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Properties

Label 252.3.c.a
Level $252$
Weight $3$
Character orbit 252.c
Analytic conductor $6.867$
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] = [252,3,Mod(197,252)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(252, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("252.197");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 252.c (of order \(2\), degree \(1\), 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: \(6.86650266188\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{7})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 8x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2\cdot 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{5} + \beta_{3} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{5} + \beta_{3} q^{7} + (\beta_{2} + 2 \beta_1) q^{11} + (4 \beta_{3} - 2) q^{13} + (2 \beta_{2} + 5 \beta_1) q^{17} + (6 \beta_{3} - 10) q^{19} + (3 \beta_{2} + 2 \beta_1) q^{23} + (4 \beta_{3} + 9) q^{25} + ( - \beta_{2} + 10 \beta_1) q^{29} + ( - 2 \beta_{3} + 10) q^{31} + (2 \beta_{2} - \beta_1) q^{35} + (12 \beta_{3} + 8) q^{37} + 3 \beta_1 q^{41} + ( - 12 \beta_{3} - 4) q^{43} + ( - 14 \beta_{2} - 6 \beta_1) q^{47} + 7 q^{49} + 11 \beta_{2} q^{53} + (2 \beta_{3} - 26) q^{55} + (2 \beta_{2} - 14 \beta_1) q^{59} + ( - 2 \beta_{3} - 68) q^{61} + (8 \beta_{2} - 6 \beta_1) q^{65} + ( - 28 \beta_{3} + 42) q^{67} + ( - 15 \beta_{2} - 22 \beta_1) q^{71} + (2 \beta_{3} + 36) q^{73} + (5 \beta_{2} + \beta_1) q^{77} + ( - 32 \beta_{3} + 58) q^{79} + ( - 20 \beta_{2} - 22 \beta_1) q^{83} + (8 \beta_{3} - 68) q^{85} + ( - 36 \beta_{2} + 3 \beta_1) q^{89} + ( - 2 \beta_{3} + 28) q^{91} + (12 \beta_{2} - 16 \beta_1) q^{95} + ( - 34 \beta_{3} + 24) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 8 q^{13} - 40 q^{19} + 36 q^{25} + 40 q^{31} + 32 q^{37} - 16 q^{43} + 28 q^{49} - 104 q^{55} - 272 q^{61} + 168 q^{67} + 144 q^{73} + 232 q^{79} - 272 q^{85} + 112 q^{91} + 96 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 8x^{2} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} + 5\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} + 4 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{2} - 5\beta_1 ) / 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

\(n\) \(29\) \(73\) \(127\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
197.1
− 2.57794i
− 1.16372i
1.16372i
2.57794i
0 0 0 − 5.15587i 0 −2.64575 0 0 0
197.2 0 0 0 − 2.32744i 0 2.64575 0 0 0
197.3 0 0 0 2.32744i 0 2.64575 0 0 0
197.4 0 0 0 5.15587i 0 −2.64575 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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 252.3.c.a ✓ 4
3.b odd 2 1 inner 252.3.c.a ✓ 4
4.b odd 2 1 1008.3.d.c 4
7.b odd 2 1 1764.3.c.f 4
7.c even 3 2 1764.3.bk.d 8
7.d odd 6 2 1764.3.bk.e 8
8.b even 2 1 4032.3.d.h 4
8.d odd 2 1 4032.3.d.e 4
9.c even 3 2 2268.3.bg.c 8
9.d odd 6 2 2268.3.bg.c 8
12.b even 2 1 1008.3.d.c 4
21.c even 2 1 1764.3.c.f 4
21.g even 6 2 1764.3.bk.e 8
21.h odd 6 2 1764.3.bk.d 8
24.f even 2 1 4032.3.d.e 4
24.h odd 2 1 4032.3.d.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.3.c.a ✓ 4 1.a even 1 1 trivial
252.3.c.a ✓ 4 3.b odd 2 1 inner
1008.3.d.c 4 4.b odd 2 1
1008.3.d.c 4 12.b even 2 1
1764.3.c.f 4 7.b odd 2 1
1764.3.c.f 4 21.c even 2 1
1764.3.bk.d 8 7.c even 3 2
1764.3.bk.d 8 21.h odd 6 2
1764.3.bk.e 8 7.d odd 6 2
1764.3.bk.e 8 21.g even 6 2
2268.3.bg.c 8 9.c even 3 2
2268.3.bg.c 8 9.d odd 6 2
4032.3.d.e 4 8.d odd 2 1
4032.3.d.e 4 24.f even 2 1
4032.3.d.h 4 8.b even 2 1
4032.3.d.h 4 24.h odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(252, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 32T^{2} + 144 \) Copy content Toggle raw display
$7$ \( (T^{2} - 7)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + 116T^{2} + 2916 \) Copy content Toggle raw display
$13$ \( (T^{2} + 4 T - 108)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 704 T^{2} + 121104 \) Copy content Toggle raw display
$19$ \( (T^{2} + 20 T - 152)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 308T^{2} + 1764 \) Copy content Toggle raw display
$29$ \( T^{4} + 3476 T^{2} + 1127844 \) Copy content Toggle raw display
$31$ \( (T^{2} - 20 T + 72)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 16 T - 944)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 288 T^{2} + 11664 \) Copy content Toggle raw display
$43$ \( (T^{2} + 8 T - 992)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 6192 T^{2} + 4359744 \) Copy content Toggle raw display
$53$ \( (T^{2} + 2178)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 7088 T^{2} + 3779136 \) Copy content Toggle raw display
$61$ \( (T^{2} + 136 T + 4596)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 84 T - 3724)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + 15668 T^{2} + 32695524 \) Copy content Toggle raw display
$73$ \( (T^{2} - 72 T + 1268)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 116 T - 3804)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 19328 T^{2} + 15116544 \) Copy content Toggle raw display
$89$ \( T^{4} + 49536 T^{2} + 601034256 \) Copy content Toggle raw display
$97$ \( (T^{2} - 48 T - 7516)^{2} \) Copy content Toggle raw display
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