Properties

Label 648.4.i.s
Level $648$
Weight $4$
Character orbit 648.i
Analytic conductor $38.233$
Analytic rank $0$
Dimension $4$
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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-11})\)
Defining polynomial: \(x^{4} - x^{3} - 2 x^{2} - 3 x + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{3} \)
Twist minimal: no (minimal twist has level 216)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( 4 + 4 \beta_{1} + \beta_{3} ) q^{5} + ( 12 \beta_{1} - \beta_{2} + \beta_{3} ) q^{7} +O(q^{10})\) \( q + ( 4 + 4 \beta_{1} + \beta_{3} ) q^{5} + ( 12 \beta_{1} - \beta_{2} + \beta_{3} ) q^{7} -\beta_{1} q^{11} + ( -16 - 16 \beta_{1} + 4 \beta_{3} ) q^{13} + ( 28 + 4 \beta_{2} ) q^{17} + ( 92 + 2 \beta_{2} ) q^{19} + ( 46 + 46 \beta_{1} - 4 \beta_{3} ) q^{23} + ( 188 \beta_{1} - 8 \beta_{2} + 8 \beta_{3} ) q^{25} + ( 168 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{29} + ( -188 - 188 \beta_{1} + 5 \beta_{3} ) q^{31} + ( -345 - 16 \beta_{2} ) q^{35} + ( 174 - 4 \beta_{2} ) q^{37} + ( -156 - 156 \beta_{1} + 10 \beta_{3} ) q^{41} + ( 40 \beta_{1} + 14 \beta_{2} - 14 \beta_{3} ) q^{43} + ( 114 \beta_{1} - 8 \beta_{2} + 8 \beta_{3} ) q^{47} + ( -98 - 98 \beta_{1} - 24 \beta_{3} ) q^{49} + ( -76 + 13 \beta_{2} ) q^{53} + ( 4 + \beta_{2} ) q^{55} + ( 340 + 340 \beta_{1} - 24 \beta_{3} ) q^{59} + ( -56 \beta_{1} + 32 \beta_{2} - 32 \beta_{3} ) q^{61} + 1124 \beta_{1} q^{65} + ( -176 - 176 \beta_{1} - 34 \beta_{3} ) q^{67} + ( -908 + 12 \beta_{2} ) q^{71} -287 q^{73} + ( 12 + 12 \beta_{1} + \beta_{3} ) q^{77} + ( -680 \beta_{1} - 32 \beta_{2} + 32 \beta_{3} ) q^{79} + ( 391 \beta_{1} + 32 \beta_{2} - 32 \beta_{3} ) q^{83} + ( 1300 + 1300 \beta_{1} + 44 \beta_{3} ) q^{85} + ( -120 - 18 \beta_{2} ) q^{89} + ( -996 - 32 \beta_{2} ) q^{91} + ( 962 + 962 \beta_{1} + 100 \beta_{3} ) q^{95} + ( -169 \beta_{1} - 8 \beta_{2} + 8 \beta_{3} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 8q^{5} - 24q^{7} + O(q^{10}) \) \( 4q + 8q^{5} - 24q^{7} + 2q^{11} - 32q^{13} + 112q^{17} + 368q^{19} + 92q^{23} - 376q^{25} - 336q^{29} - 376q^{31} - 1380q^{35} + 696q^{37} - 312q^{41} - 80q^{43} - 228q^{47} - 196q^{49} - 304q^{53} + 16q^{55} + 680q^{59} + 112q^{61} - 2248q^{65} - 352q^{67} - 3632q^{71} - 1148q^{73} + 24q^{77} + 1360q^{79} - 782q^{83} + 2600q^{85} - 480q^{89} - 3984q^{91} + 1924q^{95} + 338q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 2 x^{2} - 3 x + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} + 2 \nu^{2} - 2 \nu - 9 \)\()/6\)
\(\beta_{2}\)\(=\)\( -2 \nu^{3} + 2 \nu^{2} + 10 \nu + 3 \)
\(\beta_{3}\)\(=\)\((\)\( 7 \nu^{3} + 2 \nu^{2} + 10 \nu - 33 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{2} - 9 \beta_{1}\)\()/18\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{3} + 2 \beta_{2} + 45 \beta_{1} + 45\)\()/18\)
\(\nu^{3}\)\(=\)\((\)\(2 \beta_{3} - \beta_{2} + 36\)\()/9\)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
217.1
−1.18614 1.26217i
1.68614 + 0.396143i
−1.18614 + 1.26217i
1.68614 0.396143i
0 0 0 −6.61684 11.4607i 0 2.61684 4.53251i 0 0 0
217.2 0 0 0 10.6168 + 18.3889i 0 −14.6168 + 25.3171i 0 0 0
433.1 0 0 0 −6.61684 + 11.4607i 0 2.61684 + 4.53251i 0 0 0
433.2 0 0 0 10.6168 18.3889i 0 −14.6168 25.3171i 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.s 4
3.b odd 2 1 648.4.i.m 4
9.c even 3 1 216.4.a.e 2
9.c even 3 1 inner 648.4.i.s 4
9.d odd 6 1 216.4.a.h yes 2
9.d odd 6 1 648.4.i.m 4
36.f odd 6 1 432.4.a.o 2
36.h even 6 1 432.4.a.s 2
72.j odd 6 1 1728.4.a.bh 2
72.l even 6 1 1728.4.a.bg 2
72.n even 6 1 1728.4.a.bt 2
72.p odd 6 1 1728.4.a.bs 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
216.4.a.e 2 9.c even 3 1
216.4.a.h yes 2 9.d odd 6 1
432.4.a.o 2 36.f odd 6 1
432.4.a.s 2 36.h even 6 1
648.4.i.m 4 3.b odd 2 1
648.4.i.m 4 9.d odd 6 1
648.4.i.s 4 1.a even 1 1 trivial
648.4.i.s 4 9.c even 3 1 inner
1728.4.a.bg 2 72.l even 6 1
1728.4.a.bh 2 72.j odd 6 1
1728.4.a.bs 2 72.p odd 6 1
1728.4.a.bt 2 72.n even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 8 T_{5}^{3} + 345 T_{5}^{2} + 2248 T_{5} + 78961 \) acting on \(S_{4}^{\mathrm{new}}(648, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( 78961 + 2248 T + 345 T^{2} - 8 T^{3} + T^{4} \)
$7$ \( 23409 - 3672 T + 729 T^{2} + 24 T^{3} + T^{4} \)
$11$ \( ( 1 - T + T^{2} )^{2} \)
$13$ \( 20214016 - 143872 T + 5520 T^{2} + 32 T^{3} + T^{4} \)
$17$ \( ( -3968 - 56 T + T^{2} )^{2} \)
$19$ \( ( 7276 - 184 T + T^{2} )^{2} \)
$23$ \( 6948496 + 242512 T + 11100 T^{2} - 92 T^{3} + T^{4} \)
$29$ \( 730945296 + 9084096 T + 85860 T^{2} + 336 T^{3} + T^{4} \)
$31$ \( 779470561 + 10497544 T + 113457 T^{2} + 376 T^{3} + T^{4} \)
$37$ \( ( 25524 - 348 T + T^{2} )^{2} \)
$41$ \( 28772496 - 1673568 T + 102708 T^{2} + 312 T^{3} + T^{4} \)
$43$ \( 3204918544 - 4528960 T + 63012 T^{2} + 80 T^{3} + T^{4} \)
$47$ \( 36144144 - 1370736 T + 57996 T^{2} + 228 T^{3} + T^{4} \)
$53$ \( ( -44417 + 152 T + T^{2} )^{2} \)
$59$ \( 3077142784 + 37720960 T + 517872 T^{2} - 680 T^{3} + T^{4} \)
$61$ \( 90596184064 + 33711104 T + 313536 T^{2} - 112 T^{3} + T^{4} \)
$67$ \( 97566270736 - 109949312 T + 436260 T^{2} + 352 T^{3} + T^{4} \)
$71$ \( ( 781696 + 1816 T + T^{2} )^{2} \)
$73$ \( ( 287 + T )^{4} \)
$79$ \( 25050025984 - 215249920 T + 1691328 T^{2} - 1360 T^{3} + T^{4} \)
$83$ \( 22875655009 - 118275154 T + 762771 T^{2} + 782 T^{3} + T^{4} \)
$89$ \( ( -81828 + 240 T + T^{2} )^{2} \)
$97$ \( 91259809 - 3228914 T + 104691 T^{2} - 338 T^{3} + T^{4} \)
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