Properties

Label 3800.1.o.b
Level $3800$
Weight $1$
Character orbit 3800.o
Self dual yes
Analytic conductor $1.896$
Analytic rank $0$
Dimension $1$
Projective image $D_{3}$
CM discriminant -152
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3800,1,Mod(1101,3800)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3800, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3800.1101");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3800 = 2^{3} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3800.o (of order \(2\), degree \(1\), 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: yes
Analytic conductor: \(1.89644704801\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 152)
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.152.1
Artin image: $D_6$
Artin field: Galois closure of 6.2.23104000.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q + q^{2} - q^{3} + q^{4} - q^{6} + q^{7} + q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} - q^{3} + q^{4} - q^{6} + q^{7} + q^{8} - q^{12} - q^{13} + q^{14} + q^{16} + q^{17} - q^{19} - q^{21} + q^{23} - q^{24} - q^{26} + q^{27} + q^{28} + q^{29} + q^{32} + q^{34} + 2 q^{37} - q^{38} + q^{39} - q^{42} + q^{46} - 2 q^{47} - q^{48} - q^{51} - q^{52} - q^{53} + q^{54} + q^{56} + q^{57} + q^{58} + q^{59} + q^{64} - q^{67} + q^{68} - q^{69} + q^{73} + 2 q^{74} - q^{76} + q^{78} - q^{81} - q^{84} - q^{87} - q^{91} + q^{92} - 2 q^{94} - q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3800\mathbb{Z}\right)^\times\).

\(n\) \(401\) \(951\) \(1901\) \(1977\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1101.1
0
1.00000 −1.00000 1.00000 0 −1.00000 1.00000 1.00000 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
152.g odd 2 1 CM by \(\Q(\sqrt{-38}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3800.1.o.b 1
5.b even 2 1 152.1.g.a 1
5.c odd 4 2 3800.1.b.a 2
8.b even 2 1 3800.1.o.a 1
15.d odd 2 1 1368.1.i.b 1
19.b odd 2 1 3800.1.o.a 1
20.d odd 2 1 608.1.g.a 1
40.e odd 2 1 608.1.g.b 1
40.f even 2 1 152.1.g.b yes 1
40.i odd 4 2 3800.1.b.b 2
95.d odd 2 1 152.1.g.b yes 1
95.g even 4 2 3800.1.b.b 2
95.h odd 6 2 2888.1.l.a 2
95.i even 6 2 2888.1.l.b 2
95.o odd 18 6 2888.1.s.a 6
95.p even 18 6 2888.1.s.b 6
120.i odd 2 1 1368.1.i.a 1
152.g odd 2 1 CM 3800.1.o.b 1
285.b even 2 1 1368.1.i.a 1
380.d even 2 1 608.1.g.b 1
760.b odd 2 1 152.1.g.a 1
760.p even 2 1 608.1.g.a 1
760.t even 4 2 3800.1.b.a 2
760.z even 6 2 2888.1.l.a 2
760.bh odd 6 2 2888.1.l.b 2
760.cj even 18 6 2888.1.s.a 6
760.ck odd 18 6 2888.1.s.b 6
2280.m even 2 1 1368.1.i.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.1.g.a 1 5.b even 2 1
152.1.g.a 1 760.b odd 2 1
152.1.g.b yes 1 40.f even 2 1
152.1.g.b yes 1 95.d odd 2 1
608.1.g.a 1 20.d odd 2 1
608.1.g.a 1 760.p even 2 1
608.1.g.b 1 40.e odd 2 1
608.1.g.b 1 380.d even 2 1
1368.1.i.a 1 120.i odd 2 1
1368.1.i.a 1 285.b even 2 1
1368.1.i.b 1 15.d odd 2 1
1368.1.i.b 1 2280.m even 2 1
2888.1.l.a 2 95.h odd 6 2
2888.1.l.a 2 760.z even 6 2
2888.1.l.b 2 95.i even 6 2
2888.1.l.b 2 760.bh odd 6 2
2888.1.s.a 6 95.o odd 18 6
2888.1.s.a 6 760.cj even 18 6
2888.1.s.b 6 95.p even 18 6
2888.1.s.b 6 760.ck odd 18 6
3800.1.b.a 2 5.c odd 4 2
3800.1.b.a 2 760.t even 4 2
3800.1.b.b 2 40.i odd 4 2
3800.1.b.b 2 95.g even 4 2
3800.1.o.a 1 8.b even 2 1
3800.1.o.a 1 19.b odd 2 1
3800.1.o.b 1 1.a even 1 1 trivial
3800.1.o.b 1 152.g odd 2 1 CM

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(3800, [\chi])\):

\( T_{3} + 1 \) Copy content Toggle raw display
\( T_{7} - 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

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