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Properties

Label 453.1.t.a
Level $453$
Weight $1$
Character orbit 453.t
Analytic conductor $0.226$
Analytic rank $0$
Dimension $20$
Projective image $D_{25}$
CM discriminant -3
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [453,1,Mod(20,453)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(453, base_ring=CyclotomicField(50))
 
chi = DirichletCharacter(H, H._module([25, 34]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("453.20");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 453 = 3 \cdot 151 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 453.t (of order \(50\), degree \(20\), 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: \(0.226076450723\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\Q(\zeta_{50})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - x^{15} + x^{10} - x^{5} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{25}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{25} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{50}^{11} q^{3} - \zeta_{50}^{15} q^{4} + ( - \zeta_{50}^{23} + \zeta_{50}^{6}) q^{7} + \zeta_{50}^{22} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{50}^{11} q^{3} - \zeta_{50}^{15} q^{4} + ( - \zeta_{50}^{23} + \zeta_{50}^{6}) q^{7} + \zeta_{50}^{22} q^{9} - \zeta_{50} q^{12} + (\zeta_{50}^{24} + \zeta_{50}^{18}) q^{13} - \zeta_{50}^{5} q^{16} + ( - \zeta_{50}^{3} + \zeta_{50}^{2}) q^{19} + ( - \zeta_{50}^{17} - \zeta_{50}^{9}) q^{21} - \zeta_{50}^{19} q^{25} + \zeta_{50}^{8} q^{27} + ( - \zeta_{50}^{21} - \zeta_{50}^{13}) q^{28} + (\zeta_{50}^{20} + \zeta_{50}^{16}) q^{31} + \zeta_{50}^{12} q^{36} + (\zeta_{50}^{10} - \zeta_{50}^{7}) q^{37} + (\zeta_{50}^{10} + \zeta_{50}^{4}) q^{39} + (\zeta_{50}^{14} - \zeta_{50}^{7}) q^{43} + \zeta_{50}^{16} q^{48} + ( - \zeta_{50}^{21} + \cdots + \zeta_{50}^{4}) q^{49} + \cdots + ( - \zeta_{50}^{23} + \zeta_{50}^{8}) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 5 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 5 q^{4} - 5 q^{16} - 5 q^{31} - 5 q^{37} - 5 q^{39} - 5 q^{63} - 5 q^{64} - 5 q^{75} - 5 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/453\mathbb{Z}\right)^\times\).

\(n\) \(152\) \(157\)
\(\chi(n)\) \(-1\) \(\zeta_{50}^{12}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
20.1
0.637424 + 0.770513i
0.992115 − 0.125333i
−0.728969 + 0.684547i
−0.876307 − 0.481754i
0.637424 − 0.770513i
−0.968583 + 0.248690i
0.187381 − 0.982287i
0.425779 − 0.904827i
0.992115 + 0.125333i
0.929776 + 0.368125i
0.187381 + 0.982287i
−0.0627905 + 0.998027i
−0.728969 − 0.684547i
−0.876307 + 0.481754i
−0.535827 + 0.844328i
−0.968583 − 0.248690i
0.929776 − 0.368125i
0.425779 + 0.904827i
−0.535827 − 0.844328i
−0.0627905 − 0.998027i
0 0.968583 + 0.248690i −0.809017 − 0.587785i 0 0 0.348445 − 1.82662i 0 0.876307 + 0.481754i 0
29.1 0 −0.187381 + 0.982287i 0.309017 + 0.951057i 0 0 1.69755 − 0.435857i 0 −0.929776 − 0.368125i 0
44.1 0 −0.425779 − 0.904827i 0.309017 + 0.951057i 0 0 −0.124591 + 1.98031i 0 −0.637424 + 0.770513i 0
50.1 0 0.728969 − 0.684547i 0.309017 + 0.951057i 0 0 −0.456288 − 0.718995i 0 0.0627905 − 0.998027i 0
68.1 0 0.968583 − 0.248690i −0.809017 + 0.587785i 0 0 0.348445 + 1.82662i 0 0.876307 − 0.481754i 0
86.1 0 −0.929776 − 0.368125i −0.809017 + 0.587785i 0 0 0.939097 − 0.516273i 0 0.728969 + 0.684547i 0
98.1 0 0.876307 + 0.481754i 0.309017 + 0.951057i 0 0 −1.35556 − 0.536702i 0 0.535827 + 0.844328i 0
110.1 0 −0.992115 − 0.125333i 0.309017 − 0.951057i 0 0 0.238883 + 0.288760i 0 0.968583 + 0.248690i 0
125.1 0 −0.187381 − 0.982287i 0.309017 − 0.951057i 0 0 1.69755 + 0.435857i 0 −0.929776 + 0.368125i 0
242.1 0 0.535827 + 0.844328i −0.809017 + 0.587785i 0 0 0.0915446 + 0.0859661i 0 −0.425779 + 0.904827i 0
245.1 0 0.876307 − 0.481754i 0.309017 − 0.951057i 0 0 −1.35556 + 0.536702i 0 0.535827 − 0.844328i 0
275.1 0 −0.637424 + 0.770513i −0.809017 + 0.587785i 0 0 −1.92189 − 0.242791i 0 −0.187381 − 0.982287i 0
278.1 0 −0.425779 + 0.904827i 0.309017 − 0.951057i 0 0 −0.124591 − 1.98031i 0 −0.637424 − 0.770513i 0
299.1 0 0.728969 + 0.684547i 0.309017 − 0.951057i 0 0 −0.456288 + 0.718995i 0 0.0627905 + 0.998027i 0
311.1 0 0.0627905 + 0.998027i −0.809017 − 0.587785i 0 0 0.542804 + 1.15352i 0 −0.992115 + 0.125333i 0
374.1 0 −0.929776 + 0.368125i −0.809017 − 0.587785i 0 0 0.939097 + 0.516273i 0 0.728969 − 0.684547i 0
380.1 0 0.535827 − 0.844328i −0.809017 − 0.587785i 0 0 0.0915446 − 0.0859661i 0 −0.425779 − 0.904827i 0
383.1 0 −0.992115 + 0.125333i 0.309017 + 0.951057i 0 0 0.238883 − 0.288760i 0 0.968583 − 0.248690i 0
386.1 0 0.0627905 − 0.998027i −0.809017 + 0.587785i 0 0 0.542804 − 1.15352i 0 −0.992115 − 0.125333i 0
425.1 0 −0.637424 − 0.770513i −0.809017 − 0.587785i 0 0 −1.92189 + 0.242791i 0 −0.187381 + 0.982287i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 20.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
151.h even 25 1 inner
453.t odd 50 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 453.1.t.a ✓ 20
3.b odd 2 1 CM 453.1.t.a ✓ 20
151.h even 25 1 inner 453.1.t.a ✓ 20
453.t odd 50 1 inner 453.1.t.a ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
453.1.t.a ✓ 20 1.a even 1 1 trivial
453.1.t.a ✓ 20 3.b odd 2 1 CM
453.1.t.a ✓ 20 151.h even 25 1 inner
453.1.t.a ✓ 20 453.t odd 50 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(453, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( T^{20} + T^{15} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{20} \) Copy content Toggle raw display
$7$ \( T^{20} + 5 T^{17} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{20} \) Copy content Toggle raw display
$13$ \( T^{20} + 5 T^{17} + \cdots + 1 \) Copy content Toggle raw display
$17$ \( T^{20} \) Copy content Toggle raw display
$19$ \( T^{20} + 5 T^{18} + \cdots + 1 \) Copy content Toggle raw display
$23$ \( T^{20} \) Copy content Toggle raw display
$29$ \( T^{20} \) Copy content Toggle raw display
$31$ \( T^{20} + 5 T^{19} + \cdots + 1 \) Copy content Toggle raw display
$37$ \( T^{20} + 5 T^{19} + \cdots + 1 \) Copy content Toggle raw display
$41$ \( T^{20} \) Copy content Toggle raw display
$43$ \( T^{20} + 5 T^{17} + \cdots + 1 \) Copy content Toggle raw display
$47$ \( T^{20} \) Copy content Toggle raw display
$53$ \( T^{20} \) Copy content Toggle raw display
$59$ \( T^{20} \) Copy content Toggle raw display
$61$ \( T^{20} - 18 T^{15} + \cdots + 1 \) Copy content Toggle raw display
$67$ \( T^{20} + 5 T^{17} + \cdots + 1 \) Copy content Toggle raw display
$71$ \( T^{20} \) Copy content Toggle raw display
$73$ \( T^{20} - 18 T^{15} + \cdots + 1 \) Copy content Toggle raw display
$79$ \( T^{20} + 5 T^{17} + \cdots + 1 \) Copy content Toggle raw display
$83$ \( T^{20} \) Copy content Toggle raw display
$89$ \( T^{20} \) Copy content Toggle raw display
$97$ \( T^{20} + 7 T^{15} + \cdots + 1 \) Copy content Toggle raw display
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