Properties

Label 441.4.a.q
Level $441$
Weight $4$
Character orbit 441.a
Self dual yes
Analytic conductor $26.020$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 441.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(26.0198423125\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{19}) \)
Defining polynomial: \(x^{2} - 19\)
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 \(\beta = \sqrt{19}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + 11 q^{4} -2 \beta q^{5} + 3 \beta q^{8} +O(q^{10})\) \( q + \beta q^{2} + 11 q^{4} -2 \beta q^{5} + 3 \beta q^{8} -38 q^{10} -10 \beta q^{11} -82 q^{13} -31 q^{16} + 18 \beta q^{17} + 20 q^{19} -22 \beta q^{20} -190 q^{22} -30 \beta q^{23} -49 q^{25} -82 \beta q^{26} + 56 \beta q^{29} -156 q^{31} -55 \beta q^{32} + 342 q^{34} + 186 q^{37} + 20 \beta q^{38} -114 q^{40} + 38 \beta q^{41} + 164 q^{43} -110 \beta q^{44} -570 q^{46} -108 \beta q^{47} -49 \beta q^{50} -902 q^{52} -36 \beta q^{53} + 380 q^{55} + 1064 q^{58} -36 \beta q^{59} -790 q^{61} -156 \beta q^{62} -797 q^{64} + 164 \beta q^{65} -44 q^{67} + 198 \beta q^{68} + 102 \beta q^{71} -126 q^{73} + 186 \beta q^{74} + 220 q^{76} -712 q^{79} + 62 \beta q^{80} + 722 q^{82} + 336 \beta q^{83} -684 q^{85} + 164 \beta q^{86} -570 q^{88} + 334 \beta q^{89} -330 \beta q^{92} -2052 q^{94} -40 \beta q^{95} -798 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 22q^{4} + O(q^{10}) \) \( 2q + 22q^{4} - 76q^{10} - 164q^{13} - 62q^{16} + 40q^{19} - 380q^{22} - 98q^{25} - 312q^{31} + 684q^{34} + 372q^{37} - 228q^{40} + 328q^{43} - 1140q^{46} - 1804q^{52} + 760q^{55} + 2128q^{58} - 1580q^{61} - 1594q^{64} - 88q^{67} - 252q^{73} + 440q^{76} - 1424q^{79} + 1444q^{82} - 1368q^{85} - 1140q^{88} - 4104q^{94} - 1596q^{97} + O(q^{100}) \)

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
−4.35890
4.35890
−4.35890 0 11.0000 8.71780 0 0 −13.0767 0 −38.0000
1.2 4.35890 0 11.0000 −8.71780 0 0 13.0767 0 −38.0000
\(n\): 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 441.4.a.q 2
3.b odd 2 1 inner 441.4.a.q 2
7.b odd 2 1 63.4.a.d 2
7.c even 3 2 441.4.e.s 4
7.d odd 6 2 441.4.e.r 4
21.c even 2 1 63.4.a.d 2
21.g even 6 2 441.4.e.r 4
21.h odd 6 2 441.4.e.s 4
28.d even 2 1 1008.4.a.be 2
35.c odd 2 1 1575.4.a.t 2
84.h odd 2 1 1008.4.a.be 2
105.g even 2 1 1575.4.a.t 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.4.a.d 2 7.b odd 2 1
63.4.a.d 2 21.c even 2 1
441.4.a.q 2 1.a even 1 1 trivial
441.4.a.q 2 3.b odd 2 1 inner
441.4.e.r 4 7.d odd 6 2
441.4.e.r 4 21.g even 6 2
441.4.e.s 4 7.c even 3 2
441.4.e.s 4 21.h odd 6 2
1008.4.a.be 2 28.d even 2 1
1008.4.a.be 2 84.h odd 2 1
1575.4.a.t 2 35.c odd 2 1
1575.4.a.t 2 105.g even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(441))\):

\( T_{2}^{2} - 19 \)
\( T_{5}^{2} - 76 \)
\( T_{13} + 82 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -19 + T^{2} \)
$3$ \( T^{2} \)
$5$ \( -76 + T^{2} \)
$7$ \( T^{2} \)
$11$ \( -1900 + T^{2} \)
$13$ \( ( 82 + T )^{2} \)
$17$ \( -6156 + T^{2} \)
$19$ \( ( -20 + T )^{2} \)
$23$ \( -17100 + T^{2} \)
$29$ \( -59584 + T^{2} \)
$31$ \( ( 156 + T )^{2} \)
$37$ \( ( -186 + T )^{2} \)
$41$ \( -27436 + T^{2} \)
$43$ \( ( -164 + T )^{2} \)
$47$ \( -221616 + T^{2} \)
$53$ \( -24624 + T^{2} \)
$59$ \( -24624 + T^{2} \)
$61$ \( ( 790 + T )^{2} \)
$67$ \( ( 44 + T )^{2} \)
$71$ \( -197676 + T^{2} \)
$73$ \( ( 126 + T )^{2} \)
$79$ \( ( 712 + T )^{2} \)
$83$ \( -2145024 + T^{2} \)
$89$ \( -2119564 + T^{2} \)
$97$ \( ( 798 + T )^{2} \)
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