630.341");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 630 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 630.be (of order \(6\), degree \(2\), 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: \(37.1712033036\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 64 q^{4} - 80 q^{5} + 52 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 64 q^{4} - 80 q^{5} + 52 q^{7} + 144 q^{11} - 256 q^{16} - 252 q^{19} - 640 q^{20} + 36 q^{23} - 400 q^{25} + 48 q^{26} + 32 q^{28} + 420 q^{31} - 220 q^{35} - 1036 q^{37} + 168 q^{38} - 1968 q^{41} + 440 q^{43} + 576 q^{44} - 240 q^{46} - 456 q^{47} + 1292 q^{49} + 336 q^{52} + 1152 q^{53} - 384 q^{58} - 696 q^{59} - 672 q^{61} - 1632 q^{62} - 2048 q^{64} + 420 q^{65} - 1076 q^{67} - 756 q^{73} - 3672 q^{77} - 1820 q^{79} - 1280 q^{80} - 2688 q^{83} + 456 q^{89} + 4728 q^{91} + 936 q^{94} + 1260 q^{95} - 3672 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
341.1 −1.73205 − 1.00000i 0 2.00000 + 3.46410i −2.50000 + 4.33013i 0 14.6801 + 11.2913i − 8.00000i 0 8.66025 − 5.00000i
341.2 −1.73205 − 1.00000i 0 2.00000 + 3.46410i −2.50000 + 4.33013i 0 −10.6641 − 15.1419i − 8.00000i 0 8.66025 − 5.00000i
341.3 −1.73205 − 1.00000i 0 2.00000 + 3.46410i −2.50000 + 4.33013i 0 −15.7349 + 9.76800i − 8.00000i 0 8.66025 − 5.00000i
341.4 −1.73205 − 1.00000i 0 2.00000 + 3.46410i −2.50000 + 4.33013i 0 10.3692 − 15.3453i − 8.00000i 0 8.66025 − 5.00000i
341.5 −1.73205 − 1.00000i 0 2.00000 + 3.46410i −2.50000 + 4.33013i 0 −18.5074 + 0.689615i − 8.00000i 0 8.66025 − 5.00000i
341.6 −1.73205 − 1.00000i 0 2.00000 + 3.46410i −2.50000 + 4.33013i 0 18.5063 + 0.719622i − 8.00000i 0 8.66025 − 5.00000i
341.7 −1.73205 − 1.00000i 0 2.00000 + 3.46410i −2.50000 + 4.33013i 0 −3.48915 + 18.1886i − 8.00000i 0 8.66025 − 5.00000i
341.8 −1.73205 − 1.00000i 0 2.00000 + 3.46410i −2.50000 + 4.33013i 0 17.8399 − 4.97379i − 8.00000i 0 8.66025 − 5.00000i
341.9 1.73205 + 1.00000i 0 2.00000 + 3.46410i −2.50000 + 4.33013i 0 18.2282 − 3.27636i 8.00000i 0 −8.66025 + 5.00000i
341.10 1.73205 + 1.00000i 0 2.00000 + 3.46410i −2.50000 + 4.33013i 0 −0.713439 − 18.5065i 8.00000i 0 −8.66025 + 5.00000i
341.11 1.73205 + 1.00000i 0 2.00000 + 3.46410i −2.50000 + 4.33013i 0 13.1476 − 13.0438i 8.00000i 0 −8.66025 + 5.00000i
341.12 1.73205 + 1.00000i 0 2.00000 + 3.46410i −2.50000 + 4.33013i 0 −1.95882 + 18.4164i 8.00000i 0 −8.66025 + 5.00000i
341.13 1.73205 + 1.00000i 0 2.00000 + 3.46410i −2.50000 + 4.33013i 0 7.64044 + 16.8708i 8.00000i 0 −8.66025 + 5.00000i
341.14 1.73205 + 1.00000i 0 2.00000 + 3.46410i −2.50000 + 4.33013i 0 −18.2454 − 3.17869i 8.00000i 0 −8.66025 + 5.00000i
341.15 1.73205 + 1.00000i 0 2.00000 + 3.46410i −2.50000 + 4.33013i 0 12.5086 + 13.6577i 8.00000i 0 −8.66025 + 5.00000i
341.16 1.73205 + 1.00000i 0 2.00000 + 3.46410i −2.50000 + 4.33013i 0 −17.6072 − 5.74344i 8.00000i 0 −8.66025 + 5.00000i
521.1 −1.73205 + 1.00000i 0 2.00000 − 3.46410i −2.50000 − 4.33013i 0 14.6801 − 11.2913i 8.00000i 0 8.66025 + 5.00000i
521.2 −1.73205 + 1.00000i 0 2.00000 − 3.46410i −2.50000 − 4.33013i 0 −10.6641 + 15.1419i 8.00000i 0 8.66025 + 5.00000i
521.3 −1.73205 + 1.00000i 0 2.00000 − 3.46410i −2.50000 − 4.33013i 0 −15.7349 − 9.76800i 8.00000i 0 8.66025 + 5.00000i
521.4 −1.73205 + 1.00000i 0 2.00000 − 3.46410i −2.50000 − 4.33013i 0 10.3692 + 15.3453i 8.00000i 0 8.66025 + 5.00000i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 341.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
21.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 630.4.be.a ✓ 32
3.b odd 2 1 630.4.be.b yes 32
7.d odd 6 1 630.4.be.b yes 32
21.g even 6 1 inner 630.4.be.a ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
630.4.be.a ✓ 32 1.a even 1 1 trivial
630.4.be.a ✓ 32 21.g even 6 1 inner
630.4.be.b yes 32 3.b odd 2 1
630.4.be.b yes 32 7.d odd 6 1