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Properties

Label 882.3.n.i
Level $882$
Weight $3$
Character orbit 882.n
Analytic conductor $24.033$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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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: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.77720518656.7
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 12x^{6} + 119x^{4} + 300x^{2} + 625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{2} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} + ( - 2 \beta_1 - 2) q^{4} + ( - \beta_{7} - \beta_{6}) q^{5} + 2 \beta_{4} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{2} + ( - 2 \beta_1 - 2) q^{4} + ( - \beta_{7} - \beta_{6}) q^{5} + 2 \beta_{4} q^{8} + \beta_{2} q^{10} + ( - 9 \beta_{4} - 9 \beta_{3}) q^{11} + ( - 2 \beta_{5} + 2 \beta_{2}) q^{13} + 4 \beta_1 q^{16} + \beta_{7} q^{17} + \beta_{5} q^{19} + 2 \beta_{6} q^{20} + 18 q^{22} + 15 \beta_{3} q^{23} + (41 \beta_1 + 41) q^{25} + (4 \beta_{7} + 4 \beta_{6}) q^{26} - 24 \beta_{4} q^{29} + ( - 4 \beta_{4} - 4 \beta_{3}) q^{32} + (\beta_{5} - \beta_{2}) q^{34} + 16 \beta_1 q^{37} - 2 \beta_{7} q^{38} - 2 \beta_{5} q^{40} - 7 \beta_{6} q^{41} + 52 q^{43} + 18 \beta_{3} q^{44} + ( - 30 \beta_1 - 30) q^{46} + ( - 4 \beta_{7} - 4 \beta_{6}) q^{47} - 41 \beta_{4} q^{50} - 4 \beta_{2} q^{52} + (12 \beta_{4} + 12 \beta_{3}) q^{53} + (9 \beta_{5} - 9 \beta_{2}) q^{55} - 48 \beta_1 q^{58} - 4 \beta_{7} q^{59} - 2 \beta_{5} q^{61} + 8 q^{64} + 132 \beta_{3} q^{65} + (52 \beta_1 + 52) q^{67} + ( - 2 \beta_{7} - 2 \beta_{6}) q^{68} - 63 \beta_{4} q^{71} - 4 \beta_{2} q^{73} + ( - 16 \beta_{4} - 16 \beta_{3}) q^{74} + ( - 2 \beta_{5} + 2 \beta_{2}) q^{76} + 104 \beta_1 q^{79} + 4 \beta_{7} q^{80} + 7 \beta_{5} q^{82} + 20 \beta_{6} q^{83} - 66 q^{85} + 52 \beta_{3} q^{86} + ( - 36 \beta_1 - 36) q^{88} + (9 \beta_{7} + 9 \beta_{6}) q^{89} + 30 \beta_{4} q^{92} + 4 \beta_{2} q^{94} + ( - 66 \beta_{4} - 66 \beta_{3}) q^{95} + ( - 8 \beta_{5} + 8 \beta_{2}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{4} - 16 q^{16} + 144 q^{22} + 164 q^{25} - 64 q^{37} + 416 q^{43} - 120 q^{46} + 192 q^{58} + 64 q^{64} + 208 q^{67} - 416 q^{79} - 528 q^{85} - 144 q^{88}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 12x^{6} + 119x^{4} + 300x^{2} + 625 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -12\nu^{6} - 119\nu^{4} - 1428\nu^{2} - 3600 ) / 2975 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 44\nu^{6} + 1428\nu^{4} + 11186\nu^{2} + 48900 ) / 2975 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{7} + 533\nu ) / 595 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 12\nu^{7} + 119\nu^{5} + 1003\nu^{3} + 625\nu ) / 2125 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -194\nu^{6} - 1428\nu^{4} - 11186\nu^{2} + 13200 ) / 2975 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 22\nu^{7} + 289\nu^{5} + 3043\nu^{3} + 13825\nu ) / 2125 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 229\nu^{7} + 2023\nu^{5} + 21301\nu^{3} - 32450\nu ) / 14875 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{7} + \beta_{6} - 3\beta_{3} ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{5} - \beta_{2} - 36\beta _1 - 36 ) / 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 14\beta_{7} + 7\beta_{6} - 51\beta_{4} ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -2\beta_{5} + 4\beta_{2} + 47\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -59\beta_{7} - 118\beta_{6} + 537\beta_{4} + 537\beta_{3} ) / 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -119\beta_{5} - 119\beta_{2} + 2484 ) / 6 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -533\beta_{7} + 533\beta_{6} - 5169\beta_{3} ) / 6 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
−0.819051 + 1.41864i
1.52616 − 2.64338i
−1.52616 + 2.64338i
0.819051 − 1.41864i
−0.819051 − 1.41864i
1.52616 + 2.64338i
−1.52616 − 2.64338i
0.819051 + 1.41864i
−0.707107 + 1.22474i 0 −1.00000 − 1.73205i −7.03562 − 4.06202i 0 0 2.82843 0 9.94987 − 5.74456i
19.2 −0.707107 + 1.22474i 0 −1.00000 − 1.73205i 7.03562 + 4.06202i 0 0 2.82843 0 −9.94987 + 5.74456i
19.3 0.707107 − 1.22474i 0 −1.00000 − 1.73205i −7.03562 − 4.06202i 0 0 −2.82843 0 −9.94987 + 5.74456i
19.4 0.707107 − 1.22474i 0 −1.00000 − 1.73205i 7.03562 + 4.06202i 0 0 −2.82843 0 9.94987 − 5.74456i
325.1 −0.707107 − 1.22474i 0 −1.00000 + 1.73205i −7.03562 + 4.06202i 0 0 2.82843 0 9.94987 + 5.74456i
325.2 −0.707107 − 1.22474i 0 −1.00000 + 1.73205i 7.03562 − 4.06202i 0 0 2.82843 0 −9.94987 − 5.74456i
325.3 0.707107 + 1.22474i 0 −1.00000 + 1.73205i −7.03562 + 4.06202i 0 0 −2.82843 0 −9.94987 − 5.74456i
325.4 0.707107 + 1.22474i 0 −1.00000 + 1.73205i 7.03562 − 4.06202i 0 0 −2.82843 0 9.94987 + 5.74456i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
21.c even 2 1 inner
21.g even 6 1 inner
21.h 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.i 8
3.b odd 2 1 inner 882.3.n.i 8
7.b odd 2 1 inner 882.3.n.i 8
7.c even 3 1 126.3.c.a ✓ 4
7.c even 3 1 inner 882.3.n.i 8
7.d odd 6 1 126.3.c.a ✓ 4
7.d odd 6 1 inner 882.3.n.i 8
21.c even 2 1 inner 882.3.n.i 8
21.g even 6 1 126.3.c.a ✓ 4
21.g even 6 1 inner 882.3.n.i 8
21.h odd 6 1 126.3.c.a ✓ 4
21.h odd 6 1 inner 882.3.n.i 8
28.f even 6 1 1008.3.f.i 4
28.g odd 6 1 1008.3.f.i 4
84.j odd 6 1 1008.3.f.i 4
84.n even 6 1 1008.3.f.i 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.3.c.a ✓ 4 7.c even 3 1
126.3.c.a ✓ 4 7.d odd 6 1
126.3.c.a ✓ 4 21.g even 6 1
126.3.c.a ✓ 4 21.h odd 6 1
882.3.n.i 8 1.a even 1 1 trivial
882.3.n.i 8 3.b odd 2 1 inner
882.3.n.i 8 7.b odd 2 1 inner
882.3.n.i 8 7.c even 3 1 inner
882.3.n.i 8 7.d odd 6 1 inner
882.3.n.i 8 21.c even 2 1 inner
882.3.n.i 8 21.g even 6 1 inner
882.3.n.i 8 21.h odd 6 1 inner
1008.3.f.i 4 28.f even 6 1
1008.3.f.i 4 28.g odd 6 1
1008.3.f.i 4 84.j odd 6 1
1008.3.f.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 \(S_{3}^{\mathrm{new}}(882, [\chi])\):

\( T_{5}^{4} - 66T_{5}^{2} + 4356 \) Copy content Toggle raw display
\( T_{23}^{4} + 450T_{23}^{2} + 202500 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{4} - 66 T^{2} + 4356)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{4} + 162 T^{2} + 26244)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 528)^{4} \) Copy content Toggle raw display
$17$ \( (T^{4} - 66 T^{2} + 4356)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 132 T^{2} + 17424)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 450 T^{2} + 202500)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 1152)^{4} \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( (T^{2} + 16 T + 256)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} + 3234)^{4} \) Copy content Toggle raw display
$43$ \( (T - 52)^{8} \) Copy content Toggle raw display
$47$ \( (T^{4} - 1056 T^{2} + 1115136)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 288 T^{2} + 82944)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 1056 T^{2} + 1115136)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 528 T^{2} + 278784)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 52 T + 2704)^{4} \) Copy content Toggle raw display
$71$ \( (T^{2} - 7938)^{4} \) Copy content Toggle raw display
$73$ \( (T^{4} - 2112 T^{2} + 4460544)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 104 T + 10816)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 26400)^{4} \) Copy content Toggle raw display
$89$ \( (T^{4} - 5346 T^{2} + 28579716)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 8448)^{4} \) Copy content Toggle raw display
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