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Properties

Label 768.2.k.g
Level $768$
Weight $2$
Character orbit 768.k
Analytic conductor $6.133$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [768,2,Mod(191,768)]
 
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, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("768.191");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 768 = 2^{8} \cdot 3 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 768.k (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: \(6.13251087523\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: 16.0.12877254853348294656.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4x^{14} + 26x^{12} + 12x^{10} + 35x^{8} + 180x^{6} + 686x^{4} + 632x^{2} + 169 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{18} \)
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.

\(f(q)\) \(=\) \( q + \beta_{4} q^{3} + (\beta_{8} + \beta_{6} + \beta_1) q^{5} + (\beta_{13} + \beta_{7} + \beta_{3}) q^{7} + (\beta_{8} - \beta_{5} + \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{3} + (\beta_{8} + \beta_{6} + \beta_1) q^{5} + (\beta_{13} + \beta_{7} + \beta_{3}) q^{7} + (\beta_{8} - \beta_{5} + \beta_1) q^{9} + ( - \beta_{13} + \beta_{3}) q^{11} - \beta_{8} q^{13} + (\beta_{15} + 2 \beta_{14} + \cdots - 2 \beta_{3}) q^{15}+ \cdots + (\beta_{15} + \beta_{14} + \cdots - 5 \beta_{4}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16 q^{21} - 64 q^{33} - 64 q^{37} + 80 q^{45} + 16 q^{49} + 48 q^{69} + 16 q^{81} + 32 q^{85} + 64 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 4x^{14} + 26x^{12} + 12x^{10} + 35x^{8} + 180x^{6} + 686x^{4} + 632x^{2} + 169 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 373 \nu^{14} - 2705 \nu^{12} + 2528 \nu^{10} - 93575 \nu^{8} - 121274 \nu^{6} - 273715 \nu^{4} + \cdots - 1112488 ) / 882024 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 128 \nu^{15} + 303 \nu^{13} - 1996 \nu^{11} - 9397 \nu^{9} + 8842 \nu^{7} - 39527 \nu^{5} + \cdots - 100690 \nu ) / 40092 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1803 \nu^{15} - 9912 \nu^{13} + 58819 \nu^{11} - 52560 \nu^{9} + 59685 \nu^{7} + 258436 \nu^{5} + \cdots - 299230 \nu ) / 294008 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 12190 \nu^{15} + 53524 \nu^{13} - 336112 \nu^{11} - 22037 \nu^{9} - 361814 \nu^{7} + \cdots - 5490457 \nu ) / 882024 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -101\nu^{14} + 460\nu^{12} - 2889\nu^{10} + 425\nu^{8} - 4017\nu^{6} - 15869\nu^{4} - 61874\nu^{2} - 30849 ) / 3432 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 10775 \nu^{14} + 48779 \nu^{12} - 304578 \nu^{10} + 35353 \nu^{8} - 384632 \nu^{6} + \cdots - 3109704 ) / 294008 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 16897 \nu^{15} - 73314 \nu^{13} + 463538 \nu^{11} + 43265 \nu^{9} + 581692 \nu^{7} + \cdots + 6477347 \nu ) / 441012 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 15911 \nu^{14} + 71506 \nu^{12} - 446733 \nu^{10} + 14147 \nu^{8} - 493025 \nu^{6} + \cdots - 5253937 ) / 294008 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 49163 \nu^{14} + 229962 \nu^{12} - 1438297 \nu^{10} + 383639 \nu^{8} - 1997021 \nu^{6} + \cdots - 12314653 ) / 882024 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 27013 \nu^{14} + 122840 \nu^{12} - 767421 \nu^{10} + 93187 \nu^{8} - 980625 \nu^{6} + \cdots - 7448571 ) / 441012 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 1065 \nu^{14} - 4820 \nu^{12} + 30397 \nu^{10} - 4305 \nu^{8} + 45989 \nu^{6} + 158745 \nu^{4} + \cdots + 356857 ) / 16962 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 28901 \nu^{15} + 126474 \nu^{13} - 796348 \nu^{11} - 64501 \nu^{9} - 912014 \nu^{7} + \cdots - 11364583 \nu ) / 441012 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 60161 \nu^{15} - 276472 \nu^{13} + 1730571 \nu^{11} - 317126 \nu^{9} + 2359161 \nu^{7} + \cdots + 17538108 \nu ) / 882024 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 20624 \nu^{15} - 93314 \nu^{13} + 584388 \nu^{11} - 57367 \nu^{9} + 737538 \nu^{7} + \cdots + 6174655 \nu ) / 294008 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 6652 \nu^{15} - 30516 \nu^{13} + 190949 \nu^{11} - 31906 \nu^{9} + 249475 \nu^{7} + \cdots + 1854356 \nu ) / 40092 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -2\beta_{14} + \beta_{13} + \beta_{7} - 2\beta_{4} + \beta_{3} ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{11} + \beta_{9} + \beta_{8} - 8\beta_{5} + 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{15} - 8\beta_{14} + 11\beta_{13} + \beta_{12} + \beta_{7} + 2\beta_{4} + \beta_{3} - 8\beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -2\beta_{11} + 10\beta_{10} + 3\beta_{9} - \beta_{8} - 4\beta_{6} - 24\beta_{5} + 4\beta _1 - 18 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -9\beta_{15} + 19\beta_{14} + 24\beta_{13} + \beta_{12} - 33\beta_{7} + 3\beta_{4} - 26\beta_{3} - 17\beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 39\beta_{11} + 25\beta_{10} + 11\beta_{9} - 15\beta_{8} + 14\beta_{6} + 20\beta_{5} + 34\beta _1 - 142 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 38 \beta_{15} + 226 \beta_{14} - 81 \beta_{13} - 46 \beta_{12} - 141 \beta_{7} + 54 \beta_{4} + \cdots + 100 \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 188\beta_{11} - 116\beta_{10} - 55\beta_{9} - 95\beta_{8} + 200\beta_{6} + 672\beta_{5} + 24\beta _1 - 194 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 131 \beta_{15} + 417 \beta_{14} - 1043 \beta_{13} - 307 \beta_{12} + 116 \beta_{7} + \cdots + 847 \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( -111\beta_{11} - 1095\beta_{10} - 556\beta_{9} + 66\beta_{8} + 622\beta_{6} + 2212\beta_{5} - 878\beta _1 + 2612 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 1745 \beta_{15} - 4028 \beta_{14} - 2754 \beta_{13} - 117 \beta_{12} + 4082 \beta_{7} + \cdots + 1362 \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 2612 \beta_{11} - 934 \beta_{10} - 449 \beta_{9} + 1327 \beta_{8} - 1300 \beta_{6} - 3564 \beta_{5} + \cdots + 7704 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 4566 \beta_{15} - 27554 \beta_{14} + 13545 \beta_{13} + 7738 \beta_{12} + 15113 \beta_{7} + \cdots - 15540 \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 19838 \beta_{11} + 19760 \beta_{10} + 10061 \beta_{9} + 9941 \beta_{8} - 28720 \beta_{6} - 85168 \beta_{5} + \cdots + 674 ) / 4 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 25029 \beta_{15} - 15018 \beta_{14} + 128585 \beta_{13} + 38357 \beta_{12} - 37567 \beta_{7} + \cdots - 102610 \beta_{2} ) / 4 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(257\) \(511\) \(517\)
\(\chi(n)\) \(-1\) \(-1\) \(\beta_{5}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
191.1
1.46009 + 0.752986i
0.489978 + 1.19709i
1.90419 + 1.19709i
−0.0458788 − 0.752986i
0.0458788 + 0.752986i
−1.90419 − 1.19709i
−0.489978 − 1.19709i
−1.46009 − 0.752986i
1.46009 − 0.752986i
0.489978 − 1.19709i
1.90419 − 1.19709i
−0.0458788 + 0.752986i
0.0458788 − 0.752986i
−1.90419 + 1.19709i
−0.489978 + 1.19709i
−1.46009 + 0.752986i
0 −1.71891 + 0.212940i 0 2.90931 + 2.90931i 0 2.82843 0 2.90931 − 0.732051i 0
191.2 0 −1.45590 − 0.938266i 0 1.23931 + 1.23931i 0 −2.82843 0 1.23931 + 2.73205i 0
191.3 0 −0.938266 − 1.45590i 0 −1.23931 − 1.23931i 0 2.82843 0 −1.23931 + 2.73205i 0
191.4 0 −0.212940 + 1.71891i 0 −2.90931 − 2.90931i 0 2.82843 0 −2.90931 − 0.732051i 0
191.5 0 0.212940 − 1.71891i 0 −2.90931 − 2.90931i 0 −2.82843 0 −2.90931 − 0.732051i 0
191.6 0 0.938266 + 1.45590i 0 −1.23931 − 1.23931i 0 −2.82843 0 −1.23931 + 2.73205i 0
191.7 0 1.45590 + 0.938266i 0 1.23931 + 1.23931i 0 2.82843 0 1.23931 + 2.73205i 0
191.8 0 1.71891 − 0.212940i 0 2.90931 + 2.90931i 0 −2.82843 0 2.90931 − 0.732051i 0
575.1 0 −1.71891 − 0.212940i 0 2.90931 − 2.90931i 0 2.82843 0 2.90931 + 0.732051i 0
575.2 0 −1.45590 + 0.938266i 0 1.23931 − 1.23931i 0 −2.82843 0 1.23931 − 2.73205i 0
575.3 0 −0.938266 + 1.45590i 0 −1.23931 + 1.23931i 0 2.82843 0 −1.23931 − 2.73205i 0
575.4 0 −0.212940 − 1.71891i 0 −2.90931 + 2.90931i 0 2.82843 0 −2.90931 + 0.732051i 0
575.5 0 0.212940 + 1.71891i 0 −2.90931 + 2.90931i 0 −2.82843 0 −2.90931 + 0.732051i 0
575.6 0 0.938266 − 1.45590i 0 −1.23931 + 1.23931i 0 −2.82843 0 −1.23931 − 2.73205i 0
575.7 0 1.45590 − 0.938266i 0 1.23931 − 1.23931i 0 2.82843 0 1.23931 − 2.73205i 0
575.8 0 1.71891 + 0.212940i 0 2.90931 − 2.90931i 0 −2.82843 0 2.90931 + 0.732051i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 191.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner
16.e even 4 1 inner
16.f odd 4 1 inner
48.i odd 4 1 inner
48.k even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 768.2.k.g ✓ 16
3.b odd 2 1 inner 768.2.k.g ✓ 16
4.b odd 2 1 inner 768.2.k.g ✓ 16
8.b even 2 1 768.2.k.h yes 16
8.d odd 2 1 768.2.k.h yes 16
12.b even 2 1 inner 768.2.k.g ✓ 16
16.e even 4 1 inner 768.2.k.g ✓ 16
16.e even 4 1 768.2.k.h yes 16
16.f odd 4 1 inner 768.2.k.g ✓ 16
16.f odd 4 1 768.2.k.h yes 16
24.f even 2 1 768.2.k.h yes 16
24.h odd 2 1 768.2.k.h yes 16
48.i odd 4 1 inner 768.2.k.g ✓ 16
48.i odd 4 1 768.2.k.h yes 16
48.k even 4 1 inner 768.2.k.g ✓ 16
48.k even 4 1 768.2.k.h yes 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
768.2.k.g ✓ 16 1.a even 1 1 trivial
768.2.k.g ✓ 16 3.b odd 2 1 inner
768.2.k.g ✓ 16 4.b odd 2 1 inner
768.2.k.g ✓ 16 12.b even 2 1 inner
768.2.k.g ✓ 16 16.e even 4 1 inner
768.2.k.g ✓ 16 16.f odd 4 1 inner
768.2.k.g ✓ 16 48.i odd 4 1 inner
768.2.k.g ✓ 16 48.k even 4 1 inner
768.2.k.h yes 16 8.b even 2 1
768.2.k.h yes 16 8.d odd 2 1
768.2.k.h yes 16 16.e even 4 1
768.2.k.h yes 16 16.f odd 4 1
768.2.k.h yes 16 24.f even 2 1
768.2.k.h yes 16 24.h odd 2 1
768.2.k.h yes 16 48.i odd 4 1
768.2.k.h yes 16 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 \(S_{2}^{\mathrm{new}}(768, [\chi])\):

\( T_{5}^{8} + 296T_{5}^{4} + 2704 \) Copy content Toggle raw display
\( T_{11}^{8} + 152T_{11}^{4} + 2704 \) Copy content Toggle raw display
\( T_{13}^{4} + 36 \) Copy content Toggle raw display
\( T_{37}^{4} + 16T_{37}^{3} + 128T_{37}^{2} + 416T_{37} + 676 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} - 4 T^{12} + \cdots + 6561 \) Copy content Toggle raw display
$5$ \( (T^{8} + 296 T^{4} + 2704)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} - 8)^{8} \) Copy content Toggle raw display
$11$ \( (T^{8} + 152 T^{4} + 2704)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 36)^{4} \) Copy content Toggle raw display
$17$ \( (T^{4} + 64 T^{2} + 832)^{4} \) Copy content Toggle raw display
$19$ \( (T^{8} + 56 T^{4} + 16)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 96 T^{2} + 1872)^{4} \) Copy content Toggle raw display
$29$ \( (T^{8} + 296 T^{4} + 2704)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 48 T^{2} + 144)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} + 16 T^{3} + \cdots + 676)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} - 88 T^{2} + 208)^{4} \) Copy content Toggle raw display
$43$ \( (T^{8} + 12152 T^{4} + 29986576)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} \) Copy content Toggle raw display
$53$ \( (T^{8} + 16488 T^{4} + 219024)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + 152 T^{4} + 2704)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 2916)^{4} \) Copy content Toggle raw display
$67$ \( (T^{8} + 12152 T^{4} + 29986576)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 128 T^{2} + 208)^{4} \) Copy content Toggle raw display
$73$ \( (T^{4} + 32 T^{2} + 64)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} + 16 T^{2} + 16)^{4} \) Copy content Toggle raw display
$83$ \( (T^{8} + 35928 T^{4} + 219024)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} - 328 T^{2} + 25168)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} - 8 T - 92)^{8} \) Copy content Toggle raw display
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