Properties

Label 975.1.bd.c
Level $975$
Weight $1$
Character orbit 975.bd
Analytic conductor $0.487$
Analytic rank $0$
Dimension $8$
Projective image $D_{10}$
CM discriminant -39
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [975,1,Mod(116,975)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(975, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([5, 2, 5]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("975.116");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 975.bd (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: \(0.486588387317\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\Q(\zeta_{20})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{6} + x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{10}\)
Projective field: Galois closure of 10.0.1059009246826171875.1

$q$-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.

\(f(q)\) \(=\) \( q + (\zeta_{20}^{5} - \zeta_{20}) q^{2} + \zeta_{20}^{6} q^{3} + ( - \zeta_{20}^{6} + \zeta_{20}^{2} - 1) q^{4} - \zeta_{20}^{9} q^{5} + ( - \zeta_{20}^{7} - \zeta_{20}) q^{6} + (\zeta_{20}^{7} - \zeta_{20}^{5} - \zeta_{20}^{3} - \zeta_{20}) q^{8} - \zeta_{20}^{2} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{20}^{5} - \zeta_{20}) q^{2} + \zeta_{20}^{6} q^{3} + ( - \zeta_{20}^{6} + \zeta_{20}^{2} - 1) q^{4} - \zeta_{20}^{9} q^{5} + ( - \zeta_{20}^{7} - \zeta_{20}) q^{6} + (\zeta_{20}^{7} - \zeta_{20}^{5} - \zeta_{20}^{3} - \zeta_{20}) q^{8} - \zeta_{20}^{2} q^{9} + (\zeta_{20}^{4} - 1) q^{10} + ( - \zeta_{20}^{9} - \zeta_{20}^{7}) q^{11} + (\zeta_{20}^{8} - \zeta_{20}^{6} + \zeta_{20}^{2}) q^{12} + \zeta_{20}^{2} q^{13} + \zeta_{20}^{5} q^{15} + ( - \zeta_{20}^{8} + \zeta_{20}^{6} + \zeta_{20}^{4} + \zeta_{20}^{2} + 1) q^{16} + ( - \zeta_{20}^{7} + \zeta_{20}^{3}) q^{18} + (\zeta_{20}^{9} - \zeta_{20}^{5} + \zeta_{20}) q^{20} + (\zeta_{20}^{8} + \zeta_{20}^{4} + \zeta_{20}^{2} - 1) q^{22} + ( - \zeta_{20}^{9} + \zeta_{20}^{7} - \zeta_{20}^{3} + \zeta_{20}) q^{24} - \zeta_{20}^{8} q^{25} + (\zeta_{20}^{7} - \zeta_{20}^{3}) q^{26} - \zeta_{20}^{8} q^{27} + ( - \zeta_{20}^{6} - 1) q^{30} + (\zeta_{20}^{9} - \zeta_{20}^{7} - \zeta_{20}^{3} + \zeta_{20}) q^{32} + (\zeta_{20}^{5} + \zeta_{20}^{3}) q^{33} + (\zeta_{20}^{8} - \zeta_{20}^{4} + \zeta_{20}^{2}) q^{36} + \zeta_{20}^{8} q^{39} + (\zeta_{20}^{6} - \zeta_{20}^{4} - \zeta_{20}^{2} + 1) q^{40} + (\zeta_{20}^{9} - \zeta_{20}^{5}) q^{41} + (\zeta_{20}^{8} - \zeta_{20}^{2}) q^{43} + (\zeta_{20}^{7} - \zeta_{20}^{5} - \zeta_{20}^{3} + \zeta_{20}) q^{44} - \zeta_{20} q^{45} + (\zeta_{20}^{9} + \zeta_{20}^{3}) q^{47} + ( - \zeta_{20}^{8} + \zeta_{20}^{6} + \zeta_{20}^{4} - \zeta_{20}^{2} - 1) q^{48} + q^{49} + (\zeta_{20}^{9} + \zeta_{20}^{3}) q^{50} + ( - \zeta_{20}^{8} + \zeta_{20}^{4} - \zeta_{20}^{2}) q^{52} + (\zeta_{20}^{9} + \zeta_{20}^{3}) q^{54} + ( - \zeta_{20}^{8} - \zeta_{20}^{6}) q^{55} + ( - \zeta_{20}^{3} - \zeta_{20}) q^{59} + (\zeta_{20}^{7} - \zeta_{20}^{5} + \zeta_{20}) q^{60} + ( - \zeta_{20}^{4} + \zeta_{20}^{2}) q^{61} + (\zeta_{20}^{8} + \zeta_{20}^{4} - \zeta_{20}^{2} - 1) q^{64} + \zeta_{20} q^{65} + (\zeta_{20}^{8} - \zeta_{20}^{6} - \zeta_{20}^{4} - 1) q^{66} + (\zeta_{20}^{7} + \zeta_{20}^{5}) q^{71} + ( - \zeta_{20}^{9} + \zeta_{20}^{7} + \zeta_{20}^{5} - \zeta_{20}^{3}) q^{72} + \zeta_{20}^{4} q^{75} + ( - \zeta_{20}^{9} - \zeta_{20}^{3}) q^{78} + \zeta_{20}^{6} q^{79} + ( - \zeta_{20}^{9} - \zeta_{20}^{7} + \zeta_{20}^{5} + \zeta_{20}^{3} - \zeta_{20}) q^{80} + \zeta_{20}^{4} q^{81} + (\zeta_{20}^{6} - \zeta_{20}^{4} + 2) q^{82} + (\zeta_{20}^{7} + \zeta_{20}) q^{83} + ( - \zeta_{20}^{9} - \zeta_{20}^{7} - \zeta_{20}^{3}) q^{86} + ( - \zeta_{20}^{8} + \zeta_{20}^{6} - 2 \zeta_{20}^{2}) q^{88} + ( - \zeta_{20}^{6} + \zeta_{20}^{2}) q^{90} + (\zeta_{20}^{8} - \zeta_{20}^{4} + 1) q^{94} + (\zeta_{20}^{9} - \zeta_{20}^{7} - \zeta_{20}^{5} + \zeta_{20}^{3}) q^{96} + (\zeta_{20}^{5} - \zeta_{20}) q^{98} + (\zeta_{20}^{9} - \zeta_{20}) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{3} - 8 q^{4} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2 q^{3} - 8 q^{4} - 2 q^{9} - 10 q^{10} - 2 q^{12} + 2 q^{13} + 8 q^{16} - 10 q^{22} + 2 q^{25} + 2 q^{27} - 10 q^{30} + 2 q^{36} - 2 q^{39} + 10 q^{40} - 4 q^{43} - 8 q^{48} + 8 q^{49} - 2 q^{52} + 4 q^{61} + 2 q^{64} - 10 q^{66} - 2 q^{75} + 4 q^{79} - 2 q^{81} + 20 q^{82} + 10 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/975\mathbb{Z}\right)^\times\).

\(n\) \(301\) \(326\) \(352\)
\(\chi(n)\) \(-1\) \(-1\) \(-\zeta_{20}^{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
116.1
0.951057 0.309017i
−0.951057 + 0.309017i
0.951057 + 0.309017i
−0.951057 0.309017i
0.587785 + 0.809017i
−0.587785 0.809017i
0.587785 0.809017i
−0.587785 + 0.809017i
−0.951057 0.690983i −0.309017 0.951057i 0.118034 + 0.363271i 0.951057 + 0.309017i −0.363271 + 1.11803i 0 −0.224514 + 0.690983i −0.809017 + 0.587785i −0.690983 0.951057i
116.2 0.951057 + 0.690983i −0.309017 0.951057i 0.118034 + 0.363271i −0.951057 0.309017i 0.363271 1.11803i 0 0.224514 0.690983i −0.809017 + 0.587785i −0.690983 0.951057i
311.1 −0.951057 + 0.690983i −0.309017 + 0.951057i 0.118034 0.363271i 0.951057 0.309017i −0.363271 1.11803i 0 −0.224514 0.690983i −0.809017 0.587785i −0.690983 + 0.951057i
311.2 0.951057 0.690983i −0.309017 + 0.951057i 0.118034 0.363271i −0.951057 + 0.309017i 0.363271 + 1.11803i 0 0.224514 + 0.690983i −0.809017 0.587785i −0.690983 + 0.951057i
506.1 −0.587785 1.80902i 0.809017 0.587785i −2.11803 + 1.53884i 0.587785 0.809017i −1.53884 1.11803i 0 2.48990 + 1.80902i 0.309017 0.951057i −1.80902 0.587785i
506.2 0.587785 + 1.80902i 0.809017 0.587785i −2.11803 + 1.53884i −0.587785 + 0.809017i 1.53884 + 1.11803i 0 −2.48990 1.80902i 0.309017 0.951057i −1.80902 0.587785i
896.1 −0.587785 + 1.80902i 0.809017 + 0.587785i −2.11803 1.53884i 0.587785 + 0.809017i −1.53884 + 1.11803i 0 2.48990 1.80902i 0.309017 + 0.951057i −1.80902 + 0.587785i
896.2 0.587785 1.80902i 0.809017 + 0.587785i −2.11803 1.53884i −0.587785 0.809017i 1.53884 1.11803i 0 −2.48990 + 1.80902i 0.309017 + 0.951057i −1.80902 + 0.587785i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 116.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
39.d odd 2 1 CM by \(\Q(\sqrt{-39}) \)
3.b odd 2 1 inner
13.b even 2 1 inner
25.d even 5 1 inner
75.j odd 10 1 inner
325.q even 10 1 inner
975.bd odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 975.1.bd.c 8
3.b odd 2 1 inner 975.1.bd.c 8
13.b even 2 1 inner 975.1.bd.c 8
25.d even 5 1 inner 975.1.bd.c 8
39.d odd 2 1 CM 975.1.bd.c 8
75.j odd 10 1 inner 975.1.bd.c 8
325.q even 10 1 inner 975.1.bd.c 8
975.bd odd 10 1 inner 975.1.bd.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
975.1.bd.c 8 1.a even 1 1 trivial
975.1.bd.c 8 3.b odd 2 1 inner
975.1.bd.c 8 13.b even 2 1 inner
975.1.bd.c 8 25.d even 5 1 inner
975.1.bd.c 8 39.d odd 2 1 CM
975.1.bd.c 8 75.j odd 10 1 inner
975.1.bd.c 8 325.q even 10 1 inner
975.1.bd.c 8 975.bd odd 10 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 5T_{2}^{6} + 10T_{2}^{4} + 25 \) acting on \(S_{1}^{\mathrm{new}}(975, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 5 T^{6} + 10 T^{4} + 25 \) Copy content Toggle raw display
$3$ \( (T^{4} - T^{3} + T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} - T^{6} + T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} + 10 T^{4} + 25 T^{2} + 25 \) Copy content Toggle raw display
$13$ \( (T^{4} - T^{3} + T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} \) Copy content Toggle raw display
$19$ \( T^{8} \) Copy content Toggle raw display
$23$ \( T^{8} \) Copy content Toggle raw display
$29$ \( T^{8} \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( T^{8} \) Copy content Toggle raw display
$41$ \( T^{8} + 5 T^{6} + 10 T^{4} + 25 \) Copy content Toggle raw display
$43$ \( (T^{2} + T - 1)^{4} \) Copy content Toggle raw display
$47$ \( T^{8} + 10 T^{4} + 25 T^{2} + 25 \) Copy content Toggle raw display
$53$ \( T^{8} \) Copy content Toggle raw display
$59$ \( T^{8} + 10 T^{4} + 25 T^{2} + 25 \) Copy content Toggle raw display
$61$ \( (T^{4} - 2 T^{3} + 4 T^{2} - 3 T + 1)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} \) Copy content Toggle raw display
$71$ \( T^{8} + 5 T^{6} + 10 T^{4} + 25 \) Copy content Toggle raw display
$73$ \( T^{8} \) Copy content Toggle raw display
$79$ \( (T^{4} - 2 T^{3} + 4 T^{2} - 8 T + 16)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 10 T^{4} + 25 T^{2} + 25 \) Copy content Toggle raw display
$89$ \( T^{8} \) Copy content Toggle raw display
$97$ \( T^{8} \) Copy content Toggle raw display
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