Properties

Label 684.6.k.f
Level $684$
Weight $6$
Character orbit 684.k
Analytic conductor $109.703$
Analytic rank $0$
Dimension $18$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 684.k (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(109.702532752\)
Analytic rank: \(0\)
Dimension: \(18\)
Relative dimension: \(9\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
Defining polynomial: \(x^{18} - 7 x^{17} + 17616 x^{16} - 456301 x^{15} + 216301789 x^{14} - 6076762674 x^{13} + 1365504992251 x^{12} - 54625271848675 x^{11} + 6304911261795867 x^{10} - 242577082836320134 x^{9} + 18704695568091019759 x^{8} - 704026968823840581411 x^{7} + 38243816152807979345695 x^{6} - 1065755622605542033556002 x^{5} + 37300806349205873742869889 x^{4} - 655224799021744228503297133 x^{3} + 19168823714556940699533734824 x^{2} - 236484931518202417328772433599 x + 5519876026771720332419776049541\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{16}\cdot 3^{7}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 76)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -\beta_{1} - \beta_{2} - \beta_{4} ) q^{5} + ( 19 + \beta_{3} - \beta_{9} - \beta_{13} ) q^{7} +O(q^{10})\) \( q + ( -\beta_{1} - \beta_{2} - \beta_{4} ) q^{5} + ( 19 + \beta_{3} - \beta_{9} - \beta_{13} ) q^{7} + ( 18 - \beta_{10} ) q^{11} + ( 25 - 3 \beta_{1} - 25 \beta_{2} - 2 \beta_{5} - \beta_{11} ) q^{13} + ( -\beta_{1} - 21 \beta_{2} - \beta_{4} - 5 \beta_{5} + \beta_{7} + 5 \beta_{9} - \beta_{10} + \beta_{15} - \beta_{17} ) q^{17} + ( -170 - 2 \beta_{1} + 243 \beta_{2} + \beta_{3} - 3 \beta_{4} - 5 \beta_{5} + \beta_{6} + \beta_{7} + 3 \beta_{9} - 2 \beta_{10} + 2 \beta_{13} - \beta_{14} + \beta_{15} - \beta_{16} + \beta_{17} ) q^{19} + ( 384 - 383 \beta_{2} - 6 \beta_{3} - 4 \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} + \beta_{11} + \beta_{14} - \beta_{15} + \beta_{16} - 2 \beta_{17} ) q^{23} + ( -787 + 10 \beta_{1} + 787 \beta_{2} - 10 \beta_{3} + 11 \beta_{5} + 2 \beta_{6} + 4 \beta_{7} + \beta_{8} - \beta_{11} ) q^{25} + ( 810 + 12 \beta_{1} - 811 \beta_{2} - 6 \beta_{3} + 25 \beta_{5} - 2 \beta_{6} + 3 \beta_{7} - 4 \beta_{8} + \beta_{9} + 5 \beta_{11} - \beta_{14} + \beta_{15} - \beta_{16} + 2 \beta_{17} ) q^{29} + ( -558 + \beta_{2} + 11 \beta_{3} + 13 \beta_{4} + \beta_{5} + 2 \beta_{6} - \beta_{8} - 4 \beta_{9} + 9 \beta_{10} + 7 \beta_{11} - 2 \beta_{12} - 11 \beta_{13} - 6 \beta_{14} + 2 \beta_{15} + \beta_{17} ) q^{31} + ( -97 \beta_{1} - 1744 \beta_{2} - 97 \beta_{4} - 9 \beta_{5} + 5 \beta_{7} - 7 \beta_{8} + 9 \beta_{9} - 5 \beta_{10} + \beta_{12} - 15 \beta_{13} - 6 \beta_{14} + 7 \beta_{16} ) q^{35} + ( 1488 + 20 \beta_{3} - 42 \beta_{4} - 4 \beta_{6} - 32 \beta_{9} + 8 \beta_{10} - 3 \beta_{11} + 4 \beta_{12} - 20 \beta_{13} + 3 \beta_{14} - 2 \beta_{16} ) q^{37} + ( -55 \beta_{1} + 814 \beta_{2} - 55 \beta_{4} - 21 \beta_{5} - 16 \beta_{7} + 5 \beta_{8} + 21 \beta_{9} + 16 \beta_{10} - 5 \beta_{16} ) q^{41} + ( -16 \beta_{1} - 935 \beta_{2} - 16 \beta_{4} - 54 \beta_{5} + 11 \beta_{7} + 54 \beta_{9} - 11 \beta_{10} - 5 \beta_{12} + 29 \beta_{13} + 5 \beta_{14} + \beta_{15} - \beta_{17} ) q^{43} + ( -4183 + 19 \beta_{1} + 4183 \beta_{2} - 30 \beta_{3} - \beta_{5} - \beta_{6} + 32 \beta_{7} + 11 \beta_{8} - 12 \beta_{11} ) q^{47} + ( 6995 + \beta_{2} + 14 \beta_{3} - 44 \beta_{4} + \beta_{5} + 2 \beta_{6} - \beta_{8} - 162 \beta_{9} - 9 \beta_{10} + 9 \beta_{11} - 2 \beta_{12} - 14 \beta_{13} - 8 \beta_{14} + 2 \beta_{15} - 25 \beta_{16} + \beta_{17} ) q^{49} + ( 2352 + 228 \beta_{1} - 2353 \beta_{2} + 66 \beta_{3} - 135 \beta_{5} + 10 \beta_{6} - 13 \beta_{7} - 6 \beta_{8} + \beta_{9} - 4 \beta_{11} - \beta_{14} + \beta_{15} - \beta_{16} + 2 \beta_{17} ) q^{53} + ( -202 \beta_{1} + 159 \beta_{2} - 202 \beta_{4} + 91 \beta_{5} + 30 \beta_{7} - 2 \beta_{8} - 91 \beta_{9} - 30 \beta_{10} - 4 \beta_{12} - 71 \beta_{13} + 10 \beta_{14} + 10 \beta_{15} + 2 \beta_{16} - 10 \beta_{17} ) q^{55} + ( -102 \beta_{1} + 8331 \beta_{2} - 102 \beta_{4} + 98 \beta_{5} - 10 \beta_{7} + 14 \beta_{8} - 98 \beta_{9} + 10 \beta_{10} + 17 \beta_{12} - 129 \beta_{13} - 3 \beta_{14} - 11 \beta_{15} - 14 \beta_{16} + 11 \beta_{17} ) q^{59} + ( -817 - 19 \beta_{1} + 827 \beta_{2} + 120 \beta_{3} - 34 \beta_{5} + 4 \beta_{6} + 2 \beta_{7} + 12 \beta_{8} - 10 \beta_{9} - 6 \beta_{11} + 10 \beta_{14} - 10 \beta_{15} + 10 \beta_{16} - 20 \beta_{17} ) q^{61} + ( -10501 - 10 \beta_{2} - 186 \beta_{3} + 29 \beta_{4} - 10 \beta_{5} + 10 \beta_{6} + 10 \beta_{8} + 4 \beta_{9} + 10 \beta_{10} - 35 \beta_{11} - 10 \beta_{12} + 186 \beta_{13} + 25 \beta_{14} - 20 \beta_{15} + 30 \beta_{16} - 10 \beta_{17} ) q^{65} + ( -2870 - 103 \beta_{1} + 2860 \beta_{2} + 29 \beta_{3} - 178 \beta_{5} - 19 \beta_{6} - 27 \beta_{7} + 9 \beta_{8} + 10 \beta_{9} + 44 \beta_{11} - 10 \beta_{14} + 10 \beta_{15} - 10 \beta_{16} + 20 \beta_{17} ) q^{67} + ( 172 \beta_{1} + 6033 \beta_{2} + 172 \beta_{4} + 382 \beta_{5} - 13 \beta_{7} + 18 \beta_{8} - 382 \beta_{9} + 13 \beta_{10} + \beta_{12} + 201 \beta_{13} - 33 \beta_{14} - 11 \beta_{15} - 18 \beta_{16} + 11 \beta_{17} ) q^{71} + ( 700 \beta_{1} - 1855 \beta_{2} + 700 \beta_{4} - 433 \beta_{5} + 17 \beta_{7} + 8 \beta_{8} + 433 \beta_{9} - 17 \beta_{10} + 6 \beta_{12} + 154 \beta_{13} - 27 \beta_{14} + \beta_{15} - 8 \beta_{16} - \beta_{17} ) q^{73} + ( 2005 + 11 \beta_{2} - 6 \beta_{3} - 325 \beta_{4} + 11 \beta_{5} - 26 \beta_{6} - 11 \beta_{8} - 487 \beta_{9} - 39 \beta_{10} - 2 \beta_{11} + 26 \beta_{12} + 6 \beta_{13} + 13 \beta_{14} + 22 \beta_{15} + 11 \beta_{17} ) q^{77} + ( 540 \beta_{1} + 3557 \beta_{2} + 540 \beta_{4} + 468 \beta_{5} - 57 \beta_{7} - 6 \beta_{8} - 468 \beta_{9} + 57 \beta_{10} - 3 \beta_{12} + 21 \beta_{13} + 7 \beta_{14} + \beta_{15} + 6 \beta_{16} - \beta_{17} ) q^{79} + ( -4603 - 11 \beta_{2} + 117 \beta_{3} + 68 \beta_{4} - 11 \beta_{5} - 15 \beta_{6} + 11 \beta_{8} + 78 \beta_{9} + 41 \beta_{10} + 41 \beta_{11} + 15 \beta_{12} - 117 \beta_{13} - 52 \beta_{14} - 22 \beta_{15} + 31 \beta_{16} - 11 \beta_{17} ) q^{83} + ( -5905 - 303 \beta_{1} + 5905 \beta_{2} + 446 \beta_{5} - 12 \beta_{6} + 144 \beta_{7} + 32 \beta_{8} - 81 \beta_{11} ) q^{85} + ( 17475 + 347 \beta_{1} - 17464 \beta_{2} + 24 \beta_{3} - 348 \beta_{5} - 16 \beta_{6} - 81 \beta_{7} - 39 \beta_{8} - 11 \beta_{9} - 34 \beta_{11} + 11 \beta_{14} - 11 \beta_{15} + 11 \beta_{16} - 22 \beta_{17} ) q^{89} + ( -596 - 1170 \beta_{1} + 586 \beta_{2} - 50 \beta_{3} + 20 \beta_{5} + 40 \beta_{6} - 86 \beta_{7} - 40 \beta_{8} + 10 \beta_{9} - 18 \beta_{11} - 10 \beta_{14} + 10 \beta_{15} - 10 \beta_{16} + 20 \beta_{17} ) q^{91} + ( -11266 - 334 \beta_{1} + 16200 \beta_{2} - 123 \beta_{3} + 541 \beta_{4} + 189 \beta_{5} - 20 \beta_{6} + 97 \beta_{7} + 51 \beta_{8} + 836 \beta_{9} - 98 \beta_{10} - 69 \beta_{11} + 21 \beta_{12} + 36 \beta_{13} + 27 \beta_{14} - 35 \beta_{15} + 42 \beta_{16} - 46 \beta_{17} ) q^{95} + ( 721 \beta_{1} + 12320 \beta_{2} + 721 \beta_{4} + 316 \beta_{5} - 5 \beta_{7} + 21 \beta_{8} - 316 \beta_{9} + 5 \beta_{10} + 50 \beta_{12} + 134 \beta_{13} - 19 \beta_{14} - 45 \beta_{15} - 21 \beta_{16} + 45 \beta_{17} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18q - 11q^{5} + 336q^{7} + O(q^{10}) \) \( 18q - 11q^{5} + 336q^{7} + 320q^{11} + 227q^{13} - 179q^{17} - 868q^{19} + 3425q^{23} - 7054q^{25} + 7349q^{29} - 9960q^{31} - 15888q^{35} + 26444q^{37} + 7311q^{41} - 8283q^{43} - 37603q^{47} + 124738q^{49} + 20337q^{53} + 716q^{55} + 74455q^{59} - 7569q^{61} - 188998q^{65} - 26177q^{67} + 53463q^{71} - 14103q^{73} + 31960q^{77} + 31825q^{79} - 82600q^{83} - 50787q^{85} + 155197q^{89} - 2800q^{91} - 49315q^{95} + 111241q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{18} - 7 x^{17} + 17616 x^{16} - 456301 x^{15} + 216301789 x^{14} - 6076762674 x^{13} + 1365504992251 x^{12} - 54625271848675 x^{11} + 6304911261795867 x^{10} - 242577082836320134 x^{9} + 18704695568091019759 x^{8} - 704026968823840581411 x^{7} + 38243816152807979345695 x^{6} - 1065755622605542033556002 x^{5} + 37300806349205873742869889 x^{4} - 655224799021744228503297133 x^{3} + 19168823714556940699533734824 x^{2} - 236484931518202417328772433599 x + 5519876026771720332419776049541\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(\)\(14\!\cdots\!75\)\( \nu^{17} + \)\(24\!\cdots\!90\)\( \nu^{16} + \)\(24\!\cdots\!10\)\( \nu^{15} - \)\(40\!\cdots\!25\)\( \nu^{14} + \)\(28\!\cdots\!50\)\( \nu^{13} - \)\(54\!\cdots\!60\)\( \nu^{12} + \)\(17\!\cdots\!25\)\( \nu^{11} - \)\(54\!\cdots\!24\)\( \nu^{10} + \)\(74\!\cdots\!65\)\( \nu^{9} - \)\(22\!\cdots\!05\)\( \nu^{8} + \)\(19\!\cdots\!00\)\( \nu^{7} - \)\(58\!\cdots\!85\)\( \nu^{6} + \)\(34\!\cdots\!68\)\( \nu^{5} - \)\(57\!\cdots\!70\)\( \nu^{4} + \)\(19\!\cdots\!25\)\( \nu^{3} + \)\(19\!\cdots\!70\)\( \nu^{2} + \)\(32\!\cdots\!74\)\( \nu - \)\(55\!\cdots\!83\)\(\)\()/ \)\(31\!\cdots\!64\)\( \)
\(\beta_{2}\)\(=\)\((\)\(-\)\(14\!\cdots\!75\)\( \nu^{17} - \)\(24\!\cdots\!90\)\( \nu^{16} - \)\(24\!\cdots\!10\)\( \nu^{15} + \)\(40\!\cdots\!25\)\( \nu^{14} - \)\(28\!\cdots\!50\)\( \nu^{13} + \)\(54\!\cdots\!60\)\( \nu^{12} - \)\(17\!\cdots\!25\)\( \nu^{11} + \)\(54\!\cdots\!24\)\( \nu^{10} - \)\(74\!\cdots\!65\)\( \nu^{9} + \)\(22\!\cdots\!05\)\( \nu^{8} - \)\(19\!\cdots\!00\)\( \nu^{7} + \)\(58\!\cdots\!85\)\( \nu^{6} - \)\(34\!\cdots\!68\)\( \nu^{5} + \)\(57\!\cdots\!70\)\( \nu^{4} - \)\(19\!\cdots\!25\)\( \nu^{3} - \)\(19\!\cdots\!70\)\( \nu^{2} - \)\(75\!\cdots\!10\)\( \nu + \)\(55\!\cdots\!83\)\(\)\()/ \)\(31\!\cdots\!64\)\( \)
\(\beta_{3}\)\(=\)\((\)\(\)\(42\!\cdots\!41\)\( \nu^{17} + \)\(29\!\cdots\!74\)\( \nu^{16} + \)\(90\!\cdots\!94\)\( \nu^{15} + \)\(50\!\cdots\!69\)\( \nu^{14} + \)\(48\!\cdots\!94\)\( \nu^{13} + \)\(57\!\cdots\!36\)\( \nu^{12} + \)\(36\!\cdots\!43\)\( \nu^{11} + \)\(33\!\cdots\!32\)\( \nu^{10} - \)\(36\!\cdots\!89\)\( \nu^{9} + \)\(12\!\cdots\!85\)\( \nu^{8} - \)\(18\!\cdots\!12\)\( \nu^{7} + \)\(30\!\cdots\!97\)\( \nu^{6} - \)\(63\!\cdots\!56\)\( \nu^{5} + \)\(46\!\cdots\!50\)\( \nu^{4} - \)\(50\!\cdots\!25\)\( \nu^{3} + \)\(28\!\cdots\!62\)\( \nu^{2} - \)\(16\!\cdots\!14\)\( \nu + \)\(12\!\cdots\!35\)\(\)\()/ \)\(20\!\cdots\!00\)\( \)
\(\beta_{4}\)\(=\)\((\)\(\)\(36\!\cdots\!35\)\( \nu^{17} - \)\(23\!\cdots\!70\)\( \nu^{16} + \)\(70\!\cdots\!10\)\( \nu^{15} - \)\(51\!\cdots\!25\)\( \nu^{14} + \)\(95\!\cdots\!10\)\( \nu^{13} - \)\(65\!\cdots\!60\)\( \nu^{12} + \)\(69\!\cdots\!09\)\( \nu^{11} - \)\(46\!\cdots\!24\)\( \nu^{10} + \)\(37\!\cdots\!45\)\( \nu^{9} - \)\(21\!\cdots\!05\)\( \nu^{8} + \)\(12\!\cdots\!60\)\( \nu^{7} - \)\(60\!\cdots\!73\)\( \nu^{6} + \)\(29\!\cdots\!08\)\( \nu^{5} - \)\(10\!\cdots\!70\)\( \nu^{4} + \)\(37\!\cdots\!05\)\( \nu^{3} - \)\(66\!\cdots\!90\)\( \nu^{2} + \)\(90\!\cdots\!38\)\( \nu - \)\(27\!\cdots\!79\)\(\)\()/ \)\(10\!\cdots\!76\)\( \)
\(\beta_{5}\)\(=\)\((\)\(\)\(14\!\cdots\!03\)\( \nu^{17} - \)\(12\!\cdots\!58\)\( \nu^{16} - \)\(15\!\cdots\!98\)\( \nu^{15} - \)\(22\!\cdots\!73\)\( \nu^{14} + \)\(63\!\cdots\!02\)\( \nu^{13} - \)\(25\!\cdots\!12\)\( \nu^{12} - \)\(74\!\cdots\!31\)\( \nu^{11} - \)\(14\!\cdots\!44\)\( \nu^{10} + \)\(22\!\cdots\!13\)\( \nu^{9} - \)\(55\!\cdots\!45\)\( \nu^{8} + \)\(94\!\cdots\!04\)\( \nu^{7} - \)\(13\!\cdots\!49\)\( \nu^{6} + \)\(30\!\cdots\!52\)\( \nu^{5} - \)\(20\!\cdots\!50\)\( \nu^{4} + \)\(23\!\cdots\!25\)\( \nu^{3} - \)\(12\!\cdots\!54\)\( \nu^{2} + \)\(87\!\cdots\!38\)\( \nu - \)\(50\!\cdots\!95\)\(\)\()/ \)\(40\!\cdots\!00\)\( \)
\(\beta_{6}\)\(=\)\((\)\(\)\(50\!\cdots\!61\)\( \nu^{17} - \)\(20\!\cdots\!46\)\( \nu^{16} + \)\(26\!\cdots\!74\)\( \nu^{15} - \)\(37\!\cdots\!51\)\( \nu^{14} + \)\(74\!\cdots\!74\)\( \nu^{13} - \)\(42\!\cdots\!44\)\( \nu^{12} + \)\(40\!\cdots\!03\)\( \nu^{11} - \)\(25\!\cdots\!28\)\( \nu^{10} + \)\(55\!\cdots\!31\)\( \nu^{9} - \)\(97\!\cdots\!15\)\( \nu^{8} + \)\(20\!\cdots\!48\)\( \nu^{7} - \)\(24\!\cdots\!63\)\( \nu^{6} + \)\(58\!\cdots\!24\)\( \nu^{5} - \)\(35\!\cdots\!50\)\( \nu^{4} + \)\(43\!\cdots\!75\)\( \nu^{3} - \)\(19\!\cdots\!98\)\( \nu^{2} + \)\(16\!\cdots\!06\)\( \nu - \)\(77\!\cdots\!65\)\(\)\()/ \)\(20\!\cdots\!00\)\( \)
\(\beta_{7}\)\(=\)\((\)\(\)\(34\!\cdots\!11\)\( \nu^{17} + \)\(23\!\cdots\!54\)\( \nu^{16} + \)\(61\!\cdots\!74\)\( \nu^{15} + \)\(28\!\cdots\!99\)\( \nu^{14} + \)\(67\!\cdots\!74\)\( \nu^{13} + \)\(31\!\cdots\!56\)\( \nu^{12} + \)\(37\!\cdots\!53\)\( \nu^{11} + \)\(12\!\cdots\!72\)\( \nu^{10} + \)\(11\!\cdots\!81\)\( \nu^{9} + \)\(54\!\cdots\!35\)\( \nu^{8} + \)\(24\!\cdots\!48\)\( \nu^{7} + \)\(13\!\cdots\!87\)\( \nu^{6} + \)\(16\!\cdots\!24\)\( \nu^{5} + \)\(31\!\cdots\!50\)\( \nu^{4} - \)\(54\!\cdots\!75\)\( \nu^{3} + \)\(25\!\cdots\!02\)\( \nu^{2} + \)\(80\!\cdots\!06\)\( \nu + \)\(14\!\cdots\!85\)\(\)\()/ \)\(44\!\cdots\!00\)\( \)
\(\beta_{8}\)\(=\)\((\)\(-\)\(33\!\cdots\!69\)\( \nu^{17} + \)\(47\!\cdots\!34\)\( \nu^{16} + \)\(94\!\cdots\!54\)\( \nu^{15} + \)\(83\!\cdots\!79\)\( \nu^{14} + \)\(18\!\cdots\!54\)\( \nu^{13} + \)\(95\!\cdots\!76\)\( \nu^{12} + \)\(28\!\cdots\!13\)\( \nu^{11} + \)\(55\!\cdots\!12\)\( \nu^{10} - \)\(73\!\cdots\!99\)\( \nu^{9} + \)\(20\!\cdots\!35\)\( \nu^{8} - \)\(32\!\cdots\!92\)\( \nu^{7} + \)\(51\!\cdots\!27\)\( \nu^{6} - \)\(10\!\cdots\!96\)\( \nu^{5} + \)\(75\!\cdots\!50\)\( \nu^{4} - \)\(85\!\cdots\!75\)\( \nu^{3} + \)\(45\!\cdots\!42\)\( \nu^{2} - \)\(30\!\cdots\!74\)\( \nu + \)\(19\!\cdots\!85\)\(\)\()/ \)\(40\!\cdots\!00\)\( \)
\(\beta_{9}\)\(=\)\((\)\(\)\(17\!\cdots\!59\)\( \nu^{17} - \)\(80\!\cdots\!18\)\( \nu^{16} + \)\(28\!\cdots\!30\)\( \nu^{15} - \)\(79\!\cdots\!05\)\( \nu^{14} + \)\(33\!\cdots\!70\)\( \nu^{13} - \)\(10\!\cdots\!72\)\( \nu^{12} + \)\(19\!\cdots\!93\)\( \nu^{11} - \)\(93\!\cdots\!04\)\( \nu^{10} + \)\(81\!\cdots\!21\)\( \nu^{9} - \)\(37\!\cdots\!29\)\( \nu^{8} + \)\(21\!\cdots\!12\)\( \nu^{7} - \)\(10\!\cdots\!89\)\( \nu^{6} + \)\(40\!\cdots\!28\)\( \nu^{5} - \)\(12\!\cdots\!66\)\( \nu^{4} + \)\(26\!\cdots\!85\)\( \nu^{3} - \)\(68\!\cdots\!62\)\( \nu^{2} + \)\(10\!\cdots\!06\)\( \nu - \)\(24\!\cdots\!79\)\(\)\()/ \)\(20\!\cdots\!00\)\( \)
\(\beta_{10}\)\(=\)\((\)\(-\)\(80\!\cdots\!81\)\( \nu^{17} + \)\(16\!\cdots\!62\)\( \nu^{16} - \)\(13\!\cdots\!70\)\( \nu^{15} + \)\(55\!\cdots\!95\)\( \nu^{14} - \)\(16\!\cdots\!30\)\( \nu^{13} + \)\(69\!\cdots\!48\)\( \nu^{12} - \)\(10\!\cdots\!87\)\( \nu^{11} + \)\(54\!\cdots\!36\)\( \nu^{10} - \)\(47\!\cdots\!39\)\( \nu^{9} + \)\(22\!\cdots\!11\)\( \nu^{8} - \)\(13\!\cdots\!08\)\( \nu^{7} + \)\(63\!\cdots\!51\)\( \nu^{6} - \)\(26\!\cdots\!52\)\( \nu^{5} + \)\(89\!\cdots\!94\)\( \nu^{4} - \)\(19\!\cdots\!15\)\( \nu^{3} + \)\(49\!\cdots\!58\)\( \nu^{2} - \)\(74\!\cdots\!54\)\( \nu + \)\(23\!\cdots\!61\)\(\)\()/ \)\(54\!\cdots\!00\)\( \)
\(\beta_{11}\)\(=\)\((\)\(\)\(31\!\cdots\!17\)\( \nu^{17} - \)\(12\!\cdots\!62\)\( \nu^{16} + \)\(10\!\cdots\!78\)\( \nu^{15} - \)\(22\!\cdots\!47\)\( \nu^{14} + \)\(37\!\cdots\!78\)\( \nu^{13} - \)\(25\!\cdots\!68\)\( \nu^{12} + \)\(15\!\cdots\!91\)\( \nu^{11} - \)\(15\!\cdots\!16\)\( \nu^{10} + \)\(28\!\cdots\!07\)\( \nu^{9} - \)\(57\!\cdots\!55\)\( \nu^{8} + \)\(11\!\cdots\!56\)\( \nu^{7} - \)\(14\!\cdots\!11\)\( \nu^{6} + \)\(32\!\cdots\!28\)\( \nu^{5} - \)\(20\!\cdots\!50\)\( \nu^{4} + \)\(24\!\cdots\!75\)\( \nu^{3} - \)\(11\!\cdots\!06\)\( \nu^{2} + \)\(92\!\cdots\!82\)\( \nu - \)\(48\!\cdots\!05\)\(\)\()/ \)\(20\!\cdots\!00\)\( \)
\(\beta_{12}\)\(=\)\((\)\(-\)\(51\!\cdots\!95\)\( \nu^{17} - \)\(17\!\cdots\!34\)\( \nu^{16} - \)\(90\!\cdots\!46\)\( \nu^{15} - \)\(10\!\cdots\!31\)\( \nu^{14} - \)\(10\!\cdots\!06\)\( \nu^{13} - \)\(85\!\cdots\!96\)\( \nu^{12} - \)\(58\!\cdots\!09\)\( \nu^{11} + \)\(55\!\cdots\!08\)\( \nu^{10} - \)\(21\!\cdots\!33\)\( \nu^{9} + \)\(27\!\cdots\!21\)\( \nu^{8} - \)\(52\!\cdots\!60\)\( \nu^{7} + \)\(10\!\cdots\!13\)\( \nu^{6} - \)\(75\!\cdots\!28\)\( \nu^{5} + \)\(71\!\cdots\!94\)\( \nu^{4} - \)\(43\!\cdots\!65\)\( \nu^{3} + \)\(36\!\cdots\!10\)\( \nu^{2} - \)\(18\!\cdots\!98\)\( \nu - \)\(41\!\cdots\!29\)\(\)\()/ \)\(20\!\cdots\!00\)\( \)
\(\beta_{13}\)\(=\)\((\)\(\)\(81\!\cdots\!55\)\( \nu^{17} + \)\(24\!\cdots\!46\)\( \nu^{16} + \)\(14\!\cdots\!74\)\( \nu^{15} + \)\(11\!\cdots\!39\)\( \nu^{14} + \)\(16\!\cdots\!14\)\( \nu^{13} + \)\(78\!\cdots\!24\)\( \nu^{12} + \)\(94\!\cdots\!21\)\( \nu^{11} - \)\(11\!\cdots\!52\)\( \nu^{10} + \)\(35\!\cdots\!77\)\( \nu^{9} - \)\(56\!\cdots\!49\)\( \nu^{8} + \)\(88\!\cdots\!40\)\( \nu^{7} - \)\(19\!\cdots\!97\)\( \nu^{6} + \)\(13\!\cdots\!32\)\( \nu^{5} - \)\(16\!\cdots\!86\)\( \nu^{4} + \)\(83\!\cdots\!85\)\( \nu^{3} - \)\(60\!\cdots\!90\)\( \nu^{2} + \)\(36\!\cdots\!62\)\( \nu - \)\(54\!\cdots\!99\)\(\)\()/ \)\(20\!\cdots\!00\)\( \)
\(\beta_{14}\)\(=\)\((\)\(-\)\(31\!\cdots\!15\)\( \nu^{17} - \)\(11\!\cdots\!98\)\( \nu^{16} - \)\(55\!\cdots\!62\)\( \nu^{15} - \)\(75\!\cdots\!07\)\( \nu^{14} - \)\(62\!\cdots\!82\)\( \nu^{13} - \)\(66\!\cdots\!12\)\( \nu^{12} - \)\(35\!\cdots\!73\)\( \nu^{11} + \)\(26\!\cdots\!76\)\( \nu^{10} - \)\(12\!\cdots\!01\)\( \nu^{9} + \)\(14\!\cdots\!37\)\( \nu^{8} - \)\(31\!\cdots\!20\)\( \nu^{7} + \)\(57\!\cdots\!61\)\( \nu^{6} - \)\(44\!\cdots\!16\)\( \nu^{5} + \)\(38\!\cdots\!18\)\( \nu^{4} - \)\(25\!\cdots\!05\)\( \nu^{3} + \)\(12\!\cdots\!70\)\( \nu^{2} - \)\(10\!\cdots\!06\)\( \nu - \)\(33\!\cdots\!13\)\(\)\()/ \)\(20\!\cdots\!00\)\( \)
\(\beta_{15}\)\(=\)\((\)\(\)\(91\!\cdots\!65\)\( \nu^{17} + \)\(52\!\cdots\!66\)\( \nu^{16} + \)\(16\!\cdots\!34\)\( \nu^{15} + \)\(56\!\cdots\!49\)\( \nu^{14} + \)\(18\!\cdots\!74\)\( \nu^{13} + \)\(58\!\cdots\!04\)\( \nu^{12} + \)\(11\!\cdots\!31\)\( \nu^{11} + \)\(15\!\cdots\!08\)\( \nu^{10} + \)\(37\!\cdots\!47\)\( \nu^{9} + \)\(39\!\cdots\!81\)\( \nu^{8} + \)\(85\!\cdots\!20\)\( \nu^{7} + \)\(35\!\cdots\!13\)\( \nu^{6} + \)\(10\!\cdots\!32\)\( \nu^{5} + \)\(17\!\cdots\!34\)\( \nu^{4} + \)\(56\!\cdots\!35\)\( \nu^{3} + \)\(14\!\cdots\!30\)\( \nu^{2} + \)\(26\!\cdots\!82\)\( \nu + \)\(86\!\cdots\!31\)\(\)\()/ \)\(44\!\cdots\!00\)\( \)
\(\beta_{16}\)\(=\)\((\)\(-\)\(59\!\cdots\!13\)\( \nu^{17} + \)\(27\!\cdots\!26\)\( \nu^{16} - \)\(97\!\cdots\!10\)\( \nu^{15} + \)\(27\!\cdots\!35\)\( \nu^{14} - \)\(11\!\cdots\!90\)\( \nu^{13} + \)\(34\!\cdots\!04\)\( \nu^{12} - \)\(65\!\cdots\!51\)\( \nu^{11} + \)\(32\!\cdots\!28\)\( \nu^{10} - \)\(27\!\cdots\!47\)\( \nu^{9} + \)\(12\!\cdots\!03\)\( \nu^{8} - \)\(75\!\cdots\!84\)\( \nu^{7} + \)\(35\!\cdots\!23\)\( \nu^{6} - \)\(13\!\cdots\!96\)\( \nu^{5} + \)\(43\!\cdots\!62\)\( \nu^{4} - \)\(91\!\cdots\!95\)\( \nu^{3} + \)\(23\!\cdots\!34\)\( \nu^{2} - \)\(36\!\cdots\!42\)\( \nu + \)\(87\!\cdots\!53\)\(\)\()/ \)\(18\!\cdots\!00\)\( \)
\(\beta_{17}\)\(=\)\((\)\(-\)\(16\!\cdots\!45\)\( \nu^{17} - \)\(27\!\cdots\!46\)\( \nu^{16} - \)\(27\!\cdots\!94\)\( \nu^{15} + \)\(13\!\cdots\!91\)\( \nu^{14} - \)\(31\!\cdots\!34\)\( \nu^{13} + \)\(25\!\cdots\!76\)\( \nu^{12} - \)\(18\!\cdots\!31\)\( \nu^{11} + \)\(46\!\cdots\!52\)\( \nu^{10} - \)\(71\!\cdots\!47\)\( \nu^{9} + \)\(19\!\cdots\!59\)\( \nu^{8} - \)\(18\!\cdots\!60\)\( \nu^{7} + \)\(59\!\cdots\!47\)\( \nu^{6} - \)\(29\!\cdots\!72\)\( \nu^{5} + \)\(63\!\cdots\!26\)\( \nu^{4} - \)\(19\!\cdots\!35\)\( \nu^{3} + \)\(31\!\cdots\!10\)\( \nu^{2} - \)\(80\!\cdots\!82\)\( \nu + \)\(92\!\cdots\!09\)\(\)\()/ \)\(44\!\cdots\!00\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{2} + \beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{16} + \beta_{14} + 10 \beta_{13} - 2 \beta_{12} + 4 \beta_{10} + 11 \beta_{9} - \beta_{8} - 4 \beta_{7} - 11 \beta_{5} - 10 \beta_{4} - 3910 \beta_{2} - 8 \beta_{1} - 1\)
\(\nu^{3}\)\(=\)\(-45 \beta_{17} - 23 \beta_{16} - 90 \beta_{15} + 162 \beta_{14} + 180 \beta_{13} + 56 \beta_{12} - 204 \beta_{11} + 173 \beta_{10} + 2352 \beta_{9} + 42 \beta_{8} - 12 \beta_{7} - 62 \beta_{6} - 78 \beta_{5} + 6836 \beta_{4} - 150 \beta_{3} - 11778 \beta_{2} - 27 \beta_{1} + 59110\)
\(\nu^{4}\)\(=\)\(-5330 \beta_{17} + 2477 \beta_{16} - 2935 \beta_{15} + 3217 \beta_{14} + 660 \beta_{13} + 236 \beta_{12} - 13488 \beta_{11} + 668 \beta_{10} + 6767 \beta_{9} + 9346 \beta_{8} + 36839 \beta_{7} + 22396 \beta_{6} + 12571 \beta_{5} + 27404 \beta_{4} - 117220 \beta_{3} + 26581845 \beta_{2} + 87614 \beta_{1} - 26366473\)
\(\nu^{5}\)\(=\)\(-401865 \beta_{17} + 618262 \beta_{16} + 361890 \beta_{15} - 1450833 \beta_{14} - 1737650 \beta_{13} - 990114 \beta_{12} - 65400 \beta_{11} - 1992667 \beta_{10} - 29841333 \beta_{9} - 559337 \beta_{8} + 2178592 \beta_{7} + 112600 \beta_{6} + 29915283 \beta_{5} - 58169896 \beta_{4} - 584600 \beta_{3} - 426784388 \beta_{2} - 57800201 \beta_{1} - 132423474\)
\(\nu^{6}\)\(=\)\(31315085 \beta_{17} - 76163096 \beta_{16} + 69508015 \beta_{15} - 93640341 \beta_{14} - 1153549930 \beta_{13} + 229106402 \beta_{12} + 118343378 \beta_{11} - 335818799 \beta_{10} + 390197179 \beta_{9} - 37142582 \beta_{8} + 12518787 \beta_{7} - 234711056 \beta_{6} + 212948963 \beta_{5} + 509175164 \beta_{4} + 1141365280 \beta_{3} - 2925963682 \beta_{2} - 348115830 \beta_{1} + 224305429259\)
\(\nu^{7}\)\(=\)\(6309274755 \beta_{17} - 3586945138 \beta_{16} + 3519764325 \beta_{15} - 3665815433 \beta_{14} - 8038341850 \beta_{13} + 1624543466 \beta_{12} + 21988137508 \beta_{11} - 2308868177 \beta_{10} + 6399110487 \beta_{9} + 1692351110 \beta_{8} - 19645263215 \beta_{7} + 10553055044 \beta_{6} - 318705636015 \beta_{5} + 4786513634 \beta_{4} + 21984548660 \beta_{3} + 5455991947371 \beta_{2} + 531719887067 \beta_{1} - 3897627431452\)
\(\nu^{8}\)\(=\)\(393961777125 \beta_{17} + 341214452599 \beta_{16} - 318154178895 \beta_{15} + 320234613514 \beta_{14} + 10816039649080 \beta_{13} - 2388623579288 \beta_{12} + 172588755816 \beta_{11} + 2882899101841 \beta_{10} - 9648455892436 \beta_{9} - 353196172585 \beta_{8} - 3049471957315 \beta_{7} + 91001084328 \beta_{6} + 7133118511120 \beta_{5} - 9755043427570 \beta_{4} + 143893620840 \beta_{3} - 2016868010504734 \beta_{2} - 5515556188430 \beta_{1} - 37467138222115\)
\(\nu^{9}\)\(=\)\(-22127547800730 \beta_{17} - 14750449644371 \beta_{16} - 53881807088895 \beta_{15} + 175288367873139 \beta_{14} + 186062739185550 \beta_{13} + 116319190463462 \beta_{12} - 196959046671990 \beta_{11} + 207098998615736 \beta_{10} + 3210906350642289 \beta_{9} + 22206011818386 \beta_{8} - 26737236422001 \beta_{7} - 137436543836570 \beta_{6} + 50236839135951 \beta_{5} + 4955338649420474 \beta_{4} - 87926148797250 \beta_{3} - 18373865210941314 \beta_{2} - 68803869493341 \beta_{1} + 58104465221952634\)
\(\nu^{10}\)\(=\)\(-6823966142941505 \beta_{17} + 3129582980107109 \beta_{16} - 3816949676806120 \beta_{15} + 5030908932281944 \beta_{14} + 1373182814104950 \beta_{13} + 1270826549022602 \beta_{12} - 10800958165256670 \beta_{11} + 1941052482069281 \beta_{10} + 29251066041472994 \beta_{9} + 6784935250716436 \beta_{8} + 26070548075692274 \beta_{7} + 22827731876941840 \beta_{6} - 129381369584227874 \beta_{5} + 49992816239549168 \beta_{4} - 103653423603263200 \beta_{3} + 19489060090149136635 \beta_{2} + 116219744004274868 \beta_{1} - 18995875778841124495\)
\(\nu^{11}\)\(=\)\(-291324051049630230 \beta_{17} + 435236271386758648 \beta_{16} + 178463802818182755 \beta_{15} - 1363452816555497442 \beta_{14} - 320966313736419560 \beta_{13} - 1495747413470505876 \beta_{12} - 107956516491093864 \beta_{11} - 1606780987199050588 \beta_{10} - 33409953464240419302 \beta_{9} - 326588030905587059 \beta_{8} + 1904968307317826269 \beta_{7} + 258675484746993028 \beta_{6} + 32128845558943178556 \beta_{5} - 48280637717525080018 \beta_{4} - 1135334706363478940 \beta_{3} - 430905369741671869010 \beta_{2} - 47275288442409848807 \beta_{1} - 212155171333636760778\)
\(\nu^{12}\)\(=\)\(27741035763656749010 \beta_{17} - 56689475399047316678 \beta_{16} + 63964991954400764800 \beta_{15} - 59696515246395079068 \beta_{14} - 982555761185393393020 \beta_{13} + 225699409277359025036 \beta_{12} + 73207030204285636382 \beta_{11} - 259003676713831737362 \beta_{10} + 1111984111595987677552 \beta_{9} - 35153401283213833094 \beta_{8} + 21134575242515637564 \beta_{7} - 242138276123631385844 \beta_{6} + 424884186437601377666 \beta_{5} + 798843098743258804496 \beta_{4} + 971846820774068762380 \beta_{3} - 6429390379318354322314 \beta_{2} - 574985949666187138680 \beta_{1} + 188228988341031716909573\)
\(\nu^{13}\)\(=\)\(4130477416906616220210 \beta_{17} - 2644533563315802032062 \beta_{16} + 2691099187716892458720 \beta_{15} - 2542919457897069012092 \beta_{14} - 12747915895452788958700 \beta_{13} + 3051037830476363756444 \beta_{12} + 18201750304866532376098 \beta_{11} - 3241308101372963283818 \beta_{10} + 18942355406400660366528 \beta_{9} + 1532298375644694091694 \beta_{8} - 14083643767621838171084 \beta_{7} + 12932482726745488646564 \beta_{6} - 331042437899991773111946 \beta_{5} + 14161592341933574320016 \beta_{4} + 10395892292789461356260 \beta_{3} + 7170028968721410193814955 \beta_{2} + 468481893078233255247605 \beta_{1} - 4758240003910549640505712\)
\(\nu^{14}\)\(=\)\(341188885281995187122430 \beta_{17} + 274440679854538109523187 \beta_{16} - 254062643278178266848900 \beta_{15} + 2263651066983092781817 \beta_{14} + 9287052660975992910073750 \beta_{13} - 2429516986052749020144734 \beta_{12} + 248133785351474438335110 \beta_{11} + 2182608657327805357580398 \beta_{10} - 16497762506497425303149113 \beta_{9} - 281624632221207070086367 \beta_{8} - 2403431033177387075255548 \beta_{7} + 203160373234212491990340 \beta_{6} + 11988146778910988200861963 \beta_{5} - 15890566635588144001157530 \beta_{4} + 56794750264282724241780 \beta_{3} - 1753284143901542532201691576 \beta_{2} - 9404774397146732018364296 \beta_{1} - 83802291180889699837217581\)
\(\nu^{15}\)\(=\)\(-11452782306149694570807735 \beta_{17} - 17206515081204492088934261 \beta_{16} - 36364485602484253207354710 \beta_{15} + 145938403001377912891664274 \beta_{14} + 63350524523338461668227800 \beta_{13} + 133912001556206641065164372 \beta_{12} - 159968395058009277109437018 \beta_{11} + 156595788292382867321547191 \beta_{10} + 3087828745250362085041309044 \beta_{9} + 11749125350132427309813384 \beta_{8} - 34565588547885796354647204 \beta_{7} - 168991339029425612002004354 \beta_{6} + 198580820451991007929828164 \beta_{5} + 4384018613220557854604324708 \beta_{4} + 77050956838251253745145750 \beta_{3} - 27070543697342880031691843496 \beta_{2} - 190458953255450876301284703 \beta_{1} + 79347541599974342604552975286\)
\(\nu^{16}\)\(=\)\(-5529309966906002734616942360 \beta_{17} + 2344079061208546100795176895 \beta_{16} - 3203644915866289218667311535 \beta_{15} + 4987113500970486800693662405 \beta_{14} - 106189919347833706986907440 \beta_{13} + 2435425329245786936130787760 \beta_{12} - 7400861267318891304279675162 \beta_{11} + 2242268796322425086497632290 \beta_{10} + 48739737128504272422756884135 \beta_{9} + 5976629786443277052816886372 \beta_{8} + 20106277878392699522757969323 \beta_{7} + 21941660534647357807198002784 \beta_{6} - 183476911922537769481642621043 \beta_{5} + 72057431171717922412936025660 \beta_{4} - 89107674978664594980856639840 \beta_{3} + 18033151004386345092810152494371 \beta_{2} + 179837637377674244658196553870 \beta_{1} - 16972598059467072594968318901397\)
\(\nu^{17}\)\(=\)\(-\)\(23\!\cdots\!35\)\( \beta_{17} + \)\(41\!\cdots\!12\)\( \beta_{16} + \)\(88\!\cdots\!30\)\( \beta_{15} - \)\(11\!\cdots\!33\)\( \beta_{14} + \)\(12\!\cdots\!50\)\( \beta_{13} - \)\(17\!\cdots\!94\)\( \beta_{12} - \)\(10\!\cdots\!42\)\( \beta_{11} - \)\(10\!\cdots\!57\)\( \beta_{10} - \)\(33\!\cdots\!53\)\( \beta_{9} - \)\(27\!\cdots\!75\)\( \beta_{8} + \)\(14\!\cdots\!60\)\( \beta_{7} + \)\(39\!\cdots\!24\)\( \beta_{6} + \)\(30\!\cdots\!05\)\( \beta_{5} - \)\(44\!\cdots\!44\)\( \beta_{4} - \)\(15\!\cdots\!00\)\( \beta_{3} - \)\(56\!\cdots\!46\)\( \beta_{2} - \)\(42\!\cdots\!45\)\( \beta_{1} - \)\(29\!\cdots\!46\)\(\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/684\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(343\) \(533\)
\(\chi(n)\) \(-1 + \beta_{2}\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
505.1
−49.5928 87.6293i
−34.6401 61.7304i
−30.6056 54.7425i
−11.6095 21.8402i
16.4645 + 26.7852i
19.0963 + 31.3437i
19.6530 + 32.3080i
26.2470 + 43.7291i
48.4871 + 82.2500i
−49.5928 + 87.6293i
−34.6401 + 61.7304i
−30.6056 + 54.7425i
−11.6095 + 21.8402i
16.4645 26.7852i
19.0963 31.3437i
19.6530 32.3080i
26.2470 43.7291i
48.4871 82.2500i
0 0 0 −50.5928 + 87.6293i 0 95.5451 0 0 0
505.2 0 0 0 −35.6401 + 61.7304i 0 252.315 0 0 0
505.3 0 0 0 −31.6056 + 54.7425i 0 −80.2775 0 0 0
505.4 0 0 0 −12.6095 + 21.8402i 0 −40.7176 0 0 0
505.5 0 0 0 15.4645 26.7852i 0 132.225 0 0 0
505.6 0 0 0 18.0963 31.3437i 0 −208.885 0 0 0
505.7 0 0 0 18.6530 32.3080i 0 15.8772 0 0 0
505.8 0 0 0 25.2470 43.7291i 0 −187.942 0 0 0
505.9 0 0 0 47.4871 82.2500i 0 189.860 0 0 0
577.1 0 0 0 −50.5928 87.6293i 0 95.5451 0 0 0
577.2 0 0 0 −35.6401 61.7304i 0 252.315 0 0 0
577.3 0 0 0 −31.6056 54.7425i 0 −80.2775 0 0 0
577.4 0 0 0 −12.6095 21.8402i 0 −40.7176 0 0 0
577.5 0 0 0 15.4645 + 26.7852i 0 132.225 0 0 0
577.6 0 0 0 18.0963 + 31.3437i 0 −208.885 0 0 0
577.7 0 0 0 18.6530 + 32.3080i 0 15.8772 0 0 0
577.8 0 0 0 25.2470 + 43.7291i 0 −187.942 0 0 0
577.9 0 0 0 47.4871 + 82.2500i 0 189.860 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 577.9
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 684.6.k.f 18
3.b odd 2 1 76.6.e.a 18
12.b even 2 1 304.6.i.d 18
19.c even 3 1 inner 684.6.k.f 18
57.h odd 6 1 76.6.e.a 18
228.m even 6 1 304.6.i.d 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.6.e.a 18 3.b odd 2 1
76.6.e.a 18 57.h odd 6 1
304.6.i.d 18 12.b even 2 1
304.6.i.d 18 228.m even 6 1
684.6.k.f 18 1.a even 1 1 trivial
684.6.k.f 18 19.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(684, [\chi])\):

\(13\!\cdots\!01\)\( T_{5}^{12} - \)\(38\!\cdots\!24\)\( T_{5}^{11} + \)\(57\!\cdots\!80\)\( T_{5}^{10} - \)\(18\!\cdots\!16\)\( T_{5}^{9} + \)\(16\!\cdots\!40\)\( T_{5}^{8} - \)\(56\!\cdots\!24\)\( T_{5}^{7} + \)\(33\!\cdots\!96\)\( T_{5}^{6} - \)\(85\!\cdots\!40\)\( T_{5}^{5} + \)\(32\!\cdots\!64\)\( T_{5}^{4} - \)\(51\!\cdots\!56\)\( T_{5}^{3} + \)\(17\!\cdots\!60\)\( T_{5}^{2} - \)\(19\!\cdots\!64\)\( T_{5} + \)\(53\!\cdots\!96\)\( \)">\(T_{5}^{18} + \cdots\)
\(19\!\cdots\!68\)\( T_{7}^{3} + \)\(19\!\cdots\!52\)\( T_{7}^{2} + \)\(53\!\cdots\!40\)\( T_{7} - \)\(12\!\cdots\!00\)\( \)">\(T_{7}^{9} - \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} \)
$3$ \( T^{18} \)
$5$ \( \)\(53\!\cdots\!96\)\( - \)\(19\!\cdots\!64\)\( T + \)\(17\!\cdots\!60\)\( T^{2} - \)\(51\!\cdots\!56\)\( T^{3} + \)\(32\!\cdots\!64\)\( T^{4} - \)\(85\!\cdots\!40\)\( T^{5} + \)\(33\!\cdots\!96\)\( T^{6} - \)\(56\!\cdots\!24\)\( T^{7} + 16796878410512090940 T^{8} - 182235863003254716 T^{9} + 5792633935619680 T^{10} - 38635011538024 T^{11} + 1306014959701 T^{12} - 3086592365 T^{13} + 211569494 T^{14} - 174581 T^{15} + 17650 T^{16} + 11 T^{17} + T^{18} \)
$7$ \( ( -1233055145631744000 + 53167521425264640 T + 1928158954033152 T^{2} - 19652044114368 T^{3} - 334655586976 T^{4} + 2542736868 T^{5} + 13902400 T^{6} - 92704 T^{7} - 168 T^{8} + T^{9} )^{2} \)
$11$ \( ( -\)\(15\!\cdots\!96\)\( + 30973025188546606224 T + 2007063245270803884 T^{2} - 8001493002002373 T^{3} - 31276588367368 T^{4} + 134453704519 T^{5} + 146426924 T^{6} - 687587 T^{7} - 160 T^{8} + T^{9} )^{2} \)
$13$ \( \)\(60\!\cdots\!64\)\( + \)\(71\!\cdots\!76\)\( T + \)\(81\!\cdots\!20\)\( T^{2} + \)\(18\!\cdots\!84\)\( T^{3} + \)\(98\!\cdots\!00\)\( T^{4} + \)\(65\!\cdots\!04\)\( T^{5} + \)\(74\!\cdots\!24\)\( T^{6} + \)\(17\!\cdots\!76\)\( T^{7} + \)\(30\!\cdots\!40\)\( T^{8} + \)\(19\!\cdots\!40\)\( T^{9} + \)\(91\!\cdots\!68\)\( T^{10} + 15606742310908401088 T^{11} + 1766913125495305309 T^{12} - 49775665263507 T^{13} + 2463225262058 T^{14} - 100940479 T^{15} + 1961430 T^{16} - 227 T^{17} + T^{18} \)
$17$ \( \)\(16\!\cdots\!84\)\( + \)\(28\!\cdots\!92\)\( T + \)\(83\!\cdots\!04\)\( T^{2} + \)\(63\!\cdots\!00\)\( T^{3} + \)\(15\!\cdots\!04\)\( T^{4} + \)\(10\!\cdots\!76\)\( T^{5} + \)\(19\!\cdots\!16\)\( T^{6} + \)\(84\!\cdots\!84\)\( T^{7} + \)\(13\!\cdots\!40\)\( T^{8} + \)\(46\!\cdots\!68\)\( T^{9} + \)\(71\!\cdots\!40\)\( T^{10} + \)\(16\!\cdots\!60\)\( T^{11} + \)\(24\!\cdots\!29\)\( T^{12} + 37790308727158707 T^{13} + 63161431950210 T^{14} + 5114803983 T^{15} + 9862822 T^{16} + 179 T^{17} + T^{18} \)
$19$ \( \)\(34\!\cdots\!99\)\( + \)\(12\!\cdots\!68\)\( T + \)\(41\!\cdots\!72\)\( T^{2} + \)\(24\!\cdots\!24\)\( T^{3} + \)\(30\!\cdots\!48\)\( T^{4} + \)\(20\!\cdots\!76\)\( T^{5} + \)\(14\!\cdots\!03\)\( T^{6} + \)\(12\!\cdots\!80\)\( T^{7} + \)\(60\!\cdots\!84\)\( T^{8} + \)\(54\!\cdots\!88\)\( T^{9} + \)\(24\!\cdots\!16\)\( T^{10} + \)\(20\!\cdots\!80\)\( T^{11} + 96465687567916357097 T^{12} + 54428798269344676 T^{13} + 32274638969252 T^{14} + 10562267824 T^{15} + 7248128 T^{16} + 868 T^{17} + T^{18} \)
$23$ \( \)\(75\!\cdots\!00\)\( - \)\(45\!\cdots\!00\)\( T + \)\(32\!\cdots\!00\)\( T^{2} + \)\(24\!\cdots\!00\)\( T^{3} + \)\(50\!\cdots\!96\)\( T^{4} + \)\(59\!\cdots\!96\)\( T^{5} + \)\(22\!\cdots\!32\)\( T^{6} - \)\(75\!\cdots\!08\)\( T^{7} + \)\(75\!\cdots\!28\)\( T^{8} - \)\(10\!\cdots\!04\)\( T^{9} + \)\(13\!\cdots\!72\)\( T^{10} - \)\(16\!\cdots\!72\)\( T^{11} + \)\(15\!\cdots\!81\)\( T^{12} - 1959012574736536137 T^{13} + 1113768359604122 T^{14} - 89398709041 T^{15} + 44751186 T^{16} - 3425 T^{17} + T^{18} \)
$29$ \( \)\(96\!\cdots\!24\)\( - \)\(14\!\cdots\!80\)\( T + \)\(18\!\cdots\!96\)\( T^{2} - \)\(10\!\cdots\!80\)\( T^{3} + \)\(69\!\cdots\!64\)\( T^{4} - \)\(30\!\cdots\!76\)\( T^{5} + \)\(14\!\cdots\!88\)\( T^{6} - \)\(52\!\cdots\!68\)\( T^{7} + \)\(18\!\cdots\!24\)\( T^{8} - \)\(43\!\cdots\!64\)\( T^{9} + \)\(11\!\cdots\!88\)\( T^{10} - \)\(20\!\cdots\!64\)\( T^{11} + \)\(44\!\cdots\!49\)\( T^{12} - 61471755887734205509 T^{13} + 10053122334757166 T^{14} - 861197973469 T^{15} + 125209066 T^{16} - 7349 T^{17} + T^{18} \)
$31$ \( ( -\)\(91\!\cdots\!60\)\( - \)\(78\!\cdots\!92\)\( T - \)\(49\!\cdots\!88\)\( T^{2} - \)\(44\!\cdots\!00\)\( T^{3} + 49426332686599495056 T^{4} + 6591076477990692 T^{5} - 989114300688 T^{6} - 167296276 T^{7} + 4980 T^{8} + T^{9} )^{2} \)
$37$ \( ( -\)\(70\!\cdots\!00\)\( - \)\(14\!\cdots\!80\)\( T + \)\(38\!\cdots\!28\)\( T^{2} + \)\(31\!\cdots\!88\)\( T^{3} - 91986213539390854248 T^{4} - 4630799241237276 T^{5} + 3092826215800 T^{6} - 174702136 T^{7} - 13222 T^{8} + T^{9} )^{2} \)
$41$ \( \)\(33\!\cdots\!25\)\( + \)\(74\!\cdots\!25\)\( T + \)\(16\!\cdots\!25\)\( T^{2} + \)\(26\!\cdots\!00\)\( T^{3} + \)\(31\!\cdots\!14\)\( T^{4} + \)\(72\!\cdots\!30\)\( T^{5} + \)\(45\!\cdots\!06\)\( T^{6} + \)\(69\!\cdots\!44\)\( T^{7} + \)\(22\!\cdots\!23\)\( T^{8} + \)\(12\!\cdots\!27\)\( T^{9} + \)\(55\!\cdots\!99\)\( T^{10} + \)\(64\!\cdots\!52\)\( T^{11} + \)\(11\!\cdots\!54\)\( T^{12} - \)\(45\!\cdots\!54\)\( T^{13} + 102037756916691790 T^{14} - 1992308609992 T^{15} + 386322313 T^{16} - 7311 T^{17} + T^{18} \)
$43$ \( \)\(51\!\cdots\!00\)\( + \)\(14\!\cdots\!00\)\( T + \)\(38\!\cdots\!00\)\( T^{2} + \)\(62\!\cdots\!00\)\( T^{3} + \)\(90\!\cdots\!00\)\( T^{4} + \)\(41\!\cdots\!40\)\( T^{5} + \)\(18\!\cdots\!76\)\( T^{6} + \)\(47\!\cdots\!96\)\( T^{7} + \)\(10\!\cdots\!80\)\( T^{8} + \)\(14\!\cdots\!84\)\( T^{9} + \)\(18\!\cdots\!84\)\( T^{10} + \)\(17\!\cdots\!12\)\( T^{11} + \)\(17\!\cdots\!89\)\( T^{12} + \)\(12\!\cdots\!91\)\( T^{13} + 112961728986806062 T^{14} + 4760975065791 T^{15} + 379698838 T^{16} + 8283 T^{17} + T^{18} \)
$47$ \( \)\(16\!\cdots\!16\)\( - \)\(23\!\cdots\!44\)\( T + \)\(96\!\cdots\!24\)\( T^{2} - \)\(78\!\cdots\!88\)\( T^{3} + \)\(21\!\cdots\!36\)\( T^{4} + \)\(67\!\cdots\!76\)\( T^{5} + \)\(40\!\cdots\!32\)\( T^{6} + \)\(26\!\cdots\!32\)\( T^{7} + \)\(39\!\cdots\!12\)\( T^{8} + \)\(20\!\cdots\!40\)\( T^{9} + \)\(20\!\cdots\!08\)\( T^{10} + \)\(95\!\cdots\!48\)\( T^{11} + \)\(70\!\cdots\!29\)\( T^{12} + \)\(25\!\cdots\!99\)\( T^{13} + 1438596564851851442 T^{14} + 41903660395535 T^{15} + 1912969374 T^{16} + 37603 T^{17} + T^{18} \)
$53$ \( \)\(50\!\cdots\!64\)\( - \)\(39\!\cdots\!44\)\( T + \)\(17\!\cdots\!60\)\( T^{2} - \)\(65\!\cdots\!96\)\( T^{3} + \)\(33\!\cdots\!16\)\( T^{4} - \)\(11\!\cdots\!92\)\( T^{5} + \)\(37\!\cdots\!08\)\( T^{6} - \)\(73\!\cdots\!92\)\( T^{7} + \)\(27\!\cdots\!96\)\( T^{8} - \)\(69\!\cdots\!00\)\( T^{9} + \)\(10\!\cdots\!88\)\( T^{10} - \)\(24\!\cdots\!56\)\( T^{11} + \)\(25\!\cdots\!69\)\( T^{12} - \)\(59\!\cdots\!49\)\( T^{13} + 3460270021546904626 T^{14} - 43357881060629 T^{15} + 2267852878 T^{16} - 20337 T^{17} + T^{18} \)
$59$ \( \)\(21\!\cdots\!89\)\( - \)\(40\!\cdots\!55\)\( T + \)\(71\!\cdots\!53\)\( T^{2} - \)\(47\!\cdots\!00\)\( T^{3} + \)\(44\!\cdots\!82\)\( T^{4} - \)\(25\!\cdots\!38\)\( T^{5} + \)\(17\!\cdots\!26\)\( T^{6} - \)\(79\!\cdots\!68\)\( T^{7} + \)\(39\!\cdots\!87\)\( T^{8} - \)\(13\!\cdots\!77\)\( T^{9} + \)\(53\!\cdots\!07\)\( T^{10} - \)\(15\!\cdots\!52\)\( T^{11} + \)\(47\!\cdots\!62\)\( T^{12} - \)\(10\!\cdots\!18\)\( T^{13} + 23327287366044483930 T^{14} - 357651739695276 T^{15} + 7129477477 T^{16} - 74455 T^{17} + T^{18} \)
$61$ \( \)\(21\!\cdots\!56\)\( - \)\(20\!\cdots\!04\)\( T + \)\(11\!\cdots\!76\)\( T^{2} - \)\(16\!\cdots\!40\)\( T^{3} + \)\(57\!\cdots\!80\)\( T^{4} - \)\(63\!\cdots\!04\)\( T^{5} + \)\(81\!\cdots\!92\)\( T^{6} - \)\(33\!\cdots\!76\)\( T^{7} + \)\(26\!\cdots\!64\)\( T^{8} - \)\(57\!\cdots\!12\)\( T^{9} + \)\(57\!\cdots\!12\)\( T^{10} - \)\(69\!\cdots\!56\)\( T^{11} + \)\(74\!\cdots\!81\)\( T^{12} - \)\(16\!\cdots\!47\)\( T^{13} + 6930515824571977902 T^{14} - 1910399499327 T^{15} + 3182325786 T^{16} + 7569 T^{17} + T^{18} \)
$67$ \( \)\(92\!\cdots\!81\)\( + \)\(49\!\cdots\!29\)\( T + \)\(58\!\cdots\!85\)\( T^{2} + \)\(13\!\cdots\!60\)\( T^{3} + \)\(15\!\cdots\!62\)\( T^{4} + \)\(27\!\cdots\!90\)\( T^{5} + \)\(25\!\cdots\!86\)\( T^{6} + \)\(31\!\cdots\!92\)\( T^{7} + \)\(27\!\cdots\!35\)\( T^{8} + \)\(30\!\cdots\!31\)\( T^{9} + \)\(21\!\cdots\!79\)\( T^{10} + \)\(17\!\cdots\!60\)\( T^{11} + \)\(10\!\cdots\!10\)\( T^{12} + \)\(82\!\cdots\!10\)\( T^{13} + 38408940894285316426 T^{14} + 181133911740404 T^{15} + 7768368661 T^{16} + 26177 T^{17} + T^{18} \)
$71$ \( \)\(17\!\cdots\!00\)\( + \)\(39\!\cdots\!80\)\( T + \)\(13\!\cdots\!96\)\( T^{2} + \)\(43\!\cdots\!60\)\( T^{3} + \)\(28\!\cdots\!72\)\( T^{4} + \)\(21\!\cdots\!92\)\( T^{5} + \)\(27\!\cdots\!00\)\( T^{6} + \)\(71\!\cdots\!28\)\( T^{7} + \)\(42\!\cdots\!72\)\( T^{8} + \)\(65\!\cdots\!32\)\( T^{9} + \)\(41\!\cdots\!64\)\( T^{10} + \)\(44\!\cdots\!68\)\( T^{11} + \)\(19\!\cdots\!33\)\( T^{12} + \)\(39\!\cdots\!97\)\( T^{13} + 53973233604256978870 T^{14} - 53391804438599 T^{15} + 10847901010 T^{16} - 53463 T^{17} + T^{18} \)
$73$ \( \)\(43\!\cdots\!25\)\( + \)\(74\!\cdots\!75\)\( T + \)\(29\!\cdots\!25\)\( T^{2} + \)\(13\!\cdots\!20\)\( T^{3} + \)\(17\!\cdots\!06\)\( T^{4} + \)\(62\!\cdots\!86\)\( T^{5} + \)\(33\!\cdots\!42\)\( T^{6} + \)\(67\!\cdots\!72\)\( T^{7} + \)\(29\!\cdots\!47\)\( T^{8} + \)\(45\!\cdots\!81\)\( T^{9} + \)\(17\!\cdots\!35\)\( T^{10} + \)\(15\!\cdots\!60\)\( T^{11} + \)\(50\!\cdots\!46\)\( T^{12} + \)\(28\!\cdots\!26\)\( T^{13} + \)\(11\!\cdots\!74\)\( T^{14} + 307313940776492 T^{15} + 12536846917 T^{16} + 14103 T^{17} + T^{18} \)
$79$ \( \)\(12\!\cdots\!44\)\( + \)\(90\!\cdots\!32\)\( T + \)\(33\!\cdots\!84\)\( T^{2} - \)\(34\!\cdots\!16\)\( T^{3} + \)\(49\!\cdots\!56\)\( T^{4} - \)\(22\!\cdots\!72\)\( T^{5} + \)\(11\!\cdots\!08\)\( T^{6} - \)\(26\!\cdots\!68\)\( T^{7} + \)\(12\!\cdots\!16\)\( T^{8} - \)\(20\!\cdots\!36\)\( T^{9} + \)\(86\!\cdots\!68\)\( T^{10} - \)\(76\!\cdots\!32\)\( T^{11} + \)\(31\!\cdots\!41\)\( T^{12} - \)\(18\!\cdots\!25\)\( T^{13} + 83272908481639449190 T^{14} - 246647107958493 T^{15} + 11209186574 T^{16} - 31825 T^{17} + T^{18} \)
$83$ \( ( -\)\(24\!\cdots\!00\)\( + \)\(20\!\cdots\!20\)\( T - \)\(98\!\cdots\!28\)\( T^{2} - \)\(39\!\cdots\!37\)\( T^{3} + \)\(15\!\cdots\!20\)\( T^{4} + 38020731823258649847 T^{5} - 511917348889560 T^{6} - 11706552339 T^{7} + 41300 T^{8} + T^{9} )^{2} \)
$89$ \( \)\(86\!\cdots\!44\)\( - \)\(57\!\cdots\!40\)\( T + \)\(64\!\cdots\!92\)\( T^{2} - \)\(27\!\cdots\!72\)\( T^{3} + \)\(26\!\cdots\!88\)\( T^{4} - \)\(11\!\cdots\!40\)\( T^{5} + \)\(49\!\cdots\!48\)\( T^{6} - \)\(13\!\cdots\!36\)\( T^{7} + \)\(35\!\cdots\!68\)\( T^{8} - \)\(69\!\cdots\!20\)\( T^{9} + \)\(14\!\cdots\!24\)\( T^{10} - \)\(22\!\cdots\!88\)\( T^{11} + \)\(35\!\cdots\!61\)\( T^{12} - \)\(38\!\cdots\!45\)\( T^{13} + \)\(41\!\cdots\!70\)\( T^{14} - 3121899045395049 T^{15} + 29541644542 T^{16} - 155197 T^{17} + T^{18} \)
$97$ \( \)\(93\!\cdots\!25\)\( + \)\(95\!\cdots\!75\)\( T + \)\(10\!\cdots\!25\)\( T^{2} + \)\(26\!\cdots\!00\)\( T^{3} + \)\(17\!\cdots\!14\)\( T^{4} + \)\(38\!\cdots\!90\)\( T^{5} + \)\(22\!\cdots\!50\)\( T^{6} + \)\(28\!\cdots\!48\)\( T^{7} + \)\(71\!\cdots\!71\)\( T^{8} + \)\(31\!\cdots\!17\)\( T^{9} + \)\(10\!\cdots\!83\)\( T^{10} + \)\(19\!\cdots\!24\)\( T^{11} + \)\(12\!\cdots\!86\)\( T^{12} - \)\(12\!\cdots\!34\)\( T^{13} + \)\(85\!\cdots\!22\)\( T^{14} - 1646469577269280 T^{15} + 42695596097 T^{16} - 111241 T^{17} + T^{18} \)
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