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 $2$

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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: \( x^{4} - 111x^{2} + 756 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{3} + 24) q^{4} + (\beta_{2} - 3 \beta_1) q^{5} + (3 \beta_{2} + 32 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{3} + 24) q^{4} + (\beta_{2} - 3 \beta_1) q^{5} + (3 \beta_{2} + 32 \beta_1) q^{8} + (2 \beta_{3} - 140) q^{10} + (\beta_{2} + 45 \beta_1) q^{11} + (16 \beta_{3} - 210) q^{13} + (15 \beta_{3} + 1108) q^{16} + (31 \beta_{2} + 131 \beta_1) q^{17} + ( - 16 \beta_{3} - 1708) q^{19} + ( - 26 \beta_{2} + 36 \beta_1) q^{20} + (50 \beta_{3} + 2548) q^{22} + (47 \beta_{2} - 237 \beta_1) q^{23} + ( - 80 \beta_{3} + 711) q^{25} + (48 \beta_{2} + 430 \beta_1) q^{26} + (32 \beta_{2} - 496 \beta_1) q^{29} + (48 \beta_{3} + 1204) q^{31} + ( - 51 \beta_{2} + 684 \beta_1) q^{32} + (286 \beta_{3} + 8204) q^{34} + (240 \beta_{3} + 1634) q^{37} + ( - 48 \beta_{2} - 2348 \beta_1) q^{38} + ( - 158 \beta_{3} + 5768) q^{40} + (29 \beta_{2} + 809 \beta_1) q^{41} + (128 \beta_{3} + 2700) q^{43} + (118 \beta_{2} + 3108 \beta_1) q^{44} + ( - 2 \beta_{3} - 11956) q^{46} + ( - 114 \beta_{2} + 2582 \beta_1) q^{47} + ( - 240 \beta_{2} - 2489 \beta_1) q^{50} + (158 \beta_{3} + 32144) q^{52} + (14 \beta_{2} + 1510 \beta_1) q^{53} + (16 \beta_{3} - 2884) q^{55} + ( - 336 \beta_{3} - 26880) q^{58} + ( - 174 \beta_{2} + 1642 \beta_1) q^{59} + (240 \beta_{3} - 3206) q^{61} + (144 \beta_{2} + 3124 \beta_1) q^{62} + ( - 51 \beta_{3} + 1420) q^{64} + ( - 1010 \beta_{2} + 2358 \beta_1) q^{65} + ( - 96 \beta_{3} + 50828) q^{67} + ( - 134 \beta_{2} + 15452 \beta_1) q^{68} + (717 \beta_{2} - 1287 \beta_1) q^{71} + ( - 1120 \beta_{3} - 4942) q^{73} + (720 \beta_{2} + 11234 \beta_1) q^{74} + ( - 2076 \beta_{3} - 78176) q^{76} + ( - 672 \beta_{3} + 8264) q^{79} + (358 \beta_{2} - 1704 \beta_1) q^{80} + (954 \beta_{3} + 46116) q^{82} + (112 \beta_{2} - 9296 \beta_1) q^{83} + ( - 2032 \beta_{3} + 87556) q^{85} + (384 \beta_{2} + 7820 \beta_1) q^{86} + (2098 \beta_{3} + 95816) q^{88} + (113 \beta_{2} + 1453 \beta_1) q^{89} + ( - 1510 \beta_{2} - 4452 \beta_1) q^{92} + (2012 \beta_{3} + 141400) q^{94} + ( - 908 \beta_{2} + 3396 \beta_1) q^{95} + ( - 224 \beta_{3} - 109326) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 94 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 94 q^{4} - 564 q^{10} - 872 q^{13} + 4402 q^{16} - 6800 q^{19} + 10092 q^{22} + 3004 q^{25} + 4720 q^{31} + 32244 q^{34} + 6056 q^{37} + 23388 q^{40} + 10544 q^{43} - 47820 q^{46} + 128260 q^{52} - 11568 q^{55} - 106848 q^{58} - 13304 q^{61} + 5782 q^{64} + 203504 q^{67} - 17528 q^{73} - 308552 q^{76} + 34400 q^{79} + 182556 q^{82} + 354288 q^{85} + 379068 q^{88} + 561576 q^{94} - 436856 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - 96\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} - 56 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 56 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{2} + 96\beta_1 \) Copy content Toggle raw display

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 \(S_{6}^{\mathrm{new}}(\Gamma_0(441))\):

\( T_{2}^{4} - 111T_{2}^{2} + 756 \) Copy content Toggle raw display
\( T_{5}^{4} - 7752T_{5}^{2} + 148176 \) Copy content Toggle raw display
\( T_{13}^{2} + 436T_{13} - 547484 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 111T^{2} + 756 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 7752 T^{2} + 148176 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} - 236424 T^{2} + \cdots + 407299536 \) Copy content Toggle raw display
$13$ \( (T^{2} + 436 T - 547484)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} - 9102792 T^{2} + \cdots + 20712534557904 \) Copy content Toggle raw display
$19$ \( (T^{2} + 3400 T + 2294992)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} - 20691912 T^{2} + \cdots + 27015936275664 \) Copy content Toggle raw display
$29$ \( T^{4} + \cdots + 269206953394176 \) Copy content Toggle raw display
$31$ \( (T^{2} - 2360 T - 3962672)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 3028 T - 131584604)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} - 80977032 T^{2} + \cdots + 1390182638544 \) Copy content Toggle raw display
$43$ \( (T^{2} - 5272 T - 31132016)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 801721632 T^{2} + \cdots + 14\!\cdots\!56 \) Copy content Toggle raw display
$53$ \( T^{4} - 256630944 T^{2} + \cdots + 21\!\cdots\!56 \) Copy content Toggle raw display
$59$ \( T^{4} - 483850272 T^{2} + \cdots + 49\!\cdots\!96 \) Copy content Toggle raw display
$61$ \( (T^{2} + 6652 T - 122814524)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 101752 T + 2566947088)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} - 3718687752 T^{2} + \cdots + 11\!\cdots\!64 \) Copy content Toggle raw display
$73$ \( (T^{2} + 8764 T - 2896337276)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 17200 T - 975634112)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} - 9574483968 T^{2} + \cdots + 97\!\cdots\!56 \) Copy content Toggle raw display
$89$ \( T^{4} - 341227848 T^{2} + \cdots + 81\!\cdots\!64 \) Copy content Toggle raw display
$97$ \( (T^{2} + 218428 T + 11811076228)^{2} \) Copy content Toggle raw display
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