# Properties

 Label 16.3.f.a Level 16 Weight 3 Character orbit 16.f Analytic conductor 0.436 Analytic rank 0 Dimension 6 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$16 = 2^{4}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 16.f (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.435968422976$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(i)$$ Coefficient field: 6.0.399424.1 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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{2} + ( -1 - \beta_{2} + \beta_{5} ) q^{3} + ( -1 + \beta_{1} - \beta_{3} - \beta_{5} ) q^{4} + ( -\beta_{1} - \beta_{2} - \beta_{4} - \beta_{5} ) q^{5} + ( -2 - 2 \beta_{1} - \beta_{2} + 4 \beta_{3} + \beta_{4} ) q^{6} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} + 3 \beta_{4} - \beta_{5} ) q^{7} + ( -2 \beta_{2} - 6 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{8} + ( 2 + 2 \beta_{1} + 4 \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{9} +O(q^{10})$$ $$q + \beta_{2} q^{2} + ( -1 - \beta_{2} + \beta_{5} ) q^{3} + ( -1 + \beta_{1} - \beta_{3} - \beta_{5} ) q^{4} + ( -\beta_{1} - \beta_{2} - \beta_{4} - \beta_{5} ) q^{5} + ( -2 - 2 \beta_{1} - \beta_{2} + 4 \beta_{3} + \beta_{4} ) q^{6} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} + 3 \beta_{4} - \beta_{5} ) q^{7} + ( -2 \beta_{2} - 6 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{8} + ( 2 + 2 \beta_{1} + 4 \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{9} + ( 6 + 2 \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{10} + ( -2 - \beta_{1} + 2 \beta_{2} - 3 \beta_{3} - 3 \beta_{4} + 2 \beta_{5} ) q^{11} + ( 8 + 6 \beta_{3} + 4 \beta_{4} - 2 \beta_{5} ) q^{12} + ( -2 - 3 \beta_{1} - \beta_{2} + 2 \beta_{3} + 5 \beta_{4} - \beta_{5} ) q^{13} + ( 2 - 2 \beta_{1} - 2 \beta_{2} - 10 \beta_{3} - 2 \beta_{5} ) q^{14} + ( 1 + \beta_{1} - \beta_{2} + 5 \beta_{3} - \beta_{4} - 3 \beta_{5} ) q^{15} + ( -6 - 2 \beta_{1} + 10 \beta_{3} - 4 \beta_{4} + 2 \beta_{5} ) q^{16} + ( -2 + 4 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{5} ) q^{17} + ( -12 + 4 \beta_{1} - 8 \beta_{3} - \beta_{4} ) q^{18} + ( 5 - 2 \beta_{1} - 3 \beta_{2} - 6 \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{19} + ( -14 - 2 \beta_{1} + 4 \beta_{2} + 4 \beta_{4} ) q^{20} + ( -6 - 4 \beta_{2} + 2 \beta_{3} + 4 \beta_{5} ) q^{21} + ( -8 + 4 \beta_{1} - \beta_{2} + 10 \beta_{3} - \beta_{4} - 2 \beta_{5} ) q^{22} + ( 11 - 3 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} ) q^{23} + ( 10 - 2 \beta_{1} + 8 \beta_{2} - 10 \beta_{3} + 4 \beta_{4} - 2 \beta_{5} ) q^{24} + ( -4 - 4 \beta_{1} - 4 \beta_{2} + 3 \beta_{3} + 4 \beta_{4} + 4 \beta_{5} ) q^{25} + ( 18 - 2 \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{4} - 6 \beta_{5} ) q^{26} + ( 10 + 2 \beta_{1} - 2 \beta_{2} + 10 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{27} + ( 14 + 2 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} - 12 \beta_{4} + 2 \beta_{5} ) q^{28} + ( 7 \beta_{1} + \beta_{2} - 8 \beta_{3} - 9 \beta_{4} + \beta_{5} ) q^{29} + ( 6 + 2 \beta_{1} - 10 \beta_{3} + 2 \beta_{4} + 6 \beta_{5} ) q^{30} + ( 4 \beta_{2} - 20 \beta_{3} + 4 \beta_{5} ) q^{31} + ( -4 + 4 \beta_{1} - 4 \beta_{2} + 16 \beta_{3} + 12 \beta_{4} ) q^{32} + ( 2 - 8 \beta_{1} + 6 \beta_{2} + 6 \beta_{3} + 4 \beta_{4} - 6 \beta_{5} ) q^{33} + ( -16 - 8 \beta_{1} - 6 \beta_{2} - 4 \beta_{3} + 4 \beta_{5} ) q^{34} + ( -14 + 4 \beta_{1} + 10 \beta_{2} + 20 \beta_{3} + 4 \beta_{4} - 2 \beta_{5} ) q^{35} + ( -13 - 3 \beta_{1} - 16 \beta_{2} - 11 \beta_{3} - 8 \beta_{4} + 5 \beta_{5} ) q^{36} + ( 14 + 5 \beta_{1} + 17 \beta_{2} - 2 \beta_{3} + 5 \beta_{4} - 7 \beta_{5} ) q^{37} + ( 10 + 2 \beta_{1} + 7 \beta_{2} + 8 \beta_{3} - 7 \beta_{4} + 4 \beta_{5} ) q^{38} + ( -33 + \beta_{1} - 7 \beta_{2} - 7 \beta_{3} - 13 \beta_{4} + 7 \beta_{5} ) q^{39} + ( 6 + 2 \beta_{1} - 12 \beta_{2} - 2 \beta_{3} - 10 \beta_{5} ) q^{40} + ( -12 \beta_{2} - 4 \beta_{3} - 12 \beta_{5} ) q^{41} + ( -8 - 8 \beta_{1} - 6 \beta_{2} + 16 \beta_{3} + 6 \beta_{4} ) q^{42} + ( -24 + \beta_{1} - 8 \beta_{2} - 17 \beta_{3} + 15 \beta_{4} - 8 \beta_{5} ) q^{43} + ( -6 - 2 \beta_{1} - 12 \beta_{2} - 16 \beta_{3} + 8 \beta_{4} + 8 \beta_{5} ) q^{44} + ( 14 - \beta_{1} + 5 \beta_{2} + 10 \beta_{3} - 9 \beta_{4} + 5 \beta_{5} ) q^{45} + ( 10 + 6 \beta_{1} + 14 \beta_{2} + 6 \beta_{3} - 2 \beta_{5} ) q^{46} + ( -8 - 8 \beta_{1} - 4 \beta_{2} + 36 \beta_{3} + 8 \beta_{4} + 12 \beta_{5} ) q^{47} + ( 8 + 8 \beta_{1} + 12 \beta_{2} - 12 \beta_{3} - 12 \beta_{4} - 12 \beta_{5} ) q^{48} + ( -5 - 8 \beta_{1} - 4 \beta_{2} - 4 \beta_{3} - 16 \beta_{4} + 4 \beta_{5} ) q^{49} + ( 8 - 8 \beta_{1} + 24 \beta_{3} + 7 \beta_{4} - 8 \beta_{5} ) q^{50} + ( 28 + 6 \beta_{1} + 12 \beta_{2} - 22 \beta_{3} + 6 \beta_{4} ) q^{51} + ( 24 + 8 \beta_{1} + 20 \beta_{2} - 14 \beta_{3} - 4 \beta_{4} + 2 \beta_{5} ) q^{52} + ( 10 + \beta_{1} - 3 \beta_{2} - 14 \beta_{3} + \beta_{4} + 5 \beta_{5} ) q^{53} + ( 4 - 4 \beta_{1} + 8 \beta_{2} - 12 \beta_{3} + 8 \beta_{4} + 4 \beta_{5} ) q^{54} + ( 39 + 9 \beta_{1} - 7 \beta_{2} - 7 \beta_{3} - 5 \beta_{4} + 7 \beta_{5} ) q^{55} + ( -28 + 12 \beta_{1} + 12 \beta_{2} + 8 \beta_{3} + 4 \beta_{4} + 8 \beta_{5} ) q^{56} + ( -2 - 2 \beta_{1} - 4 \beta_{2} + 6 \beta_{3} + 2 \beta_{4} ) q^{57} + ( -34 + 2 \beta_{1} - 7 \beta_{2} - 10 \beta_{3} - 7 \beta_{4} + 14 \beta_{5} ) q^{58} + ( 36 - 3 \beta_{1} + 4 \beta_{2} + 35 \beta_{3} - 5 \beta_{4} + 4 \beta_{5} ) q^{59} + ( -22 - 10 \beta_{1} + 4 \beta_{2} + 10 \beta_{3} - 4 \beta_{4} - 6 \beta_{5} ) q^{60} + ( 2 + \beta_{1} - 5 \beta_{2} + 6 \beta_{3} + 9 \beta_{4} - 5 \beta_{5} ) q^{61} + ( -16 + 8 \beta_{3} - 16 \beta_{4} - 8 \beta_{5} ) q^{62} + ( 1 + \beta_{1} - 9 \beta_{2} - 51 \beta_{3} - \beta_{4} - 11 \beta_{5} ) q^{63} + ( 4 - 20 \beta_{1} - 8 \beta_{2} - 20 \beta_{3} + 16 \beta_{4} - 4 \beta_{5} ) q^{64} + ( -2 + 12 \beta_{1} + 12 \beta_{4} ) q^{65} + ( 40 + 16 \beta_{1} + 10 \beta_{2} - 4 \beta_{3} - 12 \beta_{5} ) q^{66} + ( -39 - 14 \beta_{1} - 23 \beta_{2} + 30 \beta_{3} - 14 \beta_{4} - 5 \beta_{5} ) q^{67} + ( 18 - 2 \beta_{1} - 8 \beta_{2} + 42 \beta_{3} - 6 \beta_{5} ) q^{68} + ( -26 - 4 \beta_{1} - 16 \beta_{2} + 14 \beta_{3} - 4 \beta_{4} + 8 \beta_{5} ) q^{69} + ( -12 + 4 \beta_{1} - 18 \beta_{2} - 32 \beta_{3} + 18 \beta_{4} - 8 \beta_{5} ) q^{70} + ( -37 - 19 \beta_{1} + 17 \beta_{2} + 17 \beta_{3} + 15 \beta_{4} - 17 \beta_{5} ) q^{71} + ( 2 - 10 \beta_{1} - 10 \beta_{2} + 48 \beta_{3} - 6 \beta_{4} + 16 \beta_{5} ) q^{72} + ( -2 - 2 \beta_{1} + 20 \beta_{2} + 2 \beta_{4} + 24 \beta_{5} ) q^{73} + ( -6 + 14 \beta_{1} + 9 \beta_{2} - 58 \beta_{3} - 9 \beta_{4} - 10 \beta_{5} ) q^{74} + ( -32 + 7 \beta_{1} + 8 \beta_{2} - 47 \beta_{3} - 23 \beta_{4} + 8 \beta_{5} ) q^{75} + ( -32 + 8 \beta_{1} + 8 \beta_{2} + 6 \beta_{3} + 12 \beta_{4} - 2 \beta_{5} ) q^{76} + ( -38 - 4 \beta_{2} - 34 \beta_{3} + 8 \beta_{4} - 4 \beta_{5} ) q^{77} + ( -30 - 2 \beta_{1} - 34 \beta_{2} + 38 \beta_{3} + 14 \beta_{5} ) q^{78} + ( 24 + 24 \beta_{1} + 32 \beta_{2} + 48 \beta_{3} - 24 \beta_{4} - 16 \beta_{5} ) q^{79} + ( 36 - 4 \beta_{1} + 4 \beta_{2} - 24 \beta_{3} - 12 \beta_{4} + 24 \beta_{5} ) q^{80} + ( 9 + 16 \beta_{1} + 6 \beta_{2} + 6 \beta_{3} + 28 \beta_{4} - 6 \beta_{5} ) q^{81} + ( 48 - 24 \beta_{3} - 16 \beta_{4} + 24 \beta_{5} ) q^{82} + ( 47 - 9 \beta_{2} - 56 \beta_{3} + 9 \beta_{5} ) q^{83} + ( 36 - 4 \beta_{1} + 24 \beta_{3} + 16 \beta_{4} - 8 \beta_{5} ) q^{84} + ( -26 - 4 \beta_{1} + 8 \beta_{2} + 38 \beta_{3} - 4 \beta_{4} - 16 \beta_{5} ) q^{85} + ( 44 - 16 \beta_{1} - 25 \beta_{2} - 34 \beta_{3} - 25 \beta_{4} + 2 \beta_{5} ) q^{86} + ( 75 - 3 \beta_{1} + 17 \beta_{2} + 17 \beta_{3} + 31 \beta_{4} - 17 \beta_{5} ) q^{87} + ( 2 - 26 \beta_{1} - 4 \beta_{2} + 34 \beta_{3} - 8 \beta_{4} - 6 \beta_{5} ) q^{88} + ( 18 + 18 \beta_{1} + 28 \beta_{2} - 16 \beta_{3} - 18 \beta_{4} - 8 \beta_{5} ) q^{89} + ( -26 + 10 \beta_{1} + 15 \beta_{2} + 22 \beta_{3} + 15 \beta_{4} - 2 \beta_{5} ) q^{90} + ( 32 - 14 \beta_{1} + 8 \beta_{2} + 38 \beta_{3} - 2 \beta_{4} + 8 \beta_{5} ) q^{91} + ( -26 + 10 \beta_{1} + 4 \beta_{2} - 38 \beta_{3} + 4 \beta_{4} - 6 \beta_{5} ) q^{92} + ( -12 - 28 \beta_{1} + 4 \beta_{2} + 12 \beta_{3} + 20 \beta_{4} + 4 \beta_{5} ) q^{93} + ( -16 \beta_{1} + 56 \beta_{3} + 48 \beta_{4} - 24 \beta_{5} ) q^{94} + ( -7 - 7 \beta_{1} - 21 \beta_{2} - 39 \beta_{3} + 7 \beta_{4} - 7 \beta_{5} ) q^{95} + ( -12 + 28 \beta_{1} - 60 \beta_{3} - 24 \beta_{4} + 20 \beta_{5} ) q^{96} + ( -2 + 4 \beta_{1} - 26 \beta_{2} - 26 \beta_{3} - 48 \beta_{4} + 26 \beta_{5} ) q^{97} + ( 16 \beta_{1} + 3 \beta_{2} + 56 \beta_{3} + 8 \beta_{5} ) q^{98} + ( -45 + 4 \beta_{1} - 5 \beta_{2} + 36 \beta_{3} + 4 \beta_{4} + 13 \beta_{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q - 2q^{2} - 2q^{3} - 8q^{4} - 2q^{5} - 8q^{6} - 4q^{7} + 4q^{8} + O(q^{10})$$ $$6q - 2q^{2} - 2q^{3} - 8q^{4} - 2q^{5} - 8q^{6} - 4q^{7} + 4q^{8} + 36q^{10} - 18q^{11} + 52q^{12} - 2q^{13} + 12q^{14} - 40q^{16} - 4q^{17} - 74q^{18} + 30q^{19} - 84q^{20} - 20q^{21} - 52q^{22} + 60q^{23} + 48q^{24} + 96q^{26} + 64q^{27} + 56q^{28} - 18q^{29} + 52q^{30} + 8q^{32} - 4q^{33} - 76q^{34} - 100q^{35} - 52q^{36} + 46q^{37} + 40q^{38} - 196q^{39} + 40q^{40} - 24q^{42} - 114q^{43} + 20q^{44} + 66q^{45} + 28q^{46} - 24q^{48} - 46q^{49} + 46q^{50} + 156q^{51} + 100q^{52} + 78q^{53} + 32q^{54} + 252q^{55} - 168q^{56} - 176q^{58} + 206q^{59} - 160q^{60} + 30q^{61} - 144q^{62} + 64q^{64} + 12q^{65} + 196q^{66} - 226q^{67} + 112q^{68} - 116q^{69} - 16q^{70} - 260q^{71} + 52q^{72} - 92q^{74} - 238q^{75} - 188q^{76} - 212q^{77} - 84q^{78} + 232q^{80} + 86q^{81} + 304q^{82} + 318q^{83} + 232q^{84} - 212q^{85} + 268q^{86} + 444q^{87} - 8q^{88} - 160q^{90} + 188q^{91} - 168q^{92} - 32q^{93} + 48q^{94} - 80q^{96} - 4q^{97} + 10q^{98} - 226q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 2 x^{5} + 3 x^{4} - 6 x^{3} + 6 x^{2} - 8 x + 8$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{5} + 3 \nu^{3} + 6 \nu - 8$$$$)/4$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{5} - 3 \nu^{3} + 2 \nu + 4$$$$)/4$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{5} + 3 \nu^{3} - 4 \nu^{2} + 2 \nu - 8$$$$)/4$$ $$\beta_{4}$$ $$=$$ $$($$$$-3 \nu^{5} + 4 \nu^{4} - 9 \nu^{3} + 12 \nu^{2} - 10 \nu + 20$$$$)/4$$ $$\beta_{5}$$ $$=$$ $$($$$$-3 \nu^{5} + 2 \nu^{4} - 5 \nu^{3} + 8 \nu^{2} - 4 \nu + 14$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} + \beta_{1} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$-2 \beta_{3} - \beta_{2} + \beta_{1} - 1$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{5} - \beta_{4} + \beta_{3} - 2 \beta_{2} + 2$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$2 \beta_{4} + 6 \beta_{3} + \beta_{2} + \beta_{1} + 3$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-3 \beta_{5} + 3 \beta_{4} - 3 \beta_{3} + 2 \beta_{1} + 4$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/16\mathbb{Z}\right)^\times$$.

 $$n$$ $$5$$ $$15$$ $$\chi(n)$$ $$-\beta_{3}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3.1
 1.40680 + 0.144584i −0.671462 + 1.24464i 0.264658 − 1.38923i 1.40680 − 0.144584i −0.671462 − 1.24464i 0.264658 + 1.38923i
−1.55139 1.26222i 2.10278 2.10278i 0.813607 + 3.91638i −4.62721 + 4.62721i −5.91638 + 0.608056i 3.04888 3.68111 7.10278i 0.156674i 13.0192 1.33804i
3.2 −0.573183 + 1.91611i 0.146365 0.146365i −3.34292 2.19656i 3.68585 3.68585i 0.196558 + 0.364346i −9.66442 6.12494 5.14637i 8.95715i 4.94981 + 9.17513i
3.3 1.12457 1.65389i −3.24914 + 3.24914i −1.47068 3.71982i −0.0586332 + 0.0586332i 1.71982 + 9.02760i 4.61555 −7.80605 1.75086i 12.1138i 0.0310355 + 0.162910i
11.1 −1.55139 + 1.26222i 2.10278 + 2.10278i 0.813607 3.91638i −4.62721 4.62721i −5.91638 0.608056i 3.04888 3.68111 + 7.10278i 0.156674i 13.0192 + 1.33804i
11.2 −0.573183 1.91611i 0.146365 + 0.146365i −3.34292 + 2.19656i 3.68585 + 3.68585i 0.196558 0.364346i −9.66442 6.12494 + 5.14637i 8.95715i 4.94981 9.17513i
11.3 1.12457 + 1.65389i −3.24914 3.24914i −1.47068 + 3.71982i −0.0586332 0.0586332i 1.71982 9.02760i 4.61555 −7.80605 + 1.75086i 12.1138i 0.0310355 0.162910i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 11.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.f odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 16.3.f.a 6
3.b odd 2 1 144.3.m.a 6
4.b odd 2 1 64.3.f.a 6
5.b even 2 1 400.3.r.c 6
5.c odd 4 1 400.3.k.c 6
5.c odd 4 1 400.3.k.d 6
8.b even 2 1 128.3.f.b 6
8.d odd 2 1 128.3.f.a 6
12.b even 2 1 576.3.m.a 6
16.e even 4 1 64.3.f.a 6
16.e even 4 1 128.3.f.a 6
16.f odd 4 1 inner 16.3.f.a 6
16.f odd 4 1 128.3.f.b 6
24.f even 2 1 1152.3.m.b 6
24.h odd 2 1 1152.3.m.a 6
32.g even 8 2 1024.3.c.j 12
32.g even 8 2 1024.3.d.k 12
32.h odd 8 2 1024.3.c.j 12
32.h odd 8 2 1024.3.d.k 12
48.i odd 4 1 576.3.m.a 6
48.i odd 4 1 1152.3.m.b 6
48.k even 4 1 144.3.m.a 6
48.k even 4 1 1152.3.m.a 6
80.j even 4 1 400.3.k.d 6
80.k odd 4 1 400.3.r.c 6
80.s even 4 1 400.3.k.c 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
16.3.f.a 6 1.a even 1 1 trivial
16.3.f.a 6 16.f odd 4 1 inner
64.3.f.a 6 4.b odd 2 1
64.3.f.a 6 16.e even 4 1
128.3.f.a 6 8.d odd 2 1
128.3.f.a 6 16.e even 4 1
128.3.f.b 6 8.b even 2 1
128.3.f.b 6 16.f odd 4 1
144.3.m.a 6 3.b odd 2 1
144.3.m.a 6 48.k even 4 1
400.3.k.c 6 5.c odd 4 1
400.3.k.c 6 80.s even 4 1
400.3.k.d 6 5.c odd 4 1
400.3.k.d 6 80.j even 4 1
400.3.r.c 6 5.b even 2 1
400.3.r.c 6 80.k odd 4 1
576.3.m.a 6 12.b even 2 1
576.3.m.a 6 48.i odd 4 1
1024.3.c.j 12 32.g even 8 2
1024.3.c.j 12 32.h odd 8 2
1024.3.d.k 12 32.g even 8 2
1024.3.d.k 12 32.h odd 8 2
1152.3.m.a 6 24.h odd 2 1
1152.3.m.a 6 48.k even 4 1
1152.3.m.b 6 24.f even 2 1
1152.3.m.b 6 48.i odd 4 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(16, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 2 T + 6 T^{2} + 8 T^{3} + 24 T^{4} + 32 T^{5} + 64 T^{6}$$
$3$ $$1 + 2 T + 2 T^{2} - 14 T^{3} - 65 T^{4} + 124 T^{5} + 476 T^{6} + 1116 T^{7} - 5265 T^{8} - 10206 T^{9} + 13122 T^{10} + 118098 T^{11} + 531441 T^{12}$$
$5$ $$1 + 2 T + 2 T^{2} - 14 T^{3} - 369 T^{4} + 636 T^{5} + 2108 T^{6} + 15900 T^{7} - 230625 T^{8} - 218750 T^{9} + 781250 T^{10} + 19531250 T^{11} + 244140625 T^{12}$$
$7$ $$( 1 + 2 T + 87 T^{2} + 332 T^{3} + 4263 T^{4} + 4802 T^{5} + 117649 T^{6} )^{2}$$
$11$ $$1 + 18 T + 162 T^{2} + 2146 T^{3} + 17759 T^{4} + 65756 T^{5} + 609308 T^{6} + 7956476 T^{7} + 260009519 T^{8} + 3801769906 T^{9} + 34726138722 T^{10} + 466873642818 T^{11} + 3138428376721 T^{12}$$
$13$ $$1 + 2 T + 2 T^{2} + 1554 T^{3} - 7825 T^{4} - 453380 T^{5} + 316348 T^{6} - 76621220 T^{7} - 223489825 T^{8} + 7500861186 T^{9} + 1631461442 T^{10} + 275716983698 T^{11} + 23298085122481 T^{12}$$
$17$ $$( 1 + 2 T + 607 T^{2} - 388 T^{3} + 175423 T^{4} + 167042 T^{5} + 24137569 T^{6} )^{2}$$
$19$ $$1 - 30 T + 450 T^{2} - 12014 T^{3} + 441215 T^{4} - 8004292 T^{5} + 113750108 T^{6} - 2889549412 T^{7} + 57499580015 T^{8} - 565209214334 T^{9} + 7642603368450 T^{10} - 183931987734030 T^{11} + 2213314919066161 T^{12}$$
$23$ $$( 1 - 30 T + 1751 T^{2} - 30772 T^{3} + 926279 T^{4} - 8395230 T^{5} + 148035889 T^{6} )^{2}$$
$29$ $$1 + 18 T + 162 T^{2} - 4894 T^{3} + 124463 T^{4} + 24625372 T^{5} + 435069308 T^{6} + 20709937852 T^{7} + 88030315103 T^{8} - 2911065332974 T^{9} + 81039918899682 T^{10} + 7572730199403618 T^{11} + 353814783205469041 T^{12}$$
$31$ $$1 - 3846 T^{2} + 7131791 T^{4} - 8361808916 T^{6} + 6586358756111 T^{8} - 3280218929998086 T^{10} + 787662783788549761 T^{12}$$
$37$ $$1 - 46 T + 1058 T^{2} + 6594 T^{3} - 356337 T^{4} - 78343460 T^{5} + 4002544124 T^{6} - 107252196740 T^{7} - 667832908257 T^{8} + 16918399940946 T^{9} + 3716203262248418 T^{10} - 221194881131221054 T^{11} + 6582952005840035281 T^{12}$$
$41$ $$1 - 5094 T^{2} + 15050223 T^{4} - 31243096276 T^{6} + 42528333194703 T^{8} - 40675209117142374 T^{10} + 22563490300366186081 T^{12}$$
$43$ $$1 + 114 T + 6498 T^{2} + 241730 T^{3} + 12357983 T^{4} + 838941724 T^{5} + 44553879452 T^{6} + 1551203247676 T^{7} + 42249484638383 T^{8} + 1528063089834770 T^{9} + 75949925403851298 T^{10} + 2463708983714404386 T^{11} + 39959630797262576401 T^{12}$$
$47$ $$1 - 4678 T^{2} + 12462287 T^{4} - 24905944212 T^{6} + 60811985090447 T^{8} - 111389199003717958 T^{10} +$$$$11\!\cdots\!41$$$$T^{12}$$
$53$ $$1 - 78 T + 3042 T^{2} - 270110 T^{3} + 31648463 T^{4} - 1389102820 T^{5} + 48555101564 T^{6} - 3901989821380 T^{7} + 249721595980703 T^{8} - 5986815584554190 T^{9} + 189393978231360162 T^{10} - 13641222688510017822 T^{11} +$$$$49\!\cdots\!41$$$$T^{12}$$
$59$ $$1 - 206 T + 21218 T^{2} - 1942462 T^{3} + 171214239 T^{4} - 11916831972 T^{5} + 708622973852 T^{6} - 41482492094532 T^{7} + 2074664742303279 T^{8} - 81934083737364142 T^{9} + 3115448225088482978 T^{10} -$$$$10\!\cdots\!06$$$$T^{11} +$$$$17\!\cdots\!81$$$$T^{12}$$
$61$ $$1 - 30 T + 450 T^{2} - 111694 T^{3} + 33268655 T^{4} - 582006980 T^{5} + 8727089468 T^{6} - 2165647972580 T^{7} + 460632507413855 T^{8} - 5754516693877534 T^{9} + 86268290848776450 T^{10} - 21400287349886478030 T^{11} +$$$$26\!\cdots\!21$$$$T^{12}$$
$67$ $$1 + 226 T + 25538 T^{2} + 2083538 T^{3} + 120508479 T^{4} + 5203289532 T^{5} + 268963196252 T^{6} + 23357566709148 T^{7} + 2428380941854959 T^{8} + 188473476667633922 T^{9} + 10370156349441497858 T^{10} +$$$$41\!\cdots\!74$$$$T^{11} +$$$$81\!\cdots\!61$$$$T^{12}$$
$71$ $$( 1 + 130 T + 11575 T^{2} + 918796 T^{3} + 58349575 T^{4} + 3303518530 T^{5} + 128100283921 T^{6} )^{2}$$
$73$ $$1 - 13126 T^{2} + 103571951 T^{4} - 653716749588 T^{6} + 2941261225338191 T^{8} - 10585595166201707206 T^{10} +$$$$22\!\cdots\!21$$$$T^{12}$$
$79$ $$1 - 70 T^{2} + 84324175 T^{4} + 17226941804 T^{6} + 3284433446508175 T^{8} - 106197616693459270 T^{10} +$$$$59\!\cdots\!41$$$$T^{12}$$
$83$ $$1 - 318 T + 50562 T^{2} - 6712846 T^{3} + 819490815 T^{4} - 81203275140 T^{5} + 6918697616348 T^{6} - 559409362439460 T^{7} + 38891658154821615 T^{8} - 2194700377608598174 T^{9} +$$$$11\!\cdots\!42$$$$T^{10} -$$$$49\!\cdots\!82$$$$T^{11} +$$$$10\!\cdots\!61$$$$T^{12}$$
$89$ $$1 - 31238 T^{2} + 466178479 T^{4} - 4433595811988 T^{6} + 29249082478431439 T^{8} -$$$$12\!\cdots\!78$$$$T^{10} +$$$$24\!\cdots\!21$$$$T^{12}$$
$97$ $$( 1 + 2 T + 10687 T^{2} + 557564 T^{3} + 100553983 T^{4} + 177058562 T^{5} + 832972004929 T^{6} )^{2}$$