Properties

Label 2304.3.h.c
Level $2304$
Weight $3$
Character orbit 2304.h
Analytic conductor $62.779$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2304,3,Mod(2177,2304)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2304, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2304.2177");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2304 = 2^{8} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2304.h (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: \(62.7794529086\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 18)
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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{5} - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{5} - 4 q^{7} + 4 \beta_{3} q^{11} + 4 \beta_1 q^{13} + 3 \beta_{2} q^{17} + 8 \beta_1 q^{19} + 4 \beta_{2} q^{23} - 7 q^{25} + \beta_{3} q^{29} - 44 q^{31} - 4 \beta_{3} q^{35} + 17 \beta_1 q^{37} + 11 \beta_{2} q^{41} - 20 \beta_1 q^{43} - 20 \beta_{2} q^{47} - 33 q^{49} - 9 \beta_{3} q^{53} + 72 q^{55} + 8 \beta_{3} q^{59} + 25 \beta_1 q^{61} + 8 \beta_{2} q^{65} - 4 \beta_1 q^{67} + 12 \beta_{2} q^{71} + 16 q^{73} - 16 \beta_{3} q^{77} + 76 q^{79} - 28 \beta_{3} q^{83} + 27 \beta_1 q^{85} + 3 \beta_{2} q^{89} - 16 \beta_1 q^{91} + 16 \beta_{2} q^{95} + 176 q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 16 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 16 q^{7} - 28 q^{25} - 176 q^{31} - 132 q^{49} + 288 q^{55} + 64 q^{73} + 304 q^{79} + 704 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( 2\zeta_{8}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 3\zeta_{8}^{3} + 3\zeta_{8} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -3\zeta_{8}^{3} + 3\zeta_{8} \) Copy content Toggle raw display
\(\zeta_{8}\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 6 \) Copy content Toggle raw display
\(\zeta_{8}^{2}\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{8}^{3}\)\(=\) \( ( -\beta_{3} + \beta_{2} ) / 6 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2304\mathbb{Z}\right)^\times\).

\(n\) \(1279\) \(1793\) \(2053\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2177.1
−0.707107 + 0.707107i
−0.707107 0.707107i
0.707107 0.707107i
0.707107 + 0.707107i
0 0 0 −4.24264 0 −4.00000 0 0 0
2177.2 0 0 0 −4.24264 0 −4.00000 0 0 0
2177.3 0 0 0 4.24264 0 −4.00000 0 0 0
2177.4 0 0 0 4.24264 0 −4.00000 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
8.b even 2 1 inner
24.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2304.3.h.c 4
3.b odd 2 1 inner 2304.3.h.c 4
4.b odd 2 1 2304.3.h.f 4
8.b even 2 1 inner 2304.3.h.c 4
8.d odd 2 1 2304.3.h.f 4
12.b even 2 1 2304.3.h.f 4
16.e even 4 1 144.3.e.b 2
16.e even 4 1 576.3.e.f 2
16.f odd 4 1 18.3.b.a 2
16.f odd 4 1 576.3.e.c 2
24.f even 2 1 2304.3.h.f 4
24.h odd 2 1 inner 2304.3.h.c 4
48.i odd 4 1 144.3.e.b 2
48.i odd 4 1 576.3.e.f 2
48.k even 4 1 18.3.b.a 2
48.k even 4 1 576.3.e.c 2
80.i odd 4 1 3600.3.c.b 4
80.j even 4 1 450.3.b.b 4
80.k odd 4 1 450.3.d.f 2
80.q even 4 1 3600.3.l.d 2
80.s even 4 1 450.3.b.b 4
80.t odd 4 1 3600.3.c.b 4
112.j even 4 1 882.3.b.a 2
112.u odd 12 2 882.3.s.b 4
112.v even 12 2 882.3.s.d 4
144.u even 12 2 162.3.d.b 4
144.v odd 12 2 162.3.d.b 4
144.w odd 12 2 1296.3.q.f 4
144.x even 12 2 1296.3.q.f 4
176.i even 4 1 2178.3.c.d 2
208.l even 4 1 3042.3.d.a 4
208.o odd 4 1 3042.3.c.e 2
208.s even 4 1 3042.3.d.a 4
240.t even 4 1 450.3.d.f 2
240.z odd 4 1 450.3.b.b 4
240.bb even 4 1 3600.3.c.b 4
240.bd odd 4 1 450.3.b.b 4
240.bf even 4 1 3600.3.c.b 4
240.bm odd 4 1 3600.3.l.d 2
336.v odd 4 1 882.3.b.a 2
336.br odd 12 2 882.3.s.d 4
336.bu even 12 2 882.3.s.b 4
528.s odd 4 1 2178.3.c.d 2
624.s odd 4 1 3042.3.d.a 4
624.v even 4 1 3042.3.c.e 2
624.bo odd 4 1 3042.3.d.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.3.b.a 2 16.f odd 4 1
18.3.b.a 2 48.k even 4 1
144.3.e.b 2 16.e even 4 1
144.3.e.b 2 48.i odd 4 1
162.3.d.b 4 144.u even 12 2
162.3.d.b 4 144.v odd 12 2
450.3.b.b 4 80.j even 4 1
450.3.b.b 4 80.s even 4 1
450.3.b.b 4 240.z odd 4 1
450.3.b.b 4 240.bd odd 4 1
450.3.d.f 2 80.k odd 4 1
450.3.d.f 2 240.t even 4 1
576.3.e.c 2 16.f odd 4 1
576.3.e.c 2 48.k even 4 1
576.3.e.f 2 16.e even 4 1
576.3.e.f 2 48.i odd 4 1
882.3.b.a 2 112.j even 4 1
882.3.b.a 2 336.v odd 4 1
882.3.s.b 4 112.u odd 12 2
882.3.s.b 4 336.bu even 12 2
882.3.s.d 4 112.v even 12 2
882.3.s.d 4 336.br odd 12 2
1296.3.q.f 4 144.w odd 12 2
1296.3.q.f 4 144.x even 12 2
2178.3.c.d 2 176.i even 4 1
2178.3.c.d 2 528.s odd 4 1
2304.3.h.c 4 1.a even 1 1 trivial
2304.3.h.c 4 3.b odd 2 1 inner
2304.3.h.c 4 8.b even 2 1 inner
2304.3.h.c 4 24.h odd 2 1 inner
2304.3.h.f 4 4.b odd 2 1
2304.3.h.f 4 8.d odd 2 1
2304.3.h.f 4 12.b even 2 1
2304.3.h.f 4 24.f even 2 1
3042.3.c.e 2 208.o odd 4 1
3042.3.c.e 2 624.v even 4 1
3042.3.d.a 4 208.l even 4 1
3042.3.d.a 4 208.s even 4 1
3042.3.d.a 4 624.s odd 4 1
3042.3.d.a 4 624.bo odd 4 1
3600.3.c.b 4 80.i odd 4 1
3600.3.c.b 4 80.t odd 4 1
3600.3.c.b 4 240.bb even 4 1
3600.3.c.b 4 240.bf even 4 1
3600.3.l.d 2 80.q even 4 1
3600.3.l.d 2 240.bm odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(2304, [\chi])\):

\( T_{5}^{2} - 18 \) Copy content Toggle raw display
\( T_{7} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} - 18)^{2} \) Copy content Toggle raw display
$7$ \( (T + 4)^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} - 288)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 64)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 162)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 256)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 288)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 18)^{2} \) Copy content Toggle raw display
$31$ \( (T + 44)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 1156)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 2178)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 1600)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 7200)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 1458)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 1152)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 2500)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 64)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 2592)^{2} \) Copy content Toggle raw display
$73$ \( (T - 16)^{4} \) Copy content Toggle raw display
$79$ \( (T - 76)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} - 14112)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 162)^{2} \) Copy content Toggle raw display
$97$ \( (T - 176)^{4} \) Copy content Toggle raw display
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