Properties

Label 1050.2.i.e
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 42)
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.

\(f(q)\) \(=\) \( q - \zeta_{6} q^{2} + ( - \zeta_{6} + 1) q^{3} + (\zeta_{6} - 1) q^{4} - q^{6} + (\zeta_{6} - 3) q^{7} + q^{8} - \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{6} q^{2} + ( - \zeta_{6} + 1) q^{3} + (\zeta_{6} - 1) q^{4} - q^{6} + (\zeta_{6} - 3) q^{7} + q^{8} - \zeta_{6} q^{9} + (3 \zeta_{6} - 3) q^{11} + \zeta_{6} q^{12} + 4 q^{13} + (2 \zeta_{6} + 1) q^{14} - \zeta_{6} q^{16} + (\zeta_{6} - 1) q^{18} + 4 \zeta_{6} q^{19} + (3 \zeta_{6} - 2) q^{21} + 3 q^{22} + ( - \zeta_{6} + 1) q^{24} - 4 \zeta_{6} q^{26} - q^{27} + ( - 3 \zeta_{6} + 2) q^{28} + 9 q^{29} + ( - \zeta_{6} + 1) q^{31} + (\zeta_{6} - 1) q^{32} + 3 \zeta_{6} q^{33} + q^{36} + 8 \zeta_{6} q^{37} + ( - 4 \zeta_{6} + 4) q^{38} + ( - 4 \zeta_{6} + 4) q^{39} + ( - \zeta_{6} + 3) q^{42} + 10 q^{43} - 3 \zeta_{6} q^{44} - 6 \zeta_{6} q^{47} - q^{48} + ( - 5 \zeta_{6} + 8) q^{49} + (4 \zeta_{6} - 4) q^{52} + (3 \zeta_{6} - 3) q^{53} + \zeta_{6} q^{54} + (\zeta_{6} - 3) q^{56} + 4 q^{57} - 9 \zeta_{6} q^{58} + (3 \zeta_{6} - 3) q^{59} + 10 \zeta_{6} q^{61} - q^{62} + (2 \zeta_{6} + 1) q^{63} + q^{64} + ( - 3 \zeta_{6} + 3) q^{66} + (10 \zeta_{6} - 10) q^{67} - 6 q^{71} - \zeta_{6} q^{72} + ( - 2 \zeta_{6} + 2) q^{73} + ( - 8 \zeta_{6} + 8) q^{74} - 4 q^{76} + ( - 9 \zeta_{6} + 6) q^{77} - 4 q^{78} + \zeta_{6} q^{79} + (\zeta_{6} - 1) q^{81} + 9 q^{83} + ( - 2 \zeta_{6} - 1) q^{84} - 10 \zeta_{6} q^{86} + ( - 9 \zeta_{6} + 9) q^{87} + (3 \zeta_{6} - 3) q^{88} - 6 \zeta_{6} q^{89} + (4 \zeta_{6} - 12) q^{91} - \zeta_{6} q^{93} + (6 \zeta_{6} - 6) q^{94} + \zeta_{6} q^{96} + q^{97} + ( - 3 \zeta_{6} - 5) q^{98} + 3 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} + q^{3} - q^{4} - 2 q^{6} - 5 q^{7} + 2 q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} + q^{3} - q^{4} - 2 q^{6} - 5 q^{7} + 2 q^{8} - q^{9} - 3 q^{11} + q^{12} + 8 q^{13} + 4 q^{14} - q^{16} - q^{18} + 4 q^{19} - q^{21} + 6 q^{22} + q^{24} - 4 q^{26} - 2 q^{27} + q^{28} + 18 q^{29} + q^{31} - q^{32} + 3 q^{33} + 2 q^{36} + 8 q^{37} + 4 q^{38} + 4 q^{39} + 5 q^{42} + 20 q^{43} - 3 q^{44} - 6 q^{47} - 2 q^{48} + 11 q^{49} - 4 q^{52} - 3 q^{53} + q^{54} - 5 q^{56} + 8 q^{57} - 9 q^{58} - 3 q^{59} + 10 q^{61} - 2 q^{62} + 4 q^{63} + 2 q^{64} + 3 q^{66} - 10 q^{67} - 12 q^{71} - q^{72} + 2 q^{73} + 8 q^{74} - 8 q^{76} + 3 q^{77} - 8 q^{78} + q^{79} - q^{81} + 18 q^{83} - 4 q^{84} - 10 q^{86} + 9 q^{87} - 3 q^{88} - 6 q^{89} - 20 q^{91} - q^{93} - 6 q^{94} + q^{96} + 2 q^{97} - 13 q^{98} + 6 q^{99}+O(q^{100}) \) 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 −2.50000 + 0.866025i 1.00000 −0.500000 0.866025i 0
751.1 −0.500000 + 0.866025i 0.500000 + 0.866025i −0.500000 0.866025i 0 −1.00000 −2.50000 0.866025i 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.e 2
5.b even 2 1 42.2.e.b 2
5.c odd 4 2 1050.2.o.b 4
7.c even 3 1 inner 1050.2.i.e 2
7.c even 3 1 7350.2.a.ce 1
7.d odd 6 1 7350.2.a.cw 1
15.d odd 2 1 126.2.g.b 2
20.d odd 2 1 336.2.q.d 2
35.c odd 2 1 294.2.e.f 2
35.i odd 6 1 294.2.a.a 1
35.i odd 6 1 294.2.e.f 2
35.j even 6 1 42.2.e.b 2
35.j even 6 1 294.2.a.d 1
35.l odd 12 2 1050.2.o.b 4
40.e odd 2 1 1344.2.q.j 2
40.f even 2 1 1344.2.q.v 2
45.h odd 6 1 1134.2.e.p 2
45.h odd 6 1 1134.2.h.a 2
45.j even 6 1 1134.2.e.a 2
45.j even 6 1 1134.2.h.p 2
60.h even 2 1 1008.2.s.n 2
105.g even 2 1 882.2.g.b 2
105.o odd 6 1 126.2.g.b 2
105.o odd 6 1 882.2.a.g 1
105.p even 6 1 882.2.a.k 1
105.p even 6 1 882.2.g.b 2
140.c even 2 1 2352.2.q.m 2
140.p odd 6 1 336.2.q.d 2
140.p odd 6 1 2352.2.a.m 1
140.s even 6 1 2352.2.a.n 1
140.s even 6 1 2352.2.q.m 2
280.ba even 6 1 9408.2.a.bm 1
280.bf even 6 1 1344.2.q.v 2
280.bf even 6 1 9408.2.a.d 1
280.bi odd 6 1 1344.2.q.j 2
280.bi odd 6 1 9408.2.a.bu 1
280.bk odd 6 1 9408.2.a.db 1
315.r even 6 1 1134.2.h.p 2
315.v odd 6 1 1134.2.e.p 2
315.bo even 6 1 1134.2.e.a 2
315.br odd 6 1 1134.2.h.a 2
420.ba even 6 1 1008.2.s.n 2
420.ba even 6 1 7056.2.a.g 1
420.be odd 6 1 7056.2.a.bz 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.2.e.b 2 5.b even 2 1
42.2.e.b 2 35.j even 6 1
126.2.g.b 2 15.d odd 2 1
126.2.g.b 2 105.o odd 6 1
294.2.a.a 1 35.i odd 6 1
294.2.a.d 1 35.j even 6 1
294.2.e.f 2 35.c odd 2 1
294.2.e.f 2 35.i odd 6 1
336.2.q.d 2 20.d odd 2 1
336.2.q.d 2 140.p odd 6 1
882.2.a.g 1 105.o odd 6 1
882.2.a.k 1 105.p even 6 1
882.2.g.b 2 105.g even 2 1
882.2.g.b 2 105.p even 6 1
1008.2.s.n 2 60.h even 2 1
1008.2.s.n 2 420.ba even 6 1
1050.2.i.e 2 1.a even 1 1 trivial
1050.2.i.e 2 7.c even 3 1 inner
1050.2.o.b 4 5.c odd 4 2
1050.2.o.b 4 35.l odd 12 2
1134.2.e.a 2 45.j even 6 1
1134.2.e.a 2 315.bo even 6 1
1134.2.e.p 2 45.h odd 6 1
1134.2.e.p 2 315.v odd 6 1
1134.2.h.a 2 45.h odd 6 1
1134.2.h.a 2 315.br odd 6 1
1134.2.h.p 2 45.j even 6 1
1134.2.h.p 2 315.r even 6 1
1344.2.q.j 2 40.e odd 2 1
1344.2.q.j 2 280.bi odd 6 1
1344.2.q.v 2 40.f even 2 1
1344.2.q.v 2 280.bf even 6 1
2352.2.a.m 1 140.p odd 6 1
2352.2.a.n 1 140.s even 6 1
2352.2.q.m 2 140.c even 2 1
2352.2.q.m 2 140.s even 6 1
7056.2.a.g 1 420.ba even 6 1
7056.2.a.bz 1 420.be odd 6 1
7350.2.a.ce 1 7.c even 3 1
7350.2.a.cw 1 7.d odd 6 1
9408.2.a.d 1 280.bf even 6 1
9408.2.a.bm 1 280.ba even 6 1
9408.2.a.bu 1 280.bi odd 6 1
9408.2.a.db 1 280.bk odd 6 1

Hecke kernels

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

\( T_{11}^{2} + 3T_{11} + 9 \) Copy content Toggle raw display
\( T_{13} - 4 \) 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
$7$ \( T^{2} + 5T + 7 \) Copy content Toggle raw display
$11$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$13$ \( (T - 4)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( (T - 9)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$37$ \( T^{2} - 8T + 64 \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( (T - 10)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$53$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$59$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$61$ \( T^{2} - 10T + 100 \) Copy content Toggle raw display
$67$ \( T^{2} + 10T + 100 \) Copy content Toggle raw display
$71$ \( (T + 6)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$79$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$83$ \( (T - 9)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$97$ \( (T - 1)^{2} \) Copy content Toggle raw display
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