Properties

Label 1344.3.f.c
Level $1344$
Weight $3$
Character orbit 1344.f
Analytic conductor $36.621$
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] = [1344,3,Mod(769,1344)]
 
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, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1344.769");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1344.f (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: \(36.6213475300\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 21)
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 \(\beta = \sqrt{-3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{3} - 4 \beta q^{5} + ( - 4 \beta + 1) q^{7} - 3 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{3} - 4 \beta q^{5} + ( - 4 \beta + 1) q^{7} - 3 q^{9} - 10 q^{11} + 4 \beta q^{13} + 12 q^{15} - 12 \beta q^{19} + (\beta + 12) q^{21} - 14 q^{23} - 23 q^{25} - 3 \beta q^{27} + 38 q^{29} - 16 \beta q^{31} - 10 \beta q^{33} + ( - 4 \beta - 48) q^{35} - 26 q^{37} - 12 q^{39} + 40 \beta q^{41} - 26 q^{43} + 12 \beta q^{45} - 16 \beta q^{47} + ( - 8 \beta - 47) q^{49} - 10 q^{53} + 40 \beta q^{55} + 36 q^{57} + 44 \beta q^{59} + 20 \beta q^{61} + (12 \beta - 3) q^{63} + 48 q^{65} - 74 q^{67} - 14 \beta q^{69} - 62 q^{71} + 24 \beta q^{73} - 23 \beta q^{75} + (40 \beta - 10) q^{77} - 46 q^{79} + 9 q^{81} + 52 \beta q^{83} + 38 \beta q^{87} + 24 \beta q^{89} + (4 \beta + 48) q^{91} + 48 q^{93} - 144 q^{95} - 32 \beta q^{97} + 30 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{7} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{7} - 6 q^{9} - 20 q^{11} + 24 q^{15} + 24 q^{21} - 28 q^{23} - 46 q^{25} + 76 q^{29} - 96 q^{35} - 52 q^{37} - 24 q^{39} - 52 q^{43} - 94 q^{49} - 20 q^{53} + 72 q^{57} - 6 q^{63} + 96 q^{65} - 148 q^{67} - 124 q^{71} - 20 q^{77} - 92 q^{79} + 18 q^{81} + 96 q^{91} + 96 q^{93} - 288 q^{95} + 60 q^{99}+O(q^{100}) \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
769.1
0.500000 0.866025i
0.500000 + 0.866025i
0 1.73205i 0 6.92820i 0 1.00000 + 6.92820i 0 −3.00000 0
769.2 0 1.73205i 0 6.92820i 0 1.00000 6.92820i 0 −3.00000 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
7.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1344.3.f.c 2
4.b odd 2 1 1344.3.f.b 2
7.b odd 2 1 inner 1344.3.f.c 2
8.b even 2 1 21.3.d.a 2
8.d odd 2 1 336.3.f.a 2
24.f even 2 1 1008.3.f.d 2
24.h odd 2 1 63.3.d.b 2
28.d even 2 1 1344.3.f.b 2
40.f even 2 1 525.3.h.a 2
40.i odd 4 2 525.3.e.a 4
56.e even 2 1 336.3.f.a 2
56.h odd 2 1 21.3.d.a 2
56.j odd 6 1 147.3.f.b 2
56.j odd 6 1 147.3.f.d 2
56.p even 6 1 147.3.f.b 2
56.p even 6 1 147.3.f.d 2
168.e odd 2 1 1008.3.f.d 2
168.i even 2 1 63.3.d.b 2
168.s odd 6 1 441.3.m.d 2
168.s odd 6 1 441.3.m.f 2
168.ba even 6 1 441.3.m.d 2
168.ba even 6 1 441.3.m.f 2
280.c odd 2 1 525.3.h.a 2
280.s even 4 2 525.3.e.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.3.d.a 2 8.b even 2 1
21.3.d.a 2 56.h odd 2 1
63.3.d.b 2 24.h odd 2 1
63.3.d.b 2 168.i even 2 1
147.3.f.b 2 56.j odd 6 1
147.3.f.b 2 56.p even 6 1
147.3.f.d 2 56.j odd 6 1
147.3.f.d 2 56.p even 6 1
336.3.f.a 2 8.d odd 2 1
336.3.f.a 2 56.e even 2 1
441.3.m.d 2 168.s odd 6 1
441.3.m.d 2 168.ba even 6 1
441.3.m.f 2 168.s odd 6 1
441.3.m.f 2 168.ba even 6 1
525.3.e.a 4 40.i odd 4 2
525.3.e.a 4 280.s even 4 2
525.3.h.a 2 40.f even 2 1
525.3.h.a 2 280.c odd 2 1
1008.3.f.d 2 24.f even 2 1
1008.3.f.d 2 168.e odd 2 1
1344.3.f.b 2 4.b odd 2 1
1344.3.f.b 2 28.d even 2 1
1344.3.f.c 2 1.a even 1 1 trivial
1344.3.f.c 2 7.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1344, [\chi])\):

\( T_{5}^{2} + 48 \) Copy content Toggle raw display
\( T_{11} + 10 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 3 \) Copy content Toggle raw display
$5$ \( T^{2} + 48 \) Copy content Toggle raw display
$7$ \( T^{2} - 2T + 49 \) Copy content Toggle raw display
$11$ \( (T + 10)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 48 \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 432 \) Copy content Toggle raw display
$23$ \( (T + 14)^{2} \) Copy content Toggle raw display
$29$ \( (T - 38)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 768 \) Copy content Toggle raw display
$37$ \( (T + 26)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 4800 \) Copy content Toggle raw display
$43$ \( (T + 26)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 768 \) Copy content Toggle raw display
$53$ \( (T + 10)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 5808 \) Copy content Toggle raw display
$61$ \( T^{2} + 1200 \) Copy content Toggle raw display
$67$ \( (T + 74)^{2} \) Copy content Toggle raw display
$71$ \( (T + 62)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 1728 \) Copy content Toggle raw display
$79$ \( (T + 46)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 8112 \) Copy content Toggle raw display
$89$ \( T^{2} + 1728 \) Copy content Toggle raw display
$97$ \( T^{2} + 3072 \) Copy content Toggle raw display
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