# Properties

 Label 30.3.b.a Level 30 Weight 3 Character orbit 30.b Analytic conductor 0.817 Analytic rank 0 Dimension 4 CM No Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$30 = 2 \cdot 3 \cdot 5$$ Weight: $$k$$ = $$3$$ Character orbit: $$[\chi]$$ = 30.b (of order $$2$$ and degree $$1$$)

## Newform invariants

 Self dual: No Analytic conductor: $$0.817440793081$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-17})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q$$ $$+ \beta_{2} q^{2}$$ $$+ ( -\beta_{1} - \beta_{2} ) q^{3}$$ $$+ 2 q^{4}$$ $$+ ( -2 \beta_{2} - \beta_{3} ) q^{5}$$ $$+ ( -1 + \beta_{3} ) q^{6}$$ $$+ ( 2 \beta_{1} + \beta_{2} ) q^{7}$$ $$+ 2 \beta_{2} q^{8}$$ $$+ ( -8 - \beta_{3} ) q^{9}$$ $$+O(q^{10})$$ $$q$$ $$+ \beta_{2} q^{2}$$ $$+ ( -\beta_{1} - \beta_{2} ) q^{3}$$ $$+ 2 q^{4}$$ $$+ ( -2 \beta_{2} - \beta_{3} ) q^{5}$$ $$+ ( -1 + \beta_{3} ) q^{6}$$ $$+ ( 2 \beta_{1} + \beta_{2} ) q^{7}$$ $$+ 2 \beta_{2} q^{8}$$ $$+ ( -8 - \beta_{3} ) q^{9}$$ $$+ ( -4 + 2 \beta_{1} + \beta_{2} ) q^{10}$$ $$+ 4 \beta_{3} q^{11}$$ $$+ ( -2 \beta_{1} - 2 \beta_{2} ) q^{12}$$ $$-2 \beta_{3} q^{14}$$ $$+ ( 2 - \beta_{1} + 8 \beta_{2} - 2 \beta_{3} ) q^{15}$$ $$+ 4 q^{16}$$ $$-8 \beta_{2} q^{17}$$ $$+ ( 2 \beta_{1} - 7 \beta_{2} ) q^{18}$$ $$+ 12 q^{19}$$ $$+ ( -4 \beta_{2} - 2 \beta_{3} ) q^{20}$$ $$+ ( 17 + \beta_{3} ) q^{21}$$ $$+ ( -8 \beta_{1} - 4 \beta_{2} ) q^{22}$$ $$+ 17 \beta_{2} q^{23}$$ $$+ ( -2 + 2 \beta_{3} ) q^{24}$$ $$+ ( -9 - 8 \beta_{1} - 4 \beta_{2} ) q^{25}$$ $$+ ( 7 \beta_{1} + 16 \beta_{2} ) q^{27}$$ $$+ ( 4 \beta_{1} + 2 \beta_{2} ) q^{28}$$ $$+ ( 17 + 4 \beta_{1} + 4 \beta_{2} + \beta_{3} ) q^{30}$$ $$-32 q^{31}$$ $$+ 4 \beta_{2} q^{32}$$ $$+ ( 4 \beta_{1} - 32 \beta_{2} ) q^{33}$$ $$-16 q^{34}$$ $$+ ( -17 \beta_{2} + 4 \beta_{3} ) q^{35}$$ $$+ ( -16 - 2 \beta_{3} ) q^{36}$$ $$+ ( 8 \beta_{1} + 4 \beta_{2} ) q^{37}$$ $$+ 12 \beta_{2} q^{38}$$ $$+ ( -8 + 4 \beta_{1} + 2 \beta_{2} ) q^{40}$$ $$-14 \beta_{3} q^{41}$$ $$+ ( -2 \beta_{1} + 16 \beta_{2} ) q^{42}$$ $$+ ( -14 \beta_{1} - 7 \beta_{2} ) q^{43}$$ $$+ 8 \beta_{3} q^{44}$$ $$+ ( -17 - 4 \beta_{1} + 14 \beta_{2} + 8 \beta_{3} ) q^{45}$$ $$+ 34 q^{46}$$ $$-25 \beta_{2} q^{47}$$ $$+ ( -4 \beta_{1} - 4 \beta_{2} ) q^{48}$$ $$+ 15 q^{49}$$ $$+ ( -9 \beta_{2} + 8 \beta_{3} ) q^{50}$$ $$+ ( 8 - 8 \beta_{3} ) q^{51}$$ $$+ 48 \beta_{2} q^{53}$$ $$+ ( 25 - 7 \beta_{3} ) q^{54}$$ $$+ ( 68 + 16 \beta_{1} + 8 \beta_{2} ) q^{55}$$ $$-4 \beta_{3} q^{56}$$ $$+ ( -12 \beta_{1} - 12 \beta_{2} ) q^{57}$$ $$+ 4 \beta_{3} q^{59}$$ $$+ ( 4 - 2 \beta_{1} + 16 \beta_{2} - 4 \beta_{3} ) q^{60}$$ $$-16 q^{61}$$ $$-32 \beta_{2} q^{62}$$ $$+ ( -16 \beta_{1} - 25 \beta_{2} ) q^{63}$$ $$+ 8 q^{64}$$ $$+ ( -68 - 4 \beta_{3} ) q^{66}$$ $$+ ( -2 \beta_{1} - \beta_{2} ) q^{67}$$ $$-16 \beta_{2} q^{68}$$ $$+ ( -17 + 17 \beta_{3} ) q^{69}$$ $$+ ( -34 - 8 \beta_{1} - 4 \beta_{2} ) q^{70}$$ $$+ ( 4 \beta_{1} - 14 \beta_{2} ) q^{72}$$ $$+ ( 40 \beta_{1} + 20 \beta_{2} ) q^{73}$$ $$-8 \beta_{3} q^{74}$$ $$+ ( -68 + 9 \beta_{1} + 9 \beta_{2} - 4 \beta_{3} ) q^{75}$$ $$+ 24 q^{76}$$ $$+ 68 \beta_{2} q^{77}$$ $$-72 q^{79}$$ $$+ ( -8 \beta_{2} - 4 \beta_{3} ) q^{80}$$ $$+ ( 47 + 16 \beta_{3} ) q^{81}$$ $$+ ( 28 \beta_{1} + 14 \beta_{2} ) q^{82}$$ $$-31 \beta_{2} q^{83}$$ $$+ ( 34 + 2 \beta_{3} ) q^{84}$$ $$+ ( 32 - 16 \beta_{1} - 8 \beta_{2} ) q^{85}$$ $$+ 14 \beta_{3} q^{86}$$ $$+ ( -16 \beta_{1} - 8 \beta_{2} ) q^{88}$$ $$-16 \beta_{3} q^{89}$$ $$+ ( 32 - 16 \beta_{1} - 25 \beta_{2} + 4 \beta_{3} ) q^{90}$$ $$+ 34 \beta_{2} q^{92}$$ $$+ ( 32 \beta_{1} + 32 \beta_{2} ) q^{93}$$ $$-50 q^{94}$$ $$+ ( -24 \beta_{2} - 12 \beta_{3} ) q^{95}$$ $$+ ( -4 + 4 \beta_{3} ) q^{96}$$ $$+ ( -56 \beta_{1} - 28 \beta_{2} ) q^{97}$$ $$+ 15 \beta_{2} q^{98}$$ $$+ ( 68 - 32 \beta_{3} ) q^{99}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q$$ $$\mathstrut +\mathstrut 8q^{4}$$ $$\mathstrut -\mathstrut 4q^{6}$$ $$\mathstrut -\mathstrut 32q^{9}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$4q$$ $$\mathstrut +\mathstrut 8q^{4}$$ $$\mathstrut -\mathstrut 4q^{6}$$ $$\mathstrut -\mathstrut 32q^{9}$$ $$\mathstrut -\mathstrut 16q^{10}$$ $$\mathstrut +\mathstrut 8q^{15}$$ $$\mathstrut +\mathstrut 16q^{16}$$ $$\mathstrut +\mathstrut 48q^{19}$$ $$\mathstrut +\mathstrut 68q^{21}$$ $$\mathstrut -\mathstrut 8q^{24}$$ $$\mathstrut -\mathstrut 36q^{25}$$ $$\mathstrut +\mathstrut 68q^{30}$$ $$\mathstrut -\mathstrut 128q^{31}$$ $$\mathstrut -\mathstrut 64q^{34}$$ $$\mathstrut -\mathstrut 64q^{36}$$ $$\mathstrut -\mathstrut 32q^{40}$$ $$\mathstrut -\mathstrut 68q^{45}$$ $$\mathstrut +\mathstrut 136q^{46}$$ $$\mathstrut +\mathstrut 60q^{49}$$ $$\mathstrut +\mathstrut 32q^{51}$$ $$\mathstrut +\mathstrut 100q^{54}$$ $$\mathstrut +\mathstrut 272q^{55}$$ $$\mathstrut +\mathstrut 16q^{60}$$ $$\mathstrut -\mathstrut 64q^{61}$$ $$\mathstrut +\mathstrut 32q^{64}$$ $$\mathstrut -\mathstrut 272q^{66}$$ $$\mathstrut -\mathstrut 68q^{69}$$ $$\mathstrut -\mathstrut 136q^{70}$$ $$\mathstrut -\mathstrut 272q^{75}$$ $$\mathstrut +\mathstrut 96q^{76}$$ $$\mathstrut -\mathstrut 288q^{79}$$ $$\mathstrut +\mathstrut 188q^{81}$$ $$\mathstrut +\mathstrut 136q^{84}$$ $$\mathstrut +\mathstrut 128q^{85}$$ $$\mathstrut +\mathstrut 128q^{90}$$ $$\mathstrut -\mathstrut 200q^{94}$$ $$\mathstrut -\mathstrut 16q^{96}$$ $$\mathstrut +\mathstrut 272q^{99}$$ $$\mathstrut +\mathstrut O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4}\mathstrut +\mathstrut$$ $$16$$ $$x^{2}\mathstrut +\mathstrut$$ $$81$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 7 \nu$$$$)/9$$ $$\beta_{3}$$ $$=$$ $$\nu^{2} + 8$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$8$$ $$\nu^{3}$$ $$=$$ $$9$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$7$$ $$\beta_{1}$$

## Character Values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/30\mathbb{Z}\right)^\times$$.

 $$n$$ $$7$$ $$11$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
29.1
 0.707107 + 2.91548i 0.707107 − 2.91548i −0.707107 + 2.91548i −0.707107 − 2.91548i
−1.41421 0.707107 2.91548i 2.00000 2.82843 4.12311i −1.00000 + 4.12311i 5.83095i −2.82843 −8.00000 4.12311i −4.00000 + 5.83095i
29.2 −1.41421 0.707107 + 2.91548i 2.00000 2.82843 + 4.12311i −1.00000 4.12311i 5.83095i −2.82843 −8.00000 + 4.12311i −4.00000 5.83095i
29.3 1.41421 −0.707107 2.91548i 2.00000 −2.82843 + 4.12311i −1.00000 4.12311i 5.83095i 2.82843 −8.00000 + 4.12311i −4.00000 + 5.83095i
29.4 1.41421 −0.707107 + 2.91548i 2.00000 −2.82843 4.12311i −1.00000 + 4.12311i 5.83095i 2.82843 −8.00000 4.12311i −4.00000 5.83095i
 $$n$$: 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
3.b Odd 1 yes
5.b Even 1 yes
15.d Odd 1 yes

## Hecke kernels

There are no other newforms in $$S_{3}^{\mathrm{new}}(30, [\chi])$$.