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Properties

Label 448.4.a.r
Level $448$
Weight $4$
Character orbit 448.a
Self dual yes
Analytic conductor $26.433$
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] = [448,4,Mod(1,448)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(448, 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("448.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Level: \( N \) \(=\) \( 448 = 2^{6} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 448.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,-2,0,-22,0,-14] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(26.4328556826\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{57}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 14 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 56)
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 \(\beta = \sqrt{57}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta - 1) q^{3} + ( - \beta - 11) q^{5} - 7 q^{7} + (2 \beta + 31) q^{9} + (6 \beta + 18) q^{11} + (5 \beta - 21) q^{13} + (12 \beta + 68) q^{15} + ( - 6 \beta + 4) q^{17} + (13 \beta - 59) q^{19}+ \cdots + (222 \beta + 1242) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} - 22 q^{5} - 14 q^{7} + 62 q^{9} + 36 q^{11} - 42 q^{13} + 136 q^{15} + 8 q^{17} - 118 q^{19} + 14 q^{21} + 104 q^{23} + 106 q^{25} - 236 q^{27} + 56 q^{29} - 20 q^{31} - 720 q^{33} + 154 q^{35}+ \cdots + 2484 q^{99}+O(q^{100}) \) 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
4.27492
−3.27492
0 −8.54983 0 −18.5498 0 −7.00000 0 46.0997 0
1.2 0 6.54983 0 −3.45017 0 −7.00000 0 15.9003 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(7\) \( +1 \)

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 448.4.a.r 2
4.b odd 2 1 448.4.a.s 2
8.b even 2 1 112.4.a.h 2
8.d odd 2 1 56.4.a.c ✓ 2
24.f even 2 1 504.4.a.i 2
24.h odd 2 1 1008.4.a.x 2
40.e odd 2 1 1400.4.a.i 2
40.k even 4 2 1400.4.g.h 4
56.e even 2 1 392.4.a.h 2
56.h odd 2 1 784.4.a.t 2
56.k odd 6 2 392.4.i.l 4
56.m even 6 2 392.4.i.i 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.4.a.c ✓ 2 8.d odd 2 1
112.4.a.h 2 8.b even 2 1
392.4.a.h 2 56.e even 2 1
392.4.i.i 4 56.m even 6 2
392.4.i.l 4 56.k odd 6 2
448.4.a.r 2 1.a even 1 1 trivial
448.4.a.s 2 4.b odd 2 1
504.4.a.i 2 24.f even 2 1
784.4.a.t 2 56.h odd 2 1
1008.4.a.x 2 24.h odd 2 1
1400.4.a.i 2 40.e odd 2 1
1400.4.g.h 4 40.k even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(448))\):

\( T_{3}^{2} + 2T_{3} - 56 \) Copy content Toggle raw display
\( T_{5}^{2} + 22T_{5} + 64 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 2T - 56 \) Copy content Toggle raw display
$5$ \( T^{2} + 22T + 64 \) Copy content Toggle raw display
$7$ \( (T + 7)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 36T - 1728 \) Copy content Toggle raw display
$13$ \( T^{2} + 42T - 984 \) Copy content Toggle raw display
$17$ \( T^{2} - 8T - 2036 \) Copy content Toggle raw display
$19$ \( T^{2} + 118T - 6152 \) Copy content Toggle raw display
$23$ \( T^{2} - 104T + 1792 \) Copy content Toggle raw display
$29$ \( T^{2} - 56T - 26804 \) Copy content Toggle raw display
$31$ \( T^{2} + 20T - 11072 \) Copy content Toggle raw display
$37$ \( T^{2} + 504T + 57804 \) Copy content Toggle raw display
$41$ \( T^{2} + 544T + 55516 \) Copy content Toggle raw display
$43$ \( T^{2} - 412T + 31264 \) Copy content Toggle raw display
$47$ \( T^{2} - 500T + 51328 \) Copy content Toggle raw display
$53$ \( T^{2} + 268T - 277532 \) Copy content Toggle raw display
$59$ \( T^{2} + 198T + 168 \) Copy content Toggle raw display
$61$ \( T^{2} - 346T - 24848 \) Copy content Toggle raw display
$67$ \( T^{2} - 1008T - 9552 \) Copy content Toggle raw display
$71$ \( T^{2} - 1224 T + 264192 \) Copy content Toggle raw display
$73$ \( T^{2} - 716T + 54292 \) Copy content Toggle raw display
$79$ \( T^{2} + 584T + 11392 \) Copy content Toggle raw display
$83$ \( T^{2} + 1230 T + 252312 \) Copy content Toggle raw display
$89$ \( T^{2} - 596 T - 1370396 \) Copy content Toggle raw display
$97$ \( T^{2} + 856T - 96116 \) Copy content Toggle raw display
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