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Properties

Label 80.11.h.c
Level $80$
Weight $11$
Character orbit 80.h
Analytic conductor $50.829$
Analytic rank $0$
Dimension $20$
Inner twists $4$

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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] = [80,11,Mod(79,80)] 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("80.79"); S:= CuspForms(chi, 11); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(80, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1])) N = Newforms(chi, 11, names="a")
 
Level: \( N \) \(=\) \( 80 = 2^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 80.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [20] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(50.8285802139\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 10 x^{19} - 214065 x^{18} + 1926870 x^{17} + 18968501725 x^{16} - 151791690812 x^{15} + \cdots + 19\!\cdots\!07 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{139}\cdot 3^{14}\cdot 5^{12}\cdot 7^{4} \)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{10} q^{3} + ( - \beta_1 - 432) q^{5} + ( - \beta_{11} - 6 \beta_{10}) q^{7} + (\beta_{2} + \beta_1 + 26599) q^{9} - \beta_{13} q^{11} + ( - \beta_{4} + 10 \beta_1 - 2) q^{13} + (\beta_{15} + \beta_{13} + \cdots - 516 \beta_{10}) q^{15}+ \cdots + (602 \beta_{19} + \cdots + 1288 \beta_{10}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 8644 q^{5} + 531980 q^{9} - 9918720 q^{21} - 27949132 q^{25} + 72610760 q^{29} - 22549160 q^{41} - 376019996 q^{45} + 1955095660 q^{49} + 1108414520 q^{61} + 1858963008 q^{65} - 3380864640 q^{69}+ \cdots - 5262787480 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 10 x^{19} - 214065 x^{18} + 1926870 x^{17} + 18968501725 x^{16} - 151791690812 x^{15} + \cdots + 19\!\cdots\!07 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 21\!\cdots\!91 \nu^{18} + \cdots - 84\!\cdots\!49 ) / 20\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 26\!\cdots\!91 \nu^{18} + \cdots + 10\!\cdots\!49 ) / 20\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 40\!\cdots\!37 \nu^{18} + \cdots + 13\!\cdots\!43 ) / 68\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 61\!\cdots\!13 \nu^{18} + \cdots + 26\!\cdots\!27 ) / 39\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 57\!\cdots\!67 \nu^{18} + \cdots + 24\!\cdots\!63 ) / 73\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 58\!\cdots\!13 \nu^{18} + \cdots - 25\!\cdots\!07 ) / 73\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 95\!\cdots\!27 \nu^{18} + \cdots - 32\!\cdots\!53 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 87\!\cdots\!43 \nu^{18} + \cdots + 38\!\cdots\!77 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 21\!\cdots\!89 \nu^{18} + \cdots + 96\!\cdots\!71 ) / 13\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 49\!\cdots\!20 \nu^{19} + \cdots + 10\!\cdots\!81 ) / 36\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 70\!\cdots\!80 \nu^{19} + \cdots - 15\!\cdots\!35 ) / 11\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 50\!\cdots\!80 \nu^{19} + \cdots - 10\!\cdots\!92 ) / 45\!\cdots\!31 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 26\!\cdots\!74 \nu^{19} + \cdots - 38\!\cdots\!77 ) / 77\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 77\!\cdots\!74 \nu^{19} + \cdots - 84\!\cdots\!93 ) / 95\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 10\!\cdots\!56 \nu^{19} + \cdots + 20\!\cdots\!67 ) / 40\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 18\!\cdots\!92 \nu^{19} + \cdots - 41\!\cdots\!69 ) / 28\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 46\!\cdots\!12 \nu^{19} + \cdots + 10\!\cdots\!09 ) / 28\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 23\!\cdots\!68 \nu^{19} + \cdots + 50\!\cdots\!01 ) / 98\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 73\!\cdots\!98 \nu^{19} + \cdots - 15\!\cdots\!61 ) / 28\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{12} - 8192\beta_{10} + 8192 ) / 16384 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - \beta_{12} - 8192 \beta_{10} - 2 \beta_{5} + 4 \beta_{4} - 2 \beta_{3} + 4096 \beta_{2} + \cdots + 350806006 ) / 16384 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 96 \beta_{19} - 6144 \beta_{18} + 4120 \beta_{17} - 7760 \beta_{16} + 14096 \beta_{15} + \cdots + 526204913 ) / 16384 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 192 \beta_{19} - 12288 \beta_{18} + 8240 \beta_{17} - 15520 \beta_{16} + 28192 \beta_{15} + \cdots + 12928708020796 ) / 16384 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 44651744 \beta_{19} - 305952768 \beta_{18} + 236934488 \beta_{17} - 491976144 \beta_{16} + \cdots + 32320893045167 ) / 16384 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 33488688 \beta_{19} - 229456896 \beta_{18} + 177695716 \beta_{17} - 368972408 \beta_{16} + \cdots + 13\!\cdots\!68 ) / 4096 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 3572686678784 \beta_{19} - 13720229093376 \beta_{18} + 12942752104736 \beta_{17} + \cdots + 19\!\cdots\!34 ) / 16384 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 14290121593408 \beta_{19} - 54876633206784 \beta_{18} + 51767691451472 \beta_{17} + \cdots + 24\!\cdots\!04 ) / 16384 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 22\!\cdots\!48 \beta_{19} + \cdots + 10\!\cdots\!97 ) / 16384 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 11\!\cdots\!84 \beta_{19} + \cdots + 10\!\cdots\!42 ) / 16384 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 12\!\cdots\!96 \beta_{19} + \cdots + 59\!\cdots\!57 ) / 16384 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 95\!\cdots\!76 \beta_{19} + \cdots + 61\!\cdots\!42 ) / 2048 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 68\!\cdots\!00 \beta_{19} + \cdots + 31\!\cdots\!92 ) / 16384 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 48\!\cdots\!88 \beta_{19} + \cdots + 22\!\cdots\!30 ) / 16384 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 35\!\cdots\!52 \beta_{19} + \cdots + 16\!\cdots\!37 ) / 16384 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( 28\!\cdots\!92 \beta_{19} + \cdots + 10\!\cdots\!28 ) / 16384 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( 18\!\cdots\!12 \beta_{19} + \cdots + 86\!\cdots\!63 ) / 16384 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( 41\!\cdots\!88 \beta_{19} + \cdots + 11\!\cdots\!12 ) / 4096 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( 92\!\cdots\!72 \beta_{19} + \cdots + 44\!\cdots\!42 ) / 16384 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(17\) \(21\) \(31\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
79.1
218.354 − 0.866025i
218.354 + 0.866025i
197.935 + 0.866025i
197.935 − 0.866025i
107.713 − 0.866025i
107.713 + 0.866025i
69.9094 + 0.866025i
69.9094 − 0.866025i
66.1248 − 0.866025i
66.1248 + 0.866025i
−65.1248 + 0.866025i
−65.1248 − 0.866025i
−68.9094 − 0.866025i
−68.9094 + 0.866025i
−106.713 + 0.866025i
−106.713 − 0.866025i
−196.935 − 0.866025i
−196.935 + 0.866025i
−217.354 + 0.866025i
−217.354 − 0.866025i
0 −435.708 0 −1949.06 − 2442.70i 0 −18935.4 0 130793. 0
79.2 0 −435.708 0 −1949.06 + 2442.70i 0 −18935.4 0 130793. 0
79.3 0 −394.871 0 540.369 − 3077.93i 0 29913.4 0 96874.0 0
79.4 0 −394.871 0 540.369 + 3077.93i 0 29913.4 0 96874.0 0
79.5 0 −214.425 0 2711.11 − 1554.19i 0 −4158.70 0 −13070.8 0
79.6 0 −214.425 0 2711.11 + 1554.19i 0 −4158.70 0 −13070.8 0
79.7 0 −138.819 0 −412.269 − 3097.69i 0 −17960.7 0 −39778.3 0
79.8 0 −138.819 0 −412.269 + 3097.69i 0 −17960.7 0 −39778.3 0
79.9 0 −131.250 0 −3051.15 − 675.357i 0 17547.3 0 −41822.5 0
79.10 0 −131.250 0 −3051.15 + 675.357i 0 17547.3 0 −41822.5 0
79.11 0 131.250 0 −3051.15 − 675.357i 0 −17547.3 0 −41822.5 0
79.12 0 131.250 0 −3051.15 + 675.357i 0 −17547.3 0 −41822.5 0
79.13 0 138.819 0 −412.269 − 3097.69i 0 17960.7 0 −39778.3 0
79.14 0 138.819 0 −412.269 + 3097.69i 0 17960.7 0 −39778.3 0
79.15 0 214.425 0 2711.11 − 1554.19i 0 4158.70 0 −13070.8 0
79.16 0 214.425 0 2711.11 + 1554.19i 0 4158.70 0 −13070.8 0
79.17 0 394.871 0 540.369 − 3077.93i 0 −29913.4 0 96874.0 0
79.18 0 394.871 0 540.369 + 3077.93i 0 −29913.4 0 96874.0 0
79.19 0 435.708 0 −1949.06 − 2442.70i 0 18935.4 0 130793. 0
79.20 0 435.708 0 −1949.06 + 2442.70i 0 18935.4 0 130793. 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 79.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.b even 2 1 inner
20.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 80.11.h.c ✓ 20
4.b odd 2 1 inner 80.11.h.c ✓ 20
5.b even 2 1 inner 80.11.h.c ✓ 20
20.d odd 2 1 inner 80.11.h.c ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
80.11.h.c ✓ 20 1.a even 1 1 trivial
80.11.h.c ✓ 20 4.b odd 2 1 inner
80.11.h.c ✓ 20 5.b even 2 1 inner
80.11.h.c ✓ 20 20.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{10} - 428240 T_{3}^{8} + 60127773840 T_{3}^{6} + \cdots - 45\!\cdots\!68 \) acting on \(S_{11}^{\mathrm{new}}(80, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( (T^{10} + \cdots - 45\!\cdots\!68)^{2} \) Copy content Toggle raw display
$5$ \( (T^{10} + \cdots + 88\!\cdots\!25)^{2} \) Copy content Toggle raw display
$7$ \( (T^{10} + \cdots - 55\!\cdots\!32)^{2} \) Copy content Toggle raw display
$11$ \( (T^{10} + \cdots + 11\!\cdots\!32)^{2} \) Copy content Toggle raw display
$13$ \( (T^{10} + \cdots + 61\!\cdots\!76)^{2} \) Copy content Toggle raw display
$17$ \( (T^{10} + \cdots + 12\!\cdots\!24)^{2} \) Copy content Toggle raw display
$19$ \( (T^{10} + \cdots + 65\!\cdots\!32)^{2} \) Copy content Toggle raw display
$23$ \( (T^{10} + \cdots - 30\!\cdots\!68)^{2} \) Copy content Toggle raw display
$29$ \( (T^{5} + \cdots - 23\!\cdots\!68)^{4} \) Copy content Toggle raw display
$31$ \( (T^{10} + \cdots + 17\!\cdots\!32)^{2} \) Copy content Toggle raw display
$37$ \( (T^{10} + \cdots + 28\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( (T^{5} + \cdots + 10\!\cdots\!68)^{4} \) Copy content Toggle raw display
$43$ \( (T^{10} + \cdots - 57\!\cdots\!68)^{2} \) Copy content Toggle raw display
$47$ \( (T^{10} + \cdots - 53\!\cdots\!32)^{2} \) Copy content Toggle raw display
$53$ \( (T^{10} + \cdots + 27\!\cdots\!24)^{2} \) Copy content Toggle raw display
$59$ \( (T^{10} + \cdots + 16\!\cdots\!32)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} + \cdots + 53\!\cdots\!24)^{4} \) Copy content Toggle raw display
$67$ \( (T^{10} + \cdots - 48\!\cdots\!68)^{2} \) Copy content Toggle raw display
$71$ \( (T^{10} + \cdots + 41\!\cdots\!68)^{2} \) Copy content Toggle raw display
$73$ \( (T^{10} + \cdots + 41\!\cdots\!24)^{2} \) Copy content Toggle raw display
$79$ \( (T^{10} + \cdots + 92\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( (T^{10} + \cdots - 11\!\cdots\!32)^{2} \) Copy content Toggle raw display
$89$ \( (T^{5} + \cdots - 17\!\cdots\!76)^{4} \) Copy content Toggle raw display
$97$ \( (T^{10} + \cdots + 88\!\cdots\!76)^{2} \) Copy content Toggle raw display
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