# Properties

 Label 300.2.h.c Level $300$ Weight $2$ Character orbit 300.h Analytic conductor $2.396$ Analytic rank $0$ Dimension $16$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$300 = 2^{2} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 300.h (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.39551206064$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 2 x^{15} + 2 x^{14} + 10 x^{13} - 42 x^{11} + 134 x^{10} + 110 x^{9} + 92 x^{8} + 142 x^{7} + 1514 x^{6} + 1102 x^{5} + 249 x^{4} - 1056 x^{3} + 392 x^{2} - 280 x + 100$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{6}\cdot 5^{2}$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} -\beta_{10} q^{3} -\beta_{4} q^{4} -\beta_{12} q^{6} + ( -\beta_{2} - \beta_{3} ) q^{7} + ( \beta_{1} + \beta_{9} - \beta_{10} - \beta_{11} ) q^{8} + ( \beta_{4} + \beta_{8} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} -\beta_{10} q^{3} -\beta_{4} q^{4} -\beta_{12} q^{6} + ( -\beta_{2} - \beta_{3} ) q^{7} + ( \beta_{1} + \beta_{9} - \beta_{10} - \beta_{11} ) q^{8} + ( \beta_{4} + \beta_{8} ) q^{9} + ( \beta_{4} - \beta_{5} + \beta_{8} - \beta_{12} + \beta_{13} + \beta_{14} ) q^{11} + ( \beta_{3} + \beta_{11} ) q^{12} + ( -\beta_{2} + \beta_{7} - \beta_{15} ) q^{13} + ( -\beta_{4} + \beta_{5} - \beta_{6} - \beta_{8} - \beta_{14} ) q^{14} + ( -2 \beta_{5} - \beta_{12} - \beta_{13} ) q^{16} + ( -\beta_{1} + \beta_{3} - \beta_{7} - \beta_{11} ) q^{17} + ( -\beta_{1} - \beta_{7} - 2 \beta_{9} + \beta_{11} + \beta_{15} ) q^{18} + ( 1 + \beta_{4} + \beta_{12} + \beta_{13} ) q^{19} + ( 1 + \beta_{4} + \beta_{5} - \beta_{8} + \beta_{12} - \beta_{13} + \beta_{14} ) q^{21} + ( \beta_{3} + \beta_{7} - 2 \beta_{9} - 2 \beta_{10} ) q^{22} + ( \beta_{1} + \beta_{3} - \beta_{7} - \beta_{9} + \beta_{10} + \beta_{11} ) q^{23} + ( -2 - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{13} ) q^{24} + ( -\beta_{4} + \beta_{5} + \beta_{6} - 2 \beta_{8} + \beta_{12} - \beta_{13} ) q^{26} + ( -\beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{7} + 2 \beta_{9} - \beta_{11} ) q^{27} + ( -2 \beta_{2} - \beta_{3} + \beta_{7} + 2 \beta_{9} + 2 \beta_{10} - \beta_{15} ) q^{28} + ( -\beta_{8} + \beta_{12} - \beta_{13} + \beta_{14} ) q^{29} + ( 1 + 2 \beta_{4} - \beta_{5} + \beta_{12} + \beta_{13} ) q^{31} + ( 2 \beta_{1} - \beta_{3} + \beta_{7} + 2 \beta_{9} - 2 \beta_{10} ) q^{32} + ( 2 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{10} + 2 \beta_{15} ) q^{33} + ( -4 + 2 \beta_{4} - 4 \beta_{5} - \beta_{12} - \beta_{13} ) q^{34} + ( -2 + \beta_{5} - \beta_{6} + \beta_{8} - \beta_{12} + 2 \beta_{13} - \beta_{14} ) q^{36} + ( 2 \beta_{2} - 2 \beta_{7} + 2 \beta_{15} ) q^{37} + ( -\beta_{3} + \beta_{7} - \beta_{9} + \beta_{10} - \beta_{11} ) q^{38} + ( 1 + \beta_{5} + \beta_{8} + 2 \beta_{13} + \beta_{14} ) q^{39} + ( \beta_{4} - \beta_{5} + 3 \beta_{8} - \beta_{12} + \beta_{13} - \beta_{14} ) q^{41} + ( -\beta_{1} + \beta_{3} - \beta_{7} - 2 \beta_{9} + 2 \beta_{10} + 2 \beta_{11} - 2 \beta_{15} ) q^{42} + ( \beta_{2} + \beta_{3} + 2 \beta_{9} + 2 \beta_{10} ) q^{43} + ( 2 \beta_{6} - \beta_{12} + \beta_{13} ) q^{44} + ( -4 - 2 \beta_{4} ) q^{46} + ( -2 \beta_{9} + 2 \beta_{10} ) q^{47} + ( -2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 3 \beta_{7} + \beta_{11} + \beta_{15} ) q^{48} + ( 2 - \beta_{4} - \beta_{5} ) q^{49} + ( 1 - 3 \beta_{4} + 4 \beta_{5} + 2 \beta_{6} - 2 \beta_{8} + 3 \beta_{12} - \beta_{13} - \beta_{14} ) q^{51} + ( 2 \beta_{2} + \beta_{3} - \beta_{7} + 2 \beta_{9} + 2 \beta_{10} - \beta_{15} ) q^{52} + ( -3 \beta_{1} + \beta_{3} - \beta_{7} - \beta_{9} + \beta_{10} + \beta_{11} ) q^{53} + ( 4 + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{8} + \beta_{12} - \beta_{13} - \beta_{14} ) q^{54} + ( -2 \beta_{4} + 2 \beta_{5} - 3 \beta_{8} + 3 \beta_{12} - 3 \beta_{13} - \beta_{14} ) q^{56} + ( 4 \beta_{1} - \beta_{3} + \beta_{7} + 2 \beta_{9} - \beta_{10} - 2 \beta_{11} - \beta_{15} ) q^{57} -2 \beta_{15} q^{58} + ( -3 - \beta_{4} - \beta_{5} ) q^{61} + ( -2 \beta_{3} + 2 \beta_{7} - \beta_{9} + \beta_{10} - \beta_{11} ) q^{62} + ( -2 \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{7} - 2 \beta_{9} - 2 \beta_{10} - 2 \beta_{11} ) q^{63} + ( 4 - 2 \beta_{4} + 2 \beta_{5} - \beta_{12} - \beta_{13} ) q^{64} + ( 4 - \beta_{4} + \beta_{5} - \beta_{6} + 3 \beta_{8} - \beta_{12} + 2 \beta_{13} - \beta_{14} ) q^{66} + ( \beta_{2} + \beta_{3} + \beta_{9} + \beta_{10} ) q^{67} + ( -2 \beta_{1} - 3 \beta_{3} + 3 \beta_{7} + 2 \beta_{9} - 2 \beta_{10} ) q^{68} + ( 2 + 2 \beta_{5} - \beta_{8} - \beta_{12} + \beta_{13} - \beta_{14} ) q^{69} + ( -4 \beta_{6} + \beta_{8} - \beta_{12} + \beta_{13} - \beta_{14} ) q^{71} + ( -3 \beta_{1} - 2 \beta_{2} + \beta_{7} - \beta_{9} + \beta_{10} + \beta_{15} ) q^{72} + ( \beta_{9} + \beta_{10} - 2 \beta_{15} ) q^{73} + ( 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + 4 \beta_{8} - 2 \beta_{12} + 2 \beta_{13} ) q^{74} + ( 4 + \beta_{4} + 2 \beta_{12} + 2 \beta_{13} ) q^{76} + ( -7 \beta_{1} + \beta_{3} - \beta_{7} - 3 \beta_{9} + 3 \beta_{10} + 5 \beta_{11} ) q^{77} + ( 2 \beta_{3} - 3 \beta_{9} - \beta_{10} - \beta_{11} ) q^{78} + ( -2 - 2 \beta_{5} - 2 \beta_{12} - 2 \beta_{13} ) q^{79} + ( -4 - \beta_{5} - \beta_{8} + 2 \beta_{12} - 2 \beta_{13} + 2 \beta_{14} ) q^{81} + ( -\beta_{3} - \beta_{7} - 2 \beta_{9} - 2 \beta_{10} + 4 \beta_{15} ) q^{82} + ( \beta_{9} - \beta_{10} ) q^{83} + ( -4 - \beta_{4} + 2 \beta_{5} - 2 \beta_{8} + \beta_{12} + \beta_{13} + 2 \beta_{14} ) q^{84} + ( \beta_{4} - \beta_{5} + \beta_{6} + \beta_{8} + 2 \beta_{12} - 2 \beta_{13} + \beta_{14} ) q^{86} + ( -\beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{7} - \beta_{9} - \beta_{10} - \beta_{11} ) q^{87} + ( 4 \beta_{2} + 3 \beta_{3} - \beta_{7} + 2 \beta_{15} ) q^{88} + ( \beta_{4} - \beta_{5} + 2 \beta_{8} ) q^{89} + ( -1 - 4 \beta_{4} + 3 \beta_{5} - \beta_{12} - \beta_{13} ) q^{91} + ( -2 \beta_{1} + 2 \beta_{9} - 2 \beta_{10} - 2 \beta_{11} ) q^{92} + ( 5 \beta_{1} + \beta_{2} - \beta_{3} + 3 \beta_{9} - \beta_{10} - 3 \beta_{11} - \beta_{15} ) q^{93} + ( 2 \beta_{12} + 2 \beta_{13} ) q^{94} + ( -6 + 3 \beta_{4} - 3 \beta_{5} - \beta_{6} + 3 \beta_{8} - 3 \beta_{12} + \beta_{14} ) q^{96} + ( -3 \beta_{2} + 3 \beta_{7} - 3 \beta_{15} ) q^{97} + ( 4 \beta_{1} - \beta_{3} + \beta_{7} + 2 \beta_{9} - 2 \beta_{10} - 2 \beta_{11} ) q^{98} + ( -3 - 3 \beta_{5} - 2 \beta_{6} - 3 \beta_{12} - 3 \beta_{13} - \beta_{14} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + 4q^{4} + 6q^{6} - 4q^{9} + O(q^{10})$$ $$16q + 4q^{4} + 6q^{6} - 4q^{9} + 20q^{16} + 8q^{21} - 26q^{24} - 44q^{34} - 42q^{36} - 56q^{46} + 40q^{49} + 56q^{54} - 40q^{61} + 76q^{64} + 58q^{66} + 24q^{69} + 36q^{76} - 60q^{81} - 80q^{84} - 24q^{94} - 78q^{96} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 2 x^{15} + 2 x^{14} + 10 x^{13} - 42 x^{11} + 134 x^{10} + 110 x^{9} + 92 x^{8} + 142 x^{7} + 1514 x^{6} + 1102 x^{5} + 249 x^{4} - 1056 x^{3} + 392 x^{2} - 280 x + 100$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$86636170065458 \nu^{15} - 139763456245741 \nu^{14} + 128073458413399 \nu^{13} + 900030238567851 \nu^{12} + 354154580025869 \nu^{11} - 3370500862960355 \nu^{10} + 10233411926272021 \nu^{9} + 13146910202250305 \nu^{8} + 14374316628170093 \nu^{7} + 19033663501635251 \nu^{6} + 137702325798693503 \nu^{5} + 153977461783007987 \nu^{4} + 92126916463550883 \nu^{3} - 45360331854531752 \nu^{2} + 14624860944800330 \nu - 10270163750249780$$$$)/ 5503939667172770$$ $$\beta_{2}$$ $$=$$ $$($$$$-226346343508237 \nu^{15} + 357846443441218 \nu^{14} - 336891024322778 \nu^{13} - 2354611158042062 \nu^{12} - 999218931322158 \nu^{11} + 8648250714000762 \nu^{10} - 26719779381795744 \nu^{9} - 34750790799689754 \nu^{8} - 39308416851880182 \nu^{7} - 54356774154451946 \nu^{6} - 368774423844636464 \nu^{5} - 414320011727745050 \nu^{4} - 273431781394961159 \nu^{3} + 61116823485676360 \nu^{2} - 75746447756853470 \nu + 60325741155284360$$$$)/ 11007879334345540$$ $$\beta_{3}$$ $$=$$ $$($$$$1929125202927577 \nu^{15} - 3174147916287646 \nu^{14} + 2646022240920770 \nu^{13} + 20733265658664390 \nu^{12} + 6180392672162110 \nu^{11} - 78221672347190694 \nu^{10} + 233101021580454792 \nu^{9} + 295038115278944338 \nu^{8} + 260557328298605406 \nu^{7} + 410585059512938258 \nu^{6} + 3044248646088284120 \nu^{5} + 3217262883760568034 \nu^{4} + 1536516552314755219 \nu^{3} - 1237980684525762656 \nu^{2} + 325847979317508630 \nu - 369876607405151400$$$$)/ 55039396671727700$$ $$\beta_{4}$$ $$=$$ $$($$$$-587554941526996 \nu^{15} + 914739690969668 \nu^{14} - 779859476137840 \nu^{13} - 6211635392505545 \nu^{12} - 2730026919251280 \nu^{11} + 23215236455682612 \nu^{10} - 68250290013865436 \nu^{9} - 94564371327666289 \nu^{8} - 97572551570538648 \nu^{7} - 128964224760195004 \nu^{6} - 943555438353894320 \nu^{5} - 1078946380058846257 \nu^{4} - 635129376362693192 \nu^{3} + 312501612828562608 \nu^{2} - 101066527757082320 \nu + 98863904592843600$$$$)/ 13759849167931925$$ $$\beta_{5}$$ $$=$$ $$($$$$604292511642446 \nu^{15} - 975202941656918 \nu^{14} + 895309719750590 \nu^{13} + 6264791576736495 \nu^{12} + 2497248966540730 \nu^{11} - 23614605152547562 \nu^{10} + 71583351140869686 \nu^{9} + 91388557154928014 \nu^{8} + 99889405018821298 \nu^{7} + 132377160938952754 \nu^{6} + 964896046730279970 \nu^{5} + 1056024008424042232 \nu^{4} + 646308845146926792 \nu^{3} - 318125533167662008 \nu^{2} + 102635821893433320 \nu - 149867394391089975$$$$)/ 13759849167931925$$ $$\beta_{6}$$ $$=$$ $$($$$$-3414934690771993 \nu^{15} + 4945550259550694 \nu^{14} - 4268579158221040 \nu^{13} - 35713662523068110 \nu^{12} - 21290293578092740 \nu^{11} + 131746283062649246 \nu^{10} - 380764546333408138 \nu^{9} - 584323894034450722 \nu^{8} - 671493580621630584 \nu^{7} - 799940874318942562 \nu^{6} - 5638045346715899910 \nu^{5} - 6879867909499226186 \nu^{4} - 4741233569077716801 \nu^{3} + 1305000111085330704 \nu^{2} - 532715906449312630 \nu + 802294040382299600$$$$)/ 55039396671727700$$ $$\beta_{7}$$ $$=$$ $$($$$$-917800794270695 \nu^{15} + 1390492292472696 \nu^{14} - 1196110432632616 \nu^{13} - 9637749939945396 \nu^{12} - 4883257079628964 \nu^{11} + 36000575239442204 \nu^{10} - 105028482082195266 \nu^{9} - 151024436856281404 \nu^{8} - 164256391137268568 \nu^{7} - 204659053273097952 \nu^{6} - 1493139717503591462 \nu^{5} - 1742910056939122056 \nu^{4} - 1108003317071291359 \nu^{3} + 448711127739636256 \nu^{2} - 160057729160681462 \nu + 206726499151380400$$$$)/ 11007879334345540$$ $$\beta_{8}$$ $$=$$ $$($$$$-1836336681975359 \nu^{15} + 2670735318417258 \nu^{14} - 2089419487333492 \nu^{13} - 19731023083720230 \nu^{12} - 10581763184612620 \nu^{11} + 72822818065112418 \nu^{10} - 206518644472271726 \nu^{9} - 319600434676759222 \nu^{8} - 326391269925404268 \nu^{7} - 423283232006252254 \nu^{6} - 3002250019579009746 \nu^{5} - 3622419863090516934 \nu^{4} - 2250220449977606159 \nu^{3} + 870435140198756592 \nu^{2} - 159234887709295082 \nu + 380659820612143220$$$$)/ 22015758668691080$$ $$\beta_{9}$$ $$=$$ $$($$$$47453684427081 \nu^{15} - 72993794820788 \nu^{14} + 60782462586860 \nu^{13} + 505465180962810 \nu^{12} + 227389497402240 \nu^{11} - 1887388853932942 \nu^{10} + 5513164517128886 \nu^{9} + 7755777214786054 \nu^{8} + 7809327047151048 \nu^{7} + 10655658220842394 \nu^{6} + 76805382087173250 \nu^{5} + 87459786900661662 \nu^{4} + 52387389441737357 \nu^{3} - 23962092350078038 \nu^{2} + 8460482668395730 \nu - 10194971362983800$$$$)/ 558775600728200$$ $$\beta_{10}$$ $$=$$ $$($$$$-10024607033433 \nu^{15} + 15519853661140 \nu^{14} - 13088749471584 \nu^{13} - 105861785006214 \nu^{12} - 48857457060196 \nu^{11} + 400226279841042 \nu^{10} - 1162254976132106 \nu^{9} - 1631072624298058 \nu^{8} - 1667197395273668 \nu^{7} - 2140329938493230 \nu^{6} - 16220541349019950 \nu^{5} - 18353859634055946 \nu^{4} - 10836885927773041 \nu^{3} + 5689896354744090 \nu^{2} - 1702682149182858 \nu + 2114870927102600$$$$)/ 111755120145640$$ $$\beta_{11}$$ $$=$$ $$($$$$1145254002757577 \nu^{15} - 1803373165163308 \nu^{14} + 1574622921644234 \nu^{13} + 12026613482603676 \nu^{12} + 5206836615561714 \nu^{11} - 45333833282593788 \nu^{10} + 134163056894225800 \nu^{9} + 181014495116556316 \nu^{8} + 189001840183997370 \nu^{7} + 249841383646932752 \nu^{6} + 1843841483182700488 \nu^{5} + 2058940421980664480 \nu^{4} + 1241373562717283671 \nu^{3} - 610626951533084724 \nu^{2} + 197520864760055710 \nu - 251189288889542480$$$$)/ 11007879334345540$$ $$\beta_{12}$$ $$=$$ $$($$$$-6321304344346553 \nu^{15} + 9909062524717134 \nu^{14} - 8459070870113610 \nu^{13} - 66701535508778410 \nu^{12} - 29080394237870590 \nu^{11} + 252175763237261366 \nu^{10} - 738833260679756268 \nu^{9} - 1011840307489741522 \nu^{8} - 1028511137770238574 \nu^{7} - 1356071467211305682 \nu^{6} - 10190050813370057620 \nu^{5} - 11380031740138452666 \nu^{4} - 6633606991105286931 \nu^{3} + 3564490225164033184 \nu^{2} - 1076335218505282650 \nu + 1336006372893832700$$$$)/ 55039396671727700$$ $$\beta_{13}$$ $$=$$ $$($$$$-1415679395091195 \nu^{15} + 2183999754967658 \nu^{14} - 1856217870781262 \nu^{13} - 14982469622939770 \nu^{12} - 6848948913343370 \nu^{11} + 56066037283580610 \nu^{10} - 164382522303202436 \nu^{9} - 229672551904476226 \nu^{8} - 237606705383002618 \nu^{7} - 315602556481814838 \nu^{6} - 2292003923942511788 \nu^{5} - 2610036989376440402 \nu^{4} - 1592179858859192745 \nu^{3} + 722782665301874912 \nu^{2} - 249416674888906222 \nu + 309863276953526780$$$$)/ 11007879334345540$$ $$\beta_{14}$$ $$=$$ $$($$$$4374911179080033 \nu^{15} - 6665594401648694 \nu^{14} + 5592021960918196 \nu^{13} + 46267388989558370 \nu^{12} + 22328159035791500 \nu^{11} - 173135027827583806 \nu^{10} + 502270511683737498 \nu^{9} + 721189421463949570 \nu^{8} + 752626077810255244 \nu^{7} + 963356512535587426 \nu^{6} + 7076860236944160150 \nu^{5} + 8207570554292642210 \nu^{4} + 4976711247918754393 \nu^{3} - 2367835990250177616 \nu^{2} + 545714415521307686 \nu - 901336837178224940$$$$)/ 22015758668691080$$ $$\beta_{15}$$ $$=$$ $$($$$$-5721678918764907 \nu^{15} + 8573258826363541 \nu^{14} - 7065908294090785 \nu^{13} - 60751710958866045 \nu^{12} - 30737718644105105 \nu^{11} + 226166284261509949 \nu^{10} - 651746171580854027 \nu^{9} - 960736540134968983 \nu^{8} - 1003873180019302111 \nu^{7} - 1282153614266501053 \nu^{6} - 9295268841203540705 \nu^{5} - 10949136728509403289 \nu^{4} - 6764496177439740744 \nu^{3} + 2932602411379671446 \nu^{2} - 669809684900877500 \nu + 1194515334992463500$$$$)/ 27519698335863850$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$3 \beta_{15} + 2 \beta_{14} + 2 \beta_{10} + 2 \beta_{9} - 2 \beta_{7} - \beta_{6} - \beta_{5} - 4 \beta_{4} + 2 \beta_{2} - 5 \beta_{1}$$$$)/10$$ $$\nu^{2}$$ $$=$$ $$($$$$7 \beta_{15} + 2 \beta_{14} - 2 \beta_{10} - 2 \beta_{9} - 10 \beta_{8} - 3 \beta_{7} + 4 \beta_{6} + 4 \beta_{5} - 4 \beta_{4} - 5 \beta_{3} - 2 \beta_{2}$$$$)/10$$ $$\nu^{3}$$ $$=$$ $$($$$$14 \beta_{15} + 25 \beta_{14} + 5 \beta_{13} + 35 \beta_{12} - 5 \beta_{11} - 39 \beta_{10} + 31 \beta_{9} - 5 \beta_{8} + 19 \beta_{7} + 20 \beta_{6} + 10 \beta_{5} + 20 \beta_{4} - 35 \beta_{3} - 24 \beta_{2} + 35 \beta_{1} - 20$$$$)/20$$ $$\nu^{4}$$ $$=$$ $$($$$$5 \beta_{13} + 5 \beta_{12} + 4 \beta_{11} - 4 \beta_{10} + 4 \beta_{9} - 4 \beta_{7} - 4 \beta_{5} + 9 \beta_{4} + 4 \beta_{3} + 12 \beta_{1} - 6$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-24 \beta_{15} - 79 \beta_{14} + 25 \beta_{13} + 195 \beta_{12} + 25 \beta_{11} - 241 \beta_{10} - 31 \beta_{9} - 25 \beta_{8} - 19 \beta_{7} - 78 \beta_{6} + 92 \beta_{5} + 108 \beta_{4} + 195 \beta_{3} + 44 \beta_{2} + 55 \beta_{1} + 260$$$$)/20$$ $$\nu^{6}$$ $$=$$ $$($$$$38 \beta_{15} - 50 \beta_{14} - 35 \beta_{13} + 35 \beta_{12} - 48 \beta_{10} - 48 \beta_{9} - 50 \beta_{8} - 12 \beta_{7} - 100 \beta_{6} + 50 \beta_{5} - 50 \beta_{4} + 115 \beta_{3} + 127 \beta_{2}$$$$)/5$$ $$\nu^{7}$$ $$=$$ $$($$$$279 \beta_{15} - 189 \beta_{14} - 485 \beta_{13} - 35 \beta_{12} - 465 \beta_{11} - 319 \beta_{10} - 389 \beta_{9} - 465 \beta_{8} + 79 \beta_{7} - 173 \beta_{6} + 567 \beta_{5} - 402 \beta_{4} + 35 \beta_{3} + 406 \beta_{2} + 260 \beta_{1} - 380$$$$)/10$$ $$\nu^{8}$$ $$=$$ $$($$$$-119 \beta_{13} - 119 \beta_{12} - 296 \beta_{11} - 88 \beta_{10} + 88 \beta_{9} + 248 \beta_{7} + 170 \beta_{5} - 69 \beta_{4} - 248 \beta_{3} + 344 \beta_{1} - 580$$$$)/2$$ $$\nu^{9}$$ $$=$$ $$($$$$-4602 \beta_{15} - 1715 \beta_{14} + 985 \beta_{13} - 4465 \beta_{12} - 4015 \beta_{11} + 1427 \beta_{10} + 3877 \beta_{9} + 4015 \beta_{8} + 2793 \beta_{7} + 1640 \beta_{6} - 170 \beta_{5} + 3100 \beta_{4} - 4465 \beta_{3} - 1808 \beta_{2} + 9085 \beta_{1} - 10740$$$$)/20$$ $$\nu^{10}$$ $$=$$ $$($$$$-9123 \beta_{15} - 3442 \beta_{14} + 3400 \beta_{13} - 3400 \beta_{12} + 2958 \beta_{10} + 2958 \beta_{9} + 8530 \beta_{8} + 2667 \beta_{7} + 696 \beta_{6} - 2544 \beta_{5} + 2544 \beta_{4} - 645 \beta_{3} - 3312 \beta_{2}$$$$)/10$$ $$\nu^{11}$$ $$=$$ $$($$$$-28892 \beta_{15} - 17087 \beta_{14} + 6585 \beta_{13} - 21405 \beta_{12} + 26495 \beta_{11} + 40697 \beta_{10} - 17273 \beta_{9} + 26495 \beta_{8} - 9237 \beta_{7} - 7194 \beta_{6} - 28584 \beta_{5} - 12476 \beta_{4} + 21405 \beta_{3} + 2652 \beta_{2} - 53475 \beta_{1} + 36100$$$$)/20$$ $$\nu^{12}$$ $$=$$ $$-1979 \beta_{13} - 1979 \beta_{12} + 2412 \beta_{11} + 4652 \beta_{10} - 4652 \beta_{9} - 1324 \beta_{7} - 2227 \beta_{5} - 3895 \beta_{4} + 1324 \beta_{3} - 9256 \beta_{1} + 6093$$ $$\nu^{13}$$ $$=$$ $$($$$$56593 \beta_{15} + 69762 \beta_{14} - 41490 \beta_{13} - 50530 \beta_{12} + 22640 \beta_{11} + 103592 \beta_{10} - 47828 \beta_{9} - 22640 \beta_{8} - 7392 \beta_{7} + 29509 \beta_{6} - 53601 \beta_{5} - 60914 \beta_{4} - 50530 \beta_{3} - 34098 \beta_{2} - 132655 \beta_{1} + 31440$$$$)/10$$ $$\nu^{14}$$ $$=$$ $$($$$$118157 \beta_{15} + 212602 \beta_{14} + 41880 \beta_{13} - 41880 \beta_{12} + 115318 \beta_{10} + 115318 \beta_{9} - 26370 \beta_{8} - 35853 \beta_{7} + 159364 \beta_{6} - 93116 \beta_{5} + 93116 \beta_{4} - 210715 \beta_{3} - 174862 \beta_{2}$$$$)/10$$ $$\nu^{15}$$ $$=$$ $$($$$$69354 \beta_{15} + 691355 \beta_{14} + 1028615 \beta_{13} + 501225 \beta_{12} + 295825 \beta_{11} - 231109 \beta_{10} + 1303021 \beta_{9} + 295825 \beta_{8} - 216751 \beta_{7} + 569300 \beta_{6} - 311570 \beta_{5} + 1098900 \beta_{4} - 501225 \beta_{3} - 811864 \beta_{2} + 517585 \beta_{1} + 485380$$$$)/20$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/300\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$151$$ $$277$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
299.1
 2.13477 − 1.10359i 1.10359 + 2.13477i 2.13477 + 1.10359i 1.10359 − 2.13477i 0.994127 + 1.77516i −1.77516 + 0.994127i 0.994127 − 1.77516i −1.77516 − 0.994127i −0.869987 − 1.05054i −1.05054 + 0.869987i −0.869987 + 1.05054i −1.05054 − 0.869987i 0.474040 − 0.0108430i −0.0108430 − 0.474040i 0.474040 + 0.0108430i −0.0108430 + 0.474040i
−1.38758 0.273147i −0.758030 1.55737i 1.85078 + 0.758030i 0 0.626440 + 2.36803i 3.56393 −2.36106 1.55737i −1.85078 + 2.36106i 0
299.2 −1.38758 0.273147i 0.758030 1.55737i 1.85078 + 0.758030i 0 −1.47722 + 1.95392i −3.56393 −2.36106 1.55737i −1.85078 2.36106i 0
299.3 −1.38758 + 0.273147i −0.758030 + 1.55737i 1.85078 0.758030i 0 0.626440 2.36803i 3.56393 −2.36106 + 1.55737i −1.85078 2.36106i 0
299.4 −1.38758 + 0.273147i 0.758030 + 1.55737i 1.85078 0.758030i 0 −1.47722 1.95392i −3.56393 −2.36106 + 1.55737i −1.85078 + 2.36106i 0
299.5 −0.569745 1.29437i −1.47492 + 0.908080i −1.35078 + 1.47492i 0 2.01572 + 1.39172i −2.50967 2.67869 + 0.908080i 1.35078 2.67869i 0
299.6 −0.569745 1.29437i 1.47492 + 0.908080i −1.35078 + 1.47492i 0 0.335062 2.42647i 2.50967 2.67869 + 0.908080i 1.35078 + 2.67869i 0
299.7 −0.569745 + 1.29437i −1.47492 0.908080i −1.35078 1.47492i 0 2.01572 1.39172i −2.50967 2.67869 0.908080i 1.35078 + 2.67869i 0
299.8 −0.569745 + 1.29437i 1.47492 0.908080i −1.35078 1.47492i 0 0.335062 + 2.42647i 2.50967 2.67869 0.908080i 1.35078 2.67869i 0
299.9 0.569745 1.29437i −1.47492 + 0.908080i −1.35078 1.47492i 0 0.335062 + 2.42647i −2.50967 −2.67869 + 0.908080i 1.35078 2.67869i 0
299.10 0.569745 1.29437i 1.47492 + 0.908080i −1.35078 1.47492i 0 2.01572 1.39172i 2.50967 −2.67869 + 0.908080i 1.35078 + 2.67869i 0
299.11 0.569745 + 1.29437i −1.47492 0.908080i −1.35078 + 1.47492i 0 0.335062 2.42647i −2.50967 −2.67869 0.908080i 1.35078 + 2.67869i 0
299.12 0.569745 + 1.29437i 1.47492 0.908080i −1.35078 + 1.47492i 0 2.01572 + 1.39172i 2.50967 −2.67869 0.908080i 1.35078 2.67869i 0
299.13 1.38758 0.273147i −0.758030 1.55737i 1.85078 0.758030i 0 −1.47722 1.95392i 3.56393 2.36106 1.55737i −1.85078 + 2.36106i 0
299.14 1.38758 0.273147i 0.758030 1.55737i 1.85078 0.758030i 0 0.626440 2.36803i −3.56393 2.36106 1.55737i −1.85078 2.36106i 0
299.15 1.38758 + 0.273147i −0.758030 + 1.55737i 1.85078 + 0.758030i 0 −1.47722 + 1.95392i 3.56393 2.36106 + 1.55737i −1.85078 2.36106i 0
299.16 1.38758 + 0.273147i 0.758030 + 1.55737i 1.85078 + 0.758030i 0 0.626440 + 2.36803i −3.56393 2.36106 + 1.55737i −1.85078 + 2.36106i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 299.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
5.b even 2 1 inner
12.b even 2 1 inner
15.d odd 2 1 inner
20.d odd 2 1 inner
60.h even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 300.2.h.c 16
3.b odd 2 1 inner 300.2.h.c 16
4.b odd 2 1 inner 300.2.h.c 16
5.b even 2 1 inner 300.2.h.c 16
5.c odd 4 1 300.2.e.d 8
5.c odd 4 1 300.2.e.e yes 8
12.b even 2 1 inner 300.2.h.c 16
15.d odd 2 1 inner 300.2.h.c 16
15.e even 4 1 300.2.e.d 8
15.e even 4 1 300.2.e.e yes 8
20.d odd 2 1 inner 300.2.h.c 16
20.e even 4 1 300.2.e.d 8
20.e even 4 1 300.2.e.e yes 8
60.h even 2 1 inner 300.2.h.c 16
60.l odd 4 1 300.2.e.d 8
60.l odd 4 1 300.2.e.e yes 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
300.2.e.d 8 5.c odd 4 1
300.2.e.d 8 15.e even 4 1
300.2.e.d 8 20.e even 4 1
300.2.e.d 8 60.l odd 4 1
300.2.e.e yes 8 5.c odd 4 1
300.2.e.e yes 8 15.e even 4 1
300.2.e.e yes 8 20.e even 4 1
300.2.e.e yes 8 60.l odd 4 1
300.2.h.c 16 1.a even 1 1 trivial
300.2.h.c 16 3.b odd 2 1 inner
300.2.h.c 16 4.b odd 2 1 inner
300.2.h.c 16 5.b even 2 1 inner
300.2.h.c 16 12.b even 2 1 inner
300.2.h.c 16 15.d odd 2 1 inner
300.2.h.c 16 20.d odd 2 1 inner
300.2.h.c 16 60.h even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(300, [\chi])$$:

 $$T_{7}^{4} - 19 T_{7}^{2} + 80$$ $$T_{17}^{4} - 59 T_{17}^{2} + 40$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 16 - 4 T^{2} - 2 T^{4} - T^{6} + T^{8} )^{2}$$
$3$ $$( 81 + 9 T^{2} + 8 T^{4} + T^{6} + T^{8} )^{2}$$
$5$ $$T^{16}$$
$7$ $$( 80 - 19 T^{2} + T^{4} )^{4}$$
$11$ $$( 200 - 29 T^{2} + T^{4} )^{4}$$
$13$ $$( 100 + 21 T^{2} + T^{4} )^{4}$$
$17$ $$( 40 - 59 T^{2} + T^{4} )^{4}$$
$19$ $$( 5 + 26 T^{2} + T^{4} )^{4}$$
$23$ $$( 32 + 28 T^{2} + T^{4} )^{4}$$
$29$ $$( 160 + 36 T^{2} + T^{4} )^{4}$$
$31$ $$( 500 + 55 T^{2} + T^{4} )^{4}$$
$37$ $$( 1600 + 84 T^{2} + T^{4} )^{4}$$
$41$ $$( 1000 + 115 T^{2} + T^{4} )^{4}$$
$43$ $$( 20 - 71 T^{2} + T^{4} )^{4}$$
$47$ $$( 512 + 52 T^{2} + T^{4} )^{4}$$
$53$ $$( 160 - 36 T^{2} + T^{4} )^{4}$$
$59$ $$T^{16}$$
$61$ $$( -4 + 5 T + T^{2} )^{8}$$
$67$ $$( 125 - 34 T^{2} + T^{4} )^{4}$$
$71$ $$( 20000 - 284 T^{2} + T^{4} )^{4}$$
$73$ $$( 100 + 61 T^{2} + T^{4} )^{4}$$
$79$ $$( 1280 + 76 T^{2} + T^{4} )^{4}$$
$83$ $$( 32 + 13 T^{2} + T^{4} )^{4}$$
$89$ $$( 640 + 51 T^{2} + T^{4} )^{4}$$
$97$ $$( 8100 + 189 T^{2} + T^{4} )^{4}$$