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Properties

Label 968.4.a.o
Level $968$
Weight $4$
Character orbit 968.a
Self dual yes
Analytic conductor $57.114$
Analytic rank $1$
Dimension $8$
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] = [968,4,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, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("968.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 968 = 2^{3} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
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: \(57.1138488856\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} - 87x^{6} + 12x^{5} + 2157x^{4} + 2939x^{3} - 5906x^{2} - 3030x + 45 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6}\cdot 11 \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{2} - 1) q^{3} + ( - \beta_{5} - 2) q^{5} + ( - \beta_{5} - \beta_{4} + \cdots - \beta_1) q^{7}+ \cdots + (\beta_{3} - \beta_1 + 4) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} - 1) q^{3} + ( - \beta_{5} - 2) q^{5} + ( - \beta_{5} - \beta_{4} + \cdots - \beta_1) q^{7}+ \cdots + ( - 44 \beta_{7} - 53 \beta_{6} + \cdots - 598) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{3} - 13 q^{5} + 9 q^{7} + 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{3} - 13 q^{5} + 9 q^{7} + 36 q^{9} - 7 q^{13} - 66 q^{15} - 94 q^{17} + 92 q^{19} - 17 q^{21} - 46 q^{23} + 101 q^{25} + 124 q^{27} - 241 q^{29} - 265 q^{31} + 664 q^{35} - 469 q^{37} - 788 q^{39} + 74 q^{41} - 35 q^{43} - 566 q^{45} + 197 q^{47} + 221 q^{49} + 608 q^{51} - 247 q^{53} + 979 q^{57} - 1208 q^{59} - 2061 q^{61} + 1617 q^{63} - 837 q^{65} - 829 q^{67} - 1860 q^{69} - 2005 q^{71} - 320 q^{73} - 2323 q^{75} + 1581 q^{79} - 3752 q^{81} - 692 q^{83} - 21 q^{85} + 131 q^{87} - 3205 q^{89} - 2724 q^{91} - 722 q^{93} + 432 q^{95} - 4702 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} - 87x^{6} + 12x^{5} + 2157x^{4} + 2939x^{3} - 5906x^{2} - 3030x + 45 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 1857162 \nu^{7} + 7310603 \nu^{6} + 142464129 \nu^{5} - 249340769 \nu^{4} + \cdots - 10004782434 ) / 1632011997 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 1028244 \nu^{7} + 1896398 \nu^{6} + 90935464 \nu^{5} - 3173169 \nu^{4} - 2285436761 \nu^{3} + \cdots + 1961828325 ) / 233144571 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 8732926 \nu^{7} + 20317527 \nu^{6} + 728311377 \nu^{5} - 199352026 \nu^{4} + \cdots + 38489516706 ) / 1632011997 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 11782151 \nu^{7} + 29812905 \nu^{6} + 976127689 \nu^{5} - 462694326 \nu^{4} + \cdots + 31028196768 ) / 1632011997 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 11853745 \nu^{7} + 33123546 \nu^{6} + 1004854302 \nu^{5} - 890230888 \nu^{4} + \cdots + 14853956190 ) / 1632011997 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3833 \nu^{7} - 7025 \nu^{6} - 338753 \nu^{5} + 45021 \nu^{4} + 8374339 \nu^{3} + 10484723 \nu^{2} + \cdots - 5793051 ) / 477057 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 24090145 \nu^{7} + 64649293 \nu^{6} + 1995783945 \nu^{5} - 1322345054 \nu^{4} + \cdots + 27147492357 ) / 1632011997 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{6} - 11\beta_{4} + 11\beta_{3} + 11\beta _1 + 5 ) / 44 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -13\beta_{6} - 22\beta_{5} - 33\beta_{4} + 11\beta_{3} + 22\beta_{2} + 121\beta _1 + 967 ) / 44 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -77\beta_{7} - 94\beta_{6} - 33\beta_{5} - 374\beta_{4} + 418\beta_{3} + 33\beta_{2} + 825\beta _1 + 2472 ) / 44 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 253 \beta_{7} - 214 \beta_{6} - 957 \beta_{5} - 2002 \beta_{4} + 1342 \beta_{3} + 1925 \beta_{2} + \cdots + 42276 ) / 44 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 542 \beta_{7} - 285 \beta_{6} - 116 \beta_{5} - 1545 \beta_{4} + 1669 \beta_{3} + 810 \beta_{2} + \cdots + 17841 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 31977 \beta_{7} + 5512 \beta_{6} - 32087 \beta_{5} - 105842 \beta_{4} + 89870 \beta_{3} + \cdots + 2137966 ) / 44 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 414315 \beta_{7} - 23725 \beta_{6} - 28061 \beta_{5} - 827585 \beta_{4} + 890659 \beta_{3} + \cdots + 12815089 ) / 44 \) 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
−0.459575
6.90062
1.40099
−3.22336
−5.59439
−4.92777
7.88904
0.0144476
0 −7.09770 0 13.4898 0 24.6683 0 23.3773 0
1.2 0 −6.24005 0 8.05999 0 25.5634 0 11.9382 0
1.3 0 −5.71590 0 −19.9654 0 −4.43442 0 5.67153 0
1.4 0 −5.64910 0 −12.1742 0 −33.8453 0 4.91235 0
1.5 0 1.03723 0 9.54657 0 −18.0044 0 −25.9242 0
1.6 0 2.31763 0 6.10149 0 −4.57313 0 −21.6286 0
1.7 0 5.54031 0 −6.21683 0 10.0967 0 3.69499 0
1.8 0 7.80759 0 −11.8414 0 9.52882 0 33.9584 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
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 968.4.a.o 8
4.b odd 2 1 1936.4.a.bv 8
11.b odd 2 1 968.4.a.n 8
11.d odd 10 2 88.4.i.a ✓ 16
44.c even 2 1 1936.4.a.bw 8
44.g even 10 2 176.4.m.e 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
88.4.i.a ✓ 16 11.d odd 10 2
176.4.m.e 16 44.g even 10 2
968.4.a.n 8 11.b odd 2 1
968.4.a.o 8 1.a even 1 1 trivial
1936.4.a.bv 8 4.b odd 2 1
1936.4.a.bw 8 44.c even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(968))\):

\( T_{3}^{8} + 8T_{3}^{7} - 94T_{3}^{6} - 820T_{3}^{5} + 2191T_{3}^{4} + 22724T_{3}^{3} - 12450T_{3}^{2} - 156300T_{3} + 148709 \) Copy content Toggle raw display
\( T_{7}^{8} - 9 T_{7}^{7} - 1442 T_{7}^{6} + 16947 T_{7}^{5} + 447082 T_{7}^{4} - 4842321 T_{7}^{3} + \cdots + 749733156 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + 8 T^{7} + \cdots + 148709 \) Copy content Toggle raw display
$5$ \( T^{8} + 13 T^{7} + \cdots + 113321764 \) Copy content Toggle raw display
$7$ \( T^{8} - 9 T^{7} + \cdots + 749733156 \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots - 31682715020 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots - 28271091339 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 706303884334059 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 47\!\cdots\!56 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots - 45\!\cdots\!36 \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots - 14\!\cdots\!64 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots - 98\!\cdots\!24 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots - 15\!\cdots\!39 \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 25\!\cdots\!84 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 53\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots - 14\!\cdots\!84 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 69\!\cdots\!19 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 65\!\cdots\!36 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots - 18\!\cdots\!20 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 99\!\cdots\!76 \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 28\!\cdots\!69 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots - 48\!\cdots\!36 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots - 34\!\cdots\!41 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 15\!\cdots\!20 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 20\!\cdots\!61 \) Copy content Toggle raw display
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