Properties

Label 8.22.a.b
Level 8
Weight 22
Character orbit 8.a
Self dual yes
Analytic conductor 22.358
Analytic rank 0
Dimension 3
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 8 = 2^{3} \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 8.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(22.3581875430\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
Defining polynomial: \(x^{3} - x^{2} - 4963 x + 96223\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{21}\cdot 3\cdot 5\cdot 7 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\) \(=\) \( q + ( 32255 - \beta_{1} ) q^{3} + ( -8037261 + 9 \beta_{1} + \beta_{2} ) q^{5} + ( 98664098 - 4014 \beta_{1} + 4 \beta_{2} ) q^{7} + ( 6281605939 - 120578 \beta_{1} - 306 \beta_{2} ) q^{9} +O(q^{10})\) \( q + ( 32255 - \beta_{1} ) q^{3} + ( -8037261 + 9 \beta_{1} + \beta_{2} ) q^{5} + ( 98664098 - 4014 \beta_{1} + 4 \beta_{2} ) q^{7} + ( 6281605939 - 120578 \beta_{1} - 306 \beta_{2} ) q^{9} + ( -13444955723 - 241515 \beta_{1} + 2856 \beta_{2} ) q^{11} + ( 44577735635 + 1219041 \beta_{1} - 8551 \beta_{2} ) q^{13} + ( -407720224330 + 24214790 \beta_{1} - 21060 \beta_{2} ) q^{15} + ( 2599270326072 - 78703794 \beta_{1} + 206750 \beta_{2} ) q^{17} + ( 11929396534227 + 10179315 \beta_{1} - 319144 \beta_{2} ) q^{19} + ( 66179872171740 - 391662132 \beta_{1} - 1323540 \beta_{2} ) q^{21} + ( 64589693893622 + 1679798214 \beta_{1} + 5214236 \beta_{2} ) q^{23} + ( 375854023120779 + 2369077164 \beta_{1} + 292556 \beta_{2} ) q^{25} + ( 1760671157218238 - 11178145762 \beta_{1} - 29609784 \beta_{2} ) q^{27} + ( 1869113920498543 + 1660970493 \beta_{1} + 33737045 \beta_{2} ) q^{29} + ( 3748921264001176 - 5920500456 \beta_{1} + 74881888 \beta_{2} ) q^{31} + ( 3338037660191858 + 36046394810 \beta_{1} - 141916374 \beta_{2} ) q^{33} + ( 1758047959172804 + 107547831684 \beta_{1} - 83942384 \beta_{2} ) q^{35} + ( -8090763662016401 - 208805051403 \beta_{1} + 10140941 \beta_{2} ) q^{37} + ( -17641746644228050 - 68445178114 \beta_{1} + 576660060 \beta_{2} ) q^{39} + ( -99386278879320634 - 272353907700 \beta_{1} + 1361738284 \beta_{2} ) q^{41} + ( -11111339343224811 + 85889919573 \beta_{1} - 5014813136 \beta_{2} ) q^{43} + ( -309137868773272277 + 2128341923353 \beta_{1} - 2549104623 \beta_{2} ) q^{45} + ( -40291335714295108 - 276404718564 \beta_{1} + 18460661896 \beta_{2} ) q^{47} + ( -283467391737079375 - 1156764401352 \beta_{1} - 5695384968 \beta_{2} ) q^{49} + ( 1318131720028028758 - 6370268425034 \beta_{1} - 29006905464 \beta_{2} ) q^{51} + ( -379481145606515761 + 43814693061 \beta_{1} + 21744109661 \beta_{2} ) q^{53} + ( 2319179746504802226 + 13244749165026 \beta_{1} - 3901779356 \beta_{2} ) q^{55} + ( 227237828911355998 - 15939600411050 \beta_{1} + 10714965606 \beta_{2} ) q^{57} + ( 3075197953846188001 + 30637171373409 \beta_{1} + 79875012000 \beta_{2} ) q^{59} + ( -2184976859120054997 + 28283296240809 \beta_{1} - 115270423775 \beta_{2} ) q^{61} + ( 7261764567978521130 - 79144304421414 \beta_{1} - 130171243644 \beta_{2} ) q^{63} + ( -6904843446032282168 - 51481464297948 \beta_{1} + 42445710468 \beta_{2} ) q^{65} + ( -5264341440545960537 - 29752893747513 \beta_{1} + 323347301080 \beta_{2} ) q^{67} + ( -24329491431145035820 + 163983745168292 \beta_{1} + 389846437380 \beta_{2} ) q^{69} + ( -13713239988715498270 + 137472678497106 \beta_{1} - 934927894380 \beta_{2} ) q^{71} + ( -6474024821894165672 - 93484221354954 \beta_{1} - 378453267642 \beta_{2} ) q^{73} + ( -25077165094461530975 - 162109742402975 \beta_{1} + 717970683600 \beta_{2} ) q^{75} + ( 22793346147343450388 + 198923328266724 \beta_{1} - 589853232572 \beta_{2} ) q^{77} + ( -43911866990278864876 + 279671030545140 \beta_{1} + 2322314505128 \beta_{2} ) q^{79} + ( 166809246069484652275 - 1942146171617678 \beta_{1} + 485482873122 \beta_{2} ) q^{81} + ( 21337821507520406283 + 529310118462219 \beta_{1} - 4513980072832 \beta_{2} ) q^{83} + ( 130085249047167064858 + 2455621500102678 \beta_{1} + 26161327462 \beta_{2} ) q^{85} + ( 33966719140332693294 - 1203447813013314 \beta_{1} - 295157018772 \beta_{2} ) q^{87} + ( 143296519800052541736 + 1887497379621558 \beta_{1} + 4599855411654 \beta_{2} ) q^{89} + ( -99060406226366062252 - 482908923309420 \beta_{1} + 2503713552880 \beta_{2} ) q^{91} + ( 213346162989254242096 - 3119957848026128 \beta_{1} - 3594910420368 \beta_{2} ) q^{93} + ( -345457140866077157474 - 984050788352274 \beta_{1} + 9966832058044 \beta_{2} ) q^{95} + ( -108019528253374798368 - 751754024242938 \beta_{1} - 21789767623018 \beta_{2} ) q^{97} + ( -316661031554270978183 + 188974335581689 \beta_{1} - 15464975405472 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 96764q^{3} - 24111774q^{5} + 295988280q^{7} + 18844697239q^{9} + O(q^{10}) \) \( 3q + 96764q^{3} - 24111774q^{5} + 295988280q^{7} + 18844697239q^{9} - 40335108684q^{11} + 133734425946q^{13} - 1223136458200q^{15} + 7797732274422q^{17} + 35788199781996q^{19} + 198539224853088q^{21} + 193770761479080q^{23} + 1127564438439501q^{25} + 5282002293508952q^{27} + 5607343422466122q^{29} + 11246757871503072q^{31} + 10014149026970384q^{33} + 5274251425350096q^{35} - 24272499791100606q^{37} - 52925308377862264q^{39} - 298159108991869602q^{41} - 33333932139754860q^{43} - 927411477977893478q^{45} - 120874283547603888q^{47} - 850403331975639477q^{49} + 3954388789815661240q^{51} - 1138443393004854222q^{53} + 6957552484263571704q^{55} + 681697547133656944q^{57} + 9225624498709937412q^{59} - 6554902294063924182q^{61} + 21785214559631141976q^{63} - 20714581819561144452q^{65} - 15793054074531629124q^{67} - 72988310309689939168q^{69} - 41139582493467997704q^{71} - 19422167949903851970q^{73} - 75231657393126995900q^{75} + 68380237365358617888q^{77} - 131735321299806049488q^{79} + 500425796062282339147q^{81} + 64013993832679681068q^{83} + 390258202763001297252q^{85} + 101898953973185066568q^{87} + 429891446897537246766q^{89} - 297181701588021496176q^{91} + 640035369009914700160q^{93} - 1036372406649019824696q^{95} - 324059336514148638042q^{97} - 949982905688477352860q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 4963 x + 96223\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 64 \nu^{2} + 640 \nu - 211989 \)
\(\beta_{2}\)\(=\)\( 8896 \nu^{2} + 662400 \nu - 29657664 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} - 139 \beta_{1} + 191193\)\()/573440\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{2} + 1035 \beta_{1} + 189750951\)\()/57344\)

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
−78.2002
57.9766
21.2235
0 −97085.2 0 −3.39292e7 0 −5.28731e8 0 −1.03483e9 0
1.2 0 −7983.67 0 3.09730e7 0 9.17385e7 0 −1.03966e10 0
1.3 0 201833. 0 −2.11555e7 0 7.32981e8 0 3.02761e10 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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 8.22.a.b 3
3.b odd 2 1 72.22.a.f 3
4.b odd 2 1 16.22.a.f 3
8.b even 2 1 64.22.a.l 3
8.d odd 2 1 64.22.a.m 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.22.a.b 3 1.a even 1 1 trivial
16.22.a.f 3 4.b odd 2 1
64.22.a.l 3 8.b even 2 1
64.22.a.m 3 8.d odd 2 1
72.22.a.f 3 3.b odd 2 1

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{3} - 96764 T_{3}^{2} - 20431242576 T_{3} - \)\(15\!\cdots\!00\)\( \) acting on \(S_{22}^{\mathrm{new}}(\Gamma_0(8))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - 96764 T + 10949817033 T^{2} - 2180811113308584 T^{3} + \)\(11\!\cdots\!99\)\( T^{4} - \)\(10\!\cdots\!76\)\( T^{5} + \)\(11\!\cdots\!27\)\( T^{6} \)
$5$ \( 1 + 24111774 T + 442162340798475 T^{2} + \)\(76\!\cdots\!00\)\( T^{3} + \)\(21\!\cdots\!75\)\( T^{4} + \)\(54\!\cdots\!50\)\( T^{5} + \)\(10\!\cdots\!25\)\( T^{6} \)
$7$ \( 1 - 295988280 T + 1306824993061424949 T^{2} - \)\(29\!\cdots\!64\)\( T^{3} + \)\(72\!\cdots\!43\)\( T^{4} - \)\(92\!\cdots\!20\)\( T^{5} + \)\(17\!\cdots\!43\)\( T^{6} \)
$11$ \( 1 + 40335108684 T + \)\(11\!\cdots\!17\)\( T^{2} + \)\(53\!\cdots\!04\)\( T^{3} + \)\(87\!\cdots\!87\)\( T^{4} + \)\(22\!\cdots\!64\)\( T^{5} + \)\(40\!\cdots\!31\)\( T^{6} \)
$13$ \( 1 - 133734425946 T + \)\(62\!\cdots\!19\)\( T^{2} - \)\(72\!\cdots\!44\)\( T^{3} + \)\(15\!\cdots\!47\)\( T^{4} - \)\(81\!\cdots\!74\)\( T^{5} + \)\(15\!\cdots\!97\)\( T^{6} \)
$17$ \( 1 - 7797732274422 T + \)\(31\!\cdots\!91\)\( T^{2} + \)\(24\!\cdots\!52\)\( T^{3} + \)\(21\!\cdots\!47\)\( T^{4} - \)\(37\!\cdots\!58\)\( T^{5} + \)\(32\!\cdots\!13\)\( T^{6} \)
$19$ \( 1 - 35788199781996 T + \)\(24\!\cdots\!41\)\( T^{2} - \)\(51\!\cdots\!12\)\( T^{3} + \)\(17\!\cdots\!79\)\( T^{4} - \)\(18\!\cdots\!56\)\( T^{5} + \)\(36\!\cdots\!59\)\( T^{6} \)
$23$ \( 1 - 193770761479080 T + \)\(32\!\cdots\!17\)\( T^{2} - \)\(21\!\cdots\!56\)\( T^{3} + \)\(12\!\cdots\!91\)\( T^{4} - \)\(30\!\cdots\!20\)\( T^{5} + \)\(61\!\cdots\!67\)\( T^{6} \)
$29$ \( 1 - 5607343422466122 T + \)\(24\!\cdots\!83\)\( T^{2} - \)\(62\!\cdots\!88\)\( T^{3} + \)\(12\!\cdots\!07\)\( T^{4} - \)\(14\!\cdots\!02\)\( T^{5} + \)\(13\!\cdots\!89\)\( T^{6} \)
$31$ \( 1 - 11246757871503072 T + \)\(97\!\cdots\!53\)\( T^{2} - \)\(49\!\cdots\!64\)\( T^{3} + \)\(20\!\cdots\!43\)\( T^{4} - \)\(48\!\cdots\!92\)\( T^{5} + \)\(90\!\cdots\!91\)\( T^{6} \)
$37$ \( 1 + 24272499791100606 T + \)\(17\!\cdots\!75\)\( T^{2} + \)\(26\!\cdots\!28\)\( T^{3} + \)\(14\!\cdots\!75\)\( T^{4} + \)\(17\!\cdots\!14\)\( T^{5} + \)\(62\!\cdots\!53\)\( T^{6} \)
$41$ \( 1 + 298159108991869602 T + \)\(47\!\cdots\!03\)\( T^{2} + \)\(50\!\cdots\!88\)\( T^{3} + \)\(35\!\cdots\!23\)\( T^{4} + \)\(16\!\cdots\!62\)\( T^{5} + \)\(40\!\cdots\!21\)\( T^{6} \)
$43$ \( 1 + 33333932139754860 T + \)\(30\!\cdots\!21\)\( T^{2} + \)\(22\!\cdots\!68\)\( T^{3} + \)\(61\!\cdots\!03\)\( T^{4} + \)\(13\!\cdots\!40\)\( T^{5} + \)\(81\!\cdots\!07\)\( T^{6} \)
$47$ \( 1 + 120874283547603888 T - \)\(88\!\cdots\!11\)\( T^{2} - \)\(47\!\cdots\!92\)\( T^{3} - \)\(11\!\cdots\!17\)\( T^{4} + \)\(20\!\cdots\!92\)\( T^{5} + \)\(21\!\cdots\!23\)\( T^{6} \)
$53$ \( 1 + 1138443393004854222 T + \)\(47\!\cdots\!87\)\( T^{2} + \)\(34\!\cdots\!56\)\( T^{3} + \)\(76\!\cdots\!11\)\( T^{4} + \)\(29\!\cdots\!98\)\( T^{5} + \)\(42\!\cdots\!77\)\( T^{6} \)
$59$ \( 1 - 9225624498709937412 T + \)\(44\!\cdots\!77\)\( T^{2} - \)\(17\!\cdots\!60\)\( T^{3} + \)\(69\!\cdots\!43\)\( T^{4} - \)\(21\!\cdots\!72\)\( T^{5} + \)\(36\!\cdots\!79\)\( T^{6} \)
$61$ \( 1 + 6554902294063924182 T + \)\(72\!\cdots\!43\)\( T^{2} + \)\(26\!\cdots\!04\)\( T^{3} + \)\(22\!\cdots\!23\)\( T^{4} + \)\(63\!\cdots\!22\)\( T^{5} + \)\(29\!\cdots\!81\)\( T^{6} \)
$67$ \( 1 + 15793054074531629124 T + \)\(60\!\cdots\!01\)\( T^{2} + \)\(66\!\cdots\!68\)\( T^{3} + \)\(13\!\cdots\!67\)\( T^{4} + \)\(78\!\cdots\!36\)\( T^{5} + \)\(11\!\cdots\!63\)\( T^{6} \)
$71$ \( 1 + 41139582493467997704 T + \)\(13\!\cdots\!17\)\( T^{2} + \)\(27\!\cdots\!68\)\( T^{3} + \)\(10\!\cdots\!07\)\( T^{4} + \)\(23\!\cdots\!64\)\( T^{5} + \)\(42\!\cdots\!11\)\( T^{6} \)
$73$ \( 1 + 19422167949903851970 T + \)\(37\!\cdots\!27\)\( T^{2} + \)\(49\!\cdots\!64\)\( T^{3} + \)\(51\!\cdots\!71\)\( T^{4} + \)\(35\!\cdots\!30\)\( T^{5} + \)\(24\!\cdots\!17\)\( T^{6} \)
$79$ \( 1 + \)\(13\!\cdots\!88\)\( T + \)\(18\!\cdots\!73\)\( T^{2} + \)\(13\!\cdots\!04\)\( T^{3} + \)\(13\!\cdots\!67\)\( T^{4} + \)\(66\!\cdots\!08\)\( T^{5} + \)\(35\!\cdots\!39\)\( T^{6} \)
$83$ \( 1 - 64013993832679681068 T + \)\(30\!\cdots\!69\)\( T^{2} - \)\(30\!\cdots\!64\)\( T^{3} + \)\(61\!\cdots\!27\)\( T^{4} - \)\(25\!\cdots\!52\)\( T^{5} + \)\(79\!\cdots\!87\)\( T^{6} \)
$89$ \( 1 - \)\(42\!\cdots\!66\)\( T + \)\(21\!\cdots\!31\)\( T^{2} - \)\(50\!\cdots\!72\)\( T^{3} + \)\(18\!\cdots\!59\)\( T^{4} - \)\(32\!\cdots\!86\)\( T^{5} + \)\(64\!\cdots\!69\)\( T^{6} \)
$97$ \( 1 + \)\(32\!\cdots\!42\)\( T + \)\(10\!\cdots\!79\)\( T^{2} + \)\(44\!\cdots\!92\)\( T^{3} + \)\(55\!\cdots\!63\)\( T^{4} + \)\(90\!\cdots\!78\)\( T^{5} + \)\(14\!\cdots\!73\)\( T^{6} \)
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