# Properties

 Label 8.5.d.b Level 8 Weight 5 Character orbit 8.d Analytic conductor 0.827 Analytic rank 0 Dimension 2 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$8 = 2^{3}$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 8.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.826959704671$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-15})$$ 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 $$\beta = \sqrt{-15}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 - \beta ) q^{2} + 6 q^{3} + ( -14 + 2 \beta ) q^{4} + 8 \beta q^{5} + ( -6 - 6 \beta ) q^{6} -16 \beta q^{7} + ( 44 + 12 \beta ) q^{8} -45 q^{9} +O(q^{10})$$ $$q + ( -1 - \beta ) q^{2} + 6 q^{3} + ( -14 + 2 \beta ) q^{4} + 8 \beta q^{5} + ( -6 - 6 \beta ) q^{6} -16 \beta q^{7} + ( 44 + 12 \beta ) q^{8} -45 q^{9} + ( 120 - 8 \beta ) q^{10} -26 q^{11} + ( -84 + 12 \beta ) q^{12} -8 \beta q^{13} + ( -240 + 16 \beta ) q^{14} + 48 \beta q^{15} + ( 136 - 56 \beta ) q^{16} + 226 q^{17} + ( 45 + 45 \beta ) q^{18} + 134 q^{19} + ( -240 - 112 \beta ) q^{20} -96 \beta q^{21} + ( 26 + 26 \beta ) q^{22} + 80 \beta q^{23} + ( 264 + 72 \beta ) q^{24} -335 q^{25} + ( -120 + 8 \beta ) q^{26} -756 q^{27} + ( 480 + 224 \beta ) q^{28} + 88 \beta q^{29} + ( 720 - 48 \beta ) q^{30} -320 \beta q^{31} + ( -976 - 80 \beta ) q^{32} -156 q^{33} + ( -226 - 226 \beta ) q^{34} + 1920 q^{35} + ( 630 - 90 \beta ) q^{36} + 456 \beta q^{37} + ( -134 - 134 \beta ) q^{38} -48 \beta q^{39} + ( -1440 + 352 \beta ) q^{40} + 994 q^{41} + ( -1440 + 96 \beta ) q^{42} -1882 q^{43} + ( 364 - 52 \beta ) q^{44} -360 \beta q^{45} + ( 1200 - 80 \beta ) q^{46} + 544 \beta q^{47} + ( 816 - 336 \beta ) q^{48} -1439 q^{49} + ( 335 + 335 \beta ) q^{50} + 1356 q^{51} + ( 240 + 112 \beta ) q^{52} -984 \beta q^{53} + ( 756 + 756 \beta ) q^{54} -208 \beta q^{55} + ( 2880 - 704 \beta ) q^{56} + 804 q^{57} + ( 1320 - 88 \beta ) q^{58} -5018 q^{59} + ( -1440 - 672 \beta ) q^{60} + 536 \beta q^{61} + ( -4800 + 320 \beta ) q^{62} + 720 \beta q^{63} + ( -224 + 1056 \beta ) q^{64} + 960 q^{65} + ( 156 + 156 \beta ) q^{66} + 8006 q^{67} + ( -3164 + 452 \beta ) q^{68} + 480 \beta q^{69} + ( -1920 - 1920 \beta ) q^{70} -144 \beta q^{71} + ( -1980 - 540 \beta ) q^{72} + 386 q^{73} + ( 6840 - 456 \beta ) q^{74} -2010 q^{75} + ( -1876 + 268 \beta ) q^{76} + 416 \beta q^{77} + ( -720 + 48 \beta ) q^{78} -2848 \beta q^{79} + ( 6720 + 1088 \beta ) q^{80} -891 q^{81} + ( -994 - 994 \beta ) q^{82} -2234 q^{83} + ( 2880 + 1344 \beta ) q^{84} + 1808 \beta q^{85} + ( 1882 + 1882 \beta ) q^{86} + 528 \beta q^{87} + ( -1144 - 312 \beta ) q^{88} -10046 q^{89} + ( -5400 + 360 \beta ) q^{90} -1920 q^{91} + ( -2400 - 1120 \beta ) q^{92} -1920 \beta q^{93} + ( 8160 - 544 \beta ) q^{94} + 1072 \beta q^{95} + ( -5856 - 480 \beta ) q^{96} + 8738 q^{97} + ( 1439 + 1439 \beta ) q^{98} + 1170 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} + 12q^{3} - 28q^{4} - 12q^{6} + 88q^{8} - 90q^{9} + O(q^{10})$$ $$2q - 2q^{2} + 12q^{3} - 28q^{4} - 12q^{6} + 88q^{8} - 90q^{9} + 240q^{10} - 52q^{11} - 168q^{12} - 480q^{14} + 272q^{16} + 452q^{17} + 90q^{18} + 268q^{19} - 480q^{20} + 52q^{22} + 528q^{24} - 670q^{25} - 240q^{26} - 1512q^{27} + 960q^{28} + 1440q^{30} - 1952q^{32} - 312q^{33} - 452q^{34} + 3840q^{35} + 1260q^{36} - 268q^{38} - 2880q^{40} + 1988q^{41} - 2880q^{42} - 3764q^{43} + 728q^{44} + 2400q^{46} + 1632q^{48} - 2878q^{49} + 670q^{50} + 2712q^{51} + 480q^{52} + 1512q^{54} + 5760q^{56} + 1608q^{57} + 2640q^{58} - 10036q^{59} - 2880q^{60} - 9600q^{62} - 448q^{64} + 1920q^{65} + 312q^{66} + 16012q^{67} - 6328q^{68} - 3840q^{70} - 3960q^{72} + 772q^{73} + 13680q^{74} - 4020q^{75} - 3752q^{76} - 1440q^{78} + 13440q^{80} - 1782q^{81} - 1988q^{82} - 4468q^{83} + 5760q^{84} + 3764q^{86} - 2288q^{88} - 20092q^{89} - 10800q^{90} - 3840q^{91} - 4800q^{92} + 16320q^{94} - 11712q^{96} + 17476q^{97} + 2878q^{98} + 2340q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/8\mathbb{Z}\right)^\times$$.

 $$n$$ $$5$$ $$7$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3.1
 0.5 + 1.93649i 0.5 − 1.93649i
−1.00000 3.87298i 6.00000 −14.0000 + 7.74597i 30.9839i −6.00000 23.2379i 61.9677i 44.0000 + 46.4758i −45.0000 120.000 30.9839i
3.2 −1.00000 + 3.87298i 6.00000 −14.0000 7.74597i 30.9839i −6.00000 + 23.2379i 61.9677i 44.0000 46.4758i −45.0000 120.000 + 30.9839i
 $$n$$: 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
8.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8.5.d.b 2
3.b odd 2 1 72.5.b.b 2
4.b odd 2 1 32.5.d.b 2
5.b even 2 1 200.5.g.d 2
5.c odd 4 2 200.5.e.c 4
8.b even 2 1 32.5.d.b 2
8.d odd 2 1 inner 8.5.d.b 2
12.b even 2 1 288.5.b.b 2
16.e even 4 2 256.5.c.i 4
16.f odd 4 2 256.5.c.i 4
20.d odd 2 1 800.5.g.d 2
20.e even 4 2 800.5.e.c 4
24.f even 2 1 72.5.b.b 2
24.h odd 2 1 288.5.b.b 2
40.e odd 2 1 200.5.g.d 2
40.f even 2 1 800.5.g.d 2
40.i odd 4 2 800.5.e.c 4
40.k even 4 2 200.5.e.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.5.d.b 2 1.a even 1 1 trivial
8.5.d.b 2 8.d odd 2 1 inner
32.5.d.b 2 4.b odd 2 1
32.5.d.b 2 8.b even 2 1
72.5.b.b 2 3.b odd 2 1
72.5.b.b 2 24.f even 2 1
200.5.e.c 4 5.c odd 4 2
200.5.e.c 4 40.k even 4 2
200.5.g.d 2 5.b even 2 1
200.5.g.d 2 40.e odd 2 1
256.5.c.i 4 16.e even 4 2
256.5.c.i 4 16.f odd 4 2
288.5.b.b 2 12.b even 2 1
288.5.b.b 2 24.h odd 2 1
800.5.e.c 4 20.e even 4 2
800.5.e.c 4 40.i odd 4 2
800.5.g.d 2 20.d odd 2 1
800.5.g.d 2 40.f even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} - 6$$ acting on $$S_{5}^{\mathrm{new}}(8, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 2 T + 16 T^{2}$$
$3$ $$( 1 - 6 T + 81 T^{2} )^{2}$$
$5$ $$1 - 290 T^{2} + 390625 T^{4}$$
$7$ $$1 - 962 T^{2} + 5764801 T^{4}$$
$11$ $$( 1 + 26 T + 14641 T^{2} )^{2}$$
$13$ $$1 - 56162 T^{2} + 815730721 T^{4}$$
$17$ $$( 1 - 226 T + 83521 T^{2} )^{2}$$
$19$ $$( 1 - 134 T + 130321 T^{2} )^{2}$$
$23$ $$1 - 463682 T^{2} + 78310985281 T^{4}$$
$29$ $$1 - 1298402 T^{2} + 500246412961 T^{4}$$
$31$ $$1 - 311042 T^{2} + 852891037441 T^{4}$$
$37$ $$1 - 629282 T^{2} + 3512479453921 T^{4}$$
$41$ $$( 1 - 994 T + 2825761 T^{2} )^{2}$$
$43$ $$( 1 + 1882 T + 3418801 T^{2} )^{2}$$
$47$ $$1 - 5320322 T^{2} + 23811286661761 T^{4}$$
$53$ $$1 - 1257122 T^{2} + 62259690411361 T^{4}$$
$59$ $$( 1 + 5018 T + 12117361 T^{2} )^{2}$$
$61$ $$1 - 23382242 T^{2} + 191707312997281 T^{4}$$
$67$ $$( 1 - 8006 T + 20151121 T^{2} )^{2}$$
$71$ $$1 - 50512322 T^{2} + 645753531245761 T^{4}$$
$73$ $$( 1 - 386 T + 28398241 T^{2} )^{2}$$
$79$ $$1 + 43766398 T^{2} + 1517108809906561 T^{4}$$
$83$ $$( 1 + 2234 T + 47458321 T^{2} )^{2}$$
$89$ $$( 1 + 10046 T + 62742241 T^{2} )^{2}$$
$97$ $$( 1 - 8738 T + 88529281 T^{2} )^{2}$$