Properties

Label 968.2.i.q
Level $968$
Weight $2$
Character orbit 968.i
Analytic conductor $7.730$
Analytic rank $0$
Dimension $8$
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [968,2,Mod(9,968)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(968, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 0, 6]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("968.9");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 968 = 2^{3} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 968.i (of order \(5\), degree \(4\), 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: \(7.72951891566\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{5})\)
Coefficient field: 8.0.1305015625.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + 5x^{6} - 9x^{5} + 29x^{4} + 36x^{3} + 80x^{2} + 64x + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 88)
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{5} q^{3} + (\beta_{7} + 2 \beta_{6} + \beta_{5} + \cdots - 2) q^{5}+ \cdots + (\beta_{3} - \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{5} q^{3} + (\beta_{7} + 2 \beta_{6} + \beta_{5} + \cdots - 2) q^{5}+ \cdots + (14 \beta_{3} + \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - q^{3} - 3 q^{5} - 2 q^{7} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - q^{3} - 3 q^{5} - 2 q^{7} - 3 q^{9} - 2 q^{13} + 7 q^{15} + 4 q^{17} - 8 q^{19} + 72 q^{21} + 36 q^{23} - 3 q^{25} - 7 q^{27} - 2 q^{29} + 7 q^{31} + 14 q^{35} + 11 q^{37} + 16 q^{39} + 6 q^{41} + 24 q^{43} - 16 q^{45} - 16 q^{47} - 22 q^{49} + 2 q^{51} - 8 q^{53} - 4 q^{57} + 5 q^{59} - 6 q^{61} - 20 q^{63} + 80 q^{65} + 60 q^{67} - 13 q^{69} + 5 q^{71} + 2 q^{73} + 24 q^{75} - 14 q^{79} + 14 q^{81} + 10 q^{83} + 6 q^{85} - 64 q^{87} - 28 q^{89} + 32 q^{91} - 5 q^{93} - 12 q^{95} - 27 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} + 5x^{6} - 9x^{5} + 29x^{4} + 36x^{3} + 80x^{2} + 64x + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{7} - 181\nu^{2} ) / 464 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{6} + 65\nu ) / 116 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{5} + 36 ) / 29 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{7} + 5\nu^{6} - 9\nu^{5} + 29\nu^{4} - 145\nu^{3} + 80\nu^{2} + 64\nu + 256 ) / 464 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -5\nu^{7} + 25\nu^{6} - 45\nu^{5} + 145\nu^{4} - 261\nu^{3} + 400\nu^{2} + 320\nu + 1280 ) / 1856 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{7} - 65\nu^{2} ) / 116 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} - 4\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 4\beta_{6} - 5\beta_{5} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -9\beta_{7} + 20\beta_{6} - 9\beta_{5} + 9\beta_{4} - 20\beta_{3} + 20\beta_{2} + 9\beta _1 - 20 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 29\beta_{4} - 36 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 116\beta_{3} - 65\beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -181\beta_{7} + 260\beta_{2} \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/968\mathbb{Z}\right)^\times\).

\(n\) \(485\) \(727\) \(849\)
\(\chi(n)\) \(1\) \(1\) \(-\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1
−0.791563 + 2.43618i
0.482546 1.48512i
−1.26332 0.917858i
2.07234 + 1.50564i
−1.26332 + 0.917858i
2.07234 1.50564i
−0.791563 2.43618i
0.482546 + 1.48512i
0 −2.07234 + 1.50564i 0 −0.173529 0.534068i 0 −4.14468 3.01129i 0 1.10058 3.38724i 0
9.2 0 1.26332 0.917858i 0 1.10058 + 3.38724i 0 2.52665 + 1.83572i 0 −0.173529 + 0.534068i 0
81.1 0 −0.482546 + 1.48512i 0 −2.88136 + 2.09343i 0 −0.965093 2.97025i 0 0.454306 + 0.330072i 0
81.2 0 0.791563 2.43618i 0 0.454306 0.330072i 0 1.58313 + 4.87236i 0 −2.88136 2.09343i 0
729.1 0 −0.482546 1.48512i 0 −2.88136 2.09343i 0 −0.965093 + 2.97025i 0 0.454306 0.330072i 0
729.2 0 0.791563 + 2.43618i 0 0.454306 + 0.330072i 0 1.58313 4.87236i 0 −2.88136 + 2.09343i 0
753.1 0 −2.07234 1.50564i 0 −0.173529 + 0.534068i 0 −4.14468 + 3.01129i 0 1.10058 + 3.38724i 0
753.2 0 1.26332 + 0.917858i 0 1.10058 3.38724i 0 2.52665 1.83572i 0 −0.173529 0.534068i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 9.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 3 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 968.2.i.q 8
11.b odd 2 1 968.2.i.r 8
11.c even 5 1 968.2.a.j 2
11.c even 5 3 inner 968.2.i.q 8
11.d odd 10 1 88.2.a.b 2
11.d odd 10 3 968.2.i.r 8
33.f even 10 1 792.2.a.h 2
33.h odd 10 1 8712.2.a.bb 2
44.g even 10 1 176.2.a.d 2
44.h odd 10 1 1936.2.a.r 2
55.h odd 10 1 2200.2.a.o 2
55.l even 20 2 2200.2.b.g 4
77.l even 10 1 4312.2.a.n 2
88.k even 10 1 704.2.a.p 2
88.l odd 10 1 7744.2.a.cl 2
88.o even 10 1 7744.2.a.by 2
88.p odd 10 1 704.2.a.m 2
132.n odd 10 1 1584.2.a.t 2
176.u odd 20 2 2816.2.c.w 4
176.x even 20 2 2816.2.c.p 4
220.o even 10 1 4400.2.a.bp 2
220.w odd 20 2 4400.2.b.v 4
264.r odd 10 1 6336.2.a.cx 2
264.u even 10 1 6336.2.a.cu 2
308.s odd 10 1 8624.2.a.cb 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
88.2.a.b 2 11.d odd 10 1
176.2.a.d 2 44.g even 10 1
704.2.a.m 2 88.p odd 10 1
704.2.a.p 2 88.k even 10 1
792.2.a.h 2 33.f even 10 1
968.2.a.j 2 11.c even 5 1
968.2.i.q 8 1.a even 1 1 trivial
968.2.i.q 8 11.c even 5 3 inner
968.2.i.r 8 11.b odd 2 1
968.2.i.r 8 11.d odd 10 3
1584.2.a.t 2 132.n odd 10 1
1936.2.a.r 2 44.h odd 10 1
2200.2.a.o 2 55.h odd 10 1
2200.2.b.g 4 55.l even 20 2
2816.2.c.p 4 176.x even 20 2
2816.2.c.w 4 176.u odd 20 2
4312.2.a.n 2 77.l even 10 1
4400.2.a.bp 2 220.o even 10 1
4400.2.b.v 4 220.w odd 20 2
6336.2.a.cu 2 264.u even 10 1
6336.2.a.cx 2 264.r odd 10 1
7744.2.a.by 2 88.o even 10 1
7744.2.a.cl 2 88.l odd 10 1
8624.2.a.cb 2 308.s odd 10 1
8712.2.a.bb 2 33.h odd 10 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(968, [\chi])\):

\( T_{3}^{8} + T_{3}^{7} + 5T_{3}^{6} + 9T_{3}^{5} + 29T_{3}^{4} - 36T_{3}^{3} + 80T_{3}^{2} - 64T_{3} + 256 \) Copy content Toggle raw display
\( T_{7}^{8} + 2T_{7}^{7} + 20T_{7}^{6} + 72T_{7}^{5} + 464T_{7}^{4} - 1152T_{7}^{3} + 5120T_{7}^{2} - 8192T_{7} + 65536 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + T^{7} + \cdots + 256 \) Copy content Toggle raw display
$5$ \( T^{8} + 3 T^{7} + \cdots + 16 \) Copy content Toggle raw display
$7$ \( T^{8} + 2 T^{7} + \cdots + 65536 \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( T^{8} + 2 T^{7} + \cdots + 65536 \) Copy content Toggle raw display
$17$ \( (T^{4} - 2 T^{3} + 4 T^{2} + \cdots + 16)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 4 T^{3} + \cdots + 256)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - 9 T + 16)^{4} \) Copy content Toggle raw display
$29$ \( T^{8} + 2 T^{7} + \cdots + 65536 \) Copy content Toggle raw display
$31$ \( T^{8} - 7 T^{7} + \cdots + 4096 \) Copy content Toggle raw display
$37$ \( T^{8} - 11 T^{7} + \cdots + 456976 \) Copy content Toggle raw display
$41$ \( T^{8} - 6 T^{7} + \cdots + 4096 \) Copy content Toggle raw display
$43$ \( (T^{2} - 6 T - 8)^{4} \) Copy content Toggle raw display
$47$ \( (T^{4} + 8 T^{3} + \cdots + 4096)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} + 8 T^{7} + \cdots + 7311616 \) Copy content Toggle raw display
$59$ \( T^{8} - 5 T^{7} + \cdots + 100000000 \) Copy content Toggle raw display
$61$ \( T^{8} + 6 T^{7} + \cdots + 4096 \) Copy content Toggle raw display
$67$ \( (T^{2} - 15 T + 52)^{4} \) Copy content Toggle raw display
$71$ \( T^{8} - 5 T^{7} + \cdots + 1048576 \) Copy content Toggle raw display
$73$ \( T^{8} - 2 T^{7} + \cdots + 65536 \) Copy content Toggle raw display
$79$ \( T^{8} + 14 T^{7} + \cdots + 1048576 \) Copy content Toggle raw display
$83$ \( T^{8} - 10 T^{7} + \cdots + 4096 \) Copy content Toggle raw display
$89$ \( (T^{2} + 7 T - 26)^{4} \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 1003875856 \) Copy content Toggle raw display
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