Properties

 Label 201.2.f.a Level 201 Weight 2 Character orbit 201.f Analytic conductor 1.605 Analytic rank 0 Dimension 2 CM discriminant -3 Inner twists 4

Related objects

Newspace parameters

 Level: $$N$$ = $$201 = 3 \cdot 67$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 201.f (of order $$6$$, degree $$2$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$1.60499308063$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

$q$-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.

 $$f(q)$$ $$=$$ $$q + ( 1 - 2 \zeta_{6} ) q^{3} + 2 \zeta_{6} q^{4} + ( 4 - 2 \zeta_{6} ) q^{7} -3 q^{9} +O(q^{10})$$ $$q + ( 1 - 2 \zeta_{6} ) q^{3} + 2 \zeta_{6} q^{4} + ( 4 - 2 \zeta_{6} ) q^{7} -3 q^{9} + ( 4 - 2 \zeta_{6} ) q^{12} + ( 1 + \zeta_{6} ) q^{13} + ( -4 + 4 \zeta_{6} ) q^{16} + ( 8 - 8 \zeta_{6} ) q^{19} -6 \zeta_{6} q^{21} -5 q^{25} + ( -3 + 6 \zeta_{6} ) q^{27} + ( 4 + 4 \zeta_{6} ) q^{28} + ( -10 + 5 \zeta_{6} ) q^{31} -6 \zeta_{6} q^{36} + ( -10 + 10 \zeta_{6} ) q^{37} + ( 3 - 3 \zeta_{6} ) q^{39} + ( -7 + 14 \zeta_{6} ) q^{43} + ( 4 + 4 \zeta_{6} ) q^{48} + ( 5 - 5 \zeta_{6} ) q^{49} + ( -2 + 4 \zeta_{6} ) q^{52} + ( -8 - 8 \zeta_{6} ) q^{57} + ( -5 - 5 \zeta_{6} ) q^{61} + ( -12 + 6 \zeta_{6} ) q^{63} -8 q^{64} + ( 2 - 9 \zeta_{6} ) q^{67} + ( 17 - 17 \zeta_{6} ) q^{73} + ( -5 + 10 \zeta_{6} ) q^{75} + 16 q^{76} + ( 6 - 3 \zeta_{6} ) q^{79} + 9 q^{81} + ( 12 - 12 \zeta_{6} ) q^{84} + 6 q^{91} + 15 \zeta_{6} q^{93} + ( -3 - 3 \zeta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{4} + 6q^{7} - 6q^{9} + O(q^{10})$$ $$2q + 2q^{4} + 6q^{7} - 6q^{9} + 6q^{12} + 3q^{13} - 4q^{16} + 8q^{19} - 6q^{21} - 10q^{25} + 12q^{28} - 15q^{31} - 6q^{36} - 10q^{37} + 3q^{39} + 12q^{48} + 5q^{49} - 24q^{57} - 15q^{61} - 18q^{63} - 16q^{64} - 5q^{67} + 17q^{73} + 32q^{76} + 9q^{79} + 18q^{81} + 12q^{84} + 12q^{91} + 15q^{93} - 9q^{97} + O(q^{100})$$

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/201\mathbb{Z}\right)^\times$$.

 $$n$$ $$68$$ $$136$$ $$\chi(n)$$ $$-1$$ $$\zeta_{6}$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
38.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 1.73205i 1.00000 + 1.73205i 0 0 3.00000 1.73205i 0 −3.00000 0
164.1 0 1.73205i 1.00000 1.73205i 0 0 3.00000 + 1.73205i 0 −3.00000 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
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
67.d odd 6 1 inner
201.f even 6 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 201.2.f.a 2
3.b odd 2 1 CM 201.2.f.a 2
67.d odd 6 1 inner 201.2.f.a 2
201.f even 6 1 inner 201.2.f.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
201.2.f.a 2 1.a even 1 1 trivial
201.2.f.a 2 3.b odd 2 1 CM
201.2.f.a 2 67.d odd 6 1 inner
201.2.f.a 2 201.f even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}$$ acting on $$S_{2}^{\mathrm{new}}(201, [\chi])$$.

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 - 2 T^{2} + 4 T^{4}$$
$3$ $$1 + 3 T^{2}$$
$5$ $$( 1 + 5 T^{2} )^{2}$$
$7$ $$( 1 - 5 T + 7 T^{2} )( 1 - T + 7 T^{2} )$$
$11$ $$1 - 11 T^{2} + 121 T^{4}$$
$13$ $$( 1 - 5 T + 13 T^{2} )( 1 + 2 T + 13 T^{2} )$$
$17$ $$1 + 17 T^{2} + 289 T^{4}$$
$19$ $$( 1 - 7 T + 19 T^{2} )( 1 - T + 19 T^{2} )$$
$23$ $$1 + 23 T^{2} + 529 T^{4}$$
$29$ $$1 + 29 T^{2} + 841 T^{4}$$
$31$ $$( 1 + 4 T + 31 T^{2} )( 1 + 11 T + 31 T^{2} )$$
$37$ $$( 1 - T + 37 T^{2} )( 1 + 11 T + 37 T^{2} )$$
$41$ $$1 - 41 T^{2} + 1681 T^{4}$$
$43$ $$( 1 - 5 T + 43 T^{2} )( 1 + 5 T + 43 T^{2} )$$
$47$ $$1 + 47 T^{2} + 2209 T^{4}$$
$53$ $$( 1 + 53 T^{2} )^{2}$$
$59$ $$( 1 - 59 T^{2} )^{2}$$
$61$ $$( 1 + T + 61 T^{2} )( 1 + 14 T + 61 T^{2} )$$
$67$ $$1 + 5 T + 67 T^{2}$$
$71$ $$1 + 71 T^{2} + 5041 T^{4}$$
$73$ $$( 1 - 10 T + 73 T^{2} )( 1 - 7 T + 73 T^{2} )$$
$79$ $$( 1 - 13 T + 79 T^{2} )( 1 + 4 T + 79 T^{2} )$$
$83$ $$1 + 83 T^{2} + 6889 T^{4}$$
$89$ $$( 1 - 89 T^{2} )^{2}$$
$97$ $$( 1 - 5 T + 97 T^{2} )( 1 + 14 T + 97 T^{2} )$$