# Properties

 Label 20.3.b.a Level 20 Weight 3 Character orbit 20.b Analytic conductor 0.545 Analytic rank 0 Dimension 4 CM No Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$20 = 2^{2} \cdot 5$$ Weight: $$k$$ = $$3$$ Character orbit: $$[\chi]$$ = 20.b (of order $$2$$ and degree $$1$$)

## Newform invariants

 Self dual: No Analytic conductor: $$0.544960528721$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{10})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{4}$$ Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q$$ $$-\beta_{2} q^{2}$$ $$+ \beta_{3} q^{3}$$ $$+ ( -2 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{4}$$ $$+ ( -1 - \beta_{1} + \beta_{2} ) q^{5}$$ $$+ ( 2 + 3 \beta_{1} - \beta_{2} - \beta_{3} ) q^{6}$$ $$+ ( 2 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{7}$$ $$+ ( -4 - 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{8}$$ $$+ ( 1 + 2 \beta_{1} - 2 \beta_{2} ) q^{9}$$ $$+O(q^{10})$$ $$q$$ $$-\beta_{2} q^{2}$$ $$+ \beta_{3} q^{3}$$ $$+ ( -2 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{4}$$ $$+ ( -1 - \beta_{1} + \beta_{2} ) q^{5}$$ $$+ ( 2 + 3 \beta_{1} - \beta_{2} - \beta_{3} ) q^{6}$$ $$+ ( 2 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{7}$$ $$+ ( -4 - 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{8}$$ $$+ ( 1 + 2 \beta_{1} - 2 \beta_{2} ) q^{9}$$ $$+ ( -2 + \beta_{1} + \beta_{3} ) q^{10}$$ $$+ ( -4 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} ) q^{11}$$ $$+ ( 8 - 4 \beta_{1} ) q^{12}$$ $$+ ( -6 - 2 \beta_{1} + 2 \beta_{2} ) q^{13}$$ $$+ ( 10 - \beta_{1} - \beta_{2} + 3 \beta_{3} ) q^{14}$$ $$+ ( -2 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{15}$$ $$+ 8 \beta_{1} q^{16}$$ $$+ ( 2 + 8 \beta_{1} - 8 \beta_{2} ) q^{17}$$ $$+ ( 4 - 2 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{18}$$ $$+ ( 4 \beta_{1} + 4 \beta_{2} ) q^{19}$$ $$+ ( 6 + 3 \beta_{1} + \beta_{2} - \beta_{3} ) q^{20}$$ $$+ ( -10 \beta_{1} + 10 \beta_{2} ) q^{21}$$ $$+ ( -28 - 10 \beta_{1} + 6 \beta_{2} - 2 \beta_{3} ) q^{22}$$ $$+ ( 2 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} ) q^{23}$$ $$+ ( -16 - 8 \beta_{2} ) q^{24}$$ $$+ 5 q^{25}$$ $$+ ( -4 + 2 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} ) q^{26}$$ $$+ ( 4 \beta_{1} + 4 \beta_{2} + 6 \beta_{3} ) q^{27}$$ $$+ ( 8 \beta_{1} - 12 \beta_{2} - 4 \beta_{3} ) q^{28}$$ $$+ ( -6 - 4 \beta_{1} + 4 \beta_{2} ) q^{29}$$ $$+ ( -10 + \beta_{1} + \beta_{2} - 3 \beta_{3} ) q^{30}$$ $$+ ( -4 \beta_{1} - 4 \beta_{2} + 10 \beta_{3} ) q^{31}$$ $$+ 32 q^{32}$$ $$+ ( 32 + 12 \beta_{1} - 12 \beta_{2} ) q^{33}$$ $$+ ( 16 - 8 \beta_{1} + 6 \beta_{2} - 8 \beta_{3} ) q^{34}$$ $$-5 \beta_{3} q^{35}$$ $$+ ( -10 - 5 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{36}$$ $$+ ( -6 - 10 \beta_{1} + 10 \beta_{2} ) q^{37}$$ $$+ ( 24 + 4 \beta_{1} - 4 \beta_{2} + 4 \beta_{3} ) q^{38}$$ $$+ ( -4 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} ) q^{39}$$ $$+ ( 12 - 2 \beta_{1} - 6 \beta_{2} + 2 \beta_{3} ) q^{40}$$ $$+ ( -22 + 6 \beta_{1} - 6 \beta_{2} ) q^{41}$$ $$+ ( -20 + 10 \beta_{1} - 10 \beta_{2} + 10 \beta_{3} ) q^{42}$$ $$+ ( -4 \beta_{1} - 4 \beta_{2} - 3 \beta_{3} ) q^{43}$$ $$+ ( -32 + 24 \beta_{2} + 8 \beta_{3} ) q^{44}$$ $$+ ( -9 + \beta_{1} - \beta_{2} ) q^{45}$$ $$+ ( 6 - 7 \beta_{1} + \beta_{2} + 5 \beta_{3} ) q^{46}$$ $$+ ( -6 \beta_{1} - 6 \beta_{2} - 13 \beta_{3} ) q^{47}$$ $$+ ( -16 - 8 \beta_{1} + 24 \beta_{2} - 8 \beta_{3} ) q^{48}$$ $$+ ( 9 + 10 \beta_{1} - 10 \beta_{2} ) q^{49}$$ $$-5 \beta_{2} q^{50}$$ $$+ ( 16 \beta_{1} + 16 \beta_{2} - 14 \beta_{3} ) q^{51}$$ $$+ ( 20 + 10 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{52}$$ $$+ ( -54 - 10 \beta_{1} + 10 \beta_{2} ) q^{53}$$ $$+ ( 36 + 22 \beta_{1} - 10 \beta_{2} - 2 \beta_{3} ) q^{54}$$ $$+ ( 8 \beta_{1} + 8 \beta_{2} + 6 \beta_{3} ) q^{55}$$ $$+ ( -24 \beta_{1} + 16 \beta_{2} - 8 \beta_{3} ) q^{56}$$ $$+ ( -16 - 16 \beta_{1} + 16 \beta_{2} ) q^{57}$$ $$+ ( -8 + 4 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} ) q^{58}$$ $$+ ( -12 \beta_{1} - 12 \beta_{2} + 12 \beta_{3} ) q^{59}$$ $$+ ( -8 \beta_{1} + 12 \beta_{2} + 4 \beta_{3} ) q^{60}$$ $$+ ( 58 + 26 \beta_{1} - 26 \beta_{2} ) q^{61}$$ $$+ ( -4 + 26 \beta_{1} - 6 \beta_{2} - 14 \beta_{3} ) q^{62}$$ $$+ ( -2 \beta_{1} - 2 \beta_{2} + 11 \beta_{3} ) q^{63}$$ $$-32 \beta_{2} q^{64}$$ $$+ ( 14 + 4 \beta_{1} - 4 \beta_{2} ) q^{65}$$ $$+ ( 24 - 12 \beta_{1} - 20 \beta_{2} - 12 \beta_{3} ) q^{66}$$ $$+ ( -20 \beta_{1} - 20 \beta_{2} + 11 \beta_{3} ) q^{67}$$ $$+ ( -36 - 18 \beta_{1} - 14 \beta_{2} + 14 \beta_{3} ) q^{68}$$ $$+ ( 16 - 14 \beta_{1} + 14 \beta_{2} ) q^{69}$$ $$+ ( -10 - 15 \beta_{1} + 5 \beta_{2} + 5 \beta_{3} ) q^{70}$$ $$+ ( 20 \beta_{1} + 20 \beta_{2} + 2 \beta_{3} ) q^{71}$$ $$+ ( -20 + 6 \beta_{1} + 10 \beta_{2} - 6 \beta_{3} ) q^{72}$$ $$+ ( 34 - 32 \beta_{1} + 32 \beta_{2} ) q^{73}$$ $$+ ( -20 + 10 \beta_{1} - 4 \beta_{2} + 10 \beta_{3} ) q^{74}$$ $$+ 5 \beta_{3} q^{75}$$ $$+ ( 16 + 8 \beta_{1} - 24 \beta_{2} - 8 \beta_{3} ) q^{76}$$ $$+ ( 80 + 20 \beta_{1} - 20 \beta_{2} ) q^{77}$$ $$+ ( -28 - 10 \beta_{1} + 6 \beta_{2} - 2 \beta_{3} ) q^{78}$$ $$+ ( 16 \beta_{1} + 16 \beta_{2} - 20 \beta_{3} ) q^{79}$$ $$+ ( -16 - 8 \beta_{2} - 8 \beta_{3} ) q^{80}$$ $$+ ( -55 + 14 \beta_{1} - 14 \beta_{2} ) q^{81}$$ $$+ ( 12 - 6 \beta_{1} + 28 \beta_{2} - 6 \beta_{3} ) q^{82}$$ $$+ ( 12 \beta_{1} + 12 \beta_{2} + 13 \beta_{3} ) q^{83}$$ $$+ ( 40 + 20 \beta_{1} + 20 \beta_{2} - 20 \beta_{3} ) q^{84}$$ $$+ ( -34 + 6 \beta_{1} - 6 \beta_{2} ) q^{85}$$ $$+ ( -30 - 13 \beta_{1} + 7 \beta_{2} - \beta_{3} ) q^{86}$$ $$+ ( -8 \beta_{1} - 8 \beta_{2} + 2 \beta_{3} ) q^{87}$$ $$+ ( 64 + 48 \beta_{1} + 16 \beta_{3} ) q^{88}$$ $$+ ( -62 - 40 \beta_{1} + 40 \beta_{2} ) q^{89}$$ $$+ ( 2 - \beta_{1} + 10 \beta_{2} - \beta_{3} ) q^{90}$$ $$+ ( -8 \beta_{1} - 8 \beta_{2} - 6 \beta_{3} ) q^{91}$$ $$+ ( -16 + 16 \beta_{1} - 12 \beta_{2} - 4 \beta_{3} ) q^{92}$$ $$+ ( -64 + 36 \beta_{1} - 36 \beta_{2} ) q^{93}$$ $$+ ( -62 - 45 \beta_{1} + 19 \beta_{2} + 7 \beta_{3} ) q^{94}$$ $$+ ( -4 \beta_{1} - 4 \beta_{2} - 8 \beta_{3} ) q^{95}$$ $$+ 32 \beta_{3} q^{96}$$ $$+ ( -78 - 12 \beta_{1} + 12 \beta_{2} ) q^{97}$$ $$+ ( 20 - 10 \beta_{1} + \beta_{2} - 10 \beta_{3} ) q^{98}$$ $$+ ( -12 \beta_{1} - 12 \beta_{2} - 10 \beta_{3} ) q^{99}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q$$ $$\mathstrut -\mathstrut 2q^{2}$$ $$\mathstrut -\mathstrut 4q^{4}$$ $$\mathstrut -\mathstrut 8q^{8}$$ $$\mathstrut -\mathstrut 4q^{9}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$4q$$ $$\mathstrut -\mathstrut 2q^{2}$$ $$\mathstrut -\mathstrut 4q^{4}$$ $$\mathstrut -\mathstrut 8q^{8}$$ $$\mathstrut -\mathstrut 4q^{9}$$ $$\mathstrut -\mathstrut 10q^{10}$$ $$\mathstrut +\mathstrut 40q^{12}$$ $$\mathstrut -\mathstrut 16q^{13}$$ $$\mathstrut +\mathstrut 40q^{14}$$ $$\mathstrut -\mathstrut 16q^{16}$$ $$\mathstrut -\mathstrut 24q^{17}$$ $$\mathstrut +\mathstrut 22q^{18}$$ $$\mathstrut +\mathstrut 20q^{20}$$ $$\mathstrut +\mathstrut 40q^{21}$$ $$\mathstrut -\mathstrut 80q^{22}$$ $$\mathstrut -\mathstrut 80q^{24}$$ $$\mathstrut +\mathstrut 20q^{25}$$ $$\mathstrut -\mathstrut 12q^{26}$$ $$\mathstrut -\mathstrut 40q^{28}$$ $$\mathstrut -\mathstrut 8q^{29}$$ $$\mathstrut -\mathstrut 40q^{30}$$ $$\mathstrut +\mathstrut 128q^{32}$$ $$\mathstrut +\mathstrut 80q^{33}$$ $$\mathstrut +\mathstrut 92q^{34}$$ $$\mathstrut -\mathstrut 36q^{36}$$ $$\mathstrut +\mathstrut 16q^{37}$$ $$\mathstrut +\mathstrut 80q^{38}$$ $$\mathstrut +\mathstrut 40q^{40}$$ $$\mathstrut -\mathstrut 112q^{41}$$ $$\mathstrut -\mathstrut 120q^{42}$$ $$\mathstrut -\mathstrut 80q^{44}$$ $$\mathstrut -\mathstrut 40q^{45}$$ $$\mathstrut +\mathstrut 40q^{46}$$ $$\mathstrut -\mathstrut 4q^{49}$$ $$\mathstrut -\mathstrut 10q^{50}$$ $$\mathstrut +\mathstrut 56q^{52}$$ $$\mathstrut -\mathstrut 176q^{53}$$ $$\mathstrut +\mathstrut 80q^{54}$$ $$\mathstrut +\mathstrut 80q^{56}$$ $$\mathstrut -\mathstrut 36q^{58}$$ $$\mathstrut +\mathstrut 40q^{60}$$ $$\mathstrut +\mathstrut 128q^{61}$$ $$\mathstrut -\mathstrut 80q^{62}$$ $$\mathstrut -\mathstrut 64q^{64}$$ $$\mathstrut +\mathstrut 40q^{65}$$ $$\mathstrut +\mathstrut 80q^{66}$$ $$\mathstrut -\mathstrut 136q^{68}$$ $$\mathstrut +\mathstrut 120q^{69}$$ $$\mathstrut -\mathstrut 72q^{72}$$ $$\mathstrut +\mathstrut 264q^{73}$$ $$\mathstrut -\mathstrut 108q^{74}$$ $$\mathstrut +\mathstrut 240q^{77}$$ $$\mathstrut -\mathstrut 80q^{78}$$ $$\mathstrut -\mathstrut 80q^{80}$$ $$\mathstrut -\mathstrut 276q^{81}$$ $$\mathstrut +\mathstrut 116q^{82}$$ $$\mathstrut +\mathstrut 160q^{84}$$ $$\mathstrut -\mathstrut 160q^{85}$$ $$\mathstrut -\mathstrut 80q^{86}$$ $$\mathstrut +\mathstrut 160q^{88}$$ $$\mathstrut -\mathstrut 88q^{89}$$ $$\mathstrut +\mathstrut 30q^{90}$$ $$\mathstrut -\mathstrut 120q^{92}$$ $$\mathstrut -\mathstrut 400q^{93}$$ $$\mathstrut -\mathstrut 120q^{94}$$ $$\mathstrut -\mathstrut 264q^{97}$$ $$\mathstrut +\mathstrut 102q^{98}$$ $$\mathstrut +\mathstrut O(q^{100})$$

Basis of coefficient ring:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$2 \zeta_{10}^{2}$$ $$\beta_{2}$$ $$=$$ $$2 \zeta_{10}^{3}$$ $$\beta_{3}$$ $$=$$ $$2 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 4 \zeta_{10} - 2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\zeta_{10}$$ $$=$$ $$($$$$\beta_{3}\mathstrut -\mathstrut$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$2$$$$)/4$$ $$\zeta_{10}^{2}$$ $$=$$ $$\beta_{1}$$$$/2$$ $$\zeta_{10}^{3}$$ $$=$$ $$\beta_{2}$$$$/2$$

## Character Values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
11.1
 −0.309017 − 0.951057i −0.309017 + 0.951057i 0.809017 + 0.587785i 0.809017 − 0.587785i
−1.61803 1.17557i 3.80423i 1.23607 + 3.80423i 2.23607 −4.47214 + 6.15537i 8.50651i 2.47214 7.60845i −5.47214 −3.61803 2.62866i
11.2 −1.61803 + 1.17557i 3.80423i 1.23607 3.80423i 2.23607 −4.47214 6.15537i 8.50651i 2.47214 + 7.60845i −5.47214 −3.61803 + 2.62866i
11.3 0.618034 1.90211i 2.35114i −3.23607 2.35114i −2.23607 4.47214 + 1.45309i 5.25731i −6.47214 + 4.70228i 3.47214 −1.38197 + 4.25325i
11.4 0.618034 + 1.90211i 2.35114i −3.23607 + 2.35114i −2.23607 4.47214 1.45309i 5.25731i −6.47214 4.70228i 3.47214 −1.38197 4.25325i
 $$n$$: 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
4.b Odd 1 yes

## Hecke kernels

There are no other newforms in $$S_{3}^{\mathrm{new}}(20, [\chi])$$.