Properties

Label 441.2.a.f
Level $441$
Weight $2$
Character orbit 441.a
Self dual yes
Analytic conductor $3.521$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [441,2,Mod(1,441)]
 
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([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("441.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 441.a (trivial)

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: yes
Analytic conductor: \(3.52140272914\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 21)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q + q^{2} - q^{4} - 2 q^{5} - 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} - q^{4} - 2 q^{5} - 3 q^{8} - 2 q^{10} - 4 q^{11} + 2 q^{13} - q^{16} - 6 q^{17} - 4 q^{19} + 2 q^{20} - 4 q^{22} - q^{25} + 2 q^{26} + 2 q^{29} + 5 q^{32} - 6 q^{34} + 6 q^{37} - 4 q^{38} + 6 q^{40} + 2 q^{41} - 4 q^{43} + 4 q^{44} - q^{50} - 2 q^{52} - 6 q^{53} + 8 q^{55} + 2 q^{58} + 12 q^{59} + 2 q^{61} + 7 q^{64} - 4 q^{65} + 4 q^{67} + 6 q^{68} + 6 q^{73} + 6 q^{74} + 4 q^{76} - 16 q^{79} + 2 q^{80} + 2 q^{82} - 12 q^{83} + 12 q^{85} - 4 q^{86} + 12 q^{88} - 14 q^{89} + 8 q^{95} - 18 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
1.00000 0 −1.00000 −2.00000 0 0 −3.00000 0 −2.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(7\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.2.a.f 1
3.b odd 2 1 147.2.a.a 1
4.b odd 2 1 7056.2.a.p 1
7.b odd 2 1 63.2.a.a 1
7.c even 3 2 441.2.e.b 2
7.d odd 6 2 441.2.e.a 2
12.b even 2 1 2352.2.a.v 1
15.d odd 2 1 3675.2.a.n 1
21.c even 2 1 21.2.a.a 1
21.g even 6 2 147.2.e.b 2
21.h odd 6 2 147.2.e.c 2
24.f even 2 1 9408.2.a.m 1
24.h odd 2 1 9408.2.a.bv 1
28.d even 2 1 1008.2.a.l 1
35.c odd 2 1 1575.2.a.c 1
35.f even 4 2 1575.2.d.a 2
56.e even 2 1 4032.2.a.k 1
56.h odd 2 1 4032.2.a.h 1
63.l odd 6 2 567.2.f.b 2
63.o even 6 2 567.2.f.g 2
77.b even 2 1 7623.2.a.g 1
84.h odd 2 1 336.2.a.a 1
84.j odd 6 2 2352.2.q.x 2
84.n even 6 2 2352.2.q.e 2
105.g even 2 1 525.2.a.d 1
105.k odd 4 2 525.2.d.a 2
168.e odd 2 1 1344.2.a.s 1
168.i even 2 1 1344.2.a.g 1
231.h odd 2 1 2541.2.a.j 1
273.g even 2 1 3549.2.a.c 1
336.v odd 4 2 5376.2.c.l 2
336.y even 4 2 5376.2.c.r 2
357.c even 2 1 6069.2.a.b 1
399.h odd 2 1 7581.2.a.d 1
420.o odd 2 1 8400.2.a.bn 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.2.a.a 1 21.c even 2 1
63.2.a.a 1 7.b odd 2 1
147.2.a.a 1 3.b odd 2 1
147.2.e.b 2 21.g even 6 2
147.2.e.c 2 21.h odd 6 2
336.2.a.a 1 84.h odd 2 1
441.2.a.f 1 1.a even 1 1 trivial
441.2.e.a 2 7.d odd 6 2
441.2.e.b 2 7.c even 3 2
525.2.a.d 1 105.g even 2 1
525.2.d.a 2 105.k odd 4 2
567.2.f.b 2 63.l odd 6 2
567.2.f.g 2 63.o even 6 2
1008.2.a.l 1 28.d even 2 1
1344.2.a.g 1 168.i even 2 1
1344.2.a.s 1 168.e odd 2 1
1575.2.a.c 1 35.c odd 2 1
1575.2.d.a 2 35.f even 4 2
2352.2.a.v 1 12.b even 2 1
2352.2.q.e 2 84.n even 6 2
2352.2.q.x 2 84.j odd 6 2
2541.2.a.j 1 231.h odd 2 1
3549.2.a.c 1 273.g even 2 1
3675.2.a.n 1 15.d odd 2 1
4032.2.a.h 1 56.h odd 2 1
4032.2.a.k 1 56.e even 2 1
5376.2.c.l 2 336.v odd 4 2
5376.2.c.r 2 336.y even 4 2
6069.2.a.b 1 357.c even 2 1
7056.2.a.p 1 4.b odd 2 1
7581.2.a.d 1 399.h odd 2 1
7623.2.a.g 1 77.b even 2 1
8400.2.a.bn 1 420.o odd 2 1
9408.2.a.m 1 24.f even 2 1
9408.2.a.bv 1 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(441))\):

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

Hecke characteristic polynomials

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