Properties

Label 2368.2.e.a
Level $2368$
Weight $2$
Character orbit 2368.e
Analytic conductor $18.909$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2368,2,Mod(2145,2368)]
 
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, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2368.2145");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2368 = 2^{6} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2368.e (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: \(18.9085751986\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.8540717056.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 31x^{4} + 625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + \beta_{4} q^{5} - \beta_{5} q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + \beta_{4} q^{5} - \beta_{5} q^{7} + 2 q^{9} + \beta_1 q^{11} - 3 \beta_{4} q^{13} + \beta_{3} q^{15} - \beta_{6} q^{17} + \beta_{7} q^{19} - \beta_{2} q^{21} + 4 \beta_{3} q^{23} - 3 q^{25} + 5 \beta_1 q^{27} + 2 \beta_{4} q^{29} + 6 \beta_{3} q^{31} - q^{33} - \beta_{7} q^{35} + ( - 3 \beta_{4} + \beta_{2}) q^{37} - 3 \beta_{3} q^{39} - 9 q^{41} - \beta_{7} q^{43} + 2 \beta_{4} q^{45} - \beta_{5} q^{47} + 12 q^{49} + \beta_{7} q^{51} - \beta_{2} q^{53} + \beta_{3} q^{55} + \beta_{6} q^{57} - 2 \beta_{7} q^{59} - 4 \beta_{4} q^{61} - 2 \beta_{5} q^{63} - 6 q^{65} - 10 \beta_1 q^{67} - 4 \beta_{4} q^{69} - 3 \beta_{5} q^{71} - 3 q^{73} - 3 \beta_1 q^{75} - \beta_{2} q^{77} - \beta_{3} q^{79} + q^{81} - 13 \beta_1 q^{83} - 2 \beta_{2} q^{85} + 2 \beta_{3} q^{87} + 3 \beta_{6} q^{89} + 3 \beta_{7} q^{91} - 6 \beta_{4} q^{93} + 2 \beta_{5} q^{95} + 2 \beta_1 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 16 q^{9} - 24 q^{25} - 8 q^{33} - 72 q^{41} + 96 q^{49} - 48 q^{65} - 24 q^{73} + 8 q^{81}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 31x^{4} + 625 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{6} + 56\nu^{2} ) / 225 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2\nu^{4} + 31 ) / 9 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -4\nu^{7} + 25\nu^{5} + \nu^{3} + 275\nu ) / 1125 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -4\nu^{7} - 25\nu^{5} + \nu^{3} - 275\nu ) / 1125 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{6} - 6\nu^{2} ) / 25 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 14\nu^{7} + 25\nu^{5} + 559\nu^{3} + 2525\nu ) / 1125 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -14\nu^{7} + 25\nu^{5} - 559\nu^{3} + 2525\nu ) / 1125 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{7} + \beta_{6} + \beta_{4} - \beta_{3} ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} + 9\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{7} + 2\beta_{6} + 7\beta_{4} + 7\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 9\beta_{2} - 31 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -11\beta_{7} - 11\beta_{6} - 101\beta_{4} + 101\beta_{3} ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -28\beta_{5} - 27\beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -\beta_{7} + \beta_{6} - 559\beta_{4} - 559\beta_{3} ) / 4 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2145.1
−1.89466 + 1.18755i
1.18755 1.89466i
1.89466 1.18755i
−1.18755 + 1.89466i
−1.89466 1.18755i
1.18755 + 1.89466i
1.89466 + 1.18755i
−1.18755 1.89466i
0 1.00000i 0 −1.41421 0 −4.35890 0 2.00000 0
2145.2 0 1.00000i 0 −1.41421 0 4.35890 0 2.00000 0
2145.3 0 1.00000i 0 1.41421 0 −4.35890 0 2.00000 0
2145.4 0 1.00000i 0 1.41421 0 4.35890 0 2.00000 0
2145.5 0 1.00000i 0 −1.41421 0 −4.35890 0 2.00000 0
2145.6 0 1.00000i 0 −1.41421 0 4.35890 0 2.00000 0
2145.7 0 1.00000i 0 1.41421 0 −4.35890 0 2.00000 0
2145.8 0 1.00000i 0 1.41421 0 4.35890 0 2.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2145.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
37.b even 2 1 inner
148.b odd 2 1 inner
296.e even 2 1 inner
296.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2368.2.e.a 8
4.b odd 2 1 inner 2368.2.e.a 8
8.b even 2 1 inner 2368.2.e.a 8
8.d odd 2 1 inner 2368.2.e.a 8
37.b even 2 1 inner 2368.2.e.a 8
148.b odd 2 1 inner 2368.2.e.a 8
296.e even 2 1 inner 2368.2.e.a 8
296.h odd 2 1 inner 2368.2.e.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2368.2.e.a 8 1.a even 1 1 trivial
2368.2.e.a 8 4.b odd 2 1 inner
2368.2.e.a 8 8.b even 2 1 inner
2368.2.e.a 8 8.d odd 2 1 inner
2368.2.e.a 8 37.b even 2 1 inner
2368.2.e.a 8 148.b odd 2 1 inner
2368.2.e.a 8 296.e even 2 1 inner
2368.2.e.a 8 296.h odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(2368, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} - 2)^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} - 19)^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} - 18)^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} + 38)^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} - 38)^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} + 32)^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} - 8)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + 72)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} + 2 T^{2} + 1369)^{2} \) Copy content Toggle raw display
$41$ \( (T + 9)^{8} \) Copy content Toggle raw display
$43$ \( (T^{2} - 38)^{4} \) Copy content Toggle raw display
$47$ \( (T^{2} - 19)^{4} \) Copy content Toggle raw display
$53$ \( (T^{2} + 19)^{4} \) Copy content Toggle raw display
$59$ \( (T^{2} - 152)^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} - 32)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} + 100)^{4} \) Copy content Toggle raw display
$71$ \( (T^{2} - 171)^{4} \) Copy content Toggle raw display
$73$ \( (T + 3)^{8} \) Copy content Toggle raw display
$79$ \( (T^{2} + 2)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 169)^{4} \) Copy content Toggle raw display
$89$ \( (T^{2} + 342)^{4} \) Copy content Toggle raw display
$97$ \( T^{8} \) Copy content Toggle raw display
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