Properties

Label 31.7.b.a
Level $31$
Weight $7$
Character orbit 31.b
Self dual yes
Analytic conductor $7.132$
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] = [31,7,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, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("31.30");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 31 \)
Weight: \( k \) \(=\) \( 7 \)
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: \(7.13167659222\)
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)
 
\(f(q)\) \(=\) \( q - 15 q^{2} + 161 q^{4} - 246 q^{5} - 430 q^{7} - 1455 q^{8} + 729 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 15 q^{2} + 161 q^{4} - 246 q^{5} - 430 q^{7} - 1455 q^{8} + 729 q^{9} + 3690 q^{10} + 6450 q^{14} + 11521 q^{16} - 10935 q^{18} + 10618 q^{19} - 39606 q^{20} + 44891 q^{25} - 69230 q^{28} - 29791 q^{31} - 79695 q^{32} + 105780 q^{35} + 117369 q^{36} - 159270 q^{38} + 357930 q^{40} - 60558 q^{41} - 179334 q^{45} + 171810 q^{47} + 67251 q^{49} - 673365 q^{50} + 625650 q^{56} + 136842 q^{59} + 446865 q^{62} - 313470 q^{63} + 458081 q^{64} - 133670 q^{67} - 1586700 q^{70} + 284178 q^{71} - 1060695 q^{72} + 1709498 q^{76} - 2834166 q^{80} + 531441 q^{81} + 908370 q^{82} + 2690010 q^{90} - 2577150 q^{94} - 2612028 q^{95} + 1807490 q^{97} - 1008765 q^{98}+O(q^{100}) \) 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
−15.0000 0 161.000 −246.000 0 −430.000 −1455.00 729.000 3690.00
\(n\): 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 31.7.b.a 1
3.b odd 2 1 279.7.d.a 1
4.b odd 2 1 496.7.e.a 1
31.b odd 2 1 CM 31.7.b.a 1
93.c even 2 1 279.7.d.a 1
124.d even 2 1 496.7.e.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
31.7.b.a 1 1.a even 1 1 trivial
31.7.b.a 1 31.b odd 2 1 CM
279.7.d.a 1 3.b odd 2 1
279.7.d.a 1 93.c even 2 1
496.7.e.a 1 4.b odd 2 1
496.7.e.a 1 124.d even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 15 \) acting on \(S_{7}^{\mathrm{new}}(31, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 15 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T + 246 \) Copy content Toggle raw display
$7$ \( T + 430 \) 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
$19$ \( T - 10618 \) Copy content Toggle raw display
$23$ \( T \) Copy content Toggle raw display
$29$ \( T \) Copy content Toggle raw display
$31$ \( T + 29791 \) Copy content Toggle raw display
$37$ \( T \) Copy content Toggle raw display
$41$ \( T + 60558 \) Copy content Toggle raw display
$43$ \( T \) Copy content Toggle raw display
$47$ \( T - 171810 \) Copy content Toggle raw display
$53$ \( T \) Copy content Toggle raw display
$59$ \( T - 136842 \) Copy content Toggle raw display
$61$ \( T \) Copy content Toggle raw display
$67$ \( T + 133670 \) Copy content Toggle raw display
$71$ \( T - 284178 \) 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
$97$ \( T - 1807490 \) Copy content Toggle raw display
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