Properties

Label 1155.2.q.e
Level $1155$
Weight $2$
Character orbit 1155.q
Analytic conductor $9.223$
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] = [1155,2,Mod(331,1155)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1155, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 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("1155.331");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1155 = 3 \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1155.q (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: \(9.22272143346\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + 2 \zeta_{6} q^{2} + (\zeta_{6} - 1) q^{3} + (2 \zeta_{6} - 2) q^{4} - \zeta_{6} q^{5} - 2 q^{6} + ( - 3 \zeta_{6} + 1) q^{7} - \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + 2 \zeta_{6} q^{2} + (\zeta_{6} - 1) q^{3} + (2 \zeta_{6} - 2) q^{4} - \zeta_{6} q^{5} - 2 q^{6} + ( - 3 \zeta_{6} + 1) q^{7} - \zeta_{6} q^{9} + ( - 2 \zeta_{6} + 2) q^{10} + ( - \zeta_{6} + 1) q^{11} - 2 \zeta_{6} q^{12} + 3 q^{13} + ( - 4 \zeta_{6} + 6) q^{14} + q^{15} + 4 \zeta_{6} q^{16} + ( - 4 \zeta_{6} + 4) q^{17} + ( - 2 \zeta_{6} + 2) q^{18} + 2 \zeta_{6} q^{19} + 2 q^{20} + (\zeta_{6} + 2) q^{21} + 2 q^{22} + \zeta_{6} q^{23} + (\zeta_{6} - 1) q^{25} + 6 \zeta_{6} q^{26} + q^{27} + (2 \zeta_{6} + 4) q^{28} + q^{29} + 2 \zeta_{6} q^{30} + ( - 8 \zeta_{6} + 8) q^{31} + (8 \zeta_{6} - 8) q^{32} + \zeta_{6} q^{33} + 8 q^{34} + (2 \zeta_{6} - 3) q^{35} + 2 q^{36} + 2 \zeta_{6} q^{37} + (4 \zeta_{6} - 4) q^{38} + (3 \zeta_{6} - 3) q^{39} + 9 q^{41} + (6 \zeta_{6} - 2) q^{42} + q^{43} + 2 \zeta_{6} q^{44} + (\zeta_{6} - 1) q^{45} + (2 \zeta_{6} - 2) q^{46} + 7 \zeta_{6} q^{47} - 4 q^{48} + (3 \zeta_{6} - 8) q^{49} - 2 q^{50} + 4 \zeta_{6} q^{51} + (6 \zeta_{6} - 6) q^{52} + (7 \zeta_{6} - 7) q^{53} + 2 \zeta_{6} q^{54} - q^{55} - 2 q^{57} + 2 \zeta_{6} q^{58} + (4 \zeta_{6} - 4) q^{59} + (2 \zeta_{6} - 2) q^{60} - 10 \zeta_{6} q^{61} + 16 q^{62} + (2 \zeta_{6} - 3) q^{63} - 8 q^{64} - 3 \zeta_{6} q^{65} + (2 \zeta_{6} - 2) q^{66} + ( - 8 \zeta_{6} + 8) q^{67} + 8 \zeta_{6} q^{68} - q^{69} + ( - 2 \zeta_{6} - 4) q^{70} - 6 q^{71} + ( - 14 \zeta_{6} + 14) q^{73} + (4 \zeta_{6} - 4) q^{74} - \zeta_{6} q^{75} - 4 q^{76} + ( - \zeta_{6} - 2) q^{77} - 6 q^{78} + 4 \zeta_{6} q^{79} + ( - 4 \zeta_{6} + 4) q^{80} + (\zeta_{6} - 1) q^{81} + 18 \zeta_{6} q^{82} - 16 q^{83} + (4 \zeta_{6} - 6) q^{84} - 4 q^{85} + 2 \zeta_{6} q^{86} + (\zeta_{6} - 1) q^{87} - 6 \zeta_{6} q^{89} - 2 q^{90} + ( - 9 \zeta_{6} + 3) q^{91} - 2 q^{92} + 8 \zeta_{6} q^{93} + (14 \zeta_{6} - 14) q^{94} + ( - 2 \zeta_{6} + 2) q^{95} - 8 \zeta_{6} q^{96} + 12 q^{97} + ( - 10 \zeta_{6} - 6) q^{98} - q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - q^{3} - 2 q^{4} - q^{5} - 4 q^{6} - q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} - q^{3} - 2 q^{4} - q^{5} - 4 q^{6} - q^{7} - q^{9} + 2 q^{10} + q^{11} - 2 q^{12} + 6 q^{13} + 8 q^{14} + 2 q^{15} + 4 q^{16} + 4 q^{17} + 2 q^{18} + 2 q^{19} + 4 q^{20} + 5 q^{21} + 4 q^{22} + q^{23} - q^{25} + 6 q^{26} + 2 q^{27} + 10 q^{28} + 2 q^{29} + 2 q^{30} + 8 q^{31} - 8 q^{32} + q^{33} + 16 q^{34} - 4 q^{35} + 4 q^{36} + 2 q^{37} - 4 q^{38} - 3 q^{39} + 18 q^{41} + 2 q^{42} + 2 q^{43} + 2 q^{44} - q^{45} - 2 q^{46} + 7 q^{47} - 8 q^{48} - 13 q^{49} - 4 q^{50} + 4 q^{51} - 6 q^{52} - 7 q^{53} + 2 q^{54} - 2 q^{55} - 4 q^{57} + 2 q^{58} - 4 q^{59} - 2 q^{60} - 10 q^{61} + 32 q^{62} - 4 q^{63} - 16 q^{64} - 3 q^{65} - 2 q^{66} + 8 q^{67} + 8 q^{68} - 2 q^{69} - 10 q^{70} - 12 q^{71} + 14 q^{73} - 4 q^{74} - q^{75} - 8 q^{76} - 5 q^{77} - 12 q^{78} + 4 q^{79} + 4 q^{80} - q^{81} + 18 q^{82} - 32 q^{83} - 8 q^{84} - 8 q^{85} + 2 q^{86} - q^{87} - 6 q^{89} - 4 q^{90} - 3 q^{91} - 4 q^{92} + 8 q^{93} - 14 q^{94} + 2 q^{95} - 8 q^{96} + 24 q^{97} - 22 q^{98} - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1155\mathbb{Z}\right)^\times\).

\(n\) \(211\) \(232\) \(386\) \(661\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-\zeta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
331.1
0.500000 0.866025i
0.500000 + 0.866025i
1.00000 1.73205i −0.500000 0.866025i −1.00000 1.73205i −0.500000 + 0.866025i −2.00000 −0.500000 + 2.59808i 0 −0.500000 + 0.866025i 1.00000 + 1.73205i
991.1 1.00000 + 1.73205i −0.500000 + 0.866025i −1.00000 + 1.73205i −0.500000 0.866025i −2.00000 −0.500000 2.59808i 0 −0.500000 0.866025i 1.00000 1.73205i
\(n\): 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 1155.2.q.e 2
7.c even 3 1 inner 1155.2.q.e 2
7.c even 3 1 8085.2.a.e 1
7.d odd 6 1 8085.2.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1155.2.q.e 2 1.a even 1 1 trivial
1155.2.q.e 2 7.c even 3 1 inner
8085.2.a.a 1 7.d odd 6 1
8085.2.a.e 1 7.c even 3 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1155, [\chi])\):

\( T_{2}^{2} - 2T_{2} + 4 \) Copy content Toggle raw display
\( T_{13} - 3 \) Copy content Toggle raw display
\( T_{17}^{2} - 4T_{17} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$3$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$5$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$7$ \( T^{2} + T + 7 \) Copy content Toggle raw display
$11$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$13$ \( (T - 3)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$19$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$23$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$29$ \( (T - 1)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 8T + 64 \) Copy content Toggle raw display
$37$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$41$ \( (T - 9)^{2} \) Copy content Toggle raw display
$43$ \( (T - 1)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$53$ \( T^{2} + 7T + 49 \) Copy content Toggle raw display
$59$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$61$ \( T^{2} + 10T + 100 \) Copy content Toggle raw display
$67$ \( T^{2} - 8T + 64 \) Copy content Toggle raw display
$71$ \( (T + 6)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 14T + 196 \) Copy content Toggle raw display
$79$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$83$ \( (T + 16)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$97$ \( (T - 12)^{2} \) Copy content Toggle raw display
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