# Properties

 Label 16.2.e.a Level 16 Weight 2 Character orbit 16.e Analytic conductor 0.128 Analytic rank 0 Dimension 2 CM No Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$16 = 2^{4}$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 16.e (of order $$4$$ and degree $$2$$)

## Newform invariants

 Self dual: No Analytic conductor: $$0.127760643234$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character Values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/16\mathbb{Z}\right)^\times$$.

 $$n$$ $$5$$ $$15$$ $$\chi(n)$$ $$i$$ $$1$$

## 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
 1.00000i − 1.00000i
−1.00000 1.00000i −1.00000 + 1.00000i 2.00000i −1.00000 1.00000i 2.00000 2.00000i 2.00000 2.00000i 1.00000i 2.00000i
13.1 −1.00000 + 1.00000i −1.00000 1.00000i 2.00000i −1.00000 + 1.00000i 2.00000 2.00000i 2.00000 + 2.00000i 1.00000i 2.00000i
 $$n$$: 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
16.e Even 1 yes

## Hecke kernels

There are no other newforms in $$S_{2}^{\mathrm{new}}(16, [\chi])$$.