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Properties

Label 847.4.a.c
Level $847$
Weight $4$
Character orbit 847.a
Self dual yes
Analytic conductor $49.975$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [847,4,Mod(1,847)] 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("847.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(847, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 847 = 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 847.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(49.9746177749\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 77)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(\beta = 2\sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta + 1) q^{2} - 2 q^{3} + (2 \beta + 1) q^{4} + (3 \beta - 2) q^{5} + ( - 2 \beta - 2) q^{6} - 7 q^{7} + ( - 5 \beta + 9) q^{8} - 23 q^{9} + (\beta + 22) q^{10} + ( - 4 \beta - 2) q^{12} + (2 \beta + 40) q^{13}+ \cdots + (49 \beta + 49) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 4 q^{3} + 2 q^{4} - 4 q^{5} - 4 q^{6} - 14 q^{7} + 18 q^{8} - 46 q^{9} + 44 q^{10} - 4 q^{12} + 80 q^{13} - 14 q^{14} + 8 q^{15} - 78 q^{16} - 48 q^{17} - 46 q^{18} + 96 q^{19} + 92 q^{20}+ \cdots + 98 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.41421
1.41421
−1.82843 −2.00000 −4.65685 −10.4853 3.65685 −7.00000 23.1421 −23.0000 19.1716
1.2 3.82843 −2.00000 6.65685 6.48528 −7.65685 −7.00000 −5.14214 −23.0000 24.8284
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(7\) \( +1 \)
\(11\) \( -1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 847.4.a.c 2
11.b odd 2 1 77.4.a.b ✓ 2
33.d even 2 1 693.4.a.i 2
44.c even 2 1 1232.4.a.m 2
55.d odd 2 1 1925.4.a.l 2
77.b even 2 1 539.4.a.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.4.a.b ✓ 2 11.b odd 2 1
539.4.a.d 2 77.b even 2 1
693.4.a.i 2 33.d even 2 1
847.4.a.c 2 1.a even 1 1 trivial
1232.4.a.m 2 44.c even 2 1
1925.4.a.l 2 55.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 2T_{2} - 7 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(847))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 2T - 7 \) Copy content Toggle raw display
$3$ \( (T + 2)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 4T - 68 \) Copy content Toggle raw display
$7$ \( (T + 7)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 80T + 1568 \) Copy content Toggle raw display
$17$ \( T^{2} + 48T - 7112 \) Copy content Toggle raw display
$19$ \( T^{2} - 96T + 2232 \) Copy content Toggle raw display
$23$ \( T^{2} + 56T - 1808 \) Copy content Toggle raw display
$29$ \( T^{2} + 4T - 64796 \) Copy content Toggle raw display
$31$ \( T^{2} - 60T - 39428 \) Copy content Toggle raw display
$37$ \( T^{2} + 380T + 34948 \) Copy content Toggle raw display
$41$ \( T^{2} + 112T - 60232 \) Copy content Toggle raw display
$43$ \( (T - 316)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 604T + 15932 \) Copy content Toggle raw display
$53$ \( T^{2} - 676T + 87332 \) Copy content Toggle raw display
$59$ \( T^{2} + 516T + 35812 \) Copy content Toggle raw display
$61$ \( T^{2} - 296T + 14704 \) Copy content Toggle raw display
$67$ \( T^{2} + 1152 T + 280576 \) Copy content Toggle raw display
$71$ \( T^{2} + 1064 T + 112496 \) Copy content Toggle raw display
$73$ \( T^{2} + 168T - 232376 \) Copy content Toggle raw display
$79$ \( T^{2} - 368T - 532192 \) Copy content Toggle raw display
$83$ \( T^{2} + 1728 T + 621496 \) Copy content Toggle raw display
$89$ \( T^{2} - 1524 T + 472996 \) Copy content Toggle raw display
$97$ \( T^{2} - 1404 T + 399492 \) Copy content Toggle raw display
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