Properties

Label 5.11.c.a
Level 5
Weight 11
Character orbit 5.c
Analytic conductor 3.177
Analytic rank 0
Dimension 8
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 5 \)
Weight: \( k \) = \( 11 \)
Character orbit: \([\chi]\) = 5.c (of order \(4\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(3.17678626337\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{2}\cdot 5^{4} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( 4 + 4 \beta_{1} - \beta_{3} ) q^{2} \) \( + ( 8 - 8 \beta_{1} + \beta_{2} - \beta_{4} ) q^{3} \) \( + ( 537 \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - \beta_{7} ) q^{4} \) \( + ( -675 + 995 \beta_{1} - 10 \beta_{2} + 15 \beta_{3} + 5 \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{5} \) \( + ( 2171 + 62 \beta_{2} - 62 \beta_{3} - 26 \beta_{4} - 26 \beta_{5} + 3 \beta_{6} ) q^{6} \) \( + ( -1785 - 1785 \beta_{1} - 157 \beta_{3} + 47 \beta_{5} - 5 \beta_{6} + 5 \beta_{7} ) q^{7} \) \( + ( 2621 - 2621 \beta_{1} - 434 \beta_{2} + 92 \beta_{4} + 5 \beta_{6} + 5 \beta_{7} ) q^{8} \) \( + ( -11613 \beta_{1} + 1047 \beta_{2} + 1047 \beta_{3} + 69 \beta_{4} - 69 \beta_{5} + 18 \beta_{7} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + ( 4 + 4 \beta_{1} - \beta_{3} ) q^{2} \) \( + ( 8 - 8 \beta_{1} + \beta_{2} - \beta_{4} ) q^{3} \) \( + ( 537 \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - \beta_{7} ) q^{4} \) \( + ( -675 + 995 \beta_{1} - 10 \beta_{2} + 15 \beta_{3} + 5 \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{5} \) \( + ( 2171 + 62 \beta_{2} - 62 \beta_{3} - 26 \beta_{4} - 26 \beta_{5} + 3 \beta_{6} ) q^{6} \) \( + ( -1785 - 1785 \beta_{1} - 157 \beta_{3} + 47 \beta_{5} - 5 \beta_{6} + 5 \beta_{7} ) q^{7} \) \( + ( 2621 - 2621 \beta_{1} - 434 \beta_{2} + 92 \beta_{4} + 5 \beta_{6} + 5 \beta_{7} ) q^{8} \) \( + ( -11613 \beta_{1} + 1047 \beta_{2} + 1047 \beta_{3} + 69 \beta_{4} - 69 \beta_{5} + 18 \beta_{7} ) q^{9} \) \( + ( -21070 - 24845 \beta_{1} - 2075 \beta_{2} + 1520 \beta_{3} - 130 \beta_{4} + 170 \beta_{5} + 14 \beta_{6} + 33 \beta_{7} ) q^{10} \) \( + ( -28158 + 2195 \beta_{2} - 2195 \beta_{3} + 65 \beta_{4} + 65 \beta_{5} - 50 \beta_{6} ) q^{11} \) \( + ( 109418 + 109418 \beta_{1} - 5620 \beta_{3} - 472 \beta_{5} + 90 \beta_{6} - 90 \beta_{7} ) q^{12} \) \( + ( 53496 - 53496 \beta_{1} - 3326 \beta_{2} - 550 \beta_{4} - 95 \beta_{6} - 95 \beta_{7} ) q^{13} \) \( + ( 201607 \beta_{1} + 8718 \beta_{2} + 8718 \beta_{3} - 814 \beta_{4} + 814 \beta_{5} - 143 \beta_{7} ) q^{14} \) \( + ( -400805 - 54445 \beta_{1} - 8305 \beta_{2} + 1650 \beta_{3} + 1105 \beta_{4} - 2130 \beta_{5} - 57 \beta_{6} - 219 \beta_{7} ) q^{15} \) \( + ( -148906 + 1714 \beta_{2} - 1714 \beta_{3} + 1028 \beta_{4} + 1028 \beta_{5} + 326 \beta_{6} ) q^{16} \) \( + ( 129477 + 129477 \beta_{1} + 2866 \beta_{3} + 1946 \beta_{5} - 670 \beta_{6} + 670 \beta_{7} ) q^{17} \) \( + ( 1602033 - 1602033 \beta_{1} + 5151 \beta_{2} - 876 \beta_{4} + 765 \beta_{6} + 765 \beta_{7} ) q^{18} \) \( + ( 1555002 \beta_{1} - 30546 \beta_{2} - 30546 \beta_{3} + 2958 \beta_{4} - 2958 \beta_{5} + 666 \beta_{7} ) q^{19} \) \( + ( -2077655 - 1909100 \beta_{1} + 52235 \beta_{2} - 14235 \beta_{3} - 970 \beta_{4} + 8150 \beta_{5} - 143 \beta_{6} + 684 \beta_{7} ) q^{20} \) \( + ( -2617580 - 45161 \beta_{2} + 45161 \beta_{3} - 4447 \beta_{4} - 4447 \beta_{5} - 864 \beta_{6} ) q^{21} \) \( + ( 3220953 + 3220953 \beta_{1} + 91528 \beta_{3} - 5660 \beta_{5} + 2465 \beta_{6} - 2465 \beta_{7} ) q^{22} \) \( + ( 3106519 - 3106519 \beta_{1} + 71257 \beta_{2} + 13971 \beta_{4} - 3225 \beta_{6} - 3225 \beta_{7} ) q^{23} \) \( + ( 7464036 \beta_{1} - 106428 \beta_{2} - 106428 \beta_{3} - 4056 \beta_{4} + 4056 \beta_{5} - 2052 \beta_{7} ) q^{24} \) \( + ( -4372175 - 3653650 \beta_{1} + 72175 \beta_{2} - 100775 \beta_{3} - 20575 \beta_{4} - 8975 \beta_{5} + 2190 \beta_{6} - 670 \beta_{7} ) q^{25} \) \( + ( -4282586 + 38451 \beta_{2} - 38451 \beta_{3} - 6548 \beta_{4} - 6548 \beta_{5} - 706 \beta_{6} ) q^{26} \) \( + ( 3428172 + 3428172 \beta_{1} - 38412 \beta_{3} + 19044 \beta_{5} - 3240 \beta_{6} + 3240 \beta_{7} ) q^{27} \) \( + ( 11208946 - 11208946 \beta_{1} - 128492 \beta_{2} - 25816 \beta_{4} + 6370 \beta_{6} + 6370 \beta_{7} ) q^{28} \) \( + ( 3626498 \beta_{1} + 232696 \beta_{2} + 232696 \beta_{3} + 8492 \beta_{4} - 8492 \beta_{5} + 4294 \beta_{7} ) q^{29} \) \( + ( -14639675 - 3161310 \beta_{1} - 250970 \beta_{2} + 329030 \beta_{3} + 90950 \beta_{4} - 4690 \beta_{5} - 8307 \beta_{6} - 1314 \beta_{7} ) q^{30} \) \( + ( -43678 - 85515 \beta_{2} + 85515 \beta_{3} + 66495 \beta_{4} + 66495 \beta_{5} + 10650 \beta_{6} ) q^{31} \) \( + ( -1350806 - 1350806 \beta_{1} - 377116 \beta_{3} - 51720 \beta_{5} - 8070 \beta_{6} + 8070 \beta_{7} ) q^{32} \) \( + ( 1433096 - 1433096 \beta_{1} + 35182 \beta_{2} - 62542 \beta_{4} + 2460 \beta_{6} + 2460 \beta_{7} ) q^{33} \) \( + ( -4069086 \beta_{1} + 423257 \beta_{2} + 423257 \beta_{3} - 52436 \beta_{4} + 52436 \beta_{5} - 3962 \beta_{7} ) q^{34} \) \( + ( 14096180 + 5790330 \beta_{1} - 735850 \beta_{2} - 139255 \beta_{3} - 118130 \beta_{4} - 43755 \beta_{5} + 11224 \beta_{6} + 1078 \beta_{7} ) q^{35} \) \( + ( 8847759 + 1072209 \beta_{2} - 1072209 \beta_{3} - 101982 \beta_{4} - 101982 \beta_{5} - 23769 \beta_{6} ) q^{36} \) \( + ( 11681656 + 11681656 \beta_{1} - 683868 \beta_{3} + 65304 \beta_{5} + 36195 \beta_{6} - 36195 \beta_{7} ) q^{37} \) \( + ( -54834366 + 54834366 \beta_{1} - 1149036 \beta_{2} + 264168 \beta_{4} - 41790 \beta_{6} - 41790 \beta_{7} ) q^{38} \) \( + ( -37111654 \beta_{1} + 279013 \beta_{2} + 279013 \beta_{3} + 109751 \beta_{4} - 109751 \beta_{5} - 13698 \beta_{7} ) q^{39} \) \( + ( 54150175 + 22931375 \beta_{1} + 1162400 \beta_{2} + 805350 \beta_{3} - 120800 \beta_{4} + 172500 \beta_{5} + 16055 \beta_{6} + 9785 \beta_{7} ) q^{40} \) \( + ( 13628922 - 1644265 \beta_{2} + 1644265 \beta_{3} - 62755 \beta_{4} - 62755 \beta_{5} - 3050 \beta_{6} ) q^{41} \) \( + ( -76691923 - 76691923 \beta_{1} + 2224424 \beta_{3} - 32812 \beta_{5} - 29355 \beta_{6} + 29355 \beta_{7} ) q^{42} \) \( + ( -32009710 + 32009710 \beta_{1} + 3102473 \beta_{2} - 185537 \beta_{4} + 76850 \beta_{6} + 76850 \beta_{7} ) q^{43} \) \( + ( -83552416 \beta_{1} - 3428728 \beta_{2} - 3428728 \beta_{3} + 19344 \beta_{4} - 19344 \beta_{5} + 68448 \beta_{7} ) q^{44} \) \( + ( 75077955 + 47846775 \beta_{1} + 840390 \beta_{2} - 3193815 \beta_{3} + 447420 \beta_{4} + 146175 \beta_{5} - 80892 \beta_{6} + 171 \beta_{7} ) q^{45} \) \( + ( 127663867 - 250930 \beta_{2} + 250930 \beta_{3} + 192790 \beta_{4} + 192790 \beta_{5} + 94915 \beta_{6} ) q^{46} \) \( + ( -83991827 - 83991827 \beta_{1} + 1108775 \beta_{3} + 117963 \beta_{5} - 82275 \beta_{6} + 82275 \beta_{7} ) q^{47} \) \( + ( -77580556 + 77580556 \beta_{1} - 1551368 \beta_{2} - 252304 \beta_{4} + 10260 \beta_{6} + 10260 \beta_{7} ) q^{48} \) \( + ( -66130911 \beta_{1} + 874105 \beta_{2} + 874105 \beta_{3} - 152365 \beta_{4} + 152365 \beta_{5} - 107710 \beta_{7} ) q^{49} \) \( + ( 118917825 + 126311225 \beta_{1} + 4028800 \beta_{2} + 2015225 \beta_{3} - 221200 \beta_{4} - 1055100 \beta_{5} + 101165 \beta_{6} - 95845 \beta_{7} ) q^{50} \) \( + ( -97428472 - 2800027 \beta_{2} + 2800027 \beta_{3} - 7829 \beta_{4} - 7829 \beta_{5} - 122208 \beta_{6} ) q^{51} \) \( + ( -9122125 - 9122125 \beta_{1} + 965158 \beta_{3} + 95092 \beta_{5} + 154475 \beta_{6} - 154475 \beta_{7} ) q^{52} \) \( + ( -164812362 + 164812362 \beta_{1} + 1920956 \beta_{2} + 182384 \beta_{4} - 209095 \beta_{6} - 209095 \beta_{7} ) q^{53} \) \( + ( 77093892 \beta_{1} + 74484 \beta_{2} + 74484 \beta_{3} - 380232 \beta_{4} + 380232 \beta_{5} - 52164 \beta_{7} ) q^{54} \) \( + ( -893350 - 146436010 \beta_{1} - 8342645 \beta_{2} - 3715595 \beta_{3} - 209875 \beta_{4} + 454885 \beta_{5} + 3198 \beta_{6} + 138146 \beta_{7} ) q^{55} \) \( + ( 110752516 + 8747068 \beta_{2} - 8747068 \beta_{3} + 470936 \beta_{4} + 470936 \beta_{5} - 32348 \beta_{6} ) q^{56} \) \( + ( 298937754 + 298937754 \beta_{1} - 10052784 \beta_{3} - 2063592 \beta_{5} + 94050 \beta_{6} - 94050 \beta_{7} ) q^{57} \) \( + ( 335089616 - 335089616 \beta_{1} - 4057064 \beta_{2} - 523168 \beta_{4} + 181360 \beta_{6} + 181360 \beta_{7} ) q^{58} \) \( + ( -131940554 \beta_{1} + 8624342 \beta_{2} + 8624342 \beta_{3} + 201934 \beta_{4} - 201934 \beta_{5} + 388958 \beta_{7} ) q^{59} \) \( + ( -535081070 - 159057770 \beta_{1} - 5448300 \beta_{2} + 13931920 \beta_{3} - 41880 \beta_{4} + 2098720 \beta_{5} - 131646 \beta_{6} + 205338 \beta_{7} ) q^{60} \) \( + ( -393994198 - 3134025 \beta_{2} + 3134025 \beta_{3} - 945675 \beta_{4} - 945675 \beta_{5} + 64350 \beta_{6} ) q^{61} \) \( + ( -172556257 - 172556257 \beta_{1} - 153812 \beta_{3} + 3533820 \beta_{5} - 303705 \beta_{6} + 303705 \beta_{7} ) q^{62} \) \( + ( 319198467 - 319198467 \beta_{1} - 9503535 \beta_{2} + 3033867 \beta_{4} + 172035 \beta_{6} + 172035 \beta_{7} ) q^{63} \) \( + ( 753019820 \beta_{1} + 7387292 \beta_{2} + 7387292 \beta_{3} + 3048184 \beta_{4} - 3048184 \beta_{5} - 275852 \beta_{7} ) q^{64} \) \( + ( -241546905 - 331244945 \beta_{1} - 227105 \beta_{2} + 5019575 \beta_{3} - 20295 \beta_{4} - 1302505 \beta_{5} + 259958 \beta_{6} - 531914 \beta_{7} ) q^{65} \) \( + ( 99931322 + 6607664 \beta_{2} - 6607664 \beta_{3} - 1571372 \beta_{4} - 1571372 \beta_{5} + 120906 \beta_{6} ) q^{66} \) \( + ( 614362562 + 614362562 \beta_{1} - 3172169 \beta_{3} + 2004991 \beta_{5} - 276070 \beta_{6} + 276070 \beta_{7} ) q^{67} \) \( + ( 827517353 - 827517353 \beta_{1} + 414830 \beta_{2} - 2217252 \beta_{4} - 126255 \beta_{6} - 126255 \beta_{7} ) q^{68} \) \( + ( 565069564 \beta_{1} - 10578001 \beta_{2} - 10578001 \beta_{3} - 4847927 \beta_{4} + 4847927 \beta_{5} - 402144 \beta_{7} ) q^{69} \) \( + ( -1027302710 + 314713525 \beta_{1} + 3700970 \beta_{2} - 34206570 \beta_{3} - 34790 \beta_{4} - 2692050 \beta_{5} - 598006 \beta_{6} - 145597 \beta_{7} ) q^{70} \) \( + ( -546489198 + 2287925 \beta_{2} - 2287925 \beta_{3} + 1502975 \beta_{4} + 1502975 \beta_{5} + 497050 \beta_{6} ) q^{71} \) \( + ( 100393101 + 100393101 \beta_{1} + 11159154 \beta_{3} - 8286948 \beta_{5} + 682965 \beta_{6} - 682965 \beta_{7} ) q^{72} \) \( + ( -88546361 + 88546361 \beta_{1} + 34520352 \beta_{2} - 4778664 \beta_{4} - 494490 \beta_{6} - 494490 \beta_{7} ) q^{73} \) \( + ( 1079624708 \beta_{1} - 28841005 \beta_{2} - 28841005 \beta_{3} - 199560 \beta_{4} + 199560 \beta_{5} + 25860 \beta_{7} ) q^{74} \) \( + ( 677531150 - 1509180050 \beta_{1} + 37439725 \beta_{2} + 12520200 \beta_{3} + 2671475 \beta_{4} + 1624800 \beta_{5} + 617130 \beta_{6} + 620910 \beta_{7} ) q^{75} \) \( + ( -730844028 - 61604244 \beta_{2} + 61604244 \beta_{3} + 5609112 \beta_{4} + 5609112 \beta_{5} - 1690716 \beta_{6} ) q^{76} \) \( + ( 927687950 + 927687950 \beta_{1} + 49319266 \beta_{3} + 951834 \beta_{5} + 683730 \beta_{6} - 683730 \beta_{7} ) q^{77} \) \( + ( 515730815 - 515730815 \beta_{1} + 12124756 \beta_{2} + 4151996 \beta_{4} + 169095 \beta_{6} + 169095 \beta_{7} ) q^{78} \) \( + ( -1550075432 \beta_{1} - 15812464 \beta_{2} - 15812464 \beta_{3} + 5171872 \beta_{4} - 5171872 \beta_{5} + 1230184 \beta_{7} ) q^{79} \) \( + ( 9336800 + 1102878770 \beta_{1} + 5716090 \beta_{2} - 6663110 \beta_{3} - 2629100 \beta_{4} + 1519980 \beta_{5} + 496864 \beta_{6} + 1054078 \beta_{7} ) q^{80} \) \( + ( -289632915 + 44667207 \beta_{2} - 44667207 \beta_{3} - 869211 \beta_{4} - 869211 \beta_{5} + 484218 \beta_{6} ) q^{81} \) \( + ( -2422378107 - 2422378107 \beta_{1} - 35417912 \beta_{3} + 3564820 \beta_{5} - 1494355 \beta_{6} + 1494355 \beta_{7} ) q^{82} \) \( + ( -1399300588 + 1399300588 \beta_{1} - 63429915 \beta_{2} + 1652107 \beta_{4} + 1372000 \beta_{6} + 1372000 \beta_{7} ) q^{83} \) \( + ( -1297699424 \beta_{1} + 37203656 \beta_{2} + 37203656 \beta_{3} - 8215088 \beta_{4} + 8215088 \beta_{5} + 804384 \beta_{7} ) q^{84} \) \( + ( 1974905430 - 123378820 \beta_{1} - 30557525 \beta_{2} - 14294055 \beta_{3} - 5120755 \beta_{4} - 3623355 \beta_{5} - 1185281 \beta_{6} - 986807 \beta_{7} ) q^{85} \) \( + ( 4548733923 + 15298358 \beta_{2} - 15298358 \beta_{3} - 10657834 \beta_{4} - 10657834 \beta_{5} + 2243947 \beta_{6} ) q^{86} \) \( + ( 247013346 + 247013346 \beta_{1} + 12179784 \beta_{3} + 9903792 \beta_{5} - 841230 \beta_{6} + 841230 \beta_{7} ) q^{87} \) \( + ( -1641774808 + 1641774808 \beta_{1} + 68034832 \beta_{2} + 8847584 \beta_{4} - 1490840 \beta_{6} - 1490840 \beta_{7} ) q^{88} \) \( + ( -857631636 \beta_{1} - 5484672 \beta_{2} - 5484672 \beta_{3} + 13020456 \beta_{4} - 13020456 \beta_{5} - 4174908 \beta_{7} ) q^{89} \) \( + ( 1159488570 + 5314871805 \beta_{1} - 53737680 \beta_{2} + 34717275 \beta_{3} + 11936730 \beta_{4} + 6177870 \beta_{5} + 591318 \beta_{6} - 3549969 \beta_{7} ) q^{90} \) \( + ( -1611026732 + 2585993 \beta_{2} - 2585993 \beta_{3} - 4809089 \beta_{4} - 4809089 \beta_{5} - 1377548 \beta_{6} ) q^{91} \) \( + ( -3193999078 - 3193999078 \beta_{1} - 98524404 \beta_{3} - 4048664 \beta_{5} + 1906570 \beta_{6} - 1906570 \beta_{7} ) q^{92} \) \( + ( -5046089144 + 5046089144 \beta_{1} - 126847858 \beta_{2} - 13353422 \beta_{4} - 583020 \beta_{6} - 583020 \beta_{7} ) q^{93} \) \( + ( -2386069205 \beta_{1} + 140943394 \beta_{2} + 140943394 \beta_{3} - 5048662 \beta_{4} + 5048662 \beta_{5} + 28301 \beta_{7} ) q^{94} \) \( + ( 1802255400 - 2495905350 \beta_{1} + 18305250 \beta_{2} + 146419350 \beta_{3} - 10830150 \beta_{4} + 9473850 \beta_{5} - 2675580 \beta_{6} + 2853990 \beta_{7} ) q^{95} \) \( + ( 4790162456 + 60969992 \beta_{2} - 60969992 \beta_{3} + 1200784 \beta_{4} + 1200784 \beta_{5} + 890328 \beta_{6} ) q^{96} \) \( + ( 3118474113 + 3118474113 \beta_{1} - 111519040 \beta_{3} - 18230032 \beta_{5} + 874960 \beta_{6} - 874960 \beta_{7} ) q^{97} \) \( + ( 1721410159 - 1721410159 \beta_{1} + 2664681 \beta_{2} - 10809940 \beta_{4} + 2040515 \beta_{6} + 2040515 \beta_{7} ) q^{98} \) \( + ( -5893566186 \beta_{1} - 61648701 \beta_{2} - 61648701 \beta_{3} - 776127 \beta_{4} + 776127 \beta_{5} + 4306086 \beta_{7} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(8q \) \(\mathstrut +\mathstrut 30q^{2} \) \(\mathstrut +\mathstrut 60q^{3} \) \(\mathstrut -\mathstrut 5340q^{5} \) \(\mathstrut +\mathstrut 17016q^{6} \) \(\mathstrut -\mathstrut 14500q^{7} \) \(\mathstrut +\mathstrut 22020q^{8} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(8q \) \(\mathstrut +\mathstrut 30q^{2} \) \(\mathstrut +\mathstrut 60q^{3} \) \(\mathstrut -\mathstrut 5340q^{5} \) \(\mathstrut +\mathstrut 17016q^{6} \) \(\mathstrut -\mathstrut 14500q^{7} \) \(\mathstrut +\mathstrut 22020q^{8} \) \(\mathstrut -\mathstrut 161290q^{10} \) \(\mathstrut -\mathstrut 233784q^{11} \) \(\mathstrut +\mathstrut 863160q^{12} \) \(\mathstrut +\mathstrut 433520q^{13} \) \(\mathstrut -\mathstrut 3188580q^{15} \) \(\mathstrut -\mathstrut 1193992q^{16} \) \(\mathstrut +\mathstrut 1045440q^{17} \) \(\mathstrut +\mathstrut 12804210q^{18} \) \(\mathstrut -\mathstrut 16739820q^{20} \) \(\mathstrut -\mathstrut 20777784q^{21} \) \(\mathstrut +\mathstrut 25939360q^{22} \) \(\mathstrut +\mathstrut 24737580q^{23} \) \(\mathstrut -\mathstrut 35382400q^{25} \) \(\mathstrut -\mathstrut 34440684q^{26} \) \(\mathstrut +\mathstrut 27386640q^{27} \) \(\mathstrut +\mathstrut 89876920q^{28} \) \(\mathstrut -\mathstrut 115784880q^{30} \) \(\mathstrut +\mathstrut 258616q^{31} \) \(\mathstrut -\mathstrut 11664120q^{32} \) \(\mathstrut +\mathstrut 11269320q^{33} \) \(\mathstrut +\mathstrut 113638860q^{35} \) \(\mathstrut +\mathstrut 66085308q^{36} \) \(\mathstrut +\mathstrut 92216120q^{37} \) \(\mathstrut -\mathstrut 435848520q^{38} \) \(\mathstrut +\mathstrut 432590700q^{40} \) \(\mathstrut +\mathstrut 115357416q^{41} \) \(\mathstrut -\mathstrut 609152160q^{42} \) \(\mathstrut -\mathstrut 262653700q^{43} \) \(\mathstrut +\mathstrut 593742420q^{45} \) \(\mathstrut +\mathstrut 1023085816q^{46} \) \(\mathstrut -\mathstrut 669481140q^{47} \) \(\mathstrut -\mathstrut 618046320q^{48} \) \(\mathstrut +\mathstrut 944762850q^{50} \) \(\mathstrut -\mathstrut 768258984q^{51} \) \(\mathstrut -\mathstrut 70856500q^{52} \) \(\mathstrut -\mathstrut 1321976040q^{53} \) \(\mathstrut +\mathstrut 2597320q^{55} \) \(\mathstrut +\mathstrut 852915600q^{56} \) \(\mathstrut +\mathstrut 2367269280q^{57} \) \(\mathstrut +\mathstrut 2687784720q^{58} \) \(\mathstrut -\mathstrut 4237774440q^{60} \) \(\mathstrut -\mathstrut 3143200184q^{61} \) \(\mathstrut -\mathstrut 1373690040q^{62} \) \(\mathstrut +\mathstrut 2578662540q^{63} \) \(\mathstrut -\mathstrut 1924527480q^{65} \) \(\mathstrut +\mathstrut 766734432q^{66} \) \(\mathstrut +\mathstrut 4912566140q^{67} \) \(\mathstrut +\mathstrut 6614874660q^{68} \) \(\mathstrut -\mathstrut 8299690440q^{70} \) \(\mathstrut -\mathstrut 4375053384q^{71} \) \(\mathstrut +\mathstrut 808889220q^{72} \) \(\mathstrut -\mathstrut 786968920q^{73} \) \(\mathstrut +\mathstrut 5379002700q^{75} \) \(\mathstrut -\mathstrut 5577898800q^{76} \) \(\mathstrut +\mathstrut 7522045800q^{77} \) \(\mathstrut +\mathstrut 4109901000q^{78} \) \(\mathstrut +\mathstrut 47717760q^{80} \) \(\mathstrut -\mathstrut 2499208992q^{81} \) \(\mathstrut -\mathstrut 19442731040q^{82} \) \(\mathstrut -\mathstrut 11064240660q^{83} \) \(\mathstrut +\mathstrut 15814282160q^{85} \) \(\mathstrut +\mathstrut 36286046616q^{86} \) \(\mathstrut +\mathstrut 2020273920q^{87} \) \(\mathstrut -\mathstrut 13252572960q^{88} \) \(\mathstrut +\mathstrut 9489047670q^{90} \) \(\mathstrut -\mathstrut 12917794184q^{91} \) \(\mathstrut -\mathstrut 25757138760q^{92} \) \(\mathstrut -\mathstrut 40141724280q^{93} \) \(\mathstrut +\mathstrut 14671558800q^{95} \) \(\mathstrut +\mathstrut 38082222816q^{96} \) \(\mathstrut +\mathstrut 24688294760q^{97} \) \(\mathstrut +\mathstrut 13744332030q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8}\mathstrut +\mathstrut \) \(1334\) \(x^{6}\mathstrut +\mathstrut \) \(456089\) \(x^{4}\mathstrut +\mathstrut \) \(43159076\) \(x^{2}\mathstrut +\mathstrut \) \(31360000\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{7} - 1334 \nu^{5} - 450489 \nu^{3} - 39423876 \nu \)\()/33454400\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{7} - 35 \nu^{6} - 1789 \nu^{5} - 37625 \nu^{4} - 963064 \nu^{3} - 7002590 \nu^{2} - 127280596 \nu - 113422400 \)\()/12545400\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{7} + 35 \nu^{6} - 1789 \nu^{5} + 37625 \nu^{4} - 963064 \nu^{3} + 7002590 \nu^{2} - 127280596 \nu + 113422400 \)\()/12545400\)
\(\beta_{4}\)\(=\)\((\)\( -5 \nu^{7} - 196 \nu^{6} - 9218 \nu^{5} - 235900 \nu^{4} - 5122865 \nu^{3} - 56022904 \nu^{2} - 839661812 \nu - 756212800 \)\()/10036320\)
\(\beta_{5}\)\(=\)\((\)\( 5 \nu^{7} - 196 \nu^{6} + 9218 \nu^{5} - 235900 \nu^{4} + 5122865 \nu^{3} - 56022904 \nu^{2} + 839661812 \nu - 756212800 \)\()/10036320\)
\(\beta_{6}\)\(=\)\((\)\( -\nu^{6} - 1585 \nu^{4} - 719464 \nu^{2} - 65776900 \)\()/35844\)
\(\beta_{7}\)\(=\)\((\)\( -29 \nu^{7} - 38414 \nu^{5} - 12978341 \nu^{3} - 1229515316 \nu \)\()/573504\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(7\) \(\beta_{3}\mathstrut +\mathstrut \) \(7\) \(\beta_{2}\mathstrut -\mathstrut \) \(4\) \(\beta_{1}\)\()/30\)
\(\nu^{2}\)\(=\)\((\)\(-\)\(6\) \(\beta_{6}\mathstrut +\mathstrut \) \(17\) \(\beta_{5}\mathstrut +\mathstrut \) \(17\) \(\beta_{4}\mathstrut +\mathstrut \) \(89\) \(\beta_{3}\mathstrut -\mathstrut \) \(89\) \(\beta_{2}\mathstrut -\mathstrut \) \(10058\)\()/30\)
\(\nu^{3}\)\(=\)\((\)\(-\)\(78\) \(\beta_{7}\mathstrut -\mathstrut \) \(629\) \(\beta_{5}\mathstrut +\mathstrut \) \(629\) \(\beta_{4}\mathstrut -\mathstrut \) \(4913\) \(\beta_{3}\mathstrut -\mathstrut \) \(4913\) \(\beta_{2}\mathstrut +\mathstrut \) \(137186\) \(\beta_{1}\)\()/30\)
\(\nu^{4}\)\(=\)\((\)\(1334\) \(\beta_{6}\mathstrut -\mathstrut \) \(5771\) \(\beta_{5}\mathstrut -\mathstrut \) \(5771\) \(\beta_{4}\mathstrut -\mathstrut \) \(33727\) \(\beta_{3}\mathstrut +\mathstrut \) \(33727\) \(\beta_{2}\mathstrut +\mathstrut \) \(2188194\)\()/10\)
\(\nu^{5}\)\(=\)\((\)\(87870\) \(\beta_{7}\mathstrut +\mathstrut \) \(515501\) \(\beta_{5}\mathstrut -\mathstrut \) \(515501\) \(\beta_{4}\mathstrut +\mathstrut \) \(3769457\) \(\beta_{3}\mathstrut +\mathstrut \) \(3769457\) \(\beta_{2}\mathstrut -\mathstrut \) \(151567154\) \(\beta_{1}\)\()/30\)
\(\nu^{6}\)\(=\)\((\)\(-\)\(3101706\) \(\beta_{6}\mathstrut +\mathstrut \) \(15210217\) \(\beta_{5}\mathstrut +\mathstrut \) \(15210217\) \(\beta_{4}\mathstrut +\mathstrut \) \(96339589\) \(\beta_{3}\mathstrut -\mathstrut \) \(96339589\) \(\beta_{2}\mathstrut -\mathstrut \) \(5141800558\)\()/30\)
\(\nu^{7}\)\(=\)\((\)\(-\)\(82080438\) \(\beta_{7}\mathstrut -\mathstrut \) \(443744629\) \(\beta_{5}\mathstrut +\mathstrut \) \(443744629\) \(\beta_{4}\mathstrut -\mathstrut \) \(3091170313\) \(\beta_{3}\mathstrut -\mathstrut \) \(3091170313\) \(\beta_{2}\mathstrut +\mathstrut \) \(139543862986\) \(\beta_{1}\)\()/30\)

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
17.5804i
0.855727i
12.6655i
29.3902i
17.5804i
0.855727i
12.6655i
29.3902i
−36.6454 36.6454i −13.8338 + 13.8338i 1661.78i −721.162 + 3040.65i 1013.89 −13891.7 13891.7i 23371.7 23371.7i 58666.3i 137853. 84998.7i
2.2 −4.63382 4.63382i 70.6389 70.6389i 981.055i 1004.90 2959.02i −654.656 2611.74 + 2611.74i −9291.07 + 9291.07i 49069.3i −18368.1 + 9055.08i
2.3 18.8402 + 18.8402i −273.023 + 273.023i 314.097i 71.2753 + 3124.19i −10287.6 12640.3 + 12640.3i 25210.0 25210.0i 90034.1i −57517.4 + 60203.0i
2.4 37.4391 + 37.4391i 246.218 246.218i 1779.37i −3025.01 + 784.185i 18436.4 −8610.35 8610.35i −28280.5 + 28280.5i 62197.5i −142613. 83894.5i
3.1 −36.6454 + 36.6454i −13.8338 13.8338i 1661.78i −721.162 3040.65i 1013.89 −13891.7 + 13891.7i 23371.7 + 23371.7i 58666.3i 137853. + 84998.7i
3.2 −4.63382 + 4.63382i 70.6389 + 70.6389i 981.055i 1004.90 + 2959.02i −654.656 2611.74 2611.74i −9291.07 9291.07i 49069.3i −18368.1 9055.08i
3.3 18.8402 18.8402i −273.023 273.023i 314.097i 71.2753 3124.19i −10287.6 12640.3 12640.3i 25210.0 + 25210.0i 90034.1i −57517.4 60203.0i
3.4 37.4391 37.4391i 246.218 + 246.218i 1779.37i −3025.01 784.185i 18436.4 −8610.35 + 8610.35i −28280.5 28280.5i 62197.5i −142613. + 83894.5i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.4
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 \(S_{11}^{\mathrm{new}}(5, [\chi])\).