Properties

Label 546.8.a.n
Level $546$
Weight $8$
Character orbit 546.a
Self dual yes
Analytic conductor $170.562$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 546.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(170.562223914\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Defining polynomial: \( x^{6} - x^{5} - 309949x^{4} - 14548431x^{3} + 25221499020x^{2} + 1862570808000x - 308009568384000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{5}\cdot 3^{4} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q - 8 q^{2} - 27 q^{3} + 64 q^{4} + ( - \beta_1 - 30) q^{5} + 216 q^{6} + 343 q^{7} - 512 q^{8} + 729 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 8 q^{2} - 27 q^{3} + 64 q^{4} + ( - \beta_1 - 30) q^{5} + 216 q^{6} + 343 q^{7} - 512 q^{8} + 729 q^{9} + (8 \beta_1 + 240) q^{10} + ( - \beta_{2} - 3 \beta_1 - 1021) q^{11} - 1728 q^{12} + 2197 q^{13} - 2744 q^{14} + (27 \beta_1 + 810) q^{15} + 4096 q^{16} + ( - \beta_{4} - 2 \beta_{3} - 2 \beta_{2} + \beta_1 - 5768) q^{17} - 5832 q^{18} + (3 \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + 31 \beta_1 - 687) q^{19} + ( - 64 \beta_1 - 1920) q^{20} - 9261 q^{21} + (8 \beta_{2} + 24 \beta_1 + 8168) q^{22} + ( - 2 \beta_{5} - 3 \beta_{4} + 4 \beta_{3} - 11 \beta_{2} + 3 \beta_1 + 257) q^{23} + 13824 q^{24} + (\beta_{5} + \beta_{4} - 7 \beta_{3} - 5 \beta_{2} + 133 \beta_1 + 26078) q^{25} - 17576 q^{26} - 19683 q^{27} + 21952 q^{28} + ( - 7 \beta_{5} + 10 \beta_{4} + 2 \beta_{3} - 23 \beta_{2} - 21 \beta_1 - 9893) q^{29} + ( - 216 \beta_1 - 6480) q^{30} + ( - 2 \beta_{5} + 7 \beta_{4} - 3 \beta_{3} - 13 \beta_{2} + 18 \beta_1 + 79703) q^{31} - 32768 q^{32} + (27 \beta_{2} + 81 \beta_1 + 27567) q^{33} + (8 \beta_{4} + 16 \beta_{3} + 16 \beta_{2} - 8 \beta_1 + 46144) q^{34} + ( - 343 \beta_1 - 10290) q^{35} + 46656 q^{36} + (\beta_{5} - 15 \beta_{4} + 29 \beta_{3} + 24 \beta_{2} - 47 \beta_1 + 95734) q^{37} + ( - 24 \beta_{5} + 8 \beta_{4} - 8 \beta_{3} - 8 \beta_{2} - 248 \beta_1 + 5496) q^{38} - 59319 q^{39} + (512 \beta_1 + 15360) q^{40} + (9 \beta_{5} + \beta_{4} - 43 \beta_{3} - 69 \beta_{2} - 34 \beta_1 + 33597) q^{41} + 74088 q^{42} + (27 \beta_{5} - 18 \beta_{4} - 4 \beta_{3} - 29 \beta_{2} - 679 \beta_1 + 121547) q^{43} + ( - 64 \beta_{2} - 192 \beta_1 - 65344) q^{44} + ( - 729 \beta_1 - 21870) q^{45} + (16 \beta_{5} + 24 \beta_{4} - 32 \beta_{3} + 88 \beta_{2} - 24 \beta_1 - 2056) q^{46} + (10 \beta_{5} - 58 \beta_{4} - 3 \beta_{3} + 32 \beta_{2} - 684 \beta_1 + 38068) q^{47} - 110592 q^{48} + 117649 q^{49} + ( - 8 \beta_{5} - 8 \beta_{4} + 56 \beta_{3} + 40 \beta_{2} + \cdots - 208624) q^{50}+ \cdots + ( - 729 \beta_{2} - 2187 \beta_1 - 744309) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 48 q^{2} - 162 q^{3} + 384 q^{4} - 181 q^{5} + 1296 q^{6} + 2058 q^{7} - 3072 q^{8} + 4374 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 48 q^{2} - 162 q^{3} + 384 q^{4} - 181 q^{5} + 1296 q^{6} + 2058 q^{7} - 3072 q^{8} + 4374 q^{9} + 1448 q^{10} - 6130 q^{11} - 10368 q^{12} + 13182 q^{13} - 16464 q^{14} + 4887 q^{15} + 24576 q^{16} - 34610 q^{17} - 34992 q^{18} - 4085 q^{19} - 11584 q^{20} - 55566 q^{21} + 49040 q^{22} + 1515 q^{23} + 82944 q^{24} + 156609 q^{25} - 105456 q^{26} - 118098 q^{27} + 131712 q^{28} - 59395 q^{29} - 39096 q^{30} + 478241 q^{31} - 196608 q^{32} + 165510 q^{33} + 276880 q^{34} - 62083 q^{35} + 279936 q^{36} + 574310 q^{37} + 32680 q^{38} - 355914 q^{39} + 92672 q^{40} + 201552 q^{41} + 444528 q^{42} + 728605 q^{43} - 392320 q^{44} - 131949 q^{45} - 12120 q^{46} + 227615 q^{47} - 663552 q^{48} + 705894 q^{49} - 1252872 q^{50} + 934470 q^{51} + 843648 q^{52} + 26321 q^{53} + 944784 q^{54} + 2115010 q^{55} - 1053696 q^{56} + 110295 q^{57} + 475160 q^{58} + 478280 q^{59} + 312768 q^{60} - 501406 q^{61} - 3825928 q^{62} + 1500282 q^{63} + 1572864 q^{64} - 397657 q^{65} - 1324080 q^{66} - 3156366 q^{67} - 2215040 q^{68} - 40905 q^{69} + 496664 q^{70} - 2003644 q^{71} - 2239488 q^{72} + 3659111 q^{73} - 4594480 q^{74} - 4228443 q^{75} - 261440 q^{76} - 2102590 q^{77} + 2847312 q^{78} + 1131065 q^{79} - 741376 q^{80} + 3188646 q^{81} - 1612416 q^{82} - 9629297 q^{83} - 3556224 q^{84} + 895068 q^{85} - 5828840 q^{86} + 1603665 q^{87} + 3138560 q^{88} - 21977377 q^{89} + 1055592 q^{90} + 4521426 q^{91} + 96960 q^{92} - 12912507 q^{93} - 1820920 q^{94} - 19325507 q^{95} + 5308416 q^{96} - 26386649 q^{97} - 5647152 q^{98} - 4468770 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} - 309949x^{4} - 14548431x^{3} + 25221499020x^{2} + 1862570808000x - 308009568384000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 2024389 \nu^{5} + 1548086629 \nu^{4} + 626570347621 \nu^{3} - 319874939700201 \nu^{2} + \cdots + 94\!\cdots\!00 ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 27588299 \nu^{5} + 2827655399 \nu^{4} + 5325442304951 \nu^{3} + \cdots + 89\!\cdots\!00 ) / 74\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 1384243 \nu^{5} + 593013523 \nu^{4} + 248055256927 \nu^{3} - 92775859244187 \nu^{2} + \cdots + 18\!\cdots\!00 ) / 117804753346800 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 83321227 \nu^{5} + 14438167357 \nu^{4} + 20223862688443 \nu^{3} + \cdots + 12\!\cdots\!00 ) / 37\!\cdots\!00 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + \beta_{4} - 7\beta_{3} - 5\beta_{2} + 73\beta _1 + 103303 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 425\beta_{5} - 771\beta_{4} - 1055\beta_{3} + 2101\beta_{2} + 147337\beta _1 + 7404281 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 105869\beta_{5} + 316233\beta_{4} - 1485023\beta_{3} - 349969\beta_{2} + 22069185\beta _1 + 15214288683 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 54491425 \beta_{5} - 154814415 \beta_{4} - 356083435 \beta_{3} + 561683925 \beta_{2} + 24495440641 \beta _1 + 2260450105145 \) 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
457.447
348.888
83.4678
−193.695
−309.836
−385.273
−8.00000 −27.0000 64.0000 −487.447 216.000 343.000 −512.000 729.000 3899.58
1.2 −8.00000 −27.0000 64.0000 −378.888 216.000 343.000 −512.000 729.000 3031.11
1.3 −8.00000 −27.0000 64.0000 −113.468 216.000 343.000 −512.000 729.000 907.742
1.4 −8.00000 −27.0000 64.0000 163.695 216.000 343.000 −512.000 729.000 −1309.56
1.5 −8.00000 −27.0000 64.0000 279.836 216.000 343.000 −512.000 729.000 −2238.69
1.6 −8.00000 −27.0000 64.0000 355.273 216.000 343.000 −512.000 729.000 −2842.18
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(1\)
\(7\) \(-1\)
\(13\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 546.8.a.n 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.8.a.n 6 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} + 181T_{5}^{5} - 296299T_{5}^{4} - 22096449T_{5}^{3} + 24869553210T_{5}^{2} - 343324745100T_{5} - 341044841259000 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(546))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 8)^{6} \) Copy content Toggle raw display
$3$ \( (T + 27)^{6} \) Copy content Toggle raw display
$5$ \( T^{6} + \cdots - 341044841259000 \) Copy content Toggle raw display
$7$ \( (T - 343)^{6} \) Copy content Toggle raw display
$11$ \( T^{6} + 6130 T^{5} + \cdots - 31\!\cdots\!36 \) Copy content Toggle raw display
$13$ \( (T - 2197)^{6} \) Copy content Toggle raw display
$17$ \( T^{6} + 34610 T^{5} + \cdots + 35\!\cdots\!08 \) Copy content Toggle raw display
$19$ \( T^{6} + 4085 T^{5} + \cdots - 37\!\cdots\!40 \) Copy content Toggle raw display
$23$ \( T^{6} - 1515 T^{5} + \cdots - 16\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{6} + 59395 T^{5} + \cdots - 21\!\cdots\!16 \) Copy content Toggle raw display
$31$ \( T^{6} - 478241 T^{5} + \cdots - 24\!\cdots\!60 \) Copy content Toggle raw display
$37$ \( T^{6} - 574310 T^{5} + \cdots + 38\!\cdots\!56 \) Copy content Toggle raw display
$41$ \( T^{6} - 201552 T^{5} + \cdots + 41\!\cdots\!08 \) Copy content Toggle raw display
$43$ \( T^{6} - 728605 T^{5} + \cdots + 64\!\cdots\!16 \) Copy content Toggle raw display
$47$ \( T^{6} - 227615 T^{5} + \cdots + 11\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{6} - 26321 T^{5} + \cdots - 48\!\cdots\!60 \) Copy content Toggle raw display
$59$ \( T^{6} - 478280 T^{5} + \cdots - 16\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{6} + 501406 T^{5} + \cdots - 13\!\cdots\!20 \) Copy content Toggle raw display
$67$ \( T^{6} + 3156366 T^{5} + \cdots + 10\!\cdots\!32 \) Copy content Toggle raw display
$71$ \( T^{6} + 2003644 T^{5} + \cdots - 16\!\cdots\!20 \) Copy content Toggle raw display
$73$ \( T^{6} - 3659111 T^{5} + \cdots + 39\!\cdots\!28 \) Copy content Toggle raw display
$79$ \( T^{6} - 1131065 T^{5} + \cdots - 28\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{6} + 9629297 T^{5} + \cdots + 29\!\cdots\!28 \) Copy content Toggle raw display
$89$ \( T^{6} + 21977377 T^{5} + \cdots - 15\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{6} + 26386649 T^{5} + \cdots - 27\!\cdots\!80 \) Copy content Toggle raw display
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