Properties

Label 1024.4.a.n
Level $1024$
Weight $4$
Character orbit 1024.a
Self dual yes
Analytic conductor $60.418$
Analytic rank $0$
Dimension $10$
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: \(0\)
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: \( x^{10} - 36x^{8} + 405x^{6} - 1380x^{4} + 420x^{2} - 32 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{22} \)
Twist minimal: no (minimal twist has level 16)
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.

\(f(q)\) \(=\) \( q + \beta_{7} q^{3} - \beta_{4} q^{5} + (\beta_{2} + 3) q^{7} + ( - \beta_{3} + \beta_{2} + 6) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{7} q^{3} - \beta_{4} q^{5} + (\beta_{2} + 3) q^{7} + ( - \beta_{3} + \beta_{2} + 6) q^{9} + (\beta_{9} - \beta_{4}) q^{11} + (\beta_{9} + \beta_{8} + 3 \beta_{7} + \beta_{4}) q^{13} + (\beta_{6} - \beta_{5} - 2 \beta_{3} + 13) q^{15} + (\beta_{6} - 2 \beta_{5} + \beta_{3} + 2 \beta_{2}) q^{17} + ( - \beta_{9} + 2 \beta_{8} - 2 \beta_{7} - 5 \beta_{4} + \beta_1) q^{19} + (\beta_{9} - 3 \beta_{8} + 11 \beta_{7} + \beta_1) q^{21} + ( - 3 \beta_{6} + \beta_{5} - 2 \beta_{3} + 2 \beta_{2} + 29) q^{23} + ( - \beta_{6} + 6 \beta_{5} + 2 \beta_{3} + 3 \beta_{2} + 6) q^{25} + ( - \beta_{9} - 2 \beta_{8} - \beta_{7} - 9 \beta_{4} - 5 \beta_1) q^{27} + ( - 2 \beta_{9} + 2 \beta_{8} + 18 \beta_{7} + \beta_{4} - 3 \beta_1) q^{29} + (3 \beta_{6} + 3 \beta_{5} + 2 \beta_{3} - 3 \beta_{2} + 36) q^{31} + ( - 2 \beta_{6} - 8 \beta_{5} - \beta_{3} + 3 \beta_{2} - 1) q^{33} + ( - 2 \beta_{8} - 2 \beta_{7} - 10 \beta_{4} + 11 \beta_1) q^{35} + ( - \beta_{9} - \beta_{8} + 29 \beta_{7} + \beta_{4} + 2 \beta_1) q^{37} + ( - 4 \beta_{6} - 2 \beta_{5} + \beta_{2} + 73) q^{39} + (\beta_{6} + 14 \beta_{5} + 4 \beta_{3} - \beta_{2} + 1) q^{41} + ( - 4 \beta_{9} - \beta_{7} - 12 \beta_{4} - 20 \beta_1) q^{43} + ( - 5 \beta_{9} - \beta_{8} + 41 \beta_{7} - \beta_{4} - \beta_1) q^{45} + (3 \beta_{6} + 5 \beta_{5} + 10 \beta_{3} - 7 \beta_{2} + 90) q^{47} + ( - 6 \beta_{6} - 16 \beta_{5} - 6 \beta_{3} - 4 \beta_{2} - 11) q^{49} + (5 \beta_{9} - 12 \beta_{8} + 7 \beta_{7} - \beta_{4} + 34 \beta_1) q^{51} + ( - 7 \beta_{9} + 13 \beta_{8} + 35 \beta_{7} - \beta_{4} - 3 \beta_1) q^{53} + (6 \beta_{6} - 8 \beta_{5} + 12 \beta_{3} - 3 \beta_{2} + 131) q^{55} + (8 \beta_{6} + 8 \beta_{5} - 9 \beta_{3} - 15 \beta_{2} + 13) q^{57} + (4 \beta_{9} + 14 \beta_{8} + 5 \beta_{7} + 18 \beta_{4} - 41 \beta_1) q^{59} + (11 \beta_{9} - 9 \beta_{8} + 41 \beta_{7} - \beta_{4} + 19 \beta_1) q^{61} + ( - 3 \beta_{6} - 21 \beta_{5} - 10 \beta_{3} + 2 \beta_{2} + 263) q^{63} + (13 \beta_{6} - 18 \beta_{5} + 4 \beta_{3} - 13 \beta_{2} - 57) q^{65} + (\beta_{9} + 12 \beta_{8} - 10 \beta_{7} + 11 \beta_{4} + 50 \beta_1) q^{67} + (5 \beta_{9} + \beta_{8} + 55 \beta_{7} + 12 \beta_{4} - 13 \beta_1) q^{69} + (3 \beta_{6} + 15 \beta_{5} + 2 \beta_{3} - 10 \beta_{2} + 347) q^{71} + ( - 6 \beta_{6} + 16 \beta_{5} + \beta_{3} - 11 \beta_{2} - 29) q^{73} + ( - 2 \beta_{9} + 2 \beta_{8} + 7 \beta_{7} + 28 \beta_{4} - 71 \beta_1) q^{75} + (5 \beta_{9} - 15 \beta_{8} + 23 \beta_{7} - 12 \beta_{4} + 3 \beta_1) q^{77} + ( - 16 \beta_{6} - 6 \beta_{5} - 8 \beta_{3} + 12 \beta_{2} + 446) q^{79} + (12 \beta_{6} + 9 \beta_{3} - 21 \beta_{2} - 56) q^{81} + ( - 2 \beta_{9} + 22 \beta_{8} + 9 \beta_{7} + 48 \beta_{4} + 71 \beta_1) q^{83} + (17 \beta_{9} - 7 \beta_{8} + 3 \beta_{7} - 8 \beta_1) q^{85} + (5 \beta_{6} + 29 \beta_{5} - 18 \beta_{3} + 2 \beta_{2} + 617) q^{87} + ( - 26 \beta_{6} + 8 \beta_{5} + \beta_{3} + 25 \beta_{2} + 15) q^{89} + (2 \beta_{9} - 34 \beta_{8} + 4 \beta_{4} - 89 \beta_1) q^{91} + ( - 14 \beta_{9} + 10 \beta_{8} + 6 \beta_{7} - 12 \beta_{4} - 48 \beta_1) q^{93} + ( - 12 \beta_{6} + 56 \beta_{5} + 8 \beta_{3} + 15 \beta_{2} + 701) q^{95} + ( - 29 \beta_{6} + 14 \beta_{5} + 3 \beta_{3} + 14 \beta_{2} + 4) q^{97} + ( - 4 \beta_{9} - 22 \beta_{8} + 17 \beta_{7} + 38 \beta_{4} + 125 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 28 q^{7} + 54 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 28 q^{7} + 54 q^{9} + 124 q^{15} + 4 q^{17} + 276 q^{23} + 50 q^{25} + 368 q^{31} - 4 q^{33} + 732 q^{39} + 944 q^{47} - 94 q^{49} + 1380 q^{55} + 108 q^{57} + 2628 q^{63} - 492 q^{65} + 3468 q^{71} - 296 q^{73} + 4416 q^{79} - 482 q^{81} + 6036 q^{87} + 88 q^{89} + 6900 q^{95} - 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 36x^{8} + 405x^{6} - 1380x^{4} + 420x^{2} - 32 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 7\nu^{9} - 250\nu^{7} + 2823\nu^{5} - 10042\nu^{3} + 4528\nu ) / 208 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 59\nu^{8} - 2018\nu^{6} + 21595\nu^{4} - 70882\nu^{2} + 17840 ) / 416 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -137\nu^{8} + 4982\nu^{6} - 55785\nu^{4} + 180342\nu^{2} - 24496 ) / 416 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 217\nu^{9} - 7750\nu^{7} + 85849\nu^{5} - 278022\nu^{3} + 18896\nu ) / 832 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -83\nu^{8} + 2994\nu^{6} - 33651\nu^{4} + 112562\nu^{2} - 18656 ) / 208 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 231\nu^{8} - 8042\nu^{6} + 86919\nu^{4} - 277098\nu^{2} + 35856 ) / 416 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 157\nu^{9} - 5622\nu^{7} + 62573\nu^{5} - 206166\nu^{3} + 34432\nu ) / 416 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -55\nu^{9} + 1978\nu^{7} - 22135\nu^{5} + 73594\nu^{3} - 13680\nu ) / 64 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 1015\nu^{9} - 36458\nu^{7} + 407255\nu^{5} - 1348970\nu^{3} + 224336\nu ) / 832 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -2\beta_{9} - 2\beta_{8} + 2\beta_{7} - \beta_1 ) / 32 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{6} - \beta_{5} + 2\beta_{3} - 6\beta_{2} + 113 ) / 16 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -26\beta_{9} - 30\beta_{8} + 2\beta_{7} + 20\beta_{4} - \beta_1 ) / 32 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 29\beta_{6} - 25\beta_{5} + 40\beta_{3} - 91\beta_{2} + 1516 ) / 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -374\beta_{9} - 454\beta_{8} - 106\beta_{7} + 384\beta_{4} + 177\beta_1 ) / 32 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 446\beta_{6} - 423\beta_{5} + 678\beta_{3} - 1362\beta_{2} + 21951 ) / 16 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -5622\beta_{9} - 6754\beta_{8} - 2066\beta_{7} + 6460\beta_{4} + 4433\beta_1 ) / 32 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 7043\beta_{6} - 6519\beta_{5} + 10952\beta_{3} - 20373\beta_{2} + 326836 ) / 16 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( -85962\beta_{9} - 99866\beta_{8} - 29462\beta_{7} + 104544\beta_{4} + 87103\beta_1 ) / 32 \) Copy content Toggle raw display

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
−2.34476
−0.357936
0.446984
3.82089
−3.94652
3.94652
−3.82089
−0.446984
0.357936
2.34476
0 −8.43597 0 −12.2748 0 1.63924 0 44.1656 0
1.2 0 −7.77277 0 −6.59550 0 24.8965 0 33.4160 0
1.3 0 −4.62644 0 17.8826 0 13.8754 0 −5.59607 0
1.4 0 −2.80518 0 −0.844070 0 −29.0828 0 −19.1310 0
1.5 0 −1.07024 0 11.6331 0 2.67171 0 −25.8546 0
1.6 0 1.07024 0 −11.6331 0 2.67171 0 −25.8546 0
1.7 0 2.80518 0 0.844070 0 −29.0828 0 −19.1310 0
1.8 0 4.62644 0 −17.8826 0 13.8754 0 −5.59607 0
1.9 0 7.77277 0 6.59550 0 24.8965 0 33.4160 0
1.10 0 8.43597 0 12.2748 0 1.63924 0 44.1656 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 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.n 10
4.b odd 2 1 1024.4.a.m 10
8.b even 2 1 inner 1024.4.a.n 10
8.d odd 2 1 1024.4.a.m 10
16.e even 4 2 1024.4.b.j 10
16.f odd 4 2 1024.4.b.k 10
32.g even 8 2 16.4.e.a 10
32.g even 8 2 128.4.e.b 10
32.h odd 8 2 64.4.e.a 10
32.h odd 8 2 128.4.e.a 10
96.o even 8 2 576.4.k.a 10
96.p odd 8 2 144.4.k.a 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
16.4.e.a 10 32.g even 8 2
64.4.e.a 10 32.h odd 8 2
128.4.e.a 10 32.h odd 8 2
128.4.e.b 10 32.g even 8 2
144.4.k.a 10 96.p odd 8 2
576.4.k.a 10 96.o even 8 2
1024.4.a.m 10 4.b odd 2 1
1024.4.a.m 10 8.d odd 2 1
1024.4.a.n 10 1.a even 1 1 trivial
1024.4.a.n 10 8.b even 2 1 inner
1024.4.b.j 10 16.e even 4 2
1024.4.b.k 10 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1024))\):

\( T_{3}^{10} - 162T_{3}^{8} + 8504T_{3}^{6} - 157552T_{3}^{4} + 893712T_{3}^{2} - 829472 \) Copy content Toggle raw display
\( T_{5}^{10} - 650T_{5}^{8} + 138664T_{5}^{6} - 11484496T_{5}^{4} + 291758672T_{5}^{2} - 202085408 \) Copy content Toggle raw display
\( T_{7}^{5} - 14T_{7}^{4} - 736T_{7}^{3} + 13376T_{7}^{2} - 46736T_{7} + 44000 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} - 162 T^{8} + 8504 T^{6} + \cdots - 829472 \) Copy content Toggle raw display
$5$ \( T^{10} - 650 T^{8} + \cdots - 202085408 \) Copy content Toggle raw display
$7$ \( (T^{5} - 14 T^{4} - 736 T^{3} + \cdots + 44000)^{2} \) Copy content Toggle raw display
$11$ \( T^{10} - 6050 T^{8} + \cdots - 3810412010528 \) Copy content Toggle raw display
$13$ \( T^{10} - 9802 T^{8} + \cdots - 11\!\cdots\!28 \) Copy content Toggle raw display
$17$ \( (T^{5} - 2 T^{4} - 11912 T^{3} + \cdots + 556317664)^{2} \) Copy content Toggle raw display
$19$ \( T^{10} - 34642 T^{8} + \cdots - 19\!\cdots\!12 \) Copy content Toggle raw display
$23$ \( (T^{5} - 138 T^{4} - 13120 T^{3} + \cdots + 1586189472)^{2} \) Copy content Toggle raw display
$29$ \( T^{10} - 89722 T^{8} + \cdots - 71\!\cdots\!12 \) Copy content Toggle raw display
$31$ \( (T^{5} - 184 T^{4} - 14912 T^{3} + \cdots + 678952960)^{2} \) Copy content Toggle raw display
$37$ \( T^{10} - 145466 T^{8} + \cdots - 69\!\cdots\!32 \) Copy content Toggle raw display
$41$ \( (T^{5} - 124096 T^{3} + \cdots - 241780654080)^{2} \) Copy content Toggle raw display
$43$ \( T^{10} - 243122 T^{8} + \cdots - 54\!\cdots\!32 \) Copy content Toggle raw display
$47$ \( (T^{5} - 472 T^{4} + \cdots - 154359955456)^{2} \) Copy content Toggle raw display
$53$ \( T^{10} - 963610 T^{8} + \cdots - 63\!\cdots\!52 \) Copy content Toggle raw display
$59$ \( T^{10} - 1095506 T^{8} + \cdots - 18\!\cdots\!48 \) Copy content Toggle raw display
$61$ \( T^{10} - 1122986 T^{8} + \cdots - 96\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{10} - 876978 T^{8} + \cdots - 15\!\cdots\!08 \) Copy content Toggle raw display
$71$ \( (T^{5} - 1734 T^{4} + \cdots + 10372402674784)^{2} \) Copy content Toggle raw display
$73$ \( (T^{5} + 148 T^{4} + \cdots - 2067386503168)^{2} \) Copy content Toggle raw display
$79$ \( (T^{5} - 2208 T^{4} + \cdots + 10448447471616)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} - 3176354 T^{8} + \cdots - 16\!\cdots\!72 \) Copy content Toggle raw display
$89$ \( (T^{5} - 44 T^{4} + \cdots - 1169744859136)^{2} \) Copy content Toggle raw display
$97$ \( (T^{5} + 2 T^{4} + \cdots + 65755091474464)^{2} \) Copy content Toggle raw display
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