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Properties

Label 80.11.p.g
Level $80$
Weight $11$
Character orbit 80.p
Analytic conductor $50.829$
Analytic rank $0$
Dimension $16$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [80,11,Mod(17,80)] 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("80.17"); S:= CuspForms(chi, 11); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(80, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 11, names="a")
 
Level: \( N \) \(=\) \( 80 = 2^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 80.p (of order \(4\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,0,-190] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(50.8285802139\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 2 x^{15} - 313727 x^{14} - 7929954 x^{13} + 37928709545 x^{12} + 1760510226354 x^{11} + \cdots + 98\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{62}\cdot 3^{2}\cdot 5^{14} \)
Twist minimal: no (minimal twist has level 40)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{2} + 12 \beta_1 - 12) q^{3} + ( - \beta_{4} - 2 \beta_{2} + \cdots - 163) q^{5} + (\beta_{8} - 16 \beta_{3} + 857 \beta_1 + 857) q^{7} + ( - \beta_{11} + 3 \beta_{7} + \cdots - 1) q^{9} + (\beta_{10} + \beta_{9} + \beta_{8} + \cdots - 2582) q^{11}+ \cdots + ( - 7349 \beta_{15} - 8928 \beta_{14} + \cdots + 688532) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 190 q^{3} - 2606 q^{5} + 13746 q^{7} - 40964 q^{11} - 203724 q^{13} + 64882 q^{15} - 2250732 q^{17} + 19181780 q^{21} + 14855882 q^{23} - 2885660 q^{25} - 28551400 q^{27} + 52289828 q^{31} + 100784980 q^{33}+ \cdots + 1378314224 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 2 x^{15} - 313727 x^{14} - 7929954 x^{13} + 37928709545 x^{12} + 1760510226354 x^{11} + \cdots + 98\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 11\!\cdots\!83 \nu^{15} + \cdots + 25\!\cdots\!00 ) / 66\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 68\!\cdots\!74 \nu^{15} + \cdots + 38\!\cdots\!00 ) / 22\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 22\!\cdots\!88 \nu^{15} + \cdots + 11\!\cdots\!00 ) / 66\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 10\!\cdots\!23 \nu^{15} + \cdots - 18\!\cdots\!00 ) / 38\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 33\!\cdots\!71 \nu^{15} + \cdots + 84\!\cdots\!00 ) / 38\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 15\!\cdots\!32 \nu^{15} + \cdots + 84\!\cdots\!00 ) / 76\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 77\!\cdots\!76 \nu^{15} + \cdots + 22\!\cdots\!00 ) / 38\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 16\!\cdots\!87 \nu^{15} + \cdots + 23\!\cdots\!00 ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 17\!\cdots\!07 \nu^{15} + \cdots - 22\!\cdots\!00 ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 59\!\cdots\!51 \nu^{15} + \cdots + 36\!\cdots\!00 ) / 38\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 20\!\cdots\!13 \nu^{15} + \cdots - 34\!\cdots\!00 ) / 38\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 21\!\cdots\!76 \nu^{15} + \cdots + 24\!\cdots\!00 ) / 38\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 42\!\cdots\!37 \nu^{15} + \cdots - 34\!\cdots\!00 ) / 76\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 30\!\cdots\!27 \nu^{15} + \cdots - 61\!\cdots\!00 ) / 38\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 51\!\cdots\!49 \nu^{15} + \cdots + 29\!\cdots\!00 ) / 38\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{3} + \beta_{2} + 2\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} + \beta_{5} + 3\beta_{4} - 43\beta_{3} + 38\beta_{2} - \beta _1 + 78423 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 5 \beta_{15} + 58 \beta_{14} - 5 \beta_{13} + 3 \beta_{11} + 29 \beta_{10} + 366 \beta_{9} + \cdots + 3191632 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 1501 \beta_{15} - 2662 \beta_{14} + 1581 \beta_{13} + 232 \beta_{12} + 504 \beta_{11} + \cdots + 11297782433 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 531996 \beta_{15} + 6896113 \beta_{14} - 539701 \beta_{13} + 15820 \beta_{12} + 548365 \beta_{11} + \cdots + 528644573528 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 163679405 \beta_{15} + 107904606 \beta_{14} + 176540569 \beta_{13} + 50440468 \beta_{12} + \cdots + 10\!\cdots\!57 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 50002324969 \beta_{15} + 709300037209 \beta_{14} - 51193202748 \beta_{13} + 3656170616 \beta_{12} + \cdots + 72\!\cdots\!01 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 13499556223225 \beta_{15} + 63576323083986 \beta_{14} + 15118924744745 \beta_{13} + \cdots + 10\!\cdots\!45 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 46\!\cdots\!94 \beta_{15} + \cdots + 93\!\cdots\!26 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 96\!\cdots\!21 \beta_{15} + \cdots + 11\!\cdots\!97 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 42\!\cdots\!91 \beta_{15} + \cdots + 11\!\cdots\!03 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 57\!\cdots\!53 \beta_{15} + \cdots + 11\!\cdots\!13 ) / 4 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 40\!\cdots\!80 \beta_{15} + \cdots + 13\!\cdots\!32 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 19\!\cdots\!81 \beta_{15} + \cdots + 12\!\cdots\!93 ) / 4 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 39\!\cdots\!81 \beta_{15} + \cdots + 16\!\cdots\!13 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/80\mathbb{Z}\right)^\times\).

\(n\) \(17\) \(21\) \(31\)
\(\chi(n)\) \(\beta_{1}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1
−291.476 + 1.00000i
−171.146 + 1.00000i
−138.604 + 1.00000i
−108.340 + 1.00000i
45.9164 + 1.00000i
161.865 + 1.00000i
168.220 + 1.00000i
334.564 + 1.00000i
−291.476 − 1.00000i
−171.146 − 1.00000i
−138.604 − 1.00000i
−108.340 − 1.00000i
45.9164 − 1.00000i
161.865 − 1.00000i
168.220 − 1.00000i
334.564 − 1.00000i
0 −303.476 + 303.476i 0 −76.8860 − 3124.05i 0 −13719.3 − 13719.3i 0 − 125146.i 0
17.2 0 −183.146 + 183.146i 0 796.240 + 3021.86i 0 −11821.9 − 11821.9i 0 − 8036.02i 0
17.3 0 −150.604 + 150.604i 0 2891.44 − 1185.42i 0 22811.8 + 22811.8i 0 13686.0i 0
17.4 0 −120.340 + 120.340i 0 −2781.37 + 1424.65i 0 11234.6 + 11234.6i 0 30085.8i 0
17.5 0 33.9164 − 33.9164i 0 −2790.75 − 1406.17i 0 −7956.94 − 7956.94i 0 56748.4i 0
17.6 0 149.865 − 149.865i 0 2207.55 + 2211.87i 0 −6074.32 − 6074.32i 0 14130.2i 0
17.7 0 156.220 − 156.220i 0 1193.03 − 2888.31i 0 −2615.28 − 2615.28i 0 10239.4i 0
17.8 0 322.564 − 322.564i 0 −2742.24 + 1498.58i 0 15014.4 + 15014.4i 0 − 149046.i 0
33.1 0 −303.476 − 303.476i 0 −76.8860 + 3124.05i 0 −13719.3 + 13719.3i 0 125146.i 0
33.2 0 −183.146 − 183.146i 0 796.240 − 3021.86i 0 −11821.9 + 11821.9i 0 8036.02i 0
33.3 0 −150.604 − 150.604i 0 2891.44 + 1185.42i 0 22811.8 − 22811.8i 0 − 13686.0i 0
33.4 0 −120.340 − 120.340i 0 −2781.37 − 1424.65i 0 11234.6 − 11234.6i 0 − 30085.8i 0
33.5 0 33.9164 + 33.9164i 0 −2790.75 + 1406.17i 0 −7956.94 + 7956.94i 0 − 56748.4i 0
33.6 0 149.865 + 149.865i 0 2207.55 − 2211.87i 0 −6074.32 + 6074.32i 0 − 14130.2i 0
33.7 0 156.220 + 156.220i 0 1193.03 + 2888.31i 0 −2615.28 + 2615.28i 0 − 10239.4i 0
33.8 0 322.564 + 322.564i 0 −2742.24 − 1498.58i 0 15014.4 − 15014.4i 0 149046.i 0
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 17.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 80.11.p.g 16
4.b odd 2 1 40.11.l.b ✓ 16
5.c odd 4 1 inner 80.11.p.g 16
20.d odd 2 1 200.11.l.d 16
20.e even 4 1 40.11.l.b ✓ 16
20.e even 4 1 200.11.l.d 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.11.l.b ✓ 16 4.b odd 2 1
40.11.l.b ✓ 16 20.e even 4 1
80.11.p.g 16 1.a even 1 1 trivial
80.11.p.g 16 5.c odd 4 1 inner
200.11.l.d 16 20.d odd 2 1
200.11.l.d 16 20.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} + 190 T_{3}^{15} + 18050 T_{3}^{14} + 3180760 T_{3}^{13} + 44958278200 T_{3}^{12} + \cdots + 17\!\cdots\!00 \) acting on \(S_{11}^{\mathrm{new}}(80, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 82\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 15\!\cdots\!04 \) Copy content Toggle raw display
$11$ \( (T^{8} + \cdots + 35\!\cdots\!64)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 44\!\cdots\!44 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 48\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 19\!\cdots\!56 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 48\!\cdots\!64 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 30\!\cdots\!96 \) Copy content Toggle raw display
$31$ \( (T^{8} + \cdots + 87\!\cdots\!00)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 14\!\cdots\!36 \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots + 34\!\cdots\!08)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 19\!\cdots\!24 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 44\!\cdots\!96 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 29\!\cdots\!16 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 40\!\cdots\!64 \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots + 15\!\cdots\!00)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 73\!\cdots\!24 \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots - 35\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 18\!\cdots\!04 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 84\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 16\!\cdots\!04 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 15\!\cdots\!16 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 37\!\cdots\!16 \) Copy content Toggle raw display
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