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Properties

Label 1792.3.d.k
Level $1792$
Weight $3$
Character orbit 1792.d
Analytic conductor $48.828$
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] = [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: \( x^{16} - 2 x^{15} + 2 x^{14} + 4 x^{13} - 3 x^{12} - 32 x^{11} + 78 x^{10} - 266 x^{9} + 196 x^{8} + \cdots + 390625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{38}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 448)
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.

\(f(q)\) \(=\) \( q + \beta_{9} q^{3} - \beta_{2} q^{5} + \beta_{11} q^{7} + (\beta_{6} - 5) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{9} q^{3} - \beta_{2} q^{5} + \beta_{11} q^{7} + (\beta_{6} - 5) q^{9} + (\beta_{15} - \beta_{12} + \beta_{9}) q^{11} + (\beta_{5} + \beta_{3} + \beta_{2} - \beta_1) q^{13} + ( - \beta_{14} + \beta_{13} + 2 \beta_{11}) q^{15} + (\beta_{8} + 4) q^{17} + ( - \beta_{12} + \beta_{9} + 2 \beta_{7}) q^{19} + (\beta_{5} + \beta_{3} + \beta_{2}) q^{21} + ( - 4 \beta_{11} + \beta_{10}) q^{23} + (\beta_{8} - \beta_{6} + \beta_{4} - 5) q^{25} + (2 \beta_{15} - \beta_{12} + \cdots + 2 \beta_{7}) q^{27}+ \cdots + (\beta_{15} + 13 \beta_{12} + \cdots - 12 \beta_{7}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 80 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 80 q^{9} + 64 q^{17} - 80 q^{25} - 224 q^{33} + 544 q^{41} - 112 q^{49} - 320 q^{57} - 160 q^{65} - 64 q^{73} + 1232 q^{81} + 512 q^{89} + 128 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 2 x^{15} + 2 x^{14} + 4 x^{13} - 3 x^{12} - 32 x^{11} + 78 x^{10} - 266 x^{9} + 196 x^{8} + \cdots + 390625 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 4041 \nu^{15} - 35998 \nu^{14} + 105503 \nu^{13} - 70174 \nu^{12} + 278528 \nu^{11} + \cdots - 1300000000 ) / 407812500 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 8053 \nu^{15} - 11734 \nu^{14} + 20099 \nu^{13} - 23942 \nu^{12} - 121776 \nu^{11} + \cdots + 481250000 ) / 815625000 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 8841 \nu^{15} + 26398 \nu^{14} + 54097 \nu^{13} - 120626 \nu^{12} - 172928 \nu^{11} + \cdots + 1075000000 ) / 815625000 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 37 \nu^{15} - 374 \nu^{14} - 851 \nu^{13} + 3098 \nu^{12} - 7236 \nu^{11} - 1634 \nu^{10} + \cdots - 19062500 ) / 3515625 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 6899 \nu^{15} + 4762 \nu^{14} - 6197 \nu^{13} + 41466 \nu^{12} - 24832 \nu^{11} + \cdots - 245000000 ) / 163125000 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 17053 \nu^{15} - 42806 \nu^{14} - 26219 \nu^{13} - 106738 \nu^{12} + 330216 \nu^{11} + \cdots - 2860625000 ) / 407812500 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 20861 \nu^{15} + 8228 \nu^{14} + 39747 \nu^{13} + 148994 \nu^{12} + 85692 \nu^{11} + \cdots - 2994687500 ) / 407812500 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 15217 \nu^{15} - 15934 \nu^{14} + 16409 \nu^{13} - 17082 \nu^{12} + 69424 \nu^{11} + \cdots - 1465937500 ) / 203906250 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 123823 \nu^{15} + 60604 \nu^{14} + 381921 \nu^{13} + 1409742 \nu^{12} + 2540956 \nu^{11} + \cdots - 21737187500 ) / 815625000 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 1722 \nu^{15} - 1389 \nu^{14} - 5868 \nu^{13} - 20674 \nu^{12} - 44996 \nu^{11} + \cdots + 391437500 ) / 8156250 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 25357 \nu^{15} + 6696 \nu^{14} + 64419 \nu^{13} + 235448 \nu^{12} + 468744 \nu^{11} + \cdots - 4316406250 ) / 90625000 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 1649 \nu^{15} + 452 \nu^{14} + 4623 \nu^{13} + 16196 \nu^{12} + 31328 \nu^{11} + 18282 \nu^{10} + \cdots - 281093750 ) / 2343750 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 16982 \nu^{15} + 2666 \nu^{14} + 44454 \nu^{13} + 162688 \nu^{12} + 316324 \nu^{11} + \cdots - 2953125000 ) / 20390625 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 695713 \nu^{15} + 203914 \nu^{14} + 1636971 \nu^{13} + 6782332 \nu^{12} + 13087296 \nu^{11} + \cdots - 120744531250 ) / 407812500 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 674037 \nu^{15} + 141876 \nu^{14} + 1687899 \nu^{13} + 6502098 \nu^{12} + 12281764 \nu^{11} + \cdots - 119040937500 ) / 271875000 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 2 \beta_{15} - 2 \beta_{14} - \beta_{13} - 2 \beta_{12} + 4 \beta_{11} + 2 \beta_{10} + 2 \beta_{9} + \cdots + 4 ) / 32 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{14} - \beta_{13} - 2\beta_{11} + \beta_{10} - 2\beta_{5} + 2\beta_{3} + 4\beta_{2} ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 4 \beta_{15} - 2 \beta_{13} + 2 \beta_{12} + 40 \beta_{11} - 4 \beta_{9} - 4 \beta_{7} - 8 \beta_{5} + \cdots - 16 ) / 16 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -2\beta_{15} + 2\beta_{12} + 18\beta_{9} - 3\beta_{8} + 12\beta_{7} + 10\beta_{6} + \beta_{4} - 14 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 22 \beta_{15} - 18 \beta_{14} - 47 \beta_{13} + 66 \beta_{12} - 124 \beta_{11} - 38 \beta_{10} + \cdots + 284 ) / 32 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 27\beta_{13} - 80\beta_{11} - 27\beta_{5} + 33\beta_{3} - 54\beta_{2} - 15\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 26 \beta_{15} + 66 \beta_{14} + 383 \beta_{13} - 786 \beta_{12} + 28 \beta_{11} - 106 \beta_{10} + \cdots + 2972 ) / 32 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -2\beta_{15} + 194\beta_{12} - 774\beta_{9} - 195\beta_{8} + 48\beta_{7} - 46\beta_{6} - \beta_{4} + 386 ) / 8 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 532 \beta_{15} - 1366 \beta_{13} - 146 \beta_{12} - 40 \beta_{11} - 428 \beta_{9} + 1012 \beta_{7} + \cdots + 10928 ) / 16 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 521 \beta_{14} - 253 \beta_{13} + 4402 \beta_{11} + 271 \beta_{10} - 1486 \beta_{5} + \cdots - 1236 \beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 6194 \beta_{15} + 7154 \beta_{14} + 4081 \beta_{13} + 3218 \beta_{12} - 10468 \beta_{11} + \cdots + 8452 ) / 32 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( -33\beta_{15} - 216\beta_{12} + 1827\beta_{9} - 963\beta_{7} + 540\beta_{4} + 4193 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 3082 \beta_{15} - 32842 \beta_{14} + 20659 \beta_{13} + 51158 \beta_{12} - 53356 \beta_{11} + \cdots - 250636 ) / 32 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 12641 \beta_{14} - 27671 \beta_{13} - 10402 \beta_{11} - 19399 \beta_{10} - 5542 \beta_{5} + \cdots - 12300 \beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 53444 \beta_{15} + 114878 \beta_{13} + 67642 \beta_{12} + 192200 \beta_{11} - 380804 \beta_{9} + \cdots + 919024 ) / 16 \) 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
1.71795 − 1.43130i
1.43130 + 1.71795i
−1.93487 + 1.12083i
−1.12083 − 1.93487i
2.23607 − 0.00323481i
−0.00323481 − 2.23607i
−2.09852 + 0.772140i
0.772140 + 2.09852i
−2.09852 − 0.772140i
0.772140 − 2.09852i
2.23607 + 0.00323481i
−0.00323481 + 2.23607i
−1.93487 − 1.12083i
−1.12083 + 1.93487i
1.71795 + 1.43130i
1.43130 − 1.71795i
0 − 5.74133i 0 −2.27722 0 − 2.64575i 0 −23.9628 0
1023.2 0 − 5.74133i 0 2.27722 0 2.64575i 0 −23.9628 0
1023.3 0 − 4.47860i 0 −7.94270 0 2.64575i 0 −11.0579 0
1023.4 0 − 4.47860i 0 7.94270 0 − 2.64575i 0 −11.0579 0
1023.5 0 − 1.62808i 0 −1.83603 0 − 2.64575i 0 6.34937 0
1023.6 0 − 1.62808i 0 1.83603 0 2.64575i 0 6.34937 0
1023.7 0 − 0.573300i 0 −2.89080 0 2.64575i 0 8.67133 0
1023.8 0 − 0.573300i 0 2.89080 0 − 2.64575i 0 8.67133 0
1023.9 0 0.573300i 0 −2.89080 0 − 2.64575i 0 8.67133 0
1023.10 0 0.573300i 0 2.89080 0 2.64575i 0 8.67133 0
1023.11 0 1.62808i 0 −1.83603 0 2.64575i 0 6.34937 0
1023.12 0 1.62808i 0 1.83603 0 − 2.64575i 0 6.34937 0
1023.13 0 4.47860i 0 −7.94270 0 − 2.64575i 0 −11.0579 0
1023.14 0 4.47860i 0 7.94270 0 2.64575i 0 −11.0579 0
1023.15 0 5.74133i 0 −2.27722 0 2.64575i 0 −23.9628 0
1023.16 0 5.74133i 0 2.27722 0 − 2.64575i 0 −23.9628 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.k 16
4.b odd 2 1 inner 1792.3.d.k 16
8.b even 2 1 inner 1792.3.d.k 16
8.d odd 2 1 inner 1792.3.d.k 16
16.e even 4 2 448.3.g.c ✓ 16
16.f odd 4 2 448.3.g.c ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
448.3.g.c ✓ 16 16.e even 4 2
448.3.g.c ✓ 16 16.f odd 4 2
1792.3.d.k 16 1.a even 1 1 trivial
1792.3.d.k 16 4.b odd 2 1 inner
1792.3.d.k 16 8.b even 2 1 inner
1792.3.d.k 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 \(S_{3}^{\mathrm{new}}(1792, [\chi])\):

\( T_{3}^{8} + 56T_{3}^{6} + 820T_{3}^{4} + 2016T_{3}^{2} + 576 \) Copy content Toggle raw display
\( T_{5}^{8} - 80T_{5}^{6} + 1156T_{5}^{4} - 5760T_{5}^{2} + 9216 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( (T^{8} + 56 T^{6} + \cdots + 576)^{2} \) Copy content Toggle raw display
$5$ \( (T^{8} - 80 T^{6} + \cdots + 9216)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 7)^{8} \) Copy content Toggle raw display
$11$ \( (T^{8} + 728 T^{6} + \cdots + 357663744)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} - 1280 T^{6} + \cdots + 1201038336)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 16 T^{3} + \cdots + 104400)^{4} \) Copy content Toggle raw display
$19$ \( (T^{8} + 1544 T^{6} + \cdots + 1011240000)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + 1736 T^{6} + \cdots + 5101387776)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} - 5408 T^{6} + \cdots + 40333492224)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + 3872 T^{6} + \cdots + 36276535296)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} - 4448 T^{6} + \cdots + 150122151936)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} - 136 T^{3} + \cdots + 249552)^{4} \) Copy content Toggle raw display
$43$ \( (T^{8} + 6296 T^{6} + \cdots + 8856576)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 4096 T^{2} + 1327104)^{4} \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots + 25304924160000)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots + 10625357238336)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots + 1998807819264)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 189534914740224)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 32438583558144)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 16 T^{3} + \cdots + 34862032)^{4} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 68797071360000)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 526399603560000)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} - 128 T^{3} + \cdots + 9378576)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} - 32 T^{3} + \cdots + 158992)^{4} \) Copy content Toggle raw display
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