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Properties

Label 1265.1.bi.b
Level $1265$
Weight $1$
Character orbit 1265.bi
Analytic conductor $0.631$
Analytic rank $0$
Dimension $20$
Projective image $D_{44}$
CM discriminant -11
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1265,1,Mod(43,1265)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1265, base_ring=CyclotomicField(44))
 
chi = DirichletCharacter(H, H._module([33, 22, 10]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1265.43");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1265 = 5 \cdot 11 \cdot 23 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1265.bi (of order \(44\), 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: \(0.631317240981\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\Q(\zeta_{44})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - x^{18} + x^{16} - x^{14} + x^{12} - x^{10} + x^{8} - x^{6} + x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{44}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{44} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q + ( - \zeta_{44}^{13} + \zeta_{44}^{8}) q^{3} - \zeta_{44}^{15} q^{4} + \zeta_{44}^{3} q^{5} + ( - \zeta_{44}^{21} + \cdots - \zeta_{44}^{4}) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{44}^{13} + \zeta_{44}^{8}) q^{3} - \zeta_{44}^{15} q^{4} + \zeta_{44}^{3} q^{5} + ( - \zeta_{44}^{21} + \cdots - \zeta_{44}^{4}) q^{9} + \cdots + (\zeta_{44}^{13} + \cdots + \zeta_{44}^{3}) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 2 q^{3}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 2 q^{3} - 2 q^{12} + 2 q^{15} + 2 q^{16} - 2 q^{20} - 2 q^{23} + 2 q^{25} - 20 q^{33} - 2 q^{36} + 2 q^{37} - 2 q^{44} + 2 q^{45} - 2 q^{47} + 2 q^{48} - 2 q^{53} + 2 q^{55} - 2 q^{60} + 2 q^{67} - 2 q^{69} - 18 q^{71} + 2 q^{75} - 2 q^{81} + 2 q^{97} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1265\mathbb{Z}\right)^\times\).

\(n\) \(166\) \(507\) \(1036\)
\(\chi(n)\) \(\zeta_{44}^{2}\) \(-\zeta_{44}^{11}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1
0.755750 + 0.654861i
−0.755750 + 0.654861i
−0.909632 − 0.415415i
0.540641 − 0.841254i
0.540641 + 0.841254i
0.989821 + 0.142315i
−0.989821 + 0.142315i
0.281733 − 0.959493i
−0.909632 + 0.415415i
−0.281733 − 0.959493i
−0.755750 − 0.654861i
−0.540641 − 0.841254i
0.755750 − 0.654861i
0.909632 + 0.415415i
−0.281733 + 0.959493i
−0.540641 + 0.841254i
0.909632 − 0.415415i
−0.989821 − 0.142315i
0.989821 − 0.142315i
0.281733 + 0.959493i
0 1.83107 − 0.682956i 0.281733 + 0.959493i −0.540641 + 0.841254i 0 0 0 2.13066 − 1.84623i 0
153.1 0 −0.148568 + 0.398326i −0.281733 + 0.959493i 0.540641 + 0.841254i 0 0 0 0.619158 + 0.536504i 0
263.1 0 −0.203743 − 0.936593i 0.989821 + 0.142315i −0.281733 − 0.959493i 0 0 0 0.0739364 − 0.0337656i 0
318.1 0 −1.05195 − 0.574406i 0.755750 + 0.654861i −0.989821 − 0.142315i 0 0 0 0.236009 + 0.367237i 0
362.1 0 −1.05195 + 0.574406i 0.755750 − 0.654861i −0.989821 + 0.142315i 0 0 0 0.236009 − 0.367237i 0
373.1 0 0.697148 − 0.0498610i 0.540641 − 0.841254i 0.909632 + 0.415415i 0 0 0 −0.506293 + 0.0727939i 0
428.1 0 0.133682 − 1.86912i −0.540641 − 0.841254i −0.909632 + 0.415415i 0 0 0 −2.48594 − 0.357424i 0
527.1 0 −0.114220 − 0.0855040i −0.909632 + 0.415415i −0.755750 + 0.654861i 0 0 0 −0.275997 − 0.939960i 0
582.1 0 −0.203743 + 0.936593i 0.989821 − 0.142315i −0.281733 + 0.959493i 0 0 0 0.0739364 + 0.0337656i 0
747.1 0 −1.19550 − 1.59700i 0.909632 + 0.415415i 0.755750 + 0.654861i 0 0 0 −0.839462 + 2.85895i 0
802.1 0 −0.148568 − 0.398326i −0.281733 − 0.959493i 0.540641 − 0.841254i 0 0 0 0.619158 − 0.536504i 0
868.1 0 0.767317 + 1.40524i −0.755750 + 0.654861i 0.989821 − 0.142315i 0 0 0 −0.845273 + 1.31527i 0
912.1 0 1.83107 + 0.682956i 0.281733 − 0.959493i −0.540641 − 0.841254i 0 0 0 2.13066 + 1.84623i 0
1022.1 0 −1.71524 + 0.373128i −0.989821 − 0.142315i 0.281733 + 0.959493i 0 0 0 1.89320 − 0.864596i 0
1033.1 0 −1.19550 + 1.59700i 0.909632 − 0.415415i 0.755750 − 0.654861i 0 0 0 −0.839462 − 2.85895i 0
1077.1 0 0.767317 − 1.40524i −0.755750 − 0.654861i 0.989821 + 0.142315i 0 0 0 −0.845273 − 1.31527i 0
1088.1 0 −1.71524 − 0.373128i −0.989821 + 0.142315i 0.281733 − 0.959493i 0 0 0 1.89320 + 0.864596i 0
1132.1 0 0.133682 + 1.86912i −0.540641 + 0.841254i −0.909632 − 0.415415i 0 0 0 −2.48594 + 0.357424i 0
1187.1 0 0.697148 + 0.0498610i 0.540641 + 0.841254i 0.909632 − 0.415415i 0 0 0 −0.506293 − 0.0727939i 0
1253.1 0 −0.114220 + 0.0855040i −0.909632 − 0.415415i −0.755750 − 0.654861i 0 0 0 −0.275997 + 0.939960i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 43.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 CM by \(\Q(\sqrt{-11}) \)
115.l even 44 1 inner
1265.bi odd 44 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1265.1.bi.b yes 20
5.c odd 4 1 1265.1.bi.a ✓ 20
11.b odd 2 1 CM 1265.1.bi.b yes 20
23.d odd 22 1 1265.1.bi.a ✓ 20
55.e even 4 1 1265.1.bi.a ✓ 20
115.l even 44 1 inner 1265.1.bi.b yes 20
253.l even 22 1 1265.1.bi.a ✓ 20
1265.bi odd 44 1 inner 1265.1.bi.b yes 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1265.1.bi.a ✓ 20 5.c odd 4 1
1265.1.bi.a ✓ 20 23.d odd 22 1
1265.1.bi.a ✓ 20 55.e even 4 1
1265.1.bi.a ✓ 20 253.l even 22 1
1265.1.bi.b yes 20 1.a even 1 1 trivial
1265.1.bi.b yes 20 11.b odd 2 1 CM
1265.1.bi.b yes 20 115.l even 44 1 inner
1265.1.bi.b yes 20 1265.bi odd 44 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{20} + 2 T_{3}^{19} + 2 T_{3}^{18} - 4 T_{3}^{16} - 8 T_{3}^{15} - 8 T_{3}^{14} + 66 T_{3}^{13} + \cdots + 1 \) acting on \(S_{1}^{\mathrm{new}}(1265, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( T^{20} + 2 T^{19} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{20} - T^{18} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{20} \) Copy content Toggle raw display
$11$ \( T^{20} - T^{18} + \cdots + 1 \) Copy content Toggle raw display
$13$ \( T^{20} \) Copy content Toggle raw display
$17$ \( T^{20} \) Copy content Toggle raw display
$19$ \( T^{20} \) Copy content Toggle raw display
$23$ \( (T^{10} + T^{9} + T^{8} + \cdots + 1)^{2} \) Copy content Toggle raw display
$29$ \( T^{20} \) Copy content Toggle raw display
$31$ \( T^{20} + 22 T^{16} + \cdots + 121 \) 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
$43$ \( T^{20} \) Copy content Toggle raw display
$47$ \( T^{20} + 2 T^{19} + \cdots + 1 \) Copy content Toggle raw display
$53$ \( T^{20} + 2 T^{19} + \cdots + 1 \) Copy content Toggle raw display
$59$ \( (T^{10} - 22 T^{5} + \cdots + 11)^{2} \) Copy content Toggle raw display
$61$ \( T^{20} \) Copy content Toggle raw display
$67$ \( T^{20} - 2 T^{19} + \cdots + 1 \) Copy content Toggle raw display
$71$ \( (T^{10} + 9 T^{9} + 37 T^{8} + \cdots + 1)^{2} \) Copy content Toggle raw display
$73$ \( T^{20} \) Copy content Toggle raw display
$79$ \( T^{20} \) Copy content Toggle raw display
$83$ \( T^{20} \) Copy content Toggle raw display
$89$ \( T^{20} + 22 T^{12} + \cdots + 121 \) Copy content Toggle raw display
$97$ \( T^{20} - 2 T^{19} + \cdots + 1 \) Copy content Toggle raw display
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