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Properties

Label 882.3.c.b
Level $882$
Weight $3$
Character orbit 882.c
Analytic conductor $24.033$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [882,3,Mod(685,882)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://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);
 
Copy content sage: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")
 
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

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,8,0,0,0,0,0,0,-24,0,0,0,0,16,0,0,0,0,0,0,48] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(23)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(24.0327593166\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 42)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content 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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + 2 q^{4} + ( - 2 \beta_{3} + 2 \beta_{2}) q^{5} - 2 \beta_1 q^{8} + (2 \beta_{3} - 4 \beta_{2}) q^{10} - 6 q^{11} + (8 \beta_{3} + \beta_{2}) q^{13} + 4 q^{16} + ( - 2 \beta_{3} + 8 \beta_{2}) q^{17}+ \cdots + ( - 4 \beta_{3} + 12 \beta_{2}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{4} - 24 q^{11} + 16 q^{16} + 48 q^{23} - 44 q^{25} - 44 q^{37} + 28 q^{43} - 48 q^{44} + 144 q^{46} + 192 q^{50} + 240 q^{53} + 192 q^{58} + 32 q^{64} + 360 q^{65} - 220 q^{67} - 312 q^{71} - 96 q^{74}+ \cdots - 264 q^{95}+O(q^{100}) \) 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

\(\beta_{1}\)\(=\) \( ( \nu^{3} ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} + 4\nu ) / 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{3} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_1 \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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
−0.707107 + 1.22474i
−0.707107 − 1.22474i
0.707107 + 1.22474i
0.707107 − 1.22474i
−1.41421 0 2.00000 − 8.36308i 0 0 −2.82843 0 11.8272i
685.2 −1.41421 0 2.00000 8.36308i 0 0 −2.82843 0 − 11.8272i
685.3 1.41421 0 2.00000 − 1.43488i 0 0 2.82843 0 − 2.02922i
685.4 1.41421 0 2.00000 1.43488i 0 0 2.82843 0 2.02922i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.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.b 4
3.b odd 2 1 294.3.c.a 4
7.b odd 2 1 inner 882.3.c.b 4
7.c even 3 1 126.3.n.a 4
7.c even 3 1 882.3.n.e 4
7.d odd 6 1 126.3.n.a 4
7.d odd 6 1 882.3.n.e 4
12.b even 2 1 2352.3.f.e 4
21.c even 2 1 294.3.c.a 4
21.g even 6 1 42.3.g.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.g.a 4
28.f even 6 1 1008.3.cg.h 4
28.g odd 6 1 1008.3.cg.h 4
84.h odd 2 1 2352.3.f.e 4
84.j odd 6 1 336.3.bh.e 4
84.n even 6 1 336.3.bh.e 4
105.o odd 6 1 1050.3.p.a 4
105.p even 6 1 1050.3.p.a 4
105.w odd 12 2 1050.3.q.a 8
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.g even 6 1
42.3.g.a ✓ 4 21.h odd 6 1
126.3.n.a 4 7.c even 3 1
126.3.n.a 4 7.d odd 6 1
294.3.c.a 4 3.b odd 2 1
294.3.c.a 4 21.c even 2 1
294.3.g.a 4 21.g even 6 1
294.3.g.a 4 21.h odd 6 1
336.3.bh.e 4 84.j odd 6 1
336.3.bh.e 4 84.n even 6 1
882.3.c.b 4 1.a even 1 1 trivial
882.3.c.b 4 7.b odd 2 1 inner
882.3.n.e 4 7.c even 3 1
882.3.n.e 4 7.d odd 6 1
1008.3.cg.h 4 28.f even 6 1
1008.3.cg.h 4 28.g odd 6 1
1050.3.p.a 4 105.o odd 6 1
1050.3.p.a 4 105.p even 6 1
1050.3.q.a 8 105.w odd 12 2
1050.3.q.a 8 105.x even 12 2
2352.3.f.e 4 12.b even 2 1
2352.3.f.e 4 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 \(S_{3}^{\mathrm{new}}(882, [\chi])\):

\( T_{5}^{4} + 72T_{5}^{2} + 144 \) Copy content Toggle raw display
\( T_{23}^{2} - 24T_{23} - 504 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 2)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 72T^{2} + 144 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( (T + 6)^{4} \) Copy content Toggle raw display
$13$ \( T^{4} + 774 T^{2} + 145161 \) Copy content Toggle raw display
$17$ \( T^{4} + 432 T^{2} + 28224 \) Copy content Toggle raw display
$19$ \( T^{4} + 342 T^{2} + 15129 \) Copy content Toggle raw display
$23$ \( (T^{2} - 24 T - 504)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 1152)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 2166 T^{2} + 423801 \) Copy content Toggle raw display
$37$ \( (T^{2} + 22 T - 167)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 4248 T^{2} + 3732624 \) Copy content Toggle raw display
$43$ \( (T^{2} - 14 T - 23)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 2952 T^{2} + 2039184 \) Copy content Toggle raw display
$53$ \( (T^{2} - 120 T + 2952)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 2448 T^{2} + 1272384 \) Copy content Toggle raw display
$61$ \( T^{4} + 1632 T^{2} + 2304 \) Copy content Toggle raw display
$67$ \( (T^{2} + 110 T - 503)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 156 T + 2556)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 19926 T^{2} + 85322169 \) Copy content Toggle raw display
$79$ \( (T^{2} - 10 T - 8687)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 17928 T^{2} + 69956496 \) Copy content Toggle raw display
$89$ \( (T^{2} + 432)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 1056 T^{2} + 112896 \) Copy content Toggle raw display
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