Properties

Label 441.2.e.f
Level $441$
Weight $2$
Character orbit 441.e
Analytic conductor $3.521$
Analytic rank $0$
Dimension $4$
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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 147)
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 basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + \beta_1 - 1) q^{2} + ( - 2 \beta_{3} + \beta_{2} - 2 \beta_1) q^{4} + ( - 2 \beta_{2} - \beta_1 - 2) q^{5} + (\beta_{3} + 3) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + \beta_1 - 1) q^{2} + ( - 2 \beta_{3} + \beta_{2} - 2 \beta_1) q^{4} + ( - 2 \beta_{2} - \beta_1 - 2) q^{5} + (\beta_{3} + 3) q^{8} + ( - \beta_{3} - \beta_1) q^{10} + 2 \beta_{2} q^{11} + ( - \beta_{3} - 4) q^{13} + ( - 3 \beta_{2} - 3) q^{16} + ( - 3 \beta_{3} + 2 \beta_{2} - 3 \beta_1) q^{17} - 2 \beta_1 q^{19} + (3 \beta_{3} - 2) q^{20} + (2 \beta_{3} + 2) q^{22} + ( - 2 \beta_{2} - 4 \beta_1 - 2) q^{23} + (4 \beta_{3} + \beta_{2} + 4 \beta_1) q^{25} + (6 \beta_{2} - 5 \beta_1 + 6) q^{26} + ( - 2 \beta_{3} + 4) q^{29} + ( - 2 \beta_{3} + 4 \beta_{2} - 2 \beta_1) q^{31} + ( - \beta_{3} - 3 \beta_{2} - \beta_1) q^{32} + (5 \beta_{3} + 8) q^{34} + (4 \beta_{2} + 4) q^{37} + (2 \beta_{3} - 4 \beta_{2} + 2 \beta_1) q^{38} + ( - 4 \beta_{2} - \beta_1 - 4) q^{40} + ( - 3 \beta_{3} + 2) q^{41} - 4 \beta_{3} q^{43} + ( - 2 \beta_{2} + 4 \beta_1 - 2) q^{44} + (2 \beta_{3} - 6 \beta_{2} + 2 \beta_1) q^{46} + 2 \beta_1 q^{47} + ( - 3 \beta_{3} - 7) q^{50} + (9 \beta_{3} - 8 \beta_{2} + 9 \beta_1) q^{52} + 2 \beta_{2} q^{53} + ( - 2 \beta_{3} + 4) q^{55} + 2 \beta_1 q^{58} + (2 \beta_{3} - 4 \beta_{2} + 2 \beta_1) q^{59} + (8 \beta_{2} + 3 \beta_1 + 8) q^{61} + (6 \beta_{3} + 8) q^{62} + ( - 2 \beta_{3} - 7) q^{64} + (6 \beta_{2} + 2 \beta_1 + 6) q^{65} + ( - 4 \beta_{3} - 4 \beta_1) q^{67} + ( - 14 \beta_{2} + 7 \beta_1 - 14) q^{68} + (8 \beta_{3} + 2) q^{71} + ( - 7 \beta_{3} - 4 \beta_{2} - 7 \beta_1) q^{73} + (4 \beta_{3} - 4 \beta_{2} + 4 \beta_1) q^{74} + ( - 2 \beta_{3} - 8) q^{76} + ( - 8 \beta_{2} - 4 \beta_1 - 8) q^{79} + (3 \beta_{3} + 6 \beta_{2} + 3 \beta_1) q^{80} + (4 \beta_{2} - \beta_1 + 4) q^{82} + ( - 8 \beta_{3} - 4) q^{83} + (4 \beta_{3} - 2) q^{85} + (8 \beta_{2} - 4 \beta_1 + 8) q^{86} + ( - 2 \beta_{3} + 6 \beta_{2} - 2 \beta_1) q^{88} + (10 \beta_{2} + 3 \beta_1 + 10) q^{89} - 14 q^{92} + ( - 2 \beta_{3} + 4 \beta_{2} - 2 \beta_1) q^{94} + (4 \beta_{3} + 4 \beta_{2} + 4 \beta_1) q^{95} + ( - \beta_{3} - 4) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} - 2 q^{4} - 4 q^{5} + 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{2} - 2 q^{4} - 4 q^{5} + 12 q^{8} - 4 q^{11} - 16 q^{13} - 6 q^{16} - 4 q^{17} - 8 q^{20} + 8 q^{22} - 4 q^{23} - 2 q^{25} + 12 q^{26} + 16 q^{29} - 8 q^{31} + 6 q^{32} + 32 q^{34} + 8 q^{37} + 8 q^{38} - 8 q^{40} + 8 q^{41} - 4 q^{44} + 12 q^{46} - 28 q^{50} + 16 q^{52} - 4 q^{53} + 16 q^{55} + 8 q^{59} + 16 q^{61} + 32 q^{62} - 28 q^{64} + 12 q^{65} - 28 q^{68} + 8 q^{71} + 8 q^{73} + 8 q^{74} - 32 q^{76} - 16 q^{79} - 12 q^{80} + 8 q^{82} - 16 q^{83} - 8 q^{85} + 16 q^{86} - 12 q^{88} + 20 q^{89} - 56 q^{92} - 8 q^{94} - 8 q^{95} - 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 2x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} \) Copy content Toggle raw display

Character values

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

\(n\) \(199\) \(344\)
\(\chi(n)\) \(-1 - \beta_{2}\) \(1\)

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
−0.707107 + 1.22474i
0.707107 1.22474i
−0.707107 1.22474i
0.707107 + 1.22474i
−1.20711 + 2.09077i 0 −1.91421 3.31552i −0.292893 + 0.507306i 0 0 4.41421 0 −0.707107 1.22474i
226.2 0.207107 0.358719i 0 0.914214 + 1.58346i −1.70711 + 2.95680i 0 0 1.58579 0 0.707107 + 1.22474i
361.1 −1.20711 2.09077i 0 −1.91421 + 3.31552i −0.292893 0.507306i 0 0 4.41421 0 −0.707107 + 1.22474i
361.2 0.207107 + 0.358719i 0 0.914214 1.58346i −1.70711 2.95680i 0 0 1.58579 0 0.707107 1.22474i
\(n\): 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.f 4
3.b odd 2 1 147.2.e.e 4
7.b odd 2 1 441.2.e.g 4
7.c even 3 1 441.2.a.j 2
7.c even 3 1 inner 441.2.e.f 4
7.d odd 6 1 441.2.a.i 2
7.d odd 6 1 441.2.e.g 4
12.b even 2 1 2352.2.q.bb 4
21.c even 2 1 147.2.e.d 4
21.g even 6 1 147.2.a.e yes 2
21.g even 6 1 147.2.e.d 4
21.h odd 6 1 147.2.a.d 2
21.h odd 6 1 147.2.e.e 4
28.f even 6 1 7056.2.a.cf 2
28.g odd 6 1 7056.2.a.cv 2
84.h odd 2 1 2352.2.q.bd 4
84.j odd 6 1 2352.2.a.bc 2
84.j odd 6 1 2352.2.q.bd 4
84.n even 6 1 2352.2.a.be 2
84.n even 6 1 2352.2.q.bb 4
105.o odd 6 1 3675.2.a.bf 2
105.p even 6 1 3675.2.a.bd 2
168.s odd 6 1 9408.2.a.ef 2
168.v even 6 1 9408.2.a.dq 2
168.ba even 6 1 9408.2.a.di 2
168.be odd 6 1 9408.2.a.dt 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
147.2.a.d 2 21.h odd 6 1
147.2.a.e yes 2 21.g even 6 1
147.2.e.d 4 21.c even 2 1
147.2.e.d 4 21.g even 6 1
147.2.e.e 4 3.b odd 2 1
147.2.e.e 4 21.h odd 6 1
441.2.a.i 2 7.d odd 6 1
441.2.a.j 2 7.c even 3 1
441.2.e.f 4 1.a even 1 1 trivial
441.2.e.f 4 7.c even 3 1 inner
441.2.e.g 4 7.b odd 2 1
441.2.e.g 4 7.d odd 6 1
2352.2.a.bc 2 84.j odd 6 1
2352.2.a.be 2 84.n even 6 1
2352.2.q.bb 4 12.b even 2 1
2352.2.q.bb 4 84.n even 6 1
2352.2.q.bd 4 84.h odd 2 1
2352.2.q.bd 4 84.j odd 6 1
3675.2.a.bd 2 105.p even 6 1
3675.2.a.bf 2 105.o odd 6 1
7056.2.a.cf 2 28.f even 6 1
7056.2.a.cv 2 28.g odd 6 1
9408.2.a.di 2 168.ba even 6 1
9408.2.a.dq 2 168.v even 6 1
9408.2.a.dt 2 168.be odd 6 1
9408.2.a.ef 2 168.s odd 6 1

Hecke kernels

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

\( T_{2}^{4} + 2T_{2}^{3} + 5T_{2}^{2} - 2T_{2} + 1 \) Copy content Toggle raw display
\( T_{5}^{4} + 4T_{5}^{3} + 14T_{5}^{2} + 8T_{5} + 4 \) Copy content Toggle raw display
\( T_{13}^{2} + 8T_{13} + 14 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 2 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 4 T^{3} + \cdots + 4 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} + 2 T + 4)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 8 T + 14)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 4 T^{3} + \cdots + 196 \) Copy content Toggle raw display
$19$ \( T^{4} + 8T^{2} + 64 \) Copy content Toggle raw display
$23$ \( T^{4} + 4 T^{3} + \cdots + 784 \) Copy content Toggle raw display
$29$ \( (T^{2} - 8 T + 8)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 8 T^{3} + \cdots + 64 \) Copy content Toggle raw display
$37$ \( (T^{2} - 4 T + 16)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 4 T - 14)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 32)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 8T^{2} + 64 \) Copy content Toggle raw display
$53$ \( (T^{2} + 2 T + 4)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} - 8 T^{3} + \cdots + 64 \) Copy content Toggle raw display
$61$ \( T^{4} - 16 T^{3} + \cdots + 2116 \) Copy content Toggle raw display
$67$ \( T^{4} + 32T^{2} + 1024 \) Copy content Toggle raw display
$71$ \( (T^{2} - 4 T - 124)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} - 8 T^{3} + \cdots + 6724 \) Copy content Toggle raw display
$79$ \( T^{4} + 16 T^{3} + \cdots + 1024 \) Copy content Toggle raw display
$83$ \( (T^{2} + 8 T - 112)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} - 20 T^{3} + \cdots + 6724 \) Copy content Toggle raw display
$97$ \( (T^{2} + 8 T + 14)^{2} \) Copy content Toggle raw display
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