Properties

Label 103.1.b.a
Level 103
Weight 1
Character orbit 103.b
Self dual yes
Analytic conductor 0.051
Analytic rank 0
Dimension 2
Projective image \(D_{5}\)
CM discriminant -103
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 103 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 103.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(0.0514036963012\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{5}\)
Projective field Galois closure of 5.1.10609.1
Artin image $D_5$
Artin field Galois closure of 5.1.10609.1

$q$-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.

\(f(q)\) \(=\) \( q + ( -1 + \beta ) q^{2} + ( 1 - \beta ) q^{4} -\beta q^{7} - q^{8} + q^{9} +O(q^{10})\) \( q + ( -1 + \beta ) q^{2} + ( 1 - \beta ) q^{4} -\beta q^{7} - q^{8} + q^{9} + ( -1 + \beta ) q^{13} - q^{14} -\beta q^{17} + ( -1 + \beta ) q^{18} + ( -1 + \beta ) q^{19} + ( -1 + \beta ) q^{23} + q^{25} + ( 2 - \beta ) q^{26} + q^{28} -\beta q^{29} + q^{32} - q^{34} + ( 1 - \beta ) q^{36} + ( 2 - \beta ) q^{38} + ( -1 + \beta ) q^{41} + ( 2 - \beta ) q^{46} + \beta q^{49} + ( -1 + \beta ) q^{50} + ( -2 + \beta ) q^{52} + \beta q^{56} - q^{58} -\beta q^{59} -\beta q^{61} -\beta q^{63} + ( -1 + \beta ) q^{64} + q^{68} - q^{72} + ( -2 + \beta ) q^{76} + ( -1 + \beta ) q^{79} + q^{81} + ( 2 - \beta ) q^{82} -\beta q^{83} - q^{91} + ( -2 + \beta ) q^{92} -\beta q^{97} + q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{2} + q^{4} - q^{7} - 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q - q^{2} + q^{4} - q^{7} - 2q^{8} + 2q^{9} - q^{13} - 2q^{14} - q^{17} - q^{18} - q^{19} - q^{23} + 2q^{25} + 3q^{26} + 2q^{28} - q^{29} + 2q^{32} - 2q^{34} + q^{36} + 3q^{38} - q^{41} + 3q^{46} + q^{49} - q^{50} - 3q^{52} + q^{56} - 2q^{58} - q^{59} - q^{61} - q^{63} - q^{64} + 2q^{68} - 2q^{72} - 3q^{76} - q^{79} + 2q^{81} + 3q^{82} - q^{83} - 2q^{91} - 3q^{92} - q^{97} + 2q^{98} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/103\mathbb{Z}\right)^\times\).

\(n\) \(5\)
\(\chi(n)\) \(-1\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
102.1
−0.618034
1.61803
−1.61803 0 1.61803 0 0 0.618034 −1.00000 1.00000 0
102.2 0.618034 0 −0.618034 0 0 −1.61803 −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
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 103.1.b.a 2
3.b odd 2 1 927.1.d.b 2
4.b odd 2 1 1648.1.c.a 2
5.b even 2 1 2575.1.d.d 2
5.c odd 4 2 2575.1.c.b 4
103.b odd 2 1 CM 103.1.b.a 2
309.c even 2 1 927.1.d.b 2
412.d even 2 1 1648.1.c.a 2
515.c odd 2 1 2575.1.d.d 2
515.f even 4 2 2575.1.c.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
103.1.b.a 2 1.a even 1 1 trivial
103.1.b.a 2 103.b odd 2 1 CM
927.1.d.b 2 3.b odd 2 1
927.1.d.b 2 309.c even 2 1
1648.1.c.a 2 4.b odd 2 1
1648.1.c.a 2 412.d even 2 1
2575.1.c.b 4 5.c odd 4 2
2575.1.c.b 4 515.f even 4 2
2575.1.d.d 2 5.b even 2 1
2575.1.d.d 2 515.c odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(103, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
$3$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$5$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$7$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
$11$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$13$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
$17$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
$19$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
$23$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
$29$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
$31$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$37$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$41$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
$43$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$47$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$53$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$59$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
$61$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
$67$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$71$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$73$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$79$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
$83$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
$89$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$97$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
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