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 $2$

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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:

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

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