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: \(\Q(\sqrt{781}) \)
Defining polynomial: \(x^{2} - x - 195\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{7} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 64\sqrt{781}\). 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})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 872q^{3} + 18476q^{5} + 110928q^{7} + 3589498q^{9} + O(q^{10}) \) \( 2q + 872q^{3} + 18476q^{5} + 110928q^{7} + 3589498q^{9} + 16474040q^{11} + 18744572q^{13} + 84830960q^{15} - 153793628q^{17} - 118747640q^{19} - 1371980736q^{21} + 718268912q^{23} - 1349419874q^{25} + 5753785616q^{27} + 309341340q^{29} + 5767504192q^{31} + 10579993952q^{33} - 16019391264q^{35} - 11621553300q^{37} - 58698760912q^{39} + 1311168276q^{41} - 29595620104q^{43} + 100107952252q^{45} + 12313617888q^{47} + 127691156146q^{49} + 177245377360q^{51} - 38006007028q^{53} + 192954931664q^{55} - 652023429728q^{57} + 253345911704q^{59} - 647244384292q^{61} - 1039453222896q^{63} - 629294375512q^{65} + 1619993806312q^{67} + 1026293771456q^{69} - 1040270142512q^{71} + 4005283908692q^{73} + 830155668760q^{75} + 159512776896q^{77} - 2521777572064q^{79} + 500597403058q^{81} - 290486230904q^{83} + 1510847254552q^{85} - 15058906809936q^{87} - 8723755657740q^{89} + 15885098476896q^{91} - 6973121519360q^{93} - 8299984202576q^{95} + 9601712299972q^{97} + 32529223326424q^{99} + O(q^{100}) \)

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
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.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8.14.a.b 2
3.b odd 2 1 72.14.a.c 2
4.b odd 2 1 16.14.a.e 2
5.b even 2 1 200.14.a.b 2
5.c odd 4 2 200.14.c.b 4
8.b even 2 1 64.14.a.j 2
8.d odd 2 1 64.14.a.l 2
12.b even 2 1 144.14.a.n 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.14.a.b 2 1.a even 1 1 trivial
16.14.a.e 2 4.b odd 2 1
64.14.a.j 2 8.b even 2 1
64.14.a.l 2 8.d odd 2 1
72.14.a.c 2 3.b odd 2 1
144.14.a.n 2 12.b even 2 1
200.14.a.b 2 5.b even 2 1
200.14.c.b 4 5.c odd 4 2

Atkin-Lehner signs

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 872 T_{3} - 3008880 \) acting on \(S_{14}^{\mathrm{new}}(\Gamma_0(8))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - 872 T + 179766 T^{2} - 1390249656 T^{3} + 2541865828329 T^{4} \)
$5$ \( 1 - 18476 T + 2066094350 T^{2} - 22553710937500 T^{3} + 1490116119384765625 T^{4} \)
$7$ \( 1 - 110928 T + 39195942926 T^{2} - 10747704146427696 T^{3} + \)\(93\!\cdots\!49\)\( T^{4} \)
$11$ \( 1 - 16474040 T + 135991936296326 T^{2} - \)\(56\!\cdots\!40\)\( T^{3} + \)\(11\!\cdots\!61\)\( T^{4} \)
$13$ \( 1 - 18744572 T + 344120051417598 T^{2} - \)\(56\!\cdots\!16\)\( T^{3} + \)\(91\!\cdots\!09\)\( T^{4} \)
$17$ \( 1 + 153793628 T + 21058111940247014 T^{2} + \)\(15\!\cdots\!36\)\( T^{3} + \)\(98\!\cdots\!69\)\( T^{4} \)
$19$ \( 1 + 118747640 T + 59473815443581782 T^{2} + \)\(49\!\cdots\!60\)\( T^{3} + \)\(17\!\cdots\!81\)\( T^{4} \)
$23$ \( 1 - 718268912 T + 1097306915486653358 T^{2} - \)\(36\!\cdots\!96\)\( T^{3} + \)\(25\!\cdots\!89\)\( T^{4} \)
$29$ \( 1 - 309341340 T + 2504177667132675934 T^{2} - \)\(31\!\cdots\!60\)\( T^{3} + \)\(10\!\cdots\!21\)\( T^{4} \)
$31$ \( 1 - 5767504192 T + 50116253247327696702 T^{2} - \)\(14\!\cdots\!72\)\( T^{3} + \)\(59\!\cdots\!81\)\( T^{4} \)
$37$ \( 1 + 11621553300 T + \)\(18\!\cdots\!38\)\( T^{2} + \)\(28\!\cdots\!00\)\( T^{3} + \)\(59\!\cdots\!09\)\( T^{4} \)
$41$ \( 1 - 1311168276 T + \)\(16\!\cdots\!30\)\( T^{2} - \)\(12\!\cdots\!96\)\( T^{3} + \)\(85\!\cdots\!41\)\( T^{4} \)
$43$ \( 1 + 29595620104 T + \)\(33\!\cdots\!74\)\( T^{2} + \)\(50\!\cdots\!72\)\( T^{3} + \)\(29\!\cdots\!49\)\( T^{4} \)
$47$ \( 1 - 12313617888 T + \)\(10\!\cdots\!90\)\( T^{2} - \)\(67\!\cdots\!76\)\( T^{3} + \)\(29\!\cdots\!29\)\( T^{4} \)
$53$ \( 1 + 38006007028 T + \)\(44\!\cdots\!42\)\( T^{2} + \)\(98\!\cdots\!44\)\( T^{3} + \)\(67\!\cdots\!29\)\( T^{4} \)
$59$ \( 1 - 253345911704 T + \)\(16\!\cdots\!06\)\( T^{2} - \)\(26\!\cdots\!16\)\( T^{3} + \)\(11\!\cdots\!41\)\( T^{4} \)
$61$ \( 1 + 647244384292 T + \)\(42\!\cdots\!62\)\( T^{2} + \)\(10\!\cdots\!52\)\( T^{3} + \)\(26\!\cdots\!61\)\( T^{4} \)
$67$ \( 1 - 1619993806312 T + \)\(17\!\cdots\!86\)\( T^{2} - \)\(88\!\cdots\!44\)\( T^{3} + \)\(30\!\cdots\!69\)\( T^{4} \)
$71$ \( 1 + 1040270142512 T + \)\(22\!\cdots\!82\)\( T^{2} + \)\(12\!\cdots\!32\)\( T^{3} + \)\(13\!\cdots\!21\)\( T^{4} \)
$73$ \( 1 - 4005283908692 T + \)\(73\!\cdots\!18\)\( T^{2} - \)\(66\!\cdots\!36\)\( T^{3} + \)\(27\!\cdots\!89\)\( T^{4} \)
$79$ \( 1 + 2521777572064 T + \)\(10\!\cdots\!58\)\( T^{2} + \)\(11\!\cdots\!96\)\( T^{3} + \)\(21\!\cdots\!21\)\( T^{4} \)
$83$ \( 1 + 290486230904 T + \)\(14\!\cdots\!54\)\( T^{2} + \)\(25\!\cdots\!52\)\( T^{3} + \)\(78\!\cdots\!69\)\( T^{4} \)
$89$ \( 1 + 8723755657740 T + \)\(61\!\cdots\!42\)\( T^{2} + \)\(19\!\cdots\!60\)\( T^{3} + \)\(48\!\cdots\!61\)\( T^{4} \)
$97$ \( 1 - 9601712299972 T + \)\(81\!\cdots\!50\)\( T^{2} - \)\(64\!\cdots\!44\)\( T^{3} + \)\(45\!\cdots\!29\)\( T^{4} \)
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