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Properties

Label 209.1.m.a
Level $209$
Weight $1$
Character orbit 209.m
Analytic conductor $0.104$
Analytic rank $0$
Dimension $4$
Projective image $D_{5}$
CM discriminant -19
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [209,1,Mod(37,209)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(209, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([2, 5]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("209.37");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 209 = 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 209.m (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: \(0.104304587640\)
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: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{5}\)
Projective field: Galois closure of 5.1.5285401.1

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{10}^{4} q^{4} + (\zeta_{10}^{4} + \zeta_{10}^{2}) q^{5} + ( - \zeta_{10}^{3} + 1) q^{7} + \zeta_{10}^{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{10}^{4} q^{4} + (\zeta_{10}^{4} + \zeta_{10}^{2}) q^{5} + ( - \zeta_{10}^{3} + 1) q^{7} + \zeta_{10}^{2} q^{9} - \zeta_{10} q^{11} - \zeta_{10}^{3} q^{16} + ( - \zeta_{10} + 1) q^{17} - \zeta_{10} q^{19} + ( - \zeta_{10}^{3} - \zeta_{10}) q^{20} + ( - \zeta_{10}^{3} + \zeta_{10}^{2}) q^{23} + (\zeta_{10}^{4} - \zeta_{10}^{3} - \zeta_{10}) q^{25} + (\zeta_{10}^{4} + \zeta_{10}^{2}) q^{28} + (\zeta_{10}^{4} + \zeta_{10}^{2} + 1) q^{35} - \zeta_{10} q^{36} + ( - \zeta_{10}^{3} + \zeta_{10}^{2}) q^{43} + q^{44} + (\zeta_{10}^{4} - \zeta_{10}) q^{45} + (\zeta_{10}^{4} - \zeta_{10}^{3}) q^{47} + ( - \zeta_{10}^{3} - \zeta_{10} + 1) q^{49} + ( - \zeta_{10}^{3} + 1) q^{55} + ( - \zeta_{10} + 1) q^{61} + (\zeta_{10}^{2} + 1) q^{63} + \zeta_{10}^{2} q^{64} + (\zeta_{10}^{4} + 1) q^{68} + \zeta_{10}^{4} q^{73} + q^{76} + (\zeta_{10}^{4} - \zeta_{10}) q^{77} + (\zeta_{10}^{2} + 1) q^{80} + \zeta_{10}^{4} q^{81} + (\zeta_{10}^{4} + \zeta_{10}^{2}) q^{83} + (\zeta_{10}^{4} - \zeta_{10}^{3} + \cdots + 1) q^{85} + \cdots - \zeta_{10}^{3} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - q^{4} - 2 q^{5} + 3 q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - q^{4} - 2 q^{5} + 3 q^{7} - q^{9} - q^{11} - q^{16} + 3 q^{17} - q^{19} - 2 q^{20} - 2 q^{23} - 3 q^{25} - 2 q^{28} + q^{35} - q^{36} - 2 q^{43} + 4 q^{44} - 2 q^{45} - 2 q^{47} + 2 q^{49} + 3 q^{55} + 3 q^{61} + 3 q^{63} - q^{64} + 3 q^{68} - 2 q^{73} + 4 q^{76} - 2 q^{77} + 3 q^{80} - q^{81} - 2 q^{83} + q^{85} - 2 q^{92} + 3 q^{95} - q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/209\mathbb{Z}\right)^\times\).

\(n\) \(78\) \(134\)
\(\chi(n)\) \(-1\) \(\zeta_{10}^{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
−0.309017 + 0.951057i
0.809017 − 0.587785i
−0.309017 − 0.951057i
0.809017 + 0.587785i
0 0 0.309017 + 0.951057i −0.500000 + 0.363271i 0 0.190983 + 0.587785i 0 −0.809017 − 0.587785i 0
75.1 0 0 −0.809017 − 0.587785i −0.500000 − 1.53884i 0 1.30902 + 0.951057i 0 0.309017 − 0.951057i 0
113.1 0 0 0.309017 − 0.951057i −0.500000 − 0.363271i 0 0.190983 − 0.587785i 0 −0.809017 + 0.587785i 0
170.1 0 0 −0.809017 + 0.587785i −0.500000 + 1.53884i 0 1.30902 − 0.951057i 0 0.309017 + 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
19.b odd 2 1 CM by \(\Q(\sqrt{-19}) \)
11.c even 5 1 inner
209.m odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 209.1.m.a ✓ 4
3.b odd 2 1 1881.1.bv.a 4
4.b odd 2 1 3344.1.bx.a 4
11.b odd 2 1 2299.1.m.a 4
11.c even 5 1 inner 209.1.m.a ✓ 4
11.c even 5 1 2299.1.b.a 2
11.c even 5 2 2299.1.m.b 4
11.d odd 10 1 2299.1.b.b 2
11.d odd 10 1 2299.1.m.a 4
11.d odd 10 2 2299.1.m.c 4
19.b odd 2 1 CM 209.1.m.a ✓ 4
19.c even 3 2 3971.1.s.a 8
19.d odd 6 2 3971.1.s.a 8
19.e even 9 6 3971.1.bc.a 24
19.f odd 18 6 3971.1.bc.a 24
33.h odd 10 1 1881.1.bv.a 4
44.h odd 10 1 3344.1.bx.a 4
57.d even 2 1 1881.1.bv.a 4
76.d even 2 1 3344.1.bx.a 4
209.d even 2 1 2299.1.m.a 4
209.k even 10 1 2299.1.b.b 2
209.k even 10 1 2299.1.m.a 4
209.k even 10 2 2299.1.m.c 4
209.m odd 10 1 inner 209.1.m.a ✓ 4
209.m odd 10 1 2299.1.b.a 2
209.m odd 10 2 2299.1.m.b 4
209.n even 15 2 3971.1.s.a 8
209.r odd 30 2 3971.1.s.a 8
209.u even 45 6 3971.1.bc.a 24
209.x odd 90 6 3971.1.bc.a 24
627.t even 10 1 1881.1.bv.a 4
836.t even 10 1 3344.1.bx.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
209.1.m.a ✓ 4 1.a even 1 1 trivial
209.1.m.a ✓ 4 11.c even 5 1 inner
209.1.m.a ✓ 4 19.b odd 2 1 CM
209.1.m.a ✓ 4 209.m odd 10 1 inner
1881.1.bv.a 4 3.b odd 2 1
1881.1.bv.a 4 33.h odd 10 1
1881.1.bv.a 4 57.d even 2 1
1881.1.bv.a 4 627.t even 10 1
2299.1.b.a 2 11.c even 5 1
2299.1.b.a 2 209.m odd 10 1
2299.1.b.b 2 11.d odd 10 1
2299.1.b.b 2 209.k even 10 1
2299.1.m.a 4 11.b odd 2 1
2299.1.m.a 4 11.d odd 10 1
2299.1.m.a 4 209.d even 2 1
2299.1.m.a 4 209.k even 10 1
2299.1.m.b 4 11.c even 5 2
2299.1.m.b 4 209.m odd 10 2
2299.1.m.c 4 11.d odd 10 2
2299.1.m.c 4 209.k even 10 2
3344.1.bx.a 4 4.b odd 2 1
3344.1.bx.a 4 44.h odd 10 1
3344.1.bx.a 4 76.d even 2 1
3344.1.bx.a 4 836.t even 10 1
3971.1.s.a 8 19.c even 3 2
3971.1.s.a 8 19.d odd 6 2
3971.1.s.a 8 209.n even 15 2
3971.1.s.a 8 209.r odd 30 2
3971.1.bc.a 24 19.e even 9 6
3971.1.bc.a 24 19.f odd 18 6
3971.1.bc.a 24 209.u even 45 6
3971.1.bc.a 24 209.x odd 90 6

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(209, [\chi])\).

Hecke characteristic polynomials

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