Properties

Label 1550.2.a.k
Level $1550$
Weight $2$
Character orbit 1550.a
Self dual yes
Analytic conductor $12.377$
Analytic rank $0$
Dimension $3$
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] = [1550,2,Mod(1,1550)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1550, 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("1550.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1550 = 2 \cdot 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1550.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: \(12.3768123133\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 310)
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 a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + (\beta_1 - 1) q^{3} + q^{4} + ( - \beta_1 + 1) q^{6} + (\beta_{2} - \beta_1) q^{7} - q^{8} + (\beta_{2} - \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + (\beta_1 - 1) q^{3} + q^{4} + ( - \beta_1 + 1) q^{6} + (\beta_{2} - \beta_1) q^{7} - q^{8} + (\beta_{2} - \beta_1 + 1) q^{9} + (\beta_{2} + 2 \beta_1 - 1) q^{11} + (\beta_1 - 1) q^{12} + ( - \beta_1 + 3) q^{13} + ( - \beta_{2} + \beta_1) q^{14} + q^{16} + (\beta_{2} - \beta_1) q^{17} + ( - \beta_{2} + \beta_1 - 1) q^{18} + (2 \beta_{2} + 2) q^{19} + ( - 2 \beta_{2} + 2 \beta_1 - 4) q^{21} + ( - \beta_{2} - 2 \beta_1 + 1) q^{22} + (\beta_{2} + \beta_1) q^{23} + ( - \beta_1 + 1) q^{24} + (\beta_1 - 3) q^{26} + ( - 2 \beta_{2} - 2) q^{27} + (\beta_{2} - \beta_1) q^{28} + ( - 3 \beta_{2} - 2 \beta_1 + 1) q^{29} + q^{31} - q^{32} + (\beta_{2} + \beta_1 + 6) q^{33} + ( - \beta_{2} + \beta_1) q^{34} + (\beta_{2} - \beta_1 + 1) q^{36} + ( - 2 \beta_{2} + \beta_1 + 3) q^{37} + ( - 2 \beta_{2} - 2) q^{38} + ( - \beta_{2} + 3 \beta_1 - 6) q^{39} + (3 \beta_{2} + \beta_1 - 2) q^{41} + (2 \beta_{2} - 2 \beta_1 + 4) q^{42} + ( - 2 \beta_{2} + 3 \beta_1 + 3) q^{43} + (\beta_{2} + 2 \beta_1 - 1) q^{44} + ( - \beta_{2} - \beta_1) q^{46} + (\beta_{2} + 3 \beta_1 - 8) q^{47} + (\beta_1 - 1) q^{48} + ( - 4 \beta_1 + 5) q^{49} + ( - 2 \beta_{2} + 2 \beta_1 - 4) q^{51} + ( - \beta_1 + 3) q^{52} + ( - 2 \beta_{2} + \beta_1 + 7) q^{53} + (2 \beta_{2} + 2) q^{54} + ( - \beta_{2} + \beta_1) q^{56} + ( - 2 \beta_{2} + 6 \beta_1 - 4) q^{57} + (3 \beta_{2} + 2 \beta_1 - 1) q^{58} + (2 \beta_1 + 6) q^{59} + ( - \beta_{2} - 2 \beta_1 - 5) q^{61} - q^{62} + (\beta_{2} - 5 \beta_1 + 12) q^{63} + q^{64} + ( - \beta_{2} - \beta_1 - 6) q^{66} + (4 \beta_{2} + 2 \beta_1 + 2) q^{67} + (\beta_{2} - \beta_1) q^{68} + (2 \beta_1 + 2) q^{69} + ( - 2 \beta_{2} - 2 \beta_1 + 4) q^{71} + ( - \beta_{2} + \beta_1 - 1) q^{72} + (3 \beta_{2} - \beta_1 + 6) q^{73} + (2 \beta_{2} - \beta_1 - 3) q^{74} + (2 \beta_{2} + 2) q^{76} - 4 \beta_{2} q^{77} + (\beta_{2} - 3 \beta_1 + 6) q^{78} + (6 \beta_1 - 2) q^{79} + ( - \beta_{2} - 3 \beta_1 + 1) q^{81} + ( - 3 \beta_{2} - \beta_1 + 2) q^{82} + ( - 2 \beta_{2} + \beta_1 - 3) q^{83} + ( - 2 \beta_{2} + 2 \beta_1 - 4) q^{84} + (2 \beta_{2} - 3 \beta_1 - 3) q^{86} + (\beta_{2} - 5 \beta_1 - 4) q^{87} + ( - \beta_{2} - 2 \beta_1 + 1) q^{88} + (2 \beta_{2} - 2 \beta_1 + 6) q^{89} + (4 \beta_{2} - 4 \beta_1 + 4) q^{91} + (\beta_{2} + \beta_1) q^{92} + (\beta_1 - 1) q^{93} + ( - \beta_{2} - 3 \beta_1 + 8) q^{94} + ( - \beta_1 + 1) q^{96} + (\beta_{2} - 3 \beta_1 + 4) q^{97} + (4 \beta_1 - 5) q^{98} + ( - 3 \beta_{2} + 2 \beta_1 - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} - 2 q^{3} + 3 q^{4} + 2 q^{6} - 3 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{2} - 2 q^{3} + 3 q^{4} + 2 q^{6} - 3 q^{8} + 3 q^{9} - 2 q^{12} + 8 q^{13} + 3 q^{16} - 3 q^{18} + 8 q^{19} - 12 q^{21} + 2 q^{23} + 2 q^{24} - 8 q^{26} - 8 q^{27} - 2 q^{29} + 3 q^{31} - 3 q^{32} + 20 q^{33} + 3 q^{36} + 8 q^{37} - 8 q^{38} - 16 q^{39} - 2 q^{41} + 12 q^{42} + 10 q^{43} - 2 q^{46} - 20 q^{47} - 2 q^{48} + 11 q^{49} - 12 q^{51} + 8 q^{52} + 20 q^{53} + 8 q^{54} - 8 q^{57} + 2 q^{58} + 20 q^{59} - 18 q^{61} - 3 q^{62} + 32 q^{63} + 3 q^{64} - 20 q^{66} + 12 q^{67} + 8 q^{69} + 8 q^{71} - 3 q^{72} + 20 q^{73} - 8 q^{74} + 8 q^{76} - 4 q^{77} + 16 q^{78} - q^{81} + 2 q^{82} - 10 q^{83} - 12 q^{84} - 10 q^{86} - 16 q^{87} + 18 q^{89} + 12 q^{91} + 2 q^{92} - 2 q^{93} + 20 q^{94} + 2 q^{96} + 10 q^{97} - 11 q^{98} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 3x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\nu^{2} + 2\nu + 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta _1 + 2 \) 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
0.311108
−1.48119
2.17009
−1.00000 −2.90321 1.00000 0 2.90321 4.42864 −1.00000 5.42864 0
1.2 −1.00000 −0.806063 1.00000 0 0.806063 −3.35026 −1.00000 −2.35026 0
1.3 −1.00000 1.70928 1.00000 0 −1.70928 −1.07838 −1.00000 −0.0783777 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(5\) \(1\)
\(31\) \(-1\)

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 1550.2.a.k 3
5.b even 2 1 310.2.a.e 3
5.c odd 4 2 1550.2.b.j 6
15.d odd 2 1 2790.2.a.bi 3
20.d odd 2 1 2480.2.a.u 3
40.e odd 2 1 9920.2.a.bx 3
40.f even 2 1 9920.2.a.bw 3
155.c odd 2 1 9610.2.a.u 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
310.2.a.e 3 5.b even 2 1
1550.2.a.k 3 1.a even 1 1 trivial
1550.2.b.j 6 5.c odd 4 2
2480.2.a.u 3 20.d odd 2 1
2790.2.a.bi 3 15.d odd 2 1
9610.2.a.u 3 155.c odd 2 1
9920.2.a.bw 3 40.f even 2 1
9920.2.a.bx 3 40.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1550))\):

\( T_{3}^{3} + 2T_{3}^{2} - 4T_{3} - 4 \) Copy content Toggle raw display
\( T_{7}^{3} - 16T_{7} - 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{3} \) Copy content Toggle raw display
$3$ \( T^{3} + 2 T^{2} - 4 T - 4 \) Copy content Toggle raw display
$5$ \( T^{3} \) Copy content Toggle raw display
$7$ \( T^{3} - 16T - 16 \) Copy content Toggle raw display
$11$ \( T^{3} - 28T - 52 \) Copy content Toggle raw display
$13$ \( T^{3} - 8 T^{2} + 16 T - 4 \) Copy content Toggle raw display
$17$ \( T^{3} - 16T - 16 \) Copy content Toggle raw display
$19$ \( T^{3} - 8 T^{2} - 16 T + 160 \) Copy content Toggle raw display
$23$ \( T^{3} - 2 T^{2} - 12 T + 8 \) Copy content Toggle raw display
$29$ \( T^{3} + 2 T^{2} - 96 T - 260 \) Copy content Toggle raw display
$31$ \( (T - 1)^{3} \) Copy content Toggle raw display
$37$ \( T^{3} - 8 T^{2} - 24 T + 92 \) Copy content Toggle raw display
$41$ \( T^{3} + 2 T^{2} - 84 T + 232 \) Copy content Toggle raw display
$43$ \( T^{3} - 10 T^{2} - 60 T + 604 \) Copy content Toggle raw display
$47$ \( T^{3} + 20 T^{2} + 80 T - 208 \) Copy content Toggle raw display
$53$ \( T^{3} - 20 T^{2} + 88 T - 4 \) Copy content Toggle raw display
$59$ \( T^{3} - 20 T^{2} + 112 T - 160 \) Copy content Toggle raw display
$61$ \( T^{3} + 18 T^{2} + 80 T + 100 \) Copy content Toggle raw display
$67$ \( T^{3} - 12 T^{2} - 112 T + 1184 \) Copy content Toggle raw display
$71$ \( T^{3} - 8 T^{2} - 32 T + 128 \) Copy content Toggle raw display
$73$ \( T^{3} - 20 T^{2} + 40 T + 464 \) Copy content Toggle raw display
$79$ \( T^{3} - 192T - 160 \) Copy content Toggle raw display
$83$ \( T^{3} + 10 T^{2} - 12 T - 124 \) Copy content Toggle raw display
$89$ \( T^{3} - 18 T^{2} + 44 T + 40 \) Copy content Toggle raw display
$97$ \( T^{3} - 10 T^{2} - 28 T + 8 \) Copy content Toggle raw display
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