# Properties

 Label 20.8.c.a Level 20 Weight 8 Character orbit 20.c Analytic conductor 6.248 Analytic rank 0 Dimension 4 CM No Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$20 = 2^{2} \cdot 5$$ Weight: $$k$$ = $$8$$ Character orbit: $$[\chi]$$ = 20.c (of order $$2$$ and degree $$1$$)

## Newform invariants

 Self dual: No Analytic conductor: $$6.24770050968$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\mathbb{Q}[x]/(x^{4} + \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{7}\cdot 5$$ 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.

 $$f(q)$$ $$=$$ $$q$$ $$+ \beta_{1} q^{3}$$ $$+ ( -39 - \beta_{1} + \beta_{2} ) q^{5}$$ $$+ ( -5 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{7}$$ $$+ ( -509 - 4 \beta_{1} + 8 \beta_{2} + \beta_{3} ) q^{9}$$ $$+O(q^{10})$$ $$q$$ $$+ \beta_{1} q^{3}$$ $$+ ( -39 - \beta_{1} + \beta_{2} ) q^{5}$$ $$+ ( -5 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{7}$$ $$+ ( -509 - 4 \beta_{1} + 8 \beta_{2} + \beta_{3} ) q^{9}$$ $$+ ( -660 - 8 \beta_{1} + 16 \beta_{2} + 2 \beta_{3} ) q^{11}$$ $$+ ( 222 \beta_{1} - 6 \beta_{2} + 3 \beta_{3} ) q^{13}$$ $$+ ( 1388 - 253 \beta_{1} - 2 \beta_{2} + 5 \beta_{3} ) q^{15}$$ $$+ ( -192 \beta_{1} + 20 \beta_{2} - 10 \beta_{3} ) q^{17}$$ $$+ ( -9084 + 24 \beta_{1} - 48 \beta_{2} - 6 \beta_{3} ) q^{19}$$ $$+ ( 15776 + 76 \beta_{1} - 152 \beta_{2} - 19 \beta_{3} ) q^{21}$$ $$+ ( 843 \beta_{1} + 58 \beta_{2} - 29 \beta_{3} ) q^{23}$$ $$+ ( 40509 - 274 \beta_{1} - 46 \beta_{2} - 55 \beta_{3} ) q^{25}$$ $$+ ( -430 \beta_{1} - 84 \beta_{2} + 42 \beta_{3} ) q^{27}$$ $$+ ( -120474 + 48 \beta_{1} - 96 \beta_{2} - 12 \beta_{3} ) q^{29}$$ $$+ ( 40336 - 240 \beta_{1} + 480 \beta_{2} + 60 \beta_{3} ) q^{31}$$ $$+ ( -4876 \beta_{1} - 168 \beta_{2} + 84 \beta_{3} ) q^{33}$$ $$+ ( 172812 + 4153 \beta_{1} + 252 \beta_{2} + 220 \beta_{3} ) q^{35}$$ $$+ ( 5746 \beta_{1} + 226 \beta_{2} - 113 \beta_{3} ) q^{37}$$ $$+ ( -591624 - 720 \beta_{1} + 1440 \beta_{2} + 180 \beta_{3} ) q^{39}$$ $$+ ( 282366 + 172 \beta_{1} - 344 \beta_{2} - 43 \beta_{3} ) q^{41}$$ $$+ ( -3163 \beta_{1} - 124 \beta_{2} + 62 \beta_{3} ) q^{43}$$ $$+ ( 597811 - 1251 \beta_{1} - 349 \beta_{2} - 275 \beta_{3} ) q^{45}$$ $$+ ( -8889 \beta_{1} - 506 \beta_{2} + 253 \beta_{3} ) q^{47}$$ $$+ ( -1078113 + 1932 \beta_{1} - 3864 \beta_{2} - 483 \beta_{3} ) q^{49}$$ $$+ ( 494672 + 208 \beta_{1} - 416 \beta_{2} - 52 \beta_{3} ) q^{51}$$ $$+ ( 10194 \beta_{1} + 1638 \beta_{2} - 819 \beta_{3} ) q^{53}$$ $$+ ( 1181660 - 2860 \beta_{1} - 340 \beta_{2} - 550 \beta_{3} ) q^{55}$$ $$+ ( 3564 \beta_{1} + 504 \beta_{2} - 252 \beta_{3} ) q^{57}$$ $$+ ( -1936788 - 632 \beta_{1} + 1264 \beta_{2} + 158 \beta_{3} ) q^{59}$$ $$+ ( 1076282 + 1356 \beta_{1} - 2712 \beta_{2} - 339 \beta_{3} ) q^{61}$$ $$+ ( 44893 \beta_{1} - 2778 \beta_{2} + 1389 \beta_{3} ) q^{63}$$ $$+ ( 847392 - 47502 \beta_{1} + 282 \beta_{2} + 1845 \beta_{3} ) q^{65}$$ $$+ ( -33319 \beta_{1} + 1916 \beta_{2} - 958 \beta_{3} ) q^{67}$$ $$+ ( -2339312 - 4996 \beta_{1} + 9992 \beta_{2} + 1249 \beta_{3} ) q^{69}$$ $$+ ( -301512 - 4224 \beta_{1} + 8448 \beta_{2} + 1056 \beta_{3} ) q^{71}$$ $$+ ( -33748 \beta_{1} - 736 \beta_{2} + 368 \beta_{3} ) q^{73}$$ $$+ ( 816472 + 72533 \beta_{1} + 3032 \beta_{2} - 440 \beta_{3} ) q^{75}$$ $$+ ( 87996 \beta_{1} - 6272 \beta_{2} + 3136 \beta_{3} ) q^{77}$$ $$+ ( 1429248 + 6768 \beta_{1} - 13536 \beta_{2} - 1692 \beta_{3} ) q^{79}$$ $$+ ( 142529 - 4676 \beta_{1} + 9352 \beta_{2} + 1169 \beta_{3} ) q^{81}$$ $$+ ( -74511 \beta_{1} + 6076 \beta_{2} - 3038 \beta_{3} ) q^{83}$$ $$+ ( -2064016 + 19696 \beta_{1} - 2036 \beta_{2} - 3410 \beta_{3} ) q^{85}$$ $$+ ( -95178 \beta_{1} + 1008 \beta_{2} - 504 \beta_{3} ) q^{87}$$ $$+ ( 3410394 - 2416 \beta_{1} + 4832 \beta_{2} + 604 \beta_{3} ) q^{89}$$ $$+ ( -1966056 + 23808 \beta_{1} - 47616 \beta_{2} - 5952 \beta_{3} ) q^{91}$$ $$+ ( -86144 \beta_{1} - 5040 \beta_{2} + 2520 \beta_{3} ) q^{93}$$ $$+ ( -3113484 + 19644 \beta_{1} - 10044 \beta_{2} + 1650 \beta_{3} ) q^{95}$$ $$+ ( 117032 \beta_{1} + 14444 \beta_{2} - 7222 \beta_{3} ) q^{97}$$ $$+ ( 11895140 + 6712 \beta_{1} - 13424 \beta_{2} - 1678 \beta_{3} ) q^{99}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q$$ $$\mathstrut -\mathstrut 156q^{5}$$ $$\mathstrut -\mathstrut 2036q^{9}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$4q$$ $$\mathstrut -\mathstrut 156q^{5}$$ $$\mathstrut -\mathstrut 2036q^{9}$$ $$\mathstrut -\mathstrut 2640q^{11}$$ $$\mathstrut +\mathstrut 5552q^{15}$$ $$\mathstrut -\mathstrut 36336q^{19}$$ $$\mathstrut +\mathstrut 63104q^{21}$$ $$\mathstrut +\mathstrut 162036q^{25}$$ $$\mathstrut -\mathstrut 481896q^{29}$$ $$\mathstrut +\mathstrut 161344q^{31}$$ $$\mathstrut +\mathstrut 691248q^{35}$$ $$\mathstrut -\mathstrut 2366496q^{39}$$ $$\mathstrut +\mathstrut 1129464q^{41}$$ $$\mathstrut +\mathstrut 2391244q^{45}$$ $$\mathstrut -\mathstrut 4312452q^{49}$$ $$\mathstrut +\mathstrut 1978688q^{51}$$ $$\mathstrut +\mathstrut 4726640q^{55}$$ $$\mathstrut -\mathstrut 7747152q^{59}$$ $$\mathstrut +\mathstrut 4305128q^{61}$$ $$\mathstrut +\mathstrut 3389568q^{65}$$ $$\mathstrut -\mathstrut 9357248q^{69}$$ $$\mathstrut -\mathstrut 1206048q^{71}$$ $$\mathstrut +\mathstrut 3265888q^{75}$$ $$\mathstrut +\mathstrut 5716992q^{79}$$ $$\mathstrut +\mathstrut 570116q^{81}$$ $$\mathstrut -\mathstrut 8256064q^{85}$$ $$\mathstrut +\mathstrut 13641576q^{89}$$ $$\mathstrut -\mathstrut 7864224q^{91}$$ $$\mathstrut -\mathstrut 12453936q^{95}$$ $$\mathstrut +\mathstrut 47580560q^{99}$$ $$\mathstrut +\mathstrut O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4}\mathstrut +\mathstrut$$ $$1348$$ $$x^{2}\mathstrut +\mathstrut$$ $$93051$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$2 \nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-2 \nu^{3} + 42 \nu^{2} - 2318 \nu + 28308$$$$)/105$$ $$\beta_{3}$$ $$=$$ $$($$$$16 \nu^{3} + 84 \nu^{2} + 19384 \nu + 56616$$$$)/105$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$$$/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3}\mathstrut +\mathstrut$$ $$8$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$4$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$2696$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$21$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$42$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$2402$$ $$\beta_{1}$$$$)/4$$

## 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
 − 35.7074i − 8.54284i 8.54284i 35.7074i
0 71.4148i 0 −279.408 7.49513i 0 860.515i 0 −2913.08 0
9.2 0 17.0857i 0 201.408 + 193.804i 0 1750.09i 0 1895.08 0
9.3 0 17.0857i 0 201.408 193.804i 0 1750.09i 0 1895.08 0
9.4 0 71.4148i 0 −279.408 + 7.49513i 0 860.515i 0 −2913.08 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 $$S_{8}^{\mathrm{new}}(20, [\chi])$$.