↑

Properties

Label 1890.2.i.g
Level $1890$
Weight $2$
Character orbit 1890.i
Analytic conductor $15.092$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1890,2,Mod(991,1890)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1890, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4, 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("1890.991");
 
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.i (of order \(3\), degree \(2\), not 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: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 3 x^{11} + 7 x^{10} - 3 x^{9} - 2 x^{8} + 24 x^{7} - 21 x^{6} + 72 x^{5} - 18 x^{4} + \cdots + 729 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 3^{4} \)
Twist minimal: no (minimal twist has level 630)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} + (\beta_{5} + 1) q^{5} + (\beta_{10} - \beta_{9} - \beta_{8} + \cdots + 2) q^{7}+ \cdots - q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{4} + (\beta_{5} + 1) q^{5} + (\beta_{10} - \beta_{9} - \beta_{8} + \cdots + 2) q^{7}+ \cdots + (2 \beta_{9} - \beta_{8} + 3 \beta_{6} + \cdots + 2) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{2} + 12 q^{4} + 6 q^{5} + 4 q^{7} - 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 12 q^{2} + 12 q^{4} + 6 q^{5} + 4 q^{7} - 12 q^{8} - 6 q^{10} - 3 q^{11} - 2 q^{13} - 4 q^{14} + 12 q^{16} - q^{17} + 8 q^{19} + 6 q^{20} + 3 q^{22} - 11 q^{23} - 6 q^{25} + 2 q^{26} + 4 q^{28} - 13 q^{29} - 42 q^{31} - 12 q^{32} + q^{34} - 4 q^{35} + 18 q^{37} - 8 q^{38} - 6 q^{40} - 5 q^{41} - 11 q^{43} - 3 q^{44} + 11 q^{46} - 46 q^{47} + 6 q^{50} - 2 q^{52} - 2 q^{53} - 6 q^{55} - 4 q^{56} + 13 q^{58} + 2 q^{59} + 2 q^{61} + 42 q^{62} + 12 q^{64} - 4 q^{65} - 4 q^{67} - q^{68} + 4 q^{70} + 30 q^{71} + 22 q^{73} - 18 q^{74} + 8 q^{76} + 31 q^{77} - 54 q^{79} + 6 q^{80} + 5 q^{82} - 6 q^{83} + q^{85} + 11 q^{86} + 3 q^{88} + 18 q^{89} + 14 q^{91} - 11 q^{92} + 46 q^{94} + 16 q^{95} - 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 3 x^{11} + 7 x^{10} - 3 x^{9} - 2 x^{8} + 24 x^{7} - 21 x^{6} + 72 x^{5} - 18 x^{4} + \cdots + 729 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 64 \nu^{11} + 12 \nu^{10} - 254 \nu^{9} + 426 \nu^{8} + 331 \nu^{7} - 978 \nu^{6} + 1302 \nu^{5} + \cdots + 53217 ) / 11421 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 83 \nu^{11} + 1368 \nu^{10} - 1931 \nu^{9} + 2034 \nu^{8} + 5539 \nu^{7} - 1935 \nu^{6} + \cdots + 299133 ) / 11421 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 91 \nu^{11} + 537 \nu^{10} - 956 \nu^{9} + 1509 \nu^{8} + 2275 \nu^{7} - 408 \nu^{6} + 4191 \nu^{5} + \cdots + 145800 ) / 11421 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 44 \nu^{11} + 67 \nu^{10} - 157 \nu^{9} + 334 \nu^{8} + 536 \nu^{7} + 109 \nu^{6} + 807 \nu^{5} + \cdots + 27783 ) / 3807 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 51 \nu^{11} - 199 \nu^{10} + 225 \nu^{9} + 56 \nu^{8} - 558 \nu^{7} + 497 \nu^{6} - 1311 \nu^{5} + \cdots - 23085 ) / 3807 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 59 \nu^{11} + 61 \nu^{10} + 17 \nu^{9} + 544 \nu^{8} + 629 \nu^{7} + 1021 \nu^{6} + 1425 \nu^{5} + \cdots + 25920 ) / 3807 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 4 \nu^{11} - 21 \nu^{10} + 37 \nu^{9} - 21 \nu^{8} - 80 \nu^{7} + 87 \nu^{6} - 156 \nu^{5} + \cdots - 4860 ) / 243 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 212 \nu^{11} + 66 \nu^{10} + 154 \nu^{9} - 1464 \nu^{8} - 1070 \nu^{7} - 1572 \nu^{6} + \cdots - 67068 ) / 11421 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 230 \nu^{11} - 327 \nu^{10} + 224 \nu^{9} + 1716 \nu^{8} - 172 \nu^{7} + 1905 \nu^{6} + 1533 \nu^{5} + \cdots + 60264 ) / 11421 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 265 \nu^{11} + 576 \nu^{10} - 865 \nu^{9} - 279 \nu^{8} + 1412 \nu^{7} - 2952 \nu^{6} + \cdots + 98901 ) / 11421 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 892 \nu^{11} - 1842 \nu^{10} + 2752 \nu^{9} + 3135 \nu^{8} - 2657 \nu^{7} + 9264 \nu^{6} + \cdots - 134136 ) / 11421 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{10} + \beta_{9} + \beta_{8} - \beta_{6} + \beta_{4} - \beta_{3} + \beta_{2} - \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - \beta_{11} - 2 \beta_{10} + 2 \beta_{9} + \beta_{8} - 2 \beta_{7} + 3 \beta_{6} - 3 \beta_{5} + \cdots - 3 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 4\beta_{10} + 2\beta_{8} + 3\beta_{6} - \beta_{5} + 4\beta_{4} + 2\beta_{3} - 4\beta_{2} + \beta _1 - 4 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - \beta_{11} - \beta_{10} - 3 \beta_{9} + 3 \beta_{8} + \beta_{7} + 2 \beta_{6} + 13 \beta_{5} + \cdots - 2 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 3 \beta_{10} - \beta_{9} - 15 \beta_{8} + 9 \beta_{7} - 5 \beta_{6} - 8 \beta_{5} - 11 \beta_{4} + \cdots - 11 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 7 \beta_{11} + 3 \beta_{10} - 8 \beta_{9} - 22 \beta_{8} + 10 \beta_{7} + 4 \beta_{6} + 26 \beta_{5} + \cdots + 50 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 18 \beta_{11} + 20 \beta_{10} - 44 \beta_{9} - 32 \beta_{8} - 34 \beta_{6} + 36 \beta_{5} - 2 \beta_{4} + \cdots + 33 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 8 \beta_{11} - 20 \beta_{10} + 38 \beta_{9} + 10 \beta_{8} - 38 \beta_{7} - 6 \beta_{6} - 48 \beta_{5} + \cdots - 138 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 54 \beta_{11} + 76 \beta_{10} - 9 \beta_{9} + 47 \beta_{8} - 27 \beta_{7} + 30 \beta_{6} - 208 \beta_{5} + \cdots + 68 ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 26 \beta_{11} - 145 \beta_{10} + 123 \beta_{9} + 309 \beta_{8} + 55 \beta_{7} - 124 \beta_{6} + \cdots - 38 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 81 \beta_{11} - 447 \beta_{10} + 197 \beta_{9} + 111 \beta_{8} - 45 \beta_{7} + 94 \beta_{6} + \cdots - 110 ) / 3 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(757\) \(1081\) \(1541\)
\(\chi(n)\) \(1\) \(\beta_{5}\) \(\beta_{5}\)

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
1.39898 − 1.02120i
0.628063 + 1.61417i
1.22323 + 1.22626i
−0.778860 − 1.54705i
−1.67391 + 0.444996i
0.702501 − 1.58319i
1.39898 + 1.02120i
0.628063 − 1.61417i
1.22323 − 1.22626i
−0.778860 + 1.54705i
−1.67391 − 0.444996i
0.702501 + 1.58319i
−1.00000 0 1.00000 0.500000 − 0.866025i 0 −2.25729 − 1.38008i −1.00000 0 −0.500000 + 0.866025i
991.2 −1.00000 0 1.00000 0.500000 − 0.866025i 0 −2.25729 − 1.38008i −1.00000 0 −0.500000 + 0.866025i
991.3 −1.00000 0 1.00000 0.500000 − 0.866025i 0 1.40545 − 2.24159i −1.00000 0 −0.500000 + 0.866025i
991.4 −1.00000 0 1.00000 0.500000 − 0.866025i 0 1.40545 − 2.24159i −1.00000 0 −0.500000 + 0.866025i
991.5 −1.00000 0 1.00000 0.500000 − 0.866025i 0 1.85185 + 1.88962i −1.00000 0 −0.500000 + 0.866025i
991.6 −1.00000 0 1.00000 0.500000 − 0.866025i 0 1.85185 + 1.88962i −1.00000 0 −0.500000 + 0.866025i
1171.1 −1.00000 0 1.00000 0.500000 + 0.866025i 0 −2.25729 + 1.38008i −1.00000 0 −0.500000 − 0.866025i
1171.2 −1.00000 0 1.00000 0.500000 + 0.866025i 0 −2.25729 + 1.38008i −1.00000 0 −0.500000 − 0.866025i
1171.3 −1.00000 0 1.00000 0.500000 + 0.866025i 0 1.40545 + 2.24159i −1.00000 0 −0.500000 − 0.866025i
1171.4 −1.00000 0 1.00000 0.500000 + 0.866025i 0 1.40545 + 2.24159i −1.00000 0 −0.500000 − 0.866025i
1171.5 −1.00000 0 1.00000 0.500000 + 0.866025i 0 1.85185 − 1.88962i −1.00000 0 −0.500000 − 0.866025i
1171.6 −1.00000 0 1.00000 0.500000 + 0.866025i 0 1.85185 − 1.88962i −1.00000 0 −0.500000 − 0.866025i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 991.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.h even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1890.2.i.g 12
3.b odd 2 1 630.2.i.g ✓ 12
7.c even 3 1 1890.2.l.g 12
9.c even 3 1 1890.2.l.g 12
9.d odd 6 1 630.2.l.g yes 12
21.h odd 6 1 630.2.l.g yes 12
63.h even 3 1 inner 1890.2.i.g 12
63.j odd 6 1 630.2.i.g ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
630.2.i.g ✓ 12 3.b odd 2 1
630.2.i.g ✓ 12 63.j odd 6 1
630.2.l.g yes 12 9.d odd 6 1
630.2.l.g yes 12 21.h odd 6 1
1890.2.i.g 12 1.a even 1 1 trivial
1890.2.i.g 12 63.h even 3 1 inner
1890.2.l.g 12 7.c even 3 1
1890.2.l.g 12 9.c even 3 1

Hecke kernels

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

\( T_{11}^{12} + 3 T_{11}^{11} + 53 T_{11}^{10} - 60 T_{11}^{9} + 1494 T_{11}^{8} - 2466 T_{11}^{7} + \cdots + 77841 \) Copy content Toggle raw display
\( T_{13}^{12} + 2 T_{13}^{11} + 38 T_{13}^{10} + 24 T_{13}^{9} + 1108 T_{13}^{8} + 1129 T_{13}^{7} + \cdots + 15625 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{12} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( (T^{2} - T + 1)^{6} \) Copy content Toggle raw display
$7$ \( (T^{6} - 2 T^{5} + \cdots + 343)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} + 3 T^{11} + \cdots + 77841 \) Copy content Toggle raw display
$13$ \( T^{12} + 2 T^{11} + \cdots + 15625 \) Copy content Toggle raw display
$17$ \( T^{12} + T^{11} + \cdots + 9801 \) Copy content Toggle raw display
$19$ \( T^{12} - 8 T^{11} + \cdots + 5929 \) Copy content Toggle raw display
$23$ \( T^{12} + 11 T^{11} + \cdots + 35390601 \) Copy content Toggle raw display
$29$ \( T^{12} + 13 T^{11} + \cdots + 729 \) Copy content Toggle raw display
$31$ \( (T^{6} + 21 T^{5} + \cdots + 34617)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 313396209 \) Copy content Toggle raw display
$41$ \( T^{12} + 5 T^{11} + \cdots + 531441 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 3018293721 \) Copy content Toggle raw display
$47$ \( (T^{6} + 23 T^{5} + \cdots + 59373)^{2} \) Copy content Toggle raw display
$53$ \( T^{12} + 2 T^{11} + \cdots + 88209 \) Copy content Toggle raw display
$59$ \( (T^{6} - T^{5} + \cdots - 20547)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} - T^{5} + \cdots + 95821)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} + 2 T^{5} + \cdots - 83727)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} - 15 T^{5} + \cdots + 4887)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} - 22 T^{11} + \cdots + 30614089 \) Copy content Toggle raw display
$79$ \( (T^{6} + 27 T^{5} + \cdots + 1983)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 265070961 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 18815334561 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 1888367924041 \) Copy content Toggle raw display
show more
show less