Properties

Label 546.4.a.n
Level $546$
Weight $4$
Character orbit 546.a
Self dual yes
Analytic conductor $32.215$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [546,4,Mod(1,546)]
 
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([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("546.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 546.a (trivial)

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: yes
Analytic conductor: \(32.2150428631\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.360321.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 153x - 224 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 q^{2} + 3 q^{3} + 4 q^{4} + ( - \beta_{2} + 4) q^{5} - 6 q^{6} + 7 q^{7} - 8 q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{2} + 3 q^{3} + 4 q^{4} + ( - \beta_{2} + 4) q^{5} - 6 q^{6} + 7 q^{7} - 8 q^{8} + 9 q^{9} + (2 \beta_{2} - 8) q^{10} + (\beta_{2} - 2 \beta_1 + 6) q^{11} + 12 q^{12} - 13 q^{13} - 14 q^{14} + ( - 3 \beta_{2} + 12) q^{15} + 16 q^{16} + (\beta_{2} + 3 \beta_1 + 30) q^{17} - 18 q^{18} + ( - 5 \beta_{2} - 3 \beta_1 + 28) q^{19} + ( - 4 \beta_{2} + 16) q^{20} + 21 q^{21} + ( - 2 \beta_{2} + 4 \beta_1 - 12) q^{22} + ( - \beta_{2} + 5 \beta_1 + 96) q^{23} - 24 q^{24} + ( - 17 \beta_{2} + \beta_1 + 23) q^{25} + 26 q^{26} + 27 q^{27} + 28 q^{28} + (13 \beta_{2} - 3 \beta_1 + 46) q^{29} + (6 \beta_{2} - 24) q^{30} + (4 \beta_{2} + 2 \beta_1 - 68) q^{31} - 32 q^{32} + (3 \beta_{2} - 6 \beta_1 + 18) q^{33} + ( - 2 \beta_{2} - 6 \beta_1 - 60) q^{34} + ( - 7 \beta_{2} + 28) q^{35} + 36 q^{36} + ( - 15 \beta_{2} - 5 \beta_1 + 66) q^{37} + (10 \beta_{2} + 6 \beta_1 - 56) q^{38} - 39 q^{39} + (8 \beta_{2} - 32) q^{40} + (22 \beta_{2} - 5 \beta_1 + 184) q^{41} - 42 q^{42} + (27 \beta_{2} + 7 \beta_1 - 16) q^{43} + (4 \beta_{2} - 8 \beta_1 + 24) q^{44} + ( - 9 \beta_{2} + 36) q^{45} + (2 \beta_{2} - 10 \beta_1 - 192) q^{46} + ( - 16 \beta_{2} + 19 \beta_1 + 94) q^{47} + 48 q^{48} + 49 q^{49} + (34 \beta_{2} - 2 \beta_1 - 46) q^{50} + (3 \beta_{2} + 9 \beta_1 + 90) q^{51} - 52 q^{52} + (4 \beta_{2} - 10 \beta_1 + 238) q^{53} - 54 q^{54} + (3 \beta_{2} + 7 \beta_1 - 28) q^{55} - 56 q^{56} + ( - 15 \beta_{2} - 9 \beta_1 + 84) q^{57} + ( - 26 \beta_{2} + 6 \beta_1 - 92) q^{58} + ( - 20 \beta_{2} - \beta_1 - 90) q^{59} + ( - 12 \beta_{2} + 48) q^{60} + (25 \beta_{2} - 15 \beta_1 - 162) q^{61} + ( - 8 \beta_{2} - 4 \beta_1 + 136) q^{62} + 63 q^{63} + 64 q^{64} + (13 \beta_{2} - 52) q^{65} + ( - 6 \beta_{2} + 12 \beta_1 - 36) q^{66} + (30 \beta_{2} + 4 \beta_1 - 164) q^{67} + (4 \beta_{2} + 12 \beta_1 + 120) q^{68} + ( - 3 \beta_{2} + 15 \beta_1 + 288) q^{69} + (14 \beta_{2} - 56) q^{70} + (24 \beta_{2} - 39 \beta_1 + 102) q^{71} - 72 q^{72} + (11 \beta_{2} + 31 \beta_1 - 230) q^{73} + (30 \beta_{2} + 10 \beta_1 - 132) q^{74} + ( - 51 \beta_{2} + 3 \beta_1 + 69) q^{75} + ( - 20 \beta_{2} - 12 \beta_1 + 112) q^{76} + (7 \beta_{2} - 14 \beta_1 + 42) q^{77} + 78 q^{78} + ( - 38 \beta_{2} + 18 \beta_1 + 408) q^{79} + ( - 16 \beta_{2} + 64) q^{80} + 81 q^{81} + ( - 44 \beta_{2} + 10 \beta_1 - 368) q^{82} + ( - 14 \beta_{2} + 3 \beta_1 + 562) q^{83} + 84 q^{84} + ( - 11 \beta_{2} - 13 \beta_1 - 132) q^{85} + ( - 54 \beta_{2} - 14 \beta_1 + 32) q^{86} + (39 \beta_{2} - 9 \beta_1 + 138) q^{87} + ( - 8 \beta_{2} + 16 \beta_1 - 48) q^{88} + ( - 32 \beta_{2} + 11 \beta_1 + 532) q^{89} + (18 \beta_{2} - 72) q^{90} - 91 q^{91} + ( - 4 \beta_{2} + 20 \beta_1 + 384) q^{92} + (12 \beta_{2} + 6 \beta_1 - 204) q^{93} + (32 \beta_{2} - 38 \beta_1 - 188) q^{94} + ( - 99 \beta_{2} + 17 \beta_1 + 892) q^{95} - 96 q^{96} + (4 \beta_{2} + 40 \beta_1 + 510) q^{97} - 98 q^{98} + (9 \beta_{2} - 18 \beta_1 + 54) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 6 q^{2} + 9 q^{3} + 12 q^{4} + 13 q^{5} - 18 q^{6} + 21 q^{7} - 24 q^{8} + 27 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 6 q^{2} + 9 q^{3} + 12 q^{4} + 13 q^{5} - 18 q^{6} + 21 q^{7} - 24 q^{8} + 27 q^{9} - 26 q^{10} + 17 q^{11} + 36 q^{12} - 39 q^{13} - 42 q^{14} + 39 q^{15} + 48 q^{16} + 89 q^{17} - 54 q^{18} + 89 q^{19} + 52 q^{20} + 63 q^{21} - 34 q^{22} + 289 q^{23} - 72 q^{24} + 86 q^{25} + 78 q^{26} + 81 q^{27} + 84 q^{28} + 125 q^{29} - 78 q^{30} - 208 q^{31} - 96 q^{32} + 51 q^{33} - 178 q^{34} + 91 q^{35} + 108 q^{36} + 213 q^{37} - 178 q^{38} - 117 q^{39} - 104 q^{40} + 530 q^{41} - 126 q^{42} - 75 q^{43} + 68 q^{44} + 117 q^{45} - 578 q^{46} + 298 q^{47} + 144 q^{48} + 147 q^{49} - 172 q^{50} + 267 q^{51} - 156 q^{52} + 710 q^{53} - 162 q^{54} - 87 q^{55} - 168 q^{56} + 267 q^{57} - 250 q^{58} - 250 q^{59} + 156 q^{60} - 511 q^{61} + 416 q^{62} + 189 q^{63} + 192 q^{64} - 169 q^{65} - 102 q^{66} - 522 q^{67} + 356 q^{68} + 867 q^{69} - 182 q^{70} + 282 q^{71} - 216 q^{72} - 701 q^{73} - 426 q^{74} + 258 q^{75} + 356 q^{76} + 119 q^{77} + 234 q^{78} + 1262 q^{79} + 208 q^{80} + 243 q^{81} - 1060 q^{82} + 1700 q^{83} + 252 q^{84} - 385 q^{85} + 150 q^{86} + 375 q^{87} - 136 q^{88} + 1628 q^{89} - 234 q^{90} - 273 q^{91} + 1156 q^{92} - 624 q^{93} - 596 q^{94} + 2775 q^{95} - 288 q^{96} + 1526 q^{97} - 294 q^{98} + 153 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - 153x - 224 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} - \nu - 104 ) / 6 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 12\beta_{2} + \beta _1 + 208 ) / 2 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
13.0450
−11.5595
−1.48548
−2.00000 3.00000 4.00000 −4.85440 −6.00000 7.00000 −8.00000 9.00000 9.70880
1.2 −2.00000 3.00000 4.00000 −2.86358 −6.00000 7.00000 −8.00000 9.00000 5.72716
1.3 −2.00000 3.00000 4.00000 20.7180 −6.00000 7.00000 −8.00000 9.00000 −41.4360
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(7\) \(-1\)
\(13\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 546.4.a.n 3
3.b odd 2 1 1638.4.a.ba 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.4.a.n 3 1.a even 1 1 trivial
1638.4.a.ba 3 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{3} - 13T_{5}^{2} - 146T_{5} - 288 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(546))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 2)^{3} \) Copy content Toggle raw display
$3$ \( (T - 3)^{3} \) Copy content Toggle raw display
$5$ \( T^{3} - 13 T^{2} - 146 T - 288 \) Copy content Toggle raw display
$7$ \( (T - 7)^{3} \) Copy content Toggle raw display
$11$ \( T^{3} - 17 T^{2} - 2310 T - 10536 \) Copy content Toggle raw display
$13$ \( (T + 13)^{3} \) Copy content Toggle raw display
$17$ \( T^{3} - 89 T^{2} - 3436 T + 16628 \) Copy content Toggle raw display
$19$ \( T^{3} - 89 T^{2} - 9756 T + 718176 \) Copy content Toggle raw display
$23$ \( T^{3} - 289 T^{2} + 12948 T + 563424 \) Copy content Toggle raw display
$29$ \( T^{3} - 125 T^{2} - 29736 T + 2752572 \) Copy content Toggle raw display
$31$ \( T^{3} + 208 T^{2} + 7760 T - 239488 \) Copy content Toggle raw display
$37$ \( T^{3} - 213 T^{2} - 54852 T + 5144564 \) Copy content Toggle raw display
$41$ \( T^{3} - 530 T^{2} + \cdots + 18905088 \) Copy content Toggle raw display
$43$ \( T^{3} + 75 T^{2} - 198672 T + 1482192 \) Copy content Toggle raw display
$47$ \( T^{3} - 298 T^{2} + \cdots + 62195328 \) Copy content Toggle raw display
$53$ \( T^{3} - 710 T^{2} + 108476 T - 1248616 \) Copy content Toggle raw display
$59$ \( T^{3} + 250 T^{2} + \cdots - 14803488 \) Copy content Toggle raw display
$61$ \( T^{3} + 511 T^{2} + \cdots - 63342772 \) Copy content Toggle raw display
$67$ \( T^{3} + 522 T^{2} - 115704 T - 7056512 \) Copy content Toggle raw display
$71$ \( T^{3} - 282 T^{2} + \cdots - 150609024 \) Copy content Toggle raw display
$73$ \( T^{3} + 701 T^{2} + \cdots - 298076364 \) Copy content Toggle raw display
$79$ \( T^{3} - 1262 T^{2} + \cdots + 144063488 \) Copy content Toggle raw display
$83$ \( T^{3} - 1700 T^{2} + \cdots - 161161584 \) Copy content Toggle raw display
$89$ \( T^{3} - 1628 T^{2} + \cdots - 32164336 \) Copy content Toggle raw display
$97$ \( T^{3} - 1526 T^{2} + \cdots + 199579224 \) Copy content Toggle raw display
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