# Properties

 Label 48.3.i.a Level 48 Weight 3 Character orbit 48.i Analytic conductor 1.308 Analytic rank 0 Dimension 8 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$48 = 2^{4} \cdot 3$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 48.i (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.30790526893$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(i)$$ Coefficient field: 8.0.629407744.1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{4} q^{2} + ( \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} ) q^{3} + ( -\beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{4} + ( \beta_{3} + 2 \beta_{4} - \beta_{5} + 2 \beta_{6} ) q^{5} + ( 2 - \beta_{1} + 2 \beta_{2} + \beta_{4} + 2 \beta_{5} - \beta_{7} ) q^{6} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{7} + ( 2 \beta_{3} - 2 \beta_{4} - 4 \beta_{6} ) q^{8} + ( -2 + 2 \beta_{1} - \beta_{2} + \beta_{3} - 5 \beta_{4} - 5 \beta_{5} - 3 \beta_{6} - 2 \beta_{7} ) q^{9} +O(q^{10})$$ $$q + \beta_{4} q^{2} + ( \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} ) q^{3} + ( -\beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{4} + ( \beta_{3} + 2 \beta_{4} - \beta_{5} + 2 \beta_{6} ) q^{5} + ( 2 - \beta_{1} + 2 \beta_{2} + \beta_{4} + 2 \beta_{5} - \beta_{7} ) q^{6} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{7} + ( 2 \beta_{3} - 2 \beta_{4} - 4 \beta_{6} ) q^{8} + ( -2 + 2 \beta_{1} - \beta_{2} + \beta_{3} - 5 \beta_{4} - 5 \beta_{5} - 3 \beta_{6} - 2 \beta_{7} ) q^{9} + ( -4 - 2 \beta_{1} + 8 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} + 3 \beta_{5} + 3 \beta_{6} + 4 \beta_{7} ) q^{10} + ( -2 \beta_{4} - 2 \beta_{6} ) q^{11} + ( 8 - 5 \beta_{3} + 3 \beta_{4} - \beta_{5} + 3 \beta_{6} + 2 \beta_{7} ) q^{12} + ( -11 - 2 \beta_{1} - 13 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} ) q^{13} + ( -\beta_{3} + \beta_{4} + 2 \beta_{5} + 2 \beta_{6} ) q^{14} + ( 14 - 4 \beta_{1} - 4 \beta_{2} + 3 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - 3 \beta_{6} - 4 \beta_{7} ) q^{15} + ( -4 \beta_{1} - 16 \beta_{2} - 4 \beta_{7} ) q^{16} + ( -9 \beta_{3} + \beta_{4} - \beta_{5} + 9 \beta_{6} ) q^{17} + ( -12 + 2 \beta_{1} - 4 \beta_{2} + 5 \beta_{3} - 2 \beta_{4} - 6 \beta_{7} ) q^{18} + ( -2 + 8 \beta_{1} + 6 \beta_{2} + 4 \beta_{3} - 4 \beta_{4} - 4 \beta_{5} - 4 \beta_{6} ) q^{19} + ( -6 \beta_{3} - 10 \beta_{4} - 4 \beta_{5} - 8 \beta_{6} ) q^{20} + ( -5 + 3 \beta_{2} + 2 \beta_{3} - 3 \beta_{4} - 2 \beta_{5} - 3 \beta_{6} - 2 \beta_{7} ) q^{21} + ( -8 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - 4 \beta_{7} ) q^{22} + ( 2 \beta_{3} + 10 \beta_{4} + 10 \beta_{5} + 2 \beta_{6} ) q^{23} + ( -8 + 8 \beta_{1} + 8 \beta_{2} + 4 \beta_{3} + 4 \beta_{4} - 2 \beta_{5} - 6 \beta_{6} + 4 \beta_{7} ) q^{24} + ( 14 - 14 \beta_{1} + 17 \beta_{2} - 14 \beta_{3} + 14 \beta_{4} + 14 \beta_{5} + 14 \beta_{6} + 14 \beta_{7} ) q^{25} + ( 11 \beta_{3} - 13 \beta_{4} - 4 \beta_{5} ) q^{26} + ( -5 + 12 \beta_{2} + 2 \beta_{3} - 3 \beta_{4} - 2 \beta_{5} - 3 \beta_{6} + 7 \beta_{7} ) q^{27} + ( 8 + 2 \beta_{1} + 8 \beta_{2} + 2 \beta_{7} ) q^{28} + ( 7 \beta_{4} - 7 \beta_{6} ) q^{29} + ( 8 - 6 \beta_{1} - 12 \beta_{2} - 5 \beta_{3} + 15 \beta_{4} - 7 \beta_{5} + 9 \beta_{6} - 4 \beta_{7} ) q^{30} + ( 9 + \beta_{1} + \beta_{2} + \beta_{7} ) q^{31} + ( 8 \beta_{3} - 8 \beta_{5} + 8 \beta_{6} ) q^{32} + ( -8 + 4 \beta_{1} + 4 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} + 4 \beta_{7} ) q^{33} + ( -4 + 18 \beta_{1} + 36 \beta_{2} + 8 \beta_{3} - 8 \beta_{4} - 8 \beta_{5} - 8 \beta_{6} + 2 \beta_{7} ) q^{34} + ( 5 \beta_{3} + \beta_{4} + 5 \beta_{5} - \beta_{6} ) q^{35} + ( 16 - 2 \beta_{1} + 16 \beta_{2} - 3 \beta_{3} - \beta_{4} + 7 \beta_{5} + 15 \beta_{6} + 4 \beta_{7} ) q^{36} + ( 9 - 19 \beta_{2} + 5 \beta_{3} - 5 \beta_{4} - 5 \beta_{5} - 5 \beta_{6} - 10 \beta_{7} ) q^{37} + ( 2 \beta_{3} + 6 \beta_{4} + 16 \beta_{5} ) q^{38} + ( 13 - 13 \beta_{1} - 19 \beta_{2} - 23 \beta_{3} + 16 \beta_{4} + 16 \beta_{5} + 3 \beta_{6} + 13 \beta_{7} ) q^{39} + ( -16 + 12 \beta_{1} - 32 \beta_{2} + 16 \beta_{3} - 16 \beta_{4} - 16 \beta_{5} - 16 \beta_{6} - 20 \beta_{7} ) q^{40} + ( -\beta_{3} + 7 \beta_{4} + 7 \beta_{5} - \beta_{6} ) q^{41} + ( -4 - 2 \beta_{1} - 8 \beta_{2} - 4 \beta_{3} - 4 \beta_{4} - \beta_{5} + 3 \beta_{6} - 4 \beta_{7} ) q^{42} + ( -34 + 26 \beta_{2} + 4 \beta_{3} - 4 \beta_{4} - 4 \beta_{5} - 4 \beta_{6} - 8 \beta_{7} ) q^{43} + ( 4 \beta_{3} + 4 \beta_{4} + 8 \beta_{6} ) q^{44} + ( -24 + 20 \beta_{1} - 4 \beta_{2} + 26 \beta_{3} - 11 \beta_{4} + 6 \beta_{5} - 9 \beta_{6} ) q^{45} + ( 40 - 4 \beta_{1} + 8 \beta_{2} - 12 \beta_{3} + 12 \beta_{4} + 12 \beta_{5} + 12 \beta_{6} + 20 \beta_{7} ) q^{46} + ( -24 \beta_{3} + 14 \beta_{4} - 14 \beta_{5} + 24 \beta_{6} ) q^{47} + ( -4 \beta_{1} - 16 \beta_{2} - 4 \beta_{3} + 4 \beta_{4} + 24 \beta_{5} + 12 \beta_{7} ) q^{48} + ( 41 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{7} ) q^{49} + ( -17 \beta_{3} - 14 \beta_{4} - 28 \beta_{5} - 28 \beta_{6} ) q^{50} + ( 6 - 20 \beta_{1} - 14 \beta_{2} - 17 \beta_{3} - 7 \beta_{4} + 3 \beta_{5} + 27 \beta_{6} ) q^{51} + ( -16 - 22 \beta_{1} + 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - 26 \beta_{7} ) q^{52} + ( 15 \beta_{3} - 8 \beta_{4} - 15 \beta_{5} - 8 \beta_{6} ) q^{53} + ( -22 - 11 \beta_{1} - 26 \beta_{2} - 4 \beta_{3} - 13 \beta_{4} - \beta_{5} - 15 \beta_{6} - 13 \beta_{7} ) q^{54} + ( -12 + 12 \beta_{1} - 28 \beta_{2} + 12 \beta_{3} - 12 \beta_{4} - 12 \beta_{5} - 12 \beta_{6} - 12 \beta_{7} ) q^{55} + ( -4 \beta_{3} + 8 \beta_{4} + 4 \beta_{5} - 4 \beta_{6} ) q^{56} + ( -6 + 6 \beta_{1} + 30 \beta_{2} - 18 \beta_{4} - 18 \beta_{5} - 12 \beta_{6} - 6 \beta_{7} ) q^{57} + ( -28 \beta_{2} - 7 \beta_{3} + 7 \beta_{4} + 7 \beta_{5} + 7 \beta_{6} + 14 \beta_{7} ) q^{58} + ( 5 \beta_{3} - 15 \beta_{4} - 5 \beta_{5} - 15 \beta_{6} ) q^{59} + ( -32 + 8 \beta_{1} + 32 \beta_{2} - 8 \beta_{3} + 16 \beta_{4} - 2 \beta_{5} + 18 \beta_{6} + 28 \beta_{7} ) q^{60} + ( 31 - 10 \beta_{1} + 21 \beta_{2} - 5 \beta_{3} + 5 \beta_{4} + 5 \beta_{5} + 5 \beta_{6} ) q^{61} + ( \beta_{3} + 9 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} ) q^{62} + ( -13 - \beta_{1} - \beta_{2} + 6 \beta_{3} + 4 \beta_{4} - 4 \beta_{5} - 6 \beta_{6} - \beta_{7} ) q^{63} + ( -32 - 16 \beta_{1} + 32 \beta_{2} - 8 \beta_{3} + 8 \beta_{4} + 8 \beta_{5} + 8 \beta_{6} ) q^{64} + ( 5 \beta_{3} - 29 \beta_{4} + 29 \beta_{5} - 5 \beta_{6} ) q^{65} + ( -8 + 2 \beta_{3} - 6 \beta_{4} + 10 \beta_{5} - 6 \beta_{6} + 4 \beta_{7} ) q^{66} + ( -44 + 30 \beta_{1} - 14 \beta_{2} + 15 \beta_{3} - 15 \beta_{4} - 15 \beta_{5} - 15 \beta_{6} ) q^{67} + ( -16 \beta_{3} + 12 \beta_{4} + 36 \beta_{5} - 4 \beta_{6} ) q^{68} + ( -12 - 12 \beta_{2} - 6 \beta_{3} - 6 \beta_{4} + 6 \beta_{5} - 6 \beta_{6} - 24 \beta_{7} ) q^{69} + ( 20 - 10 \beta_{1} - 4 \beta_{2} - 6 \beta_{3} + 6 \beta_{4} + 6 \beta_{5} + 6 \beta_{6} + 2 \beta_{7} ) q^{70} + ( 16 \beta_{3} - 26 \beta_{4} - 26 \beta_{5} + 16 \beta_{6} ) q^{71} + ( 16 + 48 \beta_{2} - 10 \beta_{3} + 6 \beta_{4} - 8 \beta_{5} - 12 \beta_{6} - 8 \beta_{7} ) q^{72} + ( -14 + 14 \beta_{1} - 22 \beta_{2} + 14 \beta_{3} - 14 \beta_{4} - 14 \beta_{5} - 14 \beta_{6} - 14 \beta_{7} ) q^{73} + ( 9 \beta_{3} + 19 \beta_{4} + 20 \beta_{6} ) q^{74} + ( 39 - 42 \beta_{2} + 3 \beta_{3} + 42 \beta_{4} - 3 \beta_{5} + 42 \beta_{6} - 3 \beta_{7} ) q^{75} + ( 64 - 4 \beta_{1} - 8 \beta_{3} + 8 \beta_{4} + 8 \beta_{5} + 8 \beta_{6} + 12 \beta_{7} ) q^{76} + ( -2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} ) q^{77} + ( 12 + 20 \beta_{1} - 40 \beta_{2} + 26 \beta_{3} - 20 \beta_{4} - 33 \beta_{5} - 33 \beta_{6} + 6 \beta_{7} ) q^{78} + ( -17 - 13 \beta_{1} - 13 \beta_{2} - 13 \beta_{7} ) q^{79} + ( 24 \beta_{3} + 16 \beta_{4} + 24 \beta_{5} + 40 \beta_{6} ) q^{80} + ( 23 + 8 \beta_{1} + 8 \beta_{2} + 6 \beta_{3} + 22 \beta_{4} - 22 \beta_{5} - 6 \beta_{6} + 8 \beta_{7} ) q^{81} + ( 28 + 2 \beta_{1} - 4 \beta_{2} - 6 \beta_{3} + 6 \beta_{4} + 6 \beta_{5} + 6 \beta_{6} + 14 \beta_{7} ) q^{82} + ( -20 \beta_{3} + 18 \beta_{4} - 20 \beta_{5} - 18 \beta_{6} ) q^{83} + ( 10 \beta_{1} + 16 \beta_{2} + 10 \beta_{3} - 10 \beta_{4} - 12 \beta_{5} - 6 \beta_{7} ) q^{84} + ( 8 + 36 \beta_{2} - 22 \beta_{3} + 22 \beta_{4} + 22 \beta_{5} + 22 \beta_{6} + 44 \beta_{7} ) q^{85} + ( -34 \beta_{3} - 26 \beta_{4} + 16 \beta_{6} ) q^{86} + ( 42 \beta_{2} + 21 \beta_{4} + 21 \beta_{5} ) q^{87} + ( -8 \beta_{1} + 32 \beta_{2} - 8 \beta_{3} + 8 \beta_{4} + 8 \beta_{5} + 8 \beta_{6} + 8 \beta_{7} ) q^{88} + ( 18 \beta_{3} + 10 \beta_{4} + 10 \beta_{5} + 18 \beta_{6} ) q^{89} + ( 64 - 32 \beta_{1} + 4 \beta_{2} + 9 \beta_{3} + 11 \beta_{4} + 55 \beta_{5} + 15 \beta_{6} - 2 \beta_{7} ) q^{90} + ( 32 - 6 \beta_{2} - 13 \beta_{3} + 13 \beta_{4} + 13 \beta_{5} + 13 \beta_{6} + 26 \beta_{7} ) q^{91} + ( 8 \beta_{3} + 16 \beta_{4} - 8 \beta_{5} - 40 \beta_{6} ) q^{92} + ( 3 + 10 \beta_{1} + 13 \beta_{2} + 7 \beta_{3} - 10 \beta_{4} - 3 \beta_{5} ) q^{93} + ( -56 + 48 \beta_{1} + 96 \beta_{2} + 10 \beta_{3} - 10 \beta_{4} - 10 \beta_{5} - 10 \beta_{6} + 28 \beta_{7} ) q^{94} + ( 26 \beta_{3} - 22 \beta_{4} + 22 \beta_{5} - 26 \beta_{6} ) q^{95} + ( 64 - 8 \beta_{1} - 32 \beta_{2} + 24 \beta_{3} - 16 \beta_{4} - 8 \beta_{5} - 24 \beta_{6} - 8 \beta_{7} ) q^{96} + ( -60 - 8 \beta_{1} - 8 \beta_{2} - 8 \beta_{7} ) q^{97} + ( -2 \beta_{3} + 41 \beta_{4} - 4 \beta_{5} + 4 \beta_{6} ) q^{98} + ( 24 - 8 \beta_{1} + 16 \beta_{2} - 14 \beta_{3} + 8 \beta_{4} - 6 \beta_{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 4q^{3} - 8q^{4} + 16q^{6} + O(q^{10})$$ $$8q + 4q^{3} - 8q^{4} + 16q^{6} - 56q^{10} + 56q^{12} - 96q^{13} + 112q^{15} - 64q^{18} + 16q^{19} - 32q^{21} + 16q^{22} - 48q^{24} - 68q^{27} + 64q^{28} + 56q^{30} + 72q^{31} - 64q^{33} + 32q^{34} + 104q^{36} + 112q^{37} - 24q^{42} - 240q^{43} - 112q^{45} + 224q^{46} - 64q^{48} + 328q^{49} - 32q^{51} - 112q^{52} - 168q^{54} - 56q^{58} - 336q^{60} + 208q^{61} - 104q^{63} - 320q^{64} - 80q^{66} - 232q^{67} + 112q^{70} + 160q^{72} + 324q^{75} + 448q^{76} + 152q^{78} - 136q^{79} + 184q^{81} + 176q^{82} + 64q^{84} - 112q^{85} - 64q^{88} + 392q^{90} + 152q^{91} + 64q^{93} - 368q^{94} + 512q^{96} - 480q^{97} + 160q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 2 x^{6} + 2 x^{4} - 8 x^{2} + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{7} + 2 \nu^{5} + 10 \nu^{3} + 24 \nu^{2} + 8 \nu$$$$)/24$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{6} + 2 \nu^{4} - 2 \nu^{2} - 4$$$$)/12$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{7} - 2 \nu^{5} + 2 \nu^{3} - 8 \nu$$$$)/12$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{7} - 2 \nu^{5} + 2 \nu^{3} + 16 \nu$$$$)/12$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{7} + 4 \nu^{5} + 2 \nu^{3} + 4 \nu$$$$)/12$$ $$\beta_{6}$$ $$=$$ $$($$$$\nu^{7} - \nu^{5} + 4 \nu^{3} - 10 \nu$$$$)/6$$ $$\beta_{7}$$ $$=$$ $$($$$$-\nu^{7} - 8 \nu^{6} - 2 \nu^{5} + 8 \nu^{4} - 10 \nu^{3} - 8 \nu^{2} - 8 \nu + 32$$$$)/24$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{4} - \beta_{3}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} + 2 \beta_{1}$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$\beta_{6} + \beta_{5} + \beta_{4}$$ $$\nu^{4}$$ $$=$$ $$\beta_{7} + 4 \beta_{2} + \beta_{1}$$ $$\nu^{5}$$ $$=$$ $$2 \beta_{5} - \beta_{4} - \beta_{3}$$ $$\nu^{6}$$ $$=$$ $$-2 \beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} + 4 \beta_{2} + 4$$ $$\nu^{7}$$ $$=$$ $$2 \beta_{6} - 2 \beta_{5} - 6 \beta_{3}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/48\mathbb{Z}\right)^\times$$.

 $$n$$ $$17$$ $$31$$ $$37$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-\beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5.1
 −1.38255 + 0.297594i 0.767178 + 1.18804i −0.767178 − 1.18804i 1.38255 − 0.297594i −1.38255 − 0.297594i 0.767178 − 1.18804i −0.767178 + 1.18804i 1.38255 + 0.297594i
−1.68014 1.08495i 2.90783 0.737922i 1.64575 + 3.64575i 1.57472 1.57472i −5.68618 1.91505i 3.64575i 1.19038 7.91094i 7.91094 4.29150i −4.35425 + 0.937254i
5.2 −0.420861 + 1.95522i −2.77809 1.13234i −3.64575 1.64575i −6.28651 + 6.28651i 3.38317 4.95522i 1.64575i 4.75216 6.43560i 6.43560 + 6.29150i −9.64575 14.9373i
5.3 0.420861 1.95522i 1.13234 + 2.77809i −3.64575 1.64575i 6.28651 6.28651i 5.90834 1.04478i 1.64575i −4.75216 + 6.43560i −6.43560 + 6.29150i −9.64575 14.9373i
5.4 1.68014 + 1.08495i 0.737922 2.90783i 1.64575 + 3.64575i −1.57472 + 1.57472i 4.39467 4.08495i 3.64575i −1.19038 + 7.91094i −7.91094 4.29150i −4.35425 + 0.937254i
29.1 −1.68014 + 1.08495i 2.90783 + 0.737922i 1.64575 3.64575i 1.57472 + 1.57472i −5.68618 + 1.91505i 3.64575i 1.19038 + 7.91094i 7.91094 + 4.29150i −4.35425 0.937254i
29.2 −0.420861 1.95522i −2.77809 + 1.13234i −3.64575 + 1.64575i −6.28651 6.28651i 3.38317 + 4.95522i 1.64575i 4.75216 + 6.43560i 6.43560 6.29150i −9.64575 + 14.9373i
29.3 0.420861 + 1.95522i 1.13234 2.77809i −3.64575 + 1.64575i 6.28651 + 6.28651i 5.90834 + 1.04478i 1.64575i −4.75216 6.43560i −6.43560 6.29150i −9.64575 + 14.9373i
29.4 1.68014 1.08495i 0.737922 + 2.90783i 1.64575 3.64575i −1.57472 1.57472i 4.39467 + 4.08495i 3.64575i −1.19038 7.91094i −7.91094 + 4.29150i −4.35425 0.937254i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 29.4 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
16.e even 4 1 inner
48.i odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 48.3.i.a 8
3.b odd 2 1 inner 48.3.i.a 8
4.b odd 2 1 192.3.i.a 8
8.b even 2 1 384.3.i.a 8
8.d odd 2 1 384.3.i.b 8
12.b even 2 1 192.3.i.a 8
16.e even 4 1 inner 48.3.i.a 8
16.e even 4 1 384.3.i.a 8
16.f odd 4 1 192.3.i.a 8
16.f odd 4 1 384.3.i.b 8
24.f even 2 1 384.3.i.b 8
24.h odd 2 1 384.3.i.a 8
48.i odd 4 1 inner 48.3.i.a 8
48.i odd 4 1 384.3.i.a 8
48.k even 4 1 192.3.i.a 8
48.k even 4 1 384.3.i.b 8

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{8} + 6272 T_{5}^{4} + 153664$$ acting on $$S_{3}^{\mathrm{new}}(48, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 4 T^{2} + 8 T^{4} + 64 T^{6} + 256 T^{8}$$
$3$ $$1 - 4 T + 8 T^{2} + 12 T^{3} - 126 T^{4} + 108 T^{5} + 648 T^{6} - 2916 T^{7} + 6561 T^{8}$$
$5$ $$1 + 372 T^{4} + 464614 T^{8} + 145312500 T^{12} + 152587890625 T^{16}$$
$7$ $$( 1 - 180 T^{2} + 12874 T^{4} - 432180 T^{6} + 5764801 T^{8} )^{2}$$
$11$ $$1 + 37380 T^{4} + 692870854 T^{8} + 8012734971780 T^{12} + 45949729863572161 T^{16}$$
$13$ $$( 1 + 48 T + 1152 T^{2} + 21264 T^{3} + 317422 T^{4} + 3593616 T^{5} + 32902272 T^{6} + 231686832 T^{7} + 815730721 T^{8} )^{2}$$
$17$ $$( 1 - 236 T^{2} + 73334 T^{4} - 19710956 T^{6} + 6975757441 T^{8} )^{2}$$
$19$ $$( 1 - 8 T + 32 T^{2} - 1160 T^{3} - 4606 T^{4} - 418760 T^{5} + 4170272 T^{6} - 376367048 T^{7} + 16983563041 T^{8} )^{2}$$
$23$ $$( 1 + 996 T^{2} + 719878 T^{4} + 278721636 T^{6} + 78310985281 T^{8} )^{2}$$
$29$ $$1 + 147060 T^{4} + 623812106854 T^{8} + 73566237490044660 T^{12} +$$$$25\!\cdots\!21$$$$T^{16}$$
$31$ $$( 1 - 18 T + 1996 T^{2} - 17298 T^{3} + 923521 T^{4} )^{4}$$
$37$ $$( 1 - 56 T + 1568 T^{2} - 79016 T^{3} + 3980078 T^{4} - 108172904 T^{5} + 2938684448 T^{6} - 143680678904 T^{7} + 3512479453921 T^{8} )^{2}$$
$41$ $$( 1 + 6060 T^{2} + 14724790 T^{4} + 17124111660 T^{6} + 7984925229121 T^{8} )^{2}$$
$43$ $$( 1 + 120 T + 7200 T^{2} + 411000 T^{3} + 20977474 T^{4} + 759939000 T^{5} + 24615367200 T^{6} + 758563565880 T^{7} + 11688200277601 T^{8} )^{2}$$
$47$ $$( 1 - 692 T^{2} + 7491686 T^{4} - 3376739252 T^{6} + 23811286661761 T^{8} )^{2}$$
$53$ $$1 + 1441524 T^{4} - 91064986479386 T^{8} + 89748837960546754164 T^{12} +$$$$38\!\cdots\!21$$$$T^{16}$$
$59$ $$1 + 15787044 T^{4} + 346992256453126 T^{8} +$$$$23\!\cdots\!24$$$$T^{12} +$$$$21\!\cdots\!41$$$$T^{16}$$
$61$ $$( 1 - 104 T + 5408 T^{2} - 491192 T^{3} + 43609454 T^{4} - 1827725432 T^{5} + 74878308128 T^{6} - 5358118933544 T^{7} + 191707312997281 T^{8} )^{2}$$
$67$ $$( 1 + 116 T + 6728 T^{2} + 350436 T^{3} + 16097858 T^{4} + 1573107204 T^{5} + 135576742088 T^{6} + 10493172331604 T^{7} + 406067677556641 T^{8} )^{2}$$
$71$ $$( 1 + 3604 T^{2} + 10180454 T^{4} + 91583698324 T^{6} + 645753531245761 T^{8} )^{2}$$
$73$ $$( 1 - 17604 T^{2} + 131615494 T^{4} - 499922634564 T^{6} + 806460091894081 T^{8} )^{2}$$
$79$ $$( 1 + 34 T + 11588 T^{2} + 212194 T^{3} + 38950081 T^{4} )^{4}$$
$83$ $$1 - 64624380 T^{4} + 2364945173919814 T^{8} -$$$$14\!\cdots\!80$$$$T^{12} +$$$$50\!\cdots\!81$$$$T^{16}$$
$89$ $$( 1 + 25444 T^{2} + 277784198 T^{4} + 1596413580004 T^{6} + 3936588805702081 T^{8} )^{2}$$
$97$ $$( 1 + 120 T + 21970 T^{2} + 1129080 T^{3} + 88529281 T^{4} )^{4}$$