Properties

Label 2268.2.k.d
Level $2268$
Weight $2$
Character orbit 2268.k
Analytic conductor $18.110$
Analytic rank $0$
Dimension $8$
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] = [2268,2,Mod(1297,2268)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2268, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2268.1297");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2268 = 2^{2} \cdot 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2268.k (of order \(3\), degree \(2\), 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: \(18.1100711784\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.310217769.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 4x^{6} - 2x^{5} + 15x^{4} - 4x^{3} + 5x^{2} + x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 3^{3} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{2} - \beta_1) q^{5} - \beta_{4} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} - \beta_1) q^{5} - \beta_{4} q^{7} + (\beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} + \beta_1 - 1) q^{11} + (\beta_{6} + \beta_{5} - \beta_{3} + 1) q^{13} + (\beta_{5} - \beta_{4} - \beta_{2} - 1) q^{17} + ( - 2 \beta_{7} + 2 \beta_{4} + 2 \beta_{2} - \beta_1) q^{19} + (\beta_{7} - \beta_{5} - 2 \beta_{4} - \beta_{3} + \beta_{2} + \beta_1) q^{23} + ( - \beta_{6} + \beta_{5} - \beta_{4} - 3 \beta_{2} - \beta_1 - 3) q^{25} + ( - 2 \beta_{6} - 2 \beta_{5} + 2 \beta_{3} - 1) q^{29} + ( - 2 \beta_{7} - 2 \beta_{6} - \beta_{5} + \beta_{4} + 2 \beta_{3} - \beta_{2} - 2 \beta_1 - 1) q^{31} + ( - \beta_{7} + 2 \beta_{5} - \beta_{3} - 2 \beta_{2} - 3) q^{35} + (\beta_{7} + \beta_{5} + \beta_{3} - \beta_{2} + \beta_1) q^{37} + (\beta_{7} - \beta_{5} + \beta_{4} + 2 \beta_{3}) q^{41} + (2 \beta_{6} + \beta_{5} - \beta_{3} + 2) q^{43} + ( - \beta_{7} + \beta_{4} + 3 \beta_{2} - 2 \beta_1) q^{47} + ( - 2 \beta_{6} + \beta_{3} + 3 \beta_{2} + \beta_1 + 1) q^{49} + ( - 2 \beta_{7} + \beta_{6} + 2 \beta_{3} + 6 \beta_{2} + \beta_1 + 6) q^{53} + ( - \beta_{7} - \beta_{6} - \beta_{4} - \beta_{3} + 1) q^{55} + ( - \beta_{6} - 2 \beta_{5} + 2 \beta_{4} - 2 \beta_{2} - \beta_1 - 2) q^{59} + ( - \beta_{7} + \beta_{5} + 2 \beta_{4} + \beta_{3} + 4 \beta_{2} + 2 \beta_1) q^{61} + ( - \beta_{7} - \beta_{5} - \beta_{3} - 3 \beta_{2} - \beta_1) q^{65} + (\beta_{7} + 5 \beta_{6} + 2 \beta_{5} - 2 \beta_{4} - \beta_{3} + 5 \beta_1) q^{67} + ( - \beta_{7} - 2 \beta_{6} + \beta_{5} - \beta_{4} - 2 \beta_{3} + 1) q^{71} + (\beta_{6} + 2 \beta_{5} - 2 \beta_{4} - 3 \beta_{2} + \beta_1 - 3) q^{73} + (3 \beta_{7} + \beta_{6} + \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + \beta_{2} + 3 \beta_1 - 2) q^{77} + ( - 2 \beta_{7} + 2 \beta_{4} + \beta_{2} + \beta_1) q^{79} + (\beta_{7} + 2 \beta_{6} + \beta_{5} + \beta_{4} + 10) q^{83} + ( - \beta_{7} - \beta_{6} + 3 \beta_{5} - \beta_{4} - 4 \beta_{3} - 5) q^{85} + ( - 2 \beta_{7} + 2 \beta_{5} + 4 \beta_{4} + 2 \beta_{3} - 6 \beta_{2} - 3 \beta_1) q^{89} + (2 \beta_{7} + 3 \beta_{6} + 3 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + 3 \beta_{2} + \cdots + 1) q^{91}+ \cdots + ( - 3 \beta_{6} + \beta_{5} - \beta_{3} + 3) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{5} + q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2 q^{5} + q^{7} - 5 q^{11} + 6 q^{13} - 2 q^{17} - 8 q^{19} - 2 q^{23} - 8 q^{25} - 4 q^{29} - 11 q^{35} + 4 q^{37} - 6 q^{41} + 10 q^{43} - 15 q^{47} + 5 q^{49} + 24 q^{53} + 16 q^{55} - 10 q^{59} - 12 q^{61} + 12 q^{65} - 7 q^{67} + 22 q^{71} - 10 q^{73} - 19 q^{77} + 70 q^{83} - 26 q^{85} + 18 q^{89} - 9 q^{91} + 10 q^{95} + 38 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 4x^{6} - 2x^{5} + 15x^{4} - 4x^{3} + 5x^{2} + x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 14\nu^{7} + 23\nu^{6} - 92\nu^{5} - 14\nu^{4} - 391\nu^{3} + 437\nu^{2} - 1586\nu + 92 ) / 289 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 64\nu^{7} - 60\nu^{6} + 240\nu^{5} - 353\nu^{4} + 1020\nu^{3} - 1140\nu^{2} + 305\nu - 240 ) / 289 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 125\nu^{7} - 63\nu^{6} + 541\nu^{5} - 414\nu^{4} + 2227\nu^{3} - 1197\nu^{2} + 1693\nu + 326 ) / 289 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -237\nu^{7} - 121\nu^{6} - 961\nu^{5} - 52\nu^{4} - 3434\nu^{3} - 854\nu^{2} - 854\nu - 773 ) / 289 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 273\nu^{7} + 15\nu^{6} + 1096\nu^{5} - 562\nu^{4} + 4080\nu^{3} - 1160\nu^{2} + 1730\nu - 229 ) / 289 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -344\nu^{7} - 111\nu^{6} - 1290\nu^{5} + 344\nu^{4} - 4471\nu^{3} - 86\nu^{2} - 86\nu - 444 ) / 289 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -412\nu^{7} + 25\nu^{6} - 1545\nu^{5} + 990\nu^{4} - 5916\nu^{3} + 1920\nu^{2} - 970\nu - 189 ) / 289 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{7} + \beta_{5} + \beta_{3} + \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{7} - \beta_{6} - 2\beta_{5} + 2\beta_{4} + 2\beta_{3} - 6\beta_{2} - \beta _1 - 6 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -3\beta_{7} + 5\beta_{6} + \beta_{5} - 3\beta_{4} - 4\beta_{3} + 3 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 9\beta_{7} + \beta_{5} - 8\beta_{4} + \beta_{3} + 21\beta_{2} + 5\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -6\beta_{7} - 20\beta_{6} - 17\beta_{5} + 17\beta_{4} + 6\beta_{3} - 18\beta_{2} - 20\beta _1 - 18 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -7\beta_{7} + 24\beta_{6} + 31\beta_{5} - 7\beta_{4} - 38\beta_{3} + 81 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 73\beta_{7} + 42\beta_{5} - 31\beta_{4} + 42\beta_{3} + 90\beta_{2} + 80\beta_1 ) / 3 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2268\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(1135\) \(1541\)
\(\chi(n)\) \(\beta_{2}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1297.1
0.882007 1.52768i
−0.198169 + 0.343239i
−1.03075 + 1.78531i
0.346911 0.600868i
0.882007 + 1.52768i
−0.198169 0.343239i
−1.03075 1.78531i
0.346911 + 0.600868i
0 0 0 −1.25300 + 2.17026i 0 −2.60333 + 0.471863i 0 0 0
1297.2 0 0 0 −0.705299 + 1.22161i 0 2.57934 + 0.589053i 0 0 0
1297.3 0 0 0 0.951526 1.64809i 0 1.17915 + 2.36846i 0 0 0
1297.4 0 0 0 2.00677 3.47583i 0 −0.655163 2.56335i 0 0 0
1621.1 0 0 0 −1.25300 2.17026i 0 −2.60333 0.471863i 0 0 0
1621.2 0 0 0 −0.705299 1.22161i 0 2.57934 0.589053i 0 0 0
1621.3 0 0 0 0.951526 + 1.64809i 0 1.17915 2.36846i 0 0 0
1621.4 0 0 0 2.00677 + 3.47583i 0 −0.655163 + 2.56335i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1297.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2268.2.k.d yes 8
3.b odd 2 1 2268.2.k.c 8
7.c even 3 1 inner 2268.2.k.d yes 8
9.c even 3 1 2268.2.i.m 8
9.c even 3 1 2268.2.l.l 8
9.d odd 6 1 2268.2.i.l 8
9.d odd 6 1 2268.2.l.m 8
21.h odd 6 1 2268.2.k.c 8
63.g even 3 1 2268.2.i.m 8
63.h even 3 1 2268.2.l.l 8
63.j odd 6 1 2268.2.l.m 8
63.n odd 6 1 2268.2.i.l 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2268.2.i.l 8 9.d odd 6 1
2268.2.i.l 8 63.n odd 6 1
2268.2.i.m 8 9.c even 3 1
2268.2.i.m 8 63.g even 3 1
2268.2.k.c 8 3.b odd 2 1
2268.2.k.c 8 21.h odd 6 1
2268.2.k.d yes 8 1.a even 1 1 trivial
2268.2.k.d yes 8 7.c even 3 1 inner
2268.2.l.l 8 9.c even 3 1
2268.2.l.l 8 63.h even 3 1
2268.2.l.m 8 9.d odd 6 1
2268.2.l.m 8 63.j odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} - 2T_{5}^{7} + 16T_{5}^{6} + 6T_{5}^{5} + 135T_{5}^{4} + 405T_{5}^{2} + 243T_{5} + 729 \) acting on \(S_{2}^{\mathrm{new}}(2268, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} - 2 T^{7} + 16 T^{6} + 6 T^{5} + \cdots + 729 \) Copy content Toggle raw display
$7$ \( T^{8} - T^{7} - 2 T^{6} + 7 T^{5} + \cdots + 2401 \) Copy content Toggle raw display
$11$ \( T^{8} + 5 T^{7} + 31 T^{6} + 42 T^{5} + \cdots + 729 \) Copy content Toggle raw display
$13$ \( (T^{4} - 3 T^{3} - 12 T^{2} + 16 T - 3)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + 2 T^{7} + 37 T^{6} + \cdots + 35721 \) Copy content Toggle raw display
$19$ \( T^{8} + 8 T^{7} + 94 T^{6} + \cdots + 97969 \) Copy content Toggle raw display
$23$ \( T^{8} + 2 T^{7} + 58 T^{6} + \cdots + 35721 \) Copy content Toggle raw display
$29$ \( (T^{4} + 2 T^{3} - 60 T^{2} - 18 T + 27)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + 75 T^{6} + 680 T^{5} + \cdots + 178929 \) Copy content Toggle raw display
$37$ \( T^{8} - 4 T^{7} + 46 T^{6} - 70 T^{5} + \cdots + 361 \) Copy content Toggle raw display
$41$ \( (T^{4} + 3 T^{3} - 72 T^{2} - 270 T - 243)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 5 T^{3} - 30 T^{2} + 94 T + 199)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 15 T^{7} + 180 T^{6} + \cdots + 6561 \) Copy content Toggle raw display
$53$ \( T^{8} - 24 T^{7} + 504 T^{6} + \cdots + 66928761 \) Copy content Toggle raw display
$59$ \( T^{8} + 10 T^{7} + 190 T^{6} + \cdots + 5774409 \) Copy content Toggle raw display
$61$ \( T^{8} + 12 T^{7} + 249 T^{6} + \cdots + 20223009 \) Copy content Toggle raw display
$67$ \( T^{8} + 7 T^{7} + 301 T^{6} + \cdots + 156925729 \) Copy content Toggle raw display
$71$ \( (T^{4} - 11 T^{3} - 141 T^{2} + 2295 T - 7803)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + 10 T^{7} + 190 T^{6} + \cdots + 2411809 \) Copy content Toggle raw display
$79$ \( T^{8} + 120 T^{6} - 454 T^{5} + \cdots + 531441 \) Copy content Toggle raw display
$83$ \( (T^{4} - 35 T^{3} + 384 T^{2} - 1350 T + 27)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} - 18 T^{7} + \cdots + 344065401 \) Copy content Toggle raw display
$97$ \( (T^{4} - 19 T^{3} - 54 T^{2} + 1780 T - 4639)^{2} \) Copy content Toggle raw display
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