Properties

Label 2352.2.b.l
Level $2352$
Weight $2$
Character orbit 2352.b
Analytic conductor $18.781$
Analytic rank $0$
Dimension $8$
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] = [2352,2,Mod(1567,2352)]
 
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, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2352.1567");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2352.b (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: \(18.7808145554\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.339738624.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{6} + 14x^{4} - 8x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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.

\(f(q)\) \(=\) \( q + q^{3} + (\beta_{6} - \beta_{4}) q^{5} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} + (\beta_{6} - \beta_{4}) q^{5} + q^{9} + (\beta_{6} - \beta_{4} - \beta_{2}) q^{11} + ( - \beta_{4} - 2 \beta_{2}) q^{13} + (\beta_{6} - \beta_{4}) q^{15} + (\beta_{6} - \beta_{4} + 2 \beta_{2}) q^{17} + ( - \beta_{3} + 2 \beta_1) q^{19} + (\beta_{7} - \beta_{6} - \beta_{4} + \beta_{2}) q^{23} + (2 \beta_{5} - \beta_1 - 3) q^{25} + q^{27} + ( - \beta_{3} - \beta_1 + 2) q^{29} + (\beta_{5} + 2 \beta_1 - 4) q^{31} + (\beta_{6} - \beta_{4} - \beta_{2}) q^{33} + (2 \beta_{3} + \beta_1) q^{37} + ( - \beta_{4} - 2 \beta_{2}) q^{39} + (\beta_{7} - \beta_{6} - \beta_{2}) q^{41} + (\beta_{7} + 6 \beta_{4} - 2 \beta_{2}) q^{43} + (\beta_{6} - \beta_{4}) q^{45} + ( - \beta_{5} + 2 \beta_1 - 2) q^{47} + (\beta_{6} - \beta_{4} + 2 \beta_{2}) q^{51} + (2 \beta_{5} + 2 \beta_1 + 2) q^{53} + (2 \beta_{5} + \beta_{3} - 2 \beta_1 - 8) q^{55} + ( - \beta_{3} + 2 \beta_1) q^{57} + ( - \beta_{5} + 2 \beta_{3} + 2 \beta_1 + 6) q^{59} + (2 \beta_{7} + 2 \beta_{6} + \beta_{4}) q^{61} + (\beta_{5} + 2 \beta_{3} - 3 \beta_1 - 2) q^{65} + (2 \beta_{7} - 2 \beta_{4} + 2 \beta_{2}) q^{67} + (\beta_{7} - \beta_{6} - \beta_{4} + \beta_{2}) q^{69} + ( - 2 \beta_{7} - \beta_{6} - 3 \beta_{4} - 3 \beta_{2}) q^{71} + ( - 2 \beta_{7} + 2 \beta_{6} + \beta_{4} - 2 \beta_{2}) q^{73} + (2 \beta_{5} - \beta_1 - 3) q^{75} + ( - \beta_{7} - 4 \beta_{4} - 4 \beta_{2}) q^{79} + q^{81} + ( - \beta_{5} - 3 \beta_{3} - 2 \beta_1) q^{83} + (2 \beta_{5} - 2 \beta_{3} + \beta_1 - 8) q^{85} + ( - \beta_{3} - \beta_1 + 2) q^{87} + (2 \beta_{7} + 3 \beta_{6} + 7 \beta_{4} + 2 \beta_{2}) q^{89} + (\beta_{5} + 2 \beta_1 - 4) q^{93} + ( - 3 \beta_{7} - 2 \beta_{4} - 8 \beta_{2}) q^{95} + (2 \beta_{7} - 2 \beta_{6} - \beta_{4} + 6 \beta_{2}) q^{97} + (\beta_{6} - \beta_{4} - \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{3} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{3} + 8 q^{9} - 24 q^{25} + 8 q^{27} + 16 q^{29} - 32 q^{31} - 16 q^{47} + 16 q^{53} - 64 q^{55} + 48 q^{59} - 16 q^{65} - 24 q^{75} + 8 q^{81} - 64 q^{85} + 16 q^{87} - 32 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{6} + 14x^{4} - 8x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{6} + 20 ) / 14 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -2\nu^{7} + 7\nu^{5} - 28\nu^{3} + 2\nu ) / 14 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{7} + 7\nu^{5} - 21\nu^{3} + 22\nu ) / 7 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -6\nu^{7} + 21\nu^{5} - 70\nu^{3} + 6\nu ) / 14 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -3\nu^{7} + 14\nu^{5} - 49\nu^{3} + 52\nu ) / 7 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{6} + 4\nu^{4} - 12\nu^{2} + 4 ) / 2 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 4\nu^{6} - 14\nu^{4} + 56\nu^{2} - 18 ) / 7 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{5} - \beta_{3} - 2\beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} + \beta_{6} - \beta _1 + 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{4} - 3\beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 3\beta_{7} + 4\beta_{6} + 4\beta _1 - 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -3\beta_{5} + 4\beta_{4} + 7\beta_{3} - 10\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 14\beta _1 - 20 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -5\beta_{5} - 7\beta_{4} + 12\beta_{3} + 17\beta_{2} \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1567.1
−0.662827 0.382683i
−1.60021 + 0.923880i
1.60021 0.923880i
0.662827 + 0.382683i
0.662827 0.382683i
1.60021 + 0.923880i
−1.60021 0.923880i
−0.662827 + 0.382683i
0 1.00000 0 4.29725i 0 0 0 1.00000 0
1567.2 0 1.00000 0 3.21486i 0 0 0 1.00000 0
1567.3 0 1.00000 0 1.68412i 0 0 0 1.00000 0
1567.4 0 1.00000 0 0.601731i 0 0 0 1.00000 0
1567.5 0 1.00000 0 0.601731i 0 0 0 1.00000 0
1567.6 0 1.00000 0 1.68412i 0 0 0 1.00000 0
1567.7 0 1.00000 0 3.21486i 0 0 0 1.00000 0
1567.8 0 1.00000 0 4.29725i 0 0 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1567.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
28.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2352.2.b.l yes 8
3.b odd 2 1 7056.2.b.w 8
4.b odd 2 1 2352.2.b.k 8
7.b odd 2 1 2352.2.b.k 8
7.c even 3 1 2352.2.bl.o 8
7.c even 3 1 2352.2.bl.q 8
7.d odd 6 1 2352.2.bl.r 8
7.d odd 6 1 2352.2.bl.t 8
12.b even 2 1 7056.2.b.x 8
21.c even 2 1 7056.2.b.x 8
28.d even 2 1 inner 2352.2.b.l yes 8
28.f even 6 1 2352.2.bl.o 8
28.f even 6 1 2352.2.bl.q 8
28.g odd 6 1 2352.2.bl.r 8
28.g odd 6 1 2352.2.bl.t 8
84.h odd 2 1 7056.2.b.w 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2352.2.b.k 8 4.b odd 2 1
2352.2.b.k 8 7.b odd 2 1
2352.2.b.l yes 8 1.a even 1 1 trivial
2352.2.b.l yes 8 28.d even 2 1 inner
2352.2.bl.o 8 7.c even 3 1
2352.2.bl.o 8 28.f even 6 1
2352.2.bl.q 8 7.c even 3 1
2352.2.bl.q 8 28.f even 6 1
2352.2.bl.r 8 7.d odd 6 1
2352.2.bl.r 8 28.g odd 6 1
2352.2.bl.t 8 7.d odd 6 1
2352.2.bl.t 8 28.g odd 6 1
7056.2.b.w 8 3.b odd 2 1
7056.2.b.w 8 84.h odd 2 1
7056.2.b.x 8 12.b even 2 1
7056.2.b.x 8 21.c even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2352, [\chi])\):

\( T_{5}^{8} + 32T_{5}^{6} + 284T_{5}^{4} + 640T_{5}^{2} + 196 \) Copy content Toggle raw display
\( T_{11}^{8} + 40T_{11}^{6} + 392T_{11}^{4} + 608T_{11}^{2} + 16 \) Copy content Toggle raw display
\( T_{13}^{4} + 20T_{13}^{2} + 98 \) Copy content Toggle raw display
\( T_{19}^{4} - 40T_{19}^{2} + 96T_{19} - 56 \) Copy content Toggle raw display
\( T_{31}^{4} + 16T_{31}^{3} + 56T_{31}^{2} - 160T_{31} - 824 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T - 1)^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + 32 T^{6} + 284 T^{4} + \cdots + 196 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} + 40 T^{6} + 392 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$13$ \( (T^{4} + 20 T^{2} + 98)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + 64 T^{6} + 860 T^{4} + \cdots + 6724 \) Copy content Toggle raw display
$19$ \( (T^{4} - 40 T^{2} + 96 T - 56)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + 88 T^{6} + 2312 T^{4} + \cdots + 38416 \) Copy content Toggle raw display
$29$ \( (T^{4} - 8 T^{3} - 4 T^{2} + 32 T + 28)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 16 T^{3} + 56 T^{2} - 160 T - 824)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 100 T^{2} + 192 T + 964)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + 80 T^{6} + 1820 T^{4} + \cdots + 6724 \) Copy content Toggle raw display
$43$ \( T^{8} + 368 T^{6} + \cdots + 20214016 \) Copy content Toggle raw display
$47$ \( (T^{4} + 8 T^{3} - 16 T^{2} - 224 T - 392)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 8 T^{3} - 88 T^{2} + 32 T + 784)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 24 T^{3} + 80 T^{2} + 1440 T - 8456)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} + 296 T^{6} + 23636 T^{4} + \cdots + 9604 \) Copy content Toggle raw display
$67$ \( T^{8} + 256 T^{6} + 18176 T^{4} + \cdots + 802816 \) Copy content Toggle raw display
$71$ \( T^{8} + 360 T^{6} + \cdots + 10265616 \) Copy content Toggle raw display
$73$ \( T^{8} + 328 T^{6} + 33236 T^{4} + \cdots + 4866436 \) Copy content Toggle raw display
$79$ \( T^{8} + 304 T^{6} + 24416 T^{4} + \cdots + 430336 \) Copy content Toggle raw display
$83$ \( (T^{4} - 256 T^{2} - 192 T + 12256)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + 832 T^{6} + \cdots + 1766857156 \) Copy content Toggle raw display
$97$ \( T^{8} + 584 T^{6} + \cdots + 325658116 \) Copy content Toggle raw display
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