# 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

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$1.41604584014$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ Defining polynomial: $$x^{8} - 10 x^{6} + 120 x^{4} - 640 x^{2} + 4096$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{8}$$ Twist minimal: yes 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})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 12q^{3} + 20q^{4} - 48q^{9} + O(q^{10})$$ $$8q - 12q^{3} + 20q^{4} - 48q^{9} - 24q^{10} + 36q^{12} - 280q^{16} + 264q^{18} + 184q^{19} - 176q^{22} + 264q^{24} + 296q^{25} - 324q^{27} + 528q^{28} - 624q^{30} - 264q^{33} - 176q^{34} - 516q^{36} - 1248q^{40} + 1320q^{42} - 152q^{43} + 1440q^{46} + 1080q^{48} - 952q^{49} + 1056q^{51} + 2112q^{52} - 1584q^{54} + 1176q^{57} - 2616q^{58} - 2640q^{60} - 1360q^{64} + 792q^{66} - 1496q^{67} + 3696q^{70} + 2640q^{72} + 1072q^{73} - 708q^{75} + 1912q^{76} - 3696q^{78} - 504q^{81} - 2816q^{82} - 4224q^{84} - 1232q^{88} + 4104q^{90} + 3168q^{91} + 4800q^{94} + 4752q^{96} - 3872q^{97} + 2112q^{99} + O(q^{100})$$

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

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

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
8.d odd 2 1 inner
24.f even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 24.4.f.b 8
3.b odd 2 1 inner 24.4.f.b 8
4.b odd 2 1 96.4.f.b 8
8.b even 2 1 96.4.f.b 8
8.d odd 2 1 inner 24.4.f.b 8
12.b even 2 1 96.4.f.b 8
16.e even 4 2 768.4.c.v 16
16.f odd 4 2 768.4.c.v 16
24.f even 2 1 inner 24.4.f.b 8
24.h odd 2 1 96.4.f.b 8
48.i odd 4 2 768.4.c.v 16
48.k even 4 2 768.4.c.v 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.4.f.b 8 1.a even 1 1 trivial
24.4.f.b 8 3.b odd 2 1 inner
24.4.f.b 8 8.d odd 2 1 inner
24.4.f.b 8 24.f even 2 1 inner
96.4.f.b 8 4.b odd 2 1
96.4.f.b 8 8.b even 2 1
96.4.f.b 8 12.b even 2 1
96.4.f.b 8 24.h odd 2 1
768.4.c.v 16 16.e even 4 2
768.4.c.v 16 16.f odd 4 2
768.4.c.v 16 48.i odd 4 2
768.4.c.v 16 48.k even 4 2

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 10 T^{2} + 120 T^{4} - 640 T^{6} + 4096 T^{8}$$
$3$ $$( 1 + 6 T + 30 T^{2} + 162 T^{3} + 729 T^{4} )^{2}$$
$5$ $$( 1 + 176 T^{2} + 38862 T^{4} + 2750000 T^{6} + 244140625 T^{8} )^{2}$$
$7$ $$( 1 - 448 T^{2} + 215646 T^{4} - 52706752 T^{6} + 13841287201 T^{8} )^{2}$$
$11$ $$( 1 - 4840 T^{2} + 9341310 T^{4} - 8574355240 T^{6} + 3138428376721 T^{8} )^{2}$$
$13$ $$( 1 - 2980 T^{2} + 4014966 T^{4} - 14383890820 T^{6} + 23298085122481 T^{8} )^{2}$$
$17$ $$( 1 - 17188 T^{2} + 120704262 T^{4} - 414876535972 T^{6} + 582622237229761 T^{8} )^{2}$$
$19$ $$( 1 - 46 T + 10254 T^{2} - 315514 T^{3} + 47045881 T^{4} )^{4}$$
$23$ $$( 1 + 31196 T^{2} + 464722854 T^{4} + 4618127593244 T^{6} + 21914624432020321 T^{8} )^{2}$$
$29$ $$( 1 + 53936 T^{2} + 1889351598 T^{4} + 32082390641456 T^{6} + 353814783205469041 T^{8} )^{2}$$
$31$ $$( 1 - 98176 T^{2} + 4116020094 T^{4} - 87131561385856 T^{6} + 787662783788549761 T^{8} )^{2}$$
$37$ $$( 1 - 124996 T^{2} + 9025572822 T^{4} - 320705538219364 T^{6} + 6582952005840035281 T^{8} )^{2}$$
$41$ $$( 1 - 82084 T^{2} + 4965540774 T^{4} - 389907556518244 T^{6} + 22563490300366186081 T^{8} )^{2}$$
$43$ $$( 1 + 38 T + 109182 T^{2} + 3021266 T^{3} + 6321363049 T^{4} )^{4}$$
$47$ $$( 1 + 93500 T^{2} - 1715710650 T^{4} + 1007856633261500 T^{6} +$$$$11\!\cdots\!41$$$$T^{8} )^{2}$$
$53$ $$( 1 + 418352 T^{2} + 87270813582 T^{4} + 9272504807039408 T^{6} +$$$$49\!\cdots\!41$$$$T^{8} )^{2}$$
$59$ $$( 1 - 799912 T^{2} + 244209482046 T^{4} - 33740715025839592 T^{6} +$$$$17\!\cdots\!81$$$$T^{8} )^{2}$$
$61$ $$( 1 - 357220 T^{2} + 77943572022 T^{4} - 18404108129236420 T^{6} +$$$$26\!\cdots\!21$$$$T^{8} )^{2}$$
$67$ $$( 1 + 374 T + 633822 T^{2} + 112485362 T^{3} + 90458382169 T^{4} )^{4}$$
$71$ $$( 1 + 776732 T^{2} + 403415373030 T^{4} + 99499589730526172 T^{6} +$$$$16\!\cdots\!41$$$$T^{8} )^{2}$$
$73$ $$( 1 - 268 T + 73158 T^{2} - 104256556 T^{3} + 151334226289 T^{4} )^{4}$$
$79$ $$( 1 - 943744 T^{2} + 544083847614 T^{4} - 229412327623210624 T^{6} +$$$$59\!\cdots\!41$$$$T^{8} )^{2}$$
$83$ $$( 1 - 1890664 T^{2} + 1510361878110 T^{4} - 618134394075327016 T^{6} +$$$$10\!\cdots\!61$$$$T^{8} )^{2}$$
$89$ $$( 1 - 175300 T^{2} - 599506777050 T^{4} - 87120820305463300 T^{6} +$$$$24\!\cdots\!21$$$$T^{8} )^{2}$$
$97$ $$( 1 + 968 T + 1792302 T^{2} + 883467464 T^{3} + 832972004929 T^{4} )^{4}$$