# Properties

 Label 20.5.f.a Level 20 Weight 5 Character orbit 20.f Analytic conductor 2.067 Analytic rank 0 Dimension 4 CM No Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$20 = 2^{2} \cdot 5$$ Weight: $$k$$ = $$5$$ Character orbit: $$[\chi]$$ = 20.f (of order $$4$$ and degree $$2$$)

## Newform invariants

 Self dual: No Analytic conductor: $$2.06739926168$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(i, \sqrt{241})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q$$ $$+ ( -2 + 2 \beta_{1} + \beta_{3} ) q^{3}$$ $$+ ( -6 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{5}$$ $$+ ( 26 + 29 \beta_{1} + 3 \beta_{2} ) q^{7}$$ $$+ ( -52 \beta_{1} - 5 \beta_{2} - 5 \beta_{3} ) q^{9}$$ $$+O(q^{10})$$ $$q$$ $$+ ( -2 + 2 \beta_{1} + \beta_{3} ) q^{3}$$ $$+ ( -6 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{5}$$ $$+ ( 26 + 29 \beta_{1} + 3 \beta_{2} ) q^{7}$$ $$+ ( -52 \beta_{1} - 5 \beta_{2} - 5 \beta_{3} ) q^{9}$$ $$+ ( -80 + 5 \beta_{1} + 5 \beta_{2} - 5 \beta_{3} ) q^{11}$$ $$+ ( -93 + 93 \beta_{1} - 6 \beta_{3} ) q^{13}$$ $$+ ( 130 - 228 \beta_{1} + 2 \beta_{2} - 9 \beta_{3} ) q^{15}$$ $$+ ( 247 + 233 \beta_{1} - 14 \beta_{2} ) q^{17}$$ $$+ ( -62 \beta_{1} + 30 \beta_{2} + 30 \beta_{3} ) q^{19}$$ $$+ ( -464 - 35 \beta_{1} - 35 \beta_{2} + 35 \beta_{3} ) q^{21}$$ $$+ ( -194 + 194 \beta_{1} + 17 \beta_{3} ) q^{23}$$ $$+ ( 455 - 339 \beta_{1} + 21 \beta_{2} + 3 \beta_{3} ) q^{25}$$ $$+ ( 532 + 528 \beta_{1} - 4 \beta_{2} ) q^{27}$$ $$+ ( -552 \beta_{1} - 40 \beta_{2} - 40 \beta_{3} ) q^{29}$$ $$+ ( -284 + 75 \beta_{1} + 75 \beta_{2} - 75 \beta_{3} ) q^{31}$$ $$+ ( -440 + 440 \beta_{1} - 50 \beta_{3} ) q^{33}$$ $$+ ( -590 - 577 \beta_{1} - 87 \beta_{2} + 14 \beta_{3} ) q^{35}$$ $$+ ( -219 - 111 \beta_{1} + 108 \beta_{2} ) q^{37}$$ $$+ ( 273 \beta_{1} - 75 \beta_{2} - 75 \beta_{3} ) q^{39}$$ $$+ ( 736 + 5 \beta_{1} + 5 \beta_{2} - 5 \beta_{3} ) q^{41}$$ $$+ ( 870 - 870 \beta_{1} + 105 \beta_{3} ) q^{43}$$ $$+ ( 365 + 1889 \beta_{1} + 89 \beta_{2} + 82 \beta_{3} ) q^{45}$$ $$+ ( -498 - 627 \beta_{1} - 129 \beta_{2} ) q^{47}$$ $$+ ( 196 \beta_{1} + 165 \beta_{2} + 165 \beta_{3} ) q^{49}$$ $$+ ( 692 - 205 \beta_{1} - 205 \beta_{2} + 205 \beta_{3} ) q^{51}$$ $$+ ( 509 - 509 \beta_{1} + 28 \beta_{3} ) q^{53}$$ $$+ ( -1800 + 1045 \beta_{1} + 45 \beta_{2} - 165 \beta_{3} ) q^{55}$$ $$+ ( -3416 - 3504 \beta_{1} - 88 \beta_{2} ) q^{57}$$ $$+ ( 4626 \beta_{1} + 110 \beta_{2} + 110 \beta_{3} ) q^{59}$$ $$+ ( 3792 + 165 \beta_{1} + 165 \beta_{2} - 165 \beta_{3} ) q^{61}$$ $$+ ( 3022 - 3022 \beta_{1} - 431 \beta_{3} ) q^{63}$$ $$+ ( -255 + 1788 \beta_{1} - 117 \beta_{2} - 261 \beta_{3} ) q^{65}$$ $$+ ( -1234 - 1171 \beta_{1} + 63 \beta_{2} ) q^{67}$$ $$+ ( -3061 \beta_{1} - 245 \beta_{2} - 245 \beta_{3} ) q^{69}$$ $$+ ( -1612 + 115 \beta_{1} + 115 \beta_{2} - 115 \beta_{3} ) q^{71}$$ $$+ ( -3383 + 3383 \beta_{1} + 264 \beta_{3} ) q^{73}$$ $$+ ( -2710 + 1558 \beta_{1} + 288 \beta_{2} + 509 \beta_{3} ) q^{75}$$ $$+ ( -280 - 230 \beta_{1} + 50 \beta_{2} ) q^{77}$$ $$+ ( -336 \beta_{1} - 240 \beta_{2} - 240 \beta_{3} ) q^{79}$$ $$+ ( 2159 - 115 \beta_{1} - 115 \beta_{2} + 115 \beta_{3} ) q^{81}$$ $$+ ( 5226 - 5226 \beta_{1} + 477 \beta_{3} ) q^{83}$$ $$+ ( 4595 - 254 \beta_{1} - 699 \beta_{2} + 303 \beta_{3} ) q^{85}$$ $$+ ( 5824 + 6576 \beta_{1} + 752 \beta_{2} ) q^{87}$$ $$+ ( 72 \beta_{1} + 280 \beta_{2} + 280 \beta_{3} ) q^{89}$$ $$+ ( -2676 - 105 \beta_{1} - 105 \beta_{2} + 105 \beta_{3} ) q^{91}$$ $$+ ( -8432 + 8432 \beta_{1} + 166 \beta_{3} ) q^{93}$$ $$+ ( -4060 - 10586 \beta_{1} + 214 \beta_{2} - 118 \beta_{3} ) q^{95}$$ $$+ ( -279 - 1311 \beta_{1} - 1032 \beta_{2} ) q^{97}$$ $$+ ( -2125 \beta_{1} + 115 \beta_{2} + 115 \beta_{3} ) q^{99}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q$$ $$\mathstrut -\mathstrut 10q^{3}$$ $$\mathstrut -\mathstrut 6q^{5}$$ $$\mathstrut +\mathstrut 110q^{7}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$4q$$ $$\mathstrut -\mathstrut 10q^{3}$$ $$\mathstrut -\mathstrut 6q^{5}$$ $$\mathstrut +\mathstrut 110q^{7}$$ $$\mathstrut -\mathstrut 300q^{11}$$ $$\mathstrut -\mathstrut 360q^{13}$$ $$\mathstrut +\mathstrut 542q^{15}$$ $$\mathstrut +\mathstrut 960q^{17}$$ $$\mathstrut -\mathstrut 1996q^{21}$$ $$\mathstrut -\mathstrut 810q^{23}$$ $$\mathstrut +\mathstrut 1856q^{25}$$ $$\mathstrut +\mathstrut 2120q^{27}$$ $$\mathstrut -\mathstrut 836q^{31}$$ $$\mathstrut -\mathstrut 1660q^{33}$$ $$\mathstrut -\mathstrut 2562q^{35}$$ $$\mathstrut -\mathstrut 660q^{37}$$ $$\mathstrut +\mathstrut 2964q^{41}$$ $$\mathstrut +\mathstrut 3270q^{43}$$ $$\mathstrut +\mathstrut 1474q^{45}$$ $$\mathstrut -\mathstrut 2250q^{47}$$ $$\mathstrut +\mathstrut 1948q^{51}$$ $$\mathstrut +\mathstrut 1980q^{53}$$ $$\mathstrut -\mathstrut 6780q^{55}$$ $$\mathstrut -\mathstrut 13840q^{57}$$ $$\mathstrut +\mathstrut 15828q^{61}$$ $$\mathstrut +\mathstrut 12950q^{63}$$ $$\mathstrut -\mathstrut 732q^{65}$$ $$\mathstrut -\mathstrut 4810q^{67}$$ $$\mathstrut -\mathstrut 5988q^{71}$$ $$\mathstrut -\mathstrut 14060q^{73}$$ $$\mathstrut -\mathstrut 11282q^{75}$$ $$\mathstrut -\mathstrut 1020q^{77}$$ $$\mathstrut +\mathstrut 8176q^{81}$$ $$\mathstrut +\mathstrut 19950q^{83}$$ $$\mathstrut +\mathstrut 16376q^{85}$$ $$\mathstrut +\mathstrut 24800q^{87}$$ $$\mathstrut -\mathstrut 11124q^{91}$$ $$\mathstrut -\mathstrut 34060q^{93}$$ $$\mathstrut -\mathstrut 15576q^{95}$$ $$\mathstrut -\mathstrut 3180q^{97}$$ $$\mathstrut +\mathstrut O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4}\mathstrut +\mathstrut$$ $$121$$ $$x^{2}\mathstrut +\mathstrut$$ $$3600$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} + 61 \nu$$$$)/60$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} + \nu + 61$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{3} - 60 \nu^{2} + 121 \nu - 3660$$$$)/60$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3}\mathstrut +\mathstrut$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$\beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$-$$$$\beta_{3}\mathstrut +\mathstrut$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$122$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-$$$$61$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$61$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$181$$ $$\beta_{1}$$$$)/2$$

## Character Values

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

 $$n$$ $$11$$ $$17$$ $$\chi(n)$$ $$1$$ $$\beta_{1}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
13.1
 − 7.26209i 8.26209i 7.26209i − 8.26209i
0 −10.2621 10.2621i 0 −24.7863 3.26209i 0 50.7863 50.7863i 0 129.621i 0
13.2 0 5.26209 + 5.26209i 0 21.7863 + 12.2621i 0 4.21374 4.21374i 0 25.6209i 0
17.1 0 −10.2621 + 10.2621i 0 −24.7863 + 3.26209i 0 50.7863 + 50.7863i 0 129.621i 0
17.2 0 5.26209 5.26209i 0 21.7863 12.2621i 0 4.21374 + 4.21374i 0 25.6209i 0
 $$n$$: 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
5.c Odd 1 yes

## Hecke kernels

There are no other newforms in $$S_{5}^{\mathrm{new}}(20, [\chi])$$.