↑

Properties

Label 1058.4.a.q
Level $1058$
Weight $4$
Character orbit 1058.a
Self dual yes
Analytic conductor $62.424$
Analytic rank $1$
Dimension $8$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1058,4,Mod(1,1058)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1058, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1058.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1058 = 2 \cdot 23^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1058.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: \(62.4240207861\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: 8.8.1182630150144.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} - 20x^{6} + 68x^{5} + 132x^{4} - 308x^{3} - 356x^{2} + 340x + 193 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 q^{2} + ( - \beta_{4} + \beta_1 - 3) q^{3} + 4 q^{4} + (\beta_{6} - 2 \beta_{3} - 3 \beta_{2}) q^{5} + ( - 2 \beta_{4} + 2 \beta_1 - 6) q^{6} + (\beta_{6} - 3 \beta_{5} + \cdots - 3 \beta_{2}) q^{7}+ \cdots + ( - 21 \beta_{6} + \cdots + 565 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 16 q^{2} - 24 q^{3} + 32 q^{4} - 48 q^{6} + 64 q^{8} + 56 q^{9} - 96 q^{12} - 200 q^{13} + 128 q^{16} + 112 q^{18} - 192 q^{24} - 344 q^{25} - 400 q^{26} - 864 q^{27} + 64 q^{29} + 96 q^{31} + 256 q^{32}+ \cdots - 432 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} - 20x^{6} + 68x^{5} + 132x^{4} - 308x^{3} - 356x^{2} + 340x + 193 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -558\nu^{7} + 2803\nu^{6} + 7616\nu^{5} - 42243\nu^{4} - 23388\nu^{3} + 146451\nu^{2} + 77920\nu - 96197 ) / 20079 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -571\nu^{7} + 3544\nu^{6} + 4299\nu^{5} - 50268\nu^{4} + 25413\nu^{3} + 140807\nu^{2} - 93523\nu - 27378 ) / 20079 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 617\nu^{7} - 3935\nu^{6} - 4575\nu^{5} + 59100\nu^{4} - 32313\nu^{3} - 197548\nu^{2} + 128345\nu + 102726 ) / 20079 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 18\nu^{7} - 106\nu^{6} - 170\nu^{5} + 1563\nu^{4} - 354\nu^{3} - 4998\nu^{2} + 1898\nu + 2126 ) / 207 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 4\nu^{7} - 21\nu^{6} - 48\nu^{5} + 309\nu^{4} + 44\nu^{3} - 965\nu^{2} + 202\nu + 337 ) / 23 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 68\nu^{7} - 380\nu^{6} - 678\nu^{5} + 5460\nu^{4} - 1092\nu^{3} - 16198\nu^{2} + 7574\nu + 4878 ) / 207 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 8388 \nu^{7} + 47533 \nu^{6} + 82532 \nu^{5} - 690063 \nu^{4} + 147162 \nu^{3} + 2152911 \nu^{2} + \cdots - 1015763 ) / 20079 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{4} - \beta_{3} + \beta_{2} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} + \beta_{6} + 2\beta_{4} + 2\beta_{2} + \beta _1 + 14 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{7} + 3\beta_{6} - 3\beta_{5} + 21\beta_{4} - 17\beta_{3} + 23\beta_{2} + 10\beta _1 + 25 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 9\beta_{7} + 11\beta_{6} - 4\beta_{5} + 29\beta_{4} - 4\beta_{3} + 44\beta_{2} + 11\beta _1 + 90 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 53\beta_{7} + 90\beta_{6} - 80\beta_{5} + 379\beta_{4} - 249\beta_{3} + 519\beta_{2} + 146\beta _1 + 521 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 329\beta_{7} + 489\beta_{6} - 278\beta_{5} + 1314\beta_{4} - 386\beta_{3} + 2264\beta_{2} + 463\beta _1 + 2978 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 1192 \beta_{7} + 2156 \beta_{6} - 1757 \beta_{5} + 7167 \beta_{4} - 4159 \beta_{3} + 11495 \beta_{2} + \cdots + 10865 ) / 2 \) 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
−0.449966
−1.86418
−1.81159
−3.22580
3.18202
4.59623
1.07953
2.49375
2.00000 −9.77825 4.00000 −13.9672 −19.5565 −21.2859 8.00000 68.6141 −27.9345
1.2 2.00000 −9.77825 4.00000 13.9672 −19.5565 21.2859 8.00000 68.6141 27.9345
1.3 2.00000 −5.57328 4.00000 −0.672044 −11.1466 22.3479 8.00000 4.06148 −1.34409
1.4 2.00000 −5.57328 4.00000 0.672044 −11.1466 −22.3479 8.00000 4.06148 1.34409
1.5 2.00000 0.314146 4.00000 −0.305569 0.628292 −8.65952 8.00000 −26.9013 −0.611137
1.6 2.00000 0.314146 4.00000 0.305569 0.628292 8.65952 8.00000 −26.9013 0.611137
1.7 2.00000 3.03738 4.00000 −11.5053 6.07477 15.3784 8.00000 −17.7743 −23.0106
1.8 2.00000 3.03738 4.00000 11.5053 6.07477 −15.3784 8.00000 −17.7743 23.0106
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1058.4.a.q ✓ 8
23.b odd 2 1 inner 1058.4.a.q ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1058.4.a.q ✓ 8 1.a even 1 1 trivial
1058.4.a.q ✓ 8 23.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1058))\):

\( T_{3}^{4} + 12T_{3}^{3} + 4T_{3}^{2} - 168T_{3} + 52 \) Copy content Toggle raw display
\( T_{5}^{8} - 328T_{5}^{6} + 26002T_{5}^{4} - 14088T_{5}^{2} + 1089 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 2)^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} + 12 T^{3} + \cdots + 52)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} - 328 T^{6} + \cdots + 1089 \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 4012969104 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 5409308304 \) Copy content Toggle raw display
$13$ \( (T^{4} + 100 T^{3} + \cdots + 865953)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 30\!\cdots\!44 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 4570513791376 \) Copy content Toggle raw display
$23$ \( T^{8} \) Copy content Toggle raw display
$29$ \( (T^{4} - 32 T^{3} + \cdots - 2250831)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 48 T^{3} + \cdots - 46412784)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 10\!\cdots\!24 \) Copy content Toggle raw display
$41$ \( (T^{4} + 580 T^{3} + \cdots + 77029881)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 13\!\cdots\!64 \) Copy content Toggle raw display
$47$ \( (T^{4} + 244 T^{3} + \cdots - 104201868)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 65\!\cdots\!89 \) Copy content Toggle raw display
$59$ \( (T^{4} + 1300 T^{3} + \cdots - 196018280364)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 15\!\cdots\!29 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 23\!\cdots\!04 \) Copy content Toggle raw display
$71$ \( (T^{4} + 1708 T^{3} + \cdots - 293804550156)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 2800 T^{3} + \cdots + 224423315961)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 19\!\cdots\!56 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 25\!\cdots\!44 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 14\!\cdots\!81 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 25\!\cdots\!09 \) Copy content Toggle raw display
show more
show less