Properties

Label 475.1.o.a
Level $475$
Weight $1$
Character orbit 475.o
Analytic conductor $0.237$
Analytic rank $0$
Dimension $4$
Projective image $D_{5}$
CM discriminant -19
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [475,1,Mod(56,475)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(475, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([4, 5]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("475.56");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 475 = 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 475.o (of order \(10\), degree \(4\), 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: no
Analytic conductor: \(0.237055881001\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{5}\)
Projective field: Galois closure of 5.1.141015625.1

$q$-expansion

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

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + \zeta_{10}^{2} q^{4} + \zeta_{10}^{4} q^{5} + (\zeta_{10}^{4} - \zeta_{10}) q^{7} + \zeta_{10}^{4} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{10}^{2} q^{4} + \zeta_{10}^{4} q^{5} + (\zeta_{10}^{4} - \zeta_{10}) q^{7} + \zeta_{10}^{4} q^{9} + (\zeta_{10}^{2} + 1) q^{11} + \zeta_{10}^{4} q^{16} + ( - \zeta_{10} + 1) q^{17} - \zeta_{10}^{3} q^{19} - \zeta_{10} q^{20} + (\zeta_{10}^{2} + 1) q^{23} - \zeta_{10}^{3} q^{25} + ( - \zeta_{10}^{3} - \zeta_{10}) q^{28} + ( - \zeta_{10}^{3} + 1) q^{35} - \zeta_{10} q^{36} + ( - \zeta_{10}^{3} + \zeta_{10}^{2}) q^{43} + (\zeta_{10}^{4} + \zeta_{10}^{2}) q^{44} - \zeta_{10}^{3} q^{45} + \zeta_{10}^{2} q^{47} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} + 1) q^{49} + (\zeta_{10}^{4} - \zeta_{10}) q^{55} + (\zeta_{10}^{4} - \zeta_{10}^{3}) q^{61} + ( - \zeta_{10}^{3} + 1) q^{63} - \zeta_{10} q^{64} + ( - \zeta_{10}^{3} + \zeta_{10}^{2}) q^{68} - \zeta_{10} q^{73} + q^{76} + (\zeta_{10}^{4} - \zeta_{10}^{3} - \zeta_{10}) q^{77} - \zeta_{10}^{3} q^{80} - \zeta_{10}^{3} q^{81} + ( - \zeta_{10} + 1) q^{83} + (\zeta_{10}^{4} + 1) q^{85} + (\zeta_{10}^{4} + \zeta_{10}^{2}) q^{92} + \zeta_{10}^{2} q^{95} + (\zeta_{10}^{4} - \zeta_{10}) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - q^{4} - q^{5} - 2 q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - q^{4} - q^{5} - 2 q^{7} - q^{9} + 3 q^{11} - q^{16} + 3 q^{17} - q^{19} - q^{20} + 3 q^{23} - q^{25} - 2 q^{28} + 3 q^{35} - q^{36} - 2 q^{43} - 2 q^{44} - q^{45} - 2 q^{47} + 2 q^{49} - 2 q^{55} - 2 q^{61} + 3 q^{63} - q^{64} - 2 q^{68} - 2 q^{73} + 4 q^{76} - 4 q^{77} - q^{80} - q^{81} + 3 q^{83} + 3 q^{85} - 2 q^{92} - q^{95} - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/475\mathbb{Z}\right)^\times\).

\(n\) \(77\) \(401\)
\(\chi(n)\) \(\zeta_{10}^{2}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
56.1
−0.309017 0.951057i
−0.309017 + 0.951057i
0.809017 + 0.587785i
0.809017 0.587785i
0 0 −0.809017 + 0.587785i 0.309017 0.951057i 0 0.618034 0 0.309017 0.951057i 0
246.1 0 0 −0.809017 0.587785i 0.309017 + 0.951057i 0 0.618034 0 0.309017 + 0.951057i 0
341.1 0 0 0.309017 + 0.951057i −0.809017 + 0.587785i 0 −1.61803 0 −0.809017 + 0.587785i 0
436.1 0 0 0.309017 0.951057i −0.809017 0.587785i 0 −1.61803 0 −0.809017 0.587785i 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
19.b odd 2 1 CM by \(\Q(\sqrt{-19}) \)
25.d even 5 1 inner
475.o odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 475.1.o.a 4
5.b even 2 1 2375.1.o.a 4
5.c odd 4 2 2375.1.m.b 8
19.b odd 2 1 CM 475.1.o.a 4
25.d even 5 1 inner 475.1.o.a 4
25.e even 10 1 2375.1.o.a 4
25.f odd 20 2 2375.1.m.b 8
95.d odd 2 1 2375.1.o.a 4
95.g even 4 2 2375.1.m.b 8
475.m odd 10 1 2375.1.o.a 4
475.o odd 10 1 inner 475.1.o.a 4
475.v even 20 2 2375.1.m.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
475.1.o.a 4 1.a even 1 1 trivial
475.1.o.a 4 19.b odd 2 1 CM
475.1.o.a 4 25.d even 5 1 inner
475.1.o.a 4 475.o odd 10 1 inner
2375.1.m.b 8 5.c odd 4 2
2375.1.m.b 8 25.f odd 20 2
2375.1.m.b 8 95.g even 4 2
2375.1.m.b 8 475.v even 20 2
2375.1.o.a 4 5.b even 2 1
2375.1.o.a 4 25.e even 10 1
2375.1.o.a 4 95.d odd 2 1
2375.1.o.a 4 475.m odd 10 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(475, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + T^{3} + T^{2} + T + 1 \) Copy content Toggle raw display
$7$ \( (T^{2} + T - 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} - 3 T^{3} + 4 T^{2} - 2 T + 1 \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} - 3 T^{3} + 4 T^{2} - 2 T + 1 \) Copy content Toggle raw display
$19$ \( T^{4} + T^{3} + T^{2} + T + 1 \) Copy content Toggle raw display
$23$ \( T^{4} - 3 T^{3} + 4 T^{2} - 2 T + 1 \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( T^{4} \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + T - 1)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 2 T^{3} + 4 T^{2} + 8 T + 16 \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( T^{4} + 2 T^{3} + 4 T^{2} + 3 T + 1 \) Copy content Toggle raw display
$67$ \( T^{4} \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( T^{4} + 2 T^{3} + 4 T^{2} + 8 T + 16 \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( T^{4} - 3 T^{3} + 4 T^{2} - 2 T + 1 \) Copy content Toggle raw display
$89$ \( T^{4} \) Copy content Toggle raw display
$97$ \( T^{4} \) Copy content Toggle raw display
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