Properties

Label 143.1.d.b
Level 143
Weight 1
Character orbit 143.d
Self dual yes
Analytic conductor 0.071
Analytic rank 0
Dimension 2
Projective image \(D_{5}\)
CM discriminant -143
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 143 = 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 143.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(0.0713662968065\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{5}\)
Projective field Galois closure of 5.1.20449.1
Artin image $D_{10}$
Artin field Galois closure of 10.2.5436100813.1

$q$-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.

\(f(q)\) \(=\) \( q + ( 1 - \beta ) q^{2} + ( -1 + \beta ) q^{3} + ( 1 - \beta ) q^{4} + ( -2 + \beta ) q^{6} + \beta q^{7} + q^{8} + ( 1 - \beta ) q^{9} +O(q^{10})\) \( q + ( 1 - \beta ) q^{2} + ( -1 + \beta ) q^{3} + ( 1 - \beta ) q^{4} + ( -2 + \beta ) q^{6} + \beta q^{7} + q^{8} + ( 1 - \beta ) q^{9} - q^{11} + ( -2 + \beta ) q^{12} - q^{13} - q^{14} + ( 2 - \beta ) q^{18} + ( 1 - \beta ) q^{19} + q^{21} + ( -1 + \beta ) q^{22} -\beta q^{23} + ( -1 + \beta ) q^{24} + q^{25} + ( -1 + \beta ) q^{26} - q^{27} - q^{28} - q^{32} + ( 1 - \beta ) q^{33} + ( 2 - \beta ) q^{36} + ( 2 - \beta ) q^{38} + ( 1 - \beta ) q^{39} + \beta q^{41} + ( 1 - \beta ) q^{42} + ( -1 + \beta ) q^{44} + q^{46} + \beta q^{49} + ( 1 - \beta ) q^{50} + ( -1 + \beta ) q^{52} + ( -1 + \beta ) q^{53} + ( -1 + \beta ) q^{54} + \beta q^{56} + ( -2 + \beta ) q^{57} - q^{63} + ( -1 + \beta ) q^{64} + ( 2 - \beta ) q^{66} - q^{69} + ( 1 - \beta ) q^{72} + ( 1 - \beta ) q^{73} + ( -1 + \beta ) q^{75} + ( 2 - \beta ) q^{76} -\beta q^{77} + ( 2 - \beta ) q^{78} - q^{82} + \beta q^{83} + ( 1 - \beta ) q^{84} - q^{88} -\beta q^{91} + q^{92} + ( 1 - \beta ) q^{96} - q^{98} + ( -1 + \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{2} - q^{3} + q^{4} - 3q^{6} + q^{7} + 2q^{8} + q^{9} + O(q^{10}) \) \( 2q + q^{2} - q^{3} + q^{4} - 3q^{6} + q^{7} + 2q^{8} + q^{9} - 2q^{11} - 3q^{12} - 2q^{13} - 2q^{14} + 3q^{18} + q^{19} + 2q^{21} - q^{22} - q^{23} - q^{24} + 2q^{25} - q^{26} - 2q^{27} - 2q^{28} - 2q^{32} + q^{33} + 3q^{36} + 3q^{38} + q^{39} + q^{41} + q^{42} - q^{44} + 2q^{46} + q^{49} + q^{50} - q^{52} - q^{53} - q^{54} + q^{56} - 3q^{57} - 2q^{63} - q^{64} + 3q^{66} - 2q^{69} + q^{72} + q^{73} - q^{75} + 3q^{76} - q^{77} + 3q^{78} - 2q^{82} + q^{83} + q^{84} - 2q^{88} - q^{91} + 2q^{92} + q^{96} - 2q^{98} - q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/143\mathbb{Z}\right)^\times\).

\(n\) \(67\) \(79\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
142.1
1.61803
−0.618034
−0.618034 0.618034 −0.618034 0 −0.381966 1.61803 1.00000 −0.618034 0
142.2 1.61803 −1.61803 1.61803 0 −2.61803 −0.618034 1.00000 1.61803 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
143.d odd 2 1 CM by \(\Q(\sqrt{-143}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 143.1.d.b yes 2
3.b odd 2 1 1287.1.g.a 2
4.b odd 2 1 2288.1.m.a 2
5.b even 2 1 3575.1.h.e 2
5.c odd 4 2 3575.1.c.c 4
11.b odd 2 1 143.1.d.a 2
11.c even 5 2 1573.1.l.a 4
11.c even 5 2 1573.1.l.c 4
11.d odd 10 2 1573.1.l.b 4
11.d odd 10 2 1573.1.l.d 4
13.b even 2 1 143.1.d.a 2
13.c even 3 2 1859.1.i.a 4
13.d odd 4 2 1859.1.c.c 4
13.e even 6 2 1859.1.i.b 4
13.f odd 12 4 1859.1.k.c 8
33.d even 2 1 1287.1.g.b 2
39.d odd 2 1 1287.1.g.b 2
44.c even 2 1 2288.1.m.b 2
52.b odd 2 1 2288.1.m.b 2
55.d odd 2 1 3575.1.h.f 2
55.e even 4 2 3575.1.c.d 4
65.d even 2 1 3575.1.h.f 2
65.h odd 4 2 3575.1.c.d 4
143.d odd 2 1 CM 143.1.d.b yes 2
143.g even 4 2 1859.1.c.c 4
143.i odd 6 2 1859.1.i.a 4
143.k odd 6 2 1859.1.i.b 4
143.l odd 10 2 1573.1.l.a 4
143.l odd 10 2 1573.1.l.c 4
143.n even 10 2 1573.1.l.b 4
143.n even 10 2 1573.1.l.d 4
143.o even 12 4 1859.1.k.c 8
429.e even 2 1 1287.1.g.a 2
572.b even 2 1 2288.1.m.a 2
715.c odd 2 1 3575.1.h.e 2
715.q even 4 2 3575.1.c.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
143.1.d.a 2 11.b odd 2 1
143.1.d.a 2 13.b even 2 1
143.1.d.b yes 2 1.a even 1 1 trivial
143.1.d.b yes 2 143.d odd 2 1 CM
1287.1.g.a 2 3.b odd 2 1
1287.1.g.a 2 429.e even 2 1
1287.1.g.b 2 33.d even 2 1
1287.1.g.b 2 39.d odd 2 1
1573.1.l.a 4 11.c even 5 2
1573.1.l.a 4 143.l odd 10 2
1573.1.l.b 4 11.d odd 10 2
1573.1.l.b 4 143.n even 10 2
1573.1.l.c 4 11.c even 5 2
1573.1.l.c 4 143.l odd 10 2
1573.1.l.d 4 11.d odd 10 2
1573.1.l.d 4 143.n even 10 2
1859.1.c.c 4 13.d odd 4 2
1859.1.c.c 4 143.g even 4 2
1859.1.i.a 4 13.c even 3 2
1859.1.i.a 4 143.i odd 6 2
1859.1.i.b 4 13.e even 6 2
1859.1.i.b 4 143.k odd 6 2
1859.1.k.c 8 13.f odd 12 4
1859.1.k.c 8 143.o even 12 4
2288.1.m.a 2 4.b odd 2 1
2288.1.m.a 2 572.b even 2 1
2288.1.m.b 2 44.c even 2 1
2288.1.m.b 2 52.b odd 2 1
3575.1.c.c 4 5.c odd 4 2
3575.1.c.c 4 715.q even 4 2
3575.1.c.d 4 55.e even 4 2
3575.1.c.d 4 65.h odd 4 2
3575.1.h.e 2 5.b even 2 1
3575.1.h.e 2 715.c odd 2 1
3575.1.h.f 2 55.d odd 2 1
3575.1.h.f 2 65.d even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - T_{2} - 1 \) acting on \(S_{1}^{\mathrm{new}}(143, [\chi])\).

Hecke characteristic polynomials

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