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Properties

Label 1792.3.d.h
Level $1792$
Weight $3$
Character orbit 1792.d
Analytic conductor $48.828$
Analytic rank $0$
Dimension $12$
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] = [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: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 6 x^{11} + 27 x^{10} - 76 x^{9} + 103 x^{8} - 178 x^{7} + 747 x^{6} - 1336 x^{5} + 260 x^{4} + \cdots + 1764 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{20} \)
Twist minimal: no (minimal twist has level 896)
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.

\(f(q)\) \(=\) \( q - \beta_{3} q^{3} + ( - \beta_{2} - 1) q^{5} - \beta_1 q^{7} + (\beta_{4} - 4) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{3} + ( - \beta_{2} - 1) q^{5} - \beta_1 q^{7} + (\beta_{4} - 4) q^{9} - \beta_{10} q^{11} + ( - \beta_{6} - 7) q^{13} + ( - \beta_{11} - 2 \beta_{10} + \cdots - 3 \beta_1) q^{15}+ \cdots + (2 \beta_{11} + 3 \beta_{10} + \cdots + 32 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 16 q^{5} - 52 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 16 q^{5} - 52 q^{9} - 80 q^{13} + 40 q^{17} + 84 q^{25} - 80 q^{29} - 64 q^{33} + 48 q^{37} - 88 q^{41} + 240 q^{45} - 84 q^{49} - 16 q^{53} + 224 q^{57} + 112 q^{61} - 64 q^{65} - 256 q^{69} + 24 q^{73} + 300 q^{81} - 224 q^{85} - 232 q^{89} + 384 q^{93} - 408 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 6 x^{11} + 27 x^{10} - 76 x^{9} + 103 x^{8} - 178 x^{7} + 747 x^{6} - 1336 x^{5} + 260 x^{4} + \cdots + 1764 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 114865 \nu^{11} - 646756 \nu^{10} + 3416824 \nu^{9} - 10574996 \nu^{8} + 21322983 \nu^{7} + \cdots + 381983616 ) / 122360112 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 11824619 \nu^{11} - 51886949 \nu^{10} + 177589280 \nu^{9} - 350051500 \nu^{8} + \cdots - 52097937948 ) / 12332761056 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 671885815 \nu^{11} - 5473101685 \nu^{10} + 26239976224 \nu^{9} - 90144395180 \nu^{8} + \cdots + 1652498616132 ) / 530308725408 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3480645 \nu^{11} - 6213617 \nu^{10} + 29757640 \nu^{9} + 22828908 \nu^{8} - 249867041 \nu^{7} + \cdots + 6362124804 ) / 2055460176 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 205567540 \nu^{11} - 1942413475 \nu^{10} + 10609598824 \nu^{9} - 39155308520 \nu^{8} + \cdots + 1593644414796 ) / 88384787568 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1251925 \nu^{11} - 6838879 \nu^{10} + 32257648 \nu^{9} - 85369700 \nu^{8} + 118358079 \nu^{7} + \cdots + 2069130924 ) / 373720032 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 10698137 \nu^{11} + 64045442 \nu^{10} - 257530592 \nu^{9} + 736644976 \nu^{8} + \cdots + 4452051744 ) / 3083190264 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 44478491 \nu^{11} - 108010817 \nu^{10} + 340469360 \nu^{9} + 190301300 \nu^{8} + \cdots + 30947848596 ) / 12332761056 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 12210400 \nu^{11} - 72195941 \nu^{10} + 335805844 \nu^{9} - 966137476 \nu^{8} + \cdots + 4923626036 ) / 2455132988 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 1295104511 \nu^{11} - 8852984423 \nu^{10} + 40362545036 \nu^{9} - 124071300796 \nu^{8} + \cdots + 437637841452 ) / 132577181352 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 5356516987 \nu^{11} + 26954533207 \nu^{10} - 118143417568 \nu^{9} + 291754123004 \nu^{8} + \cdots + 4960180117620 ) / 530308725408 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{11} - \beta_{9} - \beta_{6} + \beta_{5} - \beta_{3} - \beta_{2} - 2\beta _1 + 4 ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 2 \beta_{11} - 2 \beta_{10} + \beta_{9} + 2 \beta_{8} + 2 \beta_{5} - 4 \beta_{4} + 2 \beta_{3} + \cdots - 12 ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 10 \beta_{11} + 4 \beta_{10} + 17 \beta_{9} + 6 \beta_{8} + 2 \beta_{7} + 2 \beta_{6} - 10 \beta_{5} + \cdots - 30 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 40 \beta_{11} + 16 \beta_{10} + 60 \beta_{9} - 25 \beta_{8} - 14 \beta_{7} - 16 \beta_{6} - 40 \beta_{5} + \cdots + 185 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 92 \beta_{11} - 54 \beta_{10} - 96 \beta_{9} - 129 \beta_{8} - 40 \beta_{7} - 64 \beta_{6} + \cdots + 1129 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 752 \beta_{11} - 302 \beta_{10} - 1049 \beta_{9} + 164 \beta_{8} + 52 \beta_{7} + 100 \beta_{6} + \cdots - 1356 ) / 8 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 234 \beta_{11} + 86 \beta_{10} + 356 \beta_{9} + 2297 \beta_{8} + 604 \beta_{7} + 1186 \beta_{6} + \cdots - 20621 ) / 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 12448 \beta_{11} + 5086 \beta_{10} + 17395 \beta_{9} + 1422 \beta_{8} + 548 \beta_{7} + 896 \beta_{6} + \cdots - 12562 ) / 8 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 18158 \beta_{11} + 7854 \beta_{10} + 24672 \beta_{9} - 33677 \beta_{8} - 8952 \beta_{7} - 16658 \beta_{6} + \cdots + 288433 ) / 8 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 162492 \beta_{11} - 68298 \beta_{10} - 223739 \beta_{9} - 79322 \beta_{8} - 21544 \beta_{7} + \cdots + 672230 ) / 8 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 555702 \beta_{11} - 233022 \beta_{10} - 766178 \beta_{9} + 391195 \beta_{8} + 111368 \beta_{7} + \cdots - 3383359 ) / 8 \) 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
−1.42237 − 1.77770i
2.27591 − 0.676730i
−0.887431 + 0.147070i
1.54137 − 0.974703i
0.846062 + 3.85886i
0.646457 − 1.00963i
0.646457 + 1.00963i
0.846062 − 3.85886i
1.54137 + 0.974703i
−0.887431 − 0.147070i
2.27591 + 0.676730i
−1.42237 + 1.77770i
0 − 5.43729i 0 −8.76974 0 2.64575i 0 −20.5641 0
1023.2 0 − 5.02304i 0 −1.82758 0 − 2.64575i 0 −16.2310 0
1023.3 0 − 3.92425i 0 4.15763 0 − 2.64575i 0 −6.39971 0
1023.4 0 − 2.94543i 0 6.49853 0 2.64575i 0 0.324415 0
1023.5 0 − 0.978812i 0 −7.17285 0 2.64575i 0 8.04193 0
1023.6 0 − 0.414244i 0 −0.885998 0 − 2.64575i 0 8.82840 0
1023.7 0 0.414244i 0 −0.885998 0 2.64575i 0 8.82840 0
1023.8 0 0.978812i 0 −7.17285 0 − 2.64575i 0 8.04193 0
1023.9 0 2.94543i 0 6.49853 0 − 2.64575i 0 0.324415 0
1023.10 0 3.92425i 0 4.15763 0 2.64575i 0 −6.39971 0
1023.11 0 5.02304i 0 −1.82758 0 2.64575i 0 −16.2310 0
1023.12 0 5.43729i 0 −8.76974 0 − 2.64575i 0 −20.5641 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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1792.3.d.h 12
4.b odd 2 1 inner 1792.3.d.h 12
8.b even 2 1 1792.3.d.i 12
8.d odd 2 1 1792.3.d.i 12
16.e even 4 2 896.3.g.e ✓ 24
16.f odd 4 2 896.3.g.e ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
896.3.g.e ✓ 24 16.e even 4 2
896.3.g.e ✓ 24 16.f odd 4 2
1792.3.d.h 12 1.a even 1 1 trivial
1792.3.d.h 12 4.b odd 2 1 inner
1792.3.d.i 12 8.b even 2 1
1792.3.d.i 12 8.d odd 2 1

Hecke kernels

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

\( T_{3}^{12} + 80T_{3}^{10} + 2288T_{3}^{8} + 27776T_{3}^{6} + 128576T_{3}^{4} + 116736T_{3}^{2} + 16384 \) Copy content Toggle raw display
\( T_{5}^{6} + 8T_{5}^{5} - 64T_{5}^{4} - 448T_{5}^{3} + 920T_{5}^{2} + 4224T_{5} + 2752 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + 80 T^{10} + \cdots + 16384 \) Copy content Toggle raw display
$5$ \( (T^{6} + 8 T^{5} + \cdots + 2752)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 7)^{6} \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 25192038400 \) Copy content Toggle raw display
$13$ \( (T^{6} + 40 T^{5} + \cdots + 194240)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} - 20 T^{5} + \cdots - 38651840)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 590684416000000 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 1560700518400 \) Copy content Toggle raw display
$29$ \( (T^{6} + 40 T^{5} + \cdots + 77158400)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 324806901760000 \) Copy content Toggle raw display
$37$ \( (T^{6} - 24 T^{5} + \cdots - 91842560)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} + 44 T^{5} + \cdots + 483393600)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 563072654639104 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 22\!\cdots\!84 \) Copy content Toggle raw display
$53$ \( (T^{6} + 8 T^{5} + \cdots - 10055680)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{6} - 56 T^{5} + \cdots + 157104320)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 10\!\cdots\!44 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 87\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{6} - 12 T^{5} + \cdots + 792820800)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 40\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 28\!\cdots\!24 \) Copy content Toggle raw display
$89$ \( (T^{6} + 116 T^{5} + \cdots - 12371910592)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + 204 T^{5} + \cdots + 22310312000)^{2} \) Copy content Toggle raw display
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