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Properties

Label 82.2.b.a
Level $82$
Weight $2$
Character orbit 82.b
Analytic conductor $0.655$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [82,2,Mod(81,82)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(82, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("82.81"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 82 = 2 \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 82.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.654773296574\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + \beta q^{3} + q^{4} - \beta q^{6} + 3 \beta q^{7} - q^{8} + q^{9} - \beta q^{11} + \beta q^{12} - 3 \beta q^{14} + q^{16} - 4 \beta q^{17} - q^{18} - 3 \beta q^{19} - 6 q^{21} + \beta q^{22} + \cdots - \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} + 2 q^{9} + 2 q^{16} - 2 q^{18} - 12 q^{21} - 10 q^{25} + 16 q^{31} - 2 q^{32} + 4 q^{33} + 2 q^{36} + 16 q^{37} - 6 q^{41} + 12 q^{42} - 8 q^{43} - 22 q^{49} + 10 q^{50}+ \cdots + 22 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/82\mathbb{Z}\right)^\times\).

\(n\) \(47\)
\(\chi(n)\) \(-1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
81.1
− 1.41421i
1.41421i
−1.00000 − 1.41421i 1.00000 0 1.41421i − 4.24264i −1.00000 1.00000 0
81.2 −1.00000 1.41421i 1.00000 0 − 1.41421i 4.24264i −1.00000 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 82.2.b.a ✓ 2
3.b odd 2 1 738.2.d.f 2
4.b odd 2 1 656.2.d.b 2
5.b even 2 1 2050.2.b.f 2
5.c odd 4 2 2050.2.d.h 4
8.b even 2 1 2624.2.d.g 2
8.d odd 2 1 2624.2.d.f 2
41.b even 2 1 inner 82.2.b.a ✓ 2
41.c even 4 2 3362.2.a.l 2
123.b odd 2 1 738.2.d.f 2
164.d odd 2 1 656.2.d.b 2
205.c even 2 1 2050.2.b.f 2
205.g odd 4 2 2050.2.d.h 4
328.c odd 2 1 2624.2.d.f 2
328.g even 2 1 2624.2.d.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
82.2.b.a ✓ 2 1.a even 1 1 trivial
82.2.b.a ✓ 2 41.b even 2 1 inner
656.2.d.b 2 4.b odd 2 1
656.2.d.b 2 164.d odd 2 1
738.2.d.f 2 3.b odd 2 1
738.2.d.f 2 123.b odd 2 1
2050.2.b.f 2 5.b even 2 1
2050.2.b.f 2 205.c even 2 1
2050.2.d.h 4 5.c odd 4 2
2050.2.d.h 4 205.g odd 4 2
2624.2.d.f 2 8.d odd 2 1
2624.2.d.f 2 328.c odd 2 1
2624.2.d.g 2 8.b even 2 1
2624.2.d.g 2 328.g even 2 1
3362.2.a.l 2 41.c even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 2 \) acting on \(S_{2}^{\mathrm{new}}(82, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 2 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 18 \) Copy content Toggle raw display
$11$ \( T^{2} + 2 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 32 \) Copy content Toggle raw display
$19$ \( T^{2} + 18 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 32 \) Copy content Toggle raw display
$31$ \( (T - 8)^{2} \) Copy content Toggle raw display
$37$ \( (T - 8)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 6T + 41 \) Copy content Toggle raw display
$43$ \( (T + 4)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 50 \) Copy content Toggle raw display
$53$ \( T^{2} + 32 \) Copy content Toggle raw display
$59$ \( (T + 12)^{2} \) Copy content Toggle raw display
$61$ \( (T - 2)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 162 \) Copy content Toggle raw display
$71$ \( T^{2} + 50 \) Copy content Toggle raw display
$73$ \( (T + 4)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 18 \) Copy content Toggle raw display
$83$ \( (T - 12)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 32 \) Copy content Toggle raw display
$97$ \( T^{2} + 288 \) Copy content Toggle raw display
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