# Properties

 Label 5520.2.be.b Level $5520$ Weight $2$ Character orbit 5520.be Analytic conductor $44.077$ Analytic rank $0$ Dimension $16$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5520 = 2^{4} \cdot 3 \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5520.be (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$44.0774219157$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 4 x^{15} + 8 x^{14} + 28 x^{13} + 373 x^{12} - 920 x^{11} + 1088 x^{10} - 168 x^{9} + 16460 x^{8} - 45408 x^{7} + 62624 x^{6} - 18048 x^{5} + 2160 x^{4} - 1664 x^{3} + 6272 x^{2} - 896 x + 64$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{8}$$ 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 $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{10} q^{3} + \beta_{10} q^{5} + ( 1 + \beta_{4} ) q^{7} - q^{9} +O(q^{10})$$ $$q + \beta_{10} q^{3} + \beta_{10} q^{5} + ( 1 + \beta_{4} ) q^{7} - q^{9} -\beta_{6} q^{11} + ( 1 - \beta_{3} ) q^{13} - q^{15} + ( \beta_{1} + \beta_{14} ) q^{17} + ( \beta_{3} + \beta_{4} + \beta_{9} ) q^{19} -\beta_{14} q^{21} + ( -1 + \beta_{5} - \beta_{15} ) q^{23} - q^{25} -\beta_{10} q^{27} + ( \beta_{2} - \beta_{6} ) q^{29} + ( \beta_{1} + \beta_{11} + \beta_{12} + \beta_{13} + \beta_{14} + \beta_{15} ) q^{31} + ( -\beta_{10} + \beta_{13} ) q^{33} -\beta_{14} q^{35} + ( -\beta_{10} - \beta_{12} - \beta_{15} ) q^{37} + ( \beta_{8} + \beta_{10} ) q^{39} + ( -\beta_{5} + \beta_{6} - \beta_{7} + \beta_{9} ) q^{41} + ( -\beta_{3} - \beta_{4} - \beta_{7} + \beta_{9} ) q^{43} -\beta_{10} q^{45} + ( \beta_{1} - \beta_{12} + \beta_{13} + \beta_{14} ) q^{47} + ( 1 - \beta_{2} + \beta_{3} + \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - \beta_{7} - \beta_{9} ) q^{49} + ( 1 + \beta_{4} - \beta_{5} ) q^{51} + ( -\beta_{8} + \beta_{11} + \beta_{12} + \beta_{14} - \beta_{15} ) q^{53} + ( -\beta_{10} + \beta_{13} ) q^{55} + ( -\beta_{8} - \beta_{10} - \beta_{11} - \beta_{14} ) q^{57} + ( 2 \beta_{1} - 2 \beta_{10} + \beta_{14} + \beta_{15} ) q^{59} + ( \beta_{1} + \beta_{8} - 2 \beta_{10} + \beta_{12} + \beta_{14} ) q^{61} + ( -1 - \beta_{4} ) q^{63} + ( \beta_{8} + \beta_{10} ) q^{65} + ( 1 - \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + \beta_{7} - \beta_{9} ) q^{67} + ( \beta_{1} + \beta_{7} - \beta_{10} ) q^{69} + ( -\beta_{8} + \beta_{10} + \beta_{11} - 2 \beta_{13} - 2 \beta_{15} ) q^{71} + ( 3 + \beta_{3} - 2 \beta_{6} + 2 \beta_{7} ) q^{73} -\beta_{10} q^{75} + ( 1 - 3 \beta_{4} - 2 \beta_{6} - \beta_{7} - \beta_{9} ) q^{77} + ( -2 - \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} + 3 \beta_{6} - \beta_{7} ) q^{79} + q^{81} + ( \beta_{2} - \beta_{5} - 2 \beta_{7} ) q^{83} + ( 1 + \beta_{4} - \beta_{5} ) q^{85} + ( -\beta_{8} - \beta_{10} + \beta_{12} + \beta_{13} ) q^{87} + ( -2 \beta_{1} + 2 \beta_{10} - 2 \beta_{13} - 2 \beta_{14} - 2 \beta_{15} ) q^{89} + ( 2 - \beta_{2} + 2 \beta_{3} - \beta_{5} + \beta_{6} - \beta_{7} - \beta_{9} ) q^{91} + ( -\beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} + \beta_{9} ) q^{93} + ( -\beta_{8} - \beta_{10} - \beta_{11} - \beta_{14} ) q^{95} + ( \beta_{8} + \beta_{11} - \beta_{13} - \beta_{14} - 2 \beta_{15} ) q^{97} + \beta_{6} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + 8q^{7} - 16q^{9} + O(q^{10})$$ $$16q + 8q^{7} - 16q^{9} - 8q^{11} + 8q^{13} - 16q^{15} - 12q^{23} - 16q^{25} - 4q^{29} + 4q^{41} + 20q^{49} + 4q^{51} - 8q^{63} + 16q^{67} + 40q^{73} + 24q^{77} - 32q^{79} + 16q^{81} + 4q^{85} + 48q^{91} + 8q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 4 x^{15} + 8 x^{14} + 28 x^{13} + 373 x^{12} - 920 x^{11} + 1088 x^{10} - 168 x^{9} + 16460 x^{8} - 45408 x^{7} + 62624 x^{6} - 18048 x^{5} + 2160 x^{4} - 1664 x^{3} + 6272 x^{2} - 896 x + 64$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-22393307326694017831 \nu^{15} + 79205853823130329892 \nu^{14} - 175071232472117903174 \nu^{13} - 526628042123880972776 \nu^{12} - 9091844016900447538055 \nu^{11} + 15856188225716940095240 \nu^{10} - 27782783728011205760238 \nu^{9} + 38959469703544326596092 \nu^{8} - 449503757829354878585096 \nu^{7} + 825981882874189932557056 \nu^{6} - 1621157387025078645325152 \nu^{5} + 2003438432384731993177024 \nu^{4} - 3755942242288309796882992 \nu^{3} + 1286677423391995969163200 \nu^{2} - 1332221403888018014991392 \nu + 94236309393562936299712$$$$)/$$$$34\!\cdots\!80$$ $$\beta_{2}$$ $$=$$ $$($$$$25569479030280566412 \nu^{15} - 135876466122824388149 \nu^{14} + 345190768387595378903 \nu^{13} + 430477237088200538532 \nu^{12} + 8635195911545456863880 \nu^{11} - 35865907280266927386845 \nu^{10} + 61329501638043960405891 \nu^{9} - 42773987978022886175024 \nu^{8} + 435067467376470129669992 \nu^{7} - 1712413047279119646456252 \nu^{6} + 3217485504535708805590224 \nu^{5} - 2744439062106117518346848 \nu^{4} + 1007413334526259350866064 \nu^{3} - 99682082229361135065120 \nu^{2} + 4173357676598058424304 \nu - 311438671188616313007904$$$$)/$$$$86\!\cdots\!20$$ $$\beta_{3}$$ $$=$$ $$($$$$-57299681650542813799 \nu^{15} + 310762723733366637388 \nu^{14} - 797609807237825424606 \nu^{13} - 889272987826610412594 \nu^{12} - 19198824197009526898095 \nu^{11} + 82758888276838958876400 \nu^{10} - 141781898740870143152142 \nu^{9} + 115423269614068896266898 \nu^{8} - 965514734333909107594024 \nu^{7} + 3940945426884564607889904 \nu^{6} - 7511743716854025606700288 \nu^{5} + 6833395910542852553834256 \nu^{4} - 2355692976644302317089648 \nu^{3} + 226667030484296414298880 \nu^{2} - 8823571164277865133088 \nu - 72370372478847279136032$$$$)/$$$$17\!\cdots\!40$$ $$\beta_{4}$$ $$=$$ $$($$$$-439455554879435383139 \nu^{15} + 2241122015124087814768 \nu^{14} - 5561764669179045278226 \nu^{13} - 7995018565696568059124 \nu^{12} - 151253320213586326551555 \nu^{11} + 581239161047249858179860 \nu^{10} - 964687428832831710042042 \nu^{9} + 700416946484716590460688 \nu^{8} - 7432691074169667193750664 \nu^{7} + 27883629363643201451158784 \nu^{6} - 51191725696677861835307008 \nu^{5} + 43448370476958442988560896 \nu^{4} - 15979835659401989048171248 \nu^{3} + 1654837676509585409533440 \nu^{2} - 76921847512402003069408 \nu + 3743822249151046185922688$$$$)/$$$$10\!\cdots\!40$$ $$\beta_{5}$$ $$=$$ $$($$$$-37104260317816956443 \nu^{15} + 151526112378019231664 \nu^{14} - 321061485849858930146 \nu^{13} - 974908431881953254128 \nu^{12} - 13835641266715917580139 \nu^{11} + 34947175687175212714580 \nu^{10} - 47875325984313322571514 \nu^{9} + 16736683687098622304212 \nu^{8} - 625533344564557583278984 \nu^{7} + 1734559757843010960652096 \nu^{6} - 2643931979850558069220992 \nu^{5} + 1293457952188598334348928 \nu^{4} - 803716592544779228231536 \nu^{3} + 117111471819228528626944 \nu^{2} - 8800959934029242894816 \nu + 17380390028458691226688$$$$)/$$$$69\!\cdots\!96$$ $$\beta_{6}$$ $$=$$ $$($$$$139701703083084843466 \nu^{15} - 352863803987486548657 \nu^{14} + 350397790926717630654 \nu^{13} + 5332779684253886100096 \nu^{12} + 58315309974111227789410 \nu^{11} - 50079493872898502763745 \nu^{10} - 16471505765686500866422 \nu^{9} + 148038962812064696378968 \nu^{8} + 2321667818194567952846876 \nu^{7} - 2953484078021429244894616 \nu^{6} + 313489945615288993111112 \nu^{5} + 7777687897607258688480656 \nu^{4} - 107340536214601554404528 \nu^{3} - 309337175284090395834640 \nu^{2} + 46133565624937104022592 \nu + 844306681429309503644608$$$$)/$$$$17\!\cdots\!40$$ $$\beta_{7}$$ $$=$$ $$($$$$34009524100498954924 \nu^{15} - 93876249017534482861 \nu^{14} + 116684157278614951436 \nu^{13} + 1236916786086591809948 \nu^{12} + 13968426198284995440156 \nu^{11} - 15179392427921230632309 \nu^{10} + 3150971691525804233672 \nu^{9} + 28027087138017880071620 \nu^{8} + 566002065946927646456116 \nu^{7} - 850100044846283359262720 \nu^{6} + 430569310029225112211832 \nu^{5} + 1425441230980083855222048 \nu^{4} + 89009254303847462446576 \nu^{3} - 80106538518255115634544 \nu^{2} + 10748186561286905391968 \nu + 163776380549154061343424$$$$)/$$$$34\!\cdots\!48$$ $$\beta_{8}$$ $$=$$ $$($$$$-223356151832978209908 \nu^{15} + 880505055132253233071 \nu^{14} - 1717983330107770032102 \nu^{13} - 6445092595785024774798 \nu^{12} - 83457016995930477068460 \nu^{11} + 200933097563041183630095 \nu^{10} - 225167864663606741739234 \nu^{9} + 623219353097996059986 \nu^{8} - 3638831396467228504089708 \nu^{7} + 9878598436726013996313568 \nu^{6} - 13129113754973767703626776 \nu^{5} + 2128792998735165940548672 \nu^{4} + 1801469580584110661948304 \nu^{3} - 1858437542654510627821200 \nu^{2} - 738358944973925573408256 \nu + 55233526678960890115296$$$$)/$$$$17\!\cdots\!40$$ $$\beta_{9}$$ $$=$$ $$($$$$1467508247969570962901 \nu^{15} - 5233002171242895530902 \nu^{14} + 9692947659699732558114 \nu^{13} + 44025670310952507825176 \nu^{12} + 569272045850750951194965 \nu^{11} - 1098043412608702503400770 \nu^{10} + 1190786430340385736616338 \nu^{9} - 91948359191815807646372 \nu^{8} + 24499444218239838062427296 \nu^{7} - 56117189935335908355064496 \nu^{6} + 71186232994978237024199152 \nu^{5} - 11673278022187112926811424 \nu^{4} + 20939510598395836955100592 \nu^{3} - 4166312496714838271935200 \nu^{2} + 391637599271987046216352 \nu - 1932955519447723140567872$$$$)/$$$$10\!\cdots\!40$$ $$\beta_{10}$$ $$=$$ $$($$$$-1856365266029439028 \nu^{15} + 7283657711690606901 \nu^{14} - 14306448104402867992 \nu^{13} - 53067119951003531693 \nu^{12} - 696404637902714743190 \nu^{11} + 1653992395133468254305 \nu^{10} - 1899083034361956529764 \nu^{9} + 161278877668106868691 \nu^{8} - 30518839909115709877418 \nu^{7} + 81918401702948551124388 \nu^{6} - 110192195306696806738916 \nu^{5} + 24908556001061761521512 \nu^{4} - 970114662909385438936 \nu^{3} + 507745037561589969520 \nu^{2} - 11216975671400262226896 \nu + 817166959957868757296$$$$)/$$$$81\!\cdots\!60$$ $$\beta_{11}$$ $$=$$ $$($$$$-3161330175133885254831 \nu^{15} + 12185535405329615195492 \nu^{14} - 24185853183290763727854 \nu^{13} - 89828524897078368355296 \nu^{12} - 1195903546080911591781535 \nu^{11} + 2713239457625169804239320 \nu^{10} - 3308875375372790427186838 \nu^{9} + 486411393437322269222212 \nu^{8} - 52314610125801861039516216 \nu^{7} + 135911502779325343997747216 \nu^{6} - 189225266087830922589685312 \nu^{5} + 52211043531645188945835584 \nu^{4} - 23214628329792689250941552 \nu^{3} - 1211545795163318363457600 \nu^{2} - 25987382272721391873487392 \nu + 1876980550567798417556032$$$$)/$$$$10\!\cdots\!40$$ $$\beta_{12}$$ $$=$$ $$($$$$229762909361918816175 \nu^{15} - 901020790322433139847 \nu^{14} + 1773054069533715792634 \nu^{13} + 6545072874457425453552 \nu^{12} + 86281763985010554218963 \nu^{11} - 204495788443437004457355 \nu^{10} + 235921791902117618025326 \nu^{9} - 27015465087264750298364 \nu^{8} + 3791267880997429381455816 \nu^{7} - 10142129746831373977569608 \nu^{6} + 13689244198922517790399968 \nu^{5} - 3408341800529148310923648 \nu^{4} + 838043210186307474026992 \nu^{3} - 749426359272182604170864 \nu^{2} + 1617327804848636083628576 \nu - 117283563291563124756928$$$$)/$$$$34\!\cdots\!48$$ $$\beta_{13}$$ $$=$$ $$($$$$1647786134581718440981 \nu^{15} - 6489058375714485668502 \nu^{14} + 12726226693636926236734 \nu^{13} + 47105813531921546323536 \nu^{12} + 617241953105613212778205 \nu^{11} - 1479479863038053949806330 \nu^{10} + 1679834085428337891531678 \nu^{9} - 137215035263440896446332 \nu^{8} + 27082925975940619456063336 \nu^{7} - 73166458764427204113160696 \nu^{6} + 97753879874209701251464832 \nu^{5} - 21834891657963530120968224 \nu^{4} + 263816343263634857645072 \nu^{3} - 3441505923093088420226080 \nu^{2} + 9764350823638301822300192 \nu - 711791183647280820656192$$$$)/$$$$17\!\cdots\!40$$ $$\beta_{14}$$ $$=$$ $$($$$$10521267573052040758179 \nu^{15} - 41256382875221750700458 \nu^{14} + 81105100589352480044706 \nu^{13} + 300354967909233596033724 \nu^{12} + 3949213686776762072093155 \nu^{11} - 9362653473555985745334190 \nu^{10} + 10782076755025973164713922 \nu^{9} - 1044812632511988236449528 \nu^{8} + 173235171883211922179021424 \nu^{7} - 464019093201352332432477584 \nu^{6} + 625505202864271357582465648 \nu^{5} - 147200343497065264843608896 \nu^{4} + 18778730745673106956448528 \nu^{3} - 14771338934975639530906080 \nu^{2} + 67813411026733915122914208 \nu - 4930276316315155162179328$$$$)/$$$$10\!\cdots\!40$$ $$\beta_{15}$$ $$=$$ $$($$$$-2014702384092608461819 \nu^{15} + 7931243684916089679668 \nu^{14} - 15554783592512782860496 \nu^{13} - 57610372955257229156194 \nu^{12} - 754745724599480418409495 \nu^{11} + 1807616767329908505829800 \nu^{10} - 2054011490558481066866392 \nu^{9} + 163825187962440470325338 \nu^{8} - 33105615204951311090397084 \nu^{7} + 89399980582648206502360744 \nu^{6} - 119495653065311981325644808 \nu^{5} + 26514840899483907674201376 \nu^{4} + 80965926607595525060032 \nu^{3} + 3458967843824083491365280 \nu^{2} - 11810244640222325675946208 \nu + 861240134394725537790048$$$$)/$$$$17\!\cdots\!40$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{15} + \beta_{13} - \beta_{10} - \beta_{7} + \beta_{6}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$\beta_{15} + \beta_{14} - \beta_{12} + \beta_{11} - 6 \beta_{10} + 2 \beta_{8}$$ $$\nu^{3}$$ $$=$$ $$($$$$15 \beta_{15} + 5 \beta_{14} + 15 \beta_{13} - 7 \beta_{12} + 2 \beta_{11} - 18 \beta_{10} - 2 \beta_{9} + 6 \beta_{8} + 15 \beta_{7} - 15 \beta_{6} - \beta_{5} - 5 \beta_{4} + \beta_{3} - 7 \beta_{2} - \beta_{1} - 8$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$-23 \beta_{9} + 37 \beta_{7} - 16 \beta_{6} + 4 \beta_{5} - 35 \beta_{4} - 15 \beta_{3} - 29 \beta_{2} - 125$$ $$\nu^{5}$$ $$=$$ $$($$$$-333 \beta_{15} - 181 \beta_{14} - 297 \beta_{13} + 219 \beta_{12} - 82 \beta_{11} + 524 \beta_{10} - 82 \beta_{9} - 222 \beta_{8} + 333 \beta_{7} - 297 \beta_{6} - 29 \beta_{5} - 181 \beta_{4} - 3 \beta_{3} - 219 \beta_{2} + 29 \beta_{1} - 408$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$-1101 \beta_{15} - 923 \beta_{14} - 638 \beta_{13} + 793 \beta_{12} - 545 \beta_{11} + 2482 \beta_{10} - 1068 \beta_{8} - 82 \beta_{1}$$ $$\nu^{7}$$ $$=$$ $$($$$$-8539 \beta_{15} - 5455 \beta_{14} - 6931 \beta_{13} + 6049 \beta_{12} - 2698 \beta_{11} + 15272 \beta_{10} + 2698 \beta_{9} - 6650 \beta_{8} - 8539 \beta_{7} + 6931 \beta_{6} + 555 \beta_{5} + 5455 \beta_{4} + 601 \beta_{3} + 6049 \beta_{2} + 555 \beta_{1} + 13796$$$$)/2$$ $$\nu^{8}$$ $$=$$ $$13535 \beta_{9} - 30705 \beta_{7} + 19868 \beta_{6} - 1292 \beta_{5} + 23987 \beta_{4} + 5879 \beta_{3} + 21689 \beta_{2} + 67757$$ $$\nu^{9}$$ $$=$$ $$($$$$228693 \beta_{15} + 156049 \beta_{14} + 175233 \beta_{13} - 163999 \beta_{12} + 80530 \beta_{11} - 430600 \beta_{10} + 80530 \beta_{9} + 187902 \beta_{8} - 228693 \beta_{7} + 175233 \beta_{6} + 9441 \beta_{5} + 156049 \beta_{4} + 23903 \beta_{3} + 163999 \beta_{2} - 9441 \beta_{1} + 411416$$$$)/2$$ $$\nu^{10}$$ $$=$$ $$840209 \beta_{15} + 632431 \beta_{14} + 572066 \beta_{13} - 592981 \beta_{12} + 348673 \beta_{11} - 1685906 \beta_{10} + 731924 \beta_{8} + 17622 \beta_{1}$$ $$\nu^{11}$$ $$=$$ $$($$$$6197443 \beta_{15} + 4357551 \beta_{14} + 4599939 \beta_{13} - 4440465 \beta_{12} + 2294850 \beta_{11} - 11917632 \beta_{10} - 2294850 \beta_{9} + 5196474 \beta_{8} + 6197443 \beta_{7} - 4599939 \beta_{6} - 157907 \beta_{5} - 4357551 \beta_{4} - 756009 \beta_{3} - 4440465 \beta_{2} - 157907 \beta_{1} - 11675244$$$$)/2$$ $$\nu^{12}$$ $$=$$ $$-9198311 \beta_{9} + 22872101 \beta_{7} - 15968640 \beta_{6} + 198152 \beta_{5} - 16890359 \beta_{4} - 3493971 \beta_{3} - 16184341 \beta_{2} - 46197501$$ $$\nu^{13}$$ $$=$$ $$($$$$-168436077 \beta_{15} - 120154729 \beta_{14} - 122923041 \beta_{13} + 120385895 \beta_{12} - 63909658 \beta_{11} + 326873928 \beta_{10} - 63909658 \beta_{9} - 142418846 \beta_{8} + 168436077 \beta_{7} - 122923041 \beta_{6} - 2751249 \beta_{5} - 120154729 \beta_{4} - 22032951 \beta_{3} - 120385895 \beta_{2} + 2751249 \beta_{1} - 324105616$$$$)/2$$ $$\nu^{14}$$ $$=$$ $$-621856845 \beta_{15} - 454761563 \beta_{14} - 439706318 \beta_{13} + 441047113 \beta_{12} - 246060185 \beta_{11} + 1223537266 \beta_{10} - 532130908 \beta_{8} - 1156906 \beta_{1}$$ $$\nu^{15}$$ $$=$$ $$($$$$-4580463835 \beta_{15} - 3291035767 \beta_{14} - 3313391155 \beta_{13} + 3267676777 \beta_{12} - 1759193050 \beta_{11} + 8926247504 \beta_{10} + 1759193050 \beta_{9} - 3887314778 \beta_{8} - 4580463835 \beta_{7} + 3313391155 \beta_{6} + 51934243 \beta_{5} + 3291035767 \beta_{4} + 619638001 \beta_{3} + 3267676777 \beta_{2} + 51934243 \beta_{1} + 8903892116$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/5520\mathbb{Z}\right)^\times$$.

 $$n$$ $$1201$$ $$1381$$ $$1841$$ $$4417$$ $$4831$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1471.1
 1.49885 + 1.49885i −0.373815 − 0.373815i 1.30491 + 1.30491i −2.02116 − 2.02116i −2.64546 − 2.64546i 3.68710 + 3.68710i 0.476829 + 0.476829i 0.0727486 + 0.0727486i 1.49885 − 1.49885i −0.373815 + 0.373815i 1.30491 − 1.30491i −2.02116 + 2.02116i −2.64546 + 2.64546i 3.68710 − 3.68710i 0.476829 − 0.476829i 0.0727486 − 0.0727486i
0 1.00000i 0 1.00000i 0 −4.84428 0 −1.00000 0
1471.2 0 1.00000i 0 1.00000i 0 −1.61153 0 −1.00000 0
1471.3 0 1.00000i 0 1.00000i 0 −0.482745 0 −1.00000 0
1471.4 0 1.00000i 0 1.00000i 0 −0.279423 0 −1.00000 0
1471.5 0 1.00000i 0 1.00000i 0 1.23448 0 −1.00000 0
1471.6 0 1.00000i 0 1.00000i 0 1.58474 0 −1.00000 0
1471.7 0 1.00000i 0 1.00000i 0 3.79952 0 −1.00000 0
1471.8 0 1.00000i 0 1.00000i 0 4.59925 0 −1.00000 0
1471.9 0 1.00000i 0 1.00000i 0 −4.84428 0 −1.00000 0
1471.10 0 1.00000i 0 1.00000i 0 −1.61153 0 −1.00000 0
1471.11 0 1.00000i 0 1.00000i 0 −0.482745 0 −1.00000 0
1471.12 0 1.00000i 0 1.00000i 0 −0.279423 0 −1.00000 0
1471.13 0 1.00000i 0 1.00000i 0 1.23448 0 −1.00000 0
1471.14 0 1.00000i 0 1.00000i 0 1.58474 0 −1.00000 0
1471.15 0 1.00000i 0 1.00000i 0 3.79952 0 −1.00000 0
1471.16 0 1.00000i 0 1.00000i 0 4.59925 0 −1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1471.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5520.2.be.b yes 16
4.b odd 2 1 5520.2.be.a 16
23.b odd 2 1 5520.2.be.a 16
92.b even 2 1 inner 5520.2.be.b yes 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5520.2.be.a 16 4.b odd 2 1
5520.2.be.a 16 23.b odd 2 1
5520.2.be.b yes 16 1.a even 1 1 trivial
5520.2.be.b yes 16 92.b even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{8} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(5520, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16}$$
$3$ $$( 1 + T^{2} )^{8}$$
$5$ $$( 1 + T^{2} )^{8}$$
$7$ $$( 36 + 164 T + 37 T^{2} - 316 T^{3} + 39 T^{4} + 108 T^{5} - 25 T^{6} - 4 T^{7} + T^{8} )^{2}$$
$11$ $$( 800 + 3040 T + 3988 T^{2} + 1984 T^{3} + 60 T^{4} - 216 T^{5} - 40 T^{6} + 4 T^{7} + T^{8} )^{2}$$
$13$ $$( 592 + 816 T - 780 T^{2} - 624 T^{3} + 352 T^{4} + 92 T^{5} - 40 T^{6} - 4 T^{7} + T^{8} )^{2}$$
$17$ $$1354896 + 3447736 T^{2} + 3360361 T^{4} + 1606646 T^{6} + 401575 T^{8} + 51764 T^{10} + 3247 T^{12} + 94 T^{14} + T^{16}$$
$19$ $$( -17672 + 31760 T - 13804 T^{2} - 3528 T^{3} + 2696 T^{4} + 64 T^{5} - 102 T^{6} + T^{8} )^{2}$$
$23$ $$78310985281 + 40857905364 T + 10658584008 T^{2} + 3475625220 T^{3} + 795867804 T^{4} + 83076276 T^{5} + 11574520 T^{6} + 190532 T^{7} - 600954 T^{8} + 8284 T^{9} + 21880 T^{10} + 6828 T^{11} + 2844 T^{12} + 540 T^{13} + 72 T^{14} + 12 T^{15} + T^{16}$$
$29$ $$( 540 + 14020 T - 20403 T^{2} - 3190 T^{3} + 3563 T^{4} - 64 T^{5} - 113 T^{6} + 2 T^{7} + T^{8} )^{2}$$
$31$ $$10223636544 + 24444959440 T^{2} + 6607070457 T^{4} + 740738018 T^{6} + 43467807 T^{8} + 1436588 T^{10} + 26695 T^{12} + 258 T^{14} + T^{16}$$
$37$ $$56667802500 + 27998279316 T^{2} + 5549557933 T^{4} + 575371058 T^{6} + 34217739 T^{8} + 1197664 T^{10} + 24035 T^{12} + 250 T^{14} + T^{16}$$
$41$ $$( -127396 + 108340 T + 22381 T^{2} - 29090 T^{3} + 4375 T^{4} + 608 T^{5} - 141 T^{6} - 2 T^{7} + T^{8} )^{2}$$
$43$ $$( 2022592 + 210496 T - 329776 T^{2} - 6560 T^{3} + 14000 T^{4} + 48 T^{5} - 208 T^{6} + T^{8} )^{2}$$
$47$ $$1006281834496 + 400649540608 T^{2} + 59719495440 T^{4} + 4486194464 T^{6} + 187253328 T^{8} + 4435512 T^{10} + 57928 T^{12} + 384 T^{14} + T^{16}$$
$53$ $$1337788330384 + 1343036772264 T^{2} + 334723406649 T^{4} + 22004448414 T^{6} + 672051079 T^{8} + 11131204 T^{10} + 103103 T^{12} + 502 T^{14} + T^{16}$$
$59$ $$99130522500 + 86338855300 T^{2} + 21456226381 T^{4} + 2153638294 T^{6} + 108621347 T^{8} + 2988528 T^{10} + 44987 T^{12} + 342 T^{14} + T^{16}$$
$61$ $$713958400 + 464349666560 T^{2} + 116846287376 T^{4} + 10653617664 T^{6} + 438981792 T^{8} + 8915784 T^{10} + 93600 T^{12} + 488 T^{14} + T^{16}$$
$67$ $$( 1963932 + 180556 T - 414555 T^{2} - 51744 T^{3} + 16391 T^{4} + 1260 T^{5} - 225 T^{6} - 8 T^{7} + T^{8} )^{2}$$
$71$ $$925232372100 + 2463835005316 T^{2} + 1381219933533 T^{4} + 126576746382 T^{6} + 3489208643 T^{8} + 43210960 T^{10} + 269771 T^{12} + 830 T^{14} + T^{16}$$
$73$ $$( 36784 + 73040 T - 11628 T^{2} - 28864 T^{3} + 2576 T^{4} + 2404 T^{5} - 88 T^{6} - 20 T^{7} + T^{8} )^{2}$$
$79$ $$( -958464 - 1085440 T - 34304 T^{2} + 136704 T^{3} + 11680 T^{4} - 4304 T^{5} - 282 T^{6} + 16 T^{7} + T^{8} )^{2}$$
$83$ $$( -700572 + 1298396 T - 453931 T^{2} - 15792 T^{3} + 21231 T^{4} - 412 T^{5} - 317 T^{6} + T^{8} )^{2}$$
$89$ $$71156629504 + 188835102720 T^{2} + 64438915072 T^{4} + 7033284608 T^{6} + 335213312 T^{8} + 7763968 T^{10} + 89520 T^{12} + 488 T^{14} + T^{16}$$
$97$ $$10557973504 + 106855160832 T^{2} + 94700347136 T^{4} + 21868471936 T^{6} + 1145918016 T^{8} + 23048576 T^{10} + 202452 T^{12} + 772 T^{14} + T^{16}$$