Properties

Label 432.7.e.d
Level $432$
Weight $7$
Character orbit 432.e
Analytic conductor $99.383$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [432,7,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, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("432.161");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 432 = 2^{4} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 7 \)
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: \(99.3833641238\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4}\cdot 3 \)
Twist minimal: no (minimal twist has level 27)
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 \(\beta = 48i\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 5 \beta q^{5} - 299 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + 5 \beta q^{5} - 299 q^{7} - 13 \beta q^{11} + 2495 q^{13} - 39 \beta q^{17} + 2509 q^{19} - 299 \beta q^{23} - 41975 q^{25} - 494 \beta q^{29} - 5330 q^{31} - 1495 \beta q^{35} + 32591 q^{37} - 1378 \beta q^{41} + 70630 q^{43} + 83 \beta q^{47} - 28248 q^{49} - 3978 \beta q^{53} + 149760 q^{55} + 4945 \beta q^{59} - 61801 q^{61} + 12475 \beta q^{65} + 430261 q^{67} - 5244 \beta q^{71} + 251615 q^{73} + 3887 \beta q^{77} - 660827 q^{79} + 16622 \beta q^{83} + 449280 q^{85} + 5637 \beta q^{89} - 746005 q^{91} + 12545 \beta q^{95} + 220727 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 598 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 598 q^{7} + 4990 q^{13} + 5018 q^{19} - 83950 q^{25} - 10660 q^{31} + 65182 q^{37} + 141260 q^{43} - 56496 q^{49} + 299520 q^{55} - 123602 q^{61} + 860522 q^{67} + 503230 q^{73} - 1321654 q^{79} + 898560 q^{85} - 1492010 q^{91} + 441454 q^{97}+O(q^{100}) \) 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
1.00000i
1.00000i
0 0 0 240.000i 0 −299.000 0 0 0
161.2 0 0 0 240.000i 0 −299.000 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 432.7.e.d 2
3.b odd 2 1 inner 432.7.e.d 2
4.b odd 2 1 27.7.b.b 2
12.b even 2 1 27.7.b.b 2
36.f odd 6 2 81.7.d.b 4
36.h even 6 2 81.7.d.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
27.7.b.b 2 4.b odd 2 1
27.7.b.b 2 12.b even 2 1
81.7.d.b 4 36.f odd 6 2
81.7.d.b 4 36.h even 6 2
432.7.e.d 2 1.a even 1 1 trivial
432.7.e.d 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 \(S_{7}^{\mathrm{new}}(432, [\chi])\):

\( T_{5}^{2} + 57600 \) Copy content Toggle raw display
\( T_{7} + 299 \) 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
$5$ \( T^{2} + 57600 \) Copy content Toggle raw display
$7$ \( (T + 299)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 389376 \) Copy content Toggle raw display
$13$ \( (T - 2495)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 3504384 \) Copy content Toggle raw display
$19$ \( (T - 2509)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 205979904 \) Copy content Toggle raw display
$29$ \( T^{2} + 562258944 \) Copy content Toggle raw display
$31$ \( (T + 5330)^{2} \) Copy content Toggle raw display
$37$ \( (T - 32591)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 4375028736 \) Copy content Toggle raw display
$43$ \( (T - 70630)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 15872256 \) Copy content Toggle raw display
$53$ \( T^{2} + 36459611136 \) Copy content Toggle raw display
$59$ \( T^{2} + 56339769600 \) Copy content Toggle raw display
$61$ \( (T + 61801)^{2} \) Copy content Toggle raw display
$67$ \( (T - 430261)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 63358930944 \) Copy content Toggle raw display
$73$ \( (T - 251615)^{2} \) Copy content Toggle raw display
$79$ \( (T + 660827)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 636574196736 \) Copy content Toggle raw display
$89$ \( T^{2} + 73211371776 \) Copy content Toggle raw display
$97$ \( (T - 220727)^{2} \) Copy content Toggle raw display
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