Properties

Label 8.20.a.b
Level 8
Weight 20
Character orbit 8.a
Self dual Yes
Analytic conductor 18.305
Analytic rank 0
Dimension 3
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 8 = 2^{3} \)
Weight: \( k \) = \( 20 \)
Character orbit: \([\chi]\) = 8.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(18.3053357245\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{21}\cdot 3^{2} \)
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.

\(f(q)\) \(=\) \( q\) \( + ( 7911 - \beta_{1} ) q^{3} \) \( + ( 713429 - 70 \beta_{1} + \beta_{2} ) q^{5} \) \( + ( 18618154 - 2722 \beta_{1} - 20 \beta_{2} ) q^{7} \) \( + ( 215591919 - 21956 \beta_{1} + 150 \beta_{2} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + ( 7911 - \beta_{1} ) q^{3} \) \( + ( 713429 - 70 \beta_{1} + \beta_{2} ) q^{5} \) \( + ( 18618154 - 2722 \beta_{1} - 20 \beta_{2} ) q^{7} \) \( + ( 215591919 - 21956 \beta_{1} + 150 \beta_{2} ) q^{9} \) \( + ( -99151979 + 63333 \beta_{1} - 360 \beta_{2} ) q^{11} \) \( + ( -4954589763 + 1369882 \beta_{1} - 1615 \beta_{2} ) q^{13} \) \( + ( 97547157054 - 5829910 \beta_{1} + 12276 \beta_{2} ) q^{15} \) \( + ( 267776864140 + 1887484 \beta_{1} - 15370 \beta_{2} ) q^{17} \) \( + ( 1070753896939 + 8077147 \beta_{1} - 101080 \beta_{2} ) q^{19} \) \( + ( 3730764546324 + 25817976 \beta_{1} + 372780 \beta_{2} ) q^{21} \) \( + ( 8316140088174 + 88928218 \beta_{1} + 112820 \beta_{2} ) q^{23} \) \( + ( 24113633854987 - 538445480 \beta_{1} - 2830372 \beta_{2} ) q^{25} \) \( + ( 21364107232062 + 18297878 \beta_{1} + 3559800 \beta_{2} ) q^{27} \) \( + ( 25888630013633 + 1239207634 \beta_{1} + 10262525 \beta_{2} ) q^{29} \) \( + ( -82810767991560 + 597433112 \beta_{1} - 28652000 \beta_{2} ) q^{31} \) \( + ( -84024721099314 + 2476663124 \beta_{1} - 10139310 \beta_{2} ) q^{33} \) \( + ( -461454980102684 - 13281410860 \beta_{1} + 107486704 \beta_{2} ) q^{35} \) \( + ( -138286237201687 - 7528018766 \beta_{1} - 62935315 \beta_{2} ) q^{37} \) \( + ( -1840692212529498 + 30869912018 \beta_{1} - 208350540 \beta_{2} ) q^{39} \) \( + ( 939574252603102 + 14862604952 \beta_{1} + 260455420 \beta_{2} ) q^{41} \) \( + ( -2221090480684363 + 40602131309 \beta_{1} + 124451920 \beta_{2} ) q^{43} \) \( + ( 7608373704953061 - 148810711670 \beta_{1} - 265972791 \beta_{2} ) q^{45} \) \( + ( 500210422580428 - 133848474700 \beta_{1} + 225462040 \beta_{2} ) q^{47} \) \( + ( 13213584127233233 + 371294308848 \beta_{1} - 618686760 \beta_{2} ) q^{49} \) \( + ( -361621376282970 - 177737853890 \beta_{1} - 310419720 \beta_{2} ) q^{51} \) \( + ( 18689081009440473 + 99868809010 \beta_{1} + 1877847725 \beta_{2} ) q^{53} \) \( + ( -18939413522672806 + 383997647150 \beta_{1} + 1010165836 \beta_{2} ) q^{55} \) \( + ( -2136147609531726 - 539513269844 \beta_{1} - 1391090130 \beta_{2} ) q^{57} \) \( + ( -51439318139589743 + 691356272217 \beta_{1} - 7788999840 \beta_{2} ) q^{59} \) \( + ( -44997198970745035 - 2780969311222 \beta_{1} + 1309891465 \beta_{2} ) q^{61} \) \( + ( -26144428277991294 - 1745298490410 \beta_{1} + 20034590220 \beta_{2} ) q^{63} \) \( + ( -187966850995212512 + 7766979173640 \beta_{1} - 4717422828 \beta_{2} ) q^{65} \) \( + ( 50342944062644463 + 3361346559743 \beta_{1} - 11630042840 \beta_{2} ) q^{67} \) \( + ( -51194273240697156 - 7533465669784 \beta_{1} - 13138864380 \beta_{2} ) q^{69} \) \( + ( 403514856373642538 - 2684735998098 \beta_{1} - 35539345380 \beta_{2} ) q^{71} \) \( + ( -27293891104567556 + 5474791886508 \beta_{1} + 74922153390 \beta_{2} ) q^{73} \) \( + ( 899432685208461537 - 19977236292455 \beta_{1} + 75740081328 \beta_{2} ) q^{75} \) \( + ( 32410846285929244 + 2546433188264 \beta_{1} - 62097485660 \beta_{2} ) q^{77} \) \( + ( 479664642636798084 + 34669544655436 \beta_{1} - 114419142280 \beta_{2} ) q^{79} \) \( + ( -106219021458074961 - 10302332459900 \beta_{1} - 170761696950 \beta_{2} ) q^{81} \) \( + ( -327821367763737149 + 25744231091131 \beta_{1} + 256261270400 \beta_{2} ) q^{83} \) \( + ( -539510402881407450 - 6954020917540 \beta_{1} + 318702521350 \beta_{2} ) q^{85} \) \( + ( -1426786918155990522 - 50902371514878 \beta_{1} - 167654900700 \beta_{2} ) q^{87} \) \( + ( -437597677067787636 - 64495714179348 \beta_{1} - 1914528690 \beta_{2} ) q^{89} \) \( + ( -3829371634623831212 - 27713537850460 \beta_{1} - 764920780400 \beta_{2} ) q^{91} \) \( + ( -1436155531565376240 + 209629915861600 \beta_{1} - 140500918800 \beta_{2} ) q^{93} \) \( + ( -3642044827469186394 - 20290339711150 \beta_{1} + 1424589549364 \beta_{2} ) q^{95} \) \( + ( 4677769063243700932 - 61680013256660 \beta_{1} - 97149847570 \beta_{2} ) q^{97} \) \( + ( -3805278927355985871 + 87109073527993 \beta_{1} + 28907244960 \beta_{2} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(3q \) \(\mathstrut +\mathstrut 23732q^{3} \) \(\mathstrut +\mathstrut 2140218q^{5} \) \(\mathstrut +\mathstrut 55851720q^{7} \) \(\mathstrut +\mathstrut 646753951q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(3q \) \(\mathstrut +\mathstrut 23732q^{3} \) \(\mathstrut +\mathstrut 2140218q^{5} \) \(\mathstrut +\mathstrut 55851720q^{7} \) \(\mathstrut +\mathstrut 646753951q^{9} \) \(\mathstrut -\mathstrut 297392964q^{11} \) \(\mathstrut -\mathstrut 14862401022q^{13} \) \(\mathstrut +\mathstrut 292635653528q^{15} \) \(\mathstrut +\mathstrut 803332464534q^{17} \) \(\mathstrut +\mathstrut 3212269666884q^{19} \) \(\mathstrut +\mathstrut 11192319829728q^{21} \) \(\mathstrut +\mathstrut 24948509305560q^{23} \) \(\mathstrut +\mathstrut 72340360289109q^{25} \) \(\mathstrut +\mathstrut 64092343553864q^{27} \) \(\mathstrut +\mathstrut 77667139511058q^{29} \) \(\mathstrut -\mathstrut 248431735193568q^{31} \) \(\mathstrut -\mathstrut 252071696774128q^{33} \) \(\mathstrut -\mathstrut 1384378114232208q^{35} \) \(\mathstrut -\mathstrut 414866302559142q^{37} \) \(\mathstrut -\mathstrut 5522045976027016q^{39} \) \(\mathstrut +\mathstrut 2818737880869678q^{41} \) \(\mathstrut -\mathstrut 6663230715469860q^{43} \) \(\mathstrut +\mathstrut 22824972038174722q^{45} \) \(\mathstrut +\mathstrut 1500497644728624q^{47} \) \(\mathstrut +\mathstrut 39641123057321787q^{49} \) \(\mathstrut -\mathstrut 1085042177122520q^{51} \) \(\mathstrut +\mathstrut 56067344774978154q^{53} \) \(\mathstrut -\mathstrut 56817855560205432q^{55} \) \(\mathstrut -\mathstrut 6408983732955152q^{57} \) \(\mathstrut -\mathstrut 154317270851496852q^{59} \) \(\mathstrut -\mathstrut 134994376571654862q^{61} \) \(\mathstrut -\mathstrut 78435010097874072q^{63} \) \(\mathstrut -\mathstrut 563892790723886724q^{65} \) \(\mathstrut +\mathstrut 151032181904450292q^{67} \) \(\mathstrut -\mathstrut 153590366326625632q^{69} \) \(\mathstrut +\mathstrut 1210541848845584136q^{71} \) \(\mathstrut -\mathstrut 81876123599662770q^{73} \) \(\mathstrut +\mathstrut 2698278154129173484q^{75} \) \(\mathstrut +\mathstrut 97235023193490336q^{77} \) \(\mathstrut +\mathstrut 1439028483035907408q^{79} \) \(\mathstrut -\mathstrut 318667537468381733q^{81} \) \(\mathstrut -\mathstrut 983438102798849916q^{83} \) \(\mathstrut -\mathstrut 1618537843962618540q^{85} \) \(\mathstrut -\mathstrut 4280411824494387144q^{87} \) \(\mathstrut -\mathstrut 1312857528832070946q^{89} \) \(\mathstrut -\mathstrut 11488143382330124496q^{91} \) \(\mathstrut -\mathstrut 4308257105281185920q^{93} \) \(\mathstrut -\mathstrut 10926153348157720968q^{95} \) \(\mathstrut +\mathstrut 14033245412567998566q^{97} \) \(\mathstrut -\mathstrut 11415749644087184660q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3}\mathstrut -\mathstrut \) \(x^{2}\mathstrut -\mathstrut \) \(2519\) \(x\mathstrut +\mathstrut \) \(43659\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 64 \nu^{2} + 1664 \nu - 108053 \)
\(\beta_{2}\)\(=\)\( -896 \nu^{2} + 124160 \nu + 1463595 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2}\mathstrut +\mathstrut \) \(14\) \(\beta_{1}\mathstrut +\mathstrut \) \(49147\)\()/147456\)
\(\nu^{2}\)\(=\)\((\)\(-\)\(13\) \(\beta_{2}\mathstrut +\mathstrut \) \(970\) \(\beta_{1}\mathstrut +\mathstrut \) \(123838145\)\()/73728\)

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
37.1696
−56.8359
20.6663
0 −34307.5 0 2.59881e6 0 −1.93114e8 0 1.47441e7 0
1.2 0 3798.34 0 −8.06198e6 0 1.77174e8 0 −1.14783e9 0
1.3 0 54241.2 0 7.60339e6 0 7.17920e7 0 1.77984e9 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 \(T_{3}^{3} \) \(\mathstrut -\mathstrut 23732 T_{3}^{2} \) \(\mathstrut -\mathstrut 1785165264 T_{3} \) \(\mathstrut +\mathstrut \)\(70\!\cdots\!20\)\( \) acting on \(S_{20}^{\mathrm{new}}(\Gamma_0(8))\).