Properties

Label 1452.4.a.o
Level $1452$
Weight $4$
Character orbit 1452.a
Self dual yes
Analytic conductor $85.671$
Analytic rank $1$
Dimension $4$
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] = [1452,4,Mod(1,1452)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1452, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("1452.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1452 = 2^{2} \cdot 3 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1452.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: \(85.6707733283\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.885025.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 93x^{2} + 94x + 2189 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 132)
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 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.

\(f(q)\) \(=\) \( q - 3 q^{3} + (6 \beta_{3} + \beta_1 + 1) q^{5} + (3 \beta_{3} - 3 \beta_{2} - 2) q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 3 q^{3} + (6 \beta_{3} + \beta_1 + 1) q^{5} + (3 \beta_{3} - 3 \beta_{2} - 2) q^{7} + 9 q^{9} + ( - \beta_{3} + 6 \beta_{2} + \cdots + 39) q^{13}+ \cdots + (564 \beta_{3} - 18 \beta_{2} + \cdots + 753) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 12 q^{3} - 6 q^{5} - 11 q^{7} + 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 12 q^{3} - 6 q^{5} - 11 q^{7} + 36 q^{9} + 158 q^{13} + 18 q^{15} - 117 q^{17} - 139 q^{19} + 33 q^{21} - 174 q^{23} - 122 q^{25} - 108 q^{27} - 177 q^{29} + 104 q^{31} + 285 q^{35} + 240 q^{37} - 474 q^{39} + 534 q^{41} + 342 q^{43} - 54 q^{45} - 99 q^{47} + 1041 q^{49} + 351 q^{51} - 114 q^{53} + 417 q^{57} + 108 q^{59} + 1007 q^{61} - 99 q^{63} - 57 q^{65} + 235 q^{67} + 522 q^{69} - 273 q^{71} - 869 q^{73} + 366 q^{75} + 63 q^{79} + 324 q^{81} - 2025 q^{83} - 320 q^{85} + 531 q^{87} - 858 q^{89} - 4492 q^{91} - 312 q^{93} - 33 q^{95} + 2016 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{3} - 93x^{2} + 94x + 2189 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - \nu^{2} - 49\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{2} - \nu - 49 ) / 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 4\beta_{3} + \beta _1 + 49 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 4\beta_{3} + 4\beta_{2} + 50\beta _1 + 49 \) 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
−6.04048
−6.69181
7.04048
7.69181
0 −3.00000 0 −14.7487 0 −36.1752 0 9.00000 0
1.2 0 −3.00000 0 −1.98361 0 12.2614 0 9.00000 0
1.3 0 −3.00000 0 −1.66773 0 27.3211 0 9.00000 0
1.4 0 −3.00000 0 12.4000 0 −14.4073 0 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(3\) \( +1 \)
\(11\) \( +1 \)

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 1452.4.a.o 4
11.b odd 2 1 1452.4.a.p 4
11.c even 5 2 132.4.i.b 8
33.h odd 10 2 396.4.j.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
132.4.i.b 8 11.c even 5 2
396.4.j.b 8 33.h odd 10 2
1452.4.a.o 4 1.a even 1 1 trivial
1452.4.a.p 4 11.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1452))\):

\( T_{5}^{4} + 6T_{5}^{3} - 171T_{5}^{2} - 660T_{5} - 605 \) Copy content Toggle raw display
\( T_{7}^{4} + 11T_{7}^{3} - 1146T_{7}^{2} - 3685T_{7} + 174595 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T + 3)^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 6 T^{3} + \cdots - 605 \) Copy content Toggle raw display
$7$ \( T^{4} + 11 T^{3} + \cdots + 174595 \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} - 158 T^{3} + \cdots + 314116 \) Copy content Toggle raw display
$17$ \( T^{4} + 117 T^{3} + \cdots - 13948880 \) Copy content Toggle raw display
$19$ \( T^{4} + 139 T^{3} + \cdots + 40780 \) Copy content Toggle raw display
$23$ \( T^{4} + 174 T^{3} + \cdots - 28151636 \) Copy content Toggle raw display
$29$ \( T^{4} + 177 T^{3} + \cdots - 270422636 \) Copy content Toggle raw display
$31$ \( T^{4} - 104 T^{3} + \cdots + 147373039 \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots - 1403540784 \) Copy content Toggle raw display
$41$ \( T^{4} - 534 T^{3} + \cdots - 227885636 \) Copy content Toggle raw display
$43$ \( T^{4} - 342 T^{3} + \cdots + 819106020 \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 3184072144 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 9427551109 \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 17577710611 \) Copy content Toggle raw display
$61$ \( T^{4} + \cdots - 9607246916 \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 14033195776 \) Copy content Toggle raw display
$71$ \( T^{4} + 273 T^{3} + \cdots - 65557580 \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots + 21165828556 \) Copy content Toggle raw display
$79$ \( T^{4} - 63 T^{3} + \cdots - 487585251 \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots - 199364597009 \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots - 381112589036 \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots - 351009166905 \) Copy content Toggle raw display
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