Properties

Label 273.2.bl.b
Level $273$
Weight $2$
Character orbit 273.bl
Analytic conductor $2.180$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [273,2,Mod(88,273)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(273, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("273.88");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 273 = 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 273.bl (of order \(6\), degree \(2\), minimal)

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: no
Analytic conductor: \(2.17991597518\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-7})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - x^{2} - 2x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{3} + 1) q^{2} + q^{3} + ( - 2 \beta_{3} + \beta_{2} + \beta_1) q^{4} + ( - \beta_{2} - 1) q^{5} + ( - \beta_{3} + 1) q^{6} + ( - 2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 1) q^{7} + ( - 2 \beta_{2} + 1) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} + 1) q^{2} + q^{3} + ( - 2 \beta_{3} + \beta_{2} + \beta_1) q^{4} + ( - \beta_{2} - 1) q^{5} + ( - \beta_{3} + 1) q^{6} + ( - 2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 1) q^{7} + ( - 2 \beta_{2} + 1) q^{8} + q^{9} + (\beta_{3} + \beta_1 - 2) q^{10} + (4 \beta_{2} - 2) q^{11} + ( - 2 \beta_{3} + \beta_{2} + \beta_1) q^{12} + (4 \beta_{2} - 3) q^{13} + ( - \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 1) q^{14} + ( - \beta_{2} - 1) q^{15} + (\beta_{3} - 2 \beta_{2} - 2 \beta_1 + 1) q^{16} + (\beta_{2} - 1) q^{17} + ( - \beta_{3} + 1) q^{18} + ( - 4 \beta_{3} + 4 \beta_{2} + 4 \beta_1 - 2) q^{19} + (3 \beta_{3} - \beta_{2} - 1) q^{20} + ( - 2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 1) q^{21} + (2 \beta_{3} - 4 \beta_1 + 2) q^{22} + (2 \beta_{3} + \beta_{2} - 4 \beta_1 + 2) q^{23} + ( - 2 \beta_{2} + 1) q^{24} - 2 \beta_{2} q^{25} + (3 \beta_{3} - 4 \beta_1 + 1) q^{26} + q^{27} + ( - 3 \beta_{2} + \beta_1 - 4) q^{28} + ( - 7 \beta_{2} + 7) q^{29} + (\beta_{3} + \beta_1 - 2) q^{30} + (2 \beta_{3} + \beta_{2} - 4) q^{31} + (3 \beta_{2} - \beta_1 + 4) q^{32} + (4 \beta_{2} - 2) q^{33} + (\beta_{3} - \beta_1) q^{34} + (4 \beta_{3} - 3 \beta_{2} - 2 \beta_1 + 1) q^{35} + ( - 2 \beta_{3} + \beta_{2} + \beta_1) q^{36} + (4 \beta_{3} - \beta_{2} - 2) q^{37} + ( - 2 \beta_{3} - 4 \beta_{2} + 4 \beta_1 - 2) q^{38} + (4 \beta_{2} - 3) q^{39} + (3 \beta_{2} - 3) q^{40} + (3 \beta_{2} + 4 \beta_1 - 1) q^{41} + ( - \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 1) q^{42} + ( - 2 \beta_{3} + 3 \beta_{2} + 4 \beta_1 - 2) q^{43} + ( - 2 \beta_{2} - 6 \beta_1 + 4) q^{44} + ( - \beta_{2} - 1) q^{45} + (2 \beta_{2} - 7 \beta_1 + 9) q^{46} + (3 \beta_{2} + 2 \beta_1 + 1) q^{47} + (\beta_{3} - 2 \beta_{2} - 2 \beta_1 + 1) q^{48} - 7 q^{49} + (2 \beta_1 - 2) q^{50} + (\beta_{2} - 1) q^{51} + (2 \beta_{3} - 3 \beta_{2} - 7 \beta_1 + 4) q^{52} + (4 \beta_{3} - 9 \beta_{2} - 8 \beta_1 + 4) q^{53} + ( - \beta_{3} + 1) q^{54} + ( - 6 \beta_{2} + 6) q^{55} + (2 \beta_{3} + 2 \beta_1 - 1) q^{56} + ( - 4 \beta_{3} + 4 \beta_{2} + 4 \beta_1 - 2) q^{57} + ( - 7 \beta_{3} + 7 \beta_1) q^{58} + ( - 5 \beta_{2} - 2 \beta_1 - 3) q^{59} + (3 \beta_{3} - \beta_{2} - 1) q^{60} + (4 \beta_{3} + 4 \beta_1 + 2) q^{61} + (6 \beta_{3} + 2 \beta_{2} - 3 \beta_1 - 5) q^{62} + ( - 2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 1) q^{63} + ( - 2 \beta_{3} - 2 \beta_1 + 9) q^{64} + ( - 5 \beta_{2} + 7) q^{65} + (2 \beta_{3} - 4 \beta_1 + 2) q^{66} + ( - 8 \beta_{3} + 4 \beta_{2} + 8 \beta_1 - 2) q^{67} + (\beta_{3} - \beta_{2} - 2 \beta_1 + 1) q^{68} + (2 \beta_{3} + \beta_{2} - 4 \beta_1 + 2) q^{69} + (3 \beta_{3} + 4 \beta_{2} - 3 \beta_1 - 2) q^{70} + (6 \beta_{3} - 5 \beta_{2} + 4) q^{71} + ( - 2 \beta_{2} + 1) q^{72} + ( - 5 \beta_{2} + 10) q^{73} + (6 \beta_{3} + 4 \beta_{2} - 3 \beta_1 - 7) q^{74} - 2 \beta_{2} q^{75} + ( - 6 \beta_{2} + 2 \beta_1 - 8) q^{76} + ( - 4 \beta_{3} - 4 \beta_1 + 2) q^{77} + (3 \beta_{3} - 4 \beta_1 + 1) q^{78} + (4 \beta_{3} - 9 \beta_{2} - 2 \beta_1 + 7) q^{79} + ( - 3 \beta_{3} + 4 \beta_{2} + 3 \beta_1 - 2) q^{80} + q^{81} + (\beta_{3} + \beta_1 - 6) q^{82} + ( - 4 \beta_{2} + 2) q^{83} + ( - 3 \beta_{2} + \beta_1 - 4) q^{84} + ( - \beta_{2} + 2) q^{85} + ( - 2 \beta_{2} + 3 \beta_1 - 5) q^{86} + ( - 7 \beta_{2} + 7) q^{87} + 6 q^{88} + (9 \beta_{2} - 18) q^{89} + (\beta_{3} + \beta_1 - 2) q^{90} + ( - 2 \beta_{3} - 2 \beta_{2} - 6 \beta_1 + 3) q^{91} + ( - 5 \beta_{3} - 5 \beta_1 + 15) q^{92} + (2 \beta_{3} + \beta_{2} - 4) q^{93} + ( - \beta_{3} - \beta_1) q^{94} + (8 \beta_{3} - 6 \beta_{2} - 4 \beta_1 + 2) q^{95} + (3 \beta_{2} - \beta_1 + 4) q^{96} + ( - 4 \beta_{3} - \beta_{2} + 6) q^{97} + (7 \beta_{3} - 7) q^{98} + (4 \beta_{2} - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 3 q^{2} + 4 q^{3} + q^{4} - 6 q^{5} + 3 q^{6} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 3 q^{2} + 4 q^{3} + q^{4} - 6 q^{5} + 3 q^{6} + 4 q^{9} - 6 q^{10} + q^{12} - 4 q^{13} - 7 q^{14} - 6 q^{15} - q^{16} - 2 q^{17} + 3 q^{18} - 3 q^{20} + 6 q^{22} + 8 q^{23} - 4 q^{25} + 3 q^{26} + 4 q^{27} - 21 q^{28} + 14 q^{29} - 6 q^{30} - 12 q^{31} + 21 q^{32} + q^{36} - 6 q^{37} - 14 q^{38} - 4 q^{39} - 6 q^{40} + 6 q^{41} - 7 q^{42} + 6 q^{44} - 6 q^{45} + 33 q^{46} + 12 q^{47} - q^{48} - 28 q^{49} - 6 q^{50} - 2 q^{51} + 5 q^{52} - 6 q^{53} + 3 q^{54} + 12 q^{55} - 24 q^{59} - 3 q^{60} + 16 q^{61} - 13 q^{62} + 32 q^{64} + 18 q^{65} + 6 q^{66} + q^{68} + 8 q^{69} + 12 q^{71} + 30 q^{73} - 17 q^{74} - 4 q^{75} - 42 q^{76} + 3 q^{78} + 12 q^{79} + 4 q^{81} - 22 q^{82} - 21 q^{84} + 6 q^{85} - 21 q^{86} + 14 q^{87} + 24 q^{88} - 54 q^{89} - 6 q^{90} + 50 q^{92} - 12 q^{93} - 2 q^{94} + 21 q^{96} + 18 q^{97} - 21 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + \nu^{2} - \nu - 2 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} + \nu^{2} + \nu + 2 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{3} + \beta_{2} + \beta _1 + 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/273\mathbb{Z}\right)^\times\).

\(n\) \(92\) \(106\) \(157\)
\(\chi(n)\) \(1\) \(\beta_{2}\) \(-1 + \beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
88.1
1.39564 0.228425i
−0.895644 + 1.09445i
1.39564 + 0.228425i
−0.895644 1.09445i
−0.395644 0.228425i 1.00000 −0.895644 1.55130i −1.50000 + 0.866025i −0.395644 0.228425i 2.64575i 1.73205i 1.00000 0.791288
88.2 1.89564 + 1.09445i 1.00000 1.39564 + 2.41733i −1.50000 + 0.866025i 1.89564 + 1.09445i 2.64575i 1.73205i 1.00000 −3.79129
121.1 −0.395644 + 0.228425i 1.00000 −0.895644 + 1.55130i −1.50000 0.866025i −0.395644 + 0.228425i 2.64575i 1.73205i 1.00000 0.791288
121.2 1.89564 1.09445i 1.00000 1.39564 2.41733i −1.50000 0.866025i 1.89564 1.09445i 2.64575i 1.73205i 1.00000 −3.79129
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
91.u even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 273.2.bl.b yes 4
3.b odd 2 1 819.2.do.d 4
7.c even 3 1 273.2.t.b 4
13.e even 6 1 273.2.t.b 4
21.h odd 6 1 819.2.bm.d 4
39.h odd 6 1 819.2.bm.d 4
91.u even 6 1 inner 273.2.bl.b yes 4
273.x odd 6 1 819.2.do.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.t.b 4 7.c even 3 1
273.2.t.b 4 13.e even 6 1
273.2.bl.b yes 4 1.a even 1 1 trivial
273.2.bl.b yes 4 91.u even 6 1 inner
819.2.bm.d 4 21.h odd 6 1
819.2.bm.d 4 39.h odd 6 1
819.2.do.d 4 3.b odd 2 1
819.2.do.d 4 273.x odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 3T_{2}^{3} + 2T_{2}^{2} + 3T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(273, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 3 T^{3} + 2 T^{2} + 3 T + 1 \) Copy content Toggle raw display
$3$ \( (T - 1)^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} + 3 T + 3)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 7)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 2 T + 13)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 28)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} - 8 T^{3} + 69 T^{2} + 40 T + 25 \) Copy content Toggle raw display
$29$ \( (T^{2} - 7 T + 49)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 12 T^{3} + 53 T^{2} + 60 T + 25 \) Copy content Toggle raw display
$37$ \( T^{4} + 6 T^{3} - 13 T^{2} - 150 T + 625 \) Copy content Toggle raw display
$41$ \( T^{4} - 6 T^{3} - 13 T^{2} + 150 T + 625 \) Copy content Toggle raw display
$43$ \( T^{4} + 21T^{2} + 441 \) Copy content Toggle raw display
$47$ \( T^{4} - 12 T^{3} + 53 T^{2} - 60 T + 25 \) Copy content Toggle raw display
$53$ \( T^{4} + 6 T^{3} + 111 T^{2} + \cdots + 5625 \) Copy content Toggle raw display
$59$ \( T^{4} + 24 T^{3} + 233 T^{2} + \cdots + 1681 \) Copy content Toggle raw display
$61$ \( (T^{2} - 8 T - 68)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 248 T^{2} + 10000 \) Copy content Toggle raw display
$71$ \( T^{4} - 12 T^{3} - 3 T^{2} + \cdots + 2601 \) Copy content Toggle raw display
$73$ \( (T^{2} - 15 T + 75)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} - 12 T^{3} + 129 T^{2} + \cdots + 225 \) Copy content Toggle raw display
$83$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 27 T + 243)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} - 18 T^{3} + 107 T^{2} + 18 T + 1 \) Copy content Toggle raw display
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