Properties

Label 441.6.a.z
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: \(\Q(\sqrt{2}, \sqrt{113})\)
Defining polynomial: \(x^{4} - 2 x^{3} - 59 x^{2} + 60 x + 674\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 7 \)
Twist minimal: no (minimal twist has level 49)
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.

\(f(q)\) \(=\) \( q + ( 2 - \beta_{1} ) q^{2} -5 \beta_{1} q^{4} + ( 8 \beta_{2} - 2 \beta_{3} ) q^{5} + ( 76 + 17 \beta_{1} ) q^{8} +O(q^{10})\) \( q + ( 2 - \beta_{1} ) q^{2} -5 \beta_{1} q^{4} + ( 8 \beta_{2} - 2 \beta_{3} ) q^{5} + ( 76 + 17 \beta_{1} ) q^{8} + ( 30 \beta_{2} - 38 \beta_{3} ) q^{10} + ( 494 + 12 \beta_{1} ) q^{11} + ( 56 \beta_{2} + 64 \beta_{3} ) q^{13} + ( -324 + 135 \beta_{1} ) q^{16} + ( 152 \beta_{2} - 7 \beta_{3} ) q^{17} + ( -78 \beta_{2} - 53 \beta_{3} ) q^{19} + ( 70 \beta_{2} - 170 \beta_{3} ) q^{20} + ( 652 - 458 \beta_{1} ) q^{22} + ( 1612 - 344 \beta_{1} ) q^{23} + ( 531 - 320 \beta_{1} ) q^{25} + ( -336 \beta_{2} - 32 \beta_{3} ) q^{26} + ( 446 - 784 \beta_{1} ) q^{29} + ( 772 \beta_{2} + 246 \beta_{3} ) q^{31} + ( -6860 + 185 \beta_{1} ) q^{32} + ( 353 \beta_{2} - 629 \beta_{3} ) q^{34} + ( -2326 - 48 \beta_{1} ) q^{37} + ( 215 \beta_{2} + 153 \beta_{3} ) q^{38} + ( 370 \beta_{2} + 426 \beta_{3} ) q^{40} + ( 224 \beta_{2} + 1103 \beta_{3} ) q^{41} + ( 4622 - 980 \beta_{1} ) q^{43} + ( -1680 - 2410 \beta_{1} ) q^{44} + ( 12856 - 2644 \beta_{1} ) q^{46} + ( -1644 \beta_{2} + 590 \beta_{3} ) q^{47} + ( 10022 - 1491 \beta_{1} ) q^{50} + ( -2240 \beta_{2} - 800 \beta_{3} ) q^{52} + ( 24434 - 2592 \beta_{1} ) q^{53} + ( 3784 \beta_{2} - 580 \beta_{3} ) q^{55} + ( 22844 - 2798 \beta_{1} ) q^{58} + ( -2914 \beta_{2} + 1425 \beta_{3} ) q^{59} + ( -680 \beta_{2} - 3326 \beta_{3} ) q^{61} + ( -178 \beta_{2} - 2350 \beta_{3} ) q^{62} + ( -8532 + 3095 \beta_{1} ) q^{64} + ( 19040 + 6496 \beta_{1} ) q^{65} + ( -11540 - 11632 \beta_{1} ) q^{67} + ( 245 \beta_{2} - 3075 \beta_{3} ) q^{68} + ( 40612 + 4312 \beta_{1} ) q^{71} + ( -5080 \beta_{2} + 4407 \beta_{3} ) q^{73} + ( -3308 + 2182 \beta_{1} ) q^{74} + ( 1855 \beta_{2} + 1295 \beta_{3} ) q^{76} + ( -20540 + 4264 \beta_{1} ) q^{79} + ( -4482 \beta_{2} + 5238 \beta_{3} ) q^{80} + ( -7273 \beta_{2} + 2413 \beta_{3} ) q^{82} + ( 6230 \beta_{2} + 4425 \beta_{3} ) q^{83} + ( 66860 - 2608 \beta_{1} ) q^{85} + ( 36684 - 7562 \beta_{1} ) q^{86} + ( 43256 + 9106 \beta_{1} ) q^{88} + ( 1016 \beta_{2} - 6259 \beta_{3} ) q^{89} + ( 48160 - 9780 \beta_{1} ) q^{92} + ( -7418 \beta_{2} + 8346 \beta_{3} ) q^{94} + ( -29556 - 5000 \beta_{1} ) q^{95} + ( 7448 \beta_{2} - 5873 \beta_{3} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 10 q^{2} + 10 q^{4} + 270 q^{8} + O(q^{10}) \) \( 4 q + 10 q^{2} + 10 q^{4} + 270 q^{8} + 1952 q^{11} - 1566 q^{16} + 3524 q^{22} + 7136 q^{23} + 2764 q^{25} + 3352 q^{29} - 27810 q^{32} - 9208 q^{37} + 20448 q^{43} - 1900 q^{44} + 56712 q^{46} + 43070 q^{50} + 102920 q^{53} + 96972 q^{58} - 40318 q^{64} + 63168 q^{65} - 22896 q^{67} + 153824 q^{71} - 17596 q^{74} - 90688 q^{79} + 272656 q^{85} + 161860 q^{86} + 154812 q^{88} + 212200 q^{92} - 108224 q^{95} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 2 x^{3} - 59 x^{2} + 60 x + 674\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -2 \nu^{3} + 3 \nu^{2} + 172 \nu - 139 \)\()/105\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 51 \nu^{2} - 86 \nu - 1558 \)\()/105\)
\(\beta_{3}\)\(=\)\((\)\( 2 \nu^{3} - 3 \nu^{2} - 67 \nu + 34 \)\()/15\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + 7 \beta_{1} + 7\)\()/7\)
\(\nu^{2}\)\(=\)\(2 \beta_{2} + \beta_{1} + 31\)
\(\nu^{3}\)\(=\)\((\)\(86 \beta_{3} + 21 \beta_{2} + 245 \beta_{1} + 441\)\()/7\)

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
4.40086
7.22929
−3.40086
−6.22929
−2.81507 0 −24.0754 −45.9910 0 0 157.856 0 129.468
1.2 −2.81507 0 −24.0754 45.9910 0 0 157.856 0 −129.468
1.3 7.81507 0 29.0754 −74.2753 0 0 −22.8562 0 −580.467
1.4 7.81507 0 29.0754 74.2753 0 0 −22.8562 0 580.467
\(n\): 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
7.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.z 4
3.b odd 2 1 49.6.a.g 4
7.b odd 2 1 inner 441.6.a.z 4
12.b even 2 1 784.6.a.bf 4
21.c even 2 1 49.6.a.g 4
21.g even 6 2 49.6.c.h 8
21.h odd 6 2 49.6.c.h 8
84.h odd 2 1 784.6.a.bf 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
49.6.a.g 4 3.b odd 2 1
49.6.a.g 4 21.c even 2 1
49.6.c.h 8 21.g even 6 2
49.6.c.h 8 21.h odd 6 2
441.6.a.z 4 1.a even 1 1 trivial
441.6.a.z 4 7.b odd 2 1 inner
784.6.a.bf 4 12.b even 2 1
784.6.a.bf 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 \(S_{6}^{\mathrm{new}}(\Gamma_0(441))\):

\( T_{2}^{2} - 5 T_{2} - 22 \)
\( T_{5}^{4} - 7632 T_{5}^{2} + 11669056 \)
\( T_{13}^{4} - 1260672 T_{13}^{2} + 76158337024 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -22 - 5 T + T^{2} )^{2} \)
$3$ \( T^{4} \)
$5$ \( 11669056 - 7632 T^{2} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( ( 234076 - 976 T + T^{2} )^{2} \)
$13$ \( 76158337024 - 1260672 T^{2} + T^{4} \)
$17$ \( 1700202150724 - 2613668 T^{2} + T^{4} \)
$19$ \( 56942116 - 1359892 T^{2} + T^{4} \)
$23$ \( ( -160336 - 3568 T + T^{2} )^{2} \)
$29$ \( ( -16661788 - 1676 T + T^{2} )^{2} \)
$31$ \( 614334295349824 - 85120848 T^{2} + T^{4} \)
$37$ \( ( 5234116 + 4604 T + T^{2} )^{2} \)
$41$ \( 14370455329863556 - 251093444 T^{2} + T^{4} \)
$43$ \( ( -998756 - 10224 T + T^{2} )^{2} \)
$47$ \( 17113573085148736 - 349180624 T^{2} + T^{4} \)
$53$ \( ( 472236292 - 51460 T + T^{2} )^{2} \)
$59$ \( 111991443063303076 - 1249753044 T^{2} + T^{4} \)
$61$ \( 1187821342581020224 - 2284246736 T^{2} + T^{4} \)
$67$ \( ( -3789557552 + 11448 T + T^{2} )^{2} \)
$71$ \( ( 953601968 - 76912 T + T^{2} )^{2} \)
$73$ \( 20948903958719044 - 6121721124 T^{2} + T^{4} \)
$79$ \( ( 386672 + 45344 T + T^{2} )^{2} \)
$83$ \( 17246872858022500 - 9034370100 T^{2} + T^{4} \)
$89$ \( 13633253463731899396 - 7617937028 T^{2} + T^{4} \)
$97$ \( 114671792612888644 - 11859566628 T^{2} + T^{4} \)
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