# Properties

 Label 1792.2.m.f Level $1792$ Weight $2$ Character orbit 1792.m Analytic conductor $14.309$ Analytic rank $0$ Dimension $16$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1792 = 2^{8} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1792.m (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$14.3091920422$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(i)$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 4 x^{15} + 12 x^{14} - 48 x^{13} + 67 x^{12} - 24 x^{11} + 118 x^{10} - 176 x^{9} + 351 x^{8} - 180 x^{7} + 358 x^{6} - 336 x^{5} + 390 x^{4} - 344 x^{3} + 164 x^{2} - 40 x + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{14}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -\beta_{11} q^{3} -\beta_{10} q^{5} -\beta_{5} q^{7} + ( -\beta_{5} - \beta_{8} + \beta_{11} + \beta_{15} ) q^{9} +O(q^{10})$$ $$q -\beta_{11} q^{3} -\beta_{10} q^{5} -\beta_{5} q^{7} + ( -\beta_{5} - \beta_{8} + \beta_{11} + \beta_{15} ) q^{9} + ( -\beta_{1} - \beta_{2} + \beta_{5} - \beta_{6} + \beta_{9} + \beta_{13} - \beta_{15} ) q^{11} + ( -1 + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{11} - \beta_{12} ) q^{13} + ( 2 + 2 \beta_{1} - \beta_{3} + 2 \beta_{4} - \beta_{6} - \beta_{7} + \beta_{9} + \beta_{10} + \beta_{12} - \beta_{14} ) q^{15} + ( -1 + \beta_{1} + 2 \beta_{3} - \beta_{4} + 2 \beta_{6} + \beta_{7} - \beta_{9} + \beta_{14} ) q^{17} + ( \beta_{1} - \beta_{2} + \beta_{3} + \beta_{5} + \beta_{10} + \beta_{13} - \beta_{15} ) q^{19} + \beta_{8} q^{21} + ( 2 \beta_{2} - \beta_{3} + \beta_{6} - \beta_{9} - \beta_{10} - \beta_{13} - \beta_{14} + 3 \beta_{15} ) q^{23} + ( -\beta_{2} + \beta_{5} - \beta_{10} + \beta_{12} - \beta_{15} ) q^{25} + ( 2 - \beta_{1} - \beta_{4} + 3 \beta_{5} + \beta_{6} + \beta_{8} - \beta_{10} + \beta_{13} - \beta_{15} ) q^{27} + ( -\beta_{2} - \beta_{4} + \beta_{7} - \beta_{14} - \beta_{15} ) q^{29} + ( -1 - \beta_{1} + \beta_{3} - 2 \beta_{4} + \beta_{6} + \beta_{7} + \beta_{9} - \beta_{10} - \beta_{12} - \beta_{14} ) q^{31} + ( 3 + 3 \beta_{1} + 3 \beta_{4} - \beta_{7} - \beta_{9} + \beta_{10} + \beta_{12} + \beta_{14} ) q^{33} -\beta_{12} q^{35} + ( 2 \beta_{2} - 2 \beta_{4} + 2 \beta_{6} + 2 \beta_{7} - 2 \beta_{8} - 3 \beta_{9} - 2 \beta_{10} + 2 \beta_{15} ) q^{37} + ( 3 \beta_{2} - 2 \beta_{3} + 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{8} - 2 \beta_{9} + 2 \beta_{11} - 2 \beta_{14} + \beta_{15} ) q^{39} + ( -\beta_{3} + \beta_{6} + \beta_{8} - \beta_{10} - \beta_{11} + \beta_{12} + 2 \beta_{15} ) q^{41} + ( 1 + \beta_{5} + \beta_{6} + 2 \beta_{8} + \beta_{9} + \beta_{10} ) q^{43} + ( -1 - 2 \beta_{1} + 3 \beta_{3} - 2 \beta_{4} - \beta_{5} + 2 \beta_{7} - 2 \beta_{10} + \beta_{11} - 5 \beta_{12} - 2 \beta_{13} + 2 \beta_{14} ) q^{45} + ( 3 + \beta_{1} + 2 \beta_{4} - \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} + \beta_{12} - \beta_{14} ) q^{47} - q^{49} + ( -1 - \beta_{1} - 2 \beta_{3} - \beta_{4} - \beta_{10} + 2 \beta_{12} - \beta_{13} - 4 \beta_{14} + \beta_{15} ) q^{51} + ( 4 + 2 \beta_{1} + \beta_{2} + \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} + \beta_{9} - 2 \beta_{13} ) q^{53} + ( -2 \beta_{2} + \beta_{3} + 2 \beta_{5} - \beta_{6} + \beta_{8} - 2 \beta_{10} - \beta_{11} - 2 \beta_{13} ) q^{55} + ( -\beta_{2} - \beta_{9} + \beta_{10} - \beta_{12} - \beta_{14} - \beta_{15} ) q^{57} + ( 1 + \beta_{2} - \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} - 4 \beta_{8} - 2 \beta_{9} + \beta_{15} ) q^{59} + ( -2 \beta_{1} + \beta_{2} - \beta_{4} - 2 \beta_{5} - 2 \beta_{10} - \beta_{12} - 2 \beta_{13} + 2 \beta_{15} ) q^{61} + ( -1 + \beta_{7} - \beta_{8} - \beta_{11} ) q^{63} + ( 4 - 2 \beta_{1} - 2 \beta_{3} + 4 \beta_{4} - 2 \beta_{6} - 3 \beta_{7} + \beta_{8} + \beta_{9} + \beta_{11} - \beta_{14} ) q^{65} + ( -4 - \beta_{1} - 2 \beta_{2} + 5 \beta_{3} - 3 \beta_{4} + 3 \beta_{5} + 3 \beta_{7} - \beta_{10} - 2 \beta_{11} - 3 \beta_{12} - \beta_{13} + \beta_{14} - 2 \beta_{15} ) q^{67} + ( -5 - 2 \beta_{1} + \beta_{2} - 3 \beta_{4} - 3 \beta_{5} + \beta_{6} + 2 \beta_{7} - \beta_{8} + 2 \beta_{9} + 2 \beta_{13} ) q^{69} + ( -\beta_{2} + \beta_{3} + 2 \beta_{5} - \beta_{6} - \beta_{9} - \beta_{10} + \beta_{12} - \beta_{14} ) q^{71} + ( -4 \beta_{2} + \beta_{3} - \beta_{6} - \beta_{8} + \beta_{9} + 2 \beta_{10} + \beta_{11} + \beta_{12} + 3 \beta_{13} + \beta_{14} - 2 \beta_{15} ) q^{73} + ( 5 + 2 \beta_{1} - 2 \beta_{2} + 4 \beta_{4} + 3 \beta_{5} - 5 \beta_{6} - 3 \beta_{7} + 4 \beta_{9} + 2 \beta_{10} - 2 \beta_{13} - \beta_{15} ) q^{75} + ( -\beta_{1} + \beta_{2} - \beta_{3} - \beta_{5} - \beta_{10} - \beta_{12} - \beta_{13} - \beta_{14} + \beta_{15} ) q^{77} + ( -3 - 3 \beta_{1} - \beta_{4} + \beta_{7} + 2 \beta_{8} + 2 \beta_{9} - \beta_{10} + 2 \beta_{11} - \beta_{12} - 2 \beta_{14} ) q^{79} + ( 5 + 6 \beta_{1} - 2 \beta_{3} + 5 \beta_{4} - 2 \beta_{6} - 5 \beta_{7} - 2 \beta_{8} + \beta_{9} + 5 \beta_{10} - 2 \beta_{11} + 5 \beta_{12} - \beta_{14} ) q^{81} + ( -2 + 3 \beta_{2} - 2 \beta_{3} + 3 \beta_{4} + 2 \beta_{5} - 2 \beta_{7} + 3 \beta_{11} + 2 \beta_{12} - 4 \beta_{14} + 2 \beta_{15} ) q^{83} + ( -2 - 2 \beta_{1} + 2 \beta_{2} - 4 \beta_{4} + 2 \beta_{6} + \beta_{7} - 2 \beta_{9} + 2 \beta_{13} - \beta_{15} ) q^{85} + ( -5 \beta_{2} + 4 \beta_{3} + 6 \beta_{5} - 4 \beta_{6} + \beta_{8} + 3 \beta_{9} + 4 \beta_{10} - \beta_{11} - 3 \beta_{12} + \beta_{13} + 3 \beta_{14} - 6 \beta_{15} ) q^{87} + ( 2 \beta_{5} - \beta_{9} + \beta_{12} + \beta_{13} - \beta_{14} ) q^{89} + ( 1 + \beta_{2} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{8} + \beta_{10} ) q^{91} + ( 2 + 4 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} + \beta_{4} + 2 \beta_{5} + \beta_{7} + 4 \beta_{10} - 2 \beta_{11} + 2 \beta_{12} + 4 \beta_{13} + 2 \beta_{14} - 5 \beta_{15} ) q^{93} + ( 6 - \beta_{4} - \beta_{7} - 2 \beta_{8} - 2 \beta_{11} ) q^{95} + ( -7 - 3 \beta_{1} - \beta_{3} + \beta_{4} - \beta_{6} - \beta_{7} + 3 \beta_{8} + 3 \beta_{9} + 3 \beta_{11} - 3 \beta_{14} ) q^{97} + ( -1 - 2 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{7} - 2 \beta_{10} + 2 \beta_{11} - \beta_{12} - 2 \beta_{13} - 3 \beta_{14} + 3 \beta_{15} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q - 4q^{3} + 4q^{5} + O(q^{10})$$ $$16q - 4q^{3} + 4q^{5} + 8q^{11} - 12q^{13} - 8q^{17} - 4q^{19} + 4q^{21} + 56q^{27} + 8q^{31} + 16q^{33} + 4q^{35} + 8q^{37} + 24q^{43} + 36q^{45} + 40q^{47} - 16q^{49} - 24q^{51} + 32q^{53} + 4q^{59} + 20q^{61} - 24q^{63} + 72q^{65} - 32q^{67} - 56q^{69} + 28q^{75} + 8q^{77} - 40q^{81} - 36q^{83} + 12q^{91} - 8q^{93} + 80q^{95} - 72q^{97} + 8q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 4 x^{15} + 12 x^{14} - 48 x^{13} + 67 x^{12} - 24 x^{11} + 118 x^{10} - 176 x^{9} + 351 x^{8} - 180 x^{7} + 358 x^{6} - 336 x^{5} + 390 x^{4} - 344 x^{3} + 164 x^{2} - 40 x + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$951071408409672 \nu^{15} - 3722676599840684 \nu^{14} + 11056775295709911 \nu^{13} - 44591424589229034 \nu^{12} + 59571828256046246 \nu^{11} - 16311504998557221 \nu^{10} + 109855023226949419 \nu^{9} - 158494098831050719 \nu^{8} + 316269746406518514 \nu^{7} - 141663658022557777 \nu^{6} + 317962271773663747 \nu^{5} - 298674348575977988 \nu^{4} + 330162633044914154 \nu^{3} - 301042761683687430 \nu^{2} + 117377507327509198 \nu - 29073536866216359$$$$)/ 2586996236383195$$ $$\beta_{2}$$ $$=$$ $$($$$$-837450978918 \nu^{15} + 3268059368601 \nu^{14} - 9774129145049 \nu^{13} + 39378227440746 \nu^{12} - 52647402989314 \nu^{11} + 16633551650999 \nu^{10} - 98359361008081 \nu^{9} + 136856641603546 \nu^{8} - 285346377635636 \nu^{7} + 125536721618373 \nu^{6} - 297759021619953 \nu^{5} + 247799490169192 \nu^{4} - 315437625800966 \nu^{3} + 253762122985590 \nu^{2} - 121829545946822 \nu + 22416575382206$$$$)/ 1646719437545$$ $$\beta_{3}$$ $$=$$ $$($$$$-2704100770805192 \nu^{15} + 9848563721439349 \nu^{14} - 29239824225526546 \nu^{13} + 120429826515046084 \nu^{12} - 141252009753288336 \nu^{11} + 27728735279128781 \nu^{10} - 323008880354869584 \nu^{9} + 359150264381483384 \nu^{8} - 856904562132546654 \nu^{7} + 215321576232905127 \nu^{6} - 974932204477243482 \nu^{5} + 567587652935576758 \nu^{4} - 950165080048245094 \nu^{3} + 640181323544103480 \nu^{2} - 285278940866108728 \nu + 63352765090121984$$$$)/ 5173992472766390$$ $$\beta_{4}$$ $$=$$ $$($$$$1674708372990976 \nu^{15} - 6572201638917102 \nu^{14} + 19424222999220833 \nu^{13} - 78294768665013192 \nu^{12} + 104476272119691108 \nu^{11} - 24742665676005658 \nu^{10} + 187581053285797687 \nu^{9} - 280841928485519762 \nu^{8} + 546976894209326142 \nu^{7} - 240129108623449166 \nu^{6} + 534273028711390981 \nu^{5} - 515646310273577514 \nu^{4} + 562177999739357012 \nu^{3} - 502632258726641150 \nu^{2} + 194573129558701754 \nu - 23048604152387462$$$$)/ 2586996236383195$$ $$\beta_{5}$$ $$=$$ $$($$$$19482487322 \nu^{15} - 71829768307 \nu^{14} + 211530176114 \nu^{13} - 869648666130 \nu^{12} + 1035106924390 \nu^{11} - 152629602893 \nu^{10} + 2258419134670 \nu^{9} - 2719627979686 \nu^{8} + 6018321521990 \nu^{7} - 1643327598977 \nu^{6} + 6516717498206 \nu^{5} - 4492931472132 \nu^{4} + 6274129258728 \nu^{3} - 4768566345236 \nu^{2} + 1753345508964 \nu - 258618081490$$$$)/ 18265555126$$ $$\beta_{6}$$ $$=$$ $$($$$$7498893603309034 \nu^{15} - 28658627888228153 \nu^{14} + 84898454002472672 \nu^{13} - 344785787247940018 \nu^{12} + 440818146003185412 \nu^{11} - 101252104556846217 \nu^{10} + 863513038753683928 \nu^{9} - 1161231004486048088 \nu^{8} + 2427731722617841338 \nu^{7} - 909797757822592589 \nu^{6} + 2518330312414188084 \nu^{5} - 2042609308803956566 \nu^{4} + 2566105999528852478 \nu^{3} - 2095030105848071860 \nu^{2} + 849908894353444156 \nu - 124297662602168868$$$$)/ 5173992472766390$$ $$\beta_{7}$$ $$=$$ $$($$$$3777555552239611 \nu^{15} - 14818624514131492 \nu^{14} + 43873064609843573 \nu^{13} - 176863864075565747 \nu^{12} + 236350861493329618 \nu^{11} - 59330090045209703 \nu^{10} + 428257247412482512 \nu^{9} - 633782556466103087 \nu^{8} + 1240943173396670972 \nu^{7} - 549310814953693356 \nu^{6} + 1224404821394753001 \nu^{5} - 1173798486360141284 \nu^{4} + 1283713884161744232 \nu^{3} - 1152862769202227570 \nu^{2} + 447008615232415754 \nu - 67882571420950212$$$$)/ 2586996236383195$$ $$\beta_{8}$$ $$=$$ $$($$$$-4111165082614591 \nu^{15} + 14866011666638747 \nu^{14} - 43592489723122903 \nu^{13} + 180471639613998717 \nu^{12} - 205792432068434953 \nu^{11} + 18177688506737658 \nu^{10} - 476344981375704222 \nu^{9} + 540760231449412867 \nu^{8} - 1231844713236319142 \nu^{7} + 261476413468482641 \nu^{6} - 1363528429379028841 \nu^{5} + 857235288719333779 \nu^{4} - 1263918842903687457 \nu^{3} + 917338120456251020 \nu^{2} - 314841413375322204 \nu + 40980131050682892$$$$)/ 2586996236383195$$ $$\beta_{9}$$ $$=$$ $$($$$$-51700125227286 \nu^{15} + 181410266124727 \nu^{14} - 529691956965078 \nu^{13} + 2216205890956042 \nu^{12} - 2359965597354778 \nu^{11} + 15018673117033 \nu^{10} - 6032292554488402 \nu^{9} + 6135862988795822 \nu^{8} - 14919834611631202 \nu^{7} + 1835649209159641 \nu^{6} - 17151627123727186 \nu^{5} + 8947496261644624 \nu^{4} - 15153946647046832 \nu^{3} + 10174933047931800 \nu^{2} - 3001512409125524 \nu + 339029565430912$$$$)/ 28585593772190$$ $$\beta_{10}$$ $$=$$ $$($$$$5111890261955639 \nu^{15} - 18723422330113773 \nu^{14} + 55026059693644252 \nu^{13} - 226796911018514863 \nu^{12} + 265978766975545407 \nu^{11} - 32882698304956797 \nu^{10} + 591816357963522438 \nu^{9} - 700440973017072543 \nu^{8} + 1557941182908479308 \nu^{7} - 391651315022393869 \nu^{6} + 1698017523089689604 \nu^{5} - 1143547375046559696 \nu^{4} + 1612124569733992598 \nu^{3} - 1206798639331144120 \nu^{2} + 435161639642974356 \nu - 58837164424030928$$$$)/ 2586996236383195$$ $$\beta_{11}$$ $$=$$ $$($$$$5374657323581928 \nu^{15} - 19813878646531791 \nu^{14} + 58296781349821544 \nu^{13} - 239788425805503616 \nu^{12} + 285186819270035739 \nu^{11} - 40441701960073524 \nu^{10} + 623516248794297746 \nu^{9} - 751717640243443631 \nu^{8} + 1650583194286411736 \nu^{7} - 453843017266031758 \nu^{6} + 1789141729619286818 \nu^{5} - 1251632447632723287 \nu^{4} + 1703428508268658061 \nu^{3} - 1321246981077163610 \nu^{2} + 472716677119123542 \nu - 69616014115651576$$$$)/ 2586996236383195$$ $$\beta_{12}$$ $$=$$ $$($$$$-5522619438975152 \nu^{15} + 20318406884992724 \nu^{14} - 59803135064160756 \nu^{13} + 246097516841643734 \nu^{12} - 291627635161742131 \nu^{11} + 41301296869441296 \nu^{10} - 641433639158466729 \nu^{9} + 766246241691993109 \nu^{8} - 1697590220722951414 \nu^{7} + 458715003921674597 \nu^{6} - 1847016073682336012 \nu^{5} + 1269554917909688188 \nu^{4} - 1760055912567976994 \nu^{3} + 1351849139305683920 \nu^{2} - 493126775820972468 \nu + 76096265993090204$$$$)/ 2586996236383195$$ $$\beta_{13}$$ $$=$$ $$($$$$-1342976102346028 \nu^{15} + 4888822756910656 \nu^{14} - 14345307664530663 \nu^{13} + 59261194792100792 \nu^{12} - 68547510138283141 \nu^{11} + 7085686180956252 \nu^{10} - 155431896656409780 \nu^{9} + 180665252133307975 \nu^{8} - 405109873988720704 \nu^{7} + 94218727843551051 \nu^{6} - 443797052249781231 \nu^{5} + 291324658777882928 \nu^{4} - 415638376318740034 \nu^{3} + 308964463100290878 \nu^{2} - 105659607438284102 \nu + 13545912097023734$$$$)/ 517399247276639$$ $$\beta_{14}$$ $$=$$ $$($$$$-96105262004818 \nu^{15} + 355577051279181 \nu^{14} - 1045387256303994 \nu^{13} + 4295184660226986 \nu^{12} - 5137895187302654 \nu^{11} + 714420636220699 \nu^{10} - 11070904840803886 \nu^{9} + 13589173271460526 \nu^{8} - 29521345230669606 \nu^{7} + 8291064774952483 \nu^{6} - 31568902874141438 \nu^{5} + 22789052228712772 \nu^{4} - 30280524171409216 \nu^{3} + 23746847574694880 \nu^{2} - 8262089065435772 \nu + 1133976629719396$$$$)/ 28585593772190$$ $$\beta_{15}$$ $$=$$ $$($$$$-12901040348769007 \nu^{15} + 46882114975152344 \nu^{14} - 137516440256165211 \nu^{13} + 568446387146979339 \nu^{12} - 654936333530095316 \nu^{11} + 64154006218803931 \nu^{10} - 1493003685079757954 \nu^{9} + 1724470044458942519 \nu^{8} - 3881717238759491924 \nu^{7} + 886977360247710202 \nu^{6} - 4254657735688371947 \nu^{5} + 2775101054643465278 \nu^{4} - 3971125038086180174 \nu^{3} + 2966846397725994870 \nu^{2} - 988212428548741298 \nu + 129759186731597904$$$$)/ 2586996236383195$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{14} + \beta_{13} + \beta_{12} + 2 \beta_{10} - \beta_{9} + 2 \beta_{5} + 2 \beta_{3} - \beta_{2} + 2 \beta_{1} + 2$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$-2 \beta_{13} + 4 \beta_{12} + 2 \beta_{11} + 4 \beta_{10} + 4 \beta_{8} - 6 \beta_{7} + 2 \beta_{6} - 2 \beta_{5} + 5 \beta_{4} + 2 \beta_{3} - \beta_{2} + 8 \beta_{1} + 2$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$3 \beta_{14} - 7 \beta_{13} - 3 \beta_{12} - 2 \beta_{11} - \beta_{9} + 5 \beta_{8} + 2 \beta_{6} - 11 \beta_{5} + 4 \beta_{4} - 5 \beta_{3} + 4 \beta_{2} + 4 \beta_{1} + 11$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$18 \beta_{15} + 10 \beta_{14} - 2 \beta_{13} - 2 \beta_{12} - 2 \beta_{11} - 10 \beta_{10} - 32 \beta_{9} - 8 \beta_{8} + 30 \beta_{7} - 26 \beta_{6} + 48 \beta_{5} + 7 \beta_{4} - 6 \beta_{3} - 3 \beta_{2} + 2 \beta_{1} + 78$$$$)/4$$ $$\nu^{5}$$ $$=$$ $$($$$$74 \beta_{15} - 56 \beta_{14} + 24 \beta_{13} + 124 \beta_{12} + 88 \beta_{11} + 82 \beta_{10} - 46 \beta_{9} + 32 \beta_{8} - 118 \beta_{7} - 4 \beta_{6} + 226 \beta_{5} + 81 \beta_{4} + 98 \beta_{3} - 25 \beta_{2} + 146 \beta_{1} + 16$$$$)/4$$ $$\nu^{6}$$ $$=$$ $$($$$$21 \beta_{15} - 222 \beta_{13} + 68 \beta_{10} + 44 \beta_{9} + 190 \beta_{8} - 201 \beta_{7} + 178 \beta_{6} - 268 \beta_{5} + 111 \beta_{4} + 111 \beta_{2} + 222 \beta_{1} - 46$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$142 \beta_{15} + 592 \beta_{14} - 1292 \beta_{13} - 1282 \beta_{12} - 890 \beta_{11} - 904 \beta_{10} - 366 \beta_{9} + 212 \beta_{8} + 1092 \beta_{7} - 70 \beta_{6} - 1480 \beta_{5} - 99 \beta_{4} - 1058 \beta_{3} + 621 \beta_{2} - 460 \beta_{1} + 1868$$$$)/4$$ $$\nu^{8}$$ $$=$$ $$($$$$2330 \beta_{15} - 738 \beta_{14} + 1498 \beta_{13} + 636 \beta_{12} + 484 \beta_{11} - 880 \beta_{10} - 2740 \beta_{9} - 1810 \beta_{8} + 2206 \beta_{7} - 3154 \beta_{6} + 8300 \beta_{5} - 143 \beta_{4} + 842 \beta_{3} - 1243 \beta_{2} - 862 \beta_{1} + 3942$$$$)/4$$ $$\nu^{9}$$ $$=$$ $$($$$$2531 \beta_{15} - 3492 \beta_{14} - 210 \beta_{13} + 5181 \beta_{12} + 3691 \beta_{11} + 4159 \beta_{10} + 298 \beta_{9} + 2848 \beta_{8} - 7989 \beta_{7} + 2392 \beta_{6} + 6387 \beta_{5} + 3015 \beta_{4} + 5101 \beta_{3} - 425 \beta_{2} + 6690 \beta_{1} - 6421$$$$)/2$$ $$\nu^{10}$$ $$=$$ $$($$$$-3944 \beta_{15} + 7978 \beta_{14} - 52842 \beta_{13} - 26600 \beta_{12} - 17948 \beta_{11} - 6776 \beta_{10} + 13828 \beta_{9} + 31098 \beta_{8} - 15224 \beta_{7} + 32910 \beta_{6} - 86808 \beta_{5} + 6761 \beta_{4} - 19014 \beta_{3} + 27681 \beta_{2} + 14158 \beta_{1} - 9098$$$$)/4$$ $$\nu^{11}$$ $$=$$ $$($$$$21134 \beta_{15} + 46534 \beta_{14} - 57378 \beta_{13} - 126974 \beta_{12} - 86542 \beta_{11} - 116430 \beta_{10} - 55584 \beta_{9} - 43698 \beta_{8} + 171842 \beta_{7} - 69398 \beta_{6} - 27218 \beta_{5} - 50115 \beta_{4} - 96816 \beta_{3} + 24809 \beta_{2} - 122714 \beta_{1} + 184392$$$$)/4$$ $$\nu^{12}$$ $$=$$ $$($$$$111990 \beta_{15} - 96551 \beta_{14} + 154426 \beta_{13} + 127911 \beta_{12} + 92227 \beta_{11} + 26515 \beta_{10} - 96551 \beta_{9} - 92227 \beta_{8} - 133222 \beta_{6} + 518948 \beta_{5} + 133222 \beta_{3} - 101791 \beta_{2}$$$$)/2$$ $$\nu^{13}$$ $$=$$ $$($$$$137632 \beta_{15} - 548602 \beta_{14} - 478874 \beta_{13} + 701572 \beta_{12} + 507786 \beta_{11} + 806284 \beta_{10} + 522960 \beta_{9} + 881276 \beta_{8} - 1824304 \beta_{7} + 1017132 \beta_{6} - 329164 \beta_{5} + 541021 \beta_{4} + 744316 \beta_{3} + 214941 \beta_{2} + 1315630 \beta_{1} - 1929410$$$$)/4$$ $$\nu^{14}$$ $$=$$ $$($$$$-994874 \beta_{15} + 1834282 \beta_{14} - 5092398 \beta_{13} - 4608284 \beta_{12} - 3155984 \beta_{11} - 2431792 \beta_{10} + 1059000 \beta_{9} + 1822086 \beta_{8} + 1631110 \beta_{7} + 2088694 \beta_{6} - 9989352 \beta_{5} - 615403 \beta_{4} - 3617642 \beta_{3} + 2795925 \beta_{2} - 1364502 \beta_{1} + 1312114$$$$)/4$$ $$\nu^{15}$$ $$=$$ $$($$$$1679682 \beta_{15} + 1116243 \beta_{14} + 1730018 \beta_{13} - 3703000 \beta_{12} - 2499720 \beta_{11} - 5022618 \beta_{10} - 3827007 \beta_{9} - 4740003 \beta_{8} + 9202782 \beta_{7} - 6119880 \beta_{6} + 6935519 \beta_{5} - 3061968 \beta_{4} - 2649921 \beta_{3} - 1228613 \beta_{2} - 7112700 \beta_{1} + 8689045$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1792\mathbb{Z}\right)^\times$$.

 $$n$$ $$1023$$ $$1025$$ $$1541$$ $$\chi(n)$$ $$1$$ $$1$$ $$\beta_{5}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
449.1
 0.117630 − 0.893490i −0.709944 + 0.925217i 2.69978 + 0.355433i −0.424637 + 3.22544i 0.277956 + 0.213283i 0.792206 − 1.03242i −1.09227 − 0.838128i 0.339278 + 0.0446668i 0.117630 + 0.893490i −0.709944 − 0.925217i 2.69978 − 0.355433i −0.424637 − 3.22544i 0.277956 − 0.213283i 0.792206 + 1.03242i −1.09227 + 0.838128i 0.339278 − 0.0446668i
0 −2.41958 + 2.41958i 0 2.54136 + 2.54136i 0 1.00000i 0 8.70871i 0
449.2 0 −2.04137 + 2.04137i 0 −0.701647 0.701647i 0 1.00000i 0 5.33435i 0
449.3 0 −1.62602 + 1.62602i 0 −1.16900 1.16900i 0 1.00000i 0 2.28788i 0
449.4 0 −0.171192 + 0.171192i 0 −0.268425 0.268425i 0 1.00000i 0 2.94139i 0
449.5 0 0.328027 0.328027i 0 −1.40197 1.40197i 0 1.00000i 0 2.78480i 0
449.6 0 1.18265 1.18265i 0 1.87820 + 1.87820i 0 1.00000i 0 0.202696i 0
449.7 0 1.26274 1.26274i 0 2.95746 + 2.95746i 0 1.00000i 0 0.189043i 0
449.8 0 1.48474 1.48474i 0 −1.83598 1.83598i 0 1.00000i 0 1.40890i 0
1345.1 0 −2.41958 2.41958i 0 2.54136 2.54136i 0 1.00000i 0 8.70871i 0
1345.2 0 −2.04137 2.04137i 0 −0.701647 + 0.701647i 0 1.00000i 0 5.33435i 0
1345.3 0 −1.62602 1.62602i 0 −1.16900 + 1.16900i 0 1.00000i 0 2.28788i 0
1345.4 0 −0.171192 0.171192i 0 −0.268425 + 0.268425i 0 1.00000i 0 2.94139i 0
1345.5 0 0.328027 + 0.328027i 0 −1.40197 + 1.40197i 0 1.00000i 0 2.78480i 0
1345.6 0 1.18265 + 1.18265i 0 1.87820 1.87820i 0 1.00000i 0 0.202696i 0
1345.7 0 1.26274 + 1.26274i 0 2.95746 2.95746i 0 1.00000i 0 0.189043i 0
1345.8 0 1.48474 + 1.48474i 0 −1.83598 + 1.83598i 0 1.00000i 0 1.40890i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1345.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1792.2.m.f yes 16
4.b odd 2 1 1792.2.m.h yes 16
8.b even 2 1 1792.2.m.g yes 16
8.d odd 2 1 1792.2.m.e 16
16.e even 4 1 inner 1792.2.m.f yes 16
16.e even 4 1 1792.2.m.g yes 16
16.f odd 4 1 1792.2.m.e 16
16.f odd 4 1 1792.2.m.h yes 16
32.g even 8 1 7168.2.a.ba 8
32.g even 8 1 7168.2.a.be 8
32.h odd 8 1 7168.2.a.bb 8
32.h odd 8 1 7168.2.a.bf 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1792.2.m.e 16 8.d odd 2 1
1792.2.m.e 16 16.f odd 4 1
1792.2.m.f yes 16 1.a even 1 1 trivial
1792.2.m.f yes 16 16.e even 4 1 inner
1792.2.m.g yes 16 8.b even 2 1
1792.2.m.g yes 16 16.e even 4 1
1792.2.m.h yes 16 4.b odd 2 1
1792.2.m.h yes 16 16.f odd 4 1
7168.2.a.ba 8 32.g even 8 1
7168.2.a.bb 8 32.h odd 8 1
7168.2.a.be 8 32.g even 8 1
7168.2.a.bf 8 32.h odd 8 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1792, [\chi])$$:

 $$T_{3}^{16} + \cdots$$ $$T_{5}^{16} - \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16}$$
$3$ $$256 + 512 T + 512 T^{2} - 6912 T^{3} + 25728 T^{4} - 17984 T^{5} + 5888 T^{6} + 1280 T^{7} + 1412 T^{8} - 1072 T^{9} + 416 T^{10} + 152 T^{11} + 40 T^{12} - 16 T^{13} + 8 T^{14} + 4 T^{15} + T^{16}$$
$5$ $$16384 + 98304 T + 294912 T^{2} + 438272 T^{3} + 380416 T^{4} + 170496 T^{5} + 37376 T^{6} + 5184 T^{7} + 10500 T^{8} + 5344 T^{9} + 1024 T^{10} - 264 T^{11} + 104 T^{12} + 40 T^{13} + 8 T^{14} - 4 T^{15} + T^{16}$$
$7$ $$( 1 + T^{2} )^{8}$$
$11$ $$4096 - 24576 T + 73728 T^{2} - 120832 T^{3} + 128512 T^{4} - 119296 T^{5} + 184832 T^{6} - 264832 T^{7} + 254480 T^{8} - 147328 T^{9} + 54016 T^{10} - 11968 T^{11} + 1560 T^{12} - 128 T^{13} + 32 T^{14} - 8 T^{15} + T^{16}$$
$13$ $$16777216 - 41943040 T + 52428800 T^{2} - 11272192 T^{3} - 491520 T^{4} + 1110016 T^{5} + 2547712 T^{6} - 90368 T^{7} - 5692 T^{8} + 57808 T^{9} + 45344 T^{10} + 6376 T^{11} + 520 T^{12} + 112 T^{13} + 72 T^{14} + 12 T^{15} + T^{16}$$
$17$ $$( 116608 - 15872 T - 46272 T^{2} + 4992 T^{3} + 3528 T^{4} - 272 T^{5} - 100 T^{6} + 4 T^{7} + T^{8} )^{2}$$
$19$ $$256 - 3584 T + 25088 T^{2} + 336128 T^{3} + 2365568 T^{4} - 807616 T^{5} + 148992 T^{6} + 90816 T^{7} + 175108 T^{8} - 55488 T^{9} + 10368 T^{10} + 5096 T^{11} + 1352 T^{12} - 40 T^{13} + 8 T^{14} + 4 T^{15} + T^{16}$$
$23$ $$4983230464 + 4950327296 T^{2} + 1720396288 T^{4} + 274590208 T^{6} + 22220304 T^{8} + 938528 T^{10} + 20920 T^{12} + 232 T^{14} + T^{16}$$
$29$ $$8425771264 + 2138386432 T + 271351808 T^{2} - 1276829696 T^{3} + 3290633472 T^{4} + 43406336 T^{5} + 1785856 T^{6} - 5705728 T^{7} + 8730464 T^{8} + 16640 T^{9} + 2048 T^{10} - 5632 T^{11} + 6352 T^{12} - 64 T^{13} + T^{16}$$
$31$ $$( -251648 + 366848 T - 59584 T^{2} - 25792 T^{3} + 5304 T^{4} + 576 T^{5} - 132 T^{6} - 4 T^{7} + T^{8} )^{2}$$
$37$ $$483102843136 + 132249695232 T + 18101716992 T^{2} - 3207039488 T^{3} + 20760877312 T^{4} + 5971864320 T^{5} + 867542528 T^{6} - 234612096 T^{7} + 41581152 T^{8} + 2620352 T^{9} + 345472 T^{10} - 90336 T^{11} + 12944 T^{12} + 272 T^{13} + 32 T^{14} - 8 T^{15} + T^{16}$$
$41$ $$87310336 + 291241984 T^{2} + 329390080 T^{4} + 147362816 T^{6} + 24700224 T^{8} + 1383616 T^{10} + 31072 T^{12} + 296 T^{14} + T^{16}$$
$43$ $$2111586304 + 5140742144 T + 6257672192 T^{2} + 2619355136 T^{3} + 445188096 T^{4} - 116885504 T^{5} + 20738560 T^{6} + 12555904 T^{7} + 3493136 T^{8} - 1563008 T^{9} + 280832 T^{10} - 11072 T^{11} + 4216 T^{12} - 1568 T^{13} + 288 T^{14} - 24 T^{15} + T^{16}$$
$47$ $$( -18176 + 24832 T + 32832 T^{2} - 2880 T^{3} - 4680 T^{4} + 784 T^{5} + 68 T^{6} - 20 T^{7} + T^{8} )^{2}$$
$53$ $$133651661056 - 486103883776 T + 884003176448 T^{2} - 862648064000 T^{3} + 477213921536 T^{4} - 106174140416 T^{5} + 13721346048 T^{6} - 1343532544 T^{7} + 313823328 T^{8} - 63842816 T^{9} + 7974912 T^{10} - 532096 T^{11} + 34064 T^{12} - 4096 T^{13} + 512 T^{14} - 32 T^{15} + T^{16}$$
$59$ $$166320414976 + 95907154432 T + 27651994112 T^{2} - 13349751040 T^{3} + 11956383872 T^{4} + 4350037568 T^{5} + 1056333312 T^{6} - 372400640 T^{7} + 65216772 T^{8} + 588512 T^{9} + 336768 T^{10} - 122600 T^{11} + 21704 T^{12} - 200 T^{13} + 8 T^{14} - 4 T^{15} + T^{16}$$
$61$ $$9688858624 - 11238572032 T + 6518079488 T^{2} - 550313984 T^{3} + 395870720 T^{4} - 417754112 T^{5} + 233883648 T^{6} - 22986880 T^{7} + 1971204 T^{8} - 1097600 T^{9} + 702720 T^{10} - 58216 T^{11} + 2984 T^{12} - 520 T^{13} + 200 T^{14} - 20 T^{15} + T^{16}$$
$67$ $$2606754347155456 + 568041074851840 T + 61891267788800 T^{2} + 5089475330048 T^{3} + 2474211510272 T^{4} + 523341000704 T^{5} + 60265930752 T^{6} + 4235473408 T^{7} + 695639568 T^{8} + 130232576 T^{9} + 15464448 T^{10} + 983296 T^{11} + 49208 T^{12} + 4288 T^{13} + 512 T^{14} + 32 T^{15} + T^{16}$$
$71$ $$67108864 + 2608857088 T^{2} + 2604662784 T^{4} + 490455040 T^{6} + 38519040 T^{8} + 1498624 T^{10} + 29984 T^{12} + 288 T^{14} + T^{16}$$
$73$ $$58046357241856 + 13653050720256 T^{2} + 1216438697984 T^{4} + 53178560512 T^{6} + 1282223616 T^{8} + 17837824 T^{10} + 141696 T^{12} + 592 T^{14} + T^{16}$$
$79$ $$( -4822784 - 3951616 T - 633344 T^{2} + 107520 T^{3} + 25504 T^{4} - 704 T^{5} - 288 T^{6} + T^{8} )^{2}$$
$83$ $$401670219890944 + 220876340944384 T + 60729368986112 T^{2} + 4269696279296 T^{3} + 299011528832 T^{4} + 181577847360 T^{5} + 77333645056 T^{6} + 1337794688 T^{7} + 146493444 T^{8} + 72393072 T^{9} + 15353504 T^{10} + 633528 T^{11} + 23848 T^{12} + 4000 T^{13} + 648 T^{14} + 36 T^{15} + T^{16}$$
$89$ $$( 256 + 1536 T^{2} + 800 T^{4} + 96 T^{6} + T^{8} )^{2}$$
$97$ $$( -2571392 + 3808512 T + 3034816 T^{2} + 99840 T^{3} - 84856 T^{4} - 8256 T^{5} + 116 T^{6} + 36 T^{7} + T^{8} )^{2}$$