# Properties

 Label 231.2.j.b 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: $$\Z[a_1, a_2]$$ 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.

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

## 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
−0.500000 + 1.53884i 0.809017 0.587785i −0.500000 0.363271i 0.190983 + 0.587785i 0.500000 + 1.53884i 0.809017 + 0.587785i −1.80902 + 1.31433i 0.309017 0.951057i −1.00000
148.1 −0.500000 1.53884i 0.809017 + 0.587785i −0.500000 + 0.363271i 0.190983 0.587785i 0.500000 1.53884i 0.809017 0.587785i −1.80902 1.31433i 0.309017 + 0.951057i −1.00000
169.1 −0.500000 0.363271i −0.309017 + 0.951057i −0.500000 1.53884i 1.30902 0.951057i 0.500000 0.363271i −0.309017 0.951057i −0.690983 + 2.12663i −0.809017 0.587785i −1.00000
190.1 −0.500000 + 0.363271i −0.309017 0.951057i −0.500000 + 1.53884i 1.30902 + 0.951057i 0.500000 + 0.363271i −0.309017 + 0.951057i −0.690983 2.12663i −0.809017 + 0.587785i −1.00000
 $$n$$: 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.b 4
3.b odd 2 1 693.2.m.d 4
11.c even 5 1 inner 231.2.j.b 4
11.c even 5 1 2541.2.a.p 2
11.d odd 10 1 2541.2.a.x 2
33.f even 10 1 7623.2.a.z 2
33.h odd 10 1 693.2.m.d 4
33.h odd 10 1 7623.2.a.bo 2

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

## Hecke kernels

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

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 + 2 T + 2 T^{2} + 5 T^{3} + 11 T^{4} + 10 T^{5} + 8 T^{6} + 16 T^{7} + 16 T^{8}$$
$3$ $$1 - T + T^{2} - T^{3} + T^{4}$$
$5$ $$1 - 3 T - T^{2} + 3 T^{3} + 16 T^{4} + 15 T^{5} - 25 T^{6} - 375 T^{7} + 625 T^{8}$$
$7$ $$1 - T + T^{2} - T^{3} + T^{4}$$
$11$ $$1 - 9 T + 41 T^{2} - 99 T^{3} + 121 T^{4}$$
$13$ $$1 + 7 T + 6 T^{2} - 49 T^{3} - 181 T^{4} - 637 T^{5} + 1014 T^{6} + 15379 T^{7} + 28561 T^{8}$$
$17$ $$1 - 9 T + 44 T^{2} - 243 T^{3} + 1279 T^{4} - 4131 T^{5} + 12716 T^{6} - 44217 T^{7} + 83521 T^{8}$$
$19$ $$1 - 8 T + 45 T^{2} - 268 T^{3} + 1529 T^{4} - 5092 T^{5} + 16245 T^{6} - 54872 T^{7} + 130321 T^{8}$$
$23$ $$( 1 + 2 T + 2 T^{2} + 46 T^{3} + 529 T^{4} )^{2}$$
$29$ $$1 + 7 T - 10 T^{2} - 113 T^{3} + 59 T^{4} - 3277 T^{5} - 8410 T^{6} + 170723 T^{7} + 707281 T^{8}$$
$31$ $$1 + 3 T - 22 T^{2} - 159 T^{3} + 205 T^{4} - 4929 T^{5} - 21142 T^{6} + 89373 T^{7} + 923521 T^{8}$$
$37$ $$1 + 6 T - T^{2} - 228 T^{3} - 1331 T^{4} - 8436 T^{5} - 1369 T^{6} + 303918 T^{7} + 1874161 T^{8}$$
$41$ $$1 + T + 35 T^{2} + 99 T^{3} + 1904 T^{4} + 4059 T^{5} + 58835 T^{6} + 68921 T^{7} + 2825761 T^{8}$$
$43$ $$( 1 + 14 T + 115 T^{2} + 602 T^{3} + 1849 T^{4} )^{2}$$
$47$ $$1 - 16 T + 59 T^{2} + 738 T^{3} - 9721 T^{4} + 34686 T^{5} + 130331 T^{6} - 1661168 T^{7} + 4879681 T^{8}$$
$53$ $$1 - 25 T + 257 T^{2} - 1635 T^{3} + 10244 T^{4} - 86655 T^{5} + 721913 T^{6} - 3721925 T^{7} + 7890481 T^{8}$$
$59$ $$1 - T - 43 T^{2} + 347 T^{3} + 2540 T^{4} + 20473 T^{5} - 149683 T^{6} - 205379 T^{7} + 12117361 T^{8}$$
$61$ $$1 - 19 T + 180 T^{2} - 1781 T^{3} + 17099 T^{4} - 108641 T^{5} + 669780 T^{6} - 4312639 T^{7} + 13845841 T^{8}$$
$67$ $$( 1 - 22 T + 250 T^{2} - 1474 T^{3} + 4489 T^{4} )^{2}$$
$71$ $$1 + T + 70 T^{2} + 119 T^{3} + 5709 T^{4} + 8449 T^{5} + 352870 T^{6} + 357911 T^{7} + 25411681 T^{8}$$
$73$ $$1 + 11 T + 3 T^{2} - 815 T^{3} - 8464 T^{4} - 59495 T^{5} + 15987 T^{6} + 4279187 T^{7} + 28398241 T^{8}$$
$79$ $$1 - 13 T + 15 T^{2} - 293 T^{3} + 8624 T^{4} - 23147 T^{5} + 93615 T^{6} - 6409507 T^{7} + 38950081 T^{8}$$
$83$ $$1 + 14 T - 7 T^{2} - 640 T^{3} - 2059 T^{4} - 53120 T^{5} - 48223 T^{6} + 8005018 T^{7} + 47458321 T^{8}$$
$89$ $$( 1 + 13 T + 159 T^{2} + 1157 T^{3} + 7921 T^{4} )^{2}$$
$97$ $$1 + 3 T + 182 T^{2} - 345 T^{3} + 16591 T^{4} - 33465 T^{5} + 1712438 T^{6} + 2738019 T^{7} + 88529281 T^{8}$$