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Properties

Label 882.3.n.a
Level $882$
Weight $3$
Character orbit 882.n
Analytic conductor $24.033$
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] = [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: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 42)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + 2 \beta_{2} q^{4} + ( - 2 \beta_{3} + 2 \beta_{2} + \cdots - 2) q^{5}+ \cdots + 2 \beta_{3} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + 2 \beta_{2} q^{4} + ( - 2 \beta_{3} + 2 \beta_{2} + \cdots - 2) q^{5}+ \cdots + ( - 10 \beta_{3} - 88 \beta_{2} + \cdots - 44) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} - 12 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} - 12 q^{5} + 12 q^{10} - 12 q^{11} - 8 q^{16} + 12 q^{17} + 12 q^{19} - 24 q^{22} + 12 q^{23} - 14 q^{25} - 24 q^{26} - 120 q^{29} - 72 q^{31} + 40 q^{37} - 96 q^{38} - 24 q^{40} - 128 q^{43} - 24 q^{44} + 36 q^{46} + 168 q^{47} - 96 q^{50} + 96 q^{52} - 108 q^{53} + 168 q^{59} - 24 q^{61} + 32 q^{64} + 120 q^{65} - 88 q^{67} - 24 q^{68} + 120 q^{71} + 24 q^{73} + 144 q^{74} - 64 q^{79} + 48 q^{80} + 84 q^{82} + 216 q^{85} - 120 q^{86} + 24 q^{88} - 324 q^{89} - 48 q^{92} + 48 q^{94} - 120 q^{95}+O(q^{100}) \) 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 \(\left(\mathbb{Z}/882\mathbb{Z}\right)^\times\).

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
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 −5.12132 − 2.95680i 0 0 2.82843 0 7.24264 − 4.18154i
19.2 0.707107 − 1.22474i 0 −1.00000 − 1.73205i −0.878680 − 0.507306i 0 0 −2.82843 0 −1.24264 + 0.717439i
325.1 −0.707107 − 1.22474i 0 −1.00000 + 1.73205i −5.12132 + 2.95680i 0 0 2.82843 0 7.24264 + 4.18154i
325.2 0.707107 + 1.22474i 0 −1.00000 + 1.73205i −0.878680 + 0.507306i 0 0 −2.82843 0 −1.24264 − 0.717439i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 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.a 4
3.b odd 2 1 294.3.g.b 4
7.b odd 2 1 882.3.n.d 4
7.c even 3 1 126.3.c.b 4
7.c even 3 1 882.3.n.d 4
7.d odd 6 1 126.3.c.b 4
7.d odd 6 1 inner 882.3.n.a 4
21.c even 2 1 294.3.g.c 4
21.g even 6 1 42.3.c.a ✓ 4
21.g even 6 1 294.3.g.b 4
21.h odd 6 1 42.3.c.a ✓ 4
21.h odd 6 1 294.3.g.c 4
28.f even 6 1 1008.3.f.g 4
28.g odd 6 1 1008.3.f.g 4
84.j odd 6 1 336.3.f.c 4
84.n even 6 1 336.3.f.c 4
105.o odd 6 1 1050.3.f.a 4
105.p even 6 1 1050.3.f.a 4
105.w odd 12 2 1050.3.h.a 8
105.x even 12 2 1050.3.h.a 8
168.s odd 6 1 1344.3.f.f 4
168.v even 6 1 1344.3.f.e 4
168.ba even 6 1 1344.3.f.f 4
168.be odd 6 1 1344.3.f.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.3.c.a ✓ 4 21.g even 6 1
42.3.c.a ✓ 4 21.h odd 6 1
126.3.c.b 4 7.c even 3 1
126.3.c.b 4 7.d odd 6 1
294.3.g.b 4 3.b odd 2 1
294.3.g.b 4 21.g even 6 1
294.3.g.c 4 21.c even 2 1
294.3.g.c 4 21.h odd 6 1
336.3.f.c 4 84.j odd 6 1
336.3.f.c 4 84.n even 6 1
882.3.n.a 4 1.a even 1 1 trivial
882.3.n.a 4 7.d odd 6 1 inner
882.3.n.d 4 7.b odd 2 1
882.3.n.d 4 7.c even 3 1
1008.3.f.g 4 28.f even 6 1
1008.3.f.g 4 28.g odd 6 1
1050.3.f.a 4 105.o odd 6 1
1050.3.f.a 4 105.p even 6 1
1050.3.h.a 8 105.w odd 12 2
1050.3.h.a 8 105.x even 12 2
1344.3.f.e 4 168.v even 6 1
1344.3.f.e 4 168.be odd 6 1
1344.3.f.f 4 168.s odd 6 1
1344.3.f.f 4 168.ba even 6 1

Hecke kernels

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

\( T_{5}^{4} + 12T_{5}^{3} + 54T_{5}^{2} + 72T_{5} + 36 \) Copy content Toggle raw display
\( T_{23}^{4} - 12T_{23}^{3} + 270T_{23}^{2} + 1512T_{23} + 15876 \) 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
$5$ \( T^{4} + 12 T^{3} + \cdots + 36 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} + 12 T^{3} + \cdots + 324 \) Copy content Toggle raw display
$13$ \( T^{4} + 432 T^{2} + 28224 \) Copy content Toggle raw display
$17$ \( T^{4} - 12 T^{3} + \cdots + 509796 \) Copy content Toggle raw display
$19$ \( T^{4} - 12 T^{3} + \cdots + 138384 \) Copy content Toggle raw display
$23$ \( T^{4} - 12 T^{3} + \cdots + 15876 \) Copy content Toggle raw display
$29$ \( (T + 30)^{4} \) Copy content Toggle raw display
$31$ \( T^{4} + 72 T^{3} + \cdots + 186624 \) Copy content Toggle raw display
$37$ \( T^{4} - 40 T^{3} + \cdots + 4804864 \) Copy content Toggle raw display
$41$ \( T^{4} + 1764 T^{2} + 86436 \) Copy content Toggle raw display
$43$ \( (T^{2} + 64 T - 776)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 168 T^{3} + \cdots + 5089536 \) Copy content Toggle raw display
$53$ \( T^{4} + 108 T^{3} + \cdots + 6906384 \) Copy content Toggle raw display
$59$ \( T^{4} - 168 T^{3} + \cdots + 2304 \) Copy content Toggle raw display
$61$ \( T^{4} + 24 T^{3} + \cdots + 2304 \) Copy content Toggle raw display
$67$ \( T^{4} + 88 T^{3} + \cdots + 2715904 \) Copy content Toggle raw display
$71$ \( (T^{2} - 60 T - 5598)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} - 24 T^{3} + \cdots + 16064064 \) Copy content Toggle raw display
$79$ \( T^{4} + 64 T^{3} + \cdots + 87310336 \) Copy content Toggle raw display
$83$ \( T^{4} + 10944 T^{2} + 451584 \) Copy content Toggle raw display
$89$ \( T^{4} + 324 T^{3} + \cdots + 37234404 \) Copy content Toggle raw display
$97$ \( T^{4} + 12816 T^{2} + 27123264 \) Copy content Toggle raw display
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