Properties

Label 1183.2.e.c
Level $1183$
Weight $2$
Character orbit 1183.e
Analytic conductor $9.446$
Analytic rank $0$
Dimension $2$
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] = [1183,2,Mod(170,1183)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1183, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1183.170");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1183 = 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1183.e (of order \(3\), 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: \(9.44630255912\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 91)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{6} q^{2} + ( - 3 \zeta_{6} + 3) q^{3} + ( - \zeta_{6} + 1) q^{4} + 3 \zeta_{6} q^{5} + 3 q^{6} + ( - 3 \zeta_{6} + 1) q^{7} + 3 q^{8} - 6 \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{6} q^{2} + ( - 3 \zeta_{6} + 3) q^{3} + ( - \zeta_{6} + 1) q^{4} + 3 \zeta_{6} q^{5} + 3 q^{6} + ( - 3 \zeta_{6} + 1) q^{7} + 3 q^{8} - 6 \zeta_{6} q^{9} + (3 \zeta_{6} - 3) q^{10} + (3 \zeta_{6} - 3) q^{11} - 3 \zeta_{6} q^{12} + ( - 2 \zeta_{6} + 3) q^{14} + 9 q^{15} + \zeta_{6} q^{16} + ( - 2 \zeta_{6} + 2) q^{17} + ( - 6 \zeta_{6} + 6) q^{18} - \zeta_{6} q^{19} + 3 q^{20} + ( - 3 \zeta_{6} - 6) q^{21} - 3 q^{22} + ( - 9 \zeta_{6} + 9) q^{24} + (4 \zeta_{6} - 4) q^{25} - 9 q^{27} + ( - \zeta_{6} - 2) q^{28} + 7 q^{29} + 9 \zeta_{6} q^{30} + ( - 3 \zeta_{6} + 3) q^{31} + ( - 5 \zeta_{6} + 5) q^{32} + 9 \zeta_{6} q^{33} + 2 q^{34} + ( - 6 \zeta_{6} + 9) q^{35} - 6 q^{36} + 2 \zeta_{6} q^{37} + ( - \zeta_{6} + 1) q^{38} + 9 \zeta_{6} q^{40} - 3 q^{41} + ( - 9 \zeta_{6} + 3) q^{42} - 7 q^{43} + 3 \zeta_{6} q^{44} + ( - 18 \zeta_{6} + 18) q^{45} + \zeta_{6} q^{47} + 3 q^{48} + (3 \zeta_{6} - 8) q^{49} - 4 q^{50} - 6 \zeta_{6} q^{51} + (3 \zeta_{6} - 3) q^{53} - 9 \zeta_{6} q^{54} - 9 q^{55} + ( - 9 \zeta_{6} + 3) q^{56} - 3 q^{57} + 7 \zeta_{6} q^{58} + (4 \zeta_{6} - 4) q^{59} + ( - 9 \zeta_{6} + 9) q^{60} + 13 \zeta_{6} q^{61} + 3 q^{62} + (12 \zeta_{6} - 18) q^{63} + 7 q^{64} + (9 \zeta_{6} - 9) q^{66} + (3 \zeta_{6} - 3) q^{67} - 2 \zeta_{6} q^{68} + (3 \zeta_{6} + 6) q^{70} - 13 q^{71} - 18 \zeta_{6} q^{72} + (13 \zeta_{6} - 13) q^{73} + (2 \zeta_{6} - 2) q^{74} + 12 \zeta_{6} q^{75} - q^{76} + (3 \zeta_{6} + 6) q^{77} + 3 \zeta_{6} q^{79} + (3 \zeta_{6} - 3) q^{80} + (9 \zeta_{6} - 9) q^{81} - 3 \zeta_{6} q^{82} + (6 \zeta_{6} - 9) q^{84} + 6 q^{85} - 7 \zeta_{6} q^{86} + ( - 21 \zeta_{6} + 21) q^{87} + (9 \zeta_{6} - 9) q^{88} + 6 \zeta_{6} q^{89} + 18 q^{90} - 9 \zeta_{6} q^{93} + (\zeta_{6} - 1) q^{94} + ( - 3 \zeta_{6} + 3) q^{95} - 15 \zeta_{6} q^{96} + 5 q^{97} + ( - 5 \zeta_{6} - 3) q^{98} + 18 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} + 3 q^{3} + q^{4} + 3 q^{5} + 6 q^{6} - q^{7} + 6 q^{8} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} + 3 q^{3} + q^{4} + 3 q^{5} + 6 q^{6} - q^{7} + 6 q^{8} - 6 q^{9} - 3 q^{10} - 3 q^{11} - 3 q^{12} + 4 q^{14} + 18 q^{15} + q^{16} + 2 q^{17} + 6 q^{18} - q^{19} + 6 q^{20} - 15 q^{21} - 6 q^{22} + 9 q^{24} - 4 q^{25} - 18 q^{27} - 5 q^{28} + 14 q^{29} + 9 q^{30} + 3 q^{31} + 5 q^{32} + 9 q^{33} + 4 q^{34} + 12 q^{35} - 12 q^{36} + 2 q^{37} + q^{38} + 9 q^{40} - 6 q^{41} - 3 q^{42} - 14 q^{43} + 3 q^{44} + 18 q^{45} + q^{47} + 6 q^{48} - 13 q^{49} - 8 q^{50} - 6 q^{51} - 3 q^{53} - 9 q^{54} - 18 q^{55} - 3 q^{56} - 6 q^{57} + 7 q^{58} - 4 q^{59} + 9 q^{60} + 13 q^{61} + 6 q^{62} - 24 q^{63} + 14 q^{64} - 9 q^{66} - 3 q^{67} - 2 q^{68} + 15 q^{70} - 26 q^{71} - 18 q^{72} - 13 q^{73} - 2 q^{74} + 12 q^{75} - 2 q^{76} + 15 q^{77} + 3 q^{79} - 3 q^{80} - 9 q^{81} - 3 q^{82} - 12 q^{84} + 12 q^{85} - 7 q^{86} + 21 q^{87} - 9 q^{88} + 6 q^{89} + 36 q^{90} - 9 q^{93} - q^{94} + 3 q^{95} - 15 q^{96} + 10 q^{97} - 11 q^{98} + 36 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1183\mathbb{Z}\right)^\times\).

\(n\) \(339\) \(1016\)
\(\chi(n)\) \(-\zeta_{6}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
170.1
0.500000 0.866025i
0.500000 + 0.866025i
0.500000 0.866025i 1.50000 + 2.59808i 0.500000 + 0.866025i 1.50000 2.59808i 3.00000 −0.500000 + 2.59808i 3.00000 −3.00000 + 5.19615i −1.50000 2.59808i
508.1 0.500000 + 0.866025i 1.50000 2.59808i 0.500000 0.866025i 1.50000 + 2.59808i 3.00000 −0.500000 2.59808i 3.00000 −3.00000 5.19615i −1.50000 + 2.59808i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1183.2.e.c 2
7.c even 3 1 inner 1183.2.e.c 2
7.c even 3 1 8281.2.a.c 1
7.d odd 6 1 8281.2.a.g 1
13.b even 2 1 1183.2.e.a 2
13.e even 6 1 91.2.g.a 2
13.e even 6 1 91.2.h.a yes 2
39.h odd 6 1 819.2.n.c 2
39.h odd 6 1 819.2.s.a 2
91.k even 6 1 91.2.g.a 2
91.k even 6 1 637.2.f.b 2
91.l odd 6 1 637.2.f.a 2
91.l odd 6 1 637.2.g.a 2
91.p odd 6 1 637.2.f.a 2
91.p odd 6 1 637.2.h.a 2
91.r even 6 1 1183.2.e.a 2
91.r even 6 1 8281.2.a.i 1
91.s odd 6 1 8281.2.a.j 1
91.t odd 6 1 637.2.g.a 2
91.t odd 6 1 637.2.h.a 2
91.u even 6 1 91.2.h.a yes 2
91.u even 6 1 637.2.f.b 2
273.x odd 6 1 819.2.s.a 2
273.bp odd 6 1 819.2.n.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.g.a 2 13.e even 6 1
91.2.g.a 2 91.k even 6 1
91.2.h.a yes 2 13.e even 6 1
91.2.h.a yes 2 91.u even 6 1
637.2.f.a 2 91.l odd 6 1
637.2.f.a 2 91.p odd 6 1
637.2.f.b 2 91.k even 6 1
637.2.f.b 2 91.u even 6 1
637.2.g.a 2 91.l odd 6 1
637.2.g.a 2 91.t odd 6 1
637.2.h.a 2 91.p odd 6 1
637.2.h.a 2 91.t odd 6 1
819.2.n.c 2 39.h odd 6 1
819.2.n.c 2 273.bp odd 6 1
819.2.s.a 2 39.h odd 6 1
819.2.s.a 2 273.x odd 6 1
1183.2.e.a 2 13.b even 2 1
1183.2.e.a 2 91.r even 6 1
1183.2.e.c 2 1.a even 1 1 trivial
1183.2.e.c 2 7.c even 3 1 inner
8281.2.a.c 1 7.c even 3 1
8281.2.a.g 1 7.d odd 6 1
8281.2.a.i 1 91.r even 6 1
8281.2.a.j 1 91.s odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1183, [\chi])\):

\( T_{2}^{2} - T_{2} + 1 \) Copy content Toggle raw display
\( T_{3}^{2} - 3T_{3} + 9 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$5$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$7$ \( T^{2} + T + 7 \) Copy content Toggle raw display
$11$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$19$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( (T - 7)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$37$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$41$ \( (T + 3)^{2} \) Copy content Toggle raw display
$43$ \( (T + 7)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$53$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$59$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$61$ \( T^{2} - 13T + 169 \) Copy content Toggle raw display
$67$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$71$ \( (T + 13)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 13T + 169 \) Copy content Toggle raw display
$79$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$97$ \( (T - 5)^{2} \) Copy content Toggle raw display
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