# 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 $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.817440793081$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-17})$$ Defining polynomial: $$x^{4} + 16 x^{2} + 81$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 8q^{4} - 4q^{6} - 32q^{9} + O(q^{10})$$ $$4q + 8q^{4} - 4q^{6} - 32q^{9} - 16q^{10} + 8q^{15} + 16q^{16} + 48q^{19} + 68q^{21} - 8q^{24} - 36q^{25} + 68q^{30} - 128q^{31} - 64q^{34} - 64q^{36} - 32q^{40} - 68q^{45} + 136q^{46} + 60q^{49} + 32q^{51} + 100q^{54} + 272q^{55} + 16q^{60} - 64q^{61} + 32q^{64} - 272q^{66} - 68q^{69} - 136q^{70} - 272q^{75} + 96q^{76} - 288q^{79} + 188q^{81} + 136q^{84} + 128q^{85} + 128q^{90} - 200q^{94} - 16q^{96} + 272q^{99} + O(q^{100})$$

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

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

## 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 Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 30.3.b.a 4
3.b odd 2 1 inner 30.3.b.a 4
4.b odd 2 1 240.3.c.c 4
5.b even 2 1 inner 30.3.b.a 4
5.c odd 4 2 150.3.d.d 4
8.b even 2 1 960.3.c.f 4
8.d odd 2 1 960.3.c.e 4
9.c even 3 2 810.3.j.c 8
9.d odd 6 2 810.3.j.c 8
12.b even 2 1 240.3.c.c 4
15.d odd 2 1 inner 30.3.b.a 4
15.e even 4 2 150.3.d.d 4
20.d odd 2 1 240.3.c.c 4
20.e even 4 2 1200.3.l.t 4
24.f even 2 1 960.3.c.e 4
24.h odd 2 1 960.3.c.f 4
40.e odd 2 1 960.3.c.e 4
40.f even 2 1 960.3.c.f 4
45.h odd 6 2 810.3.j.c 8
45.j even 6 2 810.3.j.c 8
60.h even 2 1 240.3.c.c 4
60.l odd 4 2 1200.3.l.t 4
120.i odd 2 1 960.3.c.f 4
120.m even 2 1 960.3.c.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.3.b.a 4 1.a even 1 1 trivial
30.3.b.a 4 3.b odd 2 1 inner
30.3.b.a 4 5.b even 2 1 inner
30.3.b.a 4 15.d odd 2 1 inner
150.3.d.d 4 5.c odd 4 2
150.3.d.d 4 15.e even 4 2
240.3.c.c 4 4.b odd 2 1
240.3.c.c 4 12.b even 2 1
240.3.c.c 4 20.d odd 2 1
240.3.c.c 4 60.h even 2 1
810.3.j.c 8 9.c even 3 2
810.3.j.c 8 9.d odd 6 2
810.3.j.c 8 45.h odd 6 2
810.3.j.c 8 45.j even 6 2
960.3.c.e 4 8.d odd 2 1
960.3.c.e 4 24.f even 2 1
960.3.c.e 4 40.e odd 2 1
960.3.c.e 4 120.m even 2 1
960.3.c.f 4 8.b even 2 1
960.3.c.f 4 24.h odd 2 1
960.3.c.f 4 40.f even 2 1
960.3.c.f 4 120.i odd 2 1
1200.3.l.t 4 20.e even 4 2
1200.3.l.t 4 60.l odd 4 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(30, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - 2 T^{2} )^{2}$$
$3$ $$1 + 16 T^{2} + 81 T^{4}$$
$5$ $$1 + 18 T^{2} + 625 T^{4}$$
$7$ $$( 1 - 64 T^{2} + 2401 T^{4} )^{2}$$
$11$ $$( 1 + 30 T^{2} + 14641 T^{4} )^{2}$$
$13$ $$( 1 - 13 T )^{4}( 1 + 13 T )^{4}$$
$17$ $$( 1 + 450 T^{2} + 83521 T^{4} )^{2}$$
$19$ $$( 1 - 12 T + 361 T^{2} )^{4}$$
$23$ $$( 1 + 480 T^{2} + 279841 T^{4} )^{2}$$
$29$ $$( 1 - 29 T )^{4}( 1 + 29 T )^{4}$$
$31$ $$( 1 + 32 T + 961 T^{2} )^{4}$$
$37$ $$( 1 - 2194 T^{2} + 1874161 T^{4} )^{2}$$
$41$ $$( 1 - 30 T^{2} + 2825761 T^{4} )^{2}$$
$43$ $$( 1 - 2032 T^{2} + 3418801 T^{4} )^{2}$$
$47$ $$( 1 + 3168 T^{2} + 4879681 T^{4} )^{2}$$
$53$ $$( 1 + 1010 T^{2} + 7890481 T^{4} )^{2}$$
$59$ $$( 1 - 6690 T^{2} + 12117361 T^{4} )^{2}$$
$61$ $$( 1 + 16 T + 3721 T^{2} )^{4}$$
$67$ $$( 1 - 8944 T^{2} + 20151121 T^{4} )^{2}$$
$71$ $$( 1 - 71 T )^{4}( 1 + 71 T )^{4}$$
$73$ $$( 1 + 2942 T^{2} + 28398241 T^{4} )^{2}$$
$79$ $$( 1 + 72 T + 6241 T^{2} )^{4}$$
$83$ $$( 1 + 11856 T^{2} + 47458321 T^{4} )^{2}$$
$89$ $$( 1 - 11490 T^{2} + 62742241 T^{4} )^{2}$$
$97$ $$( 1 + 7838 T^{2} + 88529281 T^{4} )^{2}$$