Properties

Label 1850.2.a.b
Level $1850$
Weight $2$
Character orbit 1850.a
Self dual yes
Analytic conductor $14.772$
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] = [1850,2,Mod(1,1850)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1850, 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("1850.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1850 = 2 \cdot 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1850.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: \(14.7723243739\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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)
 
\(f(q)\) \(=\) \( q - q^{2} - 2 q^{3} + q^{4} + 2 q^{6} - q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} - 2 q^{3} + q^{4} + 2 q^{6} - q^{8} + q^{9} + 4 q^{11} - 2 q^{12} + 2 q^{13} + q^{16} - 8 q^{17} - q^{18} - 5 q^{19} - 4 q^{22} - q^{23} + 2 q^{24} - 2 q^{26} + 4 q^{27} + 10 q^{29} - 4 q^{31} - q^{32} - 8 q^{33} + 8 q^{34} + q^{36} - q^{37} + 5 q^{38} - 4 q^{39} + 7 q^{41} + 9 q^{43} + 4 q^{44} + q^{46} - 6 q^{47} - 2 q^{48} - 7 q^{49} + 16 q^{51} + 2 q^{52} + 3 q^{53} - 4 q^{54} + 10 q^{57} - 10 q^{58} - 11 q^{59} + 2 q^{61} + 4 q^{62} + q^{64} + 8 q^{66} - 2 q^{67} - 8 q^{68} + 2 q^{69} + 14 q^{71} - q^{72} - 3 q^{73} + q^{74} - 5 q^{76} + 4 q^{78} - 11 q^{79} - 11 q^{81} - 7 q^{82} - 8 q^{83} - 9 q^{86} - 20 q^{87} - 4 q^{88} - 2 q^{89} - q^{92} + 8 q^{93} + 6 q^{94} + 2 q^{96} + 8 q^{97} + 7 q^{98} + 4 q^{99}+O(q^{100}) \) 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 −2.00000 1.00000 0 2.00000 0 −1.00000 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(5\) \( +1 \)
\(37\) \( +1 \)

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 1850.2.a.b 1
5.b even 2 1 1850.2.a.n yes 1
5.c odd 4 2 1850.2.b.h 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1850.2.a.b 1 1.a even 1 1 trivial
1850.2.a.n yes 1 5.b even 2 1
1850.2.b.h 2 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1850))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 1 \) Copy content Toggle raw display
$3$ \( T + 2 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T - 4 \) Copy content Toggle raw display
$13$ \( T - 2 \) Copy content Toggle raw display
$17$ \( T + 8 \) Copy content Toggle raw display
$19$ \( T + 5 \) Copy content Toggle raw display
$23$ \( T + 1 \) Copy content Toggle raw display
$29$ \( T - 10 \) Copy content Toggle raw display
$31$ \( T + 4 \) Copy content Toggle raw display
$37$ \( T + 1 \) Copy content Toggle raw display
$41$ \( T - 7 \) Copy content Toggle raw display
$43$ \( T - 9 \) Copy content Toggle raw display
$47$ \( T + 6 \) Copy content Toggle raw display
$53$ \( T - 3 \) Copy content Toggle raw display
$59$ \( T + 11 \) Copy content Toggle raw display
$61$ \( T - 2 \) Copy content Toggle raw display
$67$ \( T + 2 \) Copy content Toggle raw display
$71$ \( T - 14 \) Copy content Toggle raw display
$73$ \( T + 3 \) Copy content Toggle raw display
$79$ \( T + 11 \) Copy content Toggle raw display
$83$ \( T + 8 \) Copy content Toggle raw display
$89$ \( T + 2 \) Copy content Toggle raw display
$97$ \( T - 8 \) Copy content Toggle raw display
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