Properties

Label 546.2.k.e
Level $546$
Weight $2$
Character orbit 546.k
Analytic conductor $4.360$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [546,2,Mod(373,546)]
 
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, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("546.373");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 546.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: \(4.35983195036\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 2x^{9} + 15x^{8} + 14x^{7} + 110x^{6} + 36x^{5} + 233x^{4} + 164x^{3} + 345x^{2} + 76x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 2x^{9} + 15x^{8} + 14x^{7} + 110x^{6} + 36x^{5} + 233x^{4} + 164x^{3} + 345x^{2} + 76x + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 503 \nu^{9} - 2241 \nu^{8} + 8466 \nu^{7} - 67528 \nu^{6} + 19422 \nu^{5} - 156870 \nu^{4} + 1003571 \nu^{3} - 301041 \nu^{2} - 66732 \nu + 438544 ) / 2044008 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 3064 \nu^{9} + 9207 \nu^{8} - 34782 \nu^{7} - 28346 \nu^{6} - 79794 \nu^{5} + 644490 \nu^{4} + 579550 \nu^{3} + 1236807 \nu^{2} + 274164 \nu + 3620228 ) / 1022004 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 3977 \nu^{9} - 10833 \nu^{8} - 12699 \nu^{7} - 334006 \nu^{6} - 625302 \nu^{5} - 1723536 \nu^{4} - 478621 \nu^{3} - 3196425 \nu^{2} - 3391749 \nu - 3118196 ) / 1022004 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 27409 \nu^{9} + 54315 \nu^{8} - 413376 \nu^{7} - 375260 \nu^{6} - 3082518 \nu^{5} - 967302 \nu^{4} - 6543167 \nu^{3} - 3491505 \nu^{2} - 9757146 \nu - 2149816 ) / 2044008 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 14545 \nu^{9} + 29941 \nu^{8} - 216152 \nu^{7} - 200758 \nu^{6} - 1521222 \nu^{5} - 440214 \nu^{4} - 2887781 \nu^{3} - 2349679 \nu^{2} - 3963994 \nu - 1074620 ) / 340668 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 45928 \nu^{9} - 96426 \nu^{8} + 695481 \nu^{7} + 561428 \nu^{6} + 5037264 \nu^{5} + 744876 \nu^{4} + 10649492 \nu^{3} + 5556402 \nu^{2} + 16896855 \nu + 165664 ) / 1022004 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 97465 \nu^{9} - 178473 \nu^{8} + 1399728 \nu^{7} + 1662752 \nu^{6} + 10555542 \nu^{5} + 4653846 \nu^{4} + 20509619 \nu^{3} + 17738031 \nu^{2} + \cdots + 5377120 ) / 2044008 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 51344 \nu^{9} - 117222 \nu^{8} + 805587 \nu^{7} + 484042 \nu^{6} + 5520312 \nu^{5} + 367938 \nu^{4} + 11604142 \nu^{3} + 5109630 \nu^{2} + 14145267 \nu + 742892 ) / 1022004 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} + 4\beta_{5} - \beta_{4} + 2\beta_{2} + 2\beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -3\beta_{8} - 3\beta_{6} - 3\beta_{4} - 2\beta_{3} + 11\beta_{2} - 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -16\beta_{9} - 16\beta_{8} + 6\beta_{7} - 18\beta_{6} - 40\beta_{5} + 2\beta_{4} - 37\beta _1 - 40 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 65 \beta_{9} + 4 \beta_{8} + 34 \beta_{7} - 4 \beta_{6} - 126 \beta_{5} + 65 \beta_{4} + 34 \beta_{3} - 168 \beta_{2} - 168 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -30\beta_{9} + 297\beta_{8} + 267\beta_{6} + 267\beta_{4} + 126\beta_{3} - 657\beta_{2} + 592 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 1080\beta_{9} + 1080\beta_{8} - 564\beta_{7} + 1176\beta_{6} + 2280\beta_{5} - 96\beta_{4} + 2797\beta _1 + 2280 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 5005 \beta_{9} - 468 \beta_{8} - 2256 \beta_{7} + 468 \beta_{6} + 9784 \beta_{5} - 5005 \beta_{4} - 2256 \beta_{3} + 11402 \beta_{2} + 11402 \beta_1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1788\beta_{9} - 20451\beta_{8} - 18663\beta_{6} - 18663\beta_{4} - 9542\beta_{3} + 47603\beta_{2} - 39726 \) Copy content Toggle raw display

Character values

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

\(n\) \(157\) \(365\) \(379\)
\(\chi(n)\) \(\beta_{5}\) \(1\) \(-1 - \beta_{5}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
373.1
2.07085 + 3.58682i
0.769836 + 1.33339i
−0.114009 0.197470i
−0.623307 1.07960i
−1.10337 1.91109i
2.07085 3.58682i
0.769836 1.33339i
−0.114009 + 0.197470i
−0.623307 + 1.07960i
−1.10337 + 1.91109i
0.500000 + 0.866025i −1.00000 −0.500000 + 0.866025i −2.07085 + 3.58682i −0.500000 0.866025i 2.11344 + 1.59166i −1.00000 1.00000 −4.14170
373.2 0.500000 + 0.866025i −1.00000 −0.500000 + 0.866025i −0.769836 + 1.33339i −0.500000 0.866025i −0.131875 2.64246i −1.00000 1.00000 −1.53967
373.3 0.500000 + 0.866025i −1.00000 −0.500000 + 0.866025i 0.114009 0.197470i −0.500000 0.866025i 0.848534 2.50599i −1.00000 1.00000 0.228019
373.4 0.500000 + 0.866025i −1.00000 −0.500000 + 0.866025i 0.623307 1.07960i −0.500000 0.866025i −2.27938 + 1.34329i −1.00000 1.00000 1.24661
373.5 0.500000 + 0.866025i −1.00000 −0.500000 + 0.866025i 1.10337 1.91109i −0.500000 0.866025i 1.44928 + 2.21350i −1.00000 1.00000 2.20674
445.1 0.500000 0.866025i −1.00000 −0.500000 0.866025i −2.07085 3.58682i −0.500000 + 0.866025i 2.11344 1.59166i −1.00000 1.00000 −4.14170
445.2 0.500000 0.866025i −1.00000 −0.500000 0.866025i −0.769836 1.33339i −0.500000 + 0.866025i −0.131875 + 2.64246i −1.00000 1.00000 −1.53967
445.3 0.500000 0.866025i −1.00000 −0.500000 0.866025i 0.114009 + 0.197470i −0.500000 + 0.866025i 0.848534 + 2.50599i −1.00000 1.00000 0.228019
445.4 0.500000 0.866025i −1.00000 −0.500000 0.866025i 0.623307 + 1.07960i −0.500000 + 0.866025i −2.27938 1.34329i −1.00000 1.00000 1.24661
445.5 0.500000 0.866025i −1.00000 −0.500000 0.866025i 1.10337 + 1.91109i −0.500000 + 0.866025i 1.44928 2.21350i −1.00000 1.00000 2.20674
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 373.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
91.g 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.k.e yes 10
3.b odd 2 1 1638.2.p.j 10
7.c even 3 1 546.2.j.e 10
13.c even 3 1 546.2.j.e 10
21.h odd 6 1 1638.2.m.k 10
39.i odd 6 1 1638.2.m.k 10
91.g even 3 1 inner 546.2.k.e yes 10
273.bm odd 6 1 1638.2.p.j 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.j.e 10 7.c even 3 1
546.2.j.e 10 13.c even 3 1
546.2.k.e yes 10 1.a even 1 1 trivial
546.2.k.e yes 10 91.g even 3 1 inner
1638.2.m.k 10 21.h odd 6 1
1638.2.m.k 10 39.i odd 6 1
1638.2.p.j 10 3.b odd 2 1
1638.2.p.j 10 273.bm odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{10} + 2 T_{5}^{9} + 15 T_{5}^{8} - 14 T_{5}^{7} + 110 T_{5}^{6} - 36 T_{5}^{5} + 233 T_{5}^{4} - 164 T_{5}^{3} + 345 T_{5}^{2} - 76 T_{5} + 16 \) acting on \(S_{2}^{\mathrm{new}}(546, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - T + 1)^{5} \) Copy content Toggle raw display
$3$ \( (T + 1)^{10} \) Copy content Toggle raw display
$5$ \( T^{10} + 2 T^{9} + 15 T^{8} - 14 T^{7} + \cdots + 16 \) Copy content Toggle raw display
$7$ \( T^{10} - 4 T^{9} + 18 T^{8} + \cdots + 16807 \) Copy content Toggle raw display
$11$ \( (T^{5} + 6 T^{4} - 12 T^{3} - 65 T^{2} + \cdots + 30)^{2} \) Copy content Toggle raw display
$13$ \( T^{10} + 4 T^{9} + 24 T^{8} + \cdots + 371293 \) Copy content Toggle raw display
$17$ \( T^{10} - 4 T^{9} + 36 T^{8} - 62 T^{7} + \cdots + 784 \) Copy content Toggle raw display
$19$ \( (T^{5} + 3 T^{4} - 69 T^{3} - 109 T^{2} + \cdots - 111)^{2} \) Copy content Toggle raw display
$23$ \( T^{10} - 6 T^{9} + 108 T^{8} + \cdots + 129600 \) Copy content Toggle raw display
$29$ \( T^{10} + 60 T^{8} - 54 T^{7} + \cdots + 236196 \) Copy content Toggle raw display
$31$ \( T^{10} + 10 T^{9} + 126 T^{8} + \cdots + 1254400 \) Copy content Toggle raw display
$37$ \( T^{10} - T^{9} + 39 T^{8} + \cdots + 319225 \) Copy content Toggle raw display
$41$ \( T^{10} + 4 T^{9} + 48 T^{8} + \cdots + 770884 \) Copy content Toggle raw display
$43$ \( T^{10} - 3 T^{9} + 111 T^{8} + \cdots + 308025 \) Copy content Toggle raw display
$47$ \( T^{10} + 15 T^{9} + \cdots + 271854144 \) Copy content Toggle raw display
$53$ \( T^{10} + 17 T^{9} + 279 T^{8} + \cdots + 256 \) Copy content Toggle raw display
$59$ \( T^{10} - 2 T^{9} + 171 T^{8} + \cdots + 1607824 \) Copy content Toggle raw display
$61$ \( (T^{5} + 11 T^{4} - 8 T^{3} - 400 T^{2} + \cdots + 80)^{2} \) Copy content Toggle raw display
$67$ \( (T^{5} - T^{4} - 137 T^{3} + 199 T^{2} + \cdots - 20)^{2} \) Copy content Toggle raw display
$71$ \( T^{10} - 18 T^{9} + 315 T^{8} + \cdots + 202500 \) Copy content Toggle raw display
$73$ \( T^{10} - 12 T^{9} + \cdots + 37941975369 \) Copy content Toggle raw display
$79$ \( T^{10} + 4 T^{9} + 87 T^{8} + \cdots + 817216 \) Copy content Toggle raw display
$83$ \( (T^{5} - 60 T^{3} - 27 T^{2} + 405 T + 486)^{2} \) Copy content Toggle raw display
$89$ \( T^{10} - 7 T^{9} + \cdots + 205435562500 \) Copy content Toggle raw display
$97$ \( T^{10} + 6 T^{9} + 279 T^{8} + \cdots + 1734489 \) Copy content Toggle raw display
show more
show less