# Properties

 Label 20.5.d.c Level 20 Weight 5 Character orbit 20.d Analytic conductor 2.067 Analytic rank 0 Dimension 8 CM No Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$20 = 2^{2} \cdot 5$$ Weight: $$k$$ = $$5$$ Character orbit: $$[\chi]$$ = 20.d (of order $$2$$ and degree $$1$$)

## Newform invariants

 Self dual: No Analytic conductor: $$2.06739926168$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.1816805376000000.2 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{14}$$ Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} -\beta_{2} q^{3} + ( -6 + \beta_{3} ) q^{4} + ( -5 + \beta_{1} - \beta_{3} - \beta_{6} - \beta_{7} ) q^{5} + ( \beta_{5} + \beta_{7} ) q^{6} + ( 6 \beta_{1} + \beta_{2} - 2 \beta_{4} ) q^{7} + ( -7 \beta_{1} - 2 \beta_{2} + \beta_{3} + 3 \beta_{4} + 2 \beta_{6} + \beta_{7} ) q^{8} + ( 39 - 3 \beta_{3} - 3 \beta_{7} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} -\beta_{2} q^{3} + ( -6 + \beta_{3} ) q^{4} + ( -5 + \beta_{1} - \beta_{3} - \beta_{6} - \beta_{7} ) q^{5} + ( \beta_{5} + \beta_{7} ) q^{6} + ( 6 \beta_{1} + \beta_{2} - 2 \beta_{4} ) q^{7} + ( -7 \beta_{1} - 2 \beta_{2} + \beta_{3} + 3 \beta_{4} + 2 \beta_{6} + \beta_{7} ) q^{8} + ( 39 - 3 \beta_{3} - 3 \beta_{7} ) q^{9} + ( -20 - 6 \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{10} -4 \beta_{5} q^{11} + ( 3 \beta_{1} + 10 \beta_{2} + 3 \beta_{3} - 3 \beta_{4} + 6 \beta_{6} + 3 \beta_{7} ) q^{12} + ( -18 \beta_{1} - \beta_{3} - 4 \beta_{4} - 2 \beta_{6} - \beta_{7} ) q^{13} + ( 80 + 8 \beta_{3} - \beta_{5} - \beta_{7} ) q^{14} + ( -18 \beta_{1} - 3 \beta_{2} - 12 \beta_{3} + 6 \beta_{4} + 4 \beta_{5} + 4 \beta_{7} ) q^{15} + ( -104 - 12 \beta_{3} + 4 \beta_{5} - 4 \beta_{7} ) q^{16} + ( 16 \beta_{1} + 2 \beta_{3} + 4 \beta_{4} + 4 \beta_{6} + 2 \beta_{7} ) q^{17} + ( 33 \beta_{1} + 12 \beta_{2} - 6 \beta_{3} + 6 \beta_{4} - 12 \beta_{6} - 6 \beta_{7} ) q^{18} + ( 24 \beta_{3} + 4 \beta_{5} - 8 \beta_{7} ) q^{19} + ( -50 - 13 \beta_{1} - 22 \beta_{2} - 2 \beta_{3} - 11 \beta_{4} - 4 \beta_{5} + 6 \beta_{6} + 7 \beta_{7} ) q^{20} + ( -120 + 15 \beta_{3} + 15 \beta_{7} ) q^{21} + ( -48 \beta_{2} - 8 \beta_{3} - 8 \beta_{4} - 16 \beta_{6} - 8 \beta_{7} ) q^{22} + ( 66 \beta_{1} + 13 \beta_{2} - 22 \beta_{4} ) q^{23} + ( 240 - 4 \beta_{5} - 28 \beta_{7} ) q^{24} + ( -255 + 90 \beta_{1} + 15 \beta_{3} + 20 \beta_{4} + 10 \beta_{6} + 15 \beta_{7} ) q^{25} + ( 312 - 12 \beta_{3} - 2 \beta_{5} + 6 \beta_{7} ) q^{26} + ( -108 \beta_{1} - 6 \beta_{2} + 36 \beta_{4} ) q^{27} + ( 69 \beta_{1} - 26 \beta_{2} + 5 \beta_{3} + 27 \beta_{4} + 10 \beta_{6} + 5 \beta_{7} ) q^{28} + ( 338 - 4 \beta_{3} - 4 \beta_{7} ) q^{29} + ( -240 + 24 \beta_{1} + 64 \beta_{2} - 24 \beta_{3} - 48 \beta_{4} + 3 \beta_{5} + 3 \beta_{7} ) q^{30} + ( -24 \beta_{3} + 16 \beta_{5} + 8 \beta_{7} ) q^{31} + ( -104 \beta_{1} + 80 \beta_{2} - 8 \beta_{3} - 8 \beta_{4} - 16 \beta_{6} - 8 \beta_{7} ) q^{32} + ( -324 \beta_{1} + 12 \beta_{3} - 60 \beta_{4} + 24 \beta_{6} + 12 \beta_{7} ) q^{33} + ( -272 + 8 \beta_{3} + 4 \beta_{5} - 12 \beta_{7} ) q^{34} + ( -60 \beta_{1} + 35 \beta_{2} + 20 \beta_{4} - 20 \beta_{5} ) q^{35} + ( -714 + 39 \beta_{3} - 24 \beta_{5} + 24 \beta_{7} ) q^{36} + ( 162 \beta_{1} - 21 \beta_{3} + 24 \beta_{4} - 42 \beta_{6} - 21 \beta_{7} ) q^{37} + ( -48 \beta_{1} + 16 \beta_{2} + 24 \beta_{3} + 120 \beta_{4} + 48 \beta_{6} + 24 \beta_{7} ) q^{38} -24 \beta_{5} q^{39} + ( 800 - 33 \beta_{1} - 46 \beta_{2} - 17 \beta_{3} - 43 \beta_{4} + 28 \beta_{5} - 18 \beta_{6} - 5 \beta_{7} ) q^{40} + ( 182 - 79 \beta_{3} - 79 \beta_{7} ) q^{41} + ( -90 \beta_{1} - 60 \beta_{2} + 30 \beta_{3} - 30 \beta_{4} + 60 \beta_{6} + 30 \beta_{7} ) q^{42} + ( 348 \beta_{1} - 77 \beta_{2} - 116 \beta_{4} ) q^{43} + ( 240 + 24 \beta_{3} + 32 \beta_{5} + 96 \beta_{7} ) q^{44} + ( 765 + 339 \beta_{1} - 24 \beta_{3} + 60 \beta_{4} - 39 \beta_{6} - 24 \beta_{7} ) q^{45} + ( 880 + 88 \beta_{3} - 13 \beta_{5} - 13 \beta_{7} ) q^{46} + ( 102 \beta_{1} - 7 \beta_{2} - 34 \beta_{4} ) q^{47} + ( 156 \beta_{1} + 8 \beta_{2} - 36 \beta_{3} + 132 \beta_{4} - 72 \beta_{6} - 36 \beta_{7} ) q^{48} + ( -1001 + 5 \beta_{3} + 5 \beta_{7} ) q^{49} + ( -1560 - 235 \beta_{1} - 40 \beta_{2} + 80 \beta_{3} - 20 \beta_{4} + 10 \beta_{5} + 40 \beta_{6} - 10 \beta_{7} ) q^{50} + ( 24 \beta_{3} + 16 \beta_{5} - 8 \beta_{7} ) q^{51} + ( 342 \beta_{1} - 12 \beta_{2} - 10 \beta_{3} - 70 \beta_{4} - 20 \beta_{6} - 10 \beta_{7} ) q^{52} + ( -502 \beta_{1} - 59 \beta_{3} - 124 \beta_{4} - 118 \beta_{6} - 59 \beta_{7} ) q^{53} + ( -1440 - 144 \beta_{3} + 6 \beta_{5} + 6 \beta_{7} ) q^{54} + ( -468 \beta_{1} - 168 \beta_{2} + 108 \beta_{3} + 156 \beta_{4} + 4 \beta_{5} - 36 \beta_{7} ) q^{55} + ( -1840 + 32 \beta_{3} + 36 \beta_{5} - 4 \beta_{7} ) q^{56} + ( 132 \beta_{1} + 84 \beta_{3} + 60 \beta_{4} + 168 \beta_{6} + 84 \beta_{7} ) q^{57} + ( 330 \beta_{1} + 16 \beta_{2} - 8 \beta_{3} + 8 \beta_{4} - 16 \beta_{6} - 8 \beta_{7} ) q^{58} + ( -216 \beta_{3} + 28 \beta_{5} + 72 \beta_{7} ) q^{59} + ( 2640 - 207 \beta_{1} + 78 \beta_{2} + 57 \beta_{3} - 81 \beta_{4} - 64 \beta_{5} - 30 \beta_{6} - 79 \beta_{7} ) q^{60} + ( 822 + 131 \beta_{3} + 131 \beta_{7} ) q^{61} + ( 48 \beta_{1} + 224 \beta_{2} + 16 \beta_{3} - 80 \beta_{4} + 32 \beta_{6} + 16 \beta_{7} ) q^{62} + ( 54 \beta_{1} + 279 \beta_{2} - 18 \beta_{4} ) q^{63} + ( 864 - 80 \beta_{3} - 96 \beta_{5} - 32 \beta_{7} ) q^{64} + ( -240 + 126 \beta_{1} - 41 \beta_{3} + 40 \beta_{4} + 74 \beta_{6} - 41 \beta_{7} ) q^{65} + ( 5760 - 288 \beta_{3} + 24 \beta_{5} - 72 \beta_{7} ) q^{66} + ( -300 \beta_{1} + 139 \beta_{2} + 100 \beta_{4} ) q^{67} + ( -316 \beta_{1} + 56 \beta_{2} + 4 \beta_{3} + 92 \beta_{4} + 8 \beta_{6} + 4 \beta_{7} ) q^{68} + ( -1560 + 171 \beta_{3} + 171 \beta_{7} ) q^{69} + ( -800 - 240 \beta_{2} - 120 \beta_{3} - 40 \beta_{4} - 35 \beta_{5} - 80 \beta_{6} - 75 \beta_{7} ) q^{70} + ( 216 \beta_{3} - 120 \beta_{5} - 72 \beta_{7} ) q^{71} + ( -681 \beta_{1} - 414 \beta_{2} + 15 \beta_{3} - 51 \beta_{4} + 30 \beta_{6} + 15 \beta_{7} ) q^{72} + ( -108 \beta_{1} + 144 \beta_{3} + 36 \beta_{4} + 288 \beta_{6} + 144 \beta_{7} ) q^{73} + ( -2952 + 180 \beta_{3} - 42 \beta_{5} + 126 \beta_{7} ) q^{74} + ( 360 \beta_{1} + 415 \beta_{2} - 120 \beta_{4} + 120 \beta_{5} ) q^{75} + ( -6000 - 216 \beta_{3} + 32 \beta_{5} - 160 \beta_{7} ) q^{76} + ( 180 \beta_{1} - 140 \beta_{3} - 20 \beta_{4} - 280 \beta_{6} - 140 \beta_{7} ) q^{77} + ( -288 \beta_{2} - 48 \beta_{3} - 48 \beta_{4} - 96 \beta_{6} - 48 \beta_{7} ) q^{78} + ( -24 \beta_{3} + 96 \beta_{5} + 8 \beta_{7} ) q^{79} + ( 2440 + 820 \beta_{1} + 344 \beta_{2} + 80 \beta_{3} + 12 \beta_{4} + 28 \beta_{5} + 104 \beta_{6} + 152 \beta_{7} ) q^{80} + ( -2439 + 9 \beta_{3} + 9 \beta_{7} ) q^{81} + ( 24 \beta_{1} + 316 \beta_{2} - 158 \beta_{3} + 158 \beta_{4} - 316 \beta_{6} - 158 \beta_{7} ) q^{82} + ( -204 \beta_{1} - 693 \beta_{2} + 68 \beta_{4} ) q^{83} + ( 3120 - 120 \beta_{3} + 120 \beta_{5} - 120 \beta_{7} ) q^{84} + ( 1120 - 216 \beta_{1} + 26 \beta_{3} - 60 \beta_{4} - 84 \beta_{6} + 26 \beta_{7} ) q^{85} + ( 4640 + 464 \beta_{3} + 77 \beta_{5} + 77 \beta_{7} ) q^{86} + ( -144 \beta_{1} - 402 \beta_{2} + 48 \beta_{4} ) q^{87} + ( 504 \beta_{1} + 144 \beta_{2} + 184 \beta_{3} - 344 \beta_{4} + 368 \beta_{6} + 184 \beta_{7} ) q^{88} + ( 3458 - 340 \beta_{3} - 340 \beta_{7} ) q^{89} + ( -6060 + 756 \beta_{1} + 18 \beta_{2} + 309 \beta_{3} + 9 \beta_{4} - 39 \beta_{5} - 18 \beta_{6} + 108 \beta_{7} ) q^{90} + ( -216 \beta_{3} - 8 \beta_{5} + 72 \beta_{7} ) q^{91} + ( 753 \beta_{1} - 306 \beta_{2} + 49 \beta_{3} + 303 \beta_{4} + 98 \beta_{6} + 49 \beta_{7} ) q^{92} + ( 1488 \beta_{1} - 144 \beta_{3} + 240 \beta_{4} - 288 \beta_{6} - 144 \beta_{7} ) q^{93} + ( 1360 + 136 \beta_{3} + 7 \beta_{5} + 7 \beta_{7} ) q^{94} + ( 1044 \beta_{1} - 536 \beta_{2} - 324 \beta_{3} - 348 \beta_{4} - 132 \beta_{5} + 108 \beta_{7} ) q^{95} + ( -9600 + 96 \beta_{3} - 80 \beta_{5} + 208 \beta_{7} ) q^{96} + ( -1224 \beta_{1} + 22 \beta_{3} - 236 \beta_{4} + 44 \beta_{6} + 22 \beta_{7} ) q^{97} + ( -991 \beta_{1} - 20 \beta_{2} + 10 \beta_{3} - 10 \beta_{4} + 20 \beta_{6} + 10 \beta_{7} ) q^{98} + ( 648 \beta_{3} - 252 \beta_{5} - 216 \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 48q^{4} - 40q^{5} + 312q^{9} + O(q^{10})$$ $$8q - 48q^{4} - 40q^{5} + 312q^{9} - 160q^{10} + 640q^{14} - 832q^{16} - 400q^{20} - 960q^{21} + 1920q^{24} - 2040q^{25} + 2496q^{26} + 2704q^{29} - 1920q^{30} - 2176q^{34} - 5712q^{36} + 6400q^{40} + 1456q^{41} + 1920q^{44} + 6120q^{45} + 7040q^{46} - 8008q^{49} - 12480q^{50} - 11520q^{54} - 14720q^{56} + 21120q^{60} + 6576q^{61} + 6912q^{64} - 1920q^{65} + 46080q^{66} - 12480q^{69} - 6400q^{70} - 23616q^{74} - 48000q^{76} + 19520q^{80} - 19512q^{81} + 24960q^{84} + 8960q^{85} + 37120q^{86} + 27664q^{89} - 48480q^{90} + 10880q^{94} - 76800q^{96} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 6 x^{6} + 31 x^{4} + 96 x^{2} + 256$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$2 \nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{7} + 10 \nu^{5} + \nu^{3} + 80 \nu$$$$)/32$$ $$\beta_{3}$$ $$=$$ $$4 \nu^{2} + 6$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{7} + 6 \nu^{5} + 31 \nu^{3} + 80 \nu$$$$)/8$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{6} + 2 \nu^{4} + \nu^{2} + 28$$$$)/2$$ $$\beta_{6}$$ $$=$$ $$($$$$-7 \nu^{7} + 8 \nu^{6} - 26 \nu^{5} + 48 \nu^{4} - 57 \nu^{3} + 120 \nu^{2} - 176 \nu + 384$$$$)/32$$ $$\beta_{7}$$ $$=$$ $$($$$$-\nu^{6} - 6 \nu^{4} - 23 \nu^{2} - 60$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$$$/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} - 6$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{7} + 2 \beta_{6} + 3 \beta_{4} + \beta_{3} - 2 \beta_{2} - 7 \beta_{1}$$$$)/8$$ $$\nu^{4}$$ $$=$$ $$($$$$-\beta_{7} + \beta_{5} - 3 \beta_{3} - 26$$$$)/4$$ $$\nu^{5}$$ $$=$$ $$($$$$-\beta_{7} - 2 \beta_{6} - \beta_{4} - \beta_{3} + 10 \beta_{2} - 13 \beta_{1}$$$$)/4$$ $$\nu^{6}$$ $$=$$ $$($$$$-2 \beta_{7} - 6 \beta_{5} - 5 \beta_{3} + 54$$$$)/4$$ $$\nu^{7}$$ $$=$$ $$($$$$-19 \beta_{7} - 38 \beta_{6} - 17 \beta_{4} - 19 \beta_{3} - 58 \beta_{2} + 53 \beta_{1}$$$$)/8$$

## Character Values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/20\mathbb{Z}\right)^\times$$.

 $$n$$ $$11$$ $$17$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
19.1
 −1.42849 − 1.39980i −1.42849 + 1.39980i −0.677813 − 1.88164i −0.677813 + 1.88164i 0.677813 − 1.88164i 0.677813 + 1.88164i 1.42849 − 1.39980i 1.42849 + 1.39980i
−2.85697 2.79959i −6.64118 0.324555 + 15.9967i −17.6491 + 17.7062i 18.9737 + 18.5926i −39.0703 43.8570 46.6107i −36.8947 99.9931 1.17570i
19.2 −2.85697 + 2.79959i −6.64118 0.324555 15.9967i −17.6491 17.7062i 18.9737 18.5926i −39.0703 43.8570 + 46.6107i −36.8947 99.9931 + 1.17570i
19.3 −1.35563 3.76328i 13.9962 −12.3246 + 10.2032i 7.64911 23.8011i −18.9737 52.6718i −35.6863 55.1050 + 32.5490i 114.895 −99.9394 + 3.47961i
19.4 −1.35563 + 3.76328i 13.9962 −12.3246 10.2032i 7.64911 + 23.8011i −18.9737 + 52.6718i −35.6863 55.1050 32.5490i 114.895 −99.9394 3.47961i
19.5 1.35563 3.76328i −13.9962 −12.3246 10.2032i 7.64911 23.8011i −18.9737 + 52.6718i 35.6863 −55.1050 + 32.5490i 114.895 −79.2008 61.0511i
19.6 1.35563 + 3.76328i −13.9962 −12.3246 + 10.2032i 7.64911 + 23.8011i −18.9737 52.6718i 35.6863 −55.1050 32.5490i 114.895 −79.2008 + 61.0511i
19.7 2.85697 2.79959i 6.64118 0.324555 15.9967i −17.6491 + 17.7062i 18.9737 18.5926i 39.0703 −43.8570 46.6107i −36.8947 −0.852871 + 99.9964i
19.8 2.85697 + 2.79959i 6.64118 0.324555 + 15.9967i −17.6491 17.7062i 18.9737 + 18.5926i 39.0703 −43.8570 + 46.6107i −36.8947 −0.852871 99.9964i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 19.8 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
4.b Odd 1 yes
5.b Even 1 yes
20.d Odd 1 yes

## Hecke kernels

This newform can be constructed as the kernel of the linear operator $$T_{3}^{4} - 240 T_{3}^{2} + 8640$$ acting on $$S_{5}^{\mathrm{new}}(20, [\chi])$$.