Properties

Label 441.2.e.c
Level 441
Weight 2
Character orbit 441.e
Analytic conductor 3.521
Analytic rank 0
Dimension 2
CM discriminant -3
Inner twists 4

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Newspace parameters

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

Self dual: no
Analytic conductor: \(3.52140272914\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 63)
Sato-Tate group: $\mathrm{U}(1)[D_{3}]$

$q$-expansion

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.

\(f(q)\) \(=\) \( q + ( 2 - 2 \zeta_{6} ) q^{4} +O(q^{10})\) \( q + ( 2 - 2 \zeta_{6} ) q^{4} + 7 q^{13} -4 \zeta_{6} q^{16} -7 \zeta_{6} q^{19} + ( 5 - 5 \zeta_{6} ) q^{25} + ( -7 + 7 \zeta_{6} ) q^{31} + \zeta_{6} q^{37} + 5 q^{43} + ( 14 - 14 \zeta_{6} ) q^{52} + 14 \zeta_{6} q^{61} -8 q^{64} + ( -11 + 11 \zeta_{6} ) q^{67} + ( -7 + 7 \zeta_{6} ) q^{73} -14 q^{76} + 13 \zeta_{6} q^{79} -14 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{4} + O(q^{10}) \) \( 2q + 2q^{4} + 14q^{13} - 4q^{16} - 7q^{19} + 5q^{25} - 7q^{31} + q^{37} + 10q^{43} + 14q^{52} + 14q^{61} - 16q^{64} - 11q^{67} - 7q^{73} - 28q^{76} + 13q^{79} - 28q^{97} + O(q^{100}) \)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
226.1
0.500000 0.866025i
0.500000 + 0.866025i
0 0 1.00000 + 1.73205i 0 0 0 0 0 0
361.1 0 0 1.00000 1.73205i 0 0 0 0 0 0
\(n\): 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 CM by \(\Q(\sqrt{-3}) \)
7.c even 3 1 inner
21.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.2.e.c 2
3.b odd 2 1 CM 441.2.e.c 2
7.b odd 2 1 63.2.e.a 2
7.c even 3 1 441.2.a.e 1
7.c even 3 1 inner 441.2.e.c 2
7.d odd 6 1 63.2.e.a 2
7.d odd 6 1 441.2.a.d 1
21.c even 2 1 63.2.e.a 2
21.g even 6 1 63.2.e.a 2
21.g even 6 1 441.2.a.d 1
21.h odd 6 1 441.2.a.e 1
21.h odd 6 1 inner 441.2.e.c 2
28.d even 2 1 1008.2.s.j 2
28.f even 6 1 1008.2.s.j 2
28.f even 6 1 7056.2.a.y 1
28.g odd 6 1 7056.2.a.bf 1
63.i even 6 1 567.2.g.d 2
63.k odd 6 1 567.2.h.c 2
63.l odd 6 1 567.2.g.d 2
63.l odd 6 1 567.2.h.c 2
63.o even 6 1 567.2.g.d 2
63.o even 6 1 567.2.h.c 2
63.s even 6 1 567.2.h.c 2
63.t odd 6 1 567.2.g.d 2
84.h odd 2 1 1008.2.s.j 2
84.j odd 6 1 1008.2.s.j 2
84.j odd 6 1 7056.2.a.y 1
84.n even 6 1 7056.2.a.bf 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.e.a 2 7.b odd 2 1
63.2.e.a 2 7.d odd 6 1
63.2.e.a 2 21.c even 2 1
63.2.e.a 2 21.g even 6 1
441.2.a.d 1 7.d odd 6 1
441.2.a.d 1 21.g even 6 1
441.2.a.e 1 7.c even 3 1
441.2.a.e 1 21.h odd 6 1
441.2.e.c 2 1.a even 1 1 trivial
441.2.e.c 2 3.b odd 2 1 CM
441.2.e.c 2 7.c even 3 1 inner
441.2.e.c 2 21.h odd 6 1 inner
567.2.g.d 2 63.i even 6 1
567.2.g.d 2 63.l odd 6 1
567.2.g.d 2 63.o even 6 1
567.2.g.d 2 63.t odd 6 1
567.2.h.c 2 63.k odd 6 1
567.2.h.c 2 63.l odd 6 1
567.2.h.c 2 63.o even 6 1
567.2.h.c 2 63.s even 6 1
1008.2.s.j 2 28.d even 2 1
1008.2.s.j 2 28.f even 6 1
1008.2.s.j 2 84.h odd 2 1
1008.2.s.j 2 84.j odd 6 1
7056.2.a.y 1 28.f even 6 1
7056.2.a.y 1 84.j odd 6 1
7056.2.a.bf 1 28.g odd 6 1
7056.2.a.bf 1 84.n even 6 1

Hecke kernels

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

\( T_{2} \)
\( T_{5} \)
\( T_{13} - 7 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 2 T^{2} + 4 T^{4} \)
$3$ 1
$5$ \( 1 - 5 T^{2} + 25 T^{4} \)
$7$ 1
$11$ \( 1 - 11 T^{2} + 121 T^{4} \)
$13$ \( ( 1 - 7 T + 13 T^{2} )^{2} \)
$17$ \( 1 - 17 T^{2} + 289 T^{4} \)
$19$ \( ( 1 - T + 19 T^{2} )( 1 + 8 T + 19 T^{2} ) \)
$23$ \( 1 - 23 T^{2} + 529 T^{4} \)
$29$ \( ( 1 + 29 T^{2} )^{2} \)
$31$ \( ( 1 - 4 T + 31 T^{2} )( 1 + 11 T + 31 T^{2} ) \)
$37$ \( ( 1 - 11 T + 37 T^{2} )( 1 + 10 T + 37 T^{2} ) \)
$41$ \( ( 1 + 41 T^{2} )^{2} \)
$43$ \( ( 1 - 5 T + 43 T^{2} )^{2} \)
$47$ \( 1 - 47 T^{2} + 2209 T^{4} \)
$53$ \( 1 - 53 T^{2} + 2809 T^{4} \)
$59$ \( 1 - 59 T^{2} + 3481 T^{4} \)
$61$ \( ( 1 - 13 T + 61 T^{2} )( 1 - T + 61 T^{2} ) \)
$67$ \( ( 1 - 5 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} ) \)
$71$ \( ( 1 + 71 T^{2} )^{2} \)
$73$ \( ( 1 - 10 T + 73 T^{2} )( 1 + 17 T + 73 T^{2} ) \)
$79$ \( ( 1 - 17 T + 79 T^{2} )( 1 + 4 T + 79 T^{2} ) \)
$83$ \( ( 1 + 83 T^{2} )^{2} \)
$89$ \( 1 - 89 T^{2} + 7921 T^{4} \)
$97$ \( ( 1 + 14 T + 97 T^{2} )^{2} \)
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