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Properties

Label 1375.1.bz.a
Level $1375$
Weight $1$
Character orbit 1375.bz
Analytic conductor $0.686$
Analytic rank $0$
Dimension $20$
Projective image $D_{50}$
CM discriminant -11
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1375,1,Mod(54,1375)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1375, base_ring=CyclotomicField(50))
 
chi = DirichletCharacter(H, H._module([11, 25]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1375.54");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1375 = 5^{3} \cdot 11 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1375.bz (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: \(0.686214392370\)
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: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{50}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{50} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q + ( - \zeta_{50}^{21} + \zeta_{50}^{7}) q^{3} - \zeta_{50}^{4} q^{4} - \zeta_{50}^{20} q^{5} + ( - \zeta_{50}^{17} + \cdots + \zeta_{50}^{3}) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{50}^{21} + \zeta_{50}^{7}) q^{3} - \zeta_{50}^{4} q^{4} - \zeta_{50}^{20} q^{5} + ( - \zeta_{50}^{17} + \cdots + \zeta_{50}^{3}) q^{9} + \cdots + (\zeta_{50}^{19} + \cdots - \zeta_{50}^{5}) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 5 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 5 q^{5} - 20 q^{12} - 5 q^{25} - 5 q^{27} + 5 q^{48} + 5 q^{49} - 5 q^{59} - 5 q^{60} + 5 q^{67} + 5 q^{69} + 5 q^{81} - 5 q^{92} - 5 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1375\mathbb{Z}\right)^\times\).

\(n\) \(376\) \(1002\)
\(\chi(n)\) \(-1\) \(-\zeta_{50}^{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
54.1
0.425779 + 0.904827i
−0.0627905 + 0.998027i
−0.0627905 − 0.998027i
−0.728969 − 0.684547i
−0.876307 − 0.481754i
−0.535827 − 0.844328i
0.637424 + 0.770513i
0.992115 + 0.125333i
0.992115 − 0.125333i
0.929776 + 0.368125i
−0.968583 − 0.248690i
−0.876307 + 0.481754i
−0.728969 + 0.684547i
−0.968583 + 0.248690i
0.929776 − 0.368125i
0.425779 − 0.904827i
0.187381 − 0.982287i
0.637424 − 0.770513i
−0.535827 + 0.844328i
0.187381 + 0.982287i
0 −0.250172 + 1.98031i 0.187381 + 0.982287i 0.809017 + 0.587785i 0 0 0 −2.89047 − 0.742148i 0
109.1 0 1.39436 − 1.15352i −0.968583 − 0.248690i −0.309017 − 0.951057i 0 0 0 0.426264 − 2.23455i 0
164.1 0 1.39436 + 1.15352i −0.968583 + 0.248690i −0.309017 + 0.951057i 0 0 0 0.426264 + 2.23455i 0
219.1 0 −1.52794 + 0.718995i 0.992115 − 0.125333i 0.809017 − 0.587785i 0 0 0 1.18023 − 1.42665i 0
329.1 0 0.503997 − 0.536702i 0.425779 − 0.904827i 0.809017 + 0.587785i 0 0 0 0.0287541 + 0.457034i 0
384.1 0 −1.36639 + 0.0859661i 0.637424 + 0.770513i −0.309017 − 0.951057i 0 0 0 0.867524 − 0.109594i 0
439.1 0 0.0623382 + 0.242791i 0.929776 + 0.368125i −0.309017 + 0.951057i 0 0 0 0.821245 − 0.451483i 0
494.1 0 1.51373 + 0.288760i −0.876307 − 0.481754i 0.809017 − 0.587785i 0 0 0 1.27822 + 0.506084i 0
604.1 0 1.51373 − 0.288760i −0.876307 + 0.481754i 0.809017 + 0.587785i 0 0 0 1.27822 − 0.506084i 0
659.1 0 −0.813516 − 0.516273i −0.0627905 − 0.998027i −0.309017 − 0.951057i 0 0 0 −0.0305086 − 0.0648341i 0
714.1 0 0.723208 − 1.82662i −0.535827 − 0.844328i −0.309017 + 0.951057i 0 0 0 −2.08452 − 1.95750i 0
769.1 0 0.503997 + 0.536702i 0.425779 + 0.904827i 0.809017 − 0.587785i 0 0 0 0.0287541 − 0.457034i 0
879.1 0 −1.52794 − 0.718995i 0.992115 + 0.125333i 0.809017 + 0.587785i 0 0 0 1.18023 + 1.42665i 0
934.1 0 0.723208 + 1.82662i −0.535827 + 0.844328i −0.309017 − 0.951057i 0 0 0 −2.08452 + 1.95750i 0
989.1 0 −0.813516 + 0.516273i −0.0627905 + 0.998027i −0.309017 + 0.951057i 0 0 0 −0.0305086 + 0.0648341i 0
1044.1 0 −0.250172 − 1.98031i 0.187381 − 0.982287i 0.809017 − 0.587785i 0 0 0 −2.89047 + 0.742148i 0
1154.1 0 −0.239615 − 0.435857i −0.728969 − 0.684547i 0.809017 + 0.587785i 0 0 0 0.403270 − 0.635452i 0
1209.1 0 0.0623382 − 0.242791i 0.929776 − 0.368125i −0.309017 − 0.951057i 0 0 0 0.821245 + 0.451483i 0
1264.1 0 −1.36639 − 0.0859661i 0.637424 − 0.770513i −0.309017 + 0.951057i 0 0 0 0.867524 + 0.109594i 0
1319.1 0 −0.239615 + 0.435857i −0.728969 + 0.684547i 0.809017 − 0.587785i 0 0 0 0.403270 + 0.635452i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 54.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 CM by \(\Q(\sqrt{-11}) \)
125.h even 50 1 inner
1375.bz odd 50 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1375.1.bz.a ✓ 20
11.b odd 2 1 CM 1375.1.bz.a ✓ 20
125.h even 50 1 inner 1375.1.bz.a ✓ 20
1375.bz odd 50 1 inner 1375.1.bz.a ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1375.1.bz.a ✓ 20 1.a even 1 1 trivial
1375.1.bz.a ✓ 20 11.b odd 2 1 CM
1375.1.bz.a ✓ 20 125.h even 50 1 inner
1375.1.bz.a ✓ 20 1375.bz odd 50 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(1375, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( T^{20} + 5 T^{17} + \cdots + 5 \) Copy content Toggle raw display
$5$ \( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{5} \) Copy content Toggle raw display
$7$ \( T^{20} \) Copy content Toggle raw display
$11$ \( T^{20} - T^{15} + \cdots + 1 \) Copy content Toggle raw display
$13$ \( T^{20} \) Copy content Toggle raw display
$17$ \( T^{20} \) Copy content Toggle raw display
$19$ \( T^{20} \) Copy content Toggle raw display
$23$ \( T^{20} - 20 T^{17} + \cdots + 5 \) Copy content Toggle raw display
$29$ \( T^{20} \) Copy content Toggle raw display
$31$ \( T^{20} + 20 T^{17} + \cdots + 1 \) Copy content Toggle raw display
$37$ \( T^{20} - 50 T^{15} + \cdots + 3125 \) Copy content Toggle raw display
$41$ \( T^{20} \) Copy content Toggle raw display
$43$ \( T^{20} \) Copy content Toggle raw display
$47$ \( T^{20} - 50 T^{15} + \cdots + 3125 \) Copy content Toggle raw display
$53$ \( T^{20} + 20 T^{17} + \cdots + 5 \) Copy content Toggle raw display
$59$ \( T^{20} + 5 T^{19} + \cdots + 1 \) Copy content Toggle raw display
$61$ \( T^{20} \) Copy content Toggle raw display
$67$ \( T^{20} - 5 T^{19} + \cdots + 5 \) Copy content Toggle raw display
$71$ \( T^{20} - 20 T^{17} + \cdots + 1 \) Copy content Toggle raw display
$73$ \( T^{20} \) Copy content Toggle raw display
$79$ \( T^{20} \) Copy content Toggle raw display
$83$ \( T^{20} \) Copy content Toggle raw display
$89$ \( T^{20} - 5 T^{17} + \cdots + 1 \) Copy content Toggle raw display
$97$ \( T^{20} + 5 T^{17} + \cdots + 5 \) Copy content Toggle raw display
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