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Properties

Label 1980.2.z.d
Level $1980$
Weight $2$
Character orbit 1980.z
Analytic conductor $15.810$
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] = [1980,2,Mod(181,1980)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1980, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1980.181");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1980 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1980.z (of order \(5\), degree \(4\), 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: \(15.8103796002\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{5})\)
Coefficient field: 8.0.159390625.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + 6x^{6} - 11x^{5} + 21x^{4} - 5x^{3} + 10x^{2} + 25x + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 220)
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{5} + ( - \beta_{5} - \beta_{4} + \beta_1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{5} + ( - \beta_{5} - \beta_{4} + \beta_1) q^{7} + ( - 2 \beta_{7} + \beta_{6} + 2 \beta_{3} + \cdots - 2) q^{11}+ \cdots + ( - 7 \beta_{6} - 6 \beta_{5} + \cdots + 7) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{5} - q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2 q^{5} - q^{7} - 5 q^{11} + 4 q^{13} - 9 q^{17} - 7 q^{19} + 10 q^{23} - 2 q^{25} + q^{29} + 22 q^{31} - 4 q^{35} + 4 q^{37} - 24 q^{41} - 22 q^{43} + 6 q^{47} - 27 q^{49} + q^{53} - 5 q^{55} + 9 q^{59} + 22 q^{61} + 26 q^{65} - 2 q^{67} + 22 q^{71} + 5 q^{73} - 16 q^{77} - 25 q^{79} - 33 q^{83} + 4 q^{85} - 8 q^{89} + 14 q^{91} + 7 q^{95} + 20 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} + 6x^{6} - 11x^{5} + 21x^{4} - 5x^{3} + 10x^{2} + 25x + 25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 555\nu^{7} - 2159\nu^{6} + 7489\nu^{5} - 18164\nu^{4} + 40069\nu^{3} - 84434\nu^{2} + 43855\nu + 375 ) / 94655 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -970\nu^{7} - 1002\nu^{6} - 6608\nu^{5} + 9063\nu^{4} - 14943\nu^{3} + 27673\nu^{2} - 68120\nu + 35160 ) / 94655 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -1604\nu^{7} + 4159\nu^{6} - 12059\nu^{5} + 28414\nu^{4} - 81659\nu^{3} + 38305\nu^{2} - 13500\nu - 13875 ) / 94655 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -2052\nu^{7} + 2252\nu^{6} - 19912\nu^{5} + 21007\nu^{4} - 82042\nu^{3} + 35785\nu^{2} - 19395\nu - 90925 ) / 94655 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -2667\nu^{7} + 6691\nu^{6} - 17466\nu^{5} + 50856\nu^{4} - 82441\nu^{3} + 72554\nu^{2} - 4035\nu - 12035 ) / 94655 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 4024\nu^{7} - 1464\nu^{6} + 21519\nu^{5} - 26434\nu^{4} + 59219\nu^{3} + 22635\nu^{2} + 54640\nu + 66675 ) / 94655 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{7} - \beta_{6} - \beta_{5} - \beta_{3} - 3\beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{6} + \beta_{5} - 4\beta_{4} - \beta_{3} - \beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 7\beta_{7} + 7\beta_{6} + 2\beta_{5} + 13\beta_{3} + 13\beta_{2} - 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -8\beta_{7} - 11\beta_{6} - 20\beta_{5} + 20\beta_{4} - 11\beta_{2} + 8\beta _1 - 12 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -19\beta_{7} - 19\beta_{4} - 68\beta_{3} - 36\beta_{2} - 24\beta _1 + 36 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 111\beta_{7} + 81\beta_{6} + 111\beta_{5} - 55\beta_{4} + 81\beta_{3} + 148\beta_{2} - 56\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1980\mathbb{Z}\right)^\times\).

\(n\) \(397\) \(541\) \(991\) \(1541\)
\(\chi(n)\) \(1\) \(-\beta_{6}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
181.1
0.453245 − 1.39494i
−0.762262 + 2.34600i
0.453245 + 1.39494i
−0.762262 − 2.34600i
1.43801 + 1.04478i
−0.628998 − 0.456994i
1.43801 − 1.04478i
−0.628998 + 0.456994i
0 0 0 −0.309017 + 0.951057i 0 −0.453245 + 0.329302i 0 0 0
181.2 0 0 0 −0.309017 + 0.951057i 0 0.762262 − 0.553816i 0 0 0
361.1 0 0 0 −0.309017 − 0.951057i 0 −0.453245 − 0.329302i 0 0 0
361.2 0 0 0 −0.309017 − 0.951057i 0 0.762262 + 0.553816i 0 0 0
1081.1 0 0 0 0.809017 + 0.587785i 0 −1.43801 + 4.42575i 0 0 0
1081.2 0 0 0 0.809017 + 0.587785i 0 0.628998 − 1.93586i 0 0 0
1621.1 0 0 0 0.809017 − 0.587785i 0 −1.43801 − 4.42575i 0 0 0
1621.2 0 0 0 0.809017 − 0.587785i 0 0.628998 + 1.93586i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 181.2
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1980.2.z.d 8
3.b odd 2 1 220.2.m.b ✓ 8
11.c even 5 1 inner 1980.2.z.d 8
12.b even 2 1 880.2.bo.c 8
15.d odd 2 1 1100.2.n.b 8
15.e even 4 2 1100.2.cb.b 16
33.f even 10 1 2420.2.a.l 4
33.h odd 10 1 220.2.m.b ✓ 8
33.h odd 10 1 2420.2.a.k 4
132.n odd 10 1 9680.2.a.co 4
132.o even 10 1 880.2.bo.c 8
132.o even 10 1 9680.2.a.cp 4
165.o odd 10 1 1100.2.n.b 8
165.v even 20 2 1100.2.cb.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
220.2.m.b ✓ 8 3.b odd 2 1
220.2.m.b ✓ 8 33.h odd 10 1
880.2.bo.c 8 12.b even 2 1
880.2.bo.c 8 132.o even 10 1
1100.2.n.b 8 15.d odd 2 1
1100.2.n.b 8 165.o odd 10 1
1100.2.cb.b 16 15.e even 4 2
1100.2.cb.b 16 165.v even 20 2
1980.2.z.d 8 1.a even 1 1 trivial
1980.2.z.d 8 11.c even 5 1 inner
2420.2.a.k 4 33.h odd 10 1
2420.2.a.l 4 33.f even 10 1
9680.2.a.co 4 132.n odd 10 1
9680.2.a.cp 4 132.o even 10 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1980, [\chi])\):

\( T_{7}^{8} + T_{7}^{7} + 21T_{7}^{6} - 29T_{7}^{5} + 96T_{7}^{4} - 45T_{7}^{3} - 15T_{7}^{2} + 25T_{7} + 25 \) Copy content Toggle raw display
\( T_{17}^{8} + 9T_{17}^{7} + 86T_{17}^{6} + 759T_{17}^{5} + 4801T_{17}^{4} + 18765T_{17}^{3} + 48350T_{17}^{2} + 78375T_{17} + 75625 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} + T^{7} + \cdots + 25 \) Copy content Toggle raw display
$11$ \( T^{8} + 5 T^{7} + \cdots + 14641 \) Copy content Toggle raw display
$13$ \( T^{8} - 4 T^{7} + \cdots + 3025 \) Copy content Toggle raw display
$17$ \( T^{8} + 9 T^{7} + \cdots + 75625 \) Copy content Toggle raw display
$19$ \( T^{8} + 7 T^{7} + \cdots + 14641 \) Copy content Toggle raw display
$23$ \( (T^{4} - 5 T^{3} - 12 T^{2} + \cdots + 11)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} - T^{7} + \cdots + 121 \) Copy content Toggle raw display
$31$ \( T^{8} - 22 T^{7} + \cdots + 75625 \) Copy content Toggle raw display
$37$ \( T^{8} - 4 T^{7} + \cdots + 546121 \) Copy content Toggle raw display
$41$ \( T^{8} + 24 T^{7} + \cdots + 15625 \) Copy content Toggle raw display
$43$ \( (T^{4} + 11 T^{3} + \cdots + 3649)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} - 6 T^{7} + \cdots + 97515625 \) Copy content Toggle raw display
$53$ \( T^{8} - T^{7} + \cdots + 4239481 \) Copy content Toggle raw display
$59$ \( T^{8} - 9 T^{7} + \cdots + 434281 \) Copy content Toggle raw display
$61$ \( T^{8} - 22 T^{7} + \cdots + 6241 \) Copy content Toggle raw display
$67$ \( (T^{4} + T^{3} - 86 T^{2} + \cdots + 995)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} - 22 T^{7} + \cdots + 841 \) Copy content Toggle raw display
$73$ \( T^{8} - 5 T^{7} + \cdots + 450241 \) Copy content Toggle raw display
$79$ \( T^{8} + 25 T^{7} + \cdots + 1692601 \) Copy content Toggle raw display
$83$ \( T^{8} + 33 T^{7} + \cdots + 21025 \) Copy content Toggle raw display
$89$ \( (T^{4} + 4 T^{3} - 194 T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} - 20 T^{7} + \cdots + 39325441 \) Copy content Toggle raw display
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