Properties

Label 216.2.n.b
Level $216$
Weight $2$
Character orbit 216.n
Analytic conductor $1.725$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 216 = 2^{3} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 216.n (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.72476868366\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - x^{15} + x^{14} + 2 x^{12} - 4 x^{11} - 8 x^{9} + 4 x^{8} - 16 x^{7} - 32 x^{5} + 32 x^{4} + 64 x^{2} - 128 x + 256\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 72)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - x^{15} + x^{14} + 2 x^{12} - 4 x^{11} - 8 x^{9} + 4 x^{8} - 16 x^{7} - 32 x^{5} + 32 x^{4} + 64 x^{2} - 128 x + 256\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)
\(\beta_{3}\)\(=\)\((\)\( \nu^{15} + \nu^{14} + 3 \nu^{13} + 4 \nu^{12} + 8 \nu^{11} + 6 \nu^{10} + 8 \nu^{9} + 4 \nu^{8} - 12 \nu^{7} - 24 \nu^{6} - 32 \nu^{5} - 72 \nu^{4} - 96 \nu^{3} - 128 \nu^{2} - 96 \nu - 256 \)\()/192\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{15} + \nu^{14} - \nu^{13} + 2 \nu^{12} + 2 \nu^{11} - 8 \nu^{9} - 8 \nu^{8} - 12 \nu^{7} - 8 \nu^{6} - 32 \nu^{5} - 32 \nu^{4} - 32 \nu^{3} + 64 \nu^{2} + 64 \nu \)\()/128\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{15} - \nu^{14} + \nu^{13} + 2 \nu^{11} - 4 \nu^{10} - 8 \nu^{8} + 4 \nu^{7} - 16 \nu^{6} - 32 \nu^{4} + 32 \nu^{3} + 64 \nu - 128 \)\()/128\)
\(\beta_{6}\)\(=\)\((\)\( -\nu^{14} - \nu^{13} - 2 \nu^{12} + \nu^{11} - 5 \nu^{10} - 4 \nu^{9} - 10 \nu^{8} + 8 \nu^{7} + 4 \nu^{6} + 32 \nu^{5} + 20 \nu^{4} + 64 \nu^{3} + 80 \nu^{2} + 80 \nu - 32 \)\()/96\)
\(\beta_{7}\)\(=\)\((\)\( \nu^{15} + 3 \nu^{14} + 5 \nu^{13} + 12 \nu^{12} + 18 \nu^{11} + 28 \nu^{10} + 24 \nu^{9} + 24 \nu^{8} + 20 \nu^{7} - 64 \nu^{6} - 96 \nu^{5} - 160 \nu^{4} - 256 \nu^{3} - 384 \nu^{2} - 448 \nu - 512 \)\()/384\)
\(\beta_{8}\)\(=\)\((\)\( -\nu^{15} - \nu^{14} - 2 \nu^{13} - 3 \nu^{12} - 5 \nu^{11} - 4 \nu^{10} - 6 \nu^{9} + 8 \nu^{8} + 4 \nu^{7} + 16 \nu^{6} + 20 \nu^{5} + 64 \nu^{4} + 64 \nu^{3} + 80 \nu^{2} + 64 \nu + 128 \)\()/96\)
\(\beta_{9}\)\(=\)\((\)\( -2 \nu^{15} + \nu^{14} - 5 \nu^{13} - 5 \nu^{12} - 16 \nu^{11} - 10 \nu^{10} - 28 \nu^{9} - 8 \nu^{8} - 32 \nu^{7} + 12 \nu^{6} + 64 \nu^{5} + 160 \nu^{4} + 96 \nu^{3} + 256 \nu^{2} + 256 \nu + 704 \)\()/192\)
\(\beta_{10}\)\(=\)\((\)\( -2 \nu^{15} - \nu^{14} - 3 \nu^{13} - 3 \nu^{12} - 8 \nu^{11} - 6 \nu^{10} + 8 \nu^{8} + 20 \nu^{6} + 32 \nu^{5} + 96 \nu^{4} + 80 \nu^{3} + 128 \nu^{2} + 256 \)\()/96\)
\(\beta_{11}\)\(=\)\((\)\( 4 \nu^{15} + \nu^{14} + 3 \nu^{13} + \nu^{12} + 8 \nu^{11} + 6 \nu^{10} + 8 \nu^{9} - 8 \nu^{8} - 12 \nu^{6} - 32 \nu^{5} - 96 \nu^{4} - 144 \nu^{3} - 128 \nu^{2} - 448 \)\()/192\)
\(\beta_{12}\)\(=\)\((\)\( -9 \nu^{15} - 7 \nu^{14} - 9 \nu^{13} - 16 \nu^{12} - 50 \nu^{11} - 12 \nu^{10} - 32 \nu^{9} + 8 \nu^{8} - 36 \nu^{7} + 80 \nu^{6} + 128 \nu^{5} + 480 \nu^{4} + 224 \nu^{3} + 512 \nu^{2} + 192 \nu + 1664 \)\()/384\)
\(\beta_{13}\)\(=\)\((\)\( -11 \nu^{15} - 3 \nu^{14} - 13 \nu^{13} - 14 \nu^{12} - 54 \nu^{11} - 32 \nu^{10} - 40 \nu^{9} + 24 \nu^{8} - 28 \nu^{7} + 120 \nu^{6} + 96 \nu^{5} + 800 \nu^{4} + 480 \nu^{3} + 576 \nu^{2} + 320 \nu + 2048 \)\()/384\)
\(\beta_{14}\)\(=\)\((\)\( -13 \nu^{15} - 7 \nu^{14} - 17 \nu^{13} - 28 \nu^{12} - 74 \nu^{11} - 28 \nu^{10} - 56 \nu^{9} + 8 \nu^{8} - 68 \nu^{7} + 192 \nu^{6} + 224 \nu^{5} + 736 \nu^{4} + 768 \nu^{3} + 896 \nu^{2} + 448 \nu + 2560 \)\()/384\)
\(\beta_{15}\)\(=\)\((\)\( 15 \nu^{15} + \nu^{14} + 15 \nu^{13} + 32 \nu^{12} + 62 \nu^{11} + 36 \nu^{10} + 64 \nu^{9} - 8 \nu^{8} + 60 \nu^{7} - 112 \nu^{6} - 128 \nu^{5} - 864 \nu^{4} - 544 \nu^{3} - 704 \nu^{2} - 576 \nu - 2944 \)\()/384\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\)
\(\nu^{3}\)\(=\)\(\beta_{14} - \beta_{12} + \beta_{7} + \beta_{5}\)
\(\nu^{4}\)\(=\)\(\beta_{13} + \beta_{11} + \beta_{7} + \beta_{6} + \beta_{3}\)
\(\nu^{5}\)\(=\)\(\beta_{15} + 2 \beta_{9} + \beta_{6} - \beta_{5} - 2 \beta_{4} + \beta_{3} + 1\)
\(\nu^{6}\)\(=\)\(\beta_{15} + \beta_{13} - \beta_{11} - 2 \beta_{10} - 2 \beta_{9} - 5 \beta_{7} - 5 \beta_{5} + 1\)
\(\nu^{7}\)\(=\)\(-2 \beta_{14} + \beta_{13} - \beta_{11} - 2 \beta_{10} + \beta_{7} - \beta_{6} - 2 \beta_{4} - 9 \beta_{3} + 2 \beta_{2}\)
\(\nu^{8}\)\(=\)\(\beta_{15} - 2 \beta_{12} + 2 \beta_{9} + 8 \beta_{8} - 3 \beta_{6} + \beta_{5} - 2 \beta_{4} + 5 \beta_{3} + 2 \beta_{2} + 5\)
\(\nu^{9}\)\(=\)\(-\beta_{15} + 2 \beta_{14} - \beta_{13} - 2 \beta_{12} + 5 \beta_{11} + 10 \beta_{10} - 2 \beta_{9} - 8 \beta_{8} + 3 \beta_{7} + 3 \beta_{5} + 8 \beta_{1} + 3\)
\(\nu^{10}\)\(=\)\(10 \beta_{14} - 3 \beta_{13} + 7 \beta_{11} - 2 \beta_{10} + 21 \beta_{7} + 7 \beta_{6} + 2 \beta_{4} - \beta_{3} + 2 \beta_{2} + 8 \beta_{1}\)
\(\nu^{11}\)\(=\)\(-3 \beta_{15} - 2 \beta_{12} - 6 \beta_{9} + 8 \beta_{8} + 17 \beta_{6} - 11 \beta_{5} + 6 \beta_{4} + 25 \beta_{3} - 2 \beta_{2} + 25\)
\(\nu^{12}\)\(=\)\(19 \beta_{15} - 2 \beta_{14} + 19 \beta_{13} + 2 \beta_{12} - 15 \beta_{11} - 6 \beta_{10} + 6 \beta_{9} - 8 \beta_{8} - 5 \beta_{7} - 5 \beta_{5} + 8 \beta_{1} + 7\)
\(\nu^{13}\)\(=\)\(-6 \beta_{14} + 13 \beta_{13} - 17 \beta_{11} - 42 \beta_{10} - 35 \beta_{7} - 17 \beta_{6} - 38 \beta_{4} + 7 \beta_{3} + 42 \beta_{2} - 8 \beta_{1}\)
\(\nu^{14}\)\(=\)\(-11 \beta_{15} - 42 \beta_{12} + 26 \beta_{9} - 24 \beta_{8} - 23 \beta_{6} - 11 \beta_{5} - 26 \beta_{4} - 47 \beta_{3} + 22 \beta_{2} - 47\)
\(\nu^{15}\)\(=\)\(19 \beta_{15} + 22 \beta_{14} + 19 \beta_{13} - 22 \beta_{12} + 65 \beta_{11} + 10 \beta_{10} + 22 \beta_{9} + 24 \beta_{8} + 35 \beta_{7} + 35 \beta_{5} - 24 \beta_{1} + 87\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/216\mathbb{Z}\right)^\times\).

\(n\) \(55\) \(109\) \(137\)
\(\chi(n)\) \(1\) \(-1\) \(-1 - \beta_{3}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
−0.722180 + 1.21592i
−1.34532 + 0.436011i
0.587625 + 1.28635i
0.820200 + 1.15207i
−1.12494 0.857038i
1.41411 0.0174668i
−0.179748 1.40274i
1.05026 0.947078i
−0.722180 1.21592i
−1.34532 0.436011i
0.587625 1.28635i
0.820200 1.15207i
−1.12494 + 0.857038i
1.41411 + 0.0174668i
−0.179748 + 1.40274i
1.05026 + 0.947078i
−1.41411 + 0.0174668i 0 1.99939 0.0493999i 3.17262 + 1.83171i 0 −0.191926 0.332426i −2.82649 + 0.104780i 0 −4.51841 2.53482i
37.2 −1.05026 + 0.947078i 0 0.206086 1.98935i −0.602794 0.348023i 0 0.795065 + 1.37709i 1.66763 + 2.28452i 0 0.962695 0.205379i
37.3 −0.820200 1.15207i 0 −0.654545 + 1.88986i −1.97542 1.14051i 0 −0.907824 1.57240i 2.71411 0.795980i 0 0.306290 + 3.21128i
37.4 −0.587625 1.28635i 0 −1.30939 + 1.51178i 1.97542 + 1.14051i 0 −0.907824 1.57240i 2.71411 + 0.795980i 0 0.306290 3.21128i
37.5 0.179748 + 1.40274i 0 −1.93538 + 0.504281i 1.19115 + 0.687709i 0 1.80469 + 3.12581i −1.05526 2.62420i 0 −0.750573 + 1.79449i
37.6 0.722180 1.21592i 0 −0.956913 1.75622i −3.17262 1.83171i 0 −0.191926 0.332426i −2.82649 0.104780i 0 −4.51841 + 2.53482i
37.7 1.12494 + 0.857038i 0 0.530970 + 1.92823i −1.19115 0.687709i 0 1.80469 + 3.12581i −1.05526 + 2.62420i 0 −0.750573 1.79449i
37.8 1.34532 0.436011i 0 1.61979 1.17315i 0.602794 + 0.348023i 0 0.795065 + 1.37709i 1.66763 2.28452i 0 0.962695 + 0.205379i
181.1 −1.41411 0.0174668i 0 1.99939 + 0.0493999i 3.17262 1.83171i 0 −0.191926 + 0.332426i −2.82649 0.104780i 0 −4.51841 + 2.53482i
181.2 −1.05026 0.947078i 0 0.206086 + 1.98935i −0.602794 + 0.348023i 0 0.795065 1.37709i 1.66763 2.28452i 0 0.962695 + 0.205379i
181.3 −0.820200 + 1.15207i 0 −0.654545 1.88986i −1.97542 + 1.14051i 0 −0.907824 + 1.57240i 2.71411 + 0.795980i 0 0.306290 3.21128i
181.4 −0.587625 + 1.28635i 0 −1.30939 1.51178i 1.97542 1.14051i 0 −0.907824 + 1.57240i 2.71411 0.795980i 0 0.306290 + 3.21128i
181.5 0.179748 1.40274i 0 −1.93538 0.504281i 1.19115 0.687709i 0 1.80469 3.12581i −1.05526 + 2.62420i 0 −0.750573 1.79449i
181.6 0.722180 + 1.21592i 0 −0.956913 + 1.75622i −3.17262 + 1.83171i 0 −0.191926 + 0.332426i −2.82649 + 0.104780i 0 −4.51841 2.53482i
181.7 1.12494 0.857038i 0 0.530970 1.92823i −1.19115 + 0.687709i 0 1.80469 3.12581i −1.05526 2.62420i 0 −0.750573 + 1.79449i
181.8 1.34532 + 0.436011i 0 1.61979 + 1.17315i 0.602794 0.348023i 0 0.795065 1.37709i 1.66763 + 2.28452i 0 0.962695 0.205379i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 181.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner
9.c even 3 1 inner
72.n even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 216.2.n.b 16
3.b odd 2 1 72.2.n.b 16
4.b odd 2 1 864.2.r.b 16
8.b even 2 1 inner 216.2.n.b 16
8.d odd 2 1 864.2.r.b 16
9.c even 3 1 inner 216.2.n.b 16
9.c even 3 1 648.2.d.k 8
9.d odd 6 1 72.2.n.b 16
9.d odd 6 1 648.2.d.j 8
12.b even 2 1 288.2.r.b 16
24.f even 2 1 288.2.r.b 16
24.h odd 2 1 72.2.n.b 16
36.f odd 6 1 864.2.r.b 16
36.f odd 6 1 2592.2.d.k 8
36.h even 6 1 288.2.r.b 16
36.h even 6 1 2592.2.d.j 8
72.j odd 6 1 72.2.n.b 16
72.j odd 6 1 648.2.d.j 8
72.l even 6 1 288.2.r.b 16
72.l even 6 1 2592.2.d.j 8
72.n even 6 1 inner 216.2.n.b 16
72.n even 6 1 648.2.d.k 8
72.p odd 6 1 864.2.r.b 16
72.p odd 6 1 2592.2.d.k 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.2.n.b 16 3.b odd 2 1
72.2.n.b 16 9.d odd 6 1
72.2.n.b 16 24.h odd 2 1
72.2.n.b 16 72.j odd 6 1
216.2.n.b 16 1.a even 1 1 trivial
216.2.n.b 16 8.b even 2 1 inner
216.2.n.b 16 9.c even 3 1 inner
216.2.n.b 16 72.n even 6 1 inner
288.2.r.b 16 12.b even 2 1
288.2.r.b 16 24.f even 2 1
288.2.r.b 16 36.h even 6 1
288.2.r.b 16 72.l even 6 1
648.2.d.j 8 9.d odd 6 1
648.2.d.j 8 72.j odd 6 1
648.2.d.k 8 9.c even 3 1
648.2.d.k 8 72.n even 6 1
864.2.r.b 16 4.b odd 2 1
864.2.r.b 16 8.d odd 2 1
864.2.r.b 16 36.f odd 6 1
864.2.r.b 16 72.p odd 6 1
2592.2.d.j 8 36.h even 6 1
2592.2.d.j 8 72.l even 6 1
2592.2.d.k 8 36.f odd 6 1
2592.2.d.k 8 72.p odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{5}^{16} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(216, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 256 + 128 T + 64 T^{2} + 32 T^{4} + 32 T^{5} + 16 T^{7} + 4 T^{8} + 8 T^{9} + 4 T^{11} + 2 T^{12} + T^{14} + T^{15} + T^{16} \)
$3$ \( T^{16} \)
$5$ \( 4096 - 11712 T^{2} + 26129 T^{4} - 18357 T^{6} + 9318 T^{8} - 2049 T^{10} + 326 T^{12} - 21 T^{14} + T^{16} \)
$7$ \( ( 16 + 36 T + 101 T^{2} - 21 T^{3} + 48 T^{4} - 3 T^{5} + 14 T^{6} - 3 T^{7} + T^{8} )^{2} \)
$11$ \( 10556001 - 6822900 T^{2} + 2895966 T^{4} - 718680 T^{6} + 129907 T^{8} - 14440 T^{10} + 1134 T^{12} - 40 T^{14} + T^{16} \)
$13$ \( 20736 - 119664 T^{2} + 587601 T^{4} - 578901 T^{6} + 467038 T^{8} - 36233 T^{10} + 2094 T^{12} - 53 T^{14} + T^{16} \)
$17$ \( ( 36 + 48 T - 2 T^{2} - 7 T^{3} + T^{4} )^{4} \)
$19$ \( ( 5184 + 7056 T^{2} + 1660 T^{4} + 83 T^{6} + T^{8} )^{2} \)
$23$ \( ( 90000 - 67500 T + 40125 T^{2} - 10875 T^{3} + 2650 T^{4} - 275 T^{5} + 60 T^{6} - 5 T^{7} + T^{8} )^{2} \)
$29$ \( 4032758016 - 1453416048 T^{2} + 356481729 T^{4} - 46463373 T^{6} + 4385038 T^{8} - 241441 T^{10} + 9246 T^{12} - 109 T^{14} + T^{16} \)
$31$ \( ( 92416 - 62320 T + 57529 T^{2} + 7415 T^{3} + 3322 T^{4} + 155 T^{5} + 76 T^{6} + 5 T^{7} + T^{8} )^{2} \)
$37$ \( ( 1327104 + 177264 T^{2} + 8080 T^{4} + 152 T^{6} + T^{8} )^{2} \)
$41$ \( ( 335241 - 264024 T + 165090 T^{2} - 38376 T^{3} + 7879 T^{4} - 616 T^{5} + 90 T^{6} - 4 T^{7} + T^{8} )^{2} \)
$43$ \( 81 - 1080 T^{2} + 13446 T^{4} - 12360 T^{6} + 8827 T^{8} - 1880 T^{10} + 294 T^{12} - 20 T^{14} + T^{16} \)
$47$ \( ( 2178576 - 225828 T + 142965 T^{2} + 3537 T^{3} + 5544 T^{4} + 63 T^{5} + 90 T^{6} + 3 T^{7} + T^{8} )^{2} \)
$53$ \( ( 451584 + 113520 T^{2} + 7744 T^{4} + 160 T^{6} + T^{8} )^{2} \)
$59$ \( 131079601 - 286866144 T^{2} + 591967766 T^{4} - 75952296 T^{6} + 7079403 T^{8} - 287928 T^{10} + 8534 T^{12} - 108 T^{14} + T^{16} \)
$61$ \( 176319369216 - 57650719680 T^{2} + 15331541409 T^{4} - 1011826485 T^{6} + 47134062 T^{8} - 1107945 T^{10} + 18846 T^{12} - 165 T^{14} + T^{16} \)
$67$ \( 9881774573841 - 1672043075100 T^{2} + 223335161334 T^{4} - 8572738680 T^{6} + 228454587 T^{8} - 3485160 T^{10} + 38646 T^{12} - 240 T^{14} + T^{16} \)
$71$ \( ( -864 - 252 T + 48 T^{2} + 18 T^{3} + T^{4} )^{4} \)
$73$ \( ( 36 - 84 T - 14 T^{2} + 11 T^{3} + T^{4} )^{4} \)
$79$ \( ( 3136 + 34440 T + 377609 T^{2} + 8445 T^{3} + 9402 T^{4} + 1065 T^{5} + 236 T^{6} + 15 T^{7} + T^{8} )^{2} \)
$83$ \( 31713911056 - 8048506380 T^{2} + 1398102029 T^{4} - 126163065 T^{6} + 8173602 T^{8} - 289605 T^{10} + 7406 T^{12} - 105 T^{14} + T^{16} \)
$89$ \( ( -504 - 156 T + 52 T^{2} + 16 T^{3} + T^{4} )^{4} \)
$97$ \( ( 1324801 + 897780 T + 454166 T^{2} + 104520 T^{3} + 19107 T^{4} + 1560 T^{5} + 134 T^{6} + T^{8} )^{2} \)
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