Properties

Label 216.3.e.c
Level $216$
Weight $3$
Character orbit 216.e
Analytic conductor $5.886$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 216 = 2^{3} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 216.e (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.88557371018\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 2 \zeta_{8} - 3 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{5} + ( -3 + 6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{7} +O(q^{10})\) \( q + ( 2 \zeta_{8} - 3 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{5} + ( -3 + 6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{7} + ( -4 \zeta_{8} + 9 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{11} + ( -2 + 12 \zeta_{8} - 12 \zeta_{8}^{3} ) q^{13} + ( -8 \zeta_{8} - 6 \zeta_{8}^{2} - 8 \zeta_{8}^{3} ) q^{17} + ( 8 + 12 \zeta_{8} - 12 \zeta_{8}^{3} ) q^{19} + ( 20 \zeta_{8} + 12 \zeta_{8}^{2} + 20 \zeta_{8}^{3} ) q^{23} + ( 8 + 12 \zeta_{8} - 12 \zeta_{8}^{3} ) q^{25} -42 \zeta_{8}^{2} q^{29} + ( -1 + 6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{31} + ( -24 \zeta_{8} + 33 \zeta_{8}^{2} - 24 \zeta_{8}^{3} ) q^{35} + ( -36 + 12 \zeta_{8} - 12 \zeta_{8}^{3} ) q^{37} + ( 36 \zeta_{8} - 12 \zeta_{8}^{2} + 36 \zeta_{8}^{3} ) q^{41} + ( -14 - 12 \zeta_{8} + 12 \zeta_{8}^{3} ) q^{43} + ( -24 \zeta_{8} + 42 \zeta_{8}^{2} - 24 \zeta_{8}^{3} ) q^{47} + ( 32 - 36 \zeta_{8} + 36 \zeta_{8}^{3} ) q^{49} + ( -34 \zeta_{8} - 51 \zeta_{8}^{2} - 34 \zeta_{8}^{3} ) q^{53} + ( 43 - 30 \zeta_{8} + 30 \zeta_{8}^{3} ) q^{55} + ( 56 \zeta_{8} + 18 \zeta_{8}^{2} + 56 \zeta_{8}^{3} ) q^{59} + ( 52 - 48 \zeta_{8} + 48 \zeta_{8}^{3} ) q^{61} + ( -40 \zeta_{8} + 54 \zeta_{8}^{2} - 40 \zeta_{8}^{3} ) q^{65} + ( -58 - 12 \zeta_{8} + 12 \zeta_{8}^{3} ) q^{67} + ( -20 \zeta_{8} - 18 \zeta_{8}^{2} - 20 \zeta_{8}^{3} ) q^{71} + ( -71 - 36 \zeta_{8} + 36 \zeta_{8}^{3} ) q^{73} + ( 66 \zeta_{8} - 75 \zeta_{8}^{2} + 66 \zeta_{8}^{3} ) q^{77} + ( -74 - 24 \zeta_{8} + 24 \zeta_{8}^{3} ) q^{79} + ( 28 \zeta_{8} - 69 \zeta_{8}^{2} + 28 \zeta_{8}^{3} ) q^{83} + ( 14 - 12 \zeta_{8} + 12 \zeta_{8}^{3} ) q^{85} + ( -12 \zeta_{8} + 90 \zeta_{8}^{2} - 12 \zeta_{8}^{3} ) q^{89} + ( 150 - 48 \zeta_{8} + 48 \zeta_{8}^{3} ) q^{91} + ( -20 \zeta_{8} + 24 \zeta_{8}^{2} - 20 \zeta_{8}^{3} ) q^{95} + ( 11 + 48 \zeta_{8} - 48 \zeta_{8}^{3} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 12q^{7} + O(q^{10}) \) \( 4q - 12q^{7} - 8q^{13} + 32q^{19} + 32q^{25} - 4q^{31} - 144q^{37} - 56q^{43} + 128q^{49} + 172q^{55} + 208q^{61} - 232q^{67} - 284q^{73} - 296q^{79} + 56q^{85} + 600q^{91} + 44q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/216\mathbb{Z}\right)^\times\).

\(n\) \(55\) \(109\) \(137\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
161.1
−0.707107 0.707107i
0.707107 + 0.707107i
0.707107 0.707107i
−0.707107 + 0.707107i
0 0 0 5.82843i 0 −11.4853 0 0 0
161.2 0 0 0 0.171573i 0 5.48528 0 0 0
161.3 0 0 0 0.171573i 0 5.48528 0 0 0
161.4 0 0 0 5.82843i 0 −11.4853 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 216.3.e.c 4
3.b odd 2 1 inner 216.3.e.c 4
4.b odd 2 1 432.3.e.h 4
8.b even 2 1 1728.3.e.p 4
8.d odd 2 1 1728.3.e.s 4
9.c even 3 2 648.3.m.f 8
9.d odd 6 2 648.3.m.f 8
12.b even 2 1 432.3.e.h 4
24.f even 2 1 1728.3.e.s 4
24.h odd 2 1 1728.3.e.p 4
36.f odd 6 2 1296.3.q.l 8
36.h even 6 2 1296.3.q.l 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
216.3.e.c 4 1.a even 1 1 trivial
216.3.e.c 4 3.b odd 2 1 inner
432.3.e.h 4 4.b odd 2 1
432.3.e.h 4 12.b even 2 1
648.3.m.f 8 9.c even 3 2
648.3.m.f 8 9.d odd 6 2
1296.3.q.l 8 36.f odd 6 2
1296.3.q.l 8 36.h even 6 2
1728.3.e.p 4 8.b even 2 1
1728.3.e.p 4 24.h odd 2 1
1728.3.e.s 4 8.d odd 2 1
1728.3.e.s 4 24.f even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 34 T_{5}^{2} + 1 \) acting on \(S_{3}^{\mathrm{new}}(216, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( 1 + 34 T^{2} + T^{4} \)
$7$ \( ( -63 + 6 T + T^{2} )^{2} \)
$11$ \( 2401 + 226 T^{2} + T^{4} \)
$13$ \( ( -284 + 4 T + T^{2} )^{2} \)
$17$ \( 8464 + 328 T^{2} + T^{4} \)
$19$ \( ( -224 - 16 T + T^{2} )^{2} \)
$23$ \( 430336 + 1888 T^{2} + T^{4} \)
$29$ \( ( 1764 + T^{2} )^{2} \)
$31$ \( ( -71 + 2 T + T^{2} )^{2} \)
$37$ \( ( 1008 + 72 T + T^{2} )^{2} \)
$41$ \( 5992704 + 5472 T^{2} + T^{4} \)
$43$ \( ( -92 + 28 T + T^{2} )^{2} \)
$47$ \( 374544 + 5832 T^{2} + T^{4} \)
$53$ \( 83521 + 9826 T^{2} + T^{4} \)
$59$ \( 35378704 + 13192 T^{2} + T^{4} \)
$61$ \( ( -1904 - 104 T + T^{2} )^{2} \)
$67$ \( ( 3076 + 116 T + T^{2} )^{2} \)
$71$ \( 226576 + 2248 T^{2} + T^{4} \)
$73$ \( ( 2449 + 142 T + T^{2} )^{2} \)
$79$ \( ( 4324 + 148 T + T^{2} )^{2} \)
$83$ \( 10195249 + 12658 T^{2} + T^{4} \)
$89$ \( 61027344 + 16776 T^{2} + T^{4} \)
$97$ \( ( -4487 - 22 T + T^{2} )^{2} \)
show more
show less