# Properties

 Label 30.3.d.a Level $30$ Weight $3$ Character orbit 30.d 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.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.817440793081$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{-5})$$ Defining polynomial: $$x^{4} - 4 x^{2} + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{2} + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{3} -2 q^{4} -\beta_{3} q^{5} + ( -1 + 2 \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{6} + ( 2 - 6 \beta_{1} + 3 \beta_{2} ) q^{7} + 2 \beta_{2} q^{8} + ( -2 + 4 \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{9} +O(q^{10})$$ $$q -\beta_{2} q^{2} + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{3} -2 q^{4} -\beta_{3} q^{5} + ( -1 + 2 \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{6} + ( 2 - 6 \beta_{1} + 3 \beta_{2} ) q^{7} + 2 \beta_{2} q^{8} + ( -2 + 4 \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{9} + ( -2 \beta_{1} + \beta_{2} ) q^{10} + 6 \beta_{2} q^{11} + ( -2 - 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{12} -10 q^{13} + ( -2 \beta_{2} + 6 \beta_{3} ) q^{14} + ( 5 - \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{15} + 4 q^{16} + ( -12 \beta_{2} - 6 \beta_{3} ) q^{17} + ( 8 + 2 \beta_{1} + \beta_{2} - 4 \beta_{3} ) q^{18} + ( 8 + 12 \beta_{1} - 6 \beta_{2} ) q^{19} + 2 \beta_{3} q^{20} + ( -13 - 4 \beta_{1} - 14 \beta_{2} + 5 \beta_{3} ) q^{21} + 12 q^{22} + ( -3 \beta_{2} + 6 \beta_{3} ) q^{23} + ( 2 - 4 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} ) q^{24} -5 q^{25} + 10 \beta_{2} q^{26} + ( 7 - 5 \beta_{1} + 20 \beta_{2} + \beta_{3} ) q^{27} + ( -4 + 12 \beta_{1} - 6 \beta_{2} ) q^{28} -12 \beta_{3} q^{29} + ( -5 - 2 \beta_{1} - 4 \beta_{2} + \beta_{3} ) q^{30} + 8 q^{31} -4 \beta_{2} q^{32} + ( 6 - 12 \beta_{1} + 12 \beta_{2} + 6 \beta_{3} ) q^{33} + ( -24 - 12 \beta_{1} + 6 \beta_{2} ) q^{34} + ( 15 \beta_{2} - 2 \beta_{3} ) q^{35} + ( 4 - 8 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} ) q^{36} + ( -22 + 24 \beta_{1} - 12 \beta_{2} ) q^{37} + ( -8 \beta_{2} - 12 \beta_{3} ) q^{38} + ( -10 - 10 \beta_{1} + 10 \beta_{2} - 10 \beta_{3} ) q^{39} + ( 4 \beta_{1} - 2 \beta_{2} ) q^{40} + ( 24 \beta_{2} + 6 \beta_{3} ) q^{41} + ( -32 + 10 \beta_{1} + 8 \beta_{2} + 4 \beta_{3} ) q^{42} + ( 14 - 18 \beta_{1} + 9 \beta_{2} ) q^{43} -12 \beta_{2} q^{44} + ( 5 + 8 \beta_{1} - 14 \beta_{2} + 2 \beta_{3} ) q^{45} + ( -6 + 12 \beta_{1} - 6 \beta_{2} ) q^{46} + ( -15 \beta_{2} + 30 \beta_{3} ) q^{47} + ( 4 + 4 \beta_{1} - 4 \beta_{2} + 4 \beta_{3} ) q^{48} + ( 45 - 24 \beta_{1} + 12 \beta_{2} ) q^{49} + 5 \beta_{2} q^{50} + ( 18 + 18 \beta_{1} - 36 \beta_{2} - 18 \beta_{3} ) q^{51} + 20 q^{52} + ( -12 \beta_{2} - 6 \beta_{3} ) q^{53} + ( 35 + 2 \beta_{1} - 8 \beta_{2} + 5 \beta_{3} ) q^{54} + ( 12 \beta_{1} - 6 \beta_{2} ) q^{55} + ( 4 \beta_{2} - 12 \beta_{3} ) q^{56} + ( 38 + 20 \beta_{1} + 16 \beta_{2} + 2 \beta_{3} ) q^{57} + ( -24 \beta_{1} + 12 \beta_{2} ) q^{58} + ( -36 \beta_{2} + 12 \beta_{3} ) q^{59} + ( -10 + 2 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} ) q^{60} + ( -16 - 24 \beta_{1} + 12 \beta_{2} ) q^{61} -8 \beta_{2} q^{62} + ( -64 + 20 \beta_{1} - 17 \beta_{2} - 22 \beta_{3} ) q^{63} -8 q^{64} + 10 \beta_{3} q^{65} + ( 12 + 12 \beta_{1} - 12 \beta_{2} + 12 \beta_{3} ) q^{66} + ( -82 - 18 \beta_{1} + 9 \beta_{2} ) q^{67} + ( 24 \beta_{2} + 12 \beta_{3} ) q^{68} + ( -33 + 12 \beta_{1} + 6 \beta_{2} + 3 \beta_{3} ) q^{69} + ( 30 - 4 \beta_{1} + 2 \beta_{2} ) q^{70} + ( 30 \beta_{2} - 12 \beta_{3} ) q^{71} + ( -16 - 4 \beta_{1} - 2 \beta_{2} + 8 \beta_{3} ) q^{72} + ( 50 + 24 \beta_{1} - 12 \beta_{2} ) q^{73} + ( 22 \beta_{2} - 24 \beta_{3} ) q^{74} + ( -5 - 5 \beta_{1} + 5 \beta_{2} - 5 \beta_{3} ) q^{75} + ( -16 - 24 \beta_{1} + 12 \beta_{2} ) q^{76} + ( 12 \beta_{2} - 36 \beta_{3} ) q^{77} + ( 10 - 20 \beta_{1} + 20 \beta_{2} + 10 \beta_{3} ) q^{78} + ( -28 - 12 \beta_{1} + 6 \beta_{2} ) q^{79} -4 \beta_{3} q^{80} + ( 7 - 32 \beta_{1} + 20 \beta_{2} + 28 \beta_{3} ) q^{81} + ( 48 + 12 \beta_{1} - 6 \beta_{2} ) q^{82} + ( 9 \beta_{2} + 6 \beta_{3} ) q^{83} + ( 26 + 8 \beta_{1} + 28 \beta_{2} - 10 \beta_{3} ) q^{84} + ( -30 - 24 \beta_{1} + 12 \beta_{2} ) q^{85} + ( -14 \beta_{2} + 18 \beta_{3} ) q^{86} + ( 60 - 12 \beta_{1} - 24 \beta_{2} - 12 \beta_{3} ) q^{87} -24 q^{88} + ( 24 \beta_{2} + 12 \beta_{3} ) q^{89} + ( -20 + 4 \beta_{1} - 7 \beta_{2} - 8 \beta_{3} ) q^{90} + ( -20 + 60 \beta_{1} - 30 \beta_{2} ) q^{91} + ( 6 \beta_{2} - 12 \beta_{3} ) q^{92} + ( 8 + 8 \beta_{1} - 8 \beta_{2} + 8 \beta_{3} ) q^{93} + ( -30 + 60 \beta_{1} - 30 \beta_{2} ) q^{94} + ( -30 \beta_{2} - 8 \beta_{3} ) q^{95} + ( -4 + 8 \beta_{1} - 8 \beta_{2} - 4 \beta_{3} ) q^{96} + ( 74 - 24 \beta_{1} + 12 \beta_{2} ) q^{97} + ( -45 \beta_{2} + 24 \beta_{3} ) q^{98} + ( -48 - 12 \beta_{1} - 6 \beta_{2} + 24 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{3} - 8q^{4} - 4q^{6} + 8q^{7} - 8q^{9} + O(q^{10})$$ $$4q + 4q^{3} - 8q^{4} - 4q^{6} + 8q^{7} - 8q^{9} - 8q^{12} - 40q^{13} + 20q^{15} + 16q^{16} + 32q^{18} + 32q^{19} - 52q^{21} + 48q^{22} + 8q^{24} - 20q^{25} + 28q^{27} - 16q^{28} - 20q^{30} + 32q^{31} + 24q^{33} - 96q^{34} + 16q^{36} - 88q^{37} - 40q^{39} - 128q^{42} + 56q^{43} + 20q^{45} - 24q^{46} + 16q^{48} + 180q^{49} + 72q^{51} + 80q^{52} + 140q^{54} + 152q^{57} - 40q^{60} - 64q^{61} - 256q^{63} - 32q^{64} + 48q^{66} - 328q^{67} - 132q^{69} + 120q^{70} - 64q^{72} + 200q^{73} - 20q^{75} - 64q^{76} + 40q^{78} - 112q^{79} + 28q^{81} + 192q^{82} + 104q^{84} - 120q^{85} + 240q^{87} - 96q^{88} - 80q^{90} - 80q^{91} + 32q^{93} - 120q^{94} - 16q^{96} + 296q^{97} - 192q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 4 x^{2} + 9$$:

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
11.1
 −1.58114 + 0.707107i 1.58114 + 0.707107i −1.58114 − 0.707107i 1.58114 − 0.707107i
1.41421i −0.581139 2.94317i −2.00000 2.23607i −4.16228 + 0.821854i 11.4868 2.82843i −8.32456 + 3.42079i 3.16228
11.2 1.41421i 2.58114 + 1.52896i −2.00000 2.23607i 2.16228 3.65028i −7.48683 2.82843i 4.32456 + 7.89292i −3.16228
11.3 1.41421i −0.581139 + 2.94317i −2.00000 2.23607i −4.16228 0.821854i 11.4868 2.82843i −8.32456 3.42079i 3.16228
11.4 1.41421i 2.58114 1.52896i −2.00000 2.23607i 2.16228 + 3.65028i −7.48683 2.82843i 4.32456 7.89292i −3.16228
 $$n$$: 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

## Twists

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

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

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 2 + T^{2} )^{2}$$
$3$ $$81 - 36 T + 12 T^{2} - 4 T^{3} + T^{4}$$
$5$ $$( 5 + T^{2} )^{2}$$
$7$ $$( -86 - 4 T + T^{2} )^{2}$$
$11$ $$( 72 + T^{2} )^{2}$$
$13$ $$( 10 + T )^{4}$$
$17$ $$11664 + 936 T^{2} + T^{4}$$
$19$ $$( -296 - 16 T + T^{2} )^{2}$$
$23$ $$26244 + 396 T^{2} + T^{4}$$
$29$ $$( 720 + T^{2} )^{2}$$
$31$ $$( -8 + T )^{4}$$
$37$ $$( -956 + 44 T + T^{2} )^{2}$$
$41$ $$944784 + 2664 T^{2} + T^{4}$$
$43$ $$( -614 - 28 T + T^{2} )^{2}$$
$47$ $$16402500 + 9900 T^{2} + T^{4}$$
$53$ $$11664 + 936 T^{2} + T^{4}$$
$59$ $$3504384 + 6624 T^{2} + T^{4}$$
$61$ $$( -1184 + 32 T + T^{2} )^{2}$$
$67$ $$( 5914 + 164 T + T^{2} )^{2}$$
$71$ $$1166400 + 5040 T^{2} + T^{4}$$
$73$ $$( 1060 - 100 T + T^{2} )^{2}$$
$79$ $$( 424 + 56 T + T^{2} )^{2}$$
$83$ $$324 + 684 T^{2} + T^{4}$$
$89$ $$186624 + 3744 T^{2} + T^{4}$$
$97$ $$( 4036 - 148 T + T^{2} )^{2}$$