Properties

Label 441.6.a.bd
Level $441$
Weight $6$
Character orbit 441.a
Self dual yes
Analytic conductor $70.729$
Analytic rank $0$
Dimension $6$
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: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Defining polynomial: \( x^{6} - 187x^{4} + 9570x^{2} - 135576 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{3} + 30) q^{4} + (\beta_{2} + 2 \beta_1) q^{5} + (\beta_{4} + 2 \beta_{2} + 21 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{3} + 30) q^{4} + (\beta_{2} + 2 \beta_1) q^{5} + (\beta_{4} + 2 \beta_{2} + 21 \beta_1) q^{8} + (8 \beta_{5} + 7 \beta_{3} + 112) q^{10} + ( - 2 \beta_{4} + 3 \beta_{2} - 24 \beta_1) q^{11} + ( - 13 \beta_{5} + 7 \beta_{3} - 28) q^{13} + (14 \beta_{5} + 43 \beta_{3} + 302) q^{16} + (10 \beta_{2} + 76 \beta_1) q^{17} + ( - \beta_{5} + 7 \beta_{3} + 1568) q^{19} + (7 \beta_{4} + 30 \beta_{2} + 249 \beta_1) q^{20} + (28 \beta_{5} - 97 \beta_{3} - 1492) q^{22} + (4 \beta_{4} + 22 \beta_{2} - 64 \beta_1) q^{23} + ( - 35 \beta_{5} + 25 \beta_{3} + 1246) q^{25} + (7 \beta_{4} - 64 \beta_{2} + 68 \beta_1) q^{26} + ( - 8 \beta_{4} + 61 \beta_{2} + 250 \beta_1) q^{29} + ( - 18 \beta_{5} + 140 \beta_{3} + 3857) q^{31} + (11 \beta_{4} + 106 \beta_{2} + 689 \beta_1) q^{32} + (80 \beta_{5} + 126 \beta_{3} + 4592) q^{34} + ( - 133 \beta_{5} - 89 \beta_{3} + 3060) q^{37} + (7 \beta_{4} + 8 \beta_{2} + 1724 \beta_1) q^{38} + ( - 30 \beta_{5} + 483 \beta_{3} + 11382) q^{40} + ( - 56 \beta_{4} - 104 \beta_{2} + 184 \beta_1) q^{41} + ( - 189 \beta_{5} - 33 \beta_{3} - 7270) q^{43} + ( - 33 \beta_{4} - 122 \beta_{2} - 2815 \beta_1) q^{44} + (168 \beta_{5} + 222 \beta_{3} - 4296) q^{46} + ( - 56 \beta_{4} + 68 \beta_{2} - 312 \beta_1) q^{47} + (25 \beta_{4} - 160 \beta_{2} + 1646 \beta_1) q^{50} + ( - 110 \beta_{5} - 168 \beta_{3} + 5768) q^{52} + ( - 175 \beta_{2} + 1202 \beta_1) q^{53} + ( - 153 \beta_{5} - 763 \beta_{3} + 8281) q^{55} + (504 \beta_{5} + 203 \beta_{3} + 14896) q^{58} + ( - 70 \beta_{4} - 83 \beta_{2} + 4440 \beta_1) q^{59} + (274 \beta_{5} - 350 \beta_{3} - 2576) q^{61} + (140 \beta_{4} + 172 \beta_{2} + 6987 \beta_1) q^{62} + (378 \beta_{5} + 327 \beta_{3} + 31606) q^{64} + (88 \beta_{4} + 414 \beta_{2} - 3972 \beta_1) q^{65} + (119 \beta_{5} - 313 \beta_{3} - 24004) q^{67} + (126 \beta_{4} + 412 \beta_{2} + 5458 \beta_1) q^{68} + ( - 48 \beta_{4} - 264 \beta_{2} + 3336 \beta_1) q^{71} + (259 \beta_{5} + 119 \beta_{3} - 16716) q^{73} + ( - 89 \beta_{4} - 976 \beta_{2} + 348 \beta_1) q^{74} + (82 \beta_{5} + 1848 \beta_{3} + 56504) q^{76} + ( - 812 \beta_{5} - 714 \beta_{3} - 16761) q^{79} + (259 \beta_{4} - 174 \beta_{2} + 14373 \beta_1) q^{80} + ( - 720 \beta_{5} - 2800 \beta_{3} + 13552) q^{82} + ( - 14 \beta_{4} - 567 \beta_{2} + 4312 \beta_1) q^{83} + (98 \beta_{5} + 642 \beta_{3} + 49982) q^{85} + ( - 33 \beta_{4} - 1200 \beta_{2} - 8974 \beta_1) q^{86} + ( - 1806 \beta_{5} - 1773 \beta_{3} - 124794) q^{88} + (112 \beta_{4} + 190 \beta_{2} + 492 \beta_1) q^{89} + (94 \beta_{4} + 748 \beta_{2} + 3698 \beta_1) q^{92} + (656 \beta_{5} - 2436 \beta_{3} - 19264) q^{94} + (52 \beta_{4} + 1602 \beta_{2} + 4056 \beta_1) q^{95} + (7 \beta_{5} - 2149 \beta_{3} + 72891) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 182 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 182 q^{4} + 686 q^{10} - 154 q^{13} + 1898 q^{16} + 9422 q^{19} - 9146 q^{22} + 7526 q^{25} + 23422 q^{31} + 27804 q^{34} + 18182 q^{37} + 69258 q^{40} - 43686 q^{43} - 25332 q^{46} + 34272 q^{52} + 48160 q^{55} + 89782 q^{58} - 16156 q^{61} + 190290 q^{64} - 144650 q^{67} - 100058 q^{73} + 342720 q^{76} - 101994 q^{79} + 75712 q^{82} + 301176 q^{85} - 752310 q^{88} - 120456 q^{94} + 433048 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 187x^{4} + 9570x^{2} - 135576 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} - 139\nu^{3} + 3318\nu ) / 84 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} - 62 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{5} + 181\nu^{3} - 6888\nu ) / 42 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{4} - 139\nu^{2} + 3388 ) / 14 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 62 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{4} + 2\beta_{2} + 85\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 14\beta_{5} + 139\beta_{3} + 5230 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 139\beta_{4} + 362\beta_{2} + 8497\beta_1 \) 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
−10.6223
−7.08933
−4.88952
4.88952
7.08933
10.6223
−10.6223 0 80.8340 −67.4751 0 0 −518.731 0 716.743
1.2 −7.08933 0 18.2586 82.2041 0 0 97.4172 0 −582.772
1.3 −4.88952 0 −8.09260 −42.7504 0 0 196.034 0 209.029
1.4 4.88952 0 −8.09260 42.7504 0 0 −196.034 0 209.029
1.5 7.08933 0 18.2586 −82.2041 0 0 −97.4172 0 −582.772
1.6 10.6223 0 80.8340 67.4751 0 0 518.731 0 716.743
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
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.bd 6
3.b odd 2 1 inner 441.6.a.bd 6
7.b odd 2 1 441.6.a.bc 6
7.d odd 6 2 63.6.e.f 12
21.c even 2 1 441.6.a.bc 6
21.g even 6 2 63.6.e.f 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.6.e.f 12 7.d odd 6 2
63.6.e.f 12 21.g even 6 2
441.6.a.bc 6 7.b odd 2 1
441.6.a.bc 6 21.c even 2 1
441.6.a.bd 6 1.a even 1 1 trivial
441.6.a.bd 6 3.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(441))\):

\( T_{2}^{6} - 187T_{2}^{4} + 9570T_{2}^{2} - 135576 \) Copy content Toggle raw display
\( T_{5}^{6} - 13138T_{5}^{4} + 51437073T_{5}^{2} - 56228247936 \) Copy content Toggle raw display
\( T_{13}^{3} + 77T_{13}^{2} - 783976T_{13} - 60080636 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} - 187 T^{4} + 9570 T^{2} + \cdots - 135576 \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} - 13138 T^{4} + \cdots - 56228247936 \) Copy content Toggle raw display
$7$ \( T^{6} \) Copy content Toggle raw display
$11$ \( T^{6} - 902578 T^{4} + \cdots - 17\!\cdots\!76 \) Copy content Toggle raw display
$13$ \( (T^{3} + 77 T^{2} - 783976 T - 60080636)^{2} \) Copy content Toggle raw display
$17$ \( T^{6} - 2284392 T^{4} + \cdots - 14\!\cdots\!76 \) Copy content Toggle raw display
$19$ \( (T^{3} - 4711 T^{2} + \cdots - 3713875508)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} - 10191432 T^{4} + \cdots - 11\!\cdots\!84 \) Copy content Toggle raw display
$29$ \( T^{6} - 66189634 T^{4} + \cdots - 75\!\cdots\!56 \) Copy content Toggle raw display
$31$ \( (T^{3} - 11711 T^{2} + \cdots + 85806884751)^{2} \) Copy content Toggle raw display
$37$ \( (T^{3} - 9091 T^{2} + \cdots + 32349623604)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} - 727900480 T^{4} + \cdots - 13\!\cdots\!64 \) Copy content Toggle raw display
$43$ \( (T^{3} + 21843 T^{2} + \cdots - 1643621929904)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} - 613454880 T^{4} + \cdots - 19\!\cdots\!64 \) Copy content Toggle raw display
$53$ \( T^{6} - 666461298 T^{4} + \cdots - 16\!\cdots\!64 \) Copy content Toggle raw display
$59$ \( T^{6} - 4695322386 T^{4} + \cdots - 31\!\cdots\!04 \) Copy content Toggle raw display
$61$ \( (T^{3} + 8078 T^{2} + \cdots - 5346133312352)^{2} \) Copy content Toggle raw display
$67$ \( (T^{3} + 72325 T^{2} + \cdots + 7112548657724)^{2} \) Copy content Toggle raw display
$71$ \( T^{6} - 3481391808 T^{4} + \cdots - 30\!\cdots\!44 \) Copy content Toggle raw display
$73$ \( (T^{3} + 50029 T^{2} + \cdots + 937824660612)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} + 50997 T^{2} + \cdots - 111559545344717)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} - 7727019426 T^{4} + \cdots - 11\!\cdots\!64 \) Copy content Toggle raw display
$89$ \( T^{6} - 2820380296 T^{4} + \cdots - 60\!\cdots\!96 \) Copy content Toggle raw display
$97$ \( (T^{3} - 216524 T^{2} + \cdots + 546802680855102)^{2} \) Copy content Toggle raw display
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