Properties

Label 138.6.a.g
Level $138$
Weight $6$
Character orbit 138.a
Self dual yes
Analytic conductor $22.133$
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: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 138.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(22.1329671342\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
Defining polynomial: \( x^{3} - x^{2} - 1383x - 16813 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
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.

\(f(q)\) \(=\) \( q - 4 q^{2} - 9 q^{3} + 16 q^{4} + ( - \beta_{2} - 6) q^{5} + 36 q^{6} + (2 \beta_{2} + \beta_1 - 17) q^{7} - 64 q^{8} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 4 q^{2} - 9 q^{3} + 16 q^{4} + ( - \beta_{2} - 6) q^{5} + 36 q^{6} + (2 \beta_{2} + \beta_1 - 17) q^{7} - 64 q^{8} + 81 q^{9} + (4 \beta_{2} + 24) q^{10} + (\beta_{2} - 15 \beta_1 + 7) q^{11} - 144 q^{12} + (3 \beta_{2} + 19 \beta_1 + 257) q^{13} + ( - 8 \beta_{2} - 4 \beta_1 + 68) q^{14} + (9 \beta_{2} + 54) q^{15} + 256 q^{16} + ( - 10 \beta_{2} + 29 \beta_1 + 373) q^{17} - 324 q^{18} + (7 \beta_{2} - 32 \beta_1 + 908) q^{19} + ( - 16 \beta_{2} - 96) q^{20} + ( - 18 \beta_{2} - 9 \beta_1 + 153) q^{21} + ( - 4 \beta_{2} + 60 \beta_1 - 28) q^{22} - 529 q^{23} + 576 q^{24} + ( - 13 \beta_{2} - 77 \beta_1 + 1010) q^{25} + ( - 12 \beta_{2} - 76 \beta_1 - 1028) q^{26} - 729 q^{27} + (32 \beta_{2} + 16 \beta_1 - 272) q^{28} + (2 \beta_{2} + 96 \beta_1 - 2002) q^{29} + ( - 36 \beta_{2} - 216) q^{30} + (52 \beta_{2} - 138 \beta_1 - 1362) q^{31} - 1024 q^{32} + ( - 9 \beta_{2} + 135 \beta_1 - 63) q^{33} + (40 \beta_{2} - 116 \beta_1 - 1492) q^{34} + (73 \beta_{2} + 123 \beta_1 - 8161) q^{35} + 1296 q^{36} + ( - 11 \beta_{2} + 231 \beta_1 - 3533) q^{37} + ( - 28 \beta_{2} + 128 \beta_1 - 3632) q^{38} + ( - 27 \beta_{2} - 171 \beta_1 - 2313) q^{39} + (64 \beta_{2} + 384) q^{40} + (50 \beta_{2} - 190 \beta_1 - 12532) q^{41} + (72 \beta_{2} + 36 \beta_1 - 612) q^{42} + ( - 61 \beta_{2} + 274 \beta_1 - 9630) q^{43} + (16 \beta_{2} - 240 \beta_1 + 112) q^{44} + ( - 81 \beta_{2} - 486) q^{45} + 2116 q^{46} + (125 \beta_{2} - 325 \beta_1 - 14341) q^{47} - 2304 q^{48} + ( - 235 \beta_{2} - 223 \beta_1 + 1054) q^{49} + (52 \beta_{2} + 308 \beta_1 - 4040) q^{50} + (90 \beta_{2} - 261 \beta_1 - 3357) q^{51} + (48 \beta_{2} + 304 \beta_1 + 4112) q^{52} + ( - 161 \beta_{2} - 460 \beta_1 - 13702) q^{53} + 2916 q^{54} + ( - 258 \beta_{2} + 542 \beta_1 - 3166) q^{55} + ( - 128 \beta_{2} - 64 \beta_1 + 1088) q^{56} + ( - 63 \beta_{2} + 288 \beta_1 - 8172) q^{57} + ( - 8 \beta_{2} - 384 \beta_1 + 8008) q^{58} + (49 \beta_{2} + 867 \beta_1 - 15809) q^{59} + (144 \beta_{2} + 864) q^{60} + ( - 459 \beta_{2} - 749 \beta_1 + 2891) q^{61} + ( - 208 \beta_{2} + 552 \beta_1 + 5448) q^{62} + (162 \beta_{2} + 81 \beta_1 - 1377) q^{63} + 4096 q^{64} + (142 \beta_{2} - 358 \beta_1 - 15074) q^{65} + (36 \beta_{2} - 540 \beta_1 + 252) q^{66} + (507 \beta_{2} - 366 \beta_1 - 5550) q^{67} + ( - 160 \beta_{2} + 464 \beta_1 + 5968) q^{68} + 4761 q^{69} + ( - 292 \beta_{2} - 492 \beta_1 + 32644) q^{70} + (356 \beta_{2} + 1044 \beta_1 - 13108) q^{71} - 5184 q^{72} + (372 \beta_{2} + 1112 \beta_1 - 4274) q^{73} + (44 \beta_{2} - 924 \beta_1 + 14132) q^{74} + (117 \beta_{2} + 693 \beta_1 - 9090) q^{75} + (112 \beta_{2} - 512 \beta_1 + 14528) q^{76} + (394 \beta_{2} - 902 \beta_1 - 7546) q^{77} + (108 \beta_{2} + 684 \beta_1 + 9252) q^{78} + (668 \beta_{2} - 381 \beta_1 + 24437) q^{79} + ( - 256 \beta_{2} - 1536) q^{80} + 6561 q^{81} + ( - 200 \beta_{2} + 760 \beta_1 + 50128) q^{82} + (609 \beta_{2} - 1439 \beta_1 - 16221) q^{83} + ( - 288 \beta_{2} - 144 \beta_1 + 2448) q^{84} + ( - 41 \beta_{2} - 1669 \beta_1 + 36867) q^{85} + (244 \beta_{2} - 1096 \beta_1 + 38520) q^{86} + ( - 18 \beta_{2} - 864 \beta_1 + 18018) q^{87} + ( - 64 \beta_{2} + 960 \beta_1 - 448) q^{88} + ( - 988 \beta_{2} + 1575 \beta_1 + 14879) q^{89} + (324 \beta_{2} + 1944) q^{90} + ( - 330 \beta_{2} + 858 \beta_1 + 40294) q^{91} - 8464 q^{92} + ( - 468 \beta_{2} + 1242 \beta_1 + 12258) q^{93} + ( - 500 \beta_{2} + 1300 \beta_1 + 57364) q^{94} + ( - 1351 \beta_{2} + 1531 \beta_1 - 32061) q^{95} + 9216 q^{96} + (138 \beta_{2} + 3592 \beta_1 - 35078) q^{97} + (940 \beta_{2} + 892 \beta_1 - 4216) q^{98} + (81 \beta_{2} - 1215 \beta_1 + 567) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 12 q^{2} - 27 q^{3} + 48 q^{4} - 18 q^{5} + 108 q^{6} - 50 q^{7} - 192 q^{8} + 243 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 12 q^{2} - 27 q^{3} + 48 q^{4} - 18 q^{5} + 108 q^{6} - 50 q^{7} - 192 q^{8} + 243 q^{9} + 72 q^{10} + 6 q^{11} - 432 q^{12} + 790 q^{13} + 200 q^{14} + 162 q^{15} + 768 q^{16} + 1148 q^{17} - 972 q^{18} + 2692 q^{19} - 288 q^{20} + 450 q^{21} - 24 q^{22} - 1587 q^{23} + 1728 q^{24} + 2953 q^{25} - 3160 q^{26} - 2187 q^{27} - 800 q^{28} - 5910 q^{29} - 648 q^{30} - 4224 q^{31} - 3072 q^{32} - 54 q^{33} - 4592 q^{34} - 24360 q^{35} + 3888 q^{36} - 10368 q^{37} - 10768 q^{38} - 7110 q^{39} + 1152 q^{40} - 37786 q^{41} - 1800 q^{42} - 28616 q^{43} + 96 q^{44} - 1458 q^{45} + 6348 q^{46} - 43348 q^{47} - 6912 q^{48} + 2939 q^{49} - 11812 q^{50} - 10332 q^{51} + 12640 q^{52} - 41566 q^{53} + 8748 q^{54} - 8956 q^{55} + 3200 q^{56} - 24228 q^{57} + 23640 q^{58} - 46560 q^{59} + 2592 q^{60} + 7924 q^{61} + 16896 q^{62} - 4050 q^{63} + 12288 q^{64} - 45580 q^{65} + 216 q^{66} - 17016 q^{67} + 18368 q^{68} + 14283 q^{69} + 97440 q^{70} - 38280 q^{71} - 15552 q^{72} - 11710 q^{73} + 41472 q^{74} - 26577 q^{75} + 43072 q^{76} - 23540 q^{77} + 28440 q^{78} + 72930 q^{79} - 4608 q^{80} + 19683 q^{81} + 151144 q^{82} - 50102 q^{83} + 7200 q^{84} + 108932 q^{85} + 114464 q^{86} + 53190 q^{87} - 384 q^{88} + 46212 q^{89} + 5832 q^{90} + 121740 q^{91} - 25392 q^{92} + 38016 q^{93} + 173392 q^{94} - 94652 q^{95} + 27648 q^{96} - 101642 q^{97} - 11756 q^{98} + 486 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 1383x - 16813 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} - 19\nu - 916 ) / 5 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 5\beta_{2} + 19\beta _1 + 916 \) 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
−27.1335
42.6589
−14.5254
−4.00000 −9.00000 16.0000 −73.1526 36.0000 90.1716 −64.0000 81.0000 292.610
1.2 −4.00000 −9.00000 16.0000 −24.6530 36.0000 62.9650 −64.0000 81.0000 98.6122
1.3 −4.00000 −9.00000 16.0000 79.8056 36.0000 −203.137 −64.0000 81.0000 −319.222
\(n\): 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 138.6.a.g 3
3.b odd 2 1 414.6.a.n 3
4.b odd 2 1 1104.6.a.k 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
138.6.a.g 3 1.a even 1 1 trivial
414.6.a.n 3 3.b odd 2 1
1104.6.a.k 3 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{3} + 18T_{5}^{2} - 6002T_{5} - 143924 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(138))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 4)^{3} \) Copy content Toggle raw display
$3$ \( (T + 9)^{3} \) Copy content Toggle raw display
$5$ \( T^{3} + 18 T^{2} - 6002 T - 143924 \) Copy content Toggle raw display
$7$ \( T^{3} + 50 T^{2} - 25430 T + 1153340 \) Copy content Toggle raw display
$11$ \( T^{3} - 6 T^{2} - 314048 T + 41102592 \) Copy content Toggle raw display
$13$ \( T^{3} - 790 T^{2} + \cdots - 17724640 \) Copy content Toggle raw display
$17$ \( T^{3} - 1148 T^{2} + \cdots + 1251288504 \) Copy content Toggle raw display
$19$ \( T^{3} - 2692 T^{2} + \cdots + 566361936 \) Copy content Toggle raw display
$23$ \( (T + 529)^{3} \) Copy content Toggle raw display
$29$ \( T^{3} + 5910 T^{2} + \cdots - 33999878760 \) Copy content Toggle raw display
$31$ \( T^{3} + 4224 T^{2} + \cdots - 140877524608 \) Copy content Toggle raw display
$37$ \( T^{3} + 10368 T^{2} + \cdots - 383183552304 \) Copy content Toggle raw display
$41$ \( T^{3} + 37786 T^{2} + \cdots + 1113643632728 \) Copy content Toggle raw display
$43$ \( T^{3} + 28616 T^{2} + \cdots - 163187296392 \) Copy content Toggle raw display
$47$ \( T^{3} + 43348 T^{2} + \cdots - 1511593331104 \) Copy content Toggle raw display
$53$ \( T^{3} + 41566 T^{2} + \cdots - 2969882763452 \) Copy content Toggle raw display
$59$ \( T^{3} + 46560 T^{2} + \cdots - 25961931296880 \) Copy content Toggle raw display
$61$ \( T^{3} - 7924 T^{2} + \cdots - 15216738764992 \) Copy content Toggle raw display
$67$ \( T^{3} + 17016 T^{2} + \cdots - 19674353996712 \) Copy content Toggle raw display
$71$ \( T^{3} + 38280 T^{2} + \cdots - 39250356203520 \) Copy content Toggle raw display
$73$ \( T^{3} + 11710 T^{2} + \cdots - 24824926300760 \) Copy content Toggle raw display
$79$ \( T^{3} - 72930 T^{2} + \cdots + 44957071020660 \) Copy content Toggle raw display
$83$ \( T^{3} + \cdots - 200825772591136 \) Copy content Toggle raw display
$89$ \( T^{3} + \cdots + 460268929768536 \) Copy content Toggle raw display
$97$ \( T^{3} + 101642 T^{2} + \cdots - 14\!\cdots\!36 \) Copy content Toggle raw display
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