Properties

Label 98.14.c.o
Level $98$
Weight $14$
Character orbit 98.c
Analytic conductor $105.086$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 98 = 2 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 14 \)
Character orbit: \([\chi]\) \(=\) 98.c (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(105.086310373\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
Defining polynomial: \(x^{8} + 209077 x^{6} - 47718852 x^{5} + 40973427094 x^{4} - 4988457209802 x^{3} + 1142094021456771 x^{2} + 65369216338084710 x + 7506311351102577225\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{14}\cdot 3^{2}\cdot 7^{4} \)
Twist minimal: no (minimal twist has level 14)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( 64 - 64 \beta_{1} ) q^{2} + ( -46 \beta_{1} - \beta_{2} ) q^{3} -4096 \beta_{1} q^{4} + ( -442 + 442 \beta_{1} - 12 \beta_{2} - 12 \beta_{3} - \beta_{5} + \beta_{7} ) q^{5} + ( -2944 + 64 \beta_{3} ) q^{6} -262144 q^{8} + ( 149791 - 149791 \beta_{1} + 338 \beta_{2} + 338 \beta_{3} + 39 \beta_{4} + 27 \beta_{5} - 27 \beta_{7} ) q^{9} +O(q^{10})\) \( q + ( 64 - 64 \beta_{1} ) q^{2} + ( -46 \beta_{1} - \beta_{2} ) q^{3} -4096 \beta_{1} q^{4} + ( -442 + 442 \beta_{1} - 12 \beta_{2} - 12 \beta_{3} - \beta_{5} + \beta_{7} ) q^{5} + ( -2944 + 64 \beta_{3} ) q^{6} -262144 q^{8} + ( 149791 - 149791 \beta_{1} + 338 \beta_{2} + 338 \beta_{3} + 39 \beta_{4} + 27 \beta_{5} - 27 \beta_{7} ) q^{9} + ( 28288 \beta_{1} - 768 \beta_{2} + 64 \beta_{7} ) q^{10} + ( -2183877 \beta_{1} - 4297 \beta_{2} - 68 \beta_{4} - 68 \beta_{6} - 17 \beta_{7} ) q^{11} + ( -188416 + 188416 \beta_{1} + 4096 \beta_{2} + 4096 \beta_{3} ) q^{12} + ( -177787 - 4030 \beta_{3} + 117 \beta_{5} + 455 \beta_{6} ) q^{13} + ( -17117703 + 17103 \beta_{3} - 393 \beta_{5} - 81 \beta_{6} ) q^{15} + ( -16777216 + 16777216 \beta_{1} ) q^{16} + ( 1964860 \beta_{1} - 41814 \beta_{2} - 3507 \beta_{4} - 3507 \beta_{6} + 2083 \beta_{7} ) q^{17} + ( -9586624 \beta_{1} + 21632 \beta_{2} + 2496 \beta_{4} + 2496 \beta_{6} - 1728 \beta_{7} ) q^{18} + ( 53931105 - 53931105 \beta_{1} - 8807 \beta_{2} - 8807 \beta_{3} - 4690 \beta_{4} + 1337 \beta_{5} - 1337 \beta_{7} ) q^{19} + ( 1810432 + 49152 \beta_{3} + 4096 \beta_{5} ) q^{20} + ( -139768128 + 275008 \beta_{3} - 1088 \beta_{5} - 4352 \beta_{6} ) q^{22} + ( -15329226 + 15329226 \beta_{1} - 305537 \beta_{2} - 305537 \beta_{3} + 18317 \beta_{4} + 2314 \beta_{5} - 2314 \beta_{7} ) q^{23} + ( 12058624 \beta_{1} + 262144 \beta_{2} ) q^{24} + ( -331837802 \beta_{1} + 246792 \beta_{2} + 6090 \beta_{4} + 6090 \beta_{6} - 49966 \beta_{7} ) q^{25} + ( -11378368 + 11378368 \beta_{1} - 257920 \beta_{2} - 257920 \beta_{3} - 29120 \beta_{4} + 7488 \beta_{5} - 7488 \beta_{7} ) q^{26} + ( 408277417 + 606119 \beta_{3} + 5085 \beta_{5} + 24612 \beta_{6} ) q^{27} + ( -791180223 + 898930 \beta_{3} + 88261 \beta_{5} - 11411 \beta_{6} ) q^{29} + ( -1095532992 + 1095532992 \beta_{1} + 1094592 \beta_{2} + 1094592 \beta_{3} + 5184 \beta_{4} - 25152 \beta_{5} + 25152 \beta_{7} ) q^{30} + ( -1529648605 \beta_{1} - 2409425 \beta_{2} + 88319 \beta_{4} + 88319 \beta_{6} - 91015 \beta_{7} ) q^{31} + 1073741824 \beta_{1} q^{32} + ( -6311165667 + 6311165667 \beta_{1} + 4351008 \beta_{2} + 4351008 \beta_{3} + 212310 \beta_{4} + 137184 \beta_{5} - 137184 \beta_{7} ) q^{33} + ( 125751040 + 2676096 \beta_{3} + 133312 \beta_{5} - 224448 \beta_{6} ) q^{34} + ( -613543936 - 1384448 \beta_{3} - 110592 \beta_{5} + 159744 \beta_{6} ) q^{36} + ( -991086713 + 991086713 \beta_{1} + 9346986 \beta_{2} + 9346986 \beta_{3} - 454311 \beta_{4} - 26676 \beta_{5} + 26676 \beta_{7} ) q^{37} + ( -3451590720 \beta_{1} - 563648 \beta_{2} - 300160 \beta_{4} - 300160 \beta_{6} - 85568 \beta_{7} ) q^{38} + ( -5889928987 \beta_{1} - 7058416 \beta_{2} - 457704 \beta_{4} - 457704 \beta_{6} + 252759 \beta_{7} ) q^{39} + ( 115867648 - 115867648 \beta_{1} + 3145728 \beta_{2} + 3145728 \beta_{3} + 262144 \beta_{5} - 262144 \beta_{7} ) q^{40} + ( -10806799001 + 18953014 \beta_{3} - 550249 \beta_{5} + 536809 \beta_{6} ) q^{41} + ( -13621164036 - 26202992 \beta_{3} + 543776 \beta_{5} - 212498 \beta_{6} ) q^{43} + ( -8945160192 + 8945160192 \beta_{1} + 17600512 \beta_{2} + 17600512 \beta_{3} + 278528 \beta_{4} - 69632 \beta_{5} + 69632 \beta_{7} ) q^{44} + ( 26240533521 \beta_{1} - 1167042 \beta_{2} + 864549 \beta_{4} + 864549 \beta_{6} + 840069 \beta_{7} ) q^{45} + ( 981070464 \beta_{1} - 19554368 \beta_{2} + 1172288 \beta_{4} + 1172288 \beta_{6} - 148096 \beta_{7} ) q^{46} + ( 785931665 - 785931665 \beta_{1} - 1261969 \beta_{2} - 1261969 \beta_{3} - 696101 \beta_{4} - 2240885 \beta_{5} + 2240885 \beta_{7} ) q^{47} + ( 771751936 - 16777216 \beta_{3} ) q^{48} + ( -21237619328 - 15794688 \beta_{3} - 3197824 \beta_{5} + 389760 \beta_{6} ) q^{50} + ( -60368996229 + 60368996229 \beta_{1} + 98858727 \beta_{2} + 98858727 \beta_{3} + 2792052 \beta_{4} + 98391 \beta_{5} - 98391 \beta_{7} ) q^{51} + ( 728215552 \beta_{1} - 16506880 \beta_{2} - 1863680 \beta_{4} - 1863680 \beta_{6} - 479232 \beta_{7} ) q^{52} + ( 37337231485 \beta_{1} - 138377218 \beta_{2} + 164941 \beta_{4} + 164941 \beta_{6} + 933086 \beta_{7} ) q^{53} + ( 26129754688 - 26129754688 \beta_{1} + 38791616 \beta_{2} + 38791616 \beta_{3} - 1575168 \beta_{4} + 325440 \beta_{5} - 325440 \beta_{7} ) q^{54} + ( -21922437026 + 116871615 \beta_{3} + 2188438 \beta_{5} - 2260251 \beta_{6} ) q^{55} + ( -16084497371 + 117122252 \beta_{3} + 1815498 \beta_{5} - 3491982 \beta_{6} ) q^{57} + ( -50635534272 + 50635534272 \beta_{1} + 57531520 \beta_{2} + 57531520 \beta_{3} + 730304 \beta_{4} + 5648704 \beta_{5} - 5648704 \beta_{7} ) q^{58} + ( 216448300886 \beta_{1} - 252055197 \beta_{2} - 1898946 \beta_{4} - 1898946 \beta_{6} + 3752480 \beta_{7} ) q^{59} + ( 70114111488 \beta_{1} + 70053888 \beta_{2} + 331776 \beta_{4} + 331776 \beta_{6} + 1609728 \beta_{7} ) q^{60} + ( 119502877813 - 119502877813 \beta_{1} - 51458534 \beta_{2} - 51458534 \beta_{3} + 72569 \beta_{4} - 759540 \beta_{5} + 759540 \beta_{7} ) q^{61} + ( -97897510720 + 154203200 \beta_{3} - 5824960 \beta_{5} + 5652416 \beta_{6} ) q^{62} + 68719476736 q^{64} + ( -274838696067 + 274838696067 \beta_{1} - 196779426 \beta_{2} - 196779426 \beta_{3} - 13144209 \beta_{4} - 10849515 \beta_{5} + 10849515 \beta_{7} ) q^{65} + ( 403914602688 \beta_{1} + 278464512 \beta_{2} + 13587840 \beta_{4} + 13587840 \beta_{6} - 8779776 \beta_{7} ) q^{66} + ( 473926436184 \beta_{1} + 102364229 \beta_{2} - 3291746 \beta_{4} - 3291746 \beta_{6} - 3985050 \beta_{7} ) q^{67} + ( 8048066560 - 8048066560 \beta_{1} + 171270144 \beta_{2} + 171270144 \beta_{3} + 14364672 \beta_{4} + 8531968 \beta_{5} - 8531968 \beta_{7} ) q^{68} + ( -437778400218 - 268673472 \beta_{3} + 7490793 \beta_{5} - 2535624 \beta_{6} ) q^{69} + ( 80041697582 - 375227132 \beta_{3} + 14165130 \beta_{5} + 31584322 \beta_{6} ) q^{71} + ( -39266811904 + 39266811904 \beta_{1} - 88604672 \beta_{2} - 88604672 \beta_{3} - 10223616 \beta_{4} - 7077888 \beta_{5} + 7077888 \beta_{7} ) q^{72} + ( 741709294201 \beta_{1} + 120492024 \beta_{2} - 13431138 \beta_{4} - 13431138 \beta_{6} - 5125102 \beta_{7} ) q^{73} + ( 63429549632 \beta_{1} + 598207104 \beta_{2} - 29075904 \beta_{4} - 29075904 \beta_{6} + 1707264 \beta_{7} ) q^{74} + ( 333042411638 - 333042411638 \beta_{1} - 544889588 \beta_{2} - 544889588 \beta_{3} + 6295464 \beta_{4} + 28358358 \beta_{5} - 28358358 \beta_{7} ) q^{75} + ( -220901806080 + 36073472 \beta_{3} - 5476352 \beta_{5} - 19210240 \beta_{6} ) q^{76} + ( -376955455168 + 451738624 \beta_{3} + 16176576 \beta_{5} - 29293056 \beta_{6} ) q^{78} + ( -1626304384762 + 1626304384762 \beta_{1} - 371069293 \beta_{2} - 371069293 \beta_{3} + 33488371 \beta_{4} - 2694762 \beta_{5} + 2694762 \beta_{7} ) q^{79} + ( -7415529472 \beta_{1} + 201326592 \beta_{2} - 16777216 \beta_{7} ) q^{80} + ( 612285301948 \beta_{1} - 37642946 \beta_{2} + 70042011 \beta_{4} + 70042011 \beta_{6} - 52517205 \beta_{7} ) q^{81} + ( -691635136064 + 691635136064 \beta_{1} + 1212992896 \beta_{2} + 1212992896 \beta_{3} - 34355776 \beta_{4} - 35215936 \beta_{5} + 35215936 \beta_{7} ) q^{82} + ( 423620056210 - 2285828428 \beta_{3} + 11946714 \beta_{5} + 16441880 \beta_{6} ) q^{83} + ( -2085756586183 + 373926750 \beta_{3} - 53785246 \beta_{5} - 81103773 \beta_{6} ) q^{85} + ( -871754498304 + 871754498304 \beta_{1} - 1676991488 \beta_{2} - 1676991488 \beta_{3} + 13599872 \beta_{4} + 34801664 \beta_{5} - 34801664 \beta_{7} ) q^{86} + ( 1319570844069 \beta_{1} + 2486105892 \beta_{2} + 7302834 \beta_{4} + 7302834 \beta_{6} + 37505775 \beta_{7} ) q^{87} + ( 572490252288 \beta_{1} + 1126432768 \beta_{2} + 17825792 \beta_{4} + 17825792 \beta_{6} + 4456448 \beta_{7} ) q^{88} + ( -2395747211569 + 2395747211569 \beta_{1} - 1806410596 \beta_{2} - 1806410596 \beta_{3} - 64454306 \beta_{4} - 51621500 \beta_{5} + 51621500 \beta_{7} ) q^{89} + ( 1679394145344 + 74690688 \beta_{3} + 53764416 \beta_{5} + 55331136 \beta_{6} ) q^{90} + ( 62788509696 + 1251479552 \beta_{3} - 9478144 \beta_{5} + 75026432 \beta_{6} ) q^{92} + ( -3548666904253 + 3548666904253 \beta_{1} - 539804942 \beta_{2} - 539804942 \beta_{3} + 79643421 \beta_{4} + 118654122 \beta_{5} - 118654122 \beta_{7} ) q^{93} + ( -50299626560 \beta_{1} - 80766016 \beta_{2} - 44550464 \beta_{4} - 44550464 \beta_{6} + 143416640 \beta_{7} ) q^{94} + ( 3594995836676 \beta_{1} - 2213395137 \beta_{2} - 133760211 \beta_{4} - 133760211 \beta_{6} + 187764464 \beta_{7} ) q^{95} + ( 49392123904 - 49392123904 \beta_{1} - 1073741824 \beta_{2} - 1073741824 \beta_{3} ) q^{96} + ( 5570640808189 - 1097788486 \beta_{3} - 128001703 \beta_{5} - 31967299 \beta_{6} ) q^{97} + ( 3091428017331 - 6251038296 \beta_{3} - 74244699 \beta_{5} + 127291050 \beta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 256q^{2} - 182q^{3} - 16384q^{4} - 1792q^{5} - 23296q^{6} - 2097152q^{8} + 599840q^{9} + O(q^{10}) \) \( 8q + 256q^{2} - 182q^{3} - 16384q^{4} - 1792q^{5} - 23296q^{6} - 2097152q^{8} + 599840q^{9} + 114688q^{10} - 8726914q^{11} - 745472q^{12} - 1438416q^{13} - 136873212q^{15} - 67108864q^{16} + 7943068q^{17} - 38389760q^{18} + 215706806q^{19} + 14680064q^{20} - 1117044992q^{22} - 61927978q^{23} + 47710208q^{24} - 1327844792q^{25} - 46029312q^{26} + 3268643812q^{27} - 6325846064q^{29} - 4379942784q^{30} - 6113775570q^{31} + 4294967296q^{32} - 25235960652q^{33} + 1016712704q^{34} - 4913889280q^{36} - 3945652880q^{37} - 13805235584q^{38} - 23545599116q^{39} + 469762048q^{40} - 86378579952q^{41} - 109074124256q^{43} - 35745439744q^{44} + 104964468168q^{45} + 3963390592q^{46} + 3141202722q^{47} + 6106906624q^{48} - 169964133376q^{50} - 241278267462q^{51} + 2945875968q^{52} + 149625680376q^{53} + 104596601984q^{54} - 174912009748q^{55} - 128207489960q^{57} - 202427074048q^{58} + 866297313938q^{59} + 280316338176q^{60} + 477908594184q^{61} - 782563272960q^{62} + 549755813888q^{64} - 1099748343120q^{65} + 1615101481728q^{66} + 1895501016278q^{67} + 32534806528q^{68} - 3503301895632q^{69} + 638832672128q^{71} - 157244456960q^{72} + 2966596192756q^{73} + 252521784320q^{74} + 1331079867376q^{75} - 1767070154752q^{76} - 3013836686848q^{78} - 6505959677634q^{79} - 30064771072q^{80} + 2449216493684q^{81} - 2764114558464q^{82} + 3379817135968q^{83} - 16684556982464q^{85} - 3490371976192q^{86} + 5273311164492q^{87} + 2287708143616q^{88} - 9586601667468q^{89} + 13435451925504q^{90} + 507313995776q^{92} - 14195747226896q^{93} - 201036974208q^{94} + 14384410136978q^{95} + 195421011968q^{96} + 44560735311568q^{97} + 24706419985464q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 209077 x^{6} - 47718852 x^{5} + 40973427094 x^{4} - 4988457209802 x^{3} + 1142094021456771 x^{2} + 65369216338084710 x + 7506311351102577225\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(-36343564573613107681343 \nu^{7} + 3911745461420541052708455 \nu^{6} - 7122353933684804467943766146 \nu^{5} + 2500869301396565104975855035246 \nu^{4} - 1582452394760617117871338276154072 \nu^{3} + 330212894100361428024557552701671966 \nu^{2} - 40284736421255101261261116490164523803 \nu + 2240733124886924787542123968883350179225\)\()/ \)\(44\!\cdots\!80\)\( \)
\(\beta_{2}\)\(=\)\((\)\(19245665182455297508653899719 \nu^{7} - 3626754114613992386988321873015 \nu^{6} + 6603452778174386406803937952075618 \nu^{5} - 1242412722272850209753755293685910478 \nu^{4} + 1467162367920194256697258163799496773976 \nu^{3} - 306155138208347124589081494569525382543678 \nu^{2} + 119920255574041349256887334773044499481407859 \nu - 2077483865088801616780561787398094760449644425\)\()/ \)\(65\!\cdots\!40\)\( \)
\(\beta_{3}\)\(=\)\((\)\(158991540582276108146 \nu^{7} + 23003238574461279327135 \nu^{6} + 31158034105189152444784412 \nu^{5} - 3793446897148813713877484196 \nu^{4} + 4871566787094386785283230406789 \nu^{3} + 49709687876945726097309999179580 \nu^{2} + 5708136264639463982283216865573050 \nu + 17141819188993673802166662001668558819\)\()/ \)\(12\!\cdots\!59\)\( \)
\(\beta_{4}\)\(=\)\((\)\(-93608922569264480644002498478 \nu^{7} + 79115040751953704941055933619090 \nu^{6} - 1268914627053407544285015556790896 \nu^{5} + 19971314700679190823759062655213385236 \nu^{4} - 5723117824547873141337768949787769005272 \nu^{3} + 2195881059189772026981665258603392553296896 \nu^{2} - 82174384326578480241921174154486784202413838 \nu - 3278742782823003071085909752932883031223643130\)\()/ \)\(15\!\cdots\!65\)\( \)
\(\beta_{5}\)\(=\)\((\)\(-4465522635862254561881 \nu^{7} - 526144798448970483216768 \nu^{6} - 875121444047450447049173582 \nu^{5} + 106544806881680408912362140306 \nu^{4} - 126128427262014521036358186558962 \nu^{3} - 1396173253137810424133263767873630 \nu^{2} - 160321810864793076285102224378732925 \nu - 1168933469354863383259043251732832255343\)\()/ \)\(36\!\cdots\!77\)\( \)
\(\beta_{6}\)\(=\)\((\)\(-746937503400178825691 \nu^{7} - 177205212226741399442619 \nu^{6} - 146379512520944935220099402 \nu^{5} + 17821500089001315078341313366 \nu^{4} - 14648707724006210310797011631984 \nu^{3} - 233534627198568820675611379959930 \nu^{2} - 26816653483343372016345280097687175 \nu + 58133725021541495462618030399052378867\)\()/ \)\(51\!\cdots\!11\)\( \)
\(\beta_{7}\)\(=\)\((\)\(31467283624009703091268243594327 \nu^{7} - 3953525311126966767581771023916895 \nu^{6} + 7198425058414187794906324691493504274 \nu^{5} - 2293095773614565869021908453926495045694 \nu^{4} + 1599353960538023981791206112217476779497368 \nu^{3} - 333739771097523430871543168640522722303539854 \nu^{2} + 59563626843577685013801475108917314873744513907 \nu - 2264665534117863957465430865691388560499971749025\)\()/ \)\(17\!\cdots\!80\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-2 \beta_{7} - \beta_{6} - \beta_{4} + 52 \beta_{2} + 26 \beta_{1}\)\()/336\)
\(\nu^{2}\)\(=\)\((\)\(890 \beta_{7} - 890 \beta_{5} - 455 \beta_{4} - 628 \beta_{3} - 628 \beta_{2} + 35124622 \beta_{1} - 35124622\)\()/336\)
\(\nu^{3}\)\(=\)\((\)\(7549 \beta_{6} + 30461 \beta_{5} + 367933 \beta_{3} + 375601993\)\()/21\)
\(\nu^{4}\)\(=\)\((\)\(-233797382 \beta_{7} + 71270609 \beta_{6} + 71270609 \beta_{4} + 1371990508 \beta_{2} - 6422569264258 \beta_{1}\)\()/336\)
\(\nu^{5}\)\(=\)\((\)\(117654471422 \beta_{7} - 117654471422 \beta_{5} + 11657352703 \beta_{4} - 1103337193564 \beta_{3} - 1103337193564 \beta_{2} + 2094600359520058 \beta_{1} - 2094600359520058\)\()/336\)
\(\nu^{6}\)\(=\)\((\)\(-96184064357 \beta_{6} + 518498001245 \beta_{5} + 3799909018822 \beta_{3} + 12410398903428382\)\()/3\)
\(\nu^{7}\)\(=\)\((\)\(-28841819630097266 \beta_{7} - 405888753844057 \beta_{6} - 405888753844057 \beta_{4} + 247288538096531956 \beta_{2} - 574706597339776303894 \beta_{1}\)\()/336\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/98\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-1 + \beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
67.1
114.279 197.937i
169.399 293.408i
−248.013 + 429.572i
−35.6648 + 61.7733i
114.279 + 197.937i
169.399 + 293.408i
−248.013 429.572i
−35.6648 61.7733i
32.0000 + 55.4256i −888.214 + 1538.43i −2048.00 + 3547.24i 13071.0 + 22639.6i −113691. 0 −262144. −780686. 1.35219e6i −836543. + 1.44893e6i
67.2 32.0000 + 55.4256i −194.343 + 336.611i −2048.00 + 3547.24i 12951.1 + 22432.0i −24875.9 0 −262144. 721623. + 1.24989e6i −828871. + 1.43565e6i
67.3 32.0000 + 55.4256i 244.702 423.837i −2048.00 + 3547.24i −34096.3 59056.5i 31321.9 0 −262144. 677403. + 1.17330e6i 2.18216e6 3.77962e6i
67.4 32.0000 + 55.4256i 746.854 1293.59i −2048.00 + 3547.24i 7178.20 + 12433.0i 95597.3 0 −262144. −318420. 551520.i −459405. + 795712.i
79.1 32.0000 55.4256i −888.214 1538.43i −2048.00 3547.24i 13071.0 22639.6i −113691. 0 −262144. −780686. + 1.35219e6i −836543. 1.44893e6i
79.2 32.0000 55.4256i −194.343 336.611i −2048.00 3547.24i 12951.1 22432.0i −24875.9 0 −262144. 721623. 1.24989e6i −828871. 1.43565e6i
79.3 32.0000 55.4256i 244.702 + 423.837i −2048.00 3547.24i −34096.3 + 59056.5i 31321.9 0 −262144. 677403. 1.17330e6i 2.18216e6 + 3.77962e6i
79.4 32.0000 55.4256i 746.854 + 1293.59i −2048.00 3547.24i 7178.20 12433.0i 95597.3 0 −262144. −318420. + 551520.i −459405. 795712.i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 79.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 98.14.c.o 8
7.b odd 2 1 14.14.c.b 8
7.c even 3 1 98.14.a.j 4
7.c even 3 1 inner 98.14.c.o 8
7.d odd 6 1 14.14.c.b 8
7.d odd 6 1 98.14.a.h 4
21.c even 2 1 126.14.g.b 8
21.g even 6 1 126.14.g.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.14.c.b 8 7.b odd 2 1
14.14.c.b 8 7.d odd 6 1
98.14.a.h 4 7.d odd 6 1
98.14.a.j 4 7.c even 3 1
98.14.c.o 8 1.a even 1 1 trivial
98.14.c.o 8 7.c even 3 1 inner
126.14.g.b 8 21.c even 2 1
126.14.g.b 8 21.g even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(77\!\cdots\!57\)\( T_{3}^{4} - \)\(79\!\cdots\!48\)\( T_{3}^{3} + \)\(14\!\cdots\!24\)\( T_{3}^{2} + \)\(10\!\cdots\!70\)\( T_{3} + \)\(25\!\cdots\!25\)\( \)">\(T_{3}^{8} + \cdots\) acting on \(S_{14}^{\mathrm{new}}(98, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 4096 - 64 T + T^{2} )^{4} \)
$3$ \( \)\(25\!\cdots\!25\)\( + \)\(10\!\cdots\!70\)\( T + 1495307469814735824 T^{2} - 796865962371948 T^{3} + 7705719718857 T^{4} - 949684092 T^{5} + 2905288 T^{6} + 182 T^{7} + T^{8} \)
$5$ \( \)\(43\!\cdots\!25\)\( - \)\(57\!\cdots\!00\)\( T + \)\(55\!\cdots\!50\)\( T^{2} - \)\(26\!\cdots\!80\)\( T^{3} + 10139379896094153831 T^{4} - 180372900060528 T^{5} + 3106934278 T^{6} + 1792 T^{7} + T^{8} \)
$7$ \( T^{8} \)
$11$ \( \)\(48\!\cdots\!25\)\( - \)\(19\!\cdots\!30\)\( T + \)\(88\!\cdots\!04\)\( T^{2} + \)\(28\!\cdots\!04\)\( T^{3} + \)\(20\!\cdots\!61\)\( T^{4} - \)\(13\!\cdots\!96\)\( T^{5} + 112179301147984 T^{6} + 8726914 T^{7} + T^{8} \)
$13$ \( ( \)\(29\!\cdots\!00\)\( + \)\(24\!\cdots\!20\)\( T - 549797025415320 T^{2} + 719208 T^{3} + T^{4} )^{2} \)
$17$ \( \)\(61\!\cdots\!41\)\( + \)\(76\!\cdots\!84\)\( T + \)\(30\!\cdots\!38\)\( T^{2} - \)\(25\!\cdots\!84\)\( T^{3} + \)\(66\!\cdots\!23\)\( T^{4} - \)\(17\!\cdots\!68\)\( T^{5} + 27198159130142842 T^{6} - 7943068 T^{7} + T^{8} \)
$19$ \( \)\(12\!\cdots\!25\)\( - \)\(35\!\cdots\!10\)\( T + \)\(85\!\cdots\!84\)\( T^{2} - \)\(53\!\cdots\!00\)\( T^{3} + \)\(39\!\cdots\!37\)\( T^{4} - \)\(11\!\cdots\!24\)\( T^{5} + 84562681108154656 T^{6} - 215706806 T^{7} + T^{8} \)
$23$ \( \)\(12\!\cdots\!09\)\( - \)\(23\!\cdots\!50\)\( T + \)\(53\!\cdots\!76\)\( T^{2} + \)\(17\!\cdots\!32\)\( T^{3} + \)\(67\!\cdots\!61\)\( T^{4} + \)\(37\!\cdots\!24\)\( T^{5} + 825289508152995676 T^{6} + 61927978 T^{7} + T^{8} \)
$29$ \( ( -\)\(94\!\cdots\!96\)\( - \)\(87\!\cdots\!92\)\( T - 18126451463113919640 T^{2} + 3162923032 T^{3} + T^{4} )^{2} \)
$31$ \( \)\(15\!\cdots\!25\)\( + \)\(92\!\cdots\!10\)\( T + \)\(45\!\cdots\!24\)\( T^{2} + \)\(10\!\cdots\!44\)\( T^{3} + \)\(25\!\cdots\!09\)\( T^{4} + \)\(31\!\cdots\!96\)\( T^{5} + 63485338461969276948 T^{6} + 6113775570 T^{7} + T^{8} \)
$37$ \( \)\(82\!\cdots\!69\)\( + \)\(46\!\cdots\!04\)\( T + \)\(19\!\cdots\!06\)\( T^{2} - \)\(12\!\cdots\!92\)\( T^{3} + \)\(32\!\cdots\!59\)\( T^{4} - \)\(56\!\cdots\!64\)\( T^{5} + \)\(61\!\cdots\!66\)\( T^{6} + 3945652880 T^{7} + T^{8} \)
$41$ \( ( -\)\(96\!\cdots\!16\)\( - \)\(79\!\cdots\!04\)\( T - \)\(11\!\cdots\!00\)\( T^{2} + 43189289976 T^{3} + T^{4} )^{2} \)
$43$ \( ( \)\(22\!\cdots\!00\)\( - \)\(67\!\cdots\!80\)\( T - \)\(15\!\cdots\!20\)\( T^{2} + 54537062128 T^{3} + T^{4} )^{2} \)
$47$ \( \)\(26\!\cdots\!21\)\( - \)\(26\!\cdots\!98\)\( T + \)\(20\!\cdots\!00\)\( T^{2} - \)\(61\!\cdots\!28\)\( T^{3} + \)\(14\!\cdots\!49\)\( T^{4} - \)\(99\!\cdots\!12\)\( T^{5} + \)\(11\!\cdots\!00\)\( T^{6} - 3141202722 T^{7} + T^{8} \)
$53$ \( \)\(91\!\cdots\!21\)\( + \)\(18\!\cdots\!32\)\( T + \)\(51\!\cdots\!22\)\( T^{2} - \)\(20\!\cdots\!48\)\( T^{3} + \)\(30\!\cdots\!03\)\( T^{4} - \)\(47\!\cdots\!24\)\( T^{5} + \)\(71\!\cdots\!78\)\( T^{6} - 149625680376 T^{7} + T^{8} \)
$59$ \( \)\(11\!\cdots\!25\)\( - \)\(83\!\cdots\!50\)\( T + \)\(68\!\cdots\!60\)\( T^{2} - \)\(13\!\cdots\!60\)\( T^{3} + \)\(83\!\cdots\!29\)\( T^{4} - \)\(21\!\cdots\!76\)\( T^{5} + \)\(68\!\cdots\!32\)\( T^{6} - 866297313938 T^{7} + T^{8} \)
$61$ \( \)\(90\!\cdots\!41\)\( - \)\(44\!\cdots\!04\)\( T + \)\(14\!\cdots\!70\)\( T^{2} - \)\(26\!\cdots\!36\)\( T^{3} + \)\(34\!\cdots\!59\)\( T^{4} - \)\(27\!\cdots\!76\)\( T^{5} + \)\(15\!\cdots\!70\)\( T^{6} - 477908594184 T^{7} + T^{8} \)
$67$ \( \)\(52\!\cdots\!21\)\( - \)\(71\!\cdots\!78\)\( T + \)\(69\!\cdots\!92\)\( T^{2} - \)\(29\!\cdots\!00\)\( T^{3} + \)\(90\!\cdots\!09\)\( T^{4} - \)\(17\!\cdots\!80\)\( T^{5} + \)\(23\!\cdots\!92\)\( T^{6} - 1895501016278 T^{7} + T^{8} \)
$71$ \( ( \)\(19\!\cdots\!56\)\( + \)\(12\!\cdots\!64\)\( T - \)\(34\!\cdots\!52\)\( T^{2} - 319416336064 T^{3} + T^{4} )^{2} \)
$73$ \( \)\(76\!\cdots\!25\)\( - \)\(82\!\cdots\!80\)\( T + \)\(64\!\cdots\!06\)\( T^{2} - \)\(20\!\cdots\!36\)\( T^{3} + \)\(46\!\cdots\!07\)\( T^{4} - \)\(62\!\cdots\!36\)\( T^{5} + \)\(60\!\cdots\!82\)\( T^{6} - 2966596192756 T^{7} + T^{8} \)
$79$ \( \)\(10\!\cdots\!25\)\( - \)\(22\!\cdots\!50\)\( T + \)\(89\!\cdots\!00\)\( T^{2} + \)\(13\!\cdots\!00\)\( T^{3} + \)\(13\!\cdots\!85\)\( T^{4} + \)\(72\!\cdots\!20\)\( T^{5} + \)\(29\!\cdots\!96\)\( T^{6} + 6505959677634 T^{7} + T^{8} \)
$83$ \( ( -\)\(29\!\cdots\!64\)\( + \)\(45\!\cdots\!44\)\( T - \)\(15\!\cdots\!12\)\( T^{2} - 1689908567984 T^{3} + T^{4} )^{2} \)
$89$ \( \)\(25\!\cdots\!61\)\( + \)\(10\!\cdots\!08\)\( T + \)\(68\!\cdots\!02\)\( T^{2} + \)\(20\!\cdots\!00\)\( T^{3} + \)\(10\!\cdots\!39\)\( T^{4} + \)\(28\!\cdots\!80\)\( T^{5} + \)\(76\!\cdots\!62\)\( T^{6} + 9586601667468 T^{7} + T^{8} \)
$97$ \( ( -\)\(20\!\cdots\!80\)\( + \)\(12\!\cdots\!68\)\( T + \)\(12\!\cdots\!20\)\( T^{2} - 22280367655784 T^{3} + T^{4} )^{2} \)
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