Properties

 Label 252.2.bf.e Level $252$ Weight $2$ Character orbit 252.bf 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.bf (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: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 28) 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.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{12}^{3} + \zeta_{12}^{2} + \zeta_{12}) q^{2} + 2 \zeta_{12} q^{4} + ( - \zeta_{12}^{2} + 2) q^{5} + (3 \zeta_{12}^{3} - 2 \zeta_{12}) q^{7} + (2 \zeta_{12}^{3} + 2) q^{8}+O(q^{10})$$ q + (-z^3 + z^2 + z) * q^2 + 2*z * q^4 + (-z^2 + 2) * q^5 + (3*z^3 - 2*z) * q^7 + (2*z^3 + 2) * q^8 $$q + ( - \zeta_{12}^{3} + \zeta_{12}^{2} + \zeta_{12}) q^{2} + 2 \zeta_{12} q^{4} + ( - \zeta_{12}^{2} + 2) q^{5} + (3 \zeta_{12}^{3} - 2 \zeta_{12}) q^{7} + (2 \zeta_{12}^{3} + 2) q^{8} + ( - 2 \zeta_{12}^{3} + \zeta_{12}^{2} + \zeta_{12} + 1) q^{10} - \zeta_{12} q^{11} + ( - 4 \zeta_{12}^{2} + 2) q^{13} + (\zeta_{12}^{3} + 3 \zeta_{12}^{2} - 3 \zeta_{12} - 2) q^{14} + 4 \zeta_{12}^{2} q^{16} + (\zeta_{12}^{2} + 1) q^{17} + ( - 3 \zeta_{12}^{3} - 3 \zeta_{12}) q^{19} + ( - 2 \zeta_{12}^{3} + 4 \zeta_{12}) q^{20} + ( - \zeta_{12}^{3} - 1) q^{22} + (\zeta_{12}^{3} - \zeta_{12}) q^{23} + (2 \zeta_{12}^{2} - 2) q^{25} + ( - 2 \zeta_{12}^{3} - 2 \zeta_{12}^{2} - 2 \zeta_{12} + 4) q^{26} + (2 \zeta_{12}^{2} - 6) q^{28} - 4 q^{29} + (2 \zeta_{12}^{3} - \zeta_{12}) q^{31} + (4 \zeta_{12}^{2} + 4 \zeta_{12} - 4) q^{32} + ( - \zeta_{12}^{3} + 2 \zeta_{12}^{2} + 2 \zeta_{12} - 1) q^{34} + (5 \zeta_{12}^{3} - \zeta_{12}) q^{35} - 3 \zeta_{12}^{2} q^{37} + ( - 6 \zeta_{12}^{3} - 3 \zeta_{12}^{2} + 3 \zeta_{12} - 3) q^{38} + (2 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 2 \zeta_{12} + 4) q^{40} + ( - 4 \zeta_{12}^{2} + 2) q^{41} + 2 \zeta_{12}^{3} q^{43} - 2 \zeta_{12}^{2} q^{44} + (\zeta_{12}^{2} - \zeta_{12} - 1) q^{46} + ( - 5 \zeta_{12}^{3} - 5 \zeta_{12}) q^{47} + ( - 8 \zeta_{12}^{2} + 3) q^{49} + (2 \zeta_{12}^{3} - 2) q^{50} + ( - 8 \zeta_{12}^{3} + 4 \zeta_{12}) q^{52} + (\zeta_{12}^{2} - 1) q^{53} + (\zeta_{12}^{3} - 2 \zeta_{12}) q^{55} + (6 \zeta_{12}^{3} - 4 \zeta_{12}^{2} - 4 \zeta_{12} - 2) q^{56} + (4 \zeta_{12}^{3} - 4 \zeta_{12}^{2} - 4 \zeta_{12}) q^{58} + ( - 6 \zeta_{12}^{3} + 3 \zeta_{12}) q^{59} + (3 \zeta_{12}^{2} - 6) q^{61} + (\zeta_{12}^{3} + 2 \zeta_{12}^{2} - 2 \zeta_{12} - 1) q^{62} + 8 \zeta_{12}^{3} q^{64} - 6 \zeta_{12}^{2} q^{65} + 3 \zeta_{12} q^{67} + (2 \zeta_{12}^{3} + 2 \zeta_{12}) q^{68} + (4 \zeta_{12}^{3} + 5 \zeta_{12}^{2} - 5 \zeta_{12} - 1) q^{70} + 14 \zeta_{12}^{3} q^{71} + (5 \zeta_{12}^{2} + 5) q^{73} + ( - 3 \zeta_{12}^{2} - 3 \zeta_{12} + 3) q^{74} + ( - 12 \zeta_{12}^{2} + 6) q^{76} + ( - \zeta_{12}^{2} + 3) q^{77} + ( - 9 \zeta_{12}^{3} + 9 \zeta_{12}) q^{79} + (4 \zeta_{12}^{2} + 4) q^{80} + ( - 2 \zeta_{12}^{3} - 2 \zeta_{12}^{2} - 2 \zeta_{12} + 4) q^{82} + ( - 8 \zeta_{12}^{3} + 16 \zeta_{12}) q^{83} + 3 q^{85} + (2 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 2 \zeta_{12}) q^{86} + ( - 2 \zeta_{12}^{2} - 2 \zeta_{12} + 2) q^{88} + (9 \zeta_{12}^{2} - 18) q^{89} + (2 \zeta_{12}^{3} + 8 \zeta_{12}) q^{91} - 2 q^{92} + ( - 10 \zeta_{12}^{3} - 5 \zeta_{12}^{2} + 5 \zeta_{12} - 5) q^{94} - 9 \zeta_{12} q^{95} + (20 \zeta_{12}^{2} - 10) q^{97} + ( - 3 \zeta_{12}^{3} - 5 \zeta_{12}^{2} - 5 \zeta_{12} + 8) q^{98} +O(q^{100})$$ q + (-z^3 + z^2 + z) * q^2 + 2*z * q^4 + (-z^2 + 2) * q^5 + (3*z^3 - 2*z) * q^7 + (2*z^3 + 2) * q^8 + (-2*z^3 + z^2 + z + 1) * q^10 - z * q^11 + (-4*z^2 + 2) * q^13 + (z^3 + 3*z^2 - 3*z - 2) * q^14 + 4*z^2 * q^16 + (z^2 + 1) * q^17 + (-3*z^3 - 3*z) * q^19 + (-2*z^3 + 4*z) * q^20 + (-z^3 - 1) * q^22 + (z^3 - z) * q^23 + (2*z^2 - 2) * q^25 + (-2*z^3 - 2*z^2 - 2*z + 4) * q^26 + (2*z^2 - 6) * q^28 - 4 * q^29 + (2*z^3 - z) * q^31 + (4*z^2 + 4*z - 4) * q^32 + (-z^3 + 2*z^2 + 2*z - 1) * q^34 + (5*z^3 - z) * q^35 - 3*z^2 * q^37 + (-6*z^3 - 3*z^2 + 3*z - 3) * q^38 + (2*z^3 - 2*z^2 + 2*z + 4) * q^40 + (-4*z^2 + 2) * q^41 + 2*z^3 * q^43 - 2*z^2 * q^44 + (z^2 - z - 1) * q^46 + (-5*z^3 - 5*z) * q^47 + (-8*z^2 + 3) * q^49 + (2*z^3 - 2) * q^50 + (-8*z^3 + 4*z) * q^52 + (z^2 - 1) * q^53 + (z^3 - 2*z) * q^55 + (6*z^3 - 4*z^2 - 4*z - 2) * q^56 + (4*z^3 - 4*z^2 - 4*z) * q^58 + (-6*z^3 + 3*z) * q^59 + (3*z^2 - 6) * q^61 + (z^3 + 2*z^2 - 2*z - 1) * q^62 + 8*z^3 * q^64 - 6*z^2 * q^65 + 3*z * q^67 + (2*z^3 + 2*z) * q^68 + (4*z^3 + 5*z^2 - 5*z - 1) * q^70 + 14*z^3 * q^71 + (5*z^2 + 5) * q^73 + (-3*z^2 - 3*z + 3) * q^74 + (-12*z^2 + 6) * q^76 + (-z^2 + 3) * q^77 + (-9*z^3 + 9*z) * q^79 + (4*z^2 + 4) * q^80 + (-2*z^3 - 2*z^2 - 2*z + 4) * q^82 + (-8*z^3 + 16*z) * q^83 + 3 * q^85 + (2*z^3 + 2*z^2 - 2*z) * q^86 + (-2*z^2 - 2*z + 2) * q^88 + (9*z^2 - 18) * q^89 + (2*z^3 + 8*z) * q^91 - 2 * q^92 + (-10*z^3 - 5*z^2 + 5*z - 5) * q^94 - 9*z * q^95 + (20*z^2 - 10) * q^97 + (-3*z^3 - 5*z^2 - 5*z + 8) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{2} + 6 q^{5} + 8 q^{8}+O(q^{10})$$ 4 * q + 2 * q^2 + 6 * q^5 + 8 * q^8 $$4 q + 2 q^{2} + 6 q^{5} + 8 q^{8} + 6 q^{10} - 2 q^{14} + 8 q^{16} + 6 q^{17} - 4 q^{22} - 4 q^{25} + 12 q^{26} - 20 q^{28} - 16 q^{29} - 8 q^{32} - 6 q^{37} - 18 q^{38} + 12 q^{40} - 4 q^{44} - 2 q^{46} - 4 q^{49} - 8 q^{50} - 2 q^{53} - 16 q^{56} - 8 q^{58} - 18 q^{61} - 12 q^{65} + 6 q^{70} + 30 q^{73} + 6 q^{74} + 10 q^{77} + 24 q^{80} + 12 q^{82} + 12 q^{85} + 4 q^{86} + 4 q^{88} - 54 q^{89} - 8 q^{92} - 30 q^{94} + 22 q^{98}+O(q^{100})$$ 4 * q + 2 * q^2 + 6 * q^5 + 8 * q^8 + 6 * q^10 - 2 * q^14 + 8 * q^16 + 6 * q^17 - 4 * q^22 - 4 * q^25 + 12 * q^26 - 20 * q^28 - 16 * q^29 - 8 * q^32 - 6 * q^37 - 18 * q^38 + 12 * q^40 - 4 * q^44 - 2 * q^46 - 4 * q^49 - 8 * q^50 - 2 * q^53 - 16 * q^56 - 8 * q^58 - 18 * q^61 - 12 * q^65 + 6 * q^70 + 30 * q^73 + 6 * q^74 + 10 * q^77 + 24 * q^80 + 12 * q^82 + 12 * q^85 + 4 * q^86 + 4 * q^88 - 54 * q^89 - 8 * q^92 - 30 * q^94 + 22 * q^98

Character values

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

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

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
19.1
 −0.866025 + 0.500000i 0.866025 − 0.500000i −0.866025 − 0.500000i 0.866025 + 0.500000i
−0.366025 1.36603i 0 −1.73205 + 1.00000i 1.50000 + 0.866025i 0 1.73205 + 2.00000i 2.00000 + 2.00000i 0 0.633975 2.36603i
19.2 1.36603 0.366025i 0 1.73205 1.00000i 1.50000 + 0.866025i 0 −1.73205 2.00000i 2.00000 2.00000i 0 2.36603 + 0.633975i
199.1 −0.366025 + 1.36603i 0 −1.73205 1.00000i 1.50000 0.866025i 0 1.73205 2.00000i 2.00000 2.00000i 0 0.633975 + 2.36603i
199.2 1.36603 + 0.366025i 0 1.73205 + 1.00000i 1.50000 0.866025i 0 −1.73205 + 2.00000i 2.00000 + 2.00000i 0 2.36603 0.633975i
 $$n$$: 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
7.d odd 6 1 inner
28.f 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.bf.e 4
3.b odd 2 1 28.2.f.a 4
4.b odd 2 1 inner 252.2.bf.e 4
7.c even 3 1 1764.2.b.a 4
7.d odd 6 1 inner 252.2.bf.e 4
7.d odd 6 1 1764.2.b.a 4
12.b even 2 1 28.2.f.a 4
15.d odd 2 1 700.2.p.a 4
15.e even 4 1 700.2.t.a 4
15.e even 4 1 700.2.t.b 4
21.c even 2 1 196.2.f.a 4
21.g even 6 1 28.2.f.a 4
21.g even 6 1 196.2.d.b 4
21.h odd 6 1 196.2.d.b 4
21.h odd 6 1 196.2.f.a 4
24.f even 2 1 448.2.p.d 4
24.h odd 2 1 448.2.p.d 4
28.f even 6 1 inner 252.2.bf.e 4
28.f even 6 1 1764.2.b.a 4
28.g odd 6 1 1764.2.b.a 4
60.h even 2 1 700.2.p.a 4
60.l odd 4 1 700.2.t.a 4
60.l odd 4 1 700.2.t.b 4
84.h odd 2 1 196.2.f.a 4
84.j odd 6 1 28.2.f.a 4
84.j odd 6 1 196.2.d.b 4
84.n even 6 1 196.2.d.b 4
84.n even 6 1 196.2.f.a 4
105.p even 6 1 700.2.p.a 4
105.w odd 12 1 700.2.t.a 4
105.w odd 12 1 700.2.t.b 4
168.s odd 6 1 3136.2.f.e 4
168.v even 6 1 3136.2.f.e 4
168.ba even 6 1 448.2.p.d 4
168.ba even 6 1 3136.2.f.e 4
168.be odd 6 1 448.2.p.d 4
168.be odd 6 1 3136.2.f.e 4
420.be odd 6 1 700.2.p.a 4
420.br even 12 1 700.2.t.a 4
420.br even 12 1 700.2.t.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.2.f.a 4 3.b odd 2 1
28.2.f.a 4 12.b even 2 1
28.2.f.a 4 21.g even 6 1
28.2.f.a 4 84.j odd 6 1
196.2.d.b 4 21.g even 6 1
196.2.d.b 4 21.h odd 6 1
196.2.d.b 4 84.j odd 6 1
196.2.d.b 4 84.n even 6 1
196.2.f.a 4 21.c even 2 1
196.2.f.a 4 21.h odd 6 1
196.2.f.a 4 84.h odd 2 1
196.2.f.a 4 84.n even 6 1
252.2.bf.e 4 1.a even 1 1 trivial
252.2.bf.e 4 4.b odd 2 1 inner
252.2.bf.e 4 7.d odd 6 1 inner
252.2.bf.e 4 28.f even 6 1 inner
448.2.p.d 4 24.f even 2 1
448.2.p.d 4 24.h odd 2 1
448.2.p.d 4 168.ba even 6 1
448.2.p.d 4 168.be odd 6 1
700.2.p.a 4 15.d odd 2 1
700.2.p.a 4 60.h even 2 1
700.2.p.a 4 105.p even 6 1
700.2.p.a 4 420.be odd 6 1
700.2.t.a 4 15.e even 4 1
700.2.t.a 4 60.l odd 4 1
700.2.t.a 4 105.w odd 12 1
700.2.t.a 4 420.br even 12 1
700.2.t.b 4 15.e even 4 1
700.2.t.b 4 60.l odd 4 1
700.2.t.b 4 105.w odd 12 1
700.2.t.b 4 420.br even 12 1
1764.2.b.a 4 7.c even 3 1
1764.2.b.a 4 7.d odd 6 1
1764.2.b.a 4 28.f even 6 1
1764.2.b.a 4 28.g odd 6 1
3136.2.f.e 4 168.s odd 6 1
3136.2.f.e 4 168.v even 6 1
3136.2.f.e 4 168.ba even 6 1
3136.2.f.e 4 168.be odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(252, [\chi])$$:

 $$T_{5}^{2} - 3T_{5} + 3$$ T5^2 - 3*T5 + 3 $$T_{11}^{4} - T_{11}^{2} + 1$$ T11^4 - T11^2 + 1 $$T_{19}^{4} + 27T_{19}^{2} + 729$$ T19^4 + 27*T19^2 + 729

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 2 T^{3} + 2 T^{2} - 4 T + 4$$
$3$ $$T^{4}$$
$5$ $$(T^{2} - 3 T + 3)^{2}$$
$7$ $$T^{4} + 2T^{2} + 49$$
$11$ $$T^{4} - T^{2} + 1$$
$13$ $$(T^{2} + 12)^{2}$$
$17$ $$(T^{2} - 3 T + 3)^{2}$$
$19$ $$T^{4} + 27T^{2} + 729$$
$23$ $$T^{4} - T^{2} + 1$$
$29$ $$(T + 4)^{4}$$
$31$ $$T^{4} + 3T^{2} + 9$$
$37$ $$(T^{2} + 3 T + 9)^{2}$$
$41$ $$(T^{2} + 12)^{2}$$
$43$ $$(T^{2} + 4)^{2}$$
$47$ $$T^{4} + 75T^{2} + 5625$$
$53$ $$(T^{2} + T + 1)^{2}$$
$59$ $$T^{4} + 27T^{2} + 729$$
$61$ $$(T^{2} + 9 T + 27)^{2}$$
$67$ $$T^{4} - 9T^{2} + 81$$
$71$ $$(T^{2} + 196)^{2}$$
$73$ $$(T^{2} - 15 T + 75)^{2}$$
$79$ $$T^{4} - 81T^{2} + 6561$$
$83$ $$(T^{2} - 192)^{2}$$
$89$ $$(T^{2} + 27 T + 243)^{2}$$
$97$ $$(T^{2} + 300)^{2}$$