Properties

Label 441.2.a.j
Level $441$
Weight $2$
Character orbit 441.a
Self dual yes
Analytic conductor $3.521$
Analytic rank $0$
Dimension $2$
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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 147)
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)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta + 1) q^{2} + (2 \beta + 1) q^{4} + ( - \beta + 2) q^{5} + (\beta + 3) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta + 1) q^{2} + (2 \beta + 1) q^{4} + ( - \beta + 2) q^{5} + (\beta + 3) q^{8} + \beta q^{10} + 2 q^{11} + ( - \beta - 4) q^{13} + 3 q^{16} + (3 \beta + 2) q^{17} - 2 \beta q^{19} + (3 \beta - 2) q^{20} + (2 \beta + 2) q^{22} + ( - 4 \beta + 2) q^{23} + ( - 4 \beta + 1) q^{25} + ( - 5 \beta - 6) q^{26} + ( - 2 \beta + 4) q^{29} + (2 \beta + 4) q^{31} + (\beta - 3) q^{32} + (5 \beta + 8) q^{34} - 4 q^{37} + ( - 2 \beta - 4) q^{38} + ( - \beta + 4) q^{40} + ( - 3 \beta + 2) q^{41} - 4 \beta q^{43} + (4 \beta + 2) q^{44} + ( - 2 \beta - 6) q^{46} + 2 \beta q^{47} + ( - 3 \beta - 7) q^{50} + ( - 9 \beta - 8) q^{52} + 2 q^{53} + ( - 2 \beta + 4) q^{55} + 2 \beta q^{58} + ( - 2 \beta - 4) q^{59} + (3 \beta - 8) q^{61} + (6 \beta + 8) q^{62} + ( - 2 \beta - 7) q^{64} + (2 \beta - 6) q^{65} + 4 \beta q^{67} + (7 \beta + 14) q^{68} + (8 \beta + 2) q^{71} + (7 \beta - 4) q^{73} + ( - 4 \beta - 4) q^{74} + ( - 2 \beta - 8) q^{76} + ( - 4 \beta + 8) q^{79} + ( - 3 \beta + 6) q^{80} + ( - \beta - 4) q^{82} + ( - 8 \beta - 4) q^{83} + (4 \beta - 2) q^{85} + ( - 4 \beta - 8) q^{86} + (2 \beta + 6) q^{88} + (3 \beta - 10) q^{89} - 14 q^{92} + (2 \beta + 4) q^{94} + ( - 4 \beta + 4) q^{95} + ( - \beta - 4) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} + 4 q^{5} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{4} + 4 q^{5} + 6 q^{8} + 4 q^{11} - 8 q^{13} + 6 q^{16} + 4 q^{17} - 4 q^{20} + 4 q^{22} + 4 q^{23} + 2 q^{25} - 12 q^{26} + 8 q^{29} + 8 q^{31} - 6 q^{32} + 16 q^{34} - 8 q^{37} - 8 q^{38} + 8 q^{40} + 4 q^{41} + 4 q^{44} - 12 q^{46} - 14 q^{50} - 16 q^{52} + 4 q^{53} + 8 q^{55} - 8 q^{59} - 16 q^{61} + 16 q^{62} - 14 q^{64} - 12 q^{65} + 28 q^{68} + 4 q^{71} - 8 q^{73} - 8 q^{74} - 16 q^{76} + 16 q^{79} + 12 q^{80} - 8 q^{82} - 8 q^{83} - 4 q^{85} - 16 q^{86} + 12 q^{88} - 20 q^{89} - 28 q^{92} + 8 q^{94} + 8 q^{95} - 8 q^{97}+O(q^{100}) \) 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
−1.41421
1.41421
−0.414214 0 −1.82843 3.41421 0 0 1.58579 0 −1.41421
1.2 2.41421 0 3.82843 0.585786 0 0 4.41421 0 1.41421
\(n\): 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.j 2
3.b odd 2 1 147.2.a.d 2
4.b odd 2 1 7056.2.a.cv 2
7.b odd 2 1 441.2.a.i 2
7.c even 3 2 441.2.e.f 4
7.d odd 6 2 441.2.e.g 4
12.b even 2 1 2352.2.a.be 2
15.d odd 2 1 3675.2.a.bf 2
21.c even 2 1 147.2.a.e yes 2
21.g even 6 2 147.2.e.d 4
21.h odd 6 2 147.2.e.e 4
24.f even 2 1 9408.2.a.dq 2
24.h odd 2 1 9408.2.a.ef 2
28.d even 2 1 7056.2.a.cf 2
84.h odd 2 1 2352.2.a.bc 2
84.j odd 6 2 2352.2.q.bd 4
84.n even 6 2 2352.2.q.bb 4
105.g even 2 1 3675.2.a.bd 2
168.e odd 2 1 9408.2.a.dt 2
168.i even 2 1 9408.2.a.di 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
147.2.a.d 2 3.b odd 2 1
147.2.a.e yes 2 21.c even 2 1
147.2.e.d 4 21.g even 6 2
147.2.e.e 4 21.h odd 6 2
441.2.a.i 2 7.b odd 2 1
441.2.a.j 2 1.a even 1 1 trivial
441.2.e.f 4 7.c even 3 2
441.2.e.g 4 7.d odd 6 2
2352.2.a.bc 2 84.h odd 2 1
2352.2.a.be 2 12.b even 2 1
2352.2.q.bb 4 84.n even 6 2
2352.2.q.bd 4 84.j odd 6 2
3675.2.a.bd 2 105.g even 2 1
3675.2.a.bf 2 15.d odd 2 1
7056.2.a.cf 2 28.d even 2 1
7056.2.a.cv 2 4.b odd 2 1
9408.2.a.di 2 168.i even 2 1
9408.2.a.dq 2 24.f even 2 1
9408.2.a.dt 2 168.e odd 2 1
9408.2.a.ef 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(441))\):

\( T_{2}^{2} - 2T_{2} - 1 \) Copy content Toggle raw display
\( T_{5}^{2} - 4T_{5} + 2 \) Copy content Toggle raw display
\( T_{13}^{2} + 8T_{13} + 14 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 2T - 1 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 4T + 2 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( (T - 2)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 8T + 14 \) Copy content Toggle raw display
$17$ \( T^{2} - 4T - 14 \) Copy content Toggle raw display
$19$ \( T^{2} - 8 \) Copy content Toggle raw display
$23$ \( T^{2} - 4T - 28 \) Copy content Toggle raw display
$29$ \( T^{2} - 8T + 8 \) Copy content Toggle raw display
$31$ \( T^{2} - 8T + 8 \) Copy content Toggle raw display
$37$ \( (T + 4)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 4T - 14 \) Copy content Toggle raw display
$43$ \( T^{2} - 32 \) Copy content Toggle raw display
$47$ \( T^{2} - 8 \) Copy content Toggle raw display
$53$ \( (T - 2)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 8T + 8 \) Copy content Toggle raw display
$61$ \( T^{2} + 16T + 46 \) Copy content Toggle raw display
$67$ \( T^{2} - 32 \) Copy content Toggle raw display
$71$ \( T^{2} - 4T - 124 \) Copy content Toggle raw display
$73$ \( T^{2} + 8T - 82 \) Copy content Toggle raw display
$79$ \( T^{2} - 16T + 32 \) Copy content Toggle raw display
$83$ \( T^{2} + 8T - 112 \) Copy content Toggle raw display
$89$ \( T^{2} + 20T + 82 \) Copy content Toggle raw display
$97$ \( T^{2} + 8T + 14 \) Copy content Toggle raw display
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