Properties

Label 441.3.d.c
Level $441$
Weight $3$
Character orbit 441.d
Analytic conductor $12.016$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 441.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(12.0163796583\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2\cdot 7 \)
Twist minimal: no (minimal twist has level 63)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q -4 q^{4} +O(q^{10})\) \( q -4 q^{4} + ( 7 - 14 \zeta_{6} ) q^{13} + 16 q^{16} + ( -21 + 42 \zeta_{6} ) q^{19} + 25 q^{25} + ( -35 + 70 \zeta_{6} ) q^{31} + 73 q^{37} + 61 q^{43} + ( -28 + 56 \zeta_{6} ) q^{52} + ( -56 + 112 \zeta_{6} ) q^{61} -64 q^{64} -13 q^{67} + ( 63 - 126 \zeta_{6} ) q^{73} + ( 84 - 168 \zeta_{6} ) q^{76} + 11 q^{79} + ( -112 + 224 \zeta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 8q^{4} + O(q^{10}) \) \( 2q - 8q^{4} + 32q^{16} + 50q^{25} + 146q^{37} + 122q^{43} - 128q^{64} - 26q^{67} + 22q^{79} + O(q^{100}) \)

Character values

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

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

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
244.1
0.500000 + 0.866025i
0.500000 0.866025i
0 0 −4.00000 0 0 0 0 0 0
244.2 0 0 −4.00000 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.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 441.3.d.c 2
3.b odd 2 1 CM 441.3.d.c 2
7.b odd 2 1 inner 441.3.d.c 2
7.c even 3 1 63.3.m.b 2
7.c even 3 1 441.3.m.c 2
7.d odd 6 1 63.3.m.b 2
7.d odd 6 1 441.3.m.c 2
21.c even 2 1 inner 441.3.d.c 2
21.g even 6 1 63.3.m.b 2
21.g even 6 1 441.3.m.c 2
21.h odd 6 1 63.3.m.b 2
21.h odd 6 1 441.3.m.c 2
28.f even 6 1 1008.3.cg.d 2
28.g odd 6 1 1008.3.cg.d 2
84.j odd 6 1 1008.3.cg.d 2
84.n even 6 1 1008.3.cg.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.3.m.b 2 7.c even 3 1
63.3.m.b 2 7.d odd 6 1
63.3.m.b 2 21.g even 6 1
63.3.m.b 2 21.h odd 6 1
441.3.d.c 2 1.a even 1 1 trivial
441.3.d.c 2 3.b odd 2 1 CM
441.3.d.c 2 7.b odd 2 1 inner
441.3.d.c 2 21.c even 2 1 inner
441.3.m.c 2 7.c even 3 1
441.3.m.c 2 7.d odd 6 1
441.3.m.c 2 21.g even 6 1
441.3.m.c 2 21.h odd 6 1
1008.3.cg.d 2 28.f even 6 1
1008.3.cg.d 2 28.g odd 6 1
1008.3.cg.d 2 84.j odd 6 1
1008.3.cg.d 2 84.n even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{3}^{\mathrm{new}}(441, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( T^{2} \)
$11$ \( T^{2} \)
$13$ \( 147 + T^{2} \)
$17$ \( T^{2} \)
$19$ \( 1323 + T^{2} \)
$23$ \( T^{2} \)
$29$ \( T^{2} \)
$31$ \( 3675 + T^{2} \)
$37$ \( ( -73 + T )^{2} \)
$41$ \( T^{2} \)
$43$ \( ( -61 + T )^{2} \)
$47$ \( T^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( 9408 + T^{2} \)
$67$ \( ( 13 + T )^{2} \)
$71$ \( T^{2} \)
$73$ \( 11907 + T^{2} \)
$79$ \( ( -11 + T )^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( 37632 + T^{2} \)
show more
show less