# Properties

 Label 252.2.j.b 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: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$3$$ 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.

 $$f(q)$$ $$=$$ $$q + ( 1 + \beta_{3} + \beta_{4} ) q^{3} + ( -\beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} ) q^{5} + ( -1 - \beta_{3} ) q^{7} + ( -1 - \beta_{1} - \beta_{2} - \beta_{3} + \beta_{5} ) q^{9} +O(q^{10})$$ $$q + ( 1 + \beta_{3} + \beta_{4} ) q^{3} + ( -\beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} ) q^{5} + ( -1 - \beta_{3} ) q^{7} + ( -1 - \beta_{1} - \beta_{2} - \beta_{3} + \beta_{5} ) q^{9} + ( 2 - \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{11} -\beta_{3} q^{13} + ( 1 + \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} ) q^{15} + ( -\beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} ) q^{17} + ( 1 + \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} ) q^{19} + ( \beta_{2} - \beta_{3} ) q^{21} + ( -1 - 3 \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{23} + ( -3 - \beta_{1} - 4 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{25} + ( -1 + 2 \beta_{2} - \beta_{3} + 2 \beta_{5} ) q^{27} + ( 6 - \beta_{1} + 2 \beta_{2} + 5 \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{29} + ( 3 \beta_{1} + 3 \beta_{2} - \beta_{3} + 3 \beta_{4} - 3 \beta_{5} ) q^{31} + ( 2 - \beta_{1} - 2 \beta_{2} + 4 \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{33} + ( -1 + \beta_{4} - \beta_{5} ) q^{35} + ( -1 + 3 \beta_{4} - 3 \beta_{5} ) q^{37} + ( 1 + \beta_{2} + \beta_{4} ) q^{39} + ( \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{5} ) q^{41} + ( -3 + \beta_{1} - 5 \beta_{2} - \beta_{3} - 4 \beta_{4} - 2 \beta_{5} ) q^{43} + ( -8 + \beta_{1} - \beta_{2} - 4 \beta_{3} + \beta_{4} - 2 \beta_{5} ) q^{45} + ( 4 + 2 \beta_{1} - \beta_{2} + 5 \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{47} + \beta_{3} q^{49} + ( -2 + \beta_{1} + 2 \beta_{2} + 5 \beta_{3} + \beta_{4} - 2 \beta_{5} ) q^{51} + ( -6 - \beta_{1} + 2 \beta_{2} - \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{53} + ( -4 + \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} ) q^{55} + ( -1 + 3 \beta_{1} - 4 \beta_{3} + 2 \beta_{4} ) q^{57} + ( -1 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 4 \beta_{4} + \beta_{5} ) q^{59} + ( 3 + \beta_{1} + 4 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{61} + ( \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{63} + ( -1 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{65} + ( -1 + \beta_{1} + \beta_{2} - 3 \beta_{3} - \beta_{4} - 2 \beta_{5} ) q^{67} + ( 7 + \beta_{1} + 2 \beta_{2} + 5 \beta_{3} + \beta_{4} - 2 \beta_{5} ) q^{69} + ( -5 - \beta_{1} + 2 \beta_{2} - \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{71} + ( 3 - \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} ) q^{73} + ( 9 + \beta_{2} + 2 \beta_{3} - 3 \beta_{4} + 3 \beta_{5} ) q^{75} + ( \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{5} ) q^{77} + ( 4 \beta_{1} + 7 \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{79} + ( -8 + 2 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{81} + ( 5 - 3 \beta_{2} + 6 \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{83} + ( -2 - 4 \beta_{1} - 4 \beta_{2} - 7 \beta_{3} - 8 \beta_{4} + 2 \beta_{5} ) q^{85} + ( -5 - 2 \beta_{1} - 4 \beta_{2} + 5 \beta_{3} + \beta_{4} + \beta_{5} ) q^{87} + ( 2 + 2 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} - 3 \beta_{4} + 3 \beta_{5} ) q^{89} - q^{91} + ( 1 - 3 \beta_{1} - 2 \beta_{2} - 6 \beta_{3} + \beta_{4} - 3 \beta_{5} ) q^{93} + ( \beta_{1} + \beta_{2} - 8 \beta_{3} + \beta_{4} - \beta_{5} ) q^{95} + ( 6 + \beta_{1} + 4 \beta_{2} + 5 \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{97} + ( -4 + 2 \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q + 2q^{3} + 3q^{5} - 3q^{7} - 4q^{9} + O(q^{10})$$ $$6q + 2q^{3} + 3q^{5} - 3q^{7} - 4q^{9} + 6q^{11} + 3q^{13} - 3q^{15} + 6q^{19} + 2q^{21} + 6q^{23} - 6q^{25} - 7q^{27} + 15q^{29} + 3q^{31} - 6q^{35} - 6q^{37} + 4q^{39} + 6q^{41} - 3q^{43} - 33q^{45} + 15q^{47} - 3q^{49} - 27q^{51} - 36q^{53} - 24q^{55} + 7q^{57} + 3q^{59} + 6q^{61} - 4q^{63} - 3q^{65} + 6q^{67} + 27q^{69} - 30q^{71} + 18q^{73} + 47q^{75} + 6q^{77} - 3q^{79} - 40q^{81} + 18q^{83} + 15q^{85} - 45q^{87} + 12q^{89} - 6q^{91} + 25q^{93} + 24q^{95} + 15q^{97} - 24q^{99} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{2} + 2$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{5} + \nu^{4} - 5 \nu^{3} - \nu^{2} - 6$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$($$$$-2 \nu^{5} + 5 \nu^{4} - 16 \nu^{3} + 19 \nu^{2} - 21 \nu + 6$$$$)/3$$ $$\beta_{4}$$ $$=$$ $$\nu^{5} - 2 \nu^{4} + 8 \nu^{3} - 8 \nu^{2} + 12 \nu - 4$$ $$\beta_{5}$$ $$=$$ $$\nu^{5} - 3 \nu^{4} + 10 \nu^{3} - 13 \nu^{2} + 16 \nu - 5$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{5} + \beta_{4} - \beta_{3} + 2 \beta_{2} + 2 \beta_{1} + 1$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$\beta_{1} - 2$$ $$\nu^{3}$$ $$=$$ $$($$$$5 \beta_{5} - 2 \beta_{4} + 8 \beta_{3} - 7 \beta_{2} - 4 \beta_{1} - 5$$$$)/3$$ $$\nu^{4}$$ $$=$$ $$\beta_{5} + \beta_{4} + 4 \beta_{3} - 2 \beta_{2} - 5 \beta_{1} + 7$$ $$\nu^{5}$$ $$=$$ $$($$$$-22 \beta_{5} + 13 \beta_{4} - 28 \beta_{3} + 20 \beta_{2} + 2 \beta_{1} + 34$$$$)/3$$

## Character values

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

 $$n$$ $$29$$ $$73$$ $$127$$ $$\chi(n)$$ $$\beta_{3}$$ $$1$$ $$1$$

## 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 −1.09097 + 1.34528i 0 1.97141 3.41458i 0 −0.500000 0.866025i 0 −0.619562 2.93533i 0
85.2 0 0.796790 + 1.53790i 0 −1.02704 + 1.77889i 0 −0.500000 0.866025i 0 −1.73025 + 2.45076i 0
85.3 0 1.29418 1.15113i 0 0.555632 0.962383i 0 −0.500000 0.866025i 0 0.349814 2.97954i 0
169.1 0 −1.09097 1.34528i 0 1.97141 + 3.41458i 0 −0.500000 + 0.866025i 0 −0.619562 + 2.93533i 0
169.2 0 0.796790 1.53790i 0 −1.02704 1.77889i 0 −0.500000 + 0.866025i 0 −1.73025 2.45076i 0
169.3 0 1.29418 + 1.15113i 0 0.555632 + 0.962383i 0 −0.500000 + 0.866025i 0 0.349814 + 2.97954i 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.b 6
3.b odd 2 1 756.2.j.a 6
4.b odd 2 1 1008.2.r.g 6
7.b odd 2 1 1764.2.j.d 6
7.c even 3 1 1764.2.i.f 6
7.c even 3 1 1764.2.l.d 6
7.d odd 6 1 1764.2.i.e 6
7.d odd 6 1 1764.2.l.g 6
9.c even 3 1 inner 252.2.j.b 6
9.c even 3 1 2268.2.a.g 3
9.d odd 6 1 756.2.j.a 6
9.d odd 6 1 2268.2.a.j 3
12.b even 2 1 3024.2.r.i 6
21.c even 2 1 5292.2.j.e 6
21.g even 6 1 5292.2.i.g 6
21.g even 6 1 5292.2.l.d 6
21.h odd 6 1 5292.2.i.d 6
21.h odd 6 1 5292.2.l.g 6
36.f odd 6 1 1008.2.r.g 6
36.f odd 6 1 9072.2.a.bt 3
36.h even 6 1 3024.2.r.i 6
36.h even 6 1 9072.2.a.bz 3
63.g even 3 1 1764.2.i.f 6
63.h even 3 1 1764.2.l.d 6
63.i even 6 1 5292.2.l.d 6
63.j odd 6 1 5292.2.l.g 6
63.k odd 6 1 1764.2.i.e 6
63.l odd 6 1 1764.2.j.d 6
63.n odd 6 1 5292.2.i.d 6
63.o even 6 1 5292.2.j.e 6
63.s even 6 1 5292.2.i.g 6
63.t odd 6 1 1764.2.l.g 6

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{6} - 3 T_{5}^{5} + 15 T_{5}^{4} + 63 T_{5}^{2} - 54 T_{5} + 81$$ acting on $$S_{2}^{\mathrm{new}}(252, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 
$3$ $$1 - 2 T + 4 T^{2} - 3 T^{3} + 12 T^{4} - 18 T^{5} + 27 T^{6}$$
$5$ $$1 - 3 T + 15 T^{3} - 27 T^{4} + 6 T^{5} + 61 T^{6} + 30 T^{7} - 675 T^{8} + 1875 T^{9} - 9375 T^{11} + 15625 T^{12}$$
$7$ $$( 1 + T + T^{2} )^{3}$$
$11$ $$1 - 6 T + 30 T^{3} + 162 T^{4} - 402 T^{5} - 821 T^{6} - 4422 T^{7} + 19602 T^{8} + 39930 T^{9} - 966306 T^{11} + 1771561 T^{12}$$
$13$ $$( 1 - T - 12 T^{2} - 13 T^{3} + 169 T^{4} )^{3}$$
$17$ $$( 1 + 18 T^{2} + 9 T^{3} + 306 T^{4} + 4913 T^{6} )^{2}$$
$19$ $$( 1 - 3 T + 21 T^{2} - 65 T^{3} + 399 T^{4} - 1083 T^{5} + 6859 T^{6} )^{2}$$
$23$ $$1 - 6 T - 18 T^{2} + 30 T^{3} + 612 T^{4} + 2172 T^{5} - 30449 T^{6} + 49956 T^{7} + 323748 T^{8} + 365010 T^{9} - 5037138 T^{10} - 38618058 T^{11} + 148035889 T^{12}$$
$29$ $$1 - 15 T + 90 T^{2} - 411 T^{3} + 2205 T^{4} - 4110 T^{5} - 17723 T^{6} - 119190 T^{7} + 1854405 T^{8} - 10023879 T^{9} + 63655290 T^{10} - 307667235 T^{11} + 594823321 T^{12}$$
$31$ $$1 - 3 T - 6 T^{2} + 221 T^{3} - 639 T^{4} - 2088 T^{5} + 61647 T^{6} - 64728 T^{7} - 614079 T^{8} + 6583811 T^{9} - 5541126 T^{10} - 85887453 T^{11} + 887503681 T^{12}$$
$37$ $$( 1 + 3 T + 33 T^{2} + 115 T^{3} + 1221 T^{4} + 4107 T^{5} + 50653 T^{6} )^{2}$$
$41$ $$1 - 6 T - 90 T^{2} + 210 T^{3} + 7812 T^{4} - 8952 T^{5} - 340301 T^{6} - 367032 T^{7} + 13131972 T^{8} + 14473410 T^{9} - 254318490 T^{10} - 695137206 T^{11} + 4750104241 T^{12}$$
$43$ $$1 + 3 T - 12 T^{2} + 473 T^{3} + 153 T^{4} - 4176 T^{5} + 165435 T^{6} - 179568 T^{7} + 282897 T^{8} + 37606811 T^{9} - 41025612 T^{10} + 441025329 T^{11} + 6321363049 T^{12}$$
$47$ $$1 - 15 T + 48 T^{2} + 3 T^{3} + 3075 T^{4} - 18798 T^{5} + 4399 T^{6} - 883506 T^{7} + 6792675 T^{8} + 311469 T^{9} + 234224688 T^{10} - 3440175105 T^{11} + 10779215329 T^{12}$$
$53$ $$( 1 + 18 T + 198 T^{2} + 1521 T^{3} + 10494 T^{4} + 50562 T^{5} + 148877 T^{6} )^{2}$$
$59$ $$1 - 3 T - 114 T^{2} + 501 T^{3} + 6567 T^{4} - 20406 T^{5} - 323957 T^{6} - 1203954 T^{7} + 22859727 T^{8} + 102894879 T^{9} - 1381379154 T^{10} - 2144772897 T^{11} + 42180533641 T^{12}$$
$61$ $$1 - 6 T - 102 T^{2} + 698 T^{3} + 6048 T^{4} - 26604 T^{5} - 259509 T^{6} - 1622844 T^{7} + 22504608 T^{8} + 158432738 T^{9} - 1412275782 T^{10} - 5067577806 T^{11} + 51520374361 T^{12}$$
$67$ $$1 - 6 T - 138 T^{2} + 446 T^{3} + 14148 T^{4} - 16668 T^{5} - 1033545 T^{6} - 1116756 T^{7} + 63510372 T^{8} + 134140298 T^{9} - 2780854698 T^{10} - 8100750642 T^{11} + 90458382169 T^{12}$$
$71$ $$( 1 + 15 T + 231 T^{2} + 1833 T^{3} + 16401 T^{4} + 75615 T^{5} + 357911 T^{6} )^{2}$$
$73$ $$( 1 - 9 T + 207 T^{2} - 1235 T^{3} + 15111 T^{4} - 47961 T^{5} + 389017 T^{6} )^{2}$$
$79$ $$1 + 3 T + 6 T^{2} + 1787 T^{3} + 2439 T^{4} + 14634 T^{5} + 1719519 T^{6} + 1156086 T^{7} + 15221799 T^{8} + 881060693 T^{9} + 233700486 T^{10} + 9231169197 T^{11} + 243087455521 T^{12}$$
$83$ $$1 - 18 T + 198 T^{3} + 23814 T^{4} - 132750 T^{5} - 711245 T^{6} - 11018250 T^{7} + 164054646 T^{8} + 113213826 T^{9} - 70902731574 T^{11} + 326940373369 T^{12}$$
$89$ $$( 1 - 6 T + 72 T^{2} + 21 T^{3} + 6408 T^{4} - 47526 T^{5} + 704969 T^{6} )^{2}$$
$97$ $$1 - 15 T - 84 T^{2} + 1139 T^{3} + 22203 T^{4} - 134028 T^{5} - 1218567 T^{6} - 13000716 T^{7} + 208908027 T^{8} + 1039534547 T^{9} - 7436459604 T^{10} - 128810103855 T^{11} + 832972004929 T^{12}$$