Properties

Label 189.3.b.b
Level $189$
Weight $3$
Character orbit 189.b
Analytic conductor $5.150$
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] = [189,3,Mod(134,189)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(189, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("189.134");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 189 = 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 189.b (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: \(5.14987699641\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{7})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 8x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 3 \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{2} q^{4} - \beta_{3} q^{5} + \beta_{2} q^{7} + (\beta_{3} + 2 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{2} q^{4} - \beta_{3} q^{5} + \beta_{2} q^{7} + (\beta_{3} + 2 \beta_1) q^{8} + (2 \beta_{2} + 1) q^{10} + ( - 4 \beta_{3} + 3 \beta_1) q^{11} + (5 \beta_{2} - 3) q^{13} + (\beta_{3} - 2 \beta_1) q^{14} + (4 \beta_{2} - 9) q^{16} + ( - \beta_{3} + \beta_1) q^{17} + ( - \beta_{2} + 6) q^{19} + ( - 2 \beta_{3} - 3 \beta_1) q^{20} + (11 \beta_{2} - 8) q^{22} + (6 \beta_{3} - 3 \beta_1) q^{23} + ( - 5 \beta_{2} + 9) q^{25} + (5 \beta_{3} - 13 \beta_1) q^{26} + 7 q^{28} + (3 \beta_{3} + 17 \beta_1) q^{29} + (7 \beta_{2} + 16) q^{31} + (8 \beta_{3} - 9 \beta_1) q^{32} + (3 \beta_{2} - 3) q^{34} + ( - 2 \beta_{3} - 3 \beta_1) q^{35} + ( - 14 \beta_{2} - 23) q^{37} + ( - \beta_{3} + 8 \beta_1) q^{38} + (9 \beta_{2} + 18) q^{40} + ( - 9 \beta_{3} - 20 \beta_1) q^{41} + ( - 15 \beta_{2} - 7) q^{43} + ( - 5 \beta_{3} - 18 \beta_1) q^{44} + ( - 15 \beta_{2} + 6) q^{46} + ( - \beta_{3} - 11 \beta_1) q^{47} + 7 q^{49} + ( - 5 \beta_{3} + 19 \beta_1) q^{50} + ( - 3 \beta_{2} + 35) q^{52} + (2 \beta_{3} + 28 \beta_1) q^{53} + ( - 14 \beta_{2} - 61) q^{55} + (4 \beta_{3} - \beta_1) q^{56} + (11 \beta_{2} - 71) q^{58} + (\beta_{3} + 41 \beta_1) q^{59} + ( - 34 \beta_{2} + 24) q^{61} + (7 \beta_{3} + 2 \beta_1) q^{62} + ( - 9 \beta_{2} - 8) q^{64} + ( - 7 \beta_{3} - 15 \beta_1) q^{65} + ( - \beta_{2} + 39) q^{67} + ( - \beta_{3} - 5 \beta_1) q^{68} + (\beta_{2} + 14) q^{70} + ( - 9 \beta_{3} + 42 \beta_1) q^{71} + ( - 23 \beta_{2} + 49) q^{73} + ( - 14 \beta_{3} + 5 \beta_1) q^{74} + (6 \beta_{2} - 7) q^{76} + ( - 5 \beta_{3} - 18 \beta_1) q^{77} + ( - 9 \beta_{2} - 121) q^{79} + (\beta_{3} - 12 \beta_1) q^{80} + ( - 2 \beta_{2} + 89) q^{82} + (25 \beta_{3} - 31 \beta_1) q^{83} + ( - 3 \beta_{2} - 15) q^{85} + ( - 15 \beta_{3} + 23 \beta_1) q^{86} + (36 \beta_{2} + 45) q^{88} + ( - 15 \beta_{3} - 10 \beta_1) q^{89} + ( - 3 \beta_{2} + 35) q^{91} + (9 \beta_{3} + 24 \beta_1) q^{92} + ( - 9 \beta_{2} + 45) q^{94} + ( - 4 \beta_{3} + 3 \beta_1) q^{95} + ( - 38 \beta_{2} + 4) q^{97} + 7 \beta_1 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{10} - 12 q^{13} - 36 q^{16} + 24 q^{19} - 32 q^{22} + 36 q^{25} + 28 q^{28} + 64 q^{31} - 12 q^{34} - 92 q^{37} + 72 q^{40} - 28 q^{43} + 24 q^{46} + 28 q^{49} + 140 q^{52} - 244 q^{55} - 284 q^{58} + 96 q^{61} - 32 q^{64} + 156 q^{67} + 56 q^{70} + 196 q^{73} - 28 q^{76} - 484 q^{79} + 356 q^{82} - 60 q^{85} + 180 q^{88} + 140 q^{91} + 180 q^{94} + 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 8x^{2} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} + 6\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - 6\beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/189\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(136\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
134.1
2.57794i
1.16372i
1.16372i
2.57794i
2.57794i 0 −2.64575 1.66471i 0 −2.64575 3.49117i 0 −4.29150
134.2 1.16372i 0 2.64575 5.40636i 0 2.64575 7.73381i 0 6.29150
134.3 1.16372i 0 2.64575 5.40636i 0 2.64575 7.73381i 0 6.29150
134.4 2.57794i 0 −2.64575 1.66471i 0 −2.64575 3.49117i 0 −4.29150
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 189.3.b.b 4
3.b odd 2 1 inner 189.3.b.b 4
4.b odd 2 1 3024.3.d.c 4
9.c even 3 2 567.3.r.b 8
9.d odd 6 2 567.3.r.b 8
12.b even 2 1 3024.3.d.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.3.b.b 4 1.a even 1 1 trivial
189.3.b.b 4 3.b odd 2 1 inner
567.3.r.b 8 9.c even 3 2
567.3.r.b 8 9.d odd 6 2
3024.3.d.c 4 4.b odd 2 1
3024.3.d.c 4 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 8T_{2}^{2} + 9 \) acting on \(S_{3}^{\mathrm{new}}(189, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 8T^{2} + 9 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 32T^{2} + 81 \) Copy content Toggle raw display
$7$ \( (T^{2} - 7)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + 536 T^{2} + 68121 \) Copy content Toggle raw display
$13$ \( (T^{2} + 6 T - 166)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 18)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 12 T + 29)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 1152 T^{2} + 263169 \) Copy content Toggle raw display
$29$ \( T^{4} + 2804 T^{2} + \cdots + 1954404 \) Copy content Toggle raw display
$31$ \( (T^{2} - 32 T - 87)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 46 T - 843)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 6512 T^{2} + \cdots + 6922161 \) Copy content Toggle raw display
$43$ \( (T^{2} + 14 T - 1526)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 1044 T^{2} + 236196 \) Copy content Toggle raw display
$53$ \( T^{4} + 6624 T^{2} + \cdots + 8928144 \) Copy content Toggle raw display
$59$ \( T^{4} + 13644 T^{2} + \cdots + 30536676 \) Copy content Toggle raw display
$61$ \( (T^{2} - 48 T - 7516)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 78 T + 1514)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + 15192 T^{2} + 729 \) Copy content Toggle raw display
$73$ \( (T^{2} - 98 T - 1302)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 242 T + 14074)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 24588 T^{2} + \cdots + 145009764 \) Copy content Toggle raw display
$89$ \( T^{4} + 8600 T^{2} + 5625 \) Copy content Toggle raw display
$97$ \( (T^{2} - 8 T - 10092)^{2} \) Copy content Toggle raw display
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