Properties

Label 1728.2.s.f
Level $1728$
Weight $2$
Character orbit 1728.s
Analytic conductor $13.798$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(13.7981494693\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.170772624.1
Defining polynomial: \( x^{8} - 3x^{7} + 5x^{6} - 6x^{5} + 6x^{4} - 12x^{3} + 20x^{2} - 24x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 36)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{2} - 1) q^{5} + ( - \beta_{6} + \beta_1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} - 1) q^{5} + ( - \beta_{6} + \beta_1) q^{7} + ( - \beta_{7} - \beta_{6} - \beta_{3}) q^{11} + (2 \beta_{5} - 2 \beta_{4} + \beta_{2} - 1) q^{13} + ( - \beta_{5} + 2 \beta_{4} - \beta_{2} + 1) q^{17} + ( - \beta_{7} - \beta_{6} - 2 \beta_{3} + 2 \beta_1) q^{19} + ( - \beta_{7} + \beta_{3}) q^{23} + (\beta_{5} - 3 \beta_{4} + 2 \beta_{2} - 2) q^{25} + ( - \beta_{5} + 1) q^{29} + ( - 3 \beta_{7} + 2 \beta_{6} - \beta_{3} + 2 \beta_1) q^{31} + ( - \beta_{7} + \beta_{6} + \beta_{3} + \beta_1) q^{35} + ( - 2 \beta_{5} + 2 \beta_{2} + 2) q^{37} + ( - \beta_{4} - 2 \beta_{2} - 4) q^{41} + ( - 3 \beta_{7} + \beta_{6} + 3 \beta_{3} + 2 \beta_1) q^{43} + ( - 4 \beta_{7} - 3 \beta_{6} - 4 \beta_{3} + \beta_1) q^{47} + (2 \beta_{5} + \beta_{4} + \beta_{2} - 1) q^{49} + ( - 2 \beta_{5} + 8 \beta_{4} - 2 \beta_{2} + 4) q^{53} + ( - \beta_{7} - \beta_{6} - \beta_{3} + \beta_1) q^{55} + ( - 3 \beta_{7} + \beta_{6} + 4 \beta_{3} - \beta_1) q^{59} + ( - \beta_{5} + 2 \beta_{4} - 2 \beta_{2} + 1) q^{61} + ( - \beta_{5} - 2 \beta_{4} + 3) q^{65} + ( - \beta_{7} + 3 \beta_{6} + 2 \beta_{3} + 3 \beta_1) q^{67} + ( - 4 \beta_{7} + 4 \beta_{6} + 2 \beta_{3} + 2 \beta_1) q^{71} + ( - \beta_{5} + \beta_{2} + 1) q^{73} + ( - 4 \beta_{4} + 3 \beta_{2} - 5) q^{77} + ( - 4 \beta_{7} + \beta_{6} + 4 \beta_{3} + 3 \beta_1) q^{79} + ( - 4 \beta_{7} - \beta_{6} - 4 \beta_{3} + 3 \beta_1) q^{83} + 2 \beta_{4} q^{85} + (2 \beta_{5} + 8 \beta_{4} + 2 \beta_{2} + 4) q^{89} + ( - 3 \beta_{7} - 3 \beta_{6} - \beta_{3} + \beta_1) q^{91} + ( - 2 \beta_{7} + 2 \beta_{6} + 4 \beta_{3} - 2 \beta_1) q^{95} + ( - 4 \beta_{5} + 7 \beta_{4} - 8 \beta_{2} + 3) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 6 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 6 q^{5} + 2 q^{13} - 6 q^{25} + 6 q^{29} + 8 q^{37} - 24 q^{41} - 10 q^{49} + 2 q^{61} + 30 q^{65} + 4 q^{73} - 30 q^{77} - 8 q^{85} + 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{7} + 5x^{6} - 6x^{5} + 6x^{4} - 12x^{3} + 20x^{2} - 24x + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{7} + \nu^{6} + \nu^{5} + 2\nu^{3} + 4\nu^{2} + 4\nu - 8 ) / 8 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{6} + \nu^{5} - \nu^{4} + 2\nu^{3} - 4\nu^{2} + 4\nu - 4 ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{7} + \nu^{6} - 3\nu^{5} + 4\nu^{4} - 2\nu^{3} + 12\nu^{2} - 12\nu + 16 ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3\nu^{7} - 5\nu^{6} + 7\nu^{5} - 6\nu^{4} + 10\nu^{3} - 24\nu^{2} + 28\nu - 32 ) / 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{7} + 3\nu^{6} - 5\nu^{5} + 4\nu^{4} - 4\nu^{3} + 10\nu^{2} - 16\nu + 20 ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3\nu^{7} - 3\nu^{6} + 9\nu^{5} - 8\nu^{4} + 10\nu^{3} - 24\nu^{2} + 20\nu - 40 ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3\nu^{7} - 5\nu^{6} + 11\nu^{5} - 6\nu^{4} + 10\nu^{3} - 28\nu^{2} + 36\nu - 48 ) / 8 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{6} + \beta_{4} - \beta_{3} - \beta_{2} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{7} - \beta_{5} + \beta_{4} + \beta_{3} - 2\beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{7} + \beta_{6} + \beta_{5} + 2\beta_{4} + 2\beta_{3} + \beta_{2} + 2\beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 4\beta_{7} - \beta_{6} + 2\beta_{5} - \beta_{4} + 3\beta_{3} + \beta_{2} - \beta _1 - 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 3\beta_{7} + 2\beta_{6} - \beta_{5} - 5\beta_{4} + 3\beta_{3} - \beta _1 + 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( \beta_{7} + \beta_{6} + 3\beta_{5} - 4\beta_{3} - 3\beta_{2} + 4\beta _1 + 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -2\beta_{7} + \beta_{6} + 7\beta_{4} + 3\beta_{3} - 13\beta_{2} - \beta _1 + 1 ) / 2 \) Copy content Toggle raw display

Character values

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

\(n\) \(325\) \(703\) \(1217\)
\(\chi(n)\) \(1\) \(-1\) \(-\beta_{4}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
575.1
−1.02187 0.977642i
0.335728 + 1.37379i
0.774115 1.18353i
1.41203 0.0786378i
−1.02187 + 0.977642i
0.335728 1.37379i
0.774115 + 1.18353i
1.41203 + 0.0786378i
0 0 0 −2.18614 + 1.26217i 0 −1.10489 0.637910i 0 0 0
575.2 0 0 0 −2.18614 + 1.26217i 0 1.10489 + 0.637910i 0 0 0
575.3 0 0 0 0.686141 0.396143i 0 −2.35143 1.35760i 0 0 0
575.4 0 0 0 0.686141 0.396143i 0 2.35143 + 1.35760i 0 0 0
1151.1 0 0 0 −2.18614 1.26217i 0 −1.10489 + 0.637910i 0 0 0
1151.2 0 0 0 −2.18614 1.26217i 0 1.10489 0.637910i 0 0 0
1151.3 0 0 0 0.686141 + 0.396143i 0 −2.35143 + 1.35760i 0 0 0
1151.4 0 0 0 0.686141 + 0.396143i 0 2.35143 1.35760i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1151.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
9.d odd 6 1 inner
36.h even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1728.2.s.f 8
3.b odd 2 1 576.2.s.f 8
4.b odd 2 1 inner 1728.2.s.f 8
8.b even 2 1 108.2.h.a 8
8.d odd 2 1 108.2.h.a 8
9.c even 3 1 576.2.s.f 8
9.c even 3 1 5184.2.c.j 8
9.d odd 6 1 inner 1728.2.s.f 8
9.d odd 6 1 5184.2.c.j 8
12.b even 2 1 576.2.s.f 8
24.f even 2 1 36.2.h.a 8
24.h odd 2 1 36.2.h.a 8
36.f odd 6 1 576.2.s.f 8
36.f odd 6 1 5184.2.c.j 8
36.h even 6 1 inner 1728.2.s.f 8
36.h even 6 1 5184.2.c.j 8
72.j odd 6 1 108.2.h.a 8
72.j odd 6 1 324.2.b.b 8
72.l even 6 1 108.2.h.a 8
72.l even 6 1 324.2.b.b 8
72.n even 6 1 36.2.h.a 8
72.n even 6 1 324.2.b.b 8
72.p odd 6 1 36.2.h.a 8
72.p odd 6 1 324.2.b.b 8
120.i odd 2 1 900.2.r.c 8
120.m even 2 1 900.2.r.c 8
120.q odd 4 2 900.2.o.a 16
120.w even 4 2 900.2.o.a 16
360.z odd 6 1 900.2.r.c 8
360.bk even 6 1 900.2.r.c 8
360.bo even 12 2 900.2.o.a 16
360.bu odd 12 2 900.2.o.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.2.h.a 8 24.f even 2 1
36.2.h.a 8 24.h odd 2 1
36.2.h.a 8 72.n even 6 1
36.2.h.a 8 72.p odd 6 1
108.2.h.a 8 8.b even 2 1
108.2.h.a 8 8.d odd 2 1
108.2.h.a 8 72.j odd 6 1
108.2.h.a 8 72.l even 6 1
324.2.b.b 8 72.j odd 6 1
324.2.b.b 8 72.l even 6 1
324.2.b.b 8 72.n even 6 1
324.2.b.b 8 72.p odd 6 1
576.2.s.f 8 3.b odd 2 1
576.2.s.f 8 9.c even 3 1
576.2.s.f 8 12.b even 2 1
576.2.s.f 8 36.f odd 6 1
900.2.o.a 16 120.q odd 4 2
900.2.o.a 16 120.w even 4 2
900.2.o.a 16 360.bo even 12 2
900.2.o.a 16 360.bu odd 12 2
900.2.r.c 8 120.i odd 2 1
900.2.r.c 8 120.m even 2 1
900.2.r.c 8 360.z odd 6 1
900.2.r.c 8 360.bk even 6 1
1728.2.s.f 8 1.a even 1 1 trivial
1728.2.s.f 8 4.b odd 2 1 inner
1728.2.s.f 8 9.d odd 6 1 inner
1728.2.s.f 8 36.h even 6 1 inner
5184.2.c.j 8 9.c even 3 1
5184.2.c.j 8 9.d odd 6 1
5184.2.c.j 8 36.f odd 6 1
5184.2.c.j 8 36.h even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1728, [\chi])\):

\( T_{5}^{4} + 3T_{5}^{3} + T_{5}^{2} - 6T_{5} + 4 \) Copy content Toggle raw display
\( T_{7}^{8} - 9T_{7}^{6} + 69T_{7}^{4} - 108T_{7}^{2} + 144 \) 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
$5$ \( (T^{4} + 3 T^{3} + T^{2} - 6 T + 4)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} - 9 T^{6} + 69 T^{4} - 108 T^{2} + \cdots + 144 \) Copy content Toggle raw display
$11$ \( T^{8} + 12 T^{6} + 141 T^{4} + 36 T^{2} + \cdots + 9 \) Copy content Toggle raw display
$13$ \( (T^{4} - T^{3} + 9 T^{2} + 8 T + 64)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 7 T^{2} + 4)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 27 T^{2} + 108)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + 15 T^{6} + 177 T^{4} + \cdots + 2304 \) Copy content Toggle raw display
$29$ \( (T^{4} - 3 T^{3} + T^{2} + 6 T + 4)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} - 69 T^{6} + 4569 T^{4} + \cdots + 36864 \) Copy content Toggle raw display
$37$ \( (T^{2} - 2 T - 32)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} + 12 T^{3} + 49 T^{2} + 12 T + 1)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} - 108 T^{6} + 8781 T^{4} + \cdots + 8311689 \) Copy content Toggle raw display
$47$ \( T^{8} + 135 T^{6} + 18033 T^{4} + \cdots + 36864 \) Copy content Toggle raw display
$53$ \( (T^{4} + 76 T^{2} + 256)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + 180 T^{6} + \cdots + 16867449 \) Copy content Toggle raw display
$61$ \( (T^{4} - T^{3} + 9 T^{2} + 8 T + 64)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} - 108 T^{6} + 8781 T^{4} + \cdots + 8311689 \) Copy content Toggle raw display
$71$ \( (T^{4} - 144 T^{2} + 432)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - T - 8)^{4} \) Copy content Toggle raw display
$79$ \( T^{8} - 201 T^{6} + \cdots + 101848464 \) Copy content Toggle raw display
$83$ \( T^{8} + 111 T^{6} + 9249 T^{4} + \cdots + 9437184 \) Copy content Toggle raw display
$89$ \( (T^{4} + 172 T^{2} + 4096)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 2 T^{3} + 135 T^{2} + 262 T + 17161)^{2} \) Copy content Toggle raw display
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