Properties

Label 3042.2.b.h
Level $3042$
Weight $2$
Character orbit 3042.b
Analytic conductor $24.290$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3042,2,Mod(1351,3042)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3042, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3042.1351");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3042 = 2 \cdot 3^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3042.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: \(24.2904922949\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
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 78)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - i q^{2} - q^{4} + 3 i q^{5} + 2 i q^{7} + i q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q - i q^{2} - q^{4} + 3 i q^{5} + 2 i q^{7} + i q^{8} + 3 q^{10} - 6 i q^{11} + 2 q^{14} + q^{16} - 3 q^{17} - 2 i q^{19} - 3 i q^{20} - 6 q^{22} - 6 q^{23} - 4 q^{25} - 2 i q^{28} - 3 q^{29} + 4 i q^{31} - i q^{32} + 3 i q^{34} - 6 q^{35} - 7 i q^{37} - 2 q^{38} - 3 q^{40} - 3 i q^{41} + 10 q^{43} + 6 i q^{44} + 6 i q^{46} - 6 i q^{47} + 3 q^{49} + 4 i q^{50} - 3 q^{53} + 18 q^{55} - 2 q^{56} + 3 i q^{58} - 7 q^{61} + 4 q^{62} - q^{64} + 10 i q^{67} + 3 q^{68} + 6 i q^{70} + 6 i q^{71} - 13 i q^{73} - 7 q^{74} + 2 i q^{76} + 12 q^{77} - 4 q^{79} + 3 i q^{80} - 3 q^{82} - 6 i q^{83} - 9 i q^{85} - 10 i q^{86} + 6 q^{88} - 18 i q^{89} + 6 q^{92} - 6 q^{94} + 6 q^{95} - 14 i q^{97} - 3 i q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{4} + 6 q^{10} + 4 q^{14} + 2 q^{16} - 6 q^{17} - 12 q^{22} - 12 q^{23} - 8 q^{25} - 6 q^{29} - 12 q^{35} - 4 q^{38} - 6 q^{40} + 20 q^{43} + 6 q^{49} - 6 q^{53} + 36 q^{55} - 4 q^{56} - 14 q^{61} + 8 q^{62} - 2 q^{64} + 6 q^{68} - 14 q^{74} + 24 q^{77} - 8 q^{79} - 6 q^{82} + 12 q^{88} + 12 q^{92} - 12 q^{94} + 12 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3042\mathbb{Z}\right)^\times\).

\(n\) \(677\) \(847\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1351.1
1.00000i
1.00000i
1.00000i 0 −1.00000 3.00000i 0 2.00000i 1.00000i 0 3.00000
1351.2 1.00000i 0 −1.00000 3.00000i 0 2.00000i 1.00000i 0 3.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3042.2.b.h 2
3.b odd 2 1 1014.2.b.c 2
13.b even 2 1 inner 3042.2.b.h 2
13.d odd 4 1 3042.2.a.h 1
13.d odd 4 1 3042.2.a.i 1
13.f odd 12 2 234.2.h.a 2
39.d odd 2 1 1014.2.b.c 2
39.f even 4 1 1014.2.a.c 1
39.f even 4 1 1014.2.a.f 1
39.h odd 6 2 1014.2.i.b 4
39.i odd 6 2 1014.2.i.b 4
39.k even 12 2 78.2.e.a 2
39.k even 12 2 1014.2.e.a 2
52.l even 12 2 1872.2.t.c 2
156.l odd 4 1 8112.2.a.c 1
156.l odd 4 1 8112.2.a.m 1
156.v odd 12 2 624.2.q.g 2
195.bc odd 12 2 1950.2.z.g 4
195.bh even 12 2 1950.2.i.m 2
195.bn odd 12 2 1950.2.z.g 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.2.e.a 2 39.k even 12 2
234.2.h.a 2 13.f odd 12 2
624.2.q.g 2 156.v odd 12 2
1014.2.a.c 1 39.f even 4 1
1014.2.a.f 1 39.f even 4 1
1014.2.b.c 2 3.b odd 2 1
1014.2.b.c 2 39.d odd 2 1
1014.2.e.a 2 39.k even 12 2
1014.2.i.b 4 39.h odd 6 2
1014.2.i.b 4 39.i odd 6 2
1872.2.t.c 2 52.l even 12 2
1950.2.i.m 2 195.bh even 12 2
1950.2.z.g 4 195.bc odd 12 2
1950.2.z.g 4 195.bn odd 12 2
3042.2.a.h 1 13.d odd 4 1
3042.2.a.i 1 13.d odd 4 1
3042.2.b.h 2 1.a even 1 1 trivial
3042.2.b.h 2 13.b even 2 1 inner
8112.2.a.c 1 156.l odd 4 1
8112.2.a.m 1 156.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3042, [\chi])\):

\( T_{5}^{2} + 9 \) Copy content Toggle raw display
\( T_{7}^{2} + 4 \) Copy content Toggle raw display
\( T_{17} + 3 \) Copy content Toggle raw display
\( T_{23} + 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 9 \) Copy content Toggle raw display
$7$ \( T^{2} + 4 \) Copy content Toggle raw display
$11$ \( T^{2} + 36 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( (T + 3)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 4 \) Copy content Toggle raw display
$23$ \( (T + 6)^{2} \) Copy content Toggle raw display
$29$ \( (T + 3)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 16 \) Copy content Toggle raw display
$37$ \( T^{2} + 49 \) Copy content Toggle raw display
$41$ \( T^{2} + 9 \) Copy content Toggle raw display
$43$ \( (T - 10)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 36 \) Copy content Toggle raw display
$53$ \( (T + 3)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( (T + 7)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 100 \) Copy content Toggle raw display
$71$ \( T^{2} + 36 \) Copy content Toggle raw display
$73$ \( T^{2} + 169 \) Copy content Toggle raw display
$79$ \( (T + 4)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 36 \) Copy content Toggle raw display
$89$ \( T^{2} + 324 \) Copy content Toggle raw display
$97$ \( T^{2} + 196 \) Copy content Toggle raw display
show more
show less