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Properties

Label 147.5.b.f
Level $147$
Weight $5$
Character orbit 147.b
Analytic conductor $15.195$
Analytic rank $0$
Dimension $8$
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] = [147,5,Mod(50,147)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(147, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("147.50");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 147 = 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 147.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: \(15.1953845733\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 103x^{6} + 3382x^{4} + 38724x^{2} + 75096 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 3^{3} \)
Twist minimal: no (minimal twist has level 21)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{2} + 1) q^{3} + (\beta_{3} - \beta_{2} - 10) q^{4} + (\beta_{5} - \beta_{2}) q^{5} + ( - \beta_{7} + \beta_{6} + \beta_{4} + \cdots - 2) q^{6}+ \cdots + (3 \beta_{7} + \beta_{4} - 4 \beta_{3} + \cdots - 5) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{2} + 1) q^{3} + (\beta_{3} - \beta_{2} - 10) q^{4} + (\beta_{5} - \beta_{2}) q^{5} + ( - \beta_{7} + \beta_{6} + \beta_{4} + \cdots - 2) q^{6}+ \cdots + ( - 123 \beta_{7} + 285 \beta_{6} + \cdots - 3053) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{3} - 78 q^{4} - 14 q^{6} - 62 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{3} - 78 q^{4} - 14 q^{6} - 62 q^{9} - 98 q^{10} + 436 q^{12} + 246 q^{13} - 428 q^{15} - 150 q^{16} + 884 q^{18} - 1326 q^{19} + 1622 q^{22} + 630 q^{24} - 2094 q^{25} - 514 q^{27} - 340 q^{30} - 2504 q^{31} - 616 q^{33} + 3864 q^{34} - 4822 q^{36} - 1342 q^{37} + 2626 q^{39} + 5754 q^{40} + 730 q^{43} - 1330 q^{45} - 8844 q^{46} - 12932 q^{48} - 7794 q^{51} + 9536 q^{52} - 18578 q^{54} + 17570 q^{55} + 5848 q^{57} + 12446 q^{58} + 18890 q^{60} + 15012 q^{61} + 3830 q^{64} - 17192 q^{66} - 8658 q^{67} - 26838 q^{69} - 21312 q^{72} + 12322 q^{73} - 19514 q^{75} + 42260 q^{76} - 16060 q^{78} + 40168 q^{79} + 23986 q^{81} + 47348 q^{82} + 11268 q^{85} - 1162 q^{87} - 56430 q^{88} - 69818 q^{90} - 8556 q^{93} + 6468 q^{94} - 40894 q^{96} + 14134 q^{97} - 23906 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 103x^{6} + 3382x^{4} + 38724x^{2} + 75096 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} + 4\nu^{6} + 69\nu^{5} + 332\nu^{4} + 812\nu^{3} + 6552\nu^{2} - 6132\nu + 13440 ) / 1344 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} + 4\nu^{6} + 69\nu^{5} + 332\nu^{4} + 812\nu^{3} + 7896\nu^{2} - 6132\nu + 48384 ) / 1344 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -3\nu^{7} + 7\nu^{6} - 235\nu^{5} + 539\nu^{4} - 4088\nu^{3} + 9660\nu^{2} + 756\nu + 17220 ) / 672 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -9\nu^{7} + 4\nu^{6} - 733\nu^{5} + 332\nu^{4} - 15260\nu^{3} + 6552\nu^{2} - 63756\nu + 13440 ) / 1344 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{7} - 8\nu^{6} + 69\nu^{5} - 664\nu^{4} + 812\nu^{3} - 13776\nu^{2} - 6132\nu - 44352 ) / 672 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} - 32\nu^{6} + 69\nu^{5} - 2488\nu^{4} + 812\nu^{3} - 45192\nu^{2} - 6132\nu - 83664 ) / 1344 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - \beta_{2} - 26 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} - \beta_{5} + 2\beta_{4} + 2\beta_{2} - 36\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 8\beta_{7} - 12\beta_{6} - 55\beta_{3} + 71\beta_{2} + 976 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -71\beta_{7} - 20\beta_{6} + 59\beta_{5} - 142\beta_{4} - 20\beta_{3} - 190\beta_{2} + 1494\beta _1 - 71 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -664\beta_{7} + 940\beta_{6} + 2871\beta_{3} - 4087\beta_{2} - 41780 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 4087 \beta_{7} + 1604 \beta_{6} - 3259 \beta_{5} + 8174 \beta_{4} + 1604 \beta_{3} + 12158 \beta_{2} + \cdots + 4087 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/147\mathbb{Z}\right)^\times\).

\(n\) \(50\) \(52\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
50.1
− 7.14374i
− 5.16081i
− 4.78777i
− 1.55250i
1.55250i
4.78777i
5.16081i
7.14374i
− 7.14374i −7.99321 − 4.13625i −35.0330 − 15.7492i −29.5483 + 57.1014i 0 135.967i 46.7828 + 66.1239i −112.508
50.2 − 5.16081i 1.83981 + 8.80994i −10.6340 36.6448i 45.4664 − 9.49493i 0 − 27.6932i −74.2302 + 32.4173i 189.117
50.3 − 4.78777i 8.79678 − 1.90177i −6.92276 − 35.0022i −9.10526 − 42.1170i 0 − 43.4597i 73.7665 − 33.4590i −167.582
50.4 − 1.55250i 1.35662 − 8.89717i 13.5897 27.0364i −13.8128 − 2.10615i 0 − 45.9381i −77.3192 − 24.1402i 41.9740
50.5 1.55250i 1.35662 + 8.89717i 13.5897 − 27.0364i −13.8128 + 2.10615i 0 45.9381i −77.3192 + 24.1402i 41.9740
50.6 4.78777i 8.79678 + 1.90177i −6.92276 35.0022i −9.10526 + 42.1170i 0 43.4597i 73.7665 + 33.4590i −167.582
50.7 5.16081i 1.83981 − 8.80994i −10.6340 − 36.6448i 45.4664 + 9.49493i 0 27.6932i −74.2302 − 32.4173i 189.117
50.8 7.14374i −7.99321 + 4.13625i −35.0330 15.7492i −29.5483 − 57.1014i 0 − 135.967i 46.7828 − 66.1239i −112.508
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 50.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 147.5.b.f 8
3.b odd 2 1 inner 147.5.b.f 8
7.b odd 2 1 147.5.b.c 8
7.c even 3 2 147.5.h.b 16
7.d odd 6 2 21.5.h.b ✓ 16
21.c even 2 1 147.5.b.c 8
21.g even 6 2 21.5.h.b ✓ 16
21.h odd 6 2 147.5.h.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.5.h.b ✓ 16 7.d odd 6 2
21.5.h.b ✓ 16 21.g even 6 2
147.5.b.c 8 7.b odd 2 1
147.5.b.c 8 21.c even 2 1
147.5.b.f 8 1.a even 1 1 trivial
147.5.b.f 8 3.b odd 2 1 inner
147.5.h.b 16 7.c even 3 2
147.5.h.b 16 21.h odd 6 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{5}^{\mathrm{new}}(147, [\chi])\):

\( T_{2}^{8} + 103T_{2}^{6} + 3382T_{2}^{4} + 38724T_{2}^{2} + 75096 \) Copy content Toggle raw display
\( T_{13}^{4} - 123T_{13}^{3} - 25046T_{13}^{2} + 1759704T_{13} + 114515728 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 103 T^{6} + \cdots + 75096 \) Copy content Toggle raw display
$3$ \( T^{8} - 8 T^{7} + \cdots + 43046721 \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 298284991704 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 485335960671744 \) Copy content Toggle raw display
$13$ \( (T^{4} - 123 T^{3} + \cdots + 114515728)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 12\!\cdots\!96 \) Copy content Toggle raw display
$19$ \( (T^{4} + 663 T^{3} + \cdots - 527256704)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 23\!\cdots\!64 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 43\!\cdots\!36 \) Copy content Toggle raw display
$31$ \( (T^{4} + 1252 T^{3} + \cdots + 6550990263)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 671 T^{3} + \cdots - 25709157612)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 27\!\cdots\!76 \) Copy content Toggle raw display
$43$ \( (T^{4} - 365 T^{3} + \cdots + 885324127312)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 27\!\cdots\!24 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 26\!\cdots\!64 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 14\!\cdots\!36 \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots + 17149395602092)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + \cdots + 174471912618172)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 10\!\cdots\!24 \) Copy content Toggle raw display
$73$ \( (T^{4} + \cdots - 175206290970978)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots - 67497920874557)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 51\!\cdots\!44 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 57\!\cdots\!84 \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots + 368225255164464)^{2} \) Copy content Toggle raw display
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