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: \(22.358187543\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{21}\cdot 3\cdot 5\cdot 7 \)
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.

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

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

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

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.

Atkin-Lehner signs

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

Hecke kernels

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