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 disc. -103
Inner twists 2

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Newspace parameters

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

Newform invariants

Self dual: Yes
Analytic conductor: \(0.0514036963012\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Projective image \(D_{5}\)
Projective field Galois closure of 5.1.10609.1
Artin image size \(10\)
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.

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

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. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
103.b Odd 1 CM by \(\Q(\sqrt{-103}) \) yes

Hecke kernels

There are no other newforms in \(S_{1}^{\mathrm{new}}(103, [\chi])\).