Properties

Label 1728.4.c.j
Level 1728
Weight 4
Character orbit 1728.c
Analytic conductor 101.955
Analytic rank 0
Dimension 12
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1728.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(101.955300490\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{32}\cdot 3^{12} \)
Twist minimal: no (minimal twist has level 108)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{2} q^{5} -\beta_{1} q^{7} +O(q^{10})\) \( q -\beta_{2} q^{5} -\beta_{1} q^{7} -\beta_{6} q^{11} + ( 6 - \beta_{5} ) q^{13} + \beta_{10} q^{17} + ( -\beta_{1} + \beta_{9} ) q^{19} + ( 2 \beta_{3} + \beta_{6} - \beta_{8} ) q^{23} + ( -32 - \beta_{4} + 2 \beta_{5} ) q^{25} + ( -3 \beta_{2} - \beta_{7} + \beta_{10} ) q^{29} + ( -4 \beta_{1} - \beta_{9} - \beta_{11} ) q^{31} + ( 5 \beta_{3} + 2 \beta_{6} - 2 \beta_{8} ) q^{35} + ( 20 - 5 \beta_{5} ) q^{37} + ( -2 \beta_{2} - 4 \beta_{7} - \beta_{10} ) q^{41} + ( 8 \beta_{1} + 2 \beta_{9} - 2 \beta_{11} ) q^{43} + ( 2 \beta_{3} - 6 \beta_{6} + 2 \beta_{8} ) q^{47} + ( 24 + \beta_{4} + 4 \beta_{5} ) q^{49} + ( 18 \beta_{2} - 5 \beta_{7} + 3 \beta_{10} ) q^{53} + ( -13 \beta_{1} + 2 \beta_{9} - 2 \beta_{11} ) q^{55} + ( 6 \beta_{6} + 8 \beta_{8} ) q^{59} + ( -12 + 2 \beta_{4} + 6 \beta_{5} ) q^{61} + ( -26 \beta_{2} + 2 \beta_{7} + 3 \beta_{10} ) q^{65} -22 \beta_{1} q^{67} + ( 10 \beta_{3} + 7 \beta_{6} + 7 \beta_{8} ) q^{71} + ( 13 + \beta_{4} - 8 \beta_{5} ) q^{73} + ( -27 \beta_{2} + 14 \beta_{7} - 4 \beta_{10} ) q^{77} + ( 29 \beta_{1} + 3 \beta_{9} + \beta_{11} ) q^{79} + ( 32 \beta_{3} - 13 \beta_{6} - 6 \beta_{8} ) q^{83} + ( 14 - 2 \beta_{4} + 11 \beta_{5} ) q^{85} + ( 64 \beta_{2} + 14 \beta_{7} - 4 \beta_{10} ) q^{89} + ( -31 \beta_{1} - 3 \beta_{9} - 2 \beta_{11} ) q^{91} + ( 50 \beta_{3} + 11 \beta_{6} - 19 \beta_{8} ) q^{95} + ( 43 + 4 \beta_{4} - 8 \beta_{5} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + O(q^{10}) \) \( 12q + 72q^{13} - 384q^{25} + 240q^{37} + 288q^{49} - 144q^{61} + 156q^{73} + 168q^{85} + 516q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} + 3 x^{10} - 12 x^{9} + 73 x^{8} - 12 x^{7} + 589 x^{6} + 84 x^{5} + 2452 x^{4} + 852 x^{3} + 6854 x^{2} - 888 x + 9496\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(-2849350 \nu^{11} + 6585588 \nu^{10} - 7900308 \nu^{9} + 8937888 \nu^{8} - 274261062 \nu^{7} + 514828262 \nu^{6} - 1348583642 \nu^{5} + 577023966 \nu^{4} - 3610618898 \nu^{3} - 4417697498 \nu^{2} - 20418808208 \nu - 3774567143\)\()/ 3046053151 \)
\(\beta_{2}\)\(=\)\((\)\(454920367261 \nu^{11} + 699408466494 \nu^{10} + 909094946747 \nu^{9} - 6055960107450 \nu^{8} + 16604234843593 \nu^{7} + 44620628131914 \nu^{6} + 261918488231045 \nu^{5} + 346379732140290 \nu^{4} + 372084361441392 \nu^{3} + 4408588811940 \nu^{2} - 876761148010650 \nu - 873663409442004\)\()/ 449256287134688 \)
\(\beta_{3}\)\(=\)\((\)\(-2201987655 \nu^{11} + 7037103366 \nu^{10} + 321745311 \nu^{9} + 50820234894 \nu^{8} - 233951534667 \nu^{7} + 379443106530 \nu^{6} - 994923167391 \nu^{5} + 4318536234906 \nu^{4} - 450964003536 \nu^{3} + 12604127190228 \nu^{2} + 566117025390 \nu + 33640324703676\)\()/ 1072210709152 \)
\(\beta_{4}\)\(=\)\((\)\(-3668351532 \nu^{11} - 35323691472 \nu^{10} - 36152667432 \nu^{9} - 188138665632 \nu^{8} + 50069628876 \nu^{7} - 1565811304500 \nu^{6} - 2253785545884 \nu^{5} - 29394681445056 \nu^{4} - 43298645321028 \nu^{3} - 89923171899804 \nu^{2} - 56510573448384 \nu - 171807057509552\)\()/ 1276296270269 \)
\(\beta_{5}\)\(=\)\((\)\(-9151812708 \nu^{11} + 12155753232 \nu^{10} + 35817493464 \nu^{9} + 135939078600 \nu^{8} - 789050014620 \nu^{7} + 300610219236 \nu^{6} - 1310314725588 \nu^{5} + 7296602696088 \nu^{4} + 3399104212836 \nu^{3} + 28479977397228 \nu^{2} + 5390654550912 \nu + 120796778623888\)\()/ 1276296270269 \)
\(\beta_{6}\)\(=\)\((\)\(-3222436257455 \nu^{11} + 5388050787734 \nu^{10} + 2723330832455 \nu^{9} + 52384868035038 \nu^{8} - 299880875358483 \nu^{7} + 246749489170834 \nu^{6} - 1028976838142151 \nu^{5} + 2676763977698314 \nu^{4} - 1992011057883600 \nu^{3} + 8892862433464884 \nu^{2} - 1768324514742466 \nu + 20917972597278300\)\()/ 449256287134688 \)
\(\beta_{7}\)\(=\)\((\)\(-18642750728255 \nu^{11} - 54280349116842 \nu^{10} - 38930579155849 \nu^{9} + 292758720471294 \nu^{8} - 854630055113507 \nu^{7} - 3576388112829486 \nu^{6} - 11660381327420215 \nu^{5} - 16340881815026454 \nu^{4} - 49527457079318736 \nu^{3} - 51882006104080140 \nu^{2} - 134713599325199970 \nu - 45157448521252068\)\()/ 2246281435673440 \)
\(\beta_{8}\)\(=\)\((\)\(-4893277942309 \nu^{11} + 8469535734322 \nu^{10} - 13632883283795 \nu^{9} + 76780435438410 \nu^{8} - 443280641637105 \nu^{7} + 639207165622022 \nu^{6} - 2619966859658157 \nu^{5} + 3875655823133870 \nu^{4} - 9916828921048944 \nu^{3} + 9414663721373244 \nu^{2} - 10536224050201622 \nu + 41995323116010612\)\()/ 449256287134688 \)
\(\beta_{9}\)\(=\)\((\)\(82140767042 \nu^{11} + 128260169172 \nu^{10} + 66239186292 \nu^{9} - 613299928464 \nu^{8} + 4292059269954 \nu^{7} + 11981142736814 \nu^{6} + 31453542055774 \nu^{5} + 69841926412242 \nu^{4} + 128188443328414 \nu^{3} + 430011600480190 \nu^{2} + 361115104649488 \nu + 579514515914887\)\()/ 6381481351345 \)
\(\beta_{10}\)\(=\)\((\)\(-24096087148205 \nu^{11} - 30785315288286 \nu^{10} - 65693209087147 \nu^{9} + 322701238859802 \nu^{8} - 1328933424436121 \nu^{7} - 1800413902806378 \nu^{6} - 14997619440051925 \nu^{5} - 12281584417815522 \nu^{4} - 53711753834599728 \nu^{3} - 25304246119887780 \nu^{2} - 157744121239963590 \nu + 4917658014297876\)\()/ 1123140717836720 \)
\(\beta_{11}\)\(=\)\((\)\(426968239932 \nu^{11} + 457010627112 \nu^{10} + 737602108632 \nu^{9} - 4752896618904 \nu^{8} + 22258024812924 \nu^{7} + 33730962757524 \nu^{6} + 222741323689284 \nu^{5} + 267153193701852 \nu^{4} + 654755938274244 \nu^{3} + 913427570025060 \nu^{2} + 1505040883452768 \nu + 749667835341342\)\()/ 6381481351345 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-2 \beta_{10} + 2 \beta_{7} - \beta_{5} + \beta_{4} + 8 \beta_{3} - 14 \beta_{2}\)\()/144\)
\(\nu^{2}\)\(=\)\((\)\(2 \beta_{11} + 6 \beta_{10} - 2 \beta_{9} - 3 \beta_{8} - 6 \beta_{7} + 6 \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} - 30 \beta_{2} - 10 \beta_{1} - 72\)\()/144\)
\(\nu^{3}\)\(=\)\((\)\(6 \beta_{11} - 2 \beta_{10} - 6 \beta_{9} - 9 \beta_{8} + 17 \beta_{7} + 18 \beta_{6} - 8 \beta_{5} - 10 \beta_{4} - 45 \beta_{3} - 197 \beta_{2} + 186 \beta_{1} + 864\)\()/288\)
\(\nu^{4}\)\(=\)\((\)\(13 \beta_{11} + 8 \beta_{10} - 28 \beta_{9} + 21 \beta_{8} + 7 \beta_{7} - 96 \beta_{6} + 41 \beta_{5} + \beta_{4} + 91 \beta_{3} - 199 \beta_{2} + 58 \beta_{1} - 3288\)\()/144\)
\(\nu^{5}\)\(=\)\((\)\(55 \beta_{11} + 122 \beta_{10} - 130 \beta_{9} + 240 \beta_{8} - 152 \beta_{7} + 150 \beta_{6} + 93 \beta_{5} - 129 \beta_{4} - 1666 \beta_{3} - 436 \beta_{2} - 380 \beta_{1} - 2880\)\()/288\)
\(\nu^{6}\)\(=\)\((\)\(-93 \beta_{11} - 480 \beta_{10} + 408 \beta_{9} + 711 \beta_{8} + 795 \beta_{7} - 936 \beta_{6} + 745 \beta_{5} - 223 \beta_{4} - 2655 \beta_{3} + 213 \beta_{2} + 2850 \beta_{1} - 44208\)\()/288\)
\(\nu^{7}\)\(=\)\((\)\(-469 \beta_{11} + 714 \beta_{10} + 154 \beta_{9} + 1554 \beta_{8} - 2454 \beta_{7} - 4242 \beta_{6} + 2261 \beta_{5} - 317 \beta_{4} - 1836 \beta_{3} + 11754 \beta_{2} - 11200 \beta_{1} - 149184\)\()/288\)
\(\nu^{8}\)\(=\)\((\)\(-634 \beta_{11} - 800 \beta_{10} + 1114 \beta_{9} + 654 \beta_{8} + 470 \beta_{7} + 2256 \beta_{6} - 769 \beta_{5} - 713 \beta_{4} - 7518 \beta_{3} + 11530 \beta_{2} - 3682 \beta_{1} + 50628\)\()/72\)
\(\nu^{9}\)\(=\)\((\)\(-8343 \beta_{11} - 13446 \beta_{10} + 17118 \beta_{9} - 6588 \beta_{8} + 11856 \beta_{7} - 22626 \beta_{6} + 10267 \beta_{5} + 7301 \beta_{4} + 73170 \beta_{3} + 143076 \beta_{2} - 14688 \beta_{1} - 632448\)\()/288\)
\(\nu^{10}\)\(=\)\((\)\(-14603 \beta_{11} + 22552 \beta_{10} + 3248 \beta_{9} - 42141 \beta_{8} - 81157 \beta_{7} + 72240 \beta_{6} - 54397 \beta_{5} + 12451 \beta_{4} + 131133 \beta_{3} + 389797 \beta_{2} - 347450 \beta_{1} + 3303696\)\()/288\)
\(\nu^{11}\)\(=\)\((\)\(-24805 \beta_{11} - 117994 \beta_{10} + 96910 \beta_{9} - 176418 \beta_{8} + 208474 \beta_{7} + 345906 \beta_{6} - 246703 \beta_{5} + 46507 \beta_{4} + 467516 \beta_{3} + 225074 \beta_{2} + 475244 \beta_{1} + 15174720\)\()/288\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(703\) \(1217\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1727.1
0.886307 1.60260i
2.61836 1.60260i
0.433705 1.36719i
−1.29835 1.36719i
−0.453986 2.07664i
−2.18604 2.07664i
−2.18604 + 2.07664i
−0.453986 + 2.07664i
−1.29835 + 1.36719i
0.433705 + 1.36719i
2.61836 + 1.60260i
0.886307 + 1.60260i
0 0 0 20.8488i 0 13.9048i 0 0 0
1727.2 0 0 0 20.8488i 0 13.9048i 0 0 0
1727.3 0 0 0 5.83890i 0 8.83113i 0 0 0
1727.4 0 0 0 5.83890i 0 8.83113i 0 0 0
1727.5 0 0 0 1.49508i 0 26.1852i 0 0 0
1727.6 0 0 0 1.49508i 0 26.1852i 0 0 0
1727.7 0 0 0 1.49508i 0 26.1852i 0 0 0
1727.8 0 0 0 1.49508i 0 26.1852i 0 0 0
1727.9 0 0 0 5.83890i 0 8.83113i 0 0 0
1727.10 0 0 0 5.83890i 0 8.83113i 0 0 0
1727.11 0 0 0 20.8488i 0 13.9048i 0 0 0
1727.12 0 0 0 20.8488i 0 13.9048i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1727.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1728.4.c.j 12
3.b odd 2 1 inner 1728.4.c.j 12
4.b odd 2 1 inner 1728.4.c.j 12
8.b even 2 1 108.4.b.b 12
8.d odd 2 1 108.4.b.b 12
12.b even 2 1 inner 1728.4.c.j 12
24.f even 2 1 108.4.b.b 12
24.h odd 2 1 108.4.b.b 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.4.b.b 12 8.b even 2 1
108.4.b.b 12 8.d odd 2 1
108.4.b.b 12 24.f even 2 1
108.4.b.b 12 24.h odd 2 1
1728.4.c.j 12 1.a even 1 1 trivial
1728.4.c.j 12 3.b odd 2 1 inner
1728.4.c.j 12 4.b odd 2 1 inner
1728.4.c.j 12 12.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(1728, [\chi])\):

\( T_{5}^{6} + 471 T_{5}^{4} + 15867 T_{5}^{2} + 33125 \)
\( T_{7}^{6} + 957 T_{7}^{4} + 201123 T_{7}^{2} + 10338975 \)
\( T_{11}^{6} - 4929 T_{11}^{4} + 6225579 T_{11}^{2} - 2112318675 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( ( 1 - 279 T^{2} + 14742 T^{4} + 1160125 T^{6} + 230343750 T^{8} - 68115234375 T^{10} + 3814697265625 T^{12} )^{2} \)
$7$ \( ( 1 - 1101 T^{2} + 652854 T^{4} - 259162985 T^{6} + 76807620246 T^{8} - 15239257208301 T^{10} + 1628413597910449 T^{12} )^{2} \)
$11$ \( ( 1 + 3057 T^{2} + 6556998 T^{4} + 9226981429 T^{6} + 11616121933878 T^{8} + 9594175547636097 T^{10} + 5559917313492231481 T^{12} )^{2} \)
$13$ \( ( 1 - 18 T + 3915 T^{2} - 8332 T^{3} + 8601255 T^{4} - 86882562 T^{5} + 10604499373 T^{6} )^{4} \)
$17$ \( ( 1 - 11946 T^{2} + 72737391 T^{4} - 332667352460 T^{6} + 1755703794142479 T^{8} - 6960005245946724906 T^{10} + \)\(14\!\cdots\!09\)\( T^{12} )^{2} \)
$19$ \( ( 1 - 8322 T^{2} + 146058135 T^{4} - 772095813500 T^{6} + 6871433638291935 T^{8} - 18419206756468591842 T^{10} + \)\(10\!\cdots\!41\)\( T^{12} )^{2} \)
$23$ \( ( 1 + 52314 T^{2} + 1249837599 T^{4} + 18511268917228 T^{6} + 185020820073590511 T^{8} + \)\(11\!\cdots\!94\)\( T^{10} + \)\(32\!\cdots\!69\)\( T^{12} )^{2} \)
$29$ \( ( 1 - 117954 T^{2} + 6353615223 T^{4} - 198355607069564 T^{6} + 3779278507301015583 T^{8} - \)\(41\!\cdots\!14\)\( T^{10} + \)\(21\!\cdots\!61\)\( T^{12} )^{2} \)
$31$ \( ( 1 - 45741 T^{2} + 1957054854 T^{4} - 51110094297449 T^{6} + 1736893386843917574 T^{8} - \)\(36\!\cdots\!01\)\( T^{10} + \)\(69\!\cdots\!41\)\( T^{12} )^{2} \)
$37$ \( ( 1 - 60 T + 83559 T^{2} + 2089640 T^{3} + 4232514027 T^{4} - 153943584540 T^{5} + 129961739795077 T^{6} )^{4} \)
$41$ \( ( 1 - 189606 T^{2} + 21529296543 T^{4} - 1820253257865236 T^{6} + \)\(10\!\cdots\!63\)\( T^{8} - \)\(42\!\cdots\!86\)\( T^{10} + \)\(10\!\cdots\!21\)\( T^{12} )^{2} \)
$43$ \( ( 1 + 27042 T^{2} + 15896713575 T^{4} + 267144985943548 T^{6} + \)\(10\!\cdots\!75\)\( T^{8} + \)\(10\!\cdots\!42\)\( T^{10} + \)\(25\!\cdots\!49\)\( T^{12} )^{2} \)
$47$ \( ( 1 + 377142 T^{2} + 74876518767 T^{4} + 9593174285133748 T^{6} + \)\(80\!\cdots\!43\)\( T^{8} + \)\(43\!\cdots\!22\)\( T^{10} + \)\(12\!\cdots\!89\)\( T^{12} )^{2} \)
$53$ \( ( 1 - 398679 T^{2} + 82394238966 T^{4} - 12639200632897475 T^{6} + \)\(18\!\cdots\!14\)\( T^{8} - \)\(19\!\cdots\!39\)\( T^{10} + \)\(10\!\cdots\!89\)\( T^{12} )^{2} \)
$59$ \( ( 1 + 275982 T^{2} + 30860919687 T^{4} - 3157264383153500 T^{6} + \)\(13\!\cdots\!67\)\( T^{8} + \)\(49\!\cdots\!42\)\( T^{10} + \)\(75\!\cdots\!21\)\( T^{12} )^{2} \)
$61$ \( ( 1 + 36 T + 351135 T^{2} + 84569384 T^{3} + 79700973435 T^{4} + 1854733476996 T^{5} + 11694146092834141 T^{6} )^{4} \)
$67$ \( ( 1 - 1341390 T^{2} + 846750752247 T^{4} - 319903491280410596 T^{6} + \)\(76\!\cdots\!43\)\( T^{8} - \)\(10\!\cdots\!90\)\( T^{10} + \)\(74\!\cdots\!09\)\( T^{12} )^{2} \)
$71$ \( ( 1 + 1203654 T^{2} + 624403431231 T^{4} + 229814346307125076 T^{6} + \)\(79\!\cdots\!51\)\( T^{8} + \)\(19\!\cdots\!14\)\( T^{10} + \)\(21\!\cdots\!61\)\( T^{12} )^{2} \)
$73$ \( ( 1 - 39 T + 955110 T^{2} - 15631571 T^{3} + 371554026870 T^{4} - 5902034825271 T^{5} + 58871586708267913 T^{6} )^{4} \)
$79$ \( ( 1 - 1716006 T^{2} + 1350264931311 T^{4} - 731914440695115860 T^{6} + \)\(32\!\cdots\!31\)\( T^{8} - \)\(10\!\cdots\!46\)\( T^{10} + \)\(14\!\cdots\!61\)\( T^{12} )^{2} \)
$83$ \( ( 1 + 1600737 T^{2} + 1256006483670 T^{4} + 724491266389094437 T^{6} + \)\(41\!\cdots\!30\)\( T^{8} + \)\(17\!\cdots\!57\)\( T^{10} + \)\(34\!\cdots\!09\)\( T^{12} )^{2} \)
$89$ \( ( 1 + 123846 T^{2} + 872233472127 T^{4} - 77031279330160556 T^{6} + \)\(43\!\cdots\!47\)\( T^{8} + \)\(30\!\cdots\!66\)\( T^{10} + \)\(12\!\cdots\!81\)\( T^{12} )^{2} \)
$97$ \( ( 1 - 129 T + 1814286 T^{2} - 533624789 T^{3} + 1655849846478 T^{4} - 107453388635841 T^{5} + 760231058654565217 T^{6} )^{4} \)
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