# Properties

 Label 252.2.j.a Level 252 Weight 2 Character orbit 252.j Analytic conductor 2.012 Analytic rank 0 Dimension 6 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$252 = 2^{2} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 252.j (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.01223013094$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: 6.0.309123.1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( \beta_{2} + \beta_{4} ) q^{3} + ( -\beta_{1} + \beta_{5} ) q^{5} + ( 1 + \beta_{4} ) q^{7} + ( 2 + \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{5} ) q^{9} +O(q^{10})$$ $$q + ( \beta_{2} + \beta_{4} ) q^{3} + ( -\beta_{1} + \beta_{5} ) q^{5} + ( 1 + \beta_{4} ) q^{7} + ( 2 + \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{5} ) q^{9} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} ) q^{11} + ( 2 - 4 \beta_{1} + 2 \beta_{3} - \beta_{4} ) q^{13} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{4} ) q^{15} + ( 2 - \beta_{1} + \beta_{2} + 3 \beta_{3} - \beta_{4} ) q^{17} + ( 1 - \beta_{1} - 3 \beta_{2} - \beta_{3} - \beta_{4} ) q^{19} + ( \beta_{2} + \beta_{3} - \beta_{5} ) q^{21} + ( -1 - \beta_{3} + 5 \beta_{4} + 2 \beta_{5} ) q^{23} + ( 2 + \beta_{1} + \beta_{3} + 2 \beta_{4} ) q^{25} + ( 1 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 3 \beta_{4} - 2 \beta_{5} ) q^{27} + ( -1 - 3 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{29} + ( 3 \beta_{1} - 3 \beta_{5} ) q^{31} + ( -2 - \beta_{1} - \beta_{2} - 4 \beta_{3} - 4 \beta_{4} ) q^{33} + ( \beta_{2} + \beta_{3} ) q^{35} + ( -2 \beta_{1} - 3 \beta_{2} + \beta_{3} - 2 \beta_{4} ) q^{37} + ( -6 + 6 \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{39} + ( -2 + \beta_{1} - 2 \beta_{3} + \beta_{4} + 3 \beta_{5} ) q^{41} + ( -\beta_{1} + 3 \beta_{2} - \beta_{3} - 3 \beta_{5} ) q^{43} + ( 2 + \beta_{1} - 3 \beta_{3} + \beta_{4} + \beta_{5} ) q^{45} + ( -8 - 2 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} - 8 \beta_{4} + 3 \beta_{5} ) q^{47} + \beta_{4} q^{49} + ( -2 + 5 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} + \beta_{4} - 2 \beta_{5} ) q^{51} + ( 1 + \beta_{1} - 2 \beta_{3} + \beta_{4} ) q^{53} + ( \beta_{1} - 3 \beta_{2} - 5 \beta_{3} + \beta_{4} ) q^{55} + ( -6 - 3 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{5} ) q^{57} + ( 1 + \beta_{3} + 10 \beta_{4} - 2 \beta_{5} ) q^{59} + ( -4 - 7 \beta_{1} - 6 \beta_{2} - 7 \beta_{3} - 4 \beta_{4} + 6 \beta_{5} ) q^{61} + ( 1 + 2 \beta_{1} + 2 \beta_{4} - \beta_{5} ) q^{63} + ( -4 + 5 \beta_{1} + 3 \beta_{2} + 5 \beta_{3} - 4 \beta_{4} - 3 \beta_{5} ) q^{65} + ( 1 + \beta_{1} + \beta_{3} + 2 \beta_{4} - 3 \beta_{5} ) q^{67} + ( 1 - \beta_{1} + \beta_{2} + 6 \beta_{3} - 9 \beta_{4} - 4 \beta_{5} ) q^{69} + ( 6 - 3 \beta_{1} - 8 \beta_{2} - 2 \beta_{3} - 3 \beta_{4} ) q^{71} + ( -1 + \beta_{1} + 3 \beta_{2} + \beta_{3} + \beta_{4} ) q^{73} + ( 2 \beta_{2} - \beta_{4} - 3 \beta_{5} ) q^{75} + ( -2 + 3 \beta_{1} - 2 \beta_{3} + \beta_{4} + \beta_{5} ) q^{77} + ( 4 + 4 \beta_{1} + 3 \beta_{2} + 4 \beta_{3} + 4 \beta_{4} - 3 \beta_{5} ) q^{79} + ( 8 - 2 \beta_{1} + \beta_{2} + 5 \beta_{3} + 6 \beta_{4} - \beta_{5} ) q^{81} + ( -7 - 2 \beta_{1} - \beta_{2} - 2 \beta_{3} - 7 \beta_{4} + \beta_{5} ) q^{83} + ( -2 + 4 \beta_{1} - 2 \beta_{3} - 3 \beta_{4} ) q^{85} + ( -4 - 2 \beta_{1} - \beta_{2} + 3 \beta_{3} - 3 \beta_{4} + 4 \beta_{5} ) q^{87} + ( 1 + 4 \beta_{1} + 3 \beta_{2} - 5 \beta_{3} + 4 \beta_{4} ) q^{89} + ( 1 - 2 \beta_{1} + 4 \beta_{3} - 2 \beta_{4} ) q^{91} + ( 3 - 3 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} + 6 \beta_{4} ) q^{93} + ( 2 - 3 \beta_{1} + 2 \beta_{3} + 5 \beta_{4} - \beta_{5} ) q^{95} + ( 1 - \beta_{1} - 6 \beta_{2} - \beta_{3} + \beta_{4} + 6 \beta_{5} ) q^{97} + ( 1 - 4 \beta_{1} - 3 \beta_{2} + 5 \beta_{4} + 8 \beta_{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q - 2q^{3} - q^{5} + 3q^{7} + 8q^{9} + O(q^{10})$$ $$6q - 2q^{3} - q^{5} + 3q^{7} + 8q^{9} - 2q^{11} - 3q^{13} + q^{15} + 4q^{17} + 6q^{19} - 4q^{21} - 14q^{23} + 6q^{25} + 7q^{27} - q^{29} + 3q^{31} + 8q^{33} - 2q^{35} - 6q^{37} - 24q^{39} - 3q^{43} + 23q^{45} - 21q^{47} - 3q^{49} + 5q^{51} + 12q^{53} + 12q^{55} - 37q^{57} - 31q^{59} - 6q^{61} + 4q^{63} - 15q^{65} - 6q^{67} + 5q^{69} + 34q^{71} - 6q^{73} - q^{75} + 2q^{77} + 9q^{79} + 8q^{81} - 20q^{83} + 15q^{85} - 23q^{87} + 24q^{89} - 6q^{91} - 3q^{93} - 20q^{95} + 9q^{97} - 8q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 3 x^{5} + 10 x^{4} - 15 x^{3} + 19 x^{2} - 12 x + 3$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{5} - \nu^{4} + 5 \nu^{3} + \nu^{2} + 6$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{5} + \nu^{4} - 5 \nu^{3} + 2 \nu^{2} - 3 \nu$$$$)/3$$ $$\beta_{4}$$ $$=$$ $$($$$$-2 \nu^{5} + 5 \nu^{4} - 16 \nu^{3} + 19 \nu^{2} - 21 \nu + 6$$$$)/3$$ $$\beta_{5}$$ $$=$$ $$($$$$2 \nu^{5} - 5 \nu^{4} + 19 \nu^{3} - 22 \nu^{2} + 33 \nu - 9$$$$)/3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + \beta_{2} + \beta_{1} - 2$$ $$\nu^{3}$$ $$=$$ $$\beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} - 3 \beta_{1} - 1$$ $$\nu^{4}$$ $$=$$ $$2 \beta_{5} + 3 \beta_{4} - 5 \beta_{3} - 3 \beta_{2} - 6 \beta_{1} + 6$$ $$\nu^{5}$$ $$=$$ $$-3 \beta_{5} - 2 \beta_{4} - 11 \beta_{3} - 6 \beta_{2} + 8 \beta_{1} + 7$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/252\mathbb{Z}\right)^\times$$.

 $$n$$ $$29$$ $$73$$ $$127$$ $$\chi(n)$$ $$\beta_{4}$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
85.1
 0.5 + 1.41036i 0.5 − 2.05195i 0.5 − 0.224437i 0.5 − 1.41036i 0.5 + 2.05195i 0.5 + 0.224437i
0 −1.71053 0.272169i 0 −0.119562 + 0.207087i 0 0.500000 + 0.866025i 0 2.85185 + 0.931107i 0
85.2 0 −0.933463 + 1.45899i 0 −1.23025 + 2.13086i 0 0.500000 + 0.866025i 0 −1.25729 2.72382i 0
85.3 0 1.64400 + 0.545231i 0 0.849814 1.47192i 0 0.500000 + 0.866025i 0 2.40545 + 1.79272i 0
169.1 0 −1.71053 + 0.272169i 0 −0.119562 0.207087i 0 0.500000 0.866025i 0 2.85185 0.931107i 0
169.2 0 −0.933463 1.45899i 0 −1.23025 2.13086i 0 0.500000 0.866025i 0 −1.25729 + 2.72382i 0
169.3 0 1.64400 0.545231i 0 0.849814 + 1.47192i 0 0.500000 0.866025i 0 2.40545 1.79272i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 169.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 252.2.j.a 6
3.b odd 2 1 756.2.j.b 6
4.b odd 2 1 1008.2.r.j 6
7.b odd 2 1 1764.2.j.e 6
7.c even 3 1 1764.2.i.g 6
7.c even 3 1 1764.2.l.e 6
7.d odd 6 1 1764.2.i.d 6
7.d odd 6 1 1764.2.l.f 6
9.c even 3 1 inner 252.2.j.a 6
9.c even 3 1 2268.2.a.i 3
9.d odd 6 1 756.2.j.b 6
9.d odd 6 1 2268.2.a.h 3
12.b even 2 1 3024.2.r.j 6
21.c even 2 1 5292.2.j.d 6
21.g even 6 1 5292.2.i.e 6
21.g even 6 1 5292.2.l.f 6
21.h odd 6 1 5292.2.i.f 6
21.h odd 6 1 5292.2.l.e 6
36.f odd 6 1 1008.2.r.j 6
36.f odd 6 1 9072.2.a.by 3
36.h even 6 1 3024.2.r.j 6
36.h even 6 1 9072.2.a.bv 3
63.g even 3 1 1764.2.i.g 6
63.h even 3 1 1764.2.l.e 6
63.i even 6 1 5292.2.l.f 6
63.j odd 6 1 5292.2.l.e 6
63.k odd 6 1 1764.2.i.d 6
63.l odd 6 1 1764.2.j.e 6
63.n odd 6 1 5292.2.i.f 6
63.o even 6 1 5292.2.j.d 6
63.s even 6 1 5292.2.i.e 6
63.t odd 6 1 1764.2.l.f 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.2.j.a 6 1.a even 1 1 trivial
252.2.j.a 6 9.c even 3 1 inner
756.2.j.b 6 3.b odd 2 1
756.2.j.b 6 9.d odd 6 1
1008.2.r.j 6 4.b odd 2 1
1008.2.r.j 6 36.f odd 6 1
1764.2.i.d 6 7.d odd 6 1
1764.2.i.d 6 63.k odd 6 1
1764.2.i.g 6 7.c even 3 1
1764.2.i.g 6 63.g even 3 1
1764.2.j.e 6 7.b odd 2 1
1764.2.j.e 6 63.l odd 6 1
1764.2.l.e 6 7.c even 3 1
1764.2.l.e 6 63.h even 3 1
1764.2.l.f 6 7.d odd 6 1
1764.2.l.f 6 63.t odd 6 1
2268.2.a.h 3 9.d odd 6 1
2268.2.a.i 3 9.c even 3 1
3024.2.r.j 6 12.b even 2 1
3024.2.r.j 6 36.h even 6 1
5292.2.i.e 6 21.g even 6 1
5292.2.i.e 6 63.s even 6 1
5292.2.i.f 6 21.h odd 6 1
5292.2.i.f 6 63.n odd 6 1
5292.2.j.d 6 21.c even 2 1
5292.2.j.d 6 63.o even 6 1
5292.2.l.e 6 21.h odd 6 1
5292.2.l.e 6 63.j odd 6 1
5292.2.l.f 6 21.g even 6 1
5292.2.l.f 6 63.i even 6 1
9072.2.a.bv 3 36.h even 6 1
9072.2.a.by 3 36.f odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{6} + T_{5}^{5} + 5 T_{5}^{4} - 2 T_{5}^{3} + 17 T_{5}^{2} + 4 T_{5} + 1$$ acting on $$S_{2}^{\mathrm{new}}(252, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 
$3$ $$1 + 2 T - 2 T^{2} - 9 T^{3} - 6 T^{4} + 18 T^{5} + 27 T^{6}$$
$5$ $$1 + T - 10 T^{2} - 7 T^{3} + 57 T^{4} + 14 T^{5} - 299 T^{6} + 70 T^{7} + 1425 T^{8} - 875 T^{9} - 6250 T^{10} + 3125 T^{11} + 15625 T^{12}$$
$7$ $$( 1 - T + T^{2} )^{3}$$
$11$ $$1 + 2 T - 4 T^{2} + 46 T^{3} + 6 T^{4} - 230 T^{5} + 1699 T^{6} - 2530 T^{7} + 726 T^{8} + 61226 T^{9} - 58564 T^{10} + 322102 T^{11} + 1771561 T^{12}$$
$13$ $$1 + 3 T + 3 T^{2} - 84 T^{3} - 195 T^{4} + 345 T^{5} + 5006 T^{6} + 4485 T^{7} - 32955 T^{8} - 184548 T^{9} + 85683 T^{10} + 1113879 T^{11} + 4826809 T^{12}$$
$17$ $$( 1 - 2 T + 32 T^{2} - 21 T^{3} + 544 T^{4} - 578 T^{5} + 4913 T^{6} )^{2}$$
$19$ $$( 1 - 3 T + 33 T^{2} - 35 T^{3} + 627 T^{4} - 1083 T^{5} + 6859 T^{6} )^{2}$$
$23$ $$1 + 14 T + 74 T^{2} + 358 T^{3} + 2628 T^{4} + 11188 T^{5} + 33943 T^{6} + 257324 T^{7} + 1390212 T^{8} + 4355786 T^{9} + 20708234 T^{10} + 90108802 T^{11} + 148035889 T^{12}$$
$29$ $$1 + T - 46 T^{2} + 149 T^{3} + 897 T^{4} - 4282 T^{5} - 13523 T^{6} - 124178 T^{7} + 754377 T^{8} + 3633961 T^{9} - 32534926 T^{10} + 20511149 T^{11} + 594823321 T^{12}$$
$31$ $$1 - 3 T - 48 T^{2} + 147 T^{3} + 1005 T^{4} - 1344 T^{5} - 24505 T^{6} - 41664 T^{7} + 965805 T^{8} + 4379277 T^{9} - 44329008 T^{10} - 85887453 T^{11} + 887503681 T^{12}$$
$37$ $$( 1 + 3 T + 81 T^{2} + 199 T^{3} + 2997 T^{4} + 4107 T^{5} + 50653 T^{6} )^{2}$$
$41$ $$1 - 90 T^{2} + 18 T^{3} + 4410 T^{4} - 810 T^{5} - 194177 T^{6} - 33210 T^{7} + 7413210 T^{8} + 1240578 T^{9} - 254318490 T^{10} + 4750104241 T^{12}$$
$43$ $$1 + 3 T - 24 T^{2} - 979 T^{3} - 1947 T^{4} + 14820 T^{5} + 386067 T^{6} + 637260 T^{7} - 3600003 T^{8} - 77837353 T^{9} - 82051224 T^{10} + 441025329 T^{11} + 6321363049 T^{12}$$
$47$ $$1 + 21 T + 180 T^{2} + 1119 T^{3} + 10053 T^{4} + 100416 T^{5} + 788551 T^{6} + 4719552 T^{7} + 22207077 T^{8} + 116177937 T^{9} + 878342580 T^{10} + 4816245147 T^{11} + 10779215329 T^{12}$$
$53$ $$( 1 - 6 T + 162 T^{2} - 627 T^{3} + 8586 T^{4} - 16854 T^{5} + 148877 T^{6} )^{2}$$
$59$ $$1 + 31 T + 476 T^{2} + 5741 T^{3} + 62553 T^{4} + 587576 T^{5} + 4781851 T^{6} + 34666984 T^{7} + 217746993 T^{8} + 1179080839 T^{9} + 5767863836 T^{10} + 22162653269 T^{11} + 42180533641 T^{12}$$
$61$ $$1 + 6 T + 48 T^{2} + 642 T^{3} + 3018 T^{4} + 35394 T^{5} + 438671 T^{6} + 2159034 T^{7} + 11229978 T^{8} + 145721802 T^{9} + 664600368 T^{10} + 5067577806 T^{11} + 51520374361 T^{12}$$
$67$ $$1 + 6 T - 150 T^{2} - 506 T^{3} + 17268 T^{4} + 28236 T^{5} - 1220289 T^{6} + 1891812 T^{7} + 77516052 T^{8} - 152186078 T^{9} - 3022668150 T^{10} + 8100750642 T^{11} + 90458382169 T^{12}$$
$71$ $$( 1 - 17 T + 119 T^{2} - 507 T^{3} + 8449 T^{4} - 85697 T^{5} + 357911 T^{6} )^{2}$$
$73$ $$( 1 + 3 T + 195 T^{2} + 359 T^{3} + 14235 T^{4} + 15987 T^{5} + 389017 T^{6} )^{2}$$
$79$ $$1 - 9 T - 114 T^{2} + 351 T^{3} + 13143 T^{4} + 15786 T^{5} - 1414609 T^{6} + 1247094 T^{7} + 82025463 T^{8} + 173056689 T^{9} - 4440309234 T^{10} - 27693507591 T^{11} + 243087455521 T^{12}$$
$83$ $$1 + 20 T + 38 T^{2} + 346 T^{3} + 32058 T^{4} + 183754 T^{5} - 606869 T^{6} + 15251582 T^{7} + 220847562 T^{8} + 197838302 T^{9} + 1803416198 T^{10} + 78780812860 T^{11} + 326940373369 T^{12}$$
$89$ $$( 1 - 12 T + 216 T^{2} - 1425 T^{3} + 19224 T^{4} - 95052 T^{5} + 704969 T^{6} )^{2}$$
$97$ $$1 - 9 T - 66 T^{2} + 2023 T^{3} - 7707 T^{4} - 73950 T^{5} + 1766073 T^{6} - 7173150 T^{7} - 72515163 T^{8} + 1846337479 T^{9} - 5842932546 T^{10} - 77286062313 T^{11} + 832972004929 T^{12}$$