Properties

Label 441.2.a.g
Level $441$
Weight $2$
Character orbit 441.a
Self dual yes
Analytic conductor $3.521$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 63)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

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

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
−1.73205
1.73205
−1.73205 0 1.00000 −3.46410 0 0 1.73205 0 6.00000
1.2 1.73205 0 1.00000 3.46410 0 0 −1.73205 0 6.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.2.a.g 2
3.b odd 2 1 inner 441.2.a.g 2
4.b odd 2 1 7056.2.a.cm 2
7.b odd 2 1 63.2.a.b 2
7.c even 3 2 441.2.e.i 4
7.d odd 6 2 441.2.e.j 4
12.b even 2 1 7056.2.a.cm 2
21.c even 2 1 63.2.a.b 2
21.g even 6 2 441.2.e.j 4
21.h odd 6 2 441.2.e.i 4
28.d even 2 1 1008.2.a.n 2
35.c odd 2 1 1575.2.a.q 2
35.f even 4 2 1575.2.d.i 4
56.e even 2 1 4032.2.a.bq 2
56.h odd 2 1 4032.2.a.bt 2
63.l odd 6 2 567.2.f.j 4
63.o even 6 2 567.2.f.j 4
77.b even 2 1 7623.2.a.bi 2
84.h odd 2 1 1008.2.a.n 2
105.g even 2 1 1575.2.a.q 2
105.k odd 4 2 1575.2.d.i 4
168.e odd 2 1 4032.2.a.bq 2
168.i even 2 1 4032.2.a.bt 2
231.h odd 2 1 7623.2.a.bi 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.a.b 2 7.b odd 2 1
63.2.a.b 2 21.c even 2 1
441.2.a.g 2 1.a even 1 1 trivial
441.2.a.g 2 3.b odd 2 1 inner
441.2.e.i 4 7.c even 3 2
441.2.e.i 4 21.h odd 6 2
441.2.e.j 4 7.d odd 6 2
441.2.e.j 4 21.g even 6 2
567.2.f.j 4 63.l odd 6 2
567.2.f.j 4 63.o even 6 2
1008.2.a.n 2 28.d even 2 1
1008.2.a.n 2 84.h odd 2 1
1575.2.a.q 2 35.c odd 2 1
1575.2.a.q 2 105.g even 2 1
1575.2.d.i 4 35.f even 4 2
1575.2.d.i 4 105.k odd 4 2
4032.2.a.bq 2 56.e even 2 1
4032.2.a.bq 2 168.e odd 2 1
4032.2.a.bt 2 56.h odd 2 1
4032.2.a.bt 2 168.i even 2 1
7056.2.a.cm 2 4.b odd 2 1
7056.2.a.cm 2 12.b even 2 1
7623.2.a.bi 2 77.b even 2 1
7623.2.a.bi 2 231.h odd 2 1

Hecke kernels

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

\( T_{2}^{2} - 3 \)
\( T_{5}^{2} - 12 \)
\( T_{13} + 2 \)

Hecke characteristic polynomials

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