# Properties

 Label 4.23.b.a Level 4 Weight 23 Character orbit 4.b Analytic conductor 12.268 Analytic rank 0 Dimension 10 CM No Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$4 = 2^{2}$$ Weight: $$k$$ = $$23$$ Character orbit: $$[\chi]$$ = 4.b (of order $$2$$ and degree $$1$$)

## Newform invariants

 Self dual: No Analytic conductor: $$12.2682973937$$ Analytic rank: $$0$$ Dimension: $$10$$ Coefficient field: $$\mathbb{Q}[x]/(x^{10} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{90}\cdot 3^{11}$$ Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{9}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q$$ $$+ ( 154 - \beta_{1} ) q^{2}$$ $$+ ( -19 \beta_{1} - \beta_{2} ) q^{3}$$ $$+ ( 226446 - 146 \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{4}$$ $$+ ( -1709110 - 3885 \beta_{1} + 8 \beta_{2} + \beta_{3} + \beta_{5} ) q^{5}$$ $$+ ( -79193586 - 2915 \beta_{1} - 65 \beta_{2} + 20 \beta_{3} + \beta_{7} ) q^{6}$$ $$+ ( 4 - 141333 \beta_{1} - 503 \beta_{2} - 9 \beta_{3} - 2 \beta_{4} + \beta_{6} - 2 \beta_{7} ) q^{7}$$ $$+ ( 980443087 - 191495 \beta_{1} - 4531 \beta_{2} + 162 \beta_{3} - 9 \beta_{4} - 37 \beta_{5} + 3 \beta_{6} - 2 \beta_{7} + \beta_{9} ) q^{8}$$ $$+ ( -10430961350 - 775646 \beta_{1} + 1486 \beta_{2} - 121 \beta_{3} + \beta_{4} - 17 \beta_{5} + 4 \beta_{6} + 57 \beta_{7} - \beta_{8} - 4 \beta_{9} ) q^{9}$$ $$+O(q^{10})$$ $$q$$ $$+ ( 154 - \beta_{1} ) q^{2}$$ $$+ ( -19 \beta_{1} - \beta_{2} ) q^{3}$$ $$+ ( 226446 - 146 \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{4}$$ $$+ ( -1709110 - 3885 \beta_{1} + 8 \beta_{2} + \beta_{3} + \beta_{5} ) q^{5}$$ $$+ ( -79193586 - 2915 \beta_{1} - 65 \beta_{2} + 20 \beta_{3} + \beta_{7} ) q^{6}$$ $$+ ( 4 - 141333 \beta_{1} - 503 \beta_{2} - 9 \beta_{3} - 2 \beta_{4} + \beta_{6} - 2 \beta_{7} ) q^{7}$$ $$+ ( 980443087 - 191495 \beta_{1} - 4531 \beta_{2} + 162 \beta_{3} - 9 \beta_{4} - 37 \beta_{5} + 3 \beta_{6} - 2 \beta_{7} + \beta_{9} ) q^{8}$$ $$+ ( -10430961350 - 775646 \beta_{1} + 1486 \beta_{2} - 121 \beta_{3} + \beta_{4} - 17 \beta_{5} + 4 \beta_{6} + 57 \beta_{7} - \beta_{8} - 4 \beta_{9} ) q^{9}$$ $$+ ( 15941402573 + 2309265 \beta_{1} + 22300 \beta_{2} + 3417 \beta_{3} + 7 \beta_{4} + 1064 \beta_{5} - 28 \beta_{6} + 12 \beta_{7} - 4 \beta_{8} + 12 \beta_{9} ) q^{10}$$ $$+ ( -4084 + 2898637 \beta_{1} + 179587 \beta_{2} + 10694 \beta_{3} + 64 \beta_{4} - 8 \beta_{5} - 18 \beta_{6} - 452 \beta_{7} - 24 \beta_{8} - 32 \beta_{9} ) q^{11}$$ $$+ ( 51902110280 + 81560200 \beta_{1} - 424004 \beta_{2} + 6052 \beta_{3} - 16 \beta_{4} - 11836 \beta_{5} - 172 \beta_{6} + 216 \beta_{7} - 80 \beta_{8} + 28 \beta_{9} ) q^{12}$$ $$+ ( -53122949836 + 92735431 \beta_{1} - 274124 \beta_{2} - 216809 \beta_{3} - 646 \beta_{4} - 1753 \beta_{5} - 24 \beta_{6} - 86 \beta_{7} - 250 \beta_{8} + 24 \beta_{9} ) q^{13}$$ $$+ ( -589400929700 - 21632298 \beta_{1} + 6653034 \beta_{2} + 119008 \beta_{3} - 3688 \beta_{4} + 32256 \beta_{5} + 864 \beta_{6} + 22 \beta_{7} - 672 \beta_{8} - 224 \beta_{9} ) q^{14}$$ $$+ ( -885532 + 246934105 \beta_{1} + 1803259 \beta_{2} + 2180645 \beta_{3} - 5518 \beta_{4} - 14128 \beta_{5} + 347 \beta_{6} + 14042 \beta_{7} - 1424 \beta_{8} + 832 \beta_{9} ) q^{15}$$ $$+ ( -2771762144564 - 1088674412 \beta_{1} - 43282956 \beta_{2} + 377560 \beta_{3} - 18740 \beta_{4} + 49260 \beta_{5} + 3532 \beta_{6} + 952 \beta_{7} - 2880 \beta_{8} - 1212 \beta_{9} ) q^{16}$$ $$+ ( 1405819335955 + 688419482 \beta_{1} - 2983058 \beta_{2} - 3828225 \beta_{3} + 4561 \beta_{4} - 26649 \beta_{5} - 1212 \beta_{6} - 12663 \beta_{7} - 4305 \beta_{8} + 1212 \beta_{9} ) q^{17}$$ $$+ ( 1628395298300 + 10553246717 \beta_{1} + 237440696 \beta_{2} + 20626 \beta_{3} - 43474 \beta_{4} - 761008 \beta_{5} - 12472 \beta_{6} + 15000 \beta_{7} - 5000 \beta_{8} + 664 \beta_{9} ) q^{18}$$ $$+ ( 698356 + 8850873975 \beta_{1} + 56779785 \beta_{2} - 1507758 \beta_{3} + 163920 \beta_{4} + 26440 \beta_{5} - 8422 \beta_{6} - 123404 \beta_{7} - 2600 \beta_{8} - 8928 \beta_{9} ) q^{19}$$ $$+ ( 23364363440972 - 15179527540 \beta_{1} - 907199852 \beta_{2} + 465674 \beta_{3} - 244032 \beta_{4} + 1275392 \beta_{5} - 37632 \beta_{6} - 14592 \beta_{7} + 9984 \beta_{8} + 16128 \beta_{9} ) q^{20}$$ $$+ ( -31313579776486 - 64545846820 \beta_{1} + 140782124 \beta_{2} + 33234670 \beta_{3} + 788186 \beta_{4} + 87038 \beta_{5} + 23912 \beta_{6} + 303114 \beta_{7} + 31654 \beta_{8} - 23912 \beta_{9} ) q^{21}$$ $$+ ( 12058970330850 + 460064083 \beta_{1} + 1733053601 \beta_{2} - 10134884 \beta_{3} - 1451952 \beta_{4} + 3613696 \beta_{5} + 112448 \beta_{6} - 157793 \beta_{7} + 74560 \beta_{8} + 19392 \beta_{9} ) q^{22}$$ $$+ ( 47985732 + 146983164741 \beta_{1} - 204629649 \beta_{2} - 118187119 \beta_{3} + 2764506 \beta_{4} + 656560 \beta_{5} + 131679 \beta_{6} + 342322 \beta_{7} + 126480 \beta_{8} + 45760 \beta_{9} ) q^{23}$$ $$+ ( -200761942857080 - 59680938952 \beta_{1} - 3759018664 \beta_{2} - 63421904 \beta_{3} - 4909112 \beta_{4} - 14064536 \beta_{5} + 225832 \beta_{6} + 264592 \beta_{7} + 157952 \beta_{8} - 106312 \beta_{9} ) q^{24}$$ $$+ ( 771017530764705 - 668819545820 \beta_{1} + 1369473940 \beta_{2} + 139147570 \beta_{3} + 9834950 \beta_{4} + 3773090 \beta_{5} - 201960 \beta_{6} - 2998250 \beta_{7} + 170810 \beta_{8} + 201960 \beta_{9} ) q^{25}$$ $$+ ( -395397295531071 + 38758351261 \beta_{1} - 2501101620 \beta_{2} - 83700819 \beta_{3} - 14157517 \beta_{4} + 14198344 \beta_{5} - 709836 \beta_{6} + 433532 \beta_{7} + 30252 \beta_{8} - 266884 \beta_{9} ) q^{26}$$ $$+ ( -128112084 + 1898813259468 \beta_{1} + 5147601768 \beta_{2} + 331566462 \beta_{3} + 29354544 \beta_{4} - 1303560 \beta_{5} - 1272330 \beta_{6} + 1916460 \beta_{7} - 224280 \beta_{8} - 53280 \beta_{9} ) q^{27}$$ $$+ ( 117795254333744 + 597252407984 \beta_{1} + 14164415400 \beta_{2} - 41775976 \beta_{3} - 45217120 \beta_{4} + 46151000 \beta_{5} - 646728 \beta_{6} - 5166704 \beta_{7} - 659680 \beta_{8} + 313320 \beta_{9} ) q^{28}$$ $$+ ( -1446447096415102 - 4203269196213 \beta_{1} + 7893338360 \beta_{2} - 827790383 \beta_{3} + 63326376 \beta_{4} - 39860207 \beta_{5} + 885408 \beta_{6} + 13759848 \beta_{7} - 1364136 \beta_{8} - 885408 \beta_{9} ) q^{29}$$ $$+ ( 1025278261130756 + 37101406410 \beta_{1} - 73905337962 \beta_{2} + 43557760 \beta_{3} - 114895096 \beta_{4} - 175453696 \beta_{5} + 3263264 \beta_{6} - 95766 \beta_{7} - 1857248 \beta_{8} + 1620064 \beta_{9} ) q^{30}$$ $$+ ( -1125657176 + 7624014103080 \beta_{1} + 24645995360 \beta_{2} + 2880997728 \beta_{3} + 116171720 \beta_{4} - 13742800 \beta_{5} + 8030472 \beta_{6} - 24284576 \beta_{7} - 2439280 \beta_{8} - 666432 \beta_{9} ) q^{31}$$ $$+ ( -7383119591837584 + 2516179621648 \beta_{1} + 68068122512 \beta_{2} + 891687136 \beta_{3} - 191877520 \beta_{4} + 81856496 \beta_{5} - 685200 \beta_{6} + 47989088 \beta_{7} - 2223360 \beta_{8} + 373712 \beta_{9} ) q^{32}$$ $$+ ( 7290695920650103 - 16528784843714 \beta_{1} + 32378864386 \beta_{2} - 24033103 \beta_{3} + 233297815 \beta_{4} + 238339081 \beta_{5} - 1329572 \beta_{6} - 17751393 \beta_{7} - 862615 \beta_{8} + 1329572 \beta_{9} ) q^{33}$$ $$+ ( -2662597405844978 - 1513327351316 \beta_{1} - 89137824232 \beta_{2} - 431461958 \beta_{3} - 246346746 \beta_{4} + 405658512 \beta_{5} - 10389528 \beta_{6} + 4897656 \beta_{7} + 1513176 \beta_{8} - 4914312 \beta_{9} ) q^{34}$$ $$+ ( 1861293000 + 24320535252720 \beta_{1} + 23734496520 \beta_{2} - 4541306980 \beta_{3} + 397509840 \beta_{4} + 27988240 \beta_{5} - 33590340 \beta_{6} + 117795160 \beta_{7} + 6550320 \beta_{8} + 3572800 \beta_{9} ) q^{35}$$ $$+ ( -3660924055656482 - 1779285826370 \beta_{1} - 162706383182 \beta_{2} - 9730403623 \beta_{3} - 292539008 \beta_{4} - 976507904 \beta_{5} + 12892672 \beta_{6} - 251810304 \beta_{7} + 12021248 \beta_{8} - 6349312 \beta_{9} ) q^{36}$$ $$+ ( 32458770666534668 - 18021578786257 \beta_{1} + 41431998308 \beta_{2} + 14262362327 \beta_{3} + 317699266 \beta_{4} - 1306437593 \beta_{5} - 6355704 \beta_{6} - 107869006 \beta_{7} + 18889150 \beta_{8} + 6355704 \beta_{9} ) q^{37}$$ $$+ ( 36914085871929670 + 1363037223209 \beta_{1} + 499801176179 \beta_{2} - 11410342636 \beta_{3} - 86371216 \beta_{4} + 1053740032 \beta_{5} + 15852480 \beta_{6} - 35669939 \beta_{7} + 23087040 \beta_{8} + 1977664 \beta_{9} ) q^{38}$$ $$+ ( 12282185436 - 23139930058883 \beta_{1} - 76796436113 \beta_{2} - 30383748111 \beta_{3} - 249593646 \beta_{4} + 175564032 \beta_{5} + 86451927 \beta_{6} - 244989102 \beta_{7} + 22884096 \beta_{8} - 3075072 \beta_{9} ) q^{39}$$ $$+ ( -115804937216696234 - 27776703691910 \beta_{1} - 1138952249918 \beta_{2} + 20273091028 \beta_{3} + 677079014 \beta_{4} + 1617065422 \beta_{5} - 40600226 \beta_{6} + 854731884 \beta_{7} + 15695872 \beta_{8} + 18098634 \beta_{9} ) q^{40}$$ $$+ ( -135545734447760192 + 52291679502692 \beta_{1} - 100539827324 \beta_{2} + 4911130362 \beta_{3} - 1238490290 \beta_{4} + 6378476970 \beta_{5} + 38652216 \beta_{6} + 546863454 \beta_{7} - 5732430 \beta_{8} - 38652216 \beta_{9} ) q^{41}$$ $$+ ( 264445547520085708 + 41303655056356 \beta_{1} + 1665980369872 \beta_{2} + 57219187340 \beta_{3} + 1859023348 \beta_{4} - 6054651680 \beta_{5} + 43141168 \beta_{6} + 11702544 \beta_{7} - 29590960 \beta_{8} + 41607440 \beta_{9} ) q^{42}$$ $$+ ( -25201251896 - 197111489212905 \beta_{1} - 364495042811 \beta_{2} + 61244182548 \beta_{3} - 3295605344 \beta_{4} - 358188720 \beta_{5} - 77677340 \beta_{6} - 288952760 \beta_{7} - 74322960 \beta_{8} - 32414400 \beta_{9} ) q^{43}$$ $$+ ( -191976148046816072 - 19214525422472 \beta_{1} + 383110977220 \beta_{2} + 47450588 \beta_{3} + 4355665424 \beta_{4} + 6029987772 \beta_{5} - 4708308 \beta_{6} - 2002620376 \beta_{7} - 112684464 \beta_{8} + 11043428 \beta_{9} ) q^{44}$$ $$+ ( 936789675110800 + 594167461995775 \beta_{1} - 1239179109188 \beta_{2} - 176280635941 \beta_{3} - 6822070250 \beta_{4} - 23154361781 \beta_{5} - 63226280 \beta_{6} - 740074650 \beta_{7} - 145093270 \beta_{8} + 63226280 \beta_{9} ) q^{45}$$ $$+ ( 613262463555171476 + 22556541579602 \beta_{1} - 889447700402 \beta_{2} - 152664857152 \beta_{3} + 8574732648 \beta_{4} + 1741341184 \beta_{5} - 352384864 \beta_{6} + 414957874 \beta_{7} - 177811808 \beta_{8} - 145832736 \beta_{9} ) q^{46}$$ $$+ ( -85770739232 - 500410208248642 \beta_{1} + 254722047650 \beta_{2} + 205174663422 \beta_{3} - 9580499884 \beta_{4} - 1538570160 \beta_{5} - 356609718 \beta_{6} + 2671081596 \beta_{7} - 138577680 \beta_{8} + 113705280 \beta_{9} ) q^{47}$$ $$+ ( -1837416965674116704 + 134180213207904 \beta_{1} + 8494225849440 \beta_{2} + 54640217408 \beta_{3} + 10226155936 \beta_{4} - 23182737248 \beta_{5} + 391267744 \beta_{6} + 2828163648 \beta_{7} - 87447040 \beta_{8} - 191143456 \beta_{9} ) q^{48}$$ $$+ ( 18355373621924773 + 554812770398600 \beta_{1} - 1070774705896 \beta_{2} + 46892660780 \beta_{3} - 11123943404 \beta_{4} + 59570399308 \beta_{5} - 128319408 \beta_{6} - 1849224076 \beta_{7} + 52752364 \beta_{8} + 128319408 \beta_{9} ) q^{49}$$ $$+ ( 2908439158724352370 - 667800636622335 \beta_{1} - 10180610915280 \beta_{2} + 673359403380 \beta_{3} + 8985932300 \beta_{4} + 36069771040 \beta_{5} + 941303760 \beta_{6} - 978895120 \beta_{7} + 244159920 \beta_{8} + 114552560 \beta_{9} ) q^{50}$$ $$+ ( 230504631308 - 889284244176236 \beta_{1} - 8684807474816 \beta_{2} - 574323820090 \beta_{3} - 10278941632 \beta_{4} + 3085699064 \beta_{5} + 1737681902 \beta_{6} - 3557421508 \beta_{7} + 457578472 \beta_{8} + 23470048 \beta_{9} ) q^{51}$$ $$+ ( -3849243299158811924 + 255631898148396 \beta_{1} + 12336999226100 \beta_{2} - 39093180870 \beta_{3} + 6598251968 \beta_{4} - 7483691520 \beta_{5} - 972566272 \beta_{6} + 1243345664 \beta_{7} + 682218240 \beta_{8} + 364235520 \beta_{9} ) q^{52}$$ $$+ ( -1142434811618976888 - 301235925226793 \beta_{1} + 904525054572 \beta_{2} + 719422436843 \beta_{3} + 6698235614 \beta_{4} - 113699549797 \beta_{5} + 719372664 \beta_{6} + 9254329326 \beta_{7} + 816887970 \beta_{8} - 719372664 \beta_{9} ) q^{53}$$ $$+ ( 7918284861424076076 + 291299975989434 \beta_{1} - 11462660324658 \beta_{2} - 1960475137704 \beta_{3} - 11613965040 \beta_{4} - 41040276480 \beta_{5} - 319942080 \beta_{6} - 624366414 \beta_{7} + 959008320 \beta_{8} + 499320000 \beta_{9} ) q^{54}$$ $$+ ( 530025956228 + 2431353517497915 \beta_{1} + 24405051224249 \beta_{2} - 1285888185945 \beta_{3} + 39321302942 \beta_{4} + 9703900032 \beta_{5} - 3549350383 \beta_{6} - 8166463138 \beta_{7} + 737725056 \beta_{8} - 907964928 \beta_{9} ) q^{55}$$ $$+ ( -10284486365450263376 - 498704483313712 \beta_{1} - 23887773420400 \beta_{2} - 415620170208 \beta_{3} - 42816630736 \beta_{4} + 129158261232 \beta_{5} - 525389712 \beta_{6} - 19844096928 \beta_{7} + 584295936 \beta_{8} + 504502992 \beta_{9} ) q^{56}$$ $$+ ( 3018851613152555541 - 4792055574617958 \beta_{1} + 9476477567142 \beta_{2} + 215400311139 \beta_{3} + 63305438325 \beta_{4} + 160406707947 \beta_{5} - 799168044 \beta_{6} - 11200958451 \beta_{7} + 12605835 \beta_{8} + 799168044 \beta_{9} ) q^{57}$$ $$+ ( 17305968061079159093 + 2096279953396953 \beta_{1} + 55778128060924 \beta_{2} + 3859313189793 \beta_{3} - 74181335873 \beta_{4} - 155048832664 \beta_{5} - 5484598012 \beta_{6} + 5323528812 \beta_{7} - 994369060 \beta_{8} - 1360405908 \beta_{9} ) q^{58}$$ $$+ ( -1237381200640 + 5665668118724863 \beta_{1} - 42003990176507 \beta_{2} + 3098281866888 \beta_{3} + 95481253936 \beta_{4} - 18792908352 \beta_{5} + 2147950296 \beta_{6} + 21011301648 \beta_{7} - 1627574976 \beta_{8} + 1479976704 \beta_{9} ) q^{59}$$ $$+ ( -37776946356330364208 - 2426747178417840 \beta_{1} - 101798966866344 \beta_{2} + 772511078760 \beta_{3} - 145793178272 \beta_{4} - 53765027672 \beta_{5} + 7180235848 \beta_{6} + 54974696048 \beta_{7} - 3030066976 \beta_{8} - 3009435112 \beta_{9} ) q^{60}$$ $$+ ( 5824173073265943308 - 8137841104378793 \beta_{1} + 14616951719236 \beta_{2} - 3130998691073 \beta_{3} + 136162300850 \beta_{4} - 68242407985 \beta_{5} - 1793398584 \beta_{6} - 21341299166 \beta_{7} - 3766281010 \beta_{8} + 1793398584 \beta_{9} ) q^{61}$$ $$+ ( 31789725536856116560 + 1170346032706456 \beta_{1} + 47735800355656 \beta_{2} - 8300015774304 \beta_{3} - 206215461184 \beta_{4} + 272899371008 \beta_{5} + 15562064640 \beta_{6} - 7704721736 \beta_{7} - 4095601920 \beta_{8} + 507064576 \beta_{9} ) q^{62}$$ $$+ ( -2852652732228 + 12222280719913455 \beta_{1} + 173848455268317 \beta_{2} + 7108843131459 \beta_{3} + 121741986078 \beta_{4} - 44123541840 \beta_{5} + 9341105085 \beta_{6} + 32395876470 \beta_{7} - 4211086320 \beta_{8} + 2929187520 \beta_{9} ) q^{63}$$ $$+ ( -41208016714210880832 + 5902067077960512 \beta_{1} - 57704245609152 \beta_{2} - 1748220963456 \beta_{3} - 146115638592 \beta_{4} - 429901354816 \beta_{5} - 10682450240 \beta_{6} - 82952683648 \beta_{7} - 3739460608 \beta_{8} + 2505222720 \beta_{9} ) q^{64}$$ $$+ ( -12153169442445960250 - 7077697923822500 \beta_{1} + 13627821229148 \beta_{2} - 687475500714 \beta_{3} + 116281530050 \beta_{4} - 506195770074 \beta_{5} + 5787004680 \beta_{6} + 82914304050 \beta_{7} - 1896238530 \beta_{8} - 5787004680 \beta_{9} ) q^{65}$$ $$+ ( 70058521702746458702 - 4737995335583830 \beta_{1} - 47552519627704 \beta_{2} + 15934670298286 \beta_{3} - 51165023086 \beta_{4} + 554711072432 \beta_{5} - 6274955848 \beta_{6} + 354122856 \beta_{7} + 697751176 \beta_{8} + 1448244328 \beta_{9} ) q^{66}$$ $$+ ( 4151944941380 - 4492740388565109 \beta_{1} - 332788792144651 \beta_{2} - 10111506802830 \beta_{3} + 93887912480 \beta_{4} + 80590075560 \beta_{5} - 34190947510 \beta_{6} - 125892546700 \beta_{7} + 2972607480 \beta_{8} - 11956364640 \beta_{9} ) q^{67}$$ $$+ ( -66789052767391254820 + 214983637637532 \beta_{1} + 247486158799556 \beta_{2} + 1368438037218 \beta_{3} + 165761954688 \beta_{4} + 18012019712 \beta_{5} - 16858873344 \beta_{6} + 69312546304 \beta_{7} + 9951659520 \beta_{8} + 8893740544 \beta_{9} ) q^{68}$$ $$+ ( 5449745983072086830 + 7722202511612852 \beta_{1} - 10488518579260 \beta_{2} + 11132076548362 \beta_{3} - 229763967634 \beta_{4} + 1789301743418 \beta_{5} - 3113374792 \beta_{6} - 56280302082 \beta_{7} + 12693054994 \beta_{8} + 3113374792 \beta_{9} ) q^{69}$$ $$+ ( 101439637377452505320 + 3729110630011340 \beta_{1} - 420447865691420 \beta_{2} - 23936619238000 \beta_{3} + 582747449760 \beta_{4} - 1405130209280 \beta_{5} - 55204105600 \beta_{6} + 31064796700 \beta_{7} + 15598602880 \beta_{8} + 1845674880 \beta_{9} ) q^{70}$$ $$+ ( 10458285822988 - 31811647108611337 \beta_{1} + 653053860106853 \beta_{2} - 26234344520101 \beta_{3} - 667012388674 \beta_{4} + 135746713168 \beta_{5} + 53966834325 \beta_{6} - 90426835994 \beta_{7} + 21116650224 \beta_{8} + 2413967680 \beta_{9} ) q^{71}$$ $$+ ( -124477324802453145865 - 907505701287535 \beta_{1} + 639072424562885 \beta_{2} + 614505916498 \beta_{3} + 785674593503 \beta_{4} + 1582507457315 \beta_{5} + 72983398667 \beta_{6} + 60530151246 \beta_{7} + 17092771840 \beta_{8} - 20588652455 \beta_{9} ) q^{72}$$ $$+ ( 8303038773032126519 + 80259301740580106 \beta_{1} - 155174183657914 \beta_{2} + 4828415819171 \beta_{3} - 984139610651 \beta_{4} - 3417344566613 \beta_{5} - 6502680684 \beta_{6} - 105865873411 \beta_{7} + 14828343835 \beta_{8} + 6502680684 \beta_{9} ) q^{73}$$ $$+ ( 80190600658331812729 - 29676777650094251 \beta_{1} - 300880400096916 \beta_{2} + 18720935169093 \beta_{3} + 1033243728283 \beta_{4} - 993927650296 \beta_{5} + 113690798484 \beta_{6} - 76719359588 \beta_{7} + 13762833612 \beta_{8} - 794228324 \beta_{9} ) q^{74}$$ $$+ ( -9028945853400 - 113805353800971645 \beta_{1} - 2014180839177495 \beta_{2} + 20815965061980 \beta_{3} - 1490062092240 \beta_{4} - 221002296240 \beta_{5} - 11354289060 \beta_{6} + 745867607640 \beta_{7} + 745405680 \beta_{8} + 45244027200 \beta_{9} ) q^{75}$$ $$+ ( 28674744383243455912 - 36064191794839320 \beta_{1} + 277882348624524 \beta_{2} - 2773444230828 \beta_{3} + 1522311967536 \beta_{4} + 1625637126388 \beta_{5} - 38850156220 \beta_{6} - 528539076872 \beta_{7} - 17119697936 \beta_{8} - 1204263060 \beta_{9} ) q^{76}$$ $$+ ( -70420174069497136010 + 91261904537057380 \beta_{1} - 186397072747340 \beta_{2} - 17152519675550 \beta_{3} - 1590786822730 \beta_{4} + 4996280235730 \beta_{5} + 8308798680 \beta_{6} + 138944308230 \beta_{7} - 22621126710 \beta_{8} - 8308798680 \beta_{9} ) q^{77}$$ $$+ ( -96441279182746680060 - 3539773339009702 \beta_{1} + 1118169340701926 \beta_{2} + 19616446250656 \beta_{3} + 1418292057768 \beta_{4} + 4146063069696 \beta_{5} + 9609844128 \beta_{6} - 8428170598 \beta_{7} - 52093154400 \beta_{8} - 43684166688 \beta_{9} ) q^{78}$$ $$+ ( -19618187034456 + 1656606770907990 \beta_{1} + 3409137238949170 \beta_{2} + 48870457213038 \beta_{3} - 1043721301220 \beta_{4} - 205241300800 \beta_{5} - 140216252638 \beta_{6} - 565060122756 \beta_{7} - 70199289280 \beta_{8} - 57230744832 \beta_{9} ) q^{79}$$ $$+ ( 150961178518173585656 + 122036669677509320 \beta_{1} - 1609100967160184 \beta_{2} + 33175091664496 \beta_{3} + 1035304708344 \beta_{4} - 8370249572040 \beta_{5} - 199673721096 \beta_{6} + 1338772416304 \beta_{7} - 54407864448 \beta_{8} + 46394304424 \beta_{9} ) q^{80}$$ $$+ ( 25157803796505967284 + 85880965847501718 \beta_{1} - 201766095872982 \beta_{2} - 78714076860027 \beta_{3} - 707173100829 \beta_{4} - 5102079825987 \beta_{5} - 25897028724 \beta_{6} - 282585685365 \beta_{7} - 79972716771 \beta_{8} + 25897028724 \beta_{9} ) q^{81}$$ $$+ ( -238957228258627660184 + 127490483181916618 \beta_{1} + 2291312411825072 \beta_{2} - 58701167527532 \beta_{3} - 153112224276 \beta_{4} - 570375351264 \beta_{5} - 290869299120 \beta_{6} + 215078706288 \beta_{7} - 74376208080 \beta_{8} + 88624041072 \beta_{9} ) q^{82}$$ $$+ ( 4688136854288 - 73785964622648471 \beta_{1} - 4909992866959901 \beta_{2} - 9255854438224 \beta_{3} + 691582343152 \beta_{4} + 197121167200 \beta_{5} + 287295685920 \beta_{6} - 1869733290080 \beta_{7} - 31044588000 \beta_{8} - 82886656640 \beta_{9} ) q^{83}$$ $$+ ( 618564076854659339264 - 243156795009748480 \beta_{1} - 2770801990840576 \beta_{2} - 36572825340800 \beta_{3} - 2846092420864 \beta_{4} - 5651936131072 \beta_{5} + 347121646592 \beta_{6} - 1554965400576 \beta_{7} - 39864667136 \beta_{8} - 42156649472 \beta_{9} ) q^{84}$$ $$+ ( -179689044629565362630 - 247232520136600230 \beta_{1} + 492944666104932 \beta_{2} + 18205469938884 \beta_{3} + 3409790403750 \beta_{4} - 2225239308876 \beta_{5} + 100067965080 \beta_{6} + 1410545560950 \beta_{7} - 9594049830 \beta_{8} - 100067965080 \beta_{9} ) q^{85}$$ $$+ ( -822183908396873745374 - 30244511776729509 \beta_{1} + 1100419782546633 \beta_{2} + 204571635281644 \beta_{3} - 4990033405600 \beta_{4} - 151800760320 \beta_{5} + 216181733760 \beta_{6} + 9692542327 \beta_{7} + 115472081280 \beta_{8} + 84171088000 \beta_{9} ) q^{86}$$ $$+ ( -9291271657564 + 562804000754464129 \beta_{1} + 7937357449832467 \beta_{2} + 21948196546541 \beta_{3} + 6108452337890 \beta_{4} - 379448327920 \beta_{5} - 209622087757 \beta_{6} + 3837670920554 \beta_{7} + 102999630640 \beta_{8} + 220089148480 \beta_{9} ) q^{87}$$ $$+ ( 1117821095278042364024 + 234334727683539144 \beta_{1} - 686640029768792 \beta_{2} + 1448021274576 \beta_{3} - 7370795203784 \beta_{4} + 32668857397912 \beta_{5} + 178261036248 \beta_{6} + 160779824240 \beta_{7} + 138224531200 \beta_{8} + 21512451144 \beta_{9} ) q^{88}$$ $$+ ( 342066814945483600207 - 772613343809157670 \beta_{1} + 1637584074138454 \beta_{2} + 291020974032075 \beta_{3} + 9530328129181 \beta_{4} + 21025426630803 \beta_{5} - 28561585548 \beta_{6} - 695194858251 \beta_{7} + 295332660579 \beta_{8} + 28561585548 \beta_{9} ) q^{89}$$ $$+ ( -2478280740524221112263 - 92785010396491195 \beta_{1} - 5765414638737940 \beta_{2} - 548601708536027 \beta_{3} - 8530781586757 \beta_{4} - 5598668804344 \beta_{5} + 327164921108 \beta_{6} - 175136972772 \beta_{7} + 168219948364 \beta_{8} - 442818293732 \beta_{9} ) q^{90}$$ $$+ ( 82521228048504 + 416266212516847896 \beta_{1} - 10001719218977584 \beta_{2} - 199254240682476 \beta_{3} + 10784665484432 \beta_{4} + 1317518181616 \beta_{5} + 99912568084 \beta_{6} - 155115400440 \beta_{7} + 199101304528 \beta_{8} + 15238190016 \beta_{9} ) q^{91}$$ $$+ ( 2147626444647357094672 - 537412199595435120 \beta_{1} + 5865339084653880 \beta_{2} - 69710172336760 \beta_{3} - 7607250100000 \beta_{4} - 1606844621368 \beta_{5} - 942483205656 \beta_{6} + 2591291984688 \beta_{7} + 363184679520 \beta_{8} - 50831193096 \beta_{9} ) q^{92}$$ $$+ ( 1549375388819582059864 - 460068001494823952 \beta_{1} + 905914463431888 \beta_{2} + 9566621767976 \beta_{3} + 10415795031640 \beta_{4} - 42593496144152 \beta_{5} - 474218077856 \beta_{6} - 6767460782184 \beta_{7} + 128407692200 \beta_{8} + 474218077856 \beta_{9} ) q^{93}$$ $$+ ( -2087454550726875145624 - 76910285854165676 \beta_{1} - 10658729939631412 \beta_{2} + 576121085058976 \beta_{3} - 10023466787856 \beta_{4} - 34887861900288 \beta_{5} - 75244318272 \beta_{6} - 1213181799372 \beta_{7} + 17689023936 \beta_{8} + 254531241792 \beta_{9} ) q^{94}$$ $$+ ( 168916751577836 + 718625021596710625 \beta_{1} + 13897807640789083 \beta_{2} - 407738106959195 \beta_{3} + 10104114949594 \beta_{4} + 3184422223424 \beta_{5} - 308457552861 \beta_{6} - 5751835882246 \beta_{7} + 178570099392 \beta_{8} - 386886305536 \beta_{9} ) q^{95}$$ $$+ ( 2925028508465485151360 + 1955918560151422848 \beta_{1} + 11918833233609600 \beta_{2} - 202162853988096 \beta_{3} - 5901390050176 \beta_{4} - 60254455136128 \beta_{5} + 223735047296 \beta_{6} - 6681941019392 \beta_{7} - 322238388224 \beta_{8} - 229129379456 \beta_{9} ) q^{96}$$ $$+ ( -2042990951613167013025 - 346377052795652878 \beta_{1} + 569912109002582 \beta_{2} - 256801127836045 \beta_{3} + 501145138621 \beta_{4} + 60299021069371 \beta_{5} + 670513232628 \beta_{6} + 9914894230037 \beta_{7} - 527708973245 \beta_{8} - 670513232628 \beta_{9} ) q^{97}$$ $$+ ( -2311041443711490087486 - 104152407201261433 \beta_{1} - 7677824998091808 \beta_{2} - 523039299611640 \beta_{3} + 2971077701368 \beta_{4} + 86278459507520 \beta_{5} - 1185043487712 \beta_{6} + 183028926304 \beta_{7} - 75122076960 \beta_{8} + 706431641440 \beta_{9} ) q^{98}$$ $$+ ( -449886184122528 - 523719790567619805 \beta_{1} - 8038457946842895 \beta_{2} + 1106860661533944 \beta_{3} - 10193981248080 \beta_{4} - 6866401610880 \beta_{5} - 659461928184 \beta_{6} + 4862590119792 \beta_{7} - 861159761280 \beta_{8} + 167656656384 \beta_{9} ) q^{99}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10q$$ $$\mathstrut +\mathstrut 1540q^{2}$$ $$\mathstrut +\mathstrut 2264464q^{4}$$ $$\mathstrut -\mathstrut 17091100q^{5}$$ $$\mathstrut -\mathstrut 791935776q^{6}$$ $$\mathstrut +\mathstrut 9804431680q^{8}$$ $$\mathstrut -\mathstrut 104309613702q^{9}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$10q$$ $$\mathstrut +\mathstrut 1540q^{2}$$ $$\mathstrut +\mathstrut 2264464q^{4}$$ $$\mathstrut -\mathstrut 17091100q^{5}$$ $$\mathstrut -\mathstrut 791935776q^{6}$$ $$\mathstrut +\mathstrut 9804431680q^{8}$$ $$\mathstrut -\mathstrut 104309613702q^{9}$$ $$\mathstrut +\mathstrut 159414035240q^{10}$$ $$\mathstrut +\mathstrut 519021175680q^{12}$$ $$\mathstrut -\mathstrut 531230356540q^{13}$$ $$\mathstrut -\mathstrut 5894008940736q^{14}$$ $$\mathstrut -\mathstrut 27717620084480q^{16}$$ $$\mathstrut +\mathstrut 14058178115540q^{17}$$ $$\mathstrut +\mathstrut 16283956279140q^{18}$$ $$\mathstrut +\mathstrut 233643631625120q^{20}$$ $$\mathstrut -\mathstrut 313135665760512q^{21}$$ $$\mathstrut +\mathstrut 120589650366240q^{22}$$ $$\mathstrut -\mathstrut 2007619616180736q^{24}$$ $$\mathstrut +\mathstrut 7710175817606670q^{25}$$ $$\mathstrut -\mathstrut 3953973318046744q^{26}$$ $$\mathstrut +\mathstrut 1177952265288960q^{28}$$ $$\mathstrut -\mathstrut 14464474185570172q^{29}$$ $$\mathstrut +\mathstrut 10252783730669760q^{30}$$ $$\mathstrut -\mathstrut 73831192092953600q^{32}$$ $$\mathstrut +\mathstrut 72906957628179840q^{33}$$ $$\mathstrut -\mathstrut 26625976920357496q^{34}$$ $$\mathstrut -\mathstrut 36609276067793136q^{36}$$ $$\mathstrut +\mathstrut 324587767823538020q^{37}$$ $$\mathstrut +\mathstrut 369140808808591200q^{38}$$ $$\mathstrut -\mathstrut 1158049295468479360q^{40}$$ $$\mathstrut -\mathstrut 1355457345814233100q^{41}$$ $$\mathstrut +\mathstrut 2644455724909770240q^{42}$$ $$\mathstrut -\mathstrut 1919761520625210240q^{44}$$ $$\mathstrut +\mathstrut 9367295763131460q^{45}$$ $$\mathstrut +\mathstrut 6132624002565203904q^{46}$$ $$\mathstrut -\mathstrut 18374169355003791360q^{48}$$ $$\mathstrut +\mathstrut 183553700661552298q^{49}$$ $$\mathstrut +\mathstrut 29084394113996178540q^{50}$$ $$\mathstrut -\mathstrut 38492433127521128800q^{52}$$ $$\mathstrut -\mathstrut 11424344766226018780q^{53}$$ $$\mathstrut +\mathstrut 79182840955393027008q^{54}$$ $$\mathstrut -\mathstrut 102844865827678657536q^{56}$$ $$\mathstrut +\mathstrut 30188516183231506560q^{57}$$ $$\mathstrut +\mathstrut 173059696836432320360q^{58}$$ $$\mathstrut -\mathstrut 377769459746631548160q^{60}$$ $$\mathstrut +\mathstrut 58241718146183220740q^{61}$$ $$\mathstrut +\mathstrut 317897221476923285760q^{62}$$ $$\mathstrut -\mathstrut 412080172429987966976q^{64}$$ $$\mathstrut -\mathstrut 121531695066047433560q^{65}$$ $$\mathstrut +\mathstrut 700585278654050000640q^{66}$$ $$\mathstrut -\mathstrut 667890522330713878240q^{68}$$ $$\mathstrut +\mathstrut 54497497397908094208q^{69}$$ $$\mathstrut +\mathstrut 1014396282552321513600q^{70}$$ $$\mathstrut -\mathstrut 1244773253376491901120q^{72}$$ $$\mathstrut +\mathstrut 83030422224875883380q^{73}$$ $$\mathstrut +\mathstrut 801906083011176262184q^{74}$$ $$\mathstrut +\mathstrut 286747421140990627200q^{76}$$ $$\mathstrut -\mathstrut 704201825495570814720q^{77}$$ $$\mathstrut -\mathstrut 964412732782594923840q^{78}$$ $$\mathstrut +\mathstrut 1509611955050789726720q^{80}$$ $$\mathstrut +\mathstrut 251577744224553978858q^{81}$$ $$\mathstrut -\mathstrut 2389572513290905036600q^{82}$$ $$\mathstrut +\mathstrut 6185640644326191863808q^{84}$$ $$\mathstrut -\mathstrut 1796890366112110859960q^{85}$$ $$\mathstrut -\mathstrut 8221838255181360278496q^{86}$$ $$\mathstrut +\mathstrut 11178210842814940531200q^{88}$$ $$\mathstrut +\mathstrut 3420669206528505615668q^{89}$$ $$\mathstrut -\mathstrut 24782809563337507735320q^{90}$$ $$\mathstrut +\mathstrut 21476264197983864733440q^{92}$$ $$\mathstrut +\mathstrut 15493754050318100597760q^{93}$$ $$\mathstrut -\mathstrut 20874543047091388369536q^{94}$$ $$\mathstrut +\mathstrut 29250284502468708163584q^{96}$$ $$\mathstrut -\mathstrut 20429910746446196146540q^{97}$$ $$\mathstrut -\mathstrut 23110416876469762175420q^{98}$$ $$\mathstrut +\mathstrut O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10}\mathstrut -\mathstrut$$ $$5$$ $$x^{9}\mathstrut -\mathstrut$$ $$63342$$ $$x^{8}\mathstrut -\mathstrut$$ $$45742928$$ $$x^{7}\mathstrut +\mathstrut$$ $$34835133568$$ $$x^{6}\mathstrut +\mathstrut$$ $$12622768560288$$ $$x^{5}\mathstrut +\mathstrut$$ $$13335465927292608$$ $$x^{4}\mathstrut -\mathstrut$$ $$404504858109047040$$ $$x^{3}\mathstrut -\mathstrut$$ $$95573988251584922880$$ $$x^{2}\mathstrut -\mathstrut$$ $$1820809561188275535801600$$ $$x\mathstrut +\mathstrut$$ $$1168838172864011361183920640$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{9}\mathstrut -\mathstrut$$ $$43$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$61708$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$43398024$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$36484258480$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$11236366738048$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$12908483991246784$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$895027249776424832$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$61562952760080779264$$ $$\nu\mathstrut -\mathstrut$$ $$1636659059392911125462272$$$$)/$$$$11\!\cdots\!24$$ $$\beta_{2}$$ $$=$$ $$($$$$12193477$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$979139351$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$4293302429244$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$3155392054892456$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$678931388885259920$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$44396490427997729664$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$142580353625449105835712$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$23161102577930174325224832$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$48064368574360474224686917632$$ $$\nu\mathstrut -\mathstrut$$ $$55187059199466748476342777147648$$$$)/$$$$27\!\cdots\!76$$ $$\beta_{3}$$ $$=$$ $$($$$$-$$$$6915365$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$91005641$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$2247044895420$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$1649325761829544$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$334950273802777744$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$10429641738256006272$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$87886170747465800167104$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$4593599511946178931213696$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$24927064383085695430121023488$$ $$\nu\mathstrut +\mathstrut$$ $$28946588768811955036170527756544$$$$)/$$$$69\!\cdots\!44$$ $$\beta_{4}$$ $$=$$ $$($$$$-$$$$1567851625$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$133933514147$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$556463209123308$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$409016929069908680$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$88588053551010598352$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$6247091218730372045184$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$18239779026117307423505856$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$3088094756747195452851651456$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$6964149343576421308934665930752$$ $$\nu\mathstrut +\mathstrut$$ $$7145480416419482982402651534132480$$$$)/$$$$27\!\cdots\!76$$ $$\beta_{5}$$ $$=$$ $$($$$$2954795719$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$1592279695507$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$438857576441772$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$92979328154984120$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$36058334367436693808$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$1097776241233167126912$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$38255428400758485907732032$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$19120799683884180390773758848$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$6869763067095753054119337166848$$ $$\nu\mathstrut -\mathstrut$$ $$4100737456350904179397805802941184$$$$)/$$$$92\!\cdots\!92$$ $$\beta_{6}$$ $$=$$ $$($$$$12059814031$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$15503143391237$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$16114776477959604$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$8489948102939117816$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$1652537251902256519088$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$142229260695441600838272$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$91990521490720802441544768$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$221999458320462660294107152512$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$195487840706234197991299109074944$$ $$\nu\mathstrut -\mathstrut$$ $$128243457687214939310629170299474688$$$$)/$$$$13\!\cdots\!88$$ $$\beta_{7}$$ $$=$$ $$($$$$-$$$$10985599895$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$2735389283747$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$963042238239252$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$1518873103244754232$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$354618914492990665264$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$19973130242086426109568$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$136181830129830949811012160$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$28086279033124733582675182464$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$6934007466044706193800405055488$$ $$\nu\mathstrut +\mathstrut$$ $$31483183425385470241121625651542784$$$$)/$$$$69\!\cdots\!44$$ $$\beta_{8}$$ $$=$$ $$($$$$-$$$$46997419903$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$22896956550571$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$4786688486201100$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$2466841500826585208$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$759670626446989619888$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$2496648121285228158336$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$606135772085440237585762368$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$248464726850720028080529717120$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$81133322843924532895487369988096$$ $$\nu\mathstrut +\mathstrut$$ $$78483132563029248768209469111681792$$$$)/$$$$13\!\cdots\!88$$ $$\beta_{9}$$ $$=$$ $$($$$$8681772593$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$10047012439493$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$5634919130649780$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$1742991710918177528$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$265307168086253747120$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$37391801177731556661888$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$122502133449736389128227776$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$128545835758382815271645913216$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$74734674879626511411481544079360$$ $$\nu\mathstrut +\mathstrut$$ $$15536148118865170391775110280991488$$$$)/$$$$11\!\cdots\!24$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{4}\mathstrut +\mathstrut$$ $$3$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$136$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$3183$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$131071$$$$)/262144$$ $$\nu^{2}$$ $$=$$ $$($$$$16$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$16$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$112$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$27$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$2993$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$25336$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$3474299$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$3321601659$$$$)/262144$$ $$\nu^{3}$$ $$=$$ $$($$$$-$$$$1024$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$2064$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$32752$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$15360$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$140400$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$31799$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$1385467$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$16322008$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$3279655767$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$3622281117033$$$$)/262144$$ $$\nu^{4}$$ $$=$$ $$($$$$-$$$$173056$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$143824$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$29099568$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$287744$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$86675632$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$163017$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$2279565189$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$39964254376$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$1239496741993$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$3418223733400553$$$$)/262144$$ $$\nu^{5}$$ $$=$$ $$($$$$-$$$$662658048$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$50467920$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$8113890224$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$957193216$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$14157888976$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$91723649$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$270378394019$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$31536679901544$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$2178093807993055$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$1294766211272704095$$$$)/262144$$ $$\nu^{6}$$ $$=$$ $$($$$$588186385408$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$39909369808$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$1521215942608$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$116501527552$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$31944420709712$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$3137287078679$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$150236658461477$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$7420110603892136$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$330165047108715657$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$2272147564306364850185$$$$)/262144$$ $$\nu^{7}$$ $$=$$ $$($$$$-$$$$213620913249280$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$43366557800016$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$2508561790606928$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$187983575589888$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$15391916965900240$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$11093536753582705$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$309118417645570445$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$784648433799572248$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$149351135913589425391$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$345364915244056276492655$$$$)/262144$$ $$\nu^{8}$$ $$=$$ $$($$$$6510433563573248$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$189098379620989744$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$806004223242620112$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$156784594785911808$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$3370665054964189776$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$306073062636384249$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$148114398009503412213$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$2566320217780703529944$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$330389491185012620781159$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$155723191263170490186335513$$$$)/262144$$ $$\nu^{9}$$ $$=$$ $$($$$$51963351995197301760$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$32113466299177618608$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$431636465317742050128$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$150204343291586288640$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$159323986847473867568$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$1179115799131348920641$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$57054922663361527919389$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$736628589008309329440408$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$179788423662949344643981343$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$344295598412894224941009467233$$$$)/262144$$

## Character Values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/4\mathbb{Z}\right)^\times$$.

 $$n$$ $$3$$ $$\chi(n)$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3.1
 −501.982 + 216.483i −501.982 − 216.483i −313.209 + 431.746i −313.209 − 431.746i 1.30158 + 510.489i 1.30158 − 510.489i 407.912 + 251.607i 407.912 − 251.607i 408.476 + 250.605i 408.476 − 250.605i
−1855.93 865.934i 142172.i 2.69462e6 + 3.21422e6i 1.40852e7 1.23111e8 2.63860e8i 7.81741e8i −2.21772e9 8.29872e9i 1.11682e10 −2.61411e10 1.21969e10i
3.2 −1855.93 + 865.934i 142172.i 2.69462e6 3.21422e6i 1.40852e7 1.23111e8 + 2.63860e8i 7.81741e8i −2.21772e9 + 8.29872e9i 1.11682e10 −2.61411e10 + 1.21969e10i
3.3 −1100.83 1726.98i 308212.i −1.77063e6 + 3.80224e6i −6.05072e7 −5.32276e8 + 3.39290e8i 1.97415e9i 8.51558e9 1.12779e9i −6.36133e10 6.66084e10 + 1.04495e11i
3.4 −1100.83 + 1726.98i 308212.i −1.77063e6 3.80224e6i −6.05072e7 −5.32276e8 3.39290e8i 1.97415e9i 8.51558e9 + 1.12779e9i −6.36133e10 6.66084e10 1.04495e11i
3.5 157.206 2041.96i 75737.5i −4.14488e6 642017.i 1.73209e7 1.54653e8 + 1.19064e7i 2.02930e9i −1.96257e9 + 8.36273e9i 2.56449e10 2.72295e9 3.53685e10i
3.6 157.206 + 2041.96i 75737.5i −4.14488e6 + 642017.i 1.73209e7 1.54653e8 1.19064e7i 2.02930e9i −1.96257e9 8.36273e9i 2.56449e10 2.72295e9 + 3.53685e10i
3.7 1783.65 1006.43i 267899.i 2.16851e6 3.59023e6i 8.57934e7 −2.69621e8 4.77838e8i 2.80689e8i 2.54549e8 8.58616e9i −4.03887e10 1.53025e11 8.63449e10i
3.8 1783.65 + 1006.43i 267899.i 2.16851e6 + 3.59023e6i 8.57934e7 −2.69621e8 + 4.77838e8i 2.80689e8i 2.54549e8 + 8.58616e9i −4.03887e10 1.53025e11 + 8.63449e10i
3.9 1785.90 1002.42i 127855.i 2.18461e6 3.58046e6i −6.52379e7 1.28165e8 + 2.28337e8i 3.27904e9i 3.12381e8 8.58425e9i 1.50341e10 −1.16509e11 + 6.53958e10i
3.10 1785.90 + 1002.42i 127855.i 2.18461e6 + 3.58046e6i −6.52379e7 1.28165e8 2.28337e8i 3.27904e9i 3.12381e8 + 8.58425e9i 1.50341e10 −1.16509e11 6.53958e10i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 3.10 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
4.b Odd 1 yes

## Hecke kernels

There are no other newforms in $$S_{23}^{\mathrm{new}}(4, [\chi])$$.