# 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

# Related objects

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 $$S_{17}^{\mathrm{new}}(4, [\chi])$$.