Properties

 Label 5.9.c.a Level 5 Weight 9 Character orbit 5.c Analytic conductor 2.037 Analytic rank 0 Dimension 6 CM No Inner twists 2

Related objects

Newspace parameters

 Level: $$N$$ = $$5$$ Weight: $$k$$ = $$9$$ Character orbit: $$[\chi]$$ = 5.c (of order $$4$$ and degree $$2$$)

Newform invariants

 Self dual: No Analytic conductor: $$2.03689305031$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(i)$$ Coefficient field: $$\mathbb{Q}[x]/(x^{6} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{6}\cdot 5^{2}$$ Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q$$ $$-\beta_{3} q^{2}$$ $$+ ( -12 - 12 \beta_{1} - \beta_{2} - \beta_{5} ) q^{3}$$ $$+ ( -10 \beta_{1} - \beta_{2} - 5 \beta_{3} + \beta_{4} + 5 \beta_{5} ) q^{4}$$ $$+ ( 48 + 39 \beta_{1} + 3 \beta_{2} - 15 \beta_{3} - 4 \beta_{4} - 20 \beta_{5} ) q^{5}$$ $$+ ( 270 + 7 \beta_{2} + 40 \beta_{3} + 7 \beta_{4} + 40 \beta_{5} ) q^{6}$$ $$+ ( -434 + 434 \beta_{1} + 119 \beta_{3} - 7 \beta_{4} ) q^{7}$$ $$+ ( -1326 - 1326 \beta_{1} + 2 \beta_{2} - 130 \beta_{5} ) q^{8}$$ $$+ ( 2823 \beta_{1} + 3 \beta_{2} - 285 \beta_{3} - 3 \beta_{4} + 285 \beta_{5} ) q^{9}$$ $$+O(q^{10})$$ $$q$$ $$-\beta_{3} q^{2}$$ $$+ ( -12 - 12 \beta_{1} - \beta_{2} - \beta_{5} ) q^{3}$$ $$+ ( -10 \beta_{1} - \beta_{2} - 5 \beta_{3} + \beta_{4} + 5 \beta_{5} ) q^{4}$$ $$+ ( 48 + 39 \beta_{1} + 3 \beta_{2} - 15 \beta_{3} - 4 \beta_{4} - 20 \beta_{5} ) q^{5}$$ $$+ ( 270 + 7 \beta_{2} + 40 \beta_{3} + 7 \beta_{4} + 40 \beta_{5} ) q^{6}$$ $$+ ( -434 + 434 \beta_{1} + 119 \beta_{3} - 7 \beta_{4} ) q^{7}$$ $$+ ( -1326 - 1326 \beta_{1} + 2 \beta_{2} - 130 \beta_{5} ) q^{8}$$ $$+ ( 2823 \beta_{1} + 3 \beta_{2} - 285 \beta_{3} - 3 \beta_{4} + 285 \beta_{5} ) q^{9}$$ $$+ ( 5308 - 4006 \beta_{1} - 37 \beta_{2} - 190 \beta_{3} + 41 \beta_{4} - 295 \beta_{5} ) q^{10}$$ $$+ ( 3792 - 85 \beta_{2} + 25 \beta_{3} - 85 \beta_{4} + 25 \beta_{5} ) q^{11}$$ $$+ ( -7596 + 7596 \beta_{1} - 76 \beta_{3} + 92 \beta_{4} ) q^{12}$$ $$+ ( -20015 - 20015 \beta_{1} + 80 \beta_{2} + 554 \beta_{5} ) q^{13}$$ $$+ ( 31626 \beta_{1} + 77 \beta_{2} + 1260 \beta_{3} - 77 \beta_{4} - 1260 \beta_{5} ) q^{14}$$ $$+ ( 39246 - 22122 \beta_{1} + 181 \beta_{2} + 1470 \beta_{3} - 108 \beta_{4} + 1585 \beta_{5} ) q^{15}$$ $$+ ( 37132 + 374 \beta_{2} - 670 \beta_{3} + 374 \beta_{4} - 670 \beta_{5} ) q^{16}$$ $$+ ( -43617 + 43617 \beta_{1} - 2366 \beta_{3} - 466 \beta_{4} ) q^{17}$$ $$+ ( -75822 - 75822 \beta_{1} - 606 \beta_{2} - 171 \beta_{5} ) q^{18}$$ $$+ ( 54264 \beta_{1} - 684 \beta_{2} + 930 \beta_{3} + 684 \beta_{4} - 930 \beta_{5} ) q^{19}$$ $$+ ( 68634 - 38088 \beta_{1} - 451 \beta_{2} - 2745 \beta_{3} - 307 \beta_{4} + 2215 \beta_{5} ) q^{20}$$ $$+ ( 40068 - 511 \beta_{2} - 2695 \beta_{3} - 511 \beta_{4} - 2695 \beta_{5} ) q^{21}$$ $$+ ( -6310 + 6310 \beta_{1} + 2068 \beta_{3} + 970 \beta_{4} ) q^{22}$$ $$+ ( 4722 + 4722 \beta_{1} + 1581 \beta_{2} + 1859 \beta_{5} ) q^{23}$$ $$+ ( 49272 \beta_{1} + 2268 \beta_{2} - 6060 \beta_{3} - 2268 \beta_{4} + 6060 \beta_{5} ) q^{24}$$ $$+ ( -51065 - 12420 \beta_{1} + 785 \beta_{2} + 4825 \beta_{3} + 2245 \beta_{4} - 18775 \beta_{5} ) q^{25}$$ $$+ ( -147684 - 1034 \beta_{2} + 20145 \beta_{3} - 1034 \beta_{4} + 20145 \beta_{5} ) q^{26}$$ $$+ ( 58572 - 58572 \beta_{1} + 20460 \beta_{3} + 156 \beta_{4} ) q^{27}$$ $$+ ( 223748 + 223748 \beta_{1} - 196 \beta_{2} - 18844 \beta_{5} ) q^{28}$$ $$+ ( -451044 \beta_{1} - 2386 \beta_{2} - 15980 \beta_{3} + 2386 \beta_{4} + 15980 \beta_{5} ) q^{29}$$ $$+ ( -422334 + 390588 \beta_{1} - 1849 \beta_{2} - 26380 \beta_{3} - 3493 \beta_{4} + 13160 \beta_{5} ) q^{30}$$ $$+ ( -99628 + 3045 \beta_{2} - 34425 \beta_{3} + 3045 \beta_{4} - 34425 \beta_{5} ) q^{31}$$ $$+ ( 516180 - 516180 \beta_{1} - 35236 \beta_{3} - 3660 \beta_{4} ) q^{32}$$ $$+ ( 697956 + 697956 \beta_{1} - 6862 \beta_{2} + 33818 \beta_{5} ) q^{33}$$ $$+ ( -631220 \beta_{1} - 5162 \beta_{2} + 47165 \beta_{3} + 5162 \beta_{4} - 47165 \beta_{5} ) q^{34}$$ $$+ ( -854112 + 233184 \beta_{1} + 2968 \beta_{2} + 47285 \beta_{3} - 3549 \beta_{4} + 28630 \beta_{5} ) q^{35}$$ $$+ ( -674778 + 3039 \beta_{2} + 22005 \beta_{3} + 3039 \beta_{4} + 22005 \beta_{5} ) q^{36}$$ $$+ ( -72653 + 72653 \beta_{1} - 7536 \beta_{3} + 1506 \beta_{4} ) q^{37}$$ $$+ ( 250116 + 250116 \beta_{1} + 10068 \beta_{2} - 18420 \beta_{5} ) q^{38}$$ $$+ ( -375300 \beta_{1} + 17417 \beta_{2} + 20785 \beta_{3} - 17417 \beta_{4} - 20785 \beta_{5} ) q^{39}$$ $$+ ( 438150 + 627450 \beta_{1} + 6400 \beta_{2} + 29250 \beta_{3} + 14550 \beta_{4} + 19000 \beta_{5} ) q^{40}$$ $$+ ( 422352 - 12505 \beta_{2} - 23675 \beta_{3} - 12505 \beta_{4} - 23675 \beta_{5} ) q^{41}$$ $$+ ( 718914 - 718914 \beta_{1} - 33292 \beta_{3} + 11522 \beta_{4} ) q^{42}$$ $$+ ( 105376 + 105376 \beta_{1} + 5323 \beta_{2} + 85519 \beta_{5} ) q^{43}$$ $$+ ( 1524720 \beta_{1} - 13872 \beta_{2} - 21760 \beta_{3} + 13872 \beta_{4} + 21760 \beta_{5} ) q^{44}$$ $$+ ( 75771 - 2545872 \beta_{1} - 33219 \beta_{2} - 81405 \beta_{3} - 6258 \beta_{4} - 80040 \beta_{5} ) q^{45}$$ $$+ ( -500818 - 11345 \beta_{2} - 47600 \beta_{3} - 11345 \beta_{4} - 47600 \beta_{5} ) q^{46}$$ $$+ ( -2583906 + 2583906 \beta_{1} + 90779 \beta_{3} - 4263 \beta_{4} ) q^{47}$$ $$+ ( -3565608 - 3565608 \beta_{1} - 15784 \beta_{2} - 157816 \beta_{5} ) q^{48}$$ $$+ ( 1195649 \beta_{1} - 8575 \beta_{2} - 219275 \beta_{3} + 8575 \beta_{4} + 219275 \beta_{5} ) q^{49}$$ $$+ ( 4991010 + 1292430 \beta_{1} + 32360 \beta_{2} - 118675 \beta_{3} - 4230 \beta_{4} - 57400 \beta_{5} ) q^{50}$$ $$+ ( 5798544 + 52723 \beta_{2} + 246835 \beta_{3} + 52723 \beta_{4} + 246835 \beta_{5} ) q^{51}$$ $$+ ( -230594 + 230594 \beta_{1} + 275554 \beta_{3} - 48362 \beta_{4} ) q^{52}$$ $$+ ( -2162457 - 2162457 \beta_{1} - 4186 \beta_{2} - 271376 \beta_{5} ) q^{53}$$ $$+ ( 5442984 \beta_{1} + 21396 \beta_{2} + 38580 \beta_{3} - 21396 \beta_{4} - 38580 \beta_{5} ) q^{54}$$ $$+ ( 710936 - 5132052 \beta_{1} + 41621 \beta_{2} + 160645 \beta_{3} - 25703 \beta_{4} - 53515 \beta_{5} ) q^{55}$$ $$+ ( -3082968 + 308 \beta_{2} + 11060 \beta_{3} + 308 \beta_{4} + 11060 \beta_{5} ) q^{56}$$ $$+ ( -5636916 + 5636916 \beta_{1} - 338880 \beta_{3} + 63132 \beta_{4} ) q^{57}$$ $$+ ( -4241136 - 4241136 \beta_{1} - 3328 \beta_{2} + 768320 \beta_{5} ) q^{58}$$ $$+ ( -1372608 \beta_{1} - 30152 \beta_{2} + 508490 \beta_{3} + 30152 \beta_{4} - 508490 \beta_{5} ) q^{59}$$ $$+ ( 2170068 + 3015924 \beta_{1} - 77052 \beta_{2} + 126760 \beta_{3} - 1064 \beta_{4} + 129180 \beta_{5} ) q^{60}$$ $$+ ( 4266032 - 95625 \beta_{2} - 466875 \beta_{3} - 95625 \beta_{4} - 466875 \beta_{5} ) q^{61}$$ $$+ ( 9144870 - 9144870 \beta_{1} - 445592 \beta_{3} + 32310 \beta_{4} ) q^{62}$$ $$+ ( 7604394 + 7604394 \beta_{1} + 27237 \beta_{2} - 367101 \beta_{5} ) q^{63}$$ $$+ ( 118376 \beta_{1} + 38548 \beta_{2} - 400060 \beta_{3} - 38548 \beta_{4} + 400060 \beta_{5} ) q^{64}$$ $$+ ( -5227299 - 2571057 \beta_{1} - 12939 \beta_{2} - 362305 \beta_{3} + 137927 \beta_{4} + 697385 \beta_{5} ) q^{65}$$ $$+ ( -8968140 + 7354 \beta_{2} - 302420 \beta_{3} + 7354 \beta_{4} - 302420 \beta_{5} ) q^{66}$$ $$+ ( -5539832 + 5539832 \beta_{1} + 101959 \beta_{3} - 103661 \beta_{4} ) q^{67}$$ $$+ ( 1400586 + 1400586 \beta_{1} + 36978 \beta_{2} - 105434 \beta_{5} ) q^{68}$$ $$+ ( -14569164 \beta_{1} + 7561 \beta_{2} + 447455 \beta_{3} - 7561 \beta_{4} - 447455 \beta_{5} ) q^{69}$$ $$+ ( -7627452 + 12563614 \beta_{1} - 20447 \beta_{2} + 1252860 \beta_{3} - 72429 \beta_{4} - 541520 \beta_{5} ) q^{70}$$ $$+ ( -2912988 + 78125 \beta_{2} + 949375 \beta_{3} + 78125 \beta_{4} + 949375 \beta_{5} ) q^{71}$$ $$+ ( 13544946 - 13544946 \beta_{1} + 738030 \beta_{3} + 74658 \beta_{4} ) q^{72}$$ $$+ ( 18480997 + 18480997 \beta_{1} + 12756 \beta_{2} + 499584 \beta_{5} ) q^{73}$$ $$+ ( -1998552 \beta_{1} + 1500 \beta_{2} - 14725 \beta_{3} - 1500 \beta_{4} + 14725 \beta_{5} ) q^{74}$$ $$+ ( -20653380 - 1097340 \beta_{1} + 197195 \beta_{2} - 1296600 \beta_{3} - 154260 \beta_{4} - 96925 \beta_{5} ) q^{75}$$ $$+ ( -9032136 + 133116 \beta_{2} - 436380 \beta_{3} + 133116 \beta_{4} - 436380 \beta_{5} ) q^{76}$$ $$+ ( 4353552 - 4353552 \beta_{1} - 41342 \beta_{3} - 110754 \beta_{4} ) q^{77}$$ $$+ ( 5459142 + 5459142 \beta_{1} - 167434 \beta_{2} - 982072 \beta_{5} ) q^{78}$$ $$+ ( -22879824 \beta_{1} - 122456 \beta_{2} - 1313680 \beta_{3} + 122456 \beta_{4} + 1313680 \beta_{5} ) q^{79}$$ $$+ ( 4670928 + 25409004 \beta_{1} - 19442 \beta_{2} - 1455290 \beta_{3} - 58494 \beta_{4} - 1112470 \beta_{5} ) q^{80}$$ $$+ ( 10214919 - 179613 \beta_{2} + 956565 \beta_{3} - 179613 \beta_{4} + 956565 \beta_{5} ) q^{81}$$ $$+ ( 6347570 - 6347570 \beta_{1} + 166228 \beta_{3} + 197410 \beta_{4} ) q^{82}$$ $$+ ( -2441256 - 2441256 \beta_{1} - 212813 \beta_{2} - 273261 \beta_{5} ) q^{83}$$ $$+ ( 1447824 \beta_{1} - 94976 \beta_{2} - 575680 \beta_{3} + 94976 \beta_{4} + 575680 \beta_{5} ) q^{84}$$ $$+ ( -3531827 - 25557811 \beta_{1} - 286147 \beta_{2} + 1708485 \beta_{3} + 200271 \beta_{4} - 1055395 \beta_{5} ) q^{85}$$ $$+ ( -22769346 - 117457 \beta_{2} + 146560 \beta_{3} - 117457 \beta_{4} + 146560 \beta_{5} ) q^{86}$$ $$+ ( -22156764 + 22156764 \beta_{1} - 284520 \beta_{3} - 150972 \beta_{4} ) q^{87}$$ $$+ ( -7348032 - 7348032 \beta_{1} + 371264 \beta_{2} + 137840 \beta_{5} ) q^{88}$$ $$+ ( 38523288 \beta_{1} + 442572 \beta_{2} + 666960 \beta_{3} - 442572 \beta_{4} - 666960 \beta_{5} ) q^{89}$$ $$+ ( 21423516 - 21678762 \beta_{1} + 160401 \beta_{2} + 419745 \beta_{3} + 398307 \beta_{4} + 3442410 \beta_{5} ) q^{90}$$ $$+ ( 29991584 + 224203 \beta_{2} - 2877665 \beta_{3} + 224203 \beta_{4} - 2877665 \beta_{5} ) q^{91}$$ $$+ ( 11498148 - 11498148 \beta_{1} + 297684 \beta_{3} - 173396 \beta_{4} ) q^{92}$$ $$+ ( -16384884 - 16384884 \beta_{1} + 679018 \beta_{2} + 1434658 \beta_{5} ) q^{93}$$ $$+ ( 24130162 \beta_{1} + 65201 \beta_{2} + 3178480 \beta_{3} - 65201 \beta_{4} - 3178480 \beta_{5} ) q^{94}$$ $$+ ( 38898900 + 17396700 \beta_{1} - 213600 \beta_{2} + 1454250 \beta_{3} - 410700 \beta_{4} + 1055250 \beta_{5} ) q^{95}$$ $$+ ( 29428560 - 328088 \beta_{2} + 1746040 \beta_{3} - 328088 \beta_{4} + 1746040 \beta_{5} ) q^{96}$$ $$+ ( -31017911 + 31017911 \beta_{1} + 586904 \beta_{3} + 861272 \beta_{4} ) q^{97}$$ $$+ ( -58292850 - 58292850 \beta_{1} - 335650 \beta_{2} + 1563051 \beta_{5} ) q^{98}$$ $$+ ( 9802896 \beta_{1} - 486789 \beta_{2} + 615855 \beta_{3} + 486789 \beta_{4} - 615855 \beta_{5} ) q^{99}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q$$ $$\mathstrut -\mathstrut 2q^{2}$$ $$\mathstrut -\mathstrut 72q^{3}$$ $$\mathstrut +\mathstrut 220q^{5}$$ $$\mathstrut +\mathstrut 1752q^{6}$$ $$\mathstrut -\mathstrut 2352q^{7}$$ $$\mathstrut -\mathstrut 8220q^{8}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$6q$$ $$\mathstrut -\mathstrut 2q^{2}$$ $$\mathstrut -\mathstrut 72q^{3}$$ $$\mathstrut +\mathstrut 220q^{5}$$ $$\mathstrut +\mathstrut 1752q^{6}$$ $$\mathstrut -\mathstrut 2352q^{7}$$ $$\mathstrut -\mathstrut 8220q^{8}$$ $$\mathstrut +\mathstrut 30870q^{10}$$ $$\mathstrut +\mathstrut 23192q^{11}$$ $$\mathstrut -\mathstrut 45912q^{12}$$ $$\mathstrut -\mathstrut 119142q^{13}$$ $$\mathstrut +\mathstrut 241440q^{15}$$ $$\mathstrut +\mathstrut 218616q^{16}$$ $$\mathstrut -\mathstrut 265502q^{17}$$ $$\mathstrut -\mathstrut 454062q^{18}$$ $$\mathstrut +\mathstrut 412260q^{20}$$ $$\mathstrut +\mathstrut 231672q^{21}$$ $$\mathstrut -\mathstrut 35664q^{22}$$ $$\mathstrut +\mathstrut 28888q^{23}$$ $$\mathstrut -\mathstrut 340350q^{25}$$ $$\mathstrut -\mathstrut 801388q^{26}$$ $$\mathstrut +\mathstrut 392040q^{27}$$ $$\mathstrut +\mathstrut 1305192q^{28}$$ $$\mathstrut -\mathstrut 2549760q^{30}$$ $$\mathstrut -\mathstrut 747648q^{31}$$ $$\mathstrut +\mathstrut 3033928q^{32}$$ $$\mathstrut +\mathstrut 4269096q^{33}$$ $$\mathstrut -\mathstrut 4971680q^{35}$$ $$\mathstrut -\mathstrut 3972804q^{36}$$ $$\mathstrut -\mathstrut 454002q^{37}$$ $$\mathstrut +\mathstrut 1443720q^{38}$$ $$\mathstrut +\mathstrut 2683500q^{40}$$ $$\mathstrut +\mathstrut 2489432q^{41}$$ $$\mathstrut +\mathstrut 4223856q^{42}$$ $$\mathstrut +\mathstrut 792648q^{43}$$ $$\mathstrut +\mathstrut 210690q^{45}$$ $$\mathstrut -\mathstrut 3149928q^{46}$$ $$\mathstrut -\mathstrut 15313352q^{47}$$ $$\mathstrut -\mathstrut 21677712q^{48}$$ $$\mathstrut +\mathstrut 29537650q^{50}$$ $$\mathstrut +\mathstrut 35567712q^{51}$$ $$\mathstrut -\mathstrut 735732q^{52}$$ $$\mathstrut -\mathstrut 13509122q^{53}$$ $$\mathstrut +\mathstrut 4448040q^{55}$$ $$\mathstrut -\mathstrut 18454800q^{56}$$ $$\mathstrut -\mathstrut 34625520q^{57}$$ $$\mathstrut -\mathstrut 23903520q^{58}$$ $$\mathstrut +\mathstrut 13688520q^{60}$$ $$\mathstrut +\mathstrut 24111192q^{61}$$ $$\mathstrut +\mathstrut 53913416q^{62}$$ $$\mathstrut +\mathstrut 44837688q^{63}$$ $$\mathstrut -\mathstrut 30943610q^{65}$$ $$\mathstrut -\mathstrut 55047936q^{66}$$ $$\mathstrut -\mathstrut 32827752q^{67}$$ $$\mathstrut +\mathstrut 8118692q^{68}$$ $$\mathstrut -\mathstrut 44156280q^{70}$$ $$\mathstrut -\mathstrut 13992928q^{71}$$ $$\mathstrut +\mathstrut 82596420q^{72}$$ $$\mathstrut +\mathstrut 111859638q^{73}$$ $$\mathstrut -\mathstrut 126793200q^{75}$$ $$\mathstrut -\mathstrut 56470800q^{76}$$ $$\mathstrut +\mathstrut 26260136q^{77}$$ $$\mathstrut +\mathstrut 31125576q^{78}$$ $$\mathstrut +\mathstrut 23045920q^{80}$$ $$\mathstrut +\mathstrut 65834226q^{81}$$ $$\mathstrut +\mathstrut 38023056q^{82}$$ $$\mathstrut -\mathstrut 14768432q^{83}$$ $$\mathstrut -\mathstrut 19713030q^{85}$$ $$\mathstrut -\mathstrut 135560008q^{86}$$ $$\mathstrut -\mathstrut 133207680q^{87}$$ $$\mathstrut -\mathstrut 44555040q^{88}$$ $$\mathstrut +\mathstrut 135147990q^{90}$$ $$\mathstrut +\mathstrut 167542032q^{91}$$ $$\mathstrut +\mathstrut 69931048q^{92}$$ $$\mathstrut -\mathstrut 96798024q^{93}$$ $$\mathstrut +\mathstrut 239661000q^{95}$$ $$\mathstrut +\mathstrut 184867872q^{96}$$ $$\mathstrut -\mathstrut 186656202q^{97}$$ $$\mathstrut -\mathstrut 345959698q^{98}$$ $$\mathstrut +\mathstrut O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6}\mathstrut -\mathstrut$$ $$2$$ $$x^{5}\mathstrut +\mathstrut$$ $$2$$ $$x^{4}\mathstrut -\mathstrut$$ $$30$$ $$x^{3}\mathstrut +\mathstrut$$ $$1089$$ $$x^{2}\mathstrut -\mathstrut$$ $$3168$$ $$x\mathstrut +\mathstrut$$ $$4608$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$39 \nu^{5} - 22 \nu^{4} - 10 \nu^{3} + 790 \nu^{2} + 41631 \nu - 62928$$$$)/66000$$ $$\beta_{2}$$ $$=$$ $$($$$$-273 \nu^{5} + 154 \nu^{4} + 70 \nu^{3} - 5530 \nu^{2} + 1028583 \nu - 21504$$$$)/66000$$ $$\beta_{3}$$ $$=$$ $$($$$$761 \nu^{5} - 2178 \nu^{4} - 4990 \nu^{3} - 45790 \nu^{2} + 838569 \nu - 2347872$$$$)/66000$$ $$\beta_{4}$$ $$=$$ $$($$$$-77 \nu^{5} + 146 \nu^{4} - 3570 \nu^{3} + 2030 \nu^{2} - 83733 \nu + 244704$$$$)/6000$$ $$\beta_{5}$$ $$=$$ $$($$$$-921 \nu^{5} - 1342 \nu^{4} - 610 \nu^{3} + 48190 \nu^{2} - 823209 \nu + 187392$$$$)/66000$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2}\mathstrut +\mathstrut$$ $$7$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$7$$$$)/20$$ $$\nu^{2}$$ $$=$$ $$($$$$10$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$10$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$446$$ $$\beta_{1}$$$$)/20$$ $$\nu^{3}$$ $$=$$ $$($$$$-$$$$31$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$20$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$283$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$283$$$$)/20$$ $$\nu^{4}$$ $$=$$ $$($$$$-$$$$35$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$5$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$35$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$5$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$1348$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-$$$$400$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$1019$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$17267$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$17267$$$$)/20$$

Character Values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/5\mathbb{Z}\right)^\times$$.

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

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2.1
 3.70505 − 3.70505i −4.23471 + 4.23471i 1.52966 − 1.52966i 3.70505 + 3.70505i −4.23471 − 4.23471i 1.52966 + 1.52966i
−11.8649 11.8649i −90.9660 + 90.9660i 25.5528i −434.373 449.383i 2158.61 508.219 + 508.219i −2734.24 + 2734.24i 9988.61i −178.084 + 10485.7i
2.2 −4.39608 4.39608i 75.2981 75.2981i 217.349i −14.1685 + 624.839i −662.032 730.992 + 730.992i −2080.88 + 2080.88i 4778.60i 2809.13 2684.56i
2.3 15.2610 + 15.2610i −20.3321 + 20.3321i 209.796i 558.542 280.457i −620.576 −2415.21 2415.21i 705.116 705.116i 5734.21i 12804.0 + 4243.86i
3.1 −11.8649 + 11.8649i −90.9660 90.9660i 25.5528i −434.373 + 449.383i 2158.61 508.219 508.219i −2734.24 2734.24i 9988.61i −178.084 10485.7i
3.2 −4.39608 + 4.39608i 75.2981 + 75.2981i 217.349i −14.1685 624.839i −662.032 730.992 730.992i −2080.88 2080.88i 4778.60i 2809.13 + 2684.56i
3.3 15.2610 15.2610i −20.3321 20.3321i 209.796i 558.542 + 280.457i −620.576 −2415.21 + 2415.21i 705.116 + 705.116i 5734.21i 12804.0 4243.86i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 3.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
5.c Odd 1 yes

Hecke kernels

There are no other newforms in $$S_{9}^{\mathrm{new}}(5, [\chi])$$.