Properties

Label 136.1.p.a
Level $136$
Weight $1$
Character orbit 136.p
Analytic conductor $0.068$
Analytic rank $0$
Dimension $4$
Projective image $D_{8}$
CM discriminant -8
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [136,1,Mod(19,136)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(136, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([4, 4, 7]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("136.19");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 136 = 2^{3} \cdot 17 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 136.p (of order \(8\), 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: \(0.0678728417181\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{8}\)
Projective field: Galois closure of 8.0.1680747204608.3

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{8}^{3} q^{2} + ( - \zeta_{8}^{3} + \zeta_{8}^{2}) q^{3} - \zeta_{8}^{2} q^{4} + (\zeta_{8}^{2} - \zeta_{8}) q^{6} + \zeta_{8} q^{8} + ( - \zeta_{8}^{2} + \zeta_{8} - 1) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{8}^{3} q^{2} + ( - \zeta_{8}^{3} + \zeta_{8}^{2}) q^{3} - \zeta_{8}^{2} q^{4} + (\zeta_{8}^{2} - \zeta_{8}) q^{6} + \zeta_{8} q^{8} + ( - \zeta_{8}^{2} + \zeta_{8} - 1) q^{9} + (\zeta_{8}^{3} - 1) q^{11} + ( - \zeta_{8} + 1) q^{12} - q^{16} - \zeta_{8}^{3} q^{17} + ( - \zeta_{8}^{3} + \zeta_{8} - 1) q^{18} + ( - \zeta_{8}^{3} - \zeta_{8}^{2}) q^{22} + (\zeta_{8}^{3} + 1) q^{24} - \zeta_{8} q^{25} + (\zeta_{8}^{3} - \zeta_{8}^{2} - \zeta_{8} - 1) q^{27} - \zeta_{8}^{3} q^{32} + (\zeta_{8}^{3} + \zeta_{8}^{2} - \zeta_{8}) q^{33} + \zeta_{8}^{2} q^{34} + ( - \zeta_{8}^{3} + \zeta_{8}^{2} - 1) q^{36} + (\zeta_{8}^{2} + \zeta_{8}) q^{41} + (\zeta_{8}^{2} + 1) q^{43} + (\zeta_{8}^{2} + \zeta_{8}) q^{44} + (\zeta_{8}^{3} - \zeta_{8}^{2}) q^{48} + \zeta_{8}^{3} q^{49} + q^{50} + ( - \zeta_{8}^{2} + \zeta_{8}) q^{51} + (\zeta_{8}^{3} - \zeta_{8}^{2} + \zeta_{8} + 1) q^{54} + ( - \zeta_{8}^{2} - 1) q^{59} + \zeta_{8}^{2} q^{64} + ( - \zeta_{8}^{2} - \zeta_{8} + 1) q^{66} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{67} - \zeta_{8} q^{68} + ( - \zeta_{8}^{3} + \zeta_{8}^{2} - \zeta_{8}) q^{72} + (\zeta_{8}^{3} + \zeta_{8}^{2}) q^{73} + ( - \zeta_{8}^{3} - 1) q^{75} + ( - \zeta_{8}^{3} + \zeta_{8}^{2} - \zeta_{8}) q^{81} + ( - \zeta_{8} - 1) q^{82} + ( - \zeta_{8}^{2} + 1) q^{83} + (\zeta_{8}^{3} - \zeta_{8}) q^{86} + ( - \zeta_{8} - 1) q^{88} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{89} + ( - \zeta_{8}^{2} + \zeta_{8}) q^{96} + (\zeta_{8} - 1) q^{97} - \zeta_{8}^{2} q^{98} + ( - \zeta_{8}^{3} + \zeta_{8}^{2} + \zeta_{8} - 1) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{9} - 4 q^{11} + 4 q^{12} - 4 q^{16} - 4 q^{18} + 4 q^{24} + 4 q^{27} - 4 q^{36} + 4 q^{43} + 4 q^{50} + 4 q^{54} - 4 q^{59} + 4 q^{66} - 4 q^{75} - 4 q^{82} + 4 q^{83} - 4 q^{88} - 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/136\mathbb{Z}\right)^\times\).

\(n\) \(69\) \(103\) \(105\)
\(\chi(n)\) \(-1\) \(-1\) \(\zeta_{8}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
0.707107 0.707107i
0.707107 + 0.707107i
−0.707107 0.707107i
−0.707107 + 0.707107i
−0.707107 0.707107i 0.707107 0.292893i 1.00000i 0 −0.707107 0.292893i 0 0.707107 0.707107i −0.292893 + 0.292893i 0
43.1 −0.707107 + 0.707107i 0.707107 + 0.292893i 1.00000i 0 −0.707107 + 0.292893i 0 0.707107 + 0.707107i −0.292893 0.292893i 0
59.1 0.707107 0.707107i −0.707107 + 1.70711i 1.00000i 0 0.707107 + 1.70711i 0 −0.707107 0.707107i −1.70711 1.70711i 0
83.1 0.707107 + 0.707107i −0.707107 1.70711i 1.00000i 0 0.707107 1.70711i 0 −0.707107 + 0.707107i −1.70711 + 1.70711i 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
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
17.d even 8 1 inner
136.p odd 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 136.1.p.a 4
3.b odd 2 1 1224.1.bv.a 4
4.b odd 2 1 544.1.bl.a 4
5.b even 2 1 3400.1.ce.a 4
5.c odd 4 1 3400.1.br.a 4
5.c odd 4 1 3400.1.br.b 4
8.b even 2 1 544.1.bl.a 4
8.d odd 2 1 CM 136.1.p.a 4
17.b even 2 1 2312.1.p.b 4
17.c even 4 1 2312.1.p.a 4
17.c even 4 1 2312.1.p.d 4
17.d even 8 1 inner 136.1.p.a 4
17.d even 8 1 2312.1.p.a 4
17.d even 8 1 2312.1.p.b 4
17.d even 8 1 2312.1.p.d 4
17.e odd 16 2 2312.1.e.b 4
17.e odd 16 2 2312.1.f.c 4
17.e odd 16 4 2312.1.j.c 8
24.f even 2 1 1224.1.bv.a 4
40.e odd 2 1 3400.1.ce.a 4
40.k even 4 1 3400.1.br.a 4
40.k even 4 1 3400.1.br.b 4
51.g odd 8 1 1224.1.bv.a 4
68.g odd 8 1 544.1.bl.a 4
85.k odd 8 1 3400.1.br.a 4
85.m even 8 1 3400.1.ce.a 4
85.n odd 8 1 3400.1.br.b 4
136.e odd 2 1 2312.1.p.b 4
136.j odd 4 1 2312.1.p.a 4
136.j odd 4 1 2312.1.p.d 4
136.o even 8 1 544.1.bl.a 4
136.p odd 8 1 inner 136.1.p.a 4
136.p odd 8 1 2312.1.p.a 4
136.p odd 8 1 2312.1.p.b 4
136.p odd 8 1 2312.1.p.d 4
136.s even 16 2 2312.1.e.b 4
136.s even 16 2 2312.1.f.c 4
136.s even 16 4 2312.1.j.c 8
408.bd even 8 1 1224.1.bv.a 4
680.bq odd 8 1 3400.1.ce.a 4
680.bw even 8 1 3400.1.br.a 4
680.bz even 8 1 3400.1.br.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
136.1.p.a 4 1.a even 1 1 trivial
136.1.p.a 4 8.d odd 2 1 CM
136.1.p.a 4 17.d even 8 1 inner
136.1.p.a 4 136.p odd 8 1 inner
544.1.bl.a 4 4.b odd 2 1
544.1.bl.a 4 8.b even 2 1
544.1.bl.a 4 68.g odd 8 1
544.1.bl.a 4 136.o even 8 1
1224.1.bv.a 4 3.b odd 2 1
1224.1.bv.a 4 24.f even 2 1
1224.1.bv.a 4 51.g odd 8 1
1224.1.bv.a 4 408.bd even 8 1
2312.1.e.b 4 17.e odd 16 2
2312.1.e.b 4 136.s even 16 2
2312.1.f.c 4 17.e odd 16 2
2312.1.f.c 4 136.s even 16 2
2312.1.j.c 8 17.e odd 16 4
2312.1.j.c 8 136.s even 16 4
2312.1.p.a 4 17.c even 4 1
2312.1.p.a 4 17.d even 8 1
2312.1.p.a 4 136.j odd 4 1
2312.1.p.a 4 136.p odd 8 1
2312.1.p.b 4 17.b even 2 1
2312.1.p.b 4 17.d even 8 1
2312.1.p.b 4 136.e odd 2 1
2312.1.p.b 4 136.p odd 8 1
2312.1.p.d 4 17.c even 4 1
2312.1.p.d 4 17.d even 8 1
2312.1.p.d 4 136.j odd 4 1
2312.1.p.d 4 136.p odd 8 1
3400.1.br.a 4 5.c odd 4 1
3400.1.br.a 4 40.k even 4 1
3400.1.br.a 4 85.k odd 8 1
3400.1.br.a 4 680.bw even 8 1
3400.1.br.b 4 5.c odd 4 1
3400.1.br.b 4 40.k even 4 1
3400.1.br.b 4 85.n odd 8 1
3400.1.br.b 4 680.bz even 8 1
3400.1.ce.a 4 5.b even 2 1
3400.1.ce.a 4 40.e odd 2 1
3400.1.ce.a 4 85.m even 8 1
3400.1.ce.a 4 680.bq odd 8 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(136, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 1 \) Copy content Toggle raw display
$3$ \( T^{4} + 2 T^{2} - 4 T + 2 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} + 4 T^{3} + 6 T^{2} + 4 T + 2 \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} + 1 \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( T^{4} \) 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
$41$ \( T^{4} + 2 T^{2} + 4 T + 2 \) Copy content Toggle raw display
$43$ \( (T^{2} - 2 T + 2)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( (T^{2} + 2 T + 2)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} - 2)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( T^{4} + 2 T^{2} - 4 T + 2 \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} - 2 T + 2)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 2)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 4 T^{3} + 6 T^{2} + 4 T + 2 \) Copy content Toggle raw display
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