Properties

Label 441.4.a.w
Level $441$
Weight $4$
Character orbit 441.a
Self dual yes
Analytic conductor $26.020$
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: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 441.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(26.0198423125\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.6257832.1
Defining polynomial: \(x^{4} - 19 x^{2} + 42\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + ( 2 + \beta_{2} ) q^{4} + ( \beta_{1} + \beta_{3} ) q^{5} + ( -\beta_{1} + \beta_{3} ) q^{8} +O(q^{10})\) \( q + \beta_{1} q^{2} + ( 2 + \beta_{2} ) q^{4} + ( \beta_{1} + \beta_{3} ) q^{5} + ( -\beta_{1} + \beta_{3} ) q^{8} + ( 8 + 5 \beta_{2} ) q^{10} + ( -\beta_{1} + 3 \beta_{3} ) q^{11} + ( 25 - \beta_{2} ) q^{13} + ( -28 - 5 \beta_{2} ) q^{16} + ( 26 \beta_{1} - 2 \beta_{3} ) q^{17} + ( 61 + 11 \beta_{2} ) q^{19} + ( 25 \beta_{1} - 3 \beta_{3} ) q^{20} + ( -16 + 11 \beta_{2} ) q^{22} + ( 46 \beta_{1} + 2 \beta_{3} ) q^{23} + ( 83 - 17 \beta_{2} ) q^{25} + ( 20 \beta_{1} - \beta_{3} ) q^{26} + ( -45 \beta_{1} - \beta_{3} ) q^{29} + ( 45 - 20 \beta_{2} ) q^{31} + ( -45 \beta_{1} - 13 \beta_{3} ) q^{32} + ( 264 + 18 \beta_{2} ) q^{34} + ( -81 + 25 \beta_{2} ) q^{37} + ( 116 \beta_{1} + 11 \beta_{3} ) q^{38} + ( 192 - 27 \beta_{2} ) q^{40} + ( 72 \beta_{1} + 16 \beta_{3} ) q^{41} + ( -223 - 27 \beta_{2} ) q^{43} + ( 47 \beta_{1} - 13 \beta_{3} ) q^{44} + ( 456 + 54 \beta_{2} ) q^{46} + ( -44 \beta_{1} - 16 \beta_{3} ) q^{47} + ( -2 \beta_{1} - 17 \beta_{3} ) q^{50} + ( 2 + 24 \beta_{2} ) q^{52} + ( -\beta_{1} + 7 \beta_{3} ) q^{53} + ( 592 - 71 \beta_{2} ) q^{55} + ( -448 - 49 \beta_{2} ) q^{58} + ( -73 \beta_{1} - 17 \beta_{3} ) q^{59} + ( 310 - 46 \beta_{2} ) q^{61} + ( -55 \beta_{1} - 20 \beta_{3} ) q^{62} + ( -200 - 57 \beta_{2} ) q^{64} + ( 2 \beta_{1} + 30 \beta_{3} ) q^{65} + ( 463 - 19 \beta_{2} ) q^{67} + ( 146 \beta_{1} + 34 \beta_{3} ) q^{68} + 36 \beta_{3} q^{71} + ( 441 + 7 \beta_{2} ) q^{73} + ( 44 \beta_{1} + 25 \beta_{3} ) q^{74} + ( 650 + 72 \beta_{2} ) q^{76} + ( 23 + 42 \beta_{2} ) q^{79} + ( -143 \beta_{1} - 3 \beta_{3} ) q^{80} + ( 688 + 136 \beta_{2} ) q^{82} + ( -189 \beta_{1} - 21 \beta_{3} ) q^{83} + ( -192 + 174 \beta_{2} ) q^{85} + ( -358 \beta_{1} - 27 \beta_{3} ) q^{86} + ( 624 - 93 \beta_{2} ) q^{88} + ( -314 \beta_{1} + 22 \beta_{3} ) q^{89} + ( 358 \beta_{1} + 38 \beta_{3} ) q^{92} + ( -408 - 108 \beta_{2} ) q^{94} + ( 314 \beta_{1} + 6 \beta_{3} ) q^{95} + ( 798 + 91 \beta_{2} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 6q^{4} + O(q^{10}) \) \( 4q + 6q^{4} + 22q^{10} + 102q^{13} - 102q^{16} + 222q^{19} - 86q^{22} + 366q^{25} + 220q^{31} + 1020q^{34} - 374q^{37} + 822q^{40} - 838q^{43} + 1716q^{46} - 40q^{52} + 2510q^{55} - 1694q^{58} + 1332q^{61} - 686q^{64} + 1890q^{67} + 1750q^{73} + 2456q^{76} + 8q^{79} + 2480q^{82} - 1116q^{85} + 2682q^{88} - 1416q^{94} + 3010q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 19 x^{2} + 42\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 10 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 15 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 10\)
\(\nu^{3}\)\(=\)\(\beta_{3} + 15 \beta_{1}\)

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
−4.05539
−1.59805
1.59805
4.05539
−4.05539 0 8.44622 −9.92039 0 0 −1.80961 0 40.2311
1.2 −1.59805 0 −5.44622 18.2917 0 0 21.4878 0 −29.2311
1.3 1.59805 0 −5.44622 −18.2917 0 0 −21.4878 0 −29.2311
1.4 4.05539 0 8.44622 9.92039 0 0 1.80961 0 40.2311
\(n\): 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.w 4
3.b odd 2 1 inner 441.4.a.w 4
7.b odd 2 1 441.4.a.v 4
7.c even 3 2 63.4.e.d 8
7.d odd 6 2 441.4.e.x 8
21.c even 2 1 441.4.a.v 4
21.g even 6 2 441.4.e.x 8
21.h odd 6 2 63.4.e.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.4.e.d 8 7.c even 3 2
63.4.e.d 8 21.h odd 6 2
441.4.a.v 4 7.b odd 2 1
441.4.a.v 4 21.c even 2 1
441.4.a.w 4 1.a even 1 1 trivial
441.4.a.w 4 3.b odd 2 1 inner
441.4.e.x 8 7.d odd 6 2
441.4.e.x 8 21.g even 6 2

Hecke kernels

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

\( T_{2}^{4} - 19 T_{2}^{2} + 42 \)
\( T_{5}^{4} - 433 T_{5}^{2} + 32928 \)
\( T_{13}^{2} - 51 T_{13} + 602 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 42 - 19 T^{2} + T^{4} \)
$3$ \( T^{4} \)
$5$ \( 32928 - 433 T^{2} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( 688128 - 3937 T^{2} + T^{4} \)
$13$ \( ( 602 - 51 T + T^{2} )^{2} \)
$17$ \( 58084992 - 15396 T^{2} + T^{4} \)
$19$ \( ( -2758 - 111 T + T^{2} )^{2} \)
$23$ \( 44731008 - 40452 T^{2} + T^{4} \)
$29$ \( 96018048 - 38185 T^{2} + T^{4} \)
$31$ \( ( -16275 - 110 T + T^{2} )^{2} \)
$37$ \( ( -21414 + 187 T + T^{2} )^{2} \)
$41$ \( 6145155072 - 190144 T^{2} + T^{4} \)
$43$ \( ( 8716 + 419 T + T^{2} )^{2} \)
$47$ \( 4556708352 - 135600 T^{2} + T^{4} \)
$53$ \( 27149472 - 21201 T^{2} + T^{4} \)
$59$ \( 7681740192 - 205665 T^{2} + T^{4} \)
$61$ \( ( 8792 - 666 T + T^{2} )^{2} \)
$67$ \( ( 205838 - 945 T + T^{2} )^{2} \)
$71$ \( 22856214528 - 557280 T^{2} + T^{4} \)
$73$ \( ( 189042 - 875 T + T^{2} )^{2} \)
$79$ \( ( -85109 - 4 T + T^{2} )^{2} \)
$83$ \( 10585989792 - 804825 T^{2} + T^{4} \)
$89$ \( 1155567856128 - 2191972 T^{2} + T^{4} \)
$97$ \( ( 166698 - 1505 T + T^{2} )^{2} \)
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