Properties

Label 550.2.a.n
Level $550$
Weight $2$
Character orbit 550.a
Self dual yes
Analytic conductor $4.392$
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] = [550,2,Mod(1,550)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(550, 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("550.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 550 = 2 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 550.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: \(4.39177211117\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 110)
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.

\(f(q)\) \(=\) \( q + q^{2} + \beta q^{3} + q^{4} + \beta q^{6} - \beta q^{7} + q^{8} + (\beta + 5) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + \beta q^{3} + q^{4} + \beta q^{6} - \beta q^{7} + q^{8} + (\beta + 5) q^{9} - q^{11} + \beta q^{12} - 2 q^{13} - \beta q^{14} + q^{16} + ( - \beta + 2) q^{17} + (\beta + 5) q^{18} + ( - \beta + 4) q^{19} + ( - \beta - 8) q^{21} - q^{22} + ( - 2 \beta + 4) q^{23} + \beta q^{24} - 2 q^{26} + (3 \beta + 8) q^{27} - \beta q^{28} + (\beta - 2) q^{29} + \beta q^{31} + q^{32} - \beta q^{33} + ( - \beta + 2) q^{34} + (\beta + 5) q^{36} + ( - \beta - 6) q^{37} + ( - \beta + 4) q^{38} - 2 \beta q^{39} + ( - 4 \beta + 2) q^{41} + ( - \beta - 8) q^{42} + 4 q^{43} - q^{44} + ( - 2 \beta + 4) q^{46} + ( - 2 \beta + 4) q^{47} + \beta q^{48} + (\beta + 1) q^{49} + (\beta - 8) q^{51} - 2 q^{52} + (3 \beta - 6) q^{53} + (3 \beta + 8) q^{54} - \beta q^{56} + (3 \beta - 8) q^{57} + (\beta - 2) q^{58} + ( - 2 \beta + 4) q^{59} + ( - \beta - 2) q^{61} + \beta q^{62} + ( - 6 \beta - 8) q^{63} + q^{64} - \beta q^{66} - 8 q^{67} + ( - \beta + 2) q^{68} + (2 \beta - 16) q^{69} + 3 \beta q^{71} + (\beta + 5) q^{72} + (4 \beta + 2) q^{73} + ( - \beta - 6) q^{74} + ( - \beta + 4) q^{76} + \beta q^{77} - 2 \beta q^{78} + (2 \beta - 8) q^{79} + (8 \beta + 9) q^{81} + ( - 4 \beta + 2) q^{82} + (2 \beta - 4) q^{83} + ( - \beta - 8) q^{84} + 4 q^{86} + ( - \beta + 8) q^{87} - q^{88} + ( - \beta + 2) q^{89} + 2 \beta q^{91} + ( - 2 \beta + 4) q^{92} + (\beta + 8) q^{93} + ( - 2 \beta + 4) q^{94} + \beta q^{96} + (2 \beta + 6) q^{97} + (\beta + 1) q^{98} + ( - \beta - 5) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + q^{3} + 2 q^{4} + q^{6} - q^{7} + 2 q^{8} + 11 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + q^{3} + 2 q^{4} + q^{6} - q^{7} + 2 q^{8} + 11 q^{9} - 2 q^{11} + q^{12} - 4 q^{13} - q^{14} + 2 q^{16} + 3 q^{17} + 11 q^{18} + 7 q^{19} - 17 q^{21} - 2 q^{22} + 6 q^{23} + q^{24} - 4 q^{26} + 19 q^{27} - q^{28} - 3 q^{29} + q^{31} + 2 q^{32} - q^{33} + 3 q^{34} + 11 q^{36} - 13 q^{37} + 7 q^{38} - 2 q^{39} - 17 q^{42} + 8 q^{43} - 2 q^{44} + 6 q^{46} + 6 q^{47} + q^{48} + 3 q^{49} - 15 q^{51} - 4 q^{52} - 9 q^{53} + 19 q^{54} - q^{56} - 13 q^{57} - 3 q^{58} + 6 q^{59} - 5 q^{61} + q^{62} - 22 q^{63} + 2 q^{64} - q^{66} - 16 q^{67} + 3 q^{68} - 30 q^{69} + 3 q^{71} + 11 q^{72} + 8 q^{73} - 13 q^{74} + 7 q^{76} + q^{77} - 2 q^{78} - 14 q^{79} + 26 q^{81} - 6 q^{83} - 17 q^{84} + 8 q^{86} + 15 q^{87} - 2 q^{88} + 3 q^{89} + 2 q^{91} + 6 q^{92} + 17 q^{93} + 6 q^{94} + q^{96} + 14 q^{97} + 3 q^{98} - 11 q^{99}+O(q^{100}) \) 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 −2.37228 1.00000 0 −2.37228 2.37228 1.00000 2.62772 0
1.2 1.00000 3.37228 1.00000 0 3.37228 −3.37228 1.00000 8.37228 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(5\) \( +1 \)
\(11\) \( +1 \)

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 550.2.a.n 2
3.b odd 2 1 4950.2.a.bw 2
4.b odd 2 1 4400.2.a.bl 2
5.b even 2 1 110.2.a.d 2
5.c odd 4 2 550.2.b.f 4
11.b odd 2 1 6050.2.a.cb 2
15.d odd 2 1 990.2.a.m 2
15.e even 4 2 4950.2.c.bc 4
20.d odd 2 1 880.2.a.n 2
20.e even 4 2 4400.2.b.p 4
35.c odd 2 1 5390.2.a.bp 2
40.e odd 2 1 3520.2.a.bj 2
40.f even 2 1 3520.2.a.bq 2
55.d odd 2 1 1210.2.a.r 2
60.h even 2 1 7920.2.a.bq 2
220.g even 2 1 9680.2.a.bt 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
110.2.a.d 2 5.b even 2 1
550.2.a.n 2 1.a even 1 1 trivial
550.2.b.f 4 5.c odd 4 2
880.2.a.n 2 20.d odd 2 1
990.2.a.m 2 15.d odd 2 1
1210.2.a.r 2 55.d odd 2 1
3520.2.a.bj 2 40.e odd 2 1
3520.2.a.bq 2 40.f even 2 1
4400.2.a.bl 2 4.b odd 2 1
4400.2.b.p 4 20.e even 4 2
4950.2.a.bw 2 3.b odd 2 1
4950.2.c.bc 4 15.e even 4 2
5390.2.a.bp 2 35.c odd 2 1
6050.2.a.cb 2 11.b odd 2 1
7920.2.a.bq 2 60.h even 2 1
9680.2.a.bt 2 220.g even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(550))\):

\( T_{3}^{2} - T_{3} - 8 \) Copy content Toggle raw display
\( T_{7}^{2} + T_{7} - 8 \) 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
$3$ \( T^{2} - T - 8 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + T - 8 \) Copy content Toggle raw display
$11$ \( (T + 1)^{2} \) Copy content Toggle raw display
$13$ \( (T + 2)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 3T - 6 \) Copy content Toggle raw display
$19$ \( T^{2} - 7T + 4 \) Copy content Toggle raw display
$23$ \( T^{2} - 6T - 24 \) Copy content Toggle raw display
$29$ \( T^{2} + 3T - 6 \) Copy content Toggle raw display
$31$ \( T^{2} - T - 8 \) Copy content Toggle raw display
$37$ \( T^{2} + 13T + 34 \) Copy content Toggle raw display
$41$ \( T^{2} - 132 \) Copy content Toggle raw display
$43$ \( (T - 4)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 6T - 24 \) Copy content Toggle raw display
$53$ \( T^{2} + 9T - 54 \) Copy content Toggle raw display
$59$ \( T^{2} - 6T - 24 \) Copy content Toggle raw display
$61$ \( T^{2} + 5T - 2 \) Copy content Toggle raw display
$67$ \( (T + 8)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} - 3T - 72 \) Copy content Toggle raw display
$73$ \( T^{2} - 8T - 116 \) Copy content Toggle raw display
$79$ \( T^{2} + 14T + 16 \) Copy content Toggle raw display
$83$ \( T^{2} + 6T - 24 \) Copy content Toggle raw display
$89$ \( T^{2} - 3T - 6 \) Copy content Toggle raw display
$97$ \( T^{2} - 14T + 16 \) Copy content Toggle raw display
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