Properties

Label 1450.2.a.s
Level $1450$
Weight $2$
Character orbit 1450.a
Self dual yes
Analytic conductor $11.578$
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] = [1450,2,Mod(1,1450)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1450, 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("1450.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1450 = 2 \cdot 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1450.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: \(11.5783082931\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.316.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} - \beta_1 q^{3} + q^{4} - \beta_1 q^{6} + ( - \beta_{2} - \beta_1 + 1) q^{7} + q^{8} + \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} - \beta_1 q^{3} + q^{4} - \beta_1 q^{6} + ( - \beta_{2} - \beta_1 + 1) q^{7} + q^{8} + \beta_{2} q^{9} + ( - \beta_{2} + 2 \beta_1 + 1) q^{11} - \beta_1 q^{12} + (\beta_{2} - \beta_1 + 1) q^{13} + ( - \beta_{2} - \beta_1 + 1) q^{14} + q^{16} + (\beta_{2} - 2 \beta_1 + 3) q^{17} + \beta_{2} q^{18} - \beta_1 q^{19} + (2 \beta_{2} + 4) q^{21} + ( - \beta_{2} + 2 \beta_1 + 1) q^{22} + ( - 3 \beta_{2} + \beta_1 - 3) q^{23} - \beta_1 q^{24} + (\beta_{2} - \beta_1 + 1) q^{26} + ( - \beta_{2} + 2 \beta_1 - 1) q^{27} + ( - \beta_{2} - \beta_1 + 1) q^{28} + q^{29} + ( - 3 \beta_{2} + 4 \beta_1) q^{31} + q^{32} + ( - \beta_{2} - 5) q^{33} + (\beta_{2} - 2 \beta_1 + 3) q^{34} + \beta_{2} q^{36} + (2 \beta_{2} - \beta_1 + 5) q^{37} - \beta_1 q^{38} + ( - 2 \beta_1 + 2) q^{39} + (2 \beta_{2} - \beta_1 + 6) q^{41} + (2 \beta_{2} + 4) q^{42} + (2 \beta_{2} + 2 \beta_1 - 2) q^{43} + ( - \beta_{2} + 2 \beta_1 + 1) q^{44} + ( - 3 \beta_{2} + \beta_1 - 3) q^{46} + (3 \beta_{2} + 4) q^{47} - \beta_1 q^{48} + (2 \beta_1 + 3) q^{49} + (\beta_{2} - 4 \beta_1 + 5) q^{51} + (\beta_{2} - \beta_1 + 1) q^{52} + (3 \beta_{2} - 3 \beta_1 - 1) q^{53} + ( - \beta_{2} + 2 \beta_1 - 1) q^{54} + ( - \beta_{2} - \beta_1 + 1) q^{56} + (\beta_{2} + 3) q^{57} + q^{58} + ( - 2 \beta_{2} + \beta_1 + 5) q^{59} + (5 \beta_1 - 1) q^{61} + ( - 3 \beta_{2} + 4 \beta_1) q^{62} + (\beta_{2} - 3 \beta_1 - 5) q^{63} + q^{64} + ( - \beta_{2} - 5) q^{66} + ( - 3 \beta_{2} + \beta_1) q^{67} + (\beta_{2} - 2 \beta_1 + 3) q^{68} + (2 \beta_{2} + 6 \beta_1) q^{69} + (3 \beta_{2} - \beta_1 + 3) q^{71} + \beta_{2} q^{72} + (5 \beta_1 - 6) q^{73} + (2 \beta_{2} - \beta_1 + 5) q^{74} - \beta_1 q^{76} + ( - 6 \beta_{2} + 2 \beta_1 - 2) q^{77} + ( - 2 \beta_1 + 2) q^{78} - 6 \beta_{2} q^{79} + ( - 4 \beta_{2} + 2 \beta_1 - 5) q^{81} + (2 \beta_{2} - \beta_1 + 6) q^{82} + ( - 3 \beta_{2} + 4 \beta_1 - 5) q^{83} + (2 \beta_{2} + 4) q^{84} + (2 \beta_{2} + 2 \beta_1 - 2) q^{86} - \beta_1 q^{87} + ( - \beta_{2} + 2 \beta_1 + 1) q^{88} + ( - 2 \beta_{2} - \beta_1 + 4) q^{89} + (2 \beta_{2} - 4 \beta_1) q^{91} + ( - 3 \beta_{2} + \beta_1 - 3) q^{92} + ( - \beta_{2} + 3 \beta_1 - 9) q^{93} + (3 \beta_{2} + 4) q^{94} - \beta_1 q^{96} + ( - \beta_{2} + 3 \beta_1 + 7) q^{97} + (2 \beta_1 + 3) q^{98} + (4 \beta_{2} - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} - q^{3} + 3 q^{4} - q^{6} + 2 q^{7} + 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{2} - q^{3} + 3 q^{4} - q^{6} + 2 q^{7} + 3 q^{8} + 5 q^{11} - q^{12} + 2 q^{13} + 2 q^{14} + 3 q^{16} + 7 q^{17} - q^{19} + 12 q^{21} + 5 q^{22} - 8 q^{23} - q^{24} + 2 q^{26} - q^{27} + 2 q^{28} + 3 q^{29} + 4 q^{31} + 3 q^{32} - 15 q^{33} + 7 q^{34} + 14 q^{37} - q^{38} + 4 q^{39} + 17 q^{41} + 12 q^{42} - 4 q^{43} + 5 q^{44} - 8 q^{46} + 12 q^{47} - q^{48} + 11 q^{49} + 11 q^{51} + 2 q^{52} - 6 q^{53} - q^{54} + 2 q^{56} + 9 q^{57} + 3 q^{58} + 16 q^{59} + 2 q^{61} + 4 q^{62} - 18 q^{63} + 3 q^{64} - 15 q^{66} + q^{67} + 7 q^{68} + 6 q^{69} + 8 q^{71} - 13 q^{73} + 14 q^{74} - q^{76} - 4 q^{77} + 4 q^{78} - 13 q^{81} + 17 q^{82} - 11 q^{83} + 12 q^{84} - 4 q^{86} - q^{87} + 5 q^{88} + 11 q^{89} - 4 q^{91} - 8 q^{92} - 24 q^{93} + 12 q^{94} - q^{96} + 24 q^{97} + 11 q^{98} - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 4x + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) 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
2.34292
0.470683
−1.81361
1.00000 −2.34292 1.00000 0 −2.34292 −3.83221 1.00000 2.48929 0
1.2 1.00000 −0.470683 1.00000 0 −0.470683 3.30777 1.00000 −2.77846 0
1.3 1.00000 1.81361 1.00000 0 1.81361 2.52444 1.00000 0.289169 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(-1\)
\(29\) \(-1\)

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 1450.2.a.s yes 3
5.b even 2 1 1450.2.a.q 3
5.c odd 4 2 1450.2.b.k 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1450.2.a.q 3 5.b even 2 1
1450.2.a.s yes 3 1.a even 1 1 trivial
1450.2.b.k 6 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1450))\):

\( T_{3}^{3} + T_{3}^{2} - 4T_{3} - 2 \) Copy content Toggle raw display
\( T_{7}^{3} - 2T_{7}^{2} - 14T_{7} + 32 \) Copy content Toggle raw display
\( T_{13}^{3} - 2T_{13}^{2} - 6T_{13} + 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{3} \) Copy content Toggle raw display
$3$ \( T^{3} + T^{2} - 4T - 2 \) Copy content Toggle raw display
$5$ \( T^{3} \) Copy content Toggle raw display
$7$ \( T^{3} - 2 T^{2} - 14 T + 32 \) Copy content Toggle raw display
$11$ \( T^{3} - 5 T^{2} - 8 T + 44 \) Copy content Toggle raw display
$13$ \( T^{3} - 2 T^{2} - 6 T + 8 \) Copy content Toggle raw display
$17$ \( T^{3} - 7T^{2} + 4 \) Copy content Toggle raw display
$19$ \( T^{3} + T^{2} - 4T - 2 \) Copy content Toggle raw display
$23$ \( T^{3} + 8 T^{2} - 34 T - 268 \) Copy content Toggle raw display
$29$ \( (T - 1)^{3} \) Copy content Toggle raw display
$31$ \( T^{3} - 4 T^{2} - 79 T + 158 \) Copy content Toggle raw display
$37$ \( T^{3} - 14 T^{2} + 41 T + 58 \) Copy content Toggle raw display
$41$ \( T^{3} - 17 T^{2} + 72 T + 2 \) Copy content Toggle raw display
$43$ \( T^{3} + 4 T^{2} - 56 T - 256 \) Copy content Toggle raw display
$47$ \( T^{3} - 12 T^{2} - 15 T + 242 \) Copy content Toggle raw display
$53$ \( T^{3} + 6 T^{2} - 54 T - 32 \) Copy content Toggle raw display
$59$ \( T^{3} - 16 T^{2} + 61 T - 68 \) Copy content Toggle raw display
$61$ \( T^{3} - 2 T^{2} - 107 T + 146 \) Copy content Toggle raw display
$67$ \( T^{3} - T^{2} - 55 T - 121 \) Copy content Toggle raw display
$71$ \( T^{3} - 8 T^{2} - 34 T + 268 \) Copy content Toggle raw display
$73$ \( T^{3} + 13 T^{2} - 52 T - 314 \) Copy content Toggle raw display
$79$ \( T^{3} - 252T - 432 \) Copy content Toggle raw display
$83$ \( T^{3} + 11 T^{2} - 44 T - 212 \) Copy content Toggle raw display
$89$ \( T^{3} - 11T^{2} + 158 \) Copy content Toggle raw display
$97$ \( T^{3} - 24 T^{2} + 158 T - 164 \) Copy content Toggle raw display
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