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: \( x^{4} - 2x^{3} - 59x^{2} + 60x + 674 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 2) q^{2} - 5 \beta_1 q^{4} + ( - 2 \beta_{3} + 8 \beta_{2}) q^{5} + (17 \beta_1 + 76) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 2) q^{2} - 5 \beta_1 q^{4} + ( - 2 \beta_{3} + 8 \beta_{2}) q^{5} + (17 \beta_1 + 76) q^{8} + ( - 38 \beta_{3} + 30 \beta_{2}) q^{10} + (12 \beta_1 + 494) q^{11} + (64 \beta_{3} + 56 \beta_{2}) q^{13} + (135 \beta_1 - 324) q^{16} + ( - 7 \beta_{3} + 152 \beta_{2}) q^{17} + ( - 53 \beta_{3} - 78 \beta_{2}) q^{19} + ( - 170 \beta_{3} + 70 \beta_{2}) q^{20} + ( - 458 \beta_1 + 652) q^{22} + ( - 344 \beta_1 + 1612) q^{23} + ( - 320 \beta_1 + 531) q^{25} + ( - 32 \beta_{3} - 336 \beta_{2}) q^{26} + ( - 784 \beta_1 + 446) q^{29} + (246 \beta_{3} + 772 \beta_{2}) q^{31} + (185 \beta_1 - 6860) q^{32} + ( - 629 \beta_{3} + 353 \beta_{2}) q^{34} + ( - 48 \beta_1 - 2326) q^{37} + (153 \beta_{3} + 215 \beta_{2}) q^{38} + (426 \beta_{3} + 370 \beta_{2}) q^{40} + (1103 \beta_{3} + 224 \beta_{2}) q^{41} + ( - 980 \beta_1 + 4622) q^{43} + ( - 2410 \beta_1 - 1680) q^{44} + ( - 2644 \beta_1 + 12856) q^{46} + (590 \beta_{3} - 1644 \beta_{2}) q^{47} + ( - 1491 \beta_1 + 10022) q^{50} + ( - 800 \beta_{3} - 2240 \beta_{2}) q^{52} + ( - 2592 \beta_1 + 24434) q^{53} + ( - 580 \beta_{3} + 3784 \beta_{2}) q^{55} + ( - 2798 \beta_1 + 22844) q^{58} + (1425 \beta_{3} - 2914 \beta_{2}) q^{59} + ( - 3326 \beta_{3} - 680 \beta_{2}) q^{61} + ( - 2350 \beta_{3} - 178 \beta_{2}) q^{62} + (3095 \beta_1 - 8532) q^{64} + (6496 \beta_1 + 19040) q^{65} + ( - 11632 \beta_1 - 11540) q^{67} + ( - 3075 \beta_{3} + 245 \beta_{2}) q^{68} + (4312 \beta_1 + 40612) q^{71} + (4407 \beta_{3} - 5080 \beta_{2}) q^{73} + (2182 \beta_1 - 3308) q^{74} + (1295 \beta_{3} + 1855 \beta_{2}) q^{76} + (4264 \beta_1 - 20540) q^{79} + (5238 \beta_{3} - 4482 \beta_{2}) q^{80} + (2413 \beta_{3} - 7273 \beta_{2}) q^{82} + (4425 \beta_{3} + 6230 \beta_{2}) q^{83} + ( - 2608 \beta_1 + 66860) q^{85} + ( - 7562 \beta_1 + 36684) q^{86} + (9106 \beta_1 + 43256) q^{88} + ( - 6259 \beta_{3} + 1016 \beta_{2}) q^{89} + ( - 9780 \beta_1 + 48160) q^{92} + (8346 \beta_{3} - 7418 \beta_{2}) q^{94} + ( - 5000 \beta_1 - 29556) q^{95} + ( - 5873 \beta_{3} + 7448 \beta_{2}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 10 q^{2} + 10 q^{4} + 270 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 10 q^{2} + 10 q^{4} + 270 q^{8} + 1952 q^{11} - 1566 q^{16} + 3524 q^{22} + 7136 q^{23} + 2764 q^{25} + 3352 q^{29} - 27810 q^{32} - 9208 q^{37} + 20448 q^{43} - 1900 q^{44} + 56712 q^{46} + 43070 q^{50} + 102920 q^{53} + 96972 q^{58} - 40318 q^{64} + 63168 q^{65} - 22896 q^{67} + 153824 q^{71} - 17596 q^{74} - 90688 q^{79} + 272656 q^{85} + 161860 q^{86} + 154812 q^{88} + 212200 q^{92} - 108224 q^{95}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( ( -2\nu^{3} + 3\nu^{2} + 172\nu - 139 ) / 105 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 51\nu^{2} - 86\nu - 1558 ) / 105 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2\nu^{3} - 3\nu^{2} - 67\nu + 34 ) / 15 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + 7\beta _1 + 7 ) / 7 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} + \beta _1 + 31 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 86\beta_{3} + 21\beta_{2} + 245\beta _1 + 441 ) / 7 \) 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
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 \(S_{6}^{\mathrm{new}}(\Gamma_0(441))\):

\( T_{2}^{2} - 5T_{2} - 22 \) Copy content Toggle raw display
\( T_{5}^{4} - 7632T_{5}^{2} + 11669056 \) Copy content Toggle raw display
\( T_{13}^{4} - 1260672T_{13}^{2} + 76158337024 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 5 T - 22)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 7632 T^{2} + \cdots + 11669056 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} - 976 T + 234076)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} - 1260672 T^{2} + \cdots + 76158337024 \) Copy content Toggle raw display
$17$ \( T^{4} - 2613668 T^{2} + \cdots + 1700202150724 \) Copy content Toggle raw display
$19$ \( T^{4} - 1359892 T^{2} + \cdots + 56942116 \) Copy content Toggle raw display
$23$ \( (T^{2} - 3568 T - 160336)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 1676 T - 16661788)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + \cdots + 614334295349824 \) Copy content Toggle raw display
$37$ \( (T^{2} + 4604 T + 5234116)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} - 251093444 T^{2} + \cdots + 14\!\cdots\!56 \) Copy content Toggle raw display
$43$ \( (T^{2} - 10224 T - 998756)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 349180624 T^{2} + \cdots + 17\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( (T^{2} - 51460 T + 472236292)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} - 1249753044 T^{2} + \cdots + 11\!\cdots\!76 \) Copy content Toggle raw display
$61$ \( T^{4} - 2284246736 T^{2} + \cdots + 11\!\cdots\!24 \) Copy content Toggle raw display
$67$ \( (T^{2} + 11448 T - 3789557552)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 76912 T + 953601968)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} - 6121721124 T^{2} + \cdots + 20\!\cdots\!44 \) Copy content Toggle raw display
$79$ \( (T^{2} + 45344 T + 386672)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} - 9034370100 T^{2} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{4} - 7617937028 T^{2} + \cdots + 13\!\cdots\!96 \) Copy content Toggle raw display
$97$ \( T^{4} - 11859566628 T^{2} + \cdots + 11\!\cdots\!44 \) Copy content Toggle raw display
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