Properties

Label 441.4.e.j
Level $441$
Weight $4$
Character orbit 441.e
Analytic conductor $26.020$
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: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 441.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(26.0198423125\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 9)
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.

\(f(q)\) \(=\) \( q + ( 8 - 8 \zeta_{6} ) q^{4} +O(q^{10})\) \( q + ( 8 - 8 \zeta_{6} ) q^{4} + 70 q^{13} -64 \zeta_{6} q^{16} + 56 \zeta_{6} q^{19} + ( 125 - 125 \zeta_{6} ) q^{25} + ( 308 - 308 \zeta_{6} ) q^{31} -110 \zeta_{6} q^{37} -520 q^{43} + ( 560 - 560 \zeta_{6} ) q^{52} + 182 \zeta_{6} q^{61} -512 q^{64} + ( 880 - 880 \zeta_{6} ) q^{67} + ( 1190 - 1190 \zeta_{6} ) q^{73} + 448 q^{76} -884 \zeta_{6} q^{79} + 1330 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 8q^{4} + O(q^{10}) \) \( 2q + 8q^{4} + 140q^{13} - 64q^{16} + 56q^{19} + 125q^{25} + 308q^{31} - 110q^{37} - 1040q^{43} + 560q^{52} + 182q^{61} - 1024q^{64} + 880q^{67} + 1190q^{73} + 896q^{76} - 884q^{79} + 2660q^{97} + O(q^{100}) \)

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 4.00000 + 6.92820i 0 0 0 0 0 0
361.1 0 0 4.00000 6.92820i 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 441.4.e.j 2
3.b odd 2 1 CM 441.4.e.j 2
7.b odd 2 1 441.4.e.i 2
7.c even 3 1 441.4.a.f 1
7.c even 3 1 inner 441.4.e.j 2
7.d odd 6 1 9.4.a.a 1
7.d odd 6 1 441.4.e.i 2
21.c even 2 1 441.4.e.i 2
21.g even 6 1 9.4.a.a 1
21.g even 6 1 441.4.e.i 2
21.h odd 6 1 441.4.a.f 1
21.h odd 6 1 inner 441.4.e.j 2
28.f even 6 1 144.4.a.d 1
35.i odd 6 1 225.4.a.d 1
35.k even 12 2 225.4.b.g 2
56.j odd 6 1 576.4.a.m 1
56.m even 6 1 576.4.a.l 1
63.i even 6 1 81.4.c.b 2
63.k odd 6 1 81.4.c.b 2
63.s even 6 1 81.4.c.b 2
63.t odd 6 1 81.4.c.b 2
77.i even 6 1 1089.4.a.g 1
84.j odd 6 1 144.4.a.d 1
91.s odd 6 1 1521.4.a.g 1
105.p even 6 1 225.4.a.d 1
105.w odd 12 2 225.4.b.g 2
168.ba even 6 1 576.4.a.m 1
168.be odd 6 1 576.4.a.l 1
231.k odd 6 1 1089.4.a.g 1
273.ba even 6 1 1521.4.a.g 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9.4.a.a 1 7.d odd 6 1
9.4.a.a 1 21.g even 6 1
81.4.c.b 2 63.i even 6 1
81.4.c.b 2 63.k odd 6 1
81.4.c.b 2 63.s even 6 1
81.4.c.b 2 63.t odd 6 1
144.4.a.d 1 28.f even 6 1
144.4.a.d 1 84.j odd 6 1
225.4.a.d 1 35.i odd 6 1
225.4.a.d 1 105.p even 6 1
225.4.b.g 2 35.k even 12 2
225.4.b.g 2 105.w odd 12 2
441.4.a.f 1 7.c even 3 1
441.4.a.f 1 21.h odd 6 1
441.4.e.i 2 7.b odd 2 1
441.4.e.i 2 7.d odd 6 1
441.4.e.i 2 21.c even 2 1
441.4.e.i 2 21.g even 6 1
441.4.e.j 2 1.a even 1 1 trivial
441.4.e.j 2 3.b odd 2 1 CM
441.4.e.j 2 7.c even 3 1 inner
441.4.e.j 2 21.h odd 6 1 inner
576.4.a.l 1 56.m even 6 1
576.4.a.l 1 168.be odd 6 1
576.4.a.m 1 56.j odd 6 1
576.4.a.m 1 168.ba even 6 1
1089.4.a.g 1 77.i even 6 1
1089.4.a.g 1 231.k odd 6 1
1521.4.a.g 1 91.s odd 6 1
1521.4.a.g 1 273.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 \(S_{4}^{\mathrm{new}}(441, [\chi])\):

\( T_{2} \)
\( T_{5} \)
\( T_{13} - 70 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( T^{2} \)
$11$ \( T^{2} \)
$13$ \( ( -70 + T )^{2} \)
$17$ \( T^{2} \)
$19$ \( 3136 - 56 T + T^{2} \)
$23$ \( T^{2} \)
$29$ \( T^{2} \)
$31$ \( 94864 - 308 T + T^{2} \)
$37$ \( 12100 + 110 T + T^{2} \)
$41$ \( T^{2} \)
$43$ \( ( 520 + T )^{2} \)
$47$ \( T^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( 33124 - 182 T + T^{2} \)
$67$ \( 774400 - 880 T + T^{2} \)
$71$ \( T^{2} \)
$73$ \( 1416100 - 1190 T + T^{2} \)
$79$ \( 781456 + 884 T + T^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( ( -1330 + T )^{2} \)
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