Properties

Label 127.1.b.a
Level 127
Weight 1
Character orbit 127.b
Self dual yes
Analytic conductor 0.063
Analytic rank 0
Dimension 2
Projective image \(D_{5}\)
CM discriminant -127
Inner twists 2

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 127 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 127.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(0.0633812566044\)
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 5.1.16129.1
Artin image $D_5$
Artin field Galois closure of 5.1.16129.1

$q$-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.

\(f(q)\) \(=\) \( q + ( -1 + \beta ) q^{2} + ( 1 - \beta ) q^{4} - q^{8} + q^{9} +O(q^{10})\) \( q + ( -1 + \beta ) q^{2} + ( 1 - \beta ) q^{4} - q^{8} + q^{9} -\beta q^{11} -\beta q^{13} + ( -1 + \beta ) q^{17} + ( -1 + \beta ) q^{18} + ( -1 + \beta ) q^{19} - q^{22} + q^{25} - q^{26} + ( -1 + \beta ) q^{31} + q^{32} + ( 2 - \beta ) q^{34} + ( 1 - \beta ) q^{36} + ( -1 + \beta ) q^{37} + ( 2 - \beta ) q^{38} -\beta q^{41} + q^{44} -\beta q^{47} + q^{49} + ( -1 + \beta ) q^{50} + q^{52} + ( -1 + \beta ) q^{61} + ( 2 - \beta ) q^{62} + ( -1 + \beta ) q^{64} + ( -2 + \beta ) q^{68} + ( -1 + \beta ) q^{71} - q^{72} -\beta q^{73} + ( 2 - \beta ) q^{74} + ( -2 + \beta ) q^{76} -\beta q^{79} + q^{81} - q^{82} + \beta q^{88} - q^{94} + ( -1 + \beta ) q^{98} -\beta q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{2} + q^{4} - 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q - q^{2} + q^{4} - 2q^{8} + 2q^{9} - q^{11} - q^{13} - q^{17} - q^{18} - q^{19} - 2q^{22} + 2q^{25} - 2q^{26} - q^{31} + 2q^{32} + 3q^{34} + q^{36} - q^{37} + 3q^{38} - q^{41} + 2q^{44} - q^{47} + 2q^{49} - q^{50} + 2q^{52} - q^{61} + 3q^{62} - q^{64} - 3q^{68} - q^{71} - 2q^{72} - q^{73} + 3q^{74} - 3q^{76} - q^{79} + 2q^{81} - 2q^{82} + q^{88} - 2q^{94} - q^{98} - q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/127\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
126.1
−0.618034
1.61803
−1.61803 0 1.61803 0 0 0 −1.00000 1.00000 0
126.2 0.618034 0 −0.618034 0 0 0 −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
127.b odd 2 1 CM by \(\Q(\sqrt{-127}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 127.1.b.a 2
3.b odd 2 1 1143.1.d.b 2
4.b odd 2 1 2032.1.b.a 2
5.b even 2 1 3175.1.d.d 2
5.c odd 4 2 3175.1.c.b 4
127.b odd 2 1 CM 127.1.b.a 2
381.c even 2 1 1143.1.d.b 2
508.d even 2 1 2032.1.b.a 2
635.c odd 2 1 3175.1.d.d 2
635.f even 4 2 3175.1.c.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
127.1.b.a 2 1.a even 1 1 trivial
127.1.b.a 2 127.b odd 2 1 CM
1143.1.d.b 2 3.b odd 2 1
1143.1.d.b 2 381.c even 2 1
2032.1.b.a 2 4.b odd 2 1
2032.1.b.a 2 508.d even 2 1
3175.1.c.b 4 5.c odd 4 2
3175.1.c.b 4 635.f even 4 2
3175.1.d.d 2 5.b even 2 1
3175.1.d.d 2 635.c odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(127, [\chi])\).

Hecke characteristic polynomials

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