# Properties

 Label 20.10.c.a Level 20 Weight 10 Character orbit 20.c Analytic conductor 10.301 Analytic rank 0 Dimension 4 CM No Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$20 = 2^{2} \cdot 5$$ Weight: $$k$$ = $$10$$ Character orbit: $$[\chi]$$ = 20.c (of order $$2$$ and degree $$1$$)

## Newform invariants

 Self dual: No Analytic conductor: $$10.3007167233$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\mathbb{Q}[x]/(x^{4} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{7}\cdot 5^{2}$$ 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.

 $$f(q)$$ $$=$$ $$q$$ $$+ \beta_{1} q^{3}$$ $$+ ( 165 + \beta_{1} - \beta_{3} ) q^{5}$$ $$+ ( 3 \beta_{1} + \beta_{2} - \beta_{3} ) q^{7}$$ $$+ ( -2261 + 7 \beta_{2} + 7 \beta_{3} ) q^{9}$$ $$+O(q^{10})$$ $$q$$ $$+ \beta_{1} q^{3}$$ $$+ ( 165 + \beta_{1} - \beta_{3} ) q^{5}$$ $$+ ( 3 \beta_{1} + \beta_{2} - \beta_{3} ) q^{7}$$ $$+ ( -2261 + 7 \beta_{2} + 7 \beta_{3} ) q^{9}$$ $$+ ( -8700 + 26 \beta_{2} + 26 \beta_{3} ) q^{11}$$ $$+ ( -282 \beta_{1} + 57 \beta_{2} - 57 \beta_{3} ) q^{13}$$ $$+ ( -24940 + 739 \beta_{1} + 125 \beta_{2} + 11 \beta_{3} ) q^{15}$$ $$+ ( 1192 \beta_{1} + 170 \beta_{2} - 170 \beta_{3} ) q^{17}$$ $$+ ( -56916 + 258 \beta_{2} + 258 \beta_{3} ) q^{19}$$ $$+ ( -71824 + 143 \beta_{2} + 143 \beta_{3} ) q^{21}$$ $$+ ( -13525 \beta_{1} + 19 \beta_{2} - 19 \beta_{3} ) q^{23}$$ $$+ ( -50475 + 11910 \beta_{1} - 625 \beta_{2} - 35 \beta_{3} ) q^{25}$$ $$+ ( 9386 \beta_{1} - 798 \beta_{2} + 798 \beta_{3} ) q^{27}$$ $$+ ( 1566414 - 2244 \beta_{2} - 2244 \beta_{3} ) q^{29}$$ $$+ ( 93616 - 1860 \beta_{2} - 1860 \beta_{3} ) q^{31}$$ $$+ ( -38548 \beta_{1} - 2964 \beta_{2} + 2964 \beta_{3} ) q^{33}$$ $$+ ( -2028540 + 12649 \beta_{1} - 500 \beta_{2} - 24 \beta_{3} ) q^{35}$$ $$+ ( -9390 \beta_{1} - 2603 \beta_{2} + 2603 \beta_{3} ) q^{37}$$ $$+ ( 5846664 + 4980 \beta_{2} + 4980 \beta_{3} ) q^{39}$$ $$+ ( -4412034 + 851 \beta_{2} + 851 \beta_{3} ) q^{41}$$ $$+ ( 122901 \beta_{1} + 11078 \beta_{2} - 11078 \beta_{3} ) q^{43}$$ $$+ ( -13310465 - 83321 \beta_{1} + 4375 \beta_{2} + 196 \beta_{3} ) q^{45}$$ $$+ ( -8137 \beta_{1} + 21013 \beta_{2} - 21013 \beta_{3} ) q^{47}$$ $$+ ( 36224703 - 381 \beta_{2} - 381 \beta_{3} ) q^{49}$$ $$+ ( -27175888 + 29084 \beta_{2} + 29084 \beta_{3} ) q^{51}$$ $$+ ( 410106 \beta_{1} - 8721 \beta_{2} + 8721 \beta_{3} ) q^{53}$$ $$+ ( -49488700 - 309780 \beta_{1} + 16250 \beta_{2} + 1030 \beta_{3} ) q^{55}$$ $$+ ( -353100 \beta_{1} - 29412 \beta_{2} + 29412 \beta_{3} ) q^{57}$$ $$+ ( 109748868 + 11846 \beta_{2} + 11846 \beta_{3} ) q^{59}$$ $$+ ( -25761118 - 85017 \beta_{2} - 85017 \beta_{3} ) q^{61}$$ $$+ ( -176939 \beta_{1} + 3381 \beta_{2} - 3381 \beta_{3} ) q^{63}$$ $$+ ( -104328960 + 386226 \beta_{1} - 85125 \beta_{2} - 6351 \beta_{3} ) q^{65}$$ $$+ ( 664065 \beta_{1} - 28918 \beta_{2} + 28918 \beta_{3} ) q^{67}$$ $$+ ( 296678752 - 92357 \beta_{2} - 92357 \beta_{3} ) q^{69}$$ $$+ ( -126020472 + 42048 \beta_{2} + 42048 \beta_{3} ) q^{71}$$ $$+ ( -1244412 \beta_{1} - 104368 \beta_{2} + 104368 \beta_{3} ) q^{73}$$ $$+ ( -259585400 + 328365 \beta_{1} + 85000 \beta_{2} + 9760 \beta_{3} ) q^{75}$$ $$+ ( -658108 \beta_{1} + 12256 \beta_{2} - 12256 \beta_{3} ) q^{77}$$ $$+ ( 448738752 + 72036 \beta_{2} + 72036 \beta_{3} ) q^{79}$$ $$+ ( -245688031 + 106127 \beta_{2} + 106127 \beta_{3} ) q^{81}$$ $$+ ( -2391607 \beta_{1} + 203338 \beta_{2} - 203338 \beta_{3} ) q^{83}$$ $$+ ( -361860880 + 2654328 \beta_{1} + 250 \beta_{2} + 3422 \beta_{3} ) q^{85}$$ $$+ ( 4142526 \beta_{1} + 255816 \beta_{2} - 255816 \beta_{3} ) q^{87}$$ $$+ ( 595329546 + 249508 \beta_{2} + 249508 \beta_{3} ) q^{89}$$ $$+ ( -202811256 - 86496 \beta_{2} - 86496 \beta_{3} ) q^{91}$$ $$+ ( 2228896 \beta_{1} + 212040 \beta_{2} - 212040 \beta_{3} ) q^{93}$$ $$+ ( -486226740 - 3044556 \beta_{1} + 161250 \beta_{2} - 19194 \beta_{3} ) q^{95}$$ $$+ ( -6424368 \beta_{1} - 196042 \beta_{2} + 196042 \beta_{3} ) q^{97}$$ $$+ ( 692415500 - 119686 \beta_{2} - 119686 \beta_{3} ) q^{99}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q$$ $$\mathstrut +\mathstrut 660q^{5}$$ $$\mathstrut -\mathstrut 9044q^{9}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$4q$$ $$\mathstrut +\mathstrut 660q^{5}$$ $$\mathstrut -\mathstrut 9044q^{9}$$ $$\mathstrut -\mathstrut 34800q^{11}$$ $$\mathstrut -\mathstrut 99760q^{15}$$ $$\mathstrut -\mathstrut 227664q^{19}$$ $$\mathstrut -\mathstrut 287296q^{21}$$ $$\mathstrut -\mathstrut 201900q^{25}$$ $$\mathstrut +\mathstrut 6265656q^{29}$$ $$\mathstrut +\mathstrut 374464q^{31}$$ $$\mathstrut -\mathstrut 8114160q^{35}$$ $$\mathstrut +\mathstrut 23386656q^{39}$$ $$\mathstrut -\mathstrut 17648136q^{41}$$ $$\mathstrut -\mathstrut 53241860q^{45}$$ $$\mathstrut +\mathstrut 144898812q^{49}$$ $$\mathstrut -\mathstrut 108703552q^{51}$$ $$\mathstrut -\mathstrut 197954800q^{55}$$ $$\mathstrut +\mathstrut 438995472q^{59}$$ $$\mathstrut -\mathstrut 103044472q^{61}$$ $$\mathstrut -\mathstrut 417315840q^{65}$$ $$\mathstrut +\mathstrut 1186715008q^{69}$$ $$\mathstrut -\mathstrut 504081888q^{71}$$ $$\mathstrut -\mathstrut 1038341600q^{75}$$ $$\mathstrut +\mathstrut 1794955008q^{79}$$ $$\mathstrut -\mathstrut 982752124q^{81}$$ $$\mathstrut -\mathstrut 1447443520q^{85}$$ $$\mathstrut +\mathstrut 2381318184q^{89}$$ $$\mathstrut -\mathstrut 811245024q^{91}$$ $$\mathstrut -\mathstrut 1944906960q^{95}$$ $$\mathstrut +\mathstrut 2769662000q^{99}$$ $$\mathstrut +\mathstrut O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4}\mathstrut -\mathstrut$$ $$x^{3}\mathstrut +\mathstrut$$ $$1095$$ $$x^{2}\mathstrut -\mathstrut$$ $$80251$$ $$x\mathstrut +\mathstrut$$ $$2230844$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{3} - 82 \nu^{2} + 99 \nu + 14468$$$$)/1000$$ $$\beta_{2}$$ $$=$$ $$($$$$3 \nu^{3} + 246 \nu^{2} + 19703 \nu - 48404$$$$)/500$$ $$\beta_{3}$$ $$=$$ $$($$$$-13 \nu^{3} - 66 \nu^{2} - 13713 \nu + 739084$$$$)/500$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2}\mathstrut +\mathstrut$$ $$6$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$10$$$$)/40$$ $$\nu^{2}$$ $$=$$ $$($$$$4$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$3$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$86$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$4378$$$$)/8$$ $$\nu^{3}$$ $$=$$ $$($$$$-$$$$1640$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$1131$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$4146$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$2374690$$$$)/40$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
9.1
 −23.7825 − 44.2040i 24.2825 + 17.1975i 24.2825 − 17.1975i −23.7825 + 44.2040i
0 188.155i 0 1126.30 827.388i 0 1842.93i 0 −15719.2 0
9.2 0 92.1183i 0 −796.301 + 1148.49i 0 2204.86i 0 11197.2 0
9.3 0 92.1183i 0 −796.301 1148.49i 0 2204.86i 0 11197.2 0
9.4 0 188.155i 0 1126.30 + 827.388i 0 1842.93i 0 −15719.2 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.b Even 1 yes

## Hecke kernels

There are no other newforms in $$S_{10}^{\mathrm{new}}(20, [\chi])$$.