Properties

Label 168.2.j.c
Level $168$
Weight $2$
Character orbit 168.j
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(155,168)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(168, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("168.155");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 168 = 2^{3} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 168.j (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: \(1.34148675396\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{2} + 1) q^{2} + ( - \beta_{3} - \beta_{2}) q^{3} + 2 \beta_{2} q^{4} + ( - \beta_{3} + \beta_1) q^{5} + ( - \beta_{3} - \beta_{2} + \beta_1 + 1) q^{6} + \beta_{2} q^{7} + (2 \beta_{2} - 2) q^{8} + (\beta_{3} - 2 \beta_{2} - \beta_1 - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} + 1) q^{2} + ( - \beta_{3} - \beta_{2}) q^{3} + 2 \beta_{2} q^{4} + ( - \beta_{3} + \beta_1) q^{5} + ( - \beta_{3} - \beta_{2} + \beta_1 + 1) q^{6} + \beta_{2} q^{7} + (2 \beta_{2} - 2) q^{8} + (\beta_{3} - 2 \beta_{2} - \beta_1 - 1) q^{9} + 2 \beta_1 q^{10} + ( - 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{11} + (2 \beta_1 + 2) q^{12} + (\beta_{3} + 2 \beta_{2} + \beta_1) q^{13} + (\beta_{2} - 1) q^{14} + (\beta_{3} - 2 \beta_{2} - \beta_1 + 2) q^{15} - 4 q^{16} + 2 \beta_{2} q^{17} + ( - 3 \beta_{2} - 2 \beta_1 + 1) q^{18} + (\beta_{3} - \beta_1 + 4) q^{19} + (2 \beta_{3} + 2 \beta_1) q^{20} + (\beta_1 + 1) q^{21} + ( - 4 \beta_{3} - 2 \beta_{2} + 2) q^{22} + (2 \beta_{3} - 2 \beta_1 - 6) q^{23} + (2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 2) q^{24} + (2 \beta_{3} - 2 \beta_1 - 1) q^{25} + (2 \beta_{3} + 2 \beta_{2} - 2) q^{26} + (3 \beta_{2} - \beta_1 - 4) q^{27} - 2 q^{28} + ( - 2 \beta_1 + 4) q^{30} + ( - 4 \beta_{3} - 4 \beta_{2} - 4 \beta_1) q^{31} + ( - 4 \beta_{2} - 4) q^{32} + (2 \beta_{3} - 4 \beta_{2} - 6) q^{33} + (2 \beta_{2} - 2) q^{34} + (\beta_{3} + \beta_1) q^{35} + ( - 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 4) q^{36} + (2 \beta_{3} + 2 \beta_1) q^{37} + (4 \beta_{2} - 2 \beta_1 + 4) q^{38} + ( - \beta_{3} + 2 \beta_{2} + \beta_1 + 4) q^{39} + 4 \beta_{3} q^{40} + (2 \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{41} + (\beta_{3} + \beta_{2} + \beta_1 + 1) q^{42} + ( - 2 \beta_{3} + 2 \beta_1 + 6) q^{43} + ( - 4 \beta_{3} + 4 \beta_1 + 4) q^{44} + ( - 3 \beta_{3} - \beta_1 - 4) q^{45} + ( - 6 \beta_{2} - 4 \beta_1 - 6) q^{46} + ( - 2 \beta_{3} + 2 \beta_1 + 4) q^{47} + (4 \beta_{3} + 4 \beta_{2}) q^{48} - q^{49} + ( - \beta_{2} - 4 \beta_1 - 1) q^{50} + (2 \beta_1 + 2) q^{51} + (2 \beta_{3} - 2 \beta_1 - 4) q^{52} + (2 \beta_{3} - 2 \beta_1 + 4) q^{53} + ( - \beta_{3} - \beta_{2} - \beta_1 - 7) q^{54} + (2 \beta_{3} - 8 \beta_{2} + 2 \beta_1) q^{55} + ( - 2 \beta_{2} - 2) q^{56} + ( - 5 \beta_{3} - 2 \beta_{2} + \beta_1 - 2) q^{57} + (\beta_{3} + 2 \beta_{2} + \beta_1) q^{59} + ( - 2 \beta_{3} + 4 \beta_{2} - 2 \beta_1 + 4) q^{60} + ( - 3 \beta_{3} + 2 \beta_{2} - 3 \beta_1) q^{61} + ( - 8 \beta_{3} - 4 \beta_{2} + 4) q^{62} + ( - \beta_{3} - \beta_{2} - \beta_1 + 2) q^{63} - 8 \beta_{2} q^{64} + 4 \beta_{2} q^{65} + (2 \beta_{3} - 10 \beta_{2} - 2 \beta_1 - 2) q^{66} + ( - 4 \beta_{3} + 4 \beta_1 + 2) q^{67} - 4 q^{68} + (4 \beta_{3} + 10 \beta_{2} + 2 \beta_1 - 4) q^{69} + 2 \beta_{3} q^{70} + (4 \beta_{3} - 4 \beta_1 - 6) q^{71} + ( - 4 \beta_{3} + 2 \beta_{2} + 6) q^{72} + ( - 2 \beta_{3} + 2 \beta_1 + 6) q^{73} + 4 \beta_{3} q^{74} + ( - \beta_{3} + 5 \beta_{2} + 2 \beta_1 - 4) q^{75} + ( - 2 \beta_{3} + 8 \beta_{2} - 2 \beta_1) q^{76} + ( - 2 \beta_{3} + 2 \beta_1 + 2) q^{77} + (6 \beta_{2} + 2 \beta_1 + 2) q^{78} + (4 \beta_{3} + 8 \beta_{2} + 4 \beta_1) q^{79} + (4 \beta_{3} - 4 \beta_1) q^{80} + (4 \beta_{3} + 4 \beta_{2} + 4 \beta_1 + 1) q^{81} + (4 \beta_{3} + 2 \beta_{2} - 2) q^{82} + (\beta_{3} - 10 \beta_{2} + \beta_1) q^{83} + (2 \beta_{3} + 2 \beta_{2}) q^{84} + (2 \beta_{3} + 2 \beta_1) q^{85} + (6 \beta_{2} + 4 \beta_1 + 6) q^{86} + (4 \beta_{2} + 8 \beta_1 + 4) q^{88} + 6 \beta_{2} q^{89} + ( - 4 \beta_{3} - 4 \beta_{2} + 2 \beta_1 - 4) q^{90} + (\beta_{3} - \beta_1 - 2) q^{91} + ( - 4 \beta_{3} - 12 \beta_{2} - 4 \beta_1) q^{92} + (4 \beta_{3} - 8 \beta_{2} - 12) q^{93} + (4 \beta_{2} + 4 \beta_1 + 4) q^{94} + ( - 6 \beta_{3} + 6 \beta_1 - 4) q^{95} + (4 \beta_{3} + 4 \beta_{2} - 4 \beta_1 - 4) q^{96} + ( - 2 \beta_{3} + 2 \beta_1 + 10) q^{97} + ( - \beta_{2} - 1) q^{98} + (4 \beta_{3} + 10 \beta_{2} - 4 \beta_1 - 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} - 2 q^{3} - 4 q^{5} - 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{2} - 2 q^{3} - 4 q^{5} - 8 q^{8} - 4 q^{10} + 4 q^{12} - 4 q^{14} + 12 q^{15} - 16 q^{16} + 8 q^{18} + 20 q^{19} + 2 q^{21} - 16 q^{23} + 8 q^{24} + 4 q^{25} - 4 q^{26} - 14 q^{27} - 8 q^{28} + 20 q^{30} - 16 q^{32} - 20 q^{33} - 8 q^{34} + 16 q^{36} + 20 q^{38} + 12 q^{39} + 8 q^{40} + 4 q^{42} + 16 q^{43} - 20 q^{45} - 16 q^{46} + 8 q^{47} + 8 q^{48} - 4 q^{49} + 4 q^{50} + 4 q^{51} - 8 q^{52} + 24 q^{53} - 28 q^{54} - 8 q^{56} - 20 q^{57} + 16 q^{60} + 8 q^{63} - 8 q^{67} - 16 q^{68} - 12 q^{69} + 4 q^{70} - 8 q^{71} + 16 q^{72} + 16 q^{73} + 8 q^{74} - 22 q^{75} + 4 q^{78} + 16 q^{80} + 4 q^{81} + 4 q^{84} + 16 q^{86} - 28 q^{90} - 4 q^{91} - 40 q^{93} + 8 q^{94} - 40 q^{95} + 32 q^{97} - 4 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 3x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} + \nu + 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} + 2\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\nu^{2} + \nu - 1 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{3} + \beta _1 - 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{3} + \beta_{2} - \beta_1 \) Copy content Toggle raw display

Character values

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

\(n\) \(73\) \(85\) \(113\) \(127\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
155.1
1.61803i
0.618034i
1.61803i
0.618034i
1.00000 1.00000i −1.61803 0.618034i 2.00000i −3.23607 −2.23607 + 1.00000i 1.00000i −2.00000 2.00000i 2.23607 + 2.00000i −3.23607 + 3.23607i
155.2 1.00000 1.00000i 0.618034 + 1.61803i 2.00000i 1.23607 2.23607 + 1.00000i 1.00000i −2.00000 2.00000i −2.23607 + 2.00000i 1.23607 1.23607i
155.3 1.00000 + 1.00000i −1.61803 + 0.618034i 2.00000i −3.23607 −2.23607 1.00000i 1.00000i −2.00000 + 2.00000i 2.23607 2.00000i −3.23607 3.23607i
155.4 1.00000 + 1.00000i 0.618034 1.61803i 2.00000i 1.23607 2.23607 1.00000i 1.00000i −2.00000 + 2.00000i −2.23607 2.00000i 1.23607 + 1.23607i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
24.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 168.2.j.c yes 4
3.b odd 2 1 168.2.j.a 4
4.b odd 2 1 672.2.j.b 4
8.b even 2 1 672.2.j.c 4
8.d odd 2 1 168.2.j.a 4
12.b even 2 1 672.2.j.c 4
24.f even 2 1 inner 168.2.j.c yes 4
24.h odd 2 1 672.2.j.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.2.j.a 4 3.b odd 2 1
168.2.j.a 4 8.d odd 2 1
168.2.j.c yes 4 1.a even 1 1 trivial
168.2.j.c yes 4 24.f even 2 1 inner
672.2.j.b 4 4.b odd 2 1
672.2.j.b 4 24.h odd 2 1
672.2.j.c 4 8.b even 2 1
672.2.j.c 4 12.b even 2 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 2 T + 2)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} + 2 T^{3} + 2 T^{2} + 6 T + 9 \) Copy content Toggle raw display
$5$ \( (T^{2} + 2 T - 4)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 20)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 12T^{2} + 16 \) Copy content Toggle raw display
$17$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 10 T + 20)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 8 T - 4)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + 80)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 48T^{2} + 256 \) Copy content Toggle raw display
$41$ \( (T^{2} + 20)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 8 T - 4)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 4 T - 16)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 12 T + 16)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 12T^{2} + 16 \) Copy content Toggle raw display
$61$ \( T^{4} + 140T^{2} + 400 \) Copy content Toggle raw display
$67$ \( (T^{2} + 4 T - 76)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 4 T - 76)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 8 T - 4)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} + 192T^{2} + 4096 \) Copy content Toggle raw display
$83$ \( T^{4} + 252 T^{2} + 13456 \) Copy content Toggle raw display
$89$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 16 T + 44)^{2} \) Copy content Toggle raw display
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