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Properties

Label 1024.2.g.c
Level $1024$
Weight $2$
Character orbit 1024.g
Analytic conductor $8.177$
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] = [1024,2,Mod(129,1024)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1024, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([0, 7]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1024.129");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1024 = 2^{10} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1024.g (of order \(8\), degree \(4\), 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: \(8.17668116698\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{8})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 8x^{14} + 32x^{12} - 64x^{10} + 127x^{8} - 576x^{6} + 2592x^{4} - 5832x^{2} + 6561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{13} q^{3} + (\beta_{8} + \beta_{5} - \beta_{3} - 1) q^{5} + (\beta_{13} + \beta_{9} + \beta_{4}) q^{7} + ( - \beta_{8} - \beta_{6} - 3 \beta_{5} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{13} q^{3} + (\beta_{8} + \beta_{5} - \beta_{3} - 1) q^{5} + (\beta_{13} + \beta_{9} + \beta_{4}) q^{7} + ( - \beta_{8} - \beta_{6} - 3 \beta_{5} + 2) q^{9} + (\beta_{15} + \beta_{9} - \beta_1) q^{11} + (\beta_{12} + \beta_{10} - \beta_{8} + \cdots - 1) q^{13}+ \cdots + ( - \beta_{15} + 3 \beta_{13} + \cdots + \beta_{4}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 8 q^{5} + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 8 q^{5} + 16 q^{9} - 24 q^{13} + 16 q^{21} + 32 q^{25} + 24 q^{29} + 80 q^{33} - 40 q^{37} + 16 q^{41} - 24 q^{45} + 56 q^{53} + 80 q^{57} - 8 q^{61} + 32 q^{65} - 32 q^{69} + 32 q^{73} + 32 q^{77} - 48 q^{85} - 32 q^{89} + 16 q^{93} + 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 8x^{14} + 32x^{12} - 64x^{10} + 127x^{8} - 576x^{6} + 2592x^{4} - 5832x^{2} + 6561 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 23 \nu^{15} + 1975 \nu^{13} - 8260 \nu^{11} + 13829 \nu^{9} - 4064 \nu^{7} + 152253 \nu^{5} + \cdots + 1533816 \nu ) / 269001 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 683 \nu^{15} - 13663 \nu^{13} + 44518 \nu^{11} - 82925 \nu^{9} + 22850 \nu^{7} + \cdots - 9086985 \nu ) / 1883007 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 928 \nu^{14} - 2582 \nu^{12} + 1895 \nu^{10} + 5156 \nu^{8} + 70408 \nu^{6} - 270972 \nu^{4} + \cdots + 654642 ) / 627669 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 998 \nu^{15} + 1630 \nu^{13} + 12497 \nu^{11} - 29215 \nu^{9} - 36395 \nu^{7} + \cdots - 2899962 \nu ) / 1883007 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 152\nu^{14} - 606\nu^{12} + 1352\nu^{10} - 828\nu^{8} + 10513\nu^{6} - 43364\nu^{4} + 119160\nu^{2} - 72738 ) / 69741 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1529 \nu^{14} - 19279 \nu^{12} + 69988 \nu^{10} - 104255 \nu^{8} + 123065 \nu^{6} - 1456047 \nu^{4} + \cdots - 10136016 ) / 627669 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 536 \nu^{14} + 4297 \nu^{12} - 11716 \nu^{10} + 13856 \nu^{8} - 29444 \nu^{6} + 308016 \nu^{4} + \cdots + 1458000 ) / 209223 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 541 \nu^{14} + 445 \nu^{12} - 5671 \nu^{10} + 23963 \nu^{8} + 27322 \nu^{6} - 19293 \nu^{4} + \cdots + 2030022 ) / 209223 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 545 \nu^{15} + 5749 \nu^{13} - 15376 \nu^{11} + 17417 \nu^{9} - 35612 \nu^{7} + \cdots + 2404728 \nu ) / 627669 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 269 \nu^{14} + 1360 \nu^{12} - 2110 \nu^{10} + 53 \nu^{8} - 22760 \nu^{6} + 121995 \nu^{4} + \cdots + 39366 ) / 89667 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 2011 \nu^{15} - 12281 \nu^{13} + 39080 \nu^{11} - 14089 \nu^{9} + 159412 \nu^{7} + \cdots - 2566080 \nu ) / 1883007 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 2069 \nu^{14} - 8245 \nu^{12} + 15790 \nu^{10} - 13121 \nu^{8} + 116027 \nu^{6} - 763695 \nu^{4} + \cdots - 1452897 ) / 627669 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 2318 \nu^{15} - 30910 \nu^{13} + 90646 \nu^{11} - 135176 \nu^{9} + 129686 \nu^{7} + \cdots - 12535155 \nu ) / 1883007 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 3323 \nu^{15} - 30895 \nu^{13} + 106156 \nu^{11} - 158573 \nu^{9} + 168104 \nu^{7} + \cdots - 16315020 \nu ) / 1883007 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 9367 \nu^{15} + 48179 \nu^{13} - 118331 \nu^{11} + 112300 \nu^{9} - 632545 \nu^{7} + \cdots + 11764602 \nu ) / 1883007 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{13} + \beta_{9} - \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{12} - \beta_{10} + \beta_{7} + \beta_{5} - 4\beta_{3} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{14} - 2\beta_{11} - 2\beta_{9} - 4\beta_{2} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -\beta_{12} - 2\beta_{10} - \beta_{8} - 2\beta_{6} + 11\beta_{5} - 13\beta_{3} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( \beta_{14} + \beta_{13} - 2\beta_{11} - 7\beta_{9} + 12\beta_{4} + \beta_{2} + 11\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -6\beta_{12} - 6\beta_{10} - 2\beta_{8} + 8\beta_{7} + 2\beta_{6} + 9\beta_{5} + 2\beta_{3} - 8 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -21\beta_{15} - 46\beta_{14} + 4\beta_{13} - 10\beta_{11} + 12\beta_{9} + 21\beta_{4} - 47\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -34\beta_{12} - 11\beta_{10} + 34\beta_{8} - 11\beta_{7} + 11\beta_{6} - 13\beta_{5} - 24\beta_{3} - 138 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 36\beta_{15} + 18\beta_{14} - 64\beta_{13} + 102\beta_{11} - 115\beta_{9} + 64\beta_{2} + 69\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( -115\beta_{12} + 115\beta_{10} + 84\beta_{8} - 7\beta_{7} + 84\beta_{6} - 31\beta_{5} + 238\beta_{3} - 7 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 69\beta_{15} + 68\beta_{14} + 344\beta_{11} + 185\beta_{9} + 69\beta_{4} + 184\beta_{2} + 160\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 92\beta_{12} + 172\beta_{10} + 92\beta_{8} + 219\beta_{7} + 172\beta_{6} - 196\beta_{5} + 368\beta_{3} - 92 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 389\beta_{14} - 415\beta_{13} + 137\beta_{11} + 643\beta_{9} - 600\beta_{4} - 415\beta_{2} - 806\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 1332 \beta_{12} + 1332 \beta_{10} + 163 \beta_{8} - 142 \beta_{7} - 163 \beta_{6} + 669 \beta_{5} + \cdots + 142 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 576\beta_{15} + 3727\beta_{14} - 1516\beta_{13} - 269\beta_{11} - 489\beta_{9} - 576\beta_{4} + 3371\beta_1 ) / 2 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(5\) \(1023\)
\(\chi(n)\) \(\beta_{3}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
129.1
1.59056 + 0.685641i
−0.639878 + 1.60952i
0.639878 − 1.60952i
−1.59056 − 0.685641i
−1.66798 − 0.466730i
−1.50947 − 0.849413i
1.50947 + 0.849413i
1.66798 + 0.466730i
−1.66798 + 0.466730i
−1.50947 + 0.849413i
1.50947 − 0.849413i
1.66798 − 0.466730i
1.59056 − 0.685641i
−0.639878 − 1.60952i
0.639878 + 1.60952i
−1.59056 + 0.685641i
0 −0.942835 + 2.27621i 0 2.49877 − 1.03503i 0 −1.37128 + 1.37128i 0 −2.17085 − 2.17085i 0
129.2 0 −0.401639 + 0.969643i 0 −3.49877 + 1.44924i 0 −3.21904 + 3.21904i 0 1.34243 + 1.34243i 0
129.3 0 0.401639 − 0.969643i 0 −3.49877 + 1.44924i 0 3.21904 − 3.21904i 0 1.34243 + 1.34243i 0
129.4 0 0.942835 − 2.27621i 0 2.49877 − 1.03503i 0 1.37128 − 1.37128i 0 −2.17085 − 2.17085i 0
385.1 0 −2.90008 + 1.20125i 0 −1.21229 + 2.92673i 0 −0.933460 − 0.933460i 0 4.84613 − 4.84613i 0
385.2 0 −1.59352 + 0.660056i 0 0.212292 − 0.512517i 0 −1.69883 − 1.69883i 0 −0.0177021 + 0.0177021i 0
385.3 0 1.59352 − 0.660056i 0 0.212292 − 0.512517i 0 1.69883 + 1.69883i 0 −0.0177021 + 0.0177021i 0
385.4 0 2.90008 − 1.20125i 0 −1.21229 + 2.92673i 0 0.933460 + 0.933460i 0 4.84613 − 4.84613i 0
641.1 0 −2.90008 − 1.20125i 0 −1.21229 − 2.92673i 0 −0.933460 + 0.933460i 0 4.84613 + 4.84613i 0
641.2 0 −1.59352 − 0.660056i 0 0.212292 + 0.512517i 0 −1.69883 + 1.69883i 0 −0.0177021 − 0.0177021i 0
641.3 0 1.59352 + 0.660056i 0 0.212292 + 0.512517i 0 1.69883 − 1.69883i 0 −0.0177021 − 0.0177021i 0
641.4 0 2.90008 + 1.20125i 0 −1.21229 − 2.92673i 0 0.933460 − 0.933460i 0 4.84613 + 4.84613i 0
897.1 0 −0.942835 − 2.27621i 0 2.49877 + 1.03503i 0 −1.37128 − 1.37128i 0 −2.17085 + 2.17085i 0
897.2 0 −0.401639 − 0.969643i 0 −3.49877 − 1.44924i 0 −3.21904 − 3.21904i 0 1.34243 − 1.34243i 0
897.3 0 0.401639 + 0.969643i 0 −3.49877 − 1.44924i 0 3.21904 + 3.21904i 0 1.34243 − 1.34243i 0
897.4 0 0.942835 + 2.27621i 0 2.49877 + 1.03503i 0 1.37128 + 1.37128i 0 −2.17085 + 2.17085i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 129.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
32.g even 8 1 inner
32.h odd 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1024.2.g.c yes 16
4.b odd 2 1 inner 1024.2.g.c yes 16
8.b even 2 1 1024.2.g.h yes 16
8.d odd 2 1 1024.2.g.h yes 16
16.e even 4 1 1024.2.g.b ✓ 16
16.e even 4 1 1024.2.g.e yes 16
16.f odd 4 1 1024.2.g.b ✓ 16
16.f odd 4 1 1024.2.g.e yes 16
32.g even 8 1 1024.2.g.b ✓ 16
32.g even 8 1 inner 1024.2.g.c yes 16
32.g even 8 1 1024.2.g.e yes 16
32.g even 8 1 1024.2.g.h yes 16
32.h odd 8 1 1024.2.g.b ✓ 16
32.h odd 8 1 inner 1024.2.g.c yes 16
32.h odd 8 1 1024.2.g.e yes 16
32.h odd 8 1 1024.2.g.h yes 16
64.i even 16 1 4096.2.a.n 8
64.i even 16 1 4096.2.a.o 8
64.j odd 16 1 4096.2.a.n 8
64.j odd 16 1 4096.2.a.o 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1024.2.g.b ✓ 16 16.e even 4 1
1024.2.g.b ✓ 16 16.f odd 4 1
1024.2.g.b ✓ 16 32.g even 8 1
1024.2.g.b ✓ 16 32.h odd 8 1
1024.2.g.c yes 16 1.a even 1 1 trivial
1024.2.g.c yes 16 4.b odd 2 1 inner
1024.2.g.c yes 16 32.g even 8 1 inner
1024.2.g.c yes 16 32.h odd 8 1 inner
1024.2.g.e yes 16 16.e even 4 1
1024.2.g.e yes 16 16.f odd 4 1
1024.2.g.e yes 16 32.g even 8 1
1024.2.g.e yes 16 32.h odd 8 1
1024.2.g.h yes 16 8.b even 2 1
1024.2.g.h yes 16 8.d odd 2 1
1024.2.g.h yes 16 32.g even 8 1
1024.2.g.h yes 16 32.h odd 8 1
4096.2.a.n 8 64.i even 16 1
4096.2.a.n 8 64.j odd 16 1
4096.2.a.o 8 64.i even 16 1
4096.2.a.o 8 64.j odd 16 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1024, [\chi])\):

\( T_{3}^{16} - 8T_{3}^{14} + 32T_{3}^{12} + 272T_{3}^{10} + 2744T_{3}^{8} - 8288T_{3}^{6} + 15488T_{3}^{4} + 34496T_{3}^{2} + 38416 \) Copy content Toggle raw display
\( T_{5}^{8} + 4T_{5}^{7} - 32T_{5}^{5} - 64T_{5}^{4} + 72T_{5}^{3} + 1008T_{5}^{2} - 432T_{5} + 324 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} - 8 T^{14} + \cdots + 38416 \) Copy content Toggle raw display
$5$ \( (T^{8} + 4 T^{7} + \cdots + 324)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} + 480 T^{12} + \cdots + 614656 \) Copy content Toggle raw display
$11$ \( T^{16} + 8 T^{14} + \cdots + 104976 \) Copy content Toggle raw display
$13$ \( (T^{8} + 12 T^{7} + \cdots + 56644)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + 72 T^{6} + \cdots + 5184)^{2} \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 35477982736 \) Copy content Toggle raw display
$23$ \( T^{16} + 2784 T^{12} + \cdots + 1679616 \) Copy content Toggle raw display
$29$ \( (T^{8} - 12 T^{7} + \cdots + 324)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 16 T^{2} + 32)^{4} \) Copy content Toggle raw display
$37$ \( (T^{8} + 20 T^{7} + \cdots + 1110916)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} - 8 T^{7} + \cdots + 104976)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + 88 T^{14} + \cdots + 45212176 \) Copy content Toggle raw display
$47$ \( (T^{8} + 128 T^{6} + \cdots + 20736)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} - 28 T^{7} + \cdots + 93636)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 2981133747216 \) Copy content Toggle raw display
$61$ \( (T^{8} + 4 T^{7} + \cdots + 3740356)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 688747536 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 49\!\cdots\!36 \) Copy content Toggle raw display
$73$ \( (T^{8} - 16 T^{7} + \cdots + 169744)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + 512 T^{6} + \cdots + 3268864)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 252047376 \) Copy content Toggle raw display
$89$ \( (T^{8} + 16 T^{7} + \cdots + 3111696)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 4 T^{3} - 36 T^{2} + \cdots + 56)^{4} \) Copy content Toggle raw display
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