Properties

Label 98.14.a.h
Level $98$
Weight $14$
Character orbit 98.a
Self dual yes
Analytic conductor $105.086$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 98 = 2 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 14 \)
Character orbit: \([\chi]\) \(=\) 98.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(105.086310373\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Defining polynomial: \(x^{4} - 209077 x^{2} - 23859426 x + 2739764835\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{7}\cdot 3\cdot 7^{2} \)
Twist minimal: no (minimal twist has level 14)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -64 q^{2} + ( -46 + \beta_{1} ) q^{3} + 4096 q^{4} + ( -442 - 12 \beta_{1} - \beta_{2} ) q^{5} + ( 2944 - 64 \beta_{1} ) q^{6} -262144 q^{8} + ( -149791 - 338 \beta_{1} - 27 \beta_{2} - 39 \beta_{3} ) q^{9} +O(q^{10})\) \( q -64 q^{2} + ( -46 + \beta_{1} ) q^{3} + 4096 q^{4} + ( -442 - 12 \beta_{1} - \beta_{2} ) q^{5} + ( 2944 - 64 \beta_{1} ) q^{6} -262144 q^{8} + ( -149791 - 338 \beta_{1} - 27 \beta_{2} - 39 \beta_{3} ) q^{9} + ( 28288 + 768 \beta_{1} + 64 \beta_{2} ) q^{10} + ( 2183877 - 4297 \beta_{1} + 17 \beta_{2} - 68 \beta_{3} ) q^{11} + ( -188416 + 4096 \beta_{1} ) q^{12} + ( 177787 + 4030 \beta_{1} - 117 \beta_{2} + 455 \beta_{3} ) q^{13} + ( -17117703 + 17103 \beta_{1} - 393 \beta_{2} + 81 \beta_{3} ) q^{15} + 16777216 q^{16} + ( 1964860 + 41814 \beta_{1} + 2083 \beta_{2} + 3507 \beta_{3} ) q^{17} + ( 9586624 + 21632 \beta_{1} + 1728 \beta_{2} + 2496 \beta_{3} ) q^{18} + ( 53931105 - 8807 \beta_{1} + 1337 \beta_{2} - 4690 \beta_{3} ) q^{19} + ( -1810432 - 49152 \beta_{1} - 4096 \beta_{2} ) q^{20} + ( -139768128 + 275008 \beta_{1} - 1088 \beta_{2} + 4352 \beta_{3} ) q^{22} + ( 15329226 + 305537 \beta_{1} - 2314 \beta_{2} - 18317 \beta_{3} ) q^{23} + ( 12058624 - 262144 \beta_{1} ) q^{24} + ( 331837802 + 246792 \beta_{1} + 49966 \beta_{2} + 6090 \beta_{3} ) q^{25} + ( -11378368 - 257920 \beta_{1} + 7488 \beta_{2} - 29120 \beta_{3} ) q^{26} + ( -408277417 - 606119 \beta_{1} - 5085 \beta_{2} + 24612 \beta_{3} ) q^{27} + ( -791180223 + 898930 \beta_{1} + 88261 \beta_{2} + 11411 \beta_{3} ) q^{29} + ( 1095532992 - 1094592 \beta_{1} + 25152 \beta_{2} - 5184 \beta_{3} ) q^{30} + ( -1529648605 + 2409425 \beta_{1} - 91015 \beta_{2} - 88319 \beta_{3} ) q^{31} -1073741824 q^{32} + ( -6311165667 + 4351008 \beta_{1} + 137184 \beta_{2} + 212310 \beta_{3} ) q^{33} + ( -125751040 - 2676096 \beta_{1} - 133312 \beta_{2} - 224448 \beta_{3} ) q^{34} + ( -613543936 - 1384448 \beta_{1} - 110592 \beta_{2} - 159744 \beta_{3} ) q^{36} + ( 991086713 - 9346986 \beta_{1} + 26676 \beta_{2} + 454311 \beta_{3} ) q^{37} + ( -3451590720 + 563648 \beta_{1} - 85568 \beta_{2} + 300160 \beta_{3} ) q^{38} + ( 5889928987 - 7058416 \beta_{1} - 252759 \beta_{2} - 457704 \beta_{3} ) q^{39} + ( 115867648 + 3145728 \beta_{1} + 262144 \beta_{2} ) q^{40} + ( 10806799001 - 18953014 \beta_{1} + 550249 \beta_{2} + 536809 \beta_{3} ) q^{41} + ( -13621164036 - 26202992 \beta_{1} + 543776 \beta_{2} + 212498 \beta_{3} ) q^{43} + ( 8945160192 - 17600512 \beta_{1} + 69632 \beta_{2} - 278528 \beta_{3} ) q^{44} + ( 26240533521 + 1167042 \beta_{1} + 840069 \beta_{2} - 864549 \beta_{3} ) q^{45} + ( -981070464 - 19554368 \beta_{1} + 148096 \beta_{2} + 1172288 \beta_{3} ) q^{46} + ( 785931665 - 1261969 \beta_{1} - 2240885 \beta_{2} - 696101 \beta_{3} ) q^{47} + ( -771751936 + 16777216 \beta_{1} ) q^{48} + ( -21237619328 - 15794688 \beta_{1} - 3197824 \beta_{2} - 389760 \beta_{3} ) q^{50} + ( 60368996229 - 98858727 \beta_{1} - 98391 \beta_{2} - 2792052 \beta_{3} ) q^{51} + ( 728215552 + 16506880 \beta_{1} - 479232 \beta_{2} + 1863680 \beta_{3} ) q^{52} + ( -37337231485 - 138377218 \beta_{1} - 933086 \beta_{2} + 164941 \beta_{3} ) q^{53} + ( 26129754688 + 38791616 \beta_{1} + 325440 \beta_{2} - 1575168 \beta_{3} ) q^{54} + ( 21922437026 - 116871615 \beta_{1} - 2188438 \beta_{2} - 2260251 \beta_{3} ) q^{55} + ( -16084497371 + 117122252 \beta_{1} + 1815498 \beta_{2} + 3491982 \beta_{3} ) q^{57} + ( 50635534272 - 57531520 \beta_{1} - 5648704 \beta_{2} - 730304 \beta_{3} ) q^{58} + ( 216448300886 + 252055197 \beta_{1} + 3752480 \beta_{2} + 1898946 \beta_{3} ) q^{59} + ( -70114111488 + 70053888 \beta_{1} - 1609728 \beta_{2} + 331776 \beta_{3} ) q^{60} + ( 119502877813 - 51458534 \beta_{1} - 759540 \beta_{2} + 72569 \beta_{3} ) q^{61} + ( 97897510720 - 154203200 \beta_{1} + 5824960 \beta_{2} + 5652416 \beta_{3} ) q^{62} + 68719476736 q^{64} + ( 274838696067 + 196779426 \beta_{1} + 10849515 \beta_{2} + 13144209 \beta_{3} ) q^{65} + ( 403914602688 - 278464512 \beta_{1} - 8779776 \beta_{2} - 13587840 \beta_{3} ) q^{66} + ( -473926436184 + 102364229 \beta_{1} + 3985050 \beta_{2} - 3291746 \beta_{3} ) q^{67} + ( 8048066560 + 171270144 \beta_{1} + 8531968 \beta_{2} + 14364672 \beta_{3} ) q^{68} + ( 437778400218 + 268673472 \beta_{1} - 7490793 \beta_{2} - 2535624 \beta_{3} ) q^{69} + ( 80041697582 - 375227132 \beta_{1} + 14165130 \beta_{2} - 31584322 \beta_{3} ) q^{71} + ( 39266811904 + 88604672 \beta_{1} + 7077888 \beta_{2} + 10223616 \beta_{3} ) q^{72} + ( 741709294201 - 120492024 \beta_{1} - 5125102 \beta_{2} + 13431138 \beta_{3} ) q^{73} + ( -63429549632 + 598207104 \beta_{1} - 1707264 \beta_{2} - 29075904 \beta_{3} ) q^{74} + ( 333042411638 - 544889588 \beta_{1} + 28358358 \beta_{2} + 6295464 \beta_{3} ) q^{75} + ( 220901806080 - 36073472 \beta_{1} + 5476352 \beta_{2} - 19210240 \beta_{3} ) q^{76} + ( -376955455168 + 451738624 \beta_{1} + 16176576 \beta_{2} + 29293056 \beta_{3} ) q^{78} + ( 1626304384762 + 371069293 \beta_{1} + 2694762 \beta_{2} - 33488371 \beta_{3} ) q^{79} + ( -7415529472 - 201326592 \beta_{1} - 16777216 \beta_{2} ) q^{80} + ( -612285301948 - 37642946 \beta_{1} + 52517205 \beta_{2} + 70042011 \beta_{3} ) q^{81} + ( -691635136064 + 1212992896 \beta_{1} - 35215936 \beta_{2} - 34355776 \beta_{3} ) q^{82} + ( -423620056210 + 2285828428 \beta_{1} - 11946714 \beta_{2} + 16441880 \beta_{3} ) q^{83} + ( -2085756586183 + 373926750 \beta_{1} - 53785246 \beta_{2} + 81103773 \beta_{3} ) q^{85} + ( 871754498304 + 1676991488 \beta_{1} - 34801664 \beta_{2} - 13599872 \beta_{3} ) q^{86} + ( 1319570844069 - 2486105892 \beta_{1} + 37505775 \beta_{2} - 7302834 \beta_{3} ) q^{87} + ( -572490252288 + 1126432768 \beta_{1} - 4456448 \beta_{2} + 17825792 \beta_{3} ) q^{88} + ( -2395747211569 - 1806410596 \beta_{1} - 51621500 \beta_{2} - 64454306 \beta_{3} ) q^{89} + ( -1679394145344 - 74690688 \beta_{1} - 53764416 \beta_{2} + 55331136 \beta_{3} ) q^{90} + ( 62788509696 + 1251479552 \beta_{1} - 9478144 \beta_{2} - 75026432 \beta_{3} ) q^{92} + ( 3548666904253 + 539804942 \beta_{1} - 118654122 \beta_{2} - 79643421 \beta_{3} ) q^{93} + ( -50299626560 + 80766016 \beta_{1} + 143416640 \beta_{2} + 44550464 \beta_{3} ) q^{94} + ( -3594995836676 - 2213395137 \beta_{1} - 187764464 \beta_{2} - 133760211 \beta_{3} ) q^{95} + ( 49392123904 - 1073741824 \beta_{1} ) q^{96} + ( -5570640808189 + 1097788486 \beta_{1} + 128001703 \beta_{2} - 31967299 \beta_{3} ) q^{97} + ( 3091428017331 - 6251038296 \beta_{1} - 74244699 \beta_{2} - 127291050 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 256q^{2} - 182q^{3} + 16384q^{4} - 1792q^{5} + 11648q^{6} - 1048576q^{8} - 599840q^{9} + O(q^{10}) \) \( 4q - 256q^{2} - 182q^{3} + 16384q^{4} - 1792q^{5} + 11648q^{6} - 1048576q^{8} - 599840q^{9} + 114688q^{10} + 8726914q^{11} - 745472q^{12} + 719208q^{13} - 68436606q^{15} + 67108864q^{16} + 7943068q^{17} + 38389760q^{18} + 215706806q^{19} - 7340032q^{20} - 558522496q^{22} + 61927978q^{23} + 47710208q^{24} + 1327844792q^{25} - 46029312q^{26} - 1634321906q^{27} - 3162923032q^{29} + 4379942784q^{30} - 6113775570q^{31} - 4294967296q^{32} - 25235960652q^{33} - 508356352q^{34} - 2456944640q^{36} + 3945652880q^{37} - 13805235584q^{38} + 23545599116q^{39} + 469762048q^{40} + 43189289976q^{41} - 54537062128q^{43} + 35745439744q^{44} + 104964468168q^{45} - 3963390592q^{46} + 3141202722q^{47} - 3053453312q^{48} - 84982066688q^{50} + 241278267462q^{51} + 2945875968q^{52} - 149625680376q^{53} + 104596601984q^{54} + 87456004874q^{55} - 64103744980q^{57} + 202427074048q^{58} + 866297313938q^{59} - 280316338176q^{60} + 477908594184q^{61} + 391281636480q^{62} + 274877906944q^{64} + 1099748343120q^{65} + 1615101481728q^{66} - 1895501016278q^{67} + 32534806528q^{68} + 1751650947816q^{69} + 319416336064q^{71} + 157244456960q^{72} + 2966596192756q^{73} - 252521784320q^{74} + 1331079867376q^{75} + 883535077376q^{76} - 1506918343424q^{78} + 6505959677634q^{79} - 30064771072q^{80} - 2449216493684q^{81} - 2764114558464q^{82} - 1689908567984q^{83} - 8342278491232q^{85} + 3490371976192q^{86} + 5273311164492q^{87} - 2287708143616q^{88} - 9586601667468q^{89} - 6717725962752q^{90} + 253656997888q^{92} + 14195747226896q^{93} - 201036974208q^{94} - 14384410136978q^{95} + 195421011968q^{96} - 22280367655784q^{97} + 12353209992732q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 209077 x^{2} - 23859426 x + 2739764835\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -25 \nu^{3} + 3414 \nu^{2} + 4572970 \nu + 90607842 \)\()/276087\)
\(\beta_{2}\)\(=\)\((\)\( 1012 \nu^{3} + 16410 \nu^{2} - 114766858 \nu - 19824781119 \)\()/828261\)
\(\beta_{3}\)\(=\)\((\)\( -268 \nu^{3} + 80772 \nu^{2} + 29364844 \nu - 3648039096 \)\()/118323\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{3} + 2 \beta_{2} + 52 \beta_{1} - 26\)\()/336\)
\(\nu^{2}\)\(=\)\((\)\(455 \beta_{3} + 890 \beta_{2} + 628 \beta_{1} + 35124622\)\()/336\)
\(\nu^{3}\)\(=\)\((\)\(-7549 \beta_{3} + 30461 \beta_{2} + 367933 \beta_{1} + 375601993\)\()/21\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−228.558
−338.799
496.027
71.3297
−64.0000 −1776.43 4096.00 26142.0 113691. 0 −262144. 1.56137e6 −1.67309e6
1.2 −64.0000 −388.685 4096.00 25902.2 24875.9 0 −262144. −1.44325e6 −1.65774e6
1.3 −64.0000 489.405 4096.00 −68192.6 −31321.9 0 −262144. −1.35481e6 4.36432e6
1.4 −64.0000 1493.71 4096.00 14356.4 −95597.3 0 −262144. 636841. −918809.
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(7\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 182 T_{3}^{3} - 2872164 T_{3}^{2} + 213475122 T_{3} + 504753851835 \) acting on \(S_{14}^{\mathrm{new}}(\Gamma_0(98))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 64 + T )^{4} \)
$3$ \( 504753851835 + 213475122 T - 2872164 T^{2} + 182 T^{3} + T^{4} \)
$5$ \( -662914029924727875 + 87405514209720 T - 3103723014 T^{2} + 1792 T^{3} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( \)\(21\!\cdots\!35\)\( + 89730790785209453418 T - 36020273184588 T^{2} - 8726914 T^{3} + T^{4} \)
$13$ \( \)\(29\!\cdots\!00\)\( - \)\(24\!\cdots\!20\)\( T - 549797025415320 T^{2} - 719208 T^{3} + T^{4} \)
$17$ \( \)\(78\!\cdots\!29\)\( + \)\(97\!\cdots\!96\)\( T - 27135066800890218 T^{2} - 7943068 T^{3} + T^{4} \)
$19$ \( -\)\(35\!\cdots\!05\)\( + \)\(99\!\cdots\!22\)\( T - 38033254953433020 T^{2} - 215706806 T^{3} + T^{4} \)
$23$ \( \)\(11\!\cdots\!03\)\( + \)\(21\!\cdots\!50\)\( T - 821454433693827192 T^{2} - 61927978 T^{3} + T^{4} \)
$29$ \( -\)\(94\!\cdots\!96\)\( - \)\(87\!\cdots\!92\)\( T - 18126451463113919640 T^{2} + 3162923032 T^{3} + T^{4} \)
$31$ \( -\)\(39\!\cdots\!45\)\( - \)\(23\!\cdots\!78\)\( T - 26107086741640452048 T^{2} + 6113775570 T^{3} + T^{4} \)
$37$ \( \)\(28\!\cdots\!37\)\( - \)\(16\!\cdots\!92\)\( T - \)\(60\!\cdots\!66\)\( T^{2} - 3945652880 T^{3} + T^{4} \)
$41$ \( -\)\(96\!\cdots\!16\)\( + \)\(79\!\cdots\!04\)\( T - \)\(11\!\cdots\!00\)\( T^{2} - 43189289976 T^{3} + T^{4} \)
$43$ \( \)\(22\!\cdots\!00\)\( - \)\(67\!\cdots\!80\)\( T - \)\(15\!\cdots\!20\)\( T^{2} + 54537062128 T^{3} + T^{4} \)
$47$ \( -\)\(51\!\cdots\!89\)\( + \)\(51\!\cdots\!82\)\( T - \)\(11\!\cdots\!16\)\( T^{2} - 3141202722 T^{3} + T^{4} \)
$53$ \( \)\(30\!\cdots\!89\)\( - \)\(60\!\cdots\!88\)\( T - \)\(49\!\cdots\!02\)\( T^{2} + 149625680376 T^{3} + T^{4} \)
$59$ \( -\)\(10\!\cdots\!05\)\( + \)\(78\!\cdots\!10\)\( T + \)\(65\!\cdots\!12\)\( T^{2} - 866297313938 T^{3} + T^{4} \)
$61$ \( \)\(95\!\cdots\!21\)\( - \)\(46\!\cdots\!24\)\( T + \)\(76\!\cdots\!86\)\( T^{2} - 477908594184 T^{3} + T^{4} \)
$67$ \( \)\(22\!\cdots\!11\)\( + \)\(31\!\cdots\!98\)\( T + \)\(12\!\cdots\!92\)\( T^{2} + 1895501016278 T^{3} + T^{4} \)
$71$ \( \)\(19\!\cdots\!56\)\( + \)\(12\!\cdots\!64\)\( T - \)\(34\!\cdots\!52\)\( T^{2} - 319416336064 T^{3} + T^{4} \)
$73$ \( \)\(87\!\cdots\!45\)\( - \)\(94\!\cdots\!44\)\( T + \)\(27\!\cdots\!54\)\( T^{2} - 2966596192756 T^{3} + T^{4} \)
$79$ \( -\)\(32\!\cdots\!25\)\( - \)\(68\!\cdots\!10\)\( T + \)\(13\!\cdots\!60\)\( T^{2} - 6505959677634 T^{3} + T^{4} \)
$83$ \( -\)\(29\!\cdots\!64\)\( - \)\(45\!\cdots\!44\)\( T - \)\(15\!\cdots\!12\)\( T^{2} + 1689908567984 T^{3} + T^{4} \)
$89$ \( -\)\(15\!\cdots\!19\)\( - \)\(65\!\cdots\!32\)\( T + \)\(15\!\cdots\!62\)\( T^{2} + 9586601667468 T^{3} + T^{4} \)
$97$ \( -\)\(20\!\cdots\!80\)\( - \)\(12\!\cdots\!68\)\( T + \)\(12\!\cdots\!20\)\( T^{2} + 22280367655784 T^{3} + T^{4} \)
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