Properties

Label 273.2.c.b
Level $273$
Weight $2$
Character orbit 273.c
Analytic conductor $2.180$
Analytic rank $0$
Dimension $6$
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(64,273)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(273, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("273.64");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 273 = 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 273.c (of order \(2\), degree \(1\), 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: \(6\)
Coefficient field: 6.0.350464.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{4} q^{2} + q^{3} + (\beta_{2} - \beta_1) q^{4} + (\beta_{5} + \beta_{4} - \beta_{3}) q^{5} - \beta_{4} q^{6} - \beta_{3} q^{7} + ( - \beta_{5} + \beta_{3}) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{2} + q^{3} + (\beta_{2} - \beta_1) q^{4} + (\beta_{5} + \beta_{4} - \beta_{3}) q^{5} - \beta_{4} q^{6} - \beta_{3} q^{7} + ( - \beta_{5} + \beta_{3}) q^{8} + q^{9} + (\beta_{2} + \beta_1 + 3) q^{10} + (\beta_{5} - \beta_{4} - 3 \beta_{3}) q^{11} + (\beta_{2} - \beta_1) q^{12} + (\beta_{5} - 2 \beta_{4} + \cdots + \beta_{2}) q^{13}+ \cdots + (\beta_{5} - \beta_{4} - 3 \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{3} - 2 q^{4} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 6 q^{3} - 2 q^{4} + 6 q^{9} + 16 q^{10} - 2 q^{12} - 2 q^{13} - 2 q^{14} - 6 q^{16} - 16 q^{22} + 4 q^{23} - 10 q^{25} - 20 q^{26} + 6 q^{27} - 4 q^{29} + 16 q^{30} - 4 q^{35} - 2 q^{36} + 24 q^{38} - 2 q^{39} + 24 q^{40} - 2 q^{42} - 32 q^{43} - 6 q^{48} - 6 q^{49} + 10 q^{52} - 36 q^{53} - 16 q^{55} + 6 q^{56} + 28 q^{61} - 8 q^{62} + 22 q^{64} + 20 q^{65} - 16 q^{66} - 24 q^{68} + 4 q^{69} + 16 q^{74} - 10 q^{75} - 20 q^{77} - 20 q^{78} - 32 q^{79} + 6 q^{81} + 8 q^{82} - 4 q^{87} + 32 q^{88} + 16 q^{90} + 8 q^{91} - 44 q^{92} + 36 q^{94} + 16 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{5} + 8\nu^{4} - 4\nu^{3} - \nu^{2} + 2\nu + 38 ) / 23 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -5\nu^{5} + 17\nu^{4} - 20\nu^{3} - 5\nu^{2} + 10\nu + 29 ) / 23 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 7\nu^{5} - 10\nu^{4} + 5\nu^{3} + 30\nu^{2} + 32\nu - 13 ) / 23 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -11\nu^{5} + 19\nu^{4} - 21\nu^{3} - 11\nu^{2} - 70\nu + 27 ) / 23 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -14\nu^{5} + 20\nu^{4} - 10\nu^{3} - 37\nu^{2} - 64\nu + 26 ) / 23 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} - \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + 2\beta_{3} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{5} - \beta_{4} + 2\beta_{3} - \beta_{2} + 2\beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{2} + 5\beta _1 - 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -8\beta_{5} + 3\beta_{4} - 9\beta_{3} - 3\beta_{2} + 8\beta _1 - 9 \) Copy content Toggle raw display

Character values

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

\(n\) \(92\) \(106\) \(157\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
64.1
−0.854638 0.854638i
0.403032 0.403032i
1.45161 + 1.45161i
1.45161 1.45161i
0.403032 + 0.403032i
−0.854638 + 0.854638i
2.17009i 1.00000 −2.70928 0.630898i 2.17009i 1.00000i 1.53919i 1.00000 1.36910
64.2 1.48119i 1.00000 −0.193937 4.15633i 1.48119i 1.00000i 2.67513i 1.00000 6.15633
64.3 0.311108i 1.00000 1.90321 1.52543i 0.311108i 1.00000i 1.21432i 1.00000 0.474572
64.4 0.311108i 1.00000 1.90321 1.52543i 0.311108i 1.00000i 1.21432i 1.00000 0.474572
64.5 1.48119i 1.00000 −0.193937 4.15633i 1.48119i 1.00000i 2.67513i 1.00000 6.15633
64.6 2.17009i 1.00000 −2.70928 0.630898i 2.17009i 1.00000i 1.53919i 1.00000 1.36910
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 64.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 273.2.c.b 6
3.b odd 2 1 819.2.c.c 6
4.b odd 2 1 4368.2.h.o 6
7.b odd 2 1 1911.2.c.h 6
13.b even 2 1 inner 273.2.c.b 6
13.d odd 4 1 3549.2.a.k 3
13.d odd 4 1 3549.2.a.q 3
39.d odd 2 1 819.2.c.c 6
52.b odd 2 1 4368.2.h.o 6
91.b odd 2 1 1911.2.c.h 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.c.b 6 1.a even 1 1 trivial
273.2.c.b 6 13.b even 2 1 inner
819.2.c.c 6 3.b odd 2 1
819.2.c.c 6 39.d odd 2 1
1911.2.c.h 6 7.b odd 2 1
1911.2.c.h 6 91.b odd 2 1
3549.2.a.k 3 13.d odd 4 1
3549.2.a.q 3 13.d odd 4 1
4368.2.h.o 6 4.b odd 2 1
4368.2.h.o 6 52.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} + 7T_{2}^{4} + 11T_{2}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(273, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + 7 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( (T - 1)^{6} \) Copy content Toggle raw display
$5$ \( T^{6} + 20 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$7$ \( (T^{2} + 1)^{3} \) Copy content Toggle raw display
$11$ \( T^{6} + 44 T^{4} + \cdots + 400 \) Copy content Toggle raw display
$13$ \( T^{6} + 2 T^{5} + \cdots + 2197 \) Copy content Toggle raw display
$17$ \( (T^{3} - 16 T + 16)^{2} \) Copy content Toggle raw display
$19$ \( T^{6} + 64 T^{4} + \cdots + 1024 \) Copy content Toggle raw display
$23$ \( (T^{3} - 2 T^{2} - 20 T + 8)^{2} \) Copy content Toggle raw display
$29$ \( (T^{3} + 2 T^{2} - 52 T - 40)^{2} \) Copy content Toggle raw display
$31$ \( T^{6} + 176 T^{4} + \cdots + 102400 \) Copy content Toggle raw display
$37$ \( T^{6} + 48 T^{4} + \cdots + 1024 \) Copy content Toggle raw display
$41$ \( T^{6} + 132 T^{4} + \cdots + 400 \) Copy content Toggle raw display
$43$ \( (T^{3} + 16 T^{2} + \cdots - 128)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} + 56 T^{4} + \cdots + 2704 \) Copy content Toggle raw display
$53$ \( (T + 6)^{6} \) Copy content Toggle raw display
$59$ \( T^{6} + 40 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$61$ \( (T^{3} - 14 T^{2} + \cdots + 2392)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + 128 T^{4} + \cdots + 1024 \) Copy content Toggle raw display
$71$ \( T^{6} + 332 T^{4} + \cdots + 547600 \) Copy content Toggle raw display
$73$ \( T^{6} + 272 T^{4} + \cdots + 43264 \) Copy content Toggle raw display
$79$ \( (T^{3} + 16 T^{2} + \cdots + 80)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 408 T^{4} + \cdots + 1567504 \) Copy content Toggle raw display
$89$ \( T^{6} + 212 T^{4} + \cdots + 5776 \) Copy content Toggle raw display
$97$ \( T^{6} + 32 T^{4} + \cdots + 256 \) Copy content Toggle raw display
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