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Properties

Label 546.4.l.g
Level $546$
Weight $4$
Character orbit 546.l
Analytic conductor $32.215$
Analytic rank $0$
Dimension $10$
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(211,546)]
 
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, 2]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("546.211");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 546.l (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: \(32.2150428631\)
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: \( x^{10} - 2 x^{9} + 295 x^{8} - 1452 x^{7} + 70650 x^{6} - 303129 x^{5} + 5566320 x^{4} + \cdots + 5103959364 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 3^{3} \)
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.

\(f(q)\) \(=\) \( q + ( - 2 \beta_{3} - 2) q^{2} + (3 \beta_{3} + 3) q^{3} + 4 \beta_{3} q^{4} + ( - \beta_{2} - 3) q^{5} - 6 \beta_{3} q^{6} + 7 \beta_{3} q^{7} + 8 q^{8} + 9 \beta_{3} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 2 \beta_{3} - 2) q^{2} + (3 \beta_{3} + 3) q^{3} + 4 \beta_{3} q^{4} + ( - \beta_{2} - 3) q^{5} - 6 \beta_{3} q^{6} + 7 \beta_{3} q^{7} + 8 q^{8} + 9 \beta_{3} q^{9} + (6 \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 6) q^{10} + (\beta_{9} + \beta_{8} + \beta_{6} + \cdots - 3) q^{11}+ \cdots + ( - 9 \beta_{9} - 9 \beta_{7} + \cdots + 36) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 10 q^{2} + 15 q^{3} - 20 q^{4} - 34 q^{5} + 30 q^{6} - 35 q^{7} + 80 q^{8} - 45 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 10 q^{2} + 15 q^{3} - 20 q^{4} - 34 q^{5} + 30 q^{6} - 35 q^{7} + 80 q^{8} - 45 q^{9} + 34 q^{10} - 17 q^{11} - 120 q^{12} - 124 q^{13} + 140 q^{14} - 51 q^{15} - 80 q^{16} - 26 q^{17} + 180 q^{18} - 76 q^{19} + 68 q^{20} - 210 q^{21} - 34 q^{22} + 157 q^{23} + 120 q^{24} + 36 q^{25} + 226 q^{26} - 270 q^{27} - 140 q^{28} + 101 q^{29} - 102 q^{30} + 10 q^{31} - 160 q^{32} + 51 q^{33} + 104 q^{34} + 119 q^{35} - 180 q^{36} - 203 q^{37} + 304 q^{38} - 339 q^{39} - 272 q^{40} - 486 q^{41} + 210 q^{42} - 343 q^{43} + 136 q^{44} + 153 q^{45} + 314 q^{46} + 320 q^{47} + 240 q^{48} - 245 q^{49} - 36 q^{50} - 156 q^{51} + 44 q^{52} - 1654 q^{53} + 270 q^{54} - 286 q^{55} - 280 q^{56} - 456 q^{57} + 202 q^{58} - 95 q^{59} + 408 q^{60} - 169 q^{61} - 10 q^{62} - 315 q^{63} + 640 q^{64} - 1979 q^{65} - 204 q^{66} - 213 q^{67} - 104 q^{68} - 471 q^{69} - 476 q^{70} + 138 q^{71} - 360 q^{72} + 2694 q^{73} - 406 q^{74} + 54 q^{75} - 304 q^{76} + 238 q^{77} - 66 q^{78} - 488 q^{79} + 272 q^{80} - 405 q^{81} - 972 q^{82} - 3048 q^{83} + 420 q^{84} - 798 q^{85} + 1372 q^{86} - 303 q^{87} - 136 q^{88} - 2168 q^{89} - 612 q^{90} + 77 q^{91} - 1256 q^{92} + 15 q^{93} - 320 q^{94} - 2042 q^{95} - 960 q^{96} - 633 q^{97} - 490 q^{98} + 306 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 2 x^{9} + 295 x^{8} - 1452 x^{7} + 70650 x^{6} - 303129 x^{5} + 5566320 x^{4} + \cdots + 5103959364 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 6756820649 \nu^{9} + 406514216399 \nu^{8} - 5855543165803 \nu^{7} + 103245457375878 \nu^{6} + \cdots + 14\!\cdots\!66 ) / 30\!\cdots\!21 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 3493362443119 \nu^{9} + 7838084288012 \nu^{8} - 979321129453831 \nu^{7} + \cdots - 81\!\cdots\!84 ) / 38\!\cdots\!46 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 93637267054165 \nu^{9} + \cdots + 36\!\cdots\!44 ) / 38\!\cdots\!46 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 56114588871274 \nu^{9} - 694798673822966 \nu^{8} + \cdots - 15\!\cdots\!05 ) / 19\!\cdots\!23 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 224924910776947 \nu^{9} + \cdots - 45\!\cdots\!22 ) / 38\!\cdots\!46 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 267002354514967 \nu^{9} + \cdots - 23\!\cdots\!56 ) / 38\!\cdots\!46 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 275901735989587 \nu^{9} + \cdots - 10\!\cdots\!40 ) / 38\!\cdots\!46 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 106768165013539 \nu^{9} + 707538759407856 \nu^{8} + \cdots + 39\!\cdots\!90 ) / 12\!\cdots\!82 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{8} - \beta_{7} - 2\beta_{5} - \beta_{4} - 118\beta_{3} + 4\beta_{2} - 3\beta _1 - 118 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -14\beta_{9} - 14\beta_{7} - \beta_{6} + 23\beta_{5} + 15\beta_{4} + 14\beta_{3} - 184\beta_{2} + 14\beta _1 + 334 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -289\beta_{9} - 599\beta_{8} + 96\beta_{7} + 193\beta_{6} + 193\beta_{4} + 20726\beta_{3} + 1150\beta _1 - 96 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4173 \beta_{9} + 7529 \beta_{8} - 730 \beta_{7} + 4173 \beta_{6} - 7529 \beta_{5} - 730 \beta_{4} + \cdots - 100327 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 32746 \beta_{9} + 32746 \beta_{7} - 67591 \beta_{6} + 150512 \beta_{5} + 34845 \beta_{4} + \cdots + 4238047 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 283654 \beta_{9} - 1986719 \beta_{8} + 1001310 \beta_{7} - 717656 \beta_{6} - 717656 \beta_{4} + \cdots - 1001310 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 9767346 \beta_{9} + 36216446 \beta_{8} - 15129856 \beta_{7} + 9767346 \beta_{6} - 36216446 \beta_{5} + \cdots - 916279852 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 144736745 \beta_{9} - 144736745 \beta_{7} - 88649461 \beta_{6} + 495025898 \beta_{5} + \cdots + 7778544688 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(157\) \(365\) \(379\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
211.1
6.80020 − 11.7783i
3.82865 − 6.63142i
2.51485 − 4.35585i
−4.40759 + 7.63417i
−7.73611 + 13.3993i
6.80020 + 11.7783i
3.82865 + 6.63142i
2.51485 + 4.35585i
−4.40759 − 7.63417i
−7.73611 − 13.3993i
−1.00000 − 1.73205i 1.50000 + 2.59808i −2.00000 + 3.46410i −16.6004 3.00000 − 5.19615i −3.50000 + 6.06218i 8.00000 −4.50000 + 7.79423i 16.6004 + 28.7527i
211.2 −1.00000 − 1.73205i 1.50000 + 2.59808i −2.00000 + 3.46410i −10.6573 3.00000 − 5.19615i −3.50000 + 6.06218i 8.00000 −4.50000 + 7.79423i 10.6573 + 18.4590i
211.3 −1.00000 − 1.73205i 1.50000 + 2.59808i −2.00000 + 3.46410i −8.02970 3.00000 − 5.19615i −3.50000 + 6.06218i 8.00000 −4.50000 + 7.79423i 8.02970 + 13.9079i
211.4 −1.00000 − 1.73205i 1.50000 + 2.59808i −2.00000 + 3.46410i 5.81518 3.00000 − 5.19615i −3.50000 + 6.06218i 8.00000 −4.50000 + 7.79423i −5.81518 − 10.0722i
211.5 −1.00000 − 1.73205i 1.50000 + 2.59808i −2.00000 + 3.46410i 12.4722 3.00000 − 5.19615i −3.50000 + 6.06218i 8.00000 −4.50000 + 7.79423i −12.4722 − 21.6025i
295.1 −1.00000 + 1.73205i 1.50000 − 2.59808i −2.00000 − 3.46410i −16.6004 3.00000 + 5.19615i −3.50000 − 6.06218i 8.00000 −4.50000 − 7.79423i 16.6004 − 28.7527i
295.2 −1.00000 + 1.73205i 1.50000 − 2.59808i −2.00000 − 3.46410i −10.6573 3.00000 + 5.19615i −3.50000 − 6.06218i 8.00000 −4.50000 − 7.79423i 10.6573 − 18.4590i
295.3 −1.00000 + 1.73205i 1.50000 − 2.59808i −2.00000 − 3.46410i −8.02970 3.00000 + 5.19615i −3.50000 − 6.06218i 8.00000 −4.50000 − 7.79423i 8.02970 − 13.9079i
295.4 −1.00000 + 1.73205i 1.50000 − 2.59808i −2.00000 − 3.46410i 5.81518 3.00000 + 5.19615i −3.50000 − 6.06218i 8.00000 −4.50000 − 7.79423i −5.81518 + 10.0722i
295.5 −1.00000 + 1.73205i 1.50000 − 2.59808i −2.00000 − 3.46410i 12.4722 3.00000 + 5.19615i −3.50000 − 6.06218i 8.00000 −4.50000 − 7.79423i −12.4722 + 21.6025i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 211.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 546.4.l.g ✓ 10
13.c even 3 1 inner 546.4.l.g ✓ 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.4.l.g ✓ 10 1.a even 1 1 trivial
546.4.l.g ✓ 10 13.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{5} + 17T_{5}^{4} - 177T_{5}^{3} - 3258T_{5}^{2} + 2727T_{5} + 103032 \) acting on \(S_{4}^{\mathrm{new}}(546, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 2 T + 4)^{5} \) Copy content Toggle raw display
$3$ \( (T^{2} - 3 T + 9)^{5} \) Copy content Toggle raw display
$5$ \( (T^{5} + 17 T^{4} + \cdots + 103032)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 7 T + 49)^{5} \) Copy content Toggle raw display
$11$ \( T^{10} + \cdots + 17\!\cdots\!36 \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots + 51\!\cdots\!57 \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots + 287153528816025 \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots + 88\!\cdots\!56 \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 1383258254400 \) Copy content Toggle raw display
$29$ \( T^{10} + \cdots + 11\!\cdots\!25 \) Copy content Toggle raw display
$31$ \( (T^{5} - 5 T^{4} + \cdots + 108463763200)^{2} \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots + 40\!\cdots\!64 \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots + 14\!\cdots\!04 \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots + 62\!\cdots\!04 \) Copy content Toggle raw display
$47$ \( (T^{5} - 160 T^{4} + \cdots - 655927356600)^{2} \) Copy content Toggle raw display
$53$ \( (T^{5} + \cdots + 7446257213319)^{2} \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 14\!\cdots\!84 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots + 11\!\cdots\!09 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 45\!\cdots\!44 \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{5} + \cdots - 15872318386996)^{2} \) Copy content Toggle raw display
$79$ \( (T^{5} + \cdots + 1809491172556)^{2} \) Copy content Toggle raw display
$83$ \( (T^{5} + \cdots + 322233857664840)^{2} \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 18\!\cdots\!04 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 77\!\cdots\!44 \) Copy content Toggle raw display
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