Properties

Label 1216.4.a.l
Level $1216$
Weight $4$
Character orbit 1216.a
Self dual yes
Analytic conductor $71.746$
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] = [1216,4,Mod(1,1216)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1216, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1216.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Level: \( N \) \(=\) \( 1216 = 2^{6} \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1216.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,1,0,-10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(71.7463225670\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{177}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 44 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 38)
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 \(\beta = \frac{1}{2}(1 + \sqrt{177})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{3} + (2 \beta - 6) q^{5} + ( - \beta - 28) q^{7} + (\beta + 17) q^{9} + (2 \beta + 4) q^{11} + (7 \beta - 10) q^{13} + ( - 4 \beta + 88) q^{15} + ( - 15 \beta - 18) q^{17} - 19 q^{19} + ( - 29 \beta - 44) q^{21}+ \cdots + (40 \beta + 156) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{3} - 10 q^{5} - 57 q^{7} + 35 q^{9} + 10 q^{11} - 13 q^{13} + 172 q^{15} - 51 q^{17} - 38 q^{19} - 117 q^{21} + 155 q^{23} + 154 q^{25} + 79 q^{27} + 79 q^{29} + 16 q^{31} + 182 q^{33} + 108 q^{35}+ \cdots + 352 q^{99}+O(q^{100}) \) 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
−6.15207
7.15207
0 −6.15207 0 −18.3041 0 −21.8479 0 10.8479 0
1.2 0 7.15207 0 8.30413 0 −35.1521 0 24.1521 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(19\) \( +1 \)

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 1216.4.a.l 2
4.b odd 2 1 1216.4.a.j 2
8.b even 2 1 304.4.a.d 2
8.d odd 2 1 38.4.a.b 2
24.f even 2 1 342.4.a.k 2
40.e odd 2 1 950.4.a.h 2
40.k even 4 2 950.4.b.g 4
56.e even 2 1 1862.4.a.b 2
152.b even 2 1 722.4.a.i 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.4.a.b 2 8.d odd 2 1
304.4.a.d 2 8.b even 2 1
342.4.a.k 2 24.f even 2 1
722.4.a.i 2 152.b even 2 1
950.4.a.h 2 40.e odd 2 1
950.4.b.g 4 40.k even 4 2
1216.4.a.j 2 4.b odd 2 1
1216.4.a.l 2 1.a even 1 1 trivial
1862.4.a.b 2 56.e even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1216))\):

\( T_{3}^{2} - T_{3} - 44 \) Copy content Toggle raw display
\( T_{5}^{2} + 10T_{5} - 152 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - T - 44 \) Copy content Toggle raw display
$5$ \( T^{2} + 10T - 152 \) Copy content Toggle raw display
$7$ \( T^{2} + 57T + 768 \) Copy content Toggle raw display
$11$ \( T^{2} - 10T - 152 \) Copy content Toggle raw display
$13$ \( T^{2} + 13T - 2126 \) Copy content Toggle raw display
$17$ \( T^{2} + 51T - 9306 \) Copy content Toggle raw display
$19$ \( (T + 19)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 155T - 1472 \) Copy content Toggle raw display
$29$ \( T^{2} - 79T - 35654 \) Copy content Toggle raw display
$31$ \( T^{2} - 16T - 11264 \) Copy content Toggle raw display
$37$ \( T^{2} + 380T + 24772 \) Copy content Toggle raw display
$41$ \( T^{2} + 790T + 154432 \) Copy content Toggle raw display
$43$ \( T^{2} - 296T - 80048 \) Copy content Toggle raw display
$47$ \( T^{2} - 200T - 60800 \) Copy content Toggle raw display
$53$ \( T^{2} + 397T + 35818 \) Copy content Toggle raw display
$59$ \( T^{2} - 201T - 212964 \) Copy content Toggle raw display
$61$ \( T^{2} - 680T + 29932 \) Copy content Toggle raw display
$67$ \( T^{2} + 939T + 138612 \) Copy content Toggle raw display
$71$ \( T^{2} + 406T + 19792 \) Copy content Toggle raw display
$73$ \( T^{2} - 123T + 3738 \) Copy content Toggle raw display
$79$ \( T^{2} + 106T - 146048 \) Copy content Toggle raw display
$83$ \( T^{2} - 2226 T + 1237176 \) Copy content Toggle raw display
$89$ \( T^{2} + 870T + 184800 \) Copy content Toggle raw display
$97$ \( T^{2} + 1864 T + 613036 \) Copy content Toggle raw display
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