Properties

 Label 266.2.f.a Level $266$ Weight $2$ Character orbit 266.f Analytic conductor $2.124$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$266 = 2 \cdot 7 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 266.f (of order $$3$$, degree $$2$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$2.12402069377$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ 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 primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\zeta_{6} - 1) q^{2} - \zeta_{6} q^{4} + (\zeta_{6} - 1) q^{5} + q^{7} + q^{8} + 3 \zeta_{6} q^{9} +O(q^{10})$$ q + (z - 1) * q^2 - z * q^4 + (z - 1) * q^5 + q^7 + q^8 + 3*z * q^9 $$q + (\zeta_{6} - 1) q^{2} - \zeta_{6} q^{4} + (\zeta_{6} - 1) q^{5} + q^{7} + q^{8} + 3 \zeta_{6} q^{9} - \zeta_{6} q^{10} - 2 q^{11} + 5 \zeta_{6} q^{13} + (\zeta_{6} - 1) q^{14} + (\zeta_{6} - 1) q^{16} - 3 q^{18} + (5 \zeta_{6} - 3) q^{19} + q^{20} + ( - 2 \zeta_{6} + 2) q^{22} - \zeta_{6} q^{23} + 4 \zeta_{6} q^{25} - 5 q^{26} - \zeta_{6} q^{28} - 6 \zeta_{6} q^{29} + 4 q^{31} - \zeta_{6} q^{32} + (\zeta_{6} - 1) q^{35} + ( - 3 \zeta_{6} + 3) q^{36} - 4 q^{37} + ( - 3 \zeta_{6} - 2) q^{38} + (\zeta_{6} - 1) q^{40} + ( - 2 \zeta_{6} + 2) q^{41} + ( - 8 \zeta_{6} + 8) q^{43} + 2 \zeta_{6} q^{44} - 3 q^{45} + q^{46} + q^{49} - 4 q^{50} + ( - 5 \zeta_{6} + 5) q^{52} + 2 \zeta_{6} q^{53} + ( - 2 \zeta_{6} + 2) q^{55} + q^{56} + 6 q^{58} + ( - 7 \zeta_{6} + 7) q^{59} + 7 \zeta_{6} q^{61} + (4 \zeta_{6} - 4) q^{62} + 3 \zeta_{6} q^{63} + q^{64} - 5 q^{65} - 12 \zeta_{6} q^{67} - \zeta_{6} q^{70} + ( - 15 \zeta_{6} + 15) q^{71} + 3 \zeta_{6} q^{72} + ( - 14 \zeta_{6} + 14) q^{73} + ( - 4 \zeta_{6} + 4) q^{74} + ( - 2 \zeta_{6} + 5) q^{76} - 2 q^{77} + ( - 4 \zeta_{6} + 4) q^{79} - \zeta_{6} q^{80} + (9 \zeta_{6} - 9) q^{81} + 2 \zeta_{6} q^{82} - 7 q^{83} + 8 \zeta_{6} q^{86} - 2 q^{88} + ( - 3 \zeta_{6} + 3) q^{90} + 5 \zeta_{6} q^{91} + (\zeta_{6} - 1) q^{92} + ( - 3 \zeta_{6} - 2) q^{95} + (12 \zeta_{6} - 12) q^{97} + (\zeta_{6} - 1) q^{98} - 6 \zeta_{6} q^{99} +O(q^{100})$$ q + (z - 1) * q^2 - z * q^4 + (z - 1) * q^5 + q^7 + q^8 + 3*z * q^9 - z * q^10 - 2 * q^11 + 5*z * q^13 + (z - 1) * q^14 + (z - 1) * q^16 - 3 * q^18 + (5*z - 3) * q^19 + q^20 + (-2*z + 2) * q^22 - z * q^23 + 4*z * q^25 - 5 * q^26 - z * q^28 - 6*z * q^29 + 4 * q^31 - z * q^32 + (z - 1) * q^35 + (-3*z + 3) * q^36 - 4 * q^37 + (-3*z - 2) * q^38 + (z - 1) * q^40 + (-2*z + 2) * q^41 + (-8*z + 8) * q^43 + 2*z * q^44 - 3 * q^45 + q^46 + q^49 - 4 * q^50 + (-5*z + 5) * q^52 + 2*z * q^53 + (-2*z + 2) * q^55 + q^56 + 6 * q^58 + (-7*z + 7) * q^59 + 7*z * q^61 + (4*z - 4) * q^62 + 3*z * q^63 + q^64 - 5 * q^65 - 12*z * q^67 - z * q^70 + (-15*z + 15) * q^71 + 3*z * q^72 + (-14*z + 14) * q^73 + (-4*z + 4) * q^74 + (-2*z + 5) * q^76 - 2 * q^77 + (-4*z + 4) * q^79 - z * q^80 + (9*z - 9) * q^81 + 2*z * q^82 - 7 * q^83 + 8*z * q^86 - 2 * q^88 + (-3*z + 3) * q^90 + 5*z * q^91 + (z - 1) * q^92 + (-3*z - 2) * q^95 + (12*z - 12) * q^97 + (z - 1) * q^98 - 6*z * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} - q^{4} - q^{5} + 2 q^{7} + 2 q^{8} + 3 q^{9}+O(q^{10})$$ 2 * q - q^2 - q^4 - q^5 + 2 * q^7 + 2 * q^8 + 3 * q^9 $$2 q - q^{2} - q^{4} - q^{5} + 2 q^{7} + 2 q^{8} + 3 q^{9} - q^{10} - 4 q^{11} + 5 q^{13} - q^{14} - q^{16} - 6 q^{18} - q^{19} + 2 q^{20} + 2 q^{22} - q^{23} + 4 q^{25} - 10 q^{26} - q^{28} - 6 q^{29} + 8 q^{31} - q^{32} - q^{35} + 3 q^{36} - 8 q^{37} - 7 q^{38} - q^{40} + 2 q^{41} + 8 q^{43} + 2 q^{44} - 6 q^{45} + 2 q^{46} + 2 q^{49} - 8 q^{50} + 5 q^{52} + 2 q^{53} + 2 q^{55} + 2 q^{56} + 12 q^{58} + 7 q^{59} + 7 q^{61} - 4 q^{62} + 3 q^{63} + 2 q^{64} - 10 q^{65} - 12 q^{67} - q^{70} + 15 q^{71} + 3 q^{72} + 14 q^{73} + 4 q^{74} + 8 q^{76} - 4 q^{77} + 4 q^{79} - q^{80} - 9 q^{81} + 2 q^{82} - 14 q^{83} + 8 q^{86} - 4 q^{88} + 3 q^{90} + 5 q^{91} - q^{92} - 7 q^{95} - 12 q^{97} - q^{98} - 6 q^{99}+O(q^{100})$$ 2 * q - q^2 - q^4 - q^5 + 2 * q^7 + 2 * q^8 + 3 * q^9 - q^10 - 4 * q^11 + 5 * q^13 - q^14 - q^16 - 6 * q^18 - q^19 + 2 * q^20 + 2 * q^22 - q^23 + 4 * q^25 - 10 * q^26 - q^28 - 6 * q^29 + 8 * q^31 - q^32 - q^35 + 3 * q^36 - 8 * q^37 - 7 * q^38 - q^40 + 2 * q^41 + 8 * q^43 + 2 * q^44 - 6 * q^45 + 2 * q^46 + 2 * q^49 - 8 * q^50 + 5 * q^52 + 2 * q^53 + 2 * q^55 + 2 * q^56 + 12 * q^58 + 7 * q^59 + 7 * q^61 - 4 * q^62 + 3 * q^63 + 2 * q^64 - 10 * q^65 - 12 * q^67 - q^70 + 15 * q^71 + 3 * q^72 + 14 * q^73 + 4 * q^74 + 8 * q^76 - 4 * q^77 + 4 * q^79 - q^80 - 9 * q^81 + 2 * q^82 - 14 * q^83 + 8 * q^86 - 4 * q^88 + 3 * q^90 + 5 * q^91 - q^92 - 7 * q^95 - 12 * q^97 - q^98 - 6 * q^99

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/266\mathbb{Z}\right)^\times$$.

 $$n$$ $$115$$ $$211$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
197.1
 0.5 − 0.866025i 0.5 + 0.866025i
−0.500000 0.866025i 0 −0.500000 + 0.866025i −0.500000 0.866025i 0 1.00000 1.00000 1.50000 2.59808i −0.500000 + 0.866025i
239.1 −0.500000 + 0.866025i 0 −0.500000 0.866025i −0.500000 + 0.866025i 0 1.00000 1.00000 1.50000 + 2.59808i −0.500000 0.866025i
 $$n$$: 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
19.c even 3 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 266.2.f.a 2
3.b odd 2 1 2394.2.o.i 2
19.c even 3 1 inner 266.2.f.a 2
19.c even 3 1 5054.2.a.b 1
19.d odd 6 1 5054.2.a.a 1
57.h odd 6 1 2394.2.o.i 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
266.2.f.a 2 1.a even 1 1 trivial
266.2.f.a 2 19.c even 3 1 inner
2394.2.o.i 2 3.b odd 2 1
2394.2.o.i 2 57.h odd 6 1
5054.2.a.a 1 19.d odd 6 1
5054.2.a.b 1 19.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}$$ acting on $$S_{2}^{\mathrm{new}}(266, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + T + 1$$
$3$ $$T^{2}$$
$5$ $$T^{2} + T + 1$$
$7$ $$(T - 1)^{2}$$
$11$ $$(T + 2)^{2}$$
$13$ $$T^{2} - 5T + 25$$
$17$ $$T^{2}$$
$19$ $$T^{2} + T + 19$$
$23$ $$T^{2} + T + 1$$
$29$ $$T^{2} + 6T + 36$$
$31$ $$(T - 4)^{2}$$
$37$ $$(T + 4)^{2}$$
$41$ $$T^{2} - 2T + 4$$
$43$ $$T^{2} - 8T + 64$$
$47$ $$T^{2}$$
$53$ $$T^{2} - 2T + 4$$
$59$ $$T^{2} - 7T + 49$$
$61$ $$T^{2} - 7T + 49$$
$67$ $$T^{2} + 12T + 144$$
$71$ $$T^{2} - 15T + 225$$
$73$ $$T^{2} - 14T + 196$$
$79$ $$T^{2} - 4T + 16$$
$83$ $$(T + 7)^{2}$$
$89$ $$T^{2}$$
$97$ $$T^{2} + 12T + 144$$