Properties

Label 912.2.bn.o
Level $912$
Weight $2$
Character orbit 912.bn
Analytic conductor $7.282$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 912.bn (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.28235666434\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - x^{15} - 6 x^{14} + 5 x^{13} + 21 x^{12} - 4 x^{11} - 94 x^{10} - 6 x^{9} + 364 x^{8} - 18 x^{7} - 846 x^{6} - 108 x^{5} + 1701 x^{4} + 1215 x^{3} - 4374 x^{2} - 2187 x + 6561\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 456)
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.

\(f(q)\) \(=\) \( q + \beta_{7} q^{3} + ( \beta_{11} - \beta_{14} ) q^{5} + ( -\beta_{5} + \beta_{13} ) q^{7} + ( 1 + \beta_{8} ) q^{9} +O(q^{10})\) \( q + \beta_{7} q^{3} + ( \beta_{11} - \beta_{14} ) q^{5} + ( -\beta_{5} + \beta_{13} ) q^{7} + ( 1 + \beta_{8} ) q^{9} + ( 1 + \beta_{2} + 2 \beta_{3} + \beta_{5} + \beta_{7} - \beta_{8} - \beta_{10} - \beta_{11} + \beta_{12} - \beta_{14} ) q^{11} + ( -\beta_{2} - \beta_{4} - \beta_{5} + \beta_{9} - \beta_{10} - \beta_{12} + \beta_{13} ) q^{13} + ( -1 - \beta_{5} - \beta_{6} - \beta_{7} + \beta_{10} - \beta_{12} + \beta_{14} + \beta_{15} ) q^{15} + ( -\beta_{1} - \beta_{5} - \beta_{6} - \beta_{7} + \beta_{10} + \beta_{13} + \beta_{14} + \beta_{15} ) q^{17} + ( -1 + 2 \beta_{1} + 2 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} - 2 \beta_{9} + \beta_{12} - \beta_{13} - \beta_{14} ) q^{19} + ( -2 \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{6} - \beta_{8} + \beta_{9} - \beta_{11} - \beta_{12} + \beta_{13} + \beta_{14} + \beta_{15} ) q^{21} + ( -2 \beta_{4} - 2 \beta_{7} + 2 \beta_{9} + \beta_{14} ) q^{23} + ( 2 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} + 2 \beta_{11} - \beta_{13} ) q^{25} + ( \beta_{4} + \beta_{5} + \beta_{7} - \beta_{9} + \beta_{10} + \beta_{12} - 2 \beta_{13} - \beta_{14} ) q^{27} + ( -2 \beta_{1} + \beta_{3} - 2 \beta_{4} - \beta_{5} - 2 \beta_{6} - \beta_{8} + \beta_{10} - \beta_{12} + 2 \beta_{13} + \beta_{14} + \beta_{15} ) q^{29} + ( -\beta_{1} - 2 \beta_{2} - \beta_{5} + \beta_{6} + \beta_{7} + 2 \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} - \beta_{12} - \beta_{13} + \beta_{14} ) q^{31} + ( \beta_{1} - 2 \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{6} - 2 \beta_{7} + \beta_{8} + 2 \beta_{9} + 2 \beta_{11} - \beta_{12} - \beta_{13} ) q^{33} + ( 1 + 3 \beta_{1} + \beta_{2} - \beta_{3} + 3 \beta_{4} + \beta_{6} - \beta_{7} + 2 \beta_{8} - 2 \beta_{9} + \beta_{11} + 2 \beta_{12} - \beta_{13} - \beta_{14} - \beta_{15} ) q^{35} + ( 1 - \beta_{1} + 2 \beta_{3} - \beta_{4} + \beta_{7} - \beta_{9} ) q^{37} + ( -\beta_{1} - 2 \beta_{2} - 2 \beta_{5} - \beta_{6} - \beta_{7} + \beta_{10} + \beta_{11} + \beta_{14} + \beta_{15} ) q^{39} + ( -1 - 2 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{5} + 3 \beta_{7} - 2 \beta_{9} - \beta_{10} - \beta_{14} ) q^{41} + ( -1 + \beta_{1} - \beta_{3} - 2 \beta_{4} - 3 \beta_{5} - 4 \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} - \beta_{12} + 2 \beta_{14} ) q^{43} + ( 2 + 2 \beta_{1} + \beta_{3} + 2 \beta_{4} + \beta_{5} - \beta_{8} - 2 \beta_{9} + \beta_{10} + \beta_{11} + \beta_{12} - \beta_{14} + \beta_{15} ) q^{45} + ( 2 - 2 \beta_{1} - 2 \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{5} - \beta_{8} - \beta_{10} - \beta_{12} + \beta_{14} + \beta_{15} ) q^{47} + ( 1 + \beta_{1} + \beta_{2} - \beta_{5} + \beta_{6} - 2 \beta_{7} + \beta_{8} + \beta_{9} + \beta_{11} + 2 \beta_{12} - 2 \beta_{14} - 2 \beta_{15} ) q^{49} + ( -3 \beta_{1} + 2 \beta_{3} - 2 \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} - 2 \beta_{8} + \beta_{10} + \beta_{13} - \beta_{14} + \beta_{15} ) q^{51} + ( \beta_{1} - \beta_{3} + \beta_{8} - \beta_{9} + 2 \beta_{11} + \beta_{12} - 2 \beta_{13} - \beta_{14} ) q^{53} + ( 3 + 3 \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{6} - 3 \beta_{7} + \beta_{8} + \beta_{9} - \beta_{11} - \beta_{12} + \beta_{13} - \beta_{14} - \beta_{15} ) q^{55} + ( 3 - \beta_{1} + 2 \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} - 2 \beta_{10} - 2 \beta_{11} - \beta_{12} - \beta_{14} ) q^{57} + ( -1 + \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{5} + \beta_{6} - 2 \beta_{7} + 2 \beta_{8} + 2 \beta_{10} + \beta_{11} - 2 \beta_{12} - \beta_{13} + 3 \beta_{14} + \beta_{15} ) q^{59} + ( \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{7} + \beta_{8} + \beta_{10} - 2 \beta_{11} + \beta_{13} + 2 \beta_{14} ) q^{61} + ( -1 - 2 \beta_{1} - \beta_{2} - 3 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{10} + 2 \beta_{11} + \beta_{13} - \beta_{14} ) q^{63} + ( 1 + \beta_{1} + \beta_{2} + 3 \beta_{4} + \beta_{5} - 2 \beta_{7} + \beta_{8} - 3 \beta_{9} + 3 \beta_{10} + \beta_{11} + 3 \beta_{12} - 4 \beta_{13} + \beta_{14} ) q^{65} + ( 2 + 2 \beta_{1} + \beta_{3} - \beta_{8} + 2 \beta_{9} - 2 \beta_{10} + \beta_{12} + \beta_{13} - 2 \beta_{14} - 2 \beta_{15} ) q^{67} + ( 1 + \beta_{2} - 3 \beta_{3} - \beta_{4} - 2 \beta_{5} - 2 \beta_{7} + \beta_{9} + \beta_{10} + \beta_{12} + \beta_{13} + \beta_{14} - \beta_{15} ) q^{69} + ( 4 - \beta_{1} + 4 \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} - \beta_{10} + \beta_{13} - \beta_{14} - \beta_{15} ) q^{71} + ( 1 + \beta_{3} - 2 \beta_{5} - 2 \beta_{7} + 2 \beta_{8} + 2 \beta_{10} - 2 \beta_{12} + 2 \beta_{14} ) q^{73} + ( 2 - 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + \beta_{5} - \beta_{6} + 2 \beta_{7} - 2 \beta_{8} - \beta_{9} - \beta_{11} + \beta_{12} + \beta_{15} ) q^{75} + ( 2 + 4 \beta_{3} - 4 \beta_{5} - \beta_{6} - 3 \beta_{7} + \beta_{9} + 2 \beta_{10} + \beta_{11} - 2 \beta_{12} - \beta_{13} + 2 \beta_{14} ) q^{77} + ( 1 + 3 \beta_{1} + 2 \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{5} + 2 \beta_{7} + \beta_{8} - 3 \beta_{9} - \beta_{10} + \beta_{11} + \beta_{12} - 2 \beta_{14} ) q^{79} + ( -1 + \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} - 2 \beta_{9} - 2 \beta_{13} ) q^{81} + ( -\beta_{1} - \beta_{4} + 2 \beta_{5} + \beta_{6} + 2 \beta_{7} - 2 \beta_{9} - 2 \beta_{11} + \beta_{13} ) q^{83} + ( 3 \beta_{1} + 2 \beta_{2} + 3 \beta_{5} + 2 \beta_{6} + 4 \beta_{7} - 3 \beta_{9} - 3 \beta_{10} + 2 \beta_{12} - 3 \beta_{14} - \beta_{15} ) q^{85} + ( -2 \beta_{1} - 3 \beta_{3} + \beta_{5} - 2 \beta_{6} - \beta_{8} + \beta_{10} + 2 \beta_{11} - \beta_{12} + 2 \beta_{13} - \beta_{14} + \beta_{15} ) q^{87} + ( -\beta_{1} + 3 \beta_{3} + 2 \beta_{4} - 2 \beta_{7} - \beta_{8} + 3 \beta_{9} - 2 \beta_{11} - \beta_{12} + 2 \beta_{13} + \beta_{14} ) q^{89} + ( 6 + 2 \beta_{1} + \beta_{2} + 3 \beta_{3} + \beta_{4} + \beta_{9} + \beta_{12} + \beta_{14} - \beta_{15} ) q^{91} + ( -4 - 2 \beta_{1} + 2 \beta_{2} - \beta_{3} + 3 \beta_{4} + 2 \beta_{5} + 3 \beta_{7} - \beta_{10} - 4 \beta_{11} + 2 \beta_{12} - \beta_{14} ) q^{93} + ( 2 + \beta_{1} + \beta_{3} + 2 \beta_{5} + 2 \beta_{7} + \beta_{8} + 3 \beta_{9} - 2 \beta_{10} - \beta_{11} - \beta_{12} + 2 \beta_{13} - \beta_{14} ) q^{95} + ( -3 - 2 \beta_{2} + 3 \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} - 2 \beta_{7} - \beta_{8} - \beta_{9} + 2 \beta_{10} + \beta_{11} - \beta_{12} - \beta_{13} + \beta_{14} + \beta_{15} ) q^{97} + ( -\beta_{1} + \beta_{2} - 3 \beta_{3} + \beta_{4} - 4 \beta_{5} - \beta_{6} - \beta_{7} + 2 \beta_{10} - 2 \beta_{11} + \beta_{12} + \beta_{13} + 3 \beta_{14} + \beta_{15} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + q^{3} - 3 q^{5} + 13 q^{9} + O(q^{10}) \) \( 16 q + q^{3} - 3 q^{5} + 13 q^{9} - 3 q^{13} - 15 q^{15} + 3 q^{17} - 11 q^{19} - 6 q^{21} - 3 q^{23} + 11 q^{25} + 4 q^{27} - 5 q^{29} + q^{33} + 24 q^{35} + 9 q^{39} - 6 q^{41} - 13 q^{43} + 33 q^{45} + 27 q^{47} + 8 q^{49} - 15 q^{51} + 7 q^{53} + 12 q^{55} + 23 q^{57} - 10 q^{59} - q^{61} + 8 q^{63} + 30 q^{65} + 24 q^{67} + 41 q^{69} + 27 q^{71} + 2 q^{73} + 21 q^{75} + 21 q^{79} - 7 q^{81} - 5 q^{85} + 23 q^{87} - 25 q^{89} + 78 q^{91} - 56 q^{93} + 13 q^{95} - 60 q^{97} + 35 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - x^{15} - 6 x^{14} + 5 x^{13} + 21 x^{12} - 4 x^{11} - 94 x^{10} - 6 x^{9} + 364 x^{8} - 18 x^{7} - 846 x^{6} - 108 x^{5} + 1701 x^{4} + 1215 x^{3} - 4374 x^{2} - 2187 x + 6561\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 1 \)
\(\beta_{3}\)\(=\)\((\)\( 3 \nu^{15} - 4 \nu^{14} + 16 \nu^{13} + 33 \nu^{12} - 50 \nu^{11} - 30 \nu^{10} + 316 \nu^{9} + 106 \nu^{8} - 1050 \nu^{7} - 1012 \nu^{6} + 2850 \nu^{5} + 2862 \nu^{4} - 7911 \nu^{3} - 1782 \nu^{2} + 18954 \nu + 2187 \)\()/23328\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{15} + 34 \nu^{14} + 18 \nu^{13} - 113 \nu^{12} - 18 \nu^{11} + 598 \nu^{10} + 124 \nu^{9} - 2142 \nu^{8} - 958 \nu^{7} + 5388 \nu^{6} + 3186 \nu^{5} - 13014 \nu^{4} - 5427 \nu^{3} + 32076 \nu^{2} + 8748 \nu - 19683 \)\()/23328\)
\(\beta_{5}\)\(=\)\((\)\( -4 \nu^{15} - 53 \nu^{14} + 153 \nu^{13} + 385 \nu^{12} - 612 \nu^{11} - 1226 \nu^{10} + 1576 \nu^{9} + 5904 \nu^{8} - 4534 \nu^{7} - 19056 \nu^{6} + 13932 \nu^{5} + 42066 \nu^{4} - 8748 \nu^{3} - 95499 \nu^{2} + 15309 \nu + 203391 \)\()/34992\)
\(\beta_{6}\)\(=\)\((\)\( 25 \nu^{15} - 76 \nu^{14} + 467 \nu^{12} - 54 \nu^{11} - 1162 \nu^{10} - 364 \nu^{9} + 4734 \nu^{8} + 2962 \nu^{7} - 20796 \nu^{6} - 4122 \nu^{5} + 12474 \nu^{4} + 8667 \nu^{3} + 2430 \nu^{2} - 71442 \nu + 111537 \)\()/69984\)
\(\beta_{7}\)\(=\)\((\)\( -\nu^{15} + \nu^{14} + 6 \nu^{13} - 5 \nu^{12} - 21 \nu^{11} + 4 \nu^{10} + 94 \nu^{9} + 6 \nu^{8} - 364 \nu^{7} + 18 \nu^{6} + 846 \nu^{5} + 108 \nu^{4} - 1701 \nu^{3} - 1215 \nu^{2} + 4374 \nu + 2187 \)\()/2187\)
\(\beta_{8}\)\(=\)\((\)\( -\nu^{15} - 2 \nu^{14} + 9 \nu^{13} + 13 \nu^{12} - 36 \nu^{11} - 59 \nu^{10} + 106 \nu^{9} + 288 \nu^{8} - 346 \nu^{7} - 1074 \nu^{6} + 900 \nu^{5} + 2646 \nu^{4} - 1377 \nu^{3} - 6318 \nu^{2} + 729 \nu + 13122 \)\()/2187\)
\(\beta_{9}\)\(=\)\((\)\( -35 \nu^{15} + 62 \nu^{14} + 174 \nu^{13} - 31 \nu^{12} - 438 \nu^{11} - 310 \nu^{10} + 3020 \nu^{9} + 3054 \nu^{8} - 11786 \nu^{7} - 8820 \nu^{6} + 20502 \nu^{5} + 29430 \nu^{4} - 33777 \nu^{3} - 113724 \nu^{2} + 137052 \nu + 247131 \)\()/69984\)
\(\beta_{10}\)\(=\)\((\)\( -2 \nu^{15} + 13 \nu^{14} + 31 \nu^{13} - 25 \nu^{12} - 86 \nu^{11} - 16 \nu^{10} + 450 \nu^{9} + 154 \nu^{8} - 1508 \nu^{7} - 514 \nu^{6} + 3018 \nu^{5} + 2844 \nu^{4} - 3564 \nu^{3} - 9315 \nu^{2} + 18225 \nu + 22599 \)\()/5832\)
\(\beta_{11}\)\(=\)\((\)\( 5 \nu^{15} - 2 \nu^{14} - 24 \nu^{13} - 2 \nu^{12} + 66 \nu^{11} + 88 \nu^{10} - 293 \nu^{9} - 348 \nu^{8} + 956 \nu^{7} + 948 \nu^{6} - 1008 \nu^{5} - 3240 \nu^{4} + 2754 \nu^{3} + 10206 \nu^{2} - 7290 \nu - 13122 \)\()/4374\)
\(\beta_{12}\)\(=\)\((\)\( -15 \nu^{15} - 8 \nu^{14} + 62 \nu^{13} + 15 \nu^{12} - 322 \nu^{11} - 30 \nu^{10} + 1232 \nu^{9} + 350 \nu^{8} - 3210 \nu^{7} - 1676 \nu^{6} + 7986 \nu^{5} + 3294 \nu^{4} - 20601 \nu^{3} - 14742 \nu^{2} + 20412 \nu + 9477 \)\()/11664\)
\(\beta_{13}\)\(=\)\((\)\( 52 \nu^{15} + 23 \nu^{14} - 171 \nu^{13} - \nu^{12} + 252 \nu^{11} + 746 \nu^{10} - 1300 \nu^{9} - 3852 \nu^{8} + 6274 \nu^{7} + 6924 \nu^{6} - 2196 \nu^{5} - 17874 \nu^{4} - 7128 \nu^{3} + 83349 \nu^{2} + 729 \nu - 107163 \)\()/34992\)
\(\beta_{14}\)\(=\)\((\)\( -53 \nu^{15} - 73 \nu^{14} + 327 \nu^{13} + 176 \nu^{12} - 1014 \nu^{11} - 508 \nu^{10} + 3056 \nu^{9} + 5178 \nu^{8} - 10184 \nu^{7} - 13644 \nu^{6} + 25254 \nu^{5} + 16956 \nu^{4} - 32643 \nu^{3} - 74601 \nu^{2} + 12393 \nu + 183708 \)\()/34992\)
\(\beta_{15}\)\(=\)\((\)\(289 \nu^{15} + 86 \nu^{14} - 1398 \nu^{13} - 679 \nu^{12} + 3570 \nu^{11} + 5738 \nu^{10} - 15868 \nu^{9} - 27354 \nu^{8} + 49774 \nu^{7} + 67308 \nu^{6} - 83970 \nu^{5} - 176634 \nu^{4} + 112995 \nu^{3} + 593892 \nu^{2} - 239112 \nu - 925101\)\()/69984\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 1\)
\(\nu^{3}\)\(=\)\(-\beta_{14} - 2 \beta_{13} + \beta_{12} + 2 \beta_{11} + \beta_{10} - \beta_{9} - \beta_{7} + \beta_{5} + \beta_{4} + 2 \beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{8} - 2 \beta_{7} - 2 \beta_{6} + 2 \beta_{3} + 2 \beta_{2} + 1\)
\(\nu^{5}\)\(=\)\(-2 \beta_{15} - 2 \beta_{14} + 6 \beta_{11} - 2 \beta_{9} + 2 \beta_{8} + 3 \beta_{7} + 2 \beta_{4} - 2 \beta_{3} + 2 \beta_{1} - 4\)
\(\nu^{6}\)\(=\)\(2 \beta_{14} + 4 \beta_{13} - 2 \beta_{12} - 4 \beta_{11} - 2 \beta_{10} + 2 \beta_{9} - 6 \beta_{8} + 2 \beta_{7} - 6 \beta_{6} + 4 \beta_{5} - 4 \beta_{4} + 2 \beta_{2} - 10 \beta_{1} + 11\)
\(\nu^{7}\)\(=\)\(-6 \beta_{15} + 4 \beta_{14} + 14 \beta_{13} - 2 \beta_{12} + 4 \beta_{11} + 2 \beta_{10} - 2 \beta_{9} + 4 \beta_{8} - 10 \beta_{7} - 2 \beta_{6} - 10 \beta_{5} - 4 \beta_{4} - 6 \beta_{3} - 2 \beta_{2} + 13 \beta_{1} - 2\)
\(\nu^{8}\)\(=\)\(4 \beta_{15} + 24 \beta_{14} + 26 \beta_{13} - 12 \beta_{12} - 20 \beta_{11} - 4 \beta_{10} + 34 \beta_{9} - 18 \beta_{8} - 2 \beta_{7} - 10 \beta_{6} - 2 \beta_{5} - 30 \beta_{4} - 22 \beta_{3} + 17 \beta_{2} - 32 \beta_{1} - 17\)
\(\nu^{9}\)\(=\)\(-8 \beta_{15} + \beta_{14} - 2 \beta_{13} + 23 \beta_{12} + 32 \beta_{11} + 31 \beta_{10} + 5 \beta_{9} - 8 \beta_{8} - 69 \beta_{7} - 21 \beta_{5} - 21 \beta_{4} + 40 \beta_{3} + 4 \beta_{1} + 40\)
\(\nu^{10}\)\(=\)\(20 \beta_{15} + 60 \beta_{14} + 24 \beta_{13} - 24 \beta_{11} + 100 \beta_{9} - 7 \beta_{8} - 120 \beta_{7} - 40 \beta_{6} - 16 \beta_{5} - 20 \beta_{4} + 28 \beta_{3} + 20 \beta_{2} + 4 \beta_{1} - 83\)
\(\nu^{11}\)\(=\)\(-56 \beta_{15} - 8 \beta_{14} + 8 \beta_{13} - 24 \beta_{12} + 112 \beta_{11} + 8 \beta_{10} + 24 \beta_{9} + 40 \beta_{8} - 77 \beta_{7} + 16 \beta_{6} - 108 \beta_{5} + 4 \beta_{4} + 220 \beta_{3} - 12 \beta_{2} - 8 \beta_{1} - 24\)
\(\nu^{12}\)\(=\)\(-16 \beta_{15} - 64 \beta_{14} + 12 \beta_{13} + 16 \beta_{12} - 60 \beta_{11} - 144 \beta_{10} + 116 \beta_{9} - 160 \beta_{8} - 20 \beta_{7} - 20 \beta_{6} + 216 \beta_{5} + 176 \beta_{4} + 48 \beta_{3} - 32 \beta_{2} - 64 \beta_{1} - 15\)
\(\nu^{13}\)\(=\)\(-104 \beta_{15} + 176 \beta_{14} + 220 \beta_{13} - 256 \beta_{12} - 160 \beta_{11} + 16 \beta_{10} - 260 \beta_{9} + 84 \beta_{8} - 128 \beta_{7} - 24 \beta_{6} - 232 \beta_{5} + 96 \beta_{4} + 500 \beta_{3} - 168 \beta_{2} + 81 \beta_{1} - 164\)
\(\nu^{14}\)\(=\)\(-68 \beta_{15} - 16 \beta_{14} + 596 \beta_{13} - 252 \beta_{12} - 524 \beta_{11} - 292 \beta_{10} + 548 \beta_{9} - 292 \beta_{8} + 388 \beta_{7} + 156 \beta_{6} + 260 \beta_{5} + 224 \beta_{4} - 1188 \beta_{3} + 129 \beta_{2} - 584 \beta_{1} - 1127\)
\(\nu^{15}\)\(=\)\(408 \beta_{15} + 595 \beta_{14} + 130 \beta_{13} - 283 \beta_{12} - 598 \beta_{11} + 781 \beta_{10} - 249 \beta_{9} - 480 \beta_{8} - 129 \beta_{7} - 40 \beta_{6} + 17 \beta_{5} - 1023 \beta_{4} + 1400 \beta_{3} - 520 \beta_{2} - 1658 \beta_{1} + 416\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/912\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(229\) \(305\) \(799\)
\(\chi(n)\) \(-\beta_{3}\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
65.1
−1.72340 0.172852i
−1.63023 + 0.585107i
−0.909329 1.47415i
−0.809132 + 1.53144i
0.956703 + 1.44386i
1.25083 1.19809i
1.66415 + 0.480229i
1.70042 0.329508i
−1.72340 + 0.172852i
−1.63023 0.585107i
−0.909329 + 1.47415i
−0.809132 1.53144i
0.956703 1.44386i
1.25083 + 1.19809i
1.66415 0.480229i
1.70042 + 0.329508i
0 −1.72340 + 0.172852i 0 −1.02997 0.594652i 0 −1.04533 0 2.94024 0.595786i 0
65.2 0 −1.63023 0.585107i 0 2.85312 + 1.64725i 0 4.86833 0 2.31530 + 1.90772i 0
65.3 0 −0.909329 + 1.47415i 0 2.16072 + 1.24749i 0 −3.30744 0 −1.34624 2.68098i 0
65.4 0 −0.809132 1.53144i 0 −3.15813 1.82335i 0 −1.32651 0 −1.69061 + 2.47827i 0
65.5 0 0.956703 1.44386i 0 −1.67415 0.966572i 0 1.30064 0 −1.16944 2.76268i 0
65.6 0 1.25083 + 1.19809i 0 −1.30557 0.753769i 0 3.02808 0 0.129140 + 2.99722i 0
65.7 0 1.66415 0.480229i 0 2.52758 + 1.45930i 0 −0.106684 0 2.53876 1.59834i 0
65.8 0 1.70042 + 0.329508i 0 −1.87360 1.08173i 0 −3.41109 0 2.78285 + 1.12060i 0
449.1 0 −1.72340 0.172852i 0 −1.02997 + 0.594652i 0 −1.04533 0 2.94024 + 0.595786i 0
449.2 0 −1.63023 + 0.585107i 0 2.85312 1.64725i 0 4.86833 0 2.31530 1.90772i 0
449.3 0 −0.909329 1.47415i 0 2.16072 1.24749i 0 −3.30744 0 −1.34624 + 2.68098i 0
449.4 0 −0.809132 + 1.53144i 0 −3.15813 + 1.82335i 0 −1.32651 0 −1.69061 2.47827i 0
449.5 0 0.956703 + 1.44386i 0 −1.67415 + 0.966572i 0 1.30064 0 −1.16944 + 2.76268i 0
449.6 0 1.25083 1.19809i 0 −1.30557 + 0.753769i 0 3.02808 0 0.129140 2.99722i 0
449.7 0 1.66415 + 0.480229i 0 2.52758 1.45930i 0 −0.106684 0 2.53876 + 1.59834i 0
449.8 0 1.70042 0.329508i 0 −1.87360 + 1.08173i 0 −3.41109 0 2.78285 1.12060i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 449.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
57.f even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 912.2.bn.o 16
3.b odd 2 1 912.2.bn.n 16
4.b odd 2 1 456.2.bf.c 16
12.b even 2 1 456.2.bf.d yes 16
19.d odd 6 1 912.2.bn.n 16
57.f even 6 1 inner 912.2.bn.o 16
76.f even 6 1 456.2.bf.d yes 16
228.n odd 6 1 456.2.bf.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
456.2.bf.c 16 4.b odd 2 1
456.2.bf.c 16 228.n odd 6 1
456.2.bf.d yes 16 12.b even 2 1
456.2.bf.d yes 16 76.f even 6 1
912.2.bn.n 16 3.b odd 2 1
912.2.bn.n 16 19.d odd 6 1
912.2.bn.o 16 1.a even 1 1 trivial
912.2.bn.o 16 57.f even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(912, [\chi])\):

\(T_{5}^{16} + \cdots\)
\( T_{7}^{8} - 30 T_{7}^{6} - 22 T_{7}^{5} + 223 T_{7}^{4} + 234 T_{7}^{3} - 278 T_{7}^{2} - 332 T_{7} - 32 \)
\(T_{17}^{16} - \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \)
$3$ \( 6561 - 2187 T - 4374 T^{2} + 1215 T^{3} + 1701 T^{4} - 108 T^{5} - 846 T^{6} - 18 T^{7} + 364 T^{8} - 6 T^{9} - 94 T^{10} - 4 T^{11} + 21 T^{12} + 5 T^{13} - 6 T^{14} - T^{15} + T^{16} \)
$5$ \( 430336 + 1275264 T + 1443392 T^{2} + 544320 T^{3} - 252000 T^{4} - 242688 T^{5} + 25760 T^{6} + 72960 T^{7} + 9624 T^{8} - 10632 T^{9} - 2224 T^{10} + 1188 T^{11} + 326 T^{12} - 72 T^{13} - 21 T^{14} + 3 T^{15} + T^{16} \)
$7$ \( ( -32 - 332 T - 278 T^{2} + 234 T^{3} + 223 T^{4} - 22 T^{5} - 30 T^{6} + T^{8} )^{2} \)
$11$ \( 2637376 + 22890544 T^{2} + 37229952 T^{4} + 13143656 T^{6} + 1945988 T^{8} + 146207 T^{10} + 5883 T^{12} + 121 T^{14} + T^{16} \)
$13$ \( 64 - 2016 T + 26016 T^{2} - 152712 T^{3} + 351032 T^{4} + 164334 T^{5} - 474235 T^{6} - 213441 T^{7} + 680559 T^{8} - 114492 T^{9} - 45991 T^{10} + 9105 T^{11} + 3117 T^{12} - 192 T^{13} - 61 T^{14} + 3 T^{15} + T^{16} \)
$17$ \( 2945449984 - 4960026624 T + 3228762112 T^{2} - 748683264 T^{3} - 150208512 T^{4} + 99373056 T^{5} + 2434048 T^{6} - 9556992 T^{7} + 1128960 T^{8} + 402432 T^{9} - 73808 T^{10} - 13344 T^{11} + 3260 T^{12} + 216 T^{13} - 69 T^{14} - 3 T^{15} + T^{16} \)
$19$ \( 16983563041 + 9832589129 T + 4939817505 T^{2} + 1797647874 T^{3} + 572109190 T^{4} + 158141104 T^{5} + 39703863 T^{6} + 9515523 T^{7} + 2164434 T^{8} + 500817 T^{9} + 109983 T^{10} + 23056 T^{11} + 4390 T^{12} + 726 T^{13} + 105 T^{14} + 11 T^{15} + T^{16} \)
$23$ \( 1328456704 - 515228928 T - 419170112 T^{2} + 188404608 T^{3} + 106375712 T^{4} - 35166336 T^{5} - 13037888 T^{6} + 3811056 T^{7} + 1232920 T^{8} - 251592 T^{9} - 71536 T^{10} + 10680 T^{11} + 3158 T^{12} - 210 T^{13} - 67 T^{14} + 3 T^{15} + T^{16} \)
$29$ \( 554696704 + 627048448 T + 1067581440 T^{2} + 595329024 T^{3} + 885616640 T^{4} + 537397248 T^{5} + 383674880 T^{6} + 102023680 T^{7} + 29191056 T^{8} + 3108008 T^{9} + 808964 T^{10} + 65592 T^{11} + 15842 T^{12} + 636 T^{13} + 153 T^{14} + 5 T^{15} + T^{16} \)
$31$ \( 5554422784 + 61498640944 T^{2} + 16672154004 T^{4} + 1677358288 T^{6} + 84484833 T^{8} + 2351020 T^{10} + 36674 T^{12} + 300 T^{14} + T^{16} \)
$37$ \( 262144 + 1018624 T^{2} + 1250404 T^{4} + 717644 T^{6} + 213681 T^{8} + 32764 T^{10} + 2386 T^{12} + 80 T^{14} + T^{16} \)
$41$ \( 27541504 + 58441728 T + 212092928 T^{2} - 121285632 T^{3} + 319632256 T^{4} - 31802688 T^{5} + 143594624 T^{6} - 53716980 T^{7} + 31383385 T^{8} - 4562790 T^{9} + 1103348 T^{10} - 13968 T^{11} + 19621 T^{12} - 60 T^{13} + 188 T^{14} + 6 T^{15} + T^{16} \)
$43$ \( 334805733376 + 665968450048 T + 1365036800576 T^{2} + 27149448576 T^{3} + 112720485480 T^{4} + 11909811440 T^{5} + 6058267489 T^{6} + 631863037 T^{7} + 192325795 T^{8} + 21327546 T^{9} + 4243425 T^{10} + 356587 T^{11} + 46497 T^{12} + 2706 T^{13} + 307 T^{14} + 13 T^{15} + T^{16} \)
$47$ \( 18653553688576 - 11153323622400 T + 776729444352 T^{2} + 864711475200 T^{3} - 82711192576 T^{4} - 54578422272 T^{5} + 11147836928 T^{6} + 389611968 T^{7} - 245084688 T^{8} + 2019096 T^{9} + 4287548 T^{10} - 285936 T^{11} - 27246 T^{12} + 3024 T^{13} + 131 T^{14} - 27 T^{15} + T^{16} \)
$53$ \( 50176 + 455168 T + 5722112 T^{2} - 17388672 T^{3} + 37527552 T^{4} - 41881184 T^{5} + 35681872 T^{6} - 16046464 T^{7} + 5963680 T^{8} - 526296 T^{9} + 422076 T^{10} - 42676 T^{11} + 11016 T^{12} - 546 T^{13} + 139 T^{14} - 7 T^{15} + T^{16} \)
$59$ \( 933720229264 - 2290030871472 T + 6026592306688 T^{2} + 773262094192 T^{3} + 448451791852 T^{4} + 29436754944 T^{5} + 16733105140 T^{6} + 983796684 T^{7} + 391587469 T^{8} + 15445562 T^{9} + 5897038 T^{10} + 233596 T^{11} + 57071 T^{12} + 1600 T^{13} + 334 T^{14} + 10 T^{15} + T^{16} \)
$61$ \( 215225477776 + 49406978152 T + 99379753636 T^{2} - 9351307320 T^{3} + 28469199738 T^{4} - 1946108732 T^{5} + 3496995873 T^{6} - 359496063 T^{7} + 301064797 T^{8} - 12157766 T^{9} + 5131093 T^{10} + 31631 T^{11} + 63135 T^{12} + 282 T^{13} + 287 T^{14} + T^{15} + T^{16} \)
$67$ \( 5605723904881 - 3845494100508 T + 465507857186 T^{2} + 283878827016 T^{3} - 56876612628 T^{4} - 20798447580 T^{5} + 7508390500 T^{6} - 246239568 T^{7} - 166047523 T^{8} + 12145140 T^{9} + 3059332 T^{10} - 385056 T^{11} - 13668 T^{12} + 3408 T^{13} + 50 T^{14} - 24 T^{15} + T^{16} \)
$71$ \( 59895709696 - 67413999616 T + 113279500288 T^{2} + 18901434368 T^{3} + 33798250496 T^{4} - 1869303808 T^{5} + 3256228864 T^{6} - 97254656 T^{7} + 183219904 T^{8} - 16819296 T^{9} + 4415776 T^{10} - 419880 T^{11} + 76552 T^{12} - 6814 T^{13} + 631 T^{14} - 27 T^{15} + T^{16} \)
$73$ \( 24812542725961 + 5409374697926 T + 4026371584194 T^{2} - 202367268528 T^{3} + 280886922092 T^{4} - 13270008162 T^{5} + 9396294764 T^{6} - 438026146 T^{7} + 217615629 T^{8} - 7582058 T^{9} + 3261260 T^{10} - 87738 T^{11} + 35588 T^{12} - 576 T^{13} + 234 T^{14} - 2 T^{15} + T^{16} \)
$79$ \( 3046449086464 - 5465570611200 T + 3481411316416 T^{2} - 381880492800 T^{3} - 212721773336 T^{4} + 37226119008 T^{5} + 11719725227 T^{6} - 2160318693 T^{7} - 210402513 T^{8} + 50750478 T^{9} + 3029431 T^{10} - 877083 T^{11} + 11797 T^{12} + 4998 T^{13} - 91 T^{14} - 21 T^{15} + T^{16} \)
$83$ \( 8981961048064 + 4414635147264 T^{2} + 567557450752 T^{4} + 32415125952 T^{6} + 961447668 T^{8} + 15513207 T^{10} + 134491 T^{12} + 585 T^{14} + T^{16} \)
$89$ \( 6885376 - 62724096 T + 449920512 T^{2} - 1044334336 T^{3} + 1874654912 T^{4} - 821744736 T^{5} + 458330832 T^{6} - 120120288 T^{7} + 58972272 T^{8} - 12917992 T^{9} + 3783708 T^{10} - 274044 T^{11} + 45204 T^{12} + 4030 T^{13} + 627 T^{14} + 25 T^{15} + T^{16} \)
$97$ \( 14736876810496 + 13411762379520 T + 3573632212096 T^{2} - 450461124480 T^{3} - 294809494224 T^{4} + 69882543552 T^{5} + 69723025688 T^{6} + 19708747680 T^{7} + 2833108337 T^{8} + 190391220 T^{9} - 926566 T^{10} - 704856 T^{11} + 61659 T^{12} + 17880 T^{13} + 1498 T^{14} + 60 T^{15} + T^{16} \)
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