# Properties

 Label 30.3.f.a Level $30$ Weight $3$ Character orbit 30.f Analytic conductor $0.817$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$30 = 2 \cdot 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 30.f (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.817440793081$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(i, \sqrt{6})$$ Defining polynomial: $$x^{4} + 9$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( 1 - \beta_{2} ) q^{2} + \beta_{1} q^{3} -2 \beta_{2} q^{4} + ( -2 \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{5} + ( \beta_{1} - \beta_{3} ) q^{6} + ( -4 + 4 \beta_{2} - 4 \beta_{3} ) q^{7} + ( -2 - 2 \beta_{2} ) q^{8} + 3 \beta_{2} q^{9} +O(q^{10})$$ $$q + ( 1 - \beta_{2} ) q^{2} + \beta_{1} q^{3} -2 \beta_{2} q^{4} + ( -2 \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{5} + ( \beta_{1} - \beta_{3} ) q^{6} + ( -4 + 4 \beta_{2} - 4 \beta_{3} ) q^{7} + ( -2 - 2 \beta_{2} ) q^{8} + 3 \beta_{2} q^{9} + ( 1 + \beta_{2} + 4 \beta_{3} ) q^{10} + ( -4 - 4 \beta_{1} + 4 \beta_{3} ) q^{11} -2 \beta_{3} q^{12} + ( 3 + 8 \beta_{1} + 3 \beta_{2} ) q^{13} + ( -4 \beta_{1} + 8 \beta_{2} - 4 \beta_{3} ) q^{14} + ( -6 - 6 \beta_{2} + \beta_{3} ) q^{15} -4 q^{16} + ( 11 - 11 \beta_{2} - 4 \beta_{3} ) q^{17} + ( 3 + 3 \beta_{2} ) q^{18} + ( -4 \beta_{1} - 16 \beta_{2} - 4 \beta_{3} ) q^{19} + ( 2 + 4 \beta_{1} + 4 \beta_{3} ) q^{20} + ( 12 - 4 \beta_{1} + 4 \beta_{3} ) q^{21} + ( -4 + 4 \beta_{2} + 8 \beta_{3} ) q^{22} + ( -4 + 12 \beta_{1} - 4 \beta_{2} ) q^{23} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{24} + ( 23 - 4 \beta_{1} - 4 \beta_{3} ) q^{25} + ( 6 + 8 \beta_{1} - 8 \beta_{3} ) q^{26} + 3 \beta_{3} q^{27} + ( 8 - 8 \beta_{1} + 8 \beta_{2} ) q^{28} + ( 4 \beta_{1} + 16 \beta_{2} + 4 \beta_{3} ) q^{29} + ( -12 + \beta_{1} + \beta_{3} ) q^{30} + ( -20 - 8 \beta_{1} + 8 \beta_{3} ) q^{31} + ( -4 + 4 \beta_{2} ) q^{32} + ( -12 - 4 \beta_{1} - 12 \beta_{2} ) q^{33} + ( -4 \beta_{1} - 22 \beta_{2} - 4 \beta_{3} ) q^{34} + ( -28 + 4 \beta_{1} + 20 \beta_{2} - 16 \beta_{3} ) q^{35} + 6 q^{36} + ( -27 + 27 \beta_{2} ) q^{37} + ( -16 - 8 \beta_{1} - 16 \beta_{2} ) q^{38} + ( 3 \beta_{1} + 24 \beta_{2} + 3 \beta_{3} ) q^{39} + ( 2 + 8 \beta_{1} - 2 \beta_{2} ) q^{40} + ( 8 + 4 \beta_{1} - 4 \beta_{3} ) q^{41} + ( 12 - 12 \beta_{2} + 8 \beta_{3} ) q^{42} + ( 12 - 20 \beta_{1} + 12 \beta_{2} ) q^{43} + ( 8 \beta_{1} + 8 \beta_{2} + 8 \beta_{3} ) q^{44} + ( -3 - 6 \beta_{1} - 6 \beta_{3} ) q^{45} + ( -8 + 12 \beta_{1} - 12 \beta_{3} ) q^{46} + ( 24 - 24 \beta_{2} + 12 \beta_{3} ) q^{47} -4 \beta_{1} q^{48} + ( 32 \beta_{1} - 31 \beta_{2} + 32 \beta_{3} ) q^{49} + ( 23 - 8 \beta_{1} - 23 \beta_{2} ) q^{50} + ( 12 + 11 \beta_{1} - 11 \beta_{3} ) q^{51} + ( 6 - 6 \beta_{2} - 16 \beta_{3} ) q^{52} + ( 25 - 36 \beta_{1} + 25 \beta_{2} ) q^{53} + ( 3 \beta_{1} + 3 \beta_{3} ) q^{54} + ( 48 + 4 \beta_{1} - 4 \beta_{2} - 12 \beta_{3} ) q^{55} + ( 16 - 8 \beta_{1} + 8 \beta_{3} ) q^{56} + ( 12 - 12 \beta_{2} - 16 \beta_{3} ) q^{57} + ( 16 + 8 \beta_{1} + 16 \beta_{2} ) q^{58} -20 \beta_{2} q^{59} + ( -12 + 2 \beta_{1} + 12 \beta_{2} ) q^{60} + ( -24 + 16 \beta_{1} - 16 \beta_{3} ) q^{61} + ( -20 + 20 \beta_{2} + 16 \beta_{3} ) q^{62} + ( -12 + 12 \beta_{1} - 12 \beta_{2} ) q^{63} + 8 \beta_{2} q^{64} + ( -51 - 12 \beta_{1} - 45 \beta_{2} + 8 \beta_{3} ) q^{65} + ( -24 - 4 \beta_{1} + 4 \beta_{3} ) q^{66} + ( -4 + 4 \beta_{2} + 36 \beta_{3} ) q^{67} + ( -22 - 8 \beta_{1} - 22 \beta_{2} ) q^{68} + ( -4 \beta_{1} + 36 \beta_{2} - 4 \beta_{3} ) q^{69} + ( -8 - 12 \beta_{1} + 48 \beta_{2} - 20 \beta_{3} ) q^{70} + ( 16 - 4 \beta_{1} + 4 \beta_{3} ) q^{71} + ( 6 - 6 \beta_{2} ) q^{72} + ( -47 + 8 \beta_{1} - 47 \beta_{2} ) q^{73} + 54 \beta_{2} q^{74} + ( 12 + 23 \beta_{1} - 12 \beta_{2} ) q^{75} + ( -32 - 8 \beta_{1} + 8 \beta_{3} ) q^{76} + ( -32 + 32 \beta_{2} - 16 \beta_{3} ) q^{77} + ( 24 + 6 \beta_{1} + 24 \beta_{2} ) q^{78} + ( -52 \beta_{1} + 12 \beta_{2} - 52 \beta_{3} ) q^{79} + ( 8 \beta_{1} - 4 \beta_{2} - 8 \beta_{3} ) q^{80} -9 q^{81} + ( 8 - 8 \beta_{2} - 8 \beta_{3} ) q^{82} + ( 48 + 28 \beta_{1} + 48 \beta_{2} ) q^{83} + ( 8 \beta_{1} - 24 \beta_{2} + 8 \beta_{3} ) q^{84} + ( -13 + 4 \beta_{1} + 35 \beta_{2} + 44 \beta_{3} ) q^{85} + ( 24 - 20 \beta_{1} + 20 \beta_{3} ) q^{86} + ( -12 + 12 \beta_{2} + 16 \beta_{3} ) q^{87} + ( 8 + 16 \beta_{1} + 8 \beta_{2} ) q^{88} + ( -12 \beta_{1} - 88 \beta_{2} - 12 \beta_{3} ) q^{89} + ( -3 - 12 \beta_{1} + 3 \beta_{2} ) q^{90} + ( 72 - 20 \beta_{1} + 20 \beta_{3} ) q^{91} + ( -8 + 8 \beta_{2} - 24 \beta_{3} ) q^{92} + ( -24 - 20 \beta_{1} - 24 \beta_{2} ) q^{93} + ( 12 \beta_{1} - 48 \beta_{2} + 12 \beta_{3} ) q^{94} + ( 16 + 36 \beta_{1} + 48 \beta_{2} + 28 \beta_{3} ) q^{95} + ( -4 \beta_{1} + 4 \beta_{3} ) q^{96} + ( 33 - 33 \beta_{2} - 40 \beta_{3} ) q^{97} + ( -31 + 64 \beta_{1} - 31 \beta_{2} ) q^{98} + ( -12 \beta_{1} - 12 \beta_{2} - 12 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{2} - 16q^{7} - 8q^{8} + O(q^{10})$$ $$4q + 4q^{2} - 16q^{7} - 8q^{8} + 4q^{10} - 16q^{11} + 12q^{13} - 24q^{15} - 16q^{16} + 44q^{17} + 12q^{18} + 8q^{20} + 48q^{21} - 16q^{22} - 16q^{23} + 92q^{25} + 24q^{26} + 32q^{28} - 48q^{30} - 80q^{31} - 16q^{32} - 48q^{33} - 112q^{35} + 24q^{36} - 108q^{37} - 64q^{38} + 8q^{40} + 32q^{41} + 48q^{42} + 48q^{43} - 12q^{45} - 32q^{46} + 96q^{47} + 92q^{50} + 48q^{51} + 24q^{52} + 100q^{53} + 192q^{55} + 64q^{56} + 48q^{57} + 64q^{58} - 48q^{60} - 96q^{61} - 80q^{62} - 48q^{63} - 204q^{65} - 96q^{66} - 16q^{67} - 88q^{68} - 32q^{70} + 64q^{71} + 24q^{72} - 188q^{73} + 48q^{75} - 128q^{76} - 128q^{77} + 96q^{78} - 36q^{81} + 32q^{82} + 192q^{83} - 52q^{85} + 96q^{86} - 48q^{87} + 32q^{88} - 12q^{90} + 288q^{91} - 32q^{92} - 96q^{93} + 64q^{95} + 132q^{97} - 124q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/3$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$3 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$3 \beta_{3}$$

## Character values

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

 $$n$$ $$7$$ $$11$$ $$\chi(n)$$ $$-\beta_{2}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
7.1
 −1.22474 + 1.22474i 1.22474 − 1.22474i −1.22474 − 1.22474i 1.22474 + 1.22474i
1.00000 + 1.00000i −1.22474 + 1.22474i 2.00000i 4.89898 1.00000i −2.44949 −8.89898 8.89898i −2.00000 + 2.00000i 3.00000i 5.89898 + 3.89898i
7.2 1.00000 + 1.00000i 1.22474 1.22474i 2.00000i −4.89898 1.00000i 2.44949 0.898979 + 0.898979i −2.00000 + 2.00000i 3.00000i −3.89898 5.89898i
13.1 1.00000 1.00000i −1.22474 1.22474i 2.00000i 4.89898 + 1.00000i −2.44949 −8.89898 + 8.89898i −2.00000 2.00000i 3.00000i 5.89898 3.89898i
13.2 1.00000 1.00000i 1.22474 + 1.22474i 2.00000i −4.89898 + 1.00000i 2.44949 0.898979 0.898979i −2.00000 2.00000i 3.00000i −3.89898 + 5.89898i
 $$n$$: 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
5.c odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 30.3.f.a 4
3.b odd 2 1 90.3.g.d 4
4.b odd 2 1 240.3.bg.b 4
5.b even 2 1 150.3.f.b 4
5.c odd 4 1 inner 30.3.f.a 4
5.c odd 4 1 150.3.f.b 4
8.b even 2 1 960.3.bg.e 4
8.d odd 2 1 960.3.bg.g 4
12.b even 2 1 720.3.bh.i 4
15.d odd 2 1 450.3.g.j 4
15.e even 4 1 90.3.g.d 4
15.e even 4 1 450.3.g.j 4
20.d odd 2 1 1200.3.bg.d 4
20.e even 4 1 240.3.bg.b 4
20.e even 4 1 1200.3.bg.d 4
40.i odd 4 1 960.3.bg.e 4
40.k even 4 1 960.3.bg.g 4
60.l odd 4 1 720.3.bh.i 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.3.f.a 4 1.a even 1 1 trivial
30.3.f.a 4 5.c odd 4 1 inner
90.3.g.d 4 3.b odd 2 1
90.3.g.d 4 15.e even 4 1
150.3.f.b 4 5.b even 2 1
150.3.f.b 4 5.c odd 4 1
240.3.bg.b 4 4.b odd 2 1
240.3.bg.b 4 20.e even 4 1
450.3.g.j 4 15.d odd 2 1
450.3.g.j 4 15.e even 4 1
720.3.bh.i 4 12.b even 2 1
720.3.bh.i 4 60.l odd 4 1
960.3.bg.e 4 8.b even 2 1
960.3.bg.e 4 40.i odd 4 1
960.3.bg.g 4 8.d odd 2 1
960.3.bg.g 4 40.k even 4 1
1200.3.bg.d 4 20.d odd 2 1
1200.3.bg.d 4 20.e even 4 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 2 - 2 T + T^{2} )^{2}$$
$3$ $$9 + T^{4}$$
$5$ $$625 - 46 T^{2} + T^{4}$$
$7$ $$256 - 256 T + 128 T^{2} + 16 T^{3} + T^{4}$$
$11$ $$( -80 + 8 T + T^{2} )^{2}$$
$13$ $$30276 + 2088 T + 72 T^{2} - 12 T^{3} + T^{4}$$
$17$ $$37636 - 8536 T + 968 T^{2} - 44 T^{3} + T^{4}$$
$19$ $$25600 + 704 T^{2} + T^{4}$$
$23$ $$160000 - 6400 T + 128 T^{2} + 16 T^{3} + T^{4}$$
$29$ $$25600 + 704 T^{2} + T^{4}$$
$31$ $$( 16 + 40 T + T^{2} )^{2}$$
$37$ $$( 1458 + 54 T + T^{2} )^{2}$$
$41$ $$( -32 - 16 T + T^{2} )^{2}$$
$43$ $$831744 + 43776 T + 1152 T^{2} - 48 T^{3} + T^{4}$$
$47$ $$518400 - 69120 T + 4608 T^{2} - 96 T^{3} + T^{4}$$
$53$ $$6959044 + 263800 T + 5000 T^{2} - 100 T^{3} + T^{4}$$
$59$ $$( 400 + T^{2} )^{2}$$
$61$ $$( -960 + 48 T + T^{2} )^{2}$$
$67$ $$14868736 - 61696 T + 128 T^{2} + 16 T^{3} + T^{4}$$
$71$ $$( 160 - 32 T + T^{2} )^{2}$$
$73$ $$17859076 + 794488 T + 17672 T^{2} + 188 T^{3} + T^{4}$$
$79$ $$258566400 + 32736 T^{2} + T^{4}$$
$83$ $$5089536 - 433152 T + 18432 T^{2} - 192 T^{3} + T^{4}$$
$89$ $$47334400 + 17216 T^{2} + T^{4}$$
$97$ $$6874884 + 346104 T + 8712 T^{2} - 132 T^{3} + T^{4}$$