Properties

Label 1890.2.d.b
Level $1890$
Weight $2$
Character orbit 1890.d
Analytic conductor $15.092$
Analytic rank $0$
Dimension $8$
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: \(8\)
Coefficient field: 8.0.77720518656.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} + 26x^{6} - 64x^{5} + 161x^{4} - 220x^{3} + 232x^{2} - 132x + 33 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} + ( - \beta_{3} - \beta_1) q^{5} + (\beta_{5} - \beta_{4}) q^{7} - q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{4} + ( - \beta_{3} - \beta_1) q^{5} + (\beta_{5} - \beta_{4}) q^{7} - q^{8} + (\beta_{3} + \beta_1) q^{10} + (\beta_{7} + \beta_{5}) q^{11} + (\beta_{5} - 2 \beta_{4} - \beta_1) q^{13} + ( - \beta_{5} + \beta_{4}) q^{14} + q^{16} + (\beta_{6} + \beta_{3}) q^{17} - 3 \beta_{3} q^{19} + ( - \beta_{3} - \beta_1) q^{20} + ( - \beta_{7} - \beta_{5}) q^{22} + 3 q^{23} + ( - 2 \beta_{5} + 1) q^{25} + ( - \beta_{5} + 2 \beta_{4} + \beta_1) q^{26} + (\beta_{5} - \beta_{4}) q^{28} + \beta_{7} q^{29} + ( - \beta_{6} + 3 \beta_{3}) q^{31} - q^{32} + ( - \beta_{6} - \beta_{3}) q^{34} + ( - \beta_{7} + \beta_{3} - \beta_{2} - \beta_1) q^{35} - 2 \beta_{5} q^{37} + 3 \beta_{3} q^{38} + (\beta_{3} + \beta_1) q^{40} + 2 \beta_1 q^{41} + 3 \beta_{7} q^{43} + (\beta_{7} + \beta_{5}) q^{44} - 3 q^{46} + ( - \beta_{6} - 2 \beta_{3}) q^{47} + ( - \beta_{6} + 4) q^{49} + (2 \beta_{5} - 1) q^{50} + (\beta_{5} - 2 \beta_{4} - \beta_1) q^{52} + (\beta_{3} + 2 \beta_{2} - 6) q^{53} + ( - \beta_{6} + \beta_{5} - 2 \beta_{4} + 3 \beta_{3} - 2 \beta_1) q^{55} + ( - \beta_{5} + \beta_{4}) q^{56} - \beta_{7} q^{58} + 2 \beta_1 q^{59} - 3 \beta_{3} q^{61} + (\beta_{6} - 3 \beta_{3}) q^{62} + q^{64} + ( - 2 \beta_{7} - \beta_{5} - \beta_{3} - 2 \beta_{2} + 3) q^{65} - 2 \beta_{5} q^{67} + (\beta_{6} + \beta_{3}) q^{68} + (\beta_{7} - \beta_{3} + \beta_{2} + \beta_1) q^{70} - 2 \beta_{7} q^{71} + (\beta_{5} - 2 \beta_{4}) q^{73} + 2 \beta_{5} q^{74} - 3 \beta_{3} q^{76} + ( - \beta_{6} + 6 \beta_{3} + \beta_{2} - 3) q^{77} + ( - \beta_{3} - 2 \beta_{2} - 1) q^{79} + ( - \beta_{3} - \beta_1) q^{80} - 2 \beta_1 q^{82} - 5 \beta_{3} q^{83} + ( - 3 \beta_{7} + \beta_{5} + \beta_{3} + 2 \beta_{2} + 2) q^{85} - 3 \beta_{7} q^{86} + ( - \beta_{7} - \beta_{5}) q^{88} + 2 \beta_1 q^{89} + ( - \beta_{6} + \beta_{3} - \beta_{2} + 11) q^{91} + 3 q^{92} + (\beta_{6} + 2 \beta_{3}) q^{94} + ( - 3 \beta_{5} - 6) q^{95} + (\beta_{5} - 2 \beta_{4} + 4 \beta_1) q^{97} + (\beta_{6} - 4) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{2} + 8 q^{4} - 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{2} + 8 q^{4} - 8 q^{8} + 8 q^{16} + 24 q^{23} + 8 q^{25} - 8 q^{32} - 24 q^{46} + 32 q^{49} - 8 q^{50} - 48 q^{53} + 8 q^{64} + 24 q^{65} - 24 q^{77} - 8 q^{79} + 16 q^{85} + 88 q^{91} + 24 q^{92} - 48 q^{95} - 32 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} + 26x^{6} - 64x^{5} + 161x^{4} - 220x^{3} + 232x^{2} - 132x + 33 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2\nu^{6} - 6\nu^{5} + 47\nu^{4} - 84\nu^{3} + 239\nu^{2} - 198\nu + 138 ) / 45 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 7\nu^{7} - 44\nu^{6} + 238\nu^{5} - 758\nu^{4} + 1762\nu^{3} - 2141\nu^{2} + 2094\nu + 708 ) / 351 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -14\nu^{7} + 49\nu^{6} - 359\nu^{5} + 775\nu^{4} - 2237\nu^{3} + 2605\nu^{2} - 3135\nu + 1158 ) / 351 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 17\nu^{7} - 79\nu^{6} + 461\nu^{5} - 1267\nu^{4} + 2825\nu^{3} - 4024\nu^{2} + 3375\nu - 1512 ) / 195 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 34\nu^{7} - 119\nu^{6} + 805\nu^{5} - 1715\nu^{4} + 4207\nu^{3} - 4655\nu^{2} + 4059\nu - 1308 ) / 195 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -8\nu^{7} + 28\nu^{6} - 194\nu^{5} + 415\nu^{4} - 1100\nu^{3} + 1249\nu^{2} - 1446\nu + 528 ) / 39 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -376\nu^{7} + 1316\nu^{6} - 9040\nu^{5} + 19310\nu^{4} - 49048\nu^{3} + 54920\nu^{2} - 48696\nu + 15807 ) / 1755 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{7} + \beta_{5} - \beta_{3} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} + 2\beta_{5} - 2\beta_{4} + 2\beta_{2} - 2\beta _1 - 9 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -8\beta_{7} - 3\beta_{6} - 9\beta_{5} - 3\beta_{4} + 14\beta_{3} + 3\beta_{2} - 3\beta _1 - 14 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -17\beta_{7} - 6\beta_{6} - 36\beta_{5} + 28\beta_{4} + 16\beta_{3} - 20\beta_{2} + 38\beta _1 + 81 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 76\beta_{7} + 40\beta_{6} + 66\beta_{5} + 75\beta_{4} - 189\beta_{3} - 55\beta_{2} + 100\beta _1 + 226 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 271\beta_{7} + 135\beta_{6} + 526\beta_{5} - 320\beta_{4} - 454\beta_{3} + 192\beta_{2} - 435\beta _1 - 777 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -701\beta_{7} - 406\beta_{6} - 258\beta_{5} - 1386\beta_{4} + 2080\beta_{3} + 868\beta_{2} - 1876\beta _1 - 3527 ) / 2 \) Copy content Toggle raw display

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
0.500000 3.59016i
0.500000 0.273539i
0.500000 + 3.59016i
0.500000 + 0.273539i
0.500000 1.14067i
0.500000 + 2.17595i
0.500000 + 1.14067i
0.500000 2.17595i
−1.00000 0 1.00000 −1.73205 1.41421i 0 −2.34521 1.22474i −1.00000 0 1.73205 + 1.41421i
1889.2 −1.00000 0 1.00000 −1.73205 1.41421i 0 2.34521 1.22474i −1.00000 0 1.73205 + 1.41421i
1889.3 −1.00000 0 1.00000 −1.73205 + 1.41421i 0 −2.34521 + 1.22474i −1.00000 0 1.73205 1.41421i
1889.4 −1.00000 0 1.00000 −1.73205 + 1.41421i 0 2.34521 + 1.22474i −1.00000 0 1.73205 1.41421i
1889.5 −1.00000 0 1.00000 1.73205 1.41421i 0 −2.34521 + 1.22474i −1.00000 0 −1.73205 + 1.41421i
1889.6 −1.00000 0 1.00000 1.73205 1.41421i 0 2.34521 + 1.22474i −1.00000 0 −1.73205 + 1.41421i
1889.7 −1.00000 0 1.00000 1.73205 + 1.41421i 0 −2.34521 1.22474i −1.00000 0 −1.73205 1.41421i
1889.8 −1.00000 0 1.00000 1.73205 + 1.41421i 0 2.34521 1.22474i −1.00000 0 −1.73205 1.41421i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1889.8
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.b 8
3.b odd 2 1 1890.2.d.c yes 8
5.b even 2 1 1890.2.d.c yes 8
7.b odd 2 1 inner 1890.2.d.b 8
15.d odd 2 1 inner 1890.2.d.b 8
21.c even 2 1 1890.2.d.c yes 8
35.c odd 2 1 1890.2.d.c yes 8
105.g even 2 1 inner 1890.2.d.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1890.2.d.b 8 1.a even 1 1 trivial
1890.2.d.b 8 7.b odd 2 1 inner
1890.2.d.b 8 15.d odd 2 1 inner
1890.2.d.b 8 105.g even 2 1 inner
1890.2.d.c yes 8 3.b odd 2 1
1890.2.d.c yes 8 5.b even 2 1
1890.2.d.c yes 8 21.c even 2 1
1890.2.d.c yes 8 35.c odd 2 1

Hecke kernels

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

\( T_{11}^{4} + 34T_{11}^{2} + 25 \) Copy content Toggle raw display
\( T_{23} - 3 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{4} - 2 T^{2} + 25)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} - 8 T^{2} + 49)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 34 T^{2} + 25)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} - 50 T^{2} + 361)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 70 T^{2} + 961)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 18)^{4} \) Copy content Toggle raw display
$23$ \( (T - 3)^{8} \) Copy content Toggle raw display
$29$ \( (T^{2} + 11)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} + 102 T^{2} + 225)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 24)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} - 12)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 99)^{4} \) Copy content Toggle raw display
$47$ \( (T^{4} + 82 T^{2} + 625)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 12 T - 30)^{4} \) Copy content Toggle raw display
$59$ \( (T^{2} - 12)^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} + 18)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} + 24)^{4} \) Copy content Toggle raw display
$71$ \( (T^{2} + 44)^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} - 22)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} + 2 T - 65)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 50)^{4} \) Copy content Toggle raw display
$89$ \( (T^{2} - 12)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} - 140 T^{2} + 676)^{2} \) Copy content Toggle raw display
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