Properties

Label 441.3.j.a
Level $441$
Weight $3$
Character orbit 441.j
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(263,441)]
 
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([1, 4]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("441.263");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 441.j (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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 9)
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.

\(f(q)\) \(=\) \( q + ( - 2 \zeta_{6} + 1) q^{2} + (3 \zeta_{6} - 3) q^{3} + q^{4} + (2 \zeta_{6} - 4) q^{5} + (3 \zeta_{6} + 3) q^{6} + ( - 10 \zeta_{6} + 5) q^{8} - 9 \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - 2 \zeta_{6} + 1) q^{2} + (3 \zeta_{6} - 3) q^{3} + q^{4} + (2 \zeta_{6} - 4) q^{5} + (3 \zeta_{6} + 3) q^{6} + ( - 10 \zeta_{6} + 5) q^{8} - 9 \zeta_{6} q^{9} + 6 \zeta_{6} q^{10} + (\zeta_{6} + 1) q^{11} + (3 \zeta_{6} - 3) q^{12} + ( - 4 \zeta_{6} + 4) q^{13} + ( - 12 \zeta_{6} + 6) q^{15} - 11 q^{16} + ( - 9 \zeta_{6} + 18) q^{17} + (9 \zeta_{6} - 18) q^{18} + (11 \zeta_{6} - 11) q^{19} + (2 \zeta_{6} - 4) q^{20} + ( - 3 \zeta_{6} + 3) q^{22} + ( - 16 \zeta_{6} + 32) q^{23} + (15 \zeta_{6} + 15) q^{24} + (13 \zeta_{6} - 13) q^{25} + ( - 4 \zeta_{6} - 4) q^{26} + 27 q^{27} + ( - 26 \zeta_{6} + 52) q^{29} - 18 q^{30} + 32 q^{31} + ( - 18 \zeta_{6} + 9) q^{32} + (3 \zeta_{6} - 6) q^{33} - 27 \zeta_{6} q^{34} - 9 \zeta_{6} q^{36} + ( - 34 \zeta_{6} + 34) q^{37} + (11 \zeta_{6} + 11) q^{38} + 12 \zeta_{6} q^{39} + 30 \zeta_{6} q^{40} + ( - 7 \zeta_{6} - 7) q^{41} + 61 \zeta_{6} q^{43} + (\zeta_{6} + 1) q^{44} + (18 \zeta_{6} + 18) q^{45} - 48 \zeta_{6} q^{46} + ( - 56 \zeta_{6} + 28) q^{47} + ( - 33 \zeta_{6} + 33) q^{48} + (13 \zeta_{6} + 13) q^{50} + (54 \zeta_{6} - 27) q^{51} + ( - 4 \zeta_{6} + 4) q^{52} + ( - 54 \zeta_{6} + 27) q^{54} - 6 q^{55} - 33 \zeta_{6} q^{57} - 78 \zeta_{6} q^{58} + ( - 58 \zeta_{6} + 29) q^{59} + ( - 12 \zeta_{6} + 6) q^{60} + 56 q^{61} + ( - 64 \zeta_{6} + 32) q^{62} - 71 q^{64} + (16 \zeta_{6} - 8) q^{65} + 9 \zeta_{6} q^{66} - 31 q^{67} + ( - 9 \zeta_{6} + 18) q^{68} + (96 \zeta_{6} - 48) q^{69} + ( - 36 \zeta_{6} + 18) q^{71} + (45 \zeta_{6} - 90) q^{72} - 65 \zeta_{6} q^{73} + ( - 34 \zeta_{6} - 34) q^{74} - 39 \zeta_{6} q^{75} + (11 \zeta_{6} - 11) q^{76} + ( - 12 \zeta_{6} + 24) q^{78} + 38 q^{79} + ( - 22 \zeta_{6} + 44) q^{80} + (81 \zeta_{6} - 81) q^{81} + (21 \zeta_{6} - 21) q^{82} + (28 \zeta_{6} - 56) q^{83} + (54 \zeta_{6} - 54) q^{85} + ( - 61 \zeta_{6} + 122) q^{86} + (156 \zeta_{6} - 78) q^{87} + ( - 15 \zeta_{6} + 15) q^{88} + ( - 72 \zeta_{6} - 72) q^{89} + ( - 54 \zeta_{6} + 54) q^{90} + ( - 16 \zeta_{6} + 32) q^{92} + (96 \zeta_{6} - 96) q^{93} - 84 q^{94} + ( - 44 \zeta_{6} + 22) q^{95} + (27 \zeta_{6} + 27) q^{96} + 115 \zeta_{6} q^{97} + ( - 18 \zeta_{6} + 9) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{3} + 2 q^{4} - 6 q^{5} + 9 q^{6} - 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 3 q^{3} + 2 q^{4} - 6 q^{5} + 9 q^{6} - 9 q^{9} + 6 q^{10} + 3 q^{11} - 3 q^{12} + 4 q^{13} - 22 q^{16} + 27 q^{17} - 27 q^{18} - 11 q^{19} - 6 q^{20} + 3 q^{22} + 48 q^{23} + 45 q^{24} - 13 q^{25} - 12 q^{26} + 54 q^{27} + 78 q^{29} - 36 q^{30} + 64 q^{31} - 9 q^{33} - 27 q^{34} - 9 q^{36} + 34 q^{37} + 33 q^{38} + 12 q^{39} + 30 q^{40} - 21 q^{41} + 61 q^{43} + 3 q^{44} + 54 q^{45} - 48 q^{46} + 33 q^{48} + 39 q^{50} + 4 q^{52} - 12 q^{55} - 33 q^{57} - 78 q^{58} + 112 q^{61} - 142 q^{64} + 9 q^{66} - 62 q^{67} + 27 q^{68} - 135 q^{72} - 65 q^{73} - 102 q^{74} - 39 q^{75} - 11 q^{76} + 36 q^{78} + 76 q^{79} + 66 q^{80} - 81 q^{81} - 21 q^{82} - 84 q^{83} - 54 q^{85} + 183 q^{86} + 15 q^{88} - 216 q^{89} + 54 q^{90} + 48 q^{92} - 96 q^{93} - 168 q^{94} + 81 q^{96} + 115 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
263.1
0.500000 0.866025i
0.500000 + 0.866025i
1.73205i −1.50000 2.59808i 1.00000 −3.00000 1.73205i 4.50000 2.59808i 0 8.66025i −4.50000 + 7.79423i 3.00000 5.19615i
275.1 1.73205i −1.50000 + 2.59808i 1.00000 −3.00000 + 1.73205i 4.50000 + 2.59808i 0 8.66025i −4.50000 7.79423i 3.00000 + 5.19615i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.j 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.j.a 2
7.b odd 2 1 441.3.j.b 2
7.c even 3 1 9.3.d.a 2
7.c even 3 1 441.3.n.b 2
7.d odd 6 1 441.3.n.a 2
7.d odd 6 1 441.3.r.a 2
9.d odd 6 1 441.3.n.b 2
21.h odd 6 1 27.3.d.a 2
28.g odd 6 1 144.3.q.a 2
35.j even 6 1 225.3.j.a 2
35.l odd 12 2 225.3.i.a 4
56.k odd 6 1 576.3.q.a 2
56.p even 6 1 576.3.q.b 2
63.g even 3 1 27.3.d.a 2
63.h even 3 1 81.3.b.a 2
63.i even 6 1 441.3.j.b 2
63.j odd 6 1 81.3.b.a 2
63.j odd 6 1 inner 441.3.j.a 2
63.n odd 6 1 9.3.d.a 2
63.o even 6 1 441.3.n.a 2
63.s even 6 1 441.3.r.a 2
84.n even 6 1 432.3.q.a 2
105.o odd 6 1 675.3.j.a 2
105.x even 12 2 675.3.i.a 4
168.s odd 6 1 1728.3.q.a 2
168.v even 6 1 1728.3.q.b 2
252.o even 6 1 144.3.q.a 2
252.u odd 6 1 1296.3.e.a 2
252.bb even 6 1 1296.3.e.a 2
252.bl odd 6 1 432.3.q.a 2
315.v odd 6 1 225.3.j.a 2
315.bo even 6 1 675.3.j.a 2
315.bx even 12 2 225.3.i.a 4
315.ch odd 12 2 675.3.i.a 4
504.w even 6 1 1728.3.q.a 2
504.ba odd 6 1 1728.3.q.b 2
504.cy even 6 1 576.3.q.a 2
504.db odd 6 1 576.3.q.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9.3.d.a 2 7.c even 3 1
9.3.d.a 2 63.n odd 6 1
27.3.d.a 2 21.h odd 6 1
27.3.d.a 2 63.g even 3 1
81.3.b.a 2 63.h even 3 1
81.3.b.a 2 63.j odd 6 1
144.3.q.a 2 28.g odd 6 1
144.3.q.a 2 252.o even 6 1
225.3.i.a 4 35.l odd 12 2
225.3.i.a 4 315.bx even 12 2
225.3.j.a 2 35.j even 6 1
225.3.j.a 2 315.v odd 6 1
432.3.q.a 2 84.n even 6 1
432.3.q.a 2 252.bl odd 6 1
441.3.j.a 2 1.a even 1 1 trivial
441.3.j.a 2 63.j odd 6 1 inner
441.3.j.b 2 7.b odd 2 1
441.3.j.b 2 63.i even 6 1
441.3.n.a 2 7.d odd 6 1
441.3.n.a 2 63.o even 6 1
441.3.n.b 2 7.c even 3 1
441.3.n.b 2 9.d odd 6 1
441.3.r.a 2 7.d odd 6 1
441.3.r.a 2 63.s even 6 1
576.3.q.a 2 56.k odd 6 1
576.3.q.a 2 504.cy even 6 1
576.3.q.b 2 56.p even 6 1
576.3.q.b 2 504.db odd 6 1
675.3.i.a 4 105.x even 12 2
675.3.i.a 4 315.ch odd 12 2
675.3.j.a 2 105.o odd 6 1
675.3.j.a 2 315.bo even 6 1
1296.3.e.a 2 252.u odd 6 1
1296.3.e.a 2 252.bb even 6 1
1728.3.q.a 2 168.s odd 6 1
1728.3.q.a 2 504.w even 6 1
1728.3.q.b 2 168.v even 6 1
1728.3.q.b 2 504.ba odd 6 1

Hecke kernels

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

\( T_{2}^{2} + 3 \) Copy content Toggle raw display
\( T_{5}^{2} + 6T_{5} + 12 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 3 \) Copy content Toggle raw display
$3$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$5$ \( T^{2} + 6T + 12 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 3T + 3 \) Copy content Toggle raw display
$13$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$17$ \( T^{2} - 27T + 243 \) Copy content Toggle raw display
$19$ \( T^{2} + 11T + 121 \) Copy content Toggle raw display
$23$ \( T^{2} - 48T + 768 \) Copy content Toggle raw display
$29$ \( T^{2} - 78T + 2028 \) Copy content Toggle raw display
$31$ \( (T - 32)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 34T + 1156 \) Copy content Toggle raw display
$41$ \( T^{2} + 21T + 147 \) Copy content Toggle raw display
$43$ \( T^{2} - 61T + 3721 \) Copy content Toggle raw display
$47$ \( T^{2} + 2352 \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 2523 \) Copy content Toggle raw display
$61$ \( (T - 56)^{2} \) Copy content Toggle raw display
$67$ \( (T + 31)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 972 \) Copy content Toggle raw display
$73$ \( T^{2} + 65T + 4225 \) Copy content Toggle raw display
$79$ \( (T - 38)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 84T + 2352 \) Copy content Toggle raw display
$89$ \( T^{2} + 216T + 15552 \) Copy content Toggle raw display
$97$ \( T^{2} - 115T + 13225 \) Copy content Toggle raw display
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