Properties

Label 138.8.a.g
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: \( x^{4} - 5167x^{2} - 24752x + 5245058 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
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.

\(f(q)\) \(=\) \( q + 8 q^{2} - 27 q^{3} + 64 q^{4} + ( - \beta_{2} + \beta_1 + 85) q^{5} - 216 q^{6} + (\beta_{3} + \beta_{2} - 7) q^{7} + 512 q^{8} + 729 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 8 q^{2} - 27 q^{3} + 64 q^{4} + ( - \beta_{2} + \beta_1 + 85) q^{5} - 216 q^{6} + (\beta_{3} + \beta_{2} - 7) q^{7} + 512 q^{8} + 729 q^{9} + ( - 8 \beta_{2} + 8 \beta_1 + 680) q^{10} + ( - \beta_{3} - 12 \beta_{2} - 11 \beta_1 - 576) q^{11} - 1728 q^{12} + ( - 2 \beta_{2} + 31 \beta_1 + 2300) q^{13} + (8 \beta_{3} + 8 \beta_{2} - 56) q^{14} + (27 \beta_{2} - 27 \beta_1 - 2295) q^{15} + 4096 q^{16} + ( - \beta_{3} + 15 \beta_{2} + 110 \beta_1 + 5417) q^{17} + 5832 q^{18} + ( - 31 \beta_{2} - 87 \beta_1 - 14555) q^{19} + ( - 64 \beta_{2} + 64 \beta_1 + 5440) q^{20} + ( - 27 \beta_{3} - 27 \beta_{2} + 189) q^{21} + ( - 8 \beta_{3} - 96 \beta_{2} - 88 \beta_1 - 4608) q^{22} + 12167 q^{23} - 13824 q^{24} + ( - 70 \beta_{3} - 82 \beta_{2} - 143 \beta_1 + 61485) q^{25} + ( - 16 \beta_{2} + 248 \beta_1 + 18400) q^{26} - 19683 q^{27} + (64 \beta_{3} + 64 \beta_{2} - 448) q^{28} + (70 \beta_{3} + 366 \beta_{2} - 346 \beta_1 + 63276) q^{29} + (216 \beta_{2} - 216 \beta_1 - 18360) q^{30} + ( - 70 \beta_{3} + 30 \beta_{2} + 40 \beta_1 + 91166) q^{31} + 32768 q^{32} + (27 \beta_{3} + 324 \beta_{2} + 297 \beta_1 + 15552) q^{33} + ( - 8 \beta_{3} + 120 \beta_{2} + 880 \beta_1 + 43336) q^{34} + (250 \beta_{3} + 812 \beta_{2} - 1827 \beta_1 - 48860) q^{35} + 46656 q^{36} + (\beta_{3} - 142 \beta_{2} + 747 \beta_1 + 89328) q^{37} + ( - 248 \beta_{2} - 696 \beta_1 - 116440) q^{38} + (54 \beta_{2} - 837 \beta_1 - 62100) q^{39} + ( - 512 \beta_{2} + 512 \beta_1 + 43520) q^{40} + (70 \beta_{3} + 308 \beta_{2} + 1504 \beta_1 + 204846) q^{41} + ( - 216 \beta_{3} - 216 \beta_{2} + 1512) q^{42} + (110 \beta_{3} - 979 \beta_{2} - 605 \beta_1 + 139777) q^{43} + ( - 64 \beta_{3} - 768 \beta_{2} - 704 \beta_1 - 36864) q^{44} + ( - 729 \beta_{2} + 729 \beta_1 + 61965) q^{45} + 97336 q^{46} + ( - 110 \beta_{3} + 740 \beta_{2} + 1279 \beta_1 + 590364) q^{47} - 110592 q^{48} + ( - 286 \beta_{3} - 1758 \beta_{2} + 4163 \beta_1 + 914315) q^{49} + ( - 560 \beta_{3} - 656 \beta_{2} - 1144 \beta_1 + 491880) q^{50} + (27 \beta_{3} - 405 \beta_{2} - 2970 \beta_1 - 146259) q^{51} + ( - 128 \beta_{2} + 1984 \beta_1 + 147200) q^{52} + ( - 182 \beta_{3} + 97 \beta_{2} - 817 \beta_1 + 1061695) q^{53} - 157464 q^{54} + ( - 140 \beta_{3} - 1692 \beta_{2} - 648 \beta_1 + 827660) q^{55} + (512 \beta_{3} + 512 \beta_{2} - 3584) q^{56} + (837 \beta_{2} + 2349 \beta_1 + 392985) q^{57} + (560 \beta_{3} + 2928 \beta_{2} - 2768 \beta_1 + 506208) q^{58} + ( - 112 \beta_{3} + 144 \beta_{2} + 2395 \beta_1 + 1266516) q^{59} + (1728 \beta_{2} - 1728 \beta_1 - 146880) q^{60} + (1191 \beta_{3} - 2030 \beta_{2} + 91 \beta_1 - 351976) q^{61} + ( - 560 \beta_{3} + 240 \beta_{2} + 320 \beta_1 + 729328) q^{62} + (729 \beta_{3} + 729 \beta_{2} - 5103) q^{63} + 262144 q^{64} + ( - 1300 \beta_{3} - 322 \beta_{2} + 1032 \beta_1 + 1287930) q^{65} + (216 \beta_{3} + 2592 \beta_{2} + 2376 \beta_1 + 124416) q^{66} + ( - 1330 \beta_{3} - 2425 \beta_{2} - 2467 \beta_1 + 909771) q^{67} + ( - 64 \beta_{3} + 960 \beta_{2} + 7040 \beta_1 + 346688) q^{68} - 328509 q^{69} + (2000 \beta_{3} + 6496 \beta_{2} - 14616 \beta_1 - 390880) q^{70} + (646 \beta_{3} + 2304 \beta_{2} - 1096 \beta_1 - 1222464) q^{71} + 373248 q^{72} + ( - 1148 \beta_{3} + 3012 \beta_{2} - 6750 \beta_1 + 1108054) q^{73} + (8 \beta_{3} - 1136 \beta_{2} + 5976 \beta_1 + 714624) q^{74} + (1890 \beta_{3} + 2214 \beta_{2} + 3861 \beta_1 - 1660095) q^{75} + ( - 1984 \beta_{2} - 5568 \beta_1 - 931520) q^{76} + (286 \beta_{3} - 6228 \beta_{2} - 28088 \beta_1 - 2764268) q^{77} + (432 \beta_{2} - 6696 \beta_1 - 496800) q^{78} + ( - 141 \beta_{3} + 1931 \beta_{2} + 13636 \beta_1 - 273021) q^{79} + ( - 4096 \beta_{2} + 4096 \beta_1 + 348160) q^{80} + 531441 q^{81} + (560 \beta_{3} + 2464 \beta_{2} + 12032 \beta_1 + 1638768) q^{82} + (1977 \beta_{3} + 9796 \beta_{2} + 7141 \beta_1 - 2598312) q^{83} + ( - 1728 \beta_{3} - 1728 \beta_{2} + 12096) q^{84} + ( - 4170 \beta_{3} + 2298 \beta_{2} + 7357 \beta_1 + 1986590) q^{85} + (880 \beta_{3} - 7832 \beta_{2} - 4840 \beta_1 + 1118216) q^{86} + ( - 1890 \beta_{3} - 9882 \beta_{2} + 9342 \beta_1 - 1708452) q^{87} + ( - 512 \beta_{3} - 6144 \beta_{2} - 5632 \beta_1 - 294912) q^{88} + ( - 3893 \beta_{3} - 9447 \beta_{2} + 4322 \beta_1 + 1153299) q^{89} + ( - 5832 \beta_{2} + 5832 \beta_1 + 495720) q^{90} + (4254 \beta_{3} + 24054 \beta_{2} + 1392 \beta_1 + 545902) q^{91} + 778688 q^{92} + (1890 \beta_{3} - 810 \beta_{2} - 1080 \beta_1 - 2461482) q^{93} + ( - 880 \beta_{3} + 5920 \beta_{2} + 10232 \beta_1 + 4722912) q^{94} + (2550 \beta_{3} + 6624 \beta_{2} - 18319 \beta_1 - 500960) q^{95} - 884736 q^{96} + ( - 392 \beta_{3} - 4762 \beta_{2} + 5750 \beta_1 - 3852968) q^{97} + ( - 2288 \beta_{3} - 14064 \beta_{2} + 33304 \beta_1 + 7314520) q^{98} + ( - 729 \beta_{3} - 8748 \beta_{2} - 8019 \beta_1 - 419904) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 32 q^{2} - 108 q^{3} + 256 q^{4} + 342 q^{5} - 864 q^{6} - 30 q^{7} + 2048 q^{8} + 2916 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 32 q^{2} - 108 q^{3} + 256 q^{4} + 342 q^{5} - 864 q^{6} - 30 q^{7} + 2048 q^{8} + 2916 q^{9} + 2736 q^{10} - 2280 q^{11} - 6912 q^{12} + 9204 q^{13} - 240 q^{14} - 9234 q^{15} + 16384 q^{16} + 21638 q^{17} + 23328 q^{18} - 58158 q^{19} + 21888 q^{20} + 810 q^{21} - 18240 q^{22} + 48668 q^{23} - 55296 q^{24} + 246104 q^{25} + 73632 q^{26} - 78732 q^{27} - 1920 q^{28} + 252372 q^{29} - 73872 q^{30} + 364604 q^{31} + 131072 q^{32} + 61560 q^{33} + 173104 q^{34} - 197064 q^{35} + 186624 q^{36} + 357596 q^{37} - 465264 q^{38} - 248508 q^{39} + 175104 q^{40} + 818768 q^{41} + 6480 q^{42} + 561066 q^{43} - 145920 q^{44} + 249318 q^{45} + 389344 q^{46} + 2359976 q^{47} - 442368 q^{48} + 3660776 q^{49} + 1968832 q^{50} - 584226 q^{51} + 589056 q^{52} + 4246586 q^{53} - 629856 q^{54} + 3314024 q^{55} - 15360 q^{56} + 1570266 q^{57} + 2018976 q^{58} + 5065776 q^{59} - 590976 q^{60} - 1403844 q^{61} + 2916832 q^{62} - 21870 q^{63} + 1048576 q^{64} + 5152364 q^{65} + 492480 q^{66} + 3643934 q^{67} + 1384832 q^{68} - 1314036 q^{69} - 1576512 q^{70} - 4894464 q^{71} + 1492992 q^{72} + 4426192 q^{73} + 2860768 q^{74} - 6644808 q^{75} - 3722112 q^{76} - 11044616 q^{77} - 1988064 q^{78} - 1095946 q^{79} + 1400832 q^{80} + 2125764 q^{81} + 6550144 q^{82} - 10412840 q^{83} + 51840 q^{84} + 7941764 q^{85} + 4488528 q^{86} - 6814044 q^{87} - 1167360 q^{88} + 4632090 q^{89} + 1994544 q^{90} + 2135500 q^{91} + 3114752 q^{92} - 9844308 q^{93} + 18879808 q^{94} - 2017088 q^{95} - 3538944 q^{96} - 15402348 q^{97} + 29286208 q^{98} - 1662120 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 5167x^{2} - 24752x + 5245058 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 4\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} + 77\nu^{2} + 2766\nu - 180439 ) / 147 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2\nu^{3} - 56\nu^{2} - 5826\nu + 107548 ) / 49 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{3} + 12\beta_{2} + 3\beta _1 + 10340 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 154\beta_{3} + 336\beta_{2} + 2997\beta _1 + 74424 ) / 4 \) 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
−53.8117
65.8209
−44.9520
32.9428
8.00000 −27.0000 64.0000 −467.042 −216.000 −746.713 512.000 729.000 −3736.34
1.2 8.00000 −27.0000 64.0000 7.78122 −216.000 1390.34 512.000 729.000 62.2498
1.3 8.00000 −27.0000 64.0000 302.129 −216.000 1118.77 512.000 729.000 2417.03
1.4 8.00000 −27.0000 64.0000 499.132 −216.000 −1792.39 512.000 729.000 3993.06
\(n\): 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.g 4
3.b odd 2 1 414.8.a.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
138.8.a.g 4 1.a even 1 1 trivial
414.8.a.h 4 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 342T_{5}^{3} - 220820T_{5}^{2} + 72169600T_{5} - 548040000 \) 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
$5$ \( T^{4} - 342 T^{3} + \cdots - 548040000 \) Copy content Toggle raw display
$7$ \( T^{4} + 30 T^{3} + \cdots + 2081845392656 \) Copy content Toggle raw display
$11$ \( T^{4} + \cdots + 134485790256000 \) Copy content Toggle raw display
$13$ \( T^{4} + \cdots + 850718415501872 \) Copy content Toggle raw display
$17$ \( T^{4} - 21638 T^{3} + \cdots + 17\!\cdots\!80 \) Copy content Toggle raw display
$19$ \( T^{4} + 58158 T^{3} + \cdots - 69\!\cdots\!28 \) Copy content Toggle raw display
$23$ \( (T - 12167)^{4} \) Copy content Toggle raw display
$29$ \( T^{4} - 252372 T^{3} + \cdots - 52\!\cdots\!56 \) Copy content Toggle raw display
$31$ \( T^{4} - 364604 T^{3} + \cdots - 42\!\cdots\!24 \) Copy content Toggle raw display
$37$ \( T^{4} - 357596 T^{3} + \cdots - 83\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{4} - 818768 T^{3} + \cdots + 12\!\cdots\!08 \) Copy content Toggle raw display
$43$ \( T^{4} - 561066 T^{3} + \cdots + 58\!\cdots\!56 \) Copy content Toggle raw display
$47$ \( T^{4} - 2359976 T^{3} + \cdots - 52\!\cdots\!20 \) Copy content Toggle raw display
$53$ \( T^{4} - 4246586 T^{3} + \cdots + 10\!\cdots\!16 \) Copy content Toggle raw display
$59$ \( T^{4} - 5065776 T^{3} + \cdots + 17\!\cdots\!12 \) Copy content Toggle raw display
$61$ \( T^{4} + 1403844 T^{3} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{4} - 3643934 T^{3} + \cdots - 38\!\cdots\!08 \) Copy content Toggle raw display
$71$ \( T^{4} + 4894464 T^{3} + \cdots + 11\!\cdots\!20 \) Copy content Toggle raw display
$73$ \( T^{4} - 4426192 T^{3} + \cdots + 28\!\cdots\!16 \) Copy content Toggle raw display
$79$ \( T^{4} + 1095946 T^{3} + \cdots + 40\!\cdots\!20 \) Copy content Toggle raw display
$83$ \( T^{4} + 10412840 T^{3} + \cdots - 49\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{4} - 4632090 T^{3} + \cdots + 11\!\cdots\!04 \) Copy content Toggle raw display
$97$ \( T^{4} + 15402348 T^{3} + \cdots + 11\!\cdots\!80 \) Copy content Toggle raw display
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