# Properties

 Label 920.2.e.c Level $920$ Weight $2$ Character orbit 920.e Analytic conductor $7.346$ Analytic rank $0$ Dimension $16$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$920 = 2^{3} \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 920.e (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.34623698596$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 2 x^{15} + 2 x^{14} + 6 x^{13} + 100 x^{12} - 196 x^{11} + 210 x^{10} + 702 x^{9} + 1572 x^{8} - 1594 x^{7} + 2464 x^{6} + 9568 x^{5} + 15457 x^{4} + 4336 x^{3} + 128 x^{2} - 512 x + 1024$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ 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.

 $$f(q)$$ $$=$$ $$q -\beta_{5} q^{3} -\beta_{7} q^{5} + ( -\beta_{2} + \beta_{13} ) q^{7} + ( -1 + \beta_{1} - \beta_{7} + \beta_{9} - \beta_{10} - \beta_{12} ) q^{9} +O(q^{10})$$ $$q -\beta_{5} q^{3} -\beta_{7} q^{5} + ( -\beta_{2} + \beta_{13} ) q^{7} + ( -1 + \beta_{1} - \beta_{7} + \beta_{9} - \beta_{10} - \beta_{12} ) q^{9} + ( 1 + \beta_{12} ) q^{11} + ( 1 - 2 \beta_{2} - \beta_{4} + \beta_{5} + \beta_{13} + \beta_{14} - \beta_{15} ) q^{13} + ( 2 \beta_{2} + \beta_{4} - \beta_{10} - \beta_{13} ) q^{15} + ( -\beta_{2} + \beta_{3} - \beta_{5} + \beta_{7} + \beta_{8} - \beta_{10} ) q^{17} + ( -1 + \beta_{8} - \beta_{9} - \beta_{11} ) q^{19} + ( -1 - 2 \beta_{1} + \beta_{4} - \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} + \beta_{11} + \beta_{14} ) q^{21} -\beta_{2} q^{23} + ( 1 + \beta_{1} + \beta_{5} - \beta_{6} - \beta_{10} - \beta_{11} + \beta_{15} ) q^{25} + ( -\beta_{2} - \beta_{3} + 2 \beta_{5} - \beta_{8} + \beta_{13} + \beta_{15} ) q^{27} + ( -3 - \beta_{1} + \beta_{6} - \beta_{8} + \beta_{11} ) q^{29} + ( 2 + \beta_{1} + \beta_{8} - \beta_{9} - \beta_{11} + \beta_{12} ) q^{31} + ( \beta_{2} - 2 \beta_{5} + 2 \beta_{7} - 2 \beta_{10} - \beta_{13} ) q^{33} + ( 2 - \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{6} + \beta_{9} - \beta_{10} + 2 \beta_{11} - \beta_{12} + \beta_{14} - \beta_{15} ) q^{35} + ( 1 - \beta_{3} - \beta_{4} + \beta_{5} + \beta_{7} - \beta_{10} + \beta_{11} + \beta_{14} - \beta_{15} ) q^{37} + ( -\beta_{1} + \beta_{4} - \beta_{6} - 2 \beta_{7} + \beta_{8} - \beta_{9} - 2 \beta_{10} - \beta_{11} - \beta_{12} + \beta_{14} ) q^{39} + ( 1 - \beta_{1} - \beta_{4} + 2 \beta_{6} + \beta_{7} + \beta_{10} - \beta_{14} ) q^{41} + ( 2 \beta_{2} - \beta_{3} - \beta_{5} + \beta_{11} + \beta_{15} ) q^{43} + ( 1 + 2 \beta_{1} + \beta_{2} - \beta_{4} + \beta_{6} + 2 \beta_{7} - 2 \beta_{8} + \beta_{9} + \beta_{12} - \beta_{13} - 2 \beta_{14} + \beta_{15} ) q^{45} + ( -1 + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{7} - \beta_{10} - \beta_{11} - \beta_{14} ) q^{47} + ( -5 + 2 \beta_{1} - \beta_{4} - \beta_{6} - 2 \beta_{8} + 2 \beta_{9} + 2 \beta_{11} + 2 \beta_{12} - \beta_{14} ) q^{49} + ( -2 - \beta_{4} + \beta_{6} + \beta_{8} + \beta_{9} - \beta_{11} - \beta_{12} - \beta_{14} ) q^{51} + ( -1 + \beta_{4} - 2 \beta_{5} - 2 \beta_{8} - 2 \beta_{11} - \beta_{14} + 2 \beta_{15} ) q^{53} + ( 2 - \beta_{2} + \beta_{3} + \beta_{5} - \beta_{6} - \beta_{7} + 3 \beta_{8} - \beta_{11} - \beta_{15} ) q^{55} + ( -2 + 2 \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{7} - \beta_{8} + \beta_{10} - 2 \beta_{11} - \beta_{13} - 2 \beta_{14} ) q^{57} + ( -4 - \beta_{7} - \beta_{8} - \beta_{10} + \beta_{11} ) q^{59} + ( 2 - \beta_{4} + \beta_{7} + \beta_{10} + \beta_{12} - \beta_{14} ) q^{61} + ( -1 + 6 \beta_{2} + 2 \beta_{3} + \beta_{4} - 4 \beta_{5} + 3 \beta_{7} + 3 \beta_{8} - 3 \beta_{10} + \beta_{11} - 3 \beta_{13} - \beta_{14} - \beta_{15} ) q^{63} + ( 1 + 2 \beta_{2} + \beta_{3} + 3 \beta_{5} + \beta_{6} - \beta_{8} - \beta_{10} - \beta_{12} + 2 \beta_{13} + \beta_{14} ) q^{65} + ( 1 + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{7} + 2 \beta_{8} - \beta_{10} + \beta_{11} + \beta_{14} - \beta_{15} ) q^{67} -\beta_{1} q^{69} + ( 3 + 3 \beta_{1} + \beta_{4} - 2 \beta_{7} - \beta_{8} - 2 \beta_{10} + \beta_{11} - 2 \beta_{12} + \beta_{14} ) q^{71} + ( -1 - \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{5} - 2 \beta_{7} - \beta_{8} + 2 \beta_{10} - \beta_{14} ) q^{73} + ( 2 - 2 \beta_{1} + \beta_{3} - 2 \beta_{5} + \beta_{7} + 2 \beta_{8} - \beta_{9} + 2 \beta_{10} - 2 \beta_{11} + \beta_{12} - \beta_{13} - 2 \beta_{14} ) q^{75} + ( 1 + 2 \beta_{2} - 4 \beta_{3} - \beta_{4} + 4 \beta_{5} - \beta_{7} - 3 \beta_{8} + \beta_{10} + \beta_{11} + \beta_{13} + \beta_{14} + \beta_{15} ) q^{77} + ( -1 - 4 \beta_{1} + \beta_{4} + \beta_{6} + \beta_{7} - 2 \beta_{9} + \beta_{10} - \beta_{12} + \beta_{14} ) q^{79} + ( 2 - 2 \beta_{1} + \beta_{4} - 2 \beta_{6} - 2 \beta_{7} - \beta_{9} - 2 \beta_{10} - \beta_{12} + \beta_{14} ) q^{81} + ( 4 \beta_{2} + 2 \beta_{3} - 2 \beta_{5} + \beta_{7} + 2 \beta_{8} - \beta_{10} ) q^{83} + ( 3 - \beta_{2} + \beta_{4} - \beta_{7} - \beta_{8} - \beta_{9} + 2 \beta_{11} - \beta_{12} - \beta_{13} + \beta_{14} - 2 \beta_{15} ) q^{85} + ( 1 + \beta_{2} - \beta_{3} - \beta_{4} + 4 \beta_{5} + \beta_{7} - \beta_{10} + \beta_{11} + \beta_{14} ) q^{87} + ( -5 - \beta_{4} + \beta_{6} + \beta_{7} + 2 \beta_{8} + \beta_{10} - 2 \beta_{11} + \beta_{12} - \beta_{14} ) q^{89} + ( -4 + 4 \beta_{1} - \beta_{4} - 2 \beta_{6} + 4 \beta_{9} + \beta_{12} - \beta_{14} ) q^{91} + ( -2 - \beta_{2} + \beta_{3} + 2 \beta_{4} - 4 \beta_{5} + \beta_{7} - 2 \beta_{8} - \beta_{10} - 3 \beta_{11} - \beta_{13} - 2 \beta_{14} + \beta_{15} ) q^{93} + ( 1 - 2 \beta_{1} - 3 \beta_{2} + \beta_{4} - \beta_{6} + \beta_{7} - 2 \beta_{8} - \beta_{10} + \beta_{13} + \beta_{14} + \beta_{15} ) q^{95} + ( 1 - \beta_{2} - \beta_{4} - \beta_{7} + \beta_{8} + \beta_{10} + \beta_{11} + \beta_{14} - \beta_{15} ) q^{97} + ( -6 + 4 \beta_{1} - 3 \beta_{4} + \beta_{6} + \beta_{7} + \beta_{8} + 3 \beta_{9} + \beta_{10} - \beta_{11} + \beta_{12} - 3 \beta_{14} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q - 2 q^{5} - 22 q^{9} + O(q^{10})$$ $$16 q - 2 q^{5} - 22 q^{9} + 14 q^{11} + 6 q^{15} - 22 q^{19} + 12 q^{25} - 44 q^{29} + 18 q^{31} + 20 q^{35} + 14 q^{41} + 14 q^{45} - 78 q^{49} - 38 q^{51} + 30 q^{55} - 64 q^{59} + 34 q^{61} + 6 q^{65} + 6 q^{69} + 30 q^{71} + 56 q^{75} + 4 q^{79} + 48 q^{81} + 52 q^{85} - 92 q^{89} - 70 q^{91} + 38 q^{95} - 122 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 2 x^{15} + 2 x^{14} + 6 x^{13} + 100 x^{12} - 196 x^{11} + 210 x^{10} + 702 x^{9} + 1572 x^{8} - 1594 x^{7} + 2464 x^{6} + 9568 x^{5} + 15457 x^{4} + 4336 x^{3} + 128 x^{2} - 512 x + 1024$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-564796151969688089 \nu^{15} + 241867657278618280682 \nu^{14} - 1533252977882876294850 \nu^{13} + 5615701405841246795546 \nu^{12} - 5798966345385152663476 \nu^{11} + 22923942005700239322180 \nu^{10} - 135549389895600500067810 \nu^{9} + 523491302527906903267042 \nu^{8} - 537061025633670808973620 \nu^{7} + 169048474594930667975626 \nu^{6} - 84234101814344138888912 \nu^{5} + 3918219876554594510372320 \nu^{4} - 2028057361772854751575673 \nu^{3} - 304907534037070840065976 \nu^{2} + 556650317363414170826464 \nu + 953326853515704207873024$$$$)/$$$$20\!\cdots\!20$$ $$\beta_{2}$$ $$=$$ $$($$$$-669081183175634857 \nu^{15} + 31617792777529048366 \nu^{14} - 64646323960557883050 \nu^{13} + 63600734450499465858 \nu^{12} + 101144014888243416932 \nu^{11} + 3167278369797318418580 \nu^{10} - 6371259042835319510930 \nu^{9} + 6577290238793402720266 \nu^{8} + 18861957671154867372740 \nu^{7} + 49351663880097415453818 \nu^{6} - 56255878203811954909816 \nu^{5} + 73974887245017554157920 \nu^{4} + 267645466920360925996151 \nu^{3} + 493638074924085942890092 \nu^{2} + 69896317461757032781472 \nu - 4340717231201691735808$$$$)/$$$$12\!\cdots\!20$$ $$\beta_{3}$$ $$=$$ $$($$$$-669081183175634857 \nu^{15} + 31617792777529048366 \nu^{14} - 64646323960557883050 \nu^{13} + 63600734450499465858 \nu^{12} + 101144014888243416932 \nu^{11} + 3167278369797318418580 \nu^{10} - 6371259042835319510930 \nu^{9} + 6577290238793402720266 \nu^{8} + 18861957671154867372740 \nu^{7} + 49351663880097415453818 \nu^{6} - 56255878203811954909816 \nu^{5} + 73974887245017554157920 \nu^{4} + 267645466920360925996151 \nu^{3} + 461763114540510134793612 \nu^{2} + 101771277845332840877952 \nu - 4340717231201691735808$$$$)/$$$$31\!\cdots\!80$$ $$\beta_{4}$$ $$=$$ $$($$$$-36082429475973024899 \nu^{15} + 51817197715454225402 \nu^{14} + 98141784974047919250 \nu^{13} + 80508305370487005526 \nu^{12} - 4150958833832665209716 \nu^{11} + 4424452632901492664220 \nu^{10} + 5800880410569448661690 \nu^{9} + 2054109647928134982382 \nu^{8} - 114373918633476906704980 \nu^{7} - 9697761543566197765634 \nu^{6} - 52052886724240319553832 \nu^{5} + 255183413268314405913440 \nu^{4} - 1232478353488625770263843 \nu^{3} - 2575592145277412776789596 \nu^{2} - 3335136930759889767274816 \nu - 93942894839672333928576$$$$)/$$$$50\!\cdots\!80$$ $$\beta_{5}$$ $$=$$ $$($$$$-318969229217745937361 \nu^{15} + 1841229998511317961538 \nu^{14} - 3285807383750098948290 \nu^{13} + 726871944630147495194 \nu^{12} - 23161774996655402727844 \nu^{11} + 167555889184928866487620 \nu^{10} - 316526821134642417546610 \nu^{9} + 52212436728277648932258 \nu^{8} + 458943318236379704058780 \nu^{7} + 952700810705768643736474 \nu^{6} - 2054926237083438692768608 \nu^{5} + 115121855831410995632480 \nu^{4} + 4286478475719333680330703 \nu^{3} + 2635056706542329997920176 \nu^{2} + 1194224413274039153395616 \nu - 140707114255972724633344$$$$)/$$$$20\!\cdots\!20$$ $$\beta_{6}$$ $$=$$ $$($$$$-328086612481608009763 \nu^{15} + 639944811506344667566 \nu^{14} - 508649117447187324278 \nu^{13} - 2037151664803358920002 \nu^{12} - 32676821886335924461052 \nu^{11} + 63267592666342922762540 \nu^{10} - 53565452553247896896982 \nu^{9} - 239291997595419620454186 \nu^{8} - 501938626773392038340220 \nu^{7} + 568205842884924476879182 \nu^{6} - 519643278737243550826480 \nu^{5} - 3264034951805109084713312 \nu^{4} - 4552141809370651565356611 \nu^{3} - 653589136267567895383144 \nu^{2} + 1227514889489791068838304 \nu - 1299074755592200570567680$$$$)/$$$$40\!\cdots\!44$$ $$\beta_{7}$$ $$=$$ $$($$$$841925339302843628619 \nu^{15} - 1535375885944134270142 \nu^{14} + 614001498766608317510 \nu^{13} + 7835326530712091726514 \nu^{12} + 81160127013367967422236 \nu^{11} - 152213113407290106228940 \nu^{10} + 76879314495638174174630 \nu^{9} + 854491089510550619162458 \nu^{8} + 1037048304461043182970780 \nu^{7} - 1382131543294845713179486 \nu^{6} + 1350815537979474984914032 \nu^{5} + 9891141608439985398920800 \nu^{4} + 10856809983417508211138923 \nu^{3} + 1549991709567331298025256 \nu^{2} - 3939967276533955427877024 \nu - 107593881030981543276544$$$$)/$$$$10\!\cdots\!60$$ $$\beta_{8}$$ $$=$$ $$($$$$-207962376041506841905 \nu^{15} + 531602863276948713442 \nu^{14} - 748886722375901601090 \nu^{13} - 858852135854606012262 \nu^{12} - 20011902708376888146916 \nu^{11} + 50823127396663014338116 \nu^{10} - 75638304146095329018290 \nu^{9} - 105935593052591979557342 \nu^{8} - 239794915024620709995940 \nu^{7} + 359794110659287610999514 \nu^{6} - 773356052025354301260832 \nu^{5} - 1555862406023457302487712 \nu^{4} - 2205874733214161441488273 \nu^{3} - 606086541272980490786928 \nu^{2} - 733530723301828313458784 \nu - 23061364425557027753728$$$$)/$$$$20\!\cdots\!72$$ $$\beta_{9}$$ $$=$$ $$($$$$2118185236141746471449 \nu^{15} - 3953155420784877654762 \nu^{14} + 2132038220718324369090 \nu^{13} + 18550348604166644455654 \nu^{12} + 206585678634484606502836 \nu^{11} - 391518309605053145121860 \nu^{10} + 242799571396014564554210 \nu^{9} + 2055560822793859187884318 \nu^{8} + 2808490629040326766600500 \nu^{7} - 3540845704199600616790986 \nu^{6} + 3423207863830156165574352 \nu^{5} + 25958054272212434904396320 \nu^{4} + 28059114061089377308872313 \nu^{3} + 4011928424236009986551736 \nu^{2} - 7551143469775208630023904 \nu + 4455281824367152369273856$$$$)/$$$$20\!\cdots\!20$$ $$\beta_{10}$$ $$=$$ $$($$$$216832476518453171567 \nu^{15} - 408368235739557435366 \nu^{14} + 230984654232606903502 \nu^{13} + 1835948719271709389514 \nu^{12} + 21089846935305423863020 \nu^{11} - 40411825872447519151324 \nu^{10} + 26651039266139952258734 \nu^{9} + 202760223167881268421202 \nu^{8} + 284665898450793362180972 \nu^{7} - 363797928128516647631366 \nu^{6} + 398765757204922242534320 \nu^{5} + 2423008249317927822153440 \nu^{4} + 2965824734178438435235855 \nu^{3} + 421969477678591882074120 \nu^{2} - 591486967727086100610080 \nu - 20422553595681348505600$$$$)/$$$$20\!\cdots\!72$$ $$\beta_{11}$$ $$=$$ $$($$$$-1282048923496956438741 \nu^{15} + 3164479609138396473898 \nu^{14} - 4285355384275934205450 \nu^{13} - 5638677744919485282366 \nu^{12} - 124348621205043592627444 \nu^{11} + 303961652938262561218260 \nu^{10} - 434567402565538232210490 \nu^{9} - 688987991591815620730262 \nu^{8} - 1585255762916027177913780 \nu^{7} + 2235828650644175579974914 \nu^{6} - 4509793508171907734061728 \nu^{5} - 10003211668266940224284960 \nu^{4} - 15001687353545491172481717 \nu^{3} - 3590288385190530566279984 \nu^{2} - 2610188589858814773945824 \nu - 120787935180359939517184$$$$)/$$$$10\!\cdots\!60$$ $$\beta_{12}$$ $$=$$ $$($$$$-2588229413723518163967 \nu^{15} + 4615436144052771597446 \nu^{14} - 1300938577750539720430 \nu^{13} - 23832307778719830030122 \nu^{12} - 247870842045704795997228 \nu^{11} + 458174412566717236259100 \nu^{10} - 184409721653063663205070 \nu^{9} - 2618657823981625025068594 \nu^{8} - 3075848072628037146480300 \nu^{7} + 4187584486409972800336038 \nu^{6} - 3641530117296416951306416 \nu^{5} - 29807609026592396802595040 \nu^{4} - 31880221946092013629388959 \nu^{3} - 4568188273249911843048328 \nu^{2} + 8597925159288225369863712 \nu - 3181384674636323243527168$$$$)/$$$$20\!\cdots\!20$$ $$\beta_{13}$$ $$=$$ $$($$$$-2642888945664910864623 \nu^{15} + 7849910266256606593854 \nu^{14} - 11457146342780235799230 \nu^{13} - 8707977648300977704858 \nu^{12} - 250389486494265538272092 \nu^{11} + 762176261837051384990140 \nu^{10} - 1161038922935840085830670 \nu^{9} - 1117119434130211646290466 \nu^{8} - 2503967682918694073411740 \nu^{7} + 6950964135040340249863142 \nu^{6} - 12018735453826353009247904 \nu^{5} - 17183680768812435879037280 \nu^{4} - 18490637950705990608063631 \nu^{3} + 10409412107504918728814928 \nu^{2} - 4246082060596735800332192 \nu - 448532962164852865847552$$$$)/$$$$20\!\cdots\!20$$ $$\beta_{14}$$ $$=$$ $$($$$$189621980412060243137 \nu^{15} - 390372336275624352541 \nu^{14} + 382730328923720987360 \nu^{13} + 1201904523584497153952 \nu^{12} + 19224867526049022000338 \nu^{11} - 38258112235964529343520 \nu^{10} + 40301528362365775227230 \nu^{9} + 139489491216171912857464 \nu^{8} + 318883474884996061259370 \nu^{7} - 323581731844098202246038 \nu^{6} + 478381537503435627193246 \nu^{5} + 1984719931099756217426160 \nu^{4} + 3248825928763831075053889 \nu^{3} + 1058982307105449261342133 \nu^{2} + 50175033137633848711128 \nu - 150605332666768364582752$$$$)/$$$$12\!\cdots\!20$$ $$\beta_{15}$$ $$=$$ $$($$$$-3214385228459778752201 \nu^{15} + 8944116696865206225138 \nu^{14} - 13659830033743327363890 \nu^{13} - 11093928427081377174326 \nu^{12} - 303190775204524701949444 \nu^{11} + 860187450071408174746660 \nu^{10} - 1367315172251677890979810 \nu^{9} - 1409871110534423896856302 \nu^{8} - 3059708196081617915726020 \nu^{7} + 6701984866603385326334154 \nu^{6} - 13689050286567661049021408 \nu^{5} - 21446083076286990646032800 \nu^{4} - 25100415896991282501011497 \nu^{3} + 4428663247793606577665776 \nu^{2} - 5841096957539534050870624 \nu - 495178165772403862599424$$$$)/$$$$20\!\cdots\!20$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{11} + \beta_{10} - \beta_{8} - \beta_{7}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{11} + \beta_{10} - \beta_{8} - \beta_{7} - 2 \beta_{3} + 8 \beta_{2}$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{15} - 2 \beta_{14} - 2 \beta_{13} + \beta_{12} - 6 \beta_{11} + 6 \beta_{10} + 2 \beta_{9} + 7 \beta_{8} - 6 \beta_{7} + \beta_{6} - \beta_{5} + \beta_{3} + 2 \beta_{2} - 2$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$2 \beta_{12} - 7 \beta_{11} - 5 \beta_{10} - 4 \beta_{9} + 7 \beta_{8} - 5 \beta_{7} - 18 \beta_{6} + 16 \beta_{1} - 54$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-11 \beta_{15} + 22 \beta_{13} + 11 \beta_{12} - 39 \beta_{11} - 39 \beta_{10} + 22 \beta_{9} + 32 \beta_{8} + 45 \beta_{7} + 13 \beta_{6} + 7 \beta_{5} - 20 \beta_{4} - 13 \beta_{3} - 10 \beta_{2} - 6 \beta_{1} + 10$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$-13 \beta_{15} - 25 \beta_{13} - 13 \beta_{11} - 18 \beta_{10} + 65 \beta_{8} + 18 \beta_{7} - 12 \beta_{5} + 78 \beta_{3} - 207 \beta_{2}$$ $$\nu^{7}$$ $$=$$ $$($$$$-106 \beta_{15} + 180 \beta_{14} + 206 \beta_{13} - 106 \beta_{12} + 363 \beta_{11} - 363 \beta_{10} - 206 \beta_{9} - 413 \beta_{8} + 269 \beta_{7} - 144 \beta_{6} + 52 \beta_{5} - 144 \beta_{3} + 12 \beta_{2} + 92 \beta_{1} - 12$$$$)/2$$ $$\nu^{8}$$ $$=$$ $$($$$$-16 \beta_{14} - 250 \beta_{12} + 113 \beta_{11} + 153 \beta_{10} + 520 \beta_{9} - 113 \beta_{8} + 153 \beta_{7} + 1364 \beta_{6} - 16 \beta_{4} - 1122 \beta_{1} + 3352$$$$)/2$$ $$\nu^{9}$$ $$=$$ $$($$$$957 \beta_{15} - 1868 \beta_{13} - 957 \beta_{12} + 1948 \beta_{11} + 1948 \beta_{10} - 1868 \beta_{9} - 1513 \beta_{8} - 3018 \beta_{7} - 1505 \beta_{6} - 435 \beta_{5} + 1574 \beta_{4} + 1505 \beta_{3} - 940 \beta_{2} + 1070 \beta_{1} - 2514$$$$)/2$$ $$\nu^{10}$$ $$=$$ $$($$$$2136 \beta_{15} + 366 \beta_{14} + 5172 \beta_{13} + 1063 \beta_{11} - 497 \beta_{10} - 10989 \beta_{8} + 497 \beta_{7} + 2304 \beta_{5} - 366 \beta_{4} - 12052 \beta_{3} + 27654 \beta_{2} + 366$$$$)/2$$ $$\nu^{11}$$ $$=$$ $$($$$$8319 \beta_{15} - 13624 \beta_{14} - 16858 \beta_{13} + 8319 \beta_{12} - 25513 \beta_{11} + 25513 \beta_{10} + 16858 \beta_{9} + 29978 \beta_{8} - 14759 \beta_{7} + 15219 \beta_{6} - 3931 \beta_{5} + 15219 \beta_{3} - 14772 \beta_{2} - 11288 \beta_{1} + 14772$$$$)/2$$ $$\nu^{12}$$ $$=$$ $$2763 \beta_{14} + 8523 \beta_{12} + 7616 \beta_{11} - 4366 \beta_{10} - 25265 \beta_{9} - 7616 \beta_{8} - 4366 \beta_{7} - 53720 \beta_{6} + 2763 \beta_{4} + 43038 \beta_{1} - 119772$$ $$\nu^{13}$$ $$=$$ $$($$$$-70750 \beta_{15} + 152444 \beta_{13} + 70750 \beta_{12} - 116569 \beta_{11} - 116569 \beta_{10} + 152444 \beta_{9} + 67759 \beta_{8} + 218389 \beta_{7} + 150630 \beta_{6} + 36830 \beta_{5} - 117752 \beta_{4} - 150630 \beta_{3} + 180488 \beta_{2} - 113800 \beta_{1} + 298240$$$$)/2$$ $$\nu^{14}$$ $$=$$ $$($$$$-129196 \beta_{15} - 69708 \beta_{14} - 488396 \beta_{13} - 80835 \beta_{11} + 214549 \beta_{10} + 884019 \beta_{8} - 214549 \beta_{7} - 195824 \beta_{5} + 69708 \beta_{4} + 964854 \beta_{3} - 2013820 \beta_{2} - 69708$$$$)/2$$ $$\nu^{15}$$ $$=$$ $$($$$$-594527 \beta_{15} + 1021262 \beta_{14} + 1383542 \beta_{13} - 594527 \beta_{12} + 1890076 \beta_{11} - 1890076 \beta_{10} - 1383542 \beta_{9} - 2423779 \beta_{8} + 954884 \beta_{7} - 1468895 \beta_{6} + 349447 \beta_{5} - 1468895 \beta_{3} + 1986426 \beta_{2} + 1119448 \beta_{1} - 1986426$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/920\mathbb{Z}\right)^\times$$.

 $$n$$ $$231$$ $$281$$ $$461$$ $$737$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-1$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
369.1
 −0.945903 + 0.945903i 1.65386 − 1.65386i 1.81400 + 1.81400i −1.13508 − 1.13508i −2.15699 + 2.15699i 0.303680 + 0.303680i 1.95202 − 1.95202i −0.485591 + 0.485591i −0.485591 − 0.485591i 1.95202 + 1.95202i 0.303680 − 0.303680i −2.15699 − 2.15699i −1.13508 + 1.13508i 1.81400 − 1.81400i 1.65386 + 1.65386i −0.945903 − 0.945903i
0 3.20935i 0 2.02615 + 0.945903i 0 4.54713i 0 −7.29996 0
369.2 0 2.89996i 0 −1.50491 1.65386i 0 0.580879i 0 −5.40975 0
369.3 0 2.54092i 0 −1.30744 + 1.81400i 0 0.780573i 0 −3.45626 0
369.4 0 2.51561i 0 −1.92655 1.13508i 0 4.64022i 0 −3.32827 0
369.5 0 1.69755i 0 0.589417 + 2.15699i 0 4.22860i 0 0.118308 0
369.6 0 0.540724i 0 2.21535 + 0.303680i 0 1.15693i 0 2.70762 0
369.7 0 0.493532i 0 1.09069 1.95202i 0 4.54439i 0 2.75643 0
369.8 0 0.296848i 0 −2.18271 + 0.485591i 0 3.46037i 0 2.91188 0
369.9 0 0.296848i 0 −2.18271 0.485591i 0 3.46037i 0 2.91188 0
369.10 0 0.493532i 0 1.09069 + 1.95202i 0 4.54439i 0 2.75643 0
369.11 0 0.540724i 0 2.21535 0.303680i 0 1.15693i 0 2.70762 0
369.12 0 1.69755i 0 0.589417 2.15699i 0 4.22860i 0 0.118308 0
369.13 0 2.51561i 0 −1.92655 + 1.13508i 0 4.64022i 0 −3.32827 0
369.14 0 2.54092i 0 −1.30744 1.81400i 0 0.780573i 0 −3.45626 0
369.15 0 2.89996i 0 −1.50491 + 1.65386i 0 0.580879i 0 −5.40975 0
369.16 0 3.20935i 0 2.02615 0.945903i 0 4.54713i 0 −7.29996 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 369.16 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 920.2.e.c 16
4.b odd 2 1 1840.2.e.h 16
5.b even 2 1 inner 920.2.e.c 16
5.c odd 4 1 4600.2.a.bj 8
5.c odd 4 1 4600.2.a.bk 8
20.d odd 2 1 1840.2.e.h 16
20.e even 4 1 9200.2.a.dd 8
20.e even 4 1 9200.2.a.de 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
920.2.e.c 16 1.a even 1 1 trivial
920.2.e.c 16 5.b even 2 1 inner
1840.2.e.h 16 4.b odd 2 1
1840.2.e.h 16 20.d odd 2 1
4600.2.a.bj 8 5.c odd 4 1
4600.2.a.bk 8 5.c odd 4 1
9200.2.a.dd 8 20.e even 4 1
9200.2.a.de 8 20.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{16} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(920, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16}$$
$3$ $$64 + 1264 T^{2} + 7441 T^{4} + 16123 T^{6} + 10815 T^{8} + 3218 T^{10} + 479 T^{12} + 35 T^{14} + T^{16}$$
$5$ $$390625 + 156250 T - 62500 T^{2} - 43750 T^{3} + 7500 T^{4} + 750 T^{5} - 4500 T^{6} + 350 T^{7} + 1462 T^{8} + 70 T^{9} - 180 T^{10} + 6 T^{11} + 12 T^{12} - 14 T^{13} - 4 T^{14} + 2 T^{15} + T^{16}$$
$7$ $$541696 + 3052288 T^{2} + 5335056 T^{4} + 3332320 T^{6} + 703096 T^{8} + 69892 T^{10} + 3621 T^{12} + 95 T^{14} + T^{16}$$
$11$ $$( 440 + 5456 T + 28 T^{2} - 2364 T^{3} + 162 T^{4} + 248 T^{5} - 29 T^{6} - 7 T^{7} + T^{8} )^{2}$$
$13$ $$196560400 + 327271176 T^{2} + 155327225 T^{4} + 32428583 T^{6} + 3541415 T^{8} + 216522 T^{10} + 7455 T^{12} + 135 T^{14} + T^{16}$$
$17$ $$30976 + 954496 T^{2} + 7831888 T^{4} + 9120800 T^{6} + 2121148 T^{8} + 194292 T^{10} + 7957 T^{12} + 147 T^{14} + T^{16}$$
$19$ $$( 11192 + 23600 T + 16488 T^{2} + 3044 T^{3} - 1116 T^{4} - 432 T^{5} - 7 T^{6} + 11 T^{7} + T^{8} )^{2}$$
$23$ $$( 1 + T^{2} )^{8}$$
$29$ $$( -400 + 4632 T - 344 T^{2} - 5614 T^{3} - 2107 T^{4} + 96 T^{5} + 146 T^{6} + 22 T^{7} + T^{8} )^{2}$$
$31$ $$( -287276 + 183760 T + 5147 T^{2} - 20781 T^{3} + 1671 T^{4} + 766 T^{5} - 79 T^{6} - 9 T^{7} + T^{8} )^{2}$$
$37$ $$259081216 + 726466304 T^{2} + 595966272 T^{4} + 149578688 T^{6} + 15894384 T^{8} + 798160 T^{10} + 19612 T^{12} + 228 T^{14} + T^{16}$$
$41$ $$( 1197584 + 601624 T - 181313 T^{2} - 44475 T^{3} + 8683 T^{4} + 1026 T^{5} - 163 T^{6} - 7 T^{7} + T^{8} )^{2}$$
$43$ $$59754824704 + 40835442432 T^{2} + 9321304384 T^{4} + 976676544 T^{6} + 53921280 T^{8} + 1671776 T^{10} + 29216 T^{12} + 268 T^{14} + T^{16}$$
$47$ $$5405190400 + 16359800384 T^{2} + 6859754432 T^{4} + 1072408548 T^{6} + 74069425 T^{8} + 2534192 T^{10} + 43630 T^{12} + 348 T^{14} + T^{16}$$
$53$ $$41160294400 + 157200080896 T^{2} + 50247134208 T^{4} + 5668106240 T^{6} + 263452352 T^{8} + 6030688 T^{10} + 71956 T^{12} + 428 T^{14} + T^{16}$$
$59$ $$( 29696 + 1536 T - 24896 T^{2} - 9024 T^{3} + 2464 T^{4} + 1872 T^{5} + 376 T^{6} + 32 T^{7} + T^{8} )^{2}$$
$61$ $$( 200 + 320 T - 592 T^{2} - 1276 T^{3} - 400 T^{4} + 208 T^{5} + 49 T^{6} - 17 T^{7} + T^{8} )^{2}$$
$67$ $$463227249664 + 197300240128 T^{2} + 32831463744 T^{4} + 2783972928 T^{6} + 130740720 T^{8} + 3460880 T^{10} + 50236 T^{12} + 364 T^{14} + T^{16}$$
$71$ $$( -8900000 + 4219360 T + 126317 T^{2} - 242207 T^{3} + 10847 T^{4} + 3982 T^{5} - 253 T^{6} - 15 T^{7} + T^{8} )^{2}$$
$73$ $$311006982400 + 335892925696 T^{2} + 85410527072 T^{4} + 8280399056 T^{6} + 381455873 T^{8} + 8909688 T^{10} + 103742 T^{12} + 552 T^{14} + T^{16}$$
$79$ $$( -5248 + 34496 T - 13136 T^{2} - 11992 T^{3} + 4108 T^{4} + 984 T^{5} - 206 T^{6} - 2 T^{7} + T^{8} )^{2}$$
$83$ $$17501839590400 + 4535098470144 T^{2} + 474639779904 T^{4} + 25914139904 T^{6} + 795198720 T^{8} + 13783616 T^{10} + 129136 T^{12} + 592 T^{14} + T^{16}$$
$89$ $$( 12804160 + 4496608 T - 440608 T^{2} - 349256 T^{3} - 36348 T^{4} + 2600 T^{5} + 710 T^{6} + 46 T^{7} + T^{8} )^{2}$$
$97$ $$8761600 + 59560576 T^{2} + 68784016 T^{4} + 29024032 T^{6} + 5390408 T^{8} + 460476 T^{10} + 17509 T^{12} + 255 T^{14} + T^{16}$$