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Properties

Label 1848.1.b.c
Level $1848$
Weight $1$
Character orbit 1848.b
Analytic conductor $0.922$
Analytic rank $0$
Dimension $4$
Projective image $D_{6}$
CM discriminant -231
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1848,1,Mod(461,1848)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1848, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 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("1848.461");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1848 = 2^{3} \cdot 3 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1848.b (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: no
Analytic conductor: \(0.922272143346\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{6}\)
Projective field: Galois closure of 6.0.81962496.1

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{12} q^{2} + \zeta_{12}^{3} q^{3} + \zeta_{12}^{2} q^{4} - \zeta_{12}^{3} q^{5} + \zeta_{12}^{4} q^{6} + q^{7} + \zeta_{12}^{3} q^{8} - q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{12} q^{2} + \zeta_{12}^{3} q^{3} + \zeta_{12}^{2} q^{4} - \zeta_{12}^{3} q^{5} + \zeta_{12}^{4} q^{6} + q^{7} + \zeta_{12}^{3} q^{8} - q^{9} - \zeta_{12}^{4} q^{10} + \zeta_{12}^{3} q^{11} + \zeta_{12}^{5} q^{12} + ( - \zeta_{12}^{4} - \zeta_{12}^{2}) q^{13} + \zeta_{12} q^{14} + q^{15} + \zeta_{12}^{4} q^{16} - \zeta_{12} q^{18} + (\zeta_{12}^{4} + \zeta_{12}^{2}) q^{19} - \zeta_{12}^{5} q^{20} + \zeta_{12}^{3} q^{21} + \zeta_{12}^{4} q^{22} - q^{24} + ( - \zeta_{12}^{5} - \zeta_{12}^{3}) q^{26} - \zeta_{12}^{3} q^{27} + \zeta_{12}^{2} q^{28} + \zeta_{12}^{3} q^{29} + \zeta_{12} q^{30} + \zeta_{12}^{5} q^{32} - q^{33} - \zeta_{12}^{3} q^{35} - \zeta_{12}^{2} q^{36} + ( - \zeta_{12}^{4} - \zeta_{12}^{2}) q^{37} + (\zeta_{12}^{5} + \zeta_{12}^{3}) q^{38} + ( - \zeta_{12}^{5} + \zeta_{12}) q^{39} + q^{40} + \zeta_{12}^{4} q^{42} + \zeta_{12}^{5} q^{44} + \zeta_{12}^{3} q^{45} + (\zeta_{12}^{5} - \zeta_{12}) q^{47} - \zeta_{12} q^{48} + q^{49} + ( - \zeta_{12}^{4} + 1) q^{52} - \zeta_{12}^{4} q^{54} + q^{55} + \zeta_{12}^{3} q^{56} + (\zeta_{12}^{5} - \zeta_{12}) q^{57} + \zeta_{12}^{4} q^{58} - \zeta_{12}^{3} q^{59} + \zeta_{12}^{2} q^{60} - q^{63} - q^{64} + (\zeta_{12}^{5} - \zeta_{12}) q^{65} - \zeta_{12} q^{66} + ( - \zeta_{12}^{4} - \zeta_{12}^{2}) q^{67} - \zeta_{12}^{4} q^{70} - \zeta_{12}^{3} q^{72} - q^{73} + ( - \zeta_{12}^{5} - \zeta_{12}^{3}) q^{74} + (\zeta_{12}^{4} - 1) q^{76} + \zeta_{12}^{3} q^{77} + (\zeta_{12}^{2} + 1) q^{78} + \zeta_{12} q^{80} + q^{81} + \zeta_{12}^{5} q^{84} - q^{87} - q^{88} + \zeta_{12}^{4} q^{90} + ( - \zeta_{12}^{4} - \zeta_{12}^{2}) q^{91} + ( - \zeta_{12}^{2} - 1) q^{94} + ( - \zeta_{12}^{5} + \zeta_{12}) q^{95} - \zeta_{12}^{2} q^{96} + \zeta_{12} q^{98} - \zeta_{12}^{3} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4} - 2 q^{6} + 4 q^{7} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{4} - 2 q^{6} + 4 q^{7} - 4 q^{9} + 2 q^{10} + 4 q^{15} - 2 q^{16} - 2 q^{22} - 4 q^{24} + 2 q^{28} - 4 q^{33} - 2 q^{36} + 4 q^{40} - 2 q^{42} + 4 q^{49} + 6 q^{52} + 2 q^{54} + 4 q^{55} - 2 q^{58} + 2 q^{60} - 4 q^{63} - 4 q^{64} + 2 q^{70} - 4 q^{73} - 6 q^{76} + 6 q^{78} + 4 q^{81} - 4 q^{87} - 4 q^{88} - 2 q^{90} - 6 q^{94} - 2 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1848\mathbb{Z}\right)^\times\).

\(n\) \(463\) \(617\) \(673\) \(925\) \(1585\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
461.1
−0.866025 − 0.500000i
−0.866025 + 0.500000i
0.866025 − 0.500000i
0.866025 + 0.500000i
−0.866025 − 0.500000i − 1.00000i 0.500000 + 0.866025i 1.00000i −0.500000 + 0.866025i 1.00000 − 1.00000i −1.00000 0.500000 − 0.866025i
461.2 −0.866025 + 0.500000i 1.00000i 0.500000 − 0.866025i − 1.00000i −0.500000 − 0.866025i 1.00000 1.00000i −1.00000 0.500000 + 0.866025i
461.3 0.866025 − 0.500000i − 1.00000i 0.500000 − 0.866025i 1.00000i −0.500000 − 0.866025i 1.00000 − 1.00000i −1.00000 0.500000 + 0.866025i
461.4 0.866025 + 0.500000i 1.00000i 0.500000 + 0.866025i − 1.00000i −0.500000 + 0.866025i 1.00000 1.00000i −1.00000 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
231.h odd 2 1 CM by \(\Q(\sqrt{-231}) \)
3.b odd 2 1 inner
8.b even 2 1 inner
24.h odd 2 1 inner
77.b even 2 1 inner
616.o even 2 1 inner
1848.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1848.1.b.c ✓ 4
3.b odd 2 1 inner 1848.1.b.c ✓ 4
7.b odd 2 1 1848.1.b.d yes 4
8.b even 2 1 inner 1848.1.b.c ✓ 4
11.b odd 2 1 1848.1.b.d yes 4
21.c even 2 1 1848.1.b.d yes 4
24.h odd 2 1 inner 1848.1.b.c ✓ 4
33.d even 2 1 1848.1.b.d yes 4
56.h odd 2 1 1848.1.b.d yes 4
77.b even 2 1 inner 1848.1.b.c ✓ 4
88.b odd 2 1 1848.1.b.d yes 4
168.i even 2 1 1848.1.b.d yes 4
231.h odd 2 1 CM 1848.1.b.c ✓ 4
264.m even 2 1 1848.1.b.d yes 4
616.o even 2 1 inner 1848.1.b.c ✓ 4
1848.b odd 2 1 inner 1848.1.b.c ✓ 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1848.1.b.c ✓ 4 1.a even 1 1 trivial
1848.1.b.c ✓ 4 3.b odd 2 1 inner
1848.1.b.c ✓ 4 8.b even 2 1 inner
1848.1.b.c ✓ 4 24.h odd 2 1 inner
1848.1.b.c ✓ 4 77.b even 2 1 inner
1848.1.b.c ✓ 4 231.h odd 2 1 CM
1848.1.b.c ✓ 4 616.o even 2 1 inner
1848.1.b.c ✓ 4 1848.b odd 2 1 inner
1848.1.b.d yes 4 7.b odd 2 1
1848.1.b.d yes 4 11.b odd 2 1
1848.1.b.d yes 4 21.c even 2 1
1848.1.b.d yes 4 33.d even 2 1
1848.1.b.d yes 4 56.h odd 2 1
1848.1.b.d yes 4 88.b odd 2 1
1848.1.b.d yes 4 168.i even 2 1
1848.1.b.d yes 4 264.m even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(1848, [\chi])\):

\( T_{5}^{2} + 1 \) Copy content Toggle raw display
\( T_{73} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

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