Properties

Label 1452.4.a.l
Level $1452$
Weight $4$
Character orbit 1452.a
Self dual yes
Analytic conductor $85.671$
Analytic rank $1$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) 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 \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 3 q^{3} + (11 \beta - 5) q^{5} + (4 \beta - 13) q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 3 q^{3} + (11 \beta - 5) q^{5} + (4 \beta - 13) q^{7} + 9 q^{9} + (14 \beta - 51) q^{13} + ( - 33 \beta + 15) q^{15} + (21 \beta - 16) q^{17} + ( - 93 \beta + 85) q^{19} + ( - 12 \beta + 39) q^{21} + 185 q^{23} + (11 \beta + 21) q^{25} - 27 q^{27} + ( - 60 \beta + 118) q^{29} + ( - 209 \beta + 154) q^{31} + ( - 119 \beta + 109) q^{35} + (154 \beta - 107) q^{37} + ( - 42 \beta + 153) q^{39} + (96 \beta - 235) q^{41} + ( - 298 \beta - 5) q^{43} + (99 \beta - 45) q^{45} + (33 \beta - 388) q^{47} + ( - 88 \beta - 158) q^{49} + ( - 63 \beta + 48) q^{51} + ( - 77 \beta + 312) q^{53} + (279 \beta - 255) q^{57} + (11 \beta + 295) q^{59} + ( - 357 \beta - 69) q^{61} + (36 \beta - 117) q^{63} + ( - 477 \beta + 409) q^{65} + (407 \beta - 575) q^{67} - 555 q^{69} + (99 \beta + 201) q^{71} + (342 \beta + 456) q^{73} + ( - 33 \beta - 63) q^{75} + ( - 336 \beta + 355) q^{79} + 81 q^{81} + (266 \beta + 219) q^{83} + ( - 50 \beta + 311) q^{85} + (180 \beta - 354) q^{87} + ( - 220 \beta + 419) q^{89} + ( - 330 \beta + 719) q^{91} + (627 \beta - 462) q^{93} + (377 \beta - 1448) q^{95} + (11 \beta + 151) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{3} + q^{5} - 22 q^{7} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 6 q^{3} + q^{5} - 22 q^{7} + 18 q^{9} - 88 q^{13} - 3 q^{15} - 11 q^{17} + 77 q^{19} + 66 q^{21} + 370 q^{23} + 53 q^{25} - 54 q^{27} + 176 q^{29} + 99 q^{31} + 99 q^{35} - 60 q^{37} + 264 q^{39} - 374 q^{41} - 308 q^{43} + 9 q^{45} - 743 q^{47} - 404 q^{49} + 33 q^{51} + 547 q^{53} - 231 q^{57} + 601 q^{59} - 495 q^{61} - 198 q^{63} + 341 q^{65} - 743 q^{67} - 1110 q^{69} + 501 q^{71} + 1254 q^{73} - 159 q^{75} + 374 q^{79} + 162 q^{81} + 704 q^{83} + 572 q^{85} - 528 q^{87} + 618 q^{89} + 1108 q^{91} - 297 q^{93} - 2519 q^{95} + 313 q^{97}+O(q^{100}) \) 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
−0.618034
1.61803
0 −3.00000 0 −11.7984 0 −15.4721 0 9.00000 0
1.2 0 −3.00000 0 12.7984 0 −6.52786 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.l 2
11.b odd 2 1 1452.4.a.m 2
11.c even 5 2 132.4.i.a 4
33.h odd 10 2 396.4.j.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
132.4.i.a 4 11.c even 5 2
396.4.j.a 4 33.h odd 10 2
1452.4.a.l 2 1.a even 1 1 trivial
1452.4.a.m 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1452))\):

\( T_{5}^{2} - T_{5} - 151 \) Copy content Toggle raw display
\( T_{7}^{2} + 22T_{7} + 101 \) 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
$5$ \( T^{2} - T - 151 \) Copy content Toggle raw display
$7$ \( T^{2} + 22T + 101 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 88T + 1691 \) Copy content Toggle raw display
$17$ \( T^{2} + 11T - 521 \) Copy content Toggle raw display
$19$ \( T^{2} - 77T - 9329 \) Copy content Toggle raw display
$23$ \( (T - 185)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} - 176T + 3244 \) Copy content Toggle raw display
$31$ \( T^{2} - 99T - 52151 \) Copy content Toggle raw display
$37$ \( T^{2} + 60T - 28745 \) Copy content Toggle raw display
$41$ \( T^{2} + 374T + 23449 \) Copy content Toggle raw display
$43$ \( T^{2} + 308T - 87289 \) Copy content Toggle raw display
$47$ \( T^{2} + 743T + 136651 \) Copy content Toggle raw display
$53$ \( T^{2} - 547T + 67391 \) Copy content Toggle raw display
$59$ \( T^{2} - 601T + 90149 \) Copy content Toggle raw display
$61$ \( T^{2} + 495T - 98055 \) Copy content Toggle raw display
$67$ \( T^{2} + 743T - 69049 \) Copy content Toggle raw display
$71$ \( T^{2} - 501T + 50499 \) Copy content Toggle raw display
$73$ \( T^{2} - 1254 T + 246924 \) Copy content Toggle raw display
$79$ \( T^{2} - 374T - 106151 \) Copy content Toggle raw display
$83$ \( T^{2} - 704T + 35459 \) Copy content Toggle raw display
$89$ \( T^{2} - 618T + 34981 \) Copy content Toggle raw display
$97$ \( T^{2} - 313T + 24341 \) Copy content Toggle raw display
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