Properties

Label 138.8.a.f
Level $138$
Weight $8$
Character orbit 138.a
Self dual yes
Analytic conductor $43.109$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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: \( x^{4} - x^{3} - 62310x^{2} - 465075x + 877928625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4}\cdot 3 \)
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.

\(f(q)\) \(=\) \( q - 8 q^{2} + 27 q^{3} + 64 q^{4} + ( - \beta_1 - 40) q^{5} - 216 q^{6} + (\beta_{2} + \beta_1 + 304) q^{7} - 512 q^{8} + 729 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 8 q^{2} + 27 q^{3} + 64 q^{4} + ( - \beta_1 - 40) q^{5} - 216 q^{6} + (\beta_{2} + \beta_1 + 304) q^{7} - 512 q^{8} + 729 q^{9} + (8 \beta_1 + 320) q^{10} + (2 \beta_{3} + 3 \beta_{2} - 4 \beta_1 + 456) q^{11} + 1728 q^{12} + ( - 4 \beta_{3} + 4 \beta_{2} - 14 \beta_1 - 1618) q^{13} + ( - 8 \beta_{2} - 8 \beta_1 - 2432) q^{14} + ( - 27 \beta_1 - 1080) q^{15} + 4096 q^{16} + (\beta_{3} - 10 \beta_{2} - 48 \beta_1 + 3241) q^{17} - 5832 q^{18} + (3 \beta_{3} - 3 \beta_{2} + 20 \beta_1 + 18607) q^{19} + ( - 64 \beta_1 - 2560) q^{20} + (27 \beta_{2} + 27 \beta_1 + 8208) q^{21} + ( - 16 \beta_{3} - 24 \beta_{2} + 32 \beta_1 - 3648) q^{22} + 12167 q^{23} - 13824 q^{24} + (15 \beta_{3} + 15 \beta_{2} + 97 \beta_1 + 48080) q^{25} + (32 \beta_{3} - 32 \beta_{2} + 112 \beta_1 + 12944) q^{26} + 19683 q^{27} + (64 \beta_{2} + 64 \beta_1 + 19456) q^{28} + ( - 5 \beta_{3} - 85 \beta_{2} + 5 \beta_1 - 23613) q^{29} + (216 \beta_1 + 8640) q^{30} + ( - 29 \beta_{3} - 81 \beta_{2} + 69 \beta_1 - 9875) q^{31} - 32768 q^{32} + (54 \beta_{3} + 81 \beta_{2} - 108 \beta_1 + 12312) q^{33} + ( - 8 \beta_{3} + 80 \beta_{2} + 384 \beta_1 - 25928) q^{34} + ( - 150 \beta_{3} + 80 \beta_{2} - 546 \beta_1 - 196750) q^{35} + 46656 q^{36} + (66 \beta_{3} - 87 \beta_{2} + 462 \beta_1 + 18698) q^{37} + ( - 24 \beta_{3} + 24 \beta_{2} - 160 \beta_1 - 148856) q^{38} + ( - 108 \beta_{3} + 108 \beta_{2} - 378 \beta_1 - 43686) q^{39} + (512 \beta_1 + 20480) q^{40} + (23 \beta_{3} + 167 \beta_{2} - 177 \beta_1 - 129681) q^{41} + ( - 216 \beta_{2} - 216 \beta_1 - 65664) q^{42} + (159 \beta_{3} + 331 \beta_{2} + 252 \beta_1 + 179031) q^{43} + (128 \beta_{3} + 192 \beta_{2} - 256 \beta_1 + 29184) q^{44} + ( - 729 \beta_1 - 29160) q^{45} - 97336 q^{46} + (39 \beta_{3} - 949 \beta_{2} + 299 \beta_1 - 3227) q^{47} + 110592 q^{48} + (350 \beta_{3} + 336 \beta_{2} + 196 \beta_1 + 413771) q^{49} + ( - 120 \beta_{3} - 120 \beta_{2} - 776 \beta_1 - 384640) q^{50} + (27 \beta_{3} - 270 \beta_{2} - 1296 \beta_1 + 87507) q^{51} + ( - 256 \beta_{3} + 256 \beta_{2} - 896 \beta_1 - 103552) q^{52} + (20 \beta_{3} + 2 \beta_{2} + 1309 \beta_1 + 262684) q^{53} - 157464 q^{54} + ( - 665 \beta_{3} - 1155 \beta_{2} - 1337 \beta_1 + 289785) q^{55} + ( - 512 \beta_{2} - 512 \beta_1 - 155648) q^{56} + (81 \beta_{3} - 81 \beta_{2} + 540 \beta_1 + 502389) q^{57} + (40 \beta_{3} + 680 \beta_{2} - 40 \beta_1 + 188904) q^{58} + (90 \beta_{3} - 70 \beta_{2} - 106 \beta_1 + 1099594) q^{59} + ( - 1728 \beta_1 - 69120) q^{60} + ( - 384 \beta_{3} - 791 \beta_{2} + 60 \beta_1 + 1332240) q^{61} + (232 \beta_{3} + 648 \beta_{2} - 552 \beta_1 + 79000) q^{62} + (729 \beta_{2} + 729 \beta_1 + 221616) q^{63} + 262144 q^{64} + (310 \beta_{3} + 3590 \beta_{2} + 2784 \beta_1 + 1590130) q^{65} + ( - 432 \beta_{3} - 648 \beta_{2} + 864 \beta_1 - 98496) q^{66} + (313 \beta_{3} - 703 \beta_{2} + 2020 \beta_1 + 2824825) q^{67} + (64 \beta_{3} - 640 \beta_{2} - 3072 \beta_1 + 207424) q^{68} + 328509 q^{69} + (1200 \beta_{3} - 640 \beta_{2} + 4368 \beta_1 + 1574000) q^{70} + ( - 107 \beta_{3} + 653 \beta_{2} + 4681 \beta_1 + 2096543) q^{71} - 373248 q^{72} + ( - 1137 \beta_{3} - 435 \beta_{2} + 1915 \beta_1 + 1776363) q^{73} + ( - 528 \beta_{3} + 696 \beta_{2} - 3696 \beta_1 - 149584) q^{74} + (405 \beta_{3} + 405 \beta_{2} + 2619 \beta_1 + 1298160) q^{75} + (192 \beta_{3} - 192 \beta_{2} + 1280 \beta_1 + 1190848) q^{76} + (512 \beta_{3} + 2918 \beta_{2} + 8396 \beta_1 + 2442836) q^{77} + (864 \beta_{3} - 864 \beta_{2} + 3024 \beta_1 + 349488) q^{78} + ( - 258 \beta_{3} + 1853 \beta_{2} - 3555 \beta_1 + 2448178) q^{79} + ( - 4096 \beta_1 - 163840) q^{80} + 531441 q^{81} + ( - 184 \beta_{3} - 1336 \beta_{2} + 1416 \beta_1 + 1037448) q^{82} + (430 \beta_{3} - 2439 \beta_{2} - 2304 \beta_1 + 607016) q^{83} + (1728 \beta_{2} + 1728 \beta_1 + 525312) q^{84} + (1910 \beta_{3} - 980 \beta_{2} + 1068 \beta_1 + 6446030) q^{85} + ( - 1272 \beta_{3} - 2648 \beta_{2} - 2016 \beta_1 - 1432248) q^{86} + ( - 135 \beta_{3} - 2295 \beta_{2} + 135 \beta_1 - 637551) q^{87} + ( - 1024 \beta_{3} - 1536 \beta_{2} + 2048 \beta_1 - 233472) q^{88} + ( - 1507 \beta_{3} - 3266 \beta_{2} + 9098 \beta_1 + 751009) q^{89} + (5832 \beta_1 + 233280) q^{90} + ( - 2324 \beta_{3} - 2382 \beta_{2} - 34770 \beta_1 + 441804) q^{91} + 778688 q^{92} + ( - 783 \beta_{3} - 2187 \beta_{2} + 1863 \beta_1 - 266625) q^{93} + ( - 312 \beta_{3} + 7592 \beta_{2} - 2392 \beta_1 + 25816) q^{94} + ( - 375 \beta_{3} - 2835 \beta_{2} - 20023 \beta_1 - 3072085) q^{95} - 884736 q^{96} + (2772 \beta_{3} + 1096 \beta_{2} - 14170 \beta_1 + 8062418) q^{97} + ( - 2800 \beta_{3} - 2688 \beta_{2} - 1568 \beta_1 - 3310168) q^{98} + (1458 \beta_{3} + 2187 \beta_{2} - 2916 \beta_1 + 332424) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 32 q^{2} + 108 q^{3} + 256 q^{4} - 162 q^{5} - 864 q^{6} + 1218 q^{7} - 2048 q^{8} + 2916 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 32 q^{2} + 108 q^{3} + 256 q^{4} - 162 q^{5} - 864 q^{6} + 1218 q^{7} - 2048 q^{8} + 2916 q^{9} + 1296 q^{10} + 1820 q^{11} + 6912 q^{12} - 6508 q^{13} - 9744 q^{14} - 4374 q^{15} + 16384 q^{16} + 12870 q^{17} - 23328 q^{18} + 74474 q^{19} - 10368 q^{20} + 32886 q^{21} - 14560 q^{22} + 48668 q^{23} - 55296 q^{24} + 192544 q^{25} + 52064 q^{26} + 78732 q^{27} + 77952 q^{28} - 94452 q^{29} + 34992 q^{30} - 39420 q^{31} - 131072 q^{32} + 49140 q^{33} - 102960 q^{34} - 788392 q^{35} + 186624 q^{36} + 75848 q^{37} - 595792 q^{38} - 175716 q^{39} + 82944 q^{40} - 519032 q^{41} - 263088 q^{42} + 716946 q^{43} + 116480 q^{44} - 118098 q^{45} - 389344 q^{46} - 12232 q^{47} + 442368 q^{48} + 1656176 q^{49} - 1540352 q^{50} + 347490 q^{51} - 416512 q^{52} + 1053394 q^{53} - 629856 q^{54} + 1155136 q^{55} - 623616 q^{56} + 2010798 q^{57} + 755616 q^{58} + 4398344 q^{59} - 279936 q^{60} + 5328312 q^{61} + 315360 q^{62} + 887922 q^{63} + 1048576 q^{64} + 6366708 q^{65} - 393120 q^{66} + 11303966 q^{67} + 823680 q^{68} + 1314036 q^{69} + 6307136 q^{70} + 8395320 q^{71} - 1492992 q^{72} + 7107008 q^{73} - 606784 q^{74} + 5198688 q^{75} + 4766336 q^{76} + 9789160 q^{77} + 1405728 q^{78} + 9785086 q^{79} - 663552 q^{80} + 2125764 q^{81} + 4152256 q^{82} + 2424316 q^{83} + 2104704 q^{84} + 25790076 q^{85} - 5735568 q^{86} - 2550204 q^{87} - 931840 q^{88} + 3019218 q^{89} + 944784 q^{90} + 1693028 q^{91} + 3114752 q^{92} - 1064340 q^{93} + 97856 q^{94} - 12329136 q^{95} - 3538944 q^{96} + 32226876 q^{97} - 13249408 q^{98} + 1326780 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 62310x^{2} - 465075x + 877928625 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - 136\nu^{2} - 31800\nu + 3849525 ) / 675 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} + 316\nu^{2} + 30270\nu - 9456750 ) / 675 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 15\beta_{3} + 15\beta_{2} + 17\beta _1 + 124605 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 510\beta_{3} + 1185\beta_{2} + 16478\beta _1 + 387045 \) 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
212.947
136.241
−167.177
−181.010
−8.00000 27.0000 64.0000 −465.894 −216.000 1570.02 −512.000 729.000 3727.15
1.2 −8.00000 27.0000 64.0000 −312.481 −216.000 −132.348 −512.000 729.000 2499.85
1.3 −8.00000 27.0000 64.0000 294.355 −216.000 995.541 −512.000 729.000 −2354.84
1.4 −8.00000 27.0000 64.0000 322.021 −216.000 −1215.22 −512.000 729.000 −2576.17
\(n\): 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.f 4
3.b odd 2 1 414.8.a.l 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
138.8.a.f 4 1.a even 1 1 trivial
414.8.a.l 4 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 162T_{5}^{3} - 239400T_{5}^{2} - 15953000T_{5} + 13799586000 \) 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
$5$ \( T^{4} + 162 T^{3} + \cdots + 13799586000 \) Copy content Toggle raw display
$7$ \( T^{4} - 1218 T^{3} + \cdots + 251384354880 \) Copy content Toggle raw display
$11$ \( T^{4} + \cdots - 243981790840800 \) Copy content Toggle raw display
$13$ \( T^{4} + 6508 T^{3} + \cdots + 60\!\cdots\!72 \) Copy content Toggle raw display
$17$ \( T^{4} - 12870 T^{3} + \cdots + 33\!\cdots\!88 \) Copy content Toggle raw display
$19$ \( T^{4} - 74474 T^{3} + \cdots + 50\!\cdots\!36 \) Copy content Toggle raw display
$23$ \( (T - 12167)^{4} \) Copy content Toggle raw display
$29$ \( T^{4} + 94452 T^{3} + \cdots + 36\!\cdots\!36 \) Copy content Toggle raw display
$31$ \( T^{4} + 39420 T^{3} + \cdots + 12\!\cdots\!40 \) Copy content Toggle raw display
$37$ \( T^{4} - 75848 T^{3} + \cdots - 91\!\cdots\!08 \) Copy content Toggle raw display
$41$ \( T^{4} + 519032 T^{3} + \cdots - 41\!\cdots\!40 \) Copy content Toggle raw display
$43$ \( T^{4} - 716946 T^{3} + \cdots - 19\!\cdots\!28 \) Copy content Toggle raw display
$47$ \( T^{4} + 12232 T^{3} + \cdots + 44\!\cdots\!52 \) Copy content Toggle raw display
$53$ \( T^{4} - 1053394 T^{3} + \cdots + 17\!\cdots\!48 \) Copy content Toggle raw display
$59$ \( T^{4} - 4398344 T^{3} + \cdots + 13\!\cdots\!68 \) Copy content Toggle raw display
$61$ \( T^{4} - 5328312 T^{3} + \cdots - 12\!\cdots\!88 \) Copy content Toggle raw display
$67$ \( T^{4} - 11303966 T^{3} + \cdots + 38\!\cdots\!12 \) Copy content Toggle raw display
$71$ \( T^{4} - 8395320 T^{3} + \cdots - 10\!\cdots\!60 \) Copy content Toggle raw display
$73$ \( T^{4} - 7107008 T^{3} + \cdots + 56\!\cdots\!32 \) Copy content Toggle raw display
$79$ \( T^{4} - 9785086 T^{3} + \cdots + 10\!\cdots\!16 \) Copy content Toggle raw display
$83$ \( T^{4} - 2424316 T^{3} + \cdots + 53\!\cdots\!04 \) Copy content Toggle raw display
$89$ \( T^{4} - 3019218 T^{3} + \cdots - 10\!\cdots\!84 \) Copy content Toggle raw display
$97$ \( T^{4} - 32226876 T^{3} + \cdots - 79\!\cdots\!20 \) Copy content Toggle raw display
show more
show less