↑

Properties

Label 1792.3.d.j
Level $1792$
Weight $3$
Character orbit 1792.d
Analytic conductor $48.828$
Analytic rank $0$
Dimension $16$
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] = [1792,3,Mod(1023,1792)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1792, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1792.1023");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1792 = 2^{8} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1792.d (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: \(48.8284633734\)
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: \( x^{16} + 5x^{14} + 48x^{12} + 180x^{10} + 1056x^{8} + 2880x^{6} + 12288x^{4} + 20480x^{2} + 65536 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{38} \)
Twist minimal: no (minimal twist has level 56)
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} + \beta_{4} q^{5} - \beta_{8} q^{7} + ( - \beta_{13} - 6) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{3} + \beta_{4} q^{5} - \beta_{8} q^{7} + ( - \beta_{13} - 6) q^{9} + ( - \beta_{7} + \beta_{3}) q^{11} + (\beta_{9} - \beta_{5} - \beta_{4} + \beta_1) q^{13} + (\beta_{11} + \beta_{10} + \cdots + \beta_{6}) q^{15}+ \cdots + (7 \beta_{7} + 5 \beta_{3} - 12 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 96 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 96 q^{9} - 160 q^{17} + 32 q^{25} + 64 q^{33} - 256 q^{41} - 112 q^{49} - 112 q^{57} - 144 q^{65} + 224 q^{73} + 96 q^{81} + 1024 q^{89} + 128 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 5x^{14} + 48x^{12} + 180x^{10} + 1056x^{8} + 2880x^{6} + 12288x^{4} + 20480x^{2} + 65536 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{13} - 5\nu^{11} - 48\nu^{9} - 180\nu^{7} - 1056\nu^{5} - 1856\nu^{3} - 7168\nu ) / 1024 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -5\nu^{15} - 81\nu^{13} - 456\nu^{11} - 3012\nu^{9} - 9984\nu^{7} - 40512\nu^{5} - 96768\nu^{3} - 323584\nu ) / 73728 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -7\nu^{15} - 27\nu^{13} + 24\nu^{11} - 300\nu^{9} + 192\nu^{7} - 3264\nu^{5} + 4608\nu^{3} - 32768\nu ) / 73728 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{15} - 13\nu^{13} - 56\nu^{11} - 276\nu^{9} - 832\nu^{7} - 4032\nu^{5} - 5120\nu^{3} - 24576\nu ) / 8192 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{15} + 5\nu^{13} - 6\nu^{11} + 36\nu^{9} + 136\nu^{7} + 480\nu^{5} + 3072\nu^{3} - 6144\nu ) / 4096 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 5\nu^{14} - 27\nu^{12} + 60\nu^{10} - 2028\nu^{8} - 3120\nu^{6} - 72960\nu^{4} - 138240\nu^{2} - 745472 ) / 36864 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -5\nu^{15} - 21\nu^{13} - 156\nu^{11} - 516\nu^{9} - 2640\nu^{7} - 3264\nu^{5} - 5376\nu^{3} + 38912\nu ) / 12288 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 23 \nu^{14} - 135 \nu^{12} - 996 \nu^{10} - 4380 \nu^{8} - 16176 \nu^{6} - 53760 \nu^{4} + \cdots - 286720 ) / 73728 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( \nu^{15} + 4\nu^{13} + 39\nu^{11} + 128\nu^{9} + 700\nu^{7} + 1808\nu^{5} + 7296\nu^{3} + 3072\nu ) / 2048 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 17\nu^{14} + 9\nu^{12} + 420\nu^{10} + 420\nu^{8} + 8688\nu^{6} - 2688\nu^{4} + 129024\nu^{2} - 57344 ) / 36864 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 19\nu^{14} - 93\nu^{12} - 300\nu^{10} - 2292\nu^{8} - 7824\nu^{6} - 38400\nu^{4} - 125952\nu^{2} - 237568 ) / 24576 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 95\nu^{15} + 99\nu^{13} + 2616\nu^{11} + 780\nu^{9} + 48000\nu^{7} + 192\nu^{5} + 373248\nu^{3} + 212992\nu ) / 73728 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( -\nu^{14} - \nu^{12} - 28\nu^{10} + 12\nu^{8} - 336\nu^{6} + 320\nu^{4} - 1792\nu^{2} + 9216 ) / 1024 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( \nu^{14} + 5\nu^{12} - 16\nu^{10} - 140\nu^{8} - 992\nu^{6} - 3520\nu^{4} - 10240\nu^{2} - 30720 ) / 1024 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( -3\nu^{14} - 23\nu^{12} - 120\nu^{10} - 604\nu^{8} - 2560\nu^{6} - 8640\nu^{4} - 18944\nu^{2} - 49152 ) / 2048 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{12} - 3\beta_{9} + 2\beta_{7} - \beta_{5} - 2\beta_{4} - \beta_{3} - 6\beta_{2} + \beta_1 ) / 32 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{15} + \beta_{14} + 2\beta_{13} - 2\beta_{11} + 10\beta_{10} - 4\beta_{8} - 3\beta_{6} - 20 ) / 32 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{12} + 15\beta_{9} + 14\beta_{7} + 5\beta_{5} + 10\beta_{4} - 15\beta_{3} - 10\beta_{2} + 11\beta_1 ) / 32 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -2\beta_{15} - \beta_{14} - 18\beta_{13} + 2\beta_{11} - 26\beta_{10} + 20\beta_{8} - 21\beta_{6} - 284 ) / 32 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -7\beta_{12} + 25\beta_{9} + 2\beta_{7} - 13\beta_{5} - 58\beta_{4} + 23\beta_{3} + 154\beta_{2} - 67\beta_1 ) / 32 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -78\beta_{15} - 23\beta_{14} + 2\beta_{13} + 14\beta_{11} + 10\beta_{10} + 364\beta_{8} + 61\beta_{6} + 220 ) / 32 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 63 \beta_{12} - 225 \beta_{9} - 210 \beta_{7} + 117 \beta_{5} + 266 \beta_{4} - 15 \beta_{3} + \cdots - 229 \beta_1 ) / 32 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 126 \beta_{15} - 17 \beta_{14} + 494 \beta_{13} + 130 \beta_{11} + 230 \beta_{10} - 1420 \beta_{8} + \cdots - 764 ) / 32 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 215 \beta_{12} + 73 \beta_{9} - 350 \beta_{7} - 253 \beta_{5} + 1382 \beta_{4} - 473 \beta_{3} + \cdots + 749 \beta_1 ) / 32 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 1234 \beta_{15} - 7 \beta_{14} - 1566 \beta_{13} - 722 \beta_{11} - 1750 \beta_{10} - 5396 \beta_{8} + \cdots + 7900 ) / 32 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 209 \beta_{12} + 1551 \beta_{9} + 1678 \beta_{7} - 2107 \beta_{5} - 7382 \beta_{4} + \cdots + 2507 \beta_1 ) / 32 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 3874 \beta_{15} + 2143 \beta_{14} - 946 \beta_{13} - 3550 \beta_{11} + 710 \beta_{10} + 20020 \beta_{8} + \cdots + 20164 ) / 32 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 2105 \beta_{12} - 3495 \beta_{9} + 3778 \beta_{7} + 13235 \beta_{5} - 20282 \beta_{4} + \cdots + 3133 \beta_1 ) / 32 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 7182 \beta_{15} + 3465 \beta_{14} + 7938 \beta_{13} + 24846 \beta_{11} + 21450 \beta_{10} + \cdots - 84580 ) / 32 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 31 \beta_{12} + 21759 \beta_{9} - 658 \beta_{7} - 30187 \beta_{5} + 43978 \beta_{4} - 216559 \beta_{3} + \cdots - 8069 \beta_1 ) / 32 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1792\mathbb{Z}\right)^\times\).

\(n\) \(1023\) \(1025\) \(1541\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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
−1.45617 + 1.37098i
1.45617 + 1.37098i
1.09337 + 1.67467i
−1.09337 + 1.67467i
1.69931 − 1.05468i
−1.69931 − 1.05468i
0.739226 − 1.85837i
−0.739226 − 1.85837i
0.739226 + 1.85837i
−0.739226 + 1.85837i
1.69931 + 1.05468i
−1.69931 + 1.05468i
1.09337 − 1.67467i
−1.09337 − 1.67467i
−1.45617 − 1.37098i
1.45617 − 1.37098i
0 − 5.22363i 0 −6.26788 0 − 2.64575i 0 −18.2863 0
1023.2 0 − 5.22363i 0 6.26788 0 2.64575i 0 −18.2863 0
1023.3 0 − 4.56747i 0 −5.73252 0 2.64575i 0 −11.8618 0
1023.4 0 − 4.56747i 0 5.73252 0 − 2.64575i 0 −11.8618 0
1023.5 0 − 3.44128i 0 −4.88287 0 2.64575i 0 −2.84239 0
1023.6 0 − 3.44128i 0 4.88287 0 − 2.64575i 0 −2.84239 0
1023.7 0 − 0.0974366i 0 −3.46547 0 2.64575i 0 8.99051 0
1023.8 0 − 0.0974366i 0 3.46547 0 − 2.64575i 0 8.99051 0
1023.9 0 0.0974366i 0 −3.46547 0 − 2.64575i 0 8.99051 0
1023.10 0 0.0974366i 0 3.46547 0 2.64575i 0 8.99051 0
1023.11 0 3.44128i 0 −4.88287 0 − 2.64575i 0 −2.84239 0
1023.12 0 3.44128i 0 4.88287 0 2.64575i 0 −2.84239 0
1023.13 0 4.56747i 0 −5.73252 0 − 2.64575i 0 −11.8618 0
1023.14 0 4.56747i 0 5.73252 0 2.64575i 0 −11.8618 0
1023.15 0 5.22363i 0 −6.26788 0 2.64575i 0 −18.2863 0
1023.16 0 5.22363i 0 6.26788 0 − 2.64575i 0 −18.2863 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1023.16
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 1792.3.d.j 16
4.b odd 2 1 inner 1792.3.d.j 16
8.b even 2 1 inner 1792.3.d.j 16
8.d odd 2 1 inner 1792.3.d.j 16
16.e even 4 1 56.3.g.b ✓ 8
16.e even 4 1 224.3.g.b 8
16.f odd 4 1 56.3.g.b ✓ 8
16.f odd 4 1 224.3.g.b 8
48.i odd 4 1 504.3.g.b 8
48.i odd 4 1 2016.3.g.b 8
48.k even 4 1 504.3.g.b 8
48.k even 4 1 2016.3.g.b 8
112.j even 4 1 392.3.g.m 8
112.j even 4 1 1568.3.g.m 8
112.l odd 4 1 392.3.g.m 8
112.l odd 4 1 1568.3.g.m 8
112.u odd 12 2 392.3.k.o 16
112.v even 12 2 392.3.k.n 16
112.w even 12 2 392.3.k.o 16
112.x odd 12 2 392.3.k.n 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.3.g.b ✓ 8 16.e even 4 1
56.3.g.b ✓ 8 16.f odd 4 1
224.3.g.b 8 16.e even 4 1
224.3.g.b 8 16.f odd 4 1
392.3.g.m 8 112.j even 4 1
392.3.g.m 8 112.l odd 4 1
392.3.k.n 16 112.v even 12 2
392.3.k.n 16 112.x odd 12 2
392.3.k.o 16 112.u odd 12 2
392.3.k.o 16 112.w even 12 2
504.3.g.b 8 48.i odd 4 1
504.3.g.b 8 48.k even 4 1
1568.3.g.m 8 112.j even 4 1
1568.3.g.m 8 112.l odd 4 1
1792.3.d.j 16 1.a even 1 1 trivial
1792.3.d.j 16 4.b odd 2 1 inner
1792.3.d.j 16 8.b even 2 1 inner
1792.3.d.j 16 8.d odd 2 1 inner
2016.3.g.b 8 48.i odd 4 1
2016.3.g.b 8 48.k even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1792, [\chi])\):

\( T_{3}^{8} + 60T_{3}^{6} + 1140T_{3}^{4} + 6752T_{3}^{2} + 64 \) Copy content Toggle raw display
\( T_{5}^{8} - 108T_{5}^{6} + 4164T_{5}^{4} - 66944T_{5}^{2} + 369664 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( (T^{8} + 60 T^{6} + \cdots + 64)^{2} \) Copy content Toggle raw display
$5$ \( (T^{8} - 108 T^{6} + \cdots + 369664)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 7)^{8} \) Copy content Toggle raw display
$11$ \( (T^{8} + 568 T^{6} + \cdots + 746496)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} - 908 T^{6} + \cdots + 133448704)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 40 T^{3} + \cdots - 752)^{4} \) Copy content Toggle raw display
$19$ \( (T^{8} + 252 T^{6} + \cdots + 1658944)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + 2488 T^{6} + \cdots + 15607005184)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} - 3344 T^{6} + \cdots + 13389266944)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + 3744 T^{6} + \cdots + 61917364224)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} - 7440 T^{6} + \cdots + 554696704)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 64 T^{3} + \cdots + 837776)^{4} \) Copy content Toggle raw display
$43$ \( (T^{8} + 5432 T^{6} + \cdots + 47503074304)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots + 7542537191424)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} - 3552 T^{6} + \cdots + 16777216)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots + 126626768156736)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots + 3223777158144)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + 14568 T^{6} + \cdots + 626959908864)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 221437256269824)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} - 56 T^{3} + \cdots - 1726704)^{4} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 89369947930624)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 10117946680384)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} - 256 T^{3} + \cdots - 618736)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} - 32 T^{3} + \cdots + 9539216)^{4} \) Copy content Toggle raw display
show more
show less