Properties

Label 6048.2.h.h
Level 6048
Weight 2
Character orbit 6048.h
Analytic conductor 48.294
Analytic rank 0
Dimension 16
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 6048 = 2^{5} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6048.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(48.2935231425\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{12} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{12} q^{5} + \beta_{3} q^{7} +O(q^{10})\) \( q -\beta_{12} q^{5} + \beta_{3} q^{7} + ( -\beta_{11} + \beta_{13} ) q^{11} -\beta_{4} q^{13} + ( -\beta_{8} - \beta_{14} ) q^{17} + ( -2 \beta_{3} + \beta_{6} ) q^{19} + ( -2 \beta_{10} - 3 \beta_{11} ) q^{23} + ( 3 + \beta_{2} ) q^{25} + \beta_{9} q^{29} + ( 2 \beta_{1} - \beta_{3} - \beta_{7} ) q^{31} + \beta_{11} q^{35} + ( 2 - \beta_{2} - \beta_{4} ) q^{37} + ( \beta_{8} - \beta_{9} - \beta_{12} + \beta_{14} ) q^{41} + ( -\beta_{1} - \beta_{3} + \beta_{7} ) q^{43} + ( 3 \beta_{10} - 2 \beta_{11} - \beta_{13} ) q^{47} - q^{49} + ( -2 \beta_{8} + 3 \beta_{12} + \beta_{14} ) q^{53} + ( -\beta_{1} + 2 \beta_{3} - \beta_{7} ) q^{55} + ( -3 \beta_{10} - 3 \beta_{11} - \beta_{13} - \beta_{15} ) q^{59} + ( 1 - 2 \beta_{2} + 2 \beta_{4} + \beta_{5} ) q^{61} + ( \beta_{9} - \beta_{12} - \beta_{14} ) q^{65} + ( -\beta_{1} - 2 \beta_{3} - \beta_{6} - \beta_{7} ) q^{67} + ( -5 \beta_{10} - 6 \beta_{11} + \beta_{15} ) q^{71} + ( -3 - 2 \beta_{2} + \beta_{4} - \beta_{5} ) q^{73} + ( -\beta_{8} + \beta_{9} ) q^{77} + ( 3 \beta_{1} - 7 \beta_{3} - 2 \beta_{6} + \beta_{7} ) q^{79} + ( -\beta_{10} - 4 \beta_{11} + \beta_{13} ) q^{83} + ( -\beta_{2} + 2 \beta_{4} + \beta_{5} ) q^{85} + ( -\beta_{8} - \beta_{9} - 6 \beta_{12} ) q^{89} + ( -\beta_{1} + \beta_{3} - \beta_{7} ) q^{91} + ( -2 \beta_{11} - \beta_{15} ) q^{95} + ( -1 + \beta_{5} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + O(q^{10}) \) \( 16q + 8q^{13} + 48q^{25} + 40q^{37} - 16q^{49} - 56q^{73} - 16q^{85} - 16q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} + 242 x^{12} + 13297 x^{8} + 201600 x^{4} + 331776\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 5 \nu^{14} + 1786 \nu^{10} + 188597 \nu^{6} + 4585536 \nu^{2} \)\()/3345408\)
\(\beta_{2}\)\(=\)\((\)\( 7 \nu^{12} + 1406 \nu^{8} + 30007 \nu^{4} - 455616 \)\()/292608\)
\(\beta_{3}\)\(=\)\((\)\( -2485 \nu^{14} - 608858 \nu^{10} - 35048677 \nu^{6} - 685063872 \nu^{2} \)\()/ 849733632 \)
\(\beta_{4}\)\(=\)\((\)\( 755 \nu^{12} + 173854 \nu^{8} + 7752299 \nu^{4} + 42152256 \)\()/8851392\)
\(\beta_{5}\)\(=\)\((\)\( 1003 \nu^{12} + 211910 \nu^{8} + 7466011 \nu^{4} + 42762240 \)\()/8851392\)
\(\beta_{6}\)\(=\)\((\)\( -25 \nu^{14} - 5602 \nu^{10} - 242441 \nu^{6} - 2339264 \nu^{2} \)\()/780288\)
\(\beta_{7}\)\(=\)\((\)\( -9721 \nu^{14} - 2320034 \nu^{10} - 120569065 \nu^{6} - 1445284032 \nu^{2} \)\()/ 283244544 \)
\(\beta_{8}\)\(=\)\((\)\( -3577 \nu^{15} - 1002 \nu^{13} - 901346 \nu^{11} - 608820 \nu^{9} - 53043433 \nu^{7} - 74677962 \nu^{5} - 679897728 \nu^{3} - 935501184 \nu \)\()/ 2549200896 \)
\(\beta_{9}\)\(=\)\((\)\( -2485 \nu^{15} - 608858 \nu^{11} - 35048677 \nu^{7} - 685063872 \nu^{3} - 849733632 \nu \)\()/ 849733632 \)
\(\beta_{10}\)\(=\)\((\)\( 58873 \nu^{15} - 148440 \nu^{13} + 13669538 \nu^{11} - 32950320 \nu^{9} + 660774121 \nu^{7} - 1373817432 \nu^{5} + 7568374464 \nu^{3} - 11174367744 \nu \)\()/ 10196803584 \)
\(\beta_{11}\)\(=\)\((\)\( -73181 \nu^{15} + 152448 \nu^{13} - 17274922 \nu^{11} + 35385600 \nu^{9} - 872947853 \nu^{7} + 1672529280 \nu^{5} - 10287965376 \nu^{3} + 14916372480 \nu \)\()/ 10196803584 \)
\(\beta_{12}\)\(=\)\((\)\( -73181 \nu^{15} - 152448 \nu^{13} - 17274922 \nu^{11} - 35385600 \nu^{9} - 872947853 \nu^{7} - 1672529280 \nu^{5} - 10287965376 \nu^{3} - 14916372480 \nu \)\()/ 10196803584 \)
\(\beta_{13}\)\(=\)\((\)\( -88693 \nu^{15} + 148440 \nu^{13} - 20975834 \nu^{11} + 32950320 \nu^{9} - 1081358245 \nu^{7} + 1373817432 \nu^{5} - 15789140928 \nu^{3} + 21371171328 \nu \)\()/ 10196803584 \)
\(\beta_{14}\)\(=\)\((\)\( 118241 \nu^{15} - 91488 \nu^{13} + 30024946 \nu^{11} - 13610688 \nu^{9} + 1868375633 \nu^{7} + 626845344 \nu^{5} + 32485445568 \nu^{3} + 40990482432 \nu \)\()/ 10196803584 \)
\(\beta_{15}\)\(=\)\((\)\( 118241 \nu^{15} + 91488 \nu^{13} + 30024946 \nu^{11} + 13610688 \nu^{9} + 1868375633 \nu^{7} - 626845344 \nu^{5} + 32485445568 \nu^{3} - 40990482432 \nu \)\()/ 10196803584 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{13} + \beta_{10} - \beta_{9}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{6} - 13 \beta_{3} - 4 \beta_{1}\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(-2 \beta_{15} - 2 \beta_{14} - 9 \beta_{13} + 2 \beta_{12} - 6 \beta_{11} - 17 \beta_{10} - 9 \beta_{9} - 8 \beta_{8}\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(17 \beta_{5} - 8 \beta_{4} - 52 \beta_{2} - 125\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(-30 \beta_{15} + 30 \beta_{14} - 97 \beta_{13} - 22 \beta_{12} - 130 \beta_{11} - 249 \beta_{10} + 97 \beta_{9} + 152 \beta_{8}\)\()/2\)
\(\nu^{6}\)\(=\)\((\)\(160 \beta_{7} - 249 \beta_{6} + 1253 \beta_{3} + 788 \beta_{1}\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(394 \beta_{15} + 394 \beta_{14} + 1145 \beta_{13} - 282 \beta_{12} + 2030 \beta_{11} + 3457 \beta_{10} + 1145 \beta_{9} + 2312 \beta_{8}\)\()/2\)
\(\nu^{8}\)\(=\)\((\)\(-3457 \beta_{5} + 2424 \beta_{4} + 7732 \beta_{2} + 17197\)\()/2\)
\(\nu^{9}\)\(=\)\((\)\(5078 \beta_{15} - 5078 \beta_{14} + 14193 \beta_{13} + 3902 \beta_{12} + 28602 \beta_{11} + 46697 \beta_{10} - 14193 \beta_{9} - 32504 \beta_{8}\)\()/2\)
\(\nu^{10}\)\(=\)\((\)\(-33680 \beta_{7} + 46697 \beta_{6} - 183333 \beta_{3} - 131076 \beta_{1}\)\()/2\)
\(\nu^{11}\)\(=\)\((\)\(-65538 \beta_{15} - 65538 \beta_{14} - 180553 \beta_{13} + 53890 \beta_{12} - 387046 \beta_{11} - 621489 \beta_{10} - 180553 \beta_{9} - 440936 \beta_{8}\)\()/2\)
\(\nu^{12}\)\(=\)\((\)\(621489 \beta_{5} - 452584 \beta_{4} - 1246516 \beta_{2} - 2788125\)\()/2\)
\(\nu^{13}\)\(=\)\((\)\(-849550 \beta_{15} + 849550 \beta_{14} - 2328065 \beta_{13} - 731238 \beta_{12} - 5145842 \beta_{11} - 8205145 \beta_{10} + 2328065 \beta_{9} + 5877080 \beta_{8}\)\()/2\)
\(\nu^{14}\)\(=\)\((\)\(5995392 \beta_{7} - 8205145 \beta_{6} + 30146533 \beta_{3} + 22104052 \beta_{1}\)\()/2\)
\(\nu^{15}\)\(=\)\((\)\(11052026 \beta_{15} + 11052026 \beta_{14} + 30227865 \beta_{13} - 9777770 \beta_{12} + 67854174 \beta_{11} + 107859809 \beta_{10} + 30227865 \beta_{9} + 77631944 \beta_{8}\)\()/2\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/6048\mathbb{Z}\right)^\times\).

\(n\) \(2593\) \(3781\) \(3809\) \(4159\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2591.1
−0.826679 0.826679i
1.53379 + 1.53379i
−1.53379 + 1.53379i
0.826679 0.826679i
−2.55743 2.55743i
1.85032 + 1.85032i
−1.85032 + 1.85032i
2.55743 2.55743i
−1.85032 1.85032i
2.55743 + 2.55743i
−2.55743 + 2.55743i
1.85032 1.85032i
−1.53379 1.53379i
0.826679 + 0.826679i
−0.826679 + 0.826679i
1.53379 1.53379i
0 0 0 1.93185i 0 1.00000i 0 0 0
2591.2 0 0 0 1.93185i 0 1.00000i 0 0 0
2591.3 0 0 0 1.93185i 0 1.00000i 0 0 0
2591.4 0 0 0 1.93185i 0 1.00000i 0 0 0
2591.5 0 0 0 0.517638i 0 1.00000i 0 0 0
2591.6 0 0 0 0.517638i 0 1.00000i 0 0 0
2591.7 0 0 0 0.517638i 0 1.00000i 0 0 0
2591.8 0 0 0 0.517638i 0 1.00000i 0 0 0
2591.9 0 0 0 0.517638i 0 1.00000i 0 0 0
2591.10 0 0 0 0.517638i 0 1.00000i 0 0 0
2591.11 0 0 0 0.517638i 0 1.00000i 0 0 0
2591.12 0 0 0 0.517638i 0 1.00000i 0 0 0
2591.13 0 0 0 1.93185i 0 1.00000i 0 0 0
2591.14 0 0 0 1.93185i 0 1.00000i 0 0 0
2591.15 0 0 0 1.93185i 0 1.00000i 0 0 0
2591.16 0 0 0 1.93185i 0 1.00000i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2591.16
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 6048.2.h.h 16
3.b odd 2 1 inner 6048.2.h.h 16
4.b odd 2 1 inner 6048.2.h.h 16
12.b even 2 1 inner 6048.2.h.h 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6048.2.h.h 16 1.a even 1 1 trivial
6048.2.h.h 16 3.b odd 2 1 inner
6048.2.h.h 16 4.b odd 2 1 inner
6048.2.h.h 16 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 \(S_{2}^{\mathrm{new}}(6048, [\chi])\):

\( T_{5}^{4} + 4 T_{5}^{2} + 1 \)
\( T_{11}^{8} - 56 T_{11}^{6} + 930 T_{11}^{4} - 5240 T_{11}^{2} + 5329 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( \)
$3$ \( \)
$5$ \( ( 1 - 16 T^{2} + 111 T^{4} - 400 T^{6} + 625 T^{8} )^{4} \)
$7$ \( ( 1 + T^{2} )^{8} \)
$11$ \( ( 1 + 32 T^{2} + 622 T^{4} + 8576 T^{6} + 99379 T^{8} + 1037696 T^{10} + 9106702 T^{12} + 56689952 T^{14} + 214358881 T^{16} )^{2} \)
$13$ \( ( 1 - 2 T + 26 T^{2} - 78 T^{3} + 434 T^{4} - 1014 T^{5} + 4394 T^{6} - 4394 T^{7} + 28561 T^{8} )^{4} \)
$17$ \( ( 1 - 24 T^{2} + 788 T^{4} - 17736 T^{6} + 292518 T^{8} - 5125704 T^{10} + 65814548 T^{12} - 579301656 T^{14} + 6975757441 T^{16} )^{2} \)
$19$ \( ( 1 - 36 T^{2} + 746 T^{4} - 8304 T^{6} + 72147 T^{8} - 2997744 T^{10} + 97219466 T^{12} - 1693651716 T^{14} + 16983563041 T^{16} )^{2} \)
$23$ \( ( 1 + 64 T^{2} + 2055 T^{4} + 33856 T^{6} + 279841 T^{8} )^{4} \)
$29$ \( ( 1 - 180 T^{2} + 15368 T^{4} - 805116 T^{6} + 28222446 T^{8} - 677102556 T^{10} + 10869494408 T^{12} - 107068197780 T^{14} + 500246412961 T^{16} )^{2} \)
$31$ \( ( 1 - 112 T^{2} + 4690 T^{4} - 84064 T^{6} + 1025299 T^{8} - 80785504 T^{10} + 4331313490 T^{12} - 99400412272 T^{14} + 852891037441 T^{16} )^{2} \)
$37$ \( ( 1 - 10 T + 146 T^{2} - 1056 T^{3} + 8039 T^{4} - 39072 T^{5} + 199874 T^{6} - 506530 T^{7} + 1874161 T^{8} )^{4} \)
$41$ \( ( 1 - 168 T^{2} + 14246 T^{4} - 874752 T^{6} + 41345307 T^{8} - 1470458112 T^{10} + 40255791206 T^{12} - 798017512488 T^{14} + 7984925229121 T^{16} )^{2} \)
$43$ \( ( 1 - 264 T^{2} + 32912 T^{4} - 2530824 T^{6} + 131089566 T^{8} - 4679493576 T^{10} + 112519578512 T^{12} - 1668839844936 T^{14} + 11688200277601 T^{16} )^{2} \)
$47$ \( ( 1 + 140 T^{2} + 12088 T^{4} + 786212 T^{6} + 39897838 T^{8} + 1736742308 T^{10} + 58985583928 T^{12} + 1509090146060 T^{14} + 23811286661761 T^{16} )^{2} \)
$53$ \( ( 1 - 264 T^{2} + 32204 T^{4} - 2520504 T^{6} + 149086662 T^{8} - 7080095736 T^{10} + 254105050124 T^{12} - 5851391338056 T^{14} + 62259690411361 T^{16} )^{2} \)
$59$ \( ( 1 + 260 T^{2} + 30952 T^{4} + 2376716 T^{6} + 147668398 T^{8} + 8273348396 T^{10} + 375056557672 T^{12} + 10966938746660 T^{14} + 146830437604321 T^{16} )^{2} \)
$61$ \( ( 1 + 80 T^{2} - 528 T^{3} + 2286 T^{4} - 32208 T^{5} + 297680 T^{6} + 13845841 T^{8} )^{4} \)
$67$ \( ( 1 - 352 T^{2} + 62320 T^{4} - 7098256 T^{6} + 563157790 T^{8} - 31864071184 T^{10} + 1255817860720 T^{12} - 31841350523488 T^{14} + 406067677556641 T^{16} )^{2} \)
$71$ \( ( 1 + 216 T^{2} + 28598 T^{4} + 2870208 T^{6} + 226923531 T^{8} + 14468718528 T^{10} + 726723253238 T^{12} + 27669661326936 T^{14} + 645753531245761 T^{16} )^{2} \)
$73$ \( ( 1 + 14 T + 278 T^{2} + 2994 T^{3} + 29954 T^{4} + 218562 T^{5} + 1481462 T^{6} + 5446238 T^{7} + 28398241 T^{8} )^{4} \)
$79$ \( ( 1 + 56 T^{2} + 14896 T^{4} + 945080 T^{6} + 108009022 T^{8} + 5898244280 T^{10} + 580200406576 T^{12} + 13612897509176 T^{14} + 1517108809906561 T^{16} )^{2} \)
$83$ \( ( 1 + 516 T^{2} + 125336 T^{4} + 18717900 T^{6} + 1875461838 T^{8} + 128947613100 T^{10} + 5948236120856 T^{12} + 168701232658404 T^{14} + 2252292232139041 T^{16} )^{2} \)
$89$ \( ( 1 - 336 T^{2} + 65102 T^{4} - 8829312 T^{6} + 891379059 T^{8} - 69936980352 T^{10} + 4084645373582 T^{12} - 166985713762896 T^{14} + 3936588805702081 T^{16} )^{2} \)
$97$ \( ( 1 + 4 T + 344 T^{2} + 1068 T^{3} + 48302 T^{4} + 103596 T^{5} + 3236696 T^{6} + 3650692 T^{7} + 88529281 T^{8} )^{4} \)
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