Properties

Label 441.2.e.i
Level $441$
Weight $2$
Character orbit 441.e
Analytic conductor $3.521$
Analytic rank $0$
Dimension $4$
CM no
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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 63)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{2} + ( -1 + \zeta_{12}^{2} ) q^{4} + ( -2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{5} + ( -2 \zeta_{12} + \zeta_{12}^{3} ) q^{8} +O(q^{10})\) \( q + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{2} + ( -1 + \zeta_{12}^{2} ) q^{4} + ( -2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{5} + ( -2 \zeta_{12} + \zeta_{12}^{3} ) q^{8} + ( -6 + 6 \zeta_{12}^{2} ) q^{10} + ( -2 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{11} -2 q^{13} + 5 \zeta_{12}^{2} q^{16} + ( 2 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{17} -4 \zeta_{12}^{2} q^{19} + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{20} + 6 q^{22} + ( 2 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{23} + ( -7 + 7 \zeta_{12}^{2} ) q^{25} + ( 2 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{26} + ( -4 + 4 \zeta_{12}^{2} ) q^{31} + ( 3 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{32} -6 q^{34} -2 \zeta_{12}^{2} q^{37} + ( -4 \zeta_{12} + 8 \zeta_{12}^{3} ) q^{38} + 6 \zeta_{12}^{2} q^{40} + ( -12 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{41} -4 q^{43} + ( -2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{44} + ( 6 - 6 \zeta_{12}^{2} ) q^{46} + ( 4 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{47} + ( 14 \zeta_{12} - 7 \zeta_{12}^{3} ) q^{50} + ( 2 - 2 \zeta_{12}^{2} ) q^{52} + ( 4 \zeta_{12} - 8 \zeta_{12}^{3} ) q^{53} + 12 q^{55} + ( -4 \zeta_{12} + 8 \zeta_{12}^{3} ) q^{59} -10 \zeta_{12}^{2} q^{61} + ( 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{62} + q^{64} + ( 4 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{65} + ( 4 - 4 \zeta_{12}^{2} ) q^{67} + ( 2 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{68} + ( -12 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{71} + ( 14 - 14 \zeta_{12}^{2} ) q^{73} + ( -2 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{74} + 4 q^{76} -8 \zeta_{12}^{2} q^{79} + ( 10 \zeta_{12} - 20 \zeta_{12}^{3} ) q^{80} + 18 \zeta_{12}^{2} q^{82} -12 q^{85} + ( 4 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{86} + ( 6 - 6 \zeta_{12}^{2} ) q^{88} + ( -2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{89} + ( -4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{92} + ( 12 - 12 \zeta_{12}^{2} ) q^{94} + ( -8 \zeta_{12} + 16 \zeta_{12}^{3} ) q^{95} -14 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{4} + O(q^{10}) \) \( 4q - 2q^{4} - 12q^{10} - 8q^{13} + 10q^{16} - 8q^{19} + 24q^{22} - 14q^{25} - 8q^{31} - 24q^{34} - 4q^{37} + 12q^{40} - 16q^{43} + 12q^{46} + 4q^{52} + 48q^{55} - 20q^{61} + 4q^{64} + 8q^{67} + 28q^{73} + 16q^{76} - 16q^{79} + 36q^{82} - 48q^{85} + 12q^{88} + 24q^{94} - 56q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(199\) \(344\)
\(\chi(n)\) \(-\zeta_{12}\) \(1\)

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
0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
−0.866025 0.500000i
−0.866025 + 1.50000i 0 −0.500000 0.866025i −1.73205 + 3.00000i 0 0 −1.73205 0 −3.00000 5.19615i
226.2 0.866025 1.50000i 0 −0.500000 0.866025i 1.73205 3.00000i 0 0 1.73205 0 −3.00000 5.19615i
361.1 −0.866025 1.50000i 0 −0.500000 + 0.866025i −1.73205 3.00000i 0 0 −1.73205 0 −3.00000 + 5.19615i
361.2 0.866025 + 1.50000i 0 −0.500000 + 0.866025i 1.73205 + 3.00000i 0 0 1.73205 0 −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
3.b odd 2 1 inner
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.i 4
3.b odd 2 1 inner 441.2.e.i 4
7.b odd 2 1 441.2.e.j 4
7.c even 3 1 441.2.a.g 2
7.c even 3 1 inner 441.2.e.i 4
7.d odd 6 1 63.2.a.b 2
7.d odd 6 1 441.2.e.j 4
21.c even 2 1 441.2.e.j 4
21.g even 6 1 63.2.a.b 2
21.g even 6 1 441.2.e.j 4
21.h odd 6 1 441.2.a.g 2
21.h odd 6 1 inner 441.2.e.i 4
28.f even 6 1 1008.2.a.n 2
28.g odd 6 1 7056.2.a.cm 2
35.i odd 6 1 1575.2.a.q 2
35.k even 12 2 1575.2.d.i 4
56.j odd 6 1 4032.2.a.bt 2
56.m even 6 1 4032.2.a.bq 2
63.i even 6 1 567.2.f.j 4
63.k odd 6 1 567.2.f.j 4
63.s even 6 1 567.2.f.j 4
63.t odd 6 1 567.2.f.j 4
77.i even 6 1 7623.2.a.bi 2
84.j odd 6 1 1008.2.a.n 2
84.n even 6 1 7056.2.a.cm 2
105.p even 6 1 1575.2.a.q 2
105.w odd 12 2 1575.2.d.i 4
168.ba even 6 1 4032.2.a.bt 2
168.be odd 6 1 4032.2.a.bq 2
231.k odd 6 1 7623.2.a.bi 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.a.b 2 7.d odd 6 1
63.2.a.b 2 21.g even 6 1
441.2.a.g 2 7.c even 3 1
441.2.a.g 2 21.h odd 6 1
441.2.e.i 4 1.a even 1 1 trivial
441.2.e.i 4 3.b odd 2 1 inner
441.2.e.i 4 7.c even 3 1 inner
441.2.e.i 4 21.h odd 6 1 inner
441.2.e.j 4 7.b odd 2 1
441.2.e.j 4 7.d odd 6 1
441.2.e.j 4 21.c even 2 1
441.2.e.j 4 21.g even 6 1
567.2.f.j 4 63.i even 6 1
567.2.f.j 4 63.k odd 6 1
567.2.f.j 4 63.s even 6 1
567.2.f.j 4 63.t odd 6 1
1008.2.a.n 2 28.f even 6 1
1008.2.a.n 2 84.j odd 6 1
1575.2.a.q 2 35.i odd 6 1
1575.2.a.q 2 105.p even 6 1
1575.2.d.i 4 35.k even 12 2
1575.2.d.i 4 105.w odd 12 2
4032.2.a.bq 2 56.m even 6 1
4032.2.a.bq 2 168.be odd 6 1
4032.2.a.bt 2 56.j odd 6 1
4032.2.a.bt 2 168.ba even 6 1
7056.2.a.cm 2 28.g odd 6 1
7056.2.a.cm 2 84.n even 6 1
7623.2.a.bi 2 77.i even 6 1
7623.2.a.bi 2 231.k odd 6 1

Hecke kernels

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

\( T_{2}^{4} + 3 T_{2}^{2} + 9 \)
\( T_{5}^{4} + 12 T_{5}^{2} + 144 \)
\( T_{13} + 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 9 + 3 T^{2} + T^{4} \)
$3$ \( T^{4} \)
$5$ \( 144 + 12 T^{2} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( 144 + 12 T^{2} + T^{4} \)
$13$ \( ( 2 + T )^{4} \)
$17$ \( 144 + 12 T^{2} + T^{4} \)
$19$ \( ( 16 + 4 T + T^{2} )^{2} \)
$23$ \( 144 + 12 T^{2} + T^{4} \)
$29$ \( T^{4} \)
$31$ \( ( 16 + 4 T + T^{2} )^{2} \)
$37$ \( ( 4 + 2 T + T^{2} )^{2} \)
$41$ \( ( -108 + T^{2} )^{2} \)
$43$ \( ( 4 + T )^{4} \)
$47$ \( 2304 + 48 T^{2} + T^{4} \)
$53$ \( 2304 + 48 T^{2} + T^{4} \)
$59$ \( 2304 + 48 T^{2} + T^{4} \)
$61$ \( ( 100 + 10 T + T^{2} )^{2} \)
$67$ \( ( 16 - 4 T + T^{2} )^{2} \)
$71$ \( ( -108 + T^{2} )^{2} \)
$73$ \( ( 196 - 14 T + T^{2} )^{2} \)
$79$ \( ( 64 + 8 T + T^{2} )^{2} \)
$83$ \( T^{4} \)
$89$ \( 144 + 12 T^{2} + T^{4} \)
$97$ \( ( 14 + T )^{4} \)
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