Properties

Label 147.4.g.d
Level $147$
Weight $4$
Character orbit 147.g
Analytic conductor $8.673$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 147 = 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 147.g (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.67328077084\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - x^{11} - 29 x^{9} + 6 x^{8} - 49 x^{7} + 1564 x^{6} - 441 x^{5} + 486 x^{4} - 21141 x^{3} - 59049 x + 531441\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3^{3} \)
Twist minimal: no (minimal twist has level 21)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \beta_{2} - \beta_{6} ) q^{2} + ( -\beta_{7} - \beta_{8} ) q^{3} + ( -2 \beta_{4} + \beta_{7} + \beta_{10} - \beta_{11} ) q^{4} -\beta_{5} q^{5} + ( -2 + \beta_{2} + \beta_{3} - 4 \beta_{4} + \beta_{5} - 2 \beta_{6} + \beta_{7} - \beta_{9} + \beta_{10} ) q^{6} + ( \beta_{1} - 2 \beta_{2} - \beta_{5} - \beta_{7} - 2 \beta_{8} - \beta_{9} + \beta_{11} ) q^{8} + ( \beta_{1} - 3 \beta_{3} + \beta_{5} - \beta_{7} + \beta_{8} - 2 \beta_{9} - 2 \beta_{11} ) q^{9} +O(q^{10})\) \( q + ( \beta_{2} - \beta_{6} ) q^{2} + ( -\beta_{7} - \beta_{8} ) q^{3} + ( -2 \beta_{4} + \beta_{7} + \beta_{10} - \beta_{11} ) q^{4} -\beta_{5} q^{5} + ( -2 + \beta_{2} + \beta_{3} - 4 \beta_{4} + \beta_{5} - 2 \beta_{6} + \beta_{7} - \beta_{9} + \beta_{10} ) q^{6} + ( \beta_{1} - 2 \beta_{2} - \beta_{5} - \beta_{7} - 2 \beta_{8} - \beta_{9} + \beta_{11} ) q^{8} + ( \beta_{1} - 3 \beta_{3} + \beta_{5} - \beta_{7} + \beta_{8} - 2 \beta_{9} - 2 \beta_{11} ) q^{9} + ( 6 \beta_{1} + 2 \beta_{3} + 5 \beta_{7} + 4 \beta_{8} - \beta_{10} + \beta_{11} ) q^{10} + ( 2 \beta_{1} - 2 \beta_{5} + 4 \beta_{6} - 2 \beta_{7} + 2 \beta_{8} + \beta_{9} - 4 \beta_{11} ) q^{11} + ( 10 - 2 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} - 10 \beta_{4} - \beta_{5} + 4 \beta_{6} + 2 \beta_{7} + \beta_{8} + 4 \beta_{10} - 5 \beta_{11} ) q^{12} + ( 9 - 2 \beta_{1} - 3 \beta_{3} + 18 \beta_{4} - 5 \beta_{7} - 3 \beta_{10} + 2 \beta_{11} ) q^{13} + ( 3 + 4 \beta_{1} + 12 \beta_{2} + \beta_{5} + \beta_{7} + 2 \beta_{8} + \beta_{9} + 4 \beta_{11} ) q^{15} + ( 24 + 3 \beta_{1} - \beta_{3} + 24 \beta_{4} + 3 \beta_{7} + 2 \beta_{8} - 2 \beta_{11} ) q^{16} + ( 9 \beta_{1} - 8 \beta_{2} + 4 \beta_{6} - 9 \beta_{7} - 9 \beta_{8} ) q^{17} + ( 6 \beta_{1} - 6 \beta_{4} - 3 \beta_{6} + 6 \beta_{7} + 6 \beta_{10} - 18 \beta_{11} ) q^{18} + ( -17 - \beta_{1} - \beta_{3} + 17 \beta_{4} + \beta_{7} + 2 \beta_{10} - 2 \beta_{11} ) q^{19} + ( -3 \beta_{1} - 12 \beta_{2} - 3 \beta_{5} + 24 \beta_{6} + 3 \beta_{7} + 3 \beta_{9} + 3 \beta_{11} ) q^{20} + ( -16 + 3 \beta_{1} - 5 \beta_{3} + 8 \beta_{7} + 16 \beta_{8} + 5 \beta_{10} + 3 \beta_{11} ) q^{22} + ( 10 \beta_{1} + 4 \beta_{2} - 4 \beta_{6} - 10 \beta_{7} - 5 \beta_{8} - 5 \beta_{11} ) q^{23} + ( -40 + 10 \beta_{1} + 20 \beta_{2} - 10 \beta_{3} - 20 \beta_{4} - 10 \beta_{6} + 3 \beta_{7} + 8 \beta_{8} + 2 \beta_{9} + 5 \beta_{10} - 5 \beta_{11} ) q^{24} + ( 11 \beta_{1} + 13 \beta_{4} - 4 \beta_{7} - 11 \beta_{8} - 15 \beta_{10} - 7 \beta_{11} ) q^{25} + ( -19 \beta_{2} + 7 \beta_{5} - 19 \beta_{6} + 9 \beta_{8} - 9 \beta_{11} ) q^{26} + ( -21 + 24 \beta_{2} - 3 \beta_{3} - 42 \beta_{4} - 3 \beta_{5} - 48 \beta_{6} + 3 \beta_{7} + 3 \beta_{9} - 3 \beta_{10} ) q^{27} + ( 7 \beta_{1} - 52 \beta_{2} + 5 \beta_{5} - 7 \beta_{7} - 14 \beta_{8} + 5 \beta_{9} + 7 \beta_{11} ) q^{29} + ( -138 + 4 \beta_{1} - 12 \beta_{2} + 18 \beta_{3} - 138 \beta_{4} - 5 \beta_{5} + 12 \beta_{6} + 4 \beta_{7} - 7 \beta_{8} + 10 \beta_{9} + 7 \beta_{11} ) q^{30} + ( 110 + 12 \beta_{1} + 4 \beta_{3} + 55 \beta_{4} + 10 \beta_{7} + 8 \beta_{8} - 2 \beta_{10} + 2 \beta_{11} ) q^{31} + ( 9 \beta_{1} + 10 \beta_{5} - 9 \beta_{7} + 9 \beta_{8} - 5 \beta_{9} - 18 \beta_{11} ) q^{32} + ( 49 + 10 \beta_{1} + 16 \beta_{2} + 10 \beta_{3} - 49 \beta_{4} + 5 \beta_{5} + 16 \beta_{6} - 10 \beta_{7} - 7 \beta_{8} - 20 \beta_{10} + 16 \beta_{11} ) q^{33} + ( 4 - 4 \beta_{1} + 14 \beta_{3} + 8 \beta_{4} + 10 \beta_{7} + 14 \beta_{10} + 4 \beta_{11} ) q^{34} + ( 66 - \beta_{1} + 48 \beta_{2} + 3 \beta_{3} - 4 \beta_{5} - 8 \beta_{7} - 16 \beta_{8} - 4 \beta_{9} - 3 \beta_{10} - \beta_{11} ) q^{36} + ( 135 + 19 \beta_{1} + 11 \beta_{3} + 135 \beta_{4} + 19 \beta_{7} + 4 \beta_{8} - 4 \beta_{11} ) q^{37} + ( 3 \beta_{1} - 26 \beta_{2} + 13 \beta_{6} - 3 \beta_{7} - 3 \beta_{8} - 3 \beta_{9} ) q^{38} + ( -5 \beta_{1} + 63 \beta_{4} + 4 \beta_{5} - 36 \beta_{6} - 16 \beta_{7} + \beta_{8} - 2 \beta_{9} - 15 \beta_{10} + 25 \beta_{11} ) q^{39} + ( -132 + 13 \beta_{1} + 13 \beta_{3} + 132 \beta_{4} - 13 \beta_{7} - 26 \beta_{8} - 26 \beta_{10} ) q^{40} + ( 6 \beta_{1} - 32 \beta_{2} + 16 \beta_{5} + 64 \beta_{6} - 6 \beta_{7} - 16 \beta_{9} - 6 \beta_{11} ) q^{41} + ( -87 + 18 \beta_{1} + 31 \beta_{3} - 13 \beta_{7} - 26 \beta_{8} - 31 \beta_{10} + 18 \beta_{11} ) q^{43} + ( -22 \beta_{1} - 12 \beta_{2} - 3 \beta_{5} + 12 \beta_{6} + 22 \beta_{7} + 11 \beta_{8} + 6 \beta_{9} + 11 \beta_{11} ) q^{44} + ( -270 + 30 \beta_{3} - 135 \beta_{4} - 6 \beta_{7} - 21 \beta_{8} - 12 \beta_{9} - 15 \beta_{10} + 15 \beta_{11} ) q^{45} + ( -10 \beta_{1} - 100 \beta_{4} + 24 \beta_{7} + 10 \beta_{8} + 34 \beta_{10} - 14 \beta_{11} ) q^{46} + ( -20 \beta_{2} - 14 \beta_{5} - 20 \beta_{6} - 27 \beta_{8} + 27 \beta_{11} ) q^{47} + ( -44 - 9 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} - 88 \beta_{4} - 5 \beta_{5} - 8 \beta_{6} - 19 \beta_{7} + 5 \beta_{9} + 4 \beta_{10} + 9 \beta_{11} ) q^{48} + ( -15 \beta_{1} - 7 \beta_{2} - 7 \beta_{5} + 15 \beta_{7} + 30 \beta_{8} - 7 \beta_{9} - 15 \beta_{11} ) q^{50} + ( -219 + 5 \beta_{1} - 12 \beta_{2} - 39 \beta_{3} - 219 \beta_{4} + 5 \beta_{5} + 12 \beta_{6} - 13 \beta_{7} + 13 \beta_{8} - 10 \beta_{9} - 22 \beta_{11} ) q^{51} + ( 308 - 58 \beta_{1} - 40 \beta_{3} + 154 \beta_{4} - 38 \beta_{7} - 18 \beta_{8} + 20 \beta_{10} - 20 \beta_{11} ) q^{52} + ( -28 \beta_{1} - 6 \beta_{5} + 40 \beta_{6} + 28 \beta_{7} - 28 \beta_{8} + 3 \beta_{9} + 56 \beta_{11} ) q^{53} + ( 228 - 15 \beta_{1} + 21 \beta_{2} - 15 \beta_{3} - 228 \beta_{4} - 3 \beta_{5} + 21 \beta_{6} + 15 \beta_{7} + 15 \beta_{8} + 30 \beta_{10} - 33 \beta_{11} ) q^{54} + ( 150 + 28 \beta_{1} - 19 \beta_{3} + 300 \beta_{4} + 9 \beta_{7} - 19 \beta_{10} - 28 \beta_{11} ) q^{55} + ( 30 - 7 \beta_{1} + 12 \beta_{2} - 6 \beta_{3} - \beta_{5} + 20 \beta_{7} + 40 \beta_{8} - \beta_{9} + 6 \beta_{10} - 7 \beta_{11} ) q^{57} + ( 436 - 93 \beta_{1} - 25 \beta_{3} + 436 \beta_{4} - 93 \beta_{7} - 34 \beta_{8} + 34 \beta_{11} ) q^{58} + ( -54 \beta_{1} + 32 \beta_{2} - 16 \beta_{6} + 54 \beta_{7} + 54 \beta_{8} + 19 \beta_{9} ) q^{59} + ( -6 \beta_{1} + 144 \beta_{4} - 24 \beta_{5} + 72 \beta_{6} - 21 \beta_{7} - 6 \beta_{8} + 12 \beta_{9} - 27 \beta_{10} + 39 \beta_{11} ) q^{60} + ( -114 - 38 \beta_{1} - 38 \beta_{3} + 114 \beta_{4} + 38 \beta_{7} + 89 \beta_{8} + 76 \beta_{10} + 13 \beta_{11} ) q^{61} + ( -6 \beta_{1} + 31 \beta_{2} - 22 \beta_{5} - 62 \beta_{6} + 6 \beta_{7} + 22 \beta_{9} + 6 \beta_{11} ) q^{62} + ( -84 - 79 \beta_{1} - 61 \beta_{3} - 18 \beta_{7} - 36 \beta_{8} + 61 \beta_{10} - 79 \beta_{11} ) q^{64} + ( 12 \beta_{1} + 84 \beta_{2} + 14 \beta_{5} - 84 \beta_{6} - 12 \beta_{7} - 6 \beta_{8} - 28 \beta_{9} - 6 \beta_{11} ) q^{65} + ( -332 - 25 \beta_{1} + 40 \beta_{2} + 28 \beta_{3} - 166 \beta_{4} - 20 \beta_{6} + 21 \beta_{7} + 7 \beta_{8} + 19 \beta_{9} - 14 \beta_{10} + 14 \beta_{11} ) q^{66} + ( -62 \beta_{1} - 181 \beta_{4} - 23 \beta_{7} + 62 \beta_{8} + 39 \beta_{10} + 85 \beta_{11} ) q^{67} + ( 48 \beta_{2} - 6 \beta_{5} + 48 \beta_{6} + 30 \beta_{8} - 30 \beta_{11} ) q^{68} + ( -143 + 15 \beta_{1} + 4 \beta_{2} - 11 \beta_{3} - 286 \beta_{4} + 19 \beta_{5} - 8 \beta_{6} - 11 \beta_{7} - 19 \beta_{9} - 11 \beta_{10} - 15 \beta_{11} ) q^{69} + ( -18 \beta_{1} + 68 \beta_{2} + 4 \beta_{5} + 18 \beta_{7} + 36 \beta_{8} + 4 \beta_{9} - 18 \beta_{11} ) q^{71} + ( -456 - 54 \beta_{1} + 66 \beta_{2} + 24 \beta_{3} - 456 \beta_{4} + 9 \beta_{5} - 66 \beta_{6} + 54 \beta_{7} + 15 \beta_{8} - 18 \beta_{9} + 39 \beta_{11} ) q^{72} + ( 290 - 38 \beta_{1} + 22 \beta_{3} + 145 \beta_{4} - 49 \beta_{7} - 60 \beta_{8} - 11 \beta_{10} + 11 \beta_{11} ) q^{73} + ( -11 \beta_{1} - 38 \beta_{5} - 67 \beta_{6} + 11 \beta_{7} - 11 \beta_{8} + 19 \beta_{9} + 22 \beta_{11} ) q^{74} + ( 147 - 3 \beta_{1} - 60 \beta_{2} - 3 \beta_{3} - 147 \beta_{4} - 18 \beta_{5} - 60 \beta_{6} + 3 \beta_{7} + \beta_{8} + 6 \beta_{10} + 75 \beta_{11} ) q^{75} + ( -18 - 6 \beta_{1} - 12 \beta_{3} - 36 \beta_{4} - 18 \beta_{7} - 12 \beta_{10} + 6 \beta_{11} ) q^{76} + ( 336 - 57 \beta_{1} - 141 \beta_{2} - 30 \beta_{3} + 21 \beta_{5} + 3 \beta_{7} + 6 \beta_{8} + 21 \beta_{9} + 30 \beta_{10} - 57 \beta_{11} ) q^{78} + ( 325 + 88 \beta_{1} - 12 \beta_{3} + 325 \beta_{4} + 88 \beta_{7} + 50 \beta_{8} - 50 \beta_{11} ) q^{79} + ( -15 \beta_{1} - 72 \beta_{2} + 36 \beta_{6} + 15 \beta_{7} + 15 \beta_{8} - 37 \beta_{9} ) q^{80} + ( -57 \beta_{1} - 144 \beta_{4} + 42 \beta_{5} - 72 \beta_{6} + 75 \beta_{7} + 6 \beta_{8} - 21 \beta_{9} + 81 \beta_{10} + 33 \beta_{11} ) q^{81} + ( -296 + 4 \beta_{1} + 4 \beta_{3} + 296 \beta_{4} - 4 \beta_{7} - 52 \beta_{8} - 8 \beta_{10} - 44 \beta_{11} ) q^{82} + ( 27 \beta_{1} + 56 \beta_{2} - 13 \beta_{5} - 112 \beta_{6} - 27 \beta_{7} + 13 \beta_{9} - 27 \beta_{11} ) q^{83} + ( 54 + 17 \beta_{1} - 12 \beta_{3} + 29 \beta_{7} + 58 \beta_{8} + 12 \beta_{10} + 17 \beta_{11} ) q^{85} + ( -62 \beta_{1} - 133 \beta_{2} - 5 \beta_{5} + 133 \beta_{6} + 62 \beta_{7} + 31 \beta_{8} + 10 \beta_{9} + 31 \beta_{11} ) q^{86} + ( -200 - 112 \beta_{1} - 224 \beta_{2} - 146 \beta_{3} - 100 \beta_{4} + 112 \beta_{6} - 15 \beta_{7} + 58 \beta_{8} + 16 \beta_{9} + 73 \beta_{10} - 73 \beta_{11} ) q^{87} + ( 74 \beta_{1} + 124 \beta_{4} - 53 \beta_{7} - 74 \beta_{8} - 127 \beta_{10} - 21 \beta_{11} ) q^{88} + ( 104 \beta_{2} + 48 \beta_{5} + 104 \beta_{6} - 51 \beta_{8} + 51 \beta_{11} ) q^{89} + ( 18 + 15 \beta_{1} - 144 \beta_{2} + 3 \beta_{3} + 36 \beta_{4} + 6 \beta_{5} + 288 \beta_{6} + 108 \beta_{7} - 6 \beta_{9} + 3 \beta_{10} - 15 \beta_{11} ) q^{90} + ( -6 \beta_{1} + 144 \beta_{2} - 14 \beta_{5} + 6 \beta_{7} + 12 \beta_{8} - 14 \beta_{9} - 6 \beta_{11} ) q^{92} + ( -276 + 8 \beta_{1} - 24 \beta_{2} + 36 \beta_{3} - 276 \beta_{4} - 10 \beta_{5} + 24 \beta_{6} - 102 \beta_{7} - 69 \beta_{8} + 20 \beta_{9} + 14 \beta_{11} ) q^{93} + ( 184 + 98 \beta_{1} + 96 \beta_{3} + 92 \beta_{4} + 50 \beta_{7} + 2 \beta_{8} - 48 \beta_{10} + 48 \beta_{11} ) q^{94} + ( -3 \beta_{1} + 32 \beta_{5} + 36 \beta_{6} + 3 \beta_{7} - 3 \beta_{8} - 16 \beta_{9} + 6 \beta_{11} ) q^{95} + ( 228 + 27 \beta_{1} - 60 \beta_{2} + 27 \beta_{3} - 228 \beta_{4} + 12 \beta_{5} - 60 \beta_{6} - 27 \beta_{7} - 42 \beta_{8} - 54 \beta_{10} - 60 \beta_{11} ) q^{96} + ( 156 - 37 \beta_{1} + 81 \beta_{3} + 312 \beta_{4} + 44 \beta_{7} + 81 \beta_{10} + 37 \beta_{11} ) q^{97} + ( -303 + 75 \beta_{1} - 132 \beta_{2} + 87 \beta_{3} - 18 \beta_{5} - 93 \beta_{7} - 186 \beta_{8} - 18 \beta_{9} - 87 \beta_{10} + 75 \beta_{11} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + 3q^{3} + 14q^{4} - 3q^{9} + O(q^{10}) \) \( 12q + 3q^{3} + 14q^{4} - 3q^{9} - 30q^{10} + 192q^{12} + 6q^{15} + 134q^{16} + 66q^{18} - 300q^{19} - 268q^{22} - 414q^{24} - 42q^{25} - 822q^{30} + 930q^{31} + 855q^{33} + 852q^{36} + 764q^{37} - 426q^{39} - 2298q^{40} - 1012q^{43} - 2367q^{45} + 608q^{46} - 1341q^{51} + 3000q^{52} + 4158q^{54} + 270q^{57} + 2870q^{58} - 918q^{60} - 2358q^{61} - 548q^{64} - 2934q^{66} + 792q^{67} - 2712q^{72} + 2904q^{73} + 2418q^{75} + 4296q^{78} + 1674q^{79} + 837q^{81} - 5040q^{82} + 348q^{85} - 1638q^{87} - 554q^{88} - 1479q^{93} + 1356q^{94} + 4410q^{96} - 3354q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - x^{11} - 29 x^{9} + 6 x^{8} - 49 x^{7} + 1564 x^{6} - 441 x^{5} + 486 x^{4} - 21141 x^{3} - 59049 x + 531441\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(-538 \nu^{11} - 22601 \nu^{10} + 146502 \nu^{9} - 1327 \nu^{8} + 161148 \nu^{7} - 632573 \nu^{6} + 6980468 \nu^{5} - 17528769 \nu^{4} + 61874280 \nu^{3} - 81309015 \nu^{2} - 145536102 \nu + 142839531\)\()/ 489398112 \)
\(\beta_{2}\)\(=\)\((\)\(-661 \nu^{11} - 17114 \nu^{10} - 13815 \nu^{9} + 226448 \nu^{8} + 617943 \nu^{7} + 441628 \nu^{6} - 2790145 \nu^{5} - 21396420 \nu^{4} - 82640331 \nu^{3} + 112284954 \nu^{2} + 410108427 \nu + 575491554\)\()/ 489398112 \)
\(\beta_{3}\)\(=\)\((\)\( 109 \nu^{11} - 3322 \nu^{10} - 26109 \nu^{9} + 69172 \nu^{8} + 83625 \nu^{7} - 33772 \nu^{6} + 26593 \nu^{5} - 1141056 \nu^{4} - 12229137 \nu^{3} + 61499898 \nu^{2} - 15136227 \nu - 271271106 \)\()/69914016\)
\(\beta_{4}\)\(=\)\((\)\( -857 \nu^{11} + 5426 \nu^{10} - 627 \nu^{9} + 6088 \nu^{8} - 24405 \nu^{7} + 289700 \nu^{6} - 658001 \nu^{5} + 2301324 \nu^{4} - 3432699 \nu^{3} - 5390226 \nu^{2} + 4113747 \nu + 173722158 \)\()/ 163132704 \)
\(\beta_{5}\)\(=\)\((\)\(-4301 \nu^{11} + 145853 \nu^{10} - 231543 \nu^{9} - 1385435 \nu^{8} - 5417499 \nu^{7} + 15945431 \nu^{6} + 12125077 \nu^{5} + 198180063 \nu^{4} - 245384073 \nu^{3} - 938815677 \nu^{2} - 3646715337 \nu + 8693961417\)\()/ 489398112 \)
\(\beta_{6}\)\(=\)\((\)\(-4345 \nu^{11} + 19933 \nu^{10} + 136449 \nu^{9} + 345029 \nu^{8} - 1557771 \nu^{7} - 3453101 \nu^{6} - 8432491 \nu^{5} + 16879851 \nu^{4} + 90039843 \nu^{3} + 303484887 \nu^{2} - 575773677 \nu - 1180448559\)\()/ 489398112 \)
\(\beta_{7}\)\(=\)\((\)\( -1523 \nu^{11} + 209 \nu^{10} + 6255 \nu^{9} + 6421 \nu^{8} - 82569 \nu^{7} - 227449 \nu^{6} - 641129 \nu^{5} + 1005399 \nu^{4} + 7836021 \nu^{3} - 1371249 \nu^{2} + 13338513 \nu - 151814979 \)\()/54377568\)
\(\beta_{8}\)\(=\)\((\)\( -1523 \nu^{11} + 209 \nu^{10} + 6255 \nu^{9} + 6421 \nu^{8} - 82569 \nu^{7} - 227449 \nu^{6} - 641129 \nu^{5} + 1005399 \nu^{4} + 7836021 \nu^{3} - 1371249 \nu^{2} - 149794191 \nu - 151814979 \)\()/54377568\)
\(\beta_{9}\)\(=\)\((\)\(22376 \nu^{11} + 116845 \nu^{10} - 61380 \nu^{9} - 2138089 \nu^{8} - 3679350 \nu^{7} + 8057845 \nu^{6} + 34199906 \nu^{5} + 126399249 \nu^{4} - 75013614 \nu^{3} - 784745901 \nu^{2} - 2180220300 \nu + 5512873689\)\()/ 489398112 \)
\(\beta_{10}\)\(=\)\((\)\( 635 \nu^{11} - 221 \nu^{10} + 4041 \nu^{9} + 4103 \nu^{8} + 30441 \nu^{7} - 164387 \nu^{6} + 335465 \nu^{5} + 2637 \nu^{4} + 2622699 \nu^{3} + 8961597 \nu^{2} + 22720743 \nu - 66193929 \)\()/13226976\)
\(\beta_{11}\)\(=\)\((\)\(-12970 \nu^{11} - 10169 \nu^{10} + 146502 \nu^{9} + 359201 \nu^{8} + 86556 \nu^{7} - 23405 \nu^{6} - 12463180 \nu^{5} - 12046257 \nu^{4} + 55832328 \nu^{3} + 181515897 \nu^{2} - 145536102 \nu + 876936699\)\()/ 244699056 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{8} + \beta_{7}\)\()/3\)
\(\nu^{2}\)\(=\)\((\)\(2 \beta_{11} + 2 \beta_{9} - \beta_{8} + \beta_{7} - \beta_{5} + 3 \beta_{3} - \beta_{1}\)\()/3\)
\(\nu^{3}\)\(=\)\((\)\(2 \beta_{11} - 3 \beta_{10} - \beta_{9} - 6 \beta_{8} - 3 \beta_{7} - \beta_{5} + 3 \beta_{3} - 24 \beta_{2} + 2 \beta_{1} + 21\)\()/3\)
\(\nu^{4}\)\(=\)\((\)\(11 \beta_{11} + 27 \beta_{10} - 7 \beta_{9} + 2 \beta_{8} + 25 \beta_{7} - 24 \beta_{6} + 14 \beta_{5} - 48 \beta_{4} - 19 \beta_{1}\)\()/3\)
\(\nu^{5}\)\(=\)\((\)\(-33 \beta_{11} + 12 \beta_{9} - 10 \beta_{8} + 52 \beta_{7} - 6 \beta_{5} + 216 \beta_{4} + 72 \beta_{3} + 138 \beta_{1} + 216\)\()/3\)
\(\nu^{6}\)\(=\)\((\)\(134 \beta_{11} - 114 \beta_{10} + 8 \beta_{9} - 280 \beta_{8} - 140 \beta_{7} + 8 \beta_{5} + 114 \beta_{3} - 360 \beta_{2} + 134 \beta_{1} - 1629\)\()/3\)
\(\nu^{7}\)\(=\)\((\)\(260 \beta_{11} + 552 \beta_{10} - 10 \beta_{9} + 681 \beta_{8} - 129 \beta_{7} - 912 \beta_{6} + 20 \beta_{5} + 984 \beta_{4} - 406 \beta_{1}\)\()/3\)
\(\nu^{8}\)\(=\)\((\)\(-1282 \beta_{11} - 1582 \beta_{9} - 493 \beta_{8} - 1013 \beta_{7} + 1176 \beta_{6} + 791 \beta_{5} + 3648 \beta_{4} - 27 \beta_{3} - 1176 \beta_{2} + 2537 \beta_{1} + 3648\)\()/3\)
\(\nu^{9}\)\(=\)\((\)\(1824 \beta_{11} + 2709 \beta_{10} + 681 \beta_{9} + 974 \beta_{8} + 487 \beta_{7} + 681 \beta_{5} - 2709 \beta_{3} + 8424 \beta_{2} + 1824 \beta_{1} - 29619\)\()/3\)
\(\nu^{10}\)\(=\)\((\)\(-10351 \beta_{11} - 4467 \beta_{10} + 3905 \beta_{9} + 8606 \beta_{8} - 13073 \beta_{7} + 11088 \beta_{6} - 7810 \beta_{5} + 114192 \beta_{4} + 7409 \beta_{1}\)\()/3\)
\(\nu^{11}\)\(=\)\((\)\(-3799 \beta_{11} - 20608 \beta_{9} - 25200 \beta_{8} - 96240 \beta_{7} + 40080 \beta_{6} + 10304 \beta_{5} - 144912 \beta_{4} - 45840 \beta_{3} - 40080 \beta_{2} - 38242 \beta_{1} - 144912\)\()/3\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/147\mathbb{Z}\right)^\times\).

\(n\) \(50\) \(52\)
\(\chi(n)\) \(-1\) \(1 + \beta_{4}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
68.1
2.70662 + 1.29391i
0.00299931 3.00000i
−2.23014 + 2.00661i
2.85284 0.928053i
−2.59957 1.49740i
−0.232749 + 2.99096i
2.70662 1.29391i
0.00299931 + 3.00000i
−2.23014 2.00661i
2.85284 + 0.928053i
−2.59957 + 1.49740i
−0.232749 2.99096i
−3.93653 2.27276i 2.24112 4.68800i 6.33084 + 10.9653i 5.80193 10.0492i −19.4769 + 13.3609i 0 21.1897i −16.9548 21.0128i −45.6790 + 26.3728i
68.2 −2.24076 1.29370i −5.19615 0.00519496i −0.652660 1.13044i −8.05907 + 13.9587i 11.6366 + 6.73392i 0 24.0767i 26.9999 + 0.0539876i 36.1169 20.8521i
68.3 −1.65310 0.954416i 3.47555 + 3.86271i −2.17818 3.77272i −0.623706 + 1.08029i −2.05878 9.70256i 0 23.5862i −2.84113 + 26.8501i 2.06209 1.19055i
68.4 1.65310 + 0.954416i −1.60743 4.94127i −2.17818 3.77272i 0.623706 1.08029i 2.05878 9.70256i 0 23.5862i −21.8323 + 15.8855i 2.06209 1.19055i
68.5 2.24076 + 1.29370i −2.59358 + 4.50260i −0.652660 1.13044i 8.05907 13.9587i −11.6366 + 6.73392i 0 24.0767i −13.5467 23.3556i 36.1169 20.8521i
68.6 3.93653 + 2.27276i 5.18049 + 0.403134i 6.33084 + 10.9653i −5.80193 + 10.0492i 19.4769 + 13.3609i 0 21.1897i 26.6750 + 4.17686i −45.6790 + 26.3728i
80.1 −3.93653 + 2.27276i 2.24112 + 4.68800i 6.33084 10.9653i 5.80193 + 10.0492i −19.4769 13.3609i 0 21.1897i −16.9548 + 21.0128i −45.6790 26.3728i
80.2 −2.24076 + 1.29370i −5.19615 + 0.00519496i −0.652660 + 1.13044i −8.05907 13.9587i 11.6366 6.73392i 0 24.0767i 26.9999 0.0539876i 36.1169 + 20.8521i
80.3 −1.65310 + 0.954416i 3.47555 3.86271i −2.17818 + 3.77272i −0.623706 1.08029i −2.05878 + 9.70256i 0 23.5862i −2.84113 26.8501i 2.06209 + 1.19055i
80.4 1.65310 0.954416i −1.60743 + 4.94127i −2.17818 + 3.77272i 0.623706 + 1.08029i 2.05878 + 9.70256i 0 23.5862i −21.8323 15.8855i 2.06209 + 1.19055i
80.5 2.24076 1.29370i −2.59358 4.50260i −0.652660 + 1.13044i 8.05907 + 13.9587i −11.6366 6.73392i 0 24.0767i −13.5467 + 23.3556i 36.1169 + 20.8521i
80.6 3.93653 2.27276i 5.18049 0.403134i 6.33084 10.9653i −5.80193 10.0492i 19.4769 13.3609i 0 21.1897i 26.6750 4.17686i −45.6790 26.3728i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 80.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.d odd 6 1 inner
21.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 147.4.g.d 12
3.b odd 2 1 inner 147.4.g.d 12
7.b odd 2 1 21.4.g.a 12
7.c even 3 1 21.4.g.a 12
7.c even 3 1 147.4.c.a 12
7.d odd 6 1 147.4.c.a 12
7.d odd 6 1 inner 147.4.g.d 12
21.c even 2 1 21.4.g.a 12
21.g even 6 1 147.4.c.a 12
21.g even 6 1 inner 147.4.g.d 12
21.h odd 6 1 21.4.g.a 12
21.h odd 6 1 147.4.c.a 12
28.d even 2 1 336.4.bc.d 12
28.g odd 6 1 336.4.bc.d 12
84.h odd 2 1 336.4.bc.d 12
84.n even 6 1 336.4.bc.d 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.4.g.a 12 7.b odd 2 1
21.4.g.a 12 7.c even 3 1
21.4.g.a 12 21.c even 2 1
21.4.g.a 12 21.h odd 6 1
147.4.c.a 12 7.c even 3 1
147.4.c.a 12 7.d odd 6 1
147.4.c.a 12 21.g even 6 1
147.4.c.a 12 21.h odd 6 1
147.4.g.d 12 1.a even 1 1 trivial
147.4.g.d 12 3.b odd 2 1 inner
147.4.g.d 12 7.d odd 6 1 inner
147.4.g.d 12 21.g even 6 1 inner
336.4.bc.d 12 28.d even 2 1
336.4.bc.d 12 28.g odd 6 1
336.4.bc.d 12 84.h odd 2 1
336.4.bc.d 12 84.n even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(147, [\chi])\):

\( T_{2}^{12} - 31 T_{2}^{10} + 723 T_{2}^{8} - 6370 T_{2}^{6} + 41020 T_{2}^{4} - 119952 T_{2}^{2} + 254016 \)
\( T_{19}^{6} + 150 T_{19}^{5} + 9753 T_{19}^{4} + 337950 T_{19}^{3} + 6428709 T_{19}^{2} + 60952662 T_{19} + 243972972 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 254016 - 119952 T^{2} + 41020 T^{4} - 6370 T^{6} + 723 T^{8} - 31 T^{10} + T^{12} \)
$3$ \( 387420489 - 43046721 T + 3188646 T^{2} - 177147 T^{3} - 144342 T^{4} + 69255 T^{5} - 36018 T^{6} + 2565 T^{7} - 198 T^{8} - 9 T^{9} + 6 T^{10} - 3 T^{11} + T^{12} \)
$5$ \( 2962842624 + 1937507040 T^{2} + 1245448953 T^{4} + 13986756 T^{6} + 121221 T^{8} + 396 T^{10} + T^{12} \)
$7$ \( T^{12} \)
$11$ \( 453620224546062336 - 2045138759862912 T^{2} + 7035594641593 T^{4} - 8503453924 T^{6} + 7487013 T^{8} - 3244 T^{10} + T^{12} \)
$13$ \( ( 82121472 + 1731204 T^{2} + 4335 T^{4} + T^{6} )^{2} \)
$17$ \( 17696515773733945344 + 61355067231370752 T^{2} + 161143714802448 T^{4} + 170413288668 T^{6} + 135747117 T^{8} + 12261 T^{10} + T^{12} \)
$19$ \( ( 243972972 + 60952662 T + 6428709 T^{2} + 337950 T^{3} + 9753 T^{4} + 150 T^{5} + T^{6} )^{2} \)
$23$ \( \)\(89\!\cdots\!76\)\( - 6200136676834232832 T^{2} + 2919015626694160 T^{4} - 746172164500 T^{6} + 139414053 T^{8} - 14311 T^{10} + T^{12} \)
$29$ \( ( 14683734245376 + 3697274560 T^{2} + 120001 T^{4} + T^{6} )^{2} \)
$31$ \( ( 33414175107 + 4755813831 T + 176555736 T^{2} - 6984765 T^{3} + 87096 T^{4} - 465 T^{5} + T^{6} )^{2} \)
$37$ \( ( 3418867564324 + 49455684446 T + 1421726885 T^{2} - 13915390 T^{3} + 119177 T^{4} - 382 T^{5} + T^{6} )^{2} \)
$41$ \( ( -4591113633792 + 4941510336 T^{2} - 172788 T^{4} + T^{6} )^{2} \)
$43$ \( ( -6662944 - 23284 T + 253 T^{2} + T^{3} )^{4} \)
$47$ \( \)\(11\!\cdots\!04\)\( + \)\(85\!\cdots\!56\)\( T^{2} + 47280242457173456640 T^{4} + 1306125205474320 T^{6} + 26259713937 T^{8} + 185553 T^{10} + T^{12} \)
$53$ \( \)\(13\!\cdots\!76\)\( - \)\(30\!\cdots\!92\)\( T^{2} + \)\(50\!\cdots\!89\)\( T^{4} - 37040421831506548 T^{6} + 198439484133 T^{8} - 531100 T^{10} + T^{12} \)
$59$ \( \)\(38\!\cdots\!24\)\( + \)\(44\!\cdots\!36\)\( T^{2} + \)\(47\!\cdots\!89\)\( T^{4} + 39639172481816796 T^{6} + 253702703277 T^{8} + 570420 T^{10} + T^{12} \)
$61$ \( ( 6477700166618112 + 28202272530048 T - 13856716944 T^{2} - 238521132 T^{3} + 261039 T^{4} + 1179 T^{5} + T^{6} )^{2} \)
$67$ \( ( 9665143560577444 + 52759337639610 T + 249067250073 T^{2} + 409138304 T^{3} + 693471 T^{4} - 396 T^{5} + T^{6} )^{2} \)
$71$ \( ( 6720226523136 + 3717765184 T^{2} + 225148 T^{4} + T^{6} )^{2} \)
$73$ \( ( 4047431204396592 + 4635670445244 T - 51563164023 T^{2} - 61084188 T^{3} + 744837 T^{4} - 1452 T^{5} + T^{6} )^{2} \)
$79$ \( ( 363201760969609 - 4951859118549 T + 83464610850 T^{2} + 179364515 T^{3} + 960402 T^{4} - 837 T^{5} + T^{6} )^{2} \)
$83$ \( ( -388952511994368 + 75760581456 T^{2} - 567987 T^{4} + T^{6} )^{2} \)
$89$ \( \)\(88\!\cdots\!24\)\( + \)\(70\!\cdots\!32\)\( T^{2} + \)\(53\!\cdots\!72\)\( T^{4} + 1923321552966019836 T^{6} + 5981516897085 T^{8} + 2594253 T^{10} + T^{12} \)
$97$ \( ( 9887068459035648 + 312291915984 T^{2} + 2159691 T^{4} + T^{6} )^{2} \)
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