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Properties

Label 1160.4.a.h
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 = [11,0,8,0,-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: \( x^{11} - 3 x^{10} - 200 x^{9} + 418 x^{8} + 13211 x^{7} - 14353 x^{6} - 314463 x^{5} + 64817 x^{4} + \cdots + 712233 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{9}\cdot 3^{3}\cdot 43 \)
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.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{3} - 5 q^{5} + ( - \beta_{5} + \beta_1 + 1) q^{7} + (\beta_{2} - \beta_1 + 11) q^{9} + ( - \beta_{10} - \beta_{6} - \beta_{5} + \cdots + 2) q^{11} + (\beta_{9} - \beta_{5} + 5) q^{13}+ \cdots + ( - 4 \beta_{10} - 4 \beta_{9} + \cdots + 409) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 11 q + 8 q^{3} - 55 q^{5} + 16 q^{7} + 117 q^{9} + 26 q^{11} + 52 q^{13} - 40 q^{15} + 14 q^{17} + 146 q^{19} - 288 q^{21} - 80 q^{23} + 275 q^{25} + 224 q^{27} + 319 q^{29} - 38 q^{31} - 152 q^{33} - 80 q^{35}+ \cdots + 4690 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{11} - 3 x^{10} - 200 x^{9} + 418 x^{8} + 13211 x^{7} - 14353 x^{6} - 314463 x^{5} + 64817 x^{4} + \cdots + 712233 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 37 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 323116042737 \nu^{10} + 1298255866276 \nu^{9} + 72170181225481 \nu^{8} + \cdots + 92\!\cdots\!86 ) / 10\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3588731856693 \nu^{10} - 31340694457096 \nu^{9} - 757976608694000 \nu^{8} + \cdots - 74\!\cdots\!39 ) / 64\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 8514438064079 \nu^{10} + 85873179894046 \nu^{9} + \cdots - 60\!\cdots\!97 ) / 64\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 21541263829425 \nu^{10} + 56707788739730 \nu^{9} + \cdots + 25\!\cdots\!99 ) / 64\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 26433150983203 \nu^{10} - 28055027975880 \nu^{9} + \cdots - 93\!\cdots\!37 ) / 64\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 11576375710827 \nu^{10} + 39386567655552 \nu^{9} + \cdots + 10\!\cdots\!39 ) / 21\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 69825857935775 \nu^{10} + 114649010492226 \nu^{9} + \cdots + 49\!\cdots\!83 ) / 64\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 36026029640398 \nu^{10} - 124807245921059 \nu^{9} + \cdots + 18\!\cdots\!36 ) / 32\!\cdots\!88 \) 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
\(\nu^{3}\)\(=\) \( - 2 \beta_{10} - 2 \beta_{9} - \beta_{8} + 2 \beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} + 2 \beta_{3} + \cdots + 31 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 6 \beta_{10} + 4 \beta_{9} - 11 \beta_{8} - 5 \beta_{7} - 8 \beta_{6} - 6 \beta_{5} - 4 \beta_{4} + \cdots + 2643 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 231 \beta_{10} - 241 \beta_{9} - 129 \beta_{8} + 271 \beta_{7} - 131 \beta_{6} - 79 \beta_{5} + \cdots + 4123 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 1098 \beta_{10} + 307 \beta_{9} - 1683 \beta_{8} - 577 \beta_{7} - 980 \beta_{6} - 1127 \beta_{5} + \cdots + 210972 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 23117 \beta_{10} - 24609 \beta_{9} - 15344 \beta_{8} + 28960 \beta_{7} - 11104 \beta_{6} + \cdots + 436734 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 139084 \beta_{10} + 10351 \beta_{9} - 198033 \beta_{8} - 49803 \beta_{7} - 82938 \beta_{6} + \cdots + 17725648 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 2239843 \beta_{10} - 2403625 \beta_{9} - 1721319 \beta_{8} + 2858822 \beta_{7} - 763165 \beta_{6} + \cdots + 43391519 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 15443852 \beta_{10} - 990063 \beta_{9} - 21374180 \beta_{8} - 3675002 \beta_{7} - 5625688 \beta_{6} + \cdots + 1534119600 \) 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
9.75401
9.05607
6.70389
1.88535
1.52858
0.291180
−1.72995
−2.12486
−5.72207
−7.27289
−9.36932
0 −8.75401 0 −5.00000 0 20.2215 0 49.6327 0
1.2 0 −8.05607 0 −5.00000 0 −3.10047 0 37.9003 0
1.3 0 −5.70389 0 −5.00000 0 11.8591 0 5.53433 0
1.4 0 −0.885350 0 −5.00000 0 12.6292 0 −26.2162 0
1.5 0 −0.528580 0 −5.00000 0 −17.5273 0 −26.7206 0
1.6 0 0.708820 0 −5.00000 0 −10.1905 0 −26.4976 0
1.7 0 2.72995 0 −5.00000 0 32.6348 0 −19.5474 0
1.8 0 3.12486 0 −5.00000 0 −23.5892 0 −17.2353 0
1.9 0 6.72207 0 −5.00000 0 19.0492 0 18.1862 0
1.10 0 8.27289 0 −5.00000 0 −31.8915 0 41.4407 0
1.11 0 10.3693 0 −5.00000 0 5.90497 0 80.5227 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.h ✓ 11
4.b odd 2 1 2320.4.a.z 11
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1160.4.a.h ✓ 11 1.a even 1 1 trivial
2320.4.a.z 11 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{11} - 8 T_{3}^{10} - 175 T_{3}^{9} + 1352 T_{3}^{8} + 9325 T_{3}^{7} - 72860 T_{3}^{6} + \cdots + 656384 \) 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
$3$ \( T^{11} - 8 T^{10} + \cdots + 656384 \) Copy content Toggle raw display
$5$ \( (T + 5)^{11} \) Copy content Toggle raw display
$7$ \( T^{11} + \cdots + 4631729065088 \) Copy content Toggle raw display
$11$ \( T^{11} + \cdots + 34\!\cdots\!72 \) Copy content Toggle raw display
$13$ \( T^{11} + \cdots + 24\!\cdots\!56 \) Copy content Toggle raw display
$17$ \( T^{11} + \cdots + 74\!\cdots\!76 \) Copy content Toggle raw display
$19$ \( T^{11} + \cdots + 10\!\cdots\!88 \) Copy content Toggle raw display
$23$ \( T^{11} + \cdots + 23\!\cdots\!64 \) Copy content Toggle raw display
$29$ \( (T - 29)^{11} \) Copy content Toggle raw display
$31$ \( T^{11} + \cdots + 13\!\cdots\!56 \) Copy content Toggle raw display
$37$ \( T^{11} + \cdots + 17\!\cdots\!92 \) Copy content Toggle raw display
$41$ \( T^{11} + \cdots - 10\!\cdots\!08 \) Copy content Toggle raw display
$43$ \( T^{11} + \cdots - 89\!\cdots\!52 \) Copy content Toggle raw display
$47$ \( T^{11} + \cdots + 76\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{11} + \cdots - 50\!\cdots\!28 \) Copy content Toggle raw display
$59$ \( T^{11} + \cdots - 10\!\cdots\!76 \) Copy content Toggle raw display
$61$ \( T^{11} + \cdots + 26\!\cdots\!56 \) Copy content Toggle raw display
$67$ \( T^{11} + \cdots - 13\!\cdots\!84 \) Copy content Toggle raw display
$71$ \( T^{11} + \cdots + 96\!\cdots\!64 \) Copy content Toggle raw display
$73$ \( T^{11} + \cdots + 17\!\cdots\!76 \) Copy content Toggle raw display
$79$ \( T^{11} + \cdots - 59\!\cdots\!52 \) Copy content Toggle raw display
$83$ \( T^{11} + \cdots + 48\!\cdots\!68 \) Copy content Toggle raw display
$89$ \( T^{11} + \cdots - 25\!\cdots\!76 \) Copy content Toggle raw display
$97$ \( T^{11} + \cdots + 76\!\cdots\!92 \) Copy content Toggle raw display
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