Properties

Label 729.2.e.g
Level $729$
Weight $2$
Character orbit 729.e
Analytic conductor $5.821$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 729 = 3^{6} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 729.e (of order \(9\), degree \(6\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.82109430735\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{18})\)
Defining polynomial: \( x^{6} - x^{3} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 243)
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{18}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \zeta_{18}^{5} + \zeta_{18}^{4} - \zeta_{18}^{3} + \zeta_{18}^{2} - \zeta_{18} + 1) q^{2} + ( - \zeta_{18}^{5} + \zeta_{18}^{4} + \zeta_{18}^{3} + \zeta_{18} - 2) q^{4} + ( - 2 \zeta_{18}^{5} - \zeta_{18}^{4} + 2 \zeta_{18}^{2} + \zeta_{18} + 1) q^{5} + (\zeta_{18}^{5} - \zeta_{18}^{4} - \zeta_{18}^{3} - 2 \zeta_{18}^{2} + 2) q^{7} + (3 \zeta_{18}^{5} - 2 \zeta_{18}^{4} + 2 \zeta_{18}^{3} - 2 \zeta_{18}^{2} + 3 \zeta_{18}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{18}^{5} + \zeta_{18}^{4} - \zeta_{18}^{3} + \zeta_{18}^{2} - \zeta_{18} + 1) q^{2} + ( - \zeta_{18}^{5} + \zeta_{18}^{4} + \zeta_{18}^{3} + \zeta_{18} - 2) q^{4} + ( - 2 \zeta_{18}^{5} - \zeta_{18}^{4} + 2 \zeta_{18}^{2} + \zeta_{18} + 1) q^{5} + (\zeta_{18}^{5} - \zeta_{18}^{4} - \zeta_{18}^{3} - 2 \zeta_{18}^{2} + 2) q^{7} + (3 \zeta_{18}^{5} - 2 \zeta_{18}^{4} + 2 \zeta_{18}^{3} - 2 \zeta_{18}^{2} + 3 \zeta_{18}) q^{8} + ( - 2 \zeta_{18}^{5} - 2 \zeta_{18}^{4} + \zeta_{18}^{2} + \zeta_{18}) q^{10} + (3 \zeta_{18}^{5} - 3 \zeta_{18}^{3} - \zeta_{18}^{2} + \zeta_{18} + 2) q^{11} + (\zeta_{18}^{4} + \zeta_{18}^{3} - \zeta_{18}^{2} + \zeta_{18} + 1) q^{13} + (3 \zeta_{18}^{4} - 4 \zeta_{18}^{3} + \zeta_{18}^{2} - 4 \zeta_{18} + 3) q^{14} + (3 \zeta_{18}^{5} - 3 \zeta_{18}^{3} - \zeta_{18} + 3) q^{16} + ( - 3 \zeta_{18}^{3} + 3) q^{17} + ( - 3 \zeta_{18}^{5} + \zeta_{18}^{3} - 3 \zeta_{18}) q^{19} + (2 \zeta_{18}^{5} + \zeta_{18}^{4} + 2 \zeta_{18}^{3}) q^{20} + (4 \zeta_{18}^{4} - 5 \zeta_{18}^{3} + \zeta_{18} + 1) q^{22} + (2 \zeta_{18}^{5} + \zeta_{18}^{4} + \zeta_{18}^{3} - 3 \zeta_{18} + 2) q^{23} + ( - 5 \zeta_{18}^{5} - \zeta_{18}^{4} - 4 \zeta_{18}^{3} + 4 \zeta_{18}^{2} + \zeta_{18} + 5) q^{25} + (\zeta_{18}^{5} - \zeta_{18}^{2} - \zeta_{18} + 4) q^{26} + ( - 6 \zeta_{18}^{5} + 3 \zeta_{18}^{4} + 3 \zeta_{18}^{2} + 3 \zeta_{18} - 4) q^{28} + (2 \zeta_{18}^{5} + 4 \zeta_{18}^{4} - \zeta_{18}^{3} + \zeta_{18}^{2} - 4 \zeta_{18} - 2) q^{29} + (4 \zeta_{18}^{5} + \zeta_{18}^{4} + \zeta_{18}^{3} - 2 \zeta_{18} + 1) q^{31} + (3 \zeta_{18}^{4} - 3 \zeta_{18} - 3) q^{32} + ( - 3 \zeta_{18}^{5} + 3 \zeta_{18}^{4} - 3 \zeta_{18}^{3}) q^{34} + ( - 3 \zeta_{18}^{5} - \zeta_{18}^{4} - 2 \zeta_{18}^{3} - \zeta_{18}^{2} - 3 \zeta_{18}) q^{35} + ( - 3 \zeta_{18}^{5} - 3 \zeta_{18}^{4} - \zeta_{18}^{3} + 3 \zeta_{18} + 1) q^{37} + ( - 3 \zeta_{18}^{5} + 3 \zeta_{18}^{3} + \zeta_{18}^{2} - 4 \zeta_{18} - 2) q^{38} + (4 \zeta_{18}^{4} + \zeta_{18}^{3} + 3 \zeta_{18}^{2} + \zeta_{18} + 4) q^{40} + ( - 3 \zeta_{18}^{4} + 5 \zeta_{18}^{3} + \zeta_{18}^{2} + 5 \zeta_{18} - 3) q^{41} + (\zeta_{18}^{5} - \zeta_{18}^{3} + \zeta_{18}^{2} + 4 \zeta_{18} + 2) q^{43} + ( - 2 \zeta_{18}^{5} - 2 \zeta_{18}^{4} + 5 \zeta_{18}^{3} - 3 \zeta_{18}^{2} + 5 \zeta_{18} - 5) q^{44} + ( - 5 \zeta_{18}^{5} + 7 \zeta_{18}^{4} - 3 \zeta_{18}^{3} + 7 \zeta_{18}^{2} - 5 \zeta_{18}) q^{46} + ( - 5 \zeta_{18}^{5} + 2 \zeta_{18}^{4} - 5 \zeta_{18}^{3}) q^{47} + ( - 2 \zeta_{18}^{4} + \zeta_{18}^{3} + \zeta_{18} + 1) q^{49} + ( - 8 \zeta_{18}^{5} - \zeta_{18}^{4} - \zeta_{18}^{3} + 3 \zeta_{18} - 2) q^{50} + (4 \zeta_{18}^{4} - 3 \zeta_{18}^{3} + 3 \zeta_{18}^{2} - 4 \zeta_{18}) q^{52} + ( - 3 \zeta_{18}^{4} + 3 \zeta_{18}^{2} + 3 \zeta_{18} - 6) q^{53} + ( - 2 \zeta_{18}^{5} + 4 \zeta_{18}^{4} - 2 \zeta_{18}^{2} - 2 \zeta_{18} + 3) q^{55} + (2 \zeta_{18}^{5} - 11 \zeta_{18}^{4} + 5 \zeta_{18}^{3} - 5 \zeta_{18}^{2} + 11 \zeta_{18} - 2) q^{56} + ( - 3 \zeta_{18}^{5} + 4 \zeta_{18}^{4} + 4 \zeta_{18}^{3} + 4 \zeta_{18} - 8) q^{58} + (7 \zeta_{18}^{5} - \zeta_{18}^{4} - 7 \zeta_{18}^{2} + \zeta_{18} + 1) q^{59} + (\zeta_{18}^{5} + 2 \zeta_{18}^{4} - 4 \zeta_{18}^{3} - 5 \zeta_{18}^{2} + 5) q^{61} + ( - 3 \zeta_{18}^{5} + 7 \zeta_{18}^{4} - 4 \zeta_{18}^{3} + 7 \zeta_{18}^{2} - 3 \zeta_{18}) q^{62} + (4 \zeta_{18}^{3} + 3 \zeta_{18}^{2} - 3 \zeta_{18} - 4) q^{64} + ( - 3 \zeta_{18}^{5} + 3 \zeta_{18}^{3} + 5 \zeta_{18}^{2} + \zeta_{18} + 2) q^{65} + (\zeta_{18}^{4} - 5 \zeta_{18}^{3} + 2 \zeta_{18}^{2} - 5 \zeta_{18} + 1) q^{67} + ( - 3 \zeta_{18}^{4} + 6 \zeta_{18}^{3} - 3 \zeta_{18}^{2} + 6 \zeta_{18} - 3) q^{68} + ( - 2 \zeta_{18}^{5} + 2 \zeta_{18}^{3} - 2 \zeta_{18}^{2} - 3 \zeta_{18} - 4) q^{70} + (6 \zeta_{18}^{5} + 6 \zeta_{18}^{4} + 3 \zeta_{18}^{3} + 3 \zeta_{18}^{2} - 9 \zeta_{18} - 3) q^{71} + (3 \zeta_{18}^{5} - 6 \zeta_{18}^{4} - 2 \zeta_{18}^{3} - 6 \zeta_{18}^{2} + 3 \zeta_{18}) q^{73} + (2 \zeta_{18}^{5} - 5 \zeta_{18}^{4} - \zeta_{18}^{3} - 3 \zeta_{18}^{2} + 3) q^{74} + ( - \zeta_{18}^{5} - \zeta_{18}^{4} - \zeta_{18}^{3} + \zeta_{18}^{2} + 2 \zeta_{18} + 2) q^{76} + (8 \zeta_{18}^{5} - 5 \zeta_{18}^{4} - 5 \zeta_{18}^{3} + 3 \zeta_{18} + 2) q^{77} + (\zeta_{18}^{5} - 2 \zeta_{18}^{4} + 5 \zeta_{18}^{3} - 5 \zeta_{18}^{2} + 2 \zeta_{18} - 1) q^{79} + ( - 2 \zeta_{18}^{5} + 3 \zeta_{18}^{4} - \zeta_{18}^{2} - \zeta_{18} + 1) q^{80} + (7 \zeta_{18}^{5} - 8 \zeta_{18}^{4} + \zeta_{18}^{2} + \zeta_{18} + 6) q^{82} + ( - 4 \zeta_{18}^{5} - 2 \zeta_{18}^{4} - \zeta_{18}^{3} + \zeta_{18}^{2} + 2 \zeta_{18} + 4) q^{83} + ( - 6 \zeta_{18}^{5} - 3 \zeta_{18}^{4} - 3 \zeta_{18}^{3} + 3) q^{85} + (\zeta_{18}^{5} - \zeta_{18}^{4} - 3 \zeta_{18}^{3} - \zeta_{18}^{2} + 4 \zeta_{18} + 4) q^{86} + (5 \zeta_{18}^{5} - 12 \zeta_{18}^{4} + 7 \zeta_{18}^{3} + 2 \zeta_{18}^{2} - 2) q^{88} + ( - 6 \zeta_{18}^{5} + 9 \zeta_{18}^{4} + 9 \zeta_{18}^{2} - 6 \zeta_{18}) q^{89} + ( - \zeta_{18}^{5} - \zeta_{18}^{4} - 2 \zeta_{18}^{3} - 4 \zeta_{18}^{2} + 5 \zeta_{18} + 2) q^{91} + ( - 6 \zeta_{18}^{5} + 6 \zeta_{18}^{3} - 5 \zeta_{18}^{2} + 8 \zeta_{18} - 11) q^{92} + ( - 5 \zeta_{18}^{4} + 7 \zeta_{18}^{3} - 12 \zeta_{18}^{2} + 7 \zeta_{18} - 5) q^{94} + ( - 6 \zeta_{18}^{4} - 2 \zeta_{18}^{3} - \zeta_{18}^{2} - 2 \zeta_{18} - 6) q^{95} + (4 \zeta_{18}^{5} - 4 \zeta_{18}^{3} + 4 \zeta_{18}^{2} - 5 \zeta_{18} + 8) q^{97} + ( - 3 \zeta_{18}^{3} + 3 \zeta_{18}^{2} - 3 \zeta_{18} + 3) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 3 q^{2} - 9 q^{4} + 6 q^{5} + 9 q^{7} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 3 q^{2} - 9 q^{4} + 6 q^{5} + 9 q^{7} + 6 q^{8} + 3 q^{11} + 9 q^{13} + 6 q^{14} + 9 q^{16} + 9 q^{17} + 3 q^{19} + 6 q^{20} - 9 q^{22} + 15 q^{23} + 18 q^{25} + 24 q^{26} - 24 q^{28} - 15 q^{29} + 9 q^{31} - 18 q^{32} - 9 q^{34} - 6 q^{35} + 3 q^{37} - 3 q^{38} + 27 q^{40} - 3 q^{41} + 9 q^{43} - 15 q^{44} - 9 q^{46} - 15 q^{47} + 9 q^{49} - 15 q^{50} - 9 q^{52} - 36 q^{53} + 18 q^{55} + 3 q^{56} - 36 q^{58} + 6 q^{59} + 18 q^{61} - 12 q^{62} - 12 q^{64} + 21 q^{65} - 9 q^{67} - 18 q^{70} - 9 q^{71} - 6 q^{73} + 15 q^{74} + 9 q^{76} - 3 q^{77} + 9 q^{79} + 6 q^{80} + 36 q^{82} + 21 q^{83} + 9 q^{85} + 15 q^{86} + 9 q^{88} + 6 q^{91} - 48 q^{92} - 9 q^{94} - 42 q^{95} + 36 q^{97} + 9 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/729\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-\zeta_{18}^{5}\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
82.1
−0.766044 + 0.642788i
−0.173648 + 0.984808i
0.939693 + 0.342020i
0.939693 0.342020i
−0.173648 0.984808i
−0.766044 0.642788i
−0.439693 2.49362i 0 −4.14543 + 1.50881i −0.358441 0.300767i 0 3.03209 + 1.10359i 3.05303 + 5.28801i 0 −0.592396 + 1.02606i
163.1 1.26604 0.460802i 0 −0.141559 + 0.118782i −0.286989 1.62760i 0 1.84730 + 1.55007i −1.47178 + 2.54920i 0 −1.11334 1.92836i
325.1 0.673648 0.565258i 0 −0.213011 + 1.20805i 3.64543 1.32683i 0 −0.379385 2.15160i 1.41875 + 2.45734i 0 1.70574 2.95442i
406.1 0.673648 + 0.565258i 0 −0.213011 1.20805i 3.64543 + 1.32683i 0 −0.379385 + 2.15160i 1.41875 2.45734i 0 1.70574 + 2.95442i
568.1 1.26604 + 0.460802i 0 −0.141559 0.118782i −0.286989 + 1.62760i 0 1.84730 1.55007i −1.47178 2.54920i 0 −1.11334 + 1.92836i
649.1 −0.439693 + 2.49362i 0 −4.14543 1.50881i −0.358441 + 0.300767i 0 3.03209 1.10359i 3.05303 5.28801i 0 −0.592396 1.02606i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 649.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
27.e even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 729.2.e.g 6
3.b odd 2 1 729.2.e.b 6
9.c even 3 1 729.2.e.a 6
9.c even 3 1 729.2.e.h 6
9.d odd 6 1 729.2.e.c 6
9.d odd 6 1 729.2.e.i 6
27.e even 9 1 243.2.a.e 3
27.e even 9 2 243.2.c.f 6
27.e even 9 1 729.2.e.a 6
27.e even 9 1 inner 729.2.e.g 6
27.e even 9 1 729.2.e.h 6
27.f odd 18 1 243.2.a.f yes 3
27.f odd 18 2 243.2.c.e 6
27.f odd 18 1 729.2.e.b 6
27.f odd 18 1 729.2.e.c 6
27.f odd 18 1 729.2.e.i 6
108.j odd 18 1 3888.2.a.bd 3
108.l even 18 1 3888.2.a.bk 3
135.n odd 18 1 6075.2.a.bq 3
135.p even 18 1 6075.2.a.bv 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
243.2.a.e 3 27.e even 9 1
243.2.a.f yes 3 27.f odd 18 1
243.2.c.e 6 27.f odd 18 2
243.2.c.f 6 27.e even 9 2
729.2.e.a 6 9.c even 3 1
729.2.e.a 6 27.e even 9 1
729.2.e.b 6 3.b odd 2 1
729.2.e.b 6 27.f odd 18 1
729.2.e.c 6 9.d odd 6 1
729.2.e.c 6 27.f odd 18 1
729.2.e.g 6 1.a even 1 1 trivial
729.2.e.g 6 27.e even 9 1 inner
729.2.e.h 6 9.c even 3 1
729.2.e.h 6 27.e even 9 1
729.2.e.i 6 9.d odd 6 1
729.2.e.i 6 27.f odd 18 1
3888.2.a.bd 3 108.j odd 18 1
3888.2.a.bk 3 108.l even 18 1
6075.2.a.bq 3 135.n odd 18 1
6075.2.a.bv 3 135.p even 18 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(729, [\chi])\):

\( T_{2}^{6} - 3T_{2}^{5} + 9T_{2}^{4} - 24T_{2}^{3} + 36T_{2}^{2} - 27T_{2} + 9 \) Copy content Toggle raw display
\( T_{5}^{6} - 6T_{5}^{5} + 9T_{5}^{4} - 3T_{5}^{3} + 36T_{5}^{2} + 27T_{5} + 9 \) Copy content Toggle raw display
\( T_{7}^{6} - 9T_{7}^{5} + 36T_{7}^{4} - 91T_{7}^{3} + 189T_{7}^{2} - 306T_{7} + 289 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} - 3 T^{5} + 9 T^{4} - 24 T^{3} + \cdots + 9 \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} - 6 T^{5} + 9 T^{4} - 3 T^{3} + \cdots + 9 \) Copy content Toggle raw display
$7$ \( T^{6} - 9 T^{5} + 36 T^{4} - 91 T^{3} + \cdots + 289 \) Copy content Toggle raw display
$11$ \( T^{6} - 3 T^{5} + 36 T^{4} - 159 T^{3} + \cdots + 9 \) Copy content Toggle raw display
$13$ \( T^{6} - 9 T^{5} + 36 T^{4} - 91 T^{3} + \cdots + 289 \) Copy content Toggle raw display
$17$ \( (T^{2} - 3 T + 9)^{3} \) Copy content Toggle raw display
$19$ \( T^{6} - 3 T^{5} + 33 T^{4} + 74 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$23$ \( T^{6} - 15 T^{5} + 90 T^{4} + \cdots + 2601 \) Copy content Toggle raw display
$29$ \( T^{6} + 15 T^{5} + 144 T^{4} + \cdots + 3249 \) Copy content Toggle raw display
$31$ \( T^{6} - 9 T^{5} + 36 T^{4} - 208 T^{3} + \cdots + 361 \) Copy content Toggle raw display
$37$ \( T^{6} - 3 T^{5} + 33 T^{4} + 74 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$41$ \( T^{6} + 3 T^{5} + 36 T^{4} + \cdots + 47961 \) Copy content Toggle raw display
$43$ \( T^{6} - 9 T^{5} + 36 T^{4} - 208 T^{3} + \cdots + 361 \) Copy content Toggle raw display
$47$ \( T^{6} + 15 T^{5} + 90 T^{4} + \cdots + 71289 \) Copy content Toggle raw display
$53$ \( (T^{3} + 18 T^{2} + 81 T + 81)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} - 6 T^{5} + 36 T^{4} + \cdots + 103041 \) Copy content Toggle raw display
$61$ \( T^{6} - 18 T^{5} + 153 T^{4} + \cdots + 2809 \) Copy content Toggle raw display
$67$ \( T^{6} + 9 T^{5} + 126 T^{4} + \cdots + 11881 \) Copy content Toggle raw display
$71$ \( T^{6} + 9 T^{5} + 243 T^{4} + \cdots + 998001 \) Copy content Toggle raw display
$73$ \( T^{6} + 6 T^{5} + 105 T^{4} + \cdots + 157609 \) Copy content Toggle raw display
$79$ \( T^{6} - 9 T^{5} + 144 T^{4} + \cdots + 2809 \) Copy content Toggle raw display
$83$ \( T^{6} - 21 T^{5} + 198 T^{4} + \cdots + 2601 \) Copy content Toggle raw display
$89$ \( T^{6} + 189 T^{4} - 1998 T^{3} + \cdots + 998001 \) Copy content Toggle raw display
$97$ \( T^{6} - 36 T^{5} + 576 T^{4} + \cdots + 361 \) Copy content Toggle raw display
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