# Properties

 Label 17.2.d.a Level 17 Weight 2 Character orbit 17.d Analytic conductor 0.136 Analytic rank 0 Dimension 4 CM No Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$17$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 17.d (of order $$8$$ and degree $$4$$)

## Newform invariants

 Self dual: No Analytic conductor: $$0.135745683436$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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.

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

## Character Values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/17\mathbb{Z}\right)^\times$$.

 $$n$$ $$3$$ $$\chi(n)$$ $$\zeta_{8}$$

## 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
 0.707107 − 0.707107i −0.707107 − 0.707107i 0.707107 + 0.707107i −0.707107 + 0.707107i
−0.292893 0.292893i −2.41421 + 1.00000i 1.82843i 0.707107 + 1.70711i 1.00000 + 0.414214i 0.414214 1.00000i −1.12132 + 1.12132i 2.70711 2.70711i 0.292893 0.707107i
8.1 −1.70711 + 1.70711i 0.414214 1.00000i 3.82843i −0.707107 0.292893i 1.00000 + 2.41421i −2.41421 + 1.00000i 3.12132 + 3.12132i 1.29289 + 1.29289i 1.70711 0.707107i
9.1 −0.292893 + 0.292893i −2.41421 1.00000i 1.82843i 0.707107 1.70711i 1.00000 0.414214i 0.414214 + 1.00000i −1.12132 1.12132i 2.70711 + 2.70711i 0.292893 + 0.707107i
15.1 −1.70711 1.70711i 0.414214 + 1.00000i 3.82843i −0.707107 + 0.292893i 1.00000 2.41421i −2.41421 1.00000i 3.12132 3.12132i 1.29289 1.29289i 1.70711 + 0.707107i
 $$n$$: 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
17.d Even 1 yes

## Hecke kernels

There are no other newforms in $$S_{2}^{\mathrm{new}}(17, [\chi])$$.