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

Related objects

Downloads

Learn more about

Newspace parameters

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

Newform invariants

Self dual: Yes
Analytic conductor: \(11.415480408\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{58}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{7}\cdot 5 \)
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.

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

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
−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.

Atkin-Lehner signs

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

Hecke kernels

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