Properties

Label 138.6.a.h
Level $138$
Weight $6$
Character orbit 138.a
Self dual yes
Analytic conductor $22.133$
Analytic rank $0$
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: \(0\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
Defining polynomial: \( x^{3} - x^{2} - 1025x - 1873 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
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.

\(f(q)\) \(=\) \( q - 4 q^{2} + 9 q^{3} + 16 q^{4} + (\beta_1 - 1) q^{5} - 36 q^{6} + ( - \beta_{2} + \beta_1 - 11) q^{7} - 64 q^{8} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 4 q^{2} + 9 q^{3} + 16 q^{4} + (\beta_1 - 1) q^{5} - 36 q^{6} + ( - \beta_{2} + \beta_1 - 11) q^{7} - 64 q^{8} + 81 q^{9} + ( - 4 \beta_1 + 4) q^{10} + (\beta_{2} + 6 \beta_1 + 102) q^{11} + 144 q^{12} + (4 \beta_1 + 334) q^{13} + (4 \beta_{2} - 4 \beta_1 + 44) q^{14} + (9 \beta_1 - 9) q^{15} + 256 q^{16} + (\beta_{2} - 17 \beta_1 + 125) q^{17} - 324 q^{18} + ( - 33 \beta_1 + 755) q^{19} + (16 \beta_1 - 16) q^{20} + ( - 9 \beta_{2} + 9 \beta_1 - 99) q^{21} + ( - 4 \beta_{2} - 24 \beta_1 - 408) q^{22} - 529 q^{23} - 576 q^{24} + (10 \beta_{2} - 389) q^{25} + ( - 16 \beta_1 - 1336) q^{26} + 729 q^{27} + ( - 16 \beta_{2} + 16 \beta_1 - 176) q^{28} + ( - 10 \beta_{2} + 90 \beta_1 - 1248) q^{29} + ( - 36 \beta_1 + 36) q^{30} + (10 \beta_{2} + 30 \beta_1 + 3830) q^{31} - 1024 q^{32} + (9 \beta_{2} + 54 \beta_1 + 918) q^{33} + ( - 4 \beta_{2} + 68 \beta_1 - 500) q^{34} + (14 \beta_{2} - 146 \beta_1 + 1658) q^{35} + 1296 q^{36} + ( - \beta_{2} - 32 \beta_1 + 9418) q^{37} + (132 \beta_1 - 3020) q^{38} + (36 \beta_1 + 3006) q^{39} + ( - 64 \beta_1 + 64) q^{40} + ( - 10 \beta_{2} - 196 \beta_1 + 3934) q^{41} + (36 \beta_{2} - 36 \beta_1 + 396) q^{42} + (70 \beta_{2} + 27 \beta_1 + 11231) q^{43} + (16 \beta_{2} + 96 \beta_1 + 1632) q^{44} + (81 \beta_1 - 81) q^{45} + 2116 q^{46} + ( - 70 \beta_{2} + 146 \beta_1 + 10510) q^{47} + 2304 q^{48} + ( - 98 \beta_{2} - 224 \beta_1 + 20525) q^{49} + ( - 40 \beta_{2} + 1556) q^{50} + (9 \beta_{2} - 153 \beta_1 + 1125) q^{51} + (64 \beta_1 + 5344) q^{52} + ( - 22 \beta_{2} + 75 \beta_1 + 8073) q^{53} - 2916 q^{54} + (56 \beta_{2} + 244 \beta_1 + 17396) q^{55} + (64 \beta_{2} - 64 \beta_1 + 704) q^{56} + ( - 297 \beta_1 + 6795) q^{57} + (40 \beta_{2} - 360 \beta_1 + 4992) q^{58} + ( - 32 \beta_{2} + 78 \beta_1 - 4818) q^{59} + (144 \beta_1 - 144) q^{60} + ( - 111 \beta_{2} - 8 \beta_1 + 13738) q^{61} + ( - 40 \beta_{2} - 120 \beta_1 - 15320) q^{62} + ( - 81 \beta_{2} + 81 \beta_1 - 891) q^{63} + 4096 q^{64} + (40 \beta_{2} + 338 \beta_1 + 10606) q^{65} + ( - 36 \beta_{2} - 216 \beta_1 - 3672) q^{66} + (130 \beta_{2} + 493 \beta_1 + 6841) q^{67} + (16 \beta_{2} - 272 \beta_1 + 2000) q^{68} - 4761 q^{69} + ( - 56 \beta_{2} + 584 \beta_1 - 6632) q^{70} + (182 \beta_{2} - 600 \beta_1 - 16176) q^{71} - 5184 q^{72} + (152 \beta_{2} - 152 \beta_1 + 23914) q^{73} + (4 \beta_{2} + 128 \beta_1 - 37672) q^{74} + (90 \beta_{2} - 3501) q^{75} + ( - 528 \beta_1 + 12080) q^{76} + (98 \beta_{2} - 700 \beta_1 - 26804) q^{77} + ( - 144 \beta_1 - 12024) q^{78} + ( - 231 \beta_{2} + 1303 \beta_1 + 4075) q^{79} + (256 \beta_1 - 256) q^{80} + 6561 q^{81} + (40 \beta_{2} + 784 \beta_1 - 15736) q^{82} + (303 \beta_{2} - 434 \beta_1 - 19834) q^{83} + ( - 144 \beta_{2} + 144 \beta_1 - 1584) q^{84} + ( - 174 \beta_{2} + 244 \beta_1 - 45532) q^{85} + ( - 280 \beta_{2} - 108 \beta_1 - 44924) q^{86} + ( - 90 \beta_{2} + 810 \beta_1 - 11232) q^{87} + ( - 64 \beta_{2} - 384 \beta_1 - 6528) q^{88} + (41 \beta_{2} + 333 \beta_1 - 43269) q^{89} + ( - 324 \beta_1 + 324) q^{90} + ( - 282 \beta_{2} - 246 \beta_1 + 2914) q^{91} - 8464 q^{92} + (90 \beta_{2} + 270 \beta_1 + 34470) q^{93} + (280 \beta_{2} - 584 \beta_1 - 42040) q^{94} + ( - 330 \beta_{2} + 722 \beta_1 - 91010) q^{95} - 9216 q^{96} + ( - 208 \beta_{2} - 1422 \beta_1 + 7412) q^{97} + (392 \beta_{2} + 896 \beta_1 - 82100) q^{98} + (81 \beta_{2} + 486 \beta_1 + 8262) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 12 q^{2} + 27 q^{3} + 48 q^{4} - 4 q^{5} - 108 q^{6} - 34 q^{7} - 192 q^{8} + 243 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 12 q^{2} + 27 q^{3} + 48 q^{4} - 4 q^{5} - 108 q^{6} - 34 q^{7} - 192 q^{8} + 243 q^{9} + 16 q^{10} + 300 q^{11} + 432 q^{12} + 998 q^{13} + 136 q^{14} - 36 q^{15} + 768 q^{16} + 392 q^{17} - 972 q^{18} + 2298 q^{19} - 64 q^{20} - 306 q^{21} - 1200 q^{22} - 1587 q^{23} - 1728 q^{24} - 1167 q^{25} - 3992 q^{26} + 2187 q^{27} - 544 q^{28} - 3834 q^{29} + 144 q^{30} + 11460 q^{31} - 3072 q^{32} + 2700 q^{33} - 1568 q^{34} + 5120 q^{35} + 3888 q^{36} + 28286 q^{37} - 9192 q^{38} + 8982 q^{39} + 256 q^{40} + 11998 q^{41} + 1224 q^{42} + 33666 q^{43} + 4800 q^{44} - 324 q^{45} + 6348 q^{46} + 31384 q^{47} + 6912 q^{48} + 61799 q^{49} + 4668 q^{50} + 3528 q^{51} + 15968 q^{52} + 24144 q^{53} - 8748 q^{54} + 51944 q^{55} + 2176 q^{56} + 20682 q^{57} + 15336 q^{58} - 14532 q^{59} - 576 q^{60} + 41222 q^{61} - 45840 q^{62} - 2754 q^{63} + 12288 q^{64} + 31480 q^{65} - 10800 q^{66} + 20030 q^{67} + 6272 q^{68} - 14283 q^{69} - 20480 q^{70} - 47928 q^{71} - 15552 q^{72} + 71894 q^{73} - 113144 q^{74} - 10503 q^{75} + 36768 q^{76} - 79712 q^{77} - 35928 q^{78} + 10922 q^{79} - 1024 q^{80} + 19683 q^{81} - 47992 q^{82} - 59068 q^{83} - 4896 q^{84} - 136840 q^{85} - 134664 q^{86} - 34506 q^{87} - 19200 q^{88} - 130140 q^{89} + 1296 q^{90} + 8988 q^{91} - 25392 q^{92} + 103140 q^{93} - 125536 q^{94} - 273752 q^{95} - 27648 q^{96} + 23658 q^{97} - 247196 q^{98} + 24300 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 1025x - 1873 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu - 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2\nu^{2} - 4\nu - 1366 ) / 5 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 5\beta_{2} + 2\beta _1 + 1368 ) / 2 \) 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
−30.5473
−1.83665
33.3840
−4.00000 9.00000 16.0000 −63.0946 −36.0000 −197.588 −64.0000 81.0000 252.378
1.2 −4.00000 9.00000 16.0000 −5.67331 −36.0000 254.708 −64.0000 81.0000 22.6932
1.3 −4.00000 9.00000 16.0000 64.7679 −36.0000 −91.1204 −64.0000 81.0000 −259.072
\(n\): 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.h 3
3.b odd 2 1 414.6.a.m 3
4.b odd 2 1 1104.6.a.j 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
138.6.a.h 3 1.a even 1 1 trivial
414.6.a.m 3 3.b odd 2 1
1104.6.a.j 3 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{3} + 4T_{5}^{2} - 4096T_{5} - 23184 \) 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
$5$ \( T^{3} + 4 T^{2} - 4096 T - 23184 \) Copy content Toggle raw display
$7$ \( T^{3} + 34 T^{2} - 55532 T - 4585832 \) Copy content Toggle raw display
$11$ \( T^{3} - 300 T^{2} + \cdots - 18434592 \) Copy content Toggle raw display
$13$ \( T^{3} - 998 T^{2} + \cdots - 16119176 \) Copy content Toggle raw display
$17$ \( T^{3} - 392 T^{2} + \cdots - 72900432 \) Copy content Toggle raw display
$19$ \( T^{3} - 2298 T^{2} + \cdots + 3608499640 \) Copy content Toggle raw display
$23$ \( (T + 529)^{3} \) Copy content Toggle raw display
$29$ \( T^{3} + 3834 T^{2} + \cdots + 26873648952 \) Copy content Toggle raw display
$31$ \( T^{3} - 11460 T^{2} + \cdots - 22999096000 \) Copy content Toggle raw display
$37$ \( T^{3} - 28286 T^{2} + \cdots - 795432448904 \) Copy content Toggle raw display
$41$ \( T^{3} - 11998 T^{2} + \cdots + 1169089349784 \) Copy content Toggle raw display
$43$ \( T^{3} - 33666 T^{2} + \cdots + 3317925025400 \) Copy content Toggle raw display
$47$ \( T^{3} - 31384 T^{2} + \cdots + 2069618590080 \) Copy content Toggle raw display
$53$ \( T^{3} - 24144 T^{2} + \cdots - 90672025008 \) Copy content Toggle raw display
$59$ \( T^{3} + 14532 T^{2} + \cdots - 206257275840 \) Copy content Toggle raw display
$61$ \( T^{3} - 41222 T^{2} + \cdots + 54318607384 \) Copy content Toggle raw display
$67$ \( T^{3} - 20030 T^{2} + \cdots - 13525331979384 \) Copy content Toggle raw display
$71$ \( T^{3} + 47928 T^{2} + \cdots - 79612179492096 \) Copy content Toggle raw display
$73$ \( T^{3} - 71894 T^{2} + \cdots + 31081392542392 \) Copy content Toggle raw display
$79$ \( T^{3} + \cdots + 357863106252424 \) Copy content Toggle raw display
$83$ \( T^{3} + 59068 T^{2} + \cdots - 18660485323872 \) Copy content Toggle raw display
$89$ \( T^{3} + 130140 T^{2} + \cdots + 50621579393712 \) Copy content Toggle raw display
$97$ \( T^{3} + \cdots + 571541083003976 \) Copy content Toggle raw display
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