Properties

Label 31.19.b.a
Level $31$
Weight $19$
Character orbit 31.b
Self dual yes
Analytic conductor $63.670$
Analytic rank $0$
Dimension $1$
CM discriminant -31
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [31,19,Mod(30,31)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(31, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 19, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("31.30");
 
S:= CuspForms(chi, 19);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 31 \)
Weight: \( k \) \(=\) \( 19 \)
Character orbit: \([\chi]\) \(=\) 31.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: yes
Analytic conductor: \(63.6697026900\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
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)
 
\(f(q)\) \(=\) \( q - 495 q^{2} - 17119 q^{4} - 3355686 q^{5} + 72260210 q^{7} + 138235185 q^{8} + 387420489 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 495 q^{2} - 17119 q^{4} - 3355686 q^{5} + 72260210 q^{7} + 138235185 q^{8} + 387420489 q^{9} + 1661064570 q^{10} - 35768803950 q^{14} - 63938773439 q^{16} - 191773142055 q^{18} - 301505744342 q^{19} + 57445988634 q^{20} + 7445931264971 q^{25} - 1237022534990 q^{28} - 26439622160671 q^{31} - 4587831484335 q^{32} - 242482575054060 q^{35} - 6632251351191 q^{36} + 149245343449290 q^{38} - 463873875011910 q^{40} + 640887818618322 q^{41} - 13\!\cdots\!54 q^{45}+ \cdots - 17\!\cdots\!45 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/31\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
30.1
0
−495.000 0 −17119.0 −3.35569e6 0 7.22602e7 1.38235e8 3.87420e8 1.66106e9
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.b odd 2 1 CM by \(\Q(\sqrt{-31}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 31.19.b.a 1
31.b odd 2 1 CM 31.19.b.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
31.19.b.a 1 1.a even 1 1 trivial
31.19.b.a 1 31.b odd 2 1 CM

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 495 \) acting on \(S_{19}^{\mathrm{new}}(31, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 495 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T + 3355686 \) Copy content Toggle raw display
$7$ \( T - 72260210 \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T \) Copy content Toggle raw display
$17$ \( T \) Copy content Toggle raw display
$19$ \( T + 301505744342 \) Copy content Toggle raw display
$23$ \( T \) Copy content Toggle raw display
$29$ \( T \) Copy content Toggle raw display
$31$ \( T + 26439622160671 \) Copy content Toggle raw display
$37$ \( T \) Copy content Toggle raw display
$41$ \( T - 640887818618322 \) Copy content Toggle raw display
$43$ \( T \) Copy content Toggle raw display
$47$ \( T + 484327216285470 \) Copy content Toggle raw display
$53$ \( T \) Copy content Toggle raw display
$59$ \( T + 14\!\cdots\!78 \) Copy content Toggle raw display
$61$ \( T \) Copy content Toggle raw display
$67$ \( T - 33\!\cdots\!90 \) Copy content Toggle raw display
$71$ \( T + 86\!\cdots\!62 \) Copy content Toggle raw display
$73$ \( T \) Copy content Toggle raw display
$79$ \( T \) Copy content Toggle raw display
$83$ \( T \) Copy content Toggle raw display
$89$ \( T \) Copy content Toggle raw display
$97$ \( T - 13\!\cdots\!70 \) Copy content Toggle raw display
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