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Properties

Label 147.2.c.b
Level $147$
Weight $2$
Character orbit 147.c
Analytic conductor $1.174$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [147,2,Mod(146,147)]
 
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, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("147.146");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 147 = 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 147.c (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: \(1.17380090971\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.3288334336.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 8x^{6} - 8x^{5} + 14x^{4} + 8x^{3} - 16x^{2} + 7 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
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.

\(f(q)\) \(=\) \( q - \beta_{3} q^{2} + ( - \beta_{5} + \beta_1) q^{3} + (\beta_{4} - 1) q^{4} + (\beta_{6} + \beta_{2} - \beta_1) q^{5} + (\beta_{6} + \beta_{5} + \beta_{2}) q^{6} + \beta_{7} q^{8} + (\beta_{7} + \beta_{4} + \beta_{3} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{2} + ( - \beta_{5} + \beta_1) q^{3} + (\beta_{4} - 1) q^{4} + (\beta_{6} + \beta_{2} - \beta_1) q^{5} + (\beta_{6} + \beta_{5} + \beta_{2}) q^{6} + \beta_{7} q^{8} + (\beta_{7} + \beta_{4} + \beta_{3} + 1) q^{9} + ( - 2 \beta_{6} + \beta_{5}) q^{10} + ( - \beta_{7} - \beta_{3}) q^{11} + ( - \beta_{6} + 2 \beta_{5} - \beta_{2}) q^{12} + ( - 2 \beta_{6} - \beta_{5}) q^{13} + ( - 2 \beta_{4} + \beta_{3} - 1) q^{15} - 3 q^{16} + (\beta_{5} - \beta_{2} - \beta_1) q^{17} + (\beta_{7} - 3 \beta_{4} + 2) q^{18} + (2 \beta_{6} - 2 \beta_{5}) q^{19} + (\beta_{5} - \beta_{2} - \beta_1) q^{20} + (3 \beta_{4} - 2) q^{22} + ( - \beta_{7} + \beta_{3}) q^{23} + (3 \beta_{6} + \beta_{5} - \beta_1) q^{24} + (\beta_{4} - 1) q^{25} + ( - 2 \beta_{6} - \beta_{5} + \cdots + 3 \beta_1) q^{26}+ \cdots + ( - \beta_{7} - 2 \beta_{4} + \beta_{3} + 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{4} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{4} + 8 q^{9} - 8 q^{15} - 24 q^{16} + 16 q^{18} - 16 q^{22} - 8 q^{25} + 24 q^{30} + 8 q^{36} + 16 q^{37} + 8 q^{39} + 16 q^{43} + 32 q^{46} - 24 q^{51} - 32 q^{57} - 24 q^{60} + 8 q^{64} - 48 q^{67} - 32 q^{72} + 40 q^{78} + 16 q^{79} - 24 q^{81} - 16 q^{85} + 32 q^{88} - 16 q^{93} + 48 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 8x^{6} - 8x^{5} + 14x^{4} + 8x^{3} - 16x^{2} + 7 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -139\nu^{7} + 647\nu^{6} - 305\nu^{5} + 7890\nu^{4} - 732\nu^{3} + 13443\nu^{2} + 5614\nu + 5957 ) / 8561 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -156\nu^{7} + 295\nu^{6} - 2190\nu^{5} + 3743\nu^{4} - 13817\nu^{3} + 13301\nu^{2} - 15687\nu + 1512 ) / 8561 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 701\nu^{7} - 1600\nu^{6} + 6219\nu^{5} - 18850\nu^{4} + 26911\nu^{3} - 27885\nu^{2} - 11991\nu + 15596 ) / 8561 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -722\nu^{7} + 158\nu^{6} - 5526\nu^{5} + 6677\nu^{4} - 9838\nu^{3} - 7038\nu^{2} + 19922\nu + 1071 ) / 8561 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -859\nu^{7} + 1350\nu^{6} - 7120\nu^{5} + 18580\nu^{4} - 25649\nu^{3} + 19515\nu^{2} + 10920\nu - 12089 ) / 8561 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1369\nu^{7} + 649\nu^{6} + 12304\nu^{5} - 5463\nu^{4} + 24393\nu^{3} + 10428\nu^{2} - 5404\nu - 98 ) / 8561 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2249\nu^{7} + 741\nu^{6} + 18731\nu^{5} - 11870\nu^{4} + 32969\nu^{3} + 25836\nu^{2} - 7133\nu - 4676 ) / 8561 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -2\beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} + \beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -3\beta_{7} + \beta_{6} - 5\beta_{5} - 7\beta_{4} - 7\beta_{3} - 6\beta_{2} + 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{7} - 6\beta_{6} + 18\beta_{5} - 7\beta_{4} + 12\beta_{3} - 6\beta_{2} - 2\beta _1 + 9 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 9\beta_{7} + 11\beta_{6} - 9\beta_{5} + 69\beta_{4} + 27\beta_{3} + 60\beta_{2} + 20\beta _1 - 90 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -47\beta_{7} + 58\beta_{6} - 136\beta_{5} + 11\beta_{4} - 116\beta_{3} + 9\beta_{2} - \beta _1 - 8 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 53\beta_{7} - 211\beta_{6} + 477\beta_{5} - 573\beta_{4} + 107\beta_{3} - 504\beta_{2} - 210\beta _1 + 812 ) / 2 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
146.1
−0.707107 + 2.79220i
−0.707107 + 0.179070i
0.707107 − 1.43164i
0.707107 − 0.349249i
0.707107 + 1.43164i
0.707107 + 0.349249i
−0.707107 − 2.79220i
−0.707107 − 0.179070i
− 2.10100i −1.13705 − 1.30656i −2.41421 −1.60804 −2.74509 + 2.38896i 0 0.870264i −0.414214 + 2.97127i 3.37849i
146.2 − 2.10100i 1.13705 + 1.30656i −2.41421 1.60804 2.74509 − 2.38896i 0 0.870264i −0.414214 + 2.97127i − 3.37849i
146.3 − 1.25928i −1.64533 + 0.541196i 0.414214 2.32685 0.681517 + 2.07193i 0 − 3.04017i 2.41421 − 1.78089i − 2.93015i
146.4 − 1.25928i 1.64533 − 0.541196i 0.414214 −2.32685 −0.681517 − 2.07193i 0 − 3.04017i 2.41421 − 1.78089i 2.93015i
146.5 1.25928i −1.64533 − 0.541196i 0.414214 2.32685 0.681517 − 2.07193i 0 3.04017i 2.41421 + 1.78089i 2.93015i
146.6 1.25928i 1.64533 + 0.541196i 0.414214 −2.32685 −0.681517 + 2.07193i 0 3.04017i 2.41421 + 1.78089i − 2.93015i
146.7 2.10100i −1.13705 + 1.30656i −2.41421 −1.60804 −2.74509 − 2.38896i 0 − 0.870264i −0.414214 − 2.97127i − 3.37849i
146.8 2.10100i 1.13705 − 1.30656i −2.41421 1.60804 2.74509 + 2.38896i 0 − 0.870264i −0.414214 − 2.97127i 3.37849i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 146.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 147.2.c.b ✓ 8
3.b odd 2 1 inner 147.2.c.b ✓ 8
4.b odd 2 1 2352.2.k.h 8
7.b odd 2 1 inner 147.2.c.b ✓ 8
7.c even 3 2 147.2.g.b 16
7.d odd 6 2 147.2.g.b 16
12.b even 2 1 2352.2.k.h 8
21.c even 2 1 inner 147.2.c.b ✓ 8
21.g even 6 2 147.2.g.b 16
21.h odd 6 2 147.2.g.b 16
28.d even 2 1 2352.2.k.h 8
84.h odd 2 1 2352.2.k.h 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
147.2.c.b ✓ 8 1.a even 1 1 trivial
147.2.c.b ✓ 8 3.b odd 2 1 inner
147.2.c.b ✓ 8 7.b odd 2 1 inner
147.2.c.b ✓ 8 21.c even 2 1 inner
147.2.g.b 16 7.c even 3 2
147.2.g.b 16 7.d odd 6 2
147.2.g.b 16 21.g even 6 2
147.2.g.b 16 21.h odd 6 2
2352.2.k.h 8 4.b odd 2 1
2352.2.k.h 8 12.b even 2 1
2352.2.k.h 8 28.d even 2 1
2352.2.k.h 8 84.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 6T_{2}^{2} + 7 \) acting on \(S_{2}^{\mathrm{new}}(147, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 6 T^{2} + 7)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} - 4 T^{6} + \cdots + 81 \) Copy content Toggle raw display
$5$ \( (T^{4} - 8 T^{2} + 14)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{4} + 12 T^{2} + 28)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 20 T^{2} + 2)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 16 T^{2} + 14)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 32 T^{2} + 128)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 20 T^{2} + 28)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 84 T^{2} + 1372)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 8 T^{2} + 8)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 4 T - 46)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} - 56 T^{2} + 686)^{2} \) Copy content Toggle raw display
$43$ \( (T - 2)^{8} \) Copy content Toggle raw display
$47$ \( (T^{4} - 176 T^{2} + 56)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 40 T^{2} + 112)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 128 T^{2} + 3584)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 116 T^{2} + 3362)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 12 T - 36)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} + 84 T^{2} + 1372)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 148 T^{2} + 4418)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 4 T - 4)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} - 224 T^{2} + 2744)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} - 176 T^{2} + 7406)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 244 T^{2} + 10082)^{2} \) Copy content Toggle raw display
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