Properties

Label 138.8.a.e
Level $138$
Weight $8$
Character orbit 138.a
Self dual yes
Analytic conductor $43.109$
Analytic rank $1$
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: \(1\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Defining polynomial: \( x^{4} - x^{3} - 47804x^{2} - 3068607x + 114119793 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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.

\(f(q)\) \(=\) \( q - 8 q^{2} - 27 q^{3} + 64 q^{4} + (\beta_1 - 23) q^{5} + 216 q^{6} + ( - \beta_{3} - 56) q^{7} - 512 q^{8} + 729 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 8 q^{2} - 27 q^{3} + 64 q^{4} + (\beta_1 - 23) q^{5} + 216 q^{6} + ( - \beta_{3} - 56) q^{7} - 512 q^{8} + 729 q^{9} + ( - 8 \beta_1 + 184) q^{10} + (7 \beta_{3} - 2 \beta_{2} - 3 \beta_1 - 1025) q^{11} - 1728 q^{12} + (3 \beta_{3} + 7 \beta_{2} - 13 \beta_1 - 1691) q^{13} + (8 \beta_{3} + 448) q^{14} + ( - 27 \beta_1 + 621) q^{15} + 4096 q^{16} + ( - 21 \beta_{3} + 2 \beta_{2} - 18 \beta_1 + 4728) q^{17} - 5832 q^{18} + ( - 18 \beta_{3} - 42 \beta_{2} - 23 \beta_1 + 4923) q^{19} + (64 \beta_1 - 1472) q^{20} + (27 \beta_{3} + 1512) q^{21} + ( - 56 \beta_{3} + 16 \beta_{2} + 24 \beta_1 + 8200) q^{22} + 12167 q^{23} + 13824 q^{24} + ( - 39 \beta_{3} + 135 \beta_{2} - 115 \beta_1 + 65364) q^{25} + ( - 24 \beta_{3} - 56 \beta_{2} + 104 \beta_1 + 13528) q^{26} - 19683 q^{27} + ( - 64 \beta_{3} - 3584) q^{28} + (276 \beta_{3} + 2 \beta_{2} + 310 \beta_1 + 3172) q^{29} + (216 \beta_1 - 4968) q^{30} + (98 \beta_{3} - 100 \beta_{2} - 80 \beta_1 + 110312) q^{31} - 32768 q^{32} + ( - 189 \beta_{3} + 54 \beta_{2} + 81 \beta_1 + 27675) q^{33} + (168 \beta_{3} - 16 \beta_{2} + 144 \beta_1 - 37824) q^{34} + (7 \beta_{3} - 147 \beta_{2} - 55 \beta_1 + 37581) q^{35} + 46656 q^{36} + (3 \beta_{3} - 252 \beta_{2} - 309 \beta_1 + 141959) q^{37} + (144 \beta_{3} + 336 \beta_{2} + 184 \beta_1 - 39384) q^{38} + ( - 81 \beta_{3} - 189 \beta_{2} + 351 \beta_1 + 45657) q^{39} + ( - 512 \beta_1 + 11776) q^{40} + ( - 556 \beta_{3} + 418 \beta_{2} - 464 \beta_1 + 68718) q^{41} + ( - 216 \beta_{3} - 12096) q^{42} + (248 \beta_{3} + 126 \beta_{2} - 701 \beta_1 + 37961) q^{43} + (448 \beta_{3} - 128 \beta_{2} - 192 \beta_1 - 65600) q^{44} + (729 \beta_1 - 16767) q^{45} - 97336 q^{46} + ( - 395 \beta_{3} + 651 \beta_{2} - 213 \beta_1 - 500273) q^{47} - 110592 q^{48} + (91 \beta_{3} - 455 \beta_{2} + 175 \beta_1 - 405880) q^{49} + (312 \beta_{3} - 1080 \beta_{2} + 920 \beta_1 - 522912) q^{50} + (567 \beta_{3} - 54 \beta_{2} + 486 \beta_1 - 127656) q^{51} + (192 \beta_{3} + 448 \beta_{2} - 832 \beta_1 - 108224) q^{52} + ( - 1792 \beta_{3} - 110 \beta_{2} - 1497 \beta_1 - 694777) q^{53} + 157464 q^{54} + ( - 532 \beta_{3} + 560 \beta_{2} - 2160 \beta_1 - 675688) q^{55} + (512 \beta_{3} + 28672) q^{56} + (486 \beta_{3} + 1134 \beta_{2} + 621 \beta_1 - 132921) q^{57} + ( - 2208 \beta_{3} - 16 \beta_{2} - 2480 \beta_1 - 25376) q^{58} + (1141 \beta_{3} - 2079 \beta_{2} + 999 \beta_1 - 914021) q^{59} + ( - 1728 \beta_1 + 39744) q^{60} + (1571 \beta_{3} + 306 \beta_{2} + 1747 \beta_1 - 247357) q^{61} + ( - 784 \beta_{3} + 800 \beta_{2} + 640 \beta_1 - 882496) q^{62} + ( - 729 \beta_{3} - 40824) q^{63} + 262144 q^{64} + (2586 \beta_{3} - 1090 \beta_{2} + 4416 \beta_1 - 1871304) q^{65} + (1512 \beta_{3} - 432 \beta_{2} - 648 \beta_1 - 221400) q^{66} + (1764 \beta_{3} + 170 \beta_{2} + 1757 \beta_1 - 1438817) q^{67} + ( - 1344 \beta_{3} + 128 \beta_{2} - 1152 \beta_1 + 302592) q^{68} - 328509 q^{69} + ( - 56 \beta_{3} + 1176 \beta_{2} + 440 \beta_1 - 300648) q^{70} + ( - 2016 \beta_{3} + 1022 \beta_{2} - 2200 \beta_1 - 2210764) q^{71} - 373248 q^{72} + ( - 462 \beta_{3} - 2034 \beta_{2} + 7346 \beta_1 - 278972) q^{73} + ( - 24 \beta_{3} + 2016 \beta_{2} + 2472 \beta_1 - 1135672) q^{74} + (1053 \beta_{3} - 3645 \beta_{2} + 3105 \beta_1 - 1764828) q^{75} + ( - 1152 \beta_{3} - 2688 \beta_{2} - 1472 \beta_1 + 315072) q^{76} + ( - 628 \beta_{3} + 3778 \beta_{2} + 32 \beta_1 - 2743700) q^{77} + (648 \beta_{3} + 1512 \beta_{2} - 2808 \beta_1 - 365256) q^{78} + ( - 4655 \beta_{3} + 6468 \beta_{2} - 700 \beta_1 + 237980) q^{79} + (4096 \beta_1 - 94208) q^{80} + 531441 q^{81} + (4448 \beta_{3} - 3344 \beta_{2} + 3712 \beta_1 - 549744) q^{82} + (2663 \beta_{3} - 112 \beta_{2} + 1381 \beta_1 - 6247005) q^{83} + (1728 \beta_{3} + 96768) q^{84} + (1449 \beta_{3} - 5453 \beta_{2} + 7809 \beta_1 - 1903539) q^{85} + ( - 1984 \beta_{3} - 1008 \beta_{2} + 5608 \beta_1 - 303688) q^{86} + ( - 7452 \beta_{3} - 54 \beta_{2} - 8370 \beta_1 - 85644) q^{87} + ( - 3584 \beta_{3} + 1024 \beta_{2} + 1536 \beta_1 + 524800) q^{88} + (2877 \beta_{3} + 376 \beta_{2} - 15734 \beta_1 - 896552) q^{89} + ( - 5832 \beta_1 + 134136) q^{90} + (6890 \beta_{3} + 2744 \beta_{2} - 3632 \beta_1 - 2353832) q^{91} + 778688 q^{92} + ( - 2646 \beta_{3} + 2700 \beta_{2} + 2160 \beta_1 - 2978424) q^{93} + (3160 \beta_{3} - 5208 \beta_{2} + 1704 \beta_1 + 4002184) q^{94} + ( - 11577 \beta_{3} - 7095 \beta_{2} - 22427 \beta_1 - 3091007) q^{95} + 884736 q^{96} + (452 \beta_{3} + 2412 \beta_{2} + 4646 \beta_1 + 3538332) q^{97} + ( - 728 \beta_{3} + 3640 \beta_{2} - 1400 \beta_1 + 3247040) q^{98} + (5103 \beta_{3} - 1458 \beta_{2} - 2187 \beta_1 - 747225) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 32 q^{2} - 108 q^{3} + 256 q^{4} - 90 q^{5} + 864 q^{6} - 222 q^{7} - 2048 q^{8} + 2916 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 32 q^{2} - 108 q^{3} + 256 q^{4} - 90 q^{5} + 864 q^{6} - 222 q^{7} - 2048 q^{8} + 2916 q^{9} + 720 q^{10} - 4120 q^{11} - 6912 q^{12} - 6796 q^{13} + 1776 q^{14} + 2430 q^{15} + 16384 q^{16} + 18918 q^{17} - 23328 q^{18} + 19682 q^{19} - 5760 q^{20} + 5994 q^{21} + 32960 q^{22} + 48668 q^{23} + 55296 q^{24} + 261304 q^{25} + 54368 q^{26} - 78732 q^{27} - 14208 q^{28} + 12756 q^{29} - 19440 q^{30} + 440892 q^{31} - 131072 q^{32} + 111240 q^{33} - 151344 q^{34} + 150200 q^{35} + 186624 q^{36} + 567212 q^{37} - 157456 q^{38} + 183492 q^{39} + 46080 q^{40} + 275056 q^{41} - 47952 q^{42} + 149946 q^{43} - 263680 q^{44} - 65610 q^{45} - 389344 q^{46} - 2000728 q^{47} - 442368 q^{48} - 1623352 q^{49} - 2090432 q^{50} - 510786 q^{51} - 434944 q^{52} - 2778518 q^{53} + 629856 q^{54} - 2706008 q^{55} + 113664 q^{56} - 531414 q^{57} - 102048 q^{58} - 3656368 q^{59} + 155520 q^{60} - 989076 q^{61} - 3527136 q^{62} - 161838 q^{63} + 1048576 q^{64} - 7481556 q^{65} - 889920 q^{66} - 5755282 q^{67} + 1210752 q^{68} - 1314036 q^{69} - 1201600 q^{70} - 8843424 q^{71} - 1492992 q^{72} - 1100272 q^{73} - 4537696 q^{74} - 7055208 q^{75} + 1259648 q^{76} - 10973480 q^{77} - 1467936 q^{78} + 959830 q^{79} - 368640 q^{80} + 2125764 q^{81} - 2200448 q^{82} - 24990584 q^{83} + 383616 q^{84} - 7601436 q^{85} - 1199568 q^{86} - 344412 q^{87} + 2109440 q^{88} - 3623430 q^{89} + 524880 q^{90} - 9436372 q^{91} + 3114752 q^{92} - 11904084 q^{93} + 16005824 q^{94} - 12385728 q^{95} + 3538944 q^{96} + 14161716 q^{97} + 12986816 q^{98} - 3003480 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 47804x^{2} - 3068607x + 114119793 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 34\nu^{3} - 2564\nu^{2} - 1118898\nu - 17857137 ) / 92517 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -34\nu^{3} + 2564\nu^{2} + 1674000\nu + 17764620 ) / 92517 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -116\nu^{3} + 16004\nu^{2} + 3487258\nu - 112321896 ) / 92517 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + \beta _1 + 1 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 153\beta_{3} + 91\beta_{2} + 613\beta _1 + 286597 ) / 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 5769\beta_{3} + 36340\beta_{2} + 72349\beta _1 + 13990561 ) / 6 \) 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
26.4468
−145.604
−122.254
242.411
−8.00000 −27.0000 64.0000 −548.447 216.000 63.4064 −512.000 729.000 4387.58
1.2 −8.00000 −27.0000 64.0000 −177.054 216.000 −891.394 −512.000 729.000 1416.44
1.3 −8.00000 −27.0000 64.0000 176.808 216.000 889.767 −512.000 729.000 −1414.46
1.4 −8.00000 −27.0000 64.0000 458.694 216.000 −283.780 −512.000 729.000 −3669.55
\(n\): 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.e 4
3.b odd 2 1 414.8.a.k 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
138.8.a.e 4 1.a even 1 1 trivial
414.8.a.k 4 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 90T_{5}^{3} - 282852T_{5}^{2} - 2871680T_{5} + 7875295680 \) 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
$5$ \( T^{4} + 90 T^{3} + \cdots + 7875295680 \) Copy content Toggle raw display
$7$ \( T^{4} + 222 T^{3} + \cdots + 14271219024 \) Copy content Toggle raw display
$11$ \( T^{4} + \cdots - 199582115587200 \) Copy content Toggle raw display
$13$ \( T^{4} + \cdots - 135949267224016 \) Copy content Toggle raw display
$17$ \( T^{4} - 18918 T^{3} + \cdots + 26\!\cdots\!12 \) Copy content Toggle raw display
$19$ \( T^{4} - 19682 T^{3} + \cdots + 65\!\cdots\!24 \) Copy content Toggle raw display
$23$ \( (T - 12167)^{4} \) Copy content Toggle raw display
$29$ \( T^{4} - 12756 T^{3} + \cdots + 15\!\cdots\!12 \) Copy content Toggle raw display
$31$ \( T^{4} - 440892 T^{3} + \cdots - 58\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{4} - 567212 T^{3} + \cdots - 38\!\cdots\!88 \) Copy content Toggle raw display
$41$ \( T^{4} - 275056 T^{3} + \cdots - 57\!\cdots\!68 \) Copy content Toggle raw display
$43$ \( T^{4} - 149946 T^{3} + \cdots + 66\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{4} + 2000728 T^{3} + \cdots - 61\!\cdots\!24 \) Copy content Toggle raw display
$53$ \( T^{4} + 2778518 T^{3} + \cdots + 67\!\cdots\!20 \) Copy content Toggle raw display
$59$ \( T^{4} + 3656368 T^{3} + \cdots + 13\!\cdots\!48 \) Copy content Toggle raw display
$61$ \( T^{4} + 989076 T^{3} + \cdots + 13\!\cdots\!32 \) Copy content Toggle raw display
$67$ \( T^{4} + 5755282 T^{3} + \cdots + 65\!\cdots\!12 \) Copy content Toggle raw display
$71$ \( T^{4} + 8843424 T^{3} + \cdots + 23\!\cdots\!72 \) Copy content Toggle raw display
$73$ \( T^{4} + 1100272 T^{3} + \cdots + 75\!\cdots\!32 \) Copy content Toggle raw display
$79$ \( T^{4} - 959830 T^{3} + \cdots + 36\!\cdots\!88 \) Copy content Toggle raw display
$83$ \( T^{4} + 24990584 T^{3} + \cdots + 13\!\cdots\!08 \) Copy content Toggle raw display
$89$ \( T^{4} + 3623430 T^{3} + \cdots + 14\!\cdots\!24 \) Copy content Toggle raw display
$97$ \( T^{4} - 14161716 T^{3} + \cdots + 57\!\cdots\!60 \) Copy content Toggle raw display
show more
show less