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Properties

Label 1160.2.g.b
Level $1160$
Weight $2$
Character orbit 1160.g
Analytic conductor $9.263$
Analytic rank $0$
Dimension $14$
Inner twists $2$

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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] = [1160,2,Mod(521,1160)] 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("1160.521"); S:= CuspForms(chi, 2); 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, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1160 = 2^{3} \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1160.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [14,0,0,0,-14] 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: no
Analytic conductor: \(9.26264663447\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 2 x^{13} - x^{12} - 4 x^{11} + 72 x^{10} - 240 x^{9} + 264 x^{8} + 96 x^{7} + 155 x^{6} + \cdots + 116 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{11} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{5} q^{3} - q^{5} + \beta_{6} q^{7} + (\beta_{3} - 1) q^{9} + \beta_{2} q^{11} - \beta_{4} q^{13} + \beta_{5} q^{15} + ( - \beta_{8} + \beta_{2}) q^{17} + (\beta_{11} + \beta_{5}) q^{19} + (\beta_{8} - \beta_{5} + \beta_{2} - \beta_1) q^{21}+ \cdots + ( - 2 \beta_{8} - 3 \beta_{2} + 2 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 14 q^{5} + 6 q^{7} - 20 q^{9} + 6 q^{13} + 10 q^{23} + 14 q^{25} - 14 q^{29} + 12 q^{33} - 6 q^{35} + 20 q^{45} + 4 q^{49} - 30 q^{53} + 48 q^{57} + 18 q^{59} - 24 q^{63} - 6 q^{65} - 4 q^{67} - 24 q^{71}+ \cdots - 24 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 2 x^{13} - x^{12} - 4 x^{11} + 72 x^{10} - 240 x^{9} + 264 x^{8} + 96 x^{7} + 155 x^{6} + \cdots + 116 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 78483545 \nu^{13} - 300411502 \nu^{12} + 423169379 \nu^{11} + 288744949 \nu^{10} + \cdots - 112195807740 ) / 87809775506 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 83348328343559 \nu^{13} + 15621667077310 \nu^{12} - 9638664411009 \nu^{11} + \cdots + 17\!\cdots\!92 ) / 38\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 205912122797799 \nu^{13} + 69825721372275 \nu^{12} - 300541818760458 \nu^{11} + \cdots + 25\!\cdots\!88 ) / 77\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 311210352917850 \nu^{13} - 593258188608205 \nu^{12} - 537922023481416 \nu^{11} + \cdots + 23\!\cdots\!68 ) / 77\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 363845198908254 \nu^{13} + 300073483807047 \nu^{12} + 504791989170328 \nu^{11} + \cdots + 62\!\cdots\!64 ) / 77\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 466434051487171 \nu^{13} - 876568964425699 \nu^{12} - 835903580213282 \nu^{11} + \cdots + 63\!\cdots\!60 ) / 77\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 474603385511853 \nu^{13} + 808234763918279 \nu^{12} + 817025124470386 \nu^{11} + \cdots - 24\!\cdots\!60 ) / 77\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 250073127359343 \nu^{13} - 170254857674546 \nu^{12} + 666235921031801 \nu^{11} + \cdots - 63\!\cdots\!92 ) / 38\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 501510899609723 \nu^{13} + \cdots + 16\!\cdots\!44 ) / 77\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 580100201558661 \nu^{13} + \cdots - 66\!\cdots\!08 ) / 77\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 586948197087924 \nu^{13} + 757713605943139 \nu^{12} + \cdots + 10\!\cdots\!48 ) / 77\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 10\!\cdots\!64 \nu^{13} + \cdots + 32\!\cdots\!96 ) / 77\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 14\!\cdots\!78 \nu^{13} + \cdots + 35\!\cdots\!64 ) / 77\!\cdots\!28 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{11} + \beta_{10} + \beta_{9} + \beta_{5} - \beta_{4} - \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 3 \beta_{12} + 3 \beta_{11} + \beta_{10} - \beta_{8} - \beta_{7} - \beta_{6} + 4 \beta_{5} + \beta_{4} + \cdots + 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - \beta_{13} + 6 \beta_{12} - 7 \beta_{11} + \beta_{10} - 2 \beta_{9} - 3 \beta_{6} - 2 \beta_{5} + \cdots + 12 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 3 \beta_{13} + \beta_{12} + 9 \beta_{11} + 9 \beta_{10} + 20 \beta_{9} - 3 \beta_{8} - 15 \beta_{7} + \cdots - 74 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 5 \beta_{13} - 22 \beta_{12} + 55 \beta_{11} - 41 \beta_{10} - 37 \beta_{9} + 4 \beta_{8} + \cdots + 232 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 32 \beta_{13} + 137 \beta_{12} - 313 \beta_{11} + 9 \beta_{10} + 60 \beta_{9} - 13 \beta_{8} + \cdots - 126 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - \beta_{13} - 498 \beta_{12} + 771 \beta_{11} + 99 \beta_{10} + 182 \beta_{9} + 40 \beta_{8} + \cdots - 948 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 63 \beta_{13} + 491 \beta_{12} - 949 \beta_{11} - 313 \beta_{10} - 1188 \beta_{9} - 17 \beta_{8} + \cdots + 6018 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 121 \beta_{13} + 2082 \beta_{12} - 3579 \beta_{11} + 1929 \beta_{10} + 3629 \beta_{9} - 440 \beta_{8} + \cdots - 18776 ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 2016 \beta_{13} - 12769 \beta_{12} + 26585 \beta_{11} - 2789 \beta_{10} - 4900 \beta_{9} + \cdots + 26382 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 5329 \beta_{13} + 42222 \beta_{12} - 83239 \beta_{11} - 6355 \beta_{10} - 11486 \beta_{9} - 6372 \beta_{8} + \cdots + 68708 ) / 4 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 5983 \beta_{13} - 68203 \beta_{12} + 124589 \beta_{11} + 42141 \beta_{10} + 102324 \beta_{9} + \cdots - 537858 ) / 4 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 12441 \beta_{13} - 117742 \beta_{12} + 223599 \beta_{11} - 143353 \beta_{10} - 347493 \beta_{9} + \cdots + 1820072 ) / 4 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1160\mathbb{Z}\right)^\times\).

\(n\) \(321\) \(581\) \(697\) \(871\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
521.1
1.86554 − 0.0866317i
1.39173 + 1.30065i
−1.70213 − 1.20290i
0.883168 + 2.02400i
−2.28874 + 2.11346i
0.0342909 − 0.218813i
0.816149 − 0.492649i
0.816149 + 0.492649i
0.0342909 + 0.218813i
−2.28874 − 2.11346i
0.883168 − 2.02400i
−1.70213 + 1.20290i
1.39173 − 1.30065i
1.86554 + 0.0866317i
0 − 3.19389i 0 −1.00000 0 −2.79013 0 −7.20092 0
521.2 0 − 2.89198i 0 −1.00000 0 5.06237 0 −5.36355 0
521.3 0 − 2.23062i 0 −1.00000 0 −0.270670 0 −1.97568 0
521.4 0 − 2.03263i 0 −1.00000 0 1.07384 0 −1.13157 0
521.5 0 − 1.61139i 0 −1.00000 0 1.15266 0 0.403433 0
521.6 0 − 0.802809i 0 −1.00000 0 2.05790 0 2.35550 0
521.7 0 − 0.295331i 0 −1.00000 0 −3.28597 0 2.91278 0
521.8 0 0.295331i 0 −1.00000 0 −3.28597 0 2.91278 0
521.9 0 0.802809i 0 −1.00000 0 2.05790 0 2.35550 0
521.10 0 1.61139i 0 −1.00000 0 1.15266 0 0.403433 0
521.11 0 2.03263i 0 −1.00000 0 1.07384 0 −1.13157 0
521.12 0 2.23062i 0 −1.00000 0 −0.270670 0 −1.97568 0
521.13 0 2.89198i 0 −1.00000 0 5.06237 0 −5.36355 0
521.14 0 3.19389i 0 −1.00000 0 −2.79013 0 −7.20092 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 521.14
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1160.2.g.b ✓ 14
4.b odd 2 1 2320.2.g.j 14
29.b even 2 1 inner 1160.2.g.b ✓ 14
116.d odd 2 1 2320.2.g.j 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1160.2.g.b ✓ 14 1.a even 1 1 trivial
1160.2.g.b ✓ 14 29.b even 2 1 inner
2320.2.g.j 14 4.b odd 2 1
2320.2.g.j 14 116.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{14} + 31T_{3}^{12} + 369T_{3}^{10} + 2128T_{3}^{8} + 6152T_{3}^{6} + 8144T_{3}^{4} + 3600T_{3}^{2} + 256 \) acting on \(S_{2}^{\mathrm{new}}(1160, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} \) Copy content Toggle raw display
$3$ \( T^{14} + 31 T^{12} + \cdots + 256 \) Copy content Toggle raw display
$5$ \( (T + 1)^{14} \) Copy content Toggle raw display
$7$ \( (T^{7} - 3 T^{6} - 21 T^{5} + \cdots + 32)^{2} \) Copy content Toggle raw display
$11$ \( T^{14} + 84 T^{12} + \cdots + 1024 \) Copy content Toggle raw display
$13$ \( (T^{7} - 3 T^{6} + \cdots - 128)^{2} \) Copy content Toggle raw display
$17$ \( T^{14} + 95 T^{12} + \cdots + 506944 \) Copy content Toggle raw display
$19$ \( T^{14} + 104 T^{12} + \cdots + 13075456 \) Copy content Toggle raw display
$23$ \( (T^{7} - 5 T^{6} + \cdots - 20752)^{2} \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots + 17249876309 \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots + 258823744 \) Copy content Toggle raw display
$37$ \( T^{14} + 252 T^{12} + \cdots + 94633984 \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots + 13687128064 \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots + 15559069696 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 2485620736 \) Copy content Toggle raw display
$53$ \( (T^{7} + 15 T^{6} + \cdots + 54784)^{2} \) Copy content Toggle raw display
$59$ \( (T^{7} - 9 T^{6} + \cdots + 19904)^{2} \) Copy content Toggle raw display
$61$ \( T^{14} + 331 T^{12} + \cdots + 2166784 \) Copy content Toggle raw display
$67$ \( (T^{7} + 2 T^{6} - 68 T^{5} + \cdots - 64)^{2} \) Copy content Toggle raw display
$71$ \( (T^{7} + 12 T^{6} + \cdots - 1045504)^{2} \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 3033978716224 \) Copy content Toggle raw display
$79$ \( T^{14} + 275 T^{12} + \cdots + 70694464 \) Copy content Toggle raw display
$83$ \( (T^{7} + 16 T^{6} + \cdots - 338432)^{2} \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 75312922624 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 2041080824896 \) Copy content Toggle raw display
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