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Properties

Label 320.4.n.g
Level $320$
Weight $4$
Character orbit 320.n
Analytic conductor $18.881$
Analytic rank $0$
Dimension $8$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [320,4,Mod(63,320)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(320, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 0, 3])) 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("320.63"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Level: \( N \) \(=\) \( 320 = 2^{6} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 320.n (of order \(4\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,-2,0,14] 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: no
Analytic conductor: \(18.8806112018\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} + 2x^{6} + 22x^{5} + 532x^{4} - 636x^{3} + 450x^{2} + 2160x + 5184 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: no (minimal twist has level 160)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{3} + (\beta_{6} + 2) q^{5} + (\beta_{5} - \beta_{4} + \beta_{2} + \cdots + 1) q^{7} + (\beta_{7} - \beta_{5} + 5 \beta_1) q^{9} + (\beta_{7} + \beta_{6} + \cdots + 6 \beta_1) q^{11}+ \cdots + ( - 3 \beta_{7} - 15 \beta_{6} + \cdots - 242) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{3} + 14 q^{5} + 10 q^{7} + 32 q^{13} + 22 q^{15} + 44 q^{17} - 80 q^{19} - 236 q^{21} + 230 q^{23} - 44 q^{25} - 80 q^{27} - 260 q^{33} + 866 q^{35} + 292 q^{37} + 1068 q^{39} + 932 q^{41} - 458 q^{43}+ \cdots - 1844 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} + 2x^{6} + 22x^{5} + 532x^{4} - 636x^{3} + 450x^{2} + 2160x + 5184 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 12827 \nu^{7} + 76486 \nu^{6} - 98830 \nu^{5} - 215714 \nu^{4} - 5831540 \nu^{3} + \cdots - 13371048 ) / 111061800 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 11087 \nu^{7} + 110714 \nu^{6} - 138830 \nu^{5} - 430606 \nu^{4} + 8851640 \nu^{3} + \cdots - 172204272 ) / 55530900 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 4549 \nu^{7} + 31262 \nu^{6} - 19214 \nu^{5} + 59186 \nu^{4} - 2042464 \nu^{3} + \cdots + 33215616 ) / 11106180 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 4361 \nu^{7} + 8822 \nu^{6} - 108590 \nu^{5} + 228422 \nu^{4} + 3185780 \nu^{3} + 4136484 \nu^{2} + \cdots + 33234264 ) / 7404120 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 7537 \nu^{7} + 25076 \nu^{6} - 99680 \nu^{5} - 37954 \nu^{4} - 4290130 \nu^{3} + 8815752 \nu^{2} + \cdots + 5405022 ) / 5553090 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 386857 \nu^{7} + 729026 \nu^{6} - 697730 \nu^{5} - 8895574 \nu^{4} - 197999140 \nu^{3} + \cdots - 839823768 ) / 111061800 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 33307 \nu^{7} - 67126 \nu^{6} + 66380 \nu^{5} + 759274 \nu^{4} + 16985890 \nu^{3} + \cdots + 61844418 ) / 9255150 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} + \beta_{6} + \beta _1 + 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} - \beta_{5} - 5\beta_{3} - 5\beta_{2} + 52\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -5\beta_{7} + 5\beta_{6} - 21\beta_{5} + 21\beta_{4} - 10\beta_{2} + 30\beta _1 - 30 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -11\beta_{7} - 27\beta_{6} - 11\beta_{5} + 27\beta_{4} + 140\beta_{3} - 140\beta_{2} - 1094 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -497\beta_{7} - 497\beta_{6} - 129\beta_{5} - 129\beta_{4} + 390\beta_{3} - 1022\beta _1 - 1022 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -379\beta_{7} - 261\beta_{6} + 379\beta_{5} - 261\beta_{4} + 1760\beta_{3} + 1760\beta_{2} - 12922\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 1499\beta_{7} - 1499\beta_{6} + 6060\beta_{5} - 6060\beta_{4} + 6130\beta_{2} - 16923\beta _1 + 16923 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/320\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(257\) \(261\)
\(\chi(n)\) \(-1\) \(\beta_{1}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
63.1
3.63424 − 3.63424i
−3.22067 + 3.22067i
−0.981077 + 0.981077i
1.56751 − 1.56751i
3.63424 + 3.63424i
−3.22067 − 3.22067i
−0.981077 − 0.981077i
1.56751 + 1.56751i
0 −4.50577 + 4.50577i 0 6.30193 − 9.23502i 0 19.0427 + 19.0427i 0 − 13.6039i 0
63.2 0 −3.64336 + 3.64336i 0 −0.725933 + 11.1567i 0 −9.23932 − 9.23932i 0 0.451865i 0
63.3 0 2.18370 − 2.18370i 0 −9.23146 − 6.30715i 0 −6.10801 − 6.10801i 0 17.4629i 0
63.4 0 4.96543 − 4.96543i 0 10.6555 + 3.38543i 0 1.30461 + 1.30461i 0 − 22.3109i 0
127.1 0 −4.50577 − 4.50577i 0 6.30193 + 9.23502i 0 19.0427 − 19.0427i 0 13.6039i 0
127.2 0 −3.64336 − 3.64336i 0 −0.725933 − 11.1567i 0 −9.23932 + 9.23932i 0 − 0.451865i 0
127.3 0 2.18370 + 2.18370i 0 −9.23146 + 6.30715i 0 −6.10801 + 6.10801i 0 − 17.4629i 0
127.4 0 4.96543 + 4.96543i 0 10.6555 − 3.38543i 0 1.30461 − 1.30461i 0 22.3109i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 63.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
20.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 320.4.n.g 8
4.b odd 2 1 320.4.n.h 8
5.c odd 4 1 320.4.n.h 8
8.b even 2 1 160.4.n.e yes 8
8.d odd 2 1 160.4.n.d ✓ 8
20.e even 4 1 inner 320.4.n.g 8
40.i odd 4 1 160.4.n.d ✓ 8
40.k even 4 1 160.4.n.e yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.4.n.d ✓ 8 8.d odd 2 1
160.4.n.d ✓ 8 40.i odd 4 1
160.4.n.e yes 8 8.b even 2 1
160.4.n.e yes 8 40.k even 4 1
320.4.n.g 8 1.a even 1 1 trivial
320.4.n.g 8 20.e even 4 1 inner
320.4.n.h 8 4.b odd 2 1
320.4.n.h 8 5.c odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(320, [\chi])\):

\( T_{3}^{8} + 2T_{3}^{7} + 2T_{3}^{6} - 8T_{3}^{5} + 2420T_{3}^{4} + 5768T_{3}^{3} + 6728T_{3}^{2} - 82592T_{3} + 506944 \) Copy content Toggle raw display
\( T_{13}^{8} - 32 T_{13}^{7} + 512 T_{13}^{6} + 63776 T_{13}^{5} + 9540264 T_{13}^{4} + \cdots + 4682705569936 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + 2 T^{7} + \cdots + 506944 \) Copy content Toggle raw display
$5$ \( T^{8} - 14 T^{7} + \cdots + 244140625 \) Copy content Toggle raw display
$7$ \( T^{8} - 10 T^{7} + \cdots + 31449664 \) Copy content Toggle raw display
$11$ \( T^{8} + 2124 T^{6} + \cdots + 92160000 \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 4682705569936 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 598290621600400 \) Copy content Toggle raw display
$19$ \( (T^{4} + 40 T^{3} + \cdots + 8404736)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 71\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{4} - 466 T^{3} + \cdots - 3094032000)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 47\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 11\!\cdots\!96 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 41\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( (T^{4} + 8 T^{3} + \cdots + 16035860224)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 1282 T^{3} + \cdots - 61321952)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 39\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 55\!\cdots\!84 \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 16\!\cdots\!04 \) Copy content Toggle raw display
$79$ \( (T^{4} - 1608 T^{3} + \cdots + 217899008)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 49\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 41\!\cdots\!76 \) Copy content Toggle raw display
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