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Properties

Label 1792.4.a.u
Level $1792$
Weight $4$
Character orbit 1792.a
Self dual yes
Analytic conductor $105.731$
Analytic rank $1$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1792,4,Mod(1,1792)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1792, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1792.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1792 = 2^{8} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1792.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(105.731422730\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} - 30x^{6} + 56x^{5} + 281x^{4} - 490x^{3} - 814x^{2} + 1248x + 240 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{14} \)
Twist minimal: no (minimal twist has level 56)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
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.

\(f(q)\) \(=\) \( q - \beta_{4} q^{3} - \beta_1 q^{5} - 7 q^{7} + ( - \beta_{6} + 4) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{3} - \beta_1 q^{5} - 7 q^{7} + ( - \beta_{6} + 4) q^{9} + (2 \beta_{4} - \beta_{2}) q^{11} + ( - \beta_{5} - 3 \beta_{4} + 2 \beta_1) q^{13} + ( - \beta_{3} + 7) q^{15} + ( - \beta_{7} + 2 \beta_{6} + \cdots - 17) q^{17}+ \cdots + (54 \beta_{5} + 140 \beta_{4} + \cdots + 2 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 56 q^{7} + 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 56 q^{7} + 36 q^{9} + 60 q^{15} - 144 q^{17} + 260 q^{23} + 28 q^{25} + 264 q^{31} - 512 q^{33} + 700 q^{39} - 712 q^{41} + 392 q^{49} - 72 q^{55} - 740 q^{57} - 252 q^{63} - 2172 q^{65} + 1312 q^{71} - 1392 q^{73} + 608 q^{79} - 188 q^{81} - 4088 q^{87} - 1848 q^{89} - 356 q^{95} - 2200 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} - 30x^{6} + 56x^{5} + 281x^{4} - 490x^{3} - 814x^{2} + 1248x + 240 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -163\nu^{7} + 714\nu^{6} + 3582\nu^{5} - 15392\nu^{4} - 19215\nu^{3} + 93586\nu^{2} + 9810\nu - 144456 ) / 7092 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 52\nu^{7} + 501\nu^{6} - 2448\nu^{5} - 10840\nu^{4} + 30060\nu^{3} + 50669\nu^{2} - 99720\nu + 12408 ) / 1773 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 59\nu^{7} + 57\nu^{6} - 2232\nu^{5} + 1612\nu^{4} + 22923\nu^{3} - 37127\nu^{2} - 47952\nu + 85953 ) / 1773 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -83\nu^{7} + 30\nu^{6} + 1998\nu^{5} - 64\nu^{4} - 12975\nu^{3} - 3034\nu^{2} + 17874\nu + 8472 ) / 2364 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -89\nu^{7} - 96\nu^{6} + 2826\nu^{5} + 2096\nu^{4} - 26445\nu^{3} - 12040\nu^{2} + 69750\nu + 19224 ) / 1182 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -55\nu^{7} + 27\nu^{6} + 1680\nu^{5} + 100\nu^{4} - 15519\nu^{3} - 7301\nu^{2} + 37008\nu + 21927 ) / 591 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 259\nu^{7} - 471\nu^{6} - 6192\nu^{5} + 6836\nu^{4} + 47979\nu^{3} - 12727\nu^{2} - 141264\nu - 14574 ) / 1773 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{6} + 2\beta_{5} - 2\beta_{4} + \beta_{3} + 2\beta _1 + 4 ) / 16 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{5} - \beta_{2} + 2\beta _1 + 64 ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 4\beta_{7} - 9\beta_{6} + 22\beta_{5} - 14\beta_{4} + 13\beta_{3} + 30\beta _1 + 40 ) / 16 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{7} + 3\beta_{6} - 16\beta_{5} + 4\beta_{3} - 16\beta_{2} + 32\beta _1 + 727 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 76\beta_{7} - 83\beta_{6} + 278\beta_{5} - 198\beta_{4} + 175\beta_{3} - 8\beta_{2} + 454\beta _1 + 712 ) / 16 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 24\beta_{7} + 72\beta_{6} - 215\beta_{5} - 80\beta_{4} + 96\beta_{3} - 223\beta_{2} + 542\beta _1 + 9256 ) / 8 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 1220 \beta_{7} - 759 \beta_{6} + 3626 \beta_{5} - 3522 \beta_{4} + 2459 \beta_{3} - 256 \beta_{2} + \cdots + 14272 ) / 16 \) Copy content Toggle raw display

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

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.

comment: embeddings in the coefficient field
 
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
3.21068
−2.14334
−3.61322
−0.174697
3.99618
1.71599
−3.37729
2.38570
0 −8.87222 0 −1.57179 0 −7.00000 0 51.7163 0
1.2 0 −5.24924 0 −9.85277 0 −7.00000 0 0.554549 0
1.3 0 −3.82805 0 −2.56837 0 −7.00000 0 −12.3460 0
1.4 0 −2.25282 0 20.1954 0 −7.00000 0 −21.9248 0
1.5 0 2.25282 0 −20.1954 0 −7.00000 0 −21.9248 0
1.6 0 3.82805 0 2.56837 0 −7.00000 0 −12.3460 0
1.7 0 5.24924 0 9.85277 0 −7.00000 0 0.554549 0
1.8 0 8.87222 0 1.57179 0 −7.00000 0 51.7163 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(7\) \(1\)

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1792.4.a.u 8
4.b odd 2 1 1792.4.a.w 8
8.b even 2 1 inner 1792.4.a.u 8
8.d odd 2 1 1792.4.a.w 8
16.e even 4 2 224.4.b.a 8
16.f odd 4 2 56.4.b.a ✓ 8
48.i odd 4 2 2016.4.c.a 8
48.k even 4 2 504.4.c.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.4.b.a ✓ 8 16.f odd 4 2
224.4.b.a 8 16.e even 4 2
504.4.c.a 8 48.k even 4 2
1792.4.a.u 8 1.a even 1 1 trivial
1792.4.a.u 8 8.b even 2 1 inner
1792.4.a.w 8 4.b odd 2 1
1792.4.a.w 8 8.d odd 2 1
2016.4.c.a 8 48.i odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1792))\):

\( T_{3}^{8} - 126T_{3}^{6} + 4340T_{3}^{4} - 50696T_{3}^{2} + 161312 \) Copy content Toggle raw display
\( T_{5}^{8} - 514T_{5}^{6} + 44188T_{5}^{4} - 367224T_{5}^{2} + 645248 \) Copy content Toggle raw display
\( T_{23}^{4} - 130T_{23}^{3} - 19220T_{23}^{2} + 2840136T_{23} - 85969984 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - 126 T^{6} + \cdots + 161312 \) Copy content Toggle raw display
$5$ \( T^{8} - 514 T^{6} + \cdots + 645248 \) Copy content Toggle raw display
$7$ \( (T + 7)^{8} \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 223860787200 \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 8031006848 \) Copy content Toggle raw display
$17$ \( (T^{4} + 72 T^{3} + \cdots - 1875824)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 61589268512 \) Copy content Toggle raw display
$23$ \( (T^{4} - 130 T^{3} + \cdots - 85969984)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 60\!\cdots\!32 \) Copy content Toggle raw display
$31$ \( (T^{4} - 132 T^{3} + \cdots + 793521152)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 64\!\cdots\!08 \) Copy content Toggle raw display
$41$ \( (T^{4} + 356 T^{3} + \cdots - 1202210128)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 18\!\cdots\!88 \) Copy content Toggle raw display
$47$ \( (T^{4} - 381312 T^{2} + \cdots + 12485394432)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 31\!\cdots\!92 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 13\!\cdots\!12 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 13\!\cdots\!48 \) Copy content Toggle raw display
$71$ \( (T^{4} - 656 T^{3} + \cdots + 31205572608)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 696 T^{3} + \cdots - 78638316912)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 304 T^{3} + \cdots - 117180334080)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 49\!\cdots\!68 \) Copy content Toggle raw display
$89$ \( (T^{4} + 924 T^{3} + \cdots + 3549530480)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots + 2140769108912)^{2} \) Copy content Toggle raw display
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