Properties

Label 1728.3.o.e
Level $1728$
Weight $3$
Character orbit 1728.o
Analytic conductor $47.085$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1728.o (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(47.0845896815\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.856615824.2
Defining polynomial: \( x^{8} + 11x^{6} + 36x^{4} + 32x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{6} \)
Twist minimal: no (minimal twist has level 144)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{4} - \beta_1 + 1) q^{5} + (\beta_{7} + \beta_{5} + \beta_{4}) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{4} - \beta_1 + 1) q^{5} + (\beta_{7} + \beta_{5} + \beta_{4}) q^{7} + (\beta_{7} + \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} + \beta_1 - 2) q^{11} + (\beta_{3} + 2 \beta_{2} + 2 \beta_1 - 2) q^{13} + (2 \beta_{7} + 2 \beta_{6} + 2 \beta_{3} + \beta_{2} - 1) q^{17} + ( - 2 \beta_{7} + 2 \beta_{6} - 3 \beta_{5} + 3 \beta_{4} - 4 \beta_1 + 2) q^{19} + ( - \beta_{6} + \beta_{5} + \beta_{3} - 7 \beta_1 - 7) q^{23} + ( - 2 \beta_{7} + 4 \beta_{6} - 7 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} - 8 \beta_1) q^{25} + (2 \beta_{5} - 3 \beta_{3} + 3 \beta_{2} + 20 \beta_1) q^{29} + (\beta_{6} - 5 \beta_{5} - 2 \beta_{4} + 3 \beta_{3} - 5 \beta_1 - 5) q^{31} + ( - 5 \beta_{7} + 5 \beta_{6} - 4 \beta_{5} + 4 \beta_{4} - 4 \beta_{2} - 36 \beta_1 + 18) q^{35} + ( - 4 \beta_{7} - 4 \beta_{6} + \beta_{5} + \beta_{4} + 2 \beta_{3} + \beta_{2} + 1) q^{37} + (8 \beta_{7} - 4 \beta_{6} + 4 \beta_{5} - 5 \beta_{4} - \beta_{3} - 2 \beta_{2} + 12 \beta_1 - 12) q^{41} + ( - \beta_{7} + 2 \beta_{5} + 5 \beta_{4} - 6 \beta_{3} - 6 \beta_{2} - 2 \beta_1 + 4) q^{43} + (7 \beta_{7} + 5 \beta_{5} + 3 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} - 18 \beta_1 + 36) q^{47} + ( - 4 \beta_{7} + 2 \beta_{6} - 2 \beta_{5} - 2 \beta_{4} - 3 \beta_{3} - 6 \beta_{2} + \cdots + 11) q^{49}+ \cdots + (8 \beta_{7} - 16 \beta_{6} + \beta_{5} - 8 \beta_{4} - 3 \beta_{3} + 3 \beta_{2} - 4 \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 3 q^{5} - 3 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 3 q^{5} - 3 q^{7} - 18 q^{11} - 5 q^{13} - 6 q^{17} - 81 q^{23} - 23 q^{25} + 69 q^{29} - 45 q^{31} + 20 q^{37} - 54 q^{41} + 207 q^{47} + 41 q^{49} - 252 q^{53} + 306 q^{59} - 7 q^{61} - 93 q^{65} + 12 q^{67} + 74 q^{73} + 207 q^{77} - 33 q^{79} - 549 q^{83} + 30 q^{85} + 168 q^{89} - 684 q^{95} - 10 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 11x^{6} + 36x^{4} + 32x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{5} + 7\nu^{3} + 10\nu + 2 ) / 4 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{5} - \nu^{3} + 14\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{5} + 6\nu^{4} + \nu^{3} + 24\nu^{2} - 14\nu - 6 ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{6} - \nu^{5} + 8\nu^{4} - 4\nu^{3} + 14\nu^{2} + 8\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{6} + \nu^{5} + 8\nu^{4} + 4\nu^{3} + 14\nu^{2} - 8\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3\nu^{7} + 2\nu^{6} + 32\nu^{5} + 22\nu^{4} + 95\nu^{3} + 70\nu^{2} + 50\nu + 42 ) / 4 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -3\nu^{7} + 2\nu^{6} - 32\nu^{5} + 22\nu^{4} - 95\nu^{3} + 70\nu^{2} - 50\nu + 42 ) / 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{5} + \beta_{4} - \beta_{2} + 2\beta _1 - 1 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} - 2\beta_{3} - \beta_{2} - 24 ) / 9 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{5} - 2\beta_{4} + 3\beta_{2} - 2\beta _1 + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -4\beta_{7} - 4\beta_{6} + 4\beta_{5} + 4\beta_{4} + 14\beta_{3} + 7\beta_{2} + 105 ) / 9 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -9\beta_{5} + 9\beta_{4} - 16\beta_{2} + 16\beta _1 - 8 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 6\beta_{7} + 6\beta_{6} - 3\beta_{5} - 3\beta_{4} - 28\beta_{3} - 14\beta_{2} - 168 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -2\beta_{7} + 2\beta_{6} + 41\beta_{5} - 41\beta_{4} + 84\beta_{2} - 124\beta _1 + 62 ) / 3 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(703\) \(1217\)
\(\chi(n)\) \(1\) \(-1\) \(-1 + \beta_{1}\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
127.1
1.07834i
0.385731i
2.33086i
2.06288i
1.07834i
0.385731i
2.33086i
2.06288i
0 0 0 −3.01729 5.22611i 0 −10.2332 5.90815i 0 0 0
127.2 0 0 0 −0.454613 0.787412i 0 6.10709 + 3.52593i 0 0 0
127.3 0 0 0 0.355304 + 0.615405i 0 −2.70480 1.56162i 0 0 0
127.4 0 0 0 4.61660 + 7.99619i 0 5.33093 + 3.07781i 0 0 0
1279.1 0 0 0 −3.01729 + 5.22611i 0 −10.2332 + 5.90815i 0 0 0
1279.2 0 0 0 −0.454613 + 0.787412i 0 6.10709 3.52593i 0 0 0
1279.3 0 0 0 0.355304 0.615405i 0 −2.70480 + 1.56162i 0 0 0
1279.4 0 0 0 4.61660 7.99619i 0 5.33093 3.07781i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1279.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
36.f odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1728.3.o.e 8
3.b odd 2 1 576.3.o.d 8
4.b odd 2 1 1728.3.o.f 8
8.b even 2 1 432.3.o.a 8
8.d odd 2 1 432.3.o.b 8
9.c even 3 1 1728.3.o.f 8
9.d odd 6 1 576.3.o.f 8
12.b even 2 1 576.3.o.f 8
24.f even 2 1 144.3.o.a 8
24.h odd 2 1 144.3.o.c yes 8
36.f odd 6 1 inner 1728.3.o.e 8
36.h even 6 1 576.3.o.d 8
72.j odd 6 1 144.3.o.a 8
72.j odd 6 1 1296.3.g.j 8
72.l even 6 1 144.3.o.c yes 8
72.l even 6 1 1296.3.g.j 8
72.n even 6 1 432.3.o.b 8
72.n even 6 1 1296.3.g.k 8
72.p odd 6 1 432.3.o.a 8
72.p odd 6 1 1296.3.g.k 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.3.o.a 8 24.f even 2 1
144.3.o.a 8 72.j odd 6 1
144.3.o.c yes 8 24.h odd 2 1
144.3.o.c yes 8 72.l even 6 1
432.3.o.a 8 8.b even 2 1
432.3.o.a 8 72.p odd 6 1
432.3.o.b 8 8.d odd 2 1
432.3.o.b 8 72.n even 6 1
576.3.o.d 8 3.b odd 2 1
576.3.o.d 8 36.h even 6 1
576.3.o.f 8 9.d odd 6 1
576.3.o.f 8 12.b even 2 1
1296.3.g.j 8 72.j odd 6 1
1296.3.g.j 8 72.l even 6 1
1296.3.g.k 8 72.n even 6 1
1296.3.g.k 8 72.p odd 6 1
1728.3.o.e 8 1.a even 1 1 trivial
1728.3.o.e 8 36.f odd 6 1 inner
1728.3.o.f 8 4.b odd 2 1
1728.3.o.f 8 9.c even 3 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1728, [\chi])\):

\( T_{5}^{8} - 3T_{5}^{7} + 66T_{5}^{6} + 189T_{5}^{5} + 3186T_{5}^{4} + 729T_{5}^{3} + 2133T_{5}^{2} - 324T_{5} + 1296 \) Copy content Toggle raw display
\( T_{7}^{8} + 3T_{7}^{7} - 114T_{7}^{6} - 351T_{7}^{5} + 12366T_{7}^{4} - 32643T_{7}^{3} - 161487T_{7}^{2} + 446958T_{7} + 2566404 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} - 3 T^{7} + 66 T^{6} + \cdots + 1296 \) Copy content Toggle raw display
$7$ \( T^{8} + 3 T^{7} - 114 T^{6} + \cdots + 2566404 \) Copy content Toggle raw display
$11$ \( T^{8} + 18 T^{7} - 36 T^{6} + \cdots + 12131289 \) Copy content Toggle raw display
$13$ \( T^{8} + 5 T^{7} + 316 T^{6} + \cdots + 10201636 \) Copy content Toggle raw display
$17$ \( (T^{4} + 3 T^{3} - 822 T^{2} - 1908 T + 84168)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + 1731 T^{6} + \cdots + 2931572736 \) Copy content Toggle raw display
$23$ \( T^{8} + 81 T^{7} + 2754 T^{6} + \cdots + 19131876 \) Copy content Toggle raw display
$29$ \( T^{8} - 69 T^{7} + \cdots + 4046639163876 \) Copy content Toggle raw display
$31$ \( T^{8} + 45 T^{7} - 324 T^{6} + \cdots + 944784 \) Copy content Toggle raw display
$37$ \( (T^{4} - 10 T^{3} - 3756 T^{2} + \cdots - 613568)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + 54 T^{7} + \cdots + 5431756955769 \) Copy content Toggle raw display
$43$ \( T^{8} - 3186 T^{6} + \cdots + 29016737649 \) Copy content Toggle raw display
$47$ \( T^{8} - 207 T^{7} + \cdots + 28643839776036 \) Copy content Toggle raw display
$53$ \( (T^{4} + 126 T^{3} + 972 T^{2} + \cdots - 6508512)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} - 306 T^{7} + \cdots + 48359409452649 \) Copy content Toggle raw display
$61$ \( T^{8} + 7 T^{7} + \cdots + 309954973696 \) Copy content Toggle raw display
$67$ \( T^{8} - 12 T^{7} + \cdots + 68036119056801 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 726110197530624 \) Copy content Toggle raw display
$73$ \( (T^{4} - 37 T^{3} - 1002 T^{2} + \cdots + 416536)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 240627852449856 \) Copy content Toggle raw display
$83$ \( T^{8} + 549 T^{7} + \cdots + 15\!\cdots\!64 \) Copy content Toggle raw display
$89$ \( (T^{4} - 84 T^{3} - 984 T^{2} + \cdots - 1161936)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + 10 T^{7} + \cdots + 30429664983481 \) Copy content Toggle raw display
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