Properties

Label 648.4.i.t
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: \(\Q(\sqrt{-3}, \sqrt{-67})\)
Defining polynomial: \(x^{4} - x^{3} - 16 x^{2} - 17 x + 289\)
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.

\(f(q)\) \(=\) \( q + ( 4 + 4 \beta_{1} - \beta_{3} ) q^{5} + ( -15 \beta_{1} + \beta_{2} - \beta_{3} ) q^{7} +O(q^{10})\) \( q + ( 4 + 4 \beta_{1} - \beta_{3} ) q^{5} + ( -15 \beta_{1} + \beta_{2} - \beta_{3} ) q^{7} + ( 23 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{11} + ( -46 - 46 \beta_{1} - \beta_{3} ) q^{13} + ( -11 + 8 \beta_{2} ) q^{17} + ( -43 - 5 \beta_{2} ) q^{19} + ( -29 - 29 \beta_{1} + \beta_{3} ) q^{23} + ( 92 \beta_{1} + 8 \beta_{2} - 8 \beta_{3} ) q^{25} + ( -54 \beta_{1} + \beta_{2} - \beta_{3} ) q^{29} + ( -68 - 68 \beta_{1} + 16 \beta_{3} ) q^{31} + ( -141 - 11 \beta_{2} ) q^{35} + ( -90 + \beta_{2} ) q^{37} + ( 336 + 336 \beta_{1} + 2 \beta_{3} ) q^{41} + ( -35 \beta_{1} - 23 \beta_{2} + 23 \beta_{3} ) q^{43} + ( -426 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{47} + ( -83 - 83 \beta_{1} - 30 \beta_{3} ) q^{49} + ( -334 - 16 \beta_{2} ) q^{53} + ( -695 + 35 \beta_{2} ) q^{55} + ( 274 + 274 \beta_{1} + 6 \beta_{3} ) q^{59} + ( 142 \beta_{1} - 17 \beta_{2} + 17 \beta_{3} ) q^{61} + ( 17 \beta_{1} - 42 \beta_{2} + 42 \beta_{3} ) q^{65} + ( -581 - 581 \beta_{1} - 11 \beta_{3} ) q^{67} + ( 403 - 3 \beta_{2} ) q^{71} + ( -755 - 18 \beta_{2} ) q^{73} + ( -258 - 258 \beta_{1} - 22 \beta_{3} ) q^{77} + ( -11 \beta_{1} + 41 \beta_{2} - 41 \beta_{3} ) q^{79} + ( -770 \beta_{1} - 38 \beta_{2} + 38 \beta_{3} ) q^{83} + ( -1652 - 1652 \beta_{1} + 43 \beta_{3} ) q^{85} -1323 q^{89} + ( -891 - 61 \beta_{2} ) q^{91} + ( 833 + 833 \beta_{1} + 23 \beta_{3} ) q^{95} + ( 236 \beta_{1} + 50 \beta_{2} - 50 \beta_{3} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 8q^{5} + 30q^{7} + O(q^{10}) \) \( 4q + 8q^{5} + 30q^{7} - 46q^{11} - 92q^{13} - 44q^{17} - 172q^{19} - 58q^{23} - 184q^{25} + 108q^{29} - 136q^{31} - 564q^{35} - 360q^{37} + 672q^{41} + 70q^{43} + 852q^{47} - 166q^{49} - 1336q^{53} - 2780q^{55} + 548q^{59} - 284q^{61} - 34q^{65} - 1162q^{67} + 1612q^{71} - 3020q^{73} - 516q^{77} + 22q^{79} + 1540q^{83} - 3304q^{85} - 5292q^{89} - 3564q^{91} + 1666q^{95} - 472q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 16 x^{2} - 17 x + 289\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} + 16 \nu^{2} - 16 \nu - 289 \)\()/272\)
\(\beta_{2}\)\(=\)\((\)\( -2 \nu^{3} + 2 \nu^{2} + 66 \nu + 17 \)\()/17\)
\(\beta_{3}\)\(=\)\((\)\( 35 \nu^{3} + 16 \nu^{2} + 528 \nu - 1139 \)\()/272\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{2} - 3 \beta_{1}\)\()/6\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{3} + 2 \beta_{2} + 99 \beta_{1} + 99\)\()/6\)
\(\nu^{3}\)\(=\)\((\)\(16 \beta_{3} - 8 \beta_{2} + 75\)\()/3\)

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
3.79436 + 1.61333i
−3.29436 2.47935i
3.79436 1.61333i
−3.29436 + 2.47935i
0 0 0 −5.08872 8.81393i 0 14.5887 25.2684i 0 0 0
217.2 0 0 0 9.08872 + 15.7421i 0 0.411277 0.712352i 0 0 0
433.1 0 0 0 −5.08872 + 8.81393i 0 14.5887 + 25.2684i 0 0 0
433.2 0 0 0 9.08872 15.7421i 0 0.411277 + 0.712352i 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.t 4
3.b odd 2 1 648.4.i.n 4
9.c even 3 1 648.4.a.c 2
9.c even 3 1 inner 648.4.i.t 4
9.d odd 6 1 648.4.a.f yes 2
9.d odd 6 1 648.4.i.n 4
36.f odd 6 1 1296.4.a.m 2
36.h even 6 1 1296.4.a.q 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
648.4.a.c 2 9.c even 3 1
648.4.a.f yes 2 9.d odd 6 1
648.4.i.n 4 3.b odd 2 1
648.4.i.n 4 9.d odd 6 1
648.4.i.t 4 1.a even 1 1 trivial
648.4.i.t 4 9.c even 3 1 inner
1296.4.a.m 2 36.f odd 6 1
1296.4.a.q 2 36.h even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 8 T_{5}^{3} + 249 T_{5}^{2} + 1480 T_{5} + 34225 \) acting on \(S_{4}^{\mathrm{new}}(648, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( 34225 + 1480 T + 249 T^{2} - 8 T^{3} + T^{4} \)
$7$ \( 576 - 720 T + 876 T^{2} - 30 T^{3} + T^{4} \)
$11$ \( 1638400 - 58880 T + 3396 T^{2} + 46 T^{3} + T^{4} \)
$13$ \( 3667225 + 176180 T + 6549 T^{2} + 92 T^{3} + T^{4} \)
$17$ \( ( -12743 + 22 T + T^{2} )^{2} \)
$19$ \( ( -3176 + 86 T + T^{2} )^{2} \)
$23$ \( 409600 + 37120 T + 2724 T^{2} + 58 T^{3} + T^{4} \)
$29$ \( 7371225 - 293220 T + 8949 T^{2} - 108 T^{3} + T^{4} \)
$31$ \( 2193236224 - 6369152 T + 65328 T^{2} + 136 T^{3} + T^{4} \)
$37$ \( ( 7899 + 180 T + T^{2} )^{2} \)
$41$ \( 12564616464 - 75325824 T + 339492 T^{2} - 672 T^{3} + T^{4} \)
$43$ \( 11046850816 + 7357280 T + 110004 T^{2} - 70 T^{3} + T^{4} \)
$47$ \( 32642371584 - 153932544 T + 545232 T^{2} - 852 T^{3} + T^{4} \)
$53$ \( ( 60100 + 668 T + T^{2} )^{2} \)
$59$ \( 4602265600 - 37176320 T + 232464 T^{2} - 548 T^{3} + T^{4} \)
$61$ \( 1438305625 - 10770700 T + 118581 T^{2} + 284 T^{3} + T^{4} \)
$67$ \( 98119297600 + 363984880 T + 1037004 T^{2} + 1162 T^{3} + T^{4} \)
$71$ \( ( 160600 - 806 T + T^{2} )^{2} \)
$73$ \( ( 504901 + 1510 T + T^{2} )^{2} \)
$79$ \( 114081817600 + 7430720 T + 338244 T^{2} - 22 T^{3} + T^{4} \)
$83$ \( 91600654336 - 466090240 T + 2068944 T^{2} - 1540 T^{3} + T^{4} \)
$89$ \( ( 1323 + T )^{4} \)
$97$ \( 199633814416 - 210891488 T + 669588 T^{2} + 472 T^{3} + T^{4} \)
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