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: \(\Q(\sqrt{2146}) \)
Defining polynomial: \(x^{2} - 2146\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: yes
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})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 952q^{3} - 53620q^{5} - 333168q^{7} + 23453338q^{9} + O(q^{10}) \) \( 2q - 952q^{3} - 53620q^{5} - 333168q^{7} + 23453338q^{9} + 430974680q^{11} + 2667521948q^{13} + 16902353840q^{15} + 60673503268q^{17} + 178629960040q^{19} + 275250928704q^{21} + 528756594608q^{23} - 511831510850q^{25} - 156001401136q^{27} - 7240660091460q^{29} - 1878351140288q^{31} - 21239581915552q^{33} + 16514472678240q^{35} - 20332464566580q^{37} + 91448192162288q^{39} - 1763041905324q^{41} + 193394525968664q^{43} - 16695526837220q^{45} + 100763837765472q^{47} - 196165218276494q^{49} - 487316066501360q^{51} - 317818146060052q^{53} - 1273620709243120q^{55} + 234031155307168q^{57} - 1262050788321736q^{59} + 2765859692723708q^{61} - 265794859238064q^{63} + 5491559693146280q^{65} + 1963006544550088q^{67} + 4879639671010496q^{69} + 483639107104528q^{71} - 2176892348591212q^{73} - 661303864041800q^{75} - 20643473818671936q^{77} - 5295839905627744q^{79} - 35853176106728462q^{81} + 9972518018887144q^{83} - 29132785359362600q^{85} + 23591579817096816q^{87} - 711099036813900q^{89} + 90233771615940576q^{91} + 47340611694588160q^{93} + 14354471748500080q^{95} + 114870546609971908q^{97} + 25078682365155064q^{99} + O(q^{100}) \)

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.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8.18.a.a 2
3.b odd 2 1 72.18.a.c 2
4.b odd 2 1 16.18.a.d 2
8.b even 2 1 64.18.a.k 2
8.d odd 2 1 64.18.a.h 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.18.a.a 2 1.a even 1 1 trivial
16.18.a.d 2 4.b odd 2 1
64.18.a.h 2 8.d odd 2 1
64.18.a.k 2 8.b even 2 1
72.18.a.c 2 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 \( T_{3}^{2} + 952 T_{3} - 140413680 \) acting on \(S_{18}^{\mathrm{new}}(\Gamma_0(8))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + 952 T + 117866646 T^{2} + 122941435176 T^{3} + 16677181699666569 T^{4} \)
$5$ \( 1 + 53620 T + 1020292760750 T^{2} + 40908813476562500 T^{3} + \)\(58\!\cdots\!25\)\( T^{4} \)
$7$ \( 1 + 333168 T + 330768623583566 T^{2} + 77505043084089781776 T^{3} + \)\(54\!\cdots\!49\)\( T^{4} \)
$11$ \( 1 - 430974680 T + 270840697861145126 T^{2} - \)\(21\!\cdots\!80\)\( T^{3} + \)\(25\!\cdots\!41\)\( T^{4} \)
$13$ \( 1 - 2667521948 T + 3798536829420038238 T^{2} - \)\(23\!\cdots\!84\)\( T^{3} + \)\(74\!\cdots\!89\)\( T^{4} \)
$17$ \( 1 - 60673503268 T + \)\(22\!\cdots\!74\)\( T^{2} - \)\(50\!\cdots\!36\)\( T^{3} + \)\(68\!\cdots\!29\)\( T^{4} \)
$19$ \( 1 - 178629960040 T + \)\(18\!\cdots\!42\)\( T^{2} - \)\(97\!\cdots\!60\)\( T^{3} + \)\(30\!\cdots\!21\)\( T^{4} \)
$23$ \( 1 - 528756594608 T + \)\(30\!\cdots\!38\)\( T^{2} - \)\(74\!\cdots\!24\)\( T^{3} + \)\(19\!\cdots\!09\)\( T^{4} \)
$29$ \( 1 + 7240660091460 T + \)\(26\!\cdots\!54\)\( T^{2} + \)\(52\!\cdots\!40\)\( T^{3} + \)\(52\!\cdots\!81\)\( T^{4} \)
$31$ \( 1 + 1878351140288 T + \)\(42\!\cdots\!42\)\( T^{2} + \)\(42\!\cdots\!68\)\( T^{3} + \)\(50\!\cdots\!21\)\( T^{4} \)
$37$ \( 1 + 20332464566580 T + \)\(11\!\cdots\!98\)\( T^{2} + \)\(92\!\cdots\!60\)\( T^{3} + \)\(20\!\cdots\!89\)\( T^{4} \)
$41$ \( 1 + 1763041905324 T + \)\(20\!\cdots\!10\)\( T^{2} + \)\(46\!\cdots\!44\)\( T^{3} + \)\(68\!\cdots\!61\)\( T^{4} \)
$43$ \( 1 - 193394525968664 T + \)\(20\!\cdots\!54\)\( T^{2} - \)\(11\!\cdots\!52\)\( T^{3} + \)\(34\!\cdots\!49\)\( T^{4} \)
$47$ \( 1 - 100763837765472 T + \)\(33\!\cdots\!70\)\( T^{2} - \)\(26\!\cdots\!64\)\( T^{3} + \)\(71\!\cdots\!69\)\( T^{4} \)
$53$ \( 1 + 317818146060052 T + \)\(20\!\cdots\!02\)\( T^{2} + \)\(65\!\cdots\!76\)\( T^{3} + \)\(42\!\cdots\!69\)\( T^{4} \)
$59$ \( 1 + 1262050788321736 T + \)\(18\!\cdots\!06\)\( T^{2} + \)\(16\!\cdots\!84\)\( T^{3} + \)\(16\!\cdots\!61\)\( T^{4} \)
$61$ \( 1 - 2765859692723708 T + \)\(50\!\cdots\!62\)\( T^{2} - \)\(62\!\cdots\!68\)\( T^{3} + \)\(50\!\cdots\!41\)\( T^{4} \)
$67$ \( 1 - 1963006544550088 T + \)\(22\!\cdots\!46\)\( T^{2} - \)\(21\!\cdots\!76\)\( T^{3} + \)\(12\!\cdots\!29\)\( T^{4} \)
$71$ \( 1 - 483639107104528 T + \)\(58\!\cdots\!42\)\( T^{2} - \)\(14\!\cdots\!48\)\( T^{3} + \)\(87\!\cdots\!81\)\( T^{4} \)
$73$ \( 1 + 2176892348591212 T + \)\(18\!\cdots\!58\)\( T^{2} + \)\(10\!\cdots\!36\)\( T^{3} + \)\(22\!\cdots\!09\)\( T^{4} \)
$79$ \( 1 + 5295839905627744 T + \)\(33\!\cdots\!98\)\( T^{2} + \)\(96\!\cdots\!96\)\( T^{3} + \)\(33\!\cdots\!81\)\( T^{4} \)
$83$ \( 1 - 9972518018887144 T + \)\(81\!\cdots\!34\)\( T^{2} - \)\(41\!\cdots\!12\)\( T^{3} + \)\(17\!\cdots\!29\)\( T^{4} \)
$89$ \( 1 + 711099036813900 T + \)\(23\!\cdots\!82\)\( T^{2} + \)\(98\!\cdots\!00\)\( T^{3} + \)\(19\!\cdots\!41\)\( T^{4} \)
$97$ \( 1 - 114870546609971908 T + \)\(15\!\cdots\!90\)\( T^{2} - \)\(68\!\cdots\!96\)\( T^{3} + \)\(35\!\cdots\!69\)\( T^{4} \)
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