# Properties

 Label 231.2.i.c Level 231 Weight 2 Character orbit 231.i Analytic conductor 1.845 Analytic rank 0 Dimension 2 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$231 = 3 \cdot 7 \cdot 11$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 231.i (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.84454428669$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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.

 $$f(q)$$ $$=$$ $$q + 2 \zeta_{6} q^{2} + ( 1 - \zeta_{6} ) q^{3} + ( -2 + 2 \zeta_{6} ) q^{4} + 2 q^{6} + ( 2 + \zeta_{6} ) q^{7} -\zeta_{6} q^{9} +O(q^{10})$$ $$q + 2 \zeta_{6} q^{2} + ( 1 - \zeta_{6} ) q^{3} + ( -2 + 2 \zeta_{6} ) q^{4} + 2 q^{6} + ( 2 + \zeta_{6} ) q^{7} -\zeta_{6} q^{9} + ( 1 - \zeta_{6} ) q^{11} + 2 \zeta_{6} q^{12} -5 q^{13} + ( -2 + 6 \zeta_{6} ) q^{14} + 4 \zeta_{6} q^{16} + ( -6 + 6 \zeta_{6} ) q^{17} + ( 2 - 2 \zeta_{6} ) q^{18} -7 \zeta_{6} q^{19} + ( 3 - 2 \zeta_{6} ) q^{21} + 2 q^{22} + 4 \zeta_{6} q^{23} + ( 5 - 5 \zeta_{6} ) q^{25} -10 \zeta_{6} q^{26} - q^{27} + ( -6 + 4 \zeta_{6} ) q^{28} -2 q^{29} + ( 7 - 7 \zeta_{6} ) q^{31} + ( -8 + 8 \zeta_{6} ) q^{32} -\zeta_{6} q^{33} -12 q^{34} + 2 q^{36} -7 \zeta_{6} q^{37} + ( 14 - 14 \zeta_{6} ) q^{38} + ( -5 + 5 \zeta_{6} ) q^{39} + 4 q^{41} + ( 4 + 2 \zeta_{6} ) q^{42} -9 q^{43} + 2 \zeta_{6} q^{44} + ( -8 + 8 \zeta_{6} ) q^{46} -6 \zeta_{6} q^{47} + 4 q^{48} + ( 3 + 5 \zeta_{6} ) q^{49} + 10 q^{50} + 6 \zeta_{6} q^{51} + ( 10 - 10 \zeta_{6} ) q^{52} + ( 2 - 2 \zeta_{6} ) q^{53} -2 \zeta_{6} q^{54} -7 q^{57} -4 \zeta_{6} q^{58} + ( -12 + 12 \zeta_{6} ) q^{59} + 2 \zeta_{6} q^{61} + 14 q^{62} + ( 1 - 3 \zeta_{6} ) q^{63} -8 q^{64} + ( 2 - 2 \zeta_{6} ) q^{66} + ( -7 + 7 \zeta_{6} ) q^{67} -12 \zeta_{6} q^{68} + 4 q^{69} + 8 q^{71} + ( 5 - 5 \zeta_{6} ) q^{73} + ( 14 - 14 \zeta_{6} ) q^{74} -5 \zeta_{6} q^{75} + 14 q^{76} + ( 3 - 2 \zeta_{6} ) q^{77} -10 q^{78} + 11 \zeta_{6} q^{79} + ( -1 + \zeta_{6} ) q^{81} + 8 \zeta_{6} q^{82} -4 q^{83} + ( -2 + 6 \zeta_{6} ) q^{84} -18 \zeta_{6} q^{86} + ( -2 + 2 \zeta_{6} ) q^{87} -6 \zeta_{6} q^{89} + ( -10 - 5 \zeta_{6} ) q^{91} -8 q^{92} -7 \zeta_{6} q^{93} + ( 12 - 12 \zeta_{6} ) q^{94} + 8 \zeta_{6} q^{96} + 2 q^{97} + ( -10 + 16 \zeta_{6} ) q^{98} - q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{2} + q^{3} - 2q^{4} + 4q^{6} + 5q^{7} - q^{9} + O(q^{10})$$ $$2q + 2q^{2} + q^{3} - 2q^{4} + 4q^{6} + 5q^{7} - q^{9} + q^{11} + 2q^{12} - 10q^{13} + 2q^{14} + 4q^{16} - 6q^{17} + 2q^{18} - 7q^{19} + 4q^{21} + 4q^{22} + 4q^{23} + 5q^{25} - 10q^{26} - 2q^{27} - 8q^{28} - 4q^{29} + 7q^{31} - 8q^{32} - q^{33} - 24q^{34} + 4q^{36} - 7q^{37} + 14q^{38} - 5q^{39} + 8q^{41} + 10q^{42} - 18q^{43} + 2q^{44} - 8q^{46} - 6q^{47} + 8q^{48} + 11q^{49} + 20q^{50} + 6q^{51} + 10q^{52} + 2q^{53} - 2q^{54} - 14q^{57} - 4q^{58} - 12q^{59} + 2q^{61} + 28q^{62} - q^{63} - 16q^{64} + 2q^{66} - 7q^{67} - 12q^{68} + 8q^{69} + 16q^{71} + 5q^{73} + 14q^{74} - 5q^{75} + 28q^{76} + 4q^{77} - 20q^{78} + 11q^{79} - q^{81} + 8q^{82} - 8q^{83} + 2q^{84} - 18q^{86} - 2q^{87} - 6q^{89} - 25q^{91} - 16q^{92} - 7q^{93} + 12q^{94} + 8q^{96} + 4q^{97} - 4q^{98} - 2q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/231\mathbb{Z}\right)^\times$$.

 $$n$$ $$155$$ $$199$$ $$211$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
67.1
 0.5 + 0.866025i 0.5 − 0.866025i
1.00000 + 1.73205i 0.500000 0.866025i −1.00000 + 1.73205i 0 2.00000 2.50000 + 0.866025i 0 −0.500000 0.866025i 0
100.1 1.00000 1.73205i 0.500000 + 0.866025i −1.00000 1.73205i 0 2.00000 2.50000 0.866025i 0 −0.500000 + 0.866025i 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
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 231.2.i.c 2
3.b odd 2 1 693.2.i.a 2
7.c even 3 1 inner 231.2.i.c 2
7.c even 3 1 1617.2.a.a 1
7.d odd 6 1 1617.2.a.b 1
21.g even 6 1 4851.2.a.s 1
21.h odd 6 1 693.2.i.a 2
21.h odd 6 1 4851.2.a.r 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.i.c 2 1.a even 1 1 trivial
231.2.i.c 2 7.c even 3 1 inner
693.2.i.a 2 3.b odd 2 1
693.2.i.a 2 21.h odd 6 1
1617.2.a.a 1 7.c even 3 1
1617.2.a.b 1 7.d odd 6 1
4851.2.a.r 1 21.h odd 6 1
4851.2.a.s 1 21.g even 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 $$S_{2}^{\mathrm{new}}(231, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 - 2 T + 2 T^{2} - 4 T^{3} + 4 T^{4}$$
$3$ $$1 - T + T^{2}$$
$5$ $$1 - 5 T^{2} + 25 T^{4}$$
$7$ $$1 - 5 T + 7 T^{2}$$
$11$ $$1 - T + T^{2}$$
$13$ $$( 1 + 5 T + 13 T^{2} )^{2}$$
$17$ $$1 + 6 T + 19 T^{2} + 102 T^{3} + 289 T^{4}$$
$19$ $$( 1 - T + 19 T^{2} )( 1 + 8 T + 19 T^{2} )$$
$23$ $$1 - 4 T - 7 T^{2} - 92 T^{3} + 529 T^{4}$$
$29$ $$( 1 + 2 T + 29 T^{2} )^{2}$$
$31$ $$( 1 - 11 T + 31 T^{2} )( 1 + 4 T + 31 T^{2} )$$
$37$ $$1 + 7 T + 12 T^{2} + 259 T^{3} + 1369 T^{4}$$
$41$ $$( 1 - 4 T + 41 T^{2} )^{2}$$
$43$ $$( 1 + 9 T + 43 T^{2} )^{2}$$
$47$ $$1 + 6 T - 11 T^{2} + 282 T^{3} + 2209 T^{4}$$
$53$ $$1 - 2 T - 49 T^{2} - 106 T^{3} + 2809 T^{4}$$
$59$ $$1 + 12 T + 85 T^{2} + 708 T^{3} + 3481 T^{4}$$
$61$ $$1 - 2 T - 57 T^{2} - 122 T^{3} + 3721 T^{4}$$
$67$ $$1 + 7 T - 18 T^{2} + 469 T^{3} + 4489 T^{4}$$
$71$ $$( 1 - 8 T + 71 T^{2} )^{2}$$
$73$ $$1 - 5 T - 48 T^{2} - 365 T^{3} + 5329 T^{4}$$
$79$ $$1 - 11 T + 42 T^{2} - 869 T^{3} + 6241 T^{4}$$
$83$ $$( 1 + 4 T + 83 T^{2} )^{2}$$
$89$ $$1 + 6 T - 53 T^{2} + 534 T^{3} + 7921 T^{4}$$
$97$ $$( 1 - 2 T + 97 T^{2} )^{2}$$