Properties

Label 961.1.e.a
Level $961$
Weight $1$
Character orbit 961.e
Analytic conductor $0.480$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -31
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [961,1,Mod(440,961)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(961, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([5]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("961.440");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 961 = 31^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 961.e (of order \(6\), degree \(2\), not 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: \(0.479601477140\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 31)
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.31.1
Artin image: $C_3\times S_3$
Artin field: Galois closure of 6.0.28629151.2

$q$-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.

\(f(q)\) \(=\) \( q - q^{2} + \zeta_{6} q^{5} - \zeta_{6}^{2} q^{7} + q^{8} - \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + \zeta_{6} q^{5} - \zeta_{6}^{2} q^{7} + q^{8} - \zeta_{6} q^{9} - \zeta_{6} q^{10} + \zeta_{6}^{2} q^{14} - q^{16} + \zeta_{6} q^{18} - \zeta_{6}^{2} q^{19} + q^{35} + \zeta_{6}^{2} q^{38} + \zeta_{6} q^{40} + \zeta_{6} q^{41} - \zeta_{6}^{2} q^{45} + 2 q^{47} - \zeta_{6}^{2} q^{56} - \zeta_{6}^{2} q^{59} - q^{63} + q^{64} - 2 \zeta_{6} q^{67} - q^{70} + \zeta_{6} q^{71} - \zeta_{6} q^{72} - \zeta_{6} q^{80} + \zeta_{6}^{2} q^{81} - \zeta_{6} q^{82} + \zeta_{6}^{2} q^{90} - 2 q^{94} + q^{95} - q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + q^{5} + q^{7} + 2 q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + q^{5} + q^{7} + 2 q^{8} - q^{9} - q^{10} - q^{14} - 2 q^{16} + q^{18} + q^{19} + 2 q^{35} - q^{38} + q^{40} + q^{41} + q^{45} + 4 q^{47} + q^{56} + q^{59} - 2 q^{63} + 2 q^{64} - 2 q^{67} - 2 q^{70} + q^{71} - q^{72} - q^{80} - q^{81} - q^{82} - q^{90} - 4 q^{94} + 2 q^{95} - 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/961\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(\zeta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
440.1
0.500000 0.866025i
0.500000 + 0.866025i
−1.00000 0 0 0.500000 0.866025i 0 0.500000 + 0.866025i 1.00000 −0.500000 + 0.866025i −0.500000 + 0.866025i
522.1 −1.00000 0 0 0.500000 + 0.866025i 0 0.500000 0.866025i 1.00000 −0.500000 0.866025i −0.500000 0.866025i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.b odd 2 1 CM by \(\Q(\sqrt{-31}) \)
31.c even 3 1 inner
31.e odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 961.1.e.a 2
31.b odd 2 1 CM 961.1.e.a 2
31.c even 3 1 31.1.b.a 1
31.c even 3 1 inner 961.1.e.a 2
31.d even 5 4 961.1.h.a 8
31.e odd 6 1 31.1.b.a 1
31.e odd 6 1 inner 961.1.e.a 2
31.f odd 10 4 961.1.h.a 8
31.g even 15 4 961.1.f.a 4
31.g even 15 4 961.1.h.a 8
31.h odd 30 4 961.1.f.a 4
31.h odd 30 4 961.1.h.a 8
93.g even 6 1 279.1.d.b 1
93.h odd 6 1 279.1.d.b 1
124.g even 6 1 496.1.e.a 1
124.i odd 6 1 496.1.e.a 1
155.i odd 6 1 775.1.d.b 1
155.j even 6 1 775.1.d.b 1
155.o odd 12 2 775.1.c.a 2
155.p even 12 2 775.1.c.a 2
217.e even 3 1 1519.1.n.b 2
217.g even 3 1 1519.1.n.b 2
217.j odd 6 1 1519.1.n.a 2
217.k odd 6 1 1519.1.n.b 2
217.l even 6 1 1519.1.n.a 2
217.o odd 6 1 1519.1.n.b 2
217.q odd 6 1 1519.1.n.a 2
217.r even 6 1 1519.1.n.a 2
217.s even 6 1 1519.1.c.a 1
217.u odd 6 1 1519.1.c.a 1
248.l odd 6 1 1984.1.e.a 1
248.m odd 6 1 1984.1.e.b 1
248.p even 6 1 1984.1.e.a 1
248.q even 6 1 1984.1.e.b 1
279.e even 3 1 2511.1.m.e 2
279.g even 3 1 2511.1.m.e 2
279.l odd 6 1 2511.1.m.e 2
279.n odd 6 1 2511.1.m.e 2
279.o even 6 1 2511.1.m.a 2
279.p odd 6 1 2511.1.m.a 2
279.q odd 6 1 2511.1.m.a 2
279.r even 6 1 2511.1.m.a 2
341.l odd 6 1 3751.1.d.b 1
341.m even 6 1 3751.1.d.b 1
341.bi even 15 4 3751.1.t.c 4
341.bp odd 30 4 3751.1.t.c 4
341.bx even 30 4 3751.1.t.a 4
341.cb odd 30 4 3751.1.t.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
31.1.b.a 1 31.c even 3 1
31.1.b.a 1 31.e odd 6 1
279.1.d.b 1 93.g even 6 1
279.1.d.b 1 93.h odd 6 1
496.1.e.a 1 124.g even 6 1
496.1.e.a 1 124.i odd 6 1
775.1.c.a 2 155.o odd 12 2
775.1.c.a 2 155.p even 12 2
775.1.d.b 1 155.i odd 6 1
775.1.d.b 1 155.j even 6 1
961.1.e.a 2 1.a even 1 1 trivial
961.1.e.a 2 31.b odd 2 1 CM
961.1.e.a 2 31.c even 3 1 inner
961.1.e.a 2 31.e odd 6 1 inner
961.1.f.a 4 31.g even 15 4
961.1.f.a 4 31.h odd 30 4
961.1.h.a 8 31.d even 5 4
961.1.h.a 8 31.f odd 10 4
961.1.h.a 8 31.g even 15 4
961.1.h.a 8 31.h odd 30 4
1519.1.c.a 1 217.s even 6 1
1519.1.c.a 1 217.u odd 6 1
1519.1.n.a 2 217.j odd 6 1
1519.1.n.a 2 217.l even 6 1
1519.1.n.a 2 217.q odd 6 1
1519.1.n.a 2 217.r even 6 1
1519.1.n.b 2 217.e even 3 1
1519.1.n.b 2 217.g even 3 1
1519.1.n.b 2 217.k odd 6 1
1519.1.n.b 2 217.o odd 6 1
1984.1.e.a 1 248.l odd 6 1
1984.1.e.a 1 248.p even 6 1
1984.1.e.b 1 248.m odd 6 1
1984.1.e.b 1 248.q even 6 1
2511.1.m.a 2 279.o even 6 1
2511.1.m.a 2 279.p odd 6 1
2511.1.m.a 2 279.q odd 6 1
2511.1.m.a 2 279.r even 6 1
2511.1.m.e 2 279.e even 3 1
2511.1.m.e 2 279.g even 3 1
2511.1.m.e 2 279.l odd 6 1
2511.1.m.e 2 279.n odd 6 1
3751.1.d.b 1 341.l odd 6 1
3751.1.d.b 1 341.m even 6 1
3751.1.t.a 4 341.bx even 30 4
3751.1.t.a 4 341.cb odd 30 4
3751.1.t.c 4 341.bi even 15 4
3751.1.t.c 4 341.bp odd 30 4

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(961, [\chi])\).

Hecke characteristic polynomials

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