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Properties

Label 1160.4.a.g
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 / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1160,4,Mod(1,1160)] 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("1160.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage: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")
 
Level: \( N \) \(=\) \( 1160 = 2^{3} \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1160.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [11,0,2,0,-55] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(68.4422156067\)
Analytic rank: \(0\)
Dimension: \(11\)
Coefficient field: \(\mathbb{Q}[x]/(x^{11} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{11} - 2 x^{10} - 179 x^{9} + 370 x^{8} + 10353 x^{7} - 19394 x^{6} - 210392 x^{5} + 267796 x^{4} + \cdots + 567808 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{12}\cdot 3 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\ldots,\beta_{10}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} - 5 q^{5} + (\beta_{7} + 5) q^{7} + (\beta_{2} + 6) q^{9} + (\beta_{3} + \beta_{2} - \beta_1) q^{11} + (\beta_{7} - \beta_{6} + \beta_{4} + \cdots + 1) q^{13} - 5 \beta_1 q^{15} + (\beta_{9} + \beta_{5} - \beta_{3} + \cdots + 6) q^{17}+ \cdots + ( - 5 \beta_{10} + 15 \beta_{9} + \cdots + 617) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 11 q + 2 q^{3} - 55 q^{5} + 56 q^{7} + 65 q^{9} - 8 q^{11} + 16 q^{13} - 10 q^{15} + 64 q^{17} + 160 q^{19} - 48 q^{21} + 140 q^{23} + 275 q^{25} - 136 q^{27} - 319 q^{29} - 144 q^{31} - 424 q^{33} - 280 q^{35}+ \cdots + 6884 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{11} - 2 x^{10} - 179 x^{9} + 370 x^{8} + 10353 x^{7} - 19394 x^{6} - 210392 x^{5} + 267796 x^{4} + \cdots + 567808 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 33 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 7619953597 \nu^{10} - 99419125230 \nu^{9} - 1440522939755 \nu^{8} + 17099864567958 \nu^{7} + \cdots - 15\!\cdots\!04 ) / 17\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 126379983379 \nu^{10} - 302580218988 \nu^{9} - 22086014520245 \nu^{8} + \cdots - 79\!\cdots\!76 ) / 17\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 15742272003 \nu^{10} + 16213437892 \nu^{9} + 2836011355901 \nu^{8} + \cdots + 66\!\cdots\!12 ) / 129500089592784 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 143570499777 \nu^{10} - 207373457908 \nu^{9} - 25832538085895 \nu^{8} + \cdots - 78\!\cdots\!32 ) / 582750403167528 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 256918137343 \nu^{10} - 303980232117 \nu^{9} - 46191297113405 \nu^{8} + \cdots - 19\!\cdots\!36 ) / 874125604751292 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 690892296023 \nu^{10} + 666618127152 \nu^{9} + 124660876149361 \nu^{8} + \cdots + 62\!\cdots\!20 ) / 17\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 734147348477 \nu^{10} - 1002508893792 \nu^{9} - 131934320929291 \nu^{8} + \cdots - 47\!\cdots\!92 ) / 17\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 274200460925 \nu^{10} + 340070249790 \nu^{9} + 49364835876883 \nu^{8} + \cdots + 21\!\cdots\!12 ) / 582750403167528 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 33 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{10} - \beta_{9} + 3\beta_{7} + 2\beta_{3} - \beta_{2} + 63\beta _1 - 14 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2 \beta_{10} + 11 \beta_{9} + 6 \beta_{8} - 2 \beta_{7} - 5 \beta_{6} - 8 \beta_{5} - 6 \beta_{4} + \cdots + 2066 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 97 \beta_{10} - 81 \beta_{9} + 23 \beta_{8} + 331 \beta_{7} + 5 \beta_{6} + 68 \beta_{5} - 30 \beta_{4} + \cdots - 2400 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 204 \beta_{10} + 1374 \beta_{9} + 676 \beta_{8} - 285 \beta_{7} - 823 \beta_{6} - 1018 \beta_{5} + \cdots + 144428 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 8695 \beta_{10} - 6427 \beta_{9} + 2590 \beta_{8} + 29964 \beta_{7} + 1814 \beta_{6} + 9938 \beta_{5} + \cdots - 286361 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 16159 \beta_{10} + 134403 \beta_{9} + 61690 \beta_{8} - 32336 \beta_{7} - 95772 \beta_{6} + \cdots + 10635472 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 755116 \beta_{10} - 550828 \beta_{9} + 213147 \beta_{8} + 2568897 \beta_{7} + 287313 \beta_{6} + \cdots - 29566899 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 1108989 \beta_{10} + 12121027 \beta_{9} + 5218458 \beta_{8} - 3591419 \beta_{7} - 9671789 \beta_{6} + \cdots + 811327584 \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−9.30253
−7.86378
−4.74161
−2.40564
−0.901365
0.728365
0.848128
2.81857
6.78682
7.40124
8.63181
0 −9.30253 0 −5.00000 0 −7.31376 0 59.5371 0
1.2 0 −7.86378 0 −5.00000 0 26.0010 0 34.8390 0
1.3 0 −4.74161 0 −5.00000 0 33.7088 0 −4.51718 0
1.4 0 −2.40564 0 −5.00000 0 −2.91555 0 −21.2129 0
1.5 0 −0.901365 0 −5.00000 0 −4.01917 0 −26.1875 0
1.6 0 0.728365 0 −5.00000 0 −29.1533 0 −26.4695 0
1.7 0 0.848128 0 −5.00000 0 28.0939 0 −26.2807 0
1.8 0 2.81857 0 −5.00000 0 −26.0684 0 −19.0556 0
1.9 0 6.78682 0 −5.00000 0 12.0509 0 19.0609 0
1.10 0 7.40124 0 −5.00000 0 −4.56279 0 27.7783 0
1.11 0 8.63181 0 −5.00000 0 30.1782 0 47.5081 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.g ✓ 11
4.b odd 2 1 2320.4.a.bb 11
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1160.4.a.g ✓ 11 1.a even 1 1 trivial
2320.4.a.bb 11 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{11} - 2 T_{3}^{10} - 179 T_{3}^{9} + 370 T_{3}^{8} + 10353 T_{3}^{7} - 19394 T_{3}^{6} + \cdots + 567808 \) 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
$3$ \( T^{11} - 2 T^{10} + \cdots + 567808 \) Copy content Toggle raw display
$5$ \( (T + 5)^{11} \) Copy content Toggle raw display
$7$ \( T^{11} + \cdots - 2661288120000 \) Copy content Toggle raw display
$11$ \( T^{11} + \cdots + 93\!\cdots\!76 \) Copy content Toggle raw display
$13$ \( T^{11} + \cdots + 23\!\cdots\!12 \) Copy content Toggle raw display
$17$ \( T^{11} + \cdots - 79\!\cdots\!76 \) Copy content Toggle raw display
$19$ \( T^{11} + \cdots - 44\!\cdots\!96 \) Copy content Toggle raw display
$23$ \( T^{11} + \cdots + 43\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T + 29)^{11} \) Copy content Toggle raw display
$31$ \( T^{11} + \cdots + 78\!\cdots\!20 \) Copy content Toggle raw display
$37$ \( T^{11} + \cdots - 56\!\cdots\!24 \) Copy content Toggle raw display
$41$ \( T^{11} + \cdots - 12\!\cdots\!20 \) Copy content Toggle raw display
$43$ \( T^{11} + \cdots - 15\!\cdots\!36 \) Copy content Toggle raw display
$47$ \( T^{11} + \cdots - 14\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( T^{11} + \cdots + 13\!\cdots\!32 \) Copy content Toggle raw display
$59$ \( T^{11} + \cdots - 82\!\cdots\!24 \) Copy content Toggle raw display
$61$ \( T^{11} + \cdots + 46\!\cdots\!96 \) Copy content Toggle raw display
$67$ \( T^{11} + \cdots + 23\!\cdots\!44 \) Copy content Toggle raw display
$71$ \( T^{11} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{11} + \cdots + 70\!\cdots\!68 \) Copy content Toggle raw display
$79$ \( T^{11} + \cdots - 33\!\cdots\!64 \) Copy content Toggle raw display
$83$ \( T^{11} + \cdots - 54\!\cdots\!88 \) Copy content Toggle raw display
$89$ \( T^{11} + \cdots + 26\!\cdots\!60 \) Copy content Toggle raw display
$97$ \( T^{11} + \cdots + 34\!\cdots\!84 \) Copy content Toggle raw display
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