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Properties

Label 1925.1.dn.b
Level $1925$
Weight $1$
Character orbit 1925.dn
Analytic conductor $0.961$
Analytic rank $0$
Dimension $16$
Projective image $D_{10}$
CM discriminant -35
Inner twists $16$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1925,1,Mod(118,1925)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1925, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([15, 10, 6]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1925.118");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1925 = 5^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1925.dn (of order \(20\), degree \(8\), 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: \(0.960700149319\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(2\) over \(\Q(\zeta_{20})\)
Coefficient field: \(\Q(\zeta_{40})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{12} + x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{10}\)
Projective field: Galois closure of 10.0.17691976269034375.2

$q$-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.

\(f(q)\) \(=\) \( q + ( - \zeta_{40}^{9} - \zeta_{40}^{5}) q^{3} - \zeta_{40}^{18} q^{4} + \zeta_{40}^{3} q^{7} + (\zeta_{40}^{18} + \cdots + \zeta_{40}^{10}) q^{9} + \zeta_{40}^{4} q^{11} + ( - \zeta_{40}^{7} - \zeta_{40}^{3}) q^{12} + \cdots + (\zeta_{40}^{18} + \cdots - \zeta_{40}^{2}) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 4 q^{11} + 4 q^{16} - 4 q^{36} - 20 q^{51} - 8 q^{71} - 16 q^{81} - 12 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1925\mathbb{Z}\right)^\times\).

\(n\) \(276\) \(1002\) \(1751\)
\(\chi(n)\) \(-1\) \(-\zeta_{40}^{10}\) \(-\zeta_{40}^{8}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
118.1
0.156434 + 0.987688i
−0.156434 − 0.987688i
0.987688 + 0.156434i
−0.987688 − 0.156434i
−0.891007 + 0.453990i
0.891007 − 0.453990i
0.987688 − 0.156434i
−0.987688 + 0.156434i
0.156434 − 0.987688i
−0.156434 + 0.987688i
−0.453990 − 0.891007i
0.453990 + 0.891007i
−0.453990 + 0.891007i
0.453990 − 0.891007i
−0.891007 − 0.453990i
0.891007 + 0.453990i
0 −1.69480 − 0.863541i −0.951057 − 0.309017i 0 0 −0.453990 − 0.891007i 0 1.53884 + 2.11803i 0
118.2 0 1.69480 + 0.863541i −0.951057 − 0.309017i 0 0 0.453990 + 0.891007i 0 1.53884 + 2.11803i 0
293.1 0 −0.863541 − 1.69480i 0.951057 − 0.309017i 0 0 0.891007 + 0.453990i 0 −1.53884 + 2.11803i 0
293.2 0 0.863541 + 1.69480i 0.951057 − 0.309017i 0 0 −0.891007 − 0.453990i 0 −1.53884 + 2.11803i 0
468.1 0 −1.16110 + 0.183900i 0.587785 + 0.809017i 0 0 −0.156434 + 0.987688i 0 0.363271 − 0.118034i 0
468.2 0 1.16110 − 0.183900i 0.587785 + 0.809017i 0 0 0.156434 − 0.987688i 0 0.363271 − 0.118034i 0
657.1 0 −0.863541 + 1.69480i 0.951057 + 0.309017i 0 0 0.891007 − 0.453990i 0 −1.53884 − 2.11803i 0
657.2 0 0.863541 − 1.69480i 0.951057 + 0.309017i 0 0 −0.891007 + 0.453990i 0 −1.53884 − 2.11803i 0
832.1 0 −1.69480 + 0.863541i −0.951057 + 0.309017i 0 0 −0.453990 + 0.891007i 0 1.53884 − 2.11803i 0
832.2 0 1.69480 − 0.863541i −0.951057 + 0.309017i 0 0 0.453990 − 0.891007i 0 1.53884 − 2.11803i 0
1007.1 0 −0.183900 − 1.16110i −0.587785 − 0.809017i 0 0 0.987688 + 0.156434i 0 −0.363271 + 0.118034i 0
1007.2 0 0.183900 + 1.16110i −0.587785 − 0.809017i 0 0 −0.987688 − 0.156434i 0 −0.363271 + 0.118034i 0
1168.1 0 −0.183900 + 1.16110i −0.587785 + 0.809017i 0 0 0.987688 − 0.156434i 0 −0.363271 − 0.118034i 0
1168.2 0 0.183900 − 1.16110i −0.587785 + 0.809017i 0 0 −0.987688 + 0.156434i 0 −0.363271 − 0.118034i 0
1707.1 0 −1.16110 − 0.183900i 0.587785 − 0.809017i 0 0 −0.156434 − 0.987688i 0 0.363271 + 0.118034i 0
1707.2 0 1.16110 + 0.183900i 0.587785 − 0.809017i 0 0 0.156434 + 0.987688i 0 0.363271 + 0.118034i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 118.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
35.c odd 2 1 CM by \(\Q(\sqrt{-35}) \)
5.b even 2 1 inner
5.c odd 4 2 inner
7.b odd 2 1 inner
11.d odd 10 1 inner
35.f even 4 2 inner
55.h odd 10 1 inner
55.l even 20 2 inner
77.l even 10 1 inner
385.v even 10 1 inner
385.bi odd 20 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1925.1.dn.b ✓ 16
5.b even 2 1 inner 1925.1.dn.b ✓ 16
5.c odd 4 2 inner 1925.1.dn.b ✓ 16
7.b odd 2 1 inner 1925.1.dn.b ✓ 16
11.d odd 10 1 inner 1925.1.dn.b ✓ 16
35.c odd 2 1 CM 1925.1.dn.b ✓ 16
35.f even 4 2 inner 1925.1.dn.b ✓ 16
55.h odd 10 1 inner 1925.1.dn.b ✓ 16
55.l even 20 2 inner 1925.1.dn.b ✓ 16
77.l even 10 1 inner 1925.1.dn.b ✓ 16
385.v even 10 1 inner 1925.1.dn.b ✓ 16
385.bi odd 20 2 inner 1925.1.dn.b ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1925.1.dn.b ✓ 16 1.a even 1 1 trivial
1925.1.dn.b ✓ 16 5.b even 2 1 inner
1925.1.dn.b ✓ 16 5.c odd 4 2 inner
1925.1.dn.b ✓ 16 7.b odd 2 1 inner
1925.1.dn.b ✓ 16 11.d odd 10 1 inner
1925.1.dn.b ✓ 16 35.c odd 2 1 CM
1925.1.dn.b ✓ 16 35.f even 4 2 inner
1925.1.dn.b ✓ 16 55.h odd 10 1 inner
1925.1.dn.b ✓ 16 55.l even 20 2 inner
1925.1.dn.b ✓ 16 77.l even 10 1 inner
1925.1.dn.b ✓ 16 385.v even 10 1 inner
1925.1.dn.b ✓ 16 385.bi odd 20 2 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{1}^{\mathrm{new}}(1925, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} + 5 T^{12} + \cdots + 625 \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( T^{16} - T^{12} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{4} \) Copy content Toggle raw display
$13$ \( T^{16} + 4 T^{12} + \cdots + 1 \) Copy content Toggle raw display
$17$ \( T^{16} - 11 T^{12} + \cdots + 1 \) Copy content Toggle raw display
$19$ \( T^{16} \) Copy content Toggle raw display
$23$ \( T^{16} \) Copy content Toggle raw display
$29$ \( (T^{8} + 10 T^{4} + \cdots + 25)^{2} \) Copy content Toggle raw display
$31$ \( T^{16} \) Copy content Toggle raw display
$37$ \( T^{16} \) Copy content Toggle raw display
$41$ \( T^{16} \) Copy content Toggle raw display
$43$ \( T^{16} \) Copy content Toggle raw display
$47$ \( T^{16} - 20 T^{12} + \cdots + 625 \) Copy content Toggle raw display
$53$ \( T^{16} \) Copy content Toggle raw display
$59$ \( T^{16} \) Copy content Toggle raw display
$61$ \( T^{16} \) Copy content Toggle raw display
$67$ \( T^{16} \) Copy content Toggle raw display
$71$ \( (T^{4} + 2 T^{3} + 4 T^{2} + \cdots + 1)^{4} \) Copy content Toggle raw display
$73$ \( T^{16} + 4 T^{12} + \cdots + 1 \) Copy content Toggle raw display
$79$ \( (T^{8} + 10 T^{4} + \cdots + 25)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + 4 T^{12} + \cdots + 1 \) Copy content Toggle raw display
$89$ \( T^{16} \) Copy content Toggle raw display
$97$ \( T^{16} + 5 T^{12} + \cdots + 625 \) Copy content Toggle raw display
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