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Properties

Label 252.3.g.a
Level $252$
Weight $3$
Character orbit 252.g
Analytic conductor $6.867$
Analytic rank $0$
Dimension $6$
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(127,252)]
 
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([1, 0, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("252.127");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 252.g (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: \(6\)
Coefficient field: 6.0.1539727.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} + 3x^{3} - 8x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: no (minimal twist has level 28)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + ( - \beta_{4} - \beta_1) q^{4} + ( - \beta_{5} - \beta_{2} + 1) q^{5} - \beta_1 q^{7} + ( - \beta_{4} - 2 \beta_{3} + \beta_1 + 2) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + ( - \beta_{4} - \beta_1) q^{4} + ( - \beta_{5} - \beta_{2} + 1) q^{5} - \beta_1 q^{7} + ( - \beta_{4} - 2 \beta_{3} + \beta_1 + 2) q^{8} + (\beta_{5} - \beta_{3} + \beta_{2} + \cdots - 5) q^{10}+ \cdots - 7 \beta_{2} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + q^{2} + q^{4} + 4 q^{5} + 13 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + q^{2} + q^{4} + 4 q^{5} + 13 q^{8} - 28 q^{10} + 12 q^{13} - 7 q^{14} + 17 q^{16} + 4 q^{17} + 32 q^{20} + 52 q^{22} - 30 q^{25} + 56 q^{26} - 35 q^{28} + 36 q^{29} + 101 q^{32} + 58 q^{34} + 28 q^{37} + 190 q^{38} + 40 q^{40} + 20 q^{41} - 164 q^{44} + 120 q^{46} - 42 q^{49} - 161 q^{50} + 292 q^{52} - 92 q^{53} + 49 q^{56} - 166 q^{58} - 164 q^{61} - 148 q^{62} - 215 q^{64} + 136 q^{65} - 62 q^{68} + 84 q^{70} - 132 q^{73} - 250 q^{74} - 78 q^{76} - 112 q^{77} - 312 q^{80} - 86 q^{82} - 232 q^{85} + 164 q^{86} - 100 q^{88} - 348 q^{89} + 104 q^{92} - 276 q^{94} + 252 q^{97} - 7 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} + 3x^{3} - 8x + 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{5} + \nu^{2} - 2\nu + 2 ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} - 2\nu^{3} + \nu^{2} + 4\nu - 4 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{5} - 2\nu^{4} + 3\nu^{2} + 4\nu - 8 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{5} + 2\nu^{4} + 2\nu^{3} - \nu^{2} - 2\nu + 6 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -\nu^{5} + \nu^{4} + \nu^{3} - 2\nu^{2} + 5 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{5} + \beta_{3} + \beta_{2} + 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{5} + 2\beta_{4} + \beta_{3} + \beta_{2} + 2\beta _1 + 3 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{4} + \beta_{3} - \beta_{2} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -\beta_{5} + 4\beta_{4} - \beta_{3} + 3\beta_{2} - 5 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -3\beta_{5} + 2\beta_{4} - \beta_{3} - \beta_{2} - 6\beta _1 + 9 ) / 4 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
127.1
−1.20134 + 0.746179i
−1.20134 − 0.746179i
1.35935 − 0.390070i
1.35935 + 0.390070i
0.841985 + 1.13625i
0.841985 − 1.13625i
−1.58777 − 1.21613i 0 1.04204 + 3.86188i 6.80536 0 2.64575i 3.04204 − 7.39905i 0 −10.8054 − 8.27622i
127.2 −1.58777 + 1.21613i 0 1.04204 − 3.86188i 6.80536 0 − 2.64575i 3.04204 + 7.39905i 0 −10.8054 + 8.27622i
127.3 0.163664 − 1.99329i 0 −3.94643 − 0.652459i −3.43742 0 − 2.64575i −1.94643 + 7.75960i 0 −0.562581 + 6.85178i
127.4 0.163664 + 1.99329i 0 −3.94643 + 0.652459i −3.43742 0 2.64575i −1.94643 − 7.75960i 0 −0.562581 − 6.85178i
127.5 1.92411 − 0.545716i 0 3.40439 − 2.10003i −1.36794 0 − 2.64575i 5.40439 − 5.89853i 0 −2.63206 + 0.746506i
127.6 1.92411 + 0.545716i 0 3.40439 + 2.10003i −1.36794 0 2.64575i 5.40439 + 5.89853i 0 −2.63206 − 0.746506i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 127.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.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.g.a 6
3.b odd 2 1 28.3.c.a ✓ 6
4.b odd 2 1 inner 252.3.g.a 6
12.b even 2 1 28.3.c.a ✓ 6
21.c even 2 1 196.3.c.g 6
21.g even 6 2 196.3.g.j 12
21.h odd 6 2 196.3.g.k 12
24.f even 2 1 448.3.d.d 6
24.h odd 2 1 448.3.d.d 6
48.i odd 4 2 1792.3.g.g 12
48.k even 4 2 1792.3.g.g 12
84.h odd 2 1 196.3.c.g 6
84.j odd 6 2 196.3.g.j 12
84.n even 6 2 196.3.g.k 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.3.c.a ✓ 6 3.b odd 2 1
28.3.c.a ✓ 6 12.b even 2 1
196.3.c.g 6 21.c even 2 1
196.3.c.g 6 84.h odd 2 1
196.3.g.j 12 21.g even 6 2
196.3.g.j 12 84.j odd 6 2
196.3.g.k 12 21.h odd 6 2
196.3.g.k 12 84.n even 6 2
252.3.g.a 6 1.a even 1 1 trivial
252.3.g.a 6 4.b odd 2 1 inner
448.3.d.d 6 24.f even 2 1
448.3.d.d 6 24.h odd 2 1
1792.3.g.g 12 48.i odd 4 2
1792.3.g.g 12 48.k even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{3} - 2T_{5}^{2} - 28T_{5} - 32 \) acting on \(S_{3}^{\mathrm{new}}(252, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} - T^{5} + \cdots + 64 \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( (T^{3} - 2 T^{2} - 28 T - 32)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 7)^{3} \) Copy content Toggle raw display
$11$ \( T^{6} + 512 T^{4} + \cdots + 3469312 \) Copy content Toggle raw display
$13$ \( (T^{3} - 6 T^{2} + \cdots + 1712)^{2} \) Copy content Toggle raw display
$17$ \( (T^{3} - 2 T^{2} + \cdots + 488)^{2} \) Copy content Toggle raw display
$19$ \( T^{6} + 1600 T^{4} + \cdots + 81683392 \) Copy content Toggle raw display
$23$ \( T^{6} + 928 T^{4} + \cdots + 114688 \) Copy content Toggle raw display
$29$ \( (T^{3} - 18 T^{2} + \cdots - 4952)^{2} \) Copy content Toggle raw display
$31$ \( T^{6} + 2048 T^{4} + \cdots + 1404928 \) Copy content Toggle raw display
$37$ \( (T^{3} - 14 T^{2} + \cdots - 4328)^{2} \) Copy content Toggle raw display
$41$ \( (T^{3} - 10 T^{2} + \cdots + 15368)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + \cdots + 1477439488 \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots + 4610486272 \) Copy content Toggle raw display
$53$ \( (T^{3} + 46 T^{2} + \cdots - 8536)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} + \cdots + 3560148928 \) Copy content Toggle raw display
$61$ \( (T^{3} + 82 T^{2} + \cdots - 7552)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots + 92073361408 \) Copy content Toggle raw display
$71$ \( T^{6} + \cdots + 1438646272 \) Copy content Toggle raw display
$73$ \( (T^{3} + 66 T^{2} + \cdots - 3688)^{2} \) Copy content Toggle raw display
$79$ \( T^{6} + \cdots + 5754585088 \) Copy content Toggle raw display
$83$ \( T^{6} + 25600 T^{4} + \cdots + 95209408 \) Copy content Toggle raw display
$89$ \( (T^{3} + 174 T^{2} + \cdots - 620888)^{2} \) Copy content Toggle raw display
$97$ \( (T^{3} - 126 T^{2} + \cdots + 196184)^{2} \) Copy content Toggle raw display
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