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Properties

Label 729.4.a.a
Level $729$
Weight $4$
Character orbit 729.a
Self dual yes
Analytic conductor $43.012$
Analytic rank $1$
Dimension $12$
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] = [729,4,Mod(1,729)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(729, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("729.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 729 = 3^{6} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 729.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: \(43.0123923942\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 6 x^{11} - 48 x^{10} + 269 x^{9} + 900 x^{8} - 4059 x^{7} - 8325 x^{6} + 23940 x^{5} + \cdots - 3392 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3}\cdot 3^{5} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{2} + (\beta_{3} + \beta_{2} + 3) q^{4} + (\beta_{11} + \beta_{5} - 1) q^{5} + (\beta_{11} + \beta_{8} + \beta_{7} + \cdots - 5) q^{7}+ \cdots + (\beta_{9} + \beta_{8} + \beta_{7} + \cdots - 3) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 1) q^{2} + (\beta_{3} + \beta_{2} + 3) q^{4} + (\beta_{11} + \beta_{5} - 1) q^{5} + (\beta_{11} + \beta_{8} + \beta_{7} + \cdots - 5) q^{7}+ \cdots + (11 \beta_{10} + 16 \beta_{9} + \cdots - 793) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 6 q^{2} + 36 q^{4} - 12 q^{5} - 42 q^{7} - 21 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 6 q^{2} + 36 q^{4} - 12 q^{5} - 42 q^{7} - 21 q^{8} - 60 q^{10} - 42 q^{11} - 78 q^{13} + 312 q^{14} + 48 q^{16} + 18 q^{17} - 228 q^{19} + 69 q^{20} - 309 q^{22} + 114 q^{23} - 18 q^{25} - 30 q^{26} - 813 q^{28} + 660 q^{29} - 708 q^{31} - 729 q^{32} - 972 q^{34} + 624 q^{35} - 354 q^{37} + 213 q^{38} - 1335 q^{40} + 1032 q^{41} - 744 q^{43} - 1653 q^{44} - 1077 q^{46} - 942 q^{47} - 192 q^{49} - 1905 q^{50} - 1371 q^{52} + 828 q^{53} - 1554 q^{55} + 3324 q^{56} - 831 q^{58} + 24 q^{59} - 1698 q^{61} + 2184 q^{62} - 933 q^{64} - 294 q^{65} - 1266 q^{67} + 4734 q^{68} - 1137 q^{70} - 3888 q^{71} - 1164 q^{73} - 5448 q^{74} - 2385 q^{76} - 3018 q^{77} - 2382 q^{79} + 4677 q^{80} - 276 q^{82} + 4008 q^{83} - 1116 q^{85} + 1176 q^{86} - 2769 q^{88} + 3582 q^{89} - 3222 q^{91} + 2958 q^{92} - 3324 q^{94} + 2784 q^{95} - 2958 q^{97} - 9567 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 6 x^{11} - 48 x^{10} + 269 x^{9} + 900 x^{8} - 4059 x^{7} - 8325 x^{6} + 23940 x^{5} + \cdots - 3392 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 1787632 \nu^{11} + 12978173 \nu^{10} + 65873086 \nu^{9} - 550719836 \nu^{8} + \cdots - 14893957296 ) / 2640155808 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1787632 \nu^{11} - 12978173 \nu^{10} - 65873086 \nu^{9} + 550719836 \nu^{8} + \cdots - 11507600784 ) / 2640155808 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 4040013 \nu^{11} - 32911423 \nu^{10} - 135719914 \nu^{9} + 1418508693 \nu^{8} + \cdots - 17571248528 ) / 2640155808 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 36044675 \nu^{11} + 281046772 \nu^{10} + 1233374120 \nu^{9} - 11955825271 \nu^{8} + \cdots + 83901439776 ) / 10560623232 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 27884599 \nu^{11} - 208518640 \nu^{10} - 990210192 \nu^{9} + 8823132011 \nu^{8} + \cdots - 43724161696 ) / 5280311616 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 57448451 \nu^{11} + 450389556 \nu^{10} + 1952375784 \nu^{9} - 19091446583 \nu^{8} + \cdots + 238717068448 ) / 10560623232 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 19753884 \nu^{11} + 150422119 \nu^{10} + 701907314 \nu^{9} - 6423757568 \nu^{8} + \cdots + 51512741936 ) / 2640155808 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 47118527 \nu^{11} + 373050362 \nu^{10} + 1569844284 \nu^{9} - 15852155131 \nu^{8} + \cdots + 92236244480 ) / 5280311616 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 44595405 \nu^{11} - 355436628 \nu^{10} - 1458449016 \nu^{9} + 15024085825 \nu^{8} + \cdots - 177325952576 ) / 2640155808 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 197505337 \nu^{11} - 1509298080 \nu^{10} - 6960650224 \nu^{9} + 64157618629 \nu^{8} + \cdots - 199672568608 ) / 10560623232 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} + 2\beta _1 + 10 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} + \beta_{8} + \beta_{7} + \beta_{6} - 5\beta_{5} + \beta_{4} + 3\beta_{2} + 21\beta _1 + 12 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2 \beta_{11} + \beta_{10} + \beta_{9} + \beta_{8} + 6 \beta_{7} - 3 \beta_{6} - 4 \beta_{5} - 2 \beta_{4} + \cdots + 196 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 3 \beta_{11} + 3 \beta_{10} + 36 \beta_{9} + 25 \beta_{8} + 40 \beta_{7} + 9 \beta_{6} - 181 \beta_{5} + \cdots + 493 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 78 \beta_{11} + 45 \beta_{10} + 80 \beta_{9} + 44 \beta_{8} + 257 \beta_{7} - 178 \beta_{6} + \cdots + 4724 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 205 \beta_{11} + 170 \beta_{10} + 1130 \beta_{9} + 599 \beta_{8} + 1443 \beta_{7} - 461 \beta_{6} + \cdots + 16363 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 2559 \beta_{11} + 1599 \beta_{10} + 3705 \beta_{9} + 1502 \beta_{8} + 9167 \beta_{7} - 7672 \beta_{6} + \cdots + 124739 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 9318 \beta_{11} + 7212 \beta_{10} + 35130 \beta_{9} + 14916 \beta_{8} + 50556 \beta_{7} - 33066 \beta_{6} + \cdots + 515050 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 82158 \beta_{11} + 54240 \beta_{10} + 144015 \beta_{9} + 47529 \beta_{8} + 311607 \beta_{7} + \cdots + 3474864 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 361470 \beta_{11} + 271695 \beta_{10} + 1103709 \beta_{9} + 388065 \beta_{8} + 1731942 \beta_{7} + \cdots + 15992410 \) Copy content Toggle raw display

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

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
−4.41458
−3.32122
−2.77788
−1.76443
−1.74514
0.0811517
0.713877
1.45995
2.39816
4.82039
4.87349
5.67624
−5.41458 0 21.3177 13.1770 0 −26.9389 −72.1096 0 −71.3478
1.2 −4.32122 0 10.6730 3.36114 0 −22.2331 −11.5505 0 −14.5243
1.3 −3.77788 0 6.27236 −17.5799 0 −32.2829 6.52680 0 66.4147
1.4 −2.76443 0 −0.357918 9.90887 0 23.9691 23.1049 0 −27.3924
1.5 −2.74514 0 −0.464188 −12.6752 0 6.95634 23.2354 0 34.7951
1.6 −0.918848 0 −7.15572 10.6825 0 23.5612 13.9258 0 −9.81560
1.7 −0.286123 0 −7.91813 −19.8713 0 −14.8337 4.55455 0 5.68565
1.8 0.459945 0 −7.78845 11.1167 0 −7.10397 −7.26182 0 5.11307
1.9 1.39816 0 −6.04514 2.49756 0 −9.14600 −19.6374 0 3.49200
1.10 3.82039 0 6.59540 −8.10095 0 9.32780 −5.36611 0 −30.9488
1.11 3.87349 0 7.00394 0.438482 0 0.200216 −3.85821 0 1.69846
1.12 4.67624 0 13.8672 −4.95489 0 6.52395 27.4363 0 −23.1702
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)

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 729.4.a.a ✓ 12
3.b odd 2 1 729.4.a.b yes 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
729.4.a.a ✓ 12 1.a even 1 1 trivial
729.4.a.b yes 12 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} + 6 T_{2}^{11} - 48 T_{2}^{10} - 321 T_{2}^{9} + 666 T_{2}^{8} + 5877 T_{2}^{7} - 870 T_{2}^{6} + \cdots - 7848 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(729))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 6 T^{11} + \cdots - 7848 \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots - 10144014741 \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 891971833024 \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 16\!\cdots\!12 \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 27\!\cdots\!31 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots - 48\!\cdots\!87 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 45\!\cdots\!92 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 43\!\cdots\!64 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 37\!\cdots\!31 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots - 21\!\cdots\!32 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 15\!\cdots\!99 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 12\!\cdots\!73 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots - 32\!\cdots\!72 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots - 81\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots - 64\!\cdots\!28 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 15\!\cdots\!16 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots - 45\!\cdots\!44 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 12\!\cdots\!32 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots - 11\!\cdots\!68 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots - 35\!\cdots\!63 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots - 22\!\cdots\!96 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots - 34\!\cdots\!28 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots - 43\!\cdots\!71 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 51\!\cdots\!49 \) Copy content Toggle raw display
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