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Properties

Label 80.5.h.c
Level $80$
Weight $5$
Character orbit 80.h
Analytic conductor $8.270$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [80,5,Mod(79,80)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(80, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("80.79");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 80 = 2^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 80.h (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(8.26959704671\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.1779622700625.10
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 41x^{6} + 1357x^{4} + 13284x^{2} + 104976 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{28}\cdot 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{3} q^{3} + \beta_{4} q^{5} + ( - \beta_{7} + 2 \beta_{3}) q^{7} + (\beta_{4} + \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{3} + \beta_{4} q^{5} + ( - \beta_{7} + 2 \beta_{3}) q^{7} + (\beta_{4} + \beta_1 + 1) q^{9} + \beta_{5} q^{11} + ( - \beta_{6} - 6 \beta_{4} + 6 \beta_1) q^{13} + ( - 3 \beta_{7} + \beta_{5} + \cdots + \beta_{2}) q^{15}+ \cdots + ( - 5 \beta_{5} + 94 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{9} - 1008 q^{21} + 1160 q^{25} + 1296 q^{29} - 8064 q^{41} + 6160 q^{45} + 3976 q^{49} - 28672 q^{61} + 23040 q^{65} + 20496 q^{69} - 40808 q^{81} + 19200 q^{85} - 1008 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 41x^{6} + 1357x^{4} + 13284x^{2} + 104976 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -295\nu^{7} + 2916\nu^{6} - 31211\nu^{5} - 839983\nu^{3} - 14031144\nu - 42382602 ) / 1978506 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 656\nu^{6} + 21712\nu^{4} + 890192\nu^{2} + 5196960 ) / 109917 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -41\nu^{7} - 1357\nu^{5} - 55637\nu^{3} - 104976\nu ) / 219834 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 295\nu^{7} + 2916\nu^{6} + 31211\nu^{5} + 839983\nu^{3} + 14031144\nu - 42382602 ) / 1978506 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -2960\nu^{6} - 119416\nu^{4} - 3137384\nu^{2} - 19975248 ) / 109917 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 77\nu^{7} + 3481\nu^{5} + 123605\nu^{3} + 2150712\nu ) / 43011 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2009\nu^{7} + 66493\nu^{5} + 1846877\nu^{3} + 5143824\nu ) / 659502 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{6} - \beta_{4} + 8\beta_{3} + \beta_1 ) / 32 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{5} - 4\beta_{4} + 11\beta_{2} - 4\beta _1 - 328 ) / 32 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -3\beta_{7} - 49\beta_{3} ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -82\beta_{5} - 164\beta_{4} - 289\beta_{2} - 164\beta _1 - 8264 ) / 32 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 492\beta_{7} - 619\beta_{6} + 2095\beta_{4} + 5444\beta_{3} - 2095\beta_1 ) / 32 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 1357\beta_{4} + 1357\beta _1 + 58138 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 16284\beta_{7} + 17927\beta_{6} - 66779\beta_{4} + 159700\beta_{3} + 66779\beta_1 ) / 32 \) 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.

comment: embeddings in the coefficient field
 
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
2.75276 + 4.76792i
2.75276 − 4.76792i
1.63472 − 2.83143i
1.63472 + 2.83143i
−1.63472 − 2.83143i
−1.63472 + 2.83143i
−2.75276 + 4.76792i
−2.75276 − 4.76792i
0 −11.0110 0 19.6214 − 15.4919i 0 −20.6325 0 40.2428 0
79.2 0 −11.0110 0 19.6214 + 15.4919i 0 −20.6325 0 40.2428 0
79.3 0 −6.53890 0 −19.6214 − 15.4919i 0 73.2823 0 −38.2428 0
79.4 0 −6.53890 0 −19.6214 + 15.4919i 0 73.2823 0 −38.2428 0
79.5 0 6.53890 0 −19.6214 − 15.4919i 0 −73.2823 0 −38.2428 0
79.6 0 6.53890 0 −19.6214 + 15.4919i 0 −73.2823 0 −38.2428 0
79.7 0 11.0110 0 19.6214 − 15.4919i 0 20.6325 0 40.2428 0
79.8 0 11.0110 0 19.6214 + 15.4919i 0 20.6325 0 40.2428 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 79.8
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 80.5.h.c ✓ 8
3.b odd 2 1 720.5.j.e 8
4.b odd 2 1 inner 80.5.h.c ✓ 8
5.b even 2 1 inner 80.5.h.c ✓ 8
5.c odd 4 2 400.5.b.k 8
8.b even 2 1 320.5.h.e 8
8.d odd 2 1 320.5.h.e 8
12.b even 2 1 720.5.j.e 8
15.d odd 2 1 720.5.j.e 8
20.d odd 2 1 inner 80.5.h.c ✓ 8
20.e even 4 2 400.5.b.k 8
40.e odd 2 1 320.5.h.e 8
40.f even 2 1 320.5.h.e 8
60.h even 2 1 720.5.j.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
80.5.h.c ✓ 8 1.a even 1 1 trivial
80.5.h.c ✓ 8 4.b odd 2 1 inner
80.5.h.c ✓ 8 5.b even 2 1 inner
80.5.h.c ✓ 8 20.d odd 2 1 inner
320.5.h.e 8 8.b even 2 1
320.5.h.e 8 8.d odd 2 1
320.5.h.e 8 40.e odd 2 1
320.5.h.e 8 40.f even 2 1
400.5.b.k 8 5.c odd 4 2
400.5.b.k 8 20.e even 4 2
720.5.j.e 8 3.b odd 2 1
720.5.j.e 8 12.b even 2 1
720.5.j.e 8 15.d odd 2 1
720.5.j.e 8 60.h even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 164T_{3}^{2} + 5184 \) acting on \(S_{5}^{\mathrm{new}}(80, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} - 164 T^{2} + 5184)^{2} \) Copy content Toggle raw display
$5$ \( (T^{4} - 290 T^{2} + 390625)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} - 5796 T^{2} + 2286144)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 37824 T^{2} + 325730304)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 98688 T^{2} + 391090176)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 314112 T^{2} + 11893211136)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 337344 T^{2} + 26885505024)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} - 425124 T^{2} + 20923043904)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 324 T - 72316)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + 602112)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} + \cdots + 1113109401600)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 2016 T - 2385796)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots + 2221077547584)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + \cdots + 56655006410304)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots + 2434798227456)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + \cdots + 1799117881344)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 7168 T + 2741116)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} + \cdots + 313967497080384)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots + 35\!\cdots\!44)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 4369152 T^{2} + 541319233536)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots + 27\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + \cdots + 15\!\cdots\!24)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 252 T - 378364)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots + 39737188417536)^{2} \) Copy content Toggle raw display
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