Properties

Label 1050.2.d.f
Level $1050$
Weight $2$
Character orbit 1050.d
Analytic conductor $8.384$
Analytic rank $0$
Dimension $4$
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(1049,1050)]
 
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([1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1050.1049");
 
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.d (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: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 210)
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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + (\beta_{3} + \beta_{2}) q^{3} + q^{4} + (\beta_{3} + \beta_{2}) q^{6} + ( - \beta_{3} - 2 \beta_{2} - 1) q^{7} + q^{8} + (\beta_{3} - 2 \beta_{2} - \beta_1 - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + (\beta_{3} + \beta_{2}) q^{3} + q^{4} + (\beta_{3} + \beta_{2}) q^{6} + ( - \beta_{3} - 2 \beta_{2} - 1) q^{7} + q^{8} + (\beta_{3} - 2 \beta_{2} - \beta_1 - 1) q^{9} + (2 \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{11} + (\beta_{3} + \beta_{2}) q^{12} + (\beta_{3} - \beta_1) q^{13} + ( - \beta_{3} - 2 \beta_{2} - 1) q^{14} + q^{16} + (\beta_{3} + 4 \beta_{2} + \beta_1) q^{17} + (\beta_{3} - 2 \beta_{2} - \beta_1 - 1) q^{18} + (2 \beta_{3} + 6 \beta_{2} + 2 \beta_1) q^{19} + ( - 2 \beta_{3} + \beta_{2} + 2 \beta_1 + 2) q^{21} + (2 \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{22} + 4 q^{23} + (\beta_{3} + \beta_{2}) q^{24} + (\beta_{3} - \beta_1) q^{26} + ( - 3 \beta_{2} + \beta_1 + 4) q^{27} + ( - \beta_{3} - 2 \beta_{2} - 1) q^{28} + (3 \beta_{3} + 4 \beta_{2} + 3 \beta_1) q^{29} + ( - \beta_{3} + 4 \beta_{2} - \beta_1) q^{31} + q^{32} + (2 \beta_{3} - 4 \beta_{2} - 6) q^{33} + (\beta_{3} + 4 \beta_{2} + \beta_1) q^{34} + (\beta_{3} - 2 \beta_{2} - \beta_1 - 1) q^{36} + (\beta_{3} - 2 \beta_{2} + \beta_1) q^{37} + (2 \beta_{3} + 6 \beta_{2} + 2 \beta_1) q^{38} + (\beta_{3} - 2 \beta_{2} - \beta_1 + 2) q^{39} + ( - 2 \beta_{3} + 2 \beta_1) q^{41} + ( - 2 \beta_{3} + \beta_{2} + 2 \beta_1 + 2) q^{42} + ( - 4 \beta_{3} - 4 \beta_1) q^{43} + (2 \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{44} + 4 q^{46} + ( - 2 \beta_{3} - 4 \beta_{2} - 2 \beta_1) q^{47} + (\beta_{3} + \beta_{2}) q^{48} + (3 \beta_{3} + 2 \beta_{2} - 3 \beta_1 - 3) q^{49} + (\beta_{3} - 2 \beta_{2} - 3 \beta_1 - 6) q^{51} + (\beta_{3} - \beta_1) q^{52} + (2 \beta_{3} - 2 \beta_1 + 2) q^{53} + ( - 3 \beta_{2} + \beta_1 + 4) q^{54} + ( - \beta_{3} - 2 \beta_{2} - 1) q^{56} + (2 \beta_{3} - 4 \beta_{2} - 4 \beta_1 - 10) q^{57} + (3 \beta_{3} + 4 \beta_{2} + 3 \beta_1) q^{58} + ( - 2 \beta_{3} + 2 \beta_1 + 2) q^{59} + ( - \beta_{3} - 6 \beta_{2} - \beta_1) q^{61} + ( - \beta_{3} + 4 \beta_{2} - \beta_1) q^{62} + (6 \beta_{2} + \beta_1 - 5) q^{63} + q^{64} + (2 \beta_{3} - 4 \beta_{2} - 6) q^{66} + 12 \beta_{2} q^{67} + (\beta_{3} + 4 \beta_{2} + \beta_1) q^{68} + (4 \beta_{3} + 4 \beta_{2}) q^{69} + ( - \beta_{3} - 6 \beta_{2} - \beta_1) q^{71} + (\beta_{3} - 2 \beta_{2} - \beta_1 - 1) q^{72} + (\beta_{3} - \beta_1 - 10) q^{73} + (\beta_{3} - 2 \beta_{2} + \beta_1) q^{74} + (2 \beta_{3} + 6 \beta_{2} + 2 \beta_1) q^{76} + ( - 6 \beta_{3} + 2 \beta_{2} + 8) q^{77} + (\beta_{3} - 2 \beta_{2} - \beta_1 + 2) q^{78} + ( - 4 \beta_{3} + 4 \beta_1 + 4) q^{79} + (4 \beta_{3} + 4 \beta_{2} + 4 \beta_1 + 1) q^{81} + ( - 2 \beta_{3} + 2 \beta_1) q^{82} + ( - 7 \beta_{3} - 6 \beta_{2} - 7 \beta_1) q^{83} + ( - 2 \beta_{3} + \beta_{2} + 2 \beta_1 + 2) q^{84} + ( - 4 \beta_{3} - 4 \beta_1) q^{86} + (3 \beta_{3} - 6 \beta_{2} - \beta_1 - 10) q^{87} + (2 \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{88} + ( - 2 \beta_{3} + 2 \beta_1 - 8) q^{89} + ( - \beta_{3} + 2 \beta_{2} + 3 \beta_1 - 2) q^{91} + 4 q^{92} + ( - \beta_{3} + 2 \beta_{2} - 5 \beta_1 - 2) q^{93} + ( - 2 \beta_{3} - 4 \beta_{2} - 2 \beta_1) q^{94} + (\beta_{3} + \beta_{2}) q^{96} + ( - \beta_{3} + \beta_1 - 2) q^{97} + (3 \beta_{3} + 2 \beta_{2} - 3 \beta_1 - 3) q^{98} + ( - 4 \beta_{3} - 10 \beta_{2} + 4 \beta_1 + 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} + 2 q^{3} + 4 q^{4} + 2 q^{6} - 6 q^{7} + 4 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{2} + 2 q^{3} + 4 q^{4} + 2 q^{6} - 6 q^{7} + 4 q^{8} + 2 q^{12} + 4 q^{13} - 6 q^{14} + 4 q^{16} + 16 q^{23} + 2 q^{24} + 4 q^{26} + 14 q^{27} - 6 q^{28} + 4 q^{32} - 20 q^{33} + 12 q^{39} - 8 q^{41} + 16 q^{46} + 2 q^{48} - 16 q^{51} + 4 q^{52} + 16 q^{53} + 14 q^{54} - 6 q^{56} - 28 q^{57} - 22 q^{63} + 4 q^{64} - 20 q^{66} + 8 q^{69} - 36 q^{73} + 20 q^{77} + 12 q^{78} + 4 q^{81} - 8 q^{82} - 32 q^{87} - 40 q^{89} - 16 q^{91} + 16 q^{92} + 2 q^{96} - 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 3x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} + \nu + 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} + 2\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\nu^{2} + \nu - 1 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{3} + \beta _1 - 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{3} + \beta_{2} - \beta_1 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1049.1
0.618034i
0.618034i
1.61803i
1.61803i
1.00000 −0.618034 1.61803i 1.00000 0 −0.618034 1.61803i −0.381966 + 2.61803i 1.00000 −2.23607 + 2.00000i 0
1049.2 1.00000 −0.618034 + 1.61803i 1.00000 0 −0.618034 + 1.61803i −0.381966 2.61803i 1.00000 −2.23607 2.00000i 0
1049.3 1.00000 1.61803 0.618034i 1.00000 0 1.61803 0.618034i −2.61803 0.381966i 1.00000 2.23607 2.00000i 0
1049.4 1.00000 1.61803 + 0.618034i 1.00000 0 1.61803 + 0.618034i −2.61803 + 0.381966i 1.00000 2.23607 + 2.00000i 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
105.g 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.d.f 4
3.b odd 2 1 1050.2.d.c 4
5.b even 2 1 1050.2.d.a 4
5.c odd 4 1 210.2.b.b yes 4
5.c odd 4 1 1050.2.b.a 4
7.b odd 2 1 1050.2.d.d 4
15.d odd 2 1 1050.2.d.d 4
15.e even 4 1 210.2.b.a 4
15.e even 4 1 1050.2.b.c 4
20.e even 4 1 1680.2.f.e 4
21.c even 2 1 1050.2.d.a 4
35.c odd 2 1 1050.2.d.c 4
35.f even 4 1 210.2.b.a 4
35.f even 4 1 1050.2.b.c 4
60.l odd 4 1 1680.2.f.i 4
105.g even 2 1 inner 1050.2.d.f 4
105.k odd 4 1 210.2.b.b yes 4
105.k odd 4 1 1050.2.b.a 4
140.j odd 4 1 1680.2.f.i 4
420.w even 4 1 1680.2.f.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.b.a 4 15.e even 4 1
210.2.b.a 4 35.f even 4 1
210.2.b.b yes 4 5.c odd 4 1
210.2.b.b yes 4 105.k odd 4 1
1050.2.b.a 4 5.c odd 4 1
1050.2.b.a 4 105.k odd 4 1
1050.2.b.c 4 15.e even 4 1
1050.2.b.c 4 35.f even 4 1
1050.2.d.a 4 5.b even 2 1
1050.2.d.a 4 21.c even 2 1
1050.2.d.c 4 3.b odd 2 1
1050.2.d.c 4 35.c odd 2 1
1050.2.d.d 4 7.b odd 2 1
1050.2.d.d 4 15.d odd 2 1
1050.2.d.f 4 1.a even 1 1 trivial
1050.2.d.f 4 105.g even 2 1 inner
1680.2.f.e 4 20.e even 4 1
1680.2.f.e 4 420.w even 4 1
1680.2.f.i 4 60.l odd 4 1
1680.2.f.i 4 140.j odd 4 1

Hecke kernels

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

\( T_{11}^{2} + 20 \) Copy content Toggle raw display
\( T_{13}^{2} - 2T_{13} - 4 \) Copy content Toggle raw display
\( T_{23} - 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - 2 T^{3} + 2 T^{2} - 6 T + 9 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 6 T^{3} + 18 T^{2} + 42 T + 49 \) Copy content Toggle raw display
$11$ \( (T^{2} + 20)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} - 2 T - 4)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 28T^{2} + 16 \) Copy content Toggle raw display
$19$ \( T^{4} + 72T^{2} + 16 \) Copy content Toggle raw display
$23$ \( (T - 4)^{4} \) Copy content Toggle raw display
$29$ \( T^{4} + 92T^{2} + 1936 \) Copy content Toggle raw display
$31$ \( T^{4} + 60T^{2} + 400 \) Copy content Toggle raw display
$37$ \( T^{4} + 28T^{2} + 16 \) Copy content Toggle raw display
$41$ \( (T^{2} + 4 T - 16)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 192T^{2} + 4096 \) Copy content Toggle raw display
$47$ \( T^{4} + 48T^{2} + 256 \) Copy content Toggle raw display
$53$ \( (T^{2} - 8 T - 4)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 20)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} + 60T^{2} + 400 \) Copy content Toggle raw display
$67$ \( (T^{2} + 144)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + 60T^{2} + 400 \) Copy content Toggle raw display
$73$ \( (T^{2} + 18 T + 76)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 80)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 492 T^{2} + 59536 \) Copy content Toggle raw display
$89$ \( (T^{2} + 20 T + 80)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 6 T + 4)^{2} \) Copy content Toggle raw display
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