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Properties

Label 2268.2.i.l
Level $2268$
Weight $2$
Character orbit 2268.i
Analytic conductor $18.110$
Analytic rank $0$
Dimension $8$
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(865,2268)]
 
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, 4, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2268.865");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2268 = 2^{2} \cdot 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2268.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: \(18.1100711784\)
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: \(\Z[a_1, \ldots, a_{13}]\)
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.

\(f(q)\) \(=\) \( q + (\beta_{2} + \beta_1) q^{5} + \beta_{5} q^{7} + ( - \beta_{7} - \beta_{6} - \beta_{5} + \cdots + 1) q^{11} + ( - \beta_{7} - \beta_{6} - \beta_{5} + \cdots - 1) q^{13} + (\beta_{7} + \beta_{5} + \cdots - \beta_{2}) q^{17}+ \cdots + (\beta_{7} - \beta_{4} + \cdots - 3 \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{5} + q^{7} + 5 q^{11} - 3 q^{13} + 2 q^{17} - 8 q^{19} + 2 q^{23} - 8 q^{25} - 2 q^{29} + 11 q^{35} + 4 q^{37} - 3 q^{41} - 5 q^{43} - 30 q^{47} - 19 q^{49} - 24 q^{53} + 16 q^{55} - 20 q^{59}+ \cdots - 19 q^{97}+O(q^{100}) \) 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

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

\(\nu\)\(=\) \( ( \beta_{7} + \beta_{5} + \beta_{3} + \beta_1 ) / 3 \) 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

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
865.1
0.346911 − 0.600868i
−1.03075 + 1.78531i
−0.198169 + 0.343239i
0.882007 − 1.52768i
0.346911 + 0.600868i
−1.03075 − 1.78531i
−0.198169 − 0.343239i
0.882007 + 1.52768i
0 0 0 −2.00677 + 3.47583i 0 −1.89234 − 1.84906i 0 0 0
865.2 0 0 0 −0.951526 + 1.64809i 0 1.46157 + 2.20541i 0 0 0
865.3 0 0 0 0.705299 − 1.22161i 0 −0.779537 + 2.52830i 0 0 0
865.4 0 0 0 1.25300 − 2.17026i 0 1.71031 − 2.01862i 0 0 0
2053.1 0 0 0 −2.00677 − 3.47583i 0 −1.89234 + 1.84906i 0 0 0
2053.2 0 0 0 −0.951526 − 1.64809i 0 1.46157 − 2.20541i 0 0 0
2053.3 0 0 0 0.705299 + 1.22161i 0 −0.779537 − 2.52830i 0 0 0
2053.4 0 0 0 1.25300 + 2.17026i 0 1.71031 + 2.01862i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 865.4
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 2268.2.i.l 8
3.b odd 2 1 2268.2.i.m 8
7.c even 3 1 2268.2.l.m 8
9.c even 3 1 2268.2.k.c ✓ 8
9.c even 3 1 2268.2.l.m 8
9.d odd 6 1 2268.2.k.d yes 8
9.d odd 6 1 2268.2.l.l 8
21.h odd 6 1 2268.2.l.l 8
63.g even 3 1 2268.2.k.c ✓ 8
63.h even 3 1 inner 2268.2.i.l 8
63.j odd 6 1 2268.2.i.m 8
63.n odd 6 1 2268.2.k.d yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2268.2.i.l 8 1.a even 1 1 trivial
2268.2.i.l 8 63.h even 3 1 inner
2268.2.i.m 8 3.b odd 2 1
2268.2.i.m 8 63.j odd 6 1
2268.2.k.c ✓ 8 9.c even 3 1
2268.2.k.c ✓ 8 63.g even 3 1
2268.2.k.d yes 8 9.d odd 6 1
2268.2.k.d yes 8 63.n odd 6 1
2268.2.l.l 8 9.d odd 6 1
2268.2.l.l 8 21.h odd 6 1
2268.2.l.m 8 7.c even 3 1
2268.2.l.m 8 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 \(S_{2}^{\mathrm{new}}(2268, [\chi])\):

\( T_{5}^{8} + 2T_{5}^{7} + 16T_{5}^{6} - 6T_{5}^{5} + 135T_{5}^{4} + 405T_{5}^{2} - 243T_{5} + 729 \) Copy content Toggle raw display
\( T_{13}^{8} + 3T_{13}^{7} + 21T_{13}^{6} - 4T_{13}^{5} + 195T_{13}^{4} + 210T_{13}^{3} + 220T_{13}^{2} + 48T_{13} + 9 \) Copy content Toggle raw display
\( T_{19}^{8} + 8T_{19}^{7} + 94T_{19}^{6} + 182T_{19}^{5} + 2275T_{19}^{4} + 1322T_{19}^{3} + 53911T_{19}^{2} - 66043T_{19} + 97969 \) 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
$5$ \( T^{8} + 2 T^{7} + \cdots + 729 \) Copy content Toggle raw display
$7$ \( T^{8} - T^{7} + \cdots + 2401 \) Copy content Toggle raw display
$11$ \( T^{8} - 5 T^{7} + \cdots + 729 \) Copy content Toggle raw display
$13$ \( T^{8} + 3 T^{7} + \cdots + 9 \) Copy content Toggle raw display
$17$ \( T^{8} - 2 T^{7} + \cdots + 35721 \) Copy content Toggle raw display
$19$ \( T^{8} + 8 T^{7} + \cdots + 97969 \) Copy content Toggle raw display
$23$ \( T^{8} - 2 T^{7} + \cdots + 35721 \) Copy content Toggle raw display
$29$ \( T^{8} + 2 T^{7} + \cdots + 729 \) Copy content Toggle raw display
$31$ \( (T^{4} - 75 T^{2} + \cdots - 423)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} - 4 T^{7} + \cdots + 361 \) Copy content Toggle raw display
$41$ \( T^{8} + 3 T^{7} + \cdots + 59049 \) Copy content Toggle raw display
$43$ \( T^{8} + 5 T^{7} + \cdots + 39601 \) Copy content Toggle raw display
$47$ \( (T^{4} + 15 T^{3} + \cdots - 81)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} + 24 T^{7} + \cdots + 66928761 \) Copy content Toggle raw display
$59$ \( (T^{4} + 10 T^{3} + \cdots + 2403)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 12 T^{3} + \cdots - 4497)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 7 T^{3} + \cdots + 12527)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 11 T^{3} + \cdots - 7803)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + 10 T^{7} + \cdots + 2411809 \) Copy content Toggle raw display
$79$ \( (T^{4} - 120 T^{2} + \cdots + 729)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} - 35 T^{7} + \cdots + 729 \) Copy content Toggle raw display
$89$ \( T^{8} + 18 T^{7} + \cdots + 344065401 \) Copy content Toggle raw display
$97$ \( T^{8} + 19 T^{7} + \cdots + 21520321 \) Copy content Toggle raw display
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