Properties

Label 1344.3.d.b
Level $1344$
Weight $3$
Character orbit 1344.d
Analytic conductor $36.621$
Analytic rank $0$
Dimension $4$
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(449,1344)]
 
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, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1344.449");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1344.d (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: \(4\)
Coefficient field: 4.0.65856.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 14x^{2} + 21 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{3} - \beta_{2} - \beta_1) q^{3} + (2 \beta_{3} - \beta_{2} + 1) q^{5} + \beta_{2} q^{7} + ( - 2 \beta_{3} + 2 \beta_{2} - \beta_1 - 6) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} - \beta_{2} - \beta_1) q^{3} + (2 \beta_{3} - \beta_{2} + 1) q^{5} + \beta_{2} q^{7} + ( - 2 \beta_{3} + 2 \beta_{2} - \beta_1 - 6) q^{9} + 2 \beta_1 q^{11} + ( - \beta_{2} + 9) q^{13} + ( - 4 \beta_{3} - 2 \beta_{2} + \beta_1 - 9) q^{15} + (4 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 2) q^{17} + ( - 5 \beta_{2} + 3) q^{19} + (\beta_{3} - \beta_{2} + 2 \beta_1 - 3) q^{21} + (4 \beta_{3} - 2 \beta_{2} - 8 \beta_1 + 2) q^{23} + ( - 10 \beta_{2} - 3) q^{25} + (\beta_{3} + 8 \beta_{2} + 5 \beta_1 + 3) q^{27} + (4 \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 2) q^{29} + ( - 2 \beta_{2} - 34) q^{31} + ( - 2 \beta_{3} - 4 \beta_{2} + 2 \beta_1 + 6) q^{33} + (6 \beta_{3} - 3 \beta_{2} - 2 \beta_1 + 3) q^{35} + ( - 14 \beta_{2} - 4) q^{37} + (8 \beta_{3} - 8 \beta_{2} - 11 \beta_1 + 3) q^{39} + (16 \beta_{3} - 8 \beta_{2} - 22 \beta_1 + 8) q^{41} + ( - 6 \beta_{2} - 40) q^{43} + ( - 4 \beta_{3} + 13 \beta_{2} - 2 \beta_1 + 33) q^{45} + ( - 8 \beta_{3} + 4 \beta_{2} - 8 \beta_1 - 4) q^{47} + 7 q^{49} + ( - 6 \beta_{3} - 24) q^{51} + (32 \beta_{3} - 16 \beta_{2} - 10 \beta_1 + 16) q^{53} + ( - 2 \beta_{2} - 14) q^{55} + ( - 2 \beta_{3} + 2 \beta_{2} - 13 \beta_1 + 15) q^{57} + ( - 2 \beta_{3} + \beta_{2} - 26 \beta_1 - 1) q^{59} + ( - 7 \beta_{2} + 39) q^{61} + ( - 8 \beta_{3} - \beta_{2} + 5 \beta_1 + 3) q^{63} + (12 \beta_{3} - 6 \beta_{2} + 2 \beta_1 + 6) q^{65} + ( - 8 \beta_{2} - 6) q^{67} + (12 \beta_{2} - 6 \beta_1 - 42) q^{69} + ( - 12 \beta_{3} + 6 \beta_{2} - 18 \beta_1 - 6) q^{71} + (26 \beta_{2} - 8) q^{73} + ( - 13 \beta_{3} + 13 \beta_{2} - 17 \beta_1 + 30) q^{75} + (4 \beta_{3} - 2 \beta_{2} - 6 \beta_1 + 2) q^{77} + (36 \beta_{2} - 32) q^{79} + (4 \beta_{3} - 22 \beta_{2} + 20 \beta_1 - 15) q^{81} + ( - 18 \beta_{3} + 9 \beta_{2} - 18 \beta_1 - 9) q^{83} + ( - 18 \beta_{2} - 42) q^{85} + ( - 10 \beta_{3} - 8 \beta_{2} + 4 \beta_1 - 12) q^{87} + ( - 32 \beta_{3} + 16 \beta_{2} + 42 \beta_1 - 16) q^{89} + (9 \beta_{2} - 7) q^{91} + ( - 36 \beta_{3} + 36 \beta_{2} + 30 \beta_1 + 6) q^{93} + ( - 24 \beta_{3} + 12 \beta_{2} + 10 \beta_1 - 12) q^{95} + (8 \beta_{2} - 2) q^{97} + (4 \beta_{3} - 4 \beta_{2} - 16 \beta_1 + 30) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{3} - 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{3} - 20 q^{9} + 36 q^{13} - 28 q^{15} + 12 q^{19} - 14 q^{21} - 12 q^{25} + 10 q^{27} - 136 q^{31} + 28 q^{33} - 16 q^{37} - 4 q^{39} - 160 q^{43} + 140 q^{45} + 28 q^{49} - 84 q^{51} - 56 q^{55} + 64 q^{57} + 156 q^{61} + 28 q^{63} - 24 q^{67} - 168 q^{69} - 32 q^{73} + 146 q^{75} - 128 q^{79} - 68 q^{81} - 168 q^{85} - 28 q^{87} - 28 q^{91} + 96 q^{93} - 8 q^{97} + 112 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 14x^{2} + 21 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} + 7 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} + \nu^{2} + 13\nu + 5 ) / 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} - 7 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 4\beta_{3} - 2\beta_{2} - 13\beta _1 + 2 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
449.1
1.30710i
1.30710i
3.50592i
3.50592i
0 −1.82288 2.38267i 0 7.37953i 0 2.64575 0 −2.35425 + 8.68663i 0
449.2 0 −1.82288 + 2.38267i 0 7.37953i 0 2.64575 0 −2.35425 8.68663i 0
449.3 0 0.822876 2.88494i 0 1.24197i 0 −2.64575 0 −7.64575 4.74789i 0
449.4 0 0.822876 + 2.88494i 0 1.24197i 0 −2.64575 0 −7.64575 + 4.74789i 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
3.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.d.b 4
3.b odd 2 1 inner 1344.3.d.b 4
4.b odd 2 1 1344.3.d.f 4
8.b even 2 1 336.3.d.c 4
8.d odd 2 1 21.3.b.a 4
12.b even 2 1 1344.3.d.f 4
24.f even 2 1 21.3.b.a 4
24.h odd 2 1 336.3.d.c 4
40.e odd 2 1 525.3.c.a 4
40.k even 4 2 525.3.f.a 8
56.e even 2 1 147.3.b.f 4
56.k odd 6 2 147.3.h.e 8
56.m even 6 2 147.3.h.c 8
72.l even 6 2 567.3.r.c 8
72.p odd 6 2 567.3.r.c 8
120.m even 2 1 525.3.c.a 4
120.q odd 4 2 525.3.f.a 8
168.e odd 2 1 147.3.b.f 4
168.v even 6 2 147.3.h.e 8
168.be odd 6 2 147.3.h.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.3.b.a 4 8.d odd 2 1
21.3.b.a 4 24.f even 2 1
147.3.b.f 4 56.e even 2 1
147.3.b.f 4 168.e odd 2 1
147.3.h.c 8 56.m even 6 2
147.3.h.c 8 168.be odd 6 2
147.3.h.e 8 56.k odd 6 2
147.3.h.e 8 168.v even 6 2
336.3.d.c 4 8.b even 2 1
336.3.d.c 4 24.h odd 2 1
525.3.c.a 4 40.e odd 2 1
525.3.c.a 4 120.m even 2 1
525.3.f.a 8 40.k even 4 2
525.3.f.a 8 120.q odd 4 2
567.3.r.c 8 72.l even 6 2
567.3.r.c 8 72.p odd 6 2
1344.3.d.b 4 1.a even 1 1 trivial
1344.3.d.b 4 3.b odd 2 1 inner
1344.3.d.f 4 4.b odd 2 1
1344.3.d.f 4 12.b even 2 1

Hecke kernels

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

\( T_{5}^{4} + 56T_{5}^{2} + 84 \) Copy content Toggle raw display
\( T_{19}^{2} - 6T_{19} - 166 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 2 T^{3} + 12 T^{2} + 18 T + 81 \) Copy content Toggle raw display
$5$ \( T^{4} + 56T^{2} + 84 \) Copy content Toggle raw display
$7$ \( (T^{2} - 7)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + 56T^{2} + 336 \) Copy content Toggle raw display
$13$ \( (T^{2} - 18 T + 74)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 168T^{2} + 3024 \) Copy content Toggle raw display
$19$ \( (T^{2} - 6 T - 166)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 672 T^{2} + 12096 \) Copy content Toggle raw display
$29$ \( T^{4} + 392 T^{2} + 27216 \) Copy content Toggle raw display
$31$ \( (T^{2} + 68 T + 1128)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 8 T - 1356)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 5432 T^{2} + \cdots + 4139856 \) Copy content Toggle raw display
$43$ \( (T^{2} + 80 T + 1348)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 2688 T^{2} + \cdots + 1741824 \) Copy content Toggle raw display
$53$ \( T^{4} + 11256 T^{2} + \cdots + 2543184 \) Copy content Toggle raw display
$59$ \( T^{4} + 10248 T^{2} + \cdots + 14606676 \) Copy content Toggle raw display
$61$ \( (T^{2} - 78 T + 1178)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 12 T - 412)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + 9576 T^{2} + \cdots + 22888656 \) Copy content Toggle raw display
$73$ \( (T^{2} + 16 T - 4668)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 64 T - 8048)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 13608 T^{2} + \cdots + 44641044 \) Copy content Toggle raw display
$89$ \( T^{4} + 20216 T^{2} + \cdots + 64754256 \) Copy content Toggle raw display
$97$ \( (T^{2} + 4 T - 444)^{2} \) Copy content Toggle raw display
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