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Properties

Label 546.4.i.h
Level $546$
Weight $4$
Character orbit 546.i
Analytic conductor $32.215$
Analytic rank $0$
Dimension $14$
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(79,546)]
 
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, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("546.79");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 546.i (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: \(14\)
Relative dimension: \(7\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 3 x^{13} + 519 x^{12} - 1172 x^{11} + 185559 x^{10} - 361626 x^{9} + 34186863 x^{8} + \cdots + 3874204890000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{2} \)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 \beta_{3} q^{2} + ( - 3 \beta_{3} + 3) q^{3} + (4 \beta_{3} - 4) q^{4} + (3 \beta_{3} - \beta_1) q^{5} - 6 q^{6} + (\beta_{10} + 1) q^{7} + 8 q^{8} - 9 \beta_{3} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 \beta_{3} q^{2} + ( - 3 \beta_{3} + 3) q^{3} + (4 \beta_{3} - 4) q^{4} + (3 \beta_{3} - \beta_1) q^{5} - 6 q^{6} + (\beta_{10} + 1) q^{7} + 8 q^{8} - 9 \beta_{3} q^{9} + ( - 6 \beta_{3} + 2 \beta_{2} + \cdots + 6) q^{10}+ \cdots + (9 \beta_{6} - 18) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 14 q^{2} + 21 q^{3} - 28 q^{4} + 18 q^{5} - 84 q^{6} + 10 q^{7} + 112 q^{8} - 63 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 14 q^{2} + 21 q^{3} - 28 q^{4} + 18 q^{5} - 84 q^{6} + 10 q^{7} + 112 q^{8} - 63 q^{9} + 36 q^{10} + 13 q^{11} + 84 q^{12} - 182 q^{13} - 10 q^{14} + 108 q^{15} - 112 q^{16} + 82 q^{17} - 126 q^{18} + 90 q^{19} - 144 q^{20} + 15 q^{21} - 52 q^{22} - 168 q^{23} + 168 q^{24} - 199 q^{25} + 182 q^{26} - 378 q^{27} - 20 q^{28} + 340 q^{29} - 108 q^{30} + 221 q^{31} - 224 q^{32} - 39 q^{33} - 328 q^{34} + 659 q^{35} + 504 q^{36} - 405 q^{37} + 180 q^{38} - 273 q^{39} + 144 q^{40} + 60 q^{41} - 60 q^{42} + 324 q^{43} + 52 q^{44} + 162 q^{45} - 336 q^{46} + 481 q^{47} - 672 q^{48} - 52 q^{49} + 796 q^{50} - 246 q^{51} + 364 q^{52} - 819 q^{53} + 378 q^{54} - 154 q^{55} + 80 q^{56} + 540 q^{57} - 340 q^{58} + 997 q^{59} - 216 q^{60} + 700 q^{61} - 884 q^{62} - 45 q^{63} + 896 q^{64} - 234 q^{65} - 78 q^{66} - 410 q^{67} + 328 q^{68} - 1008 q^{69} - 1352 q^{70} + 244 q^{71} - 504 q^{72} + 776 q^{73} - 810 q^{74} + 597 q^{75} - 720 q^{76} + 1016 q^{77} + 1092 q^{78} - 5 q^{79} + 288 q^{80} - 567 q^{81} - 60 q^{82} - 5246 q^{83} + 60 q^{84} + 300 q^{85} - 324 q^{86} + 510 q^{87} + 104 q^{88} + 121 q^{89} - 648 q^{90} - 130 q^{91} + 1344 q^{92} - 663 q^{93} + 962 q^{94} - 649 q^{95} + 672 q^{96} - 3000 q^{97} + 514 q^{98} - 234 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 3 x^{13} + 519 x^{12} - 1172 x^{11} + 185559 x^{10} - 361626 x^{9} + 34186863 x^{8} + \cdots + 3874204890000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 57\!\cdots\!40 \nu^{13} + \cdots - 11\!\cdots\!00 ) / 20\!\cdots\!51 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 69\!\cdots\!59 \nu^{13} + \cdots + 92\!\cdots\!00 ) / 49\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 18\!\cdots\!70 \nu^{13} + \cdots + 19\!\cdots\!95 ) / 68\!\cdots\!71 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 79\!\cdots\!90 \nu^{13} + \cdots + 14\!\cdots\!46 ) / 22\!\cdots\!57 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 57\!\cdots\!70 \nu^{13} + \cdots + 77\!\cdots\!86 ) / 13\!\cdots\!42 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 47\!\cdots\!31 \nu^{13} + \cdots + 55\!\cdots\!00 ) / 49\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 51\!\cdots\!31 \nu^{13} + \cdots - 10\!\cdots\!00 ) / 49\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 36\!\cdots\!13 \nu^{13} + \cdots - 31\!\cdots\!00 ) / 24\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 73\!\cdots\!83 \nu^{13} + \cdots + 10\!\cdots\!00 ) / 49\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 89\!\cdots\!83 \nu^{13} + \cdots + 48\!\cdots\!00 ) / 49\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 10\!\cdots\!51 \nu^{13} + \cdots - 26\!\cdots\!00 ) / 33\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 12\!\cdots\!37 \nu^{13} + \cdots - 20\!\cdots\!00 ) / 16\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2 \beta_{13} - \beta_{12} - \beta_{11} + 2 \beta_{10} - 2 \beta_{8} - \beta_{7} + 2 \beta_{6} + \cdots - 147 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{11} + 2\beta_{10} - 18\beta_{8} + 18\beta_{7} - 10\beta_{6} - 8\beta_{5} + 3\beta_{4} + 192\beta_{2} + 11 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 546 \beta_{13} + 375 \beta_{12} + 328 \beta_{11} - 119 \beta_{10} + 111 \beta_{9} + 209 \beta_{8} + \cdots + 338 \beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 3991 \beta_{13} - 944 \beta_{12} + 7944 \beta_{11} - 6975 \beta_{10} + 2934 \beta_{9} + 6975 \beta_{8} + \cdots - 19386 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 47509 \beta_{11} - 47509 \beta_{10} + 3473 \beta_{8} - 3473 \beta_{7} - 147122 \beta_{6} + \cdots + 6732406 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 1257390 \beta_{13} + 298377 \beta_{12} - 1983548 \beta_{11} + 2630578 \beta_{10} + \cdots + 9374911 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 39586505 \beta_{13} - 26772712 \beta_{12} + 4182457 \beta_{11} + 6921745 \beta_{10} - 13582227 \beta_{9} + \cdots - 1619779602 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 249245709 \beta_{11} - 249245709 \beta_{10} - 780792412 \beta_{8} + 780792412 \beta_{7} + \cdots + 3789144125 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 10651249377 \beta_{13} + 6891500313 \beta_{12} + 599719935 \beta_{11} + 2073927243 \beta_{10} + \cdots + 10358183006 \beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 105434516113 \beta_{13} - 30285132281 \beta_{12} + 220102577242 \beta_{11} - 139766185463 \beta_{10} + \cdots - 1286851909731 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 668317010435 \beta_{11} - 668317010435 \beta_{10} - 747232230981 \beta_{8} + 747232230981 \beta_{7} + \cdots + 103599069880138 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 29705643321069 \beta_{13} + 9290109870426 \beta_{12} - 36583815881437 \beta_{11} + \cdots + 146010506277838 \beta_1 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
79.1
7.94920 − 13.7684i
6.63081 − 11.4849i
5.20035 − 9.00727i
0.266082 − 0.460867i
−4.17721 + 7.23515i
−6.11416 + 10.5900i
−8.25506 + 14.2982i
7.94920 + 13.7684i
6.63081 + 11.4849i
5.20035 + 9.00727i
0.266082 + 0.460867i
−4.17721 − 7.23515i
−6.11416 − 10.5900i
−8.25506 − 14.2982i
−1.00000 + 1.73205i 1.50000 + 2.59808i −2.00000 − 3.46410i −6.44920 + 11.1703i −6.00000 0.292781 − 18.5179i 8.00000 −4.50000 + 7.79423i −12.8984 − 22.3407i
79.2 −1.00000 + 1.73205i 1.50000 + 2.59808i −2.00000 − 3.46410i −5.13081 + 8.88682i −6.00000 −17.3983 + 6.34825i 8.00000 −4.50000 + 7.79423i −10.2616 − 17.7736i
79.3 −1.00000 + 1.73205i 1.50000 + 2.59808i −2.00000 − 3.46410i −3.70035 + 6.40919i −6.00000 17.9434 − 4.58647i 8.00000 −4.50000 + 7.79423i −7.40070 − 12.8184i
79.4 −1.00000 + 1.73205i 1.50000 + 2.59808i −2.00000 − 3.46410i 1.23392 − 2.13721i −6.00000 15.8917 + 9.51074i 8.00000 −4.50000 + 7.79423i 2.46784 + 4.27442i
79.5 −1.00000 + 1.73205i 1.50000 + 2.59808i −2.00000 − 3.46410i 5.67721 − 9.83322i −6.00000 −16.6392 + 8.13238i 8.00000 −4.50000 + 7.79423i 11.3544 + 19.6664i
79.6 −1.00000 + 1.73205i 1.50000 + 2.59808i −2.00000 − 3.46410i 7.61416 − 13.1881i −6.00000 −0.807642 − 18.5026i 8.00000 −4.50000 + 7.79423i 15.2283 + 26.3762i
79.7 −1.00000 + 1.73205i 1.50000 + 2.59808i −2.00000 − 3.46410i 9.75506 − 16.8963i −6.00000 5.71731 + 17.6157i 8.00000 −4.50000 + 7.79423i 19.5101 + 33.7925i
235.1 −1.00000 − 1.73205i 1.50000 − 2.59808i −2.00000 + 3.46410i −6.44920 − 11.1703i −6.00000 0.292781 + 18.5179i 8.00000 −4.50000 − 7.79423i −12.8984 + 22.3407i
235.2 −1.00000 − 1.73205i 1.50000 − 2.59808i −2.00000 + 3.46410i −5.13081 − 8.88682i −6.00000 −17.3983 − 6.34825i 8.00000 −4.50000 − 7.79423i −10.2616 + 17.7736i
235.3 −1.00000 − 1.73205i 1.50000 − 2.59808i −2.00000 + 3.46410i −3.70035 − 6.40919i −6.00000 17.9434 + 4.58647i 8.00000 −4.50000 − 7.79423i −7.40070 + 12.8184i
235.4 −1.00000 − 1.73205i 1.50000 − 2.59808i −2.00000 + 3.46410i 1.23392 + 2.13721i −6.00000 15.8917 − 9.51074i 8.00000 −4.50000 − 7.79423i 2.46784 − 4.27442i
235.5 −1.00000 − 1.73205i 1.50000 − 2.59808i −2.00000 + 3.46410i 5.67721 + 9.83322i −6.00000 −16.6392 − 8.13238i 8.00000 −4.50000 − 7.79423i 11.3544 − 19.6664i
235.6 −1.00000 − 1.73205i 1.50000 − 2.59808i −2.00000 + 3.46410i 7.61416 + 13.1881i −6.00000 −0.807642 + 18.5026i 8.00000 −4.50000 − 7.79423i 15.2283 − 26.3762i
235.7 −1.00000 − 1.73205i 1.50000 − 2.59808i −2.00000 + 3.46410i 9.75506 + 16.8963i −6.00000 5.71731 − 17.6157i 8.00000 −4.50000 − 7.79423i 19.5101 − 33.7925i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 79.7
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.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.i.h ✓ 14
7.c even 3 1 inner 546.4.i.h ✓ 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.4.i.h ✓ 14 1.a even 1 1 trivial
546.4.i.h ✓ 14 7.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{14} - 18 T_{5}^{13} + 699 T_{5}^{12} - 4768 T_{5}^{11} + 194166 T_{5}^{10} + \cdots + 66502198292544 \) acting on \(S_{4}^{\mathrm{new}}(546, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 2 T + 4)^{7} \) Copy content Toggle raw display
$3$ \( (T^{2} - 3 T + 9)^{7} \) Copy content Toggle raw display
$5$ \( T^{14} + \cdots + 66502198292544 \) Copy content Toggle raw display
$7$ \( T^{14} + \cdots + 55\!\cdots\!07 \) Copy content Toggle raw display
$11$ \( T^{14} + \cdots + 43\!\cdots\!81 \) Copy content Toggle raw display
$13$ \( (T + 13)^{14} \) Copy content Toggle raw display
$17$ \( T^{14} + \cdots + 38\!\cdots\!24 \) Copy content Toggle raw display
$19$ \( T^{14} + \cdots + 15\!\cdots\!56 \) Copy content Toggle raw display
$23$ \( T^{14} + \cdots + 11\!\cdots\!36 \) Copy content Toggle raw display
$29$ \( (T^{7} + \cdots + 629696687131923)^{2} \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots + 99\!\cdots\!84 \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 52\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{7} - 30 T^{6} + \cdots + 464823594144)^{2} \) Copy content Toggle raw display
$43$ \( (T^{7} + \cdots - 96507228191232)^{2} \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 91\!\cdots\!84 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 11\!\cdots\!81 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 13\!\cdots\!21 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 38\!\cdots\!16 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 28\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( (T^{7} + \cdots + 35\!\cdots\!50)^{2} \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 16\!\cdots\!96 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 60\!\cdots\!96 \) Copy content Toggle raw display
$83$ \( (T^{7} + \cdots + 24\!\cdots\!56)^{2} \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 55\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{7} + \cdots - 16\!\cdots\!60)^{2} \) Copy content Toggle raw display
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