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Properties

Label 80.20.c.a
Level $80$
Weight $20$
Character orbit 80.c
Analytic conductor $183.053$
Analytic rank $0$
Dimension $8$
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: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 726881x^{6} + 160513523376x^{4} + 10607307647230976x^{2} + 32429098232548950016 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{34}\cdot 3^{8}\cdot 5^{13} \)
Twist minimal: no (minimal twist has level 5)
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} + (\beta_{3} - 27 \beta_{2} + \cdots + 18375) q^{5}+ \cdots + (3 \beta_{7} - 6 \beta_{6} + \cdots + 43169823) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{3} + (\beta_{3} - 27 \beta_{2} + \cdots + 18375) q^{5}+ \cdots + ( - 24185255517 \beta_{7} + \cdots + 20\!\cdots\!84) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 147000 q^{5} + 345358584 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 147000 q^{5} + 345358584 q^{9} + 3379575264 q^{11} + 242324628000 q^{15} - 4547188380640 q^{19} - 2983154334624 q^{21} + 17715709625000 q^{25} - 188222300345040 q^{29} - 72115006686976 q^{31} + 299115755916000 q^{35} - 30\!\cdots\!28 q^{39}+ \cdots + 16\!\cdots\!72 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 726881x^{6} + 160513523376x^{4} + 10607307647230976x^{2} + 32429098232548950016 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 256\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 173\nu^{7} + 111026445\nu^{5} + 20083149107568\nu^{3} + 944157442242313216\nu ) / 1505285197258752 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 24489 \nu^{7} + 1840496 \nu^{6} + 19390777353 \nu^{5} + 1808281798512 \nu^{4} + \cdots + 95\!\cdots\!04 ) / 37\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 254739 \nu^{7} + 7361984 \nu^{6} - 173282696883 \nu^{5} + 7233127194048 \nu^{4} + \cdots + 38\!\cdots\!16 ) / 75\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 294733 \nu^{7} + 14723968 \nu^{6} - 233244460461 \nu^{5} + 14466254388096 \nu^{4} + \cdots + 76\!\cdots\!32 ) / 75\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 24489 \nu^{7} + 1840496 \nu^{6} + 19390777353 \nu^{5} + 1808281798512 \nu^{4} + \cdots + 97\!\cdots\!64 ) / 37\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 8163 \nu^{7} - 51533888 \nu^{6} + 6463592451 \nu^{5} - 28030010519616 \nu^{4} + \cdots - 30\!\cdots\!52 ) / 627202165524480 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_1 ) / 256 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{6} - \beta_{3} - 23260192 ) / 128 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -32\beta_{5} + 96\beta_{4} - 64\beta_{3} + 20992\beta_{2} - 285185\beta_1 ) / 256 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 3552\beta_{7} - 342961\beta_{6} + 60384\beta_{5} + 698161\beta_{3} + 60384\beta_{2} + 6634526125088 ) / 128 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -10223584\beta_{5} - 47984736\beta_{4} + 136863808\beta_{3} - 25364308480\beta_{2} + 90488995249\beta_1 ) / 256 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 3489829344 \beta_{7} + 118717717089 \beta_{6} - 33155280864 \beta_{5} - 310669743585 \beta_{3} + \cdots - 21\!\cdots\!32 ) / 128 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 10276005538272 \beta_{5} + 19650822763104 \beta_{4} - 80405667679296 \beta_{3} + \cdots - 30\!\cdots\!17 \beta_1 ) / 256 \) 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
− 490.322i
337.134i
− 56.6657i
607.943i
− 607.943i
56.6657i
− 337.134i
490.322i
0 − 42067.9i 0 −2.12371e6 + 3.81619e6i 0 − 4.16374e7i 0 −6.07450e8 0
49.2 0 − 35433.1i 0 −4.08230e6 + 1.55186e6i 0 1.27830e8i 0 −9.32466e7 0
49.3 0 − 33157.6i 0 2.22206e6 − 3.75978e6i 0 − 2.70208e7i 0 6.28322e7 0
49.4 0 − 18754.1i 0 4.05746e6 + 1.61570e6i 0 − 1.79878e8i 0 8.10544e8 0
49.5 0 18754.1i 0 4.05746e6 − 1.61570e6i 0 1.79878e8i 0 8.10544e8 0
49.6 0 33157.6i 0 2.22206e6 + 3.75978e6i 0 2.70208e7i 0 6.28322e7 0
49.7 0 35433.1i 0 −4.08230e6 − 1.55186e6i 0 − 1.27830e8i 0 −9.32466e7 0
49.8 0 42067.9i 0 −2.12371e6 − 3.81619e6i 0 4.16374e7i 0 −6.07450e8 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.8
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.a 8
4.b odd 2 1 5.20.b.a ✓ 8
5.b even 2 1 inner 80.20.c.a 8
12.b even 2 1 45.20.b.b 8
20.d odd 2 1 5.20.b.a ✓ 8
20.e even 4 2 25.20.a.f 8
60.h even 2 1 45.20.b.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.20.b.a ✓ 8 4.b odd 2 1
5.20.b.a ✓ 8 20.d odd 2 1
25.20.a.f 8 20.e even 4 2
45.20.b.b 8 12.b even 2 1
45.20.b.b 8 60.h even 2 1
80.20.c.a 8 1.a even 1 1 trivial
80.20.c.a 8 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 4476366576 T_{3}^{6} + \cdots + 85\!\cdots\!96 \) acting on \(S_{20}^{\mathrm{new}}(80, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + \cdots + 85\!\cdots\!96 \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 13\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 66\!\cdots\!16 \) Copy content Toggle raw display
$11$ \( (T^{4} + \cdots + 34\!\cdots\!96)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 13\!\cdots\!76 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 77\!\cdots\!56 \) Copy content Toggle raw display
$19$ \( (T^{4} + \cdots + 35\!\cdots\!00)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 38\!\cdots\!56 \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots - 22\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + \cdots + 16\!\cdots\!56)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 63\!\cdots\!36 \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots + 12\!\cdots\!36)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 88\!\cdots\!16 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 65\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 67\!\cdots\!96 \) Copy content Toggle raw display
$59$ \( (T^{4} + \cdots + 41\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots - 62\!\cdots\!04)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 21\!\cdots\!56 \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots - 26\!\cdots\!24)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 14\!\cdots\!56 \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots - 16\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 18\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots + 12\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 13\!\cdots\!76 \) Copy content Toggle raw display
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