Properties

Label 138.5.b.a
Level $138$
Weight $5$
Character orbit 138.b
Analytic conductor $14.265$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 138 = 2 \cdot 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 138.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(14.2650549056\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Defining polynomial: \( x^{16} + 1428 x^{14} - 600 x^{13} + 788282 x^{12} - 529464 x^{11} + 213396724 x^{10} - 175079484 x^{9} + 29130946113 x^{8} - 26553654912 x^{7} + \cdots + 274129967370817 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{22}\cdot 3^{4}\cdot 7^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} - \beta_1 q^{3} + 8 q^{4} + \beta_{4} q^{5} + \beta_{7} q^{6} + ( - \beta_{10} - \beta_{2}) q^{7} + 8 \beta_{3} q^{8} + 27 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{2} - \beta_1 q^{3} + 8 q^{4} + \beta_{4} q^{5} + \beta_{7} q^{6} + ( - \beta_{10} - \beta_{2}) q^{7} + 8 \beta_{3} q^{8} + 27 q^{9} + (\beta_{11} - \beta_{6} - \beta_{4}) q^{10} + (\beta_{11} + \beta_{9} + \beta_{8} - \beta_{6}) q^{11} - 8 \beta_1 q^{12} + (\beta_{12} + 2 \beta_{7} + 15 \beta_{3} + 5 \beta_1 - 13) q^{13} + ( - \beta_{11} - 2 \beta_{10} + \beta_{9} - 2 \beta_{6} - \beta_{5} - 3 \beta_{4} + \beta_{2}) q^{14} + (\beta_{11} - 3 \beta_{9} - \beta_{6} - \beta_{5} + \beta_{4} - \beta_{2}) q^{15} + 64 q^{16} + (\beta_{11} - \beta_{10} + 5 \beta_{9} - \beta_{8} - \beta_{6} - 3 \beta_{5} + 3 \beta_{4} + \beta_{2}) q^{17} + 27 \beta_{3} q^{18} + ( - \beta_{11} - \beta_{10} + 4 \beta_{9} - 3 \beta_{8} - 3 \beta_{6} + 5 \beta_{5} + \cdots - 2 \beta_{2}) q^{19}+ \cdots + (27 \beta_{11} + 27 \beta_{9} + 27 \beta_{8} - 27 \beta_{6}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 128 q^{4} + 432 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 128 q^{4} + 432 q^{9} - 208 q^{13} + 1024 q^{16} + 840 q^{23} + 1056 q^{25} + 1920 q^{26} + 3600 q^{29} + 224 q^{31} - 3264 q^{35} + 3456 q^{36} - 2016 q^{39} - 6144 q^{41} + 1280 q^{46} + 8880 q^{47} - 13888 q^{49} + 7296 q^{50} - 1664 q^{52} + 832 q^{55} + 2944 q^{58} - 18240 q^{59} + 8192 q^{64} + 10584 q^{69} + 19584 q^{70} - 30048 q^{71} + 9536 q^{73} - 4176 q^{75} + 14160 q^{77} + 6912 q^{78} + 11664 q^{81} - 19584 q^{82} - 32496 q^{85} - 8064 q^{87} + 6720 q^{92} - 11952 q^{93} - 21248 q^{94} - 20064 q^{95} + 21504 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 1428 x^{14} - 600 x^{13} + 788282 x^{12} - 529464 x^{11} + 213396724 x^{10} - 175079484 x^{9} + 29130946113 x^{8} - 26553654912 x^{7} + \cdots + 274129967370817 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 83\!\cdots\!48 \nu^{15} + \cdots + 94\!\cdots\!30 ) / 19\!\cdots\!73 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 46\!\cdots\!24 \nu^{15} + \cdots - 28\!\cdots\!74 ) / 29\!\cdots\!15 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 25\!\cdots\!76 \nu^{15} + \cdots + 26\!\cdots\!84 ) / 10\!\cdots\!93 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 89\!\cdots\!10 \nu^{15} + \cdots + 25\!\cdots\!47 ) / 29\!\cdots\!15 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 15\!\cdots\!50 \nu^{15} + \cdots + 69\!\cdots\!94 ) / 29\!\cdots\!15 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 24\!\cdots\!86 \nu^{15} + \cdots + 11\!\cdots\!57 ) / 29\!\cdots\!15 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 22\!\cdots\!48 \nu^{15} + \cdots - 19\!\cdots\!56 ) / 16\!\cdots\!89 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 61\!\cdots\!76 \nu^{15} + \cdots - 52\!\cdots\!55 ) / 29\!\cdots\!15 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 61\!\cdots\!84 \nu^{15} + \cdots - 25\!\cdots\!77 ) / 29\!\cdots\!15 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 21\!\cdots\!66 \nu^{15} + \cdots - 52\!\cdots\!78 ) / 98\!\cdots\!05 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 69\!\cdots\!16 \nu^{15} + \cdots - 84\!\cdots\!76 ) / 29\!\cdots\!15 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 45\!\cdots\!68 \nu^{15} + \cdots + 19\!\cdots\!51 ) / 75\!\cdots\!21 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 10\!\cdots\!88 \nu^{15} + \cdots - 31\!\cdots\!84 ) / 75\!\cdots\!21 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 17\!\cdots\!56 \nu^{15} + \cdots + 55\!\cdots\!67 ) / 10\!\cdots\!03 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 29\!\cdots\!64 \nu^{15} + \cdots - 16\!\cdots\!87 ) / 75\!\cdots\!21 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( 4\beta_{11} - 2\beta_{10} - \beta_{9} - \beta_{8} + 2\beta_{5} - 6\beta_{4} + 6\beta_{3} + 3\beta_{2} ) / 24 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 6 \beta_{12} - \beta_{11} + \beta_{10} - 4 \beta_{9} + 2 \beta_{8} + 69 \beta_{7} - \beta_{6} + \beta_{5} + 7 \beta_{4} + 225 \beta_{3} - \beta_{2} + 178 \beta _1 - 2142 ) / 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 6 \beta_{14} - 3 \beta_{13} + 3 \beta_{12} - 314 \beta_{11} + 294 \beta_{10} - 218 \beta_{9} + 104 \beta_{8} - 86 \beta_{7} + 24 \beta_{6} - 284 \beta_{5} + 934 \beta_{4} - 1073 \beta_{3} - 444 \beta_{2} - 276 \beta _1 + 897 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 207 \beta_{15} + 225 \beta_{14} - 273 \beta_{13} - 1056 \beta_{12} + 130 \beta_{11} - 445 \beta_{10} + 1444 \beta_{9} - 695 \beta_{8} - 16942 \beta_{7} + 133 \beta_{6} + 128 \beta_{5} - 2554 \beta_{4} - 61047 \beta_{3} + \cdots + 346956 ) / 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 2640 \beta_{15} - 10710 \beta_{14} + 8175 \beta_{13} + 465 \beta_{12} + 232858 \beta_{11} - 364995 \beta_{10} + 502551 \beta_{9} - 149754 \beta_{8} + 202620 \beta_{7} - 81121 \beta_{6} + \cdots - 2450985 ) / 24 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 95779 \beta_{15} - 129999 \beta_{14} + 140836 \beta_{13} + 285468 \beta_{12} - 38759 \beta_{11} + 229578 \beta_{10} - 601124 \beta_{9} + 233506 \beta_{8} + 5079826 \beta_{7} + \cdots - 87285308 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 2616726 \beta_{15} + 6429192 \beta_{14} - 5957931 \beta_{13} - 3585225 \beta_{12} - 64199360 \beta_{11} + 166503866 \beta_{10} - 289108556 \beta_{9} + 73428808 \beta_{8} + \cdots + 1778327733 ) / 24 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 78997806 \beta_{15} + 120304872 \beta_{14} - 125610711 \beta_{13} - 191939826 \beta_{12} + 32215660 \beta_{11} - 232020092 \beta_{10} + 537300314 \beta_{9} + \cdots + 54619049958 ) / 6 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 640224756 \beta_{15} - 1301336994 \beta_{14} + 1294759767 \beta_{13} + 1135093377 \beta_{12} + 6618013706 \beta_{11} - 26451671015 \beta_{10} + \cdots - 384379493385 ) / 8 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 80923832097 \beta_{15} - 130981645953 \beta_{14} + 135072607956 \beta_{13} + 180487318938 \beta_{12} - 36500348570 \beta_{11} + 287999473976 \beta_{10} + \cdots - 48562971684354 ) / 12 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 622904220576 \beta_{15} + 1166267923854 \beta_{14} - 1188448924542 \beta_{13} - 1209542625060 \beta_{12} - 3440004036319 \beta_{11} + \cdots + 350041077130668 ) / 12 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 6704932938032 \beta_{15} + 11208157141704 \beta_{14} - 11515164797807 \beta_{13} - 14407096601019 \beta_{12} + 3369791447932 \beta_{11} + \cdots + 37\!\cdots\!57 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 755132091143586 \beta_{15} + \cdots - 40\!\cdots\!46 ) / 24 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 19\!\cdots\!18 \beta_{15} + \cdots - 10\!\cdots\!63 ) / 12 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 14\!\cdots\!92 \beta_{15} + \cdots + 77\!\cdots\!09 ) / 8 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/138\mathbb{Z}\right)^\times\).

\(n\) \(47\) \(97\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
91.1
−0.707107 + 1.93510i
−0.707107 + 11.7850i
−0.707107 11.7850i
−0.707107 1.93510i
−0.707107 + 22.2857i
−0.707107 16.9956i
−0.707107 + 16.9956i
−0.707107 22.2857i
0.707107 + 1.47108i
0.707107 + 16.3207i
0.707107 16.3207i
0.707107 1.47108i
0.707107 14.6559i
0.707107 4.53246i
0.707107 + 4.53246i
0.707107 + 14.6559i
−2.82843 −5.19615 8.00000 30.1826i 14.6969 56.6877i −22.6274 27.0000 85.3694i
91.2 −2.82843 −5.19615 8.00000 26.7045i 14.6969 66.7315i −22.6274 27.0000 75.5318i
91.3 −2.82843 −5.19615 8.00000 26.7045i 14.6969 66.7315i −22.6274 27.0000 75.5318i
91.4 −2.82843 −5.19615 8.00000 30.1826i 14.6969 56.6877i −22.6274 27.0000 85.3694i
91.5 −2.82843 5.19615 8.00000 34.5847i −14.6969 81.5113i −22.6274 27.0000 97.8202i
91.6 −2.82843 5.19615 8.00000 7.78845i −14.6969 25.8071i −22.6274 27.0000 22.0291i
91.7 −2.82843 5.19615 8.00000 7.78845i −14.6969 25.8071i −22.6274 27.0000 22.0291i
91.8 −2.82843 5.19615 8.00000 34.5847i −14.6969 81.5113i −22.6274 27.0000 97.8202i
91.9 2.82843 −5.19615 8.00000 19.9722i −14.6969 58.9751i 22.6274 27.0000 56.4898i
91.10 2.82843 −5.19615 8.00000 3.47446i −14.6969 45.9507i 22.6274 27.0000 9.82724i
91.11 2.82843 −5.19615 8.00000 3.47446i −14.6969 45.9507i 22.6274 27.0000 9.82724i
91.12 2.82843 −5.19615 8.00000 19.9722i −14.6969 58.9751i 22.6274 27.0000 56.4898i
91.13 2.82843 5.19615 8.00000 33.4782i 14.6969 19.6774i 22.6274 27.0000 94.6906i
91.14 2.82843 5.19615 8.00000 7.70519i 14.6969 72.1011i 22.6274 27.0000 21.7936i
91.15 2.82843 5.19615 8.00000 7.70519i 14.6969 72.1011i 22.6274 27.0000 21.7936i
91.16 2.82843 5.19615 8.00000 33.4782i 14.6969 19.6774i 22.6274 27.0000 94.6906i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 91.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 138.5.b.a 16
3.b odd 2 1 414.5.b.b 16
4.b odd 2 1 1104.5.c.a 16
23.b odd 2 1 inner 138.5.b.a 16
69.c even 2 1 414.5.b.b 16
92.b even 2 1 1104.5.c.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
138.5.b.a 16 1.a even 1 1 trivial
138.5.b.a 16 23.b odd 2 1 inner
414.5.b.b 16 3.b odd 2 1
414.5.b.b 16 69.c even 2 1
1104.5.c.a 16 4.b odd 2 1
1104.5.c.a 16 92.b even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(138, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 8)^{8} \) Copy content Toggle raw display
$3$ \( (T^{2} - 27)^{8} \) Copy content Toggle raw display
$5$ \( T^{16} + 4472 T^{14} + \cdots + 15\!\cdots\!84 \) Copy content Toggle raw display
$7$ \( T^{16} + 26152 T^{14} + \cdots + 93\!\cdots\!64 \) Copy content Toggle raw display
$11$ \( T^{16} + 66016 T^{14} + \cdots + 10\!\cdots\!44 \) Copy content Toggle raw display
$13$ \( (T^{8} + 104 T^{7} + \cdots + 14\!\cdots\!84)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + 465080 T^{14} + \cdots + 11\!\cdots\!56 \) Copy content Toggle raw display
$19$ \( T^{16} + 1100968 T^{14} + \cdots + 34\!\cdots\!76 \) Copy content Toggle raw display
$23$ \( T^{16} - 840 T^{15} + \cdots + 37\!\cdots\!21 \) Copy content Toggle raw display
$29$ \( (T^{8} - 1800 T^{7} + \cdots + 39\!\cdots\!32)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} - 112 T^{7} + \cdots + 33\!\cdots\!68)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} + 17417984 T^{14} + \cdots + 36\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{8} + 3072 T^{7} + \cdots - 16\!\cdots\!96)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + 37879432 T^{14} + \cdots + 56\!\cdots\!96 \) Copy content Toggle raw display
$47$ \( (T^{8} - 4440 T^{7} + \cdots - 54\!\cdots\!12)^{2} \) Copy content Toggle raw display
$53$ \( T^{16} + 44427608 T^{14} + \cdots + 58\!\cdots\!64 \) Copy content Toggle raw display
$59$ \( (T^{8} + 9120 T^{7} + \cdots - 23\!\cdots\!28)^{2} \) Copy content Toggle raw display
$61$ \( T^{16} + 130838848 T^{14} + \cdots + 33\!\cdots\!04 \) Copy content Toggle raw display
$67$ \( T^{16} + 231747432 T^{14} + \cdots + 81\!\cdots\!76 \) Copy content Toggle raw display
$71$ \( (T^{8} + 15024 T^{7} + \cdots - 14\!\cdots\!36)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} - 4768 T^{7} + \cdots - 11\!\cdots\!76)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + 314445032 T^{14} + \cdots + 11\!\cdots\!84 \) Copy content Toggle raw display
$83$ \( T^{16} + 630785248 T^{14} + \cdots + 10\!\cdots\!56 \) Copy content Toggle raw display
$89$ \( T^{16} + 693317560 T^{14} + \cdots + 31\!\cdots\!84 \) Copy content Toggle raw display
$97$ \( T^{16} + 720094784 T^{14} + \cdots + 75\!\cdots\!24 \) Copy content Toggle raw display
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