Properties

Label 1728.4.c.i
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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{36}\cdot 3^{12} \)
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.

\(f(q)\) \(=\) \( q -\beta_{6} q^{5} -\beta_{1} q^{7} +O(q^{10})\) \( q -\beta_{6} q^{5} -\beta_{1} q^{7} + \beta_{5} q^{11} + ( -3 - \beta_{2} ) q^{13} + ( -\beta_{6} - \beta_{9} ) q^{17} + ( \beta_{1} + \beta_{3} + \beta_{8} ) q^{19} + ( -\beta_{5} - \beta_{10} ) q^{23} + ( -11 + \beta_{4} ) q^{25} + ( -3 \beta_{6} + 2 \beta_{9} + \beta_{11} ) q^{29} + ( -\beta_{1} + 2 \beta_{3} + \beta_{8} ) q^{31} + ( 7 \beta_{5} - \beta_{7} ) q^{35} + ( -43 + \beta_{2} + 3 \beta_{4} ) q^{37} + ( 9 \beta_{6} - 2 \beta_{9} + \beta_{11} ) q^{41} + ( -5 \beta_{1} + 2 \beta_{3} - 3 \beta_{8} ) q^{43} + ( -3 \beta_{5} - 3 \beta_{7} - \beta_{10} ) q^{47} + ( -60 - 6 \beta_{2} + 2 \beta_{4} ) q^{49} + ( -13 \beta_{6} - \beta_{11} ) q^{53} + ( 19 \beta_{1} + 8 \beta_{3} - 3 \beta_{8} ) q^{55} + ( 3 \beta_{5} - 4 \beta_{7} - 2 \beta_{10} ) q^{59} + ( 81 + \beta_{2} - 2 \beta_{4} ) q^{61} + ( 2 \beta_{6} + 3 \beta_{9} - 5 \beta_{11} ) q^{65} + ( -4 \beta_{1} + 15 \beta_{3} ) q^{67} + ( 20 \beta_{5} - 2 \beta_{10} ) q^{71} + ( 55 + 6 \beta_{2} - 4 \beta_{4} ) q^{73} + ( 62 \beta_{6} - 2 \beta_{9} - 5 \beta_{11} ) q^{77} + ( -24 \beta_{1} + 18 \beta_{3} + \beta_{8} ) q^{79} + ( -14 \beta_{5} + 7 \beta_{7} - 2 \beta_{10} ) q^{83} + ( -88 - 8 \beta_{2} - \beta_{4} ) q^{85} + ( -45 \beta_{6} + 7 \beta_{9} + 4 \beta_{11} ) q^{89} + ( 45 \beta_{1} + 27 \beta_{3} + \beta_{8} ) q^{91} + ( 7 \beta_{5} + 6 \beta_{7} + 5 \beta_{10} ) q^{95} + ( 211 + 6 \beta_{2} - \beta_{4} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + O(q^{10}) \) \( 12q - 36q^{13} - 132q^{25} - 516q^{37} - 720q^{49} + 972q^{61} + 660q^{73} - 1056q^{85} + 2532q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 12 x^{10} + 112 x^{8} - 368 x^{6} + 928 x^{4} - 256 x^{2} + 64\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 67 \nu^{10} - 691 \nu^{8} + 6712 \nu^{6} - 18840 \nu^{4} + 63944 \nu^{2} - 9488 \)\()/3152\)
\(\beta_{2}\)\(=\)\((\)\( -153 \nu^{10} + 1428 \nu^{8} - 12540 \nu^{6} + 11832 \nu^{4} - 3264 \nu^{2} - 240608 \)\()/3152\)
\(\beta_{3}\)\(=\)\((\)\( -252 \nu^{10} + 2943 \nu^{8} - 27468 \nu^{6} + 85680 \nu^{4} - 227592 \nu^{2} + 34416 \)\()/3152\)
\(\beta_{4}\)\(=\)\((\)\( -18 \nu^{10} + 168 \nu^{8} - 1371 \nu^{6} + 1392 \nu^{4} - 384 \nu^{2} - 7448 \)\()/197\)
\(\beta_{5}\)\(=\)\((\)\( -287 \nu^{11} + 3598 \nu^{9} - 33844 \nu^{7} + 122008 \nu^{5} - 310816 \nu^{3} + 162912 \nu \)\()/3152\)
\(\beta_{6}\)\(=\)\((\)\( -190 \nu^{11} + 2233 \nu^{9} - 20710 \nu^{7} + 64600 \nu^{5} - 159552 \nu^{3} + 4560 \nu \)\()/1576\)
\(\beta_{7}\)\(=\)\((\)\( -73 \nu^{11} + 944 \nu^{9} - 8942 \nu^{7} + 33488 \nu^{5} - 84560 \nu^{3} + 44304 \nu \)\()/197\)
\(\beta_{8}\)\(=\)\((\)\( -1909 \nu^{10} + 23005 \nu^{8} - 213400 \nu^{6} + 697128 \nu^{4} - 1650872 \nu^{2} + 252272 \)\()/3152\)
\(\beta_{9}\)\(=\)\((\)\( 106 \nu^{11} - 1252 \nu^{9} + 11554 \nu^{7} - 36040 \nu^{5} + 84870 \nu^{3} - 2544 \nu \)\()/197\)
\(\beta_{10}\)\(=\)\((\)\( -1219 \nu^{11} + 14792 \nu^{9} - 138584 \nu^{7} + 470408 \nu^{5} - 1218512 \nu^{3} + 639168 \nu \)\()/1576\)
\(\beta_{11}\)\(=\)\((\)\( -2382 \nu^{11} + 27945 \nu^{9} - 259638 \nu^{7} + 809880 \nu^{5} - 2005056 \nu^{3} + 57168 \nu \)\()/1576\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{11} + 4 \beta_{10} - 4 \beta_{9} + \beta_{7} - 41 \beta_{6} - 24 \beta_{5}\)\()/576\)
\(\nu^{2}\)\(=\)\((\)\(3 \beta_{8} - \beta_{4} - 28 \beta_{3} + 4 \beta_{2} - 15 \beta_{1} + 288\)\()/144\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{11} - \beta_{9} - 17 \beta_{6}\)\()/18\)
\(\nu^{4}\)\(=\)\((\)\(9 \beta_{8} + 7 \beta_{4} - 88 \beta_{3} - 16 \beta_{2} - 81 \beta_{1} - 960\)\()/72\)
\(\nu^{5}\)\(=\)\((\)\(37 \beta_{11} - 24 \beta_{10} - 16 \beta_{9} - 45 \beta_{7} - 533 \beta_{6} + 384 \beta_{5}\)\()/144\)
\(\nu^{6}\)\(=\)\((\)\(17 \beta_{4} - 32 \beta_{2} - 1800\)\()/9\)
\(\nu^{7}\)\(=\)\((\)\(-53 \beta_{11} - 28 \beta_{10} + 12 \beta_{9} - 69 \beta_{7} + 717 \beta_{6} + 520 \beta_{5}\)\()/24\)
\(\nu^{8}\)\(=\)\((\)\(-111 \beta_{8} + 149 \beta_{4} + 1244 \beta_{3} - 260 \beta_{2} + 1563 \beta_{1} - 14208\)\()/18\)
\(\nu^{9}\)\(=\)\((\)\(-111 \beta_{11} + 16 \beta_{9} + 1463 \beta_{6}\)\()/3\)
\(\nu^{10}\)\(=\)\((\)\(-435 \beta_{8} - 629 \beta_{4} + 4992 \beta_{3} + 1064 \beta_{2} + 6531 \beta_{1} + 57408\)\()/9\)
\(\nu^{11}\)\(=\)\((\)\(-2761 \beta_{11} + 1272 \beta_{10} + 304 \beta_{9} + 3729 \beta_{7} + 35993 \beta_{6} - 26208 \beta_{5}\)\()/18\)

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
2.48442 1.43438i
−2.48442 1.43438i
−0.456937 0.263813i
0.456937 0.263813i
−1.61829 0.934317i
1.61829 0.934317i
1.61829 + 0.934317i
−1.61829 + 0.934317i
0.456937 + 0.263813i
−0.456937 + 0.263813i
−2.48442 + 1.43438i
2.48442 + 1.43438i
0 0 0 14.9230i 0 30.0528i 0 0 0
1727.2 0 0 0 14.9230i 0 30.0528i 0 0 0
1727.3 0 0 0 13.1987i 0 4.49091i 0 0 0
1727.4 0 0 0 13.1987i 0 4.49091i 0 0 0
1727.5 0 0 0 3.33155i 0 16.9016i 0 0 0
1727.6 0 0 0 3.33155i 0 16.9016i 0 0 0
1727.7 0 0 0 3.33155i 0 16.9016i 0 0 0
1727.8 0 0 0 3.33155i 0 16.9016i 0 0 0
1727.9 0 0 0 13.1987i 0 4.49091i 0 0 0
1727.10 0 0 0 13.1987i 0 4.49091i 0 0 0
1727.11 0 0 0 14.9230i 0 30.0528i 0 0 0
1727.12 0 0 0 14.9230i 0 30.0528i 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.i 12
3.b odd 2 1 inner 1728.4.c.i 12
4.b odd 2 1 inner 1728.4.c.i 12
8.b even 2 1 108.4.b.a 12
8.d odd 2 1 108.4.b.a 12
12.b even 2 1 inner 1728.4.c.i 12
24.f even 2 1 108.4.b.a 12
24.h odd 2 1 108.4.b.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.4.b.a 12 8.b even 2 1
108.4.b.a 12 8.d odd 2 1
108.4.b.a 12 24.f even 2 1
108.4.b.a 12 24.h odd 2 1
1728.4.c.i 12 1.a even 1 1 trivial
1728.4.c.i 12 3.b odd 2 1 inner
1728.4.c.i 12 4.b odd 2 1 inner
1728.4.c.i 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 \(S_{4}^{\mathrm{new}}(1728, [\chi])\):

\( T_{5}^{6} + 408 T_{5}^{4} + 43200 T_{5}^{2} + 430592 \)
\( T_{7}^{6} + 1209 T_{7}^{4} + 281979 T_{7}^{2} + 5203467 \)
\( T_{11}^{6} - 3912 T_{11}^{4} + 2591424 T_{11}^{2} - 442934784 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( ( 1 - 342 T^{2} + 73575 T^{4} - 11181908 T^{6} + 1149609375 T^{8} - 83496093750 T^{10} + 3814697265625 T^{12} )^{2} \)
$7$ \( ( 1 - 849 T^{2} + 387966 T^{4} - 141880421 T^{6} + 45643811934 T^{8} - 11751252833649 T^{10} + 1628413597910449 T^{12} )^{2} \)
$11$ \( ( 1 + 4074 T^{2} + 8337351 T^{4} + 12032309932 T^{6} + 14770125874911 T^{8} + 12785957206761354 T^{10} + 5559917313492231481 T^{12} )^{2} \)
$13$ \( ( 1 + 9 T + 1962 T^{2} - 66427 T^{3} + 4310514 T^{4} + 43441281 T^{5} + 10604499373 T^{6} )^{4} \)
$17$ \( ( 1 - 13134 T^{2} + 78982671 T^{4} - 357919940036 T^{6} + 1906449671066799 T^{8} - 7652160463775680974 T^{10} + \)\(14\!\cdots\!09\)\( T^{12} )^{2} \)
$19$ \( ( 1 - 18825 T^{2} + 202841862 T^{4} - 1656627349277 T^{6} + 9542874101470422 T^{8} - 41665653351420480825 T^{10} + \)\(10\!\cdots\!41\)\( T^{12} )^{2} \)
$23$ \( ( 1 + 6306 T^{2} + 110681535 T^{4} + 2149854095644 T^{6} + 16384839429609615 T^{8} + \)\(13\!\cdots\!26\)\( T^{10} + \)\(32\!\cdots\!69\)\( T^{12} )^{2} \)
$29$ \( ( 1 - 43326 T^{2} + 1075658103 T^{4} - 29470174509188 T^{6} + 639826525087020063 T^{8} - \)\(15\!\cdots\!66\)\( T^{10} + \)\(21\!\cdots\!61\)\( T^{12} )^{2} \)
$31$ \( ( 1 - 153750 T^{2} + 10409070351 T^{4} - 400107422263412 T^{6} + 9238088252300462031 T^{8} - \)\(12\!\cdots\!50\)\( T^{10} + \)\(69\!\cdots\!41\)\( T^{12} )^{2} \)
$37$ \( ( 1 + 129 T + 39378 T^{2} - 7527427 T^{3} + 1994613834 T^{4} + 330978706761 T^{5} + 129961739795077 T^{6} )^{4} \)
$41$ \( ( 1 - 232950 T^{2} + 25214641887 T^{4} - 1908694392304628 T^{6} + \)\(11\!\cdots\!67\)\( T^{8} - \)\(52\!\cdots\!50\)\( T^{10} + \)\(10\!\cdots\!21\)\( T^{12} )^{2} \)
$43$ \( ( 1 - 262254 T^{2} + 37283382183 T^{4} - 3613899564944228 T^{6} + \)\(23\!\cdots\!67\)\( T^{8} - \)\(10\!\cdots\!54\)\( T^{10} + \)\(25\!\cdots\!49\)\( T^{12} )^{2} \)
$47$ \( ( 1 + 179538 T^{2} + 22658226447 T^{4} + 3067316968586236 T^{6} + \)\(24\!\cdots\!63\)\( T^{8} + \)\(20\!\cdots\!58\)\( T^{10} + \)\(12\!\cdots\!89\)\( T^{12} )^{2} \)
$53$ \( ( 1 - 768750 T^{2} + 259538463591 T^{4} - 49798991602056548 T^{6} + \)\(57\!\cdots\!39\)\( T^{8} - \)\(37\!\cdots\!50\)\( T^{10} + \)\(10\!\cdots\!89\)\( T^{12} )^{2} \)
$59$ \( ( 1 + 523338 T^{2} + 180986988135 T^{4} + 40857650874211948 T^{6} + \)\(76\!\cdots\!35\)\( T^{8} + \)\(93\!\cdots\!78\)\( T^{10} + \)\(75\!\cdots\!21\)\( T^{12} )^{2} \)
$61$ \( ( 1 - 243 T + 648738 T^{2} - 106648063 T^{3} + 147251199978 T^{4} - 12519450969723 T^{5} + 11694146092834141 T^{6} )^{4} \)
$67$ \( ( 1 - 1637409 T^{2} + 1163514522486 T^{4} - 457988207385885221 T^{6} + \)\(10\!\cdots\!34\)\( T^{8} - \)\(13\!\cdots\!49\)\( T^{10} + \)\(74\!\cdots\!09\)\( T^{12} )^{2} \)
$71$ \( ( 1 + 377994 T^{2} + 13348121343 T^{4} - 1180461843095924 T^{6} + \)\(17\!\cdots\!03\)\( T^{8} + \)\(62\!\cdots\!54\)\( T^{10} + \)\(21\!\cdots\!61\)\( T^{12} )^{2} \)
$73$ \( ( 1 - 165 T + 834942 T^{2} - 185551841 T^{3} + 324806632014 T^{4} - 24970147337685 T^{5} + 58871586708267913 T^{6} )^{4} \)
$79$ \( ( 1 - 2108721 T^{2} + 2206739657838 T^{4} - 1369032710710548773 T^{6} + \)\(53\!\cdots\!98\)\( T^{8} - \)\(12\!\cdots\!61\)\( T^{10} + \)\(14\!\cdots\!61\)\( T^{12} )^{2} \)
$83$ \( ( 1 + 290850 T^{2} + 143373074583 T^{4} + 347024073827678524 T^{6} + \)\(46\!\cdots\!27\)\( T^{8} + \)\(31\!\cdots\!50\)\( T^{10} + \)\(34\!\cdots\!09\)\( T^{12} )^{2} \)
$89$ \( ( 1 - 1956702 T^{2} + 2321980636575 T^{4} - 1921108256399844452 T^{6} + \)\(11\!\cdots\!75\)\( T^{8} - \)\(48\!\cdots\!42\)\( T^{10} + \)\(12\!\cdots\!81\)\( T^{12} )^{2} \)
$97$ \( ( 1 - 633 T + 2697438 T^{2} - 1126776701 T^{3} + 2461878831774 T^{4} - 527271279120057 T^{5} + 760231058654565217 T^{6} )^{4} \)
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