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Properties

Label 147.3.d.d
Level $147$
Weight $3$
Character orbit 147.d
Analytic conductor $4.005$
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] = [147,3,Mod(97,147)]
 
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([0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("147.97");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 147 = 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 147.d (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: \(4.00545988610\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.339738624.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{6} + 14x^{4} - 8x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4}\cdot 7^{2} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{2} - 1) q^{2} + \beta_{3} q^{3} + ( - \beta_{5} + 5) q^{4} + (\beta_{6} - \beta_1) q^{5} + (\beta_{6} - \beta_{3} + \beta_1) q^{6} + (\beta_{5} + \beta_{4} - 5 \beta_{2} + 1) q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} - 1) q^{2} + \beta_{3} q^{3} + ( - \beta_{5} + 5) q^{4} + (\beta_{6} - \beta_1) q^{5} + (\beta_{6} - \beta_{3} + \beta_1) q^{6} + (\beta_{5} + \beta_{4} - 5 \beta_{2} + 1) q^{8} - 3 q^{9} + (\beta_{7} - 6 \beta_{6} - 4 \beta_{3}) q^{10} + ( - \beta_{5} + \beta_{4} + 2 \beta_{2} - 8) q^{11} + ( - 3 \beta_{7} + \beta_{6} + \cdots + \beta_1) q^{12}+ \cdots + (3 \beta_{5} - 3 \beta_{4} - 6 \beta_{2} + 24) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{2} + 40 q^{4} + 8 q^{8} - 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{2} + 40 q^{4} + 8 q^{8} - 24 q^{9} - 64 q^{11} + 120 q^{16} + 24 q^{18} - 64 q^{22} - 144 q^{23} - 56 q^{25} + 16 q^{29} + 96 q^{30} - 56 q^{32} - 120 q^{36} + 64 q^{37} + 16 q^{43} - 208 q^{44} + 176 q^{46} + 264 q^{50} + 96 q^{51} + 112 q^{53} + 192 q^{57} - 144 q^{58} + 144 q^{60} + 152 q^{64} + 112 q^{65} + 192 q^{67} - 160 q^{71} - 24 q^{72} - 528 q^{74} - 240 q^{78} - 176 q^{79} + 72 q^{81} - 544 q^{85} + 464 q^{86} - 688 q^{88} - 448 q^{92} + 288 q^{93} - 16 q^{95} + 192 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{6} + 14x^{4} - 8x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{3} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -2\nu^{7} + \nu^{6} + 7\nu^{5} - 28\nu^{3} + 30\nu + 20 ) / 14 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -2\nu^{6} + 7\nu^{4} - 28\nu^{2} + 9 ) / 7 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{7} - \nu^{6} - 7\nu^{5} + 21\nu^{3} - 22\nu - 20 ) / 7 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 8\nu^{7} + \nu^{6} - 42\nu^{5} + 140\nu^{3} - 148\nu + 20 ) / 14 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 6\nu^{7} + 7\nu^{6} - 21\nu^{5} - 28\nu^{4} + 70\nu^{3} + 84\nu^{2} - 6\nu - 28 ) / 14 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2\nu^{7} - 7\nu^{5} + 24\nu^{3} - 2\nu ) / 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} - \beta_{5} + 2\beta_{4} + 5\beta_{2} + 2\beta_1 ) / 14 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{7} - 7\beta_{6} - \beta_{5} + 2\beta_{4} - 14\beta_{3} - 2\beta_{2} - \beta _1 + 14 ) / 14 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 6\beta_{7} - 14\beta_{6} + 2\beta_{5} - 4\beta_{4} - 21\beta_{3} + 4\beta_{2} - 2\beta _1 - 21 ) / 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -\beta_{7} - \beta_{5} - 12\beta_{4} - 23\beta_{2} + 12\beta_1 ) / 7 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2\beta_{5} - 4\beta_{4} + 4\beta_{2} - 20 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 4\beta_{7} - 4\beta_{5} - 41\beta_{4} - 78\beta_{2} - 41\beta_1 ) / 7 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
97.1
0.662827 + 0.382683i
0.662827 − 0.382683i
1.60021 + 0.923880i
1.60021 − 0.923880i
−0.662827 − 0.382683i
−0.662827 + 0.382683i
−1.60021 − 0.923880i
−1.60021 + 0.923880i
−3.73987 − 1.73205i 9.98661 3.84891i 6.47764i 0 −22.3891 −3.00000 − 14.3944i
97.2 −3.73987 1.73205i 9.98661 − 3.84891i − 6.47764i 0 −22.3891 −3.00000 14.3944i
97.3 −2.78620 − 1.73205i 3.76290 − 9.52350i 4.82584i 0 0.660594 −3.00000 26.5344i
97.4 −2.78620 1.73205i 3.76290 9.52350i − 4.82584i 0 0.660594 −3.00000 − 26.5344i
97.5 −1.08856 − 1.73205i −2.81504 1.05007i 1.88544i 0 7.41857 −3.00000 − 1.14307i
97.6 −1.08856 1.73205i −2.81504 − 1.05007i − 1.88544i 0 7.41857 −3.00000 1.14307i
97.7 3.61463 − 1.73205i 9.06552 4.62452i − 6.26072i 0 18.3100 −3.00000 16.7159i
97.8 3.61463 1.73205i 9.06552 − 4.62452i 6.26072i 0 18.3100 −3.00000 − 16.7159i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 97.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 147.3.d.d ✓ 8
3.b odd 2 1 441.3.d.h 8
4.b odd 2 1 2352.3.f.l 8
7.b odd 2 1 inner 147.3.d.d ✓ 8
7.c even 3 1 147.3.f.f 8
7.c even 3 1 147.3.f.g 8
7.d odd 6 1 147.3.f.f 8
7.d odd 6 1 147.3.f.g 8
21.c even 2 1 441.3.d.h 8
21.g even 6 1 441.3.m.j 8
21.g even 6 1 441.3.m.k 8
21.h odd 6 1 441.3.m.j 8
21.h odd 6 1 441.3.m.k 8
28.d even 2 1 2352.3.f.l 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
147.3.d.d ✓ 8 1.a even 1 1 trivial
147.3.d.d ✓ 8 7.b odd 2 1 inner
147.3.f.f 8 7.c even 3 1
147.3.f.f 8 7.d odd 6 1
147.3.f.g 8 7.c even 3 1
147.3.f.g 8 7.d odd 6 1
441.3.d.h 8 3.b odd 2 1
441.3.d.h 8 21.c even 2 1
441.3.m.j 8 21.g even 6 1
441.3.m.j 8 21.h odd 6 1
441.3.m.k 8 21.g even 6 1
441.3.m.k 8 21.h odd 6 1
2352.3.f.l 8 4.b odd 2 1
2352.3.f.l 8 28.d even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 4T_{2}^{3} - 10T_{2}^{2} - 52T_{2} - 41 \) acting on \(S_{3}^{\mathrm{new}}(147, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 4 T^{3} - 10 T^{2} + \cdots - 41)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} + 3)^{4} \) Copy content Toggle raw display
$5$ \( T^{8} + 128 T^{6} + \cdots + 31684 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{4} + 32 T^{3} + \cdots - 932)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + 904 T^{6} + \cdots + 10640644 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 1039546564 \) Copy content Toggle raw display
$19$ \( T^{8} + 1264 T^{6} + \cdots + 85082176 \) Copy content Toggle raw display
$23$ \( (T^{4} + 72 T^{3} + \cdots + 67228)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 8 T^{3} + \cdots + 301468)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + 3376 T^{6} + \cdots + 851238976 \) Copy content Toggle raw display
$37$ \( (T^{4} - 32 T^{3} + \cdots - 12092)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 1143056063044 \) Copy content Toggle raw display
$43$ \( (T^{4} - 8 T^{3} + \cdots + 1217296)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 36741733758016 \) Copy content Toggle raw display
$53$ \( (T^{4} - 56 T^{3} + \cdots + 5392)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 68212902976 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 10776263729284 \) Copy content Toggle raw display
$67$ \( (T^{4} - 96 T^{3} + \cdots - 23831424)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 80 T^{3} + \cdots + 1658524)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 25117918721284 \) Copy content Toggle raw display
$79$ \( (T^{4} + 88 T^{3} + \cdots + 32034832)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 7398574081024 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 7946907588676 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 20\!\cdots\!44 \) Copy content Toggle raw display
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