Properties

Label 24.4.f.b
Level 24
Weight 4
Character orbit 24.f
Analytic conductor 1.416
Analytic rank 0
Dimension 8
CM No
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 24 = 2^{3} \cdot 3 \)
Weight: \( k \) = \( 4 \)
Character orbit: \([\chi]\) = 24.f (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(1.41604584014\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{8} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + \beta_{1} q^{2} \) \( + ( -2 + \beta_{4} ) q^{3} \) \( + ( 3 + \beta_{2} ) q^{4} \) \( + ( -\beta_{1} + \beta_{3} - \beta_{6} - \beta_{7} ) q^{5} \) \( + ( 1 - \beta_{1} - \beta_{3} + \beta_{4} + 2 \beta_{5} + \beta_{7} ) q^{6} \) \( + ( -2 - 2 \beta_{2} + \beta_{6} - \beta_{7} ) q^{7} \) \( + ( 1 + 2 \beta_{1} + 2 \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{8} \) \( + ( -6 - 3 \beta_{1} - 3 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + \beta_{1} q^{2} \) \( + ( -2 + \beta_{4} ) q^{3} \) \( + ( 3 + \beta_{2} ) q^{4} \) \( + ( -\beta_{1} + \beta_{3} - \beta_{6} - \beta_{7} ) q^{5} \) \( + ( 1 - \beta_{1} - \beta_{3} + \beta_{4} + 2 \beta_{5} + \beta_{7} ) q^{6} \) \( + ( -2 - 2 \beta_{2} + \beta_{6} - \beta_{7} ) q^{7} \) \( + ( 1 + 2 \beta_{1} + 2 \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{8} \) \( + ( -6 - 3 \beta_{1} - 3 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} ) q^{9} \) \( + ( -1 + 2 \beta_{2} - 5 \beta_{4} - 5 \beta_{5} - \beta_{6} + \beta_{7} ) q^{10} \) \( + ( -1 + 2 \beta_{1} + 2 \beta_{3} + \beta_{4} - \beta_{5} ) q^{11} \) \( + ( 2 \beta_{1} - 3 \beta_{2} - 4 \beta_{3} + 3 \beta_{4} - \beta_{5} + 3 \beta_{6} + \beta_{7} ) q^{12} \) \( + ( -4 \beta_{2} + 4 \beta_{4} + 4 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{13} \) \( + ( 5 - 2 \beta_{3} - 5 \beta_{4} + 5 \beta_{5} - 3 \beta_{6} - 3 \beta_{7} ) q^{14} \) \( + ( 2 - 10 \beta_{1} + 6 \beta_{2} + 2 \beta_{3} - 4 \beta_{4} - 4 \beta_{5} + 3 \beta_{6} + \beta_{7} ) q^{15} \) \( + ( -36 + 2 \beta_{2} + 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{16} \) \( + ( 4 + 2 \beta_{1} + 2 \beta_{3} - 4 \beta_{4} + 4 \beta_{5} ) q^{17} \) \( + ( 27 - 9 \beta_{1} - 6 \beta_{2} + 6 \beta_{3} + 3 \beta_{4} - 3 \beta_{5} - 3 \beta_{6} - 3 \beta_{7} ) q^{18} \) \( + ( 23 + 11 \beta_{4} + 11 \beta_{5} ) q^{19} \) \( + ( -10 - 8 \beta_{1} + 12 \beta_{3} + 10 \beta_{4} - 10 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{20} \) \( + ( 8 + 23 \beta_{1} + 12 \beta_{2} - 7 \beta_{3} - 4 \beta_{4} - 4 \beta_{5} - 3 \beta_{6} + \beta_{7} ) q^{21} \) \( + ( -19 + 4 \beta_{2} + 3 \beta_{4} + 3 \beta_{5} - \beta_{6} + \beta_{7} ) q^{22} \) \( + ( 28 \beta_{1} - 12 \beta_{3} + 4 \beta_{6} + 4 \beta_{7} ) q^{23} \) \( + ( 35 + 6 \beta_{1} - 6 \beta_{2} - 6 \beta_{3} + 11 \beta_{4} + 21 \beta_{5} - 3 \beta_{6} - 3 \beta_{7} ) q^{24} \) \( + ( 37 - 2 \beta_{4} - 2 \beta_{5} ) q^{25} \) \( + ( -14 + 8 \beta_{1} - 20 \beta_{3} + 14 \beta_{4} - 14 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{26} \) \( + ( -39 + 18 \beta_{1} + 18 \beta_{3} - 12 \beta_{4} - 9 \beta_{5} ) q^{27} \) \( + ( 66 + 4 \beta_{2} - 30 \beta_{4} - 30 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{28} \) \( + ( -45 \beta_{1} + 13 \beta_{3} + 3 \beta_{6} + 3 \beta_{7} ) q^{29} \) \( + ( -77 - 8 \beta_{1} - 12 \beta_{2} + 22 \beta_{3} + \beta_{4} + 19 \beta_{5} + 3 \beta_{6} - \beta_{7} ) q^{30} \) \( + ( -2 - 10 \beta_{2} + 8 \beta_{4} + 8 \beta_{5} - 3 \beta_{6} + 3 \beta_{7} ) q^{31} \) \( + ( 18 - 36 \beta_{1} + 4 \beta_{3} - 18 \beta_{4} + 18 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{32} \) \( + ( -29 - 9 \beta_{1} - 9 \beta_{3} + \beta_{4} + 9 \beta_{5} ) q^{33} \) \( + ( -24 + 4 \beta_{2} - 12 \beta_{4} - 12 \beta_{5} + 4 \beta_{6} - 4 \beta_{7} ) q^{34} \) \( + ( 2 - 42 \beta_{1} - 42 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{35} \) \( + ( -63 + 36 \beta_{1} + 3 \beta_{2} - 12 \beta_{3} - 12 \beta_{5} - 12 \beta_{6} ) q^{36} \) \( + ( -16 - 12 \beta_{2} - 4 \beta_{4} - 4 \beta_{5} + 10 \beta_{6} - 10 \beta_{7} ) q^{37} \) \( + ( 11 + 34 \beta_{1} - 22 \beta_{3} - 11 \beta_{4} + 11 \beta_{5} + 11 \beta_{6} + 11 \beta_{7} ) q^{38} \) \( + ( 4 - 74 \beta_{1} + 34 \beta_{3} + 4 \beta_{4} + 4 \beta_{5} - 18 \beta_{6} - 10 \beta_{7} ) q^{39} \) \( + ( -140 + 8 \beta_{2} + 20 \beta_{4} + 20 \beta_{5} - 12 \beta_{6} + 12 \beta_{7} ) q^{40} \) \( + ( -28 + 32 \beta_{1} + 32 \beta_{3} + 28 \beta_{4} - 28 \beta_{5} ) q^{41} \) \( + ( 169 - 8 \beta_{1} + 18 \beta_{2} + 28 \beta_{3} + \beta_{4} - 11 \beta_{5} + 9 \beta_{6} + 11 \beta_{7} ) q^{42} \) \( + ( -19 - 39 \beta_{4} - 39 \beta_{5} ) q^{43} \) \( + ( -20 \beta_{1} + 8 \beta_{6} + 8 \beta_{7} ) q^{44} \) \( + ( -24 + 75 \beta_{1} - 24 \beta_{2} - 27 \beta_{3} + 15 \beta_{6} - 9 \beta_{7} ) q^{45} \) \( + ( 180 + 8 \beta_{2} + 20 \beta_{4} + 20 \beta_{5} + 4 \beta_{6} - 4 \beta_{7} ) q^{46} \) \( + ( 72 \beta_{1} - 8 \beta_{3} - 24 \beta_{6} - 24 \beta_{7} ) q^{47} \) \( + ( 146 + 52 \beta_{1} + 6 \beta_{2} - 44 \beta_{3} - 40 \beta_{4} - 20 \beta_{5} + 12 \beta_{6} + 8 \beta_{7} ) q^{48} \) \( + ( -119 + 46 \beta_{4} + 46 \beta_{5} ) q^{49} \) \( + ( -2 + 35 \beta_{1} + 4 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{50} \) \( + ( 136 + 6 \beta_{1} + 6 \beta_{3} + 16 \beta_{4} + 24 \beta_{5} ) q^{51} \) \( + ( 268 - 16 \beta_{2} + 52 \beta_{4} + 52 \beta_{5} - 12 \beta_{6} + 12 \beta_{7} ) q^{52} \) \( + ( -89 \beta_{1} + 25 \beta_{3} + 7 \beta_{6} + 7 \beta_{7} ) q^{53} \) \( + ( -192 - 51 \beta_{1} + 36 \beta_{2} + 21 \beta_{3} + 6 \beta_{4} - 15 \beta_{5} - 9 \beta_{6} - 12 \beta_{7} ) q^{54} \) \( + ( 20 + 28 \beta_{2} - 8 \beta_{4} - 8 \beta_{5} - 6 \beta_{6} + 6 \beta_{7} ) q^{55} \) \( + ( -12 + 32 \beta_{1} + 72 \beta_{3} + 12 \beta_{4} - 12 \beta_{5} - 28 \beta_{6} - 28 \beta_{7} ) q^{56} \) \( + ( 119 - 33 \beta_{1} - 33 \beta_{3} + 23 \beta_{4} - 33 \beta_{5} ) q^{57} \) \( + ( -349 - 38 \beta_{2} + 15 \beta_{4} + 15 \beta_{5} + 3 \beta_{6} - 3 \beta_{7} ) q^{58} \) \( + ( 11 + 8 \beta_{1} + 8 \beta_{3} - 11 \beta_{4} + 11 \beta_{5} ) q^{59} \) \( + ( -322 - 64 \beta_{1} + 12 \beta_{2} - 40 \beta_{3} - 34 \beta_{4} - 10 \beta_{5} + 6 \beta_{6} - 14 \beta_{7} ) q^{60} \) \( + ( 48 + 76 \beta_{2} - 28 \beta_{4} - 28 \beta_{5} - 10 \beta_{6} + 10 \beta_{7} ) q^{61} \) \( + ( -23 + 16 \beta_{1} - 42 \beta_{3} + 23 \beta_{4} - 23 \beta_{5} + \beta_{6} + \beta_{7} ) q^{62} \) \( + ( -18 - 36 \beta_{1} - 42 \beta_{2} - 12 \beta_{3} + 24 \beta_{4} + 24 \beta_{5} + 33 \beta_{6} + 39 \beta_{7} ) q^{63} \) \( + ( -208 - 36 \beta_{2} - 44 \beta_{4} - 44 \beta_{5} + 20 \beta_{6} - 20 \beta_{7} ) q^{64} \) \( + ( 76 + 36 \beta_{1} + 36 \beta_{3} - 76 \beta_{4} + 76 \beta_{5} ) q^{65} \) \( + ( 91 - 28 \beta_{1} - 18 \beta_{2} - 10 \beta_{3} - 17 \beta_{4} - 7 \beta_{5} + 9 \beta_{6} + \beta_{7} ) q^{66} \) \( + ( -187 + 9 \beta_{4} + 9 \beta_{5} ) q^{67} \) \( + ( 20 - 40 \beta_{1} + 40 \beta_{3} - 20 \beta_{4} + 20 \beta_{5} - 12 \beta_{6} - 12 \beta_{7} ) q^{68} \) \( + ( 8 + 32 \beta_{1} - 24 \beta_{2} - 16 \beta_{3} + 32 \beta_{4} + 32 \beta_{5} - 12 \beta_{6} + 28 \beta_{7} ) q^{69} \) \( + ( 418 - 84 \beta_{2} - 6 \beta_{4} - 6 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{70} \) \( + ( 68 \beta_{1} - 52 \beta_{3} + 44 \beta_{6} + 44 \beta_{7} ) q^{71} \) \( + ( 315 - 66 \beta_{1} + 36 \beta_{2} + 6 \beta_{3} + 33 \beta_{4} - 57 \beta_{5} - 9 \beta_{6} + 15 \beta_{7} ) q^{72} \) \( + ( 134 - 148 \beta_{4} - 148 \beta_{5} ) q^{73} \) \( + ( 54 - 8 \beta_{1} + 4 \beta_{3} - 54 \beta_{4} + 54 \beta_{5} - 26 \beta_{6} - 26 \beta_{7} ) q^{74} \) \( + ( -104 + 6 \beta_{1} + 6 \beta_{3} + 37 \beta_{4} + 6 \beta_{5} ) q^{75} \) \( + ( 212 - 10 \beta_{2} + 22 \beta_{4} + 22 \beta_{5} + 22 \beta_{6} - 22 \beta_{7} ) q^{76} \) \( + ( 70 \beta_{1} - 6 \beta_{3} - 26 \beta_{6} - 26 \beta_{7} ) q^{77} \) \( + ( -478 + 8 \beta_{1} - 12 \beta_{2} - 16 \beta_{3} - 46 \beta_{4} - 94 \beta_{5} - 6 \beta_{6} + 22 \beta_{7} ) q^{78} \) \( + ( 14 + 70 \beta_{2} - 56 \beta_{4} - 56 \beta_{5} + 21 \beta_{6} - 21 \beta_{7} ) q^{79} \) \( + ( -56 - 128 \beta_{1} - 48 \beta_{3} + 56 \beta_{4} - 56 \beta_{5} + 40 \beta_{6} + 40 \beta_{7} ) q^{80} \) \( + ( 9 - 18 \beta_{1} - 18 \beta_{3} + 144 \beta_{5} ) q^{81} \) \( + ( -292 + 64 \beta_{2} + 84 \beta_{4} + 84 \beta_{5} - 28 \beta_{6} + 28 \beta_{7} ) q^{82} \) \( + ( -21 + 62 \beta_{1} + 62 \beta_{3} + 21 \beta_{4} - 21 \beta_{5} ) q^{83} \) \( + ( -526 + 152 \beta_{1} + 44 \beta_{3} + 62 \beta_{4} + 74 \beta_{5} + 18 \beta_{6} + 10 \beta_{7} ) q^{84} \) \( + ( -32 \beta_{2} + 32 \beta_{4} + 32 \beta_{5} - 16 \beta_{6} + 16 \beta_{7} ) q^{85} \) \( + ( -39 - 58 \beta_{1} + 78 \beta_{3} + 39 \beta_{4} - 39 \beta_{5} - 39 \beta_{6} - 39 \beta_{7} ) q^{86} \) \( + ( -38 + 46 \beta_{1} - 18 \beta_{2} + 10 \beta_{3} - 20 \beta_{4} - 20 \beta_{5} - 9 \beta_{6} - 67 \beta_{7} ) q^{87} \) \( + ( -180 - 36 \beta_{2} + 40 \beta_{4} + 40 \beta_{5} + 8 \beta_{6} - 8 \beta_{7} ) q^{88} \) \( + ( -112 - 106 \beta_{1} - 106 \beta_{3} + 112 \beta_{4} - 112 \beta_{5} ) q^{89} \) \( + ( 591 + 42 \beta_{2} - 24 \beta_{3} - 45 \beta_{4} + 75 \beta_{5} - 33 \beta_{6} - 39 \beta_{7} ) q^{90} \) \( + ( 396 + 116 \beta_{4} + 116 \beta_{5} ) q^{91} \) \( + ( 56 + 192 \beta_{1} - 16 \beta_{3} - 56 \beta_{4} + 56 \beta_{5} + 24 \beta_{6} + 24 \beta_{7} ) q^{92} \) \( + ( 16 - 125 \beta_{1} + 12 \beta_{2} + 61 \beta_{3} + 4 \beta_{4} + 4 \beta_{5} - 39 \beta_{6} - 19 \beta_{7} ) q^{93} \) \( + ( 680 + 112 \beta_{2} - 120 \beta_{4} - 120 \beta_{5} - 24 \beta_{6} + 24 \beta_{7} ) q^{94} \) \( + ( -276 \beta_{1} + 100 \beta_{3} - 12 \beta_{6} - 12 \beta_{7} ) q^{95} \) \( + ( 558 + 100 \beta_{1} - 12 \beta_{2} + 76 \beta_{3} + 30 \beta_{4} + 10 \beta_{5} - 6 \beta_{6} - 46 \beta_{7} ) q^{96} \) \( + ( -484 + 90 \beta_{4} + 90 \beta_{5} ) q^{97} \) \( + ( 46 - 73 \beta_{1} - 92 \beta_{3} - 46 \beta_{4} + 46 \beta_{5} + 46 \beta_{6} + 46 \beta_{7} ) q^{98} \) \( + ( 255 + 24 \beta_{1} + 24 \beta_{3} - 39 \beta_{4} - 57 \beta_{5} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(8q \) \(\mathstrut -\mathstrut 12q^{3} \) \(\mathstrut +\mathstrut 20q^{4} \) \(\mathstrut -\mathstrut 48q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(8q \) \(\mathstrut -\mathstrut 12q^{3} \) \(\mathstrut +\mathstrut 20q^{4} \) \(\mathstrut -\mathstrut 48q^{9} \) \(\mathstrut -\mathstrut 24q^{10} \) \(\mathstrut +\mathstrut 36q^{12} \) \(\mathstrut -\mathstrut 280q^{16} \) \(\mathstrut +\mathstrut 264q^{18} \) \(\mathstrut +\mathstrut 184q^{19} \) \(\mathstrut -\mathstrut 176q^{22} \) \(\mathstrut +\mathstrut 264q^{24} \) \(\mathstrut +\mathstrut 296q^{25} \) \(\mathstrut -\mathstrut 324q^{27} \) \(\mathstrut +\mathstrut 528q^{28} \) \(\mathstrut -\mathstrut 624q^{30} \) \(\mathstrut -\mathstrut 264q^{33} \) \(\mathstrut -\mathstrut 176q^{34} \) \(\mathstrut -\mathstrut 516q^{36} \) \(\mathstrut -\mathstrut 1248q^{40} \) \(\mathstrut +\mathstrut 1320q^{42} \) \(\mathstrut -\mathstrut 152q^{43} \) \(\mathstrut +\mathstrut 1440q^{46} \) \(\mathstrut +\mathstrut 1080q^{48} \) \(\mathstrut -\mathstrut 952q^{49} \) \(\mathstrut +\mathstrut 1056q^{51} \) \(\mathstrut +\mathstrut 2112q^{52} \) \(\mathstrut -\mathstrut 1584q^{54} \) \(\mathstrut +\mathstrut 1176q^{57} \) \(\mathstrut -\mathstrut 2616q^{58} \) \(\mathstrut -\mathstrut 2640q^{60} \) \(\mathstrut -\mathstrut 1360q^{64} \) \(\mathstrut +\mathstrut 792q^{66} \) \(\mathstrut -\mathstrut 1496q^{67} \) \(\mathstrut +\mathstrut 3696q^{70} \) \(\mathstrut +\mathstrut 2640q^{72} \) \(\mathstrut +\mathstrut 1072q^{73} \) \(\mathstrut -\mathstrut 708q^{75} \) \(\mathstrut +\mathstrut 1912q^{76} \) \(\mathstrut -\mathstrut 3696q^{78} \) \(\mathstrut -\mathstrut 504q^{81} \) \(\mathstrut -\mathstrut 2816q^{82} \) \(\mathstrut -\mathstrut 4224q^{84} \) \(\mathstrut -\mathstrut 1232q^{88} \) \(\mathstrut +\mathstrut 4104q^{90} \) \(\mathstrut +\mathstrut 3168q^{91} \) \(\mathstrut +\mathstrut 4800q^{94} \) \(\mathstrut +\mathstrut 4752q^{96} \) \(\mathstrut -\mathstrut 3872q^{97} \) \(\mathstrut +\mathstrut 2112q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8}\mathstrut -\mathstrut \) \(10\) \(x^{6}\mathstrut +\mathstrut \) \(120\) \(x^{4}\mathstrut -\mathstrut \) \(640\) \(x^{2}\mathstrut +\mathstrut \) \(4096\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\((\)\( \nu^{7} - 10 \nu^{5} + 120 \nu^{3} - 384 \nu \)\()/256\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{6} - 4 \nu^{5} + 10 \nu^{4} + 8 \nu^{3} - 56 \nu^{2} - 160 \nu + 384 \)\()/128\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{6} + 4 \nu^{5} + 10 \nu^{4} - 8 \nu^{3} - 56 \nu^{2} + 160 \nu + 256 \)\()/128\)
\(\beta_{6}\)\(=\)\((\)\( -\nu^{7} + 2 \nu^{6} + 2 \nu^{5} + 44 \nu^{4} + 24 \nu^{3} - 16 \nu^{2} - 192 \nu + 2048 \)\()/256\)
\(\beta_{7}\)\(=\)\((\)\( -\nu^{7} - 2 \nu^{6} + 2 \nu^{5} - 44 \nu^{4} + 24 \nu^{3} + 16 \nu^{2} - 192 \nu - 2048 \)\()/256\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(3\)
\(\nu^{3}\)\(=\)\(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(2\) \(\beta_{3}\mathstrut +\mathstrut \) \(2\) \(\beta_{1}\mathstrut +\mathstrut \) \(1\)
\(\nu^{4}\)\(=\)\(-\)\(2\) \(\beta_{7}\mathstrut +\mathstrut \) \(2\) \(\beta_{6}\mathstrut +\mathstrut \) \(2\) \(\beta_{5}\mathstrut +\mathstrut \) \(2\) \(\beta_{4}\mathstrut +\mathstrut \) \(2\) \(\beta_{2}\mathstrut -\mathstrut \) \(36\)
\(\nu^{5}\)\(=\)\(2\) \(\beta_{7}\mathstrut +\mathstrut \) \(2\) \(\beta_{6}\mathstrut +\mathstrut \) \(18\) \(\beta_{5}\mathstrut -\mathstrut \) \(18\) \(\beta_{4}\mathstrut +\mathstrut \) \(4\) \(\beta_{3}\mathstrut -\mathstrut \) \(36\) \(\beta_{1}\mathstrut +\mathstrut \) \(18\)
\(\nu^{6}\)\(=\)\(-\)\(20\) \(\beta_{7}\mathstrut +\mathstrut \) \(20\) \(\beta_{6}\mathstrut -\mathstrut \) \(44\) \(\beta_{5}\mathstrut -\mathstrut \) \(44\) \(\beta_{4}\mathstrut -\mathstrut \) \(36\) \(\beta_{2}\mathstrut -\mathstrut \) \(208\)
\(\nu^{7}\)\(=\)\(-\)\(100\) \(\beta_{7}\mathstrut -\mathstrut \) \(100\) \(\beta_{6}\mathstrut +\mathstrut \) \(60\) \(\beta_{5}\mathstrut -\mathstrut \) \(60\) \(\beta_{4}\mathstrut +\mathstrut \) \(56\) \(\beta_{3}\mathstrut -\mathstrut \) \(216\) \(\beta_{1}\mathstrut +\mathstrut \) \(60\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/24\mathbb{Z}\right)^\times\).

\(n\) \(7\) \(13\) \(17\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1
−2.58576 1.14624i
−2.58576 + 1.14624i
−1.95291 2.04601i
−1.95291 + 2.04601i
1.95291 2.04601i
1.95291 + 2.04601i
2.58576 1.14624i
2.58576 + 1.14624i
−2.58576 1.14624i 1.37228 + 5.01167i 5.37228 + 5.92778i 12.2683 2.19618 14.5319i 14.0624i −7.09677 21.4857i −23.2337 + 13.7548i −31.7228 14.0624i
11.2 −2.58576 + 1.14624i 1.37228 5.01167i 5.37228 5.92778i 12.2683 2.19618 + 14.5319i 14.0624i −7.09677 + 21.4857i −23.2337 13.7548i −31.7228 + 14.0624i
11.3 −1.95291 2.04601i −4.37228 2.80770i −0.372281 + 7.99133i −13.1715 2.79411 + 14.4289i 26.9490i 17.0773 14.8447i 11.2337 + 24.5521i 25.7228 + 26.9490i
11.4 −1.95291 + 2.04601i −4.37228 + 2.80770i −0.372281 7.99133i −13.1715 2.79411 14.4289i 26.9490i 17.0773 + 14.8447i 11.2337 24.5521i 25.7228 26.9490i
11.5 1.95291 2.04601i −4.37228 2.80770i −0.372281 7.99133i 13.1715 −14.2832 + 3.46254i 26.9490i −17.0773 14.8447i 11.2337 + 24.5521i 25.7228 26.9490i
11.6 1.95291 + 2.04601i −4.37228 + 2.80770i −0.372281 + 7.99133i 13.1715 −14.2832 3.46254i 26.9490i −17.0773 + 14.8447i 11.2337 24.5521i 25.7228 + 26.9490i
11.7 2.58576 1.14624i 1.37228 + 5.01167i 5.37228 5.92778i −12.2683 9.29295 + 11.3860i 14.0624i 7.09677 21.4857i −23.2337 + 13.7548i −31.7228 + 14.0624i
11.8 2.58576 + 1.14624i 1.37228 5.01167i 5.37228 + 5.92778i −12.2683 9.29295 11.3860i 14.0624i 7.09677 + 21.4857i −23.2337 13.7548i −31.7228 14.0624i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.8
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
3.b Odd 1 yes
8.d Odd 1 yes
24.f Even 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{5}^{4} \) \(\mathstrut -\mathstrut 324 T_{5}^{2} \) \(\mathstrut +\mathstrut 26112 \) acting on \(S_{4}^{\mathrm{new}}(24, [\chi])\).