Properties

Label 1648.1.c.a
Level $1648$
Weight $1$
Character orbit 1648.c
Self dual yes
Analytic conductor $0.822$
Analytic rank $0$
Dimension $2$
Projective image $D_{5}$
CM discriminant -103
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1648 = 2^{4} \cdot 103 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1648.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(0.822459140819\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 103)
Projective image: \(D_{5}\)
Projective field: Galois closure of 5.1.10609.1
Artin image: $D_{10}$
Artin field: Galois closure of 10.0.115252102144.4

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 - \beta ) q^{7} + q^{9} +O(q^{10})\) \( q + ( 1 - \beta ) q^{7} + q^{9} -\beta q^{13} + ( -1 + \beta ) q^{17} + \beta q^{19} + \beta q^{23} + q^{25} + ( -1 + \beta ) q^{29} -\beta q^{41} + ( 1 - \beta ) q^{49} + ( 1 - \beta ) q^{59} + ( -1 + \beta ) q^{61} + ( 1 - \beta ) q^{63} + \beta q^{79} + q^{81} + ( 1 - \beta ) q^{83} + q^{91} + ( -1 + \beta ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{7} + 2 q^{9} + O(q^{10}) \) \( 2 q + q^{7} + 2 q^{9} - q^{13} - q^{17} + q^{19} + q^{23} + 2 q^{25} - q^{29} - q^{41} + q^{49} + q^{59} - q^{61} + q^{63} + q^{79} + 2 q^{81} + q^{83} + 2 q^{91} - q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1648\mathbb{Z}\right)^\times\).

\(n\) \(207\) \(417\) \(1237\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1441.1
1.61803
−0.618034
0 0 0 0 0 −0.618034 0 1.00000 0
1441.2 0 0 0 0 0 1.61803 0 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
103.b odd 2 1 CM by \(\Q(\sqrt{-103}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1648.1.c.a 2
4.b odd 2 1 103.1.b.a 2
12.b even 2 1 927.1.d.b 2
20.d odd 2 1 2575.1.d.d 2
20.e even 4 2 2575.1.c.b 4
103.b odd 2 1 CM 1648.1.c.a 2
412.d even 2 1 103.1.b.a 2
1236.f odd 2 1 927.1.d.b 2
2060.e even 2 1 2575.1.d.d 2
2060.m odd 4 2 2575.1.c.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
103.1.b.a 2 4.b odd 2 1
103.1.b.a 2 412.d even 2 1
927.1.d.b 2 12.b even 2 1
927.1.d.b 2 1236.f odd 2 1
1648.1.c.a 2 1.a even 1 1 trivial
1648.1.c.a 2 103.b odd 2 1 CM
2575.1.c.b 4 20.e even 4 2
2575.1.c.b 4 2060.m odd 4 2
2575.1.d.d 2 20.d odd 2 1
2575.1.d.d 2 2060.e even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(1648, [\chi])\).

Hecke characteristic polynomials

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