Properties

Label 441.2.a.a
Level $441$
Weight $2$
Character orbit 441.a
Self dual yes
Analytic conductor $3.521$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 441.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(3.52140272914\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 21)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 2q^{2} + 2q^{4} - 2q^{5} + O(q^{10}) \) \( q - 2q^{2} + 2q^{4} - 2q^{5} + 4q^{10} + 2q^{11} - q^{13} - 4q^{16} - q^{19} - 4q^{20} - 4q^{22} - q^{25} + 2q^{26} - 4q^{29} - 9q^{31} + 8q^{32} + 3q^{37} + 2q^{38} - 10q^{41} + 5q^{43} + 4q^{44} - 6q^{47} + 2q^{50} - 2q^{52} - 12q^{53} - 4q^{55} + 8q^{58} - 12q^{59} - 10q^{61} + 18q^{62} - 8q^{64} + 2q^{65} - 5q^{67} + 6q^{71} + 3q^{73} - 6q^{74} - 2q^{76} - q^{79} + 8q^{80} + 20q^{82} + 6q^{83} - 10q^{86} + 16q^{89} + 12q^{94} + 2q^{95} + 6q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
−2.00000 0 2.00000 −2.00000 0 0 0 0 4.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(7\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.2.a.a 1
3.b odd 2 1 147.2.a.b 1
4.b odd 2 1 7056.2.a.m 1
7.b odd 2 1 441.2.a.b 1
7.c even 3 2 441.2.e.e 2
7.d odd 6 2 63.2.e.b 2
12.b even 2 1 2352.2.a.w 1
15.d odd 2 1 3675.2.a.c 1
21.c even 2 1 147.2.a.c 1
21.g even 6 2 21.2.e.a 2
21.h odd 6 2 147.2.e.a 2
24.f even 2 1 9408.2.a.k 1
24.h odd 2 1 9408.2.a.bz 1
28.d even 2 1 7056.2.a.bp 1
28.f even 6 2 1008.2.s.d 2
63.i even 6 2 567.2.h.f 2
63.k odd 6 2 567.2.g.f 2
63.s even 6 2 567.2.g.a 2
63.t odd 6 2 567.2.h.a 2
84.h odd 2 1 2352.2.a.d 1
84.j odd 6 2 336.2.q.f 2
84.n even 6 2 2352.2.q.c 2
105.g even 2 1 3675.2.a.a 1
105.p even 6 2 525.2.i.e 2
105.w odd 12 4 525.2.r.e 4
168.e odd 2 1 9408.2.a.cv 1
168.i even 2 1 9408.2.a.bg 1
168.ba even 6 2 1344.2.q.m 2
168.be odd 6 2 1344.2.q.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.2.e.a 2 21.g even 6 2
63.2.e.b 2 7.d odd 6 2
147.2.a.b 1 3.b odd 2 1
147.2.a.c 1 21.c even 2 1
147.2.e.a 2 21.h odd 6 2
336.2.q.f 2 84.j odd 6 2
441.2.a.a 1 1.a even 1 1 trivial
441.2.a.b 1 7.b odd 2 1
441.2.e.e 2 7.c even 3 2
525.2.i.e 2 105.p even 6 2
525.2.r.e 4 105.w odd 12 4
567.2.g.a 2 63.s even 6 2
567.2.g.f 2 63.k odd 6 2
567.2.h.a 2 63.t odd 6 2
567.2.h.f 2 63.i even 6 2
1008.2.s.d 2 28.f even 6 2
1344.2.q.c 2 168.be odd 6 2
1344.2.q.m 2 168.ba even 6 2
2352.2.a.d 1 84.h odd 2 1
2352.2.a.w 1 12.b even 2 1
2352.2.q.c 2 84.n even 6 2
3675.2.a.a 1 105.g even 2 1
3675.2.a.c 1 15.d odd 2 1
7056.2.a.m 1 4.b odd 2 1
7056.2.a.bp 1 28.d even 2 1
9408.2.a.k 1 24.f even 2 1
9408.2.a.bg 1 168.i even 2 1
9408.2.a.bz 1 24.h odd 2 1
9408.2.a.cv 1 168.e odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(441))\):

\( T_{2} + 2 \)
\( T_{5} + 2 \)
\( T_{13} + 1 \)