Properties

Label 720.6.f.n
Level 720
Weight 6
Character orbit 720.f
Analytic conductor 115.476
Analytic rank 0
Dimension 8
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 720.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(115.476350265\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
Defining polynomial: \(x^{8} + 41 x^{6} + 460 x^{4} + 969 x^{2} + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{31}\cdot 3^{2}\cdot 5^{3} \)
Twist minimal: no (minimal twist has level 40)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + \beta_{1} ) q^{5} + ( -\beta_{1} + \beta_{6} ) q^{7} +O(q^{10})\) \( q + ( -1 + \beta_{1} ) q^{5} + ( -\beta_{1} + \beta_{6} ) q^{7} + ( -92 - \beta_{1} + \beta_{4} ) q^{11} + ( -2 \beta_{1} + \beta_{2} - \beta_{3} - 4 \beta_{6} + \beta_{7} ) q^{13} + ( 9 \beta_{1} - 14 \beta_{2} - \beta_{3} - 2 \beta_{7} ) q^{17} + ( -172 + 19 \beta_{1} + 4 \beta_{2} + \beta_{4} + 4 \beta_{7} ) q^{19} + ( \beta_{1} + 6 \beta_{2} - 10 \beta_{3} + 9 \beta_{6} - 4 \beta_{7} ) q^{23} + ( -267 + \beta_{1} - 25 \beta_{2} + 9 \beta_{3} - 4 \beta_{4} + \beta_{5} + 8 \beta_{6} + 3 \beta_{7} ) q^{25} + ( -734 - 26 \beta_{1} - 4 \beta_{2} + 4 \beta_{4} + 2 \beta_{5} - 4 \beta_{7} ) q^{29} + ( -528 + 16 \beta_{1} + 4 \beta_{2} + 4 \beta_{5} + 4 \beta_{7} ) q^{31} + ( 2404 + 15 \beta_{1} + 65 \beta_{2} + 34 \beta_{3} + \beta_{4} - 4 \beta_{5} + 18 \beta_{6} - 12 \beta_{7} ) q^{35} + ( 5 \beta_{1} + 137 \beta_{2} + 26 \beta_{3} + 16 \beta_{6} + \beta_{7} ) q^{37} + ( -2950 + 34 \beta_{1} + 9 \beta_{2} + 16 \beta_{4} - 5 \beta_{5} + 9 \beta_{7} ) q^{41} + ( 54 \beta_{1} - 81 \beta_{2} + 108 \beta_{3} + 54 \beta_{6} ) q^{43} + ( -81 \beta_{1} - 168 \beta_{2} + 50 \beta_{3} - 9 \beta_{6} + 28 \beta_{7} ) q^{47} + ( -5625 - 72 \beta_{1} - 9 \beta_{2} + 24 \beta_{4} + 3 \beta_{5} - 9 \beta_{7} ) q^{49} + ( 70 \beta_{1} - 39 \beta_{2} + 191 \beta_{3} + 36 \beta_{6} + 17 \beta_{7} ) q^{53} + ( -1876 - 204 \beta_{1} - 220 \beta_{2} - 125 \beta_{3} - 15 \beta_{4} - 10 \beta_{5} + 20 \beta_{6} - 80 \beta_{7} ) q^{55} + ( 11460 - 57 \beta_{1} - 12 \beta_{2} + 13 \beta_{4} - 16 \beta_{5} - 12 \beta_{7} ) q^{59} + ( 15482 - 263 \beta_{1} - 53 \beta_{2} - 8 \beta_{4} + 6 \beta_{5} - 53 \beta_{7} ) q^{61} + ( 9008 - 50 \beta_{1} - 95 \beta_{2} - 182 \beta_{3} + 12 \beta_{4} + 7 \beta_{5} - 144 \beta_{6} + 41 \beta_{7} ) q^{65} + ( -214 \beta_{1} - 33 \beta_{2} - 124 \beta_{3} + 10 \beta_{6} + 16 \beta_{7} ) q^{67} + ( -15704 + 458 \beta_{1} + 84 \beta_{2} - 2 \beta_{4} - 36 \beta_{5} + 84 \beta_{7} ) q^{71} + ( -103 \beta_{1} - 388 \beta_{2} - 311 \beta_{3} - 8 \beta_{6} - 40 \beta_{7} ) q^{73} + ( 217 \beta_{1} + 804 \beta_{2} - 111 \beta_{3} - 468 \beta_{6} + 28 \beta_{7} ) q^{77} + ( -5408 - 628 \beta_{1} - 124 \beta_{2} + 28 \beta_{4} - 20 \beta_{5} - 124 \beta_{7} ) q^{79} + ( 302 \beta_{1} - 615 \beta_{2} - 236 \beta_{3} + 342 \beta_{6} - 176 \beta_{7} ) q^{83} + ( -36720 + 85 \beta_{1} - 90 \beta_{2} + 125 \beta_{3} - 80 \beta_{4} + 30 \beta_{5} - 60 \beta_{6} - 10 \beta_{7} ) q^{85} + ( 5238 - 270 \beta_{1} - 36 \beta_{2} + 48 \beta_{4} + 42 \beta_{5} - 36 \beta_{7} ) q^{89} + ( 60952 + 770 \beta_{1} + 140 \beta_{2} - 110 \beta_{4} + 40 \beta_{5} + 140 \beta_{7} ) q^{91} + ( 55324 - 224 \beta_{1} - 720 \beta_{2} + 55 \beta_{3} - 95 \beta_{4} + 10 \beta_{5} + 180 \beta_{6} - 20 \beta_{7} ) q^{95} + ( 107 \beta_{1} - 486 \beta_{2} + 49 \beta_{3} - 488 \beta_{6} + 86 \beta_{7} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 8q^{5} + O(q^{10}) \) \( 8q - 8q^{5} - 736q^{11} - 1376q^{19} - 2136q^{25} - 5872q^{29} - 4224q^{31} + 19232q^{35} - 23600q^{41} - 45000q^{49} - 15008q^{55} + 91680q^{59} + 123856q^{61} + 72064q^{65} - 125632q^{71} - 43264q^{79} - 293760q^{85} + 41904q^{89} + 487616q^{91} + 442592q^{95} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 41 x^{6} + 460 x^{4} + 969 x^{2} + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -176 \nu^{7} - 66 \nu^{6} - 7372 \nu^{5} - 2232 \nu^{4} - 86042 \nu^{3} - 19752 \nu^{2} - 203676 \nu - 29412 \)\()/639\)
\(\beta_{2}\)\(=\)\((\)\( -182 \nu^{7} - 7420 \nu^{5} - 82106 \nu^{3} - 160458 \nu \)\()/213\)
\(\beta_{3}\)\(=\)\((\)\( -656 \nu^{7} + 66 \nu^{6} - 26548 \nu^{5} + 2232 \nu^{4} - 295142 \nu^{3} + 19752 \nu^{2} - 646692 \nu + 29412 \)\()/639\)
\(\beta_{4}\)\(=\)\((\)\( -176 \nu^{7} - 282 \nu^{6} - 7372 \nu^{5} - 17592 \nu^{4} - 86042 \nu^{3} - 259752 \nu^{2} - 203676 \nu - 390564 \)\()/639\)
\(\beta_{5}\)\(=\)\((\)\( -176 \nu^{7} + 1710 \nu^{6} - 7372 \nu^{5} + 59688 \nu^{4} - 86042 \nu^{3} + 457848 \nu^{2} - 203676 \nu - 144180 \)\()/639\)
\(\beta_{6}\)\(=\)\((\)\( 974 \nu^{7} - 66 \nu^{6} + 40168 \nu^{5} - 2232 \nu^{4} + 454172 \nu^{3} - 19752 \nu^{2} + 991374 \nu - 29412 \)\()/639\)
\(\beta_{7}\)\(=\)\((\)\( 1426 \nu^{7} - 330 \nu^{6} + 59120 \nu^{5} - 11160 \nu^{4} + 676528 \nu^{3} - 98760 \nu^{2} + 1499754 \nu - 147060 \)\()/639\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-24 \beta_{7} + 72 \beta_{6} - 9 \beta_{3} + 64 \beta_{2} + 39 \beta_{1}\)\()/3840\)
\(\nu^{2}\)\(=\)\((\)\(-34 \beta_{7} - 13 \beta_{5} - 3 \beta_{4} - 34 \beta_{2} - 154 \beta_{1} - 19680\)\()/1920\)
\(\nu^{3}\)\(=\)\((\)\(6 \beta_{7} - 18 \beta_{6} - 3 \beta_{3} - 8 \beta_{2} - 15 \beta_{1}\)\()/48\)
\(\nu^{4}\)\(=\)\((\)\(790 \beta_{7} + 283 \beta_{5} - 87 \beta_{4} + 790 \beta_{2} + 3754 \beta_{1} + 365280\)\()/1920\)
\(\nu^{5}\)\(=\)\((\)\(-9048 \beta_{7} + 30888 \beta_{6} + 10329 \beta_{3} + 11104 \beta_{2} + 24681 \beta_{1}\)\()/3840\)
\(\nu^{6}\)\(=\)\((\)\(-230 \beta_{7} - 71 \beta_{5} + 48 \beta_{4} - 230 \beta_{2} - 1127 \beta_{1} - 91488\)\()/24\)
\(\nu^{7}\)\(=\)\((\)\(173496 \beta_{7} - 673128 \beta_{6} - 304899 \beta_{3} - 224896 \beta_{2} - 499251 \beta_{1}\)\()/3840\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/720\mathbb{Z}\right)^\times\).

\(n\) \(181\) \(271\) \(577\) \(641\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
289.1
0.0965878i
0.0965878i
3.98753i
3.98753i
1.64654i
1.64654i
4.73066i
4.73066i
0 0 0 −46.7401 30.6653i 0 179.876i 0 0 0
289.2 0 0 0 −46.7401 + 30.6653i 0 179.876i 0 0 0
289.3 0 0 0 −23.4238 50.7575i 0 10.2635i 0 0 0
289.4 0 0 0 −23.4238 + 50.7575i 0 10.2635i 0 0 0
289.5 0 0 0 13.1588 54.3309i 0 146.828i 0 0 0
289.6 0 0 0 13.1588 + 54.3309i 0 146.828i 0 0 0
289.7 0 0 0 53.0051 17.7613i 0 188.968i 0 0 0
289.8 0 0 0 53.0051 + 17.7613i 0 188.968i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 289.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 720.6.f.n 8
3.b odd 2 1 80.6.c.d 8
4.b odd 2 1 360.6.f.b 8
5.b even 2 1 inner 720.6.f.n 8
12.b even 2 1 40.6.c.a 8
15.d odd 2 1 80.6.c.d 8
15.e even 4 1 400.6.a.z 4
15.e even 4 1 400.6.a.ba 4
20.d odd 2 1 360.6.f.b 8
24.f even 2 1 320.6.c.j 8
24.h odd 2 1 320.6.c.i 8
60.h even 2 1 40.6.c.a 8
60.l odd 4 1 200.6.a.j 4
60.l odd 4 1 200.6.a.k 4
120.i odd 2 1 320.6.c.i 8
120.m even 2 1 320.6.c.j 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.6.c.a 8 12.b even 2 1
40.6.c.a 8 60.h even 2 1
80.6.c.d 8 3.b odd 2 1
80.6.c.d 8 15.d odd 2 1
200.6.a.j 4 60.l odd 4 1
200.6.a.k 4 60.l odd 4 1
320.6.c.i 8 24.h odd 2 1
320.6.c.i 8 120.i odd 2 1
320.6.c.j 8 24.f even 2 1
320.6.c.j 8 120.m even 2 1
360.6.f.b 8 4.b odd 2 1
360.6.f.b 8 20.d odd 2 1
400.6.a.z 4 15.e even 4 1
400.6.a.ba 4 15.e even 4 1
720.6.f.n 8 1.a even 1 1 trivial
720.6.f.n 8 5.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 \(S_{6}^{\mathrm{new}}(720, [\chi])\):

\( T_{7}^{8} + 89728 T_{7}^{6} + 2632172640 T_{7}^{4} + \)\(25\!\cdots\!32\)\( T_{7}^{2} + \)\(26\!\cdots\!24\)\( \)
\( T_{11}^{4} + 368 T_{11}^{3} - 468000 T_{11}^{2} - 126641408 T_{11} + 37397137664 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( 1 + 8 T + 1100 T^{2} - 113000 T^{3} - 438250 T^{4} - 353125000 T^{5} + 10742187500 T^{6} + 244140625000 T^{7} + 95367431640625 T^{8} \)
$7$ \( 1 - 44728 T^{2} + 1493128636 T^{4} - 37445733732616 T^{6} + 682894235558230726 T^{8} - \)\(10\!\cdots\!84\)\( T^{10} + \)\(11\!\cdots\!36\)\( T^{12} - \)\(10\!\cdots\!72\)\( T^{14} + \)\(63\!\cdots\!01\)\( T^{16} \)
$11$ \( ( 1 + 368 T + 176204 T^{2} + 51158896 T^{3} + 42277949270 T^{4} + 8239191359696 T^{5} + 4570277964394604 T^{6} + 1537227326344959568 T^{7} + \)\(67\!\cdots\!01\)\( T^{8} )^{2} \)
$13$ \( 1 - 1578472 T^{2} + 1074189263356 T^{4} - 437549632721743384 T^{6} + \)\(15\!\cdots\!86\)\( T^{8} - \)\(60\!\cdots\!16\)\( T^{10} + \)\(20\!\cdots\!56\)\( T^{12} - \)\(41\!\cdots\!28\)\( T^{14} + \)\(36\!\cdots\!01\)\( T^{16} \)
$17$ \( 1 - 6260872 T^{2} + 17547242668444 T^{4} - 31297293718759478968 T^{6} + \)\(45\!\cdots\!30\)\( T^{8} - \)\(63\!\cdots\!32\)\( T^{10} + \)\(71\!\cdots\!44\)\( T^{12} - \)\(51\!\cdots\!28\)\( T^{14} + \)\(16\!\cdots\!01\)\( T^{16} \)
$19$ \( ( 1 + 688 T + 5308396 T^{2} + 6058136368 T^{3} + 15069081422710 T^{4} + 15000545402668432 T^{5} + 32546127598645797196 T^{6} + \)\(10\!\cdots\!12\)\( T^{7} + \)\(37\!\cdots\!01\)\( T^{8} )^{2} \)
$23$ \( 1 - 34675896 T^{2} + 569897415616828 T^{4} - \)\(59\!\cdots\!20\)\( T^{6} + \)\(44\!\cdots\!82\)\( T^{8} - \)\(24\!\cdots\!80\)\( T^{10} + \)\(97\!\cdots\!28\)\( T^{12} - \)\(24\!\cdots\!04\)\( T^{14} + \)\(29\!\cdots\!01\)\( T^{16} \)
$29$ \( ( 1 + 2936 T + 58625996 T^{2} + 77951973928 T^{3} + 1466411094282230 T^{4} + 1598884552081323272 T^{5} + \)\(24\!\cdots\!96\)\( T^{6} + \)\(25\!\cdots\!64\)\( T^{7} + \)\(17\!\cdots\!01\)\( T^{8} )^{2} \)
$31$ \( ( 1 + 2112 T + 80187004 T^{2} + 163080265536 T^{3} + 3196344720873606 T^{4} + 4668849547150239936 T^{5} + \)\(65\!\cdots\!04\)\( T^{6} + \)\(49\!\cdots\!12\)\( T^{7} + \)\(67\!\cdots\!01\)\( T^{8} )^{2} \)
$37$ \( 1 - 251774632 T^{2} + 38631208311838780 T^{4} - \)\(41\!\cdots\!52\)\( T^{6} + \)\(32\!\cdots\!34\)\( T^{8} - \)\(19\!\cdots\!48\)\( T^{10} + \)\(89\!\cdots\!80\)\( T^{12} - \)\(27\!\cdots\!68\)\( T^{14} + \)\(53\!\cdots\!01\)\( T^{16} \)
$41$ \( ( 1 + 11800 T + 337909340 T^{2} + 2943020124776 T^{3} + 51155654972384870 T^{4} + \)\(34\!\cdots\!76\)\( T^{5} + \)\(45\!\cdots\!40\)\( T^{6} + \)\(18\!\cdots\!00\)\( T^{7} + \)\(18\!\cdots\!01\)\( T^{8} )^{2} \)
$43$ \( 1 - 283211672 T^{2} + 48021567531024796 T^{4} - \)\(54\!\cdots\!84\)\( T^{6} + \)\(43\!\cdots\!06\)\( T^{8} - \)\(11\!\cdots\!16\)\( T^{10} + \)\(22\!\cdots\!96\)\( T^{12} - \)\(28\!\cdots\!28\)\( T^{14} + \)\(21\!\cdots\!01\)\( T^{16} \)
$47$ \( 1 - 963352312 T^{2} + 473832864723586300 T^{4} - \)\(15\!\cdots\!72\)\( T^{6} + \)\(40\!\cdots\!54\)\( T^{8} - \)\(83\!\cdots\!28\)\( T^{10} + \)\(13\!\cdots\!00\)\( T^{12} - \)\(14\!\cdots\!88\)\( T^{14} + \)\(76\!\cdots\!01\)\( T^{16} \)
$53$ \( 1 - 1183385640 T^{2} + 874785099161623996 T^{4} - \)\(49\!\cdots\!80\)\( T^{6} + \)\(23\!\cdots\!06\)\( T^{8} - \)\(86\!\cdots\!20\)\( T^{10} + \)\(26\!\cdots\!96\)\( T^{12} - \)\(63\!\cdots\!60\)\( T^{14} + \)\(93\!\cdots\!01\)\( T^{16} \)
$59$ \( ( 1 - 45840 T + 3064286732 T^{2} - 94721285480976 T^{3} + 3348109683185502486 T^{4} - \)\(67\!\cdots\!24\)\( T^{5} + \)\(15\!\cdots\!32\)\( T^{6} - \)\(16\!\cdots\!60\)\( T^{7} + \)\(26\!\cdots\!01\)\( T^{8} )^{2} \)
$61$ \( ( 1 - 61928 T + 3903014764 T^{2} - 145287706763384 T^{3} + 5198153942066716726 T^{4} - \)\(12\!\cdots\!84\)\( T^{5} + \)\(27\!\cdots\!64\)\( T^{6} - \)\(37\!\cdots\!28\)\( T^{7} + \)\(50\!\cdots\!01\)\( T^{8} )^{2} \)
$67$ \( 1 - 9281919064 T^{2} + 39492482666681482588 T^{4} - \)\(10\!\cdots\!40\)\( T^{6} + \)\(16\!\cdots\!62\)\( T^{8} - \)\(18\!\cdots\!60\)\( T^{10} + \)\(13\!\cdots\!88\)\( T^{12} - \)\(56\!\cdots\!36\)\( T^{14} + \)\(11\!\cdots\!01\)\( T^{16} \)
$71$ \( ( 1 + 62816 T + 3398787356 T^{2} + 184024084124896 T^{3} + 8353296562609817510 T^{4} + \)\(33\!\cdots\!96\)\( T^{5} + \)\(11\!\cdots\!56\)\( T^{6} + \)\(36\!\cdots\!16\)\( T^{7} + \)\(10\!\cdots\!01\)\( T^{8} )^{2} \)
$73$ \( 1 - 9140679496 T^{2} + 46078306824990298588 T^{4} - \)\(15\!\cdots\!60\)\( T^{6} + \)\(37\!\cdots\!02\)\( T^{8} - \)\(66\!\cdots\!40\)\( T^{10} + \)\(85\!\cdots\!88\)\( T^{12} - \)\(72\!\cdots\!04\)\( T^{14} + \)\(34\!\cdots\!01\)\( T^{16} \)
$79$ \( ( 1 + 21632 T + 7152876604 T^{2} + 332616618908288 T^{3} + 24121899620797566790 T^{4} + \)\(10\!\cdots\!12\)\( T^{5} + \)\(67\!\cdots\!04\)\( T^{6} + \)\(63\!\cdots\!68\)\( T^{7} + \)\(89\!\cdots\!01\)\( T^{8} )^{2} \)
$83$ \( 1 - 6759897816 T^{2} + 40943120759345365468 T^{4} - \)\(86\!\cdots\!80\)\( T^{6} + \)\(39\!\cdots\!42\)\( T^{8} - \)\(13\!\cdots\!20\)\( T^{10} + \)\(98\!\cdots\!68\)\( T^{12} - \)\(25\!\cdots\!84\)\( T^{14} + \)\(57\!\cdots\!01\)\( T^{16} \)
$89$ \( ( 1 - 20952 T + 16118164796 T^{2} - 497915996461992 T^{3} + \)\(11\!\cdots\!30\)\( T^{4} - \)\(27\!\cdots\!08\)\( T^{5} + \)\(50\!\cdots\!96\)\( T^{6} - \)\(36\!\cdots\!48\)\( T^{7} + \)\(97\!\cdots\!01\)\( T^{8} )^{2} \)
$97$ \( 1 - 45263915272 T^{2} + \)\(10\!\cdots\!40\)\( T^{4} - \)\(14\!\cdots\!12\)\( T^{6} + \)\(15\!\cdots\!94\)\( T^{8} - \)\(11\!\cdots\!88\)\( T^{10} + \)\(55\!\cdots\!40\)\( T^{12} - \)\(18\!\cdots\!28\)\( T^{14} + \)\(29\!\cdots\!01\)\( T^{16} \)
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