↑

Properties

Label 1058.4.a.r
Level $1058$
Weight $4$
Character orbit 1058.a
Self dual yes
Analytic conductor $62.424$
Analytic rank $0$
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: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} - 211x^{6} + 886x^{5} + 10873x^{4} - 69190x^{3} - 32797x^{2} + 873362x - 1397066 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{4}\cdot 3 \)
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.

\(f(q)\) \(=\) \( q + 2 q^{2} + ( - \beta_{3} + 1) q^{3} + 4 q^{4} + (\beta_{4} + \beta_1) q^{5} + ( - 2 \beta_{3} + 2) q^{6} + ( - \beta_{5} + \beta_{4} + \cdots - 3 \beta_1) q^{7} + 8 q^{8} + (\beta_{7} + \beta_{6} + 24) q^{9}+ \cdots + ( - 14 \beta_{5} - 97 \beta_{4} + \cdots - 1125 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 16 q^{2} + 6 q^{3} + 32 q^{4} + 12 q^{6} + 64 q^{8} + 194 q^{9} + 24 q^{12} + 154 q^{13} + 128 q^{16} + 388 q^{18} + 48 q^{24} + 370 q^{25} + 308 q^{26} + 18 q^{27} + 1118 q^{29} - 300 q^{31} + 256 q^{32}+ \cdots + 4368 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} - 211x^{6} + 886x^{5} + 10873x^{4} - 69190x^{3} - 32797x^{2} + 873362x - 1397066 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 175523 \nu^{7} - 381803 \nu^{6} + 35403744 \nu^{5} - 7407842 \nu^{4} - 1931830337 \nu^{3} + \cdots - 58591483546 ) / 471471744 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 45187 \nu^{7} + 105553 \nu^{6} - 9052660 \nu^{5} + 1557086 \nu^{4} + 492478001 \nu^{3} + \cdots + 18072507206 ) / 117867936 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 71387 \nu^{7} + 158203 \nu^{6} - 14338608 \nu^{5} + 3210530 \nu^{4} + 779378281 \nu^{3} + \cdots + 24462835802 ) / 83200896 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 502011 \nu^{7} + 836111 \nu^{6} - 103698936 \nu^{5} + 60471402 \nu^{4} + 5817202361 \nu^{3} + \cdots + 183558301570 ) / 353603808 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1147605 \nu^{7} + 3372001 \nu^{6} - 226035336 \nu^{5} - 101237178 \nu^{4} + 12084477175 \nu^{3} + \cdots + 318781338302 ) / 353603808 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 78079 \nu^{7} + 194255 \nu^{6} - 15576848 \nu^{5} - 531206 \nu^{4} + 842760117 \nu^{3} + \cdots + 24857950898 ) / 13866816 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 5076 \nu^{7} - 14611 \nu^{6} + 1001564 \nu^{5} + 395628 \nu^{4} - 53529530 \nu^{3} + \cdots - 1413626754 ) / 866676 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} - 2\beta_{6} + 3\beta_{5} + 6\beta_{3} - 6\beta _1 + 2 ) / 12 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -7\beta_{7} + 8\beta_{6} - 15\beta_{5} + 6\beta_{4} + 12\beta_{3} - 30\beta_{2} + 120\beta _1 + 634 ) / 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 173\beta_{7} - 124\beta_{6} + 369\beta_{5} - 6\beta_{4} + 372\beta_{3} - 6\beta_{2} - 552\beta _1 - 2138 ) / 12 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -479\beta_{7} + 412\beta_{6} - 963\beta_{5} + 378\beta_{4} + 792\beta_{3} - 1098\beta_{2} + 7508\beta _1 + 20626 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 26633 \beta_{7} - 13096 \beta_{6} + 50601 \beta_{5} - 8922 \beta_{4} + 25032 \beta_{3} + 1662 \beta_{2} + \cdots - 395306 ) / 12 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 257129 \beta_{7} + 168940 \beta_{6} - 480789 \beta_{5} + 199734 \beta_{4} + 346920 \beta_{3} + \cdots + 7727414 ) / 12 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 4054273 \beta_{7} - 1766912 \beta_{6} + 7349817 \beta_{5} - 2078154 \beta_{4} + 1137840 \beta_{3} + \cdots - 64821034 ) / 12 \) 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
2.91451
9.78523
4.72618
−12.5127
4.21777
5.63588
−8.81247
−3.95435
2.00000 −9.49075 4.00000 −6.50875 −18.9815 35.7955 8.00000 63.0744 −13.0175
1.2 2.00000 −9.49075 4.00000 6.50875 −18.9815 −35.7955 8.00000 63.0744 13.0175
1.3 2.00000 −1.31679 4.00000 −3.72469 −2.63358 −23.7975 8.00000 −25.2661 −7.44938
1.4 2.00000 −1.31679 4.00000 3.72469 −2.63358 23.7975 8.00000 −25.2661 7.44938
1.5 2.00000 3.91483 4.00000 −20.0697 7.82965 −22.5771 8.00000 −11.6741 −40.1393
1.6 2.00000 3.91483 4.00000 20.0697 7.82965 22.5771 8.00000 −11.6741 40.1393
1.7 2.00000 9.89272 4.00000 −15.0324 19.7854 −10.3265 8.00000 70.8658 −30.0647
1.8 2.00000 9.89272 4.00000 15.0324 19.7854 10.3265 8.00000 70.8658 30.0647
\(n\): 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.r ✓ 8
23.b odd 2 1 inner 1058.4.a.r ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1058.4.a.r ✓ 8 1.a even 1 1 trivial
1058.4.a.r ✓ 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1058))\):

\( T_{3}^{4} - 3T_{3}^{3} - 98T_{3}^{2} + 246T_{3} + 484 \) Copy content Toggle raw display
\( T_{5}^{8} - 685T_{5}^{6} + 126967T_{5}^{4} - 5488215T_{5}^{2} + 53494596 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 2)^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} - 3 T^{3} + \cdots + 484)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} - 685 T^{6} + \cdots + 53494596 \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 39441960000 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 917043725376 \) Copy content Toggle raw display
$13$ \( (T^{4} - 77 T^{3} + \cdots + 1737084)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 561060921600 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 265632757483536 \) Copy content Toggle raw display
$23$ \( T^{8} \) Copy content Toggle raw display
$29$ \( (T^{4} - 559 T^{3} + \cdots - 174679776)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 150 T^{3} + \cdots + 2032096)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 26\!\cdots\!56 \) Copy content Toggle raw display
$41$ \( (T^{4} - 579 T^{3} + \cdots - 14482282626)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( (T^{4} - 370 T^{3} + \cdots - 5302394424)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 61\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( (T^{4} - 953 T^{3} + \cdots - 19962471636)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 23\!\cdots\!16 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 142128361497600 \) Copy content Toggle raw display
$71$ \( (T^{4} + 878 T^{3} + \cdots - 107653216680)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} - 842 T^{3} + \cdots - 3833512641)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 42\!\cdots\!76 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 10\!\cdots\!56 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 17\!\cdots\!16 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 54\!\cdots\!56 \) Copy content Toggle raw display
show more
show less