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Properties

Label 768.5.g.j
Level $768$
Weight $5$
Character orbit 768.g
Analytic conductor $79.388$
Analytic rank $0$
Dimension $8$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [768,5,Mod(511,768)] 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("768.511"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(768, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 0])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 768 = 2^{8} \cdot 3 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 768.g (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,0,96,0,0,0,-216,0,0,0,-160,0,0,0,240] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(79.3881316484\)
Analytic rank: \(0\)
Dimension: \(8\)
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: \( x^{8} - 2x^{7} - 21x^{6} - 2x^{5} + 439x^{4} - 48x^{3} - 2874x^{2} - 3312x + 19044 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{24}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 384)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{3} + (\beta_{3} + 12) q^{5} + (\beta_{7} - \beta_{5} + \beta_{4}) q^{7} - 27 q^{9} + (4 \beta_{7} - \beta_{4} + 8 \beta_{2}) q^{11} + ( - \beta_{6} + 4 \beta_{3} + \cdots - 20) q^{13}+ \cdots + ( - 108 \beta_{7} + \cdots - 216 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 96 q^{5} - 216 q^{9} - 160 q^{13} + 240 q^{17} - 72 q^{25} - 3360 q^{29} + 1440 q^{33} - 7072 q^{37} + 1776 q^{41} - 2592 q^{45} - 10040 q^{49} - 12960 q^{53} + 3744 q^{57} - 12064 q^{61} + 11520 q^{65}+ \cdots + 2544 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} - 21x^{6} - 2x^{5} + 439x^{4} - 48x^{3} - 2874x^{2} - 3312x + 19044 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 16 \nu^{7} + 5304 \nu^{6} - 1440 \nu^{5} - 12728 \nu^{4} - 330432 \nu^{3} + 340920 \nu^{2} + \cdots + 9278568 ) / 125235 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -32\nu^{7} - 81\nu^{6} + 702\nu^{5} + 2587\nu^{4} - 9234\nu^{3} - 53487\nu^{2} + 39420\nu + 303255 ) / 41745 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 37\nu^{7} + 248\nu^{6} + 465\nu^{5} - 4766\nu^{4} - 11449\nu^{3} + 27480\nu^{2} + 176424\nu + 47196 ) / 11385 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -8\nu^{7} - 32\nu^{6} + 216\nu^{5} + 1072\nu^{4} - 2360\nu^{3} - 14016\nu^{2} + 5376\nu + 86112 ) / 1035 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1192 \nu^{7} - 159 \nu^{6} - 20886 \nu^{5} - 41099 \nu^{4} + 449418 \nu^{3} + 473871 \nu^{2} + \cdots - 6180399 ) / 125235 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1946 \nu^{7} + 7792 \nu^{6} - 58254 \nu^{5} + 13220 \nu^{4} + 359518 \nu^{3} + 395664 \nu^{2} + \cdots + 19449720 ) / 125235 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 2571 \nu^{7} + 3365 \nu^{6} + 42267 \nu^{5} + 59439 \nu^{4} - 615487 \nu^{3} - 553557 \nu^{2} + \cdots + 7048557 ) / 125235 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{6} + 3\beta_{5} - 2\beta_{3} + 9\beta_{2} + 3\beta _1 + 24 ) / 96 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{6} - 3\beta_{5} + 9\beta_{4} - 4\beta_{3} - 185\beta_{2} + 3\beta _1 + 552 ) / 96 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 8\beta_{7} - \beta_{6} + 22\beta_{5} + 2\beta_{4} - 2\beta_{3} + 10\beta_{2} + 560 ) / 32 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 24\beta_{7} + 5\beta_{6} + 69\beta_{5} + 204\beta_{4} - 62\beta_{3} - 1993\beta_{2} + 69\beta _1 - 6072 ) / 96 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 504 \beta_{7} - 11 \beta_{6} + 690 \beta_{5} + 1161 \beta_{4} + 1490 \beta_{3} - 10658 \beta_{2} + \cdots + 33168 ) / 192 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 272\beta_{7} + 111\beta_{6} + 403\beta_{5} + 68\beta_{4} + 222\beta_{3} + 225\beta_{2} - 20872 ) / 16 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 7176 \beta_{7} + 3149 \beta_{6} - 6618 \beta_{5} + 22839 \beta_{4} + 27826 \beta_{3} + \cdots - 556368 ) / 192 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/768\mathbb{Z}\right)^\times\).

\(n\) \(257\) \(511\) \(517\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
511.1
3.51338 − 1.45110i
−2.39109 + 1.95785i
−3.01338 + 2.31712i
2.89109 − 1.09182i
3.51338 + 1.45110i
−2.39109 − 1.95785i
−3.01338 − 2.31712i
2.89109 + 1.09182i
0 − 5.19615i 0 −19.1142 0 6.40010i 0 −27.0000 0
511.2 0 − 5.19615i 0 4.54640 0 − 61.7048i 0 −27.0000 0
511.3 0 − 5.19615i 0 23.5183 0 − 40.3412i 0 −27.0000 0
511.4 0 − 5.19615i 0 39.0495 0 95.6459i 0 −27.0000 0
511.5 0 5.19615i 0 −19.1142 0 − 6.40010i 0 −27.0000 0
511.6 0 5.19615i 0 4.54640 0 61.7048i 0 −27.0000 0
511.7 0 5.19615i 0 23.5183 0 40.3412i 0 −27.0000 0
511.8 0 5.19615i 0 39.0495 0 − 95.6459i 0 −27.0000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 511.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 768.5.g.j 8
4.b odd 2 1 inner 768.5.g.j 8
8.b even 2 1 768.5.g.h 8
8.d odd 2 1 768.5.g.h 8
16.e even 4 2 384.5.b.d ✓ 16
16.f odd 4 2 384.5.b.d ✓ 16
48.i odd 4 2 1152.5.b.m 16
48.k even 4 2 1152.5.b.m 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.5.b.d ✓ 16 16.e even 4 2
384.5.b.d ✓ 16 16.f odd 4 2
768.5.g.h 8 8.b even 2 1
768.5.g.h 8 8.d odd 2 1
768.5.g.j 8 1.a even 1 1 trivial
768.5.g.j 8 4.b odd 2 1 inner
1152.5.b.m 16 48.i odd 4 2
1152.5.b.m 16 48.k even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 48T_{5}^{3} - 80T_{5}^{2} + 18816T_{5} - 79808 \) acting on \(S_{5}^{\mathrm{new}}(768, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{2} + 27)^{4} \) Copy content Toggle raw display
$5$ \( (T^{4} - 48 T^{3} + \cdots - 79808)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 2321893298176 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 47\!\cdots\!56 \) Copy content Toggle raw display
$13$ \( (T^{4} + 80 T^{3} + \cdots - 649371392)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 120 T^{3} + \cdots + 16654958608)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 45\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 33\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{4} + 1680 T^{3} + \cdots - 107277812672)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 76\!\cdots\!24 \) Copy content Toggle raw display
$37$ \( (T^{4} + \cdots - 6856321271552)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots + 4399896700944)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 21\!\cdots\!36 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 16\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots - 27085837717440)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 14\!\cdots\!16 \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots + 53143771552000)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 73\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 41\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{4} + \cdots - 12\!\cdots\!76)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 23\!\cdots\!04 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 27\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots - 21\!\cdots\!48)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots + 319894196086800)^{2} \) Copy content Toggle raw display
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