Properties

Label 138.8.a.d
Level $138$
Weight $8$
Character orbit 138.a
Self dual yes
Analytic conductor $43.109$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 138 = 2 \cdot 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 138.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(43.1091335168\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
Defining polynomial: \( x^{3} - x^{2} - 684x - 5052 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{3}\cdot 3 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + 8 q^{2} - 27 q^{3} + 64 q^{4} + (\beta_{2} + 31) q^{5} - 216 q^{6} - 286 q^{7} + 512 q^{8} + 729 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 8 q^{2} - 27 q^{3} + 64 q^{4} + (\beta_{2} + 31) q^{5} - 216 q^{6} - 286 q^{7} + 512 q^{8} + 729 q^{9} + (8 \beta_{2} + 248) q^{10} + ( - 21 \beta_{2} - 14 \beta_1 - 609) q^{11} - 1728 q^{12} + ( - 52 \beta_{2} + 54 \beta_1 - 1268) q^{13} - 2288 q^{14} + ( - 27 \beta_{2} - 837) q^{15} + 4096 q^{16} + (23 \beta_{2} - 131 \beta_1 + 1106) q^{17} + 5832 q^{18} + (196 \beta_{2} + 65 \beta_1 - 4701) q^{19} + (64 \beta_{2} + 1984) q^{20} + 7722 q^{21} + ( - 168 \beta_{2} - 112 \beta_1 - 4872) q^{22} - 12167 q^{23} - 13824 q^{24} + (27 \beta_{2} + 77 \beta_1 - 56723) q^{25} + ( - 416 \beta_{2} + 432 \beta_1 - 10144) q^{26} - 19683 q^{27} - 18304 q^{28} + (545 \beta_{2} - 105 \beta_1 - 57088) q^{29} + ( - 216 \beta_{2} - 6696) q^{30} + ( - 233 \beta_{2} - 51 \beta_1 - 41718) q^{31} + 32768 q^{32} + (567 \beta_{2} + 378 \beta_1 + 16443) q^{33} + (184 \beta_{2} - 1048 \beta_1 + 8848) q^{34} + ( - 286 \beta_{2} - 8866) q^{35} + 46656 q^{36} + (733 \beta_{2} - 38 \beta_1 - 226101) q^{37} + (1568 \beta_{2} + 520 \beta_1 - 37608) q^{38} + (1404 \beta_{2} - 1458 \beta_1 + 34236) q^{39} + (512 \beta_{2} + 15872) q^{40} + ( - 2369 \beta_{2} - 77 \beta_1 - 111218) q^{41} + 61776 q^{42} + ( - 5078 \beta_{2} + 1613 \beta_1 - 50871) q^{43} + ( - 1344 \beta_{2} - 896 \beta_1 - 38976) q^{44} + (729 \beta_{2} + 22599) q^{45} - 97336 q^{46} + (4507 \beta_{2} + 2371 \beta_1 - 607352) q^{47} - 110592 q^{48} - 741747 q^{49} + (216 \beta_{2} + 616 \beta_1 - 453784) q^{50} + ( - 621 \beta_{2} + 3537 \beta_1 - 29862) q^{51} + ( - 3328 \beta_{2} + 3456 \beta_1 - 81152) q^{52} + (285 \beta_{2} - 5072 \beta_1 - 1145697) q^{53} - 157464 q^{54} + ( - 2177 \beta_{2} - 2527 \beta_1 - 515942) q^{55} - 146432 q^{56} + ( - 5292 \beta_{2} - 1755 \beta_1 + 126927) q^{57} + (4360 \beta_{2} - 840 \beta_1 - 456704) q^{58} + (3600 \beta_{2} - 2016 \beta_1 - 1047348) q^{59} + ( - 1728 \beta_{2} - 53568) q^{60} + ( - 3063 \beta_{2} - 7700 \beta_1 - 1146631) q^{61} + ( - 1864 \beta_{2} - 408 \beta_1 - 333744) q^{62} - 208494 q^{63} + 262144 q^{64} + (5312 \beta_{2} - 494 \beta_1 - 840718) q^{65} + (4536 \beta_{2} + 3024 \beta_1 + 131544) q^{66} + (21354 \beta_{2} + 4995 \beta_1 - 301837) q^{67} + (1472 \beta_{2} - 8384 \beta_1 + 70784) q^{68} + 328509 q^{69} + ( - 2288 \beta_{2} - 70928) q^{70} + ( - 18009 \beta_{2} + 23895 \beta_1 + 850872) q^{71} + 373248 q^{72} + ( - 24711 \beta_{2} + 13283 \beta_1 - 192940) q^{73} + (5864 \beta_{2} - 304 \beta_1 - 1808808) q^{74} + ( - 729 \beta_{2} - 2079 \beta_1 + 1531521) q^{75} + (12544 \beta_{2} + 4160 \beta_1 - 300864) q^{76} + (6006 \beta_{2} + 4004 \beta_1 + 174174) q^{77} + (11232 \beta_{2} - 11664 \beta_1 + 273888) q^{78} + (42994 \beta_{2} - 6150 \beta_1 + 122046) q^{79} + (4096 \beta_{2} + 126976) q^{80} + 531441 q^{81} + ( - 18952 \beta_{2} - 616 \beta_1 - 889744) q^{82} + ( - 30495 \beta_{2} - 31190 \beta_1 + 1373337) q^{83} + 494208 q^{84} + ( - 14444 \beta_{2} - 6744 \beta_1 - 130004) q^{85} + ( - 40624 \beta_{2} + 12904 \beta_1 - 406968) q^{86} + ( - 14715 \beta_{2} + 2835 \beta_1 + 1541376) q^{87} + ( - 10752 \beta_{2} - 7168 \beta_1 - 311808) q^{88} + (30741 \beta_{2} - 11019 \beta_1 + 486312) q^{89} + (5832 \beta_{2} + 180792) q^{90} + (14872 \beta_{2} - 15444 \beta_1 + 362648) q^{91} - 778688 q^{92} + (6291 \beta_{2} + 1377 \beta_1 + 1126386) q^{93} + (36056 \beta_{2} + 18968 \beta_1 - 4858816) q^{94} + (2185 \beta_{2} + 19317 \beta_1 + 4175500) q^{95} - 884736 q^{96} + ( - 16514 \beta_{2} + 14628 \beta_1 - 5040156) q^{97} - 5933976 q^{98} + ( - 15309 \beta_{2} - 10206 \beta_1 - 443961) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 24 q^{2} - 81 q^{3} + 192 q^{4} + 92 q^{5} - 648 q^{6} - 858 q^{7} + 1536 q^{8} + 2187 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 24 q^{2} - 81 q^{3} + 192 q^{4} + 92 q^{5} - 648 q^{6} - 858 q^{7} + 1536 q^{8} + 2187 q^{9} + 736 q^{10} - 1820 q^{11} - 5184 q^{12} - 3698 q^{13} - 6864 q^{14} - 2484 q^{15} + 12288 q^{16} + 3164 q^{17} + 17496 q^{18} - 14234 q^{19} + 5888 q^{20} + 23166 q^{21} - 14560 q^{22} - 36501 q^{23} - 41472 q^{24} - 170119 q^{25} - 29584 q^{26} - 59049 q^{27} - 54912 q^{28} - 171914 q^{29} - 19872 q^{30} - 124972 q^{31} + 98304 q^{32} + 49140 q^{33} + 25312 q^{34} - 26312 q^{35} + 139968 q^{36} - 679074 q^{37} - 113872 q^{38} + 99846 q^{39} + 47104 q^{40} - 331362 q^{41} + 185328 q^{42} - 145922 q^{43} - 116480 q^{44} + 67068 q^{45} - 292008 q^{46} - 1824192 q^{47} - 331776 q^{48} - 2225241 q^{49} - 1360952 q^{50} - 85428 q^{51} - 236672 q^{52} - 3442448 q^{53} - 472392 q^{54} - 1548176 q^{55} - 439296 q^{56} + 384318 q^{57} - 1375312 q^{58} - 3147660 q^{59} - 158976 q^{60} - 3444530 q^{61} - 999776 q^{62} - 625482 q^{63} + 786432 q^{64} - 2527960 q^{65} + 393120 q^{66} - 921870 q^{67} + 202496 q^{68} + 985527 q^{69} - 210496 q^{70} + 2594520 q^{71} + 1119744 q^{72} - 540826 q^{73} - 5432592 q^{74} + 4593213 q^{75} - 910976 q^{76} + 520520 q^{77} + 798768 q^{78} + 316994 q^{79} + 376832 q^{80} + 1594323 q^{81} - 2650896 q^{82} + 4119316 q^{83} + 1482624 q^{84} - 382312 q^{85} - 1167376 q^{86} + 4641678 q^{87} - 931840 q^{88} + 1417176 q^{89} + 536544 q^{90} + 1057628 q^{91} - 2336064 q^{92} + 3374244 q^{93} - 14593536 q^{94} + 12543632 q^{95} - 2654208 q^{96} - 15089326 q^{97} - 17801928 q^{98} - 1326780 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 684x - 5052 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{2} + 27\nu + 448 ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} - 3\nu - 456 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + \beta _1 + 4 ) / 12 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 9\beta_{2} + \beta _1 + 1828 ) / 4 \) Copy content Toggle raw display

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
−8.33366
−20.3930
29.7267
8.00000 −27.0000 64.0000 −149.775 −216.000 −286.000 512.000 729.000 −1198.20
1.2 8.00000 −27.0000 64.0000 41.5271 −216.000 −286.000 512.000 729.000 332.217
1.3 8.00000 −27.0000 64.0000 200.248 −216.000 −286.000 512.000 729.000 1601.98
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(23\) \(1\)

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 138.8.a.d 3
3.b odd 2 1 414.8.a.c 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
138.8.a.d 3 1.a even 1 1 trivial
414.8.a.c 3 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{3} - 92T_{5}^{2} - 27896T_{5} + 1245480 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(138))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 8)^{3} \) Copy content Toggle raw display
$3$ \( (T + 27)^{3} \) Copy content Toggle raw display
$5$ \( T^{3} - 92 T^{2} - 27896 T + 1245480 \) Copy content Toggle raw display
$7$ \( (T + 286)^{3} \) Copy content Toggle raw display
$11$ \( T^{3} + 1820 T^{2} + \cdots + 39808807752 \) Copy content Toggle raw display
$13$ \( T^{3} + 3698 T^{2} + \cdots - 31329885032 \) Copy content Toggle raw display
$17$ \( T^{3} - 3164 T^{2} + \cdots - 8811060158208 \) Copy content Toggle raw display
$19$ \( T^{3} + 14234 T^{2} + \cdots - 27667220902592 \) Copy content Toggle raw display
$23$ \( (T + 12167)^{3} \) Copy content Toggle raw display
$29$ \( T^{3} + 171914 T^{2} + \cdots - 62745468718728 \) Copy content Toggle raw display
$31$ \( T^{3} + 124972 T^{2} + \cdots + 9840206494080 \) Copy content Toggle raw display
$37$ \( T^{3} + 679074 T^{2} + \cdots + 82\!\cdots\!28 \) Copy content Toggle raw display
$41$ \( T^{3} + 331362 T^{2} + \cdots - 19\!\cdots\!64 \) Copy content Toggle raw display
$43$ \( T^{3} + 145922 T^{2} + \cdots - 31\!\cdots\!92 \) Copy content Toggle raw display
$47$ \( T^{3} + 1824192 T^{2} + \cdots - 86\!\cdots\!88 \) Copy content Toggle raw display
$53$ \( T^{3} + 3442448 T^{2} + \cdots - 55\!\cdots\!52 \) Copy content Toggle raw display
$59$ \( T^{3} + 3147660 T^{2} + \cdots + 72\!\cdots\!28 \) Copy content Toggle raw display
$61$ \( T^{3} + 3444530 T^{2} + \cdots - 29\!\cdots\!16 \) Copy content Toggle raw display
$67$ \( T^{3} + 921870 T^{2} + \cdots - 21\!\cdots\!88 \) Copy content Toggle raw display
$71$ \( T^{3} - 2594520 T^{2} + \cdots + 71\!\cdots\!68 \) Copy content Toggle raw display
$73$ \( T^{3} + 540826 T^{2} + \cdots - 39\!\cdots\!04 \) Copy content Toggle raw display
$79$ \( T^{3} - 316994 T^{2} + \cdots + 11\!\cdots\!92 \) Copy content Toggle raw display
$83$ \( T^{3} - 4119316 T^{2} + \cdots + 39\!\cdots\!52 \) Copy content Toggle raw display
$89$ \( T^{3} - 1417176 T^{2} + \cdots + 79\!\cdots\!12 \) Copy content Toggle raw display
$97$ \( T^{3} + 15089326 T^{2} + \cdots + 43\!\cdots\!24 \) Copy content Toggle raw display
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