Properties

Label 546.2.e.e
Level $546$
Weight $2$
Character orbit 546.e
Analytic conductor $4.360$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [546,2,Mod(545,546)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(546, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("546.545");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 546.e (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: \(4.35983195036\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.303595776.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 5x^{6} + 16x^{4} + 45x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 5x^{6} + 16x^{4} + 45x^{2} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{6} + 8\nu^{4} + 16\nu^{2} + 45 ) / 24 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{7} + 4\nu^{5} + 2\nu^{3} - 9\nu ) / 54 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 5\nu^{6} + 16\nu^{4} + 80\nu^{2} + 153 ) / 72 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 5\nu^{7} + 16\nu^{5} + 8\nu^{3} + 81\nu ) / 216 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -5\nu^{6} - 16\nu^{4} - 8\nu^{2} - 81 ) / 72 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} + 5\nu^{5} + 16\nu^{3} + 45\nu ) / 27 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} + \beta_{4} - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{7} - 4\beta_{5} - \beta_{3} - 2\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -3\beta_{4} + 5\beta_{2} - 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{7} + 8\beta_{5} + 8\beta_{3} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -16\beta_{6} + 8\beta_{4} - 16\beta_{2} - 5 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 24\beta_{5} - 24\beta_{3} - 13\beta_1 \) Copy content Toggle raw display

Character values

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

\(n\) \(157\) \(365\) \(379\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
545.1
−1.26217 1.18614i
−1.26217 + 1.18614i
−0.396143 1.68614i
−0.396143 + 1.68614i
0.396143 1.68614i
0.396143 + 1.68614i
1.26217 1.18614i
1.26217 + 1.18614i
−1.00000 −1.26217 1.18614i 1.00000 3.37228i 1.26217 + 1.18614i 2.52434 + 0.792287i −1.00000 0.186141 + 2.99422i 3.37228i
545.2 −1.00000 −1.26217 + 1.18614i 1.00000 3.37228i 1.26217 1.18614i 2.52434 0.792287i −1.00000 0.186141 2.99422i 3.37228i
545.3 −1.00000 −0.396143 1.68614i 1.00000 2.37228i 0.396143 + 1.68614i 0.792287 2.52434i −1.00000 −2.68614 + 1.33591i 2.37228i
545.4 −1.00000 −0.396143 + 1.68614i 1.00000 2.37228i 0.396143 1.68614i 0.792287 + 2.52434i −1.00000 −2.68614 1.33591i 2.37228i
545.5 −1.00000 0.396143 1.68614i 1.00000 2.37228i −0.396143 + 1.68614i −0.792287 + 2.52434i −1.00000 −2.68614 1.33591i 2.37228i
545.6 −1.00000 0.396143 + 1.68614i 1.00000 2.37228i −0.396143 1.68614i −0.792287 2.52434i −1.00000 −2.68614 + 1.33591i 2.37228i
545.7 −1.00000 1.26217 1.18614i 1.00000 3.37228i −1.26217 + 1.18614i −2.52434 0.792287i −1.00000 0.186141 2.99422i 3.37228i
545.8 −1.00000 1.26217 + 1.18614i 1.00000 3.37228i −1.26217 1.18614i −2.52434 + 0.792287i −1.00000 0.186141 + 2.99422i 3.37228i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 545.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
39.d odd 2 1 inner
273.g even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 546.2.e.e 8
3.b odd 2 1 546.2.e.g yes 8
7.b odd 2 1 inner 546.2.e.e 8
13.b even 2 1 546.2.e.g yes 8
21.c even 2 1 546.2.e.g yes 8
39.d odd 2 1 inner 546.2.e.e 8
91.b odd 2 1 546.2.e.g yes 8
273.g even 2 1 inner 546.2.e.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.e.e 8 1.a even 1 1 trivial
546.2.e.e 8 7.b odd 2 1 inner
546.2.e.e 8 39.d odd 2 1 inner
546.2.e.e 8 273.g even 2 1 inner
546.2.e.g yes 8 3.b odd 2 1
546.2.e.g yes 8 13.b even 2 1
546.2.e.g yes 8 21.c even 2 1
546.2.e.g yes 8 91.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(546, [\chi])\):

\( T_{5}^{4} + 17T_{5}^{2} + 64 \) Copy content Toggle raw display
\( T_{17}^{4} - 7T_{17}^{2} + 4 \) Copy content Toggle raw display
\( T_{19}^{4} - 63T_{19}^{2} + 324 \) Copy content Toggle raw display
\( T_{71} - 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + 5 T^{6} + 16 T^{4} + 45 T^{2} + \cdots + 81 \) Copy content Toggle raw display
$5$ \( (T^{4} + 17 T^{2} + 64)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} - 34T^{4} + 2401 \) Copy content Toggle raw display
$11$ \( (T^{2} - 33)^{4} \) Copy content Toggle raw display
$13$ \( (T^{4} - 22 T^{2} + 169)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 7 T^{2} + 4)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 63 T^{2} + 324)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 46 T^{2} + 1)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 63 T^{2} + 324)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 51 T^{2} + 576)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 94 T^{2} + 1681)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 77 T^{2} + 484)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 11 T + 22)^{4} \) Copy content Toggle raw display
$47$ \( (T^{4} + 129 T^{2} + 2304)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 112 T^{2} + 1024)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 84 T^{2} + 576)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 74 T^{2} + 841)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 43 T^{2} + 256)^{2} \) Copy content Toggle raw display
$71$ \( (T - 2)^{8} \) Copy content Toggle raw display
$73$ \( (T^{4} - 118 T^{2} + 1369)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 13 T + 34)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + 68 T^{2} + 1024)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 100)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} - 139 T^{2} + 4624)^{2} \) Copy content Toggle raw display
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