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Properties

Label 2288.2.j.k
Level $2288$
Weight $2$
Character orbit 2288.j
Analytic conductor $18.270$
Analytic rank $0$
Dimension $12$
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] = [2288,2,Mod(1585,2288)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2288, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2288.1585");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2288 = 2^{4} \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2288.j (of order \(2\), degree \(1\), 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: \(18.2697719825\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 19x^{10} + 133x^{8} + 423x^{6} + 601x^{4} + 312x^{2} + 36 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 143)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{3} + (\beta_{6} + \beta_{4}) q^{5} + (\beta_{4} + \beta_{2}) q^{7} + ( - \beta_{9} + \beta_{3} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{3} + (\beta_{6} + \beta_{4}) q^{5} + (\beta_{4} + \beta_{2}) q^{7} + ( - \beta_{9} + \beta_{3} + 1) q^{9} - \beta_{4} q^{11} + (\beta_{11} - \beta_{8} - 1) q^{13} + (\beta_{11} + \beta_{10} + \cdots - 2 \beta_{2}) q^{15}+ \cdots + ( - \beta_{6} - \beta_{4} + \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 4 q^{3} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 4 q^{3} + 12 q^{9} - 8 q^{13} - 12 q^{17} + 4 q^{23} - 16 q^{25} + 28 q^{27} + 8 q^{29} - 16 q^{35} - 4 q^{39} - 12 q^{43} + 32 q^{49} - 20 q^{53} + 8 q^{55} - 12 q^{61} - 20 q^{65} + 52 q^{69} + 8 q^{75} + 8 q^{77} + 48 q^{79} - 36 q^{81} - 12 q^{87} + 4 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 19x^{10} + 133x^{8} + 423x^{6} + 601x^{4} + 312x^{2} + 36 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{11} + 205\nu^{9} + 2767\nu^{7} + 11877\nu^{5} + 15235\nu^{3} - 1482\nu ) / 2088 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{10} - 31\nu^{8} - 331\nu^{6} - 1437\nu^{4} - 2185\nu^{2} - 432 ) / 174 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -67\nu^{11} - 1207\nu^{9} - 7909\nu^{7} - 23199\nu^{5} - 28945\nu^{3} - 9282\nu ) / 2088 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 11\nu^{10} + 167\nu^{8} + 857\nu^{6} + 1887\nu^{4} + 1937\nu^{2} + 402 ) / 174 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -55\nu^{11} - 835\nu^{9} - 3937\nu^{7} - 4911\nu^{5} + 5627\nu^{3} + 8430\nu ) / 1044 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -41\nu^{11} - 749\nu^{9} - 4871\nu^{7} - 13677\nu^{5} - 16331\nu^{3} - 6750\nu ) / 696 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -31\nu^{10} - 439\nu^{8} - 1909\nu^{6} - 2439\nu^{4} + 647\nu^{2} + 1050 ) / 348 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 25\nu^{10} + 427\nu^{8} + 2533\nu^{6} + 6171\nu^{4} + 5557\nu^{2} + 1230 ) / 174 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 95 \nu^{11} + 59 \nu^{10} - 1727 \nu^{9} + 959 \nu^{8} - 11261 \nu^{7} + 5261 \nu^{6} - 31767 \nu^{5} + \cdots - 1830 ) / 696 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 95 \nu^{11} - 59 \nu^{10} - 1727 \nu^{9} - 959 \nu^{8} - 11261 \nu^{7} - 5261 \nu^{6} - 31767 \nu^{5} + \cdots + 1830 ) / 696 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{11} - \beta_{10} + \beta_{9} - \beta_{8} - \beta_{5} - 7 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{11} - \beta_{10} + \beta_{7} + \beta_{6} + 5\beta_{4} - 2\beta_{2} - 5\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -6\beta_{11} + 6\beta_{10} - 6\beta_{9} + 8\beta_{8} + 9\beta_{5} + 2\beta_{3} + 34 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 5\beta_{11} + 5\beta_{10} - 7\beta_{7} - 4\beta_{6} - 23\beta_{4} + 8\beta_{2} + 15\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 38\beta_{11} - 38\beta_{10} + 36\beta_{9} - 60\beta_{8} - 67\beta_{5} - 28\beta_{3} - 188 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -81\beta_{11} - 81\beta_{10} + 133\beta_{7} + 61\beta_{6} + 343\beta_{4} - 120\beta_{2} - 195\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -255\beta_{11} + 255\beta_{10} - 223\beta_{9} + 447\beta_{8} + 477\beta_{5} + 266\beta_{3} + 1127 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 610\beta_{11} + 610\beta_{10} - 1100\beta_{7} - 450\beta_{6} - 2418\beta_{4} + 894\beta_{2} + 1317\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 882\beta_{11} - 882\beta_{10} + 717\beta_{9} - 1654\beta_{8} - 1679\beta_{5} - 1100\beta_{3} - 3568 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( -4458\beta_{11} - 4458\beta_{10} + 8532\beta_{7} + 3244\beta_{6} + 16776\beta_{4} - 6616\beta_{2} - 9073\beta_1 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2288\mathbb{Z}\right)^\times\).

\(n\) \(287\) \(353\) \(1717\) \(2081\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1585.1
− 1.92437i
1.92437i
2.23753i
− 2.23753i
− 0.398488i
0.398488i
1.50594i
− 1.50594i
− 2.66546i
2.66546i
− 0.871160i
0.871160i
0 −2.13208 0 − 3.67785i 0 − 3.13208i 0 1.54577 0
1585.2 0 −2.13208 0 3.67785i 0 3.13208i 0 1.54577 0
1585.3 0 −1.75439 0 − 1.83227i 0 − 2.75439i 0 0.0778811 0
1585.4 0 −1.75439 0 1.83227i 0 2.75439i 0 0.0778811 0
1585.5 0 −0.689466 0 − 1.83517i 0 1.68947i 0 −2.52464 0
1585.6 0 −0.689466 0 1.83517i 0 − 1.68947i 0 −2.52464 0
1585.7 0 1.34063 0 − 2.54333i 0 − 0.340634i 0 −1.20270 0
1585.8 0 1.34063 0 2.54333i 0 0.340634i 0 −1.20270 0
1585.9 0 2.17072 0 − 0.458686i 0 − 1.17072i 0 1.71204 0
1585.10 0 2.17072 0 0.458686i 0 1.17072i 0 1.71204 0
1585.11 0 3.06458 0 − 3.32707i 0 2.06458i 0 6.39165 0
1585.12 0 3.06458 0 3.32707i 0 − 2.06458i 0 6.39165 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1585.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2288.2.j.k 12
4.b odd 2 1 143.2.b.a ✓ 12
12.b even 2 1 1287.2.b.b 12
13.b even 2 1 inner 2288.2.j.k 12
52.b odd 2 1 143.2.b.a ✓ 12
52.f even 4 1 1859.2.a.j 6
52.f even 4 1 1859.2.a.n 6
156.h even 2 1 1287.2.b.b 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
143.2.b.a ✓ 12 4.b odd 2 1
143.2.b.a ✓ 12 52.b odd 2 1
1287.2.b.b 12 12.b even 2 1
1287.2.b.b 12 156.h even 2 1
1859.2.a.j 6 52.f even 4 1
1859.2.a.n 6 52.f even 4 1
2288.2.j.k 12 1.a even 1 1 trivial
2288.2.j.k 12 13.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2288, [\chi])\):

\( T_{3}^{6} - 2T_{3}^{5} - 10T_{3}^{4} + 14T_{3}^{3} + 30T_{3}^{2} - 22T_{3} - 23 \) Copy content Toggle raw display
\( T_{5}^{12} + 38T_{5}^{10} + 537T_{5}^{8} + 3508T_{5}^{6} + 10720T_{5}^{4} + 13056T_{5}^{2} + 2304 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T^{6} - 2 T^{5} - 10 T^{4} + \cdots - 23)^{2} \) Copy content Toggle raw display
$5$ \( T^{12} + 38 T^{10} + \cdots + 2304 \) Copy content Toggle raw display
$7$ \( T^{12} + 26 T^{10} + \cdots + 144 \) Copy content Toggle raw display
$11$ \( (T^{2} + 1)^{6} \) Copy content Toggle raw display
$13$ \( T^{12} + 8 T^{11} + \cdots + 4826809 \) Copy content Toggle raw display
$17$ \( (T^{6} + 6 T^{5} - 34 T^{4} + \cdots - 96)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} + 118 T^{10} + \cdots + 16112196 \) Copy content Toggle raw display
$23$ \( (T^{6} - 2 T^{5} + \cdots + 17625)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} - 4 T^{5} - 42 T^{4} + \cdots - 96)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + 162 T^{10} + \cdots + 1937664 \) Copy content Toggle raw display
$37$ \( T^{12} + 226 T^{10} + \cdots + 53934336 \) Copy content Toggle raw display
$41$ \( T^{12} + 142 T^{10} + \cdots + 5062500 \) Copy content Toggle raw display
$43$ \( (T^{6} + 6 T^{5} - 86 T^{4} + \cdots + 32)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + 220 T^{10} + \cdots + 36864 \) Copy content Toggle raw display
$53$ \( (T^{6} + 10 T^{5} + \cdots + 6024)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 2477650176 \) Copy content Toggle raw display
$61$ \( (T^{6} + 6 T^{5} + \cdots + 27296)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 296941824 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 93401139456 \) Copy content Toggle raw display
$73$ \( T^{12} + 234 T^{10} + \cdots + 9216 \) Copy content Toggle raw display
$79$ \( (T^{6} - 24 T^{5} + \cdots - 32)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + 478 T^{10} + \cdots + 2762244 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 38598103296 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 152176896 \) Copy content Toggle raw display
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