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Properties

Label 1155.1.e.d
Level $1155$
Weight $1$
Character orbit 1155.e
Analytic conductor $0.576$
Analytic rank $0$
Dimension $4$
Projective image $D_{6}$
CM discriminant -231
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1155,1,Mod(1154,1155)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1155, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1155.1154");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1155 = 3 \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1155.e (of order \(2\), degree \(1\), 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: \(0.576420089591\)
Analytic rank: \(0\)
Dimension: \(4\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{6}\)
Projective field: Galois closure of 6.0.46690875.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + ( - \zeta_{12}^{4} - \zeta_{12}^{2}) q^{2} + \zeta_{12}^{3} q^{3} + (\zeta_{12}^{4} - \zeta_{12}^{2} - 1) q^{4} - \zeta_{12} q^{5} + ( - \zeta_{12}^{5} + \zeta_{12}) q^{6} + \zeta_{12}^{3} q^{7} + (\zeta_{12}^{4} + \zeta_{12}^{2}) q^{8} - q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{12}^{4} - \zeta_{12}^{2}) q^{2} + \zeta_{12}^{3} q^{3} + (\zeta_{12}^{4} - \zeta_{12}^{2} - 1) q^{4} - \zeta_{12} q^{5} + ( - \zeta_{12}^{5} + \zeta_{12}) q^{6} + \zeta_{12}^{3} q^{7} + (\zeta_{12}^{4} + \zeta_{12}^{2}) q^{8} - q^{9} + (\zeta_{12}^{5} + \zeta_{12}^{3}) q^{10} + q^{11} + ( - \zeta_{12}^{5} - \zeta_{12}^{3} - \zeta_{12}) q^{12} + \zeta_{12}^{3} q^{13} + ( - \zeta_{12}^{5} + \zeta_{12}) q^{14} - \zeta_{12}^{4} q^{15} + q^{16} + (\zeta_{12}^{4} + \zeta_{12}^{2}) q^{18} + ( - \zeta_{12}^{5} + \zeta_{12}) q^{19} + ( - \zeta_{12}^{5} + \zeta_{12}^{3} + \zeta_{12}) q^{20} - q^{21} + ( - \zeta_{12}^{4} - \zeta_{12}^{2}) q^{22} + (\zeta_{12}^{5} - \zeta_{12}) q^{24} + \zeta_{12}^{2} q^{25} + ( - \zeta_{12}^{5} + \zeta_{12}) q^{26} - \zeta_{12}^{3} q^{27} + ( - \zeta_{12}^{5} - \zeta_{12}^{3} - \zeta_{12}) q^{28} + q^{29} + ( - \zeta_{12}^{2} - 1) q^{30} + \zeta_{12}^{3} q^{33} - \zeta_{12}^{4} q^{35} + ( - \zeta_{12}^{4} + \zeta_{12}^{2} + 1) q^{36} + (\zeta_{12}^{4} + \zeta_{12}^{2}) q^{37} + ( - \zeta_{12}^{5} - \zeta_{12}^{3} - \zeta_{12}) q^{38} - q^{39} + ( - \zeta_{12}^{5} - \zeta_{12}^{3}) q^{40} + (\zeta_{12}^{4} + \zeta_{12}^{2}) q^{42} + (\zeta_{12}^{4} - \zeta_{12}^{2} - 1) q^{44} + \zeta_{12} q^{45} + \zeta_{12}^{3} q^{47} + \zeta_{12}^{3} q^{48} - q^{49} + ( - \zeta_{12}^{4} + 1) q^{50} + ( - \zeta_{12}^{5} - \zeta_{12}^{3} - \zeta_{12}) q^{52} + (\zeta_{12}^{5} - \zeta_{12}) q^{54} - \zeta_{12} q^{55} + (\zeta_{12}^{5} - \zeta_{12}) q^{56} + (\zeta_{12}^{4} + \zeta_{12}^{2}) q^{57} + ( - \zeta_{12}^{4} - \zeta_{12}^{2}) q^{58} + (\zeta_{12}^{5} - \zeta_{12}) q^{59} + (\zeta_{12}^{4} + \zeta_{12}^{2} - 1) q^{60} - \zeta_{12}^{3} q^{63} + q^{64} - \zeta_{12}^{4} q^{65} + ( - \zeta_{12}^{5} + \zeta_{12}) q^{66} + ( - \zeta_{12}^{4} - \zeta_{12}^{2}) q^{67} + ( - \zeta_{12}^{2} - 1) q^{70} + ( - \zeta_{12}^{4} - \zeta_{12}^{2}) q^{72} - \zeta_{12}^{3} q^{73} + ( - \zeta_{12}^{4} + \zeta_{12}^{2} + 2) q^{74} + \zeta_{12}^{5} q^{75} + (2 \zeta_{12}^{5} + \zeta_{12}^{3} - \zeta_{12}) q^{76} + \zeta_{12}^{3} q^{77} + (\zeta_{12}^{4} + \zeta_{12}^{2}) q^{78} - \zeta_{12} q^{80} + q^{81} + ( - \zeta_{12}^{4} + \zeta_{12}^{2} + 1) q^{84} + \zeta_{12}^{3} q^{87} + (\zeta_{12}^{4} + \zeta_{12}^{2}) q^{88} + ( - \zeta_{12}^{5} - \zeta_{12}^{3}) q^{90} - q^{91} + ( - \zeta_{12}^{5} + \zeta_{12}) q^{94} + ( - \zeta_{12}^{2} - 1) q^{95} + (\zeta_{12}^{4} + \zeta_{12}^{2}) q^{98} - q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{4} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 8 q^{4} - 4 q^{9} + 4 q^{11} + 2 q^{15} + 4 q^{16} - 4 q^{21} + 2 q^{25} + 4 q^{29} - 6 q^{30} + 2 q^{35} + 8 q^{36} - 4 q^{39} - 8 q^{44} - 4 q^{49} + 6 q^{50} - 4 q^{60} + 4 q^{64} + 2 q^{65} - 6 q^{70} + 12 q^{74} + 4 q^{81} + 8 q^{84} - 4 q^{91} - 6 q^{95} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

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

\(n\) \(211\) \(232\) \(386\) \(661\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1154.1
−0.866025 − 0.500000i
0.866025 + 0.500000i
0.866025 − 0.500000i
−0.866025 + 0.500000i
− 1.73205i − 1.00000i −2.00000 0.866025 + 0.500000i −1.73205 − 1.00000i 1.73205i −1.00000 0.866025 − 1.50000i
1154.2 − 1.73205i 1.00000i −2.00000 −0.866025 − 0.500000i 1.73205 1.00000i 1.73205i −1.00000 −0.866025 + 1.50000i
1154.3 1.73205i − 1.00000i −2.00000 −0.866025 + 0.500000i 1.73205 − 1.00000i − 1.73205i −1.00000 −0.866025 − 1.50000i
1154.4 1.73205i 1.00000i −2.00000 0.866025 − 0.500000i −1.73205 1.00000i − 1.73205i −1.00000 0.866025 + 1.50000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
231.h odd 2 1 CM by \(\Q(\sqrt{-231}) \)
5.b even 2 1 inner
7.b odd 2 1 inner
33.d even 2 1 inner
35.c odd 2 1 inner
165.d even 2 1 inner
1155.e odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1155.1.e.d yes 4
3.b odd 2 1 1155.1.e.c ✓ 4
5.b even 2 1 inner 1155.1.e.d yes 4
7.b odd 2 1 inner 1155.1.e.d yes 4
11.b odd 2 1 1155.1.e.c ✓ 4
15.d odd 2 1 1155.1.e.c ✓ 4
21.c even 2 1 1155.1.e.c ✓ 4
33.d even 2 1 inner 1155.1.e.d yes 4
35.c odd 2 1 inner 1155.1.e.d yes 4
55.d odd 2 1 1155.1.e.c ✓ 4
77.b even 2 1 1155.1.e.c ✓ 4
105.g even 2 1 1155.1.e.c ✓ 4
165.d even 2 1 inner 1155.1.e.d yes 4
231.h odd 2 1 CM 1155.1.e.d yes 4
385.h even 2 1 1155.1.e.c ✓ 4
1155.e odd 2 1 inner 1155.1.e.d yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1155.1.e.c ✓ 4 3.b odd 2 1
1155.1.e.c ✓ 4 11.b odd 2 1
1155.1.e.c ✓ 4 15.d odd 2 1
1155.1.e.c ✓ 4 21.c even 2 1
1155.1.e.c ✓ 4 55.d odd 2 1
1155.1.e.c ✓ 4 77.b even 2 1
1155.1.e.c ✓ 4 105.g even 2 1
1155.1.e.c ✓ 4 385.h even 2 1
1155.1.e.d yes 4 1.a even 1 1 trivial
1155.1.e.d yes 4 5.b even 2 1 inner
1155.1.e.d yes 4 7.b odd 2 1 inner
1155.1.e.d yes 4 33.d even 2 1 inner
1155.1.e.d yes 4 35.c odd 2 1 inner
1155.1.e.d yes 4 165.d even 2 1 inner
1155.1.e.d yes 4 231.h odd 2 1 CM
1155.1.e.d yes 4 1155.e odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(1155, [\chi])\):

\( T_{2}^{2} + 3 \) Copy content Toggle raw display
\( T_{29} - 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

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