Properties

Label 546.2.j.d
Level $546$
Weight $2$
Character orbit 546.j
Analytic conductor $4.360$
Analytic rank $0$
Dimension $8$
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] = [546,2,Mod(289,546)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(546, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("546.289");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 546.j (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: \(4.35983195036\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.447703281.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} - 2x^{6} + 2x^{5} + 3x^{4} + 4x^{3} - 8x^{2} - 8x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} - 2x^{6} + 2x^{5} + 3x^{4} + 4x^{3} - 8x^{2} - 8x + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} - \nu^{6} - 2\nu^{5} + 2\nu^{4} + 3\nu^{3} + 4\nu^{2} - 8\nu - 8 ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{6} + \nu^{5} - 2\nu^{3} - \nu^{2} + 6\nu + 4 ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{7} - \nu^{6} + 2\nu^{4} + \nu^{3} - 6\nu^{2} - 4\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{7} + \nu^{6} + 6\nu^{5} + 2\nu^{4} - 3\nu^{3} - 12\nu^{2} + 4\nu + 32 ) / 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{7} + 2\nu^{5} + \nu^{4} - \nu^{3} - 5\nu^{2} - \nu + 8 ) / 2 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2\nu^{7} + \nu^{6} - 5\nu^{5} - 4\nu^{4} + 4\nu^{3} + 17\nu^{2} + 2\nu - 24 ) / 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} - \beta_{5} - \beta_{4} + \beta_{2} + \beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + 2\beta_{5} + \beta_{4} - \beta_{3} + \beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{6} - \beta_{5} + 2\beta_{3} + 3\beta_{2} + \beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{6} + \beta_{5} - 2\beta_{4} - 2\beta_{3} + \beta_{2} + 2\beta _1 - 5 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2\beta_{7} + 2\beta_{5} + 3\beta_{4} + 4\beta_{3} - 5\beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -\beta_{7} - 4\beta_{6} + 4\beta_{5} - \beta_{3} - 2\beta _1 - 5 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/546\mathbb{Z}\right)^\times\).

\(n\) \(157\) \(365\) \(379\)
\(\chi(n)\) \(\beta_{3}\) \(1\) \(\beta_{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
289.1
1.26359 + 0.635098i
−0.571299 + 1.29368i
−1.38232 0.298668i
1.19003 0.764088i
1.26359 0.635098i
−0.571299 1.29368i
−1.38232 + 0.298668i
1.19003 + 0.764088i
1.00000 0.500000 0.866025i 1.00000 −1.97513 + 3.42102i 0.500000 0.866025i −1.15207 + 2.38175i 1.00000 −0.500000 0.866025i −1.97513 + 3.42102i
289.2 1.00000 0.500000 0.866025i 1.00000 −0.228205 + 0.395262i 0.500000 0.866025i −0.369922 2.61976i 1.00000 −0.500000 0.866025i −0.228205 + 0.395262i
289.3 1.00000 0.500000 0.866025i 1.00000 1.14553 1.98411i 0.500000 0.866025i 2.63641 0.222079i 1.00000 −0.500000 0.866025i 1.14553 1.98411i
289.4 1.00000 0.500000 0.866025i 1.00000 2.05781 3.56422i 0.500000 0.866025i −2.61442 0.405935i 1.00000 −0.500000 0.866025i 2.05781 3.56422i
529.1 1.00000 0.500000 + 0.866025i 1.00000 −1.97513 3.42102i 0.500000 + 0.866025i −1.15207 2.38175i 1.00000 −0.500000 + 0.866025i −1.97513 3.42102i
529.2 1.00000 0.500000 + 0.866025i 1.00000 −0.228205 0.395262i 0.500000 + 0.866025i −0.369922 + 2.61976i 1.00000 −0.500000 + 0.866025i −0.228205 0.395262i
529.3 1.00000 0.500000 + 0.866025i 1.00000 1.14553 + 1.98411i 0.500000 + 0.866025i 2.63641 + 0.222079i 1.00000 −0.500000 + 0.866025i 1.14553 + 1.98411i
529.4 1.00000 0.500000 + 0.866025i 1.00000 2.05781 + 3.56422i 0.500000 + 0.866025i −2.61442 + 0.405935i 1.00000 −0.500000 + 0.866025i 2.05781 + 3.56422i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 289.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
91.h even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 546.2.j.d 8
3.b odd 2 1 1638.2.m.g 8
7.c even 3 1 546.2.k.b yes 8
13.c even 3 1 546.2.k.b yes 8
21.h odd 6 1 1638.2.p.i 8
39.i odd 6 1 1638.2.p.i 8
91.h even 3 1 inner 546.2.j.d 8
273.s odd 6 1 1638.2.m.g 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.j.d 8 1.a even 1 1 trivial
546.2.j.d 8 91.h even 3 1 inner
546.2.k.b yes 8 7.c even 3 1
546.2.k.b yes 8 13.c even 3 1
1638.2.m.g 8 3.b odd 2 1
1638.2.m.g 8 273.s odd 6 1
1638.2.p.i 8 21.h odd 6 1
1638.2.p.i 8 39.i odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} - 2T_{5}^{7} + 21T_{5}^{6} - 26T_{5}^{5} + 332T_{5}^{4} - 442T_{5}^{3} + 1189T_{5}^{2} + 510T_{5} + 289 \) acting on \(S_{2}^{\mathrm{new}}(546, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{8} \) Copy content Toggle raw display
$3$ \( (T^{2} - T + 1)^{4} \) Copy content Toggle raw display
$5$ \( T^{8} - 2 T^{7} + 21 T^{6} - 26 T^{5} + \cdots + 289 \) Copy content Toggle raw display
$7$ \( T^{8} + 3 T^{7} + 2 T^{6} - 21 T^{5} + \cdots + 2401 \) Copy content Toggle raw display
$11$ \( T^{8} + 6 T^{7} + 46 T^{6} + \cdots + 4489 \) Copy content Toggle raw display
$13$ \( T^{8} + 11 T^{7} + 62 T^{6} + \cdots + 28561 \) Copy content Toggle raw display
$17$ \( (T^{4} + 4 T^{3} - 14 T^{2} + 9 T - 1)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} - 6 T^{7} + 31 T^{6} - 62 T^{5} + \cdots + 169 \) Copy content Toggle raw display
$23$ \( (T^{4} - 10 T^{3} - 10 T^{2} + 155 T - 167)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} - 2 T^{7} + 54 T^{6} + \cdots + 4489 \) Copy content Toggle raw display
$31$ \( T^{8} - 6 T^{7} + 84 T^{6} + \cdots + 210681 \) Copy content Toggle raw display
$37$ \( (T^{4} - 28 T^{3} + 260 T^{2} - 823 T + 211)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + 92 T^{6} + 198 T^{5} + \cdots + 1912689 \) Copy content Toggle raw display
$43$ \( T^{8} + 6 T^{7} + 74 T^{6} + \cdots + 4489 \) Copy content Toggle raw display
$47$ \( T^{8} - T^{7} + 57 T^{6} - 172 T^{5} + \cdots + 2601 \) Copy content Toggle raw display
$53$ \( T^{8} - 7 T^{7} + 93 T^{6} + \cdots + 6561 \) Copy content Toggle raw display
$59$ \( (T^{4} + 2 T^{3} - 143 T^{2} - 660 T + 311)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} - 24 T^{7} + 440 T^{6} + \cdots + 17438976 \) Copy content Toggle raw display
$67$ \( T^{8} + 15 T^{7} + 196 T^{6} + \cdots + 27889 \) Copy content Toggle raw display
$71$ \( T^{8} - 6 T^{7} + 131 T^{6} + \cdots + 674041 \) Copy content Toggle raw display
$73$ \( T^{8} - T^{7} + 8 T^{6} - 3 T^{5} + 53 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$79$ \( T^{8} + 12 T^{7} + 221 T^{6} + \cdots + 6985449 \) Copy content Toggle raw display
$83$ \( (T^{4} + 16 T^{3} - 86 T^{2} - 1329 T + 1919)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 25 T^{3} + 170 T^{2} + 180 T + 27)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + T^{7} + 189 T^{6} + \cdots + 45873529 \) Copy content Toggle raw display
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