# Properties

 Label 3750.2.c.k Level 3750 Weight 2 Character orbit 3750.c Analytic conductor 29.944 Analytic rank 0 Dimension 16 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$3750 = 2 \cdot 3 \cdot 5^{4}$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 3750.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$29.9439007580$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{8}\cdot 5^{4}$$ Twist minimal: no (minimal twist has level 150) 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.

 $$f(q)$$ $$=$$ $$q -\beta_{6} q^{2} + \beta_{6} q^{3} - q^{4} + q^{6} + ( -\beta_{6} - \beta_{9} ) q^{7} + \beta_{6} q^{8} - q^{9} +O(q^{10})$$ $$q -\beta_{6} q^{2} + \beta_{6} q^{3} - q^{4} + q^{6} + ( -\beta_{6} - \beta_{9} ) q^{7} + \beta_{6} q^{8} - q^{9} + ( 1 - \beta_{3} ) q^{11} -\beta_{6} q^{12} + ( \beta_{5} + \beta_{6} - \beta_{14} ) q^{13} + \beta_{8} q^{14} + q^{16} + ( \beta_{5} - \beta_{6} - \beta_{13} - \beta_{15} ) q^{17} + \beta_{6} q^{18} + ( -2 - \beta_{1} + \beta_{4} - \beta_{8} ) q^{19} -\beta_{8} q^{21} + ( -\beta_{5} - \beta_{6} - \beta_{12} ) q^{22} + ( 2 \beta_{6} - \beta_{12} - \beta_{13} ) q^{23} - q^{24} -\beta_{1} q^{26} -\beta_{6} q^{27} + ( \beta_{6} + \beta_{9} ) q^{28} + ( -1 - \beta_{2} - \beta_{7} - \beta_{10} ) q^{29} + ( 3 - \beta_{2} + \beta_{4} + \beta_{8} ) q^{31} -\beta_{6} q^{32} + ( \beta_{5} + \beta_{6} + \beta_{12} ) q^{33} + ( -1 - \beta_{2} + \beta_{7} + \beta_{10} ) q^{34} + q^{36} + ( \beta_{5} + \beta_{6} + \beta_{12} + \beta_{15} ) q^{37} + ( -\beta_{5} - \beta_{9} + \beta_{11} + \beta_{14} ) q^{38} + \beta_{1} q^{39} + ( 2 + \beta_{1} - 3 \beta_{2} + \beta_{3} - \beta_{7} ) q^{41} + ( -\beta_{6} - \beta_{9} ) q^{42} + ( -\beta_{5} + \beta_{9} - \beta_{11} + \beta_{14} ) q^{43} + ( -1 + \beta_{3} ) q^{44} + ( 2 - \beta_{2} + \beta_{3} + \beta_{7} ) q^{46} + ( \beta_{5} - 2 \beta_{6} - \beta_{9} + \beta_{11} + \beta_{12} - \beta_{13} + \beta_{14} + 2 \beta_{15} ) q^{47} + \beta_{6} q^{48} + ( -2 - 3 \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{10} ) q^{49} + ( 1 + \beta_{2} - \beta_{7} - \beta_{10} ) q^{51} + ( -\beta_{5} - \beta_{6} + \beta_{14} ) q^{52} + ( \beta_{5} + 2 \beta_{6} - \beta_{13} + \beta_{14} - \beta_{15} ) q^{53} - q^{54} -\beta_{8} q^{56} + ( \beta_{5} + \beta_{9} - \beta_{11} - \beta_{14} ) q^{57} + ( -\beta_{5} + \beta_{6} - \beta_{13} - \beta_{15} ) q^{58} + ( 1 + \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{8} ) q^{59} + ( 4 - 3 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{7} + \beta_{8} + 2 \beta_{10} ) q^{61} + ( -\beta_{5} - 2 \beta_{6} + \beta_{9} + \beta_{11} ) q^{62} + ( \beta_{6} + \beta_{9} ) q^{63} - q^{64} + ( 1 - \beta_{3} ) q^{66} + ( \beta_{5} + 2 \beta_{6} + \beta_{9} - \beta_{11} - \beta_{12} + \beta_{13} - \beta_{14} - 2 \beta_{15} ) q^{67} + ( -\beta_{5} + \beta_{6} + \beta_{13} + \beta_{15} ) q^{68} + ( -2 + \beta_{2} - \beta_{3} - \beta_{7} ) q^{69} + ( 2 - \beta_{1} - 2 \beta_{2} - \beta_{4} + \beta_{8} ) q^{71} -\beta_{6} q^{72} + ( 2 \beta_{5} - \beta_{6} - \beta_{9} - \beta_{11} + \beta_{12} + \beta_{13} + 2 \beta_{15} ) q^{73} + ( 1 - \beta_{3} - \beta_{10} ) q^{74} + ( 2 + \beta_{1} - \beta_{4} + \beta_{8} ) q^{76} + ( -\beta_{6} - 2 \beta_{11} - \beta_{12} + 2 \beta_{13} - \beta_{14} - \beta_{15} ) q^{77} + ( \beta_{5} + \beta_{6} - \beta_{14} ) q^{78} + ( 1 - 3 \beta_{2} - \beta_{3} + 2 \beta_{4} + 2 \beta_{7} - \beta_{10} ) q^{79} + q^{81} + ( -\beta_{5} - \beta_{6} + \beta_{12} - \beta_{13} - \beta_{14} ) q^{82} + ( 2 \beta_{5} + 4 \beta_{6} + 2 \beta_{11} - \beta_{12} + \beta_{13} - \beta_{14} ) q^{83} + \beta_{8} q^{84} + ( \beta_{1} + \beta_{4} - \beta_{8} ) q^{86} + ( \beta_{5} - \beta_{6} + \beta_{13} + \beta_{15} ) q^{87} + ( \beta_{5} + \beta_{6} + \beta_{12} ) q^{88} + ( -1 - \beta_{1} - 5 \beta_{2} + \beta_{4} - \beta_{7} - \beta_{8} + \beta_{10} ) q^{89} + ( 2 - \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} + \beta_{7} + \beta_{8} - 2 \beta_{10} ) q^{91} + ( -2 \beta_{6} + \beta_{12} + \beta_{13} ) q^{92} + ( \beta_{5} + 2 \beta_{6} - \beta_{9} - \beta_{11} ) q^{93} + ( \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + \beta_{7} + \beta_{8} - 2 \beta_{10} ) q^{94} + q^{96} + ( 2 \beta_{5} + 4 \beta_{6} + 2 \beta_{9} + \beta_{11} + \beta_{13} - \beta_{14} + \beta_{15} ) q^{97} + ( -2 \beta_{5} + 2 \beta_{6} - 2 \beta_{11} + \beta_{12} + \beta_{15} ) q^{98} + ( -1 + \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q - 16q^{4} + 16q^{6} - 16q^{9} + O(q^{10})$$ $$16q - 16q^{4} + 16q^{6} - 16q^{9} + 12q^{11} - 8q^{14} + 16q^{16} - 20q^{19} + 8q^{21} - 16q^{24} + 4q^{26} - 20q^{29} + 32q^{31} - 28q^{34} + 16q^{36} - 4q^{39} + 12q^{41} - 12q^{44} + 24q^{46} - 52q^{49} + 28q^{51} - 16q^{54} + 8q^{56} + 32q^{61} - 16q^{64} + 12q^{66} - 24q^{69} + 12q^{71} + 12q^{74} + 20q^{76} - 20q^{79} + 16q^{81} - 8q^{84} + 4q^{86} - 40q^{89} + 12q^{91} - 28q^{94} + 16q^{96} - 12q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 4 x^{15} - 24 x^{14} + 94 x^{13} + 262 x^{12} - 936 x^{11} - 1584 x^{10} + 4642 x^{9} + 6259 x^{8} - 11958 x^{7} - 15752 x^{6} + 14670 x^{5} + 18271 x^{4} - 10440 x^{3} + 1135 x^{2} + 21080 x + 11105$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-71443224939678170518 \nu^{15} - 923271860773063454212 \nu^{14} + 9392788625402818804922 \nu^{13} + 8383442032354171932849 \nu^{12} - 185095569687696507810996 \nu^{11} + 48631544590329079835776 \nu^{10} + 1655831298365862974566526 \nu^{9} - 1072044683108614555599200 \nu^{8} - 7511641413590227304907674 \nu^{7} + 4211625964567022275247617 \nu^{6} + 20038056733049629292467606 \nu^{5} - 4119168852298416907387809 \nu^{4} - 33453041494454969075756730 \nu^{3} + 904308497614552990462035 \nu^{2} + 30907254018073522990947460 \nu - 20348199221590624570631345$$$$)/$$$$60\!\cdots\!75$$ $$\beta_{2}$$ $$=$$ $$($$$$706683448390084 \nu^{15} - 3354965339495132 \nu^{14} - 13194180479363634 \nu^{13} + 74889870470891678 \nu^{12} + 85306799673900684 \nu^{11} - 693707483816595866 \nu^{10} + 24384455336467830 \nu^{9} + 3015117319276978331 \nu^{8} - 2335261265142210782 \nu^{7} - 6116405045925543576 \nu^{6} + 10128644433869963270 \nu^{5} + 4821033472311003142 \nu^{4} - 19455897245014461460 \nu^{3} - 3153321648753581485 \nu^{2} + 18097292700625403660 \nu + 1129571623674353675$$$$)/ 13815460204518569705$$ $$\beta_{3}$$ $$=$$ $$($$$$717569077780173820792 \nu^{15} - 4874519847612594105018 \nu^{14} - 9586903742416245658491 \nu^{13} + 116713392300509885717611 \nu^{12} + 6231595384378807145134 \nu^{11} - 1198144781809550072716533 \nu^{10} + 675850487593753514988145 \nu^{9} + 6169941324711523742519383 \nu^{8} - 4348067028873262519467363 \nu^{7} - 16785412348276723096568282 \nu^{6} + 9629468662841253199611846 \nu^{5} + 21239545131099824671329407 \nu^{4} - 6462793254498394167658110 \nu^{3} - 945898636533936270454040 \nu^{2} + 17876488134243458274418190 \nu - 16049665632396767711497440$$$$)/$$$$60\!\cdots\!75$$ $$\beta_{4}$$ $$=$$ $$($$$$104598747224765426 \nu^{15} - 492665369409056577 \nu^{14} - 1860609118043167572 \nu^{13} + 9933253670715682559 \nu^{12} + 13887839469746901817 \nu^{11} - 82481752825489061671 \nu^{10} - 45801221197687274008 \nu^{9} + 310048599187705680258 \nu^{8} + 153197059671127395164 \nu^{7} - 644079276196991518248 \nu^{6} - 490690606807926089402 \nu^{5} + 923340028190882985219 \nu^{4} + 505928391316692432855 \nu^{3} - 397177517625382665270 \nu^{2} - 138919009303557699665 \nu - 1862833702346511294005$$$$)/$$$$75\!\cdots\!75$$ $$\beta_{5}$$ $$=$$ $$($$$$380909030352847570724 \nu^{15} - 1737737565278933043200 \nu^{14} - 7730511956441966222351 \nu^{13} + 38316386095831524609617 \nu^{12} + 68978644983538847637906 \nu^{11} - 354738939816347691049970 \nu^{10} - 316359472980496769983731 \nu^{9} + 1569858428636016140385574 \nu^{8} + 1076089961330617224707707 \nu^{7} - 3506552974520914457827594 \nu^{6} - 2667496411469218513470024 \nu^{5} + 3554604937261727428258169 \nu^{4} + 2381398784429542207211400 \nu^{3} - 2089345992411439950233335 \nu^{2} + 3640725215351391999365100 \nu + 3664599585230566081028845$$$$)/$$$$27\!\cdots\!25$$ $$\beta_{6}$$ $$=$$ $$($$$$68829028972 \nu^{15} - 306672362945 \nu^{14} - 1464283156713 \nu^{13} + 7056688587646 \nu^{12} + 13604765509703 \nu^{11} - 69712193340555 \nu^{10} - 63679038351663 \nu^{9} + 349100940907062 \nu^{8} + 193696179085541 \nu^{7} - 968003603010122 \nu^{6} - 388026793715242 \nu^{5} + 1452724619642432 \nu^{4} + 243499865358825 \nu^{3} - 1209013296034805 \nu^{2} + 675632284158075 \nu + 945642292769660$$$$)/ 376857420945125$$ $$\beta_{7}$$ $$=$$ $$($$$$-1506841689306776970310 \nu^{15} + 9923165688445033798334 \nu^{14} + 16523414558856420833352 \nu^{13} - 211230149284875582045921 \nu^{12} + 30716820591105334361199 \nu^{11} + 1901308441795617258224915 \nu^{10} - 1480238771322629245276681 \nu^{9} - 8161780757123643532833575 \nu^{8} + 7375697805364392645593991 \nu^{7} + 19162674608654167193647117 \nu^{6} - 15231127451943037345380373 \nu^{5} - 24486541163167139217252887 \nu^{4} + 23167859334627856261678660 \nu^{3} + 5677896411772453374245055 \nu^{2} - 33432778368678311797757600 \nu + 1885090202319636984882240$$$$)/$$$$60\!\cdots\!75$$ $$\beta_{8}$$ $$=$$ $$($$$$1677609729609416931289 \nu^{15} - 8976624381598775387090 \nu^{14} - 27413029760740141850238 \nu^{13} + 194442054089054007646410 \nu^{12} + 151801376480970183840401 \nu^{11} - 1773797283242272695247716 \nu^{10} + 80194036246722052413897 \nu^{9} + 7770429377534851585595526 \nu^{8} - 2273792053739398877683559 \nu^{7} - 18182700173176660676936270 \nu^{6} + 5369793364478682336651421 \nu^{5} + 22259616512175786962587711 \nu^{4} - 8793329102549725875980440 \nu^{3} - 6160235615482583683328425 \nu^{2} + 17298885903098770034453385 \nu - 8000861342114749872879920$$$$)/$$$$60\!\cdots\!75$$ $$\beta_{9}$$ $$=$$ $$($$$$9039363350104095939014 \nu^{15} - 43232715776828018912515 \nu^{14} - 197526934913925264641126 \nu^{13} + 1051689330971313442779847 \nu^{12} + 1850059560756066950687916 \nu^{11} - 10786917406257666666276315 \nu^{10} - 8659223526511422378494651 \nu^{9} + 55309012061999060928872804 \nu^{8} + 25359490645632832188615307 \nu^{7} - 149198416219369764637447304 \nu^{6} - 58825705597670582663354689 \nu^{5} + 218385751755265766620723969 \nu^{4} + 48691831363329111326127175 \nu^{3} - 188879121910772381727431585 \nu^{2} + 109026697792083345345052525 \nu + 152394368917183550021745720$$$$)/$$$$30\!\cdots\!75$$ $$\beta_{10}$$ $$=$$ $$($$$$262837931740539278 \nu^{15} - 1471708972889622906 \nu^{14} - 4022774968630009901 \nu^{13} + 31863082084194063237 \nu^{12} + 17249871320965632266 \nu^{11} - 289468012713592247678 \nu^{10} + 80770827777351194341 \nu^{9} + 1246646574051873818054 \nu^{8} - 732992288018196793613 \nu^{7} - 2766600939541982046314 \nu^{6} + 2091724606383707231214 \nu^{5} + 3031581271759148501187 \nu^{4} - 3978851942084407266110 \nu^{3} - 1163746031625484625835 \nu^{2} + 4675679905232544341130 \nu - 447549436734811744865$$$$)/$$$$75\!\cdots\!75$$ $$\beta_{11}$$ $$=$$ $$($$$$501941538759325994712 \nu^{15} - 2088412231138308028260 \nu^{14} - 11828405339166718685238 \nu^{13} + 50850928287909939593456 \nu^{12} + 120897772419370389769368 \nu^{11} - 527111673957443559232120 \nu^{10} - 631583932031115211486498 \nu^{9} + 2784852874440114012858887 \nu^{8} + 1928857924714297291906126 \nu^{7} - 7959134669401921787716462 \nu^{6} - 3552183046346112408601062 \nu^{5} + 12532296972510722991960072 \nu^{4} + 1914151714904621727009420 \nu^{3} - 11434379893596687151865705 \nu^{2} + 4718581529610596287129900 \nu + 7783464929015343353535060$$$$)/$$$$12\!\cdots\!55$$ $$\beta_{12}$$ $$=$$ $$($$$$15075418459224681577203 \nu^{15} - 86035517040638338664420 \nu^{14} - 265511826832782017307822 \nu^{13} + 1997925310317870349538979 \nu^{12} + 1862094992167729727363717 \nu^{11} - 19950584801110165660287705 \nu^{10} - 4770933594886710002343567 \nu^{9} + 100822351069867859739573908 \nu^{8} + 9292020841053148230654929 \nu^{7} - 285510896921793247569734303 \nu^{6} - 38566601916112557615724873 \nu^{5} + 439264340213468028608236508 \nu^{4} + 38717568057698509568075750 \nu^{3} - 351917145919080999224531220 \nu^{2} + 214685840647872312426519425 \nu + 279650423429452050494295290$$$$)/$$$$30\!\cdots\!75$$ $$\beta_{13}$$ $$=$$ $$($$$$-15091787688528783015097 \nu^{15} + 67232453476180789061320 \nu^{14} + 349183178510023414575253 \nu^{13} - 1662327428424845927855991 \nu^{12} - 3485679352729663438421498 \nu^{11} + 17367374970051330688209155 \nu^{10} + 17886687853594229562863553 \nu^{9} - 91524508438678035085215587 \nu^{8} - 56801931403165081939929121 \nu^{7} + 253004736262063994724385362 \nu^{6} + 130320088422309089747494032 \nu^{5} - 361321957180100080878535982 \nu^{4} - 107356832393191733910813350 \nu^{3} + 264295265339677283262902080 \nu^{2} - 223374887812387772858007125 \nu - 268284921275712657490842035$$$$)/$$$$30\!\cdots\!75$$ $$\beta_{14}$$ $$=$$ $$($$$$-21515446106576623313867 \nu^{15} + 94828057362042220685660 \nu^{14} + 494439571148907072495873 \nu^{13} - 2285507927885497005202126 \nu^{12} - 5054302950849579367802458 \nu^{11} + 23471337850104055158803270 \nu^{10} + 27454404387043062976172943 \nu^{9} - 121866888964253728673825237 \nu^{8} - 94519468573524391127323461 \nu^{7} + 337041099433033041679352207 \nu^{6} + 205453261129762703468671412 \nu^{5} - 466937748220475986043734677 \nu^{4} - 144379839361317734247816300 \nu^{3} + 312301791376409423172813805 \nu^{2} - 305898325250903582089420575 \nu - 348486749456934790818460135$$$$)/$$$$30\!\cdots\!75$$ $$\beta_{15}$$ $$=$$ $$($$$$917229836170336049496 \nu^{15} - 3908542487206019642580 \nu^{14} - 20395688475714312122134 \nu^{13} + 91035705056649979482108 \nu^{12} + 197069501287986475459424 \nu^{11} - 897624501639968543469220 \nu^{10} - 971903120570677549680714 \nu^{9} + 4402281300713236969201141 \nu^{8} + 3062745611743453395511118 \nu^{7} - 11324987619190466231052966 \nu^{6} - 6756168897529186787459606 \nu^{5} + 15131016578776113359984546 \nu^{4} + 5455768682709072100604660 \nu^{3} - 11689286495545715930451665 \nu^{2} + 9564975893675672094856100 \nu + 11796582294690472088970205$$$$)/$$$$12\!\cdots\!55$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{10} + 2 \beta_{8} + 4 \beta_{7} + 5 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} - 10 \beta_{2} + 4 \beta_{1} + 10$$$$)/10$$ $$\nu^{2}$$ $$=$$ $$($$$$-2 \beta_{15} - 2 \beta_{14} + 2 \beta_{13} - 4 \beta_{12} - \beta_{11} + 4 \beta_{10} + 6 \beta_{9} - 4 \beta_{8} + 2 \beta_{7} + 8 \beta_{6} - 2 \beta_{5} + 6 \beta_{4} + 6 \beta_{3} - 15 \beta_{2} + 2 \beta_{1} + 45$$$$)/10$$ $$\nu^{3}$$ $$=$$ $$($$$$-12 \beta_{15} + 6 \beta_{14} - 6 \beta_{13} - 3 \beta_{12} + 12 \beta_{11} + 25 \beta_{10} - 3 \beta_{9} + 3 \beta_{8} + 31 \beta_{7} + 11 \beta_{6} + 56 \beta_{5} - 6 \beta_{4} + 18 \beta_{3} - 95 \beta_{2} + 16 \beta_{1} + 100$$$$)/10$$ $$\nu^{4}$$ $$=$$ $$($$$$-23 \beta_{15} - 4 \beta_{14} + 14 \beta_{13} - 38 \beta_{12} - 30 \beta_{11} + 74 \beta_{10} + 82 \beta_{9} - 56 \beta_{8} + 48 \beta_{7} + 166 \beta_{6} - 4 \beta_{5} + 40 \beta_{4} + 54 \beta_{3} - 215 \beta_{2} + 28 \beta_{1} + 330$$$$)/10$$ $$\nu^{5}$$ $$=$$ $$($$$$-200 \beta_{15} + 80 \beta_{14} - 155 \beta_{13} - 65 \beta_{12} + 105 \beta_{11} + 298 \beta_{10} - 40 \beta_{9} - 128 \beta_{8} + 239 \beta_{7} + 315 \beta_{6} + 605 \beta_{5} + 47 \beta_{4} + 167 \beta_{3} - 830 \beta_{2} + 104 \beta_{1} + 935$$$$)/10$$ $$\nu^{6}$$ $$=$$ $$($$$$-449 \beta_{15} + 63 \beta_{14} - 133 \beta_{13} - 424 \beta_{12} - 445 \beta_{11} + 999 \beta_{10} + 796 \beta_{9} - 464 \beta_{8} + 732 \beta_{7} + 2438 \beta_{6} + 348 \beta_{5} + 226 \beta_{4} + 491 \beta_{3} - 2675 \beta_{2} + 307 \beta_{1} + 2970$$$$)/10$$ $$\nu^{7}$$ $$=$$ $$($$$$-2702 \beta_{15} + 1001 \beta_{14} - 2331 \beta_{13} - 1008 \beta_{12} + 322 \beta_{11} + 3079 \beta_{10} + 77 \beta_{9} - 2222 \beta_{8} + 2051 \beta_{7} + 6151 \beta_{6} + 6326 \beta_{5} + 1312 \beta_{4} + 1668 \beta_{3} - 7115 \beta_{2} + 756 \beta_{1} + 9200$$$$)/10$$ $$\nu^{8}$$ $$=$$ $$($$$$-7934 \beta_{15} + 1906 \beta_{14} - 5116 \beta_{13} - 4938 \beta_{12} - 4907 \beta_{11} + 11192 \beta_{10} + 7242 \beta_{9} - 3196 \beta_{8} + 9008 \beta_{7} + 32126 \beta_{6} + 10096 \beta_{5} + 1296 \beta_{4} + 4444 \beta_{3} - 29745 \beta_{2} + 3418 \beta_{1} + 29065$$$$)/10$$ $$\nu^{9}$$ $$=$$ $$($$$$-35562 \beta_{15} + 11550 \beta_{14} - 29820 \beta_{13} - 14895 \beta_{12} - 6144 \beta_{11} + 30096 \beta_{10} + 10125 \beta_{9} - 25603 \beta_{8} + 19099 \beta_{7} + 99900 \beta_{6} + 67310 \beta_{5} + 17473 \beta_{4} + 16392 \beta_{3} - 61655 \beta_{2} + 6224 \beta_{1} + 91380$$$$)/10$$ $$\nu^{10}$$ $$=$$ $$($$$$-61058 \beta_{15} + 16844 \beta_{14} - 46679 \beta_{13} - 29862 \beta_{12} - 25002 \beta_{11} + 55702 \beta_{10} + 34533 \beta_{9} - 11327 \beta_{8} + 47326 \beta_{7} + 203749 \beta_{6} + 94224 \beta_{5} + 4988 \beta_{4} + 20078 \beta_{3} - 147545 \beta_{2} + 17486 \beta_{1} + 140950$$$$)/5$$ $$\nu^{11}$$ $$=$$ $$($$$$-462792 \beta_{15} + 132011 \beta_{14} - 366696 \beta_{13} - 208208 \beta_{12} - 156398 \beta_{11} + 279413 \beta_{10} + 214962 \beta_{9} - 240967 \beta_{8} + 181121 \beta_{7} + 1438976 \beta_{6} + 759501 \beta_{5} + 176350 \beta_{4} + 150483 \beta_{3} - 546935 \beta_{2} + 56386 \beta_{1} + 861390$$$$)/10$$ $$\nu^{12}$$ $$=$$ $$($$$$-1700180 \beta_{15} + 487561 \beta_{14} - 1364491 \beta_{13} - 739638 \beta_{12} - 531674 \beta_{11} + 989645 \beta_{10} + 735982 \beta_{9} - 195266 \beta_{8} + 855678 \beta_{7} + 5120436 \beta_{6} + 2836466 \beta_{5} + 110402 \beta_{4} + 351229 \beta_{3} - 2582185 \beta_{2} + 311803 \beta_{1} + 2534505$$$$)/10$$ $$\nu^{13}$$ $$=$$ $$($$$$-5921409 \beta_{15} + 1561560 \beta_{14} - 4520750 \beta_{13} - 2739425 \beta_{12} - 2409953 \beta_{11} + 2365084 \beta_{10} + 3257150 \beta_{9} - 1887423 \beta_{8} + 1609054 \beta_{7} + 19093745 \beta_{6} + 9088100 \beta_{5} + 1427499 \beta_{4} + 1226547 \beta_{3} - 4690855 \beta_{2} + 504369 \beta_{1} + 7256500$$$$)/10$$ $$\nu^{14}$$ $$=$$ $$($$$$-22036443 \beta_{15} + 6329388 \beta_{14} - 17845173 \beta_{13} - 9191079 \beta_{12} - 6219728 \beta_{11} + 7413214 \beta_{10} + 8744071 \beta_{9} - 1828144 \beta_{8} + 6349102 \beta_{7} + 63873123 \beta_{6} + 37594658 \beta_{5} + 1230656 \beta_{4} + 2721726 \beta_{3} - 18798330 \beta_{2} + 2277567 \beta_{1} + 19468025$$$$)/10$$ $$\nu^{15}$$ $$=$$ $$($$$$-74060127 \beta_{15} + 19028436 \beta_{14} - 55930176 \beta_{13} - 34220578 \beta_{12} - 30999838 \beta_{11} + 16040147 \beta_{10} + 42033302 \beta_{9} - 10984222 \beta_{8} + 11582736 \beta_{7} + 239009936 \beta_{6} + 112153531 \beta_{5} + 8425368 \beta_{4} + 7798358 \beta_{3} - 33326045 \beta_{2} + 3732696 \beta_{1} + 48033630$$$$)/10$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3750\mathbb{Z}\right)^\times$$.

 $$n$$ $$2501$$ $$3127$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1249.1
 −2.79002 − 0.809017i −0.705457 + 0.309017i 0.543374 − 0.809017i 3.42137 + 0.309017i 2.32349 + 0.309017i −1.16141 − 0.809017i −1.80334 + 0.309017i 2.17199 − 0.809017i 2.17199 + 0.809017i −1.80334 − 0.309017i −1.16141 + 0.809017i 2.32349 − 0.309017i 3.42137 − 0.309017i 0.543374 + 0.809017i −0.705457 − 0.309017i −2.79002 + 0.809017i
1.00000i 1.00000i −1.00000 0 1.00000 4.63137i 1.00000i −1.00000 0
1249.2 1.00000i 1.00000i −1.00000 0 1.00000 3.23143i 1.00000i −1.00000 0
1249.3 1.00000i 1.00000i −1.00000 0 1.00000 2.70913i 1.00000i −1.00000 0
1249.4 1.00000i 1.00000i −1.00000 0 1.00000 2.61995i 1.00000i −1.00000 0
1249.5 1.00000i 1.00000i −1.00000 0 1.00000 0.329315i 1.00000i −1.00000 0
1249.6 1.00000i 1.00000i −1.00000 0 1.00000 0.533559i 1.00000i −1.00000 0
1249.7 1.00000i 1.00000i −1.00000 0 1.00000 3.52206i 1.00000i −1.00000 0
1249.8 1.00000i 1.00000i −1.00000 0 1.00000 4.80694i 1.00000i −1.00000 0
1249.9 1.00000i 1.00000i −1.00000 0 1.00000 4.80694i 1.00000i −1.00000 0
1249.10 1.00000i 1.00000i −1.00000 0 1.00000 3.52206i 1.00000i −1.00000 0
1249.11 1.00000i 1.00000i −1.00000 0 1.00000 0.533559i 1.00000i −1.00000 0
1249.12 1.00000i 1.00000i −1.00000 0 1.00000 0.329315i 1.00000i −1.00000 0
1249.13 1.00000i 1.00000i −1.00000 0 1.00000 2.61995i 1.00000i −1.00000 0
1249.14 1.00000i 1.00000i −1.00000 0 1.00000 2.70913i 1.00000i −1.00000 0
1249.15 1.00000i 1.00000i −1.00000 0 1.00000 3.23143i 1.00000i −1.00000 0
1249.16 1.00000i 1.00000i −1.00000 0 1.00000 4.63137i 1.00000i −1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1249.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3750.2.c.k 16
5.b even 2 1 inner 3750.2.c.k 16
5.c odd 4 1 3750.2.a.u 8
5.c odd 4 1 3750.2.a.v 8
25.d even 5 1 150.2.h.b 16
25.d even 5 1 750.2.h.d 16
25.e even 10 1 150.2.h.b 16
25.e even 10 1 750.2.h.d 16
25.f odd 20 2 750.2.g.f 16
25.f odd 20 2 750.2.g.g 16
75.h odd 10 1 450.2.l.c 16
75.j odd 10 1 450.2.l.c 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
150.2.h.b 16 25.d even 5 1
150.2.h.b 16 25.e even 10 1
450.2.l.c 16 75.h odd 10 1
450.2.l.c 16 75.j odd 10 1
750.2.g.f 16 25.f odd 20 2
750.2.g.g 16 25.f odd 20 2
750.2.h.d 16 25.d even 5 1
750.2.h.d 16 25.e even 10 1
3750.2.a.u 8 5.c odd 4 1
3750.2.a.v 8 5.c odd 4 1
3750.2.c.k 16 1.a even 1 1 trivial
3750.2.c.k 16 5.b even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{16} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(3750, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} )^{8}$$
$3$ $$( 1 + T^{2} )^{8}$$
$5$ 
$7$ $$1 - 30 T^{2} + 527 T^{4} - 6940 T^{6} + 74703 T^{8} - 716280 T^{10} + 6275029 T^{12} - 50516750 T^{14} + 372187880 T^{16} - 2475320750 T^{18} + 15066344629 T^{20} - 84269625720 T^{22} + 430647929103 T^{24} - 1960378228060 T^{26} + 7294358354927 T^{28} - 20346692185470 T^{30} + 33232930569601 T^{32}$$
$11$ $$( 1 - 6 T + 45 T^{2} - 240 T^{3} + 1205 T^{4} - 4658 T^{5} + 20073 T^{6} - 68300 T^{7} + 239780 T^{8} - 751300 T^{9} + 2428833 T^{10} - 6199798 T^{11} + 17642405 T^{12} - 38652240 T^{13} + 79720245 T^{14} - 116923026 T^{15} + 214358881 T^{16} )^{2}$$
$13$ $$1 - 90 T^{2} + 3887 T^{4} - 110780 T^{6} + 2431583 T^{8} - 45139640 T^{10} + 743871389 T^{12} - 11084458050 T^{14} + 150617413480 T^{16} - 1873273410450 T^{18} + 21245710741229 T^{20} - 217880420608760 T^{22} + 1983516953761343 T^{24} - 15271963727032220 T^{26} + 90559656871083647 T^{28} - 354363874712936010 T^{30} + 665416609183179841 T^{32}$$
$17$ $$1 - 110 T^{2} + 5647 T^{4} - 191000 T^{6} + 5211463 T^{8} - 127367860 T^{10} + 2784558349 T^{12} - 53708825350 T^{14} + 943711137080 T^{16} - 15521850526150 T^{18} + 232569097866829 T^{20} - 3074350509132340 T^{22} + 36353901800746183 T^{24} - 385054834985759000 T^{26} + 3290067773636460367 T^{28} - 18521560921534102190 T^{30} + 48661191875666868481 T^{32}$$
$19$ $$( 1 + 10 T + 102 T^{2} + 670 T^{3} + 4268 T^{4} + 22710 T^{5} + 117514 T^{6} + 560450 T^{7} + 2521030 T^{8} + 10648550 T^{9} + 42422554 T^{10} + 155767890 T^{11} + 556210028 T^{12} + 1658986330 T^{13} + 4798679862 T^{14} + 8938717390 T^{15} + 16983563041 T^{16} )^{2}$$
$23$ $$1 - 120 T^{2} + 8272 T^{4} - 405160 T^{6} + 15482268 T^{8} - 487076920 T^{10} + 13245805744 T^{12} - 326418457000 T^{14} + 7636031189830 T^{16} - 172675363753000 T^{18} + 3706719525206704 T^{20} - 72104864863581880 T^{22} + 1212431661464497308 T^{24} - 16784365283322028840 T^{26} +$$$$18\!\cdots\!12$$$$T^{28} -$$$$13\!\cdots\!80$$$$T^{30} +$$$$61\!\cdots\!61$$$$T^{32}$$
$29$ $$( 1 + 10 T + 197 T^{2} + 1510 T^{3} + 17343 T^{4} + 107610 T^{5} + 913319 T^{6} + 4694150 T^{7} + 32002680 T^{8} + 136130350 T^{9} + 768101279 T^{10} + 2624500290 T^{11} + 12266374383 T^{12} + 30971834990 T^{13} + 117180194237 T^{14} + 172498763090 T^{15} + 500246412961 T^{16} )^{2}$$
$31$ $$( 1 - 16 T + 245 T^{2} - 2700 T^{3} + 25605 T^{4} - 210758 T^{5} + 1537653 T^{6} - 9946750 T^{7} + 58857900 T^{8} - 308349250 T^{9} + 1477684533 T^{10} - 6278691578 T^{11} + 23646755205 T^{12} - 77298707700 T^{13} + 217438401845 T^{14} - 440201825776 T^{15} + 852891037441 T^{16} )^{2}$$
$37$ $$1 - 450 T^{2} + 96247 T^{4} - 13039740 T^{6} + 1259916223 T^{8} - 92804354600 T^{10} + 5445063403189 T^{12} - 262329903935050 T^{14} + 10575876674386280 T^{16} - 359129638487083450 T^{18} + 10204925472784099429 T^{20} -$$$$23\!\cdots\!00$$$$T^{22} +$$$$44\!\cdots\!83$$$$T^{24} -$$$$62\!\cdots\!60$$$$T^{26} +$$$$63\!\cdots\!07$$$$T^{28} -$$$$40\!\cdots\!50$$$$T^{30} +$$$$12\!\cdots\!41$$$$T^{32}$$
$41$ $$( 1 - 6 T + 145 T^{2} - 850 T^{3} + 10255 T^{4} - 69718 T^{5} + 526063 T^{6} - 4134450 T^{7} + 23274800 T^{8} - 169512450 T^{9} + 884311903 T^{10} - 4805034278 T^{11} + 28978179055 T^{12} - 98477770850 T^{13} + 688765114945 T^{14} - 1168525643286 T^{15} + 7984925229121 T^{16} )^{2}$$
$43$ $$1 - 360 T^{2} + 63972 T^{4} - 7536360 T^{6} + 666842308 T^{8} - 47508203160 T^{10} + 2846509586204 T^{12} - 147613243096600 T^{14} + 6742598283189430 T^{16} - 272936886485613400 T^{18} + 9731649819823821404 T^{20} -$$$$30\!\cdots\!40$$$$T^{22} +$$$$77\!\cdots\!08$$$$T^{24} -$$$$16\!\cdots\!40$$$$T^{26} +$$$$25\!\cdots\!72$$$$T^{28} -$$$$26\!\cdots\!40$$$$T^{30} +$$$$13\!\cdots\!01$$$$T^{32}$$
$47$ $$1 - 120 T^{2} + 17572 T^{4} - 1601880 T^{6} + 137356228 T^{8} - 9599418920 T^{10} + 613416365084 T^{12} - 33575205819400 T^{14} + 1700876128099830 T^{16} - 74167629655054600 T^{18} + 2993276181789458204 T^{20} -$$$$10\!\cdots\!80$$$$T^{22} +$$$$32\!\cdots\!08$$$$T^{24} -$$$$84\!\cdots\!20$$$$T^{26} +$$$$20\!\cdots\!52$$$$T^{28} -$$$$30\!\cdots\!80$$$$T^{30} +$$$$56\!\cdots\!21$$$$T^{32}$$
$53$ $$1 - 560 T^{2} + 144922 T^{4} - 22765500 T^{6} + 2393957203 T^{8} - 175164644160 T^{10} + 9059081039884 T^{12} - 352699017663600 T^{14} + 14677013509359205 T^{16} - 990731540617052400 T^{18} + 71480506822664944204 T^{20} -$$$$38\!\cdots\!40$$$$T^{22} +$$$$14\!\cdots\!83$$$$T^{24} -$$$$39\!\cdots\!00$$$$T^{26} +$$$$71\!\cdots\!02$$$$T^{28} -$$$$77\!\cdots\!40$$$$T^{30} +$$$$38\!\cdots\!21$$$$T^{32}$$
$59$ $$( 1 + 247 T^{2} - 500 T^{3} + 28643 T^{4} - 123000 T^{5} + 2182549 T^{6} - 13248500 T^{7} + 135310720 T^{8} - 781661500 T^{9} + 7597453069 T^{10} - 25261617000 T^{11} + 347077571123 T^{12} - 357462149500 T^{13} + 10418591809327 T^{14} + 146830437604321 T^{16} )^{2}$$
$61$ $$( 1 - 16 T + 335 T^{2} - 3580 T^{3} + 40995 T^{4} - 303668 T^{5} + 2560793 T^{6} - 14411280 T^{7} + 131714360 T^{8} - 879088080 T^{9} + 9528710753 T^{10} - 68926866308 T^{11} + 567610251795 T^{12} - 3023654757580 T^{13} + 17259325410935 T^{14} - 50283885376336 T^{15} + 191707312997281 T^{16} )^{2}$$
$67$ $$1 - 440 T^{2} + 112372 T^{4} - 20390680 T^{6} + 2900260388 T^{8} - 337708700840 T^{10} + 33102219933324 T^{12} - 2771008225944200 T^{14} + 199830173598131830 T^{16} - 12439055926263513800 T^{18} +$$$$66\!\cdots\!04$$$$T^{20} -$$$$30\!\cdots\!60$$$$T^{22} +$$$$11\!\cdots\!08$$$$T^{24} -$$$$37\!\cdots\!20$$$$T^{26} +$$$$91\!\cdots\!92$$$$T^{28} -$$$$16\!\cdots\!60$$$$T^{30} +$$$$16\!\cdots\!81$$$$T^{32}$$
$71$ $$( 1 - 6 T + 390 T^{2} - 2050 T^{3} + 74900 T^{4} - 339378 T^{5} + 9085578 T^{6} - 35263990 T^{7} + 765200950 T^{8} - 2503743290 T^{9} + 45800398698 T^{10} - 121467119358 T^{11} + 1903334906900 T^{12} - 3698670169550 T^{13} + 49959110729190 T^{14} - 54570720950346 T^{15} + 645753531245761 T^{16} )^{2}$$
$73$ $$1 - 590 T^{2} + 178647 T^{4} - 36798660 T^{6} + 5771348543 T^{8} - 731373593640 T^{10} + 77567862900069 T^{12} - 7031544086002150 T^{14} + 551057873643279880 T^{16} - 37471098434305457350 T^{18} +$$$$22\!\cdots\!29$$$$T^{20} -$$$$11\!\cdots\!60$$$$T^{22} +$$$$46\!\cdots\!83$$$$T^{24} -$$$$15\!\cdots\!40$$$$T^{26} +$$$$40\!\cdots\!87$$$$T^{28} -$$$$72\!\cdots\!10$$$$T^{30} +$$$$65\!\cdots\!61$$$$T^{32}$$
$79$ $$( 1 + 10 T + 227 T^{2} + 1720 T^{3} + 32163 T^{4} + 256360 T^{5} + 3728769 T^{6} + 26100650 T^{7} + 323096880 T^{8} + 2061951350 T^{9} + 23271247329 T^{10} + 126395478040 T^{11} + 1252751455203 T^{12} + 5292537006280 T^{13} + 55180852403267 T^{14} + 192039089861590 T^{15} + 1517108809906561 T^{16} )^{2}$$
$83$ $$1 - 490 T^{2} + 107247 T^{4} - 14373340 T^{6} + 1475841863 T^{8} - 146194063040 T^{10} + 14859792976549 T^{12} - 1411441844814250 T^{14} + 121859257800782680 T^{16} - 9723422868925368250 T^{18} +$$$$70\!\cdots\!29$$$$T^{20} -$$$$47\!\cdots\!60$$$$T^{22} +$$$$33\!\cdots\!83$$$$T^{24} -$$$$22\!\cdots\!60$$$$T^{26} +$$$$11\!\cdots\!67$$$$T^{28} -$$$$36\!\cdots\!10$$$$T^{30} +$$$$50\!\cdots\!81$$$$T^{32}$$
$89$ $$( 1 + 20 T + 537 T^{2} + 8210 T^{3} + 124663 T^{4} + 1568070 T^{5} + 17732339 T^{6} + 192143300 T^{7} + 1812907320 T^{8} + 17100753700 T^{9} + 140457857219 T^{10} + 1105440739830 T^{11} + 7821635989783 T^{12} + 45845128076290 T^{13} + 266878953246057 T^{14} + 884626697910580 T^{15} + 3936588805702081 T^{16} )^{2}$$
$97$ $$1 - 660 T^{2} + 206842 T^{4} - 40899980 T^{6} + 5901330683 T^{8} - 716216832660 T^{10} + 83710235703084 T^{12} - 9634323674140400 T^{14} + 1005189451203013405 T^{16} - 90649351449987023600 T^{18} +$$$$74\!\cdots\!04$$$$T^{20} -$$$$59\!\cdots\!40$$$$T^{22} +$$$$46\!\cdots\!63$$$$T^{24} -$$$$30\!\cdots\!20$$$$T^{26} +$$$$14\!\cdots\!22$$$$T^{28} -$$$$43\!\cdots\!40$$$$T^{30} +$$$$61\!\cdots\!21$$$$T^{32}$$