Properties

Label 546.2.o.d
Level $546$
Weight $2$
Character orbit 546.o
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(265,546)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(546, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 2, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("546.265");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 546.o (of order \(4\), 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 \(\Q(i)\)
Coefficient field: 8.0.7442857984.4
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 26x^{6} + 205x^{4} + 540x^{2} + 324 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 26x^{6} + 205x^{4} + 540x^{2} + 324 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2\nu^{7} + 3\nu^{6} + 52\nu^{5} + 60\nu^{4} + 374\nu^{3} + 219\nu^{2} + 612\nu - 162 ) / 432 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{7} + 3\nu^{6} - 8\nu^{5} + 24\nu^{4} + 155\nu^{3} - 249\nu^{2} + 774\nu - 810 ) / 432 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2\nu^{7} - 3\nu^{6} + 52\nu^{5} - 60\nu^{4} + 374\nu^{3} - 219\nu^{2} + 612\nu + 162 ) / 432 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{7} - 3\nu^{6} - 8\nu^{5} - 24\nu^{4} + 155\nu^{3} + 249\nu^{2} + 1206\nu + 810 ) / 432 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -5\nu^{7} - 112\nu^{5} - 665\nu^{3} - 954\nu ) / 432 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{6} + 20\nu^{4} + 97\nu^{2} + 114 ) / 24 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} + 3\beta_{4} - 3\beta_{2} - 7 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -6\beta_{6} + 3\beta_{5} - 6\beta_{4} + 3\beta_{3} - 6\beta_{2} - 10\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -13\beta_{7} + 6\beta_{5} - 45\beta_{4} - 6\beta_{3} + 45\beta_{2} - 6\beta _1 + 73 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 114\beta_{6} - 45\beta_{5} + 120\beta_{4} - 45\beta_{3} + 120\beta_{2} + 118\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 187\beta_{7} - 120\beta_{5} + 609\beta_{4} + 120\beta_{3} - 609\beta_{2} + 120\beta _1 - 895 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -1842\beta_{6} + 609\beta_{5} - 1890\beta_{4} + 609\beta_{3} - 1890\beta_{2} - 1504\beta_1 \) Copy content Toggle raw display

Character values

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

\(n\) \(157\) \(365\) \(379\)
\(\chi(n)\) \(-1\) \(1\) \(\beta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
265.1
3.73923i
2.73923i
1.91681i
0.916813i
3.73923i
2.73923i
1.91681i
0.916813i
−0.707107 + 0.707107i 1.00000i 1.00000i −0.0951965 0.0951965i 0.707107 + 0.707107i −0.0951965 2.64404i 0.707107 + 0.707107i −1.00000 0.134628
265.2 −0.707107 + 0.707107i 1.00000i 1.00000i 1.80230 + 1.80230i 0.707107 + 0.707107i 1.80230 + 1.93693i 0.707107 + 0.707107i −1.00000 −2.54884
265.3 0.707107 0.707107i 1.00000i 1.00000i −2.27220 2.27220i −0.707107 0.707107i −2.27220 + 1.35539i −0.707107 0.707107i −1.00000 −3.21338
265.4 0.707107 0.707107i 1.00000i 1.00000i 2.56510 + 2.56510i −0.707107 0.707107i 2.56510 0.648285i −0.707107 0.707107i −1.00000 3.62760
307.1 −0.707107 0.707107i 1.00000i 1.00000i −0.0951965 + 0.0951965i 0.707107 0.707107i −0.0951965 + 2.64404i 0.707107 0.707107i −1.00000 0.134628
307.2 −0.707107 0.707107i 1.00000i 1.00000i 1.80230 1.80230i 0.707107 0.707107i 1.80230 1.93693i 0.707107 0.707107i −1.00000 −2.54884
307.3 0.707107 + 0.707107i 1.00000i 1.00000i −2.27220 + 2.27220i −0.707107 + 0.707107i −2.27220 1.35539i −0.707107 + 0.707107i −1.00000 −3.21338
307.4 0.707107 + 0.707107i 1.00000i 1.00000i 2.56510 2.56510i −0.707107 + 0.707107i 2.56510 + 0.648285i −0.707107 + 0.707107i −1.00000 3.62760
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 265.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
91.i even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 546.2.o.d yes 8
3.b odd 2 1 1638.2.x.b 8
7.b odd 2 1 546.2.o.a 8
13.d odd 4 1 546.2.o.a 8
21.c even 2 1 1638.2.x.d 8
39.f even 4 1 1638.2.x.d 8
91.i even 4 1 inner 546.2.o.d yes 8
273.o odd 4 1 1638.2.x.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.o.a 8 7.b odd 2 1
546.2.o.a 8 13.d odd 4 1
546.2.o.d yes 8 1.a even 1 1 trivial
546.2.o.d yes 8 91.i even 4 1 inner
1638.2.x.b 8 3.b odd 2 1
1638.2.x.b 8 273.o odd 4 1
1638.2.x.d 8 21.c even 2 1
1638.2.x.d 8 39.f even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} - 4T_{5}^{7} + 8T_{5}^{6} + 4T_{5}^{5} + 113T_{5}^{4} - 424T_{5}^{3} + 800T_{5}^{2} + 160T_{5} + 16 \) acting on \(S_{2}^{\mathrm{new}}(546, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$5$ \( T^{8} - 4 T^{7} + 8 T^{6} + 4 T^{5} + \cdots + 16 \) Copy content Toggle raw display
$7$ \( T^{8} - 4 T^{7} + 6 T^{6} - 4 T^{5} + \cdots + 2401 \) Copy content Toggle raw display
$11$ \( T^{8} + 8 T^{7} + 32 T^{6} + 16 T^{5} + \cdots + 64 \) Copy content Toggle raw display
$13$ \( (T^{2} + 4 T + 13)^{4} \) Copy content Toggle raw display
$17$ \( (T^{4} - 6 T^{3} - 35 T^{2} + 204 T - 92)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + 4 T^{7} + 8 T^{6} + \cdots + 777924 \) Copy content Toggle raw display
$23$ \( T^{8} + 102 T^{6} + 3265 T^{4} + \cdots + 135424 \) Copy content Toggle raw display
$29$ \( (T^{4} + 6 T^{3} - 67 T^{2} - 260 T + 356)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + 20 T^{7} + 200 T^{6} + \cdots + 5184 \) Copy content Toggle raw display
$37$ \( T^{8} + 8 T^{7} + 32 T^{6} + \cdots + 777924 \) Copy content Toggle raw display
$41$ \( T^{8} + 16 T^{7} + 128 T^{6} + \cdots + 3936256 \) Copy content Toggle raw display
$43$ \( T^{8} + 254 T^{6} + \cdots + 14961424 \) Copy content Toggle raw display
$47$ \( T^{8} - 16 T^{7} + 128 T^{6} + \cdots + 65536 \) Copy content Toggle raw display
$53$ \( (T^{4} + 12 T^{3} - 4 T^{2} - 128 T + 128)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + 28 T^{7} + 392 T^{6} + \cdots + 73984 \) Copy content Toggle raw display
$61$ \( T^{8} + 106 T^{6} + 3489 T^{4} + \cdots + 8464 \) Copy content Toggle raw display
$67$ \( T^{8} - 8 T^{7} + 32 T^{6} + 800 T^{5} + \cdots + 16 \) Copy content Toggle raw display
$71$ \( T^{8} - 52 T^{7} + 1352 T^{6} + \cdots + 1327104 \) Copy content Toggle raw display
$73$ \( T^{8} - 8 T^{7} + 32 T^{6} + \cdots + 777924 \) Copy content Toggle raw display
$79$ \( (T^{4} + 24 T^{3} - 2 T^{2} - 3480 T - 20744)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 32 T^{7} + 512 T^{6} + \cdots + 541696 \) Copy content Toggle raw display
$89$ \( T^{8} + 4 T^{7} + 8 T^{6} + \cdots + 45050944 \) Copy content Toggle raw display
$97$ \( T^{8} - 36 T^{7} + \cdots + 485409024 \) Copy content Toggle raw display
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