Properties

Label 546.8.a.q
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: \(x^{6} - x^{5} - 367021 x^{4} - 17702143 x^{3} + 34815194576 x^{2} + 1422988371620 x - 933871993059968\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{3}\cdot 3^{2} \)
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.

\(f(q)\) \(=\) \( q + 8 q^{2} -27 q^{3} + 64 q^{4} + ( 2 + \beta_{1} ) q^{5} -216 q^{6} + 343 q^{7} + 512 q^{8} + 729 q^{9} +O(q^{10})\) \( q + 8 q^{2} -27 q^{3} + 64 q^{4} + ( 2 + \beta_{1} ) q^{5} -216 q^{6} + 343 q^{7} + 512 q^{8} + 729 q^{9} + ( 16 + 8 \beta_{1} ) q^{10} + ( 1676 - \beta_{1} - \beta_{2} + \beta_{4} ) q^{11} -1728 q^{12} -2197 q^{13} + 2744 q^{14} + ( -54 - 27 \beta_{1} ) q^{15} + 4096 q^{16} + ( 3535 + 7 \beta_{1} + \beta_{2} + 2 \beta_{4} + \beta_{5} ) q^{17} + 5832 q^{18} + ( 1583 + 22 \beta_{1} - \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{19} + ( 128 + 64 \beta_{1} ) q^{20} -9261 q^{21} + ( 13408 - 8 \beta_{1} - 8 \beta_{2} + 8 \beta_{4} ) q^{22} + ( 5529 + 41 \beta_{1} + 6 \beta_{3} + 9 \beta_{4} - \beta_{5} ) q^{23} -13824 q^{24} + ( 44210 + 80 \beta_{1} - 4 \beta_{2} - 7 \beta_{3} - 4 \beta_{4} + 3 \beta_{5} ) q^{25} -17576 q^{26} -19683 q^{27} + 21952 q^{28} + ( 29036 - 12 \beta_{1} + 3 \beta_{2} + 12 \beta_{3} - 11 \beta_{4} - 3 \beta_{5} ) q^{29} + ( -432 - 216 \beta_{1} ) q^{30} + ( 19835 + 116 \beta_{1} - 22 \beta_{2} - 17 \beta_{3} - 18 \beta_{4} - 2 \beta_{5} ) q^{31} + 32768 q^{32} + ( -45252 + 27 \beta_{1} + 27 \beta_{2} - 27 \beta_{4} ) q^{33} + ( 28280 + 56 \beta_{1} + 8 \beta_{2} + 16 \beta_{4} + 8 \beta_{5} ) q^{34} + ( 686 + 343 \beta_{1} ) q^{35} + 46656 q^{36} + ( 9405 + 282 \beta_{1} - 29 \beta_{2} - 5 \beta_{3} - 33 \beta_{4} + \beta_{5} ) q^{37} + ( 12664 + 176 \beta_{1} - 8 \beta_{3} + 16 \beta_{4} - 8 \beta_{5} ) q^{38} + 59319 q^{39} + ( 1024 + 512 \beta_{1} ) q^{40} + ( 16867 + 385 \beta_{1} + 8 \beta_{2} + 33 \beta_{3} + 30 \beta_{4} + 3 \beta_{5} ) q^{41} -74088 q^{42} + ( 73504 + 164 \beta_{1} - 5 \beta_{2} + 2 \beta_{3} + 57 \beta_{4} - 19 \beta_{5} ) q^{43} + ( 107264 - 64 \beta_{1} - 64 \beta_{2} + 64 \beta_{4} ) q^{44} + ( 1458 + 729 \beta_{1} ) q^{45} + ( 44232 + 328 \beta_{1} + 48 \beta_{3} + 72 \beta_{4} - 8 \beta_{5} ) q^{46} + ( -148962 + 1088 \beta_{1} - 40 \beta_{2} + 13 \beta_{3} + 3 \beta_{4} + 19 \beta_{5} ) q^{47} -110592 q^{48} + 117649 q^{49} + ( 353680 + 640 \beta_{1} - 32 \beta_{2} - 56 \beta_{3} - 32 \beta_{4} + 24 \beta_{5} ) q^{50} + ( -95445 - 189 \beta_{1} - 27 \beta_{2} - 54 \beta_{4} - 27 \beta_{5} ) q^{51} -140608 q^{52} + ( 348745 + 764 \beta_{1} + 270 \beta_{2} + 7 \beta_{3} - 18 \beta_{4} + 18 \beta_{5} ) q^{53} -157464 q^{54} + ( -55691 + 2422 \beta_{1} + 247 \beta_{2} - 169 \beta_{3} - 223 \beta_{4} - 19 \beta_{5} ) q^{55} + 175616 q^{56} + ( -42741 - 594 \beta_{1} + 27 \beta_{3} - 54 \beta_{4} + 27 \beta_{5} ) q^{57} + ( 232288 - 96 \beta_{1} + 24 \beta_{2} + 96 \beta_{3} - 88 \beta_{4} - 24 \beta_{5} ) q^{58} + ( -22587 - 870 \beta_{1} + 20 \beta_{2} + 80 \beta_{3} + 107 \beta_{4} + 3 \beta_{5} ) q^{59} + ( -3456 - 1728 \beta_{1} ) q^{60} + ( -665212 + 836 \beta_{1} + 240 \beta_{2} - 80 \beta_{3} - 136 \beta_{4} + 19 \beta_{5} ) q^{61} + ( 158680 + 928 \beta_{1} - 176 \beta_{2} - 136 \beta_{3} - 144 \beta_{4} - 16 \beta_{5} ) q^{62} + 250047 q^{63} + 262144 q^{64} + ( -4394 - 2197 \beta_{1} ) q^{65} + ( -362016 + 216 \beta_{1} + 216 \beta_{2} - 216 \beta_{4} ) q^{66} + ( -369208 - 2310 \beta_{1} - 212 \beta_{2} + 240 \beta_{3} + 92 \beta_{4} ) q^{67} + ( 226240 + 448 \beta_{1} + 64 \beta_{2} + 128 \beta_{4} + 64 \beta_{5} ) q^{68} + ( -149283 - 1107 \beta_{1} - 162 \beta_{3} - 243 \beta_{4} + 27 \beta_{5} ) q^{69} + ( 5488 + 2744 \beta_{1} ) q^{70} + ( 341157 - 523 \beta_{1} - 398 \beta_{2} - 161 \beta_{3} + 330 \beta_{4} + 59 \beta_{5} ) q^{71} + 373248 q^{72} + ( -1425850 - 2104 \beta_{1} - 91 \beta_{2} + 382 \beta_{3} + 183 \beta_{4} - 7 \beta_{5} ) q^{73} + ( 75240 + 2256 \beta_{1} - 232 \beta_{2} - 40 \beta_{3} - 264 \beta_{4} + 8 \beta_{5} ) q^{74} + ( -1193670 - 2160 \beta_{1} + 108 \beta_{2} + 189 \beta_{3} + 108 \beta_{4} - 81 \beta_{5} ) q^{75} + ( 101312 + 1408 \beta_{1} - 64 \beta_{3} + 128 \beta_{4} - 64 \beta_{5} ) q^{76} + ( 574868 - 343 \beta_{1} - 343 \beta_{2} + 343 \beta_{4} ) q^{77} + 474552 q^{78} + ( -1426895 - 292 \beta_{1} + 660 \beta_{2} - 61 \beta_{3} - 210 \beta_{4} - 166 \beta_{5} ) q^{79} + ( 8192 + 4096 \beta_{1} ) q^{80} + 531441 q^{81} + ( 134936 + 3080 \beta_{1} + 64 \beta_{2} + 264 \beta_{3} + 240 \beta_{4} + 24 \beta_{5} ) q^{82} + ( 416502 - 2504 \beta_{1} - 370 \beta_{2} - 89 \beta_{3} + 507 \beta_{4} + 75 \beta_{5} ) q^{83} -592704 q^{84} + ( 887887 + 9756 \beta_{1} + 61 \beta_{2} - 447 \beta_{3} - 969 \beta_{4} + 233 \beta_{5} ) q^{85} + ( 588032 + 1312 \beta_{1} - 40 \beta_{2} + 16 \beta_{3} + 456 \beta_{4} - 152 \beta_{5} ) q^{86} + ( -783972 + 324 \beta_{1} - 81 \beta_{2} - 324 \beta_{3} + 297 \beta_{4} + 81 \beta_{5} ) q^{87} + ( 858112 - 512 \beta_{1} - 512 \beta_{2} + 512 \beta_{4} ) q^{88} + ( -408108 + 3205 \beta_{1} - 256 \beta_{2} + 46 \beta_{3} - 430 \beta_{4} - 217 \beta_{5} ) q^{89} + ( 11664 + 5832 \beta_{1} ) q^{90} -753571 q^{91} + ( 353856 + 2624 \beta_{1} + 384 \beta_{3} + 576 \beta_{4} - 64 \beta_{5} ) q^{92} + ( -535545 - 3132 \beta_{1} + 594 \beta_{2} + 459 \beta_{3} + 486 \beta_{4} + 54 \beta_{5} ) q^{93} + ( -1191696 + 8704 \beta_{1} - 320 \beta_{2} + 104 \beta_{3} + 24 \beta_{4} + 152 \beta_{5} ) q^{94} + ( 2735991 - 3322 \beta_{1} - 998 \beta_{2} - 459 \beta_{3} - 38 \beta_{4} - 99 \beta_{5} ) q^{95} -884736 q^{96} + ( 965231 + 1184 \beta_{1} - 1062 \beta_{2} + 199 \beta_{3} - 896 \beta_{4} + 252 \beta_{5} ) q^{97} + 941192 q^{98} + ( 1221804 - 729 \beta_{1} - 729 \beta_{2} + 729 \beta_{4} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 48 q^{2} - 162 q^{3} + 384 q^{4} + 13 q^{5} - 1296 q^{6} + 2058 q^{7} + 3072 q^{8} + 4374 q^{9} + O(q^{10}) \) \( 6 q + 48 q^{2} - 162 q^{3} + 384 q^{4} + 13 q^{5} - 1296 q^{6} + 2058 q^{7} + 3072 q^{8} + 4374 q^{9} + 104 q^{10} + 10054 q^{11} - 10368 q^{12} - 13182 q^{13} + 16464 q^{14} - 351 q^{15} + 24576 q^{16} + 21222 q^{17} + 34992 q^{18} + 9527 q^{19} + 832 q^{20} - 55566 q^{21} + 80432 q^{22} + 33229 q^{23} - 82944 q^{24} + 265321 q^{25} - 105456 q^{26} - 118098 q^{27} + 131712 q^{28} + 174185 q^{29} - 2808 q^{30} + 119045 q^{31} + 196608 q^{32} - 271458 q^{33} + 169776 q^{34} + 4459 q^{35} + 279936 q^{36} + 56562 q^{37} + 76216 q^{38} + 355914 q^{39} + 6656 q^{40} + 101632 q^{41} - 444528 q^{42} + 441323 q^{43} + 643456 q^{44} + 9477 q^{45} + 265832 q^{46} - 892849 q^{47} - 663552 q^{48} + 705894 q^{49} + 2122568 q^{50} - 572994 q^{51} - 843648 q^{52} + 2093965 q^{53} - 944784 q^{54} - 331222 q^{55} + 1053696 q^{56} - 257229 q^{57} + 1393480 q^{58} - 136204 q^{59} - 22464 q^{60} - 3989946 q^{61} + 952360 q^{62} + 1500282 q^{63} + 1572864 q^{64} - 28561 q^{65} - 2171664 q^{66} - 2218250 q^{67} + 1358208 q^{68} - 897183 q^{69} + 35672 q^{70} + 2045928 q^{71} + 2239488 q^{72} - 8557479 q^{73} + 452496 q^{74} - 7163667 q^{75} + 609728 q^{76} + 3448522 q^{77} + 2847312 q^{78} - 8559709 q^{79} + 53248 q^{80} + 3188646 q^{81} + 813056 q^{82} + 2496351 q^{83} - 3556224 q^{84} + 5335304 q^{85} + 3530584 q^{86} - 4702995 q^{87} + 5147648 q^{88} - 2446683 q^{89} + 75816 q^{90} - 4521426 q^{91} + 2126656 q^{92} - 3214215 q^{93} - 7142792 q^{94} + 16410211 q^{95} - 5308416 q^{96} + 5786889 q^{97} + 5647152 q^{98} + 7329366 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - x^{5} - 367021 x^{4} - 17702143 x^{3} + 34815194576 x^{2} + 1422988371620 x - 933871993059968\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 112991 \nu^{5} - 64357478 \nu^{4} - 21066200065 \nu^{3} + 12643613243992 \nu^{2} + 623871092998772 \nu - 446985120240999584 \)\()/ 45155141220000 \)
\(\beta_{3}\)\(=\)\((\)\( -83303 \nu^{5} + 2441274 \nu^{4} + 22810191145 \nu^{3} + 791497857564 \nu^{2} - 916133306279276 \nu + 3427013982382272 \)\()/ 12901468920000 \)
\(\beta_{4}\)\(=\)\((\)\( -801047 \nu^{5} + 247784626 \nu^{4} + 240345178105 \nu^{3} - 60185401146764 \nu^{2} - 15449775512953724 \nu + 3270365418802308928 \)\()/ 90310282440000 \)
\(\beta_{5}\)\(=\)\((\)\( -709123 \nu^{5} + 66211234 \nu^{4} + 212283386445 \nu^{3} - 1166557908276 \nu^{2} - 12062471864549916 \nu - 152693640305518848 \)\()/ 30103427480000 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(3 \beta_{5} - 4 \beta_{4} - 7 \beta_{3} - 4 \beta_{2} + 76 \beta_{1} + 122331\)
\(\nu^{3}\)\(=\)\(911 \beta_{5} + 362 \beta_{4} - 3169 \beta_{3} + 1682 \beta_{2} + 178698 \beta_{1} + 9003613\)
\(\nu^{4}\)\(=\)\(821445 \beta_{5} - 840260 \beta_{4} - 2372105 \beta_{3} - 1366460 \beta_{2} + 35842898 \beta_{1} + 21698255837\)
\(\nu^{5}\)\(=\)\(302029339 \beta_{5} + 36493198 \beta_{4} - 1158642641 \beta_{3} + 382517398 \beta_{2} + 39706353950 \beta_{1} + 4304734040093\)

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
−419.580
−309.150
−264.243
180.926
285.402
527.645
8.00000 −27.0000 64.0000 −417.580 −216.000 343.000 512.000 729.000 −3340.64
1.2 8.00000 −27.0000 64.0000 −307.150 −216.000 343.000 512.000 729.000 −2457.20
1.3 8.00000 −27.0000 64.0000 −262.243 −216.000 343.000 512.000 729.000 −2097.94
1.4 8.00000 −27.0000 64.0000 182.926 −216.000 343.000 512.000 729.000 1463.41
1.5 8.00000 −27.0000 64.0000 287.402 −216.000 343.000 512.000 729.000 2299.22
1.6 8.00000 −27.0000 64.0000 529.645 −216.000 343.000 512.000 729.000 4237.16
\(n\): 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.q 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.8.a.q 6 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} - 13 T_{5}^{5} - 366951 T_{5}^{4} - 14766175 T_{5}^{3} + 34912599250 T_{5}^{2} + \)\(12\!\cdots\!00\)\( T_{5} - \)\(93\!\cdots\!00\)\( \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(546))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -8 + T )^{6} \)
$3$ \( ( 27 + T )^{6} \)
$5$ \( -936578573280000 + 1283526912000 T + 34912599250 T^{2} - 14766175 T^{3} - 366951 T^{4} - 13 T^{5} + T^{6} \)
$7$ \( ( -343 + T )^{6} \)
$11$ \( \)\(16\!\cdots\!68\)\( - 6781412275063368024 T - 917226338618324 T^{2} + 673184547830 T^{3} - 44295597 T^{4} - 10054 T^{5} + T^{6} \)
$13$ \( ( 2197 + T )^{6} \)
$17$ \( -\)\(16\!\cdots\!28\)\( - \)\(27\!\cdots\!64\)\( T + 266140678897465804 T^{2} + 28878282570126 T^{3} - 1479057917 T^{4} - 21222 T^{5} + T^{6} \)
$19$ \( -\)\(16\!\cdots\!64\)\( - \)\(24\!\cdots\!44\)\( T + 473325456102939028 T^{2} - 3025888088429 T^{3} - 1876151505 T^{4} - 9527 T^{5} + T^{6} \)
$23$ \( -\)\(34\!\cdots\!52\)\( - \)\(42\!\cdots\!00\)\( T + 34413281424804396540 T^{2} + 246832081463217 T^{3} - 10546427729 T^{4} - 33229 T^{5} + T^{6} \)
$29$ \( \)\(92\!\cdots\!32\)\( - \)\(36\!\cdots\!64\)\( T + \)\(17\!\cdots\!14\)\( T^{2} + 5589489051926901 T^{3} - 35578686587 T^{4} - 174185 T^{5} + T^{6} \)
$31$ \( -\)\(77\!\cdots\!80\)\( - \)\(51\!\cdots\!08\)\( T + \)\(36\!\cdots\!76\)\( T^{2} + 18744313478857840 T^{3} - 130082373580 T^{4} - 119045 T^{5} + T^{6} \)
$37$ \( \)\(43\!\cdots\!72\)\( - \)\(94\!\cdots\!84\)\( T + \)\(25\!\cdots\!04\)\( T^{2} + 48750536762135446 T^{3} - 239344024977 T^{4} - 56562 T^{5} + T^{6} \)
$41$ \( \)\(37\!\cdots\!08\)\( + \)\(18\!\cdots\!00\)\( T + \)\(18\!\cdots\!44\)\( T^{2} - 12655318554577296 T^{3} - 305083615196 T^{4} - 101632 T^{5} + T^{6} \)
$43$ \( \)\(26\!\cdots\!00\)\( + \)\(78\!\cdots\!80\)\( T + \)\(63\!\cdots\!88\)\( T^{2} + 46555280885979651 T^{3} - 534916915253 T^{4} - 441323 T^{5} + T^{6} \)
$47$ \( \)\(70\!\cdots\!52\)\( + \)\(91\!\cdots\!32\)\( T + \)\(82\!\cdots\!36\)\( T^{2} - 592637865945161540 T^{3} - 661748750880 T^{4} + 892849 T^{5} + T^{6} \)
$53$ \( \)\(27\!\cdots\!16\)\( - \)\(56\!\cdots\!00\)\( T - \)\(28\!\cdots\!48\)\( T^{2} + 10163463538020989000 T^{3} - 3525714964198 T^{4} - 2093965 T^{5} + T^{6} \)
$59$ \( \)\(46\!\cdots\!20\)\( + \)\(32\!\cdots\!72\)\( T + \)\(47\!\cdots\!24\)\( T^{2} - 867547726719102176 T^{3} - 2045540977236 T^{4} + 136204 T^{5} + T^{6} \)
$61$ \( \)\(11\!\cdots\!20\)\( + \)\(69\!\cdots\!48\)\( T - \)\(67\!\cdots\!60\)\( T^{2} - 6019297950964593180 T^{3} + 2562905696353 T^{4} + 3989946 T^{5} + T^{6} \)
$67$ \( -\)\(48\!\cdots\!88\)\( + \)\(11\!\cdots\!68\)\( T + \)\(30\!\cdots\!40\)\( T^{2} - 13217009450631469192 T^{3} - 10526998733956 T^{4} + 2218250 T^{5} + T^{6} \)
$71$ \( -\)\(22\!\cdots\!20\)\( - \)\(16\!\cdots\!32\)\( T + \)\(13\!\cdots\!32\)\( T^{2} + 36951832557596946672 T^{3} - 21214227615948 T^{4} - 2045928 T^{5} + T^{6} \)
$73$ \( \)\(47\!\cdots\!08\)\( + \)\(95\!\cdots\!72\)\( T - \)\(12\!\cdots\!38\)\( T^{2} - 74653362397933764459 T^{3} + 6201131400745 T^{4} + 8557479 T^{5} + T^{6} \)
$79$ \( \)\(36\!\cdots\!00\)\( + \)\(25\!\cdots\!00\)\( T - \)\(74\!\cdots\!24\)\( T^{2} - \)\(30\!\cdots\!04\)\( T^{3} - 26957447904664 T^{4} + 8559709 T^{5} + T^{6} \)
$83$ \( -\)\(54\!\cdots\!48\)\( - \)\(10\!\cdots\!08\)\( T + \)\(22\!\cdots\!52\)\( T^{2} + 31631712053415657768 T^{3} - 28577962635596 T^{4} - 2496351 T^{5} + T^{6} \)
$89$ \( \)\(20\!\cdots\!96\)\( + \)\(69\!\cdots\!88\)\( T + \)\(82\!\cdots\!52\)\( T^{2} - \)\(20\!\cdots\!72\)\( T^{3} - 81258529369598 T^{4} + 2446683 T^{5} + T^{6} \)
$97$ \( -\)\(28\!\cdots\!84\)\( + \)\(75\!\cdots\!72\)\( T + \)\(19\!\cdots\!56\)\( T^{2} + \)\(43\!\cdots\!92\)\( T^{3} - 284102796212738 T^{4} - 5786889 T^{5} + T^{6} \)
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