Properties

Label 1375.1.ch.a
Level $1375$
Weight $1$
Character orbit 1375.ch
Analytic conductor $0.686$
Analytic rank $0$
Dimension $20$
Projective image $D_{25}$
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(21,1375)]
 
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([46, 25]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1375.21");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1375 = 5^{3} \cdot 11 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1375.ch (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: \(D_{25}\)
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.

\(f(q)\) \(=\) \( q + (\zeta_{50}^{18} + \zeta_{50}^{4}) q^{3} - \zeta_{50}^{21} q^{4} - \zeta_{50}^{5} q^{5} + (\zeta_{50}^{22} - \zeta_{50}^{11} + \zeta_{50}^{8}) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{50}^{18} + \zeta_{50}^{4}) q^{3} - \zeta_{50}^{21} q^{4} - \zeta_{50}^{5} q^{5} + (\zeta_{50}^{22} - \zeta_{50}^{11} + \zeta_{50}^{8}) q^{9} - \zeta_{50}^{23} q^{11} + (\zeta_{50}^{14} + 1) q^{12} + ( - \zeta_{50}^{23} - \zeta_{50}^{9}) q^{15} - \zeta_{50}^{17} q^{16} - \zeta_{50} q^{20} + (\zeta_{50}^{24} + \zeta_{50}^{2}) q^{23} + \zeta_{50}^{10} q^{25} + (\zeta_{50}^{15} + \zeta_{50}^{12} + \zeta_{50}^{4} + \zeta_{50}) q^{27} + ( - \zeta_{50}^{17} + \zeta_{50}^{16}) q^{31} + (\zeta_{50}^{16} + \zeta_{50}^{2}) q^{33} + (\zeta_{50}^{18} - \zeta_{50}^{7} + \zeta_{50}^{4}) q^{36} + ( - \zeta_{50}^{7} + \zeta_{50}^{2}) q^{37} - \zeta_{50}^{19} q^{44} + (\zeta_{50}^{16} - \zeta_{50}^{13} + \zeta_{50}^{2}) q^{45} + ( - \zeta_{50}^{11} - \zeta_{50}) q^{47} + ( - \zeta_{50}^{21} + \zeta_{50}^{10}) q^{48} + \zeta_{50}^{10} q^{49} + (\zeta_{50}^{18} + \zeta_{50}^{14}) q^{53} - \zeta_{50}^{3} q^{55} + (\zeta_{50}^{24} - \zeta_{50}^{15}) q^{59} + ( - \zeta_{50}^{19} - \zeta_{50}^{5}) q^{60} - \zeta_{50}^{13} q^{64} + ( - \zeta_{50}^{15} + \zeta_{50}^{8}) q^{67} + (\zeta_{50}^{20} - \zeta_{50}^{17} + \zeta_{50}^{6} - \zeta_{50}^{3}) q^{69} + (\zeta_{50}^{24} - \zeta_{50}^{13}) q^{71} + (\zeta_{50}^{14} - \zeta_{50}^{3}) q^{75} + \zeta_{50}^{22} q^{80} + (\zeta_{50}^{22} + \zeta_{50}^{19} + \zeta_{50}^{16} - \zeta_{50}^{8} + \zeta_{50}^{5}) q^{81} + (\zeta_{50}^{24} + \zeta_{50}^{12}) q^{89} + ( - \zeta_{50}^{23} + \zeta_{50}^{20}) q^{92} + ( - \zeta_{50}^{21} + \zeta_{50}^{20} + \zeta_{50}^{10} - \zeta_{50}^{9}) q^{93} + (\zeta_{50}^{18} - \zeta_{50}^{9}) q^{97} + (\zeta_{50}^{20} - \zeta_{50}^{9} + \zeta_{50}^{6}) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 5 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 5 q^{5} + 20 q^{12} - 5 q^{25} - 5 q^{27} - 5 q^{48} - 5 q^{49} - 5 q^{59} - 5 q^{60} - 5 q^{67} - 5 q^{69} - 5 q^{81} - 5 q^{92} - 10 q^{93} - 5 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(376\) \(1002\)
\(\chi(n)\) \(-1\) \(-\zeta_{50}^{21}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
21.1
0.992115 + 0.125333i
0.992115 0.125333i
0.637424 0.770513i
−0.535827 + 0.844328i
−0.876307 + 0.481754i
−0.728969 + 0.684547i
−0.0627905 + 0.998027i
−0.0627905 0.998027i
0.425779 0.904827i
0.187381 0.982287i
−0.535827 0.844328i
0.637424 + 0.770513i
0.187381 + 0.982287i
0.425779 + 0.904827i
0.929776 + 0.368125i
−0.968583 0.248690i
−0.728969 0.684547i
−0.876307 0.481754i
−0.968583 + 0.248690i
0.929776 0.368125i
0 0.238883 + 1.25227i 0.876307 0.481754i −0.809017 0.587785i 0 0 0 −0.581331 + 0.230165i 0
131.1 0 0.238883 1.25227i 0.876307 + 0.481754i −0.809017 + 0.587785i 0 0 0 −0.581331 0.230165i 0
186.1 0 −1.92189 + 0.493458i −0.929776 0.368125i 0.309017 0.951057i 0 0 0 2.57386 1.41499i 0
241.1 0 0.0915446 + 1.45506i −0.637424 0.770513i 0.309017 + 0.951057i 0 0 0 −1.11671 + 0.141073i 0
296.1 0 −1.35556 1.27295i −0.425779 + 0.904827i −0.809017 0.587785i 0 0 0 0.154335 + 2.45309i 0
406.1 0 −0.456288 0.969661i −0.992115 + 0.125333i −0.809017 + 0.587785i 0 0 0 −0.0946201 + 0.114376i 0
461.1 0 0.542804 0.656137i 0.968583 0.248690i 0.309017 0.951057i 0 0 0 0.0515014 + 0.269980i 0
516.1 0 0.542804 + 0.656137i 0.968583 + 0.248690i 0.309017 + 0.951057i 0 0 0 0.0515014 0.269980i 0
571.1 0 −0.124591 0.0157395i −0.187381 0.982287i −0.809017 0.587785i 0 0 0 −0.953308 0.244768i 0
681.1 0 1.69755 + 0.933237i 0.728969 0.684547i −0.809017 + 0.587785i 0 0 0 1.47492 + 2.32411i 0
736.1 0 0.0915446 1.45506i −0.637424 + 0.770513i 0.309017 0.951057i 0 0 0 −1.11671 0.141073i 0
791.1 0 −1.92189 0.493458i −0.929776 + 0.368125i 0.309017 + 0.951057i 0 0 0 2.57386 + 1.41499i 0
846.1 0 1.69755 0.933237i 0.728969 + 0.684547i −0.809017 0.587785i 0 0 0 1.47492 2.32411i 0
956.1 0 −0.124591 + 0.0157395i −0.187381 + 0.982287i −0.809017 + 0.587785i 0 0 0 −0.953308 + 0.244768i 0
1011.1 0 0.939097 + 1.47978i 0.0627905 0.998027i 0.309017 0.951057i 0 0 0 −0.882067 + 1.87449i 0
1066.1 0 0.348445 0.137959i 0.535827 0.844328i 0.309017 + 0.951057i 0 0 0 −0.626587 + 0.588404i 0
1121.1 0 −0.456288 + 0.969661i −0.992115 0.125333i −0.809017 0.587785i 0 0 0 −0.0946201 0.114376i 0
1231.1 0 −1.35556 + 1.27295i −0.425779 0.904827i −0.809017 + 0.587785i 0 0 0 0.154335 2.45309i 0
1286.1 0 0.348445 + 0.137959i 0.535827 + 0.844328i 0.309017 0.951057i 0 0 0 −0.626587 0.588404i 0
1341.1 0 0.939097 1.47978i 0.0627905 + 0.998027i 0.309017 + 0.951057i 0 0 0 −0.882067 1.87449i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 21.1
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.g even 25 1 inner
1375.ch 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.ch.a 20
11.b odd 2 1 CM 1375.1.ch.a 20
125.g even 25 1 inner 1375.1.ch.a 20
1375.ch odd 50 1 inner 1375.1.ch.a 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1375.1.ch.a 20 1.a even 1 1 trivial
1375.1.ch.a 20 11.b odd 2 1 CM
1375.1.ch.a 20 125.g even 25 1 inner
1375.1.ch.a 20 1375.ch 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
$3$ \( T^{20} + 5 T^{17} + 5 T^{16} + 2 T^{15} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( (T^{4} + T^{3} + T^{2} + T + 1)^{5} \) Copy content Toggle raw display
$7$ \( T^{20} \) Copy content Toggle raw display
$11$ \( T^{20} + T^{15} + T^{10} + T^{5} + 1 \) 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
$23$ \( T^{20} - 20 T^{17} + 5 T^{16} + 2 T^{15} + \cdots + 1 \) Copy content Toggle raw display
$29$ \( T^{20} \) Copy content Toggle raw display
$31$ \( T^{20} - 20 T^{17} + 5 T^{16} + 2 T^{15} + \cdots + 1 \) Copy content Toggle raw display
$37$ \( T^{20} - 18 T^{15} + 124 T^{10} - 7 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$41$ \( T^{20} \) Copy content Toggle raw display
$43$ \( T^{20} \) Copy content Toggle raw display
$47$ \( T^{20} - 18 T^{15} + 124 T^{10} - 7 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$53$ \( T^{20} - 20 T^{17} + 5 T^{16} + 2 T^{15} + \cdots + 1 \) Copy content Toggle raw display
$59$ \( T^{20} + 5 T^{19} + 15 T^{18} + 35 T^{17} + \cdots + 1 \) Copy content Toggle raw display
$61$ \( T^{20} \) Copy content Toggle raw display
$67$ \( T^{20} + 5 T^{19} + 15 T^{18} + 35 T^{17} + \cdots + 1 \) Copy content Toggle raw display
$71$ \( T^{20} - 20 T^{17} + 5 T^{16} + 2 T^{15} + \cdots + 1 \) 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
$89$ \( T^{20} + 5 T^{17} + 5 T^{16} + 2 T^{15} + \cdots + 1 \) Copy content Toggle raw display
$97$ \( T^{20} + 5 T^{17} + 5 T^{16} + 2 T^{15} + \cdots + 1 \) Copy content Toggle raw display
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