# Properties

 Label 31.2.d.a Level 31 Weight 2 Character orbit 31.d 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.d (of order $$5$$ and degree $$4$$)

## Newform invariants

 Self dual: No Analytic conductor: $$0.247536246266$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{10})$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ 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.

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

## Character Values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/31\mathbb{Z}\right)^\times$$.

 $$n$$ $$3$$ $$\chi(n)$$ $$-\zeta_{10}^{3}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2.1
 0.809017 + 0.587785i −0.309017 − 0.951057i −0.309017 + 0.951057i 0.809017 − 0.587785i
−1.30902 0.951057i 0.809017 0.587785i 0.190983 + 0.587785i −0.381966 −1.61803 0.927051 + 2.85317i −0.690983 + 2.12663i −0.618034 + 1.90211i 0.500000 + 0.363271i
4.1 −0.190983 0.587785i −0.309017 + 0.951057i 1.30902 0.951057i −2.61803 0.618034 −2.42705 + 1.76336i −1.80902 1.31433i 1.61803 + 1.17557i 0.500000 + 1.53884i
8.1 −0.190983 + 0.587785i −0.309017 0.951057i 1.30902 + 0.951057i −2.61803 0.618034 −2.42705 1.76336i −1.80902 + 1.31433i 1.61803 1.17557i 0.500000 1.53884i
16.1 −1.30902 + 0.951057i 0.809017 + 0.587785i 0.190983 0.587785i −0.381966 −1.61803 0.927051 2.85317i −0.690983 2.12663i −0.618034 1.90211i 0.500000 0.363271i
 $$n$$: 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.d Even 1 yes

## Hecke kernels

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