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Properties

Label 168.2.q.c
Level $168$
Weight $2$
Character orbit 168.q
Analytic conductor $1.341$
Analytic rank $0$
Dimension $4$
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] = [168,2,Mod(25,168)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(168, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("168.25");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 168 = 2^{3} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 168.q (of order \(3\), degree \(2\), 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: \(1.34148675396\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-19})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 4x^{2} - 5x + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + 1) q^{3} + ( - \beta_{3} + \beta_{2} + 2 \beta_1 - 1) q^{5} + ( - \beta_{3} + \beta_1 - 2) q^{7} - \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + 1) q^{3} + ( - \beta_{3} + \beta_{2} + 2 \beta_1 - 1) q^{5} + ( - \beta_{3} + \beta_1 - 2) q^{7} - \beta_{2} q^{9} + (2 \beta_{3} - \beta_1 + 1) q^{11} + ( - \beta_{3} - \beta_{2} - \beta_1 + 3) q^{13} + (\beta_{3} + \beta_{2} + \beta_1) q^{15} + ( - 4 \beta_{2} + 4) q^{17} + (\beta_{3} + 2 \beta_{2} - 2 \beta_1 + 1) q^{19} + (2 \beta_{2} + \beta_1 - 2) q^{21} - 4 \beta_{2} q^{23} + ( - 2 \beta_{3} + 9 \beta_{2} + \cdots - 10) q^{25}+ \cdots + ( - \beta_{3} - \beta_{2} - \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{3} + q^{5} - 6 q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{3} + q^{5} - 6 q^{7} - 2 q^{9} + q^{11} + 10 q^{13} + 2 q^{15} + 8 q^{17} + 5 q^{19} - 3 q^{21} - 8 q^{23} - 19 q^{25} - 4 q^{27} + 6 q^{29} + 2 q^{31} - q^{33} - 30 q^{35} - 3 q^{37} + 5 q^{39} + 12 q^{41} - 14 q^{43} + q^{45} + 12 q^{47} - 10 q^{49} - 8 q^{51} - 11 q^{53} + 58 q^{55} + 10 q^{57} - 5 q^{59} - 20 q^{61} + 3 q^{63} - 26 q^{65} - 7 q^{67} - 16 q^{69} + 8 q^{71} + q^{73} + 19 q^{75} + 27 q^{77} - 8 q^{79} - 2 q^{81} + 14 q^{83} + 8 q^{85} + 3 q^{87} + 6 q^{89} - 15 q^{91} - 2 q^{93} + 26 q^{95} + 50 q^{97} - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 4x^{2} - 5x + 25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 4\nu^{2} - 4\nu - 5 ) / 20 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} + 4\nu + 5 ) / 4 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 5\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -4\beta_{3} + 4\beta _1 + 5 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/168\mathbb{Z}\right)^\times\).

\(n\) \(73\) \(85\) \(113\) \(127\)
\(\chi(n)\) \(-\beta_{2}\) \(1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
25.1
−1.63746 − 1.52274i
2.13746 + 0.656712i
−1.63746 + 1.52274i
2.13746 − 0.656712i
0 0.500000 − 0.866025i 0 −1.63746 − 2.83616i 0 −1.50000 − 2.17945i 0 −0.500000 − 0.866025i 0
25.2 0 0.500000 − 0.866025i 0 2.13746 + 3.70219i 0 −1.50000 + 2.17945i 0 −0.500000 − 0.866025i 0
121.1 0 0.500000 + 0.866025i 0 −1.63746 + 2.83616i 0 −1.50000 + 2.17945i 0 −0.500000 + 0.866025i 0
121.2 0 0.500000 + 0.866025i 0 2.13746 − 3.70219i 0 −1.50000 − 2.17945i 0 −0.500000 + 0.866025i 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
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 168.2.q.c ✓ 4
3.b odd 2 1 504.2.s.i 4
4.b odd 2 1 336.2.q.g 4
7.b odd 2 1 1176.2.q.l 4
7.c even 3 1 inner 168.2.q.c ✓ 4
7.c even 3 1 1176.2.a.k 2
7.d odd 6 1 1176.2.a.n 2
7.d odd 6 1 1176.2.q.l 4
8.b even 2 1 1344.2.q.w 4
8.d odd 2 1 1344.2.q.x 4
12.b even 2 1 1008.2.s.r 4
21.c even 2 1 3528.2.s.bk 4
21.g even 6 1 3528.2.a.bd 2
21.g even 6 1 3528.2.s.bk 4
21.h odd 6 1 504.2.s.i 4
21.h odd 6 1 3528.2.a.bk 2
28.d even 2 1 2352.2.q.bf 4
28.f even 6 1 2352.2.a.ba 2
28.f even 6 1 2352.2.q.bf 4
28.g odd 6 1 336.2.q.g 4
28.g odd 6 1 2352.2.a.bf 2
56.j odd 6 1 9408.2.a.dj 2
56.k odd 6 1 1344.2.q.x 4
56.k odd 6 1 9408.2.a.dp 2
56.m even 6 1 9408.2.a.dw 2
56.p even 6 1 1344.2.q.w 4
56.p even 6 1 9408.2.a.ec 2
84.j odd 6 1 7056.2.a.ch 2
84.n even 6 1 1008.2.s.r 4
84.n even 6 1 7056.2.a.cu 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.2.q.c ✓ 4 1.a even 1 1 trivial
168.2.q.c ✓ 4 7.c even 3 1 inner
336.2.q.g 4 4.b odd 2 1
336.2.q.g 4 28.g odd 6 1
504.2.s.i 4 3.b odd 2 1
504.2.s.i 4 21.h odd 6 1
1008.2.s.r 4 12.b even 2 1
1008.2.s.r 4 84.n even 6 1
1176.2.a.k 2 7.c even 3 1
1176.2.a.n 2 7.d odd 6 1
1176.2.q.l 4 7.b odd 2 1
1176.2.q.l 4 7.d odd 6 1
1344.2.q.w 4 8.b even 2 1
1344.2.q.w 4 56.p even 6 1
1344.2.q.x 4 8.d odd 2 1
1344.2.q.x 4 56.k odd 6 1
2352.2.a.ba 2 28.f even 6 1
2352.2.a.bf 2 28.g odd 6 1
2352.2.q.bf 4 28.d even 2 1
2352.2.q.bf 4 28.f even 6 1
3528.2.a.bd 2 21.g even 6 1
3528.2.a.bk 2 21.h odd 6 1
3528.2.s.bk 4 21.c even 2 1
3528.2.s.bk 4 21.g even 6 1
7056.2.a.ch 2 84.j odd 6 1
7056.2.a.cu 2 84.n even 6 1
9408.2.a.dj 2 56.j odd 6 1
9408.2.a.dp 2 56.k odd 6 1
9408.2.a.dw 2 56.m even 6 1
9408.2.a.ec 2 56.p even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - T_{5}^{3} + 15T_{5}^{2} + 14T_{5} + 196 \) acting on \(S_{2}^{\mathrm{new}}(168, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} - T^{3} + \cdots + 196 \) Copy content Toggle raw display
$7$ \( (T^{2} + 3 T + 7)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} - T^{3} + \cdots + 196 \) Copy content Toggle raw display
$13$ \( (T^{2} - 5 T - 8)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 4 T + 16)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} - 5 T^{3} + \cdots + 64 \) Copy content Toggle raw display
$23$ \( (T^{2} + 4 T + 16)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 3 T - 12)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 3 T^{3} + \cdots + 144 \) Copy content Toggle raw display
$41$ \( (T^{2} - 6 T - 48)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 7 T - 2)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 6 T + 36)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} + 11 T^{3} + \cdots + 256 \) Copy content Toggle raw display
$59$ \( T^{4} + 5 T^{3} + \cdots + 64 \) Copy content Toggle raw display
$61$ \( (T^{2} + 10 T + 100)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 7 T^{3} + \cdots + 4 \) Copy content Toggle raw display
$71$ \( (T - 2)^{4} \) Copy content Toggle raw display
$73$ \( T^{4} - T^{3} + \cdots + 196 \) Copy content Toggle raw display
$79$ \( T^{4} + 8 T^{3} + \cdots + 1681 \) Copy content Toggle raw display
$83$ \( (T^{2} - 7 T - 2)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} - 6 T^{3} + \cdots + 2304 \) Copy content Toggle raw display
$97$ \( (T^{2} - 25 T + 142)^{2} \) Copy content Toggle raw display
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