Properties

Label 8.14.a.b
Level 8
Weight 14
Character orbit 8.a
Self dual Yes
Analytic conductor 8.578
Analytic rank 0
Dimension 2
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 8 = 2^{3} \)
Weight: \( k \) = \( 14 \)
Character orbit: \([\chi]\) = 8.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(8.57847431615\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{781}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{7} \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 64\sqrt{781}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( 436 - \beta ) q^{3} \) \( + ( 9238 - 12 \beta ) q^{5} \) \( + ( 55464 + 222 \beta ) q^{7} \) \( + ( 1794749 - 872 \beta ) q^{9} \) \(+O(q^{10})\) \( q\) \( + ( 436 - \beta ) q^{3} \) \( + ( 9238 - 12 \beta ) q^{5} \) \( + ( 55464 + 222 \beta ) q^{7} \) \( + ( 1794749 - 872 \beta ) q^{9} \) \( + ( 8237020 - 531 \beta ) q^{11} \) \( + ( 9372286 + 10452 \beta ) q^{13} \) \( + ( 42415480 - 14470 \beta ) q^{15} \) \( + ( -76896814 - 38184 \beta ) q^{17} \) \( + ( -59373820 + 93819 \beta ) q^{19} \) \( + ( -685990368 + 41328 \beta ) q^{21} \) \( + ( 359134456 - 111462 \beta ) q^{23} \) \( + ( -674709937 - 221712 \beta ) q^{25} \) \( + ( 2876892808 - 580618 \beta ) q^{27} \) \( + ( 154670670 + 2374788 \beta ) q^{29} \) \( + ( 2883752096 + 1482936 \beta ) q^{31} \) \( + ( 5289996976 - 8468536 \beta ) q^{33} \) \( + ( -8009695632 + 1385268 \beta ) q^{35} \) \( + ( -5810776650 + 10246116 \beta ) q^{37} \) \( + ( -29349380456 - 4815214 \beta ) q^{39} \) \( + ( 655584138 + 8079216 \beta ) q^{41} \) \( + ( -14797810052 - 9339771 \beta ) q^{43} \) \( + ( 50053976126 - 29592524 \beta ) q^{45} \) \( + ( 6156808944 + 16143540 \beta ) q^{47} \) \( + ( 63845578073 + 24626016 \beta ) q^{49} \) \( + ( 88622688680 + 60248590 \beta ) q^{51} \) \( + ( -19003003514 - 50827740 \beta ) q^{53} \) \( + ( 96477465832 - 103749618 \beta ) q^{55} \) \( + ( -326011714864 + 100278904 \beta ) q^{57} \) \( + ( 126672955852 - 136967391 \beta ) q^{59} \) \( + ( -323622192146 + 14150004 \beta ) q^{61} \) \( + ( -519726611448 + 350069670 \beta ) q^{63} \) \( + ( -314647187756 - 15911856 \beta ) q^{65} \) \( + ( 809996903156 - 36810993 \beta ) q^{67} \) \( + ( 513146885728 - 407731888 \beta ) q^{69} \) \( + ( -520135071256 - 352709874 \beta ) q^{71} \) \( + ( 2002641954346 - 86109192 \beta ) q^{73} \) \( + ( 415077834380 + 578043505 \beta ) q^{75} \) \( + ( 79756388448 + 1799167056 \beta ) q^{77} \) \( + ( -1260888786032 - 525520788 \beta ) q^{79} \) \( + ( 250298701529 - 1739792600 \beta ) q^{81} \) \( + ( -145243115452 - 996493749 \beta ) q^{83} \) \( + ( 755423627276 + 570017976 \beta ) q^{85} \) \( + ( -7529453404968 + 880736898 \beta ) q^{87} \) \( + ( -4361877828870 - 639382536 \beta ) q^{89} \) \( + ( 7942549238448 + 2660357220 \beta ) q^{91} \) \( + ( -3486560759680 - 2237192000 \beta ) q^{93} \) \( + ( -4149992101288 + 1579185762 \beta ) q^{95} \) \( + ( 4800856149986 + 4881770040 \beta ) q^{97} \) \( + ( 16264611663212 - 8135693159 \beta ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut +\mathstrut 872q^{3} \) \(\mathstrut +\mathstrut 18476q^{5} \) \(\mathstrut +\mathstrut 110928q^{7} \) \(\mathstrut +\mathstrut 3589498q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut +\mathstrut 872q^{3} \) \(\mathstrut +\mathstrut 18476q^{5} \) \(\mathstrut +\mathstrut 110928q^{7} \) \(\mathstrut +\mathstrut 3589498q^{9} \) \(\mathstrut +\mathstrut 16474040q^{11} \) \(\mathstrut +\mathstrut 18744572q^{13} \) \(\mathstrut +\mathstrut 84830960q^{15} \) \(\mathstrut -\mathstrut 153793628q^{17} \) \(\mathstrut -\mathstrut 118747640q^{19} \) \(\mathstrut -\mathstrut 1371980736q^{21} \) \(\mathstrut +\mathstrut 718268912q^{23} \) \(\mathstrut -\mathstrut 1349419874q^{25} \) \(\mathstrut +\mathstrut 5753785616q^{27} \) \(\mathstrut +\mathstrut 309341340q^{29} \) \(\mathstrut +\mathstrut 5767504192q^{31} \) \(\mathstrut +\mathstrut 10579993952q^{33} \) \(\mathstrut -\mathstrut 16019391264q^{35} \) \(\mathstrut -\mathstrut 11621553300q^{37} \) \(\mathstrut -\mathstrut 58698760912q^{39} \) \(\mathstrut +\mathstrut 1311168276q^{41} \) \(\mathstrut -\mathstrut 29595620104q^{43} \) \(\mathstrut +\mathstrut 100107952252q^{45} \) \(\mathstrut +\mathstrut 12313617888q^{47} \) \(\mathstrut +\mathstrut 127691156146q^{49} \) \(\mathstrut +\mathstrut 177245377360q^{51} \) \(\mathstrut -\mathstrut 38006007028q^{53} \) \(\mathstrut +\mathstrut 192954931664q^{55} \) \(\mathstrut -\mathstrut 652023429728q^{57} \) \(\mathstrut +\mathstrut 253345911704q^{59} \) \(\mathstrut -\mathstrut 647244384292q^{61} \) \(\mathstrut -\mathstrut 1039453222896q^{63} \) \(\mathstrut -\mathstrut 629294375512q^{65} \) \(\mathstrut +\mathstrut 1619993806312q^{67} \) \(\mathstrut +\mathstrut 1026293771456q^{69} \) \(\mathstrut -\mathstrut 1040270142512q^{71} \) \(\mathstrut +\mathstrut 4005283908692q^{73} \) \(\mathstrut +\mathstrut 830155668760q^{75} \) \(\mathstrut +\mathstrut 159512776896q^{77} \) \(\mathstrut -\mathstrut 2521777572064q^{79} \) \(\mathstrut +\mathstrut 500597403058q^{81} \) \(\mathstrut -\mathstrut 290486230904q^{83} \) \(\mathstrut +\mathstrut 1510847254552q^{85} \) \(\mathstrut -\mathstrut 15058906809936q^{87} \) \(\mathstrut -\mathstrut 8723755657740q^{89} \) \(\mathstrut +\mathstrut 15885098476896q^{91} \) \(\mathstrut -\mathstrut 6973121519360q^{93} \) \(\mathstrut -\mathstrut 8299984202576q^{95} \) \(\mathstrut +\mathstrut 9601712299972q^{97} \) \(\mathstrut +\mathstrut 32529223326424q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
14.4732
−13.4732
0 −1352.57 0 −12224.8 0 452526. 0 235118. 0
1.2 0 2224.57 0 30700.8 0 −341598. 0 3.35438e6 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 \(T_{3}^{2} \) \(\mathstrut -\mathstrut 872 T_{3} \) \(\mathstrut -\mathstrut 3008880 \) acting on \(S_{14}^{\mathrm{new}}(\Gamma_0(8))\).