Properties

Label 1452.4.a.j
Level $1452$
Weight $4$
Character orbit 1452.a
Self dual yes
Analytic conductor $85.671$
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] = [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: \(2\)
Coefficient field: \(\Q(\sqrt{553}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 138 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
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 \(\beta = \frac{1}{2}(1 + \sqrt{553})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 3 q^{3} + ( - \beta - 2) q^{5} + (\beta - 11) q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 3 q^{3} + ( - \beta - 2) q^{5} + (\beta - 11) q^{7} + 9 q^{9} + (\beta - 20) q^{13} + (3 \beta + 6) q^{15} + (7 \beta + 14) q^{17} + (5 \beta - 27) q^{19} + ( - 3 \beta + 33) q^{21} + ( - 6 \beta - 6) q^{23} + (5 \beta + 17) q^{25} - 27 q^{27} + ( - 11 \beta - 100) q^{29} + ( - 17 \beta + 7) q^{31} + (8 \beta - 116) q^{35} + ( - 20 \beta - 113) q^{37} + ( - 3 \beta + 60) q^{39} + (17 \beta - 188) q^{41} + ( - 20 \beta + 172) q^{43} + ( - 9 \beta - 18) q^{45} + ( - 2 \beta + 146) q^{47} + ( - 21 \beta - 84) q^{49} + ( - 21 \beta - 42) q^{51} + (\beta - 76) q^{53} + ( - 15 \beta + 81) q^{57} + (30 \beta - 378) q^{59} + (19 \beta + 481) q^{61} + (9 \beta - 99) q^{63} + (17 \beta - 98) q^{65} + ( - 13 \beta - 129) q^{67} + (18 \beta + 18) q^{69} + ( - 40 \beta + 640) q^{71} + (17 \beta - 141) q^{73} + ( - 15 \beta - 51) q^{75} + (19 \beta + 703) q^{79} + 81 q^{81} + (22 \beta + 326) q^{83} + ( - 35 \beta - 994) q^{85} + (33 \beta + 300) q^{87} + (55 \beta - 538) q^{89} + ( - 30 \beta + 358) q^{91} + (51 \beta - 21) q^{93} + (12 \beta - 636) q^{95} + (118 \beta - 71) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{3} - 5 q^{5} - 21 q^{7} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 6 q^{3} - 5 q^{5} - 21 q^{7} + 18 q^{9} - 39 q^{13} + 15 q^{15} + 35 q^{17} - 49 q^{19} + 63 q^{21} - 18 q^{23} + 39 q^{25} - 54 q^{27} - 211 q^{29} - 3 q^{31} - 224 q^{35} - 246 q^{37} + 117 q^{39} - 359 q^{41} + 324 q^{43} - 45 q^{45} + 290 q^{47} - 189 q^{49} - 105 q^{51} - 151 q^{53} + 147 q^{57} - 726 q^{59} + 981 q^{61} - 189 q^{63} - 179 q^{65} - 271 q^{67} + 54 q^{69} + 1240 q^{71} - 265 q^{73} - 117 q^{75} + 1425 q^{79} + 162 q^{81} + 674 q^{83} - 2023 q^{85} + 633 q^{87} - 1021 q^{89} + 686 q^{91} + 9 q^{93} - 1260 q^{95} - 24 q^{97}+O(q^{100}) \) 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
12.2580
−11.2580
0 −3.00000 0 −14.2580 0 1.25798 0 9.00000 0
1.2 0 −3.00000 0 9.25798 0 −22.2580 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.j 2
11.b odd 2 1 1452.4.a.k yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1452.4.a.j 2 1.a even 1 1 trivial
1452.4.a.k yes 2 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1452))\):

\( T_{5}^{2} + 5T_{5} - 132 \) Copy content Toggle raw display
\( T_{7}^{2} + 21T_{7} - 28 \) 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
$5$ \( T^{2} + 5T - 132 \) Copy content Toggle raw display
$7$ \( T^{2} + 21T - 28 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 39T + 242 \) Copy content Toggle raw display
$17$ \( T^{2} - 35T - 6468 \) Copy content Toggle raw display
$19$ \( T^{2} + 49T - 2856 \) Copy content Toggle raw display
$23$ \( T^{2} + 18T - 4896 \) Copy content Toggle raw display
$29$ \( T^{2} + 211T - 5598 \) Copy content Toggle raw display
$31$ \( T^{2} + 3T - 39952 \) Copy content Toggle raw display
$37$ \( T^{2} + 246T - 40171 \) Copy content Toggle raw display
$41$ \( T^{2} + 359T - 7734 \) Copy content Toggle raw display
$43$ \( T^{2} - 324T - 29056 \) Copy content Toggle raw display
$47$ \( T^{2} - 290T + 20472 \) Copy content Toggle raw display
$53$ \( T^{2} + 151T + 5562 \) Copy content Toggle raw display
$59$ \( T^{2} + 726T + 7344 \) Copy content Toggle raw display
$61$ \( T^{2} - 981T + 190682 \) Copy content Toggle raw display
$67$ \( T^{2} + 271T - 5004 \) Copy content Toggle raw display
$71$ \( T^{2} - 1240 T + 163200 \) Copy content Toggle raw display
$73$ \( T^{2} + 265T - 22398 \) Copy content Toggle raw display
$79$ \( T^{2} - 1425 T + 457748 \) Copy content Toggle raw display
$83$ \( T^{2} - 674T + 46656 \) Copy content Toggle raw display
$89$ \( T^{2} + 1021 T - 157596 \) Copy content Toggle raw display
$97$ \( T^{2} + 24T - 1924849 \) Copy content Toggle raw display
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