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Properties

Label 275.1.v.a
Level $275$
Weight $1$
Character orbit 275.v
Analytic conductor $0.137$
Analytic rank $0$
Dimension $4$
Projective image $D_{5}$
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] = [275,1,Mod(21,275)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(275, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([6, 5]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("275.21");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 275 = 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 275.v (of order \(10\), degree \(4\), 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: \(0.137242878474\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{5}\)
Projective field: Galois closure of 5.1.47265625.1

$q$-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.

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

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/275\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(177\)
\(\chi(n)\) \(-1\) \(-\zeta_{10}\)

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
0.809017 + 0.587785i
0.809017 − 0.587785i
−0.309017 + 0.951057i
−0.309017 − 0.951057i
0 −0.500000 − 0.363271i −0.809017 − 0.587785i 1.00000 0 0 0 −0.190983 − 0.587785i 0
131.1 0 −0.500000 + 0.363271i −0.809017 + 0.587785i 1.00000 0 0 0 −0.190983 + 0.587785i 0
186.1 0 −0.500000 + 1.53884i 0.309017 − 0.951057i 1.00000 0 0 0 −1.30902 − 0.951057i 0
241.1 0 −0.500000 − 1.53884i 0.309017 + 0.951057i 1.00000 0 0 0 −1.30902 + 0.951057i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 CM by \(\Q(\sqrt{-11}) \)
25.d even 5 1 inner
275.v odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 275.1.v.a ✓ 4
3.b odd 2 1 2475.1.bu.a 4
5.b even 2 1 1375.1.v.a 4
5.c odd 4 2 1375.1.s.b 8
11.b odd 2 1 CM 275.1.v.a ✓ 4
11.c even 5 1 3025.1.m.a 4
11.c even 5 1 3025.1.u.a 4
11.c even 5 1 3025.1.w.a 4
11.c even 5 1 3025.1.bc.a 4
11.d odd 10 1 3025.1.m.a 4
11.d odd 10 1 3025.1.u.a 4
11.d odd 10 1 3025.1.w.a 4
11.d odd 10 1 3025.1.bc.a 4
25.d even 5 1 inner 275.1.v.a ✓ 4
25.e even 10 1 1375.1.v.a 4
25.f odd 20 2 1375.1.s.b 8
33.d even 2 1 2475.1.bu.a 4
55.d odd 2 1 1375.1.v.a 4
55.e even 4 2 1375.1.s.b 8
75.j odd 10 1 2475.1.bu.a 4
275.g even 5 1 3025.1.m.a 4
275.j even 5 1 3025.1.bc.a 4
275.k even 5 1 3025.1.w.a 4
275.l even 5 1 3025.1.u.a 4
275.m odd 10 1 3025.1.u.a 4
275.s odd 10 1 1375.1.v.a 4
275.u odd 10 1 3025.1.bc.a 4
275.v odd 10 1 inner 275.1.v.a ✓ 4
275.w odd 10 1 3025.1.m.a 4
275.bc odd 10 1 3025.1.w.a 4
275.bo even 20 2 1375.1.s.b 8
825.x even 10 1 2475.1.bu.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
275.1.v.a ✓ 4 1.a even 1 1 trivial
275.1.v.a ✓ 4 11.b odd 2 1 CM
275.1.v.a ✓ 4 25.d even 5 1 inner
275.1.v.a ✓ 4 275.v odd 10 1 inner
1375.1.s.b 8 5.c odd 4 2
1375.1.s.b 8 25.f odd 20 2
1375.1.s.b 8 55.e even 4 2
1375.1.s.b 8 275.bo even 20 2
1375.1.v.a 4 5.b even 2 1
1375.1.v.a 4 25.e even 10 1
1375.1.v.a 4 55.d odd 2 1
1375.1.v.a 4 275.s odd 10 1
2475.1.bu.a 4 3.b odd 2 1
2475.1.bu.a 4 33.d even 2 1
2475.1.bu.a 4 75.j odd 10 1
2475.1.bu.a 4 825.x even 10 1
3025.1.m.a 4 11.c even 5 1
3025.1.m.a 4 11.d odd 10 1
3025.1.m.a 4 275.g even 5 1
3025.1.m.a 4 275.w odd 10 1
3025.1.u.a 4 11.c even 5 1
3025.1.u.a 4 11.d odd 10 1
3025.1.u.a 4 275.l even 5 1
3025.1.u.a 4 275.m odd 10 1
3025.1.w.a 4 11.c even 5 1
3025.1.w.a 4 11.d odd 10 1
3025.1.w.a 4 275.k even 5 1
3025.1.w.a 4 275.bc odd 10 1
3025.1.bc.a 4 11.c even 5 1
3025.1.bc.a 4 11.d odd 10 1
3025.1.bc.a 4 275.j even 5 1
3025.1.bc.a 4 275.u odd 10 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(275, [\chi])\).

Hecke characteristic polynomials

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