Properties

Label 648.3.m.d
Level $648$
Weight $3$
Character orbit 648.m
Analytic conductor $17.657$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 648 = 2^{3} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 648.m (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(17.6567211305\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} - 2 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 24)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{5} + 6 \beta_{2} q^{7} +O(q^{10})\) \( q + \beta_{1} q^{5} + 6 \beta_{2} q^{7} + ( -\beta_{1} + \beta_{3} ) q^{11} + ( -10 + 10 \beta_{2} ) q^{13} + 4 \beta_{3} q^{17} + 2 q^{19} -2 \beta_{1} q^{23} + 7 \beta_{2} q^{25} + ( -3 \beta_{1} + 3 \beta_{3} ) q^{29} + ( 22 - 22 \beta_{2} ) q^{31} + 6 \beta_{3} q^{35} -6 q^{37} + 6 \beta_{1} q^{41} -82 \beta_{2} q^{43} + ( -12 \beta_{1} + 12 \beta_{3} ) q^{47} + ( 13 - 13 \beta_{2} ) q^{49} + 11 \beta_{3} q^{53} -32 q^{55} + 13 \beta_{1} q^{59} + 86 \beta_{2} q^{61} + ( -10 \beta_{1} + 10 \beta_{3} ) q^{65} + ( -2 + 2 \beta_{2} ) q^{67} + 22 \beta_{3} q^{71} + 82 q^{73} -6 \beta_{1} q^{77} -10 \beta_{2} q^{79} + ( 13 \beta_{1} - 13 \beta_{3} ) q^{83} + ( -128 + 128 \beta_{2} ) q^{85} + 6 \beta_{3} q^{89} -60 q^{91} + 2 \beta_{1} q^{95} + 94 \beta_{2} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 12q^{7} + O(q^{10}) \) \( 4q + 12q^{7} - 20q^{13} + 8q^{19} + 14q^{25} + 44q^{31} - 24q^{37} - 164q^{43} + 26q^{49} - 128q^{55} + 172q^{61} - 4q^{67} + 328q^{73} - 20q^{79} - 256q^{85} - 240q^{91} + 188q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 2 x^{2} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 4 \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/2\)
\(\beta_{3}\)\(=\)\( 2 \nu^{3} \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/4\)
\(\nu^{2}\)\(=\)\(2 \beta_{2}\)
\(\nu^{3}\)\(=\)\(\beta_{3}\)\(/2\)

Character values

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

\(n\) \(325\) \(487\) \(569\)
\(\chi(n)\) \(1\) \(1\) \(1 - \beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
377.1
−1.22474 0.707107i
1.22474 + 0.707107i
−1.22474 + 0.707107i
1.22474 0.707107i
0 0 0 −4.89898 2.82843i 0 3.00000 + 5.19615i 0 0 0
377.2 0 0 0 4.89898 + 2.82843i 0 3.00000 + 5.19615i 0 0 0
593.1 0 0 0 −4.89898 + 2.82843i 0 3.00000 5.19615i 0 0 0
593.2 0 0 0 4.89898 2.82843i 0 3.00000 5.19615i 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
9.c even 3 1 inner
9.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 648.3.m.d 4
3.b odd 2 1 inner 648.3.m.d 4
4.b odd 2 1 1296.3.q.e 4
9.c even 3 1 24.3.e.a 2
9.c even 3 1 inner 648.3.m.d 4
9.d odd 6 1 24.3.e.a 2
9.d odd 6 1 inner 648.3.m.d 4
12.b even 2 1 1296.3.q.e 4
36.f odd 6 1 48.3.e.b 2
36.f odd 6 1 1296.3.q.e 4
36.h even 6 1 48.3.e.b 2
36.h even 6 1 1296.3.q.e 4
45.h odd 6 1 600.3.l.b 2
45.j even 6 1 600.3.l.b 2
45.k odd 12 2 600.3.c.a 4
45.l even 12 2 600.3.c.a 4
63.l odd 6 1 1176.3.d.a 2
63.o even 6 1 1176.3.d.a 2
72.j odd 6 1 192.3.e.c 2
72.l even 6 1 192.3.e.d 2
72.n even 6 1 192.3.e.c 2
72.p odd 6 1 192.3.e.d 2
144.u even 12 2 768.3.h.c 4
144.v odd 12 2 768.3.h.c 4
144.w odd 12 2 768.3.h.d 4
144.x even 12 2 768.3.h.d 4
180.n even 6 1 1200.3.l.n 2
180.p odd 6 1 1200.3.l.n 2
180.v odd 12 2 1200.3.c.i 4
180.x even 12 2 1200.3.c.i 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.3.e.a 2 9.c even 3 1
24.3.e.a 2 9.d odd 6 1
48.3.e.b 2 36.f odd 6 1
48.3.e.b 2 36.h even 6 1
192.3.e.c 2 72.j odd 6 1
192.3.e.c 2 72.n even 6 1
192.3.e.d 2 72.l even 6 1
192.3.e.d 2 72.p odd 6 1
600.3.c.a 4 45.k odd 12 2
600.3.c.a 4 45.l even 12 2
600.3.l.b 2 45.h odd 6 1
600.3.l.b 2 45.j even 6 1
648.3.m.d 4 1.a even 1 1 trivial
648.3.m.d 4 3.b odd 2 1 inner
648.3.m.d 4 9.c even 3 1 inner
648.3.m.d 4 9.d odd 6 1 inner
768.3.h.c 4 144.u even 12 2
768.3.h.c 4 144.v odd 12 2
768.3.h.d 4 144.w odd 12 2
768.3.h.d 4 144.x even 12 2
1176.3.d.a 2 63.l odd 6 1
1176.3.d.a 2 63.o even 6 1
1200.3.c.i 4 180.v odd 12 2
1200.3.c.i 4 180.x even 12 2
1200.3.l.n 2 180.n even 6 1
1200.3.l.n 2 180.p odd 6 1
1296.3.q.e 4 4.b odd 2 1
1296.3.q.e 4 12.b even 2 1
1296.3.q.e 4 36.f odd 6 1
1296.3.q.e 4 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 \(S_{3}^{\mathrm{new}}(648, [\chi])\):

\( T_{5}^{4} - 32 T_{5}^{2} + 1024 \)
\( T_{7}^{2} - 6 T_{7} + 36 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( 1024 - 32 T^{2} + T^{4} \)
$7$ \( ( 36 - 6 T + T^{2} )^{2} \)
$11$ \( 1024 - 32 T^{2} + T^{4} \)
$13$ \( ( 100 + 10 T + T^{2} )^{2} \)
$17$ \( ( 512 + T^{2} )^{2} \)
$19$ \( ( -2 + T )^{4} \)
$23$ \( 16384 - 128 T^{2} + T^{4} \)
$29$ \( 82944 - 288 T^{2} + T^{4} \)
$31$ \( ( 484 - 22 T + T^{2} )^{2} \)
$37$ \( ( 6 + T )^{4} \)
$41$ \( 1327104 - 1152 T^{2} + T^{4} \)
$43$ \( ( 6724 + 82 T + T^{2} )^{2} \)
$47$ \( 21233664 - 4608 T^{2} + T^{4} \)
$53$ \( ( 3872 + T^{2} )^{2} \)
$59$ \( 29246464 - 5408 T^{2} + T^{4} \)
$61$ \( ( 7396 - 86 T + T^{2} )^{2} \)
$67$ \( ( 4 + 2 T + T^{2} )^{2} \)
$71$ \( ( 15488 + T^{2} )^{2} \)
$73$ \( ( -82 + T )^{4} \)
$79$ \( ( 100 + 10 T + T^{2} )^{2} \)
$83$ \( 29246464 - 5408 T^{2} + T^{4} \)
$89$ \( ( 1152 + T^{2} )^{2} \)
$97$ \( ( 8836 - 94 T + T^{2} )^{2} \)
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