Properties

Label 164.1.d.b
Level $164$
Weight $1$
Character orbit 164.d
Self dual yes
Analytic conductor $0.082$
Analytic rank $0$
Dimension $2$
Projective image $D_{4}$
CM discriminant -164
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 164 = 2^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 164.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(0.0818466620718\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.2.6724.1
Artin image: $D_8$
Artin field: Galois closure of 8.0.17643776.1

$q$-expansion

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

\(f(q)\) \(=\) \( q - q^{2} -\beta q^{3} + q^{4} + \beta q^{6} + \beta q^{7} - q^{8} + q^{9} +O(q^{10})\) \( q - q^{2} -\beta q^{3} + q^{4} + \beta q^{6} + \beta q^{7} - q^{8} + q^{9} + \beta q^{11} -\beta q^{12} -\beta q^{14} + q^{16} - q^{18} -\beta q^{19} -2 q^{21} -\beta q^{22} + \beta q^{24} - q^{25} + \beta q^{28} - q^{32} -2 q^{33} + q^{36} + \beta q^{38} + q^{41} + 2 q^{42} + \beta q^{44} -\beta q^{47} -\beta q^{48} + q^{49} + q^{50} -\beta q^{56} + 2 q^{57} -2 q^{61} + \beta q^{63} + q^{64} + 2 q^{66} + \beta q^{67} -\beta q^{71} - q^{72} + \beta q^{75} -\beta q^{76} + 2 q^{77} -\beta q^{79} - q^{81} - q^{82} -2 q^{84} -\beta q^{88} + \beta q^{94} + \beta q^{96} - q^{98} + \beta q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} + 2q^{4} - 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{2} + 2q^{4} - 2q^{8} + 2q^{9} + 2q^{16} - 2q^{18} - 4q^{21} - 2q^{25} - 2q^{32} - 4q^{33} + 2q^{36} + 2q^{41} + 4q^{42} + 2q^{49} + 2q^{50} + 4q^{57} - 4q^{61} + 2q^{64} + 4q^{66} - 2q^{72} + 4q^{77} - 2q^{81} - 2q^{82} - 4q^{84} - 2q^{98} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/164\mathbb{Z}\right)^\times\).

\(n\) \(83\) \(129\)
\(\chi(n)\) \(-1\) \(-1\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
163.1
1.41421
−1.41421
−1.00000 −1.41421 1.00000 0 1.41421 1.41421 −1.00000 1.00000 0
163.2 −1.00000 1.41421 1.00000 0 −1.41421 −1.41421 −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
164.d odd 2 1 CM by \(\Q(\sqrt{-41}) \)
4.b odd 2 1 inner
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 164.1.d.b 2
3.b odd 2 1 1476.1.h.b 2
4.b odd 2 1 inner 164.1.d.b 2
8.b even 2 1 2624.1.h.b 2
8.d odd 2 1 2624.1.h.b 2
12.b even 2 1 1476.1.h.b 2
41.b even 2 1 inner 164.1.d.b 2
123.b odd 2 1 1476.1.h.b 2
164.d odd 2 1 CM 164.1.d.b 2
328.c odd 2 1 2624.1.h.b 2
328.g even 2 1 2624.1.h.b 2
492.d even 2 1 1476.1.h.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
164.1.d.b 2 1.a even 1 1 trivial
164.1.d.b 2 4.b odd 2 1 inner
164.1.d.b 2 41.b even 2 1 inner
164.1.d.b 2 164.d odd 2 1 CM
1476.1.h.b 2 3.b odd 2 1
1476.1.h.b 2 12.b even 2 1
1476.1.h.b 2 123.b odd 2 1
1476.1.h.b 2 492.d even 2 1
2624.1.h.b 2 8.b even 2 1
2624.1.h.b 2 8.d odd 2 1
2624.1.h.b 2 328.c odd 2 1
2624.1.h.b 2 328.g even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 2 \) acting on \(S_{1}^{\mathrm{new}}(164, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{2} \)
$3$ \( -2 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( -2 + T^{2} \)
$11$ \( -2 + T^{2} \)
$13$ \( T^{2} \)
$17$ \( T^{2} \)
$19$ \( -2 + T^{2} \)
$23$ \( T^{2} \)
$29$ \( T^{2} \)
$31$ \( T^{2} \)
$37$ \( T^{2} \)
$41$ \( ( -1 + T )^{2} \)
$43$ \( T^{2} \)
$47$ \( -2 + T^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( ( 2 + T )^{2} \)
$67$ \( -2 + T^{2} \)
$71$ \( -2 + T^{2} \)
$73$ \( T^{2} \)
$79$ \( -2 + T^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( T^{2} \)
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