# Properties

 Label 19.4.c.a 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: $$\Q(\sqrt{-3}, \sqrt{55})$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{2} + ( \beta_{1} + \beta_{3} ) q^{3} + ( 7 + 7 \beta_{2} ) q^{4} + ( -\beta_{1} - 7 \beta_{2} - \beta_{3} ) q^{5} -\beta_{1} q^{6} + ( -7 - \beta_{3} ) q^{7} -15 q^{8} + ( -28 - 28 \beta_{2} ) q^{9} +O(q^{10})$$ $$q + \beta_{2} q^{2} + ( \beta_{1} + \beta_{3} ) q^{3} + ( 7 + 7 \beta_{2} ) q^{4} + ( -\beta_{1} - 7 \beta_{2} - \beta_{3} ) q^{5} -\beta_{1} q^{6} + ( -7 - \beta_{3} ) q^{7} -15 q^{8} + ( -28 - 28 \beta_{2} ) q^{9} + ( 7 + \beta_{1} + 7 \beta_{2} ) q^{10} + ( 14 - 7 \beta_{3} ) q^{11} + 7 \beta_{3} q^{12} + ( 14 - 8 \beta_{1} + 14 \beta_{2} ) q^{13} + ( \beta_{1} - 7 \beta_{2} + \beta_{3} ) q^{14} + ( 55 + 7 \beta_{1} + 55 \beta_{2} ) q^{15} + 41 \beta_{2} q^{16} + ( 8 \beta_{1} - 56 \beta_{2} + 8 \beta_{3} ) q^{17} + 28 q^{18} + ( -70 + 7 \beta_{1} - 42 \beta_{2} + 8 \beta_{3} ) q^{19} + ( 49 - 7 \beta_{3} ) q^{20} + ( -7 \beta_{1} + 55 \beta_{2} - 7 \beta_{3} ) q^{21} + ( 7 \beta_{1} + 14 \beta_{2} + 7 \beta_{3} ) q^{22} + ( -57 - 7 \beta_{1} - 57 \beta_{2} ) q^{23} + ( -15 \beta_{1} - 15 \beta_{3} ) q^{24} + ( 21 - 14 \beta_{1} + 21 \beta_{2} ) q^{25} + ( -14 - 8 \beta_{3} ) q^{26} -\beta_{3} q^{27} + ( -49 + 7 \beta_{1} - 49 \beta_{2} ) q^{28} + ( -111 + 7 \beta_{1} - 111 \beta_{2} ) q^{29} + ( -55 + 7 \beta_{3} ) q^{30} + ( 133 - 7 \beta_{3} ) q^{31} + ( -161 - 161 \beta_{2} ) q^{32} + ( 14 \beta_{1} + 385 \beta_{2} + 14 \beta_{3} ) q^{33} + ( 56 - 8 \beta_{1} + 56 \beta_{2} ) q^{34} -6 \beta_{2} q^{35} -196 \beta_{2} q^{36} + ( 91 - 7 \beta_{3} ) q^{37} + ( 42 - 8 \beta_{1} - 28 \beta_{2} - \beta_{3} ) q^{38} + ( 440 + 14 \beta_{3} ) q^{39} + ( 15 \beta_{1} + 105 \beta_{2} + 15 \beta_{3} ) q^{40} + ( -34 \beta_{1} + 77 \beta_{2} - 34 \beta_{3} ) q^{41} + ( -55 + 7 \beta_{1} - 55 \beta_{2} ) q^{42} + ( 42 \beta_{1} + 134 \beta_{2} + 42 \beta_{3} ) q^{43} + ( 98 + 49 \beta_{1} + 98 \beta_{2} ) q^{44} + ( -196 + 28 \beta_{3} ) q^{45} + ( 57 - 7 \beta_{3} ) q^{46} + ( -63 - 15 \beta_{1} - 63 \beta_{2} ) q^{47} -41 \beta_{1} q^{48} + ( -239 + 14 \beta_{3} ) q^{49} + ( -21 - 14 \beta_{3} ) q^{50} + ( -440 + 56 \beta_{1} - 440 \beta_{2} ) q^{51} + ( -56 \beta_{1} + 98 \beta_{2} - 56 \beta_{3} ) q^{52} + ( 442 - 28 \beta_{1} + 442 \beta_{2} ) q^{53} + ( \beta_{1} + \beta_{3} ) q^{54} + ( -63 \beta_{1} - 483 \beta_{2} - 63 \beta_{3} ) q^{55} + ( 105 + 15 \beta_{3} ) q^{56} + ( -385 - 28 \beta_{1} - 440 \beta_{2} - 70 \beta_{3} ) q^{57} + ( 111 + 7 \beta_{3} ) q^{58} + ( 13 \beta_{1} + 56 \beta_{2} + 13 \beta_{3} ) q^{59} + ( 49 \beta_{1} + 385 \beta_{2} + 49 \beta_{3} ) q^{60} + ( -273 + 29 \beta_{1} - 273 \beta_{2} ) q^{61} + ( 7 \beta_{1} + 133 \beta_{2} + 7 \beta_{3} ) q^{62} + ( 196 - 28 \beta_{1} + 196 \beta_{2} ) q^{63} -167 q^{64} + ( -342 + 42 \beta_{3} ) q^{65} + ( -385 - 14 \beta_{1} - 385 \beta_{2} ) q^{66} + ( 370 + 63 \beta_{1} + 370 \beta_{2} ) q^{67} + ( 392 + 56 \beta_{3} ) q^{68} + ( 385 - 57 \beta_{3} ) q^{69} + ( 6 + 6 \beta_{2} ) q^{70} + ( 42 \beta_{1} + 216 \beta_{2} + 42 \beta_{3} ) q^{71} + ( 420 + 420 \beta_{2} ) q^{72} + ( 6 \beta_{1} + 175 \beta_{2} + 6 \beta_{3} ) q^{73} + ( 7 \beta_{1} + 91 \beta_{2} + 7 \beta_{3} ) q^{74} + ( 770 + 21 \beta_{3} ) q^{75} + ( -196 - 7 \beta_{1} - 490 \beta_{2} + 49 \beta_{3} ) q^{76} + ( 287 + 35 \beta_{3} ) q^{77} + ( -14 \beta_{1} + 440 \beta_{2} - 14 \beta_{3} ) q^{78} + ( -56 \beta_{1} - 76 \beta_{2} - 56 \beta_{3} ) q^{79} + ( 287 + 41 \beta_{1} + 287 \beta_{2} ) q^{80} -701 \beta_{2} q^{81} + ( -77 + 34 \beta_{1} - 77 \beta_{2} ) q^{82} + ( -952 - 35 \beta_{3} ) q^{83} + ( -385 - 49 \beta_{3} ) q^{84} + ( 48 + 48 \beta_{2} ) q^{85} + ( -134 - 42 \beta_{1} - 134 \beta_{2} ) q^{86} + ( -385 - 111 \beta_{3} ) q^{87} + ( -210 + 105 \beta_{3} ) q^{88} + ( -56 - 104 \beta_{1} - 56 \beta_{2} ) q^{89} + ( -28 \beta_{1} - 196 \beta_{2} - 28 \beta_{3} ) q^{90} + ( -538 + 70 \beta_{1} - 538 \beta_{2} ) q^{91} + ( -49 \beta_{1} - 399 \beta_{2} - 49 \beta_{3} ) q^{92} + ( 133 \beta_{1} + 385 \beta_{2} + 133 \beta_{3} ) q^{93} + ( 63 - 15 \beta_{3} ) q^{94} + ( 91 + 84 \beta_{1} + 636 \beta_{2} + 77 \beta_{3} ) q^{95} -161 \beta_{3} q^{96} + ( -62 \beta_{1} - 273 \beta_{2} - 62 \beta_{3} ) q^{97} + ( -14 \beta_{1} - 239 \beta_{2} - 14 \beta_{3} ) q^{98} + ( -392 - 196 \beta_{1} - 392 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 2q^{2} + 14q^{4} + 14q^{5} - 28q^{7} - 60q^{8} - 56q^{9} + O(q^{10})$$ $$4q - 2q^{2} + 14q^{4} + 14q^{5} - 28q^{7} - 60q^{8} - 56q^{9} + 14q^{10} + 56q^{11} + 28q^{13} + 14q^{14} + 110q^{15} - 82q^{16} + 112q^{17} + 112q^{18} - 196q^{19} + 196q^{20} - 110q^{21} - 28q^{22} - 114q^{23} + 42q^{25} - 56q^{26} - 98q^{28} - 222q^{29} - 220q^{30} + 532q^{31} - 322q^{32} - 770q^{33} + 112q^{34} + 12q^{35} + 392q^{36} + 364q^{37} + 224q^{38} + 1760q^{39} - 210q^{40} - 154q^{41} - 110q^{42} - 268q^{43} + 196q^{44} - 784q^{45} + 228q^{46} - 126q^{47} - 956q^{49} - 84q^{50} - 880q^{51} - 196q^{52} + 884q^{53} + 966q^{55} + 420q^{56} - 660q^{57} + 444q^{58} - 112q^{59} - 770q^{60} - 546q^{61} - 266q^{62} + 392q^{63} - 668q^{64} - 1368q^{65} - 770q^{66} + 740q^{67} + 1568q^{68} + 1540q^{69} + 12q^{70} - 432q^{71} + 840q^{72} - 350q^{73} - 182q^{74} + 3080q^{75} + 196q^{76} + 1148q^{77} - 880q^{78} + 152q^{79} + 574q^{80} + 1402q^{81} - 154q^{82} - 3808q^{83} - 1540q^{84} + 96q^{85} - 268q^{86} - 1540q^{87} - 840q^{88} - 112q^{89} + 392q^{90} - 1076q^{91} + 798q^{92} - 770q^{93} + 252q^{94} - 908q^{95} + 546q^{97} + 478q^{98} - 784q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 55 x^{2} + 3025$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/55$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/55$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$55 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$55 \beta_{3}$$

## 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
 3.70810 − 6.42262i −3.70810 + 6.42262i 3.70810 + 6.42262i −3.70810 − 6.42262i
−0.500000 0.866025i −3.70810 6.42262i 3.50000 6.06218i 7.20810 + 12.4848i −3.70810 + 6.42262i 0.416198 −15.0000 −14.0000 + 24.2487i 7.20810 12.4848i
7.2 −0.500000 0.866025i 3.70810 + 6.42262i 3.50000 6.06218i −0.208099 0.360438i 3.70810 6.42262i −14.4162 −15.0000 −14.0000 + 24.2487i −0.208099 + 0.360438i
11.1 −0.500000 + 0.866025i −3.70810 + 6.42262i 3.50000 + 6.06218i 7.20810 12.4848i −3.70810 6.42262i 0.416198 −15.0000 −14.0000 24.2487i 7.20810 + 12.4848i
11.2 −0.500000 + 0.866025i 3.70810 6.42262i 3.50000 + 6.06218i −0.208099 + 0.360438i 3.70810 + 6.42262i −14.4162 −15.0000 −14.0000 24.2487i −0.208099 0.360438i
 $$n$$: 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.a 4
3.b odd 2 1 171.4.f.e 4
4.b odd 2 1 304.4.i.c 4
19.c even 3 1 inner 19.4.c.a 4
19.c even 3 1 361.4.a.g 2
19.d odd 6 1 361.4.a.d 2
57.h odd 6 1 171.4.f.e 4
76.g odd 6 1 304.4.i.c 4

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + T_{2} + 1$$ acting on $$S_{4}^{\mathrm{new}}(19, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T - 7 T^{2} + 8 T^{3} + 64 T^{4} )^{2}$$
$3$ $$1 + T^{2} - 728 T^{4} + 729 T^{6} + 531441 T^{8}$$
$5$ $$1 - 14 T - 48 T^{2} + 84 T^{3} + 19411 T^{4} + 10500 T^{5} - 750000 T^{6} - 27343750 T^{7} + 244140625 T^{8}$$
$7$ $$( 1 + 14 T + 680 T^{2} + 4802 T^{3} + 117649 T^{4} )^{2}$$
$11$ $$( 1 - 28 T + 163 T^{2} - 37268 T^{3} + 1771561 T^{4} )^{2}$$
$13$ $$1 - 28 T - 286 T^{2} + 93072 T^{3} - 5404357 T^{4} + 204479184 T^{5} - 1380467374 T^{6} - 296925982444 T^{7} + 23298085122481 T^{8}$$
$17$ $$1 - 112 T + 3102 T^{2} + 43008 T^{3} + 3385123 T^{4} + 211298304 T^{5} + 74874739038 T^{6} - 13281842167664 T^{7} + 582622237229761 T^{8}$$
$19$ $$1 + 196 T + 18867 T^{2} + 1344364 T^{3} + 47045881 T^{4}$$
$23$ $$1 + 114 T - 11892 T^{2} + 63156 T^{3} + 313254323 T^{4} + 768419052 T^{5} - 1760442791988 T^{6} + 205331403406782 T^{7} + 21914624432020321 T^{8}$$
$29$ $$1 + 222 T - 9120 T^{2} + 2136972 T^{3} + 1614216419 T^{4} + 52118610108 T^{5} - 5424788687520 T^{6} + 3220586406642918 T^{7} + 353814783205469041 T^{8}$$
$31$ $$( 1 - 266 T + 74576 T^{2} - 7924406 T^{3} + 887503681 T^{4} )^{2}$$
$37$ $$( 1 - 182 T + 106892 T^{2} - 9218846 T^{3} + 2565726409 T^{4} )^{2}$$
$41$ $$1 + 154 T - 56475 T^{2} - 8878254 T^{3} + 45961804 T^{4} - 611898143934 T^{5} - 268262137010475 T^{6} + 50416817896669994 T^{7} + 22563490300366186081 T^{8}$$
$43$ $$1 + 268 T - 8126 T^{2} - 21189152 T^{3} - 5639871317 T^{4} - 1684685908064 T^{5} - 51367396136174 T^{6} + 134694819999073924 T^{7} + 39959630797262576401 T^{8}$$
$47$ $$1 + 126 T - 183364 T^{2} - 1059156 T^{3} + 27269068323 T^{4} - 109964753388 T^{5} - 1976520039586756 T^{6} + 141010439610948642 T^{7} +$$$$11\!\cdots\!41$$$$T^{8}$$
$53$ $$1 - 884 T + 331458 T^{2} - 134583696 T^{3} + 63993013963 T^{4} - 20036416909392 T^{5} + 7346554811096082 T^{6} - 2916991015153085572 T^{7} +$$$$49\!\cdots\!41$$$$T^{8}$$
$59$ $$1 + 112 T - 392055 T^{2} - 689808 T^{3} + 118943542984 T^{4} - 141672077232 T^{5} - 16537089116622255 T^{6} + 970255531689353168 T^{7} +$$$$17\!\cdots\!81$$$$T^{8}$$
$61$ $$1 + 546 T - 184120 T^{2} + 15437604 T^{3} + 113364517539 T^{4} + 3504042793524 T^{5} - 9485931327347320 T^{6} + 6385003766687440986 T^{7} +$$$$26\!\cdots\!21$$$$T^{8}$$
$67$ $$1 - 740 T + 27469 T^{2} + 60232300 T^{3} + 15380056192 T^{4} + 18115647244900 T^{5} + 2484801299800261 T^{6} - 20132835453258260780 T^{7} +$$$$81\!\cdots\!61$$$$T^{8}$$
$71$ $$1 + 432 T - 478834 T^{2} - 21757248 T^{3} + 247939283379 T^{4} - 7787158388928 T^{5} - 61338771351028114 T^{6} + 19806552310369981392 T^{7} +$$$$16\!\cdots\!41$$$$T^{8}$$
$73$ $$1 + 350 T - 684179 T^{2} + 10025750 T^{3} + 451742200252 T^{4} + 3900187187750 T^{5} - 103539699608181731 T^{6} + 20605055347893769550 T^{7} +$$$$22\!\cdots\!21$$$$T^{8}$$
$79$ $$1 - 152 T - 796270 T^{2} + 25339008 T^{3} + 416895123299 T^{4} + 12493119165312 T^{5} - 193563248207706670 T^{6} - 18217442589357984488 T^{7} +$$$$59\!\cdots\!41$$$$T^{8}$$
$83$ $$( 1 + 1904 T + 1982503 T^{2} + 1088682448 T^{3} + 326940373369 T^{4} )^{2}$$
$89$ $$1 + 112 T - 805650 T^{2} - 66275328 T^{3} + 163616999539 T^{4} - 46722051704832 T^{5} - 400392977062729650 T^{6} + 39239917215238343408 T^{7} +$$$$24\!\cdots\!21$$$$T^{8}$$
$97$ $$1 - 546 T - 1390339 T^{2} + 74742486 T^{3} + 1745825858028 T^{4} + 68215448925078 T^{5} - 1158113464360980931 T^{6} -$$$$41\!\cdots\!82$$$$T^{7} +$$$$69\!\cdots\!41$$$$T^{8}$$