Properties

 Label 3564.2.i.a Level $3564$ Weight $2$ Character orbit 3564.i Analytic conductor $28.459$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$3564 = 2^{2} \cdot 3^{4} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3564.i (of order $$3$$, degree $$2$$, not minimal)

Newform invariants

 Self dual: no Analytic conductor: $$28.4586832804$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 44) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 3 \zeta_{6} q^{5} + (2 \zeta_{6} - 2) q^{7} +O(q^{10})$$ q - 3*z * q^5 + (2*z - 2) * q^7 $$q - 3 \zeta_{6} q^{5} + (2 \zeta_{6} - 2) q^{7} + (\zeta_{6} - 1) q^{11} + 4 \zeta_{6} q^{13} - 6 q^{17} + 8 q^{19} - 3 \zeta_{6} q^{23} + (4 \zeta_{6} - 4) q^{25} - 5 \zeta_{6} q^{31} + 6 q^{35} - q^{37} + ( - 10 \zeta_{6} + 10) q^{43} + 3 \zeta_{6} q^{49} + 6 q^{53} + 3 q^{55} + 3 \zeta_{6} q^{59} + ( - 4 \zeta_{6} + 4) q^{61} + ( - 12 \zeta_{6} + 12) q^{65} + \zeta_{6} q^{67} - 15 q^{71} - 4 q^{73} - 2 \zeta_{6} q^{77} + (2 \zeta_{6} - 2) q^{79} + ( - 6 \zeta_{6} + 6) q^{83} + 18 \zeta_{6} q^{85} + 9 q^{89} - 8 q^{91} - 24 \zeta_{6} q^{95} + ( - 7 \zeta_{6} + 7) q^{97} +O(q^{100})$$ q - 3*z * q^5 + (2*z - 2) * q^7 + (z - 1) * q^11 + 4*z * q^13 - 6 * q^17 + 8 * q^19 - 3*z * q^23 + (4*z - 4) * q^25 - 5*z * q^31 + 6 * q^35 - q^37 + (-10*z + 10) * q^43 + 3*z * q^49 + 6 * q^53 + 3 * q^55 + 3*z * q^59 + (-4*z + 4) * q^61 + (-12*z + 12) * q^65 + z * q^67 - 15 * q^71 - 4 * q^73 - 2*z * q^77 + (2*z - 2) * q^79 + (-6*z + 6) * q^83 + 18*z * q^85 + 9 * q^89 - 8 * q^91 - 24*z * q^95 + (-7*z + 7) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 3 q^{5} - 2 q^{7}+O(q^{10})$$ 2 * q - 3 * q^5 - 2 * q^7 $$2 q - 3 q^{5} - 2 q^{7} - q^{11} + 4 q^{13} - 12 q^{17} + 16 q^{19} - 3 q^{23} - 4 q^{25} - 5 q^{31} + 12 q^{35} - 2 q^{37} + 10 q^{43} + 3 q^{49} + 12 q^{53} + 6 q^{55} + 3 q^{59} + 4 q^{61} + 12 q^{65} + q^{67} - 30 q^{71} - 8 q^{73} - 2 q^{77} - 2 q^{79} + 6 q^{83} + 18 q^{85} + 18 q^{89} - 16 q^{91} - 24 q^{95} + 7 q^{97}+O(q^{100})$$ 2 * q - 3 * q^5 - 2 * q^7 - q^11 + 4 * q^13 - 12 * q^17 + 16 * q^19 - 3 * q^23 - 4 * q^25 - 5 * q^31 + 12 * q^35 - 2 * q^37 + 10 * q^43 + 3 * q^49 + 12 * q^53 + 6 * q^55 + 3 * q^59 + 4 * q^61 + 12 * q^65 + q^67 - 30 * q^71 - 8 * q^73 - 2 * q^77 - 2 * q^79 + 6 * q^83 + 18 * q^85 + 18 * q^89 - 16 * q^91 - 24 * q^95 + 7 * q^97

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3564\mathbb{Z}\right)^\times$$.

 $$n$$ $$1541$$ $$1783$$ $$2917$$ $$\chi(n)$$ $$-\zeta_{6}$$ $$1$$ $$1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1189.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 0 0 −1.50000 2.59808i 0 −1.00000 + 1.73205i 0 0 0
2377.1 0 0 0 −1.50000 + 2.59808i 0 −1.00000 1.73205i 0 0 0
 $$n$$: 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
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 3564.2.i.a 2
3.b odd 2 1 3564.2.i.j 2
9.c even 3 1 396.2.a.c 1
9.c even 3 1 inner 3564.2.i.a 2
9.d odd 6 1 44.2.a.a 1
9.d odd 6 1 3564.2.i.j 2
36.f odd 6 1 1584.2.a.p 1
36.h even 6 1 176.2.a.a 1
45.h odd 6 1 1100.2.a.b 1
45.j even 6 1 9900.2.a.h 1
45.k odd 12 2 9900.2.c.g 2
45.l even 12 2 1100.2.b.c 2
63.i even 6 1 2156.2.i.c 2
63.j odd 6 1 2156.2.i.b 2
63.n odd 6 1 2156.2.i.b 2
63.o even 6 1 2156.2.a.a 1
63.s even 6 1 2156.2.i.c 2
72.j odd 6 1 704.2.a.f 1
72.l even 6 1 704.2.a.i 1
72.n even 6 1 6336.2.a.j 1
72.p odd 6 1 6336.2.a.i 1
99.g even 6 1 484.2.a.a 1
99.h odd 6 1 4356.2.a.j 1
99.n odd 30 4 484.2.e.a 4
99.p even 30 4 484.2.e.b 4
117.n odd 6 1 7436.2.a.d 1
144.u even 12 2 2816.2.c.k 2
144.w odd 12 2 2816.2.c.e 2
180.n even 6 1 4400.2.a.v 1
180.v odd 12 2 4400.2.b.k 2
252.s odd 6 1 8624.2.a.w 1
396.o odd 6 1 1936.2.a.c 1
792.s odd 6 1 7744.2.a.bc 1
792.w even 6 1 7744.2.a.m 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
44.2.a.a 1 9.d odd 6 1
176.2.a.a 1 36.h even 6 1
396.2.a.c 1 9.c even 3 1
484.2.a.a 1 99.g even 6 1
484.2.e.a 4 99.n odd 30 4
484.2.e.b 4 99.p even 30 4
704.2.a.f 1 72.j odd 6 1
704.2.a.i 1 72.l even 6 1
1100.2.a.b 1 45.h odd 6 1
1100.2.b.c 2 45.l even 12 2
1584.2.a.p 1 36.f odd 6 1
1936.2.a.c 1 396.o odd 6 1
2156.2.a.a 1 63.o even 6 1
2156.2.i.b 2 63.j odd 6 1
2156.2.i.b 2 63.n odd 6 1
2156.2.i.c 2 63.i even 6 1
2156.2.i.c 2 63.s even 6 1
2816.2.c.e 2 144.w odd 12 2
2816.2.c.k 2 144.u even 12 2
3564.2.i.a 2 1.a even 1 1 trivial
3564.2.i.a 2 9.c even 3 1 inner
3564.2.i.j 2 3.b odd 2 1
3564.2.i.j 2 9.d odd 6 1
4356.2.a.j 1 99.h odd 6 1
4400.2.a.v 1 180.n even 6 1
4400.2.b.k 2 180.v odd 12 2
6336.2.a.i 1 72.p odd 6 1
6336.2.a.j 1 72.n even 6 1
7436.2.a.d 1 117.n odd 6 1
7744.2.a.m 1 792.w even 6 1
7744.2.a.bc 1 792.s odd 6 1
8624.2.a.w 1 252.s odd 6 1
9900.2.a.h 1 45.j even 6 1
9900.2.c.g 2 45.k odd 12 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(3564, [\chi])$$:

 $$T_{5}^{2} + 3T_{5} + 9$$ T5^2 + 3*T5 + 9 $$T_{7}^{2} + 2T_{7} + 4$$ T7^2 + 2*T7 + 4 $$T_{17} + 6$$ T17 + 6

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 3T + 9$$
$7$ $$T^{2} + 2T + 4$$
$11$ $$T^{2} + T + 1$$
$13$ $$T^{2} - 4T + 16$$
$17$ $$(T + 6)^{2}$$
$19$ $$(T - 8)^{2}$$
$23$ $$T^{2} + 3T + 9$$
$29$ $$T^{2}$$
$31$ $$T^{2} + 5T + 25$$
$37$ $$(T + 1)^{2}$$
$41$ $$T^{2}$$
$43$ $$T^{2} - 10T + 100$$
$47$ $$T^{2}$$
$53$ $$(T - 6)^{2}$$
$59$ $$T^{2} - 3T + 9$$
$61$ $$T^{2} - 4T + 16$$
$67$ $$T^{2} - T + 1$$
$71$ $$(T + 15)^{2}$$
$73$ $$(T + 4)^{2}$$
$79$ $$T^{2} + 2T + 4$$
$83$ $$T^{2} - 6T + 36$$
$89$ $$(T - 9)^{2}$$
$97$ $$T^{2} - 7T + 49$$