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Properties

Label 1984.2.c.f
Level $1984$
Weight $2$
Character orbit 1984.c
Analytic conductor $15.842$
Analytic rank $0$
Dimension $20$
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] = [1984,2,Mod(993,1984)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1984, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1984.993");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1984 = 2^{6} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1984.c (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: \(15.8423197610\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 36 x^{18} + 536 x^{16} + 4280 x^{14} + 19892 x^{12} + 54784 x^{10} + 87680 x^{8} + 77472 x^{6} + \cdots + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{23} \)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} - \beta_{15} q^{5} - \beta_{6} q^{7} + (\beta_{2} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} - \beta_{15} q^{5} - \beta_{6} q^{7} + (\beta_{2} - 1) q^{9} + (\beta_{15} + \beta_{13}) q^{11} + ( - \beta_{13} + \beta_{5} + \beta_1) q^{13} + ( - \beta_{7} + 1) q^{15} + (\beta_{7} + \beta_{3} + 1) q^{17} + (\beta_{13} + \beta_{12} + \cdots - \beta_1) q^{19}+ \cdots + (2 \beta_{18} - \beta_{17} + \cdots + 4 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 12 q^{9} + 28 q^{15} + 4 q^{17} + 20 q^{23} - 16 q^{25} + 20 q^{31} + 48 q^{39} + 4 q^{41} + 16 q^{47} + 8 q^{49} + 64 q^{55} - 4 q^{57} + 8 q^{63} + 12 q^{65} + 24 q^{71} - 16 q^{73} + 52 q^{79} - 20 q^{81} + 40 q^{87} + 12 q^{89} - 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 36 x^{18} + 536 x^{16} + 4280 x^{14} + 19892 x^{12} + 54784 x^{10} + 87680 x^{8} + 77472 x^{6} + \cdots + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 623 \nu^{18} + 21542 \nu^{16} + 302430 \nu^{14} + 2207800 \nu^{12} + 8870336 \nu^{10} + \cdots - 877952 ) / 125136 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 281 \nu^{18} - 9927 \nu^{16} - 143534 \nu^{14} - 1094196 \nu^{12} - 4716192 \nu^{10} + \cdots - 163424 ) / 41712 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 3049 \nu^{19} - 108106 \nu^{17} - 1574850 \nu^{15} - 12169184 \nu^{13} - 53644612 \nu^{11} + \cdots + 17956192 \nu ) / 500544 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 2815 \nu^{18} - 99793 \nu^{16} - 1456344 \nu^{14} - 11314856 \nu^{12} - 50517088 \nu^{10} + \cdots + 195904 ) / 250272 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 952 \nu^{18} - 33709 \nu^{16} - 489792 \nu^{14} - 3768692 \nu^{12} - 16521964 \nu^{10} + \cdots - 8192 ) / 83424 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 1036 \nu^{18} + 35891 \nu^{16} + 508268 \nu^{14} + 3794620 \nu^{12} + 16054796 \nu^{10} + \cdots - 27072 ) / 83424 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 3761 \nu^{18} + 132353 \nu^{16} + 1906332 \nu^{14} + 14483824 \nu^{12} + 62305832 \nu^{10} + \cdots - 427520 ) / 250272 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 1547 \nu^{19} - 52496 \nu^{17} - 717702 \nu^{15} - 5030488 \nu^{13} - 18806900 \nu^{11} + \cdots + 10247840 \nu ) / 166848 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 4201 \nu^{18} - 148831 \nu^{16} - 2162676 \nu^{14} - 16633400 \nu^{12} - 72862312 \nu^{10} + \cdots + 444064 ) / 250272 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 1847 \nu^{19} - 64758 \nu^{17} - 928190 \nu^{15} - 7000128 \nu^{13} - 29708652 \nu^{11} + \cdots + 7322848 \nu ) / 166848 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 25 \nu^{19} - 888 \nu^{17} - 12994 \nu^{15} - 101400 \nu^{13} - 456636 \nu^{11} - 1201944 \nu^{9} + \cdots - 62176 \nu ) / 2112 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 25 \nu^{19} + 888 \nu^{17} + 12994 \nu^{15} + 101400 \nu^{13} + 456636 \nu^{11} + 1201944 \nu^{9} + \cdots + 76960 \nu ) / 2112 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 2863 \nu^{19} - 101164 \nu^{17} - 1467150 \nu^{15} - 11274056 \nu^{13} - 49413412 \nu^{11} + \cdots + 1197280 \nu ) / 166848 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 752 \nu^{18} - 26693 \nu^{16} - 389388 \nu^{14} - 3015340 \nu^{12} - 13356020 \nu^{10} + \cdots - 159808 ) / 22752 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 11003 \nu^{19} - 392972 \nu^{17} - 5781966 \nu^{15} - 45337096 \nu^{13} - 204706484 \nu^{11} + \cdots - 25484320 \nu ) / 500544 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 2687 \nu^{19} - 96137 \nu^{17} - 1418838 \nu^{15} - 11183812 \nu^{13} - 50957504 \nu^{11} + \cdots - 6868096 \nu ) / 83424 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 12611 \nu^{19} + 448025 \nu^{17} + 6548052 \nu^{15} + 50894488 \nu^{13} + 226990448 \nu^{11} + \cdots + 10750336 \nu ) / 250272 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{14} + \beta_{13} - 7\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{11} + \beta_{9} - \beta_{4} - \beta_{3} - 9\beta_{2} + 25 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{18} - 2\beta_{17} - 10\beta_{14} - 10\beta_{13} + 3\beta_{12} - \beta_{10} - 2\beta_{5} + 52\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - \beta_{16} - 14 \beta_{11} - 17 \beta_{9} + \beta_{8} + 2 \beta_{7} + \beta_{6} + 9 \beta_{4} + \cdots - 174 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 2 \beta_{19} - 19 \beta_{18} + 33 \beta_{17} + 7 \beta_{15} + 89 \beta_{14} + 94 \beta_{13} + \cdots - 406 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 12 \beta_{16} + 156 \beta_{11} + 204 \beta_{9} - 16 \beta_{8} - 36 \beta_{7} - 8 \beta_{6} - 60 \beta_{4} + \cdots + 1284 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 40 \beta_{19} + 252 \beta_{18} - 404 \beta_{17} - 128 \beta_{15} - 780 \beta_{14} - 888 \beta_{13} + \cdots + 3304 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 92 \beta_{16} - 1596 \beta_{11} - 2160 \beta_{9} + 188 \beta_{8} + 432 \beta_{7} + 12 \beta_{6} + \cdots - 9908 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 552 \beta_{19} - 2876 \beta_{18} + 4404 \beta_{17} + 1604 \beta_{15} + 6856 \beta_{14} + \cdots - 27776 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 476 \beta_{16} + 15624 \beta_{11} + 21556 \beta_{9} - 1960 \beta_{8} - 4404 \beta_{7} + 628 \beta_{6} + \cdots + 79312 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 6560 \beta_{19} + 30316 \beta_{18} - 45224 \beta_{17} - 17328 \beta_{15} - 60644 \beta_{14} + \cdots + 239224 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 340 \beta_{16} - 149064 \beta_{11} - 208324 \beta_{9} + 19260 \beta_{8} + 41424 \beta_{7} + \cdots - 654184 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 72040 \beta_{19} - 304948 \beta_{18} + 448348 \beta_{17} + 174148 \beta_{15} + 539732 \beta_{14} + \cdots - 2096456 \beta_1 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 33776 \beta_{16} + 1399776 \beta_{11} + 1975584 \beta_{9} - 183344 \beta_{8} - 373280 \beta_{7} + \cdots + 5526464 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 753664 \beta_{19} + 2978512 \beta_{18} - 4346144 \beta_{17} - 1683568 \beta_{15} + \cdots + 18598224 \beta_1 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 621040 \beta_{16} - 13012832 \beta_{11} - 18515344 \beta_{9} + 1715136 \beta_{8} + 3287376 \beta_{7} + \cdots - 47566624 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 7630880 \beta_{19} - 28534528 \beta_{18} + 41495232 \beta_{17} + 15920768 \beta_{15} + \cdots - 166396416 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1984\mathbb{Z}\right)^\times\).

\(n\) \(63\) \(65\) \(1861\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
993.1
3.02148i
2.70103i
2.47211i
2.36943i
1.86129i
1.47775i
1.03559i
0.806203i
0.683848i
0.106567i
− 0.106567i
− 0.683848i
− 0.806203i
− 1.03559i
− 1.47775i
− 1.86129i
− 2.36943i
− 2.47211i
− 2.70103i
− 3.02148i
0 − 3.02148i 0 0.522149i 0 −1.03848 0 −6.12933 0
993.2 0 − 2.70103i 0 2.08511i 0 3.39624 0 −4.29559 0
993.3 0 − 2.47211i 0 0.852912i 0 −3.99520 0 −3.11134 0
993.4 0 − 2.36943i 0 − 0.566787i 0 1.27083 0 −2.61418 0
993.5 0 − 1.86129i 0 4.17354i 0 −1.41520 0 −0.464398 0
993.6 0 − 1.47775i 0 2.74897i 0 −1.20703 0 0.816257 0
993.7 0 − 1.03559i 0 − 3.21855i 0 4.12194 0 1.92755 0
993.8 0 − 0.806203i 0 0.442058i 0 3.52782 0 2.35004 0
993.9 0 − 0.683848i 0 − 3.99129i 0 −2.49293 0 2.53235 0
993.10 0 − 0.106567i 0 − 0.933203i 0 −2.16799 0 2.98864 0
993.11 0 0.106567i 0 0.933203i 0 −2.16799 0 2.98864 0
993.12 0 0.683848i 0 3.99129i 0 −2.49293 0 2.53235 0
993.13 0 0.806203i 0 − 0.442058i 0 3.52782 0 2.35004 0
993.14 0 1.03559i 0 3.21855i 0 4.12194 0 1.92755 0
993.15 0 1.47775i 0 − 2.74897i 0 −1.20703 0 0.816257 0
993.16 0 1.86129i 0 − 4.17354i 0 −1.41520 0 −0.464398 0
993.17 0 2.36943i 0 0.566787i 0 1.27083 0 −2.61418 0
993.18 0 2.47211i 0 − 0.852912i 0 −3.99520 0 −3.11134 0
993.19 0 2.70103i 0 − 2.08511i 0 3.39624 0 −4.29559 0
993.20 0 3.02148i 0 − 0.522149i 0 −1.03848 0 −6.12933 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 993.20
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1984.2.c.f yes 20
4.b odd 2 1 1984.2.c.e ✓ 20
8.b even 2 1 inner 1984.2.c.f yes 20
8.d odd 2 1 1984.2.c.e ✓ 20
16.e even 4 1 7936.2.a.x 10
16.e even 4 1 7936.2.a.y 10
16.f odd 4 1 7936.2.a.w 10
16.f odd 4 1 7936.2.a.z 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1984.2.c.e ✓ 20 4.b odd 2 1
1984.2.c.e ✓ 20 8.d odd 2 1
1984.2.c.f yes 20 1.a even 1 1 trivial
1984.2.c.f yes 20 8.b even 2 1 inner
7936.2.a.w 10 16.f odd 4 1
7936.2.a.x 10 16.e even 4 1
7936.2.a.y 10 16.e even 4 1
7936.2.a.z 10 16.f odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1984, [\chi])\):

\( T_{3}^{20} + 36 T_{3}^{18} + 536 T_{3}^{16} + 4280 T_{3}^{14} + 19892 T_{3}^{12} + 54784 T_{3}^{10} + \cdots + 64 \) Copy content Toggle raw display
\( T_{7}^{10} - 37T_{7}^{8} - 20T_{7}^{7} + 459T_{7}^{6} + 536T_{7}^{5} - 2031T_{7}^{4} - 3796T_{7}^{3} + 744T_{7}^{2} + 4816T_{7} + 2404 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( T^{20} + 36 T^{18} + \cdots + 64 \) Copy content Toggle raw display
$5$ \( T^{20} + 58 T^{18} + \cdots + 1024 \) Copy content Toggle raw display
$7$ \( (T^{10} - 37 T^{8} + \cdots + 2404)^{2} \) Copy content Toggle raw display
$11$ \( T^{20} + 132 T^{18} + \cdots + 5308416 \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 383219776 \) Copy content Toggle raw display
$17$ \( (T^{10} - 2 T^{9} + \cdots + 240448)^{2} \) Copy content Toggle raw display
$19$ \( T^{20} + 182 T^{18} + \cdots + 1327104 \) Copy content Toggle raw display
$23$ \( (T^{10} - 10 T^{9} + \cdots - 109568)^{2} \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 4542320337984 \) Copy content Toggle raw display
$31$ \( (T - 1)^{20} \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 184638652416 \) Copy content Toggle raw display
$41$ \( (T^{10} - 2 T^{9} + \cdots + 4321252)^{2} \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 39877608376384 \) Copy content Toggle raw display
$47$ \( (T^{10} - 8 T^{9} + \cdots - 232448)^{2} \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 601083887616 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 39437998081024 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 27\!\cdots\!96 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 19\!\cdots\!64 \) Copy content Toggle raw display
$71$ \( (T^{10} - 12 T^{9} + \cdots - 180785264)^{2} \) Copy content Toggle raw display
$73$ \( (T^{10} + 8 T^{9} + \cdots + 130963456)^{2} \) Copy content Toggle raw display
$79$ \( (T^{10} - 26 T^{9} + \cdots - 10717184)^{2} \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 24\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( (T^{10} - 6 T^{9} + \cdots - 191936)^{2} \) Copy content Toggle raw display
$97$ \( (T^{10} + 6 T^{9} + \cdots - 3985052)^{2} \) Copy content Toggle raw display
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