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Properties

Label 1160.4.a.i
Level $1160$
Weight $4$
Character orbit 1160.a
Self dual yes
Analytic conductor $68.442$
Analytic rank $0$
Dimension $11$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1160,4,Mod(1,1160)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1160, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("1160.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1160 = 2^{3} \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1160.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: \(68.4422156067\)
Analytic rank: \(0\)
Dimension: \(11\)
Coefficient field: \(\mathbb{Q}[x]/(x^{11} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{11} - 3 x^{10} - 200 x^{9} + 418 x^{8} + 13963 x^{7} - 17905 x^{6} - 416127 x^{5} + 230401 x^{4} + \cdots + 1840185 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{9} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{10}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{3} + 5 q^{5} + ( - \beta_{3} - \beta_1 + 4) q^{7} + (\beta_{2} - \beta_1 + 11) q^{9} + (\beta_{4} + 10) q^{11} + (\beta_{6} - \beta_{3} - 2 \beta_1 + 3) q^{13} + ( - 5 \beta_1 + 5) q^{15}+ \cdots + (7 \beta_{10} + \beta_{9} + 14 \beta_{8} + \cdots + 355) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 11 q + 8 q^{3} + 55 q^{5} + 38 q^{7} + 117 q^{9} + 112 q^{11} + 20 q^{13} + 40 q^{15} + 122 q^{17} + 344 q^{19} + 264 q^{21} + 254 q^{23} + 275 q^{25} + 224 q^{27} - 319 q^{29} + 136 q^{31} - 24 q^{33}+ \cdots + 4352 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{11} - 3 x^{10} - 200 x^{9} + 418 x^{8} + 13963 x^{7} - 17905 x^{6} - 416127 x^{5} + 230401 x^{4} + \cdots + 1840185 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 37 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 567988984139 \nu^{10} + 5786728461593 \nu^{9} - 144908114222349 \nu^{8} + \cdots - 37\!\cdots\!76 ) / 39\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 38628969806215 \nu^{10} + 229498016835954 \nu^{9} + \cdots - 11\!\cdots\!05 ) / 23\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 23758935140794 \nu^{10} - 173488238985243 \nu^{9} + \cdots - 48\!\cdots\!74 ) / 11\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 25409973951134 \nu^{10} + 173262108408501 \nu^{9} + \cdots - 10\!\cdots\!33 ) / 11\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 26511482308880 \nu^{10} - 146827469111697 \nu^{9} + \cdots + 79\!\cdots\!38 ) / 11\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 27142944470896 \nu^{10} - 234444356809305 \nu^{9} + \cdots + 25\!\cdots\!29 ) / 11\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 74958689178601 \nu^{10} - 479702534973792 \nu^{9} + \cdots - 17\!\cdots\!05 ) / 23\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 23010207823135 \nu^{10} + 142710821940128 \nu^{9} + \cdots + 14\!\cdots\!59 ) / 39\!\cdots\!44 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 37 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{10} + \beta_{9} + \beta_{8} + 3\beta_{7} + \beta_{6} - \beta_{4} + \beta_{2} + 64\beta _1 + 35 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 6 \beta_{10} - 6 \beta_{9} + 13 \beta_{8} + 2 \beta_{7} - 8 \beta_{6} + 2 \beta_{5} + 4 \beta_{4} + \cdots + 2346 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 224 \beta_{10} + 53 \beta_{9} + 135 \beta_{8} + 356 \beta_{7} + 83 \beta_{6} + 39 \beta_{5} + \cdots + 5016 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 1054 \beta_{10} - 740 \beta_{9} + 1865 \beta_{8} + 474 \beta_{7} - 1071 \beta_{6} + 592 \beta_{5} + \cdots + 182721 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 23047 \beta_{10} + 1217 \beta_{9} + 15917 \beta_{8} + 35882 \beta_{7} + 5789 \beta_{6} + 7299 \beta_{5} + \cdots + 626693 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 140122 \beta_{10} - 75814 \beta_{9} + 213335 \beta_{8} + 80370 \beta_{7} - 114715 \beta_{6} + \cdots + 15838235 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 2357831 \beta_{10} - 141392 \beta_{9} + 1783722 \beta_{8} + 3504439 \beta_{7} + 349434 \beta_{6} + \cdots + 74765904 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 16761642 \beta_{10} - 7492624 \beta_{9} + 22846526 \beta_{8} + 11495994 \beta_{7} - 11370879 \beta_{6} + \cdots + 1465364762 \) 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
10.2720
7.41865
6.60804
4.21958
2.45739
0.0917062
−3.24511
−3.73327
−4.86025
−7.35474
−8.87403
0 −9.27204 0 5.00000 0 −11.3016 0 58.9707 0
1.2 0 −6.41865 0 5.00000 0 14.9661 0 14.1990 0
1.3 0 −5.60804 0 5.00000 0 −0.644657 0 4.45013 0
1.4 0 −3.21958 0 5.00000 0 21.1312 0 −16.6343 0
1.5 0 −1.45739 0 5.00000 0 −30.7361 0 −24.8760 0
1.6 0 0.908294 0 5.00000 0 11.7605 0 −26.1750 0
1.7 0 4.24511 0 5.00000 0 −13.8569 0 −8.97903 0
1.8 0 4.73327 0 5.00000 0 −9.04282 0 −4.59615 0
1.9 0 5.86025 0 5.00000 0 34.7868 0 7.34252 0
1.10 0 8.35474 0 5.00000 0 29.4587 0 42.8017 0
1.11 0 9.87403 0 5.00000 0 −8.52118 0 70.4965 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.11
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(5\) \( -1 \)
\(29\) \( +1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1160.4.a.i ✓ 11
4.b odd 2 1 2320.4.a.ba 11
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1160.4.a.i ✓ 11 1.a even 1 1 trivial
2320.4.a.ba 11 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{11} - 8 T_{3}^{10} - 175 T_{3}^{9} + 1352 T_{3}^{8} + 10077 T_{3}^{7} - 74572 T_{3}^{6} + \cdots + 13817344 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1160))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{11} \) Copy content Toggle raw display
$3$ \( T^{11} - 8 T^{10} + \cdots + 13817344 \) Copy content Toggle raw display
$5$ \( (T - 5)^{11} \) Copy content Toggle raw display
$7$ \( T^{11} + \cdots - 911324951584 \) Copy content Toggle raw display
$11$ \( T^{11} + \cdots - 10\!\cdots\!80 \) Copy content Toggle raw display
$13$ \( T^{11} + \cdots - 51\!\cdots\!48 \) Copy content Toggle raw display
$17$ \( T^{11} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{11} + \cdots + 32\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{11} + \cdots + 51\!\cdots\!84 \) Copy content Toggle raw display
$29$ \( (T + 29)^{11} \) Copy content Toggle raw display
$31$ \( T^{11} + \cdots + 21\!\cdots\!68 \) Copy content Toggle raw display
$37$ \( T^{11} + \cdots - 10\!\cdots\!40 \) Copy content Toggle raw display
$41$ \( T^{11} + \cdots + 58\!\cdots\!88 \) Copy content Toggle raw display
$43$ \( T^{11} + \cdots - 22\!\cdots\!60 \) Copy content Toggle raw display
$47$ \( T^{11} + \cdots + 10\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( T^{11} + \cdots + 45\!\cdots\!60 \) Copy content Toggle raw display
$59$ \( T^{11} + \cdots + 50\!\cdots\!56 \) Copy content Toggle raw display
$61$ \( T^{11} + \cdots + 30\!\cdots\!60 \) Copy content Toggle raw display
$67$ \( T^{11} + \cdots - 95\!\cdots\!56 \) Copy content Toggle raw display
$71$ \( T^{11} + \cdots - 82\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{11} + \cdots + 21\!\cdots\!08 \) Copy content Toggle raw display
$79$ \( T^{11} + \cdots - 13\!\cdots\!72 \) Copy content Toggle raw display
$83$ \( T^{11} + \cdots + 26\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( T^{11} + \cdots + 36\!\cdots\!60 \) Copy content Toggle raw display
$97$ \( T^{11} + \cdots - 74\!\cdots\!44 \) Copy content Toggle raw display
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