Properties

Label 1340.1.bb.b
Level $1340$
Weight $1$
Character orbit 1340.bb
Analytic conductor $0.669$
Analytic rank $0$
Dimension $10$
Projective image $D_{11}$
CM discriminant -20
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1340,1,Mod(59,1340)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1340, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([11, 11, 12]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1340.59");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1340 = 2^{2} \cdot 5 \cdot 67 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1340.bb (of order \(22\), degree \(10\), 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: \(0.668747116928\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\Q(\zeta_{22})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{11}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{11} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{22} q^{2} + ( - \zeta_{22}^{8} - \zeta_{22}^{4}) q^{3} + \zeta_{22}^{2} q^{4} + \zeta_{22}^{4} q^{5} + ( - \zeta_{22}^{9} - \zeta_{22}^{5}) q^{6} + (\zeta_{22}^{7} - \zeta_{22}^{6}) q^{7} + \zeta_{22}^{3} q^{8} + (\zeta_{22}^{8} - \zeta_{22}^{5} - \zeta_{22}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{22} q^{2} + ( - \zeta_{22}^{8} - \zeta_{22}^{4}) q^{3} + \zeta_{22}^{2} q^{4} + \zeta_{22}^{4} q^{5} + ( - \zeta_{22}^{9} - \zeta_{22}^{5}) q^{6} + (\zeta_{22}^{7} - \zeta_{22}^{6}) q^{7} + \zeta_{22}^{3} q^{8} + (\zeta_{22}^{8} - \zeta_{22}^{5} - \zeta_{22}) q^{9} + \zeta_{22}^{5} q^{10} + ( - \zeta_{22}^{10} - \zeta_{22}^{6}) q^{12} + (\zeta_{22}^{8} - \zeta_{22}^{7}) q^{14} + ( - \zeta_{22}^{8} + \zeta_{22}) q^{15} + \zeta_{22}^{4} q^{16} + (\zeta_{22}^{9} - \zeta_{22}^{6} - \zeta_{22}^{2}) q^{18} + \zeta_{22}^{6} q^{20} + (\zeta_{22}^{10} + \zeta_{22}^{4} - \zeta_{22}^{3} + 1) q^{21} - \zeta_{22}^{6} q^{23} + ( - \zeta_{22}^{7} + 1) q^{24} + \zeta_{22}^{8} q^{25} + (\zeta_{22}^{9} + \zeta_{22}^{5} - \zeta_{22}^{2} + \zeta_{22}) q^{27} + (\zeta_{22}^{9} - \zeta_{22}^{8}) q^{28} + (\zeta_{22}^{6} - \zeta_{22}^{5}) q^{29} + ( - \zeta_{22}^{9} + \zeta_{22}^{2}) q^{30} + \zeta_{22}^{5} q^{32} + ( - \zeta_{22}^{10} - 1) q^{35} + (\zeta_{22}^{10} - \zeta_{22}^{7} - \zeta_{22}^{3}) q^{36} + \zeta_{22}^{7} q^{40} + (\zeta_{22}^{6} - \zeta_{22}) q^{41} + (\zeta_{22}^{5} - \zeta_{22}^{4} + \zeta_{22} - 1) q^{42} + (\zeta_{22}^{7} - 1) q^{43} + ( - \zeta_{22}^{9} - \zeta_{22}^{5} - \zeta_{22}) q^{45} - 2 \zeta_{22}^{7} q^{46} + (\zeta_{22}^{9} + \zeta_{22}^{3}) q^{47} + ( - \zeta_{22}^{8} + \zeta_{22}) q^{48} + ( - \zeta_{22}^{3} + \zeta_{22}^{2} - \zeta_{22}) q^{49} + \zeta_{22}^{9} q^{50} + (\zeta_{22}^{10} + \zeta_{22}^{6} - \zeta_{22}^{3} + \zeta_{22}^{2}) q^{54} + (\zeta_{22}^{10} - \zeta_{22}^{9}) q^{56} + (\zeta_{22}^{7} - \zeta_{22}^{6}) q^{58} + ( - \zeta_{22}^{10} + \zeta_{22}^{3}) q^{60} + ( - \zeta_{22}^{3} + 1) q^{61} + ( - \zeta_{22}^{8} + \zeta_{22}^{7} - \zeta_{22}^{4} + \zeta_{22}^{3} + \zeta_{22} - 1) q^{63} + \zeta_{22}^{6} q^{64} - \zeta_{22}^{2} q^{67} + (2 \zeta_{22}^{10} - 2 \zeta_{22}^{3}) q^{69} + ( - \zeta_{22} + 1) q^{70} + ( - \zeta_{22}^{8} - \zeta_{22}^{4} - 1) q^{72} + (\zeta_{22}^{5} + \zeta_{22}) q^{75} + \zeta_{22}^{8} q^{80} + (\zeta_{22}^{10} - \zeta_{22}^{9} + \zeta_{22}^{6} - \zeta_{22}^{5} + \zeta_{22}^{2}) q^{81} + (\zeta_{22}^{7} - \zeta_{22}^{2}) q^{82} + (\zeta_{22}^{5} + \zeta_{22}^{3}) q^{83} + (\zeta_{22}^{6} - \zeta_{22}^{5} + \zeta_{22}^{2} - \zeta_{22}) q^{84} + (\zeta_{22}^{8} - \zeta_{22}) q^{86} + ( - \zeta_{22}^{10} + \zeta_{22}^{9} + \zeta_{22}^{3} - \zeta_{22}^{2}) q^{87} + (\zeta_{22}^{6} + \zeta_{22}^{4}) q^{89} + ( - \zeta_{22}^{10} - \zeta_{22}^{6} - \zeta_{22}^{2}) q^{90} - 2 \zeta_{22}^{8} q^{92} + (\zeta_{22}^{10} + \zeta_{22}^{4}) q^{94} + ( - \zeta_{22}^{9} + \zeta_{22}^{2}) q^{96} + ( - \zeta_{22}^{4} + \zeta_{22}^{3} - \zeta_{22}^{2}) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + q^{2} + 2 q^{3} - q^{4} - q^{5} - 2 q^{6} + 2 q^{7} + q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + q^{2} + 2 q^{3} - q^{4} - q^{5} - 2 q^{6} + 2 q^{7} + q^{8} - 3 q^{9} + q^{10} + 2 q^{12} - 2 q^{14} + 2 q^{15} - q^{16} + 3 q^{18} - q^{20} + 7 q^{21} + 2 q^{23} + 9 q^{24} - q^{25} + 4 q^{27} + 2 q^{28} - 2 q^{29} - 2 q^{30} + q^{32} - 9 q^{35} - 3 q^{36} + q^{40} - 2 q^{41} - 7 q^{42} - 9 q^{43} - 3 q^{45} - 2 q^{46} + 2 q^{47} + 2 q^{48} - 3 q^{49} + q^{50} - 4 q^{54} - 2 q^{56} + 2 q^{58} + 2 q^{60} + 9 q^{61} - 5 q^{63} - q^{64} + q^{67} - 4 q^{69} + 9 q^{70} - 8 q^{72} + 2 q^{75} - q^{80} - 5 q^{81} + 2 q^{82} + 2 q^{83} - 4 q^{84} - 2 q^{86} + 4 q^{87} - 2 q^{89} + 3 q^{90} + 2 q^{92} - 2 q^{94} - 2 q^{96} + 3 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1340\mathbb{Z}\right)^\times\).

\(n\) \(537\) \(671\) \(1141\)
\(\chi(n)\) \(-1\) \(-1\) \(\zeta_{22}^{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
59.1
0.142315 0.989821i
0.142315 + 0.989821i
−0.415415 + 0.909632i
0.959493 + 0.281733i
−0.841254 0.540641i
0.959493 0.281733i
0.654861 0.755750i
−0.841254 + 0.540641i
−0.415415 0.909632i
0.654861 + 0.755750i
0.142315 0.989821i −1.25667 1.45027i −0.959493 0.281733i 0.841254 + 0.540641i −1.61435 + 1.03748i −0.186393 + 1.29639i −0.415415 + 0.909632i −0.381761 + 2.65520i 0.654861 0.755750i
159.1 0.142315 + 0.989821i −1.25667 + 1.45027i −0.959493 + 0.281733i 0.841254 0.540641i −1.61435 1.03748i −0.186393 1.29639i −0.415415 0.909632i −0.381761 2.65520i 0.654861 + 0.755750i
359.1 −0.415415 + 0.909632i 1.10181 0.708089i −0.654861 0.755750i −0.142315 + 0.989821i 0.186393 + 1.29639i −0.698939 + 1.53046i 0.959493 0.281733i 0.297176 0.650724i −0.841254 0.540641i
399.1 0.959493 + 0.281733i 0.239446 1.66538i 0.841254 + 0.540641i 0.415415 + 0.909632i 0.698939 1.53046i −0.273100 0.0801894i 0.654861 + 0.755750i −1.75667 0.515804i 0.142315 + 0.989821i
679.1 −0.841254 0.540641i 0.797176 + 0.234072i 0.415415 + 0.909632i −0.654861 + 0.755750i −0.544078 0.627899i 1.61435 + 1.03748i 0.142315 0.989821i −0.260554 0.167448i 0.959493 0.281733i
759.1 0.959493 0.281733i 0.239446 + 1.66538i 0.841254 0.540641i 0.415415 0.909632i 0.698939 + 1.53046i −0.273100 + 0.0801894i 0.654861 0.755750i −1.75667 + 0.515804i 0.142315 0.989821i
799.1 0.654861 0.755750i 0.118239 + 0.258908i −0.142315 0.989821i −0.959493 + 0.281733i 0.273100 + 0.0801894i 0.544078 0.627899i −0.841254 0.540641i 0.601808 0.694523i −0.415415 + 0.909632i
819.1 −0.841254 + 0.540641i 0.797176 0.234072i 0.415415 0.909632i −0.654861 0.755750i −0.544078 + 0.627899i 1.61435 1.03748i 0.142315 + 0.989821i −0.260554 + 0.167448i 0.959493 + 0.281733i
1019.1 −0.415415 0.909632i 1.10181 + 0.708089i −0.654861 + 0.755750i −0.142315 0.989821i 0.186393 1.29639i −0.698939 1.53046i 0.959493 + 0.281733i 0.297176 + 0.650724i −0.841254 + 0.540641i
1179.1 0.654861 + 0.755750i 0.118239 0.258908i −0.142315 + 0.989821i −0.959493 0.281733i 0.273100 0.0801894i 0.544078 + 0.627899i −0.841254 + 0.540641i 0.601808 + 0.694523i −0.415415 0.909632i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 59.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
20.d odd 2 1 CM by \(\Q(\sqrt{-5}) \)
67.e even 11 1 inner
1340.bb odd 22 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1340.1.bb.b yes 10
4.b odd 2 1 1340.1.bb.a 10
5.b even 2 1 1340.1.bb.a 10
20.d odd 2 1 CM 1340.1.bb.b yes 10
67.e even 11 1 inner 1340.1.bb.b yes 10
268.k odd 22 1 1340.1.bb.a 10
335.o even 22 1 1340.1.bb.a 10
1340.bb odd 22 1 inner 1340.1.bb.b yes 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1340.1.bb.a 10 4.b odd 2 1
1340.1.bb.a 10 5.b even 2 1
1340.1.bb.a 10 268.k odd 22 1
1340.1.bb.a 10 335.o even 22 1
1340.1.bb.b yes 10 1.a even 1 1 trivial
1340.1.bb.b yes 10 20.d odd 2 1 CM
1340.1.bb.b yes 10 67.e even 11 1 inner
1340.1.bb.b yes 10 1340.bb odd 22 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{10} - 2T_{3}^{9} + 4T_{3}^{8} - 8T_{3}^{7} + 16T_{3}^{6} - 32T_{3}^{5} + 53T_{3}^{4} - 51T_{3}^{3} + 25T_{3}^{2} - 6T_{3} + 1 \) acting on \(S_{1}^{\mathrm{new}}(1340, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} - T^{9} + T^{8} - T^{7} + T^{6} - T^{5} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{10} - 2 T^{9} + 4 T^{8} - 8 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{10} + T^{9} + T^{8} + T^{7} + T^{6} + T^{5} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{10} - 2 T^{9} + 4 T^{8} - 8 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{10} \) Copy content Toggle raw display
$13$ \( T^{10} \) Copy content Toggle raw display
$17$ \( T^{10} \) Copy content Toggle raw display
$19$ \( T^{10} \) Copy content Toggle raw display
$23$ \( T^{10} - 2 T^{9} + 4 T^{8} - 8 T^{7} + \cdots + 1024 \) Copy content Toggle raw display
$29$ \( (T^{5} + T^{4} - 4 T^{3} - 3 T^{2} + 3 T + 1)^{2} \) Copy content Toggle raw display
$31$ \( T^{10} \) Copy content Toggle raw display
$37$ \( T^{10} \) Copy content Toggle raw display
$41$ \( T^{10} + 2 T^{9} + 4 T^{8} + 8 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$43$ \( T^{10} + 9 T^{9} + 37 T^{8} + 91 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$47$ \( T^{10} - 2 T^{9} + 4 T^{8} - 8 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$53$ \( T^{10} \) Copy content Toggle raw display
$59$ \( T^{10} \) Copy content Toggle raw display
$61$ \( T^{10} - 9 T^{9} + 37 T^{8} - 91 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$67$ \( T^{10} - T^{9} + T^{8} - T^{7} + T^{6} - T^{5} + \cdots + 1 \) Copy content Toggle raw display
$71$ \( T^{10} \) Copy content Toggle raw display
$73$ \( T^{10} \) Copy content Toggle raw display
$79$ \( T^{10} \) Copy content Toggle raw display
$83$ \( T^{10} - 2 T^{9} + 4 T^{8} + 3 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$89$ \( T^{10} + 2 T^{9} + 4 T^{8} + 8 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$97$ \( T^{10} \) Copy content Toggle raw display
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