↑

Properties

Label 1024.3.c.j
Level $1024$
Weight $3$
Character orbit 1024.c
Analytic conductor $27.902$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1024,3,Mod(1023,1024)]
 
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([1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1024.1023");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1024 = 2^{10} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1024.c (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: \(27.9019790705\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: 12.0.653473922154496.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 4x^{10} + 13x^{8} - 28x^{6} + 52x^{4} - 64x^{2} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{30} \)
Twist minimal: no (minimal twist has level 16)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{6} q^{3} + \beta_1 q^{5} - \beta_{2} q^{7} + (\beta_{7} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{6} q^{3} + \beta_1 q^{5} - \beta_{2} q^{7} + (\beta_{7} - 1) q^{9} - \beta_{9} q^{11} + \beta_{11} q^{13} + (\beta_{5} + \beta_{2}) q^{15} + (\beta_{4} + 1) q^{17} + (\beta_{6} - \beta_{3}) q^{19} + (\beta_{11} + \beta_{8} + \beta_1) q^{21} + ( - \beta_{10} - \beta_{2}) q^{23} + (\beta_{7} - \beta_{4} - 2) q^{25} + ( - 2 \beta_{9} + \beta_{3}) q^{27} + (2 \beta_{11} + \beta_{8} + \beta_1) q^{29} + ( - \beta_{10} + 2 \beta_{5} + 2 \beta_{2}) q^{31} + ( - \beta_{7} + 2 \beta_{4}) q^{33} + (\beta_{9} - \beta_{6} - 3 \beta_{3}) q^{35} + (3 \beta_{11} + 2 \beta_{8} - 2 \beta_1) q^{37} + ( - 3 \beta_{10} + \beta_{5} + 3 \beta_{2}) q^{39} + ( - 3 \beta_{7} - 3 \beta_{4} - 1) q^{41} + ( - \beta_{6} + 4 \beta_{3}) q^{43} + ( - \beta_{11} + 3 \beta_{8} + 4 \beta_1) q^{45} + ( - 4 \beta_{10} - \beta_{5} - 2 \beta_{2}) q^{47} + ( - 4 \beta_{7} - 2 \beta_{4} + 7) q^{49} + (5 \beta_{9} + 5 \beta_{6} - 2 \beta_{3}) q^{51} + (\beta_{11} - 3 \beta_{8} + 2 \beta_1) q^{53} + ( - \beta_{10} + 5 \beta_{5} + \beta_{2}) q^{55} + (\beta_{7} - 6) q^{57} + ( - 3 \beta_{9} - 6 \beta_{6} + 5 \beta_{3}) q^{59} + ( - \beta_{11} - \beta_{8} + 4 \beta_1) q^{61} + ( - 4 \beta_{10} + \beta_{5} - 5 \beta_{2}) q^{63} + ( - 3 \beta_{7} - 3 \beta_{4} + 1) q^{65} + (9 \beta_{9} + 4 \beta_{6} - 2 \beta_{3}) q^{67} + ( - 3 \beta_{11} - 5 \beta_{8} + \beta_1) q^{69} + (\beta_{10} + 8 \beta_{5} + \beta_{2}) q^{71} + (5 \beta_{7} + 6 \beta_{4} - 6) q^{73} + ( - 7 \beta_{9} - 14 \beta_{6} + 3 \beta_{3}) q^{75} + (\beta_{11} - 9 \beta_{8} - 3 \beta_1) q^{77} + (\beta_{10} + 9 \beta_{5} - 4 \beta_{2}) q^{79} + (7 \beta_{7} + 4 \beta_{4} - 13) q^{81} + (7 \beta_{9} - 2 \beta_{6} - 7 \beta_{3}) q^{83} + ( - 3 \beta_{11} + 8 \beta_{8} - 7 \beta_1) q^{85} + ( - 7 \beta_{10} + 2 \beta_{5} + 7 \beta_{2}) q^{87} + ( - 7 \beta_{7} + 2 \beta_{4} + 14) q^{89} + ( - \beta_{9} - 15 \beta_{6} + 5 \beta_{3}) q^{91} + ( - 6 \beta_{11} - 10 \beta_1) q^{93} + ( - 2 \beta_{10} + 2 \beta_{5} - 7 \beta_{2}) q^{95} + (12 \beta_{7} - \beta_{4} - 1) q^{97} + (3 \beta_{9} + 16 \beta_{6} - 5 \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 12 q^{9} + 8 q^{17} - 20 q^{25} - 8 q^{33} + 92 q^{49} - 72 q^{57} + 24 q^{65} - 96 q^{73} - 172 q^{81} + 160 q^{89} - 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 4x^{10} + 13x^{8} - 28x^{6} + 52x^{4} - 64x^{2} + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -3\nu^{11} - 2\nu^{9} - 3\nu^{7} + 22\nu^{5} - 72\nu^{3} + 192\nu ) / 64 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{10} + 22\nu^{8} - 25\nu^{6} + 94\nu^{4} - 80\nu^{2} + 160 ) / 32 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{11} + 4\nu^{9} + 5\nu^{7} + 28\nu^{5} + 12\nu^{3} + 80\nu ) / 16 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{8} - 4\nu^{6} + 9\nu^{4} - 12\nu^{2} + 17 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -3\nu^{10} + 2\nu^{8} - 11\nu^{6} - 38\nu^{4} + 80\nu^{2} - 160 ) / 16 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3\nu^{11} - 12\nu^{9} + 31\nu^{7} - 68\nu^{5} + 116\nu^{3} - 80\nu ) / 32 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{10} + 2\nu^{8} - 5\nu^{6} + 10\nu^{4} - 12\nu^{2} + 8 ) / 2 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -3\nu^{11} + 6\nu^{9} - 19\nu^{7} + 30\nu^{5} - 24\nu^{3} ) / 16 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 7\nu^{11} - 12\nu^{9} + 19\nu^{7} + 28\nu^{5} + 4\nu^{3} + 112\nu ) / 32 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 9\nu^{10} - 38\nu^{8} + 97\nu^{6} - 174\nu^{4} + 336\nu^{2} - 288 ) / 16 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 13\nu^{11} - 82\nu^{9} + 237\nu^{7} - 442\nu^{5} + 664\nu^{3} - 960\nu ) / 64 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -2\beta_{11} - \beta_{9} - \beta_{8} + 7\beta_{6} + \beta_{3} + 6\beta_1 ) / 32 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{10} + 2\beta_{7} + 3\beta_{5} + 2\beta_{4} + 4\beta_{2} + 22 ) / 32 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{11} + \beta_{9} + 9\beta_{8} + 17\beta_{6} + 3\beta_{3} - 2\beta_1 ) / 32 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 3\beta_{10} + 4\beta_{7} - 2\beta_{5} + 2\beta_{4} + 2\beta_{2} - 26 ) / 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 6\beta_{11} + 11\beta_{9} + 15\beta_{8} - 13\beta_{6} + 5\beta_{3} - 2\beta_1 ) / 32 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( \beta_{10} + 6\beta_{7} - 15\beta_{5} - 10\beta_{4} + 12\beta_{2} - 46 ) / 32 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 18\beta_{11} - 9\beta_{9} - 17\beta_{8} - 57\beta_{6} + 21\beta_{3} + 18\beta_1 ) / 32 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -7\beta_{10} - 12\beta_{7} + 6\beta_{5} - 10\beta_{4} + 30\beta_{2} + 2 ) / 16 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( -6\beta_{11} - 59\beta_{9} - 63\beta_{8} - 35\beta_{6} + 43\beta_{3} - 62\beta_1 ) / 32 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( -9\beta_{10} - 86\beta_{7} + 23\beta_{5} + 26\beta_{4} + 52\beta_{2} - 290 ) / 32 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( -50\beta_{11} + 41\beta_{9} - 111\beta_{8} + 25\beta_{6} - 21\beta_{3} - 242\beta_1 ) / 32 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1023.1
−1.16947 − 0.795191i
1.16947 − 0.795191i
−0.892524 − 1.09700i
0.892524 − 1.09700i
−1.35489 − 0.405301i
1.35489 − 0.405301i
−1.35489 + 0.405301i
1.35489 + 0.405301i
−0.892524 + 1.09700i
0.892524 + 1.09700i
−1.16947 + 0.795191i
1.16947 + 0.795191i
0 − 4.59498i 0 −0.0829198 0 4.61555i 0 −12.1138 0
1023.2 0 − 4.59498i 0 0.0829198 0 − 4.61555i 0 −12.1138 0
1023.3 0 − 2.97377i 0 −6.54387 0 − 3.04888i 0 0.156674 0
1023.4 0 − 2.97377i 0 6.54387 0 3.04888i 0 0.156674 0
1023.5 0 − 0.206992i 0 −5.21257 0 − 9.66442i 0 8.95715 0
1023.6 0 − 0.206992i 0 5.21257 0 9.66442i 0 8.95715 0
1023.7 0 0.206992i 0 −5.21257 0 9.66442i 0 8.95715 0
1023.8 0 0.206992i 0 5.21257 0 − 9.66442i 0 8.95715 0
1023.9 0 2.97377i 0 −6.54387 0 3.04888i 0 0.156674 0
1023.10 0 2.97377i 0 6.54387 0 − 3.04888i 0 0.156674 0
1023.11 0 4.59498i 0 −0.0829198 0 − 4.61555i 0 −12.1138 0
1023.12 0 4.59498i 0 0.0829198 0 4.61555i 0 −12.1138 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1023.12
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 1024.3.c.j 12
4.b odd 2 1 inner 1024.3.c.j 12
8.b even 2 1 inner 1024.3.c.j 12
8.d odd 2 1 inner 1024.3.c.j 12
16.e even 4 2 1024.3.d.k 12
16.f odd 4 2 1024.3.d.k 12
32.g even 8 1 16.3.f.a ✓ 6
32.g even 8 1 64.3.f.a 6
32.g even 8 1 128.3.f.a 6
32.g even 8 1 128.3.f.b 6
32.h odd 8 1 16.3.f.a ✓ 6
32.h odd 8 1 64.3.f.a 6
32.h odd 8 1 128.3.f.a 6
32.h odd 8 1 128.3.f.b 6
96.o even 8 1 144.3.m.a 6
96.o even 8 1 576.3.m.a 6
96.o even 8 1 1152.3.m.a 6
96.o even 8 1 1152.3.m.b 6
96.p odd 8 1 144.3.m.a 6
96.p odd 8 1 576.3.m.a 6
96.p odd 8 1 1152.3.m.a 6
96.p odd 8 1 1152.3.m.b 6
160.u even 8 1 400.3.k.d 6
160.v odd 8 1 400.3.k.c 6
160.y odd 8 1 400.3.r.c 6
160.z even 8 1 400.3.r.c 6
160.ba even 8 1 400.3.k.c 6
160.bb odd 8 1 400.3.k.d 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
16.3.f.a ✓ 6 32.g even 8 1
16.3.f.a ✓ 6 32.h odd 8 1
64.3.f.a 6 32.g even 8 1
64.3.f.a 6 32.h odd 8 1
128.3.f.a 6 32.g even 8 1
128.3.f.a 6 32.h odd 8 1
128.3.f.b 6 32.g even 8 1
128.3.f.b 6 32.h odd 8 1
144.3.m.a 6 96.o even 8 1
144.3.m.a 6 96.p odd 8 1
400.3.k.c 6 160.v odd 8 1
400.3.k.c 6 160.ba even 8 1
400.3.k.d 6 160.u even 8 1
400.3.k.d 6 160.bb odd 8 1
400.3.r.c 6 160.y odd 8 1
400.3.r.c 6 160.z even 8 1
576.3.m.a 6 96.o even 8 1
576.3.m.a 6 96.p odd 8 1
1024.3.c.j 12 1.a even 1 1 trivial
1024.3.c.j 12 4.b odd 2 1 inner
1024.3.c.j 12 8.b even 2 1 inner
1024.3.c.j 12 8.d odd 2 1 inner
1024.3.d.k 12 16.e even 4 2
1024.3.d.k 12 16.f odd 4 2
1152.3.m.a 6 96.o even 8 1
1152.3.m.a 6 96.p odd 8 1
1152.3.m.b 6 96.o even 8 1
1152.3.m.b 6 96.p odd 8 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1024, [\chi])\):

\( T_{3}^{6} + 30T_{3}^{4} + 188T_{3}^{2} + 8 \) Copy content Toggle raw display
\( T_{5}^{6} - 70T_{5}^{4} + 1164T_{5}^{2} - 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T^{6} + 30 T^{4} + 188 T^{2} + 8)^{2} \) Copy content Toggle raw display
$5$ \( (T^{6} - 70 T^{4} + 1164 T^{2} - 8)^{2} \) Copy content Toggle raw display
$7$ \( (T^{6} + 124 T^{4} + \cdots + 18496)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} + 286 T^{4} + \cdots + 587528)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} - 390 T^{4} + \cdots - 1286408)^{2} \) Copy content Toggle raw display
$17$ \( (T^{3} - 2 T^{2} + \cdots + 1544)^{4} \) Copy content Toggle raw display
$19$ \( (T^{6} + 382 T^{4} + \cdots + 13448)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} + 572 T^{4} + \cdots + 937024)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} - 1702 T^{4} + \cdots - 19046792)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + 1920 T^{4} + \cdots + 16777216)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} - 4070 T^{4} + \cdots - 42632)^{2} \) Copy content Toggle raw display
$41$ \( (T^{3} - 2496 T - 8192)^{4} \) Copy content Toggle raw display
$43$ \( (T^{6} + 5950 T^{4} + \cdots + 42632)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} + 8576 T^{4} + \cdots + 6056574976)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} - 1702 T^{4} + \cdots - 783752)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + 8062 T^{4} + \cdots + 8410007432)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} - 1606 T^{4} + \cdots - 151449608)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} + 19774 T^{4} + \cdots + 87233303432)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + 23996 T^{4} + \cdots + 153557394496)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} + 24 T^{2} + \cdots + 85504)^{4} \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots + 1550483193856)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + 19230 T^{4} + \cdots + 105636303368)^{2} \) Copy content Toggle raw display
$89$ \( (T^{3} - 40 T^{2} + \cdots - 160256)^{4} \) Copy content Toggle raw display
$97$ \( (T^{3} + 2 T^{2} + \cdots + 519928)^{4} \) Copy content Toggle raw display
show more
show less