# Properties

 Label 48.25.e.d Level $48$ Weight $25$ Character orbit 48.e Analytic conductor $175.184$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$48 = 2^{4} \cdot 3$$ Weight: $$k$$ $$=$$ $$25$$ Character orbit: $$[\chi]$$ $$=$$ 48.e (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$175.184233084$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ Defining polynomial: $$x^{8} + 1629189373042052 x^{6} + 272106194286045879281514214500 x^{4} + 13671756443267173943059351340058372000500000 x^{2} + 210991722585849839909329798386550369753038400000000000000$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{60}\cdot 3^{34}\cdot 17^{2}$$ Twist minimal: no (minimal twist has level 6) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + ( 16485 - \beta_{1} ) q^{3} + ( 155 \beta_{1} - 145 \beta_{2} + \beta_{3} ) q^{5} + ( 1270099330 - 1539 \beta_{1} + 5 \beta_{2} - 6 \beta_{4} + 8 \beta_{5} + \beta_{6} - \beta_{7} ) q^{7} + ( 36896131737 - 13814 \beta_{1} - 39890 \beta_{2} + 920 \beta_{3} - 37 \beta_{4} - 48 \beta_{5} - 61 \beta_{6} + 22 \beta_{7} ) q^{9} +O(q^{10})$$ $$q + ( 16485 - \beta_{1} ) q^{3} + ( 155 \beta_{1} - 145 \beta_{2} + \beta_{3} ) q^{5} + ( 1270099330 - 1539 \beta_{1} + 5 \beta_{2} - 6 \beta_{4} + 8 \beta_{5} + \beta_{6} - \beta_{7} ) q^{7} + ( 36896131737 - 13814 \beta_{1} - 39890 \beta_{2} + 920 \beta_{3} - 37 \beta_{4} - 48 \beta_{5} - 61 \beta_{6} + 22 \beta_{7} ) q^{9} + ( -3616097 \beta_{1} + 35863 \beta_{2} - 3225 \beta_{3} - 759 \beta_{4} - 694 \beta_{5} - 13 \beta_{6} - 1323 \beta_{7} ) q^{11} + ( 6321045459890 - 17433740 \beta_{1} + 41615 \beta_{2} + 15505 \beta_{3} + 910 \beta_{4} + 2185 \beta_{5} + 9725 \beta_{6} + 895 \beta_{7} ) q^{13} + ( 43504369595064 + 2249525 \beta_{1} + 56289597 \beta_{2} - 247525 \beta_{3} - 2970 \beta_{4} - 47775 \beta_{5} + 32260 \beta_{6} - 60306 \beta_{7} ) q^{15} + ( -247631906 \beta_{1} + 297209947 \beta_{2} + 1024789 \beta_{3} - 219912 \beta_{4} - 85337 \beta_{5} - 26915 \beta_{6} - 36099 \beta_{7} ) q^{17} + ( 122260453917658 - 1581912701 \beta_{1} + 4058083 \beta_{2} + 1780516 \beta_{3} + 384415 \beta_{4} + 16983 \beta_{5} - 113838 \beta_{6} + 497652 \beta_{7} ) q^{19} + ( 455001538030242 - 1227858505 \beta_{1} - 6137192178 \beta_{2} - 2541942 \beta_{3} - 4417524 \beta_{4} + 2054565 \beta_{5} + 113103 \beta_{6} - 774657 \beta_{7} ) q^{21} + ( -1866406714 \beta_{1} - 10841274802 \beta_{2} + 61999536 \beta_{3} + 5012022 \beta_{4} - 2890778 \beta_{5} + 1580560 \beta_{6} - 13684356 \beta_{7} ) q^{23} + ( 953827345876753 - 47783182660 \beta_{1} + 136703229 \beta_{2} + 58231379 \beta_{3} - 7494010 \beta_{4} + 30949615 \beta_{5} - 31312165 \beta_{6} + 22029313 \beta_{7} ) q^{25} + ( 10824316075757205 - 36254508324 \beta_{1} - 25818844335 \beta_{2} + 36982113 \beta_{3} + 42796797 \beta_{4} - 84951432 \beta_{5} + 27606579 \beta_{6} - 83244327 \beta_{7} ) q^{27} + ( -199220236339 \beta_{1} - 144545005893 \beta_{2} - 1556093219 \beta_{3} - 123377328 \beta_{4} + 15079422 \beta_{5} - 27691350 \beta_{6} + 168615594 \beta_{7} ) q^{29} + ( -386514973352838638 - 409186685695 \beta_{1} + 1002445245 \beta_{2} + 508721540 \beta_{3} + 314592080 \beta_{4} - 272050370 \beta_{5} - 33576925 \beta_{6} + 166585585 \beta_{7} ) q^{31} + ( -1019835744083566200 - 57571561732 \beta_{1} - 149043107613 \beta_{2} + 3634907063 \beta_{3} - 161210115 \beta_{4} - 316287831 \beta_{5} - 225462824 \beta_{6} - 95810739 \beta_{7} ) q^{33} + ( -651523813840 \beta_{1} - 1277684534750 \beta_{2} - 3563140761 \beta_{3} - 308628870 \beta_{4} - 9897245 \beta_{5} - 59746325 \beta_{6} + 278937135 \beta_{7} ) q^{35} + ( 2823786652999097810 - 5856855574612 \beta_{1} + 14559191689 \beta_{2} + 6362074103 \beta_{3} + 1852459994 \beta_{4} - 708703513 \beta_{5} + 625875187 \beta_{6} + 1530814145 \beta_{7} ) q^{37} + ( 5023772072735183130 - 6053833476380 \beta_{1} + 5774403008700 \beta_{2} - 16985175525 \beta_{3} - 6312467970 \beta_{4} + 1776856095 \beta_{5} - 4081298535 \beta_{6} + 695334645 \beta_{7} ) q^{39} + ( 7733813811386 \beta_{1} + 6434869762654 \beta_{2} + 22098849078 \beta_{3} + 9261084432 \beta_{4} + 607240072 \beta_{5} + 1730768872 \beta_{6} - 7439364216 \beta_{7} ) q^{41} + ( -28310933296475487110 - 4591537852101 \beta_{1} + 16078951947 \beta_{2} + 2484268644 \beta_{3} - 14389587213 \beta_{4} + 20502490851 \beta_{5} - 2580247674 \beta_{6} - 349462440 \beta_{7} ) q^{43} + ( -43463749594647188016 - 46231232636955 \beta_{1} + 47676852240927 \beta_{2} - 43187684991 \beta_{3} - 7977892830 \beta_{4} + 11264053950 \beta_{5} - 14759524500 \beta_{6} + 4432982334 \beta_{7} ) q^{45} + ( -78881621340120 \beta_{1} + 101027773460580 \beta_{2} - 151636173490 \beta_{3} + 7111341660 \beta_{4} - 13216717490 \beta_{5} + 4065611830 \beta_{6} - 46761494130 \beta_{7} ) q^{47} + ( 13085837559202091475 - 248188242741532 \beta_{1} + 653433920731 \beta_{2} + 252993257237 \beta_{3} - 38246505670 \beta_{4} + 126399503081 \beta_{5} - 2077817891 \beta_{6} + 55639359239 \beta_{7} ) q^{49} + ( -69761484367070817888 - 3732165748338 \beta_{1} - 134144508370320 \beta_{2} + 32112215085 \beta_{3} + 3559143456 \beta_{4} - 104970954501 \beta_{5} - 18111588987 \beta_{6} - 181518578811 \beta_{7} ) q^{51} + ( 480318630723431 \beta_{1} - 159521851592957 \beta_{2} + 917793197941 \beta_{3} - 85201473888 \beta_{4} + 82475578512 \beta_{5} - 33535410480 \beta_{6} + 332628209424 \beta_{7} ) q^{53} + ( 276585958031308417296 + 631546414265140 \beta_{1} - 1725230815752 \beta_{2} - 731732066732 \beta_{3} + 18030024430 \beta_{4} - 267141872050 \beta_{5} + 234030464650 \beta_{6} - 238883561254 \beta_{7} ) q^{55} + ( 448609999455095090730 - 95576415564976 \beta_{1} + 40626998283294 \beta_{2} + 1668832655976 \beta_{3} + 111367551753 \beta_{4} - 144017848206 \beta_{5} - 61894634841 \beta_{6} + 48036426840 \beta_{7} ) q^{57} + ( 277737394463069 \beta_{1} + 128388445229695 \beta_{2} + 4998593855516 \beta_{3} - 643897178427 \beta_{4} - 88264763887 \beta_{5} - 111126482908 \beta_{6} + 379102886766 \beta_{7} ) q^{59} + ( -813376347979761548206 - 698694060176068 \beta_{1} + 1285133359509 \beta_{2} + 566860681643 \beta_{3} + 712441243490 \beta_{4} - 919870803701 \beta_{5} + 1042979235431 \beta_{6} - 89562997859 \beta_{7} ) q^{61} + ( -1264122847201612422510 - 518917031266313 \beta_{1} - 4318611499290821 \beta_{2} - 5537397216064 \beta_{3} - 1414868291026 \beta_{4} - 560474626944 \beta_{5} + 26946425183 \beta_{6} - 368998290359 \beta_{7} ) q^{63} + ( 4363483986480850 \beta_{1} - 8237549054634300 \beta_{2} + 23424776775400 \beta_{3} + 758907811200 \beta_{4} + 138261727450 \beta_{5} + 124129216750 \beta_{6} - 344122628850 \beta_{7} ) q^{65} + ( 880265014768757518090 + 2709035205888255 \beta_{1} - 6699545339973 \beta_{2} - 3676770494256 \beta_{3} - 2691113439639 \beta_{4} + 2466277865169 \beta_{5} + 792516466272 \beta_{6} - 1414817128686 \beta_{7} ) q^{67} + ( -540416995921091418192 - 75669533216012 \beta_{1} + 16802936933208240 \beta_{2} + 13918357998160 \beta_{3} - 2677929473106 \beta_{4} - 1951686236544 \beta_{5} + 2805247650542 \beta_{6} - 2610276781044 \beta_{7} ) q^{69} + ( 9766616958986606 \beta_{1} + 22944359813978402 \beta_{2} + 42269654564846 \beta_{3} + 1926071970162 \beta_{4} + 666920787812 \beta_{5} + 251830236470 \beta_{6} + 74690393274 \beta_{7} ) q^{71} + ( 4532477540594822085890 + 16351641309736144 \beta_{1} - 38721079161732 \beta_{2} - 19452817522844 \beta_{3} - 13289779427840 \beta_{4} + 12494287052828 \beta_{5} - 2001020335988 \beta_{6} - 5552101603948 \beta_{7} ) q^{73} + ( 13510247306076613581909 + 7083219273605 \beta_{1} - 42940565167389948 \beta_{2} + 137542191995673 \beta_{3} + 5333044101030 \beta_{4} - 4428460714095 \beta_{5} + 9119937221955 \beta_{6} - 2795117030541 \beta_{7} ) q^{75} + ( -26937819847405826 \beta_{1} - 59866251101739226 \beta_{2} - 11962932703406 \beta_{3} - 17898141430752 \beta_{4} - 5080509968552 \beta_{5} - 2563526292440 \beta_{6} + 2656611525096 \beta_{7} ) q^{77} + ( -39600881597166251914478 - 34307928256477919 \beta_{1} + 87544024140997 \beta_{2} + 34999346238844 \beta_{3} + 644651143120 \beta_{4} + 8973744687942 \beta_{5} + 3808047603723 \beta_{6} + 7190967601353 \beta_{7} ) q^{79} + ( -33564832563049597101663 - 13123479771846972 \beta_{1} + 119933449348145823 \beta_{2} + 31922025668613 \beta_{3} + 4180278111444 \beta_{4} - 13603922721819 \beta_{5} - 4949635819005 \beta_{6} + 1200516430671 \beta_{7} ) q^{81} + ( -37816654519169679 \beta_{1} + 183183839612292669 \beta_{2} + 38411131639091 \beta_{3} - 28676135943933 \beta_{4} - 8975045677208 \beta_{5} - 3940218053345 \beta_{6} + 1750998912309 \beta_{7} ) q^{83} + ( -27814107818190525084864 + 29903656096336720 \beta_{1} - 83501504163972 \beta_{2} - 36868851069692 \beta_{3} - 655649994920 \beta_{4} - 11980971316540 \beta_{5} + 17206103457940 \beta_{6} - 13812777738724 \beta_{7} ) q^{85} + ( -56182697150225917766040 - 2360898891467045 \beta_{1} - 320105663499835185 \beta_{2} - 133332708114341 \beta_{3} + 38833533423954 \beta_{4} - 2778808552215 \beta_{5} - 57819370031386 \beta_{6} + 96667507163652 \beta_{7} ) q^{87} + ( -242166499368465104 \beta_{1} - 308397706617551015 \beta_{2} + 63169518664939 \beta_{3} - 15047033936568 \beta_{4} - 58099585105433 \beta_{5} + 8610510233773 \beta_{6} - 159251721379731 \beta_{7} ) q^{89} + ( 149604950291961346795940 - 120745937362229070 \beta_{1} + 340635222688830 \beta_{2} + 105066654081060 \beta_{3} - 109764274349040 \beta_{4} + 180229021575600 \beta_{5} - 6990893175990 \beta_{6} + 13864266676830 \beta_{7} ) q^{91} + ( 109154692181353597968210 + 392754222007451093 \beta_{1} + 246436195002472350 \beta_{2} + 465763141606050 \beta_{3} + 164969464721490 \beta_{4} - 104217390633465 \beta_{5} - 33055268130405 \beta_{6} + 42059350600785 \beta_{7} ) q^{93} + ( -241788143889951910 \beta_{1} + 262259908892643090 \beta_{2} - 451418026352392 \beta_{3} - 19655394824550 \beta_{4} - 39217010295550 \beta_{5} + 3912323094200 \beta_{6} - 97995636062100 \beta_{7} ) q^{95} + ( 102064697031025158214610 - 490904123238560300 \beta_{1} + 1292688473535599 \beta_{2} + 472852618472353 \beta_{3} - 142243651667558 \beta_{4} + 327038792939773 \beta_{5} + 38103800369729 \beta_{6} + 83317892276803 \beta_{7} ) q^{97} + ( -86982360420901115289168 + 1015579765287888369 \beta_{1} + 20805808593039027 \beta_{2} + 222565233842166 \beta_{3} + 172071962096331 \beta_{4} - 362707261319367 \beta_{5} + 113597275075752 \beta_{6} - 358926901458510 \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 131880 q^{3} + 10160794640 q^{7} + 295169053896 q^{9} + O(q^{10})$$ $$8 q + 131880 q^{3} + 10160794640 q^{7} + 295169053896 q^{9} + 50568363679120 q^{13} + 348034956760512 q^{15} + 978083631341264 q^{19} + 3640012304241936 q^{21} + 7630618767014024 q^{25} + 86594528606057640 q^{27} - 3092119786822709104 q^{31} - 8158685952668529600 q^{33} + 22590293223992782480 q^{37} + 40190176581881465040 q^{39} -$$$$22\!\cdots\!80$$$$q^{43} -$$$$34\!\cdots\!28$$$$q^{45} +$$$$10\!\cdots\!00$$$$q^{49} -$$$$55\!\cdots\!04$$$$q^{51} +$$$$22\!\cdots\!68$$$$q^{55} +$$$$35\!\cdots\!40$$$$q^{57} -$$$$65\!\cdots\!48$$$$q^{61} -$$$$10\!\cdots\!80$$$$q^{63} +$$$$70\!\cdots\!20$$$$q^{67} -$$$$43\!\cdots\!36$$$$q^{69} +$$$$36\!\cdots\!20$$$$q^{73} +$$$$10\!\cdots\!72$$$$q^{75} -$$$$31\!\cdots\!24$$$$q^{79} -$$$$26\!\cdots\!04$$$$q^{81} -$$$$22\!\cdots\!12$$$$q^{85} -$$$$44\!\cdots\!20$$$$q^{87} +$$$$11\!\cdots\!20$$$$q^{91} +$$$$87\!\cdots\!80$$$$q^{93} +$$$$81\!\cdots\!80$$$$q^{97} -$$$$69\!\cdots\!44$$$$q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 1629189373042052 x^{6} + 272106194286045879281514214500 x^{4} + 13671756443267173943059351340058372000500000 x^{2} + 210991722585849839909329798386550369753038400000000000000$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-$$$$74\!\cdots\!16$$$$\nu^{7} -$$$$58\!\cdots\!00$$$$\nu^{6} -$$$$11\!\cdots\!32$$$$\nu^{5} -$$$$92\!\cdots\!50$$$$\nu^{4} -$$$$15\!\cdots\!00$$$$\nu^{3} -$$$$11\!\cdots\!50$$$$\nu^{2} -$$$$43\!\cdots\!50$$$$\nu -$$$$30\!\cdots\!50$$$$)/$$$$71\!\cdots\!75$$ $$\beta_{2}$$ $$=$$ $$($$$$-13986119078275384896 \nu^{7} - 22314034733824269734428041721166592 \nu^{5} - 3048473705927959484497130015631016292248115232000 \nu^{3} - 83423009391397120197171153971530952598152399165401015968000000 \nu$$$$)/$$$$86\!\cdots\!25$$ $$\beta_{3}$$ $$=$$ $$($$$$-$$$$20\!\cdots\!84$$$$\nu^{7} +$$$$36\!\cdots\!00$$$$\nu^{6} -$$$$32\!\cdots\!68$$$$\nu^{5} +$$$$57\!\cdots\!50$$$$\nu^{4} -$$$$47\!\cdots\!00$$$$\nu^{3} +$$$$72\!\cdots\!50$$$$\nu^{2} -$$$$90\!\cdots\!50$$$$\nu +$$$$18\!\cdots\!50$$$$)/$$$$28\!\cdots\!75$$ $$\beta_{4}$$ $$=$$ $$($$$$-$$$$40\!\cdots\!12$$$$\nu^{7} +$$$$72\!\cdots\!00$$$$\nu^{6} -$$$$67\!\cdots\!24$$$$\nu^{5} +$$$$10\!\cdots\!50$$$$\nu^{4} -$$$$12\!\cdots\!00$$$$\nu^{3} -$$$$24\!\cdots\!50$$$$\nu^{2} -$$$$63\!\cdots\!50$$$$\nu -$$$$48\!\cdots\!50$$$$)/$$$$71\!\cdots\!75$$ $$\beta_{5}$$ $$=$$ $$($$$$-$$$$11\!\cdots\!52$$$$\nu^{7} +$$$$29\!\cdots\!00$$$$\nu^{6} -$$$$19\!\cdots\!04$$$$\nu^{5} +$$$$46\!\cdots\!00$$$$\nu^{4} -$$$$30\!\cdots\!00$$$$\nu^{3} +$$$$66\!\cdots\!00$$$$\nu^{2} -$$$$12\!\cdots\!00$$$$\nu +$$$$19\!\cdots\!00$$$$)/$$$$14\!\cdots\!75$$ $$\beta_{6}$$ $$=$$ $$($$$$-$$$$88\!\cdots\!44$$$$\nu^{7} +$$$$28\!\cdots\!00$$$$\nu^{6} -$$$$14\!\cdots\!88$$$$\nu^{5} +$$$$45\!\cdots\!50$$$$\nu^{4} -$$$$18\!\cdots\!00$$$$\nu^{3} +$$$$60\!\cdots\!50$$$$\nu^{2} -$$$$46\!\cdots\!50$$$$\nu +$$$$15\!\cdots\!50$$$$)/$$$$71\!\cdots\!75$$ $$\beta_{7}$$ $$=$$ $$($$$$-$$$$67\!\cdots\!16$$$$\nu^{7} +$$$$17\!\cdots\!00$$$$\nu^{6} -$$$$10\!\cdots\!32$$$$\nu^{5} +$$$$28\!\cdots\!00$$$$\nu^{4} -$$$$13\!\cdots\!00$$$$\nu^{3} +$$$$32\!\cdots\!00$$$$\nu^{2} -$$$$29\!\cdots\!00$$$$\nu +$$$$69\!\cdots\!00$$$$)/$$$$23\!\cdots\!25$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} - 145 \beta_{2} + 155 \beta_{1}$$$$)/12$$ $$\nu^{2}$$ $$=$$ $$($$$$22029313 \beta_{7} - 31312165 \beta_{6} + 30949615 \beta_{5} - 7494010 \beta_{4} + 58231379 \beta_{3} + 136703229 \beta_{2} - 47783182660 \beta_{1} - 58650817429513872$$$$)/144$$ $$\nu^{3}$$ $$=$$ $$($$$$-192218719815945 \beta_{7} - 23049792898825 \beta_{6} - 153733842155035 \beta_{5} - 268982806649160 \beta_{4} - 9209641247205221 \beta_{3} + 2330662123279759185 \beta_{2} - 2136351678743199810 \beta_{1}$$$$)/96$$ $$\nu^{4}$$ $$=$$ $$($$$$-5770294725436960964171 \beta_{7} + 7875752616335194606805 \beta_{6} - 7385476297801261333955 \beta_{5} + 1427697363392572520295 \beta_{4} - 15392910494384377099368 \beta_{3} - 35878355186700054040568 \beta_{2} + 12634300359882614023748220 \beta_{1} + 12660273747966376280793191650224$$$$)/24$$ $$\nu^{5}$$ $$=$$ $$($$$$921370590442788783858353830545 \beta_{7} + 98691561742080568384610444825 \beta_{6} + 707414199576595812890703027335 \beta_{5} + 1200872008286998654813755251460 \beta_{4} + 39379593699808995001548768645601 \beta_{3} - 10332220925832592850458127939360360 \beta_{2} + 9358896974979221419214015390576610 \beta_{1}$$$$)/288$$ $$\nu^{6}$$ $$=$$ $$($$$$33623682478028643046872356325296719693 \beta_{7} - 45701250989591762906127408875044188565 \beta_{6} + 42652614696213649242497595489115004515 \beta_{5} - 8032216230717190773626001534170101360 \beta_{4} + 89790194909990624703361254055790873669 \beta_{3} + 209158720017163963367965813421301099519 \beta_{2} - 73707763239117022728761037207052779459760 \beta_{1} - 72848799782201793160896313668855629581389182592$$$$)/96$$ $$\nu^{7}$$ $$=$$ $$($$$$-896201636932573250290835263497985406492230560 \beta_{7} - 94923014395701242140609293698411313435845600 \beta_{6} - 685408354455539730496940865995020986835729280 \beta_{5} - 1160023426434045941199987334487077554014957280 \beta_{4} - 37965915673948803566864455818688287161149983168 \beta_{3} + 9986280240486230040704955770262210292679153660605 \beta_{2} - 9037877672185738910146167695694951792455442412480 \beta_{1}$$$$)/192$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/48\mathbb{Z}\right)^\times$$.

 $$n$$ $$17$$ $$31$$ $$37$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
17.1
 6.54850e6i − 6.54850e6i 3.80484e7i − 3.80484e7i − 5.64115e6i 5.64115e6i − 1.03344e7i 1.03344e7i
0 −473275. 241744.i 0 7.85820e7i 0 −2.19870e9 0 1.65549e11 + 2.28823e11i 0
17.2 0 −473275. + 241744.i 0 7.85820e7i 0 −2.19870e9 0 1.65549e11 2.28823e11i 0
17.3 0 −269557. 458005.i 0 4.56581e8i 0 −1.81935e9 0 −1.37107e11 + 2.46917e11i 0
17.4 0 −269557. + 458005.i 0 4.56581e8i 0 −1.81935e9 0 −1.37107e11 2.46917e11i 0
17.5 0 317945. 425841.i 0 6.76938e7i 0 2.41596e10 0 −8.02515e10 2.70788e11i 0
17.6 0 317945. + 425841.i 0 6.76938e7i 0 2.41596e10 0 −8.02515e10 + 2.70788e11i 0
17.7 0 490828. 203759.i 0 1.24013e8i 0 −1.50611e10 0 1.99394e11 2.00021e11i 0
17.8 0 490828. + 203759.i 0 1.24013e8i 0 −1.50611e10 0 1.99394e11 + 2.00021e11i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 17.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 48.25.e.d 8
3.b odd 2 1 inner 48.25.e.d 8
4.b odd 2 1 6.25.b.a 8
12.b even 2 1 6.25.b.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.25.b.a 8 4.b odd 2 1
6.25.b.a 8 12.b even 2 1
48.25.e.d 8 1.a even 1 1 trivial
48.25.e.d 8 3.b odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{8} +$$$$23\!\cdots\!88$$$$T_{5}^{6} +$$$$56\!\cdots\!00$$$$T_{5}^{4} +$$$$40\!\cdots\!00$$$$T_{5}^{2} +$$$$90\!\cdots\!00$$ acting on $$S_{25}^{\mathrm{new}}(48, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$63\!\cdots\!21$$$$-$$$$29\!\cdots\!80$$$$T -$$$$11\!\cdots\!28$$$$T^{2} -$$$$27\!\cdots\!40$$$$T^{3} +$$$$80\!\cdots\!78$$$$T^{4} - 9783678964895640 T^{5} - 138888359748 T^{6} - 131880 T^{7} + T^{8}$$
$5$ $$90\!\cdots\!00$$$$+$$$$40\!\cdots\!00$$$$T^{2} +$$$$56\!\cdots\!00$$$$T^{4} + 234603269718055488 T^{6} + T^{8}$$
$7$ $$( -$$$$14\!\cdots\!04$$$$-$$$$14\!\cdots\!60$$$$T -$$$$39\!\cdots\!52$$$$T^{2} - 5080397320 T^{3} + T^{4} )^{2}$$
$11$ $$22\!\cdots\!36$$$$+$$$$16\!\cdots\!12$$$$T^{2} +$$$$39\!\cdots\!64$$$$T^{4} +$$$$34\!\cdots\!48$$$$T^{6} + T^{8}$$
$13$ $$( -$$$$14\!\cdots\!00$$$$+$$$$37\!\cdots\!00$$$$T -$$$$12\!\cdots\!00$$$$T^{2} - 25284181839560 T^{3} + T^{4} )^{2}$$
$17$ $$73\!\cdots\!56$$$$+$$$$61\!\cdots\!72$$$$T^{2} +$$$$46\!\cdots\!84$$$$T^{4} +$$$$11\!\cdots\!08$$$$T^{6} + T^{8}$$
$19$ $$($$$$54\!\cdots\!36$$$$-$$$$87\!\cdots\!08$$$$T -$$$$28\!\cdots\!56$$$$T^{2} - 489041815670632 T^{3} + T^{4} )^{2}$$
$23$ $$36\!\cdots\!76$$$$+$$$$62\!\cdots\!52$$$$T^{2} +$$$$22\!\cdots\!24$$$$T^{4} +$$$$26\!\cdots\!48$$$$T^{6} + T^{8}$$
$29$ $$10\!\cdots\!00$$$$+$$$$34\!\cdots\!00$$$$T^{2} +$$$$30\!\cdots\!00$$$$T^{4} +$$$$96\!\cdots\!60$$$$T^{6} + T^{8}$$
$31$ $$( -$$$$23\!\cdots\!64$$$$-$$$$72\!\cdots\!12$$$$T +$$$$35\!\cdots\!64$$$$T^{2} + 1546059893411354552 T^{3} + T^{4} )^{2}$$
$37$ $$( -$$$$34\!\cdots\!64$$$$+$$$$19\!\cdots\!60$$$$T -$$$$10\!\cdots\!92$$$$T^{2} - 11295146611996391240 T^{3} + T^{4} )^{2}$$
$41$ $$27\!\cdots\!96$$$$+$$$$36\!\cdots\!72$$$$T^{2} +$$$$13\!\cdots\!04$$$$T^{4} +$$$$20\!\cdots\!48$$$$T^{6} + T^{8}$$
$43$ $$( -$$$$18\!\cdots\!24$$$$-$$$$68\!\cdots\!20$$$$T +$$$$25\!\cdots\!32$$$$T^{2} +$$$$11\!\cdots\!40$$$$T^{3} + T^{4} )^{2}$$
$47$ $$18\!\cdots\!00$$$$+$$$$18\!\cdots\!00$$$$T^{2} +$$$$47\!\cdots\!00$$$$T^{4} +$$$$39\!\cdots\!00$$$$T^{6} + T^{8}$$
$53$ $$51\!\cdots\!36$$$$+$$$$70\!\cdots\!92$$$$T^{2} +$$$$54\!\cdots\!44$$$$T^{4} +$$$$13\!\cdots\!68$$$$T^{6} + T^{8}$$
$59$ $$14\!\cdots\!56$$$$+$$$$57\!\cdots\!72$$$$T^{2} +$$$$49\!\cdots\!84$$$$T^{4} +$$$$13\!\cdots\!08$$$$T^{6} + T^{8}$$
$61$ $$($$$$20\!\cdots\!36$$$$-$$$$26\!\cdots\!56$$$$T -$$$$11\!\cdots\!44$$$$T^{2} +$$$$32\!\cdots\!24$$$$T^{3} + T^{4} )^{2}$$
$67$ $$( -$$$$23\!\cdots\!24$$$$+$$$$27\!\cdots\!80$$$$T -$$$$66\!\cdots\!72$$$$T^{2} -$$$$35\!\cdots\!60$$$$T^{3} + T^{4} )^{2}$$
$71$ $$15\!\cdots\!00$$$$+$$$$58\!\cdots\!00$$$$T^{2} +$$$$42\!\cdots\!00$$$$T^{4} +$$$$11\!\cdots\!60$$$$T^{6} + T^{8}$$
$73$ $$($$$$15\!\cdots\!00$$$$+$$$$24\!\cdots\!00$$$$T -$$$$13\!\cdots\!40$$$$T^{2} -$$$$18\!\cdots\!60$$$$T^{3} + T^{4} )^{2}$$
$79$ $$($$$$50\!\cdots\!96$$$$+$$$$13\!\cdots\!68$$$$T +$$$$78\!\cdots\!64$$$$T^{2} +$$$$15\!\cdots\!12$$$$T^{3} + T^{4} )^{2}$$
$83$ $$78\!\cdots\!76$$$$+$$$$59\!\cdots\!08$$$$T^{2} +$$$$10\!\cdots\!64$$$$T^{4} +$$$$56\!\cdots\!92$$$$T^{6} + T^{8}$$
$89$ $$13\!\cdots\!96$$$$+$$$$72\!\cdots\!32$$$$T^{2} +$$$$34\!\cdots\!64$$$$T^{4} +$$$$35\!\cdots\!88$$$$T^{6} + T^{8}$$
$97$ $$( -$$$$78\!\cdots\!44$$$$-$$$$14\!\cdots\!80$$$$T -$$$$63\!\cdots\!48$$$$T^{2} -$$$$40\!\cdots\!40$$$$T^{3} + T^{4} )^{2}$$