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Properties

Label 882.3.c.g
Level $882$
Weight $3$
Character orbit 882.c
Analytic conductor $24.033$
Analytic rank $0$
Dimension $8$
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(685,882)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(882, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("882.685");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 882.c (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: \(24.0327593166\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.339738624.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{6} + 14x^{4} - 8x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{10}\cdot 7^{2} \)
Twist minimal: no (minimal twist has level 294)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + 2 q^{4} + ( - \beta_{4} - \beta_{2}) q^{5} + 2 \beta_1 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + 2 q^{4} + ( - \beta_{4} - \beta_{2}) q^{5} + 2 \beta_1 q^{8} + (\beta_{6} - \beta_{3}) q^{10} + (\beta_{5} + 2 \beta_1) q^{11} + (2 \beta_{4} - \beta_{3} - \beta_{2}) q^{13} + 4 q^{16} + ( - \beta_{6} + 3 \beta_{4} + \cdots + 3 \beta_{2}) q^{17}+ \cdots + ( - 4 \beta_{6} + 2 \beta_{4} + \cdots - 9 \beta_{2}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 16 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 16 q^{4} + 32 q^{16} + 32 q^{22} - 32 q^{23} - 72 q^{25} + 160 q^{29} + 128 q^{37} + 320 q^{43} - 96 q^{46} - 16 q^{50} - 384 q^{53} + 80 q^{58} + 64 q^{64} + 288 q^{65} - 352 q^{67} + 256 q^{71} - 48 q^{74} + 288 q^{79} + 768 q^{85} - 256 q^{86} + 64 q^{88} - 64 q^{92} + 32 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{6} + 14x^{4} - 8x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{6} + 20 ) / 14 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 10\nu^{7} - 35\nu^{5} + 126\nu^{3} - 10\nu ) / 14 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 8\nu^{7} - 28\nu^{5} + 91\nu^{3} - 8\nu ) / 7 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -\nu^{6} + 4\nu^{4} - 12\nu^{2} + 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 10\nu^{7} - 56\nu^{5} + 182\nu^{3} - 192\nu ) / 7 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 8\nu^{6} - 28\nu^{4} + 112\nu^{2} - 36 ) / 7 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 16\nu^{7} - 70\nu^{5} + 252\nu^{3} - 268\nu ) / 7 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -3\beta_{7} + 2\beta_{5} - 4\beta_{3} + 12\beta_{2} ) / 56 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{6} + \beta_{4} - 2\beta _1 + 4 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -5\beta_{3} + 8\beta_{2} ) / 7 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 3\beta_{6} + 4\beta_{4} + 8\beta _1 - 12 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 13\beta_{7} - 18\beta_{5} - 36\beta_{3} + 52\beta_{2} ) / 28 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 14\beta _1 - 20 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 22\beta_{7} - 31\beta_{5} + 62\beta_{3} - 88\beta_{2} ) / 14 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(199\) \(785\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
685.1
1.60021 + 0.923880i
−1.60021 − 0.923880i
−1.60021 + 0.923880i
1.60021 − 0.923880i
−0.662827 + 0.382683i
0.662827 − 0.382683i
0.662827 + 0.382683i
−0.662827 − 0.382683i
−1.41421 0 2.00000 − 7.82913i 0 0 −2.82843 0 11.0721i
685.2 −1.41421 0 2.00000 − 1.96883i 0 0 −2.82843 0 2.78434i
685.3 −1.41421 0 2.00000 1.96883i 0 0 −2.82843 0 − 2.78434i
685.4 −1.41421 0 2.00000 7.82913i 0 0 −2.82843 0 − 11.0721i
685.5 1.41421 0 2.00000 − 8.27747i 0 0 2.82843 0 − 11.7061i
685.6 1.41421 0 2.00000 − 1.52049i 0 0 2.82843 0 − 2.15029i
685.7 1.41421 0 2.00000 1.52049i 0 0 2.82843 0 2.15029i
685.8 1.41421 0 2.00000 8.27747i 0 0 2.82843 0 11.7061i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 685.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 882.3.c.g 8
3.b odd 2 1 294.3.c.b ✓ 8
7.b odd 2 1 inner 882.3.c.g 8
7.c even 3 1 882.3.n.f 8
7.c even 3 1 882.3.n.k 8
7.d odd 6 1 882.3.n.f 8
7.d odd 6 1 882.3.n.k 8
12.b even 2 1 2352.3.f.i 8
21.c even 2 1 294.3.c.b ✓ 8
21.g even 6 1 294.3.g.d 8
21.g even 6 1 294.3.g.e 8
21.h odd 6 1 294.3.g.d 8
21.h odd 6 1 294.3.g.e 8
84.h odd 2 1 2352.3.f.i 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
294.3.c.b ✓ 8 3.b odd 2 1
294.3.c.b ✓ 8 21.c even 2 1
294.3.g.d 8 21.g even 6 1
294.3.g.d 8 21.h odd 6 1
294.3.g.e 8 21.g even 6 1
294.3.g.e 8 21.h odd 6 1
882.3.c.g 8 1.a even 1 1 trivial
882.3.c.g 8 7.b odd 2 1 inner
882.3.n.f 8 7.c even 3 1
882.3.n.f 8 7.d odd 6 1
882.3.n.k 8 7.c even 3 1
882.3.n.k 8 7.d odd 6 1
2352.3.f.i 8 12.b even 2 1
2352.3.f.i 8 84.h odd 2 1

Hecke kernels

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

\( T_{5}^{8} + 136T_{5}^{6} + 5012T_{5}^{4} + 27152T_{5}^{2} + 37636 \) Copy content Toggle raw display
\( T_{23}^{4} + 16T_{23}^{3} - 1296T_{23}^{2} - 30464T_{23} - 128576 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 2)^{4} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + 136 T^{6} + \cdots + 37636 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{4} - 496 T^{2} + \cdots + 52672)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + 520 T^{6} + \cdots + 7761796 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 30806568324 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 79803990016 \) Copy content Toggle raw display
$23$ \( (T^{4} + 16 T^{3} + \cdots - 128576)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 80 T^{3} + \cdots - 17852)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 415024562176 \) Copy content Toggle raw display
$37$ \( (T^{4} - 64 T^{3} + \cdots + 2543428)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 187372973956 \) Copy content Toggle raw display
$43$ \( (T^{4} - 160 T^{3} + \cdots - 11136512)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 1408 T^{6} + \cdots + 802816 \) Copy content Toggle raw display
$53$ \( (T^{4} + 192 T^{3} + \cdots + 3218944)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 490403282944 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 7215530758276 \) Copy content Toggle raw display
$67$ \( (T^{4} + 176 T^{3} + \cdots - 3082496)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} - 128 T^{3} + \cdots + 252352)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 305317680648964 \) Copy content Toggle raw display
$79$ \( (T^{4} - 144 T^{3} + \cdots + 1033984)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 264508604416 \) Copy content Toggle raw display
$89$ \( T^{8} + 33032 T^{6} + \cdots + 526335364 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 22348518766084 \) Copy content Toggle raw display
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