Properties

Label 2352.3.m.l
Level $2352$
Weight $3$
Character orbit 2352.m
Analytic conductor $64.087$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2352,3,Mod(1471,2352)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2352, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2352.1471");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2352.m (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: \(64.0873581775\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{3} + (\beta_1 + 6) q^{5} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{3} + (\beta_1 + 6) q^{5} - 3 q^{9} + (4 \beta_{3} - 6 \beta_{2}) q^{11} + (7 \beta_1 + 12) q^{13} + (\beta_{3} - 6 \beta_{2}) q^{15} + (5 \beta_1 + 6) q^{17} + 6 \beta_{3} q^{19} + (4 \beta_{3} - 18 \beta_{2}) q^{23} + (12 \beta_1 + 13) q^{25} + 3 \beta_{2} q^{27} + (6 \beta_1 + 24) q^{29} + ( - 18 \beta_{3} + 4 \beta_{2}) q^{31} + ( - 12 \beta_1 - 18) q^{33} - 24 \beta_1 q^{37} + (7 \beta_{3} - 12 \beta_{2}) q^{39} + (7 \beta_1 - 18) q^{41} + ( - 20 \beta_{3} + 12 \beta_{2}) q^{43} + ( - 3 \beta_1 - 18) q^{45} + (6 \beta_{3} - 8 \beta_{2}) q^{47} + (5 \beta_{3} - 6 \beta_{2}) q^{51} + ( - 36 \beta_1 - 22) q^{53} + (30 \beta_{3} - 44 \beta_{2}) q^{55} - 18 \beta_1 q^{57} + ( - 18 \beta_{3} - 28 \beta_{2}) q^{59} + ( - 41 \beta_1 + 24) q^{61} + (54 \beta_1 + 86) q^{65} + ( - 4 \beta_{3} + 12 \beta_{2}) q^{67} + ( - 12 \beta_1 - 54) q^{69} + ( - 24 \beta_{3} - 18 \beta_{2}) q^{71} + (35 \beta_1 + 12) q^{73} + (12 \beta_{3} - 13 \beta_{2}) q^{75} + (20 \beta_{3} - 24 \beta_{2}) q^{79} + 9 q^{81} + ( - 12 \beta_{3} - 20 \beta_{2}) q^{83} + (36 \beta_1 + 46) q^{85} + (6 \beta_{3} - 24 \beta_{2}) q^{87} + ( - 19 \beta_1 - 30) q^{89} + (54 \beta_1 + 12) q^{93} + (36 \beta_{3} - 12 \beta_{2}) q^{95} + (11 \beta_1 + 108) q^{97} + ( - 12 \beta_{3} + 18 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 24 q^{5} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 24 q^{5} - 12 q^{9} + 48 q^{13} + 24 q^{17} + 52 q^{25} + 96 q^{29} - 72 q^{33} - 72 q^{41} - 72 q^{45} - 88 q^{53} + 96 q^{61} + 344 q^{65} - 216 q^{69} + 48 q^{73} + 36 q^{81} + 184 q^{85} - 120 q^{89} + 48 q^{93} + 432 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 2x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} + 4\nu ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2352\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1471\) \(1765\) \(2257\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1471.1
0.707107 + 1.22474i
−0.707107 1.22474i
0.707107 1.22474i
−0.707107 + 1.22474i
0 1.73205i 0 4.58579 0 0 0 −3.00000 0
1471.2 0 1.73205i 0 7.41421 0 0 0 −3.00000 0
1471.3 0 1.73205i 0 4.58579 0 0 0 −3.00000 0
1471.4 0 1.73205i 0 7.41421 0 0 0 −3.00000 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
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2352.3.m.l yes 4
4.b odd 2 1 inner 2352.3.m.l yes 4
7.b odd 2 1 2352.3.m.d 4
28.d even 2 1 2352.3.m.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2352.3.m.d 4 7.b odd 2 1
2352.3.m.d 4 28.d even 2 1
2352.3.m.l yes 4 1.a even 1 1 trivial
2352.3.m.l yes 4 4.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 12T_{5} + 34 \) acting on \(S_{3}^{\mathrm{new}}(2352, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} + 3)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} - 12 T + 34)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} + 408T^{2} + 144 \) Copy content Toggle raw display
$13$ \( (T^{2} - 24 T + 46)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 12 T - 14)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 216)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 2136 T^{2} + 767376 \) Copy content Toggle raw display
$29$ \( (T^{2} - 48 T + 504)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 3984 T^{2} + \cdots + 3594816 \) Copy content Toggle raw display
$37$ \( (T^{2} - 1152)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 36 T + 226)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 5664 T^{2} + \cdots + 3873024 \) Copy content Toggle raw display
$47$ \( T^{4} + 816T^{2} + 576 \) Copy content Toggle raw display
$53$ \( (T^{2} + 44 T - 2108)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 8592 T^{2} + 166464 \) Copy content Toggle raw display
$61$ \( (T^{2} - 48 T - 2786)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 1056 T^{2} + 112896 \) Copy content Toggle raw display
$71$ \( T^{4} + 8856 T^{2} + \cdots + 6170256 \) Copy content Toggle raw display
$73$ \( (T^{2} - 24 T - 2306)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} + 8256 T^{2} + 451584 \) Copy content Toggle raw display
$83$ \( T^{4} + 4128 T^{2} + 112896 \) Copy content Toggle raw display
$89$ \( (T^{2} + 60 T + 178)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 216 T + 11422)^{2} \) Copy content Toggle raw display
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