# Properties

 Label 38.6.c.b Level $38$ Weight $6$ Character orbit 38.c Analytic conductor $6.095$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$38 = 2 \cdot 19$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 38.c (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.09458515289$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ Defining polynomial: $$x^{8} - 2 x^{7} + 386 x^{6} + 3436 x^{5} + 128708 x^{4} + 568528 x^{3} + 7340704 x^{2} - 19430784 x + 211527936$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}\cdot 3$$ 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -4 + 4 \beta_{2} ) q^{2} + ( -3 + 3 \beta_{2} - \beta_{3} ) q^{3} -16 \beta_{2} q^{4} + ( -9 + 9 \beta_{2} + \beta_{4} + \beta_{5} ) q^{5} + ( -4 \beta_{1} - 12 \beta_{2} ) q^{6} + ( 9 + \beta_{1} + \beta_{3} - \beta_{6} - \beta_{7} ) q^{7} + 64 q^{8} + ( -14 \beta_{1} + 46 \beta_{2} - 2 \beta_{5} ) q^{9} +O(q^{10})$$ $$q + ( -4 + 4 \beta_{2} ) q^{2} + ( -3 + 3 \beta_{2} - \beta_{3} ) q^{3} -16 \beta_{2} q^{4} + ( -9 + 9 \beta_{2} + \beta_{4} + \beta_{5} ) q^{5} + ( -4 \beta_{1} - 12 \beta_{2} ) q^{6} + ( 9 + \beta_{1} + \beta_{3} - \beta_{6} - \beta_{7} ) q^{7} + 64 q^{8} + ( -14 \beta_{1} + 46 \beta_{2} - 2 \beta_{5} ) q^{9} + ( -36 \beta_{2} - 4 \beta_{5} ) q^{10} + ( 8 + 20 \beta_{1} + 20 \beta_{3} + 3 \beta_{4} - 2 \beta_{6} - 2 \beta_{7} ) q^{11} + ( 48 + 16 \beta_{1} + 16 \beta_{3} ) q^{12} + ( -25 \beta_{1} - 156 \beta_{2} - \beta_{5} - 5 \beta_{7} ) q^{13} + ( -36 + 36 \beta_{2} - 4 \beta_{3} + 4 \beta_{6} ) q^{14} + ( -46 \beta_{1} + 18 \beta_{2} - 9 \beta_{5} - \beta_{7} ) q^{15} + ( -256 + 256 \beta_{2} ) q^{16} + ( 132 - 132 \beta_{2} - 3 \beta_{3} + 15 \beta_{4} + 15 \beta_{5} + 3 \beta_{6} ) q^{17} + ( -184 + 56 \beta_{1} + 56 \beta_{3} - 8 \beta_{4} ) q^{18} + ( 23 + 21 \beta_{1} + 44 \beta_{2} - 41 \beta_{3} + 17 \beta_{4} + 3 \beta_{5} - 10 \beta_{6} - 6 \beta_{7} ) q^{19} + ( 144 - 16 \beta_{4} ) q^{20} + ( -179 + 179 \beta_{2} - 27 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - 9 \beta_{6} ) q^{21} + ( -32 + 32 \beta_{2} - 80 \beta_{3} - 12 \beta_{4} - 12 \beta_{5} + 8 \beta_{6} ) q^{22} + ( -78 \beta_{1} - 12 \beta_{2} - 19 \beta_{5} + 21 \beta_{7} ) q^{23} + ( -192 + 192 \beta_{2} - 64 \beta_{3} ) q^{24} + ( -85 \beta_{1} - 497 \beta_{2} + 45 \beta_{5} + 7 \beta_{7} ) q^{25} + ( 624 + 100 \beta_{1} + 100 \beta_{3} - 4 \beta_{4} + 20 \beta_{6} + 20 \beta_{7} ) q^{26} + ( 1675 - 61 \beta_{1} - 61 \beta_{3} - 46 \beta_{4} + 2 \beta_{6} + 2 \beta_{7} ) q^{27} + ( -16 \beta_{1} - 144 \beta_{2} + 16 \beta_{7} ) q^{28} + ( 330 \beta_{1} - 1593 \beta_{2} + 43 \beta_{5} ) q^{29} + ( -72 + 184 \beta_{1} + 184 \beta_{3} - 36 \beta_{4} + 4 \beta_{6} + 4 \beta_{7} ) q^{30} + ( -221 - 139 \beta_{1} - 139 \beta_{3} - 6 \beta_{4} - 11 \beta_{6} - 11 \beta_{7} ) q^{31} -1024 \beta_{2} q^{32} + ( -3847 + 3847 \beta_{2} - 131 \beta_{3} + 13 \beta_{4} + 13 \beta_{5} - 15 \beta_{6} ) q^{33} + ( -12 \beta_{1} + 528 \beta_{2} - 60 \beta_{5} + 12 \beta_{7} ) q^{34} + ( -1054 + 1054 \beta_{2} + 14 \beta_{3} + 22 \beta_{4} + 22 \beta_{5} + 34 \beta_{6} ) q^{35} + ( 736 - 736 \beta_{2} - 224 \beta_{3} + 32 \beta_{4} + 32 \beta_{5} ) q^{36} + ( -3035 - 307 \beta_{1} - 307 \beta_{3} + 114 \beta_{4} + 25 \beta_{6} + 25 \beta_{7} ) q^{37} + ( -268 - 248 \beta_{1} + 92 \beta_{2} - 84 \beta_{3} - 56 \beta_{4} - 68 \beta_{5} + 24 \beta_{6} - 16 \beta_{7} ) q^{38} + ( 5303 + 433 \beta_{1} + 433 \beta_{3} - 59 \beta_{4} - 44 \beta_{6} - 44 \beta_{7} ) q^{39} + ( -576 + 576 \beta_{2} + 64 \beta_{4} + 64 \beta_{5} ) q^{40} + ( -659 + 659 \beta_{2} - 422 \beta_{3} - 164 \beta_{4} - 164 \beta_{5} + 32 \beta_{6} ) q^{41} + ( -108 \beta_{1} - 716 \beta_{2} - 8 \beta_{5} - 36 \beta_{7} ) q^{42} + ( 423 - 423 \beta_{2} + 1187 \beta_{3} + 95 \beta_{4} + 95 \beta_{5} - 20 \beta_{6} ) q^{43} + ( -320 \beta_{1} - 128 \beta_{2} + 48 \beta_{5} + 32 \beta_{7} ) q^{44} + ( 6038 + 814 \beta_{1} + 814 \beta_{3} + 70 \beta_{4} ) q^{45} + ( 48 + 312 \beta_{1} + 312 \beta_{3} - 76 \beta_{4} - 84 \beta_{6} - 84 \beta_{7} ) q^{46} + ( -12 \beta_{1} - 11436 \beta_{2} - 241 \beta_{5} - 27 \beta_{7} ) q^{47} + ( -256 \beta_{1} - 768 \beta_{2} ) q^{48} + ( 697 - 1040 \beta_{1} - 1040 \beta_{3} - 122 \beta_{4} + 20 \beta_{6} + 20 \beta_{7} ) q^{49} + ( 1988 + 340 \beta_{1} + 340 \beta_{3} + 180 \beta_{4} - 28 \beta_{6} - 28 \beta_{7} ) q^{50} + ( -477 \beta_{1} + 615 \beta_{2} - 141 \beta_{5} - 42 \beta_{7} ) q^{51} + ( -2496 + 2496 \beta_{2} - 400 \beta_{3} + 16 \beta_{4} + 16 \beta_{5} - 80 \beta_{6} ) q^{52} + ( -121 \beta_{1} + 11474 \beta_{2} + 91 \beta_{5} - 23 \beta_{7} ) q^{53} + ( -6700 + 6700 \beta_{2} + 244 \beta_{3} + 184 \beta_{4} + 184 \beta_{5} - 8 \beta_{6} ) q^{54} + ( 9415 - 9415 \beta_{2} - 545 \beta_{3} + 304 \beta_{4} + 304 \beta_{5} + 29 \beta_{6} ) q^{55} + ( 576 + 64 \beta_{1} + 64 \beta_{3} - 64 \beta_{6} - 64 \beta_{7} ) q^{56} + ( -4563 - 809 \beta_{1} - 7234 \beta_{2} + 178 \beta_{3} - 84 \beta_{4} - 235 \beta_{5} - 40 \beta_{6} + 33 \beta_{7} ) q^{57} + ( 6372 - 1320 \beta_{1} - 1320 \beta_{3} + 172 \beta_{4} ) q^{58} + ( -21017 + 21017 \beta_{2} + 31 \beta_{3} - 256 \beta_{4} - 256 \beta_{5} + 2 \beta_{6} ) q^{59} + ( 288 - 288 \beta_{2} - 736 \beta_{3} + 144 \beta_{4} + 144 \beta_{5} - 16 \beta_{6} ) q^{60} + ( 1772 \beta_{1} - 4389 \beta_{2} + 311 \beta_{5} + 16 \beta_{7} ) q^{61} + ( 884 - 884 \beta_{2} + 556 \beta_{3} + 24 \beta_{4} + 24 \beta_{5} + 44 \beta_{6} ) q^{62} + ( -244 \beta_{1} - 3660 \beta_{2} - 72 \beta_{5} - 164 \beta_{7} ) q^{63} + 4096 q^{64} + ( -1270 + 1535 \beta_{1} + 1535 \beta_{3} - 217 \beta_{4} + 193 \beta_{6} + 193 \beta_{7} ) q^{65} + ( -524 \beta_{1} - 15388 \beta_{2} - 52 \beta_{5} - 60 \beta_{7} ) q^{66} + ( 939 \beta_{1} - 5231 \beta_{2} + 112 \beta_{5} + 54 \beta_{7} ) q^{67} + ( -2112 + 48 \beta_{1} + 48 \beta_{3} - 240 \beta_{4} - 48 \beta_{6} - 48 \beta_{7} ) q^{68} + ( 13089 + 1720 \beta_{1} + 1720 \beta_{3} - 327 \beta_{4} + 208 \beta_{6} + 208 \beta_{7} ) q^{69} + ( 56 \beta_{1} - 4216 \beta_{2} - 88 \beta_{5} + 136 \beta_{7} ) q^{70} + ( -13293 + 13293 \beta_{2} - 51 \beta_{3} + 13 \beta_{4} + 13 \beta_{5} + 300 \beta_{6} ) q^{71} + ( -896 \beta_{1} + 2944 \beta_{2} - 128 \beta_{5} ) q^{72} + ( -9347 + 9347 \beta_{2} - 1848 \beta_{3} - 274 \beta_{4} - 274 \beta_{5} - 312 \beta_{6} ) q^{73} + ( 12140 - 12140 \beta_{2} + 1228 \beta_{3} - 456 \beta_{4} - 456 \beta_{5} - 100 \beta_{6} ) q^{74} + ( 19244 - 184 \beta_{1} - 184 \beta_{3} + 235 \beta_{4} + 18 \beta_{6} + 18 \beta_{7} ) q^{75} + ( 704 + 656 \beta_{1} - 1072 \beta_{2} + 992 \beta_{3} - 48 \beta_{4} + 224 \beta_{5} + 64 \beta_{6} + 160 \beta_{7} ) q^{76} + ( 34735 - 1589 \beta_{1} - 1589 \beta_{3} - 214 \beta_{4} + 341 \beta_{6} + 341 \beta_{7} ) q^{77} + ( -21212 + 21212 \beta_{2} - 1732 \beta_{3} + 236 \beta_{4} + 236 \beta_{5} + 176 \beta_{6} ) q^{78} + ( -12109 + 12109 \beta_{2} - 2867 \beta_{3} - 193 \beta_{4} - 193 \beta_{5} - 304 \beta_{6} ) q^{79} + ( -2304 \beta_{2} - 256 \beta_{5} ) q^{80} + ( -2737 + 2737 \beta_{2} + 710 \beta_{3} - 194 \beta_{4} - 194 \beta_{5} - 28 \beta_{6} ) q^{81} + ( -1688 \beta_{1} - 2636 \beta_{2} + 656 \beta_{5} + 128 \beta_{7} ) q^{82} + ( 4764 + 3570 \beta_{1} + 3570 \beta_{3} + 21 \beta_{4} - 390 \beta_{6} - 390 \beta_{7} ) q^{83} + ( 2864 + 432 \beta_{1} + 432 \beta_{3} - 32 \beta_{4} + 144 \beta_{6} + 144 \beta_{7} ) q^{84} + ( -1233 \beta_{1} - 54846 \beta_{2} + 903 \beta_{5} + 207 \beta_{7} ) q^{85} + ( 4748 \beta_{1} + 1692 \beta_{2} - 380 \beta_{5} - 80 \beta_{7} ) q^{86} + ( -55326 - 3628 \beta_{1} - 3628 \beta_{3} + 1047 \beta_{4} - 43 \beta_{6} - 43 \beta_{7} ) q^{87} + ( 512 + 1280 \beta_{1} + 1280 \beta_{3} + 192 \beta_{4} - 128 \beta_{6} - 128 \beta_{7} ) q^{88} + ( 2117 \beta_{1} - 35524 \beta_{2} + 881 \beta_{5} + 91 \beta_{7} ) q^{89} + ( -24152 + 24152 \beta_{2} - 3256 \beta_{3} - 280 \beta_{4} - 280 \beta_{5} ) q^{90} + ( -6098 \beta_{1} + 80178 \beta_{2} + 528 \beta_{5} - 34 \beta_{7} ) q^{91} + ( -192 + 192 \beta_{2} - 1248 \beta_{3} + 304 \beta_{4} + 304 \beta_{5} + 336 \beta_{6} ) q^{92} + ( 27461 - 27461 \beta_{2} + 1451 \beta_{3} - 224 \beta_{4} - 224 \beta_{5} - 105 \beta_{6} ) q^{93} + ( 45744 + 48 \beta_{1} + 48 \beta_{3} - 964 \beta_{4} + 108 \beta_{6} + 108 \beta_{7} ) q^{94} + ( 43808 - 3347 \beta_{1} - 47965 \beta_{2} + 584 \beta_{3} + 991 \beta_{4} + 765 \beta_{5} + 91 \beta_{6} - 181 \beta_{7} ) q^{95} + ( 3072 + 1024 \beta_{1} + 1024 \beta_{3} ) q^{96} + ( -71281 + 71281 \beta_{2} - 3632 \beta_{3} + 486 \beta_{4} + 486 \beta_{5} - 214 \beta_{6} ) q^{97} + ( -2788 + 2788 \beta_{2} + 4160 \beta_{3} + 488 \beta_{4} + 488 \beta_{5} - 80 \beta_{6} ) q^{98} + ( -804 \beta_{1} - 34180 \beta_{2} - 1108 \beta_{5} - 364 \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 16q^{2} - 14q^{3} - 64q^{4} - 36q^{5} - 56q^{6} + 76q^{7} + 512q^{8} + 156q^{9} + O(q^{10})$$ $$8q - 16q^{2} - 14q^{3} - 64q^{4} - 36q^{5} - 56q^{6} + 76q^{7} + 512q^{8} + 156q^{9} - 144q^{10} + 144q^{11} + 448q^{12} - 674q^{13} - 152q^{14} - 20q^{15} - 1024q^{16} + 522q^{17} - 1248q^{18} + 320q^{19} + 1152q^{20} - 770q^{21} - 288q^{22} - 204q^{23} - 896q^{24} - 2158q^{25} + 5392q^{26} + 13156q^{27} - 608q^{28} - 5712q^{29} + 160q^{30} - 2324q^{31} - 4096q^{32} - 15650q^{33} + 2088q^{34} - 4188q^{35} + 2496q^{36} - 25508q^{37} - 2440q^{38} + 44156q^{39} - 2304q^{40} - 3480q^{41} - 3080q^{42} + 4066q^{43} - 1152q^{44} + 51560q^{45} + 1632q^{46} - 45768q^{47} - 3584q^{48} + 1416q^{49} + 17264q^{50} + 1506q^{51} - 10784q^{52} + 45654q^{53} - 26312q^{54} + 36570q^{55} + 4864q^{56} - 66702q^{57} + 45696q^{58} - 84006q^{59} - 320q^{60} - 14012q^{61} + 4648q^{62} - 15128q^{63} + 32768q^{64} - 4020q^{65} - 62600q^{66} - 19046q^{67} - 16704q^{68} + 111592q^{69} - 16752q^{70} - 53274q^{71} + 9984q^{72} - 41084q^{73} + 51016q^{74} + 153216q^{75} + 4640q^{76} + 271524q^{77} - 88312q^{78} - 54170q^{79} - 9216q^{80} - 9528q^{81} - 13920q^{82} + 52392q^{83} + 24640q^{84} - 221850q^{85} + 16264q^{86} - 457120q^{87} + 9216q^{88} - 137862q^{89} - 103120q^{90} + 308516q^{91} - 3264q^{92} + 112746q^{93} + 366144q^{94} + 153078q^{95} + 28672q^{96} - 292388q^{97} - 2832q^{98} - 138328q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 2 x^{7} + 386 x^{6} + 3436 x^{5} + 128708 x^{4} + 568528 x^{3} + 7340704 x^{2} - 19430784 x + 211527936$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$390471533 \nu^{7} + 3142267604 \nu^{6} + 128794858006 \nu^{5} + 2734091869688 \nu^{4} + 58739407749004 \nu^{3} + 696497843562920 \nu^{2} + 2817558974024672 \nu + 21387024882252864$$$$)/ 27200877355486656$$ $$\beta_{3}$$ $$=$$ $$($$$$-6473945 \nu^{7} + 36183422 \nu^{6} - 2297742050 \nu^{5} - 13997685940 \nu^{4} - 783009643316 \nu^{3} + 80490049760 \nu^{2} - 47812199501856 \nu + 136296431422848$$$$)/ 44885936230176$$ $$\beta_{4}$$ $$=$$ $$($$$$-18756773 \nu^{7} + 22066664 \nu^{6} - 6657181370 \nu^{5} - 40555089316 \nu^{4} - 2506296549812 \nu^{3} + 233201485664 \nu^{2} - 8478177668352 \nu + 2039876809757088$$$$)/ 22442968115088$$ $$\beta_{5}$$ $$=$$ $$($$$$-14831978953 \nu^{7} - 117204668308 \nu^{6} - 4803969780662 \nu^{5} - 120891129109120 \nu^{4} - 2190944142716108 \nu^{3} - 25978945468588840 \nu^{2} - 185235881908387744 \nu - 797723005586315328$$$$)/ 13600438677743328$$ $$\beta_{6}$$ $$=$$ $$($$$$35647275847 \nu^{7} - 1347014053248 \nu^{6} + 26507480732490 \nu^{5} - 330296195845968 \nu^{4} + 1829783642565108 \nu^{3} - 63493741971665976 \nu^{2} + 277537020774601760 \nu - 1269287639998959744$$$$)/ 9066959118495552$$ $$\beta_{7}$$ $$=$$ $$($$$$71425529135 \nu^{7} + 280447752060 \nu^{6} + 11494956177090 \nu^{5} + 561804417935112 \nu^{4} + 5242499028273060 \nu^{3} + 62162514196183800 \nu^{2} - 229139449183464992 \nu + 1908794478180000960$$$$)/ 9066959118495552$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$2 \beta_{5} + 2 \beta_{4} - 8 \beta_{3} + 188 \beta_{2} - 188$$ $$\nu^{3}$$ $$=$$ $$-2 \beta_{7} - 2 \beta_{6} + 28 \beta_{4} - 326 \beta_{3} - 326 \beta_{1} - 1414$$ $$\nu^{4}$$ $$=$$ $$-4 \beta_{7} - 820 \beta_{5} - 60100 \beta_{2} - 5044 \beta_{1}$$ $$\nu^{5}$$ $$=$$ $$772 \beta_{6} - 15008 \beta_{5} - 15008 \beta_{4} + 130764 \beta_{3} - 911516 \beta_{2} + 911516$$ $$\nu^{6}$$ $$=$$ $$5744 \beta_{7} + 5744 \beta_{6} - 351576 \beta_{4} + 2507520 \beta_{3} + 2507520 \beta_{1} + 23936064$$ $$\nu^{7}$$ $$=$$ $$282648 \beta_{7} + 7124496 \beta_{5} + 455799624 \beta_{2} + 56964328 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/38\mathbb{Z}\right)^\times$$.

 $$n$$ $$21$$ $$\chi(n)$$ $$-\beta_{2}$$

## 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
 10.6694 − 18.4800i 2.36148 − 4.09020i −5.68703 + 9.85023i −6.34386 + 10.9879i 10.6694 + 18.4800i 2.36148 + 4.09020i −5.68703 − 9.85023i −6.34386 − 10.9879i
−2.00000 3.46410i −12.1694 21.0781i −8.00000 + 13.8564i −28.6588 49.6385i −48.6777 + 84.3122i 33.7394 64.0000 −174.689 + 302.571i −114.635 + 198.554i
7.2 −2.00000 3.46410i −3.86148 6.68827i −8.00000 + 13.8564i 46.3693 + 80.3140i −15.4459 + 26.7531i 13.5472 64.0000 91.6780 158.791i 185.477 321.256i
7.3 −2.00000 3.46410i 4.18703 + 7.25215i −8.00000 + 13.8564i −12.5904 21.8073i 16.7481 29.0086i −187.086 64.0000 86.4375 149.714i −50.3618 + 87.2292i
7.4 −2.00000 3.46410i 4.84386 + 8.38982i −8.00000 + 13.8564i −23.1201 40.0451i 19.3755 33.5593i 177.800 64.0000 74.5740 129.166i −92.4803 + 160.181i
11.1 −2.00000 + 3.46410i −12.1694 + 21.0781i −8.00000 13.8564i −28.6588 + 49.6385i −48.6777 84.3122i 33.7394 64.0000 −174.689 302.571i −114.635 198.554i
11.2 −2.00000 + 3.46410i −3.86148 + 6.68827i −8.00000 13.8564i 46.3693 80.3140i −15.4459 26.7531i 13.5472 64.0000 91.6780 + 158.791i 185.477 + 321.256i
11.3 −2.00000 + 3.46410i 4.18703 7.25215i −8.00000 13.8564i −12.5904 + 21.8073i 16.7481 + 29.0086i −187.086 64.0000 86.4375 + 149.714i −50.3618 87.2292i
11.4 −2.00000 + 3.46410i 4.84386 8.38982i −8.00000 13.8564i −23.1201 + 40.0451i 19.3755 + 33.5593i 177.800 64.0000 74.5740 + 129.166i −92.4803 160.181i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 11.4 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 38.6.c.b 8
3.b odd 2 1 342.6.g.d 8
4.b odd 2 1 304.6.i.b 8
19.c even 3 1 inner 38.6.c.b 8
19.c even 3 1 722.6.a.j 4
19.d odd 6 1 722.6.a.g 4
57.h odd 6 1 342.6.g.d 8
76.g odd 6 1 304.6.i.b 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.6.c.b 8 1.a even 1 1 trivial
38.6.c.b 8 19.c even 3 1 inner
304.6.i.b 8 4.b odd 2 1
304.6.i.b 8 76.g odd 6 1
342.6.g.d 8 3.b odd 2 1
342.6.g.d 8 57.h odd 6 1
722.6.a.g 4 19.d odd 6 1
722.6.a.j 4 19.c even 3 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{8} + \cdots$$ acting on $$S_{6}^{\mathrm{new}}(38, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + 4 T + 16 T^{2} )^{4}$$
$3$ $$1 + 14 T - 466 T^{2} - 9556 T^{3} + 87593 T^{4} + 2389136 T^{5} - 10153550 T^{6} - 219863958 T^{7} + 2643549084 T^{8} - 53426941794 T^{9} - 599556973950 T^{10} + 34281490274352 T^{11} + 305417906036793 T^{12} - 8096689951837308 T^{13} - 95945267556106434 T^{14} + 700441631385995898 T^{15} + 12157665459056928801 T^{16}$$
$5$ $$1 + 36 T - 4523 T^{2} + 376256 T^{3} + 26214427 T^{4} - 1512476944 T^{5} + 68024443376 T^{6} + 5289106280720 T^{7} - 255281389260850 T^{8} + 16528457127250000 T^{9} + 664301204843750000 T^{10} - 46157133300781250000 T^{11} +$$$$25\!\cdots\!75$$$$T^{12} +$$$$11\!\cdots\!00$$$$T^{13} -$$$$42\!\cdots\!75$$$$T^{14} +$$$$10\!\cdots\!00$$$$T^{15} +$$$$90\!\cdots\!25$$$$T^{16}$$
$7$ $$( 1 - 38 T + 33982 T^{2} - 338814 T^{3} + 562116354 T^{4} - 5694446898 T^{5} + 9599073911518 T^{6} - 180407337377834 T^{7} + 79792266297612001 T^{8} )^{2}$$
$11$ $$( 1 - 72 T + 316661 T^{2} + 44034592 T^{3} + 49698680076 T^{4} + 7091815076192 T^{5} + 8213370811577261 T^{6} - 300761868197926872 T^{7} +$$$$67\!\cdots\!01$$$$T^{8} )^{2}$$
$13$ $$1 + 674 T - 122205 T^{2} + 438281134 T^{3} + 301202905985 T^{4} - 88120060372068 T^{5} + 138180616543467142 T^{6} + 97518173959653104624 T^{7} -$$$$16\!\cdots\!90$$$$T^{8} +$$$$36\!\cdots\!32$$$$T^{9} +$$$$19\!\cdots\!58$$$$T^{10} -$$$$45\!\cdots\!76$$$$T^{11} +$$$$57\!\cdots\!85$$$$T^{12} +$$$$30\!\cdots\!62$$$$T^{13} -$$$$32\!\cdots\!45$$$$T^{14} +$$$$65\!\cdots\!18$$$$T^{15} +$$$$36\!\cdots\!01$$$$T^{16}$$
$17$ $$1 - 522 T - 3419393 T^{2} + 2647918170 T^{3} + 5495287547413 T^{4} - 4447332696469860 T^{5} - 7200748427549716982 T^{6} +$$$$27\!\cdots\!60$$$$T^{7} +$$$$10\!\cdots\!78$$$$T^{8} +$$$$38\!\cdots\!20$$$$T^{9} -$$$$14\!\cdots\!18$$$$T^{10} -$$$$12\!\cdots\!80$$$$T^{11} +$$$$22\!\cdots\!13$$$$T^{12} +$$$$15\!\cdots\!90$$$$T^{13} -$$$$28\!\cdots\!57$$$$T^{14} -$$$$60\!\cdots\!46$$$$T^{15} +$$$$16\!\cdots\!01$$$$T^{16}$$
$19$ $$1 - 320 T - 3698483 T^{2} + 740497640 T^{3} + 6232925542364 T^{4} + 1833545465906360 T^{5} - 22675644326350615883 T^{6} -$$$$48\!\cdots\!80$$$$T^{7} +$$$$37\!\cdots\!01$$$$T^{8}$$
$23$ $$1 + 204 T - 4803551 T^{2} + 20538873080 T^{3} - 19379586952313 T^{4} - 128577895252239868 T^{5} +$$$$29\!\cdots\!92$$$$T^{6} +$$$$26\!\cdots\!76$$$$T^{7} -$$$$12\!\cdots\!10$$$$T^{8} +$$$$17\!\cdots\!68$$$$T^{9} +$$$$12\!\cdots\!08$$$$T^{10} -$$$$34\!\cdots\!76$$$$T^{11} -$$$$33\!\cdots\!13$$$$T^{12} +$$$$22\!\cdots\!40$$$$T^{13} -$$$$34\!\cdots\!99$$$$T^{14} +$$$$93\!\cdots\!28$$$$T^{15} +$$$$29\!\cdots\!01$$$$T^{16}$$
$29$ $$1 + 5712 T - 8141963 T^{2} + 237083772532 T^{3} + 1488021719587843 T^{4} - 2255179733329866800 T^{5} +$$$$26\!\cdots\!44$$$$T^{6} +$$$$15\!\cdots\!80$$$$T^{7} -$$$$27\!\cdots\!26$$$$T^{8} +$$$$30\!\cdots\!20$$$$T^{9} +$$$$11\!\cdots\!44$$$$T^{10} -$$$$19\!\cdots\!00$$$$T^{11} +$$$$26\!\cdots\!43$$$$T^{12} +$$$$86\!\cdots\!68$$$$T^{13} -$$$$60\!\cdots\!63$$$$T^{14} +$$$$87\!\cdots\!88$$$$T^{15} +$$$$31\!\cdots\!01$$$$T^{16}$$
$31$ $$( 1 + 1162 T + 102749710 T^{2} + 76813797026 T^{3} + 4234803816336002 T^{4} + 2199113793940704926 T^{5} +$$$$84\!\cdots\!10$$$$T^{6} +$$$$27\!\cdots\!62$$$$T^{7} +$$$$67\!\cdots\!01$$$$T^{8} )^{2}$$
$37$ $$( 1 + 12754 T + 181573786 T^{2} + 1939974331982 T^{3} + 18329643853148186 T^{4} +$$$$13\!\cdots\!74$$$$T^{5} +$$$$87\!\cdots\!14$$$$T^{6} +$$$$42\!\cdots\!22$$$$T^{7} +$$$$23\!\cdots\!01$$$$T^{8} )^{2}$$
$41$ $$1 + 3480 T - 175617710 T^{2} - 3188949547136 T^{3} + 9634643091895597 T^{4} +$$$$45\!\cdots\!68$$$$T^{5} +$$$$44\!\cdots\!74$$$$T^{6} -$$$$33\!\cdots\!96$$$$T^{7} -$$$$68\!\cdots\!56$$$$T^{8} -$$$$38\!\cdots\!96$$$$T^{9} +$$$$59\!\cdots\!74$$$$T^{10} +$$$$70\!\cdots\!68$$$$T^{11} +$$$$17\!\cdots\!97$$$$T^{12} -$$$$66\!\cdots\!36$$$$T^{13} -$$$$42\!\cdots\!10$$$$T^{14} +$$$$97\!\cdots\!80$$$$T^{15} +$$$$32\!\cdots\!01$$$$T^{16}$$
$43$ $$1 - 4066 T + 42092427 T^{2} + 1466754177546 T^{3} - 40094162896646535 T^{4} +$$$$22\!\cdots\!12$$$$T^{5} -$$$$77\!\cdots\!82$$$$T^{6} -$$$$41\!\cdots\!08$$$$T^{7} +$$$$99\!\cdots\!90$$$$T^{8} -$$$$61\!\cdots\!44$$$$T^{9} -$$$$16\!\cdots\!18$$$$T^{10} +$$$$71\!\cdots\!84$$$$T^{11} -$$$$18\!\cdots\!35$$$$T^{12} +$$$$10\!\cdots\!78$$$$T^{13} +$$$$42\!\cdots\!23$$$$T^{14} -$$$$60\!\cdots\!62$$$$T^{15} +$$$$21\!\cdots\!01$$$$T^{16}$$
$47$ $$1 + 45768 T + 806601553 T^{2} + 11152860827276 T^{3} + 173535898253598727 T^{4} +$$$$40\!\cdots\!72$$$$T^{5} -$$$$48\!\cdots\!08$$$$T^{6} -$$$$11\!\cdots\!52$$$$T^{7} -$$$$17\!\cdots\!62$$$$T^{8} -$$$$25\!\cdots\!64$$$$T^{9} -$$$$25\!\cdots\!92$$$$T^{10} +$$$$48\!\cdots\!96$$$$T^{11} +$$$$48\!\cdots\!27$$$$T^{12} +$$$$70\!\cdots\!32$$$$T^{13} +$$$$11\!\cdots\!97$$$$T^{14} +$$$$15\!\cdots\!24$$$$T^{15} +$$$$76\!\cdots\!01$$$$T^{16}$$
$53$ $$1 - 45654 T - 277597421 T^{2} + 14288789967542 T^{3} + 1160678485065770929 T^{4} -$$$$20\!\cdots\!20$$$$T^{5} -$$$$46\!\cdots\!34$$$$T^{6} -$$$$11\!\cdots\!40$$$$T^{7} +$$$$39\!\cdots\!42$$$$T^{8} -$$$$46\!\cdots\!20$$$$T^{9} -$$$$80\!\cdots\!66$$$$T^{10} -$$$$15\!\cdots\!40$$$$T^{11} +$$$$35\!\cdots\!29$$$$T^{12} +$$$$18\!\cdots\!06$$$$T^{13} -$$$$14\!\cdots\!29$$$$T^{14} -$$$$10\!\cdots\!78$$$$T^{15} +$$$$93\!\cdots\!01$$$$T^{16}$$
$59$ $$1 + 84006 T + 2018275438 T^{2} + 18454050577980 T^{3} + 2124997134843843193 T^{4} +$$$$11\!\cdots\!44$$$$T^{5} +$$$$24\!\cdots\!10$$$$T^{6} +$$$$55\!\cdots\!14$$$$T^{7} +$$$$18\!\cdots\!08$$$$T^{8} +$$$$39\!\cdots\!86$$$$T^{9} +$$$$12\!\cdots\!10$$$$T^{10} +$$$$41\!\cdots\!56$$$$T^{11} +$$$$55\!\cdots\!93$$$$T^{12} +$$$$34\!\cdots\!20$$$$T^{13} +$$$$26\!\cdots\!38$$$$T^{14} +$$$$80\!\cdots\!94$$$$T^{15} +$$$$68\!\cdots\!01$$$$T^{16}$$
$61$ $$1 + 14012 T - 1424102283 T^{2} + 57219273244216 T^{3} + 1960265998467982883 T^{4} -$$$$67\!\cdots\!40$$$$T^{5} +$$$$13\!\cdots\!76$$$$T^{6} +$$$$56\!\cdots\!12$$$$T^{7} -$$$$15\!\cdots\!54$$$$T^{8} +$$$$47\!\cdots\!12$$$$T^{9} +$$$$95\!\cdots\!76$$$$T^{10} -$$$$40\!\cdots\!40$$$$T^{11} +$$$$99\!\cdots\!83$$$$T^{12} +$$$$24\!\cdots\!16$$$$T^{13} -$$$$51\!\cdots\!83$$$$T^{14} +$$$$42\!\cdots\!12$$$$T^{15} +$$$$25\!\cdots\!01$$$$T^{16}$$
$67$ $$1 + 19046 T - 4676411334 T^{2} - 43706869689820 T^{3} + 14124551833756340293 T^{4} +$$$$67\!\cdots\!96$$$$T^{5} -$$$$28\!\cdots\!70$$$$T^{6} -$$$$34\!\cdots\!02$$$$T^{7} +$$$$44\!\cdots\!84$$$$T^{8} -$$$$46\!\cdots\!14$$$$T^{9} -$$$$52\!\cdots\!30$$$$T^{10} +$$$$16\!\cdots\!28$$$$T^{11} +$$$$46\!\cdots\!93$$$$T^{12} -$$$$19\!\cdots\!40$$$$T^{13} -$$$$28\!\cdots\!66$$$$T^{14} +$$$$15\!\cdots\!78$$$$T^{15} +$$$$11\!\cdots\!01$$$$T^{16}$$
$71$ $$1 + 53274 T - 2409470753 T^{2} - 145053452652226 T^{3} + 4361158932904192285 T^{4} +$$$$23\!\cdots\!64$$$$T^{5} -$$$$34\!\cdots\!78$$$$T^{6} -$$$$27\!\cdots\!56$$$$T^{7} -$$$$74\!\cdots\!22$$$$T^{8} -$$$$50\!\cdots\!56$$$$T^{9} -$$$$11\!\cdots\!78$$$$T^{10} +$$$$13\!\cdots\!64$$$$T^{11} +$$$$46\!\cdots\!85$$$$T^{12} -$$$$27\!\cdots\!26$$$$T^{13} -$$$$83\!\cdots\!53$$$$T^{14} +$$$$33\!\cdots\!74$$$$T^{15} +$$$$11\!\cdots\!01$$$$T^{16}$$
$73$ $$1 + 41084 T - 1658779422 T^{2} + 292442470814648 T^{3} + 14433522779770360453 T^{4} -$$$$41\!\cdots\!92$$$$T^{5} +$$$$41\!\cdots\!10$$$$T^{6} +$$$$20\!\cdots\!96$$$$T^{7} -$$$$48\!\cdots\!36$$$$T^{8} +$$$$42\!\cdots\!28$$$$T^{9} +$$$$17\!\cdots\!90$$$$T^{10} -$$$$37\!\cdots\!44$$$$T^{11} +$$$$26\!\cdots\!53$$$$T^{12} +$$$$11\!\cdots\!64$$$$T^{13} -$$$$13\!\cdots\!78$$$$T^{14} +$$$$67\!\cdots\!88$$$$T^{15} +$$$$34\!\cdots\!01$$$$T^{16}$$
$79$ $$1 + 54170 T - 3552111405 T^{2} + 270124789784518 T^{3} + 26931637455662462321 T^{4} -$$$$10\!\cdots\!96$$$$T^{5} +$$$$33\!\cdots\!66$$$$T^{6} +$$$$38\!\cdots\!84$$$$T^{7} -$$$$15\!\cdots\!14$$$$T^{8} +$$$$11\!\cdots\!16$$$$T^{9} +$$$$31\!\cdots\!66$$$$T^{10} -$$$$29\!\cdots\!04$$$$T^{11} +$$$$24\!\cdots\!21$$$$T^{12} +$$$$74\!\cdots\!82$$$$T^{13} -$$$$30\!\cdots\!05$$$$T^{14} +$$$$14\!\cdots\!30$$$$T^{15} +$$$$80\!\cdots\!01$$$$T^{16}$$
$83$ $$( 1 - 26196 T + 6227566841 T^{2} + 176717158574580 T^{3} + 11890049747683151076 T^{4} +$$$$69\!\cdots\!40$$$$T^{5} +$$$$96\!\cdots\!09$$$$T^{6} -$$$$16\!\cdots\!72$$$$T^{7} +$$$$24\!\cdots\!01$$$$T^{8} )^{2}$$
$89$ $$1 + 137862 T - 3387852065 T^{2} - 805978834080246 T^{3} + 53245997529728929381 T^{4} +$$$$48\!\cdots\!92$$$$T^{5} -$$$$34\!\cdots\!38$$$$T^{6} -$$$$16\!\cdots\!16$$$$T^{7} +$$$$12\!\cdots\!10$$$$T^{8} -$$$$91\!\cdots\!84$$$$T^{9} -$$$$10\!\cdots\!38$$$$T^{10} +$$$$84\!\cdots\!08$$$$T^{11} +$$$$51\!\cdots\!81$$$$T^{12} -$$$$43\!\cdots\!54$$$$T^{13} -$$$$10\!\cdots\!65$$$$T^{14} +$$$$23\!\cdots\!38$$$$T^{15} +$$$$94\!\cdots\!01$$$$T^{16}$$
$97$ $$1 + 292388 T + 26872716282 T^{2} + 833354453498472 T^{3} +$$$$19\!\cdots\!93$$$$T^{4} +$$$$44\!\cdots\!68$$$$T^{5} +$$$$41\!\cdots\!58$$$$T^{6} +$$$$26\!\cdots\!32$$$$T^{7} +$$$$20\!\cdots\!04$$$$T^{8} +$$$$22\!\cdots\!24$$$$T^{9} +$$$$30\!\cdots\!42$$$$T^{10} +$$$$27\!\cdots\!24$$$$T^{11} +$$$$10\!\cdots\!93$$$$T^{12} +$$$$38\!\cdots\!04$$$$T^{13} +$$$$10\!\cdots\!18$$$$T^{14} +$$$$10\!\cdots\!84$$$$T^{15} +$$$$29\!\cdots\!01$$$$T^{16}$$