# Properties

 Label 201.1.g.a Level 201 Weight 1 Character orbit 201.g Analytic conductor 0.100 Analytic rank 0 Dimension 4 Projective image $$A_{4}$$ CM/RM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$201 = 3 \cdot 67$$ Weight: $$k$$ = $$1$$ Character orbit: $$[\chi]$$ = 201.g (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.100312067539$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image $$A_{4}$$ Projective field Galois closure of 4.0.40401.1 Artin image $\SL(2,3):C_2$ Artin field Galois closure of $$\mathbb{Q}[x]/(x^{16} - \cdots)$$

## $q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q + \zeta_{12}^{5} q^{2} + \zeta_{12}^{3} q^{3} -\zeta_{12}^{2} q^{6} + \zeta_{12}^{4} q^{7} -\zeta_{12}^{3} q^{8} - q^{9} +O(q^{10})$$ $$q + \zeta_{12}^{5} q^{2} + \zeta_{12}^{3} q^{3} -\zeta_{12}^{2} q^{6} + \zeta_{12}^{4} q^{7} -\zeta_{12}^{3} q^{8} - q^{9} + \zeta_{12} q^{11} -\zeta_{12}^{2} q^{13} -\zeta_{12}^{3} q^{14} + \zeta_{12}^{2} q^{16} + \zeta_{12}^{5} q^{17} -\zeta_{12}^{5} q^{18} + \zeta_{12}^{2} q^{19} -\zeta_{12} q^{21} - q^{22} -\zeta_{12}^{5} q^{23} + q^{24} + q^{25} + \zeta_{12} q^{26} -\zeta_{12}^{3} q^{27} + \zeta_{12} q^{29} -\zeta_{12}^{4} q^{31} + \zeta_{12}^{4} q^{33} -\zeta_{12}^{4} q^{34} -\zeta_{12}^{2} q^{37} -\zeta_{12} q^{38} -\zeta_{12}^{5} q^{39} -\zeta_{12} q^{41} + q^{42} + \zeta_{12}^{4} q^{46} + \zeta_{12} q^{47} + \zeta_{12}^{5} q^{48} + \zeta_{12}^{5} q^{50} -\zeta_{12}^{2} q^{51} + \zeta_{12}^{2} q^{54} + \zeta_{12} q^{56} + \zeta_{12}^{5} q^{57} - q^{58} -\zeta_{12}^{2} q^{61} + \zeta_{12}^{3} q^{62} -\zeta_{12}^{4} q^{63} - q^{64} -\zeta_{12}^{3} q^{66} - q^{67} + \zeta_{12}^{2} q^{69} -\zeta_{12} q^{71} + \zeta_{12}^{3} q^{72} + \zeta_{12}^{2} q^{73} + \zeta_{12} q^{74} + \zeta_{12}^{3} q^{75} + \zeta_{12}^{5} q^{77} + \zeta_{12}^{4} q^{78} + \zeta_{12}^{4} q^{79} + q^{81} + q^{82} + \zeta_{12}^{5} q^{83} + \zeta_{12}^{4} q^{87} -\zeta_{12}^{4} q^{88} -2 \zeta_{12}^{3} q^{89} + q^{91} + \zeta_{12} q^{93} - q^{94} -\zeta_{12}^{2} q^{97} -\zeta_{12} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 2q^{6} - 2q^{7} - 4q^{9} + O(q^{10})$$ $$4q - 2q^{6} - 2q^{7} - 4q^{9} - 2q^{13} + 2q^{16} + 2q^{19} - 4q^{22} + 4q^{24} + 4q^{25} + 2q^{31} - 2q^{33} + 2q^{34} - 2q^{37} + 4q^{42} - 2q^{46} - 2q^{51} + 2q^{54} - 4q^{58} - 2q^{61} + 2q^{63} - 4q^{64} - 4q^{67} + 2q^{69} + 2q^{73} - 2q^{78} - 2q^{79} + 4q^{81} + 4q^{82} - 2q^{87} + 2q^{88} + 4q^{91} - 4q^{94} - 2q^{97} + O(q^{100})$$

## Character values

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

 $$n$$ $$68$$ $$136$$ $$\chi(n)$$ $$-1$$ $$\zeta_{12}^{4}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
29.1
 0.866025 − 0.500000i −0.866025 + 0.500000i 0.866025 + 0.500000i −0.866025 − 0.500000i
−0.866025 0.500000i 1.00000i 0 0 −0.500000 + 0.866025i −0.500000 0.866025i 1.00000i −1.00000 0
29.2 0.866025 + 0.500000i 1.00000i 0 0 −0.500000 + 0.866025i −0.500000 0.866025i 1.00000i −1.00000 0
104.1 −0.866025 + 0.500000i 1.00000i 0 0 −0.500000 0.866025i −0.500000 + 0.866025i 1.00000i −1.00000 0
104.2 0.866025 0.500000i 1.00000i 0 0 −0.500000 0.866025i −0.500000 + 0.866025i 1.00000i −1.00000 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 inner
67.c even 3 1 inner
201.g odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 201.1.g.a 4
3.b odd 2 1 inner 201.1.g.a 4
4.b odd 2 1 3216.1.bb.b 4
12.b even 2 1 3216.1.bb.b 4
67.c even 3 1 inner 201.1.g.a 4
201.g odd 6 1 inner 201.1.g.a 4
268.g odd 6 1 3216.1.bb.b 4
804.l even 6 1 3216.1.bb.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
201.1.g.a 4 1.a even 1 1 trivial
201.1.g.a 4 3.b odd 2 1 inner
201.1.g.a 4 67.c even 3 1 inner
201.1.g.a 4 201.g odd 6 1 inner
3216.1.bb.b 4 4.b odd 2 1
3216.1.bb.b 4 12.b even 2 1
3216.1.bb.b 4 268.g odd 6 1
3216.1.bb.b 4 804.l even 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(201, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$
$3$ $$( 1 + T^{2} )^{2}$$
$5$ $$( 1 - T )^{4}( 1 + T )^{4}$$
$7$ $$( 1 + T )^{4}( 1 - T + T^{2} )^{2}$$
$11$ $$( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$
$13$ $$( 1 + T )^{4}( 1 - T + T^{2} )^{2}$$
$17$ $$( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$
$19$ $$( 1 - T )^{4}( 1 + T + T^{2} )^{2}$$
$23$ $$( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$
$29$ $$( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$
$31$ $$( 1 - T )^{4}( 1 + T + T^{2} )^{2}$$
$37$ $$( 1 + T )^{4}( 1 - T + T^{2} )^{2}$$
$41$ $$( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$
$43$ $$( 1 + T^{2} )^{4}$$
$47$ $$( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$
$53$ $$( 1 - T )^{4}( 1 + T )^{4}$$
$59$ $$( 1 - T )^{4}( 1 + T )^{4}$$
$61$ $$( 1 + T )^{4}( 1 - T + T^{2} )^{2}$$
$67$ $$( 1 + T )^{4}$$
$71$ $$( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$
$73$ $$( 1 - T )^{4}( 1 + T + T^{2} )^{2}$$
$79$ $$( 1 + T )^{4}( 1 - T + T^{2} )^{2}$$
$83$ $$( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$
$89$ $$( 1 + T^{2} )^{4}$$
$97$ $$( 1 + T )^{4}( 1 - T + T^{2} )^{2}$$