Properties

Label 165.4.d.a
Level $165$
Weight $4$
Character orbit 165.d
Analytic conductor $9.735$
Analytic rank $0$
Dimension $2$
CM discriminant -11
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [165,4,Mod(164,165)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(165, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("165.164");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 165 = 3 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 165.d (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: \(9.73531515095\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-11}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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 \(\beta = \sqrt{-11}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta - 4) q^{3} + 8 q^{4} + (2 \beta + 9) q^{5} + (8 \beta + 5) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta - 4) q^{3} + 8 q^{4} + (2 \beta + 9) q^{5} + (8 \beta + 5) q^{9} + 11 \beta q^{11} + ( - 8 \beta - 32) q^{12} + ( - 17 \beta - 14) q^{15} + 64 q^{16} + (16 \beta + 72) q^{20} + 108 q^{23} + (36 \beta + 37) q^{25} + ( - 37 \beta + 68) q^{27} + 340 q^{31} + ( - 44 \beta + 121) q^{33} + (64 \beta + 40) q^{36} - 36 \beta q^{37} + 88 \beta q^{44} + (82 \beta - 131) q^{45} - 36 q^{47} + ( - 64 \beta - 256) q^{48} - 343 q^{49} - 738 q^{53} + (99 \beta - 242) q^{55} - 166 \beta q^{59} + ( - 136 \beta - 112) q^{60} + 512 q^{64} + 306 \beta q^{67} + ( - 108 \beta - 432) q^{69} - 310 \beta q^{71} + ( - 181 \beta + 248) q^{75} + (128 \beta + 576) q^{80} + (80 \beta - 679) q^{81} + 40 \beta q^{89} + 864 q^{92} + ( - 340 \beta - 1360) q^{93} + 576 \beta q^{97} + (55 \beta - 968) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8 q^{3} + 16 q^{4} + 18 q^{5} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 8 q^{3} + 16 q^{4} + 18 q^{5} + 10 q^{9} - 64 q^{12} - 28 q^{15} + 128 q^{16} + 144 q^{20} + 216 q^{23} + 74 q^{25} + 136 q^{27} + 680 q^{31} + 242 q^{33} + 80 q^{36} - 262 q^{45} - 72 q^{47} - 512 q^{48} - 686 q^{49} - 1476 q^{53} - 484 q^{55} - 224 q^{60} + 1024 q^{64} - 864 q^{69} + 496 q^{75} + 1152 q^{80} - 1358 q^{81} + 1728 q^{92} - 2720 q^{93} - 1936 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/165\mathbb{Z}\right)^\times\).

\(n\) \(46\) \(56\) \(67\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
164.1
0.500000 + 1.65831i
0.500000 1.65831i
0 −4.00000 3.31662i 8.00000 9.00000 + 6.63325i 0 0 0 5.00000 + 26.5330i 0
164.2 0 −4.00000 + 3.31662i 8.00000 9.00000 6.63325i 0 0 0 5.00000 26.5330i 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
11.b odd 2 1 CM by \(\Q(\sqrt{-11}) \)
15.d odd 2 1 inner
165.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 165.4.d.a 2
3.b odd 2 1 165.4.d.b yes 2
5.b even 2 1 165.4.d.b yes 2
11.b odd 2 1 CM 165.4.d.a 2
15.d odd 2 1 inner 165.4.d.a 2
33.d even 2 1 165.4.d.b yes 2
55.d odd 2 1 165.4.d.b yes 2
165.d even 2 1 inner 165.4.d.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.4.d.a 2 1.a even 1 1 trivial
165.4.d.a 2 11.b odd 2 1 CM
165.4.d.a 2 15.d odd 2 1 inner
165.4.d.a 2 165.d even 2 1 inner
165.4.d.b yes 2 3.b odd 2 1
165.4.d.b yes 2 5.b even 2 1
165.4.d.b yes 2 33.d even 2 1
165.4.d.b yes 2 55.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(165, [\chi])\):

\( T_{2} \) Copy content Toggle raw display
\( T_{23} - 108 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 8T + 27 \) Copy content Toggle raw display
$5$ \( T^{2} - 18T + 125 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 1331 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( (T - 108)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( (T - 340)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 14256 \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( (T + 36)^{2} \) Copy content Toggle raw display
$53$ \( (T + 738)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 303116 \) Copy content Toggle raw display
$61$ \( T^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 1029996 \) Copy content Toggle raw display
$71$ \( T^{2} + 1057100 \) Copy content Toggle raw display
$73$ \( T^{2} \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 17600 \) Copy content Toggle raw display
$97$ \( T^{2} + 3649536 \) Copy content Toggle raw display
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