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Properties

Label 1197.2.j.g
Level $1197$
Weight $2$
Character orbit 1197.j
Analytic conductor $9.558$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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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: \(\Z[a_1, \ldots, a_{7}]\)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - 2 \beta_1 q^{5} + (3 \beta_{2} + 2) q^{7} - 2 \beta_{3} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - 2 \beta_1 q^{5} + (3 \beta_{2} + 2) q^{7} - 2 \beta_{3} q^{8} - 4 \beta_{2} q^{10} + (2 \beta_{3} + 3 \beta_{2} + 2 \beta_1) q^{11} + ( - 3 \beta_{3} - 2) q^{13} + (3 \beta_{3} + 2 \beta_1) q^{14} + (4 \beta_{2} + 4) q^{16} - 3 \beta_{2} q^{17} + (\beta_{2} + 1) q^{19} + (3 \beta_{3} - 4) q^{22} + (3 \beta_{2} + 3) q^{23} + 3 \beta_{2} q^{25} + (6 \beta_{2} - 2 \beta_1 + 6) q^{26} + ( - 2 \beta_{3} + 6) q^{29} + ( - 3 \beta_{3} - 2 \beta_{2} - 3 \beta_1) q^{31} - 3 \beta_{3} q^{34} + ( - 6 \beta_{3} - 4 \beta_1) q^{35} + (4 \beta_{2} + 4) q^{37} + (\beta_{3} + \beta_1) q^{38} + ( - 8 \beta_{2} - 8) q^{40} + ( - 2 \beta_{3} - 6) q^{41} + ( - 6 \beta_{3} - 2) q^{43} + (3 \beta_{3} + 3 \beta_1) q^{46} + (9 \beta_{2} + 2 \beta_1 + 9) q^{47} + (3 \beta_{2} - 5) q^{49} + 3 \beta_{3} q^{50} + ( - \beta_{3} + 12 \beta_{2} - \beta_1) q^{53} + ( - 6 \beta_{3} + 8) q^{55} + (2 \beta_{3} + 6 \beta_1) q^{56} + (4 \beta_{2} + 6 \beta_1 + 4) q^{58} + ( - 3 \beta_{3} + 6 \beta_{2} - 3 \beta_1) q^{59} + (3 \beta_{2} - 6 \beta_1 + 3) q^{61} + ( - 2 \beta_{3} + 6) q^{62} + 8 q^{64} + ( - 12 \beta_{2} + 4 \beta_1 - 12) q^{65} - 2 \beta_{2} q^{67} + (4 \beta_{2} + 12) q^{70} + (5 \beta_{3} + 6) q^{71} + 3 \beta_{2} q^{73} + (4 \beta_{3} + 4 \beta_1) q^{74} + (4 \beta_{3} - 3 \beta_{2} - 2 \beta_1 - 9) q^{77} + ( - 8 \beta_{2} - 3 \beta_1 - 8) q^{79} + ( - 8 \beta_{3} - 8 \beta_1) q^{80} + (4 \beta_{2} - 6 \beta_1 + 4) q^{82} + ( - 6 \beta_{3} + 3) q^{83} + 6 \beta_{3} q^{85} + (12 \beta_{2} - 2 \beta_1 + 12) q^{86} + (6 \beta_{3} + 8 \beta_{2} + 6 \beta_1) q^{88} + ( - 6 \beta_{2} - 6) q^{89} + (3 \beta_{3} - 6 \beta_{2} + 9 \beta_1 - 4) q^{91} + (9 \beta_{3} + 4 \beta_{2} + 9 \beta_1) q^{94} + ( - 2 \beta_{3} - 2 \beta_1) q^{95} + ( - 3 \beta_{3} - 2) q^{97} + (3 \beta_{3} - 5 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{7} + 8 q^{10} - 6 q^{11} - 8 q^{13} + 8 q^{16} + 6 q^{17} + 2 q^{19} - 16 q^{22} + 6 q^{23} - 6 q^{25} + 12 q^{26} + 24 q^{29} + 4 q^{31} + 8 q^{37} - 16 q^{40} - 24 q^{41} - 8 q^{43} + 18 q^{47} - 26 q^{49} - 24 q^{53} + 32 q^{55} + 8 q^{58} - 12 q^{59} + 6 q^{61} + 24 q^{62} + 32 q^{64} - 24 q^{65} + 4 q^{67} + 40 q^{70} + 24 q^{71} - 6 q^{73} - 30 q^{77} - 16 q^{79} + 8 q^{82} + 12 q^{83} + 24 q^{86} - 16 q^{88} - 12 q^{89} - 4 q^{91} - 8 q^{94} - 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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

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

\(\nu\)\(=\) \( \beta_1 \) 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

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

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
−0.707107 − 1.22474i 0 0 1.41421 + 2.44949i 0 0.500000 + 2.59808i −2.82843 0 2.00000 − 3.46410i
172.2 0.707107 + 1.22474i 0 0 −1.41421 − 2.44949i 0 0.500000 + 2.59808i 2.82843 0 2.00000 − 3.46410i
856.1 −0.707107 + 1.22474i 0 0 1.41421 − 2.44949i 0 0.500000 − 2.59808i −2.82843 0 2.00000 + 3.46410i
856.2 0.707107 − 1.22474i 0 0 −1.41421 + 2.44949i 0 0.500000 − 2.59808i 2.82843 0 2.00000 + 3.46410i
\(n\): 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.g 4
3.b odd 2 1 133.2.f.b ✓ 4
7.c even 3 1 inner 1197.2.j.g 4
7.c even 3 1 8379.2.a.z 2
7.d odd 6 1 8379.2.a.ba 2
21.c even 2 1 931.2.f.f 4
21.g even 6 1 931.2.a.h 2
21.g even 6 1 931.2.f.f 4
21.h odd 6 1 133.2.f.b ✓ 4
21.h odd 6 1 931.2.a.i 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
133.2.f.b ✓ 4 3.b odd 2 1
133.2.f.b ✓ 4 21.h odd 6 1
931.2.a.h 2 21.g even 6 1
931.2.a.i 2 21.h odd 6 1
931.2.f.f 4 21.c even 2 1
931.2.f.f 4 21.g even 6 1
1197.2.j.g 4 1.a even 1 1 trivial
1197.2.j.g 4 7.c even 3 1 inner
8379.2.a.z 2 7.c even 3 1
8379.2.a.ba 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 \(S_{2}^{\mathrm{new}}(1197, [\chi])\):

\( T_{2}^{4} + 2T_{2}^{2} + 4 \) Copy content Toggle raw display
\( T_{5}^{4} + 8T_{5}^{2} + 64 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 2T^{2} + 4 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 8T^{2} + 64 \) Copy content Toggle raw display
$7$ \( (T^{2} - T + 7)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + 6 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$13$ \( (T^{2} + 4 T - 14)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 3 T + 9)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - 3 T + 9)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 12 T + 28)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} - 4 T^{3} + \cdots + 196 \) Copy content Toggle raw display
$37$ \( (T^{2} - 4 T + 16)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 12 T + 28)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 4 T - 68)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 18 T^{3} + \cdots + 5329 \) Copy content Toggle raw display
$53$ \( T^{4} + 24 T^{3} + \cdots + 20164 \) Copy content Toggle raw display
$59$ \( T^{4} + 12 T^{3} + \cdots + 324 \) Copy content Toggle raw display
$61$ \( T^{4} - 6 T^{3} + \cdots + 3969 \) Copy content Toggle raw display
$67$ \( (T^{2} - 2 T + 4)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 12 T - 14)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 3 T + 9)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} + 16 T^{3} + \cdots + 2116 \) Copy content Toggle raw display
$83$ \( (T^{2} - 6 T - 63)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 6 T + 36)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 4 T - 14)^{2} \) Copy content Toggle raw display
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