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Properties

Label 968.2.i.b
Level $968$
Weight $2$
Character orbit 968.i
Analytic conductor $7.730$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

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: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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 primitive root of unity \(\zeta_{10}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 2 \zeta_{10}^{3} - 2 \zeta_{10}) q^{3} + ( - 2 \zeta_{10}^{2} + \zeta_{10} - 2) q^{5} + ( - 2 \zeta_{10}^{3} - 2 \zeta_{10} + 2) q^{7} + (5 \zeta_{10}^{3} - \zeta_{10}^{2} + \cdots - 5) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 2 \zeta_{10}^{3} - 2 \zeta_{10}) q^{3} + ( - 2 \zeta_{10}^{2} + \zeta_{10} - 2) q^{5} + ( - 2 \zeta_{10}^{3} - 2 \zeta_{10} + 2) q^{7} + (5 \zeta_{10}^{3} - \zeta_{10}^{2} + \cdots - 5) q^{9}+ \cdots + (7 \zeta_{10}^{3} + \zeta_{10}^{2} + \cdots - 7) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{3} - 5 q^{5} + 4 q^{7} - 13 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{3} - 5 q^{5} + 4 q^{7} - 13 q^{9} + q^{13} - q^{17} - 16 q^{19} - 24 q^{21} + 12 q^{23} + 20 q^{27} - 9 q^{29} - 2 q^{31} - 10 q^{35} - 7 q^{37} + 14 q^{39} + 7 q^{41} + 24 q^{43} + 40 q^{45} + 16 q^{47} - 9 q^{49} - 4 q^{51} + 27 q^{53} - 24 q^{57} + 20 q^{59} - 26 q^{61} + 12 q^{63} - 20 q^{65} - 12 q^{67} - 12 q^{69} + 4 q^{71} - 18 q^{73} + 10 q^{79} + 19 q^{81} - 26 q^{83} + 5 q^{85} + 4 q^{87} + 4 q^{89} + 16 q^{91} + 32 q^{93} + 20 q^{95} - 23 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

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

\(n\) \(485\) \(727\) \(849\)
\(\chi(n)\) \(1\) \(1\) \(-\zeta_{10}^{3}\)

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
−0.309017 − 0.951057i
0.809017 − 0.587785i
0.809017 + 0.587785i
−0.309017 + 0.951057i
0 −1.00000 + 0.726543i 0 −0.690983 − 2.12663i 0 1.00000 + 0.726543i 0 −0.454915 + 1.40008i 0
81.1 0 −1.00000 + 3.07768i 0 −1.80902 + 1.31433i 0 1.00000 + 3.07768i 0 −6.04508 − 4.39201i 0
729.1 0 −1.00000 − 3.07768i 0 −1.80902 − 1.31433i 0 1.00000 − 3.07768i 0 −6.04508 + 4.39201i 0
753.1 0 −1.00000 − 0.726543i 0 −0.690983 + 2.12663i 0 1.00000 − 0.726543i 0 −0.454915 − 1.40008i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 968.2.i.b 4
11.b odd 2 1 968.2.i.a 4
11.c even 5 1 968.2.a.g yes 2
11.c even 5 1 inner 968.2.i.b 4
11.c even 5 2 968.2.i.l 4
11.d odd 10 1 968.2.a.f ✓ 2
11.d odd 10 1 968.2.i.a 4
11.d odd 10 2 968.2.i.m 4
33.f even 10 1 8712.2.a.bj 2
33.h odd 10 1 8712.2.a.bk 2
44.g even 10 1 1936.2.a.x 2
44.h odd 10 1 1936.2.a.w 2
88.k even 10 1 7744.2.a.bs 2
88.l odd 10 1 7744.2.a.br 2
88.o even 10 1 7744.2.a.cu 2
88.p odd 10 1 7744.2.a.ct 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
968.2.a.f ✓ 2 11.d odd 10 1
968.2.a.g yes 2 11.c even 5 1
968.2.i.a 4 11.b odd 2 1
968.2.i.a 4 11.d odd 10 1
968.2.i.b 4 1.a even 1 1 trivial
968.2.i.b 4 11.c even 5 1 inner
968.2.i.l 4 11.c even 5 2
968.2.i.m 4 11.d odd 10 2
1936.2.a.w 2 44.h odd 10 1
1936.2.a.x 2 44.g even 10 1
7744.2.a.br 2 88.l odd 10 1
7744.2.a.bs 2 88.k even 10 1
7744.2.a.ct 2 88.p odd 10 1
7744.2.a.cu 2 88.o even 10 1
8712.2.a.bj 2 33.f even 10 1
8712.2.a.bk 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 \(S_{2}^{\mathrm{new}}(968, [\chi])\):

\( T_{3}^{4} + 4T_{3}^{3} + 16T_{3}^{2} + 24T_{3} + 16 \) Copy content Toggle raw display
\( T_{7}^{4} - 4T_{7}^{3} + 16T_{7}^{2} - 24T_{7} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 4 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$5$ \( T^{4} + 5 T^{3} + \cdots + 25 \) Copy content Toggle raw display
$7$ \( T^{4} - 4 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} - T^{3} + \cdots + 121 \) Copy content Toggle raw display
$17$ \( T^{4} + T^{3} + T^{2} + \cdots + 1 \) Copy content Toggle raw display
$19$ \( T^{4} + 16 T^{3} + \cdots + 1936 \) Copy content Toggle raw display
$23$ \( (T^{2} - 6 T + 4)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} + 9 T^{3} + \cdots + 121 \) Copy content Toggle raw display
$31$ \( T^{4} + 2 T^{3} + \cdots + 1936 \) Copy content Toggle raw display
$37$ \( T^{4} + 7 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$41$ \( T^{4} - 7 T^{3} + \cdots + 121 \) Copy content Toggle raw display
$43$ \( (T^{2} - 12 T + 16)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 16 T^{3} + \cdots + 1936 \) Copy content Toggle raw display
$53$ \( T^{4} - 27 T^{3} + \cdots + 14641 \) Copy content Toggle raw display
$59$ \( T^{4} - 20 T^{3} + \cdots + 6400 \) Copy content Toggle raw display
$61$ \( T^{4} + 26 T^{3} + \cdots + 1936 \) Copy content Toggle raw display
$67$ \( (T^{2} + 6 T + 4)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} - 4 T^{3} + \cdots + 256 \) Copy content Toggle raw display
$73$ \( T^{4} + 18 T^{3} + \cdots + 1936 \) Copy content Toggle raw display
$79$ \( T^{4} - 10 T^{3} + \cdots + 400 \) Copy content Toggle raw display
$83$ \( T^{4} + 26 T^{3} + \cdots + 15376 \) Copy content Toggle raw display
$89$ \( (T^{2} - 2 T - 179)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 23 T^{3} + \cdots + 5041 \) Copy content Toggle raw display
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