Properties

Label 4.15.b.a
Level $4$
Weight $15$
Character orbit 4.b
Analytic conductor $4.973$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 4 = 2^{2} \)
Weight: \( k \) \(=\) \( 15 \)
Character orbit: \([\chi]\) \(=\) 4.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.97315872608\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Defining polynomial: \(x^{6} - x^{5} + 88 x^{4} - 1824 x^{3} + 325632 x^{2} + 21572352 x + 982333440\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{30}\cdot 3^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( -15 + \beta_{1} ) q^{2} + ( -\beta_{1} + \beta_{2} ) q^{3} + ( -237 - 16 \beta_{1} + \beta_{3} ) q^{4} + ( 1363 + 62 \beta_{1} - 3 \beta_{3} + \beta_{4} ) q^{5} + ( 8456 + 39 \beta_{1} - 64 \beta_{2} + 4 \beta_{4} + \beta_{5} ) q^{6} + ( -600 - 1794 \beta_{1} - 42 \beta_{2} + 40 \beta_{3} + 8 \beta_{4} - 4 \beta_{5} ) q^{7} + ( 74460 - 532 \beta_{1} + 768 \beta_{2} - 12 \beta_{3} + 16 \beta_{4} + 20 \beta_{5} ) q^{8} + ( -494453 + 16060 \beta_{1} - 6 \beta_{3} + 2 \beta_{4} - 64 \beta_{5} ) q^{9} +O(q^{10})\) \( q + ( -15 + \beta_{1} ) q^{2} + ( -\beta_{1} + \beta_{2} ) q^{3} + ( -237 - 16 \beta_{1} + \beta_{3} ) q^{4} + ( 1363 + 62 \beta_{1} - 3 \beta_{3} + \beta_{4} ) q^{5} + ( 8456 + 39 \beta_{1} - 64 \beta_{2} + 4 \beta_{4} + \beta_{5} ) q^{6} + ( -600 - 1794 \beta_{1} - 42 \beta_{2} + 40 \beta_{3} + 8 \beta_{4} - 4 \beta_{5} ) q^{7} + ( 74460 - 532 \beta_{1} + 768 \beta_{2} - 12 \beta_{3} + 16 \beta_{4} + 20 \beta_{5} ) q^{8} + ( -494453 + 16060 \beta_{1} - 6 \beta_{3} + 2 \beta_{4} - 64 \beta_{5} ) q^{9} + ( 966410 + 3642 \beta_{1} - 3072 \beta_{2} + 64 \beta_{3} - 64 \beta_{4} + 176 \beta_{5} ) q^{10} + ( -34256 - 102439 \beta_{1} + 831 \beta_{2} - 720 \beta_{3} - 144 \beta_{4} - 440 \beta_{5} ) q^{11} + ( 35528 + 7696 \beta_{1} + 1024 \beta_{2} + 24 \beta_{3} - 320 \beta_{4} + 880 \beta_{5} ) q^{12} + ( -13330889 + 395798 \beta_{1} + 897 \beta_{3} - 299 \beta_{4} - 1664 \beta_{5} ) q^{13} + ( 29219504 + 14106 \beta_{1} + 27264 \beta_{2} - 1536 \beta_{3} + 344 \beta_{4} + 2646 \beta_{5} ) q^{14} + ( -344136 - 1023502 \beta_{1} - 10262 \beta_{2} + 4920 \beta_{3} + 984 \beta_{4} - 3564 \beta_{5} ) q^{15} + ( -58840272 + 52016 \beta_{1} - 70656 \beta_{2} + 272 \beta_{3} + 2624 \beta_{4} + 4304 \beta_{5} ) q^{16} + ( -19693924 + 997740 \beta_{1} - 10494 \beta_{3} + 3498 \beta_{4} - 3136 \beta_{5} ) q^{17} + ( 266485673 - 254887 \beta_{1} - 6144 \beta_{2} + 16512 \beta_{3} - 128 \beta_{4} + 352 \beta_{5} ) q^{18} + ( 581904 + 1663419 \beta_{1} + 88269 \beta_{2} - 11760 \beta_{3} - 2352 \beta_{4} + 5784 \beta_{5} ) q^{19} + ( -841116322 + 45920 \beta_{1} + 294912 \beta_{2} - 1302 \beta_{3} - 10240 \beta_{4} - 16896 \beta_{5} ) q^{20} + ( 201670604 - 8402776 \beta_{1} + 56508 \beta_{3} - 18836 \beta_{4} + 29056 \beta_{5} ) q^{21} + ( 1653230840 + 1846169 \beta_{1} - 495552 \beta_{2} - 103424 \beta_{3} - 5892 \beta_{4} - 47553 \beta_{5} ) q^{22} + ( 5797208 + 17918266 \beta_{1} - 558222 \beta_{2} - 32040 \beta_{3} - 6408 \beta_{4} + 63620 \beta_{5} ) q^{23} + ( -4016602464 - 3841760 \beta_{1} + 198656 \beta_{2} - 1824 \beta_{3} + 9600 \beta_{4} - 74016 \beta_{5} ) q^{24} + ( 651711215 - 16285480 \beta_{1} - 145020 \beta_{3} + 48340 \beta_{4} + 77440 \beta_{5} ) q^{25} + ( 6589368058 - 7902102 \beta_{1} + 918528 \beta_{2} + 406848 \beta_{3} + 19136 \beta_{4} - 52624 \beta_{5} ) q^{26} + ( -995472 - 5913978 \beta_{1} + 2665266 \beta_{2} + 255600 \beta_{3} + 51120 \beta_{4} + 6696 \beta_{5} ) q^{27} + ( -10843208912 + 20828512 \beta_{1} - 3188736 \beta_{2} + 21520 \beta_{3} + 78976 \beta_{4} + 77728 \beta_{5} ) q^{28} + ( -1324329461 + 45071342 \beta_{1} + 21381 \beta_{3} - 7127 \beta_{4} - 182784 \beta_{5} ) q^{29} + ( 16622193488 + 14119206 \beta_{1} + 3679616 \beta_{2} - 975360 \beta_{3} + 21928 \beta_{4} + 320362 \beta_{5} ) q^{30} + ( -37573824 - 102230424 \beta_{1} - 9678024 \beta_{2} - 360640 \beta_{3} - 72128 \beta_{4} - 452384 \beta_{5} ) q^{31} + ( -14309105728 - 74843200 \beta_{1} + 2715648 \beta_{2} - 10944 \beta_{3} - 320256 \beta_{4} + 554048 \beta_{5} ) q^{32} + ( -6245024786 + 149558596 \beta_{1} + 948486 \beta_{3} - 316162 \beta_{4} - 679360 \beta_{5} ) q^{33} + ( 16345090882 - 206590 \beta_{1} - 10745856 \beta_{2} + 1026688 \beta_{3} - 223872 \beta_{4} + 615648 \beta_{5} ) q^{34} + ( -39475360 - 142965740 \beta_{1} + 26383260 \beta_{2} - 1303200 \beta_{3} - 260640 \beta_{4} - 540400 \beta_{5} ) q^{35} + ( -4488805445 + 259811440 \beta_{1} + 13172736 \beta_{2} - 199239 \beta_{3} + 241664 \beta_{4} + 293888 \beta_{5} ) q^{36} + ( 5218198335 - 10269914 \beta_{1} - 2437863 \beta_{3} + 812621 \beta_{4} + 243584 \beta_{5} ) q^{37} + ( -27555475608 - 20060613 \beta_{1} - 12874560 \beta_{2} + 1631232 \beta_{3} + 202548 \beta_{4} - 702003 \beta_{5} ) q^{38} + ( 94909464 + 329839666 \beta_{1} - 51455014 \beta_{2} + 4918680 \beta_{3} + 983736 \beta_{4} + 1425060 \beta_{5} ) q^{39} + ( 72337339672 - 777719368 \beta_{1} - 12009984 \beta_{2} + 310536 \beta_{3} + 1322656 \beta_{4} - 2147768 \beta_{5} ) q^{40} + ( 13651689734 - 730386792 \beta_{1} + 1251108 \beta_{3} - 417036 \beta_{4} + 2829440 \beta_{5} ) q^{41} + ( -138334666112 + 52049728 \beta_{1} + 57864192 \beta_{2} - 8643840 \beta_{3} + 1205504 \beta_{4} - 3315136 \beta_{5} ) q^{42} + ( 355926432 + 1003107705 \beta_{1} + 61198167 \beta_{2} - 533600 \beta_{3} - 106720 \beta_{4} + 4007024 \beta_{5} ) q^{43} + ( 153208549176 + 1842877936 \beta_{1} - 43189248 \beta_{2} + 1187304 \beta_{3} - 3542720 \beta_{4} - 3954544 \beta_{5} ) q^{44} + ( 44903529967 - 836214362 \beta_{1} + 4428633 \beta_{3} - 1476211 \beta_{4} + 2990720 \beta_{5} ) q^{45} + ( -285478254704 - 281845538 \beta_{1} + 16040832 \beta_{2} + 16696832 \beta_{3} - 2643000 \beta_{4} - 2711310 \beta_{5} ) q^{46} + ( 131493520 + 424076948 \beta_{1} - 6819660 \beta_{2} - 22673520 \beta_{3} - 4534704 \beta_{4} - 103208 \beta_{5} ) q^{47} + ( 380919317632 - 3700910976 \beta_{1} - 21487616 \beta_{2} - 2997888 \beta_{3} + 611840 \beta_{4} + 2427776 \beta_{5} ) q^{48} + ( -91610685607 + 1169801584 \beta_{1} - 8520792 \beta_{3} + 2840264 \beta_{4} - 3990784 \beta_{5} ) q^{49} + ( -273156250805 + 477518395 \beta_{1} - 148500480 \beta_{2} - 16730880 \beta_{3} - 3093760 \beta_{4} + 8507840 \beta_{5} ) q^{50} + ( -1218840528 - 3535226314 \beta_{1} - 138758174 \beta_{2} + 29252400 \beta_{3} + 5850480 \beta_{4} - 11789496 \beta_{5} ) q^{51} + ( 178522104878 + 6738926688 \beta_{1} + 238977024 \beta_{2} - 4719910 \beta_{3} + 9877504 \beta_{4} + 13571584 \beta_{5} ) q^{52} + ( 14179007655 + 4185562774 \beta_{1} + 13230273 \beta_{3} - 4410091 \beta_{4} - 17907584 \beta_{5} ) q^{53} + ( 74271035280 + 72428958 \beta_{1} - 13536384 \beta_{2} - 1557504 \beta_{3} + 13932744 \beta_{4} + 19841586 \beta_{5} ) q^{54} + ( -1508932056 - 4887241602 \beta_{1} + 329638038 \beta_{2} + 44798120 \beta_{3} + 8959624 \beta_{4} - 13990724 \beta_{5} ) q^{55} + ( -97034292800 - 11185475392 \beta_{1} + 159952896 \beta_{2} + 18287424 \beta_{3} - 13674240 \beta_{4} + 15880000 \beta_{5} ) q^{56} + ( -439985987862 + 3715770252 \beta_{1} - 34763598 \beta_{3} + 11587866 \beta_{4} - 12037440 \beta_{5} ) q^{57} + ( 747095426138 - 669389046 \beta_{1} + 21894144 \beta_{2} + 46336576 \beta_{3} + 456128 \beta_{4} - 1254352 \beta_{5} ) q^{58} + ( 709108160 + 2791293853 \beta_{1} - 546697917 \beta_{2} - 117080640 \beta_{3} - 23416128 \beta_{4} - 190816 \beta_{5} ) q^{59} + ( -1585727001904 + 15825971872 \beta_{1} - 1001412608 \beta_{2} + 11961840 \beta_{3} - 1238144 \beta_{4} - 10652576 \beta_{5} ) q^{60} + ( 1032026495719 - 8969021962 \beta_{1} + 13189713 \beta_{3} - 4396571 \beta_{4} + 34925440 \beta_{5} ) q^{61} + ( 1729852047808 + 1481292168 \beta_{1} + 397816320 \beta_{2} - 111194112 \beta_{3} - 43328288 \beta_{4} - 33913032 \beta_{5} ) q^{62} + ( 3751833096 + 10280550750 \beta_{1} + 946154406 \beta_{2} - 12929400 \beta_{3} - 2585880 \beta_{4} + 41723532 \beta_{5} ) q^{63} + ( -2405422400768 - 16751006976 \beta_{1} + 63750144 \beta_{2} - 80533248 \beta_{3} + 15811584 \beta_{4} - 73083648 \beta_{5} ) q^{64} + ( -1203588341072 - 16025165208 \beta_{1} + 166036572 \beta_{3} - 55345524 \beta_{4} + 50577280 \beta_{5} ) q^{65} + ( 2510734529600 - 4470948064 \beta_{1} + 971249664 \beta_{2} + 153681792 \beta_{3} + 20234368 \beta_{4} - 55644512 \beta_{5} ) q^{66} + ( 7139475600 + 22564537239 \beta_{1} - 1427452239 \beta_{2} + 187034000 \beta_{3} + 37406800 \beta_{4} + 94307800 \beta_{5} ) q^{67} + ( -3079723278906 + 12888487392 \beta_{1} + 1648164864 \beta_{2} - 14289438 \beta_{3} - 22974464 \beta_{4} - 43045888 \beta_{5} ) q^{68} + ( 3223837317956 - 2432999368 \beta_{1} - 273709164 \beta_{3} + 91236388 \beta_{4} + 32488576 \beta_{5} ) q^{69} + ( 2113452056800 + 3263276740 \beta_{1} - 2489214720 \beta_{2} - 121661440 \beta_{3} + 88852080 \beta_{4} - 61191780 \beta_{5} ) q^{70} + ( -2324968248 - 7719530850 \beta_{1} + 805631430 \beta_{2} - 32308920 \beta_{3} - 6461784 \beta_{4} - 28696404 \beta_{5} ) q^{71} + ( -882985550596 - 4998161524 \beta_{1} - 1181668608 \beta_{2} + 273513300 \beta_{3} + 45636496 \beta_{4} + 66220660 \beta_{5} ) q^{72} + ( -4979827917204 + 1972003948 \beta_{1} - 134328318 \beta_{3} + 44776106 \beta_{4} + 3229376 \beta_{5} ) q^{73} + ( -254838087478 + 6175721146 \beta_{1} - 2496371712 \beta_{2} - 10349760 \beta_{3} - 52007744 \beta_{4} + 143021296 \beta_{5} ) q^{74} + ( -16263822240 - 50568677835 \beta_{1} + 2025758715 \beta_{2} - 59536800 \beta_{3} - 11907360 \beta_{4} - 189010800 \beta_{5} ) q^{75} + ( 3568347623976 - 25494220080 \beta_{1} + 1921201152 \beta_{2} - 18660552 \beta_{3} - 28639296 \beta_{4} + 67551408 \beta_{5} ) q^{76} + ( 6243061431860 + 59147030360 \beta_{1} + 477690180 \beta_{3} - 159230060 \beta_{4} - 277185920 \beta_{5} ) q^{77} + ( -4954879507760 - 7922490186 \beta_{1} + 6315157888 \beta_{2} + 301856256 \beta_{3} - 142860952 \beta_{4} + 279080282 \beta_{5} ) q^{78} + ( -4606467504 - 8638783044 \beta_{1} - 4855786644 \beta_{2} - 254552240 \beta_{3} - 50910448 \beta_{4} - 70280584 \beta_{5} ) q^{79} + ( 9668034451936 + 82845800928 \beta_{1} - 8669184 \beta_{2} - 754935648 \beta_{3} - 64233856 \beta_{4} + 306347552 \beta_{5} ) q^{80} + ( -16418559874377 + 70417538196 \beta_{1} + 190923390 \beta_{3} - 63641130 \beta_{4} - 298647744 \beta_{5} ) q^{81} + ( -11982506785070 + 2311561010 \beta_{1} + 1281134592 \beta_{2} - 751026944 \beta_{3} + 26690304 \beta_{4} - 73398336 \beta_{5} ) q^{82} + ( -19227358976 - 61868726269 \beta_{1} + 4480550013 \beta_{2} - 70444800 \beta_{3} - 14088960 \beta_{4} - 223455872 \beta_{5} ) q^{83} + ( 17117066013824 - 118803644416 \beta_{1} - 11267604480 \beta_{2} + 115421568 \beta_{3} + 73867264 \beta_{4} + 169486336 \beta_{5} ) q^{84} + ( 25167908023882 - 96351331212 \beta_{1} - 275765202 \beta_{3} + 91921734 \beta_{4} + 409841280 \beta_{5} ) q^{85} + ( -16699779642952 - 14017504191 \beta_{1} - 4244526528 \beta_{2} + 1032628224 \beta_{3} + 237962588 \beta_{4} + 25340247 \beta_{5} ) q^{86} + ( 1575294072 + 9805252610 \beta_{1} - 5811078086 \beta_{2} + 666825720 \beta_{3} + 133365144 \beta_{4} + 64881972 \beta_{5} ) q^{87} + ( 8648730592608 + 159872845536 \beta_{1} + 6396770304 \beta_{2} + 1782521632 \beta_{3} - 97076608 \beta_{4} - 855525088 \beta_{5} ) q^{88} + ( -30171664582948 - 30627930036 \beta_{1} - 1615544622 \beta_{3} + 538514874 \beta_{4} + 257091776 \beta_{5} ) q^{89} + ( -14144416356550 + 30557321258 \beta_{1} + 4534920192 \beta_{2} - 860101824 \beta_{3} + 94477504 \beta_{4} - 259813136 \beta_{5} ) q^{90} + ( 111401530656 + 317469370452 \beta_{1} + 15339351612 \beta_{2} + 121390880 \beta_{3} + 24278176 \beta_{4} + 1274479024 \beta_{5} ) q^{91} + ( 13617063943440 - 282869869792 \beta_{1} + 13826377728 \beta_{2} - 217040208 \beta_{3} + 373600640 \beta_{4} - 273770528 \beta_{5} ) q^{92} + ( 49138100062192 - 88513771232 \beta_{1} + 1426504368 \beta_{3} - 475501456 \beta_{4} + 237079040 \beta_{5} ) q^{93} + ( -6735029996128 + 519549548 \beta_{1} - 13494152448 \beta_{2} + 263799808 \beta_{3} - 317499696 \beta_{4} - 1530480204 \beta_{5} ) q^{94} + ( 5221711608 + 30781392666 \beta_{1} - 15783537054 \beta_{2} + 567928440 \beta_{3} + 113585688 \beta_{4} + 99350772 \beta_{5} ) q^{95} + ( -13660178064896 + 376078203392 \beta_{1} - 1397063680 \beta_{2} - 3742818816 \beta_{3} - 143706112 \beta_{4} + 62948864 \beta_{5} ) q^{96} + ( -57706577018604 - 393968566340 \beta_{1} + 2389807866 \beta_{3} - 796602622 \beta_{4} + 1383852224 \beta_{5} ) q^{97} + ( 20208515309633 - 70483565103 \beta_{1} - 8725291008 \beta_{2} + 1203417600 \beta_{3} - 181776896 \beta_{4} + 499886464 \beta_{5} ) q^{98} + ( -58303588032 - 156110294247 \beta_{1} - 15578994297 \beta_{2} - 2390512320 \beta_{3} - 478102464 \beta_{4} - 830963232 \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 92q^{2} - 1392q^{4} + 8060q^{5} + 50784q^{6} + 446272q^{8} - 2998698q^{9} + O(q^{10}) \) \( 6q - 92q^{2} - 1392q^{4} + 8060q^{5} + 50784q^{6} + 446272q^{8} - 2998698q^{9} + 5796840q^{10} + 193920q^{12} - 80775396q^{13} + 175232064q^{14} - 353013504q^{16} - 120131764q^{17} + 1599402372q^{18} - 5047343200q^{20} + 1226658048q^{21} + 9916985760q^{22} - 24092176896q^{24} + 3942973410q^{25} + 39549467048q^{26} - 65094731520q^{28} - 8035796644q^{29} + 99698873280q^{30} - 85711465472q^{32} - 37769804160q^{33} + 98089165512q^{34} - 27478990320q^{36} + 31334118396q^{37} - 165268841760q^{38} + 435607171200q^{40} + 83362750892q^{41} - 830203906560q^{42} + 915657452160q^{44} + 271078769820q^{45} - 1712365889856q^{46} + 2292961843200q^{48} - 551978693658q^{49} - 1639579094580q^{50} + 1057159118496q^{52} + 76712275004q^{53} + 445471858368q^{54} - 560223046656q^{56} - 2647253865600q^{57} + 4483777382184q^{58} - 9544013748480q^{60} + 6210000787932q^{61} + 10375644284160q^{62} - 14398852657152q^{64} - 7189912943720q^{65} + 15071210499840q^{66} - 18507298307296q^{68} + 19348372347648q^{69} + 12679529923200q^{70} - 5286233111232q^{72} - 29882649313236q^{73} - 1536652566808q^{74} + 21457134000000q^{76} + 37339673521920q^{77} - 29727224255040q^{78} + 57843429624320q^{80} - 98651978873946q^{81} - 71900577251064q^{82} + 102961968764928q^{84} + 151199882653560q^{85} - 100164269733216q^{86} + 51557990330880q^{88} - 180966014731924q^{89} - 84934442792280q^{90} + 82241452266240q^{92} + 295002300748800q^{93} - 40381697410176q^{94} - 82703070928896q^{96} - 345459072299124q^{97} + 121406102961892q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - x^{5} + 88 x^{4} - 1824 x^{3} + 325632 x^{2} + 21572352 x + 982333440\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 4 \nu - 1 \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{5} - 253 \nu^{4} + 14692 \nu^{3} - 116112 \nu^{2} + 1274304 \nu - 52022784 \)\()/196608\)
\(\beta_{3}\)\(=\)\( 16 \nu^{2} - 64 \nu + 477 \)
\(\beta_{4}\)\(=\)\((\)\( -3 \nu^{5} + 119 \nu^{4} + 596 \nu^{3} - 12112 \nu^{2} - 798016 \nu - 30471680 \)\()/2048\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{5} + 3 \nu^{4} + 100 \nu^{3} - 1424 \nu^{2} + 291264 \nu + 18836992 \)\()/1024\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1} + 1\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} + 16 \beta_{1} - 461\)\()/16\)
\(\nu^{3}\)\(=\)\((\)\(5 \beta_{5} + 4 \beta_{4} + 9 \beta_{3} + 192 \beta_{2} - 133 \beta_{1} + 13915\)\()/16\)
\(\nu^{4}\)\(=\)\((\)\(349 \beta_{5} + 228 \beta_{4} + 65 \beta_{3} - 1344 \beta_{2} + 611 \beta_{1} - 3413693\)\()/16\)
\(\nu^{5}\)\(=\)\((\)\(14837 \beta_{5} - 1084 \beta_{4} + 329 \beta_{3} - 15168 \beta_{2} - 1130805 \beta_{1} - 294363813\)\()/16\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
−24.5778 14.3983i
−24.5778 + 14.3983i
−4.07037 30.9656i
−4.07037 + 30.9656i
29.1482 19.7881i
29.1482 + 19.7881i
−114.311 57.5931i 2948.24i 9750.07 + 13167.1i −81680.5 −169798. + 337016.i 1.09921e6i −356209. 2.06668e6i −3.90914e6 9.33699e6 + 4.70423e6i
3.2 −114.311 + 57.5931i 2948.24i 9750.07 13167.1i −81680.5 −169798. 337016.i 1.09921e6i −356209. + 2.06668e6i −3.90914e6 9.33699e6 4.70423e6i
3.3 −32.2815 123.862i 2369.41i −14299.8 + 7996.93i 113343. 293481. 76488.1i 697667.i 1.45214e6 + 1.51306e6i −831139. −3.65890e6 1.40390e7i
3.4 −32.2815 + 123.862i 2369.41i −14299.8 7996.93i 113343. 293481. + 76488.1i 697667.i 1.45214e6 1.51306e6i −831139. −3.65890e6 + 1.40390e7i
3.5 100.593 79.1526i 1241.79i 3853.74 15924.3i −27633.0 −98290.9 124915.i 784634.i −872793. 1.90690e6i 3.24093e6 −2.77967e6 + 2.18722e6i
3.6 100.593 + 79.1526i 1241.79i 3853.74 + 15924.3i −27633.0 −98290.9 + 124915.i 784634.i −872793. + 1.90690e6i 3.24093e6 −2.77967e6 2.18722e6i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4.15.b.a 6
3.b odd 2 1 36.15.d.c 6
4.b odd 2 1 inner 4.15.b.a 6
8.b even 2 1 64.15.c.d 6
8.d odd 2 1 64.15.c.d 6
12.b even 2 1 36.15.d.c 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4.15.b.a 6 1.a even 1 1 trivial
4.15.b.a 6 4.b odd 2 1 inner
36.15.d.c 6 3.b odd 2 1
36.15.d.c 6 12.b even 2 1
64.15.c.d 6 8.b even 2 1
64.15.c.d 6 8.d odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{15}^{\mathrm{new}}(4, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 4398046511104 + 24696061952 T + 80740352 T^{2} + 45056 T^{3} + 4928 T^{4} + 92 T^{5} + T^{6} \)
$3$ \( 75249278198568714240 + 70859216793600 T^{2} + 15848256 T^{4} + T^{6} \)
$5$ \( ( -255824948369000 - 10132896340 T - 4030 T^{2} + T^{3} )^{2} \)
$7$ \( \)\(36\!\cdots\!00\)\( + \)\(16\!\cdots\!00\)\( T^{2} + 2310658565376 T^{4} + T^{6} \)
$11$ \( \)\(11\!\cdots\!00\)\( + \)\(52\!\cdots\!40\)\( T^{2} + 1637312586888000 T^{4} + T^{6} \)
$13$ \( ( -\)\(42\!\cdots\!00\)\( - 8145530499153684 T + 40387698 T^{2} + T^{3} )^{2} \)
$17$ \( ( \)\(10\!\cdots\!00\)\( - 169797158875436404 T + 60065882 T^{2} + T^{3} )^{2} \)
$19$ \( \)\(11\!\cdots\!00\)\( + \)\(48\!\cdots\!40\)\( T^{2} + 569637574525834560 T^{4} + T^{6} \)
$23$ \( \)\(33\!\cdots\!40\)\( + \)\(68\!\cdots\!20\)\( T^{2} + 35572745694403312896 T^{4} + T^{6} \)
$29$ \( ( -\)\(36\!\cdots\!08\)\( - 96167830941943965844 T + 4017898322 T^{2} + T^{3} )^{2} \)
$31$ \( \)\(13\!\cdots\!00\)\( + \)\(18\!\cdots\!40\)\( T^{2} + \)\(27\!\cdots\!60\)\( T^{4} + T^{6} \)
$37$ \( ( -\)\(16\!\cdots\!00\)\( - \)\(64\!\cdots\!24\)\( T - 15667059198 T^{2} + T^{3} )^{2} \)
$41$ \( ( -\)\(22\!\cdots\!88\)\( - \)\(27\!\cdots\!68\)\( T - 41681375446 T^{2} + T^{3} )^{2} \)
$43$ \( \)\(65\!\cdots\!00\)\( + \)\(68\!\cdots\!00\)\( T^{2} + \)\(16\!\cdots\!16\)\( T^{4} + T^{6} \)
$47$ \( \)\(72\!\cdots\!40\)\( + \)\(12\!\cdots\!40\)\( T^{2} + \)\(65\!\cdots\!96\)\( T^{4} + T^{6} \)
$53$ \( ( -\)\(40\!\cdots\!00\)\( - \)\(10\!\cdots\!24\)\( T - 38356137502 T^{2} + T^{3} )^{2} \)
$59$ \( \)\(85\!\cdots\!00\)\( + \)\(12\!\cdots\!40\)\( T^{2} + \)\(22\!\cdots\!80\)\( T^{4} + T^{6} \)
$61$ \( ( \)\(24\!\cdots\!32\)\( - \)\(97\!\cdots\!88\)\( T - 3105000393966 T^{2} + T^{3} )^{2} \)
$67$ \( \)\(62\!\cdots\!40\)\( + \)\(48\!\cdots\!00\)\( T^{2} + \)\(12\!\cdots\!76\)\( T^{4} + T^{6} \)
$71$ \( \)\(50\!\cdots\!00\)\( + \)\(68\!\cdots\!40\)\( T^{2} + \)\(16\!\cdots\!40\)\( T^{4} + T^{6} \)
$73$ \( ( -\)\(34\!\cdots\!00\)\( + \)\(54\!\cdots\!96\)\( T + 14941324656618 T^{2} + T^{3} )^{2} \)
$79$ \( \)\(30\!\cdots\!00\)\( + \)\(66\!\cdots\!40\)\( T^{2} + \)\(46\!\cdots\!60\)\( T^{4} + T^{6} \)
$83$ \( \)\(10\!\cdots\!40\)\( + \)\(55\!\cdots\!20\)\( T^{2} + \)\(66\!\cdots\!76\)\( T^{4} + T^{6} \)
$89$ \( ( -\)\(12\!\cdots\!08\)\( - \)\(20\!\cdots\!84\)\( T + 90483007365962 T^{2} + T^{3} )^{2} \)
$97$ \( ( -\)\(12\!\cdots\!00\)\( - \)\(40\!\cdots\!24\)\( T + 172729536149562 T^{2} + T^{3} )^{2} \)
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