Properties

Label 322.2.g.b
Level $322$
Weight $2$
Character orbit 322.g
Analytic conductor $2.571$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 322 = 2 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 322.g (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.57118294509\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Defining polynomial: \(x^{16} + 12 x^{14} + 73 x^{12} + 312 x^{10} + 1045 x^{8} + 2808 x^{6} + 5913 x^{4} + 8748 x^{2} + 6561\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{6} q^{2} + \beta_{11} q^{3} + ( -1 - \beta_{6} ) q^{4} + ( \beta_{1} - \beta_{4} + \beta_{7} ) q^{5} + ( -\beta_{8} + \beta_{11} ) q^{6} + ( \beta_{1} + \beta_{2} - \beta_{4} - \beta_{5} - \beta_{7} + \beta_{9} + \beta_{10} - \beta_{15} ) q^{7} - q^{8} + ( \beta_{3} - 2 \beta_{6} + 2 \beta_{13} ) q^{9} +O(q^{10})\) \( q -\beta_{6} q^{2} + \beta_{11} q^{3} + ( -1 - \beta_{6} ) q^{4} + ( \beta_{1} - \beta_{4} + \beta_{7} ) q^{5} + ( -\beta_{8} + \beta_{11} ) q^{6} + ( \beta_{1} + \beta_{2} - \beta_{4} - \beta_{5} - \beta_{7} + \beta_{9} + \beta_{10} - \beta_{15} ) q^{7} - q^{8} + ( \beta_{3} - 2 \beta_{6} + 2 \beta_{13} ) q^{9} + ( \beta_{1} - \beta_{2} - \beta_{4} + \beta_{7} - \beta_{9} + \beta_{15} ) q^{10} + ( \beta_{2} + 2 \beta_{4} - \beta_{7} + 2 \beta_{9} + \beta_{10} ) q^{11} -\beta_{8} q^{12} + ( -\beta_{3} + \beta_{12} - \beta_{13} - \beta_{14} ) q^{13} + ( \beta_{1} - \beta_{4} - \beta_{7} + \beta_{10} - \beta_{15} ) q^{14} + ( \beta_{1} + \beta_{2} - \beta_{5} - \beta_{7} + \beta_{9} - \beta_{15} ) q^{15} + \beta_{6} q^{16} + ( \beta_{2} - 2 \beta_{7} + \beta_{9} - 2 \beta_{15} ) q^{17} + ( -2 + 2 \beta_{3} - 2 \beta_{6} + \beta_{13} ) q^{18} + ( \beta_{1} + 2 \beta_{2} + \beta_{4} - \beta_{7} + 2 \beta_{9} ) q^{19} + ( -\beta_{2} - \beta_{9} + \beta_{15} ) q^{20} + ( -2 \beta_{1} - \beta_{2} + \beta_{5} + \beta_{7} - 2 \beta_{9} + 3 \beta_{15} ) q^{21} + ( \beta_{1} + 2 \beta_{2} + 2 \beta_{4} + \beta_{5} - \beta_{7} ) q^{22} + ( -\beta_{1} - \beta_{2} + \beta_{3} + \beta_{5} - \beta_{6} - \beta_{8} - \beta_{10} + 2 \beta_{14} ) q^{23} -\beta_{11} q^{24} + ( 1 - 2 \beta_{3} + \beta_{6} + 2 \beta_{8} - \beta_{11} - 2 \beta_{12} ) q^{25} + ( -2 \beta_{8} + \beta_{11} + 2 \beta_{12} - \beta_{13} + \beta_{14} ) q^{26} + ( 1 - 2 \beta_{3} + 2 \beta_{6} - \beta_{8} + \beta_{11} + \beta_{12} - 2 \beta_{13} - \beta_{14} ) q^{27} + ( -\beta_{2} + \beta_{5} - \beta_{9} ) q^{28} + ( -1 + \beta_{3} - \beta_{8} + \beta_{11} + \beta_{12} - \beta_{13} + \beta_{14} ) q^{29} + ( 2 \beta_{1} + \beta_{2} - \beta_{4} - \beta_{5} - \beta_{7} + \beta_{10} - 2 \beta_{15} ) q^{30} + ( -\beta_{11} - \beta_{13} ) q^{31} + ( 1 + \beta_{6} ) q^{32} + ( -4 \beta_{2} - 3 \beta_{4} - \beta_{5} + 4 \beta_{7} - 3 \beta_{9} - \beta_{10} ) q^{33} + ( \beta_{1} - \beta_{4} - 2 \beta_{15} ) q^{34} + ( -3 \beta_{3} + 2 \beta_{8} - 2 \beta_{11} - 2 \beta_{12} - \beta_{14} ) q^{35} + ( -2 + \beta_{3} - \beta_{13} ) q^{36} + ( -\beta_{1} + 2 \beta_{2} + \beta_{4} - \beta_{5} - \beta_{7} + 3 \beta_{9} + \beta_{10} - 2 \beta_{15} ) q^{37} + ( 3 \beta_{1} + \beta_{2} - \beta_{4} - \beta_{7} - \beta_{9} - \beta_{15} ) q^{38} + ( -1 + 2 \beta_{3} - \beta_{6} + 2 \beta_{8} - \beta_{11} - 2 \beta_{12} + 2 \beta_{13} ) q^{39} + ( -\beta_{1} + \beta_{4} - \beta_{7} ) q^{40} + ( 2 + \beta_{3} + 4 \beta_{6} + \beta_{12} + \beta_{13} - \beta_{14} ) q^{41} + ( -4 \beta_{1} - \beta_{2} + \beta_{4} + \beta_{5} - \beta_{7} + \beta_{9} - \beta_{10} + 2 \beta_{15} ) q^{42} + ( 2 \beta_{1} - \beta_{2} - \beta_{4} - \beta_{5} + 3 \beta_{7} - \beta_{9} + \beta_{15} ) q^{43} + ( \beta_{1} + \beta_{2} + \beta_{5} - 2 \beta_{9} - \beta_{10} ) q^{44} + ( -4 \beta_{1} + \beta_{2} + 4 \beta_{4} + 2 \beta_{7} + \beta_{9} + 2 \beta_{15} ) q^{45} + ( -1 - \beta_{2} - \beta_{6} + \beta_{7} + 2 \beta_{8} - \beta_{10} - \beta_{11} - 2 \beta_{12} + \beta_{13} ) q^{46} + ( -2 + 4 \beta_{3} - \beta_{6} - 2 \beta_{8} + \beta_{11} + 2 \beta_{12} - \beta_{13} + \beta_{14} ) q^{47} + ( \beta_{8} - \beta_{11} ) q^{48} + ( -\beta_{3} + 3 \beta_{8} + \beta_{11} - 3 \beta_{12} + 2 \beta_{13} - \beta_{14} ) q^{49} + ( 1 - 2 \beta_{3} + 3 \beta_{8} - \beta_{11} - 2 \beta_{12} + 2 \beta_{13} - 2 \beta_{14} ) q^{50} + ( -6 \beta_{1} - 3 \beta_{2} + 3 \beta_{4} + 2 \beta_{5} + 3 \beta_{7} + \beta_{9} - 2 \beta_{10} + 6 \beta_{15} ) q^{51} + ( \beta_{3} - 2 \beta_{8} + \beta_{11} + \beta_{12} + 2 \beta_{14} ) q^{52} + ( -2 \beta_{1} - 3 \beta_{2} - \beta_{4} - \beta_{7} - 5 \beta_{9} - \beta_{10} + 2 \beta_{15} ) q^{53} + ( 2 - \beta_{3} + \beta_{6} - 3 \beta_{8} + \beta_{11} + 2 \beta_{12} - \beta_{13} + \beta_{14} ) q^{54} + ( 1 - \beta_{3} + 2 \beta_{6} - \beta_{8} + \beta_{11} + 2 \beta_{12} - \beta_{13} - 2 \beta_{14} ) q^{55} + ( -\beta_{1} - \beta_{2} + \beta_{4} + \beta_{5} + \beta_{7} - \beta_{9} - \beta_{10} + \beta_{15} ) q^{56} + ( -\beta_{1} - 5 \beta_{2} - 2 \beta_{4} - \beta_{5} + 7 \beta_{7} - 3 \beta_{9} + 3 \beta_{15} ) q^{57} + ( \beta_{6} - \beta_{8} + \beta_{11} - \beta_{13} + \beta_{14} ) q^{58} + ( 2 - 2 \beta_{6} - \beta_{11} + 3 \beta_{13} ) q^{59} + ( \beta_{1} - \beta_{4} - \beta_{9} + \beta_{10} - \beta_{15} ) q^{60} + ( -3 \beta_{1} - \beta_{2} + \beta_{4} - 2 \beta_{5} - 2 \beta_{7} + \beta_{9} - 2 \beta_{10} ) q^{61} + ( -\beta_{3} + \beta_{8} - \beta_{11} - \beta_{13} ) q^{62} + ( 3 \beta_{2} - \beta_{4} + \beta_{5} - 4 \beta_{7} - \beta_{9} - 5 \beta_{15} ) q^{63} + q^{64} + ( 2 \beta_{2} + \beta_{4} + 2 \beta_{5} - 2 \beta_{9} - 2 \beta_{10} ) q^{65} + ( -2 \beta_{1} - 2 \beta_{2} - 2 \beta_{5} + 3 \beta_{7} + \beta_{9} + \beta_{10} + 2 \beta_{15} ) q^{66} + ( \beta_{1} + 5 \beta_{2} - 2 \beta_{4} - 2 \beta_{7} + \beta_{9} + 3 \beta_{10} - \beta_{15} ) q^{67} + ( \beta_{1} - \beta_{2} - \beta_{4} + 2 \beta_{7} - \beta_{9} ) q^{68} + ( 2 - 2 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} + \beta_{4} + \beta_{5} + 4 \beta_{6} + \beta_{7} - \beta_{8} + 2 \beta_{9} - 2 \beta_{10} + \beta_{11} + \beta_{12} - 3 \beta_{13} - \beta_{14} ) q^{69} + ( -2 \beta_{3} + 3 \beta_{8} - 2 \beta_{11} - \beta_{12} + 2 \beta_{13} - 2 \beta_{14} ) q^{70} + ( -4 + \beta_{3} - 2 \beta_{8} + \beta_{12} - \beta_{13} + \beta_{14} ) q^{71} + ( -\beta_{3} + 2 \beta_{6} - 2 \beta_{13} ) q^{72} + ( 2 + 2 \beta_{3} - 2 \beta_{6} - 4 \beta_{8} + 3 \beta_{11} + 2 \beta_{12} + \beta_{13} + 4 \beta_{14} ) q^{73} + ( \beta_{1} + 2 \beta_{2} + 2 \beta_{9} + \beta_{10} - \beta_{15} ) q^{74} + ( 2 - \beta_{3} + \beta_{6} + 3 \beta_{8} ) q^{75} + ( 2 \beta_{1} - \beta_{2} - 2 \beta_{4} - 3 \beta_{9} - \beta_{15} ) q^{76} + ( 2 \beta_{3} + 5 \beta_{8} - 4 \beta_{11} - 3 \beta_{12} + 2 \beta_{13} - 3 \beta_{14} ) q^{77} + ( -1 + 3 \beta_{8} - \beta_{11} - 2 \beta_{12} - 2 \beta_{14} ) q^{78} + ( \beta_{1} - 3 \beta_{2} - 3 \beta_{4} - 2 \beta_{5} - 2 \beta_{7} + 3 \beta_{9} + 2 \beta_{10} - 4 \beta_{15} ) q^{79} + ( -\beta_{1} + \beta_{2} + \beta_{4} - \beta_{7} + \beta_{9} - \beta_{15} ) q^{80} + ( -1 - \beta_{3} - \beta_{6} + 6 \beta_{8} - 3 \beta_{11} - 3 \beta_{12} + \beta_{13} ) q^{81} + ( 4 + 2 \beta_{3} + 2 \beta_{6} - 2 \beta_{8} + \beta_{11} + 2 \beta_{12} - \beta_{13} + \beta_{14} ) q^{82} + ( 3 \beta_{1} - 2 \beta_{2} - 3 \beta_{4} - 5 \beta_{9} - \beta_{15} ) q^{83} + ( -2 \beta_{1} + \beta_{4} - 2 \beta_{7} + 3 \beta_{9} - \beta_{10} - \beta_{15} ) q^{84} + ( 5 - 2 \beta_{8} + \beta_{12} + \beta_{14} ) q^{85} + ( \beta_{1} - \beta_{5} + \beta_{7} - \beta_{9} + \beta_{10} + 2 \beta_{15} ) q^{86} + ( 2 - 2 \beta_{6} - 2 \beta_{11} + \beta_{13} ) q^{87} + ( -\beta_{2} - 2 \beta_{4} + \beta_{7} - 2 \beta_{9} - \beta_{10} ) q^{88} + ( 3 \beta_{1} + 3 \beta_{4} + 3 \beta_{7} ) q^{89} + ( -3 \beta_{1} + 4 \beta_{2} + 3 \beta_{4} + 4 \beta_{9} + 2 \beta_{15} ) q^{90} + ( 5 \beta_{2} + 5 \beta_{4} + 2 \beta_{5} - \beta_{7} + 4 \beta_{9} + 2 \beta_{10} + \beta_{15} ) q^{91} + ( -1 + \beta_{1} - \beta_{3} - \beta_{5} + \beta_{7} + 3 \beta_{8} - \beta_{11} - 2 \beta_{12} + \beta_{13} - 2 \beta_{14} ) q^{92} + ( 4 \beta_{6} - \beta_{11} - \beta_{13} + \beta_{14} ) q^{93} + ( -1 + \beta_{3} + \beta_{6} - 2 \beta_{8} + \beta_{11} + \beta_{12} - 4 \beta_{13} + 2 \beta_{14} ) q^{94} + ( -4 \beta_{3} - 2 \beta_{8} + \beta_{11} - 2 \beta_{13} ) q^{95} + \beta_{8} q^{96} + ( -8 \beta_{1} - \beta_{2} + 4 \beta_{4} + 4 \beta_{5} + 4 \beta_{7} + 2 \beta_{9} - 8 \beta_{10} + 6 \beta_{15} ) q^{97} + ( -\beta_{3} + \beta_{8} + \beta_{11} - 2 \beta_{12} + 2 \beta_{13} - 3 \beta_{14} ) q^{98} + ( 4 \beta_{1} + 4 \beta_{2} - 2 \beta_{4} - 4 \beta_{5} - 6 \beta_{7} + 5 \beta_{9} - 5 \beta_{15} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 8q^{2} - 6q^{3} - 8q^{4} - 16q^{8} + 10q^{9} + O(q^{10}) \) \( 16q + 8q^{2} - 6q^{3} - 8q^{4} - 16q^{8} + 10q^{9} + 6q^{12} - 8q^{16} - 10q^{18} + 8q^{23} + 6q^{24} + 2q^{25} - 6q^{26} - 16q^{29} + 12q^{31} + 8q^{32} - 20q^{36} - 2q^{39} - 8q^{46} - 6q^{47} - 18q^{49} + 4q^{50} - 6q^{52} + 18q^{54} - 8q^{58} + 36q^{59} + 16q^{64} - 12q^{70} - 52q^{71} - 10q^{72} + 24q^{73} + 30q^{77} - 4q^{78} - 20q^{81} + 54q^{82} + 80q^{85} + 54q^{87} - 16q^{92} - 26q^{93} - 6q^{94} - 6q^{95} - 6q^{96} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} + 12 x^{14} + 73 x^{12} + 312 x^{10} + 1045 x^{8} + 2808 x^{6} + 5913 x^{4} + 8748 x^{2} + 6561\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{15} + 12 \nu^{13} + 73 \nu^{11} + 312 \nu^{9} + 1045 \nu^{7} + 2808 \nu^{5} + 5913 \nu^{3} + 6561 \nu \)\()/2187\)
\(\beta_{3}\)\(=\)\((\)\( -100 \nu^{14} - 1713 \nu^{12} - 9244 \nu^{10} - 39975 \nu^{8} - 129853 \nu^{6} - 323595 \nu^{4} - 601668 \nu^{2} - 661203 \)\()/103518\)
\(\beta_{4}\)\(=\)\((\)\( -311 \nu^{15} - 3174 \nu^{13} - 16250 \nu^{11} - 59943 \nu^{9} - 179573 \nu^{7} - 452016 \nu^{5} - 856251 \nu^{3} - 769095 \nu \)\()/310554\)
\(\beta_{5}\)\(=\)\((\)\( -452 \nu^{15} - 4650 \nu^{13} - 23303 \nu^{11} - 95700 \nu^{9} - 286580 \nu^{7} - 673740 \nu^{5} - 1285470 \nu^{3} - 1237113 \nu \)\()/310554\)
\(\beta_{6}\)\(=\)\((\)\( -386 \nu^{14} - 3660 \nu^{12} - 19349 \nu^{10} - 74748 \nu^{8} - 227600 \nu^{6} - 541512 \nu^{4} - 985446 \nu^{2} - 1070901 \)\()/103518\)
\(\beta_{7}\)\(=\)\((\)\( -199 \nu^{15} - 2346 \nu^{13} - 12619 \nu^{11} - 49464 \nu^{9} - 149950 \nu^{7} - 367725 \nu^{5} - 668304 \nu^{3} - 737991 \nu \)\()/103518\)
\(\beta_{8}\)\(=\)\((\)\( 148 \nu^{14} + 1155 \nu^{12} + 5323 \nu^{10} + 18906 \nu^{8} + 55462 \nu^{6} + 118908 \nu^{4} + 194751 \nu^{2} + 142155 \)\()/34506\)
\(\beta_{9}\)\(=\)\((\)\( -386 \nu^{15} - 3660 \nu^{13} - 19349 \nu^{11} - 74748 \nu^{9} - 227600 \nu^{7} - 541512 \nu^{5} - 985446 \nu^{3} - 967383 \nu \)\()/103518\)
\(\beta_{10}\)\(=\)\((\)\( -149 \nu^{15} - 1525 \nu^{13} - 7784 \nu^{11} - 30151 \nu^{9} - 92585 \nu^{7} - 221938 \nu^{5} - 403254 \nu^{3} - 433269 \nu \)\()/34506\)
\(\beta_{11}\)\(=\)\((\)\( -361 \nu^{14} - 3498 \nu^{12} - 18316 \nu^{10} - 69813 \nu^{8} - 211591 \nu^{6} - 511680 \nu^{4} - 907875 \nu^{2} - 935793 \)\()/34506\)
\(\beta_{12}\)\(=\)\((\)\( 1241 \nu^{14} + 12012 \nu^{12} + 60812 \nu^{10} + 231384 \nu^{8} + 703412 \nu^{6} + 1673847 \nu^{4} + 3027618 \nu^{2} + 3031182 \)\()/103518\)
\(\beta_{13}\)\(=\)\((\)\( 1249 \nu^{14} + 11280 \nu^{12} + 56644 \nu^{10} + 216690 \nu^{8} + 651076 \nu^{6} + 1557621 \nu^{4} + 2756916 \nu^{2} + 2703132 \)\()/103518\)
\(\beta_{14}\)\(=\)\((\)\( 1493 \nu^{14} + 13236 \nu^{12} + 65627 \nu^{10} + 247134 \nu^{8} + 730286 \nu^{6} + 1700397 \nu^{4} + 2902716 \nu^{2} + 2635335 \)\()/103518\)
\(\beta_{15}\)\(=\)\((\)\( -2365 \nu^{15} - 20352 \nu^{13} - 102067 \nu^{11} - 382929 \nu^{9} - 1142809 \nu^{7} - 2673774 \nu^{5} - 4606875 \nu^{3} - 4299642 \nu \)\()/310554\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{12} + \beta_{11} - \beta_{8} - \beta_{6} + \beta_{3} - 2\)
\(\nu^{3}\)\(=\)\(\beta_{15} - 2 \beta_{9} + 2 \beta_{7} - \beta_{5} - 3 \beta_{4} - \beta_{2} + \beta_{1}\)
\(\nu^{4}\)\(=\)\(-\beta_{14} + 2 \beta_{13} - 2 \beta_{12} - 3 \beta_{11} + \beta_{8} + 6 \beta_{6} - \beta_{3} + 2\)
\(\nu^{5}\)\(=\)\(-5 \beta_{15} + 2 \beta_{10} + 11 \beta_{9} - 8 \beta_{7} + 2 \beta_{5} + 5 \beta_{4} + 9 \beta_{2} - 5 \beta_{1}\)
\(\nu^{6}\)\(=\)\(-6 \beta_{14} - \beta_{13} - 5 \beta_{12} - 4 \beta_{11} + 18 \beta_{8} - 8 \beta_{6} - 10 \beta_{3} - 4\)
\(\nu^{7}\)\(=\)\(10 \beta_{15} - 18 \beta_{10} - 19 \beta_{9} + 20 \beta_{7} + 11 \beta_{5} + 12 \beta_{4} - 13 \beta_{2} - 10 \beta_{1}\)
\(\nu^{8}\)\(=\)\(31 \beta_{14} - 3 \beta_{13} + 8 \beta_{12} + 34 \beta_{11} - 56 \beta_{8} - 27 \beta_{6} + 12 \beta_{3} + 5\)
\(\nu^{9}\)\(=\)\(-22 \beta_{15} + 52 \beta_{10} + 34 \beta_{9} - 52 \beta_{7} - 62 \beta_{5} + 16 \beta_{4} + 21 \beta_{2} + 25 \beta_{1}\)
\(\nu^{10}\)\(=\)\(-18 \beta_{14} - 56 \beta_{13} + 10 \beta_{12} - 72 \beta_{11} + 28 \beta_{8} + 60 \beta_{3} - 57\)
\(\nu^{11}\)\(=\)\(76 \beta_{15} + 12 \beta_{10} - 228 \beta_{9} + 56 \beta_{7} + 146 \beta_{5} - 168 \beta_{4} - 148 \beta_{2} + 81 \beta_{1}\)
\(\nu^{12}\)\(=\)\(60 \beta_{14} + 4 \beta_{13} + 219 \beta_{12} + 119 \beta_{11} - 101 \beta_{8} + 519 \beta_{6} - 77 \beta_{3} + 476\)
\(\nu^{13}\)\(=\)\(-9 \beta_{15} - 414 \beta_{10} + 546 \beta_{9} + 114 \beta_{7} - 183 \beta_{5} - 183 \beta_{4} - 111 \beta_{2} - 257 \beta_{1}\)
\(\nu^{14}\)\(=\)\(-729 \beta_{14} + 1134 \beta_{13} - 926 \beta_{12} - 89 \beta_{11} + 935 \beta_{8} - 1108 \beta_{6} + 145 \beta_{3} - 740\)
\(\nu^{15}\)\(=\)\(-899 \beta_{15} + 1062 \beta_{10} + 277 \beta_{9} + 506 \beta_{7} - 316 \beta_{5} + 627 \beta_{4} + 1997 \beta_{2} + 1387 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/322\mathbb{Z}\right)^\times\).

\(n\) \(185\) \(281\)
\(\chi(n)\) \(-\beta_{6}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
45.1
0.105715 + 1.72882i
−0.105715 1.72882i
−0.452119 1.67200i
0.452119 + 1.67200i
1.36749 1.06300i
−1.36749 + 1.06300i
−0.956239 1.44416i
0.956239 + 1.44416i
0.105715 1.72882i
−0.105715 + 1.72882i
−0.452119 + 1.67200i
0.452119 1.67200i
1.36749 + 1.06300i
−1.36749 1.06300i
−0.956239 + 1.44416i
0.956239 1.44416i
0.500000 + 0.866025i −2.64992 1.52993i −0.500000 + 0.866025i −0.920495 1.59434i 3.05986i −0.254505 + 2.63348i −1.00000 3.18138 + 5.51031i 0.920495 1.59434i
45.2 0.500000 + 0.866025i −2.64992 1.52993i −0.500000 + 0.866025i 0.920495 + 1.59434i 3.05986i 0.254505 2.63348i −1.00000 3.18138 + 5.51031i −0.920495 + 1.59434i
45.3 0.500000 + 0.866025i −1.07743 0.622057i −0.500000 + 0.866025i −0.668984 1.15871i 1.24411i −2.09536 + 1.61539i −1.00000 −0.726090 1.25762i 0.668984 1.15871i
45.4 0.500000 + 0.866025i −1.07743 0.622057i −0.500000 + 0.866025i 0.668984 + 1.15871i 1.24411i 2.09536 1.61539i −1.00000 −0.726090 1.25762i −0.668984 + 1.15871i
45.5 0.500000 + 0.866025i 0.0922753 + 0.0532752i −0.500000 + 0.866025i −1.78715 3.09543i 0.106550i −0.672033 2.55898i −1.00000 −1.49432 2.58824i 1.78715 3.09543i
45.6 0.500000 + 0.866025i 0.0922753 + 0.0532752i −0.500000 + 0.866025i 1.78715 + 3.09543i 0.106550i 0.672033 + 2.55898i −1.00000 −1.49432 2.58824i −1.78715 + 3.09543i
45.7 0.500000 + 0.866025i 2.13508 + 1.23269i −0.500000 + 0.866025i −0.511122 0.885289i 2.46537i 2.61593 + 0.396147i −1.00000 1.53904 + 2.66569i 0.511122 0.885289i
45.8 0.500000 + 0.866025i 2.13508 + 1.23269i −0.500000 + 0.866025i 0.511122 + 0.885289i 2.46537i −2.61593 0.396147i −1.00000 1.53904 + 2.66569i −0.511122 + 0.885289i
229.1 0.500000 0.866025i −2.64992 + 1.52993i −0.500000 0.866025i −0.920495 + 1.59434i 3.05986i −0.254505 2.63348i −1.00000 3.18138 5.51031i 0.920495 + 1.59434i
229.2 0.500000 0.866025i −2.64992 + 1.52993i −0.500000 0.866025i 0.920495 1.59434i 3.05986i 0.254505 + 2.63348i −1.00000 3.18138 5.51031i −0.920495 1.59434i
229.3 0.500000 0.866025i −1.07743 + 0.622057i −0.500000 0.866025i −0.668984 + 1.15871i 1.24411i −2.09536 1.61539i −1.00000 −0.726090 + 1.25762i 0.668984 + 1.15871i
229.4 0.500000 0.866025i −1.07743 + 0.622057i −0.500000 0.866025i 0.668984 1.15871i 1.24411i 2.09536 + 1.61539i −1.00000 −0.726090 + 1.25762i −0.668984 1.15871i
229.5 0.500000 0.866025i 0.0922753 0.0532752i −0.500000 0.866025i −1.78715 + 3.09543i 0.106550i −0.672033 + 2.55898i −1.00000 −1.49432 + 2.58824i 1.78715 + 3.09543i
229.6 0.500000 0.866025i 0.0922753 0.0532752i −0.500000 0.866025i 1.78715 3.09543i 0.106550i 0.672033 2.55898i −1.00000 −1.49432 + 2.58824i −1.78715 3.09543i
229.7 0.500000 0.866025i 2.13508 1.23269i −0.500000 0.866025i −0.511122 + 0.885289i 2.46537i 2.61593 0.396147i −1.00000 1.53904 2.66569i 0.511122 + 0.885289i
229.8 0.500000 0.866025i 2.13508 1.23269i −0.500000 0.866025i 0.511122 0.885289i 2.46537i −2.61593 + 0.396147i −1.00000 1.53904 2.66569i −0.511122 0.885289i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 229.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 322.2.g.b 16
7.c even 3 1 2254.2.c.b 16
7.d odd 6 1 inner 322.2.g.b 16
7.d odd 6 1 2254.2.c.b 16
23.b odd 2 1 inner 322.2.g.b 16
161.f odd 6 1 2254.2.c.b 16
161.g even 6 1 inner 322.2.g.b 16
161.g even 6 1 2254.2.c.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
322.2.g.b 16 1.a even 1 1 trivial
322.2.g.b 16 7.d odd 6 1 inner
322.2.g.b 16 23.b odd 2 1 inner
322.2.g.b 16 161.g even 6 1 inner
2254.2.c.b 16 7.c even 3 1
2254.2.c.b 16 7.d odd 6 1
2254.2.c.b 16 161.f odd 6 1
2254.2.c.b 16 161.g even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{3}^{8} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(322, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T + T^{2} )^{8} \)
$3$ \( ( 1 - 15 T + 68 T^{2} + 105 T^{3} + 33 T^{4} - 21 T^{5} - 4 T^{6} + 3 T^{7} + T^{8} )^{2} \)
$5$ \( 6561 + 12393 T^{2} + 16038 T^{4} + 10845 T^{6} + 5293 T^{8} + 1423 T^{10} + 270 T^{12} + 19 T^{14} + T^{16} \)
$7$ \( 5764801 + 1058841 T^{2} - 67228 T^{4} - 13671 T^{6} + 435 T^{8} - 279 T^{10} - 28 T^{12} + 9 T^{14} + T^{16} \)
$11$ \( 74805201 - 53580555 T^{2} + 28275993 T^{4} - 6128688 T^{6} + 959095 T^{8} - 62362 T^{10} + 2928 T^{12} - 64 T^{14} + T^{16} \)
$13$ \( ( 9 + 1356 T^{2} + 562 T^{4} + 55 T^{6} + T^{8} )^{2} \)
$17$ \( 194481 + 322812 T^{2} + 375300 T^{4} + 223230 T^{6} + 96187 T^{8} + 16372 T^{10} + 2037 T^{12} + 49 T^{14} + T^{16} \)
$19$ \( 639128961 + 290251161 T^{2} + 90377802 T^{4} + 14620713 T^{6} + 1708117 T^{8} + 113075 T^{10} + 5250 T^{12} + 83 T^{14} + T^{16} \)
$23$ \( 78310985281 - 27238603576 T + 11250727564 T^{2} - 1235777856 T^{3} + 396814538 T^{4} - 4477456 T^{5} + 17495088 T^{6} + 353464 T^{7} + 739315 T^{8} + 15368 T^{9} + 33072 T^{10} - 368 T^{11} + 1418 T^{12} - 192 T^{13} + 76 T^{14} - 8 T^{15} + T^{16} \)
$29$ \( ( -15 - 48 T - 11 T^{2} + 4 T^{3} + T^{4} )^{4} \)
$31$ \( ( 225 - 855 T + 1203 T^{2} - 456 T^{3} - 35 T^{4} + 48 T^{5} + 4 T^{6} - 6 T^{7} + T^{8} )^{2} \)
$37$ \( 81 - 3537 T^{2} + 145827 T^{4} - 375378 T^{6} + 893389 T^{8} - 58610 T^{10} + 2886 T^{12} - 62 T^{14} + T^{16} \)
$41$ \( ( 961 + 1820 T^{2} + 930 T^{4} + 131 T^{6} + T^{8} )^{2} \)
$43$ \( ( 729 + 8073 T^{2} + 1756 T^{4} + 80 T^{6} + T^{8} )^{2} \)
$47$ \( ( 15625 + 103125 T + 241250 T^{2} + 94875 T^{3} + 12525 T^{4} - 345 T^{5} - 112 T^{6} + 3 T^{7} + T^{8} )^{2} \)
$53$ \( 625 - 21812925 T^{2} + 761285173964 T^{4} - 25872730851 T^{6} + 604457529 T^{8} - 7595661 T^{10} + 69572 T^{12} - 315 T^{14} + T^{16} \)
$59$ \( ( 73441 + 70731 T + 5363 T^{2} - 16704 T^{3} + 2259 T^{4} + 1152 T^{5} + 44 T^{6} - 18 T^{7} + T^{8} )^{2} \)
$61$ \( 10756569837841 + 2754808213392 T^{2} + 595976680904 T^{4} + 25791389310 T^{6} + 822496839 T^{8} + 9843096 T^{10} + 85625 T^{12} + 345 T^{14} + T^{16} \)
$67$ \( 2311278643521 - 1048935557862 T^{2} + 412363216710 T^{4} - 27668146698 T^{6} + 1473483717 T^{8} - 15583914 T^{10} + 122139 T^{12} - 405 T^{14} + T^{16} \)
$71$ \( ( -15 + 36 T + 46 T^{2} + 13 T^{3} + T^{4} )^{4} \)
$73$ \( ( 10595025 - 7606935 T + 2237163 T^{2} - 299136 T^{3} + 10291 T^{4} + 1536 T^{5} - 80 T^{6} - 12 T^{7} + T^{8} )^{2} \)
$79$ \( 415557398300625 - 30278419315200 T^{2} + 1493973516744 T^{4} - 38722004682 T^{6} + 720383095 T^{8} - 8313704 T^{10} + 69393 T^{12} - 323 T^{14} + T^{16} \)
$83$ \( ( 1896129 - 266976 T^{2} + 11713 T^{4} - 196 T^{6} + T^{8} )^{2} \)
$89$ \( 121639745379681 + 12422064023505 T^{2} + 887101511958 T^{4} + 31213124253 T^{6} + 789898473 T^{8} + 9887427 T^{10} + 88614 T^{12} + 351 T^{14} + T^{16} \)
$97$ \( ( 2486119321 - 44870064 T^{2} + 302197 T^{4} - 900 T^{6} + T^{8} )^{2} \)
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