# Properties

 Label 360.2.d.e Level 360 Weight 2 Character orbit 360.d Analytic conductor 2.875 Analytic rank 0 Dimension 6 CM no Inner twists 2

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$2.87461447277$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.839056.1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 120) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{2} + ( -\beta_{1} - \beta_{2} ) q^{4} + ( \beta_{1} + \beta_{2} - \beta_{5} ) q^{5} + ( \beta_{1} - \beta_{3} + \beta_{4} ) q^{7} + ( \beta_{1} + \beta_{2} + \beta_{4} - \beta_{5} ) q^{8} +O(q^{10})$$ $$q + \beta_{3} q^{2} + ( -\beta_{1} - \beta_{2} ) q^{4} + ( \beta_{1} + \beta_{2} - \beta_{5} ) q^{5} + ( \beta_{1} - \beta_{3} + \beta_{4} ) q^{7} + ( \beta_{1} + \beta_{2} + \beta_{4} - \beta_{5} ) q^{8} + ( -2 - \beta_{1} - \beta_{4} ) q^{10} + ( -\beta_{1} + \beta_{3} - \beta_{5} ) q^{11} + ( 2 + \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} ) q^{13} + ( 2 + 2 \beta_{2} ) q^{14} + ( -\beta_{1} + \beta_{2} - \beta_{4} - \beta_{5} ) q^{16} + ( -\beta_{1} - \beta_{3} + \beta_{4} ) q^{17} + ( 2 \beta_{1} + \beta_{4} - \beta_{5} ) q^{19} + ( -2 + \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{20} + ( -2 - 2 \beta_{5} ) q^{22} + ( 2 \beta_{1} - \beta_{4} + \beta_{5} ) q^{23} + ( 1 + \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{5} ) q^{25} + ( 2 - 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} ) q^{26} + ( -2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} ) q^{28} + ( \beta_{1} + \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{29} + ( -2 + 2 \beta_{3} + \beta_{4} + \beta_{5} ) q^{31} + ( -4 + \beta_{1} - \beta_{2} - \beta_{4} - \beta_{5} ) q^{32} + ( 2 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{5} ) q^{34} + ( \beta_{1} + 2 \beta_{2} - \beta_{3} + 3 \beta_{4} ) q^{35} + ( 2 - \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{4} ) q^{37} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{38} + ( \beta_{1} + 3 \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{5} ) q^{40} + ( 2 + 4 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{41} + ( 2 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} - 2 \beta_{5} ) q^{43} + ( -4 + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{5} ) q^{44} + ( -2 \beta_{1} - 2 \beta_{2} + 4 \beta_{5} ) q^{46} + ( 2 \beta_{1} + \beta_{4} - \beta_{5} ) q^{47} + ( -1 - 2 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} + 2 \beta_{5} ) q^{49} + ( 2 - 2 \beta_{1} - 4 \beta_{2} + \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{50} + ( -4 - 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} ) q^{52} + ( 4 - \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{5} ) q^{53} + ( -2 + \beta_{1} - 3 \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{55} + ( -4 - 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{4} + 2 \beta_{5} ) q^{56} + ( -2 - 2 \beta_{1} - 4 \beta_{2} + 4 \beta_{5} ) q^{58} + ( 3 \beta_{1} - 3 \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{59} + ( 4 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} ) q^{61} + ( 4 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{62} + ( -4 - \beta_{1} + \beta_{2} - 4 \beta_{3} + \beta_{4} + \beta_{5} ) q^{64} + ( 2 - \beta_{1} + 4 \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{5} ) q^{65} + ( -2 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} + 2 \beta_{5} ) q^{67} + ( -4 - 2 \beta_{1} + 2 \beta_{3} - 2 \beta_{4} ) q^{68} + ( 6 + 2 \beta_{2} - 2 \beta_{4} - 2 \beta_{5} ) q^{70} + ( -4 - 4 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} ) q^{71} + ( -4 \beta_{1} - 4 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{73} + ( -2 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} ) q^{74} + ( 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{4} - 2 \beta_{5} ) q^{76} + 4 q^{77} + ( 2 - 2 \beta_{3} - \beta_{4} - \beta_{5} ) q^{79} + ( 4 + \beta_{1} - \beta_{2} - 3 \beta_{4} + \beta_{5} ) q^{80} + ( 8 - 4 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} ) q^{82} + ( -4 - 2 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} ) q^{83} + ( 2 + \beta_{1} - 5 \beta_{3} + \beta_{4} ) q^{85} + ( -4 - 4 \beta_{4} ) q^{86} + ( -4 + 2 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} ) q^{88} + ( 2 - 4 \beta_{1} - 8 \beta_{2} - 2 \beta_{4} + 2 \beta_{5} ) q^{89} + ( -2 \beta_{1} - 6 \beta_{3} + 4 \beta_{4} + 2 \beta_{5} ) q^{91} + ( 8 + 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{4} + 2 \beta_{5} ) q^{92} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{94} + ( -4 + 2 \beta_{1} + 4 \beta_{2} + 3 \beta_{4} + \beta_{5} ) q^{95} + ( 2 \beta_{1} + 2 \beta_{3} + 2 \beta_{4} - 4 \beta_{5} ) q^{97} + ( 4 - \beta_{3} + 4 \beta_{4} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q - q^{2} + q^{4} - q^{8} + O(q^{10})$$ $$6q - q^{2} + q^{4} - q^{8} - 11q^{10} + 8q^{13} + 10q^{14} + q^{16} - 9q^{20} - 10q^{22} + 2q^{25} + 14q^{26} + 2q^{28} - 16q^{31} - 21q^{32} + 12q^{34} - 4q^{35} + 16q^{37} - 2q^{38} - 3q^{40} + 4q^{41} - 22q^{44} - 2q^{46} - 6q^{49} + 15q^{50} - 26q^{52} + 24q^{53} - 8q^{55} - 26q^{56} - 12q^{58} + 28q^{62} - 23q^{64} + 12q^{65} - 24q^{68} + 38q^{70} - 16q^{71} - 18q^{74} + 6q^{76} + 24q^{77} + 16q^{79} + 27q^{80} + 50q^{82} - 16q^{83} + 16q^{85} - 20q^{86} - 18q^{88} + 20q^{89} + 46q^{92} - 2q^{94} - 32q^{95} + 21q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} + 6 x^{4} + 8 x^{2} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{3} + 4 \nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{5} + \nu^{4} + 5 \nu^{3} + 3 \nu^{2} + 3 \nu - 1$$$$)/2$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{5} + \nu^{4} - 5 \nu^{3} + 5 \nu^{2} - 5 \nu + 3$$$$)/2$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{5} + \nu^{4} - 7 \nu^{3} + 5 \nu^{2} - 9 \nu + 3$$$$)/2$$ $$\beta_{5}$$ $$=$$ $$($$$$3 \nu^{5} + \nu^{4} + 17 \nu^{3} + 5 \nu^{2} + 19 \nu + 3$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{4} - \beta_{3} + \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{5} + \beta_{3} - 2 \beta_{2} - \beta_{1} - 4$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$-2 \beta_{4} + 2 \beta_{3} - \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$($$$$-4 \beta_{5} + \beta_{4} - 3 \beta_{3} + 10 \beta_{2} + 5 \beta_{1} + 14$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$\beta_{5} + 16 \beta_{4} - 17 \beta_{3} + 5 \beta_{1}$$$$)/2$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
109.1
 − 1.32132i 1.32132i 2.02852i − 2.02852i 0.373087i − 0.373087i
−1.34067 0.450129i 0 1.59477 + 1.20695i −0.254102 2.22158i 0 2.64265i −1.59477 2.33596i 0 −0.659335 + 3.09278i
109.2 −1.34067 + 0.450129i 0 1.59477 1.20695i −0.254102 + 2.22158i 0 2.64265i −1.59477 + 2.33596i 0 −0.659335 3.09278i
109.3 −0.321037 1.37729i 0 −1.79387 + 0.884323i 2.11491 0.726062i 0 4.05705i 1.79387 + 2.18678i 0 −1.67896 2.67975i
109.4 −0.321037 + 1.37729i 0 −1.79387 0.884323i 2.11491 + 0.726062i 0 4.05705i 1.79387 2.18678i 0 −1.67896 + 2.67975i
109.5 1.16170 0.806504i 0 0.699104 1.87383i −1.86081 1.23992i 0 0.746175i −0.699104 2.74067i 0 −3.16170 + 0.0603290i
109.6 1.16170 + 0.806504i 0 0.699104 + 1.87383i −1.86081 + 1.23992i 0 0.746175i −0.699104 + 2.74067i 0 −3.16170 0.0603290i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 109.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
40.f even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 360.2.d.e 6
3.b odd 2 1 120.2.d.b yes 6
4.b odd 2 1 1440.2.d.f 6
5.b even 2 1 360.2.d.f 6
5.c odd 4 2 1800.2.k.u 12
8.b even 2 1 360.2.d.f 6
8.d odd 2 1 1440.2.d.e 6
12.b even 2 1 480.2.d.b 6
15.d odd 2 1 120.2.d.a 6
15.e even 4 2 600.2.k.f 12
20.d odd 2 1 1440.2.d.e 6
20.e even 4 2 7200.2.k.u 12
24.f even 2 1 480.2.d.a 6
24.h odd 2 1 120.2.d.a 6
40.e odd 2 1 1440.2.d.f 6
40.f even 2 1 inner 360.2.d.e 6
40.i odd 4 2 1800.2.k.u 12
40.k even 4 2 7200.2.k.u 12
48.i odd 4 2 3840.2.f.l 12
48.k even 4 2 3840.2.f.m 12
60.h even 2 1 480.2.d.a 6
60.l odd 4 2 2400.2.k.f 12
120.i odd 2 1 120.2.d.b yes 6
120.m even 2 1 480.2.d.b 6
120.q odd 4 2 2400.2.k.f 12
120.w even 4 2 600.2.k.f 12
240.t even 4 2 3840.2.f.m 12
240.bm odd 4 2 3840.2.f.l 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.2.d.a 6 15.d odd 2 1
120.2.d.a 6 24.h odd 2 1
120.2.d.b yes 6 3.b odd 2 1
120.2.d.b yes 6 120.i odd 2 1
360.2.d.e 6 1.a even 1 1 trivial
360.2.d.e 6 40.f even 2 1 inner
360.2.d.f 6 5.b even 2 1
360.2.d.f 6 8.b even 2 1
480.2.d.a 6 24.f even 2 1
480.2.d.a 6 60.h even 2 1
480.2.d.b 6 12.b even 2 1
480.2.d.b 6 120.m even 2 1
600.2.k.f 12 15.e even 4 2
600.2.k.f 12 120.w even 4 2
1440.2.d.e 6 8.d odd 2 1
1440.2.d.e 6 20.d odd 2 1
1440.2.d.f 6 4.b odd 2 1
1440.2.d.f 6 40.e odd 2 1
1800.2.k.u 12 5.c odd 4 2
1800.2.k.u 12 40.i odd 4 2
2400.2.k.f 12 60.l odd 4 2
2400.2.k.f 12 120.q odd 4 2
3840.2.f.l 12 48.i odd 4 2
3840.2.f.l 12 240.bm odd 4 2
3840.2.f.m 12 48.k even 4 2
3840.2.f.m 12 240.t even 4 2
7200.2.k.u 12 20.e even 4 2
7200.2.k.u 12 40.k even 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(360, [\chi])$$:

 $$T_{7}^{6} + 24 T_{7}^{4} + 128 T_{7}^{2} + 64$$ $$T_{11}^{6} + 32 T_{11}^{4} + 96 T_{11}^{2} + 64$$ $$T_{13}^{3} - 4 T_{13}^{2} - 16 T_{13} + 56$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T + 4 T^{5} + 8 T^{6}$$
$3$ 1
$5$ $$1 - T^{2} - 8 T^{3} - 5 T^{4} + 125 T^{6}$$
$7$ $$1 - 18 T^{2} + 191 T^{4} - 1532 T^{6} + 9359 T^{8} - 43218 T^{10} + 117649 T^{12}$$
$11$ $$1 - 34 T^{2} + 503 T^{4} - 5436 T^{6} + 60863 T^{8} - 497794 T^{10} + 1771561 T^{12}$$
$13$ $$( 1 - 4 T + 23 T^{2} - 48 T^{3} + 299 T^{4} - 676 T^{5} + 2197 T^{6} )^{2}$$
$17$ $$1 - 66 T^{2} + 2255 T^{4} - 47324 T^{6} + 651695 T^{8} - 5512386 T^{10} + 24137569 T^{12}$$
$19$ $$1 - 54 T^{2} + 1367 T^{4} - 25652 T^{6} + 493487 T^{8} - 7037334 T^{10} + 47045881 T^{12}$$
$23$ $$1 - 46 T^{2} + 1775 T^{4} - 40932 T^{6} + 938975 T^{8} - 12872686 T^{10} + 148035889 T^{12}$$
$29$ $$1 - 66 T^{2} + 3207 T^{4} - 111228 T^{6} + 2697087 T^{8} - 46680546 T^{10} + 594823321 T^{12}$$
$31$ $$( 1 + 8 T + 89 T^{2} + 432 T^{3} + 2759 T^{4} + 7688 T^{5} + 29791 T^{6} )^{2}$$
$37$ $$( 1 - 8 T + 111 T^{2} - 584 T^{3} + 4107 T^{4} - 10952 T^{5} + 50653 T^{6} )^{2}$$
$41$ $$( 1 - 2 T + 23 T^{2} - 220 T^{3} + 943 T^{4} - 3362 T^{5} + 68921 T^{6} )^{2}$$
$43$ $$( 1 + 65 T^{2} - 64 T^{3} + 2795 T^{4} + 79507 T^{6} )^{2}$$
$47$ $$1 - 222 T^{2} + 22367 T^{4} - 1328324 T^{6} + 49408703 T^{8} - 1083289182 T^{10} + 10779215329 T^{12}$$
$53$ $$( 1 - 12 T + 191 T^{2} - 1264 T^{3} + 10123 T^{4} - 33708 T^{5} + 148877 T^{6} )^{2}$$
$59$ $$1 - 178 T^{2} + 20567 T^{4} - 1418652 T^{6} + 71593727 T^{8} - 2156890258 T^{10} + 42180533641 T^{12}$$
$61$ $$1 - 190 T^{2} + 20039 T^{4} - 1419204 T^{6} + 74565119 T^{8} - 2630709790 T^{10} + 51520374361 T^{12}$$
$67$ $$( 1 + 137 T^{2} + 64 T^{3} + 9179 T^{4} + 300763 T^{6} )^{2}$$
$71$ $$( 1 + 8 T + 133 T^{2} + 1008 T^{3} + 9443 T^{4} + 40328 T^{5} + 357911 T^{6} )^{2}$$
$73$ $$1 - 54 T^{2} + 2367 T^{4} - 531700 T^{6} + 12613743 T^{8} - 1533505014 T^{10} + 151334226289 T^{12}$$
$79$ $$( 1 - 8 T + 233 T^{2} - 1200 T^{3} + 18407 T^{4} - 49928 T^{5} + 493039 T^{6} )^{2}$$
$83$ $$( 1 + 8 T + 185 T^{2} + 880 T^{3} + 15355 T^{4} + 55112 T^{5} + 571787 T^{6} )^{2}$$
$89$ $$( 1 - 10 T + 103 T^{2} - 396 T^{3} + 9167 T^{4} - 79210 T^{5} + 704969 T^{6} )^{2}$$
$97$ $$1 - 246 T^{2} + 39183 T^{4} - 4535476 T^{6} + 368672847 T^{8} - 21778203126 T^{10} + 832972004929 T^{12}$$