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Properties

Label 1024.4.b.l
Level $1024$
Weight $4$
Character orbit 1024.b
Analytic conductor $60.418$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1024,4,Mod(513,1024)]
 
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, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1024.513");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1024 = 2^{10} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1024.b (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(60.4179558459\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 242x^{12} + 13297x^{8} + 201600x^{4} + 331776 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{56} \)
Twist minimal: no (minimal twist has level 256)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{4} q^{3} - \beta_{10} q^{5} - \beta_{7} q^{7} + ( - \beta_{2} - \beta_1 - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{3} - \beta_{10} q^{5} - \beta_{7} q^{7} + ( - \beta_{2} - \beta_1 - 1) q^{9} + ( - \beta_{5} + 5 \beta_{4}) q^{11} + (3 \beta_{13} - \beta_{9} + \beta_{8}) q^{13} + (\beta_{15} - 3 \beta_{14} + \beta_{7}) q^{15} + (2 \beta_{2} - 3 \beta_1 + 24) q^{17} + (\beta_{12} - 5 \beta_{6} + \cdots - 2 \beta_{4}) q^{19}+ \cdots + (9 \beta_{12} - 15 \beta_{6} + \cdots - 146 \beta_{4}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16 q^{9} + 384 q^{17} - 656 q^{25} - 2432 q^{33} + 1280 q^{41} + 2832 q^{49} + 640 q^{57} + 2208 q^{65} - 32 q^{73} - 4208 q^{81} - 544 q^{89} + 4480 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 242x^{12} + 13297x^{8} + 201600x^{4} + 331776 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -1181\nu^{12} - 373066\nu^{8} - 28523501\nu^{4} - 256704192 ) / 17702784 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 7\nu^{12} + 1406\nu^{8} + 30007\nu^{4} - 455616 ) / 36576 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 35639\nu^{12} + 7900318\nu^{8} + 324482855\nu^{4} + 2138504256 ) / 17702784 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -25955\nu^{14} - 5646934\nu^{10} - 216114611\nu^{6} - 1382732352\nu^{2} ) / 849733632 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1445\nu^{14} + 330298\nu^{10} + 15381845\nu^{6} + 262588224\nu^{2} ) / 35405568 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 36115\nu^{14} + 9276086\nu^{10} + 599343715\nu^{6} + 10700541504\nu^{2} ) / 849733632 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 12881 \nu^{15} + 640320 \nu^{13} + 918830 \nu^{11} + 133378176 \nu^{9} + \cdots - 27035043840 \nu ) / 5098401792 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 10555 \nu^{15} + 14952 \nu^{13} + 2553734 \nu^{11} + 3881040 \nu^{9} + 137224459 \nu^{7} + \cdots + 2376580608 \nu ) / 463491072 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 78209 \nu^{15} - 331416 \nu^{13} + 22026034 \nu^{11} - 60171696 \nu^{9} + \cdots + 36817251840 \nu ) / 3398934528 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 26361 \nu^{15} + 36824 \nu^{13} - 6776994 \nu^{11} + 6685744 \nu^{9} + \cdots - 10133356032 \nu ) / 1132978176 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 30257 \nu^{15} - 140424 \nu^{13} + 6458770 \nu^{11} - 28079760 \nu^{9} + 236426657 \nu^{7} + \cdots - 3690358272 \nu ) / 1274600448 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( -1783\nu^{14} - 420830\nu^{10} - 21109711\nu^{6} - 234593064\nu^{2} ) / 3319272 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 58873 \nu^{15} + 148440 \nu^{13} + 13669538 \nu^{11} + 32950320 \nu^{9} + \cdots + 11174367744 \nu ) / 1274600448 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 205235 \nu^{15} - 453336 \nu^{13} + 48219382 \nu^{11} - 103721520 \nu^{9} + \cdots - 41007112704 \nu ) / 2549200896 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 103001 \nu^{15} - 152448 \nu^{13} + 24581218 \nu^{11} - 35385600 \nu^{9} + \cdots - 25113176064 \nu ) / 1274600448 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -8\beta_{15} + 6\beta_{14} - \beta_{11} - 12\beta_{10} - 4\beta_{9} ) / 128 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -7\beta_{6} + 26\beta_{5} + 25\beta_{4} ) / 64 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 64 \beta_{15} - 62 \beta_{14} - 36 \beta_{13} - 11 \beta_{11} - 68 \beta_{10} - 44 \beta_{9} + \cdots - 32 \beta_{7} ) / 128 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 13\beta_{3} - 112\beta_{2} + 71\beta _1 - 1936 ) / 32 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 656 \beta_{15} - 590 \beta_{14} + 556 \beta_{13} - 307 \beta_{11} + 564 \beta_{10} + \cdots - 480 \beta_{7} ) / 128 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 320\beta_{12} + 2575\beta_{6} - 2506\beta_{5} - 5393\beta_{4} ) / 64 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 7584 \beta_{15} + 6302 \beta_{14} + 7892 \beta_{13} + 5115 \beta_{11} + 5860 \beta_{10} + \cdots + 6304 \beta_{7} ) / 128 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -2245\beta_{3} + 17888\beta_{2} - 16431\beta _1 + 255760 ) / 32 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 93232 \beta_{15} + 74030 \beta_{14} - 108732 \beta_{13} + 74339 \beta_{11} - 68756 \beta_{10} + \cdots + 81248 \beta_{7} ) / 128 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( -67360\beta_{12} - 460399\beta_{6} + 366666\beta_{5} + 1033905\beta_{4} ) / 64 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 1182272 \beta_{15} - 918014 \beta_{14} - 1467364 \beta_{13} - 1022827 \beta_{11} + \cdots - 1048608 \beta_{7} ) / 128 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 395197\beta_{3} - 2945616\beta_{2} + 2995927\beta _1 - 40989328 ) / 32 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 15226320 \beta_{15} - 11698958 \beta_{14} + 19539788 \beta_{13} - 13722387 \beta_{11} + \cdots - 13592800 \beta_{7} ) / 128 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 11990784\beta_{12} + 79099471\beta_{6} - 60293066\beta_{5} - 183465169\beta_{4} ) / 64 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 197614816 \beta_{15} + 151082078 \beta_{14} + 258138612 \beta_{13} + 181838171 \beta_{11} + \cdots + 176832416 \beta_{7} ) / 128 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1024\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(1023\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
513.1
1.85032 + 1.85032i
−1.85032 − 1.85032i
1.53379 − 1.53379i
−1.53379 + 1.53379i
2.55743 − 2.55743i
−2.55743 + 2.55743i
−0.826679 − 0.826679i
0.826679 + 0.826679i
0.826679 − 0.826679i
−0.826679 + 0.826679i
−2.55743 − 2.55743i
2.55743 + 2.55743i
−1.53379 − 1.53379i
1.53379 + 1.53379i
−1.85032 + 1.85032i
1.85032 − 1.85032i
0 − 7.96554i 0 − 0.466324i 0 −17.2022 0 −36.4499 0
513.2 0 − 7.96554i 0 0.466324i 0 17.2022 0 −36.4499 0
513.3 0 − 5.07025i 0 − 20.4472i 0 31.2516 0 1.29254 0
513.4 0 − 5.07025i 0 20.4472i 0 −31.2516 0 1.29254 0
513.5 0 − 4.50144i 0 − 5.19053i 0 −26.2423 0 6.73704 0
513.6 0 − 4.50144i 0 5.19053i 0 26.2423 0 6.73704 0
513.7 0 − 1.60615i 0 − 14.7903i 0 −10.8978 0 24.4203 0
513.8 0 − 1.60615i 0 14.7903i 0 10.8978 0 24.4203 0
513.9 0 1.60615i 0 − 14.7903i 0 10.8978 0 24.4203 0
513.10 0 1.60615i 0 14.7903i 0 −10.8978 0 24.4203 0
513.11 0 4.50144i 0 − 5.19053i 0 26.2423 0 6.73704 0
513.12 0 4.50144i 0 5.19053i 0 −26.2423 0 6.73704 0
513.13 0 5.07025i 0 − 20.4472i 0 −31.2516 0 1.29254 0
513.14 0 5.07025i 0 20.4472i 0 31.2516 0 1.29254 0
513.15 0 7.96554i 0 − 0.466324i 0 17.2022 0 −36.4499 0
513.16 0 7.96554i 0 0.466324i 0 −17.2022 0 −36.4499 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 513.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
8.b even 2 1 inner
8.d 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.b.l 16
4.b odd 2 1 inner 1024.4.b.l 16
8.b even 2 1 inner 1024.4.b.l 16
8.d odd 2 1 inner 1024.4.b.l 16
16.e even 4 1 1024.4.a.k 8
16.e even 4 1 1024.4.a.l 8
16.f odd 4 1 1024.4.a.k 8
16.f odd 4 1 1024.4.a.l 8
32.g even 8 2 256.4.e.c ✓ 16
32.g even 8 2 256.4.e.d yes 16
32.h odd 8 2 256.4.e.c ✓ 16
32.h odd 8 2 256.4.e.d yes 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
256.4.e.c ✓ 16 32.g even 8 2
256.4.e.c ✓ 16 32.h odd 8 2
256.4.e.d yes 16 32.g even 8 2
256.4.e.d yes 16 32.h odd 8 2
1024.4.a.k 8 16.e even 4 1
1024.4.a.k 8 16.f odd 4 1
1024.4.a.l 8 16.e even 4 1
1024.4.a.l 8 16.f odd 4 1
1024.4.b.l 16 1.a even 1 1 trivial
1024.4.b.l 16 4.b odd 2 1 inner
1024.4.b.l 16 8.b even 2 1 inner
1024.4.b.l 16 8.d 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 \(S_{4}^{\mathrm{new}}(1024, [\chi])\):

\( T_{3}^{8} + 112T_{3}^{6} + 3720T_{3}^{4} + 41920T_{3}^{2} + 85264 \) Copy content Toggle raw display
\( T_{5}^{8} + 664T_{5}^{6} + 108760T_{5}^{4} + 2487648T_{5}^{2} + 535824 \) Copy content Toggle raw display
\( T_{7}^{8} - 2080T_{7}^{6} + 1398304T_{7}^{4} - 337433088T_{7}^{2} + 23637217536 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( (T^{8} + 112 T^{6} + \cdots + 85264)^{2} \) Copy content Toggle raw display
$5$ \( (T^{8} + 664 T^{6} + \cdots + 535824)^{2} \) Copy content Toggle raw display
$7$ \( (T^{8} - 2080 T^{6} + \cdots + 23637217536)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} + 4496 T^{6} + \cdots + 54110736)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + \cdots + 20270417143824)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 96 T^{3} + \cdots + 1811088)^{4} \) Copy content Toggle raw display
$19$ \( (T^{8} + \cdots + 298255525045776)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + \cdots + 20\!\cdots\!76)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + \cdots + 94\!\cdots\!76)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + \cdots + 16\!\cdots\!44)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots + 77\!\cdots\!84)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} - 320 T^{3} + \cdots + 247734528)^{4} \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots + 36\!\cdots\!84)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots + 34\!\cdots\!96)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots + 31\!\cdots\!04)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots + 34\!\cdots\!76)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots + 83\!\cdots\!96)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 69\!\cdots\!76)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 21\!\cdots\!24)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 8 T^{3} + \cdots + 38640192)^{4} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 41\!\cdots\!24)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 95\!\cdots\!16)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 136 T^{3} + \cdots + 119895761472)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} - 1120 T^{3} + \cdots - 580260467568)^{4} \) Copy content Toggle raw display
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