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Properties

Label 80.20.c.c
Level $80$
Weight $20$
Character orbit 80.c
Analytic conductor $183.053$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [80,20,Mod(49,80)]
 
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([0, 0, 1]))
 
N = Newforms(chi, 20, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("80.49");
 
S:= CuspForms(chi, 20);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 80 = 2^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 20 \)
Character orbit: \([\chi]\) \(=\) 80.c (of order \(2\), degree \(1\), not 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: \(183.053357245\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 14903918288 x^{7} + 443569659980446 x^{6} + \cdots + 53\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{69}\cdot 3^{6}\cdot 5^{18} \)
Twist minimal: no (minimal twist has level 10)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + ( - \beta_{3} + 2 \beta_{2} + \cdots + 290267) q^{5}+ \cdots + (2 \beta_{8} - 2 \beta_{7} + \cdots - 871875497) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + ( - \beta_{3} + 2 \beta_{2} + \cdots + 290267) q^{5}+ \cdots + (2757038049 \beta_{9} + \cdots - 41\!\cdots\!16) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 2902670 q^{5} - 8718754970 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 2902670 q^{5} - 8718754970 q^{9} + 2965225880 q^{11} - 225010487480 q^{15} + 4196987836200 q^{19} - 501224287480 q^{21} + 19247576437650 q^{25} + 219360620418300 q^{29} + 667121586663680 q^{31} - 14\!\cdots\!60 q^{35}+ \cdots - 41\!\cdots\!60 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 14903918288 x^{7} + 443569659980446 x^{6} + \cdots + 53\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 22\!\cdots\!37 \nu^{9} + \cdots - 15\!\cdots\!00 ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 23\!\cdots\!12 \nu^{9} + \cdots + 37\!\cdots\!00 ) / 10\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 22\!\cdots\!73 \nu^{9} + \cdots + 28\!\cdots\!00 ) / 68\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 77\!\cdots\!57 \nu^{9} + \cdots + 99\!\cdots\!00 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 97\!\cdots\!27 \nu^{9} + \cdots + 26\!\cdots\!00 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 50\!\cdots\!01 \nu^{9} + \cdots - 71\!\cdots\!00 ) / 50\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 38\!\cdots\!54 \nu^{9} + \cdots - 30\!\cdots\!00 ) / 17\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 17\!\cdots\!23 \nu^{9} + \cdots + 78\!\cdots\!00 ) / 68\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 67\!\cdots\!24 \nu^{9} + \cdots - 10\!\cdots\!00 ) / 17\!\cdots\!25 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{6} + 27\beta_{3} + 1118\beta_{2} + 8197\beta_1 ) / 163840 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2944 \beta_{9} + 5824 \beta_{8} + 8736 \beta_{7} + 5824 \beta_{6} + 5792 \beta_{5} + \cdots - 18187904 \beta_1 ) / 409600 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 134788352 \beta_{9} - 134637184 \beta_{8} - 106500256 \beta_{7} - 461715679 \beta_{6} + \cdots + 18\!\cdots\!00 ) / 4096000 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 21023858304 \beta_{9} + 197762860864 \beta_{8} - 197762860864 \beta_{7} + 133703057094 \beta_{6} + \cdots - 18\!\cdots\!00 ) / 1024000 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 309212292915200 \beta_{9} + \cdots + 83\!\cdots\!00 ) / 4096000 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 95\!\cdots\!68 \beta_{9} + \cdots + 12\!\cdots\!68 \beta_1 ) / 2048000 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 84\!\cdots\!00 \beta_{9} + \cdots - 25\!\cdots\!00 ) / 4096000 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 76\!\cdots\!60 \beta_{9} + \cdots + 39\!\cdots\!00 ) / 512000 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 50\!\cdots\!72 \beta_{9} + \cdots - 68\!\cdots\!00 ) / 4096000 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
−3428.29 − 3428.29i
2867.92 − 2867.92i
1933.50 − 1933.50i
−1192.46 − 1192.46i
−180.677 + 180.677i
−180.677 − 180.677i
−1192.46 + 1192.46i
1933.50 + 1933.50i
2867.92 + 2867.92i
−3428.29 + 3428.29i
0 − 64043.7i 0 4.32599e6 − 599382.i 0 − 1.31313e6i 0 −2.93934e9 0
49.2 0 − 61880.3i 0 −3.63002e6 − 2.42826e6i 0 − 1.58929e8i 0 −2.66691e9 0
49.3 0 − 43192.0i 0 3.01510e6 + 3.15954e6i 0 2.08285e8i 0 −7.03291e8 0
49.4 0 − 19327.1i 0 −3.24650e6 − 2.92125e6i 0 3.24295e7i 0 7.88724e8 0
49.5 0 − 908.466i 0 986770. − 4.25438e6i 0 4.96199e7i 0 1.16144e9 0
49.6 0 908.466i 0 986770. + 4.25438e6i 0 − 4.96199e7i 0 1.16144e9 0
49.7 0 19327.1i 0 −3.24650e6 + 2.92125e6i 0 − 3.24295e7i 0 7.88724e8 0
49.8 0 43192.0i 0 3.01510e6 − 3.15954e6i 0 − 2.08285e8i 0 −7.03291e8 0
49.9 0 61880.3i 0 −3.63002e6 + 2.42826e6i 0 1.58929e8i 0 −2.66691e9 0
49.10 0 64043.7i 0 4.32599e6 + 599382.i 0 1.31313e6i 0 −2.93934e9 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.10
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 80.20.c.c 10
4.b odd 2 1 10.20.b.a ✓ 10
5.b even 2 1 inner 80.20.c.c 10
12.b even 2 1 90.20.c.b 10
20.d odd 2 1 10.20.b.a ✓ 10
20.e even 4 1 50.20.a.k 5
20.e even 4 1 50.20.a.l 5
60.h even 2 1 90.20.c.b 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.20.b.a ✓ 10 4.b odd 2 1
10.20.b.a ✓ 10 20.d odd 2 1
50.20.a.k 5 20.e even 4 1
50.20.a.l 5 20.e even 4 1
80.20.c.c 10 1.a even 1 1 trivial
80.20.c.c 10 5.b even 2 1 inner
90.20.c.b 10 12.b even 2 1
90.20.c.b 10 60.h even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{10} + 10170684820 T_{3}^{8} + \cdots + 90\!\cdots\!24 \) acting on \(S_{20}^{\mathrm{new}}(80, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} + \cdots + 90\!\cdots\!24 \) Copy content Toggle raw display
$5$ \( T^{10} + \cdots + 25\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{10} + \cdots + 48\!\cdots\!76 \) Copy content Toggle raw display
$11$ \( (T^{5} + \cdots + 23\!\cdots\!32)^{2} \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots + 20\!\cdots\!24 \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots + 26\!\cdots\!76 \) Copy content Toggle raw display
$19$ \( (T^{5} + \cdots - 16\!\cdots\!00)^{2} \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 22\!\cdots\!24 \) Copy content Toggle raw display
$29$ \( (T^{5} + \cdots - 20\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{5} + \cdots + 60\!\cdots\!32)^{2} \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots + 49\!\cdots\!76 \) Copy content Toggle raw display
$41$ \( (T^{5} + \cdots - 13\!\cdots\!32)^{2} \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots + 12\!\cdots\!24 \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 45\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 45\!\cdots\!24 \) Copy content Toggle raw display
$59$ \( (T^{5} + \cdots - 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} + \cdots - 30\!\cdots\!32)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 13\!\cdots\!76 \) Copy content Toggle raw display
$71$ \( (T^{5} + \cdots + 63\!\cdots\!32)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 98\!\cdots\!24 \) Copy content Toggle raw display
$79$ \( (T^{5} + \cdots + 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 62\!\cdots\!24 \) Copy content Toggle raw display
$89$ \( (T^{5} + \cdots + 27\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 73\!\cdots\!76 \) Copy content Toggle raw display
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