# Properties

 Label 231.2.j.a Level 231 Weight 2 Character orbit 231.j Analytic conductor 1.845 Analytic rank 0 Dimension 4 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$231 = 3 \cdot 7 \cdot 11$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 231.j (of order $$5$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.84454428669$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{10})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{10}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -2 + \zeta_{10} - \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{2} -\zeta_{10}^{2} q^{3} + ( 3 - 3 \zeta_{10} - 3 \zeta_{10}^{3} ) q^{4} + ( -1 - \zeta_{10}^{2} ) q^{5} + ( 1 + \zeta_{10} + \zeta_{10}^{2} ) q^{6} -\zeta_{10}^{3} q^{7} + ( 4 \zeta_{10} + \zeta_{10}^{2} + 4 \zeta_{10}^{3} ) q^{8} + ( -1 + \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{9} +O(q^{10})$$ $$q + ( -2 + \zeta_{10} - \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{2} -\zeta_{10}^{2} q^{3} + ( 3 - 3 \zeta_{10} - 3 \zeta_{10}^{3} ) q^{4} + ( -1 - \zeta_{10}^{2} ) q^{5} + ( 1 + \zeta_{10} + \zeta_{10}^{2} ) q^{6} -\zeta_{10}^{3} q^{7} + ( 4 \zeta_{10} + \zeta_{10}^{2} + 4 \zeta_{10}^{3} ) q^{8} + ( -1 + \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{9} + ( 3 + 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{10} + ( -2 + 2 \zeta_{10} - 3 \zeta_{10}^{2} + 4 \zeta_{10}^{3} ) q^{11} + ( -3 - 3 \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{12} + ( -3 - \zeta_{10} + \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{13} + ( \zeta_{10} + \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{14} + ( -1 + \zeta_{10} + \zeta_{10}^{3} ) q^{15} + ( -3 - 5 \zeta_{10} - 3 \zeta_{10}^{2} ) q^{16} + 3 \zeta_{10} q^{17} + ( 1 - \zeta_{10} - 2 \zeta_{10}^{3} ) q^{18} + 4 \zeta_{10}^{2} q^{19} + ( -6 + 3 \zeta_{10} - 3 \zeta_{10}^{2} + 6 \zeta_{10}^{3} ) q^{20} - q^{21} + ( 3 - 3 \zeta_{10} - \zeta_{10}^{2} - 6 \zeta_{10}^{3} ) q^{22} + ( -6 - 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{23} + ( 5 - \zeta_{10} + \zeta_{10}^{2} - 5 \zeta_{10}^{3} ) q^{24} + ( \zeta_{10} - 4 \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{25} + ( 7 - 7 \zeta_{10} - 10 \zeta_{10}^{3} ) q^{26} + \zeta_{10} q^{27} + ( -3 - 3 \zeta_{10}^{2} ) q^{28} -3 \zeta_{10}^{3} q^{29} + ( -2 \zeta_{10} - \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{30} + ( 1 - 3 \zeta_{10} + 3 \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{31} + ( 9 + 3 \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{32} + ( 1 + 3 \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{33} + ( -6 - 3 \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{34} + ( -1 + \zeta_{10}^{3} ) q^{35} + ( 3 \zeta_{10} + 3 \zeta_{10}^{3} ) q^{36} + ( -4 + 4 \zeta_{10} + 6 \zeta_{10}^{3} ) q^{37} + ( -4 - 4 \zeta_{10} - 4 \zeta_{10}^{2} ) q^{38} + ( 4 - \zeta_{10} + 4 \zeta_{10}^{2} ) q^{39} + ( 5 - 5 \zeta_{10} - 9 \zeta_{10}^{3} ) q^{40} + ( -5 \zeta_{10} + 5 \zeta_{10}^{2} - 5 \zeta_{10}^{3} ) q^{41} + ( 2 - \zeta_{10} + \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{42} -9 q^{43} + ( 3 + 6 \zeta_{10} + 3 \zeta_{10}^{2} + 9 \zeta_{10}^{3} ) q^{44} + ( 1 + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{45} + ( 14 - 6 \zeta_{10} + 6 \zeta_{10}^{2} - 14 \zeta_{10}^{3} ) q^{46} + ( 5 \zeta_{10} + 2 \zeta_{10}^{2} + 5 \zeta_{10}^{3} ) q^{47} + ( -3 + 3 \zeta_{10} + 8 \zeta_{10}^{3} ) q^{48} -\zeta_{10} q^{49} + ( 2 + 3 \zeta_{10} + 2 \zeta_{10}^{2} ) q^{50} -3 \zeta_{10}^{3} q^{51} + ( 9 \zeta_{10} + 12 \zeta_{10}^{2} + 9 \zeta_{10}^{3} ) q^{52} + ( -1 - 4 \zeta_{10} + 4 \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{53} + ( -2 - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{54} + ( 3 + \zeta_{10} + 2 \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{55} + ( 5 + 4 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{56} + ( 4 - 4 \zeta_{10} + 4 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{57} + ( 3 \zeta_{10} + 3 \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{58} + ( 1 - \zeta_{10} - 12 \zeta_{10}^{3} ) q^{59} + ( 3 + 3 \zeta_{10} + 3 \zeta_{10}^{2} ) q^{60} + ( 6 - 3 \zeta_{10} + 6 \zeta_{10}^{2} ) q^{61} + ( 1 - \zeta_{10} ) q^{62} + \zeta_{10}^{2} q^{63} + ( -5 - \zeta_{10} + \zeta_{10}^{2} + 5 \zeta_{10}^{3} ) q^{64} + ( 7 + 3 \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{65} + ( -7 + \zeta_{10} - 4 \zeta_{10}^{2} + 4 \zeta_{10}^{3} ) q^{66} + ( -6 \zeta_{10}^{2} + 6 \zeta_{10}^{3} ) q^{67} + ( 9 - 9 \zeta_{10}^{3} ) q^{68} + ( 2 \zeta_{10} + 4 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{69} + ( 2 - 2 \zeta_{10} - 3 \zeta_{10}^{3} ) q^{70} + ( -2 - 9 \zeta_{10} - 2 \zeta_{10}^{2} ) q^{71} + ( -4 - \zeta_{10} - 4 \zeta_{10}^{2} ) q^{72} + ( -3 + 3 \zeta_{10} + 2 \zeta_{10}^{3} ) q^{73} + ( -10 \zeta_{10} - 6 \zeta_{10}^{2} - 10 \zeta_{10}^{3} ) q^{74} + ( -3 + 4 \zeta_{10} - 4 \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{75} + ( 12 + 12 \zeta_{10}^{2} - 12 \zeta_{10}^{3} ) q^{76} + ( -1 + 2 \zeta_{10} + 2 \zeta_{10}^{2} ) q^{77} + ( -10 - 7 \zeta_{10}^{2} + 7 \zeta_{10}^{3} ) q^{78} + ( -5 + 2 \zeta_{10} - 2 \zeta_{10}^{2} + 5 \zeta_{10}^{3} ) q^{79} + ( 8 \zeta_{10} + 3 \zeta_{10}^{2} + 8 \zeta_{10}^{3} ) q^{80} -\zeta_{10}^{3} q^{81} + ( 5 + 5 \zeta_{10}^{2} ) q^{82} + ( -8 + 10 \zeta_{10} - 8 \zeta_{10}^{2} ) q^{83} + ( -3 + 3 \zeta_{10} + 3 \zeta_{10}^{3} ) q^{84} + ( -3 \zeta_{10} - 3 \zeta_{10}^{3} ) q^{85} + ( 18 - 9 \zeta_{10} + 9 \zeta_{10}^{2} - 18 \zeta_{10}^{3} ) q^{86} -3 q^{87} + ( -13 - 3 \zeta_{10} - 15 \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{88} + ( 7 - 7 \zeta_{10}^{2} + 7 \zeta_{10}^{3} ) q^{89} + ( -3 + \zeta_{10} - \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{90} + ( 4 \zeta_{10} - \zeta_{10}^{2} + 4 \zeta_{10}^{3} ) q^{91} + ( -18 + 18 \zeta_{10} + 24 \zeta_{10}^{3} ) q^{92} + ( 2 - 3 \zeta_{10} + 2 \zeta_{10}^{2} ) q^{93} + ( -12 - 7 \zeta_{10} - 12 \zeta_{10}^{2} ) q^{94} + ( 4 - 4 \zeta_{10} - 4 \zeta_{10}^{3} ) q^{95} + ( -3 \zeta_{10} - 6 \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{96} + ( 9 + 3 \zeta_{10} - 3 \zeta_{10}^{2} - 9 \zeta_{10}^{3} ) q^{97} + ( 2 + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{98} + ( \zeta_{10} - 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{2} + q^{3} + 6q^{4} - 3q^{5} + 4q^{6} - q^{7} + 7q^{8} - q^{9} + O(q^{10})$$ $$4q - 4q^{2} + q^{3} + 6q^{4} - 3q^{5} + 4q^{6} - q^{7} + 7q^{8} - q^{9} + 8q^{10} + q^{11} - 6q^{12} - 11q^{13} + q^{14} - 2q^{15} - 14q^{16} + 3q^{17} + q^{18} - 4q^{19} - 12q^{20} - 4q^{21} + 4q^{22} - 20q^{23} + 13q^{24} + 6q^{25} + 11q^{26} + q^{27} - 9q^{28} - 3q^{29} - 3q^{30} - 3q^{31} + 30q^{32} + 9q^{33} - 18q^{34} - 3q^{35} + 6q^{36} - 6q^{37} - 16q^{38} + 11q^{39} + 6q^{40} - 15q^{41} + 4q^{42} - 36q^{43} + 24q^{44} + 2q^{45} + 30q^{46} + 8q^{47} - q^{48} - q^{49} + 9q^{50} - 3q^{51} + 6q^{52} - 11q^{53} - 6q^{54} + 8q^{55} + 12q^{56} + 4q^{57} + 3q^{58} - 9q^{59} + 12q^{60} + 15q^{61} + 3q^{62} - q^{63} - 17q^{64} + 22q^{65} - 19q^{66} + 12q^{67} + 27q^{68} + 3q^{70} - 15q^{71} - 13q^{72} - 7q^{73} - 14q^{74} - q^{75} + 24q^{76} - 4q^{77} - 26q^{78} - 11q^{79} + 13q^{80} - q^{81} + 15q^{82} - 14q^{83} - 6q^{84} - 6q^{85} + 36q^{86} - 12q^{87} - 37q^{88} + 42q^{89} - 7q^{90} + 9q^{91} - 30q^{92} + 3q^{93} - 43q^{94} + 8q^{95} + 33q^{97} + 6q^{98} + q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$155$$ $$199$$ $$211$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\zeta_{10}^{3}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
64.1
 −0.309017 − 0.951057i −0.309017 + 0.951057i 0.809017 − 0.587785i 0.809017 + 0.587785i
0.118034 0.363271i 0.809017 0.587785i 1.50000 + 1.08981i −0.190983 0.587785i −0.118034 0.363271i −0.809017 0.587785i 1.19098 0.865300i 0.309017 0.951057i −0.236068
148.1 0.118034 + 0.363271i 0.809017 + 0.587785i 1.50000 1.08981i −0.190983 + 0.587785i −0.118034 + 0.363271i −0.809017 + 0.587785i 1.19098 + 0.865300i 0.309017 + 0.951057i −0.236068
169.1 −2.11803 1.53884i −0.309017 + 0.951057i 1.50000 + 4.61653i −1.30902 + 0.951057i 2.11803 1.53884i 0.309017 + 0.951057i 2.30902 7.10642i −0.809017 0.587785i 4.23607
190.1 −2.11803 + 1.53884i −0.309017 0.951057i 1.50000 4.61653i −1.30902 0.951057i 2.11803 + 1.53884i 0.309017 0.951057i 2.30902 + 7.10642i −0.809017 + 0.587785i 4.23607
 $$n$$: 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
11.c even 5 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 231.2.j.a 4
3.b odd 2 1 693.2.m.e 4
11.c even 5 1 inner 231.2.j.a 4
11.c even 5 1 2541.2.a.bf 2
11.d odd 10 1 2541.2.a.m 2
33.f even 10 1 7623.2.a.by 2
33.h odd 10 1 693.2.m.e 4
33.h odd 10 1 7623.2.a.u 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.j.a 4 1.a even 1 1 trivial
231.2.j.a 4 11.c even 5 1 inner
693.2.m.e 4 3.b odd 2 1
693.2.m.e 4 33.h odd 10 1
2541.2.a.m 2 11.d odd 10 1
2541.2.a.bf 2 11.c even 5 1
7623.2.a.u 2 33.h odd 10 1
7623.2.a.by 2 33.f even 10 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + 4 T_{2}^{3} + 6 T_{2}^{2} - T_{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(231, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 + 4 T + 4 T^{2} - 7 T^{3} - 21 T^{4} - 14 T^{5} + 16 T^{6} + 32 T^{7} + 16 T^{8}$$
$3$ $$1 - T + T^{2} - T^{3} + T^{4}$$
$5$ $$1 + 3 T - T^{2} - 3 T^{3} + 16 T^{4} - 15 T^{5} - 25 T^{6} + 375 T^{7} + 625 T^{8}$$
$7$ $$1 + T + T^{2} + T^{3} + T^{4}$$
$11$ $$1 - T - 9 T^{2} - 11 T^{3} + 121 T^{4}$$
$13$ $$1 + 11 T + 48 T^{2} + 145 T^{3} + 491 T^{4} + 1885 T^{5} + 8112 T^{6} + 24167 T^{7} + 28561 T^{8}$$
$17$ $$1 - 3 T - 8 T^{2} + 75 T^{3} - 89 T^{4} + 1275 T^{5} - 2312 T^{6} - 14739 T^{7} + 83521 T^{8}$$
$19$ $$1 + 4 T - 3 T^{2} - 88 T^{3} - 295 T^{4} - 1672 T^{5} - 1083 T^{6} + 27436 T^{7} + 130321 T^{8}$$
$23$ $$( 1 + 10 T + 66 T^{2} + 230 T^{3} + 529 T^{4} )^{2}$$
$29$ $$1 + 3 T - 20 T^{2} - 147 T^{3} + 139 T^{4} - 4263 T^{5} - 16820 T^{6} + 73167 T^{7} + 707281 T^{8}$$
$31$ $$1 + 3 T - 12 T^{2} + 131 T^{3} + 1365 T^{4} + 4061 T^{5} - 11532 T^{6} + 89373 T^{7} + 923521 T^{8}$$
$37$ $$1 + 6 T + 39 T^{2} + 352 T^{3} + 3309 T^{4} + 13024 T^{5} + 53391 T^{6} + 303918 T^{7} + 1874161 T^{8}$$
$41$ $$1 + 15 T + 59 T^{2} + 45 T^{3} + 256 T^{4} + 1845 T^{5} + 99179 T^{6} + 1033815 T^{7} + 2825761 T^{8}$$
$43$ $$( 1 + 9 T + 43 T^{2} )^{4}$$
$47$ $$1 - 8 T + 67 T^{2} - 610 T^{3} + 6231 T^{4} - 28670 T^{5} + 148003 T^{6} - 830584 T^{7} + 4879681 T^{8}$$
$53$ $$1 + 11 T + 43 T^{2} + 565 T^{3} + 6936 T^{4} + 29945 T^{5} + 120787 T^{6} + 1637647 T^{7} + 7890481 T^{8}$$
$59$ $$1 + 9 T + 77 T^{2} + 477 T^{3} + 700 T^{4} + 28143 T^{5} + 268037 T^{6} + 1848411 T^{7} + 12117361 T^{8}$$
$61$ $$1 - 15 T + 74 T^{2} - 675 T^{3} + 8491 T^{4} - 41175 T^{5} + 275354 T^{6} - 3404715 T^{7} + 13845841 T^{8}$$
$67$ $$( 1 - 6 T + 98 T^{2} - 402 T^{3} + 4489 T^{4} )^{2}$$
$71$ $$1 + 15 T + 44 T^{2} - 945 T^{3} - 13979 T^{4} - 67095 T^{5} + 221804 T^{6} + 5368665 T^{7} + 25411681 T^{8}$$
$73$ $$1 + 7 T - 39 T^{2} + 161 T^{3} + 6764 T^{4} + 11753 T^{5} - 207831 T^{6} + 2723119 T^{7} + 28398241 T^{8}$$
$79$ $$1 + 11 T - 33 T^{2} - 557 T^{3} + 80 T^{4} - 44003 T^{5} - 205953 T^{6} + 5423429 T^{7} + 38950081 T^{8}$$
$83$ $$1 + 14 T + 193 T^{2} + 2140 T^{3} + 26421 T^{4} + 177620 T^{5} + 1329577 T^{6} + 8005018 T^{7} + 47458321 T^{8}$$
$89$ $$( 1 - 21 T + 227 T^{2} - 1869 T^{3} + 7921 T^{4} )^{2}$$
$97$ $$1 - 33 T + 452 T^{2} - 4035 T^{3} + 36031 T^{4} - 391395 T^{5} + 4252868 T^{6} - 30118209 T^{7} + 88529281 T^{8}$$