Properties

Label 141.4.c.a
Level $141$
Weight $4$
Character orbit 141.c
Analytic conductor $8.319$
Analytic rank $0$
Dimension $10$
CM discriminant -47
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [141,4,Mod(140,141)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(141, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("141.140");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 141 = 3 \cdot 47 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 141.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: \(8.31926931081\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: 10.0.2239697333984375.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 9x^{5} + 32 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2\cdot 3^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} + ( - \beta_{5} - \beta_{2} - \beta_1) q^{3} + (2 \beta_{9} - \beta_{8} + \beta_{6} + 2 \beta_{2} - \beta_1 - 8) q^{4} + ( - \beta_{9} + 2 \beta_{7} - 3 \beta_{5} - 3 \beta_{4} + \beta_{3} - 2 \beta_{2} + 7 \beta_1) q^{6} + (5 \beta_{8} + 2 \beta_{7} - 5 \beta_{6} - 9 \beta_{4} + 3 \beta_{3}) q^{7} + (2 \beta_{9} - \beta_{8} + \beta_{6} + 4 \beta_{5} - 8 \beta_{3} - 5 \beta_{2} - 5 \beta_1) q^{8} + (4 \beta_{8} + 3 \beta_{7} - 4 \beta_{6} - 7 \beta_{4} - 2 \beta_{3}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{2} + ( - \beta_{5} - \beta_{2} - \beta_1) q^{3} + (2 \beta_{9} - \beta_{8} + \beta_{6} + 2 \beta_{2} - \beta_1 - 8) q^{4} + ( - \beta_{9} + 2 \beta_{7} - 3 \beta_{5} - 3 \beta_{4} + \beta_{3} - 2 \beta_{2} + 7 \beta_1) q^{6} + (5 \beta_{8} + 2 \beta_{7} - 5 \beta_{6} - 9 \beta_{4} + 3 \beta_{3}) q^{7} + (2 \beta_{9} - \beta_{8} + \beta_{6} + 4 \beta_{5} - 8 \beta_{3} - 5 \beta_{2} - 5 \beta_1) q^{8} + (4 \beta_{8} + 3 \beta_{7} - 4 \beta_{6} - 7 \beta_{4} - 2 \beta_{3}) q^{9} + (8 \beta_{8} + 6 \beta_{7} + 3 \beta_{6} + 8 \beta_{5} + 8 \beta_{4} + 7 \beta_{3} + \cdots - 23) q^{12}+ \cdots + (82 \beta_{9} + 14 \beta_{8} + 220 \beta_{7} - 14 \beta_{6} + 164 \beta_{5} + \cdots + 235 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 80 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 80 q^{4} - 175 q^{12} + 640 q^{16} + 725 q^{18} - 805 q^{24} - 1250 q^{25} + 275 q^{42} + 1400 q^{48} + 3430 q^{49} - 2740 q^{51} + 455 q^{54} + 5000 q^{63} - 5120 q^{64} - 5800 q^{72} - 3145 q^{84} + 6440 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 9x^{5} + 32 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{9} + 9\nu^{4} - 16\nu ) / 16 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{9} + 8\nu^{6} + 7\nu^{4} - 24\nu ) / 16 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{7} - 4\nu^{3} + 9\nu^{2} ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{8} - 9\nu^{3} - 8\nu^{2} ) / 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{9} - 8\nu^{6} + 9\nu^{4} + 40\nu ) / 16 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{8} + 2\nu^{7} - 8\nu^{5} + \nu^{3} - 2\nu^{2} + 40 ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{8} + 2\nu^{7} - 5\nu^{3} - 6\nu^{2} ) / 4 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( \nu^{8} - 2\nu^{7} - 8\nu^{5} - \nu^{3} + 2\nu^{2} + 40 ) / 8 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -3\nu^{9} + \nu^{8} - 2\nu^{7} + 11\nu^{4} - \nu^{3} + 2\nu^{2} + 8\nu ) / 8 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( 2\beta_{9} - \beta_{8} + \beta_{6} + 4\beta_{5} + 4\beta_{2} - 12\beta_1 ) / 18 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{8} + 4\beta_{7} - \beta_{6} - 10\beta_{4} + 6\beta_{3} ) / 18 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 5\beta_{8} + 2\beta_{7} - 5\beta_{6} - 14\beta_{4} - 6\beta_{3} ) / 18 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -2\beta_{9} + \beta_{8} - \beta_{6} + 14\beta_{5} + 14\beta_{2} + 12\beta_1 ) / 18 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -\beta_{8} - \beta_{6} + 10 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 14\beta_{9} - 7\beta_{8} + 7\beta_{6} - 8\beta_{5} + 28\beta_{2} - 48\beta_1 ) / 18 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -11\beta_{8} + 28\beta_{7} + 11\beta_{6} - 34\beta_{4} + 6\beta_{3} ) / 18 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 53\beta_{8} + 50\beta_{7} - 53\beta_{6} - 62\beta_{4} - 6\beta_{3} ) / 18 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( -50\beta_{9} + 25\beta_{8} - 25\beta_{6} + 62\beta_{5} + 62\beta_{2} + 12\beta_1 ) / 18 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/141\mathbb{Z}\right)^\times\).

\(n\) \(52\) \(95\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
140.1
−1.24236 + 0.675681i
0.258701 1.39035i
−1.02652 + 0.972757i
1.40224 + 0.183603i
0.607934 1.27688i
0.607934 + 1.27688i
1.40224 0.183603i
−1.02652 0.972757i
0.258701 + 1.39035i
−1.24236 0.675681i
5.64034i 2.87961 + 4.32526i −23.8135 0 24.3959 16.2420i −14.7562 89.1934i −10.4157 + 24.9101i 0
140.2 4.81700i −3.22372 4.07525i −15.2035 0 −19.6305 + 15.5286i −8.03159 34.6990i −6.21529 + 26.2749i 0
140.3 4.30927i 5.00341 + 1.40209i −10.5698 0 6.04199 21.5611i 31.9076 11.0740i 23.0683 + 14.0305i 0
140.4 2.15372i −4.87197 1.80662i 3.36148 0 −3.89095 + 10.4929i 27.7516 24.4695i 20.4723 + 17.6036i 0
140.5 1.33220i 0.212671 5.19180i 6.22524 0 −6.91652 0.283321i −36.8714 18.9509i −26.9095 2.20829i 0
140.6 1.33220i 0.212671 + 5.19180i 6.22524 0 −6.91652 + 0.283321i −36.8714 18.9509i −26.9095 + 2.20829i 0
140.7 2.15372i −4.87197 + 1.80662i 3.36148 0 −3.89095 10.4929i 27.7516 24.4695i 20.4723 17.6036i 0
140.8 4.30927i 5.00341 1.40209i −10.5698 0 6.04199 + 21.5611i 31.9076 11.0740i 23.0683 14.0305i 0
140.9 4.81700i −3.22372 + 4.07525i −15.2035 0 −19.6305 15.5286i −8.03159 34.6990i −6.21529 26.2749i 0
140.10 5.64034i 2.87961 4.32526i −23.8135 0 24.3959 + 16.2420i −14.7562 89.1934i −10.4157 24.9101i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 140.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
47.b odd 2 1 CM by \(\Q(\sqrt{-47}) \)
3.b odd 2 1 inner
141.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 141.4.c.a 10
3.b odd 2 1 inner 141.4.c.a 10
47.b odd 2 1 CM 141.4.c.a 10
141.c even 2 1 inner 141.4.c.a 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
141.4.c.a 10 1.a even 1 1 trivial
141.4.c.a 10 3.b odd 2 1 inner
141.4.c.a 10 47.b odd 2 1 CM
141.4.c.a 10 141.c even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{10} + 80T_{2}^{8} + 2240T_{2}^{6} + 25600T_{2}^{4} + 102400T_{2}^{2} + 112847 \) acting on \(S_{4}^{\mathrm{new}}(141, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + 80 T^{8} + 2240 T^{6} + \cdots + 112847 \) Copy content Toggle raw display
$3$ \( T^{10} - 1540 T^{5} + \cdots + 14348907 \) Copy content Toggle raw display
$5$ \( T^{10} \) Copy content Toggle raw display
$7$ \( (T^{5} - 1715 T^{3} + 588245 T + 3869440)^{2} \) Copy content Toggle raw display
$11$ \( T^{10} \) Copy content Toggle raw display
$13$ \( T^{10} \) Copy content Toggle raw display
$17$ \( T^{10} + 49130 T^{8} + \cdots + 71\!\cdots\!72 \) Copy content Toggle raw display
$19$ \( T^{10} \) Copy content Toggle raw display
$23$ \( T^{10} \) Copy content Toggle raw display
$29$ \( T^{10} \) Copy content Toggle raw display
$31$ \( T^{10} \) Copy content Toggle raw display
$37$ \( (T^{5} - 253265 T^{3} + \cdots + 1139571232270)^{2} \) Copy content Toggle raw display
$41$ \( T^{10} \) Copy content Toggle raw display
$43$ \( T^{10} \) Copy content Toggle raw display
$47$ \( (T^{2} + 103823)^{5} \) Copy content Toggle raw display
$53$ \( T^{10} + 1488770 T^{8} + \cdots + 33\!\cdots\!28 \) Copy content Toggle raw display
$59$ \( T^{10} + 2053790 T^{8} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{5} - 1134905 T^{3} + \cdots + 45922923778102)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} \) Copy content Toggle raw display
$71$ \( T^{10} + 3579110 T^{8} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{10} \) Copy content Toggle raw display
$79$ \( (T^{5} - 2465195 T^{3} + \cdots - 283163708513936)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 699548)^{5} \) Copy content Toggle raw display
$89$ \( T^{10} + 7049690 T^{8} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{5} - 4563365 T^{3} + \cdots + 13\!\cdots\!10)^{2} \) Copy content Toggle raw display
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