# Properties

 Label 20.7.f.a Level 20 Weight 7 Character orbit 20.f Analytic conductor 4.601 Analytic rank 0 Dimension 6 CM No Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$20 = 2^{2} \cdot 5$$ Weight: $$k$$ = $$7$$ Character orbit: $$[\chi]$$ = 20.f (of order $$4$$ and degree $$2$$)

## Newform invariants

 Self dual: No Analytic conductor: $$4.6010816724$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(i)$$ Coefficient field: $$\mathbb{Q}[x]/(x^{6} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$2^{7}\cdot 5$$ Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q$$ $$+ ( 5 + 5 \beta_{1} + \beta_{4} ) q^{3}$$ $$+ ( -25 + 20 \beta_{1} + \beta_{2} - \beta_{4} - \beta_{5} ) q^{5}$$ $$+ ( -45 + 45 \beta_{1} - 6 \beta_{2} - \beta_{3} + \beta_{4} - 3 \beta_{5} ) q^{7}$$ $$+ ( 293 \beta_{1} + 21 \beta_{2} + 2 \beta_{3} + 23 \beta_{4} - 4 \beta_{5} ) q^{9}$$ $$+O(q^{10})$$ $$q$$ $$+ ( 5 + 5 \beta_{1} + \beta_{4} ) q^{3}$$ $$+ ( -25 + 20 \beta_{1} + \beta_{2} - \beta_{4} - \beta_{5} ) q^{5}$$ $$+ ( -45 + 45 \beta_{1} - 6 \beta_{2} - \beta_{3} + \beta_{4} - 3 \beta_{5} ) q^{7}$$ $$+ ( 293 \beta_{1} + 21 \beta_{2} + 2 \beta_{3} + 23 \beta_{4} - 4 \beta_{5} ) q^{9}$$ $$+ ( 350 - 27 \beta_{2} - 4 \beta_{3} + 29 \beta_{4} - 2 \beta_{5} ) q^{11}$$ $$+ ( 155 + 155 \beta_{1} - \beta_{2} + 3 \beta_{3} - 39 \beta_{4} - \beta_{5} ) q^{13}$$ $$+ ( -1245 - 1305 \beta_{1} + 65 \beta_{2} - \beta_{3} - 92 \beta_{4} + 9 \beta_{5} ) q^{15}$$ $$+ ( -595 + 595 \beta_{1} - 116 \beta_{2} + 10 \beta_{3} - 10 \beta_{4} + 30 \beta_{5} ) q^{17}$$ $$+ ( -2772 \beta_{1} + 44 \beta_{2} - 22 \beta_{3} + 22 \beta_{4} + 44 \beta_{5} ) q^{19}$$ $$+ ( 5546 + 49 \beta_{2} + 48 \beta_{3} - 73 \beta_{4} + 24 \beta_{5} ) q^{21}$$ $$+ ( 3355 + 3355 \beta_{1} + 13 \beta_{2} - 39 \beta_{3} - 34 \beta_{4} + 13 \beta_{5} ) q^{23}$$ $$+ ( -4195 - 9860 \beta_{1} - 289 \beta_{2} + 14 \beta_{3} + 217 \beta_{4} - 22 \beta_{5} ) q^{25}$$ $$+ ( -19040 + 19040 \beta_{1} + 548 \beta_{2} - 32 \beta_{3} + 32 \beta_{4} - 96 \beta_{5} ) q^{27}$$ $$+ ( -10892 \beta_{1} - 422 \beta_{2} + 86 \beta_{3} - 336 \beta_{4} - 172 \beta_{5} ) q^{29}$$ $$+ ( 17646 + 115 \beta_{2} - 220 \beta_{3} - 5 \beta_{4} - 110 \beta_{5} ) q^{31}$$ $$+ ( 29130 + 29130 \beta_{1} - 68 \beta_{2} + 204 \beta_{3} + 1066 \beta_{4} - 68 \beta_{5} ) q^{33}$$ $$+ ( -19515 - 31265 \beta_{1} + 371 \beta_{2} - 82 \beta_{3} + 990 \beta_{4} - 28 \beta_{5} ) q^{35}$$ $$+ ( -40305 + 40305 \beta_{1} - 129 \beta_{2} + 3 \beta_{3} - 3 \beta_{4} + 9 \beta_{5} ) q^{37}$$ $$+ ( -36522 \beta_{1} - 345 \beta_{2} - 90 \beta_{3} - 435 \beta_{4} + 180 \beta_{5} ) q^{39}$$ $$+ ( 58806 + 473 \beta_{2} + 396 \beta_{3} - 671 \beta_{4} + 198 \beta_{5} ) q^{41}$$ $$+ ( 11065 + 11065 \beta_{1} + 166 \beta_{2} - 498 \beta_{3} - 2337 \beta_{4} + 166 \beta_{5} ) q^{43}$$ $$+ ( -47560 - 81225 \beta_{1} - 1298 \beta_{2} + 247 \beta_{3} - 2958 \beta_{4} + 170 \beta_{5} ) q^{45}$$ $$+ ( -59625 + 59625 \beta_{1} - 720 \beta_{2} + 177 \beta_{3} - 177 \beta_{4} + 531 \beta_{5} ) q^{47}$$ $$+ ( -41829 \beta_{1} + 3047 \beta_{2} - 286 \beta_{3} + 2761 \beta_{4} + 572 \beta_{5} ) q^{49}$$ $$+ ( 105162 - 2613 \beta_{2} + 224 \beta_{3} + 2501 \beta_{4} + 112 \beta_{5} ) q^{51}$$ $$+ ( 57635 + 57635 \beta_{1} - 97 \beta_{2} + 291 \beta_{3} + 67 \beta_{4} - 97 \beta_{5} ) q^{53}$$ $$+ ( -49850 - 47700 \beta_{1} + 5035 \beta_{2} - 330 \beta_{3} - 195 \beta_{4} + 10 \beta_{5} ) q^{55}$$ $$+ ( -20020 + 20020 \beta_{1} - 4246 \beta_{2} - 198 \beta_{3} + 198 \beta_{4} - 594 \beta_{5} ) q^{57}$$ $$+ ( 46916 \beta_{1} - 1632 \beta_{2} + 766 \beta_{3} - 866 \beta_{4} - 1532 \beta_{5} ) q^{59}$$ $$+ ( -80098 + 2089 \beta_{2} - 1972 \beta_{3} - 1103 \beta_{4} - 986 \beta_{5} ) q^{61}$$ $$+ ( -725 - 725 \beta_{1} - 463 \beta_{2} + 1389 \beta_{3} + 1930 \beta_{4} - 463 \beta_{5} ) q^{63}$$ $$+ ( -2555 + 61205 \beta_{1} - 3135 \beta_{2} - 214 \beta_{3} + 2857 \beta_{4} - 654 \beta_{5} ) q^{65}$$ $$+ ( -35455 + 35455 \beta_{1} + 7401 \beta_{2} - 462 \beta_{3} + 462 \beta_{4} - 1386 \beta_{5} ) q^{67}$$ $$+ ( 2634 \beta_{1} + 1199 \beta_{2} + 88 \beta_{3} + 1287 \beta_{4} - 176 \beta_{5} ) q^{69}$$ $$+ ( 26238 - 3101 \beta_{2} + 2148 \beta_{3} + 2027 \beta_{4} + 1074 \beta_{5} ) q^{71}$$ $$+ ( -55395 - 55395 \beta_{1} + 1154 \beta_{2} - 3462 \beta_{3} + 3426 \beta_{4} + 1154 \beta_{5} ) q^{73}$$ $$+ ( 303865 + 134545 \beta_{1} - 10182 \beta_{2} + 1602 \beta_{3} + 4211 \beta_{4} - 66 \beta_{5} ) q^{75}$$ $$+ ( 271590 - 271590 \beta_{1} + 7172 \beta_{2} + 594 \beta_{3} - 594 \beta_{4} + 1782 \beta_{5} ) q^{77}$$ $$+ ( 52144 \beta_{1} - 10552 \beta_{2} - 1624 \beta_{3} - 12176 \beta_{4} + 3248 \beta_{5} ) q^{79}$$ $$+ ( -504211 + 15721 \beta_{2} + 1492 \beta_{3} - 16467 \beta_{4} + 746 \beta_{5} ) q^{81}$$ $$+ ( -364475 - 364475 \beta_{1} - 824 \beta_{2} + 2472 \beta_{3} - 4515 \beta_{4} - 824 \beta_{5} ) q^{83}$$ $$+ ( -24905 + 445145 \beta_{1} + 8632 \beta_{2} - 2689 \beta_{3} + 1540 \beta_{4} + 2029 \beta_{5} ) q^{85}$$ $$+ ( 429900 - 429900 \beta_{1} - 13630 \beta_{2} + 1274 \beta_{3} - 1274 \beta_{4} + 3822 \beta_{5} ) q^{87}$$ $$+ ( 26392 \beta_{1} - 7956 \beta_{2} - 172 \beta_{3} - 8128 \beta_{4} + 344 \beta_{5} ) q^{89}$$ $$+ ( -367246 - 5753 \beta_{2} - 3256 \beta_{3} + 7381 \beta_{4} - 1628 \beta_{5} ) q^{91}$$ $$+ ( 38930 + 38930 \beta_{1} - 540 \beta_{2} + 1620 \beta_{3} + 2626 \beta_{4} - 540 \beta_{5} ) q^{93}$$ $$+ ( 585640 + 249700 \beta_{1} + 9152 \beta_{2} + 1782 \beta_{3} - 8338 \beta_{4} + 880 \beta_{5} ) q^{95}$$ $$+ ( 560785 - 560785 \beta_{1} + 10008 \beta_{2} - 1048 \beta_{3} + 1048 \beta_{4} - 3144 \beta_{5} ) q^{97}$$ $$+ ( 1061150 \beta_{1} + 33477 \beta_{2} + 2774 \beta_{3} + 36251 \beta_{4} - 5548 \beta_{5} ) q^{99}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q$$ $$\mathstrut +\mathstrut 32q^{3}$$ $$\mathstrut -\mathstrut 156q^{5}$$ $$\mathstrut -\mathstrut 264q^{7}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$6q$$ $$\mathstrut +\mathstrut 32q^{3}$$ $$\mathstrut -\mathstrut 156q^{5}$$ $$\mathstrut -\mathstrut 264q^{7}$$ $$\mathstrut +\mathstrut 2200q^{11}$$ $$\mathstrut +\mathstrut 858q^{13}$$ $$\mathstrut -\mathstrut 7768q^{15}$$ $$\mathstrut -\mathstrut 3278q^{17}$$ $$\mathstrut +\mathstrut 33176q^{21}$$ $$\mathstrut +\mathstrut 19984q^{23}$$ $$\mathstrut -\mathstrut 24174q^{25}$$ $$\mathstrut -\mathstrut 115528q^{27}$$ $$\mathstrut +\mathstrut 104976q^{31}$$ $$\mathstrut +\mathstrut 177320q^{33}$$ $$\mathstrut -\mathstrut 116072q^{35}$$ $$\mathstrut -\mathstrut 241554q^{37}$$ $$\mathstrut +\mathstrut 351736q^{41}$$ $$\mathstrut +\mathstrut 60720q^{43}$$ $$\mathstrut -\mathstrut 287846q^{45}$$ $$\mathstrut -\mathstrut 355248q^{47}$$ $$\mathstrut +\mathstrut 641872q^{51}$$ $$\mathstrut +\mathstrut 346526q^{53}$$ $$\mathstrut -\mathstrut 310200q^{55}$$ $$\mathstrut -\mathstrut 112816q^{57}$$ $$\mathstrut -\mathstrut 492888q^{61}$$ $$\mathstrut +\mathstrut 2288q^{63}$$ $$\mathstrut -\mathstrut 5082q^{65}$$ $$\mathstrut -\mathstrut 230304q^{67}$$ $$\mathstrut +\mathstrut 174128q^{71}$$ $$\mathstrut -\mathstrut 332442q^{73}$$ $$\mathstrut +\mathstrut 1855048q^{75}$$ $$\mathstrut +\mathstrut 1618760q^{77}$$ $$\mathstrut -\mathstrut 3085166q^{81}$$ $$\mathstrut -\mathstrut 2190936q^{83}$$ $$\mathstrut -\mathstrut 164934q^{85}$$ $$\mathstrut +\mathstrut 2614304q^{87}$$ $$\mathstrut -\mathstrut 2186976q^{91}$$ $$\mathstrut +\mathstrut 242072q^{93}$$ $$\mathstrut +\mathstrut 3484184q^{95}$$ $$\mathstrut +\mathstrut 3338406q^{97}$$ $$\mathstrut +\mathstrut O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6}\mathstrut -\mathstrut$$ $$2$$ $$x^{5}\mathstrut +\mathstrut$$ $$2$$ $$x^{4}\mathstrut -\mathstrut$$ $$450$$ $$x^{3}\mathstrut +\mathstrut$$ $$23409$$ $$x^{2}\mathstrut -\mathstrut$$ $$115668$$ $$x\mathstrut +\mathstrut$$ $$285768$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$38 \nu^{5} + 17 \nu^{4} + 25 \nu^{3} - 2685 \nu^{2} + 868617 \nu - 2248722$$$$)/2216970$$ $$\beta_{2}$$ $$=$$ $$($$$$-325 \nu^{5} + 1326 \nu^{4} + 18050 \nu^{3} + 251030 \nu^{2} - 7760025 \nu + 37081044$$$$)/738990$$ $$\beta_{3}$$ $$=$$ $$($$$$1327 \nu^{5} + 14824 \nu^{4} + 12140 \nu^{3} - 844020 \nu^{2} + 38955033 \nu + 130369176$$$$)/2216970$$ $$\beta_{4}$$ $$=$$ $$($$$$-1375 \nu^{5} - 5338 \nu^{4} - 7850 \nu^{3} + 843090 \nu^{2} - 22977675 \nu + 9970128$$$$)/2216970$$ $$\beta_{5}$$ $$=$$ $$($$$$-4289 \nu^{5} + 3298 \nu^{4} - 72430 \nu^{3} + 2589630 \nu^{2} - 100691181 \nu + 349337772$$$$)/2216970$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-$$$$\beta_{5}\mathstrut +\mathstrut$$ $$7$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$3$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$10$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$10$$$$)/40$$ $$\nu^{2}$$ $$=$$ $$($$$$2$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$2$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$3$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$410$$ $$\beta_{1}$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$-$$$$429$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$143$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$143$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$1121$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$9490$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$9490$$$$)/40$$ $$\nu^{4}$$ $$=$$ $$($$$$100$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$165$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$200$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$65$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$30038$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$23659$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$157213$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$70977$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$23659$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$2401010$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$2401010$$$$)/40$$

## Character Values

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

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

## 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
 2.61494 + 2.61494i −9.34732 − 9.34732i 7.73238 + 7.73238i 2.61494 − 2.61494i −9.34732 + 9.34732i 7.73238 − 7.73238i
0 −16.7249 16.7249i 0 −22.1946 + 123.014i 0 −422.022 + 422.022i 0 169.553i 0
13.2 0 −2.90948 2.90948i 0 59.6880 109.829i 0 236.070 236.070i 0 712.070i 0
13.3 0 35.6344 + 35.6344i 0 −115.493 + 47.8149i 0 53.9521 53.9521i 0 1810.62i 0
17.1 0 −16.7249 + 16.7249i 0 −22.1946 123.014i 0 −422.022 422.022i 0 169.553i 0
17.2 0 −2.90948 + 2.90948i 0 59.6880 + 109.829i 0 236.070 + 236.070i 0 712.070i 0
17.3 0 35.6344 35.6344i 0 −115.493 47.8149i 0 53.9521 + 53.9521i 0 1810.62i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 17.3 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 $$S_{7}^{\mathrm{new}}(20, [\chi])$$.