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Properties

Label 546.4.s.a
Level $546$
Weight $4$
Character orbit 546.s
Analytic conductor $32.215$
Analytic rank $0$
Dimension $20$
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] = [546,4,Mod(43,546)]
 
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, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("546.43");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 546.s (of order \(6\), 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: \(32.2150428631\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 1388 x^{18} + 806954 x^{16} + 255183238 x^{14} + 47714604791 x^{12} + 5370647791638 x^{10} + \cdots + 42\!\cdots\!84 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 \beta_{4} q^{2} + (3 \beta_{6} - 3) q^{3} + 4 \beta_{6} q^{4} + (\beta_{5} - \beta_{4} + \beta_1) q^{5} + 6 \beta_{5} q^{6} + 7 \beta_{5} q^{7} + (8 \beta_{5} - 8 \beta_{4}) q^{8} - 9 \beta_{6} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 \beta_{4} q^{2} + (3 \beta_{6} - 3) q^{3} + 4 \beta_{6} q^{4} + (\beta_{5} - \beta_{4} + \beta_1) q^{5} + 6 \beta_{5} q^{6} + 7 \beta_{5} q^{7} + (8 \beta_{5} - 8 \beta_{4}) q^{8} - 9 \beta_{6} q^{9} + ( - 2 \beta_{8} + 2 \beta_{6} + \cdots - 2) q^{10}+ \cdots + (9 \beta_{19} + 9 \beta_{18} + \cdots - 9) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 30 q^{3} + 40 q^{4} - 90 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 30 q^{3} + 40 q^{4} - 90 q^{9} - 24 q^{10} + 18 q^{11} - 240 q^{12} + 28 q^{13} - 280 q^{14} + 18 q^{15} - 160 q^{16} - 106 q^{17} - 60 q^{19} + 24 q^{20} - 24 q^{22} - 450 q^{23} - 304 q^{25} - 60 q^{26} + 540 q^{27} + 290 q^{29} - 72 q^{30} - 54 q^{33} - 84 q^{35} + 360 q^{36} + 564 q^{37} + 160 q^{38} + 228 q^{39} - 192 q^{40} - 246 q^{41} + 420 q^{42} - 464 q^{43} - 54 q^{45} - 240 q^{46} - 480 q^{48} + 490 q^{49} + 720 q^{50} + 636 q^{51} + 416 q^{52} - 1528 q^{53} + 1384 q^{55} - 560 q^{56} - 480 q^{58} - 2496 q^{59} - 270 q^{61} - 60 q^{62} - 1280 q^{64} + 3042 q^{65} + 144 q^{66} + 1314 q^{67} + 424 q^{68} - 1350 q^{69} - 516 q^{71} - 540 q^{74} + 456 q^{75} - 240 q^{76} + 168 q^{77} + 1116 q^{78} - 4000 q^{79} + 96 q^{80} - 810 q^{81} - 476 q^{82} - 2730 q^{85} + 870 q^{87} + 96 q^{88} - 1266 q^{89} + 432 q^{90} + 1302 q^{91} - 3600 q^{92} + 1692 q^{93} + 1080 q^{94} - 3798 q^{95} - 1620 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 1388 x^{18} + 806954 x^{16} + 255183238 x^{14} + 47714604791 x^{12} + 5370647791638 x^{10} + \cdots + 42\!\cdots\!84 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 48\!\cdots\!22 \nu^{18} + \cdots - 48\!\cdots\!26 ) / 63\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 41\!\cdots\!31 \nu^{18} + \cdots - 43\!\cdots\!16 ) / 15\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 46\!\cdots\!14 \nu^{19} + \cdots - 15\!\cdots\!56 ) / 14\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 46\!\cdots\!14 \nu^{19} + \cdots - 15\!\cdots\!56 ) / 14\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 57\!\cdots\!91 \nu^{19} + \cdots + 15\!\cdots\!76 ) / 31\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 11\!\cdots\!38 \nu^{19} + \cdots + 62\!\cdots\!40 ) / 31\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 33\!\cdots\!67 \nu^{19} + \cdots + 78\!\cdots\!08 ) / 56\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 95\!\cdots\!53 \nu^{19} + \cdots + 40\!\cdots\!72 ) / 95\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 95\!\cdots\!39 \nu^{19} + \cdots + 61\!\cdots\!72 ) / 95\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 10\!\cdots\!18 \nu^{19} + \cdots - 11\!\cdots\!36 ) / 95\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 10\!\cdots\!18 \nu^{19} + \cdots + 11\!\cdots\!36 ) / 95\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 11\!\cdots\!92 \nu^{19} + \cdots - 65\!\cdots\!16 ) / 95\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 61\!\cdots\!04 \nu^{19} + \cdots + 42\!\cdots\!58 ) / 47\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 61\!\cdots\!04 \nu^{19} + \cdots - 46\!\cdots\!84 ) / 47\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 18\!\cdots\!06 \nu^{19} + \cdots - 33\!\cdots\!76 ) / 95\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 11\!\cdots\!40 \nu^{19} + \cdots + 84\!\cdots\!96 ) / 47\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 32\!\cdots\!85 \nu^{19} + \cdots + 12\!\cdots\!44 ) / 95\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 33\!\cdots\!91 \nu^{19} + \cdots + 10\!\cdots\!32 ) / 95\!\cdots\!16 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 4 \beta_{19} + 4 \beta_{18} + 4 \beta_{16} - 3 \beta_{15} - \beta_{14} + \beta_{12} + 3 \beta_{11} + \cdots - 139 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - 12 \beta_{19} + 7 \beta_{18} - 19 \beta_{17} - 7 \beta_{16} + 2 \beta_{15} + 14 \beta_{14} + \cdots + 83 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 1350 \beta_{19} - 1068 \beta_{18} + 282 \beta_{17} - 1068 \beta_{16} + 847 \beta_{15} + 503 \beta_{14} + \cdots + 31936 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 5346 \beta_{19} - 2130 \beta_{18} + 7476 \beta_{17} + 2130 \beta_{16} + 1776 \beta_{15} - 3570 \beta_{14} + \cdots - 60453 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 406317 \beta_{19} + 290687 \beta_{18} - 115630 \beta_{17} + 290687 \beta_{16} - 228450 \beta_{15} + \cdots - 8213795 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 1794809 \beta_{19} + 652558 \beta_{18} - 2447367 \beta_{17} - 652558 \beta_{16} - 917421 \beta_{15} + \cdots + 24120338 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 118635566 \beta_{19} - 81942048 \beta_{18} + 36693518 \beta_{17} - 81942048 \beta_{16} + \cdots + 2218283179 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 559693274 \beta_{19} - 196349370 \beta_{18} + 756042644 \beta_{17} + 196349370 \beta_{16} + \cdots - 8144674316 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 34152959109 \beta_{19} + 23545480217 \beta_{18} - 10607478892 \beta_{17} + 23545480217 \beta_{16} + \cdots - 614328632184 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 170279991117 \beta_{19} + 57791767035 \beta_{18} - 228071758152 \beta_{17} - 57791767035 \beta_{16} + \cdots + 2575416437300 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 9753330093323 \beta_{19} - 6827218257401 \beta_{18} + 2926111835922 \beta_{17} + \cdots + 172482091189258 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 51355990789929 \beta_{19} - 16707195471286 \beta_{18} + 68063186261215 \beta_{17} + \cdots - 790249985712175 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 27\!\cdots\!08 \beta_{19} + \cdots - 48\!\cdots\!48 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 15\!\cdots\!70 \beta_{19} + \cdots + 23\!\cdots\!66 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 78\!\cdots\!96 \beta_{19} + \cdots + 13\!\cdots\!83 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 46\!\cdots\!08 \beta_{19} + \cdots - 71\!\cdots\!77 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 22\!\cdots\!28 \beta_{19} + \cdots - 39\!\cdots\!48 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 13\!\cdots\!28 \beta_{19} + \cdots + 21\!\cdots\!55 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1
− 16.5747i
− 13.2174i
2.99163i
7.21508i
16.8533i
− 11.6957i
− 11.3369i
− 0.622533i
5.75601i
17.1671i
− 16.8533i
− 7.21508i
− 2.99163i
13.2174i
16.5747i
− 17.1671i
− 5.75601i
0.622533i
11.3369i
11.6957i
−1.73205 − 1.00000i −1.50000 + 2.59808i 2.00000 + 3.46410i − 17.5747i 5.19615 − 3.00000i 6.06218 − 3.50000i − 8.00000i −4.50000 − 7.79423i −17.5747 + 30.4403i
43.2 −1.73205 − 1.00000i −1.50000 + 2.59808i 2.00000 + 3.46410i − 14.2174i 5.19615 − 3.00000i 6.06218 − 3.50000i − 8.00000i −4.50000 − 7.79423i −14.2174 + 24.6252i
43.3 −1.73205 − 1.00000i −1.50000 + 2.59808i 2.00000 + 3.46410i 1.99163i 5.19615 − 3.00000i 6.06218 − 3.50000i − 8.00000i −4.50000 − 7.79423i 1.99163 − 3.44961i
43.4 −1.73205 − 1.00000i −1.50000 + 2.59808i 2.00000 + 3.46410i 6.21508i 5.19615 − 3.00000i 6.06218 − 3.50000i − 8.00000i −4.50000 − 7.79423i 6.21508 − 10.7648i
43.5 −1.73205 − 1.00000i −1.50000 + 2.59808i 2.00000 + 3.46410i 15.8533i 5.19615 − 3.00000i 6.06218 − 3.50000i − 8.00000i −4.50000 − 7.79423i 15.8533 − 27.4587i
43.6 1.73205 + 1.00000i −1.50000 + 2.59808i 2.00000 + 3.46410i − 10.6957i −5.19615 + 3.00000i −6.06218 + 3.50000i 8.00000i −4.50000 − 7.79423i 10.6957 − 18.5255i
43.7 1.73205 + 1.00000i −1.50000 + 2.59808i 2.00000 + 3.46410i − 10.3369i −5.19615 + 3.00000i −6.06218 + 3.50000i 8.00000i −4.50000 − 7.79423i 10.3369 − 17.9040i
43.8 1.73205 + 1.00000i −1.50000 + 2.59808i 2.00000 + 3.46410i 0.377467i −5.19615 + 3.00000i −6.06218 + 3.50000i 8.00000i −4.50000 − 7.79423i −0.377467 + 0.653792i
43.9 1.73205 + 1.00000i −1.50000 + 2.59808i 2.00000 + 3.46410i 6.75601i −5.19615 + 3.00000i −6.06218 + 3.50000i 8.00000i −4.50000 − 7.79423i −6.75601 + 11.7018i
43.10 1.73205 + 1.00000i −1.50000 + 2.59808i 2.00000 + 3.46410i 18.1671i −5.19615 + 3.00000i −6.06218 + 3.50000i 8.00000i −4.50000 − 7.79423i −18.1671 + 31.4663i
127.1 −1.73205 + 1.00000i −1.50000 − 2.59808i 2.00000 − 3.46410i − 15.8533i 5.19615 + 3.00000i 6.06218 + 3.50000i 8.00000i −4.50000 + 7.79423i 15.8533 + 27.4587i
127.2 −1.73205 + 1.00000i −1.50000 − 2.59808i 2.00000 − 3.46410i − 6.21508i 5.19615 + 3.00000i 6.06218 + 3.50000i 8.00000i −4.50000 + 7.79423i 6.21508 + 10.7648i
127.3 −1.73205 + 1.00000i −1.50000 − 2.59808i 2.00000 − 3.46410i − 1.99163i 5.19615 + 3.00000i 6.06218 + 3.50000i 8.00000i −4.50000 + 7.79423i 1.99163 + 3.44961i
127.4 −1.73205 + 1.00000i −1.50000 − 2.59808i 2.00000 − 3.46410i 14.2174i 5.19615 + 3.00000i 6.06218 + 3.50000i 8.00000i −4.50000 + 7.79423i −14.2174 − 24.6252i
127.5 −1.73205 + 1.00000i −1.50000 − 2.59808i 2.00000 − 3.46410i 17.5747i 5.19615 + 3.00000i 6.06218 + 3.50000i 8.00000i −4.50000 + 7.79423i −17.5747 − 30.4403i
127.6 1.73205 − 1.00000i −1.50000 − 2.59808i 2.00000 − 3.46410i − 18.1671i −5.19615 − 3.00000i −6.06218 − 3.50000i − 8.00000i −4.50000 + 7.79423i −18.1671 − 31.4663i
127.7 1.73205 − 1.00000i −1.50000 − 2.59808i 2.00000 − 3.46410i − 6.75601i −5.19615 − 3.00000i −6.06218 − 3.50000i − 8.00000i −4.50000 + 7.79423i −6.75601 − 11.7018i
127.8 1.73205 − 1.00000i −1.50000 − 2.59808i 2.00000 − 3.46410i − 0.377467i −5.19615 − 3.00000i −6.06218 − 3.50000i − 8.00000i −4.50000 + 7.79423i −0.377467 − 0.653792i
127.9 1.73205 − 1.00000i −1.50000 − 2.59808i 2.00000 − 3.46410i 10.3369i −5.19615 − 3.00000i −6.06218 − 3.50000i − 8.00000i −4.50000 + 7.79423i 10.3369 + 17.9040i
127.10 1.73205 − 1.00000i −1.50000 − 2.59808i 2.00000 − 3.46410i 10.6957i −5.19615 − 3.00000i −6.06218 − 3.50000i − 8.00000i −4.50000 + 7.79423i 10.6957 + 18.5255i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 43.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 546.4.s.a ✓ 20
13.e even 6 1 inner 546.4.s.a ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.4.s.a ✓ 20 1.a even 1 1 trivial
546.4.s.a ✓ 20 13.e even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{20} + 1402 T_{5}^{18} + 814579 T_{5}^{16} + 254254330 T_{5}^{14} + 46260620575 T_{5}^{12} + \cdots + 63\!\cdots\!64 \) acting on \(S_{4}^{\mathrm{new}}(546, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 4 T^{2} + 16)^{5} \) Copy content Toggle raw display
$3$ \( (T^{2} + 3 T + 9)^{10} \) Copy content Toggle raw display
$5$ \( T^{20} + \cdots + 63\!\cdots\!64 \) Copy content Toggle raw display
$7$ \( (T^{4} - 49 T^{2} + 2401)^{5} \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 21\!\cdots\!84 \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 26\!\cdots\!49 \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 10\!\cdots\!81 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 10\!\cdots\!44 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 24\!\cdots\!56 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 34\!\cdots\!21 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 59\!\cdots\!16 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 93\!\cdots\!16 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 99\!\cdots\!44 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 20\!\cdots\!16 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 19\!\cdots\!24 \) Copy content Toggle raw display
$53$ \( (T^{10} + \cdots - 95\!\cdots\!99)^{2} \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 47\!\cdots\!96 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 38\!\cdots\!49 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 32\!\cdots\!44 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 45\!\cdots\!56 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 15\!\cdots\!96 \) Copy content Toggle raw display
$79$ \( (T^{10} + \cdots + 12\!\cdots\!52)^{2} \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 82\!\cdots\!04 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 34\!\cdots\!84 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 39\!\cdots\!36 \) Copy content Toggle raw display
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