↑

Properties

Label 2254.4.a.w
Level $2254$
Weight $4$
Character orbit 2254.a
Self dual yes
Analytic conductor $132.990$
Analytic rank $1$
Dimension $11$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2254,4,Mod(1,2254)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2254, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2254.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2254 = 2 \cdot 7^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2254.a (trivial)

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: yes
Analytic conductor: \(132.990305153\)
Analytic rank: \(1\)
Dimension: \(11\)
Coefficient field: \(\mathbb{Q}[x]/(x^{11} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{11} - 5 x^{10} - 165 x^{9} + 798 x^{8} + 8769 x^{7} - 38472 x^{6} - 184213 x^{5} + 644009 x^{4} + \cdots + 2848203 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 322)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\ldots,\beta_{10}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 q^{2} + (\beta_1 - 1) q^{3} + 4 q^{4} + ( - \beta_{4} - 2) q^{5} + (2 \beta_1 - 2) q^{6} + 8 q^{8} + (\beta_{9} + \beta_{6} + \beta_{5} + \cdots + 8) q^{9} + ( - 2 \beta_{4} - 4) q^{10} + ( - \beta_{10} - \beta_{9} - \beta_{8} + \cdots + 4) q^{11}+ \cdots + (4 \beta_{10} - 5 \beta_{9} + \cdots + 24) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 11 q + 22 q^{2} - 6 q^{3} + 44 q^{4} - 27 q^{5} - 12 q^{6} + 88 q^{8} + 59 q^{9} - 54 q^{10} + 56 q^{11} - 24 q^{12} - 103 q^{13} + 62 q^{15} + 176 q^{16} - 157 q^{17} + 118 q^{18} - 266 q^{19} - 108 q^{20}+ \cdots + 101 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{11} - 5 x^{10} - 165 x^{9} + 798 x^{8} + 8769 x^{7} - 38472 x^{6} - 184213 x^{5} + 644009 x^{4} + \cdots + 2848203 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2232478627571 \nu^{10} - 837527175723 \nu^{9} - 331521715034808 \nu^{8} + \cdots + 37\!\cdots\!40 ) / 17\!\cdots\!51 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 800846482483 \nu^{10} - 1450047933992 \nu^{9} + 124366696407156 \nu^{8} + \cdots - 15\!\cdots\!72 ) / 58\!\cdots\!17 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 105108466310 \nu^{10} - 565461382633 \nu^{9} - 17792863113747 \nu^{8} + \cdots - 30\!\cdots\!75 ) / 53\!\cdots\!47 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 4124278561676 \nu^{10} - 8516597587713 \nu^{9} - 671887286268081 \nu^{8} + \cdots - 71\!\cdots\!21 ) / 17\!\cdots\!51 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 6161025989518 \nu^{10} - 13181680737972 \nu^{9} + \cdots - 16\!\cdots\!18 ) / 17\!\cdots\!51 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 6209023574063 \nu^{10} - 15336470347236 \nu^{9} + \cdots - 19\!\cdots\!04 ) / 17\!\cdots\!51 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 6786268777511 \nu^{10} + 29996492696337 \nu^{9} + \cdots + 16\!\cdots\!86 ) / 17\!\cdots\!51 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 2272241720988 \nu^{10} + 1012684232932 \nu^{9} + 373217014278411 \nu^{8} + \cdots + 25\!\cdots\!10 ) / 58\!\cdots\!17 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 2677419190947 \nu^{10} - 11493053213030 \nu^{9} - 439189534027551 \nu^{8} + \cdots - 37\!\cdots\!34 ) / 58\!\cdots\!17 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} + \beta_{6} + \beta_{5} - \beta_{4} + 34 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -3\beta_{10} + 2\beta_{9} - 3\beta_{8} + \beta_{7} + 5\beta_{2} + 58\beta _1 - 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 12 \beta_{10} + 96 \beta_{9} - 12 \beta_{8} + 13 \beta_{7} + 73 \beta_{6} + 82 \beta_{5} - 121 \beta_{4} + \cdots + 2031 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 306 \beta_{10} + 196 \beta_{9} - 291 \beta_{8} + 53 \beta_{7} + 51 \beta_{6} - 63 \beta_{5} + \cdots - 1340 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 1461 \beta_{10} + 8693 \beta_{9} - 1272 \beta_{8} + 2021 \beta_{7} + 5566 \beta_{6} + 6682 \beta_{5} + \cdots + 143912 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 26640 \beta_{10} + 15651 \beta_{9} - 24804 \beta_{8} + 2073 \beta_{7} + 6630 \beta_{6} + \cdots - 180393 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 145836 \beta_{10} + 759094 \beta_{9} - 104526 \beta_{8} + 216354 \beta_{7} + 443539 \beta_{6} + \cdots + 11101342 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 2255421 \beta_{10} + 1149425 \beta_{9} - 2044839 \beta_{8} + 35716 \beta_{7} + 617769 \beta_{6} + \cdots - 19595896 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 13605006 \beta_{10} + 64793766 \beta_{9} - 8028534 \beta_{8} + 20294305 \beta_{7} + 36146620 \beta_{6} + \cdots + 894090180 \) Copy content Toggle raw display

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−9.28444
−6.47210
−4.81403
−2.74955
−1.25414
1.05420
1.37774
5.45902
5.84446
6.87048
8.96837
2.00000 −10.2844 4.00000 −2.24614 −20.5689 0 8.00000 78.7697 −4.49228
1.2 2.00000 −7.47210 4.00000 −5.30939 −14.9442 0 8.00000 28.8323 −10.6188
1.3 2.00000 −5.81403 4.00000 −22.0021 −11.6281 0 8.00000 6.80300 −44.0042
1.4 2.00000 −3.74955 4.00000 14.1785 −7.49911 0 8.00000 −12.9409 28.3569
1.5 2.00000 −2.25414 4.00000 5.95821 −4.50828 0 8.00000 −21.9188 11.9164
1.6 2.00000 0.0542033 4.00000 13.0870 0.108407 0 8.00000 −26.9971 26.1741
1.7 2.00000 0.377744 4.00000 −19.5851 0.755487 0 8.00000 −26.8573 −39.1703
1.8 2.00000 4.45902 4.00000 −11.8862 8.91803 0 8.00000 −7.11717 −23.7724
1.9 2.00000 4.84446 4.00000 6.31640 9.68892 0 8.00000 −3.53122 12.6328
1.10 2.00000 5.87048 4.00000 −5.19815 11.7410 0 8.00000 7.46253 −10.3963
1.11 2.00000 7.96837 4.00000 −0.313034 15.9367 0 8.00000 36.4949 −0.626067
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.11
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(7\) \( +1 \)
\(23\) \( +1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2254.4.a.w 11
7.b odd 2 1 2254.4.a.x 11
7.c even 3 2 322.4.e.b ✓ 22
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
322.4.e.b ✓ 22 7.c even 3 2
2254.4.a.w 11 1.a even 1 1 trivial
2254.4.a.x 11 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{11} + 6 T_{3}^{10} - 160 T_{3}^{9} - 747 T_{3}^{8} + 8943 T_{3}^{7} + 30807 T_{3}^{6} + \cdots + 78129 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2254))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 2)^{11} \) Copy content Toggle raw display
$3$ \( T^{11} + 6 T^{10} + \cdots + 78129 \) Copy content Toggle raw display
$5$ \( T^{11} + \cdots + 694079901 \) Copy content Toggle raw display
$7$ \( T^{11} \) Copy content Toggle raw display
$11$ \( T^{11} + \cdots + 12\!\cdots\!25 \) Copy content Toggle raw display
$13$ \( T^{11} + \cdots + 33\!\cdots\!31 \) Copy content Toggle raw display
$17$ \( T^{11} + \cdots - 54\!\cdots\!59 \) Copy content Toggle raw display
$19$ \( T^{11} + \cdots - 58\!\cdots\!93 \) Copy content Toggle raw display
$23$ \( (T + 23)^{11} \) Copy content Toggle raw display
$29$ \( T^{11} + \cdots - 85\!\cdots\!05 \) Copy content Toggle raw display
$31$ \( T^{11} + \cdots - 17\!\cdots\!03 \) Copy content Toggle raw display
$37$ \( T^{11} + \cdots - 13\!\cdots\!59 \) Copy content Toggle raw display
$41$ \( T^{11} + \cdots - 58\!\cdots\!57 \) Copy content Toggle raw display
$43$ \( T^{11} + \cdots - 25\!\cdots\!69 \) Copy content Toggle raw display
$47$ \( T^{11} + \cdots + 11\!\cdots\!71 \) Copy content Toggle raw display
$53$ \( T^{11} + \cdots + 58\!\cdots\!39 \) Copy content Toggle raw display
$59$ \( T^{11} + \cdots - 10\!\cdots\!09 \) Copy content Toggle raw display
$61$ \( T^{11} + \cdots + 65\!\cdots\!13 \) Copy content Toggle raw display
$67$ \( T^{11} + \cdots - 72\!\cdots\!47 \) Copy content Toggle raw display
$71$ \( T^{11} + \cdots - 26\!\cdots\!95 \) Copy content Toggle raw display
$73$ \( T^{11} + \cdots + 23\!\cdots\!13 \) Copy content Toggle raw display
$79$ \( T^{11} + \cdots + 91\!\cdots\!25 \) Copy content Toggle raw display
$83$ \( T^{11} + \cdots + 12\!\cdots\!29 \) Copy content Toggle raw display
$89$ \( T^{11} + \cdots + 25\!\cdots\!53 \) Copy content Toggle raw display
$97$ \( T^{11} + \cdots + 82\!\cdots\!35 \) Copy content Toggle raw display
show more
show less