Properties

Label 4.17.b.b
Level 4
Weight 17
Character orbit 4.b
Analytic conductor 6.493
Analytic rank 0
Dimension 6
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 4 = 2^{2} \)
Weight: \( k \) = \( 17 \)
Character orbit: \([\chi]\) = 4.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(6.49298175427\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{30}\cdot 3^{4} \)
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.

\(f(q)\) \(=\) \( q\) \( + ( -27 - \beta_{1} ) q^{2} \) \( + ( 1 - 3 \beta_{1} + \beta_{2} ) q^{3} \) \( + ( -26741 + 39 \beta_{1} - \beta_{2} - \beta_{4} ) q^{4} \) \( + ( -84595 + 405 \beta_{1} - 5 \beta_{3} - 5 \beta_{5} ) q^{5} \) \( + ( -197869 + 76 \beta_{1} + 44 \beta_{2} + 28 \beta_{3} - 7 \beta_{4} + 9 \beta_{5} ) q^{6} \) \( + ( -930 + 2774 \beta_{1} + 110 \beta_{2} + 32 \beta_{3} + 48 \beta_{4} - 48 \beta_{5} ) q^{7} \) \( + ( -5648072 + 29444 \beta_{1} - 924 \beta_{2} + 16 \beta_{3} + 16 \beta_{4} - 196 \beta_{5} ) q^{8} \) \( + ( -22951713 + 68066 \beta_{1} + 446 \beta_{3} - 704 \beta_{4} - 258 \beta_{5} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + ( -27 - \beta_{1} ) q^{2} \) \( + ( 1 - 3 \beta_{1} + \beta_{2} ) q^{3} \) \( + ( -26741 + 39 \beta_{1} - \beta_{2} - \beta_{4} ) q^{4} \) \( + ( -84595 + 405 \beta_{1} - 5 \beta_{3} - 5 \beta_{5} ) q^{5} \) \( + ( -197869 + 76 \beta_{1} + 44 \beta_{2} + 28 \beta_{3} - 7 \beta_{4} + 9 \beta_{5} ) q^{6} \) \( + ( -930 + 2774 \beta_{1} + 110 \beta_{2} + 32 \beta_{3} + 48 \beta_{4} - 48 \beta_{5} ) q^{7} \) \( + ( -5648072 + 29444 \beta_{1} - 924 \beta_{2} + 16 \beta_{3} + 16 \beta_{4} - 196 \beta_{5} ) q^{8} \) \( + ( -22951713 + 68066 \beta_{1} + 446 \beta_{3} - 704 \beta_{4} - 258 \beta_{5} ) q^{9} \) \( + ( -23743870 + 96350 \beta_{1} + 12320 \beta_{2} - 160 \beta_{3} + 360 \beta_{4} - 600 \beta_{5} ) q^{10} \) \( + ( 118279 - 355957 \beta_{1} + 615 \beta_{2} + 192 \beta_{3} - 736 \beta_{4} - 1312 \beta_{5} ) q^{11} \) \( + ( 42300928 + 176176 \beta_{1} - 51536 \beta_{2} + 1600 \beta_{3} + 2528 \beta_{4} + 6000 \beta_{5} ) q^{12} \) \( + ( 423970625 + 314977 \beta_{1} - 6929 \beta_{3} + 1664 \beta_{4} - 5265 \beta_{5} ) q^{13} \) \( + ( 178940842 - 86360 \beta_{1} + 57576 \beta_{2} - 1528 \beta_{3} + 14718 \beta_{4} + 28766 \beta_{5} ) q^{14} \) \( + ( 3985810 - 11983590 \beta_{1} - 27710 \beta_{2} - 9120 \beta_{3} - 44400 \beta_{4} - 17040 \beta_{5} ) q^{15} \) \( + ( 22274272 + 5687600 \beta_{1} + 179632 \beta_{2} - 35392 \beta_{3} + 62336 \beta_{4} + 36240 \beta_{5} ) q^{16} \) \( + ( 258536360 + 13972442 \beta_{1} + 50502 \beta_{3} - 122048 \beta_{4} - 71546 \beta_{5} ) q^{17} \) \( + ( -3785335467 + 24691711 \beta_{1} - 918720 \beta_{2} + 14272 \beta_{3} + 148112 \beta_{4} + 53520 \beta_{5} ) q^{18} \) \( + ( 17440989 - 52522935 \beta_{1} + 155133 \beta_{2} + 74304 \beta_{3} - 51360 \beta_{4} - 274272 \beta_{5} ) q^{19} \) \( + ( 729215830 + 23579150 \beta_{1} + 1836990 \beta_{2} + 304640 \beta_{3} + 114750 \beta_{4} + 241280 \beta_{5} ) q^{20} \) \( + ( -4556507676 + 54652444 \beta_{1} - 234332 \beta_{3} - 241024 \beta_{4} - 475356 \beta_{5} ) q^{21} \) \( + ( -23107312731 + 9702036 \beta_{1} + 81332 \beta_{2} - 10428 \beta_{3} - 173521 \beta_{4} + 172191 \beta_{5} ) q^{22} \) \( + ( -4226710 + 12882130 \beta_{1} + 546906 \beta_{2} - 211488 \beta_{3} - 220976 \beta_{4} + 413488 \beta_{5} ) q^{23} \) \( + ( 67319464064 - 71074752 \beta_{1} - 8248768 \beta_{2} - 1309440 \beta_{3} - 129024 \beta_{4} - 621888 \beta_{5} ) q^{24} \) \( + ( 45356417775 - 353245100 \beta_{1} + 952300 \beta_{3} + 1865600 \beta_{4} + 2817900 \beta_{5} ) q^{25} \) \( + ( -31617190878 - 414271650 \beta_{1} + 16647072 \beta_{2} - 221728 \beta_{3} + 72904 \beta_{4} - 831480 \beta_{5} ) q^{26} \) \( + ( -313074270 + 941927994 \beta_{1} - 10656126 \beta_{2} - 267840 \beta_{3} + 2169504 \beta_{4} + 2973024 \beta_{5} ) q^{27} \) \( + ( 269712172032 - 294232928 \beta_{1} - 11278432 \beta_{2} + 2248576 \beta_{3} - 4022720 \beta_{4} - 2577888 \beta_{5} ) q^{28} \) \( + ( -193037842387 - 97353099 \beta_{1} - 3891941 \beta_{3} + 2787840 \beta_{4} - 1104101 \beta_{5} ) q^{29} \) \( + ( -777414239770 + 331597720 \beta_{1} - 24113320 \beta_{2} + 537400 \beta_{3} - 12084430 \beta_{4} - 8165550 \beta_{5} ) q^{30} \) \( + ( -530953896 + 1595365240 \beta_{1} + 61695448 \beta_{2} + 3115264 \beta_{3} + 8734080 \beta_{4} - 611712 \beta_{5} ) q^{31} \) \( + ( 777335791232 - 346646976 \beta_{1} + 82218048 \beta_{2} + 4306176 \beta_{3} - 3336704 \beta_{4} - 4368704 \beta_{5} ) q^{32} \) \( + ( -128051672430 + 191879150 \beta_{1} + 11681138 \beta_{3} - 7689536 \beta_{4} + 3991602 \beta_{5} ) q^{33} \) \( + ( -910257177286 + 106165566 \beta_{1} - 93192640 \beta_{2} + 1616064 \beta_{3} + 27608144 \beta_{4} + 6060240 \beta_{5} ) q^{34} \) \( + ( 1454680220 - 4373827220 \beta_{1} - 215712420 \beta_{2} - 5228160 \beta_{3} - 20242880 \beta_{4} - 4558400 \beta_{5} ) q^{35} \) \( + ( 2349833256267 + 2829104615 \beta_{1} + 42982335 \beta_{2} - 30057472 \beta_{3} + 27195583 \beta_{4} + 13801728 \beta_{5} ) q^{36} \) \( + ( 1429769192945 + 1389745457 \beta_{1} - 11217121 \beta_{3} - 3251072 \beta_{4} - 14468193 \beta_{5} ) q^{37} \) \( + ( -3410843716665 + 1417405212 \beta_{1} + 87597948 \beta_{2} - 6356052 \beta_{3} - 6803691 \beta_{4} + 65892069 \beta_{5} ) q^{38} \) \( + ( 5994039506 - 18019249638 \beta_{1} + 563492930 \beta_{2} - 10082592 \beta_{3} - 57296304 \beta_{4} - 27048528 \beta_{5} ) q^{39} \) \( + ( 3162608294000 - 2157760760 \beta_{1} - 446047800 \beta_{2} + 53347360 \beta_{3} + 30624800 \beta_{4} + 86396280 \beta_{5} ) q^{40} \) \( + ( 303737018702 + 8825429300 \beta_{1} - 60927732 \beta_{3} - 26285696 \beta_{4} - 87213428 \beta_{5} ) q^{41} \) \( + ( -3399795430752 + 6062118272 \beta_{1} + 639096192 \beta_{2} - 7498624 \beta_{3} + 78574048 \beta_{4} - 28119840 \beta_{5} ) q^{42} \) \( + ( -1787936097 + 5335200995 \beta_{1} - 1310087713 \beta_{2} + 42358400 \beta_{3} + 56109504 \beta_{4} - 70965696 \beta_{5} ) q^{43} \) \( + ( 599617772032 + 23125111632 \beta_{1} - 307572528 \beta_{2} + 17613760 \beta_{3} - 20055776 \beta_{4} - 67117552 \beta_{5} ) q^{44} \) \( + ( -5693643652215 - 1883440615 \beta_{1} + 246390455 \beta_{3} - 122122880 \beta_{4} + 124267575 \beta_{5} ) q^{45} \) \( + ( 839162465070 - 293450568 \beta_{1} - 299826824 \beta_{2} + 45767640 \beta_{3} - 81546998 \beta_{4} - 178649430 \beta_{5} ) q^{46} \) \( + ( -14462248124 + 43541688980 \beta_{1} + 2811268836 \beta_{2} - 15857856 \beta_{3} + 123228896 \beta_{4} + 170802464 \beta_{5} ) q^{47} \) \( + ( -9353764971008 - 63775141120 \beta_{1} + 1839389440 \beta_{2} - 236938240 \beta_{3} - 146167808 \beta_{4} - 384071424 \beta_{5} ) q^{48} \) \( + ( -909721461383 - 34594094584 \beta_{1} - 254245768 \beta_{3} + 372891904 \beta_{4} + 118646136 \beta_{5} ) q^{49} \) \( + ( 21558474548175 - 54984916675 \beta_{1} - 2824060800 \beta_{2} + 30473600 \beta_{3} - 546159200 \beta_{4} + 114276000 \beta_{5} ) q^{50} \) \( + ( -35668728926 + 107334854586 \beta_{1} - 4767481598 \beta_{2} - 95350080 \beta_{3} + 137967648 \beta_{4} + 424017888 \beta_{5} ) q^{51} \) \( + ( -17242011502378 + 39313238926 \beta_{1} + 1587684670 \beta_{2} + 428985856 \beta_{3} - 398570562 \beta_{4} + 250872960 \beta_{5} ) q^{52} \) \( + ( 21809258860025 - 93719455703 \beta_{1} - 489308761 \beta_{3} + 901036928 \beta_{4} + 411728167 \beta_{5} ) q^{53} \) \( + ( 61169347360422 - 25688558952 \beta_{1} - 252026280 \beta_{2} - 259802568 \beta_{3} + 603201906 \beta_{4} - 328390254 \beta_{5} ) q^{54} \) \( + ( -1055490 - 10927370 \beta_{1} + 5166200590 \beta_{2} + 73899040 \beta_{3} + 133704240 \beta_{4} - 87992880 \beta_{5} ) q^{55} \) \( + ( -57019919239424 - 249008012416 \beta_{1} - 4925188224 \beta_{2} - 279206400 \beta_{3} + 314077184 \beta_{4} + 318623360 \beta_{5} ) q^{56} \) \( + ( -20446952261850 + 98934205530 \beta_{1} + 1495197894 \beta_{3} - 1486792128 \beta_{4} + 8405766 \beta_{5} ) q^{57} \) \( + ( 11590009409282 + 191145161438 \beta_{1} + 8876055584 \beta_{2} - 124542112 \beta_{3} - 433467288 \beta_{4} - 467032920 \beta_{5} ) q^{58} \) \( + ( 106964991803 - 321783877937 \beta_{1} - 2049477957 \beta_{2} - 15669504 \beta_{3} - 920241536 \beta_{4} - 873233024 \beta_{5} ) q^{59} \) \( + ( -107216545203200 + 831450778720 \beta_{1} - 4239869600 \beta_{2} - 509191040 \beta_{3} + 1281506240 \beta_{4} - 784868640 \beta_{5} ) q^{60} \) \( + ( -71726648131903 + 433066704865 \beta_{1} - 652168337 \beta_{3} - 2569196416 \beta_{4} - 3221364753 \beta_{5} ) q^{61} \) \( + ( 103268548215560 - 45079188448 \beta_{1} + 8888217888 \beta_{2} + 1278874528 \beta_{3} + 2115582744 \beta_{4} + 3259308184 \beta_{5} ) q^{62} \) \( + ( 156415657374 - 471034484010 \beta_{1} - 5946241554 \beta_{2} + 579860640 \beta_{3} - 627790608 \beta_{4} - 2367372528 \beta_{5} ) q^{63} \) \( + ( -85820355318272 - 751492945152 \beta_{1} - 917359872 \beta_{2} + 2205023232 \beta_{3} + 444254208 \beta_{4} + 2254942464 \beta_{5} ) q^{64} \) \( + ( 206350100546200 - 220605924500 \beta_{1} - 1809045420 \beta_{3} + 2480666240 \beta_{4} + 671620820 \beta_{5} ) q^{65} \) \( + ( -9208979654064 + 131047569088 \beta_{1} - 26813802816 \beta_{2} + 373796416 \beta_{3} + 1127479280 \beta_{4} + 1401736560 \beta_{5} ) q^{66} \) \( + ( 55541042769 - 166593687827 \beta_{1} + 27991909489 \beta_{2} - 334531520 \beta_{3} - 639622560 \beta_{4} + 363972000 \beta_{5} ) q^{67} \) \( + ( 285837753163798 + 799893208654 \beta_{1} + 12613254654 \beta_{2} - 3576894464 \beta_{3} + 638936830 \beta_{4} + 3686855936 \beta_{5} ) q^{68} \) \( + ( -44215021593204 - 261989578124 \beta_{1} + 884781004 \beta_{3} + 1285961600 \beta_{4} + 2170742604 \beta_{5} ) q^{69} \) \( + ( -283047843490860 + 122889199440 \beta_{1} - 21281918000 \beta_{2} - 5287092720 \beta_{3} - 4195006340 \beta_{4} - 6479454660 \beta_{5} ) q^{70} \) \( + ( -277283981490 + 836230063302 \beta_{1} - 71805071202 \beta_{2} - 3486811488 \beta_{3} - 2595504144 \beta_{4} + 7864930320 \beta_{5} ) q^{71} \) \( + ( 257424484536120 - 2480235949052 \beta_{1} + 56475563364 \beta_{2} + 974865424 \beta_{3} - 2800053232 \beta_{4} - 6336152004 \beta_{5} ) q^{72} \) \( + ( -190824319116040 - 320118172918 \beta_{1} + 3635570966 \beta_{3} + 173222464 \beta_{4} + 3808793430 \beta_{5} ) q^{73} \) \( + ( -128059476679902 - 1390520245282 \beta_{1} + 28471260576 \beta_{2} - 358947872 \beta_{3} + 1639907144 \beta_{4} - 1346054520 \beta_{5} ) q^{74} \) \( + ( -809156665725 + 2431467393975 \beta_{1} + 102566129475 \beta_{2} + 4602556800 \beta_{3} + 13202510400 \beta_{4} - 605160000 \beta_{5} ) q^{75} \) \( + ( 460365907066368 + 3257974480752 \beta_{1} - 59533276176 \beta_{2} + 5727628608 \beta_{3} - 8927387040 \beta_{4} - 14757033552 \beta_{5} ) q^{76} \) \( + ( -110472616360260 - 1345259884220 \beta_{1} - 274464260 \beta_{3} + 9239807360 \beta_{4} + 8965343100 \beta_{5} ) q^{77} \) \( + ( -1170971837862554 + 493719336344 \beta_{1} - 2618561192 \beta_{2} + 17229695288 \beta_{3} - 19620563534 \beta_{4} - 3680253486 \beta_{5} ) q^{78} \) \( + ( -1057454359524 + 3175576005260 \beta_{1} - 80628767428 \beta_{2} + 6323241536 \beta_{3} + 15859409760 \beta_{4} - 3110314848 \beta_{5} ) q^{79} \) \( + ( 735056285284800 - 3514091782560 \beta_{1} - 156038543520 \beta_{2} - 10675561600 \beta_{3} - 3528177920 \beta_{4} + 1933942560 \beta_{5} ) q^{80} \) \( + ( -46988248537125 + 2143819707174 \beta_{1} - 16093982790 \beta_{3} - 5677075008 \beta_{4} - 21771057798 \beta_{5} ) q^{81} \) \( + ( -576525465852118 - 56378278914 \beta_{1} + 156855069824 \beta_{2} - 1949687424 \beta_{3} + 11115934880 \beta_{4} - 7311327840 \beta_{5} ) q^{82} \) \( + ( 1622534505325 - 4870668840007 \beta_{1} + 63602050413 \beta_{2} - 12774497280 \beta_{3} - 28614318592 \beta_{4} + 9709173248 \beta_{5} ) q^{83} \) \( + ( 597559236105216 + 3177859611136 \beta_{1} + 147058813440 \beta_{2} + 13290145792 \beta_{3} + 8343609344 \beta_{4} + 23401545216 \beta_{5} ) q^{84} \) \( + ( -322227091021850 - 321235807650 \beta_{1} + 24366608130 \beta_{3} - 11165266560 \beta_{4} + 13201341570 \beta_{5} ) q^{85} \) \( + ( 349427591586445 - 150785419724 \beta_{1} + 10261191252 \beta_{2} - 42782065564 \beta_{3} + 27770487015 \beta_{4} + 24976301783 \beta_{5} ) q^{86} \) \( + ( 2856870420754 - 8592445877094 \beta_{1} - 47078341310 \beta_{2} - 2816778144 \beta_{3} - 27468171120 \beta_{4} - 19017836688 \beta_{5} ) q^{87} \) \( + ( 120097057289088 - 669894394432 \beta_{1} + 35089385408 \beta_{2} - 10894114048 \beta_{3} + 35411146752 \beta_{4} + 11439823680 \beta_{5} ) q^{88} \) \( + ( 3404794399636808 - 989023478854 \beta_{1} + 5503737702 \beta_{3} + 3670410304 \beta_{4} + 9174148006 \beta_{5} ) q^{89} \) \( + ( 269450599067490 + 5598108420190 \beta_{1} - 575842623840 \beta_{2} + 7884494560 \beta_{3} + 13523344520 \beta_{4} + 29566854600 \beta_{5} ) q^{90} \) \( + ( 903207656988 - 2724402669716 \beta_{1} - 187796607908 \beta_{2} + 7498271872 \beta_{3} + 216844992 \beta_{4} - 22277970624 \beta_{5} ) q^{91} \) \( + ( -1633066100454400 - 171437766432 \beta_{1} + 31656432096 \beta_{2} - 13216869760 \beta_{3} + 29412300992 \beta_{4} + 29510378848 \beta_{5} ) q^{92} \) \( + ( -3397424321342640 + 9318356465840 \beta_{1} - 52055763376 \beta_{3} - 34471889408 \beta_{4} - 86527652784 \beta_{5} ) q^{93} \) \( + ( 2817213839920620 - 1207208863824 \beta_{1} + 135198096176 \beta_{2} + 80999058672 \beta_{3} + 10281929924 \beta_{4} + 11536800516 \beta_{5} ) q^{94} \) \( + ( 1618628261130 - 4869077384910 \beta_{1} + 528911990970 \beta_{2} + 5447188320 \beta_{3} - 2298224880 \beta_{4} - 18639789840 \beta_{5} ) q^{95} \) \( + ( -5469024569202688 + 11822829691904 \beta_{1} + 604074656768 \beta_{2} + 43878404096 \beta_{3} - 57348694016 \beta_{4} - 11697558528 \beta_{5} ) q^{96} \) \( + ( 697157780806640 - 236073751918 \beta_{1} + 25742619662 \beta_{3} - 12493773248 \beta_{4} + 13248846414 \beta_{5} ) q^{97} \) \( + ( 2264048021931045 + 30428430207 \beta_{1} + 531001244928 \beta_{2} - 8135864576 \beta_{3} - 77154632128 \beta_{4} - 30509492160 \beta_{5} ) q^{98} \) \( + ( -4219215747417 + 12674610703755 \beta_{1} - 514934692569 \beta_{2} + 17760238848 \beta_{3} + 52483939200 \beta_{4} - 796777344 \beta_{5} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(6q \) \(\mathstrut -\mathstrut 164q^{2} \) \(\mathstrut -\mathstrut 160368q^{4} \) \(\mathstrut -\mathstrut 506740q^{5} \) \(\mathstrut -\mathstrut 1187136q^{6} \) \(\mathstrut -\mathstrut 33829184q^{8} \) \(\mathstrut -\mathstrut 137574522q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(6q \) \(\mathstrut -\mathstrut 164q^{2} \) \(\mathstrut -\mathstrut 160368q^{4} \) \(\mathstrut -\mathstrut 506740q^{5} \) \(\mathstrut -\mathstrut 1187136q^{6} \) \(\mathstrut -\mathstrut 33829184q^{8} \) \(\mathstrut -\mathstrut 137574522q^{9} \) \(\mathstrut -\mathstrut 142269000q^{10} \) \(\mathstrut +\mathstrut 254142720q^{12} \) \(\mathstrut +\mathstrut 2544478092q^{13} \) \(\mathstrut +\mathstrut 1073417856q^{14} \) \(\mathstrut +\mathstrut 145019136q^{16} \) \(\mathstrut +\mathstrut 1579205132q^{17} \) \(\mathstrut -\mathstrut 22662764964q^{18} \) \(\mathstrut +\mathstrut 4421361440q^{20} \) \(\mathstrut -\mathstrut 27228321792q^{21} \) \(\mathstrut -\mathstrut 138624795840q^{22} \) \(\mathstrut +\mathstrut 403778497536q^{24} \) \(\mathstrut +\mathstrut 271424476050q^{25} \) \(\mathstrut -\mathstrut 190529582152q^{26} \) \(\mathstrut +\mathstrut 1617685224960q^{28} \) \(\mathstrut -\mathstrut 1158411768436q^{29} \) \(\mathstrut -\mathstrut 4663806986880q^{30} \) \(\mathstrut +\mathstrut 4663321578496q^{32} \) \(\mathstrut -\mathstrut 767957621760q^{33} \) \(\mathstrut -\mathstrut 5461346085192q^{34} \) \(\mathstrut +\mathstrut 14104690258320q^{36} \) \(\mathstrut +\mathstrut 8581446019212q^{37} \) \(\mathstrut -\mathstrut 20462346561600q^{38} \) \(\mathstrut +\mathstrut 18971054755200q^{40} \) \(\mathstrut +\mathstrut 1840369253132q^{41} \) \(\mathstrut -\mathstrut 20386577111040q^{42} \) \(\mathstrut +\mathstrut 3644055863040q^{44} \) \(\mathstrut -\mathstrut 34166370110580q^{45} \) \(\mathstrut +\mathstrut 5034653652864q^{46} \) \(\mathstrut -\mathstrut 56248898088960q^{48} \) \(\mathstrut -\mathstrut 5527245758202q^{49} \) \(\mathstrut +\mathstrut 129240587956500q^{50} \) \(\mathstrut -\mathstrut 103374802254048q^{52} \) \(\mathstrut +\mathstrut 130668269409932q^{53} \) \(\mathstrut +\mathstrut 366965883430272q^{54} \) \(\mathstrut -\mathstrut 342617610295296q^{56} \) \(\mathstrut -\mathstrut 122486852367360q^{57} \) \(\mathstrut +\mathstrut 69923529928632q^{58} \) \(\mathstrut -\mathstrut 641633781542400q^{60} \) \(\mathstrut -\mathstrut 429486008315508q^{61} \) \(\mathstrut +\mathstrut 619512054551040q^{62} \) \(\mathstrut -\mathstrut 516434037731328q^{64} \) \(\mathstrut +\mathstrut 1237661666277400q^{65} \) \(\mathstrut -\mathstrut 54995333852160q^{66} \) \(\mathstrut +\mathstrut 1716626085477152q^{68} \) \(\mathstrut -\mathstrut 265820219762688q^{69} \) \(\mathstrut -\mathstrut 1698017749451520q^{70} \) \(\mathstrut +\mathstrut 1539597157891776q^{72} \) \(\mathstrut -\mathstrut 1145601039770868q^{73} \) \(\mathstrut -\mathstrut 771134490565192q^{74} \) \(\mathstrut +\mathstrut 2768729450169600q^{76} \) \(\mathstrut -\mathstrut 665543599687680q^{77} \) \(\mathstrut -\mathstrut 7024870687386240q^{78} \) \(\mathstrut +\mathstrut 4403327011381760q^{80} \) \(\mathstrut -\mathstrut 277566121727226q^{81} \) \(\mathstrut -\mathstrut 3459247029640008q^{82} \) \(\mathstrut +\mathstrut 3591637752471552q^{84} \) \(\mathstrut -\mathstrut 1934080153645800q^{85} \) \(\mathstrut +\mathstrut 2096299590206784q^{86} \) \(\mathstrut +\mathstrut 719241463526400q^{88} \) \(\mathstrut +\mathstrut 20426758995091724q^{89} \) \(\mathstrut +\mathstrut 1627824908547000q^{90} \) \(\mathstrut -\mathstrut 9798772065277440q^{92} \) \(\mathstrut -\mathstrut 20365632048291840q^{93} \) \(\mathstrut +\mathstrut 16900683550077696q^{94} \) \(\mathstrut -\mathstrut 32790566117523456q^{96} \) \(\mathstrut +\mathstrut 4182396554403852q^{97} \) \(\mathstrut +\mathstrut 13584426279160156q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6}\mathstrut -\mathstrut \) \(x^{5}\mathstrut +\mathstrut \) \(5152\) \(x^{4}\mathstrut +\mathstrut \) \(242526\) \(x^{3}\mathstrut +\mathstrut \) \(17329473\) \(x^{2}\mathstrut +\mathstrut \) \(402444531\) \(x\mathstrut +\mathstrut \) \(64957563630\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{5} + 6 \nu^{4} + 5194 \nu^{3} + 278884 \nu^{2} + 19281661 \nu + 424169950 \)\()/4194304\)
\(\beta_{2}\)\(=\)\((\)\( -361 \nu^{5} + 1930 \nu^{4} + 275366 \nu^{3} - 20309508 \nu^{2} - 346364613 \nu + 262341750258 \)\()/37748736\)
\(\beta_{3}\)\(=\)\((\)\( -1655 \nu^{5} - 161482 \nu^{4} - 12663398 \nu^{3} - 113266044 \nu^{2} - 47656971675 \nu - 1037716791858 \)\()/37748736\)
\(\beta_{4}\)\(=\)\((\)\( -191 \nu^{5} + 34694 \nu^{4} - 1050422 \nu^{3} + 121467492 \nu^{2} + 6308187453 \nu + 313280271198 \)\()/9437184\)
\(\beta_{5}\)\(=\)\((\)\( -13 \nu^{5} - 9038 \nu^{4} - 52930 \nu^{3} - 47309076 \nu^{2} - 332488665 \nu - 104406897222 \)\()/2359296\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(108\) \(\beta_{1}\mathstrut +\mathstrut \) \(135\)\()/1024\)
\(\nu^{2}\)\(=\)\((\)\(-\)\(31\) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(32\) \(\beta_{3}\mathstrut +\mathstrut \) \(64\) \(\beta_{2}\mathstrut +\mathstrut \) \(7820\) \(\beta_{1}\mathstrut -\mathstrut \) \(1760985\)\()/1024\)
\(\nu^{3}\)\(=\)\((\)\(-\)\(1669\) \(\beta_{5}\mathstrut -\mathstrut \) \(3077\) \(\beta_{4}\mathstrut -\mathstrut \) \(896\) \(\beta_{3}\mathstrut +\mathstrut \) \(16128\) \(\beta_{2}\mathstrut +\mathstrut \) \(182372\) \(\beta_{1}\mathstrut -\mathstrut \) \(126872611\)\()/1024\)
\(\nu^{4}\)\(=\)\((\)\(-\)\(130347\) \(\beta_{5}\mathstrut -\mathstrut \) \(19019\) \(\beta_{4}\mathstrut -\mathstrut \) \(157472\) \(\beta_{3}\mathstrut -\mathstrut \) \(285760\) \(\beta_{2}\mathstrut -\mathstrut \) \(45046468\) \(\beta_{1}\mathstrut -\mathstrut \) \(2924369197\)\()/1024\)
\(\nu^{5}\)\(=\)\((\)\(-\)\(1185389\) \(\beta_{5}\mathstrut -\mathstrut \) \(3464493\) \(\beta_{4}\mathstrut -\mathstrut \) \(3325632\) \(\beta_{3}\mathstrut -\mathstrut \) \(99902848\) \(\beta_{2}\mathstrut -\mathstrut \) \(645286332\) \(\beta_{1}\mathstrut +\mathstrut \) \(730680044421\)\()/1024\)

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
−46.2446 + 35.5107i
−46.2446 35.5107i
11.8147 + 63.8186i
11.8147 63.8186i
34.9299 + 57.5840i
34.9299 57.5840i
−212.978 142.043i 2908.07i 25183.6 + 60504.2i 20884.7 413070. 619356.i 8.01505e6i 3.23062e6 1.64632e7i 3.45899e7 −4.44799e6 2.96652e6i
3.2 −212.978 + 142.043i 2908.07i 25183.6 60504.2i 20884.7 413070. + 619356.i 8.01505e6i 3.23062e6 + 1.64632e7i 3.45899e7 −4.44799e6 + 2.96652e6i
3.3 19.2587 255.275i 11183.9i −64794.2 9832.53i 389860. −2.85496e6 215387.i 1.06777e6i −3.75785e6 + 1.63509e7i −8.20322e7 7.50821e6 9.95214e7i
3.4 19.2587 + 255.275i 11183.9i −64794.2 + 9832.53i 389860. −2.85496e6 + 215387.i 1.06777e6i −3.75785e6 1.63509e7i −8.20322e7 7.50821e6 + 9.95214e7i
3.5 111.720 230.336i 8024.44i −40573.4 51466.2i −664115. 1.84832e6 + 896488.i 6.08943e6i −1.63874e7 + 3.59574e6i −2.13450e7 −7.41947e7 + 1.52970e8i
3.6 111.720 + 230.336i 8024.44i −40573.4 + 51466.2i −664115. 1.84832e6 896488.i 6.08943e6i −1.63874e7 3.59574e6i −2.13450e7 −7.41947e7 1.52970e8i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.6
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
4.b Odd 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{3}^{6} \) \(\mathstrut +\mathstrut 197927424 T_{3}^{4} \) \(\mathstrut +\mathstrut \)\(96\!\cdots\!00\)\( T_{3}^{2} \) \(\mathstrut +\mathstrut \)\(68\!\cdots\!60\)\( \) acting on \(S_{17}^{\mathrm{new}}(4, [\chi])\).