Properties

Label 1840.2.e.h
Level $1840$
Weight $2$
Character orbit 1840.e
Analytic conductor $14.692$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1840.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(14.6924739719\)
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: no (minimal twist has level 920)
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.

\(f(q)\) \(=\) \( q -\beta_{5} q^{3} -\beta_{10} q^{5} + ( -\beta_{2} + \beta_{13} ) q^{7} + ( -1 + \beta_{1} - \beta_{7} + \beta_{9} - \beta_{10} - \beta_{12} ) q^{9} +O(q^{10})\) \( q -\beta_{5} q^{3} -\beta_{10} 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} + ( -1 + 2 \beta_{2} + \beta_{7} - \beta_{13} - \beta_{14} ) 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_{7} + \beta_{8} - \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} + ( \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} + \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} + ( 2 + 2 \beta_{1} - \beta_{2} - 2 \beta_{4} + \beta_{6} + \beta_{9} + 2 \beta_{10} + 2 \beta_{11} + \beta_{12} + \beta_{13} - \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_{10} + 2 \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} + ( -2 \beta_{2} - \beta_{3} + \beta_{4} - 3 \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} + 2 \beta_{11} - \beta_{12} - 2 \beta_{13} ) 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} + ( -4 + 2 \beta_{1} + \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} - \beta_{12} - \beta_{13} ) 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} - 2 \beta_{8} - \beta_{9} - \beta_{10} + \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} - \beta_{10} - 2 \beta_{11} + \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})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 2q^{5} - 22q^{9} + O(q^{10}) \) \( 16q - 2q^{5} - 22q^{9} - 14q^{11} - 6q^{15} + 22q^{19} + 12q^{25} - 44q^{29} - 18q^{31} - 20q^{35} + 14q^{41} + 14q^{45} - 78q^{49} + 38q^{51} - 30q^{55} + 64q^{59} + 34q^{61} + 6q^{65} + 6q^{69} - 30q^{71} - 56q^{75} - 4q^{79} + 48q^{81} + 52q^{85} - 92q^{89} + 70q^{91} - 38q^{95} + 122q^{99} + O(q^{100}) \)

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 \(\left(\mathbb{Z}/1840\mathbb{Z}\right)^\times\).

\(n\) \(737\) \(1151\) \(1201\) \(1381\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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:

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

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1840, [\chi])\):

\(T_{3}^{16} + \cdots\)
\(T_{7}^{16} + \cdots\)

Hecke characteristic polynomials

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