Properties

Label 152.2.q.a
Level $152$
Weight $2$
Character orbit 152.q
Analytic conductor $1.214$
Analytic rank $0$
Dimension $6$
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] = [152,2,Mod(9,152)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(152, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 0, 8]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("152.9");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 152 = 2^{3} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 152.q (of order \(9\), degree \(6\), 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: \(1.21372611072\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{18})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{3} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-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 \(\zeta_{18}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \zeta_{18}^{5} + \zeta_{18}^{2} - 1) q^{3} + ( - \zeta_{18}^{5} + 1) q^{5} + ( - \zeta_{18}^{3} + 2 \zeta_{18}^{2} - 2 \zeta_{18} + 1) q^{7} + ( - \zeta_{18}^{5} - \zeta_{18}^{4} + \zeta_{18}^{2} + \zeta_{18} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{18}^{5} + \zeta_{18}^{2} - 1) q^{3} + ( - \zeta_{18}^{5} + 1) q^{5} + ( - \zeta_{18}^{3} + 2 \zeta_{18}^{2} - 2 \zeta_{18} + 1) q^{7} + ( - \zeta_{18}^{5} - \zeta_{18}^{4} + \zeta_{18}^{2} + \zeta_{18} + 1) q^{9} + ( - 2 \zeta_{18}^{5} + 2 \zeta_{18}^{4} - \zeta_{18}^{3} + 2 \zeta_{18}^{2} - 2 \zeta_{18}) q^{11} + ( - \zeta_{18} + 1) q^{13} + ( - \zeta_{18}^{4} + \zeta_{18}^{2} - 1) q^{15} + (\zeta_{18}^{2} + 1) q^{17} + (2 \zeta_{18}^{5} + 2 \zeta_{18}^{4} + 2 \zeta_{18}^{3} - 4 \zeta_{18}^{2} - 1) q^{19} + ( - \zeta_{18}^{5} + \zeta_{18}^{3} - 2 \zeta_{18}^{2} + 4 \zeta_{18} - 3) q^{21} + ( - 3 \zeta_{18}^{4} + 2 \zeta_{18}^{3} - 2 \zeta_{18}^{2} + 2 \zeta_{18} - 3) q^{23} + (3 \zeta_{18}^{5} - \zeta_{18} + 1) q^{25} + (3 \zeta_{18}^{5} - 3 \zeta_{18}^{4} - \zeta_{18}^{3} - 3 \zeta_{18}^{2} + 3 \zeta_{18}) q^{27} + (4 \zeta_{18}^{5} + 3 \zeta_{18}^{4} - 4 \zeta_{18}^{3} - 4 \zeta_{18}^{2} + \zeta_{18} + 1) q^{29} + (4 \zeta_{18}^{5} + 4 \zeta_{18}^{4} - \zeta_{18}^{3} - 2 \zeta_{18}^{2} - 2 \zeta_{18} + 1) q^{31} + (2 \zeta_{18}^{5} - 4 \zeta_{18}^{4} + 3 \zeta_{18}^{3} - 3 \zeta_{18}^{2} + 4 \zeta_{18} - 2) q^{33} + ( - 2 \zeta_{18}^{4} + \zeta_{18}^{3} + \zeta_{18}^{2} - 1) q^{35} + (4 \zeta_{18}^{4} - 4 \zeta_{18}^{2} - 4 \zeta_{18} - 2) q^{37} + ( - \zeta_{18}^{5} + \zeta_{18}^{2} + \zeta_{18} - 2) q^{39} + (\zeta_{18}^{5} - 4 \zeta_{18}^{4} + 4 \zeta_{18}^{3} + 3 \zeta_{18}^{2} - 3) q^{41} + ( - \zeta_{18}^{5} + 2 \zeta_{18}^{4} + 2 \zeta_{18}^{3} - 2 \zeta_{18}^{2} - 2 \zeta_{18} + 1) q^{43} + ( - 2 \zeta_{18}^{5} - 2 \zeta_{18}^{4} - \zeta_{18}^{3} + \zeta_{18}^{2} + \zeta_{18} + 1) q^{45} + ( - 2 \zeta_{18}^{5} + \zeta_{18}^{4} - 4 \zeta_{18}^{3} + 2 \zeta_{18}^{2} + 3 \zeta_{18} + 3) q^{47} + ( - 4 \zeta_{18}^{5} + 8 \zeta_{18}^{4} - 2 \zeta_{18}^{3} + 8 \zeta_{18}^{2} - 4 \zeta_{18}) q^{49} + ( - \zeta_{18}^{5} + \zeta_{18} - 1) q^{51} + ( - 7 \zeta_{18}^{4} + 4 \zeta_{18}^{3} - 4 \zeta_{18}^{2} + 4 \zeta_{18} - 7) q^{53} + ( - \zeta_{18}^{5} + \zeta_{18}^{3} + \zeta_{18}^{2} - 2 \zeta_{18}) q^{55} + ( - \zeta_{18}^{5} + 5 \zeta_{18}^{2} - 4 \zeta_{18} + 1) q^{57} + (5 \zeta_{18}^{2} + 2 \zeta_{18} + 5) q^{59} + (5 \zeta_{18}^{4} - 8 \zeta_{18}^{3} - 4 \zeta_{18}^{2} - 8 \zeta_{18} + 5) q^{61} + (\zeta_{18}^{5} - \zeta_{18}^{4} - \zeta_{18}^{3} + 1) q^{63} + ( - \zeta_{18}^{5} + \zeta_{18}^{3} - \zeta_{18}) q^{65} + (6 \zeta_{18}^{5} - 3 \zeta_{18}^{4} - 6 \zeta_{18}^{2} + 3 \zeta_{18} + 3) q^{67} + (3 \zeta_{18}^{5} + 3 \zeta_{18}^{4} - 5 \zeta_{18}^{3} + \zeta_{18}^{2} - 4 \zeta_{18} + 5) q^{69} + ( - 5 \zeta_{18}^{5} - 6 \zeta_{18}^{4} + 2 \zeta_{18}^{3} - 2 \zeta_{18}^{2} + 6 \zeta_{18} + 5) q^{71} + (5 \zeta_{18}^{5} - 8 \zeta_{18}^{4} + 8 \zeta_{18}^{3} + 3 \zeta_{18}^{2} - 3) q^{73} + ( - 4 \zeta_{18}^{5} + 3 \zeta_{18}^{4} + \zeta_{18}^{2} + \zeta_{18} - 2) q^{75} + ( - 8 \zeta_{18}^{5} + 4 \zeta_{18}^{4} + 4 \zeta_{18}^{2} + 4 \zeta_{18} - 9) q^{77} + (3 \zeta_{18}^{5} + 8 \zeta_{18}^{4} - 3 \zeta_{18}^{2} + 3) q^{79} + ( - 6 \zeta_{18}^{5} + 3 \zeta_{18}^{4} - 5 \zeta_{18}^{3} + 5 \zeta_{18}^{2} - 3 \zeta_{18} + 6) q^{81} + ( - 8 \zeta_{18}^{5} - 8 \zeta_{18}^{4} - 5 \zeta_{18}^{3} + 10 \zeta_{18}^{2} - 2 \zeta_{18} + 5) q^{83} + ( - \zeta_{18}^{5} - \zeta_{18}^{4} + \zeta_{18}^{2} + \zeta_{18} + 1) q^{85} + ( - 5 \zeta_{18}^{5} + \zeta_{18}^{4} + 7 \zeta_{18}^{3} + \zeta_{18}^{2} - 5 \zeta_{18}) q^{87} + (3 \zeta_{18} - 3) q^{89} + (\zeta_{18}^{4} - 3 \zeta_{18}^{3} + 4 \zeta_{18}^{2} - 3 \zeta_{18} + 1) q^{91} + ( - 5 \zeta_{18}^{5} + 5 \zeta_{18}^{3} + 2 \zeta_{18}^{2} - 3) q^{93} + (\zeta_{18}^{5} + 6 \zeta_{18}^{4} + 2 \zeta_{18}^{3} - 2 \zeta_{18}^{2} - 2 \zeta_{18} + 1) q^{95} + (5 \zeta_{18}^{2} + 5) q^{97} + ( - \zeta_{18}^{3} + \zeta_{18}^{2} - \zeta_{18}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{3} + 6 q^{5} + 3 q^{7} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 6 q^{3} + 6 q^{5} + 3 q^{7} + 6 q^{9} - 3 q^{11} + 6 q^{13} - 6 q^{15} + 6 q^{17} - 15 q^{21} - 12 q^{23} + 6 q^{25} - 3 q^{27} - 6 q^{29} + 3 q^{31} - 3 q^{33} - 3 q^{35} - 12 q^{37} - 12 q^{39} - 6 q^{41} + 12 q^{43} + 3 q^{45} + 6 q^{47} - 6 q^{49} - 6 q^{51} - 30 q^{53} + 3 q^{55} + 6 q^{57} + 30 q^{59} + 6 q^{61} + 3 q^{63} + 3 q^{65} + 18 q^{67} + 15 q^{69} + 36 q^{71} + 6 q^{73} - 12 q^{75} - 54 q^{77} + 18 q^{79} + 21 q^{81} + 15 q^{83} + 6 q^{85} + 21 q^{87} - 18 q^{89} - 3 q^{91} - 3 q^{93} + 12 q^{95} + 30 q^{97} - 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/152\mathbb{Z}\right)^\times\).

\(n\) \(39\) \(77\) \(97\)
\(\chi(n)\) \(1\) \(1\) \(\zeta_{18}^{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1
−0.173648 0.984808i
−0.173648 + 0.984808i
−0.766044 + 0.642788i
−0.766044 0.642788i
0.939693 0.342020i
0.939693 + 0.342020i
0 −1.17365 + 0.984808i 0 1.76604 + 0.642788i 0 −1.03209 + 1.78763i 0 −0.113341 + 0.642788i 0
17.1 0 −1.17365 0.984808i 0 1.76604 0.642788i 0 −1.03209 1.78763i 0 −0.113341 0.642788i 0
25.1 0 −1.76604 0.642788i 0 0.0603074 + 0.342020i 0 2.37939 4.12122i 0 0.407604 + 0.342020i 0
73.1 0 −1.76604 + 0.642788i 0 0.0603074 0.342020i 0 2.37939 + 4.12122i 0 0.407604 0.342020i 0
81.1 0 −0.0603074 + 0.342020i 0 1.17365 + 0.984808i 0 0.152704 + 0.264490i 0 2.70574 + 0.984808i 0
137.1 0 −0.0603074 0.342020i 0 1.17365 0.984808i 0 0.152704 0.264490i 0 2.70574 0.984808i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 9.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.e even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 152.2.q.a 6
4.b odd 2 1 304.2.u.d 6
19.e even 9 1 inner 152.2.q.a 6
19.e even 9 1 2888.2.a.q 3
19.f odd 18 1 2888.2.a.p 3
76.k even 18 1 5776.2.a.bm 3
76.l odd 18 1 304.2.u.d 6
76.l odd 18 1 5776.2.a.bl 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.2.q.a 6 1.a even 1 1 trivial
152.2.q.a 6 19.e even 9 1 inner
304.2.u.d 6 4.b odd 2 1
304.2.u.d 6 76.l odd 18 1
2888.2.a.p 3 19.f odd 18 1
2888.2.a.q 3 19.e even 9 1
5776.2.a.bl 3 76.l odd 18 1
5776.2.a.bm 3 76.k even 18 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} + 6T_{3}^{5} + 15T_{3}^{4} + 19T_{3}^{3} + 12T_{3}^{2} + 3T_{3} + 1 \) acting on \(S_{2}^{\mathrm{new}}(152, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} + 6 T^{5} + 15 T^{4} + 19 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{6} - 6 T^{5} + 15 T^{4} - 19 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{6} - 3 T^{5} + 18 T^{4} + 21 T^{3} + \cdots + 9 \) Copy content Toggle raw display
$11$ \( T^{6} + 3 T^{5} + 18 T^{4} - 21 T^{3} + \cdots + 9 \) Copy content Toggle raw display
$13$ \( T^{6} - 6 T^{5} + 15 T^{4} - 19 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$17$ \( T^{6} - 6 T^{5} + 15 T^{4} - 19 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$19$ \( T^{6} + 9 T^{4} - 64 T^{3} + \cdots + 6859 \) Copy content Toggle raw display
$23$ \( T^{6} + 12 T^{5} + 99 T^{4} + \cdots + 2601 \) Copy content Toggle raw display
$29$ \( T^{6} + 6 T^{5} + 111 T^{4} + \cdots + 5329 \) Copy content Toggle raw display
$31$ \( T^{6} - 3 T^{5} + 42 T^{4} + \cdots + 11449 \) Copy content Toggle raw display
$37$ \( (T^{3} + 6 T^{2} - 36 T - 152)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} + 6 T^{5} - 9 T^{4} + \cdots + 12321 \) Copy content Toggle raw display
$43$ \( T^{6} - 12 T^{5} + 39 T^{4} + \cdots + 289 \) Copy content Toggle raw display
$47$ \( T^{6} - 6 T^{5} + 9 T^{4} + \cdots + 25281 \) Copy content Toggle raw display
$53$ \( T^{6} + 30 T^{5} + 543 T^{4} + \cdots + 494209 \) Copy content Toggle raw display
$59$ \( T^{6} - 30 T^{5} + 345 T^{4} + \cdots + 289 \) Copy content Toggle raw display
$61$ \( T^{6} - 6 T^{5} + 3 T^{4} + \cdots + 1456849 \) Copy content Toggle raw display
$67$ \( T^{6} - 18 T^{5} + 189 T^{4} + \cdots + 729 \) Copy content Toggle raw display
$71$ \( T^{6} - 36 T^{5} + 711 T^{4} + \cdots + 927369 \) Copy content Toggle raw display
$73$ \( T^{6} - 6 T^{5} - 153 T^{4} + \cdots + 1814409 \) Copy content Toggle raw display
$79$ \( T^{6} - 18 T^{5} + 207 T^{4} + \cdots + 72361 \) Copy content Toggle raw display
$83$ \( T^{6} - 15 T^{5} + 402 T^{4} + \cdots + 6265009 \) Copy content Toggle raw display
$89$ \( T^{6} + 18 T^{5} + 135 T^{4} + \cdots + 729 \) Copy content Toggle raw display
$97$ \( T^{6} - 30 T^{5} + 375 T^{4} + \cdots + 15625 \) Copy content Toggle raw display
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