Properties

Label 648.4.i.q
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: \(\Q(\sqrt{-3}, \sqrt{-43})\)
Defining polynomial: \(x^{4} - x^{3} - 10 x^{2} - 11 x + 121\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + ( 2 + 2 \beta_{1} + \beta_{3} ) q^{5} + ( -3 \beta_{1} + \beta_{2} - \beta_{3} ) q^{7} +O(q^{10})\) \( q + ( 2 + 2 \beta_{1} + \beta_{3} ) q^{5} + ( -3 \beta_{1} + \beta_{2} - \beta_{3} ) q^{7} + ( -5 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{11} + ( 20 + 20 \beta_{1} + 5 \beta_{3} ) q^{13} + ( 17 + 4 \beta_{2} ) q^{17} + ( 17 - 5 \beta_{2} ) q^{19} + ( -49 - 49 \beta_{1} - 7 \beta_{3} ) q^{23} + ( 8 \beta_{1} - 4 \beta_{2} + 4 \beta_{3} ) q^{25} + ( 60 \beta_{1} + 11 \beta_{2} - 11 \beta_{3} ) q^{29} + ( 100 + 100 \beta_{1} - 8 \beta_{3} ) q^{31} + ( 135 + 5 \beta_{2} ) q^{35} + ( 240 + 7 \beta_{2} ) q^{37} + ( -96 - 96 \beta_{1} + 10 \beta_{3} ) q^{41} + ( -167 \beta_{1} + 25 \beta_{2} - 25 \beta_{3} ) q^{43} + ( 150 \beta_{1} - 38 \beta_{2} + 38 \beta_{3} ) q^{47} + ( 205 + 205 \beta_{1} - 6 \beta_{3} ) q^{49} + ( 34 - 32 \beta_{2} ) q^{53} + ( 397 + 11 \beta_{2} ) q^{55} + ( -310 - 310 \beta_{1} + 6 \beta_{3} ) q^{59} + ( 100 \beta_{1} - 35 \beta_{2} + 35 \beta_{3} ) q^{61} + ( 685 \beta_{1} - 30 \beta_{2} + 30 \beta_{3} ) q^{65} + ( 703 + 703 \beta_{1} - 11 \beta_{3} ) q^{67} + ( 695 - 3 \beta_{2} ) q^{71} + ( 901 - 18 \beta_{2} ) q^{73} + ( -402 - 402 \beta_{1} - 14 \beta_{3} ) q^{77} + ( -167 \beta_{1} - 79 \beta_{2} + 79 \beta_{3} ) q^{79} + ( -250 \beta_{1} + 74 \beta_{2} - 74 \beta_{3} ) q^{83} + ( 550 + 550 \beta_{1} + 25 \beta_{3} ) q^{85} + ( 45 + 24 \beta_{2} ) q^{89} + ( 705 + 35 \beta_{2} ) q^{91} + ( -611 - 611 \beta_{1} + 7 \beta_{3} ) q^{95} + ( -100 \beta_{1} + 122 \beta_{2} - 122 \beta_{3} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{5} + 6q^{7} + O(q^{10}) \) \( 4q + 4q^{5} + 6q^{7} + 10q^{11} + 40q^{13} + 68q^{17} + 68q^{19} - 98q^{23} - 16q^{25} - 120q^{29} + 200q^{31} + 540q^{35} + 960q^{37} - 192q^{41} + 334q^{43} - 300q^{47} + 410q^{49} + 136q^{53} + 1588q^{55} - 620q^{59} - 200q^{61} - 1370q^{65} + 1406q^{67} + 2780q^{71} + 3604q^{73} - 804q^{77} + 334q^{79} + 500q^{83} + 1100q^{85} + 180q^{89} + 2820q^{91} - 1222q^{95} + 200q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 10 x^{2} - 11 x + 121\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} + 10 \nu^{2} - 10 \nu - 121 \)\()/110\)
\(\beta_{2}\)\(=\)\((\)\( -2 \nu^{3} + 2 \nu^{2} + 42 \nu + 11 \)\()/11\)
\(\beta_{3}\)\(=\)\((\)\( 23 \nu^{3} + 10 \nu^{2} + 210 \nu - 473 \)\()/110\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{2} - 3 \beta_{1}\)\()/6\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{3} + 2 \beta_{2} + 63 \beta_{1} + 63\)\()/6\)
\(\nu^{3}\)\(=\)\((\)\(10 \beta_{3} - 5 \beta_{2} + 48\)\()/3\)

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
−2.58945 2.07237i
3.08945 + 1.20635i
−2.58945 + 2.07237i
3.08945 1.20635i
0 0 0 −4.67891 8.10411i 0 −4.17891 + 7.23808i 0 0 0
217.2 0 0 0 6.67891 + 11.5682i 0 7.17891 12.4342i 0 0 0
433.1 0 0 0 −4.67891 + 8.10411i 0 −4.17891 7.23808i 0 0 0
433.2 0 0 0 6.67891 11.5682i 0 7.17891 + 12.4342i 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.q 4
3.b odd 2 1 648.4.i.p 4
9.c even 3 1 648.4.a.d 2
9.c even 3 1 inner 648.4.i.q 4
9.d odd 6 1 648.4.a.e yes 2
9.d odd 6 1 648.4.i.p 4
36.f odd 6 1 1296.4.a.n 2
36.h even 6 1 1296.4.a.p 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
648.4.a.d 2 9.c even 3 1
648.4.a.e yes 2 9.d odd 6 1
648.4.i.p 4 3.b odd 2 1
648.4.i.p 4 9.d odd 6 1
648.4.i.q 4 1.a even 1 1 trivial
648.4.i.q 4 9.c even 3 1 inner
1296.4.a.n 2 36.f odd 6 1
1296.4.a.p 2 36.h even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 4 T_{5}^{3} + 141 T_{5}^{2} + 500 T_{5} + 15625 \) acting on \(S_{4}^{\mathrm{new}}(648, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( 15625 + 500 T + 141 T^{2} - 4 T^{3} + T^{4} \)
$7$ \( 14400 + 720 T + 156 T^{2} - 6 T^{3} + T^{4} \)
$11$ \( 1290496 + 11360 T + 1236 T^{2} - 10 T^{3} + T^{4} \)
$13$ \( 7980625 + 113000 T + 4425 T^{2} - 40 T^{3} + T^{4} \)
$17$ \( ( -1775 - 34 T + T^{2} )^{2} \)
$19$ \( ( -2936 - 34 T + T^{2} )^{2} \)
$23$ \( 15366400 - 384160 T + 13524 T^{2} + 98 T^{3} + T^{4} \)
$29$ \( 144216081 - 1441080 T + 26409 T^{2} + 120 T^{3} + T^{4} \)
$31$ \( 3041536 - 348800 T + 38256 T^{2} - 200 T^{3} + T^{4} \)
$37$ \( ( 51279 - 480 T + T^{2} )^{2} \)
$41$ \( 13571856 - 707328 T + 40548 T^{2} + 192 T^{3} + T^{4} \)
$43$ \( 2781085696 + 17613824 T + 164292 T^{2} - 334 T^{3} + T^{4} \)
$47$ \( 26822578176 - 49132800 T + 253776 T^{2} + 300 T^{3} + T^{4} \)
$53$ \( ( -130940 - 68 T + T^{2} )^{2} \)
$59$ \( 8364199936 + 56702720 T + 292944 T^{2} + 620 T^{3} + T^{4} \)
$61$ \( 21911400625 - 29605000 T + 188025 T^{2} + 200 T^{3} + T^{4} \)
$67$ \( 229057960000 - 672911600 T + 1498236 T^{2} - 1406 T^{3} + T^{4} \)
$71$ \( ( 481864 - 1390 T + T^{2} )^{2} \)
$73$ \( ( 770005 - 1802 T + T^{2} )^{2} \)
$79$ \( 604039840000 + 259584800 T + 888756 T^{2} - 334 T^{3} + T^{4} \)
$83$ \( 414612361216 + 321952000 T + 893904 T^{2} - 500 T^{3} + T^{4} \)
$89$ \( ( -72279 - 90 T + T^{2} )^{2} \)
$97$ \( 3648237521296 + 382007200 T + 1950036 T^{2} - 200 T^{3} + T^{4} \)
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