Properties

Label 252.4.k.f
Level $252$
Weight $4$
Character orbit 252.k
Analytic conductor $14.868$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [252,4,Mod(37,252)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(252, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 2]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("252.37");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 252.k (of order \(3\), degree \(2\), minimal)

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: no
Analytic conductor: \(14.8684813214\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{193})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 49x^{2} + 48x + 2304 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 84)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + (6 \beta_{2} - \beta_1) q^{5} + (3 \beta_{3} + \beta_{2} + 2 \beta_1 - 1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (6 \beta_{2} - \beta_1) q^{5} + (3 \beta_{3} + \beta_{2} + 2 \beta_1 - 1) q^{7} + ( - 7 \beta_{3} + 6 \beta_{2} - 7 \beta_1 + 1) q^{11} + 5 \beta_{3} q^{13} + ( - 4 \beta_{3} - 48 \beta_{2} - 4 \beta_1 + 52) q^{17} + ( - 35 \beta_{2} + 3 \beta_1) q^{19} + (48 \beta_{2} - 20 \beta_1) q^{23} + ( - 11 \beta_{3} - 41 \beta_{2} - 11 \beta_1 + 52) q^{25} + (11 \beta_{3} - 143) q^{29} + ( - 20 \beta_{3} + 191 \beta_{2} - 20 \beta_1 - 171) q^{31} + (9 \beta_{3} + 66 \beta_{2} - 8 \beta_1 + 81) q^{35} + (25 \beta_{2} - 45 \beta_1) q^{37} + (18 \beta_{3} + 72) q^{41} + ( - 3 \beta_{3} + 362) q^{43} + ( - 90 \beta_{2} - 36 \beta_1) q^{47} + (11 \beta_{3} - 379 \beta_{2} - 2 \beta_1 + 232) q^{49} + (9 \beta_{3} - 252 \beta_{2} + 9 \beta_1 + 243) q^{53} + ( - 41 \beta_{3} - 331) q^{55} + ( - 53 \beta_{3} - 60 \beta_{2} - 53 \beta_1 + 113) q^{59} + (286 \beta_{2} - 40 \beta_1) q^{61} + (270 \beta_{2} - 30 \beta_1) q^{65} + (77 \beta_{3} + 17 \beta_{2} + 77 \beta_1 - 94) q^{67} + ( - 44 \beta_{3} - 778) q^{71} + (53 \beta_{3} + 581 \beta_{2} + 53 \beta_1 - 634) q^{73} + ( - 20 \beta_{3} + 1020 \beta_{2} - 11 \beta_1 - 334) q^{77} + ( - 761 \beta_{2} + 62 \beta_1) q^{79} + ( - 101 \beta_{3} + 755) q^{83} + (28 \beta_{3} + 68) q^{85} + ( - 1008 \beta_{2} + 42 \beta_1) q^{89} + (10 \beta_{3} - 475 \beta_{2} - 5 \beta_1 + 720) q^{91} + (50 \beta_{3} - 354 \beta_{2} + 50 \beta_1 + 304) q^{95} + ( - 29 \beta_{3} + 295) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 11 q^{5} + 6 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 11 q^{5} + 6 q^{7} - 5 q^{11} + 10 q^{13} + 100 q^{17} - 67 q^{19} + 76 q^{23} + 93 q^{25} - 550 q^{29} - 362 q^{31} + 466 q^{35} + 5 q^{37} + 324 q^{41} + 1442 q^{43} - 216 q^{47} + 190 q^{49} + 495 q^{53} - 1406 q^{55} + 173 q^{59} + 532 q^{61} + 510 q^{65} - 111 q^{67} - 3200 q^{71} - 1215 q^{73} + 653 q^{77} - 1460 q^{79} + 2818 q^{83} + 328 q^{85} - 1974 q^{89} + 1945 q^{91} + 658 q^{95} + 1122 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} + 49x^{2} + 48x + 2304 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} + 49\nu^{2} - 49\nu + 2304 ) / 2352 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} + 97 ) / 49 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 48\beta_{2} + \beta _1 - 49 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 49\beta_{3} - 97 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/252\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(73\) \(127\)
\(\chi(n)\) \(1\) \(-\beta_{2}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
3.72311 6.44862i
−3.22311 + 5.58259i
3.72311 + 6.44862i
−3.22311 5.58259i
0 0 0 −0.723111 + 1.25246i 0 −12.3924 13.7633i 0 0 0
37.2 0 0 0 6.22311 10.7787i 0 15.3924 + 10.2992i 0 0 0
109.1 0 0 0 −0.723111 1.25246i 0 −12.3924 + 13.7633i 0 0 0
109.2 0 0 0 6.22311 + 10.7787i 0 15.3924 10.2992i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 252.4.k.f 4
3.b odd 2 1 84.4.i.a 4
7.b odd 2 1 1764.4.k.q 4
7.c even 3 1 inner 252.4.k.f 4
7.c even 3 1 1764.4.a.o 2
7.d odd 6 1 1764.4.a.y 2
7.d odd 6 1 1764.4.k.q 4
12.b even 2 1 336.4.q.i 4
21.c even 2 1 588.4.i.j 4
21.g even 6 1 588.4.a.f 2
21.g even 6 1 588.4.i.j 4
21.h odd 6 1 84.4.i.a 4
21.h odd 6 1 588.4.a.i 2
84.j odd 6 1 2352.4.a.bx 2
84.n even 6 1 336.4.q.i 4
84.n even 6 1 2352.4.a.bt 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.4.i.a 4 3.b odd 2 1
84.4.i.a 4 21.h odd 6 1
252.4.k.f 4 1.a even 1 1 trivial
252.4.k.f 4 7.c even 3 1 inner
336.4.q.i 4 12.b even 2 1
336.4.q.i 4 84.n even 6 1
588.4.a.f 2 21.g even 6 1
588.4.a.i 2 21.h odd 6 1
588.4.i.j 4 21.c even 2 1
588.4.i.j 4 21.g even 6 1
1764.4.a.o 2 7.c even 3 1
1764.4.a.y 2 7.d odd 6 1
1764.4.k.q 4 7.b odd 2 1
1764.4.k.q 4 7.d odd 6 1
2352.4.a.bt 2 84.n even 6 1
2352.4.a.bx 2 84.j odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(252, [\chi])\):

\( T_{5}^{4} - 11T_{5}^{3} + 139T_{5}^{2} + 198T_{5} + 324 \) Copy content Toggle raw display
\( T_{13}^{2} - 5T_{13} - 1200 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 11 T^{3} + 139 T^{2} + \cdots + 324 \) Copy content Toggle raw display
$7$ \( T^{4} - 6 T^{3} - 77 T^{2} + \cdots + 117649 \) Copy content Toggle raw display
$11$ \( T^{4} + 5 T^{3} + 2383 T^{2} + \cdots + 5560164 \) Copy content Toggle raw display
$13$ \( (T^{2} - 5 T - 1200)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} - 100 T^{3} + 8272 T^{2} + \cdots + 2985984 \) Copy content Toggle raw display
$19$ \( T^{4} + 67 T^{3} + 3801 T^{2} + \cdots + 473344 \) Copy content Toggle raw display
$23$ \( T^{4} - 76 T^{3} + \cdots + 318836736 \) Copy content Toggle raw display
$29$ \( (T^{2} + 275 T + 13068)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 362 T^{3} + \cdots + 181198521 \) Copy content Toggle raw display
$37$ \( T^{4} - 5 T^{3} + \cdots + 9545290000 \) Copy content Toggle raw display
$41$ \( (T^{2} - 162 T - 9072)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 721 T + 129526)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 216 T^{3} + \cdots + 2587553424 \) Copy content Toggle raw display
$53$ \( T^{4} - 495 T^{3} + \cdots + 3288793104 \) Copy content Toggle raw display
$59$ \( T^{4} - 173 T^{3} + \cdots + 16397314704 \) Copy content Toggle raw display
$61$ \( T^{4} - 532 T^{3} + \cdots + 41525136 \) Copy content Toggle raw display
$67$ \( T^{4} + 111 T^{3} + \cdots + 80085604036 \) Copy content Toggle raw display
$71$ \( (T^{2} + 1600 T + 546588)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 1215 T^{3} + \cdots + 54532524484 \) Copy content Toggle raw display
$79$ \( T^{4} + 1460 T^{3} + \cdots + 120705520329 \) Copy content Toggle raw display
$83$ \( (T^{2} - 1409 T + 4122)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 1974 T^{3} + \cdots + 790420571136 \) Copy content Toggle raw display
$97$ \( (T^{2} - 561 T + 38102)^{2} \) Copy content Toggle raw display
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