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Properties

Label 968.6.a.q
Level $968$
Weight $6$
Character orbit 968.a
Self dual yes
Analytic conductor $155.252$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [968,6,Mod(1,968)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(968, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("968.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 968 = 2^{3} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 968.a (trivial)

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: yes
Analytic conductor: \(155.251537579\)
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: \( x^{16} - 4 x^{15} - 2635 x^{14} + 10644 x^{13} + 2721739 x^{12} - 11107836 x^{11} + \cdots + 53\!\cdots\!20 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{26}\cdot 11^{8} \)
Twist minimal: no (minimal twist has level 88)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_1 + 1) q^{3} + ( - \beta_{5} + 5) q^{5} + (\beta_{6} - \beta_{2} + 2) q^{7} + (\beta_{3} + 2 \beta_{2} + \beta_1 + 90) q^{9} + ( - \beta_{9} + \beta_{6} + \beta_{5} + \cdots - 52) q^{13} + ( - \beta_{10} - \beta_{9} - 2 \beta_{6} + \cdots - 47) q^{15}+ \cdots + ( - 20 \beta_{15} + 3 \beta_{14} + \cdots + 11731) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 12 q^{3} + 81 q^{5} + 47 q^{7} + 1416 q^{9} - 859 q^{13} - 738 q^{15} - 1226 q^{17} - 616 q^{19} - 1141 q^{21} + 2258 q^{23} + 10307 q^{25} + 564 q^{27} - 1613 q^{29} + 18511 q^{31} + 23544 q^{35}+ \cdots + 171314 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 4 x^{15} - 2635 x^{14} + 10644 x^{13} + 2721739 x^{12} - 11107836 x^{11} + \cdots + 53\!\cdots\!20 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 20\!\cdots\!29 \nu^{15} + \cdots - 15\!\cdots\!60 ) / 30\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 20\!\cdots\!29 \nu^{15} + \cdots + 14\!\cdots\!60 ) / 28\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 97\!\cdots\!01 \nu^{15} + \cdots - 27\!\cdots\!40 ) / 77\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 20\!\cdots\!16 \nu^{15} + \cdots + 63\!\cdots\!40 ) / 15\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 24\!\cdots\!32 \nu^{15} + \cdots + 42\!\cdots\!00 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 30\!\cdots\!49 \nu^{15} + \cdots - 30\!\cdots\!20 ) / 27\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 33\!\cdots\!45 \nu^{15} + \cdots - 14\!\cdots\!20 ) / 27\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 36\!\cdots\!57 \nu^{15} + \cdots + 15\!\cdots\!00 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 42\!\cdots\!97 \nu^{15} + \cdots - 38\!\cdots\!20 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 43\!\cdots\!39 \nu^{15} + \cdots + 49\!\cdots\!00 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 87\!\cdots\!13 \nu^{15} + \cdots + 10\!\cdots\!00 ) / 16\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 10\!\cdots\!71 \nu^{15} + \cdots + 18\!\cdots\!60 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 13\!\cdots\!74 \nu^{15} + \cdots - 65\!\cdots\!80 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 16\!\cdots\!91 \nu^{15} + \cdots - 22\!\cdots\!20 ) / 16\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 37\!\cdots\!59 \nu^{15} + \cdots - 35\!\cdots\!20 ) / 27\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{2} + 11\beta _1 + 6 ) / 11 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{4} + 11\beta_{3} + 23\beta_{2} + \beta _1 + 3649 ) / 11 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 14 \beta_{15} - 14 \beta_{14} - 16 \beta_{13} + 3 \beta_{12} - 11 \beta_{10} - 11 \beta_{9} + \cdots + 1613 ) / 11 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 86 \beta_{15} - 50 \beta_{14} - 112 \beta_{13} - 118 \beta_{12} + 481 \beta_{11} - 22 \beta_{10} + \cdots + 2070581 ) / 11 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 16360 \beta_{15} - 14289 \beta_{14} - 14919 \beta_{13} + 3286 \beta_{12} + 3541 \beta_{11} + \cdots + 896230 ) / 11 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 80100 \beta_{15} - 35489 \beta_{14} - 109901 \beta_{13} - 77894 \beta_{12} + 540559 \beta_{11} + \cdots + 1305606147 ) / 11 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 14233192 \beta_{15} - 11923064 \beta_{14} - 11762716 \beta_{13} + 2752214 \beta_{12} + \cdots + 1173802456 ) / 11 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 57895680 \beta_{15} - 23701634 \beta_{14} - 87244208 \beta_{13} - 42550746 \beta_{12} + \cdots + 855837391975 ) / 11 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 11233153028 \beta_{15} - 9268631135 \beta_{14} - 8820173389 \beta_{13} + 2102923450 \beta_{12} + \cdots + 1389135395578 ) / 11 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 39771429124 \beta_{15} - 17454103611 \beta_{14} - 65800025875 \beta_{13} - 21605764158 \beta_{12} + \cdots + 573922922803927 ) / 11 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 8486951484528 \beta_{15} - 6950351199814 \beta_{14} - 6480825394054 \beta_{13} + 1531120812430 \beta_{12} + \cdots + 14\!\cdots\!64 ) / 11 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 27666473523240 \beta_{15} - 14098244508668 \beta_{14} - 48969656825462 \beta_{13} + \cdots + 39\!\cdots\!07 ) / 11 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 62\!\cdots\!24 \beta_{15} + \cdots + 13\!\cdots\!22 ) / 11 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 19\!\cdots\!56 \beta_{15} + \cdots + 26\!\cdots\!27 ) / 11 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 45\!\cdots\!40 \beta_{15} + \cdots + 11\!\cdots\!92 ) / 11 \) Copy content Toggle raw display

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−26.5066
−23.8164
−25.8955
−21.8511
−10.4080
−6.91330
−4.22794
−5.39811
3.21970
6.63850
10.0842
13.3325
19.5696
22.3288
26.8952
26.9484
0 −27.1247 0 88.7213 0 184.080 0 492.748 0
1.2 0 −24.4344 0 −57.4032 0 −156.812 0 354.040 0
1.3 0 −24.2774 0 −51.2310 0 25.3165 0 346.393 0
1.4 0 −20.2331 0 59.3479 0 −110.064 0 166.379 0
1.5 0 −11.0260 0 −30.6908 0 62.4872 0 −121.427 0
1.6 0 −5.29527 0 6.33197 0 −24.6791 0 −214.960 0
1.7 0 −4.84597 0 11.7645 0 −55.0471 0 −219.517 0
1.8 0 −3.78008 0 89.9154 0 257.151 0 −228.711 0
1.9 0 4.83773 0 −82.0306 0 11.3366 0 −219.596 0
1.10 0 6.02047 0 76.6139 0 −112.762 0 −206.754 0
1.11 0 11.7022 0 −24.8620 0 −76.4943 0 −106.058 0
1.12 0 12.7145 0 −55.8973 0 194.771 0 −81.3426 0
1.13 0 21.1877 0 85.4976 0 −215.502 0 205.918 0
1.14 0 21.7107 0 −83.2917 0 −240.887 0 228.355 0
1.15 0 26.2772 0 56.0244 0 77.2326 0 447.491 0
1.16 0 28.5665 0 −7.81027 0 226.871 0 573.042 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.16
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(11\) \( +1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 968.6.a.q 16
11.b odd 2 1 968.6.a.p 16
11.c even 5 2 88.6.i.b ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
88.6.i.b ✓ 32 11.c even 5 2
968.6.a.p 16 11.b odd 2 1
968.6.a.q 16 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(968))\):

\( T_{3}^{16} - 12 T_{3}^{15} - 2580 T_{3}^{14} + 29404 T_{3}^{13} + 2591498 T_{3}^{12} + \cdots + 52\!\cdots\!25 \) Copy content Toggle raw display
\( T_{7}^{16} - 47 T_{7}^{15} - 180613 T_{7}^{14} + 4592698 T_{7}^{13} + 12459543981 T_{7}^{12} + \cdots + 30\!\cdots\!84 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} + \cdots + 52\!\cdots\!25 \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 86\!\cdots\!24 \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 30\!\cdots\!84 \) Copy content Toggle raw display
$11$ \( T^{16} \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 36\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 30\!\cdots\!05 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 19\!\cdots\!95 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots - 34\!\cdots\!96 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 46\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 37\!\cdots\!80 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots - 16\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 23\!\cdots\!25 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots - 75\!\cdots\!80 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 55\!\cdots\!96 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots - 91\!\cdots\!80 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 32\!\cdots\!31 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 99\!\cdots\!80 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots - 24\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 24\!\cdots\!45 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots - 30\!\cdots\!20 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots - 62\!\cdots\!25 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots - 34\!\cdots\!20 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 53\!\cdots\!05 \) Copy content Toggle raw display
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