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Properties

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

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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: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Defining polynomial: \(x^{2} + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{5} -3 q^{7} +O(q^{10})\) \( q + \beta q^{5} -3 q^{7} + 7 \beta q^{11} + 7 q^{13} + 5 \beta q^{17} -19 q^{19} + \beta q^{23} + 17 q^{25} + 18 \beta q^{29} -10 q^{31} -3 \beta q^{35} + 63 q^{37} -18 \beta q^{41} -50 q^{43} + 15 \beta q^{47} -40 q^{49} -26 \beta q^{53} -56 q^{55} -35 \beta q^{59} + 79 q^{61} + 7 \beta q^{65} + 77 q^{67} -28 \beta q^{71} -17 q^{73} -21 \beta q^{77} -11 q^{79} + 14 \beta q^{83} -40 q^{85} -15 \beta q^{89} -21 q^{91} -19 \beta q^{95} -97 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 6q^{7} + O(q^{10}) \) \( 2q - 6q^{7} + 14q^{13} - 38q^{19} + 34q^{25} - 20q^{31} + 126q^{37} - 100q^{43} - 80q^{49} - 112q^{55} + 158q^{61} + 154q^{67} - 34q^{73} - 22q^{79} - 80q^{85} - 42q^{91} - 194q^{97} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

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
− 1.41421i
1.41421i
0 0 0 − 2.82843i 0 −3.00000 0 0 0
161.2 0 0 0 2.82843i 0 −3.00000 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.a ✓ 2
3.b odd 2 1 inner 216.3.e.a ✓ 2
4.b odd 2 1 432.3.e.f 2
8.b even 2 1 1728.3.e.h 2
8.d odd 2 1 1728.3.e.j 2
9.c even 3 2 648.3.m.c 4
9.d odd 6 2 648.3.m.c 4
12.b even 2 1 432.3.e.f 2
24.f even 2 1 1728.3.e.j 2
24.h odd 2 1 1728.3.e.h 2
36.f odd 6 2 1296.3.q.g 4
36.h even 6 2 1296.3.q.g 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
216.3.e.a ✓ 2 1.a even 1 1 trivial
216.3.e.a ✓ 2 3.b odd 2 1 inner
432.3.e.f 2 4.b odd 2 1
432.3.e.f 2 12.b even 2 1
648.3.m.c 4 9.c even 3 2
648.3.m.c 4 9.d odd 6 2
1296.3.q.g 4 36.f odd 6 2
1296.3.q.g 4 36.h even 6 2
1728.3.e.h 2 8.b even 2 1
1728.3.e.h 2 24.h odd 2 1
1728.3.e.j 2 8.d odd 2 1
1728.3.e.j 2 24.f even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 8 \) acting on \(S_{3}^{\mathrm{new}}(216, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( 8 + T^{2} \)
$7$ \( ( 3 + T )^{2} \)
$11$ \( 392 + T^{2} \)
$13$ \( ( -7 + T )^{2} \)
$17$ \( 200 + T^{2} \)
$19$ \( ( 19 + T )^{2} \)
$23$ \( 8 + T^{2} \)
$29$ \( 2592 + T^{2} \)
$31$ \( ( 10 + T )^{2} \)
$37$ \( ( -63 + T )^{2} \)
$41$ \( 2592 + T^{2} \)
$43$ \( ( 50 + T )^{2} \)
$47$ \( 1800 + T^{2} \)
$53$ \( 5408 + T^{2} \)
$59$ \( 9800 + T^{2} \)
$61$ \( ( -79 + T )^{2} \)
$67$ \( ( -77 + T )^{2} \)
$71$ \( 6272 + T^{2} \)
$73$ \( ( 17 + T )^{2} \)
$79$ \( ( 11 + T )^{2} \)
$83$ \( 1568 + T^{2} \)
$89$ \( 1800 + T^{2} \)
$97$ \( ( 97 + T )^{2} \)
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