Properties

Label 1197.2.j.e
Level $1197$
Weight $2$
Character orbit 1197.j
Analytic conductor $9.558$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1197,2,Mod(172,1197)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1197, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1197.172");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1197 = 3^{2} \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1197.j (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: \(9.55809312195\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 133)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + \beta_1 - 1) q^{2} + ( - 2 \beta_{3} + \beta_{2} - 2 \beta_1) q^{4} + (\beta_{2} + 1) q^{5} + (2 \beta_{3} - \beta_{2} + \beta_1 - 1) q^{7} + (\beta_{3} + 3) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + \beta_1 - 1) q^{2} + ( - 2 \beta_{3} + \beta_{2} - 2 \beta_1) q^{4} + (\beta_{2} + 1) q^{5} + (2 \beta_{3} - \beta_{2} + \beta_1 - 1) q^{7} + (\beta_{3} + 3) q^{8} + (\beta_{3} - \beta_{2} + \beta_1) q^{10} + ( - \beta_{3} - \beta_{2} - \beta_1) q^{11} + ( - 3 \beta_{3} + 2) q^{13} + ( - 2 \beta_{3} - \beta_{2} - 4) q^{14} + ( - 3 \beta_{2} - 3) q^{16} + 4 \beta_{2} q^{17} + (\beta_{2} + 1) q^{19} + ( - 2 \beta_{3} - 1) q^{20} + q^{22} + ( - 7 \beta_{2} - \beta_1 - 7) q^{23} - 4 \beta_{2} q^{25} + (4 \beta_{2} - \beta_1 + 4) q^{26} + (\beta_{3} + 8 \beta_{2} - 2 \beta_1 + 5) q^{28} + ( - \beta_{3} + 8) q^{29} + (3 \beta_{3} - 2 \beta_{2} + 3 \beta_1) q^{31} + ( - \beta_{3} - 3 \beta_{2} - \beta_1) q^{32} + (4 \beta_{3} + 4) q^{34} + (\beta_{3} - \beta_{2} - \beta_1) q^{35} + ( - 2 \beta_{2} + 5 \beta_1 - 2) q^{37} + (\beta_{3} - \beta_{2} + \beta_1) q^{38} + (3 \beta_{2} - \beta_1 + 3) q^{40} + ( - 2 \beta_{3} + 6) q^{41} + (5 \beta_{3} + 1) q^{43} + ( - 3 \beta_{2} - \beta_1 - 3) q^{44} + ( - 6 \beta_{3} + 5 \beta_{2} - 6 \beta_1) q^{46} + ( - 3 \beta_{2} + \beta_1 - 3) q^{47} + ( - 2 \beta_{3} - 5 \beta_{2} + 2 \beta_1) q^{49} + ( - 4 \beta_{3} - 4) q^{50} + ( - \beta_{3} - 10 \beta_{2} - \beta_1) q^{52} + ( - 3 \beta_{3} - 3 \beta_1) q^{53} + ( - \beta_{3} + 1) q^{55} + (6 \beta_{3} - 5 \beta_{2} + 4 \beta_1 - 1) q^{56} + ( - 6 \beta_{2} + 7 \beta_1 - 6) q^{58} + (2 \beta_{3} - 4 \beta_{2} + 2 \beta_1) q^{59} + ( - 9 \beta_{2} + 2 \beta_1 - 9) q^{61} + ( - 5 \beta_{3} - 8) q^{62} + ( - 2 \beta_{3} - 7) q^{64} + (2 \beta_{2} + 3 \beta_1 + 2) q^{65} + (2 \beta_{3} + 4 \beta_{2} + 2 \beta_1) q^{67} + ( - 4 \beta_{2} + 8 \beta_1 - 4) q^{68} + ( - 4 \beta_{2} + 2 \beta_1 - 3) q^{70} + (3 \beta_{3} + 10) q^{71} + ( - 10 \beta_{3} + \beta_{2} - 10 \beta_1) q^{73} + ( - 7 \beta_{3} + 12 \beta_{2} - 7 \beta_1) q^{74} + ( - 2 \beta_{3} - 1) q^{76} + (2 \beta_{3} + 4 \beta_{2} + 2 \beta_1 + 1) q^{77} + \beta_1 q^{79} - 3 \beta_{2} q^{80} + ( - 2 \beta_{2} + 4 \beta_1 - 2) q^{82} + ( - 3 \beta_{3} - 5) q^{83} - 4 q^{85} + ( - 11 \beta_{2} + 6 \beta_1 - 11) q^{86} + ( - 2 \beta_{3} - \beta_{2} - 2 \beta_1) q^{88} + ( - 10 \beta_{2} + \beta_1 - 10) q^{89} + (4 \beta_{3} + 4 \beta_{2} - \beta_1 - 8) q^{91} + (13 \beta_{3} + 3) q^{92} + ( - 4 \beta_{3} + 5 \beta_{2} - 4 \beta_1) q^{94} + \beta_{2} q^{95} + (2 \beta_{3} - 6) q^{97} + ( - 7 \beta_{3} + 8 \beta_{2} - 4 \beta_1 - 1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} - 2 q^{4} + 2 q^{5} - 2 q^{7} + 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{2} - 2 q^{4} + 2 q^{5} - 2 q^{7} + 12 q^{8} + 2 q^{10} + 2 q^{11} + 8 q^{13} - 14 q^{14} - 6 q^{16} - 8 q^{17} + 2 q^{19} - 4 q^{20} + 4 q^{22} - 14 q^{23} + 8 q^{25} + 8 q^{26} + 4 q^{28} + 32 q^{29} + 4 q^{31} + 6 q^{32} + 16 q^{34} + 2 q^{35} - 4 q^{37} + 2 q^{38} + 6 q^{40} + 24 q^{41} + 4 q^{43} - 6 q^{44} - 10 q^{46} - 6 q^{47} + 10 q^{49} - 16 q^{50} + 20 q^{52} + 4 q^{55} + 6 q^{56} - 12 q^{58} + 8 q^{59} - 18 q^{61} - 32 q^{62} - 28 q^{64} + 4 q^{65} - 8 q^{67} - 8 q^{68} - 4 q^{70} + 40 q^{71} - 2 q^{73} - 24 q^{74} - 4 q^{76} - 4 q^{77} + 6 q^{80} - 4 q^{82} - 20 q^{83} - 16 q^{85} - 22 q^{86} + 2 q^{88} - 20 q^{89} - 40 q^{91} + 12 q^{92} - 10 q^{94} - 2 q^{95} - 24 q^{97} - 20 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1197\mathbb{Z}\right)^\times\).

\(n\) \(514\) \(533\) \(1009\)
\(\chi(n)\) \(-1 - \beta_{2}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
172.1
−0.707107 1.22474i
0.707107 + 1.22474i
−0.707107 + 1.22474i
0.707107 1.22474i
−1.20711 2.09077i 0 −1.91421 + 3.31552i 0.500000 + 0.866025i 0 1.62132 2.09077i 4.41421 0 1.20711 2.09077i
172.2 0.207107 + 0.358719i 0 0.914214 1.58346i 0.500000 + 0.866025i 0 −2.62132 + 0.358719i 1.58579 0 −0.207107 + 0.358719i
856.1 −1.20711 + 2.09077i 0 −1.91421 3.31552i 0.500000 0.866025i 0 1.62132 + 2.09077i 4.41421 0 1.20711 + 2.09077i
856.2 0.207107 0.358719i 0 0.914214 + 1.58346i 0.500000 0.866025i 0 −2.62132 0.358719i 1.58579 0 −0.207107 0.358719i
\(n\): 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 1197.2.j.e 4
3.b odd 2 1 133.2.f.c 4
7.c even 3 1 inner 1197.2.j.e 4
7.c even 3 1 8379.2.a.bi 2
7.d odd 6 1 8379.2.a.bl 2
21.c even 2 1 931.2.f.i 4
21.g even 6 1 931.2.a.e 2
21.g even 6 1 931.2.f.i 4
21.h odd 6 1 133.2.f.c 4
21.h odd 6 1 931.2.a.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
133.2.f.c 4 3.b odd 2 1
133.2.f.c 4 21.h odd 6 1
931.2.a.e 2 21.g even 6 1
931.2.a.f 2 21.h odd 6 1
931.2.f.i 4 21.c even 2 1
931.2.f.i 4 21.g even 6 1
1197.2.j.e 4 1.a even 1 1 trivial
1197.2.j.e 4 7.c even 3 1 inner
8379.2.a.bi 2 7.c even 3 1
8379.2.a.bl 2 7.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1197, [\chi])\):

\( T_{2}^{4} + 2T_{2}^{3} + 5T_{2}^{2} - 2T_{2} + 1 \) Copy content Toggle raw display
\( T_{5}^{2} - T_{5} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 2 T^{3} + 5 T^{2} - 2 T + 1 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + 2 T^{3} - 3 T^{2} + 14 T + 49 \) Copy content Toggle raw display
$11$ \( T^{4} - 2 T^{3} + 5 T^{2} + 2 T + 1 \) Copy content Toggle raw display
$13$ \( (T^{2} - 4 T - 14)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 4 T + 16)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 14 T^{3} + 149 T^{2} + \cdots + 2209 \) Copy content Toggle raw display
$29$ \( (T^{2} - 16 T + 62)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} - 4 T^{3} + 30 T^{2} + 56 T + 196 \) Copy content Toggle raw display
$37$ \( T^{4} + 4 T^{3} + 62 T^{2} + \cdots + 2116 \) Copy content Toggle raw display
$41$ \( (T^{2} - 12 T + 28)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 2 T - 49)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 6 T^{3} + 29 T^{2} + 42 T + 49 \) Copy content Toggle raw display
$53$ \( T^{4} + 18T^{2} + 324 \) Copy content Toggle raw display
$59$ \( T^{4} - 8 T^{3} + 56 T^{2} - 64 T + 64 \) Copy content Toggle raw display
$61$ \( T^{4} + 18 T^{3} + 251 T^{2} + \cdots + 5329 \) Copy content Toggle raw display
$67$ \( T^{4} + 8 T^{3} + 56 T^{2} + 64 T + 64 \) Copy content Toggle raw display
$71$ \( (T^{2} - 20 T + 82)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 2 T^{3} + 203 T^{2} + \cdots + 39601 \) Copy content Toggle raw display
$79$ \( T^{4} + 2T^{2} + 4 \) Copy content Toggle raw display
$83$ \( (T^{2} + 10 T + 7)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 20 T^{3} + 302 T^{2} + \cdots + 9604 \) Copy content Toggle raw display
$97$ \( (T^{2} + 12 T + 28)^{2} \) Copy content Toggle raw display
show more
show less