Properties

Label 8.16.a.c
Level 8
Weight 16
Character orbit 8.a
Self dual yes
Analytic conductor 11.415
Analytic rank 0
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 8 = 2^{3} \)
Weight: \( k \) \(=\) \( 16 \)
Character orbit: \([\chi]\) \(=\) 8.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(11.4154804080\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{58}) \)
Defining polynomial: \(x^{2} - 58\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{7}\cdot 5 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 640\sqrt{58}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -2036 + \beta ) q^{3} + ( -70130 + 36 \beta ) q^{5} + ( 63096 + 90 \beta ) q^{7} + ( 13553189 - 4072 \beta ) q^{9} +O(q^{10})\) \( q + ( -2036 + \beta ) q^{3} + ( -70130 + 36 \beta ) q^{5} + ( 63096 + 90 \beta ) q^{7} + ( 13553189 - 4072 \beta ) q^{9} + ( 10341316 + 17307 \beta ) q^{11} + ( 249903238 + 2340 \beta ) q^{13} + ( 998029480 - 143426 \beta ) q^{15} + ( 1569758450 + 175896 \beta ) q^{17} + ( -237334276 + 285813 \beta ) q^{19} + ( 2009648544 - 120144 \beta ) q^{21} + ( -20388001304 - 407250 \beta ) q^{23} + ( 5189451575 - 5049360 \beta ) q^{25} + ( -95117607752 + 7494874 \beta ) q^{27} + ( 17626578678 + 19501524 \beta ) q^{29} + ( -17194596640 - 30813336 \beta ) q^{31} + ( 390104018224 - 24895736 \beta ) q^{33} + ( 72547109520 - 4040244 \beta ) q^{35} + ( 435114282222 + 65307348 \beta ) q^{37} + ( -453212080568 + 245138998 \beta ) q^{39} + ( 450042726042 - 384323472 \beta ) q^{41} + ( 250353999268 - 376557597 \beta ) q^{43} + ( -4433041970170 + 773484164 \beta ) q^{45} + ( 604029559632 - 356149188 \beta ) q^{47} + ( -4551150324727 + 11357280 \beta ) q^{49} + ( 982697888600 + 1211634194 \beta ) q^{51} + ( 618367101022 - 1167361452 \beta ) q^{53} + ( 14076485262520 - 841452534 \beta ) q^{55} + ( 7273214864336 - 819249544 \beta ) q^{57} + ( 7220987952532 + 884641527 \beta ) q^{59} + ( 9168151708630 + 1334372868 \beta ) q^{61} + ( -7851240050856 + 962860098 \beta ) q^{63} + ( -15524441248940 + 8832412368 \beta ) q^{65} + ( -38300710757428 - 8202370383 \beta ) q^{67} + ( 31835013854944 - 19558840304 \beta ) q^{69} + ( -72938586943432 + 1679200362 \beta ) q^{71} + ( -13208634962054 + 17753485176 \beta ) q^{73} + ( -130522359054700 + 15469948535 \beta ) q^{75} + ( 37656800058336 + 2022720912 \beta ) q^{77} + ( 128953916694064 + 12009529764 \beta ) q^{79} + ( 177240223511849 - 51948421912 \beta ) q^{81} + ( 127756403291324 - 35231817627 \beta ) q^{83} + ( 40346979242300 + 44175717720 \beta ) q^{85} + ( 427406091174792 - 22078524186 \beta ) q^{87} + ( -359897305856406 + 61727093496 \beta ) q^{89} + ( 20771076784848 + 22638936060 \beta ) q^{91} + ( -697018061925760 + 45541355456 \beta ) q^{93} + ( 261084334798280 - 28588099626 \beta ) q^{95} + ( 203817795209378 - 259618417416 \beta ) q^{97} + ( -1534081383650476 + 192455203271 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 4072q^{3} - 140260q^{5} + 126192q^{7} + 27106378q^{9} + O(q^{10}) \) \( 2q - 4072q^{3} - 140260q^{5} + 126192q^{7} + 27106378q^{9} + 20682632q^{11} + 499806476q^{13} + 1996058960q^{15} + 3139516900q^{17} - 474668552q^{19} + 4019297088q^{21} - 40776002608q^{23} + 10378903150q^{25} - 190235215504q^{27} + 35253157356q^{29} - 34389193280q^{31} + 780208036448q^{33} + 145094219040q^{35} + 870228564444q^{37} - 906424161136q^{39} + 900085452084q^{41} + 500707998536q^{43} - 8866083940340q^{45} + 1208059119264q^{47} - 9102300649454q^{49} + 1965395777200q^{51} + 1236734202044q^{53} + 28152970525040q^{55} + 14546429728672q^{57} + 14441975905064q^{59} + 18336303417260q^{61} - 15702480101712q^{63} - 31048882497880q^{65} - 76601421514856q^{67} + 63670027709888q^{69} - 145877173886864q^{71} - 26417269924108q^{73} - 261044718109400q^{75} + 75313600116672q^{77} + 257907833388128q^{79} + 354480447023698q^{81} + 255512806582648q^{83} + 80693958484600q^{85} + 854812182349584q^{87} - 719794611712812q^{89} + 41542153569696q^{91} - 1394036123851520q^{93} + 522168669596560q^{95} + 407635590418756q^{97} - 3068162767300952q^{99} + O(q^{100}) \)

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
−7.61577
7.61577
0 −6910.09 0 −245597. 0 −375573. 0 3.34005e7 0
1.2 0 2838.09 0 105337. 0 501765. 0 −6.29412e6 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 8.16.a.c 2
3.b odd 2 1 72.16.a.g 2
4.b odd 2 1 16.16.a.f 2
8.b even 2 1 64.16.a.n 2
8.d odd 2 1 64.16.a.l 2
12.b even 2 1 144.16.a.v 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.16.a.c 2 1.a even 1 1 trivial
16.16.a.f 2 4.b odd 2 1
64.16.a.l 2 8.d odd 2 1
64.16.a.n 2 8.b even 2 1
72.16.a.g 2 3.b odd 2 1
144.16.a.v 2 12.b even 2 1

Atkin-Lehner signs

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 4072 T_{3} - 19611504 \) acting on \(S_{16}^{\mathrm{new}}(\Gamma_0(8))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + 4072 T + 9086310 T^{2} + 58428749304 T^{3} + 205891132094649 T^{4} \)
$5$ \( 1 + 140260 T + 35164560350 T^{2} + 4280395507812500 T^{3} + \)\(93\!\cdots\!25\)\( T^{4} \)
$7$ \( 1 - 126192 T + 9306674045102 T^{2} - 599104282062727056 T^{3} + \)\(22\!\cdots\!49\)\( T^{4} \)
$11$ \( 1 - 20682632 T + 1345511422399958 T^{2} - \)\(86\!\cdots\!32\)\( T^{3} + \)\(17\!\cdots\!01\)\( T^{4} \)
$13$ \( 1 - 499806476 T + 164693331656986158 T^{2} - \)\(25\!\cdots\!32\)\( T^{3} + \)\(26\!\cdots\!49\)\( T^{4} \)
$17$ \( 1 - 3139516900 T + 7453966489546885286 T^{2} - \)\(89\!\cdots\!00\)\( T^{3} + \)\(81\!\cdots\!49\)\( T^{4} \)
$19$ \( 1 + 474668552 T + 28477910697117701574 T^{2} + \)\(72\!\cdots\!48\)\( T^{3} + \)\(23\!\cdots\!01\)\( T^{4} \)
$23$ \( 1 + 40776002608 T + \)\(94\!\cdots\!30\)\( T^{2} + \)\(10\!\cdots\!56\)\( T^{3} + \)\(71\!\cdots\!49\)\( T^{4} \)
$29$ \( 1 - 35253157356 T + \)\(85\!\cdots\!82\)\( T^{2} - \)\(30\!\cdots\!44\)\( T^{3} + \)\(74\!\cdots\!01\)\( T^{4} \)
$31$ \( 1 + 34389193280 T + \)\(24\!\cdots\!02\)\( T^{2} + \)\(80\!\cdots\!80\)\( T^{3} + \)\(55\!\cdots\!01\)\( T^{4} \)
$37$ \( 1 - 870228564444 T + \)\(75\!\cdots\!70\)\( T^{2} - \)\(29\!\cdots\!92\)\( T^{3} + \)\(11\!\cdots\!49\)\( T^{4} \)
$41$ \( 1 - 900085452084 T - \)\(19\!\cdots\!34\)\( T^{2} - \)\(13\!\cdots\!84\)\( T^{3} + \)\(24\!\cdots\!01\)\( T^{4} \)
$43$ \( 1 - 500707998536 T + \)\(30\!\cdots\!38\)\( T^{2} - \)\(15\!\cdots\!52\)\( T^{3} + \)\(10\!\cdots\!49\)\( T^{4} \)
$47$ \( 1 - 1208059119264 T + \)\(21\!\cdots\!10\)\( T^{2} - \)\(14\!\cdots\!52\)\( T^{3} + \)\(14\!\cdots\!49\)\( T^{4} \)
$53$ \( 1 - 1236734202044 T + \)\(11\!\cdots\!98\)\( T^{2} - \)\(90\!\cdots\!08\)\( T^{3} + \)\(53\!\cdots\!49\)\( T^{4} \)
$59$ \( 1 - 14441975905064 T + \)\(76\!\cdots\!22\)\( T^{2} - \)\(52\!\cdots\!36\)\( T^{3} + \)\(13\!\cdots\!01\)\( T^{4} \)
$61$ \( 1 - 18336303417260 T + \)\(12\!\cdots\!02\)\( T^{2} - \)\(11\!\cdots\!60\)\( T^{3} + \)\(36\!\cdots\!01\)\( T^{4} \)
$67$ \( 1 + 76601421514856 T + \)\(47\!\cdots\!70\)\( T^{2} + \)\(18\!\cdots\!08\)\( T^{3} + \)\(60\!\cdots\!49\)\( T^{4} \)
$71$ \( 1 + 145877173886864 T + \)\(16\!\cdots\!26\)\( T^{2} + \)\(85\!\cdots\!64\)\( T^{3} + \)\(34\!\cdots\!01\)\( T^{4} \)
$73$ \( 1 + 26417269924108 T + \)\(10\!\cdots\!30\)\( T^{2} + \)\(23\!\cdots\!56\)\( T^{3} + \)\(79\!\cdots\!49\)\( T^{4} \)
$79$ \( 1 - 257907833388128 T + \)\(71\!\cdots\!94\)\( T^{2} - \)\(75\!\cdots\!72\)\( T^{3} + \)\(84\!\cdots\!01\)\( T^{4} \)
$83$ \( 1 - 255512806582648 T + \)\(10\!\cdots\!90\)\( T^{2} - \)\(15\!\cdots\!36\)\( T^{3} + \)\(37\!\cdots\!49\)\( T^{4} \)
$89$ \( 1 + 719794611712812 T + \)\(38\!\cdots\!34\)\( T^{2} + \)\(12\!\cdots\!88\)\( T^{3} + \)\(30\!\cdots\!01\)\( T^{4} \)
$97$ \( 1 - 407635590418756 T - \)\(29\!\cdots\!30\)\( T^{2} - \)\(25\!\cdots\!08\)\( T^{3} + \)\(40\!\cdots\!49\)\( T^{4} \)
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