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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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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [138,5,Mod(91,138)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(138, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("138.91");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 138 = 2 \cdot 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 138.b (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(14.2650549056\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 1428 x^{14} - 600 x^{13} + 788282 x^{12} - 529464 x^{11} + 213396724 x^{10} + \cdots + 274129967370817 \) 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

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

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 \( q + \beta_{3} q^{2} - \beta_1 q^{3} + 8 q^{4} + \beta_{4} q^{5} + \beta_{7} q^{6} + ( - \beta_{10} - \beta_{2}) q^{7} + 8 \beta_{3} q^{8} + 27 q^{9} + (\beta_{11} - \beta_{6} - \beta_{4}) q^{10} + (\beta_{11} + \beta_{9} + \cdots - \beta_{6}) q^{11}+ \cdots + (27 \beta_{11} + 27 \beta_{9} + \cdots - 27 \beta_{6}) q^{99}+O(q^{100}) \) 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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 1428 x^{14} - 600 x^{13} + 788282 x^{12} - 529464 x^{11} + 213396724 x^{10} + \cdots + 274129967370817 \) : 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

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( 4\beta_{11} - 2\beta_{10} - \beta_{9} - \beta_{8} + 2\beta_{5} - 6\beta_{4} + 6\beta_{3} + 3\beta_{2} ) / 24 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 6 \beta_{12} - \beta_{11} + \beta_{10} - 4 \beta_{9} + 2 \beta_{8} + 69 \beta_{7} - \beta_{6} + \cdots - 2142 ) / 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 6 \beta_{14} - 3 \beta_{13} + 3 \beta_{12} - 314 \beta_{11} + 294 \beta_{10} - 218 \beta_{9} + \cdots + 897 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 207 \beta_{15} + 225 \beta_{14} - 273 \beta_{13} - 1056 \beta_{12} + 130 \beta_{11} - 445 \beta_{10} + \cdots + 346956 ) / 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 2640 \beta_{15} - 10710 \beta_{14} + 8175 \beta_{13} + 465 \beta_{12} + 232858 \beta_{11} + \cdots - 2450985 ) / 24 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 95779 \beta_{15} - 129999 \beta_{14} + 140836 \beta_{13} + 285468 \beta_{12} - 38759 \beta_{11} + \cdots - 87285308 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 2616726 \beta_{15} + 6429192 \beta_{14} - 5957931 \beta_{13} - 3585225 \beta_{12} - 64199360 \beta_{11} + \cdots + 1778327733 ) / 24 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 78997806 \beta_{15} + 120304872 \beta_{14} - 125610711 \beta_{13} - 191939826 \beta_{12} + \cdots + 54619049958 ) / 6 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 640224756 \beta_{15} - 1301336994 \beta_{14} + 1294759767 \beta_{13} + 1135093377 \beta_{12} + \cdots - 384379493385 ) / 8 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 80923832097 \beta_{15} - 130981645953 \beta_{14} + 135072607956 \beta_{13} + 180487318938 \beta_{12} + \cdots - 48562971684354 ) / 12 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 622904220576 \beta_{15} + 1166267923854 \beta_{14} - 1188448924542 \beta_{13} - 1209542625060 \beta_{12} + \cdots + 350041077130668 ) / 12 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 6704932938032 \beta_{15} + 11208157141704 \beta_{14} - 11515164797807 \beta_{13} - 14407096601019 \beta_{12} + \cdots + 37\!\cdots\!57 ) / 2 \) 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

Display \(\beta_i\) in terms of \(\nu^j\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
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
$5$ \( T^{16} + \cdots + 15\!\cdots\!84 \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 93\!\cdots\!64 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 10\!\cdots\!44 \) Copy content Toggle raw display
$13$ \( (T^{8} + \cdots + 14\!\cdots\!84)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 11\!\cdots\!56 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 34\!\cdots\!76 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 37\!\cdots\!21 \) Copy content Toggle raw display
$29$ \( (T^{8} + \cdots + 39\!\cdots\!32)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + \cdots + 33\!\cdots\!68)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 36\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots - 16\!\cdots\!96)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 56\!\cdots\!96 \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots - 54\!\cdots\!12)^{2} \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 58\!\cdots\!64 \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots - 23\!\cdots\!28)^{2} \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 33\!\cdots\!04 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 81\!\cdots\!76 \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots - 14\!\cdots\!36)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots - 11\!\cdots\!76)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 11\!\cdots\!84 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 10\!\cdots\!56 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 31\!\cdots\!84 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 75\!\cdots\!24 \) Copy content Toggle raw display
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