# Properties

 Label 30.2.a.a Level $30$ Weight $2$ Character orbit 30.a Self dual yes Analytic conductor $0.240$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [30,2,Mod(1,30)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(30, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("30.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$30 = 2 \cdot 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 30.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$0.239551206064$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{3} + q^{4} - q^{5} - q^{6} - 4 q^{7} - q^{8} + q^{9}+O(q^{10})$$ q - q^2 + q^3 + q^4 - q^5 - q^6 - 4 * q^7 - q^8 + q^9 $$q - q^{2} + q^{3} + q^{4} - q^{5} - q^{6} - 4 q^{7} - q^{8} + q^{9} + q^{10} + q^{12} + 2 q^{13} + 4 q^{14} - q^{15} + q^{16} + 6 q^{17} - q^{18} - 4 q^{19} - q^{20} - 4 q^{21} - q^{24} + q^{25} - 2 q^{26} + q^{27} - 4 q^{28} - 6 q^{29} + q^{30} + 8 q^{31} - q^{32} - 6 q^{34} + 4 q^{35} + q^{36} + 2 q^{37} + 4 q^{38} + 2 q^{39} + q^{40} - 6 q^{41} + 4 q^{42} - 4 q^{43} - q^{45} + q^{48} + 9 q^{49} - q^{50} + 6 q^{51} + 2 q^{52} - 6 q^{53} - q^{54} + 4 q^{56} - 4 q^{57} + 6 q^{58} - q^{60} - 10 q^{61} - 8 q^{62} - 4 q^{63} + q^{64} - 2 q^{65} - 4 q^{67} + 6 q^{68} - 4 q^{70} - q^{72} + 2 q^{73} - 2 q^{74} + q^{75} - 4 q^{76} - 2 q^{78} + 8 q^{79} - q^{80} + q^{81} + 6 q^{82} + 12 q^{83} - 4 q^{84} - 6 q^{85} + 4 q^{86} - 6 q^{87} + 18 q^{89} + q^{90} - 8 q^{91} + 8 q^{93} + 4 q^{95} - q^{96} + 2 q^{97} - 9 q^{98}+O(q^{100})$$ q - q^2 + q^3 + q^4 - q^5 - q^6 - 4 * q^7 - q^8 + q^9 + q^10 + q^12 + 2 * q^13 + 4 * q^14 - q^15 + q^16 + 6 * q^17 - q^18 - 4 * q^19 - q^20 - 4 * q^21 - q^24 + q^25 - 2 * q^26 + q^27 - 4 * q^28 - 6 * q^29 + q^30 + 8 * q^31 - q^32 - 6 * q^34 + 4 * q^35 + q^36 + 2 * q^37 + 4 * q^38 + 2 * q^39 + q^40 - 6 * q^41 + 4 * q^42 - 4 * q^43 - q^45 + q^48 + 9 * q^49 - q^50 + 6 * q^51 + 2 * q^52 - 6 * q^53 - q^54 + 4 * q^56 - 4 * q^57 + 6 * q^58 - q^60 - 10 * q^61 - 8 * q^62 - 4 * q^63 + q^64 - 2 * q^65 - 4 * q^67 + 6 * q^68 - 4 * q^70 - q^72 + 2 * q^73 - 2 * q^74 + q^75 - 4 * q^76 - 2 * q^78 + 8 * q^79 - q^80 + q^81 + 6 * q^82 + 12 * q^83 - 4 * q^84 - 6 * q^85 + 4 * q^86 - 6 * q^87 + 18 * q^89 + q^90 - 8 * q^91 + 8 * q^93 + 4 * q^95 - q^96 + 2 * q^97 - 9 * q^98

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
−1.00000 1.00000 1.00000 −1.00000 −1.00000 −4.00000 −1.00000 1.00000 1.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$-1$$
$$5$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 30.2.a.a 1
3.b odd 2 1 90.2.a.c 1
4.b odd 2 1 240.2.a.b 1
5.b even 2 1 150.2.a.b 1
5.c odd 4 2 150.2.c.a 2
7.b odd 2 1 1470.2.a.d 1
7.c even 3 2 1470.2.i.o 2
7.d odd 6 2 1470.2.i.q 2
8.b even 2 1 960.2.a.e 1
8.d odd 2 1 960.2.a.p 1
9.c even 3 2 810.2.e.l 2
9.d odd 6 2 810.2.e.b 2
11.b odd 2 1 3630.2.a.w 1
12.b even 2 1 720.2.a.j 1
13.b even 2 1 5070.2.a.w 1
13.d odd 4 2 5070.2.b.k 2
15.d odd 2 1 450.2.a.d 1
15.e even 4 2 450.2.c.b 2
16.e even 4 2 3840.2.k.y 2
16.f odd 4 2 3840.2.k.f 2
17.b even 2 1 8670.2.a.g 1
20.d odd 2 1 1200.2.a.k 1
20.e even 4 2 1200.2.f.e 2
21.c even 2 1 4410.2.a.z 1
24.f even 2 1 2880.2.a.q 1
24.h odd 2 1 2880.2.a.a 1
35.c odd 2 1 7350.2.a.ct 1
40.e odd 2 1 4800.2.a.d 1
40.f even 2 1 4800.2.a.cq 1
40.i odd 4 2 4800.2.f.p 2
40.k even 4 2 4800.2.f.w 2
60.h even 2 1 3600.2.a.f 1
60.l odd 4 2 3600.2.f.i 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.2.a.a 1 1.a even 1 1 trivial
90.2.a.c 1 3.b odd 2 1
150.2.a.b 1 5.b even 2 1
150.2.c.a 2 5.c odd 4 2
240.2.a.b 1 4.b odd 2 1
450.2.a.d 1 15.d odd 2 1
450.2.c.b 2 15.e even 4 2
720.2.a.j 1 12.b even 2 1
810.2.e.b 2 9.d odd 6 2
810.2.e.l 2 9.c even 3 2
960.2.a.e 1 8.b even 2 1
960.2.a.p 1 8.d odd 2 1
1200.2.a.k 1 20.d odd 2 1
1200.2.f.e 2 20.e even 4 2
1470.2.a.d 1 7.b odd 2 1
1470.2.i.o 2 7.c even 3 2
1470.2.i.q 2 7.d odd 6 2
2880.2.a.a 1 24.h odd 2 1
2880.2.a.q 1 24.f even 2 1
3600.2.a.f 1 60.h even 2 1
3600.2.f.i 2 60.l odd 4 2
3630.2.a.w 1 11.b odd 2 1
3840.2.k.f 2 16.f odd 4 2
3840.2.k.y 2 16.e even 4 2
4410.2.a.z 1 21.c even 2 1
4800.2.a.d 1 40.e odd 2 1
4800.2.a.cq 1 40.f even 2 1
4800.2.f.p 2 40.i odd 4 2
4800.2.f.w 2 40.k even 4 2
5070.2.a.w 1 13.b even 2 1
5070.2.b.k 2 13.d odd 4 2
7350.2.a.ct 1 35.c odd 2 1
8670.2.a.g 1 17.b even 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(\Gamma_0(30))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 1$$
$3$ $$T - 1$$
$5$ $$T + 1$$
$7$ $$T + 4$$
$11$ $$T$$
$13$ $$T - 2$$
$17$ $$T - 6$$
$19$ $$T + 4$$
$23$ $$T$$
$29$ $$T + 6$$
$31$ $$T - 8$$
$37$ $$T - 2$$
$41$ $$T + 6$$
$43$ $$T + 4$$
$47$ $$T$$
$53$ $$T + 6$$
$59$ $$T$$
$61$ $$T + 10$$
$67$ $$T + 4$$
$71$ $$T$$
$73$ $$T - 2$$
$79$ $$T - 8$$
$83$ $$T - 12$$
$89$ $$T - 18$$
$97$ $$T - 2$$