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Properties

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

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1050,2,Mod(101,1050)]
 
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([3, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1050.101");
 
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.s (of order \(6\), 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 \(\Q(\zeta_{6})\)
Coefficient field: 8.0.303595776.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 5x^{6} + 16x^{4} + 45x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 210)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{5} q^{2} + (\beta_{7} - \beta_1) q^{3} + (\beta_{4} + 1) q^{4} + ( - \beta_{4} + \beta_{2}) q^{6} + (2 \beta_{5} + 3 \beta_{3}) q^{7} + (\beta_{5} + \beta_{3}) q^{8} + ( - \beta_{6} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{5} q^{2} + (\beta_{7} - \beta_1) q^{3} + (\beta_{4} + 1) q^{4} + ( - \beta_{4} + \beta_{2}) q^{6} + (2 \beta_{5} + 3 \beta_{3}) q^{7} + (\beta_{5} + \beta_{3}) q^{8} + ( - \beta_{6} + 2) q^{9} + ( - \beta_{6} + \beta_{4} + \beta_{2} - 2) q^{11} + \beta_{7} q^{12} + ( - 2 \beta_{5} - 2 \beta_{3}) q^{13} + (2 \beta_{4} - 1) q^{14} + \beta_{4} q^{16} + (4 \beta_{7} + 2 \beta_{3}) q^{17} + (3 \beta_{5} - \beta_1) q^{18} + (2 \beta_{4} + 4) q^{19} + (3 \beta_{6} + \beta_{4} - \beta_{2} + 3) q^{21} + (\beta_{7} - \beta_{5} + 2 \beta_{3} - \beta_1) q^{22} + (2 \beta_{7} - \beta_{5} + \cdots - \beta_1) q^{23}+ \cdots + ( - \beta_{6} - \beta_{4} + 4 \beta_{2} - 7) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{4} + 2 q^{6} + 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{4} + 2 q^{6} + 20 q^{9} - 18 q^{11} - 16 q^{14} - 4 q^{16} + 24 q^{19} + 10 q^{21} + 4 q^{24} + 8 q^{26} + 18 q^{31} + 10 q^{36} - 8 q^{39} - 36 q^{41} - 18 q^{44} - 6 q^{46} - 44 q^{49} + 44 q^{51} + 16 q^{54} - 20 q^{56} - 18 q^{59} - 54 q^{61} - 8 q^{64} + 22 q^{66} + 30 q^{69} - 12 q^{74} + 2 q^{79} + 28 q^{81} + 2 q^{84} - 42 q^{86} + 6 q^{89} + 40 q^{91} - 36 q^{94} + 2 q^{96} - 56 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 5x^{6} + 16x^{4} + 45x^{2} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{6} + 32\nu^{4} + 16\nu^{2} + 45 ) / 144 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} + 32\nu^{5} + 16\nu^{3} + 45\nu ) / 432 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -5\nu^{6} - 16\nu^{4} - 80\nu^{2} - 225 ) / 144 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{7} + 13\nu ) / 48 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{6} - 13 ) / 16 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 5\nu^{7} + 16\nu^{5} + 80\nu^{3} + 225\nu ) / 144 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} - 2\beta_{4} - \beta_{2} - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{7} - 3\beta_{5} - 3\beta_{3} - 2\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} + 5\beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{7} + 15\beta_{3} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -16\beta_{6} - 13 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 48\beta_{5} - 13\beta_1 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

\(n\) \(127\) \(451\) \(701\)
\(\chi(n)\) \(1\) \(-\beta_{4}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
101.1
0.396143 + 1.68614i
−1.26217 − 1.18614i
1.26217 + 1.18614i
−0.396143 − 1.68614i
0.396143 − 1.68614i
−1.26217 + 1.18614i
1.26217 − 1.18614i
−0.396143 + 1.68614i
−0.866025 + 0.500000i −1.65831 − 0.500000i 0.500000 − 0.866025i 0 1.68614 − 0.396143i 0.866025 + 2.50000i 1.00000i 2.50000 + 1.65831i 0
101.2 −0.866025 + 0.500000i 1.65831 − 0.500000i 0.500000 − 0.866025i 0 −1.18614 + 1.26217i 0.866025 + 2.50000i 1.00000i 2.50000 − 1.65831i 0
101.3 0.866025 − 0.500000i −1.65831 + 0.500000i 0.500000 − 0.866025i 0 −1.18614 + 1.26217i −0.866025 − 2.50000i − 1.00000i 2.50000 − 1.65831i 0
101.4 0.866025 − 0.500000i 1.65831 + 0.500000i 0.500000 − 0.866025i 0 1.68614 − 0.396143i −0.866025 − 2.50000i − 1.00000i 2.50000 + 1.65831i 0
551.1 −0.866025 − 0.500000i −1.65831 + 0.500000i 0.500000 + 0.866025i 0 1.68614 + 0.396143i 0.866025 − 2.50000i − 1.00000i 2.50000 − 1.65831i 0
551.2 −0.866025 − 0.500000i 1.65831 + 0.500000i 0.500000 + 0.866025i 0 −1.18614 − 1.26217i 0.866025 − 2.50000i − 1.00000i 2.50000 + 1.65831i 0
551.3 0.866025 + 0.500000i −1.65831 − 0.500000i 0.500000 + 0.866025i 0 −1.18614 − 1.26217i −0.866025 + 2.50000i 1.00000i 2.50000 + 1.65831i 0
551.4 0.866025 + 0.500000i 1.65831 − 0.500000i 0.500000 + 0.866025i 0 1.68614 + 0.396143i −0.866025 + 2.50000i 1.00000i 2.50000 − 1.65831i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 101.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
21.g even 6 1 inner
105.p even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1050.2.s.e 8
3.b odd 2 1 1050.2.s.d 8
5.b even 2 1 inner 1050.2.s.e 8
5.c odd 4 1 210.2.t.a ✓ 4
5.c odd 4 1 210.2.t.d yes 4
7.d odd 6 1 1050.2.s.d 8
15.d odd 2 1 1050.2.s.d 8
15.e even 4 1 210.2.t.b yes 4
15.e even 4 1 210.2.t.c yes 4
21.g even 6 1 inner 1050.2.s.e 8
35.i odd 6 1 1050.2.s.d 8
35.k even 12 1 210.2.t.b yes 4
35.k even 12 1 210.2.t.c yes 4
35.k even 12 1 1470.2.d.b 4
35.k even 12 1 1470.2.d.c 4
35.l odd 12 1 1470.2.d.a 4
35.l odd 12 1 1470.2.d.d 4
105.p even 6 1 inner 1050.2.s.e 8
105.w odd 12 1 210.2.t.a ✓ 4
105.w odd 12 1 210.2.t.d yes 4
105.w odd 12 1 1470.2.d.a 4
105.w odd 12 1 1470.2.d.d 4
105.x even 12 1 1470.2.d.b 4
105.x even 12 1 1470.2.d.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.t.a ✓ 4 5.c odd 4 1
210.2.t.a ✓ 4 105.w odd 12 1
210.2.t.b yes 4 15.e even 4 1
210.2.t.b yes 4 35.k even 12 1
210.2.t.c yes 4 15.e even 4 1
210.2.t.c yes 4 35.k even 12 1
210.2.t.d yes 4 5.c odd 4 1
210.2.t.d yes 4 105.w odd 12 1
1050.2.s.d 8 3.b odd 2 1
1050.2.s.d 8 7.d odd 6 1
1050.2.s.d 8 15.d odd 2 1
1050.2.s.d 8 35.i odd 6 1
1050.2.s.e 8 1.a even 1 1 trivial
1050.2.s.e 8 5.b even 2 1 inner
1050.2.s.e 8 21.g even 6 1 inner
1050.2.s.e 8 105.p even 6 1 inner
1470.2.d.a 4 35.l odd 12 1
1470.2.d.a 4 105.w odd 12 1
1470.2.d.b 4 35.k even 12 1
1470.2.d.b 4 105.x even 12 1
1470.2.d.c 4 35.k even 12 1
1470.2.d.c 4 105.x even 12 1
1470.2.d.d 4 35.l odd 12 1
1470.2.d.d 4 105.w odd 12 1

Hecke kernels

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

\( T_{11}^{4} + 9T_{11}^{3} + 31T_{11}^{2} + 36T_{11} + 16 \) Copy content Toggle raw display
\( T_{37}^{8} + 204T_{37}^{6} + 32400T_{37}^{4} + 1880064T_{37}^{2} + 84934656 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - T^{2} + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T^{4} - 5 T^{2} + 9)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} + 11 T^{2} + 49)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 9 T^{3} + 31 T^{2} + \cdots + 16)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 4)^{4} \) Copy content Toggle raw display
$17$ \( (T^{4} + 44 T^{2} + 1936)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 6 T + 12)^{4} \) Copy content Toggle raw display
$23$ \( T^{8} - 21 T^{6} + \cdots + 1296 \) Copy content Toggle raw display
$29$ \( (T^{2} + 11)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} - 9 T^{3} + \cdots + 324)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + 204 T^{6} + \cdots + 84934656 \) Copy content Toggle raw display
$41$ \( (T^{2} + 9 T + 12)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} - 123 T^{2} + 144)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 76 T^{6} + \cdots + 65536 \) Copy content Toggle raw display
$53$ \( T^{8} - 21 T^{6} + \cdots + 1296 \) Copy content Toggle raw display
$59$ \( (T^{4} + 9 T^{3} + \cdots + 2916)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 27 T^{3} + \cdots + 1296)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + 63 T^{6} + \cdots + 104976 \) Copy content Toggle raw display
$71$ \( (T^{4} + 76 T^{2} + 256)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} - 4 T^{2} + 16)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - T^{3} + 75 T^{2} + \cdots + 5476)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} - 142 T^{2} + 289)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} - 3 T^{3} + 15 T^{2} + \cdots + 36)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 233 T^{2} + 1024)^{2} \) Copy content Toggle raw display
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