Properties

Label 882.3.n.e
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: \(\Z[a_1, \ldots, a_{19}]\)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{3} - \beta_1) q^{2} + ( - 2 \beta_{2} - 2) q^{4} + ( - 2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 4) q^{5} + 2 \beta_{3} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} - \beta_1) q^{2} + ( - 2 \beta_{2} - 2) q^{4} + ( - 2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 4) q^{5} + 2 \beta_{3} q^{8} + ( - 4 \beta_{3} - 4 \beta_{2} - 2 \beta_1 + 4) q^{10} + (6 \beta_{2} + 6) q^{11} + (8 \beta_{3} - 2 \beta_{2} + 16 \beta_1 - 1) q^{13} + 4 \beta_{2} q^{16} + (4 \beta_{3} + 8 \beta_{2} + 2 \beta_1 - 8) q^{17} + ( - 2 \beta_{3} + 7 \beta_{2} + 2 \beta_1 + 14) q^{19} + ( - 4 \beta_{3} - 8 \beta_{2} - 8 \beta_1 - 4) q^{20} - 6 \beta_{3} q^{22} + ( - 18 \beta_{3} + 12 \beta_{2} - 18 \beta_1) q^{23} + (11 \beta_{2} + 24 \beta_1 + 11) q^{25} + (\beta_{3} + 16 \beta_{2} - \beta_1 + 32) q^{26} + 24 \beta_{3} q^{29} + ( - 12 \beta_{3} + 17 \beta_{2} - 6 \beta_1 - 17) q^{31} + 4 \beta_1 q^{32} + (8 \beta_{3} + 8 \beta_{2} + 16 \beta_1 + 4) q^{34} + (12 \beta_{3} - 11 \beta_{2} + 12 \beta_1) q^{37} + ( - 14 \beta_{3} - 4 \beta_{2} - 7 \beta_1 + 4) q^{38} + (4 \beta_{3} - 8 \beta_{2} - 4 \beta_1 - 16) q^{40} + ( - 4 \beta_{3} + 52 \beta_{2} - 8 \beta_1 + 26) q^{41} + ( - 6 \beta_{3} + 7) q^{43} - 12 \beta_{2} q^{44} + ( - 36 \beta_{2} + 12 \beta_1 - 36) q^{46} + ( - 2 \beta_{3} - 22 \beta_{2} + 2 \beta_1 - 44) q^{47} + ( - 11 \beta_{3} + 48) q^{50} + ( - 32 \beta_{3} + 2 \beta_{2} - 16 \beta_1 - 2) q^{52} + ( - 60 \beta_{2} + 18 \beta_1 - 60) q^{53} + (12 \beta_{3} + 24 \beta_{2} + 24 \beta_1 + 12) q^{55} + 48 \beta_{2} q^{58} + ( - 28 \beta_{3} + 4 \beta_{2} - 14 \beta_1 - 4) q^{59} + ( - 8 \beta_{3} + 12 \beta_{2} + 8 \beta_1 + 24) q^{61} + (17 \beta_{3} - 24 \beta_{2} + 34 \beta_1 - 12) q^{62} + 8 q^{64} + (42 \beta_{3} + 90 \beta_{2} + 42 \beta_1) q^{65} + (55 \beta_{2} + 42 \beta_1 + 55) q^{67} + ( - 4 \beta_{3} + 16 \beta_{2} + 4 \beta_1 + 32) q^{68} + ( - 42 \beta_{3} - 78) q^{71} + ( - 80 \beta_{3} - 11 \beta_{2} - 40 \beta_1 + 11) q^{73} + (24 \beta_{2} - 11 \beta_1 + 24) q^{74} + ( - 4 \beta_{3} - 28 \beta_{2} - 8 \beta_1 - 14) q^{76} + ( - 66 \beta_{3} + 5 \beta_{2} - 66 \beta_1) q^{79} + (16 \beta_{3} + 8 \beta_{2} + 8 \beta_1 - 8) q^{80} + ( - 26 \beta_{3} - 8 \beta_{2} + 26 \beta_1 - 16) q^{82} + (38 \beta_{3} - 20 \beta_{2} + 76 \beta_1 - 10) q^{83} + (60 \beta_{3} - 72) q^{85} + ( - 7 \beta_{3} - 12 \beta_{2} - 7 \beta_1) q^{86} - 12 \beta_1 q^{88} + ( - 12 \beta_{2} - 24) q^{89} + (36 \beta_{3} + 24) q^{92} + (44 \beta_{3} - 4 \beta_{2} + 22 \beta_1 + 4) q^{94} + (66 \beta_{2} + 54 \beta_1 + 66) q^{95} + ( - 4 \beta_{3} - 24 \beta_{2} - 8 \beta_1 - 12) 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 \( 4 q - 4 q^{4} + 12 q^{5} + 24 q^{10} + 12 q^{11} - 8 q^{16} - 48 q^{17} + 42 q^{19} - 24 q^{23} + 22 q^{25} + 96 q^{26} - 102 q^{31} + 22 q^{37} + 24 q^{38} - 48 q^{40} + 28 q^{43} + 24 q^{44} - 72 q^{46} - 132 q^{47} + 192 q^{50} - 12 q^{52} - 120 q^{53} - 96 q^{58} - 24 q^{59} + 72 q^{61} + 32 q^{64} - 180 q^{65} + 110 q^{67} + 96 q^{68} - 312 q^{71} + 66 q^{73} + 48 q^{74} - 10 q^{79} - 48 q^{80} - 48 q^{82} - 288 q^{85} + 24 q^{86} - 72 q^{89} + 96 q^{92} + 24 q^{94} + 132 q^{95}+O(q^{100}) \) Copy content Toggle raw display

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
\(\nu\)\(=\) \( \beta_1 \) 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

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 −1.24264 0.717439i 0 0 2.82843 0 1.75736 1.01461i
19.2 0.707107 1.22474i 0 −1.00000 1.73205i 7.24264 + 4.18154i 0 0 −2.82843 0 10.2426 5.91359i
325.1 −0.707107 1.22474i 0 −1.00000 + 1.73205i −1.24264 + 0.717439i 0 0 2.82843 0 1.75736 + 1.01461i
325.2 0.707107 + 1.22474i 0 −1.00000 + 1.73205i 7.24264 4.18154i 0 0 −2.82843 0 10.2426 + 5.91359i
\(n\): 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.e 4
3.b odd 2 1 294.3.g.a 4
7.b odd 2 1 126.3.n.a 4
7.c even 3 1 126.3.n.a 4
7.c even 3 1 882.3.c.b 4
7.d odd 6 1 882.3.c.b 4
7.d odd 6 1 inner 882.3.n.e 4
21.c even 2 1 42.3.g.a 4
21.g even 6 1 294.3.c.a 4
21.g even 6 1 294.3.g.a 4
21.h odd 6 1 42.3.g.a 4
21.h odd 6 1 294.3.c.a 4
28.d even 2 1 1008.3.cg.h 4
28.g odd 6 1 1008.3.cg.h 4
84.h odd 2 1 336.3.bh.e 4
84.j odd 6 1 2352.3.f.e 4
84.n even 6 1 336.3.bh.e 4
84.n even 6 1 2352.3.f.e 4
105.g even 2 1 1050.3.p.a 4
105.k odd 4 2 1050.3.q.a 8
105.o odd 6 1 1050.3.p.a 4
105.x even 12 2 1050.3.q.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.3.g.a 4 21.c even 2 1
42.3.g.a 4 21.h odd 6 1
126.3.n.a 4 7.b odd 2 1
126.3.n.a 4 7.c even 3 1
294.3.c.a 4 21.g even 6 1
294.3.c.a 4 21.h odd 6 1
294.3.g.a 4 3.b odd 2 1
294.3.g.a 4 21.g even 6 1
336.3.bh.e 4 84.h odd 2 1
336.3.bh.e 4 84.n even 6 1
882.3.c.b 4 7.c even 3 1
882.3.c.b 4 7.d odd 6 1
882.3.n.e 4 1.a even 1 1 trivial
882.3.n.e 4 7.d odd 6 1 inner
1008.3.cg.h 4 28.d even 2 1
1008.3.cg.h 4 28.g odd 6 1
1050.3.p.a 4 105.g even 2 1
1050.3.p.a 4 105.o odd 6 1
1050.3.q.a 8 105.k odd 4 2
1050.3.q.a 8 105.x even 12 2
2352.3.f.e 4 84.j odd 6 1
2352.3.f.e 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 \(S_{3}^{\mathrm{new}}(882, [\chi])\):

\( T_{5}^{4} - 12T_{5}^{3} + 36T_{5}^{2} + 144T_{5} + 144 \) Copy content Toggle raw display
\( T_{23}^{4} + 24T_{23}^{3} + 1080T_{23}^{2} - 12096T_{23} + 254016 \) 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
$5$ \( T^{4} - 12 T^{3} + 36 T^{2} + \cdots + 144 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} - 6 T + 36)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 774 T^{2} + 145161 \) Copy content Toggle raw display
$17$ \( T^{4} + 48 T^{3} + 936 T^{2} + \cdots + 28224 \) Copy content Toggle raw display
$19$ \( T^{4} - 42 T^{3} + 711 T^{2} + \cdots + 15129 \) Copy content Toggle raw display
$23$ \( T^{4} + 24 T^{3} + 1080 T^{2} + \cdots + 254016 \) Copy content Toggle raw display
$29$ \( (T^{2} - 1152)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 102 T^{3} + 4119 T^{2} + \cdots + 423801 \) Copy content Toggle raw display
$37$ \( T^{4} - 22 T^{3} + 651 T^{2} + \cdots + 27889 \) Copy content Toggle raw display
$41$ \( T^{4} + 4248 T^{2} + \cdots + 3732624 \) Copy content Toggle raw display
$43$ \( (T^{2} - 14 T - 23)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 132 T^{3} + 7236 T^{2} + \cdots + 2039184 \) Copy content Toggle raw display
$53$ \( T^{4} + 120 T^{3} + 11448 T^{2} + \cdots + 8714304 \) Copy content Toggle raw display
$59$ \( T^{4} + 24 T^{3} - 936 T^{2} + \cdots + 1272384 \) Copy content Toggle raw display
$61$ \( T^{4} - 72 T^{3} + 1776 T^{2} + \cdots + 2304 \) Copy content Toggle raw display
$67$ \( T^{4} - 110 T^{3} + 12603 T^{2} + \cdots + 253009 \) Copy content Toggle raw display
$71$ \( (T^{2} + 156 T + 2556)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} - 66 T^{3} - 7785 T^{2} + \cdots + 85322169 \) Copy content Toggle raw display
$79$ \( T^{4} + 10 T^{3} + 8787 T^{2} + \cdots + 75463969 \) Copy content Toggle raw display
$83$ \( T^{4} + 17928 T^{2} + \cdots + 69956496 \) Copy content Toggle raw display
$89$ \( (T^{2} + 36 T + 432)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 1056 T^{2} + 112896 \) Copy content Toggle raw display
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