Properties

Label 1975.1.d.c
Level $1975$
Weight $1$
Character orbit 1975.d
Self dual yes
Analytic conductor $0.986$
Analytic rank $0$
Dimension $2$
Projective image $D_{5}$
CM discriminant -79
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 1975 = 5^{2} \cdot 79 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1975.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(0.985653399950\)
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: no (minimal twist has level 79)
Projective image: \(D_{5}\)
Projective field: Galois closure of 5.1.6241.1
Artin image: $D_{10}$
Artin field: Galois closure of 10.2.121719003125.1

$q$-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.

\(f(q)\) \(=\) \( q + ( 1 - \beta ) q^{2} + ( 1 - \beta ) q^{4} + q^{8} + q^{9} +O(q^{10})\) \( q + ( 1 - \beta ) q^{2} + ( 1 - \beta ) q^{4} + q^{8} + q^{9} + ( -1 + \beta ) q^{11} + ( 1 - \beta ) q^{13} + ( 1 - \beta ) q^{18} -\beta q^{19} + ( -2 + \beta ) q^{22} + \beta q^{23} + ( 2 - \beta ) q^{26} + ( -1 + \beta ) q^{31} - q^{32} + ( 1 - \beta ) q^{36} + q^{38} + ( -2 + \beta ) q^{44} - q^{46} + q^{49} + ( 2 - \beta ) q^{52} + ( -2 + \beta ) q^{62} + ( -1 + \beta ) q^{64} + \beta q^{67} + q^{72} + \beta q^{73} + q^{76} + q^{79} + q^{81} -2 q^{83} + ( -1 + \beta ) q^{88} + ( -1 + \beta ) q^{89} - q^{92} + \beta q^{97} + ( 1 - \beta ) q^{98} + ( -1 + \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{2} + q^{4} + 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q + q^{2} + q^{4} + 2q^{8} + 2q^{9} - q^{11} + q^{13} + q^{18} - q^{19} - 3q^{22} + q^{23} + 3q^{26} - q^{31} - 2q^{32} + q^{36} + 2q^{38} - 3q^{44} - 2q^{46} + 2q^{49} + 3q^{52} - 3q^{62} - q^{64} + q^{67} + 2q^{72} + q^{73} + 2q^{76} + 2q^{79} + 2q^{81} - 4q^{83} - q^{88} - q^{89} - 2q^{92} + q^{97} + q^{98} - q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1975\mathbb{Z}\right)^\times\).

\(n\) \(951\) \(1502\)
\(\chi(n)\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1026.1
1.61803
−0.618034
−0.618034 0 −0.618034 0 0 0 1.00000 1.00000 0
1026.2 1.61803 0 1.61803 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
79.b odd 2 1 CM by \(\Q(\sqrt{-79}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1975.1.d.c 2
5.b even 2 1 79.1.b.a 2
5.c odd 4 2 1975.1.c.a 4
15.d odd 2 1 711.1.d.b 2
20.d odd 2 1 1264.1.e.a 2
35.c odd 2 1 3871.1.c.c 2
35.i odd 6 2 3871.1.m.b 4
35.j even 6 2 3871.1.m.c 4
79.b odd 2 1 CM 1975.1.d.c 2
395.c odd 2 1 79.1.b.a 2
395.f even 4 2 1975.1.c.a 4
1185.h even 2 1 711.1.d.b 2
1580.b even 2 1 1264.1.e.a 2
2765.c even 2 1 3871.1.c.c 2
2765.bf odd 6 2 3871.1.m.c 4
2765.bm even 6 2 3871.1.m.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
79.1.b.a 2 5.b even 2 1
79.1.b.a 2 395.c odd 2 1
711.1.d.b 2 15.d odd 2 1
711.1.d.b 2 1185.h even 2 1
1264.1.e.a 2 20.d odd 2 1
1264.1.e.a 2 1580.b even 2 1
1975.1.c.a 4 5.c odd 4 2
1975.1.c.a 4 395.f even 4 2
1975.1.d.c 2 1.a even 1 1 trivial
1975.1.d.c 2 79.b odd 2 1 CM
3871.1.c.c 2 35.c odd 2 1
3871.1.c.c 2 2765.c even 2 1
3871.1.m.b 4 35.i odd 6 2
3871.1.m.b 4 2765.bm even 6 2
3871.1.m.c 4 35.j even 6 2
3871.1.m.c 4 2765.bf odd 6 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(1975, [\chi])\):

\( T_{2}^{2} - T_{2} - 1 \)
\( T_{3} \)

Hecke characteristic polynomials

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