Properties

Label 136.3.p.a
Level $136$
Weight $3$
Character orbit 136.p
Analytic conductor $3.706$
Analytic rank $0$
Dimension $4$
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] = [136,3,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, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("136.19");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 136 = 2^{3} \cdot 17 \)
Weight: \( k \) \(=\) \( 3 \)
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: \(3.70573159530\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{8}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 \zeta_{8}^{3} q^{2} + (3 \zeta_{8}^{3} - 3 \zeta_{8}^{2} - 2 \zeta_{8} - 2) q^{3} - 4 \zeta_{8}^{2} q^{4} + (4 \zeta_{8}^{3} + 6 \zeta_{8}^{2} - 6 \zeta_{8} - 4) q^{6} - 8 \zeta_{8} q^{8} + (7 \zeta_{8}^{2} + 17 \zeta_{8} + 7) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - 2 \zeta_{8}^{3} q^{2} + (3 \zeta_{8}^{3} - 3 \zeta_{8}^{2} - 2 \zeta_{8} - 2) q^{3} - 4 \zeta_{8}^{2} q^{4} + (4 \zeta_{8}^{3} + 6 \zeta_{8}^{2} - 6 \zeta_{8} - 4) q^{6} - 8 \zeta_{8} q^{8} + (7 \zeta_{8}^{2} + 17 \zeta_{8} + 7) q^{9} + (13 \zeta_{8}^{3} - 6 \zeta_{8}^{2} + 6 \zeta_{8} - 13) q^{11} + (8 \zeta_{8}^{3} + 8 \zeta_{8}^{2} + 12 \zeta_{8} - 12) q^{12} - 16 q^{16} + ( - \zeta_{8}^{3} - 12 \zeta_{8}^{2} - 12) q^{17} + ( - 14 \zeta_{8}^{3} + 14 \zeta_{8} + 34) q^{18} + ( - 12 \zeta_{8}^{2} + 12) q^{19} + (26 \zeta_{8}^{3} + 26 \zeta_{8}^{2} - 12 \zeta_{8} + 12) q^{22} + (24 \zeta_{8}^{3} + 16 \zeta_{8}^{2} + 16 \zeta_{8} + 24) q^{24} - 25 \zeta_{8} q^{25} + ( - 17 \zeta_{8}^{3} - 51 \zeta_{8}^{2} - 51 \zeta_{8} - 17) q^{27} + 32 \zeta_{8}^{3} q^{32} + ( - 71 \zeta_{8}^{3} + 71 \zeta_{8} + 16) q^{33} + (24 \zeta_{8}^{3} - 2 \zeta_{8}^{2} - 24 \zeta_{8}) q^{34} + ( - 68 \zeta_{8}^{3} - 28 \zeta_{8}^{2} + 28) q^{36} + ( - 24 \zeta_{8}^{3} - 24 \zeta_{8}) q^{38} + ( - 24 \zeta_{8}^{3} + \zeta_{8}^{2} + \zeta_{8} - 24) q^{41} + (7 \zeta_{8}^{2} - 60 \zeta_{8} + 7) q^{43} + ( - 24 \zeta_{8}^{3} + 52 \zeta_{8}^{2} + 52 \zeta_{8} - 24) q^{44} + ( - 48 \zeta_{8}^{3} + 48 \zeta_{8}^{2} + 32 \zeta_{8} + 32) q^{48} + 49 \zeta_{8}^{3} q^{49} - 50 q^{50} + ( - 10 \zeta_{8}^{3} + 63 \zeta_{8}^{2} + 57 \zeta_{8} - 14) q^{51} + (34 \zeta_{8}^{3} - 34 \zeta_{8}^{2} - 102 \zeta_{8} - 102) q^{54} + (60 \zeta_{8}^{3} - 12 \zeta_{8}^{2} + 12 \zeta_{8} - 60) q^{57} + (41 \zeta_{8}^{2} - 60 \zeta_{8} + 41) q^{59} + 64 \zeta_{8}^{2} q^{64} + ( - 32 \zeta_{8}^{3} - 142 \zeta_{8}^{2} + 142) q^{66} + ( - 31 \zeta_{8}^{3} + 31 \zeta_{8} + 84) q^{67} + (48 \zeta_{8}^{2} - 4 \zeta_{8} - 48) q^{68} + ( - 56 \zeta_{8}^{3} - 136 \zeta_{8}^{2} - 56 \zeta_{8}) q^{72} + ( - 59 \zeta_{8}^{3} - 59 \zeta_{8}^{2} - 12 \zeta_{8} + 12) q^{73} + (75 \zeta_{8}^{3} + 50 \zeta_{8}^{2} + 50 \zeta_{8} + 75) q^{75} + ( - 48 \zeta_{8}^{2} - 48) q^{76} + (175 \zeta_{8}^{3} + 153 \zeta_{8}^{2} + 175 \zeta_{8}) q^{81} + (48 \zeta_{8}^{3} - 48 \zeta_{8}^{2} + 2 \zeta_{8} + 2) q^{82} + (36 \zeta_{8}^{3} - 79 \zeta_{8}^{2} + 79) q^{83} + ( - 14 \zeta_{8}^{3} + 14 \zeta_{8} - 120) q^{86} + (48 \zeta_{8}^{3} - 48 \zeta_{8}^{2} + 104 \zeta_{8} + 104) q^{88} + ( - 73 \zeta_{8}^{3} + 72 \zeta_{8}^{2} - 73 \zeta_{8}) q^{89} + ( - 64 \zeta_{8}^{3} - 96 \zeta_{8}^{2} + 96 \zeta_{8} + 64) q^{96} + ( - 60 \zeta_{8}^{3} - 60 \zeta_{8}^{2} - 107 \zeta_{8} + 107) q^{97} + 98 \zeta_{8}^{2} q^{98} + (31 \zeta_{8}^{3} - 31 \zeta_{8}^{2} - 270 \zeta_{8} - 270) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{3} - 16 q^{6} + 28 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 8 q^{3} - 16 q^{6} + 28 q^{9} - 52 q^{11} - 48 q^{12} - 64 q^{16} - 48 q^{17} + 136 q^{18} + 48 q^{19} + 48 q^{22} + 96 q^{24} - 68 q^{27} + 64 q^{33} + 112 q^{36} - 96 q^{41} + 28 q^{43} - 96 q^{44} + 128 q^{48} - 200 q^{50} - 56 q^{51} - 408 q^{54} - 240 q^{57} + 164 q^{59} + 568 q^{66} + 336 q^{67} - 192 q^{68} + 48 q^{73} + 300 q^{75} - 192 q^{76} + 8 q^{82} + 316 q^{83} - 480 q^{86} + 416 q^{88} + 256 q^{96} + 428 q^{97} - 1080 q^{99}+O(q^{100}) \) 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
1.41421 + 1.41421i −5.53553 + 2.29289i 4.00000i 0 −11.0711 4.58579i 0 −5.65685 + 5.65685i 19.0208 19.0208i 0
43.1 1.41421 1.41421i −5.53553 2.29289i 4.00000i 0 −11.0711 + 4.58579i 0 −5.65685 5.65685i 19.0208 + 19.0208i 0
59.1 −1.41421 + 1.41421i 1.53553 3.70711i 4.00000i 0 3.07107 + 7.41421i 0 5.65685 + 5.65685i −5.02082 5.02082i 0
83.1 −1.41421 1.41421i 1.53553 + 3.70711i 4.00000i 0 3.07107 7.41421i 0 5.65685 5.65685i −5.02082 + 5.02082i 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 136.3.p.a 4
8.d odd 2 1 CM 136.3.p.a 4
17.d even 8 1 inner 136.3.p.a 4
136.p odd 8 1 inner 136.3.p.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
136.3.p.a 4 1.a even 1 1 trivial
136.3.p.a 4 8.d odd 2 1 CM
136.3.p.a 4 17.d even 8 1 inner
136.3.p.a 4 136.p odd 8 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 8T_{3}^{3} + 18T_{3}^{2} + 68T_{3} + 578 \) acting on \(S_{3}^{\mathrm{new}}(136, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 16 \) Copy content Toggle raw display
$3$ \( T^{4} + 8 T^{3} + 18 T^{2} + 68 T + 578 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} + 52 T^{3} + 1398 T^{2} + \cdots + 167042 \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} + 48 T^{3} + 1152 T^{2} + \cdots + 83521 \) Copy content Toggle raw display
$19$ \( (T^{2} - 24 T + 288)^{2} \) 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
$41$ \( T^{4} + 96 T^{3} + 3362 T^{2} + \cdots + 555458 \) Copy content Toggle raw display
$43$ \( T^{4} - 28 T^{3} + 392 T^{2} + \cdots + 12264004 \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( T^{4} - 164 T^{3} + 13448 T^{2} + \cdots + 56644 \) Copy content Toggle raw display
$61$ \( T^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} - 168 T + 5134)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( T^{4} - 48 T^{3} + 10658 T^{2} + \cdots + 7380482 \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( T^{4} - 316 T^{3} + \cdots + 125126596 \) Copy content Toggle raw display
$89$ \( T^{4} + 31684 T^{2} + \cdots + 29964676 \) Copy content Toggle raw display
$97$ \( T^{4} - 428 T^{3} + \cdots + 856069442 \) Copy content Toggle raw display
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