Properties

Label 1344.4.a.be
Level $1344$
Weight $4$
Character orbit 1344.a
Self dual yes
Analytic conductor $79.299$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1344,4,Mod(1,1344)]
 
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, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1344.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1344.a (trivial)

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: yes
Analytic conductor: \(79.2985670477\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{137}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 34 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 672)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(\beta = \sqrt{137}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 3 q^{3} + ( - \beta - 5) q^{5} + 7 q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 3 q^{3} + ( - \beta - 5) q^{5} + 7 q^{7} + 9 q^{9} + ( - \beta - 11) q^{11} + (2 \beta - 12) q^{13} + (3 \beta + 15) q^{15} + ( - \beta + 15) q^{17} + ( - 4 \beta - 16) q^{19} - 21 q^{21} + (3 \beta + 41) q^{23} + (10 \beta + 37) q^{25} - 27 q^{27} + (20 \beta - 18) q^{29} + (8 \beta + 56) q^{31} + (3 \beta + 33) q^{33} + ( - 7 \beta - 35) q^{35} + ( - 14 \beta - 24) q^{37} + ( - 6 \beta + 36) q^{39} + (\beta - 35) q^{41} + (20 \beta - 20) q^{43} + ( - 9 \beta - 45) q^{45} + ( - 34 \beta + 210) q^{47} + 49 q^{49} + (3 \beta - 45) q^{51} + (18 \beta - 88) q^{53} + (16 \beta + 192) q^{55} + (12 \beta + 48) q^{57} + ( - 2 \beta - 202) q^{59} + (40 \beta + 78) q^{61} + 63 q^{63} + (2 \beta - 214) q^{65} + (30 \beta + 2) q^{67} + ( - 9 \beta - 123) q^{69} + (5 \beta + 407) q^{71} + (46 \beta - 108) q^{73} + ( - 30 \beta - 111) q^{75} + ( - 7 \beta - 77) q^{77} + ( - 22 \beta + 790) q^{79} + 81 q^{81} + (4 \beta - 232) q^{83} + ( - 10 \beta + 62) q^{85} + ( - 60 \beta + 54) q^{87} + ( - 87 \beta - 579) q^{89} + (14 \beta - 84) q^{91} + ( - 24 \beta - 168) q^{93} + (36 \beta + 628) q^{95} + ( - 62 \beta - 880) q^{97} + ( - 9 \beta - 99) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{3} - 10 q^{5} + 14 q^{7} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 6 q^{3} - 10 q^{5} + 14 q^{7} + 18 q^{9} - 22 q^{11} - 24 q^{13} + 30 q^{15} + 30 q^{17} - 32 q^{19} - 42 q^{21} + 82 q^{23} + 74 q^{25} - 54 q^{27} - 36 q^{29} + 112 q^{31} + 66 q^{33} - 70 q^{35} - 48 q^{37} + 72 q^{39} - 70 q^{41} - 40 q^{43} - 90 q^{45} + 420 q^{47} + 98 q^{49} - 90 q^{51} - 176 q^{53} + 384 q^{55} + 96 q^{57} - 404 q^{59} + 156 q^{61} + 126 q^{63} - 428 q^{65} + 4 q^{67} - 246 q^{69} + 814 q^{71} - 216 q^{73} - 222 q^{75} - 154 q^{77} + 1580 q^{79} + 162 q^{81} - 464 q^{83} + 124 q^{85} + 108 q^{87} - 1158 q^{89} - 168 q^{91} - 336 q^{93} + 1256 q^{95} - 1760 q^{97} - 198 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
6.35235
−5.35235
0 −3.00000 0 −16.7047 0 7.00000 0 9.00000 0
1.2 0 −3.00000 0 6.70470 0 7.00000 0 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(1\)
\(7\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1344.4.a.be 2
4.b odd 2 1 1344.4.a.bm 2
8.b even 2 1 672.4.a.m yes 2
8.d odd 2 1 672.4.a.h 2
24.f even 2 1 2016.4.a.i 2
24.h odd 2 1 2016.4.a.j 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
672.4.a.h 2 8.d odd 2 1
672.4.a.m yes 2 8.b even 2 1
1344.4.a.be 2 1.a even 1 1 trivial
1344.4.a.bm 2 4.b odd 2 1
2016.4.a.i 2 24.f even 2 1
2016.4.a.j 2 24.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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1344))\):

\( T_{5}^{2} + 10T_{5} - 112 \) Copy content Toggle raw display
\( T_{11}^{2} + 22T_{11} - 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T + 3)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 10T - 112 \) Copy content Toggle raw display
$7$ \( (T - 7)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 22T - 16 \) Copy content Toggle raw display
$13$ \( T^{2} + 24T - 404 \) Copy content Toggle raw display
$17$ \( T^{2} - 30T + 88 \) Copy content Toggle raw display
$19$ \( T^{2} + 32T - 1936 \) Copy content Toggle raw display
$23$ \( T^{2} - 82T + 448 \) Copy content Toggle raw display
$29$ \( T^{2} + 36T - 54476 \) Copy content Toggle raw display
$31$ \( T^{2} - 112T - 5632 \) Copy content Toggle raw display
$37$ \( T^{2} + 48T - 26276 \) Copy content Toggle raw display
$41$ \( T^{2} + 70T + 1088 \) Copy content Toggle raw display
$43$ \( T^{2} + 40T - 54400 \) Copy content Toggle raw display
$47$ \( T^{2} - 420T - 114272 \) Copy content Toggle raw display
$53$ \( T^{2} + 176T - 36644 \) Copy content Toggle raw display
$59$ \( T^{2} + 404T + 40256 \) Copy content Toggle raw display
$61$ \( T^{2} - 156T - 213116 \) Copy content Toggle raw display
$67$ \( T^{2} - 4T - 123296 \) Copy content Toggle raw display
$71$ \( T^{2} - 814T + 162224 \) Copy content Toggle raw display
$73$ \( T^{2} + 216T - 278228 \) Copy content Toggle raw display
$79$ \( T^{2} - 1580 T + 557792 \) Copy content Toggle raw display
$83$ \( T^{2} + 464T + 51632 \) Copy content Toggle raw display
$89$ \( T^{2} + 1158 T - 701712 \) Copy content Toggle raw display
$97$ \( T^{2} + 1760 T + 247772 \) Copy content Toggle raw display
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