# Properties

 Label 201.2.e.a Level 201 Weight 2 Character orbit 201.e Analytic conductor 1.605 Analytic rank 1 Dimension 2 CM no Inner twists 2

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$1.60499308063$$ Analytic rank: $$1$$ 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: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( -2 + 2 \zeta_{6} ) q^{2} + q^{3} -2 \zeta_{6} q^{4} -4 q^{5} + ( -2 + 2 \zeta_{6} ) q^{6} -4 \zeta_{6} q^{7} + q^{9} +O(q^{10})$$ $$q + ( -2 + 2 \zeta_{6} ) q^{2} + q^{3} -2 \zeta_{6} q^{4} -4 q^{5} + ( -2 + 2 \zeta_{6} ) q^{6} -4 \zeta_{6} q^{7} + q^{9} + ( 8 - 8 \zeta_{6} ) q^{10} -2 \zeta_{6} q^{11} -2 \zeta_{6} q^{12} + ( -5 + 5 \zeta_{6} ) q^{13} + 8 q^{14} -4 q^{15} + ( 4 - 4 \zeta_{6} ) q^{16} + ( -6 + 6 \zeta_{6} ) q^{17} + ( -2 + 2 \zeta_{6} ) q^{18} + ( -4 + 4 \zeta_{6} ) q^{19} + 8 \zeta_{6} q^{20} -4 \zeta_{6} q^{21} + 4 q^{22} + ( 6 - 6 \zeta_{6} ) q^{23} + 11 q^{25} -10 \zeta_{6} q^{26} + q^{27} + ( -8 + 8 \zeta_{6} ) q^{28} + 2 \zeta_{6} q^{29} + ( 8 - 8 \zeta_{6} ) q^{30} -5 \zeta_{6} q^{31} + 8 \zeta_{6} q^{32} -2 \zeta_{6} q^{33} -12 \zeta_{6} q^{34} + 16 \zeta_{6} q^{35} -2 \zeta_{6} q^{36} + ( -2 + 2 \zeta_{6} ) q^{37} -8 \zeta_{6} q^{38} + ( -5 + 5 \zeta_{6} ) q^{39} + 8 q^{42} -11 q^{43} + ( -4 + 4 \zeta_{6} ) q^{44} -4 q^{45} + 12 \zeta_{6} q^{46} -8 \zeta_{6} q^{47} + ( 4 - 4 \zeta_{6} ) q^{48} + ( -9 + 9 \zeta_{6} ) q^{49} + ( -22 + 22 \zeta_{6} ) q^{50} + ( -6 + 6 \zeta_{6} ) q^{51} + 10 q^{52} -2 q^{53} + ( -2 + 2 \zeta_{6} ) q^{54} + 8 \zeta_{6} q^{55} + ( -4 + 4 \zeta_{6} ) q^{57} -4 q^{58} + 8 \zeta_{6} q^{60} + ( 5 - 5 \zeta_{6} ) q^{61} + 10 q^{62} -4 \zeta_{6} q^{63} -8 q^{64} + ( 20 - 20 \zeta_{6} ) q^{65} + 4 q^{66} + ( -2 + 9 \zeta_{6} ) q^{67} + 12 q^{68} + ( 6 - 6 \zeta_{6} ) q^{69} -32 q^{70} + ( -5 + 5 \zeta_{6} ) q^{73} -4 \zeta_{6} q^{74} + 11 q^{75} + 8 q^{76} + ( -8 + 8 \zeta_{6} ) q^{77} -10 \zeta_{6} q^{78} -\zeta_{6} q^{79} + ( -16 + 16 \zeta_{6} ) q^{80} + q^{81} + ( 8 - 8 \zeta_{6} ) q^{83} + ( -8 + 8 \zeta_{6} ) q^{84} + ( 24 - 24 \zeta_{6} ) q^{85} + ( 22 - 22 \zeta_{6} ) q^{86} + 2 \zeta_{6} q^{87} -8 q^{89} + ( 8 - 8 \zeta_{6} ) q^{90} + 20 q^{91} -12 q^{92} -5 \zeta_{6} q^{93} + 16 q^{94} + ( 16 - 16 \zeta_{6} ) q^{95} + 8 \zeta_{6} q^{96} + ( -9 + 9 \zeta_{6} ) q^{97} -18 \zeta_{6} q^{98} -2 \zeta_{6} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} + 2q^{3} - 2q^{4} - 8q^{5} - 2q^{6} - 4q^{7} + 2q^{9} + O(q^{10})$$ $$2q - 2q^{2} + 2q^{3} - 2q^{4} - 8q^{5} - 2q^{6} - 4q^{7} + 2q^{9} + 8q^{10} - 2q^{11} - 2q^{12} - 5q^{13} + 16q^{14} - 8q^{15} + 4q^{16} - 6q^{17} - 2q^{18} - 4q^{19} + 8q^{20} - 4q^{21} + 8q^{22} + 6q^{23} + 22q^{25} - 10q^{26} + 2q^{27} - 8q^{28} + 2q^{29} + 8q^{30} - 5q^{31} + 8q^{32} - 2q^{33} - 12q^{34} + 16q^{35} - 2q^{36} - 2q^{37} - 8q^{38} - 5q^{39} + 16q^{42} - 22q^{43} - 4q^{44} - 8q^{45} + 12q^{46} - 8q^{47} + 4q^{48} - 9q^{49} - 22q^{50} - 6q^{51} + 20q^{52} - 4q^{53} - 2q^{54} + 8q^{55} - 4q^{57} - 8q^{58} + 8q^{60} + 5q^{61} + 20q^{62} - 4q^{63} - 16q^{64} + 20q^{65} + 8q^{66} + 5q^{67} + 24q^{68} + 6q^{69} - 64q^{70} - 5q^{73} - 4q^{74} + 22q^{75} + 16q^{76} - 8q^{77} - 10q^{78} - q^{79} - 16q^{80} + 2q^{81} + 8q^{83} - 8q^{84} + 24q^{85} + 22q^{86} + 2q^{87} - 16q^{89} + 8q^{90} + 40q^{91} - 24q^{92} - 5q^{93} + 32q^{94} + 16q^{95} + 8q^{96} - 9q^{97} - 18q^{98} - 2q^{99} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
37.1
 0.5 − 0.866025i 0.5 + 0.866025i
−1.00000 1.73205i 1.00000 −1.00000 + 1.73205i −4.00000 −1.00000 1.73205i −2.00000 + 3.46410i 0 1.00000 4.00000 + 6.92820i
163.1 −1.00000 + 1.73205i 1.00000 −1.00000 1.73205i −4.00000 −1.00000 + 1.73205i −2.00000 3.46410i 0 1.00000 4.00000 6.92820i
 $$n$$: 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
67.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 201.2.e.a 2
3.b odd 2 1 603.2.g.b 2
67.c even 3 1 inner 201.2.e.a 2
201.g odd 6 1 603.2.g.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
201.2.e.a 2 1.a even 1 1 trivial
201.2.e.a 2 67.c even 3 1 inner
603.2.g.b 2 3.b odd 2 1
603.2.g.b 2 201.g odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 2 T_{2} + 4$$ acting on $$S_{2}^{\mathrm{new}}(201, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 + 2 T + 2 T^{2} + 4 T^{3} + 4 T^{4}$$
$3$ $$( 1 - T )^{2}$$
$5$ $$( 1 + 4 T + 5 T^{2} )^{2}$$
$7$ $$( 1 - T + 7 T^{2} )( 1 + 5 T + 7 T^{2} )$$
$11$ $$1 + 2 T - 7 T^{2} + 22 T^{3} + 121 T^{4}$$
$13$ $$( 1 - 2 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} )$$
$17$ $$1 + 6 T + 19 T^{2} + 102 T^{3} + 289 T^{4}$$
$19$ $$1 + 4 T - 3 T^{2} + 76 T^{3} + 361 T^{4}$$
$23$ $$1 - 6 T + 13 T^{2} - 138 T^{3} + 529 T^{4}$$
$29$ $$1 - 2 T - 25 T^{2} - 58 T^{3} + 841 T^{4}$$
$31$ $$1 + 5 T - 6 T^{2} + 155 T^{3} + 961 T^{4}$$
$37$ $$1 + 2 T - 33 T^{2} + 74 T^{3} + 1369 T^{4}$$
$41$ $$1 - 41 T^{2} + 1681 T^{4}$$
$43$ $$( 1 + 11 T + 43 T^{2} )^{2}$$
$47$ $$1 + 8 T + 17 T^{2} + 376 T^{3} + 2209 T^{4}$$
$53$ $$( 1 + 2 T + 53 T^{2} )^{2}$$
$59$ $$( 1 + 59 T^{2} )^{2}$$
$61$ $$1 - 5 T - 36 T^{2} - 305 T^{3} + 3721 T^{4}$$
$67$ $$1 - 5 T + 67 T^{2}$$
$71$ $$1 - 71 T^{2} + 5041 T^{4}$$
$73$ $$1 + 5 T - 48 T^{2} + 365 T^{3} + 5329 T^{4}$$
$79$ $$1 + T - 78 T^{2} + 79 T^{3} + 6241 T^{4}$$
$83$ $$1 - 8 T - 19 T^{2} - 664 T^{3} + 6889 T^{4}$$
$89$ $$( 1 + 8 T + 89 T^{2} )^{2}$$
$97$ $$1 + 9 T - 16 T^{2} + 873 T^{3} + 9409 T^{4}$$