Properties

Label 8.18.a.a
Level 8
Weight 18
Character orbit 8.a
Self dual Yes
Analytic conductor 14.658
Analytic rank 0
Dimension 2
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 8 = 2^{3} \)
Weight: \( k \) = \( 18 \)
Character orbit: \([\chi]\) = 8.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(14.6577669876\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2146}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{8} \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 256\sqrt{2146}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( -476 + \beta ) q^{3} \) \( + ( -26810 + 60 \beta ) q^{5} \) \( + ( -166584 + 978 \beta ) q^{7} \) \( + ( 11726669 - 952 \beta ) q^{9} \) \(+O(q^{10})\) \( q\) \( + ( -476 + \beta ) q^{3} \) \( + ( -26810 + 60 \beta ) q^{5} \) \( + ( -166584 + 978 \beta ) q^{7} \) \( + ( 11726669 - 952 \beta ) q^{9} \) \( + ( 215487340 - 74781 \beta ) q^{11} \) \( + ( 1333760974 + 329628 \beta ) q^{13} \) \( + ( 8451176920 - 55370 \beta ) q^{15} \) \( + ( 30336751634 - 1629816 \beta ) q^{17} \) \( + ( 89314980020 + 1134309 \beta ) q^{19} \) \( + ( 137625464352 - 632112 \beta ) q^{21} \) \( + ( 264378297304 + 18242742 \beta ) q^{23} \) \( + ( -255915755425 - 3217200 \beta ) q^{25} \) \( + ( -78000700568 - 116960342 \beta ) q^{27} \) \( + ( -3620330045730 + 71618988 \beta ) q^{29} \) \( + ( -939175570144 + 165125256 \beta ) q^{31} \) \( + ( -10619790957776 + 251083096 \beta ) q^{33} \) \( + ( 8257236339120 - 36215220 \beta ) q^{35} \) \( + ( -10166232283290 - 2527244916 \beta ) q^{37} \) \( + ( 45724096081144 + 1176858046 \beta ) q^{39} \) \( + ( -881520952662 + 4750902096 \beta ) q^{41} \) \( + ( 96697262984332 - 1772936949 \beta ) q^{43} \) \( + ( -8347763418610 + 729123260 \beta ) q^{45} \) \( + ( 50381918882736 - 12557431380 \beta ) q^{47} \) \( + ( -98082609138247 - 325838304 \beta ) q^{49} \) \( + ( -243658033250680 + 31112544050 \beta ) q^{51} \) \( + ( -158909073030026 - 40341828660 \beta ) q^{53} \) \( + ( -636810354621560 + 14934119010 \beta ) q^{55} \) \( + ( 117015577653584 + 88775048936 \beta ) q^{57} \) \( + ( -631025394160868 - 86330682801 \beta ) q^{59} \) \( + ( 1382929846361854 - 97871720196 \beta ) q^{61} \) \( + ( -132897429619032 + 11627270250 \beta ) q^{63} \) \( + ( 2745779846573140 + 71188331760 \beta ) q^{65} \) \( + ( 981503272275044 + 62855476593 \beta ) q^{67} \) \( + ( 2439819835505248 + 255694752112 \beta ) q^{69} \) \( + ( 241819553552264 + 82368492066 \beta ) q^{71} \) \( + ( -1088446174295606 - 741325171608 \beta ) q^{73} \) \( + ( -330651932020900 - 254384368225 \beta ) q^{75} \) \( + ( -10321736909335968 + 223203936624 \beta ) q^{77} \) \( + ( -2647919952813872 + 468379685172 \beta ) q^{79} \) \( + ( -17926588053364231 + 100613857400 \beta ) q^{81} \) \( + ( 4986259009443572 + 631666459029 \beta ) q^{83} \) \( + ( -14566392679681300 + 1863900465000 \beta ) q^{85} \) \( + ( 11795789908548408 - 3654420684018 \beta ) q^{87} \) \( + ( -355549518406950 - 1713360438936 \beta ) q^{89} \) \( + ( 45116885807970288 + 1249507481820 \beta ) q^{91} \) \( + ( 23670305847294080 - 1017775192000 \beta ) q^{93} \) \( + ( 7177235874250040 + 5328487976910 \beta ) q^{95} \) \( + ( 57435273304985954 - 48030348120 \beta ) q^{97} \) \( + ( 12539341182577532 - 1082075982169 \beta ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut -\mathstrut 952q^{3} \) \(\mathstrut -\mathstrut 53620q^{5} \) \(\mathstrut -\mathstrut 333168q^{7} \) \(\mathstrut +\mathstrut 23453338q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut -\mathstrut 952q^{3} \) \(\mathstrut -\mathstrut 53620q^{5} \) \(\mathstrut -\mathstrut 333168q^{7} \) \(\mathstrut +\mathstrut 23453338q^{9} \) \(\mathstrut +\mathstrut 430974680q^{11} \) \(\mathstrut +\mathstrut 2667521948q^{13} \) \(\mathstrut +\mathstrut 16902353840q^{15} \) \(\mathstrut +\mathstrut 60673503268q^{17} \) \(\mathstrut +\mathstrut 178629960040q^{19} \) \(\mathstrut +\mathstrut 275250928704q^{21} \) \(\mathstrut +\mathstrut 528756594608q^{23} \) \(\mathstrut -\mathstrut 511831510850q^{25} \) \(\mathstrut -\mathstrut 156001401136q^{27} \) \(\mathstrut -\mathstrut 7240660091460q^{29} \) \(\mathstrut -\mathstrut 1878351140288q^{31} \) \(\mathstrut -\mathstrut 21239581915552q^{33} \) \(\mathstrut +\mathstrut 16514472678240q^{35} \) \(\mathstrut -\mathstrut 20332464566580q^{37} \) \(\mathstrut +\mathstrut 91448192162288q^{39} \) \(\mathstrut -\mathstrut 1763041905324q^{41} \) \(\mathstrut +\mathstrut 193394525968664q^{43} \) \(\mathstrut -\mathstrut 16695526837220q^{45} \) \(\mathstrut +\mathstrut 100763837765472q^{47} \) \(\mathstrut -\mathstrut 196165218276494q^{49} \) \(\mathstrut -\mathstrut 487316066501360q^{51} \) \(\mathstrut -\mathstrut 317818146060052q^{53} \) \(\mathstrut -\mathstrut 1273620709243120q^{55} \) \(\mathstrut +\mathstrut 234031155307168q^{57} \) \(\mathstrut -\mathstrut 1262050788321736q^{59} \) \(\mathstrut +\mathstrut 2765859692723708q^{61} \) \(\mathstrut -\mathstrut 265794859238064q^{63} \) \(\mathstrut +\mathstrut 5491559693146280q^{65} \) \(\mathstrut +\mathstrut 1963006544550088q^{67} \) \(\mathstrut +\mathstrut 4879639671010496q^{69} \) \(\mathstrut +\mathstrut 483639107104528q^{71} \) \(\mathstrut -\mathstrut 2176892348591212q^{73} \) \(\mathstrut -\mathstrut 661303864041800q^{75} \) \(\mathstrut -\mathstrut 20643473818671936q^{77} \) \(\mathstrut -\mathstrut 5295839905627744q^{79} \) \(\mathstrut -\mathstrut 35853176106728462q^{81} \) \(\mathstrut +\mathstrut 9972518018887144q^{83} \) \(\mathstrut -\mathstrut 29132785359362600q^{85} \) \(\mathstrut +\mathstrut 23591579817096816q^{87} \) \(\mathstrut -\mathstrut 711099036813900q^{89} \) \(\mathstrut +\mathstrut 90233771615940576q^{91} \) \(\mathstrut +\mathstrut 47340611694588160q^{93} \) \(\mathstrut +\mathstrut 14354471748500080q^{95} \) \(\mathstrut +\mathstrut 114870546609971908q^{97} \) \(\mathstrut +\mathstrut 25078682365155064q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
−46.3249
46.3249
0 −12335.2 0 −738361. 0 −1.17649e7 0 2.30166e7 0
1.2 0 11383.2 0 684741. 0 1.14317e7 0 436725. 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 \(T_{3}^{2} \) \(\mathstrut +\mathstrut 952 T_{3} \) \(\mathstrut -\mathstrut 140413680 \) acting on \(S_{18}^{\mathrm{new}}(\Gamma_0(8))\).