# Properties

 Label 19.4.c.b Level 19 Weight 4 Character orbit 19.c Analytic conductor 1.121 Analytic rank 0 Dimension 4 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$19$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 19.c (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.12103629011$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{73})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{2} -\beta_{2} q^{3} + ( -11 + \beta_{1} + 10 \beta_{2} + \beta_{3} ) q^{4} + ( -\beta_{1} - 9 \beta_{2} ) q^{5} + ( -1 + \beta_{1} + \beta_{3} ) q^{6} + ( 12 - 4 \beta_{3} ) q^{7} + ( 21 - 3 \beta_{3} ) q^{8} + ( 26 - 26 \beta_{2} ) q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{2} -\beta_{2} q^{3} + ( -11 + \beta_{1} + 10 \beta_{2} + \beta_{3} ) q^{4} + ( -\beta_{1} - 9 \beta_{2} ) q^{5} + ( -1 + \beta_{1} + \beta_{3} ) q^{6} + ( 12 - 4 \beta_{3} ) q^{7} + ( 21 - 3 \beta_{3} ) q^{8} + ( 26 - 26 \beta_{2} ) q^{9} + ( -28 + 10 \beta_{1} + 18 \beta_{2} + 10 \beta_{3} ) q^{10} + ( -15 - 3 \beta_{3} ) q^{11} + ( 11 - \beta_{3} ) q^{12} + ( -48 - 5 \beta_{1} + 53 \beta_{2} - 5 \beta_{3} ) q^{13} + ( -12 \beta_{1} - 72 \beta_{2} ) q^{14} + ( -10 + \beta_{1} + 9 \beta_{2} + \beta_{3} ) q^{15} + ( -13 \beta_{1} + 26 \beta_{2} ) q^{16} + ( 21 \beta_{1} + 27 \beta_{2} ) q^{17} + ( -26 + 26 \beta_{3} ) q^{18} + ( 19 - 19 \beta_{1} ) q^{19} + ( 128 - 20 \beta_{3} ) q^{20} + ( -4 \beta_{1} - 8 \beta_{2} ) q^{21} + ( 15 \beta_{1} - 54 \beta_{2} ) q^{22} + ( 8 - 17 \beta_{1} + 9 \beta_{2} - 17 \beta_{3} ) q^{23} + ( -3 \beta_{1} - 18 \beta_{2} ) q^{24} + ( 7 + 19 \beta_{1} - 26 \beta_{2} + 19 \beta_{3} ) q^{25} + ( -42 - 48 \beta_{3} ) q^{26} -53 q^{27} + ( -204 + 52 \beta_{1} + 152 \beta_{2} + 52 \beta_{3} ) q^{28} + ( 22 + 41 \beta_{1} - 63 \beta_{2} + 41 \beta_{3} ) q^{29} + ( 28 - 10 \beta_{3} ) q^{30} + ( 14 - 6 \beta_{3} ) q^{31} + ( -53 - 37 \beta_{1} + 90 \beta_{2} - 37 \beta_{3} ) q^{32} + ( -3 \beta_{1} + 18 \beta_{2} ) q^{33} + ( 426 - 48 \beta_{1} - 378 \beta_{2} - 48 \beta_{3} ) q^{34} + ( -48 \beta_{1} - 144 \beta_{2} ) q^{35} + ( 26 \beta_{1} + 260 \beta_{2} ) q^{36} + ( 206 + 36 \beta_{3} ) q^{37} + ( -361 + 342 \beta_{2} + 19 \beta_{3} ) q^{38} + ( 48 + 5 \beta_{3} ) q^{39} + ( -48 \beta_{1} - 216 \beta_{2} ) q^{40} + ( 2 \beta_{1} - 63 \beta_{2} ) q^{41} + ( -84 + 12 \beta_{1} + 72 \beta_{2} + 12 \beta_{3} ) q^{42} + ( 21 \beta_{1} + 145 \beta_{2} ) q^{43} + ( 111 + 15 \beta_{1} - 126 \beta_{2} + 15 \beta_{3} ) q^{44} + ( -260 + 26 \beta_{3} ) q^{45} + ( -314 + 8 \beta_{3} ) q^{46} + ( -186 - 39 \beta_{1} + 225 \beta_{2} - 39 \beta_{3} ) q^{47} + ( 13 + 13 \beta_{1} - 26 \beta_{2} + 13 \beta_{3} ) q^{48} + ( 89 - 80 \beta_{3} ) q^{49} + ( 335 + 7 \beta_{3} ) q^{50} + ( 48 - 21 \beta_{1} - 27 \beta_{2} - 21 \beta_{3} ) q^{51} + ( 2 \beta_{1} - 440 \beta_{2} ) q^{52} + ( -180 + 99 \beta_{1} + 81 \beta_{2} + 99 \beta_{3} ) q^{53} + 53 \beta_{1} q^{54} + ( -12 \beta_{1} + 108 \beta_{2} ) q^{55} + ( 468 - 108 \beta_{3} ) q^{56} + ( -19 + 19 \beta_{1} - 19 \beta_{2} + 19 \beta_{3} ) q^{57} + ( 716 + 22 \beta_{3} ) q^{58} + ( 102 \beta_{1} - 153 \beta_{2} ) q^{59} + ( -20 \beta_{1} - 108 \beta_{2} ) q^{60} + ( -262 - 7 \beta_{1} + 269 \beta_{2} - 7 \beta_{3} ) q^{61} + ( -14 \beta_{1} - 108 \beta_{2} ) q^{62} + ( 312 - 104 \beta_{1} - 208 \beta_{2} - 104 \beta_{3} ) q^{63} + ( -509 + 51 \beta_{3} ) q^{64} + ( 390 - 3 \beta_{3} ) q^{65} + ( -39 - 15 \beta_{1} + 54 \beta_{2} - 15 \beta_{3} ) q^{66} + ( -197 - 162 \beta_{1} + 359 \beta_{2} - 162 \beta_{3} ) q^{67} + ( -906 + 258 \beta_{3} ) q^{68} + ( -8 + 17 \beta_{3} ) q^{69} + ( -1056 + 192 \beta_{1} + 864 \beta_{2} + 192 \beta_{3} ) q^{70} + ( 3 \beta_{1} - 783 \beta_{2} ) q^{71} + ( 546 - 78 \beta_{1} - 468 \beta_{2} - 78 \beta_{3} ) q^{72} + ( 88 \beta_{1} + 73 \beta_{2} ) q^{73} + ( -206 \beta_{1} + 648 \beta_{2} ) q^{74} + ( -7 - 19 \beta_{3} ) q^{75} + ( 342 + 19 \beta_{1} + 190 \beta_{2} - 190 \beta_{3} ) q^{76} + ( 36 + 36 \beta_{3} ) q^{77} + ( -48 \beta_{1} + 90 \beta_{2} ) q^{78} + ( -31 \beta_{1} + 181 \beta_{2} ) q^{79} + ( -104 + 104 \beta_{1} + 104 \beta_{3} ) q^{80} -649 \beta_{2} q^{81} + ( -25 + 61 \beta_{1} - 36 \beta_{2} + 61 \beta_{3} ) q^{82} + ( 649 + 161 \beta_{3} ) q^{83} + ( 204 - 52 \beta_{3} ) q^{84} + ( 858 - 237 \beta_{1} - 621 \beta_{2} - 237 \beta_{3} ) q^{85} + ( 544 - 166 \beta_{1} - 378 \beta_{2} - 166 \beta_{3} ) q^{86} + ( -22 - 41 \beta_{3} ) q^{87} + ( -153 - 9 \beta_{3} ) q^{88} + ( -340 + 79 \beta_{1} + 261 \beta_{2} + 79 \beta_{3} ) q^{89} + ( 260 \beta_{1} + 468 \beta_{2} ) q^{90} + ( -216 + 152 \beta_{1} + 64 \beta_{2} + 152 \beta_{3} ) q^{91} + ( 178 \beta_{1} + 216 \beta_{2} ) q^{92} + ( -6 \beta_{1} - 8 \beta_{2} ) q^{93} + ( -516 - 186 \beta_{3} ) q^{94} + ( -532 + 171 \beta_{1} + 171 \beta_{2} + 190 \beta_{3} ) q^{95} + ( 53 + 37 \beta_{3} ) q^{96} + ( 274 \beta_{1} + 25 \beta_{2} ) q^{97} + ( -89 \beta_{1} - 1440 \beta_{2} ) q^{98} + ( -390 - 78 \beta_{1} + 468 \beta_{2} - 78 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - q^{2} - 2q^{3} - 21q^{4} - 19q^{5} - q^{6} + 40q^{7} + 78q^{8} + 52q^{9} + O(q^{10})$$ $$4q - q^{2} - 2q^{3} - 21q^{4} - 19q^{5} - q^{6} + 40q^{7} + 78q^{8} + 52q^{9} - 46q^{10} - 66q^{11} + 42q^{12} - 101q^{13} - 156q^{14} - 19q^{15} + 39q^{16} + 75q^{17} - 52q^{18} + 57q^{19} + 472q^{20} - 20q^{21} - 93q^{22} - q^{23} - 39q^{24} + 33q^{25} - 264q^{26} - 212q^{27} - 356q^{28} + 85q^{29} + 92q^{30} + 44q^{31} - 143q^{32} + 33q^{33} + 804q^{34} - 336q^{35} + 546q^{36} + 896q^{37} - 722q^{38} + 202q^{39} - 480q^{40} - 124q^{41} - 156q^{42} + 311q^{43} + 237q^{44} - 988q^{45} - 1240q^{46} - 411q^{47} + 39q^{48} + 196q^{49} + 1354q^{50} + 75q^{51} - 878q^{52} - 261q^{53} + 53q^{54} + 204q^{55} + 1656q^{56} - 57q^{57} + 2908q^{58} - 204q^{59} - 236q^{60} - 531q^{61} - 230q^{62} + 520q^{63} - 1934q^{64} + 1554q^{65} - 93q^{66} - 556q^{67} - 3108q^{68} + 2q^{69} - 1920q^{70} - 1563q^{71} + 1014q^{72} + 234q^{73} + 1090q^{74} - 66q^{75} + 1387q^{76} + 216q^{77} + 132q^{78} + 331q^{79} - 104q^{80} - 1298q^{81} + 11q^{82} + 2918q^{83} + 712q^{84} + 1479q^{85} + 922q^{86} - 170q^{87} - 630q^{88} - 601q^{89} + 1196q^{90} - 280q^{91} + 610q^{92} - 22q^{93} - 2436q^{94} - 1235q^{95} + 286q^{96} + 324q^{97} - 2969q^{98} - 858q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} + 19 x^{2} + 18 x + 324$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{3} + 19 \nu^{2} - 19 \nu + 324$$$$)/342$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{3} + 37$$$$)/19$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + 18 \beta_{2} + \beta_{1} - 19$$ $$\nu^{3}$$ $$=$$ $$19 \beta_{3} - 37$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/19\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$-1 + \beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
7.1
 2.38600 + 4.13267i −1.88600 − 3.26665i 2.38600 − 4.13267i −1.88600 + 3.26665i
−2.38600 4.13267i −0.500000 0.866025i −7.38600 + 12.7929i −6.88600 11.9269i −2.38600 + 4.13267i 27.0880 32.3160 13.0000 22.5167i −32.8600 + 56.9152i
7.2 1.88600 + 3.26665i −0.500000 0.866025i −3.11400 + 5.39360i −2.61400 4.52758i 1.88600 3.26665i −7.08801 6.68399 13.0000 22.5167i 9.86001 17.0780i
11.1 −2.38600 + 4.13267i −0.500000 + 0.866025i −7.38600 12.7929i −6.88600 + 11.9269i −2.38600 4.13267i 27.0880 32.3160 13.0000 + 22.5167i −32.8600 56.9152i
11.2 1.88600 3.26665i −0.500000 + 0.866025i −3.11400 5.39360i −2.61400 + 4.52758i 1.88600 + 3.26665i −7.08801 6.68399 13.0000 + 22.5167i 9.86001 + 17.0780i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 19.4.c.b 4
3.b odd 2 1 171.4.f.d 4
4.b odd 2 1 304.4.i.d 4
19.c even 3 1 inner 19.4.c.b 4
19.c even 3 1 361.4.a.f 2
19.d odd 6 1 361.4.a.e 2
57.h odd 6 1 171.4.f.d 4
76.g odd 6 1 304.4.i.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.4.c.b 4 1.a even 1 1 trivial
19.4.c.b 4 19.c even 3 1 inner
171.4.f.d 4 3.b odd 2 1
171.4.f.d 4 57.h odd 6 1
304.4.i.d 4 4.b odd 2 1
304.4.i.d 4 76.g odd 6 1
361.4.a.e 2 19.d odd 6 1
361.4.a.f 2 19.c even 3 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + T_{2}^{3} + 19 T_{2}^{2} - 18 T_{2} + 324$$ acting on $$S_{4}^{\mathrm{new}}(19, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T + 3 T^{2} - 18 T^{3} - 68 T^{4} - 144 T^{5} + 192 T^{6} + 512 T^{7} + 4096 T^{8}$$
$3$ $$( 1 + T - 26 T^{2} + 27 T^{3} + 729 T^{4} )^{2}$$
$5$ $$1 + 19 T + 39 T^{2} + 1368 T^{3} + 42934 T^{4} + 171000 T^{5} + 609375 T^{6} + 37109375 T^{7} + 244140625 T^{8}$$
$7$ $$( 1 - 20 T + 494 T^{2} - 6860 T^{3} + 117649 T^{4} )^{2}$$
$11$ $$( 1 + 33 T + 2770 T^{2} + 43923 T^{3} + 1771561 T^{4} )^{2}$$
$13$ $$1 + 101 T + 3713 T^{2} + 211494 T^{3} + 14855738 T^{4} + 464652318 T^{5} + 17921941817 T^{6} + 1071054436673 T^{7} + 23298085122481 T^{8}$$
$17$ $$1 - 75 T + 2441 T^{2} + 498150 T^{3} - 41635338 T^{4} + 2447410950 T^{5} + 58919805929 T^{6} - 8894090737275 T^{7} + 582622237229761 T^{8}$$
$19$ $$1 - 57 T + 7942 T^{2} - 390963 T^{3} + 47045881 T^{4}$$
$23$ $$1 + T - 19059 T^{2} - 5274 T^{3} + 215235544 T^{4} - 64168758 T^{5} - 2821416008451 T^{6} + 1801152661463 T^{7} + 21914624432020321 T^{8}$$
$29$ $$1 - 85 T - 12681 T^{2} + 2454120 T^{3} - 374785010 T^{4} + 59853532680 T^{5} - 7542954533601 T^{6} - 1233107407948865 T^{7} + 353814783205469041 T^{8}$$
$31$ $$( 1 - 22 T + 59046 T^{2} - 655402 T^{3} + 887503681 T^{4} )^{2}$$
$37$ $$( 1 - 448 T + 127830 T^{2} - 22692544 T^{3} + 2565726409 T^{4} )^{2}$$
$41$ $$1 + 124 T - 126237 T^{2} + 467604 T^{3} + 14244408232 T^{4} + 32227735284 T^{5} - 599638909071117 T^{6} + 40595359864851164 T^{7} + 22563490300366186081 T^{8}$$
$43$ $$1 - 311 T - 78425 T^{2} - 5017052 T^{3} + 16664761720 T^{4} - 398890753364 T^{5} - 495752897117825 T^{6} - 156306302312358173 T^{7} + 39959630797262576401 T^{8}$$
$47$ $$1 + 411 T - 53197 T^{2} + 5947992 T^{3} + 21019305612 T^{4} + 617538373416 T^{5} - 573421917856813 T^{6} + 459962624445237237 T^{7} +$$$$11\!\cdots\!41$$$$T^{8}$$
$53$ $$1 + 261 T - 67795 T^{2} - 42239718 T^{3} - 13832852190 T^{4} - 6288522496686 T^{5} - 1502632862740555 T^{6} + 861238297460356713 T^{7} +$$$$49\!\cdots\!41$$$$T^{8}$$
$59$ $$1 + 204 T - 189673 T^{2} - 36611676 T^{3} + 2767015416 T^{4} - 7519269405204 T^{5} - 8000508357289393 T^{6} + 1767251147005607556 T^{7} +$$$$17\!\cdots\!81$$$$T^{8}$$
$61$ $$1 + 531 T - 241597 T^{2} + 36955476 T^{3} + 158592815262 T^{4} + 8388190897956 T^{5} - 12447167884494517 T^{6} + 6209591575294928871 T^{7} +$$$$26\!\cdots\!21$$$$T^{8}$$
$67$ $$1 + 556 T + 109279 T^{2} - 223327964 T^{3} - 143492232488 T^{4} - 67168788436532 T^{5} + 9885201545046151 T^{6} + 15126833124339990532 T^{7} +$$$$81\!\cdots\!61$$$$T^{8}$$
$71$ $$1 + 1563 T + 1116569 T^{2} + 954333414 T^{3} + 756871198320 T^{4} + 341566426538154 T^{5} + 143032805917387049 T^{6} + 71661206622935835453 T^{7} +$$$$16\!\cdots\!41$$$$T^{8}$$
$73$ $$1 - 234 T - 595639 T^{2} + 29867526 T^{3} + 250378414884 T^{4} + 11618975361942 T^{5} - 90140567212553671 T^{6} - 13775951289734691642 T^{7} +$$$$22\!\cdots\!21$$$$T^{8}$$
$79$ $$1 - 331 T - 886369 T^{2} - 3261012 T^{3} + 694771263500 T^{4} - 1607806095468 T^{5} - 215465184862693249 T^{6} - 39670878270246663589 T^{7} +$$$$59\!\cdots\!41$$$$T^{8}$$
$83$ $$( 1 - 1459 T + 1202686 T^{2} - 834237233 T^{3} + 326940373369 T^{4} )^{2}$$
$89$ $$1 + 601 T - 1025139 T^{2} - 14182398 T^{3} + 1170321796870 T^{4} - 9998150935662 T^{5} - 509474903634468579 T^{6} +$$$$21\!\cdots\!09$$$$T^{7} +$$$$24\!\cdots\!21$$$$T^{8}$$
$97$ $$1 - 324 T - 376477 T^{2} + 435421332 T^{3} - 696983774568 T^{4} + 397397293340436 T^{5} - 313594801499655133 T^{6} -$$$$24\!\cdots\!08$$$$T^{7} +$$$$69\!\cdots\!41$$$$T^{8}$$