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Properties

Label 448.4.i.l
Level $448$
Weight $4$
Character orbit 448.i
Analytic conductor $26.433$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [448,4,Mod(65,448)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(448, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2])) 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("448.65"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Level: \( N \) \(=\) \( 448 = 2^{6} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 448.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,1,0,13,0,-20] 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: no
Analytic conductor: \(26.4328556826\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.11163123.4
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 14x^{4} + 49x^{2} + 27 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 56)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{3} q^{3} + (2 \beta_{5} + \beta_{4} + \beta_{3} + \cdots + 5) q^{5} + ( - \beta_{5} + 2 \beta_{4} - \beta_{3} + \cdots - 2) q^{7} + ( - 6 \beta_{5} - 3 \beta_{4} + \cdots - 18) q^{9} + ( - \beta_{5} - 2 \beta_{4} + \cdots - 4 \beta_1) q^{11}+ \cdots + (11 \beta_{5} - 11 \beta_{4} + \cdots - 324) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + q^{3} + 13 q^{5} - 20 q^{7} - 50 q^{9} - 11 q^{11} - 140 q^{13} - 82 q^{15} - 97 q^{17} - 81 q^{19} - 345 q^{21} - 191 q^{23} - 12 q^{25} - 230 q^{27} + 324 q^{29} - 597 q^{31} - 343 q^{33} - 815 q^{35}+ \cdots - 2188 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 14x^{4} + 49x^{2} + 27 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} + 7\nu + 3 ) / 6 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\nu^{2} + 9 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2\nu^{5} + 23\nu^{3} + 6\nu^{2} + 63\nu + 27 ) / 6 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -2\nu^{5} - 4\nu^{4} - 23\nu^{3} - 34\nu^{2} - 51\nu - 27 ) / 6 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -2\nu^{5} + 4\nu^{4} - 23\nu^{3} + 34\nu^{2} - 51\nu + 27 ) / 6 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{5} + \beta_{4} + 2\beta_{3} - \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{2} - 9 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -7\beta_{5} - 7\beta_{4} - 14\beta_{3} + 7\beta_{2} + 24\beta _1 - 12 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 3\beta_{5} - 3\beta_{4} - 17\beta_{2} + 126 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 49\beta_{5} + 49\beta_{4} + 110\beta_{3} - 55\beta_{2} - 276\beta _1 + 138 ) / 4 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/448\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(129\) \(197\)
\(\chi(n)\) \(1\) \(-1 + \beta_{1}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
65.1
0.821510i
2.13755i
− 2.95906i
− 0.821510i
− 2.13755i
2.95906i
0 −3.82512 − 6.62530i 0 7.30445 − 12.6517i 0 −10.7631 − 15.0717i 0 −15.7631 + 27.3025i 0
65.2 0 0.0691041 + 0.119692i 0 −5.04289 + 8.73453i 0 18.4904 + 1.05037i 0 13.4904 − 23.3661i 0
65.3 0 4.25602 + 7.37164i 0 4.23844 − 7.34119i 0 −17.7274 + 5.36106i 0 −22.7274 + 39.3649i 0
193.1 0 −3.82512 + 6.62530i 0 7.30445 + 12.6517i 0 −10.7631 + 15.0717i 0 −15.7631 − 27.3025i 0
193.2 0 0.0691041 − 0.119692i 0 −5.04289 − 8.73453i 0 18.4904 − 1.05037i 0 13.4904 + 23.3661i 0
193.3 0 4.25602 − 7.37164i 0 4.23844 + 7.34119i 0 −17.7274 − 5.36106i 0 −22.7274 − 39.3649i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 65.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 448.4.i.l 6
4.b odd 2 1 448.4.i.k 6
7.c even 3 1 inner 448.4.i.l 6
8.b even 2 1 56.4.i.a ✓ 6
8.d odd 2 1 112.4.i.f 6
24.h odd 2 1 504.4.s.i 6
28.g odd 6 1 448.4.i.k 6
56.h odd 2 1 392.4.i.n 6
56.j odd 6 1 392.4.a.j 3
56.j odd 6 1 392.4.i.n 6
56.k odd 6 1 112.4.i.f 6
56.k odd 6 1 784.4.a.bc 3
56.m even 6 1 784.4.a.bd 3
56.p even 6 1 56.4.i.a ✓ 6
56.p even 6 1 392.4.a.k 3
168.s odd 6 1 504.4.s.i 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.4.i.a ✓ 6 8.b even 2 1
56.4.i.a ✓ 6 56.p even 6 1
112.4.i.f 6 8.d odd 2 1
112.4.i.f 6 56.k odd 6 1
392.4.a.j 3 56.j odd 6 1
392.4.a.k 3 56.p even 6 1
392.4.i.n 6 56.h odd 2 1
392.4.i.n 6 56.j odd 6 1
448.4.i.k 6 4.b odd 2 1
448.4.i.k 6 28.g odd 6 1
448.4.i.l 6 1.a even 1 1 trivial
448.4.i.l 6 7.c even 3 1 inner
504.4.s.i 6 24.h odd 2 1
504.4.s.i 6 168.s odd 6 1
784.4.a.bc 3 56.k odd 6 1
784.4.a.bd 3 56.m even 6 1

Hecke kernels

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

\( T_{3}^{6} - T_{3}^{5} + 66T_{3}^{4} + 47T_{3}^{3} + 4234T_{3}^{2} - 585T_{3} + 81 \) Copy content Toggle raw display
\( T_{11}^{6} + 11T_{11}^{5} + 698T_{11}^{4} + 9851T_{11}^{3} + 422018T_{11}^{2} + 4673123T_{11} + 65593801 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} - T^{5} + \cdots + 81 \) Copy content Toggle raw display
$5$ \( T^{6} - 13 T^{5} + \cdots + 1560001 \) Copy content Toggle raw display
$7$ \( T^{6} + 20 T^{5} + \cdots + 40353607 \) Copy content Toggle raw display
$11$ \( T^{6} + 11 T^{5} + \cdots + 65593801 \) Copy content Toggle raw display
$13$ \( (T^{3} + 70 T^{2} + \cdots - 17752)^{2} \) Copy content Toggle raw display
$17$ \( T^{6} + \cdots + 344049114249 \) Copy content Toggle raw display
$19$ \( T^{6} + \cdots + 15221884129 \) Copy content Toggle raw display
$23$ \( T^{6} + \cdots + 12196012613841 \) Copy content Toggle raw display
$29$ \( (T^{3} - 162 T^{2} + \cdots + 1324296)^{2} \) Copy content Toggle raw display
$31$ \( T^{6} + \cdots + 21744865333321 \) Copy content Toggle raw display
$37$ \( T^{6} + \cdots + 228323372199129 \) Copy content Toggle raw display
$41$ \( (T^{3} - 698 T^{2} + \cdots + 6465192)^{2} \) Copy content Toggle raw display
$43$ \( (T^{3} - 308 T^{2} + \cdots - 38848)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots + 350054034092361 \) Copy content Toggle raw display
$53$ \( T^{6} + \cdots + 8496286395921 \) Copy content Toggle raw display
$59$ \( T^{6} + \cdots + 13\!\cdots\!69 \) Copy content Toggle raw display
$61$ \( T^{6} + \cdots + 98846286969609 \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots + 454704744973249 \) Copy content Toggle raw display
$71$ \( (T^{3} - 224 T^{2} + \cdots + 7551488)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + \cdots + 52\!\cdots\!61 \) Copy content Toggle raw display
$79$ \( T^{6} + \cdots + 88\!\cdots\!69 \) Copy content Toggle raw display
$83$ \( (T^{3} - 1380 T^{2} + \cdots - 4797376)^{2} \) Copy content Toggle raw display
$89$ \( T^{6} + \cdots + 24\!\cdots\!41 \) Copy content Toggle raw display
$97$ \( (T^{3} - 2602 T^{2} + \cdots - 528741144)^{2} \) Copy content Toggle raw display
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