Properties

Label 648.4.i.j
Level $648$
Weight $4$
Character orbit 648.i
Analytic conductor $38.233$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 648 = 2^{3} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 648.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(38.2332376837\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 5 \zeta_{6} q^{5} + ( -36 + 36 \zeta_{6} ) q^{7} +O(q^{10})\) \( q + 5 \zeta_{6} q^{5} + ( -36 + 36 \zeta_{6} ) q^{7} + ( -64 + 64 \zeta_{6} ) q^{11} + 65 \zeta_{6} q^{13} + 59 q^{17} -28 q^{19} -160 \zeta_{6} q^{23} + ( 100 - 100 \zeta_{6} ) q^{25} + ( 57 - 57 \zeta_{6} ) q^{29} -164 \zeta_{6} q^{31} -180 q^{35} -321 q^{37} + 246 \zeta_{6} q^{41} + ( 8 - 8 \zeta_{6} ) q^{43} + ( -84 + 84 \zeta_{6} ) q^{47} -953 \zeta_{6} q^{49} + 478 q^{53} -320 q^{55} + 32 \zeta_{6} q^{59} + ( -415 + 415 \zeta_{6} ) q^{61} + ( -325 + 325 \zeta_{6} ) q^{65} + 220 \zeta_{6} q^{67} + 884 q^{71} -77 q^{73} -2304 \zeta_{6} q^{77} + ( 80 - 80 \zeta_{6} ) q^{79} + ( -1268 + 1268 \zeta_{6} ) q^{83} + 295 \zeta_{6} q^{85} + 123 q^{89} -2340 q^{91} -140 \zeta_{6} q^{95} + ( -1346 + 1346 \zeta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 5q^{5} - 36q^{7} + O(q^{10}) \) \( 2q + 5q^{5} - 36q^{7} - 64q^{11} + 65q^{13} + 118q^{17} - 56q^{19} - 160q^{23} + 100q^{25} + 57q^{29} - 164q^{31} - 360q^{35} - 642q^{37} + 246q^{41} + 8q^{43} - 84q^{47} - 953q^{49} + 956q^{53} - 640q^{55} + 32q^{59} - 415q^{61} - 325q^{65} + 220q^{67} + 1768q^{71} - 154q^{73} - 2304q^{77} + 80q^{79} - 1268q^{83} + 295q^{85} + 246q^{89} - 4680q^{91} - 140q^{95} - 1346q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/648\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(487\) \(569\)
\(\chi(n)\) \(1\) \(1\) \(-\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
217.1
0.500000 + 0.866025i
0.500000 0.866025i
0 0 0 2.50000 + 4.33013i 0 −18.0000 + 31.1769i 0 0 0
433.1 0 0 0 2.50000 4.33013i 0 −18.0000 31.1769i 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
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 648.4.i.j 2
3.b odd 2 1 648.4.i.c 2
9.c even 3 1 648.4.a.a 1
9.c even 3 1 inner 648.4.i.j 2
9.d odd 6 1 648.4.a.b yes 1
9.d odd 6 1 648.4.i.c 2
36.f odd 6 1 1296.4.a.c 1
36.h even 6 1 1296.4.a.f 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
648.4.a.a 1 9.c even 3 1
648.4.a.b yes 1 9.d odd 6 1
648.4.i.c 2 3.b odd 2 1
648.4.i.c 2 9.d odd 6 1
648.4.i.j 2 1.a even 1 1 trivial
648.4.i.j 2 9.c even 3 1 inner
1296.4.a.c 1 36.f odd 6 1
1296.4.a.f 1 36.h even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 5 T_{5} + 25 \) acting on \(S_{4}^{\mathrm{new}}(648, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( 25 - 5 T + T^{2} \)
$7$ \( 1296 + 36 T + T^{2} \)
$11$ \( 4096 + 64 T + T^{2} \)
$13$ \( 4225 - 65 T + T^{2} \)
$17$ \( ( -59 + T )^{2} \)
$19$ \( ( 28 + T )^{2} \)
$23$ \( 25600 + 160 T + T^{2} \)
$29$ \( 3249 - 57 T + T^{2} \)
$31$ \( 26896 + 164 T + T^{2} \)
$37$ \( ( 321 + T )^{2} \)
$41$ \( 60516 - 246 T + T^{2} \)
$43$ \( 64 - 8 T + T^{2} \)
$47$ \( 7056 + 84 T + T^{2} \)
$53$ \( ( -478 + T )^{2} \)
$59$ \( 1024 - 32 T + T^{2} \)
$61$ \( 172225 + 415 T + T^{2} \)
$67$ \( 48400 - 220 T + T^{2} \)
$71$ \( ( -884 + T )^{2} \)
$73$ \( ( 77 + T )^{2} \)
$79$ \( 6400 - 80 T + T^{2} \)
$83$ \( 1607824 + 1268 T + T^{2} \)
$89$ \( ( -123 + T )^{2} \)
$97$ \( 1811716 + 1346 T + T^{2} \)
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