Properties

Label 648.4.i.u
Level $648$
Weight $4$
Character orbit 648.i
Analytic conductor $38.233$
Analytic rank $0$
Dimension $8$
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: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.897122304.10
Defining polynomial: \(x^{8} - 8 x^{6} + 51 x^{4} - 104 x^{2} + 169\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{6} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 2 \beta_{1} + \beta_{7} ) q^{5} + ( \beta_{2} + \beta_{3} - \beta_{5} - \beta_{7} ) q^{7} +O(q^{10})\) \( q + ( 2 \beta_{1} + \beta_{7} ) q^{5} + ( \beta_{2} + \beta_{3} - \beta_{5} - \beta_{7} ) q^{7} + ( -2 - 2 \beta_{1} + \beta_{2} + 2 \beta_{3} - 2 \beta_{5} + \beta_{6} - \beta_{7} ) q^{11} + ( \beta_{1} + 2 \beta_{4} + \beta_{5} - 2 \beta_{6} + 3 \beta_{7} ) q^{13} + ( 4 + 2 \beta_{2} - 4 \beta_{3} - \beta_{4} ) q^{17} + ( 20 - 3 \beta_{2} + 5 \beta_{3} ) q^{19} + ( 50 \beta_{1} - 7 \beta_{4} - 6 \beta_{5} + 7 \beta_{6} + 5 \beta_{7} ) q^{23} + ( -2 - 2 \beta_{1} + 6 \beta_{2} - 2 \beta_{3} + 2 \beta_{5} - 16 \beta_{6} - 6 \beta_{7} ) q^{25} + ( -54 - 54 \beta_{1} - 9 \beta_{2} - 4 \beta_{3} + 4 \beta_{5} - 4 \beta_{6} + 9 \beta_{7} ) q^{29} + ( 20 \beta_{1} + 16 \beta_{4} - 2 \beta_{5} - 16 \beta_{6} - 2 \beta_{7} ) q^{31} + ( 102 - 3 \beta_{2} + 2 \beta_{3} - 7 \beta_{4} ) q^{35} + ( -69 + \beta_{2} - 17 \beta_{3} - 22 \beta_{4} ) q^{37} + ( 96 \beta_{1} + 20 \beta_{4} + 4 \beta_{5} - 20 \beta_{6} - 12 \beta_{7} ) q^{41} + ( -40 - 40 \beta_{1} - 19 \beta_{2} - 3 \beta_{3} + 3 \beta_{5} - 56 \beta_{6} + 19 \beta_{7} ) q^{43} + ( -192 - 192 \beta_{1} + 16 \beta_{2} + 12 \beta_{6} - 16 \beta_{7} ) q^{47} + ( -67 \beta_{1} + 44 \beta_{4} - 8 \beta_{5} - 44 \beta_{6} - 4 \beta_{7} ) q^{49} + ( 236 + 10 \beta_{2} + 24 \beta_{3} + 34 \beta_{4} ) q^{53} + ( 76 - \beta_{2} - \beta_{3} - 32 \beta_{4} ) q^{55} + ( 248 \beta_{1} - 28 \beta_{4} + 24 \beta_{5} + 28 \beta_{6} - 12 \beta_{7} ) q^{59} + ( 137 + 137 \beta_{1} - 3 \beta_{2} - \beta_{3} + \beta_{5} - 114 \beta_{6} + 3 \beta_{7} ) q^{61} + ( -332 - 332 \beta_{1} + 6 \beta_{2} - 21 \beta_{6} - 6 \beta_{7} ) q^{65} + ( 116 \beta_{1} + 96 \beta_{4} + 21 \beta_{5} - 96 \beta_{6} - 19 \beta_{7} ) q^{67} + ( 430 - 27 \beta_{2} - 6 \beta_{3} - 43 \beta_{4} ) q^{71} + ( -191 + 44 \beta_{2} + 40 \beta_{3} - 36 \beta_{4} ) q^{73} + ( 432 \beta_{1} + 84 \beta_{4} - 4 \beta_{5} - 84 \beta_{6} - 4 \beta_{7} ) q^{77} + ( -172 - 172 \beta_{1} + 23 \beta_{2} + 23 \beta_{3} - 23 \beta_{5} - 160 \beta_{6} - 23 \beta_{7} ) q^{79} + ( -532 - 532 \beta_{1} - 38 \beta_{2} - 20 \beta_{3} + 20 \beta_{5} + 6 \beta_{6} + 38 \beta_{7} ) q^{83} + ( -331 \beta_{1} + 126 \beta_{4} + 7 \beta_{5} - 126 \beta_{6} - 11 \beta_{7} ) q^{85} + ( 528 - 40 \beta_{2} - 56 \beta_{3} + 55 \beta_{4} ) q^{89} + ( 444 - 5 \beta_{2} + 3 \beta_{3} - 64 \beta_{4} ) q^{91} + ( 514 \beta_{1} - 163 \beta_{4} - 6 \beta_{5} + 163 \beta_{6} + 37 \beta_{7} ) q^{95} + ( 454 + 454 \beta_{1} - 40 \beta_{2} - 4 \beta_{3} + 4 \beta_{5} - 204 \beta_{6} + 40 \beta_{7} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 8q^{5} + O(q^{10}) \) \( 8q - 8q^{5} - 8q^{11} - 4q^{13} + 32q^{17} + 160q^{19} - 200q^{23} - 8q^{25} - 216q^{29} - 80q^{31} + 816q^{35} - 552q^{37} - 384q^{41} - 160q^{43} - 768q^{47} + 268q^{49} + 1888q^{53} + 608q^{55} - 992q^{59} + 548q^{61} - 1328q^{65} - 464q^{67} + 3440q^{71} - 1528q^{73} - 1728q^{77} - 688q^{79} - 2128q^{83} + 1324q^{85} + 4224q^{89} + 3552q^{91} - 2056q^{95} + 1816q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 8 x^{6} + 51 x^{4} - 104 x^{2} + 169\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -8 \nu^{6} + 51 \nu^{4} - 408 \nu^{2} + 169 \)\()/663\)
\(\beta_{2}\)\(=\)\((\)\( 28 \nu^{7} - 13 \nu^{6} - 510 \nu^{5} + 2754 \nu^{3} - 9542 \nu - 1300 \)\()/663\)
\(\beta_{3}\)\(=\)\((\)\( -32 \nu^{7} - 13 \nu^{6} + 204 \nu^{5} - 1632 \nu^{3} + 5980 \nu - 1300 \)\()/663\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{6} + 100 \)\()/17\)
\(\beta_{5}\)\(=\)\((\)\( 32 \nu^{7} - 32 \nu^{6} - 204 \nu^{5} + 204 \nu^{4} + 1632 \nu^{3} - 969 \nu^{2} + 1976 \nu + 676 \)\()/663\)
\(\beta_{6}\)\(=\)\((\)\( -19 \nu^{6} + 204 \nu^{4} - 969 \nu^{2} + 1976 \)\()/221\)
\(\beta_{7}\)\(=\)\((\)\( -106 \nu^{7} - 32 \nu^{6} + 510 \nu^{5} + 204 \nu^{4} - 2754 \nu^{3} - 969 \nu^{2} - 2236 \nu + 676 \)\()/663\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{6} + 3 \beta_{5} + 2 \beta_{4} + 3 \beta_{3}\)\()/36\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{6} - \beta_{4} - 12 \beta_{1}\)\()/3\)
\(\nu^{3}\)\(=\)\((\)\(12 \beta_{7} - 14 \beta_{6} + 30 \beta_{5} + 7 \beta_{4} - 15 \beta_{3} - 6 \beta_{2}\)\()/36\)
\(\nu^{4}\)\(=\)\((\)\(8 \beta_{6} - 57 \beta_{1} - 57\)\()/3\)
\(\nu^{5}\)\(=\)\((\)\(48 \beta_{7} - 43 \beta_{6} + 81 \beta_{5} - 43 \beta_{4} - 162 \beta_{3} - 96 \beta_{2}\)\()/36\)
\(\nu^{6}\)\(=\)\(17 \beta_{4} - 100\)
\(\nu^{7}\)\(=\)\((\)\(-306 \beta_{7} + 253 \beta_{6} - 453 \beta_{5} - 506 \beta_{4} - 453 \beta_{3} - 306 \beta_{2}\)\()/36\)

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
1.30421 + 0.752986i
−2.07341 1.19709i
2.07341 + 1.19709i
−1.30421 0.752986i
1.30421 0.752986i
−2.07341 + 1.19709i
2.07341 1.19709i
−1.30421 + 0.752986i
0 0 0 −8.99236 15.5752i 0 −3.64155 + 6.30734i 0 0 0
217.2 0 0 0 −3.16966 5.49001i 0 −9.59728 + 16.6230i 0 0 0
217.3 0 0 0 2.90171 + 5.02591i 0 13.0614 22.6230i 0 0 0
217.4 0 0 0 5.26031 + 9.11112i 0 0.177444 0.307342i 0 0 0
433.1 0 0 0 −8.99236 + 15.5752i 0 −3.64155 6.30734i 0 0 0
433.2 0 0 0 −3.16966 + 5.49001i 0 −9.59728 16.6230i 0 0 0
433.3 0 0 0 2.90171 5.02591i 0 13.0614 + 22.6230i 0 0 0
433.4 0 0 0 5.26031 9.11112i 0 0.177444 + 0.307342i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 433.4
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.u 8
3.b odd 2 1 648.4.i.v 8
9.c even 3 1 648.4.a.j yes 4
9.c even 3 1 inner 648.4.i.u 8
9.d odd 6 1 648.4.a.g 4
9.d odd 6 1 648.4.i.v 8
36.f odd 6 1 1296.4.a.bb 4
36.h even 6 1 1296.4.a.x 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
648.4.a.g 4 9.d odd 6 1
648.4.a.j yes 4 9.c even 3 1
648.4.i.u 8 1.a even 1 1 trivial
648.4.i.u 8 9.c even 3 1 inner
648.4.i.v 8 3.b odd 2 1
648.4.i.v 8 9.d odd 6 1
1296.4.a.x 4 36.h even 6 1
1296.4.a.bb 4 36.f odd 6 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( T^{8} \)
$5$ \( 48455521 - 2617336 T + 1686718 T^{2} - 27904 T^{3} + 45331 T^{4} - 1024 T^{5} + 286 T^{6} + 8 T^{7} + T^{8} \)
$7$ \( 1679616 - 4478976 T + 12659328 T^{2} + 1907712 T^{3} + 303408 T^{4} + 6912 T^{5} + 552 T^{6} + T^{8} \)
$11$ \( 3729789184 - 384997888 T + 135012736 T^{2} + 8857088 T^{3} + 2422960 T^{4} + 128 T^{5} + 1624 T^{6} + 8 T^{7} + T^{8} \)
$13$ \( 1531200831889 - 4212167468 T + 2825473474 T^{2} - 2158640 T^{3} + 3947275 T^{4} - 2288 T^{5} + 2290 T^{6} + 4 T^{7} + T^{8} \)
$17$ \( ( 14319721 + 69680 T - 7782 T^{2} - 16 T^{3} + T^{4} )^{2} \)
$19$ \( ( 31790992 + 530368 T - 10824 T^{2} - 80 T^{3} + T^{4} )^{2} \)
$23$ \( 2393099086553344 - 53826897379840 T + 1704984830848 T^{2} - 8450091520 T^{3} + 273235504 T^{4} + 179840 T^{5} + 50104 T^{6} + 200 T^{7} + T^{8} \)
$29$ \( 3009972579589089 - 95234205503016 T + 3267623879550 T^{2} - 15650046720 T^{3} + 341590995 T^{4} + 2469888 T^{5} + 51294 T^{6} + 216 T^{7} + T^{8} \)
$31$ \( 2382014703468544 - 16692394590208 T + 894745575424 T^{2} - 2358575104 T^{3} + 232511488 T^{4} - 590848 T^{5} + 22336 T^{6} + 80 T^{7} + T^{8} \)
$37$ \( ( 948334041 - 23154588 T - 99282 T^{2} + 276 T^{3} + T^{4} )^{2} \)
$41$ \( 4183346207054168064 + 43241204554924032 T + 410638254759936 T^{2} + 1946281181184 T^{3} + 10479078144 T^{4} + 35463168 T^{5} + 165216 T^{6} + 384 T^{7} + T^{8} \)
$43$ \( 263225950184694016 - 782011512528896 T + 106432481673856 T^{2} + 145117772800 T^{3} + 40907346736 T^{4} - 29418752 T^{5} + 228520 T^{6} + 160 T^{7} + T^{8} \)
$47$ \( 1304686485344157696 - 968463762849792 T + 180441725165568 T^{2} + 1887870910464 T^{3} + 25248197376 T^{4} + 119144448 T^{5} + 432480 T^{6} + 768 T^{7} + T^{8} \)
$53$ \( ( -22310316032 + 112215040 T + 84288 T^{2} - 944 T^{3} + T^{4} )^{2} \)
$59$ \( 351967793115037696 - 49294348059148288 T + 6885852321415168 T^{2} - 3697645453312 T^{3} + 82751696896 T^{4} + 196272128 T^{5} + 953728 T^{6} + 992 T^{7} + T^{8} \)
$61$ \( \)\(10\!\cdots\!81\)\( + 18536597548535186380 T + 91573627200592546 T^{2} + 11821109108656 T^{3} + 324101933035 T^{4} - 40752272 T^{5} + 875410 T^{6} - 548 T^{7} + T^{8} \)
$67$ \( \)\(94\!\cdots\!84\)\( + 11048998541913347072 T + 107906079380037760 T^{2} + 281978284166144 T^{3} + 693772243120 T^{4} + 392994560 T^{5} + 919624 T^{6} + 464 T^{7} + T^{8} \)
$71$ \( ( -9807661424 - 110047648 T + 880872 T^{2} - 1720 T^{3} + T^{4} )^{2} \)
$73$ \( ( -194677777919 - 951038212 T - 912378 T^{2} + 764 T^{3} + T^{4} )^{2} \)
$79$ \( \)\(10\!\cdots\!96\)\( - \)\(36\!\cdots\!32\)\( T + 1835312684912414848 T^{2} + 1546372208530432 T^{3} + 3559957053616 T^{4} + 1043060992 T^{5} + 2235208 T^{6} + 688 T^{7} + T^{8} \)
$83$ \( 37632232766838931456 + 1011994578789597184 T + 19630227884474368 T^{2} + 177838765244416 T^{3} + 1171222829056 T^{4} + 2300886016 T^{5} + 3292096 T^{6} + 2128 T^{7} + T^{8} \)
$89$ \( ( -1312975087623 + 2534079168 T - 109206 T^{2} - 2112 T^{3} + T^{4} )^{2} \)
$97$ \( 8938629745893040384 + 6233177652285278720 T + 4349794338096916096 T^{2} - 2227915820598784 T^{3} + 4936206181936 T^{4} - 2219613568 T^{5} + 4371688 T^{6} - 1816 T^{7} + T^{8} \)
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