Properties

 Label 930.2.d.b Level $930$ Weight $2$ Character orbit 930.d Analytic conductor $7.426$ 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] = [930,2,Mod(559,930)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(930, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

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

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

chi := DirichletCharacter("930.559");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$930 = 2 \cdot 3 \cdot 5 \cdot 31$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 930.d (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: $$7.42608738798$$ 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.

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

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/930\mathbb{Z}\right)^\times$$.

 $$n$$ $$187$$ $$311$$ $$871$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
559.1
 − 1.00000i 1.00000i
1.00000i 1.00000i −1.00000 −1.00000 + 2.00000i 1.00000 1.00000i 1.00000i −1.00000 2.00000 + 1.00000i
559.2 1.00000i 1.00000i −1.00000 −1.00000 2.00000i 1.00000 1.00000i 1.00000i −1.00000 2.00000 1.00000i
 $$n$$: 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 930.2.d.b 2
3.b odd 2 1 2790.2.d.g 2
5.b even 2 1 inner 930.2.d.b 2
5.c odd 4 1 4650.2.a.f 1
5.c odd 4 1 4650.2.a.bq 1
15.d odd 2 1 2790.2.d.g 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.d.b 2 1.a even 1 1 trivial
930.2.d.b 2 5.b even 2 1 inner
2790.2.d.g 2 3.b odd 2 1
2790.2.d.g 2 15.d odd 2 1
4650.2.a.f 1 5.c odd 4 1
4650.2.a.bq 1 5.c odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(930, [\chi])$$:

 $$T_{7}^{2} + 1$$ T7^2 + 1 $$T_{11} + 5$$ T11 + 5

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 1$$
$3$ $$T^{2} + 1$$
$5$ $$T^{2} + 2T + 5$$
$7$ $$T^{2} + 1$$
$11$ $$(T + 5)^{2}$$
$13$ $$T^{2} + 16$$
$17$ $$T^{2}$$
$19$ $$(T - 5)^{2}$$
$23$ $$T^{2} + 81$$
$29$ $$(T - 2)^{2}$$
$31$ $$(T - 1)^{2}$$
$37$ $$T^{2} + 64$$
$41$ $$(T - 6)^{2}$$
$43$ $$T^{2} + 1$$
$47$ $$T^{2} + 144$$
$53$ $$T^{2} + 169$$
$59$ $$(T + 10)^{2}$$
$61$ $$(T + 14)^{2}$$
$67$ $$T^{2} + 196$$
$71$ $$(T + 9)^{2}$$
$73$ $$T^{2} + 81$$
$79$ $$(T + 5)^{2}$$
$83$ $$T^{2} + 36$$
$89$ $$(T + 3)^{2}$$
$97$ $$T^{2} + 324$$