Properties

Label 1344.2.k.f
Level $1344$
Weight $2$
Character orbit 1344.k
Analytic conductor $10.732$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1344,2,Mod(1217,1344)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1344, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("1344.1217");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1344.k (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: \(10.7318940317\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.342102016.5
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + x^{6} + 4x^{4} + 4x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: no (minimal twist has level 168)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} - \beta_{2} q^{5} + ( - \beta_{5} - 1) q^{7} + ( - \beta_{7} - \beta_{5}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} - \beta_{2} q^{5} + ( - \beta_{5} - 1) q^{7} + ( - \beta_{7} - \beta_{5}) q^{9} + ( - \beta_{7} + \beta_{6} - \beta_{3}) q^{11} + ( - \beta_{7} - \beta_{6} + \beta_{4} - \beta_1) q^{13} + (\beta_{6} + \beta_{5} + \beta_{3} + 1) q^{15} + (\beta_{4} + \beta_{2} + \beta_1) q^{17} + (2 \beta_{7} + 2 \beta_{6} - 3 \beta_{4} + 3 \beta_1) q^{19} + ( - \beta_{6} + 2 \beta_{4} + \beta_{3} + \beta_{2} - \beta_1 + 1) q^{21} - \beta_{3} q^{23} + (\beta_{7} + \beta_{6} + 2 \beta_{5} + 1) q^{25} + ( - \beta_{7} - \beta_{6} + \beta_{4} + 2 \beta_{2}) q^{27} + ( - \beta_{7} + \beta_{6} + 2 \beta_{3}) q^{29} + ( - \beta_{7} - \beta_{6} - 2 \beta_{4} + 2 \beta_1) q^{31} + ( - 2 \beta_{7} - 2 \beta_{6} - \beta_{4} + \beta_{2} - \beta_1) q^{33} + ( - \beta_{4} - \beta_{3} + 2 \beta_{2} - \beta_1) q^{35} + 2 q^{37} + (\beta_{6} + \beta_{5} - 2 \beta_{3} + 1) q^{39} + (\beta_{4} - 3 \beta_{2} + \beta_1) q^{41} + 4 q^{43} + (\beta_{7} + \beta_{6} - 4 \beta_{4} + \beta_{2} + 2 \beta_1) q^{45} + (2 \beta_{4} + 4 \beta_{2} + 2 \beta_1) q^{47} + (3 \beta_{7} + 3 \beta_{6} + 2 \beta_{5} - 2 \beta_{4} + 2 \beta_1 + 3) q^{49} + ( - \beta_{7} - \beta_{6} - 2 \beta_{5} - \beta_{3} + 2) q^{51} + (\beta_{7} - \beta_{6} - 2 \beta_{3}) q^{53} + (2 \beta_{7} + 2 \beta_{6} - 4 \beta_{4} + 4 \beta_1) q^{55} + ( - \beta_{7} - 2 \beta_{6} - 3 \beta_{5} + 4 \beta_{3} - 5) q^{57} + (3 \beta_{4} + 3 \beta_1) q^{59} + (3 \beta_{7} + 3 \beta_{6} - 3 \beta_{4} + 3 \beta_1) q^{61} + (2 \beta_{7} + \beta_{6} - 2 \beta_{3} + 2 \beta_1 + 4) q^{63} + (\beta_{7} - \beta_{6}) q^{65} + (2 \beta_{7} + 2 \beta_{6} + 4 \beta_{5}) q^{67} + ( - \beta_{7} - \beta_{6} + \beta_{4} - \beta_{2} - \beta_1) q^{69} + ( - \beta_{7} + \beta_{6} - 5 \beta_{3}) q^{71} + (4 \beta_{4} - 4 \beta_1) q^{73} + (\beta_{7} + \beta_{6} - 4 \beta_{4} - 2 \beta_{2} + \beta_1) q^{75} + (3 \beta_{4} - 4 \beta_{3} + \beta_{2} + 3 \beta_1) q^{77} + ( - \beta_{7} - \beta_{6} - 2 \beta_{5} + 6) q^{79} + ( - \beta_{7} - \beta_{6} - 2 \beta_{5} - 4 \beta_{3} - 1) q^{81} + (\beta_{4} + \beta_1) q^{83} - 4 q^{85} + (\beta_{7} + \beta_{6} - 4 \beta_{4} + 4 \beta_{2} + 2 \beta_1) q^{87} + ( - 5 \beta_{4} + \beta_{2} - 5 \beta_1) q^{89} + ( - 2 \beta_{7} - 2 \beta_{6} - 2 \beta_{5} - \beta_{4} + \beta_1 + 2) q^{91} + ( - 3 \beta_{7} + \beta_{6} - 2 \beta_{5} - 2 \beta_{3} - 8) q^{93} + ( - 3 \beta_{7} + 3 \beta_{6} + 2 \beta_{3}) q^{95} + (2 \beta_{7} + 2 \beta_{6} + 2 \beta_{4} - 2 \beta_1) q^{97} + ( - \beta_{7} + \beta_{6} - 5 \beta_{3} - 8) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{7} + 4 q^{9} + 4 q^{15} + 8 q^{21} + 16 q^{37} + 4 q^{39} + 32 q^{43} + 16 q^{49} + 24 q^{51} - 28 q^{57} + 32 q^{63} - 16 q^{67} + 56 q^{79} - 32 q^{85} + 24 q^{91} - 56 q^{93} - 64 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( ( -\nu^{7} + \nu^{6} + \nu^{5} + 3\nu^{4} - 6\nu^{3} + 10\nu^{2} + 8\nu + 8 ) / 16 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} + 3\nu^{5} + 2\nu^{3} ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{7} + \nu^{5} + 2\nu^{3} ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{7} - \nu^{6} + \nu^{5} - 3\nu^{4} - 6\nu^{3} - 10\nu^{2} + 8\nu - 8 ) / 16 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{6} - 3\nu^{4} + 6\nu^{2} - 8 ) / 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{7} - 2\nu^{6} + 3\nu^{5} + 2\nu^{4} + 10\nu^{3} - 12\nu^{2} + 24\nu ) / 16 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{7} - 2\nu^{6} - 3\nu^{5} + 2\nu^{4} - 10\nu^{3} - 12\nu^{2} - 24\nu ) / 16 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{7} + \beta_{6} + \beta_{4} - \beta_{3} - \beta_{2} + \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} - \beta_{4} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{7} + \beta_{6} - 3\beta_{4} + 3\beta_{3} - \beta_{2} - 3\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2\beta_{7} + 2\beta_{6} - \beta_{5} - 3\beta_{4} + 3\beta _1 - 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( \beta_{7} - \beta_{6} + 3\beta_{4} + 5\beta_{3} + 9\beta_{2} + 3\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -6\beta_{7} - 6\beta_{6} - 7\beta_{5} + 3\beta_{4} - 3\beta _1 - 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -\beta_{7} + \beta_{6} - 3\beta_{4} - 21\beta_{3} + 7\beta_{2} - 3\beta_1 ) / 4 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1344\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(449\) \(577\) \(1093\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1217.1
−0.599676 + 1.28078i
−0.599676 1.28078i
−1.17915 + 0.780776i
−1.17915 0.780776i
1.17915 0.780776i
1.17915 + 0.780776i
0.599676 1.28078i
0.599676 + 1.28078i
0 −1.66757 0.468213i 0 0.936426 0 1.56155 + 2.13578i 0 2.56155 + 1.56155i 0
1217.2 0 −1.66757 + 0.468213i 0 0.936426 0 1.56155 2.13578i 0 2.56155 1.56155i 0
1217.3 0 −0.848071 1.51022i 0 −3.02045 0 −2.56155 + 0.662153i 0 −1.56155 + 2.56155i 0
1217.4 0 −0.848071 + 1.51022i 0 −3.02045 0 −2.56155 0.662153i 0 −1.56155 2.56155i 0
1217.5 0 0.848071 1.51022i 0 3.02045 0 −2.56155 + 0.662153i 0 −1.56155 2.56155i 0
1217.6 0 0.848071 + 1.51022i 0 3.02045 0 −2.56155 0.662153i 0 −1.56155 + 2.56155i 0
1217.7 0 1.66757 0.468213i 0 −0.936426 0 1.56155 + 2.13578i 0 2.56155 1.56155i 0
1217.8 0 1.66757 + 0.468213i 0 −0.936426 0 1.56155 2.13578i 0 2.56155 + 1.56155i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1217.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1344.2.k.f 8
3.b odd 2 1 inner 1344.2.k.f 8
4.b odd 2 1 1344.2.k.i 8
7.b odd 2 1 inner 1344.2.k.f 8
8.b even 2 1 168.2.k.a 8
8.d odd 2 1 336.2.k.c 8
12.b even 2 1 1344.2.k.i 8
21.c even 2 1 inner 1344.2.k.f 8
24.f even 2 1 336.2.k.c 8
24.h odd 2 1 168.2.k.a 8
28.d even 2 1 1344.2.k.i 8
56.e even 2 1 336.2.k.c 8
56.h odd 2 1 168.2.k.a 8
56.j odd 6 2 1176.2.u.a 16
56.p even 6 2 1176.2.u.a 16
84.h odd 2 1 1344.2.k.i 8
168.e odd 2 1 336.2.k.c 8
168.i even 2 1 168.2.k.a 8
168.s odd 6 2 1176.2.u.a 16
168.ba even 6 2 1176.2.u.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.2.k.a 8 8.b even 2 1
168.2.k.a 8 24.h odd 2 1
168.2.k.a 8 56.h odd 2 1
168.2.k.a 8 168.i even 2 1
336.2.k.c 8 8.d odd 2 1
336.2.k.c 8 24.f even 2 1
336.2.k.c 8 56.e even 2 1
336.2.k.c 8 168.e odd 2 1
1176.2.u.a 16 56.j odd 6 2
1176.2.u.a 16 56.p even 6 2
1176.2.u.a 16 168.s odd 6 2
1176.2.u.a 16 168.ba even 6 2
1344.2.k.f 8 1.a even 1 1 trivial
1344.2.k.f 8 3.b odd 2 1 inner
1344.2.k.f 8 7.b odd 2 1 inner
1344.2.k.f 8 21.c even 2 1 inner
1344.2.k.i 8 4.b odd 2 1
1344.2.k.i 8 12.b even 2 1
1344.2.k.i 8 28.d even 2 1
1344.2.k.i 8 84.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1344, [\chi])\):

\( T_{5}^{4} - 10T_{5}^{2} + 8 \) Copy content Toggle raw display
\( T_{43} - 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - 2 T^{6} + 2 T^{4} - 18 T^{2} + \cdots + 81 \) Copy content Toggle raw display
$5$ \( (T^{4} - 10 T^{2} + 8)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} + 2 T^{3} - 2 T^{2} + 14 T + 49)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 36 T^{2} + 256)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 14 T^{2} + 32)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 20 T^{2} + 32)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 74 T^{2} + 1352)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 4)^{4} \) Copy content Toggle raw display
$29$ \( (T^{4} + 84 T^{2} + 64)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 92 T^{2} + 2048)^{2} \) Copy content Toggle raw display
$37$ \( (T - 2)^{8} \) Copy content Toggle raw display
$41$ \( (T^{4} - 116 T^{2} + 32)^{2} \) Copy content Toggle raw display
$43$ \( (T - 4)^{8} \) Copy content Toggle raw display
$47$ \( (T^{4} - 184 T^{2} + 8192)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 84 T^{2} + 64)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 126 T^{2} + 2592)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 126 T^{2} + 2592)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 4 T - 64)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} + 196 T^{2} + 4096)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 160 T^{2} + 2048)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 14 T + 32)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} - 14 T^{2} + 32)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} - 380 T^{2} + 32768)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 184 T^{2} + 8192)^{2} \) Copy content Toggle raw display
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