Properties

Label 1440.4.a.bg
Level $1440$
Weight $4$
Character orbit 1440.a
Self dual yes
Analytic conductor $84.963$
Analytic rank $0$
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] = [1440,4,Mod(1,1440)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1440, 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("1440.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1440 = 2^{5} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1440.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: \(84.9627504083\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{41}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 10 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 480)
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 \(\beta = 2\sqrt{41}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 5 q^{5} + (\beta + 6) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + 5 q^{5} + (\beta + 6) q^{7} + (4 \beta + 12) q^{11} + ( - 3 \beta + 40) q^{13} + ( - 9 \beta - 20) q^{17} + (9 \beta - 18) q^{19} + (\beta - 54) q^{23} + 25 q^{25} + (12 \beta + 54) q^{29} + ( - 5 \beta + 258) q^{31} + (5 \beta + 30) q^{35} + (9 \beta + 224) q^{37} + (30 \beta + 106) q^{41} + ( - 12 \beta + 228) q^{43} + ( - 13 \beta - 378) q^{47} + (12 \beta - 143) q^{49} + ( - 42 \beta - 126) q^{53} + (20 \beta + 60) q^{55} + (4 \beta - 396) q^{59} + ( - 6 \beta + 82) q^{61} + ( - 15 \beta + 200) q^{65} + ( - 44 \beta - 108) q^{67} + (8 \beta - 528) q^{71} + ( - 6 \beta - 18) q^{73} + (36 \beta + 728) q^{77} + ( - 19 \beta + 1014) q^{79} + ( - 32 \beta - 180) q^{83} + ( - 45 \beta - 100) q^{85} + ( - 30 \beta + 794) q^{89} + (22 \beta - 252) q^{91} + (45 \beta - 90) q^{95} + ( - 72 \beta + 770) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 10 q^{5} + 12 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 10 q^{5} + 12 q^{7} + 24 q^{11} + 80 q^{13} - 40 q^{17} - 36 q^{19} - 108 q^{23} + 50 q^{25} + 108 q^{29} + 516 q^{31} + 60 q^{35} + 448 q^{37} + 212 q^{41} + 456 q^{43} - 756 q^{47} - 286 q^{49} - 252 q^{53} + 120 q^{55} - 792 q^{59} + 164 q^{61} + 400 q^{65} - 216 q^{67} - 1056 q^{71} - 36 q^{73} + 1456 q^{77} + 2028 q^{79} - 360 q^{83} - 200 q^{85} + 1588 q^{89} - 504 q^{91} - 180 q^{95} + 1540 q^{97}+O(q^{100}) \) 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
−2.70156
3.70156
0 0 0 5.00000 0 −6.80625 0 0 0
1.2 0 0 0 5.00000 0 18.8062 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(3\) \( -1 \)
\(5\) \( -1 \)

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 1440.4.a.bg 2
3.b odd 2 1 480.4.a.q yes 2
4.b odd 2 1 1440.4.a.z 2
12.b even 2 1 480.4.a.m 2
15.d odd 2 1 2400.4.a.x 2
24.f even 2 1 960.4.a.bo 2
24.h odd 2 1 960.4.a.bm 2
60.h even 2 1 2400.4.a.bc 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
480.4.a.m 2 12.b even 2 1
480.4.a.q yes 2 3.b odd 2 1
960.4.a.bm 2 24.h odd 2 1
960.4.a.bo 2 24.f even 2 1
1440.4.a.z 2 4.b odd 2 1
1440.4.a.bg 2 1.a even 1 1 trivial
2400.4.a.x 2 15.d odd 2 1
2400.4.a.bc 2 60.h even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1440))\):

\( T_{7}^{2} - 12T_{7} - 128 \) Copy content Toggle raw display
\( T_{11}^{2} - 24T_{11} - 2480 \) Copy content Toggle raw display
\( T_{17}^{2} + 40T_{17} - 12884 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( (T - 5)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 12T - 128 \) Copy content Toggle raw display
$11$ \( T^{2} - 24T - 2480 \) Copy content Toggle raw display
$13$ \( T^{2} - 80T + 124 \) Copy content Toggle raw display
$17$ \( T^{2} + 40T - 12884 \) Copy content Toggle raw display
$19$ \( T^{2} + 36T - 12960 \) Copy content Toggle raw display
$23$ \( T^{2} + 108T + 2752 \) Copy content Toggle raw display
$29$ \( T^{2} - 108T - 20700 \) Copy content Toggle raw display
$31$ \( T^{2} - 516T + 62464 \) Copy content Toggle raw display
$37$ \( T^{2} - 448T + 36892 \) Copy content Toggle raw display
$41$ \( T^{2} - 212T - 136364 \) Copy content Toggle raw display
$43$ \( T^{2} - 456T + 28368 \) Copy content Toggle raw display
$47$ \( T^{2} + 756T + 115168 \) Copy content Toggle raw display
$53$ \( T^{2} + 252T - 273420 \) Copy content Toggle raw display
$59$ \( T^{2} + 792T + 154192 \) Copy content Toggle raw display
$61$ \( T^{2} - 164T + 820 \) Copy content Toggle raw display
$67$ \( T^{2} + 216T - 305840 \) Copy content Toggle raw display
$71$ \( T^{2} + 1056 T + 268288 \) Copy content Toggle raw display
$73$ \( T^{2} + 36T - 5580 \) Copy content Toggle raw display
$79$ \( T^{2} - 2028 T + 968992 \) Copy content Toggle raw display
$83$ \( T^{2} + 360T - 135536 \) Copy content Toggle raw display
$89$ \( T^{2} - 1588 T + 482836 \) Copy content Toggle raw display
$97$ \( T^{2} - 1540 T - 257276 \) Copy content Toggle raw display
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