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Properties

Label 441.2.g.a
Level $441$
Weight $2$
Character orbit 441.g
Analytic conductor $3.521$
Analytic rank $0$
Dimension $2$
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(67,441)]
 
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([2, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("441.67");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 441.g (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 63)
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.

\(f(q)\) \(=\) \( q + (\zeta_{6} - 1) q^{2} + ( - \zeta_{6} + 2) q^{3} + \zeta_{6} q^{4} + q^{5} + (2 \zeta_{6} - 1) q^{6} - 3 q^{8} + ( - 3 \zeta_{6} + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{6} - 1) q^{2} + ( - \zeta_{6} + 2) q^{3} + \zeta_{6} q^{4} + q^{5} + (2 \zeta_{6} - 1) q^{6} - 3 q^{8} + ( - 3 \zeta_{6} + 3) q^{9} + (\zeta_{6} - 1) q^{10} + 5 q^{11} + (\zeta_{6} + 1) q^{12} + (5 \zeta_{6} - 5) q^{13} + ( - \zeta_{6} + 2) q^{15} + ( - \zeta_{6} + 1) q^{16} + ( - 3 \zeta_{6} + 3) q^{17} + 3 \zeta_{6} q^{18} + \zeta_{6} q^{19} + \zeta_{6} q^{20} + (5 \zeta_{6} - 5) q^{22} + 3 q^{23} + (3 \zeta_{6} - 6) q^{24} - 4 q^{25} - 5 \zeta_{6} q^{26} + ( - 6 \zeta_{6} + 3) q^{27} + \zeta_{6} q^{29} + (2 \zeta_{6} - 1) q^{30} - 5 \zeta_{6} q^{32} + ( - 5 \zeta_{6} + 10) q^{33} + 3 \zeta_{6} q^{34} + 3 q^{36} - 3 \zeta_{6} q^{37} - q^{38} + (10 \zeta_{6} - 5) q^{39} - 3 q^{40} + (5 \zeta_{6} - 5) q^{41} + \zeta_{6} q^{43} + 5 \zeta_{6} q^{44} + ( - 3 \zeta_{6} + 3) q^{45} + (3 \zeta_{6} - 3) q^{46} + ( - 2 \zeta_{6} + 1) q^{48} + ( - 4 \zeta_{6} + 4) q^{50} + ( - 6 \zeta_{6} + 3) q^{51} - 5 q^{52} + ( - 9 \zeta_{6} + 9) q^{53} + (3 \zeta_{6} + 3) q^{54} + 5 q^{55} + (\zeta_{6} + 1) q^{57} - q^{58} + (\zeta_{6} + 1) q^{60} + (14 \zeta_{6} - 14) q^{61} + 7 q^{64} + (5 \zeta_{6} - 5) q^{65} + (10 \zeta_{6} - 5) q^{66} - 4 \zeta_{6} q^{67} + 3 q^{68} + ( - 3 \zeta_{6} + 6) q^{69} - 12 q^{71} + (9 \zeta_{6} - 9) q^{72} + ( - 3 \zeta_{6} + 3) q^{73} + 3 q^{74} + (4 \zeta_{6} - 8) q^{75} + (\zeta_{6} - 1) q^{76} + ( - 5 \zeta_{6} - 5) q^{78} + (8 \zeta_{6} - 8) q^{79} + ( - \zeta_{6} + 1) q^{80} - 9 \zeta_{6} q^{81} - 5 \zeta_{6} q^{82} - 9 \zeta_{6} q^{83} + ( - 3 \zeta_{6} + 3) q^{85} - q^{86} + (\zeta_{6} + 1) q^{87} - 15 q^{88} - 13 \zeta_{6} q^{89} + 3 \zeta_{6} q^{90} + 3 \zeta_{6} q^{92} + \zeta_{6} q^{95} + ( - 5 \zeta_{6} - 5) q^{96} - 9 \zeta_{6} q^{97} + ( - 15 \zeta_{6} + 15) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} + 3 q^{3} + q^{4} + 2 q^{5} - 6 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} + 3 q^{3} + q^{4} + 2 q^{5} - 6 q^{8} + 3 q^{9} - q^{10} + 10 q^{11} + 3 q^{12} - 5 q^{13} + 3 q^{15} + q^{16} + 3 q^{17} + 3 q^{18} + q^{19} + q^{20} - 5 q^{22} + 6 q^{23} - 9 q^{24} - 8 q^{25} - 5 q^{26} + q^{29} - 5 q^{32} + 15 q^{33} + 3 q^{34} + 6 q^{36} - 3 q^{37} - 2 q^{38} - 6 q^{40} - 5 q^{41} + q^{43} + 5 q^{44} + 3 q^{45} - 3 q^{46} + 4 q^{50} - 10 q^{52} + 9 q^{53} + 9 q^{54} + 10 q^{55} + 3 q^{57} - 2 q^{58} + 3 q^{60} - 14 q^{61} + 14 q^{64} - 5 q^{65} - 4 q^{67} + 6 q^{68} + 9 q^{69} - 24 q^{71} - 9 q^{72} + 3 q^{73} + 6 q^{74} - 12 q^{75} - q^{76} - 15 q^{78} - 8 q^{79} + q^{80} - 9 q^{81} - 5 q^{82} - 9 q^{83} + 3 q^{85} - 2 q^{86} + 3 q^{87} - 30 q^{88} - 13 q^{89} + 3 q^{90} + 3 q^{92} + q^{95} - 15 q^{96} - 9 q^{97} + 15 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
67.1
0.500000 + 0.866025i
0.500000 − 0.866025i
−0.500000 + 0.866025i 1.50000 − 0.866025i 0.500000 + 0.866025i 1.00000 1.73205i 0 −3.00000 1.50000 − 2.59808i −0.500000 + 0.866025i
79.1 −0.500000 − 0.866025i 1.50000 + 0.866025i 0.500000 − 0.866025i 1.00000 − 1.73205i 0 −3.00000 1.50000 + 2.59808i −0.500000 − 0.866025i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.g 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.g.a 2
3.b odd 2 1 1323.2.g.a 2
7.b odd 2 1 63.2.g.a ✓ 2
7.c even 3 1 441.2.f.a 2
7.c even 3 1 441.2.h.a 2
7.d odd 6 1 63.2.h.a yes 2
7.d odd 6 1 441.2.f.b 2
9.c even 3 1 441.2.h.a 2
9.d odd 6 1 1323.2.h.a 2
21.c even 2 1 189.2.g.a 2
21.g even 6 1 189.2.h.a 2
21.g even 6 1 1323.2.f.a 2
21.h odd 6 1 1323.2.f.b 2
21.h odd 6 1 1323.2.h.a 2
28.d even 2 1 1008.2.t.d 2
28.f even 6 1 1008.2.q.c 2
63.g even 3 1 inner 441.2.g.a 2
63.g even 3 1 3969.2.a.f 1
63.h even 3 1 441.2.f.a 2
63.i even 6 1 567.2.e.b 2
63.i even 6 1 1323.2.f.a 2
63.j odd 6 1 1323.2.f.b 2
63.k odd 6 1 63.2.g.a ✓ 2
63.k odd 6 1 3969.2.a.d 1
63.l odd 6 1 63.2.h.a yes 2
63.l odd 6 1 567.2.e.a 2
63.n odd 6 1 1323.2.g.a 2
63.n odd 6 1 3969.2.a.a 1
63.o even 6 1 189.2.h.a 2
63.o even 6 1 567.2.e.b 2
63.s even 6 1 189.2.g.a 2
63.s even 6 1 3969.2.a.c 1
63.t odd 6 1 441.2.f.b 2
63.t odd 6 1 567.2.e.a 2
84.h odd 2 1 3024.2.t.d 2
84.j odd 6 1 3024.2.q.b 2
252.n even 6 1 1008.2.t.d 2
252.s odd 6 1 3024.2.q.b 2
252.bi even 6 1 1008.2.q.c 2
252.bn odd 6 1 3024.2.t.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.g.a ✓ 2 7.b odd 2 1
63.2.g.a ✓ 2 63.k odd 6 1
63.2.h.a yes 2 7.d odd 6 1
63.2.h.a yes 2 63.l odd 6 1
189.2.g.a 2 21.c even 2 1
189.2.g.a 2 63.s even 6 1
189.2.h.a 2 21.g even 6 1
189.2.h.a 2 63.o even 6 1
441.2.f.a 2 7.c even 3 1
441.2.f.a 2 63.h even 3 1
441.2.f.b 2 7.d odd 6 1
441.2.f.b 2 63.t odd 6 1
441.2.g.a 2 1.a even 1 1 trivial
441.2.g.a 2 63.g even 3 1 inner
441.2.h.a 2 7.c even 3 1
441.2.h.a 2 9.c even 3 1
567.2.e.a 2 63.l odd 6 1
567.2.e.a 2 63.t odd 6 1
567.2.e.b 2 63.i even 6 1
567.2.e.b 2 63.o even 6 1
1008.2.q.c 2 28.f even 6 1
1008.2.q.c 2 252.bi even 6 1
1008.2.t.d 2 28.d even 2 1
1008.2.t.d 2 252.n even 6 1
1323.2.f.a 2 21.g even 6 1
1323.2.f.a 2 63.i even 6 1
1323.2.f.b 2 21.h odd 6 1
1323.2.f.b 2 63.j odd 6 1
1323.2.g.a 2 3.b odd 2 1
1323.2.g.a 2 63.n odd 6 1
1323.2.h.a 2 9.d odd 6 1
1323.2.h.a 2 21.h odd 6 1
3024.2.q.b 2 84.j odd 6 1
3024.2.q.b 2 252.s odd 6 1
3024.2.t.d 2 84.h odd 2 1
3024.2.t.d 2 252.bn odd 6 1
3969.2.a.a 1 63.n odd 6 1
3969.2.a.c 1 63.s even 6 1
3969.2.a.d 1 63.k odd 6 1
3969.2.a.f 1 63.g even 3 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
\( T_{5} - 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} - 3T + 3 \) Copy content Toggle raw display
$5$ \( (T - 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( (T - 5)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$17$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$19$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$23$ \( (T - 3)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$41$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$43$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} - 9T + 81 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 14T + 196 \) Copy content Toggle raw display
$67$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$71$ \( (T + 12)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$79$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$83$ \( T^{2} + 9T + 81 \) Copy content Toggle raw display
$89$ \( T^{2} + 13T + 169 \) Copy content Toggle raw display
$97$ \( T^{2} + 9T + 81 \) Copy content Toggle raw display
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