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Properties

Label 273.2.k.d
Level $273$
Weight $2$
Character orbit 273.k
Analytic conductor $2.180$
Analytic rank $0$
Dimension $6$
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(22,273)]
 
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, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("273.22");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 273 = 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 273.k (of order \(3\), 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: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.771147.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} + 5x^{4} + 6x^{3} + 15x^{2} + 4x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} + 5x^{4} + 6x^{3} + 15x^{2} + 4x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{5} + 5\nu^{4} - 25\nu^{3} + 15\nu^{2} + 4\nu - 20 ) / 79 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -5\nu^{5} + 25\nu^{4} - 46\nu^{3} + 75\nu^{2} + 20\nu + 216 ) / 79 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 20\nu^{5} - 21\nu^{4} + 105\nu^{3} + 95\nu^{2} + 315\nu + 5 ) / 79 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 38\nu^{5} - 32\nu^{4} + 160\nu^{3} + 299\nu^{2} + 480\nu + 128 ) / 79 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} - 2\beta_{4} - 2\beta_{2} + 2\beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - 5\beta_{2} - 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -5\beta_{5} + 11\beta_{4} + 5\beta_{3} + 5\beta_{2} - 15\beta _1 - 5 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -10\beta_{5} + 25\beta_{4} + 41\beta_{2} - 41\beta _1 + 25 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

\(n\) \(92\) \(106\) \(157\)
\(\chi(n)\) \(1\) \(-1 - \beta_{4}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
22.1
−0.136945 + 0.237196i
−0.688601 + 1.19269i
1.32555 − 2.29591i
−0.136945 − 0.237196i
−0.688601 − 1.19269i
1.32555 + 2.29591i
−0.825547 + 1.42989i 0.500000 − 0.866025i −0.363055 − 0.628829i 2.92498 0.825547 + 1.42989i 0.500000 + 0.866025i −2.10331 −0.500000 − 0.866025i −2.41471 + 4.18240i
22.2 0.636945 − 1.10322i 0.500000 − 0.866025i 0.188601 + 0.326667i 1.10331 −0.636945 − 1.10322i 0.500000 + 0.866025i 3.02830 −0.500000 − 0.866025i 0.702750 − 1.21720i
22.3 1.18860 − 2.05872i 0.500000 − 0.866025i −1.82555 − 3.16194i −4.02830 −1.18860 − 2.05872i 0.500000 + 0.866025i −3.92498 −0.500000 − 0.866025i −4.78804 + 8.29313i
211.1 −0.825547 − 1.42989i 0.500000 + 0.866025i −0.363055 + 0.628829i 2.92498 0.825547 − 1.42989i 0.500000 − 0.866025i −2.10331 −0.500000 + 0.866025i −2.41471 − 4.18240i
211.2 0.636945 + 1.10322i 0.500000 + 0.866025i 0.188601 − 0.326667i 1.10331 −0.636945 + 1.10322i 0.500000 − 0.866025i 3.02830 −0.500000 + 0.866025i 0.702750 + 1.21720i
211.3 1.18860 + 2.05872i 0.500000 + 0.866025i −1.82555 + 3.16194i −4.02830 −1.18860 + 2.05872i 0.500000 − 0.866025i −3.92498 −0.500000 + 0.866025i −4.78804 − 8.29313i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 211.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 273.2.k.d ✓ 6
3.b odd 2 1 819.2.o.d 6
13.c even 3 1 inner 273.2.k.d ✓ 6
13.c even 3 1 3549.2.a.h 3
13.e even 6 1 3549.2.a.s 3
39.i odd 6 1 819.2.o.d 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.k.d ✓ 6 1.a even 1 1 trivial
273.2.k.d ✓ 6 13.c even 3 1 inner
819.2.o.d 6 3.b odd 2 1
819.2.o.d 6 39.i odd 6 1
3549.2.a.h 3 13.c even 3 1
3549.2.a.s 3 13.e even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} - 2T_{2}^{5} + 7T_{2}^{4} - 4T_{2}^{3} + 19T_{2}^{2} - 15T_{2} + 25 \) acting on \(S_{2}^{\mathrm{new}}(273, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} - 2 T^{5} + 7 T^{4} - 4 T^{3} + \cdots + 25 \) Copy content Toggle raw display
$3$ \( (T^{2} - T + 1)^{3} \) Copy content Toggle raw display
$5$ \( (T^{3} - 13 T + 13)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} - T + 1)^{3} \) Copy content Toggle raw display
$11$ \( T^{6} - 8 T^{5} + 47 T^{4} - 126 T^{3} + \cdots + 25 \) Copy content Toggle raw display
$13$ \( T^{6} + 65T^{3} + 2197 \) Copy content Toggle raw display
$17$ \( T^{6} + 4 T^{5} + 15 T^{4} + 6 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$19$ \( T^{6} - 7 T^{5} + 50 T^{4} + \cdots + 2209 \) Copy content Toggle raw display
$23$ \( T^{6} + 9 T^{5} + 67 T^{4} + 124 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$29$ \( T^{6} - 7 T^{5} + 63 T^{4} + 108 T^{3} + \cdots + 25 \) Copy content Toggle raw display
$31$ \( (T^{3} + 7 T^{2} - 40 T - 281)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + 13 T^{4} - 26 T^{3} + \cdots + 169 \) Copy content Toggle raw display
$41$ \( T^{6} + 2 T^{5} + 72 T^{4} + \cdots + 40000 \) Copy content Toggle raw display
$43$ \( T^{6} - 19 T^{5} + 245 T^{4} + \cdots + 52441 \) Copy content Toggle raw display
$47$ \( (T^{3} + 17 T^{2} + 14 T - 547)^{2} \) Copy content Toggle raw display
$53$ \( (T^{3} - 13 T^{2} + 39 T + 13)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} - 3 T^{5} + 19 T^{4} - 20 T^{3} + \cdots + 625 \) Copy content Toggle raw display
$61$ \( T^{6} + 13 T^{5} + 130 T^{4} + \cdots + 169 \) Copy content Toggle raw display
$67$ \( T^{6} + 5 T^{5} + 99 T^{4} + \cdots + 156025 \) Copy content Toggle raw display
$71$ \( T^{6} + 8 T^{5} + 177 T^{4} + \cdots + 265225 \) Copy content Toggle raw display
$73$ \( (T^{3} + 2 T^{2} - 81 T + 73)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} + T^{2} - 290 T - 337)^{2} \) Copy content Toggle raw display
$83$ \( (T^{3} - 2 T^{2} - 185 T - 229)^{2} \) Copy content Toggle raw display
$89$ \( T^{6} - 19 T^{5} + 362 T^{4} + \cdots + 1038361 \) Copy content Toggle raw display
$97$ \( T^{6} + 27 T^{5} + 499 T^{4} + \cdots + 358801 \) Copy content Toggle raw display
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