Properties

Label 1175.1.d.c
Level $1175$
Weight $1$
Character orbit 1175.d
Self dual yes
Analytic conductor $0.586$
Analytic rank $0$
Dimension $2$
Projective image $D_{5}$
CM discriminant -47
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1175,1,Mod(751,1175)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1175, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1175.751");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1175 = 5^{2} \cdot 47 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1175.d (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: \(0.586401389844\)
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 47)
Projective image: \(D_{5}\)
Projective field: Galois closure of 5.1.2209.1
Artin image: $D_{10}$
Artin field: Galois closure of 10.2.15249003125.1

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta + 1) q^{2} + \beta q^{3} + ( - \beta + 1) q^{4} - q^{6} + ( - \beta + 1) q^{7} + q^{8} + \beta q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta + 1) q^{2} + \beta q^{3} + ( - \beta + 1) q^{4} - q^{6} + ( - \beta + 1) q^{7} + q^{8} + \beta q^{9} - q^{12} + ( - \beta + 2) q^{14} + \beta q^{17} - q^{18} - q^{21} + \beta q^{24} + q^{27} + ( - \beta + 2) q^{28} - q^{32} - q^{34} - q^{36} + \beta q^{37} + (\beta - 1) q^{42} - q^{47} + ( - \beta + 1) q^{49} + (\beta + 1) q^{51} + ( - \beta + 1) q^{53} + ( - \beta + 1) q^{54} + ( - \beta + 1) q^{56} + (\beta - 1) q^{59} + (\beta - 1) q^{61} - q^{63} + (\beta - 1) q^{64} - q^{68} - \beta q^{71} + \beta q^{72} - q^{74} - \beta q^{79} - 2 q^{83} + (\beta - 1) q^{84} + (\beta - 1) q^{89} + (\beta - 1) q^{94} - \beta q^{96} + ( - \beta + 1) q^{97} + ( - \beta + 2) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} + q^{3} + q^{4} - 2 q^{6} + q^{7} + 2 q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} + q^{3} + q^{4} - 2 q^{6} + q^{7} + 2 q^{8} + q^{9} - 2 q^{12} + 3 q^{14} + q^{17} - 2 q^{18} - 2 q^{21} + q^{24} + 2 q^{27} + 3 q^{28} - 2 q^{32} - 2 q^{34} - 2 q^{36} + q^{37} - q^{42} - 2 q^{47} + q^{49} + 3 q^{51} + q^{53} + q^{54} + q^{56} - q^{59} - q^{61} - 2 q^{63} - q^{64} - 2 q^{68} - q^{71} + q^{72} - 2 q^{74} - q^{79} - 4 q^{83} - q^{84} - q^{89} - q^{94} - q^{96} + q^{97} + 3 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1175\mathbb{Z}\right)^\times\).

\(n\) \(377\) \(851\)
\(\chi(n)\) \(1\) \(-1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
751.1
1.61803
−0.618034
−0.618034 1.61803 −0.618034 0 −1.00000 −0.618034 1.00000 1.61803 0
751.2 1.61803 −0.618034 1.61803 0 −1.00000 1.61803 1.00000 −0.618034 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
47.b odd 2 1 CM by \(\Q(\sqrt{-47}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1175.1.d.c 2
5.b even 2 1 47.1.b.a 2
5.c odd 4 2 1175.1.b.b 4
15.d odd 2 1 423.1.d.a 2
20.d odd 2 1 752.1.g.a 2
35.c odd 2 1 2303.1.d.c 2
35.i odd 6 2 2303.1.f.b 4
35.j even 6 2 2303.1.f.c 4
40.e odd 2 1 3008.1.g.a 2
40.f even 2 1 3008.1.g.b 2
45.h odd 6 2 3807.1.f.a 4
45.j even 6 2 3807.1.f.b 4
47.b odd 2 1 CM 1175.1.d.c 2
235.b odd 2 1 47.1.b.a 2
235.e even 4 2 1175.1.b.b 4
235.i even 46 22 2209.1.d.a 44
235.j odd 46 22 2209.1.d.a 44
705.g even 2 1 423.1.d.a 2
940.h even 2 1 752.1.g.a 2
1645.d even 2 1 2303.1.d.c 2
1645.r even 6 2 2303.1.f.b 4
1645.t odd 6 2 2303.1.f.c 4
1880.f even 2 1 3008.1.g.a 2
1880.l odd 2 1 3008.1.g.b 2
2115.o even 6 2 3807.1.f.a 4
2115.t odd 6 2 3807.1.f.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
47.1.b.a 2 5.b even 2 1
47.1.b.a 2 235.b odd 2 1
423.1.d.a 2 15.d odd 2 1
423.1.d.a 2 705.g even 2 1
752.1.g.a 2 20.d odd 2 1
752.1.g.a 2 940.h even 2 1
1175.1.b.b 4 5.c odd 4 2
1175.1.b.b 4 235.e even 4 2
1175.1.d.c 2 1.a even 1 1 trivial
1175.1.d.c 2 47.b odd 2 1 CM
2209.1.d.a 44 235.i even 46 22
2209.1.d.a 44 235.j odd 46 22
2303.1.d.c 2 35.c odd 2 1
2303.1.d.c 2 1645.d even 2 1
2303.1.f.b 4 35.i odd 6 2
2303.1.f.b 4 1645.r even 6 2
2303.1.f.c 4 35.j even 6 2
2303.1.f.c 4 1645.t odd 6 2
3008.1.g.a 2 40.e odd 2 1
3008.1.g.a 2 1880.f even 2 1
3008.1.g.b 2 40.f even 2 1
3008.1.g.b 2 1880.l odd 2 1
3807.1.f.a 4 45.h odd 6 2
3807.1.f.a 4 2115.o even 6 2
3807.1.f.b 4 45.j even 6 2
3807.1.f.b 4 2115.t odd 6 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - T_{2} - 1 \) acting on \(S_{1}^{\mathrm{new}}(1175, [\chi])\). 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
$7$ \( T^{2} - T - 1 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - T - 1 \) 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
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - T - 1 \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( (T + 1)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} - T - 1 \) Copy content Toggle raw display
$59$ \( T^{2} + T - 1 \) Copy content Toggle raw display
$61$ \( T^{2} + T - 1 \) Copy content Toggle raw display
$67$ \( T^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + T - 1 \) Copy content Toggle raw display
$73$ \( T^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + T - 1 \) Copy content Toggle raw display
$83$ \( (T + 2)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + T - 1 \) Copy content Toggle raw display
$97$ \( T^{2} - T - 1 \) Copy content Toggle raw display
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