↑

Properties

Label 2975.1.h.d
Level $2975$
Weight $1$
Character orbit 2975.h
Self dual yes
Analytic conductor $1.485$
Analytic rank $0$
Dimension $2$
Projective image $D_{5}$
CM discriminant -119
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2975,1,Mod(951,2975)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2975, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2975.951");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2975 = 5^{2} \cdot 7 \cdot 17 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2975.h (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(1.48471841258\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 119)
Projective image: \(D_{5}\)
Projective field: Galois closure of 5.1.14161.1
Artin image: $D_{10}$
Artin field: Galois closure of 10.2.626668503125.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

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.

\(f(q)\) \(=\) \( q + ( - \beta + 1) q^{2} + ( - \beta + 1) q^{3} + ( - \beta + 1) q^{4} + ( - \beta + 2) q^{6} - q^{7} + q^{8} + ( - \beta + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta + 1) q^{2} + ( - \beta + 1) q^{3} + ( - \beta + 1) q^{4} + ( - \beta + 2) q^{6} - q^{7} + q^{8} + ( - \beta + 1) q^{9} + ( - \beta + 2) q^{12} + (\beta - 1) q^{14} - q^{17} + ( - \beta + 2) q^{18} + (\beta - 1) q^{21} + ( - \beta + 1) q^{24} + q^{27} + (\beta - 1) q^{28} - \beta q^{31} - q^{32} + (\beta - 1) q^{34} + ( - \beta + 2) q^{36} + (\beta - 1) q^{41} + (\beta - 2) q^{42} + ( - \beta + 1) q^{43} + q^{49} + (\beta - 1) q^{51} + \beta q^{53} + ( - \beta + 1) q^{54} - q^{56} + (\beta - 1) q^{61} + q^{62} + (\beta - 1) q^{63} + (\beta - 1) q^{64} + \beta q^{67} + (\beta - 1) q^{68} + ( - \beta + 1) q^{72} + ( - \beta + 1) q^{73} + (\beta - 2) q^{82} + (\beta - 2) q^{84} + ( - \beta + 2) q^{86} + q^{93} + (\beta - 1) q^{96} + \beta q^{97} + ( - \beta + 1) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} + q^{3} + q^{4} + 3 q^{6} - 2 q^{7} + 2 q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} + q^{3} + q^{4} + 3 q^{6} - 2 q^{7} + 2 q^{8} + q^{9} + 3 q^{12} - q^{14} - 2 q^{17} + 3 q^{18} - q^{21} + q^{24} + 2 q^{27} - q^{28} - q^{31} - 2 q^{32} - q^{34} + 3 q^{36} - q^{41} - 3 q^{42} + q^{43} + 2 q^{49} - q^{51} + q^{53} + q^{54} - 2 q^{56} - q^{61} + 2 q^{62} - q^{63} - q^{64} + q^{67} - q^{68} + q^{72} + q^{73} - 3 q^{82} - 3 q^{84} + 3 q^{86} + 2 q^{93} - q^{96} + q^{97} + q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2975\mathbb{Z}\right)^\times\).

\(n\) \(477\) \(2451\) \(2551\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
951.1
1.61803
−0.618034
−0.618034 −0.618034 −0.618034 0 0.381966 −1.00000 1.00000 −0.618034 0
951.2 1.61803 1.61803 1.61803 0 2.61803 −1.00000 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
119.d odd 2 1 CM by \(\Q(\sqrt{-119}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2975.1.h.d 2
5.b even 2 1 119.1.d.a ✓ 2
5.c odd 4 2 2975.1.b.b 4
7.b odd 2 1 2975.1.h.c 2
15.d odd 2 1 1071.1.h.b 2
17.b even 2 1 2975.1.h.c 2
20.d odd 2 1 1904.1.n.b 2
35.c odd 2 1 119.1.d.b yes 2
35.f even 4 2 2975.1.b.a 4
35.i odd 6 2 833.1.h.a 4
35.j even 6 2 833.1.h.b 4
85.c even 2 1 119.1.d.b yes 2
85.g odd 4 2 2975.1.b.a 4
85.j even 4 2 2023.1.c.e 4
85.m even 8 4 2023.1.f.b 8
85.p odd 16 8 2023.1.l.b 16
105.g even 2 1 1071.1.h.a 2
119.d odd 2 1 CM 2975.1.h.d 2
140.c even 2 1 1904.1.n.a 2
255.h odd 2 1 1071.1.h.a 2
340.d odd 2 1 1904.1.n.a 2
595.b odd 2 1 119.1.d.a ✓ 2
595.p even 4 2 2975.1.b.b 4
595.u odd 4 2 2023.1.c.e 4
595.z even 6 2 833.1.h.a 4
595.bb odd 6 2 833.1.h.b 4
595.be odd 8 4 2023.1.f.b 8
595.by even 16 8 2023.1.l.b 16
1785.p even 2 1 1071.1.h.b 2
2380.p even 2 1 1904.1.n.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
119.1.d.a ✓ 2 5.b even 2 1
119.1.d.a ✓ 2 595.b odd 2 1
119.1.d.b yes 2 35.c odd 2 1
119.1.d.b yes 2 85.c even 2 1
833.1.h.a 4 35.i odd 6 2
833.1.h.a 4 595.z even 6 2
833.1.h.b 4 35.j even 6 2
833.1.h.b 4 595.bb odd 6 2
1071.1.h.a 2 105.g even 2 1
1071.1.h.a 2 255.h odd 2 1
1071.1.h.b 2 15.d odd 2 1
1071.1.h.b 2 1785.p even 2 1
1904.1.n.a 2 140.c even 2 1
1904.1.n.a 2 340.d odd 2 1
1904.1.n.b 2 20.d odd 2 1
1904.1.n.b 2 2380.p even 2 1
2023.1.c.e 4 85.j even 4 2
2023.1.c.e 4 595.u odd 4 2
2023.1.f.b 8 85.m even 8 4
2023.1.f.b 8 595.be odd 8 4
2023.1.l.b 16 85.p odd 16 8
2023.1.l.b 16 595.by even 16 8
2975.1.b.a 4 35.f even 4 2
2975.1.b.a 4 85.g odd 4 2
2975.1.b.b 4 5.c odd 4 2
2975.1.b.b 4 595.p even 4 2
2975.1.h.c 2 7.b odd 2 1
2975.1.h.c 2 17.b even 2 1
2975.1.h.d 2 1.a even 1 1 trivial
2975.1.h.d 2 119.d odd 2 1 CM

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(2975, [\chi])\):

\( T_{2}^{2} - T_{2} - 1 \) Copy content Toggle raw display
\( T_{3}^{2} - T_{3} - 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

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