Properties

Label 6.21.b.a
Level $6$
Weight $21$
Character orbit 6.b
Analytic conductor $15.211$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 6 = 2 \cdot 3 \)
Weight: \( k \) \(=\) \( 21 \)
Character orbit: \([\chi]\) \(=\) 6.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(15.2108259062\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} + \cdots)\)
Defining polynomial: \(x^{6} + 3396 x^{4} + 2813589 x^{2} + 548136050\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{31}\cdot 3^{17} \)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + ( 3141 - 11 \beta_{1} - \beta_{2} ) q^{3} -524288 q^{4} + ( -1706 \beta_{1} + 31 \beta_{2} + 22 \beta_{3} + 7 \beta_{5} ) q^{5} + ( 5842944 + 3121 \beta_{1} - 8 \beta_{2} + 40 \beta_{3} - 24 \beta_{4} + 56 \beta_{5} ) q^{6} + ( -94445302 - 257 \beta_{1} + 819 \beta_{2} - 164 \beta_{3} - 163 \beta_{4} + 70 \beta_{5} ) q^{7} -524288 \beta_{1} q^{8} + ( 324156897 + 1027539 \beta_{1} - 8586 \beta_{2} - 15849 \beta_{3} + 1053 \beta_{4} + 702 \beta_{5} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} + ( 3141 - 11 \beta_{1} - \beta_{2} ) q^{3} -524288 q^{4} + ( -1706 \beta_{1} + 31 \beta_{2} + 22 \beta_{3} + 7 \beta_{5} ) q^{5} + ( 5842944 + 3121 \beta_{1} - 8 \beta_{2} + 40 \beta_{3} - 24 \beta_{4} + 56 \beta_{5} ) q^{6} + ( -94445302 - 257 \beta_{1} + 819 \beta_{2} - 164 \beta_{3} - 163 \beta_{4} + 70 \beta_{5} ) q^{7} -524288 \beta_{1} q^{8} + ( 324156897 + 1027539 \beta_{1} - 8586 \beta_{2} - 15849 \beta_{3} + 1053 \beta_{4} + 702 \beta_{5} ) q^{9} + ( 893497344 - 8144 \beta_{1} + 43344 \beta_{2} - 11408 \beta_{3} + 2288 \beta_{4} + 976 \beta_{5} ) q^{10} + ( -6722624 \beta_{1} - 176174 \beta_{2} - 69263 \beta_{3} - 6323 \beta_{5} ) q^{11} + ( -1646788608 + 5767168 \beta_{1} + 524288 \beta_{2} ) q^{12} + ( -8316424270 + 84290 \beta_{1} - 669690 \beta_{2} + 197030 \beta_{3} - 118430 \beta_{4} + 5690 \beta_{5} ) q^{13} + ( -94569054 \beta_{1} + 1765120 \beta_{2} + 428800 \beta_{3} - 95744 \beta_{5} ) q^{14} + ( 81284691888 + 292874325 \beta_{1} - 343335 \beta_{2} + 177108 \beta_{3} - 120699 \beta_{4} - 363534 \beta_{5} ) q^{15} + 274877906944 q^{16} + ( -798126640 \beta_{1} + 6580256 \beta_{2} - 4684108 \beta_{3} - 4126516 \beta_{5} ) q^{17} + ( -539854958592 + 324020385 \beta_{1} - 8430480 \beta_{2} - 4807728 \beta_{3} - 1631664 \beta_{4} - 1542672 \beta_{5} ) q^{18} + ( 2654599547882 - 15326008 \beta_{1} + 57731310 \beta_{2} - 12955321 \beta_{3} - 5910026 \beta_{4} + 3539339 \beta_{5} ) q^{19} + ( 894435328 \beta_{1} - 16252928 \beta_{2} - 11534336 \beta_{3} - 3670016 \beta_{5} ) q^{20} + ( -3446259021198 - 9609515362 \beta_{1} + 106224349 \beta_{2} - 11344374 \beta_{3} - 5871852 \beta_{4} - 5459319 \beta_{5} ) q^{21} + ( 3530698272768 + 5775856 \beta_{1} - 50215536 \beta_{2} + 15046192 \beta_{3} - 9969232 \beta_{4} + 698896 \beta_{5} ) q^{22} + ( 16280328080 \beta_{1} - 6505084 \beta_{2} - 80265538 \beta_{3} - 46858306 \beta_{5} ) q^{23} + ( -3063385423872 - 1636302848 \beta_{1} + 4194304 \beta_{2} - 20971520 \beta_{3} + 12582912 \beta_{4} - 29360128 \beta_{5} ) q^{24} + ( 56251752051409 + 82714386 \beta_{1} - 365487066 \beta_{2} + 89173902 \beta_{3} + 8791458 \beta_{4} - 15250974 \beta_{5} ) q^{25} + ( -8387908910 \beta_{1} + 1085895680 \beta_{2} + 438748160 \beta_{3} + 46069760 \beta_{5} ) q^{26} + ( -169919088174027 + 32976509643 \beta_{1} - 548786097 \beta_{2} + 85777785 \beta_{3} + 19844838 \beta_{4} + 162328617 \beta_{5} ) q^{27} + ( 49516538494976 + 134742016 \beta_{1} - 429391872 \beta_{2} + 85983232 \beta_{3} + 85458944 \beta_{4} - 36700160 \beta_{5} ) q^{28} + ( -104623278398 \beta_{1} - 4289732051 \beta_{2} - 225602582 \beta_{3} + 722584861 \beta_{5} ) q^{29} + ( -153468214382592 + 81649188528 \beta_{1} - 936998448 \beta_{2} + 997358832 \beta_{3} + 26737776 \beta_{4} + 39508560 \beta_{5} ) q^{30} + ( 305424237732938 - 1313586195 \beta_{1} + 7586290845 \beta_{2} - 2052596490 \beta_{3} + 624095115 \beta_{4} + 114915180 \beta_{5} ) q^{31} + 274877906944 \beta_{1} q^{32} + ( -583927085724624 - 353564046843 \beta_{1} - 527660826 \beta_{2} - 2628105027 \beta_{3} + 488122803 \beta_{4} + 307955058 \beta_{5} ) q^{33} + ( 418118850330624 + 5066024768 \beta_{1} - 23695560000 \beta_{2} + 5929687616 \beta_{3} - 23190464 \beta_{4} - 840472384 \beta_{5} ) q^{34} + ( 593126156744 \beta_{1} + 1951885376 \beta_{2} + 770218262 \beta_{3} + 71753882 \beta_{5} ) q^{35} + ( -169951571214336 - 538726367232 \beta_{1} + 4501536768 \beta_{2} + 8309440512 \beta_{3} - 552075264 \beta_{4} - 368050176 \beta_{5} ) q^{36} + ( 70722054478802 - 6364664474 \beta_{1} + 20682929106 \beta_{2} - 4204428158 \beta_{3} - 3865216474 \beta_{4} + 1704980158 \beta_{5} ) q^{37} + ( 2649704042386 \beta_{1} + 68357036288 \beta_{2} + 12727721216 \beta_{3} - 6034774528 \beta_{5} ) q^{38} + ( 2309157224272890 - 6307839950200 \beta_{1} + 14064755860 \beta_{2} - 27571490370 \beta_{3} - 2982064410 \beta_{4} - 3110404050 \beta_{5} ) q^{39} + ( -468449935491072 + 4269801472 \beta_{1} - 22724739072 \beta_{2} + 5981077504 \beta_{3} - 1199570944 \beta_{4} - 511705088 \beta_{5} ) q^{40} + ( 6276770098900 \beta_{1} - 173744114654 \beta_{2} - 74388551708 \beta_{3} - 9884308094 \beta_{5} ) q^{41} + ( 5028251198164992 - 3439740523006 \beta_{1} + 18993681392 \beta_{2} + 24442544720 \beta_{3} + 1809826512 \beta_{4} - 8165547152 \beta_{5} ) q^{42} + ( -8248125388991926 - 48958919508 \beta_{1} + 233609552442 \beta_{2} - 58952523261 \beta_{3} + 2200540602 \beta_{4} + 7793063151 \beta_{5} ) q^{43} + ( 3524591091712 \beta_{1} + 92365914112 \beta_{2} + 36313759744 \beta_{3} + 3315073024 \beta_{5} ) q^{44} + ( 10716322761572832 - 19205364928644 \beta_{1} - 46071579411 \beta_{2} - 2255534100 \beta_{3} - 4565878146 \beta_{4} + 19075843197 \beta_{5} ) q^{45} + ( -8536437339119616 + 56660401568 \beta_{1} - 275137863840 \beta_{2} + 69933322016 \beta_{3} - 4595424224 \beta_{4} - 8677496224 \beta_{5} ) q^{46} + ( 33912733620240 \beta_{1} + 255115412880 \beta_{2} + 158968157460 \beta_{3} + 44357811900 \beta_{5} ) q^{47} + ( 863391505711104 - 3023656976384 \beta_{1} - 274877906944 \beta_{2} ) q^{48} + ( -30994188764917245 + 120769564846 \beta_{1} - 476034658470 \beta_{2} + 109627595602 \beta_{3} + 37524276062 \beta_{4} - 26382306818 \beta_{5} ) q^{49} + ( 56263108660273 \beta_{1} - 145143702528 \beta_{2} + 9187544064 \beta_{3} + 34541266944 \beta_{5} ) q^{50} + ( 19076989325548608 - 167674409633988 \beta_{1} + 368450036352 \beta_{2} + 182005744788 \beta_{3} + 84121068564 \beta_{4} + 39090398616 \beta_{5} ) q^{51} + ( 4360201447669760 - 44192235520 \beta_{1} + 351110430720 \beta_{2} - 103300464640 \beta_{3} + 62091427840 \beta_{4} - 2983198720 \beta_{5} ) q^{52} + ( 316328417666350 \beta_{1} - 962580777701 \beta_{2} - 221625421682 \beta_{3} + 59540902531 \beta_{5} ) q^{53} + ( -17261244125153280 - 170113716386991 \beta_{1} + 739134738792 \beta_{2} - 293723087112 \beta_{3} - 14554065096 \beta_{4} + 38921070888 \beta_{5} ) q^{54} + ( 78083931113345952 - 14528778282 \beta_{1} - 73908862398 \beta_{2} + 33660411426 \beta_{3} - 60732783306 \beta_{4} + 12543593598 \beta_{5} ) q^{55} + ( 49581420183552 \beta_{1} - 925431234560 \beta_{2} - 224814694400 \beta_{3} + 50197430272 \beta_{5} ) q^{56} + ( -208354598184897198 - 444936482547763 \beta_{1} - 2165228573096 \beta_{2} + 17153095329 \beta_{3} - 241368306273 \beta_{4} - 216921520584 \beta_{5} ) q^{57} + ( 55021780829245440 - 920566546928 \beta_{1} + 3915010738800 \beta_{2} - 937934847536 \beta_{3} - 163271348656 \beta_{4} + 180639649264 \beta_{5} ) q^{58} + ( 540213395512904 \beta_{1} + 6171949753142 \beta_{2} + 1109342405849 \beta_{3} - 568784507119 \beta_{5} ) q^{59} + ( -42616588540575744 - 153550494105600 \beta_{1} + 180006420480 \beta_{2} - 92855599104 \beta_{3} + 63281037312 \beta_{4} + 190596513792 \beta_{5} ) q^{60} + ( 550305002145348914 + 1072457321694 \beta_{1} - 5415634159110 \beta_{2} + 1397926395978 \beta_{3} - 176071424802 \beta_{4} - 149397649482 \beta_{5} ) q^{61} + ( 305741823886658 \beta_{1} - 5091781367040 \beta_{2} - 2720126388480 \beta_{3} - 613719559680 \beta_{5} ) q^{62} + ( -120669881455215414 - 748735004507571 \beta_{1} + 5007891160881 \beta_{2} + 596489540940 \beta_{3} + 89353705851 \beta_{4} - 674698091886 \beta_{5} ) q^{63} -144115188075855872 q^{64} + ( 1054902372825500 \beta_{1} + 5694090028550 \beta_{2} + 1205787364100 \beta_{3} - 415345587250 \beta_{5} ) q^{65} + ( 185157902366502912 - 584033120040336 \beta_{1} - 3033132802032 \beta_{2} - 2226810700368 \beta_{3} - 257277603024 \beta_{4} - 265996813680 \beta_{5} ) q^{66} + ( -463204378751426038 - 208380488070 \beta_{1} + 1038487520196 \beta_{2} - 266698156035 \beta_{3} + 28305103944 \beta_{4} + 30012564021 \beta_{5} ) q^{67} + ( 418448219832320 \beta_{1} - 3449949257728 \beta_{2} + 2455821615104 \beta_{3} + 2163482820608 \beta_{5} ) q^{68} + ( 111322401220362528 - 1859574828173538 \beta_{1} + 3485646840012 \beta_{2} + 1908684060918 \beta_{3} + 446084903394 \beta_{4} + 2105606037036 \beta_{5} ) q^{69} + ( -311036550339895296 - 66049922464 \beta_{1} + 566314195104 \beta_{2} - 169230108448 \beta_{3} + 110606238688 \beta_{4} - 7426052704 \beta_{5} ) q^{70} + ( 2093637286782368 \beta_{1} - 32132008960684 \beta_{2} - 3832841460238 \beta_{3} + 4126696915994 \beta_{5} ) q^{71} + ( 283039476530282496 - 169879999610880 \beta_{1} + 4419999498240 \beta_{2} + 2520634097664 \beta_{3} + 855461855232 \beta_{4} + 808804417536 \beta_{5} ) q^{72} + ( -1524900256132083550 - 1141857242352 \beta_{1} + 11336787861840 \beta_{2} - 3477924186624 \beta_{3} + 2574908884656 \beta_{4} - 238841940384 \beta_{5} ) q^{73} + ( 67769136153522 \beta_{1} + 42052413254656 \beta_{2} + 10041207135232 \beta_{3} - 2385758369792 \beta_{5} ) q^{74} + ( 1521224597252841813 + 295338021020923 \beta_{1} - 57600926762443 \beta_{2} - 4298768416458 \beta_{3} + 642960222606 \beta_{4} + 511956795870 \beta_{5} ) q^{75} + ( -1391774687759958016 + 8035242082304 \beta_{1} - 30267833057280 \beta_{2} + 6792319336448 \beta_{3} + 3098555711488 \beta_{4} - 1855632965632 \beta_{5} ) q^{76} + ( -2853822506409124 \beta_{1} - 1714286856394 \beta_{2} - 3649705031908 \beta_{3} - 1846965647866 \beta_{5} ) q^{77} + ( 3302542722480721920 + 2312673990443570 \beta_{1} + 3097373341040 \beta_{2} + 14001401593040 \beta_{3} - 1938410900400 \beta_{4} - 4866206646800 \beta_{5} ) q^{78} + ( 15009341400542858 - 7527108047827 \beta_{1} + 25168027149405 \beta_{2} - 5225027100874 \beta_{3} - 4267918745909 \beta_{4} + 1965837798956 \beta_{5} ) q^{79} + ( -468941709246464 \beta_{1} + 8521215115264 \beta_{2} + 6047313952768 \beta_{3} + 1924145348608 \beta_{5} ) q^{80} + ( -2218393021297788063 + 9715683455787024 \beta_{1} + 152035437851010 \beta_{2} - 8502457872996 \beta_{3} - 2460574428156 \beta_{4} - 85710765114 \beta_{5} ) q^{81} + ( -3284896978581086208 + 10113331776160 \beta_{1} - 70908118402464 \beta_{2} + 20267662112800 \beta_{3} - 10162530048736 \beta_{4} + 8199712096 \beta_{5} ) q^{82} + ( -21756374237598288 \beta_{1} - 33883669156914 \beta_{2} - 22302757045773 \beta_{3} - 6604920396081 \beta_{5} ) q^{83} + ( 1806832249705857024 + 5038153590112256 \beta_{1} - 55692151488512 \beta_{2} + 5947719155712 \beta_{3} + 3078541541376 \beta_{4} + 2862255439872 \beta_{5} ) q^{84} + ( 13037314343755667328 - 32876865319128 \beta_{1} + 144965052417528 \beta_{2} - 35334800346696 \beta_{3} - 3625851030744 \beta_{4} + 6083786058312 \beta_{5} ) q^{85} + ( -8250019843045966 \beta_{1} + 14198567039232 \beta_{2} - 30395341744896 \beta_{3} - 21076918454784 \beta_{5} ) q^{86} + ( -14555732551457923440 + 27259630195895175 \beta_{1} - 81703062817413 \beta_{2} - 86908178697468 \beta_{3} - 5037371822457 \beta_{4} - 11163972186714 \beta_{5} ) q^{87} + ( -1851102736032989184 - 3028211990528 \beta_{1} + 26327402938368 \beta_{2} - 7888537911296 \beta_{3} + 5226748706816 \beta_{4} - 366422786048 \beta_{5} ) q^{88} + ( -27024222103944076 \beta_{1} + 13083280910282 \beta_{2} - 11456000349496 \beta_{3} - 9490256391754 \beta_{5} ) q^{89} + ( 10067830628444909568 + 10691488642737456 \beta_{1} + 143971187843664 \beta_{2} - 11323793998224 \beta_{3} - 2408979150864 \beta_{4} + 816208603344 \beta_{5} ) q^{90} + ( 22077127781702501620 + 37237859243850 \beta_{1} - 163601755088310 \beta_{2} + 39810268623900 \beta_{3} + 4360680592710 \beta_{4} - 6933089972760 \beta_{5} ) q^{91} + ( -8535580648407040 \beta_{1} + 3410537480192 \beta_{2} + 42082258386944 \beta_{3} + 24567247536128 \beta_{5} ) q^{92} + ( -26179320591023430222 + 26072690447351792 \beta_{1} - 326702502608633 \beta_{2} + 207300579668460 \beta_{3} + 11595047586030 \beta_{4} + 13644581128875 \beta_{5} ) q^{93} + ( -17788065966905671680 - 50981293202880 \beta_{1} + 279044538978240 \beta_{2} - 74168117319360 \beta_{3} + 17627930299200 \beta_{4} + 5558893817280 \beta_{5} ) q^{94} + ( -4977367547098192 \beta_{1} + 205873882524092 \beta_{2} + 176749337683154 \beta_{3} + 64874826105074 \beta_{5} ) q^{95} + ( 1606096217111003136 + 857893947572224 \beta_{1} - 2199023255552 \beta_{2} + 10995116277760 \beta_{3} - 6597069766656 \beta_{4} + 15393162788864 \beta_{5} ) q^{96} + ( 15338150853331262306 + 167418825680510 \beta_{1} - 750477532850262 \beta_{2} + 184318277463410 \beta_{3} + 13204422996622 \beta_{4} - 30103874779522 \beta_{5} ) q^{97} + ( -30961510606515293 \beta_{1} - 451033312787456 \beta_{2} - 69800644078592 \beta_{3} + 48326276110336 \beta_{5} ) q^{98} + ( -37390428468498807072 + 37970978140961586 \beta_{1} + 514983524756004 \beta_{2} + 46272402631539 \beta_{3} + 20562285600666 \beta_{4} + 913132240983 \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 18846q^{3} - 3145728q^{4} + 35057664q^{6} - 566671812q^{7} + 1944941382q^{9} + O(q^{10}) \) \( 6q + 18846q^{3} - 3145728q^{4} + 35057664q^{6} - 566671812q^{7} + 1944941382q^{9} + 5360984064q^{10} - 9880731648q^{12} - 49898545620q^{13} + 487708151328q^{15} + 1649267441664q^{16} - 3239129751552q^{18} + 15927597287292q^{19} - 20677554127188q^{21} + 21184189636608q^{22} - 18380312543232q^{24} + 337510512308454q^{25} - 1019514529044162q^{27} + 297099230969856q^{28} - 920809286295552q^{30} + 1832545426397628q^{31} - 3503562514347744q^{33} + 2508713101983744q^{34} - 1019709427286016q^{36} + 424332326872812q^{37} + 13854943345637340q^{39} - 2810699612946432q^{40} + 30169507188989952q^{42} - 49488752333951556q^{43} + 64297936569436992q^{45} - 51218624034717696q^{46} + 5180349034266624q^{48} - 185965132589503470q^{49} + 114461935953291648q^{51} + 26161208686018560q^{52} - 103567464750919680q^{54} + 468503586680075712q^{55} - 1250127589109383188q^{57} + 330130684975472640q^{58} - 255699531243454464q^{60} + 3301830012872093484q^{61} - 724019288731292484q^{63} - 864691128455135232q^{64} + 1110947414199017472q^{66} - 2779226272508556228q^{67} + 667934407322175168q^{69} - 1866219302039371776q^{70} + 1698236859181694976q^{72} - 9149401536792501300q^{73} + 9127347583517050878q^{75} - 8350648126559748096q^{76} + 19815256334884331520q^{78} + 90056048403257148q^{79} - 13310358127786728378q^{81} - 19709381871486517248q^{82} + 10840993498235142144q^{84} + 78223886062534003968q^{85} - 87334395308747540640q^{87} - 11106616416197935104q^{88} + 60406983770669457408q^{90} + 132462766690215009720q^{91} - 157075923546140581332q^{93} - 106728395801434030080q^{94} + 9636577302666018816q^{96} + 92028905119987573836q^{97} - 224342570810992842432q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} + 3396 x^{4} + 2813589 x^{2} + 548136050\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 128 \nu^{5} + 2553728 \nu^{3} + 3960388352 \nu \)\()/75540465\)
\(\beta_{2}\)\(=\)\((\)\( -238684 \nu^{5} - 8939700 \nu^{4} - 682799374 \nu^{3} - 23346920520 \nu^{2} - 397366323826 \nu - 8830503352440 \)\()/75540465\)
\(\beta_{3}\)\(=\)\((\)\( -47804 \nu^{5} + 5363820 \nu^{4} - 137900582 \nu^{3} + 14008152312 \nu^{2} - 133766038058 \nu + 5298302011464 \)\()/15108093\)
\(\beta_{4}\)\(=\)\((\)\( 1192876 \nu^{5} + 19667340 \nu^{4} + 3403143526 \nu^{3} + 155790363960 \nu^{2} + 1447864274554 \nu + 137638628515080 \)\()/75540465\)
\(\beta_{5}\)\(=\)\((\)\( 334052 \nu^{5} - 2979900 \nu^{4} + 953812298 \nu^{3} - 7782306840 \nu^{2} + 431213871014 \nu - 2943501117480 \)\()/5036031\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-16 \beta_{5} - 104 \beta_{3} - 232 \beta_{2} - 471 \beta_{1}\)\()/746496\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{5} + 5 \beta_{4} - 3 \beta_{3} + 7 \beta_{2} + \beta_{1} - 7824384\)\()/6912\)
\(\nu^{3}\)\(=\)\((\)\(27152 \beta_{5} + 179944 \beta_{3} + 404072 \beta_{2} + 26586231 \beta_{1}\)\()/746496\)
\(\nu^{4}\)\(=\)\((\)\(765 \beta_{5} - 6529 \beta_{4} + 7703 \beta_{3} - 24283 \beta_{2} + 1939 \beta_{1} + 6803294976\)\()/3456\)
\(\nu^{5}\)\(=\)\((\)\(-46661008 \beta_{5} - 372247208 \beta_{3} - 883436584 \beta_{2} - 75296911287 \beta_{1}\)\()/746496\)

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
29.2393i
47.5078i
16.8544i
29.2393i
47.5078i
16.8544i
724.077i −57693.2 + 12580.8i −524288. 8.94833e6i 9.10944e6 + 4.17744e7i −3.40159e7 3.79625e8i 3.17023e9 1.45165e9i 6.47928e9
5.2 724.077i 25323.8 + 53343.1i −524288. 6.05191e6i 3.86246e7 1.83364e7i 1.14251e8 3.79625e8i −2.20420e9 + 2.70170e9i −4.38205e9
5.3 724.077i 41792.5 41715.4i −524288. 805527.i −3.02052e7 3.02610e7i −3.63571e8 3.79625e8i 6.43531e6 3.48678e9i 5.83264e8
5.4 724.077i −57693.2 12580.8i −524288. 8.94833e6i 9.10944e6 4.17744e7i −3.40159e7 3.79625e8i 3.17023e9 + 1.45165e9i 6.47928e9
5.5 724.077i 25323.8 53343.1i −524288. 6.05191e6i 3.86246e7 + 1.83364e7i 1.14251e8 3.79625e8i −2.20420e9 2.70170e9i −4.38205e9
5.6 724.077i 41792.5 + 41715.4i −524288. 805527.i −3.02052e7 + 3.02610e7i −3.63571e8 3.79625e8i 6.43531e6 + 3.48678e9i 5.83264e8
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.6
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 6.21.b.a 6
3.b odd 2 1 inner 6.21.b.a 6
4.b odd 2 1 48.21.e.b 6
12.b even 2 1 48.21.e.b 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.21.b.a 6 1.a even 1 1 trivial
6.21.b.a 6 3.b odd 2 1 inner
48.21.e.b 6 4.b odd 2 1
48.21.e.b 6 12.b even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{21}^{\mathrm{new}}(6, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 524288 + T^{2} )^{3} \)
$3$ \( \)\(42\!\cdots\!01\)\( - \)\(22\!\cdots\!46\)\( T - 2771592036295890033 T^{2} + 357049764630684 T^{3} - 794884833 T^{4} - 18846 T^{5} + T^{6} \)
$5$ \( \)\(19\!\cdots\!00\)\( + \)\(30\!\cdots\!00\)\( T^{2} + 117347038767648 T^{4} + T^{6} \)
$7$ \( ( -\)\(14\!\cdots\!76\)\( - 33057498484621716 T + 283335906 T^{2} + T^{3} )^{2} \)
$11$ \( \)\(18\!\cdots\!68\)\( + \)\(79\!\cdots\!72\)\( T^{2} + \)\(90\!\cdots\!96\)\( T^{4} + T^{6} \)
$13$ \( ( -\)\(24\!\cdots\!00\)\( - \)\(31\!\cdots\!00\)\( T + 24949272810 T^{2} + T^{3} )^{2} \)
$17$ \( \)\(38\!\cdots\!48\)\( + \)\(17\!\cdots\!12\)\( T^{2} + \)\(23\!\cdots\!96\)\( T^{4} + T^{6} \)
$19$ \( ( \)\(67\!\cdots\!52\)\( - \)\(84\!\cdots\!88\)\( T - 7963798643646 T^{2} + T^{3} )^{2} \)
$23$ \( \)\(46\!\cdots\!68\)\( + \)\(90\!\cdots\!32\)\( T^{2} + \)\(36\!\cdots\!76\)\( T^{4} + T^{6} \)
$29$ \( \)\(54\!\cdots\!00\)\( + \)\(19\!\cdots\!00\)\( T^{2} + \)\(84\!\cdots\!40\)\( T^{4} + T^{6} \)
$31$ \( ( \)\(68\!\cdots\!28\)\( - \)\(10\!\cdots\!68\)\( T - 916272713198814 T^{2} + T^{3} )^{2} \)
$37$ \( ( \)\(52\!\cdots\!56\)\( - \)\(34\!\cdots\!96\)\( T - 212166163436406 T^{2} + T^{3} )^{2} \)
$41$ \( \)\(35\!\cdots\!08\)\( + \)\(33\!\cdots\!52\)\( T^{2} + \)\(10\!\cdots\!76\)\( T^{4} + T^{6} \)
$43$ \( ( -\)\(96\!\cdots\!52\)\( - \)\(37\!\cdots\!64\)\( T + 24744376166975778 T^{2} + T^{3} )^{2} \)
$47$ \( \)\(19\!\cdots\!00\)\( + \)\(73\!\cdots\!00\)\( T^{2} + \)\(71\!\cdots\!00\)\( T^{4} + T^{6} \)
$53$ \( \)\(10\!\cdots\!08\)\( + \)\(79\!\cdots\!72\)\( T^{2} + \)\(17\!\cdots\!36\)\( T^{4} + T^{6} \)
$59$ \( \)\(58\!\cdots\!08\)\( + \)\(56\!\cdots\!72\)\( T^{2} + \)\(13\!\cdots\!36\)\( T^{4} + T^{6} \)
$61$ \( ( \)\(94\!\cdots\!16\)\( + \)\(54\!\cdots\!48\)\( T - 1650915006436046742 T^{2} + T^{3} )^{2} \)
$67$ \( ( \)\(92\!\cdots\!56\)\( + \)\(63\!\cdots\!24\)\( T + 1389613136254278114 T^{2} + T^{3} )^{2} \)
$71$ \( \)\(12\!\cdots\!00\)\( + \)\(97\!\cdots\!00\)\( T^{2} + \)\(40\!\cdots\!40\)\( T^{4} + T^{6} \)
$73$ \( ( \)\(19\!\cdots\!00\)\( - \)\(71\!\cdots\!60\)\( T + 4574700768396250650 T^{2} + T^{3} )^{2} \)
$79$ \( ( \)\(79\!\cdots\!68\)\( - \)\(42\!\cdots\!68\)\( T - 45028024201628574 T^{2} + T^{3} )^{2} \)
$83$ \( \)\(92\!\cdots\!72\)\( + \)\(18\!\cdots\!72\)\( T^{2} + \)\(85\!\cdots\!44\)\( T^{4} + T^{6} \)
$89$ \( \)\(37\!\cdots\!48\)\( + \)\(46\!\cdots\!12\)\( T^{2} + \)\(12\!\cdots\!96\)\( T^{4} + T^{6} \)
$97$ \( ( \)\(17\!\cdots\!92\)\( - \)\(56\!\cdots\!64\)\( T - 46014452559993786918 T^{2} + T^{3} )^{2} \)
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