Properties

 Label 231.2.i.d Level $231$ Weight $2$ Character orbit 231.i 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.i (of order $$3$$, degree $$2$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$1.84454428669$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} + 2 x^{2} + 4$$ 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 basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 2 x^{2} + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/2$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$2 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{3}$$

Character values

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

 $$n$$ $$155$$ $$199$$ $$211$$ $$\chi(n)$$ $$1$$ $$-1 - \beta_{2}$$ $$1$$

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
 −0.707107 − 1.22474i 0.707107 + 1.22474i −0.707107 + 1.22474i 0.707107 − 1.22474i
−0.207107 0.358719i −0.500000 + 0.866025i 0.914214 1.58346i 1.00000 + 1.73205i 0.414214 1.00000 + 2.44949i −1.58579 −0.500000 0.866025i 0.414214 0.717439i
67.2 1.20711 + 2.09077i −0.500000 + 0.866025i −1.91421 + 3.31552i 1.00000 + 1.73205i −2.41421 1.00000 2.44949i −4.41421 −0.500000 0.866025i −2.41421 + 4.18154i
100.1 −0.207107 + 0.358719i −0.500000 0.866025i 0.914214 + 1.58346i 1.00000 1.73205i 0.414214 1.00000 2.44949i −1.58579 −0.500000 + 0.866025i 0.414214 + 0.717439i
100.2 1.20711 2.09077i −0.500000 0.866025i −1.91421 3.31552i 1.00000 1.73205i −2.41421 1.00000 + 2.44949i −4.41421 −0.500000 + 0.866025i −2.41421 4.18154i
 $$n$$: 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.d 4
3.b odd 2 1 693.2.i.f 4
7.c even 3 1 inner 231.2.i.d 4
7.c even 3 1 1617.2.a.n 2
7.d odd 6 1 1617.2.a.m 2
21.g even 6 1 4851.2.a.bd 2
21.h odd 6 1 693.2.i.f 4
21.h odd 6 1 4851.2.a.be 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.i.d 4 1.a even 1 1 trivial
231.2.i.d 4 7.c even 3 1 inner
693.2.i.f 4 3.b odd 2 1
693.2.i.f 4 21.h odd 6 1
1617.2.a.m 2 7.d odd 6 1
1617.2.a.n 2 7.c even 3 1
4851.2.a.bd 2 21.g even 6 1
4851.2.a.be 2 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} - 2 T_{2}^{3} + 5 T_{2}^{2} + 2 T_{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(231, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 2 T + 5 T^{2} - 2 T^{3} + T^{4}$$
$3$ $$( 1 + T + T^{2} )^{2}$$
$5$ $$( 4 - 2 T + T^{2} )^{2}$$
$7$ $$( 7 - 2 T + T^{2} )^{2}$$
$11$ $$( 1 + T + T^{2} )^{2}$$
$13$ $$( -4 - 4 T + T^{2} )^{2}$$
$17$ $$49 - 42 T + 29 T^{2} - 6 T^{3} + T^{4}$$
$19$ $$81 - 54 T + 45 T^{2} + 6 T^{3} + T^{4}$$
$23$ $$( 49 - 7 T + T^{2} )^{2}$$
$29$ $$( -17 + 2 T + T^{2} )^{2}$$
$31$ $$1024 + 32 T^{2} + T^{4}$$
$37$ $$5041 + 142 T + 75 T^{2} - 2 T^{3} + T^{4}$$
$41$ $$( 8 + 8 T + T^{2} )^{2}$$
$43$ $$( 31 + 14 T + T^{2} )^{2}$$
$47$ $$1681 + 574 T + 155 T^{2} + 14 T^{3} + T^{4}$$
$53$ $$8464 - 1840 T + 308 T^{2} - 20 T^{3} + T^{4}$$
$59$ $$529 + 138 T + 59 T^{2} - 6 T^{3} + T^{4}$$
$61$ $$( 16 - 4 T + T^{2} )^{2}$$
$67$ $$784 - 336 T + 116 T^{2} - 12 T^{3} + T^{4}$$
$71$ $$( 41 + 14 T + T^{2} )^{2}$$
$73$ $$16 + 48 T + 140 T^{2} + 12 T^{3} + T^{4}$$
$79$ $$15376 - 496 T + 140 T^{2} + 4 T^{3} + T^{4}$$
$83$ $$( -8 + T^{2} )^{2}$$
$89$ $$40000 + 200 T^{2} + T^{4}$$
$97$ $$( -63 + 6 T + T^{2} )^{2}$$