Properties

Label 441.3.m.d
Level $441$
Weight $3$
Character orbit 441.m
Analytic conductor $12.016$
Analytic rank $0$
Dimension $2$
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] = [441,3,Mod(19,441)]
 
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, 5]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("441.19");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 441.m (of order \(6\), 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: \(12.0163796583\)
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 21)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{6} q^{2} + ( - 3 \zeta_{6} + 3) q^{4} + (4 \zeta_{6} - 8) q^{5} + 7 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{6} q^{2} + ( - 3 \zeta_{6} + 3) q^{4} + (4 \zeta_{6} - 8) q^{5} + 7 q^{8} + ( - 4 \zeta_{6} - 4) q^{10} + ( - 10 \zeta_{6} + 10) q^{11} + ( - 8 \zeta_{6} + 4) q^{13} - 5 \zeta_{6} q^{16} + ( - 12 \zeta_{6} + 24) q^{19} + (24 \zeta_{6} - 12) q^{20} + 10 q^{22} - 14 \zeta_{6} q^{23} + ( - 23 \zeta_{6} + 23) q^{25} + ( - 4 \zeta_{6} + 8) q^{26} + 38 q^{29} + (16 \zeta_{6} + 16) q^{31} + ( - 33 \zeta_{6} + 33) q^{32} - 26 \zeta_{6} q^{37} + (12 \zeta_{6} + 12) q^{38} + (28 \zeta_{6} - 56) q^{40} + ( - 80 \zeta_{6} + 40) q^{41} + 26 q^{43} - 30 \zeta_{6} q^{44} + ( - 14 \zeta_{6} + 14) q^{46} + ( - 16 \zeta_{6} + 32) q^{47} + 23 q^{50} + ( - 12 \zeta_{6} - 12) q^{52} + ( - 10 \zeta_{6} + 10) q^{53} + (80 \zeta_{6} - 40) q^{55} + 38 \zeta_{6} q^{58} + ( - 44 \zeta_{6} - 44) q^{59} + (20 \zeta_{6} - 40) q^{61} + (32 \zeta_{6} - 16) q^{62} + 13 q^{64} + 48 \zeta_{6} q^{65} + (74 \zeta_{6} - 74) q^{67} + 62 q^{71} + ( - 24 \zeta_{6} - 24) q^{73} + ( - 26 \zeta_{6} + 26) q^{74} + ( - 72 \zeta_{6} + 36) q^{76} + 46 \zeta_{6} q^{79} + (20 \zeta_{6} + 20) q^{80} + ( - 40 \zeta_{6} + 80) q^{82} + (104 \zeta_{6} - 52) q^{83} + 26 \zeta_{6} q^{86} + ( - 70 \zeta_{6} + 70) q^{88} + (24 \zeta_{6} - 48) q^{89} - 42 q^{92} + (16 \zeta_{6} + 16) q^{94} + (144 \zeta_{6} - 144) q^{95} + ( - 64 \zeta_{6} + 32) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} + 3 q^{4} - 12 q^{5} + 14 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} + 3 q^{4} - 12 q^{5} + 14 q^{8} - 12 q^{10} + 10 q^{11} - 5 q^{16} + 36 q^{19} + 20 q^{22} - 14 q^{23} + 23 q^{25} + 12 q^{26} + 76 q^{29} + 48 q^{31} + 33 q^{32} - 26 q^{37} + 36 q^{38} - 84 q^{40} + 52 q^{43} - 30 q^{44} + 14 q^{46} + 48 q^{47} + 46 q^{50} - 36 q^{52} + 10 q^{53} + 38 q^{58} - 132 q^{59} - 60 q^{61} + 26 q^{64} + 48 q^{65} - 74 q^{67} + 124 q^{71} - 72 q^{73} + 26 q^{74} + 46 q^{79} + 60 q^{80} + 120 q^{82} + 26 q^{86} + 70 q^{88} - 72 q^{89} - 84 q^{92} + 48 q^{94} - 144 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
0.500000 0.866025i
0.500000 + 0.866025i
0.500000 0.866025i 0 1.50000 + 2.59808i −6.00000 3.46410i 0 0 7.00000 0 −6.00000 + 3.46410i
325.1 0.500000 + 0.866025i 0 1.50000 2.59808i −6.00000 + 3.46410i 0 0 7.00000 0 −6.00000 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
7.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.3.m.d 2
3.b odd 2 1 147.3.f.d 2
7.b odd 2 1 441.3.m.f 2
7.c even 3 1 63.3.d.b 2
7.c even 3 1 441.3.m.f 2
7.d odd 6 1 63.3.d.b 2
7.d odd 6 1 inner 441.3.m.d 2
21.c even 2 1 147.3.f.b 2
21.g even 6 1 21.3.d.a 2
21.g even 6 1 147.3.f.d 2
21.h odd 6 1 21.3.d.a 2
21.h odd 6 1 147.3.f.b 2
28.f even 6 1 1008.3.f.d 2
28.g odd 6 1 1008.3.f.d 2
84.j odd 6 1 336.3.f.a 2
84.n even 6 1 336.3.f.a 2
105.o odd 6 1 525.3.h.a 2
105.p even 6 1 525.3.h.a 2
105.w odd 12 2 525.3.e.a 4
105.x even 12 2 525.3.e.a 4
168.s odd 6 1 1344.3.f.c 2
168.v even 6 1 1344.3.f.b 2
168.ba even 6 1 1344.3.f.c 2
168.be odd 6 1 1344.3.f.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.3.d.a 2 21.g even 6 1
21.3.d.a 2 21.h odd 6 1
63.3.d.b 2 7.c even 3 1
63.3.d.b 2 7.d odd 6 1
147.3.f.b 2 21.c even 2 1
147.3.f.b 2 21.h odd 6 1
147.3.f.d 2 3.b odd 2 1
147.3.f.d 2 21.g even 6 1
336.3.f.a 2 84.j odd 6 1
336.3.f.a 2 84.n even 6 1
441.3.m.d 2 1.a even 1 1 trivial
441.3.m.d 2 7.d odd 6 1 inner
441.3.m.f 2 7.b odd 2 1
441.3.m.f 2 7.c even 3 1
525.3.e.a 4 105.w odd 12 2
525.3.e.a 4 105.x even 12 2
525.3.h.a 2 105.o odd 6 1
525.3.h.a 2 105.p even 6 1
1008.3.f.d 2 28.f even 6 1
1008.3.f.d 2 28.g odd 6 1
1344.3.f.b 2 168.v even 6 1
1344.3.f.b 2 168.be odd 6 1
1344.3.f.c 2 168.s odd 6 1
1344.3.f.c 2 168.ba even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(441, [\chi])\):

\( T_{2}^{2} - T_{2} + 1 \) Copy content Toggle raw display
\( T_{5}^{2} + 12T_{5} + 48 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 12T + 48 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 10T + 100 \) Copy content Toggle raw display
$13$ \( T^{2} + 48 \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} - 36T + 432 \) Copy content Toggle raw display
$23$ \( T^{2} + 14T + 196 \) Copy content Toggle raw display
$29$ \( (T - 38)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 48T + 768 \) Copy content Toggle raw display
$37$ \( T^{2} + 26T + 676 \) Copy content Toggle raw display
$41$ \( T^{2} + 4800 \) Copy content Toggle raw display
$43$ \( (T - 26)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 48T + 768 \) Copy content Toggle raw display
$53$ \( T^{2} - 10T + 100 \) Copy content Toggle raw display
$59$ \( T^{2} + 132T + 5808 \) Copy content Toggle raw display
$61$ \( T^{2} + 60T + 1200 \) Copy content Toggle raw display
$67$ \( T^{2} + 74T + 5476 \) Copy content Toggle raw display
$71$ \( (T - 62)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 72T + 1728 \) Copy content Toggle raw display
$79$ \( T^{2} - 46T + 2116 \) Copy content Toggle raw display
$83$ \( T^{2} + 8112 \) Copy content Toggle raw display
$89$ \( T^{2} + 72T + 1728 \) Copy content Toggle raw display
$97$ \( T^{2} + 3072 \) Copy content Toggle raw display
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