Properties

Label 138.8.a.h
Level $138$
Weight $8$
Character orbit 138.a
Self dual yes
Analytic conductor $43.109$
Analytic rank $0$
Dimension $4$
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: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Defining polynomial: \( x^{4} - 2x^{3} - 8367x^{2} - 89140x + 11077220 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{5}\cdot 3 \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 8 q^{2} + 27 q^{3} + 64 q^{4} + (\beta_{2} - \beta_1 + 68) q^{5} + 216 q^{6} + (\beta_{3} + \beta_1 + 506) q^{7} + 512 q^{8} + 729 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 8 q^{2} + 27 q^{3} + 64 q^{4} + (\beta_{2} - \beta_1 + 68) q^{5} + 216 q^{6} + (\beta_{3} + \beta_1 + 506) q^{7} + 512 q^{8} + 729 q^{9} + (8 \beta_{2} - 8 \beta_1 + 544) q^{10} + (3 \beta_{3} - 7 \beta_{2} - 4 \beta_1 + 1028) q^{11} + 1728 q^{12} + ( - 6 \beta_{3} + 20 \beta_{2} + 7 \beta_1 + 2016) q^{13} + (8 \beta_{3} + 8 \beta_1 + 4048) q^{14} + (27 \beta_{2} - 27 \beta_1 + 1836) q^{15} + 4096 q^{16} + ( - 7 \beta_{3} - 80 \beta_{2} - 19 \beta_1 + 9252) q^{17} + 5832 q^{18} + ( - 32 \beta_{3} + 113 \beta_{2} + 31 \beta_1 + 1466) q^{19} + (64 \beta_{2} - 64 \beta_1 + 4352) q^{20} + (27 \beta_{3} + 27 \beta_1 + 13662) q^{21} + (24 \beta_{3} - 56 \beta_{2} - 32 \beta_1 + 8224) q^{22} - 12167 q^{23} + 13824 q^{24} + ( - 20 \beta_{3} - 230 \beta_{2} - 115 \beta_1 + 30245) q^{25} + ( - 48 \beta_{3} + 160 \beta_{2} + 56 \beta_1 + 16128) q^{26} + 19683 q^{27} + (64 \beta_{3} + 64 \beta_1 + 32384) q^{28} + ( - 198 \beta_{3} - 304 \beta_{2} + 214 \beta_1 + 54178) q^{29} + (216 \beta_{2} - 216 \beta_1 + 14688) q^{30} + (118 \beta_{3} - 200 \beta_{2} + 648 \beta_1 + 55672) q^{31} + 32768 q^{32} + (81 \beta_{3} - 189 \beta_{2} - 108 \beta_1 + 27756) q^{33} + ( - 56 \beta_{3} - 640 \beta_{2} - 152 \beta_1 + 74016) q^{34} + (340 \beta_{3} + 92 \beta_{2} - 247 \beta_1 + 17326) q^{35} + 46656 q^{36} + (287 \beta_{3} - 561 \beta_{2} + 1464 \beta_1 + 121470) q^{37} + ( - 256 \beta_{3} + 904 \beta_{2} + 248 \beta_1 + 11728) q^{38} + ( - 162 \beta_{3} + 540 \beta_{2} + 189 \beta_1 + 54432) q^{39} + (512 \beta_{2} - 512 \beta_1 + 34816) q^{40} + ( - 206 \beta_{3} + 786 \beta_{2} - 448 \beta_1 + 84874) q^{41} + (216 \beta_{3} + 216 \beta_1 + 109296) q^{42} + ( - 490 \beta_{3} - 1201 \beta_{2} + 599 \beta_1 + 181898) q^{43} + (192 \beta_{3} - 448 \beta_{2} - 256 \beta_1 + 65792) q^{44} + (729 \beta_{2} - 729 \beta_1 + 49572) q^{45} - 97336 q^{46} + ( - 60 \beta_{3} - 492 \beta_{2} - 3671 \beta_1 + 84286) q^{47} + 110592 q^{48} + (716 \beta_{3} + 990 \beta_{2} - 439 \beta_1 - 76785) q^{49} + ( - 160 \beta_{3} - 1840 \beta_{2} - 920 \beta_1 + 241960) q^{50} + ( - 189 \beta_{3} - 2160 \beta_{2} - 513 \beta_1 + 249804) q^{51} + ( - 384 \beta_{3} + 1280 \beta_{2} + 448 \beta_1 + 129024) q^{52} + ( - 806 \beta_{3} + 4709 \beta_{2} - 381 \beta_1 - 91924) q^{53} + 157464 q^{54} + (1160 \beta_{3} + 1340 \beta_{2} - 1630 \beta_1 + 107460) q^{55} + (512 \beta_{3} + 512 \beta_1 + 259072) q^{56} + ( - 864 \beta_{3} + 3051 \beta_{2} + 837 \beta_1 + 39582) q^{57} + ( - 1584 \beta_{3} - 2432 \beta_{2} + 1712 \beta_1 + 433424) q^{58} + (2338 \beta_{3} + 5346 \beta_{2} + 3229 \beta_1 + 21690) q^{59} + (1728 \beta_{2} - 1728 \beta_1 + 117504) q^{60} + ( - 2379 \beta_{3} - 4903 \beta_{2} - 3478 \beta_1 + 521650) q^{61} + (944 \beta_{3} - 1600 \beta_{2} + 5184 \beta_1 + 445376) q^{62} + (729 \beta_{3} + 729 \beta_1 + 368874) q^{63} + 262144 q^{64} + ( - 2440 \beta_{3} - 206 \beta_{2} - 484 \beta_1 + 393372) q^{65} + (648 \beta_{3} - 1512 \beta_{2} - 864 \beta_1 + 222048) q^{66} + ( - 306 \beta_{3} - 5799 \beta_{2} + 10509 \beta_1 + 1081238) q^{67} + ( - 448 \beta_{3} - 5120 \beta_{2} - 1216 \beta_1 + 592128) q^{68} - 328509 q^{69} + (2720 \beta_{3} + 736 \beta_{2} - 1976 \beta_1 + 138608) q^{70} + (2430 \beta_{3} - 12294 \beta_{2} - 2388 \beta_1 + 567072) q^{71} + 373248 q^{72} + ( - 212 \beta_{3} + 7240 \beta_{2} + 2072 \beta_1 - 273318) q^{73} + (2296 \beta_{3} - 4488 \beta_{2} + 11712 \beta_1 + 971760) q^{74} + ( - 540 \beta_{3} - 6210 \beta_{2} - 3105 \beta_1 + 816615) q^{75} + ( - 2048 \beta_{3} + 7232 \beta_{2} + 1984 \beta_1 + 93824) q^{76} + (2862 \beta_{3} - 3218 \beta_{2} - 7946 \beta_1 + 1456372) q^{77} + ( - 1296 \beta_{3} + 4320 \beta_{2} + 1512 \beta_1 + 435456) q^{78} + (4019 \beta_{3} - 9176 \beta_{2} - 7373 \beta_1 + 12574) q^{79} + (4096 \beta_{2} - 4096 \beta_1 + 278528) q^{80} + 531441 q^{81} + ( - 1648 \beta_{3} + 6288 \beta_{2} - 3584 \beta_1 + 678992) q^{82} + ( - 2903 \beta_{3} + 13259 \beta_{2} + 11034 \beta_1 + 376544) q^{83} + (1728 \beta_{3} + 1728 \beta_1 + 874368) q^{84} + ( - 780 \beta_{3} + 32494 \beta_{2} - 18529 \beta_1 - 2195098) q^{85} + ( - 3920 \beta_{3} - 9608 \beta_{2} + 4792 \beta_1 + 1455184) q^{86} + ( - 5346 \beta_{3} - 8208 \beta_{2} + 5778 \beta_1 + 1462806) q^{87} + (1536 \beta_{3} - 3584 \beta_{2} - 2048 \beta_1 + 526336) q^{88} + ( - 4239 \beta_{3} + 1350 \beta_{2} + 657 \beta_1 + 369828) q^{89} + (5832 \beta_{2} - 5832 \beta_1 + 396576) q^{90} + ( - 1130 \beta_{3} + 8968 \beta_{2} + 21054 \beta_1 - 720684) q^{91} - 778688 q^{92} + (3186 \beta_{3} - 5400 \beta_{2} + 17496 \beta_1 + 1503144) q^{93} + ( - 480 \beta_{3} - 3936 \beta_{2} - 29368 \beta_1 + 674288) q^{94} + ( - 13140 \beta_{3} - 12272 \beta_{2} + 6407 \beta_1 + 2123594) q^{95} + 884736 q^{96} + (9616 \beta_{3} - 33518 \beta_{2} - 6966 \beta_1 + 471910) q^{97} + (5728 \beta_{3} + 7920 \beta_{2} - 3512 \beta_1 - 614280) q^{98} + (2187 \beta_{3} - 5103 \beta_{2} - 2916 \beta_1 + 749412) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 32 q^{2} + 108 q^{3} + 256 q^{4} + 270 q^{5} + 864 q^{6} + 2022 q^{7} + 2048 q^{8} + 2916 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 32 q^{2} + 108 q^{3} + 256 q^{4} + 270 q^{5} + 864 q^{6} + 2022 q^{7} + 2048 q^{8} + 2916 q^{9} + 2160 q^{10} + 4120 q^{11} + 6912 q^{12} + 8036 q^{13} + 16176 q^{14} + 7290 q^{15} + 16384 q^{16} + 37182 q^{17} + 23328 q^{18} + 5702 q^{19} + 17280 q^{20} + 54594 q^{21} + 32960 q^{22} - 48668 q^{23} + 55296 q^{24} + 121480 q^{25} + 64288 q^{26} + 78732 q^{27} + 129408 q^{28} + 217716 q^{29} + 58320 q^{30} + 222852 q^{31} + 131072 q^{32} + 111240 q^{33} + 297456 q^{34} + 68440 q^{35} + 186624 q^{36} + 486428 q^{37} + 45616 q^{38} + 216972 q^{39} + 138240 q^{40} + 338336 q^{41} + 436752 q^{42} + 730974 q^{43} + 263680 q^{44} + 196830 q^{45} - 389344 q^{46} + 338248 q^{47} + 442368 q^{48} - 310552 q^{49} + 971840 q^{50} + 1003914 q^{51} + 514304 q^{52} - 375502 q^{53} + 629856 q^{54} + 424840 q^{55} + 1035264 q^{56} + 153954 q^{57} + 1741728 q^{58} + 71392 q^{59} + 466560 q^{60} + 2101164 q^{61} + 1782816 q^{62} + 1474038 q^{63} + 1048576 q^{64} + 1578780 q^{65} + 889920 q^{66} + 4337162 q^{67} + 2379648 q^{68} - 1314036 q^{69} + 547520 q^{70} + 2288016 q^{71} + 1492992 q^{72} - 1107328 q^{73} + 3891424 q^{74} + 3279960 q^{75} + 364928 q^{76} + 5826200 q^{77} + 1735776 q^{78} + 60610 q^{79} + 1105920 q^{80} + 2125764 q^{81} + 2706688 q^{82} + 1485464 q^{83} + 3494016 q^{84} - 8843820 q^{85} + 5847792 q^{86} + 5878332 q^{87} + 2109440 q^{88} + 1485090 q^{89} + 1574640 q^{90} - 2898412 q^{91} - 3114752 q^{92} + 6017004 q^{93} + 2705984 q^{94} + 8545200 q^{95} + 3538944 q^{96} + 1935444 q^{97} - 2484416 q^{98} + 3003480 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{3} - 8367x^{2} - 89140x + 11077220 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 4\nu - 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 8\nu^{2} - 5551\nu - 110450 ) / 684 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} + 68\nu^{2} + 4411\nu - 207382 ) / 76 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta _1 + 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 4\beta_{3} + 36\beta_{2} + 15\beta _1 + 16758 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -32\beta_{3} + 2448\beta_{2} + 5431\beta _1 + 318838 ) / 4 \) 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
33.2734
90.4389
−65.5856
−56.1267
8.00000 27.0000 64.0000 −427.795 216.000 345.429 512.000 729.000 −3422.36
1.2 8.00000 27.0000 64.0000 −10.0669 216.000 971.172 512.000 729.000 −80.5353
1.3 8.00000 27.0000 64.0000 340.987 216.000 1267.12 512.000 729.000 2727.89
1.4 8.00000 27.0000 64.0000 366.876 216.000 −561.721 512.000 729.000 2935.00
\(n\): 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.h 4
3.b odd 2 1 414.8.a.i 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
138.8.a.h 4 1.a even 1 1 trivial
414.8.a.i 4 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 270T_{5}^{3} - 180540T_{5}^{2} + 51728000T_{5} + 538752000 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(138))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 8)^{4} \) Copy content Toggle raw display
$3$ \( (T - 27)^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 270 T^{3} + \cdots + 538752000 \) Copy content Toggle raw display
$7$ \( T^{4} - 2022 T^{3} + \cdots - 238777204176 \) Copy content Toggle raw display
$11$ \( T^{4} - 4120 T^{3} + \cdots + 25210679155200 \) Copy content Toggle raw display
$13$ \( T^{4} - 8036 T^{3} + \cdots + 10\!\cdots\!84 \) Copy content Toggle raw display
$17$ \( T^{4} - 37182 T^{3} + \cdots - 69\!\cdots\!08 \) Copy content Toggle raw display
$19$ \( T^{4} - 5702 T^{3} + \cdots + 12\!\cdots\!64 \) Copy content Toggle raw display
$23$ \( (T + 12167)^{4} \) Copy content Toggle raw display
$29$ \( T^{4} - 217716 T^{3} + \cdots - 19\!\cdots\!68 \) Copy content Toggle raw display
$31$ \( T^{4} - 222852 T^{3} + \cdots - 32\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{4} - 486428 T^{3} + \cdots - 26\!\cdots\!28 \) Copy content Toggle raw display
$41$ \( T^{4} - 338336 T^{3} + \cdots + 11\!\cdots\!52 \) Copy content Toggle raw display
$43$ \( T^{4} - 730974 T^{3} + \cdots - 11\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{4} - 338248 T^{3} + \cdots + 25\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{4} + 375502 T^{3} + \cdots + 12\!\cdots\!20 \) Copy content Toggle raw display
$59$ \( T^{4} - 71392 T^{3} + \cdots + 23\!\cdots\!28 \) Copy content Toggle raw display
$61$ \( T^{4} - 2101164 T^{3} + \cdots + 18\!\cdots\!72 \) Copy content Toggle raw display
$67$ \( T^{4} - 4337162 T^{3} + \cdots + 44\!\cdots\!92 \) Copy content Toggle raw display
$71$ \( T^{4} - 2288016 T^{3} + \cdots + 92\!\cdots\!92 \) Copy content Toggle raw display
$73$ \( T^{4} + 1107328 T^{3} + \cdots - 25\!\cdots\!88 \) Copy content Toggle raw display
$79$ \( T^{4} - 60610 T^{3} + \cdots + 83\!\cdots\!48 \) Copy content Toggle raw display
$83$ \( T^{4} - 1485464 T^{3} + \cdots + 11\!\cdots\!08 \) Copy content Toggle raw display
$89$ \( T^{4} - 1485090 T^{3} + \cdots - 10\!\cdots\!56 \) Copy content Toggle raw display
$97$ \( T^{4} - 1935444 T^{3} + \cdots + 94\!\cdots\!60 \) Copy content Toggle raw display
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