Properties

Label 2268.2.w.i
Level $2268$
Weight $2$
Character orbit 2268.w
Analytic conductor $18.110$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2268,2,Mod(269,2268)]
 
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, 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("2268.269");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2268 = 2^{2} \cdot 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2268.w (of order \(6\), 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: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: 12.0.17213603549184.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 5x^{10} + 19x^{8} - 28x^{6} + 31x^{4} - 6x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 3^{9} \)
Twist minimal: no (minimal twist has level 756)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{5} + \beta_{9} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{5} + \beta_{9} q^{7} + (\beta_{5} + \beta_{3}) q^{11} + ( - \beta_{10} + \beta_{9}) q^{13} + (\beta_{7} - 2 \beta_{5} - 2 \beta_{3} + \beta_{2} + \beta_1) q^{17} + ( - \beta_{10} + \beta_{8} - \beta_{6} + \beta_{4} - 2) q^{19} + ( - \beta_{11} - \beta_{3} + \beta_{2}) q^{23} + ( - \beta_{9} + \beta_{8} - \beta_{4}) q^{25} + (\beta_{11} + 3 \beta_{3} - 2 \beta_{2}) q^{29} + ( - \beta_{10} - \beta_{9} - \beta_{8} - \beta_{4}) q^{31} + (\beta_{11} + \beta_{7} - 2 \beta_{3} + \beta_{2}) q^{35} + ( - 2 \beta_{10} - \beta_{9} - \beta_{8} + 4 \beta_{6} + \beta_{4}) q^{37} + ( - 2 \beta_{11} - \beta_{5} + 2 \beta_{3} - \beta_{2}) q^{41} + (\beta_{10} - \beta_{9} + \beta_{6} - \beta_{4} + 1) q^{43} + \beta_{3} q^{47} + ( - 3 \beta_{10} + \beta_{4}) q^{49} + (\beta_{11} - \beta_{7} - \beta_{3} + \beta_1) q^{53} + ( - \beta_{10} + 2 \beta_{9} - \beta_{8} - 4 \beta_{6} + 2 \beta_{4} - 2) q^{55} + (\beta_{11} - \beta_{7} - \beta_{5} - 2 \beta_{3}) q^{59} + ( - 2 \beta_{10} - 2 \beta_{8} - 2 \beta_{6} - 1) q^{61} + (\beta_{11} + \beta_{5} - \beta_{3} + 2 \beta_{2}) q^{65} + ( - 2 \beta_{10} + \beta_{9} + 2 \beta_{8} - \beta_{4}) q^{67} + ( - \beta_{11} - \beta_{7} + \beta_{5} + 2 \beta_{3} - 2 \beta_1) q^{71} + ( - \beta_{10} - \beta_{9} - 3 \beta_{6} + \beta_{4} + 3) q^{73} + (\beta_{11} + 2 \beta_{5} + 2 \beta_{3} - 4 \beta_{2} + \beta_1) q^{77} + (2 \beta_{10} + \beta_{9} - 2 \beta_{8} - \beta_{4} - 4) q^{79} - 5 \beta_{2} q^{83} + ( - 3 \beta_{10} - 4 \beta_{9} + \beta_{8} - \beta_{6} - \beta_{4}) q^{85} + ( - \beta_{11} + \beta_{2} - \beta_1) q^{89} + ( - 3 \beta_{10} + 7 \beta_{6} + \beta_{4} + 7) q^{91} + ( - \beta_{7} + 2 \beta_{5} + 4 \beta_{2} - 2 \beta_1) q^{95} + (2 \beta_{8} - 2 \beta_{6} - 2 \beta_{4} + 2) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 18 q^{19} - 24 q^{37} + 6 q^{43} + 54 q^{73} - 48 q^{79} + 6 q^{85} + 42 q^{91} + 36 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 5x^{10} + 19x^{8} - 28x^{6} + 31x^{4} - 6x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -99\nu^{11} + 488\nu^{9} - 1519\nu^{7} + 1061\nu^{5} + 1670\nu^{3} - 3551\nu ) / 559 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 114\nu^{11} - 545\nu^{9} + 2071\nu^{7} - 2831\nu^{5} + 3379\nu^{3} + 464\nu ) / 559 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -114\nu^{11} + 545\nu^{9} - 2071\nu^{7} + 2831\nu^{5} - 3379\nu^{3} + 1213\nu ) / 559 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 49\nu^{10} - 298\nu^{8} + 1356\nu^{6} - 2987\nu^{4} + 4419\nu^{2} - 1811 ) / 559 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -171\nu^{11} + 538\nu^{9} - 1709\nu^{7} - 1064\nu^{5} + 3037\nu^{3} - 7963\nu ) / 559 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 114\nu^{10} - 545\nu^{8} + 2071\nu^{6} - 2831\nu^{4} + 3379\nu^{2} - 654 ) / 559 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 402\nu^{11} - 2422\nu^{9} + 9539\nu^{7} - 17809\nu^{5} + 19712\nu^{3} - 7043\nu ) / 559 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -187\nu^{10} + 1046\nu^{8} - 3863\nu^{6} + 6414\nu^{4} - 6038\nu^{2} + 1926 ) / 559 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 259\nu^{10} - 1096\nu^{8} + 4053\nu^{6} - 4289\nu^{4} + 4671\nu^{2} + 809 ) / 559 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -262\nu^{10} + 1331\nu^{8} - 4946\nu^{6} + 6879\nu^{4} - 6128\nu^{2} - 527 ) / 559 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -1449\nu^{11} + 7295\nu^{9} - 27721\nu^{7} + 41294\nu^{5} - 45229\nu^{3} + 9313\nu ) / 559 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{10} - \beta_{9} + 5\beta_{6} - \beta_{4} + 5 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{11} - \beta_{7} + 3\beta_{5} - 5\beta_{3} + 14\beta_{2} - 2\beta_1 ) / 9 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -\beta_{10} - 4\beta_{9} + 3\beta_{8} + 13\beta_{6} - 3\beta_{4} ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 7\beta_{11} + 5\beta_{7} - 47\beta_{3} + 20\beta_{2} - 5\beta_1 ) / 9 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -14\beta_{10} - 5\beta_{9} + 14\beta_{8} + 5\beta_{4} - 38 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 19\beta_{11} + 38\beta_{7} - 42\beta_{5} - 122\beta_{3} - 61\beta_{2} + 19\beta_1 ) / 9 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -28\beta_{10} + 28\beta_{9} + 19\beta_{8} - 117\beta_{6} + 47\beta_{4} - 117 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( -3\beta_{11} + 22\beta_{7} - 47\beta_{5} + 20\beta_{3} - 128\beta_{2} + 44\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 66\beta_{10} + 155\beta_{9} - 89\beta_{8} - 370\beta_{6} + 89\beta_{4} ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( -244\beta_{11} - 221\beta_{7} + 1472\beta_{3} - 614\beta_{2} + 221\beta_1 ) / 9 \) Copy content Toggle raw display

Character values

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

\(n\) \(325\) \(1135\) \(1541\)
\(\chi(n)\) \(-\beta_{6}\) \(1\) \(-\beta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
269.1
1.56052 0.900969i
1.07992 0.623490i
0.385418 0.222521i
−0.385418 + 0.222521i
−1.07992 + 0.623490i
−1.56052 + 0.900969i
1.56052 + 0.900969i
1.07992 + 0.623490i
0.385418 + 0.222521i
−0.385418 0.222521i
−1.07992 0.623490i
−1.56052 0.900969i
0 0 0 −1.56052 + 2.70291i 0 −0.167563 + 2.64044i 0 0 0
269.2 0 0 0 −1.07992 + 1.87047i 0 −2.20291 1.46533i 0 0 0
269.3 0 0 0 −0.385418 + 0.667563i 0 2.37047 1.17511i 0 0 0
269.4 0 0 0 0.385418 0.667563i 0 2.37047 1.17511i 0 0 0
269.5 0 0 0 1.07992 1.87047i 0 −2.20291 1.46533i 0 0 0
269.6 0 0 0 1.56052 2.70291i 0 −0.167563 + 2.64044i 0 0 0
1349.1 0 0 0 −1.56052 2.70291i 0 −0.167563 2.64044i 0 0 0
1349.2 0 0 0 −1.07992 1.87047i 0 −2.20291 + 1.46533i 0 0 0
1349.3 0 0 0 −0.385418 0.667563i 0 2.37047 + 1.17511i 0 0 0
1349.4 0 0 0 0.385418 + 0.667563i 0 2.37047 + 1.17511i 0 0 0
1349.5 0 0 0 1.07992 + 1.87047i 0 −2.20291 + 1.46533i 0 0 0
1349.6 0 0 0 1.56052 + 2.70291i 0 −0.167563 2.64044i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 269.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
63.i even 6 1 inner
63.t odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2268.2.w.i 12
3.b odd 2 1 inner 2268.2.w.i 12
7.d odd 6 1 2268.2.bm.i 12
9.c even 3 1 756.2.t.e 12
9.c even 3 1 2268.2.bm.i 12
9.d odd 6 1 756.2.t.e 12
9.d odd 6 1 2268.2.bm.i 12
21.g even 6 1 2268.2.bm.i 12
63.h even 3 1 5292.2.f.e 12
63.i even 6 1 inner 2268.2.w.i 12
63.i even 6 1 5292.2.f.e 12
63.j odd 6 1 5292.2.f.e 12
63.k odd 6 1 756.2.t.e 12
63.s even 6 1 756.2.t.e 12
63.t odd 6 1 inner 2268.2.w.i 12
63.t odd 6 1 5292.2.f.e 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
756.2.t.e 12 9.c even 3 1
756.2.t.e 12 9.d odd 6 1
756.2.t.e 12 63.k odd 6 1
756.2.t.e 12 63.s even 6 1
2268.2.w.i 12 1.a even 1 1 trivial
2268.2.w.i 12 3.b odd 2 1 inner
2268.2.w.i 12 63.i even 6 1 inner
2268.2.w.i 12 63.t odd 6 1 inner
2268.2.bm.i 12 7.d odd 6 1
2268.2.bm.i 12 9.c even 3 1
2268.2.bm.i 12 9.d odd 6 1
2268.2.bm.i 12 21.g even 6 1
5292.2.f.e 12 63.h even 3 1
5292.2.f.e 12 63.i even 6 1
5292.2.f.e 12 63.j odd 6 1
5292.2.f.e 12 63.t odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2268, [\chi])\):

\( T_{5}^{12} + 15T_{5}^{10} + 171T_{5}^{8} + 756T_{5}^{6} + 2511T_{5}^{4} + 1458T_{5}^{2} + 729 \) Copy content Toggle raw display
\( T_{13}^{6} - 21T_{13}^{4} + 441T_{13}^{2} + 1323T_{13} + 1323 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + 15 T^{10} + 171 T^{8} + \cdots + 729 \) Copy content Toggle raw display
$7$ \( (T^{6} - 7 T^{3} + 343)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} - 45 T^{10} + 1539 T^{8} + \cdots + 531441 \) Copy content Toggle raw display
$13$ \( (T^{6} - 21 T^{4} + 441 T^{2} + 1323 T + 1323)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + 114 T^{10} + \cdots + 2492305929 \) Copy content Toggle raw display
$19$ \( (T^{6} + 9 T^{5} + 15 T^{4} - 108 T^{3} + \cdots + 27)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} - 54 T^{10} + 2511 T^{8} + \cdots + 531441 \) Copy content Toggle raw display
$29$ \( T^{12} - 153 T^{10} + \cdots + 15178486401 \) Copy content Toggle raw display
$31$ \( (T^{6} + 126 T^{4} + 3969 T^{2} + \cdots + 1323)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + 12 T^{5} + 159 T^{4} + \cdots + 142129)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + 186 T^{10} + \cdots + 18525115449 \) Copy content Toggle raw display
$43$ \( (T^{6} - 3 T^{5} + 27 T^{4} + 28 T^{3} + \cdots + 169)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} - 15 T^{4} + 54 T^{2} - 27)^{2} \) Copy content Toggle raw display
$53$ \( T^{12} - 234 T^{10} + \cdots + 15178486401 \) Copy content Toggle raw display
$59$ \( (T^{6} - 303 T^{4} + 23274 T^{2} + \cdots - 344763)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + 177 T^{4} + 4059 T^{2} + \cdots + 22707)^{2} \) Copy content Toggle raw display
$67$ \( (T^{3} - 147 T - 497)^{4} \) Copy content Toggle raw display
$71$ \( (T^{6} + 189 T^{4} + 7938 T^{2} + \cdots + 35721)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} - 27 T^{5} + 303 T^{4} - 1620 T^{3} + \cdots + 4563)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} + 12 T^{2} - 15 T - 377)^{4} \) Copy content Toggle raw display
$83$ \( T^{12} + 375 T^{10} + \cdots + 177978515625 \) Copy content Toggle raw display
$89$ \( (T^{4} + 27 T^{2} + 729)^{3} \) Copy content Toggle raw display
$97$ \( (T^{6} - 18 T^{5} + 60 T^{4} + 864 T^{3} + \cdots + 1728)^{2} \) Copy content Toggle raw display
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