# Properties

 Label 32.2.g.a Level 32 Weight 2 Character orbit 32.g Analytic conductor 0.256 Analytic rank 0 Dimension 4 CM No Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$32 = 2^{5}$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 32.g (of order $$8$$ and degree $$4$$)

## Newform invariants

 Self dual: No Analytic conductor: $$0.255521286468$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{8}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character Values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/32\mathbb{Z}\right)^\times$$.

 $$n$$ $$5$$ $$31$$ $$\chi(n)$$ $$\zeta_{8}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5.1
 0.707107 + 0.707107i 0.707107 − 0.707107i −0.707107 − 0.707107i −0.707107 + 0.707107i
1.41421i 0.707107 + 1.70711i −2.00000 −3.12132 1.29289i 2.41421 1.00000i 1.00000 + 1.00000i 2.82843i −0.292893 + 0.292893i −1.82843 + 4.41421i
13.1 1.41421i 0.707107 1.70711i −2.00000 −3.12132 + 1.29289i 2.41421 + 1.00000i 1.00000 1.00000i 2.82843i −0.292893 0.292893i −1.82843 4.41421i
21.1 1.41421i −0.707107 + 0.292893i −2.00000 1.12132 2.70711i −0.414214 1.00000i 1.00000 + 1.00000i 2.82843i −1.70711 + 1.70711i 3.82843 + 1.58579i
29.1 1.41421i −0.707107 0.292893i −2.00000 1.12132 + 2.70711i −0.414214 + 1.00000i 1.00000 1.00000i 2.82843i −1.70711 1.70711i 3.82843 1.58579i
 $$n$$: 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
32.g Even 1 yes

## Hecke kernels

This newform can be constructed as the kernel of the linear operator $$T_{3}^{4}$$ $$\mathstrut +\mathstrut 2 T_{3}^{2}$$ $$\mathstrut +\mathstrut 4 T_{3}$$ $$\mathstrut +\mathstrut 2$$ acting on $$S_{2}^{\mathrm{new}}(32, [\chi])$$.