# Properties

 Label 5.8.b.a Level 5 Weight 8 Character orbit 5.b Analytic conductor 1.562 Analytic rank 0 Dimension 2 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 5.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.56192512742$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-29})$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 2\sqrt{-29}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + 3 \beta q^{3} + 12 q^{4} + ( 75 - 25 \beta ) q^{5} -348 q^{6} -39 \beta q^{7} + 140 \beta q^{8} + 1143 q^{9} +O(q^{10})$$ $$q + \beta q^{2} + 3 \beta q^{3} + 12 q^{4} + ( 75 - 25 \beta ) q^{5} -348 q^{6} -39 \beta q^{7} + 140 \beta q^{8} + 1143 q^{9} + ( 2900 + 75 \beta ) q^{10} -6828 q^{11} + 36 \beta q^{12} -942 \beta q^{13} + 4524 q^{14} + ( 8700 + 225 \beta ) q^{15} -14704 q^{16} + 1456 \beta q^{17} + 1143 \beta q^{18} + 6860 q^{19} + ( 900 - 300 \beta ) q^{20} + 13572 q^{21} -6828 \beta q^{22} + 2713 \beta q^{23} -48720 q^{24} + ( -66875 - 3750 \beta ) q^{25} + 109272 q^{26} + 9990 \beta q^{27} -468 \beta q^{28} + 25590 q^{29} + ( -26100 + 8700 \beta ) q^{30} + 82112 q^{31} + 3216 \beta q^{32} -20484 \beta q^{33} -168896 q^{34} + ( -113100 - 2925 \beta ) q^{35} + 13716 q^{36} -20754 \beta q^{37} + 6860 \beta q^{38} + 327816 q^{39} + ( 406000 + 10500 \beta ) q^{40} -533118 q^{41} + 13572 \beta q^{42} + 65823 \beta q^{43} -81936 q^{44} + ( 85725 - 28575 \beta ) q^{45} -314708 q^{46} + 541 \beta q^{47} -44112 \beta q^{48} + 647107 q^{49} + ( 435000 - 66875 \beta ) q^{50} -506688 q^{51} -11304 \beta q^{52} -54722 \beta q^{53} -1158840 q^{54} + ( -512100 + 170700 \beta ) q^{55} + 633360 q^{56} + 20580 \beta q^{57} + 25590 \beta q^{58} + 1438980 q^{59} + ( 104400 + 2700 \beta ) q^{60} + 1381022 q^{61} + 82112 \beta q^{62} -44577 \beta q^{63} -2255168 q^{64} + ( -2731800 - 70650 \beta ) q^{65} + 2376144 q^{66} -252069 \beta q^{67} + 17472 \beta q^{68} -944124 q^{69} + ( 339300 - 113100 \beta ) q^{70} -481608 q^{71} + 160020 \beta q^{72} + 137988 \beta q^{73} + 2407464 q^{74} + ( 1305000 - 200625 \beta ) q^{75} + 82320 q^{76} + 266292 \beta q^{77} + 327816 \beta q^{78} -1059760 q^{79} + ( -1102800 + 367600 \beta ) q^{80} -976779 q^{81} -533118 \beta q^{82} -241757 \beta q^{83} + 162864 q^{84} + ( 4222400 + 109200 \beta ) q^{85} -7635468 q^{86} + 76770 \beta q^{87} -955920 \beta q^{88} + 5644170 q^{89} + ( 3314700 + 85725 \beta ) q^{90} -4261608 q^{91} + 32556 \beta q^{92} + 246336 \beta q^{93} -62756 q^{94} + ( 514500 - 171500 \beta ) q^{95} -1119168 q^{96} + 1115016 \beta q^{97} + 647107 \beta q^{98} -7804404 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 24q^{4} + 150q^{5} - 696q^{6} + 2286q^{9} + O(q^{10})$$ $$2q + 24q^{4} + 150q^{5} - 696q^{6} + 2286q^{9} + 5800q^{10} - 13656q^{11} + 9048q^{14} + 17400q^{15} - 29408q^{16} + 13720q^{19} + 1800q^{20} + 27144q^{21} - 97440q^{24} - 133750q^{25} + 218544q^{26} + 51180q^{29} - 52200q^{30} + 164224q^{31} - 337792q^{34} - 226200q^{35} + 27432q^{36} + 655632q^{39} + 812000q^{40} - 1066236q^{41} - 163872q^{44} + 171450q^{45} - 629416q^{46} + 1294214q^{49} + 870000q^{50} - 1013376q^{51} - 2317680q^{54} - 1024200q^{55} + 1266720q^{56} + 2877960q^{59} + 208800q^{60} + 2762044q^{61} - 4510336q^{64} - 5463600q^{65} + 4752288q^{66} - 1888248q^{69} + 678600q^{70} - 963216q^{71} + 4814928q^{74} + 2610000q^{75} + 164640q^{76} - 2119520q^{79} - 2205600q^{80} - 1953558q^{81} + 325728q^{84} + 8444800q^{85} - 15270936q^{86} + 11288340q^{89} + 6629400q^{90} - 8523216q^{91} - 125512q^{94} + 1029000q^{95} - 2238336q^{96} - 15608808q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/5\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
4.1
 − 5.38516i 5.38516i
10.7703i 32.3110i 12.0000 75.0000 + 269.258i −348.000 420.043i 1507.85i 1143.00 2900.00 807.775i
4.2 10.7703i 32.3110i 12.0000 75.0000 269.258i −348.000 420.043i 1507.85i 1143.00 2900.00 + 807.775i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5.8.b.a 2
3.b odd 2 1 45.8.b.a 2
4.b odd 2 1 80.8.c.a 2
5.b even 2 1 inner 5.8.b.a 2
5.c odd 4 2 25.8.a.d 2
8.b even 2 1 320.8.c.d 2
8.d odd 2 1 320.8.c.c 2
15.d odd 2 1 45.8.b.a 2
15.e even 4 2 225.8.a.n 2
20.d odd 2 1 80.8.c.a 2
20.e even 4 2 400.8.a.y 2
40.e odd 2 1 320.8.c.c 2
40.f even 2 1 320.8.c.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.8.b.a 2 1.a even 1 1 trivial
5.8.b.a 2 5.b even 2 1 inner
25.8.a.d 2 5.c odd 4 2
45.8.b.a 2 3.b odd 2 1
45.8.b.a 2 15.d odd 2 1
80.8.c.a 2 4.b odd 2 1
80.8.c.a 2 20.d odd 2 1
225.8.a.n 2 15.e even 4 2
320.8.c.c 2 8.d odd 2 1
320.8.c.c 2 40.e odd 2 1
320.8.c.d 2 8.b even 2 1
320.8.c.d 2 40.f even 2 1
400.8.a.y 2 20.e even 4 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{8}^{\mathrm{new}}(5, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 140 T^{2} + 16384 T^{4}$$
$3$ $$1 - 3330 T^{2} + 4782969 T^{4}$$
$5$ $$1 - 150 T + 78125 T^{2}$$
$7$ $$1 - 1470650 T^{2} + 678223072849 T^{4}$$
$11$ $$( 1 + 6828 T + 19487171 T^{2} )^{2}$$
$13$ $$1 - 22562810 T^{2} + 3937376385699289 T^{4}$$
$17$ $$1 - 574764770 T^{2} + 168377826559400929 T^{4}$$
$19$ $$( 1 - 6860 T + 893871739 T^{2} )^{2}$$
$23$ $$1 - 5955848090 T^{2} + 11592836324538749809 T^{4}$$
$29$ $$( 1 - 25590 T + 17249876309 T^{2} )^{2}$$
$31$ $$( 1 - 82112 T + 27512614111 T^{2} )^{2}$$
$37$ $$1 - 139899246410 T^{2} +$$$$90\!\cdots\!89$$$$T^{4}$$
$41$ $$( 1 + 533118 T + 194754273881 T^{2} )^{2}$$
$43$ $$1 - 41047812050 T^{2} +$$$$73\!\cdots\!49$$$$T^{4}$$
$47$ $$1 - 1013212289930 T^{2} +$$$$25\!\cdots\!69$$$$T^{4}$$
$53$ $$1 - 2002060594730 T^{2} +$$$$13\!\cdots\!69$$$$T^{4}$$
$59$ $$( 1 - 1438980 T + 2488651484819 T^{2} )^{2}$$
$61$ $$( 1 - 1381022 T + 3142742836021 T^{2} )^{2}$$
$67$ $$1 - 4750924642370 T^{2} +$$$$36\!\cdots\!29$$$$T^{4}$$
$71$ $$( 1 + 481608 T + 9095120158391 T^{2} )^{2}$$
$73$ $$1 - 19886077213490 T^{2} +$$$$12\!\cdots\!09$$$$T^{4}$$
$79$ $$( 1 + 1059760 T + 19203908986159 T^{2} )^{2}$$
$83$ $$1 - 47492314121570 T^{2} +$$$$73\!\cdots\!29$$$$T^{4}$$
$89$ $$( 1 - 5644170 T + 44231334895529 T^{2} )^{2}$$
$97$ $$1 - 17378330046530 T^{2} +$$$$65\!\cdots\!69$$$$T^{4}$$