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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [448,6,Mod(1,448)] 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("448.1"); S:= CuspForms(chi, 6); N := Newforms(S);
 
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, 6, names="a")
 
Level: \( N \) \(=\) \( 448 = 2^{6} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 448.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,-6,0,18] 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: \(71.8519512762\)
Analytic rank: \(0\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 7)
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.

\(f(q)\) \(=\) \( q + ( - 3 \beta - 3) q^{3} + ( - 5 \beta + 9) q^{5} - 49 q^{7} + (18 \beta + 279) q^{9} + ( - 62 \beta + 198) q^{11} + ( - 63 \beta + 175) q^{13} + ( - 12 \beta + 828) q^{15} + ( - 38 \beta + 900) q^{17}+ \cdots + ( - 13734 \beta - 8370) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{3} + 18 q^{5} - 98 q^{7} + 558 q^{9} + 396 q^{11} + 350 q^{13} + 1656 q^{15} + 1800 q^{17} - 3266 q^{19} + 294 q^{21} - 2088 q^{23} - 3238 q^{25} - 6372 q^{27} - 6696 q^{29} + 20 q^{31} + 20016 q^{33}+ \cdots - 16740 q^{99}+O(q^{100}) \) 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 −25.6495 0 −28.7492 0 −49.0000 0 414.897 0
1.2 0 19.6495 0 46.7492 0 −49.0000 0 143.103 0
\(n\): e.g. 2-40 or 80-90
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 448.6.a.u 2
4.b odd 2 1 448.6.a.w 2
8.b even 2 1 112.6.a.h 2
8.d odd 2 1 7.6.a.b ✓ 2
24.f even 2 1 63.6.a.f 2
24.h odd 2 1 1008.6.a.bq 2
40.e odd 2 1 175.6.a.c 2
40.k even 4 2 175.6.b.c 4
56.e even 2 1 49.6.a.f 2
56.h odd 2 1 784.6.a.v 2
56.k odd 6 2 49.6.c.e 4
56.m even 6 2 49.6.c.d 4
88.g even 2 1 847.6.a.c 2
168.e odd 2 1 441.6.a.l 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.6.a.b ✓ 2 8.d odd 2 1
49.6.a.f 2 56.e even 2 1
49.6.c.d 4 56.m even 6 2
49.6.c.e 4 56.k odd 6 2
63.6.a.f 2 24.f even 2 1
112.6.a.h 2 8.b even 2 1
175.6.a.c 2 40.e odd 2 1
175.6.b.c 4 40.k even 4 2
441.6.a.l 2 168.e odd 2 1
448.6.a.u 2 1.a even 1 1 trivial
448.6.a.w 2 4.b odd 2 1
784.6.a.v 2 56.h odd 2 1
847.6.a.c 2 88.g even 2 1
1008.6.a.bq 2 24.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 \(S_{6}^{\mathrm{new}}(\Gamma_0(448))\):

\( T_{3}^{2} + 6T_{3} - 504 \) Copy content Toggle raw display
\( T_{5}^{2} - 18T_{5} - 1344 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 6T - 504 \) Copy content Toggle raw display
$5$ \( T^{2} - 18T - 1344 \) Copy content Toggle raw display
$7$ \( (T + 49)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 396T - 179904 \) Copy content Toggle raw display
$13$ \( T^{2} - 350T - 195608 \) Copy content Toggle raw display
$17$ \( T^{2} - 1800 T + 727692 \) Copy content Toggle raw display
$19$ \( T^{2} + 3266 T + 2662072 \) Copy content Toggle raw display
$23$ \( T^{2} + 2088 T - 3507456 \) Copy content Toggle raw display
$29$ \( T^{2} + 6696 T + 10304172 \) Copy content Toggle raw display
$31$ \( T^{2} - 20T - 4155200 \) Copy content Toggle raw display
$37$ \( T^{2} + 6232 T + 5554156 \) Copy content Toggle raw display
$41$ \( T^{2} + 6048 T - 7848036 \) Copy content Toggle raw display
$43$ \( T^{2} + 3020 T - 324400352 \) Copy content Toggle raw display
$47$ \( T^{2} + 11700 T - 165954432 \) Copy content Toggle raw display
$53$ \( T^{2} + 9468 T + 21794244 \) Copy content Toggle raw display
$59$ \( T^{2} + 43938 T + 422751336 \) Copy content Toggle raw display
$61$ \( T^{2} - 64754 T + 719128816 \) Copy content Toggle raw display
$67$ \( T^{2} - 24784 T + 99708976 \) Copy content Toggle raw display
$71$ \( T^{2} + \cdots + 2121099264 \) Copy content Toggle raw display
$73$ \( T^{2} - 17452 T - 317520812 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots - 2508546944 \) Copy content Toggle raw display
$83$ \( T^{2} - 117558 T - 79919784 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots - 5252421468 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots - 1000631156 \) Copy content Toggle raw display
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