Properties

Label 252.5.z.c
Level $252$
Weight $5$
Character orbit 252.z
Analytic conductor $26.049$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [252,5,Mod(73,252)]
 
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, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("252.73");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 252.z (of order \(6\), 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: \(26.0492306971\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{14})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 14x^{2} + 196 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{5} + ( - 35 \beta_1 - 56) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{5} + ( - 35 \beta_1 - 56) q^{7} + (2 \beta_{3} + 2 \beta_{2}) q^{11} + (114 \beta_1 + 57) q^{13} + ( - \beta_{3} + \beta_{2}) q^{17} + (157 \beta_1 + 314) q^{19} + (5 \beta_{3} - 10 \beta_{2}) q^{23} + (887 \beta_1 + 887) q^{25} + ( - 24 \beta_{3} + 12 \beta_{2}) q^{29} + (163 \beta_1 - 163) q^{31} + ( - 21 \beta_{3} - 35 \beta_{2}) q^{35} + 2041 \beta_1 q^{37} + 34 \beta_{2} q^{41} + 1723 q^{43} + 19 \beta_{3} q^{47} + (2695 \beta_1 + 1911) q^{49} + (73 \beta_{3} + 73 \beta_{2}) q^{53} + (6048 \beta_1 + 3024) q^{55} + (163 \beta_{3} - 163 \beta_{2}) q^{59} + (2408 \beta_1 + 4816) q^{61} + ( - 57 \beta_{3} + 114 \beta_{2}) q^{65} + (5035 \beta_1 + 5035) q^{67} + ( - 250 \beta_{3} + 125 \beta_{2}) q^{71} + (2287 \beta_1 - 2287) q^{73} + (28 \beta_{3} - 182 \beta_{2}) q^{77} - 235 \beta_1 q^{79} - 59 \beta_{2} q^{83} - 1512 q^{85} - 372 \beta_{3} q^{89} + ( - 4389 \beta_1 + 798) q^{91} + (157 \beta_{3} + 157 \beta_{2}) q^{95} + (7504 \beta_1 + 3752) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 154 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 154 q^{7} + 942 q^{19} + 1774 q^{25} - 978 q^{31} - 4082 q^{37} + 6892 q^{43} + 2254 q^{49} + 14448 q^{61} + 10070 q^{67} - 13722 q^{73} + 470 q^{79} - 6048 q^{85} + 11970 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 14x^{2} + 196 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{2} ) / 14 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 3\nu^{3} + 84\nu ) / 7 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -3\nu^{3} + 42\nu ) / 7 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 18 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 14\beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -14\beta_{3} + 7\beta_{2} ) / 9 \) Copy content Toggle raw display

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
73.1
−1.87083 + 3.24037i
1.87083 3.24037i
−1.87083 3.24037i
1.87083 + 3.24037i
0 0 0 −33.6749 + 19.4422i 0 −38.5000 + 30.3109i 0 0 0
73.2 0 0 0 33.6749 19.4422i 0 −38.5000 + 30.3109i 0 0 0
145.1 0 0 0 −33.6749 19.4422i 0 −38.5000 30.3109i 0 0 0
145.2 0 0 0 33.6749 + 19.4422i 0 −38.5000 30.3109i 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
3.b odd 2 1 inner
7.d odd 6 1 inner
21.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 252.5.z.c 4
3.b odd 2 1 inner 252.5.z.c 4
7.d odd 6 1 inner 252.5.z.c 4
21.g even 6 1 inner 252.5.z.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.5.z.c 4 1.a even 1 1 trivial
252.5.z.c 4 3.b odd 2 1 inner
252.5.z.c 4 7.d odd 6 1 inner
252.5.z.c 4 21.g even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 1512T_{5}^{2} + 2286144 \) acting on \(S_{5}^{\mathrm{new}}(252, [\chi])\). 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
$5$ \( T^{4} - 1512 T^{2} + \cdots + 2286144 \) Copy content Toggle raw display
$7$ \( (T^{2} + 77 T + 2401)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + 18144 T^{2} + \cdots + 329204736 \) Copy content Toggle raw display
$13$ \( (T^{2} + 9747)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} - 1512 T^{2} + \cdots + 2286144 \) Copy content Toggle raw display
$19$ \( (T^{2} - 471 T + 73947)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 113400 T^{2} + \cdots + 12859560000 \) Copy content Toggle raw display
$29$ \( (T^{2} - 653184)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 489 T + 79707)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 2041 T + 4165681)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 1747872)^{2} \) Copy content Toggle raw display
$43$ \( (T - 1723)^{4} \) Copy content Toggle raw display
$47$ \( T^{4} - 545832 T^{2} + \cdots + 297932572224 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 584302214454336 \) Copy content Toggle raw display
$59$ \( T^{4} - 40172328 T^{2} + \cdots + 16\!\cdots\!84 \) Copy content Toggle raw display
$61$ \( (T^{2} - 7224 T + 17395392)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 5035 T + 25351225)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 70875000)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 6861 T + 15691107)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 235 T + 55225)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 5263272)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} - 209236608 T^{2} + \cdots + 43\!\cdots\!64 \) Copy content Toggle raw display
$97$ \( (T^{2} + 42232512)^{2} \) Copy content Toggle raw display
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