Properties

Label 3800.1.b.b
Level $3800$
Weight $1$
Character orbit 3800.b
Analytic conductor $1.896$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -152
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3800,1,Mod(949,3800)]
 
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, 1, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3800.949");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3800 = 2^{3} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3800.b (of order \(2\), degree \(1\), 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: \(1.89644704801\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
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: $C_4\times D_6$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{24} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q - i q^{2} - i q^{3} - q^{4} - q^{6} + i q^{7} + i q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q - i q^{2} - i q^{3} - q^{4} - q^{6} + i q^{7} + i q^{8} + i q^{12} - i q^{13} + q^{14} + q^{16} + i q^{17} - q^{19} + q^{21} - i q^{23} + q^{24} - q^{26} - i q^{27} - i q^{28} + q^{29} - i q^{32} + q^{34} - i q^{37} + i q^{38} - q^{39} - i q^{42} - q^{46} - i q^{47} - i q^{48} + q^{51} + i q^{52} - i q^{53} - q^{54} - q^{56} + i q^{57} - i q^{58} + q^{59} - q^{64} + i q^{67} - i q^{68} - q^{69} - i q^{73} - 2 q^{74} + q^{76} + i q^{78} - q^{81} - q^{84} - i q^{87} + q^{91} + i q^{92} - 2 q^{94} - q^{96} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} - 2 q^{6}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{4} - 2 q^{6} + 2 q^{14} + 2 q^{16} - 2 q^{19} + 2 q^{21} + 2 q^{24} - 2 q^{26} + 2 q^{29} + 2 q^{34} - 2 q^{39} - 2 q^{46} + 2 q^{51} - 2 q^{54} - 2 q^{56} + 2 q^{59} - 2 q^{64} - 2 q^{69} - 4 q^{74} + 2 q^{76} - 2 q^{81} - 2 q^{84} + 2 q^{91} - 4 q^{94} - 2 q^{96}+O(q^{100}) \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
949.1
1.00000i
1.00000i
1.00000i 1.00000i −1.00000 0 −1.00000 1.00000i 1.00000i 0 0
949.2 1.00000i 1.00000i −1.00000 0 −1.00000 1.00000i 1.00000i 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 \(\Q(\sqrt{-38}) \)
5.b even 2 1 inner
760.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3800.1.b.b 2
5.b even 2 1 inner 3800.1.b.b 2
5.c odd 4 1 152.1.g.b yes 1
5.c odd 4 1 3800.1.o.a 1
8.b even 2 1 3800.1.b.a 2
15.e even 4 1 1368.1.i.a 1
19.b odd 2 1 3800.1.b.a 2
20.e even 4 1 608.1.g.b 1
40.f even 2 1 3800.1.b.a 2
40.i odd 4 1 152.1.g.a 1
40.i odd 4 1 3800.1.o.b 1
40.k even 4 1 608.1.g.a 1
95.d odd 2 1 3800.1.b.a 2
95.g even 4 1 152.1.g.a 1
95.g even 4 1 3800.1.o.b 1
95.l even 12 2 2888.1.l.b 2
95.m odd 12 2 2888.1.l.a 2
95.q odd 36 6 2888.1.s.a 6
95.r even 36 6 2888.1.s.b 6
120.w even 4 1 1368.1.i.b 1
152.g odd 2 1 CM 3800.1.b.b 2
285.j odd 4 1 1368.1.i.b 1
380.j odd 4 1 608.1.g.a 1
760.b odd 2 1 inner 3800.1.b.b 2
760.t even 4 1 152.1.g.b yes 1
760.t even 4 1 3800.1.o.a 1
760.y odd 4 1 608.1.g.b 1
760.bp even 12 2 2888.1.l.a 2
760.br odd 12 2 2888.1.l.b 2
760.cq odd 36 6 2888.1.s.b 6
760.cs even 36 6 2888.1.s.a 6
2280.bw odd 4 1 1368.1.i.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.1.g.a 1 40.i odd 4 1
152.1.g.a 1 95.g even 4 1
152.1.g.b yes 1 5.c odd 4 1
152.1.g.b yes 1 760.t even 4 1
608.1.g.a 1 40.k even 4 1
608.1.g.a 1 380.j odd 4 1
608.1.g.b 1 20.e even 4 1
608.1.g.b 1 760.y odd 4 1
1368.1.i.a 1 15.e even 4 1
1368.1.i.a 1 2280.bw odd 4 1
1368.1.i.b 1 120.w even 4 1
1368.1.i.b 1 285.j odd 4 1
2888.1.l.a 2 95.m odd 12 2
2888.1.l.a 2 760.bp even 12 2
2888.1.l.b 2 95.l even 12 2
2888.1.l.b 2 760.br odd 12 2
2888.1.s.a 6 95.q odd 36 6
2888.1.s.a 6 760.cs even 36 6
2888.1.s.b 6 95.r even 36 6
2888.1.s.b 6 760.cq odd 36 6
3800.1.b.a 2 8.b even 2 1
3800.1.b.a 2 19.b odd 2 1
3800.1.b.a 2 40.f even 2 1
3800.1.b.a 2 95.d odd 2 1
3800.1.b.b 2 1.a even 1 1 trivial
3800.1.b.b 2 5.b even 2 1 inner
3800.1.b.b 2 152.g odd 2 1 CM
3800.1.b.b 2 760.b odd 2 1 inner
3800.1.o.a 1 5.c odd 4 1
3800.1.o.a 1 760.t even 4 1
3800.1.o.b 1 40.i odd 4 1
3800.1.o.b 1 95.g even 4 1

Hecke kernels

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

\( T_{3}^{2} + 1 \) Copy content Toggle raw display
\( T_{29} - 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

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