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Properties

Label 1587.4.a.p
Level $1587$
Weight $4$
Character orbit 1587.a
Self dual yes
Analytic conductor $93.636$
Analytic rank $0$
Dimension $10$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1587,4,Mod(1,1587)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1587, 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("1587.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1587 = 3 \cdot 23^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1587.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: \(93.6360311791\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 4 x^{9} - 54 x^{8} + 220 x^{7} + 710 x^{6} - 3044 x^{5} - 2384 x^{4} + 13204 x^{3} - 2975 x^{2} + \cdots + 334 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3} \)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - 3 q^{3} + (\beta_{7} + 2) q^{4} + (\beta_{4} + 2 \beta_{3} + \beta_{2}) q^{5} - 3 \beta_1 q^{6} + (\beta_{6} - 4 \beta_{2}) q^{7} + (\beta_{9} + 2 \beta_1 - 4) q^{8} + 9 q^{9} + (\beta_{8} + 2 \beta_{6} + \cdots - 14 \beta_{2}) q^{10}+ \cdots + (18 \beta_{8} + 9 \beta_{6} + \cdots + 36 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 4 q^{2} - 30 q^{3} + 24 q^{4} - 12 q^{6} - 36 q^{8} + 90 q^{9} - 72 q^{12} - 80 q^{13} + 40 q^{16} + 36 q^{18} + 108 q^{24} + 518 q^{25} - 836 q^{26} - 270 q^{27} + 924 q^{29} - 556 q^{31} - 124 q^{32}+ \cdots - 228 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 4 x^{9} - 54 x^{8} + 220 x^{7} + 710 x^{6} - 3044 x^{5} - 2384 x^{4} + 13204 x^{3} - 2975 x^{2} + \cdots + 334 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 431730 \nu^{9} - 1127289 \nu^{8} - 24634656 \nu^{7} + 62055019 \nu^{6} + 372035034 \nu^{5} + \cdots - 911864662 ) / 629620952 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 431730 \nu^{9} - 1127289 \nu^{8} - 24634656 \nu^{7} + 62055019 \nu^{6} + 372035034 \nu^{5} + \cdots - 911864662 ) / 629620952 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 599631 \nu^{9} + 1321236 \nu^{8} + 32925581 \nu^{7} - 65506734 \nu^{6} - 464114267 \nu^{5} + \cdots - 1115044084 ) / 629620952 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1846473 \nu^{9} - 6860665 \nu^{8} - 106610543 \nu^{7} + 388989263 \nu^{6} + 1704644037 \nu^{5} + \cdots - 17716990846 ) / 1259241904 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 474233 \nu^{9} + 1195929 \nu^{8} - 34643037 \nu^{7} - 68578324 \nu^{6} + 805321065 \nu^{5} + \cdots + 3312206336 ) / 314810476 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1152319 \nu^{9} - 2214189 \nu^{8} - 65433413 \nu^{7} + 104740913 \nu^{6} + 1012847351 \nu^{5} + \cdots + 11769906774 ) / 629620952 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 599631 \nu^{9} - 1321236 \nu^{8} - 32925581 \nu^{7} + 65506734 \nu^{6} + 464114267 \nu^{5} + \cdots - 2662681628 ) / 314810476 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 2906631 \nu^{9} - 10084945 \nu^{8} - 154719781 \nu^{7} + 527138527 \nu^{6} + 1930396267 \nu^{5} + \cdots + 3527709762 ) / 1259241904 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 3675816 \nu^{9} + 16400103 \nu^{8} + 194918238 \nu^{7} - 877751533 \nu^{6} + \cdots + 16507488138 ) / 629620952 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( -\beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} + 2\beta_{3} + 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} + 3\beta_{8} + 3\beta_{4} - 32\beta_{2} + 24\beta _1 - 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{9} + 35\beta_{7} + 8\beta_{6} - 2\beta_{5} + 84\beta_{3} + 16\beta_{2} - 5\beta _1 + 306 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 46 \beta_{9} + 125 \beta_{8} + 10 \beta_{6} - 4 \beta_{5} + 145 \beta_{4} - 10 \beta_{3} - 1114 \beta_{2} + \cdots - 216 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 60 \beta_{9} - 24 \beta_{8} + 1243 \beta_{7} + 392 \beta_{6} - 116 \beta_{5} + 24 \beta_{4} + \cdots + 9758 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 1811 \beta_{9} + 4613 \beta_{8} - 16 \beta_{7} + 560 \beta_{6} - 284 \beta_{5} + 5677 \beta_{4} + \cdots - 9732 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 2639 \beta_{9} - 1504 \beta_{8} + 44715 \beta_{7} + 15536 \beta_{6} - 4970 \beta_{5} + 1248 \beta_{4} + \cdots + 336790 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 67860 \beta_{9} + 166809 \beta_{8} - 2040 \beta_{7} + 23310 \beta_{6} - 13000 \beta_{5} + 210573 \beta_{4} + \cdots - 401048 \) 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
−3.20919
−6.03761
0.0402482
−2.78818
−0.706075
2.12235
4.14545
1.31702
3.14378
5.97221
−4.62340 −3.00000 13.3758 −2.55125 13.8702 0.718142 −24.8546 9.00000 11.7954
1.2 −4.62340 −3.00000 13.3758 2.55125 13.8702 −0.718142 −24.8546 9.00000 −11.7954
1.3 −1.37397 −3.00000 −6.11222 −18.8553 4.12190 25.1105 19.3897 9.00000 25.9066
1.4 −1.37397 −3.00000 −6.11222 18.8553 4.12190 −25.1105 19.3897 9.00000 −25.9066
1.5 0.708138 −3.00000 −7.49854 −10.5719 −2.12442 0.777483 −10.9751 9.00000 −7.48633
1.6 0.708138 −3.00000 −7.49854 10.5719 −2.12442 −0.777483 −10.9751 9.00000 7.48633
1.7 2.73124 −3.00000 −0.540354 −7.83986 −8.19371 −13.8023 −23.3257 9.00000 −21.4125
1.8 2.73124 −3.00000 −0.540354 7.83986 −8.19371 13.8023 −23.3257 9.00000 21.4125
1.9 4.55799 −3.00000 12.7753 −18.6746 −13.6740 −14.2070 21.7657 9.00000 −85.1186
1.10 4.55799 −3.00000 12.7753 18.6746 −13.6740 14.2070 21.7657 9.00000 85.1186
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( +1 \)
\(23\) \( +1 \)

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 1587.4.a.p ✓ 10
23.b odd 2 1 inner 1587.4.a.p ✓ 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1587.4.a.p ✓ 10 1.a even 1 1 trivial
1587.4.a.p ✓ 10 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1587))\):

\( T_{2}^{5} - 2T_{2}^{4} - 24T_{2}^{3} + 46T_{2}^{2} + 59T_{2} - 56 \) Copy content Toggle raw display
\( T_{5}^{10} - 884T_{5}^{8} + 258564T_{5}^{6} - 27961312T_{5}^{4} + 1022988672T_{5}^{2} - 5543623808 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{5} - 2 T^{4} - 24 T^{3} + \cdots - 56)^{2} \) Copy content Toggle raw display
$3$ \( (T + 3)^{10} \) Copy content Toggle raw display
$5$ \( T^{10} + \cdots - 5543623808 \) Copy content Toggle raw display
$7$ \( T^{10} - 1024 T^{8} + \cdots - 7558272 \) Copy content Toggle raw display
$11$ \( T^{10} + \cdots - 122754179072 \) Copy content Toggle raw display
$13$ \( (T^{5} + 40 T^{4} + \cdots + 350477696)^{2} \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots - 25\!\cdots\!92 \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots - 47\!\cdots\!88 \) Copy content Toggle raw display
$23$ \( T^{10} \) Copy content Toggle raw display
$29$ \( (T^{5} - 462 T^{4} + \cdots - 460966091392)^{2} \) Copy content Toggle raw display
$31$ \( (T^{5} + 278 T^{4} + \cdots - 46337134592)^{2} \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots - 96\!\cdots\!28 \) Copy content Toggle raw display
$41$ \( (T^{5} - 370 T^{4} + \cdots - 240200416928)^{2} \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots - 23\!\cdots\!72 \) Copy content Toggle raw display
$47$ \( (T^{5} - 692 T^{4} + \cdots - 335728672768)^{2} \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots - 12\!\cdots\!32 \) Copy content Toggle raw display
$59$ \( (T^{5} + \cdots - 1672042639104)^{2} \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots - 10\!\cdots\!12 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots - 83\!\cdots\!52 \) Copy content Toggle raw display
$71$ \( (T^{5} + \cdots + 1032640745472)^{2} \) Copy content Toggle raw display
$73$ \( (T^{5} + \cdots - 27364736086016)^{2} \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots - 89\!\cdots\!92 \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots - 31\!\cdots\!88 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots - 54\!\cdots\!88 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots - 17\!\cdots\!88 \) Copy content Toggle raw display
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