Properties

Label 1900.2.s.a
Level $1900$
Weight $2$
Character orbit 1900.s
Analytic conductor $15.172$
Analytic rank $0$
Dimension $4$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1900,2,Mod(49,1900)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1900, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1900.49");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1900 = 2^{2} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1900.s (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: \(15.1715763840\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 76)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{12} q^{3} - 2 \zeta_{12}^{2} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{12} q^{3} - 2 \zeta_{12}^{2} q^{9} - 4 q^{11} + (\zeta_{12}^{3} - \zeta_{12}) q^{13} + 3 \zeta_{12} q^{17} + ( - 2 \zeta_{12}^{2} + 5) q^{19} + ( - 5 \zeta_{12}^{3} + 5 \zeta_{12}) q^{23} - 5 \zeta_{12}^{3} q^{27} + 7 \zeta_{12}^{2} q^{29} + 4 q^{31} - 4 \zeta_{12} q^{33} - 10 \zeta_{12}^{3} q^{37} - q^{39} + ( - 5 \zeta_{12}^{2} + 5) q^{41} + 5 \zeta_{12} q^{43} + ( - 7 \zeta_{12}^{3} + 7 \zeta_{12}) q^{47} + 7 q^{49} + 3 \zeta_{12}^{2} q^{51} + ( - 11 \zeta_{12}^{3} + 11 \zeta_{12}) q^{53} + ( - 2 \zeta_{12}^{3} + 5 \zeta_{12}) q^{57} + ( - 3 \zeta_{12}^{2} + 3) q^{59} - 11 \zeta_{12}^{2} q^{61} + ( - 3 \zeta_{12}^{3} + 3 \zeta_{12}) q^{67} + 5 q^{69} + (11 \zeta_{12}^{2} - 11) q^{71} - 15 \zeta_{12} q^{73} + (13 \zeta_{12}^{2} - 13) q^{79} + (\zeta_{12}^{2} - 1) q^{81} + 7 \zeta_{12}^{3} q^{87} + 3 \zeta_{12}^{2} q^{89} + 4 \zeta_{12} q^{93} - 5 \zeta_{12} q^{97} + 8 \zeta_{12}^{2} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{9} - 16 q^{11} + 16 q^{19} + 14 q^{29} + 16 q^{31} - 4 q^{39} + 10 q^{41} + 28 q^{49} + 6 q^{51} + 6 q^{59} - 22 q^{61} + 20 q^{69} - 22 q^{71} - 26 q^{79} - 2 q^{81} + 6 q^{89} + 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1900\mathbb{Z}\right)^\times\).

\(n\) \(77\) \(401\) \(951\)
\(\chi(n)\) \(-1\) \(-\zeta_{12}^{2}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
−0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
0 −0.866025 0.500000i 0 0 0 0 0 −1.00000 1.73205i 0
49.2 0 0.866025 + 0.500000i 0 0 0 0 0 −1.00000 1.73205i 0
349.1 0 −0.866025 + 0.500000i 0 0 0 0 0 −1.00000 + 1.73205i 0
349.2 0 0.866025 0.500000i 0 0 0 0 0 −1.00000 + 1.73205i 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
5.b even 2 1 inner
19.c even 3 1 inner
95.i even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1900.2.s.a 4
5.b even 2 1 inner 1900.2.s.a 4
5.c odd 4 1 76.2.e.a 2
5.c odd 4 1 1900.2.i.a 2
15.e even 4 1 684.2.k.b 2
19.c even 3 1 inner 1900.2.s.a 4
20.e even 4 1 304.2.i.a 2
40.i odd 4 1 1216.2.i.c 2
40.k even 4 1 1216.2.i.g 2
60.l odd 4 1 2736.2.s.g 2
95.g even 4 1 1444.2.e.b 2
95.i even 6 1 inner 1900.2.s.a 4
95.l even 12 1 1444.2.a.c 1
95.l even 12 1 1444.2.e.b 2
95.m odd 12 1 76.2.e.a 2
95.m odd 12 1 1444.2.a.b 1
95.m odd 12 1 1900.2.i.a 2
285.v even 12 1 684.2.k.b 2
380.v even 12 1 304.2.i.a 2
380.v even 12 1 5776.2.a.k 1
380.w odd 12 1 5776.2.a.f 1
760.br odd 12 1 1216.2.i.c 2
760.bw even 12 1 1216.2.i.g 2
1140.bu odd 12 1 2736.2.s.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.2.e.a 2 5.c odd 4 1
76.2.e.a 2 95.m odd 12 1
304.2.i.a 2 20.e even 4 1
304.2.i.a 2 380.v even 12 1
684.2.k.b 2 15.e even 4 1
684.2.k.b 2 285.v even 12 1
1216.2.i.c 2 40.i odd 4 1
1216.2.i.c 2 760.br odd 12 1
1216.2.i.g 2 40.k even 4 1
1216.2.i.g 2 760.bw even 12 1
1444.2.a.b 1 95.m odd 12 1
1444.2.a.c 1 95.l even 12 1
1444.2.e.b 2 95.g even 4 1
1444.2.e.b 2 95.l even 12 1
1900.2.i.a 2 5.c odd 4 1
1900.2.i.a 2 95.m odd 12 1
1900.2.s.a 4 1.a even 1 1 trivial
1900.2.s.a 4 5.b even 2 1 inner
1900.2.s.a 4 19.c even 3 1 inner
1900.2.s.a 4 95.i even 6 1 inner
2736.2.s.g 2 60.l odd 4 1
2736.2.s.g 2 1140.bu odd 12 1
5776.2.a.f 1 380.w odd 12 1
5776.2.a.k 1 380.v even 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - T_{3}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(1900, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( (T + 4)^{4} \) Copy content Toggle raw display
$13$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$17$ \( T^{4} - 9T^{2} + 81 \) Copy content Toggle raw display
$19$ \( (T^{2} - 8 T + 19)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} - 25T^{2} + 625 \) Copy content Toggle raw display
$29$ \( (T^{2} - 7 T + 49)^{2} \) Copy content Toggle raw display
$31$ \( (T - 4)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 100)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 5 T + 25)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} - 25T^{2} + 625 \) Copy content Toggle raw display
$47$ \( T^{4} - 49T^{2} + 2401 \) Copy content Toggle raw display
$53$ \( T^{4} - 121 T^{2} + 14641 \) Copy content Toggle raw display
$59$ \( (T^{2} - 3 T + 9)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 11 T + 121)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 9T^{2} + 81 \) Copy content Toggle raw display
$71$ \( (T^{2} + 11 T + 121)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} - 225 T^{2} + 50625 \) Copy content Toggle raw display
$79$ \( (T^{2} + 13 T + 169)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} \) Copy content Toggle raw display
$89$ \( (T^{2} - 3 T + 9)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} - 25T^{2} + 625 \) Copy content Toggle raw display
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