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Properties

Label 1344.3.d.e
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: \(\Q(\sqrt{-2}, \sqrt{7})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 8x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 42)
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.

\(f(q)\) \(=\) \( q + (\beta_{3} + \beta_1) q^{3} + (\beta_{2} - 2 \beta_1) q^{5} + \beta_{3} q^{7} + (2 \beta_{2} + 5) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} + \beta_1) q^{3} + (\beta_{2} - 2 \beta_1) q^{5} + \beta_{3} q^{7} + (2 \beta_{2} + 5) q^{9} + ( - 2 \beta_{2} - 5 \beta_1) q^{11} + (4 \beta_{3} - 10) q^{13} + ( - 2 \beta_{3} - 2 \beta_{2} + \cdots + 4) q^{15}+ \cdots + (20 \beta_{3} - 10 \beta_{2} + \cdots + 56) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 20 q^{9} - 40 q^{13} + 16 q^{15} - 64 q^{19} + 28 q^{21} + 12 q^{25} + 128 q^{31} + 40 q^{33} - 80 q^{37} + 112 q^{39} + 80 q^{43} - 112 q^{45} + 28 q^{49} + 16 q^{51} + 32 q^{55} + 56 q^{61} + 240 q^{67} + 8 q^{69} + 120 q^{73} + 224 q^{75} + 128 q^{79} - 124 q^{81} + 248 q^{85} + 160 q^{87} + 112 q^{91} + 280 q^{93} - 360 q^{97} + 224 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 8x^{2} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} + 5\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 11\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} + 4 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{2} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -5\beta_{2} + 11\beta_1 ) / 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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
2.57794i
− 2.57794i
− 1.16372i
1.16372i
0 −2.64575 − 1.41421i 0 6.57008i 0 −2.64575 0 5.00000 + 7.48331i 0
449.2 0 −2.64575 + 1.41421i 0 − 6.57008i 0 −2.64575 0 5.00000 − 7.48331i 0
449.3 0 2.64575 − 1.41421i 0 − 0.913230i 0 2.64575 0 5.00000 − 7.48331i 0
449.4 0 2.64575 + 1.41421i 0 0.913230i 0 2.64575 0 5.00000 + 7.48331i 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.e 4
3.b odd 2 1 inner 1344.3.d.e 4
4.b odd 2 1 1344.3.d.c 4
8.b even 2 1 336.3.d.b 4
8.d odd 2 1 42.3.b.a ✓ 4
12.b even 2 1 1344.3.d.c 4
24.f even 2 1 42.3.b.a ✓ 4
24.h odd 2 1 336.3.d.b 4
40.e odd 2 1 1050.3.e.a 4
40.k even 4 2 1050.3.c.a 8
56.e even 2 1 294.3.b.h 4
56.k odd 6 2 294.3.h.d 8
56.m even 6 2 294.3.h.g 8
72.l even 6 2 1134.3.q.a 8
72.p odd 6 2 1134.3.q.a 8
120.m even 2 1 1050.3.e.a 4
120.q odd 4 2 1050.3.c.a 8
168.e odd 2 1 294.3.b.h 4
168.v even 6 2 294.3.h.d 8
168.be odd 6 2 294.3.h.g 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.3.b.a ✓ 4 8.d odd 2 1
42.3.b.a ✓ 4 24.f even 2 1
294.3.b.h 4 56.e even 2 1
294.3.b.h 4 168.e odd 2 1
294.3.h.d 8 56.k odd 6 2
294.3.h.d 8 168.v even 6 2
294.3.h.g 8 56.m even 6 2
294.3.h.g 8 168.be odd 6 2
336.3.d.b 4 8.b even 2 1
336.3.d.b 4 24.h odd 2 1
1050.3.c.a 8 40.k even 4 2
1050.3.c.a 8 120.q odd 4 2
1050.3.e.a 4 40.e odd 2 1
1050.3.e.a 4 120.m even 2 1
1134.3.q.a 8 72.l even 6 2
1134.3.q.a 8 72.p odd 6 2
1344.3.d.c 4 4.b odd 2 1
1344.3.d.c 4 12.b even 2 1
1344.3.d.e 4 1.a even 1 1 trivial
1344.3.d.e 4 3.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 \(S_{3}^{\mathrm{new}}(1344, [\chi])\):

\( T_{5}^{4} + 44T_{5}^{2} + 36 \) Copy content Toggle raw display
\( T_{19} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - 10T^{2} + 81 \) Copy content Toggle raw display
$5$ \( T^{4} + 44T^{2} + 36 \) Copy content Toggle raw display
$7$ \( (T^{2} - 7)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + 212T^{2} + 36 \) Copy content Toggle raw display
$13$ \( (T^{2} + 20 T - 12)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 716 T^{2} + 116964 \) Copy content Toggle raw display
$19$ \( (T + 16)^{4} \) Copy content Toggle raw display
$23$ \( T^{4} + 2804 T^{2} + 1954404 \) Copy content Toggle raw display
$29$ \( T^{4} + 1712 T^{2} + 553536 \) Copy content Toggle raw display
$31$ \( (T^{2} - 64 T + 324)^{2} \) Copy content Toggle raw display
$37$ \( (T + 20)^{4} \) Copy content Toggle raw display
$41$ \( T^{4} + 5868 T^{2} + 443556 \) Copy content Toggle raw display
$43$ \( (T^{2} - 40 T - 608)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 72)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 2592)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 3392 T^{2} + 9216 \) Copy content Toggle raw display
$61$ \( (T^{2} - 28 T - 2604)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 120 T + 3488)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + 7988 T^{2} + 2232036 \) Copy content Toggle raw display
$73$ \( (T^{2} - 60 T - 892)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 64 T - 1776)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 21200 T^{2} + 360000 \) Copy content Toggle raw display
$89$ \( T^{4} + 3852 T^{2} + 2802276 \) Copy content Toggle raw display
$97$ \( (T^{2} + 180 T + 7652)^{2} \) Copy content Toggle raw display
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