Properties

Label 850.2.a.g
Level $850$
Weight $2$
Character orbit 850.a
Self dual yes
Analytic conductor $6.787$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [850,2,Mod(1,850)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(850, 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("850.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 850 = 2 \cdot 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 850.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: \(6.78728417181\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 170)
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)
 
\(f(q)\) \(=\) \( q + q^{2} - q^{3} + q^{4} - q^{6} - 2 q^{7} + q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} - q^{3} + q^{4} - q^{6} - 2 q^{7} + q^{8} - 2 q^{9} - q^{12} - 5 q^{13} - 2 q^{14} + q^{16} + q^{17} - 2 q^{18} - q^{19} + 2 q^{21} - 6 q^{23} - q^{24} - 5 q^{26} + 5 q^{27} - 2 q^{28} - 9 q^{29} - q^{31} + q^{32} + q^{34} - 2 q^{36} + 4 q^{37} - q^{38} + 5 q^{39} - 6 q^{41} + 2 q^{42} - 2 q^{43} - 6 q^{46} + 9 q^{47} - q^{48} - 3 q^{49} - q^{51} - 5 q^{52} + 9 q^{53} + 5 q^{54} - 2 q^{56} + q^{57} - 9 q^{58} + 3 q^{59} - 7 q^{61} - q^{62} + 4 q^{63} + q^{64} - 14 q^{67} + q^{68} + 6 q^{69} + 3 q^{71} - 2 q^{72} - 11 q^{73} + 4 q^{74} - q^{76} + 5 q^{78} + 8 q^{79} + q^{81} - 6 q^{82} + 2 q^{84} - 2 q^{86} + 9 q^{87} - 9 q^{89} + 10 q^{91} - 6 q^{92} + q^{93} + 9 q^{94} - q^{96} + 7 q^{97} - 3 q^{98}+O(q^{100}) \) 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 −1.00000 1.00000 0 −1.00000 −2.00000 1.00000 −2.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(1\)
\(17\) \(-1\)

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 850.2.a.g 1
3.b odd 2 1 7650.2.a.i 1
4.b odd 2 1 6800.2.a.s 1
5.b even 2 1 170.2.a.c 1
5.c odd 4 2 850.2.c.c 2
15.d odd 2 1 1530.2.a.l 1
20.d odd 2 1 1360.2.a.e 1
35.c odd 2 1 8330.2.a.d 1
40.e odd 2 1 5440.2.a.p 1
40.f even 2 1 5440.2.a.i 1
85.c even 2 1 2890.2.a.e 1
85.j even 4 2 2890.2.b.h 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
170.2.a.c 1 5.b even 2 1
850.2.a.g 1 1.a even 1 1 trivial
850.2.c.c 2 5.c odd 4 2
1360.2.a.e 1 20.d odd 2 1
1530.2.a.l 1 15.d odd 2 1
2890.2.a.e 1 85.c even 2 1
2890.2.b.h 2 85.j even 4 2
5440.2.a.i 1 40.f even 2 1
5440.2.a.p 1 40.e odd 2 1
6800.2.a.s 1 4.b odd 2 1
7650.2.a.i 1 3.b odd 2 1
8330.2.a.d 1 35.c odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(850))\):

\( T_{3} + 1 \) Copy content Toggle raw display
\( T_{7} + 2 \) Copy content Toggle raw display
\( T_{13} + 5 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 1 \) Copy content Toggle raw display
$3$ \( T + 1 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T + 2 \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T + 5 \) Copy content Toggle raw display
$17$ \( T - 1 \) Copy content Toggle raw display
$19$ \( T + 1 \) Copy content Toggle raw display
$23$ \( T + 6 \) Copy content Toggle raw display
$29$ \( T + 9 \) Copy content Toggle raw display
$31$ \( T + 1 \) Copy content Toggle raw display
$37$ \( T - 4 \) Copy content Toggle raw display
$41$ \( T + 6 \) Copy content Toggle raw display
$43$ \( T + 2 \) Copy content Toggle raw display
$47$ \( T - 9 \) Copy content Toggle raw display
$53$ \( T - 9 \) Copy content Toggle raw display
$59$ \( T - 3 \) Copy content Toggle raw display
$61$ \( T + 7 \) Copy content Toggle raw display
$67$ \( T + 14 \) Copy content Toggle raw display
$71$ \( T - 3 \) Copy content Toggle raw display
$73$ \( T + 11 \) Copy content Toggle raw display
$79$ \( T - 8 \) Copy content Toggle raw display
$83$ \( T \) Copy content Toggle raw display
$89$ \( T + 9 \) Copy content Toggle raw display
$97$ \( T - 7 \) Copy content Toggle raw display
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