# Properties

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

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [30,2,Mod(19,30)]

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, 1]))

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

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

chi := DirichletCharacter("30.19");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

## 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: no Analytic conductor: $$0.239551206064$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
19.1
 − 1.00000i 1.00000i
1.00000i 1.00000i −1.00000 −2.00000 1.00000i 1.00000 2.00000i 1.00000i −1.00000 −1.00000 + 2.00000i
19.2 1.00000i 1.00000i −1.00000 −2.00000 + 1.00000i 1.00000 2.00000i 1.00000i −1.00000 −1.00000 2.00000i
 $$n$$: 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.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 30.2.c.a 2
3.b odd 2 1 90.2.c.a 2
4.b odd 2 1 240.2.f.a 2
5.b even 2 1 inner 30.2.c.a 2
5.c odd 4 1 150.2.a.a 1
5.c odd 4 1 150.2.a.c 1
7.b odd 2 1 1470.2.g.g 2
7.c even 3 2 1470.2.n.h 4
7.d odd 6 2 1470.2.n.a 4
8.b even 2 1 960.2.f.h 2
8.d odd 2 1 960.2.f.i 2
9.c even 3 2 810.2.i.e 4
9.d odd 6 2 810.2.i.b 4
12.b even 2 1 720.2.f.f 2
15.d odd 2 1 90.2.c.a 2
15.e even 4 1 450.2.a.b 1
15.e even 4 1 450.2.a.f 1
16.e even 4 1 3840.2.d.g 2
16.e even 4 1 3840.2.d.y 2
16.f odd 4 1 3840.2.d.j 2
16.f odd 4 1 3840.2.d.x 2
20.d odd 2 1 240.2.f.a 2
20.e even 4 1 1200.2.a.g 1
20.e even 4 1 1200.2.a.m 1
24.f even 2 1 2880.2.f.c 2
24.h odd 2 1 2880.2.f.e 2
35.c odd 2 1 1470.2.g.g 2
35.f even 4 1 7350.2.a.bg 1
35.f even 4 1 7350.2.a.cc 1
35.i odd 6 2 1470.2.n.a 4
35.j even 6 2 1470.2.n.h 4
40.e odd 2 1 960.2.f.i 2
40.f even 2 1 960.2.f.h 2
40.i odd 4 1 4800.2.a.l 1
40.i odd 4 1 4800.2.a.cg 1
40.k even 4 1 4800.2.a.m 1
40.k even 4 1 4800.2.a.cj 1
45.h odd 6 2 810.2.i.b 4
45.j even 6 2 810.2.i.e 4
60.h even 2 1 720.2.f.f 2
60.l odd 4 1 3600.2.a.o 1
60.l odd 4 1 3600.2.a.bg 1
80.k odd 4 1 3840.2.d.j 2
80.k odd 4 1 3840.2.d.x 2
80.q even 4 1 3840.2.d.g 2
80.q even 4 1 3840.2.d.y 2
120.i odd 2 1 2880.2.f.e 2
120.m even 2 1 2880.2.f.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.2.c.a 2 1.a even 1 1 trivial
30.2.c.a 2 5.b even 2 1 inner
90.2.c.a 2 3.b odd 2 1
90.2.c.a 2 15.d odd 2 1
150.2.a.a 1 5.c odd 4 1
150.2.a.c 1 5.c odd 4 1
240.2.f.a 2 4.b odd 2 1
240.2.f.a 2 20.d odd 2 1
450.2.a.b 1 15.e even 4 1
450.2.a.f 1 15.e even 4 1
720.2.f.f 2 12.b even 2 1
720.2.f.f 2 60.h even 2 1
810.2.i.b 4 9.d odd 6 2
810.2.i.b 4 45.h odd 6 2
810.2.i.e 4 9.c even 3 2
810.2.i.e 4 45.j even 6 2
960.2.f.h 2 8.b even 2 1
960.2.f.h 2 40.f even 2 1
960.2.f.i 2 8.d odd 2 1
960.2.f.i 2 40.e odd 2 1
1200.2.a.g 1 20.e even 4 1
1200.2.a.m 1 20.e even 4 1
1470.2.g.g 2 7.b odd 2 1
1470.2.g.g 2 35.c odd 2 1
1470.2.n.a 4 7.d odd 6 2
1470.2.n.a 4 35.i odd 6 2
1470.2.n.h 4 7.c even 3 2
1470.2.n.h 4 35.j even 6 2
2880.2.f.c 2 24.f even 2 1
2880.2.f.c 2 120.m even 2 1
2880.2.f.e 2 24.h odd 2 1
2880.2.f.e 2 120.i odd 2 1
3600.2.a.o 1 60.l odd 4 1
3600.2.a.bg 1 60.l odd 4 1
3840.2.d.g 2 16.e even 4 1
3840.2.d.g 2 80.q even 4 1
3840.2.d.j 2 16.f odd 4 1
3840.2.d.j 2 80.k odd 4 1
3840.2.d.x 2 16.f odd 4 1
3840.2.d.x 2 80.k odd 4 1
3840.2.d.y 2 16.e even 4 1
3840.2.d.y 2 80.q even 4 1
4800.2.a.l 1 40.i odd 4 1
4800.2.a.m 1 40.k even 4 1
4800.2.a.cg 1 40.i odd 4 1
4800.2.a.cj 1 40.k even 4 1
7350.2.a.bg 1 35.f even 4 1
7350.2.a.cc 1 35.f even 4 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 1$$
$3$ $$T^{2} + 1$$
$5$ $$T^{2} + 4T + 5$$
$7$ $$T^{2} + 4$$
$11$ $$(T - 2)^{2}$$
$13$ $$T^{2} + 36$$
$17$ $$T^{2} + 4$$
$19$ $$T^{2}$$
$23$ $$T^{2} + 16$$
$29$ $$T^{2}$$
$31$ $$(T + 8)^{2}$$
$37$ $$T^{2} + 4$$
$41$ $$(T - 2)^{2}$$
$43$ $$T^{2} + 16$$
$47$ $$T^{2} + 64$$
$53$ $$T^{2} + 36$$
$59$ $$(T + 10)^{2}$$
$61$ $$(T - 2)^{2}$$
$67$ $$T^{2} + 64$$
$71$ $$(T - 12)^{2}$$
$73$ $$T^{2} + 16$$
$79$ $$T^{2}$$
$83$ $$T^{2} + 16$$
$89$ $$(T - 10)^{2}$$
$97$ $$T^{2} + 64$$