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Properties

Label 768.3.l.f
Level $768$
Weight $3$
Character orbit 768.l
Analytic conductor $20.926$
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] = [768,3,Mod(319,768)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(768, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 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("768.319");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 768 = 2^{8} \cdot 3 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 768.l (of order \(4\), degree \(2\), 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: \(20.9264843029\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 52 x^{14} + 1743 x^{12} - 34996 x^{10} + 513409 x^{8} - 5039424 x^{6} + 36142848 x^{4} + \cdots + 429981696 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{24} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{7} q^{3} + ( - \beta_{6} + \beta_1 + 1) q^{5} + (\beta_{13} + 2 \beta_{8} + 2 \beta_{7}) q^{7} + 3 \beta_1 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{7} q^{3} + ( - \beta_{6} + \beta_1 + 1) q^{5} + (\beta_{13} + 2 \beta_{8} + 2 \beta_{7}) q^{7} + 3 \beta_1 q^{9} + ( - \beta_{13} + \beta_{12} + \cdots - \beta_{4}) q^{11}+ \cdots + (3 \beta_{14} - 3 \beta_{13} + \cdots + 3 \beta_{4}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 8 q^{5} - 32 q^{13} + 112 q^{17} + 72 q^{21} - 56 q^{29} - 48 q^{33} + 272 q^{37} - 24 q^{45} + 304 q^{49} + 504 q^{53} - 176 q^{61} + 624 q^{65} + 288 q^{69} - 848 q^{77} - 144 q^{81} + 880 q^{85} + 216 q^{93} - 160 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 52 x^{14} + 1743 x^{12} - 34996 x^{10} + 513409 x^{8} - 5039424 x^{6} + 36142848 x^{4} + \cdots + 429981696 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 160465 \nu^{14} + 9422452 \nu^{12} - 318819327 \nu^{10} + 6769384180 \nu^{8} + \cdots + 18371762233344 ) / 2491322904576 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 5414461 \nu^{14} + 253287040 \nu^{12} - 8226683907 \nu^{10} + 151061140360 \nu^{8} + \cdots + 449896465838592 ) / 41496097129344 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 693625847 \nu^{14} + 9684122444 \nu^{12} + 8291865543 \nu^{10} - 11759165031796 \nu^{8} + \cdots - 78\!\cdots\!60 ) / 13\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 64812599 \nu^{15} - 5320265600 \nu^{13} + 172800564105 \nu^{11} - 3767555194520 \nu^{9} + \cdots - 59\!\cdots\!40 \nu ) / 19\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 961 \nu^{14} - 42484 \nu^{12} + 1306383 \nu^{10} - 21657844 \nu^{8} + 264492865 \nu^{6} + \cdots - 2717245440 ) / 1193647104 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 1280466785 \nu^{14} + 64144492052 \nu^{12} - 1961897503215 \nu^{10} + \cdots + 18\!\cdots\!04 ) / 13\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 2419885 \nu^{15} + 104570212 \nu^{13} - 2939977443 \nu^{11} + 43298224996 \nu^{9} + \cdots - 49169767550976 \nu ) / 43182930345984 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 22900681 \nu^{15} + 799330948 \nu^{13} - 21854146119 \nu^{11} + \cdots - 601474078752768 \nu ) / 388646373113856 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 1021690297 \nu^{15} + 51519009364 \nu^{13} - 1601054358423 \nu^{11} + \cdots + 16\!\cdots\!16 \nu ) / 15\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 860602975 \nu^{14} + 36650246620 \nu^{12} - 1086119673873 \nu^{10} + \cdots + 98\!\cdots\!72 ) / 663937554069504 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 590093377 \nu^{14} + 34455716692 \nu^{12} - 1119241542159 \nu^{10} + \cdots + 45\!\cdots\!56 ) / 442625036046336 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 7743625 \nu^{15} - 526561972 \nu^{13} + 18131157063 \nu^{11} - 391520735092 \nu^{9} + \cdots - 650114397499392 \nu ) / 102144239087616 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 443523605 \nu^{15} + 16918372796 \nu^{13} - 481633493211 \nu^{11} + \cdots - 15\!\cdots\!08 \nu ) / 26\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 6686137 \nu^{15} + 346392516 \nu^{13} - 10907857751 \nu^{11} + \cdots + 368313104719872 \nu ) / 34048079695872 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 794919979 \nu^{15} + 42147813760 \nu^{13} - 1368947819733 \nu^{11} + \cdots + 47\!\cdots\!04 \nu ) / 13\!\cdots\!08 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -2\beta_{14} - \beta_{12} + 2\beta_{9} + 4\beta_{8} - 2\beta_{7} - 2\beta_{4} ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 4\beta_{10} - 2\beta_{6} - 2\beta_{3} - 13\beta_{2} - 8\beta _1 + 50 ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -4\beta_{15} + \beta_{14} - \beta_{12} + 104\beta_{8} - 104\beta_{7} - 56\beta_{4} ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 56\beta_{11} + 52\beta_{10} - 110\beta_{6} - 13\beta_{5} - 2\beta_{3} - 194\beta_{2} - 312\beta _1 - 782 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 80 \beta_{15} + 883 \beta_{14} + 240 \beta_{13} + 638 \beta_{12} - 446 \beta_{9} + \cdots - 446 \beta_{4} ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 587\beta_{11} - 691\beta_{10} - 691\beta_{6} - 532\beta_{5} + 587\beta_{3} + 691\beta _1 - 12465 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 2238 \beta_{15} + 10867 \beta_{14} + 6714 \beta_{13} + 18614 \beta_{12} - 7804 \beta_{9} + \cdots + 7804 \beta_{4} ) / 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 3486 \beta_{11} - 51746 \beta_{10} + 20644 \beta_{6} - 15171 \beta_{5} + 27616 \beta_{3} + \cdots - 207940 ) / 8 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 108944\beta_{15} - 195623\beta_{14} + 195623\beta_{12} - 869908\beta_{8} + 869908\beta_{7} + 290788\beta_{4} ) / 8 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 585730 \beta_{11} - 393122 \beta_{10} + 1075156 \beta_{6} + 374984 \beta_{5} + 96304 \beta_{3} + \cdots + 4564612 ) / 8 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 1239394 \beta_{15} - 7860878 \beta_{14} - 3718182 \beta_{13} - 3319525 \beta_{12} + \cdots + 2819884 \beta_{4} ) / 8 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 4966195 \beta_{11} + 7361237 \beta_{10} + 7361237 \beta_{6} + 8633885 \beta_{5} - 4966195 \beta_{3} + \cdots + 82575801 ) / 4 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 27200160 \beta_{15} - 60522278 \beta_{14} - 81600480 \beta_{13} - 161608147 \beta_{12} + \cdots - 56098430 \beta_{4} ) / 8 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 55940664 \beta_{11} + 460915204 \beta_{10} - 146546606 \beta_{6} + 191171292 \beta_{5} + \cdots + 1492908926 ) / 8 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 1169659708 \beta_{15} + 2196408187 \beta_{14} - 2196408187 \beta_{12} + 7127887208 \beta_{8} + \cdots - 2268192728 \beta_{4} ) / 8 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/768\mathbb{Z}\right)^\times\).

\(n\) \(257\) \(511\) \(517\)
\(\chi(n)\) \(1\) \(-1\) \(\beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
319.1
−2.78422 − 1.60747i
−3.95096 + 2.28109i
3.23251 + 1.86629i
2.27793 − 1.31516i
2.78422 + 1.60747i
3.95096 − 2.28109i
−3.23251 − 1.86629i
−2.27793 + 1.31516i
−2.78422 + 1.60747i
−3.95096 − 2.28109i
3.23251 − 1.86629i
2.27793 + 1.31516i
2.78422 − 1.60747i
3.95096 + 2.28109i
−3.23251 + 1.86629i
−2.27793 − 1.31516i
0 −1.22474 − 1.22474i 0 −5.27866 − 5.27866i 0 −12.7431 0 3.00000i 0
319.2 0 −1.22474 − 1.22474i 0 −3.71984 − 3.71984i 0 5.63959 0 3.00000i 0
319.3 0 −1.22474 − 1.22474i 0 4.54661 + 4.54661i 0 1.15196 0 3.00000i 0
319.4 0 −1.22474 − 1.22474i 0 6.45189 + 6.45189i 0 −8.74541 0 3.00000i 0
319.5 0 1.22474 + 1.22474i 0 −5.27866 − 5.27866i 0 12.7431 0 3.00000i 0
319.6 0 1.22474 + 1.22474i 0 −3.71984 − 3.71984i 0 −5.63959 0 3.00000i 0
319.7 0 1.22474 + 1.22474i 0 4.54661 + 4.54661i 0 −1.15196 0 3.00000i 0
319.8 0 1.22474 + 1.22474i 0 6.45189 + 6.45189i 0 8.74541 0 3.00000i 0
703.1 0 −1.22474 + 1.22474i 0 −5.27866 + 5.27866i 0 −12.7431 0 − 3.00000i 0
703.2 0 −1.22474 + 1.22474i 0 −3.71984 + 3.71984i 0 5.63959 0 − 3.00000i 0
703.3 0 −1.22474 + 1.22474i 0 4.54661 − 4.54661i 0 1.15196 0 − 3.00000i 0
703.4 0 −1.22474 + 1.22474i 0 6.45189 − 6.45189i 0 −8.74541 0 − 3.00000i 0
703.5 0 1.22474 − 1.22474i 0 −5.27866 + 5.27866i 0 12.7431 0 − 3.00000i 0
703.6 0 1.22474 − 1.22474i 0 −3.71984 + 3.71984i 0 −5.63959 0 − 3.00000i 0
703.7 0 1.22474 − 1.22474i 0 4.54661 − 4.54661i 0 −1.15196 0 − 3.00000i 0
703.8 0 1.22474 − 1.22474i 0 6.45189 − 6.45189i 0 8.74541 0 − 3.00000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 319.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
16.e even 4 1 inner
16.f odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 768.3.l.f yes 16
4.b odd 2 1 inner 768.3.l.f yes 16
8.b even 2 1 768.3.l.e ✓ 16
8.d odd 2 1 768.3.l.e ✓ 16
16.e even 4 1 768.3.l.e ✓ 16
16.e even 4 1 inner 768.3.l.f yes 16
16.f odd 4 1 768.3.l.e ✓ 16
16.f odd 4 1 inner 768.3.l.f yes 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
768.3.l.e ✓ 16 8.b even 2 1
768.3.l.e ✓ 16 8.d odd 2 1
768.3.l.e ✓ 16 16.e even 4 1
768.3.l.e ✓ 16 16.f odd 4 1
768.3.l.f yes 16 1.a even 1 1 trivial
768.3.l.f yes 16 4.b odd 2 1 inner
768.3.l.f yes 16 16.e even 4 1 inner
768.3.l.f yes 16 16.f odd 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} - 4T_{5}^{7} + 8T_{5}^{6} + 208T_{5}^{5} + 5392T_{5}^{4} - 9984T_{5}^{3} + 18432T_{5}^{2} + 442368T_{5} + 5308416 \) acting on \(S_{3}^{\mathrm{new}}(768, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( (T^{4} + 9)^{4} \) Copy content Toggle raw display
$5$ \( (T^{8} - 4 T^{7} + \cdots + 5308416)^{2} \) Copy content Toggle raw display
$7$ \( (T^{8} - 272 T^{6} + \cdots + 524176)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 72\!\cdots\!36 \) Copy content Toggle raw display
$13$ \( (T^{8} + 16 T^{7} + \cdots + 129686544)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 28 T^{3} + \cdots + 1872)^{4} \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 13\!\cdots\!96 \) Copy content Toggle raw display
$23$ \( (T^{4} - 1008 T^{2} + 5184)^{4} \) Copy content Toggle raw display
$29$ \( (T^{8} + 28 T^{7} + \cdots + 109502751744)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + 3376 T^{6} + \cdots + 36638553744)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots + 97388397525136)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + 7520 T^{6} + \cdots + 132834007296)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 86\!\cdots\!56 \) Copy content Toggle raw display
$47$ \( (T^{4} + 4464 T^{2} + 2742336)^{4} \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots + 85131180942336)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 69\!\cdots\!36 \) Copy content Toggle raw display
$61$ \( (T^{8} + 88 T^{7} + \cdots + 278622864)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 28\!\cdots\!16 \) Copy content Toggle raw display
$71$ \( (T^{8} - 2752 T^{6} + \cdots + 15912308736)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 5580782567424)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 137219389108624)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 40\!\cdots\!56 \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 70\!\cdots\!76)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 40 T^{3} + \cdots - 26948352)^{4} \) Copy content Toggle raw display
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