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Properties

Label 1890.2.d.f
Level $1890$
Weight $2$
Character orbit 1890.d
Analytic conductor $15.092$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1890,2,Mod(1889,1890)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1890, 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("1890.1889");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1890 = 2 \cdot 3^{3} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1890.d (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: \(15.0917259820\)
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: \( x^{16} - 3 x^{15} + 3 x^{14} + 5 x^{12} + 15 x^{11} - 12 x^{10} + 381 x^{9} - 1356 x^{8} + 1905 x^{7} + \cdots + 390625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{8} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + q^{4} - \beta_{8} q^{5} - \beta_{11} q^{7} + q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + q^{4} - \beta_{8} q^{5} - \beta_{11} q^{7} + q^{8} - \beta_{8} q^{10} - \beta_{10} q^{11} + (\beta_{15} - \beta_{8} + \beta_{3}) q^{13} - \beta_{11} q^{14} + q^{16} + ( - \beta_{15} + \beta_{13} + \cdots - \beta_1) q^{17}+ \cdots + (\beta_{11} - 2 \beta_{10} + \beta_{7} + \cdots + 1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 16 q^{2} + 16 q^{4} + 16 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 16 q^{2} + 16 q^{4} + 16 q^{8} + 16 q^{16} - 8 q^{23} - 6 q^{25} + 16 q^{32} + q^{35} - 8 q^{46} + 2 q^{49} - 6 q^{50} + 16 q^{53} + 16 q^{64} + 40 q^{65} + q^{70} + 14 q^{77} - 8 q^{79} - 44 q^{85} - 40 q^{91} - 8 q^{92} + 36 q^{95} + 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 3 x^{15} + 3 x^{14} + 5 x^{12} + 15 x^{11} - 12 x^{10} + 381 x^{9} - 1356 x^{8} + 1905 x^{7} + \cdots + 390625 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2889 \nu^{15} - 5237 \nu^{14} - 7698 \nu^{13} + 8140 \nu^{12} + 91095 \nu^{11} + 176235 \nu^{10} + \cdots - 2265625 ) / 320625000 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 1157 \nu^{15} + 13675 \nu^{14} + 18367 \nu^{13} + 31537 \nu^{12} - 23010 \nu^{11} + \cdots + 1051156250 ) / 96187500 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 92 \nu^{15} - 534 \nu^{14} - 896 \nu^{13} - 1155 \nu^{12} - 2235 \nu^{11} - 8430 \nu^{10} + \cdots - 38671875 ) / 4218750 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 36657 \nu^{15} - 175754 \nu^{14} - 267996 \nu^{13} - 399200 \nu^{12} - 1119135 \nu^{11} + \cdots - 9946562500 ) / 961875000 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 9464 \nu^{15} + 71635 \nu^{14} + 56284 \nu^{13} + 191314 \nu^{12} + 164370 \nu^{11} + \cdots + 6282546875 ) / 192375000 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 52446 \nu^{15} - 5267 \nu^{14} - 113298 \nu^{13} - 12740 \nu^{12} - 545430 \nu^{11} + \cdots + 4601796875 ) / 961875000 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 76979 \nu^{15} - 271048 \nu^{14} - 397982 \nu^{13} - 508330 \nu^{12} - 1787895 \nu^{11} + \cdots - 16908125000 ) / 961875000 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 94123 \nu^{15} + 19986 \nu^{14} + 21004 \nu^{13} + 71340 \nu^{12} + 225465 \nu^{11} + \cdots - 7737187500 ) / 961875000 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 6621 \nu^{15} + 3433 \nu^{14} - 3678 \nu^{13} + 7300 \nu^{12} + 5505 \nu^{11} + \cdots + 731703125 ) / 64125000 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 26091 \nu^{15} - 35008 \nu^{14} - 3147 \nu^{13} + 545 \nu^{12} - 64545 \nu^{11} + \cdots - 5057421875 ) / 240468750 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 105171 \nu^{15} - 214468 \nu^{14} - 115272 \nu^{13} - 189040 \nu^{12} + 156405 \nu^{11} + \cdots - 24921406250 ) / 961875000 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 118747 \nu^{15} + 53886 \nu^{14} - 94876 \nu^{13} - 71340 \nu^{12} - 348585 \nu^{11} + \cdots + 13508437500 ) / 961875000 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 60471 \nu^{15} - 52273 \nu^{14} + 14268 \nu^{13} - 49030 \nu^{12} + 87555 \nu^{11} + \cdots - 7834296875 ) / 480937500 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 25828 \nu^{15} - 33429 \nu^{14} - 9806 \nu^{13} - 42960 \nu^{12} - 7860 \nu^{11} + \cdots - 4724296875 ) / 192375000 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 25828 \nu^{15} - 33429 \nu^{14} - 9806 \nu^{13} - 42960 \nu^{12} - 7860 \nu^{11} + \cdots - 4724296875 ) / 192375000 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{15} + \beta_{14} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{10} - \beta_{9} + \beta_{7} - 3\beta_{6} + \beta_{5} + \beta_{4} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{13} + \beta_{11} + \beta_{8} + \beta_{6} - \beta_{5} - 2\beta_{3} + \beta_{2} + 2\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 6 \beta_{15} + 2 \beta_{14} - 7 \beta_{13} - 2 \beta_{12} - 8 \beta_{11} + 6 \beta_{10} - 2 \beta_{9} + \cdots - 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - \beta_{15} - \beta_{14} - 2 \beta_{13} - 6 \beta_{12} + 4 \beta_{11} - 10 \beta_{10} - 18 \beta_{9} + \cdots - 18 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 6 \beta_{14} - 30 \beta_{12} - 8 \beta_{11} - 6 \beta_{10} + 9 \beta_{9} + 8 \beta_{6} - 3 \beta_{5} + \cdots - 18 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 12 \beta_{15} - 12 \beta_{14} + 148 \beta_{13} + 31 \beta_{12} - 20 \beta_{11} - 24 \beta_{10} + \cdots - 432 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 402 \beta_{15} - 134 \beta_{14} - 131 \beta_{13} + 26 \beta_{12} - 22 \beta_{11} - 24 \beta_{10} + \cdots + 173 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 233 \beta_{15} - 80 \beta_{13} + 154 \beta_{11} + 158 \beta_{10} + 54 \beta_{8} - 158 \beta_{7} + \cdots - 208 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 612 \beta_{15} - 204 \beta_{14} + 1008 \beta_{13} - 12 \beta_{12} - 998 \beta_{11} + 1021 \beta_{10} + \cdots + 144 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 846 \beta_{15} - 846 \beta_{14} + 858 \beta_{13} + 1649 \beta_{12} + 2730 \beta_{11} - 1086 \beta_{10} + \cdots + 1422 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 624 \beta_{14} + 996 \beta_{12} + 2610 \beta_{11} + 18 \beta_{10} + 1764 \beta_{9} - 2610 \beta_{6} + \cdots + 4447 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 4429 \beta_{15} + 4429 \beta_{14} - 1110 \beta_{13} + 10170 \beta_{12} + 2130 \beta_{11} + \cdots + 5850 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 720 \beta_{15} - 240 \beta_{14} - 54720 \beta_{13} - 10560 \beta_{12} - 5760 \beta_{11} + 6551 \beta_{10} + \cdots + 48240 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 39150 \beta_{15} + 46732 \beta_{13} + 1519 \beta_{11} - 8400 \beta_{10} + 15949 \beta_{8} + \cdots + 14468 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1890\mathbb{Z}\right)^\times\).

\(n\) \(757\) \(1081\) \(1541\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1889.1
0.599527 − 2.15420i
0.599527 + 2.15420i
1.98399 + 1.03140i
1.98399 − 1.03140i
−1.24491 − 1.85747i
−1.24491 + 1.85747i
2.02291 − 0.952801i
2.02291 + 0.952801i
1.83661 − 1.27549i
1.83661 + 1.27549i
−2.23107 − 0.149392i
−2.23107 + 0.149392i
0.0987778 − 2.23389i
0.0987778 + 2.23389i
−1.56583 + 1.59630i
−1.56583 − 1.59630i
1.00000 0 1.00000 −2.16535 − 0.557894i 0 −1.39492 + 2.24815i 1.00000 0 −2.16535 − 0.557894i
1889.2 1.00000 0 1.00000 −2.16535 + 0.557894i 0 −1.39492 − 2.24815i 1.00000 0 −2.16535 + 0.557894i
1889.3 1.00000 0 1.00000 −1.88521 − 1.20249i 0 2.61764 − 0.384656i 1.00000 0 −1.88521 − 1.20249i
1889.4 1.00000 0 1.00000 −1.88521 + 1.20249i 0 2.61764 + 0.384656i 1.00000 0 −1.88521 + 1.20249i
1889.5 1.00000 0 1.00000 −0.986159 − 2.00686i 0 −2.28525 − 1.33328i 1.00000 0 −0.986159 − 2.00686i
1889.6 1.00000 0 1.00000 −0.986159 + 2.00686i 0 −2.28525 + 1.33328i 1.00000 0 −0.986159 + 2.00686i
1889.7 1.00000 0 1.00000 −0.186305 − 2.22829i 0 0.479366 − 2.60196i 1.00000 0 −0.186305 − 2.22829i
1889.8 1.00000 0 1.00000 −0.186305 + 2.22829i 0 0.479366 + 2.60196i 1.00000 0 −0.186305 + 2.22829i
1889.9 1.00000 0 1.00000 0.186305 − 2.22829i 0 −0.479366 + 2.60196i 1.00000 0 0.186305 − 2.22829i
1889.10 1.00000 0 1.00000 0.186305 + 2.22829i 0 −0.479366 − 2.60196i 1.00000 0 0.186305 + 2.22829i
1889.11 1.00000 0 1.00000 0.986159 − 2.00686i 0 2.28525 + 1.33328i 1.00000 0 0.986159 − 2.00686i
1889.12 1.00000 0 1.00000 0.986159 + 2.00686i 0 2.28525 − 1.33328i 1.00000 0 0.986159 + 2.00686i
1889.13 1.00000 0 1.00000 1.88521 − 1.20249i 0 −2.61764 + 0.384656i 1.00000 0 1.88521 − 1.20249i
1889.14 1.00000 0 1.00000 1.88521 + 1.20249i 0 −2.61764 − 0.384656i 1.00000 0 1.88521 + 1.20249i
1889.15 1.00000 0 1.00000 2.16535 − 0.557894i 0 1.39492 − 2.24815i 1.00000 0 2.16535 − 0.557894i
1889.16 1.00000 0 1.00000 2.16535 + 0.557894i 0 1.39492 + 2.24815i 1.00000 0 2.16535 + 0.557894i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1889.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
15.d odd 2 1 inner
105.g even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1890.2.d.f yes 16
3.b odd 2 1 1890.2.d.e ✓ 16
5.b even 2 1 1890.2.d.e ✓ 16
7.b odd 2 1 inner 1890.2.d.f yes 16
15.d odd 2 1 inner 1890.2.d.f yes 16
21.c even 2 1 1890.2.d.e ✓ 16
35.c odd 2 1 1890.2.d.e ✓ 16
105.g even 2 1 inner 1890.2.d.f yes 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1890.2.d.e ✓ 16 3.b odd 2 1
1890.2.d.e ✓ 16 5.b even 2 1
1890.2.d.e ✓ 16 21.c even 2 1
1890.2.d.e ✓ 16 35.c odd 2 1
1890.2.d.f yes 16 1.a even 1 1 trivial
1890.2.d.f yes 16 7.b odd 2 1 inner
1890.2.d.f yes 16 15.d odd 2 1 inner
1890.2.d.f yes 16 105.g even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1890, [\chi])\):

\( T_{11}^{8} + 50T_{11}^{6} + 825T_{11}^{4} + 4496T_{11}^{2} + 100 \) Copy content Toggle raw display
\( T_{23}^{4} + 2T_{23}^{3} - 59T_{23}^{2} - 138T_{23} + 360 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{16} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} + 3 T^{14} + \cdots + 390625 \) Copy content Toggle raw display
$7$ \( T^{16} - T^{14} + \cdots + 5764801 \) Copy content Toggle raw display
$11$ \( (T^{8} + 50 T^{6} + \cdots + 100)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} - 51 T^{6} + \cdots + 1600)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + 78 T^{6} + \cdots + 25600)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + 71 T^{6} + \cdots + 1296)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 2 T^{3} + \cdots + 360)^{4} \) Copy content Toggle raw display
$29$ \( (T^{8} + 98 T^{6} + \cdots + 16384)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + 89 T^{6} + \cdots + 60516)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + 100 T^{6} + \cdots + 230400)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} - 169 T^{6} + \cdots + 360000)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + 49 T^{6} + \cdots + 8100)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + 391 T^{6} + \cdots + 20793600)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 4 T^{3} + \cdots + 1011)^{4} \) Copy content Toggle raw display
$59$ \( (T^{8} - 274 T^{6} + \cdots + 6471936)^{2} \) Copy content Toggle raw display
$61$ \( T^{16} \) Copy content Toggle raw display
$67$ \( (T^{8} + 457 T^{6} + \cdots + 4796100)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + 182 T^{6} + \cdots + 732736)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} - 278 T^{6} + \cdots + 16842816)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 2 T^{3} + \cdots - 1152)^{4} \) Copy content Toggle raw display
$83$ \( (T^{8} + 78 T^{6} + \cdots + 64)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} - 523 T^{6} + \cdots + 459684)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} - 269 T^{6} + \cdots + 1299600)^{2} \) Copy content Toggle raw display
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