# 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 $$2$$ and degree $$1$$)

## 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: $$2$$ 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.

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

## 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. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
8.d Odd 1 yes

## Hecke kernels

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