Properties

Label 1050.2.i.f
Level $1050$
Weight $2$
Character orbit 1050.i
Analytic conductor $8.384$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1050,2,Mod(151,1050)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1050, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1050.151");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1050.i (of order \(3\), degree \(2\), 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: \(8.38429221223\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 210)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \zeta_{6} q^{2} + ( - \zeta_{6} + 1) q^{3} + (\zeta_{6} - 1) q^{4} - q^{6} + (2 \zeta_{6} - 3) q^{7} + q^{8} - \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{6} q^{2} + ( - \zeta_{6} + 1) q^{3} + (\zeta_{6} - 1) q^{4} - q^{6} + (2 \zeta_{6} - 3) q^{7} + q^{8} - \zeta_{6} q^{9} + ( - 5 \zeta_{6} + 5) q^{11} + \zeta_{6} q^{12} - q^{13} + (\zeta_{6} + 2) q^{14} - \zeta_{6} q^{16} + (2 \zeta_{6} - 2) q^{17} + (\zeta_{6} - 1) q^{18} - 7 \zeta_{6} q^{19} + (3 \zeta_{6} - 1) q^{21} - 5 q^{22} + 3 \zeta_{6} q^{23} + ( - \zeta_{6} + 1) q^{24} + \zeta_{6} q^{26} - q^{27} + ( - 3 \zeta_{6} + 1) q^{28} + ( - 6 \zeta_{6} + 6) q^{31} + (\zeta_{6} - 1) q^{32} - 5 \zeta_{6} q^{33} + 2 q^{34} + q^{36} - 5 \zeta_{6} q^{37} + (7 \zeta_{6} - 7) q^{38} + (\zeta_{6} - 1) q^{39} - 9 q^{41} + ( - 2 \zeta_{6} + 3) q^{42} - 10 q^{43} + 5 \zeta_{6} q^{44} + ( - 3 \zeta_{6} + 3) q^{46} - 13 \zeta_{6} q^{47} - q^{48} + ( - 8 \zeta_{6} + 5) q^{49} + 2 \zeta_{6} q^{51} + ( - \zeta_{6} + 1) q^{52} + (\zeta_{6} - 1) q^{53} + \zeta_{6} q^{54} + (2 \zeta_{6} - 3) q^{56} - 7 q^{57} + (4 \zeta_{6} - 4) q^{59} + 2 \zeta_{6} q^{61} - 6 q^{62} + (\zeta_{6} + 2) q^{63} + q^{64} + (5 \zeta_{6} - 5) q^{66} + ( - 6 \zeta_{6} + 6) q^{67} - 2 \zeta_{6} q^{68} + 3 q^{69} - 2 q^{71} - \zeta_{6} q^{72} + ( - 4 \zeta_{6} + 4) q^{73} + (5 \zeta_{6} - 5) q^{74} + 7 q^{76} + (15 \zeta_{6} - 5) q^{77} + q^{78} + 14 \zeta_{6} q^{79} + (\zeta_{6} - 1) q^{81} + 9 \zeta_{6} q^{82} + 10 q^{83} + ( - \zeta_{6} - 2) q^{84} + 10 \zeta_{6} q^{86} + ( - 5 \zeta_{6} + 5) q^{88} - 10 \zeta_{6} q^{89} + ( - 2 \zeta_{6} + 3) q^{91} - 3 q^{92} - 6 \zeta_{6} q^{93} + (13 \zeta_{6} - 13) q^{94} + \zeta_{6} q^{96} - 8 q^{97} + (3 \zeta_{6} - 8) q^{98} - 5 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} + q^{3} - q^{4} - 2 q^{6} - 4 q^{7} + 2 q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} + q^{3} - q^{4} - 2 q^{6} - 4 q^{7} + 2 q^{8} - q^{9} + 5 q^{11} + q^{12} - 2 q^{13} + 5 q^{14} - q^{16} - 2 q^{17} - q^{18} - 7 q^{19} + q^{21} - 10 q^{22} + 3 q^{23} + q^{24} + q^{26} - 2 q^{27} - q^{28} + 6 q^{31} - q^{32} - 5 q^{33} + 4 q^{34} + 2 q^{36} - 5 q^{37} - 7 q^{38} - q^{39} - 18 q^{41} + 4 q^{42} - 20 q^{43} + 5 q^{44} + 3 q^{46} - 13 q^{47} - 2 q^{48} + 2 q^{49} + 2 q^{51} + q^{52} - q^{53} + q^{54} - 4 q^{56} - 14 q^{57} - 4 q^{59} + 2 q^{61} - 12 q^{62} + 5 q^{63} + 2 q^{64} - 5 q^{66} + 6 q^{67} - 2 q^{68} + 6 q^{69} - 4 q^{71} - q^{72} + 4 q^{73} - 5 q^{74} + 14 q^{76} + 5 q^{77} + 2 q^{78} + 14 q^{79} - q^{81} + 9 q^{82} + 20 q^{83} - 5 q^{84} + 10 q^{86} + 5 q^{88} - 10 q^{89} + 4 q^{91} - 6 q^{92} - 6 q^{93} - 13 q^{94} + q^{96} - 16 q^{97} - 13 q^{98} - 10 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1050\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(451\) \(701\)
\(\chi(n)\) \(1\) \(-\zeta_{6}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
151.1
0.500000 + 0.866025i
0.500000 0.866025i
−0.500000 0.866025i 0.500000 0.866025i −0.500000 + 0.866025i 0 −1.00000 −2.00000 + 1.73205i 1.00000 −0.500000 0.866025i 0
751.1 −0.500000 + 0.866025i 0.500000 + 0.866025i −0.500000 0.866025i 0 −1.00000 −2.00000 1.73205i 1.00000 −0.500000 + 0.866025i 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
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1050.2.i.f 2
5.b even 2 1 1050.2.i.o 2
5.c odd 4 2 210.2.n.a 4
7.c even 3 1 inner 1050.2.i.f 2
7.c even 3 1 7350.2.a.bn 1
7.d odd 6 1 7350.2.a.ch 1
15.e even 4 2 630.2.u.c 4
20.e even 4 2 1680.2.di.a 4
35.f even 4 2 1470.2.n.i 4
35.i odd 6 1 7350.2.a.b 1
35.j even 6 1 1050.2.i.o 2
35.j even 6 1 7350.2.a.t 1
35.k even 12 2 1470.2.g.a 2
35.k even 12 2 1470.2.n.i 4
35.l odd 12 2 210.2.n.a 4
35.l odd 12 2 1470.2.g.f 2
105.x even 12 2 630.2.u.c 4
140.w even 12 2 1680.2.di.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.n.a 4 5.c odd 4 2
210.2.n.a 4 35.l odd 12 2
630.2.u.c 4 15.e even 4 2
630.2.u.c 4 105.x even 12 2
1050.2.i.f 2 1.a even 1 1 trivial
1050.2.i.f 2 7.c even 3 1 inner
1050.2.i.o 2 5.b even 2 1
1050.2.i.o 2 35.j even 6 1
1470.2.g.a 2 35.k even 12 2
1470.2.g.f 2 35.l odd 12 2
1470.2.n.i 4 35.f even 4 2
1470.2.n.i 4 35.k even 12 2
1680.2.di.a 4 20.e even 4 2
1680.2.di.a 4 140.w even 12 2
7350.2.a.b 1 35.i odd 6 1
7350.2.a.t 1 35.j even 6 1
7350.2.a.bn 1 7.c even 3 1
7350.2.a.ch 1 7.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1050, [\chi])\):

\( T_{11}^{2} - 5T_{11} + 25 \) Copy content Toggle raw display
\( T_{13} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 4T + 7 \) Copy content Toggle raw display
$11$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$13$ \( (T + 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$19$ \( T^{2} + 7T + 49 \) Copy content Toggle raw display
$23$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$37$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$41$ \( (T + 9)^{2} \) Copy content Toggle raw display
$43$ \( (T + 10)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 13T + 169 \) Copy content Toggle raw display
$53$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$59$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$61$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$67$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$71$ \( (T + 2)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$79$ \( T^{2} - 14T + 196 \) Copy content Toggle raw display
$83$ \( (T - 10)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 10T + 100 \) Copy content Toggle raw display
$97$ \( (T + 8)^{2} \) Copy content Toggle raw display
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