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Properties

Label 1024.4.a.o
Level $1024$
Weight $4$
Character orbit 1024.a
Self dual yes
Analytic conductor $60.418$
Analytic rank $1$
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] = [1024,4,Mod(1,1024)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1024, 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("1024.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1024 = 2^{10} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1024.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: \(60.4179558459\)
Analytic rank: \(1\)
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: \( x^{12} - 40x^{10} + 504x^{8} - 2280x^{6} + 3248x^{4} - 480x^{2} + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{33} \)
Twist minimal: no (minimal twist has level 512)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{3} q^{3} + (\beta_{7} - 3) q^{5} + ( - \beta_{9} + \beta_{3}) q^{7} + (\beta_{8} - \beta_{2} + 9) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{3} + (\beta_{7} - 3) q^{5} + ( - \beta_{9} + \beta_{3}) q^{7} + (\beta_{8} - \beta_{2} + 9) q^{9} + ( - \beta_{10} + 2 \beta_{9} + \cdots - \beta_{3}) q^{11}+ \cdots + ( - 7 \beta_{11} + 8 \beta_{10} + \cdots - 51 \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 40 q^{5} + 108 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 40 q^{5} + 108 q^{9} - 104 q^{13} - 336 q^{21} + 300 q^{25} - 696 q^{29} + 464 q^{33} - 296 q^{37} + 80 q^{41} - 1080 q^{45} + 588 q^{49} - 1272 q^{53} - 688 q^{57} - 1464 q^{61} - 488 q^{65} - 5200 q^{69} - 296 q^{73} - 5328 q^{77} - 868 q^{81} - 6832 q^{85} + 88 q^{89} - 9312 q^{93} - 2976 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 40x^{10} + 504x^{8} - 2280x^{6} + 3248x^{4} - 480x^{2} + 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -8\nu^{10} + 346\nu^{8} - 4873\nu^{6} + 24904\nu^{4} - 31268\nu^{2} - 8911 ) / 729 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -22\nu^{10} + 884\nu^{8} - 11234\nu^{6} + 51584\nu^{4} - 73648\nu^{2} + 5512 ) / 729 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 17\nu^{11} - 688\nu^{9} + 8887\nu^{7} - 42742\nu^{5} + 72533\nu^{3} - 27494\nu ) / 729 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2\nu^{11} - 77\nu^{9} + 898\nu^{7} - 3408\nu^{5} + 3348\nu^{3} - 1096\nu ) / 81 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -19\nu^{11} + 725\nu^{9} - 8249\nu^{7} + 28214\nu^{5} - 5497\nu^{3} - 55826\nu ) / 729 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 4\nu^{10} - 164\nu^{8} + 2162\nu^{6} - 10688\nu^{4} + 18136\nu^{2} - 3757 ) / 81 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 31\nu^{10} - 1232\nu^{8} + 15311\nu^{6} - 67004\nu^{4} + 88000\nu^{2} - 9214 ) / 486 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 116\nu^{10} - 4612\nu^{8} + 57415\nu^{6} - 252892\nu^{4} + 335612\nu^{2} - 10556 ) / 729 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -26\nu^{11} + 1039\nu^{9} - 13072\nu^{7} + 59068\nu^{5} - 85214\nu^{3} + 18404\nu ) / 162 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 622\nu^{11} - 24863\nu^{9} + 312800\nu^{7} - 1409192\nu^{5} + 1976218\nu^{3} - 226756\nu ) / 729 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -730\nu^{11} + 29165\nu^{9} - 366530\nu^{7} + 1647296\nu^{5} - 2299876\nu^{3} + 273160\nu ) / 729 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{11} + 8\beta_{9} + \beta_{5} + 4\beta_{4} + 9\beta_{3} ) / 64 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -6\beta_{8} + 12\beta_{7} - 7\beta_{2} + 2\beta _1 + 218 ) / 32 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{11} + 8\beta_{10} + 56\beta_{9} + 23\beta_{5} + 42\beta_{4} + 31\beta_{3} ) / 32 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -62\beta_{8} + 108\beta_{7} - 15\beta_{6} - 129\beta_{2} + 16\beta _1 + 1625 ) / 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 107\beta_{11} + 288\beta_{10} + 936\beta_{9} + 503\beta_{5} + 822\beta_{4} + 191\beta_{3} ) / 32 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -622\beta_{8} + 972\beta_{7} - 210\beta_{6} - 1635\beta_{2} + 182\beta _1 + 14268 ) / 8 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 1884\beta_{11} + 3708\beta_{10} + 8408\beta_{9} + 5111\beta_{5} + 8179\beta_{4} + 151\beta_{3} ) / 16 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -12478\beta_{8} + 18228\beta_{7} - 4863\beta_{6} - 36739\beta_{2} + 4168\beta _1 + 268073 ) / 8 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 24002\beta_{11} + 42428\beta_{10} + 79032\beta_{9} + 51453\beta_{5} + 82295\beta_{4} - 7395\beta_{3} ) / 8 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( -251438\beta_{8} + 352668\beta_{7} - 105756\beta_{6} - 786991\beta_{2} + 91626\beta _1 + 5210550 ) / 8 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 547051 \beta_{11} + 920372 \beta_{10} + 1530984 \beta_{9} + 1038242 \beta_{5} + 1665365 \beta_{4} - 249598 \beta_{3} ) / 8 \) 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
−0.380956
2.25392
1.51967
−0.138214
3.47452
−4.51375
4.51375
−3.47452
0.138214
−1.51967
−2.25392
0.380956
0 −9.32315 0 1.51080 0 26.3526 0 59.9211 0
1.2 0 −7.23522 0 −2.33804 0 5.40314 0 25.3484 0
1.3 0 −6.38533 0 −18.9781 0 −26.2604 0 13.7725 0
1.4 0 −4.95293 0 −18.5500 0 19.2842 0 −2.46849 0
1.5 0 −3.36468 0 12.3022 0 −15.0403 0 −15.6789 0
1.6 0 −0.324689 0 6.05304 0 18.4566 0 −26.8946 0
1.7 0 0.324689 0 6.05304 0 −18.4566 0 −26.8946 0
1.8 0 3.36468 0 12.3022 0 15.0403 0 −15.6789 0
1.9 0 4.95293 0 −18.5500 0 −19.2842 0 −2.46849 0
1.10 0 6.38533 0 −18.9781 0 26.2604 0 13.7725 0
1.11 0 7.23522 0 −2.33804 0 −5.40314 0 25.3484 0
1.12 0 9.32315 0 1.51080 0 −26.3526 0 59.9211 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1024.4.a.o 12
4.b odd 2 1 inner 1024.4.a.o 12
8.b even 2 1 1024.4.a.p 12
8.d odd 2 1 1024.4.a.p 12
16.e even 4 2 1024.4.b.m 24
16.f odd 4 2 1024.4.b.m 24
32.g even 8 2 512.4.e.q ✓ 24
32.g even 8 2 512.4.e.r yes 24
32.h odd 8 2 512.4.e.q ✓ 24
32.h odd 8 2 512.4.e.r yes 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
512.4.e.q ✓ 24 32.g even 8 2
512.4.e.q ✓ 24 32.h odd 8 2
512.4.e.r yes 24 32.g even 8 2
512.4.e.r yes 24 32.h odd 8 2
1024.4.a.o 12 1.a even 1 1 trivial
1024.4.a.o 12 4.b odd 2 1 inner
1024.4.a.p 12 8.b even 2 1
1024.4.a.p 12 8.d odd 2 1
1024.4.b.m 24 16.e even 4 2
1024.4.b.m 24 16.f odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1024))\):

\( T_{3}^{12} - 216T_{3}^{10} + 16984T_{3}^{8} - 604032T_{3}^{6} + 9555648T_{3}^{4} - 52524544T_{3}^{2} + 5431808 \) Copy content Toggle raw display
\( T_{5}^{6} + 20T_{5}^{5} - 250T_{5}^{4} - 3952T_{5}^{3} + 24108T_{5}^{2} + 34640T_{5} - 92600 \) Copy content Toggle raw display
\( T_{7}^{12} - 2352 T_{7}^{10} + 2133856 T_{7}^{8} - 936958976 T_{7}^{6} + 203113524224 T_{7}^{4} + \cdots + 400648785920000 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} - 216 T^{10} + \cdots + 5431808 \) Copy content Toggle raw display
$5$ \( (T^{6} + 20 T^{5} + \cdots - 92600)^{2} \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 400648785920000 \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 43\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{6} + 52 T^{5} + \cdots - 1393286200)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} - 20248 T^{4} + \cdots - 1298547200)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 53\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 52\!\cdots\!72 \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots + 2350068244552)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 40\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( (T^{6} + \cdots - 1989440875000)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots + 8212668551168)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 32\!\cdots\!68 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 57\!\cdots\!48 \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots - 156729558527800)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{6} + \cdots - 111027905674232)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 48\!\cdots\!52 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{6} + \cdots + 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 88\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots - 40\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots + 17\!\cdots\!00)^{2} \) Copy content Toggle raw display
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