# Properties

 Label 231.2.j.d 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.

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

## 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 −0.809017 2.48990i 0.500000 + 1.53884i −0.809017 0.587785i 1.80902 1.31433i 0.309017 0.951057i −4.23607
148.1 0.500000 + 1.53884i −0.809017 0.587785i −0.500000 + 0.363271i −0.809017 + 2.48990i 0.500000 1.53884i −0.809017 + 0.587785i 1.80902 + 1.31433i 0.309017 + 0.951057i −4.23607
169.1 0.500000 + 0.363271i 0.309017 0.951057i −0.500000 1.53884i 0.309017 0.224514i 0.500000 0.363271i 0.309017 + 0.951057i 0.690983 2.12663i −0.809017 0.587785i 0.236068
190.1 0.500000 0.363271i 0.309017 + 0.951057i −0.500000 + 1.53884i 0.309017 + 0.224514i 0.500000 + 0.363271i 0.309017 0.951057i 0.690983 + 2.12663i −0.809017 + 0.587785i 0.236068
 $$n$$: 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.d 4
3.b odd 2 1 693.2.m.b 4
11.c even 5 1 inner 231.2.j.d 4
11.c even 5 1 2541.2.a.bb 2
11.d odd 10 1 2541.2.a.s 2
33.f even 10 1 7623.2.a.bp 2
33.h odd 10 1 693.2.m.b 4
33.h odd 10 1 7623.2.a.ba 2

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

$p$ $F_p(T)$
$2$ $$1 - 2 T + 2 T^{2} - 5 T^{3} + 11 T^{4} - 10 T^{5} + 8 T^{6} - 16 T^{7} + 16 T^{8}$$
$3$ $$1 + T + T^{2} + T^{3} + T^{4}$$
$5$ $$1 + T + T^{2} + 11 T^{3} + 36 T^{4} + 55 T^{5} + 25 T^{6} + 125 T^{7} + 625 T^{8}$$
$7$ $$1 + T + T^{2} + T^{3} + T^{4}$$
$11$ $$1 + T + 21 T^{2} + 11 T^{3} + 121 T^{4}$$
$13$ $$1 - T - 12 T^{2} + 25 T^{3} + 131 T^{4} + 325 T^{5} - 2028 T^{6} - 2197 T^{7} + 28561 T^{8}$$
$17$ $$1 - 7 T + 2 T^{2} + 65 T^{3} - 169 T^{4} + 1105 T^{5} + 578 T^{6} - 34391 T^{7} + 83521 T^{8}$$
$19$ $$1 - 19 T^{2} + 361 T^{4} - 6859 T^{6} + 130321 T^{8}$$
$23$ $$( 1 + 2 T + 42 T^{2} + 46 T^{3} + 529 T^{4} )^{2}$$
$29$ $$1 - 15 T + 106 T^{2} - 675 T^{3} + 4171 T^{4} - 19575 T^{5} + 89146 T^{6} - 365835 T^{7} + 707281 T^{8}$$
$31$ $$1 - 13 T + 48 T^{2} + 319 T^{3} - 3835 T^{4} + 9889 T^{5} + 46128 T^{6} - 387283 T^{7} + 923521 T^{8}$$
$37$ $$1 - 22 T + 207 T^{2} - 1220 T^{3} + 6701 T^{4} - 45140 T^{5} + 283383 T^{6} - 1114366 T^{7} + 1874161 T^{8}$$
$41$ $$1 - 13 T + 53 T^{2} - 331 T^{3} + 3380 T^{4} - 13571 T^{5} + 89093 T^{6} - 895973 T^{7} + 2825761 T^{8}$$
$43$ $$( 1 + T + 43 T^{2} )^{4}$$
$47$ $$1 + 18 T + 137 T^{2} + 990 T^{3} + 7951 T^{4} + 46530 T^{5} + 302633 T^{6} + 1868814 T^{7} + 4879681 T^{8}$$
$53$ $$1 + 9 T - 7 T^{2} - 525 T^{3} - 3884 T^{4} - 27825 T^{5} - 19663 T^{6} + 1339893 T^{7} + 7890481 T^{8}$$
$59$ $$1 - 15 T + 31 T^{2} + 885 T^{3} - 10424 T^{4} + 52215 T^{5} + 107911 T^{6} - 3080685 T^{7} + 12117361 T^{8}$$
$61$ $$1 - 3 T + 18 T^{2} + 209 T^{3} + 675 T^{4} + 12749 T^{5} + 66978 T^{6} - 680943 T^{7} + 13845841 T^{8}$$
$67$ $$( 1 + 14 T + 178 T^{2} + 938 T^{3} + 4489 T^{4} )^{2}$$
$71$ $$1 - 13 T - 2 T^{2} + 349 T^{3} + 405 T^{4} + 24779 T^{5} - 10082 T^{6} - 4652843 T^{7} + 25411681 T^{8}$$
$73$ $$1 - 21 T + 233 T^{2} - 2595 T^{3} + 26956 T^{4} - 189435 T^{5} + 1241657 T^{6} - 8169357 T^{7} + 28398241 T^{8}$$
$79$ $$1 + 15 T + 11 T^{2} - 1185 T^{3} - 13064 T^{4} - 93615 T^{5} + 68651 T^{6} + 7395585 T^{7} + 38950081 T^{8}$$
$83$ $$1 - 6 T - 47 T^{2} + 780 T^{3} - 779 T^{4} + 64740 T^{5} - 323783 T^{6} - 3430722 T^{7} + 47458321 T^{8}$$
$89$ $$( 1 + 15 T + 233 T^{2} + 1335 T^{3} + 7921 T^{4} )^{2}$$
$97$ $$1 - 17 T + 192 T^{2} - 1615 T^{3} + 8831 T^{4} - 156655 T^{5} + 1806528 T^{6} - 15515441 T^{7} + 88529281 T^{8}$$