Properties

Label 2368.2.g.e
Level $2368$
Weight $2$
Character orbit 2368.g
Analytic conductor $18.909$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2368,2,Mod(961,2368)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2368, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2368.961");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2368 = 2^{6} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2368.g (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: \(18.9085751986\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 148)
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 \(\beta = 2\sqrt{-3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} - \beta q^{5} - q^{7} - 2 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} - \beta q^{5} - q^{7} - 2 q^{9} + 3 q^{11} + \beta q^{13} - \beta q^{15} - \beta q^{17} - \beta q^{19} - q^{21} - 2 \beta q^{23} - 7 q^{25} - 5 q^{27} + 2 \beta q^{29} + 3 q^{33} + \beta q^{35} + ( - \beta + 5) q^{37} + \beta q^{39} - 3 q^{41} - 3 \beta q^{43} + 2 \beta q^{45} - 9 q^{47} - 6 q^{49} - \beta q^{51} + 3 q^{53} - 3 \beta q^{55} - \beta q^{57} + 2 \beta q^{59} - 4 \beta q^{61} + 2 q^{63} + 12 q^{65} + 4 q^{67} - 2 \beta q^{69} - 15 q^{71} + q^{73} - 7 q^{75} - 3 q^{77} + \beta q^{79} + q^{81} - 9 q^{83} - 12 q^{85} + 2 \beta q^{87} + 3 \beta q^{89} - \beta q^{91} - 12 q^{95} - 2 \beta q^{97} - 6 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - 2 q^{7} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} - 2 q^{7} - 4 q^{9} + 6 q^{11} - 2 q^{21} - 14 q^{25} - 10 q^{27} + 6 q^{33} + 10 q^{37} - 6 q^{41} - 18 q^{47} - 12 q^{49} + 6 q^{53} + 4 q^{63} + 24 q^{65} + 8 q^{67} - 30 q^{71} + 2 q^{73} - 14 q^{75} - 6 q^{77} + 2 q^{81} - 18 q^{83} - 24 q^{85} - 24 q^{95} - 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2368\mathbb{Z}\right)^\times\).

\(n\) \(705\) \(1407\) \(1925\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
961.1
0.500000 + 0.866025i
0.500000 0.866025i
0 1.00000 0 3.46410i 0 −1.00000 0 −2.00000 0
961.2 0 1.00000 0 3.46410i 0 −1.00000 0 −2.00000 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
37.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2368.2.g.e 2
4.b odd 2 1 2368.2.g.c 2
8.b even 2 1 592.2.g.a 2
8.d odd 2 1 148.2.d.a 2
24.f even 2 1 1332.2.e.a 2
24.h odd 2 1 5328.2.h.d 2
37.b even 2 1 inner 2368.2.g.e 2
40.e odd 2 1 3700.2.h.b 2
40.k even 4 2 3700.2.e.c 4
148.b odd 2 1 2368.2.g.c 2
296.e even 2 1 592.2.g.a 2
296.h odd 2 1 148.2.d.a 2
296.j even 4 2 5476.2.a.b 2
888.c even 2 1 1332.2.e.a 2
888.i odd 2 1 5328.2.h.d 2
1480.h odd 2 1 3700.2.h.b 2
1480.bh even 4 2 3700.2.e.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
148.2.d.a 2 8.d odd 2 1
148.2.d.a 2 296.h odd 2 1
592.2.g.a 2 8.b even 2 1
592.2.g.a 2 296.e even 2 1
1332.2.e.a 2 24.f even 2 1
1332.2.e.a 2 888.c even 2 1
2368.2.g.c 2 4.b odd 2 1
2368.2.g.c 2 148.b odd 2 1
2368.2.g.e 2 1.a even 1 1 trivial
2368.2.g.e 2 37.b even 2 1 inner
3700.2.e.c 4 40.k even 4 2
3700.2.e.c 4 1480.bh even 4 2
3700.2.h.b 2 40.e odd 2 1
3700.2.h.b 2 1480.h odd 2 1
5328.2.h.d 2 24.h odd 2 1
5328.2.h.d 2 888.i odd 2 1
5476.2.a.b 2 296.j even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2368, [\chi])\):

\( T_{3} - 1 \) Copy content Toggle raw display
\( T_{5}^{2} + 12 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T - 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 12 \) Copy content Toggle raw display
$7$ \( (T + 1)^{2} \) Copy content Toggle raw display
$11$ \( (T - 3)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 12 \) Copy content Toggle raw display
$17$ \( T^{2} + 12 \) Copy content Toggle raw display
$19$ \( T^{2} + 12 \) Copy content Toggle raw display
$23$ \( T^{2} + 48 \) Copy content Toggle raw display
$29$ \( T^{2} + 48 \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 10T + 37 \) Copy content Toggle raw display
$41$ \( (T + 3)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 108 \) Copy content Toggle raw display
$47$ \( (T + 9)^{2} \) Copy content Toggle raw display
$53$ \( (T - 3)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 48 \) Copy content Toggle raw display
$61$ \( T^{2} + 192 \) Copy content Toggle raw display
$67$ \( (T - 4)^{2} \) Copy content Toggle raw display
$71$ \( (T + 15)^{2} \) Copy content Toggle raw display
$73$ \( (T - 1)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 12 \) Copy content Toggle raw display
$83$ \( (T + 9)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 108 \) Copy content Toggle raw display
$97$ \( T^{2} + 48 \) Copy content Toggle raw display
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