# 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 $$3$$ and degree $$2$$)

## Newform invariants

 Self dual: No Analytic conductor: $$0.247536246266$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-3})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ 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.

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

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

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

## 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. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
31.c Even 1 yes

## Hecke kernels

There are no other newforms in $$S_{2}^{\mathrm{new}}(31, [\chi])$$.