Properties

Label 441.2.e.a
Level $441$
Weight $2$
Character orbit 441.e
Analytic conductor $3.521$
Analytic rank $0$
Dimension $2$
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] = [441,2,Mod(226,441)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(441, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("441.226");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 441.e (of order \(3\), degree \(2\), not 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: \(3.52140272914\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 21)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \zeta_{6} q^{2} + ( - \zeta_{6} + 1) q^{4} - 2 \zeta_{6} q^{5} - 3 q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{6} q^{2} + ( - \zeta_{6} + 1) q^{4} - 2 \zeta_{6} q^{5} - 3 q^{8} + (2 \zeta_{6} - 2) q^{10} + ( - 4 \zeta_{6} + 4) q^{11} - 2 q^{13} + \zeta_{6} q^{16} + (6 \zeta_{6} - 6) q^{17} - 4 \zeta_{6} q^{19} - 2 q^{20} - 4 q^{22} + ( - \zeta_{6} + 1) q^{25} + 2 \zeta_{6} q^{26} + 2 q^{29} + (5 \zeta_{6} - 5) q^{32} + 6 q^{34} - 6 \zeta_{6} q^{37} + (4 \zeta_{6} - 4) q^{38} + 6 \zeta_{6} q^{40} - 2 q^{41} - 4 q^{43} - 4 \zeta_{6} q^{44} - q^{50} + (2 \zeta_{6} - 2) q^{52} + ( - 6 \zeta_{6} + 6) q^{53} - 8 q^{55} - 2 \zeta_{6} q^{58} + ( - 12 \zeta_{6} + 12) q^{59} + 2 \zeta_{6} q^{61} + 7 q^{64} + 4 \zeta_{6} q^{65} + (4 \zeta_{6} - 4) q^{67} + 6 \zeta_{6} q^{68} + ( - 6 \zeta_{6} + 6) q^{73} + (6 \zeta_{6} - 6) q^{74} - 4 q^{76} + 16 \zeta_{6} q^{79} + ( - 2 \zeta_{6} + 2) q^{80} + 2 \zeta_{6} q^{82} + 12 q^{83} + 12 q^{85} + 4 \zeta_{6} q^{86} + (12 \zeta_{6} - 12) q^{88} - 14 \zeta_{6} q^{89} + (8 \zeta_{6} - 8) q^{95} + 18 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} + q^{4} - 2 q^{5} - 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} + q^{4} - 2 q^{5} - 6 q^{8} - 2 q^{10} + 4 q^{11} - 4 q^{13} + q^{16} - 6 q^{17} - 4 q^{19} - 4 q^{20} - 8 q^{22} + q^{25} + 2 q^{26} + 4 q^{29} - 5 q^{32} + 12 q^{34} - 6 q^{37} - 4 q^{38} + 6 q^{40} - 4 q^{41} - 8 q^{43} - 4 q^{44} - 2 q^{50} - 2 q^{52} + 6 q^{53} - 16 q^{55} - 2 q^{58} + 12 q^{59} + 2 q^{61} + 14 q^{64} + 4 q^{65} - 4 q^{67} + 6 q^{68} + 6 q^{73} - 6 q^{74} - 8 q^{76} + 16 q^{79} + 2 q^{80} + 2 q^{82} + 24 q^{83} + 24 q^{85} + 4 q^{86} - 12 q^{88} - 14 q^{89} - 8 q^{95} + 36 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/441\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(344\)
\(\chi(n)\) \(-\zeta_{6}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
226.1
0.500000 0.866025i
0.500000 + 0.866025i
−0.500000 + 0.866025i 0 0.500000 + 0.866025i −1.00000 + 1.73205i 0 0 −3.00000 0 −1.00000 1.73205i
361.1 −0.500000 0.866025i 0 0.500000 0.866025i −1.00000 1.73205i 0 0 −3.00000 0 −1.00000 + 1.73205i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.2.e.a 2
3.b odd 2 1 147.2.e.b 2
7.b odd 2 1 441.2.e.b 2
7.c even 3 1 63.2.a.a 1
7.c even 3 1 inner 441.2.e.a 2
7.d odd 6 1 441.2.a.f 1
7.d odd 6 1 441.2.e.b 2
12.b even 2 1 2352.2.q.x 2
21.c even 2 1 147.2.e.c 2
21.g even 6 1 147.2.a.a 1
21.g even 6 1 147.2.e.c 2
21.h odd 6 1 21.2.a.a 1
21.h odd 6 1 147.2.e.b 2
28.f even 6 1 7056.2.a.p 1
28.g odd 6 1 1008.2.a.l 1
35.j even 6 1 1575.2.a.c 1
35.l odd 12 2 1575.2.d.a 2
56.k odd 6 1 4032.2.a.k 1
56.p even 6 1 4032.2.a.h 1
63.g even 3 1 567.2.f.b 2
63.h even 3 1 567.2.f.b 2
63.j odd 6 1 567.2.f.g 2
63.n odd 6 1 567.2.f.g 2
77.h odd 6 1 7623.2.a.g 1
84.h odd 2 1 2352.2.q.e 2
84.j odd 6 1 2352.2.a.v 1
84.j odd 6 1 2352.2.q.e 2
84.n even 6 1 336.2.a.a 1
84.n even 6 1 2352.2.q.x 2
105.o odd 6 1 525.2.a.d 1
105.p even 6 1 3675.2.a.n 1
105.x even 12 2 525.2.d.a 2
168.s odd 6 1 1344.2.a.g 1
168.v even 6 1 1344.2.a.s 1
168.ba even 6 1 9408.2.a.bv 1
168.be odd 6 1 9408.2.a.m 1
231.l even 6 1 2541.2.a.j 1
273.w odd 6 1 3549.2.a.c 1
336.bt odd 12 2 5376.2.c.r 2
336.bu even 12 2 5376.2.c.l 2
357.q odd 6 1 6069.2.a.b 1
399.w even 6 1 7581.2.a.d 1
420.ba even 6 1 8400.2.a.bn 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.2.a.a 1 21.h odd 6 1
63.2.a.a 1 7.c even 3 1
147.2.a.a 1 21.g even 6 1
147.2.e.b 2 3.b odd 2 1
147.2.e.b 2 21.h odd 6 1
147.2.e.c 2 21.c even 2 1
147.2.e.c 2 21.g even 6 1
336.2.a.a 1 84.n even 6 1
441.2.a.f 1 7.d odd 6 1
441.2.e.a 2 1.a even 1 1 trivial
441.2.e.a 2 7.c even 3 1 inner
441.2.e.b 2 7.b odd 2 1
441.2.e.b 2 7.d odd 6 1
525.2.a.d 1 105.o odd 6 1
525.2.d.a 2 105.x even 12 2
567.2.f.b 2 63.g even 3 1
567.2.f.b 2 63.h even 3 1
567.2.f.g 2 63.j odd 6 1
567.2.f.g 2 63.n odd 6 1
1008.2.a.l 1 28.g odd 6 1
1344.2.a.g 1 168.s odd 6 1
1344.2.a.s 1 168.v even 6 1
1575.2.a.c 1 35.j even 6 1
1575.2.d.a 2 35.l odd 12 2
2352.2.a.v 1 84.j odd 6 1
2352.2.q.e 2 84.h odd 2 1
2352.2.q.e 2 84.j odd 6 1
2352.2.q.x 2 12.b even 2 1
2352.2.q.x 2 84.n even 6 1
2541.2.a.j 1 231.l even 6 1
3549.2.a.c 1 273.w odd 6 1
3675.2.a.n 1 105.p even 6 1
4032.2.a.h 1 56.p even 6 1
4032.2.a.k 1 56.k odd 6 1
5376.2.c.l 2 336.bu even 12 2
5376.2.c.r 2 336.bt odd 12 2
6069.2.a.b 1 357.q odd 6 1
7056.2.a.p 1 28.f even 6 1
7581.2.a.d 1 399.w even 6 1
7623.2.a.g 1 77.h odd 6 1
8400.2.a.bn 1 420.ba even 6 1
9408.2.a.m 1 168.be odd 6 1
9408.2.a.bv 1 168.ba even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(441, [\chi])\):

\( T_{2}^{2} + T_{2} + 1 \) Copy content Toggle raw display
\( T_{5}^{2} + 2T_{5} + 4 \) Copy content Toggle raw display
\( T_{13} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$13$ \( (T + 2)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$19$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( (T - 2)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$41$ \( (T + 2)^{2} \) Copy content Toggle raw display
$43$ \( (T + 4)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$59$ \( T^{2} - 12T + 144 \) Copy content Toggle raw display
$61$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$67$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$79$ \( T^{2} - 16T + 256 \) Copy content Toggle raw display
$83$ \( (T - 12)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 14T + 196 \) Copy content Toggle raw display
$97$ \( (T - 18)^{2} \) Copy content Toggle raw display
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