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Properties

Label 1984.2.b.b
Level $1984$
Weight $2$
Character orbit 1984.b
Analytic conductor $15.842$
Analytic rank $0$
Dimension $16$
CM discriminant -248
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1984,2,Mod(991,1984)]
 
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([1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1984.991");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1984 = 2^{6} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1984.b (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: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 24x^{14} + 368x^{12} - 3384x^{10} + 22416x^{8} - 96192x^{6} + 251168x^{4} - 408432x^{2} + 487204 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{31}]\)
Coefficient ring index: \( 2^{22} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{5} q^{3} - \beta_{3} q^{7} + (\beta_{8} - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{5} q^{3} - \beta_{3} q^{7} + (\beta_{8} - 3) q^{9} - \beta_{2} q^{11} + \beta_{6} q^{13} + (\beta_{11} - \beta_{9}) q^{21} + 5 q^{25} + (\beta_{12} + 3 \beta_{5} - \beta_{2}) q^{27} + ( - \beta_{11} - \beta_1) q^{29} - \beta_{10} q^{31} + ( - \beta_{15} - \beta_{13} - \beta_{8}) q^{33} + ( - \beta_{11} + \beta_1) q^{37} + (\beta_{14} - 2 \beta_{10} + \beta_{3}) q^{39} + (\beta_{13} - \beta_{8}) q^{41} + ( - \beta_{12} - \beta_{2}) q^{43} + ( - \beta_{14} + \beta_{3}) q^{47} + ( - \beta_{15} - 7) q^{49} + (\beta_{9} + 2 \beta_{6}) q^{53} + (\beta_{11} + 2 \beta_1) q^{61} + ( - 2 \beta_{10} - \beta_{7} + 3 \beta_{3}) q^{63} + 3 \beta_{7} q^{71} - 5 \beta_{5} q^{75} + (\beta_{11} - 2 \beta_{9} + \cdots - 2 \beta_1) q^{77}+ \cdots + ( - \beta_{12} - 2 \beta_{5} + \cdots + 6 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 48 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 48 q^{9} + 80 q^{25} - 112 q^{49} + 144 q^{81}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 24x^{14} + 368x^{12} - 3384x^{10} + 22416x^{8} - 96192x^{6} + 251168x^{4} - 408432x^{2} + 487204 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 34460 \nu^{15} + 891718 \nu^{13} - 12907398 \nu^{11} + 102662655 \nu^{9} + \cdots - 35195362018 \nu ) / 13569628842 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 65326 \nu^{15} - 786173 \nu^{13} + 8063790 \nu^{11} + 20350191 \nu^{9} + \cdots + 11386659386 \nu ) / 15077365380 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 74731 \nu^{14} - 506858 \nu^{12} + 1307040 \nu^{10} + 143558376 \nu^{8} - 1559556258 \nu^{6} + \cdots + 64878228896 ) / 12336026220 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 17899 \nu^{15} + 396783 \nu^{13} - 9310560 \nu^{11} + 160314309 \nu^{9} + \cdots + 16634729934 \nu ) / 2153909340 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 80141 \nu^{15} + 1850447 \nu^{13} - 27548976 \nu^{11} + 240892275 \nu^{9} + \cdots + 12879122218 \nu ) / 9046419228 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2672 \nu^{15} - 52847 \nu^{13} + 689472 \nu^{11} - 4669533 \nu^{9} + 20915508 \nu^{7} + \cdots + 1133128370 \nu ) / 274133916 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 3184 \nu^{14} + 64367 \nu^{12} - 962430 \nu^{10} + 7912776 \nu^{8} - 52346748 \nu^{6} + \cdots + 683593696 ) / 194407290 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 207265 \nu^{14} + 4267084 \nu^{12} - 58468830 \nu^{10} + 451064508 \nu^{8} + \cdots + 48144388064 ) / 9046419228 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 676454 \nu^{15} + 14551099 \nu^{13} - 207736788 \nu^{11} + 1702312107 \nu^{9} + \cdots + 144042779450 \nu ) / 27139257684 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 2\nu^{14} - 31\nu^{12} + 390\nu^{10} - 1968\nu^{8} + 8304\nu^{6} + 17790\nu^{4} - 144884\nu^{2} + 211792 ) / 53460 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 124889 \nu^{15} + 2653633 \nu^{13} - 37989462 \nu^{11} + 304416069 \nu^{9} + \cdots + 3270784814 \nu ) / 2467205244 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 1198949 \nu^{15} - 25714487 \nu^{13} + 383182590 \nu^{11} - 3216605016 \nu^{9} + \cdots - 164786140996 \nu ) / 22616048070 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 117713 \nu^{14} + 2717636 \nu^{12} - 39692856 \nu^{10} + 338714232 \nu^{8} + \cdots + 5892462952 ) / 1507736538 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 581201 \nu^{14} - 11938798 \nu^{12} + 173449290 \nu^{10} - 1374197784 \nu^{8} + \cdots - 81687420224 ) / 6168013110 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 8 \nu^{14} - 180 \nu^{12} + 2674 \nu^{10} - 23061 \nu^{8} + 141100 \nu^{6} - 514248 \nu^{4} + \cdots - 755928 ) / 65457 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{9} + \beta_{6} - 2\beta_{5} + \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{14} + \beta_{13} + 6\beta_{7} - 2\beta_{3} + 12 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -5\beta_{12} + 2\beta_{11} + 5\beta_{9} - 3\beta_{6} - 57\beta_{5} + 3\beta_{4} - 4\beta_{2} - 3\beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 6\beta_{14} + 6\beta_{10} + 12\beta_{8} + 27\beta_{7} - 12\beta_{3} - 40 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -6\beta_{12} + 24\beta_{11} - 32\beta_{9} - 26\beta_{6} - 95\beta_{5} + 15\beta_{4} - 54\beta_{2} - 101\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -54\beta_{15} - 31\beta_{14} - 125\beta_{13} + 108\beta_{10} + 222\beta_{8} - 90\beta_{7} - 110\beta_{3} - 1260 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 203 \beta_{12} + 328 \beta_{11} - 539 \beta_{9} + 135 \beta_{6} + 1125 \beta_{5} + 3 \beta_{4} + \cdots - 717 \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -189\beta_{15} - 210\beta_{14} - 324\beta_{13} - 120\beta_{10} + 48\beta_{8} - 1260\beta_{7} + 420\beta_{3} - 1256 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 1173 \beta_{12} + 762 \beta_{11} - 1180 \beta_{9} + 1490 \beta_{6} + 8294 \beta_{5} - 714 \beta_{4} + \cdots + 335 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 630 \beta_{15} - 1895 \beta_{14} + 32 \beta_{13} - 8460 \beta_{10} - 5538 \beta_{8} - 21432 \beta_{7} + \cdots + 21936 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 9107 \beta_{12} - 17714 \beta_{11} + 30577 \beta_{9} + 2157 \beta_{6} + 81015 \beta_{5} + \cdots + 55425 \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 17820 \beta_{15} + 4410 \beta_{14} + 34290 \beta_{13} - 27126 \beta_{10} - 23904 \beta_{8} + \cdots + 208000 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 31410 \beta_{12} - 172764 \beta_{11} + 283262 \beta_{9} - 137632 \beta_{6} - 195715 \beta_{5} + \cdots + 265649 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 415422 \beta_{15} + 176383 \beta_{14} + 714839 \beta_{13} + 348516 \beta_{10} - 134262 \beta_{8} + \cdots + 2893092 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 346243 \beta_{12} - 476183 \beta_{11} + 760171 \beta_{9} - 547944 \beta_{6} - 2812002 \beta_{5} + \cdots + 436836 \beta_1 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
991.1
−2.23678 − 1.68803i
2.23678 − 1.68803i
2.76799 − 1.46799i
−2.76799 − 1.46799i
1.15682 − 0.919237i
−1.15682 − 0.919237i
−1.99920 − 0.388029i
1.99920 − 0.388029i
1.99920 + 0.388029i
−1.99920 + 0.388029i
−1.15682 + 0.919237i
1.15682 + 0.919237i
−2.76799 + 1.46799i
2.76799 + 1.46799i
2.23678 + 1.68803i
−2.23678 + 1.68803i
0 − 3.37605i 0 0 0 − 1.69247i 0 −8.39773 0
991.2 0 − 3.37605i 0 0 0 1.69247i 0 −8.39773 0
991.3 0 − 2.93599i 0 0 0 − 5.01353i 0 −5.62001 0
991.4 0 − 2.93599i 0 0 0 5.01353i 0 −5.62001 0
991.5 0 − 1.83847i 0 0 0 − 5.01353i 0 −0.379986 0
991.6 0 − 1.83847i 0 0 0 5.01353i 0 −0.379986 0
991.7 0 − 0.776058i 0 0 0 − 1.69247i 0 2.39773 0
991.8 0 − 0.776058i 0 0 0 1.69247i 0 2.39773 0
991.9 0 0.776058i 0 0 0 − 1.69247i 0 2.39773 0
991.10 0 0.776058i 0 0 0 1.69247i 0 2.39773 0
991.11 0 1.83847i 0 0 0 − 5.01353i 0 −0.379986 0
991.12 0 1.83847i 0 0 0 5.01353i 0 −0.379986 0
991.13 0 2.93599i 0 0 0 − 5.01353i 0 −5.62001 0
991.14 0 2.93599i 0 0 0 5.01353i 0 −5.62001 0
991.15 0 3.37605i 0 0 0 − 1.69247i 0 −8.39773 0
991.16 0 3.37605i 0 0 0 1.69247i 0 −8.39773 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 991.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
248.g odd 2 1 CM by \(\Q(\sqrt{-62}) \)
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
31.b odd 2 1 inner
124.d even 2 1 inner
248.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.b.b ✓ 16
4.b odd 2 1 inner 1984.2.b.b ✓ 16
8.b even 2 1 inner 1984.2.b.b ✓ 16
8.d odd 2 1 inner 1984.2.b.b ✓ 16
31.b odd 2 1 inner 1984.2.b.b ✓ 16
124.d even 2 1 inner 1984.2.b.b ✓ 16
248.b even 2 1 inner 1984.2.b.b ✓ 16
248.g odd 2 1 CM 1984.2.b.b ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1984.2.b.b ✓ 16 1.a even 1 1 trivial
1984.2.b.b ✓ 16 4.b odd 2 1 inner
1984.2.b.b ✓ 16 8.b even 2 1 inner
1984.2.b.b ✓ 16 8.d odd 2 1 inner
1984.2.b.b ✓ 16 31.b odd 2 1 inner
1984.2.b.b ✓ 16 124.d even 2 1 inner
1984.2.b.b ✓ 16 248.b even 2 1 inner
1984.2.b.b ✓ 16 248.g odd 2 1 CM

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 24T_{3}^{6} + 180T_{3}^{4} + 432T_{3}^{2} + 200 \) acting on \(S_{2}^{\mathrm{new}}(1984, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( (T^{8} + 24 T^{6} + \cdots + 200)^{2} \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( (T^{4} + 28 T^{2} + 72)^{4} \) Copy content Toggle raw display
$11$ \( (T^{8} + 88 T^{6} + \cdots + 52488)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} - 104 T^{6} + \cdots + 78408)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} \) Copy content Toggle raw display
$19$ \( T^{16} \) Copy content Toggle raw display
$23$ \( T^{16} \) Copy content Toggle raw display
$29$ \( (T^{8} - 232 T^{6} + \cdots + 1800)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 31)^{8} \) Copy content Toggle raw display
$37$ \( (T^{8} - 296 T^{6} + \cdots + 6181128)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} - 164 T^{2} + 648)^{4} \) Copy content Toggle raw display
$43$ \( (T^{8} + 344 T^{6} + \cdots + 11347848)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 188 T^{2} + 8712)^{4} \) Copy content Toggle raw display
$53$ \( (T^{8} - 424 T^{6} + \cdots + 4980168)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} \) Copy content Toggle raw display
$61$ \( (T^{8} - 488 T^{6} + \cdots + 55377288)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} \) Copy content Toggle raw display
$71$ \( (T^{2} + 36)^{8} \) Copy content Toggle raw display
$73$ \( T^{16} \) Copy content Toggle raw display
$79$ \( T^{16} \) Copy content Toggle raw display
$83$ \( (T^{8} + 664 T^{6} + \cdots + 159204168)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} \) Copy content Toggle raw display
$97$ \( (T^{4} - 388 T^{2} + 1800)^{4} \) Copy content Toggle raw display
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