Properties

Label 2550.2.a.bm
Level $2550$
Weight $2$
Character orbit 2550.a
Self dual yes
Analytic conductor $20.362$
Analytic rank $0$
Dimension $2$
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] = [2550,2,Mod(1,2550)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2550, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("2550.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2550 = 2 \cdot 3 \cdot 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2550.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: \(20.3618525154\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{33}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{33})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + q^{3} + q^{4} + q^{6} + \beta q^{7} + q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + q^{3} + q^{4} + q^{6} + \beta q^{7} + q^{8} + q^{9} + ( - \beta + 2) q^{11} + q^{12} + 2 q^{13} + \beta q^{14} + q^{16} - q^{17} + q^{18} + \beta q^{19} + \beta q^{21} + ( - \beta + 2) q^{22} + (\beta + 1) q^{23} + q^{24} + 2 q^{26} + q^{27} + \beta q^{28} + ( - 2 \beta + 4) q^{29} + ( - 3 \beta + 2) q^{31} + q^{32} + ( - \beta + 2) q^{33} - q^{34} + q^{36} - q^{37} + \beta q^{38} + 2 q^{39} + (\beta + 1) q^{41} + \beta q^{42} + (\beta + 6) q^{43} + ( - \beta + 2) q^{44} + (\beta + 1) q^{46} + ( - \beta - 4) q^{47} + q^{48} + (\beta + 1) q^{49} - q^{51} + 2 q^{52} + ( - 2 \beta + 1) q^{53} + q^{54} + \beta q^{56} + \beta q^{57} + ( - 2 \beta + 4) q^{58} + ( - 3 \beta + 3) q^{59} + (3 \beta - 1) q^{61} + ( - 3 \beta + 2) q^{62} + \beta q^{63} + q^{64} + ( - \beta + 2) q^{66} + ( - \beta - 2) q^{67} - q^{68} + (\beta + 1) q^{69} + (\beta + 1) q^{71} + q^{72} + 8 q^{73} - q^{74} + \beta q^{76} + (\beta - 8) q^{77} + 2 q^{78} + (3 \beta - 4) q^{79} + q^{81} + (\beta + 1) q^{82} + ( - \beta - 1) q^{83} + \beta q^{84} + (\beta + 6) q^{86} + ( - 2 \beta + 4) q^{87} + ( - \beta + 2) q^{88} + (2 \beta + 2) q^{89} + 2 \beta q^{91} + (\beta + 1) q^{92} + ( - 3 \beta + 2) q^{93} + ( - \beta - 4) q^{94} + q^{96} + (2 \beta - 8) q^{97} + (\beta + 1) q^{98} + ( - \beta + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{6} + q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{6} + q^{7} + 2 q^{8} + 2 q^{9} + 3 q^{11} + 2 q^{12} + 4 q^{13} + q^{14} + 2 q^{16} - 2 q^{17} + 2 q^{18} + q^{19} + q^{21} + 3 q^{22} + 3 q^{23} + 2 q^{24} + 4 q^{26} + 2 q^{27} + q^{28} + 6 q^{29} + q^{31} + 2 q^{32} + 3 q^{33} - 2 q^{34} + 2 q^{36} - 2 q^{37} + q^{38} + 4 q^{39} + 3 q^{41} + q^{42} + 13 q^{43} + 3 q^{44} + 3 q^{46} - 9 q^{47} + 2 q^{48} + 3 q^{49} - 2 q^{51} + 4 q^{52} + 2 q^{54} + q^{56} + q^{57} + 6 q^{58} + 3 q^{59} + q^{61} + q^{62} + q^{63} + 2 q^{64} + 3 q^{66} - 5 q^{67} - 2 q^{68} + 3 q^{69} + 3 q^{71} + 2 q^{72} + 16 q^{73} - 2 q^{74} + q^{76} - 15 q^{77} + 4 q^{78} - 5 q^{79} + 2 q^{81} + 3 q^{82} - 3 q^{83} + q^{84} + 13 q^{86} + 6 q^{87} + 3 q^{88} + 6 q^{89} + 2 q^{91} + 3 q^{92} + q^{93} - 9 q^{94} + 2 q^{96} - 14 q^{97} + 3 q^{98} + 3 q^{99}+O(q^{100}) \) 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
−2.37228
3.37228
1.00000 1.00000 1.00000 0 1.00000 −2.37228 1.00000 1.00000 0
1.2 1.00000 1.00000 1.00000 0 1.00000 3.37228 1.00000 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(5\) \(-1\)
\(17\) \(1\)

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 2550.2.a.bm yes 2
3.b odd 2 1 7650.2.a.cv 2
5.b even 2 1 2550.2.a.bg 2
5.c odd 4 2 2550.2.d.v 4
15.d odd 2 1 7650.2.a.df 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2550.2.a.bg 2 5.b even 2 1
2550.2.a.bm yes 2 1.a even 1 1 trivial
2550.2.d.v 4 5.c odd 4 2
7650.2.a.cv 2 3.b odd 2 1
7650.2.a.df 2 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2550))\):

\( T_{7}^{2} - T_{7} - 8 \) Copy content Toggle raw display
\( T_{11}^{2} - 3T_{11} - 6 \) Copy content Toggle raw display
\( T_{13} - 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{2} \) Copy content Toggle raw display
$3$ \( (T - 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - T - 8 \) Copy content Toggle raw display
$11$ \( T^{2} - 3T - 6 \) Copy content Toggle raw display
$13$ \( (T - 2)^{2} \) Copy content Toggle raw display
$17$ \( (T + 1)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} - T - 8 \) Copy content Toggle raw display
$23$ \( T^{2} - 3T - 6 \) Copy content Toggle raw display
$29$ \( T^{2} - 6T - 24 \) Copy content Toggle raw display
$31$ \( T^{2} - T - 74 \) Copy content Toggle raw display
$37$ \( (T + 1)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 3T - 6 \) Copy content Toggle raw display
$43$ \( T^{2} - 13T + 34 \) Copy content Toggle raw display
$47$ \( T^{2} + 9T + 12 \) Copy content Toggle raw display
$53$ \( T^{2} - 33 \) Copy content Toggle raw display
$59$ \( T^{2} - 3T - 72 \) Copy content Toggle raw display
$61$ \( T^{2} - T - 74 \) Copy content Toggle raw display
$67$ \( T^{2} + 5T - 2 \) Copy content Toggle raw display
$71$ \( T^{2} - 3T - 6 \) Copy content Toggle raw display
$73$ \( (T - 8)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 5T - 68 \) Copy content Toggle raw display
$83$ \( T^{2} + 3T - 6 \) Copy content Toggle raw display
$89$ \( T^{2} - 6T - 24 \) Copy content Toggle raw display
$97$ \( T^{2} + 14T + 16 \) Copy content Toggle raw display
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