Properties

Label 6.23.b.a
Level $6$
Weight $23$
Character orbit 6.b
Analytic conductor $18.402$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 6 = 2 \cdot 3 \)
Weight: \( k \) \(=\) \( 23 \)
Character orbit: \([\chi]\) \(=\) 6.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(18.4024460905\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - 4 x^{7} - 12878974 x^{6} - 2567056924 x^{5} + 47744458496177 x^{4} + 16683879235776608 x^{3} - 38709208854294504224 x^{2} - 8131330234969024044480 x + 9835296816172879412979984\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{53}\cdot 3^{29} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} + ( -8715 + 9 \beta_{1} - \beta_{2} ) q^{3} -2097152 q^{4} + ( 2752 \beta_{1} - \beta_{2} + \beta_{3} ) q^{5} + ( 19544064 + 8716 \beta_{1} - 2 \beta_{3} - \beta_{4} ) q^{6} + ( 705697730 - 913 \beta_{1} - 2855 \beta_{2} + \beta_{3} + 3 \beta_{5} - \beta_{7} ) q^{7} + 2097152 \beta_{1} q^{8} + ( -6699041127 - 3311065 \beta_{1} + 8622 \beta_{2} + 208 \beta_{3} - 18 \beta_{4} + \beta_{5} + 15 \beta_{6} - 23 \beta_{7} ) q^{9} +O(q^{10})\) \( q -\beta_{1} q^{2} + ( -8715 + 9 \beta_{1} - \beta_{2} ) q^{3} -2097152 q^{4} + ( 2752 \beta_{1} - \beta_{2} + \beta_{3} ) q^{5} + ( 19544064 + 8716 \beta_{1} - 2 \beta_{3} - \beta_{4} ) q^{6} + ( 705697730 - 913 \beta_{1} - 2855 \beta_{2} + \beta_{3} + 3 \beta_{5} - \beta_{7} ) q^{7} + 2097152 \beta_{1} q^{8} + ( -6699041127 - 3311065 \beta_{1} + 8622 \beta_{2} + 208 \beta_{3} - 18 \beta_{4} + \beta_{5} + 15 \beta_{6} - 23 \beta_{7} ) q^{9} + ( 5772595200 - 66858 \beta_{1} - 209120 \beta_{2} + 36 \beta_{3} + 58 \beta_{4} - 16 \beta_{5} + 112 \beta_{6} + 32 \beta_{7} ) q^{10} + ( -30189228 \beta_{1} + 1008686 \beta_{2} - 2041 \beta_{3} + 746 \beta_{4} + 307 \beta_{5} + 152 \beta_{6} + 132 \beta_{7} ) q^{11} + ( 18276679680 - 18874368 \beta_{1} + 2097152 \beta_{2} ) q^{12} + ( -359460297670 - 1814710 \beta_{1} - 5673310 \beta_{2} + 760 \beta_{3} + 3700 \beta_{4} - 100 \beta_{5} + 2620 \beta_{6} + 310 \beta_{7} ) q^{13} + ( -705512082 \beta_{1} + 589280 \beta_{2} + 20688 \beta_{3} - 6656 \beta_{4} - 2416 \beta_{5} + 1552 \beta_{6} - 1824 \beta_{7} ) q^{14} + ( -91483580160 + 3079898313 \beta_{1} + 1945539 \beta_{2} - 52041 \beta_{3} + 5664 \beta_{4} + 5157 \beta_{5} - 10854 \beta_{6} - 3429 \beta_{7} ) q^{15} + 4398046511104 q^{16} + ( -6032431796 \beta_{1} - 78248420 \beta_{2} - 276822 \beta_{3} - 31624 \beta_{4} - 14222 \beta_{5} - 17314 \beta_{6} - 3180 \beta_{7} ) q^{17} + ( -6949462855680 + 6694716977 \beta_{1} - 13508064 \beta_{2} + 161596 \beta_{3} - 25722 \beta_{4} - 72080 \beta_{5} + 4848 \beta_{6} + 2848 \beta_{7} ) q^{18} + ( -39350494693942 + 49316040 \beta_{1} + 154731338 \beta_{2} - 99687 \beta_{3} + 407846 \beta_{4} - 129275 \beta_{5} - 64762 \beta_{6} - 37718 \beta_{7} ) q^{19} + ( -5771362304 \beta_{1} + 2097152 \beta_{2} - 2097152 \beta_{3} ) q^{20} + ( 82699220760042 + 14998548888 \beta_{1} - 726307127 \beta_{2} - 3736395 \beta_{3} + 631152 \beta_{4} - 104922 \beta_{5} + 414882 \beta_{6} - 151416 \beta_{7} ) q^{21} + ( -63987013939200 - 152877170 \beta_{1} - 476669024 \beta_{2} - 9484 \beta_{3} + 1221314 \beta_{4} + 286384 \beta_{5} - 35536 \beta_{6} - 287584 \beta_{7} ) q^{22} + ( 95765451216 \beta_{1} + 934143700 \beta_{2} - 4436978 \beta_{3} - 2070596 \beta_{4} - 726658 \beta_{5} + 707164 \beta_{6} - 617280 \beta_{7} ) q^{23} + ( -40986872905728 - 18278776832 \beta_{1} + 4194304 \beta_{3} + 2097152 \beta_{4} ) q^{24} + ( -783052276610615 - 1121057950 \beta_{1} - 3495862470 \beta_{2} - 708320 \beta_{3} + 10220420 \beta_{4} - 1686380 \beta_{5} + 1499420 \beta_{6} - 479970 \beta_{7} ) q^{25} + ( 358196955910 \beta_{1} - 3998549120 \beta_{2} - 44000960 \beta_{3} - 2385920 \beta_{4} - 1009600 \beta_{5} - 735680 \beta_{6} - 366720 \beta_{7} ) q^{26} + ( 1469597146856685 - 1104423642741 \beta_{1} + 5633385111 \beta_{2} - 67817607 \beta_{3} + 715338 \beta_{4} - 3514839 \beta_{5} - 4124772 \beta_{6} - 1850874 \beta_{7} ) q^{27} + ( -1479955405864960 + 1914699776 \beta_{1} + 5987368960 \beta_{2} - 2097152 \beta_{3} - 6291456 \beta_{5} + 2097152 \beta_{7} ) q^{28} + ( 2062306845496 \beta_{1} - 11020914499 \beta_{2} + 68647903 \beta_{3} + 5799408 \beta_{4} + 1754004 \beta_{5} - 4511892 \beta_{6} + 2291400 \beta_{7} ) q^{29} + ( 6457682101800960 + 89044909470 \beta_{1} - 7523327904 \beta_{2} + 184521780 \beta_{3} - 24713262 \beta_{4} + 435024 \beta_{5} - 5395248 \beta_{6} - 4395168 \beta_{7} ) q^{30} + ( -13786653531151822 - 3387656315 \beta_{1} - 10646226945 \beta_{2} + 11814785 \beta_{3} - 49768220 \beta_{4} + 26926655 \beta_{5} - 2079740 \beta_{6} - 2257215 \beta_{7} ) q^{31} -4398046511104 \beta_{1} q^{32} + ( 32011439870931840 + 7329027834309 \beta_{1} - 4345210494 \beta_{2} + 63801570 \beta_{3} + 18229578 \beta_{4} + 39069837 \beta_{5} + 53842563 \beta_{6} + 33748299 \beta_{7} ) q^{33} + ( -12598735349366784 + 28095434440 \beta_{1} + 87769162112 \beta_{2} - 4189648 \beta_{3} - 113041416 \beta_{4} + 7490880 \beta_{5} - 36965568 \beta_{6} + 2612608 \beta_{7} ) q^{34} + ( -28812212854836 \beta_{1} + 365516040268 \beta_{2} + 1186245232 \beta_{3} + 167556220 \beta_{4} + 73690940 \beta_{5} + 76771690 \beta_{6} + 20174340 \beta_{7} ) q^{35} + ( 14048907497570304 + 6943806586880 \beta_{1} - 18081644544 \beta_{2} - 436207616 \beta_{3} + 37748736 \beta_{4} - 2097152 \beta_{5} - 31457280 \beta_{6} + 48234496 \beta_{7} ) q^{36} + ( -94061295820410550 - 23237775218 \beta_{1} - 73182253802 \beta_{2} + 65294264 \beta_{3} - 393540868 \beta_{4} + 11674180 \beta_{5} + 79163252 \beta_{6} + 79332722 \beta_{7} ) q^{37} + ( 39094607992134 \beta_{1} - 801005949920 \beta_{2} + 853718576 \beta_{3} - 27452928 \beta_{4} - 37006224 \beta_{5} - 236970768 \beta_{6} + 46559520 \beta_{7} ) q^{38} + ( 179655265286535810 - 62479935934380 \beta_{1} + 273625909060 \beta_{2} + 3175331490 \beta_{3} - 392307300 \beta_{4} + 106918650 \beta_{5} - 47084490 \beta_{6} - 134945730 \beta_{7} ) q^{39} + ( -12106009568870400 + 140211388416 \beta_{1} + 438556426240 \beta_{2} - 75497472 \beta_{3} - 121634816 \beta_{4} + 33554432 \beta_{5} - 234881024 \beta_{6} - 67108864 \beta_{7} ) q^{40} + ( 175099277363040 \beta_{1} + 458474738426 \beta_{2} - 6455261818 \beta_{3} - 85900000 \beta_{4} - 16217024 \beta_{5} + 154696784 \beta_{6} - 53465952 \beta_{7} ) q^{41} + ( 31939488225914880 - 82716602868616 \beta_{1} - 61435721376 \beta_{2} - 4644223828 \beta_{3} - 967415858 \beta_{4} - 600027696 \beta_{5} - 822341808 \beta_{6} - 158733216 \beta_{7} ) q^{42} + ( -162626945430259270 - 823637865924 \beta_{1} - 2572543316946 \beta_{2} + 95287617 \beta_{3} + 3653054286 \beta_{4} - 110661675 \beta_{5} + 988620846 \beta_{6} - 193073514 \beta_{7} ) q^{43} + ( 63311399878656 \beta_{1} - 2115367862272 \beta_{2} + 4280287232 \beta_{3} - 1564475392 \beta_{4} - 643825664 \beta_{5} - 318767104 \beta_{6} - 276824064 \beta_{7} ) q^{44} + ( -614551064261218560 - 294608523614262 \beta_{1} + 74884637997 \beta_{2} - 8729505021 \beta_{3} + 5811142068 \beta_{4} + 382875234 \beta_{5} + 1996647822 \beta_{6} + 156474582 \beta_{7} ) q^{45} + ( 200204874605912064 + 1377832318340 \beta_{1} + 4310078875328 \beta_{2} - 1219013672 \beta_{3} + 198952156 \beta_{4} - 2068409440 \beta_{5} - 1263616352 \beta_{6} + 262190272 \beta_{7} ) q^{46} + ( 173348779873720 \beta_{1} + 9870179977800 \beta_{2} - 18810460600 \beta_{3} + 431081320 \beta_{4} + 489691040 \beta_{5} + 2898434740 \beta_{6} - 548300760 \beta_{7} ) q^{47} + ( -38328975344271360 + 39582418599936 \beta_{1} - 4398046511104 \beta_{2} ) q^{48} + ( 1503673921707572595 - 2158943905490 \beta_{1} - 6744444885866 \beta_{2} + 1079011904 \beta_{3} + 6912424988 \beta_{4} + 3648691180 \beta_{5} + 919995524 \beta_{6} - 1950665134 \beta_{7} ) q^{49} + ( 777701343899895 \beta_{1} - 16791448652160 \beta_{2} - 25468325440 \beta_{3} - 3897128960 \beta_{4} - 1993165120 \beta_{5} - 4298534720 \beta_{6} + 89201280 \beta_{7} ) q^{50} + ( -2316518993519764992 - 1249462537977240 \beta_{1} - 130124027556 \beta_{2} + 10990511046 \beta_{3} - 9751880376 \beta_{4} - 2542378158 \beta_{5} + 2838895290 \beta_{6} - 748686456 \beta_{7} ) q^{51} + ( 753842882179235840 + 3805722705920 \beta_{1} + 11897793413120 \beta_{2} - 1593835520 \beta_{3} - 7759462400 \beta_{4} + 209715200 \beta_{5} - 5494538240 \beta_{6} - 650117120 \beta_{7} ) q^{52} + ( 2645469017455744 \beta_{1} + 5251721742155 \beta_{2} + 115492682293 \beta_{3} + 9262886976 \beta_{4} + 3571665408 \beta_{5} - 275115744 \beta_{6} + 2119556160 \beta_{7} ) q^{53} + ( -2319960273043691520 - 1467448944063474 \beta_{1} + 6750606069216 \beta_{2} + 60104638650 \beta_{3} - 6749946891 \beta_{4} - 654447984 \beta_{5} - 12014536176 \beta_{6} - 32331552 \beta_{7} ) q^{54} + ( 7057010697097996800 + 709822830998 \beta_{1} + 2159255994870 \beta_{2} + 4453954984 \beta_{3} - 47371167148 \beta_{4} - 6712166204 \beta_{5} + 7811267828 \beta_{6} + 11714442258 \beta_{7} ) q^{55} + ( 1479566073790464 \beta_{1} - 1235809730560 \beta_{2} - 43385880576 \beta_{3} + 13958643712 \beta_{4} + 5066719232 \beta_{5} - 3254779904 \beta_{6} + 3825205248 \beta_{7} ) q^{56} + ( -4479097878041641710 - 12247004399120091 \beta_{1} + 33215563659184 \beta_{2} - 84774232962 \beta_{3} + 9276190794 \beta_{4} + 24623270055 \beta_{5} - 8278186311 \beta_{6} + 6088591179 \beta_{7} ) q^{57} + ( 4332392509767905280 - 8241382510302 \beta_{1} - 25789279212704 \beta_{2} + 7414821420 \beta_{3} - 6655832306 \beta_{4} + 8111239376 \beta_{5} + 10353396304 \beta_{6} + 1541705312 \beta_{7} ) q^{58} + ( 17945224795400872 \beta_{1} + 29057578246678 \beta_{2} + 263941061253 \beta_{3} + 5157716186 \beta_{4} + 2875897429 \beta_{5} + 7831070210 \beta_{6} - 594078672 \beta_{7} ) q^{59} + ( 191854973099704320 - 6459014906904576 \beta_{1} - 4080091004928 \beta_{2} + 109137887232 \beta_{3} - 11878268928 \beta_{4} - 10815012864 \beta_{5} + 22762487808 \beta_{6} + 7191134208 \beta_{7} ) q^{60} + ( -15551910176478535846 - 2135066050570 \beta_{1} - 6638362666146 \beta_{2} + 2876620984 \beta_{3} + 20828835788 \beta_{4} + 31137767380 \beta_{5} - 14628497116 \beta_{6} - 18070276854 \beta_{7} ) q^{61} + ( 13815482375415102 \beta_{1} + 90419292643680 \beta_{2} + 156792374800 \beta_{3} - 22148907520 \beta_{4} - 5056562480 \beta_{5} + 32012114000 \beta_{6} - 12035782560 \beta_{7} ) q^{62} + ( 27913211752949648370 - 40260507365354795 \beta_{1} - 116396367704397 \beta_{2} + 289758823919 \beta_{3} - 22869050256 \beta_{4} - 22194236467 \beta_{5} + 15920054184 \beta_{6} - 12391694527 \beta_{7} ) q^{63} -9223372036854775808 q^{64} + ( 64178352005099080 \beta_{1} - 265612472499290 \beta_{2} - 1510003464210 \beta_{3} - 10696186000 \beta_{4} - 12914706500 \beta_{5} - 78795707500 \beta_{6} + 15133227000 \beta_{7} ) q^{65} + ( 15373119505064251392 - 31998593248506378 \beta_{1} + 39862505773536 \beta_{2} - 831918290364 \beta_{3} + 134797812282 \beta_{4} + 46281865104 \beta_{5} + 62409181968 \beta_{6} - 22478372640 \beta_{7} ) q^{66} + ( -43709142157848770710 + 100785335908146 \beta_{1} + 315463581413556 \beta_{2} - 108414482271 \beta_{3} + 170735948274 \beta_{4} - 152486999247 \beta_{5} - 110303284686 \beta_{6} - 9190322952 \beta_{7} ) q^{67} + ( 12650926405844992 \beta_{1} + 164098830499840 \beta_{2} + 580537810944 \beta_{3} + 66320334848 \beta_{4} + 29825695744 \beta_{5} + 36310089728 \beta_{6} + 6668943360 \beta_{7} ) q^{68} + ( 27220316704895077632 - 64812857230946370 \beta_{1} - 48367571569524 \beta_{2} - 341559026256 \beta_{3} - 4804767204 \beta_{4} - 263114925102 \beta_{5} - 162112386690 \beta_{6} - 457802334 \beta_{7} ) q^{69} + ( -60667426759674716160 - 133795257598516 \beta_{1} - 417975078262720 \beta_{2} + 23787325192 \beta_{3} + 527337862996 \beta_{4} - 16180478752 \beta_{5} + 167505255904 \beta_{6} - 19997498816 \beta_{7} ) q^{70} + ( 91765398069688456 \beta_{1} - 510501979099364 \beta_{2} - 509978764362 \beta_{3} - 645434450332 \beta_{4} - 253488619586 \beta_{5} - 22377000112 \beta_{6} - 138457211160 \beta_{7} ) q^{71} + ( 14574079926715023360 - 14039839097749504 \beta_{1} + 28328463433728 \beta_{2} - 338891374592 \beta_{3} + 53942943744 \beta_{4} + 151162716160 \beta_{5} - 10166992896 \beta_{6} - 5972688896 \beta_{7} ) q^{72} + ( -8197047398560007950 - 86843553463088 \beta_{1} - 271620851730096 \beta_{2} + 72751282160 \beta_{3} + 20064056416 \beta_{4} + 129619163984 \beta_{5} + 76167295936 \beta_{6} - 22200247632 \beta_{7} ) q^{73} + ( 94306365295012726 \beta_{1} + 766594424887424 \beta_{2} - 1910912293696 \beta_{3} + 278611601408 \beta_{4} + 127752490432 \beta_{5} + 174631808960 \beta_{6} + 23106620544 \beta_{7} ) q^{74} + ( 115475498594882625885 - 256254044001909525 \beta_{1} + 596441551946045 \beta_{2} + 1081422894330 \beta_{3} - 124285261140 \beta_{4} + 485877638970 \beta_{5} - 109798740810 \beta_{6} - 58237837650 \beta_{7} ) q^{75} + ( 82523968648389853184 - 103423231918080 \beta_{1} - 324495134949376 \beta_{2} + 209058791424 \beta_{3} - 855315054592 \beta_{4} + 271109324800 \beta_{5} + 135815757824 \beta_{6} + 79100379136 \beta_{7} ) q^{76} + ( 340174078201403488 \beta_{1} + 374597990518150 \beta_{2} + 5830568656458 \beta_{3} + 1024371090176 \beta_{4} + 387805977328 \beta_{5} - 95045019664 \beta_{6} + 248759135520 \beta_{7} ) q^{77} + ( -131211712408320430080 - 179680332804373960 \beta_{1} - 75396415800960 \beta_{2} + 3397027189580 \beta_{3} + 42223732870 \beta_{4} - 362861449920 \beta_{5} + 403267740480 \beta_{6} + 52380190080 \beta_{7} ) q^{78} + ( -40654070429041294318 + 294121470028245 \beta_{1} + 917930790307247 \beta_{2} - 6707435823 \beta_{3} - 1789805808796 \beta_{4} - 210820545425 \beta_{5} - 169393406908 \beta_{6} + 282102377233 \beta_{7} ) q^{79} + ( 12103423998558208 \beta_{1} - 4398046511104 \beta_{2} + 4398046511104 \beta_{3} ) q^{80} + ( -347521275415049455503 - 416767042038658896 \beta_{1} - 1646214685667298 \beta_{2} - 4029611285844 \beta_{3} - 886526848224 \beta_{4} - 249670057878 \beta_{5} + 429261008598 \beta_{6} + 369518529456 \beta_{7} ) q^{81} + ( 366899182870486179840 + 507318152328676 \beta_{1} + 1587335496684736 \beta_{2} - 374226380776 \beta_{3} + 108048789884 \beta_{4} - 144816256352 \beta_{5} - 780424399456 \beta_{6} - 213424305472 \beta_{7} ) q^{82} + ( 378005490351390868 \beta_{1} + 112324311177450 \beta_{2} + 722904724337 \beta_{3} + 1733467221454 \beta_{4} + 638375569997 \beta_{5} - 321755145116 \beta_{6} + 456716081460 \beta_{7} ) q^{83} + ( -173432836215363600384 - 31454236797566976 \beta_{1} + 1523176444002304 \beta_{2} + 7835788247040 \beta_{3} - 1323621679104 \beta_{4} + 220037382144 \beta_{5} - 870070616064 \beta_{6} + 317542367232 \beta_{7} ) q^{84} + ( 871338342918203688960 + 175938079700104 \beta_{1} + 549246057627240 \beta_{2} + 183879281312 \beta_{3} - 1423076054384 \beta_{4} + 1006281951728 \beta_{5} - 630722643536 \beta_{6} - 323162067336 \beta_{7} ) q^{85} + ( 160884789718529878 \beta_{1} - 5473543316948448 \beta_{2} - 14056730301648 \beta_{3} - 2450399233536 \beta_{4} - 1081886538384 \beta_{5} - 1160581528080 \beta_{6} - 286626156768 \beta_{7} ) q^{86} + ( -383799133845580458240 + 364888754430241251 \beta_{1} + 336261661817193 \beta_{2} + 4108170974601 \beta_{3} + 2365928867184 \beta_{4} + 1256840731899 \beta_{5} + 118048741794 \beta_{6} - 322816474359 \beta_{7} ) q^{87} + ( 134190494256621158400 + 320606662819840 \beta_{1} + 999647397019648 \beta_{2} + 19889389568 \beta_{3} - 2561281097728 \beta_{4} - 600590778368 \beta_{5} + 74524393472 \beta_{6} + 603107360768 \beta_{7} ) q^{88} + ( -1169976603047619668 \beta_{1} + 6327554495376478 \beta_{2} + 18657598666728 \beta_{3} - 5375528034664 \beta_{4} - 1754009888846 \beta_{5} + 3028259121710 \beta_{6} - 1867508256972 \beta_{7} ) q^{89} + ( -617886201281425551360 + 612654916761423462 \beta_{1} - 5957787424622688 \beta_{2} - 22209407459004 \beta_{3} + 1292001517674 \beta_{4} - 709145241168 \beta_{5} - 1982616425424 \beta_{6} - 1043442832224 \beta_{7} ) q^{90} + ( -795466945217062074380 - 4365086530243470 \beta_{1} - 13640308600193610 \beta_{2} + 1172841289530 \beta_{3} + 14052357773160 \beta_{4} - 438668393370 \beta_{5} + 5792676900360 \beta_{6} - 111987477750 \beta_{7} ) q^{91} + ( -200834707548536832 \beta_{1} - 1959041328742400 \beta_{2} + 9305017286656 \beta_{3} + 4342354542592 \beta_{4} + 1523912278016 \beta_{5} - 1483030396928 \beta_{6} + 1294529986560 \beta_{7} ) q^{92} + ( 452203447293208182810 + 1217534651855215542 \beta_{1} + 14529654767781427 \beta_{2} - 19978258401045 \beta_{3} + 3399434449740 \beta_{4} - 3149027054820 \beta_{5} + 2926900323900 \beta_{6} - 882446049270 \beta_{7} ) q^{93} + ( 356921640253600481280 + 1782743631164840 \beta_{1} + 5588518298037120 \beta_{2} - 2834965039760 \beta_{3} + 10075495087640 \beta_{4} - 3124999624640 \beta_{5} - 2569824479680 \beta_{6} - 1260018215040 \beta_{7} ) q^{94} + ( -1564907138335715504 \beta_{1} + 10392500333748652 \beta_{2} - 130814490430302 \beta_{3} + 6788410536260 \beta_{4} + 2830729869970 \beta_{5} + 1717131952820 \beta_{6} + 1126950796320 \beta_{7} ) q^{95} + ( 85955702487993286656 + 38333373390782464 \beta_{1} - 8796093022208 \beta_{3} - 4398046511104 \beta_{4} ) q^{96} + ( -2990312940585397636990 - 671316953904802 \beta_{1} - 2093409851389882 \beta_{2} - 322931068208 \beta_{3} + 5880363010492 \beta_{4} - 447554948836 \beta_{5} + 642689247172 \beta_{6} - 519651811166 \beta_{7} ) q^{97} + ( -1507025669713210419 \beta_{1} - 10505615886382720 \beta_{2} + 17536928894528 \beta_{3} - 13268648495104 \beta_{4} - 5210565424832 \beta_{5} - 454819090624 \beta_{6} - 2847517645440 \beta_{7} ) q^{98} + ( 4566914065767286582272 + 3847819721723008674 \beta_{1} - 31113000582440832 \beta_{2} + 99219174322413 \beta_{3} + 1522508932926 \beta_{4} + 1715199436713 \beta_{5} + 2053646636922 \beta_{6} + 203410407054 \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 69720q^{3} - 16777216q^{4} + 156352512q^{6} + 5645581840q^{7} - 53592329016q^{9} + O(q^{10}) \) \( 8q - 69720q^{3} - 16777216q^{4} + 156352512q^{6} + 5645581840q^{7} - 53592329016q^{9} + 46180761600q^{10} + 146213437440q^{12} - 2875682381360q^{13} - 731868641280q^{15} + 35184372088832q^{16} - 55595702845440q^{18} - 314803957551536q^{19} + 661593766080336q^{21} - 511896111513600q^{22} - 327894983245824q^{24} - 6264418212884920q^{25} + 11756777174853480q^{27} - 11839643246919680q^{28} + 51661456814407680q^{30} - 110293228249214576q^{31} + 256091518967454720q^{33} - 100789882794934272q^{34} + 112391259980562432q^{36} - 752490366563284400q^{37} + 1437242122292286480q^{39} - 96848076550963200q^{40} + 255515905807319040q^{42} - 1301015563442074160q^{43} - 4916408514089748480q^{45} + 1601638996847296512q^{46} - 306631802754170880q^{48} + 12029391373660580760q^{49} - 18532151948158119936q^{51} + 6030743057433886720q^{52} - 18559682184349532160q^{54} + 56456085576783974400q^{55} - 35832783024333133680q^{57} + 34659140078143242240q^{58} + 1534839784797634560q^{60} - 124415281411828286768q^{61} + 223305694023597186960q^{63} - 73786976294838206464q^{64} + 122984956040514011136q^{66} - 349673137262790165680q^{67} + 217762533639160621056q^{69} - 485339414077397729280q^{70} + 116592639413720186880q^{72} - 65576379188480063600q^{73} + 923803988759061007080q^{75} + 660191749187118825472q^{76} - 1049693699266563440640q^{78} - 325232563432330354544q^{79} - 2780170203320395644024q^{81} + 2935193462963889438720q^{82} - 1387462689722908803072q^{84} + 6970706743345629511680q^{85} - 3070393070764643665920q^{87} + 1073523954052969267200q^{88} - 4943089610251404410880q^{90} - 6363735561736496595040q^{91} + 3617627578345665462480q^{93} + 2855373122028803850240q^{94} + 687645619903946293248q^{96} - 23922503524683181095920q^{97} + 36535312526138292658176q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 4 x^{7} - 12878974 x^{6} - 2567056924 x^{5} + 47744458496177 x^{4} + 16683879235776608 x^{3} - 38709208854294504224 x^{2} - 8131330234969024044480 x + 9835296816172879412979984\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(293131103712509514112 \nu^{7} - 85576662350617574243072 \nu^{6} - 3463857046441066847962512896 \nu^{5} + 157021124379758766383067854080 \nu^{4} + 11180801407143914690721621436309120 \nu^{3} + 1863483255659735157972422100812053504 \nu^{2} - 4926396799510809641815252512603360155136 \nu - 42041842823900274091884311603579792695296\)\()/ \)\(77\!\cdots\!37\)\( \)
\(\beta_{2}\)\(=\)\((\)\(-32240812203505011589724451 \nu^{7} + 162087179405169029938292245568 \nu^{6} + 334167361611076829252840187038063 \nu^{5} - 1635712878562505137075844925047097244 \nu^{4} - 1124614154885599222015950891215908168936 \nu^{3} + 4425691481546163814076906359942591227932990 \nu^{2} + 1207616321653089621272253510734917503017460848 \nu - 1990185393869566151329254299401107289774446469584\)\()/ \)\(56\!\cdots\!90\)\( \)
\(\beta_{3}\)\(=\)\((\)\(-43943738740687906340769207861 \nu^{7} + 50082364308085952921883624086748 \nu^{6} + 555499601208997962380228653499927153 \nu^{5} - 569163978166934668409791225345781101224 \nu^{4} - 1957578547650358773860346191288896875037816 \nu^{3} + 1758700025847811786601488620609739705057341450 \nu^{2} + 1248616856095256784313087210390303300298202420368 \nu - 952280158528042850479659471246758206317257263113744\)\()/ \)\(56\!\cdots\!90\)\( \)
\(\beta_{4}\)\(=\)\((\)\(136460087200299379792360463135 \nu^{7} + 107417050156783012386594365083244 \nu^{6} - 1797012189740965809119145592053675859 \nu^{5} - 1675469319749854972477616247737951750920 \nu^{4} + 6200045274324726337422871935108663771549416 \nu^{3} + 6724155138813532129138159899826806228198757346 \nu^{2} - 1814281904513321068990495285360447421480711518128 \nu - 2528958758064042355814443879554983461439114921412048\)\()/ \)\(85\!\cdots\!35\)\( \)
\(\beta_{5}\)\(=\)\((\)\(213032029264646893999371804513 \nu^{7} + 209831433844827524858894779067720 \nu^{6} - 2834020118563748908520277763136279733 \nu^{5} - 2974787890327410641979026850761213839588 \nu^{4} + 9811526652486142408425172390768653995237304 \nu^{3} + 11222542772180860844839348048597571268745965446 \nu^{2} - 2843193790266732786426760278473242686486494661072 \nu - 4195173819054262528704243238822052652359313583376976\)\()/ \)\(56\!\cdots\!90\)\( \)
\(\beta_{6}\)\(=\)\((\)\(-387406867377046949940124821496 \nu^{7} + 325349650201371614925591776908514 \nu^{6} + 4796639334558847136027106382843913732 \nu^{5} - 2003649887207110500377975911193884779214 \nu^{4} - 16474487046852588251044228648619948751072112 \nu^{3} + 688071107532008966033408400617447612301588404 \nu^{2} + 7812047137855714503521185107720915793667578151616 \nu - 1301592975966242152012038183888771189218086496598096\)\()/ \)\(85\!\cdots\!35\)\( \)
\(\beta_{7}\)\(=\)\((\)\(2060093341803808760918014042093 \nu^{7} + 641349008310747508107768084796528 \nu^{6} - 26683890225116394635225668375684436801 \nu^{5} - 14379321253758656055392931643959928402028 \nu^{4} + 92029399787128161980971223593434790777779736 \nu^{3} + 68273201000632761080250182630199238119041069758 \nu^{2} - 32277267079396478460314552157206766872111559059088 \nu - 23926290440988716816929261563348407589408221397346512\)\()/ \)\(17\!\cdots\!70\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-48 \beta_{7} - 40 \beta_{6} - 104 \beta_{5} + 473 \beta_{4} - 86 \beta_{3} + 69456 \beta_{2} + 100763 \beta_{1} + 40310784\)\()/80621568\)
\(\nu^{2}\)\(=\)\((\)\(3938 \beta_{7} + 945 \beta_{6} - 12319 \beta_{5} + 3114 \beta_{4} - 4015 \beta_{3} + 9365466 \beta_{2} + 2997436 \beta_{1} + 10815872115456\)\()/3359232\)
\(\nu^{3}\)\(=\)\((\)\(-62893506 \beta_{7} - 128082667 \beta_{6} - 175640243 \beta_{5} + 545626679 \beta_{4} - 149457569 \beta_{3} + 287930544222 \beta_{2} + 281964525479 \beta_{1} + 19791885786610176\)\()/20155392\)
\(\nu^{4}\)\(=\)\((\)\(11357565712 \beta_{7} + 1984929015 \beta_{6} - 42506203637 \beta_{5} + 22869213600 \beta_{4} - 15718198949 \beta_{3} + 32208103893660 \beta_{2} + 16734980085116 \beta_{1} + 29561242810587431040\)\()/1679616\)
\(\nu^{5}\)\(=\)\((\)\(-337030034441106 \beta_{7} - 785245564002859 \beta_{6} - 1146935811750419 \beta_{5} + 3165711526199396 \beta_{4} - 919854725119751 \beta_{3} + 1874657852997624174 \beta_{2} + 2330219168928859622 \beta_{1} + 212257895593114001696256\)\()/20155392\)
\(\nu^{6}\)\(=\)\((\)\(120069612078611702 \beta_{7} - 4740677149366761 \beta_{6} - 547102774369202473 \beta_{5} + 403623210334768968 \beta_{4} - 225698005639285573 \beta_{3} + 450178992700816035630 \beta_{2} + 351100230659976229666 \beta_{1} + 351145206617137924682859264\)\()/3359232\)
\(\nu^{7}\)\(=\)\((\)\(-1798242457078791138738 \beta_{7} - 4618805970899112051043 \beta_{6} - 7505838577094208982955 \beta_{5} + 19025311998546337180184 \beta_{4} - 5671475639720003704823 \beta_{3} + 11686036785814447008847662 \beta_{2} + 18131262364196364276268418 \beta_{1} + 1768853004802271151769914223104\)\()/20155392\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/6\mathbb{Z}\right)^\times\).

\(n\) \(5\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
−865.062 + 1.41421i
2543.52 + 1.41421i
−2296.98 + 1.41421i
620.515 + 1.41421i
−865.062 1.41421i
2543.52 1.41421i
−2296.98 1.41421i
620.515 1.41421i
1448.15i −128493. + 121945.i −2.09715e6 8.15211e7i 1.76595e8 + 1.86078e8i 2.43000e9 3.03700e9i 1.64002e9 3.13382e10i −1.18055e11
5.2 1448.15i −92471.5 151096.i −2.09715e6 4.00267e6i −2.18811e8 + 1.33913e8i −1.86396e9 3.03700e9i −1.42791e10 + 2.79442e10i −5.79649e9
5.3 1448.15i 33954.9 + 173862.i −2.09715e6 7.14623e7i 2.51780e8 4.91720e7i −1.07711e9 3.03700e9i −2.90752e10 + 1.18070e10i 1.03488e11
5.4 1448.15i 152150. 90727.5i −2.09715e6 3.00062e7i −1.31387e8 2.20337e8i 3.33386e9 3.03700e9i 1.49181e10 2.76083e10i 4.34536e10
5.5 1448.15i −128493. 121945.i −2.09715e6 8.15211e7i 1.76595e8 1.86078e8i 2.43000e9 3.03700e9i 1.64002e9 + 3.13382e10i −1.18055e11
5.6 1448.15i −92471.5 + 151096.i −2.09715e6 4.00267e6i −2.18811e8 1.33913e8i −1.86396e9 3.03700e9i −1.42791e10 2.79442e10i −5.79649e9
5.7 1448.15i 33954.9 173862.i −2.09715e6 7.14623e7i 2.51780e8 + 4.91720e7i −1.07711e9 3.03700e9i −2.90752e10 1.18070e10i 1.03488e11
5.8 1448.15i 152150. + 90727.5i −2.09715e6 3.00062e7i −1.31387e8 + 2.20337e8i 3.33386e9 3.03700e9i 1.49181e10 + 2.76083e10i 4.34536e10
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6.23.b.a 8
3.b odd 2 1 inner 6.23.b.a 8
4.b odd 2 1 48.23.e.d 8
12.b even 2 1 48.23.e.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.23.b.a 8 1.a even 1 1 trivial
6.23.b.a 8 3.b odd 2 1 inner
48.23.e.d 8 4.b odd 2 1
48.23.e.d 8 12.b even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{23}^{\mathrm{new}}(6, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 2097152 + T^{2} )^{4} \)
$3$ \( \)\(96\!\cdots\!61\)\( + \)\(21\!\cdots\!80\)\( T + \)\(28\!\cdots\!48\)\( T^{2} - \)\(62\!\cdots\!00\)\( T^{3} + \)\(84\!\cdots\!38\)\( T^{4} - 1994213728445400 T^{5} + 29226603708 T^{6} + 69720 T^{7} + T^{8} \)
$5$ \( \)\(48\!\cdots\!00\)\( + \)\(31\!\cdots\!00\)\( T^{2} + \)\(44\!\cdots\!00\)\( T^{4} + 12668952270504960 T^{6} + T^{8} \)
$7$ \( ( \)\(16\!\cdots\!76\)\( + \)\(12\!\cdots\!40\)\( T - 6842915651563898088 T^{2} - 2822790920 T^{3} + T^{4} )^{2} \)
$11$ \( \)\(96\!\cdots\!56\)\( + \)\(26\!\cdots\!92\)\( T^{2} + \)\(48\!\cdots\!04\)\( T^{4} + \)\(46\!\cdots\!88\)\( T^{6} + T^{8} \)
$13$ \( ( \)\(13\!\cdots\!00\)\( + \)\(14\!\cdots\!00\)\( T - \)\(60\!\cdots\!00\)\( T^{2} + 1437841190680 T^{3} + T^{4} )^{2} \)
$17$ \( \)\(58\!\cdots\!96\)\( + \)\(48\!\cdots\!88\)\( T^{2} + \)\(19\!\cdots\!44\)\( T^{4} + \)\(25\!\cdots\!92\)\( T^{6} + T^{8} \)
$19$ \( ( -\)\(13\!\cdots\!24\)\( - \)\(48\!\cdots\!68\)\( T - \)\(30\!\cdots\!96\)\( T^{2} + 157401978775768 T^{3} + T^{4} )^{2} \)
$23$ \( \)\(20\!\cdots\!56\)\( + \)\(12\!\cdots\!48\)\( T^{2} + \)\(51\!\cdots\!64\)\( T^{4} + \)\(46\!\cdots\!72\)\( T^{6} + T^{8} \)
$29$ \( \)\(98\!\cdots\!00\)\( + \)\(15\!\cdots\!00\)\( T^{2} + \)\(99\!\cdots\!00\)\( T^{4} + \)\(17\!\cdots\!20\)\( T^{6} + T^{8} \)
$31$ \( ( -\)\(29\!\cdots\!44\)\( - \)\(44\!\cdots\!08\)\( T + \)\(30\!\cdots\!04\)\( T^{2} + 55146614124607288 T^{3} + T^{4} )^{2} \)
$37$ \( ( -\)\(75\!\cdots\!24\)\( - \)\(86\!\cdots\!60\)\( T + \)\(10\!\cdots\!12\)\( T^{2} + 376245183281642200 T^{3} + T^{4} )^{2} \)
$41$ \( \)\(10\!\cdots\!36\)\( + \)\(10\!\cdots\!52\)\( T^{2} + \)\(20\!\cdots\!04\)\( T^{4} + \)\(86\!\cdots\!08\)\( T^{6} + T^{8} \)
$43$ \( ( \)\(50\!\cdots\!56\)\( - \)\(66\!\cdots\!00\)\( T - \)\(20\!\cdots\!92\)\( T^{2} + 650507781721037080 T^{3} + T^{4} )^{2} \)
$47$ \( \)\(12\!\cdots\!00\)\( + \)\(13\!\cdots\!00\)\( T^{2} + \)\(19\!\cdots\!00\)\( T^{4} + \)\(28\!\cdots\!00\)\( T^{6} + T^{8} \)
$53$ \( \)\(10\!\cdots\!76\)\( + \)\(47\!\cdots\!68\)\( T^{2} + \)\(21\!\cdots\!04\)\( T^{4} + \)\(27\!\cdots\!32\)\( T^{6} + T^{8} \)
$59$ \( \)\(56\!\cdots\!36\)\( + \)\(19\!\cdots\!72\)\( T^{2} + \)\(46\!\cdots\!24\)\( T^{4} + \)\(37\!\cdots\!88\)\( T^{6} + T^{8} \)
$61$ \( ( \)\(38\!\cdots\!36\)\( - \)\(35\!\cdots\!96\)\( T - \)\(48\!\cdots\!24\)\( T^{2} + 62207640705914143384 T^{3} + T^{4} )^{2} \)
$67$ \( ( -\)\(14\!\cdots\!64\)\( - \)\(45\!\cdots\!00\)\( T - \)\(20\!\cdots\!48\)\( T^{2} + \)\(17\!\cdots\!40\)\( T^{3} + T^{4} )^{2} \)
$71$ \( \)\(39\!\cdots\!00\)\( + \)\(21\!\cdots\!00\)\( T^{2} + \)\(40\!\cdots\!00\)\( T^{4} + \)\(33\!\cdots\!20\)\( T^{6} + T^{8} \)
$73$ \( ( -\)\(30\!\cdots\!00\)\( - \)\(49\!\cdots\!00\)\( T - \)\(18\!\cdots\!20\)\( T^{2} + 32788189594240031800 T^{3} + T^{4} )^{2} \)
$79$ \( ( \)\(21\!\cdots\!56\)\( - \)\(76\!\cdots\!52\)\( T - \)\(62\!\cdots\!36\)\( T^{2} + \)\(16\!\cdots\!72\)\( T^{3} + T^{4} )^{2} \)
$83$ \( \)\(74\!\cdots\!76\)\( + \)\(22\!\cdots\!12\)\( T^{2} + \)\(17\!\cdots\!84\)\( T^{4} + \)\(34\!\cdots\!88\)\( T^{6} + T^{8} \)
$89$ \( \)\(46\!\cdots\!36\)\( + \)\(82\!\cdots\!72\)\( T^{2} + \)\(12\!\cdots\!24\)\( T^{4} + \)\(61\!\cdots\!88\)\( T^{6} + T^{8} \)
$97$ \( ( \)\(41\!\cdots\!96\)\( + \)\(77\!\cdots\!60\)\( T + \)\(48\!\cdots\!88\)\( T^{2} + \)\(11\!\cdots\!60\)\( T^{3} + T^{4} )^{2} \)
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