Properties

Label 385.1.q.b
Level $385$
Weight $1$
Character orbit 385.q
Analytic conductor $0.192$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -55
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [385,1,Mod(109,385)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(385, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 4, 3]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("385.109");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 385 = 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 385.q (of order \(6\), degree \(2\), 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: \(0.192140029864\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.2695.1
Artin image: $C_3\times S_3$
Artin field: Galois closure of 6.0.8152375.1

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{6} q^{2} - \zeta_{6} q^{5} + q^{7} + q^{8} - \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{6} q^{2} - \zeta_{6} q^{5} + q^{7} + q^{8} - \zeta_{6} q^{9} - \zeta_{6}^{2} q^{10} + \zeta_{6}^{2} q^{11} - q^{13} + \zeta_{6} q^{14} + \zeta_{6} q^{16} + \zeta_{6}^{2} q^{17} - \zeta_{6}^{2} q^{18} - q^{22} + \zeta_{6}^{2} q^{25} - \zeta_{6} q^{26} - \zeta_{6}^{2} q^{31} - 2 q^{34} - \zeta_{6} q^{35} - \zeta_{6} q^{40} - q^{43} + \zeta_{6}^{2} q^{45} + q^{49} - q^{50} + q^{55} + q^{56} - \zeta_{6}^{2} q^{59} + q^{62} - \zeta_{6} q^{63} + q^{64} + \zeta_{6} q^{65} - \zeta_{6}^{2} q^{70} - q^{71} - \zeta_{6} q^{72} - \zeta_{6}^{2} q^{73} + \zeta_{6}^{2} q^{77} - \zeta_{6}^{2} q^{80} + \zeta_{6}^{2} q^{81} - q^{83} + 2 q^{85} - \zeta_{6} q^{86} + \zeta_{6}^{2} q^{88} + \zeta_{6} q^{89} - q^{90} - q^{91} + \zeta_{6} q^{98} + q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - q^{5} + 2 q^{7} + 2 q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} - q^{5} + 2 q^{7} + 2 q^{8} - q^{9} + q^{10} - q^{11} - 2 q^{13} + q^{14} + q^{16} - 2 q^{17} + q^{18} - 2 q^{22} - q^{25} - q^{26} + q^{31} - 4 q^{34} - q^{35} - q^{40} - 2 q^{43} - q^{45} + 2 q^{49} - 2 q^{50} + 2 q^{55} + 2 q^{56} + q^{59} + 2 q^{62} - q^{63} + 2 q^{64} + q^{65} + q^{70} - 2 q^{71} - q^{72} + q^{73} - q^{77} + q^{80} - q^{81} - 2 q^{83} + 4 q^{85} - q^{86} - q^{88} + q^{89} - 2 q^{90} - 2 q^{91} + q^{98} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/385\mathbb{Z}\right)^\times\).

\(n\) \(211\) \(232\) \(276\)
\(\chi(n)\) \(-1\) \(-1\) \(-\zeta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
109.1
0.500000 + 0.866025i
0.500000 0.866025i
0.500000 + 0.866025i 0 0 −0.500000 0.866025i 0 1.00000 1.00000 −0.500000 0.866025i 0.500000 0.866025i
219.1 0.500000 0.866025i 0 0 −0.500000 + 0.866025i 0 1.00000 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
55.d odd 2 1 CM by \(\Q(\sqrt{-55}) \)
7.c even 3 1 inner
385.q odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 385.1.q.b yes 2
3.b odd 2 1 3465.1.cd.a 2
5.b even 2 1 385.1.q.a 2
5.c odd 4 2 1925.1.w.c 4
7.b odd 2 1 2695.1.q.d 2
7.c even 3 1 inner 385.1.q.b yes 2
7.c even 3 1 2695.1.g.b 1
7.d odd 6 1 2695.1.g.a 1
7.d odd 6 1 2695.1.q.d 2
11.b odd 2 1 385.1.q.a 2
15.d odd 2 1 3465.1.cd.b 2
21.h odd 6 1 3465.1.cd.a 2
33.d even 2 1 3465.1.cd.b 2
35.c odd 2 1 2695.1.q.a 2
35.i odd 6 1 2695.1.g.d 1
35.i odd 6 1 2695.1.q.a 2
35.j even 6 1 385.1.q.a 2
35.j even 6 1 2695.1.g.e 1
35.l odd 12 2 1925.1.w.c 4
55.d odd 2 1 CM 385.1.q.b yes 2
55.e even 4 2 1925.1.w.c 4
77.b even 2 1 2695.1.q.a 2
77.h odd 6 1 385.1.q.a 2
77.h odd 6 1 2695.1.g.e 1
77.i even 6 1 2695.1.g.d 1
77.i even 6 1 2695.1.q.a 2
105.o odd 6 1 3465.1.cd.b 2
165.d even 2 1 3465.1.cd.a 2
231.l even 6 1 3465.1.cd.b 2
385.h even 2 1 2695.1.q.d 2
385.o even 6 1 2695.1.g.a 1
385.o even 6 1 2695.1.q.d 2
385.q odd 6 1 inner 385.1.q.b yes 2
385.q odd 6 1 2695.1.g.b 1
385.bc even 12 2 1925.1.w.c 4
1155.bo even 6 1 3465.1.cd.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
385.1.q.a 2 5.b even 2 1
385.1.q.a 2 11.b odd 2 1
385.1.q.a 2 35.j even 6 1
385.1.q.a 2 77.h odd 6 1
385.1.q.b yes 2 1.a even 1 1 trivial
385.1.q.b yes 2 7.c even 3 1 inner
385.1.q.b yes 2 55.d odd 2 1 CM
385.1.q.b yes 2 385.q odd 6 1 inner
1925.1.w.c 4 5.c odd 4 2
1925.1.w.c 4 35.l odd 12 2
1925.1.w.c 4 55.e even 4 2
1925.1.w.c 4 385.bc even 12 2
2695.1.g.a 1 7.d odd 6 1
2695.1.g.a 1 385.o even 6 1
2695.1.g.b 1 7.c even 3 1
2695.1.g.b 1 385.q odd 6 1
2695.1.g.d 1 35.i odd 6 1
2695.1.g.d 1 77.i even 6 1
2695.1.g.e 1 35.j even 6 1
2695.1.g.e 1 77.h odd 6 1
2695.1.q.a 2 35.c odd 2 1
2695.1.q.a 2 35.i odd 6 1
2695.1.q.a 2 77.b even 2 1
2695.1.q.a 2 77.i even 6 1
2695.1.q.d 2 7.b odd 2 1
2695.1.q.d 2 7.d odd 6 1
2695.1.q.d 2 385.h even 2 1
2695.1.q.d 2 385.o even 6 1
3465.1.cd.a 2 3.b odd 2 1
3465.1.cd.a 2 21.h odd 6 1
3465.1.cd.a 2 165.d even 2 1
3465.1.cd.a 2 1155.bo even 6 1
3465.1.cd.b 2 15.d odd 2 1
3465.1.cd.b 2 33.d even 2 1
3465.1.cd.b 2 105.o odd 6 1
3465.1.cd.b 2 231.l even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - T_{2} + 1 \) acting on \(S_{1}^{\mathrm{new}}(385, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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