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Properties

Label 882.3.n.c
Level $882$
Weight $3$
Character orbit 882.n
Analytic conductor $24.033$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [882,3,Mod(19,882)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(882, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 5]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("882.19");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 882.n (of order \(6\), degree \(2\), not 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: \(24.0327593166\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 126)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{3} - \beta_1) q^{2} + ( - 2 \beta_{2} - 2) q^{4} + ( - 2 \beta_{3} + 2 \beta_1) q^{5} + 2 \beta_{3} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} - \beta_1) q^{2} + ( - 2 \beta_{2} - 2) q^{4} + ( - 2 \beta_{3} + 2 \beta_1) q^{5} + 2 \beta_{3} q^{8} + ( - 4 \beta_{2} + 4) q^{10} - 12 \beta_1 q^{11} + ( - 2 \beta_{2} - 1) q^{13} + 4 \beta_{2} q^{16} + (4 \beta_{3} + 2 \beta_1) q^{17} + ( - 17 \beta_{2} - 34) q^{19} + ( - 4 \beta_{3} - 8 \beta_1) q^{20} - 24 q^{22} + (6 \beta_{3} + 6 \beta_1) q^{23} + ( - \beta_{2} - 1) q^{25} + (\beta_{3} - \beta_1) q^{26} - 24 \beta_{3} q^{29} + ( - 7 \beta_{2} + 7) q^{31} + 4 \beta_1 q^{32} + (8 \beta_{2} + 4) q^{34} - 47 \beta_{2} q^{37} + (34 \beta_{3} + 17 \beta_1) q^{38} + ( - 8 \beta_{2} - 16) q^{40} + ( - 28 \beta_{3} - 56 \beta_1) q^{41} + 31 q^{43} + (24 \beta_{3} + 24 \beta_1) q^{44} + (12 \beta_{2} + 12) q^{46} + (34 \beta_{3} - 34 \beta_1) q^{47} + \beta_{3} q^{50} + (2 \beta_{2} - 2) q^{52} - 54 \beta_1 q^{53} + ( - 96 \beta_{2} - 48) q^{55} - 48 \beta_{2} q^{58} + (68 \beta_{3} + 34 \beta_1) q^{59} + (48 \beta_{2} + 96) q^{61} + ( - 7 \beta_{3} - 14 \beta_1) q^{62} + 8 q^{64} + ( - 6 \beta_{3} - 6 \beta_1) q^{65} + (31 \beta_{2} + 31) q^{67} + ( - 4 \beta_{3} + 4 \beta_1) q^{68} + 42 \beta_{3} q^{71} + ( - 47 \beta_{2} + 47) q^{73} - 47 \beta_1 q^{74} + (68 \beta_{2} + 34) q^{76} + 41 \beta_{2} q^{79} + (16 \beta_{3} + 8 \beta_1) q^{80} + ( - 56 \beta_{2} - 112) q^{82} + (2 \beta_{3} + 4 \beta_1) q^{83} - 24 q^{85} + ( - 31 \beta_{3} - 31 \beta_1) q^{86} + (48 \beta_{2} + 48) q^{88} + ( - 24 \beta_{3} + 24 \beta_1) q^{89} - 12 \beta_{3} q^{92} + (68 \beta_{2} - 68) q^{94} - 102 \beta_1 q^{95} + (48 \beta_{2} + 24) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} + 24 q^{10} - 8 q^{16} - 102 q^{19} - 96 q^{22} - 2 q^{25} + 42 q^{31} + 94 q^{37} - 48 q^{40} + 124 q^{43} + 24 q^{46} - 12 q^{52} + 96 q^{58} + 288 q^{61} + 32 q^{64} + 62 q^{67} + 282 q^{73} - 82 q^{79} - 336 q^{82} - 96 q^{85} + 96 q^{88} - 408 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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

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

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

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

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/882\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(785\)
\(\chi(n)\) \(-\beta_{2}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
−0.707107 − 1.22474i
0.707107 + 1.22474i
−0.707107 + 1.22474i
0.707107 − 1.22474i
−0.707107 + 1.22474i 0 −1.00000 − 1.73205i −4.24264 − 2.44949i 0 0 2.82843 0 6.00000 − 3.46410i
19.2 0.707107 − 1.22474i 0 −1.00000 − 1.73205i 4.24264 + 2.44949i 0 0 −2.82843 0 6.00000 − 3.46410i
325.1 −0.707107 − 1.22474i 0 −1.00000 + 1.73205i −4.24264 + 2.44949i 0 0 2.82843 0 6.00000 + 3.46410i
325.2 0.707107 + 1.22474i 0 −1.00000 + 1.73205i 4.24264 − 2.44949i 0 0 −2.82843 0 6.00000 + 3.46410i
\(n\): 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 882.3.n.c 4
3.b odd 2 1 inner 882.3.n.c 4
7.b odd 2 1 126.3.n.b ✓ 4
7.c even 3 1 126.3.n.b ✓ 4
7.c even 3 1 882.3.c.c 4
7.d odd 6 1 882.3.c.c 4
7.d odd 6 1 inner 882.3.n.c 4
21.c even 2 1 126.3.n.b ✓ 4
21.g even 6 1 882.3.c.c 4
21.g even 6 1 inner 882.3.n.c 4
21.h odd 6 1 126.3.n.b ✓ 4
21.h odd 6 1 882.3.c.c 4
28.d even 2 1 1008.3.cg.i 4
28.g odd 6 1 1008.3.cg.i 4
84.h odd 2 1 1008.3.cg.i 4
84.n even 6 1 1008.3.cg.i 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.3.n.b ✓ 4 7.b odd 2 1
126.3.n.b ✓ 4 7.c even 3 1
126.3.n.b ✓ 4 21.c even 2 1
126.3.n.b ✓ 4 21.h odd 6 1
882.3.c.c 4 7.c even 3 1
882.3.c.c 4 7.d odd 6 1
882.3.c.c 4 21.g even 6 1
882.3.c.c 4 21.h odd 6 1
882.3.n.c 4 1.a even 1 1 trivial
882.3.n.c 4 3.b odd 2 1 inner
882.3.n.c 4 7.d odd 6 1 inner
882.3.n.c 4 21.g even 6 1 inner
1008.3.cg.i 4 28.d even 2 1
1008.3.cg.i 4 28.g odd 6 1
1008.3.cg.i 4 84.h odd 2 1
1008.3.cg.i 4 84.n even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(882, [\chi])\):

\( T_{5}^{4} - 24T_{5}^{2} + 576 \) Copy content Toggle raw display
\( T_{23}^{4} + 72T_{23}^{2} + 5184 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 2T^{2} + 4 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 24T^{2} + 576 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} + 288 T^{2} + 82944 \) Copy content Toggle raw display
$13$ \( (T^{2} + 3)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} - 24T^{2} + 576 \) Copy content Toggle raw display
$19$ \( (T^{2} + 51 T + 867)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 72T^{2} + 5184 \) Copy content Toggle raw display
$29$ \( (T^{2} - 1152)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 21 T + 147)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 47 T + 2209)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 4704)^{2} \) Copy content Toggle raw display
$43$ \( (T - 31)^{4} \) Copy content Toggle raw display
$47$ \( T^{4} - 6936 T^{2} + 48108096 \) Copy content Toggle raw display
$53$ \( T^{4} + 5832 T^{2} + 34012224 \) Copy content Toggle raw display
$59$ \( T^{4} - 6936 T^{2} + 48108096 \) Copy content Toggle raw display
$61$ \( (T^{2} - 144 T + 6912)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 31 T + 961)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 3528)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 141 T + 6627)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 41 T + 1681)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 24)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} - 3456 T^{2} + 11943936 \) Copy content Toggle raw display
$97$ \( (T^{2} + 1728)^{2} \) Copy content Toggle raw display
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