Properties

Label 576.7.e.c
Level $576$
Weight $7$
Character orbit 576.e
Analytic conductor $132.511$
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] = [576,7,Mod(449,576)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(576, 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("576.449");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 576 = 2^{6} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 576.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: \(132.511152165\)
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: \( 3^{2} \)
Twist minimal: no (minimal twist has level 36)
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 \(\beta = 9\sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 5 \beta q^{5} - 244 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + 5 \beta q^{5} - 244 q^{7} + 188 \beta q^{11} + 2728 q^{13} - 463 \beta q^{17} + 5392 q^{19} - 196 \beta q^{23} + 11575 q^{25} - 3301 \beta q^{29} + 10172 q^{31} - 1220 \beta q^{35} - 65006 q^{37} + 5479 \beta q^{41} + 49480 q^{43} - 12948 \beta q^{47} - 58113 q^{49} + 1955 \beta q^{53} - 152280 q^{55} + 28216 \beta q^{59} - 100610 q^{61} + 13640 \beta q^{65} + 435736 q^{67} + 4724 \beta q^{71} + 619568 q^{73} - 45872 \beta q^{77} + 514340 q^{79} - 19164 \beta q^{83} + 375030 q^{85} + 33647 \beta q^{89} - 665632 q^{91} + 26960 \beta q^{95} + 42704 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 488 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 488 q^{7} + 5456 q^{13} + 10784 q^{19} + 23150 q^{25} + 20344 q^{31} - 130012 q^{37} + 98960 q^{43} - 116226 q^{49} - 304560 q^{55} - 201220 q^{61} + 871472 q^{67} + 1239136 q^{73} + 1028680 q^{79} + 750060 q^{85} - 1331264 q^{91} + 85408 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/576\mathbb{Z}\right)^\times\).

\(n\) \(65\) \(127\) \(325\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
449.1
1.41421i
1.41421i
0 0 0 63.6396i 0 −244.000 0 0 0
449.2 0 0 0 63.6396i 0 −244.000 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 576.7.e.c 2
3.b odd 2 1 inner 576.7.e.c 2
4.b odd 2 1 576.7.e.j 2
8.b even 2 1 36.7.c.a 2
8.d odd 2 1 144.7.e.c 2
12.b even 2 1 576.7.e.j 2
24.f even 2 1 144.7.e.c 2
24.h odd 2 1 36.7.c.a 2
72.j odd 6 2 324.7.g.e 4
72.n even 6 2 324.7.g.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.7.c.a 2 8.b even 2 1
36.7.c.a 2 24.h odd 2 1
144.7.e.c 2 8.d odd 2 1
144.7.e.c 2 24.f even 2 1
324.7.g.e 4 72.j odd 6 2
324.7.g.e 4 72.n even 6 2
576.7.e.c 2 1.a even 1 1 trivial
576.7.e.c 2 3.b odd 2 1 inner
576.7.e.j 2 4.b odd 2 1
576.7.e.j 2 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{7}^{\mathrm{new}}(576, [\chi])\):

\( T_{5}^{2} + 4050 \) Copy content Toggle raw display
\( T_{7} + 244 \) 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
$5$ \( T^{2} + 4050 \) Copy content Toggle raw display
$7$ \( (T + 244)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 5725728 \) Copy content Toggle raw display
$13$ \( (T - 2728)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 34727778 \) Copy content Toggle raw display
$19$ \( (T - 5392)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 6223392 \) Copy content Toggle raw display
$29$ \( T^{2} + 1765249362 \) Copy content Toggle raw display
$31$ \( (T - 10172)^{2} \) Copy content Toggle raw display
$37$ \( (T + 65006)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 4863149442 \) Copy content Toggle raw display
$43$ \( (T - 49480)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 27159414048 \) Copy content Toggle raw display
$53$ \( T^{2} + 619168050 \) Copy content Toggle raw display
$59$ \( T^{2} + 128975110272 \) Copy content Toggle raw display
$61$ \( (T + 100610)^{2} \) Copy content Toggle raw display
$67$ \( (T - 435736)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 3615220512 \) Copy content Toggle raw display
$73$ \( (T - 619568)^{2} \) Copy content Toggle raw display
$79$ \( (T - 514340)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 59495941152 \) Copy content Toggle raw display
$89$ \( T^{2} + 183403538658 \) Copy content Toggle raw display
$97$ \( (T - 42704)^{2} \) Copy content Toggle raw display
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