Properties

Label 1440.4.a.bf
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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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: \(\Q(\sqrt{201}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 50 \) 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 \(\beta = 2\sqrt{201}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 5 q^{5} + (\beta + 2) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + 5 q^{5} + (\beta + 2) q^{7} - 20 q^{11} + (\beta - 12) q^{13} + (3 \beta - 16) q^{17} + (\beta + 58) q^{19} + (\beta + 6) q^{23} + 25 q^{25} + (4 \beta - 74) q^{29} + ( - 9 \beta + 38) q^{31} + (5 \beta + 10) q^{35} + (5 \beta + 140) q^{37} + ( - 2 \beta - 286) q^{41} + (4 \beta + 260) q^{43} + ( - 13 \beta + 170) q^{47} + (4 \beta + 465) q^{49} + (6 \beta - 262) q^{53} - 100 q^{55} + ( - 8 \beta + 276) q^{59} + ( - 22 \beta + 314) q^{61} + (5 \beta - 60) q^{65} + (20 \beta + 84) q^{67} + (8 \beta + 624) q^{71} + (18 \beta + 454) q^{73} + ( - 20 \beta - 40) q^{77} + ( - 7 \beta - 526) q^{79} + (16 \beta - 20) q^{83} + (15 \beta - 80) q^{85} + ( - 30 \beta - 414) q^{89} + ( - 10 \beta + 780) q^{91} + (5 \beta + 290) q^{95} + ( - 56 \beta - 158) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 10 q^{5} + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 10 q^{5} + 4 q^{7} - 40 q^{11} - 24 q^{13} - 32 q^{17} + 116 q^{19} + 12 q^{23} + 50 q^{25} - 148 q^{29} + 76 q^{31} + 20 q^{35} + 280 q^{37} - 572 q^{41} + 520 q^{43} + 340 q^{47} + 930 q^{49} - 524 q^{53} - 200 q^{55} + 552 q^{59} + 628 q^{61} - 120 q^{65} + 168 q^{67} + 1248 q^{71} + 908 q^{73} - 80 q^{77} - 1052 q^{79} - 40 q^{83} - 160 q^{85} - 828 q^{89} + 1560 q^{91} + 580 q^{95} - 316 q^{97}+O(q^{100}) \) 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
−6.58872
7.58872
0 0 0 5.00000 0 −26.3549 0 0 0
1.2 0 0 0 5.00000 0 30.3549 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.bf 2
3.b odd 2 1 480.4.a.p yes 2
4.b odd 2 1 1440.4.a.ba 2
12.b even 2 1 480.4.a.n 2
15.d odd 2 1 2400.4.a.z 2
24.f even 2 1 960.4.a.bp 2
24.h odd 2 1 960.4.a.bl 2
60.h even 2 1 2400.4.a.ba 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
480.4.a.n 2 12.b even 2 1
480.4.a.p yes 2 3.b odd 2 1
960.4.a.bl 2 24.h odd 2 1
960.4.a.bp 2 24.f even 2 1
1440.4.a.ba 2 4.b odd 2 1
1440.4.a.bf 2 1.a even 1 1 trivial
2400.4.a.z 2 15.d odd 2 1
2400.4.a.ba 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1440))\):

\( T_{7}^{2} - 4T_{7} - 800 \) Copy content Toggle raw display
\( T_{11} + 20 \) Copy content Toggle raw display
\( T_{17}^{2} + 32T_{17} - 6980 \) 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
$7$ \( T^{2} - 4T - 800 \) Copy content Toggle raw display
$11$ \( (T + 20)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 24T - 660 \) Copy content Toggle raw display
$17$ \( T^{2} + 32T - 6980 \) Copy content Toggle raw display
$19$ \( T^{2} - 116T + 2560 \) Copy content Toggle raw display
$23$ \( T^{2} - 12T - 768 \) Copy content Toggle raw display
$29$ \( T^{2} + 148T - 7388 \) Copy content Toggle raw display
$31$ \( T^{2} - 76T - 63680 \) Copy content Toggle raw display
$37$ \( T^{2} - 280T - 500 \) Copy content Toggle raw display
$41$ \( T^{2} + 572T + 78580 \) Copy content Toggle raw display
$43$ \( T^{2} - 520T + 54736 \) Copy content Toggle raw display
$47$ \( T^{2} - 340T - 106976 \) Copy content Toggle raw display
$53$ \( T^{2} + 524T + 39700 \) Copy content Toggle raw display
$59$ \( T^{2} - 552T + 24720 \) Copy content Toggle raw display
$61$ \( T^{2} - 628T - 290540 \) Copy content Toggle raw display
$67$ \( T^{2} - 168T - 314544 \) Copy content Toggle raw display
$71$ \( T^{2} - 1248 T + 337920 \) Copy content Toggle raw display
$73$ \( T^{2} - 908T - 54380 \) Copy content Toggle raw display
$79$ \( T^{2} + 1052 T + 237280 \) Copy content Toggle raw display
$83$ \( T^{2} + 40T - 205424 \) Copy content Toggle raw display
$89$ \( T^{2} + 828T - 552204 \) Copy content Toggle raw display
$97$ \( T^{2} + 316 T - 2496380 \) Copy content Toggle raw display
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