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Properties

Label 1191.1.bb.a
Level $1191$
Weight $1$
Character orbit 1191.bb
Analytic conductor $0.594$
Analytic rank $0$
Dimension $20$
Projective image $D_{33}$
CM discriminant -3
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1191,1,Mod(110,1191)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1191, base_ring=CyclotomicField(66))
 
chi = DirichletCharacter(H, H._module([33, 64]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1191.110");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1191 = 3 \cdot 397 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1191.bb (of order \(66\), degree \(20\), 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: \(0.594386430046\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\Q(\zeta_{33})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - x^{19} + x^{17} - x^{16} + x^{14} - x^{13} + x^{11} - x^{10} + x^{9} - x^{7} + x^{6} - x^{4} + x^{3} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{33}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{33} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{66}^{17} q^{3} + \zeta_{66}^{24} q^{4} - \zeta_{66}^{32} q^{7} - \zeta_{66} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{66}^{17} q^{3} + \zeta_{66}^{24} q^{4} - \zeta_{66}^{32} q^{7} - \zeta_{66} q^{9} + \zeta_{66}^{8} q^{12} + ( - \zeta_{66}^{21} + \zeta_{66}^{16}) q^{13} - \zeta_{66}^{15} q^{16} + ( - \zeta_{66}^{19} + \zeta_{66}^{6}) q^{19} - \zeta_{66}^{16} q^{21} + \zeta_{66}^{2} q^{25} + \zeta_{66}^{18} q^{27} + \zeta_{66}^{23} q^{28} + ( - \zeta_{66}^{13} + \zeta_{66}^{8}) q^{31} - \zeta_{66}^{25} q^{36} + ( - \zeta_{66}^{23} + \zeta_{66}^{20}) q^{37} + ( - \zeta_{66}^{5} + 1) q^{39} + ( - \zeta_{66}^{13} - \zeta_{66}^{11}) q^{43} + \zeta_{66}^{32} q^{48} + (\zeta_{66}^{12} - \zeta_{66}^{7}) q^{52} + ( - \zeta_{66}^{23} - \zeta_{66}^{3}) q^{57} + ( - \zeta_{66}^{15} + \zeta_{66}^{14}) q^{61} - q^{63} + \zeta_{66}^{6} q^{64} + ( - \zeta_{66}^{29} + \zeta_{66}^{14}) q^{67} + (\zeta_{66}^{28} + \zeta_{66}^{24}) q^{73} - \zeta_{66}^{19} q^{75} + (\zeta_{66}^{30} + \zeta_{66}^{10}) q^{76} + (\zeta_{66}^{28} - \zeta_{66}^{27}) q^{79} + \zeta_{66}^{2} q^{81} + \zeta_{66}^{7} q^{84} + ( - \zeta_{66}^{20} + \zeta_{66}^{15}) q^{91} + (\zeta_{66}^{30} - \zeta_{66}^{25}) q^{93} + ( - \zeta_{66}^{17} - \zeta_{66}^{3}) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + q^{3} - 2 q^{4} - q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + q^{3} - 2 q^{4} - q^{7} + q^{9} + q^{12} - q^{13} - 2 q^{16} - q^{19} - q^{21} + q^{25} - 2 q^{27} - q^{28} + 2 q^{31} + q^{36} + 2 q^{37} + 21 q^{39} - 9 q^{43} + q^{48} - q^{52} - q^{57} - q^{61} - 20 q^{63} - 2 q^{64} + 2 q^{67} - q^{73} + q^{75} - q^{76} - q^{79} + q^{81} - q^{84} + q^{91} - q^{93} - q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1191\mathbb{Z}\right)^\times\).

\(n\) \(398\) \(799\)
\(\chi(n)\) \(-1\) \(\zeta_{66}^{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
110.1
−0.995472 + 0.0950560i
0.981929 + 0.189251i
−0.888835 + 0.458227i
0.0475819 + 0.998867i
−0.995472 − 0.0950560i
0.723734 − 0.690079i
−0.888835 − 0.458227i
−0.786053 − 0.618159i
0.235759 + 0.971812i
0.981929 − 0.189251i
0.235759 − 0.971812i
0.928368 − 0.371662i
−0.327068 + 0.945001i
0.580057 + 0.814576i
0.928368 + 0.371662i
0.723734 + 0.690079i
−0.786053 + 0.618159i
0.0475819 − 0.998867i
−0.327068 − 0.945001i
0.580057 − 0.814576i
0 0.0475819 + 0.998867i −0.654861 − 0.755750i 0 0 0.995472 + 0.0950560i 0 −0.995472 + 0.0950560i 0
140.1 0 −0.995472 − 0.0950560i −0.142315 − 0.989821i 0 0 −0.981929 + 0.189251i 0 0.981929 + 0.189251i 0
206.1 0 0.235759 + 0.971812i 0.415415 + 0.909632i 0 0 0.888835 + 0.458227i 0 −0.888835 + 0.458227i 0
260.1 0 0.723734 + 0.690079i 0.415415 − 0.909632i 0 0 −0.0475819 + 0.998867i 0 0.0475819 + 0.998867i 0
314.1 0 0.0475819 − 0.998867i −0.654861 + 0.755750i 0 0 0.995472 − 0.0950560i 0 −0.995472 − 0.0950560i 0
332.1 0 0.928368 − 0.371662i 0.841254 + 0.540641i 0 0 −0.723734 − 0.690079i 0 0.723734 − 0.690079i 0
503.1 0 0.235759 − 0.971812i 0.415415 − 0.909632i 0 0 0.888835 − 0.458227i 0 −0.888835 − 0.458227i 0
548.1 0 −0.327068 + 0.945001i −0.959493 − 0.281733i 0 0 0.786053 − 0.618159i 0 −0.786053 − 0.618159i 0
569.1 0 −0.786053 − 0.618159i 0.841254 + 0.540641i 0 0 −0.235759 + 0.971812i 0 0.235759 + 0.971812i 0
587.1 0 −0.995472 + 0.0950560i −0.142315 + 0.989821i 0 0 −0.981929 − 0.189251i 0 0.981929 − 0.189251i 0
764.1 0 −0.786053 + 0.618159i 0.841254 − 0.540641i 0 0 −0.235759 − 0.971812i 0 0.235759 − 0.971812i 0
767.1 0 0.981929 − 0.189251i −0.959493 − 0.281733i 0 0 −0.928368 − 0.371662i 0 0.928368 − 0.371662i 0
902.1 0 0.580057 + 0.814576i −0.142315 + 0.989821i 0 0 0.327068 + 0.945001i 0 −0.327068 + 0.945001i 0
914.1 0 −0.888835 − 0.458227i −0.654861 − 0.755750i 0 0 −0.580057 + 0.814576i 0 0.580057 + 0.814576i 0
941.1 0 0.981929 + 0.189251i −0.959493 + 0.281733i 0 0 −0.928368 + 0.371662i 0 0.928368 + 0.371662i 0
965.1 0 0.928368 + 0.371662i 0.841254 − 0.540641i 0 0 −0.723734 + 0.690079i 0 0.723734 + 0.690079i 0
1028.1 0 −0.327068 − 0.945001i −0.959493 + 0.281733i 0 0 0.786053 + 0.618159i 0 −0.786053 + 0.618159i 0
1049.1 0 0.723734 − 0.690079i 0.415415 + 0.909632i 0 0 −0.0475819 − 0.998867i 0 0.0475819 − 0.998867i 0
1055.1 0 0.580057 − 0.814576i −0.142315 − 0.989821i 0 0 0.327068 − 0.945001i 0 −0.327068 − 0.945001i 0
1148.1 0 −0.888835 + 0.458227i −0.654861 + 0.755750i 0 0 −0.580057 − 0.814576i 0 0.580057 − 0.814576i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 110.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
397.k even 33 1 inner
1191.bb odd 66 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1191.1.bb.a ✓ 20
3.b odd 2 1 CM 1191.1.bb.a ✓ 20
397.k even 33 1 inner 1191.1.bb.a ✓ 20
1191.bb odd 66 1 inner 1191.1.bb.a ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1191.1.bb.a ✓ 20 1.a even 1 1 trivial
1191.1.bb.a ✓ 20 3.b odd 2 1 CM
1191.1.bb.a ✓ 20 397.k even 33 1 inner
1191.1.bb.a ✓ 20 1191.bb odd 66 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(1191, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( T^{20} - T^{19} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{20} \) Copy content Toggle raw display
$7$ \( T^{20} + T^{19} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{20} \) Copy content Toggle raw display
$13$ \( T^{20} + T^{19} + \cdots + 1 \) Copy content Toggle raw display
$17$ \( T^{20} \) Copy content Toggle raw display
$19$ \( T^{20} + T^{19} + \cdots + 1 \) Copy content Toggle raw display
$23$ \( T^{20} \) Copy content Toggle raw display
$29$ \( T^{20} \) Copy content Toggle raw display
$31$ \( T^{20} - 2 T^{19} + \cdots + 1 \) Copy content Toggle raw display
$37$ \( T^{20} - 2 T^{19} + \cdots + 1 \) Copy content Toggle raw display
$41$ \( T^{20} \) Copy content Toggle raw display
$43$ \( T^{20} + 9 T^{19} + \cdots + 1 \) Copy content Toggle raw display
$47$ \( T^{20} \) Copy content Toggle raw display
$53$ \( T^{20} \) Copy content Toggle raw display
$59$ \( T^{20} \) Copy content Toggle raw display
$61$ \( T^{20} + T^{19} + \cdots + 1 \) Copy content Toggle raw display
$67$ \( T^{20} - 2 T^{19} + \cdots + 1 \) Copy content Toggle raw display
$71$ \( T^{20} \) Copy content Toggle raw display
$73$ \( T^{20} + T^{19} + \cdots + 1 \) Copy content Toggle raw display
$79$ \( T^{20} + T^{19} + \cdots + 1 \) Copy content Toggle raw display
$83$ \( T^{20} \) Copy content Toggle raw display
$89$ \( T^{20} \) Copy content Toggle raw display
$97$ \( T^{20} + T^{19} + \cdots + 1 \) Copy content Toggle raw display
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