Properties

Label 1050.2.j.b
Level $1050$
Weight $2$
Character orbit 1050.j
Analytic conductor $8.384$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1050,2,Mod(407,1050)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1050, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 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("1050.407");
 
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.j (of order \(4\), 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: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.40960000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 7x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 7x^{4} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{5} + 5\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{6} + 8\nu^{2} ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{6} + \nu^{4} + 5\nu^{2} + 2 ) / 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{7} + 8\nu^{3} + 3\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{6} + \nu^{4} - 5\nu^{2} + 2 ) / 3 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{7} + \nu^{5} - 8\nu^{3} + 8\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -2\nu^{7} - 13\nu^{3} ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{6} + \beta_{4} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} - \beta_{3} + 2\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} - \beta_{6} + \beta_{4} + \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 3\beta_{5} + 3\beta_{3} - 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -5\beta_{6} - 5\beta_{4} + 11\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -4\beta_{5} + 4\beta_{3} - 5\beta_{2} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -16\beta_{7} + 13\beta_{6} - 13\beta_{4} - 13\beta_1 ) / 2 \) Copy content Toggle raw display

Character values

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

\(n\) \(127\) \(451\) \(701\)
\(\chi(n)\) \(\beta_{2}\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
407.1
1.14412 + 1.14412i
−0.437016 0.437016i
0.437016 + 0.437016i
−1.14412 1.14412i
1.14412 1.14412i
−0.437016 + 0.437016i
0.437016 0.437016i
−1.14412 + 1.14412i
−0.707107 + 0.707107i −1.58114 0.707107i 1.00000i 0 1.61803 0.618034i 0.707107 + 0.707107i 0.707107 + 0.707107i 2.00000 + 2.23607i 0
407.2 −0.707107 + 0.707107i 1.58114 0.707107i 1.00000i 0 −0.618034 + 1.61803i 0.707107 + 0.707107i 0.707107 + 0.707107i 2.00000 2.23607i 0
407.3 0.707107 0.707107i −1.58114 + 0.707107i 1.00000i 0 −0.618034 + 1.61803i −0.707107 0.707107i −0.707107 0.707107i 2.00000 2.23607i 0
407.4 0.707107 0.707107i 1.58114 + 0.707107i 1.00000i 0 1.61803 0.618034i −0.707107 0.707107i −0.707107 0.707107i 2.00000 + 2.23607i 0
743.1 −0.707107 0.707107i −1.58114 + 0.707107i 1.00000i 0 1.61803 + 0.618034i 0.707107 0.707107i 0.707107 0.707107i 2.00000 2.23607i 0
743.2 −0.707107 0.707107i 1.58114 + 0.707107i 1.00000i 0 −0.618034 1.61803i 0.707107 0.707107i 0.707107 0.707107i 2.00000 + 2.23607i 0
743.3 0.707107 + 0.707107i −1.58114 0.707107i 1.00000i 0 −0.618034 1.61803i −0.707107 + 0.707107i −0.707107 + 0.707107i 2.00000 + 2.23607i 0
743.4 0.707107 + 0.707107i 1.58114 0.707107i 1.00000i 0 1.61803 + 0.618034i −0.707107 + 0.707107i −0.707107 + 0.707107i 2.00000 2.23607i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 407.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
15.e even 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1050.2.j.b yes 8
3.b odd 2 1 1050.2.j.a 8
5.b even 2 1 inner 1050.2.j.b yes 8
5.c odd 4 2 1050.2.j.a 8
15.d odd 2 1 1050.2.j.a 8
15.e even 4 2 inner 1050.2.j.b yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1050.2.j.a 8 3.b odd 2 1
1050.2.j.a 8 5.c odd 4 2
1050.2.j.a 8 15.d odd 2 1
1050.2.j.b yes 8 1.a even 1 1 trivial
1050.2.j.b yes 8 5.b even 2 1 inner
1050.2.j.b yes 8 15.e even 4 2 inner

Hecke kernels

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

\( T_{11}^{4} + 12T_{11}^{2} + 16 \) Copy content Toggle raw display
\( T_{17}^{4} + 16 \) Copy content Toggle raw display
\( T_{29}^{2} + 6T_{29} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T^{4} - 4 T^{2} + 9)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 12 T^{2} + 16)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + 1792 T^{4} + 65536 \) Copy content Toggle raw display
$17$ \( (T^{4} + 16)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 28 T^{2} + 16)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + 1792 T^{4} + 65536 \) Copy content Toggle raw display
$29$ \( (T^{2} + 6 T + 4)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} - 6 T + 4)^{4} \) Copy content Toggle raw display
$37$ \( T^{8} + 2800 T^{4} + 160000 \) Copy content Toggle raw display
$41$ \( (T^{4} + 108 T^{2} + 1936)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + 112T^{4} + 256 \) Copy content Toggle raw display
$47$ \( T^{8} + 32752 T^{4} + \cdots + 236421376 \) Copy content Toggle raw display
$53$ \( T^{8} + 16672 T^{4} + \cdots + 33362176 \) Copy content Toggle raw display
$59$ \( (T^{2} + 14 T + 4)^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} + 8 T - 64)^{4} \) Copy content Toggle raw display
$67$ \( T^{8} + 36592 T^{4} + \cdots + 181063936 \) Copy content Toggle raw display
$71$ \( (T^{4} + 368 T^{2} + 30976)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + 9072 T^{4} + \cdots + 1679616 \) Copy content Toggle raw display
$79$ \( (T^{4} + 192 T^{2} + 4096)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 2800 T^{4} + 160000 \) Copy content Toggle raw display
$89$ \( (T^{2} + 14 T + 44)^{4} \) Copy content Toggle raw display
$97$ \( T^{8} + 44912 T^{4} + \cdots + 181063936 \) Copy content Toggle raw display
show more
show less