Properties

Label 1160.4.a.d
Level $1160$
Weight $4$
Character orbit 1160.a
Self dual yes
Analytic conductor $68.442$
Analytic rank $1$
Dimension $9$
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: \(1\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - 2x^{8} - 153x^{7} + 229x^{6} + 7393x^{5} - 8331x^{4} - 115371x^{3} + 125775x^{2} + 306882x + 29241 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{7}\cdot 3 \)
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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{3} + 5 q^{5} + ( - \beta_{4} - \beta_1 - 4) q^{7} + (\beta_{2} - \beta_1 + 8) q^{9} + (\beta_{8} - \beta_1 - 12) q^{11} + ( - \beta_{8} + \beta_{6} + \beta_{5} + \cdots - 2) q^{13}+ \cdots + ( - 14 \beta_{8} + 6 \beta_{7} + \cdots - 316) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q - 7 q^{3} + 45 q^{5} - 39 q^{7} + 72 q^{9} - 108 q^{11} - 19 q^{13} - 35 q^{15} + 91 q^{17} - 36 q^{19} - 200 q^{21} - 209 q^{23} + 225 q^{25} - 316 q^{27} + 261 q^{29} - 599 q^{31} - 192 q^{33} - 195 q^{35}+ \cdots - 2946 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - 2x^{8} - 153x^{7} + 229x^{6} + 7393x^{5} - 8331x^{4} - 115371x^{3} + 125775x^{2} + 306882x + 29241 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 34 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 28 \nu^{8} - 975 \nu^{7} - 1044 \nu^{6} + 117166 \nu^{5} - 14679 \nu^{4} - 3863979 \nu^{3} + \cdots + 38859669 ) / 940248 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 181 \nu^{8} - 175 \nu^{7} + 27270 \nu^{6} + 41381 \nu^{5} - 1262884 \nu^{4} - 2051001 \nu^{3} + \cdots - 25186491 ) / 1880496 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 92 \nu^{8} - 113 \nu^{7} + 13380 \nu^{6} + 32914 \nu^{5} - 532169 \nu^{4} - 2243565 \nu^{3} + \cdots + 28309851 ) / 940248 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 121 \nu^{8} + 606 \nu^{7} - 20058 \nu^{6} - 83507 \nu^{5} + 1004139 \nu^{4} + 3331332 \nu^{3} + \cdots + 30254310 ) / 940248 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 151 \nu^{8} - 335 \nu^{7} - 23664 \nu^{6} + 37675 \nu^{5} + 1167610 \nu^{4} - 1259181 \nu^{3} + \cdots + 33459831 ) / 470124 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 771 \nu^{8} + 1403 \nu^{7} + 112698 \nu^{6} - 142149 \nu^{5} - 4990984 \nu^{4} + \cdots - 32740713 ) / 1880496 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 34 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{6} - \beta_{5} + 4\beta_{4} + \beta_{3} + \beta_{2} + 59\beta _1 + 12 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3\beta_{8} + 8\beta_{6} + 5\beta_{5} - 5\beta_{4} + 7\beta_{3} + 78\beta_{2} + 115\beta _1 + 1940 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 33 \beta_{8} - 18 \beta_{7} + 207 \beta_{6} - 45 \beta_{5} + 429 \beta_{4} + 84 \beta_{3} + \cdots + 2129 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 90 \beta_{8} - 288 \beta_{7} + 1009 \beta_{6} + 598 \beta_{5} - 340 \beta_{4} + 851 \beta_{3} + \cdots + 124104 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 4413 \beta_{8} - 3132 \beta_{7} + 18700 \beta_{6} - 530 \beta_{5} + 35891 \beta_{4} + 6476 \beta_{3} + \cdots + 227797 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 10650 \beta_{8} - 44478 \beta_{7} + 102783 \beta_{6} + 56766 \beta_{5} - 8676 \beta_{4} + \cdots + 8364403 \) Copy content Toggle raw display

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
−8.24176
−6.79757
−5.39224
−1.18047
−0.0997306
2.82150
4.19773
7.85421
8.83834
0 −9.24176 0 5.00000 0 27.4318 0 58.4102 0
1.2 0 −7.79757 0 5.00000 0 −15.5907 0 33.8021 0
1.3 0 −6.39224 0 5.00000 0 −27.1625 0 13.8607 0
1.4 0 −2.18047 0 5.00000 0 2.96567 0 −22.2456 0
1.5 0 −1.09973 0 5.00000 0 9.86968 0 −25.7906 0
1.6 0 1.82150 0 5.00000 0 −22.7027 0 −23.6822 0
1.7 0 3.19773 0 5.00000 0 17.9152 0 −16.7745 0
1.8 0 6.85421 0 5.00000 0 −8.52252 0 19.9802 0
1.9 0 7.83834 0 5.00000 0 −23.2039 0 34.4396 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
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.d 9
4.b odd 2 1 2320.4.a.y 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1160.4.a.d 9 1.a even 1 1 trivial
2320.4.a.y 9 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{9} + 7 T_{3}^{8} - 133 T_{3}^{7} - 814 T_{3}^{6} + 5568 T_{3}^{5} + 26700 T_{3}^{4} + \cdots + 345664 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1160))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{9} \) Copy content Toggle raw display
$3$ \( T^{9} + 7 T^{8} + \cdots + 345664 \) Copy content Toggle raw display
$5$ \( (T - 5)^{9} \) Copy content Toggle raw display
$7$ \( T^{9} + \cdots + 27349061184 \) Copy content Toggle raw display
$11$ \( T^{9} + \cdots - 609992679168 \) Copy content Toggle raw display
$13$ \( T^{9} + \cdots - 12955239884160 \) Copy content Toggle raw display
$17$ \( T^{9} + \cdots + 65\!\cdots\!56 \) Copy content Toggle raw display
$19$ \( T^{9} + \cdots + 44\!\cdots\!08 \) Copy content Toggle raw display
$23$ \( T^{9} + \cdots + 75\!\cdots\!56 \) Copy content Toggle raw display
$29$ \( (T - 29)^{9} \) Copy content Toggle raw display
$31$ \( T^{9} + \cdots - 58\!\cdots\!56 \) Copy content Toggle raw display
$37$ \( T^{9} + \cdots - 15\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{9} + \cdots - 10\!\cdots\!60 \) Copy content Toggle raw display
$43$ \( T^{9} + \cdots + 13\!\cdots\!56 \) Copy content Toggle raw display
$47$ \( T^{9} + \cdots - 51\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{9} + \cdots + 10\!\cdots\!68 \) Copy content Toggle raw display
$59$ \( T^{9} + \cdots - 44\!\cdots\!28 \) Copy content Toggle raw display
$61$ \( T^{9} + \cdots + 84\!\cdots\!04 \) Copy content Toggle raw display
$67$ \( T^{9} + \cdots - 22\!\cdots\!68 \) Copy content Toggle raw display
$71$ \( T^{9} + \cdots + 11\!\cdots\!40 \) Copy content Toggle raw display
$73$ \( T^{9} + \cdots - 77\!\cdots\!80 \) Copy content Toggle raw display
$79$ \( T^{9} + \cdots - 42\!\cdots\!40 \) Copy content Toggle raw display
$83$ \( T^{9} + \cdots + 65\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( T^{9} + \cdots + 17\!\cdots\!92 \) Copy content Toggle raw display
$97$ \( T^{9} + \cdots - 91\!\cdots\!92 \) Copy content Toggle raw display
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