Properties

Label 441.6.a.x
Level $441$
Weight $6$
Character orbit 441.a
Self dual yes
Analytic conductor $70.729$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 441.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(70.7292645375\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.358541904.1
Defining polynomial: \(x^{4} - 111 x^{2} + 756\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2\cdot 3^{2} \)
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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + ( 24 + \beta_{3} ) q^{4} + ( -3 \beta_{1} + \beta_{2} ) q^{5} + ( 32 \beta_{1} + 3 \beta_{2} ) q^{8} +O(q^{10})\) \( q + \beta_{1} q^{2} + ( 24 + \beta_{3} ) q^{4} + ( -3 \beta_{1} + \beta_{2} ) q^{5} + ( 32 \beta_{1} + 3 \beta_{2} ) q^{8} + ( -140 + 2 \beta_{3} ) q^{10} + ( 45 \beta_{1} + \beta_{2} ) q^{11} + ( -210 + 16 \beta_{3} ) q^{13} + ( 1108 + 15 \beta_{3} ) q^{16} + ( 131 \beta_{1} + 31 \beta_{2} ) q^{17} + ( -1708 - 16 \beta_{3} ) q^{19} + ( 36 \beta_{1} - 26 \beta_{2} ) q^{20} + ( 2548 + 50 \beta_{3} ) q^{22} + ( -237 \beta_{1} + 47 \beta_{2} ) q^{23} + ( 711 - 80 \beta_{3} ) q^{25} + ( 430 \beta_{1} + 48 \beta_{2} ) q^{26} + ( -496 \beta_{1} + 32 \beta_{2} ) q^{29} + ( 1204 + 48 \beta_{3} ) q^{31} + ( 684 \beta_{1} - 51 \beta_{2} ) q^{32} + ( 8204 + 286 \beta_{3} ) q^{34} + ( 1634 + 240 \beta_{3} ) q^{37} + ( -2348 \beta_{1} - 48 \beta_{2} ) q^{38} + ( 5768 - 158 \beta_{3} ) q^{40} + ( 809 \beta_{1} + 29 \beta_{2} ) q^{41} + ( 2700 + 128 \beta_{3} ) q^{43} + ( 3108 \beta_{1} + 118 \beta_{2} ) q^{44} + ( -11956 - 2 \beta_{3} ) q^{46} + ( 2582 \beta_{1} - 114 \beta_{2} ) q^{47} + ( -2489 \beta_{1} - 240 \beta_{2} ) q^{50} + ( 32144 + 158 \beta_{3} ) q^{52} + ( 1510 \beta_{1} + 14 \beta_{2} ) q^{53} + ( -2884 + 16 \beta_{3} ) q^{55} + ( -26880 - 336 \beta_{3} ) q^{58} + ( 1642 \beta_{1} - 174 \beta_{2} ) q^{59} + ( -3206 + 240 \beta_{3} ) q^{61} + ( 3124 \beta_{1} + 144 \beta_{2} ) q^{62} + ( 1420 - 51 \beta_{3} ) q^{64} + ( 2358 \beta_{1} - 1010 \beta_{2} ) q^{65} + ( 50828 - 96 \beta_{3} ) q^{67} + ( 15452 \beta_{1} - 134 \beta_{2} ) q^{68} + ( -1287 \beta_{1} + 717 \beta_{2} ) q^{71} + ( -4942 - 1120 \beta_{3} ) q^{73} + ( 11234 \beta_{1} + 720 \beta_{2} ) q^{74} + ( -78176 - 2076 \beta_{3} ) q^{76} + ( 8264 - 672 \beta_{3} ) q^{79} + ( -1704 \beta_{1} + 358 \beta_{2} ) q^{80} + ( 46116 + 954 \beta_{3} ) q^{82} + ( -9296 \beta_{1} + 112 \beta_{2} ) q^{83} + ( 87556 - 2032 \beta_{3} ) q^{85} + ( 7820 \beta_{1} + 384 \beta_{2} ) q^{86} + ( 95816 + 2098 \beta_{3} ) q^{88} + ( 1453 \beta_{1} + 113 \beta_{2} ) q^{89} + ( -4452 \beta_{1} - 1510 \beta_{2} ) q^{92} + ( 141400 + 2012 \beta_{3} ) q^{94} + ( 3396 \beta_{1} - 908 \beta_{2} ) q^{95} + ( -109326 - 224 \beta_{3} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 94 q^{4} + O(q^{10}) \) \( 4 q + 94 q^{4} - 564 q^{10} - 872 q^{13} + 4402 q^{16} - 6800 q^{19} + 10092 q^{22} + 3004 q^{25} + 4720 q^{31} + 32244 q^{34} + 6056 q^{37} + 23388 q^{40} + 10544 q^{43} - 47820 q^{46} + 128260 q^{52} - 11568 q^{55} - 106848 q^{58} - 13304 q^{61} + 5782 q^{64} + 203504 q^{67} - 17528 q^{73} - 308552 q^{76} + 34400 q^{79} + 182556 q^{82} + 354288 q^{85} + 379068 q^{88} + 561576 q^{94} - 436856 q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 111 x^{2} + 756\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} - 96 \nu \)\()/3\)
\(\beta_{3}\)\(=\)\( \nu^{2} - 56 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + 56\)
\(\nu^{3}\)\(=\)\(3 \beta_{2} + 96 \beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−10.1838
−2.69991
2.69991
10.1838
−10.1838 0 71.7105 4.37743 0 0 −404.405 0 −44.5790
1.2 −2.69991 0 −24.7105 87.9366 0 0 153.113 0 −237.421
1.3 2.69991 0 −24.7105 −87.9366 0 0 −153.113 0 −237.421
1.4 10.1838 0 71.7105 −4.37743 0 0 404.405 0 −44.5790
\(n\): 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 441.6.a.x 4
3.b odd 2 1 inner 441.6.a.x 4
7.b odd 2 1 63.6.a.h 4
21.c even 2 1 63.6.a.h 4
28.d even 2 1 1008.6.a.by 4
84.h odd 2 1 1008.6.a.by 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.6.a.h 4 7.b odd 2 1
63.6.a.h 4 21.c even 2 1
441.6.a.x 4 1.a even 1 1 trivial
441.6.a.x 4 3.b odd 2 1 inner
1008.6.a.by 4 28.d even 2 1
1008.6.a.by 4 84.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 \(S_{6}^{\mathrm{new}}(\Gamma_0(441))\):

\( T_{2}^{4} - 111 T_{2}^{2} + 756 \)
\( T_{5}^{4} - 7752 T_{5}^{2} + 148176 \)
\( T_{13}^{2} + 436 T_{13} - 547484 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 756 - 111 T^{2} + T^{4} \)
$3$ \( T^{4} \)
$5$ \( 148176 - 7752 T^{2} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( 407299536 - 236424 T^{2} + T^{4} \)
$13$ \( ( -547484 + 436 T + T^{2} )^{2} \)
$17$ \( 20712534557904 - 9102792 T^{2} + T^{4} \)
$19$ \( ( 2294992 + 3400 T + T^{2} )^{2} \)
$23$ \( 27015936275664 - 20691912 T^{2} + T^{4} \)
$29$ \( 269206953394176 - 32917248 T^{2} + T^{4} \)
$31$ \( ( -3962672 - 2360 T + T^{2} )^{2} \)
$37$ \( ( -131584604 - 3028 T + T^{2} )^{2} \)
$41$ \( 1390182638544 - 80977032 T^{2} + T^{4} \)
$43$ \( ( -31132016 - 5272 T + T^{2} )^{2} \)
$47$ \( 140373649856998656 - 801721632 T^{2} + T^{4} \)
$53$ \( 2170530954982656 - 256630944 T^{2} + T^{4} \)
$59$ \( 49715173391498496 - 483850272 T^{2} + T^{4} \)
$61$ \( ( -122814524 + 6652 T + T^{2} )^{2} \)
$67$ \( ( 2566947088 - 101752 T + T^{2} )^{2} \)
$71$ \( 118112942107725264 - 3718687752 T^{2} + T^{4} \)
$73$ \( ( -2896337276 + 8764 T + T^{2} )^{2} \)
$79$ \( ( -975634112 - 17200 T + T^{2} )^{2} \)
$83$ \( 9751577272444256256 - 9574483968 T^{2} + T^{4} \)
$89$ \( 8194654066720464 - 341227848 T^{2} + T^{4} \)
$97$ \( ( 11811076228 + 218428 T + T^{2} )^{2} \)
show more
show less