Properties

Label 1050.2.g.a
Level $1050$
Weight $2$
Character orbit 1050.g
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(799,1050)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1050, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1050.799");
 
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.g (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: \(8.38429221223\)
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 42)
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.

\(f(q)\) \(=\) \( q + i q^{2} + i q^{3} - q^{4} - q^{6} - i q^{7} - i q^{8} - q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + i q^{2} + i q^{3} - q^{4} - q^{6} - i q^{7} - i q^{8} - q^{9} - 4 q^{11} - i q^{12} - 6 i q^{13} + q^{14} + q^{16} + 2 i q^{17} - i q^{18} + 4 q^{19} + q^{21} - 4 i q^{22} - 8 i q^{23} + q^{24} + 6 q^{26} - i q^{27} + i q^{28} + 2 q^{29} + i q^{32} - 4 i q^{33} - 2 q^{34} + q^{36} - 10 i q^{37} + 4 i q^{38} + 6 q^{39} - 6 q^{41} + i q^{42} + 4 i q^{43} + 4 q^{44} + 8 q^{46} + i q^{48} - q^{49} - 2 q^{51} + 6 i q^{52} - 6 i q^{53} + q^{54} - q^{56} + 4 i q^{57} + 2 i q^{58} - 4 q^{59} + 6 q^{61} + i q^{63} - q^{64} + 4 q^{66} + 4 i q^{67} - 2 i q^{68} + 8 q^{69} + 8 q^{71} + i q^{72} - 10 i q^{73} + 10 q^{74} - 4 q^{76} + 4 i q^{77} + 6 i q^{78} + q^{81} - 6 i q^{82} + 4 i q^{83} - q^{84} - 4 q^{86} + 2 i q^{87} + 4 i q^{88} + 6 q^{89} - 6 q^{91} + 8 i q^{92} - q^{96} - 14 i q^{97} - i q^{98} + 4 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9} - 8 q^{11} + 2 q^{14} + 2 q^{16} + 8 q^{19} + 2 q^{21} + 2 q^{24} + 12 q^{26} + 4 q^{29} - 4 q^{34} + 2 q^{36} + 12 q^{39} - 12 q^{41} + 8 q^{44} + 16 q^{46} - 2 q^{49} - 4 q^{51} + 2 q^{54} - 2 q^{56} - 8 q^{59} + 12 q^{61} - 2 q^{64} + 8 q^{66} + 16 q^{69} + 16 q^{71} + 20 q^{74} - 8 q^{76} + 2 q^{81} - 2 q^{84} - 8 q^{86} + 12 q^{89} - 12 q^{91} - 2 q^{96} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(127\) \(451\) \(701\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
799.1
1.00000i
1.00000i
1.00000i 1.00000i −1.00000 0 −1.00000 1.00000i 1.00000i −1.00000 0
799.2 1.00000i 1.00000i −1.00000 0 −1.00000 1.00000i 1.00000i −1.00000 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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1050.2.g.a 2
3.b odd 2 1 3150.2.g.r 2
5.b even 2 1 inner 1050.2.g.a 2
5.c odd 4 1 42.2.a.a 1
5.c odd 4 1 1050.2.a.i 1
15.d odd 2 1 3150.2.g.r 2
15.e even 4 1 126.2.a.a 1
15.e even 4 1 3150.2.a.bo 1
20.e even 4 1 336.2.a.d 1
20.e even 4 1 8400.2.a.k 1
35.f even 4 1 294.2.a.g 1
35.f even 4 1 7350.2.a.f 1
35.k even 12 2 294.2.e.a 2
35.l odd 12 2 294.2.e.c 2
40.i odd 4 1 1344.2.a.q 1
40.k even 4 1 1344.2.a.i 1
45.k odd 12 2 1134.2.f.g 2
45.l even 12 2 1134.2.f.j 2
55.e even 4 1 5082.2.a.d 1
60.l odd 4 1 1008.2.a.j 1
65.h odd 4 1 7098.2.a.f 1
80.i odd 4 1 5376.2.c.bc 2
80.j even 4 1 5376.2.c.e 2
80.s even 4 1 5376.2.c.e 2
80.t odd 4 1 5376.2.c.bc 2
105.k odd 4 1 882.2.a.b 1
105.w odd 12 2 882.2.g.j 2
105.x even 12 2 882.2.g.h 2
120.q odd 4 1 4032.2.a.m 1
120.w even 4 1 4032.2.a.e 1
140.j odd 4 1 2352.2.a.l 1
140.w even 12 2 2352.2.q.i 2
140.x odd 12 2 2352.2.q.n 2
280.s even 4 1 9408.2.a.n 1
280.y odd 4 1 9408.2.a.bw 1
420.w even 4 1 7056.2.a.k 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.2.a.a 1 5.c odd 4 1
126.2.a.a 1 15.e even 4 1
294.2.a.g 1 35.f even 4 1
294.2.e.a 2 35.k even 12 2
294.2.e.c 2 35.l odd 12 2
336.2.a.d 1 20.e even 4 1
882.2.a.b 1 105.k odd 4 1
882.2.g.h 2 105.x even 12 2
882.2.g.j 2 105.w odd 12 2
1008.2.a.j 1 60.l odd 4 1
1050.2.a.i 1 5.c odd 4 1
1050.2.g.a 2 1.a even 1 1 trivial
1050.2.g.a 2 5.b even 2 1 inner
1134.2.f.g 2 45.k odd 12 2
1134.2.f.j 2 45.l even 12 2
1344.2.a.i 1 40.k even 4 1
1344.2.a.q 1 40.i odd 4 1
2352.2.a.l 1 140.j odd 4 1
2352.2.q.i 2 140.w even 12 2
2352.2.q.n 2 140.x odd 12 2
3150.2.a.bo 1 15.e even 4 1
3150.2.g.r 2 3.b odd 2 1
3150.2.g.r 2 15.d odd 2 1
4032.2.a.e 1 120.w even 4 1
4032.2.a.m 1 120.q odd 4 1
5082.2.a.d 1 55.e even 4 1
5376.2.c.e 2 80.j even 4 1
5376.2.c.e 2 80.s even 4 1
5376.2.c.bc 2 80.i odd 4 1
5376.2.c.bc 2 80.t odd 4 1
7056.2.a.k 1 420.w even 4 1
7098.2.a.f 1 65.h odd 4 1
7350.2.a.f 1 35.f even 4 1
8400.2.a.k 1 20.e even 4 1
9408.2.a.n 1 280.s even 4 1
9408.2.a.bw 1 280.y odd 4 1

Hecke kernels

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

\( T_{11} + 4 \) Copy content Toggle raw display
\( T_{13}^{2} + 36 \) Copy content Toggle raw display
\( T_{17}^{2} + 4 \) Copy content Toggle raw display
\( T_{19} - 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{2} + 1 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 1 \) Copy content Toggle raw display
$11$ \( (T + 4)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 36 \) Copy content Toggle raw display
$17$ \( T^{2} + 4 \) Copy content Toggle raw display
$19$ \( (T - 4)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 64 \) Copy content Toggle raw display
$29$ \( (T - 2)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 100 \) Copy content Toggle raw display
$41$ \( (T + 6)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 16 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 36 \) Copy content Toggle raw display
$59$ \( (T + 4)^{2} \) Copy content Toggle raw display
$61$ \( (T - 6)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 16 \) Copy content Toggle raw display
$71$ \( (T - 8)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 100 \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 16 \) Copy content Toggle raw display
$89$ \( (T - 6)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 196 \) Copy content Toggle raw display
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