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Properties

Label 784.5.c.g
Level $784$
Weight $5$
Character orbit 784.c
Analytic conductor $81.042$
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] = [784,5,Mod(97,784)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(784, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 5, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("784.97"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Level: \( N \) \(=\) \( 784 = 2^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 784.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,0,0,0,0,0,-352,0,-120] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(81.0420510577\)
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: \( x^{8} - 4x^{7} - 66x^{6} + 212x^{5} + 2021x^{4} - 4400x^{3} - 25028x^{2} + 27264x + 127778 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2\cdot 7^{6} \)
Twist minimal: no (minimal twist has level 49)
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.

\(f(q)\) \(=\) \( q - \beta_{3} q^{3} + ( - \beta_{6} + \beta_{3}) q^{5} + ( - \beta_{4} + \beta_{2} + 9 \beta_1 - 44) q^{9} + ( - \beta_{4} + 7 \beta_{2} + \cdots - 15) q^{11} + ( - \beta_{7} + 5 \beta_{6} + \cdots + 8 \beta_{3}) q^{13}+ \cdots + (167 \beta_{4} - 156 \beta_{2} + \cdots - 3768) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 352 q^{9} - 120 q^{11} + 632 q^{15} - 1752 q^{23} - 2192 q^{25} + 1248 q^{29} + 2368 q^{37} + 7672 q^{39} + 8552 q^{43} + 11976 q^{51} + 5496 q^{53} - 9200 q^{57} + 30240 q^{65} + 7440 q^{67} - 9984 q^{71}+ \cdots - 30144 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} - 66x^{6} + 212x^{5} + 2021x^{4} - 4400x^{3} - 25028x^{2} + 27264x + 127778 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -12\nu^{6} + 36\nu^{5} + 475\nu^{4} - 1010\nu^{3} - 6245\nu^{2} + 6756\nu - 53158 ) / 8647 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 5824 \nu^{7} - 20384 \nu^{6} - 314032 \nu^{5} + 836040 \nu^{4} + 9000124 \nu^{3} + \cdots + 83321259 ) / 50783831 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 9708 \nu^{7} + 171577 \nu^{6} - 1219847 \nu^{5} - 10775638 \nu^{4} + 39790506 \nu^{3} + \cdots - 2400548024 ) / 50783831 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 19754 \nu^{7} - 104377 \nu^{6} + 435732 \nu^{5} + 742640 \nu^{4} - 44219994 \nu^{3} + \cdots - 679070112 ) / 50783831 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 29366 \nu^{7} + 308336 \nu^{6} + 1166068 \nu^{5} - 17082323 \nu^{4} - 17174395 \nu^{3} + \cdots - 2600778684 ) / 50783831 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 99621 \nu^{7} - 345737 \nu^{6} - 5759093 \nu^{5} + 17247149 \nu^{4} + 145325670 \nu^{3} + \cdots + 625456972 ) / 50783831 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 199363 \nu^{7} - 700707 \nu^{6} - 11478351 \nu^{5} + 28462571 \nu^{4} + 303036972 \nu^{3} + \cdots + 972632654 ) / 50783831 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 2\beta_{7} + 2\beta_{6} + 8\beta_{5} - 8\beta_{3} - 49\beta_{2} + 49 ) / 98 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -16\beta_{7} + 40\beta_{6} + 20\beta_{5} + 8\beta_{3} - 49\beta_{2} + 84\beta _1 + 1813 ) / 98 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 92\beta_{7} + 176\beta_{6} + 774\beta_{5} - 56\beta_{4} - 732\beta_{3} - 847\beta_{2} + 154\beta _1 + 2695 ) / 98 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -112\beta_{7} + 336\beta_{6} + 256\beta_{5} - 16\beta_{4} - 200\beta_{3} - 235\beta_{2} + 190\beta _1 + 3003 ) / 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 256\beta_{7} + 1356\beta_{6} + 4812\beta_{5} + 58\beta_{4} - 4682\beta_{3} - 87\beta_{2} + 396\beta _1 + 6867 ) / 14 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 23948 \beta_{7} + 87072 \beta_{6} + 100936 \beta_{5} + 1498 \beta_{4} - 100796 \beta_{3} + \cdots - 600593 ) / 98 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 670 \beta_{7} + 361048 \beta_{6} + 987708 \beta_{5} + 129752 \beta_{4} - 990354 \beta_{3} + \cdots - 2267181 ) / 98 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/784\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(687\) \(689\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
97.1
3.71722 + 0.765367i
−2.71722 − 0.765367i
−4.99086 − 1.84776i
5.99086 − 1.84776i
5.99086 + 1.84776i
−4.99086 + 1.84776i
−2.71722 + 0.765367i
3.71722 − 0.765367i
0 − 16.1337i 0 − 7.83726i 0 0 0 −179.298 0
97.2 0 − 12.5695i 0 33.6557i 0 0 0 −76.9935 0
97.3 0 − 7.89087i 0 − 22.0095i 0 0 0 18.7341 0
97.4 0 − 4.40939i 0 43.7887i 0 0 0 61.5573 0
97.5 0 4.40939i 0 − 43.7887i 0 0 0 61.5573 0
97.6 0 7.89087i 0 22.0095i 0 0 0 18.7341 0
97.7 0 12.5695i 0 − 33.6557i 0 0 0 −76.9935 0
97.8 0 16.1337i 0 7.83726i 0 0 0 −179.298 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 97.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 784.5.c.g 8
4.b odd 2 1 49.5.b.b ✓ 8
7.b odd 2 1 inner 784.5.c.g 8
12.b even 2 1 441.5.d.g 8
28.d even 2 1 49.5.b.b ✓ 8
28.f even 6 2 49.5.d.c 16
28.g odd 6 2 49.5.d.c 16
84.h odd 2 1 441.5.d.g 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
49.5.b.b ✓ 8 4.b odd 2 1
49.5.b.b ✓ 8 28.d even 2 1
49.5.d.c 16 28.f even 6 2
49.5.d.c 16 28.g odd 6 2
441.5.d.g 8 12.b even 2 1
441.5.d.g 8 84.h odd 2 1
784.5.c.g 8 1.a even 1 1 trivial
784.5.c.g 8 7.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 500T_{3}^{6} + 76514T_{3}^{4} + 3866688T_{3}^{2} + 49787136 \) acting on \(S_{5}^{\mathrm{new}}(784, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + 500 T^{6} + \cdots + 49787136 \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 64623740944 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{4} + 60 T^{3} + \cdots - 27217952)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 11\!\cdots\!16 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 2775229473604 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 56\!\cdots\!56 \) Copy content Toggle raw display
$23$ \( (T^{4} + 876 T^{3} + \cdots - 38047796288)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 624 T^{3} + \cdots - 124444104944)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 16\!\cdots\!24 \) Copy content Toggle raw display
$37$ \( (T^{4} - 1184 T^{3} + \cdots + 374530303744)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 51\!\cdots\!16 \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots - 4296135337184)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 52\!\cdots\!64 \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots + 4515238365184)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 85\!\cdots\!36 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 24\!\cdots\!36 \) Copy content Toggle raw display
$67$ \( (T^{4} - 3720 T^{3} + \cdots + 327162240000)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots + 29541809698816)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 19\!\cdots\!16 \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots + 166340550164224)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 10\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 43\!\cdots\!96 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 57\!\cdots\!76 \) Copy content Toggle raw display
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