Properties

Label 1183.2.e.e
Level $1183$
Weight $2$
Character orbit 1183.e
Analytic conductor $9.446$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \zeta_{12}^{3} - \zeta_{12}) q^{2} + ( - \zeta_{12}^{2} + 1) q^{3} + (\zeta_{12}^{2} - 1) q^{4} + (\zeta_{12}^{3} + \zeta_{12}) q^{5} + (\zeta_{12}^{3} - 2 \zeta_{12}) q^{6} + (2 \zeta_{12}^{3} - 3 \zeta_{12}) q^{7} + (\zeta_{12}^{3} - 2 \zeta_{12}) q^{8} + 2 \zeta_{12}^{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{12}^{3} - \zeta_{12}) q^{2} + ( - \zeta_{12}^{2} + 1) q^{3} + (\zeta_{12}^{2} - 1) q^{4} + (\zeta_{12}^{3} + \zeta_{12}) q^{5} + (\zeta_{12}^{3} - 2 \zeta_{12}) q^{6} + (2 \zeta_{12}^{3} - 3 \zeta_{12}) q^{7} + (\zeta_{12}^{3} - 2 \zeta_{12}) q^{8} + 2 \zeta_{12}^{2} q^{9} + ( - 3 \zeta_{12}^{2} + 3) q^{10} + ( - 6 \zeta_{12}^{3} + 3 \zeta_{12}) q^{11} + \zeta_{12}^{2} q^{12} + (4 \zeta_{12}^{2} + 1) q^{14} + ( - \zeta_{12}^{3} + 2 \zeta_{12}) q^{15} + 5 \zeta_{12}^{2} q^{16} + ( - 6 \zeta_{12}^{2} + 6) q^{17} + ( - 4 \zeta_{12}^{3} + 2 \zeta_{12}) q^{18} + (\zeta_{12}^{3} + \zeta_{12}) q^{19} + (\zeta_{12}^{3} - 2 \zeta_{12}) q^{20} + (3 \zeta_{12}^{3} - \zeta_{12}) q^{21} - 9 q^{22} + (2 \zeta_{12}^{3} - \zeta_{12}) q^{24} + ( - 2 \zeta_{12}^{2} + 2) q^{25} + 5 q^{27} + ( - 3 \zeta_{12}^{3} + \zeta_{12}) q^{28} + 3 q^{29} - 3 \zeta_{12}^{2} q^{30} + ( - 2 \zeta_{12}^{3} + \zeta_{12}) q^{31} + ( - 6 \zeta_{12}^{3} + 3 \zeta_{12}) q^{32} + ( - 3 \zeta_{12}^{3} - 3 \zeta_{12}) q^{33} + (6 \zeta_{12}^{3} - 12 \zeta_{12}) q^{34} + ( - 4 \zeta_{12}^{2} - 1) q^{35} - 2 q^{36} + ( - 3 \zeta_{12}^{2} + 3) q^{38} - 3 \zeta_{12}^{2} q^{40} + (3 \zeta_{12}^{3} - 6 \zeta_{12}) q^{41} + ( - \zeta_{12}^{2} + 5) q^{42} - 11 q^{43} + (3 \zeta_{12}^{3} + 3 \zeta_{12}) q^{44} + (4 \zeta_{12}^{3} - 2 \zeta_{12}) q^{45} + ( - 5 \zeta_{12}^{3} - 5 \zeta_{12}) q^{47} + 5 q^{48} + ( - 3 \zeta_{12}^{2} + 8) q^{49} + (2 \zeta_{12}^{3} - 4 \zeta_{12}) q^{50} - 6 \zeta_{12}^{2} q^{51} + ( - 9 \zeta_{12}^{2} + 9) q^{53} + ( - 5 \zeta_{12}^{3} - 5 \zeta_{12}) q^{54} + 9 q^{55} + ( - \zeta_{12}^{2} + 5) q^{56} + ( - \zeta_{12}^{3} + 2 \zeta_{12}) q^{57} + ( - 3 \zeta_{12}^{3} - 3 \zeta_{12}) q^{58} + (4 \zeta_{12}^{3} - 2 \zeta_{12}) q^{59} + (2 \zeta_{12}^{3} - \zeta_{12}) q^{60} - 7 \zeta_{12}^{2} q^{61} - 3 q^{62} + ( - 2 \zeta_{12}^{3} - 4 \zeta_{12}) q^{63} + q^{64} + (9 \zeta_{12}^{2} - 9) q^{66} + ( - 10 \zeta_{12}^{3} + 5 \zeta_{12}) q^{67} + 6 \zeta_{12}^{2} q^{68} + (9 \zeta_{12}^{3} - 3 \zeta_{12}) q^{70} + (\zeta_{12}^{3} - 2 \zeta_{12}) q^{71} + ( - 2 \zeta_{12}^{3} - 2 \zeta_{12}) q^{72} + ( - 10 \zeta_{12}^{3} + 5 \zeta_{12}) q^{73} - 2 \zeta_{12}^{2} q^{75} + (\zeta_{12}^{3} - 2 \zeta_{12}) q^{76} + (15 \zeta_{12}^{2} - 12) q^{77} + 5 \zeta_{12}^{2} q^{79} + (10 \zeta_{12}^{3} - 5 \zeta_{12}) q^{80} + (\zeta_{12}^{2} - 1) q^{81} + 9 \zeta_{12}^{2} q^{82} + ( - 2 \zeta_{12}^{3} + 4 \zeta_{12}) q^{83} + ( - \zeta_{12}^{3} - 2 \zeta_{12}) q^{84} + ( - 6 \zeta_{12}^{3} + 12 \zeta_{12}) q^{85} + (11 \zeta_{12}^{3} + 11 \zeta_{12}) q^{86} + ( - 3 \zeta_{12}^{2} + 3) q^{87} + (9 \zeta_{12}^{2} - 9) q^{88} + (4 \zeta_{12}^{3} + 4 \zeta_{12}) q^{89} + 6 q^{90} + ( - \zeta_{12}^{3} - \zeta_{12}) q^{93} + (15 \zeta_{12}^{2} - 15) q^{94} + (3 \zeta_{12}^{2} - 3) q^{95} + ( - 3 \zeta_{12}^{3} - 3 \zeta_{12}) q^{96} + ( - 3 \zeta_{12}^{3} + 6 \zeta_{12}) q^{97} + ( - 2 \zeta_{12}^{3} - 11 \zeta_{12}) q^{98} + ( - 6 \zeta_{12}^{3} + 12 \zeta_{12}) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{3} - 2 q^{4} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{3} - 2 q^{4} + 4 q^{9} + 6 q^{10} + 2 q^{12} + 12 q^{14} + 10 q^{16} + 12 q^{17} - 36 q^{22} + 4 q^{25} + 20 q^{27} + 12 q^{29} - 6 q^{30} - 12 q^{35} - 8 q^{36} + 6 q^{38} - 6 q^{40} + 18 q^{42} - 44 q^{43} + 20 q^{48} + 26 q^{49} - 12 q^{51} + 18 q^{53} + 36 q^{55} + 18 q^{56} - 14 q^{61} - 12 q^{62} + 4 q^{64} - 18 q^{66} + 12 q^{68} - 4 q^{75} - 18 q^{77} + 10 q^{79} - 2 q^{81} + 18 q^{82} + 6 q^{87} - 18 q^{88} + 24 q^{90} - 30 q^{94} - 6 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(339\) \(1016\)
\(\chi(n)\) \(-\zeta_{12}^{2}\) \(1\)

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
0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
−0.866025 0.500000i
−0.866025 + 1.50000i 0.500000 + 0.866025i −0.500000 0.866025i 0.866025 1.50000i −1.73205 −2.59808 0.500000i −1.73205 1.00000 1.73205i 1.50000 + 2.59808i
170.2 0.866025 1.50000i 0.500000 + 0.866025i −0.500000 0.866025i −0.866025 + 1.50000i 1.73205 2.59808 + 0.500000i 1.73205 1.00000 1.73205i 1.50000 + 2.59808i
508.1 −0.866025 1.50000i 0.500000 0.866025i −0.500000 + 0.866025i 0.866025 + 1.50000i −1.73205 −2.59808 + 0.500000i −1.73205 1.00000 + 1.73205i 1.50000 2.59808i
508.2 0.866025 + 1.50000i 0.500000 0.866025i −0.500000 + 0.866025i −0.866025 1.50000i 1.73205 2.59808 0.500000i 1.73205 1.00000 + 1.73205i 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
13.b even 2 1 inner
91.r even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1183.2.e.e 4
7.c even 3 1 inner 1183.2.e.e 4
7.c even 3 1 8281.2.a.s 2
7.d odd 6 1 8281.2.a.w 2
13.b even 2 1 inner 1183.2.e.e 4
13.f odd 12 1 91.2.k.a 2
13.f odd 12 1 91.2.u.a yes 2
39.k even 12 1 819.2.bm.a 2
39.k even 12 1 819.2.do.c 2
91.r even 6 1 inner 1183.2.e.e 4
91.r even 6 1 8281.2.a.s 2
91.s odd 6 1 8281.2.a.w 2
91.w even 12 1 637.2.k.b 2
91.w even 12 1 637.2.q.b 2
91.x odd 12 1 91.2.u.a yes 2
91.x odd 12 1 637.2.q.c 2
91.ba even 12 1 637.2.q.b 2
91.ba even 12 1 637.2.u.a 2
91.bc even 12 1 637.2.k.b 2
91.bc even 12 1 637.2.u.a 2
91.bd odd 12 1 91.2.k.a 2
91.bd odd 12 1 637.2.q.c 2
273.bv even 12 1 819.2.do.c 2
273.bw even 12 1 819.2.bm.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.k.a 2 13.f odd 12 1
91.2.k.a 2 91.bd odd 12 1
91.2.u.a yes 2 13.f odd 12 1
91.2.u.a yes 2 91.x odd 12 1
637.2.k.b 2 91.w even 12 1
637.2.k.b 2 91.bc even 12 1
637.2.q.b 2 91.w even 12 1
637.2.q.b 2 91.ba even 12 1
637.2.q.c 2 91.x odd 12 1
637.2.q.c 2 91.bd odd 12 1
637.2.u.a 2 91.ba even 12 1
637.2.u.a 2 91.bc even 12 1
819.2.bm.a 2 39.k even 12 1
819.2.bm.a 2 273.bw even 12 1
819.2.do.c 2 39.k even 12 1
819.2.do.c 2 273.bv even 12 1
1183.2.e.e 4 1.a even 1 1 trivial
1183.2.e.e 4 7.c even 3 1 inner
1183.2.e.e 4 13.b even 2 1 inner
1183.2.e.e 4 91.r even 6 1 inner
8281.2.a.s 2 7.c even 3 1
8281.2.a.s 2 91.r even 6 1
8281.2.a.w 2 7.d odd 6 1
8281.2.a.w 2 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 \(S_{2}^{\mathrm{new}}(1183, [\chi])\):

\( T_{2}^{4} + 3T_{2}^{2} + 9 \) Copy content Toggle raw display
\( T_{3}^{2} - T_{3} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$3$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$7$ \( T^{4} - 13T^{2} + 49 \) Copy content Toggle raw display
$11$ \( T^{4} + 27T^{2} + 729 \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} - 6 T + 36)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( (T - 3)^{4} \) Copy content Toggle raw display
$31$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$37$ \( T^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} - 27)^{2} \) Copy content Toggle raw display
$43$ \( (T + 11)^{4} \) Copy content Toggle raw display
$47$ \( T^{4} + 75T^{2} + 5625 \) Copy content Toggle raw display
$53$ \( (T^{2} - 9 T + 81)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 12T^{2} + 144 \) Copy content Toggle raw display
$61$ \( (T^{2} + 7 T + 49)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 75T^{2} + 5625 \) Copy content Toggle raw display
$71$ \( (T^{2} - 3)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 75T^{2} + 5625 \) Copy content Toggle raw display
$79$ \( (T^{2} - 5 T + 25)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 48T^{2} + 2304 \) Copy content Toggle raw display
$97$ \( (T^{2} - 27)^{2} \) Copy content Toggle raw display
show more
show less