# Properties

 Label 31.2.c.a Level $31$ Weight $2$ Character orbit 31.c Analytic conductor $0.248$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$31$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 31.c (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.247536246266$$ 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.

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

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 $$\left(\mathbb{Z}/31\mathbb{Z}\right)^\times$$.

 $$n$$ $$3$$ $$\chi(n)$$ $$\beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5.1
 0.707107 − 1.22474i −0.707107 + 1.22474i 0.707107 + 1.22474i −0.707107 − 1.22474i
−2.41421 1.20711 2.09077i 3.82843 −0.500000 0.866025i −2.91421 + 5.04757i −1.20711 + 2.09077i −4.41421 −1.41421 2.44949i 1.20711 + 2.09077i
5.2 0.414214 −0.207107 + 0.358719i −1.82843 −0.500000 0.866025i −0.0857864 + 0.148586i 0.207107 0.358719i −1.58579 1.41421 + 2.44949i −0.207107 0.358719i
25.1 −2.41421 1.20711 + 2.09077i 3.82843 −0.500000 + 0.866025i −2.91421 5.04757i −1.20711 2.09077i −4.41421 −1.41421 + 2.44949i 1.20711 2.09077i
25.2 0.414214 −0.207107 0.358719i −1.82843 −0.500000 + 0.866025i −0.0857864 0.148586i 0.207107 + 0.358719i −1.58579 1.41421 2.44949i −0.207107 + 0.358719i
 $$n$$: 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
31.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 31.2.c.a 4
3.b odd 2 1 279.2.h.c 4
4.b odd 2 1 496.2.i.h 4
5.b even 2 1 775.2.e.e 4
5.c odd 4 2 775.2.o.d 8
31.b odd 2 1 961.2.c.a 4
31.c even 3 1 inner 31.2.c.a 4
31.c even 3 1 961.2.a.a 2
31.d even 5 4 961.2.g.o 16
31.e odd 6 1 961.2.a.c 2
31.e odd 6 1 961.2.c.a 4
31.f odd 10 4 961.2.g.r 16
31.g even 15 4 961.2.d.l 8
31.g even 15 4 961.2.g.o 16
31.h odd 30 4 961.2.d.i 8
31.h odd 30 4 961.2.g.r 16
93.g even 6 1 8649.2.a.k 2
93.h odd 6 1 279.2.h.c 4
93.h odd 6 1 8649.2.a.l 2
124.i odd 6 1 496.2.i.h 4
155.j even 6 1 775.2.e.e 4
155.o odd 12 2 775.2.o.d 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
31.2.c.a 4 1.a even 1 1 trivial
31.2.c.a 4 31.c even 3 1 inner
279.2.h.c 4 3.b odd 2 1
279.2.h.c 4 93.h odd 6 1
496.2.i.h 4 4.b odd 2 1
496.2.i.h 4 124.i odd 6 1
775.2.e.e 4 5.b even 2 1
775.2.e.e 4 155.j even 6 1
775.2.o.d 8 5.c odd 4 2
775.2.o.d 8 155.o odd 12 2
961.2.a.a 2 31.c even 3 1
961.2.a.c 2 31.e odd 6 1
961.2.c.a 4 31.b odd 2 1
961.2.c.a 4 31.e odd 6 1
961.2.d.i 8 31.h odd 30 4
961.2.d.l 8 31.g even 15 4
961.2.g.o 16 31.d even 5 4
961.2.g.o 16 31.g even 15 4
961.2.g.r 16 31.f odd 10 4
961.2.g.r 16 31.h odd 30 4
8649.2.a.k 2 93.g even 6 1
8649.2.a.l 2 93.h odd 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(31, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + 2 T + T^{2} )^{2}$$
$3$ $$1 + 2 T + 5 T^{2} - 2 T^{3} + T^{4}$$
$5$ $$( 1 + T + T^{2} )^{2}$$
$7$ $$1 - 2 T + 5 T^{2} + 2 T^{3} + T^{4}$$
$11$ $$289 + 34 T + 21 T^{2} - 2 T^{3} + T^{4}$$
$13$ $$49 + 14 T + 11 T^{2} - 2 T^{3} + T^{4}$$
$17$ $$1 - 6 T + 35 T^{2} - 6 T^{3} + T^{4}$$
$19$ $$49 + 42 T + 29 T^{2} + 6 T^{3} + T^{4}$$
$23$ $$( 4 + T )^{4}$$
$29$ $$( 8 + 8 T + T^{2} )^{2}$$
$31$ $$( 31 + 10 T + T^{2} )^{2}$$
$37$ $$( 1 + T + T^{2} )^{2}$$
$41$ $$5041 - 142 T + 75 T^{2} + 2 T^{3} + T^{4}$$
$43$ $$9409 + 194 T + 101 T^{2} - 2 T^{3} + T^{4}$$
$47$ $$( -16 - 8 T + T^{2} )^{2}$$
$53$ $$1 + 6 T + 35 T^{2} + 6 T^{3} + T^{4}$$
$59$ $$1681 + 246 T + 77 T^{2} - 6 T^{3} + T^{4}$$
$61$ $$( -8 + T^{2} )^{2}$$
$67$ $$289 + 34 T + 21 T^{2} - 2 T^{3} + T^{4}$$
$71$ $$1 + 14 T + 197 T^{2} - 14 T^{3} + T^{4}$$
$73$ $$49 - 14 T + 11 T^{2} + 2 T^{3} + T^{4}$$
$79$ $$10609 - 2266 T + 381 T^{2} - 22 T^{3} + T^{4}$$
$83$ $$1681 + 246 T + 77 T^{2} - 6 T^{3} + T^{4}$$
$89$ $$( -56 + 8 T + T^{2} )^{2}$$
$97$ $$( 56 - 16 T + T^{2} )^{2}$$