Properties

Label 1452.4.a.q
Level $1452$
Weight $4$
Character orbit 1452.a
Self dual yes
Analytic conductor $85.671$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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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: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.20959101.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 90x^{2} + 91x + 2026 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{5}\cdot 3 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + 3 q^{3} + (\beta_{3} + 3) q^{5} - \beta_1 q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 3 q^{3} + (\beta_{3} + 3) q^{5} - \beta_1 q^{7} + 9 q^{9} + \beta_{2} q^{13} + (3 \beta_{3} + 9) q^{15} + ( - \beta_{2} - \beta_1) q^{17} + (\beta_{2} + 3 \beta_1) q^{19} - 3 \beta_1 q^{21} + (11 \beta_{3} + 39) q^{23} + (6 \beta_{3} + 61) q^{25} + 27 q^{27} + (\beta_{2} - 5 \beta_1) q^{29} + ( - 18 \beta_{3} + 46) q^{31} + ( - 4 \beta_{2} - 10 \beta_1) q^{35} + ( - 6 \beta_{3} - 44) q^{37} + 3 \beta_{2} q^{39} + ( - 3 \beta_{2} + 9 \beta_1) q^{41} + (\beta_{2} - 11 \beta_1) q^{43} + (9 \beta_{3} + 27) q^{45} + (21 \beta_{3} + 213) q^{47} + (30 \beta_{3} + 395) q^{49} + ( - 3 \beta_{2} - 3 \beta_1) q^{51} + ( - 15 \beta_{3} + 231) q^{53} + (3 \beta_{2} + 9 \beta_1) q^{57} + (22 \beta_{3} - 90) q^{59} + (\beta_{2} - 10 \beta_1) q^{61} - 9 \beta_1 q^{63} + ( - 4 \beta_{2} + 32 \beta_1) q^{65} + (24 \beta_{3} - 44) q^{67} + (33 \beta_{3} + 117) q^{69} + ( - 13 \beta_{3} + 819) q^{71} + ( - 2 \beta_{2} + 2 \beta_1) q^{73} + (18 \beta_{3} + 183) q^{75} + (2 \beta_{2} + 27 \beta_1) q^{79} + 81 q^{81} + ( - 4 \beta_{2} + 38 \beta_1) q^{83} - 42 \beta_1 q^{85} + (3 \beta_{2} - 15 \beta_1) q^{87} + ( - 20 \beta_{3} - 786) q^{89} + ( - 132 \beta_{3} - 36) q^{91} + ( - 54 \beta_{3} + 138) q^{93} + (8 \beta_{2} + 62 \beta_1) q^{95} + ( - 30 \beta_{3} + 692) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 12 q^{3} + 12 q^{5} + 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 12 q^{3} + 12 q^{5} + 36 q^{9} + 36 q^{15} + 156 q^{23} + 244 q^{25} + 108 q^{27} + 184 q^{31} - 176 q^{37} + 108 q^{45} + 852 q^{47} + 1580 q^{49} + 924 q^{53} - 360 q^{59} - 176 q^{67} + 468 q^{69} + 3276 q^{71} + 732 q^{75} + 324 q^{81} - 3144 q^{89} - 144 q^{91} + 552 q^{93} + 2768 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{3} - 90x^{2} + 91x + 2026 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 4\nu^{3} - 6\nu^{2} - 174\nu + 88 ) / 7 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -4\nu^{3} + 6\nu^{2} + 258\nu - 130 ) / 7 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\nu^{2} - 2\nu - 91 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + \beta _1 + 6 ) / 12 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 6\beta_{3} + \beta_{2} + \beta _1 + 552 ) / 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3\beta_{3} + 15\beta_{2} + 22\beta _1 + 275 ) / 4 \) 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
−5.75283
6.75283
7.73893
−6.73893
0 3.00000 0 −10.3041 0 −18.4086 0 9.00000 0
1.2 0 3.00000 0 −10.3041 0 18.4086 0 9.00000 0
1.3 0 3.00000 0 16.3041 0 −33.7213 0 9.00000 0
1.4 0 3.00000 0 16.3041 0 33.7213 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

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1452.4.a.q 4
11.b odd 2 1 inner 1452.4.a.q 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1452.4.a.q 4 1.a even 1 1 trivial
1452.4.a.q 4 11.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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1452))\):

\( T_{5}^{2} - 6T_{5} - 168 \) Copy content Toggle raw display
\( T_{7}^{4} - 1476T_{7}^{2} + 385344 \) 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
$5$ \( (T^{2} - 6 T - 168)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} - 1476 T^{2} + 385344 \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} - 11556 T^{2} + 24662016 \) Copy content Toggle raw display
$17$ \( T^{4} - 13176 T^{2} + 42484176 \) Copy content Toggle raw display
$19$ \( T^{4} - 25272 T^{2} + 34777296 \) Copy content Toggle raw display
$23$ \( (T^{2} - 78 T - 19896)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} - 47736 T^{2} + 458655696 \) Copy content Toggle raw display
$31$ \( (T^{2} - 92 T - 55232)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 88 T - 4436)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + \cdots + 4127901264 \) Copy content Toggle raw display
$43$ \( T^{4} + \cdots + 8844511824 \) Copy content Toggle raw display
$47$ \( (T^{2} - 426 T - 32688)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 462 T + 13536)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 180 T - 77568)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} + \cdots + 6215213376 \) Copy content Toggle raw display
$67$ \( (T^{2} + 88 T - 100016)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 1638 T + 640848)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} - 51552 T^{2} + 75527424 \) Copy content Toggle raw display
$79$ \( T^{4} + \cdots + 99443414016 \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 1315989835776 \) Copy content Toggle raw display
$89$ \( (T^{2} + 1572 T + 546996)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 1384 T + 319564)^{2} \) Copy content Toggle raw display
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