# Properties

 Label 360.6.f.b Level 360 Weight 6 Character orbit 360.f Analytic conductor 57.738 Analytic rank 0 Dimension 8 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$360 = 2^{3} \cdot 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 360.f (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$57.7381751327$$ 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.

 $$f(q)$$ $$=$$ $$q + ( -1 + \beta_{1} ) q^{5} + ( \beta_{1} + \beta_{5} ) q^{7} +O(q^{10})$$ $$q + ( -1 + \beta_{1} ) q^{5} + ( \beta_{1} + \beta_{5} ) q^{7} + ( 92 + \beta_{1} + \beta_{3} ) q^{11} + ( -2 \beta_{1} + \beta_{2} + \beta_{4} + 4 \beta_{5} + \beta_{7} ) q^{13} + ( 9 \beta_{1} - 14 \beta_{2} + \beta_{4} - 2 \beta_{7} ) q^{17} + ( 172 - 19 \beta_{1} - 4 \beta_{2} + \beta_{3} - 4 \beta_{7} ) q^{19} + ( -\beta_{1} - 6 \beta_{2} - 10 \beta_{4} + 9 \beta_{5} + 4 \beta_{7} ) q^{23} + ( -267 + \beta_{1} - 25 \beta_{2} + 4 \beta_{3} - 9 \beta_{4} - 8 \beta_{5} - \beta_{6} + 3 \beta_{7} ) q^{25} + ( -734 - 26 \beta_{1} - 4 \beta_{2} - 4 \beta_{3} - 2 \beta_{6} - 4 \beta_{7} ) q^{29} + ( 528 - 16 \beta_{1} - 4 \beta_{2} + 4 \beta_{6} - 4 \beta_{7} ) q^{31} + ( -2404 - 15 \beta_{1} - 65 \beta_{2} + \beta_{3} + 34 \beta_{4} + 18 \beta_{5} - 4 \beta_{6} + 12 \beta_{7} ) q^{35} + ( 5 \beta_{1} + 137 \beta_{2} - 26 \beta_{4} - 16 \beta_{5} + \beta_{7} ) q^{37} + ( -2950 + 34 \beta_{1} + 9 \beta_{2} - 16 \beta_{3} + 5 \beta_{6} + 9 \beta_{7} ) q^{41} + ( -54 \beta_{1} + 81 \beta_{2} + 108 \beta_{4} + 54 \beta_{5} ) q^{43} + ( 81 \beta_{1} + 168 \beta_{2} + 50 \beta_{4} - 9 \beta_{5} - 28 \beta_{7} ) q^{47} + ( -5625 - 72 \beta_{1} - 9 \beta_{2} - 24 \beta_{3} - 3 \beta_{6} - 9 \beta_{7} ) q^{49} + ( 70 \beta_{1} - 39 \beta_{2} - 191 \beta_{4} - 36 \beta_{5} + 17 \beta_{7} ) q^{53} + ( 1876 + 204 \beta_{1} + 220 \beta_{2} - 15 \beta_{3} - 125 \beta_{4} + 20 \beta_{5} - 10 \beta_{6} + 80 \beta_{7} ) q^{55} + ( -11460 + 57 \beta_{1} + 12 \beta_{2} + 13 \beta_{3} - 16 \beta_{6} + 12 \beta_{7} ) q^{59} + ( 15482 - 263 \beta_{1} - 53 \beta_{2} + 8 \beta_{3} - 6 \beta_{6} - 53 \beta_{7} ) q^{61} + ( 9008 - 50 \beta_{1} - 95 \beta_{2} - 12 \beta_{3} + 182 \beta_{4} + 144 \beta_{5} - 7 \beta_{6} + 41 \beta_{7} ) q^{65} + ( 214 \beta_{1} + 33 \beta_{2} - 124 \beta_{4} + 10 \beta_{5} - 16 \beta_{7} ) q^{67} + ( 15704 - 458 \beta_{1} - 84 \beta_{2} - 2 \beta_{3} - 36 \beta_{6} - 84 \beta_{7} ) q^{71} + ( -103 \beta_{1} - 388 \beta_{2} + 311 \beta_{4} + 8 \beta_{5} - 40 \beta_{7} ) q^{73} + ( 217 \beta_{1} + 804 \beta_{2} + 111 \beta_{4} + 468 \beta_{5} + 28 \beta_{7} ) q^{77} + ( 5408 + 628 \beta_{1} + 124 \beta_{2} + 28 \beta_{3} - 20 \beta_{6} + 124 \beta_{7} ) q^{79} + ( -302 \beta_{1} + 615 \beta_{2} - 236 \beta_{4} + 342 \beta_{5} + 176 \beta_{7} ) q^{83} + ( -36720 + 85 \beta_{1} - 90 \beta_{2} + 80 \beta_{3} - 125 \beta_{4} + 60 \beta_{5} - 30 \beta_{6} - 10 \beta_{7} ) q^{85} + ( 5238 - 270 \beta_{1} - 36 \beta_{2} - 48 \beta_{3} - 42 \beta_{6} - 36 \beta_{7} ) q^{89} + ( -60952 - 770 \beta_{1} - 140 \beta_{2} - 110 \beta_{3} + 40 \beta_{6} - 140 \beta_{7} ) q^{91} + ( -55324 + 224 \beta_{1} + 720 \beta_{2} - 95 \beta_{3} + 55 \beta_{4} + 180 \beta_{5} + 10 \beta_{6} + 20 \beta_{7} ) q^{95} + ( 107 \beta_{1} - 486 \beta_{2} - 49 \beta_{4} + 488 \beta_{5} + 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 $$x^{8} + 41 x^{6} + 460 x^{4} + 969 x^{2} + 9$$:

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/360\mathbb{Z}\right)^\times$$.

 $$n$$ $$181$$ $$217$$ $$271$$ $$281$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$ $$1$$

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 360.6.f.b 8
3.b odd 2 1 40.6.c.a 8
4.b odd 2 1 720.6.f.n 8
5.b even 2 1 inner 360.6.f.b 8
12.b even 2 1 80.6.c.d 8
15.d odd 2 1 40.6.c.a 8
15.e even 4 1 200.6.a.j 4
15.e even 4 1 200.6.a.k 4
20.d odd 2 1 720.6.f.n 8
24.f even 2 1 320.6.c.i 8
24.h odd 2 1 320.6.c.j 8
60.h even 2 1 80.6.c.d 8
60.l odd 4 1 400.6.a.z 4
60.l odd 4 1 400.6.a.ba 4
120.i odd 2 1 320.6.c.j 8
120.m even 2 1 320.6.c.i 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.6.c.a 8 3.b odd 2 1
40.6.c.a 8 15.d odd 2 1
80.6.c.d 8 12.b even 2 1
80.6.c.d 8 60.h even 2 1
200.6.a.j 4 15.e even 4 1
200.6.a.k 4 15.e even 4 1
320.6.c.i 8 24.f even 2 1
320.6.c.i 8 120.m even 2 1
320.6.c.j 8 24.h odd 2 1
320.6.c.j 8 120.i odd 2 1
360.6.f.b 8 1.a even 1 1 trivial
360.6.f.b 8 5.b even 2 1 inner
400.6.a.z 4 60.l odd 4 1
400.6.a.ba 4 60.l odd 4 1
720.6.f.n 8 4.b odd 2 1
720.6.f.n 8 20.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{8} + 89728 T_{7}^{6} + 2632172640 T_{7}^{4} +$$$$25\!\cdots\!32$$$$T_{7}^{2} +$$$$26\!\cdots\!24$$ acting on $$S_{6}^{\mathrm{new}}(360, [\chi])$$.

## Hecke characteristic polynomials

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