Properties

Label 2368.2.g.k
Level $2368$
Weight $2$
Character orbit 2368.g
Analytic conductor $18.909$
Analytic rank $0$
Dimension $6$
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: \(6\)
Coefficient field: 6.0.3356224.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 8x^{4} + 16x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 1184)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{3} - 1) q^{3} + ( - \beta_{5} + \beta_{2} - \beta_1) q^{5} + ( - \beta_{4} + \beta_{3} + 2) q^{7} + (\beta_{4} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} - 1) q^{3} + ( - \beta_{5} + \beta_{2} - \beta_1) q^{5} + ( - \beta_{4} + \beta_{3} + 2) q^{7} + (\beta_{4} + 1) q^{9} + ( - \beta_{4} - 4) q^{11} + (2 \beta_{2} + \beta_1) q^{13} + ( - \beta_{5} - 2 \beta_{2} + \beta_1) q^{15} + ( - 2 \beta_{5} + \beta_{2} + \beta_1) q^{17} - 2 \beta_{5} q^{19} + (3 \beta_{4} + 3 \beta_{3} + 2) q^{21} + ( - 2 \beta_{5} - \beta_{2} - 2 \beta_1) q^{23} + (3 \beta_{4} + 2 \beta_{3} - 6) q^{25} + ( - 2 \beta_{4} - 2 \beta_{3} + 1) q^{27} + ( - 2 \beta_{2} - \beta_1) q^{29} + ( - \beta_{5} + 3 \beta_{2} + 3 \beta_1) q^{31} + (2 \beta_{4} - 4 \beta_{3} + 5) q^{33} + ( - 6 \beta_{5} + 4 \beta_{2}) q^{35} + (\beta_{5} + 3 \beta_{4} + \beta_{3} + \beta_{2} + 2 \beta_1) q^{37} + (3 \beta_{5} - 2 \beta_{2}) q^{39} + ( - 2 \beta_{4} - 3 \beta_{3} + 1) q^{41} + (3 \beta_{2} - \beta_1) q^{43} + (\beta_{5} - 2 \beta_{2} - 3 \beta_1) q^{45} + (3 \beta_{4} + 3 \beta_{3} + 2) q^{47} + ( - \beta_{4} + 5 \beta_{3} + 5) q^{49} + ( - 3 \beta_{2} - 5 \beta_1) q^{51} + ( - 3 \beta_{4} + \beta_{3}) q^{53} + (2 \beta_{5} - \beta_{2} + 6 \beta_1) q^{55} + ( - 2 \beta_{5} - 2 \beta_{2} - 4 \beta_1) q^{57} + (4 \beta_{5} - 3 \beta_{2} - \beta_1) q^{59} + ( - 3 \beta_{5} - 2 \beta_{2} - 2 \beta_1) q^{61} + (2 \beta_{3} - 2) q^{63} + (7 \beta_{4} + 5 \beta_{3} - 5) q^{65} + ( - 6 \beta_{4} - 3 \beta_{3}) q^{67} + ( - 5 \beta_{5} - \beta_{2} - \beta_1) q^{69} + ( - 3 \beta_{4} - \beta_{3} + 10) q^{71} + ( - \beta_{4} + 4 \beta_{3} - 2) q^{73} + ( - 4 \beta_{4} - 4 \beta_{3} + 9) q^{75} + (3 \beta_{4} - 5 \beta_{3} - 4) q^{77} + (4 \beta_{5} - 4 \beta_{2} - 3 \beta_1) q^{79} + ( - \beta_{4} - \beta_{3} - 8) q^{81} + (5 \beta_{4} + 3 \beta_{3} - 4) q^{83} + (6 \beta_{4} - 2 \beta_{3} - 2) q^{85} + ( - 3 \beta_{5} + 2 \beta_{2}) q^{87} + (2 \beta_{5} - 4 \beta_{2} - 6 \beta_1) q^{89} + (2 \beta_{5} + 7 \beta_{2} + 3 \beta_1) q^{91} + (5 \beta_{5} - 4 \beta_{2} - 5 \beta_1) q^{93} + (2 \beta_{4} - 4 \beta_{3} - 2) q^{95} + (4 \beta_{5} + 3 \beta_{2} - 3 \beta_1) q^{97} + ( - 5 \beta_{4} + \beta_{3} - 7) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 4 q^{3} + 14 q^{7} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 4 q^{3} + 14 q^{7} + 6 q^{9} - 24 q^{11} + 18 q^{21} - 32 q^{25} + 2 q^{27} + 22 q^{33} + 2 q^{37} + 18 q^{47} + 40 q^{49} + 2 q^{53} - 8 q^{63} - 20 q^{65} - 6 q^{67} + 58 q^{71} - 4 q^{73} + 46 q^{75} - 34 q^{77} - 50 q^{81} - 18 q^{83} - 16 q^{85} - 20 q^{95} - 40 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 8x^{4} + 16x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{3} + 3\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} + 5\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} + 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} + 4\nu^{2} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{5} + 7\nu^{3} + 12\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -3\beta_{2} + 5\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} - 4\beta_{3} + 12 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 2\beta_{5} + 9\beta_{2} - 23\beta_1 ) / 2 \) 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
2.11491i
2.11491i
1.86081i
1.86081i
0.254102i
0.254102i
0 −2.47283 0 2.75698i 0 −1.58774 0 3.11491 0
961.2 0 −2.47283 0 2.75698i 0 −1.58774 0 3.11491 0
961.3 0 −1.46260 0 4.18421i 0 3.39821 0 −0.860806 0
961.4 0 −1.46260 0 4.18421i 0 3.39821 0 −0.860806 0
961.5 0 1.93543 0 2.42723i 0 5.18953 0 0.745898 0
961.6 0 1.93543 0 2.42723i 0 5.18953 0 0.745898 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 961.6
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.k 6
4.b odd 2 1 2368.2.g.l 6
8.b even 2 1 1184.2.g.f yes 6
8.d odd 2 1 1184.2.g.e 6
37.b even 2 1 inner 2368.2.g.k 6
148.b odd 2 1 2368.2.g.l 6
296.e even 2 1 1184.2.g.f yes 6
296.h odd 2 1 1184.2.g.e 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1184.2.g.e 6 8.d odd 2 1
1184.2.g.e 6 296.h odd 2 1
1184.2.g.f yes 6 8.b even 2 1
1184.2.g.f yes 6 296.e even 2 1
2368.2.g.k 6 1.a even 1 1 trivial
2368.2.g.k 6 37.b even 2 1 inner
2368.2.g.l 6 4.b odd 2 1
2368.2.g.l 6 148.b odd 2 1

Hecke kernels

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

\( T_{3}^{3} + 2T_{3}^{2} - 4T_{3} - 7 \) Copy content Toggle raw display
\( T_{5}^{6} + 31T_{5}^{4} + 281T_{5}^{2} + 784 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( (T^{3} + 2 T^{2} - 4 T - 7)^{2} \) Copy content Toggle raw display
$5$ \( T^{6} + 31 T^{4} + 281 T^{2} + \cdots + 784 \) Copy content Toggle raw display
$7$ \( (T^{3} - 7 T^{2} + 4 T + 28)^{2} \) Copy content Toggle raw display
$11$ \( (T^{3} + 12 T^{2} + 44 T + 49)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} + 35 T^{4} + 277 T^{2} + \cdots + 196 \) Copy content Toggle raw display
$17$ \( T^{6} + 48 T^{4} + 704 T^{2} + \cdots + 3136 \) Copy content Toggle raw display
$19$ \( T^{6} + 44 T^{4} + 336 T^{2} + \cdots + 256 \) Copy content Toggle raw display
$23$ \( T^{6} + 79 T^{4} + 353 T^{2} + \cdots + 16 \) Copy content Toggle raw display
$29$ \( T^{6} + 35 T^{4} + 277 T^{2} + \cdots + 196 \) Copy content Toggle raw display
$31$ \( T^{6} + 107 T^{4} + 3249 T^{2} + \cdots + 21904 \) Copy content Toggle raw display
$37$ \( T^{6} - 2 T^{5} - 17 T^{4} + \cdots + 50653 \) Copy content Toggle raw display
$41$ \( (T^{3} - 46 T + 53)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + 140 T^{4} + 4912 T^{2} + \cdots + 33856 \) Copy content Toggle raw display
$47$ \( (T^{3} - 9 T^{2} - 30 T + 196)^{2} \) Copy content Toggle raw display
$53$ \( (T^{3} - T^{2} - 50 T + 148)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} + 176 T^{4} + 9408 T^{2} + \cdots + 153664 \) Copy content Toggle raw display
$61$ \( T^{6} + 171 T^{4} + 1149 T^{2} + \cdots + 196 \) Copy content Toggle raw display
$67$ \( (T^{3} + 3 T^{2} - 135 T - 756)^{2} \) Copy content Toggle raw display
$71$ \( (T^{3} - 29 T^{2} + 248 T - 644)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} + 2 T^{2} - 100 T - 199)^{2} \) Copy content Toggle raw display
$79$ \( T^{6} + 251 T^{4} + 17973 T^{2} + \cdots + 268324 \) Copy content Toggle raw display
$83$ \( (T^{3} + 9 T^{2} - 76 T + 112)^{2} \) Copy content Toggle raw display
$89$ \( T^{6} + 360 T^{4} + 19856 T^{2} + \cdots + 200704 \) Copy content Toggle raw display
$97$ \( T^{6} + 608 T^{4} + 105984 T^{2} + \cdots + 5271616 \) Copy content Toggle raw display
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