# Properties

 Label 252.2.bi.b Level 252 Weight 2 Character orbit 252.bi Analytic conductor 2.012 Analytic rank 0 Dimension 4 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$252 = 2^{2} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 252.bi (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.01223013094$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( \zeta_{12} + \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{2} + ( -2 \zeta_{12} + \zeta_{12}^{3} ) q^{3} + 2 \zeta_{12} q^{4} + ( 1 + \zeta_{12}^{2} ) q^{5} + ( -2 - \zeta_{12} + \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{6} + ( -2 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{7} + ( 2 + 2 \zeta_{12}^{3} ) q^{8} + 3 q^{9} +O(q^{10})$$ $$q + ( \zeta_{12} + \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{2} + ( -2 \zeta_{12} + \zeta_{12}^{3} ) q^{3} + 2 \zeta_{12} q^{4} + ( 1 + \zeta_{12}^{2} ) q^{5} + ( -2 - \zeta_{12} + \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{6} + ( -2 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{7} + ( 2 + 2 \zeta_{12}^{3} ) q^{8} + 3 q^{9} + ( -1 + 2 \zeta_{12} + 2 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{10} + ( \zeta_{12} - \zeta_{12}^{3} ) q^{11} + ( -2 - 2 \zeta_{12}^{2} ) q^{12} + ( -1 - \zeta_{12}^{2} ) q^{13} + ( -2 - 3 \zeta_{12} + 3 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{14} -3 \zeta_{12} q^{15} + 4 \zeta_{12}^{2} q^{16} + ( -2 + 4 \zeta_{12}^{2} ) q^{17} + ( 3 \zeta_{12} + 3 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{18} + ( 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{19} + ( 2 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{20} + ( 5 - 4 \zeta_{12}^{2} ) q^{21} + ( 1 + \zeta_{12} - \zeta_{12}^{2} ) q^{22} -5 \zeta_{12} q^{23} + ( 2 - 4 \zeta_{12} - 4 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{24} -2 \zeta_{12}^{2} q^{25} + ( 1 - 2 \zeta_{12} - 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{26} + ( -6 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{27} + ( -6 + 2 \zeta_{12}^{2} ) q^{28} -5 \zeta_{12}^{2} q^{29} + ( -3 - 3 \zeta_{12}^{3} ) q^{30} + ( 5 \zeta_{12} - 10 \zeta_{12}^{3} ) q^{31} + ( -4 + 4 \zeta_{12} + 4 \zeta_{12}^{2} ) q^{32} + ( -2 + \zeta_{12}^{2} ) q^{33} + ( -4 + 2 \zeta_{12} + 2 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{34} + ( -5 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{35} + 6 \zeta_{12} q^{36} + ( 8 + 4 \zeta_{12} - 4 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{38} + 3 \zeta_{12} q^{39} + ( 2 - 2 \zeta_{12} + 2 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{40} + ( -5 - 5 \zeta_{12}^{2} ) q^{41} + ( 4 + \zeta_{12} + \zeta_{12}^{2} - 5 \zeta_{12}^{3} ) q^{42} + ( -\zeta_{12} + \zeta_{12}^{3} ) q^{43} + 2 q^{44} + ( 3 + 3 \zeta_{12}^{2} ) q^{45} + ( -5 - 5 \zeta_{12}^{3} ) q^{46} + ( -3 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{47} + ( -4 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{48} + ( 3 - 8 \zeta_{12}^{2} ) q^{49} + ( 2 - 2 \zeta_{12} - 2 \zeta_{12}^{2} ) q^{50} -6 \zeta_{12}^{3} q^{51} + ( -2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{52} + 4 q^{53} + ( -6 - 3 \zeta_{12} + 3 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{54} + ( 2 \zeta_{12} - \zeta_{12}^{3} ) q^{55} + ( -2 - 4 \zeta_{12} - 4 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{56} -12 q^{57} + ( 5 - 5 \zeta_{12} - 5 \zeta_{12}^{2} ) q^{58} + ( -5 \zeta_{12} + 10 \zeta_{12}^{3} ) q^{59} -6 \zeta_{12}^{2} q^{60} + ( 6 - 3 \zeta_{12}^{2} ) q^{61} + ( 5 + 10 \zeta_{12} - 10 \zeta_{12}^{2} - 5 \zeta_{12}^{3} ) q^{62} + ( -6 \zeta_{12} + 9 \zeta_{12}^{3} ) q^{63} + 8 \zeta_{12}^{3} q^{64} -3 \zeta_{12}^{2} q^{65} + ( -1 - \zeta_{12} - \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{66} + 15 \zeta_{12} q^{67} + ( -4 \zeta_{12} + 8 \zeta_{12}^{3} ) q^{68} + ( 5 + 5 \zeta_{12}^{2} ) q^{69} + ( -5 - 4 \zeta_{12} + 4 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{70} -4 \zeta_{12}^{3} q^{71} + ( 6 + 6 \zeta_{12}^{3} ) q^{72} + ( 6 - 12 \zeta_{12}^{2} ) q^{73} + ( 2 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{75} + ( 8 + 8 \zeta_{12}^{2} ) q^{76} + ( -2 + 3 \zeta_{12}^{2} ) q^{77} + ( 3 + 3 \zeta_{12}^{3} ) q^{78} + ( -3 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{79} + ( -4 + 8 \zeta_{12}^{2} ) q^{80} + 9 q^{81} + ( 5 - 10 \zeta_{12} - 10 \zeta_{12}^{2} + 5 \zeta_{12}^{3} ) q^{82} + ( \zeta_{12} + \zeta_{12}^{3} ) q^{83} + ( 10 \zeta_{12} - 8 \zeta_{12}^{3} ) q^{84} + ( -6 + 6 \zeta_{12}^{2} ) q^{85} + ( -1 - \zeta_{12} + \zeta_{12}^{2} ) q^{86} + ( 5 \zeta_{12} + 5 \zeta_{12}^{3} ) q^{87} + ( 2 \zeta_{12} + 2 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{88} + ( -10 + 20 \zeta_{12}^{2} ) q^{89} + ( -3 + 6 \zeta_{12} + 6 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{90} + ( 5 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{91} -10 \zeta_{12}^{2} q^{92} + ( -15 + 15 \zeta_{12}^{2} ) q^{93} + ( -3 + 3 \zeta_{12} - 3 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{94} + 12 \zeta_{12} q^{95} + ( -4 + 4 \zeta_{12} - 4 \zeta_{12}^{2} - 8 \zeta_{12}^{3} ) q^{96} + ( 6 - 3 \zeta_{12}^{2} ) q^{97} + ( 8 - 5 \zeta_{12} - 5 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{98} + ( 3 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{2} + 6q^{5} - 6q^{6} + 8q^{8} + 12q^{9} + O(q^{10})$$ $$4q + 2q^{2} + 6q^{5} - 6q^{6} + 8q^{8} + 12q^{9} - 12q^{12} - 6q^{13} - 2q^{14} + 8q^{16} + 6q^{18} + 12q^{21} + 2q^{22} - 4q^{25} - 20q^{28} - 10q^{29} - 12q^{30} - 8q^{32} - 6q^{33} - 12q^{34} + 24q^{38} + 12q^{40} - 30q^{41} + 18q^{42} + 8q^{44} + 18q^{45} - 20q^{46} - 4q^{49} + 4q^{50} + 16q^{53} - 18q^{54} - 16q^{56} - 48q^{57} + 10q^{58} - 12q^{60} + 18q^{61} - 6q^{65} - 6q^{66} + 30q^{69} - 12q^{70} + 24q^{72} + 48q^{76} - 2q^{77} + 12q^{78} + 36q^{81} - 12q^{85} - 2q^{86} + 4q^{88} - 20q^{92} - 30q^{93} - 18q^{94} - 24q^{96} + 18q^{97} + 22q^{98} + O(q^{100})$$

## Character values

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

 $$n$$ $$29$$ $$73$$ $$127$$ $$\chi(n)$$ $$-1 + \zeta_{12}^{2}$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
139.1
 −0.866025 − 0.500000i 0.866025 + 0.500000i −0.866025 + 0.500000i 0.866025 − 0.500000i
−0.366025 + 1.36603i 1.73205 −1.73205 1.00000i 1.50000 + 0.866025i −0.633975 + 2.36603i 1.73205 2.00000i 2.00000 2.00000i 3.00000 −1.73205 + 1.73205i
139.2 1.36603 + 0.366025i −1.73205 1.73205 + 1.00000i 1.50000 + 0.866025i −2.36603 0.633975i −1.73205 + 2.00000i 2.00000 + 2.00000i 3.00000 1.73205 + 1.73205i
223.1 −0.366025 1.36603i 1.73205 −1.73205 + 1.00000i 1.50000 0.866025i −0.633975 2.36603i 1.73205 + 2.00000i 2.00000 + 2.00000i 3.00000 −1.73205 1.73205i
223.2 1.36603 0.366025i −1.73205 1.73205 1.00000i 1.50000 0.866025i −2.36603 + 0.633975i −1.73205 2.00000i 2.00000 2.00000i 3.00000 1.73205 1.73205i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
63.l odd 6 1 inner
252.bi even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 252.2.bi.b yes 4
3.b odd 2 1 756.2.bi.a 4
4.b odd 2 1 inner 252.2.bi.b yes 4
7.b odd 2 1 252.2.bi.a 4
9.c even 3 1 252.2.bi.a 4
9.d odd 6 1 756.2.bi.b 4
12.b even 2 1 756.2.bi.a 4
21.c even 2 1 756.2.bi.b 4
28.d even 2 1 252.2.bi.a 4
36.f odd 6 1 252.2.bi.a 4
36.h even 6 1 756.2.bi.b 4
63.l odd 6 1 inner 252.2.bi.b yes 4
63.o even 6 1 756.2.bi.a 4
84.h odd 2 1 756.2.bi.b 4
252.s odd 6 1 756.2.bi.a 4
252.bi even 6 1 inner 252.2.bi.b yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.2.bi.a 4 7.b odd 2 1
252.2.bi.a 4 9.c even 3 1
252.2.bi.a 4 28.d even 2 1
252.2.bi.a 4 36.f odd 6 1
252.2.bi.b yes 4 1.a even 1 1 trivial
252.2.bi.b yes 4 4.b odd 2 1 inner
252.2.bi.b yes 4 63.l odd 6 1 inner
252.2.bi.b yes 4 252.bi even 6 1 inner
756.2.bi.a 4 3.b odd 2 1
756.2.bi.a 4 12.b even 2 1
756.2.bi.a 4 63.o even 6 1
756.2.bi.a 4 252.s odd 6 1
756.2.bi.b 4 9.d odd 6 1
756.2.bi.b 4 21.c even 2 1
756.2.bi.b 4 36.h even 6 1
756.2.bi.b 4 84.h odd 2 1

## Hecke kernels

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

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 - 2 T + 2 T^{2} - 4 T^{3} + 4 T^{4}$$
$3$ $$( 1 - 3 T^{2} )^{2}$$
$5$ $$( 1 - 3 T + 8 T^{2} - 15 T^{3} + 25 T^{4} )^{2}$$
$7$ $$1 + 2 T^{2} + 49 T^{4}$$
$11$ $$1 + 21 T^{2} + 320 T^{4} + 2541 T^{6} + 14641 T^{8}$$
$13$ $$( 1 - 2 T + 13 T^{2} )^{2}( 1 + 5 T + 13 T^{2} )^{2}$$
$17$ $$( 1 - 22 T^{2} + 289 T^{4} )^{2}$$
$19$ $$( 1 - 10 T^{2} + 361 T^{4} )^{2}$$
$23$ $$1 + 21 T^{2} - 88 T^{4} + 11109 T^{6} + 279841 T^{8}$$
$29$ $$( 1 + 5 T - 4 T^{2} + 145 T^{3} + 841 T^{4} )^{2}$$
$31$ $$( 1 - 46 T^{2} + 961 T^{4} )( 1 + 59 T^{2} + 961 T^{4} )$$
$37$ $$( 1 + 37 T^{2} )^{4}$$
$41$ $$( 1 + 15 T + 116 T^{2} + 615 T^{3} + 1681 T^{4} )^{2}$$
$43$ $$1 + 85 T^{2} + 5376 T^{4} + 157165 T^{6} + 3418801 T^{8}$$
$47$ $$1 - 67 T^{2} + 2280 T^{4} - 148003 T^{6} + 4879681 T^{8}$$
$53$ $$( 1 - 4 T + 53 T^{2} )^{4}$$
$59$ $$1 - 43 T^{2} - 1632 T^{4} - 149683 T^{6} + 12117361 T^{8}$$
$61$ $$( 1 - 9 T + 88 T^{2} - 549 T^{3} + 3721 T^{4} )^{2}$$
$67$ $$1 - 91 T^{2} + 3792 T^{4} - 408499 T^{6} + 20151121 T^{8}$$
$71$ $$( 1 - 126 T^{2} + 5041 T^{4} )^{2}$$
$73$ $$( 1 - 38 T^{2} + 5329 T^{4} )^{2}$$
$79$ $$1 + 149 T^{2} + 15960 T^{4} + 929909 T^{6} + 38950081 T^{8}$$
$83$ $$1 - 163 T^{2} + 19680 T^{4} - 1122907 T^{6} + 47458321 T^{8}$$
$89$ $$( 1 + 122 T^{2} + 7921 T^{4} )^{2}$$
$97$ $$( 1 - 14 T + 97 T^{2} )^{2}( 1 + 5 T + 97 T^{2} )^{2}$$