Properties

Label 648.4.i.o
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: \(\Q(\sqrt{-3}, \sqrt{5})\)
Defining polynomial: \(x^{4} - x^{3} + 2 x^{2} + x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{2} \)
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.

\(f(q)\) \(=\) \( q + ( -2 \beta_{1} - \beta_{2} ) q^{5} + ( 3 - 3 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{7} +O(q^{10})\) \( q + ( -2 \beta_{1} - \beta_{2} ) q^{5} + ( 3 - 3 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{7} + ( -26 + 26 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} ) q^{11} + ( -13 \beta_{1} + 2 \beta_{2} ) q^{13} + ( 94 + \beta_{3} ) q^{17} + ( -37 + 8 \beta_{3} ) q^{19} + ( -74 \beta_{1} - 5 \beta_{2} ) q^{23} + ( -59 + 59 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} ) q^{25} + ( -144 + 144 \beta_{1} + 8 \beta_{2} + 8 \beta_{3} ) q^{29} + ( 124 \beta_{1} - 8 \beta_{2} ) q^{31} + ( 354 - \beta_{3} ) q^{35} + ( 171 - 10 \beta_{3} ) q^{37} + 32 \beta_{2} q^{41} + ( 128 - 128 \beta_{1} - 4 \beta_{2} - 4 \beta_{3} ) q^{43} + ( -66 + 66 \beta_{1} + \beta_{2} + \beta_{3} ) q^{47} + ( -386 \beta_{1} + 12 \beta_{2} ) q^{49} + ( 476 - 14 \beta_{3} ) q^{53} + ( -488 - 20 \beta_{3} ) q^{55} + ( 502 \beta_{1} - 21 \beta_{2} ) q^{59} + ( 17 - 17 \beta_{1} - 34 \beta_{2} - 34 \beta_{3} ) q^{61} + ( 334 - 334 \beta_{1} + 9 \beta_{2} + 9 \beta_{3} ) q^{65} + ( 433 \beta_{1} + 16 \beta_{2} ) q^{67} + ( 388 + 30 \beta_{3} ) q^{71} + ( 937 + 12 \beta_{3} ) q^{73} + ( 1158 \beta_{1} - 61 \beta_{2} ) q^{77} + ( -91 + 91 \beta_{1} - 26 \beta_{2} - 26 \beta_{3} ) q^{79} + ( 668 - 668 \beta_{1} - 46 \beta_{2} - 46 \beta_{3} ) q^{83} + ( -8 \beta_{1} - 92 \beta_{2} ) q^{85} + ( 438 + 21 \beta_{3} ) q^{89} + ( -759 - 32 \beta_{3} ) q^{91} + ( 1514 \beta_{1} + 53 \beta_{2} ) q^{95} + ( 19 - 19 \beta_{1} + 88 \beta_{2} + 88 \beta_{3} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{5} + 6q^{7} + O(q^{10}) \) \( 4q - 4q^{5} + 6q^{7} - 52q^{11} - 26q^{13} + 376q^{17} - 148q^{19} - 148q^{23} - 118q^{25} - 288q^{29} + 248q^{31} + 1416q^{35} + 684q^{37} + 256q^{43} - 132q^{47} - 772q^{49} + 1904q^{53} - 1952q^{55} + 1004q^{59} + 34q^{61} + 668q^{65} + 866q^{67} + 1552q^{71} + 3748q^{73} + 2316q^{77} - 182q^{79} + 1336q^{83} - 16q^{85} + 1752q^{89} - 3036q^{91} + 3028q^{95} + 38q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} + 2 x^{2} + x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{3} + 2 \nu^{2} - 2 \nu + 1 \)\()/2\)
\(\beta_{2}\)\(=\)\( 3 \nu^{3} - 6 \nu^{2} + 18 \nu - 3 \)
\(\beta_{3}\)\(=\)\( 6 \nu^{3} + 12 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} + 6 \beta_{1}\)\()/12\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} + \beta_{2} + 18 \beta_{1} - 18\)\()/12\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{3} - 12\)\()/6\)

Character values

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

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

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
0.809017 + 1.40126i
−0.309017 0.535233i
0.809017 1.40126i
−0.309017 + 0.535233i
0 0 0 −7.70820 13.3510i 0 −11.9164 + 20.6398i 0 0 0
217.2 0 0 0 5.70820 + 9.88690i 0 14.9164 25.8360i 0 0 0
433.1 0 0 0 −7.70820 + 13.3510i 0 −11.9164 20.6398i 0 0 0
433.2 0 0 0 5.70820 9.88690i 0 14.9164 + 25.8360i 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.o 4
3.b odd 2 1 648.4.i.r 4
9.c even 3 1 216.4.a.g yes 2
9.c even 3 1 inner 648.4.i.o 4
9.d odd 6 1 216.4.a.f 2
9.d odd 6 1 648.4.i.r 4
36.f odd 6 1 432.4.a.r 2
36.h even 6 1 432.4.a.p 2
72.j odd 6 1 1728.4.a.bq 2
72.l even 6 1 1728.4.a.br 2
72.n even 6 1 1728.4.a.bi 2
72.p odd 6 1 1728.4.a.bj 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
216.4.a.f 2 9.d odd 6 1
216.4.a.g yes 2 9.c even 3 1
432.4.a.p 2 36.h even 6 1
432.4.a.r 2 36.f odd 6 1
648.4.i.o 4 1.a even 1 1 trivial
648.4.i.o 4 9.c even 3 1 inner
648.4.i.r 4 3.b odd 2 1
648.4.i.r 4 9.d odd 6 1
1728.4.a.bi 2 72.n even 6 1
1728.4.a.bj 2 72.p odd 6 1
1728.4.a.bq 2 72.j odd 6 1
1728.4.a.br 2 72.l even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 4 T_{5}^{3} + 192 T_{5}^{2} - 704 T_{5} + 30976 \) acting on \(S_{4}^{\mathrm{new}}(648, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( 30976 - 704 T + 192 T^{2} + 4 T^{3} + T^{4} \)
$7$ \( 505521 + 4266 T + 747 T^{2} - 6 T^{3} + T^{4} \)
$11$ \( 891136 - 49088 T + 3648 T^{2} + 52 T^{3} + T^{4} \)
$13$ \( 303601 - 14326 T + 1227 T^{2} + 26 T^{3} + T^{4} \)
$17$ \( ( 8656 - 188 T + T^{2} )^{2} \)
$19$ \( ( -10151 + 74 T + T^{2} )^{2} \)
$23$ \( 952576 + 144448 T + 20928 T^{2} + 148 T^{3} + T^{4} \)
$29$ \( 84934656 + 2654208 T + 73728 T^{2} + 288 T^{3} + T^{4} \)
$31$ \( 14868736 - 956288 T + 57648 T^{2} - 248 T^{3} + T^{4} \)
$37$ \( ( 11241 - 342 T + T^{2} )^{2} \)
$41$ \( 33973862400 + 184320 T^{2} + T^{4} \)
$43$ \( 182358016 - 3457024 T + 52032 T^{2} - 256 T^{3} + T^{4} \)
$47$ \( 17438976 + 551232 T + 13248 T^{2} + 132 T^{3} + T^{4} \)
$53$ \( ( 191296 - 952 T + T^{2} )^{2} \)
$59$ \( 29799045376 - 173314496 T + 835392 T^{2} - 1004 T^{3} + T^{4} \)
$61$ \( 43177099681 + 7064894 T + 208947 T^{2} - 34 T^{3} + T^{4} \)
$67$ \( 19996505281 - 122460194 T + 608547 T^{2} - 866 T^{3} + T^{4} \)
$71$ \( ( -11456 - 776 T + T^{2} )^{2} \)
$73$ \( ( 852049 - 1874 T + T^{2} )^{2} \)
$79$ \( 12859333201 - 20638618 T + 146523 T^{2} + 182 T^{3} + T^{4} \)
$83$ \( 4269838336 - 87299584 T + 1719552 T^{2} - 1336 T^{3} + T^{4} \)
$89$ \( ( 112464 - 876 T + T^{2} )^{2} \)
$97$ \( 1942006686481 + 52955242 T + 1395003 T^{2} - 38 T^{3} + T^{4} \)
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