Properties

Label 546.8.a.q
Level $546$
Weight $8$
Character orbit 546.a
Self dual yes
Analytic conductor $170.562$
Analytic rank $0$
Dimension $6$
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] = [546,8,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, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("546.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 8 \)
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: \(170.562223914\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 367021x^{4} - 17702143x^{3} + 34815194576x^{2} + 1422988371620x - 933871993059968 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{3}\cdot 3^{2} \)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 8 q^{2} - 27 q^{3} + 64 q^{4} + (\beta_1 + 2) q^{5} - 216 q^{6} + 343 q^{7} + 512 q^{8} + 729 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 8 q^{2} - 27 q^{3} + 64 q^{4} + (\beta_1 + 2) q^{5} - 216 q^{6} + 343 q^{7} + 512 q^{8} + 729 q^{9} + (8 \beta_1 + 16) q^{10} + (\beta_{4} - \beta_{2} - \beta_1 + 1676) q^{11} - 1728 q^{12} - 2197 q^{13} + 2744 q^{14} + ( - 27 \beta_1 - 54) q^{15} + 4096 q^{16} + (\beta_{5} + 2 \beta_{4} + \beta_{2} + 7 \beta_1 + 3535) q^{17} + 5832 q^{18} + ( - \beta_{5} + 2 \beta_{4} - \beta_{3} + 22 \beta_1 + 1583) q^{19} + (64 \beta_1 + 128) q^{20} - 9261 q^{21} + (8 \beta_{4} - 8 \beta_{2} - 8 \beta_1 + 13408) q^{22} + ( - \beta_{5} + 9 \beta_{4} + 6 \beta_{3} + 41 \beta_1 + 5529) q^{23} - 13824 q^{24} + (3 \beta_{5} - 4 \beta_{4} - 7 \beta_{3} - 4 \beta_{2} + 80 \beta_1 + 44210) q^{25} - 17576 q^{26} - 19683 q^{27} + 21952 q^{28} + ( - 3 \beta_{5} - 11 \beta_{4} + 12 \beta_{3} + 3 \beta_{2} - 12 \beta_1 + 29036) q^{29} + ( - 216 \beta_1 - 432) q^{30} + ( - 2 \beta_{5} - 18 \beta_{4} - 17 \beta_{3} - 22 \beta_{2} + \cdots + 19835) q^{31}+ \cdots + (729 \beta_{4} - 729 \beta_{2} - 729 \beta_1 + 1221804) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 48 q^{2} - 162 q^{3} + 384 q^{4} + 13 q^{5} - 1296 q^{6} + 2058 q^{7} + 3072 q^{8} + 4374 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 48 q^{2} - 162 q^{3} + 384 q^{4} + 13 q^{5} - 1296 q^{6} + 2058 q^{7} + 3072 q^{8} + 4374 q^{9} + 104 q^{10} + 10054 q^{11} - 10368 q^{12} - 13182 q^{13} + 16464 q^{14} - 351 q^{15} + 24576 q^{16} + 21222 q^{17} + 34992 q^{18} + 9527 q^{19} + 832 q^{20} - 55566 q^{21} + 80432 q^{22} + 33229 q^{23} - 82944 q^{24} + 265321 q^{25} - 105456 q^{26} - 118098 q^{27} + 131712 q^{28} + 174185 q^{29} - 2808 q^{30} + 119045 q^{31} + 196608 q^{32} - 271458 q^{33} + 169776 q^{34} + 4459 q^{35} + 279936 q^{36} + 56562 q^{37} + 76216 q^{38} + 355914 q^{39} + 6656 q^{40} + 101632 q^{41} - 444528 q^{42} + 441323 q^{43} + 643456 q^{44} + 9477 q^{45} + 265832 q^{46} - 892849 q^{47} - 663552 q^{48} + 705894 q^{49} + 2122568 q^{50} - 572994 q^{51} - 843648 q^{52} + 2093965 q^{53} - 944784 q^{54} - 331222 q^{55} + 1053696 q^{56} - 257229 q^{57} + 1393480 q^{58} - 136204 q^{59} - 22464 q^{60} - 3989946 q^{61} + 952360 q^{62} + 1500282 q^{63} + 1572864 q^{64} - 28561 q^{65} - 2171664 q^{66} - 2218250 q^{67} + 1358208 q^{68} - 897183 q^{69} + 35672 q^{70} + 2045928 q^{71} + 2239488 q^{72} - 8557479 q^{73} + 452496 q^{74} - 7163667 q^{75} + 609728 q^{76} + 3448522 q^{77} + 2847312 q^{78} - 8559709 q^{79} + 53248 q^{80} + 3188646 q^{81} + 813056 q^{82} + 2496351 q^{83} - 3556224 q^{84} + 5335304 q^{85} + 3530584 q^{86} - 4702995 q^{87} + 5147648 q^{88} - 2446683 q^{89} + 75816 q^{90} - 4521426 q^{91} + 2126656 q^{92} - 3214215 q^{93} - 7142792 q^{94} + 16410211 q^{95} - 5308416 q^{96} + 5786889 q^{97} + 5647152 q^{98} + 7329366 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} - 367021x^{4} - 17702143x^{3} + 34815194576x^{2} + 1422988371620x - 933871993059968 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 112991 \nu^{5} - 64357478 \nu^{4} - 21066200065 \nu^{3} + 12643613243992 \nu^{2} + 623871092998772 \nu - 44\!\cdots\!84 ) / 45155141220000 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 83303 \nu^{5} + 2441274 \nu^{4} + 22810191145 \nu^{3} + 791497857564 \nu^{2} - 916133306279276 \nu + 34\!\cdots\!72 ) / 12901468920000 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 801047 \nu^{5} + 247784626 \nu^{4} + 240345178105 \nu^{3} - 60185401146764 \nu^{2} + \cdots + 32\!\cdots\!28 ) / 90310282440000 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 709123 \nu^{5} + 66211234 \nu^{4} + 212283386445 \nu^{3} - 1166557908276 \nu^{2} + \cdots - 15\!\cdots\!48 ) / 30103427480000 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 3\beta_{5} - 4\beta_{4} - 7\beta_{3} - 4\beta_{2} + 76\beta _1 + 122331 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 911\beta_{5} + 362\beta_{4} - 3169\beta_{3} + 1682\beta_{2} + 178698\beta _1 + 9003613 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 821445\beta_{5} - 840260\beta_{4} - 2372105\beta_{3} - 1366460\beta_{2} + 35842898\beta _1 + 21698255837 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 302029339 \beta_{5} + 36493198 \beta_{4} - 1158642641 \beta_{3} + 382517398 \beta_{2} + 39706353950 \beta _1 + 4304734040093 \) 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
−419.580
−309.150
−264.243
180.926
285.402
527.645
8.00000 −27.0000 64.0000 −417.580 −216.000 343.000 512.000 729.000 −3340.64
1.2 8.00000 −27.0000 64.0000 −307.150 −216.000 343.000 512.000 729.000 −2457.20
1.3 8.00000 −27.0000 64.0000 −262.243 −216.000 343.000 512.000 729.000 −2097.94
1.4 8.00000 −27.0000 64.0000 182.926 −216.000 343.000 512.000 729.000 1463.41
1.5 8.00000 −27.0000 64.0000 287.402 −216.000 343.000 512.000 729.000 2299.22
1.6 8.00000 −27.0000 64.0000 529.645 −216.000 343.000 512.000 729.000 4237.16
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
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 546.8.a.q 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.8.a.q 6 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} - 13T_{5}^{5} - 366951T_{5}^{4} - 14766175T_{5}^{3} + 34912599250T_{5}^{2} + 1283526912000T_{5} - 936578573280000 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(546))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 8)^{6} \) Copy content Toggle raw display
$3$ \( (T + 27)^{6} \) Copy content Toggle raw display
$5$ \( T^{6} + \cdots - 936578573280000 \) Copy content Toggle raw display
$7$ \( (T - 343)^{6} \) Copy content Toggle raw display
$11$ \( T^{6} - 10054 T^{5} + \cdots + 16\!\cdots\!68 \) Copy content Toggle raw display
$13$ \( (T + 2197)^{6} \) Copy content Toggle raw display
$17$ \( T^{6} - 21222 T^{5} + \cdots - 16\!\cdots\!28 \) Copy content Toggle raw display
$19$ \( T^{6} - 9527 T^{5} + \cdots - 16\!\cdots\!64 \) Copy content Toggle raw display
$23$ \( T^{6} - 33229 T^{5} + \cdots - 34\!\cdots\!52 \) Copy content Toggle raw display
$29$ \( T^{6} - 174185 T^{5} + \cdots + 92\!\cdots\!32 \) Copy content Toggle raw display
$31$ \( T^{6} - 119045 T^{5} + \cdots - 77\!\cdots\!80 \) Copy content Toggle raw display
$37$ \( T^{6} - 56562 T^{5} + \cdots + 43\!\cdots\!72 \) Copy content Toggle raw display
$41$ \( T^{6} - 101632 T^{5} + \cdots + 37\!\cdots\!08 \) Copy content Toggle raw display
$43$ \( T^{6} - 441323 T^{5} + \cdots + 26\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{6} + 892849 T^{5} + \cdots + 70\!\cdots\!52 \) Copy content Toggle raw display
$53$ \( T^{6} - 2093965 T^{5} + \cdots + 27\!\cdots\!16 \) Copy content Toggle raw display
$59$ \( T^{6} + 136204 T^{5} + \cdots + 46\!\cdots\!20 \) Copy content Toggle raw display
$61$ \( T^{6} + 3989946 T^{5} + \cdots + 11\!\cdots\!20 \) Copy content Toggle raw display
$67$ \( T^{6} + 2218250 T^{5} + \cdots - 48\!\cdots\!88 \) Copy content Toggle raw display
$71$ \( T^{6} - 2045928 T^{5} + \cdots - 22\!\cdots\!20 \) Copy content Toggle raw display
$73$ \( T^{6} + 8557479 T^{5} + \cdots + 47\!\cdots\!08 \) Copy content Toggle raw display
$79$ \( T^{6} + 8559709 T^{5} + \cdots + 36\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{6} - 2496351 T^{5} + \cdots - 54\!\cdots\!48 \) Copy content Toggle raw display
$89$ \( T^{6} + 2446683 T^{5} + \cdots + 20\!\cdots\!96 \) Copy content Toggle raw display
$97$ \( T^{6} - 5786889 T^{5} + \cdots - 28\!\cdots\!84 \) Copy content Toggle raw display
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