# Properties

 Label 19.8.a.a Level 19 Weight 8 Character orbit 19.a Self dual Yes Analytic conductor 5.935 Analytic rank 1 Dimension 4 CM No Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ = $$19$$ Weight: $$k$$ = $$8$$ Character orbit: $$[\chi]$$ = 19.a (trivial)

## Newform invariants

 Self dual: Yes Analytic conductor: $$5.9353154842$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: $$\mathbb{Q}[x]/(x^{4} - \cdots)$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2\cdot 3$$ Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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.

 $$f(q)$$ $$=$$ $$q$$ $$+ ( -2 + \beta_{1} ) q^{2}$$ $$+ ( -4 - \beta_{1} - \beta_{2} ) q^{3}$$ $$+ ( 8 - 10 \beta_{1} + 4 \beta_{2} - \beta_{3} ) q^{4}$$ $$+ ( -59 - 11 \beta_{1} + 3 \beta_{2} + 6 \beta_{3} ) q^{5}$$ $$+ ( -156 - 18 \beta_{1} - 6 \beta_{2} - 3 \beta_{3} ) q^{6}$$ $$+ ( -311 - 20 \beta_{1} - 4 \beta_{2} - 26 \beta_{3} ) q^{7}$$ $$+ ( -892 + 50 \beta_{1} - 36 \beta_{2} + 27 \beta_{3} ) q^{8}$$ $$+ ( -1201 + 77 \beta_{1} + 23 \beta_{2} + 6 \beta_{3} ) q^{9}$$ $$+O(q^{10})$$ $$q$$ $$+ ( -2 + \beta_{1} ) q^{2}$$ $$+ ( -4 - \beta_{1} - \beta_{2} ) q^{3}$$ $$+ ( 8 - 10 \beta_{1} + 4 \beta_{2} - \beta_{3} ) q^{4}$$ $$+ ( -59 - 11 \beta_{1} + 3 \beta_{2} + 6 \beta_{3} ) q^{5}$$ $$+ ( -156 - 18 \beta_{1} - 6 \beta_{2} - 3 \beta_{3} ) q^{6}$$ $$+ ( -311 - 20 \beta_{1} - 4 \beta_{2} - 26 \beta_{3} ) q^{7}$$ $$+ ( -892 + 50 \beta_{1} - 36 \beta_{2} + 27 \beta_{3} ) q^{8}$$ $$+ ( -1201 + 77 \beta_{1} + 23 \beta_{2} + 6 \beta_{3} ) q^{9}$$ $$+ ( -1598 + 83 \beta_{1} - 14 \beta_{2} + 17 \beta_{3} ) q^{10}$$ $$+ ( -2121 + 177 \beta_{1} + 87 \beta_{2} + 30 \beta_{3} ) q^{11}$$ $$+ ( -1564 - 10 \beta_{1} + 32 \beta_{2} - 3 \beta_{3} ) q^{12}$$ $$+ ( -1054 - 99 \beta_{1} + 123 \beta_{2} - 240 \beta_{3} ) q^{13}$$ $$+ ( -586 - 187 \beta_{1} - 192 \beta_{2} + 30 \beta_{3} ) q^{14}$$ $$+ ( -690 + 60 \beta_{1} + 90 \beta_{2} + 150 \beta_{3} ) q^{15}$$ $$+ ( 4588 - 858 \beta_{1} - 276 \beta_{2} - 93 \beta_{3} ) q^{16}$$ $$+ ( -1175 + 778 \beta_{1} - 510 \beta_{2} + 528 \beta_{3} ) q^{17}$$ $$+ ( 12942 - 1323 \beta_{1} + 378 \beta_{2} + 9 \beta_{3} ) q^{18}$$ $$+ 6859 q^{19}$$ $$+ ( 20236 - 1196 \beta_{1} - 12 \beta_{2} - 924 \beta_{3} ) q^{20}$$ $$+ ( 9100 + 1183 \beta_{1} + 619 \beta_{2} - 474 \beta_{3} ) q^{21}$$ $$+ ( 28590 - 1683 \beta_{1} + 1002 \beta_{2} + 141 \beta_{3} ) q^{22}$$ $$+ ( -7042 + 4217 \beta_{1} - 429 \beta_{2} + 1428 \beta_{3} ) q^{23}$$ $$+ ( 22980 + 1530 \beta_{1} + 780 \beta_{2} + 525 \beta_{3} ) q^{24}$$ $$+ ( -5722 + 887 \beta_{1} - 1931 \beta_{2} - 2062 \beta_{3} ) q^{25}$$ $$+ ( 7376 + 2924 \beta_{1} - 1110 \beta_{2} + 831 \beta_{3} ) q^{26}$$ $$+ ( -17926 + 569 \beta_{1} + 2495 \beta_{2} - 174 \beta_{3} ) q^{27}$$ $$+ ( 8352 - 814 \beta_{1} - 500 \beta_{2} + 2717 \beta_{3} ) q^{28}$$ $$+ ( -63172 + 953 \beta_{1} - 2907 \beta_{2} - 3510 \beta_{3} ) q^{29}$$ $$+ ( 3180 + 510 \beta_{1} + 1020 \beta_{2} + 150 \beta_{3} ) q^{30}$$ $$+ ( -79472 - 7694 \beta_{1} - 2428 \beta_{2} + 4306 \beta_{3} ) q^{31}$$ $$+ ( -11508 - 834 \beta_{1} + 252 \beta_{2} - 3609 \beta_{3} ) q^{32}$$ $$+ ( -92226 - 6330 \beta_{1} - 372 \beta_{2} - 162 \beta_{3} ) q^{33}$$ $$+ ( 57046 - 19675 \beta_{1} + 4204 \beta_{2} - 3346 \beta_{3} ) q^{34}$$ $$+ ( -140419 - 2431 \beta_{1} + 6363 \beta_{2} + 1746 \beta_{3} ) q^{35}$$ $$+ ( -35236 + 21968 \beta_{1} - 7444 \beta_{2} + 2058 \beta_{3} ) q^{36}$$ $$+ ( -69986 - 18722 \beta_{1} + 9572 \beta_{2} + 334 \beta_{3} ) q^{37}$$ $$+ ( -13718 + 6859 \beta_{1} ) q^{38}$$ $$+ ( -66206 - 1085 \beta_{1} + 1453 \beta_{2} - 5112 \beta_{3} ) q^{39}$$ $$+ ( 61256 + 20764 \beta_{1} - 6712 \beta_{2} - 104 \beta_{3} ) q^{40}$$ $$+ ( -207854 + 17284 \beta_{1} - 6306 \beta_{2} + 13830 \beta_{3} ) q^{41}$$ $$+ ( 186204 + 14202 \beta_{1} + 4074 \beta_{2} + 1767 \beta_{3} ) q^{42}$$ $$+ ( 2927 - 15777 \beta_{1} - 12495 \beta_{2} - 2526 \beta_{3} ) q^{43}$$ $$+ ( 15756 + 41160 \beta_{1} - 15300 \beta_{2} + 1710 \beta_{3} ) q^{44}$$ $$+ ( 41613 + 17277 \beta_{1} - 7941 \beta_{2} - 10422 \beta_{3} ) q^{45}$$ $$+ ( 471320 - 53072 \beta_{1} + 21722 \beta_{2} - 7361 \beta_{3} ) q^{46}$$ $$+ ( 88385 - 30067 \beta_{1} + 33855 \beta_{2} - 14082 \beta_{3} ) q^{47}$$ $$+ ( 349652 + 28130 \beta_{1} + 5684 \beta_{2} + 1449 \beta_{3} ) q^{48}$$ $$+ ( -16918 + 29344 \beta_{1} - 15400 \beta_{2} + 14056 \beta_{3} ) q^{49}$$ $$+ ( 190456 - 51176 \beta_{1} - 8562 \beta_{2} - 6549 \beta_{3} ) q^{50}$$ $$+ ( 260712 + 7365 \beta_{1} - 1551 \beta_{2} + 10284 \beta_{3} ) q^{51}$$ $$+ ( 420748 - 29426 \beta_{1} - 2944 \beta_{2} + 22525 \beta_{3} ) q^{52}$$ $$+ ( 3542 - 61405 \beta_{1} + 6159 \beta_{2} + 7314 \beta_{3} ) q^{53}$$ $$+ ( 201240 + 32760 \beta_{1} + 6570 \beta_{2} + 9585 \beta_{3} ) q^{54}$$ $$+ ( 261861 + 33129 \beta_{1} - 23757 \beta_{2} - 29154 \beta_{3} ) q^{55}$$ $$+ ( -228164 + 22366 \beta_{1} + 31188 \beta_{2} - 7743 \beta_{3} ) q^{56}$$ $$+ ( -27436 - 6859 \beta_{1} - 6859 \beta_{2} ) q^{57}$$ $$+ ( 369716 - 127730 \beta_{1} - 16042 \beta_{2} - 9071 \beta_{3} ) q^{58}$$ $$+ ( -767012 + 122233 \beta_{1} + 10575 \beta_{2} + 9522 \beta_{3} ) q^{59}$$ $$+ ( 172920 + 13560 \beta_{1} - 6840 \beta_{2} - 15780 \beta_{3} ) q^{60}$$ $$+ ( 82493 - 30189 \beta_{1} - 13935 \beta_{2} - 27546 \beta_{3} ) q^{61}$$ $$+ ( -1192720 - 79948 \beta_{1} - 18408 \beta_{2} - 6324 \beta_{3} ) q^{62}$$ $$+ ( -24521 - 17087 \beta_{1} - 14729 \beta_{2} + 41502 \beta_{3} ) q^{63}$$ $$+ ( -449732 + 117750 \beta_{1} + 18060 \beta_{2} + 17355 \beta_{3} ) q^{64}$$ $$+ ( -1127764 - 49516 \beta_{1} + 48468 \beta_{2} - 2964 \beta_{3} ) q^{65}$$ $$+ ( -653292 - 49446 \beta_{1} - 26712 \beta_{2} + 5004 \beta_{3} ) q^{66}$$ $$+ ( -33310 + 100737 \beta_{1} + 74733 \beta_{2} + 38388 \beta_{3} ) q^{67}$$ $$+ ( -2225504 + 214042 \beta_{1} - 18396 \beta_{2} - 27747 \beta_{3} ) q^{68}$$ $$+ ( -391902 - 68841 \beta_{1} - 27687 \beta_{2} + 18624 \beta_{3} ) q^{69}$$ $$+ ( 58802 + 15523 \beta_{1} + 9986 \beta_{2} + 26137 \beta_{3} ) q^{70}$$ $$+ ( -542294 - 70856 \beta_{1} - 85524 \beta_{2} - 53328 \beta_{3} ) q^{71}$$ $$+ ( 951984 - 209520 \beta_{1} + 32832 \beta_{2} - 54954 \beta_{3} ) q^{72}$$ $$+ ( -37495 + 204564 \beta_{1} + 23496 \beta_{2} - 38004 \beta_{3} ) q^{73}$$ $$+ ( -2045068 + 289706 \beta_{1} - 54408 \beta_{2} + 56676 \beta_{3} ) q^{74}$$ $$+ ( 1541030 + 110945 \beta_{1} + 28115 \beta_{2} - 40170 \beta_{3} ) q^{75}$$ $$+ ( 54872 - 68590 \beta_{1} + 27436 \beta_{2} - 6859 \beta_{3} ) q^{76}$$ $$+ ( -893121 - 91371 \beta_{1} - 36297 \beta_{2} + 94302 \beta_{3} ) q^{77}$$ $$+ ( 342408 - 15336 \beta_{1} - 21882 \beta_{2} + 12009 \beta_{3} ) q^{78}$$ $$+ ( 4023872 - 24108 \beta_{1} + 39378 \beta_{2} - 54354 \beta_{3} ) q^{79}$$ $$+ ( -180416 - 99224 \beta_{1} + 70752 \beta_{2} + 70764 \beta_{3} ) q^{80}$$ $$+ ( 603053 - 292348 \beta_{1} - 63676 \beta_{2} - 25968 \beta_{3} ) q^{81}$$ $$+ ( 1665604 - 512518 \beta_{1} + 111844 \beta_{2} - 56338 \beta_{3} ) q^{82}$$ $$+ ( 2086224 - 173070 \beta_{1} + 40110 \beta_{2} + 46272 \beta_{3} ) q^{83}$$ $$+ ( 361804 + 7258 \beta_{1} - 7208 \beta_{2} + 60999 \beta_{3} ) q^{84}$$ $$+ ( 1093417 + 199973 \beta_{1} - 99089 \beta_{2} + 50402 \beta_{3} ) q^{85}$$ $$+ ( -2336698 - 140695 \beta_{1} - 98202 \beta_{2} - 31677 \beta_{3} ) q^{86}$$ $$+ ( 2621958 + 233283 \beta_{1} + 101565 \beta_{2} - 67848 \beta_{3} ) q^{87}$$ $$+ ( 1149888 - 438120 \beta_{1} + 12624 \beta_{2} - 122118 \beta_{3} ) q^{88}$$ $$+ ( 803576 + 204626 \beta_{1} + 1716 \beta_{2} - 108870 \beta_{3} ) q^{89}$$ $$+ ( 2568546 - 250461 \beta_{1} + 11538 \beta_{2} - 38619 \beta_{3} ) q^{90}$$ $$+ ( 5714162 + 75005 \beta_{1} - 230993 \beta_{2} + 176582 \beta_{3} ) q^{91}$$ $$+ ( -5910004 + 848726 \beta_{1} - 143376 \beta_{2} - 35463 \beta_{3} ) q^{92}$$ $$+ ( 3312760 + 318976 \beta_{1} + 150508 \beta_{2} + 120792 \beta_{3} ) q^{93}$$ $$+ ( -2217334 + 1101895 \beta_{1} - 108886 \beta_{2} + 179569 \beta_{3} ) q^{94}$$ $$+ ( -404681 - 75449 \beta_{1} + 20577 \beta_{2} + 41154 \beta_{3} ) q^{95}$$ $$+ ( 167364 + 50922 \beta_{1} + 29844 \beta_{2} - 74043 \beta_{3} ) q^{96}$$ $$+ ( -969756 + 482594 \beta_{1} + 216394 \beta_{2} - 188896 \beta_{3} ) q^{97}$$ $$+ ( 2571084 - 618582 \beta_{1} + 142800 \beta_{2} - 105000 \beta_{3} ) q^{98}$$ $$+ ( 6353907 - 146937 \beta_{1} - 42663 \beta_{2} - 48906 \beta_{3} ) q^{99}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q$$ $$\mathstrut -\mathstrut 9q^{2}$$ $$\mathstrut -\mathstrut 14q^{3}$$ $$\mathstrut +\mathstrut 37q^{4}$$ $$\mathstrut -\mathstrut 222q^{5}$$ $$\mathstrut -\mathstrut 603q^{6}$$ $$\mathstrut -\mathstrut 1246q^{7}$$ $$\mathstrut -\mathstrut 3555q^{8}$$ $$\mathstrut -\mathstrut 4898q^{9}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$4q$$ $$\mathstrut -\mathstrut 9q^{2}$$ $$\mathstrut -\mathstrut 14q^{3}$$ $$\mathstrut +\mathstrut 37q^{4}$$ $$\mathstrut -\mathstrut 222q^{5}$$ $$\mathstrut -\mathstrut 603q^{6}$$ $$\mathstrut -\mathstrut 1246q^{7}$$ $$\mathstrut -\mathstrut 3555q^{8}$$ $$\mathstrut -\mathstrut 4898q^{9}$$ $$\mathstrut -\mathstrut 6444q^{10}$$ $$\mathstrut -\mathstrut 8718q^{11}$$ $$\mathstrut -\mathstrut 6281q^{12}$$ $$\mathstrut -\mathstrut 4480q^{13}$$ $$\mathstrut -\mathstrut 1935q^{14}$$ $$\mathstrut -\mathstrut 2760q^{15}$$ $$\mathstrut +\mathstrut 19393q^{16}$$ $$\mathstrut -\mathstrut 4440q^{17}$$ $$\mathstrut +\mathstrut 52722q^{18}$$ $$\mathstrut +\mathstrut 27436q^{19}$$ $$\mathstrut +\mathstrut 81228q^{20}$$ $$\mathstrut +\mathstrut 34124q^{21}$$ $$\mathstrut +\mathstrut 115182q^{22}$$ $$\mathstrut -\mathstrut 30528q^{23}$$ $$\mathstrut +\mathstrut 90135q^{24}$$ $$\mathstrut -\mathstrut 23906q^{25}$$ $$\mathstrut +\mathstrut 28521q^{26}$$ $$\mathstrut -\mathstrut 74942q^{27}$$ $$\mathstrut +\mathstrut 37439q^{28}$$ $$\mathstrut -\mathstrut 254244q^{29}$$ $$\mathstrut +\mathstrut 11340q^{30}$$ $$\mathstrut -\mathstrut 303460q^{31}$$ $$\mathstrut -\mathstrut 49059q^{32}$$ $$\mathstrut -\mathstrut 362364q^{33}$$ $$\mathstrut +\mathstrut 240309q^{34}$$ $$\mathstrut -\mathstrut 563862q^{35}$$ $$\mathstrut -\mathstrut 153410q^{36}$$ $$\mathstrut -\mathstrut 270460q^{37}$$ $$\mathstrut -\mathstrut 61731q^{38}$$ $$\mathstrut -\mathstrut 270304q^{39}$$ $$\mathstrut +\mathstrut 230868q^{40}$$ $$\mathstrut -\mathstrut 828564q^{41}$$ $$\mathstrut +\mathstrut 728307q^{42}$$ $$\mathstrut +\mathstrut 37454q^{43}$$ $$\mathstrut +\mathstrut 38874q^{44}$$ $$\mathstrut +\mathstrut 146694q^{45}$$ $$\mathstrut +\mathstrut 1909269q^{46}$$ $$\mathstrut +\mathstrut 335670q^{47}$$ $$\mathstrut +\mathstrut 1366243q^{48}$$ $$\mathstrut -\mathstrut 67560q^{49}$$ $$\mathstrut +\mathstrut 815013q^{50}$$ $$\mathstrut +\mathstrut 1047318q^{51}$$ $$\mathstrut +\mathstrut 1737887q^{52}$$ $$\mathstrut +\mathstrut 76728q^{53}$$ $$\mathstrut +\mathstrut 775215q^{54}$$ $$\mathstrut +\mathstrut 1008918q^{55}$$ $$\mathstrut -\mathstrut 973953q^{56}$$ $$\mathstrut -\mathstrut 96026q^{57}$$ $$\mathstrut +\mathstrut 1613565q^{58}$$ $$\mathstrut -\mathstrut 3191334q^{59}$$ $$\mathstrut +\mathstrut 669180q^{60}$$ $$\mathstrut +\mathstrut 346550q^{61}$$ $$\mathstrut -\mathstrut 4678848q^{62}$$ $$\mathstrut -\mathstrut 24766q^{63}$$ $$\mathstrut -\mathstrut 1917383q^{64}$$ $$\mathstrut -\mathstrut 4512972q^{65}$$ $$\mathstrut -\mathstrut 2532006q^{66}$$ $$\mathstrut -\mathstrut 270322q^{67}$$ $$\mathstrut -\mathstrut 9125409q^{68}$$ $$\mathstrut -\mathstrut 1452456q^{69}$$ $$\mathstrut +\mathstrut 235836q^{70}$$ $$\mathstrut -\mathstrut 2066124q^{71}$$ $$\mathstrut +\mathstrut 3929670q^{72}$$ $$\mathstrut -\mathstrut 416044q^{73}$$ $$\mathstrut -\mathstrut 8358894q^{74}$$ $$\mathstrut +\mathstrut 5984890q^{75}$$ $$\mathstrut +\mathstrut 253783q^{76}$$ $$\mathstrut -\mathstrut 3350514q^{77}$$ $$\mathstrut +\mathstrut 1418859q^{78}$$ $$\mathstrut +\mathstrut 16025864q^{79}$$ $$\mathstrut -\mathstrut 622428q^{80}$$ $$\mathstrut +\mathstrut 2742268q^{81}$$ $$\mathstrut +\mathstrut 7006752q^{82}$$ $$\mathstrut +\mathstrut 8524128q^{83}$$ $$\mathstrut +\mathstrut 1508165q^{84}$$ $$\mathstrut +\mathstrut 4323186q^{85}$$ $$\mathstrut -\mathstrut 9139572q^{86}$$ $$\mathstrut +\mathstrut 10085136q^{87}$$ $$\mathstrut +\mathstrut 4902930q^{88}$$ $$\mathstrut +\mathstrut 2899092q^{89}$$ $$\mathstrut +\mathstrut 10474488q^{90}$$ $$\mathstrut +\mathstrut 23189218q^{91}$$ $$\mathstrut -\mathstrut 24380829q^{92}$$ $$\mathstrut +\mathstrut 12902348q^{93}$$ $$\mathstrut -\mathstrut 9682776q^{94}$$ $$\mathstrut -\mathstrut 1522698q^{95}$$ $$\mathstrut +\mathstrut 514647q^{96}$$ $$\mathstrut -\mathstrut 4766908q^{97}$$ $$\mathstrut +\mathstrut 10655118q^{98}$$ $$\mathstrut +\mathstrut 25556322q^{99}$$ $$\mathstrut +\mathstrut O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4}\mathstrut -\mathstrut$$ $$x^{3}\mathstrut -\mathstrut$$ $$255$$ $$x^{2}\mathstrut +\mathstrut$$ $$475$$ $$x\mathstrut +\mathstrut$$ $$500$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} + 4 \nu^{2} - 235 \nu - 300$$$$)/50$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{3} + \nu^{2} + 305 \nu - 375$$$$)/25$$ $$\beta_{3}$$ $$=$$ $$($$$$3 \nu^{3} - 8 \nu^{2} - 685 \nu + 1700$$$$)/50$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3}\mathstrut +\mathstrut$$ $$2$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$2$$$$)/6$$ $$\nu^{2}$$ $$=$$ $$($$$$-$$$$7$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$23$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$391$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$($$$$97$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$154$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$117$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$286$$$$)/2$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 2.64844 −0.751230 −16.3077 15.4104
−19.5150 −3.33349 252.835 170.463 65.0531 31.4807 −2436.16 −2175.89 −3326.58
1.2 −4.43255 22.5580 −108.352 160.438 −99.9896 −1314.43 1047.64 −1678.14 −711.151
1.3 3.18411 20.6576 −117.861 −477.630 65.7761 883.696 −782.850 −1760.26 −1520.83
1.4 11.7634 −53.8822 10.3786 −75.2709 −633.840 −846.751 −1383.63 716.286 −885.446
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Atkin-Lehner signs

$$p$$ Sign
$$19$$ $$-1$$

## Hecke kernels

This newform can be constructed as the kernel of the linear operator $$T_{2}^{4}$$ $$\mathstrut +\mathstrut 9 T_{2}^{3}$$ $$\mathstrut -\mathstrut 234 T_{2}^{2}$$ $$\mathstrut -\mathstrut 396 T_{2}$$ $$\mathstrut +\mathstrut 3240$$ acting on $$S_{8}^{\mathrm{new}}(\Gamma_0(19))$$.