Properties

Label 432.9.e.g
Level $432$
Weight $9$
Character orbit 432.e
Analytic conductor $175.988$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [432,9,Mod(161,432)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(432, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("432.161");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 432 = 2^{4} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 432.e (of order \(2\), degree \(1\), not 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: \(175.987559546\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{5}\cdot 3 \)
Twist minimal: no (minimal twist has level 54)
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 \(\beta = 96\sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 5 \beta q^{5} + 2065 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + 5 \beta q^{5} + 2065 q^{7} - 49 \beta q^{11} + 8063 q^{13} - 159 \beta q^{17} + 226609 q^{19} + 2713 \beta q^{23} - 70175 q^{25} - 6902 \beta q^{29} - 826370 q^{31} + 10325 \beta q^{35} + 1344575 q^{37} - 38242 \beta q^{41} + 6147742 q^{43} - 43537 \beta q^{47} - 1500576 q^{49} - 5658 \beta q^{53} + 4515840 q^{55} - 3491 \beta q^{59} - 14985697 q^{61} + 40315 \beta q^{65} + 10023697 q^{67} - 335028 \beta q^{71} - 23261569 q^{73} - 101185 \beta q^{77} - 14267183 q^{79} - 266578 \beta q^{83} + 14653440 q^{85} + 847701 \beta q^{89} + 16650095 q^{91} + 1133045 \beta q^{95} - 40571617 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4130 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4130 q^{7} + 16126 q^{13} + 453218 q^{19} - 140350 q^{25} - 1652740 q^{31} + 2689150 q^{37} + 12295484 q^{43} - 3001152 q^{49} + 9031680 q^{55} - 29971394 q^{61} + 20047394 q^{67} - 46523138 q^{73} - 28534366 q^{79} + 29306880 q^{85} + 33300190 q^{91} - 81143234 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/432\mathbb{Z}\right)^\times\).

\(n\) \(271\) \(325\) \(353\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
161.1
1.41421i
1.41421i
0 0 0 678.823i 0 2065.00 0 0 0
161.2 0 0 0 678.823i 0 2065.00 0 0 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
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 432.9.e.g 2
3.b odd 2 1 inner 432.9.e.g 2
4.b odd 2 1 54.9.b.a 2
12.b even 2 1 54.9.b.a 2
36.f odd 6 2 162.9.d.c 4
36.h even 6 2 162.9.d.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
54.9.b.a 2 4.b odd 2 1
54.9.b.a 2 12.b even 2 1
162.9.d.c 4 36.f odd 6 2
162.9.d.c 4 36.h even 6 2
432.9.e.g 2 1.a even 1 1 trivial
432.9.e.g 2 3.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{9}^{\mathrm{new}}(432, [\chi])\):

\( T_{5}^{2} + 460800 \) Copy content Toggle raw display
\( T_{7} - 2065 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 460800 \) Copy content Toggle raw display
$7$ \( (T - 2065)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 44255232 \) Copy content Toggle raw display
$13$ \( (T - 8063)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 465979392 \) Copy content Toggle raw display
$19$ \( (T - 226609)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 135666321408 \) Copy content Toggle raw display
$29$ \( T^{2} + 878056316928 \) Copy content Toggle raw display
$31$ \( (T + 826370)^{2} \) Copy content Toggle raw display
$37$ \( (T - 1344575)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 26955888795648 \) Copy content Toggle raw display
$43$ \( (T - 6147742)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 34937309841408 \) Copy content Toggle raw display
$53$ \( T^{2} + 590062952448 \) Copy content Toggle raw display
$59$ \( T^{2} + 224632276992 \) Copy content Toggle raw display
$61$ \( (T + 14985697)^{2} \) Copy content Toggle raw display
$67$ \( (T - 10023697)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 20\!\cdots\!88 \) Copy content Toggle raw display
$73$ \( (T + 23261569)^{2} \) Copy content Toggle raw display
$79$ \( (T + 14267183)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 13\!\cdots\!88 \) Copy content Toggle raw display
$89$ \( T^{2} + 13\!\cdots\!32 \) Copy content Toggle raw display
$97$ \( (T + 40571617)^{2} \) Copy content Toggle raw display
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