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Properties

Label 441.6.c.c
Level $441$
Weight $6$
Character orbit 441.c
Analytic conductor $70.729$
Analytic rank $0$
Dimension $40$
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] = [441,6,Mod(440,441)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(441, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("441.440");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 441.c (of order \(2\), degree \(1\), 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: \(70.7292645375\)
Analytic rank: \(0\)
Dimension: \(40\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 640 q^{4}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q - 640 q^{4} + 13584 q^{16} - 16208 q^{22} + 14792 q^{25} + 52448 q^{37} - 43168 q^{43} + 58192 q^{46} + 383664 q^{58} - 278432 q^{64} + 228480 q^{67} + 154592 q^{79} - 357824 q^{85} + 1796656 q^{88}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
440.1 − 1.31309i 0 30.2758 110.304 0 0 − 81.7738i 0 − 144.839i
440.2 1.31309i 0 30.2758 110.304 0 0 81.7738i 0 144.839i
440.3 − 5.47542i 0 2.01979 −93.2652 0 0 − 186.273i 0 510.666i
440.4 5.47542i 0 2.01979 −93.2652 0 0 186.273i 0 − 510.666i
440.5 − 4.84889i 0 8.48823 −56.6454 0 0 − 196.323i 0 274.668i
440.6 4.84889i 0 8.48823 −56.6454 0 0 196.323i 0 − 274.668i
440.7 − 1.43734i 0 29.9340 51.8726 0 0 − 89.0205i 0 − 74.5588i
440.8 1.43734i 0 29.9340 51.8726 0 0 89.0205i 0 74.5588i
440.9 − 8.96302i 0 −48.3358 46.3628 0 0 146.418i 0 − 415.551i
440.10 8.96302i 0 −48.3358 46.3628 0 0 − 146.418i 0 415.551i
440.11 − 11.0155i 0 −89.3406 −50.4010 0 0 631.634i 0 555.191i
440.12 11.0155i 0 −89.3406 −50.4010 0 0 − 631.634i 0 − 555.191i
440.13 − 5.81487i 0 −1.81276 42.5471 0 0 − 175.535i 0 − 247.406i
440.14 5.81487i 0 −1.81276 42.5471 0 0 175.535i 0 247.406i
440.15 − 3.51435i 0 19.6494 28.2666 0 0 − 181.514i 0 − 99.3387i
440.16 3.51435i 0 19.6494 28.2666 0 0 181.514i 0 99.3387i
440.17 − 9.48494i 0 −57.9641 28.9542 0 0 246.268i 0 − 274.629i
440.18 9.48494i 0 −57.9641 28.9542 0 0 − 246.268i 0 274.629i
440.19 − 9.21487i 0 −52.9139 6.77222 0 0 192.719i 0 − 62.4052i
440.20 9.21487i 0 −52.9139 6.77222 0 0 − 192.719i 0 62.4052i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 440.40
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.6.c.c ✓ 40
3.b odd 2 1 inner 441.6.c.c ✓ 40
7.b odd 2 1 inner 441.6.c.c ✓ 40
21.c even 2 1 inner 441.6.c.c ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
441.6.c.c ✓ 40 1.a even 1 1 trivial
441.6.c.c ✓ 40 3.b odd 2 1 inner
441.6.c.c ✓ 40 7.b odd 2 1 inner
441.6.c.c ✓ 40 21.c even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{20} + 480 T_{2}^{18} + 95582 T_{2}^{16} + 10257336 T_{2}^{14} + 645834433 T_{2}^{12} + \cdots + 78084509863936 \) acting on \(S_{6}^{\mathrm{new}}(441, [\chi])\). Copy content Toggle raw display