Properties

Label 8450.2.a.y
Level $8450$
Weight $2$
Character orbit 8450.a
Self dual yes
Analytic conductor $67.474$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8450,2,Mod(1,8450)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8450, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8450.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8450 = 2 \cdot 5^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8450.a (trivial)

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: yes
Analytic conductor: \(67.4735897080\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 26)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q + q^{2} + 3 q^{3} + q^{4} + 3 q^{6} + q^{7} + q^{8} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + 3 q^{3} + q^{4} + 3 q^{6} + q^{7} + q^{8} + 6 q^{9} + 2 q^{11} + 3 q^{12} + q^{14} + q^{16} + 3 q^{17} + 6 q^{18} - 6 q^{19} + 3 q^{21} + 2 q^{22} + 4 q^{23} + 3 q^{24} + 9 q^{27} + q^{28} + 2 q^{29} - 4 q^{31} + q^{32} + 6 q^{33} + 3 q^{34} + 6 q^{36} + 3 q^{37} - 6 q^{38} + 3 q^{42} + 5 q^{43} + 2 q^{44} + 4 q^{46} + 13 q^{47} + 3 q^{48} - 6 q^{49} + 9 q^{51} - 12 q^{53} + 9 q^{54} + q^{56} - 18 q^{57} + 2 q^{58} + 10 q^{59} - 8 q^{61} - 4 q^{62} + 6 q^{63} + q^{64} + 6 q^{66} - 2 q^{67} + 3 q^{68} + 12 q^{69} + 5 q^{71} + 6 q^{72} - 10 q^{73} + 3 q^{74} - 6 q^{76} + 2 q^{77} - 4 q^{79} + 9 q^{81} + 3 q^{84} + 5 q^{86} + 6 q^{87} + 2 q^{88} - 6 q^{89} + 4 q^{92} - 12 q^{93} + 13 q^{94} + 3 q^{96} + 14 q^{97} - 6 q^{98} + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
1.00000 3.00000 1.00000 0 3.00000 1.00000 1.00000 6.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(5\) \( +1 \)
\(13\) \( +1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8450.2.a.y 1
5.b even 2 1 338.2.a.a 1
13.b even 2 1 650.2.a.g 1
15.d odd 2 1 3042.2.a.l 1
20.d odd 2 1 2704.2.a.n 1
39.d odd 2 1 5850.2.a.bn 1
52.b odd 2 1 5200.2.a.c 1
65.d even 2 1 26.2.a.b 1
65.g odd 4 2 338.2.b.a 2
65.h odd 4 2 650.2.b.a 2
65.l even 6 2 338.2.c.c 2
65.n even 6 2 338.2.c.g 2
65.s odd 12 4 338.2.e.d 4
195.e odd 2 1 234.2.a.b 1
195.n even 4 2 3042.2.b.f 2
195.s even 4 2 5850.2.e.v 2
260.g odd 2 1 208.2.a.d 1
260.u even 4 2 2704.2.f.j 2
455.h odd 2 1 1274.2.a.o 1
455.bf odd 6 2 1274.2.f.a 2
455.bh even 6 2 1274.2.f.l 2
520.b odd 2 1 832.2.a.a 1
520.p even 2 1 832.2.a.j 1
585.be even 6 2 2106.2.e.h 2
585.bo odd 6 2 2106.2.e.t 2
715.c odd 2 1 3146.2.a.a 1
780.d even 2 1 1872.2.a.m 1
1040.be even 4 2 3328.2.b.g 2
1040.cb odd 4 2 3328.2.b.k 2
1105.h even 2 1 7514.2.a.i 1
1235.e odd 2 1 9386.2.a.f 1
1560.n even 2 1 7488.2.a.v 1
1560.y odd 2 1 7488.2.a.w 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.2.a.b 1 65.d even 2 1
208.2.a.d 1 260.g odd 2 1
234.2.a.b 1 195.e odd 2 1
338.2.a.a 1 5.b even 2 1
338.2.b.a 2 65.g odd 4 2
338.2.c.c 2 65.l even 6 2
338.2.c.g 2 65.n even 6 2
338.2.e.d 4 65.s odd 12 4
650.2.a.g 1 13.b even 2 1
650.2.b.a 2 65.h odd 4 2
832.2.a.a 1 520.b odd 2 1
832.2.a.j 1 520.p even 2 1
1274.2.a.o 1 455.h odd 2 1
1274.2.f.a 2 455.bf odd 6 2
1274.2.f.l 2 455.bh even 6 2
1872.2.a.m 1 780.d even 2 1
2106.2.e.h 2 585.be even 6 2
2106.2.e.t 2 585.bo odd 6 2
2704.2.a.n 1 20.d odd 2 1
2704.2.f.j 2 260.u even 4 2
3042.2.a.l 1 15.d odd 2 1
3042.2.b.f 2 195.n even 4 2
3146.2.a.a 1 715.c odd 2 1
3328.2.b.g 2 1040.be even 4 2
3328.2.b.k 2 1040.cb odd 4 2
5200.2.a.c 1 52.b odd 2 1
5850.2.a.bn 1 39.d odd 2 1
5850.2.e.v 2 195.s even 4 2
7488.2.a.v 1 1560.n even 2 1
7488.2.a.w 1 1560.y odd 2 1
7514.2.a.i 1 1105.h even 2 1
8450.2.a.y 1 1.a even 1 1 trivial
9386.2.a.f 1 1235.e odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8450))\):

\( T_{3} - 3 \) Copy content Toggle raw display
\( T_{7} - 1 \) Copy content Toggle raw display
\( T_{11} - 2 \) Copy content Toggle raw display
\( T_{17} - 3 \) Copy content Toggle raw display
\( T_{31} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 1 \) Copy content Toggle raw display
$3$ \( T - 3 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T - 1 \) Copy content Toggle raw display
$11$ \( T - 2 \) Copy content Toggle raw display
$13$ \( T \) Copy content Toggle raw display
$17$ \( T - 3 \) Copy content Toggle raw display
$19$ \( T + 6 \) Copy content Toggle raw display
$23$ \( T - 4 \) Copy content Toggle raw display
$29$ \( T - 2 \) Copy content Toggle raw display
$31$ \( T + 4 \) Copy content Toggle raw display
$37$ \( T - 3 \) Copy content Toggle raw display
$41$ \( T \) Copy content Toggle raw display
$43$ \( T - 5 \) Copy content Toggle raw display
$47$ \( T - 13 \) Copy content Toggle raw display
$53$ \( T + 12 \) Copy content Toggle raw display
$59$ \( T - 10 \) Copy content Toggle raw display
$61$ \( T + 8 \) Copy content Toggle raw display
$67$ \( T + 2 \) Copy content Toggle raw display
$71$ \( T - 5 \) Copy content Toggle raw display
$73$ \( T + 10 \) Copy content Toggle raw display
$79$ \( T + 4 \) Copy content Toggle raw display
$83$ \( T \) Copy content Toggle raw display
$89$ \( T + 6 \) Copy content Toggle raw display
$97$ \( T - 14 \) Copy content Toggle raw display
show more
show less