Properties

Label 3042.2.b.f
Level $3042$
Weight $2$
Character orbit 3042.b
Analytic conductor $24.290$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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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 26)
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.

\(f(q)\) \(=\) \( q - i q^{2} - q^{4} + i q^{5} - i q^{7} + i q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q - i q^{2} - q^{4} + i q^{5} - i q^{7} + i q^{8} + q^{10} - 2 i q^{11} - q^{14} + q^{16} - 3 q^{17} + 6 i q^{19} - i q^{20} - 2 q^{22} - 4 q^{23} + 4 q^{25} + i q^{28} - 2 q^{29} + 4 i q^{31} - i q^{32} + 3 i q^{34} + q^{35} - 3 i q^{37} + 6 q^{38} - q^{40} + 5 q^{43} + 2 i q^{44} + 4 i q^{46} + 13 i q^{47} + 6 q^{49} - 4 i q^{50} - 12 q^{53} + 2 q^{55} + q^{56} + 2 i q^{58} - 10 i q^{59} - 8 q^{61} + 4 q^{62} - q^{64} - 2 i q^{67} + 3 q^{68} - i q^{70} + 5 i q^{71} + 10 i q^{73} - 3 q^{74} - 6 i q^{76} - 2 q^{77} - 4 q^{79} + i q^{80} - 3 i q^{85} - 5 i q^{86} + 2 q^{88} + 6 i q^{89} + 4 q^{92} + 13 q^{94} - 6 q^{95} + 14 i q^{97} - 6 i q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{4} + 2 q^{10} - 2 q^{14} + 2 q^{16} - 6 q^{17} - 4 q^{22} - 8 q^{23} + 8 q^{25} - 4 q^{29} + 2 q^{35} + 12 q^{38} - 2 q^{40} + 10 q^{43} + 12 q^{49} - 24 q^{53} + 4 q^{55} + 2 q^{56} - 16 q^{61} + 8 q^{62} - 2 q^{64} + 6 q^{68} - 6 q^{74} - 4 q^{77} - 8 q^{79} + 4 q^{88} + 8 q^{92} + 26 q^{94} - 12 q^{95}+O(q^{100}) \) 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 1.00000i 0 1.00000i 1.00000i 0 1.00000
1351.2 1.00000i 0 −1.00000 1.00000i 0 1.00000i 1.00000i 0 1.00000
\(n\): 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.f 2
3.b odd 2 1 338.2.b.a 2
12.b even 2 1 2704.2.f.j 2
13.b even 2 1 inner 3042.2.b.f 2
13.d odd 4 1 234.2.a.b 1
13.d odd 4 1 3042.2.a.l 1
39.d odd 2 1 338.2.b.a 2
39.f even 4 1 26.2.a.b 1
39.f even 4 1 338.2.a.a 1
39.h odd 6 2 338.2.e.d 4
39.i odd 6 2 338.2.e.d 4
39.k even 12 2 338.2.c.c 2
39.k even 12 2 338.2.c.g 2
52.f even 4 1 1872.2.a.m 1
65.f even 4 1 5850.2.e.v 2
65.g odd 4 1 5850.2.a.bn 1
65.k even 4 1 5850.2.e.v 2
104.j odd 4 1 7488.2.a.w 1
104.m even 4 1 7488.2.a.v 1
117.y odd 12 2 2106.2.e.t 2
117.z even 12 2 2106.2.e.h 2
156.h even 2 1 2704.2.f.j 2
156.l odd 4 1 208.2.a.d 1
156.l odd 4 1 2704.2.a.n 1
195.j odd 4 1 650.2.b.a 2
195.n even 4 1 650.2.a.g 1
195.n even 4 1 8450.2.a.y 1
195.u odd 4 1 650.2.b.a 2
273.o odd 4 1 1274.2.a.o 1
273.cb odd 12 2 1274.2.f.a 2
273.cd even 12 2 1274.2.f.l 2
312.w odd 4 1 832.2.a.a 1
312.y even 4 1 832.2.a.j 1
429.l odd 4 1 3146.2.a.a 1
624.s odd 4 1 3328.2.b.k 2
624.u even 4 1 3328.2.b.g 2
624.bm even 4 1 3328.2.b.g 2
624.bo odd 4 1 3328.2.b.k 2
663.q even 4 1 7514.2.a.i 1
741.p odd 4 1 9386.2.a.f 1
780.bb odd 4 1 5200.2.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.2.a.b 1 39.f even 4 1
208.2.a.d 1 156.l odd 4 1
234.2.a.b 1 13.d odd 4 1
338.2.a.a 1 39.f even 4 1
338.2.b.a 2 3.b odd 2 1
338.2.b.a 2 39.d odd 2 1
338.2.c.c 2 39.k even 12 2
338.2.c.g 2 39.k even 12 2
338.2.e.d 4 39.h odd 6 2
338.2.e.d 4 39.i odd 6 2
650.2.a.g 1 195.n even 4 1
650.2.b.a 2 195.j odd 4 1
650.2.b.a 2 195.u odd 4 1
832.2.a.a 1 312.w odd 4 1
832.2.a.j 1 312.y even 4 1
1274.2.a.o 1 273.o odd 4 1
1274.2.f.a 2 273.cb odd 12 2
1274.2.f.l 2 273.cd even 12 2
1872.2.a.m 1 52.f even 4 1
2106.2.e.h 2 117.z even 12 2
2106.2.e.t 2 117.y odd 12 2
2704.2.a.n 1 156.l odd 4 1
2704.2.f.j 2 12.b even 2 1
2704.2.f.j 2 156.h even 2 1
3042.2.a.l 1 13.d odd 4 1
3042.2.b.f 2 1.a even 1 1 trivial
3042.2.b.f 2 13.b even 2 1 inner
3146.2.a.a 1 429.l odd 4 1
3328.2.b.g 2 624.u even 4 1
3328.2.b.g 2 624.bm even 4 1
3328.2.b.k 2 624.s odd 4 1
3328.2.b.k 2 624.bo odd 4 1
5200.2.a.c 1 780.bb odd 4 1
5850.2.a.bn 1 65.g odd 4 1
5850.2.e.v 2 65.f even 4 1
5850.2.e.v 2 65.k even 4 1
7488.2.a.v 1 104.m even 4 1
7488.2.a.w 1 104.j odd 4 1
7514.2.a.i 1 663.q even 4 1
8450.2.a.y 1 195.n even 4 1
9386.2.a.f 1 741.p odd 4 1

Hecke kernels

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

\( T_{5}^{2} + 1 \) Copy content Toggle raw display
\( T_{7}^{2} + 1 \) Copy content Toggle raw display
\( T_{17} + 3 \) Copy content Toggle raw display
\( T_{23} + 4 \) 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
$5$ \( T^{2} + 1 \) Copy content Toggle raw display
$7$ \( T^{2} + 1 \) Copy content Toggle raw display
$11$ \( T^{2} + 4 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( (T + 3)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 36 \) Copy content Toggle raw display
$23$ \( (T + 4)^{2} \) Copy content Toggle raw display
$29$ \( (T + 2)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 16 \) Copy content Toggle raw display
$37$ \( T^{2} + 9 \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( (T - 5)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 169 \) Copy content Toggle raw display
$53$ \( (T + 12)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 100 \) Copy content Toggle raw display
$61$ \( (T + 8)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 4 \) Copy content Toggle raw display
$71$ \( T^{2} + 25 \) Copy content Toggle raw display
$73$ \( T^{2} + 100 \) Copy content Toggle raw display
$79$ \( (T + 4)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 36 \) Copy content Toggle raw display
$97$ \( T^{2} + 196 \) Copy content Toggle raw display
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