Properties

Label 441.2.c.a
Level $441$
Weight $2$
Character orbit 441.c
Analytic conductor $3.521$
Analytic rank $0$
Dimension $4$
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [441,2,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, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("441.440");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
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: \(3.52140272914\)
Analytic rank: \(0\)
Dimension: \(4\)
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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 63)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + \beta_{3} q^{5} - 2 \beta_1 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + \beta_{3} q^{5} - 2 \beta_1 q^{8} - 2 \beta_{2} q^{10} - \beta_1 q^{11} + 3 \beta_{2} q^{13} - 4 q^{16} + 2 \beta_{3} q^{17} - \beta_{2} q^{19} - 2 q^{22} - 4 \beta_1 q^{23} + q^{25} + 3 \beta_{3} q^{26} + 2 \beta_1 q^{29} + \beta_{2} q^{31} - 4 \beta_{2} q^{34} - q^{37} - \beta_{3} q^{38} - 4 \beta_{2} q^{40} - 3 \beta_{3} q^{41} - q^{43} - 8 q^{46} - 5 \beta_{3} q^{47} - \beta_1 q^{50} + 2 \beta_1 q^{53} - 2 \beta_{2} q^{55} + 4 q^{58} + 2 \beta_{3} q^{59} + 2 \beta_{2} q^{61} + \beta_{3} q^{62} - 8 q^{64} + 9 \beta_1 q^{65} + 11 q^{67} + 5 \beta_1 q^{71} + \beta_{2} q^{73} + \beta_1 q^{74} + 5 q^{79} - 4 \beta_{3} q^{80} + 6 \beta_{2} q^{82} + 3 \beta_{3} q^{83} + 12 q^{85} + \beta_1 q^{86} - 4 q^{88} - 2 \beta_{3} q^{89} + 10 \beta_{2} q^{94} - 3 \beta_1 q^{95} + 6 \beta_{2} q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 16 q^{16} - 8 q^{22} + 4 q^{25} - 4 q^{37} - 4 q^{43} - 32 q^{46} + 16 q^{58} - 32 q^{64} + 44 q^{67} + 20 q^{79} + 48 q^{85} - 16 q^{88}+O(q^{100}) \) 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

\(\beta_{1}\)\(=\) \( ( \nu^{3} ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} + 4\nu ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_1 \) Copy content Toggle raw display

Character values

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

\(n\) \(199\) \(344\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
440.1
−1.22474 + 0.707107i
1.22474 + 0.707107i
−1.22474 0.707107i
1.22474 0.707107i
1.41421i 0 0 −2.44949 0 0 2.82843i 0 3.46410i
440.2 1.41421i 0 0 2.44949 0 0 2.82843i 0 3.46410i
440.3 1.41421i 0 0 −2.44949 0 0 2.82843i 0 3.46410i
440.4 1.41421i 0 0 2.44949 0 0 2.82843i 0 3.46410i
\(n\): e.g. 2-40 or 990-1000
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 441.2.c.a 4
3.b odd 2 1 inner 441.2.c.a 4
4.b odd 2 1 7056.2.k.b 4
7.b odd 2 1 inner 441.2.c.a 4
7.c even 3 1 63.2.p.a 4
7.c even 3 1 441.2.p.a 4
7.d odd 6 1 63.2.p.a 4
7.d odd 6 1 441.2.p.a 4
12.b even 2 1 7056.2.k.b 4
21.c even 2 1 inner 441.2.c.a 4
21.g even 6 1 63.2.p.a 4
21.g even 6 1 441.2.p.a 4
21.h odd 6 1 63.2.p.a 4
21.h odd 6 1 441.2.p.a 4
28.d even 2 1 7056.2.k.b 4
28.f even 6 1 1008.2.bt.b 4
28.g odd 6 1 1008.2.bt.b 4
35.i odd 6 1 1575.2.bk.c 4
35.j even 6 1 1575.2.bk.c 4
35.k even 12 2 1575.2.bc.a 8
35.l odd 12 2 1575.2.bc.a 8
63.g even 3 1 567.2.s.d 4
63.h even 3 1 567.2.i.d 4
63.i even 6 1 567.2.i.d 4
63.j odd 6 1 567.2.i.d 4
63.k odd 6 1 567.2.s.d 4
63.n odd 6 1 567.2.s.d 4
63.s even 6 1 567.2.s.d 4
63.t odd 6 1 567.2.i.d 4
84.h odd 2 1 7056.2.k.b 4
84.j odd 6 1 1008.2.bt.b 4
84.n even 6 1 1008.2.bt.b 4
105.o odd 6 1 1575.2.bk.c 4
105.p even 6 1 1575.2.bk.c 4
105.w odd 12 2 1575.2.bc.a 8
105.x even 12 2 1575.2.bc.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.p.a 4 7.c even 3 1
63.2.p.a 4 7.d odd 6 1
63.2.p.a 4 21.g even 6 1
63.2.p.a 4 21.h odd 6 1
441.2.c.a 4 1.a even 1 1 trivial
441.2.c.a 4 3.b odd 2 1 inner
441.2.c.a 4 7.b odd 2 1 inner
441.2.c.a 4 21.c even 2 1 inner
441.2.p.a 4 7.c even 3 1
441.2.p.a 4 7.d odd 6 1
441.2.p.a 4 21.g even 6 1
441.2.p.a 4 21.h odd 6 1
567.2.i.d 4 63.h even 3 1
567.2.i.d 4 63.i even 6 1
567.2.i.d 4 63.j odd 6 1
567.2.i.d 4 63.t odd 6 1
567.2.s.d 4 63.g even 3 1
567.2.s.d 4 63.k odd 6 1
567.2.s.d 4 63.n odd 6 1
567.2.s.d 4 63.s even 6 1
1008.2.bt.b 4 28.f even 6 1
1008.2.bt.b 4 28.g odd 6 1
1008.2.bt.b 4 84.j odd 6 1
1008.2.bt.b 4 84.n even 6 1
1575.2.bc.a 8 35.k even 12 2
1575.2.bc.a 8 35.l odd 12 2
1575.2.bc.a 8 105.w odd 12 2
1575.2.bc.a 8 105.x even 12 2
1575.2.bk.c 4 35.i odd 6 1
1575.2.bk.c 4 35.j even 6 1
1575.2.bk.c 4 105.o odd 6 1
1575.2.bk.c 4 105.p even 6 1
7056.2.k.b 4 4.b odd 2 1
7056.2.k.b 4 12.b even 2 1
7056.2.k.b 4 28.d even 2 1
7056.2.k.b 4 84.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 2 \) acting on \(S_{2}^{\mathrm{new}}(441, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 2)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} - 6)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} + 2)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 27)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 24)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 3)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 32)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 8)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 3)^{2} \) Copy content Toggle raw display
$37$ \( (T + 1)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} - 54)^{2} \) Copy content Toggle raw display
$43$ \( (T + 1)^{4} \) Copy content Toggle raw display
$47$ \( (T^{2} - 150)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 8)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 24)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$67$ \( (T - 11)^{4} \) Copy content Toggle raw display
$71$ \( (T^{2} + 50)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 3)^{2} \) Copy content Toggle raw display
$79$ \( (T - 5)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} - 54)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 24)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 108)^{2} \) Copy content Toggle raw display
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