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Properties

Label 136.4.b.b
Level $136$
Weight $4$
Character orbit 136.b
Analytic conductor $8.024$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [136,4,Mod(33,136)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(136, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("136.33");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 136 = 2^{3} \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 136.b (of order \(2\), degree \(1\), 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: \(8.02425976078\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 95x^{6} + 756x^{4} + 1780x^{2} + 1152 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{13} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} - \beta_{4} q^{5} + ( - \beta_{4} - \beta_{2} - \beta_1) q^{7} + (\beta_{5} - 16) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{3} - \beta_{4} q^{5} + ( - \beta_{4} - \beta_{2} - \beta_1) q^{7} + (\beta_{5} - 16) q^{9} + (\beta_{6} + \beta_{4} - 2 \beta_{2}) q^{11} + (\beta_{7} + 5) q^{13} + ( - \beta_{7} - \beta_{5} - 2 \beta_{3} + 2) q^{15} + (\beta_{6} - 2 \beta_{4} + \beta_{3} - \beta_{2} - 2 \beta_1 + 4) q^{17} + (\beta_{7} - \beta_{5} - 2 \beta_{3} + 4) q^{19} + ( - 2 \beta_{7} - 3 \beta_{5} - 2 \beta_{3} + 37) q^{21} + (\beta_{6} - 5 \beta_1) q^{23} + ( - \beta_{7} + 3 \beta_{5} - 63) q^{25} + (3 \beta_{6} - 3 \beta_{4} - 19 \beta_{2} - 6 \beta_1) q^{27} + ( - 2 \beta_{6} + 5 \beta_{4} + 18 \beta_{2}) q^{29} + (3 \beta_{6} - 4 \beta_{4} + 14 \beta_{2} + 5 \beta_1) q^{31} + (3 \beta_{7} - 4 \beta_{5} + 2 \beta_{3} + 99) q^{33} + (2 \beta_{7} + 8 \beta_{5} - 130) q^{35} + (\beta_{4} - 28 \beta_{2} + 4 \beta_1) q^{37} + ( - 3 \beta_{6} - 21 \beta_{4} + 3 \beta_{2} - 10 \beta_1) q^{39} + ( - 6 \beta_{6} - 16 \beta_{4} + 2 \beta_{2} + 4 \beta_1) q^{41} + ( - \beta_{7} + 3 \beta_{5} - 2 \beta_{3} + 2) q^{43} + (2 \beta_{6} + 43 \beta_{4} + 10 \beta_{2} + 20 \beta_1) q^{45} + ( - 3 \beta_{7} + \beta_{5} + 2 \beta_{3} + 42) q^{47} + (\beta_{7} + 10 \beta_{5} + 8 \beta_{3} - 132) q^{49} + ( - 2 \beta_{7} - \beta_{6} - 8 \beta_{5} - 23 \beta_{4} - 4 \beta_{3} + 16 \beta_{2} + \cdots + 46) q^{51}+ \cdots + (4 \beta_{6} - 70 \beta_{4} + 183 \beta_{2} - 10 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 132 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 132 q^{9} + 44 q^{13} + 24 q^{15} + 28 q^{17} + 48 q^{19} + 308 q^{21} - 520 q^{25} + 812 q^{33} - 1064 q^{35} + 8 q^{43} + 312 q^{47} - 1124 q^{49} + 408 q^{51} + 472 q^{53} + 1416 q^{55} - 72 q^{59} - 624 q^{67} - 180 q^{69} + 1660 q^{77} + 3156 q^{81} + 2472 q^{83} - 2160 q^{85} - 6664 q^{87} + 68 q^{89} - 4036 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 95x^{6} + 756x^{4} + 1780x^{2} + 1152 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 4\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -3\nu^{7} - 284\nu^{5} - 2159\nu^{3} - 3402\nu ) / 172 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 3\nu^{6} + 327\nu^{4} + 6158\nu^{2} + 17420 ) / 172 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 59\nu^{7} + 5485\nu^{5} + 33588\nu^{3} + 47900\nu ) / 1032 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -41\nu^{6} - 3781\nu^{4} - 20634\nu^{2} - 22844 ) / 172 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -109\nu^{7} - 9989\nu^{5} - 48702\nu^{3} - 15160\nu ) / 1032 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 45\nu^{6} + 4217\nu^{4} + 28386\nu^{2} + 35292 ) / 172 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -3\beta_{7} - 3\beta_{5} + 4\beta_{3} - 188 ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 7\beta_{6} + 23\beta_{4} + 33\beta_{2} - 78\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 277\beta_{7} + 279\beta_{5} - 342\beta_{3} + 14856 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -619\beta_{6} - 2075\beta_{4} - 3053\beta_{2} + 6708\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -24035\beta_{7} - 24253\beta_{5} + 29526\beta_{3} - 1279856 ) / 8 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 53561\beta_{6} + 179881\beta_{4} + 265039\beta_{2} - 580024\beta_1 ) / 4 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

\(n\) \(69\) \(103\) \(105\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
33.1
− 2.20783i
1.03229i
− 1.60125i
9.30031i
− 9.30031i
1.60125i
− 1.03229i
2.20783i
0 − 9.26065i 0 16.4090i 0 34.5010i 0 −58.7597 0
33.2 0 − 8.52350i 0 − 18.2701i 0 − 13.8757i 0 −45.6501 0
33.3 0 − 2.95309i 0 − 4.89575i 0 4.46235i 0 18.2793 0
33.4 0 − 2.62097i 0 11.5318i 0 − 23.0485i 0 20.1305 0
33.5 0 2.62097i 0 − 11.5318i 0 23.0485i 0 20.1305 0
33.6 0 2.95309i 0 4.89575i 0 − 4.46235i 0 18.2793 0
33.7 0 8.52350i 0 18.2701i 0 13.8757i 0 −45.6501 0
33.8 0 9.26065i 0 − 16.4090i 0 − 34.5010i 0 −58.7597 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 33.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 136.4.b.b ✓ 8
3.b odd 2 1 1224.4.c.e 8
4.b odd 2 1 272.4.b.f 8
17.b even 2 1 inner 136.4.b.b ✓ 8
17.c even 4 2 2312.4.a.k 8
51.c odd 2 1 1224.4.c.e 8
68.d odd 2 1 272.4.b.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
136.4.b.b ✓ 8 1.a even 1 1 trivial
136.4.b.b ✓ 8 17.b even 2 1 inner
272.4.b.f 8 4.b odd 2 1
272.4.b.f 8 68.d odd 2 1
1224.4.c.e 8 3.b odd 2 1
1224.4.c.e 8 51.c odd 2 1
2312.4.a.k 8 17.c even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 174T_{3}^{6} + 8760T_{3}^{4} + 106624T_{3}^{2} + 373248 \) acting on \(S_{4}^{\mathrm{new}}(136, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + 174 T^{6} + 8760 T^{4} + \cdots + 373248 \) Copy content Toggle raw display
$5$ \( T^{8} + 760 T^{6} + \cdots + 286466048 \) Copy content Toggle raw display
$7$ \( T^{8} + 1934 T^{6} + \cdots + 2424307712 \) Copy content Toggle raw display
$11$ \( T^{8} + 7406 T^{6} + \cdots + 1063158272 \) Copy content Toggle raw display
$13$ \( (T^{4} - 22 T^{3} - 5836 T^{2} + \cdots + 8525216)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 582622237229761 \) Copy content Toggle raw display
$19$ \( (T^{4} - 24 T^{3} - 21056 T^{2} + \cdots + 44946176)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + 36734 T^{6} + \cdots + 2935871578112 \) Copy content Toggle raw display
$29$ \( T^{8} + 106232 T^{6} + \cdots + 14\!\cdots\!52 \) Copy content Toggle raw display
$31$ \( T^{8} + 155918 T^{6} + \cdots + 32\!\cdots\!52 \) Copy content Toggle raw display
$37$ \( T^{8} + 166072 T^{6} + \cdots + 54\!\cdots\!12 \) Copy content Toggle raw display
$41$ \( T^{8} + 419392 T^{6} + \cdots + 55\!\cdots\!32 \) Copy content Toggle raw display
$43$ \( (T^{4} - 4 T^{3} - 60592 T^{2} + \cdots + 112195072)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - 156 T^{3} - 57344 T^{2} + \cdots + 134217728)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 236 T^{3} + \cdots - 14761769616)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 36 T^{3} - 60112 T^{2} + \cdots + 70465536)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} + 1597240 T^{6} + \cdots + 18\!\cdots\!28 \) Copy content Toggle raw display
$67$ \( (T^{4} + 312 T^{3} + \cdots + 30967766784)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + 1607086 T^{6} + \cdots + 14\!\cdots\!28 \) Copy content Toggle raw display
$73$ \( T^{8} + 1779648 T^{6} + \cdots + 21\!\cdots\!68 \) Copy content Toggle raw display
$79$ \( T^{8} + 1578638 T^{6} + \cdots + 17\!\cdots\!72 \) Copy content Toggle raw display
$83$ \( (T^{4} - 1236 T^{3} + \cdots - 54225864704)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} - 34 T^{3} - 1202908 T^{2} + \cdots + 21605388512)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + 5185024 T^{6} + \cdots + 13\!\cdots\!92 \) Copy content Toggle raw display
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