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Properties

Label 1980.2.z.g
Level $1980$
Weight $2$
Character orbit 1980.z
Analytic conductor $15.810$
Analytic rank $0$
Dimension $16$
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: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 2 x^{15} + 7 x^{14} - 6 x^{13} + 478 x^{12} + 1440 x^{11} + 8131 x^{10} + 9262 x^{9} + \cdots + 11881 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{5} + \beta_{3} - \beta_{2} - 1) q^{5} - \beta_{10} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{5} + \beta_{3} - \beta_{2} - 1) q^{5} - \beta_{10} q^{7} + ( - \beta_{15} + \beta_{7} + \beta_{6} + \cdots + 1) q^{11}+ \cdots + (3 \beta_{15} + \beta_{14} - \beta_{11} + \cdots - 2) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 4 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 4 q^{5} + 10 q^{13} + 4 q^{17} + 4 q^{19} - 4 q^{25} + 14 q^{29} - 10 q^{31} - 10 q^{37} - 6 q^{41} - 12 q^{43} - 12 q^{47} - 6 q^{49} + 22 q^{53} + 10 q^{55} + 12 q^{59} + 18 q^{61} - 20 q^{65} + 12 q^{67} - 26 q^{71} + 42 q^{73} + 26 q^{77} + 36 q^{79} - 8 q^{83} + 4 q^{85} - 20 q^{89} - 10 q^{91} + 4 q^{95} - 10 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 2 x^{15} + 7 x^{14} - 6 x^{13} + 478 x^{12} + 1440 x^{11} + 8131 x^{10} + 9262 x^{9} + \cdots + 11881 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 29\!\cdots\!94 \nu^{15} + \cdots - 27\!\cdots\!33 ) / 38\!\cdots\!89 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 53\!\cdots\!52 \nu^{15} + \cdots + 34\!\cdots\!16 ) / 38\!\cdots\!89 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 12\!\cdots\!26 \nu^{15} + \cdots + 49\!\cdots\!15 ) / 38\!\cdots\!89 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 18\!\cdots\!18 \nu^{15} + \cdots - 18\!\cdots\!59 ) / 38\!\cdots\!89 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 29\!\cdots\!98 \nu^{15} + \cdots + 15\!\cdots\!46 ) / 38\!\cdots\!89 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 48\!\cdots\!40 \nu^{15} + \cdots - 38\!\cdots\!77 ) / 38\!\cdots\!89 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 49\!\cdots\!39 \nu^{15} + \cdots + 87\!\cdots\!60 ) / 38\!\cdots\!89 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 64\!\cdots\!07 \nu^{15} + \cdots + 13\!\cdots\!77 ) / 38\!\cdots\!89 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 60\!\cdots\!20 \nu^{15} + \cdots + 77\!\cdots\!56 ) / 35\!\cdots\!99 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 14\!\cdots\!24 \nu^{15} + \cdots + 10\!\cdots\!73 ) / 35\!\cdots\!99 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 23\!\cdots\!96 \nu^{15} + \cdots - 11\!\cdots\!38 ) / 35\!\cdots\!99 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 30\!\cdots\!77 \nu^{15} + \cdots + 83\!\cdots\!77 ) / 38\!\cdots\!89 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 43\!\cdots\!08 \nu^{15} + \cdots + 58\!\cdots\!84 ) / 38\!\cdots\!89 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 48\!\cdots\!28 \nu^{15} + \cdots + 19\!\cdots\!12 ) / 38\!\cdots\!89 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{15} + \beta_{14} + 2 \beta_{13} - \beta_{12} + \beta_{11} - 2 \beta_{10} - 2 \beta_{9} - 2 \beta_{8} + \cdots - 9 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 4 \beta_{15} + 4 \beta_{14} + 7 \beta_{13} + \beta_{11} - \beta_{10} - 3 \beta_{9} - 4 \beta_{8} + \cdots - 36 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 31 \beta_{15} + 49 \beta_{14} + 56 \beta_{13} + 17 \beta_{12} + 25 \beta_{11} + 17 \beta_{9} + \cdots - 218 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 105 \beta_{15} + 85 \beta_{14} + 151 \beta_{13} - 20 \beta_{12} + 90 \beta_{11} + 85 \beta_{10} + \cdots - 922 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 84 \beta_{13} + 84 \beta_{12} + 1342 \beta_{10} + 1067 \beta_{9} + 749 \beta_{8} - 2067 \beta_{7} + \cdots - 833 \beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 3409 \beta_{15} - 3541 \beta_{14} - 6230 \beta_{13} + 868 \beta_{12} - 2689 \beta_{11} + 6230 \beta_{10} + \cdots + 28139 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 31503 \beta_{15} - 40086 \beta_{14} - 60261 \beta_{13} - 23685 \beta_{11} + 23685 \beta_{10} + \cdots + 246256 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 174320 \beta_{15} - 209344 \beta_{14} - 299707 \beta_{13} - 19760 \beta_{12} - 125387 \beta_{11} + \cdots + 1401903 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 636714 \beta_{15} - 760630 \beta_{14} - 1015900 \beta_{13} - 123916 \beta_{12} - 493056 \beta_{11} + \cdots + 5044985 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 641148 \beta_{13} - 641148 \beta_{12} - 6886636 \beta_{10} - 6357398 \beta_{9} - 4205036 \beta_{8} + \cdots + 4671411 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 20421302 \beta_{15} + 24562505 \beta_{14} + 40037935 \beta_{13} - 3036065 \beta_{12} + 15475430 \beta_{11} + \cdots - 162287264 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 186196237 \beta_{15} + 224630689 \beta_{14} + 340104946 \beta_{13} + 138176365 \beta_{11} + \cdots - 1485369184 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 1064385468 \beta_{15} + 1290112415 \beta_{14} + 1859387970 \beta_{13} + 89980426 \beta_{12} + \cdots - 8466096395 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 3743305722 \beta_{15} + 4504311944 \beta_{14} + 6048550853 \beta_{13} + 761006222 \beta_{12} + \cdots - 29839697385 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(397\) \(541\) \(991\) \(1541\)
\(\chi(n)\) \(1\) \(\beta_{2}\) \(1\) \(1\)

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
−0.391893 + 0.284727i
0.356538 − 0.259040i
4.60677 − 3.34701i
−2.95338 + 2.14575i
−0.391893 − 0.284727i
0.356538 + 0.259040i
4.60677 + 3.34701i
−2.95338 − 2.14575i
−1.10361 + 3.39655i
0.509879 − 1.56925i
0.618861 − 1.90466i
−0.643169 + 1.97947i
−1.10361 − 3.39655i
0.509879 + 1.56925i
0.618861 + 1.90466i
−0.643169 − 1.97947i
0 0 0 0.309017 − 0.951057i 0 −3.94036 + 2.86284i 0 0 0
181.2 0 0 0 0.309017 − 0.951057i 0 −1.36290 + 0.990206i 0 0 0
181.3 0 0 0 0.309017 − 0.951057i 0 2.27804 − 1.65509i 0 0 0
181.4 0 0 0 0.309017 − 0.951057i 0 3.02523 − 2.19796i 0 0 0
361.1 0 0 0 0.309017 + 0.951057i 0 −3.94036 − 2.86284i 0 0 0
361.2 0 0 0 0.309017 + 0.951057i 0 −1.36290 − 0.990206i 0 0 0
361.3 0 0 0 0.309017 + 0.951057i 0 2.27804 + 1.65509i 0 0 0
361.4 0 0 0 0.309017 + 0.951057i 0 3.02523 + 2.19796i 0 0 0
1081.1 0 0 0 −0.809017 − 0.587785i 0 −1.24868 + 3.84304i 0 0 0
1081.2 0 0 0 −0.809017 − 0.587785i 0 −0.361738 + 1.11332i 0 0 0
1081.3 0 0 0 −0.809017 − 0.587785i 0 0.177595 − 0.546581i 0 0 0
1081.4 0 0 0 −0.809017 − 0.587785i 0 1.43282 − 4.40977i 0 0 0
1621.1 0 0 0 −0.809017 + 0.587785i 0 −1.24868 − 3.84304i 0 0 0
1621.2 0 0 0 −0.809017 + 0.587785i 0 −0.361738 − 1.11332i 0 0 0
1621.3 0 0 0 −0.809017 + 0.587785i 0 0.177595 + 0.546581i 0 0 0
1621.4 0 0 0 −0.809017 + 0.587785i 0 1.43282 + 4.40977i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 181.4
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.g ✓ 16
3.b odd 2 1 1980.2.z.h yes 16
11.c even 5 1 inner 1980.2.z.g ✓ 16
33.h odd 10 1 1980.2.z.h yes 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1980.2.z.g ✓ 16 1.a even 1 1 trivial
1980.2.z.g ✓ 16 11.c even 5 1 inner
1980.2.z.h yes 16 3.b odd 2 1
1980.2.z.h yes 16 33.h odd 10 1

Hecke kernels

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

\( T_{7}^{16} + 17 T_{7}^{14} + 8 T_{7}^{13} + 367 T_{7}^{12} - 1028 T_{7}^{11} + 7639 T_{7}^{10} + \cdots + 1185921 \) Copy content Toggle raw display
\( T_{17}^{16} - 4 T_{17}^{15} + 18 T_{17}^{14} + 8 T_{17}^{13} + 1633 T_{17}^{12} + 2908 T_{17}^{11} + \cdots + 6405961 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{4} \) Copy content Toggle raw display
$7$ \( T^{16} + 17 T^{14} + \cdots + 1185921 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 214358881 \) Copy content Toggle raw display
$13$ \( T^{16} - 10 T^{15} + \cdots + 192721 \) Copy content Toggle raw display
$17$ \( T^{16} - 4 T^{15} + \cdots + 6405961 \) Copy content Toggle raw display
$19$ \( T^{16} - 4 T^{15} + \cdots + 229441 \) Copy content Toggle raw display
$23$ \( (T^{8} - 81 T^{6} + \cdots - 6525)^{2} \) Copy content Toggle raw display
$29$ \( T^{16} - 14 T^{15} + \cdots + 116281 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 4341360321 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 2135456521 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 330646850361 \) Copy content Toggle raw display
$43$ \( (T^{8} + 6 T^{7} + \cdots + 68981)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 23696946938401 \) Copy content Toggle raw display
$53$ \( T^{16} - 22 T^{15} + \cdots + 11978521 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 172423161 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 75785134681 \) Copy content Toggle raw display
$67$ \( (T^{8} - 6 T^{7} + \cdots - 17879)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 43873072681 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 1095222947841 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 332328037441 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 1417359299841 \) Copy content Toggle raw display
$89$ \( (T^{8} + 10 T^{7} + \cdots - 31141)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 9006895320201 \) Copy content Toggle raw display
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