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

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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:

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 \(S_{9}^{\mathrm{new}}(5, [\chi])\).