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Properties

Label 1925.1.dn.c
Level $1925$
Weight $1$
Character orbit 1925.dn
Analytic conductor $0.961$
Analytic rank $0$
Dimension $32$
Projective image $D_{30}$
CM discriminant -7
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: \(32\)
Relative dimension: \(4\) over \(\Q(\zeta_{20})\)
Coefficient field: \(\Q(\zeta_{120})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{32} + x^{28} - x^{20} - x^{16} - x^{12} + x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{30}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{30} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q + ( - \zeta_{120}^{53} + \zeta_{120}) q^{2} + ( - \zeta_{120}^{54} + \cdots + \zeta_{120}^{2}) q^{4}+ \cdots + \zeta_{120}^{42} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{120}^{53} + \zeta_{120}) q^{2} + ( - \zeta_{120}^{54} + \cdots + \zeta_{120}^{2}) q^{4}+ \cdots + \zeta_{120}^{10} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32 q+O(q^{10}) \) Copy content Toggle raw display \( 32 q + 4 q^{11} + 12 q^{16} - 40 q^{46} - 8 q^{56} - 12 q^{71} + 8 q^{81} - 28 q^{86}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

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

\(n\) \(276\) \(1002\) \(1751\)
\(\chi(n)\) \(-1\) \(\zeta_{120}^{30}\) \(-\zeta_{120}^{24}\)

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
−0.998630 + 0.0523360i
−0.544639 − 0.838671i
0.544639 + 0.838671i
0.998630 − 0.0523360i
0.0523360 − 0.998630i
−0.838671 − 0.544639i
0.838671 + 0.544639i
−0.0523360 + 0.998630i
−0.777146 + 0.629320i
−0.933580 − 0.358368i
0.933580 + 0.358368i
0.777146 − 0.629320i
0.0523360 + 0.998630i
−0.838671 + 0.544639i
0.838671 − 0.544639i
−0.0523360 − 0.998630i
−0.998630 − 0.0523360i
−0.544639 + 0.838671i
0.544639 − 0.838671i
0.998630 + 0.0523360i
−1.93221 − 0.306032i 0 2.68872 + 0.873619i 0 0 0.891007 − 0.453990i −3.18475 − 1.62271i −0.587785 − 0.809017i 0
118.2 −1.32178 − 0.209350i 0 0.752232 + 0.244415i 0 0 −0.891007 + 0.453990i 0.249279 + 0.127014i −0.587785 − 0.809017i 0
118.3 1.32178 + 0.209350i 0 0.752232 + 0.244415i 0 0 0.891007 − 0.453990i −0.249279 − 0.127014i −0.587785 − 0.809017i 0
118.4 1.93221 + 0.306032i 0 2.68872 + 0.873619i 0 0 −0.891007 + 0.453990i 3.18475 + 1.62271i −0.587785 − 0.809017i 0
293.1 −0.306032 − 1.93221i 0 −2.68872 + 0.873619i 0 0 −0.453990 + 0.891007i 1.62271 + 3.18475i 0.587785 − 0.809017i 0
293.2 −0.209350 − 1.32178i 0 −0.752232 + 0.244415i 0 0 0.453990 − 0.891007i −0.127014 − 0.249279i 0.587785 − 0.809017i 0
293.3 0.209350 + 1.32178i 0 −0.752232 + 0.244415i 0 0 −0.453990 + 0.891007i 0.127014 + 0.249279i 0.587785 − 0.809017i 0
293.4 0.306032 + 1.93221i 0 −2.68872 + 0.873619i 0 0 0.453990 − 0.891007i −1.62271 − 3.18475i 0.587785 − 0.809017i 0
468.1 −0.829482 + 1.62795i 0 −1.37440 − 1.89169i 0 0 0.987688 + 0.156434i 2.41502 − 0.382502i −0.951057 + 0.309017i 0
468.2 −0.0949099 + 0.186271i 0 0.562096 + 0.773659i 0 0 −0.987688 − 0.156434i −0.403942 + 0.0639781i −0.951057 + 0.309017i 0
468.3 0.0949099 − 0.186271i 0 0.562096 + 0.773659i 0 0 0.987688 + 0.156434i 0.403942 − 0.0639781i −0.951057 + 0.309017i 0
468.4 0.829482 − 1.62795i 0 −1.37440 − 1.89169i 0 0 −0.987688 − 0.156434i −2.41502 + 0.382502i −0.951057 + 0.309017i 0
657.1 −0.306032 + 1.93221i 0 −2.68872 − 0.873619i 0 0 −0.453990 − 0.891007i 1.62271 − 3.18475i 0.587785 + 0.809017i 0
657.2 −0.209350 + 1.32178i 0 −0.752232 − 0.244415i 0 0 0.453990 + 0.891007i −0.127014 + 0.249279i 0.587785 + 0.809017i 0
657.3 0.209350 − 1.32178i 0 −0.752232 − 0.244415i 0 0 −0.453990 − 0.891007i 0.127014 − 0.249279i 0.587785 + 0.809017i 0
657.4 0.306032 − 1.93221i 0 −2.68872 − 0.873619i 0 0 0.453990 + 0.891007i −1.62271 + 3.18475i 0.587785 + 0.809017i 0
832.1 −1.93221 + 0.306032i 0 2.68872 − 0.873619i 0 0 0.891007 + 0.453990i −3.18475 + 1.62271i −0.587785 + 0.809017i 0
832.2 −1.32178 + 0.209350i 0 0.752232 − 0.244415i 0 0 −0.891007 − 0.453990i 0.249279 − 0.127014i −0.587785 + 0.809017i 0
832.3 1.32178 − 0.209350i 0 0.752232 − 0.244415i 0 0 0.891007 + 0.453990i −0.249279 + 0.127014i −0.587785 + 0.809017i 0
832.4 1.93221 − 0.306032i 0 2.68872 − 0.873619i 0 0 −0.891007 − 0.453990i 3.18475 − 1.62271i −0.587785 + 0.809017i 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 118.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)
5.b even 2 1 inner
5.c odd 4 2 inner
11.d odd 10 1 inner
35.c odd 2 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.c ✓ 32
5.b even 2 1 inner 1925.1.dn.c ✓ 32
5.c odd 4 2 inner 1925.1.dn.c ✓ 32
7.b odd 2 1 CM 1925.1.dn.c ✓ 32
11.d odd 10 1 inner 1925.1.dn.c ✓ 32
35.c odd 2 1 inner 1925.1.dn.c ✓ 32
35.f even 4 2 inner 1925.1.dn.c ✓ 32
55.h odd 10 1 inner 1925.1.dn.c ✓ 32
55.l even 20 2 inner 1925.1.dn.c ✓ 32
77.l even 10 1 inner 1925.1.dn.c ✓ 32
385.v even 10 1 inner 1925.1.dn.c ✓ 32
385.bi odd 20 2 inner 1925.1.dn.c ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1925.1.dn.c ✓ 32 1.a even 1 1 trivial
1925.1.dn.c ✓ 32 5.b even 2 1 inner
1925.1.dn.c ✓ 32 5.c odd 4 2 inner
1925.1.dn.c ✓ 32 7.b odd 2 1 CM
1925.1.dn.c ✓ 32 11.d odd 10 1 inner
1925.1.dn.c ✓ 32 35.c odd 2 1 inner
1925.1.dn.c ✓ 32 35.f even 4 2 inner
1925.1.dn.c ✓ 32 55.h odd 10 1 inner
1925.1.dn.c ✓ 32 55.l even 20 2 inner
1925.1.dn.c ✓ 32 77.l even 10 1 inner
1925.1.dn.c ✓ 32 385.v even 10 1 inner
1925.1.dn.c ✓ 32 385.bi odd 20 2 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{32} - 22T_{2}^{28} + 273T_{2}^{24} - 2549T_{2}^{20} + 36050T_{2}^{16} - 153299T_{2}^{12} + 273903T_{2}^{8} + 323T_{2}^{4} + 1 \) acting on \(S_{1}^{\mathrm{new}}(1925, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{32} - 22 T^{28} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{32} \) Copy content Toggle raw display
$5$ \( T^{32} \) Copy content Toggle raw display
$7$ \( (T^{16} - T^{12} + T^{8} + \cdots + 1)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} - T^{7} + T^{5} + \cdots + 1)^{4} \) Copy content Toggle raw display
$13$ \( T^{32} \) Copy content Toggle raw display
$17$ \( T^{32} \) Copy content Toggle raw display
$19$ \( T^{32} \) Copy content Toggle raw display
$23$ \( (T^{16} + 21 T^{12} + \cdots + 1)^{2} \) Copy content Toggle raw display
$29$ \( (T^{16} + 6 T^{14} + \cdots + 1)^{2} \) Copy content Toggle raw display
$31$ \( T^{32} \) Copy content Toggle raw display
$37$ \( T^{32} - 23 T^{28} + \cdots + 1 \) Copy content Toggle raw display
$41$ \( T^{32} \) Copy content Toggle raw display
$43$ \( (T^{16} + 29 T^{12} + \cdots + 1)^{2} \) Copy content Toggle raw display
$47$ \( T^{32} \) Copy content Toggle raw display
$53$ \( (T^{16} + 5 T^{12} + \cdots + 625)^{2} \) Copy content Toggle raw display
$59$ \( T^{32} \) Copy content Toggle raw display
$61$ \( T^{32} \) Copy content Toggle raw display
$67$ \( (T^{16} + 21 T^{12} + \cdots + 1)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + 3 T^{7} + 8 T^{6} + \cdots + 1)^{4} \) Copy content Toggle raw display
$73$ \( T^{32} \) Copy content Toggle raw display
$79$ \( (T^{16} + T^{14} + 12 T^{12} + \cdots + 1)^{2} \) Copy content Toggle raw display
$83$ \( T^{32} \) Copy content Toggle raw display
$89$ \( T^{32} \) Copy content Toggle raw display
$97$ \( T^{32} \) Copy content Toggle raw display
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